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 $e-\mu$, $e-\tau$ and $\mu-\tau$ 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.


I. 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, 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 II the symmetric parametrization of the lepton mixing matrix is reviewed. In section III 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 IV 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 V we discuss the phenomenological implications of our scheme for neutrinoless double beta decay. Further summary and conclusions are given in section VI.
Notice that only the phases φ 12 and φ 13 , but not φ 23 , appear in m ee .

III. 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 where m ψ is written in a basis with left-handed (right-handed) fields on the righthand (left-hand) 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. (9), Eq. (10), Eq. (11) and Eq. (12), 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 2 Even though the X-matrix is symmetric, we prefer to use X † instead of X * when dealing with Dirac neutrinos. 3 If neutrinos are Majorana particles and the lightest one is massless (this possibility is still allowed by current experimental data) one "±" entry would be a complex phase.
Inserting Eq. (14) into Eq. (13), we find that the combination P − 1 2 U † ψ Σ is a real orthogonal matrix, i.e., which implies where O 3 is a generic 3 × 3 real orthogonal matrix. As we will discuss in detail in Section IV, 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 IV. Throughout this paper we will parameterize the orthogonal matrix O 3 as Notice that the matrix O 3 has the following properties 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.  [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. (16), 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 , 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.

IV. 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.
A. 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. (15), 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 one label, Θ. Using the explicit form of the lepton mixing matrix in Eq. (22), 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 sin 2φ 13 = − sin 2θ 3 sin Θ cos 2θ 3 cos Θ + 1 .
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. (23)-Eq. (29), one can obtain the allowed ranges for the CP label Θ and the parameters   Table I. 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. and Moreover, Eq. (23) 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. (23) we have cos Θ cos 2θ 3 = cos 2θ 12 .
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 0.242 < | cos Θ| ≤ 1 .
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. (27), 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. (23)-Eq. (29) lead to several important correlations among the various neutrino mixing and CP violation parameters, e.g.
• φ 13 − δ CP − θ 12 correlation sin 2δ CP sin 2 θ 12 sin 2φ 13 = cos Θ, or sin 2 2φ 13 = sin 2 2δ CP sin 4 θ 12 1 − sin 2 2θ 12 sin 2 δ CP . 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 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. (35) and Eq. (36) 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 .
The results are summarized in table II. As an example, for the representative value Θ = π/9, the best fit values of the three lepton mixing angles [3] can be reproduced and we have besides the CP label Θ, the mixing angles plus CP phases are correlated with each other. After tedious algebra calculations, we find the following relations: One sees from Eq. (54) that sin 2 Θ should be quite small in this case as it is proportional to sin 2 θ 13 at leading order. Detailed numerical analysis shows that the phenomenologically viable ranges of Θ are Note that all three CP phases would be trivial for Θ = 0, π. Moreover, one sees that the solar mixing angle θ 12 and the Dirac CP phase δ CP are strongly correlated.
For example, taking Θ = π/12 and Θ = π/9, the allowed regions of δ CP and θ 12 are determined as shown in Fig. 4, where both θ 13 and θ 23 are required to lie within their current 3σ global ranges [3].
Concerning the Majorana CP phase φ 12 , from Eq. (55) one sees that in this case sin 2φ 12 is of order sin 2 θ 13 , so that φ 12 lies close to 0, π/2 or π. Moreover, Eq. (56) indicates that the Majorana phase φ 13 is the same as δ CP up to π at leading order. The strong correlation between φ 13 and δ CP is displayed in Fig. 5, as expected from the approximation of Eq. (56). The important implications of respectively.
these correlations for neutrinoless double beta decay experiments will be discussed in section V.

C. 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 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].
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 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. (38) 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 III. As an example, taking Θ = π/6, we have  Table III. 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.
As in the previous two cases, here we also find that the relations Eq.
Taking into account the current allowed 3σ range from [3], Eq. (71) gives rise to the following viable range of Θ, help us to fix the value of the CP label Θ. In addition, Eq. (72) 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. (73) 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 Fig. 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 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 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.

V. 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 Fig. 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 Fig. 9, the Majorana phases φ 12 and φ 13 are strongly correlated with the generalized CP label Θ. This in turn implies that The effective Majorana mass m ee characterizing the neutrinoless double beta decay amplitude for case of generalized e − µ symmetry is shown in Fig. 11.
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.
Similar neutrinoless double beta decay predictions can also be obtained for the cases of e − τ and µ − τ symmetries, as shown in Fig. 12 and Fig. 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.

VI. 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. (34,54,71). Specific benchmark model examples were given in the tables. Our predictions for the leptonic CP violating phase δ CP , summarized in Fig. 8, provides 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. 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 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. study, which goes far beyond the scope of this present paper and that will be tackled elsewhere.