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Constraints on 3- and 4-loop \(\beta\)-functions in a general four-dimensional quantum field theory. (English) Zbl 1423.81128

Summary: 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 \(\overline{\mathrm{MS}}\) scheme, including kinetic mixing.

MSC:

81T10 Model quantum field theories
81T17 Renormalization group methods applied to problems in quantum field theory

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References:

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