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

• Published in: The journal of high energy physics 2019-09, Vol.2019 (9), p.1-63, Article 55
• Main Authors: Poole, Colin, Thomsen, Anders Eller
• Format: Article
• Language: English
• Subjects: Classical and Quantum Gravitation; Couplings; Elementary Particles; Field theory; Gauge Symmetry; High energy physics; Physics; Physics and Astronomy; Polynomials; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Quantum theory; Regular Article - Theoretical Physics; Relativity Theory; Renormalization Group; String Theory
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP09(2019)055

A bstract 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 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 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.