Nearly Tri-Bimaximal Neutrino Mixing and CP Violation

We point out two simple but instructive possibilities to modify the tri-bimaximal neutrino mixing ansatz, such that leptonic CP violation can naturally be incorporated into the resultant scenarios of nearly tri-bimaximal flavor mixing. The consequences of two new ansaetze on solar, atmospheric and reactor neutrino oscillations are analyzed. We also discuss an interesting approach to construct lepton mass matrices under permutation symmetry, from which one may derive another nearly tri-bimaximal neutrino mixing scenario with no intrinsic CP violation in neutrino oscillations.


Introduction
The atmospheric and solar neutrino oscillations observed in the Super-Kamiokande experiment [1] have provided robust evidence that neutrinos are massive and lepton flavors are mixed. All analyses of the atmospheric neutrino deficit favor ν µ → ν τ as the dominant oscillation mode with the mass-squared difference ∆m 2 atm ∼ 10 −3 eV 2 and the mixing factor sin 2 2θ atm > 0.85 at the 99% confidence level. In addition, the present Super-Kamiokande and SNO [2] data indicate that the solar neutrino anomaly is most likely attributed to the matter-enhanced ν e → ν µ oscillation via the Mikheyev-Smirnov-Wolfenstein (MSW) mechanism [3] with ∆m 2 sun ∼ 10 −5 eV 2 and sin 2 2θ sun ∼ 0.6 − 0.98 at the 3σ confidence level (large-angle MSW solution). The strong hierarchy between ∆m 2 atm and ∆m 2 sun , together with the small ν 3 -component in ν e configuration restricted by the CHOOZ reactor neutrino oscillation experiment [4], implies that atmospheric and solar neutrino oscillations decouple approximately from each other. Each of them is dominated by a single mass scale, which can be set as ∆m 2 sun ≡ |m 2 2 − m 2 1 | or ∆m 2 atm ≡ |m 2 3 − m 2 2 |. The mixing factors of solar, atmospheric and CHOOZ neutrino oscillations are simply given by where V is the 3 × 3 lepton flavor mixing matrix linking the neutrino mass eigenstates (ν 1 , ν 2 , ν 3 ) to the neutrino flavor eigenstates (ν e , ν µ , ν τ ). As current experimental data favor sin 2 2θ chz ≪ sin 2 2θ sun ∼ sin 2 2θ atm ∼ O(1), two large flavor mixing angles can be drawn from Eq. (1) in a specific parametrization of V : one between the 2nd and 3rd lepton families and the other between the 1st and 2nd lepton families. So far a number of phenomenological ansätze of lepton flavor mixing with two large rotation angles, including the "democratic" ansatz [5] and the "bimaximal" ansatz [6], have been proposed and discussed [7]. In this paper we pay our particular attention to a new ansatz of the form (up to a trivial sign or phase rearrangement) proposed recently by Harrison,Perkins and Scott [8]. This so-called "tri-bimaximal" flavor mixing pattern predicts sin 2 2θ atm = 1 and sin 2 2θ sun = 8/9, consistent very well with the atmospheric neutrino oscillation data and the large-angle MSW solution to the solar neutrino problem. However, it leads also to sin 2 2θ chz = 0, implying the absence of both high-energy matter resonances and intrinsic CP violation in neutrino oscillations.
The main purpose of this paper is to discuss two simple but instructive possibilities to modify the tri-bimaximal neutrino mixing pattern in Eq. (2), such that CP violation can naturally be incorporated into the resultant scenarios of nearly tri-bimaximal flavor mixing. Two specific textures of the charged lepton mass matrix are taken into account, in order to obtain small but non-vanishing |V e3 | or sin 2 2θ chz . We find that two new ansätze have practically indistinguishable consequences on the atmospheric neutrino oscillation, but their predictions for sin 2 2θ sun , sin 2 2θ chz and leptonic CP violation are rather different. We also discuss an interesting approach to construct lepton mass matrices under permutation symmetry, from which one may derive another nearly tri-bimaximal neutrino mixing scenario with |V e3 | = 0 but with no intrinsic CP violation in neutrino oscillations.

