Transcript Chiara Benedetta Mezzetti Claudio Ciofi degli Atti Ph.D. student INT09

Chiara Benedetta Mezzetti
Ph.D. student
Claudio Ciofi degli Atti
INT09
“The Jefferson Laboratory
Upgrade to 12 GeV”
October 26 – November 20, 2009
Seattle, WA - USA -
Chiara Benedetta Mezzetti
Seattle,05/11/2009
Outline





The realistic many-body description
of nuclei
The spectral function and the nucleon
momentum distributions
A new approach to the treatment of
inclusive cross sections
Calculation of the inclusive plateaux
Conclusions
2
Chiara Benedetta Mezzetti
Seattle,05/11/2009
The realistic many-body
description of nuclei
3
Many-body problem
Hˆ A  
2
2m
Chiara Benedetta Mezzetti
Seattle,05/11/2009
2
ˆ

i i  i j vˆ(i, j )  i j k vˆ(i, j, k )  ...
ˆvij ( xi , x j )   n v ( n ) (rij ) Oˆ ij( n )
rij  ri  rj
(n)
ˆ
Oij  1,  i   j , Sˆij , ( L  S )ij ,...  1, i  j 
4
Shell model
Chiara Benedetta Mezzetti
Seattle,05/11/2009
1963 Nobel Prize
Maria Goeppert Mayer
Hˆ A  
2
2m
2
ˆ
i i  i V i (ri )
Nucleons move independently in the
average potential generated by the
mutual interaction.
Nucleons occupy all available states
below the Fermi level, leaving empty
the states above it.
Successfull interpretation of many
nuclear low energy properties, but:
how can we explain the missing
percentage?
5
Chiara Benedetta Mezzetti
Seattle,05/11/2009
High momentum components
At short and intermediate distances
nucleons feel the strong central
repulsion and the tensor attraction
M. Alvioli, C. Ciofi degli Atti, H. Morita
Phys. Rev. C72 (2005) 054310
Phys. Rev. Lett. 100 (2008) 162503
 0  F̂0
Mean field
wave function
Correlation operator
The correlation functions fij
Central, Spin-Isospin, Tensorial, …
Variational Monte Carlo
(Urbana Group)
Cluster expansion techniques
(Alvioli, Ciofi degli Atti, Morita)
Fˆ  Sˆ  fij  Sˆ   f ( n ) (rij )Oˆ ij( n )
i j
i j
n
6
A cartoon of SRCs in nuclei
What is the
percentage of
correlated nucleons
in nuclei?
Chiara Benedetta Mezzetti
Seattle,05/11/2009
7
Chiara Benedetta Mezzetti
Seattle,05/11/2009
Semi-esclusive scattering
12
12
of f C(p,p’pN) and C(e,e’pN)
These experiments provide quantitative information on 2NC
only. What about 3NC?
8
Cross section ratios at CLAS
Original idea:
 A  Q , xB    A
A
2
j 2
a j  A
j
Chiara Benedetta Mezzetti
Seattle,05/11/2009
Experimental data:
 j  Q , xB 
2
K.S. Egiyan et al, PRL 96, 082501 (2006)
Frankfurt & Strikman, Phys. Rep. 5
(1988) 235
xB  1.5
Mean Field
2NC
1.5  xB  2
3NC
2  xB  3
But: no direct microscopic many-body calculation of the ratio
Plateaux: 2NC
and
areis the
in A and in 3He.
3He)3NC
r(A/
This
justsame
our aim
9
Chiara Benedetta Mezzetti
Seattle,05/11/2009
The Spectral Function
and
the Nucleon Momentum
Distributions
10
Spectral Function
P(k , E )   A0 ak†  E   H  E A   ak  A0 
  
f
A1
f
| ak |    E   EAf 1  EA0 
0
A
E  Emin  E A* 1
P(k , E )  P0 (k , E )  P1 (k , E )
f
A1
| ak |     E   E Af 1  E A0  
f
A1
| ak |     E   E Af 1  E A0  
P0 (k , E ) 


