Determining the Dirac CP Violation Phase in the Neutrino Mixing Matrix from Sum Rules

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 analyse the sum rules which the Dirac phase $\delta$ present in $U$ satisfies when $U_{\nu}$ has a form dictated by flavour symmetries and $U_e$ has a"minimal"form (in terms of angles and phases it contains) 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. The following symmetry forms are considered: i) tri-bimaximal (TBM), ii) bimaximal (BM) (or corresponding to the conservation of the lepton charge $L' = L_e - L_\mu - L_{\tau}$ (LC)), iii) golden ratio type A (GRA), iv) golden ratio type B (GRB), and v) hexagonal (HG). We investigate the predictions for $\delta$ in the cases of TBM, BM (LC), GRA, GRB and HG forms using the exact and the leading order sum rules for $\cos\delta$ proposed in the literature, taking into account also the uncertainties in the measured values of $\sin^2\theta_{12}$, $\sin^2\theta_{23}$ and $\sin^2\theta_{13}$. This allows us, in particular, to assess the accuracy of the predictions for $\cos\delta$ based on the leading order sum rules and its dependence on the values of the indicated neutrino mixing parameters when the latter are varied in their respective 3$\sigma$ experimentally allowed ranges.


Introduction
One of the major goals of the future experimental studies in neutrino physics is the searches for CP violation (CPV) effects in neutrino oscillations (see, e.g., [1,2]). It is part of a more general and ambitious program of research aiming to determine the status of the CP symmetry in the lepton sector.
In the case of the reference 3-neutrino mixing scheme 1 , CPV effects in the flavour neutrino oscillations, i.e., a difference between the probabilities of ν l → ν l andν l →ν l oscillations in vacuum [3,4], P (ν l → ν l ) and P (ν l →ν l ), l = l = e, µ, τ , can be caused, as is well known, by the Dirac phase present in the Pontecorvo, Maki, Nakagawa and Sakata (PMNS) neutrino mixing matrix U PMNS ≡ U . If the neutrinos with definite masses ν i , i = 1, 2, 3, are Majorana particles, the 3-neutrino mixing matrix contains two additional Majorana CPV phases [4]. However, the flavour neutrino oscillation probabilities P (ν l → ν l ) and P (ν l →ν l ), l, l = e, µ, τ , do not depend on the Majorana phases 2 [4,8]. Our interest in the CPV phases present in the neutrino mixing matrix is stimulated also by the intriguing possibility that the Dirac phase and/or the Majorana phases in U PMNS can provide the CP violation necessary for the generation of the observed baryon asymmetry of the Universe [9,10].
As was explained earlier, the requirement that U e has a "minimal" form in terms of angles and phases it contains, needed to provide the requisite corrections to U ν , makes not necessary the inclusion inŨ e of the orthogonal matrix describing the rotation in the 1-3 plane, R 13 (θ e 13 ). Effectively, this is equivalent to the assumption that the angle θ e 13 , if nonzero, is sufficiently small and thus is either negligible, or leads to sub-dominant effects in the observable of interest in the present analysis, cos δ. We will use θ e 13 ∼ = 0 to denote values of θ e 13 which satisfy the indicated condition.
We note that θ e 13 ∼ = 0 is a feature of many theories of charged lepton and neutrino mass generation (see, e.g., [22,27,28,[30][31][32]). The assumption that θ e 13 ∼ = 0 was also used in a large 3 The diagonal phase matrix Ψ, as we see, can originate from the charged lepton or the neutrino sector, or else can receive contributions from both sectors [29].
number of studies dedicated to the problem of understanding the origins of the observed pattern of lepton mixing (see, e.g., [15,29,[33][34][35][36][37]). In large class of GUT inspired models of flavour, the matrix U e is directly related to the quark mixing matrix (see, e.g., [28,30,31,38]). As a consequence, in this class of models we have θ e 13 ∼ = 0. We will comment later on the possible effects of θ e 13 = 0, | sin θ e 13 | 1, on the predictions for cos δ, which are of principal interest of the present study.
More generally, the approach to understanding the observed pattern of neutrino mixing on the basis of discrete symmetries employed in the present article, which leads to the sum rule of interest for cos δ, is by no means unique -it is one of the several possible approaches discussed in the literature on the subject (see, e.g., [18]). It is employed in a large number of phenomenological studies (see, e.g., [15,29,[33][34][35][36][37]) as well as in a class of models (see, e.g., [27,28,30,31,38]) of neutrino mixing based on discrete symmetries. However, it should be clear that the conditions which define the approach used in the present article are not fulfilled in all models with discrete flavour symmetries. For example, they are not fulfilled in the models with discrete flavour symmetry ∆(6n 2 ) studied in [39,40], with the S 4 flavour symmetry constructed in [41] and in the models discussed in [42].
