$[SU(2)]^3$ Dark Matter
Ernest Ma
hep-ph
  
 
 
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         UCRHEP-T586
        
         Jan 2018
        
        
        [SU(2)]^3 Dark Matter
        
         Ernest Ma
        
         Physics and Astronomy Department,
         
        University of California, Riverside, California 92521, USA
        
         Jockey Club Institute for Advanced Study,
         
        Hong Kong University of Science and Technology, Hong Kong, China
        
        
        An extra SU(2)_D gauge factor is added to the well-known left-right 
        extension of the standard model (SM) of quarks and leptons.  Under 
        SU(2)_L × SU(2)_R × SU(2)_D, two fermion bidoublets 
        (2,1,2) and (1,2,2) are assumed.  The resulting model has an 
        automatic dark U(1) symmetry, in the same way that the SM has automatic 
        baryon and lepton U(1) symmetries.  Phenomenological implications are 
        discussed, as well as the possible theoretical origin of this proposal.
         24pt
        
         Introduction : 
        In the standard model (SM) of quarks and leptons, the choice of the gauge 
        symmetry, i.e. SU(3)_C × SU(2)_L × U(1)_Y, and the particle 
        content, i.e. quarks and leptons:
        
            (u,d)_L ∼ (3,2,1/6),     u_R ∼ (3,1,2/3),     d_R ∼ (3,1,-1/3), 
            
                   (ν,l)_L ∼ (1,2,-1/2),     l_R ∼ (1,1,-1),
        
        together with the one Higgs scalar doublet
        
            Φ = (ϕ^+,ϕ^0) ∼ (1,2,1/2),
        
        automatically imply the existence of two global U(1) symmetries, i.e. 
        baryon number (B) under which quarks have charge 1/3, and lepton 
        number (L) under which leptons have charge 1.  Is there a corresponding 
        scenario for the existence of dark matter?  Consider for example the 
        conventional left-right extension of the SM.  Because of the implied 
        U(1)_B-L gauge factor, a discrete Z_2 parity, i.e. 
        R = (-1)^3B+L+2j, may be used to distinguish some new particles from 
        those of the SM automatically.  The importance of this observation is 
        that this parity is not imposed, as is necessary in the minimal 
        supersymmetric standard model, or in models of dark matter <cit.> 
        assuming only the SM gauge symmetry.  Whereas this idea 
        of an automatic R parity has been implemented in some recent 
        studies <cit.>, I look instead 
        in this paper for a dark U(1) symmetry (and not just a dark parity) 
        which is also unrelated to B or L, but on the same footing, i.e. 
        its emergence as the result of gauge symmetry and particle content.   
        In the following I show how it may be achieved by inserting an extra 
        SU(2)_D gauge factor to the well-known 
        SU(3)_C × U(1)_B-L× SU(2)_L × SU(2)_R model.  Its 
        theoretical origin is a possible SU(6) generalization of the Pati-Salam 
        SU(4) symmetry <cit.>.
        
         Particle Content : 
        Under 
        SU(3)_C × U(1)_B-L× SU(2)_L × SU(2)_R × SU(2)_D, 
        the quarks and leptons transform as expected, i.e. as singlets under 
        SU(2)_D:
        
            (u,d)_L ∼ (3,1/6,2,1,1),     (u,d)_R ∼ (3,1/6,1,2,1), 
                   (ν,l)_L ∼ (1,-1/2,2,1,1),     (ν,l)_R ∼ (1,-1/2,1,2,1),
        
        and the new fermions transform as bidoublets:
        
            ψ_1^0    ψ_2^+ ψ_1^-    ψ_2^0_L ∼ (1,0,2,1,2),    ψ_3^0    ψ_4^+ ψ_3^-    ψ_4^0_R ∼ (1,0,1,2,2),
        
        where SU(2)_L,R act vertically, and SU(2)_D horizontally.  The 
        electric charge is given by
        
            Q = 1  2(B-L) + I_3L + I_3R + I_3D.
        
