Predictions for the leptonic Dirac CP violation phase: a systematic phenomenological analysis

We derive predictions for the Dirac phase \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}δ present in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3\times 3$$\end{document}3×3 unitary neutrino mixing matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U = U_e^{\dagger } \, U_{\nu }$$\end{document}U=Ue†Uν, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_e$$\end{document}Ue and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\nu }$$\end{document}Uν are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3\times 3$$\end{document}3×3 unitary matrices which arise from the diagonalisation, respectively, of the charged lepton and the neutrino mass matrices. We consider forms of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_e$$\end{document}Ue and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\nu }$$\end{document}Uν allowing us to express \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}δ as a function of three neutrino mixing angles, present in U, and the angles contained in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\nu }$$\end{document}Uν. We consider several forms of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\nu }$$\end{document}Uν determined by, or associated with, symmetries, tri-bimaximal, bimaximal, etc., for which the angles in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\nu }$$\end{document}Uν are fixed. For each of these forms and forms of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_e$$\end{document}Ue allowing one to reproduce the measured values of the neutrino mixing angles, we construct the likelihood function for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cos \delta $$\end{document}cosδ, 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.


Introduction
Understanding the origin of the observed pattern of neutrino mixing, establishing the status of the CP symmetry in the lepton sector, determining the type of spectrum the neutrino masses obey and determining the nature-Dirac or Majorana-of massive neutrinos are among the highest priority goals of the programme of future research in neutrino physics (see, e.g., [1]). One of the major experimental efforts S. T. Petcov: Also at: Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria. a e-mail: igirardi@sissa.it within this programme will be dedicated to the searches for CP-violating effects in neutrino oscillations (see, e.g., [2][3][4]). In the reference three neutrino mixing scheme with three light massive neutrinos we are going to consider (see, e.g., [1]), the CP-violating effects in neutrino oscillations can be caused, as is well known, by the Dirac CP violation (CPV) phase present in the Pontecorvo, Maki, Nakagawa, Sakata (PMNS) neutrino mixing matrix. Predictions for the Dirac CPV phase in the lepton sector can be, and were, obtained, in particular, combining the phenomenological approach, developed in [5][6][7][8][9][10] and further exploited in various versions by many authors with the aim of understanding the pattern of neutrino mixing emerging from the data (see, e.g., [11][12][13][14][15][16][17][18][19][20][21]), with symmetry considerations. In this approach one exploits the fact that the PMNS mixing matrix U has the form [8]: where U e and U ν are 3 × 3 unitary matrices originating from the diagonalisation, respectively, of the charged lepton 1 and neutrino mass matrices. In Eq. (1)Ũ e andŨ ν are CKM-like 3 × 3 unitary matrices, and and Q 0 are diagonal phase matrices each containing in the general case two physical CPV phases 2 : 21 2 , e i ξ 31 It is further assumed that, up to subleading perturbative corrections (and phase matrices), the PMNS matrix U has a specific known formŨ ν , which is dictated by continuous 1 If the charged lepton mass term is written in the right-left convention, the matrix U e diagonalises the hermitian matrix M † E M E , U † e M † E M E U e = diag(m 2 e , m 2 μ , m 2 τ ), M E being the charged lepton mass matrix. 2 The phases in the matrix Q 0 contribute to the Majorana phases in the PMNS matrix [22]. and/or discrete symmetries, or by arguments related to symmetries. This assumption seems very natural in view of the observation that the measured values of the three neutrino mixing angles differ from certain possible symmetry values by subdominant corrections. Indeed, the best fit values and the 3σ allowed ranges of the three neutrino mixing parameters sin 2 θ 12 , sin 2 θ 23 and sin 2 θ 13 in the standard parametrisation of the PMNS matrix (see, e.g., [1]), derived in the global analysis of the neutrino oscillation data performed in [23] read (sin 2 θ 12 ) BF = 0.308, 0.259 ≤ sin 2 θ 12 ≤ 0.359, where the value (the value in parentheses) corresponds to m 2 31(32) > 0 ( m 2 31(32) < 0), i.e., neutrino mass spectrum with normal (inverted) ordering 3 (see, e.g., [1]). In terms of angles, the best fit values quoted above imply: θ 12 ∼ = π/5.34, θ 13 ∼ = π/20 and θ 23 ∼ = π/4.35. Thus, for instance, θ 12 deviates from the possible symmetry value π/4, corresponding to the bimaximal mixing [25][26][27][28], by approximately 0.2, θ 13 deviates from 0 (or from 0.32) by approximately 0.16 and θ 23 deviates from the symmetry value π/4 by approximately 0.06, where we used sin 2 θ 23 = 0.437.
