Leptonic Dirac CP violation predictions from residual discrete symmetries

Abstract 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 ν of the charged lepton and neutrino mass terms, we derive sum rules for the cosine of the Dirac phase δ of the neutrino mixing matrix U . The residual symmetries considered are: i) G e = Z 2 and G ν = Z n , n > 2 or Z n × Z m , n , m ≥ 2 ; ii) G e = Z n , n > 2 or Z n × Z m , n , m ≥ 2 and G ν = Z 2 ; iii) G e = Z 2 and G ν = Z 2 ; iv) G e is fully broken and G ν = Z n , n > 2 or Z n × Z m , n , m ≥ 2 ; and v) G e = Z n , n > 2 or Z n × Z m , n , m ≥ 2 and G ν is fully broken. For given G e and G ν , the sum rules for cos ⁡ δ thus derived are exact, within the approach employed, and are valid, in particular, for any G f containing G e and G ν as subgroups. We identify the cases when the value of cos ⁡ δ 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 ⁡ δ can be unambiguously predicted once the flavour symmetry G f is fixed. We present predictions for cos ⁡ δ in these cases for the flavour symmetry groups G f = S 4 , A 4 , T ′ and A 5 , requiring that the measured values of the 3-neutrino mixing parameters sin 2 ⁡ θ 12 , sin 2 ⁡ θ 13 and sin 2 ⁡ θ 23 , taking into account their respective 3 σ uncertainties, are successfully reproduced.