Friedberg-Lee Symmetry Breaking and Its Prediction for theta_13

We consider an effective Majorana neutrino mass operator with the Friedberg-Lee symmetry; i.e., it is invariant under the transformation \nu_\alpha ->\nu_\alpha + z (for \alpha = e, \mu, \tau) with z being a space-time independent constant element of the Grassmann algebra. We show that this new flavor symmetry can be broken in such a nontrivial way that the lightest neutrino remains massless but an experimentally-favored neutrino mixing pattern is achievable. In particular, we get a novel prediction for the unknown neutrino mixing angle \theta_13 in terms of two known angles: \sin\theta_13 = \tan\theta_12 |(1- \tan\theta_23)/ (1+\tan\theta_23)|. The model can simply be generalized to accommodate CP violation and be combined with the seesaw mechanism.

Recent solar [1], atmospheric [2], reactor [3] and accelerator [4] neutrino experiments have convincingly verified the hypothesis of neutrino oscillations. The latter can naturally happen if neutrinos are slightly massive and lepton flavors are not conserved. The mixing of three lepton families is described by a 3 × 3 unitary matrix V , whose nine elements are usually parameterized in terms of three rotation angles (θ 12 , θ 23 , θ 13 ) and three CP-violating phases (δ, ρ, σ) [5]: with c ij ≡ cos θ ij and s ij ≡ sin θ ij (for ij = 12, 23 and 13). A global analysis of current neutrino oscillation data yields 30 • ≤ θ 12 ≤ 38 • , 36 • ≤ θ 23 ≤ 54 • and 0 • ≤ θ 13 < 10 • at the 99% confidence level [6], but three phases of V remain entirely unconstrained. While the absolute mass scale of three neutrinos is not yet fixed, their two mass-squared differences have already been determined to a quite good degree of accuracy [6]: ∆m 2 21 ≡ m 2 2 − m 2 1 = (7.2···8.9)×10 −5 eV 2 and ∆m 2 32 ≡ m 2 3 −m 2 2 = ±(2.1···3.1)×10 −3 eV 2 at the 99% confidence level. The on-going and forthcoming neutrino oscillation experiments aim to measure the sign of ∆m 2 32 , the magnitude of θ 13 and even the CP-violating phase δ. How to understand the smallness of θ 13 and the largeness of θ 12 and θ 23 is a real challenge. So far many neutrino mass models have been proposed [7]. Some of them follow such a guiding principle: there exists an underlying flavor symmetry in the lepton sector and its spontaneous or explicit breaking gives rise to the observed pattern of V . This is certainly a reasonable starting point for model building, and it might even shed light on the true flavor structures of leptons and quarks.
The purpose of this paper is just to follow the above-mentioned guideline to explore a simple and testable correlation between the neutrino mass spectrum and the neutrino mixing pattern. In the basis where the charged-lepton mass matrix is diagonal, we hypothesize that the effective Majorana neutrino mass operator is of the form where a, b and c are in general complex, and ν c αL ≡ Cν αL T (for α = e, µ, τ ). A salient feature of L mass is its translational symmetry; i.e., L mass is invariant under the transformation ν α → ν α + z with z being a space-time independent constant element of the Grassmann algebra. Note that this new kind of flavor symmetry was first introduced by Friedberg and Lee in Ref. [8] to describe the Dirac neutrino mass operator. Here we apply the same symmetry to the case of Majorana neutrinos and then reveal a completely new way to break it. Corresponding to Eq. (2), the Majorana neutrino mass matrix M ν reads The diagonalization of M ν is straightforward: This result, which is an immediate consequence of the Friedberg-Lee (FL) symmetry in L mass , implies that one of the three neutrinos must be massless. In the FL model [8] such a flavor symmetry is broken by an extra term of the form m 0 (ν e ν e + ν µ ν µ + ν τ ν τ ) added into the Dirac neutrino mass operator, hence all the three neutrinos are massive. One may wonder whether it is possible to break the FL symmetry in L mass but keep m 1 = 0 or m 3 = 0 unchanged? 