Double type-II Dirac seesaw accompanied by Dirac fermionic dark matter

A TeV-scale Higgs doublet with a small mixing to the standard model Higgs doublet can have the sizable Yukawa couplings to several right-handed neutrinos and the standard model lepton doublets. This provides a testable Dirac neutrino mass generation. We further consider a seesaw mechanism involving a $U(1)_{B-L}^{}$ gauge symmetry, which predicts the existence of two right-handed neutrinos and a stable Dirac fermionic dark matter, to simultaneously explain the small mixing between the two Higgs doublets and the generation of the cosmic baryon asymmetry.


I. INTRODUCTION
The discovery of neutrino oscillations indicates that three flavors of neutrinos should be massive and mixed [1]. Meanwhile, the cosmological observation requires that the neutrinos should be extremely light [1]. The tiny neutrino masses can be naturally induced in various seesaw [2] extensions of the SU (3) c × SU (2) L × U (1) Y standard model (SM). In these popular seesaw scenarios [2][3][4][5], the neutrino mass generation is accompanied by certain lepton-number-violating interactions and hence the neutrinos have a Majorana nature. Meanwhile, the interactions for realizing the seesaw can produce a lepton asymmetry stored in the SM leptons and then the produced lepton asymmetry can be partially converted to a baryon symmetry by the sphaleron processes [6]. This is the so-called leptogenesis mechanism [7] to explain the observed baryon asymmetry in the universe [5,[7][8][9][10][11][12][13][14][15][16].
However, we should keep in mind that the theoretical assumption of the lepton number violation and then the Majorana neutrinos has not been confirmed by any experiments yet. So, it is worth studying the possibility of Dirac neutrinos. In analogy to the usual seesaw models for the Majorana neutrino mass generation, we can construct some Dirac seesaw models [17][18][19][20] for the Dirac neutrino mass generation. The interactions for the Dirac seesaw can induce a lepton asymmetry stored in the SM left-handed leptons and an opposite lepton asymmetry stored in the right-handed neutrinos although the total lepton asymmetry is exactly zero. The righthanded neutrinos will go into equilibrium with the lefthanded neutrinos at a very low temperature, where the sphalerons have already stopped working. Therefore, the sphalerons will never affect the right-handed neutrino asymmetry, but it can still transfer the SM lepton asymmetry. This type of leptogenesis is named as the neutrinogenesis mechanism [21] and has been studied in liter- * Electronic address: peihong.gu@sjtu.edu.cn atures [18][19][20][21][22][23]. In the Dirac seesaw models, the renormalizable Yukawa couplings of the right-handed neutrinos to the SM lepton and Higgs doublets can appear until an additionally discrete, global or gauge symmetry is spontaneously broken. This new symmetry breaking scale may be constrained by other new physics. For example, in a class of mirror models, the additional symmetry is a mirror electroweak symmetry so that it can be fixed by the dark matter mass [24].
In this paper we shall realize the Dirac seesaw by introducing a U (1) B−L gauge symmetry with four righthanded neutrinos. Because of their Yukawa couplings to a Higgs singlet for spontaneously breaking the U (1) B−L symmetry, two right-handed neutrinos can form a Dirac fermion and then become a stable dark matter particle. Furthermore, additionally heavy Higgs doublet(s) and/or fermion singlet(s) can mediate a dimension-5 operator among the other two right-handed neutrinos, the SM lepton and Higgs doublets as well as the U (1) B−L Higgs singlet. This means a highly suppressed Dirac neutrino mass matrix with two nonzero eigenvalues. Finally, through the interactions for the neutrino mass generation, the heavy Higgs doublet(s) and/or fermion singlet(s) can decay to realize a leptogenesis mechanism.

II. THE MODEL
The SM fermions and scalar are denoted as follows, Here and thereafter the first and second brackets following the fields respectively describe the transformations under the SM SU (3) c ×SU (2) L ×U (1) Y gauge groups and the U (1) B−L gauge group. In order to cancel the gauge anomalies, we need some right-handed neutrinos [25]. In the present work, we consider the following right-handed neutrinos, For spontaneously breaking the U (1) B−L symmetry, we can introduce a Higgs singlet, The mass of the U (1) B−L gauge boson Z B−L then should be with g B−L being the U (1) B−L gauge coupling. The experimental constraints on the U (1) B−L gauge symmetry is [26], It is easy to see the Higgs singlet ξ can have a Yukawa interaction with the third and forth right-handed neutrinos ν R3,4 , i.e.
Furthermore, in association with the Higgs singlet ξ, we can construct the following dimension-5 operators involving the first and second right-handed neutrinos ν R1,2 , i.e.
As shown later, the above effective operators can be induced by the renormalizable terms as below, where η and N L,R are additionally heavy Higgs doublet(s) and fermion singlet(s), For convenience and without loss of generality, we have chosen the mass matrices M 2 η and M N to be real and diagonal. In this basis, we can further rotate the parameters ρ a to be real, i.e. ρ a = ρ * a .

