Minimal inverse seesaw accompanied by Dirac fermionic dark matter
Pei-Hong Gu
hep-ph







peihong.gu@sjtu.edu.cn



School of Physics and Astronomy, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, China







We present a minimal inverse seesaw mechanism by resorting to a U(1)_B-L^ gauge symmetry. In order to cancel the gauge anomalies, we introduce seven neutral fermions among which four participate in the inverse seesaw to induce two nonzero neutrino mass eigenvalues, two forms a stable Dirac fermion to become a dark matter, while the last one keeps massless but decouples early. In this inverse seesaw, two neutral fermions are the usual right-handed neutrinos while the other two have a small Majorana mass term. An additional seesaw mechanism for generating these small Majorana masses also explains the cosmic baryon asymmetry in association with the sphaleron processes.









98.80.Cq, 14.60.Pq, 95.35.+d, 12.60.Cn, 12.60.Fr




Minimal inverse seesaw accompanied by Dirac fermionic dark matter
    Pei-Hong Gu
    Received 15 May 2019 / Accepted 18 July 2019
=================================================================






§ INTRODUCTION


The fact that three flavors of neutrinos should be massive and mixed has been confirmed by various neutrino oscillation experiments <cit.>. Moreover, the neutrinos should be extremely light to satisfy the cosmological constraints <cit.>.  The tiny but nonzero neutrino masses can be naturally explained by the famous seesaw mechanism <cit.>. Some seesaw models such as the type-I  <cit.>, type-II  <cit.> and type-III <cit.> seesaw can further accommodate a leptogenesis mechanism <cit.> to understand the cosmic baryon asymmetry <cit.>, which is much much bigger than the value induced in the standard model (SM). In this seesaw-leptogenesis framework, additionally heavy particles couple to the SM lepton and Higgs doublets so that their decays can produce a lepton asymmetry stored in the SM leptons <cit.>. In association with the sphaleron processes <cit.>, the produced lepton asymmetry can be partially converted to a baryon asymmetry. Meanwhile, the small neutrino masses can be induced by integrating out the heavy particles from the same interactions.



The inverse seesaw mechanism is another attractive seesaw scenario <cit.>. In the inverse seesaw models, several right-handed neutrinos can have a sizable mass term with the same number of other neutral fermions which are assigned to a small Majorana mass term. Due to such small Majorana masses, the right-handed neutrinos can be allowed to significantly mix with the left-handed neutrinos. The right-handed neutrinos thus may be verified experimentally. Remarkably, the small Majorana mass term of the neutral fermions is crucial to the testability of the inverse seesaw. It has been shown such small Majorana masses can be suppressed by additionally heavy fermion <cit.> and/or scalar <cit.> singlets after a U(1)_X^ global symmetry is spontaneously broken. The same interactions can also account for the generation of the baryon asymmetry.  


In this paper we shall develop our recent work <cit.> to solve the dark matter puzzle, which is another big challenge to the SM. Specifically, the SM SU(3)_c^× SU(2)_L^× U(1)_Y^ gauge symmetries will be extended by a U(1)_B-L^ gauge symmetry. In order to cancel the gauge anomalies, we introduce seven neutral fermions including two usual right-handed neutrinos. After a Higgs singlet develops its vacuum expectation value (VEV) for spontaneously breaking the U(1)_B-L^ symmetry, two neutral fermions can obtain their small Majorana masses by integrating out some heavy scalar or fermion singlets, meanwhile, they can have a sizable mass term with the right-handed neutrinos. After the SM Higgs doublet drives the electroweak symmetry breaking, the two right-handed neutrinos and the three left-handed neutrinos can acquire their Dirac masses. We thus can realize an inverse seesaw to give a rank-2 neutrino mass matrix with two nonzero eigenvalues. Within this framework, the heavy scalar or fermion singlet decays can explain the cosmic baryon asymmetry in association with the sphaleron processes. On the other hand, another two neutral fermions can form a Dirac particle due to their Yukawa coupling with the U(1)_B-L^ Higgs singlet. This Dirac fermion can serve as a stable dark matter particle. As for the seventh neutral fermion, it keeps massless but decouples early. 








