Deviation from Tri-Bimaximal Mixing and Large Reactor Mixing Angle

Recent observations for a non-zero $\theta_{13}$ have come from various experiments. We study a model of lepton mixing with a 2-3 flavor symmetry to accommodate the sizable $\theta_{13}$ measurement. In this work, we derive deviations from the tri-bimaximal (TBM) pattern arising from breaking the flavor symmetry in the neutrino sector, while the charged leptons contribution has been discussed in a previous work. Contributions from both sectors towards accommodating the non-zero $\theta_{13}$ measurement are presented.


Introduction
Neutrino oscillations can be parametrized in terms of three mixing angles θ 12 , θ 13 where c ij ≡ cos θ ij , s ij ≡ sin θ ij , and P ν ≡ {1, e iζ 1 , e iζ 2 } is a diagonal phase matrix, which is physically relevant if neutrinos are Majorana particles. Experiments have put no constraints on the Majorana phases, therefore, we usually ignore them. The experiments have shown that the angles of lepton mixing are relatively larger than their counterparts in the quark sector. The best fit values for the mixing angles are given as [1] θ 12 = 34.5 ± 1.0( +3.2 −2.8 ) • , θ 23 = 42.8 +4.7 −2.9 ( +10.7 at 1σ(3σ) level. Recent data from the T2K [2] and MINOS [3,4] experiments have indicated a nonzero values for θ 13 . The results of the T2K are given at the 90% confidence level as 5.0 • θ 13 16.0 • (NH), 5.8 • θ 13 17.8 with δ = 0 • , where "NH" and "IH" stand for the normal and inverted neutrino mass hierarchies, respectively. The best fit values are θ 13 = 9.7 • (NH) and 11.0 • (IH). The distribution of the flavors in the mass eigenstates, corresponding to the best fit values of the mixing angles, has shown that the leading order mixing method is a successful way to describe the lepton mixing. The most common patterns that have been discussed in the literatures to describe the lepton mixing, which may arise from discrete symmetries, are called; democratic (DC) [5], bimaximal (BM) [6], and tri-bimaximal (TBM) [7] mixing matrix. The realistic mixing matrix is well approximated by the tri-bimaximal pattern Some previous studies have considered the TBM form in the symmetric limit of various flavor symmetries [8,9]. Contributions from the charged lepton sector to the leptonic mixing have been studied previously [10]. Several papers have discussed the recent T2K data [11,12]. Some of those studies have considered deviations from the charged lepton sector [11].
In Ref. [13], they have considered the decoupled 2 − 3 symmetry as the flavor symmetry of the charged lepton sector. The contributions of the charged lepton and neutrino sector have been discussed in Ref. [13] with the Bimaximal (BM) pattern being the leading order term of the lepton mixing. In this work, we assume a certain texture for the neutrino mass matrix with the third generation decoupled from the first two generations. Requiring the elements of the unitary matrix that diagonalizes the neutrino mass matrix to be independent of the mass parameters causes the leptonic mixing to have the TBM form in the symmetric limit under a certain condition.
In our model, we introduce a Lagrangian that extends the SM particle content by three right-handed neutrinos, three complex singlet scalar fields, and an additional Higgs doublet. 2 The symmetry group of the SM is extended by the product of the symmetries Z 2 × Z 4 × U(1). The Z 2 symmetry serves to have a 2 − 3 symmetric Yukawa matrix in the charged lepton sector. The Z 4 yields the mass matrices in the charged lepton and neutrino sector to have decoupled structures. We present a global U(1) symmetry which compels only the extra Higgs to generate the Dirac neutrino mass. Also, the U(1) symmetry equates certain couplings of the righthanded neutrinos as we relate the couplings to the U(1) charges. The U(1) symmetry forbids the Majorana masses of the right-handed neutrinos. The Majorana neutrino masses are generated via the v.e.v of the singlet scalars and the U(1) gets broken spontaneously. Without altering the lepton mixing, an additional Majorana mass term is introduced to protect one of the neutrino masses from blowing up.
Breaking the symmetry in the charged lepton sector has been studied in Ref. [13]. In the neutrino sector we generate a deviation to the TBM mixing by explicitly breaking the S 3 symmetry in the effective potential. The symmetry breaking term violates the alignment of the v.e.v's of the singlet scalar fields. The contribution of the neutrino sector to the deviation of the lepton mixing goes as ∼ v 2 w 2 where v is the v.e.v of the SM Higgs and w is the scale of the v.e.v of singlet scalars. Assuming w ∼ TeV, the deviations are sufficient to generate the realistic lepton mixing which is a hint that the TBM is not an accidental symmetry [15].
The paper is organized in the following manner: In Sec. 2 we study the TBM mixing in the flavor symmetric limit. In Sec. 3 we discuss the Lagrangian that describes the flavor symmetry in the charged lepton and neutrino sector. In Sec. 4 we break the flavor symmetry to generate the realistic leptonic mixing. In Sec. 5 we show the numerical results due to the symmetry breaking, and, finally, in Sec. 6 we summarize the results reported in this work.

