Clarifying Some Remaining Questions in the Anomaly Puzzle

TL;DR

This paper investigates the anomaly puzzle in N=1 supersymmetric Yang-Mills theories, providing explicit expressions for supercurrents and clarifying the incompatibility of combining the R-current and stress tensor into a single supercurrent.

Contribution

It offers a detailed analysis of the supercurrents in supersymmetric theories, including explicit formulas and clarifications on the anomaly structure and the role of infrared effects.

Findings

The supercurrent with the energy-momentum tensor has an anomaly consistent with the trace anomaly.

The R-current satisfies the Adler-Bardeen theorem and remains separate from the supercurrent with the energy-momentum tensor.

No consistent supercurrent contains both the R-current and the stress tensor in the quantum theory.

Abstract

We discuss several points that may help to clarify some questions that remain about the anomaly puzzle in supersymmetric theories. In particular, we consider a general N=1 supersymmetric Yang-Mills theory. The anomaly puzzle concerns the question of whether there is a consistent way to put the R-current and the stress tensor in a single supercurrent, even though in the classical theory they are in the same supermultiplet. As is well known, the classically conserved supercurrent bifurcates into two supercurrents having different anomalies in the quantum regime. The most interesting result we obtain is an explicit expression for the lowest component of one of the two supercurrents in 4-dimensional spacetime, namely the supercurrent that has the energy-momentum tensor as one of its components. This expression for the lowest component is an energy-dependent linear combination of two chiral currents, which itself does not correspond to a classically conserved chiral current. The lowest component of the other supercurrent, namely, the R-current, satisfies the Adler-Bardeen theorem. The lowest component of the first supercurrent has an anomaly that we show is consistent with the anomaly of the trace of the energy-momentum tensor. Therefore, we conclude that there is no consistent way to put the R-current and the stress tensor in a single supercurrent in the quantized theory. We also discuss and try to clarify some technical points in the derivations of the two-supercurrents in the literature. These latter points concern the significance of infrared contributions to the NSVZ beta-function and the role of the equations of motion in deriving the two supercurrents.