§ FERMIONS AND SCALARS





The SM fermions now are gauged by a U(1)_B-L^ symmetry besides the SU(3)_c^× SU(2)^_L× U(1)_Y^ groups, 

    [ q^_L(3,2,+1/6,+1/3), ][ d^_R(3,1,-1/3,+1/3), ][ u^_R(3,1,+2/3,+1/3); ]
    
    [2mm]
          [ l^_L(1,2,-1/2,-1), ][ e^_R(1,1,-1,-1). ]

Here and thereafter the brackets following the fields describe the transformations under the SU(3)_c^× SU(2)^_L× U(1)_Y^× U(1)_B-L^ gauge groups. For simplicity, we do not show the indices of the three generations of fermions. In order to cancel the gauge anomalies, we can introduce some neutral fermions with appropriate U(1)_B-L^ charges <cit.>. In the present work, we take seven neutral fermions including two usual right-handed neutrinos,

    [ ν^_R1,2(1,1,0,-1), ][ S^_R1,2(1,1,0,+1/2), ]
          [ χ^_R1(1,1,0,1+√(153)/4), ][ χ^_R2(1,1,0,1-√(153)/4), ]
          [ ζ^_R(1,1,0,-5/2). ]

 


 

While the SM Higgs doublet, 

    [ ϕ^(1,2,-1/2,0), ]

is responsible for the electroweak symmetry breaking as usual, we introduce a Higgs singlet, 

    [ ξ(1,1,0,+1/2), ]

to spontaneously break the U(1)_B-L^ symmetry. Furthermore, our model contains some heavy SM-singlet scalars or fermions,

    [ X^_Ra(1,1,0,0) ,  Σ_b^(1,1,0,+1) ,  (a,b=1,...) . ]






For simplicity, we do not write down the full Yukawa and scalar interactions. Indeed, we will show later the inverse seesaw and the leptogenesis only depend on the following terms, 

    ℒ   ⊃   ℒ_I/II^- (y l̅_L^ϕν_R^ + f ξν̅_R^c S_R^ +H.c.)  with
          ℒ_I⊃ -1/2M_X^X̅_R^X_R^c -g_X^ξS̅_R^ X_R^c +H.c. ,
          ℒ_II^⊃ -M_Σ^2 Σ^†_Σ + ρ_Σ^Σξξ+ 1/2g_Σ^ΣS̅_R^c S_R^ +H.c. .

Without loss of generality and for convenience, we can take the mass matrices M_Σ^ and M_X^ to be real and diagonal. Accordingly, we can define the Majorana fermions as below,

    X_a^= X_Ra^ + X_Ra^c = X^c_a .

Note if the heavy scalar singlet(s) Σ_b^ and the heavy fermion singlet(s) X_Ra^ are both introduced to the models, they should have the Yukawa interactions of the form Σν̅_R^cX_R^ +H.c.. However, such terms are harmful for the realisation of an inverse seesaw. So, we will not consider this case. Instead, we will focus on the case with either the heavy scalar singlet(s) Σ_b^ or the heavy fermion singlet(s) X_Ra^.





We then move to the other neutral fermions χ_R1,2^ and ζ_R^. We can easily find that because of the U(1)_B-L^ symmetry, the ζ_R^ fermion is forbidden to have any Yukawa coupling, but the χ_R1,2^ fermions are allowed to have a Yukawa coupling as below,

    ℒ   ⊃    - y_χ^(ξχ̅_R1^χ_R2^c +H.c.) .

We will show in the following that the χ_R1,2^ fermions provide a dark matter candidate while the ζ_R^ fermion keeps massless and decouples safely. 





§ MINIMAL INVERSE SEESAW















When the Higgs singlet ξ acquires its VEV for the spontaneous U(1)_B-L^ symmetry breaking, the two neutral fermions S_R1,2^ can obtain a Majorana mass term by integrating out the Majorana fermions X_a^ or the Higgs singlets Σ_b^ in Eq. (<ref>), i.e. 

    ℒ   ⊃   - 1/2∑_i,j=1,2^(μ_S^)_ij^S̅_Ri^c S_Rj^ +H.c.  with
          (μ_S^)_ij^=- ∑_b=1^n≥ 2(g_X^∗)_ib^⟨ξ⟩^2_/M_X_b^(g_X^†)_bj^  or
          (μ_S^)_ij^ = - ∑_a=1^n≥ 2(g_Σ_a^)_ij^ρ_Σ_a^⟨ξ⟩^2_/M_Σ_a^2 .

Clearly, as the Majorana fermions X_a^ or the Higgs singlets Σ_b^ are assumed to be much heavier than the U(1)_B-L^ breaking scale ⟨ξ⟩, the above Majorana masses μ_S^ can be highly suppressed in a natural way. This Majorana mass generation is similar to the conventional type-I or type-II seesaw mechanism. 

At this stage, the two neutral fermions S_R1,2^ can also mix with the two right-handed neutrinos ν_R1,2^ to form the quasi-Dirac fermions, i.e.

    ℒ   ⊃    -∑_i,j=1,2^(m_N^)_ijN̅_Ri^ N_Lj^ +H.c.^  with
          N=ν_R^+S_R^c ,  m_N^ = f⟨ξ⟩ .

Subsequently, the Higgs doublet ϕ develops its VEV for the electroweak symmetry breaking. The two right-handed neutrinos ν_R1,2^ can further have a Dirac mass term with and the three left-handed neutrinos ν_Lα^(α=e,μ,τ), i.e.
 
