{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/constraints-on-3-and-4-loop-beta-functions-in","title":"Constraints on 3- and 4-loop $\\beta$-functions in a general four-dimensional Quantum Field Theory","arxiv_id":"1906.04625","date":"2019-06-11","proceeding":null,"authors":["Colin Poole","Anders Eller Thomsen"],"abstract":"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 $\\bar{\\text{MS}}$ scheme, including kinetic mixing.","url_abs":"http://arxiv.org/abs/1906.04625v3","url_pdf":"http://arxiv.org/pdf/1906.04625v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"constraints-on-3-and-4-loop-beta-functions-in","repo_url":"https://github.com/LSartore/pyrate","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"constraints-on-3-and-4-loop-beta-functions-in","repo_url":"https://github.com/apik/RGBeta","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}