Constraints on 3-and 4-loop β-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 (A) over tilde 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 (MS) over bar scheme, including kinetic mixing.