albert einstein center for fundamental physics, institute for … · 2020-06-17 · sidlerstrasse...

MITP20-033

Forward doubly-virtual Compton scattering off the nucleon in

chiral perturbation theory II Spin polarizabilities and moments

of polarized structure functions

Jose Manuel Alarcon

Departamento de Fısica Teorica amp IPARCOS

Universidad Complutense de Madrid 28040 Madrid Spain

Franziska Hagelstein

Albert Einstein Center for Fundamental Physics

Institute for Theoretical Physics University of Bern

Sidlerstrasse 5 CH-3012 Bern Switzerland

Vadim Lensky and Vladimir Pascalutsa

Institut fur Kernphysik amp Cluster of Excellence PRISMA

Johannes Gutenberg-Universitat Mainz D-55128 Mainz Germany

(Dated June 17 2020)

Abstract

We examine the polarized doubly-virtual Compton scattering (VVCS) off the nucleon using chi-

ral perturbation theory (χPT) The polarized VVCS contains a wealth of information on the spin

structure of the nucleon which is relevant to the calculation of the two-photon-exchange effects

in atomic spectroscopy and electron scattering We report on a complete next-to-leading-order

(NLO) calculation of the polarized VVCS amplitudes S1(νQ2) and S2(νQ2) and the corre-

sponding polarized spin structure functions g1(xQ2) and g2(xQ2) Our results for the moments

of polarized structure functions partially related to different spin polarizabilities are compared

to other theoretical predictions and ldquodata-drivenrdquo evaluations as well as to the recent Jefferson

Lab measurements By expanding the results in powers of the inverse nucleon mass we repro-

duce the known ldquoheavy-baryonrdquo expressions This serves as a check of our calculation as well as

demonstrates the differences between the manifestly Lorentz-invariant baryon χPT (BχPT) and

heavy-baryon (HBχPT) frameworks

1

arX

iv2

006

0862

6v1

[he

p-ph

] 1

5 Ju

n 20

20

CONTENTS

I Introduction 3

II Calculation of unpolarized VVCS at NLO 5

A VVCS amplitudes and relations to structure functions 5

B Low-energy expansions and relations to polarizabilities 6

C Details on χPT calculation and uncertainty estimate 8

III Results and discussion 11

A γ0(Q2) mdash generalized forward spin polarizability 11

B δLT (Q2) mdash longitudinal-transverse polarizability 15

C IA(Q2) mdash a generalized GDH integral 16

D Γ1(Q2) and I1(Q

2) mdash the first moment of the structure function g1(xQ2) 20

E d2(Q2) mdash a measure of color polarizability 22

F γ0(Q2) mdash fifth-order generalized forward spin polarizability 24

G Summary 26

IV Conclusions 27

Acknowledgements 28

A Tensor decompositions of the VVCS amplitudes 28

B Photoabsorption cross sections 29

1 πN -production channel 30

2 ∆-production channel 36

C Polarizabilities at Q2 = 0 38

1 πN -loop contribution 38

2 ∆-exchange contribution 40

References 41

2

I INTRODUCTION

In the studies of nucleon structure the forward doubly-virtual Compton scattering

(VVCS) amplitude Fig 1 is playing a central role (see eg Refs [1ndash4] for reviews)

Traditionally its general properties such as unitarity analyticity and crossing are used to

establish various useful sum rules for the nucleon magnetic moment (GerasimovndashDrellndashHearn

[5 6] and Schwinger sum rules [7ndash9]) and polarizabilities (eg Baldin [10] and Gell-Mannndash

GoldbergerndashThirring sum rules [11]) More recently the interest in nucleon VVCS has been

renewed in connection with precision atomic spectroscopy where this amplitude enters in

the form of two-photon exchange (TPE) corrections As the TPE corrections in atomic do-

main are dominated by low-energy VVCS it makes sense to calculate them systematically

using chiral perturbation theory (χPT) which is a low-energy effective-field theory of the

Standard Model

In this paper we present a state-of-the-art χPT calculation of the polarized nucleon

VVCS relevant to TPE corrections to hyperfine structure of hydrogen and muonic hydro-

gen This will extend the leading-order χPT evaluation of the TPE effects in hyperfine

splittings [12ndash17] Here we however do not discuss the TPE evaluation but rather focus on

testing the χPT framework against the available empirical information on low-energy spin

structure of the nucleon

It is especially interesting to confront the χPT predictions with the recent measurements

coming from the ongoing ldquoSpin Physics Programrdquo at Jefferson Laboratory [18ndash27] with

the exception of a recent measurement of the deuteron spin polarizability by the CLAS

Collaboration [28] which does not treat correctly complications due to deuteron spin [29]

Our present calculation is extending Ref [30] to the case of polarized VVCS We therefore

use a manifestly-covariant extension of SU(2) χPT to the baryon sector called Baryon χPT

(BχPT) First attempts to calculate VVCS in the straightforward BχPT framework (rather

than the heavy-baryon expansion or the ldquoinfrared regularizationrdquo) were done by Bernard

et al [31] and our group [32] The two works obtained somewhat different results most

notably for the proton spin polarizability δLT Here we improve on [32] in three important

aspects appreciable at finite Q2 1) inclusion of the Coulomb-quadrupole (C2) N rarr ∆ tran-

sition [33 34] 2) correct inclusion of the elastic form-factor contributions to the integrals

Γ1(Q2) I1(Q2) and IA(Q2) (see Sections III C and III D for details) and 3) cancellations

between different orders in the chiral prediction and their effect on the convergence of the

effective-field-theory calculation and thus the error estimate These improvements how-

ever do not bring us closer to the results of [31] and the source of discrepancies most likely

3

FIG 1 The forward Compton scattering or VVCS in case of virtual photons q2 = minusQ2

lies in the different counting and renormalization of the π∆-loop contributions Bernard

et al [31] use the so-called small-scale expansion [35] for the ∆(1232) contributions whereas

we are using the δ-counting scheme [36] (see also Ref [37 Sec 4] for review)

This paper is organized as follows In Sec II A we introduce the polarized VVCS am-

plitudes and their relations to spin structure functions In Sec II B we introduce the spin

polarizabilities appearing in the low-energy expansion (LEX) of the polarized VVCS am-

plitudes In Sec II C we briefly describe our χPT calculation focusing mainly on the

uncertainty estimate In Sec III we show our predictions for the proton and neutron po-

larizabilities as well as some interesting moments of their structure functions In Sec III G

we summarize the results obtained herein comment on the improvements done with respect

to previous calculations and give an outlook to future applications In App B we discuss

the structure functions in particular we define the longitudinal-transverse response func-

tion discuss the ∆-pole contribution and give analytical results for the tree-level πN - and

∆-production channels of the photoabsorption cross sections In App C we give analytical

expressions for the πN -loop and ∆-exchange contributions to the static values and slopes of

the polarizabilities and moments of structure functions The complete expressions also for

the π∆-loop contributions can be found in the Supplemented material

TABLE I Parameters (fundamental and low-energy constants) [38] at the order they appear first

The πN∆ coupling constant hA is fit to the experimental Delta width and the γlowastN∆ coupling

constants gM gE and gC are taken from the pion photoproduction study of Ref [33]

O(p2) α = 1(13704) MN = Mp = 93827 MeV

O(p3) gA = 1270 fπ = 9221 MeV mπ = 13957 MeV

O(p4∆) M∆ = 1232 MeV hA equiv 2gπN∆ = 285 gM = 297 gE = minus10 gC = minus26

4

II CALCULATION OF UNPOLARIZED VVCS AT NLO

A VVCS amplitudes and relations to structure functions

The polarized part of forward VVCS can be described in terms of two independent

Lorentz-covariant and gauge-invariant tensor structures and two scalar amplitudes [3]

T microν(p q) = minus 1

MN

γmicroναqα S1(νQ2)minus 1

M2N

(γmicroνq2 + qmicroγναqα minus qνγmicroαqα

)S2(νQ2) (1)

Here q and p are the photon and nucleon four-momenta (cf Fig 1) ν is the photon

lab-frame energy Q2 = minusq2 is the photon virtuality and γmicroν = 12

[γmicro γν ] and γmicroνα =12

(γmicroγνγα minus γαγνγmicro) are the usual Dirac matrices Alternatively one can use the following

laboratory-frame amplitudes

gTT (νQ2) =ν

MN

[S1(νQ2)minus Q2

MN νS2(νQ2)

] (2a)

gLT (νQ2) =Q

MN

[S1(νQ2) +

ν

MN

S2(νQ2)] (2b)

introduced in Eq (A2) The optical theorem relates the absorptive parts of the forward

VVCS amplitudes to the nucleon structure functions or the cross sections of virtual pho-

toabsorption

ImS1(νQ2) =4π2α

νg1(xQ2) =

MNνK(νQ2)

ν2 +Q2

[Q

νσLT (νQ2) + σTT (νQ2)

] (3a)

ImS2(νQ2) =4π2αMN

ν2g2(xQ2) =

M2NK(νQ2)

ν2 +Q2

[ν

QσLT (νQ2)minus σTT (νQ2)

] (3b)

with α the fine structure constant andK(νQ2) the photon flux factor Note that the photon

flux factor in the optical theorem and the cross sections measured in electroproduction

processes is a matter of convention and has to be chosen in both quantities consistently In

what follows we use Gilmanrsquos flux factor

K(νQ2) equiv |~q | =radicν2 +Q2 (4)

The helicity-difference photoabsorption cross section is defined as σTT = 12 (σ12 minus σ32)

where the photons are transversely polarized and the subscripts on the right-hand side

indicate the total helicities of the γlowastN states The cross section σLT corresponds to a

simultaneous helicity change of the photon and nucleon spin flip such that the total helicity

5

is conserved A detailed derivation of the connection between this response function and

the photoabsorption cross sections can be found in App B The forward VVCS amplitudes

satisfy dispersion relations derived from the general principles of analyticity and causality1

S1(νQ2) =16παMN

Q2

int 1

0

dxg1(xQ2)

1minus x2(ννel)2 minus i0+(5a)

=2MN

π

int infinνel

dν primeν prime 2[QνprimeσLT + σTT

](ν prime Q2)radic

ν prime 2 +Q2(ν prime 2 minus ν2 minus i0+)

νS2(νQ2) =16παM2

N

Q2

int 1

0

dxg2(xQ2)

1minus x2(ννel)2 minus i0+(5b)

=2M2

N

π

int infinνel

dν primeν prime 2[νprime

QσLT minus σTT

](ν prime Q2)radic


with νel = Q22MN the elastic threshold

B Low-energy expansions and relations to polarizabilities

The VVCS amplitudes naturally split into nucleon-pole (Spolei ) and non-pole (Snonpole

i )

parts or Born (SBorni ) and non-Born (Si) parts

Si = Spolei + Snonpole

i = SBorni + Si (6)

The Born amplitudes are given uniquely in terms of the nucleon form factors [1]

SBorn1 (νQ2) =

2πα

M

Q2GM(Q2)F1(Q2)

ν2el minus ν2

minus F 22 (Q2)

(7a)

SBorn2 (νQ2) = minus 2παν

ν2el minus ν2

GM(Q2)F2(Q2) (7b)

The same is true for the nucleon-pole amplitudes which are related to the Born amplitudes

in the following way

Spole1 (νQ2) = SBorn

1 (νQ2) +2πα

MN

F 22 (Q2) (8a)


2 (νQ2) (8b)

1 The dispersion relation for νS2(νQ2) is used because it is pole-free in the limit Q2 rarr 0 and then ν rarr 0

cf Eq (7b)

6

Here we used the elastic Dirac and Pauli form factors F1(Q2) and F2(Q2) related to the

electric and magnetic Sachs form factors GE(Q2) and GM(Q2) through

F1(Q2) =GE(Q2) + τGM(Q2)

1 + τ (9a)

F2(Q2) =GM(Q2)minusGE(Q2)

1 + τ (9b)

where τ = Q24M2N

A low-energy expansion (LEX) of Eq (5) in combination with the unitarity relations

given in Eq (3) establishes various sum rules relating the nucleon properties (electromag-

netic moments polarizabilities) to experimentally observable response functions [1 3] The

leading terms in the LEX of the RCS amplitudes are determined uniquely by charge mass

and anomalous magnetic moment as the global properties of the nucleon These lowest-

order terms represent the celebrated low-energy theorem (LET) of Low Gell-Mann and

Goldberger [39 40] The polarizabilities related to the internal structure of the nucleon

enter the LEX at higher orders They make up the non-Born amplitudes and can be related

to moments of inelastic structure functions

The process of VVCS can be realized experimentally in electron-nucleon scattering where

a virtual photon is exchanged between the electron and the nucleon This virtual photon

acts as a probe whose resolution depends on its virtuality Q2 In this way one can access

the so-called generalized polarizabilities which extend the notion of polarizabilities to the

case of response to finite momentum transfer The generalized forward spin polarizability

γ0(Q2) and the longitudinal-transverse polarizability δLT (Q2) are most naturally defined via

the LEX of the non-Born part of the lab-frame VVCS amplitudes [1]

1

4πgnonpoleTT (νQ2) =

2α

M2N

IA(Q2) ν + γ0(Q2)ν3 + γ0(Q2)ν5 + (10a)

1

4πgnonpoleLT (νQ2) =

2α

M2N

I3(Q2)Q+ δLT (Q2)ν2Q+ (10b)

Their definitions in terms of integrals over structure functions are postponed to Eqs (19)

and (22) Here we only give the definition of the moment I3(Q2)

I3(Q2) =M2

N

4π2α

int infinν0

dνK(νQ2)

νQσLT (νQ2) =

2M2N

Q2

int x0

0

dx[g1(xQ2) + g2(xQ2)

] (11)

which is related to the Schwinger sum rule in the real photon limit [7ndash9 41 42] The LEX

of the non-pole part of the covariant VVCS amplitudes can be described entirely in terms

7

of moments of inelastic spin structure functions (up to O(ν4) [43])

1

4πSnonpole

1 (νQ2) =2α

MN

I1(Q2) +

2α

MNQ2

[IA(Q2)minus I1(Q2)

]+MNδLT (Q2)

ν2 (12a)

1

4πνSnonpole

2 (νQ2) = 2αI2(Q2) +2α

Q2

[I1(Q2)minus IA(Q2)

]ν2 (12b)

I1(Q2) and IA(Q2) are generalizations of the famous GerasimovndashDrellndashHearn (GDH) sum

rule [5 6] from RCS to the case of virtual photons [1] Their definitions are given in Eqs (26)

and (32) I2(Q2) is the well-known Burkhardt-Cottingham (BC) sum rule [44]

I2(Q2) equiv 2M2N

Q2

int x0

0

dx g2(x Q2) =1

4F2(Q2)GM(Q2) (13)

which can be written as a ldquosuperconvergence sum rulerdquo

Q2

16παM2N

limνrarr0

νS2(νQ2) =

int 1

0

dx g2 (x Q2) = 0 (14)

The latter is valid for any value of Q2 provided that the integral converges for x rarr 0

Combining Eq (5) with the above LEXs of the VVCS amplitudes we can relate IA(Q2)

I1(Q2) γ0(Q2) and δLT (Q2) to moments of inelastic structure functions see Sec III It is

important to note that only γ0(Q2) and δLT (Q2) are generalized polarizabilities The relation

of the inelastic moments IA(Q2) and I1(Q2) to polarizabilities will be discussed in details in

Secs III C and III D The difference between S1(νQ2) and Snonpole1 (νQ2) cf Eq (8a) will

be important in this context

C Details on χPT calculation and uncertainty estimate

In this work we calculated the NLO prediction of BχPT for the polarized non-Born

VVCS amplitudes This includes the leading pion-nucleon (πN) loops see Ref [32 Fig 1]

as well as the subleading tree-level Delta-exchange (∆-exchange) see Ref [30 Fig 2] and

the pion-Delta (π∆) loops see Ref [32 Fig 2] In the δ-power-counting scheme [36] the

LO and NLO non-Born VVCS amplitudes and polarizabilities are of O(p3) and O(p4∆)

respectively2 The LECs are listed in Table I sorted by the order at which they appear in

our calculation At the given orders there are no ldquonewrdquo LECs that would need to be fitted

from Compton processes For more details on the BχPT formalism we refer to Ref [30]

2 In the full Compton amplitude there is a lower order contribution coming from the Born terms leading

to a shift in nomenclature by one order the LO contribution referred to as the NLO contribution etc

see eg Ref [45]

8

where power counting predictive orders (Sec III A) and the renormalization procedure

(Sec III B) are discussed

A few remarks are in order for the inclusion of the ∆(1232) and the tree-level ∆-exchange

contribution In contrast to Ref [32] we include the Coulomb-quadrupole (C2) N rarr ∆

transition described by the LEC gC The relevant Lagrangian describing the non-minimal

γlowastN∆ coupling [33 34] (note that in these references the overall sign of gC is inconsistent

between the Lagrangian and Feynman rules) reads

L(2)∆ =

3e

2MNM+

N T3

igM F

microν partmicro∆ν minus gEγ5Fmicroν partmicro∆ν (15)

+igCM∆

γ5γα(partα∆ν minus partν∆α)partmicroF

microν

+ Hc

with M+ = MN + M∆ and the dual of the electromagnetic field strength tensor F microν =12εmicroνρλFρλ Even though the Coulomb coupling is subleading compared with the electric

and magnetic couplings (gE and gM) its relatively large magnitude cf Table I makes it

numerically important for instance in γ0(Q2) Furthermore we study the effect of modifying

the magnetic coupling using a dipole form factor

gM rarrgM[

1 + (QΛ)2 ]2 (16)

where Λ2 = 071 GeV2 The inclusion of this Q2 dependence mimics the form expected from

vector-meson dominance It is motivated by observing the importance of this form factor

for the correct description of the electroproduction data [33]

To estimate the uncertainties of our NLO predictions we define

δ(Q2) =

radic(∆

MN

)2

+

(Q2

2MN∆

)2

(17)

such that the neglected next-to-next-to-leading order terms are expected to be of relative

size δ2 [33] The uncertainties in the values of the parameters in Table I have a much

smaller impact compared to the truncation uncertainty and can be neglected Unfortunately

∆IA(Q2) γ0(Q2) and γ0(Q2) ie the sum rules involving the cross section σTT (νQ2)

as well as the polarizability ∆I1(Q2) turn out to be numerically small Their smallness

suggests a cancellation of leading orders (which can indeed be confirmed by looking at

separate contributions as shown below) Therefore an error of δ2(Q2)P (Q2) where P (Q2)

is a generalized polarizability might underestimate the theoretical uncertainty for some of

the NLO predictions To avoid this we estimate the uncertainty of our NLO polarizability

9

predictions by

σP (Q2) =

Max[δ4(0)P (0)2 δ4(0)P LO(0)2 δ2(0)PNLO(0)2

]+Max

[δ4(Q2)

[P (Q2)minus P (0)

]2 δ4(Q2)

[P LO(Q2)minus P LO(0)

]2

δ2(Q2)[PNLO(Q2)minus PNLO(0)

]2]12

(18)

where P LO(Q2) is the πN -loop contribution PNLO(Q2) are the ∆-exchange and π∆-loop

contributions and P (Q2) = P LO(Q2) + PNLO(Q2) This error prescription is similar to the

one used in eg Refs [46ndash49] Here since we are interested in the generalized polarizabil-

ities we added in quadrature the error due to the static piece P (0) and the Q2-dependent

remainder P (Q2) minus P (0) Note that the static values of IA(0) and I1(0) are given by the

elastic Pauli form factor which is not part of our BχPT prediction and is considered to be

exact

Note that our result for the spin polarizabilities (and the unpolarized moments [30]) are

NLO predictions only at low momentum transfers Q mπ At larger values of Q amp ∆ they

become incomplete LO predictions Indeed in this regime the ∆ propagators do not carry

additional suppression compared to the nucleon propagators and the π∆ loops are promoted

to LO In general we only expect a rather small contribution from omitted π∆ loops to the

Q2 dependence of the polarizabilities since π∆ loops show rather weak dependence on Q2

compared with the ∆ exchange or πN loops Nevertheless this issue has to be reflected in

the error estimate Since the static polarizabilities P (0) are not affected it is natural to

separate the error on the Q2-dependent remainder P (Q2) minus P (0) as done in Eq (18) To

accommodate for the potential loss of precision above Q amp ∆ we define the relative error

δ(Q2) as growing with increasing Q2 see Eq (17)

Upon expanding our results in powers of the inverse nucleon mass Mminus1N we are able to

reproduce existing results of heavy-baryon χPT (HBχPT) at LO We however do not see

a rationale to drop the higher-order Mminus1N terms when they are not negligible (ie when

their actual size exceeds by far the natural estimate for the size of higher-order terms)

Comparing our BχPT predictions to HBχPT we will also see a deficiency of HBχPT in

the description of the Q2 behaviour of the polarizabilities Note that the O(p4) HBχPT

results from Ref [50 51] which we use here for comparison do not include the ∆ These

references studied the leading effect of the latter in the HBχPT framework using the small-

scale expansion [35] observing no qualitative improvement in the HBχPT description of the

empirical data [50 51] when including it We therefore choose to use the O(p4) results as

the representative HBχPT curves

10

Another approach used in the literature to calculate the polarizabilities in χPT is the

infrared regularization (IR) scheme introduced in Ref [52] This covariant approach tries to

solve the power counting violation observed in Ref [53] by dropping the regular parts of the

loop integrals that contain the power-counting-breaking terms However this subtraction

scheme modifies the analytic structure of the loop contributions and may lead to unexpected

problems as was shown in Ref [54] As we will see in the next section the IR approach

also fails to describe the Q2 behaviour of the polarizabilities

III RESULTS AND DISCUSSION

We now present the NLO BχPT predictions for the nucleon polarizabilities and selected

moments of the nucleon spin structure functions Our results are obtained from the calcu-

lated non-Born VVCS amplitudes and the LEXs in Eqs (10) and (12) For a cross-check

we used the photoabsorption cross sections described in App B In addition to the full NLO

results we also analyse the individual contributions from the πN loops the ∆ exchange

and the π∆ loops

A γ0(Q2) mdash generalized forward spin polarizability

The forward spin polarizability

γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν3(19)

=16αM2

N

Q6

int x0

0

dx x2

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]

provides information about the spin-dependent response of the nucleon to transversal pho-

ton probes The RCS analogue of the above generalized forward spin polarizability sum

rule is sometimes referred to as the Gell-Mann Goldberger and Thirring (GGT) sum rule

[11] At Q2 = 0 the forward spin polarizability is expressed through the lowest-order spin

polarizabilities of RCS as γ0 = minus(γE1E1 + γM1M1 + γE1M2 + γM1E2) The forward spin po-

larizability of the proton is relevant for an accurate knowledge of the (muonic-)hydrogen

hyperfine splitting as it controls the leading proton-polarizability correction [16 62]

The πN -loop ∆-exchange and π∆-loop contributions to the NLO BχPT prediction of

11

000 005 010 015 020 025 030

-4

-3

-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

4D

proton

000 005 010 015 020 025 030

-4

-3

-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

4D

neutron

000 005 010 015 020 025 030

0

1

2

Q2 GeV2D

∆LT

10-

4fm

4D

proton

000 005 010 015 020 025 030

0

1

2

3

Q2 GeV2D

∆LT

10-

4fm

4D

neutron

FIG 2 Upper panel Generalized forward spin polarizability for the proton (left) and neutron

(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the

blue solid line and the blue band The red line represents the LO BχPT result The purple short-

dashed line is the O(p4) HB result from Ref [50] note that the corresponding proton curve is

outside of the plotted range The black dotted line is the MAID model prediction [55ndash57] which

is taken from Ref [1] (proton) and Ref [20] (neutron) The pink band is the IR+∆ result from

Ref [58] and the gray band is the BχPT+∆ result from Ref [31] Empirical extractions for the

proton Ref [18] (blue dots) Ref [59] (purple square) and Ref [60] (orange triangle uncertainties

added in quadrature) and neutron Ref [20] (blue diamonds) and Ref [61] (green dots statistical

and systematic uncertainties added in quadrature) Lower Panel Longitudinal-transverse spin

polarizability for the proton (left) and neutron (right) The orange dot-dashed and purple short-

dashed lines are the O(p3) and O(p4) HB results from Ref [50] The pink band is the IR result

from Ref [58] and the gray band is the covariant BχPT+∆ result from Ref [31] The black dotted

line is the MAID model prediction [55ndash57] note that for the proton we use the updated estimate

from Ref [1] obtained using the π η ππ channels

12

the static forward spin polarizability amount to in units of 10minus4 fm4

γ0p = minus093(92) asymp 201minus 284minus 010 (20a)

γ0n = 003(92) asymp 298minus 284minus 010 (20b)

while the slope is composed as follows in units of 10minus4 fm6

dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

= minus022(4) asymp minus033 + 011 + 001 (21a)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

= minus061(7) asymp minus073 + 011 + 001 (21b)

Figure 2 upper panel shows our NLO prediction as well as the LO πN loops compared

to different experimental and theoretical results For the proton we have one determination

at the real-photon point by the GDH collaboration [19] γ0p = minus100(8)(12)times10minus4 fm4 and

further Jefferson Laboratory data [18 60] at very low Q2 For the neutron only data at finite

Q2 are available [20 61] The experimental data for the proton are fairly well reproduced

in the whole Q2 range considered here while for the neutron the agreement improves with

increasing Q2 The HB limit of our πN -loop contribution reproduces the results published

in Refs [50 63] for arbitrary Q2 In addition our prediction is compared to the MAID

model [1 20] the IR+∆ calculation of Ref [58] and the BχPT+∆ result of Ref [31]

The πN -production channel gives a positive contribution to the photoabsorption cross

section σTT (νQ2) at low Q2 cf Fig 10 Accordingly one observes that the πN loops give

a sizeable positive contribution to γ0(Q2) The Delta on the other hand has a very large

effect by cancelling the πN loops and bringing the result close to the empirical data From

Fig 3 upper panel one can see that it is the ∆ exchange which dominates while π∆

loops are negligible This was expected since the forward spin polarizability sum rule is an

integral over the helicity-difference cross section in which σ32 is governed by the Delta at

low energies (the relevant energy region for the sum rule)

To elucidate the difference between the present calculation and the one from Ref [31] we

note that the two calculations differ in the following important aspects Firstly Ref [31] uses

the small-scale counting [64] that considers ∆ and mπ as being of the same size ∆ sim mπ

In practice this results in a set of π∆-loop graphs which contains graphs with one or two

γ∆∆ couplings and hence two or three Delta propagators Such graphs are suppressed in

the δ-counting and thus omitted from our calculation while present in that of Ref [31]

Secondly the Lagrangians describing the interaction of the Delta are constructed differently

and assume slightly different values for the coupling constants In particular we employ

13

000 005 010 015 020 025 030

-3

-2

-1

0

1

2

Q2GeV

2D

Γ01

0-

4fm

4D

proton

000 005 010 015 020 025 030

-3

-2

-1

0

1

2

3

Q2GeV

2D

Γ01

0-

4fm

4D

neutron

000 005 010 015 020 025 030

00

05

10

15

Q2GeV

2D

∆LT1

0-

4fm

4D

proton

000 005 010 015 020 025 030

00

05

10

15

20

25

Q2GeV

2D

∆LT1

0-

4fm

4D

neutron

FIG 3 Contributions of the different orders to the chiral predictions of γ0(Q2) upper paneland δLT (Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop

contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop con-

tribution blue long-dashed line total result purple dot-dot-dashed line total result without gCcontribution black short-dashed line total result without gM dipole

(where possible) the so-called ldquoconsistentrdquo couplings to the Delta field ie those couplings

that project out the spurious degree of freedom see Refs [37 65 66] The authors of

Ref [31] on the other hand use couplings where the consistency in this sense is not enforced

The effects of these differences are of higher order in the δ-counting expansion and their

contribution to the Q2 dependence of the considered polarizabilities is expected to be rather

small however the differences at Q2 = 0 could be noticeable [67]

Finally as mentioned in Sec II C the inclusion of the dipole form factor in the magnetic

coupling gM is expected to be important to generate the correct Q2 behaviour of the polar-

izabilities Comparing our predictions for the forward spin polarizability with and without

inclusion of the form factor see Fig 3 upper panel confirms this Without the dipole

our results for the proton and neutron are closer to the ones from Ref [31] where the form

factor is not included For the neutron our prediction without the dipole is able to describe

14

the experimental points at very low Q2 deviating from the data with increasing Q2 The

π∆-loop contribution does not modify the Q2 behavior of γ0(Q2) and only differs from

Ref [31] by a small global shift Note also the relatively large effect of gC which generates

a sign change for virtualities above sim 02 GeV2 see Fig 3 upper panel

B δLT (Q2) mdash longitudinal-transverse polarizability

The longitudinal-transverse spin polarizability

δLT (Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σLT (νQ2)

Qν2(22)

=16αM2

N

Q6

int x0

0

dx x2[g1(xQ2) + g2(xQ2)

]

contains information about the spin structure of the nucleon and is another important

input in the determination of the (muonic-)hydrogen hyperfine splitting [16 62] It is also

relevant in studies of higher-twist corrections to the structure function g2(xQ2) given by

the moment d2(Q2) [51] see Section III E The peculiarity of the response encoded in this

polarizability is that it involves a spin flip of the nucleon and a polarization change of the

photon see App B and Fig 11

It is expected that the Delta isobar gives only a small contribution to δLT (Q2) what makes

this polarizability a potentially clean test case for chiral calculations Consequently there are

relatively many different theoretical calculations of δLT (Q2) coming from different versions

of χPT with baryons (HB IR and covariant) Ref [50] found a systematic deviation of the

