Leptonic mixing angle $\theta_{13}$ and ruling out of minimal texture for Dirac neutrinos

Implications of recently measured leptonic mixing angle $\theta_{13}$ as well as the other two mixing angles have been examined for Fritzsch-like mass matrices with minimal texture for Dirac neutrinos. Interestingly, the existing data seems to rule out this texture specific case of Dirac neutrinos for normal, inverted hierarchy as well as degenerate scenario of masses.


Introduction
The recent measurements [1]- [5] regarding the neutrino mixing angle θ 13 have undoubtedly improved our knowledge of neutrino oscillation phenomenology. Interestingly, this θ 13 value which is unexpectedly 'large', being almost near the Cabibbo angle, would have important implications for flavor physics. Also, it may be mentioned that before the measurement of θ 13 , assuming it to be zero or nearly equal to zero and considering the canonical values of the other two neutrino mixing angles, the effort was to discover some underlying symmetry [6] in the leptonic sector. The non zero value of θ 13 leads to parallelism between the mixings of quarks and leptons as well as signifies the difference between the mixing angles of quarks and leptons as the leptonic mixing angles are large compared with the corresponding quark mixing angles.
Ever since the observations regarding θ 13 there has been a good deal of activity on the theoretical front in understanding the pattern of neutrino masses and mixings. Noting that there is a similarity between quark and lepton mixing phenomena [7], it becomes desirable to understand these from the same perspective as far as possible. However, there are some important differences which have to be kept in mind before considering a unified framework for formulating quark and lepton mass matrices on the same footing. For example, one may note that unlike the case of quark mixings which show a hierarchical structure, the pattern of neutrino mixings do not show any explicit hierarchy. Further, at present there is no consensus about neutrino masses which may show normal/inverted hierarchy or may even be degenerate. Furthermore, the situation becomes complicated when one realizes that it is yet not clear whether neutrinos are Dirac or Majorana particles.
It may be mentioned that in the absence of any viable theory for flavor physics, one usually resorts to phenomenological models. In this context, texture specific mass matrices have got good deal of attention in the literature, for details and extensive references we refer the readers to a recent review article [8]. In particular, Fritzsch-like texture specific mass matrices seem to be very helpful in understanding the pattern of quark mixings and CP violation [9,10]. Keeping in mind quark lepton parallelism [7] and taking clue from the success of these texture specific mass matrices in the context of quarks, several attempts [9,11] have also been made to consider similar lepton mass matrices. However, noting the above mentioned complexities of neutrino masses and mixings, it seems necessary to carry out a detailed and case by case analysis of texture specific mass matrices for their compatibility with the mixing data. In particular, for any given texture, the analysis needs to be carried out for all the neutrino mass hierarchies as well as for both Majorana and Dirac neutrinos since the latter have not yet been ruled out experimentally [12].
Considering neutrinos to be Majorana particles, after the recent measurements of θ 13 , a few analyses have been carried out for texture specific mass matrices in the non flavor basis. In particular, Fukugita et al. [13] have investigated the implications of angle θ 13 on minimal texture mass matrices (Fritzsch-like texture 6 zero) for normal hierarchy of masses. This analysis has been extended further by Fakay et al. [14] wherein for all the hierarchies of neutrino masses, texture 6 and 5 zero mass matrices have been examined in detail. For the case of Dirac neutrinos, although several authors have examined the possibility of these having small masses [15] as well as their compatibility with the supersymmetric GUTs [16], however, similar attempts have not yet been carried out after the measurements of θ 13 . In this context, it may be added that the original texture 6 zero Fritzsch mass matrices have been ruled out in the case of quarks, therefore, in the light of similarity between the mixing patterns of quarks and leptons, it becomes desirable to examine similar mass matrices for the cases of neutrinos.
In the present paper, for the case of Dirac neutrinos, we have carried out detailed calculations pertaining to mass matrices with minimal texture for the three possibilities of neutrino masses having normal/inverted hierarchy or being degenerate. In particular, the analysis has been carried out by imposing Fritzsch-like texture 6 zero structure on Dirac neutrino mass matrices as well as on charged lepton mass matrices. The compatibility of these texture specific mass matrices have been examined by plotting the parameter space corresponding to the recently measured mixing angle s 13 along with the other two mixing angles s 12 and s 23 . Further, for the normal hierarchy case, the implications of mixing angles on the lightest neutrino mass m ν 1 have also been investigated.
The detailed plan of the paper is as follows. To set notations and conventions as well as to make the paper self contained, in Section (2) we present some of the essentials regarding texture 6 zero Dirac neutrino mass matrices. Inputs used in the present analysis have been given in Section (3). The analysis pertaining to inverted, normal hierarchy and degenerate scenario of neutrino masses have been respectively presented in Sections (4), (5) and (6). Finally, Section (7) summarizes our conclusions.

