The recent measurement of reactor mixing angle θ13 and its subsequent refinements [1–4], along with the precision measurement of the solar and atmospheric mixing angles θ12 and θ23 and the neutrino mass-squared differences, have given a new impetus to the neutrino oscillation phenomenology. The observation of a non-zero value of θ13, on the one hand, restored the parallelism between quark and lepton mixing, while on the other hand it has triggered a great deal of interest in the exploration of CP violation in the leptonic sector. It has also brought into focus the issue of constraining the CP-violating phases of the mixing matrix from the textures of the mass matrices. This has further been strengthened by a very recent constraint on the sum of absolute neutrino masses provided by the Planck experiment [5]. In particular, efforts have been made to carry out fine-grained analysis regarding the compatibility of texture-specific mass matrices in the flavor [6–14] as well as non-flavor basis [15–19]. Specifically, in the flavor basis, wherein the charged lepton mass matrix is considered to be diagonal, a good number of attempts have been made to explore the compatibility of texture zero mass matrices for Majorana neutrinos with the neutrino oscillation data.

Present refinements in the data warrant a reconsideration of the compatability of the texture zero neutrino mass matrices. Therefore, the purpose of the present paper is to reinvestigate all the possible structures of texture three- and two-zero neutrino mass matrices in the flavor basis. In particular, we have carried out an analysis, similar to the one carried out in Ref. [20], with an emphasis on obtaining useful constraints on the CP-violating phases. Specifically, for the viable possibilities of texture zero mass matrices, we have examined the implications of a relatively large and refined θ13 on the parameter space of the Dirac-like CP-violating phase δ and the Majorana phases ρ, σ.

The plan of the paper is as follows. To begin with, some essential details pertaining to the construction of the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix [21–24] from the mass matrices are presented in Sect. 2. Using the inputs given in Sect. 3, and keeping focus on the CP-violating phases ρ, σ, and δ, the results pertaining to the analyses of texture three-zero Majorana mass matrices are presented in Sect. 4. For the texture two-zero case, results of the analysis for Majorana neutrino mass matrices are given in Sect. 5. Finally, Sect. 6 summarizes our conclusions.

Before proceeding further, we briefly underline the methodology relating the elements of the mass matrices to those of the mixing matrix. In the flavor basis, wherein the charged lepton mass matrix Ml is diagonal, the Majorana neutrino mass matrix Mν can be expressed in terms of three neutrino masses m1, m2, m3 and the flavor-mixing matrix V as (1)Mν=V(m1000m2000m3)VT. The mixing matrix V can be written as V=UP, where U denotes the PMNS [21–24] neutrino mixing matrix consisting of three flavor mixing angles and one Dirac-like CP-violating phase, whereas the matrix P is a diagonal phase matrix, i.e., P = diag(eiρ,eiσ,1), with ρ and σ being the two Majorana CP-violating phases. The neutrino mass matrix Mν can then be rewritten as (2)Mν=(MeeMe&mgr;Me&tgr;Me&mgr;M&mgr;&mgr;M&mgr;&tgr;Me&tgr;M&mgr;&tgr;M&tgr;&tgr;)=U(λ1000λ2000λ3)UT, where λ1=m1e2iρ, λ2=m2e2iσ, λ3=m3.

For the calculations, we have adopted the parameterization of the mixing matrix U considered by Ref. [20], i.e., (3)U=(c12c13s12c13s13−c12s23s13−s12c23e−iδ−s12s23s13+c12c23e−iδs23c13−c12c23s13+s12s23e−iδ−s12c23s13−c12s23e−iδc23c13)​, where cij=cosθij, sij=sinθij for i,j=1,2,3 and δ is the CP-violating phase.

Before discussing the results of the analysis, we summarize the experimental information for various neutrino oscillation parameters. For both normal mass ordering (NO) and inverted mass ordering (IO), the best-fit values and the latest experimental constraints on neutrino parameters at 1 σ, 2 σ, and 3 σ confidence levels (CLs), following Ref. [25], are given in Table 1.

In the flavor basis, there are 20 possible texture three-zero patterns; the texture structures of these have been elaborated in Ref. [26], and it has been shown that they are all found to be incompatible with the neutrino oscillation data. Our present analysis, carried out with the current refined data, reinforces this earlier conclusion. To elaborate, we explicitly examine one of the possibilities and bring forward its incompatibility with the latest neutrino mixing data. In particular, we consider the case wherein Mee=0, Me&tgr;=0, M&mgr;&mgr;=0, i.e., (4)Mν=(0×0×0×0××)​.

