Twenty years after the discovery of $\mu-\tau$ symmetry

It has passed 20 years after we proposed $\mu-\tau$ symmetry in light neutrino mass matrix. This model is simple but reproduced the characterestic properties of lepton sector. After that, during the experimental developments, there have appeared so many extensions but most of those phenomenological models are lacking systematic outlooks towards more fundamental theories. In this paper, we try to consider rather systematic model extensions and application to GUT model.

Indeed there have appeared a vast variety of papers during new experimental developments. Unfortunately, most of those phenomenological models are lacking systematic analyses valid for theoretical developments from phenomenological model to more fundamental one.
In this paper we reconsider µ − τ symmetry in these experimental backgrounds and try to fill this deficit. This paper is organized as follows. In section 2 we review the original µ − τ symmetric model. This model is extended in section 3. In section 4 we argue the correlaton with GUTs. Section 5 is devoted to discussions. and we obtain the mass matrix of (1). Neutrino masses are expressed in terms of A, B, C, The double sign corresponds to (1). Eq. (11) indicates that neutrinoless double beta decay does not happen in this limit, since (11) favored the small mixing angle solution for θ 12 which still had survived at that time. Also the vanishing (1,1) component is interested in connection with seesaw invariant mass matrix [1].
(c) is transformed to (d) by the interchange of C to −C and these are physically equivalent as follows. If we leave θ 23 as a free parameter and keep the assumtion (11), then M ν is reduced to where c 2 (s 2 ) ≡ cos θ 23 (sin θ 23 ) for brevity. Therefore, (c) and (d) are corresponding to s 23 = −π/4 and π/4, repecetively. θ 23 has been determined from the mixing factor sin 2 2θ 23 and they are equivalent. (a) and (b) are also substantially same and from Eq.(10) they are enforced to m 3 ≈ 0. This is the case of inverted hierarchy. This symmetric and seesaw invariant concepts is extended to two-zero texture [6]. Eq.(11) was imposed as it enables us to fix all three masses by the same three parameters as (12). You can easily generalize this simple model as Eqs. (11) and (12) in this case are generalized to If B ± C − D = A, U goes to tri-bi-maximal case [7], 3 Extension of µ − τ symmetric model The original µ−τ symmetric model was real and can not involve CP phases. So a naive extension is to extend it to a complex and symmetric mass matrix retaining µ−τ symmetry. The reasoning why we adhere to a symmetric matrix will be explained in section 4. Eq.(1) is a real symmetric matrix. Its naive extension is where D, C ∈ R, A, B ∈ C. This matrix is diagonalized by the unitary matrix [8], as Then In Eqs. (20), (21), and (23), double sign corresponds. Here x, z ∈ R, y, w ∈ C. In this complex form we have, in addition to θ 23 = π 4 , θ 13 = 0, another solution, is interesting since it is the global minimum (though 1 σ) [9]. One strategy for extending µ − τ symmetry is the following: The extensions not only explain the leptonic CP phase but also must include quark sector. This is because we are considering GUT as its more fundamental final correspondent. One of such examples preserves µ − τ symmetry up to phase but breaks the symmetric property of mass matrices [10] like, where and f includes up-type and down-type quark. First we diagonalizeM f by two orthogonal where U T B is a tri-bi-maximal mixing matrix (19). We found that these matrices are consistent with the experimental data of CKM mixing matrix. This is the extension to quark sector but is left on the same phenomenological level as the original work of lepton sector. Hereafter we restrict ourselves in symmetric mass matrices again. If we involve quark sector, it must reveal some higher symmetric (more fundamental) new character. In this sence, though there are a vast variety of these extensions, most of these phenomenological extensions have no systematic idea leading to more fundamental theoretical models. For the route from (low energy) phenomenological model to (high energy) more fundamental one, some symmetry must play an important role. Let us explain it in the well known example: QED lagrangian has U (1) and Lorentz invariances, Under T transformation, E → −E and B → B, and therefore θ → −θ, if we preserve T invariance. In another word, T invariance requires θ = 0. Lorentz invariance breaks to spatial rotation invariance for dielctics, Thus, permeability (µ) and permittivity (ǫ) characterize this symmetry breaking. Eq.