Quarks mass function at finite density in real-time formalism

The chiral symmetry restoration of quarks is investigated at finite density in quantum chromodynamics. Quark mass functions are calculated with the Schwinger-Dyson equation in the real-time formalism without the instantaneous exchange approximation. We present some properties of real and imaginary parts of the mass functions.


§1. Introduction
Understanding of phase transitions, such as chiral phase transition, in quantum chromodynamics (QCD) is one of important problems in physics. In order to study the chiral phase transitions, various methods are implemented. One of useful tools is the Schwinger-Dyson equation (SDE) [1,2], which can evaluate nonperturbative phenomena.
At finite temperature and density for equilibrium systems, the imaginary-time formalism (ITF) is implemented, which continues to Euclidean space at zero temperature limit. On the other hand, the real-time formalism (RTF) for non-equilibrium systems is formulated in Minkowski space. The SDE in RTF has been studied with the instantaneous exchange approximation (IEA) [3], in which gauge boson energy is neglected. In the IEA, it has been observed that the mass function does not dependent on the energy. Furthermore, the critical coupling of chiral symmetry breaking in quantum electrodynamics (QED) is about half that calculated with four momentum integration in Euclidean space at zero temperature. [3] In the previous papers [4,5,6], we formulated the SDE for QED and QCD, in which the momentum integration is performed in Minkowski space without the IEA. In Minkowski space, if the imaginary part of the mass function is small, the propagator varies rapidly near resonance peaks, which is one of the difficulties for numerical calculation. In our method, the resonance contributions in momentum integration in Minkowski space are efficiently evaluated. Furthermore, we can directory evaluate real and imaginary parts of the mass function.
In this paper, we study properties of the quark mass function with SDE in the RTF beyond the IEA for finite density at zero temperature, which corresponds to a high density matter at low temperature, such as internal structure of neutron stars.
In section 2, we formulate the SDE in the RTF without the IEA. In section 3, some numerical results are shown. Section 4 is devoted to the summary and some comments. §2. SDE for quark mass function In the RTF, two types of fields specified by 1 and 2 are implemented in the theory, in which the type-1 field is the usual field and the type-2 field corresponds to a ghost filed in the heat bath.
We calculate the 1-1 component of a self-energy of quark Σ 11 (P ) in QCD , which is given by in one-loop order, where S 11 (Q) and D 11 µν (K) are the 1-1 components of thermal propagators for a quark with momentum Q = (q 0 , q) and a gluon with momentum K = P − Q = (k 0 , k), respectively. Here, P = (p 0 , p) is an external momentum of the quark. * ) The strong coupling constant and the color factor are denoted by g s and C F = 4/3, respectively.
The 1-1 component of the quark propagator in the RTF is given as with a temperature T and a chemical potential µ, where we define ǫ(z) = θ(z)−θ(−z) with the step function θ(z). Here, (S F (Q)) R and (S F (Q)) I are the real and imaginary parts of the quark propagator S F (Q). * ) We define the quark propagator as The 1-1 component of the gluon propagator in RTF is given as and where the longitudinal and transverse components of the gluon propagator are given as N denote the hamiltonian and number operator of the quark, respectively. The energy eigenvalues for H ′ are denoted by p0 and q0. * ) The real and imaginary terms of a propagator G are defined by respectively. Here, m L and m T denote the longitudinal and transverse gluon masses, respectively. The quark-gluon vertex is defined by Γ µ = γ µ . Our model corresponds to a mass function in the Landau gauge at T = µ = 0.
Integrating over the azimuthal angle of the momentum q, the trace of the selfenergy Σ 11 is given by and with η ± = |p ± q| and k = |k|, respectively. respectively.
In this paper, we study cases with µ > 0 for T → 0. Therefore, N F and N B can be approximated as (2 . 17) and respectively. In Minkowski space, if the imaginary part of the mass function (M 2 ) I is small, As implemented in the previous works [4,5,6], we divide the q 0 integration into small ranges and integrate the quark propagator over q In this section, some numerical results are presented. We solve the SDE presented in Eq. (2·19) by a recursion method starting from a constant mass at µ = 0. * ) For each iteration, the mass function is normalized by a current quark mass at large ζ 2 = p 2 0 − p 2 , in which perturbative calculations are reliable. In the iteration, the mass function M (p 0 , p) in integrand of the SDE is replaced by the renormalized one obtained by the previous iteration. * * ) In this paper, we evaluate and as order parameters. In Fig.1, the µ dependences of |M | for Λ QCD = 0.30, 0.32, 0.35GeV and 0.40GeV with the massless gluon are presented at T = 0. The transition of the chiral symmetry restoration seems to be the first order at T = 0. The critical chemical potentials of the phase transition µ C depend on the QCD parameter Λ QCD . In our model, 0.30GeV ≤ Λ QCD ≤ 0.40GeV gives 0.27GeV ≤ µ C ≤ 0.36GeV, roughly µ C ∼ 0.9Λ QCD .
