Abstract
We study the phenomenological consequences of recent results from atmospheric and accelerator neutrino experiments, favoring normal neutrino mass ordering , a near maximal lepton Dirac CP phase along with , for possible realization of natural structure in the lepton mass matrices characterized by for . It is observed that deviations from parallel texture structures for and are essential for realizing such structures. In particular, such hierarchical neutrino mass matrices are not supportive for a vanishing neutrino mass characterized by and predict meV, , , , and , respectively, indicating that the task of observing a decay may be rather challenging for near future experiments.
1. Introduction
One of the intriguing phenomena in particle physics is the origin of fermion masses which appear to span several orders of magnitudes starting with neutrinos to the top quark. The masses and flavor mixing schemes of quarks and leptons are significantly different with the quark sector exhibiting strong mass hierarchy, small mixing angles, and relatively heavier mass spectrum whereas the neutrinos are extremely light while two of their mixing angles are still large. In the current scenario, there is also a lack of consensus on the nature of neutrinos, that is, Dirac or Majorana along with doubts on the possible ordering of neutrino masses, namely, normal, that is, (NO), or inverted, that is, (IO). This nevertheless makes the task of constructing the fermion mass matrices nontrivial especially in the context of quark-lepton complementarity.
The confirmation of Higgs Boson by the ATLAS and CMS collaborations [1] completes the Standard Model (SM) of particle physics. Within this model, the quark mass terms in the Lagrangian are expressible aswhere and are the left (right) handed quark fields and are the quark mass matrices with , for the “up” type and “down” type quarks. The resulting weak charged current quark interactions are given bywhere is the Cabibbo-Kobayashi-Maskawa (CKM) matrix [2, 3] or the quark mixing matrix measuring the nontrivial mismatch between the flavor and mass eigenstates of quarks; for example, where are unitary matrices.
Interestingly, the quark masses as well as the elements of CKM matrix observe a hierarchical pattern: namely, and . It is natural to expect this hierarchy to be embedded within the corresponding quark mass matrices: namely, (for )with . Recent investigations [4] in this regard indicate that the current quark mixing data indeed permit the quark mass matrices to have such a natural and hierarchical structure provided for , and . Such hierarchical mass matrices have been referred to as natural mass matrices [5]. In particular, naturalness provides a rationale framework to correlate the observed fermion mass ratios, the corresponding mass matrices, and observed mixing angles. Specifically, for , the observed strong hierarchy among the quark masses and CKM elements gets naturally translated onto the structure of the corresponding mass matrices.
A concomitant of such naturalness in mass matrices is the absence of parallel texture structure for the “up” and “down” type quark mass matrices [4]. A parallel texture structure corresponds, for example, to the mass matrices and with texture zeros at identical positions in both the mass matrices. Hierarchical structures have fetched greater importance in the literature as these predict certain very simple yet compelling relations among the CKM elements and the quark mass ratios [4, 6–14].
However, the mass spectrum for leptons is quite distinguished from the quark sector, wherein the charged leptons masses are strongly hierarchical, that is, , while at least two of the neutrinos are allowed to have the same order of mass. It should be interesting to investigate if naturalness can provide a unique explanation for the fermion mass matrices, corresponding observed mass spectra, and the mixing angles both for the quark and the lepton sectors.
Since the neutrinos are massless within the SM, one has to explore beyond the realms of SM to comprehend the origin of neutrino masses and observe neutrino oscillations phenomenon. A simplistic way to achieve this is to extend the SM theory by assuming neutrinos as Dirac-like particles. In this case, the neutrinos acquire mass through the Higgs mechanism in the similar way as quarks and charged leptons do within the SM, through a Dirac mass term: for example,where and represent the charged-lepton and Dirac neutrino mass matrix, respectively. Indeed, the current experiments have not ruled out such a possibility. In this context, it is also observed that highly suppressed Yukawa couplings for Dirac neutrinos can naturally be achieved using models with extra spatial dimensions [15, 16] or through radiative mechanisms [17–22]. However, such a possibility is perceived to be highly unlikely due to several orders of magnitude difference among () and ().
A rather convincing and natural explanation of neutrino masses can be obtained if neutrinos are assumed to be Majorana particles. This usually involves adding the lepton number (and flavor) violating Majorana mass terms for neutrinos in the Lagrangian; for example,where and correspond, respectively, to the left and right-handed Majorana neutrino mass matrices and the latter usually has an extremely high mass scale. This facilitates generating the light neutrino masses through Type I or Type II seesaw mechanisms: namely,where is usually a complex symmetric matrix: for example,
This allows writing the corresponding charged weak current term for leptons aswhere is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix [23] or the neutrino mixing matrix and emerges through the diagonalization of the matrices and : for example,This mixing matrix relates the neutrino flavor states to the neutrino mass eigenstates throughIn the standard parametrization [24], the PMNS matrix is expressed as , where with , being two Majorana CP violating phases and can be parametrized in terms of three mixing angles , , and and one Dirac CP violating phase : namely,with and for . The neutrino oscillation experiments provide constraints on the three mixing angles , , and along with the two mass square differences: namely, and with for NO and for IO cases.