1 We find that the simplest way to make this possibility realizable is to transform one of the neutrino fields ν α into κ * ν α with κ = 1. Taking ν e → κ * ν e for example, the resultant Majorana neutrino mass operator takes the form Accordingly, the neutrino mass matrix is given by We are then left with Det(M ′ ν ) = κ 2 Det(M ν ) = 0, which is independent of the magnitude and phase of κ. Thus we obtain either m 1 = 0 or m 3 = 0 from M ′ ν . Our next step is to show that a generic bi-large neutrino mixing pattern, which is compatible very well with current experimental data, can be derived from M ′ ν . Let us focus on the m 1 = 0 case 2 , in which M ′ ν is diagonalized by the transformation As the best-fit values of the atmospheric and CHOOZ neutrino mixing angles are θ 23 = π/4 and θ 13 = 0 respectively [6], we may decompose the neutrino mixing matrix V into a product of three special unitary matrices: and P = 1e iγ with the definitionsĉ ≡ cos θ e iφ ands ≡ sin θ e iϕ . Note that u * 1 is a column vector associated with m 1 = 0 (i.e., M ′ ν u * 1 = 0 holds). This observation, together with the unitarity of U, allows us to obtain One can see that U is only dependent on κ, a free parameter characterizing the strength of FL symmetry breaking in L ′ mass . Apparently, If a, b and c are all complex, the phases of U † M ′ ν U * will finally be absorbed into the phases of P (γ) and R (φ and ϕ).
For simplicity, we assume that a, b and c are all real. In this case, U † M ′ ν U * is a real symmetric matrix. Hence the phases of R and P (i.e., γ, φ and ϕ) can be switched off and the rotation angle of R is determined by One may observe that b = c, which is a clear reflection of the µ-τ permutation symmetry in L ′ mass or M ′ ν [9], simply leads to θ = 0 or equivalently θ 23 = π/4 and θ 13 = 0. In addition, two non-vanishing neutrino masses are obtained from Eq. (9) as follows: Nine elements of the neutrino mixing matrix V = URP can be given in terms of κ and θ: After rephasing this expression of V with the transformations of charged-lepton and neutrino fields e → e iφ κ e, µ → −µ, ν 1 → e iφ κ ν 1 and ν 3 → −ν 3 , where φ κ ≡ arg(κ), we may directly compare it with the parametrization given in Eq. (1). Then we arrive at tan θ 12 = √ 2|κ| cos θ , tan θ 23 = 2|κ| 2 + 1 − tan θ 2|κ| 2 + 1 + tan θ , sin θ 13 = √ 2|κ|| sin θ| where θ 12 , θ 23 and θ 13 are required to lie in the first quadrant, θ is close to zero due to the smallness of θ 13 but it may be either positive (in the first quadrant) or negative (in the fourth quadrant). Furthermore, we have δ = 0 (when θ < 0) or δ = π (when θ > 0) together with σ = π, while the CP-violating phase ρ is not well-defined in the m 1 = 0 case. Thus we conclude that there is no CP violation in this simple neutrino mass model, although its mass operator involves a complex parameter κ. Now that the mixing angles θ 12 and θ 23 are already known to a reasonable degree of accuracy, we can use them to determine the unknown mixing angle θ 13 and the unknown magnitude of κ from Eq. (13). Indeed, This interesting expression indicates that the deviation of θ 13 from zero is closely correlated with the deviation of θ 23 from π/4. It is a novel prediction of our model, which can easily be tested in the near future. On the other hand, |κ| = sin θ 12 √ cos 2θ 12 + sin 2θ 23 .