HB result for δLTn(Q2) from the MAID model prediction This disagreement was identified

by the authors of Ref [68] as a puzzle involving the neutron δLT polarizabilitymdashthe δLT

puzzle The IR calculation in Ref [58] also showed a deviation from the data and predicted

a rapid rise of δLT (Q2) with growing Q2 The problem is solved by keeping the relativistic

structure of the theory as the BχPT+∆ result of Ref [31] showed

As expected already the leading πN loops provide a reasonable agreement with the

experimental data cf Fig 2 lower panel Since the ∆-exchange contribution to δLT (Q2)

is small the effect of the gM form factor is negligible in this polarizability as is that of the gC

coupling cf Fig 3 lower panel In fact we predict both the ∆-exchange and the π∆-loop

contributions to be small and negative This is in agreement with the MAID model which

predicts a small and negative contribution of the P33 wave to δLT (Q2) However in the

calculation of Ref [31] which is different from the one presented here only in the way the

∆(1232) is included the contribution of this resonance to δLTp(Q2) is sizeable and positive

15

The authors of that work attributed this large contribution to diagrams where the photons

couple directly to the Delta inside a loop As mentioned in Sec III F the effect of such loop

diagrams does not change the Q2 behaviour of the polarizabilities On the other hand it

can produce a substantial shift of the δLT (Q2) as a whole A higher-order calculation should

resolve the discrepancy between the two covariant approaches however it will partially lose

the predictive power since the LECs appearing at higher orders will have to be fitted to

experimental data


the static longitudinal-transverse polarizability are in units of 10minus4 fm4

δLTp = 132(15) asymp 150minus 016minus 002 (23a)

δLTn = 218(23) asymp 235minus 016minus 002 (23b)

while the slopes are in units of 10minus4 fm6

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

= minus085(8) asymp minus080minus 004minus 001 (24a)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

= minus124(12) asymp minus119minus 004minus 001 (24b)

C IA(Q2) mdash a generalized GDH integral

The helicity-difference cross section σTT exhibits a faster fall-off in ν than its spin-averaged

counterpart σT This is due to a cancellation between the leading (constant) terms of σ12

and σ32 at large ν3 The resulting 1ν fall-off of the helicity-difference cross section allows

one to write an unsubtracted dispersion relation for the VVCS amplitude gnonpoleTT (νQ2) cf

Eq (10a) This is the origin of the GDH sum rule [5 6]

minus α

2M2N

κ2 =1

2π2

int infinν0

dνσTT (ν)

ν (25)

which establishes a relation to the anomalous magnetic moment κ It is experimentally

verified for the nucleon by MAMI (Mainz) and ELSA (Bonn) [71 72]

There are two extensions of the GDH sum rule to finite Q2 the generalized GDH integrals

3 Notice that a constant term in σTT at ν rarrinfin is forbidden by crossing symmetry

16

000 005 010 015 020 025 030

-30

-25

-20

-15

-10

-05

00

Q2 GeV2D

I Aproton

000 005 010 015 020 025 030

-25

-20

-15

-10

-05

00

Q2 GeV2D

I A

neutron

000 005 010 015 020 025 030

-05

00

05

Q2 GeV2D

I 1

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

I 1

neutron

FIG 4 Upper panel The generalized GDH integral IA(Q2) for the proton (left) and neutron

(right) as function of Q2 The result of this work the NLO BχPT prediction is shown by the blue

solid line and the blue band The red line represents the LO BχPT result The purple short-dashed

line is the O(p4) HB result from Ref [50 51] The gray band is the BχPT+∆ result from Ref [31]

The black dotted line is the MAID model prediction [69] Experimental extractions for the proton

Ref [60] (orange triangle uncertainties added in quadrature) and neutron Refs [21][27] where

magenta dotsorange diagonal crosses correspond to data and red squareslilac crosses correspond

to data plus extrapolation to unmeasured energy regions The green stars at the real-photon point

are derived from the anomalous magnetic moments κp asymp 1793 and κn asymp minus1913 [70] Lower

panel The generalized GDH integral I1(Q2) for the proton (left) and neutron (right) as function

of Q2 The purple short-dashed line is the HB result from Ref [51] Experimental extractions for

the proton Ref [18] (blue dots) and Ref [60] (orange triangle uncertainties added in quadrature)

and neutron Ref [61][27] (uncertainties added in quadrature) where green dotsorange diagonal

crosses correspond to data and gray squareslilac crosses correspond to data plus extrapolation to

unmeasured energy regions

17

IA(Q2) and I1(Q2) The latter will be discussed in Sec III D The former is defined as4

minus α

2M2N

IA(Q2) = minus 1

8π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν(26)

=α

Q2

int x0

0

dx[4M2

Nx2

Q2g2(xQ2)minus g1(xQ2)

]

Due to its energy weighting the integral in Eq (26) converges slower than the one in the

generalized forward spin polarizability sum rule (19) Therefore knowledge of the cross

section at higher energies is required and the evaluation of the generalized GDH integral

IA(Q2) is not as simple as the evaluation of γ0(Q2)

The generalized GDH integral IA(Q2) is directly related to the non-pole amplitude

gnonpoleTT (νQ2) which differs from non-Born amplitude gTT (νQ2) by a term involving the

elastic Pauli form factor

gnonpoleTT (νQ2) = gTT (νQ2)minus 2παν

M2N

F 22 (Q2) (27)

cf Eqs (2a) and (8a) Consequently IA(Q2) is not a pure polarizability but also contains

an elastic contribution The ldquonon-polarizabilityrdquo or the Born part of IA(Q2) is given by

IBornA (Q2) = IA(Q2)minus∆IA(Q2) = minus1

4F 2

2 (Q2) (28)

where we refer to the polarizability part as ∆IA(Q2) The same is true for the generalized

GDH integral I1(Q2) which is directly related to Snonpole1 (νQ2)

IBorn1 (Q2) = I1(Q2)minus∆I1(Q2) = minus1

4F 2

2 (Q2) (29)

In the following we will add the Born parts to our LO and NLO BχPT predictions for

the polarizabilities ∆IA(Q2) and ∆I1(Q2) employing an empirical parametrization for the

elastic Pauli form factor [73] This allows us to compare to the experimental results for

IA(Q2) and I1(Q2) cf Fig 4 Note that the blue error bands only describe the uncertainties

of our BχPT predictions of the polarizabilities while the elastic contributions are considered

to be exact as explained in Sec II C The uncertainties of the polarizability predictions are

therefore better reflected in Fig 5 where we show the contributions of the different orders

to the BχPT predictions of ∆IA(Q2) and ∆I1(Q2) as well as the total results with error

bands

4 Note that IA(Q2) is sometimes called ITT (Q2)

18

000 005 010 015 020 025 030

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

neutron

000 005 010 015 020 025 030-005

000

005

010

015

020

025

Q2 GeV2D

DI 1

proton

000 005 010 015 020 025 030-015

-010

-005

000

005

010

015

Q2 GeV2D

DI 1

neutron

FIG 5 Contributions of the different orders to the chiral predictions of ∆IA(Q2) upper paneland ∆I1(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop

contribution green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contri-

bution blue solid line and blue band total result purple dot-dot-dashed line total result without

gC contribution black short-dashed line total result without gM dipole

The E97-110 experiment at Jefferson Lab has recently published their data for IAn(Q2) in

the region of 0035 GeV2 lt Q2 lt 024 GeV2 [27] In addition there are results for IAn(Q2)

from the earlier E94-010 experiment [21] and for IAp(Q2) from the E08-027 experiment [60]

The O(p4) HB calculation gives a large negative effect [51] which does not describe the data

The BχPT+∆ result from Ref [31] which mainly differs from our work by the absence of

the dipole form factor in gM looks similar to this HB result and only describes the data

points at lowest Q2 Our NLO prediction however follows closely the Q2 evolution of the

data In Fig 5 upper panel we show the polarizability ∆IA(Q2) whose Q2 evolution is

clearly dominated by the ∆ exchange Similar to the case of γ0p(Q2) inclusion of the dipole

in gM and the Coulomb coupling gC is very important in order to describe the experimental

data The LO prediction on the other hand slightly overestimates the data cf Fig 4

upper panel

19

At the real-photon point IA(0) = minusκ2

4and ∆IA(0) = 0 Therefore we give only the

slope of the polarizability ∆IA(Q2) [showing also the separate contributions from πN loops

∆ exchange and π∆ loops] in units of GeVminus2

d∆IAp(Q2)

dQ2

∣∣∣∣Q2=0

= minus858(343) asymp 238minus 1121 + 025 (30a)

d∆IAn(Q2)

dQ2

∣∣∣∣Q2=0

= minus955(343) asymp 141minus 1121 + 025 (30b)

Including the empirical Pauli form factor [73] we find in units of GeVminus2

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

= minus318dIAn(Q2)

dQ2

∣∣∣∣Q2=0

= minus300 (31)

D Γ1(Q2) and I1(Q2) mdash the first moment of the structure function g1(xQ2)

The second variant for a generalization of the GDH sum rule to finite Q2 is defined as

minus α

2M2N

I1(Q2) = minus 1

8π2

int infinν0

dν1radic

ν2 +Q2

[σTT (νQ2) +

Q

νσLT (νQ2)

](32)

= minus α

Q2

int x0

0

dx g1(xQ2)

where I1(0) = minusκ2

4 This generalized GDH integral directly stems from the amplitude

Snonpole1 (νQ2) with the LEX from Eq (12a) It is given by the first moment of the struc-

ture function g1(xQ2) Γ1(Q2) =int x0

0dx g1(xQ2) as follows I1(Q2) =

2M2N

Q2 Γ1(Q2) The

isovector combination

Γ1(pminusn)(Q2) =

int x0

0

dx[g1p(xQ

2)minus g1n(xQ2)] (33)

is related to the axial coupling of the nucleon through the Bjorken sum rule [74 75]

limQ2rarrinfin

Γ1(pminusn)(Q2) =

gA6 (34)

As explained in Eq (28) the moment I1(Q2) splits into a polarizability part ∆I1(Q2) and

a Born part IBorn1 (Q2) Figure 4 lower panel shows the Q2 dependence of I1(Q2) which in

contrast to IA(Q2) shown in Figure 4 upper panel is clearly dominated by its Born part

and the elastic Pauli form factor The πN -loop ∆-exchange and π∆-loop contributions to

the polarizability ∆I1(Q2) are shown in Fig 5 lower panel Comparing to Fig 5 upper

20

000 005 010 015 020 025 030

-006

-004

-002

000

002

004

Q2 GeV2D

G1

proton

000 005 010 015 020 025 030

-006

-005

-004

-003

-002

-001

000

001

Q2 GeV2D

G1

neutron

FIG 6 First moment of the structure function g1(xQ2) for the proton (left) and neutron (right)

as function of Q2 The legend is the same as in Fig 4

panel one sees that ∆I1(Q2) is less sensitive to gC and the dipole form factor in gM than

∆IA(Q2)

For the proton our NLO BχPT prediction gives a very good description of the ex-

perimental data [18 60] and is in reasonable agreement with the MAID prediction [69]

For the neutron one observes good agreement with the empirical evaluations including ex-

trapolations to unmeasured energy regions starting from Q2 gt 01 GeV2 [27 61] In the

region of Q2 lt 005 GeV2 one observes an interesting tension between the recent E97-

110 experiment [27] and the data from CLAS [61] While the newest measurement finds

I1n(0035 GeV2) lt κ2n4 thus suggesting a negative slope at low Q the older measurement

found a rather large value for I1n(00496 GeV2) A similar but milder behaviour is seen in

the E97-110 [27] and E94-010 [21] data for IAn The MAID predictions do not agree with

the CODATA recommended values for the anomalous magnetic moments of the proton and

neutron [70] which in our work are imposed by using empirical parametrizations for the

elastic Pauli form factors [73] The slope of the HB result from Ref [51] is too large and

therefore only reproduces the data at very low Q2

Figure 6 shows the moment Γ1(Q2) for the proton and neutron while Fig 7 shows the

isovector combination Γ1 pminusn(Q2) The LO and NLO BχPT predictions are identical be-

cause our calculation produces the same Delta contributions for the proton and the neutron

For the isovector combination the MAID model only agrees with the data at very low

Q2 lt 010 GeV2 The same is true for the IR result [58 76] while all other chiral results

describe the data NLO BχPT (this work) BχPT+∆ [31] and HBχPT [51]

At the real-photon point I1(0) = minusκ2

4and ∆I1(0) = 0 Therefore we give only the

slope of the polarizability ∆I1(Q2) [showing also the separate contributions from πN loops

21

000 005 010 015 020 025 030-002

000

002

004

006

008

010

Q2 GeV2D

G1Hp-

nL

FIG 7 Isovector combination of Γ1(Q2) as function of Q2 The legend is the same as in Fig 6

The pink curve is the IR result from Ref [58 76] The experimental points are from Ref [77]

(brown dots) and Ref [23] (orange squares)


d∆I1p(Q2)

dQ2

∣∣∣∣Q2=0

= 039(4) asymp 034minus 053 + 058 (35a)

d∆I1n(Q2)

dQ2

∣∣∣∣Q2=0

= minus101(10) asymp minus107minus 053 + 058 (35b)


dI1p(Q2)

dQ2

∣∣∣∣Q2=0

= 580dI1n(Q2)

dQ2

∣∣∣∣Q2=0

= 553 (36)

E d2(Q2) mdash a measure of color polarizability

Another interesting moment to consider is d2(Q2) which is related to the twist-3 part of

the spin structure function g2(xQ2) [79 80]

d2(Q2) equiv 3

int 1

0

dx x2[g2(xQ2)minus gWW2 (xQ2)] (37)

where gWW2 (xQ2) is the twist-2 part of g2(xQ2) Using the Wandzura-Wilczek relation [81]

one can relate d2(Q2) to moments of the spin structure functions g1(xQ2) and g2(xQ2)

d2(Q2) =

int 1

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)

22

000 005 010 015 020 025 030

0000

0001

0002

0003

0004

0005

0006

0007

Q2 GeV2D

d2

proton

000 005 010 015 020 025 030

0000

0002

0004

0006

0008

0010

0012

Q2 GeV2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

Q2 GeV2D

Γ 010

-4

fm6

D

proton

000 005 010 015 020 025 030-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron

FIG 8 Upper panel The inelastic moment d2(Q2) for the proton (left) and neutron (right) as

function of Q2 The result of this work the NLO BχPT prediction is shown by the blue solid line

and the blue band The red line represents the LO BχPT result The purple short-dashed line is

the O(p4) HB result from Ref [50 51] The black dotted line is the MAID model prediction [69]

The experimental points for the neutron (cyan dots) are from Ref [22] Lower panel Fifth-order

generalized forward spin polarizability γ0(Q2) for the proton (left) and neutron (right) as function

of Q2 The experimental points for the proton are from Ref [59] (purple square) and Ref [78]

(orange dot)

This relation however only holds for asymptotically large Q2 It is also in the high-Q2

region where d2(Q2) is a measure of color polarizability [82 83] through its relation to the

gluon field strength tensor [80] We refer to Ref [84] for a recent review on the spin structure

of the nucleon including a discussion of sum rules for deep inelastic scattering and color

polarizabilities

What we consider in the following is the inelastic part of d2(Q2) defined as the moment

of g1(xQ2) and g2(xQ2) spin structure functions cf Eq (38)

d2(Q2) =

int x0

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)

23

This moment provides another testing ground for our BχPT predictions through comparison

with experiments on the neutron [22] Going towards the low-Q2 region the interpretation

of d2(Q2) in terms of color polarizabilities will fade out The above definition however

implies it is related to other VVCS polarizabilities

d2(Q2) =Q4

8M4N

[M2

NQ2

αδLT (Q2) + I1(Q2)minus IA(Q2)

] (40)

Note that d2(Q2) and its first two derivatives with respect to Q2 vanish at Q2 = 0 The

considerations in Eqs (28) and (29) have no effect on d2(Q2) since the Born contribution

from IA(Q2) and I1(Q2) cancel out Therefore d2(Q2) is a pure polarizability

In Fig 8 upper panel we show our NLO BχPT prediction and other results for d2(Q2)

While MAID [69] and BχPT describe the experimental data for the neutron [22] very well

the HB limit [50 51] is showing a fast growth with Q2 This illustrates the importance

of keeping the relativistic result Note also that even though the πN -loop contribution

is dominant both gC and the form factor in gM are essential to obtain a curvature that

reproduces the data cf Fig 9 upper panel For the proton there are to our knowledge

no experimental results to compare with However the agreement between the NLO BχPT

prediction and the MAID prediction at low energies is reasonable

F γ0(Q2) mdash fifth-order generalized forward spin polarizability

It is interesting to compare the generalized fifth-order forward spin polarizability sum

rule

γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν5(41)

=64αM4

N

Q10

int x0

0

dx x4

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]

to the sum rule integrals for IA(Q2) and γ0(Q2) since they differ merely by their energy

weighting of σTT (νQ2) and a constant prefactor cf Eqs (19) (26) and (41) From IA(Q2)

to γ0(Q2) to γ0(Q2) the energy suppression is increasing by a factor of νminus2 respectively

Therefore the description of γ0(Q2) should be easiest in a low-energy effective-field theory

such as χPT whereas γ0(Q2) and IA(Q2) receive larger contributions from higher energies

In Fig 8 lower panel we show our LO and NLO BχPT predictions for γ0(Q2) One

can see that the πN -loop contribution is positive (in accordance to what we see for the cross

section σTT see Fig 10) The Delta shifts it substantially especially in the low Q2 region

24

000 005 010 015 020 025 030

-0004

-0002

0000

0002

0004

0006

0008

0010

Q2GeV

2D

d2

proton

000 005 010 015 020 025 030

-0005

0000

0005

0010

0015

Q2GeV

2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

proton

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron

FIG 9 Contributions of the different orders to the chiral predictions of d2(Q2) upper panel and

γ0(Q2) lower panel for the proton (left) and neutron (right) Red solid line πN -loop contribu-

tion green dot-dashed line ∆-exchange contribution orange dotted line π∆-loop contribution

blue long-dashed line total result purple dot-dot-dashed line total result without gC contribution

black short-dashed line total result without gM dipole

bringing it into a better agreement with data In general the BχPT curves start above the

empirical data points at the real-photon point and then decrease asymptotically to zero

above Q2 gt 01 GeV2 On the other hand the MAID prediction reproduces the empirical

data at the real-photon point then decreases to negative values until about Q2 gt 006 GeV2

from where it also starts to asymptotically approach zero Consequently our NLO BχPT

prediction of γ0(Q2) is consistently above the MAID prediction This is very different to

what we saw for IA(Q2) in Fig 4 upper panel where the MAID prediction at the real-

photon point is above the experimental value While the agreement of our predictions with

the empirical data is in general quite good for all moments of σTT (νQ2) one should point

out that both for γ0n(Q2) and γ0p(Q2) we overestimate the data at low Q2 For IA(Q2) such

observation cannot be made because ∆IA(0) = 0 and thus IA(0) is given by the empirical

Pauli form factor only From IA(Q2) γ0(Q2) and γ0(Q2) the latter has the smallest however

25

non-negligible dependence on gC and the dipole in gM cf Fig 9 lower panel


the static fifth-order forward spin polarizability amount to in units of 10minus4 fm6

γ0p = 112(30) asymp 208minus 096minus 001 (42a)



dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0


Note that the HB prediction of the static polarizability γ0p (423 at O(p3) and 365 at O(ε3)

[78 85]) is almost one order of magnitude larger than the empirical value and therefore not

shown in Fig 8

G Summary

Our results are summarized in Table II where we give the contributions of the different

orders to the chiral predictions of the polarizabilities and their slopes at the real-photon

point A quantitative comparison of our predictions for the static spin polarizabilities to

the work of Bernard et al [31] and different empirical evaluations is shown in Table III We

can see that the inclusion of the Delta turns out to be very important for all moments of

the helicity-difference cross section To describe the Q2 behavior of the polarizabilities the

magnetic coupling of the N rarr ∆ transition should be modified by a dipole form factor as

has been observed previously in the description of electroproduction data [33] This dipole

form factor effectively takes account of vector-meson exchanges The Coulomb-quadrupole

N rarr ∆ transition despite its subleading order is important in the description of some

moments of spin structure functions This is contrary to what we saw for the moments of

unpolarized structure functions [30] where the Coulomb coupling had a negligible effect

The π∆ loops are mainly relevant for the generalized GDH integrals

26

TABLE II The NLO BχPT predictions for the forward VVCS polarizabilities and their slopes at

Q2 = 0 The contributions of the πN loops the ∆ exchange and the π∆ loops are shown together

with the combined total result Note that IA(0) = I1(0) = d2(0) = 0 and (d2)prime = 0

πN loops ∆ exchange π∆ loops Total

γ0 p 201 minus284 minus010minus093(92)

(10minus4 fm4) n 298 003(92)

δLT p 150 minus016 minus002132(15)

(10minus4 fm4) n 235 218(23)

γ0 p 208 minus096 minus001112(30)

(10minus4 fm6) n 292 195(30)

(γ0)prime p minus033011 001

minus022(4)

(10minus4 fm6) n minus073 minus061(7)

(δLT )prime p minus080 minus004 minus001minus085(8)



minus084(10)


(∆IA)prime p 238 minus1121 025minus858(343)

(GeVminus2) n 141 minus955(343)

(∆I1)prime p 034 minus053 058039(4)

(GeVminus2) n minus107 minus101(10)

TABLE III Our NLO BχPT predictions for the spin polarizabilities at Q2 = 0 compared with

the BχPT+∆ predictions from Bernard et al [31] and the available empirical information Where

the reference is not given the empirical number is provided by the MAID analysis [55 69] with

unspecified uncertainty

Proton Neutron

This work BχPT+∆ Empirical This work BχPT+∆ Empirical

γ0 minus093(92) minus174(40) minus100(8)(12) [19] 003(92) minus077(40) minus0005

(10minus4 fm4) minus090(8)(11) [78] [MAID]

minus0929(105) [59]

δLT 132(15) 240(1) 134 218(23) 238(3) 203

(10minus4 fm4) [MAID] [MAID]

γ0 112(30) 060(7)(7) [78] 195(30) 123

(10minus4 fm6) 0484(82) [59] [MAID]

IV CONCLUSIONS

We have presented a complete NLO calculation of the polarized non-Born VVCS ampli-

tudes in covariant BχPT with pion nucleon and ∆(1232) fields The dispersion relations

between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a

27

cross-check of these calculations

The obtained moments of the proton and neutron spin structure functions related to gen-

eralized polarizabilities and GDH-type integrals agree well with the available experimental

data The description of their Q2 evolution is improved compared to the previous χPT pre-

dictions In particular the NLO BχPT predictions obtained here give a better description

of the empirical data (eg from the Jefferson Laboratory ldquoSpin Physics Programrdquo) than

the HB [50 51] and IR [58] calculations

The demonstrated predictive power of the χPT framework amplitudes makes it well

suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of

(muonic-)hydrogen [15ndash17] to next-to-leading order

ACKNOWLEDGEMENTS

We thank Lothar Tiator and Marc Vanderhaeghen for helpful discussions This work

is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative

Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)] JMA ac-

knowledges support from the Community of Madrid through the ldquoPrograma de atraccion de

talento investigador 2017 (Modalidad 1)rdquo and the Spanish MECD grants FPA2016-77313-P

FH gratefully acknowledges financial support from the Swiss National Science Foundation

Appendix A Tensor decompositions of the VVCS amplitudes

In this appendix we review the decomposition of the forward VVCS process into tensor struc-

tures and scalar amplitudes In particular we consider the connection between the covariant and

the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional

transverse longitudinal transverse-transverse and transverse-longitudinal amplitudes

As explained in Sec II A the process of forward VVCS off the nucleon can be described in

terms of four explicitly covariant amplitudes S1 2 and T1 2 [3]

T (νQ2) =

(minusgmicroν +

qmicroqν

q2

)T1(νQ2) +

1

M2N

(pmicro minus p middot q

q2qmicro)(

pν minus p middot qq2

qν)T2(νQ2) (A1)

minus 1

MNγmicroναqα S1(νQ2)minus 1

M2N


)S2(νQ2)

εprimelowastmicro εν

where εmicro (εprimelowastmicro ) are the incoming (outgoing) photon polarization vectors ν is the photon lab-frame

energy and Q2 is the photon virtuality Alternatively the decomposition in the laboratory frame

28

(which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon

Pauli matrices ~σ and the four scalar functions fL fT gTT and gLT

T (νQ2) = ε0 εprimelowast0 fL(νQ2) + (~ε primelowast middot ~ε ) fT (νQ2) + i~σ middot (~ε primelowast times ~ε ) gTT (νQ2) (A2)

minusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)

Here ~q and q = ~q|~q | are the photon three-momentum in the lab system and its unit vector The

modified polarization vector components are given by

ε0 =

[ε0 minus

ν

|~q | (~ε middot q )

] |~q |Q ~ε = ~εminus q (~ε middot q ) (A3)

where ε = (ε0~ε ) is the usual incoming photon polarization vector and εprimelowast the outgoing polarization

vector The LEX of the lab frame amplitudes [Eq (10)] can serve in particular as the definition

of the generalized polarizabilities The lab frame amplitudes are also conveniently used for the

definition of the response functions see the example of the scalar amplitude gLT (νQ2) and the

corresponding response function σLT (νQ2) below in App B

Appendix B Photoabsorption cross sections

In the forward kinematics the spin-dependent VVCS amplitudes and the spin polarizabilities

can be described in terms of the polarized structure functions g1(xQ2) and g2(xQ2) or equiv-

alently the helicity-difference cross section σTT (νQ2) and the longitudinal-transverse response

function σLT (νQ2) with the help of dispersion relations (5) and the optical theorem (3) In this

way the photoabsorption cross sections measured in electroproduction processes form the basis

for most empirical evaluations shown throughout Sec III In the following we present the BχPT

predictions for the tree-level cross sections of πN - π∆- and ∆-production through photoabsorption

on the nucleon cf Figs 8 9 and 10 in Ref [30] In Secs B 1 and B 2 we will discuss the leading

πN -production channel and the ∆-production channel respectively We used these cross sections

to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS

amplitudes Due to the bad high-energy behavior of the π∆-production cross sections in BχPT

cf Fig 10 the dispersion relations in Eq (5) require further subtractions for a reconstruction of

the π∆-loop contribution to the spin-dependent VVCS amplitudes Therefore not all polarizabil-

ities could be verified but only those appearing as higher-order terms in the LEX of the VVCS

amplitudes such as γ0 [16]

29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

FIG 10 Photoabsorption cross sections for πN (red) and π∆ production (orange) with Q2 = 0

(solid) and Q2 = 01 GeV2 (dashed for πN and dotted for π∆ channel)

1 πN-production channel

In order to extract the response function σLT (νQ2) we have developed a method similar to

the one used to calculate σTT (νQ2) see for example Ref [86] For σLT (νQ2) however the

calculation is more complicated because one has to take into account that the associated Compton

process involves a spin-flip of the nucleon as illustrated in Fig 11 When calculating the cross

section the product of the incoming nucleon spinors has to reflect this flip

The forward VVCS amplitude related to σLT (νQ2) mdash and δLT (Q2) mdash is gLT (νQ2) It can

be extracted from Eq (A2) if one takes the modified polarization vector components in Eq (A3)

with ε = εL and εprimelowast = εlowastplusmn as input where εL = 1Q(|~q | 0 0 q0) and εplusmn = ∓ 1radic

2(0 1plusmni 0) are the

standard longitudinal and transverse polarization vectors respectively For εL and εlowastplusmn only the

choice of helicities hprime = plusmn12 and h = ∓12 gives a non-zero contribution and one obtains

χdaggerhprime T (νQ2)χh = χdaggerhprimeminusi~σ middot [(ε0~εprimelowast minus ~ε ε primelowast0 )times q] gLT (νQ2)χh =

radic2 gLT (νQ2) (B1)

where χh and χdaggerhprime are two-component Pauli spinors with opposite helicities or here spins

Let us now consider the related photoabsorption process and in particular the tree-level γlowastN rarr

30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1

rprimerprime hacute=frac12h=-frac12

λacute=1λ=0

FIG 11 Relation between the forward Compton process and the photoabsorption process given

by the optical theorem In particular we show the longitudinal-transverse contribution The

double-line arrows represent the spin of the external particles while the dot represents the scalar

(longitudinal) polarization of the incoming photon Inside the blob the intermediate states are

represented eg nucleons with spins rprime (which are averaged in the calculation of the cross section)

and pions

πN channel see diagrams in Fig 8 of Ref [30] We define the πN -production amplitude as

T = uhB (PB)sumi

Ai(s t)Γi uhA(PA) (B2)

with the Dirac structures

Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)

where uhA(PA) and udaggerhB (PB) are the Dirac spinors and PA and PB are the four-momenta of

the incoming and outgoing nucleons respectively When calculating the photoabsorption cross

section related to the VVCS amplitude in Eq (B1) the nucleon spin flip should be implemented

by uhprime(PA) in T dagger and uh(PA) in T together with the appropriate transverse and longitudinal

photon polarization vectors εlowastplusmn and εL

However if one wants to use the properties of the Dirac matrices it is more useful to construct

an operator to produce this spin flip in the external nucleons of Fig 11 This is accomplished by

introducing the projector ΓLT equiv 12radic

2(γ1 + iγ2)γ5 which also takes into account the extra factor

radic2 in Eq (B1) We checked that with this projector one correctly extracts δLT by comparing the