Widely discussed symmetry forms ofŨ ν include: (i) tribimaximal (TBM) form [7,[29][30][31][32], (ii) bimaximal (BM) form, or due to a symmetry corresponding to the conservation of the lepton charge L = L e − L μ − L τ (LC) [25][26][27][28], (iii) golden ratio type A (GRA) form [33,34], (iv) golden ratio type B (GRB) form [35,36], and (v) hexagonal (HG) form [21,37]. For all these forms the matrixŨ ν represents a product of two orthogonal matrices describing rotations in the 1-2 and 2-3 planes on fixed angles θ ν 12 and θ ν 23 : Thus,Ũ ν does not include a rotation in the 1-3 plane, i.e., θ ν 13 = 0. Moreover, for all the symmetry forms quoted above 3 Similar results were obtained in the global analysis of the neutrino oscillation data performed in [24]. one has also θ ν 23 = − π/4. The forms differ by the value of the angle θ ν 12 , and, correspondingly, of sin 2 θ ν 12 : for the TBM, BM (LC), GRA, GRB and HG forms we have, respectively, sin 2 θ ν 12 = 1/3, 1/2, (2 + r ) −1 ∼ = 0.276, (3 − r )/4 ∼ = 0.345, and 1/4, r being the golden ratio, r = (1 + √ 5)/2. As is clear from the preceding discussion, the values of the angles in the matrixŨ ν , which are fixed by symmetry arguments, typically differ from the values determined experimentally by relatively small perturbative corrections. In the approach we are following, the requisite corrections are provided by the angles in the matrixŨ e . The matrixŨ e in the general case depends on three angles and one phase [8]. 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., [38][39][40][41][42][43][44]),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, 4 U e = R −1 13 (θ e 13 ), (10) or (ii) two rotations in any other two of the three planes, e.g., The use of the inverse matrices in Eqs. (8)- (12) is a matter of convenience-this allows us to lighten the notations in expressions which will appear further in the text. It was shown in [45] (see also [46]) that forŨ ν andŨ e given in Eqs. (6) and (9), the Dirac phase δ present in the PMNS matrix satisfies a sum rule by which it is expressed in terms of the three neutrino mixing angles measured in the neutrino oscillation experiments and the angle θ ν 12 . In the standard parametrisation of the PMNS matrix (see, e.g., [1]) the sum rule reads [45] cos δ = tan θ 23 sin 2θ 12 sin θ 13 [cos 2θ ν 12 + (sin 2 θ 12 − cos 2 θ ν 12 ) For the specific values of θ ν 12 = π/4 and θ ν 12 = sin −1 (1/ √ 3), i.e., for the BM (LC) and TBM forms ofŨ ν , Eq. (13) reduces to the expressions for cos δ derived first in [46]. On the basis of the analysis performed and the results obtained using the best fit values of sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 , it was concluded in [45], in particular, that the measurement of cos δ can allow one to distinguish between the different symmetry forms of the matrixŨ ν considered.
Within the approach employed, the expression for cos δ given in Eq. (13) is exact. In [45] the correction to the sum rule Eq. (13) due to a non-zero angle θ e 13 1 inŨ e , corresponding tõ with | sin θ e 13 | 1, was also derived. Using the best fit values of the neutrino mixing parameters sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 , found in the global analysis in [47], predictions for cos δ, δ and the rephasing invariant sin δ sin 2θ 13 sin 2θ 23 sin 2θ 12 cos θ 13 , which controls the magnitude of CP-violating effects in neutrino oscillations [48], were presented in [45] for each of the five symmetry forms ofŨ ν -TBM, BM (LC), GRA, GRB and HG-considered. Statistical analysis of the sum rule Eq. (13) predictions for δ and J CP (for cos δ) using the current (the prospective) uncertainties in the determination of the three neutrino mixing parameters, sin 2 θ 12 , sin 2 θ 13 , sin 2 θ 23 , and δ (sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 ), was performed in [49,50] for the five symmetry forms-BM (LC), TBM, GRA, GRB and HG-of U ν . Using the current uncertainties in the measured values of sin 2 θ 12 , sin 2 θ 13 , sin 2 θ 23 and δ 5 , it was found, in particular, that for the TBM, GRA, GRB and HG forms, J CP = 0 at 5σ , 4σ , 4σ and 3σ , respectively. For all these four forms |J CP | is predicted at 3σ to lie in the following narrow interval [49,50]: 0.020 ≤ |J CP | ≤ 0.039. As a consequence, in all these cases the CP-violating effects in neutrino oscillations are predicted to be relatively large. In contrast, for the BM (LC) form, the predicted best fit value is J CP ∼ = 0, and the CP-violating effects in neutrino oscillations can be strongly suppressed. The statistical analysis of the sum rule predictions for cos δ, performed in [49,50] by employing prospective uncertainties of 0.7, 3 and 5 % in the determination of sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 , revealed that with a precision in the measurement of δ, δ ∼ = (12 • -16 • ), which is planned to be achieved in the future neutrino experiments like T2HK and ESSνSB 5 We would like to note that the recent statistical analyses performed in [23,24] showed indications/hints that δ ∼ = 3π/2. As for sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 , in the case of δ we utilise as "data" the results obtained in Ref. [23]. [4], it will be possible to distinguish at 3σ between the BM (LC), TBM/GRB and GRA/HG forms ofŨ ν . Distinguishing between the TBM and GRB forms, and between the GRA and HG forms, requires a measurement of δ with an uncertainty of a few degrees.
In the present article we derive new sum rules for cos δ using the general approach employed, in particular, in [45,49,50]. We perform a systematic study of the forms of the matricesŨ e andŨ ν , for which it is possible to derive sum rules for cos δ of the type of Eq. (13), but for which the sum rules of interest do not exist in the literature. More specifically, we consider the following forms ofŨ e andŨ ν : with θ ν 23 = −π/4 and θ ν 12 corresponding to the TBM, BM (LC), GRA, GRB and HG mixing, and (i) with θ ν 23 , θ ν 13 and θ ν 12 fixed by arguments associated with symmetries, and (iv) In each of these cases we obtain the respective sum rule for cos δ. This is done first for θ ν 23 = − π/4 in the cases listed in point A, and for the specific values of (some of) the angles inŨ ν , characterising the cases listed in point B. For each of the cases listed in points A and B we derive also generalised sum rules for cos δ for arbitrary fixed values of all angles contained inŨ ν (i.e., without setting θ ν 23 = − π/4 in the cases listed in point A, etc.). Next we derive predictions for cos δ and J CP (cos δ), performing a statistical analysis using the current (the prospective) uncertainties in the determination of the neutrino mixing parameters sin 2 θ 12 , sin 2 θ 13 , sin 2 θ 23 and δ (sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 ).