The phase α in the matrix P 1 can be absorbed in the τ lepton field and, thus, is unphysical. The phase β gives a contribution to the matrixQ = Q 1 Q 0 ; the diagonal phase matrixQ contributes to the matrix of physical Majorana phases. In the setting considered the PMNS matrix takes the form: where θ ν 12 has a fixed value which depends on the symmetry form ofŨ ν used. For the angles θ 13 , θ 23 and θ 12 of the standard parametrisation of the PMNS matrix U we get in terms of the parameters in the expression eq. (18) for U [13]: where eq. (19) was used in order to obtain the expression for sin 2 θ 23 in terms ofθ 23 and θ 13 , and eqs. (19) and (20) were used to get the last expression for sin 2 θ 12 . Within the approach employed, the expressions in eqs. (19) - (21) are exact. It follows from eqs. (1), (2) and (18) that the four observables θ 12 , θ 23 , θ 13 and δ are functions of three parameters θ e 12 ,θ 23 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 [13], leading to a new "sum rule" relating δ and θ 12 , θ 23 and θ 13 . For an arbitrary fixed value of the angle θ ν 12 the sum rule for cos δ reads [14]: For θ ν 12 = π/4 and θ ν 12 = sin −1 (1/ √ 3) the expression eq. (22) for cos δ reduces to those found in [13] in the BM (LC) and TBM cases, respectively. A similar sum rule for an arbitrary θ ν 12 can be derived for the phase φ [13,14]. It proves convenient for our further discussion to cast the sum rules for cos δ and cos φ of interest in the form: sin 2 θ 12 = cos 2 θ ν 12 + sin 2θ 12 sin θ 13 cos δ − tan θ 23 cos 2θ ν 12 tan θ 23 (1 − cot 2 θ 23 sin 2 θ 13 ) , sin 2 θ 12 = cos 2 θ ν 12 + 1 2 sin 2θ 23 sin 2θ ν 12 sin θ 13 cos φ − tan θ 23 cos 2θ ν
Within the scheme considered the sum rules eqs. (22) - (24) and the relations eqs. (25) and (26) are exact. In a complete self-consistent theory of (lepton) flavour based on discrete flavour symmetry, the indicated sum rules and relations are expected to get corrections due to, e.g., θ e 13 = 0, renormalisation group (RG) effects, etc. Analytic expression for the correction in the expression for cos δ, eq. (22), due to | sin θ e 13 | 1 was derived in [14]. As was shown in [14], for the best fit values of the lepton mixing angles θ 12 , θ 13 and θ 23 , a nonzero θ e 13 ∼ < 10 −3 produces a correction to the value of cos δ obtained from the "exact" sum rule eq. (22), which does not exceed 11% (4.9%) in the TBM (GRB) cases and is even smaller in the other three cases of symmetry forms ofŨ ν analysed in the present article. A value of θ e 13 ∼ < 10 −3 is a feature of many theories and models of charged lepton and neutrino mass generation (see, e.g., [22,27,28,[30][31][32]). The RG effects on the lepton mixing angles and the CPV phases are known to be negligible for hierarchical neutrino mass spectrum (see, e.g., [43,44] and the references quoted therein); these effects are relatively small for values of the lightest neutrino mass not exceeding approximately 0.05 eV 4 . We will call the sum rules and the relations given in eqs. (22) -(24), (25) and (26) "exact", keeping in mind that they can be subject to corrections, which, however, in a number of physically interesting cases, if not absent, can only be sub-dominant.
4 In supersymmetric theories this result is valid for moderate values of the parameter tan β ∼ < 10 (see [43,44]); for tan β = 50 the same statement is true for values of the lightest neutrino mass smaller than approximately 0.01 eV. 5 In contrast to θ ν 23 = π/4 employed in [15], we use θ ν 23 = −π/4. The effect of the difference in the signs of sin θ e 12 and sin θ e 23 utilised by us and in [15] is discussed in Appendix A.
The first equation leads (in the leading order approximation used to derive it and using sin 2θ 12 ∼ = sin 2θ ν 12 ) to eq. (29), while from the second equation we find: sin θ 12 ∼ = sin θ ν 12 + sin 2θ ν 12 2 sin θ ν 12 sin θ 13 cos φ , and correspondingly, This implies that in the leading order approximation adopted in ref. [15] we have [14] cos δ = cos φ. Note, however, that the sum rules for cos δ and cos φ given in eqs. (31) and (32), differ somewhat by the factors multiplying the terms ∼ sin θ 13 .
As was shown in [14], the leading order sum rule (29) leads in the cases of the TBM, GRA, GRB and HG forms ofŨ ν to largely imprecise predictions for the value of cos δ: for the best fit values of sin 2 θ 12 = 0.308, sin 2 θ 13 = 0.0234 and sin 2 θ 23 = 0.425 used in [14], they differ approximately by factors (1.4 -1.9) from the values found from the exact sum rule. The same result holds for cos φ. Moreover, the predicted values of cos δ and cos φ differ approximately by factors of (1.5 -2.0), in contrast to the prediction cos δ ∼ = cos φ following from the leading order sum rules. The large differences between the results for cos δ and cos φ, obtained using the leading order and the exact sum rules, are a consequence [14] of the quantitative importance of the next-to-leading order terms which are neglected in the leading order sum rules (29) - (34). The next-to-leading order terms are significant for the TBM, GRA, GRB and HG forms ofŨ ν because in all these cases the "dominant" terms |θ 12 − θ ν 12 | ∼ sin 2 θ 13 , or equivalently 6 | sin 2 θ 12 − sin 2 θ ν 12 | ∼ sin 2 θ 13 . It was shown also in [14] that in the case of the BM (LC) form ofŨ ν we have |θ 12 − θ ν 12 | ∼ sin θ 13 and the leading order sum rules provide rather precise predictions for cos δ and cos φ.
The results quoted above were obtained in [14] for the best fit values of the neutrino mixing parameters sin 2 θ 12 , sin 2 θ 23 and sin 2 θ 13 . In the present article we investigate in detail the predictions for cos δ and cos φ in the cases of the TBM, BM (LC), GRA, GRB and HG forms ofŨ ν using the exact sum rules given in eqs. (23) (or (22)) and (24) and the leading order sum rules in eqs. (31) and (32), taking into account also the uncertainties in the measured values of sin 2 θ 12 , sin 2 θ 23 and sin 2 θ 13 . This allows us to better assess the accuracy of the predictions for cos δ and cos φ based on the leading order sum rules and its dependence on the values of the neutrino mixing angles. We investigate also how the predictions for cos δ and cos φ, obtained using the exact and the leading order sum rules, vary when the PMNS neutrino mixing parameters sin 2 θ 12 , sin 2 θ 23 and sin 2 θ 13 are varied in their respective experimentally allowed 3σ ranges.
