Predictions for the Majorana CP Violation Phases in the Neutrino Mixing Matrix and Neutrinoless Double Beta Decay

We obtain predictions for the Majorana phases $\alpha_{21}/2$ and $\alpha_{31}/2$ of the $3\times 3$ unitary neutrino mixing matrix $U = U_e^{\dagger} \, U_{\nu}$, $U_e$ and $U_{\nu}$ being the $3\times 3$ unitary matrices resulting from the diagonalisation of the charged lepton and neutrino Majorana mass matrices, respectively. We focus on forms of $U_e$ and $U_{\nu}$ permitting to express $\alpha_{21}/2$ and $\alpha_{31}/2$ in terms of the Dirac phase $\delta$ and the three neutrino mixing angles of the standard parametrisation of $U$, and the angles and the two Majorana-like phases $\xi_{21}/2$ and $\xi_{31}/2$ present, in general, in $U_{\nu}$. The concrete forms of $U_{\nu}$ considered are fixed by, or associated with, symmetries (tri-bimaximal, bimaximal, etc.), so that the angles in $U_{\nu}$ 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 $\theta_{12}$, $\theta_{23}$ and $\theta_{13}$, we derive predictions for phase differences $(\alpha_{21}/2 - \xi_{21}/2)$, $(\alpha_{31}/2 - \xi_{31}/2)$, 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 $\xi_{21} = 0$ or $\pi$ and $\xi_{31} = 0$ or $\pi$. For these values of $\xi_{21}$ and $\xi_{31}$ and the best fit values of $\theta_{12}$, $\theta_{23}$ and $\theta_{13}$, we present predictions for the effective Majorana mass in neutrinoless double beta decay for both neutrino mass spectra with normal and inverted ordering.


Introduction
Determining the status of the CP symmetry in the lepton sector, discerning the type of spectrum the neutrino masses obey, identifying the nature -Dirac or Majorana -of massive neutrinos and determining the absolute neutrino mass scale are among the highest priority goals of the programme of future research in neutrino physics (see, e.g., [1]). The results obtained within this ambitious research programme can shed light, in particular, on the origin of the observed pattern of neutrino mixing. Comprehending the origin of the patterns of neutrino masses and mixing is one of the most challenging problems in neutrino physics. It is an integral part of the more general fundamental problem in particle physics of deciphering the origins of flavour, i.e., of the patterns of quark, charged lepton and neutrino masses and of the quark and neutrino mixing.
In refs. [2][3][4][5] (see also [6]), working in the framework of the reference 3-neutrino mixing scheme (see, e.g., [1]), we have derived predictions for the Dirac CP violation (CPV) phase in the Pontecorvo, Maki, Nakagawa and Sakata (PMNS) neutrino mixing matrix within the discrete flavour symmetry approach to neutrino mixing. This approach provides a natural explanation of the observed pattern of neutrino mixing and is widely explored at present (see, e.g., [7,8] and references therein). In the present article, using the method developed and utilised in [2], we derive predictions for the Majorana CPV phases in the PMNS matrix [9] within the same approach based on discrete flavour symmetries. Our study is a natural continuation of the studies performed in [2][3][4][5][6].
As is well known, the PMNS matrix will contain physical CPV Majorana phases if the massive neutrinos are Majorana particles [9]. The massive neutrinos are predicted to be Majorana fermions by a large number of theories of neutrino mass generation (see, e.g., [7,10,11]), most notably, by the theories based on the seesaw mechanism [12]. The flavour neutrino oscillation probabilities do not depend on the Majorana phases [9,13]. The Majorana phases play particularly important role in processes involving real or virtual neutrinos, which are characteristic of Majorana nature of massive neutrinos and in which the total lepton charge L changes by two units, |∆L| = 2 (see, e.g., [14]). One widely discussed and experimentally relevant example is neutrinoless double beta ((ββ) 0ν -) decay of even-even nuclei (see, e.g., [10,15,16]) 48 Ca, 76 Ge, 82 Se, 100 M o, 130 T e, 136 Xe, etc.: (A, Z) → (A, Z + 2) + e − + e − . The predictions for the rates of the lepton flavour violating processes, µ → e+γ and µ → 3e decays, µ−e conversion in nuclei, etc., in theories of neutrino mass generation with massive Majorana neutrinos (e.g., TeV scale type I seesaw model, the Higgs triplet model, etc.) depend on the Majorana phases (see, e.g., [17,18]). And the Majorana phases in the PMNS matrix can provide the CP violation necessary for the generation of the observed baryon asymmetry of the Universe [19] 1 .
In the reference case of 3-neutrino mixing, which we are going to consider in the present article, there can be two physical Majorana CPV phases in the PMNS neutrino mixing matrix in addition to the Dirac CPV phase [9]. The PMNS matrix in this case is given by where α 21,31 are the two Majorana CPV phases and V is a CKM-like matrix containing the Dirac CPV phase. The matrix V has the following form in the standard parametrisation of the PMNS matrix [1], which we are going to employ in what follows: Here 0 ≤ δ ≤ 2π is the Dirac CPV phase and we have used the standard notation c ij = cos θ ij , s ij = sin θ ij with 0 ≤ θ ij ≤ π/2. In the case of CP invariance we have δ = 0, π, 2π, 0 and 2π being physically indistinguishable, and [22] α 21 = kπ, α 31 = k π, k, k = 0, 1, 2 2 . The neutrino mixing parameters sin 2 θ 12 , sin 2 θ 23 and sin 2 θ 13 play important role in our further considerations. They were determined with relatively small uncertainties in the most recent analysis of the global neutrino oscillation data performed in [24] (for earlier analyses see, e.g., [25,26]). The authors of ref. [24], using, in particular, the first NOνA (LID) data on ν µ → ν e oscillations from [27], find the following best fit values and 3σ allowed ranges of sin 2 θ 12 , sin 2 θ 23 and sin 2 θ 13 : (sin 2 θ 12 ) BF = 0.297 , 0.250 ≤ sin 2 θ 12 ≤ 0.354 , (sin 2 θ 13 ) BF = 0.0214 (0.0218) , 0.0185 (0.0186) ≤ sin 2 θ 13 ≤ 0.0246 (0.0248) .
The values (values in brackets) correspond to neutrino mass spectrum with normal ordering (inverted ordering) (see, e.g., [1]), denoted further as the NO (IO) spectrum. Note, in particular, that sin 2 θ 23 can differ significantly from 0.5 and that sin 2 θ 23 = 0.5 lies in the 2σ interval of allowed values. Using the same set of data the authors of [24] find also the following best fit value and 2σ allowed range of the Dirac phase δ: δ = 1.35 π (1.32 π) , 0.92 π (0.83 π) ≤ δ ≤ 1.99 π .
