Higgs portal, fermionic dark matter, and a Standard Model like Higgs at 125 GeV

We show that fermionic dark matter (DM) which communicates with the Standard Model (SM) via the Higgs portal is a viable scenario, even if a SM-like Higgs is found at around 125 GeV. Using effective field theory we show that for DM with a mass in the range from about 60 GeV to 2 TeV the Higgs portal needs to be parity violating in order to be in agreement with direct detection searches. For parity conserving interactions we identify two distinct options that remain viable: a resonant Higgs portal, and an indirect Higgs portal. We illustrate both possibilities using a simple renormalizable toy model.


Introduction
How dark matter (DM) couples to Standard Model (SM) particles is an open question. An interesting possibility is that the coupling of dark and visible sectors is through a Higgs portal [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. The operator (H † H) is one of the two lowest dimensional gauge invariant operators that one can write in the SM (the other one being the hyper-charge gauge field strength B µν ). Therefore, it is quite likely that also (H † H)-(dark sector) will be the lowest dimension operator connecting dark and visible sectors, and thus potentially the most important one.
Experimentally the Higgs portal is probed from two complementary directions. On the one hand, the new generation of direct DM detection experiments [19,20] is starting to probe DM-nucleon scattering cross sections of roughly the size given by a single Higgs exchange with the SM Yukawa couplings to the quarks. On the other hand, the first hints of a SM-like Higgs boson signal were reported by the ATLAS and CMS collaborations. The hints of a signal are seen in several channels, pointing to a Higgs mass of roughly m h ∼ 124 − 126 GeV [21,22], with the SM Higgs boson consistent with the current data at 82% C.L. [23], see also [24,25]. Those hints are supported by recent results from D0 and CDF. For updates from ATLAS, CMS, D0, and CDF see [26].
In view of these experimental developments we revisit the Higgs portal to DM. In particular we focus on fermionic DM. The Higgs portals for the fermionic DM and the scalar DM are qualitatively different. For instance, if DM is a scalar, φ DM , then the Higgs portal operator (H † H)(φ † DM φ DM ) is renormalizable. The same is true if DM is a spin-1 particle. In contrast, if DM is a fermion, χ, then the Higgs portal necessarily proceeds through non-renormalizable interactions. The lowest dimensional Higgs portal in that case consists of two dim = 5 operators which enter the effective Hamiltonian The mass scales Λ 1,5 are roughly the masses of the mediators for O(1) couplings between DM and the mediators. Since the DM-Higgs effective Hamiltonian is non-renormalizable, this means that a Higgs portal for fermionic dark matter necessarily requires a UV completion. In this paper we also consider situations when such UV completions are required in order to obtain a correct description of the DM phenomenology. We first use the effective field theory (EFT) description of the Higgs portal (2) and derive consequences for each of the two operators Q 1,5 . The parity conserving interaction, Q 1 , is severely constrained by direct detection experiments. If only Q 1 is present in H eff then one cannot obtain a small enough relic density consistent with the bound from XENON100 for DM masses below about 2 TeV [14]. In contrast, as we will show in section 2, the parity violating operator Q 5 is allowed by direct detection searches and the observed relic density can be obtained, see also [27]. Hence, when DM interactions are mediated by fields much heavier than 2m χ and 2m h the EFT description is valid and we must conclude that the Higgs portal interactions for fermionic DM need to be parity violating ("pseudo-scalar Higgs portal"). Yet viable scenarios with parity conserving operators can be found when EFT breaks down. We identify two distinct options: • "resonant Higgs portal" -where the dominant contribution is due to a resonant annihilation either through the Higgs or the mediator, • "indirect Higgs portal" -where the DM annihilations into the mediator set the relic density and the Higgs portal only provides the link between the visible and dark sector thermal baths.
In section 3 we illustrate both of these two possibilities using a toy model -the minimal extension of the SM with a DM fermion χ and a real singlet φ see e.g. [4,28].

