Construction of one-loop ${\cal N}=4$ SYM effective action on the mixed branch in the harmonic superspace approach

TL;DR

This paper develops a systematic method to construct the one-loop effective action of ${ m abla}4$ SYM theory in harmonic superspace, incorporating both vector multiplet and hypermultiplet backgrounds, and expresses it in terms of harmonic superfields.

Contribution

It introduces a new approach to derive the one-loop effective action in ${ m abla}4$ SYM using covariant harmonic supergraphs and provides explicit formulas for constant background fields.

Findings

Effective action expressed as integral over harmonic superspace

Each term in the Schwinger-De Witt expansion written as integral over full ${ m abla}2$ superspace

Hypermultiplet-dependent effective action explicitly derived

Abstract

We develop a systematic approach to construct the one-loop ${\cal N}=4$ SYM effective action depending on both ${\cal N}=2$ vector multiplet and hypermultiplet background fields. Beginning with the formulation of ${\cal N}=4$ SYM theory in terms of ${\cal N}=2$ harmonic superfields, we construct the one-loop effective action using the covariant ${\cal N}=2$ harmonic supergraphs and calculate it in ${\cal N}=2$ harmonic superfield form for constant Abelian strength $F_{mn}$ and corresponding constant hypermultiplet fields. The hypermultiplet-dependent effective action is derived and given by integral over the analytic subspace of harmonic superspace. We show that each term in the Schwinger-De Witt expansion of the low-energy effective action is written as integral over full ${\cal N}=2$ superspace.