Printer Friendly

Investigating the Hybrid Textures of Neutrino Mass Matrix for Near Maximal Atmospheric Neutrino Mixing.

1. Introduction

In leptonic sector, the reactor mixing angle ([[theta].sub.13]) has been established to a reasonably good degree of precision [16], and its nonzero and relatively large value has not only provided an opportunity in exploring CP violation and the neutrino mass ordering in the future experiments but has also highlighted the puzzle of neutrino mass and mixing pattern. In spite of the significant developments made over the years, there are still several intriguing questions in the neutrino sector which remain unsettled. For instance, the present available data is unable to throw any light on the neutrino mass spectrum, which maybe normal/inverted and may even be degenerate. Another important issue is the determination of octant of atmospheric mixing angle [[theta].sub.23], which may be greater than or less than or equal to 45[degrees]. The determination of the nature of neutrinos whether Dirac or Majorana also remains an open question. The observation of neutrinoless double beta (0v[beta][beta]) decay would eventually establish the Majorana nature of neutrinos.

The effective Majorana mass term related to 0v[beta][beta] decay can be expressed as

[[absolute value of M].sub.ee] = [absolute value of ([m.sub.1] [c.sup.2.sub.12] [c.sup.2.sub.13] [e.sup.2i[rho]] + [m.sub.2] [s.sup.2.sub.12] [c.sup.2.sub.13] [e.sup.2i[sigma]] + [m.sub.3] [s.sup.2.sub.13])]. (1)

Data from KamLAND-Zen experiment has presented an improved search for neutrinoless double-beta (0v[beta][beta]) decay [7] and it is found that [[absolute value of M].sub.ee] < (0.061-0.165) eV at 90% (or <2[sigma]) CL. For recent reviews on 0v[beta][beta] decay see [8-13].

In the lack of any convincing theory, several phenomenological ideas have been proposed in the literature so as to restrict the form of neutrino mass matrix, such as some elements of neutrino mass matrix that are considered to be zero or equal [14-21] or some cofactors of neutrino mass matrix to be either zero or equal [19, 22-27]. Specifically, mass matrices with zero textures (or cofactors) have been extensively studied [14-18, 22-24] due to their connections to flavor symmetries. In addition, texture structures with one zero element (or minor) and an equality between two independent elements (or cofactors) in neutrino mass matrix have also been studied in the literature [20, 21, 26, 27]. Such form of texture structures sets to one constraint equation and thus reduces the number of real free parameters of neutrino mass matrix to seven. Hence they are considered as predictive as the well-known two-zero textures and can also be realised within the framework of seesaw mechanism. Out of sixty possibilities, only fifty-four are found to be compatible with the neutrino oscillation data [21] for texture structures having one zero element and equal matrix elements in the vanishing minor and equal cofactors in the neutrino mass matrix (1TEC) only fifty-two cases are able to survive the data [26, 27].

The purpose of present paper is to investigate the implication of large effective neutrino mass [[absolute value of M].sub.ee] on 1TEE and 1TEC structures of neutrino mass matrix, while taking into account the assumptions of [28, 29]. The consideration of large [[absolute value of M].sub.ee] is motivated by the extensive search for this parameter in the ongoing 0v[beta][beta] experiments. The implication of large [[absolute value of M].sub.ee] has earlier been studied for the viable cases of texture two-zero and two-vanishing minor, respectively [28,29]. Grimus et al. [30] also predicted the near maximal atmospheric mixing for two-zero textures when supplemented with the assumption of quasi-degenerate mass spectrum. However, the observation made in all these analyses is independent of solar and reactor mixing angles. Motivated by these works, we find that only four out of sixty cases are able to predict near maximal [[theta].sub.23] for 1TEE and 1TEC, respectively. In addition, the analysis also hints towards the indistinguishable feature of 1TEE and 1TEC. To present the indistinguishable nature of the 1TEE and 1TEC texture structures, we have then carried out a comparative study of all the viable cases of 1TEE and 1TEC at 3[sigma] CL. The similarity between texture zero structures with one mass ordering and corresponding cofactor zero structures with the opposite mass ordering has earlier been noted in [31-33]. In [19], the strong similarities have also been noted between the texture structures with two equalities of elements and structures with two equalities of cofactors in neutrino mass matrix, with opposite mass ordering.

The rest of the paper is planned in the following manner. In Section 2, we shall discuss the methodology to obtain the constraint equations. Section 3 is devoted to numerical analysis. In the end we will summarize our result.

2. Methodology

The effective Majorana neutrino mass matrix ([M.sub.v]) contains nine parameters which include three neutrino masses ([m.sub.1], [m.sub.2], [m.sub.3]), three mixing angles ([[theta].sub.12], [[theta].sub.23], [[theta].sub.13]), and three CP violating phases ([delta], [rho], [alpha]). In the flavor basis, the Majorana neutrino mass matrix can be expressed as follows:

[M.sub.v] = [P.sub.l]U[P.sub.v][M.sup.diag][P.sup.T.sub.u][U.sup.T][P.sup.T.sub.l], (2)

where [M.sup.diag] = diag([m.sub.1], [m.sub.2], [m.sub.3]) is the diagonal matrix of neutrino masses and U is the flavor mixing matrix, and

[mathematical expression not reproducible], (3)

where [P.sub.v] is diagonal phase matrix containing Majorana neutrinos [rho], [sigma]. [P.sub.l] is unobservable phase matrix and depends on phase convention. Equation (2) can be rewritten as

[mathematical expression not reproducible], (4)

where [[lambda].sub.1] = [m.sub.1][e.sup.2i[rho]], [[lambda].sub.2] = [m.sub.2][e.sup.2i[sigma]], [[lambda].sub.3] = [m.sup.3]. For the present analysis, we consider the following parameterization of U [20]:

[mathematical expression not reproducible], (5)

where [c.sub.ij] = cos [[theta].sub.ij], [s.sub.ij] =sin [[theta].sub.ij]. Here, U is a 3 x 3 unitary matrix consisting of three favor mixing angles ([[theta].sub.12], [[theta].sub.23], [[theta].sub.13]) and one Dirac CP-violating phase [delta].

For hybrid texture structure (1TEE) of [M.sub.v], we can express the ratios of neutrinomass eigenvalues in terms of the mixing matrix elements as [21]

[mathematical expression not reproducible], (6)

where [mathematical expression not reproducible] is a phase factor. Similarly, in case of inverse hybrid texture structure (1TEC) of [M.sub.v], we can express the ratios of mass eigenvalues as [26, 27] follows:

[mathematical expression not reproducible], (7)

where

[A.sub.i] = ([U.sub.pj][U.sub.qj][U.sub.rk][U.sub.sk] - [U.sub.tj][U.sub.uj][U.sub.vk][U.sub.wk]) + (j [left right arrow] k),

[mathematical expression not reproducible], (8)

with (i, j, k) a cyclic permutation of (1, 2, 3) and [mathematical expression not reproducible], (7) is phase factor.

Using the above expressions, we can obtain the magnitude of neutrino mass ratios, [alpha] [equivalent to] [absolute value of [[lambda].sub.1]/[absolute value of [[lambda].sub.3]] and [beta] [equivalent to] [absolute value of [[lambda].sub.2]/[absolute value of [[lambda].sub.3]] in each texture structure, and the Majorana phases ([rho], [sigma]) can be given as [rho] = (1/2) arg([[lambda].sub.1]/[[lambda].sub.3]) and [sigma] = (1/2) arg([[lambda].sub.2]/[[lambda].sub.3]).

The solar and atmospheric mass squared differences ([delta][m.sup.2], [DELTA][m.sup.2]), where [delta][m.sup.2] corresponds to solar mass-squared difference and [DELTA][m.sup.2] corresponds to atmospheric mass-squared difference, can be defined as [20]

[delta][m.sup.2] = ([m.sup.2.sub.2] - [m.sup.2.sub.1]), [DELTA][m.sup.2] = [m.sup.2.sub.3] - [1/2] ([m.sup.2.sub.1] + [m.sup.2.sub.2]) (9)

The experimentally determined solar and atmospheric neutrino mass-squared differences can be related to neutrino mass ratios ([alpha], [beta]) as

[R.sub.v] [equivalent to] [delta][m.sup.2]/[absolute value of ([DELTA][m.sup.2])] = 2([[beta].sup.2] - [[alpha].sup.2]) / [absolute value of (2 - ([[beta].sup.2] - [[alpha].sup.2])], (10)

and the three neutrino masses can be determined in terms of [alpha], [beta] as

[mathematical expression not reproducible], (11)

Among the sixty logically possible cases of 1TEE or 1TEC texture structures, there are certain pair, which exhibit similar phenomenological implications and are related via permutation symmetry [21, 26, 27]. This corresponds to permutation of the 2-3 rows and 2-3 columns of [M.sub.v]. The corresponding permutation matrix can be given by

[mathematical expression not reproducible]. (12)

With the help of permutation symmetry, one obtains the following relations among the neutrino oscillation parameters:

[mathematical expression not reproducible], (13)

where X and Y denote the cases related to 2-3 permutation. The following pair among sixty cases are related via permutation symmetry:

[mathematical expression not reproducible], (14).

Clearly we are left with only thirty-two independent cases. It is worthwhile to mention that cases [A.sub.1], [A.sub.5], [E.sub.5], and [E.sub.10] are invariant under the permutations of 2 and 3 rows and columns.

