Lepto-quark portal dark matter

We consider the extension of the Standard Model with scalar leptoquarks as a portal to dark matter (DM), motivated by the recent anomalies in semi-leptonic B-meson decays. Taking singlet and triplet scalar leptoquarks as the best scenarios for explaining B-meson anomalies, we discuss the phenomenological constraints from rare meson decays, muon (g − 2)μ, and leptoquark searches at the Large Hadron Collider (LHC). Introducing leptoquark couplings to scalar dark matter, we find that the DM annihilations into a pair of leptoquarks open a wide parameter space, being compatible with XENON1T bound, and show that there is an interesting interplay between LHC leptoquark searches and distinct signatures from cascade annihilations of dark matter.


JHEP10(2018)104
Dark matter (DM) is known to occupy about 85% of the total matter density in the Universe, and there are a variety of evidences for the existence of dark matter such as galaxy rotation curves, gravitational lensing, large scale structure, etc. The Weakly Interacting Massive Particles (WIMPs) paradigm has driven forces for searching particle dark matter with non-gravitational interactions beyond the Standard Model (SM) for more than three decades. Various direct detection experiments [17][18][19][20][21] have put stringent bounds on the cross section of DM-nucleon elastic scattering, and forthcoming XENON-nT and largescale experiments such as DARWIN [22] and LZ [23] will push the limits further to the neutrino floor where there are irreducible backgrounds due to neutrino coherent scattering. In particular, Higgs-portal type models for dark matter have been strongly constrained, apart from the resonance region or the heavy DM masses.
In this article, we consider a leptoquark-portal model for dark matter where scalar dark matter communicates with the SM through the quartic couplings of scalar leptoquarks, S 1 and S 3 . We show that sizable leptoquark couplings to dark matter lead to new annihilation channels of dark matter into a pair of leptoquarks, opening a wide parameter space where the correct relic density can be explained, being compatible with the direct detection bounds from XENON1T. Moreover, we also discuss that the cascade annihilations of dark matter can lead to distinct signatures for cosmic ray observation, in correlation to leptoquark searches at the LHC. We argue that our models with scalar leptoquarks are consistent with the current bounds from rare meson decays, mixings and lepton flavor violation, whereas the loop corrections of leptoquarks to DM-nucleon couplings and Higgs couplings can be negligible in most of the parameter space of our interest.
The paper is organized as follows. We first give a brief overview on the R K ( * ) and R D ( * ) anomalies and the necessary corrections to the effective Hamiltonians. Then, in models with scalar leptoquarks, we derive the effective interactions for the semi-leptonic B-meson decays and discuss the conditions for B-meson anomalies and various constraints from rare meson decays, mixings, muon (g − 2) µ and leptoquark searches at the LHC. Next we describe leptoquark-portal models for dark matter and consider various constraints on the models, coming from the relic density, direct and indirect detection of dark matter and Higgs data. There are two appendices dealing with the details on effective Hamiltonians for B-meson decays and effective interactions for dark matter and Higgs due to leptoquarks, respectively. Finally, conclusions are drawn. which again differs from the SM prediction by 2.1-2.3σ and 2.4-2.5σ, depending on the energy bins. The deviation in R K * is supported by the reduction in the angular distribution of B → K * µ + µ − , the so called P 5 variable [5,6]. The effective Hamiltonian for b → sµ + µ − is given by , and α em is the electromagnetic coupling. In the SM, the Wilson coefficients are given by C µ,SM Taking the results of BaBar [7,8], Belle [9,10] and LHCb [11] for R D = B(B → Dτ ν)/B(B → Dlν) and R D * = B(B → D * τ ν)/B(B → D * lν) with l = e, µ for BaBar and Belle and l = µ for LHCb, the Heavy Flavor Averaging Group [40] reported the experimental world averages as follows, On the other hand, taking into account the lattice calculation of R D , which is R D = 0.299 ± 0.011 [41], and the uncertainties in R D * in various groups [42][43][44][45][46][47], we take the SM predictions for these ratios as follows,

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Then, the combined derivation between the measurements and the SM predictions for R D and R D * is about 4.1σ. We quote the best fit values for R D and R D * including the new physics contributions [48], The effective Hamiltonian for b → cτ ν in the SM is given by where C cb = 1 in the SM with V cb ≈ 0.04. The new physics contribution may contain the dimension-6 four-fermion vector operators, O V R,L = (cγ µ P R,L b)(τ γ µ P L ν τ ) and/or scalar