It should be noted that the approach to understanding the experimentally determined pattern of lepton mixing and to obtaining predictions for cos δ and J CP employed in the present work and in the earlier related studies [45,49,50] is by no means unique-it is one of a number of approaches discussed in the literature on the problem (see, e.g., [51][52][53][54]). It is used in a large number of phenomenological studies (see, e.g., [5,6,8,10,[16][17][18][19][20]55]) as well as in a class of models (see [38][39][40][41][42][43][44]56]) of neutrino mixing based on discrete symmetries. However, it should be clear that the conditions of the validity of the approach employed in the present work are not fulfilled in all theories with discrete flavour symmetries. For example, they are not fulfilled in the theories with discrete flavour symmetry (6n 2 ) studied in [57,58], with the S 4 flavour symmetry constructed in [59] and in the models discussed in [60][61][62]. Further, the conditions of our analysis are also not fulfilled in the phenomenological approach developed and exploited in [52][53][54]. In these articles, in particular, the matrices U e and U ν are assumed to have specific given fixed forms, in which all three mixing angles in each of the two matrices are fixed to some numerical values, typically, but not only, π/4, or some integer powers n of the parameter ∼ = θ C , θ C being the Cabibbo angle. The angles θ ν i j ∼ = (θ C ) n with n > 2 are set to zero. For example, in [54] the following sets of values of the angles in U e and U ν have been used: (θ e 12 , θ e 13 , θ e 23 , θ ν 12 , θ ν 13 , θ ν 23 ) = ( * , π/4, π/4, * , π/4, * ) and ( * , * , π/4, π/4, * , * ), where " * " means angles not exceeding θ C . None of these sets correspond to the cases studied by us. As a consequence, the sum rules for cos δ derived in our work and in [54] are very different. In [54] the authors have also considered specific textures of the neutrino Majorana mass matrix leading to the two sets of values of the angles in U e and U ν quoted above. However, these textures lead to values of sin 2 θ 23 or of sin 2 θ 12 which are strongly disfavoured by the current data. Although in [52,53] a large variety of forms of U e and U ν have been investigated, none of them corresponds to the forms studied by us, as can be inferred from the results on the values of the PMNS angles θ 12 , θ 13 and θ 23 obtained in [52,53] and summarised in Table 2 in each of the two articles we have cited in [52,53].
Our article is organised as follows. In Sect. 2 we consider the models which contain one rotation from the charged lepton sector, i.e.,Ũ e = R −1 12 (θ e 12 ), orŨ e = R −1 13 (θ e 13 ), and two rotations from the neutrino sector:Ũ ν = R 23 (θ ν 23 ) R 12 (θ ν 12 ). In these cases the PMNS matrix reads with (i j) = (12), (13). The matrixŨ ν is assumed to have the following symmetry forms: TBM, BM (LC), GRA, GRB and HG. As we have already noted, for all these forms θ ν 23 = −π/4, but we discuss also the general case of an arbitrary fixed value of θ ν 23 . The forms listed above differ by the value of the angle θ ν 12 , which for each of the forms of interest was given earlier. In Sect. 3 we analyse the models which contain two rotations from the charged lepton sector, i.e.,Ũ e = R −1 23 (θ e 23 ) R −1 13 (θ e 13 ), orŨ e = R −1 13 (θ e 13 ) R −1 12 (θ e 12 ), and 6 two rotations from the neutrino sector, i.e., with (i j)-(kl) = (13)-(23), (12)- (13). First we assume the angle θ ν 23 to correspond to the TBM, BM (LC), GRA, GRB and HG symmetry forms ofŨ ν . After that we give the formulae for an arbitrary fixed value of this angle. Further, in Sect. 4, we generalise the schemes considered in Sect. 2 by allowing also a third rotation matrix to be present inŨ ν : 6 We consider only the "standard" ordering of the two rotations inŨ e ; see [46]. The case withŨ e = R −1 23 (θ e 23 ) R −1 12 (θ e 12 ) has been analysed in detail in [45,46,49,50] and will not be discussed by us.
Using the sum rules for cos δ derived in Sects. 2-4, in Sect. 5 we obtain predictions for cos δ, δ and J CP for each of the models considered in the preceding sections. Section 6 contains summary of the results of the present study and conclusions.
We note finally that the titles of Sects. 2-4 and of their subsections reflect the rotations contained in the corresponding parametrisation, Eqs. (16)- (18). In this section we derive the sum rules for cos δ of interest in the case when the matrixŨ 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. (16). We do not consider the cases of Eq. (16) (i) with (i j) = (23), because the reactor angle θ 13 does not get corrected and remains zero, and (ii) with (i j) = (12) and θ ν 23 = −π/4, which has been already analysed in detail in [45,49]. For θ ν 23 = −π/4 the sum rule for cos δ in this case was derived in Ref. [45] and is given in Eq. (50) therein. Here we consider the case of an arbitrary fixed value of the angle θ ν 23 . Using Eq. (16) with (i j) = (12), one finds the following expressions for the mixing angles θ 13 and θ 23 of the standard parametrisation of the PMNS matrix: Although Eq. (13) was derived in [45] for θ ν 23 = −π/4 and , it is not difficult to convince oneself that it holds also in the case under discussion for an arbitrary fixed value of θ ν 23 . The sum rule for cos δ of interest, expressed in terms of the angles θ 12 , θ 13 , θ ν 12 and θ ν 23 , can be obtained from Eq. (13) by using the expression for sin 2 θ 23 given in Eq. (20). The result reads cos δ = (cos 2θ 13 − cos 2θ ν 23 ) Setting θ ν 23 = −π/4 in Eq. (21), one reproduces the sum rule given in Eq. (50) in Ref. [45].