In the limiting case of negligible θ e 23 the exact sum rules for cos δ and cos φ take the following form [14]: From the above equations, to leading order in sin θ 13 we get: cos φ = 1 sin 2θ ν 12 sin θ 13 or equivalently, The last two equations coincide with eqs. (31) and (32) which were derived from the exact sum rules keeping the leading order corrections in both sin θ 13 and sin θ e 23 . This implies, in particular, that the correction due to | sin θ e 23 | 1 appears in the sum rules of interest only in the next-to-leading order terms. Casting the results obtained in a form we are going to use in our numerical analysis, we obtain: We have replaced sin 2θ 12 with sin 2θ ν 12 in eq. (43), so that it corresponds to eqs. (29) and (30). In the cases of the TBM, GRA, GRB and HG symmetry forms ofŨ ν we are considering and for the best fit value of sin 2 θ 12 = 0.308 we indeed have | sin θ 12 − sin θ ν 12 | ∼ sin 2 θ 13 . Thus, if one applies consistently the approximations employed in [15], which lead to eqs. (29) -(34) (or to eqs. (38) and (39)), one should neglect also the difference between θ 12 and θ ν 12 . This leads to cos δ = cos φ = 0.
In Fig. 1 we show predictions for cos δ and cos φ in the cases of the TBM, GRA, GRB and HG forms of the matrixŨ ν , as functions of sin θ 13 which is varied in the 3σ interval given in eq. (5) and corresponding to NO neutrino mass spectrum. The predictions are obtained for the best fit value of sin 2 θ 12 = 0.308 using the exact sum rules eqs. (36) and (37) for cos δ (solid lines) and cos φ (dashed lines) and the leading order sum rules eqs. (43) and (44)  The unphysical value of cos δ in the case of the BM (LC) form ofŨ ν is a reflection of the fact that the scheme under discussion with the BM (LC) form of the matrixŨ ν does not provide a good description of the current data on θ 12 , θ 23 and θ 13 [13]. One gets a physical result for cos δ, cos δ = −0.973, for, e.g., values of sin 2 θ 12 = 0.32, and sin θ 13 = 0.16, lying in the 2σ experimentally allowed intervals of these neutrino mixing parameters. We have checked that for the best fit value of sin 2 θ 13 , physical values of (cos δ) E , (cos δ) LO and (cos φ) E in the BM (LC) case can be obtained for relatively large values of sin 2 θ 12 . For, e.g., sin 2 θ 12 = 0.359 and sin 2 θ 13 = 0.0234 we find (cos δ) E = −0.915, (cos δ) LO = −0.998 and (cos φ) E = −0.922. In this case the differences between the exact and leading order sum rule results for cos δ and cos φ are relatively small.  Table 1: The predicted values of cos δ and cos φ, obtained from the exact sum rules in eqs. (36) and (37) The above results imply that it would be possible to distinguish between the different symmetry forms ofŨ ν considered by measuring cos δ [14], provided sin 2 θ 12 is known with sufficiently high precision. Even determining the sign of cos δ will be sufficient to eliminate some of the possible symmetry forms ofŨ ν .
The leading order sum rules eqs. (43) and (44) (37)). The dash-dotted line in each of the 4 panels represents (cos δ) LO = (cos φ) LO obtained from the leading order sum rule in eq. (43). The vertical dash-dotted line corresponds to the best fit value of sin 2 θ 13 = 0.0234; the three coloured vertical bands indicate the 1σ, 2σ and 3σ experimentally allowed ranges of sin θ 13 (see text for further details).
As Fig. 1 indicates, the differences |(cos δ) E − (cos δ) LO | and |(cos φ) E − (cos φ) LO | exhibit weak dependence on the value of sin θ 13 when it is varied in the 3σ interval quoted in eq. (5). The values of cos δ, obtained using the exact sum rule eq. (36) in the TBM, GRA, GRB and HG cases, differ from those calculated using the approximate sum rule eq. (43) by the factors 0.638, 1.29, 0.756 and 1.15, respectively. The largest difference is found to hold in the TBM case. As was shown in [14], the correction to (cos δ) LO -the leading order sum rule result for cos δ -is given approximately by cos 2θ ν 12 sin θ 13 /(sin 2θ 12 ). For given θ ν 12 , the relative magnitude of the correction depends on the magnitude of the ratio | sin 2 θ 12 −sin 2 θ ν 12 |/ sin θ 13 .
The largest correction occurs for the symmetry form ofŨ ν , for which this ratio has the smallest value. For the best fit value of sin 2 θ 12 , the smallest value of the ratio of interest corresponds to the TBM form ofŨ ν and is equal approximately to 0.166.
The absolute values of the difference |(cos δ) E − (cos δ) LO | for the TBM, GRB, GRA and HG symmetry forms, as it follows from Table 1, lie in the narrow interval (0.061 -0.065). These differences seem to be rather small. However, they are sufficiently large to lead to misleading results. Indeed, suppose cos δ is measured and the value determined experimentally reads: cos δ = −0.18 ± 0.025. If one compares this value with the value of cos δ predicted using the leading order sum rule, (cos δ) LO , one would conclude that data are compatible with the TBM form ofŨ ν and that all the other forms considered by us are ruled out. Using the prediction based on the exact sum rule, i.e., (cos δ) E , would lead to a completely different conclusion, namely, that the data are compatible only with the GRB form ofŨ ν 7 . In this hypothetical example, which is included to illustrate the significance of the difference between the predictions for cos δ obtained using the exact and the leading order sum rules, we have assumed that the prospective uncertainties in the predicted values of (cos δ) LO and (cos δ) E due to the uncertainties in the measured values of sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 are sufficiently small. These uncertainties will be discussed in Section 5 (see Fig. 13). The relative difference between (cos δ) E and (cos δ) LO , i.e., the ratio |(cos δ) E −(cos δ) LO |/|(cos δ) E |, is also significant. For the TBM, GRA, GRB and HG symmetry forms it reads: 57.0%, 22.1%, 32.5% and 12.8%, respectively.
