Constraining Higgs mediated dark matter interactions

We perform an analysis of Higgs portal models of dark matter (DM), where DM is light enough to contribute to invisible Higgs decays. Using effective field theory we show that DM can be a thermal relic only if there are additional light particles present with masses below a few 100 GeV. We give three concrete examples of viable Higgs portal models of light DM: (i) the SM extended by DM scalar along with an electroweak triplet and a singlet, (ii) a Two Higgs Doublet Model of type II with additional scalar DM, (iii) SM with DM and an extra scalar singlet that is lighter than DM. In all three examples the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathcal{B} $\end{document}(h → invisible) constraint is not too restrictive, because it is governed by different parameters than the relic abundance. Additional light particles can have implications for flavor violation and collider searches.


Introduction
The narrow resonance with mass m h 125 GeV that was recently discovered at the LHC [1,2] is a scalar [3][4][5][6][7] and has interactions consistent with those of the standard model (SM) Higgs boson [8,9]. At present the experimental uncertainties are still relatively large and even O(1) deviations with respect to the SM couplings are possible. One of the more intriguing possibilities is that the Higgs could couple to dark matter (DM).
The argument in favor of this possibility is quite general. Assuming that the discovered scalar is part of the Higgs electroweak doublet H, then H † H is the only gauge and Lorentz invariant relevant operator in the SM. As such it can act as the "Higgs portal" to DM [10]. The experimental searches place a number of nontrivial constraints on this idea. A pivotal parameter in the constraints is the DM mass. If DM is light, m DM < m h /2, then Higgs can decay into DM. The resulting invisible decay width of the Higgs is bounded at 95% CL to B(h → invisible) < 0.19(0.38) from global fits with the Higgs couplings to the SM fermions fixed to their SM values (varied freely while also allowing new particles in loops) [11] (see also [12][13][14][15][16][17][18][19][20][21][22][23][24][25]). This is a nontrivial constraint, since the SM Higgs decay width is so narrow. It essentially requires -with some caveats to be discussed belowthat the Higgs coupling to DM needs to be smaller than roughly the SM bottom Yukawa coupling, y b ∼ O(0.02). This then insures that the invisible branching ratio is smaller than the dominant channel, h → bb.

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On one hand we thus have a requirement that the Higgs should not couple too strongly to light DM. On the other hand, one needs O(1) couplings of Higgs to DM in order to obtain the correct thermal relic density. The tension between the two requirements leads to the apparent conclusion that the Higgs portal models with light DM are excluded. This was shown quantitatively in [26] for the simplest models by assuming that Γ invisible h 0.2Γ SM h 0.8 MeV. Relaxing this bound by a factor of a few does not change the conclusion.
For heavier DM, m DM > m h /2, the bound on the invisible decay width of the Higgs is irrelevant. In this case one can search for DM using direct and indirect detection experiments. Existing constraints from direct DM detection are not stringent enough, but the next generation experiments are expected to cover most of the remaining viable parameter space [27,28], with the exception of the parity violating Higgs portal where DM is a fermion [29]. This, on the other hand, can be covered in the future using indirect DM searches [29].
In this work we are primarily interested in the implications of an invisible Higgs decay signal (and the absence thereof so far) for light thermal relic DM. Are there still viable Higgs portal models with light DM? What modifications of the simplest models [26] are needed? The conclusion that the simplest versions of the Higgs portal are excluded by the bound on B(h → invisible) utilizes effective field theory (EFT). The conclusion therefore relies on the assumption that an EFT description with the SM particles and DM as the only relevant dynamical degrees of freedom is valid both for the relic abundance calculation as well as for direct DM detection and Higgs phenomenology. For viable DM Higgs portals then either the EFT description (with naïve power counting) must be violated, or the invisible decay width of the Higgs is naturally suppressed. As we will show below this implies that given present experimental constraints, the Higgs can couple significantly to thermal relic DM with mass less than half of the Higgs only if there are other light particles in the theory (barring fine-tuned situations). In turn, should a nonzero invisible Higgs decay eventually be found and interpreted as a decay to thermal relic DM particles, then other new light particles need to be discovered.
To demonstrate this we first show in section 2 that extending the EFT description to higher dimensional operators but not enlarging the field content does not change the conclusions about the minimal DM Higgs portals if h →DM+DM decay is allowed. In section 3 we then show that for models where the two body Higgs decays to dark sector are forbidden, the scale of the EFT is small, Λ ∼ O(few 100 GeV). This again implies that viable Higgs portals of DM require new light degrees of freedom beyond SM+DM. In section 4 we in turn give three examples of viable Higgs portal models of DM. Two models, described in subsections 4.1 and 4.2, can be matched onto EFT since the additional degrees of freedom are heavier -though not much heavier -than the Higgs. The two models do require fine-tuned cancellations in order to avoid experimental constraints. A model discussed in subsection 4.3, on the other hand, requires no such tunings. It contains, however, a particle lighter than DM and therefore violates the EFT assumptions. We summarize our conclusions in section 5. Details on direct DM detection, relic abundance calculations, and the fits to the Higgs data are relegated to the appendices A and B, respectively.

