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Hadron Propagation in Hadron Propagation in the Medium 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. Pa lli, I. Chiara, Univ. of Perugia L.P. Kaptari, BLTP JINP, Dubna H. Morita, Sapporo Gakuin Univ., Sapporo

Hadron Propagation in the Medium

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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 - PowerPoint PPT Presentation

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Page 1: Hadron Propagation in the Medium

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

Page 2: Hadron Propagation in the Medium

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

Page 3: Hadron Propagation in the Medium

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

Page 4: Hadron Propagation in the Medium

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 >

Page 5: Hadron Propagation in the Medium

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)

Page 6: Hadron Propagation in the Medium

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

Page 7: Hadron Propagation in the Medium

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

Page 8: Hadron Propagation in the Medium

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

Page 9: Hadron Propagation in the Medium

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

Page 10: Hadron Propagation in the Medium

)(

),(),(),()(),(

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

Page 11: Hadron Propagation in the Medium

)(

),()(),(

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

Page 12: Hadron Propagation in the Medium

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

Page 13: Hadron Propagation in the Medium

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

Page 14: Hadron Propagation in the Medium

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.

Page 15: Hadron Propagation in the Medium

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

Page 16: Hadron Propagation in the Medium

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

Page 17: Hadron Propagation in the Medium

Effect of the GEA on GA

GEA effect is small at this (parallel) kinematics

M.S.

Page 18: Hadron Propagation in the Medium

Results at CQω 2 -Kinem.

agreement is quite good.

Perpendicular Kinem.

Q2=1.78(GeV/c) ω=0.525(GeV)

Page 19: Hadron Propagation in the Medium

Effect of the GEA on GA

~ 30%   effect on conventional Glauber

Page 20: Hadron Propagation in the Medium

Multiple Scattering Contributions

Single Domain

Double Domain

Triple Domain

Ref: GEA Effect at higher pm

Page 21: Hadron Propagation in the Medium

factor 3.4 difference!

Comment on the choice of Z-axis

correct choice of z-axis is important!

factor ~ 8.0

Page 22: Hadron Propagation in the Medium

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

Page 23: Hadron Propagation in the Medium

Effect of the FFT

FFT effect is small at this region

Q2=1.78(GeV/c)2

Page 24: Hadron Propagation in the Medium

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

Page 25: Hadron Propagation in the Medium

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

Page 26: Hadron Propagation in the Medium

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

Page 27: Hadron Propagation in the Medium

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

Page 28: Hadron Propagation in the Medium

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

Page 29: Hadron Propagation in the Medium

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.

Page 30: Hadron Propagation in the Medium

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) .

Page 31: Hadron Propagation in the Medium

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)).

Page 32: Hadron Propagation in the Medium

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(

Page 33: Hadron Propagation in the Medium

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

Page 34: Hadron Propagation in the Medium

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.

Page 35: Hadron Propagation in the Medium

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%

Page 36: Hadron Propagation in the Medium

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

Page 37: Hadron Propagation in the Medium

Multiple Scattering Contributions

Single rescattering contribution is dominant!

“Double” starts to contribute!

Page 38: Hadron Propagation in the Medium

Multiple Scattering Contributions 2

“Triple” starts to contribute!

Single DomainDouble Domain

Triple Domain

PWIA Domain

Page 39: Hadron Propagation in the Medium

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

Page 40: Hadron Propagation in the Medium

GEA Effect at higher Pm

Discussion data 1

Change over

Effect on the Triple Rescattering

Page 41: Hadron Propagation in the Medium

Discussion data 2

GEA Effect on the Single Rescatt.

GEA effect →small   at Pm >600(MeV/c)

Single Domain

Page 42: Hadron Propagation in the Medium

Discussion data 3

GEA Effect on the Double Rescatt.

GEA effect →small at Pm > 1100 (MeV/c)

Double Domain

Page 43: Hadron Propagation in the Medium

Discussion data 4

GEA Effect on the Triple Rescatt.

Full Calculation

GEA effect on the Triple Rescatt. becomes large!

Triple Domain

Page 44: Hadron Propagation in the Medium

§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

Page 45: Hadron Propagation in the Medium

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

Page 46: Hadron Propagation in the Medium

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

Page 47: Hadron Propagation in the Medium

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

Page 48: Hadron Propagation in the Medium

Para. Kinem. at Q2=2(GeV/c)2 ,x=1

Parallel KinematicsParallel Kinematics

q

mk

Dip →Dip →   masked by FSImasked by FSI

Page 49: Hadron Propagation in the Medium

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

Page 50: Hadron Propagation in the Medium

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

Page 51: Hadron Propagation in the Medium