nucleon, delta andnucleon todelta electromagnetic...

Nucleon, Delta and Nucleon to Delta

Electromagnetic Form Factors in DSEs

Jorge Segovia, Ian C. Cloet and Craig D. Roberts

Argonne National Laboratory

Chen Chen and Shaolong Wan

University of Science and Technology of China

International Workshop on Nonperturbative Phenomena

in Hadron and Particle Physics

May 5-10, 2014, Ubatuba, Sao Paulo, Brazil

Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 1/34

The challenge of QCD

Quantum Chromodynamics is the only known example in nature of a nonperturvativefundamental quantum field theory

☞ QCD have profound implications for our understanding of the real-world:

Explain how quarks and gluons bind together to form hadrons.

Origin of the 98% of the mass in the visible universe.

☞ Given QCD’s complexity:

The best promise for progress is a strong interplay between experiment and theory.

Emergent phenomena

ւ ցQuark and gluon confinement Dynamical chiral symmetry breaking

↓ ↓Colored particles

have never been seenisolated

Hadrons do notfollow the chiralsymmetry pattern

Neither of these phenomena is apparent in QCD’s Lagrangianyet!

They play a dominant role determining characteristics of real-world QCD


Emergent phenomena: Confinement

Confinement is associated with dramatic, dynamically-driven changes in the analyticstructure of QCD’s propagators and vertices (QCD’s Schwinger functions)

☞ Dressed-propagator for a colored state:

An observable particle is associated with a poleat timelike-P2.

When the dressing interaction is confining:

Real-axis mass-pole splits, moving into a pairof complex conjugate singularities.

No mass-shell can be associated with a particlewhose propagator exhibits such singularity.

☞ Dressed-gluon propagator:

Confined gluon.

IR-massive but UV-massless.

mG ∼ 2− 4ΛQCD (ΛQCD ≃ 200MeV).

Any 2-point Schwinger function with an inflexionpoint at p2 > 0:→ Breaks the axiom of reflexion positivity→ No physical observable related with


Emergent phenomena: Dynamical chiral symmetry breaking

Spectrum of a theory invariant under chiral transformations should exhibit degenerateparity doublets

π JP = 0− m = 140MeV cf. σ JP = 0+ m = 500MeV

ρ JP = 1− m = 775MeV cf. a1 JP = 1+ m = 1260MeV

N JP = 1/2+ m = 938MeV cf. N(1535) JP = 1/2− m = 1535MeV

Splittings between parity partners are greater than 100-times the light quark massscale: mu/md ∼ 0.5, md = 4MeV

☞ Dynamical chiral symmetry breaking

Mass generated from the interaction of quarks withthe gluon-medium.

Quarks acquire a HUGE constituent mass.

Responsible of the 98% of the mass of the proton.

☞ (Not) spontaneous chiral symmetry breaking

Higgs mechanism.

Quarks acquire a TINY current mass.

Responsible of the 2% of the mass of the proton. 0 1 2 3

p [GeV]

0

0.1

0.2

0.3

0.4

M(p

) [G

eV

] m = 0 (Chiral limit)m = 30 MeVm = 70 MeV

effect of gluon cloudRapid acquisition of mass is


Theory tool: Dyson-Schwinger equations

Confinement and Dynamical Chiral Symmetry Breaking (DCSB) can be identified withproperties of dressed-quark and -gluon propagators and vertices (Schwinger functions)

Dyson-Schwinger equations (DSEs)

The quantum equations of motion of QCD whosesolutions are the Schwinger functions.

→ Propagators and vertices.

Generating tool for perturbation theory.

→ No model-dependence.

Nonperturbative tool for the study of continuum strongQCD.

→ Any model-dependence should be incorporated here.

Allows the study of the interaction between light quarksin the whole range of momenta.

→ Analysis of the infrared behaviour is crucial todisentangle confinement and DCSB.

Connect quark-quark interaction with experimentalobservables.

→ Elastic and transition form factors can be used toilluminate QCD (at infrared momenta).