    ℒ   ⊃    -∑_α=e,μ,τ i=1,2^ (m_D^)_α iν̅_Lα^ν_Ri^ +H.c.^  with  m_D^ = y⟨ϕ⟩ .



In the limiting case,
 
    m_N^≫ m_D^ , μ_S^ ,

the left-handed neutrinos ν_L^ can obtain a tiny Majorana mass term, 
 
    ℒ   ⊃    -1/2m_ν^ν̅_L^ν_L^c +H.c.  with
          m_ν^=m_D^1/m_N^†μ_S^1/m_N^∗m_D^T .

This is an application of the so-called inverse seesaw mechanism. 


The neutrino masses (<ref>) can be understood in Fig. <ref>. It should be noted that only two right-handed neutrinos ν_R1,2^ and the same number of fermion singlets S_R1,2^ participate in the neutrino mass generation. Such rank-2 neutrino mass matrix can only have two nonzero eigenvalues. In this sense, we may refer to our model as a minimal inverse seesaw <cit.>. 






§ BARYON ASYMMETRY



In this section we shall demonstrate how to generate the baryon asymmetry in our models.




 §.§ The heavy fermion or Higgs singlet decays








As shown in  Fig. <ref>, the heavy fermion singlet X_b^ have two decay modes, 

    X_b^→ S_R^c+ξ ,   X_b^→ S_R^+ ξ^∗_ .

As long as the CP is not conserved, we can expect a CP asymmetry in the above decays,

    ε_X_b^   =   Γ(X_b^→ S_R^c+ξ)- Γ(X_b^→ S_R^+ξ^∗_)/Γ_X_b^^ ,

with Γ_Σ_a^ being the total decay width,

    Γ_X_b^   =   Γ(X_b^→ S_R^c+ξ)+Γ( X_b^→ S_R^+ ξ^∗_) .

We can calculate the decay width at tree level and the CP asymmetry at one-loop order, i.e.

    Γ_X_b^   =   1/16π(g_X^† g_X^)_bb^M_X_b^
    
    [2mm]
    ε_X_b^   =   -1/16π∑_d≠ b^Im{[(g_X^T g_X^∗)_bd]^2_}/(g_X^† g_X^)_bb^{M_X_b^M_X_d^/M_X_d^2- M_X_b^2.
          .+2M_X_d^/M_X_b^[1-(1+M_X_d^2/M_X_b^2)ln(1+M_X_b^2/M_X_d^2)]} .

For a nonzero CP asymmetry ε_X_b^, we need at least two heavy fermion singlets X_1,...,n≥ 2^.  














As for the heavy Higgs singlet Σ_a^, their decay modes are 

    Σ_a^→  S_R^c+S_R^c ,  Σ_a^→ξ^∗_+ξ^∗_ .

The relevant diagrams are shown in Fig. <ref>. The tree-level decay width and the one-loop CP asymmetry can be calculated by

    Γ_Σ_a^   =   Γ(Σ_a^→ S_R^c+S_R^c)  + Γ( Σ_a^→ξ^∗_+ξ^∗_)
       =   Γ( Σ_a^∗→  S_R^+S_R^  )+Γ( Σ^∗_a→ξ+ξ)
       =   1/8π[Tr(g_Σ_a^^† g_Σ_a^)+ρ_Σ_a^2/M_Σ_b^2]M_Σ_a^ ,


    ε_Σ_a^   =   2 Γ(Σ_a^→ S_R^c+S_R^c  )-Γ( Σ_a^∗→  S_R^+S_R^  )/Γ_Σ_a^^
       =   2Γ( Σ^∗_a→ξ+ξ)-Γ( Σ_a^→ξ^∗_+ξ^∗_)/Γ_Σ_a^^
       =    -1/π∑_c≠ a^Im[Tr(g_Σ_a^† g_Σ_c^)ρ_Σ_a^ρ_Σ_c^]/Tr(g^†_Σ_a g_Σ_a^)+ ρ_Σ_a^2/M_Σ_a^21/M_Σ_c^2-M_Σ_a^2 .

A nonzero CP asymmetry ε_Σ_a^ requires the existence of at least two heavy Higgs singlets Σ_1,...,n≥ 2^.




After the heavy fermion singlets X_b^ or the heavy Higgs singlets Σ_a^ go out of equilibrium, their decays can generate an asymmetry A_S^ stored in the fermion singlets S_R^. For demonstration, we can simply assume the lightest heavy fermion singlet X_1^ or the lightest heavy Higgs singlet Σ_1^ to be much lighter than the other heavy Higgs or fermion singlets. The A_S^ asymmetry then should mainly come from the X_1^ or Σ_1^  decays, i.e.