HB limit of our result to the HB result of Ref [50] where the authors calculate this polarizability

from the Compton amplitude directly With all those ingredients the longitudinal-transverse cross

section is calculated in the following way

σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with

Xij = Tr[(PB +MN )Γi(PA +MN )ΓLTγ0Γdaggerjγ

0] (B5)

where θ is the scattering angle in the center-of-mass (cm) frame and |~pi|cm (|~pf |cm) is the three-

momentum of an incoming (outgoing) particle in the cm frame An explicit calculation of the

matrix Xij leads to

X = MNQ

0 2(PB minus PA) middot εLminusradic

2|~qf |cm sin θ (sminus u)

(B6)

where |~qi|cm (|~qf |cm) is the relative three-momentum of the incoming (outgoing) particles in the

cm frame Here s t and u are the usual Mandelstam variables For the different γlowastN rarr πN

channels we obtain the following amplitudes Ai where we introduce qA as the four-momentum of

the incoming photon and qB as the four-momentum of the outgoing pion

bull γlowastprarr π0p

A1 =e gAMN

fπ

[2PA middot ε+ qA middot ε

sminusM2N

+2PB middot εminus qA middot ε

uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)

bull γlowastprarr π+n

A1 =

radic2 e gAMN

fπ


sminusM2N

+2(PA minus PB) middot ε+ qA middot ε

tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)

bull γlowastnrarr π0n

A1 = 0 (B9a)

A2 = 0 (B9b)

bull γlowastnrarr πminusp

A1 =

radic2 e gAMN

fπ

[2PB middot εminus qA middot ε

uminusM2N

minus 2(PA minus PB) middot ε+ qA middot εtminusm2

π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)

The analytical expressions shown above were checked with the amplitudes given in Ref [87] An-

alytical expressions for the tree-level γlowastN rarr πN channel of the σLT (νQ2) and σTT (νQ2) cross

32

sections are given below (proton channels π+n and π0p neutron channel πminusp) We checked that

they reproduce the known results in the real-photon limit [45 86] To shorten the final expressions

for the cross sections which are considerably longer for finite Q2 than in the real-photon limit we

define the following dimensionless kinematic variables

αγ = (ENi )cmradics =

s+M2N +Q2

2s (B11)

απ = (ENf )cmradics =

s+M2N minusm2

π

2s (B12)

βγ = Eγcmradics =

sminusM2N minusQ2

2s (B13)

βπ = Eπcmradics =

sminusM2N +m2

π

2s (B14)

λγ = |~qi|cmradics =

radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)

λπ = |~qf |cmradics =

radic(sminusM2

N +m2π)2 minus 4sm2

π

2s (B16)

Here (ENi )cm and (ENf )cm are the energies of the incoming and outgoing nucleon Eγcm is the energy

of the incoming photon Eπcm is the energy of the outgoing pion all in the cm frame

σ(π+n)TT = minus e2g2

AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2

N minus s)(M2N minusQ2 minus s)(Q2 + 2sβγβπ)

+ 2s(m2πQ

2 minus (M2N minus s)

(M2N + s(minus1 + 2βγβπ)

))λ2γ

]+ (M2

N minus s)(M2N minusQ2 minus s)

times (Q2 + 2sβγβπ minus 2sλπλγ)(Q2 + 2s(βγβπ + λπλγ)

)times log

(Q2 + 2sβγβπ minus 2sλγλπQ2 + 2sβγβπ + 2sλγλπ

) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2

N + s(2βγβπ minus 3))(M2N

+ s(2βγβπ minus 1))

+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2

N minus s)minus 2(minusm2

πQ2 + (sminusM2

N )2)

times sβγβπ + 4s2(sminusM2N )β2

γβ2π

)λ2γ

]arctan

(2sλπλγ

M2N + s(2βγβπ minus 1)

)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2

N ) + 4sβγβπ +(sminusM2

N +Q2)(3M2N minus 3s+ 2sβγβπ)

sλ2γ

+m2πQ

2(sminusM2N )

(M2N minus s+ 2sβγβπ)(M2

N + s(minus1 + 2βγβπ minus 2λπλγ))

+m2πQ

2(sminusM2N )


N + s(minus1 + 2βγβπ + 2λπλγ))

)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2

N minus s)(M2N minus s+ 2sβγβπ)

)minus (M2

N minus s)(M2N minusQ2 minus s)λ2

π

)log

(M2N + s(minus1 + 2βγβπ + 2λπλγ)

M2N + s(minus1 + 2βγβπ minus 2λπλγ)

)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2

N + s(2βπβγ minus 1))2 minus 4λ2πs

2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N

+ s(2βπβγ minus 1))

2 minus 2sλ2γ

(2λ2

πs(minusM2N +Q2 + s) +m2

πQ2))

+((M2N + s(2βπβγ minus 1)

)2

minus 4λ2πs

2λ2γ

)((M2N + s(2βπβγ minus 1)

)(M2N minus 2Q2 minus s(2βπβγ + 1)

)+ 4λ2

πs2λ2

γ

)times log

(minusM2

N minus 2βπsβγ minus 2λπsλγ + s)minus(M2N + s(2βπβγ minus 2λπλγ minus 1)

)(M2N + s(minus1

+ 2βπβγ + 2λπλγ))[((

M2N + s(2βπβγ minus 1)


)+ 4λ2

πs2λ2

γ

)times log

(minusM2

N minus 2βπsβγ + 2λπsλγ + s)

+((Q2 + 2βπsβγ)2 minus 4λ2

πs2λ2

γ

)times log

(Q2 + 2βπsβγ minus 2λπsλγQ2 + 2sβπβγ + 2λπsλγ

)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ

M2N + 2βπsβγ minus s

) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2

N minus s)(Q2 + 2sβ2γ)(Q2 + 2sβγβπ)

minus 4s

((M2

N minus s)(Q2 minus 2s(απ minus 1)βγ

)+Q2sβγβπ

)λ2γ + 8s3(minus1 + απ)λ4

γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2

γ)(Q2 + 2sβγβπ)2 + 4s2(2(απ minus 1)βγ(Q2 + 2sβγβπ)

minusQ2λ2π

)λ2γ + 8s3(απ minus 1)2λ4

γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1

minus2s(M2N + s(minus1 + 2βγβπ))2λ2

γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2

γ)(minus3 + 2βγβπ) +(Q2(minus3 + 2βγβπ + 6λ2

π) + 2sβγ

times (minus3 + 12απ + 2βγβπ minus 12απβγβπ + 4βγλ2π))λ2γ minus 4s(α2

π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2

+ 2sβ2γ)(minus1 + 2βγβπ)

(6 + βγβπ(minus9 + 2βγβπ)

)+

(Q2(3minus 12λ2

π + 4βγβπ(minus1 + 2βγβπ

+ λ2π))minus 2sβγ

(minus 3 + 4απ(3 + 2απ(3 + 2βγβπ(minus3 + βγβπ)) + 4βγ(βπ + (2minus βγβπ)

times λ2π)))λ2γ + 8s

((απ minus 1)(1 + απ minus 2βγβπ) + 2απβγλ

2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)

times (1minus 2βγβπ)2(2βγβπ minus 3) + 2(Q2(minus1 + 6λ2

π + 2βγβπ((1minus 2βγβπ)2 minus 2λ2π)) + 2sβγ

times (minus1 + 2βγβπ + 4(minus1 + βγβπ)(minusαπ + 2απβγβπ minus βγλ2π)))λ2γ minus 8

(s(απ minus 1)(απ

+ (1minus 2βγβπ)2) + 2βγ(2sαπ +Q2βγ)λ2π

)λ4γ + 32s(απ minus 1)λ2

πλ6γ

)+ 2M6

Ns

(Q2(6

minus 7βγβπ + λ2γ) + 2sβγ(βγ(6minus 7βγβπ) + (1minus 4απ)λ2

γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2

N + s(2βγβπ minus 3))(M2N + s(2βγβπ minus 1)

)+ 2s

(minusM2

N (Q2 + 2sβγ minus 8sαπβγ)

+ s(2sβγ(1minus 4απ + 4απβγβπ) +Q2(1minus 2λ2π)))λ2γ + 8s3(απ minus 1)λ4

γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2

N minusQ2 minus s)(M2N + s(2βπβγ + 2λπλγ minus 1))

times

1

M2N + s(2βπβγ minus 2λπλγ minus 1)

[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2

+ 2(2απ minus 1)sβγ(2βπβγ minus 1))minusM2N (Q2 + 2(1minus 2απ)sβγ)

)+ (Q2 + 2sβ2

γ)(M2N + s(2βπβγ minus 1)

)2 + 8(απ minus 1)απs

3λ4γ

)]minus(M2N + s(2βπβγ + 2λπλγ minus 1)

)times[

log

(minusM2N minus 2βπsβγ minus 2λπsλγ + s

minusM2N minus 2βπsβγ + 2λπsλγ + s

)(minus (Q2 + 2sβ2


)(minusM2

N + 2Q2 + 2βπsβγ + s)minus 2sλ2γ

(minusM2

N (Q2 + 2sβγ) +Q4 +Q2s(4απβγ

minus 2βγ minus 2λ2π + 1) + 2s2βγ(4(απ minus 1)βπβγ + 1)

)minus 8(απ minus 1)2s3λ4

γ

)+ 2

(4s2λ2

γ

(2(απ minus 1)βγ(Q2 + 2βπsβγ)minus λ2

πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)

+ 8(απ minus 1)2s3λ4γ

)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)

2 ∆-production channel

The tree-level ∆-exchange diagram in Fig 2 of Ref [30] contributes to the non-Born part of

the VVCS amplitudes The contribution of the ∆ exchange to the VVCS amplitudes can be split

into [17]

S∆-exch1 (νQ2) = S∆-pole

1 (νQ2) + S∆-exch1 (νQ2) (B23a)

νS∆-exch2 (νQ2) = νS∆-pole

2 (νQ2) + νS2∆-exch

(νQ2) (B23b)

and similarly for the unpolarized VVCS amplitudes discussed in Ref [30] Here we introduced the

∆-pole contributions S∆-polei and the ∆-non-pole contributions S∆-exch

i The former amplitudes

feature a pole at the ∆(1232)-production threshold and thus are proportional to

1

[sminusM2∆][uminusM2

∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36

They can be reconstructed from the dispersion relations in Eq (5) with the tree-level ∆-production

cross sections as input cf Fig 10 in Ref [30]

σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2

MMN (M+ + ν)|~q |2 +g2E(∆minus ν)(Q2 minusMNν)2

MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆

minus 4gMgE(Q2 minusMNν)|~q |2 minus 4gMgCQ2|~q |2

+2gEgCQ

2[minusMNM∆ |~q |2 + s(Q2 +∆ν)

]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |

g2E(MNν minusQ2) [M∆(MN + ν)minus s]

MN(B25b)

+g2CQ

2[MNM∆|~q |2 minus s(Q2 +∆ν)

]MNM2

∆

+ gMgEM∆|~q |2 minusgMgC(Q2 minusMNν)|~q |2

M∆

+gEgC(ν minus∆)(M2

N |~q |2 minus 2Q2s)

MNM∆

δ(ν minus ν∆)

with ∆ = M∆ minusMN M+ = M∆ + MN and the Mandelstam variable s = M2N + 2MNν minus Q2

Analytical expressions for the spin structure functions g1(xQ2) and g2(xQ2) can be constructed

from Eq (3) with the flux factor K(νQ2) = |~q | =radicν2 +Q2

In the ∆-non-pole contributions to S1(νQ2) and νS2(νQ2) the pole in ν at the ∆(1232)-

production threshold has canceled out

S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆

minus 8gMgEM∆ωminus (B26a)

minus2gMgCQ2(MN minus 4M∆)

M∆+

2gEgCQ2(3MN minus 2M∆)

M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+

[g2EM∆∆ωminus +

g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)

+gEgM M∆(M∆ω+ minus 4MNωminus)minus gEgC ∆(2Q2 +MNω+)

+gMgC Q2(4MN minus ω+)

]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic

(M∆ +MN )2 +Q2 and ωplusmn = (M2∆minusM2

N plusmnQ2)2M∆ and the non-pole contribution

to S2(νQ2)

S∆-exch2 (νQ2) = minus2παMNν

M∆M2+

[gM + gE

]gC (B27)

These amplitudes to the contrary are not described by the tree-level ∆-production cross sections

in the standard dispersive approach [17] This peculiarity has been previously missed e g in the

calculation of the ∆-exchange contribution to the hydrogen hyperfine splitting in Ref [88] The

importance of including the ∆-non-pole contribution is also evident when considering the BC sum

37

rule in Eq (14) The ∆-pole terms by themselves violate the BC sum rule but cancel exactly with

the ∆-non-pole terms

limνrarr0

νS∆-pole2 (νQ2) + lim

νrarr0νS2

∆-exch(νQ2) = 0 (B28)

Appendix C Polarizabilities at Q2 = 0

In this section we give analytical expressions for the static values and slopes at Q2 = 0 of

the spin polarizabilities and moments of polarized structure functions In particular we give the

HB expansion of the πN -loop contributions and the ∆-exchange contributions The complete

expressions also for the π∆-loop contributions can be found in the Supplemented material Recall

that IA(0) = I1(0) = d2(0) = 0 and dd2(Q2)dQ2

∣∣∣Q2=0

= 0

1 πN-loop contribution

Here we give analytical expressions for the πN -loop contributions to the proton and neutron

spin polarizabilities expanded in powers of micro = mπMN viz the HB expansion Note that we

choose to expand here to a high order in micro the strict HB expansion would only retain the leading

term in an analogous NLO calculation

bull Static polarizabilities (Q2 = 0)

γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro

16minus 2micro2 logmicro+

75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro

)micro2 minus 465πmicro3

64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro

16+ (1 + 2 log micro)micro2 minus 105πmicro3

128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192

5(1 + log micro)micro4

+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3

384minus 32micro4 logmicro

15

+1323πmicro5

2048+

(C7)

bull Slopes of polarizabilities at Q2 = 0

dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro

4+ 223micro2 minus 28515πmicro3

64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32

minus3 (1 + 90 logmicro)micro4 +84315πmicro5

1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15

8(11 + 48 logmicro)micro4 minus 267015πmicro5

8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1

2(115 + 88 logmicro)micro2 +

1839πmicro3

32

+5 (5 + 34 logmicro)micro4 +

(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π

8+ 2 (4 + 3 logmicro)microminus 537πmicro2

64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1

4(3 + 4 logmicro)microminus 57πmicro2

128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2

210minus 15551πmicro3

6144+

3371micro4

105

minus1640457πmicro5

32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)

2 ∆-exchange contribution

Here we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon

spin polarizabilities and their slopes at Q2 = 0 Note that the ∆-exchange contributes equally to

proton and neutron polarizabilities Recall that for the magnetic γlowastN∆ coupling we introduced a

dipole form factor to mimic vector-meson dominance gM rarr gM(1 +Q2Λ2)2


γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE

∆M+minus gEgCM∆M+

) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E

MNM∆minus gMgE∆MN

minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2

∆ + 2M∆MN minus 9M2N

∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E

∆M+[7M∆ minus 9MN ]

minus8gMgC∆

+4gEgCM+

) (C25)

[1] D Drechsel B Pasquini and M Vanderhaeghen Dispersion relations in real and virtual

Compton scattering Phys Rept 378 99 (2003) hep-ph0212124

[2] S E Kuhn J-P Chen and E Leader Spin structure of the nucleon mdash status and recent

results Prog Part Nucl Phys 63 1 (2009) arXiv08123535 [hep-ph]

[3] F Hagelstein R Miskimen and V Pascalutsa Nucleon polarizabilities from Compton scat-

tering to hydrogen atom Prog Part Nucl Phys 88 29 (2016) arXiv151203765 [nucl-th]

[4] B Pasquini and M Vanderhaeghen Dispersion theory in electromagnetic interactions Ann

Rev Nucl Part Sci 68 75 (2018) arXiv180510482 [hep-ph]

[5] S Gerasimov A Sum rule for magnetic moments and the damping of the nucleon magnetic

moment in nuclei Sov J Nucl Phys 2 430 (1966)

[6] S Drell and A C Hearn Exact sum rule for nucleon magnetic moments Phys Rev Lett

16 908 (1966)

[7] J S Schwinger Source theory viewpoints in deep inelastic scattering Proc Natl Acad Sci

USA 72 1 (1975)

[8] J S Schwinger Source theory viewpoints in deep inelastic scattering Electromagnetic In-

teractions and Field Theory Proceedings 14 Internationale Universitatswochen Schladming

Austria February 24-March 7 1975 Acta Phys Austriaca Suppl 14 471 (1975)

41

[9] J Schwinger Source theory discussion of deep inelastic scattering with polarized particles

Proc Natl Acad Sci USA 72 1559 (1975)

[10] A M Baldin Polarizability of nucleons Nucl Phys 18 310 (1960)

[11] M Gell-Mann M L Goldberger and W E Thirring Use of causality conditions in quantum

theory Phys Rev 95 1612 (1954)

[12] A Pineda Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic

hydrogen Phys Rev C 67 025201 (2003)

[13] C Peset and A Pineda The two-photon exchange contribution to muonic hydrogen from

chiral perturbation theory Nucl Phys B 887 69 (2014) arXiv14064524 [hep-ph]

[14] C Peset and A Pineda Model-independent determination of the two-photon exchange contri-

bution to hyperfine splitting in muonic hydrogen JHEP 04 060 arXiv161205206 [nucl-th]

[15] F Hagelstein and V Pascalutsa Proton structure in the hyperfine splitting of muonic hydro-

gen PoS CD15 077 (2016) arXiv151104301 [nucl-th]

[16] F Hagelstein Exciting Nucleons in Compton Scattering and Hydrogen-Like Atoms PhD

thesis Mainz U Inst Kernphys (2017) arXiv171000874 [nucl-th]

[17] F Hagelstein ∆(1232)-Resonance in the hydrogen spectrum Proceedings 11th International

Workshop on the Physics of Excited Nucleons (NSTAR 2017) Columbia SC USA August

20-23 2017 Few Body Syst 59 93 (2018) arXiv180109790 [nucl-th]

[18] Y Prok et al (CLAS) Moments of the spin structure functions gp1 and gd1 for 005 lt Q2 lt 30

GeV2 Phys Lett B 672 12 (2009) arXiv08022232 [nucl-ex]

[19] H Dutz et al (GDH) First measurement of the Gerasimov-Drell-Hearn sum rule for 1H from

07 GeV to 18 GeV at ELSA Phys Rev Lett 91 192001 (2003)

[20] M Amarian et al (Jefferson Lab E94010) Measurement of the generalized forward spin po-

larizabilities of the neutron Phys Rev Lett 93 152301 (2004) arXivnucl-ex0406005

[21] M Amarian et al The Q2 evolution of the generalized Gerasimov-Drell-Hearn integral for

the neutron using a 3He target Phys Rev Lett 89 242301 (2002) arXivnucl-ex0205020

[22] M Amarian et al (Jefferson Lab E94-010)Q2 evolution of the neutron spin structure moments

using a 3He target Phys Rev Lett 92 022301 (2004) arXivhep-ex0310003

[23] A Deur et al Experimental determination of the evolution of the Bjorken integral at low Q2

Phys Rev Lett 93 212001 (2004) arXivhep-ex0407007

[24] K Slifer Low Q2 measurement of gp2 and the δLT spin polarizability Spin structure at long dis-

tance Proceedings Workshop Newport News USA March 12ndash13 2009 nucl-ex09064775

(2009) AIP Conf Proc 1155 10106313203293 (2009) arXiv09064775 [nucl-ex]

[25] P Solvignon et al (E01-012) Moments of the neutron g2 structure function at intermediate

42

Q2 Phys Rev C 92 015208 (2015) arXiv13044497 [nucl-ex]

[26] A Deur Experimental studies at low Q2 of the spin structure of the nucleon at Jefferson Lab

in 9th International Workshop on Chiral Dynamics (CD18) Durham NC USA September

17-21 2018 (2019) arXiv190305661 [nucl-ex]

[27] V Sulkosky et al (Jefferson Lab E97-110) Measurement of the 3He spin-structure functions

and of neutron (3He) spin-dependent sum rules at 0035 le Q2 le 024 GeV2 Phys Lett B

805 135428 (2020) arXiv190805709 [nucl-ex]

[28] K Adhikari et al (CLAS) Measurement of the Q2 Dependence of the Deuteron Spin Structure

Function g1 and its Moments at Low Q2 with CLAS Phys Rev Lett 120 062501 (2018)

arXiv171101974 [nucl-ex]

[29] V Lensky F Hagelstein A Hiller Blin and V Pascalutsa Comment on rdquoMeasurement of

the Q2 Dependence of the Deuteron Spin Structure Function g1 and its Moments at Low Q2

with CLASrdquo (2018) arXiv180603219 [nucl-th]

[30] J M Alarcn F Hagelstein V Lensky and V Pascalutsa Forward doubly-virtual Compton

scattering off the nucleon in chiral perturbation theory at NLO the subtraction function and

moments of unpolarized structure functions (2020) arXiv200509518 [hep-ph]

[31] V Bernard E Epelbaum H Krebs and U-G Meiszligner New insights into the spin structure

of the nucleon Phys Rev D 87 054032 (2013) arXiv12092523 [hep-ph]

[32] V Lensky J M Alarcon and V Pascalutsa Moments of nucleon structure functions at

next-to-leading order in baryon chiral perturbation theory Phys Rev C 90 055202 (2014)

arXiv14072574 [hep-ph]

[33] V Pascalutsa and M Vanderhaeghen Chiral effective-field theory in the ∆(1232) region

I Pion electroproduction on the nucleon Phys Rev D 73 034003 (2006) arXivhep-

ph0512244

[34] V Pascalutsa and M Vanderhaeghen Electromagnetic nucleon-to-Delta transition in chiral

effective field theory Phys Rev Lett 95 232001 (2005) arXivhep-ph0508060

[35] T R Hemmert B R Holstein and J Kambor Systematic 1M expansion for spin 32

particles in baryon chiral perturbation theory Phys Lett B 395 89 (1997) arXivhep-

ph9606456

[36] V Pascalutsa and D R Phillips Effective theory of the ∆(1232) in Compton scattering off

the nucleon Phys Rev C 67 055202 (2003) arXivnucl-th0212024

[37] V Pascalutsa M Vanderhaeghen and S N Yang Electromagnetic excitation of the ∆(1232)-

resonance Phys Rept 437 125 (2007) arXivhep-ph0609004

[38] K A Olive et al (Particle Data Group) Review of Particle Physics Chin Phys C 38 090001

43

(2014)

[39] F E Low Scattering of light of very low frequency by systems of spin 12 Phys Rev 96

1428 (1954)

[40] M Gell-Mann and M L Goldberger Scattering of low-energy photons by particles of spin

12 Phys Rev 96 1433 (1954)

[41] A M Harun ar-Rashid A simple derivation of Schwingerrsquos sum rule for spin dependent

structure functions Nuovo Cim A 33 447 (1976)

[42] F Hagelstein and V Pascalutsa Dissecting the hadronic contributions to (g minus 2)micro by

Schwingerrsquos sum rule Phys Rev Lett 120 072002 (2018) arXiv171004571 [hep-ph]

[43] V Lensky V Pascalutsa M Vanderhaeghen and C Kao Spin-dependent sum rules con-

necting real and virtual Compton scattering verified Phys Rev D 95 074001 (2017)


[44] H Burkhardt and W N Cottingham Sum rules for forward virtual Compton scattering

Annals Phys 56 453 (1970)

[45] V Lensky and V Pascalutsa Predictive powers of chiral perturbation theory in Compton

scattering off protons Eur Phys J C 65 195 (2010) arXiv09070451 [hep-ph]

[46] H Grieszlighammer J McGovern D Phillips and G Feldman Using effective field theory to

analyse low-energy Compton scattering data from protons and light nuclei Prog Part Nucl

Phys 67 841 (2012) arXiv12036834 [nucl-th]

[47] H W Grieszlighammer J A McGovern and D R Phillips Nucleon polarisabilities at and

beyond physical pion masses Eur Phys J A 52 139 (2016) arXiv151101952 [nucl-th]

[48] E Epelbaum H Krebs and U-G Meiszligner Improved chiral nucleon-nucleon potential up to

next-to-next-to-next-to-leading order Eur Phys J A 51 53 (2015) arXiv14120142 [nucl-

th]

[49] E Epelbaum H Krebs and U-G Meiszligner Precision nucleon-nucleon potential at fifth order

in the chiral expansion Phys Rev Lett 115 122301 (2015) arXiv14124623 [nucl-th]

[50] C W Kao T Spitzenberg and M Vanderhaeghen Burkhardt-Cottingham sum rule and

forward spin polarizabilities in heavy baryon chiral perturbation theory Phys Rev D 67

016001 (2003) arXivhep-ph0209241

[51] C-W Kao D Drechsel S Kamalov and M Vanderhaeghen Higher moments of nucleon

spin structure functions in heavy baryon chiral perturbation theory and in a resonance model

Phys Rev D 69 056004 (2004) arXivhep-ph0312102

[52] T Becher and H Leutwyler Baryon chiral perturbation theory in manifestly Lorentz invariant

form Eur Phys J C 9 643 (1999) arXivhep-ph9901384

44

[53] J Gasser M E Sainio and A Svarc Nucleons with chiral loops Nucl Phys B 307 779

(1988)

[54] L S Geng J Martin Camalich L Alvarez-Ruso and M J Vicente Vacas Leading SU(3)-

breaking corrections to the baryon magnetic moments in chiral perturbation theory Phys

Rev Lett 101 222002 (2008) arXiv08051419 [hep-ph]

[55] D Drechsel S S Kamalov and L Tiator The GDH sum rule and related integrals Phys

Rev D 63 114010 (2001) arXivhep-ph0008306

[56] D Drechsel O Hanstein S S Kamalov and L Tiator A unitary isobar model for pion

photo- and electroproduction on the proton up to 1 GeV Nucl Phys A 645 145 (1999)

arXivnucl-th9807001

[57] L Tiator private communication (2020)

[58] V Bernard T R Hemmert and U-G Meiszligner Spin structure of the nucleon at low energies


[59] O Gryniuk F Hagelstein and V Pascalutsa Evaluation of the forward Compton scattering

off protons II Spin-dependent amplitude and observables Phys Rev D 94 034043 (2016)

arXiv160400789 [nucl-th]

[60] R Zielinski The g2p Experiment A Measurement of the Protonrsquos Spin Structure Functions

PhD thesis New Hampshire U (2010) arXiv170808297 [nucl-ex]

[61] N Guler et al (CLAS) Precise determination of the deuteron spin structure at low to mod-

erate Q2 with CLAS and extraction of the neutron contribution Phys Rev C 92 055201

(2015) arXiv150507877 [nucl-ex]

[62] C E Carlson V Nazaryan and K Griffioen Proton structure corrections to electronic

and muonic hydrogen hyperfine splitting Phys Rev A 78 022517 (2008) arXiv08052603

[physicsatom-ph]

[63] V Bernard N Kaiser and U-G Meiszligner Chiral dynamics in nucleons and nuclei Int J

Mod Phys E 4 193 (1995) arXivhep-ph9501384

[64] T R Hemmert B R Holstein and J Kambor ∆(1232) and the polarizabilities of the nucleon


[65] V Pascalutsa and R Timmermans Field theory of nucleon to higher-spin baryon transitions

Phys Rev C 60 042201 (1999) arXivnucl-th9905065

[66] V Pascalutsa Quantization of an interacting spin-32 field and the Delta isobar Phys Rev

D 58 096002 (1998) arXivhep-ph9802288

[67] H Krebs Double Virtual Compton Scattering and SpinStructure of the Nucleon PoS

CD2018 031 (2019)

45

[68] N Kochelev and Y Oh Axial anomaly and the δLT puzzle Phys Rev D 016012 (2012)


[69] D Drechsel S Kamalov and L Tiator Unitary isobar model ndash MAID2007 Eur Phys J A

69 (2007) available at httpsmaidkphuni-mainzde arXiv07100306 [nucl-th]

[70] P J Mohr B N Taylor and D B Newell CODATA recommended values of the fundamental

physical constants 2010 Rev Mod Phys 84 1527 (2012)

[71] J Ahrens et al (GDH A2) First measurement of the Gerasimov-Drell-Hearn integral for 1H

from 200 to 800 MeV Phys Rev Lett 87 022003 (2001) arXivhep-ex0105089 [hep-ex]

[72] K Helbing (GDH) Experimental verification of the GDH sum rule at ELSA and MAMI Nucl

Phys Proc Suppl 105 113 (2002)

[73] R Bradford A Bodek H S Budd and J Arrington A New parameterization of the nucleon

elastic form-factors NuInt05 proceedings of the 4th International Workshop on Neutrino-

Nucleus Interactions in the Few-GeV Region Okayama Japan 26-29 September 2005 Nucl

Phys Proc Suppl 159 127 (2006) arXivhep-ex0602017

[74] J D Bjorken Applications of the chiral U(6)timesU(6) algebra of current densities Phys Rev

148 1467 (1966)

[75] J D Bjorken Inelastic scattering of polarized leptons from polarized nucleons Phys Rev D

1 1376 (1970)