Further, one can find 7 a relation between sin δ (cos δ) and sin ω (cos ω) by comparing the imaginary (the real) part of the quantity U * e1 U * μ3 U e3 U μ1 , written by using Eq. (16) with (i j) = (13) and in the standard parametrisation of U . For the relation between sin δ and sin ω we get sin δ = − sin 2θ ν 12 sin 2θ 12 sin ω.
We note that the expression for cos δ thus found differs only by an overall minus sign from the analogous expression for cos δ derived in [45] in the case of (i j) = (12) rotation in the charged lepton sector (see Eq. (50) in [45]). In Eq. (15) we have given the expression for the rephasing invariant J CP in the standard parametrisation of the PMNS matrix. Below and in the next sections we give for completeness also the expressions of the J CP factor in terms of the independent parameters of the set-up considered. In terms of the parameters ω, θ e 13 and θ ν 12 of the set-up discussed in the present subsection, J CP is given by In the case of an arbitrary fixed value of the angle θ ν 23 the expressions for the mixing angles θ 13 and θ 23 take the form The sum rule for cos δ in this case can be obtained with a simpler procedure, namely, by using the expressions for the absolute value of the element U μ1 of the PMNS matrix in the two parametrisations employed in the present subsection: |U μ1 | = | cos θ 23 sin θ 12 + e iδ cos θ 12 sin θ 13 sin θ 23 | = | cos θ ν 23 sin θ ν 12 |.
From Eq. (31) we get cos δ = − (cos 2θ 13 + cos 2θ ν 23 ) We will use the sum rules for cos δ derived in the present and the next two sections to obtain predictions for cos δ, δ and for the J CP factor in Sect. 5.
3 The cases of (θ e i j , θ e kl ) − (θ ν 23 , θ ν 12 ) rotations As we have seen in the preceding section, in the case of one rotation from the charged lepton sector and for 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 matrix U e has to contain at least two rotations in order to be possible to reproduce the current best fit values of the neutrino mixing parameters, quoted in Eqs. (3)-(5). This conclusion will remain valid if higher precision measurements of sin 2 θ 23 confirm that θ 23 deviates significantly from π/4. In what follows we investigate different combinations of two rotations from the charged lepton sector and derive a sum rule for cos δ in each set-up. We will not consider the case (θ e 12 , θ e 23 )-(θ ν 23 , θ ν 12 ), because it has been thoroughly analysed in Refs. [45,46,49,50], and, as we have already noted, the resulting sum rule Eq. (13) derived in [45] holds for an arbitrary fixed value of θ ν 23 .
The sum rule expression for cos δ as a function of the mixing angles θ 12 , θ 13 , θ 23 and θ ν 12 , with θ ν 12 having an arbitrary fixed value, reads cos δ = − cot θ 23 sin 2θ 12 sin θ 13 [cos 2θ ν 12 + (sin 2 θ 12 − cos 2 θ ν 12 ) This sum rule for cos δ can be obtained formally from the r.h.s. of Eq. (13) by interchanging tan θ 23 and cot θ 23 and by multiplying it by (−1). Thus, in the case of θ 23 = π/4, the predictions for cos δ in the case under consideration will differ from those obtained using Eq. (13) only by a sign. We would like to emphasise that, as the sum rule in Eq. (13), the sum rule in Eq. (45) is valid for any fixed value of θ ν 23 . The J CP factor has the following form in the parametrisation of the PMNS matrix employed in the present subsection: sin 2θ e 13 sin 2θ ν 12 sin 2θ 23 cosθ 23 sin α.
3.2 The scheme with (θ e 12 , θ e 13 ) -(θ ν 23 , θ ν 12 ) rotations In this subsection we consider the parametrisation of the matrix U defined in Eq. (17) with (i j)-(kl) = (12)- (13) under the assumption of vanishing ω, i.e., = diag(1, e −iψ , 1). In the case of non-fixed ω it is impossible to express cos δ only in terms of the independent angles of the scheme. We will comment more on this case later.
Finally, we generalise Eq. (60) to the case of an arbitrary fixed value of θ ν 23 . In this case and Eqs. (57) and (60) It follows from the results for cos δ obtained for cos ω = 0, Eqs. (60) and (63), that in the case analysed in the present subsection one can obtain predictions for cos δ only in theoretical models in which the value of the phase ω is fixed by the model. We consider next a generalisation of the cases analysed in Sect. 2 in 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. (18). Non-zero values of θ ν 13 are inspired by certain types of flavour symmetries (see, e.g., [63][64][65][66][67]). In the case of θ ν 12 = θ ν 23 = −π/4 and θ ν 13 = sin −1 (1/3), for instance, we have the so-called tri-permuting (TP) pattern, which was proposed and studied in [63]. In the statistical analysis of the predictions for cos δ, δ and the J CP factor we will perform in Sect. 5, 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).
For the parametrisation of the matrix U given in Eq. (18) with (i j) = (23), no constraints on the phase δ can be obtained. Indeed, after we recast U in the form where sin 2θ 23 and Q 1 are given in Eqs. (35) and (36), respectively, we find employing a similar procedure used in the previous sections: Thus, there is no correlation between the Dirac CPV phase δ and the mixing angles in this set-up.

