neutrino-nucleus scattering at high energies tus k. saito

Neutrino-Nucleus scattering at high energies

TUS K. Saito

東海研究会 5『レプトン原子核反応型模型の構築に向けて』 1/17

DIS kinematics 　―　 what can we see in DIS ? Neutrino scattering at high Q2 Neutrino scattering at low Q2

Summary


0. Lepton reactions

(GeV)

(GeV2)

2

1

DISRES

Regge region

GeV2

BFKL evolution

DGLAP evolution

QE

atmospheric

T2K

1. Kinematics of Deep Inelastic Scattering (DIS)　　　　　　　　　　　　　　　　　　　　　　　　　　　Initial and final lepton 4-momentum:

Virtual photon or boson 4-momentum squared:

Initial nucleon (nucleus) 4-momentum:

Final hadronic 4-momenyum squared:

y variable (--> energy loss at T rest fr.):

Bjorken variable:

0,, 222 mkkkk

0)( 222 Qkkq

22),,( TT MppEp

222 )( Wqpp f

EEkpqpy

1)/()(

12/0 2 TMQx


High momentum flow

High Q2: high resolutionPartons in target

W, Z,

p

pf

2. Charged current differential cross section:

lepton tensor current:

hadronic tensor (including anti-symmetric part):


V - A

Non-conservation of axial current

Non-conservation of parity

𝜈

ℓ−

𝑊+¿¿

left-handed

Cabibbo angle


The structure functions: 3 , because

: VV + AA contributions, : VA contributions.

difference between weak and EM interactions

Virtual boson helicity cross sections:

Def: , where

, where ,

average over T-polarizations

the right-left asymmetry

Note;

T: transverse

L: longitudinal

γ=|𝑞|𝑞0

𝐹 1=𝑀𝑁𝑊 1∝𝑊+1+𝑊 −1∝𝐹 𝑇

𝐹 2=𝜈𝑊 2∝𝑊 +1+𝑊− 1+2𝑊 0 ,𝐹𝐿=2𝑥𝑊 0

𝐹 3=𝜈𝑊 3∝𝑊+1−𝑊 −1

𝑒 ∙𝑞=0

3. Neutral-current differential cross section: 3

where


NC

4. What can we see in the target in the Bjorken limit

Bjorken limit The approximate Q^2-independence of the structure functions → the virtual photon sees point-like constituents in the target – quarks → using distributions of quarks and anti-quarks,

(Callan-Gross relation)

The small scaling violation is calculated by pQCD.

DIS probes a current-current correlation in the target ground state. In the Bjorken limit, the probed correlation is light-like:

~ 2.0(fm) for x ~ 0.1 ~ 1.0(fm) for x ~ 0.2 ~ 0.4(fm) for x ~ 0.5 ~ 0.2(fm) for x ~ 1.0

f

fff xqxqexF )],()([21)( 2

1

,2/)( 3yty xMyyy T/2,0,0

cxfmyt /)(2.0|||,| 3

)(2)( 12 xxFxF


)(),( 2,1:,,2

2,1

2

xFQxF fixedxQ

5. Simple consideration on the ν / ν reactions


The cross section may naively be given in terms of the incoherent sum of ν/ν-bar scattering off a quark:

1. neutrino-quark scattering (CC)

Then, average over the quark probability distribution q(x’) in a target,

2. neutrino-anti-quark scattering (CC)

𝜈

ℓ−

𝑊+¿¿

𝑝1=𝑥𝑝

𝑝2=𝑥𝑝+𝑞

3. Scattering-angle (or y-variable) dependence

4. Mixing-angle dependence


d’ term s’ term

h=-1/2

h=+1/2h=-1/2

5. Results (Leading order)

Average


𝜈𝑁

𝜈𝑁

isospin singlet (u d)


6. Parameterization of Nuclear PDF by HKN


7. Neutrino reactions at low Q2

(GeV)

(GeV2)

2

1

DISRES

?

GeV2

1. Kulagin prescription

only for low momentum transfer• the transverse cross section finite as Q2 0 (photo-absorption); FT 0 shadowing, VMD model, etc.

• the longitudinal cross section contains the VV and AA parts. the vector-current part FL

VC 0, because of the CVC; . only the axial-current part remains, because of the PCAC;


Separate the axial current as

(= pion current + heavy hadron current -- axial vector meson, ρπ continuum, etc.)

But, the pion derivative does not contribute, because

Thus,

(interference main term between jπ and A’)

πN (or A) scattering


Adding the form factor to cut off the large-Q2 contribution, we finally obtain

The πN forward scattering amplitude (the total cross section) is given by the Regge parameterization:

𝜋

𝐴𝑥𝑖𝑎𝑙𝑚𝑒𝑠𝑜𝑛

𝑁


shad

owin

g


2. A la Bodek and Yang

Their parameterization (for N) is very messy:

i = valence – up, down sea – up, down, strange j = sea – up, down, starnge

(for all Q2)

2

2

• At large Q2, we can see the quark-gluon structure of a target – pQCD + higher order corrections.

-- relatively easy to handle the structure functions (the quark-gluon distributions) even for a nucleus.• At low Q2, we need non-perturbative treatment: -- the Regge, the BFKL, BK and/or CGC approach, -- need a careful treatment on the axial current, -- nuclear (shadowing) effects.• How do we connect the two pictures ? • How do we connect to the resonance region ?