        The gauge symmetry is broken by one SU(2)_R doublet, and two 
        SU(2)_L × SU(2)_R bidoublets:
        
            ϕ_R^+ ϕ_R^0∼ (1,1/2,1,2,1),    ϕ_1^0    ϕ_2^+ ϕ_1^-    ϕ_2^0∼ (1,0,2,2,1),    ϕ_3^0    ϕ_4^+ ϕ_3^-    ϕ_4^0∼ (1,0,2,2,1).
        
        Whereas the gauge U(1)_B-L 
        is broken, the global U(1) symmetries of baryon number (B) and lepton 
        number (L) remain. 
        
        What about the extra fermion bidoublets?  The crucial observation is that 
        they have built-in invariant masses because of the allowed terms
        
            ψ^0_1 ψ^0_2 - ψ^-_1 ψ^+_2,    ψ^0_3 ψ^0_4 - ψ^-_3 ψ^+_4.
        
        At the same time, ψ̅_1Lψ_3R and 
        ψ̅_2Lψ_4R acquire mass terms from the ϕ_1,2,3,4 
        vacuum expectation values.  This means that an automatic global U(1)_D 
        symmetry emerges, i.e.
        
            ψ_1L,  ψ_3R∼ -1,    ψ_2L,  ψ_4R∼ 1,
        
        whereas all particles which are singlets under SU(2)_D are trivial under it.  
        It thus serves as a possible dark U(1) symmetry unrelated to B or L.  
        The lighter of the two neutral Dirac fermion eigenstates is then 
        a possible candidate for dark matter.  Since ψ_1,2 
        have SU(2)_L interactions, they may scatter off nuclei with a large 
        elastic cross section and are thus ruled out by direct-search 
        experiments.  It is hence assumed that the dark matter is predominantly 
        ψ^0_3,4.  At this stage, SU(2)_D remains unbroken.  To break it, 
        one SU(2)_D Higgs triplet is added, i.e.
        
            ϕ_D^++ϕ_D^+ ϕ_D^0∼ (1,1,1,1,3).
        
        This choice ensures that there is no coupling between Φ_D and the 
        SM fermions, which would not be the case if it were a doublet.
        
        
         Gauge Bosons : 
        Masses of the gauge bosons come from the vacuum expectation values of the 
        appropriate neutral scalar bosons.  Let
        
            ⟨ϕ^0_R,D,1,2,3,4⟩ = v_R,D,1,2,3,4.
        
        The charged gauge bosons W_D^± have mass g_D^2v_D^2 and 
        does not mix with W^±_L,R, the 2 × 2 mass-squared matrix of 
        which is given by
        
            M^2_W_L - W_R = (1/2) g_L^2 (v_1^2 + v_2^2+v_3^2+v_4^2)     
            -g_L g_R (v_1 v_2+v_3v_4)  -g_L g_R (v_1 v_2 + v_3 v_4)     
            (1/2) g_R^2 (v_R^2 + v_1^2 + v_2^2+v_3^2+v_4^2).
        
        Since W_D^+ takes ψ_1,3 to ψ_2,4, it has charge +2 under 
        U(1)_D to conform with Eq. (10) and ϕ_D^++ has charge +4.  This 
        shows that U(1)_D is not broken by ϕ_D.  Note that the mass 
        degeneracy of ψ^0_3R/ψ^0_4R 
        with ψ^-_3R/ψ^+_4R is broken by a small finite radiative 
        correction <cit.> through the exchange of neutral gauge bosons. 
        Hence ψ^-_3R decays to the invisible ψ^0_3R and a virtual 
        W^-_R which may convert to u̅ d.  Its lifetime is presumably 
        quite long and the outgoing lepton has rather low momentum because of 
        the kinematics.  This kind of signature may be searched for at the 
        Large Hadron Collider (LHC) as already pointed out <cit.>.
        
