The $\beta$-function of supersymmetric theories from vacuum supergraphs: a three-loop example

TL;DR

This paper verifies a method to compute the three-loop $eta$-function in supersymmetric theories using vacuum supergraphs, confirming its correctness through detailed calculations and gauge independence.

Contribution

It introduces and validates a novel technique for calculating the $eta$-function in supersymmetric gauge theories using modified vacuum supergraphs at three loops.

Findings

The method successfully reproduces known results in Feynman gauge.

Gauge-dependent terms cancel as expected, confirming gauge invariance.

The approach simplifies multiloop calculations in supersymmetric theories.

Abstract

We verify a method which allows to obtain the $\beta$-function of supersymmetric theories regularized by higher covariant derivatives by calculating only specially modified vacuum supergraphs. With the help of this method for a general renormalizable ${\cal N}=1$ supersymmetric gauge theory a part of the three-loop $\beta$-function depending on the Yukawa couplings is constructed in the general $\xi$-gauge. The result is written in the form of an integral of double total derivatives with respect to the loop momenta. It is demonstrated that all gauge dependent terms cancel each other in agreement with the general statements. Taking into account that the result in the Feynman gauge (found earlier) coincides with the one obtained by the standard technique, this fact confirms the correctness of the considered method by a highly nontrivial multiloop calculation.