The discrete flavour symmetry approach to neutrino mixing is based on the observation that the PMNS neutrino mixing angles θ 12 , θ 23 and θ 13 have values which differ from those of specific symmetry forms of the mixing matrix by subleading perturbative corrections (see further). The fact that the PMNS matrix in the case of 3-neutrino mixing is a product of two 3 × 3 unitary matrices U e and U ν , originating from the diagonalisation of the charged lepton and neutrino mass matrices, is also widely exploited. In terms of the parameters of U e and U ν , in the absence of constraints the PMNS matrix can be parametrised as [28] HereŨ e andŨ ν are CKM-like 3 × 3 unitary matrices, and Ψ and Q 0 are given by Ψ = diag 1, e −iψ , e −iω , Q 0 = diag 1, e i ξ 21 2 , e i ξ 31 where ψ, ω, ξ 21 and ξ 31 are phases which contribute to physical CPV phases. The phases in Q 0 result from the diagonalisation of the neutrino Majorana mass term and contribute to the Majorana phases in the PMNS matrix.
In the approach of interest one assumes the existence at certain energy scale of a (lepton) flavour symmetry corresponding to a non-Abelian discrete group G f . The symmetry group G f can be broken, in general, to different symmetry subgroups, or "residual symmetries", G e and G ν of the charged lepton and neutrino mass terms, respectively. Given a discrete symmetry G f , there are more than one (but still a finite number of) possible residual symmetries G e and G ν . The subgroup G e , in particular, can be trivial. Non-trivial residual symmetries G e and G ν (of a given G f ) constrain the forms of the matrices U e and U ν , and thus the form of U .
The symmetry forms ofŨ ν considered above do not include rotation in the 1-3 plane, i.e., θ ν 13 = 0. However, forms ofŨ ν of the typẽ with non-zero values of θ ν 13 are inspired by certain types of flavour symmetries (see, e.g., [41][42][43][44]). In [41], for example, the so-called tri-permuting pattern, corresponding to θ ν 12 = θ ν 23 = −π/4 and θ ν 13 = sin −1 (1/3), was proposed and investigated. In the study we will perform we will consider also the form in eq. (12) for three representative values of θ ν 13 discussed in the literature: θ ν 13 = π/20, π/10 and sin −1 (1/3). The symmetry values of the angles in the matrixŨ ν typically, and in all cases considered above, differ by relatively small perturbative corrections from the experimentally determined values of at least some of the angles θ 12 , θ 23 and θ 13 . The requisite corrections are provided by the matrix U e , or equivalently, byŨ e . In the approach followed in [2][3][4]6] we are going to adopt, the matrixŨ e is unconstrained and was chosen on phenomenological grounds. This corresponds to the case of trivial subgroup G e , i.e., of the charged lepton mass term breaking the symmetry G f completely. The matrixŨ e in the general case depends on three angles and one phase [28]. However, in a class of theories of (lepton) flavour and neutrino mass generation, based on a GUT and/or a discrete symmetry (see, e.g., [45][46][47][48][49][50]),Ũ e is an orthogonal matrix which describes one rotation in the 1-2 plane, or two rotations in the planes 1-2 and 2-3, θ e 12 and θ e 23 being the corresponding rotation angles. Other possibilities includeŨ e being an orthogonal matrix which describes i) one rotation in the 1-3 plane 5 , or ii) two rotations in any other two of the three planes, e.g., on which the magnitude of CP-violating effects in neutrino oscillations depends [51]. The results of these studies showed that the predictions for cos δ exhibit strong dependence on the symmetry form ofŨ ν . This led to the conclusion that a sufficiently precise measurement of cos δ combined with high precision measurements of sin 2 θ 12 , sin 2 θ 23 and sin 2 θ 13 can allow to test critically the idea of existence of an underlying discrete symmetry form of the PMNS matrix and, thus, of existence of a new symmetry in particle physics. In ref. [2] predictions for the Majorana phases of the PMNS matrix α 21 and α 31 in the case ofŨ ν = R 23 (θ ν 23 )R 12 (θ ν 12 ), corresponding to the TBM, BM (LC), GRA, GRB and HG symmetry forms, andŨ e = R −1 23 (θ e 23 )R −1 12 (θ e 12 ) were derived under the assumption that the phases ξ 21 and ξ 31 in eqs. (8) and (9), which originate from the diagonalisation of the neutrino Majorana mass term, are known (i.e., are fixed by symmetry or other arguments). In the present article we extend the analysis performed in [2] to obtain predictions for the phases α 21 and α 31 in the cases of the forms of the matricesŨ ν andŨ e listed in items A and B above. This allows us to obtain predictions for the phase differences (α 21 −ξ 21 ) and (α 31 −ξ 31 ). We further employ the generalised CP symmetry constraint in the neutrino sector [52][53][54], which allows us to fix the values of the phases ξ 21 and ξ 31 , and thus to predict the values of α 21 and α 31 . We use these results together with the sum rule results on cos δ to derive (in graphic form) predictions for the dependence of the absolute value of the (ββ) 0ν -decay effective Majorana mass (see, e.g., [10]), | m |, on the lightest neutrino mass in all cases considered for both the NO and IO spectra.
B the reported results are for θ ν 23 = − π/4 and five sets of values of θ ν 13 and θ ν 12 associated with symmetries. We then set (ξ 21 , ξ 31 ) = (0, 0), (0, π), (π, 0) and (π, π) and use the resulting values of α 21 /2 and α 31 /2 to derive graphical predictions for the absolute value of the effective Majorana mass in (ββ) 0ν -decay, | m |, as a function of the lightest neutrino mass in the schemes of mixing studied. We show in Section 7 that the requirement of generalised CP invariance of the neutrino Majorana mass term in the cases of S 4 , A 4 , T and A 5 lepton flavour symmetries leads indeed to ξ 21 = 0 or π, ξ 31 = 0 or π. In the first two cases (third case) studied in Section 3, B1 and B2 (B3), the phase α 31 /2 (the phases δ, α 21 /2 and α 31 /2) depends (depend) on an additional phase, β (ω), which, in general, is not constrained. For schemes B1 and B2, the predictions for | m | are obtained in Section 6 by varying β in the interval [0, π]. In the case of scheme B3 the results for the Majorana phases and | m | are derived for the value of ω = 0, for which the Dirac phase δ has a value in its 2σ allowed interval quoted in eq. (6). Section 8 contains summary of the results of the present study and conclusions.
We note finally that the titles of Sections 2 -4 and of their subsections reflect the rotations contained in the corresponding parametrisation, eqs. (19) -(21).