Effective field theory considerations
Let us first assume that the mediators are heavy so that they can be integrated out. The Higgs portal is then given by eq. (2). We will be interested in the direct detection of DM and in the annihilation of DM in the galactic halo. In both cases the DM particles entering the process are non-relativistic, with velocities typical of DM in the galactic halo, ν ∼ 10 −3 . For annihilations in the early universe, responsible for obtaining the thermal relic density, DM is moderately relativistic. The indirect and direct DM detection signals are given by the annihilation of two non-relativistic DM particles and the scattering of non-relativistic DM on the nuclei, respectively. For these two processes the two effective operators Q 1,5 behave in exactly the opposite way. For instance, the annihilation cross section is (for a Majorana fermion χ) for the decays to Higgs and top, respectively. For m χ very heavy f h → 1 and f t → 0. In the early universe, around the freeze-out temperature T F we have ν 2 ∼ T F /m χ 1/20, whereas in the galactic halo we have ν 2 ∼ 10 −6 . As a consequence, for parity conserving interactions the annihilation cross section relevant for indirect detection signals is significantly suppressed compared to the one relevant for thermal freeze-out. In contrast, for parity violating interactions the annihilation cross section is independent of the velocity.
For direct detection the situation is exactly opposite. Integrating out the Higgs field, the scattering of DM on matter is given by an effective Hamiltonian where the two operators and Wilson coefficients are where v = 246 GeV is the Higgs vacuum expectation value (VEV). The cross section for χ scattering on the proton induced by the operator O 1 is then Figure 1: Proton-dark matter scattering cross section as a function of the dark matter mass in the effective field theory of eq. (2), as predicted by requiring that the correct relic density is obtained by thermal freeze-out. The scattering cross section is shown for several ratios of pseudo scalar coupling to scalar coupling Λ 1 /Λ 5 , and compared to the limit from XENON100 [19].
where m red = m p m χ /(m p + m χ ) is the reduced mass of the DM-proton system and see e.g., [29], where also values for f (p) q are given. The operator O 5 , on the other hand, induces a velocity suppressed scattering where typically ν ∼ 10 −3 . Hence in direct detection one obtains a velocity suppressed scattering cross section for parity violating interactions, but unsuppressed scattering for parity conserving ones. This means that it is possible to find regions of C 1,5 parameter space with correct relic density but small enough direct detection signals. In fig. 1 we show the scattering cross section for different ratios of pseudo scalar coupling to scalar coupling Λ 1 /Λ 5 . For a fixed ratio and given DM mass we determine the sizes of Λ i by requiring that the thermal relic density is obtained in the interval 0.09 < Ωh 2 < 0.13. For the thermal freeze-out calculations we use the micrOMEGAs [29,30] public code, and we assume m h = 125 GeV.
The curve for Λ 1 /Λ 5 = 0 corresponds to parity conservation, i.e., pure scalar coupling. This case was considered for example in [2,10,14] and is incompatible with the bound from XENON100 [19] for DM masses m χ 2 TeV. However, already for Λ 1 /Λ 5 1 the predicted cross section is below the XENON100 bound for m χ 100 GeV, while Λ 1 /Λ 5 10 leads to cross sections of about two to three orders of magnitudes below the limit.
Around m χ ≈ m h /2 = 63 GeV the effect of the s-channel resonance due to Higgs exchange is clearly visible in fig. 1. For DM masses below the resonance Higgs decays h → 2χ become possible. The shape of the curves below the resonance is due to the Breit-Wigner form of the annihilation cross section. The contribution of h → 2χ to the Higgs decay width allows for two solutions for Λ i giving rise to the correct relic density at a given DM mass above a certain minimal mass and below the resonance. Note however, that in some cases Λ i may become even smaller than m χ and the EFT description may no longer be valid. Moreover, typically large branching fractions of the Higgs into DM are obtained in those cases. Therefore, the region below m h /2 would be excluded by observing a Higgs with SM-like decay branching fractions.
Let us briefly mention constraints from indirect detection. In the case of pure scalar interactions DM annihilations are ν 2 suppressed. In the early universe at freeze-out ν 2 1/20, whereas in the galactic halo we have ν 2 ∼ 10 −6 . This leads to a negligible signal for indirect detection experiments. As soon as the pseudo-scalar coupling becomes comparable to the scalar one, the annihilation cross section is dominated by pseudo-scalar interactions with σν independent of the velocity, see eq. (4). Therefore, in the latter case, the annihilation cross section will be the "thermal" one with σν 3 × 10 −26 cm 3 s −1 . Current data from FERMI-LAT and BESS-Polar II disfavour such cross sections for m χ 30 GeV [31,32]. Since here we are restricted to DM masses m χ 60 GeV current limits from indirect detection do not constrain this model.
