Double type-II Dirac 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






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. 











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







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






§ INTRODUCTION


The concept of three massive and mixing neutrinos has been established by the atmospheric and solar neutrino oscillations and then has been confirmed by the accelerator and reactor neutrino oscillations <cit.>. Meanwhile, the cosmological observations have constrained the neutrinos to be extremely light <cit.>. Currently, various seesaw <cit.> mechanisms at tree or loop level <cit.> are considered the best explanation for the tiny but nonzero neutrino masses. In this popular seesaw scenario, the neutrino masses are induced by certain lepton-number-violating interactions so that the neutrinos should have a Majorana nature. However, the lepton number violation and then the Majorana neutrinos are just a theoretical assumption and have not been seen in any experiments. So, we should also consider the possibility of Dirac neutrinos <cit.>. Actually, the so-called type-I <cit.>, type-II <cit.> and type-III <cit.> Dirac seesaw as well as the radiative Dirac seesaw <cit.> for the Dirac neutrinos have been constructed in analogy to the conventional seesaw mechanisms for the Majorana neutrinos. In the Majorana or Dirac seesaw models, the cosmic baryon asymmetry, which is another big challenge to the standard model (SM), can be understood in a natural way <cit.>. This is the famous leptogenesis mechanism <cit.>. 



In a recent work <cit.>, we have proposed a double Dirac seesaw model where a TeV-scale Higgs doublet with a small mixing to the SM Higgs doublet can have the sizable Yukawa couplings to several right-handed neutrinos and the SM lepton doublets. This Dirac neutrino mass generation could be tested experimentally <cit.>. We showed the small mixing between the two Higgs doublets could come from certain interactions for generating the cosmic baryon asymmetry. 









In this paper we shall develop our idea of the double Dirac seesaw to naturally include a dark matter particle, which is absent from the SM. Specifically we shall introduce a U(1)_B-L^ gauge symmetry with four right-handed neutrinos. Through 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 to be a stable dark matter particle. As for the other two right-handed neutrinos, they can couple to the SM lepton doublets with a second Higgs doublet. After the U(1)_B-L^ symmetry breaking, two or more heavy Higgs singlets can acquire their suppressed vacuum expectation values (VEVs) to result in a small mixing between the second Higgs doublet and the SM Higgs doublet. Three left-handed neutrinos and two right-handed neutrinos thus can have a mass matrix with two nonzero eigenvalues. The heavy Higgs singlet decays can generate an asymmetry stored in the second Higgs doublet. This asymmetry can lead to a baryon asymmetry in association with the sphaleron processes <cit.>. 







§ FERMIONS AND SCALARS


Under the SU(3)_c^× SU(2)^_L× U(1)_Y^× U(1)_B-L^ gauge symmetries, we need extend the SM fermions,

    [ q^_L(3,2,+1/6)(+1/3) , ]  [ d^_R(3,1,-1/3)(+1/3) , ]
    
    [1mm]
           [ u^_R(3,1,+2/3)(+1/3) , ]  [ l^_L(1,2,-1/2)(-1) , ]
    
    [1mm]
          [ e^_R(1,1,-1)(-1) , ]
 
by some right-handed neutrinos to cancel the U(1)_B-L^ gauge anomalies <cit.>. Here and thereafter the first and second brackets following the fields respectively describe the transformations under the SU(3)_c^× SU(2)^_L× U(1)_Y^ gauge groups and the U(1)_B-L^ gauge group. In the present work, we consider the following four right-handed neutrinos,

    [ ν^_R1,2(1,1,0)(-8/5) , ][ ν^_R3(1,1,0)(1+√(865)/10) , ]
          [ ν^_R4(1,1,0)(1-√(865)/10) . ]




While the SM Higgs doublet

    [ ϕ(1,2,-1/2)(0) , ]
 
is responsible for the spontaneous electroweak symmetry breaking, we introduce a Higgs singlet, 

    [ ξ(1,1,0)(+1/5), ]

to drive the spontaneous U(1)_B-L^ symmetry breaking. 




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.

    ℒ⊃ - y_34^(ξν̅_R3^ν_R4^c +H.c.) .

Furthermore, in association with the Higgs singlet ξ, we can construct the following dimension-5 operators involving the SM lepton and Higgs doublets as well as the first and second right-handed neutrinos ν_R1,2^, i.e.