0

0.1

0.2

0.3

0.4

M(p)

(GeV)

0 1 2 3 4

p (GeV)

α = 2.0

α = 1.8

α = 1.4

α = 1.0

−0.2

0

0.2

0.4

0.6

0.8

1.0

µpG

Ep/G

Mp

0 2 4 6 8 10

Q2 (GeV2)

α = 2.0α = 1.8α = 1.4α = 1.0

Ian C. Cloet and Craig D. RobertsarXiv:nul-th/1310.2651


The simplest example of DSEs: The gap equation

The quark propagator is given by the gap equation:

S−1(p) = Z2(iγ · p +mbm) + Σ(p)

Σ(p) = Z1

∫ Λ

qg2Dµν(p − q)

λa

2γµS(q)

λa

2Γν(q, p)

General solution:

S(p) =Z(p2)

iγ · p +M(p2)

Kernel involves:Dµν (p − q) - dressed gluon propagatorΓν(q, p) - dressed-quark-gluon vertex

0 1 2 3

p [GeV]

0

0.1

0.2

0.3

0.4

M(p

) [G

eV

]

m = 0 (Chiral limit)m = 30 MeVm = 70 MeV

effect of gluon cloudRapid acquisition of mass is

M(p2) exhibits dynamicalmass generation

Each of which satisfies its own Dyson-Schwinger equation

↓Infinitely many coupled equations

↓Coupling between equations necessitates truncation


Vector and Axial-vector Ward-Green-Takahashi identities

Symmetries should be preserved by any truncation

↓Highly nontrivial constraint → failure implies loss of any connection with QCD

↓Symmetries in QFT are implemented by WTIs which relate different Schwinger functions

For instance, axial-vector Ward-Takahashi identity:

These observations show that symmetries relate the kernel of the gap equation – aone-body problem – with that of the Bethe-Salpeter equation – a two-body problem –


Bethe-Salpeter and Faddeev equations

Hadrons are studied via Poincare covariant bound-state equations

☞ Mesons

A 2-body bound state problem in quantumfield theory.

Properties emerge from solutions ofBethe-Salpeter equation:

Γ(k;P) =

∫

d4q

(2π)4K(q, k;P)S(q+P) Γ(q;P) S(q)

The kernel is that of the gap equation.

=

iΓ

iS

iΓ

K

iS

☞ Baryons

A 3-body bound state problem in quantumfield theory.

Structure comes from solving the Faddeevequation.

Faddeev equation: Sums all possible quantumfield theoretical interactions that can takeplace between the three quarks that define itsvalence quark content.

=aΨ

P

pq

pd Γb

Γ−a

pd

pq

bΨP

q


Diquarks inside baryons

The attractive nature of quark-antiquark correlations in a color-singlet meson is alsoattractive for 3c quark-quark correlations within a color-singlet baryon

☞ Diquark correlations:

Empirical evidence in support of strong diquarkcorrelations inside the nucleon.

A dynamical prediction of Faddeev equationstudies.

In our approach: Non-pointlike color-antitripletand fully interacting. Thanks to G. Eichmann.

Diquark composition of the nucleon and ∆

Positive parity states

ւ ցpseudoscalar and vector diquarks scalar and axial-vector diquarks

↓ ↓Ignored

wrong paritylarger mass-scales

Dominantright parity

shorter mass-scales

→ N ⇒ 0+, 1+ diquarks∆ ⇒ only 1+ diquark


Study of electromagnetic form factors - Motivation

A central goal of Nuclear Physics: understand the structure and properties of protonsand neutrons, and ultimately atomic nuclei, in terms of the quarks and gluons of QCD.

Elastic and transition form factors

ւ ցUnique window into its

quark and gluon structureHigh-Q2 reach by

experiments

↓ ↓Distinctive information on theroles played by confinement

and DCSB in QCD

Probe the excited nucleonstructures at perturbative andnon-perturbative QCD scales

CEBAF Large Acceptance Spectrometer (CLAS)

☞ Most accurate results for the electroexcitation amplitudesof the four lowest excited states.

☞ They have been measured in a range of Q2 up to:

8.0GeV2 for ∆(1232)P33 and N(1535)S11 .

4.5GeV2 for N(1440)P11 and N(1520)D13 .

☞ The majority of new data was obtained at JLab.

