Neutrino predictions from generalized CP symmetries of charged leptons

We study the implications of generalized CP transformations acting on the mass matrices of charged leptons in a model-independent way. Generalized e − μ, e − τ and μ−τ symmetries are considered in detail. In all cases the physical parameters of the lepton mixing matrix, three mixing angles and three CP phases can be expressed in terms of a restricted set of independent “theory parameters” that characterize a given choice of CP transformation. This leads to implications for neutrino oscillations as well as neutrinoless double beta decay experiments.


Introduction
Following the discovery of neutrino oscillations [1,2], particle physics has entered probably its most exciting phase in many decades. The observed pattern of the lepton mixing angles, at odds with those of the quark sector, has added a further challenge to the so-called flavour problem, a long-standing mystery in particle physics. Over the last two decades, neutrino oscillation experiments have made considerable progress in the determination of the neutrino oscillation parameters, four of which are rather well measured. These include the solar and atmospheric mass splittings, as well as two of the lepton mixing angles θ 12 and θ 13 [3]. So far the octant of the angle θ 23 is not yet well determined, nor the phase parameters characterizing CP violation. Concerning CP violation there are three physical phases in the simplest unitary mixing matrix, one Dirac and the two Majorana phases [4,5]. Although latest results of T2K [6] and NOνA [7] provide a first positive hint for CP violation, characterized by the Dirac CP phase δ CP around 3π/2, its value is not yet measured with high significance [3]. Finally, negative searches for neutrinoless double beta decay do not allow us to decide whether neutrinos are Majorana or Dirac particles nor determine the two associated Majorana phases.
Understanding the pattern of neutrino mixing and CP violation constitutes a fundamental problem in particle physics. Flavor symmetries provide an attractive framework for explaining the leptonic mixing angles and phases. In particular, non-Abelian discrete flavor symmetries have been widely studied in the literature see, for example, [8][9][10][11][12][13][14][15].
It has been noted in recent years that discrete flavor symmetries can be extended so as to include a CP symmetry [16][17][18][19][20][21][22][23][24]. The possible lepton mixing patterns which can be obtained from the breaking of flavor and CP symmetry have been widely explored (see, JHEP07(2018)077 e.g., [25][26][27][28][29][30][31][32][33][34] and references quoted therein). It is remarkable that the observed patterns of quark and lepton flavor mixings can be simultaneously understood from a flavor group and CP symmetry [35]. In this approach the mixing is determined in terms of a few free parameters, so that certain sum rules relate the mixing angles and CP violation phases. These sum rules are sensitive probes to test the discrete flavor symmetry approach in current and future neutrino experiments [36][37][38].
The imposition of a CP symmetry may allow us to obtain predictions for CP violating phases. A simple and interesting example is the so called µ − τ reflection symmetry which exchanges a muon (tau) neutrino with a tau (muon) antineutrino. If the neutrino mass matrix is invariant under the action of µ − τ reflection in the charged lepton diagonal basis, both atmospheric mixing angle θ 23 and Dirac CP phase δ CP are maximal and the Majorana phases are trivial [39][40][41]. The above prediction for maximal θ 23 may be at variance with current experiments [6,7]. A generalized µ − τ reflection in the charged lepton diagonal mass basis is suggested in [23]. This can accommodate the observed non-maximal θ 23 together with maximal δ CP . Moreover, all possible CP transformations can be classified according to the number of zero entries [24].
In this work we shall study three kinds of simple and attractive CP symmetries: generalized e − µ, generalized e − τ and generalized µ − τ symmetries in the neutrino mass diagonal basis. The rest of this paper is organized as follows: in section 2 the symmetric parametrization of the lepton mixing matrix is reviewed. In section 3 we demonstrate how to extract the lepton mixing matrix from an imposed CP transformation, explaining the difference between Dirac and Majorana neutrino cases. In section 4 we perform a detailed study of the generalized e − µ, e − τ and µ − τ symmetries acting on the charged lepton fields. The predictions for lepton mixing parameters are discussed, and the correlations among mixing angles and CP phases are analyzed. In section 5 we discuss the phenomenological implications of our scheme for neutrinoless double beta decay. Further summary and conclusions are given in section 6.