        
        There are four neutral gauge bosons, i.e. B from U(1)_B-L, W_3L from 
        SU(2)_L, W_3R from SU(2)_R, W_3D from SU(2)_D, with couplings 
        g_B, g_L, g_R, g_D respectively.  Let them be rotated to the following four 
        orthonormal states:
        
            A    =   e  g_B B + e  g_L W_3L + e  g_R W_3R + 
            e  g_D W_3D, 
             
            Z    =   e  g_Y W_3L - e  g_L( g_Y  g_B B + 
            g_Y  g_R W_3R + g_Y  g_D W_3D), 
             
            Z_R    =   g_R √(g_R^2+g_B^2) W_3R - g_B √(g_R^2+g_B^2) 
            B, 
             
            Z_D    =   √(1-g_Y^2  g_D^2) W_3D - g_Y  g_D( 
            g_B √(g_R^2+g_B^2) W_3R + g_R √(g_R^2+g_B^2) B ),
        
        where
        
            1  e^2 = 1  g_L^2 + 1  g_Y^2,  
               1  g_Y^2 = 1  g_D^2 + 1  g_R^2 + 1  g_B^2.
        
        The mass terms are given by
        
            1  2 (g_B B - g_R W_3R)^2 v_R^2 + 2 (g_B B - g_D W_3D)^2 v_D^2 
            
                   +  1  2 (g_L W_3L - g_R W_3R)^2 (v_1^2+ v_2^2+ v_3^2 + v_4^2).
        
        It is easily shown that the photon A is massless and decouples from 
        Z,Z_R,Z_D as it should.  The 3 × 3 mass-squared matrix spanning 
        the latter is given by
        
            M^2_ZZ   =   1  2 (g_L^2+g_Y^2)(v_1^2 + v_2^2 + v_3^2 + v_4^2), 
             M^2_Z_R Z_R   =   1  2 (g_R^2+g_B^2) v_R^2 + 4g_B^4 v_D^2 + 
            g_R^4 (v_1^2+v_2^2 + v_3^2 + v_4^2)  2(g_R^2+g_B^2), 
             M^2_Z_D Z_D   =   g_D^2 g_R^2 g_B^2  2g_Y^2 (g_R^2+g_B^2) (4v_D^2)
             + g_Y^2 g_R^2 g_B^2  2g_D^2 (g_R^2+g_B^2) (v_1^2+v_2^2+v_3^2+v_4^2), 
             M^2_ZZ_R   =    -g_L g_Y g_R^2 2e √(g_R^2+g_B^2)  
            (v_1^2+v_2^2+v_3^2+v_4^2), 
             M^2_ZZ_D   =   e g_Y^2 g_R g_B  2 g_L g_D 
            √(g_R^2+g_B^2) (v_1^2 + v_2^2+v_3^2+v_4^2), 
             M^2_Z_R Z_D   =   g_R g_B  2(g_R^2+g_B^2)[ g_D g_B^2 
             g_Y (4v_D^2) - g_Y g_R^2 g_D (v_1^2+v_2^2+v_3^2+v_4^2) 
            ].
        
        To ensure that SU(2)_L is broken at a scale significantly lower than that 
        of SU(2)_R or SU(2)_D, it is assumed that
        
            v_1^2+v_2^2+v_3^2+v_4^2 << v_R^2,   v_D^2.
        
        Hence Z decouples effectively from Z_R and Z_D, with negligible 
        mixing to the latter.  In the remaining Z_R-Z_D sector, if the 
        v_1^2+v_2^2+v_3^2+v_4^2 terms are neglected, then the 2 × 2 
        mass-squared matrix is of the form
        
            M^2_Z_R-Z_D = A+B    √(BC)√(BC)    C,
        
        where
        
            A = 1  2(g_R^2+g_B^2)v_R^2,    B = g_B^4  2(g_R^2+g_B^2) (4 v_D^2), 
               C = g_R^2 g_D^2  g_Y^2 g_B^2 B.
        
        There are two interesting limits.
        
