Nuclear structure functions
TUS
K. Saito
 DIS kinematics ― what can we see in DIS ?
 Experiments ― what is the nuclear EMC effect ?
 Theoretical approaches ― can we understand it ?
 Summary
東海研究会『レプトン原子核反応型模型の構築に向けて』 1/26
1. Kinematics of Deep Inelastic Scattering (DIS)
Initial and final lepton 4-momentum:
k  , k  , k 2  k 2  m2  0
Virtual photon 4-momentum squared:
High
momentum
flow
q 2  (k  k )2  Q2  0
Initial nucleon (nucleus) 4momentum:

P  ( ET , P), P2  MT2

Final hadronic 4-momenyum squared:
PX2  ( P  q ) 2  W 2
Inelasticity (energy transfer in Lab):
  ( P  q) / M T
Bjorken variable:
High Q2: high resolution
Partons in target
0  x  Q 2 / 2M T  1
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The differential cross section (unpolarized):
d 2
 2 E
 4 L W 
ddE  Q E
lepton tensor (symmetric part):
L  2(k  k   k  k  k  k g  )
hadronic tensor (symmetric part):
W

 W1 ( x, Q )e
2

~  ~
 [W2 ( x, Q ) / M ]P P ,
2
2
T
~
(e  g   q  q / q 2 , P   P   ( P  q)q  / q 2 )
Structure functions F1 and F2:
F1 ( x, Q2 )  MTW1 ( x, Q2 ), F2 ( x, Q2 )  ( P  q / MT )W2 ( x, Q2 )
Bjorken limit:
Q 2  ,  , x: fixed
F1, 2 ( x, Q )     
 F1, 2 ( x)
2
Bjorken scaling
x: Bjorken variable
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What can we see in DIS ?
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
東海研究会『レプトン原子核反応型模型の構築に向けて』 4/26
Nucleus target in DIS
High momentum flow
P-QCD : N str. func. F2
Convolution form
Low momentum
flow
Non-perturbative : spect. func.
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2. Experiments
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F2A/F2D
Slope of the EMC ratio
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SLAC
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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)
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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.
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Nonrelativistic calculation (by Li, Liu, Brown)
(by Atti, Liuti)
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Relativistic calculation (by Smith, Miller)
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What is missing ?
Final state interaction:
q
2
k
pQCD (OPE)
di-quark
p
MF
(light-cone exp.)
≅
A-1
A
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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).
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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
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NJL model calculation
I.C. Cloet, W. Bentz, A.W.T.,
Phys.Lett.B642, 210-217 (2006).
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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
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Shadowing effect (by Piller et al.)
NMC+FNAL (  ,  ,  )
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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 
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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 !
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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
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 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)
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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
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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.
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By Atti and West
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