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2 Higgs portals in effective field theory We start by reviewing the minimal Higgs portal scenarios. The SM is enlarged by a single neutral (DM) field, odd under a Z 2 symmetry. In the following we consider DM with spins up to and including spin 1, i.e. the possibility that DM is a scalar, φ, a fermion, ψ or a vector, V µ . The dominant interactions of DM with the SM are in each case, respectively, After electroweak (EW) symmetry breaking where v EW 246 GeV is the electroweak condensate and h the Higgs boson. We see that the scalar and vector DM have renormalizable Higgs portal interactions with the SM. For fermion DM these interactions start only at dimension 5. In eq. (2.1b) Λ is the scale at which the non-renormalizable DM-Higgs interactions are generated. In principle one can also write down higher dimensional operators that supplement (2.1a)-(2.1c), but are suppressed by more powers of Λ. The minimal Higgs portal models of DM assume Λ v EW , m DM , such that the expansion v EW /Λ makes sense and (2.1a)-(2.1c) are the dominant contributions to DM-SM interactions in the early universe and current experiments. As shown in [26], in all such models with light DM (m DM m h /2), the observed DM relic abundance is in conflict with the experimental bounds on the invisible decay width of the Higgs, while in the region m DM > m h /2, direct DM detection experiments are beginning to exclude the remaining parameter space.
But would the situation change if the v EW /Λ expansion would not start at the lowest order, eqs. (2.1a)-(2.1c)? Can higher dimensional Higgs-DM operators [30] open new possibilities to reconcile Higgs portal DM with current experimental constraints? To answer this question we first perform a naïve dimensional analysis of the relevant processes based solely on the canonical dimension (d = 4 + n) of the relevant interaction operator. For m DM m h /2 the invisible Higgs branching fraction scales as where the overall normalization, 10 3 ∼ 1/y 2 b , is set by the total width of the SM Higgs. In (2.3) we used v EW ∼ m h , assumed that all dimensionless DM-Higgs couplings are O(1), and also assumed two-body h → invisible decay kinematics. In comparison, the current constraints from direct DM detection experiments give

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where m, m are non-negative integers, while the numerical pre-factor is simply the translation of the experimental limit due to XENON100 [31] and will increase in the future. Note that (2.4) assumes spin independent scattering since this is stronger than spin dependent one. The suppression in terms of m h /Λ is the same as for B(h → invisible), but depending on the operator structure there may be additional suppressions from typical DM velocity in the galactic halo, β ∼ 10 −3 , or from DM mass insertions, m DM /m h . Both of these factors are smaller than one, therefore we conclude that at present for light DM the Higgs constraints are stronger than direct DM detection constraints for any operator dimension. If DM is a thermal relic, then its abundance is fixed by thermal DM annihilation cross-section at the time of freeze-out, where y f is the SM Yukawa coupling for the heaviest open SM fermion channel, and k k min = 0(2) for scalar and vector (fermion) DM with the equality sign for the lowest dimensional operators. In (2.5) we neglected relative velocity suppressions, v r ∼ 0.4, and as before set all Wilson coefficients to be O(1). In order to obtain the correct relic density, σ ann. v 3 · 10 −26 cm 3 /s, with Ω DM ∝ 1/ σ ann. v . From eq. (2.5) we then see that the correct relic density requires the scale Λ to be lower if the dimensionality n of the operator setting the annihilation cross section is higher. The scaling of Br(h → invisible) in terms of Λ is the same as for σ ann. v , so that for the correct relic density one has where n min = 4(5) for scalar and vector (fermion) DM. Since k − k min 0, the Higgs constraints can only become stronger if the Higgs portal proceeds through higher dimensional operators. As a result, the higher dimensional operators cannot reconcile Higgs portal DM with the bounds on invisible Higgs branching ratio as long as h → DM + DM is possible and all couplings are O(1).

