Implications of LHC searches for Higgs--portal dark matter

The search for the a Standard Model Higgs boson at the LHC is reaching a critical stage as the possible mass range for the particle has become extremely narrow and some signal at a mass of about 125 GeV is starting to emerge. We study the implications of these LHC Higgs searches for Higgs portal models of dark matter in a rather model independent way. Their impact on the cosmological relic density and on the direct detection rates are studied in the context of generic scalar, vector and fermionic thermal dark matter particles. Assuming a sufficiently small invisible Higgs decay branching ratio, we find that current data, in particular from the XENON experiment, essentially exclude fermionic dark matter as well as light, i.e. with masses below 50 GeV, scalar and vector dark matter particles. Possible observation of these particles at the planned upgrade of the XENON experiment as well in collider searches is discussed.


INTRODUCTION
The ATLAS and CMS collaborations have recently reported on the search of the Standard Model (SM) Higgs boson with 5 fb −1 data [1]. Higgs bosons have been excluded in a significant mass range and, ignoring the unlikely possibility of a very heavy particle, only the very narrow window m h ≈ 115-130 GeV is now left over. There is even a slight excess of events in the data which could correspond to a SM like Higgs boson with a mass of 125 ± 1 GeV. Although the statistics are not sufficient for the experiments to claim discovery, one is tempted to take this piece of evidence seriously and analyze its consequences.
In this Letter, we study the implications of these LHC results for Higgs-portal models of dark matter (DM). The Higgs sector of the SM enjoys a special status since it allows for a direct coupling to the hidden sector that is renormalizable. Hence, determination of the properties of the Higgs boson would allow us to gain information about the hidden world. The latter is particularly important in the context of dark matter since hidden sector particles can be stable and couple very weakly to the SM sector, thereby offering a viable dark matter candidate [2]. In principle, the Higgs boson could decay into light DM particles which escape detection [3]. However, given the fact that the ATLAS and CMS signal is close to what one expects for a Standard Model-like Higgs particle, there is little room for invisible decays. In what follows, we will assume that 10% is the upper bound on the invisible Higgs decay branching ratio, although values up to 20% will not significantly change our conclusions. * abdelhak.djouadi@th.u-psud.fr † lebedev@mail.desy.de ‡ yann.mambrini@th.u-psud.fr § jeremie.quevillon@th.u-psud.fr We adopt a model independent approach and study generic scenarios in which the Higgs-portal DM is a scalar, a vector or a Majorana fermion. We first discuss the available constraints on the thermal DM from WMAP and current direct detection experiments, and show that the fermionic DM case is excluded while in the scalar and vector cases, one needs DM particles that are heavier than about 60 GeV. We then derive the direct DM detection rates to be probed by the XENON100upgrade and XENON1T experiments. Finally, we discuss the possibility of observing directly or indirectly these DM particles in collider experiments and, in particular, we determine the rate for the pair production of scalar particles at the LHC and a high-energy e + e − collider.

