Predicting the Values of the Leptonic CP Violation Phases

Using the fact that the neutrino mixing matrix $U = U^\dagger_{e}U_{\nu}$, where $U_{e}$ and $U_{\nu}$ result from the diagonalisation of the charged lepton and neutrino mass matrices, we consider a number of forms of $U_{\nu}$ associated with a variety of flavour symmetries: i) bimaximal (BM) and ii) tri-bimaximal (TBM) forms, the forms corresponding iii) to the conservation of the lepton charge $L' = L_e - L_\mu - L_{\tau}$ (LC), iv) to golden ratio type A (GRA) mixing, v) golden ratio type B (GRB) mixing, and iii) to hexagonal (HG) mixing. Employing the minimal form of $U_e$ that can provide the requisite corrections to $U_{\nu}$ so that reactor, atmospheric and solar neutrino mixing angles $\theta_{13}$, $\theta_{23}$ and $\theta_{12}$ have values compatible with the current data, including a possible sizable deviation of $\theta_{23}$ from $\pi/4$, we discuss the possibility to obtain predictions for the CP violation phases in the neutrino mixing matrix. Following the approach developed in [1] we derive predictions for the Dirac phase $\delta$ and the rephasing invariant $J_{\rm CP}$ in the cases of GRA, GRB and HG forms of $U_{\nu}$ (results for the TBM and BM (LC) forms were obtained in [1]). We show also that under rather general conditions within the scheme considered the values of the Majorana phases in the PMNS matrix can be predicted for each of the forms of $U_{\nu}$ discussed. We give examples of these predictions and of their implications for neutrinoless double beta decay. In the GRA, GRB and HG cases, as in the TBM one, relatively large CP violation effects in neutrino oscillations are predicted ($|J_{CP}| \sim (0.031 - 0.034)$). Distinguishing between the TBM, BM (LC), GRA, GRB and HG forms of $U_{\nu}$ requires a measurement of $\cos\delta$ or a relatively high precision measurement of $J_{\rm CP}$.


Introduction
Determining the status of the CP symmetry in the lepton sector is one of the highest priority principal goals of the program of future research in neutrino physics (see, e.g., [2,3]).As in the case of the quark sector, the CP symmetry can be violated in the lepton sector by the presence of physical phases in the Pontecorvo, Maki, Nakagawa and Sakata (PMNS) neutrino mixing matrix.In the case of 3-neutrino mixing and massive Majorana neutrinos we are going to consider 1 , the 3 × 3 unitary PMNS matrix U PMNS ≡ U contains, as is well known, one Dirac and two Majorana [4] CP violation (CPV) phases which can be the source of CP violation in the lepton sector.In the widely used standard parametrisation [2] of the PMNS matrix we also are going to employ, U PMNS is expressed in terms of the solar, atmospheric and reactor neutrino mixing angles θ 12 , θ 23 and θ 13 , respectively, and the Dirac and Majorana CPV phases, as follows: 2 , e i α 31 2 ) , where α 21,31 are the two Majorana CPV phases and V is a CKM-like matrix, In eq. ( 2), δ is the Dirac CPV phase, 0 ≤ δ ≤ 2π, we have used the standard notation c ij = cos θ ij , s ij = sin θ ij , and 0 ≤ θ ij ≤ π/2.In what concerns the Majorana CPV phases, for the purpose of the present study it is sufficient to consider that they vary in the intervals 0 ≤ α 21,31 ≤ 2π 2 .If CP invariance holds, we have δ = 0, π, 2π, the values 0 and 2π being physically indistinguishable, and [7] α 21 (31) = k ( ) π, k ( ) = 0, 1, 2, ....The CP symmetry will not hold in the lepton sector if the Dirac and/or Majorana phases possess CP-nonconserving values.If the Dirac phase δ has a CP-nonconserving value, this will induce, as is well known, CP violation effects in neutrino oscillations, i.e., a difference between the 3-flavour neutrino oscillation probabilities P (ν l → ν l ) and P (ν l → νl ), l = l = e, µ, τ .
The flavour neutrino oscillation probabilities P (ν l → ν l ) and P (ν l → νl ), l, l = e, µ, τ , do not depend on the Majorana phases [4,8].The Majorana phases can play important role in processes which are characteristic for Majorana neutrinos, in which the total lepton charge L changes by two units, like neutrinoless double beta ((ββ) 0ν -) decay (A, Z) → (A, Z + 2) + e − + e − (see, e.g., [9][10][11]), etc.The rates of the processes of emission of two different Majorana neutrinos, an example of which is the radiative emission of neutrino pair in atomic physics [12], depend in the threshold region on the Majorana phases [13].The phases α 21,31 can affect significantly the predictions for the rates of the lepton flavour violating (LFV) 1 All compelling data on neutrino masses, mixing and oscillations are compatible with the existence of mixing of three light massive neutrinos νi, i = 1, 2, 3, in the weak charged lepton current (see, e.g., [2]).It follows also from the data that the masses mi of the three light neutrinos νi do not exceed approximately 1 eV, mi ∼ < 1 eV, i.e., they are significantly smaller than the masses of the charged leptons and quarks. 2 One should keep in mind, however, that in the case of the seesaw mechanism of neutrino mass generation the Majorana phases α21 and α31 vary in the interval [5] 0 ≤ α21,31 ≤ 4π.The interval beyond 2π, 2π ≤ α21,31 ≤ 4π, is relevant, e.g., in the calculations of the baryon asymmetry within the leptogenesis scenario [5], in the calculation of the neutrinoless double beta decay effective Majorana mass in the TeV scale version of the type I seesaw model of neutrino mass generation [6], etc. decays µ → e + γ, τ → µ + γ, etc. in a large class of supersymmetric theories incorporating the see-saw mechanism [14].
