Regularization by Dimensional Reduction: Consistency, Quantum Action Principle, and Supersymmetry

TL;DR

The paper addresses the inconsistency of dimensional reduction in SUSY gauge theories and its impact on symmetry relations. It establishes a mathematically consistent formulation of DRED and proves the quantum action principle (QAP), expressing the relation between the regularized Lagrangian ${\cal L}$ and Ward/Slavnov-Taylor identities via $i\,\delta\langle T\phi_1\ldots\phi_n\rangle = \langle T\phi_1\ldots\phi_n\Delta\rangle$ with $\Delta = \int d^D x\, \delta{\cal L}$. The analysis shows that in DRED the SUSY variation ${\rm \delta}_{\rm SUSY}{\cal L}$ does not, in general, vanish due to the introduction of a quasi-4-dimensional space (Q4S), so SUSY identities hold only to the extent that the right-hand side vanishes. The approach is demonstrated by checking several one- and two-loop identities, illustrating its practical utility for SUSY calculations.

Abstract

It is proven by explicit construction that regularization by dimensional reduction can be formulated in a mathematically consistent way. In this formulation the quantum action principle is shown to hold. This provides an intuitive and elegant relation between the D-dimensional Lagrangian and Ward or Slavnov-Taylor identities, and it can be used in particular to study to what extent dimensional reduction preserves supersymmetry. We give several examples of previously unchecked cases.