Suppressed Higgs decays to dark sector
In the previous section we saw that B(h → invisible) places strong constraints on Higgs portals of DM. The analysis relied on two assumptions, i) that h →DM+DM decay is possible, and ii) that DM is the only light new physics particle. In this section we investigate in more details the first assumption, while the second assumption will be relaxed in the subsequent section. In the remainder of this section we therefore assume that h →DM+DM decay is forbidden either accidentally or due to the structure of the theory. There are three possibilities to suppress the h →DM+DM decay. The first one is to assume DM annihilation to SM particles proceeds predominantly through operators not involving the Higgs. This possibility is orthogonal to the basic idea of a Higgs portal. It has also been studied extensively (cf. [32][33][34][35][36]) and we do not pursue it any further. The second possibility is that the h →DM+DM decay is kinematically forbidden simply because

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DM is heavy enough, m DM > m h /2. The final possibility is that DM couples through a special subset of Higgs portal operators, such that h →DM+DM decay is forbidden, while h →DM+DM+X SM is allowed, where X SM denotes one or more SM particles in the final state. We set aside the model building question of how this is arranged in the UV theory and work within EFT. The y 2 b suppression of the SM Higgs decay width is roughly of the same size as the phase space suppression from one or two additional final state particles. One may thus expect that O(1) couplings between DM and the Higgs would give at the same time the correct relic abundance as well as small enough B(h → 2DM + X). Below we go through a list of possible operators, and as we will see a number of them are not excluded by direct and indirect DM detection constraints.
The simplest effective interactions generating h → DM + DM + X SM decays are built from the Higgs vector current where c W = cos θ W , with θ W the weak mixing angle. The operators of the lowest dimension are [30] H For example, they appear in models where the DM is charged under a hidden U(1) gauge symmetry (spontaneously broken above the weak scale), exhibiting kinetic mixing with the SM hypercharge [37]. These operators induce a three body decay h → DM + DM + Z, where for Z → νν the decay would be completely invisible. They do not lead, however, to two body invisible decay h → DM + DM. The three body Higgs decay is kinematically allowed if m DM < (m h − m Z )/2 17 GeV. Such a light DM is subject to bounds from Z → E miss measurements at LEP [39]. Requiring the correct relic density this constrains m DM > 24(34) GeV for scalar (vector) DM, and m DM > 14(31) GeV for fermionic DM with vector (axial-vector) interaction.
The operators in eq. (3.2) are also subject to severe direct DM detection constraints from Z-mediated DM scattering on nuclei (for details see appendix A). In figure 1 we show the predicted spin independent DM-nucleon cross sections (dashed blue lines) after requiring the correct thermal relic density Ω DM h 2 = 0.1186 ± 0.0031 [38]. The shaded blue regions indicate the validity of EFT, i.e., that Λ ≥ 2m DM . With the exception of fermionic DM with purely axial-vector interaction (c V ψ = 0) all parameter space allowed by relic density is excluded by XENON100 [31] (dot-dashed red lines). For fermionic DM with purely axial-vector interactions the spin-dependent cross section is plotted in figure 1, bottom right panel, since the SI cross-section is velocity suppressed. The result is compared to recent XENON100 bound on SD DM-neutron cross section [40], which excludes m DM < 35 GeV and 50 GeV< m DM <150 GeV. Note that the XENON1T [ expected to cover almost completely the remaining low DM mass window. In summary, the combination of invisible Z decay and direct DM detection constraints excludes any appreciable B(h → invisible) from operators in eq. (3.2). Another possibility is to couple DM to scalar or tensor fermionic currents. These automatically involve a Higgs field, The lowest dimensional operators are then where the dependence of couplings on SM fermion flavors is implicit. Operators involving Γ S can be generated for example in models with extended scalar sectors, as we will discuss

