Comparison of quasi-elastic cross sections using spectral functions
with (e,e') data from 0.5 GeV to 1.5 GeV
Hiroki Nakamura (Waseda U).
Makoto Sakuda(Okayama U.)
Ryoichi Seki(CSUN,Caltech)
Introduction
• Goal is to calculate -A (mainly quasi-elastic) cross sections with appropriate Nuclear Effects and Form Factors.
• Nuclear Effects and Form Factors are verified with comparing C,O(e,e’) data.
• Spectral function vs. Fermi Gas model (NuInt04 hep-ph/0409300 )• The latest form factors are compared with dipole form f
actor.• Pauli blocking and Final State Interaction.
Vertex Correction Final State InteractionInitial State
Nuclear Effect on QE -A
-A reaction ~ -N with Nuclear Effect
• 3 Stages of Nuclear Effect
`Quasi-elastic
Fermi gas, spectral function Pauli blocking, optical potential
Quasielastic -A and e-A
• Comparison Nuclear Effect between -A and e-A– Initial State of Nucleons: Same
• Fermi gas, Spectral function
– Final State Interaction: Same • Pauli Blocking, Optical potential,…
• Information obtained from e-A – Vector Form Factors
– Initial State of Nucleons
– FSI
Differential Cross Section
• A(e,e’) cross section
p: initial nucleon momentum, q: momentum transfer, : energy transfer
d¾dE 0d
=k0
8(2¼)4MAE
Zd3pF(p;q;! )
X
spin
jM eN j2
1
d¾dE`d `
=k`
8(2¼)4MAEº
Zd3pF(p;q; ! )
X
spin
jM ºN j2
p: initial nucleonmomentum, q: momentum transferImaginary part of 1h1pGreen'sfunction ( involvingall nuclear e®ects)
F (p;q;! ) =hAjaypap+q±(H ¡ MA ¡ ! )ayp+qapjAi
Approximation : 1p1h! convolution of 1pand 1h
F (p;q;! ) =1
2MA
Zd! 0Ph(p; ! 0)Pp(p+q;! ¡ ! 0)
Ph =hAjayp±(H ¡ MA ¡ ! )apjAi à Initial Stateof Nucleon (1h)Pp =hAjap+q±(H ¡ MA ¡ ! )ayp+qjAi à Final State Interaction (1p)
2
Form Factors
• The latest form factors are used.Brash et al., PRC65,051001(2002). Bosted PRC51,409(199
5)
• Axial form factor: dipole
GpM (Q2) = [1+0:116Q+2:874Q2 +0:241Q3+1:006Q4 +0:345Q5)]¡ 1
GpE (Q
2) = (1¡ 0:130(Q2 ¡ 0:04D0))GpM (Q2)
(Q in GeV)
4
FA(Q2) = ¡ 1:26£ (1+Q2=(1:07GeV)2)¡ 2
4
Fermi Gas Model
• Non-interacting and uniform Fermi Gas Model (Moniz)
• Initial State : Fermi Gas
• Final State Interaction: Pauli BlockingPh(p;! ) = 1
Epµ(PF ¡ jpj)±(Ep +! )
Pp(p0; ! ) = 1E 0
pµ(jp0j ¡ PF )±(E 0
p ¡ ! )
Ep =pp2 +M 2 ¡ EB ; E 0
p =qp02+M 2
PF : Fermi momentum (225MeV for oxygen),EB : B̀inding' Energy (27MeV for oxygen)
5
Fermi Gas Pauli Blocking
Spectral Function
• More realistic model than FG
• Initial State: realistic spectral function (Benhar et al.)
(single particle + correlation with local density approx.)
0. 300. P (MeV/c)
20.
40.
E(MeV)
Ph (p; ! ) = 1Ep
P (p; ! )
Ph(p;! ) = 1Ep
P (p;! )Pp(p0;! ) = 1
E 0p±(E 0
p ¡ ! )
4
Probability of removing a nucleon of momentum p with excitation energy E.
Momentum Distribution
• Momentum distribution of a nucleon in nucleus.
