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Dimensional Reduction is Supersymmetric at Three Loops

Mrigankamauli Chakraborty, Sven-Olaf Moch

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 $β$ 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.

Dimensional Reduction is Supersymmetric at Three Loops

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 functions in 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 SYM only at five loops, consistent with existing four-loop results. Our findings confirm that DR preserves supersymmetry for and SYM through the loop orders examined.
Paper Structure (4 sections, 16 equations, 3 figures, 1 table)

This paper contains 4 sections, 16 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The generalised Lagrangian
  3. Restoring SUSY
  4. Conclusions

Figures (3)

  • Figure 1: Left: fermion loop connected to gluons only (arrows indicate fermion flow). Right: fermion loop with scalars.
  • Figure 2: Sample topologies of diagrams that yield a non‑vanishing Levi-Civita contribution in the integrand with gluons, fermions (solid lines, arrows indicate fermion flow), and scalars (dashed lines).
  • Figure 3: Diagram where Levi‑Civita contributions appear for the first time in $\mathcal{N}=4$ SYM, resembling the Urdhwa Pundra (upward lotus) symbol of Lord Vishnu in Hindu mythology.