Nearly tri-bimaximal neutrino mixing
The fact that masses of three active neutrinos are extremely small is presumably attributed to the Majorana nature of neutrino fields [9]. In this picture, the light (left-handed) neutrino mass matrix M ν must be symmetric and can be diagonalized by a single unitary transformation: The charged lepton mass matrix M l is in general non-Hermitian, hence the diagonalization of M l needs a bi-unitary transformation: The lepton flavor mixing matrix V , defined to link the neutrino mass eigenstates (ν 1 , ν 2 , ν 3 ) to the neutrino flavor eigenstates (ν e , ν µ , ν τ ), measures the mismatch between the diagonalization of M l and that of M ν : (3) and (m e , m µ , m τ ) in Eq. (4) are physical (real and positive) masses of light neutrinos and charged leptons, respectively.
In the flavor basis where M l is diagonal (i.e., U l = 1 being a unity matrix), the flavor mixing matrix is simplified to V = U ν . The tri-bimaximal neutrino mixing pattern U ν = V 0 can then be constructed from the product of two Euler rotation matrices: where s x ≡ sin θ x , c y ≡ cos θ y , and so on.
The vanishing of the (1,3) element in V 0 assures an exact decoupling between solar (ν e → ν µ ) and atmospheric (ν µ → ν τ ) neutrino oscillations. The corresponding neutrino mass matrix M ν takes the form where If m 1 ≈ m 2 holds, one may arrive at a simpler texture of Let us comment briefly on the mathematical structure of M ν obtained in Eq. (7). Indeed M ν can be decomposed as where Such a structure of the neutrino mass matrix is very similar to that giving rise to the bimaximal flavor mixing [10]. Note that the diagonalization of M ν requires an orthogonal matrix which is able to diagonalize I B and I C simultaneously. This orthogonal matrix is just V 0 given in Eq. (6). Although the decomposition of M ν shown above is by no means unique, it might have a meaningful interpretation in an underlying theory of neutrino masses with specific flavor symmetries. The tri-bimaximal neutrino mixing pattern will be modified, if U l deviates somehow from the unity matrix. This can certainly happen, provided that the charged lepton mass matrix M l is not diagonal in the flavor basis where the neutrino mass matrix M ν takes the form given in Eq. (7). As U ν = V 0 describes a product of two special Euler rotations in the real (2,3) and (1,2) planes, the simplest form of U l which allows V = U † l U ν to cover the whole 3 × 3 space should be U l = R 12 (θ x ) or U l = R 31 (θ z ) (see Ref. [11] for a detailed discussion). To make CP violation incorporated into V , we adopt the complex rotation matrices: In this case, we arrive at lepton flavor mixing of the pattern or of the pattern It is obvious that V (x) and V (z) represent two nearly tri-bimaximal flavor mixing scenarios, if the rotation angles θ x and θ z are small in magnitude. The complex phase φ x in V (x) or φ z in V (z) is the source of leptonic CP violation in neutrino oscillations.

Constraints on mixing factors and CP violation
As the mixing angle θ x or θ z arises from the diagonalization of M l , it is expected to be a simple function of the ratios of charged lepton masses. Then the strong mass hierarchy of charged leptons naturally assures the smallness of θ x or θ z , as one can see later on. Indeed a proper texture of M l which may lead to the flavor mixing pattern V (x) is where A l = m τ , B l = m µ − m e , and C l = √ m e m µ e iφx . The mixing angle θ x in V (x) is then given by On the other hand, a proper texture of M l which may give rise to the mixing pattern V (z) reads as follows: where A l = m τ − m e , B l = m µ , and C l = √ m e m τ e iφz . The mixing angle θ z in V (z) turns out to be tan(2θ In view of the hierarchy of three charged lepton masses (i.e., m e ≪ m µ ≪ m τ ), we obtain s x ≈ m e /m µ and s z ≈ m e /m τ to a good degree of accuracy. Numerically, we find θ x ≈ 3.978 • and θ z ≈ 0.972 • by using the inputs m e = 0.511 MeV, m µ = 105.658 MeV, and m τ = 1.777 GeV [12]. Now let us calculate the mixing factors of solar, atmospheric and reactor neutrino oscillations. With the help of Eqs. (1) and (12) or (13), we arrive straightforwardly at and Allowing φ x and φ z to take arbitrary values, we find that the minimal and maximal magnitudes of sin 2 2θ (x) sun and sin 2 2θ (z) sun are and sin 2 2θ (z) sun min respectively. A numerical illustration of sin 2 2θ (x) sun and sin 2 2θ (z) sun as functions of φ x and φ z is shown in Fig. 1, from which 0.816 ≤ sin 2 2θ (x) sun ≤ 0.938 and 0.873 ≤ sin 2 2θ (z) sun ≤ 0.903 can be obtained. Note that sin 2 2θ atm = 1.000 holds in both scenarios. In addition, we get sin 2 2θ (x) chz ≈ 0.01 and sin 2 2θ (z) chz ≈ 0.0006. Therefore two nearly tri-bimaximal neutrino mixing patterns are practically indistinguishable in the atmospheric neutrino oscillation experiment. It is possible to distinguish between them in the solar neutrino oscillation experiment. They can unambiguously be distinguished with the measurements of |V e3 | and CP or T violation in a variety of long-baseline neutrino oscillation experiments.
The strength of CP or T violation in neutrino oscillations, no matter whether neutrinos are Dirac or Majorana particles, is measured by a universal and rephasing-invariant parameter J [13], defined through the following equation: in which the Greek subscripts run over (e, µ, τ ), and the Latin subscripts run over (1,2,3). Considering two lepton mixing scenarios proposed above, we obtain For illustration, we typically take φ x = φ z = 90 • . Then we arrive at J (x) ≈ 0.0115 and J (z) ≈ 0.0028, respectively. The former could be determined from the CP-violating asymmetry between ν µ → ν e andν µ →ν e transitions or from the T-violating asymmetry between ν µ → ν e and ν e → ν µ transitions in a long-baseline neutrino oscillation experiment, if the terrestrial matter effects are insignificant or under control.