Upgrade of CLAS up to 12GeV → CLAS12 (New generation experiments in 2015)


The γ∗N → ∆ reaction

Two ways in order to analyze the structure of the ∆-resonances

ւ ցπ-mesons as a probe photons as a probe

↓ ↓complex relatively simple

BUT: B(∆ → γN) . 1%

This became possible with the advent of intense, energetic electron-beam facilities

Reliable data on the γ∗p → ∆+ transition:

☞ Available on the entire domain 0 ≤ Q2 . 8GeV2.

Isospin symmetry implies γ∗n → ∆0 is simply related with γ∗p → ∆+.

γ∗p → ∆+ data has stimulated a great deal of theoretical analysis:

Deformation of hadrons.

The relevance of pQCD to processes involving moderate momentum transfers.

The role that experiments on resonance electroproduction can play in exposingnon-perturbative phenomena in QCD:

☞ The nature of confinement and Dynamical Chiral Symmetry Breaking.


The electromagnetic current

☞ The electromagnetic current can be generally written as:

Jµλ(K ,Q) = Λ+(Pf )Rλα(Pf ) iγ5 Γαµ(K ,Q) Λ+(Pi )

Incoming nucleon: P2i = −m2

N , and outgoing delta: P2f = −m2

∆.

Photon momentum: Q = Pf − Pi , and total momentum: K = (Pi + Pf )/2.

The on-shell structure is ensured by the N- and ∆-baryon projection operators.

☞ The composition of the 4-point function Γαµ is determined by Poincare covariance:

Convenient to work with orthogonal momenta ↔ Simplify its structure considerably

↓Not yet the case for K and Q ↔ ∆(m∆ −mN) 6= 0 ⇒ K · Q 6= 0

↓We take instead K⊥

µ = TQµν Kν and Q

☞ Vertex decomposes in terms of three (Jones-Scadron) form factors:

Γαµ = k

[

λm

2λ+(G∗

M − G∗

E )γ5εαµγδK⊥γ Qδ − G∗

ETQαγT

Kγµ − iς

λmG∗

C QαK⊥µ

]

,

Magnetic dipole ⇒ G∗

M Electric quadrupole ⇒ G∗

E Coulomb quadrupole ⇒ G∗

C


Extraction of the form factors

The Jones-Scadron form factors are:

G ∗M = 3(s2 + s1),

G ∗E = s2 − s1,

G ∗C = s3.

G∗M,Ash

vs G∗M,J−S

G∗M,Ash = G∗

M,J−S

(

1 +Q2

(m∆ +mN)2

)− 12

The scalars are obtained from the following Dirac traces and momentum contractions:

s1 = n

√

ς(1 + 2d )

d − ςTKµν K

⊥λ Tr[γ5Jµλγν ],

s2 = nλ+

λm

TKµλTr[γ5Jµλ],

s3 = 3nλ+

λm

(1 + 2d )

d − ςK⊥

µ K⊥λ Tr[γ5Jµλ].

We have used the following notation:

n =

√

1 − 4d 2

4ik λm

, λ± =(m∆±mN )2+Q2

2(m2∆+m2

N)

, ς =Q2

2(m2∆ + m2

N),

d =m2

∆ − m2N

2(m2∆ + m2

N ), λm =

√

λ+λ−, k =

√

3

2

(

1 +m∆

mN

)

.

G. Eichmann et al., Phys. Rev. D 85, 093004 (2012).


Experimental results and theoretical expectations

I.G. Aznauryan and V.D. Burkert Prog. Part. Nucl Phys. 67, 1-54 (2012)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

10-1

1Q2 (GeV2)

G* M

,Ash

/3G

D

-7-6-5-4-3-2-1012

RE

M (

%)

-35

-30

-25

-20

-15

-10

-5

0

10-1

1Q2 (GeV2)

RS

M (

%)

☞ The REM ratio is measured to beminus a few percent.

☞ The RSM ratio does not seem to

settle to a constant at large Q2.

pQCD predictions

For Q2 → ∞G∗

M → 1/Q4.

REM → +100%.

RSM → constant.

CQM predictions

Without quark orbital angularmomentum:

REM → 0.

RSM → 0.

SU(6) predictions

〈p|µ|∆+〉 = 〈n|µ|∆0〉〈p|µ|∆+〉 = −

√2 〈n|µ|n〉

Data do not support these predictions

↓Our aim: try to understand this longstanding puzzle


Electromagnetic current description in the quark-diquark picture

To compute the electromagnetic properties of the γ∗N∆ reaction in a givenframework, one must specify how the photon couples to its constituents.

