N. PaarN. Paar1,21,2

11Department of Physics, University of Basel, SwitzerlandDepartment of Physics, University of Basel, Switzerland22Department of Physics, Faculty of Science, University of Zagreb, CroatiaDepartment of Physics, Faculty of Science, University of Zagreb, Croatia

International Workshop on Neutrino Physics and Astrophysics, 16-20.3.2015, Istanbul

Relativistic Nuclear Energy Density Functional for Astrophysical Applications

• There are about 3000 stable and radioactive nuclides that exist in nature or have been synthesized in laboratory; about 7000 nuclides may exist in total

• Exotic nuclei play an important role in various astrophysical scenarios; e.g. supernova evolution, nucleosynthesis, etc.

• Energy density functional (EDF) based on self-consistent mean-field approach, i.e. effective interaction between nucleons is represented by a functional which depends on densities and currents

(UNEDF)

ENERGY DENSITY FUNCTIONAL

• EDF can be applied through the entire nuclear chart, consistent approach to • nuclear structure and excitations• nuclear matter equation of state• astrophysically relevant processes involving nuclei (electron capture,

neutrino induced reactions, beta decays,…)• neutron star properties• …

Nuclear EnergyDensity Functional

Nuclear ground state propertiesand excitations; EOS

Stellar weak Interaction processesinvolving nuclei

NEXT GENERATIONRIB FACILITIES

“Final breakthrough in our understanding of how supernova explosions work, based on self-consistent models with all relevant physics included, has not been achieved yet.”

OUR GOAL: Universal relativistic nuclear energy density functional (RNEDF) for •properties of finite nuclei (masses, radii, excitations)•neutron star properties (mass/radius, …)•electron capture in presupernova collapse•neutrino-nucleus reactions and beta decays for the nucleosynthesis•other astrophysically relevant phenomena…

NUCLEAR PROCESSES IN STELLAR SYSTEMS

H.-Th. Janka et al., Phys. Rep. 442, 38 (2007):

RNEDFSUPERNOVAE ,…NEUTRON STARS,NS MERGERS, ……

NUCLEOSYNTHESIS

THEORY FRAMEWORK

• In the relativistic framework the basis an effective Lagrangian with relativistic symmetries; it is used in a mean field concept (Hartree-level)

1) Nucleons are considered as Dirac particles coupled by the exchange mesons and the photon field FRAMEWORK

sigma-meson:

attractive scalar field omega-meson:

short-range repulsive rho-meson:isovector field

Extensions: +pairing correlations(Relativistic Hartree-Bogoliubov model)

ρσ ω

2) Interaction Lagrangian with four-fermion (contact) interaction terms

T. Niksic, et al., Comp. Phys. Comm. 185, 1808 (2014).

• many-body correlations encoded in density-dependent coupling functions. To establish the density dependence of the couplings one could start from a microscopic equation of state of symmetric and asymmetric nuclear matter.

• The model parameters are constrained directly by many-body observables using minimization

• Calculated values are compared to experimental, observational, and/or pseudo-data, e.g.

• properties of finite nuclei – binding energies, charge radii, diffraction radii, surface thicknesses, pairing gaps, excitations,…

• nuclear matter properties – equation of state,binding energy and density at saturation point, symmetry energy, incompressibility…

• neutron star properties – neutron star mass and radiusJ. Antoniadis, P. C. C. Freire, N. Wex et al. Science 340, 448 (2013) 2.01(4) MsunP. B. Demorest et al., Nature 467, 1081 (2010) 1.97(4) Msun

• Mass-radius relations of cold neutron stars for different EOS – observational constraints on the neutron star mass rule out many models for EOS.

M. Hempel et al., Astr.J. 748,70 (2012) P. B. Demorest et al., Nature 467, 1081 (2010)

THEORY FRAMEWORK

B. Tsang, NSCL

Nuclear matter equation of state:

Symmetry energy term:

J – symmetry energy at saturation densityL – slope of the symmetry energy (related to the pressure of neutron matter)

EQUATION OF STATE

• The same pressure that pushes the neutrons against the surface tension in nuclei, and determines the neutron skin thickness also supports a neutron star against gravity

αD A. Tamii et al., PRL 107, 062502 (2011). PDR O. Wieland, A. Bracco, F. Camera et al., PRL 102, 092502 (2009). A. Klimkiewicz et al., PRC 76, 051603(R) (2007). IVGQR S. S. Henshaw, M. W. Ahmed G. Feldman et al, PRL 107, 222501 (2011). AGDR A. Krasznahorkay et al., arXiv:1311.1456 (2013)

• Symmetry energy at saturation density (J) vs. slope of the symmetry energy (L). Calculations of various modes are based on the same set of energy density functionals.

