Three-loop contribution of the Faddeev-Popov ghosts to the $\beta$-function of ${\cal N}=1$ supersymmetric gauge theories and the NSVZ relation

TL;DR

This paper calculates the three-loop contribution of Faddeev-Popov ghosts to the beta function in ${ m N}=1$ supersymmetric gauge theories, confirming the NSVZ relation using a new computational algorithm.

Contribution

It introduces a simplified algorithm for three-loop calculations of ghost contributions and verifies the NSVZ relation in this context.

Findings

The three-loop ghost contribution to the beta function is expressed as an integral of double total derivatives.

The NSVZ relation holds for the beta function and anomalous dimensions at this order.

Nonlinear renormalization of the quantum gauge superfield is confirmed as necessary.

Abstract

We find the three-loop contribution to the $\beta$-function of ${\cal N}=1$ supersymmetric gauge theories regularized by higher covariant derivatives produced by the supergraphs containing loops of the Faddeev--Popov ghosts. This is done using a recently proposed algorithm, which essentially simplifies such multiloop calculations. The result is presented in the form of an integral of double total derivatives in the momentum space. The considered contribution to the $\beta$-function is compared with the two-loop anomalous dimension of the Faddeev--Popov ghosts. This allows verifying the validity of the NSVZ equation written as a relation between the $\beta$-function and the anomalous dimensions of the quantum superfields. It is demonstrated that in the considered approximation the NSVZ equation is satisfied for the renormalization group functions defined in terms of the bare couplings. The necessity of the nonlinear renormalization for the quantum gauge superfield is also confirmed.