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Hadron Propagation in the Medium ICTP Workshop, Trieste, Italy 23 May, 2006 The Exclusive Process A(e,ep)B in few-nucleon systems H. Morita for 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

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A(e,ep)B Reaction k e q,ωq,ω e e k=k 1 + q p p B=(A-1) keke e e p 1 (k) k1k1 A A SGSG S G :Glauber Operator pmpm 1. Introduction Study of the (knocked out) p propagation in the process of 3 He(e,ep)d(pn) and 4 He(e,ep) 3 H reaction. How can we describe the p propagation in medium? FSI

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Outline of my talk 1.Introduction 2.Framework of the calculation for the A(e,ep)B Reaction -Generalized Eikonal Approximation- 3.Results of 3 He(e,ep) 2 H(pn) Reaction 4.Results of 4 He(e,ep) 3 H Reaction 5.Finite Formation Time Effect on 4 He(e,ep) 3 H Reaction -at higher Q 2 region- 6.Summary

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2.Framework of the Calculation for the A(e,ep)B Reaction nuclear distorted spectral functione-N cross section If B(recoil system) is bound state We use Factorization Ansatz. nucleon distorted momentum distribution distorted by FSI (here only unpolarized cross section will be discussed)

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Nuclear Distorted Spectral Function in 3 He(e,ep) 2 H(pn) process 3 He(e,ep) 2 H 3 He(e,ep)pn S G : Glauber operator P P n ρ t=(P p -P n )/2 Pisa Groups w.f. with AV18 pot. A. Kievsky et al., Nucl. Phys. A551(1993) 241 C. Ciofi degli Atti, L.P. Kaptari, Phys. Rev. C71, (2005)

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Glauber Operator S G 1st nucleon Struck proton σ tot Particle Data Group (http://pdg.lbl.gov/) α Partial-Wave Analysis Facility (http://lux2.phys.va.gwu.edu/) Usual Parameterization not q !! n P z z i -z 1 i |b1-bi||b1-bi| 1 p1 p1 ejected protons Momentum Choice of Z-axis p 1 // z

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

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Diagramatic representation of the process 3 He(e,ep) 2 H(pn) f NN Double resc. Single resc. f NN PWIA ++ Single Resc. Amp. Double Resc. Amp. IA Amplitude

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Single Rescattering Amplitude sd f NN Effect of residual nucleons Fermi motion.

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

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

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

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

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

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4. Results of 4 He(e,ep) 3 H Reaction 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 We extend the same calculation as 3 He-case to 4 He(e,ep) 3 H reaction But, for the realistic wave function we adopted the ATMS method NN force :Reid Soft Core :ATMS wave function JLab E97-111: B. Reitz et al., Eur. Phys. J. A S19(2004) Py2 Kinematics Parallel Kinematics 2.CQω2 Kinematics Perpendicular Kinematics

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

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

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Results at CQω -Kinem. agreement is quite good. Perpendicular Kinem. Q 2 =1.78(GeV/c) ω=0.525(GeV )

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Effect of the GEA on GA 30% effect on conventional Glauber

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Multiple Scattering Contributions Single Domain Double Domain Triple Domain Ref: GEA Effect at higher p m

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

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5. Finite Formation Time Effect on 4 He(e,ep) 3 H Reaction at High Q 2 region M.A.Braun et al. Phys. Rev. C62, (2000) One must take into account the effect of p* Finite Formation Time (FFT) effect k e q,ν S G (FSI) 4 He e e p p 3H3H 3H3H keke e e kiki p* Excited state Formation length Q 2 higher, FSIweaker damping factor

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

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

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

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Glauber + FFT FFT effect is prominent! large Q 2 -dep. Q 2 -dep. -Parallel Kinem.- Glauber +FFT

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Q 2 -dep. -Parallel Kinem.- Glauber Glauber little Q 2 -dep

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P ab -dep. of σ tot σ σ σ tot almost const. at high P Lab P Lab

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6. Summary 1. Ability of the GA/GEA to describe the FSI Without any free parameter, the data of 3 He(e,ep) 2 H(pn) and 4 He(e,ep) 3 H 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 3 He(e,ep) 2 H(pn), the effect of GEA on GA (conventional Glauber approximaion) is rather small at considered kinematics. In the 4 He(e,ep) 3 H reaction it gives 30 % effect at (JLab E97-111) CQω2 kinematics.