The neutrino mass matrix Mν can be diagonalized by using flavor mixing matrix U. For the present case, the constraints Mee=0 and Me&tgr;=0 yield the following expression for the ratios of the complex neutrino mass eigenvalues: (5)λ1λ3=−s13c132(s12c23c12s23eiδ+s13),  λ2λ3=+s13c132(c12c23s12s23eiδ−s13). For M&mgr;&mgr;=0, we obtain (6)λ3(U&mgr;1U&mgr;1.λ1λ3+U&mgr;2U&mgr;2.λ2λ3+U&mgr;3U&mgr;3)=0. From the real and imaginary parts of the above equations, we obtain the following constraints on θ12. From the real part, we get (7)θ12=12acot(s23c232s132(2+cot2δ)−s233cos2θ132c232.s13cosδ), whereas, from the imaginary part, we obtain (8)θ12=12acot(s13cos(2δ)s232sinδ). To check the compatibility of the above two solutions, in Fig. 1(a) and (b) we present the correlation plots between θ12 and θ23 for the constraints obtained from the real and imaginary parts respectively. These graphs have been obtained by giving full allowed variation to the phase δ and 3 σ variation to θ13. From Fig. 1(a), we find that the range obtained for θ12 here has no overlap with its experimental range, i.e., 31.8°–37.8°, whereas the plot given in Fig. 1(b) shows that θ12 found here includes its experimental range. For compatibility with the texture three-zero possibility considered here, both the graphs should include the experimental range of θ12, therefore ruling out the present case. A similar analysis corresponding to the other possibilities of texture three-zero Majorana neutrino mass matrices yields the same result, therefore ruling these out.

Coming to the case of texture two-zero Majorana neutrino mass matrices, it is well known that there are 15 possible structures of neutrino mass matrix. Adopting the classification scheme of Ref. [20], these can be divided into the six classes A, B, C, D, E, and F. For the purpose of examining whether a possible pattern of texture two-zero mass matrices is viable or not, one needs to examine the compatability of the corresponding PMNS matrix constructed using them. To execute the present analysis, to begin with we vary the input neutrino oscillation parameters, i.e., the three mixing angles θ12, θ23, and θ13, as well as the mass-squared differences δm2 and Δm2, within their 3 σ CL ranges as summarized in Table 1. In the absence of any experimental constraint on Dirac-like CP-violating phase δ, we give full variation of 0° to 360° to it. In order to obtain constraints on the Majorana phases ρ and σ, these can be expressed in terms of the mixing matrix elements: (9)ρ=12arg(Ua3Ub3Ul2Um2−Ua2Ub2Ul3Um3Ua2Ub2Ul1Um1−Ua1Ub1Ul2Um2) and (10)σ=12arg(Ua1Ub1Ul3Um3−Ua3Ub3Ul1Um1Ua2Ub2Ul1Um1−Ua1Ub1Ul2Um2)​. Coming to the results of the analysis, it may be mentioned that even with the present refinements of the mixing angle θ13, out of the 15 possible texture two-zero Majorana neutrino mass matrices, the texture patterns A1,2, B1,2,3,4 and C remain viable for the normal mass ordering (NO). The possibilities D1,2 have already been ruled out by Ref. [20] due to mixing angle θ12 being less than 38°; the conclusions of our analysis in this regard are the same. Similarly, patterns E1,2,3 and F1,2,3 remain phenomenologically disfavored. In the following, for NO, we present results for the patterns of categories A, B, and C. In particular, we present the implications of the relatively large and refined θ13 on the parameter space of the three CP-violating phases δ, ρ, and σ.

To summarize, in the light of recent refinements of mixing angle θ13, we have reinvestigated all the possible texture structures of three-zero and two-zero neutrino mass matrices for both Majorana and Dirac neutrinos in the flavor basis. In particular, we have examined the implications of angle θ13 on the parameter space of three CP-violating phases: ρ, σ, and δ. For the case of texture three-zero mass matrices, confirming earlier results, we find that all possible structures remain ruled out even with the recent data. For a particular texture structure, this has been shown using correlation plots between angles θ12 and θ23 for the constraints obtained from the real and imaginary parts of ratios of complex neutrino mass eigenvalues. These graphs show that θ12 obtained here has no overlap with its experimental range, therefore ruling out the present texture-specific case.

For the case of texture two-zero mass matrices, out of the 15 possible structures, the matrices belonging to classes D, E, and F remain ruled out even with the latest data. For the classes A, B, and C, analysis has been carried out for both NO and IO. The analysis shows that the matrices of class A are ruled out for IO, whereas for NO one does not obtain any useful constraints on the CP-violating phases. For the class B matrices, all these are viable for both NO and IO. For both the mass orderings, phase δ takes values close to 90° and 270°; however, the Majorana phases ρ and σ both acquire quite small values. The matrix belonging to class C is again viable for both NO and IO. For NO, phase δ has lower limit 0°, whereas for IO the lower limit is 40.5°. The Majorana phases also acquire different constraints for different mass orderings.