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Hiroki Nakamura (Waseda U). Makoto Sakuda (Okayama U.) Ryoichi Seki (CSUN,Caltech)

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.

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Hiroki Nakamura (Waseda U). Makoto Sakuda (Okayama U.) Ryoichi Seki (CSUN,Caltech)

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  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 n-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. Nuclear Effect on QE n-A • n-A reaction ~ n-N with Nuclear Effect • 3 Stages of Nuclear Effect Quasi-elastic n+A ` n Final State Interaction Initial State Vertex Correction Fermi gas, spectral function Pauli blocking, optical potential

  4. Quasielastic n-A and e-A • Comparison Nuclear Effect between n-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, w: 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.) 1 P ( p ; ! ) = P ( p ; ! ) h E (MeV) E p 40. Probability of removing a nucleon of momentum p with excitation energy E. 20. 0. 300. P (MeV/c)

  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 r~ 0.4 r0 Sum rule for uniform Nuclear Matter

  11. Experimental Data • 16O(e,e’) : E=700-1500 MeV q=32 deg Anghinolfi et al., NPA602(’96),405. • 12C(e,e’) : E=780 MeV q=50.4 deg Garino et al., PRC45(’92),780. E=500 MeV q=60 deg Whittney et al., PRC9(’74),2230.

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

  13. 16O(e,e’) q =32 degE=700,880,1080,1200 MeV

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

  15. 16O(nm,m -) QE E=800 MeV • ds/dQ2 E=800MeV • Blue:Fermi Gas • Red: Spectral Function+PWIA • Green: Spectral Function + Pauli Blocking • Pauli Blocking has large effect at small Q. E = 800 MeV 16 SF 14 SF+PB ] 2 FG 12 /MeV 2 10 fm -18 8 [10 6 2 /dQ 4 s d 2 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 2 2 Q [GeV ]

  16. E = 800 MeV 3 SF SF+PB 2.5 FG /MeV] 2 2 fm -14 1.5 [10 lep 1 /dE s 0.5 d 0 0 100 200 300 400 500 600 700 800 E [MeV] lep 16O(nm,m -) QE E=800 MeV • ds/dEm 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.

  17. 16O(nm,m -) QE E=2000 MeV • ds/dEm ds/dQ2 E = 2000 MeV E = 2000 MeV 16 2.5 SF SF 14 SF+PB ] SF+PB 2 FG FG 2 /MeV] 12 /MeV 2 2 fm 10 fm 1.5 -14 -18 8 [10 [10 1 6 2 lep /dQ /dE 4 s 0.5 s d d 2 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 0 500 1000 1500 2000 2 2 Q [GeV ] E [MeV] lep

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

  19. Pauli Blocking for Spectral function model • PWIA (no Pauli blocking) • Simple Pauli Blocking ( same as FG) • Modified Pauli Blocking r~ 0.4 r0 Sum rule for uniform NM

  20. Comparison of Pauli Blocking • Simple PB suppresses cross section at small Q2, more strongly than Modified PB. O(n,m)

  21. Final State Interaction • Simple approach is tried here. • Optical Potential Model Imaginary part of potential On-shell condition of recoiled nucleon is changed: r =0.16 fm-3 Nuclear Matter density sNN= 40 mb Typical value of NN cross section

  22. 16O(e,e’) q =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. • (n,m): More than 20 % difference between FG and SF shows at ds/dEm peak. • Pauli blocking should be verified by forward e-A scattering data. • Appropriate FSI is necessary.

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

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