International Workshop on Fundamental Symmetries: From Nuclei and Neutrinos to the Universe ECT*, Trento, June 2007 Charged-Current Neutrino-Nucleus Reactions in the Relativistic QRPA Charged-Current Neutrino-Nucleus Reactions in the Relativistic QRPA N. Paar Physics Department Faculty of Science University of Zagreb Croatia Croatiawww.phy.hr/~npaar

A  ν e,e - )B A(ν μ,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 E l >> m l c 2 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 ground state 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 model space which is problematic to treat numerically Different interactions in various mass regions employed, only lower mass nuclei 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 or zero-range Landau Migdal force PARTIAL OCCUPATION OF ORBITS ( 12 C) 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 or zero-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, (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, Surman, Engel, Phys. Rev. C 58, (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, (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: attractive scalar 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 RADII DIFFERENCES r n -r p SnSn RELATIVISTIC HARTREE-BOGOLIUBOV MODEL (RHB) Unified description of mean-field and pairing correlations Dirac hamiltonian Pairing field BINDING ENERGIES σ RMS =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-RQRPA 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, (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, (2006). 208 Pb Exp MeV Exp MeV MONOPOLE EXCITATIONS DIPOLE 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 ISOBARIC ANALOG AND GAMOW-TELLER RESONANCES (RQRPA)

→ single-particle reduced matrix elements Back to reduced matrix elements for the neutrino-nucleus cross 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 momentum approximation RHB+RQRPA NEUTRINO-NUCLEUS ( 12 C) CROSS SECTION Engel, Phys. Rev. C 57, 2004 (1998) RHB+RQRPA

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

(ν μ,μ - ) ( cm 2 ) (ν e,μ - ) ( cm 2 ) RPA(Volpe et al.) QRPA(Volpe et al.) CRPA(Kolbe) PN-RQRPA EXP.(LSND)12 ± 0.3 ± ± 1.6 EXP.(KARMEN)14.0 ± 1.2 Flux averaged cross section RHB+RQRPA CROSS SECTION AVERAGED OVER NEUTRINO FLUX

Supernova neutrino flux is given 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. Marketin T. Nikšić P. Ring D. Vretenar COLLABORATORS