CP Violation in Dual Dark Matter

Abstract

In this paper we study the consequences of extending dual symmetry (DuSy) to include a generic $C_{\mu }$ vector field as a dual partner of the photon $A_{\mu }$. A new combined field, the complex $Z_{\mu }^{\prime }$, is obtained from $C_{\mu }$ and $A_{\mu }$. The promotion of dual symmetry to a local symmetry for $Z_{\mu }^{\prime }$ implies in the inclusion of an extra complex vector field $W_{\mu }$ with a complex gauge transformation. A dual dark matter (DM) Lagrangian ${\mathcal{L}}_{\mathrm{DDM}}$ is obtained from the general DuSy invariant Lagrangian ${\mathcal{L}}_{\mathrm{DuSy}}$. Our tentative conjecture is to interpret $W_{\mu }$ as the actual weak interaction charged gauge boson $W_{\mu }^{\pm }$, which leads us to speculate about a possible extra CP violation scenarios for future calculations.