Nonlinear Schwinger mechanism in QCD, and Fredholm alternatives theorem

TL;DR

This paper introduces a novel nonlinear approach to the Schwinger mechanism in QCD, utilizing the Fredholm alternatives theorem to determine the gluon mass scale and align results with lattice data.

Contribution

It presents a new method that fixes the gluon mass scale in QCD using nonlinear equations and Fredholm theorem, improving the understanding of gluon mass generation.

Findings

Numerical results agree with lattice QCD data.

The approach uniquely determines the gluon mass scale.

Nonlinear treatment allows for nonzero gluon mass.

Abstract

We present a novel implementation of the Schwinger mechanism in QCD, which fixes uniquely the scale of the effective gluon mass and streamlines considerably the procedure of multiplicative renormalization. The key advantage of this method stems from the nonlinear nature of the dynamical equation that generates massless poles in the longitudinal sector of the three-gluon vertex. An exceptional feature of this approach is an extensive cancellation involving the components of the integral expression that determines the gluon mass; it is triggered once the Schwinger-Dyson equation of the pole-free part of the three-gluon vertex has been appropriately exploited. It turns out that this cancellation is driven by the so-called Fredholm alternatives theorem, which operates among the set of integral equations describing this system. Quite remarkably, in the linearized approximation this theorem enforces the exact masslessness of the gluon. Instead, the nonlinearity induced by the full treatment of the relevant kernel evades this theorem, allowing for the emergence of a nonvanishing mass. The numerical results obtained from the resulting equations are compatible with the lattice findings, and may be further refined through the inclusion of the remaining fundamental vertices of the theory.