3. Numerical Analysis

The experimental constraints on neutrino parameters at 3a confidence levels (CL) are given in Table 1. The classification of sixty phenomenologically possible cases of 1TEE and 1TEC is done in the nomenclature, given by Wang et al. in [26, 27]. All the sixty cases are divided into six categories A, B, C, D, and E (Table 2). In [26, 27], it is found that the phenomenological results of cases belonging to 1TEC (or 1TEE) are almost similar to each other due to permutation symmetry. For the purpose of calculation, we have used the latest experimental data on neutrino mixing angles ([[theta].sub.12], [[theta].sub.23], [[theta].sub.13], [delta][m.sup.2]) and mass squared differences ([DELTA][m.sup.2], [delta]) at 3[sigma] CL [5, 6].

3.1. Near Maximal Atmospheric Mixing for 1TEE and 1TEC Texture Structures. As a first step of the analysis, all the sixty cases of 1TEE and 1TEC have been investigated in the limit of large [[absolute value of M].sub.ee]. For the analysis, we have incorporated the assumptions of [28,29], wherein authors have considered the

lower bound on [[absolute value of M].sub.ee] to be large (i.e., [[absolute value of M].sub.ee] > 0.08 eV). The upper bound on [[absolute value of M].sub.ee] is chosen to be more conservative; that is, [[absolute value of M].sub.ee] < 0.5 eV at 3[sigma] CL [10]. The input parameters ([[theta].sub.12], [[theta].sub.23], [[theta].sub.13], [delta][m.sup.2], [DELTA][m.sup.2] [delta]) are generated by the method of random number generation. The three neutrino mixing angles and Dirac-type CP-violating phase [delta] are varied between 0[degrees] to 90[degrees] and 0[degrees] to 360[degrees], respectively. However, the mass-squared differences ([delta][m.sup.2], [DELTA][m.sup.2]) are varied randomly within their 3a experimental range [5, 6]. For the numerical analysis, we follow the same procedure as discussed in [20]. The main results and discussion are summarized as follows.

In Figures 1, 2, 3, 4, 5, and 6, it is explicitly shown that the octant of [[theta].sub.23] is well restricted for [B.sub.2], [C.sub.7], [D.sub.3], [F.sub.4] of 1TEE and 1TEC texture structures, respectively. However, for the remaining cases, the value of [[theta].sub.23] is unconstrained like other oscillation parameters. Apart from restricting the octant of [[theta].sub.23], the analysis also ensures the quasi-degenerate mass ordering for these cases similar to the observation of [28-30]. From Figures 1(a), 1(b), 3(a), and 3(b), it is clear that, for increasing value of [[absolute value of M].sub.ee], atmospheric mixing angle [[theta].sub.23] approaches to maximal value for the structure [B.sub.2] of 1TEE and 1TEC for both normal ordering (NO) and inverted ordering (IO). In Figures 2 and 4, it is explicitly shown that for cases [B.sub.2] and [C.sub.7] the quadrant of [[theta].sub.23] is already decided without the experimental input of the mixing angles. For 1TEE, we have [[theta].sub.23] < 45[degrees] for NO and [[theta].sub.23] > 45[degrees] for IO, whereas for 1TEC, [[theta].sub.23] > 45[degrees] for NO, while [[theta].sub.23] < 45[degrees] for IO (Figures 2(a), 2(b), 4(a), and 4(b)). Clearly the correlation plots of case [B.sub.2] are indistinguishable for 1TEE and 1TEC, if neutrino mass ordering is not considered as also pointed out earlier. Similar conclusion can be drawn for structure [C.sub.7] since both are related through 2-3 exchange symmetry (Figures 2(c), 2(d), 4(c), and 4(d)). Apart from the prediction of near maximality of [[theta].sub.23], cases [B.sub.2] and [C.sub.7] also predict [delta] [equivalent] 90[degrees], 270[degrees] for 1TEE and 1TEC, respectively, if experimental range of mixing angles is considered as in Table 2. Figures 2(a) and 2(c) for NO and Figures 2(b) and 2(d) for IO depict the 2-3 interchange symmetry between cases [B.sub.2] and [C.sub.7] for 1TEE. Similar phenomenological observation is shown for 1TEC in Figures 4(a), 4(c), 4(b), and 4(d), respectively.

Similarly, cases [D.sub.3] and [F.sub.4] of 1TEE also predict near maximal atmospheric mixing angle ([[theta].sub.23]) for IO (Figures 5(a) and 5(b)). Interestingly the parameter space of reactor mixing angle [[theta].sub.13] is found to be constrained between 0[degrees] and 35[degrees] (Figures 5(c) and 5(d)). In Figures 5(c) and 5(d), it is clear that, for the allowed experimental range of [[theta].sub.13] (8.5[degrees]-9.8[degrees]), [[theta].sub.23] inches closer to 45[degrees]. Similar predictions have been noted for cases [D.sub.3] and [F.sub.4] of 1TEC, however, for normal mass ordering (NO) (Figures 6(a), 6(b), 6(c), and 6(d)).

3.2. Comparing the Results for 1TEE and 1TEC Texture Structures. In this subsection, we compare the results of all the viable structures of 1TEE and 1TEC in neutrino mass matrix. It is worthwhile to mention that the present refinements of the experimental data do not limit the number of viable cases in 1TEE and 1TEC textures respectively. The number of viable cases obtained is the same as predicted in [21, 26, 27] for 1TEE and 1TEC, respectively. For executing the analysis, we vary the allowed ranges of three neutrino mixing angles ([[theta].sub.12], [[theta].sub.23], [[theta].sub.13]) and mass squared differences ([delta][m.sup.2], [DELTA][m.sup.2]) within their 3[sigma] confidence level. To facilitate the comparison, we have encapsulated the the predictions regarding three CP violating phases ([rho], [alpha], [delta]) and neutrino masses [m.sub.1,2,3] for all the allowed texture structures of 1TEE and 1TEC, respectively (Tables 3, 4, 5, and 6).

Category A. In Category A, there are 10 possible cases out of which only four ([A.sub.1,4,5,6]) are allowed for 1TEE at 3[sigma] CL, and in addition inverted mass ordering (IO) is ruled out for all these cases. On the other hand, only three ([A.sub.1,4,6]) are allowed for 1TEC with current oscillation data, while normal mass ordering (NO) is ruled out for these cases. For 1TEE, [rho], [alpha], [delta] remain unconstrained; however, for 1TEC, only [delta] remains unconstrained, while Majorana phases ([rho], [sigma]) are restricted near 0[degrees] pertaining to viable cases. From Table 3, it is clear that lower bound on lowest neutrino mass ([m.sub.1] (NO) or [m.sub.3] (IO)) is nearly equal or less than 1 meV for 1TEE and 1TEC.

Category B (C). In Category B, all the ten possible cases are allowed for both 1TEE and 1TEC, respectively, at 3[sigma] CL; however, cases [B.sub.1,6,7] allow only NO for 1TEE, while the same allow only IO for 1TEC (Table 4). Cases [B.sub.2,3,4,5,8,9,10] allow both NO and IO for 1TEE and 1TEC, respectively. As mentioned in [26,27], cases of Category B are related to the cases belonging to Category C through permutation symmetry; therefore, we can obtain the results for Category C from B. We find that cases [C.sub.1,2,3] allow only NO for 1TEE, while the same allow IO for 1TEC.

Textures [B.sub.1] (NO), [B.sub.3] (NO, IO), [B.sub.5] (NO, IO), [B.sub.6] (NO), [B.sub.7] (NO), [B.sub.8] (NO), [B.sub.10] (NO), [C.sub.1] (NO), [C.sub.2] (NO), [C.sub.3] (NO), [C.sub.4] (NO, IO), [C.sub.6] (NO, IO), [C.sub.8] (NO), and [C.sup.10] (NO) held nearly no constraint on Dirac CP violating phase ([delta]) for 1TEE and 1TEC, respectively, but with opposite neutrino mass ordering (Table 4). Only cases [B.sub.2] (NO), [B.sub.4] (IO), [C.sub.7] (NO), and [C.sub.5] (IO) for 1TEE and [B.sub.2] (IO), [B.sub.4] (NO), [C.sub.7] (IO), and [C.sub.5] (NO) for 1TEC show significant reduction in the parameter space of [delta]. It is found that [delta] is restricted near 90[degrees] and 270[degrees] for 1TEE and 1TEC, respectively (Table 4). These predictions are significant considering the latest hint on [delta] near 270[degrees] [5, 6]. Therefore all the above cases discussed are almost indistinguishable for 1TEE and 1TEC, if neutrino mass ordering is not considered.

Category D(F). All the ten possible cases belonging to Category D are acceptable with neutrino oscillation data at 3a CL for 1TEE and 1TEC, respectively (Table 5). However cases [D.sub.1,2,4,5,6,7,9] favor both NO and IO for 1TEE and 1TEC, while [D.sub.3], [D.sub.8], and [D.sub.10] are acceptable only for IO in case of 1TEE; however, the same cases allowed NO in case of 1TEC. Similarly, the results for cases belonging to Category F can be derived from Category D.