    ℒ⊃ - ∑_i=1,2 α=e,μ,τc_α i^/Λ^3_l̅_Lα^ϕν_Ri^ξ^3_ +H.c. .

The above effective operators can originate from two renormalizable models as below, 

    ℒ_A^   ⊃    - (m_η^2 +λ_ηξ^ξ^†_ξ+  λ_ηϕ^ϕ^†_ϕ)η^†_η - λ'^_ηϕη^†_ϕϕ^†_η
          -∑_a=1^n≥ 2(M_σ_a^2 σ^†_aσ_a^ + ρ_σ_a^σ_a^η^†_ϕ + κ_σ_a^σ_a^ξ^†_ξ^†_ξ^†_)
            - ∑_ i=1,2 α=e,μ,τ^ f_α i^l̅_Lα^ην_Ri^ + H.c. ,


    ℒ_B^   ⊃     - (m_η^2 +λ_ηξ^ξ^†_ξ+  λ_ηϕ^ϕ^†_ϕ)η^†_η - λ'^_ηϕη^†_ϕϕ^†_η
          -∑_b=1^n≥ 2(M_ω_b^2 ω^†_bω_b^ + κ_ω_b^ω_b^ξη^†_ϕ + ρ_ω_b^ω_b^ξ^†_ξ^†_)
           - ∑_ i=1,2 α=e,μ,τ^ f_α i^l̅_Lα^ην_Ri^ + H.c. ,

where η is a second Higgs doublet,

    [ η(1,2,-1/2)(+3/5) , ]

while σ_a^ and ω_b^ are two types of heavy Higgs singlets,  

    [ σ_a^(1,2,-1/2)(+3/5) , ][ ω_b^(1,2,-1/2)(+2/5) . ]

Note the model (<ref>) or (<ref>) can always contains a relative phase among the four parameters (κ_σ_a^,κ_σ_c≠ a^,ρ_σ_a^,ρ_σ_c≠ a^) or (κ_ω_b^,κ_ω_d≠ b^,ρ_ω_b^,ρ_ω_d≠ b^).












§ DIRAC NEUTRINO MASS
















As shown in Fig. <ref>, we can obtain the effective operators (<ref>) by integrating out the heavy Higgs singlets σ_a^/ω_b^ and the second Higgs doublet η, i.e.

    c_α i^/Λ^3_ = -∑_a/b^ρ_σ_a/ω_b^κ_σ_a/ω_b^/M_σ_a/ω_b^2 m_η^2 f_α i^ .

The three left-handed neutrinos ν_Lα^ and the two right-handed neutrinos ν_R1,2^ thus can acquire a tiny Dirac mass term after the SU(2)_L^ and U(1)_B-L^ symmetry breaking,

    ℒ   ⊃    -∑_i=1,2 α=e,μ,τ^(m_ν^)_α i^ν̅_Lα^ν_Ri^ +H.c.  with
          (m_ν^)_α i^ = -∑_a/b^ρ_σ_a/ω_b^κ_σ_a/ω_b^⟨ξ⟩^3_/M_σ_a/ω_b^2 m_η^2 f_α i^⟨ϕ⟩ .
 
This Dirac neutrino mass generation actually is a two-step seesaw mechanism. We demonstrate the details in the following. 


When the Higgs singlet develops its VEV ⟨ξ⟩ for the U(1)_B-L^ symmetry breaking, the heavy Higgs singlets σ_a^ or ω_b^ can acquire a suppressed VEV, 

    ⟨σ_a^⟩   ≃    -κ_σ_a^^⟨ξ⟩^3_/M_σ_a^^2≪⟨ξ⟩  for  M_σ_a^^≫⟨ξ⟩ ,
    ⟨ω_b^⟩   ≃    -ρ_ω_b^^⟨ξ⟩^2_/M_ω_b^^2  for  M_ω_b^^≳ρ_ω_b^^ , M_ω_b^^≫⟨ξ⟩ .

In consequence, the second Higgs doublet η can obtain a small mixing with the SM Higgs doublet ϕ, i.e.

    ℒ   ⊃    - μ_ηϕ^2 η^†_ϕ +H.c.  with
          μ_ηϕ^2=∑_a^ρ_σ_a^^⟨σ_a^⟩  or  μ_ηϕ^2=∑_b^κ_ω_b^^⟨ω_b^⟩⟨ξ⟩ .