2 The Cases of θ e ij − (θ ν 23 , θ ν 12 ) Rotations In this section we derive the sum rules for α 21 and α 31 of interest in the case when the matrix U ν = R 23 (θ ν 23 ) R 12 (θ ν 12 ) with fixed (e.g., symmetry) values of the angles θ ν 23 and θ ν 12 , gets correction only due to one rotation from the charged lepton sector. The neutrino mixing matrix U has the form given in eq. (19). We do not consider the case of eq. (19) with (ij) = (23), because in this case the reactor angle θ 13 = 0 and thus the measured value of θ 13 ∼ = 0.15 cannot be reproduced.

2.1
The Scheme with θ e 12 − (θ ν 23 , θ ν 12 ) Rotations (Case A1) In the present subsection we consider the parametrisation of the neutrino mixing matrix given in eq. (19) with (ij) = (12). In this parametrisation the PMNS matrix has the form The phase ω in the phase matrix Ψ is unphysical. We are interested in deriving analytic expressions for the Majorana phases α 21 and α 31 i) in terms of the parameters of the parametrisation in eq. (22), θ e 12 , ψ, θ ν 23 , θ ν 12 , ξ 21 and ξ 31 , and possibly ii) in terms of the angles θ 12 , θ 13 , θ 23 and the Dirac phase δ of the standard parametrisation of the PMNS matrix, the fixed angles θ ν 23 and θ ν 12 , and the phases ξ 21 and ξ 31 . The values of the phases α 21 and α 31 in the latter case, as we will see, indeed depend on the value of the Dirac phase δ. Thus, we first recall the sum rule satisfied by the Dirac phase δ in the case under study, by which cos δ is expressed in terms of the angles θ 12 , θ 13 and θ 23 . The sum rule of interest reads [2]: cos δ = tan θ 23 sin 2θ 12 sin θ 13 cos 2θ ν 12 + sin 2 θ 12 − cos 2 θ ν and cos θ ν 23 in terms of the measured angles θ 13 and θ 23 of the standard parametrisation of the PMNS matrix using the relation in eq. (24).
As can be shown employing the formalism developed in [2] and taking into account the possibility of negative signs of c e 12 s ν 12 and c e 12 c ν 12 , the expressions for the phases β e2 and β e1 in terms of the angles θ 12 , θ 13 , θ 23 and the Dirac phase δ of the standard parametrisation of the PMNS matrix have the form: For sgn(c e 12 s ν 12 ) = 1 and sgn(c e 12 c ν 12 ) = 1, eqs. (32) and (33) reduce respectively to eqs. (100) and (101) in ref. [2].
It follows from eqs. (32) and (33) that the phases β e1 and β e2 are determined by the values of the standard parametrisation mixing angles θ 12 , θ 13 , θ 23 and of the Dirac phase δ. The phase δ is also determined (up to a sign ambiguity of sin δ) by the values of "standard" angles θ 12 , θ 13 , θ 23 via the sum rule given in eq. (23). Since the relations in eqs. (28) and (29) between the Majorana phases α 21 and α 31 and the phases β e1 and β e2 involve the phases ξ 21 and ξ 31 originating from the diagonalisation of the neutrino Majorana mass term, α 21 and α 31 will be determined by the values of the "standard" neutrino mixing angles θ 12 , θ 13 , θ 23 (up to the mentioned ambiguity related to the undetermined so far sign of sin δ), provided the values of ξ 21 and ξ 31 are known. Thus, predictions for the Majorana phases α 21 and α 31 can be obtained when the phases ξ 21 and ξ 31 are fixed by additional considerations of, e.g., generalised CP invariance, symmetries, etc. In theories with discrete lepton flavour symmetries the phases ξ 21 and ξ 31 are often determined by the employed symmetries of the theory (see, e.g., [45,49,50,55,56] and references quoted therein). We will show in Section 7 how the phases ξ 21 and ξ 31 are fixed by the requirement of generalised CP invariance of the neutrino Majorana mass term in the cases of the non-Abelian discrete flavour symmetries S 4 , A 4 , T and A 5 . In all these cases the generalised CP invariance constraint fixes the values of ξ 21 and ξ 31 , which allows us to obtain predictions for the Majorana phases α 21 and α 31 .
We note that within the approach employed in our analysis, the results presented in eqs. (28) -(36) are exact and are valid for arbitrary fixed values of θ ν 12 and θ ν 23 and for arbitrary signs of sin θ e 12 and cos θ e 12 (| sin θ e 12 | and | cos θ e 12 | can be expressed in terms of θ 13 and θ ν 23 ). Although the sum rules derived above allow to determine the values of the Majorana phases α 21 and α 31 (up to a two-fold ambiguity related to the ambiguity of sgn(sin δ) or of sgn(sin ψ)) if the phases ξ 21 and ξ 31 are known, we will present below an alternative method of determination of α 21 and α 31 , which can be used in the cases when the method developed in [2] cannot be applied. The alternative method makes use of the rephasing invariants associated with the two Majorana phases of the PMNS matrix.
In the case of 3-neutrino mixing under discussion there are, in principle, three independent CPV rephasing invariants. The first is associated with the Dirac phase δ and is given by the well-known expression in eq. (18), where we have shown also the expression of the J CP factor in the standard parametrisation. The other two, I 1 and I 2 , are related to the two Majorana CPV phases in the PMNS matrix and can be chosen as [15,57,58] 9 : The rephasing invariants associated with the Majorana phases are not uniquely determined. Instead of I 1 defined above we could have chosen, e.g., However, the three invariants -J CP and any two chosen Majorana phase invariants -form a complete set in the case of 3-neutrino mixing: any other two rephasing invariants associated with the Majorana phases can be expressed in terms of the two chosen Majorana phase invariants and the J CP factor [57]. We note also that CP violation due to the Majorana phase α 21 requires that both I 1 = Im {U * e1 U e2 } = 0 and Re {U * e1 U e2 } = 0 [58]. Similarly, I 2 = Im {U * e1 U e3 } = 0 would imply violation of the CP symmetry only if in addition Re {U * e1 U e3 } = 0.
It follows from eqs. (45) and (47)   In the present subsection we consider the parametrisation of the neutrino mixing matrix given in eq. (19) with (ij) = (13). In this parametrisation the PMNS matrix has the form Now the phase ψ in the phase matrix Ψ is unphysical. We employ the approaches used in the preceding subsection, which are based on the method developed in [2] and on the relevant rephasing invariants, for determining the Majorana phases α 21 and α 31 . We first give the expressions for sin 2 θ 13 , sin 2 θ 23 and sin 2 θ 12 in terms of the parameters of the parametrisation in eq. (48), which will be used in our analysis: sin 2 θ ν 23 sin 2 θ e 13 cos 2 θ ν 12 + cos 2 θ e 13 sin 2 θ ν 12 − 1 2 sin 2θ e 13 sin 2θ ν 12 sin θ ν 23 cos ω .
