Dark Scalar Doublets and Neutrino Tribimaximal Mixing from A4 Symmetry

Abstract In the context of A 4 symmetry, neutrino tribimaximal mixing is achieved through the breaking of A 4 to Z 3 ( Z 2 ) in the charged-lepton (neutrino) sector respectively. The implied vacuum misalignment of the ( 1 , 1 , 1 ) and ( 1 , 0 , 0 ) directions in A 4 space is a difficult technical problem, and cannot be treated without many auxiliary fields and symmetries (and perhaps extra dimensions). It is pointed out here that an alternative scenario exists with A 4 alone and no redundant fields, if neutrino masses are “scotogenic”, i.e. radiatively induced by dark scalar doublets as recently proposed.

Such a pattern is best understood as the result of a family symmetry and the non-Abelian finite group A 4 has proved to be useful in this regard [3,4,5]. Specifically, it was shown [6,7,8] how this may be achieved by the breaking of A 4 in a prescribed manner [9], i.e.
A 4 → Z 3 in the charged-lepton sector and A 4 → Z 2 in the neutrino sector. The grouptheoretical framework of how this works in general has also been discussed [10,11]. For a brief history, see Ref. [12].
In another development, it has been proposed recently [13] that neutrino mass is radiative in origin such that the particles in the loop are odd under a new discrete Z ′ 2 symmetry, thereby accommodating a dark-matter candidate. The simplest realization of this "scotogenic" neutrino mass is depicted in Fig. 1. Here N k are heavy Majorana fermion singlets odd under Z ′ 2 and (η + , η 0 ) is a scalar doublet also odd under Z ′ 2 [14], whereas the standardmodel (φ + , φ 0 ) is even. Exact conservation of Z ′ 2 means of course that η 0 has no vacuum expectation value, so that N is not the Dirac mass partner of ν as usually assumed. The allowed quartic coupling (λ 5 /2)(Φ † η) 2 + H.c. splits Re(η 0 ) and Im(η 0 ) so that whichever is lighter is a good dark-matter candidate [13,15,16,17]. The collider signatures of Re(η 0 ) and Im(η 0 ) have also been discussed [18]. For a brief review of the further developments of this idea, see Ref. [19].
Going back to A 4 , let (ν i , l i ) ∼ 3 and either (I) l c i ∼ 1, 1 ′ , 1 ′′ , or (II) l c i ∼ 3, then with the where This is a natural minimum of the Higgs potential [3] because it corresponds to a Z 3 residual symmetry with which requires effective scalar triplet fields (ξ ++ i , ξ + i , ξ 0 i ) transforming as 3 with ξ 0 1 = 0 and ξ 0 2,3 = 0, thereby breaking A 4 → Z 2 . Let the eigenvalues of M ν be denoted by then the mixing matrix linking ν e,µ,τ to ν 1,2,3 is given by [12] ( i.e. tribimaximal mixing. Because the scalar fields Φ i and ξ i are both 3 under A 4 , the requirement that they break the vacuum in different directions is incompatible with the most general Higgs potential allowed by A 4 alone. Complicated sets of auxiliary fields and symmetries (and/or possible extra dimensions) are then needed [7,8,20,21,22] for it to happen. This is perhaps the one stumbling block of the application of A 4 to tribimaximal mixing.
The reason that the two breaking directions are incompatible is because A 4 allows 3 × 3 to be invariant, so if one 3 has a vacuum expectation value along a certain direction, the other is forced to as well. This is of course not a problem if only one 3 is required to have vacuum expectation values and not the other, because that corresponds to having an exactly conserved Z ′ 2 under which the second 3 is odd. Specifically, let the charged leptons acquire mass from Φ i , but the neutrino masses are obtained radiatively as discussed earlier, without any vacuum expectation value for η 0 . Instead of having three N's (which would have been necessary in the canonical seesaw mechanism), assume just one N but three scalar η doublets, as shown in Fig. 2 which will lead to tribimaximal mixing [6], with Since the origin of M ν is the mass-squared matrix of η 0 1,2,3 , this model may be tested at least in principle. Note that b = 0 cannot be a solution here as in Ref. [7] because that would require a negative mass-squared eigenvalue for η 0 i . As it is, ∆m 2 sol << ∆m 2 atm implies d ≃ 3b or −2a − b in this scenario.
Consider now the scalar sector in more detail. Since η i are odd under the new exactly conserved Z ′ 2 for dark matter, and have no vacuum expectation value. The bilinear terms Φ † i η j are forbidden, and the quartic terms must contain an even number of Φ i and η j . The scalar potential consisting of only Φ i is given by [3] The parameters m 2 and λ 1,2,3 are automatically real, and λ 4 may be chosen real by rotating the overall phase of Φ i . The vacuum solution is protected by the residual symmetry Z 3 , under which as already mentioned. The scalar doublet Φ has the properties of the standard-model Higgs doublet with mass-squared eigenvalues 2(3λ 1 + 2λ 3 + λ 4 )v 2 , 0, and 0 for √ 2Reφ 0 , √ 2Imφ 0 , and φ ± respectively. The charged scalars φ ′ ± and φ ′′ ± have m 2 and φ ′′ 0 are not mass eigenstates, but rather i.e.
This subtlety in the mass spectrum of φ ′ 0 and φ ′′ 0 was not recognized in Ref. [3], where τ − → µ − µ + e − and µ → eγ were thought to be nonzero. In fact, they are forbidden by the residual Z 3 symmetry.
The addition of η i to the scalar potential does not change the above because η 0 i = 0 and Z ′ 2 remains exactly conserved. However, the breaking of A 4 → Z 3 by φ 0 i generates additional contributions to the η 0 i mass-squared matrix of the form In other words, except for soft terms, the complete Higgs potential remains invariant under Z 3 after spontaneous symmetry breaking. The induced neutrino mass matrix of Eq. (7) is then modified: Since the one-loop neutrino mass of Fig. 1 is proportional to ∆ 2 3 and ∆ 2 4 which split Re(η 0 i ) and Im(η 0 i ), these parameters should be relatively small. Assuming that ∆ 2 2 is also small, then e should be small compared to a, b, d in Eq. (17). This means that [23] sin 2 2θ 23 = 1 and θ 13 = 0 as before, but the solar mixing angle is now given by where ǫ = e/(d − 3b). Thus tan 2 θ 12 = 0.47 is obtained for ǫ = 0.01.
One possible explanation of the smallness of the terms in Eq. (16) is that Φ and η are separated in an extra dimension so that they communicate only through a singlet in the bulk. In the limit this effect vanishes, there would be no mass splitting between Re(η 0 ) and Im(η 0 ), resulting in zero neutrino mass and no viable dark-matter candidate. With it, neutrinos acquire small radiative Majorana seesaw masses, Re(η 0 ) is a good dark-matter candidate, and near tribimaximal mixing is possible.
In conclusion, it has been shown how A 4 symmetry may be implemented in a model of "scotogenic" neutrino mass with dark scalar doublets. The neutrino mass matrix is induced by the neutral scalar mass-squared matrix spanning Re(η 0 1,2,3 ) and Im(η 0 1,2,3 ). This scheme allows the neutrino mixing angles θ 23 and θ 13 to be exactly π/4 and 0, whereas tan 2 θ 12 should not be exactly 1/2. Suppose the lightest Re(η 0 ) is dark matter, then its possible discovery [18] at the LHC together with the other η particles in accordance with