Scotogenic model for co-bimaximal mixing
We present a scotogenic model, i.e. a one-loop neutrino mass model with dark right-handed neutrino gauge singlets and one inert dark scalar gauge doublet , which has symmetries that lead to co-bimaximal mixing, i.e. to an atmospheric mixing angle and to a -violating phase , while the mixing angle remains arbitrary. The symmetries consist of softly broken lepton numbers (), a non-standard symmetry, and three symmetries. We indicate two possibilities for extending the model to the quark sector. Since the model has, besides , three scalar gauge doublets, we perform a thorough discussion of its scalar sector. We demonstrate that it can accommodate a Standard Model-like scalar with mass , with all the other charged and neutral scalars having much higher masses.
With the experimental finding that the lepton mixing angle is nonzero, many theoretical neutrino mass models fell into disfavour. An exception is the model in ref. , in which there is a relation
and is the (symmetric) light-neutrino Majorana mass matrix in the basis where the charged-lepton mass matrix is . The condition (1) leads to and (provided ) , which is in agreement with the phenomenology ; this situation has recently been dubbed ‘co-bimaximal mixing’ .333A different way for obtaining co-bimaximal mixing, not involving the condition (1), has been recently proposed in ref. . A relevant point is that the condition (1) does not restrict the neutrino masses; it only restricts lepton mixing. Actually, as a consequence of the condition (1), the lepton mixing matrix has the form 
The atmospheric mixing angle is maximal, i.e. .
The -violating phase is .
The Majorana phase factors in effective neutrinoless decay are .
Because the predictions of condition (1) do not depend on the neutrino masses, it is possible that, in some multi-Higgs-doublet models, co-bimaximal mixing is not disturbed by the one-loop corrections to the neutrino mass matrix . This may, in particular, be the case in a ‘scotogenic’ model . In such a model, the masses of the light neutrinos have radiative origin and the particles in the loop that generates them belong to the dark sector of astrophysics, i.e. they are prevented from mixing with the ordinary particles by an unbroken (usually ) symmetry.
The purpose of this paper is to propose a scotogenic model for the neutrino masses which predicts co-bimaximal mixing.444Recently, another such model, but which employes a completely different mechanism, has been proposed in ref. . The model of ref.  is more complicated than the one presented in this paper for several reasons: (1) It has two types of dark matter, one of them protected by a symmetry and the other one by a symmetry. (2) It has several more fields in the dark sector. (3) The masses of the charged leptons are of radiative origin, just as those of the neutrinos. (4) The soft breaking of the symmetries occurs in two steps, with an symmetry in the dimension-four terms being softly broken to through dimension-three terms and that being softly broken through dimension-two terms. This is done in section 2. In section 3 we expose two possible extensions of that model to the quark sector. An analysis of the scalar potential of the model and of its compatibility with the recently discovered scalar of mass is performed in section 4. We summarize our findings in section 5. Appendix A collects some formulae from ref.  which are used in section 2.
2 The model for the lepton sector
Our model is an extension of the Standard Model with gauge symmetry . The usual fermion multiplets are three and three (). Besides, we introduce three right-handed neutrinos ; they belong to the dark sector of the model. Our model has four scalar doublets:
The doublet gives mass to the electron, gives mass to the muon, and gives mass to the lepton; the doublet belongs to the dark sector. We shall also use the conjugate doublets and .
The symmetries of our model are the following:
, and . This is an exact symmetry that prevents dark matter from mixing with ordinary matter. It is broken neither softly nor spontaneously, because the vacuum expectation value (VEV) of is zero.555Such scalar doublets have been dubbed ‘inert’ in ref. .
The flavour lepton numbers . They are broken only softly by the Majorana mass terms of the :
where is the charge-conjugation matrix in Dirac space and is a symmetric matrix in flavour space.
, , and . Because of these symmetries and of the , the lepton Yukawa Lagrangian is
The are broken spontaneously, through the VEVs , to give mass to the charged leptons:
Besides, the are also broken softly666 We recall that in a renormalizable theory a symmetry is said to be broken softly when all the symmetry-breaking terms have dimension smaller than four. This leaves open two possibilities: either they have both dimension two and dimension three or they have only dimension two. Soft symmetry breaking is consistent in quantum field-theoretic terms because, when using it, the dimension-four symmetry-violating terms generated by loops are finite. The soft breaking of (super)symmetries is extensively used in model-building; in particular, all supersymmetric models contain soft supersymmetry-breaking terms. through quadratic terms in the scalar potential.
Because of this symmetry, in equation (4)
with real and , i.e. ; moreover, in equation (5) and are real and . Therefore,
i.e. the small ratio of muon to -lepton mass is explained through a small ratio of VEVs . The symmetry is not broken softly777We might accept the soft breaking of by quadratic terms in the scalar potential; that soft breaking by terms of dimension two would not disturb the dimension-three terms in . But, for the sake of simplicity, we shall refrain in this paper from such a soft breaking. but it is broken spontaneously through the VEVs , especially through .888Ours is a model of ‘real violation’, i.e. violation originates in the inequality of two VEVs, even if those VEVs are real .
