# Peccei-Quinn symmetry for Dirac seesaw and leptogenesis

###### Abstract

We extend the DFSZ invisible axion model to simultaneously explain small Dirac neutrino masses and cosmic matter-antimatter asymmetry. After the Peccei-Quinn and electroweak symmetry breaking, the effective Yukawa couplings of the Dirac neutrinos to the standard model Higgs scalar can be highly suppressed by the ratio of the vacuum expectation value of an iso-triplet Higgs scalar over the masses of some heavy gauge-singlet fermions, iso-doublet Higgs scalars or iso-triplet fermions. The iso-triplet fields can carry a zero or nonzero hypercharge. Through the decays of the heavy gauge-singlet fermions, iso-doublet scalars or iso-triplet fermions, we can obtain a lepton asymmetry in the left-handed leptons and an opposite lepton asymmetry in the right-handed neutrinos. Since the right-handed neutrinos do not participate in the sphaleron processes, the left-handed lepton asymmetry can be partially converted to a baryon asymmetry.

###### pacs:

98.80.Cq, 14.60.Pq, 14.80.Va## I Introduction

The phenomena of neutrino oscillations have been established by the atmospheric, solar, accelerator and reactor neutrino experiments olive2014 (). This means three flavors of neutrinos should be massive and mixed. Since the neutrinos are massless in the standard model (SM), we need new physics. Currently the most popular scheme for the neutrino mass generation is the so-called seesaw minkowski1977 () mechanism which can highly suppress the neutrino masses by a small ratio of the electroweak scale over a newly high scale. Remarkably the neutrinos have a Majorana nature in the usual seesaw models minkowski1977 (); mw1980 (); flhj1989 (); ma1998 (); barr2003 (). Such Majorana neutrino masses are induced by some lepton-number-violating interactions which can also generate a lepton asymmetry fy1986 () and then give a baryon asymmetry in association with the sphaleron krs1985 () processes. We hence can understand the cosmic matter-antimatter asymmetry which is the same as a baryon asymmetry. This baryogensis scenario in the lepton-number-violating seesaw context is the well known leptogenesis fy1986 () mechanism and has been widely studied lpy1986 (); fps1995 (); ms1998 (); bcst1999 (); hambye2001 (); di2002 (); gnrrs2003 (); hs2004 (); bbp2005 (); ma2006 (); dnn2008 (); dhh2014 (); ksy2015 (); fmmn2015 ().

However, one should keep in mind that the theoretical assumption of the lepton number violation and then the Majorana neutrinos have not been confirmed by any experiments. So it is worth studying the Dirac neutrinos rw1983 (); dlrw1999 (); mp2002 (); gh2006 (); tt2006 (); dsz2007 (); gu2012 (). In particular, we can construct some lepton-number-conserving Dirac seesaw models rw1983 (); mp2002 (); gh2006 (); gu2012 () to generate the small Dirac neutrino masses. The key of the Dirac seesaw models is that the effective Yukawa couplings of the right-handed neutrinos to the SM leptons and Higgs scalar can be suppressed by a ratio of ceratin symmetry breaking scale over some heavy field masses. Through the out-of-equilibrium and CP-violating decays of these heavy fields, we can obtain a lepton asymmetry in the SM left-handed leptons and an opposite lepton asymmetry in the right-handed neutrinos although the lepton number is totally zero dlrw1999 (); mp2002 (); gh2006 (); tt2006 (). The right-handed neutrinos then will go into equilibrium with the left-handed neutrinos at a very low temperature where the sphalerons have already stopped working. Therefore, the sphalerons can partially convert the induced lepton asymmetry in the SM leptons to a baryon asymmetry. This type of leptogenesis is named as the neutrinogenesis dlrw1999 () mechanism.