[76] V Bernard T R Hemmert and U-G Meiszligner Novel analysis of chiral loop effects in

the generalized Gerasimov-Drell-Hearn sum rule Phys Lett B 545 105 (2002) arXivhep-

ph0203167

[77] A Deur et al Experimental study of isovector spin sum rules Phys Rev D 78 032001

(2008) arXiv08023198 [nucl-ex]

[78] B Pasquini P Pedroni and D Drechsel Higher order forward spin polarizability Phys Lett

B 687 160 (2010) arXiv10014230 [hep-ph]

[79] R Jaffe g2ndashThe nucleonrsquos other spin-dependent structure function Comments Nucl Part

Phys 19 239 (1990)

[80] E V Shuryak and A Vainshtein Theory of power corrections to deep inelastic scattering in

quantum chromodynamics (II) Qminus4 effects polarized target Nucl Phys B 201 141 (1982)

[81] S Wandzura and F Wilczek Sum rules for spin dependent electroproduction Test of rela-

tivistic constituent quarks Phys Lett B 72 195 (1977)

[82] B W Filippone and X-D Ji The spin structure of the nucleon Adv Nucl Phys 26 1

(2001) arXivhep-ph0101224

[83] M Burkardt The g2 structure function Proceedings Workshop on Spin structure at long

46

distance Newport News USA March 12-13 2009 AIP Conf Proc 1155 26 (2009)


[84] A Deur S J Brodsky and G F de Teramond The spin structure of the nucleon Rept

Prog Phys 82 076201 (2019) arXiv180705250 [hep-ph]

[85] B R Holstein D Drechsel B Pasquini and M Vanderhaeghen Higher order polarizabilities

of the proton Phys Rev C 61 034316 (2000) arXivhep-ph9910427

[86] B R Holstein V Pascalutsa and M Vanderhaeghen Sum rules for magnetic moments and

polarizabilities in QED and chiral effective-field theory Phys Rev D 72 094014 (2005)

arXivhep-ph0507016

[87] B Pasquini D Drechsel and L Tiator Invariant amplitudes for pion electroproduction Eur

Phys J A 34 387 (2007) arXiv07122327 [hep-ph]

[88] A J Buchmann Non-spherical proton shape and hydrogen hyperfine splitting Proceedings

International Workshop on Precision Physics of Simple Atomic Systems (PSAS 2008) Wind-

sor Ontario Canada July 21-26 2008 Can J Phys 87 773 (2009) arXiv09104747

[physicsatom-ph]

47

Forward doubly-virtual Compton scattering off the nucleon in chiral perturbation theory II Spin polarizabilities and moments of polarized structure functions

Abstract

Contents

I Introduction

II Calculation of unpolarized VVCS at NLO



C Details on PT calculation and uncertainty estimate

III Results and discussion

A 0(Q2) mdash generalized forward spin polarizability

B LT(Q2) mdash longitudinal-transverse polarizability


D 1(Q2) and I1(Q2) mdash the first moment of the structure function g1(xQ2)

E 2(Q2) mdash a measure of color polarizability

F 0(Q2) mdash fifth-order generalized forward spin polarizability

G Summary

IV Conclusions

Acknowledgements

A Tensor decompositions of the VVCS amplitudes

B Photoabsorption cross sections

1 N-production channel

2 -production channel

C Polarizabilities at Q2=0

1 N-loop contribution

2 -exchange contribution

References

CONTENTS

I Introduction 3

II Calculation of unpolarized VVCS at NLO 5

A VVCS amplitudes and relations to structure functions 5

B Low-energy expansions and relations to polarizabilities 6

C Details on χPT calculation and uncertainty estimate 8

III Results and discussion 11

A γ0(Q2) mdash generalized forward spin polarizability 11

B δLT (Q2) mdash longitudinal-transverse polarizability 15

C IA(Q2) mdash a generalized GDH integral 16

D Γ1(Q2) and I1(Q

2) mdash the first moment of the structure function g1(xQ2) 20

E d2(Q2) mdash a measure of color polarizability 22

F γ0(Q2) mdash fifth-order generalized forward spin polarizability 24

G Summary 26

IV Conclusions 27

Acknowledgements 28

A Tensor decompositions of the VVCS amplitudes 28

B Photoabsorption cross sections 29

1 πN -production channel 30

2 ∆-production channel 36

C Polarizabilities at Q2 = 0 38

1 πN -loop contribution 38

2 ∆-exchange contribution 40

References 41

2

I INTRODUCTION











Standard Model























3






















O(p2) α = 1(13704) MN = Mp = 93827 MeV

O(p3) gA = 1270 fπ = 9221 MeV mπ = 13957 MeV


4






MN


M2N


)S2(νQ2) (1)






gTT (νQ2) =ν

MN

[S1(νQ2)minus Q2

MN νS2(νQ2)

] (2a)

gLT (νQ2) =Q

MN

[S1(νQ2) +

ν

MN

S2(νQ2)] (2b)



toabsorption

ImS1(νQ2) =4π2α

νg1(xQ2) =

MNνK(νQ2)

ν2 +Q2

[Q


] (3a)


ν2g2(xQ2) =

M2NK(νQ2)

ν2 +Q2

[ν


] (3b)










5




S1(νQ2) =16παMN

Q2

int 1

0

dxg1(xQ2)


=2MN

π

int infinνel


](ν prime Q2)radic



N

Q2

int 1

0

dxg2(xQ2)


=2M2

N

π

int infinνel


QσLT minus σTT

](ν prime Q2)radic





i )



i = SBorni + Si (6)


SBorn1 (νQ2) =

2πα

M

Q2GM(Q2)F1(Q2)

ν2el minus ν2

minus F 22 (Q2)

(7a)


ν2el minus ν2

GM(Q2)F2(Q2) (7b)




1 (νQ2) +2πα

MN

F 22 (Q2) (8a)


2 (νQ2) (8b)


cf Eq (7b)

6



F1(Q2) =GE(Q2) + τGM(Q2)

1 + τ (9a)


1 + τ (9b)

where τ = Q24M2N

















1


2α

M2N

IA(Q2) ν + γ0(Q2)ν3 + γ0(Q2)ν5 + (10a)

1


2α

M2N

I3(Q2)Q+ δLT (Q2)ν2Q+ (10b)



I3(Q2) =M2

N

4π2α

int infinν0

dνK(νQ2)

νQσLT (νQ2) =

2M2N

Q2

int x0

0


] (11)



7


1

4πSnonpole

1 (νQ2) =2α

MN

I1(Q2) +

2α

MNQ2

[IA(Q2)minus I1(Q2)

]+MNδLT (Q2)

ν2 (12a)

1

4πνSnonpole

2 (νQ2) = 2αI2(Q2) +2α

Q2

[I1(Q2)minus IA(Q2)

]ν2 (12b)




I2(Q2) equiv 2M2N

Q2

int x0

0

dx g2(x Q2) =1

4F2(Q2)GM(Q2) (13)


Q2

16παM2N

limνrarr0

νS2(νQ2) =

int 1

0

dx g2 (x Q2) = 0 (14)



















see eg Ref [45]

8








L(2)∆ =

3e

2MNM+

N T3

igM F


+igCM∆


microν

+ Hc





gM rarrgM[

1 + (QΛ)2 ]2 (16)





δ(Q2) =

radic(∆

MN

)2

+

(Q2

2MN∆

)2

(17)










9

predictions by

σP (Q2) =

Max[δ4(0)P (0)2 δ4(0)P LO(0)2 δ2(0)PNLO(0)2

]+Max

[δ4(Q2)

[P (Q2)minus P (0)

]2 δ4(Q2)


]2


]2]12

(18)







exact























10














and the π∆ loops



γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν3(19)

=16αM2

N

Q6

int x0

0

dx x2

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









11

000 005 010 015 020 025 030

-4

-3

-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

4D

proton

000 005 010 015 020 025 030

-4

-3

-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

4D

neutron

000 005 010 015 020 025 030

0

1

2

Q2 GeV2D

∆LT

10-

4fm

4D

proton

000 005 010 015 020 025 030

0

1

2

3

Q2 GeV2D

∆LT

10-

4fm

4D

neutron
















12





dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0



























13

000 005 010 015 020 025 030

-3

-2

-1

0

1

2

Q2GeV

2D

Γ01

0-

4fm

4D

proton

000 005 010 015 020 025 030

-3

-2

-1

0

1

2

3

Q2GeV

2D

Γ01

0-

4fm

4D

neutron

000 005 010 015 020 025 030

00

05

10

15

Q2GeV

2D

∆LT1

0-

4fm

4D

proton

000 005 010 015 020 025 030

00

05

10

15

20

25

Q2GeV

2D

∆LT1

0-

4fm

4D

neutron
















14







δLT (Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σLT (νQ2)

Qν2(22)

=16αM2

N

Q6

int x0

0

dx x2[g1(xQ2) + g2(xQ2)

]
























15







experimental data






dδLTp(Q2)

dQ2

∣∣∣∣Q2=0


dδLTn(Q2)

dQ2

∣∣∣∣Q2=0








minus α

2M2N

κ2 =1

2π2

int infinν0

dνσTT (ν)

ν (25)





16

000 005 010 015 020 025 030

-30

-25

-20

-15

-10

-05

00

Q2 GeV2D

I Aproton

000 005 010 015 020 025 030

-25

-20

-15

-10

-05

00

Q2 GeV2D

I A

neutron

000 005 010 015 020 025 030

-05

00

05

Q2 GeV2D

I 1

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

I 1

neutron
















17


minus α

2M2N

IA(Q2) = minus 1

8π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν(26)

=α

Q2

int x0

0

dx[4M2

Nx2


]









M2N

F 22 (Q2) (27)




4F 2

2 (Q2) (28)




4F 2

2 (Q2) (29)









bands


18

000 005 010 015 020 025 030

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

neutron

000 005 010 015 020 025 030-005

000

005

010

015

020

025

Q2 GeV2D

DI 1

proton

000 005 010 015 020 025 030-015

-010

-005

000

005

010

015

Q2 GeV2D

DI 1

neutron
















upper panel

19





d∆IAp(Q2)

dQ2

∣∣∣∣Q2=0


d∆IAn(Q2)

dQ2

∣∣∣∣Q2=0



dIAp(Q2)

dQ2

∣∣∣∣Q2=0

= minus318dIAn(Q2)

dQ2

∣∣∣∣Q2=0

= minus300 (31)



minus α

2M2N

I1(Q2) = minus 1

8π2

int infinν0

dν1radic

ν2 +Q2

[σTT (νQ2) +

Q

νσLT (νQ2)

](32)

= minus α

Q2

int x0

0

dx g1(xQ2)






2M2N

Q2 Γ1(Q2) The


Γ1(pminusn)(Q2) =

int x0

0

dx[g1p(xQ



limQ2rarrinfin

Γ1(pminusn)(Q2) =

gA6 (34)






20

000 005 010 015 020 025 030

-006

-004

-002

000

002

004

Q2 GeV2D

G1

proton

000 005 010 015 020 025 030

-006

-005

-004

-003

-002

-001

000

001

Q2 GeV2D

G1

neutron




∆IA(Q2)























21

000 005 010 015 020 025 030-002

000

002

004

006

008

010

Q2 GeV2D

G1Hp-

nL





d∆I1p(Q2)

dQ2

∣∣∣∣Q2=0

= 039(4) asymp 034minus 053 + 058 (35a)

d∆I1n(Q2)

dQ2

∣∣∣∣Q2=0



dI1p(Q2)

dQ2

∣∣∣∣Q2=0

= 580dI1n(Q2)

dQ2

∣∣∣∣Q2=0

= 553 (36)




d2(Q2) equiv 3

int 1

0




d2(Q2) =

int 1

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)

22

000 005 010 015 020 025 030

0000

0001

0002

0003

0004

0005

0006

0007

Q2 GeV2D

d2

proton

000 005 010 015 020 025 030

0000

0002

0004

0006

0008

0010

0012

Q2 GeV2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

Q2 GeV2D

Γ 010

-4

fm6

D

proton

000 005 010 015 020 025 030-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron








(orange dot)





polarizabilities



d2(Q2) =

int x0

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)

23





d2(Q2) =Q4

8M4N

[M2

NQ2


] (40)














rule

γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν5(41)

=64αM4

N

Q10

int x0

0

dx x4

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









24

000 005 010 015 020 025 030

-0004

-0002

0000

0002

0004

0006

0008

0010

Q2GeV

2D

d2

proton

000 005 010 015 020 025 030

-0005

0000

0005

0010

0015

Q2GeV

2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

proton

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron


















25







dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0




shown in Fig 8

G Summary














26






(10minus4 fm4) n 298 003(92)


(10minus4 fm4) n 235 218(23)

γ0 p 208 minus096 minus001112(30)

(10minus4 fm6) n 292 195(30)


minus022(4)





minus084(10)




(∆I1)prime p 034 minus053 058039(4)






Proton Neutron




minus0929(105) [59]

δLT 132(15) 240(1) 134 218(23) 238(3) 203


γ0 112(30) 060(7)(7) [78] 195(30) 123

(10minus4 fm6) 0484(82) [59] [MAID]

IV CONCLUSIONS




27











ACKNOWLEDGEMENTS














T (νQ2) =

(minusgmicroν +

qmicroqν

q2

)T1(νQ2) +

1

M2N


q2qmicro)(


qν)T2(νQ2) (A1)

minus 1


M2N


)S2(νQ2)




28







ε0 =

[ε0 minus

ν
























29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References

I INTRODUCTION











Standard Model























3






















O(p2) α = 1(13704) MN = Mp = 93827 MeV

O(p3) gA = 1270 fπ = 9221 MeV mπ = 13957 MeV


4






MN


M2N


)S2(νQ2) (1)






gTT (νQ2) =ν

MN

[S1(νQ2)minus Q2

MN νS2(νQ2)

] (2a)

gLT (νQ2) =Q

MN

[S1(νQ2) +

ν

MN

S2(νQ2)] (2b)



toabsorption

ImS1(νQ2) =4π2α

νg1(xQ2) =

MNνK(νQ2)

ν2 +Q2

[Q


] (3a)


ν2g2(xQ2) =

M2NK(νQ2)

ν2 +Q2

[ν


] (3b)










5




S1(νQ2) =16παMN

Q2

int 1

0

dxg1(xQ2)


=2MN

π

int infinνel


](ν prime Q2)radic



N

Q2

int 1

0

dxg2(xQ2)


=2M2

N

π

int infinνel


QσLT minus σTT

](ν prime Q2)radic





i )



i = SBorni + Si (6)


SBorn1 (νQ2) =

2πα

M

Q2GM(Q2)F1(Q2)

ν2el minus ν2

minus F 22 (Q2)

(7a)


ν2el minus ν2

GM(Q2)F2(Q2) (7b)




1 (νQ2) +2πα

MN

F 22 (Q2) (8a)


2 (νQ2) (8b)


cf Eq (7b)

6



F1(Q2) =GE(Q2) + τGM(Q2)

1 + τ (9a)


1 + τ (9b)

where τ = Q24M2N

















1


2α

M2N

IA(Q2) ν + γ0(Q2)ν3 + γ0(Q2)ν5 + (10a)

1


2α

M2N

I3(Q2)Q+ δLT (Q2)ν2Q+ (10b)



I3(Q2) =M2

N

4π2α

int infinν0

dνK(νQ2)

νQσLT (νQ2) =

2M2N

Q2

int x0

0


] (11)



7


1

4πSnonpole

1 (νQ2) =2α

MN

I1(Q2) +

2α

MNQ2

[IA(Q2)minus I1(Q2)

]+MNδLT (Q2)

ν2 (12a)

1

4πνSnonpole

2 (νQ2) = 2αI2(Q2) +2α

Q2

[I1(Q2)minus IA(Q2)

]ν2 (12b)




I2(Q2) equiv 2M2N

Q2

int x0

0

dx g2(x Q2) =1

4F2(Q2)GM(Q2) (13)


Q2

16παM2N

limνrarr0

νS2(νQ2) =

int 1

0

dx g2 (x Q2) = 0 (14)



















see eg Ref [45]

8








L(2)∆ =

3e

2MNM+

N T3

igM F


+igCM∆


microν

+ Hc





gM rarrgM[

1 + (QΛ)2 ]2 (16)





δ(Q2) =

radic(∆

MN

)2

+

(Q2

2MN∆

)2

(17)










9

predictions by

σP (Q2) =

Max[δ4(0)P (0)2 δ4(0)P LO(0)2 δ2(0)PNLO(0)2

]+Max

[δ4(Q2)

[P (Q2)minus P (0)

]2 δ4(Q2)


]2


]2]12

(18)







exact























10














and the π∆ loops



γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν3(19)

=16αM2

N

Q6

int x0

0

dx x2

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









11

000 005 010 015 020 025 030

-4

-3

-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

4D

proton

000 005 010 015 020 025 030

-4

-3

-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

4D

neutron

000 005 010 015 020 025 030

0

1

2

Q2 GeV2D

∆LT

10-

4fm

4D

proton

000 005 010 015 020 025 030

0

1

2

3

Q2 GeV2D

∆LT

10-

4fm

4D

neutron
















12





dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0



























13

000 005 010 015 020 025 030

-3

-2

-1

0

1

2

Q2GeV

2D

Γ01

0-

4fm

4D

proton

000 005 010 015 020 025 030

-3

-2

-1

0

1

2

3

Q2GeV

2D

Γ01

0-

4fm

4D

neutron

000 005 010 015 020 025 030

00

05

10

15

Q2GeV

2D

∆LT1

0-

4fm

4D

proton

000 005 010 015 020 025 030

00

05

10

15

20

25

Q2GeV

2D

∆LT1

0-

4fm

4D

neutron
















14







δLT (Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σLT (νQ2)

Qν2(22)

=16αM2

N

Q6

int x0

0

dx x2[g1(xQ2) + g2(xQ2)

]
























15







experimental data






dδLTp(Q2)

dQ2

∣∣∣∣Q2=0


dδLTn(Q2)

dQ2

∣∣∣∣Q2=0








minus α

2M2N

κ2 =1

2π2

int infinν0

dνσTT (ν)

ν (25)





16

000 005 010 015 020 025 030

-30

-25

-20

-15

-10

-05

00

Q2 GeV2D

I Aproton

000 005 010 015 020 025 030

-25

-20

-15

-10

-05

00

Q2 GeV2D

I A

neutron

000 005 010 015 020 025 030

-05

00

05

Q2 GeV2D

I 1

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

I 1

neutron
















17


minus α

2M2N

IA(Q2) = minus 1

8π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν(26)

=α

Q2

int x0

0

dx[4M2

Nx2


]









M2N

F 22 (Q2) (27)




4F 2

2 (Q2) (28)




4F 2

2 (Q2) (29)









bands


18

000 005 010 015 020 025 030

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

neutron

000 005 010 015 020 025 030-005

000

005

010

015

020

025

Q2 GeV2D

DI 1

proton

000 005 010 015 020 025 030-015

-010

-005

000

005

010

015

Q2 GeV2D

DI 1

neutron
















upper panel

19





d∆IAp(Q2)

dQ2

∣∣∣∣Q2=0


d∆IAn(Q2)

dQ2

∣∣∣∣Q2=0



dIAp(Q2)

dQ2

∣∣∣∣Q2=0

= minus318dIAn(Q2)

dQ2

∣∣∣∣Q2=0

= minus300 (31)



minus α

2M2N

I1(Q2) = minus 1

8π2

int infinν0

dν1radic

ν2 +Q2

[σTT (νQ2) +

Q

νσLT (νQ2)

](32)

= minus α

Q2

int x0

0

dx g1(xQ2)






2M2N

Q2 Γ1(Q2) The


Γ1(pminusn)(Q2) =

int x0

0

dx[g1p(xQ



limQ2rarrinfin

Γ1(pminusn)(Q2) =

gA6 (34)






20

000 005 010 015 020 025 030

-006

-004

-002

000

002

004

Q2 GeV2D

G1

proton

000 005 010 015 020 025 030

-006

-005

-004

-003

-002

-001

000

001

Q2 GeV2D

G1

neutron




∆IA(Q2)























21

000 005 010 015 020 025 030-002

000

002

004

006

008

010

Q2 GeV2D

G1Hp-

nL





d∆I1p(Q2)

dQ2

∣∣∣∣Q2=0

= 039(4) asymp 034minus 053 + 058 (35a)

d∆I1n(Q2)

dQ2

∣∣∣∣Q2=0



dI1p(Q2)

dQ2

∣∣∣∣Q2=0

= 580dI1n(Q2)

dQ2

∣∣∣∣Q2=0

= 553 (36)




d2(Q2) equiv 3

int 1

0




d2(Q2) =

int 1

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)

22

000 005 010 015 020 025 030

0000

0001

0002

0003

0004

0005

0006

0007

Q2 GeV2D

d2

proton

000 005 010 015 020 025 030

0000

0002

0004

0006

0008

0010

0012

Q2 GeV2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

Q2 GeV2D

Γ 010

-4

fm6

D

proton

000 005 010 015 020 025 030-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron








(orange dot)





polarizabilities



d2(Q2) =

int x0

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)

23





d2(Q2) =Q4

8M4N

[M2

NQ2


] (40)














rule

γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν5(41)

=64αM4

N

Q10

int x0

0

dx x4

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









24

000 005 010 015 020 025 030

-0004

-0002

0000

0002

0004

0006

0008

0010

Q2GeV

2D

d2

proton

000 005 010 015 020 025 030

-0005

0000

0005

0010

0015

Q2GeV

2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

proton

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron


















25







dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0




shown in Fig 8

G Summary














26






(10minus4 fm4) n 298 003(92)


(10minus4 fm4) n 235 218(23)

γ0 p 208 minus096 minus001112(30)

(10minus4 fm6) n 292 195(30)


minus022(4)





minus084(10)




(∆I1)prime p 034 minus053 058039(4)






Proton Neutron




minus0929(105) [59]

δLT 132(15) 240(1) 134 218(23) 238(3) 203


γ0 112(30) 060(7)(7) [78] 195(30) 123

(10minus4 fm6) 0484(82) [59] [MAID]

IV CONCLUSIONS




27











ACKNOWLEDGEMENTS














T (νQ2) =

(minusgmicroν +

qmicroqν

q2

)T1(νQ2) +

1

M2N


q2qmicro)(


qν)T2(νQ2) (A1)

minus 1


M2N


)S2(νQ2)




28







ε0 =

[ε0 minus

ν
























29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References






















O(p2) α = 1(13704) MN = Mp = 93827 MeV

O(p3) gA = 1270 fπ = 9221 MeV mπ = 13957 MeV


4






MN


M2N


)S2(νQ2) (1)






gTT (νQ2) =ν

MN

[S1(νQ2)minus Q2

MN νS2(νQ2)

] (2a)

gLT (νQ2) =Q

MN

[S1(νQ2) +

ν

MN

S2(νQ2)] (2b)



toabsorption

ImS1(νQ2) =4π2α

νg1(xQ2) =

MNνK(νQ2)

ν2 +Q2

[Q


] (3a)


ν2g2(xQ2) =

M2NK(νQ2)

ν2 +Q2

[ν


] (3b)










5




S1(νQ2) =16παMN

Q2

int 1

0

dxg1(xQ2)


=2MN

π

int infinνel


](ν prime Q2)radic



N

Q2

int 1

0

dxg2(xQ2)


=2M2

N

π

int infinνel


QσLT minus σTT

](ν prime Q2)radic





i )



i = SBorni + Si (6)


SBorn1 (νQ2) =

2πα

M

Q2GM(Q2)F1(Q2)

ν2el minus ν2

minus F 22 (Q2)

(7a)


ν2el minus ν2

GM(Q2)F2(Q2) (7b)




1 (νQ2) +2πα

MN

F 22 (Q2) (8a)


2 (νQ2) (8b)


cf Eq (7b)

6



F1(Q2) =GE(Q2) + τGM(Q2)

1 + τ (9a)


1 + τ (9b)

where τ = Q24M2N

















1


2α

M2N

IA(Q2) ν + γ0(Q2)ν3 + γ0(Q2)ν5 + (10a)

1


2α

M2N

I3(Q2)Q+ δLT (Q2)ν2Q+ (10b)



I3(Q2) =M2

N

4π2α

int infinν0

dνK(νQ2)

νQσLT (νQ2) =

2M2N

Q2

int x0

0


] (11)



7


1

4πSnonpole

1 (νQ2) =2α

MN

I1(Q2) +

2α

MNQ2

[IA(Q2)minus I1(Q2)

]+MNδLT (Q2)

ν2 (12a)

1

4πνSnonpole

2 (νQ2) = 2αI2(Q2) +2α

Q2

[I1(Q2)minus IA(Q2)

]ν2 (12b)




I2(Q2) equiv 2M2N

Q2

int x0

0

dx g2(x Q2) =1

4F2(Q2)GM(Q2) (13)


Q2

16παM2N

limνrarr0

νS2(νQ2) =

int 1

0

dx g2 (x Q2) = 0 (14)



















see eg Ref [45]

8








L(2)∆ =

3e

2MNM+

N T3

igM F


+igCM∆


microν

+ Hc





gM rarrgM[

1 + (QΛ)2 ]2 (16)





δ(Q2) =

radic(∆

MN

)2

+

(Q2

2MN∆

)2

(17)










9

predictions by

σP (Q2) =

Max[δ4(0)P (0)2 δ4(0)P LO(0)2 δ2(0)PNLO(0)2

]+Max

[δ4(Q2)

[P (Q2)minus P (0)

]2 δ4(Q2)


]2


]2]12

(18)







exact























10














and the π∆ loops



γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν3(19)

=16αM2

N

Q6

int x0

0

dx x2

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









11

000 005 010 015 020 025 030

-4

-3

-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

4D

proton

000 005 010 015 020 025 030

-4

-3

-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

4D

neutron

000 005 010 015 020 025 030

0

1

2

Q2 GeV2D

∆LT

10-

4fm

4D

proton

000 005 010 015 020 025 030

0

1

2

3

Q2 GeV2D

∆LT

10-

4fm

4D

neutron
















12





dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0



























13

000 005 010 015 020 025 030

-3

-2

-1

0

1

2

Q2GeV

2D

Γ01

0-

4fm

4D

proton

000 005 010 015 020 025 030

-3

-2

-1

0

1

2

3

Q2GeV

2D

Γ01

0-

4fm

4D

neutron

000 005 010 015 020 025 030

00

05

10

15

Q2GeV

2D

∆LT1

0-

4fm

4D

proton

000 005 010 015 020 025 030

00

05

10

15

20

25

Q2GeV

2D

∆LT1

0-

4fm

4D

neutron
















14







δLT (Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σLT (νQ2)

Qν2(22)

=16αM2

N

Q6

int x0

0

dx x2[g1(xQ2) + g2(xQ2)

]
























15







experimental data






dδLTp(Q2)

dQ2

∣∣∣∣Q2=0


dδLTn(Q2)

dQ2

∣∣∣∣Q2=0








minus α

2M2N

κ2 =1

2π2

int infinν0

dνσTT (ν)

ν (25)





16

000 005 010 015 020 025 030

-30

-25

-20

-15

-10

-05

00

Q2 GeV2D

I Aproton

000 005 010 015 020 025 030

-25

-20

-15

-10

-05

00

Q2 GeV2D

I A

neutron

000 005 010 015 020 025 030

-05

00

05

Q2 GeV2D

I 1

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

I 1

neutron
















17


minus α

2M2N

IA(Q2) = minus 1

8π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν(26)

=α

Q2

int x0

0

dx[4M2

Nx2


]









M2N

F 22 (Q2) (27)




4F 2

2 (Q2) (28)




4F 2

2 (Q2) (29)









bands


18

000 005 010 015 020 025 030

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

neutron

000 005 010 015 020 025 030-005

000

005

010

015

020

025

Q2 GeV2D

DI 1

proton

000 005 010 015 020 025 030-015

-010

-005

000

005

010

015

Q2 GeV2D

DI 1

neutron
















upper panel

19





d∆IAp(Q2)

dQ2

∣∣∣∣Q2=0


d∆IAn(Q2)

dQ2

∣∣∣∣Q2=0



dIAp(Q2)

dQ2

∣∣∣∣Q2=0

= minus318dIAn(Q2)

dQ2

∣∣∣∣Q2=0

= minus300 (31)



minus α

2M2N

I1(Q2) = minus 1

8π2

int infinν0

dν1radic

ν2 +Q2

[σTT (νQ2) +

Q

νσLT (νQ2)

](32)

= minus α

Q2

int x0

0

dx g1(xQ2)






2M2N

Q2 Γ1(Q2) The


Γ1(pminusn)(Q2) =

int x0

0

dx[g1p(xQ



limQ2rarrinfin

Γ1(pminusn)(Q2) =

gA6 (34)






20

000 005 010 015 020 025 030

-006

-004

-002

000

002

004

Q2 GeV2D

G1

proton

000 005 010 015 020 025 030

-006

-005

-004

-003

-002

-001

000

001

Q2 GeV2D

G1

neutron




∆IA(Q2)























21

000 005 010 015 020 025 030-002

000

002

004

006

008

010

Q2 GeV2D

G1Hp-

nL





d∆I1p(Q2)

dQ2

∣∣∣∣Q2=0

= 039(4) asymp 034minus 053 + 058 (35a)

d∆I1n(Q2)

dQ2

∣∣∣∣Q2=0



dI1p(Q2)

dQ2

∣∣∣∣Q2=0

= 580dI1n(Q2)

dQ2

∣∣∣∣Q2=0

= 553 (36)




d2(Q2) equiv 3

int 1

0




d2(Q2) =

int 1

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)