          * (1) B,C << A, then A and C are eigenvalues with Z_R and Z_D 
        as eigenstates.
          * (2) A << B,C, then B+C and AC/(B+C) are eigenvalues with 
        Z_1 = (g_Yg_B Z_R + g_R g_D Z_D)/√(g_Y^2 g_B^2 + g_R^2 g_D^2) and 
        Z_2 = (g_Rg_D Z_R - g_Y g_B Z_D)/√(g_Y^2 g_B^2 + g_R^2 g_D^2) as 
        eigenstates. Gauge Interactions : 
        The neutral-current gauge interactions are given by
        
            L_NC   =    e A j_em + g_Z Z (j_3L - sin^2 θ_W j_em) + 
            1 √(g_R^2+g_B^2) Z_R (g_R^2 j_3R - g_B^2 j_B) 
               +    g_Y Z_D ( g_D √(g_R^2+g_B^2) g_R g_B j_3D - 
            g_R g_B  g_D √(g_R^2+g_B^2) (j_3R + j_B) ).
        
        In particular Z_2 couples to 
        
            g_R √(g_Y^2 g_B^2 + g_R^2 g_D^2) g_D √(g_R^2+g_B^2) j_3R   
            - g_Y^2 g_D √(g_R^2+g_B^2) g_R √(g_Y^2 g_B^2 + g_R^2 g_D^2) 
            (j_3D+j_B).
        
        
        If v_D^2 << v_R^2, then Z_D is the much lighter mass eigenstate with mass 
        given by Eq. (22).  It couples to quarks and leptons according to Eq. (29) 
        with 
        
            j_3R   =   1  2u̅_R γ u_R - 1  2d̅_R 
            γ d_R + 1  2ν̅_R γν_R - 1  2l̅_R γ l_R, 
            
            j_B   =   1  6 (u̅γ u + d̅γ d) 
            - 1  2 (ν̅γν + l̅γ l), 
             
             j_3D   =    0.
        
        For the dark Dirac fermion ψ_3/ψ_4,
        
            j_3R = -j_3D = 1  2ψ̅_3Rγψ_3R - 1  2ψ̅_4Rγψ_4R,     j_B=0.
        
        At the LHC, Z_D may be observed through its 
        production by u and d quarks, with its subsequent decay to lepton 
        pairs.  The c_u,d coefficients <cit.> used in the data 
        analysis are 
        
            c_u    =    (g^2_uL + g^2_uR)B = g_Y^2 g_R^2 g_B^2  g_D^2 (g_R^2+g_B^2)[ ( 1  6)^2 + ( 2  3)^2 
            ] B, 
             
            c_d    =    (g^2_dL + g^2_dR)B = g_Y^2 g_R^2 g_B^2  g_D^2 (g_R^2+g_B^2)[ ( 1  6)^2 + ( -1  3)^2 ] B,
        
        where B is the Z_D branching fraction to e^-e^+ and μ^-μ^+. 
        To estimate c_u,d, let g_D = g_R = g_L, then
        
            e^2  g_B^2 = 1 - 3 e^2  g_L^2 = 1 - 3(0.23) = 0.31.
        
        Assuming that Z_D decays to 3 copies of the dark fermions of Eq. (6) 
        in addition to all the quarks and leptons, B is estimated to be about 0.07,  
        and c_u =1.8 × 10^-3, c_d = 5.4 × 10^-4.  Based on the 
        13 TeV LHC data from ATLAS <cit.>, this translates to a bound of 
        about 3.5 TeV on the Z_D mass. 
        
        If v_R^2 << v_D^2, then Z_2 is the much lighter mass eigenstate with mass 
        given by
        
            M^2_Z_2 = g_R^2 g_D^2 (g_R^2 + g_B^2)  2(g_Y^2 g_B^2 + g_R^2 g_D^2) 
            v_R^2 = 0.304 v_R^2.
        