Six contributions to the current

1 Coupling of the photon to thedressed quark.

2 Coupling of the photon to thedressed diquark:

➫ Elastic transition.

➫ Induced transition.

3 Exchange and seagull terms.

Ingredients in the contributions

1 Ψi,f ≡ Faddeev amplitudes.

2 Single line ≡ Quark prop.

3 Double line ≡ Diquark prop.

4 Γ ≡ Diquark BS amplitudes.

One-loop diagrams

i

iΨ ΨPf

f

P

Q

i

iΨ ΨPf

f

P

Q

scalaraxial vector

i

iΨ ΨPf

f

P

Q

Two-loop diagrams

i

iΨ ΨPPf

f

Q

Γ−

Γ

µ

i

i

X

Ψ ΨPf

f

Q

P Γ−

µi

i

X−

Ψ ΨPf

f

P

Q

ΓJorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 15/34

Elastic form factor of the proton in the quark-diquark picture (I)

☞ Each diagram can be expressed in a similar way:

Γµ = Λ+(Pf )Jµ(K ,Q)Λ+(Pi )

Photon coupling directly to a dressed-quark with the diquark acting as a bystander

Initial state and final state: Proton

Two axial-vector diquark isospin states:

(I , Iz ) = (1, 1) → flavor content: {uu}(I , Iz ) = (1, 0) → flavor content: {ud}

In the isospin limit, they appear with relative

weighting: (−√

2/3) : (√

1/3)

Therefore

Jscalar

µ =

√

1

3

√

1

3eu I

{ud}µ 6= 0

Jaxial

µ =

√

2

3

√

2

3ed I

{uu}µ +

√

1

3

√

1

3eu I

{ud}µ = 0

ΨiΨfPf Pi

Q

Pf

Pi

Q

ΨiΨf

axial − vector

scalar

Hard contributions appear in the microscopic description of the elastic form factor ofthe proton


Elastic form factor of the proton in the quark-diquark picture (II)

Remaining diagrams: Photon interacting with diquarks

H.L.L. Roberts et al. Phys. Rev. C 83, 065206 (2011)I.C. Cloet et al., Few Body Syst. 46, 1-36 (2012)

ΨiΨf PiPf

Q

Q

ΨiΨfPf Pi

axial scalar 0 2 4 6 8 10-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Q2HGeV2

L

GE

,M,Q

1+

,G

M1+

�GQ1+

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

Q2HGeV2

L

G0+Γ

1+

,GΠΓΡ

Composite object

↓Electromagnetic radius is nonzero (rqq & rπ)

↓Softer contribution to the form factors

Soft contributions appear in the microscopic description of the elastic form factor ofthe proton


Transition form factor of γ∗N∆ in the quark-diquark picture

☞ Diagrams in which the photon interact with diquarks appear

Photon coupling directly to a dressed-quark with the diquark acting as a bystander

Initial state: Proton

Two axial-vector diquark isospin states:

(I , Iz ) = (1, 1) → flavor content: {uu}(I , Iz ) = (1, 0) → flavor content: {ud}

In the isospin limit, they appear with relative

weighting: (−√

2/3) : (√

1/3)

Final state: ∆+

Same isospin states of axial-vector diquark.

Different weighting due to I∆ = 3/2:

(√

1/3) : (√

2/3)

Therefore

J1,axialµα = −

√

2

3

√

1

3ed I

1{uu}µα +

√

1

3

√

2

3eu I

1{ud}µα =

√2

3I1{qq}µα (K ,Q)

ΨiΨfPf Pi

Q

Pf

Pi

Q

ΨiΨf

axial − vector

scalar

Soft and still hard contributions appear in the microscopic description of the γ∗N∆electromagnetic reaction


General observation

GpM vs G∗p

M

☞ Similar contributions in both cases:

G∗pM should fall asymptotically at the same rate as Gp

M .

☞ By isospin considerations:

G∗nM should fall asymptotically at the same rate as G∗p

M .

☞ Hold SU(6):

〈p|µ|∆+〉 ∝ 〈n|µ|∆0〉 ∝ 〈p|µ|p〉 .


