$\sin{\theta}_{13}$ and neutrino mass matrix with an approximate flavor symmetry

For a neutrino mass matrix whose texture has an approximate flavor symmetry and where one has near degenerate neutrino mass, it is shown that the tribimaximal values for atmospheric angle $\sin^{2}{\theta_{23}}=1/2$ and solar angle $\sin^{2}{\theta_{12}}=1/3$ can be maintained even when the reactor angle $\theta_{13}\neq0$. The non zero $\sin\theta_{13}$ implies approximate $\nu_{\mu}\rightarrow-\nu_{\tau}$ symmetry instead of $\nu_{\mu}\rightarrow\nu_{\tau}$ symmetry.


Contents 1 Introduction
There is compelling evidence that neutrinos change flavor, have non zero masses and that neutrino mass eigenstates are different from weak eigenstates. As such they undergo oscillations. The flavor and mass eigenstates are related by the so called Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix [1], where the matrix U has been parameterized by the Particle Data Group (PDG) as [2]    Here c ij = cos θ ij , s ij = sin θ ij and P is a diagonal matrix which contains (Majorana) CP violating phases in addition to the (Dirac) CP violating phase δ. θ 12 , θ 23 and θ 13 are respectively known as solar, atmospheric and reactor angles. Current global fits allow the following ranges for the mass squared differences and mixing angles [2]: 7.05 × 10 −5 eV 2 ≤ ∆m 2 12 ≤ 8.34 × 10 −5 eV 2 , 0.25 ≤ sin 2 θ 12 ≤ 0.37, 2.70 × 10 −3 eV 2 ≤ ∆m 2 31 ≤ 2.75 × 10 −3 eV 2 , 0.36 ≤ sin 2 θ 23 ≤ 0.67, (1.3) with the following best fit (BF) values ∆m 2 12 = 7.65 × 10 −5 eV 2 , sin 2 θ 12 = 0.304, ∆m 2 31 = 2.40 × 10 −3 eV 2 , sin 2 θ 23 = 0.5. (1.4) Recent results from T2K collaboration [3] and MINOS indicate a relatively large θ 13 and when combined with the global fit gives [4] sin 2 θ 13 = 0.025 ± 0.007. (1.5) There is further evidence for nonzero reactor also θ 13 from DAYA BAY [5] and RENO [6] collaborations which respectively give It is interesting on its own right to consider non-zero value for sin 2 θ 13 in the above range. In fact it has been shown by the author [7] that nonzero value of sin 2 θ 13 has important implications for the leptogenesis asymmetry parameter; its contribution to this parameter may even dominate. Before we proceed further it is instructive to summarize the theoretical framework needed. The effective Majorana neutrino mass matrix M ν constructed directly or in seesaw mechanism, can be symbolically written as [8] where L ℓ = (e L , v L ) are lepton doublets, e R charged lepton SU L (2) singlets with nonvanishing hypercharge, N R are SU L (2) × U (1) singlets. It is convenient to have a basis in which M ℓ and M R are simultaneously diagonal We can select a basis in which U L is diagonal i.e. M ℓ =diag(m e , m µ , m τ ). One may remark that the so called 2 − 3(µ − τ ) symmetry can not be simultaneously valid for lefthanded charged leptons and left-handed neutrinos. In the above basis it is obvious since m µ = m τ but in fact it is independent of what basis one chooses [9]. Thus 2 − 3(µ − τ ) symmetry can only be regarded as an effective (approximate) symmetry in the neutrino sector and was inspired by maximal atmospheric angle and θ r = 0. If θ r = 0, it has to be violated.