As we will see, the expressions for the Majorana phases α 21 and α 31 we will obtain depend on the Dirac phase δ. Therefore we give also the sum rule for the Dirac phase δ in the considered case by which cos δ is expressed in terms of the measured angles θ 12 and θ 13 of the standard parametrisation of the PMNS matrix [4]: Equating the expressions for the rephasing invariant associated with the Dirac phase in the PMNS matrix, J CP , obtained in the standard parametrisation and in the parametrisation given in eq. (48) allows us to get a relation between sin δ and sin ω: As can be shown using the method developed in [2] and employed in the preceding subsection, the phases δ, α 21 /2 and α 31 /2 are related with the phase ω and the phases β e1 and β e2 , in the following way: From eqs. (57) -(59) we get a relation analogous to that in eq. (35) in the preceding subsection: where we took into account that 2 arg (c e 13 s ν 23 c ν 23 ) = 0 or 2π. Equation (49) allows one to express s e 13 and c e 13 (given their signs) in terms of sin θ 13 and cos θ ν 23 . The phase ω is determined by the angles θ 12 , θ 13 , θ ν 12 and θ ν 23 via eq. (52) (up to an ambiguity of the sign of sin ω). Thus, using eqs. (55) and (56), the phases β e1 and β e2 can be expressed in terms of the measured mixing angles θ 12 and θ 13 and the angles θ ν 12 and θ ν 23 fixed by symmetry arguments. It is not difficult to derive expressions for β e1 and β e2 in terms of the angles θ 12 , θ 13 , θ 23 and the phase δ of the standard parametrisation of the PMNS matrix. They read: β e2 = arg U µ1 e iπ sgn (c e 13 s ν 12 ) = arg s 12 c 23 + c 12 s 23 s 13 e iδ sgn (c e 13 s ν 12 ) .
3 The Cases of (θ e ij , θ e kl ) − (θ ν 23 , θ ν 12 ) Rotations As it follows from eqs. (24) and (50) in the preceding Section, in the cases when the matrix U e originating from the charged lepton sector contains one rotation angle (θ e 12 or θ e 13 ) and θ ν 23 = −π/4, the mixing angle θ 23 cannot deviate significantly from π/4 due to the smallness of the angle θ 13 . If the matrixŨ ν has one of the symmetry forms considered in this study, the matrixŨ e has to contain at least two rotation angles in order to be possible to reproduce the current best fit values of the neutrino mixing parameters quoted in eqs. (3) -(5), or more generally, in order to be possible to account for deviations of sin 2 θ 23 from 0.5 which are bigger than sin 2 θ 13 , i.e., for sin 2 θ 23 = 0.5(1 ∓ sin 2 θ 13 ). In this Section we consider the determination of the Majorana phases α 21 and α 31 in the cases when the matrixŨ e contains two rotation angles.

The Scheme with
The PMNS matrix in this scheme has the form The scheme has been analysed in detail in [2], where a sum rule for cos δ and analytic expressions for α 21 and α 31 were derived for θ ν 23 = −π/4. As was shown in [4], the sum rule for cos δ found in [2] holds for an arbitrary fixed value of θ ν 23 . The sum rule under discussion, eq. (30) in [2], coincides with the sum rule given in eq. (23) in subsection 2.1. However, in contrast to the case considered in subsection 2.1, the PMNS mixing angle θ 23 in the scheme under discussion can differ significantly from θ ν 23 and from π/4: where sinθ 23 = e −iψ cos θ e 23 sin θ ν 23 + e −iω sin θ e 23 cos θ ν 23 , In the preceding equations sinθ 23 and cosθ 23 are expressed in terms of the parameters of the scheme considered, defined in eq. (71) for the PMNS matrix. Obviously, sinθ 23 > 0 and cosθ 23 > 0. The parameter sin 2θ 23 enters also into the expression for sin 2 θ 13 : The angleθ 23 results from the rearrangement of the product of matrices R 23 (θ e 23 )ΨR 23 (θ ν 23 ) in the expression for U given in eq. (71): Here where Equations (73), (78) and (79) have been derived in [6]. The phase α in the matrix P 1 is unphysical. The phase β contributes to the matrix of physical Majorana phases, which now is equal toQ = Q 1 Q 0 . The phase φ serves as source for the Dirac phase δ and gives contributions also to the Majorana phases α 21 and α 31 [2]. The PMNS matrix takes the form where θ ν 12 has a fixed value which depends on the symmetry form ofŨ ν used. Before continuing further we note that we can consider both sin θ e 12 and cos θ e 12 to be positive without loss of generality. Only their relative sign is physical. If sin θ e 12 > 0 ( sin θ e 12 < 0) and cos θ e 12 < 0 ( cos θ e 12 > 0), the negative sign can be absorbed in the phase φ by adding ±π to φ. Similarly, we can consider both sin θ ν 12 and cos θ ν 12 to be positive: the negative signs of sin θ ν 12 and/or cos θ ν 12 can be absorbed in the phases ξ 21 /2, ξ 31 /2 and φ 10 . Nevertheless, for convenience of using our results for making predictions in theoretical models in which the value of, e.g., | sin θ e 12 | and the signs of sin θ e 12 and cos θ e 12 are specified, we will present the results for arbitrary signs of sin θ e 12 and cos θ e 12 .
The analytic results on the Majorana phases α 21 and α 31 , on the relation between the Dirac phase δ and the phase φ, etc., derived in [2], do not depend explicitly on the value of the angle θ ν 23 and are valid in the case under consideration. Thus, generalising eqs. (88) -(91), (94) and (102) in [2] for arbitrary sings of s e 12 , c e 12 , s ν 12 and c ν 12 , we have: where withĉ 23 ≡ cosθ 23 . The preceding results can be obtained by casting U in eq. (80) in the standard parametrisation form. This leads, in particular, to additional contribution to the matrix Q of the Majorana phases, which takes the formQ = Q 2 Q 1 Q 0 , where the generalisation of the corresponding expression for Q 2 in [2] reads: Note that we got rid of the common unphysical phase factor e −i(β e2 +β µ3 −φ) in the matrix Q 2 .