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below. On the other hand, the generation of tensorial Γ T µν interactions is typically more involved. One possibility is to introduce a SM-DM mediator sector with a gauge symmetry under which both SM and DM are neutral. The appropriate irrelevant couplings to generate the tensorial SM-DM interaction can then possibly be obtained at the loop level. A complete model construction is thus quite intricate and beyond our scope, so we do not pursue it any further.
We first assume the couplings in eqs. (3.4) to be proportional to the fermion masses, so that possible flavor changing neutral currents (FCNCs) are automatically suppressed. The operators in eq. (3.4) lead to four body Higgs decays, that are unobservably small. For instance, assuming thermal relic DM with m DM = 20 GeV one has B(h → DM + DM+bb) ∼ O(10 −7 ) for both purely pseudo scalar and purely tensorial DM interactions. Figure 2 shows the predictions for the spin-independent DM-nucleon cross sections in the upper four panels, for scalar DM, vector DM, and fermion DM with scalar and pseudoscalar interactions, respectively (blue dashed lines), requiring correct thermal relic DM abundance. The spin-dependent cross section for fermion DM with purely tensorial interaction is shown in the lower panel in figure 2. For the chosen flavor structure of the relevant couplings, eq. (3.5), XENON100 bounds (dot-dashed red lines) exclude almost all possibilities except for fermionic DM with parity-violating or tensorial interactions. The parity violating fermionic DM evades the current XENON100 and also the projected XENON1T bound (red solid line) because the scattering cross section is velocity suppressed. The direct detection cross section for the tensorial interactions is strongly suppressed by the assumption that the coupling to light quarks is suppressed by light quark masses, eq. (3.5) (unlike for scalar interactions this suppression carries over for tensor interactions when matching from quark to nucleon level operators, see refs. [42] and [43] for further details).
The remaining two possibilities are constrained by indirect DM searches. In figure 3 we compare the bounds on annihilation cross sections σv for bb (blue lines) and τ + τ − (red lines) channels [44,45] with the predictions from the last two operators in eq. (3.4), when correct relic density is assumed in the predictions. We see that the fermionic DM with pseudo-scalar or tensorial interactions is constrained to be heavier than m DM > 15 GeV. For reference we also show in figure 3 the possibility of Higgs portal coupling to DM through the axial-vector operator from eq. (3.2), which is not excluded by direct detection. It demonstrates that for Z mediated channels, the constraints from indirect detection are not as significant. The reason lies in the assumed flavor structure. This is fixed for operators in eq. (3.2) by the couplings of the Z. DM then annihilates to all fermions democratically, reducing the signal in the bb and τ + τ − final states. For the flavor structure assumed in eq. (3.4) these are the dominant channels, however, making the constraints more powerful.
This also highlights the fact that the bounds on operators in eq. (3.4) depend strongly on the assumed flavor structure of the Wilson coefficients. We do not attempt to cover all possibilities but rather only entertain a few representative cases. For instance, increasing the couplings to light quarks, u, d, s, the direct DM detection bounds would become sig-   nificantly stronger, while the relic density would remain practically unaffected. Note that in the limit where DM does not couple to the light quarks but only to 3 rd generation, the direct detection bounds are still relevant since one induces interactions to gluons at loop level. An interesting possibility is to have Wilson coefficients differ in sign such that the DM-nucleon elastic scattering cross-section vanishes. This possibility was pointed out in the context of type II Two-Higgs-Doublet Model (2HDM-II) in ref. [46], to be discussed in more detail in subsection 4.2. Another possibility where direct detection bounds are weak or completely irrelevant is the case of leptophilic DM, where the Wilson coefficients for operators coupling to quarks in eq. (3.4) are suppressed [47]. Finally, DM can couple to the Higgs through Weinberg-like operator, O dark the DM operator. The lowest dimensional interactions are explicitly, and similar operators with φ † φ → φφ,ψψ →ψ C ψ andψγ 5 ψ →ψ C γ 5 ψ replacements. The operators in eqs. (3.7) contribute to neutrino masses at one loop. Modulo cancellations, this suppresses all the operators well below the level required for the thermal scattering cross-section to give the observed DM relic density. The only exception is the fermionic DM operator with purely pseudo-scalar interaction (g P ψ ) whose loop contributions to neutrino masses vanish identically by parity invariance, and the φφ,ψ C ψ,ψ C γ 5 ψ type operators if DM carries (conserved) lepton number. The resulting invisible Higgs decay governed by the g P ψ interaction is very suppressed, that is, B(h → DM + DM +νν) 10 −7 for m DM = 20 GeV and assuming correct relic DM abundance. Note that the operator L i L j H k H l ik jl × iψγ 5 ψ does induce DM-nucleon scattering, but only at loop level and the contribution is furthermore proportional to neutrino mass. The DM-nucleon cross section, therefore, is very suppressed.
The DM annihilation cross section induced by the L i L j H k H l ik jl × iψγ 5 ψ operator is given by

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with β(M 2 ) ≡ 4M 2 /s and s 4m 2 DM is the energy in the center of mass frame. The value of Λ required to obtain the correct relic density is shown in figure 4 (red solid line), assuming only one neutrino flavor in the final state and setting g p ψ = 1. We observe that the required scale is again low, i.e. for m DM = 40 GeV, Λ 300 GeV.
In conclusion, our discussion in this section shows that even if the invisible branching ratio of the Higgs is suppressed, viable Higgs portals to light thermal relic DM require new particles with masses of a few 100 GeV.

