Leptonic Dirac CP Violation Predictions from Residual Discrete Symmetries

Assuming that the observed pattern of 3-neutrino mixing is related to the existence of a (lepton) flavour symmetry, corresponding to a non-Abelian discrete symmetry group $G_f$, and that $G_f$ is broken to specific residual symmetries $G_e$ and $G_{\nu }$ of the charged lepton and neutrino mass terms, we derive sum rules for the cosine of the Dirac phase $\delta$ of the neutrino mixing matrix $U$. The residual symmetries considered are: i) $G_e=Z_2$ and $G_{\nu }=Z_n$, $n>2$ or $Z_n\times Z_m$, $n,m\ge 2$; ii) $G_e=Z_n$, $n

2$or$Z_n \times Z_m$,$n,m \geq 2$and$G_{\nu} = Z_2$;iii)$G_e = Z_2$and$G_{\nu} = Z_2$;iv)$G_e$isfullybrokenand$G_{\nu} = Z_n$,$n > 2$or$Z_n \times Z_m$,$n,m \geq 2$;andv)$G_e = Z_n$,$n > 2$or$Z_n \times Z_m$,$n,m \geq 2$and$G_{\nu}$isfullybroken\mathrm{.}Forgiven$G_e$and$G_\nu$, the sum rules for $\cos \delta$ thus derived are exact, within the approach employed, and are valid, in particular, for any $G_f$ containing $G_e$ and $G_{\nu }$ as subgroups. We identify the cases when the value of $\cos \delta$ cannot be determined, or cannot be uniquely determined, without making additional assumptions on unconstrained parameters. In a large class of cases considered the value of $\cos \delta$ can be unambiguously predicted once the flavour symmetry $G_f$ is fixed. We present predictions for $\cos \delta$ in these cases for the flavour symmetry groups $G_f=S_4$, $A_4$, $T^{\mathrm{\prime }}$ and $A_5$, requiring that the measured values of the 3-neutrino mixing parameters ${\sin }^2{\theta }_{12}$, ${\sin }^2{\theta }_{13}$ and ${\sin }^2{\theta }_{23}$, taking into account their respective $3\sigma$ uncertainties, are successfully reproduced.