We can see the suppressed VEV ⟨σ_a^⟩ or ⟨ω_b^⟩ and then small quadratic coupling μ_ηϕ^2 are induced by a seesaw mechanism like the traditional type-II seesaw <cit.>.

Subsequently, the SM Higgs doublet ϕ acquires a VEV ⟨ϕ⟩≃ 174 GeV for the electroweak symmetry breaking. The neutral and charged components of the second Higgs doublet η=(η^0_, η^-_)^T_ have their masses as below,  

    m_η^0_^2   =   m_η^2 +λ_ηξ^⟨ξ⟩^2_+ ( λ_ηϕ^+ λ'^_ηϕ)⟨ϕ⟩^2_ ,
    
    m_η^±_^2   =    m_η^2 +λ_ηξ^⟨ξ⟩^2_+ λ_ηϕ^⟨ϕ⟩^2_ .

The neutral component η^0_ can pick up a small VEV, 

    ⟨η⟩≃ -μ_ηϕ^2⟨ϕ⟩/m_η^0_^2≪⟨ϕ⟩  for  |μ_ηϕ^2|≪ m_η^0_^2 .

The left-handed neutrinos ν_L^ and the first and second right-handed neutrinos ν_R1,2^ can naturally obtain a tiny Dirac mass term,

    ℒ   ⊃    -∑_i=1,2 α=e,μ,τ^(m_ν^)_α i^ν̅_Lα^ν_Ri^ +H.c.  with  m_ν^ = f ⟨η⟩ .
 
The above Dirac neutrino mass generation by the small VEV ⟨η⟩ is again a seesaw mechanism like the traditional type-II seesaw <cit.> for the Majorana neutrinos.  


In conclusion, we have realized a two-step seesaw mechanism to generate the tiny Dirac neutrino masses. Such Dirac neutrino mass generation may be entitled as a double type-II Dirac seesaw <cit.>, in analogy to our double type-II seesaw <cit.> for the Majorana neutrinos. Note this Dirac neutrino mass matrix can only have two nonzero eigenvalues because it is a 3× 2 matrix with two right-handed neutrinos. In this sense, we may further name our neutrino mass generation as a minimal double type-II Dirac seesaw.

For a numerical estimation, we choose

    ⟨ξ⟩=𝒪(100 TeV) ,

and

    M_σ_a^/ω_b^^=𝒪(10^16_ GeV) ,  ρ_σ_a^/ω_b^^=𝒪(10^14_ GeV) ,
          κ_σ_a^/ω_b^^=𝒪(0.01) .

We then obtain

    μ_ηϕ^2 =  𝒪(10^-5_ GeV^2_)  for
          ⟨σ_a^⟩ = 𝒪(10^-19_ GeV)  or  ⟨ω_b^⟩ =𝒪(10^-8_ GeV) .

By further taking 

    m_η^0_^= 𝒪(1-10 TeV) ,

we realize

    ⟨η⟩ = 𝒪(0.01-1 eV) .

and hence 

    m_ν^=⟨η⟩ = 𝒪(0.01-0.1 eV)  for  f_α i^ =𝒪(0.1-1) .





§ BARYON ASYMMETRY









As shown in Fig. <ref>, the heavy Higgs singlets σ_a^ have two decay modes,

    σ_a^→ξξξ ,  σ_a^→ηϕ^∗_ .

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

    ε_σ_a^   =   Γ(σ_a^→ηϕ^∗_) - Γ(σ_a^∗→η^∗_ϕ ) /Γ_a^
       =   - Γ(σ_a^→ξξξ) - Γ(σ_a^∗→ξ^∗_ξ^∗_ξ^∗_) /Γ_a^≠ 0 ,

where Γ_a^ is the total decay width,

    Γ_σ_a^   =   Γ(σ_a^→ξξξ) + Γ(σ_a^→ηϕ^∗_) 
       =   Γ(σ_a^∗→ξ^∗_ξ^∗_ξ^∗_) + Γ(σ_a^∗→η^∗_ϕ )  .