Let us mention that monojet searches for dark matter at colliders [33,34] will apriori not constrain further this EFT model of dark matter. Limits on spin-independent interactions from recent LHC data [35,36] are much weaker than Xenon bounds in the region of interest.
3 Beyond the EFT framework 3

.1 The toy model
Now we move to situations which cannot be described by the EFT. In order to illustrate when it is possible to have a viable fermionic DM Higgs portal we consider a simple UV completion by introducing a real scalar singlet that will act as mediator particle 1 . For simplicity we consider a Majorana fermion χ as DM, with χ = χ L + χ c L in 4-component notation. (All our arguments will equally apply to the Dirac case.) We denote the SM Higgs doublet by H and the real singlet scalar by ϕ. The relevant terms in the Lagrangian are Here D µ is the SM gauge-covariant derivative, V (ϕ, H) is the Higgs potential, and we allow the coupling constant g and the mass parameter µ χ to be complex. Let us work in unitary gauge and expand H and ϕ around their VEVs: where v 1 = 246 GeV. By performing a phase transformation χ L → e iα/2 χ L with α = Arg(µ χ + gv 2 ) we find that the physical mass of χ corresponds to m χ = |µ χ + gv 2 |. The phase of g relative to the mass term determines the scalar (S) or pseudo-scalar (P ) nature of the Yukawa coupling: A non-zero value of g P violates parity. The mass term and the interaction terms for the Majorana fermion χ become thus: As discussed in the previous section the pseudo-scalar coupling leads to suppressed rates in direct detection. Therefore, it is always possible to consider the situation of g S g P in order to reconcile the annihilation cross section required for the relic density with stringent bounds on the DM-nucleon scattering cross section. In the following we discuss alternative ways to achieve this goal, and therefore we assume in this section parity conservation, g P = 0, keeping always in mind the possibility of parity violation on top of the mechanisms discussed here.
The Higgs potential is where the λ 4 and µ terms provide the Higgs portal between the dark and SM sectors. In order to keep the expressions simple we set in the following always µ 1 = µ 3 = 0. Those terms will not introduce new physical effects and therefore all features relevant for our discussion can be captured within this restricted framework.
In general mixing between h and φ will be induced, with physical mass states H 1 and H 2 and a mixing angle α with We adopt the convention that for α → 0, H 1 corresponds to h. Hence, for small mixing and m H 1 = 125 GeV, H 1 ≈ h becomes a SM-like Higgs. All direct processes coupling χ to the SM are proportional to sin 2 2α and therefore the mixing angle plays a crucial role for DM signals.

Direct detection
DM scattering on nuclei relevant for direct detection is mediated via t-channel exchange of the Higgs mass eigenstates H 1 and H 2 . Hence, scattering is spin-independent. The elastic scattering cross section σ p of χ off a proton p is obtained as The typical size of the scattering cross sections is This number has to be compared to the limit from XENON100, which is σ p 10 −44 cm 2 for m χ 50 GeV [19]. Hence, couplings of order one and large mixing are in tension with the bound. In eq. (18) we take into account only the scalar coupling g S . Similar to the EFT case discussed above, for pseudo-scalar interactions the cross section is suppressed by ν 2 ∼ 10 −6 .

LHC Higgs signatures
In order to define a SM-like Higgs h with m H 1 = 125 GeV, we will use the notion of signal strength reduction factor in the event number of a specific final state of the Standard Model, X, in the Higgs boson decay, see e.g. [15,28,39]. The latter is defined as: where α denotes the Higgs mixing angle, Γ SM H i is the total decay width of a SM Higgs of mass m h = m H i and Γ H i is the total decay width of H i including the decay into H j =i and χ. In order to have a SM-like Higgs we require small mixing α and identify H 1 with the SM Higgs h with m H 1 = 125 GeV. In practice, we will require that r 1 > 0.9 and r 2 < 0.1. The latter constraint is imposed to respect the fact that no indication of a second Higgslike particle is seen at LHC. In the model under consideration typically requiring r 1 > 0.9 automatically leads to r 2 < 0.1. Note that eq. (21) is independent of the Higgs decay channel X. Therefore, we can compare r i directly with the ATLAS/CMS results on the signal strength reduction factor obtained from a combination of all search channels.

