Relativistic Models of Quasielastic Electron and Neutrino-Nucleus Scattering Carlotta Giusti

Relativistic Models of Quasielastic Electron and Neutrino-Nucleus

Scattering

Carlotta GiustiUniversity 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

both e’ and N detected one-nucleon-knockout (e,e’p)

(A-1) is a discrete eigenstate n exclusive (e,e’p)


both e’ and N detected one-nucleon-knockout (e,e’p)


only e’ detected inclusive (e,e’)


NC

CC

both e’ and N detected one-nucleon knockout (e,e’p)




QE -nucleus scattering

only N detected semi-inclusive NC and CC

NC

CC






only N detected semi-inclusive NC and CC

only final lepton detected inclusive CC

NC

CC






electron electron scatteringscattering

NCNC

neutrino neutrino scatteringscattering

CCCC

PVESPVES


NCNC


CCCC

PVESPVES

kinkin factorfactor


NCNC


CCCC

PVESPVES

lepton tensor contains lepton lepton tensor contains lepton

kinematicskinematics


NCNC


CCCC

PVESPVES

hadron hadron tensortensor

e’

eq

p n

0

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

|i >

|f >

e’

eq

p n

0

Direct knockout DWIA (e,e’p)

|i >

|f >

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

rel RDWIA

nonrel DWIA

RDWIA diff opt.pot.

RDWIA: (e,e’p) comparison to data

NIKHEF parallel kin e=520 MeV Tp = 90 MeV

16O(e,e’p)

n = 0.7

n =0.65

JLab (,q) const kin e=2445 MeV =439 MeV Tp= 435 MeV

n = 0.7

k’

kq

N n

0

RDWIA: NC and CC –nucleus scattering

|i >

|f >

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

One-body nuclear weak current

NC

K anomalous part of the magnetic moment

CC

induced pseudoscalar form factor

The axial form factor

possible strange-quark contribution

CC

NC



The weak isovector Dirac and Pauli FF are related to the Dirac and Pauli elm FF by the CVC hypothesis

strange FF

CC

NC

–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

pure Shell Model description: n one-hole states in the target with an unitary spectral strengthn over all occupied states in the SM: all the nucleons are included but correlations are neglectedthe cross section for the -nucleus scattering where one nucleon is detected is obtained from the sum of all the integrated one-nucleon knockout channelsFSI are described by a complex optical potential with an imaginary absorptive part

calculationscalculations

RPWIARPWIA

RDWIARDWIA

RPWIARPWIA

RDWIARDWIA

CC CC

NCNC

FSIFSI

FSIFSI

the imaginary part of the the imaginary part of the optical potential gives an optical potential gives an absorption that reduces the absorption that reduces the calculated cross sectionscalculated cross sections

FSIFSI

FSI for the inclusive scattering : Green’s Function Approach

(e,e’) nonrelativistic(e,e’) nonrelativistic

F. Capuzzi, C. Giusti, F.D. Pacati, Nucl. Phys. A 524 (1991) 281F. 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) 492F. Capuzzi, C. Giusti, F.D. Pacati, D.N. Kadrev Ann. Phys. 317 (2005) 492 (AS CORR)(AS CORR)

(e,e’) relativistic(e,e’) relativistic

A.A. MeucciMeucci, , F. Capuzzi, C. Giusti, F.D. Pacati, PRC (2003) 67 054601 F. Capuzzi, C. Giusti, F.D. Pacati, PRC (2003) 67 054601

A. Meucci, C. Giusti, F.D. Pacati Nucl. Phys. A 756 (2005) 359A. Meucci, C. Giusti, F.D. Pacati Nucl. Phys. A 756 (2005) 359 (PVES)(PVES)

CC relativisticCC relativistic

A. Meucci, C. Giusti, F.D. Pacati Nucl. Phys. A739 (2004) 277A. Meucci, C. Giusti, F.D. Pacati Nucl. Phys. A739 (2004) 277


the components of the inclusive response are expressed in terms of thethe components of the inclusive response are expressed in terms of the Green’s operatorsGreen’s operators

under suitable approximations can be written in terms of theunder suitable approximations can be written in terms of the s.p. optical s.p. optical model Green’s function model Green’s function

the explicit calculation of the s.p. Green’s function can be avoided by its 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 spectral representation which is based on a biorthogonal expansion in terms of aterms of a non Herm optical potential V and Vnon Herm optical potential V and V++

matrix elements similar to RDWIA matrix elements similar to RDWIA

scattering states eigenfunctions of V and Vscattering states eigenfunctions of V and V++ (absorption and gain of (absorption and gain of flux): flux): the imaginary part redistributes the flux and the total flux is the imaginary part redistributes the flux and the total flux is conservedconserved

consistent treatmentconsistent treatment of FSI in the exclusive and in the inclusive of FSI in the exclusive and in the inclusive scatteringscattering


interference between different channels


eigenfunctions of V and V+


Flux redistributed and conserved

The imaginary part of the optical potential is responsible for the redistribution of the flux among the different channels

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)

gain of flux loss of flux

16O(e,e’)

data from Frascati NPA 602 405 (1996)