As compared to the model in ref. , the present model has an extra doublet , whose vanishing VEV causes neutrino mass generation to occur only at the one-loop level. However, as we will show below, the very same mechanism that produces co-bimaximal mixing at the tree level in the model of ref.  is effective at the one-loop level in the model of this paper.
In our model, just as in the original model of Ma , dark matter may be either spin-one half—the lightest particle arising from the mixture of , , and —or spin-zero—the lightest of the two components of —depending on which of them is lighter. No other fields are needed in principle to account for the dark matter.
In the scalar potential, a crucial role is played by the -invariant terms
where and because of Hermiticity. Let us write
where the fields and are real and the phase is defined such that
is real and positive. Then, the terms (10) generate a mass term
which means that and are mass eigenfields with distinct masses. The term (13) is the only one that makes the masses of and different; all other terms in the scalar potential contain .
Now we make use of the results in appendix A. In the notation of equation (A1), equation (5b) means that and ; notice that . In the notation of equation (A2), equation (11) reads and . Then, according to equation (A3), and . Applying equation (A4) we find the one-loop contribution to :
where the matrices and are defined through equation (A5). Note that there is no contribution to from a loop with because the VEV of is assumed to vanish; therefore, the Dirac neutrino mass matrix in line (A4b) also vanishes.
In the limit , the masses of and become equal and the contributions of and to exactly cancel each other; the light neutrinos then remain massless at the one-loop level . This happens in the limit where all the terms in equation (10) vanish. Indeed, in that limit the full Lagrangian is invariant under the symmetry
which forbids light-neutrino masses . We remark that there are, in the scotogenic model of this paper, several mechanisms for potentially suppressing the light-neutrino masses, viz.
a large seesaw scale, i.e. large heavy-neutrino masses in ;
small Yukawa couplings of , i.e. small ;
the factor in equation (14) from the loop integral.
Let us present a benchmark for all these suppressing factors. Let both and be of order . With one then obtains . Assuming , one requires in order to obtain . One concludes that the main suppression still originates in the high seesaw scale. However, with small and , of order or , the seesaw scale could easily be in the TeV range and thus accessible to the LHC.
Next we exploit the -invariance properties, viz. and . Equation (14) may be rewritten
where is a diagonal sign matrix . This is because, according to the assumptions of the seesaw mechanism, all the diagonal matrix elements of , i.e. all the heavy-neutrino masses, are nonzero. Using equation (17) we derive
Thus, after a physically meaningless rephasing, displays the defining feature (1) of co-bimaximal mixing.
2.1 Approximation to the Higgs boson
is orthogonal. The last row of corresponds to . For definiteness, we let the last two columns of correspond to and , which belong to the dark sector and do not mix with all the other scalars. Therefore, for practical purposes is just a matrix. By definition, is the Goldstone boson and 
The couplings of () to the gauge bosons are given by 
Therefore, a given couples to the gauge bosons with exactly the same strength as the Higgs boson of the Standard Model if
Notice that, because both three-vectors and have unit modulus, . Therefore, equation (23) holds in a limit situation.
3 Extension of the model to the quark sector
It is non-trivial to extend our model to the quark sector because the symmetry relates the Yukawa couplings of to those of ; moreover, some quarks must couple to —and, correspondingly, other quarks must couple to —in order that violation, which is generated through , manifests itself in the CKM matrix .
We firstly expound some notation. The quark Yukawa Lagrangian is
where . The mass matrices are and . They are diagonalized as
where the matrices are unitary. The physical quarks are given by
and analogously for the left-handed fields. The quark mixing matrix is .
3.1 Extension 1
One may include the quarks in the symmetries as follows:
where and are row matrices. Notice that both and are in this extension rank 1 matrices. The quark mass matrices are
We define the matrices
(We do not use the summation convention.) The Yukawa couplings of the neutral scalars—see equation (A2)— are
Defining the Hermitian matrices
the Yukawa couplings of a given to the third-generation quarks are given by
Thus, couples to the third-generation quarks in the same way as the Higgs boson if
We have not yet specified the way in which the symmetry is to be extended to the quark sector. This may be chosen to be
The symmetry (39) enforces real and and
3.2 Extension 2
The extension of our model to the quark sector expounded in the previous subsection treats the down-type and up-type quarks in similar fashion. It possesses flavour-changing neutral Yukawa interactions (FCNYI) in both quark sectors. In this subsection we suggest a different extension, in which FCNYI are restricted to the up-type-quark sector.
Let the quarks be included in the symmetries as
With this extension, the Yukawa-coupling matrices and vanish outright. In extension 2, as distinct from extension 1, the matrices are rank 0 while is rank 3. Without loss of generality, one may rotate the and the so that is equal to from the outset. Then, and the CKM matrix .
Analogously to equation (24), the couplings of the neutral scalars to the down-type quarks are given by
A given couples to the bottom quark in the same way as the Higgs boson if
where are column vectors. The up-type-quark mass matrix is
We define the column matrices . Let