Accidental scale-invariant Majorana dark matter in leptoquark-Higgs portals
aa r X i v : . [ h e p - ph ] O c t Accidental scale-invariant Ma jorana dark matter inleptoquark-Higgs portals
Ahmad Mohamadnejad ∗ Young Researchers and Elite Club, Islamshahr Branch, Islamic Azad University,Islamshahr 3314767653, IranOctober 15, 2019
Abstract
We study a classically accidental scale-invariant extension of the Standard Model (SM)containing three additional fields, a vector leptoquark ( V µ ), a real scalar ( φ ), and a neutralMajorana fermion ( χ ) as a dark matter (DM) candidate. The scalar φ (scalon) and Majoranafermion χ are both singlets under the SM gauge group, while V µ has ( , , 2/3) quantumnumbers under the SU (3) c × SU (2) L × U (1) Y . The Majorana DM couples to the SM sectorvia both Higgs and leptoquark portals. We perform a scan over the independent parametersto determine the viable parameter space consistent with the Planck data for DM relic den-sity, and with the PandaX-II and LUX direct detection limits for the spin-independent (SI)and spin-dependent (SD) DM-nucleon cross section. The model generally evades indirectdetection constraints while being consistent with collider data. Cosmological observations implies that DM is the majority of matter in the Universe. It is notmade of SM particles and understanding its nature is one of the most important issues at thefrontier of particle physics [1].On the other hand, SM is expected to be valid up to energies of the order of the Planck scalewhere vacuum stability problem arises. One solution to this problem is the supersymmetricextensions of SM where the Higgs mass is radiatively stable down to the scale of supersym-metry breaking. However, the results from LHC have been negative for supersymmetry sofar. Another solution is scale-invariant extensions of SM with no dimensionful parameter [2]. Inscale-invariant extensions of SM, all physical masses arise via Coleman-Weinberg mechanism [3].This mechanism works only if extra bosonic degrees of freedom are added to SM with sizablecouplings. Scale-invariant extensions of SM are also a generic feature of many DM models withbosonic [4–19] and fermionic [20–26] DM candidates. ∗ [email protected]
1n this paper, we study an accidental scale-invariant extension of SM with vector leptoquarksas extra bosonic degrees of freedom mediating lepton-quark interactions. Leptoquarks are anatural result of unification of quarks and leptons [27] initially proposed in the Pati-Salam model[28]. Leptoquarks would turn leptons into quarks generating new physical effects. Leptoquarksalso appear in SUSY models with R-parity violation [29–31], and in composite models of leptonsand quarks [32]. Besides vector leptoquarks, we also introduce a Majorana DM candidate anda real scalar field which is needed in order to get mass term for Majorana DM after symmetrybreaking. In this model, DM mediates with SM via Higgs and leptoquark portals.Leptoquarks can explain some deviations from the SM such as anomalous B decays observedin BaBar [33, 34], Belle [35] and LHCb [36–38], a violation in lepton universality [39] and adeviation from the SM prediction of ( g − µ [40, 41]. It is also shown that all three anomaliescould be interpreted via the addition of a single scalar leptoquark [42]. DM models with scalarleptoquark portal can be found in [43–47]. Vector leptoquark portal is also studied in [48].In our scenario, vector leptoquark is not a gauge field, however, it gets mass via its couplingto scalar fields. Particularly, we study the case in which vector leptoquark couples to scalonand the spontaneous symmetry breaking makes it massive. Some