Most importantly, the Dirac phases δ and/or the Majorana phases α 21,31 in the PMNS neutrino mixing matrix can provide the CP violation necessary for the generation of the observed baryon asymmetry of the Universe [15].
The theoretical predictions for the values of the CPV phases in the neutrino mixing matrix depend on the approach and the type of symmetries one uses in the attempts to understand the pattern of neutrino mixing (see, e.g., [1,[20][21][22] and references quoted therein).In the case of the Dirac phase δ, the predictions vary considerably: they include the values 0, π/2, π, 3π/2, but not only; in certain cases 0, π/2, π and 3π/2 are approximate values, the exact predictions being slightly different from these values.Obviously, a sufficiently precise measurement of δ will serve as an additional very useful constraint for identifying the approaches and/or the symmetries, if any, at the origin of the observed pattern of neutrino mixing.Understanding the origin of the patterns of neutrino masses and mixing, emerging from the neutrino oscillation, 3 H β−decay, cosmological, etc. data is one of the most challenging problems in neutrino physics.It is part of the more general fundamental problem in particle physics of understanding the origins of flavour, i.e., of the patterns of the quark, charged lepton and neutrino masses and of the quark and lepton mixing.
perturbative corrections to the TBM and BM (LC) mixing angles are provided by the matrix U e , and that U e has a minimal form in terms of angles and phases it contains that can provide the corrections to U ν so that the angles θ 13 , θ 23 and θ 12 in the PMNS matrix have values compatible with the current data, we have obtained in [1] predictions for the Dirac phase δ present in the PMNS matrix U4 .An important requirement is that the corrections due to the matrix U e should allow sizable deviations of the angle θ 23 from the BM and TBM value ±π/4.These requirements imply that U e should be a product of two rotations in the 12 and 23 planes, R 12 (θ e 12 ) and R 23 (θ e 23 ), and a diagonal phase matrix which contains, in general, two physical CP violation phases.In the case of "standard" ordering with U e ∝ R 23 (θ e 23 )R 12 (θ e 12 ), which we are going to consider and which is related to the hierarchy of the charged lepton masses, m 2 e m 2 µ m 2 τ , and is a common feature of the overwhelming majority of the existing models of the charged lepton (and neutrino) masses and the associated mixing, cos δ was shown to satisfy in the cases of the TBM and BM (LC) forms of U ν a new sum rule [1] by which it is expressed in terms of the three angles θ 12 , θ 23 and θ 13 of the PMNS matrix.For the current best fit values of sin 2 θ 12 , sin 2 θ 23 and sin 2 θ 13 , the following predictions for δ were obtained for the two forms of U ν [1]: i) δ ∼ = π in the BM (or LC) case, ii) δ ∼ = 3π/2 or π/2 in the TBM case 5 , the CP conserving values δ = 0, π, 2π being excluded in the TBM case at more than 5σ.A model based on the T flavour symmetry leading to the TBM form of U ν , in which the conditions of the general phenomenological approach followed in [1] are realised and thus which predicts, in particular, δ ∼ = 3π/2 or π/2, was constructed in [20].