Lepton mixing matrix parametrization
Throughout this paper we will use the so-called "symmetric parametrization" of the lepton mixing matrix. In this parametrization [4], the lepton mixing matrix U can be written as [42] where P 0 = e diag(iδ 1 ,iδ 2 ,iδ 3 ) and the U ij are the complex rotation matrices in the ij-axis. For example:

General discussion
In this section we begin with a general discussion of the generalized CP transformations, highlighting key concepts as well as setting up our notation and conventions. Following refs. [20,23,24] we start by defining the generalized remnant CP transformations for each fermionic field as follows: Such generalized CP transformations acting on the chiral fermions will be a symmetry of the mass term in the Lagrangian provided they satisfy the following conditions, 2

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where m ψ is written in a basis with left-handed (right-handed) fields on the right-hand (lefthand) side. Note that the mass matrices m ψ and M 2 ψ can be diagonalized by a unitary transformation U ψ , with m 1 = m 2 = m 3 . From eq. (3.2), eq. (3.3), eq. (3.4) and eq. (3.5), after straightforward algebra, we find that the unitary transformation U ψ is subject to the following constraint from the imposed CP symmetry X ψ , diag(e iδe , e iδµ , e iδτ ), for Dirac fields , where δ e , δ µ and δ τ are arbitrary real parameters. 3 Because X ψ is a symmetric matrix, one can use Takagi decomposition (note that this decomposition is not unique) to express X ψ as Inserting eq. (3.7) into eq. (3.6), we find that the combination P − 1 2 U † ψ Σ is a real orthogonal matrix, i.e., where O 3 is a generic 3 × 3 real orthogonal matrix. As we will discuss in detail in section 4, the matrix Σ can be expressed in terms of three independent "CP labels", namely two of the three phases α, β, γ and the CP angle Θ. These CP labels characterize a given generalized CP transformation. The resulting explicit form of Σ will depend on the generalized CP symmetry under consideration, and will be discussed in detail in section 4. Throughout this paper we will parameterize the orthogonal matrix O 3 as Notice that the matrix O 3 has the following properties

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where the diagonal matrices can be absorbed into the matrix P . As a consequence, the range of variation of the free parameters θ 1,2,3 describing a given theory, can be taken to be [0, π). At this point we would like to remark that, the generalized CP symmetries do not impose any constraint on the fermion masses. These can always be chosen to match the required experimental values. The predictive power of generalized CP symmetries lies in the mixing matrix elements and their phases. Before going to particular cases of generalized CP transformations and their implications we would like to briefly comment about the differences one should expect in case neutrinos are Dirac or Majorana particles. In previous works on generalized CP symmetries [20,23,24], the neutrinos where assumed to be Majorana particles but that may not be the case in nature. We now comment on the differences that arise if neutrinos are Dirac particles.
As clear from our previous discussion, and in particular from eq. (3.9), the only difference between Majorana and Dirac neutrino mixing matrices (for a given CP symmetry) is in the diagonal unitary matrix P that appears in the right side of the lepton mixing matrix. For the case of Majorana fields, P is real, while in the Dirac case it is a general diagonal matrix of phases. Using the technique described in [4], it is easy to show that δ CP of a given unitary matrix U is the same as δ CP associated to another unitary matrix given by U · P , where P is a diagonal matrix of phases. This implies that for any choice of a given generalized CP symmetry: • All the mixing parameters characterizing neutrino oscillations, i.e., θ 12 , θ 13 , θ 23 and δ CP are identical both for Majorana or Dirac neutrinos, irrespective of the choice of the set of generalized CP symmetries imposed on either the charged leptons or neutrinos.
• For Majorana neutrinos, the imposition of a given generalized CP symmetry leads to interesting correlations between the Majorana phases, as we will discuss later. In contrast, if neutrinos are Dirac fields, the Majorana phases are unphysical and can be rotated away by appropriate field redefinitions.
• The imposition of generalized CP symmetries for Majorana neutrinos leads to important implications for neutrinoless double beta decays, as we discuss in section 5. 4 In contrast, if neutrinos are Dirac in nature then neutrinoless double beta decay is simply forbidden.
In what follows we will primarily discuss the implications of generalized CP symmetries acting on charged leptons, assuming neutrinos to be Majorana particles. All of the resulting predictions for neutrino oscillation parameters hold equally well if neutrinos are Dirac-type. In other words, all of the correlations between oscillation observables remain unchanged irrespective of whether neutrinos are Majorana or Dirac-type. In the latter case the Majorana phases are unphysical and neutrinoless double beta decay is forbidden.