Numerical results
We have performed a numerical scan over the parameters of this model using micrOMEGAs [29,30]. We assume m H 1 = 125 GeV and set g P = 0. Then we scan randomly over m χ , g S , v 2 , µ, λ 4 , and m H 2 or λ φ as free parameters. In order to ensure perturbativity, we impose that the absolute value of the couplings λ 4 , λ φ , λ H and g S are smaller than π. For the scalar potential to be bounded from below, we imposed λ φ , λ H > 0 and λ 4 > −2 λ φ λ H . We also assume that χ is the only dark matter candidate that gives rise to a relic density 0.09 < Ωh 2 < 0.13 obtained by thermal freeze-out. If not mentioned otherwise, we scanned the following range of parameters: 5 GeV≤ m H 2 , m χ ≤ 10 4 GeV, 10 −4 GeV ≤ |µ|, v 2 ≤ 10 4 GeV, and 10 −5 ≤ |λ 4 |, |g S | ≤ π. We identify two viable parity conserving Higgs portals.

Resonant Higgs portal
We first assume m H 1 m H 2 and fix m H 2 = 2000 GeV. Requiring the correct relic abundance we show the predicted direct detection scattering cross section in fig. 2 compared to the bound from XENON100. For DM masses m χ 500 GeV the mediator mass is still "heavy" and we recover roughly the EFT behaviour from fig. 1. However, we clearly observe the suppression of the direct detection rate when m χ ≈ m H 1 /2 or m H 2 /2, where there is an s-channel resonance for annihilations, allowing for small coupling constants while maintaining the correct relic abundance. The red dots in fig. 2 correspond to a signal strength modifier for the Higgs signal at LHC of r 1 < 0.9. Hence, those points would be excluded by confirming a SM-like Higgs at 125 GeV, while for the green points we have r 1 > 0.9, showing that close to the resonances we can easily have parity conserving fermionic Higgs portal DM consistent with a SM-like Higgs.

Indirect Higgs portal
Let us now discuss the region m χ > m H 2 = 2 TeV in fig. 2. In this case annihilation of χ into the mediator becomes kinematically allowed. There are t-and u-channel diagrams contributing to this annihilation channel, which are independent of the mixing angle α with an LHC Higgs signal strength modifier r 1 > 0.9, while the red points have r 1 < 0.9. The points above the blue line are excluded at 95% CL by the XENON100 experiment [19]. This exclusion limit has been extended for m χ > 1 TeV assuming a linear dependence in m χ . and only depend on the coupling g S . Assuming pure scalar coupling and m H 1,2 m χ we find where ν is the χ velocity in the CMS. The relic density is obtained when the reaction χχ ↔ φφ freezes out. (Note that for small mixing we have φ ∼ H 2 .) This fixes essentially the coupling g S , while leaving the Higgs mixing α unconstrained. Since the direct detection cross section is proportional to sin 2 (2α), essentially any value for σ p below the XENON100 bound can be obtained 2 , as confirmed in fig. 2 for m χ > 2 TeV. We study this situation in more detail in the following. For m H 1,2 < m χ the exchange of light scalar fields H 1,2 between the two annihilating dark matter particles creates a long range attractive potential (long range compared to 2 At 1-loop DM-nucleus scattering is induced also for zero Higgs mixing, if λ 4 = 0, giving a lower bound on the scattering cross section. The Wilson coefficient in eq. (8) is in this case  with m H1 = 125 GeV and g P = 0. Green and red points correspond to m H2 < m χ with a more (r 1 > 0. 9) or less (r 1 < 0.9) SM Higgs-like H 1 , respectively. We show the scalar coupling g S as a function of the dark matter mass without (left) and with (right) Sommerfeld enhancement for the relic density computation.