Since the oscillation data are only sensitive to mass squared differences, they allow for three possible arrangements of different mass levels [8]. It is customary to order the mass eigenstates such that m 2 1 < m 2 2 . We have two squared mass differences: , where ∆m 2 > 0 for normal hierarchy (m 1 m 2 ≤ m 3 ) and < 0 for inverted hierarchy (m 1 ≃ m 2 ≫ m 3 ). For degenerate case (m 1 ≃ m 2 ≃ m 3 ), one can write neutrino mass matrix as in case of opposite CP-parity of ν 2 and ν 3 (m 2 and m 3 are of opposite sign). We have three mixing angles θ 12 = θ s (solar), θ 23 = θ a (atmospheric) and θ 13 = θ r (reactor). Some new ingredients are needed to describe correctly the three mixing angles. However it is well known that the best fit values given in Eq. (1.4) are consistent with the so called tribimaximal (TB) mixing [10] corresponding to For our discussion it is convenient to state various symmetries and/or conditions on M ν which lead to θ 13 = 0 and TB mixing. In an obvious notation if one has 2-3 symmetry Recently, a possibility has been discussed [11](named TBR) which allows the extension of TB mixing, so as to have a non-zero value of θ 13 , preserving at the same time the predictions for the TB solar angle [sin 2 θ 12 = 1 3 ] and the maximal atmospheric angle [sin 2 θ 23 = 1 2 ]. To implement TBR, it is generally assumed that satisfies the conditions mentioned above. The various recent attempts in this aspect differ in the treatment of δM ν . In general δM ν contains six parameters. In [12], the conditions on these parameters are put so as to give θ r = 0 but |θ 23 − 45| ≪ 1, fixing θ s at the TB value. However these conditions are not unique. A simple form for δM ν for normal hierarchy is also invented [12]. In [13,14], δM ν is realized in specific flavor symmetry models based on S 4 and A 4 symmetries respectively.
In this paper we attempt to realize TBR in degenerate mass spectrum given in Eq. (1.8) which as we shall see has some attractive features.

Approximate flavor symmetry and diagonalization of neutrino mass matrix
For degenerate case a particularly attractive Majorana neutrino mass matrix, which is a multiple of unit matrix supplemented by three far smaller off-diagonal entries, is given by [15,16] with ǫ ij ≪ 1. This implies that in the limit of off-diagonals going to zero, M ν is flavor blind or preserves flavor symmetry. When they are switched on, but being at least an order of magnitude smaller than diagonal elements, they violate it as small perturbations A particularly attractive realization of δM ν given in Eq. (2.1)is provided by Zee model [17] where diagonal elements are vanishing and off-diagonal elements arise from radiative corrections [18].
Another realization of δM ν in Eq.(2.1) is provided by a simple extension of the standard electroweak group to [16] In addition to usual fermions there are three right-handed SU L (2) singlet neutrinos which carry appropriate U i (1) quantum numbers. Further in addition to SU L (2) Higgs doublets, there are three Higgs SU L (2) singlets Σ i with appropriate U i (1) quantum numbers. The fermions and Higgs bosons are assigned to the following representations of the group G: The Yukawa couplings of neutrinos with Higgs are given by The symmetry is spontaneously broken by giving vacuum expectation values to Higgs bosons φ (i) , Σ (i) : where Λ i ≫ v i so that extra gauge bosons which break the e − µ − τ universality become super heavy and so do the right handed neutrinos. For simplicity we shall take, v 1 = v 2 = v 3 and Λ 1 = Λ 2 = Λ 3 (any differences can be absorbed in the corresponding Yukawa coupling constants with the Higgs bosons). Then the charged lepton and Dirac neutrino mass matrices are Then the effective Majorana mass matrix for the light neutrinos is Here In order to generate M 0 ν , we introduce a right handed neutrino N and a corresponding Higgs boson Σ, both of which are SU L (2) and [U (1)] 3 singlets, with the Yukawa coupling Although a term M N N T CN is allowed in (2.9) but it can be absorbed in f Λ √ 2 after the symmetry breaking. After spontaneous symmetry breaking Σ = Λ 2 (so that M = f Λ/ √ 2) and in the basisN The effective neutrino mass matrix is then  .17)]. We may remark here that although the Lagrangian (2.9) as it stands is not flavour blind, but if one takes h 1 = h 2 = h 3 = h, then it is flavor singlet. This is easy to see as follows. In three dimensional flavor space, introduce two vectors L = (L 1 , L 2 , L 3 ) and Φ = (φ 1 , φ 2 , φ 3 ), then the Lagrangian (2.9) takes the form which is flavor single, just as (Σ.π)Λ is singlet under isospin in hadron physics. When the symmetry is broken, M 0 ν is then a multiple of unit matrix (flavor blind) and since it dominates over δM ν , M ν has approximate flavor symmetry. It is in this limited sense that we talk about approximate falvor symmetry.
