Democratic Neutrino Mixing Reexamined

We reexamine the democratic neutrino mixing ansatz, in which the mass matrices of charged leptons and Majorana neutrinos arise respectively from the explicit breaking of S(3)_L x S(3)_R and S(3) flavor symmetries. It is shown that a democracy term in the neutrino sector can naturally allow the ansatz to fit the solar neutrino mixing angle \theta_sun \approx 33^\circ. We predict \sin^2 2\theta_atm \approx 0.95 for atmospheric neutrino mixing and J \approx 1.2% for leptonic CP violation in neutrino oscillations without any fine-tuning. Direct relations between the model parameters and experimental observables are also discussed.

The recent solar [1], atmospheric [2], KamLAND [3] and K2K [4] neutrino oscillation experiments provide us with very compelling evidence that neutrinos are massive and lepton flavors are mixed. To account for the observed neutrino mass-squred differences (∆m 2 sun ∼ 6.9 × 10 −5 eV 2 and ∆m 2 atm ∼ 2.3 × 10 −3 eV 2 ) and mixing factors (sin 2 2θ sun ∼ 0.84 and sin 2 2θ atm ∼ 1.0 [5]), many phenomenological models of lepton mass matrices have been proposed in the literature [6]. Some of them take advantage of the idea of flavor democracy, from which the largeness of two lepton mixing angles, the smallness of three quark mixing angles, and the wide mass gaps between (m τ , m t , m b ) and their lighter counterparts can simultaneously be understood.
The original ansatz of democratic neutrino mixing [7] is based on the phenomenological conjecture that charged lepton and Majorana neutrino mass matrices may arise from the breaking of S where c l and c ν measure the corresponding mass scales of charged leptons and neutrinos. The explicit symmetry-breaking term ∆M l is responsible for the generation of muon and electron masses, and ∆M ν is responsible for the breaking of neutrino mass degeneracy. A very simple form of ∆M l and ∆M ν reads [7] ∆M l = c l 3 where (δ l , ε l ) and (δ ν , ε ν ) are real dimensionless perturbation parameters of small magnitude, and the imaginary phase of ∆M l is a natural source of leptonic CP violation in neutrino oscillations. Because M ν is already diagonal, we only need to diagonalize M l by means of the orthogonal transformation V M l V T = Diag{m e , m µ , m τ }, in order to express the leptonic charged-current interactions in terms of the mass eigenstates of charged leptons and neutrinos. The lepton flavor mixing matrix is just given by the unitary matrix V ; i.e., Given m e /m µ ≈ 0.00484 and m µ /m τ ≈ 0.0594 [8], the mixing factors of solar and atmospheric neutrino oscillations turn out to be The result of sin 2 2θ sun is obviously disfavored by current solar neutrino data, and that of sin 2 2θ atm apparently deviates from the maximal atmospheric neutrino mixing. A simple way to suppress the afore-obtained value of sin 2 2θ sun and enhance that of sin 2 2θ atm is to add another S(3)-symmetry term, which was not included in Eq. (1), into the neutrino mass matrix M ν [9]. In this case, we have where r ν is in principle an arbitrary parameter. To get large lepton mixing angles, however, |r ν | ≪ 1 must be satisfied. It is shown in Ref. [9] that the r ν -induced corrections to sin 2 2θ sun and sin 2 2θ atm can both be constructive, and ∆m 2 sun ∼ (1 − 2) × 10 −4 eV 2 is predicted by taking the appropriate parameter space of (c ν , r ν , δ ν , ε ν ) 1 . Note that ∆m 2 sun ∼ O(10 −4 ) eV 2 is no more favored by today's experimental data on solar neutrino oscillations. It is therefore necessary to reexamine whether a favorable bi-large neutrino mixing pattern can naturally be derived from the explicit breaking of S(3) L × S(3) R symmetry of charged leptons and S(3) symmetry of Majorana neutrinos. If the answer remains affirmative, then direct and testable relations between the model parameters and experimental observables should be established.