JHEP07(2018)077 4 Imposing a particular CP symmetry
In this section we consider the implications of the mass matrix having symmetry under certain generalized CP transformations, in the same spirit as the well studied case of generalized µ − τ symmetry [23] for the neutrino sector. From now on we will focus in the basis in which the neutrino mass matrix is diagonal and the CP symmetry is imposed in the charged lepton sector.
As already explained, all of our predictions for neutrino oscillation parameters, namely the mixing angles and δ CP , would remain unchanged should neutrinos be Dirac type. Moreover, in the Majorana case we obtain predictions for neutrinoless double beta decay, that would be absent in the case of Dirac neutrinos.

Generalized e − µ reflection symmetry for charged leptons
In analogy with the generalized µ−τ symmetry in the neutrino sector one can consider other possibilities, for example, the case of a e − µ symmetry imposed on the charged leptons. The CP transformation corresponding to the generalized e − µ symmetry is given by where the three CP labels α, β and Θ characterize a given CP transformation. The phases α and β can take any value between 0 and 2π, and Θ can be limited in the range 0 ≤ Θ ≤ π without loss of generality. Notice that the X 33 entry can be taken to be real without loss of generality, as a global phase in the CP matrix will be unphysical. The Takagi factorization for X eµ gives us Given the above assumptions and using eq. (3.8), the charged lepton mixing matrix takes the form Since we are in the diagonal neutrino mass basis, the lepton mixing matrix U arises solely from the charged lepton sector, and it is simply given by where the phases in the matrix P can be absorbed into the charged lepton fields. It follows that, for a given CP transformation X eµ , the lepton mixing matrix i.e. all mixing angles and CP phases depend on three free parameters θ 1,2,3 plus the three CP labels. We first discuss the interesting subclass of generalized e − µ symmetry in which α = β = 0, leaving only JHEP07(2018)077 one label, Θ. Using the explicit form of the lepton mixing matrix in eq. (4.4), the mixing and phase parameters characterizing the lepton mixing matrix can be extracted as follows and for the CP invariants we get which implies the following CP violation phases For the most general case, in which α = β = 0, the Majorana CP phases φ 12 and φ 13 would change into φ 12 + (α − β)/2 and φ 13 + α/2, respectively, while all the mixing angles as well as the Dirac CP phase δ CP would remain intact.
In the simplifying case of α = β = 0, different models are characterized by a single label, namely Θ. Different values of Θ correspond to different models of this class. In the remaining part of the discussion we shall treat the generalized CP label Θ as a free parameter of the theory, however it should be kept in mind that it is really a label and any given model will have a given fixed Θ value.
Contrasting the current ranges for the lepton mixing parameters obtained from general neutrino oscillation global fits [3] with the predicted relations in eq. (4.5)-eq. (4.11), one can obtain the allowed ranges for the CP label Θ and the parameters θ 1 , θ 2 , θ 3 . This can in turn be used to obtain predictions for the values of the Majorana CP phases φ 12 , φ 13 determining the neutrinoless double beta decay amplitude.
For example, taking the current best fit values of the lepton mixing angles of [3], the resulting values for the CP label Θ and the parameters θ 1 , θ 2 , θ 3 and the Majorana CP phases φ 12 , φ 13 are shown in table 1, for both normal (NO) as well as inverted (IO) mass ordering. In particular, if Θ is a rational angle, i.e. a rational multiple of π, the experimental data on lepton flavor mixing can be accommodated as well. For example, for Θ = 3π/8, we have