It follows from the results presented in Tables 1 -3 7 The same hypothetical example can be used to illustrate the significance of the difference between the exact and the leading order sum rule predictions for cos δ also in the case of θ e 23 = 0 (see Table 4).  Table 2: The same as in Table 1, but for sin 2 θ 12 = 0.259. the symmetry forms ofŨ ν considered, the exact sum rule predictions for cos δ not only change significantly in magnitude when sin 2 θ 12 is varied in its 3σ allowed range, but also the sign of cos δ changes in the TBM, GRA and GRB cases (see Fig. 4). We observe also that for sin 2 θ 12 = 0.259, the values of cos δ, obtained using the exact sum rule eq. (36) in the TBM, GRA, GRB and HG cases differ from those calculated using the leading order sum rule in eq.  For sin 2 θ 12 = 0.259, the largest difference between the exact and leading order sum rule results for cos δ occurs for the GRA and HG forms ofŨ ν , while if sin 2 θ 12 = 0.359, the largest difference holds for the TBM and GRB forms.
As Figs. 1 -3 and Tables 1 -3 show, similar results are valid for cos φ obtained from the exact and the leading order sum rules.
It is worth noting also that the values of cos φ and cos δ, derived from the respective exact sum rules differ significantly for the TBM, GRA, GRB and HG forms ofŨ ν considered. As pointed out in [14], for the best fit values of sin 2 θ 13 and sin 2 θ 12 they differ by factors (1.4 -2.0), as can be seen also from Table 1 For θ e 23 = 0 we have in the scheme we are considering: θ 23 ∼ = π/4 − 0.5 sin 2 θ 13 . A nonzero value of θ e 23 allows for a significant deviation of θ 23 from π/4. Such deviation is not excluded by the current data on sin 2 θ 23 , eq. (4): at 3σ, values of sin 2 θ 23 in the interval (0.37 -0.64) are allowed, the best fit value being sin 2 θ 23 = 0.437 (0.455). The exact sum rules for cos δ and cos φ, eqs. (22), (23) and (24), depend on θ 23 , while the leading order sum rules, eqs. (29) and (34), are independent of θ 23 . In this Section we are going to investigate how the dependence on θ 23 affects the predictions for cos δ and cos φ, based on the exact sum rules.
In Fig. 5 we show the predictions for cos δ and cos φ in the cases of the TBM, GRA, GRB and HG forms of the matrixŨ ν , derived from the exact sum rules in eqs. (23) and (24), (cos δ) E (solid line) and (cos φ) E (dashed line), and from the leading order sum rule in eq. (30) (eq. (33)), (cos δ) LO = (cos φ) LO (dash-dotted line). The results presented in Fig. 5 are obtained for the best fit values of sin 2 θ 12 = 0.308 and sin 2 θ 23 = 0.437. The parameter sin 2 θ 13 is varied in its 3σ allowed range, eq. (5). In Table 4 we give the values of (cos δ) E , (cos δ) LO , (cos φ) E and of their ratios, corresponding to the best fit values of sin 2 θ 12 , sin 2 θ 23 and sin 2 θ 13 . We see from Table 4 that for the TBM, GRA, GRB and HG forms ofŨ ν , cos δ determined from the exact sum rule takes respectively the values (−0.091), 0.275, (−0.169) and 0.445. The values of cos δ, found using the exact sum rule, eq. (23), differ in the TBM, GRA, GRB and HG cases from those calculated using the leading order sum rule, eq. (30), by the factors 0.506, 1.22, 0.636 and 1.07, respectively. Thus, the largest difference between the predictions of the exact and the leading order sum rules occurs for the TBM form ofŨ ν .
Since the predictions of the sum rules depend on the value of θ 12 , we show in Fig. 6 and Fig. 7 also results for the values of sin 2 θ 12 , corresponding to the lower and the upper bounds of the 3σ allowed range of sin 2 θ 12 , sin 2 θ 12 = 0.259 and 0.359, keeping sin 2 θ 23 fixed to its best fit value. The predictions for (cos δ) E , (cos φ) E , (cos δ) LO = (cos φ) LO and their ratios, obtained for the best fit values of sin 2 θ 13 = 0.0234 and sin 2 θ 23 = 0.437, and for sin 2 θ 12 = 0.259 (sin 2 θ 12 = 0.359) are given in Table 5 (Table 6). For sin 2 θ 12 = 0.259, the exact sum rule predictions of cos δ for the TBM, GRA, GRB and HG forms ofŨ ν read (see Table 5): (cos δ) E = (−0.408), (−0.022), (−0.490) and 0.156. As in the case of negligible θ e 23 analysed in the preceding Section, these values differ drastically (in general, both in magnitude and sign) from the exact sum rule values of cos δ corresponding to the best fit value and the 3σ upper bound of sin 2 θ 12 = 0.308 and 0.359. The dependence of (cos δ) E , (cos δ) LO and (cos φ) E on sin 2 θ 12 under discussion is shown graphically in Fig. 8.
Further, for sin 2 θ 12 = 0.259, the ratio (cos δ) E /(cos δ) LO in the TBM, GRA, GRB and HG cases reads, respectively, 0.744, 0.172, 0.769 and 2.32 (see Table 5). Thus, the predictions for cos δ of the exact and the leading order sum rules differ by the factors of 5.8 and 2.3 in the GRA and HG cases. For the upper bound of the 3σ range of sin 2 θ 12 = 0.359, the ratio (cos δ) E /(cos δ) LO takes the values 1.2, 0.996, 1.46 and 0.969 for the TBM, GRA, GRB and HG forms ofŨ ν , respectively (see Table 6). For the GRA and HG symmetry forms the leading order sum rule prediction for cos δ is very close to the exact sum rule prediction, which can also be seen in Fig. 7.
