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

## TL;DR

This paper develops a supergraph technique in 6D harmonic superspace to compute one-loop divergences in supersymmetric gauge theories, confirming previous results with a manifestly gauge-invariant approach.

## Contribution

It introduces an off-shell supergraph method for 6D supersymmetric gauge theories and calculates divergences, providing a new computational framework that aligns with earlier findings.

## Key findings

- Calculated one-loop divergences for 6D supersymmetric gauge theories.
- Matched previous results obtained by proper time superfield method.
- Provided explicit Green functions for gauge and hypermultiplet fields.

## Abstract

We study the one-loop effective action for $6D,$ ${\cal N}=(1,0)$ supersymmetric Yang--Mills (SYM) theory with hypermultiplets and $6D,$ ${\cal N}=(1,1)$ SYM theory as a subclass of the former, using the off-shell formulation of these theories in $6D,$ ${\cal N}=(1,0)$ harmonic superspace. We develop the corresponding supergraph technique and apply it to compute the one-loop divergences in the background field method ensuring the manifest gauge invariance. We calculate the two-point Green functions of the gauge superfield and the hypermultiplet, as well as the three-point gauge-hypermultipet Green function. Using these Green functions and exploiting gauge invariance of the theory, we find the full set of the off-shell one-loop divergent contributions, including the logarithmic and power ones. Our results precisely match with those obtained earlier in [1,2] within the proper time superfield method.