P1 (k , E ) 


f F
f F
2
Chiara Benedetta Mezzetti
Seattle,05/11/2009
0
A
0
A
2
2
Nucleon Momentum Distribution
1
ik ( z  z ')
n( k ) 
e
 ( z , z ')dzdz '   dEP(k , E )
3 
(2 )
n0 (k )   dEP0 (k , E )
n1 (k )   dEP1 (k , E )
11
Chiara Benedetta Mezzetti
Seattle,05/11/2009
Spectral function of
3He and NM
P(k , E )[ fm4 ]
Spectral functions
calculated within many-body
theories exhibit a common
feature:
E  100MeV
3He
k  1.5  2.0 fm1
P (k , E )
has maxima, for fixed k, at
( A  2)k 2
E
2( A  1)mN
1
k[ fm ]
This represents evidence of two-nucleon correlations
Frankfurt, & Strikman
Two-nucleon
(1988)
correlation model
k1
k2
Momentum
conservation
Chiara Benedetta Mezzetti
Seattle,05/11/2009
k1  k2  k A2  0
k1  k2  k , k A2  0
MA-2
k A 2
Energy
conservation
Can easily be
generalized to
relativistic
kinematics
2
2
k
k
EA* 1 

2M A1 2mN
2
2
2
k
k
A

2
k
EA* 1 


2mN 2M A1 A  1 2mN
2


A

2
k
(2)
P(k , E )  n(k )  E  E(thr ) 

A  1 2mN 

Improved two-nucleon
correlation model
Chiara Benedetta Mezzetti
Ciofi, Simula, Frankfurft,
Strikman, PRC 44 (1991)R1;
Ciofi, Simula, PRC 53 (1996)
1689;
k1  k2  k A2  0
k1  k
k2
k A2  0 , k1  k
kCM  k1  k2  k A2
MA-2
k A 2

kCM
P (k , E )   dkCM nrel  k1 

2

E
*
A1
A 2 
A 1 

k1 
k3 

2M ( A 1) 
A2 

nCM kCM



2

A2 
A 1
 
(2)
   E  Ethr 
k1 
kCM  



2
m
(
A

1)
A

2


N


2
Can easily be
generalized to
relativistic
kinematics and
light cone
variables
Many-body validation of the
convolution formula from BBG theory
Chiara Benedetta Mezzetti
Seattle,05/11/2009
2 | k  k0 |
m
P (k , E ) 
32k
 1 2 1 2 1 2
dk kcm n (kcm )nrel 
k  kcm  k0 

4
2 
| k  k0 |
 2
FG
cm
Baldo, Borromeo,
Ciofi degli Atti,
Nucl. Phys. A604
(1996) 429-440
Brueckner –
Bethe Goldstone
Theory leads
explicitly to the
convolution
formula
Chiara Benedetta Mezzetti
Seattle,05/11/2009
Points: numerical
calculation of the spectral
functions of 3He (Ciofi degli Atti, Pace,
Salmè, PRC 21 (1980)805) and NM
(Benhar, Fabrocini, Fantoni, Nucl. Phys.
A550(1992)201)
Curves: 2N correlation model
A
A
P1A (k , E )   d 3kcm nrel
(| k  kcm / 2 |) ncm
(| kcm |)
2

( A  1)kcm  
( A  2) 
(2)
  E  Ethr 
 k 
 
2 M ( A  1) 
( A  2)  


Recently (Massimiliano’s talk)
A
cm
n (kcm )
k  1.5 fm
1
k  3.0 fm1
k  2.2 fm
1
k  3.5 fm1
A
nrel
(krel )
have been calculated by manybody approach
 no free parameters!!
The macroscopic origin of
the convolution formula:
wave function factorization
 3 He  x, y  5,  90 
 3 He  x, y '  6,  90 
*
 3 He  x, y  2,  90 
 3 He  x, y '  5,  90 
c
 3 He  x, y  3,  90 
 3 He  x, y '  4.5,  90 
3He
Ciofi, Simula, unpublished;
Alvioli, Ciofi, Kaptari,
Mezzetti, Scopetta, in
preparation
– MT/V’ (s-wave)
 3 He  x, y, 
 3 He  x, y ', 

from
Faddeev
equation
x
y

Large y, small x  Wave function factorizes!
17
Small y, small x  Wave function does not factorize!
3He
– MT/V’ (s-wave)
 3 He  x, y  0,  90 
 3 He  x, y '  0.5,  90 
c
 3 He  x, y  0.5,  90 
 3 He  x, y '  1.5,  90 
x
y
 3 He  x, y, 
 3 He  x, y ', 

