Charged Lepton Masses as a Possible CPV Source

Received: September 18, 2020 Accepted: January 23, 2021 Published Online: February 10, 2021 We realize a model-independent study of the so-called Tri-Bi-Maximal pattern of leptonic flavor mixing. Different charged lepton mass matrix textures are studied. In particular, we are interested in those textures with a minimum number of parameters and that are able to reproduce the current experimental data on neutrino oscillation. The textures studied here form an equivalent class with two texture zeros. We obtain a Tri-Bi-Maximal pattern deviation in terms of the charged leptons masses, leading to a reactor angle and three CP violation phases non-zero. These lastest are one CP violation phase Dirac-like and two phases Majorana-like. Also, we can test the phenomenological implications of the numerical values obtained for the mixing angles and CP violation phases, on the neutrinoless double beta decay, and in the present and upcoming experiments on long-base neutrino oscillation, such as T2K, NOvA, and DUNE.


Introduction
Detecting neutrinos requires the use of very large and precise devices, in addition to avoiding interference caused by other natural phenomena. Therefore, experiments to detect neutrinos must be installed deep within the Earth or Sea [1,2]. Some examples of experiments located in mines are Homestake [3], Super-Kamiokande [4], and SNO [5,6], while the ANTARES experiment is located in the depths of the Mediterranean Sea [7]. In the aforementioned experiments, the quantum mechanical phenomenon of neutrino oscillation can be detected. This phenomenon consists in that the flavor of the neutrino (electron neutrino ν e , muon neutrino ν µ , and tau neutrino ν τ ) changes as it travels through space, therefore the flavor of the neutrino that is emitted at the source is not necessarily the same as detected in the experiment. Furthermore, to have this flavor oscillation it is necessary that the neutrinos have a non-zero mass [2].
In the theoretical framework of the Standard Model (SM), which governs the dynamics of fundamental particles and their interactions [8], there are three massless neutrinos, which are treated as Dirac particles, and these neutrinos have the flavors ν e , ν µ , and ν τ [1,2]. However, a wellestablished fact is that from solar, atmospheric, and reactor neutrino oscillation experiments, the neutrino changes flavor as it travels due to its small mixing of mass and flavor [9]. So, this is clear evidence for physics beyond SM.
One of the main characteristics of neutrinos is to have a zero-electric charge, which is why, in the context of Quantum Field Theory, these particles can be represented as Dirac or Majorana particles. Thus, neutrinos apart from being chameleonic as they change their flavor while traveling from the source to the detector, also have an identity problem. In other words, until now, if the existing experimental data are considered, the nature of the neutrinos cannot be determined. However, in minimum extensions of the SM, considering neutrinos as Majorana particles explains very well the smallness of their mass [10]. The smallness in the neutrino mass scale is well explained by the seesaw mechanism, which links it to a new physical scale in nature [9].
The neutrino flavor mixing exhibits a very interesting pattern, in which two mixing angles from a three-flavor scenario appear to be maximum, while the third is still very small. Different lepton flavor mixing schemes such as Tri-Bi-Maximum (TBM) [11], Bi-maximum (BM) [12], and democratic mixing (DC) [13]  > > ), in the mass spectrum. However, the Daya Bay experiment presented the first conclusive results to have a reactor angle different to zero [15]. The mixing angle value θ 13 at 90% C.L. is sin 2 syst . From the previous results, it is evident that the BM, TBM, and DC scenarios cannot be considered at their nominal value. Therefore, they must be analyzed to determine possible deviations from them.
In Table 1 we show the last results of a global fit of neutrino oscillation data in the simplest three-neutrino framework. From these values, one can easily conclude that neutrino physics is in its precision stage with respect to its fundamental parameter determination. However, the CPV phase factors are in their first stage of predicting values, since only for the phase Dirac-like there exists a value range obtained from the global fit of neutrino data. But in the case when the neutrino is a Majorana particle, the other two CPV phases associated with the effective mass in the neutrinoless double beta decay, do not have experimental evidence for obtaining their values. = ν ν is the difference of the squares of the neutrino masses, and is the Dirac-like CPV phase. The latter is the just phase associated with CPV involved in the transition amplitudes of neutrino oscillations.  The goal of Long-baseline experiments is to obtain precise measurements of neutrino oscillation parameters. In particular, the T2K, DUNE, and NOνA experiments take measurements of the transition amplitude between solar and atmospheric neutrinos, ν ν µ → e , to accurately determine the numerical value of the CPV phase Dirac-like. In the T2K experiment, the neutrino beam has a mean energy of 0.6 GeV and a width of about 0.3 GeV, traveling from the J-PARC accelerator to the Super-Kamiokande detector which is 295 km away [16]. In the NOνA experiment, the neutrino beam travels a distance of 810 km, and has an energy between 1 and 3 GeV, however, the maximum signal is around 2 GeV [17]. In the DUNE experiment, the neutrino beam is of high intensity with an average energy of 2.8 GeV, the particles travel a distance of 1300 km to the detector [18]. p.163

PARAMETER
The neutrinoless double beta decay, 0νββ , is a secondorder nuclear reaction that has not yet been observed, but it makes it possible to elucidate whether the neutrinos are Dirac or Majorana particles, since only in the latter case does this decay exist. For this decay, the physical observable to be measured is the amplitude T 1 2 0 / ν , which is sensitive to the Majorana phases associated with CP violation, and proportional to the effective Majorana mass, where the U ej are elements of the PMNS matrix [19].
In this work, an independent study of models is proposed, in which the neutrinos mass terms are of Majorana type. In particular, the neutrino mass matrix form is fixed with the so-called TBM flavor pattern. On the other hand, to establish the form of the charged leptons mass matrix, an equivalence class whose elements are matrices with two texture zeros is proposed. The leptonic flavor mixing matrix, PMNS, is expressed in terms of the charged leptons masses, which allow us to obtain a deviation from the TBM standard pattern. In addition, it predicts a range of values for the Charge-Parity violation (CPV) phases and the effective Majorana mass in the neutrinoless double beta decay.