(31) ((30)) may be considered as a phenomenological (a fundamental) model. According to this general idea, how the phenomenological µ − τ symmetry is incorporated to higher symmetry group or more comprehensive model ? It is natural to incorporate quark sector in this higher symmetric world. We consider here A 4 group as a candidate for it [11]. A 4 is the four degreed symmetry group with even permutation whose elements we denote as (a 1 , a 2 , a 3 , a 4 ) [12]. A 4 is generated by the S and T and their products, which satisfy The three-dimensional unitary representation, in a basis where the element S is diagonal, is built up from: Let us practice the transformation, for instance, The rule of the game for reading the permutation group of four degree from three dimensional vector is to make plus element change to a 4 and do minus signs interchange, and ST corresponds to (a 4 , a 1 , a 3 , a 2 ). Thus S means the 2 ↔ 3 symmetry and T does cyclic permutation or equivalently Z 3 . Mathematically this is the elementary example of Sylow's theorem [13]. The order of A 4 is 12 = 2 2 × 3, and it is the product of normal subgroup V 4 , composed of (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3), 1 and Z 3 . Thus A 4 ⊃ (2 ↔ 3) symmetry × Z 3 . Namely, µ − τ symmetry may be considered the residual symmetry broken from A 4 . This fact is very important for the model buiding of more fundamental theories. So far we have not considered Z 3 , and let us consider how Z 3 symmetry appears. Corresponding to this extension, we generalize from the lepton sector to the quark-lepton sector, denoting their fields as ψ i (i: generation), and call 2-3 symmetry instead of µ − τ summetry in that case. We assign Z 3 charge of each generation of fermions so as to be compatible with 2-3 symmetry [14], where ω 3 = +1. Then, the bilinear terms q Li u Rj , q Li d Rj , l Li ν Rj , l Li e Rj and ν Ri c ν Rj are transformed as follows: where Therefore, if we assume two SU(2) doublet Higgs scalars H 1 and H 2 , which are transformed as the Yukawa interactions are given as follows where Therefore, In (41), the symbol * denotes non-zero quantities. Here, in order to give heavy Majorana masses of the right-handed neutrinos ν R , we have assumed an SU(2) singlet Higgs scalar Φ 0 , which is transformed as H 1 . Mass matrices are sums of Y (1) and Y (2) and their (1,1) element must be vanished: Thus zero-texture model becomes another useful character as well as symmetric property. Then how far can we go along this line of thought ? In this case, neutrino mass matrix may have the special property of seesaw mechanism [15], and the concept of the seesaw invariance plays an important role [1,6]. Two-zero texture is interesting from the parameter counting. Mass matrix M ν is determined by 9 out-put parameters, 3 masses, 3 angles and 3 CP phases (one Dirac δ and 2 Majorana phases) in the charged lepton flavour diagonal base. Two-zero texture gives four constraints and 9-4=5 in-put free parameters [16,17,18]. Among others, the following textures are very important, They are related by µ − τ excahange, M Here As mentioned above, five parameters remain free in two-zero texture model. Therefore, if θ ij , ∆m 2 sol , ∆m 2 atm are determined, δ, ρ, σ are predicted. give similar results. This fact is used in GUT formulation as as will be discussed in the next section.

Mass matrix model and GUT
GUT models basically search the vertical structure of quark-lepton of one generation. Interfamily (horizontal) relations like Yukawa structure are not predicted. µ − τ symmetry may be clue to this extension.
In the previous section, we considered that 2 ↔ 3 symmetry and Z 3 suggest zero texture solution. So we consider GUT implimented witbh texture. For that purpose we set two requirements • GUT model itself must have few ambiguities and be predictive enough.
• The reliable mass texture model should be adopted.