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Summary Multiple Scattering Feature of the GA/GEA In the 3 He(e,ep) 2 H reaction, the p m -dep. of dσ exp exhibits different slopes (p m <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 4 He(e,ep) 3 H (theoretical) results, but we also found that the triple rescattering stars to contribute at p m >800(MeV/c).

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Summary FFT Effect on 4 He(e,ep) 3 H FFT effects are small at JLab E CQω2 Kinematics (Q 2 =1.78(GeV/c) 2 ) We extend theoretical calculation to higher Q 2 region and found.. FSI is almost diminished by FFT effect at around Q 2 10(GeV/c) 2 region. FFT effect will be appeared as a prominent Q 2 -dep. of n D (p m ) around dip region (p m 2.2(fm-1)).

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Future Developments Calculations without factorization are now in progress. Under such calculation we will derive each nuclear response functions: R L,R T,R TT,R TL Then we will calculate A TL (left-right asymmetry), which is consider to be sensitive to the theoretical prescription.

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

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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 :mean-field w.f.

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Introduction of the State dependence ^ ^ ^ ^ w(ij)= 1 w S (ij)P 1E (ij)+ 3 w S (ij)P 3E (ij) + 3 w D (ij)S ij P 3E (ij) Euler-Lagrange eq. δ u [ - E ] = 0 δ u :performed respect to {w,u} The radial form of {w(r),.u(r)} are directly determined from Euler-Lagrange eq. Results V NN (r): Reid Soft core V8 model potential =-21.2 MeV, 1/2 =1.57 fm PS=87.94%, PS=0.24%, PD=11.82%

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Experimental Data JLab E B. Reitz et al., Eur. Phys. J. A S19(2004) 165 Kinem.E e (GeV)ω(GeV)q(Gev/c)Q 2 (GeV/c) 2 p m (MeV/c) CQω CQω CQω PY2a PY2b PY2c PY2d PY2e PY2f Py2 Kinematics Parallel Kinematics 2.CQω2 Kinematics Perpendicular Kinematics

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Multiple Scattering Contributions Single rescattering contribution is dominant! Double starts to contribute!

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Multiple Scattering Contributions 2 Triple starts to contribute! Single Domain Double Domain Triple Domain PWIA Domain

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Note: Choice of z-axis q = p 1 + p m Momentum transfer Mom. of ejected proton Missing Mom. However, in the Glauber theory z-axis should be the direction of the incident particle p 1 direction In many cases q-direction is taken to be z-axis assuming q >> p m The deviation by θ will make some different results especially at perpendicular kinematics (q p m ) pmpm p1p1 θ

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GEA Effect at higher P m Discussion data 1 Change over Effect on the Triple Rescattering

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Discussion data 2 GEA Effect on the Single Rescatt. GEA effect small at P m 600(MeV/c) Single Domain

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Discussion data 3 GEA Effect on the Double Rescatt. GEA effect small at P m 1100 (MeV/c) Double Domain

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

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§5. Formulation of FFT effect M.A.Braun et al. Phys. Rev. C62, (2000). Virtuality of particle 1 m:Nucleon Mass k2k2 k 2 =k 2 +q 2 k3k3 k 3 =k 3 +q A k 1 (1) =k 1 +qk 1 (2) =k 1 (1) -q 2 f 2 (v 2 )f 3 (v 3 ) k 1 (3) k 1 A A-1e Bjorken variable

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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)

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Here if we take then Formation length In the following calculation… (Braun et al.) M 2 Average Virtuality

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z-axis effects Ref) 2 H(e,ep)n case Calculation by L.Kaptari

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

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

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

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