$CP$ Violation from Five-Dimensional QED

It is shown that QED in $(1+4)$-dimensional space-time, with the fifth dimension compactified on a circle, is, in general, a $CP$ violating theory. Depending on the fermionic boundary conditions, $CP$ violation may be either explicit (through the Scherk-Schwarz mechanism) or spontaneous (via the Hosotani mechanism). The fifth component of the gauge field acquires (at the one-loop level) a nonzero vacuum expectation value which, in the presence of two fermionic fields, leads to spontaneous $CP$ violation when the boundary conditions are $CP$ symmetric. Phenomenological consequences are illustrated by a calculation of the electric dipole moment for the fermionic zero modes.

Figure 1

The effective potential $V_{\mathrm{e}\mathrm{f}\mathrm{f}}$ (in units of ${10}^{-3}{\mathrm{T}\mathrm{e}\mathrm{V}}^4$) as a function of $a$ (in units of TeV) for $L^{-1}=0.3\mathrm{T}\mathrm{e}\mathrm{V}$, $M_1=0.2\mathrm{T}\mathrm{e}\mathrm{V}$, $M_2=0.005\mathrm{T}\mathrm{e}\mathrm{V}$, $q_1=2/3$, $q_2=-1/3$, and four choices of the twist angle $\alpha =0,\pi /2,\pi ,3\pi /2$.

Figure 2

Left: the fermionic EDM, $d_1(L)$ for ${\psi }_{i=1,n=0}$ for $\alpha =0$. Right: the same, as a function of $\alpha$ for $L=1.5{\mathrm{T}\mathrm{e}\mathrm{V}}^{-1}$. Curves correspond to the number of modes included in (18), from $|n|=1$ (bottom) to $|n|=4$ (top), indicating fast convergence. Note that the mass of the zero mode also varies with $L$. We used $e=\sqrt{4\pi {\alpha }_{\mathrm{Q}\mathrm{E}\mathrm{D}}}$ and the same parameters as in Fig. 1.