attempts to write a modelwith gauge leptoquarks can be found in [49–51]. Lately, the vector leptoquark has also beenconsidered as a possible explanation of the anomalies observed in charged-current and neutralcurrent transitions of B mesons [52–56].Majorana DM can leave detectable signals at direct detection experiments. Both spin-independent (SI) and spin-dependent (SD) DM-nucleon scattering occur in our model. It isbecause Majorana DM interacts with SM via Higgs portal (with SI DM-nucleon scattering) aswell as vector leptoquark portal (with SD DM-nucleon scattering). Hence, our model providemore opportunity to be probed compared to a Majorana fermion DM with one portal eitherHiggs or vector leptoquark. The PICO [57] and LUX [58] data for DM-nucleon SD cross sectionallows the region compatible with relic density and does not constrain the model. However, thedirect detection experiments such as XENON1T [59, 60], LUX [61], and PandaX-II [62] imposebounds on the SI DM-nuclei cross section. We also show that indirect detection experiments suchas Fermi Large Area Telescope (Fermi-LAT) [63] and Alpha Magnetic Spectrometer (AMS) [64]do not constrain our model. Finally, our model is compatible with collider physics.The paper is organized as follows. In section 2, we introduce the model. Section 3 containsthe calculation of DM relic density. In section 4, the SI and SD DM-nucleon cross section fordirect detection experiments as well as DM indirect detection are studied. Finally, our conclusionincluding a discussion on recent collider bounds comes in section 5.2 The model
We begin with constructing a model in which all couplings are diemnsionless. The fields gainmass via radiative Coleman-Weinberg symmetry breaking at one-loop level [3]. Therefore, themodel is a scale-invariant extension of SM without Higgs mass term.Apart from SM fields, the model contains three new fields which two of them are singletsunder SM gauge transformation. These two fields are the real scalar φ and the Majorana spinor χ . The other field, V µ , is a vector leptoquark which has ( , , 2/3) quantum numbers under the SU (3) c × SU (2) L × U (1) Y gauge group. This vector leptoquark does not lead to proton decayand it is also a part of the gauge sector of the Pati-Salam model [28].Putting together these fields, and regarding scale invariance, gauge invariance, and renor-malization conditions, we get L ⊃ ∂ µ φ ∂ µ φ + 12 iχγ µ ∂ µ χ − V † µν V µν − λ H ( H † H ) − λ φ φ − λ φH φ H † H − λ HV H † HV † µ V µ − λ φV φ V † µ V µ − g φ φχχ − X generations ( g L q L γ µ V µ l L + g R d R γ µ V µ l R + g χ u R γ µ V µ χ + h.c.) , (2.1)where V µν = D µ V ν − D ν V µ ,D µ = ∂ µ − ig s λ a G aµ − ig Y Y B µ . (2.2)In DM models with leptoquark portal, we do not necessarily need the real scalar field, howeverin our scenario, because of scale invariance condition, we need this field in order to get massterm for Majorana spinor after symmetry breaking. Therefore, DM interacts with SM particlesvia both Higgs and vector leptoquark portals.Note that the model can not be fundamental and might be regarded as an effective theory. Aswe mentioned in Introduction, V µ is not a gauge field, i.e., Lagrangian (2.1) is not invariant undergauge transformation V µ → U V µ U † − ig ( ∂ µ U ) U † where U presents some gauge group. Instead itis invariant under V µ → U V µ U † where for U being SM gauge group, symmetry properties of V µ ismentioned in table 1. Nonetheless, there will be an accidental gauge symmetry when neglectingall the interactions. In this case, the internal degrees of freedom (dof) of V µ is 2 × × × × SU (3) c , SU (2) L , U (1) Y )Scalon φ ( , , H ( , , )Left-handed leptons l L ( , , − )Right-handed leptons l R ( , , − q L ( , , )Right-handed quarks (up-type) u R ( , , )Right-handed quarks (down-type) d R ( , , − )Majorana DM χ ( , , V µ ( , , ) U (1) Y electroweak boson field B µ ( , , G µ ( , , g L , g R , and g χ is independent of generations.In our model Majorana spinor can be a DM candidate if M χ < M V , otherwise the two-bodydecay of χ to vector leptoquark V and up-type anti-quarks occurs at tree level and Majoranaparticle will be unstable. Even if M χ < M V , in the case of non-zero couplings g L and g R ,still tree level three-body decay and one-loop induced decay of χ can occur (see figure 1). Toevade such decays, we impose a discrete Z symmetry under which only vector leptoquark andMajorana spinor are odd [48]. In this case, g L and g R are zero and Majorana particle can serveas a cosmological stable DM candidate. Moreover, relaxing Z symmetry, the constraint on DMlifetime leads to highly suppressed g L and g R for O (1) g χ values [65–67]. For the rest of thepaper, we assume g L and g R are zero. Therefore, in the last line of Lagrangian (2.1), the termwith g χ coupling plays the important role in linking the visible and dark sector to each other.In unitary gauge, we have H = √ (cid:18) h (cid:19) , and the potential terms (line 2 in (2.1)) become: − V ( h, φ ) = − λ H h − λ φ φ − λ φH h φ . (2.3)Vacuum expectation values, h h i = ν h and h φ i = ν φ , correspond to local minimum of V ( h, φ ).4 Vχ u, c, t ¯ u, ¯ c, ¯ tν e , ν µ , ν τ ✁ Vχ u, c, t ¯ d, ¯ s, ¯ be, µ, τ ✁ χ u, c, t u, c, t ¯ V Zν e , ν µ , ν τ ✁ χ u, c, t d, s, b ¯ V W + e, µ, τ Figure 1: Some decay modes of Majorana particle.The potential V ( h, φ ) has local minimum if ∂V ( h, φ ) ∂h (cid:12)(cid:12)(cid:12)(cid:12) ν h ,ν φ = ∂V ( h, φ ) ∂φ (cid:12)(cid:12)(cid:12)(cid:12) ν h ,ν φ = 0 , (2.4) ∂ V ( h, φ ) ∂h (cid:12)(cid:12)(cid:12)(cid:12) ν h ,ν φ > , (2.5) ∂ V ( h, φ ) ∂h (cid:12)(cid:12)(cid:12)(cid:12) ν h ,ν φ ! ∂ V ( h, φ ) ∂φ (cid:12)(cid:12)(cid:12)(cid:12) ν h ,ν φ ! − ∂ V ( h, φ ) ∂h ∂φ (cid:12)(cid:12)(cid:12)(cid:12) ν h ,ν φ ! > . (2.6)Eq. (2.4) leads to λ H λ φ = (3! λ φH ) and the following constraint ν h ν φ = s − λ φH λ H . (2.7)Vacuum stability, constraints (2.4) and (2.5), implies that λ H > λ φ >
0, and λ φH < V ( ν h , ν φ ) = 0. Therefore, the one-loop effective potential dominates along the flatdirection. In this direction, due to one-loop corrections, a small curvature appears with aminimum as the vacuum expectation value ν = ν h + ν φ characterized by a RG scale Λ. Therefore,we substitute h → ν h + h and φ → ν φ + φ as a result of spontaneous symmetry breaking where ν h = 246 GeV.We define the mass eigenstates H and H as (cid:18) H H (cid:19) = (cid:18) cosα − sinαsinα cosα (cid:19) (cid:18) hφ (cid:19) . (2.8)The scalar field H is along the flat direction, thus M H = 0 at the tree level, while H isperpendicular to the flat direction and we consider it as the SM-like Higgs observed at the LHC5ith M H = 125 GeV. We have these constraints following the symmetry breaking: ν φ = M χ g φ , tanα = ν h ν φ , λ H = 3 M H ν h cos α, λ φ = 3 M H ν φ sin α,λ φH = − M H ν h ν φ sin α, λ φV = − λ HV ν h ν φ − M V ν φ , (2.9)where M χ and M V are the mass of Majorana DM and vector leptoquark after symmetry break-ing. In the next sections, to get a more minimal model, we put λ HV = 0. Note that non-zero λ HV does not add any new vertex to our model. Therefore, according to constraints (2.9), weconsider four free parameters in our model: M χ , M V , g χ , g φ . As we mentioned before, the scalon field H is massless in tree level. However, using