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

    Γ_σ_a^   =   1/8π(|ρ_σ_a^|^2/M_σ_a^2+3|κ_σ_a^|^2_/32π^2_) M_σ_a^ ,


    ε_σ_a^   =   - 3/64π^3_∑_c≠ a^Im(κ^∗_σ_aκ_σ_c^ρ_σ_a^ρ_σ_c^∗)/|ρ_σ_a^|^2/M_σ_a^2+3|κ_σ_a^|^2_/32π^2_1/M_σ_c^2-M_σ_a^2
       =   - 3/64π^3_∑_c≠ a^|κ^_σ_aκ_σ_c^ρ_σ_a^ρ_σ_c^|sinσ_ac^/|ρ_σ_a^|^2/M_σ_a^2+3|κ_σ_a^|^2_/32π^2_1/M_σ_c^2-M_σ_a^2 .

Here σ_ac^ is the relative phase among the parameters ρ_σ_a,σ_c^ and κ_σ_a,σ_c^. 


Similarly, the heavy Higgs singlets ω_b^ also have a two-body decay and a three-body decay,

    ω_a^→ξξ ,  ω_b^→ηϕ^∗_ξ^∗_ .

We can calculate the decay width and the CP asymmetry,

    Γ_ω_b^   =   Γ(ω_b^→ξξ) + Γ(ω_b^→ηϕ^∗_ξ^∗_) 
       =   Γ(ω_b^∗→ξ^∗_ξ^∗_) + Γ(ω_b^∗→η^∗_ϕξ ) 
       =   1/8π(|ρ_ω_b^|^2/M_ω_b^2+|κ_ω_b^|^2_/32π^2_) M_ω_b^ ,


    ε_ω_b^   =   Γ(ω_b^→ηϕ^∗_ξ^∗_) - Γ(ω_b^∗→η^∗_ϕξ) /Γ_ω_b^^
       =   - Γ(ω_b^→ξξ) - Γ(ω_b^∗→ξ^∗_ξ^∗_) /Γ_ω_b^^
       =   - 1/64π^3_∑_d≠ b^Im(κ^∗_ω_bκ_ω_d^ρ_ω_b^ρ_ω_d^∗)/|ρ_ω_b^|^2/M_ω_b^2+|κ_ω_b^|^2_/32π^2_1/M_ω_d^2-M_ω_b^2
       =   - 1/64π^3_∑_d≠ b^|κ^_ω_bκ_ω_c^ρ_ω_b^ρ_ω_d^|sinω_ab^/|ρ_ω_b^|^2/M_ω_b^2+|κ_ω_b^|^2_/32π^2_1/M_ω_d^2-M_ω_b^2 .

Here ω_bd^ is the relative phase among the parameters ρ_ω_b,ω_d^ and κ_ω_b,ω_d^. 



After the heavy Higgs singlets σ_a^/ω_b^ go out of equilibrium, their decays can generate an asymmetry A_η^ stored in the second Higgs doublet η. For demonstration, we simply assume a hierarchical spectrum of the heavy Higgs singlets σ_a^/ω_b^, i.e. M_σ_1^/ω_1^^2≪ M_σ_2,...^/ω_2,...^^2. In this case, the decays of the lightest σ_1^ should dominate the final A_η^ asymmetry, i.e.

    A_η^   =   ε_σ_1^/ω_1^^(n^eq_σ_1^/ω_1^/s)|_T=T_D^^. ,

where the symbols n^eq_σ_1^/ω_1^ and T_D^ respectively are the equilibrium number density and the decoupled temperature of the heavy Higgs singlets σ_1^/ω_1^, while the character s is the entropy density of the universe <cit.>. The sphaleron processes eventually will partially transfer this A_η^ asymmetry to a baryon asymmetry <cit.>,

    B= -28/79ε_σ_1^/ω_1^^(n^eq_σ_1^/ω_1^/s)|_T=T_D^^. .




For a numerical example, we consider the case with two heavy Higgs singlets σ_1,2^/ω_1,2 . We then take 

    M_σ_1/ω_1^=10^16_ GeV , |ρ_σ_1/ω_1^|=10^14_ GeV , 
           |κ_σ_1/ω_1^|=0.02 ;  M_σ_2/ω_2^^=3× 10^16_ GeV ,
          |ρ_σ_2/ω_2^| =3× 10^14_ GeV , |κ_σ_2/ω_2^|=0.02 ,

and hence obtain

    Γ_σ_1^   =   4× 10^10_ ,GeV ,
    ε_σ_1^   =   -2.3× 10^-7sinσ_12^ ;  or
    Γ_ω_1^   =    4× 10^10_ GeV ,
    ε_ω_1^   =   -7.6× 10^-8sinω_12^ .