Predictions
In this section we present results of a statistical analysis, performed using the procedure described in Appendix A (see also [49,50]), which allows us to get the dependence of the χ 2 function on the value of δ and on the value of the J CP factor. In what follows we always assume that θ ν 23 = −π/4. We find that in the case corresponding to Eq. (16) with (i j) = (12), analysed in [45], the results for χ 2 as a function of δ or J CP are rather similar to those obtained in [49,50] in the case of the parametrisation defined by Eq. (17) with (i j)-(kl) = (12)- (23). The main difference between these two cases is the predictions for sin 2 θ 23 , which can deviate only by approximately 0.5 sin 2 θ 13 from 0.5 in the first case and by a significantly larger amount in the second. As a consequence, the Table 3 The predicted values of cos δ using the current best fit values of the mixing angles, quoted in Eqs. (3)-(5) and corresponding to neutrino mass spectrum with NO, except for the case (θ e 12 , θ e 13 )-(θ ν 23 , θ ν 12 ) with ω = 0 and κ = 1, in which sin 2 θ 23 = 0.48802 is used. We have defined a = sin −1 (1/3) predictions in the first case are somewhat less favoured by the current data than in the second case, which is reflected in the higher value of χ 2 at the minimum, χ 2 min . Similar conclusions hold on comparing the results in the case of θ e 13 − (θ ν 23 , θ ν 12 ) rotations, described in Sect. 2.2, and in the corresponding case defined by Eq. (17) with (i j)-(kl) = (13)- (23) and discussed in Sect. 3.1. Therefore, in what concerns these four schemes, in what follows we will present results of the statistical analysis of the predictions for δ and the J CP factor only for the scheme with (θ e 13 , θ e 23 )-(θ ν 23 , θ ν 12 ) rotations, considered in Sect. 3.1.
We show in Tables 3 and 4 the predictions for cos δ and δ for all the schemes considered in the present study using the current 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), which enter into the sum rule expressions for cos δ, Eqs. (13), (27), (45), (58), (77), (93) and Eq. (50) in Ref. [45], unless other values of the indicated mixing parameters are explicitly specified. We present results only for the NO neutrino mass spectrum, since the results for the IO spectrum differ insignificantly. Several comments are in order.
Results for the scheme (θ e 12 , θ e 23 ) − (θ ν 23 , θ ν 12 ) in the cases of the TBM and BM symmetry forms of the matrixŨ ν were presented first in [46], while results for the same scheme and the GRA, GRB and HG symmetry forms ofŨ ν , as well as for the scheme θ e 12 − (θ ν 23 , θ ν 12 ) for all symmetry forms considered, were obtained first in [45]. The predictions for cos δ and δ were derived in [45,46] for the best fit values of the relevant neutrino mixing parameters found in an earlier global analysis performed in [47] and differ somewhat (albeit not much) from those quoted in Tables 3 and 4. The values under discussion given in these tables are from [49] and correspond to the best fit values quoted in Eqs. (3)-(5).
In the schemes with three rotations inŨ ν we consider, cos δ has values which differ significantly (being larger in absolute value) from the values predicted by the schemes with two rotations inŨ ν discussed by us, the only exceptions being (i) the θ e 12 (13)  ) differ for each of the symmetry forms ofŨ ν considered both by sign and magnitude. If the best fit value of θ 23 were π/4, these predictions would differ only by sign.
In the case of the (θ e 12 , θ e 13 ) − (θ ν 23 , θ ν 12 ) scheme with ω = 0, the predictions for cos δ are very sensitive to the value of sin 2 θ 23 . Using the best fit values of sin 2 θ 12 and sin 2 θ 13 for the NO neutrino mass spectrum, quoted in Eqs. (3) and (5), we find from the constraints (−1 < cos ψ < 1) and (0 < sin 2 θ e 13 < 1) ∧ (0 < sin 2 θ e 12 < 1), where sin 2 θ e 13 , sin 2 θ e 12 and cos ψ are given in Eqs. (53)- (55), that sin 2 θ 23 should lie in the following intervals: Obviously, the quoted intervals with sin 2 θ 23 ≥ 0.78 are ruled out by the current data. We observe that a small increase of sin 2 θ 23 from the value 0.48802 9 produces a relatively large variation of cos δ. The strong dependence of cos δ on sin 2 θ 23 takes place for values of ω satisfying roughly cos ω 0.01. In contrast, for cos ω = 0, cos δ exhibits a relatively weak dependence on sin 2 θ 23 . For the reasons related to the dependence of cos δ on ω we are not going to present results of the statistical analysis in this case. This can be done in specific models of neutrino mixing, in which the value of the phase ω is fixed by the model.

The scheme with
In the left panel of Fig. 1 we show the likelihood function, defined as versus cos δ for the NO neutrino mass spectrum for the scheme with (θ e 13 , θ e 23 )-(θ ν 23 , θ ν 12 ) rotations. 10 This function represents the most probable values of cos δ for each of the symmetry forms considered. In the analysis performed by us we use as input the current global neutrino oscillation data on sin 2 θ 12 , sin 2 θ 23 , sin 2 θ 13 and δ [23]. The maxima of L(cos δ), L(χ 2 = χ 2 min ), for the different symmetry forms ofŨ ν considered, correspond to the values of cos δ given in Table 3. The results shown are obtained by marginalising over sin 2 θ 13 and sin 2 θ 23 for a fixed value of δ (for details of the statistical analysis see Appendix A and [49,50]). The nσ confidence level (CL) region corresponds to the interval of values of cos δ for which L(cos δ) ≥ L(χ 2 = χ 2 min ) · L(χ 2 = n 2 ). Here χ 2 min is the value of χ 2 in the minimum. As can be observed from the left panel of Fig. 1, for the TBM and GRB forms there is a substantial overlap of the corresponding likelihood functions. The same observation holds also for the GRA and HG forms. However, the likelihood functions of these two sets of symmetry forms overlap only at 3σ and in a small interval of values of cos δ. Thus, the TBM/GRB, GRA/HG and BM (LC) symmetry forms might be distinguished with a not very demanding (in terms of precision) measurement of cos δ. At the maximum, the non-normalised likelihood function equals exp(−χ 2 min /2), and this value allows one to judge quantitatively about the compatibility of a given symmetry form with the global neutrino oscillation data, as we have pointed out.
