# Three-loop NSVZ relation for terms quartic in the Yukawa couplings with the higher covariant derivative regularization

## TL;DR

This paper proves that the NSVZ relation holds for quartic Yukawa terms in non-Abelian ${\
m N}=1$ supersymmetric gauge theories at three loops when using higher covariant derivative regularization, linking loop integrals to anomalous dimensions.

## Contribution

It demonstrates the validity of the NSVZ relation for quartic Yukawa terms at three loops with higher covariant derivative regularization and clarifies scheme dependence.

## Key findings

- Three-loop beta function terms are integrals of double total derivatives.
- The three-loop contribution reduces to two-loop anomalous dimensions.
- NSVZ scheme boundary conditions ensure the relation holds.

## Abstract

We demonstrate that in non-Abelian ${\cal N}=1$ supersymmetric gauge theories the NSVZ relation is valid for terms quartic in the Yukawa couplings independently of the subtraction scheme if the renormalization group functions are defined in terms of the bare couplings and the theory is regularized by higher covariant derivatives. The terms quartic in the Yukawa couplings appear in the three-loop $\beta$-function and in the two-loop anomalous dimension of the matter superfields. We have obtained that the three-loop contribution to the $\beta$-function quartic in the Yukawa couplings is given by an integral of double total derivatives. Consequently, one of the loop integrals can be taken and the three-loop contribution to the $\beta$-function is reduced to the two-loop contribution to the anomalous dimension. The remaining loop integrals have been calculated for the simplest form of the higher derivative regularizing term. Then we construct the renormalization group functions defined in terms of the renormalized couplings. In the considered approximation they do not satisfy the NSVZ relation for a general renormalization prescription. However, we verify that the recently proposed boundary conditions defining the NSVZ scheme in the non-Abelian case really lead to the NSVZ relation between the terms of the considered structure.