On ambiguities and divergences in perturbative renormalization group functions

Abstract

There is an ambiguity in choosing field-strength renormalization factors in the $$ \overline{\mathrm{MS}} $$ 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.

General gauge-Yukawa-quartic β-functions at 4-3-2-loop order

“…In ref. [42], we found that there always exists a choice of renormalization constants for which β I is identical to B I (meaning that υ = 0). Thus it is perfectly valid to use B I in place of β I for the RG, a choice that also carries the benefits discussed above.…”

“…The MS β-functions are generally not unique beyond the second loop order. The flow in the direction of flavor symmetry transformations (the symmetry of the kinetic term of Lagrangian (2.1)) is fixed by the choice of renormalization constants, which are themselves ambiguous [42]. It can even be divergent in the dimensional expansion, although physics is insensitive to any such flavor rotations along the flow due to the symmetry of the action under combined field and coupling transformations.…”

“…The flavor-improved β-function B I is a particular construction of the β-function engineered to be invariant under the ambiguity [31,32,44] and has been shown to be independent of the choice of renormalization constants [42]. It factors out the flow along the flavorsymmetric direction, thereby avoiding the appearance of non-physical limit cycles in the RG flow.…”

“…To this end, we follow methods outlined in refs. [12,39,42]. In the following, we present our computational setup and discuss issues arising in such calculations.…”

“…[12] and subsequently improved for the computations in ref. [42]. It relies on QGRAF [45] for diagram generation, followed by q2e and exp [46,47] for translating Feynman diagrams into FORM [48] expressions and mapping them onto integral families, as well as COLOR [49] to compute SU(N ) color factors.…”