On two-loop divergences of effective action in $6D$, ${\cal N}=(1,1)$ SYM theory

TL;DR

This paper investigates the two-loop divergences in the effective action of 6D, ${ m N}=(1,1)$ supersymmetric Yang-Mills theory using harmonic superspace, revealing non-local harmonic-dependent divergences.

Contribution

It develops a harmonic superfield background field method for off-shell two-loop calculations in 6D, ${ m N}=(1,1)$ SYM, and explicitly computes non-local divergences.

Findings

Identifies gauge and hypermultiplet divergences at two loops.

Demonstrates non-locality of divergences in harmonic variables.

Provides explicit divergence coefficients in dimensional regularization.

Abstract

We study the off-shell structure of the two-loop effective action in $6D, {\cal N}=(1,1)$ supersymmetric gauge theories formulated in ${\cal N}=(1,0)$ harmonic superspace. The off-shell effective action involving all fields of $6D, {\cal N}=(1,1)$ supermultiplet is constructed by the harmonic superfield background field method, which ensures both manifest gauge covariance and manifest ${\cal N}=(1,0)$ supersymmetry. We analyze the off-shell divergences dependent on both gauge and hypermultiplet superfields and argue that the gauge invariance of the divergences is consistent with the non-locality in harmonics. The two-loop contributions to the effective action are given by harmonic supergraphs with the background gauge and hypermultiplet superfields. The procedure is developed to operate with the harmonic-dependent superpropagators in the two-loop supergraphs within the superfield dimensional regularization. We explicitly calculate the gauge and the hypermultiplet-mixed divergences as the coefficients of $\frac{1}{{\varepsilon}^2}$ and demonstrate that the corresponding expressions are non-local in harmonics.