SM + DM with an extra triplet and a singlet
In this section, we present a model that could generate the operator L i L j H k H l ik jl × iψγ 5 ψ. As we will see shortly, it can be done by extending SM particle content by a Dirac fermion DM (ψ), an electroweak singlet scalar (φ), and an electroweak triplet scalar (∆). The extra fields therefore transform under the SM gauge group SU(3) C ×SU(2) L ×U(1) Y as ψ ∼ (1, 1, 0), φ ∼ (1, 1, 0), ∆ ∼ (1, 3, 1). (4.1) We use the notation in which ∆ is represented by the 2 × 2 matrix, We introduce the following interactions 3) where H is the usual SM Higgs doublet, a, b = 1, 2, 3 are generation indices, i, j, k are SU(2) L indices, and ij is the antisymmetric tensor. In the above Lagrangian, the φ is JHEP11(2013)190 assumed to be a real scalar. Note that we have written only terms relevant to generate the L i L j H k H l ik jl × iψγ 5 ψ operator, which is obtained after integrating out φ and ∆. It is worth mentioning that one could also consider a variation of the above model in which lepton number is preserved. In this case, the dark matter fermion carries a lepton number -1 and the Lagrangian is modified to Some of them are already phenomenologically constrained to be small. For instance, H T ∆ † H would generate neutrino masses once ∆ is integrated out [48][49][50]. Its coefficient therefore must be very small, much smaller than m ∆ .
By the same reasoning, the term µH † Hφ should be suppressed too. The simultaneous presences of f ab L a L b ∆, λφH T H∆ † , and µH † Hφ terms breaks lepton number by two units, and as a result the neutrino masses are generated at tree level. To generate unsuppressed Weinberg-like operator (3.7) we require f ∼ λ ∼ 1 and m φ ∼ few hundreds GeV, so that µ needs to be very small, i.e., µ 1 eV. Consequently, the φ − h mixing is extremely suppressed and cannot induce sizeable h → DM + DM decay nor DM-nucleon elastic cross section. The invisible Higgs decay can thus only occur through the 4-body mode h → νν + DM + DM with branching ratio of ∼ 10 −6 for m DM = 40 GeV. This number is much too small to be measured in the near future.
The correct DM relic density is obtained fromψψ →νν annihilation that can proceed through s-channel φ and ∆ 0 virtual states. The annihilation is unsuppressed as long as there is significant mixing between φ and ∆ 0 states through the λφH i H j ik ∆ * jk term (after electroweak symmetry breaking). In figure 5 we show as a function of m DM the required m ∆ and the masses m 1,2 of the two φ-∆ 0 mixed physical states such that the observed DM relic density is generated. The numerical example shown is for maximal mixing, where m φ = m ∆ , and we set f ab = y = λ = 1. As anticipated, the required extra states are light, with masses of the order of the weak scale.
The fact that viable Higgs portal models with light DM require additional light states can have phenomenological implications beyond dark matter searches. In the present model, for instance, there are two charged scalars, ∆ ++ and ∆ + . These can mediate lepton flavor violating (LFV) processes such as a → b γ and − The radiative decays can arise at one-loop mediated by either ∆ + or ∆ ++ particles, with the rate where α em is the QED fine-structure constant.
where δ cd encodes the symmetry factor for two identical particles in the final state [51].
The resulting bounds on f ab from various LFV processes are given in table 1 for the case of m ∆ + = m ∆ ++ = m ∆ . (For previous study of LFV in the triplet model, see refs. [53][54][55].) For m ∆ = 220 − 350 GeV as required by the relic abundance, the off-diagonal f ab are severely constrained. There are also bounds on diagonal couplings from collider searches. For flavor degenerate case, with f aa = 1 for a = 1, 2, 3, the CMS Collaboration [56] reports a bound m ∆ > 403 GeV, which is inconsistent with the relic DM density requirement. The search is less effective for f τ τ = 1 and f ee = f µµ = 0, in which case ∆ −− decays exclusively into same-sign tau pairs. The lower limit on ∆ ++ mass is then m ∆ > 204 GeV [56], so that correct relic density can still be obtained.

2HDM-II + DM
Our next example of a viable Higgs portal DM is a type II Two-Higgs-Doublet-Model (2HDM-II) supplemented by an extra singlet scalar -the DM. This is the simplest realization of the fermionic operators in eq. (3.4), discussed in the previous section assuming EFT. While phenomenologically viable, the model does have two ad-hoc features. The invisible Higgs decay width is suppressed by dialling down the appropriate dimensionless parameter, while direct DM detection bounds are avoided by fine-tuning the parameters so that two competing operator contributions cancel to a large extent.