The expressions for the phases (β e2 + β µ3 − φ) and (β e1 + β µ3 − φ) in terms of the angles θ 12 , θ 13 , θ 23 and the Dirac phases δ of the standard parametrisation of the PMNS matrix have the form (cf. eqs. (100) and (101) in ref. [2]): where We also have A few comments are in order. As like the cosine of the Dirac phase δ, cos φ satisfies a sum rule by which it is expressed in terms of the three measured neutrino mixing angles θ 12 , θ 13 and θ 23 , and is uniquely determined by the values of θ 12 , θ 13 and θ 23 [2]. The values of sin δ and sin φ, however, are fixed up to a sign. Through eq. (90) the signs sin δ and sin φ are correlated. Thus, δ and φ are predicted with an ambiguity related to the ambiguity of the sign of sin δ (or of sin φ). Together with eqs. (87) and (88) this implies that also the phases β e1 and β e2 are determined by the values of θ 12 , θ 13 , θ 23 and δ with a two-fold ambiguity. The knowledge of the difference (β e2 − β e1 ) allows to determine the Majorana phase α 21 (up to the discussed two-fold ambiguity) if the value of the phase ξ 21 is known. In contrast, the knowledge of β e2 and ξ 31 is not enough to predict the value of the Majorana phase α 31 since it receives a contribution also from the phase β that cannot be fixed on general phenomenological grounds. It is possible to determine the phase β in certain specific cases (see [2] for a detailed discussion of the cases when β can be fixed). It should be noted, however, that the term involving the phase α 31 in the (ββ) 0ν -decay effective Majorana mass m gives practically a negligible contribution in | m | in the cases of neutrino mass spectrum with IO or of quasi-degenerate (QD) type [2,15]. In these cases we have [59] | m | ∼ > 0.014 eV (see also, e.g., [1,14]). Values of | m | ∼ > 0.014 eV are in the range of planned sensitivity of the future large scale (ββ) 0ν -decay experiments (see, e.g., [60]).
where the angleθ 23 and the matrix P 1 are given by eqs. (73) and (76), respectively, and Q = Q 1 Q 0 with Q 1 as in eq. (76). In explicit form eq. (100) reads: To bring this matrix to the standard parametrisation form, we first rewrite it as follows: where We recall that the angleθ 23 belongs to the first quadrant by construction (see eq. (73)). Further, comparing the expressions for the J CP invariant in the standard parametrisation and in the parametrisation given in eq. (100), we have 11 It is not difficult to check that this relation holds if which, in turn, suggests what rearrangement of the phases in the PMNS matrix in eq. (102) one has to do to bring it to the standard parametrisation form. Namely, the required rearrangement should be made in the following way: where The phases in the matrix P 2 are unphysical. The phases (β e2 − β e1 ) and (β e2 + β τ 3 + α) in the matrix Q 2 contribute to the Majorana phases α 21 and α 31 , respectively, while the common phase (−β e2 − β τ 3 − α) in this matrix is unphysical and we will not keep it further. Thus, the Majorana phases in the PMNS matrix are determined by the phases in the product Q 2Q : In terms of the standard parametrisation mixing angles θ 12 , θ 23 , θ 13 and the Dirac phase δ the phases (β e1 + β τ 3 + α) and (β e2 + β τ 3 + α) read: The relevant expressions for the parameters sin 2 θ e 13 , sin 2θ 23 and cos α in terms of the neutrino mixing angles θ 12 , θ 13 , θ 23 and the angles contained inŨ ν have been derived in [4]: sin 2θ 23 = sin 2 θ 23 cos 2 θ 13 , sin(α 31 /2) = 1 |U e2 | cos θ e 13 sin θ ν 12 sin(β + ξ 31 /2) + sin θ e 13 sinθ 23 cos θ ν 12 sin(α − β − ξ 31 /2) .
In explicit form this matrix reads: Comparing the expressions for the absolute value of the element U τ 3 in the standard parametrisation of the PMNS matrix and the parametrisation we are considering here, we have [4] cos 2 θ e 13 = cos 2 θ 23 cos 2 θ 13 cos 2 θ ν
Hence, the angle θ e 13 is expressed in terms of the known angles and can be determined up to a quadrant. The phase ω is a free phase parameter, which enters, e.g., the sum rule for cos δ (see eq. (63) in ref. [4]), so its presence is expected as well in the sum rules for the Majorana phases we are going to derive.
We note finally that sin 2 θ 23 is constrained by the requirements that cos ψ, sin 2 θ e 12 and sin 2 θ e 13 possess physically acceptable values, to lie for both the NO and IO spectra in the following narrow intervals [4]: 4 The Cases of θ e ij − (θ ν 23 , θ ν 13 , θ ν 12 ) Rotations We consider next a generalisation of the cases analysed in Section 2 with the presence of a third rotation matrix inŨ ν arising from the neutrino sector, i.e., we employ the parametrisation of U given in eq. (21). Non-zero values of θ ν 13 are inspired by certain types of flavour symmetries [41,42]. In the numerical analysis of the predictions for α 21 , α 31 and | m | we will perform in Section 6, we will consider three representative values of θ ν 13 discussed in the literature: θ ν 13 = π/20, π/10 and sin −1 (1/3). We are not going to consider the case in which the U matrix is parametrised as in eq. (21) with (ij) = (23) for the reasons explained in [4], i.e., the absence of a correlation between the Dirac CPV phase δ and the mixing angles. It should be noted that for this and other cases for which it is not possible to derive such a correlation, different symmetry forms ofŨ ν can still be tested with an improvement of the precision in the measurement of the neutrino mixing angles. For instance, in the case corresponding to eq. (21) with (ij) = (23), one has, as was shown in [4], sin 2 θ 13 = sin 2 θ ν 13 and sin 2 θ 12 = sin 2 θ ν 12 , i.e., the angles θ 13 and θ 12 are predicted to have particular values when the angles θ ν 13 and θ ν 23 are fixed by a symmetry.

26
The phases in the matrix P 2 are unphysical. The Majorana phases get contribution from the matrix Q 2 Q 0 and read: In terms of the standard parametrisation mixing angles θ 12 , θ 23 , θ 13 and the Dirac phase δ we have: where β τ 1 and β τ 2 can be 0 or π and are known when the angles θ ν 12 , θ ν 23 and θ ν 13 are fixed (see eqs. (171) and (172)).

