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

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.