The absence of global anomalies of CP symmetry

Some solutions to the strong CP problem assume that CP symmetry is a gauge symmetry, which is then spontaneously broken. For this scenario to be possible, the CP symmetry should not have any nonperturbative (global) anomalies. In this paper, we study anomalies of CP symmetry of fermions which are coupled to gravity and gauge fields with a gauge group $G$. When $G$ is connected and simply connected, we show that gauging a CP symmetry does not produce any new anomaly beyond the one before gauging it. In particular, the standard model matter content does not have anomalies.

💡 Research Summary

The paper addresses a crucial consistency condition for a class of solutions to the strong CP problem in which the CP symmetry is promoted to an exact gauge symmetry that is later spontaneously broken. For such a scenario to be viable, the CP gauge symmetry must be free of both perturbative and non‑perturbative (global) anomalies. The authors systematically study anomalies associated with CP transformations of fermions coupled to gravity and to gauge fields with a gauge group (G).

1. Definition of CP in four dimensions
The authors begin by constructing the CP transformation for a single neutral Weyl fermion. They show that the transformation can be written as a spatial reflection (R_n) combined with a phase factor (\alpha = \pm i) and a (\gamma^5) matrix, yielding (R_n^2 = 1). This property identifies the relevant spacetime symmetry group as the Pin(^+)(4) group, whose defining relation is ((R_n)^2 = 1). The alternative Pin(^-)(4) group, for which ((R_n)^2 = (-1)^F), would require an even number of Weyl fermions and is not considered here.

2. Inclusion of a gauge group
When a gauge group (G) is present, the CP transformation must also act on the internal gauge indices. The authors introduce an involutive automorphism (\sigma: G \to G) satisfying (\sigma^2 = 1) and (\rho(\sigma(g)) = \rho^*(g)) for any representation (\rho). This automorphism implements charge conjugation on the gauge sector. The combined CP operation (denoted again by (R_n) for brevity) acts as:

• a spatial reflection on coordinates,
• a gauge‑field transformation (A_\mu \to (\delta_\nu^\mu - 2 n^\mu n_\nu),\sigma(A_\nu)),
• a fermion transformation (\Psi \to \alpha \gamma^5 !!\not! n ,\Psi).

Consequently, the full symmetry group is the semidirect product \