Goldstone bosons and fermions in QCD

TL;DR

This paper explores the implications of the Gribov region restriction in Euclidean Landau gauge QCD, revealing spontaneous symmetry breaking and the emergence of Goldstone bosons and fermions, which may relate to confinement.

Contribution

It demonstrates that the Gribov restriction leads to spontaneous symmetry breaking, resulting in massless Goldstone modes and providing insights into long-range quark interactions.

Findings

Gluon propagator vanishes as $k^2$ in the infrared

Spontaneous symmetry breaking produces Goldstone bosons and fermions

Goldstone bosons may mediate long-range confining forces

Abstract

We consider the version of QCD in Euclidean Landau gauge in which the restriction to the Gribov region is implemented by a local, renormalizable action. This action depends on the Gribov parameter $\gamma$, with dimensions of (mass)$^4$, whose value is fixed in terms of $\Lambda_{QCD}$, by the gap equation, known as the horizon condition, ${\p \Gamma \over \p \gamma} = 0$, where $\Gamma$ is the quantum effective action. The restriction to the Gribov region suppresses gluons in the infrared, which nicely explains why gluons are not in the physical spectrum, but this only makes makes more mysterious the origin of the long-range force between quarks. In the present article we exhibit the symmetries of $\Gamma$, and show that the solution to the gap equation, which defines the classical vacuum, spontaneously breaks some of the symmetries $\Gamma$. This implies the existence of massless Goldstone bosons and fermions that do not appear in the physical spectrum. Some of the Goldstone bosons may be exchanged between quarks, and are candidates for a long-range confining force. As an exact result we also find that in the infrared limit the gluon propagator vanishes like $k^2$.