Gildener-Weinberg mechanism [68], the radiative corrections give a mass to H .The one-loop effective potential, Along the flat direction, takes the form V − loopeff = aH + bH ln H Λ , (2.10)where a and b are dimensionless constants given by a = 164 π ( ν h + ν φ ) n X k =1 g k M k ln M k ν ,b = 164 π ( ν h + ν φ ) n X k =1 g k M k . (2.11)and M k ( g k ) is the tree-level mass (the internal degrees of freedom) of the particle k . Note that g k is positive (negative) for bosons (fermions).In terms of the one-loop VEV ν , effective potential along the flat direction is given by V − loopeff = bH (cid:18) ln H ν − (cid:19) , (2.12)and the scalon mass will be M H = d V − loopeff dH (cid:12)(cid:12)(cid:12)(cid:12) ν = 8 bν . (2.13)According to (2.11) and (2.13), the mass of scalon can be expressed as M H = 18 π ( ν h + ν φ ) (cid:0) M H + 6 M W + 3 M Z + 18 M V − M t − M χ (cid:1) , (2.14)where M W , M Z are the masses of W and Z gauge bosons, respectively, and M t is the mass oftop quark.To get spontaneous symmetry breaking, the minimum of the one-loop potential V − loopeff should be negative, thus, b should be positive. Note that the presence of vector leptoquark is6 M V [ G e V ] M χ [GeV] b > M V = M χ Figure 2: Symmetry breaking ocuures if b > b . Indeed, to get a positive b we should have M V &
150 GeV (see figure2). According to figure 2, for M χ .
150 GeV, the constraint M χ < M V , which avoids DM decay,is automatically satisfied duo to symmetry breaking condition, i.e., b >
0. For M χ &
150 GeVwe put aside a part of mass parameter space by hand in order to avoid DM decay.
If DM does not interact sufficiently in the early Universe, it will fall out of local thermodynamicequilibrium and it is said to be decoupled. This happens when DM interaction rate drops belowthe expansion rate of the Universe. To calculate DM relic density one should use Boltzmannequation in which DM annihilation cross sections is needed. Feynman diagrams for all possibleDM annihilation channels is depicted in figure 3 (a). DM annihilates through s-channel in Higgsportal and t-channel in leptoquark portal. Since Majorana particle is its own antiparticle, forevery t-channel annihilation there is also a u-channel diagram. In our model, coannihilationchannels also exist (see figure 3 (b)).Coannihilation channels are relevant if there is some other particle nearly degenerate in masswith the DM such that it annihilates with DM more efficiently than DM with itself. In thiscase, coannihilation channels primarily determines DM relic density. To quantitatively accountDM relic density, one should solve Boltzmann equation [69] dn χ dt + 3 Hn χ = −h σ eff | v rel |i ( n χ − n χ,eq ) , (3.1)where n χ is the number density of Majorana DM, H is the Hubble parameter, and h σ eff | v rel |i is the thermally averaged of effective annihilation cross section (multiplied by relative velocity).7 u,c,tχχ V ¯ V ✁ Vχχ u,c,t ¯ u, ¯ c, ¯ t ✁ χχχ H ,H H ,H ✁ H ,H χχ MPMP ✁ V,χ,u,c,tχin out out ✁ V,χ,u,c,tχin out out (a) (b)Figure 3: (a) DM annihilation and (b) coannihilation channels. In this figure MP stands formassive particle.Effective annihilation σ eff is given by [69] σ eff = X i,j σ i,j g i g j g eff (1 + ∆ i ) / (1 + ∆ j ) / e − MχT (∆ i +∆ j ) , (3.2)where the double sum is over all particle species with σ , being Majorana DM annihilationcross section and σ i,j is the cross section for the coannihilation of species i and j (or self-annihilation in the case of i = j ) into Standard Model particles. The quantities ∆ i = ( M i − M χ ) /M χ is the fractional mass splittings between the species i and the Majorana DM and g eff = P i g i (1 + ∆ i ) / e − MχT ∆ i . In order to calculate Majorana DM relic density includingcoannihilations channels, we use the public numerical code micrOMEGAs [70]. The Lagrangian(2.1) has been implemented through LanHEP [71] package. We use DM relic density (Ω DM h =0 . ± . M V [ G e V ] M χ [GeV] M χ = M V . . . g χ M V [ G e V ] M χ [GeV] M χ = M V . . . g φ Figure 4: The parameter space of the model constrained by DM relic density as reported byPlanck collaboration [72]. 8