    A_S^   =   ε_X_1/Σ_1 ^(n^eq_Σ_1/X_1/s)|_T=T_D^^.,

where the symbols n^eq_X_1/Σ_1 and T_D^ respectively are the equilibrium number density and the decoupled temperature of the lightest heavy Higgs or fermion singlets, while the character s is the entropy density of the universe. In this case, the CP asymmetries ε_X_1/Σ_1^ can be simplified by 

    ε_X_1^   ≃   1/8πIm[(g_X^T μ_S^ g_X^)_11] M_X_1^/(g_X^† g_X^)_11^⟨ξ⟩^2_
       ≲   1/8πμ_max^ M_X_1^/⟨ξ⟩^2_ ,


    ε_Σ_1^   ≃   1/πIm[Tr(g_Σ_1^†μ_S^)] ρ_Σ_1^/[Tr(g^†_Σ_1 g_Σ_1^)+ ρ_Σ_1^^2/M_Σ_1^2] ⟨ξ⟩^2_
       ≤   1/πIm[Tr(g_Σ_1^†μ_S^)] ρ_Σ_1^/2√(Tr(g^†_Σ_1 g_Σ_1^)ρ_Σ_1^2/M_Σ_1^2)⟨ξ⟩^2_
       ≲   1/2πμ_max^ M_Σ_1^/⟨ξ⟩^2_ ,

with μ_max^ being the largest eigenvalue of the Majorana mass matrix μ_S^.




Note the Majorana masses μ_S^ can lead to an S_R^-S_R^c oscillation, which tends to wash out the A_S^ asymmetry. However, this oscillation will not go into equilibrium before the sphalerons stop working, i.e.
    [Γ_S_R^-S_R^c^>H(T)]|_T≪  100 GeV^.
          with  Γ_S_R^-S_R^c^∼{[          μ_S^2/T    for  T>m_N^ ,; [2mm]
    μ_S^2/m_N^    for  T<m_N^ . ].

Here H(T) is the Hubble constant to be given later. This means we need not take the Majorana masses μ_S^ into account before the electroweak symmetry breaking. 





 §.§ The lepton-to-baryon conversion




The asymmetry A_S^ stored in the fermion singlets S_R^ will lead to a lepton asymmetry stored in the SM lepton doublets l_L^ because of the related Yukawa interactions in Eq. (<ref>). The sphaleron processes then can partially transfer this lepton asymmetry to a baryon asymmetry. We can impose a global symmetry of lepton number under which the right-handed neutrinos ν_R^ and the fermion singlets S_R^ carry the lepton numbers +1 and -1, respectively. The global lepton number in our models (<ref>) then is softly broken by the Majorana masses of the heavy fermion singlets X_R^ or the cubic couplings of the heavy Higgs singlets Σ to the U(1)_B-L^ Higgs singlet ξ. For such global lepton number, the A_S^ asymmetry equals to a lepton asymmetry L_S^ stored in the fermion singlets S_R^. The lepton-to-baryon conversion thus will not be affected by the U(1)_B-L^ gauge symmetry breaking. Therefore, we can conveniently analysize the lepton-to-baryon conversion after the U(1)_B-L^ symmetry breaking. For this purpose, we denote μ_q^, μ_d^, μ_u^, μ_l^, μ_e^, μ_ν^, μ_S^ and μ_ϕ^ for the chemical potentials of the fields q_L^, d_R^, u_R^, l_L^, e_R^, ν_R^, S_R^ and ϕ. The SM Yukawa interactions are in equilibrium and hence yield <cit.>,

    -μ_q^+μ_d^-μ_ϕ^   =   0 ,
    
     -μ_q^+μ_u^+μ_ϕ^   =   0 ,
     
     -μ_l^+μ_e^-μ_ϕ^   =   0 ,
    
     -μ_l^+μ_ν^ + μ_ϕ^   =   0 ,

the fast sphalerons constrain <cit.>,

    3μ_q^+μ_l^   =   0 ,

while the neutral hypercharge in the universe requires <cit.>,

    3( μ_q^ -μ_d^+2μ_u^-μ_l^ -μ_e^)-2μ_ϕ^ =0 .

In addition, the mass term between the right-handed neutrinos ν_R^ and the fermion singlets S_R^ are also in equilibrium. This means

    μ_S^+μ_ν^   =   0 .

In the above Eqs. (<ref>-<ref>), we have identified the chemical potentials of the different-generation fermions because the Yukawa interactions establish an equilibrium between the different generations. By solving Eqs. (<ref>-<ref>), we can express the chemical potentials by

    μ_ϕ^=-4/7μ_l^ ,  μ_q^=-1/3μ_l^ ,  μ_d^=-19/21μ_l^ ,  μ_u^=5/21μ_l^ ,
          μ_e^=3/7μ_l^ ,  μ_ν^=11/7μ_l^ ,  μ_S^=-11/7μ_l^ .


