The Universal RG Machine

TL;DR

The paper introduces a systematic algorithm using off-diagonal heat-kernel techniques to compute functional renormalization group equations, enabling complex calculations in theories like gravity and Yang-Mills.

Contribution

It presents a novel computational method for deriving flow equations in quantum field theories, applicable to background-independent scenarios.

Findings

Re-derivation of gravitational beta-functions demonstrating background-independence

Development of a computer-implementable algorithm for flow equation expansion

Calculation of heat-kernel coefficients for specific tensor fields to second order in curvature

Abstract

Functional Renormalization Group Equations constitute a powerful tool to encode the perturbative and non-perturbative properties of a physical system. We present an algorithm to systematically compute the expansion of such flow equations in a given background quantity specified by the approximation scheme. The method is based on off-diagonal heat-kernel techniques and can be implemented on a computer algebra system, opening access to complex computations in, e.g., Gravity or Yang-Mills theory. In a first illustrative example, we re-derive the gravitational $\beta$-functions of the Einstein-Hilbert truncation, demonstrating their background-independence. As an additional result, the heat-kernel coefficients for transverse vectors and transverse-traceless symmetric matrices are computed to second order in the curvature.