The SM encounters other challenges besides the small neutrino masses and the cosmic baryon asymmetry. In order to solve those problems, people have also extended the SM in other ways except for the seesaw scenario. For example, the invisible axion models kim1979 (); dfs1981 () based on the Peccei-Quinn (PQ) symmetry pq1977 (); weinberg1978 (); wilczek1978 () have been studied widely by theorists and experimentalists since they can solve the strong CP problem. Due to the unobserved axion, the PQ symmetry breaking scale now has a low limit far above the electroweak scale olive2014 (). Furthermore, for a proper choice of the breaking scale of the PQ symmetry and the initial value of the strong CP phase, the invisible axion can account for the dark matter relic density in the universe olive2014 (). In some interesting models for the neutrino mass generation, the PQ symmetry also plays an essential role shin1987 ().

We would like to point out the usual Dirac seesaw models contain an arbitrary breaking scale of the additional discrete, global or gauge symmetry. To fix or constrain this symmetry breaking scale, we can connect it to other new physics. For example, in a class of mirror models gu2012 (), the additional symmetry is a mirror electroweak symmetry so that it can be fixed by the dark matter mass.

In this paper we shall make use of the PQ symmetry to forbid the Yukawa couplings of the right-handed neutrinos to the SM leptons and Higgs scalar. Specifically we shall extend the DFSZ dfs1981 () invisible axion model by three gauge-singlet right-handed neutrinos, an iso-triplet Higgs scalar with or without hypercharge, as well as some heavy gauge-singlet fermions, iso-doublet Higgs scalars or iso-triplet fermions. After the PQ and electroweak symmetry breaking, the iso-triplet Higgs scalar can acquire an induced vacuum expectation value (VEV) constrained by the parameter. This VEV can help us to naturally suppress the Dirac neutrino masses by its ratio over the masses of the heavy gauge-singlet fermions, iso-doublet Higgs scalars or iso-triplet fermions. Meanwhile, the decays of the heavy gauge-singlet fermions, iso-doublet Higgs scalars or iso-triplet fermions can realize a neutrinogenesis to explain the cosmic matter-antimatter asymmetry.

## Ii The DFSZ model

Before introducing our models, we briefly review the DFSZ invisible axion model which contains three generations of fermions,

(1) |

as well as three Higgs scalars,

(2) |

Here and thereafter the first brackets following the fields describe the transformations under the gauge groups while the second ones denote the charges under a global symmetry.

We write down the kinetic terms of the above fermions and scalars,

(3) | |||||

with the covariant derivatives,

(4) |

Here , and are the , and gauge couplings, , and are the corresponding gauge fields, while and are the Gell-Mann and Pauli matrices. Under the and symmetries, we can give the Yukawa interactions,

(5) |

and the scalar potential,

(6) | |||||

After the gauge-singlet scalar develops a VEV,

(7) |

to spontaneously break the global symmetry, it can be rewritten by

(8) |

where is a massive Higgs boson while is a Nambu-Goldstone boson. By making the following phase rotation,

(9) |

the kinetic terms (3) can give us the axial couplings of the Nambu-Goldstone boson to the SM fermions , and ,

(10) | |||||

The non-perturbative QCD Lagrangian then should be

(11) |

where is a constant from the quark mass matrices and the QCD -vacuum. Clearly, the physical strong CP phase now can naturally roll into a tiny value to solve the strong CP problem since it now has become a dynamical field. Therefore, the global symmetry is the PQ symmetry while the Nambu-Goldstone boson is the axion. The PQ symmetry should be broken at a high scale to fulfill the experimental constraints olive2014 (). From the color anomaly the axion can pick up a tiny mass. For an appropriate PQ symmetry breaking scale , the axion can serve as a cold dark matter particle if the strong CP phase has an initial value of the order of olive2014 ().

The -doublet Higgs scalars are responsible for the spontaneous electroweak symmetry breaking. Their VEVs should be

(12) |

We can conveniently define

(13) |

and then obtain

(14) |

This means the newly defined will drive the electroweak symmetry breaking. It is easy to see the perturbation requirement in the Yukawa interactions can constrain the rotation angle by

(15) |

By inputting olive2014 ()

(16) |

we can read

(17) |

## Iii Higgs triplets and right-handed neutrinos

We now introduce the Higgs triplets with or without hypercharge,

(18a) | |||||

(18b) |

which have the kinetic terms as below,

(19a) | |||||

(19b) | |||||

The supplement of the potential (6) should be

(20a) | |||||

(20b) | |||||

After the Higgs doublets develop their VEVs for the electroweak symmetry breaking, the Higgs triplets can acquire the induced VEVs,