Now the global baryon number can be given by

    B= 3(2μ_q^ + μ_d^ + μ_u^) = -4 μ_l^ .

As for the global lepton number, it should be 

    L= 3(2μ_L^+μ_e^) + 2(μ_ν^-μ_S^) = 95/7μ_l^ .

The baryon and lepton numbers then can be transferred from a conserved B-L number, i.e.  

    B=28/123(B-L) ,  L=-95/123(B-L) .

If the quasi-Dirac fermions N=ν_R^+S_R^c keep relativistic before the electroweak symmetry breaking, the final baryon number can be given by (<ref>). Alternatively, the N fermions have become non-relativistic and have completely decayed before the electroweak symmetry breaking. In this case, the global lepton number (<ref>) should be modified as 

    L= 3(2μ_L^+μ_e^) = 51/7μ_l^ ,

and hence the baryon and lepton numbers from the conserved B-L number should be   

    B=28/79(B-L) ,  L=-51/79(B-L) .






 §.§ A numerical example


For simplicity, we consider the weak washout condition <cit.>,

    .[Γ_Σ_1/X_1^< H(T)=[8π^3_g_∗^(T)/90]^1/2_T^2_/M_Pl^]|_T=M_Σ_1^/X_1^^ .

The final baryon asymmetry then can approximate to <cit.>

    B^f_∼ c ε_Σ_1/X_1^/g_∗^  with  c=-28/79 or -28/123 .

Here H(T) is the Hubble constant with M_Pl^≃ 1.22× 10^19_ GeV being the Planck mass and g_∗^(T)=123 being the relativistic degrees of freedom (the SM fields plus the seven neutral fermions (ν_R1,2^,S_R1,2^,χ_R1,2^,ζ_R^), the U(1)_B-L^ Higgs singlet ξ and the U(1)_B-L^ gauge boson.). 





As an example, we choose 

    ⟨ξ⟩= 40 TeV ,

and then take

    M_Σ_1^^=10^14_ GeV , ρ_Σ_1^^=3× 10^12_ GeV ,  g_Σ_1^^= 𝒪(0.03) ; 
          or  M_X_1^^=10^14_ GeV , g_X^= 𝒪(0.03) .

With these inputting, we can have

    μ_S^= 𝒪(10 eV) ,

as well as 

    Γ_Σ_1^/X_1^^/H(T)|_T=M_Σ_1^/X_1^^^.=𝒪(0.1) ,  ε_Σ_1^/X_1^^max=𝒪( 10^-5)  .

So, the final baryon asymmetry (<ref>) can arrive at the observed value B^f_∼ 10^-10_. 



We further take 

    f∼𝒪(10^-3_-0.1) , y∼𝒪(10^-2_-1) ,

to give  

    m_N^∼𝒪(10-1000 GeV) ,   m_D^∼𝒪(1-100 GeV) .

By inserting the outputs (<ref>) and (<ref>) into the inverse seesaw (<ref>), we can obtain the neutrino masses m_ν^=𝒪(0.1 eV). 


For the above parameter choice, we also check the S_R^-S_R^c oscillation (<ref>) induced by the Majorana masses μ_S^. We find this oscillation can not go into equilibrium before the electroweak symmetry breaking and hence it will not affect the production of the baryon asymmetry.  












§ DARK FERMIONS






As shown in Eq. (<ref>), the two neutral fermions χ_R1,2^ have a Yukawa coupling with the U(1)_B-L^ Higgs singlet ξ so that they can form a Dirac particle <cit.> after the U(1)_B-L^ symmetry breaking, i.e.

    ℒ   ⊃     i χ̅γ^μ_∂_μ^χ - m_χ^χ̅χ
          with  χ = χ_R1^+χ_R2^c ,   m_χ^= y_χ^⟨ξ⟩ .

Meanwhile, the another neutral fermion ζ_R^ has no Yukawa couplings so that it should be massless. 




Now the Dirac fermion χ is stable and hence leaves a dark matter relic density. The dark matter annihilation and scattering could be determined by the gauge interactions,

    ℒ   ⊃   g_B-L^ Z_B-L^μ[∑_i=1^3(1/3d̅_i^γ_μ^ d_i^ + 1/3u̅_i^γ_μ^u_i^-e̅_i^γ_μ^ e_i^..
          ..-ν̅_Li^γ_μ^ν_Li^)-ν̅_R1^γ_μ^ν_R1^ -ν̅_R1^γ_μ^ν_R1^.
          .+1/2S̅_R1^γ_μ^ S_R1^+1/2S̅_R1^γ_μ^ S_R1^-5/2ζ̅_R^γ_μ^ζ_R^.
          . + 1/4χ̅γ_μ^(√(153)+γ_5^)χ] .