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Then, after integrating out the leptoquark scalars, we obtain the effective Lagrangian for the SM fermions in the following, where σ I (I = 1, 2, 3) are the Pauli matrices. There, we find that there are both SU(2) L singlet and triplet V −A operators. As compared to the case with U(2) flavor symmetry [28], the effective interactions for either singlet or triplet leptoquark can be written as (3.6) So, we obtain C S = −C T for the singlet leptoquark and C S = 3C T for the triplet leptoquark. A fit to low-energy data including the R K ( * ) and R D ( * ) anomalies has been done with four free parameters, C T , C S , λ q sb and λ l µµ , under the assumption that the CKM matrix stems solely from the mixing between up-type quarks [28]. As a result, the best-fit values are given by C S ≈ C T ≈ 0.02 for |λ q sb | < 5|V cb | [28].

Singlet scalar leptoquark
After integrating out the leptoquark S 1 , from the results in eq. (A.2), we obtain the effective Hamiltonian relevant for b → cτν τ as As a consequence, the singlet leptoquark gives rise to the effective operator for explaining the R D ( * ) anomalies and the effective cutoff scale is to be Λ D ∼ 3.5 TeV [49]. Thus, for m S 1 1 TeV, we need λ * 33 λ 23 0.4. In the left plot of figure 1, we depict the parameter space for m S 1 and the effective leptoquark coupling, λ eff = |λ * 33 λ 23 |, in which the R D ( * ) anomalies can be explained within 2σ(1σ) errors in green(yellow) region from the conditions below eq. (2.9).
From the couplings of the singlet scalar leptoquark necessary for R D ( * ) anomalies, (3.8) the decay modes of the singlet scalar leptoquark are given by S 1 →tτ ,cτ and S 1 →bν τ ,sν τ , which are summarized together with the corresponding LHC bounds on leptoquark masses in table 1.

LQs
BRs m LQ,min BRs m LQ,min

Triplet scalar leptoquark
After integrating out the leptoquark φ 1 with Q = + 4 3 , from the results in eq. (A.8), we also obtain the effective Hamiltonian relevant for b → sµ + µ − as As a consequence, the triplet leptoquark gives rise to the effective operator of the (V − A) form for the quark current, that is, C µ,NP 9 = −C µ,NP 10 = 0, as favored by the R K ( * ) anomalies, and the effective cutoff scale is to be Λ K ∼ 30 TeV [49]. The result is in contrast to the case for Z models with family-dependent charges such as Q = x(B 3 − L 3 ) + y(L µ − L τ ) with x, y being arbitrary parameters where C µ,NP 9 = 0 and C µ,NP 10 = 0 [60,61]. Then, for m φ 1 1 TeV, we need κ * 32 κ 22 0.03. Therefore, we can combine scalar leptoquarks, S 1 and S 3 , to explain R D ( * ) and R K ( * ) anomalies, respectively.
In the right plot of figure 1, we depict the parameter space for m S 3 and the effective leptoquark coupling, λ eff = |κ * 32 κ 22 |, in which the R K ( * ) anomalies can be explained within 2σ(1σ) errors in green(yellow) region from the conditions below eq. (2.3). Likewise as for the singlet scalar leptoquark, from the triplet leptoquark couplings necessary for R K ( * ) anomalies, the decay modes of the singlet scalar leptoquark are given by φ 1 →bμ,sμ, φ 2 → tμ,cμ,bν µ ,sν µ , and φ 3 →tν µ ,cν µ . As will be discussed in the next section, the bounds from B → Kνν could require κ 33 and κ 23 to be sizable. In this case, the decay modes containingτ orν τ are relevant too. The decay branching ratios of the triplet leptoquark and the corresponding LHC bounds on the mass of triplet scalar leptoquark are also summarized in table 1.

Constraints on leptoquarks
We discuss the constraints on scalar leptoquark models, due to other rare meson decays, muon (g−2) µ , lepton flavor violation as well as the LHC searches. The constraints discussed in this section can give rise to important implications for the indirect signatures of DM annihilation into a leptoquark pair in the later discussion.

