The NSVZ $\beta$-function for theories regularized by higher covariant derivatives: the all-loop sum of matter and ghost singularities

TL;DR

This paper calculates the all-loop contributions of matter superfields and ghosts to the beta function in N=1 supersymmetric gauge theories with higher covariant derivative regularization, confirming the NSVZ relation through singularity analysis.

Contribution

It provides a comprehensive all-order calculation of the beta function contributions from matter and ghost singularities, extending the NSVZ formula in a regularization-independent manner.

Findings

Beta function expressed as sum of singular contributions from matter and ghosts.

Confirmed the all-loop NSVZ relation using singularity analysis.

Derived the beta function in terms of anomalous dimensions of superfields.

Abstract

The contributions of the matter superfields and of the Faddeev--Popov ghosts to the $\beta$-function of ${\cal N}=1$ supersymmetric gauge theories defined in terms of the bare couplings are calculated in all orders in the case of using the higher covariant derivative regularization. For this purpose we use the recently proved statement that the $\beta$-function in these theories is given by integrals of double total derivatives with respect to the loop momenta. These integrals do not vanish due to singularities of the integrands. This implies that the $\beta$-function beyond the one-loop approximation is given by the sum of the singular contributions, which is calculated in all orders for singularities produced by the matter superfields and by the Faddeev--Popov ghosts. The result is expressed in terms of the anomalous dimensions of these superfields. It coincides with the corresponding part of the new form of the NSVZ equation, which can be reduced to the original one with the help of the non-renormalization theorem for the triple gauge-ghost vertices.