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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