1. Relativistic Models of Quasielastic Electron and Neutrino-Nucleus Scattering
Carlotta Giusti
University and INFN Pavia
in collaboration with: A. Meucci (Pavia)
F.D. Pacati (Pavia)
J.A. Caballero (Sevilla)
J.M. Udías (Madrid)
XXVIII International Workshop in Nuclear Theory, Rila Mountains June 21st-27th 2009

2. nuclear response to the electroweak probe
QE-peak
QE-peak dominated by one-nucleon knockout

3. QE e-nucleus scattering

4. QE e-nucleus scattering
both e’ and N detected one-nucleon-knockout (e,e’p)
(A-1) is a discrete eigenstate n exclusive (e,e’p)

5. QE e-nucleus scattering
both e’ and N detected one-nucleon-knockout (e,e’p)
(A-1) is a discrete eigenstate n exclusive (e,e’p)
only e’ detected inclusive (e,e’)

6. QE e-nucleus scattering
both e’ and N detected one-nucleon knockout (e,e’p)
(A-1) is a discrete eigenstate n exclusive (e,e’p)
only e’ detected inclusive (e,e’)
QE -nucleus scattering NC CC

7. QE e-nucleus scattering
both e’ and N detected one-nucleon knockout (e,e’p)
(A-1) is a discrete eigenstate n exclusive (e,e’p)
only e’ detected inclusive (e,e’)
QE -nucleus scattering NC CC
only N detected semi-inclusive NC and CC

8. QE e-nucleus scattering
both e’ and N detected one-nucleon knockout (e,e’p)
(A-1) is a discrete eigenstate n exclusive (e,e’p)
only e’ detected inclusive (e,e’)
QE -nucleus scattering NC CC
only N detected semi-inclusive NC and CC
only final lepton detected inclusive CC

9. electron scattering
PVES
neutrino scattering NC CC

10. electron scattering
PVES
neutrino scattering NC CC
kin factor

11. lepton tensor contains lepton kinematics
electron scattering
PVES
neutrino scattering NC CC
lepton tensor contains lepton kinematics

12. electron scattering
PVES
neutrino scattering NC CC
hadron tensor

13. Direct knockout DWIA (e,e’p)
exclusive reaction: n
DKO mechanism: the probe interacts through a one-body current with one nucleon which is then emitted the remaining nucleons are spectators
|f > e’
,q e
|i > 0

14. Direct knockout DWIA (e,e’p)
|f > e’
j one-body nuclear current
(-) = < n|f > s.p. scattering w.f. H+(+Em)
n = <n|0> one-nucleon overlap H(-Em)
n spectroscopic factor
(-) and  consistently derived as eigenfunctions of a Feshbach-type optical model Hamiltonian
phenomenological ingredients used in the calculations for (-) and 
,q e
|i > 0

15. RDWIA: (e,e’p) comparison to data
16O(e,e’p)
NIKHEF parallel kin e=520 MeV Tp = 90 MeV
n = 0.7
n =0.65
rel RDWIA
nonrel DWIA
JLab (,q) const kin e=2445 MeV  =439 MeV Tp= 435 MeV
RDWIA diff opt.pot.
n = 0.7

16. RDWIA: NC and CC  –nucleus scattering
|f > k’
transition amplitudes calculated with the same model used for (e,e’p)
the same phenomenological ingredients are used for (-) and 
j one-body nuclear weak current
,q k
|i > 0

17. One-body nuclear weak currentNC
K anomalous part of the magnetic moment

18. One-body nuclear weak currentCC
induced pseudoscalar form factor CC
The axial form factor NC
possible strange-quark contribution

19. One-body nuclear weak current
The weak isovector Dirac and Pauli FF are related to the Dirac and Pauli elm FF by the CVC hypothesis CC NC
strange FF

20.  –nucleus scattering
only the outgoing nucleon is detected: semi inclusive process
cross section integrated over the energy and angle of the outgoing lepton
integration over the angle of the outgoing nucleon
final nuclear state is not determined : sum over n

21. calculations
pure Shell Model description: n one-hole states in the target with an unitary spectral strength
n over all occupied states in the SM: all the nucleons are included but correlations are neglected
the cross section for the -nucleus scattering where one nucleon is detected is obtained from the sum of all the integrated one-nucleon knockout channels
FSI are described by a complex optical potential with an imaginary absorptive part

22. CC NC
FSI
the imaginary part of the optical potential gives an absorption that reduces the calculated cross sections
RPWIA
RDWIA
FSI
FSI

23. FSI for the inclusive scattering : Green’s Function Approach
(e,e’) nonrelativistic
F. Capuzzi, C. Giusti, F.D. Pacati, Nucl. Phys. A 524 (1991) 281
F. Capuzzi, C. Giusti, F.D. Pacati, D.N. Kadrev Ann. Phys. 317 (2005) 492 (AS CORR)
(e,e’) relativistic
Meucci, F. Capuzzi, C. Giusti, F.D. Pacati, PRC (2003)
A. Meucci, C. Giusti, F.D. Pacati Nucl. Phys. A 756 (2005) 359 (PVES)
CC relativistic
A. Meucci, C. Giusti, F.D. Pacati Nucl. Phys. A739 (2004) 277