We will investigate next the dependence of the predictions for cos δ and cos φ on the value of θ 23 given the facts that i) sin 2 θ 23 is determined experimentally with a relatively large uncertainty, and ii) in contrast to the leading order sum rule predictions for cos δ and cos φ,  (24)). The dash-dotted line in each of the 4 panels represents (cos δ) LO = (cos φ) LO obtained from the leading order sum rule in eq. (30) (eq. (33)). The vertical dash-dotted line corresponds to the best fit value of sin 2 θ 13 = 0.0234; the three coloured vertical bands indicate the 1σ, 2σ and 3σ experimentally allowed ranges of sin θ 13 (see text for further details). the exact sum rule predictions depend on θ 23 . In Figs. 9 and 10 we show the dependence of predictions for cos δ and cos φ on sin θ 13 for the best fit value of sin 2 θ 12 = 0.308 and the 3σ lower and upper bounds of sin 2 θ 23 = 0.374 and 0.626, respectively. For sin 2 θ 23 = 0.374 (0.626) and the best fit values of sin 2 θ 13 and sin 2 θ 12 , the exact and the leading order sum rule results (cos δ) E , (cos φ) E , (cos δ) LO = (cos φ) LO and their ratios are given in Tables  7 and 8. Comparing the values of (cos δ) E quoted in Tables 7 and 8 with the values given in Table 4 we note that the exact sum rule predictions for cos δ for sin 2 θ 23 = 0.374 (lower 3σ bound) and sin 2 θ 23 = 0.437 (best fit value) do not differ significantly in the cases of the TBM, GRA, GRB and HG forms ofŨ ν considered. However, the differences between the predictions for sin 2 θ 23 = 0.437 and sin 2 θ 23 = 0.626 are rather large -by factors of 2.05, 1.25, 1.77 and 1.32 in the TBM, GRA, GRB and HG cases, respectively.
In what concerns the difference between the exact and leading order sum rules predictions  Table 4: The predicted values of cos δ and cos φ, obtained from the exact sum rules in eqs. (23) and (24) Table 5: The same as in Table 4, but for sin 2 θ 13 = 0.0234 (best fit value), sin 2 θ 12 = 0.259 (lower bound of the 3σ range) and sin 2 θ 23 = 0.437 (best fit value).  Table 6: The same as in Table 4, but for sin 2 θ 13 = 0.0234 (best fit value), sin 2 θ 12 = 0.359 (upper bound of the 3σ range) and sin 2 θ 23 = 0.437 (best fit value).

Statistical Analysis
In the present Section we perform a statistical analysis of the predictions for δ, cos δ and the rephasing invariant J CP which controls the magnitude of CPV effects in neutrino oscillations [47], in the cases of the TBM, BM (LC), GRA, GRB and HG symmetry forms of the matrix U ν (see eq. (8)). In this analysis we use as input the latest results on sin 2 θ 12 , sin 2 θ 13 , sin 2 θ 23  Table 7: The same as in Table 4, but for sin 2 θ 13 = 0.0234 (best fit value), sin 2 θ 12 = 0.308 (best fit value) and sin 2 θ 23 = 0.374 (lower bound of the 3σ range). and δ, obtained in the global analysis of the neutrino oscillation data performed in [11]. Our goal is to derive the allowed ranges for δ, cos δ and J CP , predicted on the basis of the current data on the neutrino mixing parameters for each of the symmetry forms ofŨ ν considered. We recall that in the standard parametrisation of the PMNS matrix, the J CP factor reads    Table 4, but for sin 2 θ 13 = 0.0234 (best fit value), sin 2 θ 12 = 0.308 (best fit value) and sin 2 θ 23 = 0.626 (upper bound of the 3σ range).
We construct χ 2 for the schemes considered -TBM, BM (LC), GRA, GRB and HG -as  Figure 9: The same as in Fig. 5, but for sin 2 θ 12 = 0.308 (best fit value) and sin 2 θ 23 = 0.374 (lower bound of the 3σ interval in eq. (4)). described in Appendix B. We will focus on the general case of non-vanishing θ e 23 in order to allow for possible sizeable deviations of θ 23 from the symmetry value π/4.
In the five panels in Fig. 11 we show N σ ≡ χ 2 as a function of δ for the five symmetry forms ofŨ ν we have studied. The dashed lines correspond to the results of the global fit [11]. The solid lines represent the results we obtain by minimising the value of χ 2 in sin 2 θ 13 and sin 2 θ 23 (or, equivalently, in sin 2 θ e 12 and sin 2θ 23 ) for a fixed value of δ 8 . The blue (red) lines correspond to NO (IO) neutrino mass spectrum. The value of χ 2 at the minimum, χ 2 min , which determines the best fit value of δ predicted for each symmetry form ofŨ ν , allows us to make conclusions about the compatibility of a given symmetry form ofŨ ν with the current global neutrino oscillation data.
It follows from the results shown in Fig. 11 that the BM (LC) symmetry form is disfavoured by the data at approximately 1.8σ, all the other symmetry forms considered being compatible with the data. We note that for the TBM, GRA, GRB and HG symmetry forms, a value of δ in the vicinity of 3π/2 is preferred statistically. For the TBM symmetry form this result was first obtained in [13] while for the GRA, GRB and HG symmetry forms it was first found in [14]. In contrast, in the case of the BM (LC) form the best fit value is very close to π [13,14]. The somewhat larger value of χ 2 at the second local minimum in the vicinity of π/2 in the TBM, GRA, GRB and HG cases, is a consequence of the fact that the best fit value of δ obtained in the global analysis of the current neutrino oscillation data is close to 3π/2 and that the value of δ = π/2 is statistically disfavoured (approximately at 2.5σ). In the absence of any information on δ, the two minima would have exactly the same value of χ 2 , because they correspond to the same value of cos δ. In the schemes considered, as we have discussed, cos δ is determined by the values of θ 12 , θ 13 and θ 23 . The degeneracy in the sign of sin δ can only be solved by an experimental input on δ. In Table 9 we give the best fit values of δ and the corresponding 3σ ranges for the TBM, BM (LC), GRA, GRB and HG forms of U ν , found by fixing χ 2 − χ 2 min = 3. In Fig. 12 we show the likelihood function versus cos δ for NO neutrino mass spectrum. The results shown are obtained by marginalising over all the other relevant parameters of the scheme considered (see Appendix B for details). The dependence of the likelihood function on cos δ in the case of IO neutrino mass spectrum differs little from that shown in Fig. 12.