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numerical benchmark for generalized e − µ reflection Table 1. Numerical examples for the generalized e − µ reflection, where α = β = 0 is assumed. The values of Θ and θ 1,2,3 are fixed by the best fit value of mixing angles and the favored value δ CP = 3π/2 [6,7]. Notice that the two Majorana CP phases are predicted, as shown in the table.
Moreover, eq. (4.5) implies that the solar neutrino mixing angle θ 12 depends on the generalized CP label Θ and the angle θ 3 of the rotation matrix O 3 . Using the current experimental 3σ range of the angle θ 12 from [3], one can obtain the allowed regions for the Θ and θ 3 , as shown in figure 1. To be more specific, from eq. (4.5) we have cos Θ cos 2θ 3 = cos 2θ 12 . (4.14) Thus, the value of Θ cannot be arbitrary and is restricted by the experimental measurement of θ 12 . Inputting the 3σ experiment range 0.273 < sin 2 θ 12 < 0.379 given in [3], we find that cos Θ is constrained to be Notice that a residual CP symmetry characterized by Θ = π/2 is disfavored by current oscillation data. Since the Dirac phase δ CP depends on the CP label Θ and the parameter θ 3 as well, as shown in eq. (4.9), we display the contour plot of | sin δ CP | in the θ 3 − Θ plane in figure 1, where only the values of | sin δ CP | in the phenomenologically viable regions are shown.
Notice also that the above relations eq. (4.5)-eq. (4.11) lead to several important correlations among the various neutrino mixing and CP violation parameters, e.g.
The correlation in eq. (4.16) implies that for the generalized e − µ symmetry in the charged lepton sector, the mixing angle θ 12 and the CP violating phase δ CP are correlated with each other, for a given choice of the CP label Θ, as shown in figure 2. We see that the measurement of δ CP in next generation long baseline experiments, together with high precision measurement of θ 12 , could help us to determine the CP label JHEP07(2018)077 Θ characterizing the residual CP symmetry. Notice that this prediction is analogous to the prediction for the atmospheric mixing angle θ 23 and the Dirac phase δ CP obtained if we impose a generalized µ − τ reflection in the neutrino sector [23].
Given the current experimental range of θ 12 , the other two correlations in eq. (4.17) and eq. (4.18) can be used to obtain the allowed ranges for the Dirac CP phase δ CP and the two Majorana phases φ 12 and φ 13 . These relations together also imply correlations between the two Majorana phases φ 12 and φ 13 .
Notice that all three parameters θ 1,2,3 and the CP label Θ are varied randomly between 0 and π, keeping only the points for which the lepton mixing angles are consistent with experimental data at 3σ level. The resulting predictions for the three CP phases δ CP , φ 12 and φ 13 are displayed in figure 3. We notice that the values of φ 13 is around 0, π/2 and π. The implications of the correlations between the Majorana phases and other mixing parameters for neutrinoless double beta decay will be further discussed in section 5.

Generalized e − τ reflection symmetry for charged leptons
In this section we look at the implications of imposing a generalized e − τ symmetry on charged leptons. The corresponding CP transformation matrix X eτ is of the form 5 where the two phase labels α and γ take values in the range 0 ≤ α, γ ≤ 2π. The angle Θ labeling the residual CP tranformation lies in [0, π]. The X 22 element is taken as real 5 Note that, although the CP label Θ characterizing the e−τ symmetry differs from that of the e−µ case, we denote it by the same symbol (similarly also for the µ − τ case, discussed in subsection 4.3). However, they have different ranges of variation, see eqs. (4.15), (4.39) and (4.56).