Direct and indirect detection
Majorana DM can elastically scatter off the nucleus. The momentum transfer gives rise to anuclear recoil which might produce a signal in direct detection experiments. In our model, thissignal can arise from the Feynman diagrams shown in figure 5. ✁ Vqχ qχ ✁ H , H qχ qχ Figure 5: Feynman diagrams responsible for DM-nucleon scattering.Both spin-dependent (SD) and spin-independent (SI) DM-nucleon scattering exist in ourmodel. The left (right) diagram of figure 5 leads to SD (SI) scattering which can be describedby effective axial-vector (scalar) Lagrangian L A = c A χγ µ γ χuγ µ γ u ( L S = c S,q χχqq ). To obtain c A and c S,q we should integrate out the intermediate particles shown in Feynman diagrams 5.The result is c A = − g χ M V − M χ ) , c S,q = − m q ν h g φ sin 2 α M H − M H ! , (4.1)Having coefficients c A and c S,q , SD and SI DM-nucleon cross sections become [73] σ SD = 16 µ Nχ π c A (∆ Nµ ) J N ( J N + 1) , (4.2) σ SI = 4 µ Nχ M N π c S,q m q f N , (4.3)where µ Nχ = M χ M N / ( M χ + M N ) is the reduced mass of DM and nucleon, J N = is the angularmomentum of the nucleon, ∆ Nµ = 0 . ± .
02 (∆ Nµ = − . ± .
02) is the u-quark spin fractionin the proton (neutron) [74, 75], and f N ≃ . micrOMEGAs package. In order to constrain the model, we usePandaX-II [62] and LUX [58] experiments for SI and SD DM-nucleon scattering, respectively.In figure 6 we have depicted SD DM-nucleon cross section for the parameter space compatiblewith DM relic density. As it is seen in this figure, the model can evade the upper limit ofSD DM-nucleon cross section. However, a small part of the parameter space can be probedby future LZ experiment [76]. Unlike the upper limit of SD DM-nucleon cross section, as it isdepicted in 7 (a), PandaX-II upper limit of SI DM-nucleon scattering excludes some part of the9 − −
500 1000 1500 2000 2500 3000 σ S D N [ z b ] M χ [GeV] LUX 2016LZ Projected 00 . . . g χ −
500 1000 1500 2000 2500 3000 σ S D P [ z b ] M χ [GeV] LUX 2016LZ Projected 00 . . . g χ (a) (b)Figure 6: The direct detection SD cross section vs DM mass for (a) DM-neutron and (b) DM-proton scattering. − − −
500 1000 1500 2000 2500 3000 σ S I [ z b ] M χ [GeV]PandaX-II ν -floor 00 . . . g φ (a) − − − − − − − −
500 1000 1500 2000 2500 3000 < σ v > t o t [ c m / s ] M χ [GeV] 00 . . . g χ − − − − − − − −
500 1000 1500 2000 2500 3000 < σ v > t o t [ c m / s ] M χ [GeV] 00 . . . g φ (b) (c)Figure 7: (a) SI DM-nucleon cross section. (b) and (c) DM total velocity-averaged annihilationcross section vs DM mass.parameter space already constrained by DM relic density. In this figure, we have also shown theneutrino floor [77] which limits the parameter space from below from the irreducible backgroundof coherent neutrino-nucleus scattering. 