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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
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
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
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