18
Chiara Benedetta Mezzetti
Seattle,05/11/2009
nA (k )  C AnD (k )
Factorization leads to an
important consequence:
A tensor Deuteron-like
(T=0, S=1) dominance at large k,
both in light and complex nuclei
“At high momenta, the properties
of the nuclear wave function are
mainly determined by the
properties of a np pair, moving
Experimentally confirmed by Jlab data with soft CM nmd.”
Alvioli, Ciofi degli Atti, Morita, Phys. Rev. Lett 100 (2008) 162503
Chiara Benedetta Mezzetti
Seattle,05/11/2009
nA (k )
cA 
nD ( k )
A
3
CA
1.9
4
12
16
3.8
4
4.2
40
56
208
4.4
4.5
4.8
NM
4.9
20
Mezzetti
Probability of SM and SRC high Chiara Benedetta
Seattle,05/11/2009
momentum components
in
1and
2Alvioli, Ciofi,

Morita, to
2
body nmd S  S0  S1   dk k n(k )  1
appear

S0 

0
dk k 2 n0 (k )  0.8
0

S1 

0
dk k 2 n1 (k )  0.2

P0 
 dk k
2
n0 (k )
k*

P1 

dk k 2 n1 (k )
k*
k* [fm-1]
P0
P1
0.75
0.3
0.14
1.5
1.4 * 10-2
8.2 * 10-2
2
6.2 * 10-4
6.2 * 10-2
Chiara Benedetta Mezzetti
Seattle,05/11/2009
A new approach to the treatment of
inclusive cross sections
22
Mezzetti
D. Day,Chiara
AIPBenedetta
Conference
Proceedings,Seattle,05/11/2009
Vol. 1056
(2008)
FSI
Chiara Benedetta Mezzetti
Seattle,05/11/2009
k1
k
2
( q, )
k
k+q
Nuclear
Structure
Function

F A (q, )  2
d 2
 F A (q, ) K (q, )  Z ep  N en 


d d  '

  MA 
M
A1
E
 k
2
*
A1
2

 m  k q
2

2


dE
Emin
kdkP A (k , E )
kmin ( q, , E )

f (Y )  2  kdknA (k )
A
Let us introduce a
generic scaling variable
Y  Y (q, )
F A (q, )  F A (q, Y )  f A (Y )  B A (q, Y )
Binding
B A (q, Y )  2
correction
Longitudinal momentum
distribution

k min ( q ,Y , E )
Emin
|Y |
 dE
|Y |
n A (k )  c A n D (k )
A
k
dk
P
(k , E )
1

Our aim is to find Y
A
B
(q, Y )  0
such that
24
Chiara Benedetta Mezzetti
Seattle,05/11/2009
Yy
  M A  (M A1  E*A1)2  y 2  m2   y  q 
2
Minimum longitudinal momentum of a nucleon
having the minimum value of the removal
energy E = Emin
B A (q, y)  0
FexA (q, y) 
F A (q, y)  f A ( y)
 2,Aex
[ Z ep  N en ]K
F A (q, y)  f D ( y)
25
Chiara Benedetta Mezzetti
Seattle,05/11/2009
Y  yCW
  MA 

Minimum longitudinal momentum of a nucleon
with removal energy E=Emin+< E∗A−1 (k)>2NC
M A1  E*A1( yCW ) 2 NC