Rare meson decays and mixing
In leptoquark models explaining the B-meson anomalies, there is no B −B mixing at tree level, but instead it appears at one-loop level. Therefore, the resulting new contribution to the B s −B s mixing is about 1% level [26], which can be ignored. Both singlet and triplet leptoquarks contribute to B → K ( * ) νν at tree level, so their couplings are severely constrained in this case [26,28]. The effective Hamiltonian relevant forb →sνν [62] is Here, the SM contribution C SM L is given by C SM where s W ≡ sin θ W and X t = 1.469 ± 0.017. From the result in eq. (A.9), the scalar leptoquarks leads to additional contributions to the effective Hamiltonian for B → Kνν as Therefore, the ratio of the branching ratios are given by

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Comparing the experimental bounds on B(B → K ( * ) νν) [63] given by to the SM values [64] given by and ignoring the imaginary part of C l,NP ν , we get the R K * ν bound as Taking into account κ 32 and κ 22 , which are necessary for B → K ( * ) µ + µ − , the triplet scalar leptoquark contributes only to B → K ( * ) ν µνµ . In this case, as the triplet leptoquark contribution to C µ,NP ν is about the same as C µ,NP 9 = −0.61, it satisfies the R K * ν bound on its own easily.
In summary, taking account of bounds from B → K ( * ) νν, the necessary flavor structure for leptoquark couplings is given by the following, If the extra couplings for B → K ( * ) νν are sizable, namely, λ 32 ∼ λ 23 , λ 33 for the singlet leptoquark, and κ 23 , κ 33 κ 22 , κ 32 for the triplet leptoquark, the decay branching ratios of leptoquarks are changed, so that the LHC searches for leptoquarks as well as the indirect searches for leptoquark portal dark matter will be affected. In particular, we will discuss the impact of extra couplings on the signatures of DM annihilations into a leptoquark pair in detail in the later section.
On the other hand, the additional coupling also contributes to the branching ratio of τ → µγ as follows, ) and the lifetime of tau is given by τ τ = (290.3 ± 0.5) × 10 −15 s [70]. The current experimental bound is given [71] by in green(yellow) region at 2σ(1σ) level. The gray region is excluded by the bound on B(τ → µγ). We have taken λ 32 = λ 33 = 1(0.1) on left(right) plot and λ 33 = 0. Therefore, for m LQ 10 − 50 TeV under perturbativity and leptoquark couplings less than unity, the (g − 2) µ anomaly can be explained in our model, being compatible with B(τ → µγ).

Leptoquark searches
There are two main production channels for leptoquarks at the LHC, one is pair production via gluon fusion and the other is single production via gluon-quark fusion [25,[65][66][67][68].
In the case of R K ( * ) anomalies, the triplet scalar leptoquark (φ 1 ) couples to b/s, µ. The other components of the triplet leptoquark couple to b/s, ν µ and t/c, µ for φ 2 and t/c, ν µ for φ 3 . On the other hand, in the case of R D ( * ) anomalies, the singlet scalar leptoquark (S 1 ) couples to b/s, ν τ and t/c, τ . When the leptoquark pair production via gluon fusion is dominant, the current limits on leptoquark masses listed in table 1 apply. The current LHC bounds on leptoquarks depend on decay modes, but the leptoquark masses are constrained to be greater than about 1 TeV in most cases.
When the Yukawa couplings, φ 1 -b/s-µ, S 1 -b-ν τ and S 1 -c-τ couplings, present in models explaining the B-anomalies, are sizable, the leptoquarks can be singly produced by b/s/c quark fusions with gluons. For instance, in the case of φ 1 , the relevant production/decay channels are pp → φ *

Leptoquarks and scalar dark matter
We introduce a scalar dark matter that have direct interactions to scalar leptoquarks and the SM Higgs doublet H by quartic couplings. Thus, this is the minimal dark matter model without a need of extra mediator particle. In this section, we regard scalar leptoquarks as portals to scalar dark matter and discuss the impacts of leptoquarks on direct and indirect detection of dark matter as well as Higgs data. We can also consider leptoquark-portal models for fermion or vector dark matter too. But, in this case, there is a need of mediator particles [72,73] and/or non-renormalizable interactions [74,75], leading to more parameters in the model, so this case is postponed to a future publication for comparison [76].