(21a) | |||||

(21b) | |||||

where the Higgs triplet masses have been given by

(22a) | |||||

(22b) |

It is well known the VEV of a Higgs triplet will affect the parameter olive2014 (),

(23) |

In the presence of two Higgs doublets and a Higgs triplet or , we can express the parameter by

(24a) | |||||

(24b) |

By inserting

(25a) | |||||

we can derive the upper bounds on the VEVs of the Higgs triplets,

(26a) | |||||

(26b) |

Our models also contain three right-handed neutrinos, which are the singlets but carry a charge as below,

(27) |

Therefore, the right-handed neutrinos are forbidden to have the following gauge-invariant Yukawa couplings and Majorana masses, i.e.

except for their kinetic terms,

(29) |

Meanwhile, the gauge-invariant Yukawa couplings of the Higgs triplet with hypercharge to the lepton doublets are also absent from the Lagrangian due to the PQ symmetry, i.e.

(30) |

In consequence, the neutrinos should keep massless in the present context.

## Iv Dirac seesaw models

In this section we will draw the outline of our models with the heavy fermion singlets, the heavy Higgs doublets or the heavy fermion triplets. The generation of the neutrino masses and the baryon asymmetry will be discussed in the later sections. According to the usual type-I, II and III seesaw models for the Majorana neutrinos, we would like to name our models with the heavy fermion singlets, the heavy Higgs doublets and the heavy fermion triplets as the type-I, type-II and type-III Dirac seesaw, respectively.

### iv.1 Type-I Dirac seesaw

The type-I Dirac seesaw contains the gauge-singlet fermions and scalar as follows,

(31) |

The allowed kinetic, Yukawa and scalar interactions are

(32a) | |||||

(32b) | |||||

Here we have prevented the fermion singlets from the gauge-invariant Majorana masses by imposing a conserved global symmetry of lepton number, under which the singlet fermions and , the right-handed neutrinos and the SM leptons and all carry a lepton number of one unit.

After the PQ symmetry breaking, we can obtain a mass term between the fermion singlets and , i.e.

(33) |

Without loss of generality, it is convenient to choose a basis where the masses of the fermion singlets are real and diagonal, i.e.

(34) |

and then define the following vector-like fermions,

(35) |

As for the scalar singlet , its mass is dominated by

### iv.2 Type-II Dirac seesaw

The heavy Higgs doublets for the type-II Dirac seesaw are denoted by

(37) |

which have the kinetic, Yukawa and scalar interactions,

(38a) | |||||

(38b) | |||||

with the covariant derivative,

(39) |

The masses of the Higgs doublets should be

(40) |

### iv.3 Type-III Dirac seesaw

In the type-III Dirac seesaw, we have the fermion triplets with or without hypercharge, i.e.

(41a) | |||||

(41b) |

The kinetic and Yukawa terms are

(42a) | |||||

(42b) | |||||

where the covariant derivatives are given by

(43a) | |||||

(43b) |

After the PQ symmetry breaking, the fermion triplets can obtain the gauge-invariant masses, i.e.

(44a) | |||||

Without loss of generality and for convenience, we can choose a basis where the masses of the fermion triplets are real and diagonal, i.e.

(45a) | |||||

(45b) |

In this basis, we can define the vector-like fermions as below,

(46a) | |||||

(46b) |

## V Neutrino masses

In this section we will demonstrate the neutrino mass generation in the type-I, II and III Dirac seesaw models. Specifically, we will show the Dirac neutrino masses can be highly suppressed by the ratio of the constrained VEVs of the Higgs triplets over the heavy masses of the fermion singlets, the Higgs doublets or the fermion triplets.

### v.1 Neutrino masses from the type-I Dirac seesaw

In the type-I Dirac seesaw model, the scalar singlet has a quartic coupling with two Higgs doublets and one Higgs triplet. See the -term and -term in Eq. (32). Accordingly, this scalar singlet can acquire an induced VEV,

(47a) | |||||

(47b) | |||||