C. CIOFI degli ATTI

NN CORRELATIONS :

(e, e′p) EXPERIMENTS AND THEORETICAL SPECTRALFUNCTIONS

The short range structure of nuclei at 12 GeV

JLAB WORKSHOP

October 25-26, 2007

C. Ciofi degli Atti 1 Jlab, October07

New dedicated experiments:

• high Q2

• back-to-back detected nucleons

• Em ≃ p2m/2mN

q ν

Μ p

p

mA

if FSI etc. could be disregarded:

q νp

Μ A

k

=k+q

pm=P = kA−1

in this case the central quantity is the spectral function

C. Ciofi degli Atti 2 Jlab, October07

P (k,E) = P0(k,E) + P1(k,E)

P (k,E) =∑

f

∣

∣

∣< Ψ

fA−1| ak|Ψ

0A >

∣

∣

∣

2δ(

E −(

EfA−1 − E0

A

))

P0(k,E) =∑

f<F

∣

∣

∣< Ψ

fA−1| ak|Ψ

0A >

∣

∣

∣

2δ(

E −(

EfA−1 − E0

A

))

P1(k,E) =∑

f>F

∣

∣

∣< Ψ

fA−1| ak|Ψ

0A >

∣

∣

∣

2δ(

E −(

EfA−1 − E0

A

))

P (k,E) = −1

πIm G(k,E) =

1

π

W (k,E)

(−E − k2/2m − V (k,E))2 + W (k,E)2

G(k,E) =1

−E − k2/2m − V (k,E) − ıW (k,E)

DO WE KNOW IT?

C. Ciofi degli Atti 3 Jlab, October07

Spectral Function of A = 3 and A = ∞

Spectral Functions calculated from many-body theory exhibita common feature:

at high E (> 40 MeV ) and k (> 1.5 − 2.0 fm−1)

P (k,E) has maxima at E =(A−2)k2

2(A−1)mN

explanation in terms of a simple and physically sound model:two-nucleon correlations ( Frankfurt & Strikman 1988)

C. Ciofi degli Atti 4 Jlab, October07

C. Ciofi degli Atti 5 Jlab, October07

C. Ciofi degli Atti 6 Jlab, October07

C. Ciofi degli Atti 7 Jlab, October07

MANY BODY vs CONVOLUTION MODEL

W (k,E) =1

2

∑

hh′p

Im| < kp | G

(

e(h) + e(h′))

| hh′ >a |2

E − e(p) + e(h) + e(h′) − ıη

=π

2

∑

hh′p

| < kp | G(

e(h) + e(h′))

| hh′ >a |2 δ(

− E + e(p) − e(h) − e(h′))

P (k,E) =1

2

∑

q,P

| ξ(

k −1

2P,q)

|2 θ(kF − | q +1

2P|)θ(kF − | q −

1

2P|)

× θ(|P − k| − kF ) δ(

E − e(p) + e(h) + e(h′))

|ξ >= |Ψ > −|φ >

P (k,E) =mρ2

32k

∫ k+k0

|k−k0|dP P nFG

cm (P ) nrel

(

√

1

2k2 −

1

4P 2 +

1

2k2

0

)

C. Ciofi degli Atti 8 Jlab, October07

k=1.5 fm-1

k=2.2 fm-1

k=3.0 fm-1

k=3.5 fm-1

Points are numerical calculation of the spectral functions of 3He and nuclear matter - curves two nucleon approximation from CSFS 91

48

P A1 (|k|, E) =

∫

d3Pcm nArel (|k − Pcm/2|) nA

cm(|Pcm|)

δ

[

E − E(2)thr −

(A − 2)

2M(A − 1)·

(

k −(A − 1)Pcm

(A − 2)

)2]

CdA, Simula , 1996

C. Ciofi degli Atti 9 Jlab, October07

BBG Theory leads explicitlyto the convolution formula

Baldo,Borromeo,CdA 1996

C. Ciofi degli Atti 10 Jlab, October07

Factorization of nNN (krel,KCM)

Many Body:Alvioli, CdA, Morita

arXiv:0709.3989

nNN (krel,KCM)

CS: Ciofi, SimulaPRC53, (1996)

CA n2H(krel) nCS(KCM )

2 3 4

100

101

102

103

104 MB CS KCM= 0.0 fm-1

KCM= 0.5 fm-1

KCM= 1.0 fm-1

n pn(

k rel,K

CM)

[fm6 ]

krel [fm-1]

16O

C. Ciofi degli Atti 11 Jlab, October07

• nNN (krel,KCM) factorization in the Two-Nucleon correlationmodel Spectral Function implies that nNN (krel,KCM , θ) ∝nNN (krel,K

′CM , θ)

1 2 3 40

1

2

3

4

5

6

7

krel [fm-1]

KCM - K'CM

0.0 - 0.2 0.0 - 0.5 0.0 - 1.0 0.5 - 1.0

n(k re

l,KC

M,

) / n

(kre

l,K' C

M) 16O - kK=90o

• high krel and low KCM factorization validated by many-bodycalculations

C. Ciofi degli Atti 12 Jlab, October07

C. Ciofi degli Atti 13 Jlab, October07

C. Ciofi degli Atti 14 Jlab, October07

Momentum distributions and occupation numbers:Many Body vs 2NC

0 1 2 3 410-5

10-4

10-3

10-2

10-1

100

n(k)

[fm

3 ]

k [fm-1]

no(k)

nH(k) - nS(k) n(k)

16O - HO - V8'

ρo(r, r′)

1 1’

∆ρH(r, r′)

1 1’

22

1’ 1

∆ρS(r, r′)

1’ 1

2 3 32

1 1’

∆nS reduces the SM occupation; ∆nH mostly generates corre-lations (populates states above the Fermi sea)

C. Ciofi degli Atti 15 Jlab, October07

M. Alvioli, CdA, H. Morita , to appear

0 1 2 3

10-4

10-3

10-2

10-1

16O

FNC model no

n1

fulln(

k) [fm

3 ]

k [fm-1]

many-body nSM

nH

nS

full

0 1 2 3

10-4

10-3

10-2

10-1

FNC model no

n1

fulln(k)

[fm

3 ]

k [fm-1]

many-body no

n1

full

16O

no(k) = nSM (k) + ∆nS(k) + CA nHSM (k)

n1(k) = ∆nH(k) − CA nHSM (k)

So =

∫

dk no(k)

S1 =

∫

dk n1(k)

A So S13 0.65 0.3516 0.8 0.2

no ”external” quantities in the convolution model

C. Ciofi degli Atti 16 Jlab, October07

C. Ciofi degli Atti 17 Jlab, October07

C. Ciofi degli Atti 18 Jlab, October07

Tentative Summary

• the (parameter free) convolution model is physically soundand theoretically validated by many body calculations. Anymany-body approach to the spectral function should lead forE ≃ k2/2mN to a convolution integral;

• representing an effective three body problem, the model canbe readily extended to accommodate missing effects: rela-tivistic effects (LC variables), three-nucleon correlations, FSIeffects, isospin dependence. Work is in progress;

• isospin dependence is governed by the isospin dependenceof nrel which has been recently calculated (Schiavilla et al,Perugia (Alvioli’s talk));

• a careful comparison with other models of SF for finite nuclei(e.g. LDA) is order; at E ≃ k2/2mN all of them should predictthe convolution model.

C. Ciofi degli Atti 19 Jlab, October07