Quark-hadron Phase Transition in Proto-Neutron Stars Cores based on a Non-local NJL Model

We study the QCD phase diagram using a non-local SU(3) NJL model with vector interactions among quarks. We analyze several thermodynamic quantities such as entropy and specific heat, and study the influence of vector interactions on the thermodynamic properties of quark matter. Upon imposing electric charge neutrality and baryon number conservation on the field equations, we compute models for the equation of state of the inner cores of proto-neutron stars providing a non-local treatment of quark matter for astrophysics.


Introduction
It is known that quantum chromodynamics (QCD), has two very important properties, namely asymptotic freedom and confinement. The former implies that at high momentum transfers, the quarks behave as quasi free particles, i.e., the interaction between two quarks due to gluon interchange can be treated using perturbation theory. For this momentum range, the dispersion processes can be then be calculated with a very good precision. By contrast, at low momentum exchange among the quarks ( 1 GeV) QCD is highly nonlinear and leads to quark confinement. This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 3.0 (CC-BY) License. Further distribution of this work is permitted, provided the original work is properly cited.
Several approximate methods have been developed to study the physical processes among quarks in the low momentum range of QCD. Among them is lattice QCD, which tries to solve the QCD equations of motion numerically on a discretized spacetime grid 1 . This method, however, presents problems if extended to finite chemical potentials 2 . Another option is to use effective QCD models, like the Nambu-Jona-Lasinio model (NJL), which is described by a Lagrangian which accounts for the main features of QCD at low energies. The advantage of NJL models is that they can be extended to finite chemical potentials easily. Local as well as non-local extensions of the model have been studied in the literature (see, for instance, Refs. 3, and references therein).
Non-local extensions of the NJL model at zero temperature have been used to study hybrid stars with and/or without a quark-hadron mixed phases in their inner cores (see Ref. 4, 5, 3, and references therein). Here, we present an extension of such studies to finite temperatures in order to explore the role of a quark-hadron phase transition for proto-neutron stars or core collapse supernovae.

n3PNJL model and phase diagram
The Euclidean effective action of the model presented in this paper is for the nonlocal 3-flavor Polyakov NJL model, including the vector interactions among quarks. This action is given by where j S a (x), j P a (x) and j µ v (x) are the scalar, pseudoscalar and vector currents, respectively, and U[A(x)] is the effective potential related with the Polyakov loop. To extend the model to finite temperatures we use the Matsubara imaginary time formalism 6 . After the bosonization of Eq. (1) we obtain the grand canonical potential for the mean field approximation, from which the phase diagram (Fig. (1)) and thermodynamic quantities such as the specific heat, C v , and specific entropy, s, (Fig. (2)) can be calculated.
The quark current masses and coupling constants in Eq. (1) can be chosen so as to reproduce the phenomenological values of the pion decay constant, f π , and the meson masses m π , m η , and m η , as described in Ref. 7 Because of their short mean-free paths, the neutrinos produced in the core of a proto-neutron star are prevented from leaving the star on a dynamical time scale. The number of lepton-to-baryon ratio of such matter is around Y Le 0.4, but the exact value depends on the efficiency of electron capture reactions during the gravitational collapse of the supernova. The number of muons per baryon is Y Lµ = 0, because no muons are present in the stellar matter prior to the trapping. On a timescale of 10 to 20 seconds, the neutrinos diffuse from the star, but leave behind much of their energy which causes significant heating of the ambient matter 8 . The temperatures generally achieved in the inner 50% of the stellar core at the peak of the heating are in the range of 30 to 50 MeV. Following the heating, the star cools by radiating neutrino-anti-neutrino pairs, and consequently the temperature drops off to ∼ 1 MeV within minutes. A model for the equation of state of such matter,