Cobimaximal Neutrino Mixing from $S_3 \times Z_2$

It has recently been shown that the phenomenologically successful pattern of cobimaximal neutrino mixing ($\theta_{13} \neq 0$, $\theta_{23} = \pi/4$, and $\delta_{CP} = \pm \pi/2$) may be achieved in the context of the non-Abelian discrete symmetry $A_4$. In this paper, the same goal is achieved with $S_3 \times Z_2$. The residual lepton $Z_3$ triality in the case of $A_4$ is replaced here by $Z_2 \times Z_2$. The associated phenomenology of the scalar sector is discussed.

Soft breaking of S 3 to Z 2 with two Higgs doublets: then the most general S 3 invariant scalar potential for Φ 1,2 is given by [25] The soft breaking of V 0 to a residual Z 2 symmetry may be accomplished in three ways, The soft term required is then Two lepton families under S 3 : Let (ν 1 , l 1 ) L , (ν 2 , l 2 ) L ∼ 2 under S 3 , and l 1R , l 2R ∼ 1 ′ , 1 under S 3 . The S 3 invariant Yukawa terms for charged-lepton masses are then Choosing the third option for Z 2 with iµ 2 Three lepton families under S 3 × Z 2 : A third lepton family may be added which transforms as (1, −) under S 3 × Z 2 , so that it couples to a third Higgs doublet which transforms as (1, +). The 3 × 3 unitary matrix linking the diagonal charged-lepton mass matrix to the neutrino mass matrix is then This serves the same purpose as U ω of Eq. (3), because also yields |U µi | = |U τ i | which leads to cobimaximal mixing. In fact, since U ei = O 1i , the θ 12 and θ 13 angles are the same in both U lν and O.

More about O:
The 3 × 3 Majorana neutrino mass matrix M ν is symmetric but also complex in general with three physical phases. It is thus not diagonalized by an orthogonal matrix. However, if the origin of this mass matrix is radiative and comes from a set of three real scalars, and there are no extraneous phases from the additional interactions, then it is possible [9,14,15,16] to achieve this result.
In gneral M ν is diagonalized by a unitary matrix with 3 angles and 3 phases: where c ij = cos θ ij and s ij = sin θ ij . If δ = 0, then it is equal to an orthogonal matrix times a diagonal matrix involving only Majorana phases. Upon multiplification on the left by U 2 of Eq. (7), it will still lead to cobimaximal neutrino mixing. Now The diagonal matrix of phases on the left may be absorbed into the charged leptons, and the remaining part of U 2 U becomes This means that if δ = 0, cobimaximal mixing is achieved with e −iδ CP = e iπ/2 = i as expected. However, even if δ = 0, so that δ CP deviates from −π/2, θ 23 remains at π/4. This is a remarkable result and it is only true because of U 2 of Eq. (7), and does not hold for Soft breaking of S 3 to Z 2 with three Higgs doublets: Adding Φ 3 ∼ 1 under S 3 , the scalar potential of our model becomes The µ 2 12 and µ 2 30 terms break S 3 softly to Z 2 , under which Φ 3 and Φ + = (Φ 1 +iΦ 2 )/ √ 2 are even is forbidden by making Φ 3 odd under an extra Z 2 symmetry, which is then broken softly by The Z 3 triality [27,28] coming from A 4 has now been replaced by the above under Z 2 × Z 2 .
Phenomenology of scalar interactions: The leptonic Yukawa interactions are given by The scalar interactions are given by Assuming v, v 3 to be real, the conditions for minimizing V are v[(µ 2 0 + µ 2 12 ) + (λ 1 + For The states are the wouldbe massless Goldstone modes for the Z and W ± bosons. The states have masses given by The states h = √ 2Re(φ 0 + ) and H = √ 2Re(φ 0 3 ) are approximate mass eigenstates with and h − H mixing given by The Φ − doublet has odd Z 2 and does not mix with Φ + or Φ 3 . The masses of its components are given by Phenomenology of lepton interactions: From Eq. (14), the lepton interactions of this model are given by to a very good approximation. Since v 3 << v is assumed, the heavy H and A couple predominantly to e − e + . If they are produced, through a virtual Z for example, at the Large Hadron Collider (LHC), the e − e + e − e + final state is very distinctive and potentially measurable. In the same way, √ 2Re(φ 0 − ) + √ 2Im(φ 0 − ) may be produced. They decay to µ − τ + and µ + τ − which are again rather distinctive if τ ± can be reconstructed experimentally.

Conclusion:
The notion of cobimaximal neutrino mixing, i.e. θ 13 = 0, θ 23 = π/4, and δ CP = −π/2, is shown to be a consequence of the residual Z 2 × Z 2 symmetry of an S 3 × Z 2 model of lepton masses. This is an alternative theoretical understanding from the usual A 4 realization. It has verifiable decay signatures in its three Higgs doublets, as well as the prediction that even if δ CP deviates from −π/2, θ 23 will remain at π/4, in contrast to other models.