• Spectral function has long tail due to correlation.
dEEpPpnh ),()(
Pauli Blocking for Spectral function model
• PWIA (no Pauli blocking)
• Simple Pauli Blocking ( same as FG)
• Modified Pauli BlockingPp(! ;p) =
1E0
p
np(p)±(E 0p ¡ ! )
np(p) =1¡ (2¼)3½nh(p)
1
Pp(! ;p) =1E0
p
np(p)±(E 0p ¡ ! )
np(p) =1¡ (2¼)3¹½nh(p)
1
Pp(! ;p) =1E0
p
np(p)±(E 0p ¡ ! )
np(p) =1¡ (2¼)3¹½nh(p)Z(Ph(! ;p) +Pp(! ;p))d! =
1(2¼)3½
1
Sum rule for uniform Nuclear Matter
~ 0.4 0
0
0.2
0.4
0.6
0.8
1
0 100 200 300 400 500 600
n p(p
)
p [MeV]
Experimental Data
• 16O(e,e’) : E=700-1500 MeV =32 deg Anghinolfi et al., NPA602(’96),405.
• 12C(e,e’) : E=780 MeV =50.4 deg Garino et al., PRC45(’92),780.
E=500 MeV =60 deg Whittney et al., PRC9(’74),2230.
QE Resonance
(e,e’): Fermi Gas vs. Spectral function
• Data: 16O(e,e’)E=1080 MeV=32 deg• FG > SF at peak.
SF agrees better with data.
• SF can explain ‘dip region’, because of ‘correlation’.
0 2 4 6 8
10 12 14 16 18
0 100 200 300 400 500 600d/d
d [
10-7
fm2 /M
eV]
[MeV]
E = 1080 MeV = 32 deg
Spectral func.Fermi Gas
O(e,e')
16O(e,e’) =32 degE=700,880,1080,1200 MeV
0 10 20 30 40 50 60 70 80 90
0 100 200 300 400 500 600d/d
d [
10-7
fm2 /M
eV]
[MeV]
E = 700 MeV = 32 deg
Spectral func.Fermi Gas
O(e,e')
0
10
20
30
40
50
0 100 200 300 400 500 600d/d
d [
10-7
fm2 /M
eV]
[MeV]
E = 880 MeV = 32 deg
Spectral func.Fermi Gas
O(e,e')
0 2 4 6 8
10 12 14 16 18
0 100 200 300 400 500 600d/d
d [
10-7
fm2 /M
eV]
[MeV]
E = 1080 MeV = 32 deg
Spectral func.Fermi Gas
O(e,e')
0 2
4 6
8 10
12
0 100 200 300 400 500 600d/d
d [
10-7
fm2 /M
eV]
[MeV]
E = 1200 MeV = 32 deg
Spectral func.Fermi Gas
O(e,e')
12C(e,e’) quasielastic
E=500MeV =60 deg
E=780 MeV =50.4 deg
Red: spectral func
Blue: Fermi Gas
0
5
10
0 100 200 300d/d
d [
10-7
fm2 /M
eV]
[MeV]
E = 500 MeV = 60 deg
SFFG
C(e,e')
0
5
0 100 200 300d/d
d [
10-7
fm2 /M
eV]
[MeV]
E = 780 MeV = 50.4 deg
Spectral func.Fermi GasC(e,e')
16O(-) QE E=800 MeV
• d/dQ2
E=800MeV
– Blue:Fermi Gas
– Red: Spectral
Function+PWIA
– Green: Spectral
Function + Pauli
Blocking
• Pauli Blocking has large
effect at small Q.
0
2
4
6
8
10
12
14
16
0 0.2 0.4 0.6 0.8 1 1.2 1.4
d/d
Q2 [1
0-18 fm
2 /MeV
2 ]
Q2 [GeV2]
E = 800 MeV
SFSF+PB
FG
16O(-) QE E=800 MeV
• d/dE
E=800MeV
– Blue:Fermi Gas
– Red: Spectral Function +PWIA
– Green: Spectral Function + Pauli
Blocking
• Clear difference at peak
(FG > SP).
– FG has low-energy-transfer
nucleons more than SF.
0
0.5
1
1.5
2
2.5
3
0 100 200 300 400 500 600 700 800
d/d
Ele
p [10
-14 fm
2 /MeV
]
Elep [MeV]
E = 800 MeV
SFSF+PB
FG
16O(-) QE E=2000 MeV
• d/dEd/dQ2
0
0.5
1
1.5
2
2.5
0 500 1000 1500 2000
d/d
Ele
p [1
0-14 fm
2 /MeV
]
Elep [MeV]
E = 2000 MeV
SFSF+PB
FG
0
2
4
6
8
10
12
14
16
0 0.5 1 1.5 2 2.5 3 3.5 4
d/d
Q2 [1
0-18 fm
2 /MeV
2 ]
Q2 [GeV2]
E = 2000 MeV
SFSF+PB
FG
Form Factor: Dipole vs. Latest
• The latest form factor make smaller cross sections at QE peak than dipole.
