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Hadron Propagation in the Medium Hadron Propagation in the Medium Hadron Propagation in the Medium Hadron Propagation in the Medium
ICTP Workshop, Trieste, Italy 23 May, 2006
- The Exclusive Process A(e,e’p)B in few-nucleon
systems ー
< Perugia-Dubna-Sapporo Collab. >
H. Morita for
C. Ciofi degli Atti, M. Alvioli , V. Palli, I. Chiara, Univ. of PerugiaL.P. Kaptari, BLTP JINP, Dubna
H. Morita, Sapporo Gakuin Univ., Sapporo
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A(e,ep’)B Reaction
ke’
q,ω
e’e’
k’=k1+qpp
B=(A-1) B=(A-1)
ke
ee
p1(≠k’)
k1
AA
SG
SG:Glauber Operator
pm
1. Introduction
Study of the (knocked out) p propagation in the process of 3He(e,e’p)d(pn) and 4He(e,e’p)3H reaction.
< Subject of this talk >
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,e’p)B Reaction -Generalized Eikonal Approximation-
3. Results of 3He(e,e’p)2H(pn) Reaction
4. Results of 4He(e,e’p)3H Reaction
5. Finite Formation Time Effect on 4He(e,e’p)3H Reaction -at higher Q2 region-
6. Summary
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2.Framework of the Calculation for the A(e,e’p)B Reaction
)E,( 6
mmDeppe
pSKdpddd
d
nuclear distorted spectral functione-N cross section
25
)( σ
mDeppe
pKddd
d
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)
< Cross Section of A(e,e’p)B >
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Nuclear Distorted Spectral Function in 3He(e,e’p)2H(pn) process
)(
),(),()(),(
3
33
2
2
HeDm
f
M
HeGMDmmD
EEE
ddSEpS
rρρrρrr
3He(e,e’p)2H
)(
),(),()()2(
),(
3
33
121
2
2
3
3
HeN
m
f
M
HeGnpsi
mmD
EM
E
ddSed
EpS m
t
rρρrρrrt tpρ
3He(e,e’p)pn
SG: Glauber operator
P
P
nr
ρt=(Pp-Pn)/2
33
M
He : Pisa Group’s 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 SG
1st nucleon Struck proton
),1(3
2iGS
iG
σtot ← Particle Data Group (http://pdg.lbl.gov/) α ← Partial-Wave Analysis Facility (http://lux2.phys.va.gwu.edu/)
),()(1)1( 11 ii ZZiG bb
)2/exp(4
)1()( 2
02
20
bb
itot bb
Usual Parameterization
not q !!
n
P
zzi-z1
i
|b1-bi|
1 p1
~~ejected proton’s Momentum
< Choice of Z-axis>
p1 // 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 motionP
P
n
q,ω
P
P
n
q,ω
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Diagramatic representation of the process 3He(e,e’p)2H(pn)
fNN
Double resc.Single resc.
fNN
PWIA
)(),(12
1),( 3
3
2
2
1
2
0
)(
He
fm
f nf
n
He
mmD EEEsMJ
EpS
+ +
)1( : Single Resc. Amp.
)2( : Double Resc. Amp.
)0( : IA Amplitude
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Single Rescattering Amplitude sd
)(
),,,()'(
)(
),,,,,(22'
1
323'2)23(
22'1
1122
1
321321)23(123
)1(
N
f
N
NN
N
He
Mk
sskkG
Mk
kpf
Mk
ssskkkGd
fNN
),,(|,, 3213213
3 kkkM
Hesss ),(|, 322332
23 kkMss
)(),(12
1),( 3
3
2
2
1
2
0
)(
He
fm
f nf
n
He
mmD EEEsMJ
EpS
mmz EE
EEE
||||2 11
111
ppppqk
Effect of residual
nucleon’s Fermi motion.
)1(
);,(||4/)(
),,(|)2( 233
'2)23(
132113
3)1( 3
3 Si
Mfs
d f
zz
NNNM
Hekk
pkkk
κ
'11 kp
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)(
),(),(),()(),(
3
33
23
2)1()0(
23)10(
He
fm
f
M
HeGEAf
mmD
EEE
ddSSEpS
rρρrρrρrr
Single Rescattering Amplitude sd (cont.))1(
Using the coordinate space representation of the propagator
);,(||4/)(
),,(|)2( 233
'2)23(
132113
3)1( 3
3 Si
Mfs
d f
zz
NNNM
Hekk
pkkk
κ
dzezii
zi
zz
zz )()(1
Finally we get
3
21
)(1
)1( )()(),( 1
ii
zziiGEA
izezzS bbρr
GEA Operator
1),()0( ρrS
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)(
),()(),(
3
33
23
2)2()1()0(
23
He
fm
f
M
HeGEAGEAf
mmD
EEE
ddSSSEpS
rρρrr
Double Rescattering Case
),(),(])()(
)()([),(
3121))(()(
3213
))(()(2312
)2(
122132
133123
bbbb
ρr
zzzzizzi
zzizziGEA
z
z
eezzzz
eezzzzS
)(),( 11 ii
izze bbbb )( '
0ii
EEq
i pkq
Finally we get the distorted spectral function
Ref) L.L.Frankfurt et al., Phys. Rev. C56, 1124(1997)
In the following calculation, we put2/zi for the double rescattering
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3.Results of 3He(e,e’p) 2H(pn) ReactionC. Ciofi degli Atti, L.P. Kaptari, Phys. Rev. Lett. 95, 052502 (2005)
Q2~ 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.
