The all-loop perturbative derivation of the NSVZ $\beta$-function and the NSVZ scheme in the non-Abelian case by summing singular contributions

TL;DR

This paper provides a comprehensive all-loop derivation of the NSVZ $eta$-function for ${ m extbf{N}=1}$ supersymmetric gauge theories, connecting it with anomalous dimensions and establishing the NSVZ scheme within a specific regularization framework.

Contribution

It presents a novel all-loop derivation of the NSVZ $eta$-function using singularity summation in higher covariant derivative regularization, clarifying scheme dependence.

Findings

Derived the NSVZ $eta$-function in terms of anomalous dimensions.

Established the HD+MSL scheme as an NSVZ scheme for renormalized functions.

Proved the scheme independence of the NSVZ relation in the bare coupling formulation.

Abstract

The perturbative all-loop derivation of the NSVZ $\beta$-function for ${\cal N}=1$ supersymmetric gauge theories regularized by higher covariant derivatives is finalized by calculating the sum of singularities produced by quantum superfields. These singularities originate from integrals of double total derivatives and determine all contributions to the $\beta$-function starting from the two-loop approximation. Their sum is expressed in terms of the anomalous dimensions of the quantum gauge superfield, of the Faddeev--Popov ghosts, and of the matter superfields. This allows obtaining the NSVZ equation in the form of a relation between the $\beta$-function and these anomalous dimensions for the renormalization group functions defined in terms of the bare couplings. It holds for an arbitrary renormalization prescription supplementing the higher covariant derivative regularization. For the renormalization group functions defined in terms of the renormalized couplings we prove that in all loops one of the NSVZ schemes is given by the HD+MSL prescription.