A gauge invariant flow equation

TL;DR

This paper develops a gauge-invariant flow equation framework for quantum field theories, extending it from scalar fields to SU(N) Yang-Mills theories, ensuring universality and gauge invariance in beta function calculations.

Contribution

It introduces a generalized flow equation that maintains gauge invariance throughout the renormalization group flow, applicable to complex gauge theories without gauge fixing.

Findings

Derived a gauge-invariant flow equation for scalar fields.

Calculated the one-loop beta function for SU(N) Yang-Mills without gauge fixing.

Demonstrated universality and gauge invariance at finite N.

Abstract

Given a Quantum Field Theory, with a particular content of fields and a symmetry associated with them, if one wants to study the evolution of the couplings via a Wilsonian renormalisation group, there is still a freedom on the construction of a flow equation, allowed by scheme independence. In the present thesis, making use of this choice, we first build up a generalisation of the Polchinski flow equation for the massless scalar field, and, applying it to the calculation of the beta function at one loop for the characteristic self-interaction, we test its universality beyond the already known cutoff independence. Doing so we also develop a method to perform the calculation with this generalised flow equation for more complex cases. In the second part of the thesis, the method is extended to SU(N) Yang-Mills gauge theory, regulated by incorporating it in a spontaneously broken SU(N|N) supergauge group. Making use of the freedom allowed by scheme independence, we develop a flow equation for a SU(N|N) gauge theory, which preserves the invariance step by step throughout the flow and demonstrate the technique with a compact calculation of the one-loop beta function for the SU(N) Yang-Mills physical sector of SU(N|N), achieving a manifestly universal result, and without gauge fixing, for the first time at finite N.