Texture 6 zero Dirac neutrino mass matrices
In the Standard Model (SM), the mass terms corresponding to the charged leptons and Dirac neutrinos having non zero masses are respectively given by and where L stands for left handedness, M l denotes the charged lepton mass matrix, M νD is the complex 3 × 3 Dirac neutrino mass matrix and The three flavor fields are ν aL (a = e, µ, τ ) and ν aR are the right-handed singlets which are sterile and do not mix with the active neutrinos. The mass matrices M l and M νD are arbitrary in the SM with a total of 36 real, free parameters, these being quite large in number in comparison with the 10 physical observables. Using the polar decomposition theorem any general mass matrix M can be expressed as M=HU, where H denotes a Hermitian and U a unitary matrix. In the present case, the matrix U can be absorbed by redefining the right handed singlet neutrino fields, therefore, enabling one to bring down the number of free parameters from 36 to 18, which are further brought down by considering textures, discussed below. After defining the charged lepton and neutrino mass matrices, their texture 6 zero Fritzsch structures are given as M l and M νD respectively corresponding to charged lepton and Dirac neutrino mass matrices. It may be noted that each of the above matrix is texture 3 zero type with A l(ν) = |A l(ν) |e iα l(ν) and B l(ν) = |B l(ν) |e iβ l(ν) . The formalism connecting the mass matrix to the neutrino mixing matrix [17] involves diagonalization of the mass matrices M l and M νD , details in this regard can be looked up in [8]. In general, to facilitate diagonalization, the mass matrix M k , where k = l, νD, can be expressed as where M r k is a real symmetric matrix with real eigenvalues and Q k and P k are diagonal phase matrices Diag{e iα k , 1, e −iβ k } and Diag{e −iα k , 1, e iβ k } respectively. The real matrix M r k is diagonalized by the orthogonal transformation O k , e.g., which on using equation (6) can be rewritten as The elements of the general diagonalizing transformation O k can figure with different phase possibilities, however these possibilities are related to each other through the phase matrices [8]. For the present work, we have chosen the possibility where m 1 , −m 2 , m 3 being the eigenvalues of M k . It may be added that without loss of generality, we can always choose phase of one of the mass eigenvalue relative to the other two. For details, we refer the reader to [8,18].
In the case of charged leptons, because of the hierarchy m e ≪ m µ ≪ m τ , the mass eigenstates can be approximated respectively to the flavor eigenstates as has been considered by several authors [19,20]. Using the approximation, m l1 ≃ m e , m l2 ≃ m µ and m l3 ≃ m τ , the first element of the matrix O l can be obtained from the corresponding element of equation (10) by replacing m 1 , −m 2 , m 3 with m e , −m µ , m τ , e.g., In the case of neutrinos, for normal hierarchy of neutrino masses defined as m ν 1 < m ν 2 ≪ m ν 3 , as well as for the corresponding degenerate case given by m ν 1 m ν 2 ∼ m ν 3 , equation (10) can also be used to obtain the elements of diagonalizing transformation for Dirac neutrinos. The first element can be obtained from the corresponding element of equation (10) by replacing m 1 , −m 2 , m 3 with m ν1 , −m ν2 , m ν3 and is given by where m ν 1 , m ν 2 and m ν 3 are neutrino masses.
In the same manner, one can obtain the elements of diagonalizing transformation for the inverted hierarchy case defined as m ν 3 ≪ m ν 1 < m ν 2 as well as for the corresponding degenerate case given by m ν 3 ∼ m ν 1 m ν 2 . The corresponding first element, obtained by replacing m 1 , −m 2 , m 3 with m ν1 , −m ν2 , −m ν3 in equation (10), is given by As already mentioned, one can choose the sign of one eigenvalue relative to the other two, therefore, to facilitate calculations for the inverted hierarchy case we have chosen m ν1 to be positive and both m ν2 and m ν3 to be negative. The other elements of diagonalizing transformations in the case of neutrinos as well as charged leptons can similarly be found. After the elements of diagonalizing transformations O l and O νD are known, the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [17] can be obtained through the relation where Q l P νD , without loss of generality, can be taken as Diag{e −iφ 1 , 1, e iφ 2 }. The parameters φ 1 and φ 2 are related to the phases of mass matrices, i.e., φ 1 = α νD − α l , φ 2 = β νD − β l and can be treated as free parameters.

Inputs used for the analysis
In the present analysis, we have made use of the results of the latest global three neutrino oscillation analysis carried out by Fogli et al. [21]. At 1σ C.L. the allowed ranges of the various input parameters are where ∆m 2 ij 's correspond to the solar and atmospheric neutrino mass square differences and s ij 's correspond to the sine of the mixing angle θ ij where i, j = 1, 2, 3. At 3σ C.L. the allowed ranges are given as For the purpose of the calculations, the masses and mixing angles have been constrained by the data given in the above equations. In the case of normal hierarchy, the explored range for the lightest neutrino mass corresponding to m ν 1 is taken to be 0.0001 eV − 1.0 eV, essentially governed by the mixing angle s 12 related to the ratio mν 1 mν 2 . For the inverted hierarchy case also we have taken the same range for the lightest neutrino mass corresponding to m ν 3 . It may be mentioned that our conclusions remain unaffected even if the range is extended further. In the absence of any constraint on the phases, φ 1 and φ 2 have been given full variation from 0 to 2π.