In the current scenario, the global picture of neutrino oscillation parameters for NO at suggests [25]Moreover, the above data does not seem to forbid for NO or for IO cases, the signatures for which are obtained through . The Planck collaboration measurements of the cosmic microwave background [26] provide further insight into the sum of absolute neutrino masses: for example,More recent results from long-baseline accelerator neutrino experiments T2K [27] and NOA [28] are indicative of a near maximal Dirac CP phase: that is,along with preference for the normal ordering (NO) of neutrino masses. These results are also supported by the preliminary results from the atmospheric neutrino experiment at Super-Kamiokande [28]. In addition, a statistical analysis of the cosmological data [29] also indicates preference for NO providing maximum likelihood for Majorana effective mass: that is, in neutrinoless double beta decay at 1 where
As the mixing angles are related to the corresponding mass matrices, it therefore becomes desirable to study the implications of a combination of NO, along with for lepton mass matrices assuming quarks and lepton mass matrices have similar origins and investigate the conditions affecting the possibility of obtaining natural lepton mass matrices, synchronous with the quark sector. Nevertheless, from a top-down prospective, it should be more economical to have a common framework explaining the fermion masses and mixing for the quark and lepton sectors.
2. Lepton Mass Matrices
Phenomenologically, the problem of constructing the fermion mass matrices has always been a difficult task within the framework of Standard Model (SM) and its possible extensions, wherein the flavor structure of these matrices is usually not constrained by the gauge symmetry. As a result, the matrices and remain arbitrary complex matrices, thereby involving several free parameters as compared to the number of physical observables, namely, six lepton masses, three mixing angles, and one Dirac-like CP phase along with two Majorana phases and .
In this regard, the “texture zero” ansatz initiated by Weinberg [30] and Fritzsch [31, 32] has been quite successful in explaining the fermion masses and mixing patterns [33–52]. However, one requires handling all possible texture structures on a case to case basis. In this context, a common framework allowing for the study of such possibilities is more desirable. This is addressed in the following section.
3. Constructing the PMNS Matrix
In order to reconstruct the PMNS matrix, one requires obtaining the diagonalizing transformations for the corresponding mass matrices. To start with, for , we consider the following two possibilities of texture one-zero mass matrices, as referred to as Type I and Type II structures, respectively, in the following text.
One may also consider these matrices to be Hermitian for Dirac neutrinos. Using standard procedures, it is not possible to obtain the exact diagonalizing transformations for the latter case. In order to avoid a large number of free parameters in these matrices, we assume that the phases are factorizable in these, requiring for symmetric and and for Hermitian and .
The diagonalization of above is realized usingwith for and for . Hereand (symmetric case) and (Hermitian case). Considering and as free parameters, one can write [45]such that The above constraints on the parameters and nevertheless allow hierarchical mass matrices: that is, . Texture rotation from and positions in to position in is realized by rotating element in to and positions in through a unitary transformation on usingfor symmetric mass matrices andfor Hermitian case, where is a complex rotation matrix in the 1–3-generation plane: for example,where for symmetric matrices and for Hermitian matrices.
The condition of a texture zero rotation from positions in to position in requireswhich can be translated towhere and is always negative. Note that the rotation angle is not a free parameter and is completely fixed through and due to repositioning of texture zeros as a result of the rotation . One can now relate the matrix elements in with the corresponding elements in : for example,
The texture rotation in 1–3-generation plane allows . Note that , while the other off-diagonal elements essentially get rescaled due to texture rotation. Furthermore, for , one expects allowing hierarchical structures in Type II possibility: namely, along with since are allowed by oscillation data. Henceforth, it is trivial to obtain the orthogonal transformation for (symmetric case) as and (Hermitian case) aswithwith . Note that in the absence of texture rotation, (unit matrix) for whilefor signifying the corresponding effect of such rotation on real diagonalizing transformation . The resulting mixing matrix for and/or may be constructed asAlso , , and (symmetrics case) or (Hermitian case). Note that a change in sign for and can always be accommodated in the redefinition of phases and which only appear implicitly in the PMNS matrix through and . Considering the six lepton masses, , , , and as free parameters, one can reconstruct the unitary mixing matrix using the above procedure and confront it with the current oscillation data. In lieu of this, we restrict our investigation to only texture four-zero mass matrices involving ten free parameters. Furthermore, the condition of naturalness forbids a texture zero at matrix elements.
Recent works [53–56] in this regard suggest that there exist several viable texture structures of lepton mass matrices. Most of these investigations work in the flavor basis with diagonal charged-lepton mass matrix or enforce parallel texture structures for lepton mass matrices and . In this letter, we investigate all possible structures for four-zero lepton mass matrices, both symmetric and/or Hermitian, assuming factorizable phases (for simplicity) in these. The resulting structures are summarized in Tables 1 and 2 wherein we enlist all texture five and four zeros in agreement with current data at 3. and in the tables represent the position of texture zeros in the corresponding mass matrices. It is observed that the constraints of naturalness, near maximal , , and normal ordering for neutrino masses, taken together, greatly reduce the number of possible viable structures and only a few possibilities seem to survive the test. The possibility of a vanishing neutrino mass is also studied for these texture structures.