In the present article we first generalise in Section 3 the analytic results for the sum rule involving the cosine of the Dirac phase δ, obtained in [1] for the specific BM (LC) and TBM values π/4 and sin −1 (1/ √ 3) of the angle θ ν 12 in the matrix U ν , to the case of arbitrary fixed value of θ ν 12 .This allows us to obtain new predictions for the phase δ and the J CP factor, which controls the magnitude of CP violation effects due to δ in neutrino oscillations, in the cases of i) golden ratio type A (GRA) mixing [40,41] with sin 2 θ ν 12 = (2 + r) −1 ∼ = 0.276, r being the golden ratio, r = (1 + √ 5)/2, ii) golden ratio type B (GRB) mixing [42] with sin 2 θ ν 12 = (3 − r)/4 ∼ = 0.345, and iii) hexagonal (HG) mixing [43] in which θ ν 12 = π/6.As like the TBM and BM forms of U ν , the GRA form can be obtained from discrete family symmetry in the lepton sector, while the GRB and HG forms are considered on general phenomenological grounds (see, e.g., the reviews [44][45][46] and [42,43]).In section 3 we derive also analytic expression for the correction in the new sum rule for cos δ due to the possible presence in U e of the 13 rotation matrix R 13 (θ e 13 ) with angle θ e 13 1 and determine the conditions under which this correction is sub-dominant.In Section 4 we show that the approximate sum rule for δ proposed in [34] can be obtained in the leading order approximation from the "exact" sum rule for cos δ derived in Section 3. We compare the predictions for δ in the cases of the TBM, BM (LC), GRA, GRB and HG forms of of the matrix U ν , obtained using the "exact" and the leading order sum rules and determine the origin of the difference in the predictions.We next analyse in Section 5 the possibility to obtain predictions for the values of the Majorana phases in the PMNS matrix, α 21 and α 31 , using the same approach which allowed us to get predictions for the Dirac phase δ.For the TBM, BM (LC), GRA, GRB and HG forms of U ν considered by us, we obtain analytic expressions for the contribution to the phases α 21 and α 31 , generated by the CPV phases which serve in the approach employed as a "source" for the Dirac phase δ and which are present in the PMNS matrix due to the non-trivial form of the charged lepton "correction" matrix U e .We determine the cases when the phases α 21,31 can be predicted and give example of prediction of their values.We show in Section 6 that the results obtained on the Majorana phases for the different symmetry forms of the matrix U ν , can lead, in particular, to specific predictions for the (ββ) 0ν -decay effective Majorana mass in the physically important cases of neutrino mass spectrum with inverted ordering or of quasi-degenerate type.The results of the present study are summarised in Section 7.

The Framework
In what follows we consider 3-neutrino mixing of the three left-handed (LH) flavour neutrinos and antineutrinos, ν l and νl , l = e, µ, τ .The neutrino mixing matrix in this case receives contributions from the diagonalisation of the charged lepton and neutrino Majorana mass terms.Taking into account the contributions from the charged lepton and neutrino sectors, the PMNS neutrino mixing matrix can be written as [27]: Here U e and U ν are 3 × 3 unitary matrices originating from the diagonalisation respectively of the charged lepton 6 and neutrino mass matrices, Ũe and Ũν are CKM-like 3 × 3 unitary matrices and Ψ and Q 0 are diagonal phase matrices each containing in the general case two physical CPV phases, The phase matrix Q 0 contributes to the Majorana phases in the PMNS matrix and can appear in eq. ( 6) as a result of the diagonalisation of the neutrino Majorana mass term, while Ψ can originate from the charged lepton sector (U † e = ( Ũe ) † Ψ), or from the neutrino sector (U ν = Ψ Ũν Q 0 ), or can receive contributions from both sectors.
Following the results of the analysis performed in [1], we will assume that the matrix Ũe is a product of two orthogonal matrices describing rotations in the 12 and 23 planes and that the two rotations in Ũe are in the "standard ordering".It proves convenient to adopt for Ũe the notation used in [1]: R −1 23 are two arbitrary (real) angles.The fact that Ũe does not include the matrix R 13 (θ e 13 ) describing rotation in the 13 plane, i.e., that θ e 13 ∼ = 0, follows from the requirement that 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 the mixing angles θ 13 , θ 23 and θ 12 in U have values compatible with the current data, including 6 For charged lepton mass term written in the left-right convention, the matrix Ue diagonalises the hermitian matrix , ME being the charged lepton mass matrix.
the possibility of a sizable deviation of θ 23 from π/4.As will be discussed briefly in Section 3, a nonzero θ e 13 ∼ < 10 −3 generates a correction to cos δ derived from the exact sum rule, which does not exceed 11% (4.9%) in the TBM (GRB) cases and is even smaller in the other three cases of symmetry form of Ũν analysed in the present article.We note that θ e 13 ∼ = 0 is a feature of many theories and models of charged lepton mass generation (see, e.g., [20,36,37,40,44,47]) and was used in a large number of articles dedicated to the problem of understanding the origins of the observed pattern of neutrino mixing (see, e.g., [22,26,27,29,34,35,38,39,48,49]).In large class of GUT inspired models of flavour, for instance, the matrix U e is directly related to the quark mixing matrix (see, e.g., [31,36,37,44,45]).As a consequence, in this class of models, in particular, θ e 13 is negligibly small.We will assume further that the matrix Ũν has one of the following symmetry forms: TBM, BM, LC, GRA, GRB and HG.For all symmetry forms of interest, Ũν is also a product 23 and 12 rotations in the plane: In the case of the TBM, BM, GRA, GRB and HG forms of Ũν we have θ ν 23 = − π/4, while θ ν i) θ 13 = 0 in all six cases of interest of Ũν ; ii) θ 23 = −π/4, if Ũν has any of the forms TBM, BM, GRA, GRB and HG, while θ 23 can have an arbitrary value if U ν has the LC form; iii) sin 2 θ 12 = 0.5 for the BM and LC forms of Ũν ; sin 2 θ 12 = 1/3 in the TBM case; sin 2 θ 12 ∼ = 0.276 and 0.345 for the GRA and GRB mixing and sin 2 θ 12 = 0.25 for HG mixing.Thus, the matrix U e has to generate corrections i) leading to θ 13 = 0 compatible with the observations in all six cases of U ν considered; ii) leading to the observed deviation of θ 12 from a) π/4, b) from the two golden ratio values7 and c) from π/6, in the cases of a) BM and LC, b) GRA and GRB, and c) HG, mixing; iii) leading to the sizable deviation of θ 23 from π/4 for all cases considered except the LC one, if it is confirmed by further data that sin 2 θ 23 ∼ = 0.40 − 0.44.The minimal form of U e in terms of angles and phases it contains, which can produce the requisite corrections discussed above, is the one with Ũe given in eq. ( 8).The presence of R −1 12 (θ e 12 ) in Ũe allows to correct the symmetry values of θ 12 and θ 13 , while the presence of R −1 23 (θ e 23 ) allows to have sizable deviations (bigger than 0.5 sin 2 θ 13 ) of sin 2 θ 23 from the symmetry value of 0.5.