Annihilation cross sections for scalar dark matter
We consider a scalar leptoquark S LQ = S 1,3 and a singlet real scalar dark matter S. Then, the most general renormalizable Lagrangian consistent with S → −S is The above Lagrangian generalizes the Higgs portal interactions to those for leptoquarks. After electroweak symmetry breaking with H = (0, v + h) T / √ 2, the new interactions relevant for SS → S LQ S * LQ , hh are Scalar dark matter S annihilates through three channels at tree level, SS → ff with f being the SM fermions, SS → hh, with h being the SM Higgs boson, SS → V V with V being electroweak gauge bosons, and SS → S LQ S * LQ for m LQ < m S . Feynman diagrams for the annihilation channels of scalar dark matter are shown in figure 3. Depending on the quartic couplings, a heavy scalar dark matter may annihilate into a pair of leptoquarks dominantly, leaving the signatures in both anti-proton and positron from cosmic rays, due to the decay products of leptoquarks, as will be discussed later.
For m LQ > m S , SS → S LQ S * LQ channels are kinematically closed, so instead leptoquark loops make corrections to SS → V V with V being electroweak gauge bosons and contribute to new annihilations such as SS → gg, Zγ, γγ. In this case, depending on the relative contributions of SS → ff , hh, V V channels, the loop-induced annihilation channels can be relevant.

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We obtain the effective interactions between scalar dark matter and the SM gauge bosons due to leptoquarks with m LQ > m S , as follows, The details on the above effective interactions are given in appendix B. Then, in the basis of mass eigenstates, the above effective interactions become First, the tree-level annihilation cross sections are with f being all the SM fermions satisfying m f < m S . Here, we note that N c = 3 is the number of colors and N LQ = 1, 3 for S LQ = S 1 , S 3 , respectively. On the other hand, for m LQ > m S , instead of SS → S LQ S * LQ , we need to consider the loop-induced annihilation cross sections [77][78][79][80][81] for SS → gg, γγ, Zγ, as follows, Adding loop corrections of leptoquarks to tree-level contributions coming from the Higgs portal coupling λ 4 , we also obtain the annihilation cross sections for SS → W W, ZZ,  respectively, For light dark matter with m S < m LQ , we find that the tree-level annihilation processes such as W W, hh, ff , ZZ are dominant and the loop-induced processes due to leptoquarks are suppressed, except the gg channel, which can be as large as 1-10% of the total annihilation cross section, depending on whether the scalar leptoquark is singlet or triplet. In the case of triplet scalar leptoquark, the Zγ, γγ channels can be as large as 1% or 0.1% of the total annihilation cross section, so they could be probed by Fermi-LAT [83] or HESS [84,85] line searches. On the other hand, for heavy dark matter with m S > m LQ , the S LQ S * LQ channel becomes dominant while the other tree-level processes are negligible as far as |λ 3 | |λ 4 |.

Direct detection bounds
For scalar dark matter, the effective DM-quark interaction is induced due to the SM Higgs exchange at tree level, as follows, where l 3 (S LQ ) is the Dynkin index of a leptoquark S LQ under SU(3) C . Then, the spinindependent cross section for DM-nucleon elastic scattering is given by where Z, A − Z are the numbers of protons and neutrons in the detector nucleus, µ N = m N m S /(m N + m S ) is the reduced mass of DM-nucleon system, and with f p,n T G = 1 − q=u,d,s f p,n T q . Here, the mass fractions are f p Tu = 0.023, f p T d = 0.032 and f p Ts = 0.020 for a proton and f n Tu = 0.017, f n T d = 0.041 and f n Ts = 0.020 for a neutron [82]. Therefore, the quartic coupling λ 4 between scalar dark matter and SM Higgs is strongly constrained by direct detection experiments such as XENON1T [17,18]. Consequently, tree-level annihilations of scalar dark matter into hh, ff , W W, ZZ are constrained, while the leptoquark-induced annihilations at tree or loop levels can be relevant.
In figure 6, we show the DM relic density as a function of DM mass in red solid(dashed) lines for triplet(singlet) scalar leptoquarks. We also show the DM-nucleon scattering cross section in blue lines as can be read from the right vertical axis, and the XENON1T bound in purple dot-dashed lines. We find that the extra annihilation of dark matter into a pair of leptoquarks opens a new parameter space at m S > m LQ due to a sizable leptoquark portal coupling, λ 3 , avoiding the direct detection bound from XENON1T.

