Telling compositeness at a distance with outer automorphisms and CP

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 $\mathrm{SU}(N)$.

arXiv comment29+10 pages, 1 table. Incl. rebuttal of claims of existence of non-standard outer automorphism transformation of $\mathrm{SU}(2N)$, as recently proposed by 1903.06662, 2111.04728, 2212.02413

References in corpus (24)

• Theory and phenomenology of two-Higgs-doublet models
• Non-Abelian Discrete Symmetries in Particle Physics
• Unification of Flavor, CP, and Modular Symmetries
• CP and Discrete Flavour Symmetries
• Generalised CP Symmetry in Modular-Invariant Models of Flavour
• CP Violation from Finite Groups
• Automorphisms in Gauge Theories and the Definition of CP and P
• CP Violation in the General Two-Higgs-Doublet Model: a Geometric View
• Group Theoretical Origin of CP Violation
• A CP-conserving multi-Higgs model without real basis
• Gauge theories from principally extended disconnected gauge groups
• Natural Vacuum Alignment from Group Theory: The Minimal Case
• CP4 miracle: shaping Yukawa sector with CP symmetry of order four
• Discrete gauge theories of charge conjugation
• Symmetries of symmetries and geometrical CP violation
• Multi-particle Representations of the Poincaré Group
• When the C in CP does not matter: anatomy of order-4 CP eigenstates and their Yukawa interactions
• Multi-Higgs models with CP symmetries of increasingly high order
• Outer Automorphism Anomalies
• 23, 381, 6242, 103268, 1743183, ...: Hilbert series for CP-violating operators in SMEFT
• Exotic hadron naming convention
• Zwanziger's pairwise little group on the celestial sphere
• Simultaneous Block Diagonalization of Matrices of Finite Order
• CP as a Symmetry of Symmetries

Cited by in corpus (2)

• Symmetries from outer automorphisms and unorthodox group extensions
• CP-like Symmetry with Discrete and Continuous Groups and CP Violation/Restoration