In the approach adopted by us following [1] the PMNS neutrino mixing matrix has the form: where θ ν 23 = − π/4 and θ ν 12 has a known value.As a consequence, the three angles θ 12 , θ 23 and θ 13 and the Dirac CPV phase δ of the PMNS mixing matrix, eqs.( 1) -( 2), can be expressed as functions of the two real angles, θ e 12 and θ e 23 , and the two phases, ψ and ω, of the phase matrix Ψ.The results will depend on the specific value of the angle θ ν 12 , i.e., on the assumed symmetry form of Ũν .We will discuss how the Majorana phases in the PMNS matrix, α 21 and α 31 , are expressed in terms of these parameters later.
As was shown in [1], the product of matrices R 23 (θ e 23 )ΨR 23 (θ ν 23 = −π/4) in the expression (12) for U PMNS can be rearranged as follows: Here the angle θ23 is determined by and where 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 to Q = Q 1 Q 0 .The PMNS matrix takes the form: where θ ν 12 has a fixed value which depends on the symmetry form of Ũν used.Thus, the four observables θ 12 , θ 23 , θ 13 and δ are functions of three parameters θ e 12 , θ23 and φ.As a consequence, the Dirac phase δ can be expressed as a function of the three PMNS angles θ 12 , θ 23 and θ 13 , leading to a new "sum rule" relating δ and θ 12 , θ 23 and θ 13 [1].Using the measured values of θ 12 , θ 23 and θ 13 , we have obtained in [1] predictions for the values of δ and of the rephasing invariant ), which controls the magnitude of CP violating effects in neutrino oscillations [50], in the cases of the TBM, BM (LC) forms of Ũν .Here we will first obtain predictions for δ and J CP in the cases of GRA, GRB and HG forms of Ũν .After that we will analyse the possibility to obtain predictions for the Majorana phases in the PMNS matrix within the framework described above.
As it follows from eqs. ( 13), ( 15) and ( 16), the phase φ contributes to the Majorana phase α 31 , in particular, via the phase β.Thus, we will give next the values of cos φ and | sin φ| for the different symmetry forms of the matrix Ũν we are considering, TBM, BM (LC), GRA, GRB and HG 8 .These values will be relevant in the discussion of the Majorana phases determination.Using the best fit values of the neutrino mixing parameters sin 2 θ 12 , sin 2 θ 23 and sin 2 θ 13 quoted in eqs.( 3) -( 5), for ∆m 2  31 > 0 and the specific value of θ ν 12 characterising a given case of Ũν , we get: GRA : GRB : HG : The same procedure leads in the BM (LC) case to the unphysical value of cos φ ∼ = −1.13.This reflects the fact that the scheme under discussion with BM (LC) form of the matrix Ũν does not provide a good description of the current data on θ 12 , θ 23 and θ 13 [1].Thus, we will calculate cos φ using the best values of sin 2 θ 12 = 0.32, sin 2 θ 23 = 0.41 (0.42) and sin θ 13 = 0.158, determined for ∆m 2 31 > 0 (∆m 2 31 < 0) in the statistical analysis performed in [1].For these values of sin 2 θ 12 , sin 2 θ 23 and sin θ 13 in the case of ∆m 2  31 > 0 we get: We do not give the results on cos φ for ∆m 2 31 < 0 since they differ little from those shown.Comparing the imaginary and real parts of U * e1 U * µ3 U e3 U µ1 , obtained using eq.( 18) and the standard parametrisation of U PMNS , one gets the following relation between φ and δ: For θ ν 12 = π/4 and θ ν 12 = sin −1 (1/ √ 3) the expression (30) for cos δ we have derived reduces to those found in [1] in the BM (LC) and TBM cases, respectively.
From eq. ( 30) we find in the cases of TBM and BM (LC) forms of Ũν The value of cos δ corresponds in the TBM case to the best fit values of sin 2 θ ij given in eqs.
