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The $ \beta $-functions of marginal couplings are known to be closely relatedto the $ A $-function through Osborn's equation, derived using the localrenormalization group. It is possible to derive strong constraints on the$\beta$-functions by parametrizing the terms in Osborn's equation aspolynomials in the couplings, then eliminating unknown $\tilde{A}$ and $T_{IJ}$coefficients. In this paper we extend this program to completely general gaugetheories with arbitrarily many Abelian and non-Abelian factors. We detail thecomputational strategy used to extract consistency conditions on $ \beta$-functions, and discuss our automation of the procedure. Finally, we implementthe procedure up to 4-, 3-, and 2-loops for the gauge, Yukawa and quarticcouplings respectively, corresponding to the present forefront of general $\beta $-function computations. We find an extensive collection of highlynon-trivial constraints, and argue that they constitute an useful supplement totraditional perturbative computations; as a corollary, we present the complete3-loop gauge $\beta$-function of a general QFT in the $\bar{\text{MS}}$ scheme,including kinetic mixing.

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