Neutrino-Nucleus scattering
at high energies
TUS
K. Saito
 DIS kinematics ― what can we see in DIS ?
 Neutrino scattering at high Q2
 Neutrino scattering at low Q2
 Summary
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
0. Lepton reactions
2 (GeV2)
atmospheric
 2 = 4 GeV2
QE
DGLAP evolution
T2K
RES
DIS
BFKL evolution
1
Regge region
 (GeV)
2
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1. Kinematics of Deep Inelastic Scattering (DIS)
Initial and final lepton 4-momentum:
k  , k  , k 2  k 2  m2  0
W, Z,
Virtual photon or boson 4-momentum
squared:
High
momentum
flow
q 2  (k  k )2  Q2  0
Initial nucleon (nucleus) 4-momentum:

p  ( ET , p ), p 2  M T2

pf
Final hadronic 4-momenyum squared:
p 2f  ( p  q ) 2  W 2
y variable (--> energy loss at T rest fr.):
p
High Q2: high resolution
Partons in target
y  ( p  q ) /( p  k )  1 
E
E
Bjorken variable:
0  x  Q 2 / 2M T  1
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2. Charged current differential cross section:
ℓ−
+
lepton tensor current:
V-A
left-handed

hadronic tensor (including anti-symmetric part):
Cabibbo angle
Non-conservation of parity
Non-conservation of axial current
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The structure functions:
3
, because 4,5 ∝

∙
≈0.
1,2 : VV + AA contributions, 3 : VA contributions.
 difference between weak and EM interactions
Virtual boson helicity cross sections:
 T: transverse
Def:
, where
 L: longitudinal

, where γ =
, ∙ =0
0
1 =  1 ∝ +1 + −1 ∝   average over T-polarizations
2 = 2 ∝ +1 + −1 + 20 ,
3 = 3 ∝ +1 − −1
 = 20
 the right-left asymmetry
Note; |1,2  | = |1,2  |
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3. Neutral-current differential cross section:
3
NC
where
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4. What can we see in the target in the Bjorken limit
 ,  , x: fixed
F1, 2 ( x, Q 2 ) Q

 F1, 2 ( x)
2
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,
1
F1 ( x)   e 2f [q f ( x)  q f ( x)], F2 ( x)  2 xF1 ( x) (Callan-Gross relation)
2 f
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:

y   (t  y3 ) / 2, y   0, y  0, y   2 / MT x
| t |, | y3 | 0.2( fm) / x   c
~ 2.0(fm)
~ 1.0(fm)
~ 0.4(fm)
~ 0.2(fm)
for x ~ 0.1
for x ~ 0.2
for x ~ 0.5
for x ~ 1.0
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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:
ℓ−
1. neutrino-quark scattering (CC)

2 =  + 
+
1 = 
Then, average over the quark probability distribution q(x’) in a target,
2. neutrino-anti-quark scattering (CC)
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3. Scattering-angle (or y-variable) dependence
h=-1/2
h=-1/2
h=+1/2
4. Mixing-angle dependence
 d’ term
 s’ term
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5. Results (Leading order)


Average
isospin singlet (u ↔ d)
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6. Parameterization of Nuclear PDF by HKN
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7. Neutrino reactions at low Q2
2
 2 = 4 GeV2
(GeV2)
RES
DIS
1
?
 (GeV)
2
1. Kulagin prescription
, ~2 ,  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 FLVC  0, because of the CVC;   = 0.
only the axial-current part remains, because of the PCAC;
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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
between jπ and A’)
main term
 πN (or A) scattering
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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:

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shadowing
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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)
 
 = 0.00001, 2 = 0 |+
2
= 0.33 ± 0.16
2
2  = 0.00001, 2 = 0 |
= 0.25 ± 0.11
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8. Summary
• 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 ?
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』 17/17
F2A/F2D
Slope of the EMC ratio
SLAC
3. Theoretical approaches
3-1. Effect of the conventional nuclear physics ― Binding and Fermi motion
3-2. Shadowing effect at small x
3-3. Anti-shadowing ?
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
 c  d  2 fm  x  0.1
The light-cone momentum distribution of N in A:

d 4 p 
p  q M A 
2
2




D j (  p , n ) / A ( y, p )  y 
S
(
p
)

y


p

p
j
4

(2 )
PA  q M 


p

2
y  p / PA

S  ( p)  ( A 1) , p |ˆ (0) | A

 2 ( p0  M    TR ) |  ( p) |2
Spectral function
Quasi-elastic reaction A(e,e’p)A’ →  
Koltun sum rule: E/A = (T-e)/2 (2body force only)

Convolution form:
f a / A ( x)    dydz ( x  yz) dp2 Dj / A ( y, p 2 ) f a / j ( z, p 2 )
j ,
Assumptions in the convolution model:
 on-mass shell approximation → p 2  M 2 → if the binding is weak, OK?
 impulse approximation ― final state interactions and interference terms are ignored.
If OK, we get F2A ( x) 
 
j,
A
x
dyDj / A ( y ) F2j ( x / y )
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.
Nonrelativistic calculation (by Li, Liu, Brown)
(by Atti, Liuti)
Relativistic calculation (by Smith, Miller)
What is missing ?
Final state interaction:
q
2
k
di-quark
p
MF
A-1
A
pQCD (OPE)
(light-cone exp.)
≅
Naïve Bag model calculation – include not only FSI but also SRC
Quark picture
with FSI
Quark picture, but
no FSI
No fermi motion,
no c.m. correction
K. Saito, A.W.T., N.P.A574, 659 (1994).
Chiral Quark Soliton model calculation
R.S.Jason, G.A. Miller,
P.R.L.91, 212301 (2003).
SLAC-E139
Fe & Ag
Drell-Yan exp.
FNAL-E772
W
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 →  c  d  2 fm  x  0.1
DIS occurs coherently: F2A ( x)     A  A     N (   A  A0.8     N )
Aa / Ab >> 1 for x > 0.1
<< 1 for x < 0.1
for small x, the photon is supposed to be
converted into vector mesons
2/3
 A0.8
VMD → surface interaction A
Shadowing effect (by Piller et al.)
NMC+FNAL (  ,  ,  )
3-3. Anti-shadowing ?
Anti-shadowing region → 0.1  x  0.2
An enhancement at small x region → pion field enhancement ???
Recent data of the giant Gamow-Teller states → the Landau-Migdal parameters

g NN  0.59, g N  0.18  0.05g 
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 = Q^2/2Mν, Q^2 fixed
ν  large, x  small
very low Q^2
σ
elastic
1
A
x
very low Q^2
σ
elastic + excited states
1
A
x
low Q^2
σ
QE peak
displacement energy
1
A
x
mid Q^2
σ
Δ
N*
QE
1
A
x
mid Q^2
σ
QE peak of quark
Δ, N*
duality
1/3
1
A
x
high Q^2
σ
valence quark
1/3
1
A
x
very high Q^2
σ
sea + glue
BK region
1/3
1
A
x
Comment on the QE peak in
e-A scattering
T. Suzuki, P.L.B101 (1981), 298
R. Rosenfelder, P.L.B79 (1978), 15
 QE peak in e-A scattering at low energy
Differential 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.
y scaling
By Atti and West
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