Given the global fit results, the likelihood function, i.e., represents the most probable value of cos δ for each of the considered symmetry forms of U ν . The nσ confidence level region corresponds to the interval of values of cos δ in which L(cos δ) ≥ L(χ 2 = χ 2 min ) · L(χ 2 = n 2 ). As can be observed from Fig. 12, a rather precise measurement of cos δ would allow one to distinguish between the different symmetry forms ofŨ ν considered by us. For the TBM and GRB forms there is a significant overlap of the corresponding likelihood functions. The same observation is valid for the GRA and HG forms. However, the overlap of the likelihood functions of these two groups of symmetry forms occurs only at 3σ level in a very small interval of values of cos δ, as can also be seen from Table 9. This implies that in order to distinguish between TBM/GRB, GRA/HG and BM symmetry forms a not very demanding measurement (in terms of accuracy) of cos δ might be sufficient. The value of the non-normalised likelihood function at the maximum in Fig. 12 is equal to exp(−χ 2 min /2), which allows us to make conclusions about the compatibility of the symmetry schemes with the current global data, as has already been pointed out.
In the left panel of Fig. 13 we present the likelihood function versus cos δ within the Gaussian approximation (see Appendix B for details), using the current best fit values of the mixing angles for NO neutrino mass spectrum in eqs. (3) -(5) and the prospective 1σ uncertainties in the determination of sin 2 θ 12 (0.7% from JUNO [48]), sin 2 θ 13 (almost 3% derived from an expected error on sin 2 2θ 13 of 3% from Daya Bay, see A. de Gouvea et al. in [2]) and sin 2 θ 23 (5% 9 derived from the potential sensitivity of NOvA and T2K on sin 2 2θ 23 of 2%, see A. de Gouvea et al. in [2]). The BM case is very sensitive to the best fit values of sin 2 θ 12 and sin 2 θ 23 and is disfavoured at more than 2σ for the current best fit values quoted in eqs. (3) -(5). This case might turn out to be compatible with the data for larger (smaller) measured values of sin 2 θ 12 (sin 2 θ 23 ), as can be seen from the right panel of Fig. 13, which was obtained for sin 2 θ 12 = 0.332. With the increase of the value of sin 2 θ 23 the BM form becomes increasingly disfavoured, while the TBM/GRB (GRA/HG) predictions for cos δ are shifted somewhat -approximately by 0.1 -to the left (right) with respect to those shown in the left panel of Fig. 13. This shift is illustrated in Fig. 14, which is obtained for sin 2 θ 23 = 0.579, more precisely, for the best fit values found in [12] and corresponding to IO neutrino mass spectrum. The measurement of sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 with the quoted precision will open up the possibility to distinguish between the BM, TBM/GRB, GRA and HG forms of U ν . Distinguishing between the TBM and GRB forms would require relatively high precision measurement of cos δ.
We have performed also a statistical analysis in order to derive predictions for J CP . In Fig. 15 we present N σ ≡ χ 2 as a function of J CP for NO and IO neutrino mass spectra. Similarly to the case of δ, we minimise the value of χ 2 for a fixed value of J CP by varying sin 2 θ 13 and sin 2 θ 23 (or, equivalently, sin 2 θ e 12 and sin 2θ 23 ). The best fit value of J CP and the corresponding 3σ range for each of the considered symmetry forms ofŨ ν are summarised in Table 9. As Fig. 15 shows, the CP-conserving value of J CP = 0 is excluded in the cases of the TBM, GRA, GRB and HG neutrino mixing symmetry forms, respectively, at approximately 5σ, 4σ, 4σ and 3σ confidence levels with respect to the confidence level of the corresponding best fit values 10 . These results correspond to those we have obtained for δ, more specifically to the confidence levels at which the CP-conserving values of δ = 0, π, 2π, are excluded (see Fig. 11).
In contrast, for the BM (LC) symmetry form, the CP-conserving value of δ, namely, δ ∼ = π, is preferred and therefore the CP-violating effects in neutrino oscillations are predicted to be suppressed. At the best fit point we obtain a value of J CP = −0.005 (−0.002) for NO (IO) neutrino mass spectrum, which corresponds to the best fit value of δ/π = 1.04 (1.02). The allowed range of the J CP factor in the BM (LC) includes the CP-conserving value J CP = 0   Figure 13: The same as in Fig. 12, but using the prospective 1σ uncertainties in the determination of the neutrino mixing angles within the Gaussian approximation (see text for further details). In the left (right) panel sin 2 θ 12 = 0.308 (0.332), the other mixing angles being fixed to their NO best fit values. at practically any confidence level. As can be seen from Table 9, the 3σ allowed intervals of values of δ and J CP are rather narrow for all the symmetry forms considered, except for the BM (LC) form.
Finally, for completeness, we present in Appendix C also results of a statistical analysis of the predictions for the values of sin 2 θ 23 for the TBM, BM (LC), GRA, GRB and HG neutrino mixing symmetry forms considered. We recall that of the three neutrino mixing parameters, sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 , sin 2 θ 23 is determined in the global analyses of the neutrino oscillation data with the largest uncertainty.  Figure 14: The same as in Fig. 13, but using the IO best fit values taken from [12].

