Journal of High Energy Physics

, 2019:55 | Cite as

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

  • Colin PooleEmail author
  • Anders Eller Thomsen
Open Access
Regular Article - Theoretical Physics


The β-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 β-functions by parametrizing the terms in Osborn’s equation as polynomials in the couplings, then eliminating unknown à and TIJ 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 β-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 β-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 β-function of a general QFT in the \( \overline{\mathrm{MS}} \) scheme, including kinetic mixing.


Gauge Symmetry Renormalization Group 


Open Access

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© The Author(s) 2019

Authors and Affiliations

  1. 1.CP3-Origins, University of Southern DenmarkOdense MDenmark

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