We can see the decay width Γ_σ_1/ω_1^ is smaller than the Hubble constant H(T) at the temperature T=M_σ_1/ω_1^, i.e.

    .[H(T)=(8π^3_g_∗^/90)^1/2_T^2_/M_Pl^]|_T=M_σ_1^/ω_1^^
          = 1.5× 10^14_ GeV ,

with M_Pl^≃ 1.22× 10^19_ GeV being the Planck mass and g_∗^=121.75 being the relativistic degrees of freedom (the SM fields plus the four right-handed neutrinos ν_R^, the second Higgs doublet η, the Higgs singlet ξ and the U(1)_B-L^ gauge boson.). Therefore, the weak washout condition <cit.>,

    Γ_σ_1/ω_1^/H(T=M_σ_1^/ω_1^)<1 ,

can be satisfied in the heavy Higgs singlet decays. The baryon asymmetry (<ref>) hence can be simply given by 

    B   ∼     -28/79ε_σ_1^^/g_∗^  =10^-10_(sinσ_12^/0.15)  or
    
    B   ∼     -28/79ε_ω_1^^/g_∗^  = 10^-10_(sinσ_12^/0.45) .






§ DARK MATTER AND RIGHT-HANDED NEUTRINOS




After the U(1)_B-L^ symmetry breaking, the gauge boson Z_B-L^ can obtain its mass 

    M_Z_B-L^^≃√(2)/5g_B-L^⟨ξ⟩ .

Meanwhile, the third and forth right-handed neutrinos ν_R3,4^ can form a Dirac particle <cit.>, i.e.

    ℒ   ⊃     i χ̅γ^μ_∂_μ^χ - m_χ^χ̅χ
          with  χ = ν_R3^+ν_R4^c ,   m_χ^= y_34^⟨ξ⟩ .

This Dirac fermion has no decay channels so that it can be a stable dark matter particle. The dark matter annihilation and scattering could be dominated 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_Liγ_μ^ν_Li^)-8/5ν̅_R1^γ_μ^ν_R1^ - 8/5ν̅_R2^γ_μ^ν_R2^.
          .-i3/5[(∂_μ^η)^†_η - H.c.]  + 1/10χ̅γ_μ^(√(865)+γ_5^)χ} .

The perturbation requirement then constrain the gauge coupling g_B-L^ by

    √(865)/10g_B-L^ < √(4π)⇒ g_B-L^<  √(80π/173) ,
 
while the experimental results constrain the U(1)_B-L^ symmetry breaking scale by <cit.>

    M_Z_B-L^^/g_B-L^≳ 7  TeV⇒⟨ξ⟩≳ 25 TeV .




We can calculate the thermally averaging dark matter annihilating cross section <cit.>,  

    ⟨σ_A^ v_rel^⟩   =   ∑_f=d,u,e,ν_L^,ν_R1,2^^⟨σ(χ+χ^c_→ f+f^c_) v_rel^⟩
          + ⟨σ(χ+χ^c_→η+η^∗_) v_rel^⟩
       ≃   39963 g_B-L^4/500πm_χ^2/M_Z_B-L^^4
       =   199815/16πm_χ^2/⟨ξ⟩^4_ =  199815/16πy_χ^2/⟨ξ⟩^2_ ,

by simply assuming 

    4m_χ^2 ≪ M_Z_B-L^^2   ⇒    y_χ^2≪1/50 g_B-L^2 < 8π/865
       ⇒    y_χ^< √(8π/865) .

The dark matter relic density then can approximately given by <cit.>

    Ω_χ^ h^2 ≃0.1 pb/⟨σ_A^ v_rel^⟩   =   0.1 pb×16π⟨ξ⟩^4_/199815m_χ^2
       =   0.1 pb×16π⟨ξ⟩^2_/199815 y_χ^2 .





From Eqs. (<ref>) and (<ref>), we can put a constraint on the the VEV ⟨ξ⟩, i.e.