The Scheme with
In this subsection we derive the formulae for the Majorana phases in the case when the PMNS matrix U is parametrised as in eq. (21) with (ij) = (13), i.e., In this case the phase ψ in the matrix Ψ is unphysical, and Ψ = diag (1, 1, e −iω ). We will proceed in analogy with the previous subsection. We start by writing the matrix U in explicit form: where where we have used the equality sin 2 θ 23 cos 2 θ 13 = sin 2 θ ν 23 cos 2 θ ν 13 valid in this scheme. As can be shown, the relation between sin δ and sin ω in eq. (210) takes place if where β µ3 = arg(s ν 23 c ν 13 ). Knowing the expression for δ allows us to rearrange the phases in eq. (200) in such a way as to render U in the standard parametrisation form: with The matrix P 2 contains unphysical phases which can be removed. The Majorana phases are determined by the phases in the product Q 2 Q 0 : In terms of the "standard" mixing angles θ 12 , θ 23 , θ 13 and the Dirac phase δ one has: where β µ1 = arg(−s ν 12 c ν 23 − c ν 12 s ν 23 s ν 13 ) and β µ2 = arg(c ν 12 c ν 23 − s ν 12 s ν 23 s ν 13 ) can take values of 0 or π and are known when the angles θ ν 12 , θ ν 23 and θ ν 13 are fixed. The mixing angles θ 12 , θ 23 and θ 13 of the standard parametrisation are related with the angles θ e 13 , θ ν ij and the phase ω present in the parametrisation of U given in eq. (199) in the 14 For θ ν 23 = −π/4 this relation reduces to eq. (91) in ref. [4].
Thus, we have at our disposal expressions for sin 2 θ e 13 and cos ω in terms of the known angles.

Summary of the Sum Rules for the Majorana Phases
In the present Section we summarise the sum rules for the Majorana phases obtained in the previous Sections. Throughout this Section the neutrino mixing matrix U is assumed to be in the standard parametrisation.
In schemes A1, B1, B3 and C1 the sum rules for α 21 /2 and α 31 /2 can be cast in the form: where the expressions for the phases κ 21 and κ 31 , which should be used in these sum rules in each particular case, are given in Table 1. In schemes A1 and C1 the phases κ 21 and κ 31 take values 0 or π and are known once the angles θ ν ij are fixed. In scheme B1 (B3), κ 31 (κ 21 and κ 31 ) depends (depend) on the free phase parameter β (ω).
In schemes A2, B2 and C2 we similarly have: where the corresponding expressions for κ 21 and κ 31 are given again in Table 1. In cases A2 and C2 the phases κ 21 and κ 31 can take values 0 or π. They are fixed when the angles θ ν ij are given. The phase β, which is a free parameter as long as it is not fixed by additional arguments, enters the sum rule for α 31 /2 in scheme B2. In all schemes considered, A1, A2, B1, B2, B3, C1 and C2, the phases (α 21 /2−ξ 21 /2−κ 21 ) and (α 31 /2 − ξ 31 /2 − κ 31 ) are determined by the values of the neutrino mixing angles θ 12 , θ 23 and θ 13 , and of the Dirac phase δ. The Dirac phase is determined in each scheme by a corresponding sum rule. In schemes A1, A2, C1 and C2 there is a correlation between the values of sin 2 θ 23 and sin 2 θ 13 . The sum rules for cos δ and the relevant expressions for sin 2 θ 23 in the cases of interest, which should be used in eqs. (235) -(238), are given, e.g., in Tables 1 and 2 of ref. [4]. In the following Section we use the sum rules given in eqs. (235) -(238) to obtain the numerical predictions for the Majorana phases in the PMNS matrix.

Dirac Phase
In Table 2 15 we show predictions for the Dirac phase δ, obtained from the sum rules, derived in refs. [2,4] and summarised in Table 1 in ref. [4]. The numerical values are obtained using the best fit values of the neutrino mixing parameters given in eqs. (3) -(5) for both the NO and IO spectra. In the BM (LC) case, the sum rules for cos δ lead to unphysical values of | cos δ| > 1 if one uses as input the current best fit values of sin 2 θ 12 , sin 2 θ 23 and sin 2 θ 13 [2][3][4]6]. This is an indication of the fact that the current data disfavours the BM (LC) form ofŨ ν . In the case of the B1 scheme and the NO spectrum, for example, the BM (LC) form is disfavoured at approximately 2σ confidence level. Physical values of cos δ are found for larger (smaller) values of sin 2 θ 12 (sin 2 θ 23 ) [2][3][4]. For, e.g., sin 2 θ 12 = 0.354, which is the 3σ upper bound of sin 2 θ 12 , and the best fit values of sin 2 θ 23 and sin 2 θ 13 , we get | cos δ| ≤ 1 in most of the schemes considered in the present article, the exceptions being the schemes B1 with the IO spectrum, B2 with the NO spectrum and B3. The values of the Dirac phase corresponding to the BM (LC) form quoted in Table 2 are obtained for sin 2 θ 12 = 0.354 and the best fit values of sin 2 θ 23 and sin 2 θ 13 .
In each of cases C1 and C2 we report results for θ ν 23 = − π/4 and five sets of values of [θ ν 13 , θ ν 12 ], associated with, or inspired by, models of neutrino mixing. These sets include the three values of θ ν 13 = π/20, π/10 and a ≡ sin −1 (1/3) and selected values of θ ν 12 from the set: The values in square brackets in Table 2    As can be seen from Table 2, the values of δ for the IO spectrum differ insignificantly from the values obtained for the NO one in all the schemes considered, except for the B1 and B2 ones. The difference between the NO and IO values of δ in the B1 and B2 schemes is a consequence of the difference between the best fit values of sin 2 θ 23 corresponding to the NO and IO spectra 16 . We use the values of δ from Table 2 to obtain predictions for the Majorana phases in the next subsection.

Majorana Phases
In this subsection we present results of the numerical analysis of the predictions for the Majorana phases, performed using the best fit values of the neutrino mixing parameters given in eqs. (3) -(5). These predictions are obtained from the sum rules in eqs. (235) -(238), in which we have used the proper expressions for sin 2 θ 23 and cos δ from [2,4]. We summarise the predictions for all the cases considered in the present study in Tables 3 and 4, in which we give, respectively, the values of the phase differences (α 21 /2 − ξ 21 /2) and (α 31 /2 − ξ 31 /2) found in schemes A1, A2, B3, C1 and C2. In the cases of schemes B1 and B2 we present in Table 4 results for the difference (α 31 /2 − ξ 31 /2 − β), since the phase β, in general, is not fixed, unless some additional arguments are used that fix it. In the case of the B3 scheme the results are obtained for ω = 0, sgn (sin 2θ e 13 ) = 1, and for sin 2 θ 23 = 0.48907 (0.48886) for the NO (IO) spectrum (see subsection 3.3 and ref. [4] for details).     Table 4: The same as in Table 3, but for the phase difference (α 31 /2 − ξ 31 /2) given in degrees.
In cases B1 and B2 the presented numbers correspond to (α 31 /2 − ξ 31 /2 − β), where β is a free phase parameter. See text for further details.
