# On gauge dependence of the one-loop divergences in $6D$, ${\cal N} = (1,0)$ and ${\cal N} = (1,1)$ SYM theories

## TL;DR

This paper investigates how gauge choices affect one-loop divergence calculations in 6D supersymmetric gauge theories, developing a gauge-invariant method and revealing gauge dependence of divergences off-shell.

## Contribution

It introduces a gauge-invariant approach to compute one-loop divergences in 6D supersymmetric theories and analyzes their gauge dependence explicitly.

## Key findings

- Divergences depend on gauge parameter $\xi_0$ off-shell.
- Divergences vanish on-shell, ensuring gauge independence of physical S-matrix.
- Non-minimal gauges show non-vanishing divergences off-shell in ${
m N}=(1,1)$ SYM.

## Abstract

We study the gauge dependence of one-loop divergences in a general matter-coupled $6D$, ${\cal N}=(1,0)$ supersymmetric gauge theory in the harmonic superspace formulation. Our analysis is based on the effective action constructed by the background superfield method, with the gauge-fixing term involving one real parameter $\xi_0$. A manifestly gauge invariant and ${\cal N}=(1,0)$ supersymmetric procedure for calculating the one-loop effective action is developed. It yields the one-loop divergences in an explicit form and allows one to investigate their gauge dependence. As compared to the minimal gauge, $\xi_0=1$, the divergent part of the general-gauge effective action contains a new term depending on $\xi_0\,$. This term vanishes for the background superfields satisfying the classical equations of motion, so that the $S$-matrix divergences are gauge-independent. In the case of $6D$, ${\cal N} = (1,1)$ SYM theory we demonstrate that some divergent contributions in the non-minimal gauges do not vanish off shell, as opposed to the minimal gauge.