It follows from the results derived and quoted above that, in general, the predicted values of cos δ and δ vary significantly with the assumed symmetry form of the matrix Ũν .One exception are the predictions of δ in the cases of TBM and GRB forms of Ũν : they differ only by approximately 5 • .We note also that, except for the BM (LC) case, the values of cos δ and cos φ differ significantly for a given assumed form of the symmetry mixing, TBM, GRA, etc.
If we consider the indications obtained in [16,17] that δ ∼ = 3π/2, only the case of BM (LC) mixing is weakly disfavoured for ∆m 2  31 > 0 at approximately 1.4σ, while for ∆m 2  31 < 0 all cases of the form of Ũν considered by us are statistically compatible with the results on δ found in [16,17] (see, e.g., Fig. 3 in [16]).
As was mentioned in Section 2, a nonzero | sin θ e 13 | 1, θ e 13 being the angle of rotation in the 13 plane, generates a correction to the value of cos δ derived from the exact sum rule.In this case we have: cos δ(θ e 13 ) = cos δ − ∆(cos δ), where cos δ is the value obtained from the exact sum rule and ∆(cos δ) is the correction due to | sin θ e 13 | = 0.As can be shown using the parametrisation Ũe = R − where κ = arg(c e 23 e − i ω − s e 23 e − i ψ ).The result (36) for ∆(cos δ) can be derived by taking into account, in particular, that | sin θ e 13 | 1 and that in the approximation employed by us cos δ(θ e 13 ) sin θ 13 ∼ = cos δ sin θ 13 .It is not difficult to convince oneself that for the best fit values of the neutrino mixing parameters and the symmetry forms of Ũν considered, the correction satisfies the inequality: , where the constant C = 9.0, 12.7, 7.9, 9.2, and 7.3 for the TBM, BM, GRA, GRB and HG forms of Ũν , respectively.Thus, for | sin θ e 13 | ∼ < 10 −3 , the correction |∆(cos δ)| to the exact sum rule result for cos δ does not exceed 11% (4.9%) in the case of the TBM (GRB) form and is even smaller for the BM, GRA and HG forms of Ũν .In what follows we concentrate on the case of negligibly small sin θ e 13 ∼ = 0.
The fact that the value of the Dirac CPV phase δ is determined (up to an ambiguity of the sign of sin δ) by the values of the three mixing angles θ 12 , θ 23 and θ 13 of the PMNS matrix and the value of θ ν 12 of the matrix Ũν , eq. ( 11), is the most striking prediction of the model considered.This result implies also that in the scheme under discussion, the rephasing invariant J CP associated with the Dirac phase δ, which determines the magnitude of CP violation effects in neutrino oscillations [50] and in the standard parametrisation of the PMNS matrix has the well known form, is also a function of the three angles θ 12 , θ 23 and θ 13 of the PMNS matrix and of θ ν 12 : we have for ∆m 2 31 > 0 (∆m 2 31 < 0): TBM : BM (LC) : GRA : GRB : HG : where the results in the TBM and BM cases were obtained in [1]  The case of negligible θ e 23 ∼ = 0 was analysed by many authors (see, e.g., [26][27][28][29][30][31][32]34] as well as [22]).It corresponds to a large number of theories and models of charged lepton and neutrino mass generation (see, e.g., [31,32,36,37,39,44]).In the limit of negligibly small θ e 23 we find from eqs. ( 14), ( 16) and ( 17): The phase ω is unphysical.All results obtained in the previous section are valid also in the case of negligibly small θ e 23 : one has to set sin 2 θ23 = 0.5 in the expressions derived for arbitrary sin 2 θ23 in the preceding Section.From eqs. ( 19) -( 21), using the fact that sin 2 θ23 = 0.5, we get the well known results for sin θ 13 and sin 2 θ 23 , and the following new exact expression for sin 2 θ 12 : In the case of θ e 23 = 0, as is well known, sin 2 θ 23 can deviate only by 0.5 sin 2 θ 13 from 0.5.Let us emphasise that the exact sum rules in eqs.( 22) and (30) correspond to sin θ e 23 = 0, including the case of a relatively small but non-negligible sin θ e 23 .Equation (46) represents an exact sum rule connecting the value of the CPV phase φ with the values of the angles θ 13 and θ 12 for θ e 23 = 0. From eq. ( 46) we can get approximate sum etc.), e.g., in [26][27][28][30][31][32][34][35][36] 12 .For arbitrary fixed value of θ ν 12 the sum rule in eq. ( 49) was proposed in [34], where the approximate relation cos φ ∼ = cos δ, which holds to leading order in sin θ 13 , was implicitly used (see further).,31,36].It was suggested in ref. [35] that the sum rule (55) should be used to obtain the value of cos δ using the experimentally determined values of sin 2 θ 12 and sin θ 13 , e.g., in the case of the TBM form of Ũν .The same sum rule ( 55) is given also, e.g., in the review articles [44,51].