Generalized µ − τ reflection symmetry for charged leptons
In this section we proceed to study the implications of imposing generalized µ−τ symmetry on the charged lepton sector. The analogous case of generalized µ − τ symmetry acting on neutrinos has been studied previously in [23]. The CP transformation corresponding to the generalized µ − τ symmetry, X µτ is given by where the phase labels β and γ can take any value between 0 and 2π, and the fundamental range for the CP label Θ is [0, π]. The Takagi factorization of X µτ is found to be (4.41) Figure 4. Predicted correlation between the Dirac CP phase δ CP and the solar mixing angle θ 12 in the case of generalized e − τ reflection, when θ 13 and θ 23 are in the 3σ ranges given in [3]. The vertical orange and yellow bands are the currently allowed 1σ and 3σ θ 12 and δ CP regions respectively for normal ordering, while the star denotes the best fit point. The green and light blue hatched regions correspond to Θ = π/12 and Θ = π/9 respectively. Figure 5. Strong correlation between φ 13 and δ CP for the case of generalized e − τ reflection in the charged lepton sector, assuming α = γ = 0 and the three mixing angles in the experimentally preferred 3σ regions [3].

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As a result, the total lepton mixing matrix U in the diagonal neutrino basis is simply given by Notice that the matrix Σ µτ is related to Σ eτ as follows, where α → β implies replacement of phase α by β in eq. (4.20) and P 12 is a permutation matrix given by Thus it is straightforward to see that the mixing parameters predicted by the generalized e − τ and µ − τ symmetries obey the following relations From the predictions of generalized e − τ symmetry in previous section, the above relations can be used to obtain the values of the mixing angles and CP phases corresponding the generalized µ − τ symmetry. As before, the values of the parameters θ 1,2,3 and the CP label Θ can be determined from the measured values of three lepton mixing angles and the favored δ CP 3π/2 value [6,7]. The predicted Majorana CP phases are given in table 3. As an example, taking Θ = π/6, we have  Table 3. Numerical examples for the eneralized µ − τ reflection, where we assume β = γ = 0. The values of Θ and θ 1,2,3 are fixed by the best fitted value of mixing angles and experimentally favored value δ CP = 3π/2 [6,7]. Notice that the two Majorana CP phases are predicted, as shown in the table.
lengthy and not very illuminating. By expanding in the small quantity sin θ 13 , one can display the predictions in a simple form sin 2 Θ 4 sin 2 θ 13 sin 2 δ CP sin 2 θ 12 [1 + 4 sin θ 13 cot θ 12 cot 2θ 23 cos δ CP ] ,  Taking into account the current allowed 3σ range from [3], eq. (4.53) gives rise to the following viable range of Θ, (4.56) Notice that the relation between mixing angles and δ CP in eq. (4.53) is different from those in eq. (4.16) and eq. (4.36) corresponding to the cases of generalized e − µ and e − τ reflection, respectively. Taking, for example, Θ = π/9 and Θ = π/6, the possible values of δ CP as a function of the solar mixing angle θ 12 are shown in figure 6. The measurement of δ CP in future neutrino oscillation experiments can help us to fix the value of the CP label Θ. In addition, eq. (4.54) implies that sin 2φ 12 is quite small, so that φ 12 lies close to 0, π/2 and π. In addition, one can see from eq. (4.55) that the Majorana phase φ 13 is close to δ CP (mod π) or δ CP + π/2 (mod π), up to higher order terms in sin θ 13 . The predicted correlation between φ 13 and δ CP in shown in figure 7.
To conclude this section we stress that the imposition of generalized CP symmetries implies correlations involving the lepton mixing angles, the Dirac CP violating phase relevant for neutrino oscillations. We have already discussed some of the phenomenological JHEP07(2018)077 Figure 6. Predicted correlation between the Dirac CP phase δ CP as a function of the solar mixing angle θ 12 in the case of generalized µ − τ reflection, when θ 13 and θ 23 are required to lie in their 3σ ranges. The vertical orange and yellow bands are the currently allowed 1σ and 3σ (NO case) respectively and the star denotes the best fit point [3]. The green and light blue hatched regions correspond to Θ = π/9 and Θ = π/6 respectively. Figure 7. The correlation between φ 13 and δ CP for the generalized µ − τ reflection acting on the charged lepton sector, taking β = γ = 0 and the three mixing angles in their 3σ regions [3].
implications of these correlations. Here we mention the generalized CP symmetry predictions for the CP violation phase δ CP . Figure 8 shows how the correlations between the Dirac phase δ CP characterizing neutrino oscillations and the CP label Θ. Note that the predictions for this phase do not depend on the values of the phase labels α, β and γ. This is as expected, since oscillation probabilities can not depend on Majorana phases [5]. In principle, by fixing a given CP JHEP07(2018)077 symmetry or Θ value one can make predictions for the upcoming generation of neutrino oscillation experiments such as DUNE [47,48]. This exercise has been carried out in the analogous case of generalized CP symmetries acting on neutrinos in refs. [23,24]. In the remaining of this paper we prefer to focus on the rather significant implications of the above predictions for the upcoming neutrinoless double beta decay experiments, as discussed in the next section.

