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

Colin Poole*, Anders Eller Thomsen

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

49 Downloads (Pure)


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 M S ¯ scheme, including kinetic mixing.

Original languageEnglish
Article number55
JournalJournal of High Energy Physics
Issue number9
Number of pages63
Publication statusPublished - Sep 2019


  • Gauge Symmetry
  • Renormalization Group


Dive into the research topics of 'Constraints on 3- and 4-loop β-functions in a general four-dimensional Quantum Field Theory'. Together they form a unique fingerprint.

Cite this