4. Fritzsch-Like Four Zeros
It has been observed [36, 47] that, in the absence of constraint, the Fritzsch-like texture four-zero mass matrices are physically equivalent to the generic lepton mass matrices. Interestingly, these matrices can be obtained from the above structures using the assumption of and . In particular, , where is a unit matrix, for this case. The predictions from these matrices and their experimental tests can be found in previous works. To start with, using (13), (35) and allowing free variations to the parameters , , , , and , we first reconstruct the viable structures for (in units of GeV) and (in units of eV) for using the available oscillation data and obtain the following best-fits:along with and . The corresponding predictions for the absolute neutrino masses and read , , , , and , respectively. In the context of agreement with along with , it is observed that naturalness is allowed in independent of octant. This is depicted in Figure 1 where one observes that is still consistent with . However, one finds that the near maximal constraint of requires large deviation of from a possible natural structure. In particular, we identify three vital sources for CP violation in these matrices: namely, the two nontrivial phases , along with the free parameter as elaborated in Figure 2, indicating GeV is required to obtain . This also implies that Fritzsch-like texture five-zero matrices () should be ruled out by . Our study reveals this conclusion to hold true for all possible texture five-zero structures, all of which seem to be ruled out by a near maximal ; see Table 1. This calls upon investigating alternate texture structures, which on the one hand account for near maximal and at the same time allow possible natural structures for and (i.e., ).
5. Natural Lepton Mass Matrices
In this context, and/or in the corresponding mass matrices provide greater possibility of realizing naturalness in corresponding mass matrices as compared to the Fritzsch-like structures wherein interactions between the first and third generations of leptons are suppressed due to texture zeros invoked at and matrix elements. At least, for the quark sector, nonvanishing elements are observed to be crucial in effectuating the natural structures of corresponding mass matrices. A careful analysis of all possible texture four-zero structures reveals that only four possibilities for natural structures are allowed by recent data; see Table 2. We categorize these as Type I and Type II, based on the texture structure of .
5.1. Type I
Case A (). The viable best-fit structures of the lepton mass matrices are summarized below:with , and meV, , , , and , respectively. Like the Fritzsch-like texture four zeros, [36, 38, 47, 57] as depicted in Figure 3. However, the other two mixing angles are fixed by the free parameter illustrated in Figures 4 and 5. The latter also indicates that natural structure for is allowed independent of octant, with also accounting for . Finally, the parameter accounts for near maximal as shown in Figure 6. In particular, a small deviation of provides greater agreement of with the notion of naturalness in the corresponding mass matrix.
Case B (). We obtain the following viable best-fit structures for these lepton mass matrices: namely,wherein , and , , , , and , respectively. dependence for remains the same as before while the other two mixing angles are fixed by the parameter . Furthermore, apart from phases and , is now fixed by the parameter as shown in Figure 7. Naturalness in and seems to be in good agreement with and compatible with GeV and , respectively. A greater agreement with naturalness in is achieved for up to GeV.
Case C (). The viable best-fit structures so obtained for these lepton mass matrices are shown below:wherein , and , , , , and , respectively. It is noteworthy that the condition of texture zero at , that is, , fixes the parameter and hencethrough (30) with This results in only one free parameter in . This is depicted in Figure 8. This parameter also determines the Dirac-like CP phase as shown in Figure 9. Other observations pertaining to the dependence of mixing angles remain the same as in previous cases. It is clear that naturalness in and is in good agreement with and compatible with and , respectively.
5.2. Type II
Case D (). The best-fit values obtained for this possibility are summarized below:wherein , and , , , , and , respectively. It is observed that naturalness is in good agreement with and compatible with ; see Figure 10 and , respectively. Again a greater agreement with naturalness in can be achieved for up to .
6. Conclusions
Assuming factorizable phases in lepton mass matrices, we show that natural mass matrices characterized by for , , and provide a reasonable explanation for the observed fermion masses and flavor mixing patterns in the quark as well as the lepton sectors. It is also observed that deviations from parallel texture structures for and are essential for establishing such natural structures. Such phenomenological textures have also been observed to be stable under the renormalization group running from the lightest right-handed neutrino mass scale to the electroweak scale [39, 40, 53, 58, 59].
Interestingly, naturalness in the lepton sector implies and or such that the observed large values of these mixing angles are perhaps indicative of the possible realization of the neutrino mass ratios as obtained above: that is, , , , , and , respectively. In particular, the possibility of a vanishing neutrino mass, that is, , is not supported by natural lepton matrices. From the point of view of decays, these results seem to indicate that multiton scale detectors may be required to possibly observe signals for such processes.
Competing Interests
The author declares that they have no competing interests.
Acknowledgments
The author would like to thank Shun Zhou, IHEP, Beijing, for discussions and valuable suggestions. This work was supported in part by the Department of Science and Technology under SERB Research Grant no. SB/FTP/PS-140/2013.