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Relatively large theta13 and nearly maximal theta23 from the approximate S3 symmetry of lepton mass matrices

We apply the permutation symmetry S_3 to both charged-lepton and neutrino mass matrices, and suggest a useful symmetry-breaking scheme, in which the flavor symmetry is explicitly broken down via S_3 ->Z_3 ->nothing in the charged-lepton sector and via S_3 ->Z_2 ->nothing in the neutrino sector. Such a two-stage breaking scenario is reasonable in the sense that both Z_3 and Z_2 are the subgroups of S_3, while Z_3 and Z_2 only have a trivial subgroup. In this scenario, we can obtain a relatively large value of the smallest neutrino mixing angle, e.g., theta_{13} ~ 9^\circ, which is compatible with the recent result from T2K experiment and will be precisely measured in the ongoing Double Chooz and Daya Bay reactor neutrino experiments. Moreover, the maximal atmospheric mixing angle theta_{23} ~ 45^\circ can also be obtained while the best-fit value of solar mixing angle theta_{12} ~ 34^\circ is assumed, which cannot be achieved in previous S_3 symmetry models.


I. INTRODUCTION
Flavor symmetry is currently a promising and widely-adopted approach to understanding lepton mass spectra and neutrino mixing pattern [1]. In particular, a lot of attention has recently been paid to discrete flavor symmetries, such as A 4 [2][3][4][5] and S 4 [6][7][8][9][10][11][12][13][14], which are able to predict the tri-bimaximal neutrino mixing with θ T 12 = 35.3 • , θ T 23 = 45 • and θ T 13 = 0 that is well compatible with neutrino oscillation experiments [15][16][17][18]. However, the latest result from T2K experiment indicates that θ 13 is likely to be not vanishing but relatively large. At the 90% confidence level, the T2K data are consistent with in the case of normal neutrino mass hierarchy; and in the case of inverted neutrino mass hierarchy, for a vanishing Dirac CP-violating phase δ = 0 [19]. As a matter of fact, the global-fit analyses of neutrino oscillation experiments before the T2K result have already shown some hint on a nonzero θ 13 [20][21][22]. For instance, the latest best-fit values of three neutrino mixing angles are θ 12 = 34 • , θ 23 = 46 • and θ 13 = 6 • [22]. No doubt the symmetry-breaking terms in the A 4 or S 4 models can account for a relatively large θ 13 , but one has to avoid the resultant large corrections to θ T 12 and θ T 23 , which are already in excellent agreement with experimental data [23][24][25][26].
In this paper, we reconsider the S 3 symmetry and its explicit breaking for lepton mass matrices, and demonstrate that a relatively large θ 13 can be achieved while both θ 12 and θ 23 are in good agreement with neutrino oscillation experiments. Note that we shall follow a phenomenological approach and work at the mass-matrix level, however, the derived patterns of lepton mass matrices and the proposed symmetry-breaking scheme may be helpful for the model building at the field-theory level. Our work differs from previous ones in several aspects. First, we apply the S 3 symmetry to both charged-lepton and neutrino mass matrices. In Refs. [41,42,45], the S 3 symmetry has only been applied to the neutrino mass matrix, and the exactly or nearly tri-bimaximal neutrino mixing can be derived. Second, we propose an interesting symmetry-breaking scheme, i.e., S 3 → Z 3 → ∅ for charged leptons and S 3 → Z 2 → ∅ for neutrinos. Such a two-stage breaking scenario is quite natural, because both Z 3 and Z 2 are the subgroups of S 3 , while Z 3 and Z 2 only have a trivial subgroup. As a consequence of this breaking scheme, the charged-lepton mass matrix is non-symmetric, while neutrino mass matrix is still symmetric as it should be, because neutrinos are assumed to be Majorana particles. Third, we can get both a relatively large θ 13 and a nearly maximal θ 23 , which cannot be reached in the previous S 3 models [37,44].
In Sec. II, the lepton mass matrices in the S 3 -symmetry limit and the symmetry-breaking terms are constructed. The phenomenological implications for lepton mass spectra and neutrino mixing angles are explored in Sec. III. We summarize our conclusions in Sec. IV.