Cases [D.sub.1] (IO), [D.sub.2] (IO), [D.sub.3] (IO), [D.sub.4] (IO), [D.sub.5] (NO, IO), [D.sub.6] (IO), [D.sub.7] (NO, IO), [D.sub.8] (IO), [D.sub.9] (NO, IO), [D.sub.10] (IO), [F.sub.1] (IO), [F.sub.2] (IO), [F.sub.3] (IO), [F.sub.4] (IO), [F.sub.5] (NO, IO), [F.sub.6] (IO), [F.sub.7] (NO, IO), [F.sub.8] (IO), [F.sub.9] (NO, IO), and [F.sub.10] (IO) predict literally no constraints on [delta] for 1TEE. These cases give identical predictions for 1TEC, but for opposite mass ordering. For cases [D.sub.1] (NO), [D.sub.2] (NO), [D.sub.4] (NO), [D.sub.6] (NO), [F.sub.1] (NO), [F.sub.2] (NO), [F.sub.4] (NO), and [F.sub.6] (NO), the parameter space of [delta] is found to be well constrained for 1TEE. These cases give similar predictions regarding the parameter space of [delta] for 1TEC, but for IO.

Category E. In Category E, all the ten possible cases are allowed for 1TEE at 3[sigma] CL, while only nine other than [E.sub.5] are acceptable in case of 1TEC (Table 6). Cases [E.sub.1,2,3,4,6,7,8,9] allow only inverted mass ordering (IO) for 1TEE, while the same textures allow only normal mass ordering (NO) for 1TEC. Cases [E.sub.5] and [E.sub.10] allow both NO and IO for 1TEE; however [E.sub.5] is ruled out for both NO and IO for 1TEC at 3[sigma] CL. Similar to cases belonging to Category D, [E.sub.1,2,3,4,5,6,7,8,9] (IO) cover full range of [rho], [alpha], [delta] for 1TEE, whereas the same cases (except [E.sub.5]) give identical predictions for 1TEC, but for NO. For [E.sub.5] (NO) and [E.sub.10] (NO), phases [rho], [alpha], [delta] are somewhat restricted at 3[sigma] CL for 1TEE, while only for [E.sub.10] (IO), the parameter space of [rho], [alpha], [sigma] seems to be restricted for 1TEC (Table 6).

To summarize our discussion, we have investigated all the viable cases of 1TEE and 1TEC texture structures in the limit of large effective neutrino mass [[absolute value of M].sub.ee]. It is found that only four cases are able to produce near maximal atmospheric mixing for 1TEE and 1TEC, respectively. However, the predictions remain true irrespective of the experimental data on solar and reactor mixing angle. The observation also hints towards the indistinguishable feature of 1TEE and 1TEC texture structures, but for opposite mass ordering. In order to depict the indistinguishability, we have carried out a comparative study of 1TEE and 1TEC texture structures using the current experimental data at 3[sigma] CL. From our discussion we find that most of the cases belonging to 1TEE and 1TEC are almost indistinguishable as far as the neutrino oscillation parameters are concerned, but with opposite neutrino mass ordering. The indistinguishable nature of 1TEE and 1TEC is more prominent for quasi-degenerate mass ordering. For the cases where lower bound on lowest neutrino mass is very small (<1meV), there is noticeable deviation in the predictions for 1TEE and 1TEC (Tables 3, 4, 5, and 6). This point is also discussed by Liao et. al. in [31]. In addition, the parameter space of [delta] for most of the cases belonging to 1TEE and 1TEC remains unrestricted, while only eight cases show maximal restriction for [delta]. Since no presently feasible experiment has been able to determine the neutrino mass ordering, therefore, we cannot distinguish 1TEE and 1TEC structures using the present oscillation data. However, the currently running and forthcoming neutrino experiments aimed at distinguishing the mass ordering of neutrinos will test our phenomenological results. Also the ongoing and future neutrinoless double beta decay experiments are capable of measuring [[absolute value of M].sub.ee] term, which would, in turn, either confirm or rule out our assumption of large [[absolute value of M].sub.ee].

https://doi.org/10.1155/2018/5806743

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The author would like to thank the Director, National Institute of Technology Kurukshetra, for providing necessary facilities to work.

References

[1] Y. Abe, C. Aberle, T. Akiriet al., "Indication of Reactor [[bar.v].sub.e] Disappearance in the Double Chooz Experiment," Physical Review Letters, vol. 108, no. 13, Article ID 131801, 2012.

[2] F. P. An, J. Z. Bai, A. B. Balantekin et al., "Observation of Electron-Antineutrino Disappearance at Daya Bay," Physical Review Letters, vol. 108, no. 17, Article ID 171803, 2012.

[3] J. K. Ahn, S. Chebotaryov, J. H. Choi et al., "Observation of reactor electron antineutrinos disappearance in the RENO experiment," Physical Review Letters, vol. 108, no. 19, Article ID 191802, 2012.

[4] K. Abe, J. Adam, H. Aihara et al., "Observation of Electron Neutrino Appearance in a Muon Neutrino Beam," Physical Review Letters, vol. 112, no. 6, Article ID 061802, 8 pages, 2014.

[5] I. Esteban, M. C. Gonzalez-Garcia, and M. Maltoni, "Updated fit to three neutrino mixing: exploring the accelerator-reactor complementarity," Journal of High Energy Physics, vol. 87, 2017.

[6] P. F. de Salas, D. V. Forero, C. A. Ternes, M. Tortola, and J. W. F. Valle, "Status of neutrino oscillations 2017," https://arxiv.org/ abs/1708.01186.

[7] A. Gando, Y. Gando, T. Hachiya et al., "Search for Majorana Neutrinos Near the Inverted Mass Hierarchy Region with KamLAND-Zen," Physical Review Letters, vol. 117, no. 8, Article ID 082503, 2016.

[8] J. B. Albert, D. J. Auty, and P. S. Barbeau, "Search for Majorana neutrinos with the first two years of EXO-200 data," Nature, vol. 510, pp. 229-234, 2014.

[9] S. Dell'Oro, S. Marcocci, M. Viel, and F. Vissani, "Neutrinoless Double Beta Decay: 2015 Review," Advances in High Energy Physics, vol. 2016, Article ID 2162659, 37 pages, 2016.

[10] W. Rodejohann, "Neutrino-less double beta decay and particlE," International Journal of Modern Physics E, vol. 20, article 1833, 2011.

[11] F. T. Avignone III, S. R. Elliott, and J. Engel, "Double beta decay, Majorana neutrinos, and neutrino mass," Reviews of Modern Physics, vol. 80, article 481, 2008.

[12] J. J. Gomez-Cadenas, J. Martin-Albo, M. Mezzetto, F. Monrabal, and M. Sorel, "The search for neutrinoless double beta decay," La Rivista del Nuovo Cimento, vol. 29, no. 2, 2012.

[13] S. M. Bilenky and C. Giunti, "Neutrinoless double-beta decay: a brief review," Modern Physics Letters A, vol. 27, no. 13, Article ID 1230015, 22 pages, 2012.

[14] P. H. Frampton, S. L. Glashow, and D. Marfatia, "Zeroes of the neutrino mass matrix," Physics Letters B, vol. 536, no. 1-2, pp. 79-82, 2002.

[15] Z.-Z. Xing, "Texture zeros and majorana phases of the neutrino mass matrix," Physics Letters B, vol. 530, no. 1-4, pp. 159-166, 2002.

[16] J. Liao, D. Marfatia, and K. Whisnant, "Texture and cofactor zeros of the neutrino mass matrix," Journal of High Energy Physics, vol. 13, 2014.

[17] D. Meloni, A. Meroni, and E. Peinado, "Two-zero Majorana textures in the light of the Planck results," Physical Review D, vol. 89, no. 5, Article ID 053009, 2014.

[18] P. O. Ludl and W. Grimus, "A complete survey of texture zeros in the lepton mass matrices," Journal ofHigh Energy Physics, vol. 9, 2014.

[19] S. Dev, R. R. Gautam, and L. Singh, "Neutrino mass matrices with two equalities between the elements or cofactors," Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 87, Article ID 073011, 2013.

[20] J.-Y. Liu and S. Zhou, "Hybrid textures of Majorana neutrino mass matrix and current experimental tests," Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 87, no. 9-1, Article ID 093010, 2013.

[21] S. Kaneko, H. Sawanaka, and M. Tanimoto, Journal of High Energy Physics, vol. 73, Article ID 0508, 2005.

[22] L. Lavoura, "Zeros of the inverted neutrino mass matrix," Physics Letters B, vol. 609, no. 3-4, pp. 317-322, 2005.

[23] E. I. Lashin and N. Chamoun, "Zero minors of the neutrino mass matrix," Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 78, no. 7, Article ID 073002, 2008.

[24] E. I. Lashin and N. Chamoun, "One vanishing minor in the neutrino mass matrix," Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 80, no. 9, Article ID 093004, 2009.

[25] J. Liao, D. Marfatia, and K. Whisnant, "Texture and cofactor zeros of the neutrino mass matrix," Journal of High Energy Physics, vol. 13, Article ID 1409, 2014.

[26] W. Wang, "Neutrino mass textures with one vanishing minor and two equal cofactors," The European Physical Journal C, vol. 73, article 2551, 2013.

[27] S. Dev, R. R. Gautam, and L. Singh, "Hybrid textures of the right-handed Majorana neutrino mass matrix," Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 88, Article ID 033008, 2013.

[28] S. Dev, R. R. Gautam, L. Singh, and M. Gupta, "Hybrid textures of the right-handed Majorana neutrino mass matrix," Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 90, Article ID 013021, 2014.

[29] S. Dev, S. Gupta, R. R. Gautam, and L. Singh, "Near maximal atmospheric mixing in neutrino mass matrices with two vanishing minors," Physics Letters B, vol. 706, no. 2-3, pp. 168-176, 2011.

