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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, 024005 (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. C71 044614(2005) C. Ciofi degli Atti, L.P. Kaptari, Phys. Rev. C71, 024005 (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, 052502 (2005) Q 2 1.55(GeV/c) 2, x=1 Data: JLab E89044 M.M. Rvachev et al., Phys. Rev. Lett. 94(2005) 192302 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, 052502 (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, 052502 (2005) Q 2 1.55(GeV/c) 2, x=1 Data: JLab E89044 F. Benmokhtav et al., Phys. Rev. Lett. 94(2005) 082325 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) 165 1.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 2 0.90 (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, 034606(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 -2 2. 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 -3 3. FFT Effect on 4 He(e,ep) 3 H FFT effects are small at JLab E97-111 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, 024005 (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 E97-111 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) CQω23.9520.5251.431.78395 CQω23.9520.5251.431.78446 CQω23.9520.5251.431.78495 PY2a3.170.5371.090.90824 PY2b3.170.6531.140.868124 PY2c3.170.7981.210.818223 PY2d3.170.9851.310.753323 PY2e3.171.2391.480.666423 PY2f3.171.4811.670.582493 1.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, 034606(2000). Virtuality of particle 1 m:Nucleon Mass k2k2 k 2 =k 2 +q 2 k3k3 k 3 =k 3 +q 3 1 2 3 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) 192302 Data

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