The SO(10) grand unified theory (GUT) can provide the most promising framework to unify quarks and leptons, because the entire SM matter contents of each generation (including a right-handed neutrino) can be unified in a single irreducible representation, 16. A particular attention has been paid to the renormalizable minimal SO(10) model, where two Higgs multiplets {10 ⊕ 126} are utilized for the Yukawa couplings with the matter representation [19]. The couplings to the 10 and 126 Higgs fields can reproduce realistic charged fermion and neutrino mass matrices using their phases thoroughly [20,21]. 126 Higgs is selected since it includes (10, 1, 3) and (10, 1, 3) under the Pati-Salam subgroup which induce type I and type II seesaw mechanism, respectively. Yukawa coupling is given by Here c 10 , c 126 , c T , c R are comlex constants. It should be remarked that mass matrices are complex and symmetric matrices because of the group property of 10, 126 representations. Here we proceed to incorporate two-zero texture i n this model. We adopted the two-zero texture mass M (2) ν of (43). Unfortunately, the data fittings have been performed by inputting quark sector spectrum and outputted the lepton sector. In this approach, neutrino mass texture is contaminated by the special base adopted in quark sector and shows no clear texture in M ν . The reason why we adopted the quark sector as input data is that the leptoin sector had been more ambiguous than the quark sector. Nowaday, however, the situation changed. The lepton parameters are more accurate rather than the quark ones, and a large threshold correction is expected in the quark sector in SUSY models. In that sence, it is better to perform the fitting by inputting the parameters in the lepton sector. The formulation is presented in not only such a practical purpose, but also to make clear the property of the solution with v R ≈ 10 16 GeV. Here v R is the typical intermediate enrygy scale, and usualy adopted 10 13 GeV spoils the gauge coupling unifications [22]. Real data fitting revealed that in v R ≈ 10 16 GeV solution, the down quark mass is smaller than the observation, and (1,1) and (1,2) elements of (M ν ) are smaller than the other elements in the fit result. The deficit of down quark mass can be considered as the threshold correction. Under the assumtion of (M ν ) 11 = (M ν ) 12 lead us to [23] cos δ PMNS = In Fig.1 we plot the relation between δ P M N S and θ 23 in the assumption. Of course those two mass matrix elements are not exactly zero in the fits, and provides a guide to understand the fit results for the prediction of the PMNS phase depending on the mixing angle. In Fig.2, we show the plot of proton decay, τ (p → Kν), in the δ P M N S − θ 23 plane. As expected , the partial proton lifetime is larger near the curve of zero-texture in Fig.1. Near the curve, the lifetime is about 10 times bigger than the current experimental bound τ (p → Kν) > 0.59 × 10 34 years. Please see [23,24] for the detail. The dotted lines are drawn using 3σ range of the mixing parameters. We also overlap the current 1σ (red), 2σ (blue), 3σ (orange) region of the global analysis in Ref. [9]. The star symbol shows the current best fit of the global analysis. This figure is cited from [24]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Discussion
We have not connected 2 − 3 symmetry or A 4 symmetry directly with GUT symmetry SO(10). Naively it seems to be natural to consider SO(10)×A 4 symmetry. However, the merit of renormalizable minimum SO(10) GUT is that all mass matrices, as you see in (48), are represented by only two mass matrices M 10 and M 126 . So it makes this model very predictive and few room to add any assumption. If we incorporate A 4 into SO(10) GUT naively, it brings about many ambiguities on how to specify A 4 to Higgs and inflavons et.al. [26]. This spoils the high predictivity of the minimal SO(10) GUT. On the other hand, two-zero texture is very useful for full data fitting scan because we know (M ν ) 11 ≈ (M ν ) 12 ≈ 0 phenomenologically. So we can scan around this neighbourhood. And indeed we have found very good fitting around this solution.
It is very interesting that long proton decay is obtained along this solution like Fig.2. Such collaboration of GUT (fundamental theory) with phenomenological model is unprecedented. 2 − 3 symmetry and two-zero texture make clear the GUT solution as well as the practical usefulness of (43). τ (p → Kν) EXP > 0.59 × 10 34 years [25]. This figure is cited from [24].
of comprehensive data fittings. Thus we can not only fix all mass matrices but also predict many unobserved parameters, like proton decay and lepton flavour violation. Moreover, using the SUSY breaking boundary condition indicateded in [23], we can fit the other almost all known bounds like LFV and Ω DM BR(B 0 s → µ + µ − ) etc. [27]