Dimensional Reduction is Supersymmetric at Three Loops

TL;DR

This paper demonstrates that dimensional reduction (DR) preserves supersymmetry in Super Yang-Mills theories up to three loops, correcting previous misconceptions caused by algebraic subtleties, and extends this validity to higher supersymmetries.

Contribution

The authors identify and correct algebraic subtleties in three-loop calculations, confirming DR's validity for supersymmetry preservation in SYM theories up to three loops.

Findings

DR preserves supersymmetry at three loops in SYM theories.

Previous discrepancies were due to Clifford algebra treatment, not DR itself.

Supersymmetry breaking effects in DR appear only at five loops in $ =4$ SYM.

Abstract

We resolve the long-standing claim that regularisation by dimensional reduction (DR) fails to preserve supersymmetry in Super Yang-Mills (SYM) theories at three loops. Earlier results reported a mismatch between the Yukawa and ghost-gluon $\beta$ functions in $\mathcal{N}=2$ SYM, suggesting a breakdown of supersymmertry. We show that this discrepancy does not originate from DR itself but from subtleties in the treatment of the Clifford algebra. A corrected three-loop calculation restores full supersymmetric behaviour, and we demonstrate that the same issue would first affect $\mathcal{N}=4$ SYM only at five loops, consistent with existing four-loop results. Our findings confirm that DR preserves supersymmetry for $\mathcal{N}=1, 2$ and $4$ SYM through the loop orders examined.