In order to chose the QCD parameter Λ QCD , we need another condition. Our model roughly gives the real part of the squared quark mass function (M 2 ) R ≃ Λ 2 QCD at µ = T = 0. Here, (M 2 ) R is determined by the resonance peak of the quark propagator. In our calcularion, Λ QCD = 0.32GeV gives (M 2 ) R ≃ 0.32GeV at T = µ = 0 and the critical temperature of the chiral symmetry restoration T C ≃ 0.175GeV with µ = 0. [6] In Fig.2, the µ dependences of the integrated mass functions |M | , M R and M I with the massless gluon are presented at Λ QCD = 0.32GeV, which gives µ C ≃ 0.29GeV.
As shown in Fig.2, the imaginary part of the mass function M I is non-zero value for broken chiral symmetric phase below the critical chemical potential µ C , which means the massive quark state may be unstable if energy scale rapidly changes. Furthermore, the real and imaginary parts vanish at the same critical chemical potential. * ) The initial input parameters are MR = ΛQCD and MI = 0 at µ = 0 with Λ0 = Λ = 10ΛQCD and δ = 0.1ΛQCD with ε = 10 −6 . In evaluation of the quark mass function at µ + ∆µ, we implement the solution of M obtained at µ as the initial input. Here, we define M = MR + iMI. * * ) We take the renormarized mass m(ζ 2 ) = 0 at ζ = 10ΛQCD.  In Fig.3, the gluon mass dependences are presented with different values of Λ QCD with Λ QCD = 0.30(GeV),0.32(GeV),0.35(GeV) and 0.40(GeV), respectively. The gluon mass is defined as m T = 0 and m L = aΛ QCD , respectively. Though the critical chemical potential µ C depends on the gluon mass, the gluon mass dependence on the effective quark mass seems to be weaker as larger gluon mass.  In this paper, we have investigated quark mass functions solved by the Schwinger-Dyson equation (SDE) at finite density with zero temperature in the real-time formalism (RTF).
In our calculations, we improved the four momentum integration of the SDE near the resonance peaks, which is one of the difficulties with numerical calculation in Minkowski space.
In our model, the critical chemical potential µ C , in which the chiral symmetry is restored, depends on the QCD scale parameter Λ QCD and the gluon masses (m L and m T ). Here,m L and m T denote the masses for a longitudinal component and a transverse component of the gluon propagator, respectively. Our model roughly gives the critical chemical potential for the chiral symmetry restoration µ C ∼ 0.9Λ QCD at T = 0 with a massless gluon (m L = m T = 0).
The gluon mass at finite density in deep infrared region is nontrivial. In this paper, in order to compare the massless gluon case, we investigated the gluon mass dependence of the quark mass function with a simple ansatz. We presented the results with m L = aΛ QCD for 0 ≤ a ≤ 2 and m T = 0 for different values of Λ QCD . Though the critical chemical potential decreases as increasing the gluon mass m L , the gluon mass dependence is weak for large m L .
We found that the imaginary part of the mass function M I , which is the imaginary part of the mass function M I integrated over the energy and momentum, is non-zero value for broken chiral symmetric phase below µ C , which means that the massive quark state may be unstable if the energy scale of the system rapidly changes. Furthermore, the real and imaginary parts of the integrated mass functions vanish at the same critical point.
In our calculations, the chemical potential dependences are not smooth curves. Therefore, we present crude dependences in order to search for the critical point of the chiral symmetry restoration. Within present accuracy, the phase transition at T = 0 seems to be the first order.
Our study suggests that the SDE in the RTF seems to be useful to investigate the chiral phase transition in QCD at finite density, particularly for direct evaluation of the instability of the massive quark state.
Further studies are needed for the real and imaginary parts of the mass function separately in entire range of the phase diagram, in order to know behaviors of the mass function at strong coupling region.

Acknowledgements
This work was partially supported by MEXT-Supported Program for the Strategic Research Foundation at Private Universities, 2014-2017 (S1411024).