The derivation of the sum rule of interest, given in ref. [34], is based on the following expression for sin 2 θ 12 : the leading order approximation in sin θ 13 is consistent with taking into account the difference between θ 12 and θ ν 12 in the sum rules given in eqs.( 49) and (55), and in eqs.( 48) and ( 54).We will show next that the sum rules in eqs.( 48) and (54), and the equivalent "leading order sum rules" in eqs.( 49) and ( 55), give imprecise, and in some cases -largely incorrect, results for both cos φ and cos δ in the cases of TBM, GRA, GRB and HG forms of Ũν .
Indeed, using the "leading order sum rules" in eqs.( 48) and ( 54), we get for the best fit values of sin 2 θ 12 = 0.308 and sin 2 θ 13 = 0.0234 in the TBM, GRA, GRB and HG cases15 : TBM, eqs.( 54) and ( 48 Clearly, in all these cases we have cos δ ∼ = cos φ.The slight differences in the values of cos δ and cos φ are caused by the differences between the factors sin 2θ ν 12 and sin 2θ 12 in eqs.( 48) and ( 54).In the approximation in which eqs. ( 49) and ( 55) are derived, these differences should be neglected and we would have sin 2θ ν 12 = sin 2θ 12 .But in this case, as we have already have noticed, we would have also θ ν 12 = θ 12 , and thus cos δ = cos φ = 0. Using the exact sum rules for cos φ and cos δ, given in eqs.( 50) and ( 46 71) -( 74), the values of cos δ, obtained using the exact sum rule (50) in the TBM, GRA, GRB and HG cases differ from those calculated using the "leading order sum rule" (54), by the factors 1.57, 0.78, 1.31 and 0.86, respectively.In the case of cos φ, the corresponding factors are 0.76, 1.53, 0.83 and 1.26.The higher order corrections have opposite effect on the leading order results for | cos δ| and | cos φ|: if the exact sum rule value of | cos δ| is smaller (larger) than the "leading order sum rule" value, as in the TBM and GRB (GRA and HG) cases, the corresponding exact sum rule value of | cos φ| is larger (smaller) than the "leading order sum rule" value.We see also from eqs. ( 71) -( 74) that the values of cos δ and cos φ, derived from the exact sum rules in the cases of TBM, GRA, GRB and HG forms of the matrix Ũν indeed differ approximately by factors (1.5 -2.0).As we have seen, for finite values of θ e 23 , for which we have sin 2 θ 23 ∼ = (0.43 − 0.44), cos φ and cos δ in all cases we are considering with the exception of the BM (LC) one, differ approximately by the same factor of (1.5 -2.0).
The origin of these significant differences between the results derived using the exact and the "leading order sum rules" for cos δ and cos φ for the TBM, GRA, GRB and HG forms of the matrix Ũν can be traced to the importance of the next-to-leading order corrections

The Majorana Phases
We will analyse next the possibility to obtain predictions for the values of Majorana phases α 21 and α 31 in the PMNS matrix using the approach described above, We will show in what follows that in many cases of interest it is possible to determine the phases α 21 and α 31 if the values of the phase φ, or δ, and of the phases ξ 21 and ξ 31 in the diagonal matrix Q 0 in eq. ( 6) are known.The matrix Ũν Q 0 , as we have already briefly discussed, originates from the diagonalisation of the flavour neutrino Majorana mass term.In many theories and models of neutrino mixing the values of the phases ξ 21 and ξ 31 are fixed by the form of flavour neutrino Majorana mass term, which is dictated by the chosen discrete (or continuous) flavour symmetry (see, e.g., [20,37]), or on phenomenological grounds (see, e.g., [38]).Typical values of the phases ξ 21 and ξ 31 are 0, π/2 and π.In the model with T flavour symmetry in the lepton sector constructed in [20], for instance, ξ 21 and ξ 31 can take two sets of values: (ξ 21 , ξ 31 ) = (0, 0) and (0, π).
In what follows we will assume that the phases ξ 21 and ξ 31 are known.Under this condition the Majorana phases α 21 and α 31 can be determined, as we will discuss in greater detail below, i) if the angles θ e where, as we have shown, (−φ + β e1 + β e2 ) = δ and The phases in the diagonal matrix P 2 are unphysical -they can be absorbed by the electron and muon fields in the weak charged lepton current.The phases (β e2 − β e1 ) and β e2 in the diagonal matrix Q 2 give contribution to the Majorana phases α 21 /2 and α 31 /2, respectively, while the common phase (−β e2 ) in Q 2 is also unphysical and we will not keep it in our further analysis.One can show further (analytically or numerically) that we have: This implies that the matrix in eq. ( 95) is in the standard parametrisation form.Correspondingly, the Majorana phases α 21 /2 and α 31 /2 in the matrix Q in eq. ( 1) are determined by the phases in the matrix The expressions we have obtained for the phases β e1 and β e2 , eqs.(90), ( 101) and ( 91), (100), are exact.It follows from these expressions that the phases β e1 , β e2 can be determined knowing the values of θ e 12 and φ, or, alternatively, of δ and of θ 12 , θ 23 and θ 13 .In what concerns the phases ξ 21 and ξ 31 in eq. ( 102), they are assumed to be fixed by the symmetry which determines the TBM, BM, GRA, etc. form of the matrix Ũν .