The gauge coupling g_B-L^ then should have an upper bound from the perturbation requirement, i.e.

    √(153)/4g_B-L^ < √(4π)⇒ g_B-L^< √(64 π/153) ,
 
while the gauge boson mass M_Z_B-L^ should be

    M_Z_B-L^^   =    1/√(2) g_B-L^⟨ξ⟩ .

Currently the experimental constraints on the U(1)_B-L^ symmetry breaking is <cit.>

    M_Z_B-L^^/g_B-L^≳ 7  TeV⇒⟨ξ⟩≳ 10  TeV .





The thermally averaging dark matter annihilating cross section can be given by <cit.>

    ⟨σ_A^ v_rel^⟩   =   ∑_f=d,u,e,ν_L^,ν_R^,S_R^,ζ_R^^⟨σ(χ+χ^c_→ f+f^c_) v_rel^⟩
       ≃    13311 g_B-L^4/128πm_χ^2/M_Z_B-L^^4
       =   13311/32πm_χ^2/⟨ξ⟩^4_ =  13311/32πy_χ^2/⟨ξ⟩^2_ ,

where we have assumed

    4m_χ^2 ≪ M_Z_B-L^^2    ⇒    y_χ^2≪1/8 g_B-L^2 < 8π/153
       ⇒    y_χ^< √(8π/153) .

The dark matter relic density then can approximate to <cit.>

    Ω_χ^ h^2 ≃0.1 pb/⟨σ_A^ v_rel^⟩   =   0.1 pb×32π⟨ξ⟩^4_/13311 m_χ^2
       =   0.1 pb×32π⟨ξ⟩^2_/13311 y_χ^2 .






Due to Eq. (<ref>), the VEV ⟨ξ⟩ should have an upper bound,

    ⟨ξ⟩   ≃   (13311 y_χ^2 Ω_χ^ h^2/32 π×  0.1 pb)^1/2_
       =    97  TeV(y_χ^/√(8π/153)) (Ω_χ^ h^2/0.11)^1/2_
       <   97 TeV(Ω_χ^ h^2/0.11)^1/2_ ,

besides the experimental limit (<ref>). The dark matter mass, 

    m_χ^   ≃   (0.1 pb×32π⟨ξ⟩^4_/13311 Ω_χ^ h^2)^1/2_
       =    5 TeV(⟨ξ⟩/34.5 TeV)^2_(0.11/Ω_χ^ h^2)^1/2_ ,

thus should be in the range, 

    420 GeV(0.11/Ω_χ^ h^2)^1/2_≲
    m_χ^ < 40 TeV(0.11/Ω_χ^ h^2)^1/2_
          for  10 TeV≲⟨ξ⟩ < 97 TeV .






The gauge interactions can also result in a dark matter scattering off nucleons. The spin-independent cross section is given by <cit.>

    σ_χ N^   =   153 g_B-L^4/16 πμ_r^2 /M_Z_B-L^^4
       =   153 /4πμ_r^2 /⟨ξ⟩^4_= 8/87μ_r^2 /m_χ^20.1 pb/Ω_χ^ h^2_ .

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, we can simply read 

    σ_χ N^   =    3× 10^-45_ cm^2_(μ_r^/940 MeV)^2_(0.11/Ω_χ^ h^2_)
          ×( 5 TeV/m_χ^)^2_ .

To satisfy the dark matter direct detection results <cit.>, the dark matter mass should have a low limit,

    m_χ^≳ 5 TeV .

Therefore, the allowed dark matter mass should be

    5 TeV(0.11/Ω_χ^ h^2)^1/2_≲
    m_χ^ < 40 TeV(0.11/Ω_χ^ h^2)^1/2_
          for  34.5 TeV≲⟨ξ⟩ < 97 TeV .




The fermion ζ_R^ is massless so that it will affect the effective neutrino number which is stringently constrained by the BBN. We hope the massless Z_R^ can decouple above the QCD scale and hence give a negligible contribution to the effective neutrino number. For this purpose, we need consider the annihilations of the ζ_R^ fermion into the light species,   

    σ_ζ^   =   ∑_f=d,u,s,e,μ,ν_L^^σ(ζ_R^+ζ_R^c→ f+f^c) 
       =   75/32πg_B-L^4/M_Z_B-L^4s =  75/8πs/⟨ξ⟩^4_ ,

with s being the Mandelstam variable. The interaction rate then should be <cit.>

    Γ_ζ^   =   T/32π^4_∫^∞_0 s^3/2_ K_1^(√(s)/T) σ_ζ^ds /2/π^2_T^3_= 18/π^3_T^5_/⟨ξ⟩^4_ ,

with K_1^ being a Bessel function. We take g_∗^(300 MeV)≃ 59.75 and then find 

    [Γ_ζ^ < H(T)]_T≳ 300 MeV^  for  ⟨ξ⟩≳ 37 TeV .

