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 Br(h to 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.


I. 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]). 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.
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 [13] for the simplest models by assuming that Γ invisible 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 [14], with the exception of the parity violating Higgs portal where DM is a fermion [15]. This, on the other hand, can be covered in the future using indirect DM searches [15].
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 [13] 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 II 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 III 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 IV we in turn give three examples of viable Higgs portal models of DM. Two models, described in subsections IV A and IV B, 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 IV C, 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 V. Details on direct DM detection, relic abundance calculations, and the fits to the Higgs data are relegated to the Appendices A and B, respectively.

II. 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 where the overall normalization, 10 3 ∼ 1/y 2 b , is set by the total width of the SM Higgs. In (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 where m, m are non-negative integers, while the numerical pre-factor is simply the translation of the experimental limit due to XENON100 [17] and will increase in the future. Note that (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 (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. (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).

III. 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 (c.f. [18]) and we do not pursue it any further.  and projected future XENON1T bounds [14] are denoted by dot-dashed and solid red lines, respectively.
The shaded blue regions indicate where the EFT description breaks down (Λ < 2m DM ).
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 [16]  The operators in Eq. (8) are also subject to severe direct DM detection constraints from Zmediated DM scattering on nuclei (for details see Appendix A). In Fig. 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 [20]. 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 [17] (dot-dashed red lines). For fermionic DM with purely axial-vector interactions the spin-dependent cross section is plotted in Fig. 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 [22], which excludes m DM < 35 GeV and 50 GeV< m DM <150 GeV. Note that the XENON1T [23] is 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. (8).
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 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. (10) to be proportional to the fermion masses,  (unlike for scalar interactions this suppression carries over for tensor interactions when matching from quark to nucleon level operators, see Refs. [24] and [25] for further details).
The remaining two possibilities are constrained by indirect DM searches. In Fig. 3 we compare the bounds on annihilation cross sections σv for bb (blue lines) and τ + τ − (red lines) channels  [ 26,27] with the predictions from the last two operators in Eq. (10), 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 Fig. 3 the possibility of Higgs portal coupling to DM through the axial-vector operator from Eq. (8), Finally, DM can couple to the Higgs through Weinberg-like operator, where i, j, k, l are SU (2) L indices, ij is the antisymmetric tensor with 12 = − 21 = 1, and 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. (13) 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 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 Fig. 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. 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), We use the notation in which ∆ is represented by the 2 × 2 matrix, We introduce the following interactions 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 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 with φ complex in this case.
From now on, we shall focus on the model given in Eq. (17). The Lagrangian (17) could be supplemented by several other gauge-invariant terms such as Some of them are already phenomenologically constrained to be small. For instance, H T ∆ † H would generate neutrino masses once ∆ is integrated out [30][31][32]. 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 (13) we require f ∼ λ ∼ 1 and m φ ∼ few hundreds GeV, so that µ needs to be very small, i.e., 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 Fig. 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. by either ∆ + or ∆ ++ particles, with the rate where α em is the QED fine-structure constant. The − a → + b − c − d decay can proceed through tree-level ∆ ++ exchange, giving where δ cd encodes the symmetry factor for two identical particles in the final state [33]. The resulting bounds on f ab from various LFV processes are given in Table I  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 [38] 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 [38], so that correct relic density can still be obtained.  [21], except for µ → eγ which is from [34]. We set m ∆ + = m ∆ ++ = m ∆ .

Process
Branching ratio bound Bounds on f ab 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. [39,40]). The remaining 5 degrees of freedom consist of two CP-even scalars h and H, a CP-odd scalar A ≡ −χ 1 sin β+χ 2 cos β, and a pair of charged scalars H ± ≡ −H ± 1 sin β+H ± 2 cos β.
It is 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, 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 Fig. 6.
Similarly, DM-nucleon scattering cross section also receives contributions from both h and H exchanges, where c u,c,t = g SSh (m H /m h ) 2 cos α csc β + g SSH sin α csc β, while the relevant nuclear form factors f p q are listed in Eq. (A7) . The h and H contributions may interfere destructively. In fact, σ SI p vanishes completely, if Note that it is possible to fulfill this requirement even if g SSh = 0. Then B(h → SS) = 0, while Eq. (31) gives 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, c.f. Eq. (26), is preferred by recent Higgs data. In this limit Eq. (32) then completely fixes the value of tan β; i.e., using the values of nuclear form factors in Eq. (A7) 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 Fig. 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 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 [45], while SM branching ratios for m h = 125 GeV Higgs boson are taken from [46]. In our model all the Higgs signal strengths µ i depend on three parameters, α, β and B(h → SS). Fig. 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 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 DMnucleon cross section from XENON100 [17] 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 .
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 [47]). 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 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 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 [48], so that at  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 Fig. 8  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 [41], 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 Fig. 9 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 [49]) combining to the Higgs mass.
Similarly, for operators in Eq. (10) we have In above equations, |p| ∼ 1 MeV is the DM momentum in the center of mass frame, µ χN is the DMnucleon 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 [24] We use XENON100 bounds from Ref. [17] for spin-independent (SI) case and Ref. [22] for spin-dependent (SD) case to constrain the parameter space given by the relic density. We always use the more constraining choice. In our fitting procedures we follow the method adopted in references [52][53][54][55]. The latest available LHC Higgs data are presented in Table II. 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 sepa-ration 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 Best-fit values (μ), variances (σ) and correlations (ρ) are obtained from the plots provided by the experiments and listed in Table II.
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.