Smallness of Leptonic $\theta_{13}$ and Discrete Symmetry

The leptonic mixing angle $\theta_{13}$ is known to be small. If it is indeed tiny, the simplest explanation is that charged leptons mix only in the $\mu-\tau$ sector and neutrinos only in the 1-2 sector. We show that this pattern may be explained by the discrete symmetry $Z_2 \times Z_2$ of a complete Lagrangian, which has 2 Higgs doublets and 2 Higgs triplets (or 2 heavy right-handed singlet neutrinos). In the case of Higgs triplets, the Majorana neutrino masses are arbitrary, whereas in the case of heavy singlet neutrinos, an inverted hierarchy is predicted. Lepton-Flavor-Violation effects, present only in the $\mu-\tau$ sector, are analyzed in detail: the LFV $\tau$-decay rates are predicted below the present bounds by a few orders of magnitude, whereas LFV Higgs decays could allow for a direct test of the model.

Recent experimental advances in measuring the neutrino oscillation parameters in atmospheric and solar data [1] have now fixed the 3 × 3 lepton mixing matrix U to a large extent.
Assuming that the neutrino mass matrix M ν is Majorana and it is written in the basis where the charged-lepton mass matrix is diagonal, then for present data imply that θ 23 is close to π/4, θ 12 is large but far from π/4, and θ 13 is small and consistent with zero (sin 2 θ 13 ≤ 0.047 at 3σ C.L. [2]).
If data will significantly strengthen the upper bound on θ 13 , this will imply a very special pattern for the violation of lepton flavor, which begs for a theoretical rationale. In fact, it is possible to define quantitatively and experimentally when the 1 −3 mixing can be considered negligible: a value as tiny as sin 2 θ 13 ≤ 10 −4 can be generated by gravity effects alone [3] and neutrino factories could be sensitive to such small mixing [4].
The question of whether the origin of the lepton mixing U = U † l U ν is in the neutrino or the charged-lepton sector has been discussed in many recent papers, e.g. [5,6,7,8,9,10].
Just from the form of Eq. (2), it is apparent which is the most simple-minded realization of zero 1 − 3 mixing: besides the diagonal contributions to the neutrino and charged-lepton mass matrices M ν and M l , one needs to generate off-diagonal entries only in the 1 −2 sector of M ν and in the µ − τ sector of M l . In this case the atmospheric mixing originates in the charged-lepton sector and the solar mixing in the neutrino sector. In particular, this hybrid scenario has been shown to be generically associated with small values of θ 13 [11].
We point out in this paper that the above-mentioned hybrid scenario with θ 13 = 0 is realized by a discrete symmetry of the Lagrangian of a complete theory, with distinct experimentally verifiable predictions. Other models predicting θ 13 = 0 have also been proposed [12,13,14,15].
Consider the discrete symmetry Z 2 × Z 2 , also known as the Klein group. There are 4 possible representations, i.e. (+, +), (+, −), (−, +), and (−, −). Suppose the 3 lepton families transform as follows: with 2 Higgs doublets and 2 Higgs triplets Then the charged-lepton mass matrix linking l i to l c j is given by where the diagonal entries a, b, c are induced by φ 0 1 , and d, e by φ 0 2 , and the Majorana neutrino mass matrix is given by where A, B, C come from ξ 0 1 , and D from ξ 0 2 . The Higgs triplets are assumed to be very heavy (∼ M ξ ), so that they acquire naturally small vacuum expectation values (∼ φ 0 i 2 /M ξ ) [16].
Then M l is diagonalized by a rotation in the 2 − 3 sector and M ν by a rotation in the 1 − 2 sector. Hence U is exactly of the form desired with θ 13 = 0 (models predicting Eqs. (6) and (7) by using different discrete symmetries can be found in [13]). In particular, with s L = s 23 , c L = c 23 . As for θ 12 , it is determined by Eq. (7) which also allows for arbitrary m 1,2,3 . In other words, this model does not constrain any mass or mixing other than θ 13 = 0, but it identifies this particular limit as the result of a well-defined symmetry. Of course CP violation is not observable in oscillations, but it can appear in neutrinoless 2β decay, since m i are in general complex parameters.
One can ask the question if it is crucial for the above scenario to use Higgs triplets ξ i (type II seesaw) instead of right-handed neutrinos N i (type I seesaw). In this last case the predictions depend on the source of the N i Majorana masses. For definiteness, one can assume this source to be given by Higgs singlets S i which acquire super-heavy vacuum expectation values. In order to reproduce as closely as possible the above pattern, let us make the following assignments: Then the neutrino mass matrix is given by (a general method to obtain texture zeros in type I seesaw matrices using flavor symmetries can be found in [17]). In this case the diagonalization of M D requires also a right-handed rotation, analogously to Eq. (8). Therefore a non-zero θ 13 is in general induced. However, an interesting physical limit exists, such that θ 13 is maintained to be zero, i.e. M 3 ≡ C R → ∞, so that the heaviest N 3 decouples, then it is easy to check that θ 13 → 0 and, at the same time, m 3 → 0. The smallness of θ 13 is now related to the inverted hierarchy of the spectrum.
Alternatively, we can simply eliminate N 3 from the beginning, i.e. keep only two right-handed neutrinos as in Ref. [18], which obtained the same result using a U(1) flavor symmetry. In this scenario, since φ 2 contributes both to M l and to M D , the observable left-handed 2 − 3 mixing angle receives contributions from both U l and U ν .
There are only two other ways for θ 13 to be zero in Eq. (10).  Table 1. The physical charged Higgs boson is given by where φ − 1,2 couple to leptons as in Table 1, with µ replaced by ν µ and τ by ν τ respectively.
In the case of M ν generated by Higgs triplets as in Eq. (7), we have θ L = θ 23 . In the limit θ L = θ 23 = π/4 (which is preferred by the data) and neglecting m µ versus m τ , the coupling of h − to leptons is given by where tan β = v 2 /v 1 . This implies that instead of m 2 µ /m 2 τ , as in the usual (MSSM like) two Higgs doublet models. Thus this ratio is, in general, not suppressed and is a good experimental test of this model.