For illustration, we also show the points with m H2 > m χ (blue points).
the Compton wavelength of χ). As a result there is a Sommerfeld enhancement of the dark matter annihilation cross-section [40]. This velocity dependent effect has been studied in detail in several references, see e.g. [41][42][43] (see also [44] for a very similar framework). In the calculations of the dark matter relic density we estimate the Sommerfeld enhancement averaged over a thermal distribution at freeze-out temperature T f following [45]. We assume that the cross section determining the dark matter relic abundance is p-wave suppressed. The thermally averaged Sommerfeld factorsS Φ (x f ) due to Φ = H 1 and H 2 exchanges are functions of the couplings α H 1 = (g S sin α) 2 /(4π) and α H 2 = (g S cos α) 2 /(4π), respectively, and of the dimensionless parameters Φ = m χ /(α Φ m Φ ) and x f = m χ /T f (all in the notation of [45]). Let us emphasize that in our toy model dark matter does couple to two mediators, in which case the computation of the exact Sommerfeld factor is more involved [46] than the results in [45]. In most of the cases considered here, only one of the two scalars leads to a non-negligible Sommerfeld correctionS and the relic density is taken to be Ω χ h 2 ∝ 1/(S σv ), where the thermal averaged annihilation cross-section σv is obtained with micrOMEGAs. If both H 1 and H 2 lead to a non negligible thermally averaged Sommerfeld factor, thenS is taken to be the largest of the two 3 .
For masses m χ 100 GeV, no Sommerfeld enhancement of the thermal averaged annihilation cross-section is observed. For m χ 100 GeV,S H 2 can become larger than one and take values up to 4.5. Above m χ ∼ 1 TeV, we observe values ofS H 1 ≥ 1 going with m H1 = 125 GeV and g P = 0. Green and red points correspond to m H2 < m χ with a more (r 1 > 0. 9) or less (r 1 < 0.9) SM Higgs-like H 1 , respectively. We show the DM-proton scattering cross section as a function of the dark matter mass for m H2 < m χ only. The points above the blue line are excluded at 95% CL by the XENON100 experiment [19]. This exclusion limit has been extended for m χ > 1 TeV assuming a linear dependence in m χ . up to 2. The main impact of the Sommerfeld enhancement is to allow for smaller values of the couplings g S at a given mass m χ in order to account for the correct relic density. This is illustrated in fig. 3 where we show the DM coupling to the scalar singlet, g S , as a function of the DM mass with and without Sommerfeld enhancement. For the case m H 2 < m χ (red and green points) we observe a clear correlation consistent with g 2 S ∝ m χ . This is expected when the relic density is driven by the process χχ ↔ φφ according to eq. (22). Also notice the relative flattening of the g 2 S − m χ correlation for m χ in the right panel of fig. 3. This is due to the presence of the Sommerfeld enhancement factor allowing for smaller coupling at a given value m χ in order to still be consistent with WMAP data.
In fig. 4, we show the direct detection cross section requiring m H 2 < m χ . We see that the χ-nucleon scattering cross section can have vastly varying values. It can be well below the XENON100 bound, while still accounting for the correct relic density. This confirms that the size of the DM annihilation cross section (which controls the relic density) is no longer related to the strength of DM interactions with the SM. In fig. 4 the green points have r 1 > 0.9, which means that H 1 will look like a SM Higgs at the LHC, while the red points have a suppressed Higgs signal, r 1 < 0.9. We observe that the r 1 > 0.9 requirement tends to keep the scattering cross section below the XENON100 limit.
In the relic density calculation we have assumed that the thermal bath of χ and φ has the same temperature as the SM thermal bath. The contact between those two sectors is provided by the Higgs portal λ 4 ϕ 2 (H † H), providing interactions like φφ ↔ hh, φ ↔ hh, φφ ↔ h. If those interactions freeze out before χ decouples from φ, in principle the dark and visible sectors may acquire different temperatures due to a change in the number of relativistic degrees of freedom in the visible sector. Unless both, λ 4 and α, are extremely tiny, this may change the relic abundance by factors of order one compared to the situation presented above, while maintaining the qualitative picture. Various possibilities to obtain the relic abundance for various cases of DM and mediator properties have been discussed recently in [47]. Note that as long as the scalar mixing angle α is not exactly zero, H 2 is not stable and decays via the Higgs h into SM particles.