Implications for neutrinoless double beta decay
We shall proceed to investigate the implications of generalized CP symmetries of charged leptons in theories with Majorana neutrinos, assuming the basis in which their mass matrix is diagonal. 6 As in previous sections, we shall set α = β = γ = 0 in this section.
As we already saw at length, the imposition of generalized e − µ, e − τ and µ − τ symmetries leads to correlations amongst the lepton mixing angles, the Dirac CP violating phase δ CP , and also the two Majorana phases φ 12 , φ 13 . We have already discussed some features of these correlations. Here we focus on the generalized CP symmetry predictions involving the Majorana CP violation phases.
By treating θ 1,2,3 and the CP label Θ as random numbers in the range of 0 and π we perform a comprehensive numerical scan of the parameter space. The three lepton mixing angles are required to lie within their 3σ ranges, as determimed in [3]. In figure 9 we show the correlations between the Majorana CP phases φ 12 , φ 13 with the CP label Θ characterizing the three generalized CP symmetries discussed in previous section. As clear from figure 9, the Majorana phases φ 12 and φ 13 are strongly correlated with the generalized JHEP07(2018)077  Correlations between the two Majorana phases φ 12 and φ 13 of the lepton mixing matrix. The light blue, magenta and green regions are for the generalized e − µ, e − τ and µ − τ symmetries respectively. The magenta regions are quite narrow and hardly visible, and they are closely distributed around φ 12 ∼ 0, φ 12 ∼ π/2 and φ 12 ∼ π. Here we take α = β = γ = 0 for illustration.
CP label Θ. This in turn implies that they are correlated with other mixing parameters as discussed in previous section, as well as mutually correlated, as shown in figure 10.
These correlations have interesting consequences for the effective Majorana mass m ee characterizing the neutrinoless double beta decay amplitude, as we discuss next. We first consider the case of generalized e − µ reflection. The relevant results and correlations for this case have already been discussed in section 4.1. The effective Majorana mass m ee characterizing the neutrinoless double beta decay amplitude for case of generalized e − µ symmetry is shown in figure 11. JHEP07(2018)077 Figure 11. Attainable values of the effective Majorana mass m ee as a function of the lightest neutrino mass in the generalized e − µ reflection scheme. In the left panel the CP label Θ is varied over all its values, while we fix Θ = 2π/5 and Θ = 3π/5 in the right panel. Notice that the expected m ee for 0 ≤ Θ < π/2 and π/2 ≤ Θ ≤ π almost coincide in the case of IO. The red (blue) dashed lines delimit the most general allowed regions for IO (NO) neutrino mass spectrum obtained by varying the mixing parameters over their 3σ ranges [3]. The present most stringent upper limits m ee < 0.061 eV from KamLAND-ZEN [49] and EXO-200 [50] is shown by horizontal grey band. The vertical grey band indicates the current sensitivity of cosmological data from the Planck collaboration [51].
As is clear from this figure, almost all the 3σ range values of the effective mass m ee can be reproduced if the variation of the CP label Θ is taken into account. However, the allowed range of m ee for a given fixed Θ is quite restricted, for both NO and IO. We note that due to the constraints on mixing angles and CP phases, the range of m lightest in which m ee can be very small for NO is considerably reduced. Along with the predictions for the neutrino oscillation parameters, the predicted range for m ee can also be used to test the generalized e − µ reflection. We also indicate by the vertical grey band the sensitivity m i < 0.230 eV at 95% C.L. limit claimed by the Planck collaboration [51].
Similar neutrinoless double beta decay predictions can also be obtained for the cases of e − τ and µ − τ symmetries, as shown in figure 12 and figure 13 respectively. One sees that the generalized CP symmetry allows only a restricted range for m ee , and thus can be used to test the predictions of the generalized CP symmetries. In particular, if we fix the CP label Θ, then the allowed range for m ee becomes much narrower for both mass orderings, and also for both cases of e − τ and µ − τ symmetries. Future neutrinoless double decay experiments will probe almost all of the IO region, thus allowing to distinguish between 0 ≤ Θ < π/2 and π/2 ≤ Θ ≤ π. JHEP07(2018)077 Figure 12. The same as figure 11 for the generalized e − τ reflection. The CP label Θ is free on the left panel, and we fix Θ = π/9 and Θ = 8π/9 on the right panel. Notice that the IO: 0 ≤ Θ ≤ π 2 region (cyan) is very narrow and is close to the maximum m ee for IO. This region can be tested very soon in neutrinoless double beta decay experiments. Figure 13. The same as figure 11 for the generalized µ − τ reflection. The CP label Θ is free on the left panel, and we fix Θ = π/6 and Θ = 5π/6 on the right panel. Notice that the IO: 0 ≤ Θ ≤ π 2 region (cyan) is very narrow and is close to the maximum m ee for IO. This region can be tested very soon in neutrinoless double beta decay experiments.