We now explore the TBR possibility for the diagonalization of the neutrino mass matrix given in Eq. (2.1). In spite of its attractiveness, its diagonalization is in conflict with the neutrino data. It is instructive to show it as it would provide us a guidance for possible modification of M ν in Eq. (2.1) to obtain agreement with the experimental data. The diagonalization of M ν (we need to consider the diagonalization of δM ν as m 0 I commutes with any diagonalyzing matrix) give among others the following relation [9] where ǫ i are the eigenvalues of δM ν . Now The first of relations (2.13) give ǫ + (1 − 2 tan 2 θ r ) = ǫ − cos 2θ s (2.14) while We require ǫ − ≪ ǫ + and the relation (2.14) then implies that tan 2 θ r ≃ 1/2, contrary to the experimental data. The relation (2.12) is the consequence of det|δM ν | = 0. To avoid this, the simplest extension is that where a ≪ 1. Then the relations in Eq. (2.12) are replaced by Further while the relation (2.15a) remains the same but (2.15b) is changed to On using Eq. (2.15a) Since |∆m 2 | ≫ ∆m 2 12 , it follows that The other relations which diagonalization give are cos 2θ a ǫ + − sin 2θ a sin 2θ s sin θ r ǫ − = 0 (2.22) ǫ 12 + ǫ 13 = cos θ r sin 2θ s (cos θ a − sin θ a )ǫ − − 2(cos θ a + sin θ a ) tan θ r ǫ + (2.23) ǫ 12 − ǫ 13 = cos θ r sin 2θ s (cos θ a + sin θ a )ǫ − + 2(cos θ a − sin θ a ) tan θ r ǫ + (2.24) To proceed further it is convenient to use the expansion about the maximum atmospheric angle sin 2 θ a = 1 Then the relations (2.22-2.25) simplify to − 2tǫ + + sin 2θ s sin θ r ǫ − = 0 (2.27) We note that if ν µ ↔ ν τ (2 ↔ 3) symmetry is imposed so that ǫ 12 = ǫ 13 , t → 0, θ r → 0, the relations (2.27), (2.28) are identically satisfied. Further if a − ǫ 23 = −ǫ 12 , then Eq. (2.32) gives tan 2θ s = 2 √ 2 i.e. the TB solar angle. However if sin θ r = 0, Eq. (2.27) implies that as given in the third relation of Eq.(2.26) in Eq.(2.33), we plot θ r as a function of θ a in Fig.1 which is if sign is taken negative in Eq.(2.33). If the sign is positive, θ a is greater than 45 • and for θ r = 10 • it is 45.1 • . It is clear that one can achieve θ r ≃ 7 • to 10 • , keeping θ a around 45 • . Thus it covers recently measured values of θ r by T 2K, DAYA BAY [5] and RENO collaboration [6] On the other hand from Eqs. (1.5) and (2.28) Thus it is possible to have TB solar angle sin 2 θ s = 1 3 and almost maximal atmospheric angle sin 2 θ a ≃ 1 2 and non zero sin 2 θ r but at the cost of ν µ → ν τ symmetry as the relations if the sign is chosen to be positive for ǫ − ǫ + . If the sign is negative, ǫ 12 and ǫ 13 should be interchanged. This is meant for the rest of the manuscript.
We may remark here that there are four parameters, apart from m o , in Eq.(2.17). There are two mass differences and three mixing angles. Thus the prediction one gets is a relationship between θ r and θ a which is shown in Fig.1. All above parameters as well Fig.1 are obtained by using best fit values given in Eq.(1.4).

Summary and Conclusion
We have considered a model of approximate flavor symmetry which has near degenerate neutrino mass. It is shown that it is possible to have nonzero reactor angle while preserving the TB solar angle sin 2 θ s = 1 3 and near maximal atmospheric angle sin 2 θ a ≃ 0.5.The nonzero sin θ r implies approximate ν µ → −ν τ symmetry rather than that ν µ → ν τ symmetry.