[30] W. Grimus and P. O. Ludl, "Maximal atmospheric neutrino mixing from texture zeros and quasi-degenerate neutrino masses," Physics Letters B, vol. 700, no. 5, pp. 356-361, 2011.

[31] J. Liao, D. Marfatia, and K. Whisnant, "Dual models of the neutrino mass spectrum," Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 89, no. 1, Article ID 013009, 2014.

[32] J. Liao, D. Marfatia, and K. Whisnant, "Seesaw mechanism with four texture zeros in the neutrino Yukawa matrix," Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 87, no. 7, Article ID 073013, 2013.

[33] J. Liao, D. Marfatia, and K. Whisnant, "One diagonal texture or cofactor zero of the neutrino mass matrix," Physical Review D: covering particles, fields, gravitation, and cosmology, vol. 88, no. 3, Article ID 033011, 2013.

Madan Singh (iD)

Department of Physics, National Institute of Technology, Kurukshetra, Haryana 136119, India

Correspondence should be addressed to Madan Singh; singhmadan179@gmail.com

Received 22 December 2017; Accepted 5 April 2018; Published 14 May 2018

Academic Editor: Jose W. F. Valle

Caption: Figure 1: Correlation plots for textures [B.sub.2] ((a) NO and (b) IO) and [C.sub.7] for ((c) NO and (d) IO) at 3[sigma] CL for 1TEE. The symbols have their usual meaning. The horizontal line indicates the upper limit on effective neutrino mass term [[absolute value of M].sub.ee] (i.e., [[absolute value of M].sub.ee] < 0.165 eV) at 90% CL, given in KamLAND-Zen experiment [7].

Caption: Figure 2: Correlation plots for textures [B.sub.2] ((a) NO and (b) IO) and [C.sub.7] for ((c) NO and (d) IO) at 3[sigma] CL for 1TEE. The symbols have their usual meaning. The horizontal line indicates the upper limit on reactor mixing angle [[theta].sub.13] < 8.9[degrees], as given in Table 1.

Caption: Figure 3: Correlation plots for textures [B.sub.2] ((a) NO and (b) IO) and [C.sub.7] for ((c) NO and (d) IO) at 3[sigma] CL for 1TEC. The symbols have their usual meaning. The horizontal line indicates the upper limit on effective neutrino mass term [[absolute value of M].sub.ee] (i.e., [[absolute value of M].sub.ee] < 0.165 eV) at 90% (<2[sigma])CL, given in KamLAND-Zen experiment [7].

Caption: Figure 4: Correlation plots for textures [B.sub.2] ((a) NO and (b) IO) and [C.sub.7] for ((c) NO and (d) IO) at 3[sigma] CL for 1TEC. The symbols have their usual meaning. The horizontal line indicates the upper limit on reactor mixing angle [[theta].sub.13] < 8.9[degrees], as given in Table 1.

Caption: Figure 5: Correlation plots for textures [D.sub.3] ((a), (c)) and [F.sub.4] ((b), (d)) with IO for 1TEE at 3[sigma] CL. The symbols have their usual meaning. In (a) and (b), colored horizontal line indicates the upper limit on effective neutrino mass term [[absolute value of M].sub.ee] (i.e., [[absolute value of M].sub.ee] < 0.165 eV) at 90 (<2[sigma])% CL, given in KamLAND-Zen experiment [7]. In (c) and (d), we have shown the upper limit on reactor mixing angle [[theta].sub.13].

Caption: Figure 6: Correlation plots for textures [D.sub.3] ((a), (c)) and [F.sub.4] ((b), (d)) with NO for 1TEC at 3[sigma] CL. The symbols have their usual meaning. In (a) and (b), colored horizontal line indicates the upper limit on effective neutrino mass term [[absolute value of M].sub.ee] (i.e., [[absolute value of M].sub.ee] < 0.165 eV) at 90 (<2[sigma])% CL, given in KamLAND experiment [7]. In (c) and (d), we have shown the upper limit on reactor mixing angle [[theta].sub.13].

Caption: Table 2: Sixty phenomenologically possible hybrid texture structures of [M.sub.v] at 3[sigma] C.L where the triangles "[DELTA]" denote equal and nonzero elements (or cofactors) and "0" denotes the vanishing element (or minor). "x" denotes the nonzero elements or cofactor.
Table 1: Current neutrino oscillation parameters from global fits at
3[sigma] confidence level [5, 6]. NO (IO) refers to normal (inverted)
neutrino mass ordering.

Parameter                   Best fit                3[sigma]

[delta][m.sup.2]              7.50                  703-8.09
[[10.sup.-5]
e[V.sup.2]]

[absolute value of            2.52                 2.407-2.643
([DELTA]
[m.sup.2.sub.31])]
[[10.sup.-3]
e[V.sup.2]] (NO)

[absolute value of            2.52                  2.39-2.63
([DELTA]
[m.sup.2.sub.31])]
[[10.sup.-3]
e[V.sup.2]] (IO)

[[theta].sub.12]         33.56[degrees]   31.3[degrees]-35.99[degrees]

[[theta].sub.23] (NO)    41.6[degrees]     38.4[degrees]-52.8[degrees]

[[theta].sub.23] (IO)    50.0[degrees]     38.8[degrees]-53.1[degrees]

[[theta].sub.13] (NO)    8.46[degrees]     7.99[degrees]-8.90[degrees]

[[theta].sub.13] (IO)    8.49[degrees]     8.03[degrees]-8.93[degrees]

[delta] (NO)              261[degrees]       0[degrees]-360[degrees]

[delta] (IO)              277[degrees]      145[degrees]-391[degrees]

Table 3: The allowed ranges of Dirac CP-violating phase [delta], the
Majorana phases [rho], [sigma], and three neutrino masses [m.sub.1],
[m.sub.2], [m.sub.3] for the experimentally allowed cases of Category
A. Masses are in eV. "x" denotes the nonviability of case for a
particular mass ordering.

Cases                                        1TEE

                                           NO                     IO

                           [rho] = -90[degrees] - 90[degrees]     x

                          [sigma] = -90[degrees] - 90[degrees]    x

[A.sub.1]                  [delta] = 0[degrees] - 360[degrees]    x
                               [m.sub.1] = 0.00147-0.0106         x
                               [m.sub.2] = 0.00849-0.0139         x
                                [m.sub.3] = 0.0437-0.0551         x

[A.sub.2] ([A.sub.8])                       x                     x

[A.sub.3] ([A.sub.7])                       x                     x

                           [rho] = -90[degrees] - 90[degrees]     x

                          [sigma] = -90[degrees] - 90[degrees]    x

[A.sub.4] ([A.sub.6])      [delta] = 0[degrees] - 360[degrees]    x
                               [m.sub.1] = 0.00148-0.0106         x
                               [m.sub.2] = 0.00850-0.0139         x
                                [m.sub.3] = 0.0437-0.0551         x

                           [rho] = -90[degrees] - 90[degrees]     x
                          [sigma] = -90[degrees] - 90[degrees]    x
[A.sub.5] ([A.sub.5])      [delta] = 0[degrees] - 360[degrees]    x
                               [m.sub.1] = 0.00163-0.0105         x
                               [m.sub.2] = 0.00854-0.0137         x
                                [m.sub.3] = 0.0438-0.0545         x
[A.sub.9] ([A.sub.10])                      x                     x

Cases                                       1TEC

                          NO                     IO

                          x          [rho] = -0.0277[degrees] -
                                   -0.0220[degrees] [direct sum]
                                  0.0214[degrees] - 0.0274[degrees]
                          x          [sigma] = -0.0273[degrees]-
                                           0.0271[degrees]
[A.sub.1]                 x       [delta] = 0[degrees]-360[degrees]
                          x           [m.sub.1] = 0.0430-0.0534
                          x           [m.sub.2] = 0.0439-0.0541
                          x         [m.sub.3] = 0.000904-0.00504

[A.sub.2] ([A.sub.8])     x                       x

[A.sub.3] ([A.sub.7])     x                       x

                          x          [rho] = -0.0275[degrees] -
                                   -0.0222[degrees] [direct sum]
                                  0.0215[degrees] - 0.0269[degrees]
                          x         [sigma] = -0.0250[degrees] -
                                           0.0268[degrees]
[A.sub.4] ([A.sub.6])     x      [delta] = 0[degrees] - 360[degrees]
                          x           [m.sub.1] = 0.0431-0.0534
                          x           [m.sub.2] = 0.0439-0.0540
                          x         [m.sub.3] = 0.000950-0.00504

                          x                       x
                          x                       x
[A.sub.5] ([A.sub.5])     x                       x
                          x                       x
                          x                       x
                          x                       x
[A.sub.9] ([A.sub.10])    x                       x

Table 4: The allowed ranges of Dirac CP-violating phase [delta], the
Majorana phases [rho], [alpha], and three neutrino masses [m.sub.1],
[m.sub.2], [m.sub.3] for the experimentally allowed cases of Category
B (C). Masses are in eV. "x" denotes the nonviability of case for a
particular mass ordering.