So, if the massless ζ_R^ is harmless, the dark matter mass (<ref>) should be modified, i.e.

    5.7 TeV(0.11/Ω_χ^ h^2)^1/2_≲
    m_χ^ < 40 TeV(0.11/Ω_χ^ h^2)^1/2_
          for  37 TeV≲⟨ξ⟩ < 97 TeV .

 










§ CONCLUSION



In this paper, we have simultaneously explained the neutrino mass, the baryon asymmetry and the dark matter by resorting to a U(1)_B-L^ gauge symmetry. In order to cancel the gauge anomalies, we introduce seven neutral fermions including two usual right-handed neutrinos. After a Higgs singlet develops its VEV for spontaneously breaking the U(1)_B-L^ symmetry, two neutral fermions can obtain their small Majorana masses by integrating out some heavy scalar or fermion singlets, meanwhile, they can have a sizable mass term with the right-handed neutrinos. After the SM Higgs doublet drives the electroweak symmetry breaking, the two right-handed neutrinos and the three left-handed neutrinos can acquire their Dirac masses. We thus can realize an inverse seesaw to give a neutrino mass matrix with two nonzero eigenvalues. Within this framework, the heavy scalar or fermion singlet decays can generate the cosmic baryon asymmetry in association with the sphaleron processes. On the other hand, another two neutral fermions can form a stable Dirac particle due to their Yukawa coupling with the U(1)_B-L^ Higgs singlet. This Dirac fermion can serve as a dark matter particle. As for the seventh neutral fermion, it keeps massless but decouples early. 





Acknowledgement: This work was supported by the National Natural Science Foundation of China under Grant No. 11675100 and the Recruitment Program for Young Professionals under Grant No. 15Z127060004. 









§ THE U(1)_B-L^ GAUGE ANOMALIES 






The SU(3)_c^-SU(3)_c^-U(1)_B-L^ anomaly is

    3× 3×[2×(+1/3) -(+ 1/3) - (+1/3) ]=0 .


The SU(2)_L^-SU(2)_L^-U(1)_B-L^ anomaly is 

    3× 2 ×[3×(+ 1/3) +(-1)]=0 .


The U(1)_Y^-U(1)_Y^-U(1)_B-L^ anomaly is 

    3×{3×[2×(+1/6)^2_ - (-1/3)^2_ - (+2/3)^2_] ×(+ 1/3) .
    
    [2mm]
          .+[2×(-1/2)^2_ - (-1)^2_]×(-1) }  =0 .


The U(1)_Y^-U(1)_B-L^-U(1)_B-L^ anomaly is 

    3×{3×[2×(+1/6) -(-1/3)-(+2/3)]×(+1/3)^2_.
    
    [2mm]
          .+[2×(-1/2) -(-1)]×(-1) }=0 .


The U(1)_B-L^-U(1)_B-L^-U(1)_B-L^ anomaly is 

    3×{3×[2×(+1/3)^3_-(+1/3)^3_-(+1/3)^3_] .
    
    [2mm]
          .+[2×(-1)^3_ -(-1)^3_] } -2×(-1)^3_ -2×(+1/2)^3_
    
    [2mm]
          -(1-√(153)/4)^3_ -(1+√(153)/4)^3_ - (-5/2)^3_=0 .


The graviton-graviton-U(1)_B-L^ anomaly is

    3×{3×[2×(+1/3)-(+1/3)-(+1/3)] .
    
    [2mm]
          .+[2×(-1) -(-1)] } - -2×(-1) -2×(+1/2) 
    
    [2mm]
          -(1-√(153)/4) -(1+√(153)/4)- (-5/2)=0 .
 






99



 
pdg2018
M. Tanabashi et al., (Particle Data Group), Phys. Rev. D 98, 030001 (2018).





minkowski1977
P. Minkowski, Phys. Lett. B 67, 421 (1977).

yanagida1979
T. Yanagida, Proceedings of the Workshop on Unified Theory and the Baryon Number of the Universe, ed. O. Sawada and A. Sugamoto (Tsukuba 1979).

grs1979
M. Gell-Mann, P. Ramond, and R. Slansky, Supergravity, ed. F. van Nieuwenhuizen and D. Freedman
(North Holland 1979).

ms1980
R.N. Mohapatra and G. Senjanović, Phys. Rev. Lett. 44, 912 (1980).






mw1980
M. Magg and C. Wetterich, Phys. Lett. B 94, 61 (1980).

sv1980
J. Schechter and J.W.F. Valle, Phys. Rev. D 22, 2227 (1980).