In the above discussions we have taken ν e → κ * ν e to break the FL symmetry and achieve a realistic pattern of the neutrino mass matrix. One may similarly consider ν µ → κ * ν µ or ν τ → κ * ν τ with κ = 1. In either possibility it is easy to show that m 1 = 0 or m 3 = 0 holds, but the neutrino mixing pattern turns out to be disfavored by current experimental data. We find that there is no way to simultaneously obtain large θ 23 and tiny θ 13 in the m 3 = 0 case, no matter whether ν µ → κ * ν µ or ν τ → κ * ν τ is taken. As for the m 1 = 0 case, it is straightforward to get the neutrino mixing matrix from Eq. (12) with the interchange of its first and second rows (when ν µ → κ * ν µ is concerned) or its first and third rows (when ν τ → κ * ν τ is concerned). We observe that |κ| ∼ 1 is required to assure θ 23 ∼ π/4 and θ 13 ∼ 0 in the leading-order approximation, either for ν µ → κ * ν µ or for ν τ → κ * ν τ . But |κ| ∼ 1 will give rise to an excessively large value of θ 12 (e.g., θ 12 > π/4), which has been ruled out by the solar neutrino oscillation data. Hence neither ν µ → κ * ν µ nor ν τ → κ * ν τ with κ = 1, which automatically breaks the µ-τ permutation symmetry, is favored to reproduce the exactly or approximately tri-bimaximal neutrino mixing pattern.
It is worth mentioning that the present scenario has the same number of free parameters as the previous one. Taking account of current experimental data on ∆m 2 21 , ∆m 2 32 , θ 12 and θ 13 , we arrive at 0.026 eV < ∼ a < ∼ 0.032 eV, −0.010 eV < ∼ Re(b) < ∼ −0.005 eV and −0.013 eV < ∼ Im(b) < ∼ 0.013 eV from a straightforward calculation. Finally we point out that it is possible to derive the Majorana neutrino mass operator L ′ mass from the minimal seesaw model (MSM) [12], a canonical extension of the standard model with only two heavy right-handed Majorana neutrinos. The neutrino mass term in the MSM can be written as where ν L and N R denote the column vectors of (ν e , ν µ , ν τ ) L and (N 1 , N 2 ) R fields, respectively. Provided the mass scale of M R is considerably higher than that of M D , one may obtain the effective (left-handed) Majorana neutrino mass matrix M ′ ν from Eq. (17) via the well-known seesaw mechanism [13]: As M R is of rank 2, Det(M ′ ν ) = 0 holds and m 1 = 0 (or m 3 = 0) is guaranteed. We find that the expression of M ′ ν given in Eq. (6) can be reproduced from M D and M R if they take the following forms: where Λ D characterizes the mass scale of M D . For simplicity, we require a, b and c to be real and get the mass eigenvalues of M R Given Λ D ∼ 174 GeV (i.e., the scale of electroweak symmetry breaking), a ∼ 0.022 eV and b ∼ c ∼ 0.006 eV as the typical inputs, the masses of two right-handed Majorana neutrinos turn out to be M 1 ∼ 1 × 10 15 GeV and M 2 ∼ 3 × 10 15 GeV, which are quite close to the energy scale of grand unified theories Λ GUT ∼ 10 16 GeV. Note that the textures of M D and M R taken in Eq. (18) are by no means unique, but they may serve as a good example to illustrate how the seesaw mechanism works to give rise to M ′ ν or L ′ mass in the MSM. To summarize, we emphasize that the FL symmetry is a new kind of flavor symmetry applicable to the building of neutrino mass models. Imposing this symmetry on the effective Majorana neutrino mass operator, we have shown that it can be broken in such a novel way that the lightest neutrino remains massless but an experimentally-favored bi-large neutrino mixing pattern is achievable. This phenomenological scenario predicts a testable relationship between the unknown neutrino mixing angle θ 13 and the known angles θ 12 and θ 23 in the CPconserving case: sin θ 13 = tan θ 12 |(1 − tan θ 23 )/(1 + tan θ 23 )|. Such a result is suggestive and interesting because it directly correlates the deviation of θ 13 from zero with the deviation of θ 23 from π/4. We have discussed a simple but instructive possibility of introducing CP violation into the Majorana neutrino mass operator, in which the soft breaking of µ-τ permutation symmetry yields δ = π/2 (or δ = −π/2) but keeps θ 23 = π/4. We have also discussed the possibility of incorporating our scenario in the MSM.
In conclusion, the FL symmetry and its breaking mechanism may have a wealth of implications in neutrino phenomenology. The physics behind this new flavor symmetry remains unclear and deserves a further study.
One of us (Z.Z.X.) likes to thank Z. Chang, X.D. Ji, M. Li, J.X. Lu and S. Zhou for very useful discussions. This work was supported in part by the National Natural Science Foundation of China.