    ⟨ξ⟩   ≃   (199815 y_χ^2  Ω_χ^ h^2/16π×  0.1 pb)^1/2_
       =     222 TeV(y_χ^/√(8π/865)) (Ω_χ^ h^2/0.11)^1/2_
       <   222 TeV(Ω_χ^ h^2/0.11)^1/2_ .

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

    m_χ^   ≃   (0.1 pb×16π⟨ξ⟩^4_/199815 Ω_χ^ h^2)^1/2_
       =    5 TeV(⟨ξ⟩/81 TeV)^2_(0.11/Ω_χ^ h^2)^1/2_ ,

thus should be in the range, 

    480 GeV(0.11/Ω_χ^ h^2)^1/2_≲
    m_χ^ < 38 TeV(0.11/Ω_χ^ h^2)^1/2_
          for  25 TeV≲⟨ξ⟩ < 222 TeV .






The gauge interactions can also mediate the dark matter scattering off nucleons. The dominant scattering cross section is spin independent <cit.>, 

    σ_χ N^   =   173g_B-L^4/20 πμ_r^2 /M_Z_B-L^^4
       =   21625 /16πμ_r^2 /⟨ξ⟩^4_ =   25/231μ_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, the above dark matter scattering cross section indeed should be inversely proportional to the squared dark matter mass, 

    σ_χ N^   =    3.5× 10^-45_ cm^2_(μ_r^/940 MeV)^2_(0.11/Ω_χ^ h^2_)
          ×( 5 TeV/m_χ^)^2_ .

The dark matter direct detection results <cit.> then put a low limit on the dark matter mass, i.e. 

    m_χ^≳ 5 TeV .

So, the dark matter masses (<ref>) should be modified by

    5 TeV(0.11/Ω_χ^ h^2)^1/2_≲
    m_χ^ < 38 TeV(0.11/Ω_χ^ h^2)^1/2_
          for  81 TeV≲⟨ξ⟩ < 222 TeV .






The two right-handed neutrinos ν_R1,2^ would significantly contribute to the effective neutrino number <cit.> if they did not decouple above the QCD scale. This is not allowed by the BBN. So, we check the annihilations of the right-handed neutrinos into the relativistic species at the QCD scale,   

    σ_ν_R^^   =   ∑_f=d,u,s,e,μ,ν_L^^σ(ν_R^+ν_R^c→ f+f^c) 
       =   24 g_B-L^4/25πs/M_Z_B-L^4 =  150/πs/⟨ξ⟩^4_ ,

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

    Γ_ν_R^^ =T/32π^4_∫^∞_0 s^3/2_ K_1^(√(s)/T) σ_ζ^ds /2/π^2_T^3_= 1800/π^3_T^5_/⟨ξ⟩^4_ ,

with K_1^ being a Bessel function. We take g_∗^(300 MeV)≃ 61.75 and then find 

    [Γ_ν_R^^ < H(T)]_T≳ 300 MeV^  for  ⟨ξ⟩≳ 51 TeV ,

to fulfil the parameter space (<ref>). 




§ CONCLUSION




In this paper we have shown the U(1)_B-L^ gauge symmetry with four neutral fermions can predict two right-handed neutrinos for the Dirac neutrino mass generation and one stable Dirac fermion for the dark matter relic density. Specifically, two neutral fermions can form a Dirac fermion for a stable dark matter particle through their Yukawa couplings to the Higgs singlet for spontaneously breaking the U(1)_B-L^ symmetry. The other two neutral fermions as the right-handed neutrinos can couple to the SM lepton doublets with a second Higgs doublet. After the U(1)_B-L^ symmetry breaking, some heavy Higgs singlets can acquire their suppressed VEVs to highly suppress the mixing between the second Higgs doublet and the SM Higgs doublet. Consequently, the second Higgs doublet can naturally pick up a tiny VEV even if it is at the TeV scale. This means a testable Dirac neutrino mass generation. Remarkably, the neutrino mass matrix can only have two nonzero eigenvalues since it just involoves two right-handed neutrinos. The interactions for generating the neutrino masses can also explain the observed baryon asymmetry in association with the sphaleron processes. 





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×(-8/5)^3_
    
    [2mm]
          -(1-√(865)/10)^3_ -(1+√(865)/10)^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×(-8/5) 
    
    [2mm]
          -(1-√(865)/10) -(1+√(865)/10)=0 .
 





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