The main purpose of this short paper is to demonstrate that the r ν -modified version of our phenomenological ansatz is actually compatible with current neutrino oscillation data. We find that the experimentally-favored value of sin 2 2θ sun can naturally be achieved. We derive a simple relation between sin 2 2θ atm and cos 2θ sun , and then arrive at the prediction sin 2 2θ atm ≈ 0.95 without any fine-tuning. We also show how to relate the model parameters to the relevant observables. Our analytical results will be very useful to test the democratic neutrino mixing scenario, when more accurate experimental data are available in the near future.
For simplicity, we take c ν , r ν , δ ν and ε ν in Eq. (5) to be real and positive. Then M ν can be diagonalized by means of a real orthogonal transformation U T M ν U = Diag{m 1 , m 2 , m 3 }. It is obvious that three neutrino masses must be nearly degenerate. Taking the convention m 1 < m 2 < m 3 , we obtain The near degeneracy of three neutrino masses implies that the effective mass-squared term of the tritium beta decay, defined as m 2 e ≡ (m 2 i |V ei | 2 ) for i = 1, 2 and 3, approximately amounts to c 2 ν . In other words, c ν ≈ m e holds. Then we obtain c ν < 2.2 eV from the direct-mass-search experiments [8] and c ν < 0.23 eV from the recent WMAP observational data [10]. In view of ∆m 2 atm ≫ ∆m 2 sun , we require ε ν ≫ r ν and ε ν ≫ δ ν . Therefore, As for the orthogonal matrix U, its nine elements U 1i , U 2i and U 3i (for i = 1, 2, 3) have the following relations: To the accuracy of O(r ν /ε ν ), the expression of U is found to be where tan 2θ ≡ r ν /δ ν . Without loss of generality, θ is required to lie in the first quadrant. It is clear that U becomes the unity matrix in the limit r ν = 0. In the case of r ν = 0, the lepton flavor mixing matrix takes the form V = V U, where V has been given in Eq. (3).
We explicitly obtain V ≈ V 0 + V 1 as a good approximation, in which and Comparing V with V , we see that V e3 ≈ V e3 holds. This result implies that the mixing angle θ 13 in the standard parametrization of V [8] is rather small: or θ 13 ≈ 3.2 • . On the other hand, eight other elements of V may get appreciable r ν -induced corrections.
With the help of Eqs. (10) and (11), the solar neutrino mixing factor is obtained as It follows that θ ≈ (45 • − θ sun ) holds. In other words, θ measures the deviation of θ sun from 45 • . As observed in Ref. [11], the sum θ sun + θ C ≈ 45 • with θ C being the Cabibbo angle of quark mixing is favored by current experimental data. In this case, we are then left with θ ≈ θ C ≈ 12 • . The ratio r ν /δ ν can in turn be determined in terms of the mixing angle θ sun : r ν /δ ν ≈ cot 2θ sun . Typically taking the best-fit value θ sun ≈ 33 • , we arrive at r ν /δ ν ≈ 0.44. One may also estimate the magnitude of r ν /ε ν with the help of Eq. (7). The result is where ∆m 2 sun /∆m 2 atm ≈ 3 × 10 −2 and θ sun ≈ 33 • have been used. It is then clear that ε ν ≫ δ ν ∼ r ν holds. Now let us calculate the atmospheric neutrino mixing factor sin 2 2θ atm by using Eqs. (10) and (11). We obtain Comparing between Eqs. (4) and (15), we find that the r ν -induced correction to sin 2 2θ atm is constructive but suppressed by ∆m 2 sun /∆m 2 atm ∼ O(10 −2 ). We conclude that the maximal atmospheric neutrino mixing cannot be achieved in a simple and natural way, unless the ratio ∆m 2 sun /∆m 2 atm is as large as of O(10 −1 ). A more precise determination of ∆m 2 sun , ∆m 2 atm , θ sun and θ atm will test the validity of Eq. (15). The consequences of this phenomenological ansatz on the neutrinoless double beta decay and CP violation in neutrino oscillations are interesting. A straightforward calculation yields m ee ≡ | (m i V 2 ei )| ≈ c ν for the effective mass of the neutrinoless double beta decay. It becomes obvious that m ee ≈ m e ≈ c ν holds. The absolute neutrino mass scale in our ansatz can be fixed either from a measurement of the tritium beta decay or from a positive signal of the neutrinoless double beta decay. The Jarlskog invariant of CP violation [12] is found to be Such a strength of leptonic CP violation is likely to be observed in a long-baseline neutrino oscillation experiment.
