Quark running mass and vacuum energy density in truncated Coulomb gauge QCD for five orders of magnitude of current masses

Abstract

We study in detail the effect of the finite current quark mass on chiral symmetry breaking, in the framework of truncated Coulomb gauge QCD with a linear confining quark-antiquark potential. In the chiral limit of massless current quarks, the breaking of chiral symmetry is spontaneous. But for a finite current quark mass, some dynamical symmetry breaking continues to add to the explicit breaking caused by the quark mass. Moreover, using as order parameter the mass gap, i. e. the quark mass at vanishing moment or the quark condensate, a finite quark mass transforms the chiral symmetry breaking from a phase transition into a crossover. For the study of the QCD phase diagram it thus is relevant to determine how the current quark mass affects chiral symmetry breaking. Since the current quark masses of the six standard flavours u, d, s, c, b, t span over five orders of magnitude from 1.5 MeV to 171 GeV, we develop an accurate numerical method to study the running quark mass gap and the quark vacuum energy density from very small to very large current quark masses. IntroductionChiral symmetry breaking has been studied in detail with chiral invariant and confining quark models in the chiral limit of a vanishing current quark mass m 0 . For a finite current quark mass m 0 = 0, studies exist with an approximate Email address: bicudo@ist.utl.pt (P. Bicudo).

Chiral symmetry crossover with a linear confining and temperature dependent quark-antiquark potential

“…6. In the vanishing temperature limit T = 0 the different regularizations lead to the same physical results since any constant term in a density-density interaction has no effect in the quark running mass m(p) or in the hadron spectrum [4].…”

“…To solve the coupled systems of non-linear Equations ( 18), we apply the techniques detailed in [4], i. e. we utilize a Padé ansatz for the quark running mass m T (p), we solve the mass gap variationally, we compute the one quark energy E T (p), we fit is with a Padé approximant, we feed it back into the mass gap equation, and then we repeat this cycle iteratively until the solution converges. The solution of the system of Eqs.…”

“…Thus at finite T , the solution of the system of Eqs. (18) depends on a constant shift of the potential, in contradistinction with vanishing temperature where M T (p) ≃ 1 decouples the energy E T (p) from the mass [4]. At finite T the one quark energy E T (p) con- I, ranging from T = 0 to T = Tc.…”

“…To ensure convergence, we utilize the technique [4] of translating the mass gap equation into the variational equation for the vacuum energy density,…”

“…Here we utilize the linear confining potential for the quark-antiquark interaction, in the Coulomb Gauge Chiral Quark Model (CGχQM ), including both confinement and chiral symmetry [4]. While this model, inspired from the framework of Coulomb gauge Hamiltonian formalism is not yet full QCD, it is presently the only model able to include explicitly both the quark-antiquark confining potential and the quark-antiquark vacuum condensation.…”