Simple framework

Symmetry preserving Dyson-Schwinger equation treatment of a vector × vectorcontact interaction

☞ Gluon propagator: Contact interaction.

g2Dµν(p − q) = δµν4παIR

m2G

☞ Truncation scheme: Rainbow-ladder.

Γaν(q, p) = (λa/2)γν

☞ Quark propagator: Gap equation.

S−1(p) = iγ · p +m+ Σ(p)

= iγ · p +M

M ∼ 0.4GeV. Implies momentumindependent constituent quark mass.

Implies momentum independent BSAs.

☞ Baryons: Faddeev equation.

mN = 1.14GeV m∆ = 1.39GeV

(masses reduced by meson-cloud effects)

☞ Form Factors: Two-loop diagrams notincorporated.

Exchange diagram

It is zero because our treatment of thecontact interaction model

i

iΨ ΨPPf

f

Q

Γ−

Γ

Seagull diagrams

They are zero

µ

i

i

X

Ψ ΨPf

f

Q

P Γ−

µi

i

X−

Ψ ΨPf

f

P

Q

Γ


Series of papers establishes strengths and limitations

Used judiciously, produces results indistinguishable from most-sophisticatedinteractions employed in the rainbow ladder truncation of QCDs DSEs

1 Features and flaws of a contact interaction treatment of the kaonC. Chen, L. Chang, C.D. Roberts, S.M. Schmidt S. Wan and D.J. WilsonPhys. Rev. C 87 045207 (2013). arXiv:1212.2212 [nucl-th]

2 Spectrum of hadrons with strangenessC. Chen, L. Chang, C.D. Roberts, S. Wan and D.J. WilsonFew Body Syst. 53 293-326 (2012). arXiv:1204.2553 [nucl-th]

3 Nucleon and Roper electromagnetic elastic and transition form factorsD.J. Wilson, I.C. Cloet, L. Chang and C.D. RobertsPhys. Rev. C 85, 025205 (2012). arXiv:1112.2212 [nucl-th]

4 π- and ρ-mesons, and their diquark partners, from a contact interactionH.L.L. Roberts, A. Bashir, L.X. Gutierrez-Guerrero, C.D. Roberts and D.J. WilsonPhys. Rev. C 83, 065206 (2011). arXiv:1102.4376 [nucl-th]

5 Masses of ground and excited-state hadronsH.L.L. Roberts, L. Chang, I.C. Cloet and C.D. RobertsFew Body Syst. 51, 1-25 (2011). arXiv:1101.4244 [nucl-th]

6 Abelian anomaly and neutral pion productionH.L.L. Roberts, C.D. Roberts, A. Bashir, L.X. Gutierrez-Guerrero and P.C. TandyPhys. Rev. C 82, 065202 (2010). arXiv:1009.0067 [nucl-th]

7 Pion form factor from a contact interactionL.X. Gutierrez-Guerrero, A. Bashir, I.C. Cloet and C.D. RobertsPhys. Rev. C 81, 065202 (2010). arXiv:1002.1968 [nucl-th]


Weakness of contact-interaction

A truncation which produces Faddeev amplitudes that are independent of relativemomentum:

Underestimates the quark orbital angular momentum content of the bound-state.

Eliminates two-loop diagram contributions.

G∗

M (Q2 = 0) ind.-p DSE kernels dep.-p DSE kernelsaxial-diquark(∆) − axial-diquark(p) 0.85 0.83axial-diquark(∆) − scalar-diquark(p) 0.18 1.10

Improved version

Rescale the axial-diquark(∆) − scalar-diquark(p) diagram using:

1 +gas/aa

1 +Q2/m2ρ

axial(∆)-scalar(p) = axial(∆)-axial(p) for G∗

M(Q2 = 0)

Incorporate dressed quark-anomalous magnetic moment ⇔ Consequence of theDCSB.


More sophisticated framework

☞ Gluon propagator: 1/k2-behaviour.

☞ Truncation scheme: Rainbow-ladder.

Γaν(q, p) = (λa/2)γν

☞ Quark propagator: Gap equation.

S−1(p) = Z2(iγ · p +mbm) + Σ(p)

=[

1/Z(p2)] [

iγ · p +M(p2)]

M ∼ 0.33GeV. Implies momentum dependentconstituent quark mass.