10 sin α g φ g χ M χ (GeV) M V (GeV) M H (GeV) h σv rel i ( cm /s )1 0.001 0.001 3.000 240.9 4835 46.36 1 . × − . × − . × − Table 2: Total DM annihilation cross section in galactic halos for three benchmark pointsconstrained by the measured cosmological DM relic density, i.e., Ω h ≃ . χ, χ → t, tχ, χ → c, tχ, χ → t, cχ, χ → u, tχ, χ → t, u χ, χ → t, tχ, χ → c, tχ, χ → t, cχ, χ → u, tχ, χ → t, u χ, χ → H , H χ, χ → t, tχ, χ → c, tχ, χ → t, cχ, χ → u, tχ, χ → t, u χ, χ → W + , W − χ, χ → Z, Z χ, χ → W + , W − χ, χ → Z, Z g χ where DM mostlyannihilates via leptoquark portal.Using micrOMEGAs , we have also calculated DM total annihilation cross section in the modernUniverse for the parameters which are already constrained by DM relic density. The result isdepicted in figure 7 (b) and (c). Furthermore, for the benchmark points shown in table 2,different DM annihilation channels, both at freez-out and today temperatures, are reported intable 3. Today DM annihilation cross section is relevant in indirect searches for DM which includeattempts to detect the gamma rays, positrons, antiprotons, neutrinos, and other particles thatare produced in DM annihilations or decays. Stable DM particles with a thermally averagedannihilation cross section of h σv rel i ∼ O (10 − ) cm /s is predicted to freeze out of thermalequilibrium with an abundance equal to the measured cosmological density of DM. Indirectsearches for DM, especially gamma ray and cosmic ray searches for DM annihilation products,have recently become sensitive to this benchmark cross section for masses up to around the weakscale, i.e., O (10 ) GeV. However, our model generally evades indirect detection constraints. As11gure 7 (b) and (c) shows, Majorana DM mostly annihilates with a smaller cross section than h σv rel i ∼ O (10 − ) cm /s in the universe today. This is because of some velocity dependentDM annihilation cross sections. Velocity dependence of h σv rel i can have significant effects onthe late-time DM annihilation while leaving freeze-out largely unaffected. In appendix A someDM annihilation cross sections are written as a Taylor series expansion in powers of v rel . Inthis expansion, p-wave amplitudes only contribute to the v rel and higher order terms, whiles-wave annihilation amplitudes contribute to all orders. In our model, the s-wave annihilation ofMajorana DM to SM products via Higgs portal is absent. Therefore, these annihilation channelsare p-wave suppressed and DM annihilation cross section is only large in vector leptoquark portalwith large g χ . The velocities of DM particles today are around v rel ∼ − c, while it is v rel ∼ . (cid:16) − . (cid:17) ∼ − . In this paper we discussed a classically accidental scale-invariant extension of SM containinga Majorana DM candidate. DM interacts with SM via two portals, namely, Higgs and vectorleptoquark portals. In the leptoquark sector, we have assumed the DM couples to all generationsof up-type quarks with equal coupling. We have also avoid interactions which lead to DM decayby constraining M χ < M V and g L = g R = 0. To get a minimal theory, we further assume λ HV = 0. Therefore, we left with four independent parameters which we choose M χ , M V , g χ ,and g φ . To put a constraint on these parameters we used Planck data for DM relic density, andLUX and PandaX-II upper bounds for SD and SI DM-nucleon cross sections, respectively. Theparameter space constrained by DM relic density evade SD DM-nucleon cross section upper limit,however SI DM-nucleon cross section excludes some part of the parameter space. We have alsoshown that our model generally can evade indirect detection constraints because of dominationof p-wave DM annihilation cross section in Higgs portal. Therefore, for the parameter spacewhich is already constrained by DM relic density, the annihilation cross section is not largeenough to give a signal in indirect detection experiments.Finally, collider searches for vector leptoquarks via pair and/or single production also imposebound on the leptoquark mass. CMS 13 TeV data [78] excludes M V . α . .
