$\theta_{13}$ in Neutrino Mass Matrix with the Minimal Texture

We show that the neutrino mass matrix with the minimal texture leads to the neutrino mass and mixing that are empirically determined, including $\theta_{13}$ that is measured recently at a high precision. This mass matrix, where the Majorana nature of the neutrino is essential, only allows the normal hierarchy of the neutrino mass, excluding either inverse mass hierarchy or degenerate mass neutrinos, and hence predicts the effective mass of double beta decay to lie within the range $m_{ee}=3.7-5.6$ eV.

Weinberg [1] has pointed out, within the two generations of quarks, that the relation is described between the quark mass and the generation mixing if the quark mass matrix takes a simple texture. Namely, with the 2×2 mass matrix the vanishing (1,1) element gives the mixing angle written as a square root of the quark mass ratio of the down and the strange quarks, (m d /m s ) 1/2 in agreement with experiment. Fritzsch extended this argument to three generations assuming the minimal texture of the 3×3 matrix, having off-diagonal (1,2) and (2,3) elements in addition to one diagonal (3,3) matrix element retained [2].
Following the discovery of the finite mass of neutrinos and, at the same time surprisingly large mixing in the neutrino sector, the present authors [3] proposed a mass matrix with the minimal texture, where neutrinos are assumed to be of the Majorana type so that the mixing angle is a quartic root of the neutrino mass ratio and hence takes a large value unlike that in the quark sector. This matrix was shown to give empirically derived mixing angles at a good accuracy, and even more accurately as experiments develop [4]. The novel prediction characteristics of this mass matrix is that the mixing between the first and third components cannot be very small, for which only an upper limit was known at the time. Using all available data relevant to neutrino oscillation in the year 2003, the θ 13 is predicted to lie in the range sinθ 13 ≃ 0.04 − 0.2 (sin 2 2θ 13 = 0.006 − 0.15). This means that vanishingly small θ 13 , if measured to be so, would have falsified the assumption of the minimal texture mass matrix.
θ 13 was the last mixing angle to be measured to determine the three Kobayashi-Maskawa angles in the neutrino sector. It was within the last year that θ 13 was measured to be finite, 0.03 < sin 2 (2θ 13 ) < 0.28 at the 90% C.L. (for normal neutrino mass hierarchy) by the T2K experiment [6]. This was immediately confirmed by MINOS [7] and Double Chooz [8] experiments. Most recently it was measured at the Daya Bay reactor neutrino experiment [5] at a high precision to give sin 2 2θ 13 = 0.092 ± 0.016 ± 0.005, which is 5.2 σ to vanishing.
Motivated by the agreement of the minimal texture mass matrix with experiments, we scrutinise further the model. We show that the neutrino mass is well constrained within the present model, and there are no uncertainties that allow the mass matrix to give the inverted mass hierarchy or degenerate mass neutrinos. This allows us to predict the size of double beta decay without ambiguities. It would be important to see if this minimal mass matrix ansatz would be falsified in future experiments.
Here, we briefly recapitulate our minimal texture hypothesis [3,4] and the resulting conse-quences. Our model consists of the mass matrices of the charged leptons and the Dirac neutrinos of the form [2] where each entry is complex. We assume that neutrinos are of the Majorana type and take, for simplicity, the right-handed Majorana mass matrix to be the unit matrix, as where M 0 is much larger than the Dirac neutrino mass scale. We may consider a more general case where M R has three different eigenvalues. In such a case, however, the differences in the eigenvalues can be absorbed into the wave functions of the right-handed neutrinos, which leads to the violation of the symmetric matrix structure (the minimal structure) of the neutrino mass.
Therefore, it amounts to the increase of the complexity of the structure in the mass matrix.
We remark that our neutrino mass is stable against radiative corrections. For the heavy right-handed neutrino of mass O(10 10 )GeV the Yukawa coupling for the Dirac neutrino mass is smaller than 10 −2.5 . A calculation with the renormalisation group equation (e.g., [9]) gives the radiative correction to be 10 −6 relative to the leading term, which is negligible in our argument.
We obtain the three light neutrino masses, m i (i = 1, 2, 3), as The lepton mixing matrix is given by where the expressions of U ℓ and U ν are given in [4], and Q is a phase matrix written which is a reflection of phases contained in the charged lepton mass matrix and the Dirac mass matrix of neutrinos. The Majorana phases can be neglected in the right-handed neutrino mass matrix, because, in the presence of mass hierarchy of the left-handed neutrinos, the effect of phases in the right-handed mass matrix is suppressed by the mass hierarchy, so that the inclusion of phases changes the analysis only a little.