We note that by writing, 2β e2 = ±r • 2 and 2(β e2 − β e1 ) = ±r • 21 we imply, in the convention used by us for the intervals in which the phases α 21 and α 31 vary, where k 2 = 1 (k 2 = 2) and k 21 = 1 (k 21 = 2) has to be taken into account in certain cases [5] when the flavour neutrino Majorana mass term is generated by the type I seesaw mechanism [53].
We will consider next the possibility to calculate also the phase β = γ − φ determined in eqs.( 16) and (17).We note first that the phase β enters only in the expression for the Majorana phase α 31 .The latter plays a subdominant role in a number of cases of processes, characteristic of the Majorana nature of massive neutrinos ν j .More specifically, the term involving the Majorana phase α 31 gives a subdominant contribution in the (ββ) 0ν -decay rate in the cases of neutrino mass spectrum i) with inverted ordering (IO), corresponding to ∆m 2 31(32) < 0, and ii) of quasi-degenerate (QD) type (see, e.g., [2,11]), the reason being that the term of interest involves the suppression factor sin 2 θ 13 ∼ = (0.023 − 0.024).For the same reason the rate of the process of radiative emission of two different Majorana neutrinos in atomic physics depends weakly on the Majorana phase α 31 [13].The value of the phase α 31 plays important role, for example, for the prediction of the (ββ) 0ν -decay rate if neutrino mass spectrum is with normal ordering (NO) but is not quasi-degenerate, i.e., if ∆m 2  31(32) > 0, m 1 < ( )m 2,3 and m 1 ∼ < ∆m 2 31 ∼ = 0.05 eV (see, e.g., [2]).
In the case of negligibly small θ e 23 , as we have seen, γ = −ψ + π, φ = −ψ, and β = π.In the "counter-intuitive" case [27] of | sin θ e 23 | = 1 we have γ = φ = −ω, and β = 0.In these cases we get, e.g., for (ξ 21 , ξ 31 ) = (0, 0) using eqs.( 108 where the values (values in brackets) correspond to β = π (β = 0).In the general case of non-negligible θ e 23 we get from eq. ( 17), using eq.( 14): This constraint reduces the number of the unknown parameters in terms of which the phase β is expressed to one.The sign of sin(ψ −ω) is also undetermined.Obviously, it is impossible to determine the phase β without some additional input.In what follows we will exploit several possibilities.The first possibility corresponds to the phase ψ or the phase ω having one of the following specific values: 0, π/2, π and 3π/2.In any of these cases the phase γ is determined (up to a possible sign ambiguity either of sin γ or of cos γ) by the phase φ, which allows to determine also the phase β (again up to a possible sign ambiguity of cos β or of sin β).This possibility is realised in certain models of neutrino mixing based on discrete flavour symmetries.
To be more specific, assume first that ψ = 0.In this case we get from eqs.
It is clear from eq. ( 123) that the value of sin γ can be determined knowing the values of sin φ and of cot θ23 , independently of the values of θ e 23 and ω.This, obviously, allows to find also | cos γ|, but not the sign of cos γ.If, however, the inequality √ 2| sin θ e 23 cos ω| < | cos φ cos θ23 | is fulfilled, eq. ( 124) would allow to correlate the sign of cos γ with the sign of cos φ and thus to determine γ for a given φ: we would have cos γ < 0 if cos φ > 0, and cos γ > 0 for cos φ < 0. In the case of √ 2| sin θ e 23 cos ω| > | cos φ cos θ23 |, the sign of cos γ will coincide with the sign of sin θ e 23 cos ω, and if the latter cannot be fixed, the two possible signs of cos γ have to be considered.
These relations hold in the model with T family symmetry proposed in [20] 17 .Now the value of cos γ can be determined knowing the values of cos φ and of cot θ23 , independently of the values of θ e 23 and ω.This allows to find also | sin γ|, leaving the sign of sin γ undetermined.Depending on the relative magnitude of the terms | sin φ cos θ23 | and | √ 2 sin θ e 23 sin ω|, the sign of sin γ will be anti-correlated either with the sign of sin φ, or with the sign of sin θ e 23 sin ω.In the latter case both signs of sin γ have to be considered if the sign of sin θ e 23 sin ω is undetermined.