Indirect detection bounds
For relatively light scalar dark matter with m S < m LQ , the DM annihilation cross sections into hh, W W, ZZ, tt, bb are dominant. In this case, Fermi-LAT dwarf galaxies [86] and HESS gamma-rays [87] and AMS-02 antiprotons [88,89] can constrain the model.
In figure 7, we depict the parameter space in λ 4 vs m S in black and red solid lines, satisfying the correct relic density for models with singlet and triplet scalar leptoquarks, respectively. In the same plots, we superimpose the indirect detection bounds on the DM annihilations into a W W pair from Fermi-LAT and HESS gamma-ray and AMS-02 antiproton searches, and include the direct detection bounds from XENON1T. Moreover, the region with m S < m h /2 can be also constrained by Higgs data such as Higgs invisible decay and the signal strength for gg → h → γγ, as will be discussed in the next subsection.
As a result, the Higgs data as well as indirect detection constrains the region with light and weak-scale dark matter, but the XENON1T experiment constrains most, ruling out most of the DM masses below m S = 1 TeV, except the resonance region near m S = m h /2. However, we find that the correct relic density can be obtained for a small value of λ 4 due to the contribution of DM annihilation channels into a leptoquark pair with a sizable leptoquark-portal coupling λ 3 for m S > m LQ . Therefore, there is a wide parameter space above m S = 1 TeV that is consistent with the XENON1T bound.
We remark the indirect signatures of dark matter and the leptoquark decays in the case of heavy scalar dark matter. For m S > m LQ , dark matter can annihilate into a pair of leptoquarks, each of which decays into a pair of quark and lepton in cascade. In the case with leptoquark couplings to explain the B-meson anomalies, the branching ratios of final products of DM annihilations are shown in table 2, depending on the decay branching ratios of leptoquarks. In the case where the extra leptoquark couplings, λ 32 , κ 23 and κ 33 , introduced for accommodating B → K ( * )ν ν bounds and/or the (g − 2) µ excess, are dominant, as discussed in section 4.1, we also show the corresponding branching ratios of final products of DM annihilations in table 3.
In particular, for m S m LQ , a leptoquark pair is produced with almost zero velocities, so each leptoquark decays into a pair of quark and lepton such asql or q l , back-to-back.
Relic, SLQ=S3 Relic, SLQ=S3  In this case, a pair of two quarks (q q) or a pair of leptons (l l ) carry about the energy of DM mass, so we take them as if they are produced from the direct annihilations of dark matter with mass m S /2 and impose the indirect detection bounds on the annihilation cross section. But, if m S m LQ , leptoquarks produced from the DM annihilations are boosted so the full energy spectra for quarks or leptons carry the energy spectra of wide box rather than a monochromatic energy. In this case, we need to take more care before imposing the indirect detection bounds. Henceforth, ignoring the boost effects of leptoquarks, in particular, for m S m LQ , we discuss the indirect detection bounds for the direct annihilations of dark matter to cascade annihilations.
First, we consider the case in table 2 with leptoquark couplings necessary to explain the B-meson anomalies. In this case, for a singlet leptoquark with λ 33 λ 23 or β ≈ 1,
In figure 8, in the parameter space in λ 4 vs m S , in addition to the correct relic density conditions for models with singlet and triplet scalar leptoquarks, respectively, in black and red solid lines and the direct detection bounds from XENON1T, we superimpose the Fermi-LAT constraints from bb and ττ on the products of DM cascade annihilations into a leptoquark pair. Here, we assume that each leptoquark decays into a pair of quark and lepton, according to table 2 with β ≈ 1 and γ ≈ 1. Then, as explained in the caption of figure 8, the resulting Fermi-LAT bounds are shown to constrain the parameter space as strong as or stronger than the XENON1T bounds, depending on the value of leptoquarkportal coupling λ 3 .
In summary, the leptoquark-portal couplings lead to potentially distinct signatures with quarks and leptons mixed from the cascade annihilations of dark matter, as compared to the case with direct annihilations into a quark pair or a lepton pair. Our lepto-quark portal scenario is different from the Higgs portal scenario with additional SU(2) L singlet or triplet scalars, because the final states in the cascade DM annihilations contain quarks and leptons together due to the leptoquark decays in our case. In other words, the region with m S > m LQ can be constrained by indirect detection experiments too. The more general cases that the boost effects of leptoquarks cannot be ignored will be discussed in a future work.