Implies momentum dependent BSAs.

☞ Baryons: Faddeev equation.

mN = 1.18GeV m∆ = 1.33GeV

(masses reduced by meson-cloud effects)

☞ Form Factors: Two-loop diagramsincorporated.

Exchange diagram

Play an important role

i

iΨ ΨPPf

f

Q

Γ−

Γ

Seagull diagrams

They are less important

µ

i

i

X

Ψ ΨPf

f

Q

P Γ−

µi

i

X−

Ψ ΨPf

f

P

Q

ΓJorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 23/34

Electromagnetic form factors of the Nucleon

☞ Bibliography:Current quark mass dependence of nucleon magnetic moments and radii

I.C. Cloet et al., Few Body Syst. 42, 91-113 (2008). arXiv:0804.3118 [nucl-th].

Survey of nucleon electromagnetic form factors

I.C. Cloet et al., Few Body Syst. 46, 1-36 (2012). arXiv:0812.0416 [nucl-th].

Revealing dressed-quarks via the proton’s charge distribution

I.C. Cloet et al., Phys. Rev. Lett. 111, 101803 (2013). arXiv:1304.0855 [nucl-th].

☞ Observations:

Model parameters revisited in order to accomodate ∆ and N → ∆.Axial-vector diquark properties.Axial-scalar transition.

Anomalous magnetic moment of the dressed quark.

☞ Results:

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4

GE

Q2 (GeV2)

(a) CloetSegovia

Segovia-AMM

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3 3.5 4

GM

Q2 (GeV2)

(b) CloetSegovia

Segovia-AMM

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10µ

GE/G

M

Q2 (GeV2)

(c) Gayou et al. 2002Punjabi et al. 2005Puckett et al. 2010Puckett et al. 2011

cloet-raa040cloet-raa080

SegoviaSegovia-AMM


Q2-behaviour of G ∗M,Jones−Scadron

(I)

G∗

M,J−S cf. Experimental data and dynamical models

æ

æ

æ

æ

æææ

ææ

æææææææææææææææææææ

ææææ ææ æ æ

æ

0 0.5 1 1.50

1

2

3

x=Q 2�mD

2

GM*

Solid-black:Sophisticated interaction.

Dashed-blue:Contact interaction.

Dot-Dashed-green:Dynamical + no meson-cloud

☞ Observations:

All curves are in marked disagreement at infrared momenta.

Similarity between Solid-black and Dot-Dashed-green.

The discrepancy at infrared comes from omission of meson-cloud effects.

Both curves are consistent with data for Q2 & 0.75m2∆ ∼ 1.14GeV

2.


Q2-behaviour of G ∗M,Jones−Scadron

(II)

Transition cf. elastic magnetic form factors

0 2 40.5

0.7

0.9

1.1

x=Q 2�mD

2

HΜ�Μ*LH

GM*�G

ML

Solid-black:Sophisticated interaction.

Dashed-blue:Contact interaction.

Fall-off rate of G∗

M,J−S (Q2) in the γ∗p → ∆+ must much that of GM(Q2).

With isospin symmetry:〈p|µ|∆+〉 = −〈n|µ|∆0〉

so same is true of the γ∗n → ∆0 magnetic form factor.

These are statements about the dressed quark core contributions

→ Outside the domain of meson-cloud effects, Q2 & 1GeV2


Q2-behaviour of G ∗M,Ash

Presentations of experimental data typically use the Ash convention– G∗

M,Ash(Q2) falls faster than a dipole –

æææææææææ

æ

ææææææ

æ

ææ

ææ

æ

0 1 2 3 4 50.0

0.5

1.0

x=Q 2�mD

2

GM

,Ash

*�3G

D

No sound reason to expect:

G∗

M,Ash/GM ∼ constant

Jones-Scadron should exhibit:

G∗

M,J−S/GM ∼ constant

Meson-cloud effects

More than 30% for Q2 . 0.75m2∆.

Very soft → disappear rapidly.

G∗

M,Ashvs G∗

M,J−S

A factor 1/Q of difference.


Electric and coulomb quadrupoles

☞ REM = RSM = 0 in SU(6)-symmetric CQM.

Deformation of the hadrons involved.

Modification of the structure of the transitioncurrent. ⇔

☞ RSM : Good description of the rapid fallat large momentum transfer.