III. NEUTRINO MASS
We can integrate out the heavy Higgs doublet(s) η and fermion singlet(s) N = N L + N R to realize the effective operator (7), When the Higgs singlet ξ develops its VEV ξ for the U (1) B−L symmetry breaking, the two right-handed neutrinos ν R1,2 can acquire the Yukawa couplings to the SM lepton and Higgs doublets l L and φ, i.e.
Therefore, we can obtain a Dirac neutrino mass matrix, The above Dirac neutrino mass generation can be also understood by Fig. 1.
The experimental limit on the U (1) B−L symmetry breaking scale is a few TeV, as shown in Eq. (5). On the other hand, we will show later the right-handed neutrinos ν R3,4 are expected to form a dark matter particle. To account for the observed dark matter relic density, the annihilations of the dark matter right-handed neutrinos into the light species should have a right cross section. The upper bound of the U (1) B−L symmetry breaking scale thus should not be far above the TeV scale unless a fine-tuned resonant enhancement [27] is introduced in the s-channel dark matter annihilations. Therefore, the Dirac neutrino masses can be highly suppressed in a natural way as long as the masses of the heavy Higgs doublet(s) η and/or fermion singlet(s) N are much larger than the TeV scale. For example, we can take and then obtain by further inputting In Eq. (12), the Dirac neutrino mass matrix only involves two right-handed neutrinos so that it can have at most two nonzero eigenvalues. This fact is independent on the number of the heavy Higgs doublet(s) η and/or fermion singlet(s). Since the current experimental data indicate the existence of at least two massive neutrinos, we would like to name such 3 × 2 neutrino mass matrix as a minimal Dirac neutrino mass matrix.
Note if we do not introduce the heavy Higgs doublet(s) η, we should have at least two heavy fermion singlets N to guarantee at least two nonzero eigenvalues of the neutrino mass matrix. We will show in the following that a successful leptogenesis needs at least two heavy Higgs doublets η, or at least two heavy fermion singlets N , or at least one heavy Higgs doublet η and at least one heavy fermion singlet N . We also check if the right-handed neutrinos ν R can decouple above the QCD scale to satisfy the BBN constraint on the effective neutrino number. For this purpose, we need consider the annihilations of the right-handed neutrinos into the relativistic species at the QCD scale, with s being the Mandelstam variable. The interaction rate then should be [14] Γ with K 1 being a Bessel function. We take g * (300 MeV) ≃ 61.75 and then find