        It couples to fermions according to Eq. (30).  The branching fraction B 
        is then about 0.03, and the c_u,d coefficients are 1.6 × 10^-3 
        and 2.8 × 10^-3 respectively.  This translates to a bound 
        of about 3.6 TeV on the Z_2 mass.  Note that this bound depends on 
        c_u more than c_d because the LHC is a proton collider.
        
         Dark Matter Interactions : 
        The particles beyond the conventional left-right model are the SU(2)_D 
        gauge bosons, the ψ fermions and the one Higgs scalar Φ_D triplet.  
        Whereas SU(2)_D is completely broken by Φ_D, a residual global 
        U(1)_D symmetry remains, under which
        
            ψ_1L,ψ_3R∼ -1,    ψ_2L,ψ_4R∼ +1,     
            W_D^±∼± 2,    ϕ_D^±±∼± 4,
        
        and the neutral W_3D and the physical neutral scalar h_D are trivial, 
        which allow them to mix with the other neutral gauge bosons and scalar 
        bosons.  The dark Dirac fermion ψ is assumed to be dominantly 
        composed of ψ_3R and ψ_4R.  To be specific, the outgoing 
        ψ_4R may be redefined as an incoming ψ_3L, in which case 
        the Dirac fermion ψ has a vector coupling to 
        g_R W_3R - g_D W_3D.
        
        The elastic scattering of ψ off nuclei in underground direct-search 
        experiments is possible through Z_D or Z_2.  The spin-independent 
        cross section σ_0 is enhanced by coherence and depends only on 
        their vector couplings to the u and d quarks.  For Z_D which couples to 
        0.547 j_3D - 0.233 (j_3R+j_B),
        
            u_V = -0.0971,     d_V = 0.0194,    ψ_V = -0.390.
        
        For Z_2 which couples to 0.547 j_3R - 0.233 (j_3D+j_B),
        
            u_V = 0.0979,     d_V = -0.1756,    ψ_V = 0.390.
        
        The cross section σ_0 is then given by 
        
            σ_0 = 4 μ^2 π A^2 [Z f_p + (A-Z)f_n]^2,
        
        where μ is the reduced mass of the effective interaction and equal to 
        the nucleon mass for large m_ψ. 
        In the case of Z_D as the mediator,
        
            f_p = ψ_V (2u_V + d_V)  M^2_Z_D = 0.0682  M^2_Z_D,     
            f_n = ψ_V (u_V + 2d_V)  M^2_Z_D = 0.0227  M^2_Z_D.
        
        In the case of Z_2 as the mediator,
        
            f_p = ψ_V (2u_V + d_V)  M^2_Z_2 = 0.0079  M^2_Z_2,     
            f_n = ψ_V (u_V + 2d_V)  M^2_Z_2 = -0.0988  M^2_Z_2.
        
        Assuming m_ψ = 150 GeV for example, σ_0 is bounded by the latest 
        experimental result <cit.> to be below 2 × 10^-46 cm^2. 
        Using Z=54 and A=131 for xenon, this translates to M_Z_D > 7.8 TeV 
        and M_Z_2 > 9.0 TeV, which are 
        stronger than the LHC bounds discussed earlier.
        
        Instead of Z_D or Z_2, if the lightest new neutral gauge boson is 
        Z_3 = (g_B g_D Z_R + g_Y g_R Z_D)/√(g_Y^2 g_R^2 + g_B^2 g_D^2), then 
        it is easily shown from Eq. (29) that it couples to
        
            g_Y^2 g_D √(g_R^2+g_B^2) g_R √(g_Y^2 g_R^2 + g_B^2 g_D^2) 
            (j_3R+j_3D) - g_B √(g_Y^2 g_R^2 + g_B^2 g_D^2) g_D 
            √(g_R^2+g_B^2) j_B.
        
        This means that ψ_V = 0 and there would be no interaction through 
        Z_3 with nuclei and no bound on the mass of Z_3 from direct-search 
        experiments.  In other words, if the lightest new neutral gauge boson 
        has a dominant Z_3 component, its bound may be lowered to a value 
        comparable to that from the LHC.
        