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Process Branching ratio bound Bounds on f ab Table 1. The bounds on LFV couplings f ab of ∆ in eq. (4.3), following from leptonic LFV decays.
The experimental 95% C.L. upper bounds are from [39], except for µ → eγ which is from [52]. We The detailed structure of the model is as follows. The particle content consists of SM fermions, two Higgs doublets, H 1 and H 2 , and an extra real scalar S. Under SM gauge group, these scalars transform as The singlet S is assumed to be Z 2 odd and is identified as DM. The Yukawa interactions of the two doublets are assumed to be the same as in type II 2HDM; H 1 couples to d R and e R , while H 2 only couples to u R , DM couples directly to the two Higgs doublets, For suitable choices of parameters, these interactions allow for large enough DM annihilation cross section and as a result can accommodate the observed relic abundance. After electroweak symmetry breaking three out of eight real degrees of freedom in H 1 and H 2 are absorbed as longitudinal components of W ± and Z bosons (for reviews see e.g. [57,58] h that we identify as the newly discovered particle with 125 GeV mass. The interactions of the CP-even scalars, h, H, with the SM fermions and gauge bosons are given by with r u = cos α csc β, r d = r = − sin α sec β, R u = sin α csc β, R d = R = cos α sec β. After electroweak symmetry breaking there are also trilinear couplings of h, H with the DM, where g SSh = λ S1 sin α cos β − λ S2 cos α sin β, g SSH = −λ S1 cos α cos β − λ S2 sin α sin β. (4.14) DM annihilation into a pair of SM fermions, SS →f f , is mediated by both CP-even scalars, h and H and is proportional to σ ann ∝ (g SSh /m 2 h + g SSH /m 2 H ) 2 . For light DM the g SSh coupling also leads to B(h → SS). As we show below the bounds on invisible decay width of the Higgs require g SSh < 0.01. Correct relic abundance then requires g SSH ∼ O(1), see figure 6.
Similarly, DM-nucleon scattering cross section also receives contributions from both h and H exchanges, Note that it is possible to fulfill this requirement even if g SSh = 0. Then B(h → SS) = 0, while eq. (4.17) gives (4.18) As we will show below the pseudo-decoupling limit, β − α = π/2, where the couplings of the Higgs to W and Z are the SM ones, cf. eq. (4.12), is preferred by recent Higgs data. In this limit eq. (4.18) then completely fixes the value of tan β; i.e., using the values of nuclear form factors in eq. (A.7) one obtains tan β 0.61.
In the limit where B(h → SS) vanishes, the relic abundance is set by DM annihilation with the heavy CP-even Higgs boson H in the s-channel. In figure 6, we plot the coupling g SSH giving the correct relic abundance as a function of DM mass, m DM , for two sample values of heavy CP-even Higgs boson masses, m H = 200, 300 GeV. We also set tan β = 0.61 such that σ SI p vanishes. For heavier H a larger value of g SSH coupling is needed. Perturbativity therefore bounds m H from above, with m H 850 GeV for g SHH 4π (and m H 450 GeV for g SHH 4). Note that in this case H decays invisibly practically 100% of the time. In principle H can be directly searched in the process of associated production with a Z boson (see, e.g., a recent ATLAS analysis of pp → Zh → l + l − invisible [59]). The challenge is that in the limit β − α = π/2, the couplings of H to gauge bosons vanish. As a result, the heavy Higgs boson in this scenario can easily escape such collider searches. On the other hand, H also couples to SM fermions with roughly SM strengths, thus making gg → H(tt) the dominant production mechanisms at the LHC. Especially in the second case, the dominant decay mode H →DM+DM then leads to the interesting tt + E miss T signature. For m H = 200, 300 GeV, we find using [60] the cross-section estimates of σ ttE miss T = 29 fb, 7.7 fb at 8 TeV and σ ttE miss T = 150 fb, 51 fb at 14 TeV LHC, respectively. Given these small cross-sections, also compared to irreducible SM (tt + Z) backgrounds [61][62][63], the search remains challenging for the foreseeable future. On the other hand, interesting mono-jet plus missing transverse energy signature would come from gg → H+jet. Using this particular signature, a dedicated analysis for the SM Higgs boson invisible decay was performed in [64]. The upper limit on µ Hj ≡ σ gg→Hj × B(H → inv)/σ SM gg→Hj at 95% C.L. was found to be µ Hj < 25 (50) for 200 GeV (300 GeV) Higgs boson using just 1 fb −1 of data at 7 TeV. It might be possible for 14 TeV LHC to probe the prediction of this model, µ Hj = R 2 u = 2.7. Finally, we assess the quantitative impact of existing Higgs measurements on the model's parameter space by performing a fit to the latest LHC Higgs data assuming that h is the newly discovered Higgs resonance (for details see appendix B). The partial decay widths normalized to the SM ones are given by while the normalized production rates are In the Higgs signal strengths, µ i , one measures the product of cross section and Higgs branching ratios. Therefore in all the signal strengths the total Higgs decay width enters. This can be modified by the invisible decay width of the Higgs, and as a result one is quite sensitive to it. Normalized to the SM the total width is given bŷ Numerical values for loop functions in h → γγ and h → gg are taken from [65], while SM branching ratios for m h = 125 GeV Higgs boson are taken from [66]. In our model all the Higgs signal strengths µ i depend on three parameters, α, β and B(h → SS). Figure 7 shows the 68.3% and 95.5% C.L. allowed region in the parameter space (α, tan β) obtained from a global fit after marginalizing over B(h → SS). The allowed parameter

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space is constrained to a very narrow region around β − α = π/2. We also derive the bound on invisible branching ratio of the Higgs by marginalizing over α and tan β. We get B(h → SS) < 0.3 at 95.5% C.L., which implies that g SSh < 0.01 for DM mass up to m h /2. We emphasize that B(h → invisible) is a free parameter in this model, and can be both close to present experimental bound or much smaller, depending on the derived dimensionless parameter g SSh . Finally, we combine the Higgs data and 90% C.L. upper bound on spin-independent DM-nucleon cross section from XENON100 [31] into a single χ 2 . For illustration we fix m H = 200 GeV, m S = 40 GeV and g SSH to value determined by relic density. The DM scattering cross section σ SI p and the signal strength rates µ i are expressed in terms of three fitting parameters α, β and g SSh . After marginalizing over g SSh , we obtain the 95.5% C.L. allowed region in (α, tan β) plane, shown as cyan region in figure 7. Marginalizing analogously over α and g SSh , we find tan β = (0.61 ± 0.03).