44 [79,80]. This boundis compatible with parameter space satisfying DM relic density.12
DM annihilation cross sections
The leading order s-wave annihilation cross sections of Majorana DM through leptoquark portal(see figure 3 (a) top right) expanded in powers of v rel are given by σv rel ( χ, χ → q u , q u ) ≈ g χ M q u q M χ − M q u (cid:0) − M q u + 2 M V + M χ (cid:1) πM V M χ (cid:0) − M q u + M V + M χ (cid:1) , (A.1)where q u represents up-type quarks { u, c, t } .The leading order suppressed p-wave annihilation cross sections of Majorana DM throughs-channel Higgs portal (see figure 3 (a) bottom right) expanded in powers of v rel are given by σv rel ( χ, χ → ℓ, ℓ ) ≈ n c e g φ sin (2 α ) M ℓ (cid:0) M H − M H (cid:1) (cid:0) M χ − M ℓ (cid:1) / π sin ( θ W ) M W M χ (cid:16) M H − M χ (cid:17) (cid:16) M H − M χ (cid:17) v rel , (A.2) σv rel ( χ, χ → V , V ) ≈ n e e g φ sin (2 α ) (cid:0) M H − M H (cid:1) q M χ − M V π sin ( θ W ) M W M χ (cid:16) M H − M χ (cid:17) (cid:16) M H − M χ (cid:17) × (cid:0) M χ − M χ M V + 3 M V (cid:1) v rel , (A.3) σv rel ( χ, χ → H , H ) ≈ g φ q M χ − M H πe M χ (cid:16) M H − M χ (cid:17) (cid:16) M H − M χ (cid:17) × (cid:0) M χ ( A H sin α − B H cos α ) − A H sin αM H + B H cos αM H (cid:1) v rel , (A.4)where ℓ , V , and H represent massive fermions, gauge bosons, and scalars, respectively. In theseformulas θ W is the Weinberg angle, n c = 3 ( n c = 1) for quarks (leptons), n e = 2 ( n e = 1) forcharged (neutral) gauge bosons, and A H =2 cos ( α ) sin( θ W ) λ H M W − ( α ) e ν sin( α ) λ φH + 12 cos( α ) sin ( α ) sin( θ W ) λ φH M W − e ν sin ( α ) λ φ , (A.5) B H =2 cos ( α ) sin( α ) sin( θ W ) λ H M W + 2 cos( α ) e ν λ φH + cos( α ) e ν sin ( α ) λ φ − α ) e ν sin ( α ) λ φH + 12 sin ( α ) sin( θ W ) λ φH M W − α ) sin( θ W ) λ φH M W , (A.6) A H = − cos ( α ) e ν sin( α ) λ φ + 2 cos( α ) sin ( α ) sin( θ W ) λ H M W −
12 cos( α ) sin ( α ) sin( θ W ) λ φH M W + 4 cos( α ) sin( θ W ) λ φH M W − e ν sin ( α ) λ φH + 4 e ν sin( α ) λ φH , (A.7) B H = cos ( α ) e ν λ φ + 12 cos ( α ) sin( α ) sin( θ W ) λ φH M W + 6 cos( α ) e ν sin ( α ) λ φH + 2 sin ( α ) sin( θ W ) λ H M W . (A.8)Finally, the leading order suppressed p-wave annihilation cross sections of Majorana DMthrough t-channel Higgs portal (see figure 3 (a) bottom left) expanded in powers of v rel is given13y σv rel ( χ, χ → H , H ) ≈ n H g φ M χ q M χ − M H (cid:0) − M H M χ + 2 M H + 9 M χ (cid:1) π (cid:0) M H − M χ (cid:1) v rel (A.9)where n H = sin α and n H = cos α . Acknowledgment
This work is supported financially by the Young Researchers and Elite Club of IslamshahrBranch of Islamic Azad University.
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