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Hadron Propagation in the Medium

ICTP Workshop, Trieste, Italy 23 May, 2006. Hadron Propagation in the Medium. － The Exclusive Process A(e,e’p)B in few-nucleon systems ー. H. Morita for. < Perugia-Dubna-Sapporo Collab. >. C. Ciofi degli Atti , M. Alvioli , V. Palli, I. Chiara, Univ. of Perugia

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Hadron Propagation in the Medium

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  1. ICTP Workshop, Trieste, Italy 23 May, 2006 Hadron Propagation in the Medium － The Exclusive Process A(e,e’p)B in few-nucleon systems ー H. Morita for < Perugia-Dubna-Sapporo Collab. > C. Ciofi degli Atti, M. Alvioli , V. Palli, I. Chiara, Univ. of Perugia L.P. Kaptari, BLTP JINP, Dubna H. Morita, Sapporo Gakuin Univ., Sapporo

  2. How can we describe the p propagation in medium? SG SG:Glauber Operator < Subject of this talk > Study of the (knocked out) p propagation in the process of 3He(e,e’p)d(pn) and 4He(e,e’p)3H reaction. 1. Introduction A(e,ep’)B Reaction ke’ e’ ke e q,ω p1(≠k’) k’=k1+q p k1 FSI pm A B=(A-1)

  3. Outline of my talk • Introduction • Framework of the calculation for the A(e,e’p)B Reaction -Generalized Eikonal Approximation- • Results of 3He(e,e’p)2H(pn) Reaction • Results of 4He(e,e’p)3H Reaction • Finite Formation Time Effect on 4He(e,e’p)3H Reaction -at higher Q2 region- • Summary

  4. 2.Framework of the Calculation for the A(e,e’p)B Reaction < Cross Section of A(e,e’p)B > ＊We use Factorization Ansatz. (here only unpolarized cross section will be discussed) e-N cross section nuclear distorted spectral function If “B”(recoil system) is bound state distorted by FSI nucleon distorted momentum distribution

  5. ：Pisa Group’s w.f. with AV18 pot. A. Kievsky et al., Nucl. Phys. A551(1993) 241 P ρ P t=(Pp-Pn)/2 ｒ n Nuclear Distorted Spectral Function in 3He(e,e’p)2H(pn) process C. Ciofi degli Atti, L.P. Kaptari, Phys. Rev. C71, 024005 (2005) SG: Glauber operator 3He(e,e’p)2H 3He(e,e’p)pn

  6. n P ～ ～ ＜Choice of Z-axis＞ i p1 // z |b1-bi| p1 ejected proton’s Momentum 1 z zi-z1 Glauber Operator SG 1st nucleonStruck proton Usual Parameterization σtot ←Particle Data Group (http://pdg.lbl.gov/) α　←　Partial-Wave Analysis Facility (http://lux2.phys.va.gwu.edu/) not q !!

  7. P q,ω P n Generalized Eikonal Approximation L.L.Frankfurt et al., Phys. Rev. C56, 1124(1997) and M.M. Sargsian et al, Phys. Rev.C71 044614(2005) C. Ciofi degli Atti, L.P. Kaptari, Phys. Rev. C71, 024005 (2005) Conventional Glauber Approximation Frozen Approximation Generalized Eikonal Approximation P q,ω Consider the Fermi motion P n

  8. fNN fNN ：IAAmplitude ：Double Resc. Amp. PWIA Single resc. Double resc. ：Single Resc. Amp. Diagramatic representation of the process 3He(e,e’p)2H(pn) + +

  9. fNN Effect of residual nucleon’s Fermi motion. Single Rescattering Amplitude sd

  10. Single Rescattering Amplitude sd (cont.) Using the coordinate space representation of the propagator Finally we get GEA Operator

  11. Double Rescattering Case Ref) L.L.Frankfurt et al., Phys. Rev. C56, 1124(1997) In the following calculation, we put for the double rescattering Finally we get the distorted spectral function

  12. 3.Results of 3He(e,e’p) 2H(pn) Reaction C. Ciofi degli Atti, L.P. Kaptari, Phys. Rev. Lett. 95, 052502 (2005) Data: JLab E89044 Q2～1.55(GeV/c)2, x=1 M.M. Rvachev et al., Phys. Rev. Lett.94(2005) 192302 GEA(GA) Calculation reproduces the data almost perfectly.