• Difference: < 10%
0 2 4 6 8
10 12 14 16 18
0 100 200 300 400 500 600d/d
d [
10-7
fm2 /M
eV]
[MeV]
E = 1080 MeV = 32 deg SF
Latest FFDipole FF
O(e,e')
0
2
4
6
8
10
12
14
0 0.2 0.4 0.6 0.8 1 1.2 1.4
d/d
Q2 [
10-1
8 fm2 /M
eV2 ]
Q2 [GeV2]
E = 800 MeV
FGFG(Dipole)
(e,e’) ()
Pauli Blocking for Spectral function model
• PWIA (no Pauli blocking)
• Simple Pauli Blocking ( same as FG)
• Modified Pauli BlockingPp(! ;p) =
1E0
p
np(p)±(E 0p ¡ ! )
np(p) =1¡ (2¼)3½nh(p)
1
Pp(! ;p) =1E0
p
np(p)±(E 0p ¡ ! )
np(p) =1¡ (2¼)3¹½nh(p)
1
Pp(! ;p) =1E0
p
np(p)±(E 0p ¡ ! )
np(p) =1¡ (2¼)3¹½nh(p)Z(Ph(! ;p) +Pp(! ;p))d! =
1(2¼)3½
1
Sum rule for uniform NM
~ 0.4 0
0
0.2
0.4
0.6
0.8
1
0 100 200 300 400 500 600
n p(p
)
p [MeV]
Comparison of Pauli Blocking• Simple PB suppresses cross section at small Q2, more strongly than
Modified PB.
2
4
6
8
10
12
14
16
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
d/d
Q2 [1
0-18 fm
2 /MeV2 ]
Q2 [GeV2]
E = 800 MeV
SF+PWIASF+PBFG+PB
SF+MPB
O()
Final State Interaction
• Simple approach is tried here.• Optical Potential Model Imaginary part of potential
On-shell condition of recoiled nucleon is changed:
=0.16 fm-3 Nuclear Matter density
NN= 40 mb Typical value of NN cross section
±(! ¡ E0p) !
W=¼(! ¡ E 0
p)2 +W2=4
13
±(! ¡ E 0p) !
W=¼(! ¡ E 0
p)2 +W2=4
W =12v½¾N N
13
16O(e,e’) =32 deg: QE with FSI
• E=700,1080 MeV
Red: Spectral Function
Green: Fermi Gas
Blue: SF+FSI• SP +FSI < SP only• SP+FSI: broader width. • Difference 10%
at peak
0
10
20
30
40
50
60
70
80
90
0 100 200 300 400 500 600
d/d
d [1
0-7fm
2 /MeV
]
[MeV]
E = 700 MeV = 32 deg
Spectral func.Fermi Gas
Spectral func. +FSIO(e,e')
0
2
4
6
8
10
12
14
16
18
0 100 200 300 400 500 600
d/d
d [1
0-7fm
2 /MeV
]
[MeV]
E = 1080 MeV = 32 deg
Spectral func.Fermi Gas
Spectral func.+FSIO(e,e')
Summary• Systematic comparison of the model calculation wi
th A(e,e’) data in the wide energy range with the latest form factors.
• (e,e’): SF agrees better with the experimental data than FG, in particular, at dip region.
• (,): More than 20 % difference between FG and SF shows at d/dE peak.
• Pauli blocking should be verified by forward e-A scattering data.
• Appropriate FSI is necessary.
N- Form Factors
CV3 =
h(1+Q2=M 2
V)2(1+Q2=(4M 2
V ))i ¡ 1
(MV =840MeV)
CV4 =¡ M=WCV
3 ;CVi =0 (i 6=3;4)
7
CV3 =
h(1+Q2=M 2
V)2(1+Q2=(4M 2
V ))i ¡ 1
(MV =840MeV)
CV4 =¡ M=WCV
3 ;CVi =0 (i 6=3;4)
CA5 (Q
2) =1:2£h(1+Q2=M 2
V )2(1+Q2=(3M 2
V ))i ¡ 1
7
Paschos et al. PRD69,014013(2004),