Notes on Operator Equations of Supercurrent Multiplets and the Anomaly Puzzle in Supersymmetric Field Theories

TL;DR

This paper explores the quantum properties of a new supercurrent multiplet in supersymmetric field theories, offering a simple resolution to the longstanding anomaly puzzle related to R-symmetry and beta functions.

Contribution

It proposes an all-orders operator equation for the supercurrent that addresses the anomaly puzzle and aligns with previous formulations when interpreted correctly.

Findings

The new supercurrent resolves the anomaly puzzle in supersymmetric theories.

The proposed operator equation is consistent with known results and previous proposals.

The work provides a framework for understanding supercurrent behavior at all perturbation orders.

Abstract

Recently, Komargodski and Seiberg have proposed a new type of supercurrent multiplet which contains the energy-momentum tensor and the supersymmetry current consistently. In this paper we study quantum properties of the supercurrent in renormalizable field theories. We point out that the new supercurrent gives a quite simple resolution to the classic problem, called the anomaly puzzle, that the Adler-Bardeen theorem applied to an R-symmetry current is inconsistent with all order corrections to $\beta$ functions. We propose an operator equation for the supercurrent in all orders of perturbation theory, and then perform several consistency checks of the equation. The operator equation we propose is consisitent with the one proposed by Shifman and Vainshtein, if we take some care in interpreting the meaning of non-conserved currents.