II. S 3 SYMMETRY AND ITS BREAKING
From the phenomenological point of view, the lepton masses and mixing angles at low energies are determined by lepton mass terms where M ℓ stands for the mass matrix of charged leptons, and M ν for the effective mass matrix of Majorana neutrinos. The latter can be realized in various neutrino mass models, such as seesaw models, which extend the standard model by introducing singlet or triplet fermions, or triplet scalars [49].
As usual, we can decompose the lepton mass matrices into a symmetry-limit part and a symmetry-breaking perturbation term: In the S 3 -symmetry limit, the Lagrangian in Eq. (3) is invariant under the transformation The three-dimensional representations of all six group elements are Thus M where the real and positive parameters c ℓ and c ν set the mass scales of charged leptons and neutrinos, respectively. Since both M the neutrino mixing matrix turns out to be an identity matrix. In other words, the democratic mixing arising from the charged-lepton sector gets too large corrections from the neutrino sector, or vice versa. Additionally, the first two generations of leptons are exactly degenerate in mass. In the limit of r ν = 0, the neutrino mass matrix is diagonal and we obtain the democratic mixing. But neutrinos are exactly degenerate in mass. The pertur-bation terms explicitly breaking the S 3 symmetry are necessary to generate realistic lepton mass spectra and neutrino mixing angles.
Note that the group elements of S 3 can be categorized into three conjugacy classes C 0 = S (123) , C 1 = S (231) , S (312) and C 2 = S (213) , S (132) , S (321) . It is straightforward to show that the invariant subgroup of S 3 is the cyclic group of order three where we have defined the identity element as e ≡ S (123) and the generator of Z 3 group as a ≡ S (231) . With the explicit representations in Eq. (5), one can immediately verify a 3 = e.
The S 3 group has three Z 2 subgroups The Z (23) 2 group can be identified with the µ-τ symmetry, which has been extensively discussed in connection with the maximal atmospheric mixing angle and the small reactor mixing angle [50,51].
Obviously, it is natural to explicitly break S 3 symmetry to its subgroups. Along this line, we propose to construct the perturbation terms ∆M ℓ and ∆M ν as such that the flavor symmetry is explicitly broken down via the chain in the charged-lepton sector, while via a distinct chain in the neutrino sector. It is worthwhile to remark that the mass term breaking S 3 to Z 3 is proportional to S (231) or S (312) , which is a non-symmetric matrix, thus the symmetrybreaking chain in Eq. (11) is only allowed for charged leptons. Neutrino mass matrix must be symmetric because we have assumed neutrinos to be Majorana particles as in a class of seesaw models.
It is easy to show that ∆M can always be cast into the following form by redefining the parameters c ℓ and r ℓ in M ℓ . The first-order perturbation term in the neutrino mass matrix can be written as which reduces the S 3 symmetry to Z . Nevertheless, the second-stage perturbation terms ∆M (2) ℓ and ∆M (2) ν , which respectively break down the residual Z 3 and Z 2 symmetries, could have many different forms. As both of them are intended for breaking the mass degeneracy between the first and second generations, we choose the diagonal form for simplicity [37] ∆M Now that ∆M

III. LEPTON MASSES AND MIXING ANGLES
Now we are ready to figure out the lepton mass spectra and neutrino mixing angles.
In general, the charged-lepton mass matrix M ℓ is an arbitrary complex matrix and can be diagonalized by a bi-unitary transformation U † ℓ M ℓŨℓ = Diag{m e , m µ , m τ }, where m α (for α = e, µ, τ ) are charged-lepton masses and the matrices U ℓ andŨ ℓ are unitary. SinceŨ ℓ is associated with the right-handed fields of charged leptons and has nothing to do with the lepton flavor mixing, it is more convenient to consider the Hermitian matrix H ℓ ≡ M ℓ M † ℓ , which can be diagonalized as U † ℓ H ℓ U ℓ = Diag{m 2 e , m 2 µ , m 2 τ }. Furthermore, we shall work in the so-called hierarchy basis, where the relevant matrix is In the leadingorder approximation, we arrive at with a rational assumption of |r ℓ |, |δ ℓ |, |ǫ ℓ | ≪ |ε ℓ | < 1. After diagonalizing the above matrix via V † ℓ H ′ V ℓ = Diag{m 2 e , m 2 µ , m 2 τ }, one obtains three charged-lepton masses Defining m 0 = c ℓ (2r ℓ − δ ℓ )/6, we have |m 0 | < m µ . The small parameters ε ℓ , δ ℓ and ǫ ℓ can be expressed in terms of charged-lepton masses and the m 0 parameter For δ ℓ = 0, one immediately reproduces the same results in Ref. [44], where the perturbation term ∆M (1) ℓ is absent in the charged-lepton mass matrix. The unitary matrix U ℓ = V D V ℓ is found to be where φ ≡ arctan[2ǫ ℓ /(3δ ℓ )] gives rise to the Dirac CP-violating phase. Comparing Eq. (19) with the counterpart in Ref. [44], we can observe that the additional symmetry-breaking term ∆M (1) ℓ or the parameter δ ℓ can be determined by measuring the CP violation in neutrino oscillations, which is indeed to be performed in the long-baseline neutrino experiments. It is worth mentioning that there are five real parameters in the charged-lepton mass matrix (i.e., c ℓ , r ℓ , δ ℓ , ǫ ℓ and ε ℓ ), which can be expressed in terms of charged-lepton masses (m e , m µ , m τ ) and (m 0 , φ). The latter two enter into the neutrino mixing matrix, and can be determined by neutrino mixing angles and the Dirac CP-violating phase, as we shall show later.
2. The smallest neutrino mixing angle θ 13 is given by while the Dirac CP-violating phase by Note that θ 13 receives contributions both from charged-lepton and neutrino sectors. If m 0 = 0 and δ ν = 0 are taken, as in Ref. [37], we get sin θ 13 ≈ 2m e /3m µ ≈ 0.057 or θ 13 ≈ 3.2 • by inputting m e = 0.4866 MeV and m µ = 102.718 MeV at the electroweak scale [53]. As observed in Ref. [44], when m 0 is switched on and set to m 0 < 0 and |m 0 | > m e , one can get relatively large values of θ 13 and saturate the upper bound for m 0 ≈ −14 m e . In our scenario, the sizable θ 13 can be obtained even for somewhat smaller |m 0 | due to the δ ν /ε ν term.
3. The atmospheric mixing angle θ 23 is given by which can also be enhanced due to the symmetry-breaking term ∆M can be nearly maximal in our scenario with a nonvanishing δ ν .