One can see that the democratic neutrino mixing ansatz under discussion is compatible with all of current neutrino data. Its prediction for sin 2 2θ atm , which apparently deviates from the maximal atmospheric neutrino mixing, can easily be tested in the near future. Of course, part of our results depend on the explicit symmetry breaking patterns (i.e., ∆M l and ∆M ν ). Let us comment on the effects of S(3) flavor symmetry breaking terms in some detail: (a) The lepton flavor mixing matrix V is insensitive to the form of ∆M l , as already observed in Ref. [7]. The point is simply that the strong mass hierarchy of three charged leptons makes the contribution of ∆M l to V insignificant, no matter whether ∆M l is diagonal or off-diagonal.
(b) If a contrived and fine-tuned pattern of ∆M ν is taken, it should be possible to obtain a "proper" (2,3)-rotation angle from M ν in order to arrive at θ atm ∼ 45 • . However, it is more natural to consider the simple forms of ∆M ν such as the diagonal perturbation given in Eq. (2), at least from the point of view of model building [13].
(c) A remarkable advantage of the diagonal perturbation ∆M ν is that it guarantees M ν to be stable against radiative corrections [9,14], although three mass eigenvalues of M ν are almost degenerate. This feature makes sense for model building too, because the S(3) L × S(3) R symmetry of M l and the S(3) symmetry of M ν are most likely to manifest themselves at a high energy scale (e.g., the seesaw scale [15], where three heavy right-handed neutrinos might also have an approximate flavor democracy [16]).
Finally, it is worth emphasizing that four free parameters of M ν may all be determined in terms of the relevant observable quantities. We obtain c ν ≈ m e ≈ m ee . Then ε ν ≈ ∆m 2 atm /(2 m 2 e ) can straightforwardly be derived from Eq. (7). With the help of Eq. (14), we further arrive at Note that the magnitudes of ε ν and δ ν should be of or below O(0.1), because they are perturbative parameters of ∆M ν . Taking ε ν ∼ 0.1, for instance, we may get m e ≈ m ee ∼ 0.1 eV. To measure such a small m e in the tritium beta decay is practically difficult (but not impossible) in the near future [17]. In comparison, m ee ∼ 0.1 eV is definitely accessible in a number of planned experiments for the neutrinoless double beta decay [18]. This numerical example indicates that ε ν ∼ O(0.1) is most plausible. A much smaller ε ν would make m i ≈ c ν (for i = 1, 2, 3) too large to be compatible with the WMAP upper limit on m i , while a much bigger ε ν would loss its physical meaning as a perturbative parameter.
In summary, we have reexamined the democratic neutrino mixing ansatz by taking into account an extra S(3)-symmetry term in the Majorana neutrino mass matrix. After explicit symmetry breaking induced by the diagonal perturbations, we obtain the mass spectrum of charged leptons with a strong hierarchy and that of neutrinos with a near degeneracy. The suppressed democracy term in the neutrino sector can naturally permit the model to fit current solar neutrino oscillation data with θ sun ≈ 33 • . We have derived a simple relation between sin 2 2θ atm and cos 2θ sun , and achieved the prediction sin 2 2θ atm ≈ 0.95 without any fine-tuning. Whether this atmospheric neutrino mixing factor is really maximal or not will provide a sensitive test of our phenomenological ansatz. We have also established the direct relations between the model parameters and relevant experimental observables. We remark that the democratic neutrino mixing scenario is simple, viable and suggestive. It could be useful for model building, in particular at a high energy scale at which the S(3) L × S(3) R symmetry of charged leptons and the S(3) symmetry of Majorana neutrinos are expected to become relevant.
One of us (Z.Z.X.) is grateful to S. Zhou for useful discussions and partial involvement at the early stage of this work, which was supported in part by the National Natural Science Foundation of China.