IV. BARYON ASYMMETRY
As shown in Fig. 2, there are two decay modes of the heavy Higgs doublet(s) η, i.e.
If the CP is not conserved, we can expect a CP asymmetry in the above decays, where Γ η a is the total decay width, We can calculate the decay width at tree level and the CP asymmetry at one-loop order, Here the first term in the CP asymmetry is the self-energy correction mediated by the heavy Higgs doublet(s) while the second term is the vertex correction mediated by the heavy fermion singlet(s). A nonzero CP asymmetry ε η a needs at least two heavy Higgs doublets η, or at least one heavy Higgs doublet η and at least one heavy fermion singlet N . As for the heavy fermion singlet(s) N , their decay modes are The relevant diagrams are shown in Fig. 3. The decay width and CP asymmetry can be calculated by The lepton-number-conserving decays of the heavy fermion singlets.
Here the first term in the CP asymmetry is the self-energy correction mediated by the heavy fermion singlet(s) while the second term is the vertex correction mediated by the heavy Higgs doublet(s). A nonzero CP asymmetry ε N b needs at least two heavy fermion singlets N , or at least one heavy fermion singlet N and at least one heavy Higgs doublet η.
When the heavy Higgs doublets η a and/or the heavy fermion singlets N b go out of equilibrium, their decays can generate a lepton number L l L stored in the SM lepton doublets l L and an opposite lepton number L ν R1,2 +ξ stored in the right-handed neutrinos ν R1,2 and the Higgs singlet ξ. For example, if the Higgs doublet η 1 is much lighter than the other heavy Higgs doublet(s) η a =1 and the heavy fermion singlet(s) N b , its decays will dominate the final lepton numbers [28], Here the CP asymmetry ε η 1 can be simplified by with m max being the largest eigenvalue of the neutrino mass matrix m ν . Alternatively, we can consider another simple case where the fermion singlet N 1 is much lighter than the other heavy fermion singlet(s) N b =1 and the heavy Higgs doublet(s) η a . The final lepton numbers then should be [28], with the CP asymmetry ε η 1 being simplified as In Eqs. (27) and (29), n eq η 1 ,N 1 and T D respectively are the equilibrium number density and the decoupled temperature of the decaying heavy particles, while s is the entropy density of the universe. The decay-produced lepton number in the SM lepton doublets can be partially converted to a baryon asymmetry by the sphaleron processes [29], In the weak washout region [28], we can approximately obtain the lepton numbers (27) and (29) by Here H(T ) is the Hubble constant with M Pl ≃ 1.22 × 10 19 GeV being the Planck mass and g * = 117.75 being the relativistic degrees of freedom (the SM fields plus the right-handed neutrinos ν R1,2,3,4 , the Higgs singlet ξ and the U (1) B−L gauge field.). The baryon number (31) then can be given by For a numerical estimation, we take the CP asymmetry (28) or (30) can arrive at a value around O(10 −7 ) as its maximal value is of the order of O(10 −4 ). The final baryon asymmetry (34) thus can match the observed value, i.e. B ∼ 10 −10 .

V. DARK MATTER
Due to the Yukawa interaction (6), the third and forth right-handed neutrinos ν R3,4 can form a Dirac particle after the U (1) B−L symmetry breaking, i.e.
Clearly, the Dirac fermion χ will keep stable to leave a dark matter relic density. The dark matter annihilation and scattering can be determined by the related gauge and/or Yukawa interactions, where h ξ is the Higgs boson from the Higgs scalar ξ. The gauge boson Z B−L also couples to the SM fields as well as the first and second right-handed neutrinos ν R1,2 , The perturbation requirement then should put an upper bound on the gauge coupling g B−L , i.e.
As for the Higgs boson h ξ , it can interact with the SM through a Higgs portal as below, For demonstration, we shall focus on the case that the gauge interactions dominate the dark matter annihilations and scatterings. In this case, the thermally averaging dark matter annihilating cross section is given by [30] The dark matter relic density then can well approximate to [1] It should be noted that Eqs. (42) and (43) are based on the assumption, This indeed means . (46) besides the experimental limit (5). The dark matter mass, The gauge interactions can also mediate the dark matter scattering off nucleons. The dominant scattering cross section is spin independent [31], Here µ r = m N m χ /(m N + m χ ) is a reduced mass with m N being the nucleon mass. As the dark matter is much heavier than the nucleon, the above dark matter scattering cross section indeed should be inversely proportional to the squared dark matter mass, By taking the dark matter direct detection results [32,33] into account, we can put a more stringent low limit on the dark matter mass 1 , m χ 5 TeV . In this paper we have shown a U (1) B−L gauge symmetry can predict the existence of the Dirac neutrinos and the stable dark matter. Specifically, we have extended the SM SU (3) c × SU (2) L × U (1) Y gauge symmetries by a U (1) B−L gauge symmetry. We then introduced four right-handed neutrinos in order to cancel the gauge anomalies. Because of their Yukawa couplings to the Higgs singlet for spontaneously breaking the U (1) B−L symmetry, two right-handed neutrinos can form a stable Dirac fermion and hence can account for the dark matter relic density. Furthermore, mediated by additionally heavy Higgs doublet(s) and/or fermion singlet(s), the other two right-handed neutrinos can have a dimension-5 operator with the SM lepton and Higgs doublets as well as the U (1) B−L Higgs singlet. We hence can obtain a Dirac neutrino mass matrix with two nonzero eigenvalues. Finally, the interactions for the neutrino mass generation can also allow the decays of the heavy Higgs doublet(s) and/or fermion singlet(s) to produce a lepton asymmetry motivated by the leptogenesis mechanism.