One finds β = π + 2πk, k = 0, 1, if the equality ψ = ω holds.This possibility is realised in a scheme considered in [38], in which also the phases ξ In a general analysis in which one attempts to reproduce the values of the three neutrino mixing parameters sin 2 θ 12 , sin 2 θ 23 and sin 2 θ 13 in the cases of the TBM, BM, GRA, etc. forms of the matrix Ũν with the help of the "correcting" matrix ( Ũe ) † Ψ = R 12 (θ e 12 )R 23 (θ e 23 )Ψ, the four parameters θ e 12 , θ e 23 , ψ and ω will have to satisfy three constraints.This implies that the values of any two parameters, say, θ e 23 and (ψ − ω), will have to be correlated 18 .In addition, θ e 23 and (ψ − ω) have to satisfy the constraint given in eq. ( 14).This can allow to limit significantly the range of possible values of, or even to determine, | sin θ e 23 |.As a consequence, cos β (and therefore | sin β|) will either be constrained to lie in a relatively narrow interval, or its value will be determined, which will lead to a similar information about the phase β (up to the possible ambiguity related to the two possible signs of sin β).Such an analysis, however, is outside the scope of the present investigation; we intend to perform it elsewhere.corrections to (55), or to the equivalent sum rule (54), derived in eq. ( 53) (and in eq. ( 47) for cos φ), are significant and should be taken into account.For the TBM GRA, GRB and HG forms of Ũν , the predictions for cos δ (and cos φ) derived using the exact sum rule eq. ( 50) (eq.( 46)), or the next to leading order sum rule eq. ( 53) (eq.( 47)), differ by factors of (1.2 -1.6) from the predictions obtained from the leading order sum rule eq. ( 55) (eq.( 49)), or the equivalent one eq.( 54) (eq.( 48)).As we have shown in subsection 4.2, this difference can be further amplified by an additional factor of 1.2 by the next-to-leading order correction due to θ e 23 = 0, sin θ e 23 1, if sin 2 θ 23 ∼ = 0.4.Using the exact sum rules eqs.( 30) and ( 22) leads for θ e 23 = 0 to practically the same results respectively for cos δ and cos φ as the next-to-leading order sum rules eq. ( 86) and eq.( 87).We have shown also that the leading order sum rule (55) provides a rather accurate prediction for cos δ only in the case of BM (LC) form of the matrix Ũν .
In Section 5 we have analysed the possibility to obtain predictions for the values of the Majorana phases α 21 /2 and α 31 /2 in the PMNS matrix.We have shown that α 21 /2 β e2 − β e1 + ξ 21 /2 and α 31 /2 = β e2 + β + ξ 31 /2, where ξ 21 and ξ 31 are the phases of the matrix Q 0 , and β e1 , β e2 and β are real calculable phases.In many theories and models of neutrino mixing the values of the phases ξ 21 and ξ 31 are fixed by the form of the neutrino Majorana mass term, which is dictated by the chosen discrete (or continuous) flavour symmetry or on phenomenological grounds.Typical values of ξ 21 /2 and ξ 31 /2 are 0, π/2 and π20 .Within the approach adopted in the present article, the phases β e1 and β e2 can be calculated exactly for each of the five symmetry forms of Ũν considered by us.We have first derived exact analytic expressions for β e1 and β e2 in terms of the three neutrino mixing angles, θ 12 , θ 23 , θ 13 , and the Dirac phases δ (eqs.( 101) and (100)).Given θ 12 , θ 23 , θ 13 and θ ν 12 (i.e., the symmetry form of Ũν ), these expressions allow to get predictions for the values of β e1 and β e2 .We give such predictions for β e1 , β e2 and (β e2 − β e1 ) for each of the five symmetry forms of Ũν considered using the the best fit values of sin 2 θ 12 , sin 2 θ 23 and sin 2 θ 13 (eqs.(103) -(117)).In what concerns the phase β entering into the expression for the Majorana phase α 31 /2, we have discussed a number of cases in which it can be calculated exactly.
Finally, in Section 6 we have analysed the implications of the results obtained on the leptonic CPV phases for the predictions of the effective Majorana mass in (ββ) 0ν -decay.This was done on the examples of the neutrino mass spectra with inverted ordering and of quasidegenerate type.
The predictions for the leptonic CP violation phases in the PMNS neutrino mixing matrix derived in the present article will be tested in the experiments on CP violation in neutrino oscillations and possibly in the neutrinoless double beta decay experiments.

Note Added.
After this study was completed, results of an updated global analysis of the neutrino oscillation data were published in [54], in which the latest T2K data on sin 2 θ 23 [55], sin 2 θ 23 = 0.514 + 0.055/ − 0.056 (0.511 ± 0.055) for the NO (IO) neutrino mass spectrum, were taken into account.As a consequence, the authors of [54] find a somewhat larger central value of sin 2 θ 23 than the one used by us in the numerical predictions for the Dirac and Majorana phases, namely sin 2 θ 23 = 0.437 (0.455) in the NO (IO) case.At the same time, the MINOS collaboration finds for the best fit value of sin 2 θ 23 = 0.41, performing a 3-neutrino oscillation analysis of their data [56].Obviously, high precision measurement of sin 2 θ 23 is lacking at present.Our numerical predictions for the values of the Dirac and Majorana phases should be updated when a sufficiently precise determination of sin 2 θ 23 will be available.However, if sin 2 θ 23 is found to lie in the interval (0.40 -0.50), the numerical predictions obtained in this study will not change significantly.
10.It follows from eqs. (39) -(43) that, apart from the BM (LC) case, the |J CP | factor has rather similar values in the TBM, GRA, GRB and HG mixing cases.As our results show, distinguishing between these cases requires a measurement of cos δ or a very high precision measurement of |J CP |.