Realistic Tri-Bi-Maximal neutrino mixing

We propose a generalized version of the Tri-Bi-Maximal (TBM) ansatz for lepton mixing, leading to non-zero reactor angle $\theta_{13}$ and CP violation. The latter is characterized by two CP phases. The Dirac phase affecting neutrino oscillations is nearly maximal ($\delta_{CP} \sim \pm \pi/2$), while the Majorana phase implies narrow allowed ranges for the neutrinoless double beta decay amplitude. The solar angle $\theta_{12}$ lies nearly at its TBM value, while the atmospheric angle $\theta_{23}$ has the TBM value for maximal $\delta_{CP}$. Neutrino oscillation predictions can be tested in present and upcoming experiments.


A. TBM Limit
The first is the limit θ, ρ, σ → 0, in which case our gTBM mixing matrix in Eq. (2) reduces to the simplest celebrated TBM form, U 0 in Eq. (1). This is unrealistic, as it can not describe reactor neutrino data.

B. Complex TBM Limit
In the limit of θ → 0 and any arbitrary value of ρ, σ, the matrix reduces to "complex TBM" matrix which is TBM matrix with additional CP phases. This matrix is given by The phases ρ and σ are physical parameters only if neutrinos are Majorana-type, and can be rotated away otherwise. Indeed, for Dirac neutrinos there is no difference between TBM and complex TBM. For the Majorana neutrino case the phases in the symmetric parametrization are given as φ 12 = ρ and φ 23 = σ, while the Dirac phase δ CP is unphysical, since θ 13 = 0.

C. The µ − τ Symmetric Limit
We now discuss realistic limits of gTBM that lead to θ 13 = 0, as required by current data [4][5][6]. One of the properties of the TBM matrix was the so-called µ − τ symmetry, i.e. |U µj | = |U τ j |; j = 1, 2, 3 [1,10]. For σ → 0 and any arbitrary values of θ, ρ, the gTBM matrix also retains this symmetry, reducing to Indeed, one sees that the matrix in (5) also has an inherent µ − τ symmetry, leading to maximal atmospheric angle θ 23 = π 4 and maximal CP violating value of CP phase δ CP = ± π 2 . The other two angles are also nonzero and are correlated with each other, as follows Figure 1. Correlation between sin 2 θ13 and sin 2 θ12 given in Eq. 6. Notice that in the whole experimentally allowed range [2], the value of sin 2 θ12 remains very close to 1/3.
Using the 3σ range of the reactor mixing angle 1.96 × 10 −2 ≤ sin 2 θ 13 ≤ 2.41 × 10 −2 [2,3], we obtain for the solar mixing angle 0.346 ≤ sin 2 θ 12 ≤ 0.349. This is illustrated in Fig. 1, in which the shaded boxes highlight the 1 and 3σ regions indicated by the current neutrino oscillation global fit. This correlation is rather different from the one predicted in [11]. The additional CP phases are physical, both Majorana and Dirac, since θ 13 = 0 makes φ 13 also well defined. This µ − τ symmetric case has implications for m ee , shown in the Fig. 5.
In the µ − τ symmetric matrix of Eq. (5), one can further take the ρ → 0 limit, in which case we get an even simpler matrix given by Notice that this matrix shares many properties of matrix in Eq. (5) e.g. maximal atmospheric angle, maximal CP violation and the correlation given in Eq. (6). In addition, the Majorana phase is fixed, since now ρ = 0, leading to very sharp predictions for m ee as shown in Fig.  2. For example, for the case of inverse ordering (IO) the neutrinoless double beta decay amplitude is nearly maximal, while for the NO case there is a lower bound for this amplitude, since destructive interference is prevented. Here θ is taken as a free parameter, and we require the three mixing angles to lie in their allowed 3σ regions [2,3]. Note that mee does not depend on σ.

D. The ρ → 0 Limit
So far the limits we have discussed all lead to maximal atmospheric mixing angle i.e. they all predict θ 23 = π/4. While this is consistent with current data, there is a slight preference for the second octant [2,3]. Our proposed gTBM matrix is flexible enough to allow for deviations from maximal θ 23 . The possibility of non-maximal θ 23 can be seen in the limiting case where ρ → 0, where the mixing matrix is given by This matrix still shares some of the properties of the µ − τ symmetric matrix of (5). For example, the correlation in (6) still holds, relating solar and reactor angles as shown in Fig. 1. However, in contrast to the µ − τ symmetric limit, we can now have deviations from maximal atmospheric mixing, as well as deviations from maximal CP violation. In fact, these departures are correlated with each other, as shown in Fig. 3, which also highlights the 1 and 3σ regions indicated by the current neutrino oscillation global fit [2,3]. The mixing matrix of (8) also leads to fixed Majorana phase values given by φ 12 = 0, φ 13 = π 2 implying sharp predictions for m ee , as shown in Fig. 2.
From these mixing angles and phases in Eq. (10), one can further obtain two non-trivial relations given by tan 2θ 23 cos δ CP = 5 sin 2 θ 13 − 1 4 tan θ 12 sin θ 13 The first is a correlation between θ 12 and θ 13 , shown in Fig. 1 while the second is a correlation between θ 23 and δ CP , depicted in Fig. 3. Owing to the constrained nature of the mixing angles and phases of our ansatz, one also gets predictions for m ee shown in Fig. 5.
The predictions made by the gTBM ansatz can also be tested in currently running and upcoming neutrino oscillation experiments. The predictions made by gTBM to oscillation experiments is illustrated in Fig. 6. This estimate is for the T2K setup, neglecting matter effects, as an approximation. Clearly the allowed range of electron neutrino appearance probability at T2K is substantially restricted w.r.t. the generic expectation.
In conclusion we have proposed a realistic generalization of the TBM ansatz which not only accounts for nonzero measured value of θ 13 but also makes definite and testable predictions for the other parameters of the lepton mixing matrix, including CP phases. Our gTBM matrix is characterized in terms of three independent parameters, which determine all six mixing parameters, leading to several testable predictions as we discussed at length. Apart from correcting for θ 13 , the gTBM matrix retains many of the features of the original TBM matrix from  Figure 6. The allowed range of electron neutrino appearance probability at T2K covers a more restricted region, thanks to the gTBM predictions. Here black line corresponds to the best fit, the cyan region is the general three-neutrino result, while the yellow region is the gTBM prediction. point of basic underlying symmetries, as we showed by discussing various limits of the gTBM matrix.
Before closing we comment on the theoretical origin of the gTBM matrix. We note that this ansatz may be derived systematically by the method of generalized CP symmetries [12][13][14]. In this approach one starts from the TBM matrix and exploits various associated CP symmetries. For example, the mixing matrix in Eq. 7 can be derived from the S 4 flavor symmetry and generalized CP [15,16]. A detailed derivation of the gTBM ansatz from the generalized CP approach, as well as other consequences of this methodology will be discussed elsewhere.