22

000 005 010 015 020 025 030

0000

0001

0002

0003

0004

0005

0006

0007

Q2 GeV2D

d2

proton

000 005 010 015 020 025 030

0000

0002

0004

0006

0008

0010

0012

Q2 GeV2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

Q2 GeV2D

Γ 010

-4

fm6

D

proton

000 005 010 015 020 025 030-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron








(orange dot)





polarizabilities



d2(Q2) =

int x0

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)

23





d2(Q2) =Q4

8M4N

[M2

NQ2


] (40)














rule

γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν5(41)

=64αM4

N

Q10

int x0

0

dx x4

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









24

000 005 010 015 020 025 030

-0004

-0002

0000

0002

0004

0006

0008

0010

Q2GeV

2D

d2

proton

000 005 010 015 020 025 030

-0005

0000

0005

0010

0015

Q2GeV

2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

proton

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron


















25







dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0




shown in Fig 8

G Summary














26






(10minus4 fm4) n 298 003(92)


(10minus4 fm4) n 235 218(23)

γ0 p 208 minus096 minus001112(30)

(10minus4 fm6) n 292 195(30)


minus022(4)





minus084(10)




(∆I1)prime p 034 minus053 058039(4)






Proton Neutron




minus0929(105) [59]

δLT 132(15) 240(1) 134 218(23) 238(3) 203


γ0 112(30) 060(7)(7) [78] 195(30) 123

(10minus4 fm6) 0484(82) [59] [MAID]

IV CONCLUSIONS




27











ACKNOWLEDGEMENTS














T (νQ2) =

(minusgmicroν +

qmicroqν

q2

)T1(νQ2) +

1

M2N


q2qmicro)(


qν)T2(νQ2) (A1)

minus 1


M2N


)S2(νQ2)




28







ε0 =

[ε0 minus

ν
























29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References






MN


M2N


)S2(νQ2) (1)






gTT (νQ2) =ν

MN

[S1(νQ2)minus Q2

MN νS2(νQ2)

] (2a)

gLT (νQ2) =Q

MN

[S1(νQ2) +

ν

MN

S2(νQ2)] (2b)



toabsorption

ImS1(νQ2) =4π2α

νg1(xQ2) =

MNνK(νQ2)

ν2 +Q2

[Q


] (3a)


ν2g2(xQ2) =

M2NK(νQ2)

ν2 +Q2

[ν


] (3b)










5




S1(νQ2) =16παMN

Q2

int 1

0

dxg1(xQ2)


=2MN

π

int infinνel


](ν prime Q2)radic



N

Q2

int 1

0

dxg2(xQ2)


=2M2

N

π

int infinνel


QσLT minus σTT

](ν prime Q2)radic





i )



i = SBorni + Si (6)


SBorn1 (νQ2) =

2πα

M

Q2GM(Q2)F1(Q2)

ν2el minus ν2

minus F 22 (Q2)

(7a)


ν2el minus ν2

GM(Q2)F2(Q2) (7b)




1 (νQ2) +2πα

MN

F 22 (Q2) (8a)


2 (νQ2) (8b)


cf Eq (7b)

6



F1(Q2) =GE(Q2) + τGM(Q2)

1 + τ (9a)


1 + τ (9b)

where τ = Q24M2N

















1


2α

M2N

IA(Q2) ν + γ0(Q2)ν3 + γ0(Q2)ν5 + (10a)

1


2α

M2N

I3(Q2)Q+ δLT (Q2)ν2Q+ (10b)



I3(Q2) =M2

N

4π2α

int infinν0

dνK(νQ2)

νQσLT (νQ2) =

2M2N

Q2

int x0

0


] (11)



7


1

4πSnonpole

1 (νQ2) =2α

MN

I1(Q2) +

2α

MNQ2

[IA(Q2)minus I1(Q2)

]+MNδLT (Q2)

ν2 (12a)

1

4πνSnonpole

2 (νQ2) = 2αI2(Q2) +2α

Q2

[I1(Q2)minus IA(Q2)

]ν2 (12b)




I2(Q2) equiv 2M2N

Q2

int x0

0

dx g2(x Q2) =1

4F2(Q2)GM(Q2) (13)


Q2

16παM2N

limνrarr0

νS2(νQ2) =

int 1

0

dx g2 (x Q2) = 0 (14)



















see eg Ref [45]

8








L(2)∆ =

3e

2MNM+

N T3

igM F


+igCM∆


microν

+ Hc





gM rarrgM[

1 + (QΛ)2 ]2 (16)





δ(Q2) =

radic(∆

MN

)2

+

(Q2

2MN∆

)2

(17)










9

predictions by

σP (Q2) =

Max[δ4(0)P (0)2 δ4(0)P LO(0)2 δ2(0)PNLO(0)2

]+Max

[δ4(Q2)

[P (Q2)minus P (0)

]2 δ4(Q2)


]2


]2]12

(18)







exact























10














and the π∆ loops



γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν3(19)

=16αM2

N

Q6

int x0

0

dx x2

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









11

000 005 010 015 020 025 030

-4

-3

-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

4D

proton

000 005 010 015 020 025 030

-4

-3

-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

4D

neutron

000 005 010 015 020 025 030

0

1

2

Q2 GeV2D

∆LT

10-

4fm

4D

proton

000 005 010 015 020 025 030

0

1

2

3

Q2 GeV2D

∆LT

10-

4fm

4D

neutron
















12





dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0



























13

000 005 010 015 020 025 030

-3

-2

-1

0

1

2

Q2GeV

2D

Γ01

0-

4fm

4D

proton

000 005 010 015 020 025 030

-3

-2

-1

0

1

2

3

Q2GeV

2D

Γ01

0-

4fm

4D

neutron

000 005 010 015 020 025 030

00

05

10

15

Q2GeV

2D

∆LT1

0-

4fm

4D

proton

000 005 010 015 020 025 030

00

05

10

15

20

25

Q2GeV

2D

∆LT1

0-

4fm

4D

neutron
















14







δLT (Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σLT (νQ2)

Qν2(22)

=16αM2

N

Q6

int x0

0

dx x2[g1(xQ2) + g2(xQ2)

]
























15







experimental data






dδLTp(Q2)

dQ2

∣∣∣∣Q2=0


dδLTn(Q2)

dQ2

∣∣∣∣Q2=0








minus α

2M2N

κ2 =1

2π2

int infinν0

dνσTT (ν)

ν (25)





16

000 005 010 015 020 025 030

-30

-25

-20

-15

-10

-05

00

Q2 GeV2D

I Aproton

000 005 010 015 020 025 030

-25

-20

-15

-10

-05

00

Q2 GeV2D

I A

neutron

000 005 010 015 020 025 030

-05

00

05

Q2 GeV2D

I 1

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

I 1

neutron
















17


minus α

2M2N

IA(Q2) = minus 1

8π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν(26)

=α

Q2

int x0

0

dx[4M2

Nx2


]









M2N

F 22 (Q2) (27)




4F 2

2 (Q2) (28)




4F 2

2 (Q2) (29)









bands


18

000 005 010 015 020 025 030

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

neutron

000 005 010 015 020 025 030-005

000

005

010

015

020

025

Q2 GeV2D

DI 1

proton

000 005 010 015 020 025 030-015

-010

-005

000

005

010

015

Q2 GeV2D

DI 1

neutron
















upper panel

19





d∆IAp(Q2)

dQ2

∣∣∣∣Q2=0


d∆IAn(Q2)

dQ2

∣∣∣∣Q2=0



dIAp(Q2)

dQ2

∣∣∣∣Q2=0

= minus318dIAn(Q2)

dQ2

∣∣∣∣Q2=0

= minus300 (31)



minus α

2M2N

I1(Q2) = minus 1

8π2

int infinν0

dν1radic

ν2 +Q2

[σTT (νQ2) +

Q

νσLT (νQ2)

](32)

= minus α

Q2

int x0

0

dx g1(xQ2)






2M2N

Q2 Γ1(Q2) The


Γ1(pminusn)(Q2) =

int x0

0

dx[g1p(xQ



limQ2rarrinfin

Γ1(pminusn)(Q2) =

gA6 (34)






20

000 005 010 015 020 025 030

-006

-004

-002

000

002

004

Q2 GeV2D

G1

proton

000 005 010 015 020 025 030

-006

-005

-004

-003

-002

-001

000

001

Q2 GeV2D

G1

neutron




∆IA(Q2)























21

000 005 010 015 020 025 030-002

000

002

004

006

008

010

Q2 GeV2D

G1Hp-

nL





d∆I1p(Q2)

dQ2

∣∣∣∣Q2=0

= 039(4) asymp 034minus 053 + 058 (35a)

d∆I1n(Q2)

dQ2

∣∣∣∣Q2=0



dI1p(Q2)

dQ2

∣∣∣∣Q2=0

= 580dI1n(Q2)

dQ2

∣∣∣∣Q2=0

= 553 (36)




d2(Q2) equiv 3

int 1

0




d2(Q2) =

int 1

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)

22

000 005 010 015 020 025 030

0000

0001

0002

0003

0004

0005

0006

0007

Q2 GeV2D

d2

proton

000 005 010 015 020 025 030

0000

0002

0004

0006

0008

0010

0012

Q2 GeV2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

Q2 GeV2D

Γ 010

-4

fm6

D

proton

000 005 010 015 020 025 030-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron








(orange dot)





polarizabilities



d2(Q2) =

int x0

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)

23





d2(Q2) =Q4

8M4N

[M2

NQ2


] (40)














rule

γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν5(41)

=64αM4

N

Q10

int x0

0

dx x4

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









24

000 005 010 015 020 025 030

-0004

-0002

0000

0002

0004

0006

0008

0010

Q2GeV

2D

d2

proton

000 005 010 015 020 025 030

-0005

0000

0005

0010

0015

Q2GeV

2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

proton

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron


















25







dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0




shown in Fig 8

G Summary














26






(10minus4 fm4) n 298 003(92)


(10minus4 fm4) n 235 218(23)

γ0 p 208 minus096 minus001112(30)

(10minus4 fm6) n 292 195(30)


minus022(4)





minus084(10)




(∆I1)prime p 034 minus053 058039(4)






Proton Neutron




minus0929(105) [59]

δLT 132(15) 240(1) 134 218(23) 238(3) 203


γ0 112(30) 060(7)(7) [78] 195(30) 123

(10minus4 fm6) 0484(82) [59] [MAID]

IV CONCLUSIONS




27











ACKNOWLEDGEMENTS














T (νQ2) =

(minusgmicroν +

qmicroqν

q2

)T1(νQ2) +

1

M2N


q2qmicro)(


qν)T2(νQ2) (A1)

minus 1


M2N


)S2(νQ2)




28







ε0 =

[ε0 minus

ν
























29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References




S1(νQ2) =16παMN

Q2

int 1

0

dxg1(xQ2)


=2MN

π

int infinνel


](ν prime Q2)radic



N

Q2

int 1

0

dxg2(xQ2)


=2M2

N

π

int infinνel


QσLT minus σTT

](ν prime Q2)radic





i )



i = SBorni + Si (6)


SBorn1 (νQ2) =

2πα

M

Q2GM(Q2)F1(Q2)

ν2el minus ν2

minus F 22 (Q2)

(7a)


ν2el minus ν2

GM(Q2)F2(Q2) (7b)




1 (νQ2) +2πα

MN

F 22 (Q2) (8a)


2 (νQ2) (8b)


cf Eq (7b)

6



F1(Q2) =GE(Q2) + τGM(Q2)

1 + τ (9a)


1 + τ (9b)

where τ = Q24M2N

















1


2α

M2N

IA(Q2) ν + γ0(Q2)ν3 + γ0(Q2)ν5 + (10a)

1


2α

M2N

I3(Q2)Q+ δLT (Q2)ν2Q+ (10b)



I3(Q2) =M2

N

4π2α

int infinν0

dνK(νQ2)

νQσLT (νQ2) =

2M2N

Q2

int x0

0


] (11)



7


1

4πSnonpole

1 (νQ2) =2α

MN

I1(Q2) +

2α

MNQ2

[IA(Q2)minus I1(Q2)

]+MNδLT (Q2)

ν2 (12a)

1

4πνSnonpole

2 (νQ2) = 2αI2(Q2) +2α

Q2

[I1(Q2)minus IA(Q2)

]ν2 (12b)




I2(Q2) equiv 2M2N

Q2

int x0

0

dx g2(x Q2) =1

4F2(Q2)GM(Q2) (13)


Q2

16παM2N

limνrarr0

νS2(νQ2) =

int 1

0

dx g2 (x Q2) = 0 (14)



















see eg Ref [45]

8








L(2)∆ =

3e

2MNM+

N T3

igM F


+igCM∆


microν

+ Hc





gM rarrgM[

1 + (QΛ)2 ]2 (16)





δ(Q2) =

radic(∆

MN

)2

+

(Q2

2MN∆

)2

(17)










9

predictions by

σP (Q2) =

Max[δ4(0)P (0)2 δ4(0)P LO(0)2 δ2(0)PNLO(0)2

]+Max

[δ4(Q2)

[P (Q2)minus P (0)

]2 δ4(Q2)


]2


]2]12

(18)







exact























10














and the π∆ loops



γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν3(19)

=16αM2

N

Q6

int x0

0

dx x2

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









11

000 005 010 015 020 025 030

-4

-3

-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

4D

proton

000 005 010 015 020 025 030

-4

-3

-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

4D

neutron

000 005 010 015 020 025 030

0

1

2

Q2 GeV2D

∆LT

10-

4fm

4D

proton

000 005 010 015 020 025 030

0

1

2

3

Q2 GeV2D

∆LT

10-

4fm

4D

neutron
















12





dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0



























13

000 005 010 015 020 025 030

-3

-2

-1

0

1

2

Q2GeV

2D

Γ01

0-

4fm

4D

proton

000 005 010 015 020 025 030

-3

-2

-1

0

1

2

3

Q2GeV

2D

Γ01

0-

4fm

4D

neutron

000 005 010 015 020 025 030

00

05

10

15

Q2GeV

2D

∆LT1

0-

4fm

4D

proton

000 005 010 015 020 025 030

00

05

10

15

20

25

Q2GeV

2D

∆LT1

0-

4fm

4D

neutron
















14







δLT (Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σLT (νQ2)

Qν2(22)

=16αM2

N

Q6

int x0

0

dx x2[g1(xQ2) + g2(xQ2)

]
























15







experimental data






dδLTp(Q2)

dQ2

∣∣∣∣Q2=0


dδLTn(Q2)

dQ2

∣∣∣∣Q2=0








minus α

2M2N

κ2 =1

2π2

int infinν0

dνσTT (ν)

ν (25)





16

000 005 010 015 020 025 030

-30

-25

-20

-15

-10

-05

00

Q2 GeV2D

I Aproton

000 005 010 015 020 025 030

-25

-20

-15

-10

-05

00

Q2 GeV2D

I A

neutron

000 005 010 015 020 025 030

-05

00

05

Q2 GeV2D

I 1

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

I 1

neutron
















17


minus α

2M2N

IA(Q2) = minus 1

8π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν(26)

=α

Q2

int x0

0

dx[4M2

Nx2


]









M2N

F 22 (Q2) (27)




4F 2

2 (Q2) (28)




4F 2

2 (Q2) (29)









bands


18

000 005 010 015 020 025 030

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

neutron

000 005 010 015 020 025 030-005

000

005

010

015

020

025

Q2 GeV2D

DI 1

proton

000 005 010 015 020 025 030-015

-010

-005

000

005

010

015

Q2 GeV2D

DI 1

neutron
















upper panel

19





d∆IAp(Q2)

dQ2

∣∣∣∣Q2=0


d∆IAn(Q2)

dQ2

∣∣∣∣Q2=0



dIAp(Q2)

dQ2

∣∣∣∣Q2=0

= minus318dIAn(Q2)

dQ2

∣∣∣∣Q2=0

= minus300 (31)



minus α

2M2N

I1(Q2) = minus 1

8π2

int infinν0

dν1radic

ν2 +Q2

[σTT (νQ2) +

Q

νσLT (νQ2)

](32)

= minus α

Q2

int x0

0

dx g1(xQ2)






2M2N

Q2 Γ1(Q2) The


Γ1(pminusn)(Q2) =

int x0

0

dx[g1p(xQ



limQ2rarrinfin

Γ1(pminusn)(Q2) =

gA6 (34)






20

000 005 010 015 020 025 030

-006

-004

-002

000

002

004

Q2 GeV2D

G1

proton

000 005 010 015 020 025 030

-006

-005

-004

-003

-002

-001

000

001

Q2 GeV2D

G1

neutron




∆IA(Q2)























21

000 005 010 015 020 025 030-002

000

002

004

006

008

010

Q2 GeV2D

G1Hp-

nL





d∆I1p(Q2)

dQ2

∣∣∣∣Q2=0

= 039(4) asymp 034minus 053 + 058 (35a)

d∆I1n(Q2)

dQ2

∣∣∣∣Q2=0



dI1p(Q2)

dQ2

∣∣∣∣Q2=0

= 580dI1n(Q2)

dQ2

∣∣∣∣Q2=0

= 553 (36)




d2(Q2) equiv 3

int 1

0




d2(Q2) =

int 1

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)

22

000 005 010 015 020 025 030

0000

0001

0002

0003

0004

0005

0006

0007

Q2 GeV2D

d2

proton

000 005 010 015 020 025 030

0000

0002

0004

0006

0008

0010

0012

Q2 GeV2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

Q2 GeV2D

Γ 010

-4

fm6

D

proton

000 005 010 015 020 025 030-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron








(orange dot)





polarizabilities



d2(Q2) =

int x0

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)

23





d2(Q2) =Q4

8M4N

[M2

NQ2


] (40)














rule

γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν5(41)

=64αM4

N

Q10

int x0

0

dx x4

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









24

000 005 010 015 020 025 030

-0004

-0002

0000

0002

0004

0006

0008

0010

Q2GeV

2D

d2

proton

000 005 010 015 020 025 030

-0005

0000

0005

0010

0015

Q2GeV

2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

proton

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron


















25







dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0




shown in Fig 8

G Summary














26






(10minus4 fm4) n 298 003(92)


(10minus4 fm4) n 235 218(23)

γ0 p 208 minus096 minus001112(30)

(10minus4 fm6) n 292 195(30)


minus022(4)





minus084(10)




(∆I1)prime p 034 minus053 058039(4)






Proton Neutron




minus0929(105) [59]

δLT 132(15) 240(1) 134 218(23) 238(3) 203


γ0 112(30) 060(7)(7) [78] 195(30) 123

(10minus4 fm6) 0484(82) [59] [MAID]

IV CONCLUSIONS




27











ACKNOWLEDGEMENTS














T (νQ2) =

(minusgmicroν +

qmicroqν

q2

)T1(νQ2) +

1

M2N


q2qmicro)(


qν)T2(νQ2) (A1)

minus 1


M2N


)S2(νQ2)




28







ε0 =

[ε0 minus

ν
























29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References



F1(Q2) =GE(Q2) + τGM(Q2)

1 + τ (9a)


1 + τ (9b)

where τ = Q24M2N

















1


2α

M2N

IA(Q2) ν + γ0(Q2)ν3 + γ0(Q2)ν5 + (10a)

1


2α

M2N

I3(Q2)Q+ δLT (Q2)ν2Q+ (10b)



I3(Q2) =M2

N

4π2α

int infinν0

dνK(νQ2)

νQσLT (νQ2) =

2M2N

Q2

int x0

0


] (11)



7


1

4πSnonpole

1 (νQ2) =2α

MN

I1(Q2) +

2α

MNQ2

[IA(Q2)minus I1(Q2)

]+MNδLT (Q2)

ν2 (12a)

1

4πνSnonpole

2 (νQ2) = 2αI2(Q2) +2α

Q2

[I1(Q2)minus IA(Q2)

]ν2 (12b)




I2(Q2) equiv 2M2N

Q2

int x0

0

dx g2(x Q2) =1

4F2(Q2)GM(Q2) (13)


Q2

16παM2N

limνrarr0

νS2(νQ2) =

int 1

0

dx g2 (x Q2) = 0 (14)



















see eg Ref [45]

8








L(2)∆ =

3e

2MNM+

N T3

igM F


+igCM∆


microν

+ Hc





gM rarrgM[

1 + (QΛ)2 ]2 (16)





δ(Q2) =

radic(∆

MN

)2

+

(Q2

2MN∆

)2

(17)










9

predictions by

σP (Q2) =

Max[δ4(0)P (0)2 δ4(0)P LO(0)2 δ2(0)PNLO(0)2

]+Max

[δ4(Q2)

[P (Q2)minus P (0)

]2 δ4(Q2)


]2


]2]12

(18)







exact























10














and the π∆ loops



γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν3(19)

=16αM2

N

Q6

int x0

0

dx x2

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









11

000 005 010 015 020 025 030

-4

-3

-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

4D

proton

000 005 010 015 020 025 030

-4

-3

-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

4D

neutron

000 005 010 015 020 025 030

0

1

2

Q2 GeV2D

∆LT

10-

4fm

4D

proton

000 005 010 015 020 025 030

0

1

2

3

Q2 GeV2D

∆LT

10-

4fm

4D

neutron
















12





dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0



























13

000 005 010 015 020 025 030

-3

-2

-1

0

1

2

Q2GeV

2D

Γ01

0-

4fm

4D

proton

000 005 010 015 020 025 030

-3

-2

-1

0

1

2

3

Q2GeV

2D

Γ01

0-

4fm

4D

neutron

000 005 010 015 020 025 030

00

05

10

15

Q2GeV

2D

∆LT1

0-

4fm

4D

proton

000 005 010 015 020 025 030

00

05

10

15

20

25

Q2GeV

2D

∆LT1

0-

4fm

4D

neutron
















14







δLT (Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σLT (νQ2)

Qν2(22)

=16αM2

N

Q6

int x0

0

dx x2[g1(xQ2) + g2(xQ2)

]
























15







experimental data






dδLTp(Q2)

dQ2

∣∣∣∣Q2=0


dδLTn(Q2)

dQ2

∣∣∣∣Q2=0








minus α

2M2N

κ2 =1

2π2

int infinν0

dνσTT (ν)

ν (25)





16

000 005 010 015 020 025 030

-30

-25

-20

-15

-10

-05

00

Q2 GeV2D

I Aproton

000 005 010 015 020 025 030

-25

-20

-15

-10

-05

00

Q2 GeV2D

I A

neutron

000 005 010 015 020 025 030

-05

00

05

Q2 GeV2D

I 1

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

I 1

neutron
















17


minus α

2M2N

IA(Q2) = minus 1

8π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν(26)

=α

Q2

int x0

0

dx[4M2

Nx2


]









M2N

F 22 (Q2) (27)




4F 2

2 (Q2) (28)




4F 2

2 (Q2) (29)









bands


18

000 005 010 015 020 025 030

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

neutron

000 005 010 015 020 025 030-005

000

005

010

015

020

025

Q2 GeV2D

DI 1

proton

000 005 010 015 020 025 030-015

-010

-005

000

005

010

015

Q2 GeV2D

DI 1

neutron
















upper panel

19





d∆IAp(Q2)

dQ2

∣∣∣∣Q2=0


d∆IAn(Q2)

dQ2

∣∣∣∣Q2=0



dIAp(Q2)

dQ2

∣∣∣∣Q2=0

= minus318dIAn(Q2)

dQ2

∣∣∣∣Q2=0

= minus300 (31)



minus α

2M2N

I1(Q2) = minus 1

8π2

int infinν0

dν1radic

ν2 +Q2

[σTT (νQ2) +

Q

νσLT (νQ2)

](32)

= minus α

Q2

int x0

0

dx g1(xQ2)






2M2N

Q2 Γ1(Q2) The


Γ1(pminusn)(Q2) =

int x0

0

dx[g1p(xQ



limQ2rarrinfin

Γ1(pminusn)(Q2) =

gA6 (34)






20

000 005 010 015 020 025 030

-006

-004

-002

000

002

004

Q2 GeV2D

G1

proton

000 005 010 015 020 025 030

-006

-005

-004

-003

-002

-001

000

001

Q2 GeV2D

G1

neutron




∆IA(Q2)























21

000 005 010 015 020 025 030-002

000

002

004

006

008

010

Q2 GeV2D

G1Hp-

nL





d∆I1p(Q2)

dQ2

∣∣∣∣Q2=0

= 039(4) asymp 034minus 053 + 058 (35a)

d∆I1n(Q2)

dQ2

∣∣∣∣Q2=0



dI1p(Q2)

dQ2

∣∣∣∣Q2=0

= 580dI1n(Q2)

dQ2

∣∣∣∣Q2=0

= 553 (36)




d2(Q2) equiv 3

int 1

0




d2(Q2) =

int 1

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)

22

000 005 010 015 020 025 030

0000

0001

0002

0003

0004

0005

0006

0007

Q2 GeV2D

d2

proton

000 005 010 015 020 025 030

0000

0002

0004

0006

0008

0010

0012

Q2 GeV2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

Q2 GeV2D

Γ 010

-4

fm6

D

proton

000 005 010 015 020 025 030-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron








(orange dot)





polarizabilities



d2(Q2) =

int x0

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)

23





d2(Q2) =Q4

8M4N

[M2

NQ2


] (40)














rule

γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν5(41)

=64αM4

N

Q10

int x0

0

dx x4

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









24

000 005 010 015 020 025 030

-0004

-0002

0000

0002

0004

0006

0008

0010

Q2GeV

2D

d2

proton

000 005 010 015 020 025 030

-0005

0000

0005

0010

0015

Q2GeV

2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

proton

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron


















25







dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0




shown in Fig 8

G Summary














26






(10minus4 fm4) n 298 003(92)


(10minus4 fm4) n 235 218(23)

γ0 p 208 minus096 minus001112(30)

(10minus4 fm6) n 292 195(30)


minus022(4)





minus084(10)




(∆I1)prime p 034 minus053 058039(4)






Proton Neutron




minus0929(105) [59]

δLT 132(15) 240(1) 134 218(23) 238(3) 203


γ0 112(30) 060(7)(7) [78] 195(30) 123

(10minus4 fm6) 0484(82) [59] [MAID]

IV CONCLUSIONS




27











ACKNOWLEDGEMENTS














T (νQ2) =

(minusgmicroν +

qmicroqν

q2

)T1(νQ2) +

1

M2N


q2qmicro)(


qν)T2(νQ2) (A1)

minus 1


M2N


)S2(νQ2)




28







ε0 =

[ε0 minus

ν
























29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References


1

4πSnonpole

1 (νQ2) =2α

MN

I1(Q2) +

2α

MNQ2

[IA(Q2)minus I1(Q2)

]+MNδLT (Q2)

ν2 (12a)

1

4πνSnonpole

2 (νQ2) = 2αI2(Q2) +2α

Q2

[I1(Q2)minus IA(Q2)

]ν2 (12b)




I2(Q2) equiv 2M2N

Q2

int x0

0

dx g2(x Q2) =1

4F2(Q2)GM(Q2) (13)


Q2

16παM2N

limνrarr0

νS2(νQ2) =

int 1

0

dx g2 (x Q2) = 0 (14)



















see eg Ref [45]

8








L(2)∆ =

3e

2MNM+

N T3

igM F


+igCM∆


microν

+ Hc





gM rarrgM[

1 + (QΛ)2 ]2 (16)





δ(Q2) =

radic(∆

MN

)2

+

(Q2

2MN∆

)2

(17)










9

predictions by

σP (Q2) =

Max[δ4(0)P (0)2 δ4(0)P LO(0)2 δ2(0)PNLO(0)2

]+Max

[δ4(Q2)

[P (Q2)minus P (0)

]2 δ4(Q2)


]2


]2]12

(18)







exact























10














and the π∆ loops



γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν3(19)

=16αM2

N

Q6

int x0

0

dx x2

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









11

000 005 010 015 020 025 030

-4

-3

-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

4D

proton

000 005 010 015 020 025 030

-4

-3

-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

4D

neutron

000 005 010 015 020 025 030

0

1

2

Q2 GeV2D

∆LT

10-

4fm

4D

proton

000 005 010 015 020 025 030

0

1

2

3

Q2 GeV2D

∆LT

10-

4fm

4D

neutron
















12





dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0



























13

000 005 010 015 020 025 030

-3

-2

-1

0

1

2

Q2GeV

2D

Γ01

0-

4fm

4D

proton

000 005 010 015 020 025 030

-3

-2

-1

0

1

2

3

Q2GeV

2D

Γ01

0-

4fm

4D

neutron

000 005 010 015 020 025 030

00

05

10

15

Q2GeV

2D

∆LT1

0-

4fm

4D

proton

000 005 010 015 020 025 030

00

05

10

15

20

25

Q2GeV

2D

∆LT1

0-

4fm

4D

neutron
















14







δLT (Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σLT (νQ2)

Qν2(22)

=16αM2

N

Q6

int x0

0

dx x2[g1(xQ2) + g2(xQ2)

]
























15







experimental data






dδLTp(Q2)

dQ2

∣∣∣∣Q2=0


dδLTn(Q2)

dQ2

∣∣∣∣Q2=0








minus α

2M2N

κ2 =1

2π2

int infinν0

dνσTT (ν)

ν (25)





16

000 005 010 015 020 025 030

-30

-25

-20

-15

-10

-05

00

Q2 GeV2D

I Aproton

000 005 010 015 020 025 030

-25

-20

-15

-10

-05

00

Q2 GeV2D

I A

neutron

000 005 010 015 020 025 030

-05

00

05

Q2 GeV2D

I 1

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

I 1

neutron
















17


minus α

2M2N

IA(Q2) = minus 1

8π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν(26)