The TBM matrix from flavor symmetry
In Ref. [13], it was assumed the Yukawa matrix of the charged lepton sector to be invariant under the µ − τ interchange. Since the leptonic mixing matrix is composed of pure numbers, Ref. [13] naturally supposed the mass matrices to be diagonalized by a unitary matrices composed of pure numbers. This results in the Yukawa matrix of the charged lepton sector to be decoupled as This Yukawa matrix leads to zero muon mass m µ = 0, which will be later the source of symmetry breaking. The Yukawa matrix is invariant under a Z 2 symmetry and diagonalized by the unitary matrix W l 23 given by In the neutrino sector we assume that, in the symmetric limit, M ν has a certain texture Here, we consider all the parameters are real. This matrix can be diagonalized by the matrix where tan 2θ 12 = 2d (a − b) .
The leptonic mixing can be described, in the symmetric limit, by the matrix where If we require θ 12 in Eq. 9 to be independent of the parameters a, b and d, then, we have d = ±k(a − b) and in particular we obtain the tri-bimaximal mixing with k = √ 2. By choosing the positive sign, where M ν is given as We see that the neutrino mass matrix exhibits decoupling of the first two generations from the third one. The mass eigenvalues are given by 3 The origin of the µ − τ symmetry in this model The Lagrangian that describes this model will be discussed in this section. It is assumed to be invariant under the product of the symmetries Z 2 × Z 4 × U(1). The Yukawa Lagrangian exhibits µ − τ symmetry, which is a Z 2 symmetry. We use the see-saw mechanism to generate the neutrino masses. The particle content of the model is given as • three left-handed lepton doublets D α L , where α is denoted by e, µ, and τ , • three right-handed charged-lepton singlets α R , and • three right-handed neutrino singlets ν αR .
In the scalar sector, we employ • three complex singlet scalar fields ǫ k with v.e.v's 0|ǫ 0 k |0 = w k , k = 1, 2, 3 . The symmetries of the Lagrangians are assumed as (The rest of the fields remain the same.) The most general Lagrangian invariant under the underlined symmetries which is similar to the Lagrangian that describes the BM structure in Ref. [13]. Here, φ j ≡ iσ 2 φ * j is the conjugate Higgs doublet. We can simplify the above Lagrangian in many ways; We can assign U(1) charges, ν (e,µ,τ )R , ǫ (1,2,3) , D (e,µ,τ ) L , (e, µ, τ ) R , φ 1 , φ 2 = {1/3, 2/3, 4/3, 4/3, 0, −1}, to compel an individual Higgs to each sector. In our model we relate the couplings to the U(1) charges as y = cq where y is a coupling, q is a U(1) charge, and c is a constant. This leads to a universal coupling to the righthanded neutrinos. Secondly, we choose to work in the basis which has the Dirac neutrino mass matrix to be diagonal. Thirdly, to reduce the number of parameters we can impose an approximate symmetry of the Lagrangian. A SU (3) symmetry where the right handed singlet fields and the left handed doublet fields transform as the SU (3) triplets leads to y 4 = y 5 . The SU(3) symmetry is only satisfied by the Dirac mass term for the neutrinos and is broken by the other terms in the Lagrangian. Fourthly, imposing a transformation on the right-handed charged leptons µ R ↔ τ R requires y 2 = y 3 leading to vanishing the muon mass. Finally, we can redefine aǫ 1 → ǫ 1 and bǫ 2 → ǫ 2 . We can then rewrite the Lagrangian as When the singlet scalar fields acquire their v.e.v's, the U(1) symmetry gets broken and the neutrinos obtain their Majorana masses [16]. One of the neutrino masses blows up, therefore, we need to introduce a Majorana mass term as a U(1) symmetry breaking term, which is not going to change the mixing, The most general scalar potential V that is invariant under the above symmetries is [13] where V 2HD (φ 1 , φ 2 ) is the potential of the two Higgs doublets, If we impose an additional S 3 symmetry on the three singlet scalar fields in the effective potential, it will be give by One can easily verify that the v.e.v's of the Higgs doublets are different and non-zero in the symmetric limit [13]. We can minimize the potential to get the v.e.v's ( 0|ǫ 0 k |0 = w k ) as follows One can notice that the three equations are not independent. Thus, we have the three v.e.v's are the same and equal to where w k = w for k = 1, 2, 3. The explicit form of the Yukawa matrix, Majorana and Dirac neutrino mass matrices can be written from the Lagrangian (17) as follows Using the seesaw formula [17], the neutrino mass matrix is given as Then M ν has the structure where One can easily verifies that the relation G = √ 2(X − Y ) in Eq. 13 is satisfied. The mass eigenvalues (2X − Y, 2Y − X, Z) can be written as From the above equations one can estimate the range of the v.e.v, v 2 . As the absolute neutrino masses are in the eV scale, therefore, v 2 has to be in the MeV scale if the see-saw scale (M) is in the TeV range. The mass eigenvalues satisfy the relation Similar relations among the masses are discussed in Ref. [18]. Thus, we can use the above sum-rule to obtain an upper limit for the heaviest mass |m 3 | 3|m 1 ||m 2 | |2|m 1 |+|m 2 || for the normal hierarchy or |m 2 | 2|m 1 ||m 3 | |3|m 1 |−|m 3 || for the inverted hierarchy.

