International Workshop on Fundamental Symmetries: From Nuclei and Neutrinos to the Universe

ECT*, Trento, 25-29. June 2007

Charged-Current Neutrino-Nucleus Reactionsin the Relativistic QRPA

Charged-Current Neutrino-Nucleus Reactionsin the Relativistic QRPA

N. PaarN. Paar

Physics DepartmentPhysics Department

Faculty of ScienceFaculty of Science

University of ZagrebUniversity of Zagreb

CroatiaCroatia

www.phy.hr/~npaarwww.phy.hr/~npaar

Aνe,e-)B A(νμ,μ-)B

neutrino nucleus charged current reactions

neutrino-nucleus cross section

NEUTRINO-NUCLEUS REACTIONS

Hamiltonian for the weak interaction

transition nuclear matrix elements (assuming plane waves for leptons)

NEUTRINO-NUCLEUS CROSS SECTION

Coulomb & longitudinal

multipole operators

Transverse magnetic & Transverse electric multipole operators

Extreme relativistic limit : final lepton energy El >> mlc2

Walecka, Donnely

multipole operators are given in terms of spherical Bessel functions, spherical harmonics, and vector spherical harmonics

→ Coulomb multipole operator

→Longitudinal multipole operator

→ transverse electric multipole operator

→ transverse magnetic multipole operator

→ nucleon form factors

→ 7 fundamental operators:

Reduced matrix elements for various transition operators include the details of the nuclear structure (nuclear groundstate and excitations)

Recent microscopic studies improved significantly description of weak interaction rates, however, there are several problems:

1. SHELL MODEL → accurate in description of the ground state wave functions, description of high-lying states necessitates a large modelspace which is problematic to treat numerically

Different interactions in various mass regions employed, only lower massnuclei can be studied

NEUTRINO-NUCLEUS REACTIONS IN OPEN-SHELL NUCLEI

2. (Continuum) Random Phase Approximation → includes high lying particle-hole configurations, but configuration mixing was included by putting by hand the fractional occupancies of single-particle states

GROUND STATE:

Woods-Saxon potential fitted to single-particle energies

RPA RESIDUAL INTERACTION:G-matrix from Bonn potential orzero-range Landau Migdal force

PARTIAL OCCUPATION OF ORBITS (12C) determined in shell model (e.g. OXBASH with Cohen Kurath interaction)

Kolbe, Langanke, Vogel, Nucl. Phys. A 652, 91 (1999).

+ adjustment factor to rescale the neutrino cross section

CONTINUUM RPA → forbidden transitions

3. HYBRID MODEL

RPA RESIDUAL INTERACTION:G-matrix from Bonn potential orzero-range Landau Migdal force

SHELL MODEL – Strasbourg-Madrid codes by Caurier et al.→Gamow-Teller strength (with quenching factor)( + rescaled by a factor to account finite momentum transfer)

Kolbe, Langanke, Martinez-Pinedo, Phys. Rev. C 60, 052801 (1999).

GROUND STATE:

Woods-Saxon potential fitted to single-particle energies

Occupation numbers

4. Skyrme Hartree-Fock + Quasiparticle RPA → towards a consistent description of weak interaction rates along the nuclide chart

GROUND STATE: based on Skyrme energy density functional, BCS pairing

RPA RESIDUAL INTERACTION:Derived from the same Skyrme functional, renormalized strength in the pairing channel (0.7)

Volpe, Auerbach, Colò, Suzuki, Van Giai, Phys. Rev. C 62, 015501

Surman, Engel, Phys. Rev. C 58, 2526 - 2530 (1998)

QRPA for neutrino capture by r-process waiting-point nuclei (includes Gamow-Teller and first forbidden transitions)

Different transtion operators are employed in different approaches

RELATIVISTIC HARTREE-BOGOLIUBOV MODEL (RHB)+ RELATIVISTIC QUASIPARTICLE RPA (RQRPA)

BOTH THE GROUND STATE EQUATIONS & RESIDUAL QRPA INTERACTION ARE DERIVED FROM THE SAME EFFECTIVE LAGRANGIAN DENSITY

PAIRING CORRELATIONS ARE DESCRIBED BY THE FINITE RANGE GOGNY FORCE

NEW → Neutrino-nucleus cross sections based on self-consistent Relativistic Quasiparticle Random Phase Approximation (RQRPA)

RHB → Vretenar, Afanasjev, Lalazissis, Ring, Phys. Rep. 409, 101 (2005).

RQRPA → Paar, Ring, Niksic, Vretenar, Phys. Rev. C 67, 034312 (2003).

system of Dirac nucleons coupled to the exchange mesons and the photon field through an effective Lagrangian.