TBM Pattern of Matrix PMNS
The low energy neutrinos oscillations are described through the Lagrangian density [9]  = - The first Lagrangian term represents charged currents, M ν is the neutrino mass matrix Majorana-like, and M  is the charged lepton mass matrix Dirac-like. In general, M ν is a symmetric matrix, while M  has no special characteristics, both 3 3 � × complex matrices. These matrices are diagonalized by the following unit transformations:  (1) and (2), the charged current term takes the form L cc is the lepton flavor mixing matrix, known as the PMNS matrix that governs the neutrinos and leptons couplings.
In symmetric parameterization, the PMNS matrix has the form [19]: 12 . The mixing angles in terms of the matrix PMNS entries have the following form [19,20]: These expressions are invariant before reparametrizations of the matrix PMNS. The CPV phases φ φ δ 12 13 , and CP have the expressions: where J Im are invariant related to CPV phases Majorana-like [19]. p.164 In order to obtain the TBM pattern in lepton flavor mixing, we should take into account that charged lepton mass matrix M  is diagonal. Moreover, solar, atmospheric and reactor mixing angles have the values sinθ 12 In this scenario, neutrino mass matrix M ν is expressed, by considering (2) and (6), as: where a m m However, in concordance with current experimental data of neutrino oscillations, TBM pattern is not realistic as the reactor angle θ 13 and CPV phase δ CP are non nulls. Motivated by the need to deviate from the simplest form, to first order, for the TBM pattern (6), and considering that from a theoretical point of view, the PMNS lepton flavor mixing matrix comes from the discordance between the charged lepton mass matrix and neutrinos diagonalization, we propose a generalized version of TBM pattern. For that, neutrino mass matrix is given by (7), whereas to fix the charged lepton mass matrix form we propose several equivalence classes. These are different one and other by the texture zeros number in their matrices. Particularly, one takes that the charged lepton mass matrix is constructed through a Hermitian matrix, which in terms of an equivalence class is expressed as Here, T i are the elements of S 3 real representation [19,20]. P  is the diagonal matrix of phase factors, which is obtained to write down M  i in a polar form. Finally, O  is a real and orthogonal matrix. So, matrix PMNS takes the form We fix the form of M

Numerical Results
Our main goal is found a form of M  0 that provides a reactor angle deviation of null value, which is in TBM pattern, and at the same time, provides a current numerical values for solar and atmospheric angles. One numerical analysis is done, in which is taken into account the following values of charged lepton masses (GeV): , . Finally, supported on above statement and beginning with (9), (12) and (4), first result obtained is the theoretical expression of reactor mixing angle, which more depends on φ a that � φ c . Furthermore, the only mass matrix in the equivalence class (10)  In Figure 1 are shown the shape of the theoretical mixing angles expressions. To obtain these plots, phase factors take the numerical values φ a = 0 9 . rad and φ c = 1 78 . rad. On Figure 1(a), we conclude the BFP is obtained for large values of δ  as δ  = 0 9 . , whereas solar angle can be reproduced with any value of δ  , Figure 1(b). On Figure 1(c), atmospheric angle, BFP is obtained δ  = 0 06 0 98 . , . (NH) and δ  = 0 07 0 99 . , . (IH). Then, we conclude that δ  = 0 9 . , φ a = 0 9 . rad and φ c = 1 78 . rad reproduce the current experimental values of the three lepton flavor mixing angles. Figure 2 show CPV phases φ φ δ 12 13 , and CP & � δ  . Transition probability in matter for solar and atmospheric oscillations, ν ν µ → e and ν ν µ → e respectively, can be described through the expressions [19,20], where φ 12 and φ 13 are given in (5). To analyze the effective mass m ee , neutrino masses are expressed in terms of ∆m ij  . rad and φ c = 1 78 . rad .  . rad and φ c = 1 78 . rad.

Conclusion
In theoretical frame of an independent model study, one considered that Majorana neutrino mass matrix is represented through one matrix with a mixing TBM pattern. In case of the charged lepton mass matrix, we explored six mass matrices corresponding to one equivalence class with two zeros. One obtained that just M  0 and M  3 , reproduced the current numerical values of mixing angles. Moreover, we obtain that the reactor, atmospheric, and solar mixing angles just have three different forms for an equivalent class. Furthermore, one can obtained predictions for CPV phases like-Dirac and Majorana as well. Likewise, phenomenological implications were shown for the DUNE, T2K and NOνA experiments. Finally, a numerical range values is provided for the Majorana effective mass in the double beta decay without neutrinos.