SM + DM with extra scalar singlet
In our final example of a viable Higgs portal model of DM we add to the SM two real scalars, φ and S (for existing studies of similar models see [67,68]). Under the SM gauge group both scalars therefore transform as The singlet S is the DM candidate, odd under Z 2 , while φ is even. The resulting scalar potential is while the Yukawa interactions take the usual form For simplicity, we assume that φ does not acquire a vacuum expectation value by appropriately adjusting the parameter κ (this has no relevant phenomenological consequences apart from simplifying our discussion). The scalar mass matrix is given by where m 2 h = λ 1 v 2 EW and m 2 φ = m 2 2 + λ 4 v 2 EW /2. Parameter µ 2 induces mixing between h and φ, so that the physical neutral scalars h 1 , h 2 are given by  with the mixing angle given by (4.27) We will assume that m h 1 /2 > m S > m h 2 with m h 1 = 125 GeV. The couplings of h 1 (h 2 ) to the SM fields are the same as for the SM Higgs boson except that they are rescaled by cos α (sin α). The mixing angle α has been constrained by LEP [69], so that at 95% C.L. |sin α| < 0.13 for m h 2 = 20 GeV and |sin α| < 0.2 for m h 2 = 50 GeV. On the other hand, sin α also has to be greater than 10 −8 , otherwise h 2 is sufficiently long lived that it escapes the detector. For sin α ∼ 10 −4 the h 2 particle travels less than a few µm before decaying and can be searched for using displaced vertices. Note that the branching ratios of h 2 are not affected by sin α and are the same as they would be for the SM Higgs with m h 2 mass. For instance, for m h 2 = 20 GeV the dominant branching ratio is B(h 2 → bb) ∼ 85%.
The relic abundance is set by the dominant DM annihilation process SS → h 2 h 2 , with the annihilation cross section given by where λ p = λ 6 cos 2 α + λ 5 sin 2 α. The values of λ p for which the correct relic abundance is obtained are shown in figure 8 as a function of DM mass, m DM , for three choices of light scalar mass m h 2 . Note that λ p that governs the relic abundance is different from λ h = λ 5 cos α − λ 6 sin α that governs the invisible Higgs branching ratio, B(h → invisible). The relic abundance and invisible decay width of the Higgs are thus decoupled in this Higgs portal model. Next, we perform fit to the latest available LHC Higgs data. Unlike the 2HDM-II case, section 4.2, here all the Higgs (h 1 ) signal strengths are rescaled by common factor JHEP11(2013)190 cos 2 α. Also, there are additional contributions to the total Higgs decay width coming from h 1 → h 2 h 2 and h 1 → SS. The Higgs signal strengths, therefore, are given by . A direct bound on invisible Higgs decay width from ATLAS analysis of pp → Zh → l + l − invisible [59], is added to χ 2 as assuming that sin α 10 −4 so that h 2 decays instantaneously. We then take sin α, B(h 1 → SS) and B(h 1 → h 2 h 2 ) as fitting parameters. We obtain 95.5% C.L. bounds on each parameter to be |sin α| < 0.5, B(h 1 → h 2 h 2 ) < 0.24 and B(h 1 → SS) < 0.22. Note that the bound on sin α obtained from this fit is less stringent than the LEP limit. In figure 9, we show 68.3% and 95.5% C.L. allowed region in the parameter space of sin α and B(h 1 → SS), after marginalizing over B(h 1 → h 2 h 2 ). If sin α is very small, so that h 2 escapes the detector, then we obtain B(h 1 → invisible) < 0.22.
Since there is an extra light scalar state, there are interesting collider signatures beside the invisible decay width of the Higgs. The Higgs can also decay to two light scalars, h 1 → h 2 h 2 , where h 2 decays to bb pairs. These decay chains can then be searched for using associated hZ or hW production with four b-tagged jets in the final state (possibly originating from two displaced secondary vertices, see also [70]) combining to the Higgs mass. As discussed above, the h 1 → h 2 h 2 branching ratio can be sizeable, of O(20%).

