Background Fields Meet the Heat Kernel: Gauge Invariance and RGEs without diagrams

TL;DR

This paper presents a novel method combining the Heat Kernel and Background Field techniques to compute gauge-invariant quantities like RGEs without relying on diagrammatic methods, validated on Standard Model examples.

Contribution

It introduces a background-field-based approach for calculating gauge-invariant RG quantities directly from background dynamics, avoiding traditional diagram calculations.

Findings

Successfully computes gauge-invariant beta functions and anomalous dimensions.

Demonstrates the method on Scalar QED, Yukawa theory, and Standard Model bosonic sector.

Ensures gauge-parameter independence when background fields are on-shell.

Abstract

We introduce a new method that exploits the combination of the Heat Kernel (HK) and Background Field Method to compute gauge-invariant and gauge parameter-independent quantities such as the effective potential, anomalous dimensions, and renormalization group equations. In contrast to currently employed techniques, these results are obtained exclusively from the dynamics of the background fields, without relying on supplementary input from, e.g., traditional diagrammatic calculations. This is achieved by a consistent treatment of open and closed derivatives in the HK expansions. In this way, we compute the standard quantities such as $\beta$ functions and their gauge-parameter independence when background fields are on-shell. We demonstrate this formalism for instructive examples such as Scalar QED and Yukawa theory. Full results for the bosonic part of the Standard Model provide further validation of our approach.