=α

Q2

int x0

0

dx[4M2

Nx2


]









M2N

F 22 (Q2) (27)




4F 2

2 (Q2) (28)




4F 2

2 (Q2) (29)









bands


18

000 005 010 015 020 025 030

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

neutron

000 005 010 015 020 025 030-005

000

005

010

015

020

025

Q2 GeV2D

DI 1

proton

000 005 010 015 020 025 030-015

-010

-005

000

005

010

015

Q2 GeV2D

DI 1

neutron
















upper panel

19





d∆IAp(Q2)

dQ2

∣∣∣∣Q2=0


d∆IAn(Q2)

dQ2

∣∣∣∣Q2=0



dIAp(Q2)

dQ2

∣∣∣∣Q2=0

= minus318dIAn(Q2)

dQ2

∣∣∣∣Q2=0

= minus300 (31)



minus α

2M2N

I1(Q2) = minus 1

8π2

int infinν0

dν1radic

ν2 +Q2

[σTT (νQ2) +

Q

νσLT (νQ2)

](32)

= minus α

Q2

int x0

0

dx g1(xQ2)






2M2N

Q2 Γ1(Q2) The


Γ1(pminusn)(Q2) =

int x0

0

dx[g1p(xQ



limQ2rarrinfin

Γ1(pminusn)(Q2) =

gA6 (34)






20

000 005 010 015 020 025 030

-006

-004

-002

000

002

004

Q2 GeV2D

G1

proton

000 005 010 015 020 025 030

-006

-005

-004

-003

-002

-001

000

001

Q2 GeV2D

G1

neutron




∆IA(Q2)























21

000 005 010 015 020 025 030-002

000

002

004

006

008

010

Q2 GeV2D

G1Hp-

nL





d∆I1p(Q2)

dQ2

∣∣∣∣Q2=0

= 039(4) asymp 034minus 053 + 058 (35a)

d∆I1n(Q2)

dQ2

∣∣∣∣Q2=0



dI1p(Q2)

dQ2

∣∣∣∣Q2=0

= 580dI1n(Q2)

dQ2

∣∣∣∣Q2=0

= 553 (36)




d2(Q2) equiv 3

int 1

0




d2(Q2) =

int 1

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)

22

000 005 010 015 020 025 030

0000

0001

0002

0003

0004

0005

0006

0007

Q2 GeV2D

d2

proton

000 005 010 015 020 025 030

0000

0002

0004

0006

0008

0010

0012

Q2 GeV2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

Q2 GeV2D

Γ 010

-4

fm6

D

proton

000 005 010 015 020 025 030-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron








(orange dot)





polarizabilities



d2(Q2) =

int x0

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)

23





d2(Q2) =Q4

8M4N

[M2

NQ2


] (40)














rule

γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν5(41)

=64αM4

N

Q10

int x0

0

dx x4

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









24

000 005 010 015 020 025 030

-0004

-0002

0000

0002

0004

0006

0008

0010

Q2GeV

2D

d2

proton

000 005 010 015 020 025 030

-0005

0000

0005

0010

0015

Q2GeV

2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

proton

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron


















25







dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0




shown in Fig 8

G Summary














26






(10minus4 fm4) n 298 003(92)


(10minus4 fm4) n 235 218(23)

γ0 p 208 minus096 minus001112(30)

(10minus4 fm6) n 292 195(30)


minus022(4)





minus084(10)




(∆I1)prime p 034 minus053 058039(4)






Proton Neutron




minus0929(105) [59]

δLT 132(15) 240(1) 134 218(23) 238(3) 203


γ0 112(30) 060(7)(7) [78] 195(30) 123

(10minus4 fm6) 0484(82) [59] [MAID]

IV CONCLUSIONS




27











ACKNOWLEDGEMENTS














T (νQ2) =

(minusgmicroν +

qmicroqν

q2

)T1(νQ2) +

1

M2N


q2qmicro)(


qν)T2(νQ2) (A1)

minus 1


M2N


)S2(νQ2)




28







ε0 =

[ε0 minus

ν
























29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References








L(2)∆ =

3e

2MNM+

N T3

igM F


+igCM∆


microν

+ Hc





gM rarrgM[

1 + (QΛ)2 ]2 (16)





δ(Q2) =

radic(∆

MN

)2

+

(Q2

2MN∆

)2

(17)










9

predictions by

σP (Q2) =

Max[δ4(0)P (0)2 δ4(0)P LO(0)2 δ2(0)PNLO(0)2

]+Max

[δ4(Q2)

[P (Q2)minus P (0)

]2 δ4(Q2)


]2


]2]12

(18)







exact























10














and the π∆ loops



γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν3(19)

=16αM2

N

Q6

int x0

0

dx x2

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









11

000 005 010 015 020 025 030

-4

-3

-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

4D

proton

000 005 010 015 020 025 030

-4

-3

-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

4D

neutron

000 005 010 015 020 025 030

0

1

2

Q2 GeV2D

∆LT

10-

4fm

4D

proton

000 005 010 015 020 025 030

0

1

2

3

Q2 GeV2D

∆LT

10-

4fm

4D

neutron
















12





dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0



























13

000 005 010 015 020 025 030

-3

-2

-1

0

1

2

Q2GeV

2D

Γ01

0-

4fm

4D

proton

000 005 010 015 020 025 030

-3

-2

-1

0

1

2

3

Q2GeV

2D

Γ01

0-

4fm

4D

neutron

000 005 010 015 020 025 030

00

05

10

15

Q2GeV

2D

∆LT1

0-

4fm

4D

proton

000 005 010 015 020 025 030

00

05

10

15

20

25

Q2GeV

2D

∆LT1

0-

4fm

4D

neutron
















14







δLT (Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σLT (νQ2)

Qν2(22)

=16αM2

N

Q6

int x0

0

dx x2[g1(xQ2) + g2(xQ2)

]
























15







experimental data






dδLTp(Q2)

dQ2

∣∣∣∣Q2=0


dδLTn(Q2)

dQ2

∣∣∣∣Q2=0








minus α

2M2N

κ2 =1

2π2

int infinν0

dνσTT (ν)

ν (25)





16

000 005 010 015 020 025 030

-30

-25

-20

-15

-10

-05

00

Q2 GeV2D

I Aproton

000 005 010 015 020 025 030

-25

-20

-15

-10

-05

00

Q2 GeV2D

I A

neutron

000 005 010 015 020 025 030

-05

00

05

Q2 GeV2D

I 1

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

I 1

neutron
















17


minus α

2M2N

IA(Q2) = minus 1

8π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν(26)

=α

Q2

int x0

0

dx[4M2

Nx2


]









M2N

F 22 (Q2) (27)




4F 2

2 (Q2) (28)




4F 2

2 (Q2) (29)









bands


18

000 005 010 015 020 025 030

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

neutron

000 005 010 015 020 025 030-005

000

005

010

015

020

025

Q2 GeV2D

DI 1

proton

000 005 010 015 020 025 030-015

-010

-005

000

005

010

015

Q2 GeV2D

DI 1

neutron
















upper panel

19





d∆IAp(Q2)

dQ2

∣∣∣∣Q2=0


d∆IAn(Q2)

dQ2

∣∣∣∣Q2=0



dIAp(Q2)

dQ2

∣∣∣∣Q2=0

= minus318dIAn(Q2)

dQ2

∣∣∣∣Q2=0

= minus300 (31)



minus α

2M2N

I1(Q2) = minus 1

8π2

int infinν0

dν1radic

ν2 +Q2

[σTT (νQ2) +

Q

νσLT (νQ2)

](32)

= minus α

Q2

int x0

0

dx g1(xQ2)






2M2N

Q2 Γ1(Q2) The


Γ1(pminusn)(Q2) =

int x0

0

dx[g1p(xQ



limQ2rarrinfin

Γ1(pminusn)(Q2) =

gA6 (34)






20

000 005 010 015 020 025 030

-006

-004

-002

000

002

004

Q2 GeV2D

G1

proton

000 005 010 015 020 025 030

-006

-005

-004

-003

-002

-001

000

001

Q2 GeV2D

G1

neutron




∆IA(Q2)























21

000 005 010 015 020 025 030-002

000

002

004

006

008

010

Q2 GeV2D

G1Hp-

nL





d∆I1p(Q2)

dQ2

∣∣∣∣Q2=0

= 039(4) asymp 034minus 053 + 058 (35a)

d∆I1n(Q2)

dQ2

∣∣∣∣Q2=0



dI1p(Q2)

dQ2

∣∣∣∣Q2=0

= 580dI1n(Q2)

dQ2

∣∣∣∣Q2=0

= 553 (36)




d2(Q2) equiv 3

int 1

0




d2(Q2) =

int 1

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)

22

000 005 010 015 020 025 030

0000

0001

0002

0003

0004

0005

0006

0007

Q2 GeV2D

d2

proton

000 005 010 015 020 025 030

0000

0002

0004

0006

0008

0010

0012

Q2 GeV2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

Q2 GeV2D

Γ 010

-4

fm6

D

proton

000 005 010 015 020 025 030-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron








(orange dot)





polarizabilities



d2(Q2) =

int x0

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)

23





d2(Q2) =Q4

8M4N

[M2

NQ2


] (40)














rule

γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν5(41)

=64αM4

N

Q10

int x0

0

dx x4

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









24

000 005 010 015 020 025 030

-0004

-0002

0000

0002

0004

0006

0008

0010

Q2GeV

2D

d2

proton

000 005 010 015 020 025 030

-0005

0000

0005

0010

0015

Q2GeV

2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

proton

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron


















25







dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0




shown in Fig 8

G Summary














26






(10minus4 fm4) n 298 003(92)


(10minus4 fm4) n 235 218(23)

γ0 p 208 minus096 minus001112(30)

(10minus4 fm6) n 292 195(30)


minus022(4)





minus084(10)




(∆I1)prime p 034 minus053 058039(4)






Proton Neutron




minus0929(105) [59]

δLT 132(15) 240(1) 134 218(23) 238(3) 203


γ0 112(30) 060(7)(7) [78] 195(30) 123

(10minus4 fm6) 0484(82) [59] [MAID]

IV CONCLUSIONS




27











ACKNOWLEDGEMENTS














T (νQ2) =

(minusgmicroν +

qmicroqν

q2

)T1(νQ2) +

1

M2N


q2qmicro)(


qν)T2(νQ2) (A1)

minus 1


M2N


)S2(νQ2)




28







ε0 =

[ε0 minus

ν
























29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References

predictions by

σP (Q2) =

Max[δ4(0)P (0)2 δ4(0)P LO(0)2 δ2(0)PNLO(0)2

]+Max

[δ4(Q2)

[P (Q2)minus P (0)

]2 δ4(Q2)


]2


]2]12

(18)







exact























10














and the π∆ loops



γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν3(19)

=16αM2

N

Q6

int x0

0

dx x2

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









11

000 005 010 015 020 025 030

-4

-3

-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

4D

proton

000 005 010 015 020 025 030

-4

-3

-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

4D

neutron

000 005 010 015 020 025 030

0

1

2

Q2 GeV2D

∆LT

10-

4fm

4D

proton

000 005 010 015 020 025 030

0

1

2

3

Q2 GeV2D

∆LT

10-

4fm

4D

neutron
















12





dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0



























13

000 005 010 015 020 025 030

-3

-2

-1

0

1

2

Q2GeV

2D

Γ01

0-

4fm

4D

proton

000 005 010 015 020 025 030

-3

-2

-1

0

1

2

3

Q2GeV

2D

Γ01

0-

4fm

4D

neutron

000 005 010 015 020 025 030

00

05

10

15

Q2GeV

2D

∆LT1

0-

4fm

4D

proton

000 005 010 015 020 025 030

00

05

10

15

20

25

Q2GeV

2D

∆LT1

0-

4fm

4D

neutron
















14







δLT (Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σLT (νQ2)

Qν2(22)

=16αM2

N

Q6

int x0

0

dx x2[g1(xQ2) + g2(xQ2)

]
























15







experimental data






dδLTp(Q2)

dQ2

∣∣∣∣Q2=0


dδLTn(Q2)

dQ2

∣∣∣∣Q2=0








minus α

2M2N

κ2 =1

2π2

int infinν0

dνσTT (ν)

ν (25)





16

000 005 010 015 020 025 030

-30

-25

-20

-15

-10

-05

00

Q2 GeV2D

I Aproton

000 005 010 015 020 025 030

-25

-20

-15

-10

-05

00

Q2 GeV2D

I A

neutron

000 005 010 015 020 025 030

-05

00

05

Q2 GeV2D

I 1

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

I 1

neutron
















17


minus α

2M2N

IA(Q2) = minus 1

8π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν(26)

=α

Q2

int x0

0

dx[4M2

Nx2


]









M2N

F 22 (Q2) (27)




4F 2

2 (Q2) (28)




4F 2

2 (Q2) (29)









bands


18

000 005 010 015 020 025 030

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

neutron

000 005 010 015 020 025 030-005

000

005

010

015

020

025

Q2 GeV2D

DI 1

proton

000 005 010 015 020 025 030-015

-010

-005

000

005

010

015

Q2 GeV2D

DI 1

neutron
















upper panel

19





d∆IAp(Q2)

dQ2

∣∣∣∣Q2=0


d∆IAn(Q2)

dQ2

∣∣∣∣Q2=0



dIAp(Q2)

dQ2

∣∣∣∣Q2=0

= minus318dIAn(Q2)

dQ2

∣∣∣∣Q2=0

= minus300 (31)



minus α

2M2N

I1(Q2) = minus 1

8π2

int infinν0

dν1radic

ν2 +Q2

[σTT (νQ2) +

Q

νσLT (νQ2)

](32)

= minus α

Q2

int x0

0

dx g1(xQ2)






2M2N

Q2 Γ1(Q2) The


Γ1(pminusn)(Q2) =

int x0

0

dx[g1p(xQ



limQ2rarrinfin

Γ1(pminusn)(Q2) =

gA6 (34)






20

000 005 010 015 020 025 030

-006

-004

-002

000

002

004

Q2 GeV2D

G1

proton

000 005 010 015 020 025 030

-006

-005

-004

-003

-002

-001

000

001

Q2 GeV2D

G1

neutron




∆IA(Q2)























21

000 005 010 015 020 025 030-002

000

002

004

006

008

010

Q2 GeV2D

G1Hp-

nL





d∆I1p(Q2)

dQ2

∣∣∣∣Q2=0

= 039(4) asymp 034minus 053 + 058 (35a)

d∆I1n(Q2)

dQ2

∣∣∣∣Q2=0



dI1p(Q2)

dQ2

∣∣∣∣Q2=0

= 580dI1n(Q2)

dQ2

∣∣∣∣Q2=0

= 553 (36)




d2(Q2) equiv 3

int 1

0




d2(Q2) =

int 1

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)

22

000 005 010 015 020 025 030

0000

0001

0002

0003

0004

0005

0006

0007

Q2 GeV2D

d2

proton

000 005 010 015 020 025 030

0000

0002

0004

0006

0008

0010

0012

Q2 GeV2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

Q2 GeV2D

Γ 010

-4

fm6

D

proton

000 005 010 015 020 025 030-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron








(orange dot)





polarizabilities



d2(Q2) =

int x0

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)

23





d2(Q2) =Q4

8M4N

[M2

NQ2


] (40)














rule

γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν5(41)

=64αM4

N

Q10

int x0

0

dx x4

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









24

000 005 010 015 020 025 030

-0004

-0002

0000

0002

0004

0006

0008

0010

Q2GeV

2D

d2

proton

000 005 010 015 020 025 030

-0005

0000

0005

0010

0015

Q2GeV

2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

proton

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron


















25







dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0




shown in Fig 8

G Summary














26






(10minus4 fm4) n 298 003(92)


(10minus4 fm4) n 235 218(23)

γ0 p 208 minus096 minus001112(30)

(10minus4 fm6) n 292 195(30)


minus022(4)





minus084(10)




(∆I1)prime p 034 minus053 058039(4)






Proton Neutron




minus0929(105) [59]

δLT 132(15) 240(1) 134 218(23) 238(3) 203


γ0 112(30) 060(7)(7) [78] 195(30) 123

(10minus4 fm6) 0484(82) [59] [MAID]

IV CONCLUSIONS




27











ACKNOWLEDGEMENTS














T (νQ2) =

(minusgmicroν +

qmicroqν

q2

)T1(νQ2) +

1

M2N


q2qmicro)(


qν)T2(νQ2) (A1)

minus 1


M2N


)S2(νQ2)




28







ε0 =

[ε0 minus

ν
























29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References














and the π∆ loops



γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν3(19)

=16αM2

N

Q6

int x0

0

dx x2

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









11

000 005 010 015 020 025 030

-4

-3

-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

4D

proton

000 005 010 015 020 025 030

-4

-3

-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

4D

neutron

000 005 010 015 020 025 030

0

1

2

Q2 GeV2D

∆LT

10-

4fm

4D

proton

000 005 010 015 020 025 030

0

1

2

3

Q2 GeV2D

∆LT

10-

4fm

4D

neutron
















12





dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0



























13

000 005 010 015 020 025 030

-3

-2

-1

0

1

2

Q2GeV

2D

Γ01

0-

4fm

4D

proton

000 005 010 015 020 025 030

-3

-2

-1

0

1

2

3

Q2GeV

2D

Γ01

0-

4fm

4D

neutron

000 005 010 015 020 025 030

00

05

10

15

Q2GeV

2D

∆LT1

0-

4fm

4D

proton

000 005 010 015 020 025 030

00

05

10

15

20

25

Q2GeV

2D

∆LT1

0-

4fm

4D

neutron
















14







δLT (Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σLT (νQ2)

Qν2(22)

=16αM2

N

Q6

int x0

0

dx x2[g1(xQ2) + g2(xQ2)

]
























15







experimental data






dδLTp(Q2)

dQ2

∣∣∣∣Q2=0


dδLTn(Q2)

dQ2

∣∣∣∣Q2=0








minus α

2M2N

κ2 =1

2π2

int infinν0

dνσTT (ν)

ν (25)





16

000 005 010 015 020 025 030

-30

-25

-20

-15

-10

-05

00

Q2 GeV2D

I Aproton

000 005 010 015 020 025 030

-25

-20

-15

-10

-05

00

Q2 GeV2D

I A

neutron

000 005 010 015 020 025 030

-05

00

05

Q2 GeV2D

I 1

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

I 1

neutron
















17


minus α

2M2N

IA(Q2) = minus 1

8π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν(26)

=α

Q2

int x0

0

dx[4M2

Nx2


]









M2N

F 22 (Q2) (27)




4F 2

2 (Q2) (28)




4F 2

2 (Q2) (29)









bands


18

000 005 010 015 020 025 030

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

neutron

000 005 010 015 020 025 030-005

000

005

010

015

020

025

Q2 GeV2D

DI 1

proton

000 005 010 015 020 025 030-015

-010

-005

000

005

010

015

Q2 GeV2D

DI 1

neutron
















upper panel

19





d∆IAp(Q2)

dQ2

∣∣∣∣Q2=0


d∆IAn(Q2)

dQ2

∣∣∣∣Q2=0



dIAp(Q2)

dQ2

∣∣∣∣Q2=0

= minus318dIAn(Q2)

dQ2

∣∣∣∣Q2=0

= minus300 (31)



minus α

2M2N

I1(Q2) = minus 1

8π2

int infinν0

dν1radic

ν2 +Q2

[σTT (νQ2) +

Q

νσLT (νQ2)

](32)

= minus α

Q2

int x0

0

dx g1(xQ2)






2M2N

Q2 Γ1(Q2) The


Γ1(pminusn)(Q2) =

int x0

0

dx[g1p(xQ



limQ2rarrinfin

Γ1(pminusn)(Q2) =

gA6 (34)






20

000 005 010 015 020 025 030

-006

-004

-002

000

002

004

Q2 GeV2D

G1

proton

000 005 010 015 020 025 030

-006

-005

-004

-003

-002

-001

000

001

Q2 GeV2D

G1

neutron




∆IA(Q2)























21

000 005 010 015 020 025 030-002

000

002

004

006

008

010

Q2 GeV2D

G1Hp-

nL





d∆I1p(Q2)

dQ2

∣∣∣∣Q2=0

= 039(4) asymp 034minus 053 + 058 (35a)

d∆I1n(Q2)

dQ2

∣∣∣∣Q2=0



dI1p(Q2)

dQ2

∣∣∣∣Q2=0

= 580dI1n(Q2)

dQ2

∣∣∣∣Q2=0

= 553 (36)




d2(Q2) equiv 3

int 1

0




d2(Q2) =

int 1

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)

22

000 005 010 015 020 025 030

0000

0001

0002

0003

0004

0005

0006

0007

Q2 GeV2D

d2

proton

000 005 010 015 020 025 030

0000

0002

0004

0006

0008

0010

0012

Q2 GeV2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

Q2 GeV2D

Γ 010

-4

fm6

D

proton

000 005 010 015 020 025 030-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron








(orange dot)





polarizabilities



d2(Q2) =

int x0

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)

23





d2(Q2) =Q4

8M4N

[M2

NQ2


] (40)














rule

γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν5(41)

=64αM4

N

Q10

int x0

0

dx x4

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









24

000 005 010 015 020 025 030

-0004

-0002

0000

0002

0004

0006

0008

0010

Q2GeV

2D

d2

proton

000 005 010 015 020 025 030

-0005

0000

0005

0010

0015

Q2GeV

2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

proton

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron


















25







dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0




shown in Fig 8

G Summary














26






(10minus4 fm4) n 298 003(92)


(10minus4 fm4) n 235 218(23)

γ0 p 208 minus096 minus001112(30)

(10minus4 fm6) n 292 195(30)


minus022(4)





minus084(10)




(∆I1)prime p 034 minus053 058039(4)






Proton Neutron




minus0929(105) [59]

δLT 132(15) 240(1) 134 218(23) 238(3) 203


γ0 112(30) 060(7)(7) [78] 195(30) 123

(10minus4 fm6) 0484(82) [59] [MAID]

IV CONCLUSIONS




27











ACKNOWLEDGEMENTS














T (νQ2) =

(minusgmicroν +

qmicroqν

q2

)T1(νQ2) +

1

M2N


q2qmicro)(


qν)T2(νQ2) (A1)

minus 1


M2N


)S2(νQ2)




28







ε0 =

[ε0 minus

ν
























29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References

000 005 010 015 020 025 030

-4

-3

-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

4D

proton

000 005 010 015 020 025 030

-4

-3

-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

4D

neutron

000 005 010 015 020 025 030

0

1

2

Q2 GeV2D

∆LT

10-

4fm

4D

proton

000 005 010 015 020 025 030

0

1

2

3

Q2 GeV2D

∆LT

10-

4fm

4D

neutron
















12





dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0



























13

000 005 010 015 020 025 030

-3

-2

-1

0

1

2

Q2GeV

2D

Γ01

0-

4fm

4D

proton

000 005 010 015 020 025 030

-3

-2

-1

0

1

2

3

Q2GeV

2D

Γ01

0-

4fm

4D

neutron

000 005 010 015 020 025 030

00

05

10

15

Q2GeV

2D

∆LT1

0-

4fm

4D

proton

000 005 010 015 020 025 030

00

05

10

15

20

25

Q2GeV

2D

∆LT1

0-

4fm

4D

neutron
















14







δLT (Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σLT (νQ2)

Qν2(22)

=16αM2

N

Q6

int x0

0

dx x2[g1(xQ2) + g2(xQ2)

]
























15







experimental data






dδLTp(Q2)

dQ2

∣∣∣∣Q2=0


dδLTn(Q2)

dQ2

∣∣∣∣Q2=0








minus α

2M2N

κ2 =1

2π2

int infinν0

dνσTT (ν)

ν (25)





16

000 005 010 015 020 025 030

-30

-25

-20

-15

-10

-05

00

Q2 GeV2D

I Aproton

000 005 010 015 020 025 030

-25

-20

-15

-10

-05

00

Q2 GeV2D

I A

neutron

000 005 010 015 020 025 030

-05

00

05

Q2 GeV2D

I 1

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

I 1

neutron
















17


minus α

2M2N

IA(Q2) = minus 1

8π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν(26)

=α

Q2

int x0

0

dx[4M2

Nx2


]









M2N

F 22 (Q2) (27)




4F 2

2 (Q2) (28)




4F 2

2 (Q2) (29)









bands


18

000 005 010 015 020 025 030

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

neutron

000 005 010 015 020 025 030-005

000

005

010

015

020

025

Q2 GeV2D

DI 1

proton

000 005 010 015 020 025 030-015

-010

-005

000

005

010

015

Q2 GeV2D

DI 1

neutron
















upper panel

19





d∆IAp(Q2)

dQ2

∣∣∣∣Q2=0


d∆IAn(Q2)

dQ2

∣∣∣∣Q2=0



dIAp(Q2)

dQ2

∣∣∣∣Q2=0

= minus318dIAn(Q2)

dQ2

∣∣∣∣Q2=0

= minus300 (31)



minus α

2M2N

I1(Q2) = minus 1

8π2

int infinν0

dν1radic

ν2 +Q2

[σTT (νQ2) +

Q

νσLT (νQ2)

](32)

= minus α

Q2

int x0

0

dx g1(xQ2)






2M2N

Q2 Γ1(Q2) The


Γ1(pminusn)(Q2) =

int x0

0

dx[g1p(xQ



limQ2rarrinfin

Γ1(pminusn)(Q2) =

gA6 (34)






20

000 005 010 015 020 025 030

-006

-004

-002

000

002

004

Q2 GeV2D

G1

proton

000 005 010 015 020 025 030

-006

-005

-004

-003

-002

-001

000

001

Q2 GeV2D

G1

neutron




∆IA(Q2)























21

000 005 010 015 020 025 030-002

000

002

004

006

008

010

Q2 GeV2D

G1Hp-

nL





d∆I1p(Q2)

dQ2

∣∣∣∣Q2=0

= 039(4) asymp 034minus 053 + 058 (35a)

d∆I1n(Q2)

dQ2

∣∣∣∣Q2=0



dI1p(Q2)

dQ2

∣∣∣∣Q2=0

= 580dI1n(Q2)

dQ2

∣∣∣∣Q2=0

= 553 (36)




d2(Q2) equiv 3

int 1

0




d2(Q2) =

int 1

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)

22

000 005 010 015 020 025 030

0000

0001

0002

0003

0004

0005

0006

0007

Q2 GeV2D

d2

proton

000 005 010 015 020 025 030

0000

0002

0004

0006

0008

0010

0012

Q2 GeV2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

Q2 GeV2D

Γ 010

-4

fm6

D

proton

000 005 010 015 020 025 030-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron








(orange dot)





polarizabilities



d2(Q2) =

int x0

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)

23





d2(Q2) =Q4

8M4N

[M2

NQ2


] (40)














rule

γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν5(41)

=64αM4

N

Q10

int x0

0

dx x4

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









24

000 005 010 015 020 025 030

-0004

-0002

0000

0002

0004

0006

0008

0010

Q2GeV

2D

d2

proton

000 005 010 015 020 025 030

-0005

0000

0005

0010

0015

Q2GeV

2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

proton

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron


















25







dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0




shown in Fig 8

G Summary














26






(10minus4 fm4) n 298 003(92)


(10minus4 fm4) n 235 218(23)

γ0 p 208 minus096 minus001112(30)

(10minus4 fm6) n 292 195(30)


minus022(4)





minus084(10)




(∆I1)prime p 034 minus053 058039(4)






Proton Neutron




minus0929(105) [59]

δLT 132(15) 240(1) 134 218(23) 238(3) 203


γ0 112(30) 060(7)(7) [78] 195(30) 123

(10minus4 fm6) 0484(82) [59] [MAID]

IV CONCLUSIONS




27











ACKNOWLEDGEMENTS














T (νQ2) =

(minusgmicroν +

qmicroqν

q2

)T1(νQ2) +

1

M2N


q2qmicro)(


qν)T2(νQ2) (A1)

minus 1


M2N


)S2(νQ2)




28







ε0 =

[ε0 minus

ν
























29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References





dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0



























13

000 005 010 015 020 025 030

-3

-2

-1

0

1

2

Q2GeV

2D

Γ01

0-

4fm

4D

proton

000 005 010 015 020 025 030

-3

-2

-1

0

1

2

3

Q2GeV

2D

Γ01

0-

4fm

4D

neutron

000 005 010 015 020 025 030

00

05

10

15

Q2GeV

2D

∆LT1

0-

4fm

4D

proton

000 005 010 015 020 025 030

00

05

10

15

20

25

Q2GeV

2D

∆LT1

0-

4fm

4D

neutron
















14







δLT (Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σLT (νQ2)

Qν2(22)

=16αM2

N

Q6

int x0

0

dx x2[g1(xQ2) + g2(xQ2)

]
























15







experimental data






dδLTp(Q2)

dQ2

∣∣∣∣Q2=0


dδLTn(Q2)

dQ2

∣∣∣∣Q2=0








minus α

2M2N

κ2 =1

2π2

int infinν0

dνσTT (ν)

ν (25)





16

000 005 010 015 020 025 030

-30

-25

-20

-15

-10

-05

00

Q2 GeV2D

I Aproton

000 005 010 015 020 025 030

-25

-20

-15

-10

-05

00

Q2 GeV2D

I A

neutron

000 005 010 015 020 025 030

-05

00

05

Q2 GeV2D

I 1

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

I 1

neutron
















17


minus α

2M2N

IA(Q2) = minus 1

8π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν(26)

