The all-order equation of the effective gluon mass

TL;DR

This paper derives a comprehensive non-perturbative equation for the evolution of the gluon mass, incorporating novel vertices and simplifying the Schwinger-Dyson equation within the PT-BFM framework, leading to physically meaningful solutions.

Contribution

It introduces a new form of the gluon mass equation using the PT-BFM formalism with two-loop contributions, enabling the calculation of positive, decreasing gluon masses.

Findings

Two-loop contributions significantly alter the mass equation.

Numerical solutions produce positive, monotonically decreasing gluon masses.

The approach simplifies the calculation of the gluon mass evolution.

Abstract

We present the general derivation of the full non-perturbative equation that governs the momentum evolution of the dynamically generated gluon mass, in the Landau gauge. The entire construction hinges crucially on the inclusion of longitudinally coupled vertices containing massless poles of non-perturbative origin, which preserve the form of the fundamental Slavnov-Taylor identities of the theory. The mass equation is obtained from a previously unexplored version of the Schwinger-Dyson equation for the gluon propagator, particular to the PT-BFM formalism, which involves a reduced number of "two-loop dressed" diagrams, thus simplifying the calculational task considerably. The two-loop contributions turn out to be of paramount importance, modifying the qualitative features of the full mass equation, and enabling the emergence of physically meaningful solutions. Specifically, the resulting homogeneous integral equation is solved numerically, subject to certain approximations, for the entire range of physical momenta, yielding positive-definite and monotonically decreasing gluon masses.