Green’s function approach GF

A. Meucci, F. Capuzzi, C. Giusti, F.D. Pacati, PRC 67 (2003) 054601

16O(e,e’)

data from Frascati NPA 602 405 (1996)

e = 1080 MeV

=320

e = 1200 MeV

=320

RPWIARPWIA

GFGF

1NKO1NKO

FSI FSI

RPWIARPWIA

GFGF

rROPrROP

1NKO1NKO

GFGF

FSI FSI FSI FSI

PAVIA MADRID-SEVILLA

COMPARISON OF RELATIVISTIC MODELS

consistency of numerical results

comparison of different descriptions of FSI

12C(e,e’)

RPWIARPWIA

rROPrROP

GF1GF1

GF2GF2

RMFRMF

e = 1 GeV

relativistic models

FSI FSI

Relativistic Mean Field: same real energy-independent potential of bound states

Orthogonalization, fulfills dispersion relations and maintains the continuity equation

12C(e,e’)

GF1GF1

GF2GF2

RMFRMF

e = 1 GeV

relativistic models

q=500 MeV/cq=500 MeV/c



FSI FSI

12C(e,e’)

GF1GF1

GF2GF2

RMFRMF

relativistic models

GF RMF

SCALING PROPERTIES

SCALING FUNCTION

Scaling properties of the electron scattering data Scaling properties of the electron scattering data

At sufficiently high q the At sufficiently high q the scaling functionscaling function

depends only upon one kinematical variable (scaling variable) depends only upon one kinematical variable (scaling variable)

(SCALING OF I KIND)(SCALING OF I KIND)

is the same for all nucleiis the same for all nuclei

(SCALING OF II KIND)(SCALING OF II KIND)

I+II SUPERSCALING I+II SUPERSCALING

Scaling variable (QE)Scaling variable (QE)

+ (-) for + (-) for lower (higher) than the QEP, where lower (higher) than the QEP, where =0 =0

Reasonable scaling of I kind at the left of QEPReasonable scaling of I kind at the left of QEP

Excellent scaling of II kind in the same regionExcellent scaling of II kind in the same region

Breaking of scaling particularly of I kind at the Breaking of scaling particularly of I kind at the right of QEP (effects beyond IA) right of QEP (effects beyond IA)

The longitudinal contribution superscalesThe longitudinal contribution superscales

ffQEQE extracted from the data extracted from the data

Experimental QE superscaling functionExperimental 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

The properties of the experimental scaling function should be The properties of the experimental scaling function should be accounted for by microscopic calculationsaccounted for by microscopic calculations

The asymmetric shape of fThe asymmetric shape of fQEQE should be explained should be explained

The scaling properties of different models can be verified The scaling properties of different models can be verified

The associated scaling functions compared with the experimental The associated scaling functions compared with the experimental ffQEQE

SCALING FUNCTION

QE SUPERSCALING FUNCTION: RFGQE SUPERSCALING FUNCTION: RFG

Relativistic Fermi Gas

J.A. Caballero J.E. Amaro, M.B. Barbaro, T.W. Donnelly, C. Maieron, and J.M.

Udias PRL 95 (2005) 252502

only RMF gives only RMF gives an asymmetric an asymmetric

shapeshape

QE SUPERSCALING FUNCTION: RPWIA, rROP, RMFQE SUPERSCALING FUNCTION: RPWIA, rROP, RMF

QE SCALING FUNCTION: GF, RMFQE SCALING FUNCTION: GF, RMF

GF1GF1

GF2GF2

RMFRMF

asymmetric asymmetric shapeshape




Analysis first-kind scaling : GF RMFAnalysis first-kind scaling : GF RMF

GF1GF1

GF2GF2

RMFRMF



q=1000 q=1000 MeV/cMeV/c

RMF GF

DIFFERENT DESCRIPTIONS OF FSI

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

RESULTS :RESULTS :

GF1, GF2, RMF similar results for the cross section and QE scaling GF1, GF2, RMF similar results for the cross section and QE scaling function for q=500-700 MeV/c, differences increase increasing qfunction for q=500-700 MeV/c, differences increase increasing q

WHY ?WHY ?

At higher q and energies the phenomenological OP can include At higher q and energies the phenomenological OP can include contributions from non nucleonic d.o.f. (flux lost into inelastic contributions from non nucleonic d.o.f. (flux lost into inelastic excitation with or without real excitation with or without real production) production)

GF can give better description of (e,e’) experimental c.s. at higher GF can give better description of (e,e’) experimental c.s. at higher q, where QE and q, where QE and peak approach and overlap peak approach and overlap

Non nucleonic contributions in the OP break scaling, they should Non nucleonic contributions in the OP break scaling, they should not be included in the QE longitudinal scaling function (purely not be included in the QE longitudinal scaling function (purely nucleonic) nucleonic)

COMPARISON RMF-GF

FUTURE WORK…..

deeper understanding of the differences deeper understanding of the differences

disentangle nucleonic and non nucleonic contributions in the OP, disentangle nucleonic and non nucleonic contributions in the OP, extract purely nucleonic OP, use it in the GF approach and extract purely nucleonic OP, use it in the GF approach and compare the results with the QE experimental scaling functioncompare the results with the QE experimental scaling function

extend the comparison to different situationsextend the comparison to different situations, neutrino scattering…

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Relativistic Models of Quasielastic Electron and Neutrino-Nucleus Scattering Carlotta Giusti