JHEP07(2018)077 6 Conclusion
The imposition of generalized CP symmetries provides a powerful framework for predicting the lepton mixing angles and phases. We have investigated the theory of generalized CP transformations acting on the mass matrices of charged leptons. We have considered in detail the case of generalized e − µ, e − τ and µ − τ symmetries. The basic tool is the Takagi factorization which is used to express the physical parameters of the lepton mixing matrix, three mixing angles, and three CP phases, in terms of a restricted set of independent "theory parameters" (labels) associated with a given choice of the CP transformation. Current best fit values of the mixing angles and the favored value for δ CP = 3π/2 constrain the allowed "theory" values of the labels characterizing the CP transformation, for example Θ. In each case we have obtained strong correlations involving the mixing angles and CP phases, valid both for Majorana and Dirac neutrinos see, for example, in eqs. (4.16), (4.36), (4.53). Specific benchmark model examples were given in the tables. Our predictions for the leptonic CP violating phase δ CP , summarized in figure 8, provides model-independent probes of our underlying generalized CP symmetry approach at upcoming long baseline oscillation experiments, such as DUNE, aimed at the measurement of CP violation. For the case of Majorana neutrinos, our predictions for the neutrino mixing angles and phases also include the two Majorana phases, as seen in figures 9 and 10. We have also derived the resulting predictions for the effective mass parameter m ee characterizing the neutrinoless double beta decay rates. Predicted ranges for m ee in each case can be used to test the residual CP symmetry hypothesis at the uponming generation of sensitive experiments, such as KamLAND-ZEN, CUORE, LEGEND, nEXO and NEXT. We would like to remind the readers that the effect of the CP labels α, β and γ is to shift the Majorana phases φ 12 and φ 13 . The numerical results in figures 11, 12 and 13 for m ee are obtained under the assumption of α = β = γ = 0.
Notice that, although we have treated in full generality the implications of generalized CP symmetries, we have not attempted to obtain a rationale for their possible origin. In fact this is an interesting open issue that deserves a dedicated study, which goes far beyond the scope of this present paper and that will be tackled elsewhere.