24. FSI for the inclusive scattering : Green’s Function Approach
the components of the inclusive response are expressed in terms of the Green’s operators
under suitable approximations can be written in terms of the s.p. optical model Green’s function
the explicit calculation of the s.p. Green’s function can be avoided by its spectral representation which is based on a biorthogonal expansion in terms of a non Herm optical potential V and V+
matrix elements similar to RDWIA
scattering states eigenfunctions of V and V+ (absorption and gain of flux): the imaginary part redistributes the flux and the total flux is conserved
consistent treatment of FSI in the exclusive and in the inclusive scattering

25. FSI for the inclusive scattering : Green’s Function Approach
interference between different channels

26. FSI for the inclusive scattering : Green’s Function Approach
eigenfunctions of V and V+

27. FSI for the inclusive scattering : Green’s Function Approach
gain of flux
loss of flux
Flux redistributed and conserved
The imaginary part of the optical potential is responsible for the redistribution of the flux among the different channels

28. FSI for the inclusive scattering : Green’s Function Approach
gain of flux
loss of flux
For a real optical potential V=V+ the second term vanishes and the nuclear response is given by the sum of all the integrated one-nucleon knockout processes (without absorption)

29. Green’s function approach GF
16O(e,e’)
Green’s function approach GF
data from Frascati NPA (1996)
A. Meucci, F. Capuzzi, C. Giusti, F.D. Pacati, PRC 67 (2003)

30. data from Frascati NPA 602 405 (1996)
16O(e,e’)
e = 1080 MeV
 =320
RPWIA GF
1NKO
FSI
e = 1200 MeV
 =320
data from Frascati NPA (1996)

31. FSI
FSI
RPWIA GF
rROP
1NKO GF

32. COMPARISON OF RELATIVISTIC MODELS
PAVIA MADRID-SEVILLA
consistency of numerical results
comparison of different descriptions of FSI

33. 12C(e,e’) relativistic models FSI RPWIA rROP GF1 GF2 RMF
e = 1 GeV
FSI
RPWIA
rROP
GF1
GF2
RMF
Relativistic Mean Field: same real energy-independent potential of bound states
Orthogonalization, fulfills dispersion relations and maintains the continuity equation

34. 12C(e,e’) relativistic models FSI GF1 GF2 RMF e = 1 GeV q=500 MeV/c

35. 12C(e,e’)
relativistic models
GF1
GF2
RMF

36. SCALING PROPERTIES
GF RMF

37. SCALING FUNCTION Scaling properties of the electron scattering data
At sufficiently high q the scaling function
depends only upon one kinematical variable (scaling variable)
(SCALING OF I KIND)
is the same for all nuclei
(SCALING OF II KIND)
I+II SUPERSCALING

38. fQE extracted from the data
Scaling variable (QE)
+ (-) for  lower (higher) than the QEP, where =0
Reasonable scaling of I kind at the left of QEP
Excellent scaling of II kind in the same region
Breaking of scaling particularly of I kind at the right of QEP (effects beyond IA)
The longitudinal contribution superscales
fQE extracted from the data

39. Experimental QE superscaling function
M.B. Barbaro, J.E. Amaro, J.A. Caballero, T.W. Donnelly, A. Molinari, and I. Sick, Nucl. Phys Proc. Suppl 155 (2006) 257

40. SCALING FUNCTION
The properties of the experimental scaling function should be accounted for by microscopic calculations
The asymmetric shape of fQE should be explained
The scaling properties of different models can be verified
The associated scaling functions compared with the experimental fQE

41. QE SUPERSCALING FUNCTION: RFG
Relativistic Fermi Gas

42. QE SUPERSCALING FUNCTION: RPWIA, rROP, RMF
only RMF gives an asymmetric shape
J.A. Caballero J.E. Amaro, M.B. Barbaro, T.W. Donnelly, C. Maieron, and J.M. Udias PRL 95 (2005)

43. QE SCALING FUNCTION: GF, RMF
q=500 MeV/c
q=800 MeV/c
q=1000 MeV/c
GF1
GF2
RMF
asymmetric shape

44. Analysis first-kind scaling : GF RMF
q=500 MeV/c
q=800 MeV/c
q=1000 MeV/c

45. DIFFERENT DESCRIPTIONS OF FSI
RMF GF
real energy-independent MF reproduces nuclear saturation properties, purely nucleonic contribution, no information from scattering reactions explicitly incorporated
complex energy-dependent phenomenological OP fitted to elastic p-A scattering information from scattering reactions
the imaginary part includes the overall effect of inelastic channels not univocally determined from elastic phenomenology
different OP reproduce elastic p-A scattering but can give different predictions for non elastic observables

46. COMPARISON RMF-GF RESULTS :
GF1, GF2, RMF similar results for the cross section and QE scaling function for q= MeV/c, differences increase increasing q
WHY ?
At higher q and energies the phenomenological OP can include contributions from non nucleonic d.o.f. (flux lost into inelastic  excitation with or without real  production)
GF can give better description of (e,e’) experimental c.s. at higher q, where QE and  peak approach and overlap
Non nucleonic contributions in the OP break scaling, they should not be included in the QE longitudinal scaling function (purely nucleonic)

47. FUTURE WORK….. deeper understanding of the differences
disentangle nucleonic and non nucleonic contributions in the OP, extract purely nucleonic OP, use it in the GF approach and compare the results with the QE experimental scaling function
extend the comparison to different situations, neutrino scattering…