Holomorphy, rescaling anomalies and exact β functions in supersymmetric gauge theories

Abstract

There have long been known "exact" β functions for the gauge coupling in N = 1 supersymmetric gauge theories, the so-called Novikov-Shifman-Vainshtein-Zakharov (NSVZ) β functions. Shifman and Vainshtein further related these β functions to the exact 1-loop running of the gauge coupling in a "Wilsonian" action. All these results, however, remain somewhat mysterious. We attempt to clarify these issues by presenting new perspectives on the NSVZ β function. Our interpretation of the results is somewhat different than the one given by Shifman and Vainshtein, having nothing to do with the distinction between "Wilsonian" and "1PI" effective actions. Throughout we work in the context of the Wilsonian Renormalization Group; namely, as the cutoff of the theory is changed from M to M ′ , we seek to determine the appropriate changes in the bare couplings needed in order to keep the low energy physics fixed. The entire analysis is therefore free of infrared subtleties. When the bare Lagrangian given at the cutoff is manifestly holomorphic in the gauge coupling, we show that the required change in the holomorphic gauge coupling is exhausted at 1-loop to all orders of perturbation theory, and even non-perturbatively in some cases. On the other hand, when the bare Lagrangian at the cutoff has canonically normalized kinetic terms, we find that the required change in the gauge coupling is given by the NSVZ β function. The higher order contributions in the NSVZ β function are due to anomalous Jacobians under the rescaling of the fields done in passing from holomorphic to canonical normalization. We also give prescriptions for regularizing certain N = 1 theories with an ultraviolet cutoff M preserving manifest holomorphy, starting from finite N = 4 and N = 2 theories. It is then at least in principle possible to check the validity of the exact β function by higher order calculations in these theories. * This work was supported in part by the

Running and matching from 5 to 4 dimensions

“…Then one sees directly that in eq. (21), the term ∝ C(A) represents the well known rescaling anomaly of the gauge multiplet wave function [18,19] while the matter contribution is just the Konishi anomaly [20].…”

Duality of nonsupersymmetric largeNgauge theories

“…The contents of this subsection are not new and can be found in the literature [11,15,16,17]. We start with the Lagrangian in the customary normalization…”

One-loop divergences in non-Abelian supersymmetric theories regularized by BRST-invariant version of the higher derivative regularization

“…In the early papers the NSVZ β-function was obtained from general arguments, such as the structure of instanton contributions to the effective action [2,8], anomalies [3,5,9],…”