Predictions via large theta13 from cascades

We investigate a relation among neutrino observables, three mixing angles and two mass squared differences, from a cascade texture of neutrino mass matrix. We show an allowed region of the correlation by use of current data of neutrino oscillation experiments. The relation predicts sharp correlations among neutrino mixing angles as $0.315\lesssim\sin^2\theta_{12}\lesssim0.332$ and $0.480\lesssim\sin^2\theta_{23}\lesssim0.500$ with a large $\theta_{13}$ ($0.03<\sin^22\theta_{13}<0.28$). These magnitudes are modified $0.310\lesssim\sin^2\theta_{12}\lesssim0.330$ and $0.540\lesssim\sin^2\theta_{23}\lesssim0.560$ when the charged lepton mass matrix also has the cascade form.


Introduction
Current neutrino oscillation experiments suggest an existence of two large mixing angles among three generations in lepton sector [1]. It is well known that the two large mixing angles is suitably approximated by so-called tri-bimaximal mixing (TBM) [2,3], which induces mixing angle, Such a characteristic form of mixing matrix strongly motivates a study of flavor structure in the lepton sector. Actually, there are a large number of models which try to realize the TBM based on a flavor symmetry, neutrino mass textures, and so on.
In a study of suitably realization of the TBM, one peculiar relation among the neutrino observables, three mixing angles and two mass squared differences, has been proposed in [4], that is where r ≡ ∆m 2 21 /∆m 2 31 , and θ ij (i, j = 1, 2, 3), ∆m 2 21 and ∆m 2 31 are the leptonic mixing angles and two mass squared differences of neutrinos, respectively. Notice that the exact TBM satisfies this relation independently of the mass squared differences. Moreover, this relation also shows correlations among deviations from the TBM. In fact, this attractive relation is derived from so-called cascade texture [4] with hierarchical neutrino masses.
A typical cascade texture is represented by where v denotes an overall mass scale. In ref. [4], it has been pointed out that the TBM can be realized at a leading order in type-I seesaw mechanism [5]- [7] if the neutrino Dirac mass matrix is taken as the cascade form. 1 Realizations of such a cascade texture have also been discussed in terms of flavor symmetries and extra-dimensions [4,12]. We call the model which induces the cascade texture "cascade model". We here comment on a slightly modified cascade texture, called hybrid cascade texture, which is given by This can naturally fit a quark sector, masses and mixing angles. Thus, there have been some researches, where the (hybrid) cascade textures can really reproduce the suitable masses and mixing angles of the SM fermions at a low energy regime in the frameworks of SUSY SU(5) [12] and SUSY SO(10) [13] GUTs. However, it should be noticed that the TBM in the lepton sector is hardly realized by any seesaw mechanism [4,12,13].
In this letter, we investigate the relation (3) at 1σ level for normal neutrino mass hierarchy (NH). In addition there are recent observations of ν µ → ν e oscillation by T2K experiment [14], which suggested 0.03 < sin 2 2θ 13 < 0.28, at 90% C.L. for NH with δ CP = 0 [15]. This experimental result of (a non-vanishing or) large sin θ 13 motivates studies of deviation from the TBM to search a true physics which determine the lepton flavor structure, and screens a large number of neutrino (lepton) flavor models. 2 The letter is organized as follows: In section 2, we will give a brief review of the cascade model and investigate predictions from the model as focusing on the recent data of neutrino oscillation experiments. The section 3 is devoted to a summary.

Cascade model and probing a relation among neutrino observables
In this section, we present a brief review of the cascade texture and investigate predictions from it as focusing on recent data of neutrino oscillation experiments.

Cascade neutrino texture
At first, we give a brief review of the cascade neutrino texture [4]. In the cascade neutrino model, the neutrino Dirac mass matrix takes the following cascade form: This mass matrix can lead to experimentally favored (nearly TBM) mixing angles with NH in the context of type-I seesaw mechanism. Such types of mass texture have often been seen in the lepton mass models, e.g. with the vacuum alignments and non-Abelian flavor symmetry (see, for example, refs. [7] in [4]). Mass eigenvalues of light neutrinos, m i , in the model are given by in the diagonal basis of right-handed neutrino mass matrix, . The cascade neutrino model leads to the NH in order to realize the tri-bimaximal mixing at the leading order [4]. Thus, we perturbatively computed to give eqs. (9)-(11) around m 1 /m 2,3 and δ/λ, which are small quantities to be consistent with experimental values. In the same perturbation, the mixing angle are evaluated as in the diagonal basis of the charged lepton mass matrix. Notice again that this cascade neutrino model leads to the TBM at leading order. In other words, the corrections of next-leading order shift the mixing angles form the exact TBM. We will focus on this point in the following subsections.
It can be seen that there are four combinations of independent model parameters, m i and δ/λ, while the five observables exist (three mixing angles and two mass squared differences can be expressed by m i and δ/λ). Therefore, one parameter independent relation among neutrino observables must exist, that is just (3), with real parameters in the model.