Constraining the nuclear symmetry energy

Exp. datafor various excitations:

Isovector dipole transition strength

E1

Lattimer & Lim, ApJ 771, 51 (2013)

K. Hebeler et al. AJ 773, 11 (2013)

Weighted average (αD,AGDR, PDR,IVGQR):

J = (32.35 ± 0.47) MeVL = (49.95 ± 4.72) MeV

Constraining the nuclear symmetry energy

Important constraints for the EDFs

Covariance analysis in energy density functionals – a framework to assess statistical uncertainties and correlations of calculated quantities

Focus issue:

Enhancing the interaction betweenNuclear experiment and theory through Information and statistics, Eds. D.G. Ireland, W. Nazarewicz

X. Roca Maza et al., J. Phys. G 42, 034033 (2015)T. Niksic et al., J. Phys. G 42, 034008 (2015)

COVARIANCE ANALYSIS IN ENERGY DENSITY FUNCTIONALS

Statistical model uncertainties – when the model is based on parametersthat were constrained by Χ2 minimization to large datasets, the quality of thatfit is an indicator of the statistical uncertainty

•Assume that is a well behaved hyper-function of the parameters around their optimal value

•Near the minimum, can be approximated by a Taylor expansion as an hyper-parabola in the parameter space

Curvature matrix: Covariance between two quantities A and B:

• Variance and define statistical uncertainties of each quantity.

Pearson product-moment correlation coefficientprovides a measure of the correlation (linear dependence)between two variables A and B.

J. Dobaczewski, W. Nazarewicz, P.-G. Reinhard, JPG 41, 074001 (2014).

STATISTICAL UNCERTAINTIES

CORRELATIONS: NUCLEAR MATTER AND PROPERTIES OF NUCLEI

Correlation matrix between nuclear matter properties and several quantities in 208Pb(neutron skin thickness, properties of giant resonances) (DDME-min1)

cAB=1: A & B strongly correlated

cAB=0: A & B uncorrelated

208Pb cAB

Correlation matrix indicates important correlations between various quantities.

SYSTEMATIC UNCERTAINTIES : NEUTRON SKIN THICKNESS

J. Piekarewicz,et al., PRC 85, 041302 (R) (2012)

Systematical model uncertainties – limitations of the model, deficient parametrizations, wrong assumptions, and missing physics due to our lack of knowledge. These uncertainties are difficult to estimate – systematic errors can be determined by comparing a variety of theoretical frameworks and parameterizations

• Electric dipole polarizability αD and neutron skin thickness (rskin) for 208Pb using both nonrelativistic and relativistic EDFs:

Model averaged value:

rskin(208Pb) = (0.168 ± 0.022) fm

By using only models consistent with measured αD (48 EDFs 25 EDFs),systematic model uncertaintyin rskin is reduced.

Towards a universal relativistic nuclear energy density functional for astrophysical applications (N.P., M. Hempel et al. 2015)

The goal is to•achieve reasonable accuracy in description od nuclear properties (use exp. data on binding energies, charge radii, diffraction radii, surface thickness)

•constrain the symmetry energy from experimental data on giant resonances and dipole polarizability (208Pb) A. Tamii et al., PRL 107, 062502 (2011)

•constrain the nuclear matter incompressibility from exp. data on compression modes (208Pb) J. Ritman et al., PRL 70, 533 (1993)

•constrain the equation of state using the saturation point and point at twice the saturation density from the FOPI data on heavy ion collisions A. Le Fevre et al., arXiv:1501.05246v1 (2015)

•constrain the maximal neutron star mass on observational data J. Antoniadis, et al. Science 340, 448 (2013); P. B. Demorest et al., Nature 467, 1081 (2010)

by solving the Tolman-Oppenheimer-Volkov (TOV) equations within theχ2 minimization.

EQUATION OF STATE

SYMMETRIC NUCLEAR MATTER NEUTRON MATTER

TEST CASE

Constraining the EDF from collective modes of excitation:

ISOSCALAR GIANT MONOPOLE RESONANCE

ISOVECTOR GIANT DIPOLE RESONANCE

ISGMR energy determines the nuclear matter incompressibility: K=232.18 MeV

Dipole polarizability:αD=20.62 fm3

Exp.αD= 20.1(6) fm3

A.Tamii et al.,PRL 107, 062502 (2011).

