Revised variational approach to QCD in Coulomb gauge

TL;DR

This paper revisits the variational approach to QCD in Coulomb gauge, deriving UV-finite equations for the quark wave functional, and computes the quark condensate and effective mass using input from Yang-Mills theory.

Contribution

It introduces a new variational framework with a Slater determinant ansatz for quarks and Gaussian gluonic wave functional, improving the treatment of UV divergences in Coulomb gauge QCD.

Findings

Reproduces the phenomenological quark condensate value.

Derives UV-finite variational equations for quark kernels.

Numerically solves for quark mass and condensate using input from Yang-Mills sector.

Abstract

The variational approach to QCD in Coulomb gauge is revisited. By assuming the non-Abelian Coulomb potential to be given by the sum of its infrared and ultraviolet parts, i.e.~by a linearly rising potential and an ordinary Coulomb potential, and by using a Slater determinant ansatz for the quark wave functional, which contains the coupling of the quarks and the gluons with two different Dirac structures, we obtain variational equations for the kernels of the fermionic vacuum wave functional, which are free of ultraviolet divergences. Thereby, a Gaussian type wave functional is assumed for the gluonic part of the vacuum. By using the results of the pure Yang--Mills sector for the gluon propagator as input, we solve the equations for the fermionic kernels numerically and calculate the quark condensate and the effective quark mass in leading order. Assuming a value of $\sigma_{\mathrm{C}} = 2.5 \sigma$ for the Coulomb string tension (where $\sigma$ is the usual Wilsonian string tension) the phenomenological value of the quark condensate $\langle \bar{\psi} \psi \rangle \simeq (-235 \, \mathrm{MeV})^3$ is reproduced with a value of $g \simeq 2.1$ for the strong coupling constant of the quark-gluon vertex.