                                              1TEE
Cases
                                               NO

                                [rho] = -90[degrees]-90[degrees]
                               [sigma] = -90[degrees]-90[degrees]
[B.sub.1] ([C.sub.1])          [delta] = 0[degrees]-360[degrees]
                                   [m.sub.1] = 0.00550-0.0298
                                   [m.sub.2] = 0.0101-0.0310
                                   [m.sub.3] = 0.0443-0.0605

                              [rho] = -2.76[degrees]-2.81[degrees]
                            [sigma] = -10.48[degrees]-10.22[degrees]
[B.sub.2] ([C.sub.7])       [delta] = 77.3[degrees]-94.27[degrees]
                            [direct sum] 266.3[degrees]-284[degrees]
                                    [m.sub.1] = 0.0235-0.314
                                   [m.sub.2] = 0.0242-0.0310
                                   [m.sub.3] = 0.0485-0.0315

                               [rho] = -90[degrees]--5.9[degrees]
                              [direct sum] 5.9[degrees]-90[degrees]
                            [sigma]' = -90[degrees] - -5.9[degrees]
                             [direct sum] 5.9[degrees]-90[degrees]
[B.sub.3] ([C.sub.6])         [delta] = 7.47[degrees]-356[degrees]

                                    [m.sub.1] = 0.0153-0.331
                                    [m.sub.2] = 0.0161-0.331
                                   [m.sub.3] = 0.0445-0.0335

                             [rho] = -19.07[degrees]-19.75[degrees]
                               [sigma] = -90[degrees]-90[degrees]
[B.sub.4] ([C.sub.5])          [delta] = 0[degrees]-100[degrees]
                            [direct sum] 260.2[degrees]-360[degrees]
                                   [m.sub.1] = 0.00681-0.318
                                   [m.sub.2] = 0.00969-0.0317
                                   [m.sub.3] = 0.0421-0.0321

                             [rho] = -20.83[degrees]-20.79[degrees]

                              [sigma] = -36.3[degrees]-36[degrees]

[B.sub.5] ([C.sub.6])         [delta] = 0[degrees]-177.5[degrees]
                             [direct sum] 182[degrees]-360[degrees]
                                   [m.sub.1] = 0.00965-0.330
                                   [m.sub.2] = 0.0113-0.0330
                                   [m.sub.3] = 0.0421-0.0332

                                [rho] = -90[degrees]-90[degrees]
                               [sigma] = -90[degrees]-90[degrees]
[B.sub.6] ([C.sub.3])          [delta] = 0[degrees]-360[degrees]
                                    [m.sub.1] = 0.0147-0.324
                                   [m.sub.2] = 0.00161-0.324
                                    [m.sub.3] = 0.0445-0.324

                             [rho] = -26.25[degrees]-26.43[degrees]
                            [sigma] = -50.28[degrees]-50.2[degrees]
[B.sub.7] ([C.sub.2])         [delta] = 0[degrees]-177.5[degrees]
                             [direct sum] 182[degrees]-360[degrees]
                                   [m.sub.1] = 0.00883-0.331
                                   [m.sub.2] = 0.00121-0.329
                                    [m.sub.3] = 0.0421-0.334

                                [rho] = -90[degrees]-90[degrees]
                               [sigma] = -90[degrees]-90[degrees]
[B.sub.8] ([C.sub.10])        [delta] = 0[degrees]-166.5[degrees]
                            [direct sum] 196.5[degrees]-360[degrees]
                                   [m.sub.1] = 0.00127-0.279
                                   [m.sub.2] = 0.00691-0.277
                                    [m.sub.3] = 0.0409-0.280

                                [rho] = -90[degrees]-90[degrees]
                               [sigma] = -90[degrees]-90[degrees]
[B.sub.9] ([C.sub.9])         [delta] = 5.55[degrees]-354[degrees]

                                   [m.sub.1] = 0.00338-0.279
                                   [m.sub.2] = 0.00831-0.277
                                    [m.sub.3] = 0.0423-0.282

                                [rho] = -90[degrees]-90[degrees]
                               [sigma] = -90[degrees]-90[degrees]
[B.sub.10] ([C.sub.8])         [delta] = 0[degrees]-360[degrees]

                                   [m.sub.1] = 0.00163-0.294
                                   [m.sub.2] = 0.00691-0.293
                                    [m.sub.3] = 0.0409-0.299

                                              1TEE
Cases
                                               IO

                                               X
                                               X
[B.sub.1] ([C.sub.1])                          X
                                               X
                                               X
                                               X

                              [rho] = -7.74[degrees]-7.75[degrees]
                             [sigma] = -8.84[degrees]-9.22[degrees]
[B.sub.2] ([C.sub.7])       [delta] = 84.01[degrees]-98.7[degrees]
                          [direct sum] 262.19[degrees]-276.6[degrees]
                                    [m.sub.1] = 0.0453-0.277
                                    [m.sub.2] = 0.0465-0.274
                                    [m.sub.3] = 0.0117-0.271

                                [rho] = -90[degrees]-90[degrees]
                               [sigma] = -90[degrees]-90[degrees]

[B.sub.3] ([C.sub.6])         [delta] = 8.3[degrees]-173[degrees]
                             [direct sum] 187[degrees]-351[degrees]
                                    [m.sub.1] = 0.0459-0.335
                                    [m.sub.2] = 0.0461-0.334
                                    [m.sub.3] = 0.0136-0.330

                              [rho] = -4.80[degrees]-4.66[degrees]
                             [sigma] = -2.83[degrees]-3.12[degrees]
[B.sub.4] ([C.sub.5])        [delta] = 84.68[degrees]-94.7[degrees]
                          [direct sum] 265.08[degrees]-275.5[degrees]
                                    [m.sub.1] = 0.0488-0.314
                                    [m.sub.2] = 0.0486-0.312
                                    [m.sub.3] = 0.0202-0.310

                             [rho] = -90[degrees]- -11.3[degrees]
                             [direct sum] 11.3[degrees]-90[degrees]
                           [sigma]- = -90[degrees] - -14.3[degrees]
                             [direct sum] 14.8[degrees]-90[degrees]
[B.sub.5] ([C.sub.6])          [delta] = 0[degrees]-168[degrees]
                            [direct sum] 192.5[degrees]-360[degrees]
                                    [m.sub.1] = 0.0761-0.330
                                    [m.sub.2] = 0.0762-0.330
                                    [m.sub.3] = 0.0603-0.328

                                               X
                                               X
[B.sub.6] ([C.sub.3])                          X
                                               X
                                               X
                                               X

                                               X
                                               X
[B.sub.7] ([C.sub.2])                          X

                                               X
                                               X
                                               X

                                [rho] = -90[degrees]-90[degrees]
                               [sigma] = -90[degrees]-90[degrees]
[B.sub.8] ([C.sub.10])       [delta] = 45.05[degrees]-135[degrees]
                           [direct sum] 224.5[degrees]-314.6[degrees]
                                    [m.sub.1] = 0.0418-0.279
                                    [m.sub.2] = 0.0429-0.307
                                   [m.sub.3] = 0.00398-0.305

                                [rho] = -90[degrees]-90[degrees]
                               [sigma] = -90[degrees]-90[degrees]
[B.sub.9] ([C.sub.9])        [delta] = 12[degrees]-171.7[degrees]
                           [direct sum] 189.7[degrees]-352.7[degrees]
                                    [m.sub.1] = 0.0435-0.319
                                    [m.sub.2] = 0.0445-0.319
                                   [m.sub.3] = 0.00398-0.317

                                [rho] = -90[degrees]-90[degrees]
                               [sigma] = -90[degrees]-90[degrees]
[B.sub.10] ([C.sub.8])      [delta] = 44.64[degrees]-136.5[degrees]
                           [direct sum] 224.6[degrees]-314.2[degrees]
                                    [m.sub.1] = 0.0425-0.277
                                    [m.sub.2] = 0.0423-0.275
                                   [m.sub.3] = 0.00380-0.272

                                              1TEC
Cases
                                               NO

                                               X
                                               X
[B.sub.1] ([C.sub.1])                          X
                                               X
                                               X
                                               X

                                [rho] = -90[degrees]-90[degrees]
                               [sigma] = -90[degrees]-90[degrees]
[B.sub.2] ([C.sub.7])       [delta] = 86.22[degrees]-273.3[degrees]

                                   [m.sub.1] = 0.00548-0.267
                                   [m.sub.2] = 0.00806-0.316
                                    [m.sub.3] = 0.0428-0.320

                                [rho] = -90[degrees]-90[degrees]
                               [sigma] = -90[degrees]-90[degrees]

[B.sub.3] ([C.sub.6])        [delta] = 5.16[degrees]-178[degrees]
                           [direct sum] 183.7[degrees]-353.6[degrees]
                                   [m.sub.1] = 0.00504-0.485
                                   [m.sub.2] = 0.00920-0.483
                                    [m.sub.3] = 0.0416-0.480

                             [rho] = -43.13[degrees]-40.12[degrees]
                              [sigma] = -43[degrees]-40.1[degrees]
[B.sub.4] ([C.sub.5])       [delta] = 88.04[degrees]-124.4[degrees]
                           [direct sum] 230.09[degrees]-272[degrees]
                                    [m.sub.1] = 0.0105-0.373
                                    [m.sub.2] = 0.0129-0.369
                                    [m.sub.3] = 0.0435-0.374

                                [rho] = -90[degrees]-90[degrees]

                               [sigma] = -90[degrees]-90[degrees]

[B.sub.5] ([C.sub.6])       [delta] = 9.33[degrees]-172.4[degrees]
                               [direct sum] 187.20-350.4[degrees]
                                    [m.sub.1] = 0.0603-0.498
                                    [m.sub.2] = 0.0608-0.496
                                    [m.sub.3] = 0.0629-0.500