Hence, in this situation the Higgs portal acts indirectly, providing the link between the dark and visible thermal baths in the early universe. We call this "indirect Higgs portal". This situation is similar to secluded DM models [48], where DM annihilations into light mediator particles have been discussed, see also [49,50]. A particular version of the indirect Higgs portal has been obtained in the model from [51]. That model respects a global U (1) symmetry with a complex mediator φ, and the relic density may be set by the annihilation of DM into the massless Goldstone boson from the spontaneous breaking of the U (1).
Let us mention that the inclusion of Sommerfeld enhancement in the computation of the DM relic density does not change qualitatively the general picture presented here. With respect to indirect detection searches, notice that the Sommerfeld correction is a velocity dependent effect that becomes larger when smaller velocities are involved. The Sommerfeld enhancement that affect dark matter annihilations in the galactic halo (v ∼ 10 −6 ) is larger thanS by several orders of magnitude. The annihilation cross sections involved in the scalar case are however always p-wave suppressed and we have checked that they for the model under study, they stay unconstrained by indirect detection searches.

Conclusions
Motivated by recent hints from LHC experiments for a SM-like Higgs particle around 125 GeV we have revisited here the possibility that the operator (H † H) acts as a portal between the SM and the dark sector. We adopt the assumption that DM is a fermion, which necessarily requires additional degrees of freedom to couple it to the Higgs portal. We consider configurations where those additional particles are heavy and an EFT description is possible, as well as situations with light mediators. In the latter case we adopt a simple renormalizable toy model where a real scalar φ plays the role of the mediator particle. Assuming further that the DM relic abundance is obtained by thermal freeze-out in the early universe, the most simple realization of the fermionic Higgs portal DM is under pressure from constraints on the DM-nucleon scattering cross section from XENON100.
We have identified three simple ways to make thermal fermionic DM consistent with a SM-like Higgs at 125 GeV and XENON100 bounds: • Pseudo-scalar Higgs portal. If DM couples to the Higgs portal viaχγ 5 χ the direct detection cross section is suppressed by the DM velocity ν 2 ∼ 10 −6 , whereas the annihilation cross section responsible for the relic abundance is unsuppressed.
• Resonant Higgs portal. If the DM mass m χ is close to half of the Higgs mass m h or the mediator mass m φ , then the annihilation cross section is enhanced by an schannel resonance, allowing for small couplings and a suppressed direct detection cross section.
• Indirect Higgs portal. If the mediator φ is lighter than the DM χ, the relic density can be obtained by χχ ↔ φφ annihilations, where the t-and u-channel diagrams are independent of the Higgs portal strength. The Higgs portal only acts indirectly to provide thermal contact between the dark and the visible sector thermal baths.
In all cases it is possible to have a SM-like Higgs, with an LHC signal strength modifier r 1 > 0.9 (where r 1 = 1 corresponds to the SM Higgs). This framework is sometimes called "LHC nightmare scenario", with no new-physics signal at LHC apart from a SM-like Higgs. Also, by construction, the models discussed here can have unobservably small signals in direct detection experiments. However, in general a signal can be expected for indirect detection. For the pseudo-scalar and the indirect Higgs portals we predict a conventional indirect detection signal (dominated by annihilations intobb or gauge bosons), with an annihilation cross section determined by the thermal freeze-out of σν 3 × 10 −26 cm 3 s −1 .
In the case of resonant Higgs portal there might be also an enhancement of the annihilation cross section today compared to the one in the early universe [52,53] if the resonance is combined with a pseudo-scalar coupling. However, the enhancement effect may be not enough to overcome the velocity suppressed annihilation rate for pure scalar couplings.
In conclusion, fermionic Higgs portal DM remains a viable option if a SM-like Higgs should be established at the currently hinted mass of around 125 GeV. We have outlined simple mechanisms to obtain a classic "WIMP" DM candidate, whose relic abundance is set by thermal freeze-out, with no DM related signal at the LHC and highly suppressed rates in direct detection experiments, but still potentially observable in indirect detection.