All the quoted phases are determined with a two-fold ambiguity owing to the fact that the Dirac phase δ, which enters into the expressions for all the phases under discussion, is determined with a two-fold ambiguity from the sum rules it satisfies in the schemes of interest (see [2,4]). The absolute values of the sines of the phases quoted in Tables 3 and 4 are all proportional to sin θ 13 , and thus are relatively small. The results in cases A1 and B2 for the TBM, BM (LC), GRA, GRB and HG symmetry forms ofŨ ν considered were first obtained in [2] using the best fit values of sin 2 θ 12 , sin 2 θ 23 and sin 2 θ 13 from (the first e-archive version of) ref. [25]. Here, in particular, we update the results derived in [2].
As we have already noticed, in the BM (LC) case, the sum rules for cos δ lead to unphysical values of | cos δ| > 1 if one uses as input the current best fit values of sin 2 θ 12 , sin 2 θ 23 and sin 2 θ 13 [2,4,6]. Physical values of cos δ are found for larger (smaller) values of sin 2 θ 12 (sin 2 θ 23 ) [2][3][4]. The values of the phases given in Tables 3 and 4 and corresponding to the BM (LC) mixing are obtained for the 3σ upper bound of sin 2 θ 12 = 0.354 and the best fit values of sin 2 θ 23 and sin 2 θ 13 . For these values of the three mixing parameters | cos δ| has an unphysical value greater than one only for schemes B1 with the IO spectrum, B2 with the NO spectrum and B3.
A few comments on the results presented in Tables 3 and 4 are in order. These results show that for a given scheme and fixed form of the matrixŨ ν , the difference between the  Tables 3 and 4 and within the group of the last two ones, but change significantly -approximately by π -when switching from a case of one of the groups to a case in the second group. For a given symmetry form ofŨ ν -TBM, GRA, GRB and HG -the phase difference (α 21 /2 − ξ 21 /2) has very similar values for the A1, B1 and B3 schemes, they differ approximately by at most 2 • , and for the A2 and B2 schemes, for which the difference does not exceed 3 • . However, the predictions for (α 21 /2 − ξ 21 /2) for schemes A1, B1, B3 and A2, B2 differ significantly -the sum of the values of (α 21 /2 − ξ 21 /2) for any of the A1, B1, B3 schemes and for any of the A2, B2 schemes being roughly equal to 2π. In contrast, for a given are drastically different. At the same time, the values of (α 31 /2 − ξ 31 /2) for the A1 and B3 schemes practically coincide. Finally, for any given of the five cases of schemes C1 and C2, the values of the phase difference (α 21 /2 − ξ 21 /2) for schemes C1 and C2 differ drastically. The same conclusion is valid for the C1 and C2 values of the phase difference (α 31 /2 − ξ 31 /2) for any of the first three cases of these schemes listed in Table 4. For the last two cases in Table 4 the difference between the C1 and C2 values of (α 31 /2 − ξ 31 /2) is approximately 12 • and 18 • .
Further, we show how the predictions for the phase differences presented in Tables 3 and 4 change when the uncertainties in determination of the neutrino mixing parameters are taken into account. As an example, we consider the cases B1 and B2 with the TBM form of the matrixŨ ν . We fix two of sin 2 θ ij to their best fit values for the NO neutrino mass spectrum and vary the third one in its 3σ allowed range given in eqs. (3) -(5). We show the results for cases B1 and B2 in Figs. 1 and 2, respectively. As can be seen, the phase differences of interest depend weakly on sin 2 θ 12 and sin 2 θ 13 . When these parameters are varied in their 3σ ranges, the variation of the phase differences is within a few degrees. The dependence on sin 2 θ 23 is stronger: the maximal variations of (α 21 /2 − ξ 21 /2) and (α 31 /2 − ξ 31 /2 − β) are approximately of 9 • and 6 • in both cases. Another example, corresponding to the cases A1 and A2 with the TBM form of the matrixŨ ν , is considered in Appendix A.

Neutrinoless Double Beta Decay
If the light neutrinos with definite mass ν j are Majorana fermions, their exchange can trigger processes in which the total lepton charge changes by two units, |∆L| = 2: K + → π − + µ + + µ + , e − + (A, Z) → e + + (A, Z − 2), etc. The experimental searches for (ββ) 0ν -decay, (A, Z) → (A, Z + 2) + e − + e − , of even-even nuclei 48 Ca, 76 Ge, 82 Se, 100 M o, 116 Cd, 130 T e, 136 Xe, 150 N d, etc., are unique in reaching the sensitivity that might allow to observe this process if it is triggered by the exchange of the light neutrinos ν j (see, e.g., refs. [14,16]). In (ββ) 0ν -decay, two neutrons of the initial nucleus (A, Z) transform by exchanging virtual ν 1,2,3 into two protons of the final state nucleus (A, Z +2) and two free electrons. The corresponding (ββ) 0ν -decay amplitude has the form (see, e.g., refs. [10,16] where G F is the Fermi constant, m is the (ββ) 0ν -decay effective Majorana mass and M (A, Z) is the nuclear matrix element (NME) of the process. The (ββ) 0ν -decay effective Majorana mass m contains all the dependence of A((ββ) 0ν ) on the neutrino mixing parameters. The current experimental limits on | m | are in the range of (0.1−0.7) eV. Most importantly, a large number of experiments of a new generation aim at sensitivity to | m | ∼ (0.01 − 0.05) eV (for a detailed discussion of the current limits on | m | and of the currently running and future planned (ββ) 0ν -decay experiments and their prospective sensitivities see, e.g., the recent review article [60]).
The predictions for | m | (see, e.g., [10,15,16]), = m 1 cos 2 θ 12 cos 2 θ 13 + m 2 sin 2 θ 12 cos 2 θ 13 e iα 21 + m 3 sin 2 θ 13 e i(α 31 −2δ) , . In what follows we will derive predictions for | m | as a function of the lightest neutrino mass m min ≡ min(m j ), j = 1, 2, 3, for both the NO and IO neutrino mass spectra 17 and for two values of each of the phases ξ 21 and ξ 31 : ξ 21 = 0 or π, ξ 31 = 0 or π. The choice of the two values of the phases ξ 21 and ξ 31 will be justified in the next Section where we show that the requirement of generalised CP invariance of the neutrino Majorana mass term in the cases of the S 4 , A 4 , T and A 5 lepton flavour symmetries leads to the constraints ξ 21 = 0 or π, ξ 31 = 0 or π. We use the standard convention for numbering the neutrinos with definite masses in the cases of the NO and IO spectra (see, e.g., [1]): m 1 < m 2 < m 3 for the NO spectrum and m 3 < m 1 < m 2 for the IO one. We recall that the two heavier neutrino masses are expressed in terms of the lightest neutrino mass and the two independent neutrino mass squared differences grey band indicates the upper bound on | m | of (0.2 − 0.4) eV obtained in [62]. The vertical dashed line represents the prospective upper limit on m min of 0.2 eV from the KATRIN experiment [63].