8. Summary


F2A/F2

D

Slope of the EMC ratio

3-1. Effect of the conventional nuclear physics ― Binding and Fermi motion

How does the conventional nuclear physics affect F2(x) ?

The nucleon is scattered incoherently in case of

The light-cone momentum distribution of N in A:

Spectral function　　　　　　　　　　　　　　　　　　　　　　　　　 Quasi-elastic reaction A(e,e’p)A’ → Koltun sum rule: E/A = (T-e)/2 (2body force only)

3. Theoretical approaches

1.02 xfmdc

22

4

42

/),( )()2(

),( ppMM

qPqpypSpdypyD A

AjAnpj

3-1. Effect of the conventional nuclear physics ― Binding and Fermi motion3-2. Shadowing effect at small x3-3. Anti-shadowing ?

APpy /

ApApS |)0(ˆ|,)1()(

2

0 |)(|)(2 pTMp R

p

Convolution form:

Assumptions in the convolution model: on-mass shell approximation 　→ → if the binding is weak, OK? impulse approximation ― final state interactions and interference terms are ignored.

If OK, we get

Model-dependent calculations:① Off-mass shell effect by Kulagin et al. ↓② Off-mass shell (↓) + final state interaction (MFA) by Saito et al. ↑

Ignored diagrams

Note: Deuteron is also different from the average of proton and neutron ― small EMC effect.

22 Mp

),(),()()( 2/

2

,/

2/ pzfpyDdpyzxdydzxf ja

jAjAa

,2/2 )/()()(

j

jAj

A

x

A yxFyDdyxF

Nonrelativistic calculation (by Li, Liu, Brown)

(by Atti, Liuti)

Relativistic calculation (by Smith, Miller)

What is missing ?Final state interaction:

q 2 pQCD (OPE) k di-quark (light-cone exp.)

p MF

A-1

A

K. Saito, A.W.T., N.P.A574, 659 (1994).

No fermi motion, no c.m. correction

Quark picture, but no FSI

Quark picture with FSI

Naïve Bag model calculation – include not only FSI but also SRC

SLAC-E139Fe & Ag

Drell-Yan exp.

FNAL-E772W

Chiral Quark Soliton model calculation

R.S.Jason, G.A. Miller, P.R.L.91, 212301 (2003).

NJL model calculation

I.C. Cloet, W. Bentz, A.W.T., Phys.Lett.B642, 210-217 (2006).

3-2. Shadowing effect at small x

Shadowing region →

DIS occurs coherently:

>> 1 for x > 0.1 << 1 for x < 0.1 for small x, the photon is supposed to be converted into vector mesons VMD 　→ surface interaction

1.02 xfmdc

)()( 8.02 NANAA AAxF

ba AA /

8.03/2 AA

Shadowing effect (by Piller et al.)

NMC+FNAL ( ) ,,

3-3. Anti-shadowing ?

Anti-shadowing region →

An enhancement at small x region → pion field enhancement ???

Recent data of the giant Gamow-Teller states → the Landau-Migdal parameters

2.01.0 x

ggg NNN 05.018.0,59.0

4. Summary

1. The quark distribution in a nucleus is different from that in the free nucleon: ― about 20% reduction at x ~ 0.7-0.8 ― at small x, the structure function is reduced due to shadowing ― for large x, the EMC ratio is very enhanced because of Fermi motion and short-range correlation

2. The energy-momentum distribution of a nucleon in a nucleus is vital to explain the EMC effect, but its effect is insufficient ? ― the internal structure of a nucleon is modified in a nucleus ?

3. The sea quark is enhanced in a nucleus around x ~ 0.15 ? ― cf. the Drell-Yan result

4. At large x (>1), what happens ? new JLab data !

x

σx = Q^2/2Mν, Q^2 fixedν large, x smallvery low Q^2

A1

elastic

x

σvery low Q^2

A1

elastic + excited states

x

σlow Q^2

A1

QE peak displacement energy

x

σmid Q^2

A1

QEΔ

N*

x

σmid Q^2

A1

Δ, N* duality

QE peak of quark

1/3

x

σhigh Q^2

A1

valence quark

1/3

x

σvery high Q^2

A1

sea + glue BK region

1/3

Comment on the QE peak in e-A scattering

T. Suzuki, P.L.B101 (1981), 298R. Rosenfelder, P.L.B79 (1978), 15

QE peak in e-A scattering at low energyDifferential cross section:

The response functions (structure functions):

S = W(L) or W(T) for longitudinal mode

The characteristic function:

(k-th energy weighted moment)

The characteristic function is described in terms of the cumulants;

The displacement energy at the peak of QE cross section can be given by the cumulants; (0).

　　　　　　　　　　 σ

ω

The 1st moment is then given by

If we take Hamiltonian as

,then we get (as an example, for longitudinal mode)

,which implies that the Wigner and Bartlett forces do not contribute to the displacement energy (for longitudinal mode) !

Summary: • the displacement of QE peak is caused by some specific forces in nuclear force.• the binding effect appears when FSI is ignored, while, if it is include, the binding is cancelled by FSI – Wigner force does not contribute. • the energy shift is also caused by a non-local (energy dependent) one-body potential.

By Atti and West

y scaling

neutrino-nucleus scattering at high energies tus k. saito

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