Quantum properties of gauge theories with extended supersymmetry formulated in ${\cal N}=1$ superspace

TL;DR

This paper investigates the quantum properties and renormalization behavior of ${ m N}=2$ and ${ m N}=4$ supersymmetric gauge theories formulated in ${ m N}=1$ superspace, focusing on loop contributions and scheme dependence.

Contribution

It provides a detailed analysis of the conditions under which non-renormalization theorems hold in ${ m N}=2$ and ${ m N}=4$ theories within ${ m N}=1$ superspace, including explicit multi-loop calculations.

Findings

Two-loop anomalous dimensions vanish under compatible subtraction schemes.

Three-loop beta functions generally do not vanish, indicating scheme dependence.

The NSVZ scheme yields the purely one-loop beta function in these theories.

Abstract

We analyse quantum properties of ${\cal N}=2$ and ${\cal N}=4$ supersymmetric gauge theories formulated in terms of ${\cal N}=1$ superfields and investigate the conditions imposed on a renormalization prescription under which the non-renormalization theorems are valid. For this purpose in these models we calculate the two-loop contributions to the anomalous dimensions of all chiral matter superfields and the three-loop contributions to the $\beta$-functions for an arbitrary ${\cal N}=1$ supersymmetric subtraction scheme supplementing the higher covariant derivative regularization. We demonstrate that, in general, the results do not vanish due to the scheme dependence, which becomes essential in the considered approximations. However, the two-loop anomalous dimensions vanish if a subtraction scheme is compatible with the structure of quantum corrections and does not break the relation between the Yukawa and gauge couplings which follows from ${\cal N}=2$ supersymmetry. Nevertheless, even under these conditions the three-loop contribution to the $\beta$-function does not in general vanishes for ${\cal N}=2$ supersymmetric theories. To obtain the purely one-loop $\beta$-function, one should also chose an NSVZ renormalization prescription. The similar statements for the higher loop contributions are proved in all orders.