Four-loop beta function in N=2 supersymmetric Yang-Mills theory, dimensional reduction, and treatment of ${\gamma }_5$

Abstract

Adopting the dimensional-reduction scheme, we diagrammatically compute, at four loops, the beta function of N=2 supersymmetric Yang-Mills theory, which is well known to vanish beyond one loop. This provides a formidable testbed for both the treatment of the problematic ${\gamma }_5$ Dirac matrix in D dimensions and the implementation of the dimensional-reduction scheme. In the use of the reading-point method of ${\gamma }_5$, being preferable for convenience, specific details of implementation start to matter at four loops. We thus rigorously validate these implementation issues, which have previously been advocated for the calculation of the gauge-coupling beta functions at four loops in the Standard Model.

5 pages, 1 figure