# The three-loop Adler $D$-function for ${\cal N}=1$ SQCD regularized by dimensional reduction

## TL;DR

This paper computes the three-loop Adler D-function for ${
m extbf{N=1}}$ SQCD using dimensional reduction, compares it with higher derivative regularization results, and discusses scheme dependence and NSVZ-like relations.

## Contribution

It provides the three-loop D-function in the $ar{ ext{DR}}$ scheme for ${
m extbf{N=1}}$ SQCD and analyzes scheme dependence and NSVZ-like equations.

## Key findings

- The three-loop D-function does not satisfy the NSVZ-like relation in the $ar{ ext{DR}}$ scheme.
- A scheme can be constructed where the NSVZ-like relation holds via finite renormalization.
- The three-loop D-function in terms of bare couplings does not satisfy the NSVZ-like equation for arbitrary schemes.

## Abstract

The three-loop Adler $D$-function for ${\cal N}=1$ SQCD in the $\overline{\mbox{DR}}$ scheme is calculated starting from the three-loop result recently obtained with the higher covariant derivative regularization. For this purpose, for the theory regularized by higher derivatives we find a subtraction scheme in which the Green functions coincide with the ones obtained with the dimensional reduction and the modified minimal subtraction prescription for the renormalization of the SQCD coupling constant and of the matter superfields. Also we calculate the $D$-function in the $\overline{\mbox{DR}}$ scheme for all renormalization constants (including the one for the electromagnetic coupling constant which appears due to the SQCD corrections). It is shown that the results do not satisfy the NSVZ-like equation relating the $D$-function to the anomalous dimension of the matter superfields. However, the NSVZ-like scheme can be constructed with the help of a properly tuned finite renormalization. It is also demonstrated that the three-loop $D$-function defined in terms of the bare couplings with the dimensional reduction does not satisfy the NSVZ-like equation for an arbitrary renormalization prescription. We also investigate a possibility to present the results in the form of the $\beta$-expansion and the scheme dependence of this expansion.