IVGDR and dipole polarizabilityconstrain the symmetry energy

J=32.62 MeV L=56.30 MeV

TEST CASE

PC-PK1: P.W.Zhao et al., PRC 82, 054319 (2010).(K=238 MeV, J=35.6, L = 113 MeV)

Next step: include nuclear deformation in constraining the EDF.

Nuclear binding energies (calculated – experimental)

DD-PCtest:Maximal neutronstar mass: 2.18 Msun

TEST CASE

Important weak processes during the star collapse and explosion:

K. Langanke, G. Martinez-Pinedo RMP 75, 819 (2003).:

APPLICATION OF THE EDFs IN DESCRIPTION OF ASTROPHYSICAL PROCESSES

K. Langanke, G. Martinez-Pinedo RMP 75, 819 (2003).

NEUTRINO-NUCLEUS CROSS SECTIONS

Transition matrix elements are described in a self-consistent way usingrelativistic Hartree-Bogoliubov model for the initial (ground) state and relativisticquasiparticle random phase approximation for excited states (RHB+RQRPA)

• Nuclear ground state properties and excitations determine the ν-nucleus cross sections.

• Nuclear transitions induced by neutrinos involve operators with finite momentum transfer.

• Multipole composition of the neutrino-nucleus cross sections: RNEDF (DD-ME2) vs. shell model + RPA (SGII) (T. Suzuki et al.)

CHARGED-CURRENT NEUTRINO-NUCLEUS CROSS SECTIONS

Agreement betweenmodels based on differentfoundations and effective interactions !

In addition to GT, at largerneutrino energies exitationsof higher multipolarities (forbidden transitions)contribute.

•The RNEDF allows systematic calculations of high multipole excitations (forbidden transitions), enables extrapolations toward nuclei away from the valley of stability.

Neutrino-nucleus cross sections averaged over neutrino fluxes for varioustemperatures (Fermi-Dirac distribution, T=2-10 MeV)

LARGE-SCALE CALCULATIONS OF νe-NUCLEUS CROSS SECTIONS

The cross sections are averaged over the neutrino spectrum from muon DAR.

• The model calculations reasonably reproduce the only two experimental cases, 12C and 56Fe.• The cross sections become considerably enhanced in neutron-rich nuclei, while those in neutron-deficient and proton-rich nuclei are small (blocking).

Model calculations include all multipoles (both parities) up toJ=5.

e FLUX(Michel)

LARGE-SCALE CALCULATIONS OF ν-NUCLEUS CROSS SECTIONS

• The ν-nucleus cross sections averaged over the Fermi-Dirac distribution (T=4 MeV)• Comparison of the RNEDF results with the ETFSI+CQRPA (only IAS & GT

transitions; I. N. Borzov and S. Goriely, Phys. Rev. C 62, 035501 (2000).):

• RNEDF calculations provide considerable improvement: include all relevant transition operators at finite momentum transfer and multipoles up to J=5

• Flux-averaged cross sections using the supernova simulation spectra at different postbounce times t=1s, 5s, 20s• Supernova model is based on general relativistic radiation hydrodynamics and threeflavor Boltzmann neutrino transport in spherical symmetry (T. Fischer)

LARGE-SCALE CALCULATIONS OF ν-NUCLEUS CROSS SECTIONS

CONCLUDING REMARKS

Towards a universal self-consistent framework based on the relativistic nuclear energy density functional for astrophysical applications

• constraints on nuclear masses and radii, pairing gaps, dipole polarizability, compression modes, nuclear matter and neutron star properties

• systematic calculations necessitate inclusion of nuclear deformation

APPLICATIONS IN PROGRESS• neutrino-nucleus cross sections, both for neutral-current and charged current

reactions• modeling neutrino response in neutrino detectors – constraining neutrino mass

hierarchy from supernova neutrino signal• systematic calculations of presupernova electron capture rates at finite temperature• neutron star properties – mass/radius relationship, liquid-to-solid core-crust transition

density and pressure

Acknowledgements: D. Vale, T. Marketin, T. Nikšić, D. Vretenar (U. Zagreb) X. Roca-Maza, G. Colò, Y.F. Niu (U. Milano)

Ch. Moustakidis, G.A. Lalazissis (U. Thessaloniki) P. G.-Reinhard (U. Erlangen-Nurnberg)

T. Suzuki (U. Nihon) M. Hempel, F.-K. Thielemann (U. Basel)