        Consider now the relic abundance of ψ.  Its annihilation cross section 
        through any new neutral gauge boson is much below 1 pb for a gauge-boson 
        mass greater than 3.5 TeV.  Hence a different process is required.  
        Consider then the Yukawa sector.  Note first that there is no scalar 
        singlet, so if the dark 
        fermion ψ is composed of only ψ^0_3R and ψ^0_4R with 
        the invariant mass term ψ^0_3Rψ^0_4R, it has 
        no ψ̅ψ coupling to any scalar.  However, as pointed out 
        already, there are also the allowed 
        ψ̅^0_3R(ϕ̅_1^0 ψ^0_1L + ϕ_1^+ ψ^-_1L) + 
        ψ̅^0_4R(ϕ̅_2^0 ψ^0_2L + ϕ_2^- ψ^+_2L) 
        and 
        ψ̅^0_3R(ϕ̅_3^0 ψ^0_1L + ϕ_3^+ ψ^-_1L) + 
        ψ̅^0_4R(ϕ̅_4^0 ψ^0_2L + ϕ_4^- ψ^+_2L) 
        terms.  Hence ψ annihilation to scalars is possible and it may remain 
        in thermal equilibrium in the early Universe until the temperature drops 
        below m_ψ.
        
        There are several diagrams for 
        ψ annihilation to scalars.  As an estimate, consider Fig. 1 which 
        depicts the process ψψ̅→ϕ^+ ϕ^- through ψ^- 
        exchange.
        
        The cross section × relative velocity is given by
        
            σ v_rel = f^4  16 π( 1 - m_ϕ^2  m_ψ^2)^3/2m_ψ^2  (M^2+m_ψ^2-m_ϕ^2)^2,
        
        where f is the ψ̅^0 ψ^- ϕ^+ coupling and M is 
        the mass of the exchanged ψ^-.  As an example, let m_ψ = 150 GeV, 
        m_ϕ = 100 GeV, and M = 200 GeV, then σ v_rel = 1 pb is 
        obtained for f=0.442.  This shows that the proper relic abundance of 
        dark matter in the Universe is possible within this framework.
        
        
        
         Theoretical Origin of SU(2)_D : 
        As presented, the introduction of SU(2)_D and the new fermions of 
        Eq. (6) seems rather  ad hoc.  However, there is a possible 
        unifying theoretical framework underlying their existence.  Consider 
        the well-known Pati-Salam partial unification symmetry 
        SU(4) × SU(2)_L × SU(2)_R <cit.>, under which quarks and 
        leptons are organized according to
        
            u     u     u    ν d     d     d     l_L ∼ (4,2,1),  
               u     u     u    ν d     d     d     l_R ∼ (4,1,2),
        
        where SU(4) contains SU(3)_C × U(1)_B-L. 
        If this is extended to SU(6) × SU(2)_L × SU(2)_R, the new 
        fermions introduced are naturally included, i.e.
        
            u     u     u    ν   ψ^0_1    ψ_2^+  d     d     d     l    ψ_1^-    ψ_2^0_L ∼ (6,2,1),    u     u     u    ν   ψ_3^0    ψ_4^+  d     d     d     l    ψ_3^-    ψ_4^0_R ∼ (6,1,2).
        
        This points to the possible unity of matter with dark matter, as 
        discussed previously <cit.>.
        
        The only other possible (and very intriguing) SU(6) assignment is
        
            u     u     u    ν    x_1     x_2  d     d     d     l     
            y_1     y_2_L ∼ (6,2,1),    u     u     u    ν    x_3     x_4  d     d     d     l     
            y_3     y_4_R ∼ (6,1,2),
        
        where x_i and y_i have charges 1/2 and -1/2 respectively, and 
        SU(2)_D is unbroken.  This is a realization of an idea proposed many 
        years ago <cit.>, where color SU(3)_q for quarks is matched with 
        a parallel color SU(3)_l for leptons.  Whereas SU(3)_q is unbroken, 
        SU(3)_l is broken to SU(2)_l, thereby confining only two components 
        of the fundamental fermion triplet, leaving the third component free 
        as the observed lepton.  This notion of leptonic color may be 
        unified <cit.> under [SU(3)]^4, with interesting 
        predictions <cit.> for a future e^-e^+ collider.
        