% few a)()( 55 GAdGEAd
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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(pm) has different (three) slopes
different Multiple Scattering Component
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Results of 3He(e,e’p) pn C. Ciofi degli Atti, L.P. Kaptari, Phys. Rev. Lett. 95, 052502 (2005)
Q2~ 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 4He(e,e’p)3H 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 3He-case to 4He(e,e’p)3H reaction
But, for the realistic wave function we adopted the ATMS method
NN force :Reid Soft CoreHeH 43 , :ATMS wave function
< Extension to 4He(e,e’p)3H Reaction >
< Experimental Data >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
eppe
m Kddd
dp
σ
)(n 5
D
0.58 ≦Q2 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 2 -Kinem.
agreement is quite good.
Perpendicular Kinem.
Q2=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 pm
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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 4He(e,e’p)3H Reaction
at High Q2 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
ke’
q,ν
SG(FSI)4He
4He
e’e’
pp
3H3H
keee
ki
p*( Excited state)
),()(1)( 11 bbzzJrS jjj
),)
)(exp(1)(()(
2Ql
zzzJ
Formation lengthFormation length2
22 )(
xmM
QQl Q2→higher, FSI→weaker
damping factor
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Effect of the FFT
FFT effect is small at this region
Q2=1.78(GeV/c)2
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0.0 0.5 1.0 1.5 2.0 2.5 3.010
-6
10-5
10-4
10-3
10-2
10-1
100
n(k) at Para. Kinem.
4He(e,e'p)
3H Q
2=5(GeV/c)
2
FSI(Glauber) FSI(Glauber+FFT) No FSI
n(k)
(fm
3 )
k (fm-1)
FFT effect becomes sizable
Para. Kinem. at Q2=5(GeV/c)2 ,x=1
Parallel KinematicsParallel Kinematics
q
mk
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0.0 0.5 1.0 1.5 2.0 2.5 3.010
-6
10-5
10-4
10-3
10-2
10-1
100
n(k) at Para. Kinem.
4He(e,e'p)
3H Q
2=10(GeV/c)
2
FSI(Glauber) FSI(Glauber+FFT) No FSI
n(k)
(fm
3 )
k (fm-1)
dip appears!dip appears!
Q2-dep. -Parallel Kinem.- at Q2=10(GeV/c)2
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0.0 0.5 1.0 1.5 2.0 2.5 3.010
-6
10-5
10-4
10-3
10-2
10-1
100
4He(e,e'p)
3H
n(k) at Para. Kinem.
FSI(G+FFT) Q2-Dep.
Q2=20(GeV/c)
Q2=10(GeV/c)
Q2= 5(GeV/c)
Q2= 2(GeV/c)
No FSI
n(k)
(fm
3 )
k(fm-1)
Glauber + FFTGlauber + FFT
FFT effect is prominent!FFT effect is prominent!
large Qlarge Q22-dep.-dep.
Q2-dep. -Parallel Kinem.- Glauber +FF
T
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0.0 0.5 1.0 1.5 2.0 2.5 3.010
-6
10-5
10-4
10-3
10-2
10-1
100
4He(e,e'p)
3H
n(k) at Para. Kinem.
FSI(G) Q2-Dep.
Q2=20(GeV/c)
Q2=10(GeV/c)
Q2= 5(GeV/c)
Q2= 2(GeV/c)
No FSI
n(k
) (f
m3 )
k(fm-1)
Q2-dep. -Parallel Kinem.- Glaube
r GlauberGlauber
little Qlittle Q22-dep-dep
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P l ab-dep. of σtot
σ pNσ pN
0 1 2 3 4 5 6 7 8 9 1020
30
40
50
60
70pN Cross Secttion (P
Lab)
=(np
+pp
)/2
(m
b)
PLab
(GeV/c)
σtot → almost const. at high PLab
NP PLab
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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 rather small at considered kinematics.
• In the 4He(e,e’p)3H 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 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 (theo
retical) results, but we also found that the triple rescattering stars to contribute at pm>800(MeV/c) .
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Summary -3
3. FFT Effect on 4He(e,e’p)3H
• 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 around Q2 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)).
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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.
TTTTTTLL
TLTLTL vvv
vA
)180()0(
)180()0(
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Results of 3He(e,e’p) pn ②
0 20 40 60 80 1000
5
10
15
20
25
30
0 20 40 60 80 1000
5
10
15
20
25
0 20 40 60 80 1000
5
10
15
20
0 40 80 120
2
4
6
8
10
Pm = 620 MeV/c
Pm = 340 MeV/c
Pm = 440 MeV/c
PWA
PWIA
FSI
d6 /d
Ee'
de'
dp'
dEm
[10
-2pb
/MeV
2 /sr2 ]
JLAB 89044
Pm = 550 MeV/c
Erel = Em - E3 [MeV]
C. Ciofi degli Atti, L.P. Kaptari, Phys. Rev. C71, 024005 (2005)
0369
1215
0
1
2
3
0 20 40 60 80 1000
1
2
PWA PWIA FSI
p' = 60o
Em [MeV]
d6/
dEe'd
e' d p'
dE
m [p
b M
ev -
2sr
-2]
p' = 90.5o
PWA PWIA FSI
3He(e,e'p)pn (SACLAY kinem.)