  13. PWIA Domain different Multiple Scattering Component Single Domain Double Domain Multiple Scattering Contributions C. Ciofi degli Atti, L.P. Kaptari, Phys. Rev. Lett. 95, 052502 (2005) dσ(pm) has different (three) slopes

  14. Results of 3He(e,e’p) pn C. Ciofi degli Atti, L.P. Kaptari, Phys. Rev. Lett. 95, 052502 (2005) Data: JLab E89044 Q2～1.55(GeV/c)2, x=1 F. Benmokhtav et al., Phys. Rev. Lett.94(2005) 082325 GEA(GA) Calculation reproduces the data quite well.

  15. :ATMS wave function NN force :Reid Soft Core 4. Results of 4He(e,e’p)3H Reaction < Extension to 4He(e,e’p)3H Reaction > We extend the same calculation as 3He-case to 4He(e,e’p)3H reaction But, for the realistic wave function we adopted the ATMS method ATMS method M. Sakai et al., Prog.Theor.Phys.Suppl.56(1974)108. H. Morita et al., Prog.Theor.Phys.78(1987)1117. Variational wave function < Experimental Data > JLab E97-111:B. Reitz et al., Eur. Phys. J. A S19(2004) 165 • Py2 Kinematics－Parallel Kinematics • CQω2 Kinematics－Perpendicular Kinematics

  16. Dip is masked by FSI Results at Py2-Kinem. Parallel Kinem. 0.58 ≦Q2 ≦ 0.90 (GeV/c)2 agreement is reasonably well. Preliminary slightly underestimate

  17. Effect of the GEA on GA GEA effect is small at this (parallel) kinematics M.S.

  18. Results at CQω２-Kinem. Perpendicular Kinem. Q2=1.78(GeV/c) ω=0.525(GeV) agreement is quite good.

  19. Effect of the GEA on GA ～30%effect on conventional Glauber

  20. Double Domain Triple Domain Single Domain Multiple Scattering Contributions Ref: GEA Effect at higher pm

  21. factor 3.4difference! factor ～8.0 Comment on the choice of Z-axis correct choice of z-axis is important!

  22. ke’ e’ ke e q,ν p ki SG(FSI) 4He 3H p*（Excited state） damping factor Formation length Q2→higher, FSI→weaker 5. Finite Formation Time Effect on 4He(e,e’p)3H Reaction at High Q2 region One must take into account the effect of p* Finite Formation Time (FFT) effect M.A.Braun et al.Phys. Rev. C62, 034606(2000)

  23. Effect of the FFT Q2=1.78(GeV/c)2 FFT effect is small at this region

  24. Parallel Kinematics Para. Kinem. at Q2=5(GeV/c)2 ,x=1 FFT effect becomes sizable

  25. Q2-dep. -Parallel Kinem.-at Q2=10(GeV/c)2 dip appears!

  26. Q2-dep. -Parallel Kinem.-Glauber +FFT Glauber + FFT FFT effect is prominent! large Q2-dep.

  27. Q2-dep. -Parallel Kinem.-Glauber Glauber little Q2-dep

  28. PLab Ｐ Ｎ Pｌab-dep. of σtot σｐＮ σtot → almost const. at high PLab

  29. 6. Summary 1. Ability of the GA/GEA to describe the FSI • Without any free parameter, the data of 3He(e,e’p)2H(pn) and 4He(e,e’p)3H were well reproduced by the Glauber/GEA calculation. • This means that in the energy-momentum range covered by the data, FSI (propagation of knocked out proton in medium) can be described within the GA/GEA. • In the case of 3He(e,e’p)2H(pn), the effect of GEA on GA (conventional Glauber approximaion) is rathersmall at considered kinematics. • In the 4He(e,e’p)3H reaction it gives ～30 % effect at (JLab E97-111) CQω2 kinematics.

  30. Summary -2 • In the 3He(e,e’p)2H reaction, the pm-dep. of dσexp exhibits different slopes (pm<1200(MeV/c)), which correspond to PWIA, single rescattering and double rescattering contributions produced by the GA/GEA. • Such features correspond to the data very well. • Rather similar features are seen in 4He(e,e’p)3H (theoretical) results, but we also found that the triple rescattering stars to contribute at pm>800(MeV/c) . 2. Multiple Scattering Feature of the GA/GEA

  31. Summary -3 • FFT effects are small at JLab E97-111 CQω2 Kinematics (Q2=1.78(GeV/c)2) • We extend theoretical calculation to higher Q2 region and found.. • FSI is almost diminished by FFT effect at aroundQ2≧10(GeV/c)2 region. • FFT effect will be appeared as a prominent Q2-dep. of nD(pm) around dip region (pm～2.2(fm-1)). 3. FFT Effect on 4He(e,e’p)3H

  32. Future Developments • Calculations without factorization are now in progress. • Under such calculation we will derive each nuclear response functions: RL,RT,RTT,RTL • Then we will calculate ATL(left-right asymmetry) , which is consider to be sensitive to the theoretical prescription.