In the right panel of Fig. 1 we present L versus cos δ within the Gaussian approximation (see [49,50] for details), using the current best fit values of sin 2 θ 12 , sin 2 θ 23 , sin 2 θ 13 for the NO spectrum, given in Eqs. (3)-(5), and the prospective 1σ uncertainties in the measurement of these mixing parameters. More specifically, we use as 1σ uncertainties (i) 0.7 % for sin 2 θ 12 , which is the prospective sensitivity of the JUNO experiment [69], (ii) 5 % for sin 2 θ 23 , 11 obtained from the prospective uncertainty of 2 % [4] on sin 2 2θ 23 expected to be reached in the NOvA and T2K experiments, and (iii) 3 % for sin 2 θ 13 , deduced from the error of 3 % on sin 2 2θ 13 planned to be reached in the Daya Bay experiment [4,71]. The BM (LC) case is quite sensitive to the values of sin 2 θ 12 and sin 2 θ 23 and for the current best fit values is disfavoured at more than 2σ .
That the BM (LC) case is disfavoured by the current data can be understood, in particular, from the following observation. Using the best fit values of sin 2 θ 13 and sin 2 θ 12 as well as the constraint −1 ≤ cos α ≤ 1, where cos α is defined in Eq. (42), one finds that sin 2 θ 23 should satisfy sin 2 θ 23 ≥ 0.63, which practically coincides with the currently allowed maximal value of sin 2 θ 23 at 3σ (see Eq. (4)).
It is interesting to compare the results described above and obtained in the scheme denoted by (θ e 13 , θ e 23 ) − (θ ν 23 , θ ν 12 ) with those obtained in [49,50] in the (θ e 12 , θ e 23 ) − (θ ν 23 , θ ν 12 ) set-up. We recall that for each of the symmetry forms we have considered-TBM, BM, GRA, GRB and HG-θ ν 12 has a specific fixed value and θ ν 23 = −π/4. The first thing to 11 This sensitivity is planned to be achieved in future neutrino facilities [70].
note is that for a given symmetry form, cos δ is predicted to have opposite signs in the two schemes. In the scheme (θ e 13 , θ e 23 )-(θ ν 23 , θ ν 12 ) analysed in the present article, one has cos δ > 0 in the TBM, GRB and BM (LC) cases, while cos δ < 0 in the cases of the GRA and HG symmetry forms. As in the (θ e 12 , θ e 23 )-(θ ν 23 , θ ν 12 ) set-up, there are significant overlaps between the TBM/GRB and GRA/HG forms ofŨ ν , respectively. The BM (LC) case is disfavoured at more than 2σ confidence level. It is also important to notice that due to the fact that the best fit value of sin 2 θ 23 < 0.5, the predictions for cos δ for each symmetry form, obtained in the two setups differ not only by sign but also in absolute value, as was already pointed out in Sect. 3.1. Thus, a precise measurement of cos δ would allow one to distinguish not only between the symmetry forms ofŨ ν , but also could provide an indication about the structure of the matrixŨ e .
For the rephasing invariant J CP , using the current global neutrino oscillation data, we find for the symmetry forms considered the following best fit values and the 3σ ranges for the NO neutrino mass spectrum:  Ranges of sin 2 θ 12 obtained from the requirements (0 < sin 2 θ e 12 < 1) ∧ (−1 < cos φ < 1) allowing sin 2 θ 13 to vary in the 3σ allowed range for the NO neutrino mass spectrum, quoted in Eq. (5). The cases for which the best fit value of sin 2 θ 12 = 0.308 is within the corresponding allowed ranges are marked with the subscripts I, II, III, IV, V. The cases marked with an asterisk contain values of sin 2 θ 12 allowed at 2σ [23] θ ν 12 θ ν 13 = π/20 θ ν 13 = π/10 θ ν 13 = sin −1 (1/3) For the scheme with θ e 12 −(θ ν 23 , θ ν 13 , θ ν 12 ) rotations we find that only for particular values of θ ν 12 and θ ν 13 , among those considered by us, the allowed intervals of values of sin 2 θ 12 satisfy the requirement that they contain in addition to the best fit value of sin 2 θ 12 also the 1.5σ experimentally allowed range of sin 2 θ 12 . Indeed, combining the conditions 0 < sin 2 θ e 12 < 1 and | cos φ| < 1, where sin 2 θ e 12 and cos φ are given in Eqs. (73) and (74), respectively, and allowing sin 2 θ 13 to vary in the 3σ range for NO spectrum, we get restrictions on the value of sin 2 θ 12 , presented in Table 5. We see from the Table  that only five out of 18 combinations of the angles θ ν 12 and θ ν 13 considered by us satisfy the requirement formulated above. In Table 5 these cases are marked with the subscripts I, II, III, IV, V, while the ones marked with an asterisk contain values of sin 2 θ 12 allowed at 2σ [23]. Equation (67) implies that sin 2 θ 23 is fixed by the value of θ ν 13 , and for the best fit value of sin 2 θ 13 and the values of θ ν 13 = 0, π/20, π/10, sin −1 (1/3), considered by us, we get, respectively: sin 2 θ 23 = 0.488, 0.501, 0.537, 0.545. Therefore a measurement of sin 2 θ 23 with a sufficiently high precision would rule out at least some of the cases with fixed values of θ ν 13 considered in the literature. We will perform a statistical analysis of the predictions for cos δ in the five cases-I, II, III, IV, V-listed above. The analysis is similar to the one discussed in Sect. 5.1. The only difference is that when we consider the prospective sensitivities on the PMNS mixing angles we will assume sin 2 θ 23 to have the following potential best fit values: sin 2 θ 23 = 0.488, 0.501, 0.537, 0.545. Note that for the best fit value of sin 2 θ 13 , sin 2 θ 23 = 0.488 does not correspond to any of the values of θ ν 13 in the five cases-I, II, III, IV, V-of interest. Thus, sin 2 θ 23 = 0.488 is not the most probable value in any of the five cases considered: depending on the case, the most probable value is one of the other three values of sin 2 θ 23 listed above. We include results for sin 2 θ 23 = 0.488 to illustrate how the likelihood function changes when the best fit value of sin 2 θ 23 , determined in a global analysis, differs from the value of sin 2 θ 23 predicted in a given case.