Summary and Conclusions
Using the fact that the neutrino mixing matrix U = U † e U ν , where U e and U ν result from the diagonalisation of the charged lepton and neutrino mass matrices, we have analysed the sum rules which the Dirac phase δ present in U satisfies when U ν has a form dictated by, or associated with, discrete symmetries and U e has a "minimal" form (in terms of angles and phases it contains) that can provide the requisite corrections to U ν , so that the reactor, atmospheric and solar neutrino mixing angles θ 13 , θ 23 and θ 12 have values compatible with the current data.
In this scheme the four observables θ 12 , θ 23 , θ 13 and the Dirac phase δ in the PMNS matrix are functions of three parameters θ e 12 ,θ 23 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 . This sum rule is exact within the scheme considered. Its explicit form depends on the symmetry form of the matrixŨ ν , i.e., on the value of the angle θ ν 12 . For arbitrary fixed value of θ ν 12 the sum rule of interest is given in eq. (22) (or the equivalent eq. (23)) [14]. A similar exact sum rule can be derived for the phase φ (eq. (24)) [14].  Table 9: Best fit values of J CP , δ and cos δ and corresponding 3σ ranges (found fixing χ 2 − χ 2 min = 3) in our setup using the data from [11]. cos δ discussed above and given in eqs. (23) and (30). It was shown in [14], in particular, using the best fit values of the neutrino mixing parameters sin 2 θ 12 , sin 2 θ 23 and sin 2 θ 13 and the exact sum rule results for cos δ derived for the TBM, GRA, GRB and HG forms ofŨ ν , that the leading order sum rule provides largely imprecise predictions for cos δ. Here we have performed a thorough study of the exact and leading order sum rule predictions for cos δ in the TBM, BM (LC), GRA, GRB and HG cases taking into account the uncertainties in the measured values of sin 2 θ 12 , sin 2 θ 23 and sin 2 θ 13 . This allowed us, in particular, to assess the accuracy of the predictions for cos δ based on the leading order sum rules and its dependence on the values of the indicated neutrino mixing parameters when the latter are varied in their respective 3σ experimentally allowed ranges. In contrast to the leading order sum rule, the exact sum rule for cos δ depends not only on θ 12 and θ 13 , but also on θ 23 , and we have investigated this dependence as well.
We confirm the result found in [14] that the exact sum rule predictions for cos δ vary significantly with the symmetry form ofŨ ν . This result implies that the measurement of cos δ can allow us to distinguish between the different symmetry forms ofŨ ν [14] provided sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 are known with a sufficiently good precision. Even determining the sign of cos δ will be sufficient to eliminate some of the possible symmetry forms ofŨ ν .
We find also that the exact sum rule predictions for cos δ exhibit strong dependence on the value of sin 2 θ 12 when the latter is varied in its 3σ experimentally allowed range (0.259 -0.359) (Tables 1 -6). The predictions for cos δ change significantly not only in magnitude, but in the cases of TBM, GRA and GRB forms ofŨ ν also the sign of cos δ can change. These significant changes take place both for θ e 23 = 0 and θ e 23 = 0. We have investigated the dependence of the exact sum rule predictions for cos δ in the cases of the symmetry forms ofŨ ν considered on the value of sin 2 θ 23 varying the latter in the respective 3σ allowed interval 0.374 ≤ sin 2 θ 23 ≤ 0.626 (Figs. 9 and 10, and Tables 7  and 8). The results we get for sin 2 θ 23 = 0.374 and sin 2 θ 23 = 0.437, setting sin 2 θ 12 and sin 2 θ 13 to their best fit values, do not differ significantly. However, the differences between the predictions for cos δ obtained for sin 2 θ 23 = 0.437 and for sin 2 θ 23 = 0.626 are relatively large (they differ by the factors of 2.05, 1.25, 1.77 and 1.32 in the TBM, GRA, GRB and HG cases, respectively).
In all cases considered, having the exact sum rule results for cos δ, we could investigate the precision of the leading order sum rule predictions for cos δ. We found that the leading order sum rule predictions for cos δ are, in general, imprecise and in many cases are largely incorrect, the only exception being the case of the BM (LC) form ofŨ ν [14].
We have performed a similar analysis of the predictions for the cosine of the phase φ. The phase φ is related to, but does not coincide with, the Dirac phase δ. The parameter cos φ obeys a leading order sum rule which is almost identical to the leading order sum rule satisfied by cos δ. This leads to the confusing identification of φ with δ: the exact sum rules satisfied by cos φ and cos δ differ significantly. Correspondingly, the predicted values of cos φ and cos δ in the cases of the TBM, GRA, GRB and HG symmetry forms ofŨ ν considered by us also differ significantly (see Figs. 1 -10 and Tables 1 -8). This conclusion is not valid for the BM (LC) form: for this form the exact sum rule predictions for cos φ and cos δ are rather similar. The phase φ appears in a large class of models of neutrino mixing and neutrino mass generation and serves as a "source" for the Dirac phase δ in these models. Finally, we have performed a statistical analysis of the predictions for δ, cos δ and the rephasing invariant J CP which controls the magnitude of CPV effects in neutrino oscillations [47], in the cases of the TBM, BM (LC), GRA, GRB and HG symmetry forms of the matrix U ν considered. In this analysis we have used as input the latest results on sin 2 θ 12 , sin 2 θ 13 , sin 2 θ 23 and δ, obtained in the global analysis of the neutrino oscillation data performed in [11]. Our goal was to derive the allowed ranges for δ, cos δ and J CP , predicted on the basis of the current data on the neutrino mixing parameters for each of the symmetry forms ofŨ ν considered. The results of this analysis are shown in Figs. 11, 12 and 15, and are summarised in Table 9, in which we give the predicted best fit values and 3σ ranges of J CP , δ and cos δ for each of the symmetry forms ofŨ ν considered. We have shown, in particular, that the CP-conserving value of J CP = 0 is excluded in the cases of the TBM, GRA, GRB and HG neutrino mixing symmetry forms, respectively, at approximately 5σ, 4σ, 4σ and 3σ confidence levels with respect to the confidence level of the corresponding best fit values (Fig. 15). These results reflect the predictions we have obtained for δ, more specifically, the confidence levels at which the CP-conserving values of δ = 0, π, 2π, are excluded in the discussed cases (see Fig. 11). We have found also that the 3σ allowed intervals of values of δ and J CP are rather narrow for all the symmetry forms considered, except for the BM (LC) form (Table 9). More specifically, for the TBM, GRA, GRB and HG symmetry forms we have obtained at 3σ: 0.020 ≤ |J CP | ≤ 0.039. For the best fit values of J CP we have found, respectively: J CP = (−0.034), (−0.033), (−0.034), and (−0.031). Our results indicate that distinguishing between the TBM, GRA, GRB and HG symmetry forms of the neutrino mixing would require extremely high precision measurement of the J CP factor.
