###### Abstract

The model of radiative neutrino mass with dark matter proposed by one of us is extended to include a real singlet scalar field. There are then two important new consequences. One is the realistic possibility of having the lightest neutral singlet fermion (instead of the lightest neutral component of the dark scalar doublet) as the dark matter of the Universe. The other is a modification of the effective Higgs potential of the Standard Model, consistent with electroweak baryogenesis.

OSU-HEP-07-03

UCRHEP-T438

August 2007

Singlet fermion dark matter and electroweak

baryogenesis with radiative neutrino mass

K. S. Babu

Department of Physics, Oklahoma State University,

Stillwater, Oklahoma 74078, USA

Ernest Ma

Department of Physics and Astronomy, University of California,

Riverside, California 92521, USA

With the addition of a second scalar doublet [] to the Standard Model (SM) of quark and lepton interactions, a cornucopia of new opportunities opens up for the understanding of physics phenomena beyond the SM. One possibility [1] is that is odd with respect to an exactly conserved discrete symmetry, allowing [2, 3, 4, 5, 6, 7, 8, 9, 10] or to be a candidate for the dark matter [11] of the Universe.

If three heavy neutral singlet Majorana fermions are added as well [2, 4, 5, 9], which are odd under the aforementioned , then neutrinos acquire radiative seesaw masses through the Yukawa interactions

(1) |

and the mass splitting of and from the quartic scalar term

(2) |

where is the usual SM Higgs doublet with , whereas . Thus the neutrinos of this model have no Dirac masses linking them to , but they obtain Majorana masses in one loop given by [2]

(3) |

Instead of or , the lightest among the fermions may now be considered [2, 12, 13] as a dark-matter candidate. However, if Eq. (1) is the only interaction of , the requirement of a realistic dark-matter relic abundance is generally in conflict [13] with flavor changing radiative decays such as , which cannot be alleviated without some degree of fine tuning.

One way to evade the above constraint is to endow with some other interaction, such as an extra gauge [14]. Here we propose instead the minimal addition of a real singlet scalar to allow another channel for annihilation, thus freeing the constraint of relic abundance from Eq. (1). With smaller values of , flavor changing radiative decays such as are suitably suppressed. At the same time, with the addition of , the effective Higgs potential involving the SM Higgs doublet is now such [15, 16, 17] that electroweak baryogenesis [18] may also become possible, as elaborated below.

There are two possible scenarios in which successful baryogenesis could be realized within the present model. In the first scenario the singlet scalar remains light down to the scale of electroweak symmetry breaking. Its interactions with the SM Higgs doublet modifies the condition for strongly first-order phase transition and leads to successful baryogenesis without contradicting the lower limit on the SM Higgs boson mass [16]. We have nothing new to add for this scenario. There is a second possibility where the singlet scalar has a mass larger than the electroweak scale. One can then integrate out from the effective low energy theory. The potential for the SM Higgs doublet gets modified in this process, enabling successful baryogenesis [17].

Consider first the most general renormalizable Higgs potential of two doublets , , and a singlet , where is odd under an extra as proposed in Ref. [2]:

(4) | |||||

where has been chosen real without any loss of generality. To obtain the tree-level effective potential containing only and , we eliminate by its own equation of motion in powers of and . We assume that at this point. To eighth order in the fields, we then obtain

We seek a solution where is not broken, i.e. , in which case the effective Higgs potential for alone is given by

(6) | |||||

This is of the form obtained by Ref. [17] where the coefficient of the term may be chosen negative to allow for a strong first-order phase transition required by electroweak baryogenesis. The numerical conditions have been analyzed fully in Ref. [17] and we have nothing to add here.