Further discussions and remarks
We have discussed two simple possibilities to construct the charged lepton and neutrino mass matrices, from which two nearly tri-bimaximal neutrino mixing patterns can naturally emerge. Both scenarios are compatible with the large-angle MSW solution to the solar neutrino problem, although their numerical predictions for the mixing factor sin 2 2θ sun may be different from each other. Two lepton mixing patterns are practically indistinguishable in the atmospheric neutrino oscillation experiment, but their consequences on |V e3 | and leptonic CP violation are different and distinguishable. Only one of them is likely to yield an observable CP-or T-violating asymmetry in long-baseline neutrino oscillation experiments.
There are certainly other possibilities to modify the tri-bimaximal neutrino mixing ansatz, such that non-vanishing |V e3 | (and CP violation) can naturally be incorporated into the resultant scenarios of nearly tri-bimaximal mixing. For illustration, we follow an interesting approach proposed in Ref. [8] to consider charged lepton and neutrino mass matrices of the form Clearly M l M † l is invariant under cyclic permutation of three generation indices [14], and M ν M † ν has four texture zeros [15]. Note that y was assumed to be real in Ref. [8]. One will see later on that |V e3 | = 0 may non-trivially result from the phase of y. The Hermitian matrices M l M † l and M ν M † ν can be diagonalized as follows: It is straightfoward to obtain where ω = exp(+i2π/3) andω = exp(−i2π/3); and where ϕ = arg(y). Then the lepton flavor mixing matrix V = U † l U ν , which describes the mismatch between the diagonalization of M l M † l and that of M ν M † ν , takes the form The tri-bimaximal neutrino mixing (up to a trivial sign or phase rearrangement [8]) can then be reproduced from V in the limit ϕ = 0. The nearly tri-bimaximal mixing scenario obtained in Eq. (28) leads to sin 2 2θ sun = 8 9 cos 2 ϕ 2 , We plot the changes of sin 2 2θ sun , sin 2 2θ atm and sin 2 2θ chz as functions of ϕ in Fig. 2, where the experimental upper bound sin 2 2θ chz < 0.1 has been taken into account. One can see that the allowed ranges of ϕ are 0 ≤ ϕ ≤ 0.125π (or 22.5 • ) and 2π ≥ ϕ ≥ 1.875π (or 337.5 • ). Accordingly, we obtain 0.939 ≤ sin 2 2θ atm ≤ 1 and 0.962 ≤ sin 2 2θ atm ≤ 1. We also obtain 0.855 ≤ sin 2 2θ sun ≤ 0.889 for both ranges of ϕ. Note that J = 0 holds exactly, although V is complex. Therefore no intrinsic CP violation could be observed in neutrino oscillation experiments, if the tri-bimaximal neutrino mixing pattern or its revised version in Eq. (28) were correct. The result J = 0 makes such a nearly tri-bimaximal ansatz less interesting. Of course, the complex phases in V may have significant effects on the neutrinoless double beta decay, if neutrinos are Majorana particles.
Finally let us remark that both the tri-bimaximal mixing pattern and its possible extensions require some peculiar flavor symmetries to be imposed on the charged lepton and neutrino mass matrices. It is likely that one of the three nearly tri-bimaximal neutrino mixing patterns under discussion serves as the leading-order approximation of a more complicated flavor mixing matrix. For the time being, however, such simple ansätze are very instructive and useful to explore the main features of lepton flavor mixing and CP violation through neutrino oscillations. We expect that more delicate neutrino oscillation experiments in the near future can help pin down the explicit pattern of neutrino mixing, from which one may get some insight into the underlying flavor symmetry and its breaking mechanism responsible for the origin of both lepton masses and leptonic CP violation.
The author would like to thank H. Fritzsch for useful discussions and comments. This work was supported in part by the National Natural Science Foundation of China.  Figure 1: The mixing factors sin 2 2θ (x) sun and sin 2 2θ (z) sun against arbitrary values of φ x and φ z in two nearly tri-bimaximal neutrino mixing patterns.