Gauge Symmetry Beyond Perturbation Theory: BRST and anti-BRST Structure, Background Fields, and Infrared Dynamics of Yang--Mills Theory

Abstract: We present a pedagogical and self contained account of the functional formulation of non-Abelian gauge theories, aimed at the construction of a process independent effective charge for Yang--Mills theory. Starting from the path integral quantization of gauge fields, we review gauge fixing and the emergence of Faddeev--Popov ghosts, illustrating how gauge invariance is preserved at the quantum level through Becchi--Rouet--Stora--Tyutin (BRST) symmetry. We then develop the BRST and anti-BRST formalisms and show how their simultaneous implementation leads to powerful functional identities that severely constrain the ghost and gluon sectors. Background field gauges are introduced as a natural framework in which these symmetries manifest themselves through Abelian like Ward identities, allowing for a transparent separation between quantum and background degrees of freedom. This structure makes it possible to define renormalization group invariant combinations of Green functions that generalize the QED effective charge to the non-Abelian case. The resulting effective charge is shown to be unique, gauge invariant, and process independent, providing a unified description of the theory from the ultraviolet down to the infrared. The interplay between functional identities, Dyson--Schwinger equations, and lattice results is discussed in detail, highlighting how dynamical mass generation and infrared saturation naturally emerge within this framework.