# Constraints on 3- and 4-loop $\beta$-functions in a general four-dimensional Quantum Field Theory

## TL;DR

This paper derives and automates constraints on 3- and 4-loop $eta$-functions in general four-dimensional quantum field theories, providing new consistency conditions and the complete 3-loop gauge $eta$-function including kinetic mixing.

## Contribution

It extends the Osborn equation approach to general gauge theories and automates the derivation of high-loop $eta$-function constraints, including the first complete 3-loop gauge $eta$-function with kinetic mixing.

## Key findings

- Derived extensive non-trivial constraints on $eta$-functions.
- Automated the computational procedure for high-loop order calculations.
- Presented the complete 3-loop gauge $eta$-function for general QFTs.

## Abstract

The $ \beta $-functions of marginal couplings are known to be closely related to the $ A $-function through Osborn's equation, derived using the local renormalization group. It is possible to derive strong constraints on the $\beta$-functions by parametrizing the terms in Osborn's equation as polynomials in the couplings, then eliminating unknown $\tilde{A}$ and $T_{IJ}$ coefficients. In this paper we extend this program to completely general gauge theories with arbitrarily many Abelian and non-Abelian factors. We detail the computational strategy used to extract consistency conditions on $ \beta $-functions, and discuss our automation of the procedure. Finally, we implement the procedure up to 4-, 3-, and 2-loops for the gauge, Yukawa and quartic couplings respectively, corresponding to the present forefront of general $ \beta $-function computations. We find an extensive collection of highly non-trivial constraints, and argue that they constitute an useful supplement to traditional perturbative computations; as a corollary, we present the complete 3-loop gauge $\beta$-function of a general QFT in the $\bar{\text{MS}}$ scheme, including kinetic mixing.