=α

Q2

int x0

0

dx[4M2

Nx2


]









M2N

F 22 (Q2) (27)




4F 2

2 (Q2) (28)




4F 2

2 (Q2) (29)









bands


18

000 005 010 015 020 025 030

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

neutron

000 005 010 015 020 025 030-005

000

005

010

015

020

025

Q2 GeV2D

DI 1

proton

000 005 010 015 020 025 030-015

-010

-005

000

005

010

015

Q2 GeV2D

DI 1

neutron
















upper panel

19





d∆IAp(Q2)

dQ2

∣∣∣∣Q2=0


d∆IAn(Q2)

dQ2

∣∣∣∣Q2=0



dIAp(Q2)

dQ2

∣∣∣∣Q2=0

= minus318dIAn(Q2)

dQ2

∣∣∣∣Q2=0

= minus300 (31)



minus α

2M2N

I1(Q2) = minus 1

8π2

int infinν0

dν1radic

ν2 +Q2

[σTT (νQ2) +

Q

νσLT (νQ2)

](32)

= minus α

Q2

int x0

0

dx g1(xQ2)






2M2N

Q2 Γ1(Q2) The


Γ1(pminusn)(Q2) =

int x0

0

dx[g1p(xQ



limQ2rarrinfin

Γ1(pminusn)(Q2) =

gA6 (34)






20

000 005 010 015 020 025 030

-006

-004

-002

000

002

004

Q2 GeV2D

G1

proton

000 005 010 015 020 025 030

-006

-005

-004

-003

-002

-001

000

001

Q2 GeV2D

G1

neutron




∆IA(Q2)























21

000 005 010 015 020 025 030-002

000

002

004

006

008

010

Q2 GeV2D

G1Hp-

nL





d∆I1p(Q2)

dQ2

∣∣∣∣Q2=0

= 039(4) asymp 034minus 053 + 058 (35a)

d∆I1n(Q2)

dQ2

∣∣∣∣Q2=0



dI1p(Q2)

dQ2

∣∣∣∣Q2=0

= 580dI1n(Q2)

dQ2

∣∣∣∣Q2=0

= 553 (36)




d2(Q2) equiv 3

int 1

0




d2(Q2) =

int 1

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)

22

000 005 010 015 020 025 030

0000

0001

0002

0003

0004

0005

0006

0007

Q2 GeV2D

d2

proton

000 005 010 015 020 025 030

0000

0002

0004

0006

0008

0010

0012

Q2 GeV2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

Q2 GeV2D

Γ 010

-4

fm6

D

proton

000 005 010 015 020 025 030-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron








(orange dot)





polarizabilities



d2(Q2) =

int x0

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)

23





d2(Q2) =Q4

8M4N

[M2

NQ2


] (40)














rule

γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν5(41)

=64αM4

N

Q10

int x0

0

dx x4

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









24

000 005 010 015 020 025 030

-0004

-0002

0000

0002

0004

0006

0008

0010

Q2GeV

2D

d2

proton

000 005 010 015 020 025 030

-0005

0000

0005

0010

0015

Q2GeV

2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

proton

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron


















25







dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0




shown in Fig 8

G Summary














26






(10minus4 fm4) n 298 003(92)


(10minus4 fm4) n 235 218(23)

γ0 p 208 minus096 minus001112(30)

(10minus4 fm6) n 292 195(30)


minus022(4)





minus084(10)




(∆I1)prime p 034 minus053 058039(4)






Proton Neutron




minus0929(105) [59]

δLT 132(15) 240(1) 134 218(23) 238(3) 203


γ0 112(30) 060(7)(7) [78] 195(30) 123

(10minus4 fm6) 0484(82) [59] [MAID]

IV CONCLUSIONS




27











ACKNOWLEDGEMENTS














T (νQ2) =

(minusgmicroν +

qmicroqν

q2

)T1(νQ2) +

1

M2N


q2qmicro)(


qν)T2(νQ2) (A1)

minus 1


M2N


)S2(νQ2)




28







ε0 =

[ε0 minus

ν
























29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References

000 005 010 015 020 025 030

-3

-2

-1

0

1

2

Q2GeV

2D

Γ01

0-

4fm

4D

proton

000 005 010 015 020 025 030

-3

-2

-1

0

1

2

3

Q2GeV

2D

Γ01

0-

4fm

4D

neutron

000 005 010 015 020 025 030

00

05

10

15

Q2GeV

2D

∆LT1

0-

4fm

4D

proton

000 005 010 015 020 025 030

00

05

10

15

20

25

Q2GeV

2D

∆LT1

0-

4fm

4D

neutron
















14







δLT (Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σLT (νQ2)

Qν2(22)

=16αM2

N

Q6

int x0

0

dx x2[g1(xQ2) + g2(xQ2)

]
























15







experimental data






dδLTp(Q2)

dQ2

∣∣∣∣Q2=0


dδLTn(Q2)

dQ2

∣∣∣∣Q2=0








minus α

2M2N

κ2 =1

2π2

int infinν0

dνσTT (ν)

ν (25)





16

000 005 010 015 020 025 030

-30

-25

-20

-15

-10

-05

00

Q2 GeV2D

I Aproton

000 005 010 015 020 025 030

-25

-20

-15

-10

-05

00

Q2 GeV2D

I A

neutron

000 005 010 015 020 025 030

-05

00

05

Q2 GeV2D

I 1

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

I 1

neutron
















17


minus α

2M2N

IA(Q2) = minus 1

8π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν(26)

=α

Q2

int x0

0

dx[4M2

Nx2


]









M2N

F 22 (Q2) (27)




4F 2

2 (Q2) (28)




4F 2

2 (Q2) (29)









bands


18

000 005 010 015 020 025 030

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

neutron

000 005 010 015 020 025 030-005

000

005

010

015

020

025

Q2 GeV2D

DI 1

proton

000 005 010 015 020 025 030-015

-010

-005

000

005

010

015

Q2 GeV2D

DI 1

neutron
















upper panel

19





d∆IAp(Q2)

dQ2

∣∣∣∣Q2=0


d∆IAn(Q2)

dQ2

∣∣∣∣Q2=0



dIAp(Q2)

dQ2

∣∣∣∣Q2=0

= minus318dIAn(Q2)

dQ2

∣∣∣∣Q2=0

= minus300 (31)



minus α

2M2N

I1(Q2) = minus 1

8π2

int infinν0

dν1radic

ν2 +Q2

[σTT (νQ2) +

Q

νσLT (νQ2)

](32)

= minus α

Q2

int x0

0

dx g1(xQ2)






2M2N

Q2 Γ1(Q2) The


Γ1(pminusn)(Q2) =

int x0

0

dx[g1p(xQ



limQ2rarrinfin

Γ1(pminusn)(Q2) =

gA6 (34)






20

000 005 010 015 020 025 030

-006

-004

-002

000

002

004

Q2 GeV2D

G1

proton

000 005 010 015 020 025 030

-006

-005

-004

-003

-002

-001

000

001

Q2 GeV2D

G1

neutron




∆IA(Q2)























21

000 005 010 015 020 025 030-002

000

002

004

006

008

010

Q2 GeV2D

G1Hp-

nL





d∆I1p(Q2)

dQ2

∣∣∣∣Q2=0

= 039(4) asymp 034minus 053 + 058 (35a)

d∆I1n(Q2)

dQ2

∣∣∣∣Q2=0



dI1p(Q2)

dQ2

∣∣∣∣Q2=0

= 580dI1n(Q2)

dQ2

∣∣∣∣Q2=0

= 553 (36)




d2(Q2) equiv 3

int 1

0




d2(Q2) =

int 1

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)

22

000 005 010 015 020 025 030

0000

0001

0002

0003

0004

0005

0006

0007

Q2 GeV2D

d2

proton

000 005 010 015 020 025 030

0000

0002

0004

0006

0008

0010

0012

Q2 GeV2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

Q2 GeV2D

Γ 010

-4

fm6

D

proton

000 005 010 015 020 025 030-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron








(orange dot)





polarizabilities



d2(Q2) =

int x0

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)

23





d2(Q2) =Q4

8M4N

[M2

NQ2


] (40)














rule

γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν5(41)

=64αM4

N

Q10

int x0

0

dx x4

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









24

000 005 010 015 020 025 030

-0004

-0002

0000

0002

0004

0006

0008

0010

Q2GeV

2D

d2

proton

000 005 010 015 020 025 030

-0005

0000

0005

0010

0015

Q2GeV

2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

proton

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron


















25







dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0




shown in Fig 8

G Summary














26






(10minus4 fm4) n 298 003(92)


(10minus4 fm4) n 235 218(23)

γ0 p 208 minus096 minus001112(30)

(10minus4 fm6) n 292 195(30)


minus022(4)





minus084(10)




(∆I1)prime p 034 minus053 058039(4)






Proton Neutron




minus0929(105) [59]

δLT 132(15) 240(1) 134 218(23) 238(3) 203


γ0 112(30) 060(7)(7) [78] 195(30) 123

(10minus4 fm6) 0484(82) [59] [MAID]

IV CONCLUSIONS




27











ACKNOWLEDGEMENTS














T (νQ2) =

(minusgmicroν +

qmicroqν

q2

)T1(νQ2) +

1

M2N


q2qmicro)(


qν)T2(νQ2) (A1)

minus 1


M2N


)S2(νQ2)




28







ε0 =

[ε0 minus

ν
























29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References







δLT (Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σLT (νQ2)

Qν2(22)

=16αM2

N

Q6

int x0

0

dx x2[g1(xQ2) + g2(xQ2)

]
























15







experimental data






dδLTp(Q2)

dQ2

∣∣∣∣Q2=0


dδLTn(Q2)

dQ2

∣∣∣∣Q2=0








minus α

2M2N

κ2 =1

2π2

int infinν0

dνσTT (ν)

ν (25)





16

000 005 010 015 020 025 030

-30

-25

-20

-15

-10

-05

00

Q2 GeV2D

I Aproton

000 005 010 015 020 025 030

-25

-20

-15

-10

-05

00

Q2 GeV2D

I A

neutron

000 005 010 015 020 025 030

-05

00

05

Q2 GeV2D

I 1

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

I 1

neutron
















17


minus α

2M2N

IA(Q2) = minus 1

8π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν(26)

=α

Q2

int x0

0

dx[4M2

Nx2


]









M2N

F 22 (Q2) (27)




4F 2

2 (Q2) (28)




4F 2

2 (Q2) (29)









bands


18

000 005 010 015 020 025 030

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

neutron

000 005 010 015 020 025 030-005

000

005

010

015

020

025

Q2 GeV2D

DI 1

proton

000 005 010 015 020 025 030-015

-010

-005

000

005

010

015

Q2 GeV2D

DI 1

neutron
















upper panel

19





d∆IAp(Q2)

dQ2

∣∣∣∣Q2=0


d∆IAn(Q2)

dQ2

∣∣∣∣Q2=0



dIAp(Q2)

dQ2

∣∣∣∣Q2=0

= minus318dIAn(Q2)

dQ2

∣∣∣∣Q2=0

= minus300 (31)



minus α

2M2N

I1(Q2) = minus 1

8π2

int infinν0

dν1radic

ν2 +Q2

[σTT (νQ2) +

Q

νσLT (νQ2)

](32)

= minus α

Q2

int x0

0

dx g1(xQ2)






2M2N

Q2 Γ1(Q2) The


Γ1(pminusn)(Q2) =

int x0

0

dx[g1p(xQ



limQ2rarrinfin

Γ1(pminusn)(Q2) =

gA6 (34)






20

000 005 010 015 020 025 030

-006

-004

-002

000

002

004

Q2 GeV2D

G1

proton

000 005 010 015 020 025 030

-006

-005

-004

-003

-002

-001

000

001

Q2 GeV2D

G1

neutron




∆IA(Q2)























21

000 005 010 015 020 025 030-002

000

002

004

006

008

010

Q2 GeV2D

G1Hp-

nL





d∆I1p(Q2)

dQ2

∣∣∣∣Q2=0

= 039(4) asymp 034minus 053 + 058 (35a)

d∆I1n(Q2)

dQ2

∣∣∣∣Q2=0



dI1p(Q2)

dQ2

∣∣∣∣Q2=0

= 580dI1n(Q2)

dQ2

∣∣∣∣Q2=0

= 553 (36)




d2(Q2) equiv 3

int 1

0




d2(Q2) =

int 1

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)

22

000 005 010 015 020 025 030

0000

0001

0002

0003

0004

0005

0006

0007

Q2 GeV2D

d2

proton

000 005 010 015 020 025 030

0000

0002

0004

0006

0008

0010

0012

Q2 GeV2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

Q2 GeV2D

Γ 010

-4

fm6

D

proton

000 005 010 015 020 025 030-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron








(orange dot)





polarizabilities



d2(Q2) =

int x0

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)

23





d2(Q2) =Q4

8M4N

[M2

NQ2


] (40)














rule

γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν5(41)

=64αM4

N

Q10

int x0

0

dx x4

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









24

000 005 010 015 020 025 030

-0004

-0002

0000

0002

0004

0006

0008

0010

Q2GeV

2D

d2

proton

000 005 010 015 020 025 030

-0005

0000

0005

0010

0015

Q2GeV

2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

proton

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron


















25







dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0




shown in Fig 8

G Summary














26






(10minus4 fm4) n 298 003(92)


(10minus4 fm4) n 235 218(23)

γ0 p 208 minus096 minus001112(30)

(10minus4 fm6) n 292 195(30)


minus022(4)





minus084(10)




(∆I1)prime p 034 minus053 058039(4)






Proton Neutron




minus0929(105) [59]

δLT 132(15) 240(1) 134 218(23) 238(3) 203


γ0 112(30) 060(7)(7) [78] 195(30) 123

(10minus4 fm6) 0484(82) [59] [MAID]

IV CONCLUSIONS




27











ACKNOWLEDGEMENTS














T (νQ2) =

(minusgmicroν +

qmicroqν

q2

)T1(νQ2) +

1

M2N


q2qmicro)(


qν)T2(νQ2) (A1)

minus 1


M2N


)S2(νQ2)




28







ε0 =

[ε0 minus

ν
























29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References







experimental data






dδLTp(Q2)

dQ2

∣∣∣∣Q2=0


dδLTn(Q2)

dQ2

∣∣∣∣Q2=0








minus α

2M2N

κ2 =1

2π2

int infinν0

dνσTT (ν)

ν (25)





16

000 005 010 015 020 025 030

-30

-25

-20

-15

-10

-05

00

Q2 GeV2D

I Aproton

000 005 010 015 020 025 030

-25

-20

-15

-10

-05

00

Q2 GeV2D

I A

neutron

000 005 010 015 020 025 030

-05

00

05

Q2 GeV2D

I 1

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

I 1

neutron
















17


minus α

2M2N

IA(Q2) = minus 1

8π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν(26)

=α

Q2

int x0

0

dx[4M2

Nx2


]









M2N

F 22 (Q2) (27)




4F 2

2 (Q2) (28)




4F 2

2 (Q2) (29)









bands


18

000 005 010 015 020 025 030

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

neutron

000 005 010 015 020 025 030-005

000

005

010

015

020

025

Q2 GeV2D

DI 1

proton

000 005 010 015 020 025 030-015

-010

-005

000

005

010

015

Q2 GeV2D

DI 1

neutron
















upper panel

19





d∆IAp(Q2)

dQ2

∣∣∣∣Q2=0


d∆IAn(Q2)

dQ2

∣∣∣∣Q2=0



dIAp(Q2)

dQ2

∣∣∣∣Q2=0

= minus318dIAn(Q2)

dQ2

∣∣∣∣Q2=0

= minus300 (31)



minus α

2M2N

I1(Q2) = minus 1

8π2

int infinν0

dν1radic

ν2 +Q2

[σTT (νQ2) +

Q

νσLT (νQ2)

](32)

= minus α

Q2

int x0

0

dx g1(xQ2)






2M2N

Q2 Γ1(Q2) The


Γ1(pminusn)(Q2) =

int x0

0

dx[g1p(xQ



limQ2rarrinfin

Γ1(pminusn)(Q2) =

gA6 (34)






20

000 005 010 015 020 025 030

-006

-004

-002

000

002

004

Q2 GeV2D

G1

proton

000 005 010 015 020 025 030

-006

-005

-004

-003

-002

-001

000

001

Q2 GeV2D

G1

neutron




∆IA(Q2)























21

000 005 010 015 020 025 030-002

000

002

004

006

008

010

Q2 GeV2D

G1Hp-

nL





d∆I1p(Q2)

dQ2

∣∣∣∣Q2=0

= 039(4) asymp 034minus 053 + 058 (35a)

d∆I1n(Q2)

dQ2

∣∣∣∣Q2=0



dI1p(Q2)

dQ2

∣∣∣∣Q2=0

= 580dI1n(Q2)

dQ2

∣∣∣∣Q2=0

= 553 (36)




d2(Q2) equiv 3

int 1

0




d2(Q2) =

int 1

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)

22

000 005 010 015 020 025 030

0000

0001

0002

0003

0004

0005

0006

0007

Q2 GeV2D

d2

proton

000 005 010 015 020 025 030

0000

0002

0004

0006

0008

0010

0012

Q2 GeV2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

Q2 GeV2D

Γ 010

-4

fm6

D

proton

000 005 010 015 020 025 030-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron








(orange dot)





polarizabilities



d2(Q2) =

int x0

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)

23





d2(Q2) =Q4

8M4N

[M2

NQ2


] (40)














rule

γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν5(41)

=64αM4

N

Q10

int x0

0

dx x4

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









24

000 005 010 015 020 025 030

-0004

-0002

0000

0002

0004

0006

0008

0010

Q2GeV

2D

d2

proton

000 005 010 015 020 025 030

-0005

0000

0005

0010

0015

Q2GeV

2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

proton

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron


















25







dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0




shown in Fig 8

G Summary














26






(10minus4 fm4) n 298 003(92)


(10minus4 fm4) n 235 218(23)

γ0 p 208 minus096 minus001112(30)

(10minus4 fm6) n 292 195(30)


minus022(4)





minus084(10)




(∆I1)prime p 034 minus053 058039(4)






Proton Neutron




minus0929(105) [59]

δLT 132(15) 240(1) 134 218(23) 238(3) 203


γ0 112(30) 060(7)(7) [78] 195(30) 123

(10minus4 fm6) 0484(82) [59] [MAID]

IV CONCLUSIONS




27











ACKNOWLEDGEMENTS














T (νQ2) =

(minusgmicroν +

qmicroqν

q2

)T1(νQ2) +

1

M2N


q2qmicro)(


qν)T2(νQ2) (A1)

minus 1


M2N


)S2(νQ2)




28







ε0 =

[ε0 minus

ν
























29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References

000 005 010 015 020 025 030

-30

-25

-20

-15

-10

-05

00

Q2 GeV2D

I Aproton

000 005 010 015 020 025 030

-25

-20

-15

-10

-05

00

Q2 GeV2D

I A

neutron

000 005 010 015 020 025 030

-05

00

05

Q2 GeV2D

I 1

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

I 1

neutron
















17


minus α

2M2N

IA(Q2) = minus 1

8π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν(26)

=α

Q2

int x0

0

dx[4M2

Nx2


]









M2N

F 22 (Q2) (27)




4F 2

2 (Q2) (28)




4F 2

2 (Q2) (29)









bands


18

000 005 010 015 020 025 030

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

neutron

000 005 010 015 020 025 030-005

000

005

010

015

020

025

Q2 GeV2D

DI 1

proton

000 005 010 015 020 025 030-015

-010

-005

000

005

010

015

Q2 GeV2D

DI 1

neutron
















upper panel

19





d∆IAp(Q2)

dQ2

∣∣∣∣Q2=0


d∆IAn(Q2)

dQ2

∣∣∣∣Q2=0



dIAp(Q2)

dQ2

∣∣∣∣Q2=0

= minus318dIAn(Q2)

dQ2

∣∣∣∣Q2=0

= minus300 (31)



minus α

2M2N

I1(Q2) = minus 1

8π2

int infinν0

dν1radic

ν2 +Q2

[σTT (νQ2) +

Q

νσLT (νQ2)

](32)

= minus α

Q2

int x0

0

dx g1(xQ2)






2M2N

Q2 Γ1(Q2) The


Γ1(pminusn)(Q2) =

int x0

0

dx[g1p(xQ



limQ2rarrinfin

Γ1(pminusn)(Q2) =

gA6 (34)






20

000 005 010 015 020 025 030

-006

-004

-002

000

002

004

Q2 GeV2D

G1

proton

000 005 010 015 020 025 030

-006

-005

-004

-003

-002

-001

000

001

Q2 GeV2D

G1

neutron




∆IA(Q2)























21

000 005 010 015 020 025 030-002

000

002

004

006

008

010

Q2 GeV2D

G1Hp-

nL





d∆I1p(Q2)

dQ2

∣∣∣∣Q2=0

= 039(4) asymp 034minus 053 + 058 (35a)

d∆I1n(Q2)

dQ2

∣∣∣∣Q2=0



dI1p(Q2)

dQ2

∣∣∣∣Q2=0

= 580dI1n(Q2)

dQ2

∣∣∣∣Q2=0

= 553 (36)




d2(Q2) equiv 3

int 1

0




d2(Q2) =

int 1

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)

22

000 005 010 015 020 025 030

0000

0001

0002

0003

0004

0005

0006

0007

Q2 GeV2D

d2

proton

000 005 010 015 020 025 030

0000

0002

0004

0006

0008

0010

0012

Q2 GeV2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

Q2 GeV2D

Γ 010

-4

fm6

D

proton

000 005 010 015 020 025 030-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron








(orange dot)





polarizabilities



d2(Q2) =

int x0

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)

23





d2(Q2) =Q4

8M4N

[M2

NQ2


] (40)














rule

γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν5(41)

=64αM4

N

Q10

int x0

0

dx x4

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









24

000 005 010 015 020 025 030

-0004

-0002

0000

0002

0004

0006

0008

0010

Q2GeV

2D

d2

proton

000 005 010 015 020 025 030

-0005

0000

0005

0010

0015

Q2GeV

2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

proton

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron


















25







dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0




shown in Fig 8

G Summary














26






(10minus4 fm4) n 298 003(92)


(10minus4 fm4) n 235 218(23)

γ0 p 208 minus096 minus001112(30)

(10minus4 fm6) n 292 195(30)


minus022(4)





minus084(10)




(∆I1)prime p 034 minus053 058039(4)






Proton Neutron




minus0929(105) [59]

δLT 132(15) 240(1) 134 218(23) 238(3) 203


γ0 112(30) 060(7)(7) [78] 195(30) 123

(10minus4 fm6) 0484(82) [59] [MAID]

IV CONCLUSIONS




27











ACKNOWLEDGEMENTS














T (νQ2) =

(minusgmicroν +

qmicroqν

q2

)T1(νQ2) +

1

M2N


q2qmicro)(


qν)T2(νQ2) (A1)

minus 1


M2N


)S2(νQ2)




28







ε0 =

[ε0 minus

ν
























29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References


minus α

2M2N

IA(Q2) = minus 1

8π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν(26)

=α

Q2

int x0

0

dx[4M2

Nx2


]









M2N

F 22 (Q2) (27)




4F 2

2 (Q2) (28)




4F 2

2 (Q2) (29)









bands


18

000 005 010 015 020 025 030

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

neutron

000 005 010 015 020 025 030-005

000

005

010

015

020

025

Q2 GeV2D

DI 1

proton

000 005 010 015 020 025 030-015

-010

-005

000

005

010

015

Q2 GeV2D

DI 1

neutron
















upper panel

19





d∆IAp(Q2)

dQ2

∣∣∣∣Q2=0


d∆IAn(Q2)

dQ2

∣∣∣∣Q2=0



dIAp(Q2)

dQ2

∣∣∣∣Q2=0

= minus318dIAn(Q2)

dQ2

∣∣∣∣Q2=0

= minus300 (31)



minus α

2M2N

I1(Q2) = minus 1

8π2

int infinν0

dν1radic

ν2 +Q2

[σTT (νQ2) +

Q

νσLT (νQ2)

](32)

= minus α

Q2

int x0

0

dx g1(xQ2)






2M2N

Q2 Γ1(Q2) The


Γ1(pminusn)(Q2) =

int x0

0

dx[g1p(xQ



limQ2rarrinfin

Γ1(pminusn)(Q2) =

gA6 (34)






20

000 005 010 015 020 025 030

-006

-004

-002

000

002

004

Q2 GeV2D

G1

proton

000 005 010 015 020 025 030

-006

-005

-004

-003

-002

-001

000

001

Q2 GeV2D

G1

neutron




∆IA(Q2)























21

000 005 010 015 020 025 030-002

000

002

004

006

008

010

Q2 GeV2D

G1Hp-

nL





d∆I1p(Q2)

dQ2

∣∣∣∣Q2=0

= 039(4) asymp 034minus 053 + 058 (35a)

d∆I1n(Q2)

dQ2

∣∣∣∣Q2=0



dI1p(Q2)

dQ2

∣∣∣∣Q2=0

= 580dI1n(Q2)

dQ2

∣∣∣∣Q2=0

= 553 (36)




d2(Q2) equiv 3

int 1

0




d2(Q2) =

int 1

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)

22

000 005 010 015 020 025 030

0000

0001

0002

0003

0004

0005

0006

0007

Q2 GeV2D

d2

proton

000 005 010 015 020 025 030

0000

0002

0004

0006

0008

0010

0012

Q2 GeV2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

Q2 GeV2D

Γ 010

-4

fm6

D

proton

000 005 010 015 020 025 030-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron








(orange dot)





polarizabilities



d2(Q2) =

int x0

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)

23





d2(Q2) =Q4

8M4N

[M2

NQ2


] (40)














rule

γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν5(41)

=64αM4

N

Q10

int x0

0

dx x4

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









24

000 005 010 015 020 025 030

-0004

-0002

0000

0002

0004

0006

0008

0010

Q2GeV

2D

d2

proton

000 005 010 015 020 025 030

-0005

0000

0005

0010

0015

Q2GeV

2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

proton

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron


















25







dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0




shown in Fig 8

G Summary














26






(10minus4 fm4) n 298 003(92)


(10minus4 fm4) n 235 218(23)

γ0 p 208 minus096 minus001112(30)

(10minus4 fm6) n 292 195(30)


minus022(4)





minus084(10)




(∆I1)prime p 034 minus053 058039(4)






Proton Neutron




minus0929(105) [59]

δLT 132(15) 240(1) 134 218(23) 238(3) 203


γ0 112(30) 060(7)(7) [78] 195(30) 123

(10minus4 fm6) 0484(82) [59] [MAID]

IV CONCLUSIONS




27











ACKNOWLEDGEMENTS














T (νQ2) =

(minusgmicroν +

qmicroqν

q2

)T1(νQ2) +

1

M2N


q2qmicro)(


qν)T2(νQ2) (A1)

minus 1


M2N


)S2(νQ2)




28







ε0 =

[ε0 minus

ν
























29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References

000 005 010 015 020 025 030

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

proton

000 005 010 015 020 025 030

-10

-08

-06

-04

-02

00

02

Q2 GeV2D

DI A

neutron

000 005 010 015 020 025 030-005

000

005

010

015

020

025

Q2 GeV2D

DI 1

proton

000 005 010 015 020 025 030-015

-010

-005

000

005

010

015

Q2 GeV2D

DI 1

neutron
















upper panel

19





d∆IAp(Q2)

dQ2

∣∣∣∣Q2=0


d∆IAn(Q2)

dQ2

∣∣∣∣Q2=0



dIAp(Q2)

dQ2

∣∣∣∣Q2=0

= minus318dIAn(Q2)

dQ2

∣∣∣∣Q2=0

= minus300 (31)



minus α

2M2N

I1(Q2) = minus 1

8π2

int infinν0

dν1radic

ν2 +Q2

[σTT (νQ2) +

Q

νσLT (νQ2)

](32)

= minus α

Q2

int x0

0

dx g1(xQ2)






2M2N

Q2 Γ1(Q2) The


Γ1(pminusn)(Q2) =

int x0

0

dx[g1p(xQ



limQ2rarrinfin

Γ1(pminusn)(Q2) =

gA6 (34)






20

000 005 010 015 020 025 030

-006

-004

-002

000

002

004

Q2 GeV2D

G1

proton

000 005 010 015 020 025 030

-006

-005

-004

-003

-002

-001

000

001

Q2 GeV2D

G1

neutron




∆IA(Q2)























21

000 005 010 015 020 025 030-002

000

002

004

006

008

010

Q2 GeV2D

G1Hp-

nL





d∆I1p(Q2)

dQ2

∣∣∣∣Q2=0

= 039(4) asymp 034minus 053 + 058 (35a)

d∆I1n(Q2)

dQ2

∣∣∣∣Q2=0



dI1p(Q2)

dQ2

∣∣∣∣Q2=0

= 580dI1n(Q2)

dQ2

∣∣∣∣Q2=0

= 553 (36)




d2(Q2) equiv 3

int 1

0




d2(Q2) =

int 1

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)

22

000 005 010 015 020 025 030

0000

0001

0002

0003

0004

0005

0006

0007

Q2 GeV2D

d2

proton

000 005 010 015 020 025 030

0000

0002

0004

0006

0008

0010

0012

Q2 GeV2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

Q2 GeV2D

Γ 010

-4

fm6

D

proton

000 005 010 015 020 025 030-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron








