1. 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)

2. 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 factor. Pauli blocking and Final State Interaction.

3. Vertex Correction Final State Interaction Initial 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

4. 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

5. Differential Cross Section A(e,e’) cross section p : initial nucleon momentum, q : momentum transfer,  : energy transfer

6. Form Factors The latest form factors are used. Brash et al., PRC65,051001(2002). Bosted PRC51,409(1995) Axial form factor: dipole

7. Fermi Gas Model Non-interacting and uniform Fermi Gas Model (Moniz) Initial State : Fermi Gas Final State Interaction: Pauli Blocking Fermi Gas Pauli Blocking

8. 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) P h ( p;! )= 1 E p P ( p;! ) Probability of removing a nucleon of momentum p with excitation energy E.

9. Momentum Distribution Momentum distribution of a nucleon in nucleus. Spectral function has long tail due to correlation.

10. Pauli Blocking for Spectral function model PWIA (no Pauli blocking) Simple Pauli Blocking ( same as FG) Modified Pauli Blocking Sum rule for uniform Nuclear Matter  ~ 0.4  0

11. Experimental Data 16 O(e,e’) : E=700-1500 MeV  =32 deg Anghinolfi et al., NPA602(’96),405. 12 C(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.

12. QE  Resonance (e,e’): Fermi Gas vs. Spectral function Data: 16 O(e,e’) E=1080 MeV  =32 deg FG > SF at peak. SF agrees better with data. SF can explain ‘dip region’, because of ‘correlation’.

13. 16 O(e,e’)  =32 deg E=700,880,1080,1200 MeV

14. 12 C(e,e’) quasielastic E=500MeV  =60 deg E=780 MeV  =50.4 deg Red: spectral func Blue: Fermi Gas

15. 16 O(   - ) QE E=800 MeV d  /dQ 2 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 00.20.40.60.811.21.4 d  /dQ 2 [10 -18 fm 2 /MeV 2 ] Q 2 [GeV 2 ] E = 800 MeV SF SF+PB FG

16. 16 O(   - ) 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 0100200300400500600700800 d  /dE lep [10 -14 fm 2 /MeV] E lep [MeV] E = 800 MeV SF SF+PB FG

17. 16 O(   - ) QE E=2000 MeV d  /dE  d  /dQ 2 0 0.5 1 1.5 2 2.5 0500100015002000 d  /dE lep [10 -14 fm 2 /MeV] E lep [MeV] E = 2000 MeV SF SF+PB FG 0 2 4 6 8 10 12 14 16 00.511.522.533.54 d  /dQ 2 [10 -18 fm 2 /MeV 2 ] Q 2 [GeV 2 ] E = 2000 MeV SF SF+PB FG

18. Form Factor: Dipole vs. Latest The latest form factor make smaller cross sections at QE peak than dipole. Difference: < 10% (e,e’) (  )

19. Pauli Blocking for Spectral function model PWIA (no Pauli blocking) Simple Pauli Blocking ( same as FG) Modified Pauli Blocking Sum rule for uniform NM  ~ 0.4  0

20. Comparison of Pauli Blocking Simple PB suppresses cross section at small Q 2, more strongly than Modified PB. O(  )

21. 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

22. 16 O(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

23. Summary Systematic comparison of the model calculation with 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.

24. N-  Form Factors Paschos et al. PRD69,014013(2004),