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

Understand

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.