Symmetry Breaking
The breaking of flavor symmetries in the charged lepton and neutrino sector cause deviations from the TBM form. Symmetry breaking in the charged lepton sector has been considered in Ref. [13]. They evaluated the correction angles numerically. In this section we are going to consider deviations of the TBM structure from the neutrino sector.
We are going to break the S 3 symmetry, which has led to equal v.e.v's in the symmetric limit, and maintain the symmetries of the Lagrangian. We will break the symmetry by introducing symmetry breaking terms of dimension four. We can present a large number of symmetry breaking terms. But for simplicity, we will shift the v.e.v of the two singlet scalars (ǫ 1 , ǫ 2 ) to violate the decoupling in the neutrino mass matrix. Here, we introduce the most general form of symmetry breaking terms The most general symmetry breaking terms can be expressed in terms of the form in Eq. 30 and symmetry conserving terms that can be absorbed in the symmetric potential. Thus, the scalar potential including all the terms of the form in Eq. 30 is given as follows We can parametrize the v.e.v's of the singlet scalars as We require that all terms in the symmetry breaking potential are of the same size which results in, from Eq. 30, ̺ ∼ v 2 is the EW scale. The only terms that depend on γ are f (γ) = ξβ 4 1 cos 2 2γ + β 2 1 cos 2γ After minimizing the potential, one can get the parameters of v.e.v's as follows where . (35) Then, we find the following relation is satisfied In the above equations, since ̺ ∼ v 2 β 2 1 ρ i that leads to cos 2γ ≈ 0 up to corrections of v 2 /β 2 1 where we assume β 1 in the TeV range where v is the EW scale. Then Hence, the Majorana mass matrix remains has the same structure as in Eq. 24, where v β i = yβ i . Now, we consider in our analysis the first order correction to cos 2γ (cos 2γ ≈ τ ) where the symmetry breaking term is defined by This leads to shifting the v.e.v's of the two singlet scalars ( 0 |ǫ 1 | 0 = 0 |ǫ 2 | 0 ) up to the first order of τ . Then, the Majorana neutrino mass matrix in Eq. 38 takes the form where We write the v.e.v's of the singlet scalars after symmetry breaking as then Note that, from the discussion below Eq. 32 and using Eq. 39 one finds which leads to ξ = 10̺ for τ = 0.1. One finds that breaking the S 3 symmetry to generate different v.e.v's for the singlet scalars is not sufficient to break the almost degeneracy of (m 1 , m 2 ) to satisfy the squared mass difference measurements. Therefore, we introduce an additional U(1) symmetry breaking term, Thus where M ′ = M + M 1 . Using the see-saw formula, the neutrino mass matrix is given by where From Eqs. 47, one gets the mass eigenvalues We can diagonalize the mass matrix in Eq. 47 using the unitary matrix U ν = W ν 12 R ν 23 R ν 12 with, On can find relations between the mass matrix elements in Eq. 48 Applying the above relations to the corresponding mass matrix elements, , one can get the two mixing angles Following Ref. [19], we expand the angles in Eq. 1 as where the three real parameters r, s, a describe the deviations of the reactor, solar, and atmospheric angles from their tri-bimaximal values. We use global fits of the conventional mixing parameters (s, a) [20] that can be translated into 3σ ranges and the mixing parameter r with 2.5σ significance (90% C.L.) [2] 0.12 < r < 0.39, −0.13 < s < 0.05, −0.15 < a < 0.16.
To first order in r, s, a the lepton mixing matrix can be written as [19], We are not going to consider CP violation in this work, thus, we assume that δ = 0. The present data prefers a negative value for s and r is positive [19]. We can write the parameters (r, s, a) in terms of the elements of the mixing matrix, Now, we can calculate the full deviation of the leptonic mixing coming from the charged lepton and neutrino sector. We obtain the elements of the lepton mixing matrix with U ℓ = W l 23 R l 23 R l 13 R l 12 and U ν = W ν 12 R ν 23 R ν 12 . Thus, up to the first order one can get In Ref. [13], it was found that the contribution of the charged lepton sector, with δ = 0, is give as The above results do not match the experimental values where the charged lepton sector introduces a large correction to the mixing angle θ 12 . Thus, it becomes necessary to combine the contributions come from the charged lepton and neutrino sector in order to calculate the full deviation from the TBM mixing.