The neutral Higgs boson of the Standard Model (with the usual Yukawa couplings to leptons) is
but it is not in general a mass eigenstate in a two-Higgs-doublet model. It mixes with which couples to leptons, in the same limit as in Eq. (12), according to In general, H 0 may also mix with which couples to leptons according to If the Higgs potential has exact Z 2 × Z 2 symmetry, then one can check that CP is conserved and A 0 is a mass eigenstate (with odd CP ) and does not mix with h 0 and H 0 which are even under CP . The decay of A 0 is thus another distinct signature of this model: the branching fractions of A to τ + τ − , τ + µ − + µ + τ − , and µ + µ − are proportional to cos 2 2β, cos 2 2θ R , and sin 2 2θ R respectively. If Z 2 × Z 2 is allowed to be broken by soft terms of the Higgs potential, then CP is violated and all 3 neutral Higgs bosons A 0 , h 0 , H 0 mix with one another. In the following we assume, for simplicity, that A 0 is a mass eigenstate and that where h 0 1,2 are the eigenstates with masses m 1,2 .
Consider first τ → 3µ. It proceeds through A 0 and h 0 exchange. Although h 0 mixes with H 0 , the latter does not couple toμτ and its coupling toμµ is proportional to m µ . We obtain where m h 0 is the effective contribution of h 0 exchange: Numerically, for m A = m h 0 = 100 GeV and sin 2θ R cos 2θ R / sin 2 2β = 1, this implies a branching fraction of 4.5 × 10 −9 , well below the present experimental upper bound [20] of 1.9 × 10 −6 .
In the same approximation as above, the radiative decay rate of τ → µγ is given by where and with k 1 ≡ sin 2 α − sin α cos α tan 2β, k 2 ≡ cos 2 α + sin α cos α tan 2β. Numerically (using We computed also the contribution to the anomalous magnetic moment of the muon a µ ≡ (g µ − 2)/2 from 1-loop diagrams mediated by h − , h 0 and A 0 : where k 1 ≡ sin 2 α, k 2 ≡ cos 2 α. Using the parameter values given above, δa µ ≈ 6.2 × 10 −13 , that is much smaller than the present uncertainty (∼ 10 −9 ) and therefore negligible as a possible explanation of the discrepancy (∼ 3 × 10 −9 ) between the Standard Model prediction [21] and the experimental value [22].
As shown in Ref. [15], this matrix by itself has a Z 2 symmetry. This may also be understood by its form invariance [23], i.e. where The matrix of Eq. (26) was in fact obtained previously as the remnant of a complete D 4 × Z 2 model [14]. Another model [24] based on the quaternion group Q 8 also obtains this structure (if one CP phase is put to zero) with the further restriction If c 23 = s 23 in Eq. (26), then M ν has the Z 2 symmetry proposed in Ref. [25], which is realized in the A 4 model [26], with m 1 = m 2 = −m 3 (before radiative corrections). These examples and others in Ref. [13] show that our present proposal of Z 2 × Z 2 is not unique for obtaining θ 13 = 0, but is rather the simplest scenario and it is also consistent with arbitrary charged-lepton and Majorana neutrino masses. It should also be noted that after the heavy Higgs triplets (or the right-handed neutrinos) are integrated away, the effective Lagrangian of this model (including the Higgs doublets) conserves L e and L µ + L τ separately, broken only by the very small Majorana neutrino masses.
Quarks can be incorporated into this model, for example, assigning Since left-handed and right-handed fermions transform in the same way under Z 2 × Z 2 , one could embed this model in a left-right symmetric theory. In particular, theories based on SO(10) are a natural framework to provide both types of seesaw mechanism, since their particle spectrum may include both super-heavy right-handed neutrinos N i and scalar isotriplets ξ i .
In conclusion, we pointed out that the absence of 1 − 3 mixing in the lepton sector can be explained if the Standard Model Lagrangian is extended to include two Higgs doublets and an appropriate source for neutrino Majorana masses, in such a way to respect a Z 2 × Z 2 family symmetry. In this scenario the atmospheric mixing angle originates in the µ−τ sector of the charged lepton mass matrix and it relates with predictable Lepton-Flavor-Violation effects: the physical Higgs bosons have specific decay rates into muons and taus, while their indirect contributions to τ → µγ, τ → 3µ and g µ − 2 are in general negligible. The solar mixing angle originates in the neutrino mass matrix, that can be generated either by two Higgs triplets or by two right-handed neutrinos.
This work was supported in part by the U. S. Department of Energy under Grant No. DE-FG03-94ER40837.