Telling compositeness at a distance with outer automorphisms and CP

TL;DR

This paper explores how outer automorphisms of groups can distinguish composite from elementary states through their transformation properties, providing a new method to identify compositeness without scattering experiments.

Contribution

It introduces a framework for using outer automorphisms to differentiate composite and elementary states based on their quantum numbers, clarifies conditions for this distinction, and corrects misconceptions about $ ext{SU}(N)$ automorphisms.

Findings

Outer automorphisms can distinguish composite from elementary states.

Antisymmetric matrices in $ ext{SU}(N)$ do not produce new automorphisms.

Composite states in $ ext{SU}(N)$ transform trivially under outer automorphisms.

Abstract

We investigate charge-parity (CP) and non-CP outer automorphism of groups and the transformation behavior of group representations under them. We identify situations where composite and elementary states that transform in exactly the same representation of the group, transform differently under outer automorphisms. This can be instrumental in discriminating composite from elementary states solely by their quantum numbers with respect to the outer automorphism, i.e. without the need for explicit short distance scattering experiments. We discuss under what conditions such a distinction is unequivocally possible. We cleanly separate the case of symmetry constrained (representation) spaces from the case of multiple copies of identical representations in flavor space, and identify conditions under which non-trivial transformation in flavor space can be enforced for composite states. Next to composite product states, we also discuss composite states in non-product representations. Comprehensive examples are given based on the finite groups $\Sigma(72)$ and $D_8$. The discussion also applies to $\mathrm{SU}(N)$ and we scrutinize recent claims in the literature that $\mathrm{SU}(2N)$ outer automorphism with antisymmetric matrices correspond to distinct outer automorphisms. We show that outer automorphism transformations with antisymmetric matrices are related by an inner automorphism to the standard $\mathbb{Z}_2$ outer automorphism of $\mathrm{SU}(N)$. As a direct implication, no non-trivial transformation behavior can arise for composite product states under the outer automorphism of $\mathrm{SU}(N)$.