  33. Results of 3He(e,e’p) pn　② C. Ciofi degli Atti, L.P. Kaptari, Phys. Rev. C71, 024005 (2005) C. Marchand et al., Phys. Rev. Lett.60(1988) 1703

  34. :mean-field w.f. ATMS Few-body Wave Function ATMS method M. Sakai et al., Prog.Theor.Phys.Suppl.56(1974)108. H. Morita et al., Prog.Theor.Phys.78(1987)1117. w(ij): on-shell correlation function u(ij) : off-shell correlation function

  35. Introduction of the State dependence ^^ ^ ^ w(ij)=1wS(ij)P1E(ij)+ 3wS(ij)P3E(ij) + 3wD(ij)SijP3E(ij) Euler-Lagrange eq. δu[<ψATMS|H| ψATMS> - E <ψATMS|ψATMS>] = 0 δu:performed respect to {w,u} The radial form of {w(r),.u(r)} are directly determined from Euler-Lagrange eq. Results VNN(r): Reid Soft core V8 model potential <H>=-21.2 MeV, <r2>1/2=1.57 fm PS=87.94%, PS’=0.24%, PD=11.82%

  36. Experimental Data JLab E97-111： B. Reitz et al., Eur. Phys. J. A S19(2004) 165 • Py2 Kinematics－Parallel Kinematics • CQω2 Kinematics－Perpendicular Kinematics

  37. Multiple Scattering Contributions Single rescattering contribution is dominant! “Double” starts to contribute!

  38. PWIA Domain “Triple” starts to contribute! Triple Domain Double Domain Single Domain Multiple Scattering Contributions 2

  39. p1 pm θ ｑ Note: Choice of z-axis Momentum transfer Missing Mom. q = p1 + pm Mom. of ejected proton In many cases q-direction is taken to be z-axis assuming q >> pm However, in the Glauber theory z-axis should be the direction of the incident particle ～p1 direction The deviation by θ will make some different results especially at perpendicular kinematics (q ⊥pm)

  40. Discussion data 1 Change over Effect on the Triple Rescattering GEA Effect at higher Pm

  41. Discussion data 2 GEA effect →smallat Pm＞600(MeV/c) Single Domain GEA Effect on the Single Rescatt.

  42. Discussion data 3 GEA effect →small at Pm＞1100 (MeV/c) Double Domain GEA Effect on the Double Rescatt.

  43. Discussion data 4 GEA effect on the Triple Rescatt. becomes large! Triple Domain GEA Effect on the Triple Rescatt. Full Calculation

  44. e k1(1)=k1+q k1(2)=k1(1)-q2 k1(3) k1’ f2(v2) f3(v3) 1 k2 k2’=k2+q2 2 k3 k3’=k3+q3 3 A A A-1 §5. Formulation of FFT effect M.A.Braun et al.Phys. Rev. C62, 034606(2000). Virtuality of particle “1” Bjorken variable m:Nucleon Mass

  45. Scattering Amplitude f :on-shall Amplitude F(v)　→　F.F. for Vertuality dep. F(0)=1 Scattering Matrix Vertuality dep. J(z) (Virtuality dep. Factor)

  46. Here if we take M2：Average Virtuality then Formation length In the following calculation… (Braun et al.)

  47. Ref)2H(e,e’p)n case “z-axis” effects Calculation by L.Kaptari

  48. Parallel Kinematics Para. Kinem. at Q2=2(GeV/c)2 ,x=1 Dip →masked by FSI

  49. FSI → diminished by FFT effect! Q2-dep. -Parallel Kinem.-at Q2=20(GeV/c)2 dip → apparent!

  50. ATL within Factorization Calc. Data M.M. Rvachev et al., Phys.Rev.Lett.94 (2005) 192302 Factorization calculation can not reproduce these structures. Calculations with no factorization are now in progress.

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