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 3σ uncertainties, are successfully reproduced.Within the approach employed this sum rule is exact. 5 It is valid, in particular, for any value of the angle θ ν 23 [14]. 6 In [11], by using the sum rule in eq. (12), predictions for cos δ and δ were obtained in the TBM, BM, GRA, GRB and HG cases for the best fit values of sin 2 θ 12 , sin 2 θ 23 and sin 2 θ 13 . The results thus obtained permitted to conclude that a sufficiently precise measurement of cos δ would allow to discriminate between the different forms ofŨ ν considered.Statistical analyses of predictions of the sum rule given in eq. (12) i) for δ and for the J CP factor, which determines the magnitude of CP-violating effects in neutrino oscillations [38], using the current uncertainties in the determination of sin 2 θ 12 , sin 2 θ 13 , sin 2 θ 23 and δ from [28], and ii) for cos δ using the prospective uncertainties on sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 , were performed in [13] for the five symmetry forms -BM (LC), TBM, GRA, GRB and HG -of U ν .In [14] we extended the analyses performed in [11,13] by obtaining sum rules for cos δ for the following forms of the matricesŨ e andŨ ν : 7 A.Ũ ν = R 23 (θ ν 23 )R 12 (θ ν 12 ) with θ ν 23 = −π/4 and θ ν 12 as dictated by TBM, BM, GRA, GRB or HG mixing, and i)Ũ e = R −1 13 (θ e 13 ), ii)Ũ e = R −1 23 (θ e 23 )R −1 13 (θ e 13 ), and iii)Ũ e = R −1 13 (θ e 13 )R −1 12 (θ e 12 ); B.Ũ ν = R 23 (θ ν 23 )R 13 (θ ν 13 )R 12 (θ ν 12 ) with θ ν 23 , θ ν 13 and θ ν 12 fixed by arguments associated with symmetries, and iv)Ũ e = R −1 12 (θ e 12 ), and v)Ũ e = R −1 13 (θ e 13 ).The sum rules for ...

We derive predictions for the Dirac phase present in the unitary neutrino mixing matrix , where and are unitary matrices which arise from the diagonalisation, respectively, of the charged lepton and the neutrino mass matrices. We consider forms of and allowing us to express as a function of three neutrino mixing angles, present in U, and the angles contained in . We consider several forms of determined by, or associated with, symmetries, tri-bimaximal, bimaximal, etc., for which the angles in are fixed. For each of these forms and forms of allowing one to reproduce the measured values of the neutrino mixing angles, we construct the likelihood function for , using (i) the latest results of the global fit analysis of neutrino oscillation data, and (ii) the prospective sensitivities on the neutrino mixing angles. Our results, in particular, confirm the conclusion, reached in earlier similar studies, that the measurement of the Dirac phase in the neutrino mixing matrix, together with an improvement of the precision on the mixing angles, can provide unique information as regards the possible existence of symmetry in the lepton sector.

Using the fact that the neutrino mixing matrix U = U † e U ν , where U e and U ν result from the diagonalisation of the charged lepton and neutrino mass matrices, we analyse the sum rules which the Dirac phase δ present in U satisfies when U ν has a form dictated by, or associated with, discrete symmetries and U e has a "minimal" form (in terms of angles and phases it contains) that can provide the requisite corrections to U ν , so that reactor, atmospheric and solar neutrino mixing angles θ 13 , θ 23 and θ 12 have values compatible with the current data. The following symmetry forms are considered: i) tri-bimaximal (TBM), ii) bimaximal (BM) (or corresponding to the conservation of the lepton charge L = L e − L μ − L τ (LC)), iii) golden ratio type A (GRA), iv) golden ratio type B (GRB), and v) hexagonal (HG). We investigate the predictions for δ in the cases of TBM, BM (LC), GRA, GRB and HG forms using the exact and the leading order sum rules for cos δ proposed in the literature, taking into account also the uncertainties in the measured values of sin 2 θ 12 , sin 2 θ 23 and sin 2 θ 13 . This allows us, in particular, to assess the accuracy of the predictions for cos δ based on the leading order sum rules and its dependence on the values of the indicated neutrino mixing parameters when the latter are varied in their respective 3σ experimentally allowed ranges.

Abstract:We analyse the interplay of generalised CP transformations and the nonAbelian discrete group T and use the semi-direct product G f = T H CP , as family symmetry acting in the lepton sector. The family symmetry is shown to be spontaneously broken in a geometrical manner. In the resulting flavour model, naturally small Majorana neutrino masses for the light active neutrinos are obtained through the type I see-saw mechanism. The known masses of the charged leptons, lepton mixing angles and the two neutrino mass squared differences are reproduced by the model with a good accuracy. The model allows for two neutrino mass spectra with normal ordering (NO) and one with inverted ordering (IO). For each of the three spectra the absolute scale of neutrino masses is predicted with relatively small uncertainty. The value of the Dirac CP violation (CPV) phase δ in the lepton mixing matrix is predicted to be δ ∼ = π/2 or 3π/2. Thus, the CP violating effects in neutrino oscillations are predicted to be maximal (given the values of the neutrino mixing angles) and experimentally observable. We present also predictions for the sum of the neutrino masses, for the Majorana CPV phases and for the effective Majorana mass in neutrinoless double beta decay. The predictions of the model can be tested in a variety of ongoing and future planned neutrino experiments.

We obtain predictions for the Majorana phases α 21 /2 and α 31 /2 of the 3 × 3 unitary neutrino mixing matrix U = U † e U ν , U e and U ν being the 3 × 3 unitary matrices resulting from the diagonalisation of the charged lepton and neutrino Majorana mass matrices, respectively. We focus on forms of U e and U ν permitting to express α 21 /2 and α 31 /2 in terms of the Dirac phase δ and the three neutrino mixing angles of the standard parametrisation of U , and the angles and the two Majorana-like phases ξ 21 /2 and ξ 31 /2 present, in general, in U ν . The concrete forms of U ν considered are fixed by, or associated with, symmetries (tri-bimaximal, bimaximal, etc.), so that the angles in U ν are fixed. For each of these forms and forms of U e that allow to reproduce the measured values of the three neutrino mixing angles θ 12 , θ 23 and θ 13 , we derive predictions for phase differences (α 21 /2 − ξ 21 /2), (α 31 /2 − ξ 31 /2), etc., which are completely determined by the values of the mixing angles. We show that the requirement of generalised CP invariance of the neutrino Majorana mass term implies ξ 21 = 0 or π and ξ 31 = 0 or π. For these values of ξ 21 and ξ 31 and the best fit values of θ 12 , θ 23 and θ 13 , we present predictions for the effective Majorana mass in neutrinoless double beta decay for both neutrino mass spectra with normal and inverted ordering.B.Ũ ν = R 23 (θ ν 23 )R 13 (θ ν 13 )R 12 (θ ν 12 ) and vi)Ũ e = R −1 12 (θ e 12 ), vii)Ũ e = R −1 13 (θ e 13 ).The sum rules thus found allowed us in the cases of the TBM, BM (LC), GRA, GRB and HG mixing forms ofŨ ν in item A and for certain fixed values of θ ν ij in item B to obtain predictions for cos δ (see refs. [2-4, 6]) as well as for the rephasing invariant7 The relation between cos δ and cos ψ can be deduced from eq. (29) in [2]. 8 In [2] both sin 2θ e 12 and cos θ ν 23 could be and were considered to be positive without loss of generality. 9 The expressions for the invariants I1,2 we give and will use further correspond to Majorana conditions satisfied by the fields of the light massive Majorana neutrinos, which do not contain phase factors, see, e.g., [15]. 9