In Fig. 2 we show the likelihood function versus cos δ for all the cases marked with the subscripts in Table 5. The maxima of the likelihood function in the five cases considered take place at the corresponding values of cos δ cited in Table  3. As Fig. 2 clearly indicates, the cases differ not only in the predictions for sin 2 θ 23 , which in the considered set-up is a function of sin 2 θ ν 13 and sin 2 θ 13 , but also in the predictions for cos δ. Given the values of θ 12 and θ 13 , the positions of the peaks are determined by the values of θ ν 12 and θ ν 13 . The Cases I and IV are disfavoured by the current data because the corresponding values of sin 2 θ 23 = 0.537 and 0.545 are disfavoured. The Cases II, III and V are less favoured for the NO neutrino mass spectrum than for the IO spectrum since sin 2 θ 23 = 0.501 is less favoured for the first than for the second spectrum.
In Fig. 3 we show the predictions for cos δ using the prospective precision in the measurement of sin 2 θ 12 , sin 2 θ 13 , sin 2 θ 23 , the best fit values for sin 2 θ 12 and sin 2 θ 13 as in Eqs. (3) and (5) and the potential best fit values of sin 2 θ 23 = 0.488, 0.501, 0.537, 0.545. The values of sin 2 θ 23 correspond in the scheme discussed to the best fit value of sin 2 θ 13 in the cases which are compatible with the current 1.5σ range of allowed values of sin 2 θ 12 . The position of the peaks, obviously, does not depend explicitly on the assumed experimentally determined best fit value of sin 2 θ 23 . For the best fit value of sin 2 θ 13 used, the corresponding sum rule for cos δ depends on the given fixed value of θ ν 13 , and via it, on the predicted value of sin 2 θ 23 (see Eqs. (67) and   (77)). Therefore, the compatibility of a given case with the considered hypothetical data on sin 2 θ 23 clearly depends on the assumed best fit value of sin 2 θ 23 determined from the data.
As the results shown in Fig. 3 indicate, distinguishing between the Cases I/IV and the other three cases would not require exceedingly high precision measurement of cos δ. Distinguishing between the Cases II, III and V would be more challenging in terms of the requisite precision on cos δ. In both cases the precision required will depend, in particular, on the experimentally determined best fit value of cos δ. As Fig. 3 also indicates, one of the discussed two groups of Cases might be strongly disfavoured by the best fit value of sin 2 θ 23 determined in the future high precision experiments.
We have performed also a statistical analysis of the predictions for the rephasing invariant J CP , minimising χ 2 for fixed values of J CP . We give N σ ≡ χ 2 as a function of J CP in Fig. 4. The dashed lines represent the results of the global fit [23], while the solid ones represent the results we obtain for each of the considered cases, minimising the value of χ 2 in θ e 12 for a fixed value of J CP using Eq. (78). The blue lines correspond to the NO neutrino mass spectrum, while the red ones are for the IO spectrum. The value of χ 2 in the minimum, which corresponds to the best fit value of J CP predicted in the model, allows one to conclude about compatibility of this model with the global neutrino oscillation data. As it can be observed from Fig. 4, the zero value of J CP in the Cases III and V is excluded at more than 3σ with respect to the confidence level of the corresponding minimum. Although in the other three cases the best fit values of J CP are relatively large, as their numerical values quoted below show, J CP = 0 is only weakly disfavoured statistically.
The best fit values and the 3σ ranges of the rephasing invariant J CP , obtained for the NO neutrino mass spectrum using the current global neutrino oscillation data, in the five cases considered by us are given by Table 6 Ranges of sin 2 θ 12 obtained from the requirements (0 < sin 2 θ e 13 < 1) ∧ (−1 < cos ω < 1) allowing sin 2 θ 13 to vary in the 3σ allowed range for the NO neutrino mass spectrum, quoted in Eq. (5). The cases for which the best fit value of sin 2 θ 12 = 0.308 is within the corresponding allowed ranges are marked with the subscripts I, II, III, IV, V. The case marked with an asterisk contains values of sin 2 θ 12 allowed at 2σ [23] θ ν 12 θ ν 13 = π/20 θ ν 13 = π/10 θ ν 13 = sin −1 (1/3)  As in the set-up discussed in Sect. 5.2, we find for the scheme with θ e 13 − (θ ν 23 , θ ν 13 , θ ν 12 ) rotations that only particular values of θ ν 12 and θ ν 13 allow one to obtain the current best fit value of sin 2 θ 12 . Combining the requirements 0 < sin 2 θ e 13 < 1 and | cos ω| < 1, where sin 2 θ e 13 and cos ω are given in Eqs. (89) and (90), respectively, and allowing sin 2 θ 13 to vary in its 3σ allowed range corresponding to the NO spectrum, we get restrictions on the value of sin 2 θ 12 , presented in Table 6. It follows from the results in Table 6 that only for five out of 18 combinations of the angles θ ν 12 and θ ν 13 , the best fit value of sin 2 θ 12 = 0.308 and the 1.5σ experimentally allowed interval of values of sin 2 θ 12 are inside the allowed ranges. In Table 6 these cases are marked with the subscripts I-V, while in the case marked with an asterisk, the allowed range contains values of sin 2 θ 12 allowed at 2σ [23].