IV. REMARKS AND CONCLUSIONS
How to understand lepton mass spectra and neutrino mixing pattern remains an open question in elementary particle physics. Flavor symmetry is currently a powerful tool to tackle this longstanding problem. In this paper, we apply the S 3 symmetry to both chargedlepton and neutrino mass matrices. In order to explain realistic lepton mass spectra and neutrino mixing angles, the S 3 symmetry is explicitly broken down via S 3 → Z 3 → ∅ in the charged-lepton sector, while via S 3 → Z 2 → ∅ in the neutrino sector. Along this line, the mass matrices of charged leptons and neutrinos are constructed step by step. Some interesting features of this model have been explored: • It seems reasonable that the flavor symmetry first breaks down to its subgroups. The permutation group S 3 contains only two kinds of non-trivial subgroups, i.e., Z 3 and Z 2 .
For the breaking chain S 3 → Z 3 → ∅, the mass matrix has to be non-symmetric, so it is only allowed for charged leptons. Neutrinos are assumed to be Majorana particles, which are actually realized in various seesaw models. Therefore, neutrino mass matrix should be symmetric, which is not spoiled in the S 3 → Z 2 → ∅ breaking chain.
• After the flavor symmetry breaking, lepton mass matrices are determined by ten parameters, i.e., c f , r f , δ f , ǫ f and ε f for f = ℓ, ν. It has been shown that all of them are completely fixed by the ten observables in the lepton sector, namely three chargedlepton masses (m e , m µ , m τ ), three neutrino masses (m 1 , m 2 , m 3 ), three neutrino mixing angles (θ 12 , θ 23 , θ 13 ) and one Dirac CP-violating phase δ. If leptonic CP violation is finally measured in the future long-baseline neutrino experiments, the model parameters will be fully determined. In light of the recent T2K indication of relatively-large θ 13 , the discovery of CP violation in neutrino oscillations seems very promising.
• In our symmetry-breaking scheme, it has been found that θ 13 and θ 23 can receive large corrections from the S 3 symmetry-breaking terms. More explicitly, both corrections from charged leptons and neutrinos are significant for obtaining a relatively-large θ 13 , while θ 23 is mainly sensitive to the breaking term in the neutrino mass matrix. We show that both a relatively-large θ 13 and a nearly-maximal θ 23 can be accommodated simultaneously, which is not the case for previous S 3 symmetry models.
Finally, it is worth mentioning that the S 3 → Z 2 → ∅ chain can be applied to charged leptons. In addition, the two-stage breaking scheme could also be applicable to quarks, and may be helpful in understanding quark mass spectra, mixing angles and CP violation.
It should be very interesting if the mass spectra and mixing patterns for both quarks and leptons can be understood in this way, and a renormalizable field-theory model with the S 3 symmetry can be constructed to realize the derived fermion mass matrices. We leave these issues for future works.