# Vanishing OPE Coefficients in 4d N=2 SCFTs

## TL;DR

This paper analyzes 4d N=2 superconformal field theories, revealing that certain operator product expansion coefficients vanish in specific Argyres-Douglas theories, indicating a form of operator decoupling.

## Contribution

It introduces a method to compute superconformal characters and shows the vanishing of specific OPE coefficients in various AD theories, expanding understanding of their operator spectra.

## Key findings

- Certain short multiplets are absent in specific AD theories.
- OPE coefficients involving these multiplets vanish at particular central charges.
- The Macdonald index computations support the vanishing of multiplets in generalized AD theories.

## Abstract

We compute the superconformal characters of various short multiplets in 4d N=2 superconformal algebra, from which selection rules for operator products are obtained. Combining with the superconformal index, we show that a particular short multiplet appearing in the n-fold product of stress-tensor multiplet is absent in the $(A_1, A_{2n})$ Argyres-Douglas (AD) theory. This implies that the operator product expansion (OPE) coefficients involving this multiplet vanish whenever the central charge $c$ is identical to that of the AD theory. Similarly, by considering the n-th power of the current multiplet, we show that a particular short multiplet and OPE coefficients vanish for a class of AD theories with ADE flavor symmetry. We also consider the generalized AD theory of type $(A_{k-1}, A_{n-1})$ for coprime k, n and compute its Macdonald index using the associated W-algebra under a mild assumption. This allows us to show that a number of short multiplets and OPE coefficients vanish in this theory. We also provide a Mathematica file along with this paper, where we implement the algorithm by Cordova-Dumitrescu-Intriligator to compute the spectrum of 4d N=2 superconformal multiplets as well as their superconformal character.