Gauge dependence of the one-loop divergences in $6D$, ${\cal N} = (1,0)$ abelian theory

TL;DR

This paper investigates how the choice of gauge affects the one-loop divergences in a six-dimensional supersymmetric abelian gauge theory, revealing gauge-dependent divergences and their gauge invariance on-shell.

Contribution

It introduces the superfield -gauge, constructs the gauge superfield propagator, and calculates the one-loop Green functions, demonstrating gauge dependence and gauge invariance on-shell.

Findings

Two-point hypermultiplet Green function is divergent in general -gauge.

Three-point Green function with hypermultiplets and gauge superfield is divergent.

Gauge dependence of Green functions vanishes on shell.

Abstract

We study the gauge dependence of the one-loop effective action for the abelian $6D$, ${\cal N}=(1,0)$ supersymmetric gauge theory formulated in harmonic superspace. We introduce the superfield $\xi$-gauge, construct the corresponding gauge superfield propagator, and calculate the one-loop two-and three-point Green functions with two external hypermultiplet legs. We demonstrate that in the general $\xi$-gauge the two-point Green function of the hypermultiplet is divergent, as opposed to the Feynman gauge $\xi =1$. The three-point Green function with two external hypermultiplet legs and one leg of the gauge superfield is also divergent. We verified that the Green functions considered satisfy the Ward identity formulated in ${\cal N}=(1,0)$ harmonic superspace and that their gauge dependence vanishes on shell. Using the result for the two- and three-point Green functions and arguments based on the gauge invariance, we present the complete divergent part of the one-loop effective action in the general $\xi$-gauge.