2


2
2
2
 yCW  m  yCW  q
 E A* 1 (k )  2 NC

A
*
*
*
P
(
k
,
E
)
E
dE
A1
A1
A1
1
n A (k )
E
*
A1
(k ) 2 NC 
A2
TN  bA  cA k
A 1
26
Chiara Benedetta Mezzetti
Seattle,05/11/2009
C. Ciofi degli Atti,
C.B. Mezzetti,
Phys. Rev. C79,
051392(R), (2009)
A
B (q, yCW ) 0
A
A
F ( q, y
)  f (y
)
CW
CW
A
1 dF (q, yCW )
A
n (k )  
, k | yCW |
2 yCW
dyCW
27
Chiara Benedetta Mezzetti
Seattle,05/11/2009
C. Ciofi degli
Atti, G.B. West,
PLB 458 (1999)
447;
C. Ciofi degli
Atti, C.B.
Mezzetti, Phys.
Rev. C79,
051392(R),
(2009)
Confirmation of the
theoretically prediction
of
Deuteron scaling
F A (q, yCW )
C A f D ( yCW )
n A (k ) C An D (k )
28
Chiara Benedetta Mezzetti
Seattle,05/11/2009
 Deuteron scaling
again
demonstrated
 CA in agreement
with Frankfurt,
Strikman, Day,
Sargsyan,PRC 48
(1993) 2451
 FSI important
but similar in
Deuteron and in A
 In the SRC
region FSI acts
mainly within the
correlated pair
29
Chiara Benedetta Mezzetti
Seattle,05/11/2009
P2 NC (k , E )  P3 NC (k , E )
High k, high E
Under
investigation
High k, low E


k
2
k2
k
k3
( k 2  k3 ) 2
E
mN
k Calculated
2
Present
A(e,e’)X
k
kinematics
at 2<xB<3
sensitive to high k, low E
3NC configuration
30
Chiara Benedetta Mezzetti
Seattle,05/11/2009
Y  y3
  MA 

M A1  E*A1( y3 ) 3NC

2
 y32  m2   y3  q 
Minimum longitudinal
momentum of a nucleon
with removal energy
E=Emin+< E∗A−1 (k)>3NC
2
A

3
k
*
 EA1 (k ) 3 NC 
A  1 4m

2
k
2
k

k
2
31
Chiara Benedetta Mezzetti
Seattle,05/11/2009
  MA 

M A1  E*A1

2
 Y 2  m 2  Y  q 
2
E A* 1  0
xBj  A
E A* 1  E A* 1 (k )  2 NC
xBj  2
xBj  A
E A* 1  E A* 1 (k ) 3 NC
xBj  3
32
Chiara Benedetta Mezzetti
Seattle,05/11/2009
d 

d d
2
kmax ( q, , E )
Emax

Emin


| y ( q , )|

dE
kdkP A (k , E ) 
kmin ( q, , E )

n0A (k )kdk 
n2A (k )kdk 
| yCW ( q , )|
Mean
Field
n2 (k ) 



2NC

n3A (k )kdk
| y3 ( q , )|
3NC

soft
dk
n
k

k
n
CM
CM (kCM )
 CM rel
n3 (k ) 
hard
dk
n
(
k

k
)
n
CM
CM ( kCM )
 CM rel
33
Chiara Benedetta Mezzetti
Seattle,05/11/2009
Calculation of the inclusive plateaux
34
Chiara Benedetta Mezzetti
Seattle,05/11/2009
SRCs vs. xB
PWIA results
r ( A, He) 
3
A  2 ep   en 
3  Z ep  N en 
R( A,3 He)
Chiara Benedetta Mezzetti
Seattle,05/11/2009
Distorted nucleon momentum
Alvioli, Ciofi, Kaptari, Mezzetti,
distributions Morita,
Palli, Scopetta, to be published
Final results
C.B. Mezzetti,
C. Ciofi degli
Atti,
ArXiv:
0906.5564
(2009)
Mean Field
2NC
1.5  xB  2
Chiara Benedetta Mezzetti
Seattle,05/11/2009
3NC
2  xB  3
In
progress
xB  1.5
37
Searching for n3(k)
Chiara Benedetta Mezzetti
Seattle,05/11/2009
What we have
found:
n3 (k )  n2 (k )
We expect that
n3 (k )   dEP3 NC (k , E )
A
3
n (k )
A
3
He
C n3 (k )
Explanation of the
second plateaux at
2  xB  3
38
Conclusions
Chiara Benedetta Mezzetti
Seattle,05/11/2009
 Proper scaling variables have been introduced which
include E*A-1 of the residual system and allows to describe
A(e,e’)X cross section in terms of momentum distributions
of 2N and 3N SRC.
 The experimental scaling function in the 2NC region
scales to the Deuteron scaling function and exhibits
A-independent FSI effects mostly due to the FSI in the
correlated pair.
The experimental ratio r(A/3He) in the 2NC region has
been successfully reproduced. Calculations in the 3NC
region are in progress, using the proper scaling variable and
modelling the 3NC momentum distributions.
 12 GeV upgrade at Jlab: higher statistics in the 3NC
region?
39