Neutrino emissivity of the dense superfluid nuclear matter and problems of neutron star cooling Evgeni Kolomeitsev Matej Bel University, Banska Bystrica We want to learn about properties of microscopic excitations in dense matter How to study response function of the NS? How to look inside the NS?

important for supernova At temperatures smaller than the opacity temperature (Topac~1-few MeV) mean free path of neutrinos and antineutrinos is larger than the neutron star radius white body radiation problem After >105 yr –black body radiation of photons At temperatures T>Topac neutrino transport problem important for supernova

Cooling scenario [neutrino production] Given: EoS Cooling scenario [neutrino production] Uncertainties: Data extraction; Tin-Tsurf relation Mass of NS Cooling curve Uncertainty: spin-down age vs. kinematic age

neutron star is transparent for neutrino CV - specific heat, L - luminosity emissivity each leg on a Fermi surface / T neutrino phase space ´ neutrino energy

~T6 ~T8 Cooling: role of crust and interior? most important are reactions in the interior (The baryon density is where n0 is the nuclear saturation density) one-nucleon reactions: direct URCA (DU) ~T6 modified URCA (MU) two-nucleon reactions: ~T8 nucleon bremsstrahlung (NB) URCA=Gamow’s acronym for “Un-Recordable Coolant Agent”

volume neutrino radiation DU: neutrino cooling MU: DATA Tn photon cooling

How to calculate reaction rates in medium? Let a lepton pair (l1,l2) be coupled to a boson (B) or to a fermion pair (F1,F2) Lepton production rate in medium consisting of the bosons and fermions is given by Feynman diagrams summation over the phase space of initial particle (occupation factors) Introduce a coupling among boson and fermions: Background Blue 124

consisting of fermions F1 the boson propagator is resummed diagrams polarization operator Spectrum of excitations with the quantum numbers of bosons B two characteristic mass gaps

To calculate the lepton production rates we cannot use Feynman diagrams with in-medium (dressed propagators). It can lead to double counting!! + …. additional complications due to vertex corrections

In general case one should deal with Perturbative diagrams are irrelevant for calculation of in-medium processes. In general case one should deal with closed diagrams in terms of dressed Green's functions [Voskresensky, Senatorov, Sov. Nucl. Phys. 45 (1987); Knoll, Voskresensky, Ann. Phys. 249 (1996)] one-nucleon reactions two-nucleon reactions for superfluid systems: [Kolomeitsev, Voskresensky, Phys. Atom. Nucl.

weak current susceptibility vector (V) and axial (A) currents non-relativistic limit relativistic corrections can be large !

In medium there exists a single-particle excitation mechanism Green’s function of interacting nucleon possesses a pole pole residue q.p. energy q.p. width small for T<<F complicated background part

T<<eF Only particles on the Fermi surface take part in reactions. particle-hole interaction: particle-particle interaction: Interaction in this two channels can be essentially different ! spin zero spin one expansion in Legendre polynomials Landau-Migdal constants: empirical info., calculated from NN potential

Graphically, the resummation is straightforward and yields: full pion propagator Poles yield zero-sound modes in scalar and spin channels dressed vertex

Pion modes in nuclear medium A.B. Migdal, G.E. Brown dressed vertices quasi-particle modes “pion gap” n=1n0 pion propagator has a complex pole when instability pion condensation [A.B.Migdal et al, Phys. Rept. 192 (1990) 179]

|*|~ amplitude of the condensate reconstruction of pion spectrum on top of the pion condensate pion gap for n<ncPU no pion condensate LM parameters increase with density saturation of pion softenning no pion condensate 1st-order phase transition |*|~ amplitude of the condensate [Blaschke, Grigorian, Voskresensky, A&A 424, 979–992 (2004)]

emissivity: larger smaller Very strong density dependence [Voskresensky, Senatorov, Sov. Nucl. Phys. 45 (1987)]

1S0 proton pairing 1S0 neutron pairing 3P2 neutron pairing [Schwenk Friman]

Ground state Excited state unpaired fermions paired fermions pair breaking “exciton” D pairing gap excitation spectrum emission spectrum

minimal standard exotic

Urbana-Argonne A18+d v +UIX* based EoS. Blaschke, Grigorian, Voskresensky, A&A 424, 979 (2004). Grigorian,Voskresensky, A&A 444, 913 (2005). Urbana-Argonne A18+d v +UIX* based EoS. Medium effects included in all processes. 3P2 gaps from Schwenk & Friman model. Passed log N -log S (population synthesis) control [S.Popov et al., astro-ph/0411618]

particle hole normal Green’s function particle  hole hole particle anomalous Green’s function Number of excitations is not conserved ! amplitude of 2 particle annihilation amplitude of 2 particle creation Fs, Gs and Ds are connected by the Gorkov’s and gap equations S wave pairing

bare vertices: ext. field create 2 particles dressed vertices ext. field create 2 holes Without vertex corrections the current conservation is violated !

Larkin-Migdal equations bare vertices dressed vertices Larkin-Migdal equations For T=0 and S-pairing written by Larkin, Migdal 1963 [Sov.Phys.JETP 17, 1146]. For finite T and S-pairing re-derived by Leggett 1965-6 (no Gw terms!). [Phys.Rev. 140, A1869 (1965), Phys.Rev. 147, 119 (1966)]. Applied to weak interactions in [EEK, Voskresensky, Phys.Rev.C 77, 065808 (2008)]. Equivalence of Leggett’s and Larkin-Migdal’s approaches for finite T [EEK, Voskresensky, Phys.Rev.C 81, 065801 (2010)]. General structure for arbitrary pairing and non-equlibrium systems [EEK, Voskresensky, Phys. Atom. Nucl. (2010)].

Main contribution is due to the axial current. moderate suppression strong suppression Leinson,Perez (2006), Kolomeitsev,Voskresensky (2008) Kolomeitsev,Voskresensky (2008) with free vertices Main contribution is due to the axial current. Suppression is of the order ~0.1

Consider axial weak currents (dominate the emissivity) Gamov-Teller transitions in nuclei particle-particle interaction: s-wave paring: spin zero channel next possible harmonics , which is expected to be much smaller! spin one channel next possible harmonics , which is expected to be much smaller! particle-hole interaction: does not contribute to the axial channel drop here for simplicity  corrections neglect density states at Fermi surface

C<0 exciton modes for w<2D C>gT(0) diffusive modes for w>2D [Kolomeitsev, Voskresensky, PRC 84, 068801 (2011)]

We need neutrino emissivity for description of neutron star cooling Closed diagram technique is the convenient scheme to calculate reaction rates in medium. (No double counting!) Pion modes are softened in dense nuclear matter. enhancement of a nucleon-nucleon interaction enhanced emissivity Pair formation and breaking reactions are important neutrino source in superfluid nuclear matter . In the case of pairing, vertex corrections are important: conservation of vector current new excitation modes