PWA PWIA FSI
p' = 112o
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.
ij ijklp
p kluijun
nijwDF ),())(
)1()((1
,)())()1(
1(
ijklij p
p kluijun
nD 2/)1( AAnp
, FATMS
w(ij): on-shell correlation functionu(ij) : off-shell correlation function
AS TS }0,0{ :mean-field w.f.
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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%
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Experimental Data JLab E97-111 : B. Reitz et al., Eur. Phys. J. A S19(2004) 165
Kinem. Ee(GeV) ω(GeV) q(Gev/c) Q2(GeV/c)2 pm(MeV/c)
CQω2 3.952 0.525 1.43 1.78 395
CQω2 3.952 0.525 1.43 1.78 446
CQω2 3.952 0.525 1.43 1.78 495
PY2a 3.17 0.537 1.09 0.908 24
PY2b 3.17 0.653 1.14 0.868 124
PY2c 3.17 0.798 1.21 0.818 223
PY2d 3.17 0.985 1.31 0.753 323
PY2e 3.17 1.239 1.48 0.666 423
PY2f 3.17 1.481 1.67 0.582 493
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 DomainDouble Domain
Triple Domain
PWIA Domain
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Note: Choice of z-axis
q = p1 + pm
Momentum transfer
Mom. of ejected proton
Missing Mom.
However, in the Glauber theory z-axis should be the direction of the incident particle ~ p1 direction
In many cases q-direction is taken to be z-axis assuming q >> pm
The deviation by θ will make some different results especially at perpendicular kinematics (q ⊥pm)
qpm
p1
θ
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GEA Effect at higher Pm
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 Pm >600(MeV/c)
Single Domain
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Discussion data 3
GEA Effect on the Double Rescatt.
GEA effect →small at Pm > 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
),()11
()( 12
2222)(
1 z
i
jjz
ii kq
xm
Q
xQmkv
,222 qQ
M.A.Braun et al. Phys. Rev. C62, 034606(2000).
)2/( 12 qkQx
Virtuality of particle “1”Virtuality of particle “1”
m:Nucleon Mass
k2 k2’=k2+q2
k3 k3’=k3+q3
12
3
A
k1(1)=k1+q k1
(2)=k1(1)-q2
f2(v2) f3(v3)
k1(3) k1
’
A A-1
),( qqee
Bjorken variable
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Scattering AmplitudeScattering Amplitude
f :on-shall f :on-shall AmplitudeAmplitude
F(v)F(v) → → F.F. for Vertuality F.F. for Vertuality dep.dep.
F(0)=1F(0)=1
Scattering MatrixScattering Matrix
,)(),,,(2
32
A
jjA rSrrrS
)()(1)( 11 bbzzJrS jjj
Vertuality deVertuality dep.p.
J(z) (Virtuality dep. Factor)J(z) (Virtuality dep. Factor)
),exp(0
)(
2)(
2
2
zQ
xmvi
iv
vFdvizJ
0
2
0'
)'('')(
ivv
vvdvvF
))exp(1)(()()(0
21 zQ
xmvvdvzzJ
)()(1 iviv
1)(0
vdv
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Here if we takeHere if we take
)*(),()( 22221 mmMMvv
thenthen
),))(
exp(1)(()(2Ql
zzzJ
Formation lengthFormation length
In the following calculation…In the following calculation…
,)( 22*2 mmM Av )(8.1* GeVmAv (Braun et al.)(Braun et al.)
2
22 )(
xmM
QQl
M2 : Average Virtuality
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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
10-2
10-1
100
101
102
103
104
PWIA FSI (z || p')
pm [GeV/c]
Nef
f ( p
m )
[(G
eV /c
) -3 ]
2 H(e,e'p)n
FSI (z || q)
“z-axis” effects Ref) 2H(e,e’p)n case
Calculation by L.Kaptari
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Para. Kinem. at Q2=2(GeV/c)2 ,x=1
Parallel KinematicsParallel Kinematics
q
mk
Dip →Dip → masked by FSImasked by FSI
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0.0 0.5 1.0 1.5 2.0 2.5 3.010-6
10-5
10-4
10-3
10-2
10-1
100
n(k) at Para. Kinem.
4He(e,e'p)3H Q2=20(GeV/c)2
FSI(Glauber) FSI(Glauber+FFT) No FSI
n(k)
(fm
3 )
k (fm-1)
dip → apparent!dip → apparent!
FSI → diminished by FFT effect!FSI → diminished by FFT effect!
Q2-dep. -Parallel Kinem.- at Q2=20(GeV/c)2
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ATL 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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