# The $\beta$-function of ${\cal N}=1$ supersymmetric gauge theories regularized by higher covariant derivatives as an integral of double total derivatives

## TL;DR

This paper proves that in ${
m f N}=1$ supersymmetric gauge theories regularized by higher covariant derivatives, the $eta$-function can be expressed as integrals of double total derivatives, simplifying higher-order calculations and aiding in deriving the NSVZ relation.

## Contribution

The paper introduces a proof that the $eta$-function is given by double total derivatives and presents a method to simplify loop integral calculations in supersymmetric gauge theories.

## Key findings

- Derived the three-loop $eta$-function contribution with Yukawa couplings.
- Compared the new integral expression with standard calculations.
- Discussed implications for all-loop derivation of the NSVZ relation.

## Abstract

For a general ${\cal N}=1$ supersymmetric gauge theory regularized by higher covariant derivatives we prove in all orders that the $\beta$-function defined in terms of the bare couplings is given by integrals of double total derivatives with respect to loop momenta. With the help of the technique used for this proof it is possible to construct a method for obtaining these loop integrals, which essentially simplifies the calculations. As an illustration of this method, we find the expression for the three-loop contribution to the $\beta$-function containing the Yukawa couplings and compare it with the result of the standard calculations made earlier. Also we briefly discuss, how the structure of the loop integrals for the $\beta$-function considered in this paper can be used for the all-loop perturbative derivation of the NSVZ relation in the non-Abelian case.