                                               X
                                               X
[B.sub.6] ([C.sub.3])                          X
                                               X
                                               X
                                               X

                                               X
                                               X
[B.sub.7] ([C.sub.2])                          X

                                               X
                                               X
                                               X

                                [rho] = -90[degrees]-90[degrees]
                               [sigma] = -90[degrees]-90[degrees]
[B.sub.8] ([C.sub.10])      [delta] = 45.86[degrees]-315.05[degrees]

                                   [m.sub.1] = 0.00498-0.381
                                   [m.sub.2] = 0.00736-0.382
                                    [m.sub.3] = 0.0407-0.384

                                [rho] = -90[degrees]-90[degrees]
                               [sigma] = -90[degrees]-90[degrees]
[B.sub.9] ([C.sub.9])       [delta] = 8.32[degrees]-175.9[degrees]
                           [direct sum] 186[degrees]-350.47[degrees]
                                   [m.sub.1] = 0.00622-0.487
                                   [m.sub.2] = 0.00681-0.486
                                   [m.sub.3] = 0.00914-0.486

                                [rho] = -90[degrees]-90[degrees]
                               [sigma] = -90[degrees]-90[degrees]
[B.sub.10] ([C.sub.8])      [delta] = 1.92[degrees]-313.04[degrees]

                                   [m.sub.1] = 0.00516-0.348
                                   [m.sub.2] = 0.00806-0.346
                                    [m.sub.3] = 0.0416-0.348

                                              1TEC
Cases
                                               IO

                            [rho] = -0.0277[degrees]-0.0279[degrees]
                           [sigma] = -0.0284[degrees]-0.0284[degrees]
[B.sub.1] ([C.sub.1])          [delta] = 0[degrees]-360[degrees]
                                   [m.sub.1] = 0.0439-0.0597
                                   [m.sub.2] = 0.0447-0.0604
                                  [m.sub.3] = 0.000754-0.0297

                              [rho] = -4.46[degrees]-4.40[degrees]
                             [sigma] = -5.41[degrees]-5.39[degrees]
[B.sub.2] ([C.sub.7])       [delta] = 81.56[degrees]-96.48[degrees]
                           [direct sum] 263.7[degrees]-278.4[degrees]
                                    [m.sub.1] = 0.0507-0.419
                                    [m.sub.2] = 0.0508-0.419
                                    [m.sub.3] = 0.0249-0.414

                                [rho] = -90[degrees]-90[degrees]
                               [sigma] = -90[degrees]-90[degrees]

[B.sub.3] ([C.sub.6])          [delta] = 0[degrees]-360[degrees]

                                    [m.sub.1] = 0.0427-0.505
                                    [m.sub.2] = 0.0445-0.503
                                   [m.sub.3] = 0.00884-0.500

                             [rho] = -16.38[degrees]-18.8[degrees]
                            [sigma] = -41.43[degrees]-40.59[degrees]
[B.sub.4] ([C.sub.5])       [delta] = 5.56[degrees]-13.82[degrees]
                           [direct sum] 255.7[degrees]-307.3[degrees]
                                    [m.sub.1] = 0.0444-0.420
                                    [m.sub.2] = 0.0447-0.416
                                    [m.sub.3] = 0.0103-0.418

                             [rho] = -24.19[degrees]-24.45[degrees]

                             [sigma] = -27.7[degrees]-27.4[degrees]

[B.sub.5] ([C.sub.6])         [delta] = 0[degrees]-179.3[degrees]
                            [direct sum] 180.6[degrees]-360[degrees]
                                    [m.sub.1] = 0.0440-0.498
                                    [m.sub.2] = 0.0447-0.499
                                    [m.sub.3] = 0.0103-0.499

                                [rho] = -90[degrees]-90[degrees]
                               [sigma] = -90[degrees]-90[degrees]
[B.sub.6] ([C.sub.3])          [delta] = 0[degrees]-360[degrees]
                                    [m.sub.1] = 0.0452-0.454
                                    [m.sub.2] = 0.0451-0.451
                                   [m.sub.3] = 0.00161-0.450

                             [rho] = -30.92[degrees]-30.42[degrees]
                             [sigma] = -41.9[degrees]-43.8[degrees]
[B.sub.7] ([C.sub.2])          [delta] = 0[degrees]-360[degrees]

                                    [m.sub.1] = 0.0447-0.498
                                    [m.sub.2] = 0.0451-0.498
                                   [m.sub.3] = 0.00804-0.498

                                [rho] = -90[degrees]-90[degrees]
                               [sigma] = -90[degrees]-90[degrees]
[B.sub.8] ([C.sub.10])         [delta] = 0[degrees]-360[degrees]

                                    [m.sub.1] = 0.0427-0.303
                                    [m.sub.2] = 0.0428-0.302
                                   [m.sub.3] = 0.00342-0.298

                                [rho] = -90[degrees]-90[degrees]
                               [sigma] = -90[degrees]-90[degrees]
[B.sub.9] ([C.sub.9])          [delta] = 0[degrees]-360[degrees]

                                    [m.sub.1] = 0.0447-0.486
                                    [m.sub.2] = 0.0451-0.487
                                   [m.sub.3] = 0.00458-0.480

                                [rho] = -90[degrees]-90[degrees]
                               [sigma] = -90[degrees]-90[degrees]
[B.sub.10] ([C.sub.8])         [delta] = 0[degrees]-360[degrees]

                                    [m.sub.1] = 0.0427-0.365
                                    [m.sub.2] = 0.0453-0.365
                                   [m.sub.3] = 0.00879-0.391

Table 5: The allowed ranges of Dirac CP-violating phase [delta], the
Majorana phases [rho], [alpha], and three neutrino masses [m.sub.1],
[m.sub.2], [m.sub.3] for the experimentally allowed cases of Category
D (F). Masses are in eV. "x" denotes the nonviability of case for a
particular mass ordering.

                                               1TEE
Cases
                                                NO

                                 [rho] = -90[degrees]-90[degrees]
                                [sigma] = -90[degrees]-90[degrees]
[D.sub.1] ([F.sub.2])        [delta] = 30.185[degrees]-322.6[degrees]
                                     [m.sub.1] = 0.0541-0.224
                                     [m.sub.2] = 0.0538-0.224
                                     [m.sub.3] = 0.0679-0.230

                              [rho] = -50.28[degrees]-60.48[degrees]
                                [sigma] = -90[degrees]-90[degrees]
[D.sub.2] ([F.sub.1])        [delta] = 87.19[degrees]-297.4[degrees]

                                     [m.sub.1] = 0.124-0.293
                                     [m.sub.2] = 0.125-0.294
                                     [m.sub.3] = 0.132-0.295

                                                X
                                                X
[D.sub.3] ([F.sub.4])                           X

                                                X
                                                X
                                                X

                              [rho] = -58.71[degrees]-59.35[degrees]
                             [sigma] = -88.90[degrees]-87.18[degrees]
[D.sub.4] ([F.sub.3])        [delta] = 75.87[degrees]-285.58[degrees]
                                     [m.sub.1] = 0.0574-0.287
                                     [m.sub.2] = 0.0575-0.286
                                     [m.sub.3] = 0.0679-0.291

                                 [rho] = -90[degrees]-90[degrees]
                                [sigma] = -90[degrees]-90[degrees]
[D.sub.5] ([F.sub.5])           [delta] = 0[degrees]-360[degrees]
                                     [m.sub.1] = 0.0337-0.334
                                     [m.sub.2] = 0.0338-0.334
                                     [m.sub.3] = 0.0541-0.334

                              [rho] = -65.09[degrees]-58.48[degrees]

                                [sigma] = -90[degrees]-90[degrees]

[D.sub.6] ([F.sub.9])         [delta] = 64.2[degrees]-288.1[degrees]

                                     [m.sub.1] = 0.0974-0.323
                                    [m.sub.2] = 0.0965-0.0321
                                     [m.sub.3] = 0.107-0.0326

                                 [rho] = -90[degrees]-90[degrees]
                                [sigma] = -90[degrees]-90[degrees]
[D.sub.7] ([F.sub.8])           [delta] = 0[degrees]-360[degrees]
                                     [m.sub.1] = 0.0478-0.273
                                     [m.sub.2] = 0.0479-0.273
                                     [m.sub.3] = 0.0644-0.300

                                                X
                                                X
[D.sub.8] ([F.sub.7])                           X

                                                X
                                                X
                                                X

                                 [rho] = -90[degrees]-90[degrees]
                                [sigma] = -90[degrees]-90[degrees]
[D.sub.9] ([F.sub.6])           [delta] = 0[degrees]-360[degrees]
                                     [m.sub.1] = 0.0492-0.300
                                     [m.sub.2] = 0.0496-0.300
                                     [m.sub.3] = 0.0638-0.303

                                                X
                                                X
                                                X
[D.sub.10] ([F.sub.10])                         X
                                                X
                                                X

                                               1TEE
Cases
                                                IO

                                 [rho] = -90[degrees]-90[degrees]
                                [sigma] = -90[degrees]-90[degrees]
[D.sub.1] ([F.sub.2])           [delta] = 0[degrees]-360[degrees]
                                    [m.sub.1] = 0.00432-0.252
                                    [m.sub.2] = 0.00437-0.251
                                    [m.sub.3] = 0.00343-0.248

                                 [rho] = -90[degrees]-90[degrees]
                                [sigma] = -90[degrees]-90[degrees]
[D.sub.2] ([F.sub.1])           [delta] = 0[degrees]-360[degrees]