Numerical results
In the case of degenerate neutrino masses m 1 ≈ m 2 ≈ m 3 , one can find from Eq. 14 that a ≈ b ≈ c. This leads to a diagonalized neutrino mass matrix M ν ≈ diag (a, a, a). That means the lepton mixing matrix does not include a contribution from the neutrino sector, which is inconsistent with the experimental data. Thus, in the symmetric limit our model excludes the case of the degenerate neutrino masses.
The numerics goes as following; we choose random values of the masses (m 1 , m 2 , m 3 ) which satisfy the experimental values of the squared mass differences

Conclusion
In this paper, we extended our model in Ref. [13] to treat the leptonic mixing in the flavor symmetric limit as the tri-bimaximal pattern. The charged lepton sector was considered in Ref. [13] in a basis where the Yukawa matrix is non-diagonal with a 2 − 3 symmetric structure except for one breaking by the muon mass. We fixed the neutrino mass matrix to have a decoupling of the first two generations from the third one, and under a certain condition we generated the lepton mixing with the TBM structure. This model was described by the Lagrangian that extended the SM by three right-handed neutrinos, an extra Higgs doublet, and three complex scalar fields. Also, the symmetry group of the SM was extended by the product of the symmetries Z 2 × Z 4 × U(1). The symmetry breaking in the charged lepton sector did not fix the data by introducing a large contribution to the mixing angle θ 12 . Thereafter, by breaking the S 3 symmetry in the effective potential and violating the alignment of the v.e.v's of the singlet scalars, the contribution of the neutrino sector was introduced to accommodate the measurements. The analysis of our model to fit the experimental constraints of the mixing angles showed that this model supported the mass normal hierarchy with masses in the few meV to ∼ 50 meV range. Also, the v.e.v of the additional Higgs was obtained in the MeV scale. The v.e.v of the singlet scalars and the see-saw scale were found to be in the TeV range. The graphs showed that Σm i ≈ 0.06 eV and M ee < M β and M ee < 0.35 eV.

Acknowledgements
The author gratefully thanks A. Datta for many enlightening discussions and comments. Also, the author thanks Yue-Liang Wu for suggesting useful references. This work was supported in part by the US-Egypt Joint Board on Scientific and Technological Co-operation award (Project ID: 1855) administered by the US Department of Agriculture, summer grant from the College of Liberal Arts, University of Mississippi, and in part by the National Science Foundation under Grant No. 1068052.