Using the likelihood method, we have derived also the ranges of the predicted values of cos δ for the different forms ofŨ ν considered, using the prospective 1σ uncertainties in the determination of sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 respectively in JUNO, Daya Bay and accelerator and atmospheric neutrino experiments (Fig. 13). In this analysis the current best fit values of sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 have been utilised (left panel of Fig. 13). The results thus obtained show that i) the measurement of the sign of cos δ will allow to distinguish between the TBM/GRB, BM and GRA/HG forms ofŨ ν , ii) for a best fit value of cos δ = −1 (−0.1) distinguishing at 3σ between the BM (TBM/GRB) and the other forms ofŨ ν would be possible if cos δ is measured with 1σ uncertainty of 0.3 (0.1).
The predictions for δ, cos δ and J CP in the case of the BM (LC) symmetry form ofŨ ν , as the results of the statistical analysis performed by us showed, differ significantly from those found for the TBM, GRA, GRB and HG forms: the best fit value of δ ∼ = π, and, correspondingly, of J CP ∼ = 0. For the 3σ range of J CP we have obtained in the case of NO (IO) neutrino mass spectrum: −0.026 (−0.025) ≤ J CP ≤ 0.021 (0.023), i.e., it includes a sub-interval of values centred on zero, which does not overlap with the 3σ allowed intervals of values of J CP in the TBM, GRA, GRB and HG cases.
The results obtained in the present study, in particular, reinforce the conclusion reached in ref. [14] that the experimental measurement of the cosine of the Dirac phase δ of the PMNS neutrino mixing matrix can provide unique information about the possible discrete symmetry origin of the observed pattern of neutrino mixing.  Figure 16: Confidence regions at 1σ, 2σ and 3σ for 1 degree of freedom in the planes (sin 2 θ 23 , δ), (sin 2 θ 13 , δ) and (sin 2 θ 23 , sin 2 θ 13 ) in the blue (dashed lines), purple (solid lines) and light-purple (dash-dotted lines) for NO (IO) neutrino mass spectrum, respectively, obtained using eq. (55). The best fit points are indicated with a cross (NO) and an asterisk (IO).
in which we have neglected the correlations among the oscillation parameters, since the functions χ 2 i have been extracted from the 1-dimensional projections in [11]. In order to quantify the accuracy of our approximation we show in Fig. 16 the confidence regions at 1σ, 2σ and 3σ for 1 degree of freedom in the planes (sin 2 θ 23 , δ), (sin 2 θ 13 , δ) and (sin 2 θ 23 , sin 2 θ 13 ) in blue (dashed lines), purple (solid lines) and light-purple (dash-dotted lines) for NO (IO) neutrino mass spectrum, respectively, obtained using eq. (55). The parameters not shown in the plot have been marginalised. It should be noted that what is also used in the literature is the Gaussian approximation, in which χ 2 can be simplified using the best fit values and the 1σ uncertainties as follows: Here x i = {sin 2 θ 12 , sin 2 θ 13 , sin 2 θ 23 , δ}, x i and σ x i being the best fit values and the 1σ uncertainties 11 taken from [11]. We present in Fig. 17 the results of a similar two-dimensional analysis for the confidence level regions in the planes shown in Fig. 16, but using the approximation for χ 2 given in eq. (56). It follows from these figures that the Gaussian approximation does not allow to reproduce the confidence regions of [11] with sufficiently good accuracy. For this reason in our analysis we use the more accurate procedure defined through eq. (55). In both the figures the best fit points are indicated with a cross and an asterisk for NO and IO spectra, respectively. Each symmetry scheme considered in our analysis, which we label with an index m, depends on a set of parameters y m j , which are related to the standard oscillation parameters through expressions of the form x i = x m i (y m j ). In order to produce the 1-dimensional figures we minimise 11 In the case of asymmetric errors we take the mean value of the two errors.  Figure 17: The same as in Fig. 16, but using eq. (56).
for a fixed value of the corresponding observable α, i.e., with α = {δ, J CP , sin 2 θ 23 }. The likelihood function for cos δ has been computed by taking which was used to produce the likelihood function for the different symmetry forms in Fig. 12.
It is worth noticing that in the case of flat priors on the mixing parameters, the posterior probability density function reduces to the likelihood function. Although we did not use the Gaussian approximation for obtaining Figs. 11, 12, 15 and 18, we employed it to obtain Figs. 13 and 14.

C Results for the Atmospheric Angle
For completeness in Fig. 18 we give N σ ≡ χ 2 as a function of sin 2 θ 23 . The best fit values and the 3σ regions are summarised in Table 10.   Table 10: Best fit values of sin 2 θ 23 and corresponding 3σ ranges (found fixing χ 2 − χ 2 min = 3) in our setup using the data from [11].