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 $Z_n \times Z_m$, $n,m \geq 2$; 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) $G_e$ is fully broken and $G_{\nu} = Z_n$, $n > 2$ or $Z_n \times Z_m$, $n,m \geq 2$; and v) $G_e = Z_n$, $n > 2$ or $Z_n \times Z_m$, $n,m \geq 2$ and $G_{\nu}$ is fully broken. For given $G_e$ 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 $G_f = S_4$, $A_4$, $T^\prime$ 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.