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0.0 1.0 2.0 3.0 4.0

0

-5

-10

-15

-20

-25

-30

x=Q 2�mD

2

RS

MH%L

☞ REM : A particularly sensitive measure oforbital angular momentum correlations.

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x=Q 2�mD

2

RE

MH%L

Zero Crossing in the transition electric form factor

Contact interaction → at Q2 ∼ 0.75m2∆ ∼ 1.14GeV

2

Sophisticated interaction → at Q2 ∼ 1.25m2∆ ∼ 1.90GeV

2


Large Q2-behaviour of the quadrupole ratios

Helicity conservation arguments in pQCD should apply equally to aninternally-consistent symmetry-preserving treatment of a contact interaction

REMQ2

→∞= 1, RSM

Q2→∞= constant

0 20 40 60 80 100-0.5

0.0

0.5

1.0

x=Q 2�m Ρ

2

RS

M,R

EM

Observations:

Truly asymptotic Q2 is required before predictions are realized.

REM = 0 at an empirical accessible momentum and then REM → 1.

RSM → constant. Curve contains the logarithmic corrections expected in QCD.


The ∆ elastic form factors

The small-Q2 behaviour of the ∆ elastic form factors is a necessary element in

computing the γ∗N → ∆ transition form factors

☞ The electromagnetic current can be generally written as:

Jµ,λω(K ,Q) = Λ+(Pf )Rλα(Pf ) Γµ,αβ(K ,Q) Λ+(Pi )Rβω(Pi )

☞ Vertex decomposes in terms of four form factors:

Γµ,αβ(K ,Q) =

[

(F∗

1 + F∗

2 )iγµ − F∗

2

m∆Kµ

]

δαβ −[

(F∗

3 + F∗

4 )iγµ − F∗

4

m∆Kµ

]

QαQβ

4m2∆

☞ The multipole form factors:

GE0(Q2), GM1(Q

2), GE2(Q2), GM3(Q

2),

are functions of F∗

1 , F∗

2 , F∗

3 and F∗

4 .

☞ They are obtained by any four sensible projection operators. Physical interpretation:

GE0 and GM1 → Momentum space distribution of ∆’s charge and magnetization.

GE2 and GM3 → Shape deformation of the ∆-baryon.


No experimental data

☞ Since one must deal with the very short ∆-lifetime (τ∆ ∼ 10−16τπ+ ):

Little is experimentally known about the elastic form factors.

Lattice-regularised QCD results are usually used as a guide.

☞ Lattice-regularised QCD produce ∆-resonance masses that are very large:

Approach mπ mρ m∆

Unquenched I 0.691 0.986 1.687Unquenched II 0.509 0.899 1.559Unquenched III 0.384 0.848 1.395Hybrid 0.353 0.959 1.533Quenched I 0.563 0.873 1.470Quenched II 0.490 0.835 1.425Quenched III 0.411 0.817 1.382

☞ We artificially inflate the mass of the ∆-baryon (right panels on the next...)

Black solid line: DSEs + sophisticated interaction (with/without m∆ = 1.8GeV)

Blue dashed line: DSEs + contact interaction (with/without m∆ = 1.8GeV)


GE0 and GM1 with(out) an inflated quark core mass

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GE2 and GM3 with(out) an inflated quark core mass

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Conclusions

☞ The γ∗N → ∆ transition form factors

Jones-Scadron G∗pM :

☞ G∗pM falls asymptotically at the same rate as Gp

M .

☞ Compatible with isospin symmetry and pQCD predictions.

☞ Data do not fall unexpectedly rapid once the kinematic relation betweenJones-Scadron and Ash conventions is properly account for.

REM and RSM :

☞ Limits of pQCD, REM → 1 and RSM → constant, are apparent in ourcalculation but truly asymptotic Q2 is required before the predictions are realized.

☞ Strong diquark correlations within baryons produce a zero in the transitionelectric quadrupole at Q2 ∼ 2GeV

2.

☞ The ∆ elastic form factors

The ∆ elastic form factors are very sensitive to m∆.

Lattice-regularised QCD produce ∆-resonance masses that are very large, theform factors obtained therewith should be interpreted carefully.


nucleon, delta andnucleon todelta electromagnetic...

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