https://doi.org/10.1140/epjp/s13360-023-04629-3
Regular Article
Dyons with phase 
School of Physical Sciences, The Open University, Walton Hall, MK7 6AA, Milton Keynes, UK
Received:
22
September
2023
Accepted:
24
October
2023
Published online:
7
November
2023
In a recent paper (Heras in 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 
and thereby connected with CP violation. We show that this phase becomes quantised in a vacuum state and that the most general vacuum state associated with this quantisation identifies with an Abelian form of the -vacua. We discuss two hypothetical interference effects in the vacuum where the angle could manifest.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
