On Ambiguities and Divergences in Perturbative Renormalization Group Functions

Abstract

There is an ambiguity in choosing field-strength renormalization factors in the MS scheme starting from the 3-loop order in perturbation theory. More concerning, trivially choosing Hermitian factors has been shown to produce divergent renormalization group functions, which are commonly understood to be finite quantities. We demonstrate that the divergences of the RG functions are such that they vanish in the RG equation due to the Ward identity associated with the flavor symmetry. It turns out that any such divergences can be removed using the renormalization ambiguity and that the use of the flavor-improved β-function is preferred. We show how our observations resolve the issue of divergences appearing in previous calculations of the 3-loop SM Yukawa β-functions and provide the first calculation of the flavor-improved 3-loop SM β-functions in the gaugeless limit.

Divergences in anomalous dimension matrices of quarks at three loops: Explanation and simple solution

“…( 9), as their generalization to other vertices is trivial once the well-known structure of counterterms of non-Abelian gauge theories [6,7] in a gauge-symmetry-preserving regularization is taken into account. 4 Note that R is a first order partial differential operator and, importantly, the coefficients that multiply derivatives are at most linear in quantum (i.e. propagating) fields.…”

“…More recently, Ref. [4] proved that these divergent anomalous dimensions and beta functions still lead to a finite RG flow of renormalized correlation functions. At the same time a generic prescription for finite ('flavor improved') beta and gamma functions is given in [4], however these functions are defined in terms of additional diagrams involving external fields.…”

“…The set {Ψ I } contains both, Weyl fermions χ and their conjugates χ. One can assume that anomalous dimensions of χ are related by complex conjugation to those corresponding to χ 4. In fact, in[10] an inductive proof was given that, in a variant of smooth cutoff regularization, the complete action (containing counterterms restoring finiteness as well as counterterms restoring BRST-invariance of correlation functions) also obeys Eq.…”

“…I am also grateful to Florian Herren and Anders Eller Thomsen for helpful correspondence regarding Refs. [3,9] and [4].…”

Four-Loop Gauge and Three-Loop Yukawa Beta-Functions in a General Renormalizable Theory

“…There is also an (unphysical) ambiguity in RG functions due to possible unitary contributions [20,30,31] to fields renormalization in models with flavor symmetries. In what follows, we assume a natural choice of hermitian anomalous dimension and refer to recent work [32] for more details.…”