(J,T)=(0+,0) (J,T)=(1-,0) (J,T)=(1-,1)

Sigma-meson: attractivescalar field

Omega-meson: short-range repulsive field

Rho-meson:isovector field

RELATIVISTIC MEAN FIELD THEORY

the Lagrangian of the free nucleon:

the Lagrangian of the free meson fields and the electromagnetic field:

minimal set of interaction terms:

with the vertices:

LAGRANGIAN DENSITY

The parameters are determined from properties of nuclear matter and bulk properties of finite nuclei (binding energies, charge radii, neutron radii, surface thickeness ...)

the meson-nucleon couplings g, g, g -> functions of vector density:

MODELS WITH DENSITY DEPENDENT COUPLINGS

Density-dependent meson-nucleon couplings – connection to Dirac-Brueckner calculations based on realistic NN interactions

RMS CHARGE RADIIRMS CHARGE RADII

DIFFERENCES rn-rp

DIFFERENCES rn-rp

SnSn SnSn

RELATIVISTIC HARTREE-BOGOLIUBOV MODEL (RHB)Unified description of mean-field and pairing correlations

Dirac hamiltonian

Pairing field

BINDING ENERGIESBINDING ENERGIES

σRMS=0.9 MeV

Accurate description of ground state properties along nuclide chart

Paar, Niksic, Vretenar, Ring, Phys. Rev. C 69, 1 (2004)

π and ρ-meson exchange generate the spin-isospin dependent interaction terms

The ρ-nucleon and π-nucleon interaction Lagrangian ( )

plus the Landau-Migdal zero-range force in the spin-isospin channel

PN-RQRPAPN-RQRPA

f2 4 0.08m 138 MeV

PROTON-NEUTRON RELATIVISTIC QUASI-PARTICLE RPA (RQRPA)

Derived from the time-dependent RHB equation in the limit of small oscillations around the RHB ground state generalized density

RQRPA DESCRIPTION OF COLLECTIVE EXCITATIONS IN NUCLEI

Paar, Vretenar, Ring, Phys. Rev. Lett. 94, 182501 (2005).

Paar, Papakonstantinou, Ponomarev, Wambach, Phys. Lett. B 624, 195 (2005).

Paar, Nikšić Vretenar, Ring, Phys. Lett. B 606, 288 (2005).

Paar, Vretenar, Nikšić, Ring, Phys. Rev. C 74, 037303 (2006).

208Pb

Exp. 7.37 MeV

Exp. 13.3 MeV

MONOPOLE EXCITATIONSDIPOLE EXCITATIONS

EXOTIC MODES IN NUCLEI AWAY FROM THE VALLEY OF STABILITY

PYGMY DIPOLE RESONANCE

EXOTIC MODES IN NUCLEI AWAY FROM THE VALLEY OF STABILITY

REVIEW PAPER:

Paar, Vretenar, Khan, Colo"Exotic modes of excitation in atomic nuclei far from stability", Rep. Prog. Phys. 70, 691 (2007).

S=0 T=1 J = 0+

ISOSPIN-FLIP EXCITATIONS

S=1 T=1 J = 1+

SPIN-FLIP & ISOSPIN-FLIP EXCITATIONS

SPIN-FLIP & ISOSPIN-FLIP EXCITATIONS

ISOBARIC ANALOG AND GAMOW-TELLER RESONANCES (RQRPA)

→ single-particle reduced matrix elements

Back to reduced matrix elements for the neutrino-nucleuscross section →

RHB+RQRPA NEUTRINO-NUCLEUS CROSS SECTIONS

RQRPA amplitudes

RHB occupation probabilities

Coulomb correction from numerical solution of the Dirac equation for an extended nuclear charge

Effective momentumapproximation

RHB+RQRPA NEUTRINO-NUCLEUS (12C) CROSS SECTION

Engel, Phys. Rev. C 57, 2004 (1998)

RHB+RQRPA

RHB+RQRPA NEUTRINO-NUCLEUS (208Pb) CROSS SECTION

RHB+RQRPA

RHB+RQRPA

RHB+RQRPA

DISTRIBUTION OF CROSS SECTIONS OVER MULTIPOLARITIES

RHB+RQRPA

→talk Lazauskas

CROSSING BETWEEN THE CROSS SECTIONS WITH ALLOWED AND FORBIDDEN TRANSITIONS !

CONTRIBUTIONS FROM ALLOWED AND FORBIDDEN TRANSITIONS

RHB+RQRPA

(νμ,μ-)

<σ>(10-40cm2)

(νe,μ-)

<σ>(10-42cm2)

RPA(Volpe et al.) 19.23 55.10

QRPA(Volpe et al.) 20.29 52.0

CRPA(Kolbe) 18.18 19.28

PN-RQRPA 19.59 12.14

EXP.(LSND) 12±0.3±1.8 14.1±1.6

EXP.(KARMEN) 14.0±1.2

Flux averaged cross section

RHB+RQRPA CROSS SECTION AVERAGED OVER NEUTRINO FLUX

Supernova neutrino flux isgiven by Fermi-Dirac spectrum

Cross section averaged over Supernova neutrino flux

CROSS SECTIONS AVERAGED OVER SUPERNOVA NEUTRINO FLUX

CROSS SECTIONS AVERAGED OVER SUPERNOVA NEUTRINO FLUX

Relativistic framework for nuclear charge-exchange excitations provides consistent description of neutrino-nucleus charged current reactions

NEUTRINO-NUCLEUS CROSS SECTIONS FOR NEUTRINO DETECTORS, BETA BEAMS…

CONSISTENT DESCRIPTION OF NEUTRINO-NUCLEUS REACTIONS AT THE LIMITS OF STABILITY:

Relevance for nuclear astrophysics

NEUTRINO NUCLEOSYNTHESIS

CHARGED CURRENT NEUTRINO-NUCLEUS CROSS SECTIONS IN RQRPA

The same effective interactions can be employed in a systematic way to evaluate reduced transition matrix elements, for nuclei throughout the nuclide chart

Better knowledge on spin-dipole and excitations of higher multipolarities can improve description of neutrino-nucleus reactions

T. MarketinT. NikšićP. RingD. Vretenar

COLLABORATORS

www.phy.hr/~npaar