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),