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

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

QE e-nucleus scattering

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)

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 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

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

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

electron scattering PVES neutrino scattering NC CC

electron scattering PVES neutrino scattering NC CC kin factor

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

electron scattering PVES neutrino scattering NC CC hadron tensor

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

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

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

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

One-body nuclear weak current NC K anomalous part of the magnetic moment

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

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

 –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

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

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

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) 67 054601 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

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

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

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

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

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)

Green’s function approach GF 16O(e,e’) Green’s function approach GF data from Frascati NPA 602 405 (1996) A. Meucci, F. Capuzzi, C. Giusti, F.D. Pacati, PRC 67 (2003) 054601

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 602 405 (1996)

FSI FSI RPWIA GF rROP 1NKO GF

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

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

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

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

SCALING PROPERTIES GF RMF

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

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

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

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

QE SUPERSCALING FUNCTION: RFG Relativistic Fermi Gas

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) 252502

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

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

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

COMPARISON RMF-GF RESULTS : GF1, GF2, RMF similar results for the cross section and QE scaling function for q=500-700 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)

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…