Path integral regularization of pure Yang-Mills theory

TL;DR

This paper introduces a gauge-invariant regularization method for pure Yang-Mills theory using a cutoff-dependent scalar field, ensuring a well-defined path integral and preserving Ward-Takahashi identities.

Contribution

The authors develop a novel gauge-invariant regularization scheme for Yang-Mills theory by enlarging the field content with a scalar field and constructing an invariant operator that converges to the original gauge field.

Findings

Constructed a cutoff-dependent invariant operator for Yang-Mills fields.

Established a mathematically well-defined, gauge-invariant path integral.

Maintained the original Ward-Takahashi identities after regularization.

Abstract

In enlarging the field content of pure Yang-Mills theory to a cutoff dependent matrix valued complex scalar field, we construct a vectorial operator, which is by definition invariant with respect to the gauge transformation of the Yang-Mills field and with respect to a Stueckelberg type gauge transformation of the scalar field. This invariant operator converges to the original Yang-Mills field as the cutoff goes to infinity. With the help of cutoff functions, we construct with this invariant a regularized action for the pure Yang-Mills theory. In order to be able to define both the gauge and scalar fields kinetic terms, other invariant terms are added to the action. Since the scalar fields flat measure is invariant under the Stueckelberg type gauge transformation, we obtain a regularized gauge-invariant path integral for pure Yang-Mills theory that is mathematically well defined. Moreover, the regularized Ward-Takahashi identities describing the dynamics of the gauge fields are exactly the same as the formal Ward-Takahashi identities of the unregularized theory.