cl1980

T.P. Cheng and L.F. Li, Phys. Rev. D 22, 2860 (1980). 

lsw1981
G. Lazarides, Q. Shafi, and C. Wetterich, Nucl. Phys. B 181,
287 (1981).

ms1981
R.N. Mohapatra and G. Senjanović, Phys. Rev. D
23, 165 (1981).





flhj1989
R. Foot, H. Lew, X.G. He, and G.C. Joshi, Z. Phys. C 44,
441 (1989).





fy1986
M. Fukugita and T. Yanagida, Phys. Lett. B 174, 45 (1986).



lpy1986
P. Langacker, R.D. Peccei, and T. Yanagida, Mod. Phys. Lett. A
1, 541 (1986); M.A. Luty, Phys. Rev. D 45, 455
(1992); R.N. Mohapatra and X. Zhang, Phys. Rev. D 46, 5331 (1992).





fps1995
M. Flanz, E.A. Paschos, and U. Sarkar, Phys. Lett. B 345,
248 (1995); M. Flanz, E.A. Paschos, U. Sarkar, and J. Weiss, Phys.
Lett. B 389, 693 (1996); L. Covi, E. Roulet, and F.
Vissani, Phys. Lett. B 384, 169 (1996); A. Pilaftsis, Phys.
Rev. D 56, 5431 (1997).


ms1998
E. Ma and U. Sarkar, Phys. Rev. Lett. 80, 5716 (1998).


bcst1999
R. Barbieri, P. Creminelli, A. Strumia, and N. Tetradis, Nucl. Phys. B 575, 61 (2000).




hambye2001
T. Hambye, Nucl. Phys. B 633, 171 (2002).


di2002
S. Davidson and A. Ibarra, Phys. Lett. B 535, 25 (2002); W.
Buchmüller, P. Di Bari, and M. Plümacher, Nucl. Phys. B
665, 445 (2003).

gnrrs2003
G.F. Giudice, A. Notari, M. Raidal, A. Riotto, and A. Strumia, Nucl. Phys. B 685, 89 (2004).


hs2004
T. Hambye and G. Senjanović, Phys. Lett. B 582, 73
(2004); S. Antusch and S.F. King, Phys. Lett. B 597, 199
(2004).


bbp2005
W. Buchmuller, P. Di Bari, and M. Plumacher, Annals Phys. 315, 305 (2005).



krs1985
V.A. Kuzmin, V.A. Rubakov, and M.E. Shaposhnikov, Phys. Lett. B
155, 36 (1985).



mv1986
R. Mohapatra and J. W. F. Valle, Phys. Rev. D 34, 1642 (1986).






adefhv2018 
K. Agashe, P. Du, M. Ekhterachian, C.S. Fong, S. Hong, and L. Vecchi, Phys. Lett. B 785, 489 (2018); K. Agashe, P. Du, M. Ekhterachian, C.S. Fong, S. Hong, and L. Vecchi, JHEP 1904, 029 (2019).


gu2019
P.H. Gu, arXiv:1907.09444 [hep-ph]. 




rsv2019
N. Rojas, R. Srivastava, and J.W.F. Valle, arXiv:1907.07728 [hep-ph].














mp2007
J.C. Montero and V. Pleitez, Phys. Lett. B 675, 64 (2009). 


pry2016 
S. Patra, W. Rodejohann, and C.E. Yaguna, JHEP 1609, 076 (2016).









gu2019-2
P.H. Gu, arXiv:1907.10018 [hep-ph].




ht1990
J.A. Harvey and M.S. Turner, Phys. Rev. D 42, 3344 (1990).




kt1990
E.W. Kolb and M.S. Turner, The Early Universe,
Addison-Wesley, 1990.

 
 





gu2019-3
P.H. Gu, arXiv:1907.10019 [hep-ph].
  
 
 
afpr2017
S. Alioli, M. Farina, D. Pappadopulo, and J.T. Ruderman, Phys. Rev. Lett. 120, 101801 (2018).




klq2016
M. Klasen, F. Lyonnet, and F.S. Queiroz, Eur. Phys. J. C 77, 348 (2017).

 



bhkk2009
M. Beltran, D. Hooper, E.W. Kolb, and Z.C. Krusberg, Phys. Rev. D 80, 043509 (2009);
K. Cheung, P.Y. Tseng, and T.C. Yuan, JCAP 1101, 004 (2011).




jkg1996
G. Jungman, M. Kamionkowski, and K. Griest, Phys. Rept. 267, 195 (1996).




cui2017

X. Cui et. al., (PandaX-II Collaboration), Phys. Rev. Lett. 119, 181302 (2017). 



aprile2018

E. Aprile et. al., (XENON Collaboration), Phys. Rev. Lett. 121, 111302 (2018).