The values of sin 2 θ 23 in this model depend on the reactor angle θ 13 and θ ν 13 through Eq. (83). Using the best fit value of sin 2 θ 13 for the NO spectrum and Eq. (83), we find sin 2 θ 23 = 0.512, 0.499, 0.463, 0.455 for θ ν 13 = 0, π/20, π/10, sin −1 (1/3), respectively. Thus, in the scheme under discussion sin 2 θ 23 decreases with the increase of θ ν 13 , which is in contrast to the behaviour of sin 2 θ 23 in the set-up discussed in the preceding subsection. As we have already remarked, a measurement of sin 2 θ 23 with a sufficiently high precision, or at least the determination of the octant of θ 23 , would allow one to exclude some of the values of θ ν 13 considered in the literature.
The statistical analyses for δ and J CP performed in the present subsection are similar to those performed in the previous subsections. In particular, we show in Fig. 5 the dependence of the likelihood function on cos δ using the current knowledge on the PMNS mixing angles and the Dirac CPV phase from the latest global fit results. Due to the very narrow prediction for sin 2 θ 23 in this set-up, the prospective sensitivity likelihood curve depends strongly on the assumed best fit value of sin 2 θ 23 . For this reason we present in Fig. 6 the predictions for cos δ using the prospective sensitivities on the mixing angles, the best fit values for sin 2 θ 12 and sin 2 θ 13 as in Eqs. (3) and (5) and the potential best fit values of sin 2 θ 23 = 0.512, 0.499, 0.463, 0.455. We use the value of sin 2 θ 23 = 0.512, corresponding to θ ν 13 = 0, for the same reason we used the value of sin 2 θ 23 = 0.488 in the analysis in the preceding subsection, where we gave also a detailed explanation.
As Fig. 6 clearly shows, the position of the peaks does not depend on the assumed best fit value of sin 2 θ 23 . However, the height of the peaks reflects to what degree the model is disfavoured due to the difference between the assumed best fit value of sin 2 θ 23 and the value predicted in the corresponding set-up.
The results shown in Fig. 6 clearly indicate that (i) the measurement of cos δ can allow one to distinguish between the Case I and the other four cases; (ii) distinguishing between the Cases II/III and the Cases IV/V might be possible, but is very challenging in terms of the precision on cos δ required to achieve that; and (iii) distinguishing between the Cases II and III (the Cases IV and V) seems practically impossible. Some of, or even all, these cases would be strongly disfavoured if the best fit value of sin 2 θ 23 determined with the assumed high precision in the future experiments were relatively large, say, sin 2 θ 23 0.54.
The results on the predictions for the rephasing invariant J CP are presented in Fig. 7, where we show the dependence of N σ ≡ χ 2 on J CP . It follows from the results presented   Fig. 7, in particular, that J CP = 0 is excluded at more than 3σ with respect to the confidence level of the corresponding minimum only in the Case I. For the rephasing invariant J CP , using the current global neutrino oscillation data, we find for the different cases considered the following best fit values and 3σ ranges for the NO neutrino mass spectrum: (113)

Summary and conclusions
In the present article we have derived predictions for the Dirac phase δ present in the 3 × 3 unitary neutrino mixing matrix U = U † e U ν = (Ũ e ) † Ũ ν Q 0 , where U e (Ũ e ) and U ν (Ũ ν ) are 3 × 3 unitary (CKM-like) matrices which arise from the diagonalisation, respectively, of the charged lepton and the neutrino mass matrices, and and Q 0 are diagonal phase matrices each containing in the general case two physical CPV phases. The phases in the matrix Q 0 contribute to the Majorana phases in the PMNS matrix. After performing a systematic search, we have considered forms ofŨ e andŨ ν allowing us to express δ as a function of the PMNS mixing angles, θ 12 , θ 13 and θ 23 , present in U , and the angles contained inŨ ν . We have derived such sum rules for cos δ in the cases of forms for which the sum rules of interest do not exist in the literature. More specifically, we have derived new sum rules for cos δ in the following cases: (i) U = R 12 (θ e five symmetry forms considered -TBM, BM (LC), GRA, each scheme considered. For this purpose we construct the χ 2 function in the following way: χ 2 ({x i }) = i χ 2 i (x i ), with x i = {sin 2 θ 12 , sin 2 θ 13 , sin 2 θ 23 , δ}. The functions χ 2 i have been extracted from the 1-dimensional projections given in [23] and, thus, the correlations between the oscillation parameters have been neglected. This approximation is sufficiently precise since it allows one to reproduce the contours in the planes (sin 2 θ 23 , δ), (sin 2 θ 13 , δ) and (sin 2 θ 23 , sin 2 θ 13 ), given in [23], with a rather high accuracy (see [49,50]). We construct, e.g., χ 2 (cos δ) by marginalising χ 2 ({x i }) over the free parameters, e.g., sin 2 θ 13 and sin 2 θ 23 , for a fixed value of cos δ. Given the global fit results, the likelihood function, represents the most probable values of cos δ in each considered case. When we present the likelihood function versus cos δ within the Gaussian approximation we use χ 2 G = i (y i − y i ) 2 /σ 2 y i , with y i = {sin 2 θ 12 , sin 2 θ 13 , sin 2 θ 23 }, y i are the potential best fit values of the indicated mixing parameters and σ y i are the prospective 1σ uncertainties in the determination of these mixing parameters. More specifically, we use as 1σ uncertainties (i) 0.7 % for sin 2 θ 12 , which is the prospective sensitivity of the JUNO experiment [69], (ii) 5 % for sin 2 θ 23 , obtained from the prospective uncertainty of 2 % [4] on sin 2 2θ 23 expected to be reached in the NOvA and T2K experiments, and (iii) 3 % for sin 2 θ 13 , deduced from the error of 3 % on sin 2 2θ 13 planned to be reached in the Daya Bay experiment [4,71].