Since the charged lepton masses are known, the number of parameters contained in our model is six: m 1D , m 2D , m 3D , σ, τ and M 0 , which are to be determined by empirical neutrino masses and mixing angles. We note that this is the minimum texture of the 3 × 3 neutrino mass matrix in the sense that reducing one more matrix element (i.e., letting A, B or C to zero) leads to the neutrino mixing that is in a gross disagreement with experiment. An antisymmetric mass matrix with 3 finite elements also leads to a gross disagreement with experiment, as one can readily see.
The lepton mixing matrix elements are written approximately, where the charged lepton mass is denoted as m e , m µ and m τ , and terms with (m e /m τ ) 1/2 are neglected. Approximate characteristics of the mixing angles can be seen from these expressions.
For instance, we see from these equations the relations between neutrino masses and mixing angles, The relation among the mixing angles is With |U µ3 | ∼ 1/ √ 2 and |U e2 | ∼ 1/ √ 3 we see that |U e3 | ∼ 1/(2 √ 6) ≃ 0.2, which means that |U e3 | cannot be too small.
We emphasize that only the normal neutrino mass hierarchy is allowed in our model, which allows us to predict uniquely, up to errors, the effective mass that appears in double beta decay.
We can see that this matrix does not allow the inverted hierarchy, which is seen easily from the expression of |U µ3 | that should empirically be around 1/ √ 2 [4]. It can also be seen that this matrix does not allow degenerate neutrinos, since they require |U e2 | close to 1/ √ 2, which is excluded by experiment for θ 12 (see Figure 9 below).
We now present our results using the accurate expression of the lepton mixing angles given in [4]. We take [5, 10] where θ ij are the lepton mixing angles defined in the standard manner, and we leave out for the moment θ 13 for the input. ∆m 2 atm and ∆m 2 sol represent mass difference squares relevant to atmospheric neutrino (θ 23 ) and solar neutrino (θ 12 ) experiments. We take the error at the 90% confidence level throughout our analysis. We take the lowest neutrino mass m 1 as a free parameter; m 2 and m 3 are given by ∆m 2 atm and ∆m 2 sol . The phase parameters σ and τ are constrained reasonably well to give empirical sin 2 θ 23 and sin 2 θ 12 .
We consider the allowed region of |U e3 | = sin θ 13 . Figures 1 and 2 show sin θ 13 versus sin 2 2θ 12 and sin 2 2θ 23 , respectively, where the other parameters are marginalised. The ranges of sin 2 2θ 12 and sin 2 2θ 23 are cut at the boundary of the region experimentally allowed at the 90% confidence level. We stress that the predicted range of θ 13 coincides with the value obtained by the Daya Bay experiment.
We see that the maximum mixing, θ 23 = π/4 (i.e., sin 2 (2θ 23 ) = 1), is excluded. Figure 3 shows sin θ 13 as a function of m 3 . It is noted that only the normal mass hierarchy, m 3 ≫ m 2 , m 1 , is allowed in our mass matrix. hierarchy and the degenerate neutrinos are not allowed with our mass matrix, the effective mass that appears in neutrinoless double beta decay is uniquely predicted: We may also calculate the rephasinginvariant CP violation measure J CP defined by in Figure 8 as a function of sin θ 13 . The figure shows that vanishing J CP is still marginally allowed, but some further improvement for the allowed range of θ 13 will lead to non-vanishing J CP . (If we would take the one sigma range for θ 13 , vanishing J CP would not be allowed any more.) We said earlier that the inverted hierarchy and the degenerate neutrino masses are not allowed. While the former is seen from the expression given above, the latter is seen in Figure   9, where sin 2 (2θ 12 ) is shown as a function of m 1 /m 3 . A large m 1 /m 3 requires maximum mixing between the first and the second generation, sin 2 (2θ 12 ) = 1, which is already not allowed by experiment. This excludes the degenerate case.  Figure 9: sin 2 θ 12 as a function of the mass ratio m 1 /m 3 . Data points are limited to the empirical 90% confidence limit of θ 12 , and light shade indicates how the region would be extended if we do not limit the range of θ 12 .
We have shown that the neutrino mass and mixing derived from the neutrino mass matrix of the minimal texture satisfy all experiments, without adding any further matrix elements or extending the assumption. In particular, θ 13 falls in the middle of the range that is measured recently. The matrix only allows the normal hierarchy of the neutrino mass, excluding either inverse hierarchy or degenerate mass cases. This predicts the effective mass of double beta decay to lie within the range m ee = 3.7 −5.6 meV and the total mass of the three neutrinos to be 61 ±2 meV. Further improvement on the allowed range of θ 13 would exclude vanishing CP violation.
It would be interesting to see if this simple texture mass matrix model will be falsified in future experiments when the accuracy is improved.