As for CP violation needed for electroweak baryogenesis, the SM contribution from the CKM phase is known to be too small. The new Yukawa couplings of leptons to the doublet (see Eq. (1)) contain new CP violating phases. However, since does not directly participate in electroweak symmetry breaking, these phases are unlikely to be significant for baryogenesis. There is however another source of CP violation in the model – the strong CP violation parameter [19]. This parameter can be of order unity at temperatures of order 100 GeV. The Peccei-Quinn [20] mechanism which solves the strong CP problem indeed assumes the initial value of to be of order unity. Once the QCD phase transition is turned on, at temperatures of order 1 GeV, the PQ mechanism ensures that is relaxed dynamically to zero. The effect of on electroweak baryogenesis has been studied in Ref. [21], where it is shown that this might be sufficient for baryogenesis, but in this case the axion cannot be the dark matter. In our model of course, the lightest remains our choice for dark matter. (It is also possible to introduce higher dimensional operators which violate CP, as in Ref. [22].)

Going back to Eq. (4), suppose is the remnant of a spontaneously broken symmetry, then it is easy to show that the following parameters are related:

(7) |

In that case, Eq. (5) reduces exactly to

(8) | |||||

i.e. to all orders in and . The proof is very simple. It merely comes from the fact that the combination always appears together in . This also means that the singlet Majoron model of spontaneous lepton number violation [23] will not generate a nonzero term (or any other beyond ) in Eq. (6).

However, if the lepton symmetry is spontaneously broken at the TeV scale, with but with at the electroweak scale, then electroweak baryogenesis can be successful [16]. Even though our model has a second Higgs doublet , TeV scale lepton number violation is consistent with all experimental and astrophysical data. This is because and the Majoron resides entirely in the (complex) singlet [23]. However, our model is different from the singlet Majoron model in one important respect. The charged scalar induces charged-lepton couplings to the Majoron in our model, which is stronger than those in the singlet Majoron model. We find these couplings to be

(9) |

where with being the mass of , and stands for the mass of the th charged lepton. Note that these couplings can lead to decays such as . The relevant Yukawa coupling can be written, for , as which is of order . (Note that the light neutrino mass is proportional to , while these induced couplings are not.) The branching ratio for the decay is of order , which is below the current experimental limit for . Since the branching ratio is close to the current experimental limit, there is some hope that this decay may be accessible to the next round of experiments.

Going back to the case of a heavy real singlet , consider the electroweak symmetry breaking due to Eq. (6) up to order . Let , then [15]

(10) |

and the cubic interaction

(11) |

appears. The SM is recovered if . Since , Eq. (4) implies that

(12) |

and mixes with .

Consider now the Yukawa couplings . Unlike the of Eq. (1), these are not constrained by flavor changing radiative decays such as . The process

(13) |

will contribute to the relic abundance of the lightest as dark matter. Furthermore, the term in will mix with . As a result, the processes

(14) |

are also possible. Presumably, the direct detection of the lightest will be from the elastic scattering of off nuclei through exchange.

Let be the lightest singlet fermion, then its nonrelativistic annihilation cross section in the early Universe from Eq. (13) multiplied by its relative velocity is given by

(15) |

Assuming this to be the dominant contribution to the dark-matter relic density of the Universe, we need of order 1 pb [24], which may be obtained for example with GeV, GeV, GeV, GeV, and . The heavier will decay through Eq. (1) to and to .

If there is a lepton family symmetry such as [25, 26] which makes in Eq. (1), then the neutrino mass matrix of Eq. (3) is diagonalized by the same unitary transformation which diagonalizes the mass matrix . In that case, we have the prediction that the neutrino mass eigenvalues responsible for neutrino oscillations are given by

(16) |

This may be verifiable experimentally from decay.

With as dark matter, its direct detection becomes very difficult, because its elastic scattering cross section with nuclei is only through exchange [3, 6], with the small effective coupling . Our model has many other testable predictions, including the production of neutral and charged scalar particles and at the Large Hadron Collider [27].

This work was supported in part by the U. S. Department of Energy under Grant Nos. DE-FG03-94ER40837 and DE-FG03-98ER41076.

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