Note on gauge and gravitational anomalies of discrete $Z_N$ symmetries

TL;DR

This paper examines the conditions under which discrete $Z_N$ symmetries in supersymmetric models are free from gauge and gravitational anomalies, identifying specific non-anomalous symmetries and their violations due to gravitational instantons.

Contribution

It systematically analyzes anomaly conditions for $Z_N$ symmetries in the MSSM, revealing which are non-anomalous and how gravitational instantons affect their viability.

Findings

Certain $Z_N$ symmetries forbid Baryon number violation up to dimension 6.

No non-anomalous $Z_3$ ${ m R}$-symmetry exists in the MSSM.

Gravitational instantons violate many $Z_N$ ${ m R}$-symmetries in supergravity.

Abstract

In this note, we discuss the consistency conditions which a discrete $Z_N$ symmetry should satisfy in order that it is not violated by gauge and gravitational instantons. As examples, we enlist all the $Z_N$ ${\cal R}$-symmetries as well as non-${\cal R}$ $Z_N$ symmetries (N=2,3,4) in the minimally supersymmetric standard model (MSSM) that are free from gauge and gravitational anomalies. We show that there exists non-anomalous discrete symmetries that forbid Baryon number violation up to dimension 6 level (in superspace). We also observe that there exists no non-anomalous $Z_3$ ${\cal R}$-symmetry in the MSSM. Furthermore, we point out that in a theory with one Majorana spin 3/2 gravitino, a large class of $Z_4$ ${\cal R}$-symmetries are violated in the presence of Eguchi-Hanson (EH) gravitational instanton. This is also in general true for higher $Z_N$ ${\cal R}$-symmetries. We also notice that in 4 dimensional ${\cal N}=1$ supergravity, the global $U(1)$ ${\cal R}$-symmetry is always violated by the EH instanton irrespective of the matter content of the theory.