Diffusion of Neutrinos in Proto-Neutron Star Matter with Quarks

Neutrino opacities important in the evolution of a proto-neutron star containing quark matter are studied. The results for pure quark matter are compared with limiting expressions previously derived, and are generalized to the temperatures, neutrino degeneracies and lepton contents encountered in a proto-neutron star’s evolution. We find that the appearance of quarks in baryonic matter drastically reduces the neutrino opacity for a given entropy, the reduction being sensitive to the thermodynamic conditions in the mixed quark-hadron phase. PACS: 97.60.Jd, 21.65.+f, 13.15.+g, 26.60.+c A general picture of the early evolution of a proto–neutron star (PNS) is becoming well established [1–6]. Neutrinos are produced in large quantities by electron capture as the progenitor star collapses, but most are temporarily prevented from escaping because their mean free paths are considerably smaller than the radius of the star. During this trapped-neutrino era, the entropy per baryon s is about 1 through most of the star and the total number of leptons per baryon YL = Ye+Yνe ≃ 0.4. The neutrinos trapped in the core strongly inhibit the appearance of exotic matter, whether in the form of hyperons, a Bose (pion or kaon) condensate or quarks, due to the large values of the electron chemical potential. As the star cools, the neutrino mean free path increases, and the neutrinos eventually leak out of the star, on a timescale of 20-60 s. During deleptonization, neutrino diffusion heats the matter to an approximately uniform entropy per baryon of 2. If the strongly interacting components consist only of nucleons, the maximum supportable mass increases. In the case that hyperons, a Bose condensate 1 (pion, kaon) or quarks appear in the core as the neutrinos leave, the maximum mass decreases with decreasing leptonic content. Neutron stars which have masses above the maximum mass for completely deleptonized matter are thus metastable, and will collapse into a black hole during deleptonization. Alternatively, if the mass of the neutron star is sufficiently small, the star remains stable and cools within a minute or so to temperatures below 1 MeV as the neutrinos continue to carry energy away from the star. The way in which this picture is modified when the core of a PNS contains deconfined quark matter is only beginning to be investigated [2,7–9]. In his seminal paper, Iwamoto [10] noted that the non-degenerate ν mean free path in cold quark matter is about ten times larger than in nucleonic matter. We find that in PNS matter, in which quarks appear towards the end of deleptonization, similarly large enhancements persist even up to the largest relevant temperatures (∼ 30−40 MeV [9]), inasmuch as quarks remain largely degenerate. On this basis, it can be anticipated that the presence of quark matter increases the neutrino fluxes while simultaneously decreasing the deleptonization time, relative to matter without quarks. In work to be reported elsewhere, we explore the possibility that such a change might be detected from a Galactic supernova in current and planned neutrino detectors. This would have direct implications for the theoretical understanding of the high-density regime of QCD which is inaccessible to high energy Relativistic Heavy-Ion Collider experiments, and, currently, to lattice QCD calculations at finite baryon density. To perform detailed simulations of the neutrino signal from a PNS containing quark matter, as has already been done for matter containing nucleons, hyperons and/or a kaon condensate [1–6], consistent calculations of neutino interactions in hot lepton-rich matter containing quarks are required. It is most likely that quarks exist in a mixed phase with hadrons [7,9,11]. Steiner, et. al. [9], recently showed that the temperature of an adiabat decreases as a function of density in a mixed phase of quarks and nucleons. Because ν−cross sections usually scale with T , this suggests that the presence of quark matter might influence the neutrino signal of a PNS with quarks. In this work, we calculate the diffusion coefficients of neutrinos in a mixed phase of hadrons and quarks for the temperatures, neutrino degeneracies, and lepton contents likely to be encountered in the evolution of a PNS with quarks. We demonstrate that the cross sections for scattering and absorption 2 of neutrinos by nucleons, leptons, and quarks are reduced to two integrals, whose integrands are products of simple polynomials and thermal distribution functions. The limiting behaviors of the cross sections, for non-degenerate and degenerate neutrinos, respectively, are compared with previous calculations [10] in the case of pure quark matter. For simulations of PNS evolution using the diffusion approximation, diffusion coefficients, which are energy weighted averages of neutrino cross sections, are required for matter in which quarks exist in a mixed phase with hadrons. We examine the relevant diffusion coefficients for two thermodynamic conditions especially germane to PNS evolution. The first situation is when neutrinos are trapped, s ≈ 1, and the total lepton content of the matter YL = Ye+Yνe (which measures the concentrations of the leptons per baryon) is approximately 0.4. We also consider the situation when neutrinos have mostly left the star and the matter has been diffusively heated (Yν ≈ 0, s ≈ 2). We discuss the impact these results might have upon the evolution of a PNS which contains quarks in a mixed phase. For the ν−energies of interest, the neutral and charged current ν−interactions with the matter in a PNS are well described by a current-current Lagrangian [12] L = G 2 F √ 2 ( ψν(1)(1− γ5)ψp3(3) ) ( ψp2(2)(V − Aγ5)ψp4(4) ) +H.C. , (1) where V and A are the vector and axial–vector coupling constants (See Table 1) and GF ≃ 1.17 GeV is the Fermi weak coupling constant. The subscripts i = 1, 2, 3, and 4 on the four-momenta pi denote the incoming neutrino, the incoming lepton, baryon or quark, the outgoing neutrino (or electron), and the outgoing lepton, baryon or quark, respectively. The charged current reactions contribute to absorption of neutrinos by baryons or quarks and scattering of neutrinos with leptons of the same generation. The neutral current interactions contribute to scattering of neutrinos with leptons and baryons or quarks. The charged current contribution to neutrino–lepton scattering in the same generation can be transformed into the neutral current form, which modifies the constants V and A for that case. For completeness, the values of V and A for electron neutrinos are given in Table 1. The corresponding values for reactions with electron anti-neutrinos are obtained by the replacement A → −A. From Fermi’s golden rule, the cross section per unit volume (or inverse mean free path) is σ V = g ∫ dp2 (2π) ∫ dp3 (2π) ∫ dp4 (2π) Wfif2 (1− f3) (1− f4) (2π) δ (p1 + p2 − p3 − p4) , (2) 3 where the degeneracy factor g is 6 (3 colors × 2 spins) for reactions involving quarks while it is 2 (2 spins) for baryons of a single species. The Fermi–Dirac distribution functions are denoted by fi = [