                                     [m.sub.1] = 0.0422-0.263
                                     [m.sub.2] = 0.0430-0.262
                                    [m.sub.3] = 0.00552-0.257

                                 [rho] = -90[degrees]-90[degrees]
                                [sigma] = -90[degrees]-90[degrees]
[D.sub.3] ([F.sub.4])           [delta] = 0[degrees]-360[degrees]

                                     [m.sub.1] = 0.0439-0.308
                                     [m.sub.2] = 0.0445-0.308
                                    [m.sub.3] = 0.00888-0.304

                                 [rho] = -90[degrees]-90[degrees]
                                [sigma] = -90[degrees]-90[degrees]
[D.sub.4] ([F.sub.3])           [delta] = 0[degrees]-360[degrees]
                                    [m.sub.1] = 0.00429-0.254
                                    [m.sub.2] = 0.00430-0.254
                                    [m.sub.3] = 0.00969-0.202

                                 [rho] = -90[degrees]-90[degrees]
                                [sigma] = -90[degrees]-90[degrees]
[D.sub.5] ([F.sub.5])           [delta] = 0[degrees]-360[degrees]
                                     [m.sub.1] = 0.0431-0.335
                                     [m.sub.2] = 0.0434-0.334
                                    [m.sub.3] = 0.00940-0.330

                                 [rho] = -90[degrees]-90[degrees]

                                [sigma] = -90[degrees]-90[degrees]

[D.sub.6] ([F.sub.9])           [delta] = 0[degrees]-360[degrees]

                                     [m.sub.1] = 0.0446-0.277
                                     [m.sub.2] = 0.0448-0.277
                                    [m.sub.3] = 0.00888-0.324

                                 [rho] = -90[degrees]-90[degrees]
                                [sigma] = -90[degrees]-90[degrees]
[D.sub.7] ([F.sub.8])           [delta] = 0[degrees]-360[degrees]
                                     [m.sub.1] = 0.0439-0.333
                                     [m.sub.2] = 0.0454-0.333
                                    [m.sub.3] = 0.00806-0.330

                                 [rho] = -90[degrees]-90[degrees]
                                [sigma] = -90[degrees]-90[degrees]
[D.sub.8] ([F.sub.7])           [delta] = 0[degrees]-360[degrees]

                                    [m.sub.1] = 0.0451-0.0938
                                    [m.sub.2] = 0.0459-0.0941
                                    [m.sub.3] = 0.00121-0.0790

                                 [rho] = -90[degrees]-90[degrees]
                                [sigma] = -90[degrees]-90[degrees]
[D.sub.9] ([F.sub.6])           [delta] = 0[degrees]-360[degrees]
                                     [m.sub.1] = 0.0455-0.310
                                     [m.sub.2] = 0.0456-0.310
                                    [m.sub.3] = 0.00969-0.306

                                 [rho] = -90[degrees]-90[degrees]
                                [sigma] = -90[degrees]-90[degrees]
                                [delta] = 0[degrees]-360[degrees]

[D.sub.10] ([F.sub.10])              [m.sub.1] = 0.0432-0.102
                                     [m.sub.2] = 0.0442-0.102
                                    [m.sub.3] = 0.00301-0.0901

                                               1TEC
Cases
                                                NO

                                 [rho] = -90[degrees]-90[degrees]
                                [sigma] = -90[degrees]-90[degrees]
[D.sub.1] ([F.sub.2])           [delta] = 0[degrees]-360[degrees]
                                    [m.sub.1] = 0.00331-0.603
                                    [m.sub.2] = 0.00800-0.606
                                     [m.sub.3] = 0.0371-0.604

                                 [rho] = -90[degrees]-90[degrees]
                                [sigma] = -90[degrees]-90[degrees]
[D.sub.2] ([F.sub.1])          [delta] = 0[degrees]-64.34[degrees]
                             [direct sum] 295.7[degrees]-360[degrees]
                                    [m.sub.1] = 0.00542-0.0277
                                    [m.sub.2] = 0.00993-0.0291
                                    [m.sub.3] = 0.0440-0.0600

                                 [rho] = -90[degrees]-90[degrees]
                                [sigma] = -90[degrees]-90[degrees]
[D.sub.3] ([F.sub.4])          [delta] = 0[degrees]-64.34[degrees]
                             [direct sum] 295.7[degrees]-360[degrees]
                                     [m.sub.1] = 0.0161-0.474
                                     [m.sub.2] = 0.0164-0.471
                                     [m.sub.3] = 0.0416-0.469

                                 [rho] = -90[degrees]-90[degrees]
                                [sigma] = -90[degrees]-90[degrees]
[D.sub.4] ([F.sub.3])        [delta] = 18.12[degrees]-318.75[degrees]
                                     [m.sub.1] = 0.0102-0.795
                                     [m.sub.2] = 0.0147-0.796
                                     [m.sub.3] = 0.0369-0.790

                                 [rho] = -90[degrees]-90[degrees]
                                [sigma] = -90[degrees]-90[degrees]
[D.sub.5] ([F.sub.5])           [delta] = 0[degrees]-360[degrees]
                                    [m.sub.1] = 0.00255-0.509
                                    [m.sub.2] = 0.00370-0.505
                                     [m.sub.3] = 0.0351-0.510

                             [rho] = -23.03[degrees]- -7.74[degrees]
                            [direct sum] 7.66[degrees]-22.14[degrees]
                           [sigma] = -68.53[degrees] - -38.75[degrees]
                            [direct sum] 37.86[degrees]-69.0[degrees]
[D.sub.6] ([F.sub.9])       [delta] = 109.68[degrees]-150.05[degrees]
                            [direct sum] 210.4[degrees]-250.2[degrees]
                                     [m.sub.1] = 0.0202-0.456
                                     [m.sub.2] = 0.0204-0.456
                                     [m.sub.3] = 0.0451-0.456

                                 [rho] = -90[degrees]-90[degrees]
                                [sigma] = -90[degrees]-90[degrees]
[D.sub.7] ([F.sub.8])           [delta] = 0[degrees]-360[degrees]
                                    [m.sub.1] = 0.00663-0.436
                                     [m.sub.2] = 0.0103-0.434
                                     [m.sub.3] = 0.0416-0.438

                                 [rho] = -90[degrees]-90[degrees]
                                [sigma] = -90[degrees]-90[degrees]
[D.sub.8] ([F.sub.7])          [delta] = 0[degrees]-64.34[degrees]
                             [direct sum] 295.7[degrees]-360[degrees]
                                    [m.sub.1] = 0.00941-0.0803
                                    [m.sub.2] = 0.0125-0.0804
                                    [m.sub.3] = 0.0445-0.0953

                                 [rho] = -90[degrees]-90[degrees]
                                [sigma] = -90[degrees]-90[degrees]
[D.sub.9] ([F.sub.6])           [delta] = 0[degrees]-360[degrees]
                                    [m.sub.1] = 0.00769-0.436
                                    [m.sub.2] = 0.00932-0.434
                                     [m.sub.3] = 0.0426-0.438

                                 [rho] = -90[degrees]-90[degrees]
                                [sigma] = -90[degrees]-90[degrees]
                               [delta] = 0[degrees]-64.34[degrees]
                             [direct sum] 295.7[degrees]-360[degrees]
[D.sub.10] ([F.sub.10])             [m.sub.1] = 0.00201-0.0793
                                    [m.sub.2] = 0.00870-0.0795
                                    [m.sub.3] = 0.0435-0.0932

                                               1TEC
Cases
                                                IO

                                 [rho] = -90[degrees]-90[degrees]
                                [sigma] = -90[degrees]-90[degrees]
[D.sub.1] ([F.sub.2])        [delta] = 31.79[degrees]-318.66[degrees]
                                     [m.sub.1] = 0.0731-0.501
                                     [m.sub.2] = 0.0719-0.500
                                     [m.sub.3] = 0.0593-0.496

                                [rho] = -63.3[degrees]-64[degrees]
                             [sigma] = -89.37[degrees]-86.25[degrees]
[D.sub.2] ([F.sub.1])        [delta] = 70.50[degrees]-295.77[degrees]

                                     [m.sub.1] = 0.128-0.581
                                     [m.sub.2] = 0.130-0.582
                                     [m.sub.3] = 0.120-0.579

                                                X
                                                X
[D.sub.3] ([F.sub.4])                           X
                                                X
                                                X
                                                X

                               [rho] = -61.4[degrees]-61.5[degrees]
                                [sigma] = -90[degrees]-90[degrees]
[D.sub.4] ([F.sub.3])        [delta] = 77.86[degrees]-286.15[degrees]
                                     [m.sub.1] = 0.0711-0.424
                                     [m.sub.2] = 0.0740-0.424
                                     [m.sub.3] = 0.0564-0.418

                                 [rho] = -90[degrees]-90[degrees]
                                [sigma] = -90[degrees]-90[degrees]
[D.sub.5] ([F.sub.5])           [delta] = 0[degrees]-360[degrees]
                                     [m.sub.1] = 0.0538-0.505
                                     [m.sub.2] = 0.0539-0.506
                                     [m.sub.3] = 0.0305-0.500