Inverted hierarchy of neutrino masses
To examine the compatibility of texture 6 zero Dirac neutrino mass matrices with the recent mixing data, we first discuss the implications of mixing angle θ 13 for the case pertaining to inverted hierarchy of neutrino masses. To this end, in Figures 1(a) and 1(b) we present the plots of the parameter space corresponding to s 13 along with the other two mixing angles s 12 and s 23 respectively. Giving full allowed variation to other parameters, Figure 1(a) has been obtained by constraining the angle s 23 by its experimental bound given in equation (18) and similarly while plotting Figure 1(b) the angle s 12 has been constrained by its experimental limits. Also included in the figures are blank rectangular regions indicating the experimentally allowed 3σ C.L. region of the plotted angles. Interestingly, a general look at these figures reveals that pertaining to inverted hierarchy of neutrino masses, the texture 6 zero Dirac neutrino mass matrices are clearly ruled out at 3σ C.L.. This can be understood by noting that that the plotted parameter space of the two angles has no overlap with their experimentally allowed 3σ C.L. region.

Normal hierarchy of neutrino masses
After ruling out texture 6 zero Dirac neutrino mass matrices for inverted hierarchy, we now examine the compatibility of these matrices for the case of normal hierarchy.
To this end, in Figure 2(a) we present the graph of s 13 versus m ν 1 , in the graph the solid horizontal lines and the dashed lines depict respectively the 3σ C.L. and 1σ C.L. range of this angle. The graph depicts an interesting result that the 1σ C.L. range of s 13 has no overlap with the plotted values of the angle s 13 indicating towards the ruling out of texture 6 zero mass matrices at 1σ C.L. for normal hierarchy of neutrinos. However, a look at the figure also reveals that corresponding to the 3σ C.L. range of s 13 , one gets get a lower bound on mass m ν 1 ∼ 0.001eV. One may add that refinements in the measurement of angle s 13 would have interesting implications for this case. To sharpen the above mentioned conclusions, in Figure 2(b) we present the graph of angle s 23 w.r.t. mass m ν 1 , with the solid horizontal lines and the dashed lines depicting respectively the 3σ C.L. and 1σ C.L. range of this angle. Interestingly, from this figure one can conclude that not only the 1σ C.L. range of s 23 again confirms the ruling out of this case of texture 6 zero mass matrices, but also corresponding to the 3σ C.L. range of this angle, one finds that again the ruling out is largely confirmed. It may be added in case we plot a graph of angle s 12 versus m ν 1 , it indicates towards compatibility of these mass matrices with the data. However, it needs to be noted that to rule out the matrices it is sufficient to do so from any one of the mixing angle versus the mass m ν 1 graph.

Degenerate scenario of neutrino masses
The degenerate scenario of neutrino masses can be characterized by either m ν 1 m ν 2 ∼ m ν 3 ∼ 0.1 eV or m ν 3 ∼ m ν 1 m ν 2 ∼ 0.1 eV corresponding to normal hierarchy and inverted hierarchy respectively. As mentioned earlier, the diagonalizing transformations for the above two cases are respectively the same as the ones obtained for normal hierarchy of masses, equation (12) and for inverted hierarchy of masses, equation (13). Therefore, the conclusions regarding the texture 6 zero Dirac neutrino mass matrices corresponding to both normal and inverted hierarchy remain valid for this case also. This can be understood from Figures (1) and (2). While plotting Figures 1(a) and 1(b) the range of the lightest neutrino mass is taken to be 0.0001 eV − 1.0 eV, which includes the neutrino masses corresponding to degenerate scenario, therefore by discussion similar to the one given for ruling out texture specific mass matrices for inverted hierarchy, these are ruled out for degenerate scenario of neutrino masses as well. Similarly, for degenerate scenario corresponding to normal hierarchy of neutrino masses, Figure 2(b) clearly shows that the values of s 23 corresponding to m ν 1 0.1 eV lie outside the experimentally allowed range, thereby ruling out the mass matrices for degenerate scenario.

Summary and conclusions
To summarize, for Dirac neutrinos, we have carried out detailed calculations pertaining to minimal texture characterized by texture 6 zero Fritzsch-like mass matrices. Corresponding to these, we have considered neutrino masses having normal, inverted hierarchy as well as degenerate scenario. The compatibility of these texture specific mass matrices have been examined by plotting the parameter space corresponding to the recently measured mixing angle s 13 along with the other two mixing angles s 12 and s 23 . Further, for the normal hierarchy case, the implications of mixing angles on the lightest neutrino mass m ν 1 have also been investigated.
Interestingly, the analysis reveals that using 1σ C.L.inputs, all the texture 6 zero cases of Dirac neutrino mass matrices pertaining to normal, inverted hierarchy and degenerate scenario of the neutrino masses seem to be completely ruled out, for 3σ C.L. inputs, again these are largely ruled out.