Conclusions
In this paper we have extended the analysis of Higgs portal models of DM by including higher dimensional operators. We focused on the case where DM is light, so that h → JHEP11(2013)190 DM+DM decays are kinematically allowed. The main difference between the minimal Higgs portals and the case where higher dimensional operators dominate, is that there is now a new scale Λ in the problem. In fact, already for minimal Higgs portal with fermionic DM one is forced to introduce a dimensionful scale Λ since the Higgs couplings then require at least dimension 5 operators. We arrive at the following general conclusions • First assume that an EFT description of SM+DM as the only weak scale dynamical degrees of freedom is valid and all dimensionless coefficients are O(1). If h → DM + DM is discovered close to its present experimental limit, at the order of O(few 10%), then DM cannot be a thermal relic, or its relic density must be controlled by interactions not involving the Higgs field.
• Higgs portal to DM is still possible if either EFT is not valid or if B(h → invisible) is suppressed below naive counting estimate (or both). In both cases there need to be other light particles, with masses below O(few 100 GeV).
We demonstrate this with three examples of viable Higgs portal models of light DM, (i) the SM extended by DM scalar along with electroweak triplet and singlet (subsection 4.1), (ii) a Two Higgs Doublet Model of type II with an addition of scalar DM field (subsection 4.2), (iii) SM with DM and an extra scalar singlet that is lighter than DM (subsection 4.3). All the examples share the feature that the invisible Higgs branching ratio and the relic abundance are decoupled and are governed by different parameters. Furthermore, only in example (ii) the dominant DM annihilation channel is to bb pairs as in the simplest Higgs portal models. As a result this model also requires tuned cancellation to avoid direct DM detection constraints.
Since the Higgs portals of DM require additional light particles, there may be interesting phenomenological consequences. Indeed, non-trivial dynamics taking place below the TeV could leave significant footprints in low energy precision observables, or could be directly detected at high-energy collider experiments. For instance, the charged scalars in example (i) can lead to lepton flavor violating decays, in example (ii) the heavy Higgs decay is dominated by the invisible channels, while in (iii) the Higgs decays to four b jets at the level of a few tens of percent are possible.

A Relic density and direct detection
The DM relic abundance is found by solving the following Boltzmann equation,

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where H is the Hubble constant, x ≡ m DM /T with m DM the DM mass, and Y ≡ n/s with n and s the number density and entropy density respectively. The thermal average of the annihilation cross section is given by [71] σv = In the early universe, DM is assumed to be in equilibrium. Once the temperature drops below the DM mass, Y eq is exponentially suppressed. When the freeze-out temperature is reached, the equilibrium is no longer maintained. As the result, one can integrate the Boltzmann equation to determine relic abundance [71,72] where M Pl = 1.22 × 10 19 GeV is the Planck mass, and g * is the number of effective relativistic degrees of freedom at freeze-out. The freeze-out temperature T f is determined with g the number of DM degrees of freedom.
We review next the calculation of direct DM detection bounds. The operators given in eqs. (3.2) and (3.4) lead to the DM-quark interactions which then induce the scattering of DM on nuclei. For operators in eq. (3.2), the DM-nucleon cross sections are found to be (N = p, n) (A. 6) In above equations, |p| ∼ 1 MeV is the DM momentum in the center of mass frame, µ χN is the DM-nucleon reduced masses (with χ = φ, ψ, V ), and the relevant quark-Z couplings are Y u = 1 2 − 4 3 s 2 W , and Y d = − 1 2 + 2 3 s 2 W . The parameters f N q ≡ m −1 N N | m qψq ψ q |N , ∆ N q , and δ N q indicate the nucleon form factors for scalar, axial-vector, and tensor interactions, respectively. Their values are given by [42]  We use XENON100 bounds from ref. [31] for spin-independent (SI) case and ref. [40] for spin-dependent (SD) case to constrain the parameter space given by the relic density. We always use the more constraining choice.

B Analysis of Higgs data
In our fitting procedures we follow the method adopted in references [73][74][75][76]. The latest available LHC Higgs data are presented in table 2. Measurements are reported in terms of signal strengths normalized to the SM predictions where index i represents the decay mode, while k denotes different production channels. ATLAS and CMS also combine different production sub-channels for a given decay mode to provide separation into production mechanisms. Results are presented in 2D plots in which gluon-gluon fusion (ggF) and associated production with a top pair (ttH) are combined as one signal (µ (ggF +ttH) ), while vector boson fusion (VBF) and associated production with a gauge boson (VH) as another, (µ (V BF +V H) ). In this case, we parametrize the likelihood with where the correlation matrices are given by

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Other data are given in terms of signal strengths with specified production mechanism. In this case, we parametrize the likelihood with The total χ 2 function is given by the sum of all the contributions. In order to confront the DM model to the data, we express all signal strengths (µ) in terms of model parameters and minimize χ 2 to find the best fit point. The best fit regions are defined by appropriate cumulative distribution functions.
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