Higgs data
The decay rate of the Higgs boson into a pair of dark matter particles is The decay rate of the Higgs boson into a diphoton or a digluon is also modified due to leptoquarks, as given in eqs. (B.7) and (B.8). Corrections to h → W W, ZZ are small because they are already present at tree level in the SM, so we can ignore them. Then, the total Higgs decay width is modified to where Γ h,SM = 4 MeV in the SM. The bound from invisible Higgs decay, BR(h → SS) < 0.24, leads to the following condition [90,91], The diphoton signal strength for gluon-fusion production is given by The other visible decays, h → ij, are similarly modified to µ ij = R gg R ij , through the modified total decay width of Higgs boson, with R ij = BR(h → ij)/BR(h → ij) SM = Γ h,SM /Γ h . The measurements of gg → h → γγ show µ γγ = 1.10 +0.23 −0.22 from the combined fit of LHC 7 TeV + 8 TeV data [92], and µ γγ = 0.81 +0. 19 −0.18 and µ γγ = 1.10 +0.20 −0.18 from the ATLAS and CMS 13 TeV data, respectively [93,94].
In our model, as far as |λ 5 | 10, the decay rate into a diphoton or a digluon can be ignored, but the diphoton signal strength is modified by the enhanced total decay width of Higgs boson due to the invisible decay mode. This result can be read from figure 7 in the purple dot-dashed lines the region above which is excluded by Higgs invisible decay and in the green region which is excluded by the Higgs diphoton signal strength.

Conclusions
We have presented leptoquark models where scalar leptoquarks not only lead to the effective operators necessary for the B-meson anomalies but also become a portal to scalar dark matter through quartic couplings. We showed that the annihilations of dark matter into a leptoquark pair allow for a wide parameter space that is consistent with both the correct relic density and the XENON1T bound. These new annihilation channels lead to fourbody final states in cascade with quarks and leptons mixed, due to the leptoquark decays. Therefore, there is an interesting interplay between the cascade annihilations of dark matter and the leptoquark search channels at the LHC, which can be tested in the current and future experiments. (A.1) Then, after integrating out the leptoquark S 1 , we obtain the effective Hamiltonian relevant for b → cτν τ as where use is made of Fierz identity in the second line.
In particular, in MSSM, down-type squarks (b * Rk ) [95] belong to singlet scalar leptoquarks. We introduce the R-parity violating (RPV) superpotential as follows, resulting in the component field Lagrangian for doublet scalar leptoquarks S 2 ≡ũ Lk with Y = + 1 6 or singlet scalar leptoquarks S 1 =b * Rk with Y = + 1 3 as Picking up the necessary terms for R K ( * ) and R D ( * ) anomalies, we get, in terms of two component spinors, Then, after integrating out the up-type squarks,ũ Lk , and down-type squarks,b * Rk , we obtain the effective Hamiltonian for the semi-leptonic B-decays in terms of four-component Therefore, the effective Hamiltonian for the b-to-s transition is of the (V + A) form, which was originally proposed to explain R K anomalies [96,97] but is not consistent with R K * anomalies as it favors (V − A) form. On the other hand, the effective Hamiltonian for the b-to-c transition is consistent with the R D ( * ) anomalies [48,98,99]. From eq. (3.3), we obtain the relevant Yukawa couplings for the triplet leptoquark S 3 in components, (A.7) Then, after integrating out the leptoquark φ 1 with Q = + 4 3 , we obtain the effective Hamiltonian relevant for b → sµ + µ − as Here, we note that use is made of the Fierz identity in the second line and (s C ) R γ µ (b C ) R = b † Lσ µ s L =b L γ µ s L is used in the third line.
The Yukawa couplings for the singlet scalar leptoquark also lead to effective Hamiltonian for b → sν iνj as follows, A similar effective interactions can be obtained for the triplet scalar leptoquark, as discussed in the text. For heavy leptoquarks, we the effective interactions between scalar dark matter and SM gauge bosons, induced by leptoquarks, as follows, Moreover, leptoquark couplings to the SM Higgs can modify the decay rates of Higgs boson into a diphoton or a digluon, as follows, where g LQ ≡ λ 5 v 2 /(2m 2 LQ ), x i = m 2 h /(4m 2 i ) and the loop functions are