Probing a relation among neutrino observables
Now let us investigate a predicted relation (3)  collaborations [34] thanks to the above strong correlations among mixing angles.

Cascade charged lepton mass texture
It might be more natural that the charged lepton mass matrix also takes the cascade form in the sense that the neutrino Dirac mass matrix of the cascade form is obtained from an U(1) flavor symmetry and/or other dynamics [4]. Thus, here we research the case when the charged lepton mass matrix also has the cascade form, Note that the magnitudes of cascade parameters, δ e and λ e , should be evaluated from the experimentally observed values of charged lepton masses, m e , m µ and m τ , which are given by It can be found from (15)-(17) that the contributions to the mixing angles from the charged lepton sector are small. Therefore, the total lepton mixing angles can be estimated at the first order perturbation as One can see that the solar neutrino mixing is little affected, on the other hand, as for the atmospheric neutrino mixing, the charged lepton effect often dominates. And the magnitude of the contribution to the reactor neutrino mixing is of negligible order. Since the hierarchy in the charged lepton mass matrix can be expressed by the observables as (16) and (17), the strong correlation among neutrino observables still holds but (3) is slightly modified as by including the charged lepton contributions in the first order approximation.
We show numerical plots with the relation (21)  correlations slightly weaken and mixing angles are predicted as 0.320 sin 2 θ 12 0.345 and 0.530 sin 2 θ 23 0.580 for 0.008 | sin θ 13 | 2 , which are shown in Fig 4. Therefore, we conclude that all leptonic Dirac mass textures of the cascade model, also predict explicit deviations from the exact TBM, and the deviations are strongly correlated with each other. These would be also checked in the future neutrino oscillation experiments with higher sensitivities.

Summary
We have studied a relation among neutrino observables, which are three mixing angles and two mass squared differences. This relation is predicted from a cascade texture with hierarchical neutrino masses. The neutrino cascade model is favored by the current neutrino oscillation experiments and is supported by theoretical studies of new physics such as the realizations from flavor symmetry, extra-dimensional theory, and embedding the model into GUTs. Since recent data of the neutrino oscillation experiments including the latest T2K result might suggest the deviations from the exact TBM, we have motivated for the precise investigations of the relation. The relation gives strong correlations among each deviation of leptonic mixing angle from the TBM.
We have numerically shown predicted regions of the relation and scatter plots from a cascade model by use of recent data of neutrino oscillation experiments in two cases. One is the model with the diagonal charged lepton mass matrix, and the other is the case of cascade form of charged lepton mass matrix also. In both cases, we can see that predictions of the cascade model and deviations from the TBM are strongly correlated among three lepton mixing angles. This is a strong advantage of the cascade model for the verifiability of the model compared with other neutrino flavor models. We have predicted 0.315 sin 2 θ 12 0.332, 0.480 sin 2 θ 23 0.500 in the case of the diagonal charged lepton mass matrix, and 0.310 sin 2 θ 12 0.330, 0.540 sin 2 θ 23 0.560 in the case of the cascade charged lepton mass matrix. Hence, we conclude that the cascade model has predicted deviations of all mixing angles from the exact TBM with a relatively large θ 13 . These would be also checked in the future neutrino oscillation experiments with higher sensitivities.
At the end of this letter, we comment on recent MINOS result, where θ 13 can be still consistent with zero [35]. Figures 1-4 suggest the correlations among θ ij even if θ 13 = 0, where the cascade predictions are slightly modified as 0.315 sin 2 θ 12 0.335. Anyhow, the cascade model predicts the strong correlation in wide range of θ 13 (as sin 2 2θ 13 < 0.28).