Title: Dyons with phase $δ_θ=nθ$

Abstract: In a recent paper (Heras in $\href{this https URL}{{\rm Eur. Phys. J. Plus\, 138: 329, 2023}}$), we have demonstrated that when a dyon encircles an infinitely long solenoid enclosing electric and magnetic fluxes, its wave function accumulates a quantum phase invariant under electromagnetic duality transformations. In this paper, we show that this phase, in conjunction with the Witten effect, gives rise to a topological phase proportional to the vacuum angle $\theta$ and thereby connected with CP violation. We show that this phase becomes quantised in a vacuum state $\delta_{\theta}=n\theta$ and that the most general vacuum state associated with this quantisation identifies with an Abelian form of the $\theta$-vacua. We discuss two hypothetical interference effects in the vacuum where the angle $\theta$ could manifest.