                            [rho] = -18.42[degrees] - -6.56[degrees]
                            [direct sum] 6.77[degrees]-18.27[degrees]
                             [sigma] = --71.1[degrees]-- 46.2[degrees]
                            [direct sum] 46.2[degrees]-70.48[degrees]
[D.sub.6] ([F.sub.9])       [delta] = 118.19[degrees]-154.2[degrees]
                            [direct sum] 206.7[degrees]-242.3[degrees]
                                     [m.sub.1] = 0.108-0.398
                                     [m.sub.2] = 0.109-0.396
                                     [m.sub.3] = 0.0962-0.396

                                 [rho] = -90[degrees]-90[degrees]
                                [sigma] = -90[degrees]-90[degrees]
[D.sub.7] ([F.sub.8])           [delta] = 0[degrees]-360[degrees]
                                     [m.sub.1] = 0.0647-0.437
                                     [m.sub.2] = 0.0649-0.438
                                     [m.sub.3] = 0.0439-0.491

                                                X
                                                X
[D.sub.8] ([F.sub.7])                           X
                                                X
                                                X
                                                X

                                 [rho] = -90[degrees]-90[degrees]
                                [sigma] = -90[degrees]-90[degrees]
[D.sub.9] ([F.sub.6])           [delta] = 0[degrees]-360[degrees]
                                     [m.sub.1] = 0.0647-0.411
                                     [m.sub.2] = 0.0638-0.412
                                     [m.sub.3] = 0.0458-0.407

                                                X
                                                X
                                                X
[D.sub.10] ([F.sub.10])                         X
                                                X
                                                X

Table 6: The allowed ranges of Dirac CP-violating phase [delta], the
Majorana phases [rho], [alpha], and three neutrino masses [m.sub.1],
[m.sub.2], [m.sub.3] for the experimentally allowed cases of Category
E. Masses are in eV. "x" denotes the nonviability of case for a
particular mass ordering.

                                              1TEE
Cases
                                               NO

                                               X
                                               X
[E.sub.1] ([E.sub.2])                          X

                                               X
                                               X
                                               X

                                               X
                                               X
[E.sub.3] ([E.sub.4])                          X

                                               X
                                               X
                                               X

                                  [rho] = -90[degrees] - -156
                                 [direct sum] 29.3-90[degrees]
                                 [rho] = -90[degrees] - - 156
                                 [direct sum] 29.3-90[degrees]
[E.sub.5]                   [delta] = 29.3[degrees]-153.5[degrees]
                            [direct sum] 206.5[degrees]-334[degrees]
                                    [m.sub.1] = 0.130-0.332
                                    [m.sub.2] = 0.132-0.330
                                    [m.sub.3] = 0.137-0.334

                                               X
                                               X
[E.sub.6] ([E.sub.9])                          X
                                               X
                                               X
                                               X

                                               X
                                               X
[E.sub.7] ([E.sub.8])                          X
                                               X
                                               X
                                               X

                             [rho] = -90[degrees] - -28.6[degrees]
                            [direct sum] 26.01[degrees]-90[degrees]
                            [sigma] = -90[degrees] - -28.6[degrees]
                            [direct sum] 26.01[degrees]-90[degrees]
[E.sub.10]                 [delta] = 25.35[degrees]-151.92[degrees]
                          [direct sum] 206.97[degrees]-332.3[degrees]
                                    [m.sub.1] = 0.136-0.331
                                    [m.sub.2] = 0.137-0.331
                                    [m.sub.3] = 0.143-0.334

                                              1TEE
Cases
                                               IO

                                [rho] = -90[degrees]-90[degrees]
                               [sigma] = -90[degrees]-90[degrees]
[E.sub.1] ([E.sub.2])          [delta] = 0[degrees]-360[degrees]

                                   [m.sub.1] = 0.0511-0.0804
                                   [m.sub.2] = 0.0516-0.0805
                                   [m.sub.3] = 0.0264-0.0678

                                [rho] = -90[degrees]-90[degrees]
                               [sigma] = -90[degrees]-90[degrees]
[E.sub.3] ([E.sub.4])          [delta] = 0[degrees]-360[degrees]

                                    [m.sub.1] = 0.0439-0.333
                                    [m.sub.2] = 0.0457-0.333
                                   [m.sub.3] = 0.00888-0.333

                                [rho] = -90[degrees]-90[degrees]

                               [sigma] = -90[degrees]-90[degrees]

[E.sub.5]                      [delta] = 0[degrees]-360[degrees]

                                    [m.sub.1] = 0.0439-0.269
                                    [m.sub.2] = 0.0451-0.270
                                    [m.sub.3] = 0.0103-0.262

                                [rho] = -90[degrees]-90[degrees]
                               [sigma] = -90[degrees]-90[degrees]
[E.sub.6] ([E.sub.9])          [delta] = 0[degrees]-360[degrees]
                                    [m.sub.1] = 0.0511-0.333
                                    [m.sub.2] = 0.0517-0.333
                                    [m.sub.3] = 0.0273-0.333

                                [rho] = -90[degrees]-90[degrees]
                               [sigma] = -90[degrees]-90[degrees]
[E.sub.7] ([E.sub.8])          [delta] = 0[degrees]-360[degrees]
                                   [m.sub.1] = 0.0511-0.0908
                                   [m.sub.2] = 0.0517-0.0911
                                   [m.sub.3] = 0.0273-0.0746

                             [rho] = -18.15[degrees]-18.08[degrees]

                            [sigma] = -90[degrees] - -6.41[degrees]
                             [direct sum] 6.09[degrees]-90[degrees]
[E.sub.10]                     [delta] = 0[degrees]-177[degrees]
                           [direct sum] 184.3[degrees]-357.5[degrees]
                                    [m.sub.1] = 0.0431-0.334
                                    [m.sub.2] = 0.0445-0.334
                                   [m.sub.3] = 0.00969-0.330

                                              1TEC
Cases
                                               NO

                                [rho] = -90[degrees]-90[degrees]
                               [sigma] = -90[degrees]-90[degrees]
[E.sub.1] ([E.sub.2])         [delta] = 0[degrees]-64.34[degrees]
                            [direct sum] 295.7[degrees]-360[degrees]
                                   [m.sub.1] = 0.00283-0.0629
                                   [m.sub.2] = 0.00878-0.0634
                                   [m.sub.3] = 0.0440-0.0809

                                [rho] = -90[degrees]-90[degrees]
                               [sigma] = -90[degrees]-90[degrees]
[E.sub.3] ([E.sub.4])         [delta] = 0[degrees]-64.34[degrees]
                            [direct sum] 295.7[degrees]-360[degrees]
                                   [m.sub.1] = 0.00914-0.499
                                   [m.sub.2] = 0.00115-0.504
                                    [m.sub.3] = 0.0432-0.500

                                               X

                                               X

[E.sub.5]                                      X

                                               X
                                               X
                                               X

                                [rho] = -90[degrees]-90[degrees]
                               [sigma] = -90[degrees]-90[degrees]
[E.sub.6] ([E.sub.9])          [delta] = 0[degrees]-360[degrees]
                                   [m.sub.1] = 0.00914-0.499
                                   [m.sub.2] = 0.00115-0.504
                                    [m.sub.3] = 0.0432-0.500

                                [rho] = -90[degrees]-90[degrees]
                               [sigma] = -90[degrees]-90[degrees]
[E.sub.7] ([E.sub.8])          [delta] = 0[degrees]-360[degrees]
                                   [m.sub.1] = 0.0225-0.0469
                                   [m.sub.2] = 0.0238-0.0480
                                   [m.sub.3] = 0.0494-0.0708

                                [rho] = -90[degrees]-90[degrees]

                               [sigma] = -90[degrees]-90[degrees]

[E.sub.10]                     [delta] = 0[degrees]-360[degrees]

                                   [m.sub.1] = 0.00156-0.481
                                   [m.sub.2] = 0.00571-0.480
                                    [m.sub.3] = 0.0404-0.484

                                              1TEC
Cases
                                               IO

                                               X
                                               X
[E.sub.1] ([E.sub.2])                          X

                                               X
                                               X
                                               X

                                               X
                                               X
[E.sub.3] ([E.sub.4])                          X

                                               X
                                               X
                                               X

                                               X

                                               X

[E.sub.5]                                      X

                                               X
                                               X
                                               X

                                               X
                                               X
[E.sub.6] ([E.sub.9])                          X
                                               X
                                               X
                                               X

                                               X
                                               X
[E.sub.7] ([E.sub.8])                          X
                                               X
                                               X
                                               X

                              [rho] = -90[degrees]- -19[degrees]
                             [direct sum] 19.5[degrees]-90[degrees]
                             [sigma] = -90[degrees] - -19[degrees]
                             [direct sum] 19.5[degrees]-90[degrees]
[E.sub.10]                 [delta] = 23.75[degrees]-161.97[degrees]
                            [direct sum] 198.9[degrees]-339[degrees]
                                    [m.sub.1] = 0.141-0.481
                                    [m.sub.2] = 0.142-0.492
                                    [m.sub.3] = 0.133-0.491
COPYRIGHT 2018 Hindawi Limited
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2018 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
Title Annotation:Research Article
Author:Singh, Madan
Publication:Advances in High Energy Physics
Article Type:Report
Geographic Code:1USA
Date:Jan 1, 2018
Words:9941
Previous Article:Nonlocal Black Hole Evaporation and Quantum Metric Fluctuations via Inhomogeneous Vacuum Density.
Next Article:Neutrino Emission from Cooper Pairs at Finite Temperatures.
Topics:

Terms of use | Privacy policy | Copyright © 2020 Farlex, Inc. | Feedback | For webmasters