46
In the present Section we derive constraints on the phases ξ 21 and ξ 31 in the matrix U ν , which diagonalises the neutrino Majorana mass matrix M ν , within the approach in which a lepton flavour symmetry G f is combined with a generalised CP symmetry H CP . We examine successively the cases of G f = A 4 (T ), S 4 and A 5 with the three LH charged leptons and three LH flavour neutrinos transforming under a 3-dimensional representation ρ of G f . At low energies the flavour symmetry G f has necessarily to be broken down to residual symmetries G e and G ν in the charged lepton and neutrino sectors, respectively. All the cases considered in the present study fall into the class of residual symmetries with trivial G e (G f being fully broken in the charged lepton sector) and G ν = Z 2 × Z 2 19 . The residual symmetry G ν alone does not provide any information on the phases ξ 21 and ξ 31 of interest. Indeed, letŪ ν be a unitary matrix which diagonalises the complex symmetric neutrino Majorana mass matrix: where m i are non-negative non-degenerate masses 20 and ξ i are phases contributing to the Majorana phases in the PMNS matrix. Let us introduce the matrices Q 0 = diag e i ξ 1 2 , e i ξ 2 2 , e i ξ 3 and (α 31 /2−ξ 31 /2) for a fixed ω in scheme B3, are determined completely by the values of the measured neutrino mixing angles θ 12 , θ 13 and θ 23 and the angles in the matrixŨ ν . If the value of the Dirac phase δ is measured, that will allow to fix the value of ω in scheme B3. Using the best fit values of θ 12 , θ 13 and θ 23 , we have obtained predictions for the phase differences listed above, which are summarised in Tables 3 and 4. In the case of scheme B3, we have set ω = 0. For this value of ω the predicted value of the Dirac phase δ lies in the 2σ interval of allowed values quoted in eq. (6). The results reported in Tables 3 and 4 show that the phase differences of interest involving the Majorana phases α 21 /2 and α 31 /2 are determined with a two-fold ambiguity by the values of θ 12 , θ 13 and θ 23 . This is a consequence of the fact that, as long as the sign of sin δ is not fixed by the data, the Dirac phase δ, on which the phase differences under discussion depend, is determined by the values of θ 12 , θ 13 and θ 23 in the schemes studied by us with a two-fold ambiguity [2][3][4], as Table 2 also shows. It follows from eq. (6) that the current data appear to favour negative values of sin δ. The predictions for the BM (LC) symmetry form ofŨ ν in Tables 3 and 4 correspond to the current 3σ upper bound of allowed values of sin 2 θ 12 = 0.354 and the best fit values of sin 2 θ 23 and sin 2 θ 13 , since using the best fit values of the three neutrino mixing angles one gets unphysical values of | cos δ| > 1 [2,4,6]. Physical values of cos δ are found for larger (smaller) values of sin 2 θ 12 (sin 2 θ 23 ) [2][3][4]. For sin 2 θ 12 = 0.354 and the best fit values of sin 2 θ 23 and sin 2 θ 13 , | cos δ| has an unphysical value greater than one only for schemes B1 with the IO spectrum, B2 with the NO spectrum and B3, and for these cases we do not present results for the relevant phase differences.
Extracting the values of the Majorana phases α 21 /2 and α 31 /2 from the results presented in Tables 3 and 4 for two fixed values of each of the phases ξ 21 and ξ 31 , ξ 21 = 0 and π, ξ 31 = 0 and π (altogether four cases), and using also the predicted values of the Dirac phase δ from Table 2 and the best fit values of sin 2 θ 12 , sin 2 θ 23 and sin 2 θ 13 , we derived (in graphic form) predictions for the absolute value of the neutrinoless double beta decay effective Majorana mass | m | as a function of the lightest neutrino mass m min ≡ min(m j ), j = 1, 2, 3, for both the NO and IO neutrino mass spectra (Figs. 3 -6). For schemes B1 and B2 the predictions are obtained by varying the phase β in the interval [0, π]. As a possible justification of the choice of the two values of the phases ξ 21 and ξ 31 used for the predictions of | m |, we show that the requirement of generalised CP invariance of the neutrino Majorana mass term in the cases of the S 4 , A 4 , T and A 5 lepton flavour symmetries leads to the constraints ξ 21 = 0 or π, ξ 31 = 0 or π.
The results derived in the present article for the Majorana CPV phases in the PMNS neutrino mixing matrix U complement the results obtained in [2][3][4] on the predictions for the Dirac phase δ in U in schemes in which the underlying form of U is determined by, or is associated with, in particular, discrete (lepton) flavour symmetries. A Impact of the sin 2 θ ij Uncertainties in Cases A1 and A2 In this Appendix we illustrate the impact of the uncertainties in determination of the neutrino mixing parameters on the predictions for the phase differences (α 21 /2 − ξ 21 /2) and (α 31 /2 − ξ 31 /2) in cases A1 and A2 with the TBM symmetry form of the matrixŨ ν . In Fig. 7 we show the dependence of (α 21 /2 − ξ 21 /2) and (α 31 /2 − ξ 31 /2) on sin 2 θ 12 (sin 2 θ 13 ) in case A1, fixing sin 2 θ 13 (sin 2 θ 12 ) to its best fit value for the NO spectrum. We recall that in this setup sin 2 θ 23 is correlated with sin 2 θ 13 by eq. (24) and, hence, is not a free parameter. In Fig. 8 we present results for case A2. Also in this scheme sin 2 θ 23 is correlated with sin 2 θ 13 and is not a free parameter (see eq. (50)). As can be seen from Figs. 7 and 8, in both cases A1 and A2 the variation of (α 21 /2 − ξ 21 /2) is within 3 • , while that of (α 31 /2 − ξ 31 /2) is within 2 • .  (13) in case A1 and for the TBM form of the matrixŨ ν , fixing sin 2 θ 13 (12) to its best fit value for the NO spectrum. The upper panels correspond to δ = cos −1 (cos δ), while the lower panels correspond to δ = 2π − cos −1 (cos δ). The vertical line and the three coloured vertical bands indicate the best fit value and the 1σ, 2σ and 3σ allowed ranges of sin 2 θ 12 (13) . 0.020 0.022 0.024 sin 2 θ 13 Figure 8: The same as in Fig. 7, but for case A2.