        Since SU(4) is isomorphic to SO(6) and SU(2) × SU(2) is 
        isomorphic to SO(4), it is well-known that 
        SU(4) × SU(2)_L × SU(2)_R may be embedded into SO(10). 
        As for SU(6) × SU(2)_L × SU(2)_R, it is not clear which  
        simple group may be a possible unification symmetry.  It must of 
        course be at least rank 7.
        
         Concluding Remarks : 
        The notion is put forward that dark matter is intimately related to 
        matter and the global U(1) symmetry which allows it to be stable is an 
        automatic consequence of gauge symmetry and particle content in the 
        same way that baryon and lepton numbers are so in the standard model. 
        A specific proposal is the addition of an SU(2)_D gauge symmetry 
        with new fermions which are bidoublets under SU(2)_L,R× SU(2)_D. 
        It is shown that with the complete breaking of the SU(2)_D gauge symmetry 
        by an SU(2)_D × U(1)_B-L scalar triplet, a global U(1)_D symmetry 
        remains for the new particles.  Dark matter thus emerges naturally within 
        this framework.  Its phenomenology is discussed, as well as the intriguing 
        possibility that it may have a theoretical origin in 
        SU(6) × SU(2)_L × SU(2)_R, where SU(6) is a generalization 
        of the well-known Pati-Salam SU(4) which unifies quarks and leptons.
        
         Acknowledgement : 
        This work was supported in part by the U. S. Department of Energy Grant 
        No. DE-SC0008541.
        
        
        unsrt99m15 E. Ma, Phys. Rev. Lett.  115, 011801 (2015).
        hp15 J. Heeck and S. Patra, Phys. Rev. Lett.  115, 121804 
        (2015).
        gh16 C. Garcia-Cely and J. Heeck, JCAP  1603, 021 (2016). 
        ahr17 C. Arbelaez, M. Hirsch, and D. Restrepo, Phys. Rev. 
         D95, 095034 (2017).
        kmppz18 C. Kownacki, E. Ma, N. Pollard, O. Popov, and M. Zakeri, 
        Phys. Lett.  B777, 121 (2018).
        dhqvv18 P. V. Dong, D. T. Huong, F. Queiroz, J. W. F. Valle, and 
        C. A. Vaquera-Araujo, arXiv:1710.06951.
        ps74 J. C. Pati and A. Salam, Phys. Rev.  D10, 275 (1974), 
        Erratum: Phys. Rev.  D11, 703 (1975).
        s95 M. Sher, Phys. Rev.  D52, 3136 (1995).
        ATLAS14 G. Aad  et al. (ATLAS Collaboration), Phys. Rev. 
         D90, 052005 (2014).
        CMS15 S. Khachatryan  et al. (CMS Collaboration), JHEP 
         1504, 025 (2015).
        ATLAS17 M. Aaboud  et al. (ATLAS Collaboration), JHEP 
         1710, 182 (2017).
        x1t17 E. Aprile  et al. (XENON Collaboration), Phys. Rev. 
        Lett.  119, 181301 (2017).
        b12 S. M. Barr, Phys. Rev.  D85, 013001 (2012).
        m13 E. Ma, Phys. Rev.  D88, 117702 (2013).
        fl90 R. Foot and H. Lew, Phys. Rev.  D41, 3502 (1990).
        bmw04 K. S. Babu, E. Ma, and S. Willenbrock, Phys. Rev.  D69, 
        051301(R) (2004).
        kmppz17 C. Kownacki, E. Ma, N. Pollard, O. Popov, and M. Zakeri, 
        Phys. Lett.  B769, 267 (2017).