(orange dot)





polarizabilities



d2(Q2) =

int x0

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)

23





d2(Q2) =Q4

8M4N

[M2

NQ2


] (40)














rule

γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν5(41)

=64αM4

N

Q10

int x0

0

dx x4

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









24

000 005 010 015 020 025 030

-0004

-0002

0000

0002

0004

0006

0008

0010

Q2GeV

2D

d2

proton

000 005 010 015 020 025 030

-0005

0000

0005

0010

0015

Q2GeV

2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

proton

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron


















25







dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0




shown in Fig 8

G Summary














26






(10minus4 fm4) n 298 003(92)


(10minus4 fm4) n 235 218(23)

γ0 p 208 minus096 minus001112(30)

(10minus4 fm6) n 292 195(30)


minus022(4)





minus084(10)




(∆I1)prime p 034 minus053 058039(4)






Proton Neutron




minus0929(105) [59]

δLT 132(15) 240(1) 134 218(23) 238(3) 203


γ0 112(30) 060(7)(7) [78] 195(30) 123

(10minus4 fm6) 0484(82) [59] [MAID]

IV CONCLUSIONS




27











ACKNOWLEDGEMENTS














T (νQ2) =

(minusgmicroν +

qmicroqν

q2

)T1(νQ2) +

1

M2N


q2qmicro)(


qν)T2(νQ2) (A1)

minus 1


M2N


)S2(νQ2)




28







ε0 =

[ε0 minus

ν
























29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References





d∆IAp(Q2)

dQ2

∣∣∣∣Q2=0


d∆IAn(Q2)

dQ2

∣∣∣∣Q2=0



dIAp(Q2)

dQ2

∣∣∣∣Q2=0

= minus318dIAn(Q2)

dQ2

∣∣∣∣Q2=0

= minus300 (31)



minus α

2M2N

I1(Q2) = minus 1

8π2

int infinν0

dν1radic

ν2 +Q2

[σTT (νQ2) +

Q

νσLT (νQ2)

](32)

= minus α

Q2

int x0

0

dx g1(xQ2)






2M2N

Q2 Γ1(Q2) The


Γ1(pminusn)(Q2) =

int x0

0

dx[g1p(xQ



limQ2rarrinfin

Γ1(pminusn)(Q2) =

gA6 (34)






20

000 005 010 015 020 025 030

-006

-004

-002

000

002

004

Q2 GeV2D

G1

proton

000 005 010 015 020 025 030

-006

-005

-004

-003

-002

-001

000

001

Q2 GeV2D

G1

neutron




∆IA(Q2)























21

000 005 010 015 020 025 030-002

000

002

004

006

008

010

Q2 GeV2D

G1Hp-

nL





d∆I1p(Q2)

dQ2

∣∣∣∣Q2=0

= 039(4) asymp 034minus 053 + 058 (35a)

d∆I1n(Q2)

dQ2

∣∣∣∣Q2=0



dI1p(Q2)

dQ2

∣∣∣∣Q2=0

= 580dI1n(Q2)

dQ2

∣∣∣∣Q2=0

= 553 (36)




d2(Q2) equiv 3

int 1

0




d2(Q2) =

int 1

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)

22

000 005 010 015 020 025 030

0000

0001

0002

0003

0004

0005

0006

0007

Q2 GeV2D

d2

proton

000 005 010 015 020 025 030

0000

0002

0004

0006

0008

0010

0012

Q2 GeV2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

Q2 GeV2D

Γ 010

-4

fm6

D

proton

000 005 010 015 020 025 030-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron








(orange dot)





polarizabilities



d2(Q2) =

int x0

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)

23





d2(Q2) =Q4

8M4N

[M2

NQ2


] (40)














rule

γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν5(41)

=64αM4

N

Q10

int x0

0

dx x4

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









24

000 005 010 015 020 025 030

-0004

-0002

0000

0002

0004

0006

0008

0010

Q2GeV

2D

d2

proton

000 005 010 015 020 025 030

-0005

0000

0005

0010

0015

Q2GeV

2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

proton

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron


















25







dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0




shown in Fig 8

G Summary














26






(10minus4 fm4) n 298 003(92)


(10minus4 fm4) n 235 218(23)

γ0 p 208 minus096 minus001112(30)

(10minus4 fm6) n 292 195(30)


minus022(4)





minus084(10)




(∆I1)prime p 034 minus053 058039(4)






Proton Neutron




minus0929(105) [59]

δLT 132(15) 240(1) 134 218(23) 238(3) 203


γ0 112(30) 060(7)(7) [78] 195(30) 123

(10minus4 fm6) 0484(82) [59] [MAID]

IV CONCLUSIONS




27











ACKNOWLEDGEMENTS














T (νQ2) =

(minusgmicroν +

qmicroqν

q2

)T1(νQ2) +

1

M2N


q2qmicro)(


qν)T2(νQ2) (A1)

minus 1


M2N


)S2(νQ2)




28







ε0 =

[ε0 minus

ν
























29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References

000 005 010 015 020 025 030

-006

-004

-002

000

002

004

Q2 GeV2D

G1

proton

000 005 010 015 020 025 030

-006

-005

-004

-003

-002

-001

000

001

Q2 GeV2D

G1

neutron




∆IA(Q2)























21

000 005 010 015 020 025 030-002

000

002

004

006

008

010

Q2 GeV2D

G1Hp-

nL





d∆I1p(Q2)

dQ2

∣∣∣∣Q2=0

= 039(4) asymp 034minus 053 + 058 (35a)

d∆I1n(Q2)

dQ2

∣∣∣∣Q2=0



dI1p(Q2)

dQ2

∣∣∣∣Q2=0

= 580dI1n(Q2)

dQ2

∣∣∣∣Q2=0

= 553 (36)




d2(Q2) equiv 3

int 1

0




d2(Q2) =

int 1

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)

22

000 005 010 015 020 025 030

0000

0001

0002

0003

0004

0005

0006

0007

Q2 GeV2D

d2

proton

000 005 010 015 020 025 030

0000

0002

0004

0006

0008

0010

0012

Q2 GeV2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

Q2 GeV2D

Γ 010

-4

fm6

D

proton

000 005 010 015 020 025 030-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron








(orange dot)





polarizabilities



d2(Q2) =

int x0

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)

23





d2(Q2) =Q4

8M4N

[M2

NQ2


] (40)














rule

γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν5(41)

=64αM4

N

Q10

int x0

0

dx x4

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









24

000 005 010 015 020 025 030

-0004

-0002

0000

0002

0004

0006

0008

0010

Q2GeV

2D

d2

proton

000 005 010 015 020 025 030

-0005

0000

0005

0010

0015

Q2GeV

2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

proton

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron


















25







dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0




shown in Fig 8

G Summary














26






(10minus4 fm4) n 298 003(92)


(10minus4 fm4) n 235 218(23)

γ0 p 208 minus096 minus001112(30)

(10minus4 fm6) n 292 195(30)


minus022(4)





minus084(10)




(∆I1)prime p 034 minus053 058039(4)






Proton Neutron




minus0929(105) [59]

δLT 132(15) 240(1) 134 218(23) 238(3) 203


γ0 112(30) 060(7)(7) [78] 195(30) 123

(10minus4 fm6) 0484(82) [59] [MAID]

IV CONCLUSIONS




27











ACKNOWLEDGEMENTS














T (νQ2) =

(minusgmicroν +

qmicroqν

q2

)T1(νQ2) +

1

M2N


q2qmicro)(


qν)T2(νQ2) (A1)

minus 1


M2N


)S2(νQ2)




28







ε0 =

[ε0 minus

ν
























29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References

000 005 010 015 020 025 030-002

000

002

004

006

008

010

Q2 GeV2D

G1Hp-

nL





d∆I1p(Q2)

dQ2

∣∣∣∣Q2=0

= 039(4) asymp 034minus 053 + 058 (35a)

d∆I1n(Q2)

dQ2

∣∣∣∣Q2=0



dI1p(Q2)

dQ2

∣∣∣∣Q2=0

= 580dI1n(Q2)

dQ2

∣∣∣∣Q2=0

= 553 (36)




d2(Q2) equiv 3

int 1

0




d2(Q2) =

int 1

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (38)

22

000 005 010 015 020 025 030

0000

0001

0002

0003

0004

0005

0006

0007

Q2 GeV2D

d2

proton

000 005 010 015 020 025 030

0000

0002

0004

0006

0008

0010

0012

Q2 GeV2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

Q2 GeV2D

Γ 010

-4

fm6

D

proton

000 005 010 015 020 025 030-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron








(orange dot)





polarizabilities



d2(Q2) =

int x0

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)

23





d2(Q2) =Q4

8M4N

[M2

NQ2


] (40)














rule

γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν5(41)

=64αM4

N

Q10

int x0

0

dx x4

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









24

000 005 010 015 020 025 030

-0004

-0002

0000

0002

0004

0006

0008

0010

Q2GeV

2D

d2

proton

000 005 010 015 020 025 030

-0005

0000

0005

0010

0015

Q2GeV

2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

proton

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron


















25







dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0




shown in Fig 8

G Summary














26






(10minus4 fm4) n 298 003(92)


(10minus4 fm4) n 235 218(23)

γ0 p 208 minus096 minus001112(30)

(10minus4 fm6) n 292 195(30)


minus022(4)





minus084(10)




(∆I1)prime p 034 minus053 058039(4)






Proton Neutron




minus0929(105) [59]

δLT 132(15) 240(1) 134 218(23) 238(3) 203


γ0 112(30) 060(7)(7) [78] 195(30) 123

(10minus4 fm6) 0484(82) [59] [MAID]

IV CONCLUSIONS




27











ACKNOWLEDGEMENTS














T (νQ2) =

(minusgmicroν +

qmicroqν

q2

)T1(νQ2) +

1

M2N


q2qmicro)(


qν)T2(νQ2) (A1)

minus 1


M2N


)S2(νQ2)




28







ε0 =

[ε0 minus

ν
























29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References

000 005 010 015 020 025 030

0000

0001

0002

0003

0004

0005

0006

0007

Q2 GeV2D

d2

proton

000 005 010 015 020 025 030

0000

0002

0004

0006

0008

0010

0012

Q2 GeV2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

Q2 GeV2D

Γ 010

-4

fm6

D

proton

000 005 010 015 020 025 030-2

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron








(orange dot)





polarizabilities



d2(Q2) =

int x0

0

dx x2 [3g2(xQ2) + 2g1(xQ2)] (39)

23





d2(Q2) =Q4

8M4N

[M2

NQ2


] (40)














rule

γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν5(41)

=64αM4

N

Q10

int x0

0

dx x4

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









24

000 005 010 015 020 025 030

-0004

-0002

0000

0002

0004

0006

0008

0010

Q2GeV

2D

d2

proton

000 005 010 015 020 025 030

-0005

0000

0005

0010

0015

Q2GeV

2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

proton

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron


















25







dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0




shown in Fig 8

G Summary














26






(10minus4 fm4) n 298 003(92)


(10minus4 fm4) n 235 218(23)

γ0 p 208 minus096 minus001112(30)

(10minus4 fm6) n 292 195(30)


minus022(4)





minus084(10)




(∆I1)prime p 034 minus053 058039(4)






Proton Neutron




minus0929(105) [59]

δLT 132(15) 240(1) 134 218(23) 238(3) 203


γ0 112(30) 060(7)(7) [78] 195(30) 123

(10minus4 fm6) 0484(82) [59] [MAID]

IV CONCLUSIONS




27











ACKNOWLEDGEMENTS














T (νQ2) =

(minusgmicroν +

qmicroqν

q2

)T1(νQ2) +

1

M2N


q2qmicro)(


qν)T2(νQ2) (A1)

minus 1


M2N


)S2(νQ2)




28







ε0 =

[ε0 minus

ν
























29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References





d2(Q2) =Q4

8M4N

[M2

NQ2


] (40)














rule

γ0(Q2) =1

2π2

int infinν0

dν

radic1 +

Q2

ν2

σTT (νQ2)

ν5(41)

=64αM4

N

Q10

int x0

0

dx x4

[g1(xQ2)minus 4M2

Nx2

Q2g2(xQ2)

]









24

000 005 010 015 020 025 030

-0004

-0002

0000

0002

0004

0006

0008

0010

Q2GeV

2D

d2

proton

000 005 010 015 020 025 030

-0005

0000

0005

0010

0015

Q2GeV

2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

proton

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron


















25







dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0




shown in Fig 8

G Summary














26






(10minus4 fm4) n 298 003(92)


(10minus4 fm4) n 235 218(23)

γ0 p 208 minus096 minus001112(30)

(10minus4 fm6) n 292 195(30)


minus022(4)





minus084(10)




(∆I1)prime p 034 minus053 058039(4)






Proton Neutron




minus0929(105) [59]

δLT 132(15) 240(1) 134 218(23) 238(3) 203


γ0 112(30) 060(7)(7) [78] 195(30) 123

(10minus4 fm6) 0484(82) [59] [MAID]

IV CONCLUSIONS




27











ACKNOWLEDGEMENTS














T (νQ2) =

(minusgmicroν +

qmicroqν

q2

)T1(νQ2) +

1

M2N


q2qmicro)(


qν)T2(νQ2) (A1)

minus 1


M2N


)S2(νQ2)




28







ε0 =

[ε0 minus

ν
























29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References

000 005 010 015 020 025 030

-0004

-0002

0000

0002

0004

0006

0008

0010

Q2GeV

2D

d2

proton

000 005 010 015 020 025 030

-0005

0000

0005

0010

0015

Q2GeV

2D

d2

neutron

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

proton

000 005 010 015 020 025 030

-1

0

1

2

3

Q2 GeV2D

Γ0

10-

4fm

6D

neutron


















25







dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0




shown in Fig 8

G Summary














26






(10minus4 fm4) n 298 003(92)


(10minus4 fm4) n 235 218(23)

γ0 p 208 minus096 minus001112(30)

(10minus4 fm6) n 292 195(30)


minus022(4)





minus084(10)




(∆I1)prime p 034 minus053 058039(4)






Proton Neutron




minus0929(105) [59]

δLT 132(15) 240(1) 134 218(23) 238(3) 203


γ0 112(30) 060(7)(7) [78] 195(30) 123

(10minus4 fm6) 0484(82) [59] [MAID]

IV CONCLUSIONS




27











ACKNOWLEDGEMENTS














T (νQ2) =

(minusgmicroν +

qmicroqν

q2

)T1(νQ2) +

1

M2N


q2qmicro)(


qν)T2(νQ2) (A1)

minus 1


M2N


)S2(νQ2)




28







ε0 =

[ε0 minus

ν
























29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References







dγ0p(Q2)

dQ2

∣∣∣∣Q2=0


dγ0n(Q2)

dQ2

∣∣∣∣Q2=0




shown in Fig 8

G Summary














26






(10minus4 fm4) n 298 003(92)


(10minus4 fm4) n 235 218(23)

γ0 p 208 minus096 minus001112(30)

(10minus4 fm6) n 292 195(30)


minus022(4)





minus084(10)




(∆I1)prime p 034 minus053 058039(4)






Proton Neutron




minus0929(105) [59]

δLT 132(15) 240(1) 134 218(23) 238(3) 203


γ0 112(30) 060(7)(7) [78] 195(30) 123

(10minus4 fm6) 0484(82) [59] [MAID]

IV CONCLUSIONS




27











ACKNOWLEDGEMENTS














T (νQ2) =

(minusgmicroν +

qmicroqν

q2

)T1(νQ2) +

1

M2N


q2qmicro)(


qν)T2(νQ2) (A1)

minus 1


M2N


)S2(νQ2)




28







ε0 =

[ε0 minus

ν
























29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References






(10minus4 fm4) n 298 003(92)


(10minus4 fm4) n 235 218(23)

γ0 p 208 minus096 minus001112(30)

(10minus4 fm6) n 292 195(30)


minus022(4)





minus084(10)




(∆I1)prime p 034 minus053 058039(4)






Proton Neutron




minus0929(105) [59]

δLT 132(15) 240(1) 134 218(23) 238(3) 203


γ0 112(30) 060(7)(7) [78] 195(30) 123

(10minus4 fm6) 0484(82) [59] [MAID]

IV CONCLUSIONS




27











ACKNOWLEDGEMENTS














T (νQ2) =

(minusgmicroν +

qmicroqν

q2

)T1(νQ2) +

1

M2N


q2qmicro)(


qν)T2(νQ2) (A1)

minus 1


M2N


)S2(νQ2)




28







ε0 =

[ε0 minus

ν
























29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References











ACKNOWLEDGEMENTS














T (νQ2) =

(minusgmicroν +

qmicroqν

q2

)T1(νQ2) +

1

M2N


q2qmicro)(


qν)T2(νQ2) (A1)

minus 1


M2N


)S2(νQ2)




28







ε0 =

[ε0 minus

ν
























29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References







ε0 =

[ε0 minus

ν
























29

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References

proton neutron

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Neutron

02 04 06 08 100

50

100

150

200

250

[GeV]

T [b]

02 04 06 08 10

-50

0

50

100

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

5

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

08

[GeV]

LTQ

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02040608100

50

100

150

200

250

300

[GeV]

T[b]

0204060810

-50

0

50

[GeV]

TT[b]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02040608100

1

2

3

4

[GeV]

LQ2

[GeV-4]

0204060810

00

02

04

06

[GeV]

LTQ

[GeV-3]

Proton

02 04 06 08 100

50

100

150

200

250

300

[GeV]

T [b]

02 04 06 08 10

-50

0

50

[GeV]

TT [b]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 100

1

2

3

4

[GeV]

LQ2

[GeV-4]

02 04 06 08 10

00

02

04

06

[GeV]

LTQ

[GeV-3]



















30

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References

s = 12

sprime = minus12

λ = 0 λprime = 1

rprime rprime

s=12sprime=minus1

2

λ=0λprime=1

rprimerprime

s=12

sprime=minus12

λ=0λprime=1


λacute=1λ=0






and pions


T = uhB (PB)sumi



Γ1 = γ5 (B3a)

Γ2 =1

2

[qA ε

]γ5 (B3b)














σLT (νQ2) =1

64π2 s

|~pf |cm

|~pi|cm

int 1

minus1dcos θ

sumij

AiAdaggerjXij (B4)

31

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References

with


0] (B5)



matrix Xij leads to

X = MNQ



(B6)






A1 =e gAMN

fπ


sminusM2N


uminusM2N

] (B7a)

A2 =e gAMN

fπ

[1

sminusM2N

+1

uminusM2N

] (B7b)


A1 =

radic2 e gAMN

fπ


sminusM2N


tminusm2π

] (B8a)

A2 =

radic2 e gAMN

fπ(sminusM2N )

(B8b)


A1 = 0 (B9a)

A2 = 0 (B9b)


A1 =

radic2 e gAMN

fπ


uminusM2N


π

] (B10a)

A2 =

radic2 e gAMN

fπ(uminusM2N )

(B10b)



32






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References






s+M2N +Q2

2s (B11)


s+M2N minusm2

π

2s (B12)


sminusM2N minusQ2

2s (B13)


sminusM2N +m2

π

2s (B14)


radic(sminusM2

N minusQ2)2 + 4sQ2

2s (B15)


radic(sminusM2


π

2s (B16)




AM2N

64πf2πs

3(sminusM2N )2λ4

γ

4sλπλγ

[(M2


+ 2s(m2πQ



))λ2γ

]+ (M2



)times log


) (B17)

33

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References

σ(π0p)TT =

e2g2AM

2Nλπ

64πf2πs(sminusM2

N )2λγ

1

s2λπλ3γ

[(M2


(3M2



+ 2s

((2m2

πQ2 + (M2

N minus s)2)(M2


πQ2 + (sminusM2

N )2)


γβ2π

)λ2γ

]arctan

(2sλπλγ


)+ 2

[minus 2m2

πQ2

+ (M2N minus s)

(2(sminusM2



sλ2γ

+m2πQ

2(sminusM2N )



+m2πQ

2(sminusM2N )



)

+1

λπλγ

(βγβπ

(minusm2

πQ2 + (M2


)minus (M2


π

)log



)] (B18)

σ(πminusp)TT =

e2g2AM

2N

64πf2πs

3λ4γ(M2


2λ2γ)

4sλπλγ

((minusM2

N +Q2 + s)(M2N


2 minus 2sλ2γ

(2λ2


πQ2))


)2

minus 4λ2πs

2λ2γ



)+ 4λ2

πs2λ2

γ

)times log

(minusM2


)(M2N + s(minus1




)+ 4λ2

πs2λ2

γ

)times log

(minusM2



πs2λ2

γ

)times log


)]+ 4sλ2

γ

((M2


2λ2γ

)times (M2

N minus s) arctanh

(2λπsλγ


) (B19)

34

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References

σ(π+n)LT =

e2g2AM

3Nλπ

32πf2πQs

3(sminusM2N )2λ4

γ

2sλγ

[(M2


minus 4s

((M2


)+Q2sβγβπ


γ

]minus M2

N minus sλπ

[(Q2 + 2sβ2


minusQ2λ2π


γ

]arctan

(2sλπλγ

Q2 + 2sβγβπ

) (B20)

σ(π0p)LT =

e2g2AM

3Nλπ

16πf2πQs(sminusM2

N )2λγ

1


γ + 8s3λ2πλ

4γ

[minus 3M8

N (Q2

+ 2sβ2γ) + 2M4

Ns2

(minus (Q2 + 2sβ2


π) + 2sβγ


π minus 1)λ4π

)+ 2M2

Ns3

[minus (Q2



)+

(Q2(3minus 12λ2






2π

)λ4γ

)+ s4

((Q2 + 2sβ2

γ)







πλ6γ

)+ 2M6

Ns

(Q2(6


γ)

)]+sminusM2

N

4s2λπλ3γ

[(Q2 + 2sβ2

γ)

times(3M2


)+ 2s

(minusM2



γ

]

times arctan

(2sλπλγ


) (B21)

35

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References

σ(πminusp)LT =

e2g2AM

3N

64πf2πQs

3λ4γ(M2


times

1


[4λπsλγ(M2

N minusQ2 minus s)(

2sλ2γ

(s((1minus 2λ2

π)Q2


)+ (Q2 + 2sβ2



3λ4γ


)times[

log



)(minus (Q2 + 2sβ2


)(minusM2


(minusM2




γ

)+ 2

(4s2λ2

γ


πQ2)

+ (Q2 + 2βπsβγ)2(Q2 + 2sβ2γ)


)arctanh

(2λπsλγ

Q2 + 2βπsβγ

)] (B22)




into [17]


1 (νQ2) + S∆-exch1 (νQ2) (B23a)



(νQ2) (B23b)





1


∆]=

1

4M2N

1

ν2∆ minus ν2

(B24)

36



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References



σTT (νQ2) =π2α

M2NM

2+|~q |

minus g2


MN(B25a)

+g2CQ

4s(∆minus ν)

MNM2∆


+2gEgCQ


]MNM∆

δ(ν minus ν∆)

σLT (νQ2) =Qπ2α

M2NM

2+|~q |


MN(B25b)

+g2CQ


]MNM2

∆


M∆



MNM∆

δ(ν minus ν∆)






S∆-exch1 (νQ2) =

πα

MNM2+

[g2MQ

2+ + g2

E

(∆2 minus 3Q2

)+

4g2CQ

4

M2∆



M∆+


M∆

]

νS2∆-exch

(νQ2) =2πα

MNM2+


g2M MNQ

2+

2+g2C Q

2(Q2 minus∆2)

2M∆(B26b)



]+S∆-exch

2 (νQ2)

ν

[M2

∆ ω2+

M2N

+ ν2

]

with Q+ =radic



to S2(νQ2)


M∆M2+

[gM + gE

]gC (B27)





37



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References



limνrarr0


νrarr0νS2

∆-exch(νQ2) = 0 (B28)







∣∣∣Q2=0

= 0







γ0p =e2g2

A

96π3f2πm

2π

1minus 21πmicro

8minus(

59

2+ 26 logmicro

)micro2 +

1875πmicro3

64

+3

(3

2+ 26 logmicro

)micro4 +

(C1)

γ0n =e2g2

A

48π3f2πm

2π

1

2minus 9πmicro


75πmicro3

128minus 3micro4

4+

(C2)

δLTp =e2g2

A

192π3f2πm

2π

1minus 9πmicro

8+

(13

2minus 2 logmicro


64

minus(

47

2+ 42 logmicro

)micro4 +

(C3)

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C4)

38

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References

δLTn =e2g2

A

96π3f2πm

2π

1

2+

3πmicro


128+

5micro4

4+

(C5)

γ0p =e2g2

A

16π3f2πm

4π

4

45minus 3πmicro

16+

14micro2

5minus 1813πmicro3

384minus 192


+80703πmicro5

2048+

(C6)

γ0n =e2g2

A

16π3f2πm

4π

4

45minus 5πmicro

48+

4micro2

5minus 245πmicro3


15

+1323πmicro5

2048+

(C7)


dγ0p(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 45πmicro


64

minus9

(1953

4+ 449 logmicro

)micro4 +

570255πmicro5

128+

(C8)

dγ0n(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

2minus 81πmicro


32


1024+

(C9)

dδLTp(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

2880π3f2πm

4π

minus5

2minus 27πmicro


256

+3

(617

4+ 36 logmicro


4096+

(C10)

dδLTn(Q2)

dQ2

∣∣∣∣Q2=0

=e2g2

A

1440π3f2πm

4π

minus5

4minus 81πmicro

64minus 11micro2 +

10005πmicro3

512

+15


8192+

(C11)

dIAp(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

2

1minus 15πmicro

4minus 1


1839πmicro3

32


(C12)

dIAn(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

2

1

2minus 11πmicro

8minus 1


99πmicro3

64

minus25micro4

12+

(C13)

dI1p(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

96π2f2πmicro

3π


64

minus1


(C14)

39

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References

dI1n(Q2)

dQ2

∣∣∣∣Q2=0

=g2A

48π2f2πmicro

minus π

16+

1


128+

2micro3

3+

(C15)

γ0p(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 23πmicro

256+

377micro2


6144+

3371micro4

105


32768+

(C16)

γ0n(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2g2

A

16π3f2πm

6π

1

105minus 153πmicro

1792+

69micro2


6144+

172micro4

35

minus120897πmicro5

32768+

(C17)







γ0 = minus e2

4πM2+

(g2M

∆2+

g2E

M2+

minus 4gMgEM+∆

) (C18)

δLT =e2M∆

4πM3+

(g2E

MNM++gMgE∆MN

minus gEgCM2

∆

) (C19)

γ0 =e2M2

N

π∆2M4+

(minusg

2M

∆2+

g2E

M2+

+4gMgE∆M+

) (C20)


dγ0(Q2)

dQ2

∣∣∣∣∣Q2=0

= minus e2

πM2+∆

(g2M

∆

[1

4∆2minus 1

∆M++

1

2M2+

]minus 1

Λ2

g2M

∆+

g2E

2M2+

[1

2∆minus 3

M+

]minusgMgE

M+

[1

∆2minus 5

∆M++

1

M2+

]+

1

Λ2

2gMgEM+

+2gMgC∆M2

+

minus gEgCM3

+

) (C21)

40

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References

d δLT (Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M∆∆

4πMNM2+

(g2E

∆2M2+

[1

∆minus 4

M+

]minus g2

C

∆M2∆M

2+

+gMgE∆2M+

[1

∆2

minus 3

∆M++

1

M2+

]minus 2

Λ2

gMgE∆2M+

+gMgC∆M2

∆

[1

2∆2minus 2

∆M++

1

2M2+

]minus gEgC

2M2∆M

2+

[7

∆+

1

M+

]) (C22)

dIA(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM2N

M2+

(g2M

2∆2+

g2E

MNM+minus 2gMgE


) (C23)

dI1(Q2)

dQ2

∣∣∣∣∣Q2=0

= minusM∆M2N

2M3+

(g2E


minus gEgCM2

∆

) (C24)

γ0(Q2)

dQ2

∣∣∣∣∣Q2=0

=e2M2

N

π∆3M6+

(g2M

∆

[3M2


∆2+

4M2+

Λ2

]

minus4gMgE

[5M2

∆ minus 9M2N

M+∆2+

2M+

Λ2

]+

g2E


minus8gMgC∆

+4gEgCM+

) (C25)












16 908 (1966)


USA 72 1 (1975)




41



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References



































42




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References




17-21 2018 (2019) arXiv190305661 [nucl-ex]



805 135428 (2020) arXiv190805709 [nucl-ex]

















ph0512244





ph9606456






43

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References

(2014)


1428 (1954)


12 Phys Rev 96 1433 (1954)



















th]





016001 (2003) arXivhep-ph0209241






44


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References


(1988)





Rev D 63 114010 (2001) arXivhep-ph0008306



arXivnucl-th9807001











(2015) arXiv150507877 [nucl-ex]



[physicsatom-ph]








D 58 096002 (1998) arXivhep-ph9802288


CD2018 031 (2019)

45
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References
















148 1467 (1966)


1 1376 (1970)



ph0203167


(2008) arXiv08023198 [nucl-ex]


B 687 160 (2010) arXiv10014230 [hep-ph]


Phys 19 239 (1990)








46









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References









arXivhep-ph0507016






[physicsatom-ph]

47


Abstract

Contents

I Introduction












G Summary

IV Conclusions

Acknowledgements








References

albert einstein center for fundamental physics, institute for … · 2020-06-17 · sidlerstrasse...

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