THE MODELS
Following the model independent approach of Ref. [4], we consider the three possibilities that dark matter consists of real scalars S, vectors V or Majorana fermions χ which interact with the SM fields only through the Higgs-portal. The stability of the DM particle is ensured by a Z 2 parity, whose origin is model-dependent. For example, in the vector case it stems from a natural parity symmetry of abelian gauge sectors with minimal field content [5]. The relevant terms in the Lagrangians are Although in the fermionic case above the Higgs-DM coupling is not renormalizable, we still include it for completeness. The self-interaction terms S 4 in the scalar case and the (V µ V µ ) 2 term in the vector case are not essential for our discussion and we will ignore them. After electroweak symmetry breaking, the neutral component of the doublet field H is shifted to H 0 → v + h/ √ 2 with v = 174 GeV and the physical masses of the DM particles will be given by In what follows, we summarize the most important formulas relevant to our study. Related ideas and analyses can be found in [6][7][8][9] and more recent studies of Higgsportal scenarios have appeared in [10,11].
The relic abundance of the DM particles is obtained through the s-channel annihilation via the exchange of the Higgs boson. For instance, the annihilation cross section into light fermions of mass m ferm is given by where v r is the DM relative velocity. (The cross section for Majorana fermion annihilation was computed in [12] in a similar framework.) We should note that in our numerical analysis, we take into account the full set of relevant diagrams and channels, and we have adapted the program micrOMEGAs [13] to calculate the relic DM density.
The properties of the dark matter particles can be studied in direct detection experiments. The DM interacts elastically with nuclei through the Higgs boson exchange. The resulting nuclear recoil is then interpreted in terms of the DM mass and DM-nucleon cross section. The spin-independent DM-nucleon interaction can be expressed as [4] where m N is the nucleon mass and f N parameterizes the Higgs-nucleon coupling. The latter subsumes contributions of the light quarks (f L ) and heavy quarks (f H ), There exist different estimations of this factor and in what follows we will use the lattice result f N = 0.326 [14] as well as the MILC results [15] which provide the minimal value f N = 0.260 and the maximal value f N = 0.629. We note that the most recent lattice evaluation of the strangeness content of the nucleon [16] favors f N values closer to the lower end of the above range. In our numerical analysis, we have taken into account these lattice results, which appear more reliable than those extracted from the pion-nucleon cross section.
If the DM particles are light enough, M DM ≤ 1 2 m h , they will appear as invisible decay products of the Higgs boson. For the various cases, the Higgs partial decay widths into invisible DM particles are given by where We have adapted the program HDECAY [17] which calculates all Higgs decay widths and branching ratios to include invisible decays.

ASTROPHYSICAL CONSEQUENCES
The first aim of our study is to derive constraints on the various DM particles from the WMAP satellite [18] and from the current direct detection experiment XENON100 [19], and to make predictions for future upgrades of the latter experiment, assuming that the Higgs boson has a mass m h = 125 GeV and is approximately SM-like such that its invisible decay branching ratio is smaller than 10%; we have checked that increasing this fraction to 20% does not change our results significantly.
In Fig. 1, we delineate the viable parameter space for the Higgs-portal scalar DM particle. The area between the two solid (red) curves satisfies the WMAP constraint, with the dip corresponding to resonant DM annihilation mediated by the Higgs exchange. We display three versions of the XENON100 direct DM detection bound corresponding to the three values of f N discussed above. The dash-dotted (brown) curve around the Higgs pole region represents BR inv = 10% such that the area to the left of this line is excluded by our constraint BR inv < 10%. The prospects for the upgrade of XENON100 (with a projected sensitivity corresponding to 60,000 kg-d, 5-30 keV and 45% efficiency) and XENON1T are shown by the dotted lines.
We find that light dark matter, M DM < ∼ 60 GeV, violates the bound on the invisible Higgs decay branching ratio and thus is excluded. This applies in particular to the case of scalar DM with a mass of 5-10 GeV considered, for instance, in Ref. [8]. On the other hand, heavier dark matter, particularly for M DM > ∼ 80 GeV, is allowed by both BR inv and XENON100. We note that almost the entire available parameter space will be probed by the XENON100 upgrade. The exception is a small resonant region around 62 GeV, where the Higgs-DM coupling is extremely small.
In the case of vector Higgs-portal DM, the results are shown in Fig. 2 and are quite similar to the scalar case. WMAP requires the Higgs-DM coupling to be almost twice as large as that in the scalar case. This is because only opposite polarization states can annihilate through the Higgs channel, which reduces the annihilation cross section by a factor of 3. The resulting direct detection rates are therefore somewhat higher in the vector case. Note that for DM masses below m h /2, only very small values λ hV V < O(10 −2 ) are allowed if BR inv < 10%.
Similarly, the fermion Higgs-portal results are shown in Fig. 3. We find no parameter regions satisfying the constraints, most notably the XENON100 bound, and this scenario is thus ruled out for λ hf f /Λ > ∼ 10 −3 .
This can also be seen from Fig. 4, which displays predictions for the spin-independent DM-nucleon cross section σ SI (based on the lattice f N ) subject to the WMAP and BR inv < 10% bounds. The upper band corresponds to the fermion Higgs-portal DM and is excluded by XENON100. On the other hand, scalar and vector DM are both allowed for a wide range of masses. Apart from a very small region around 1 2 m h , this parameter space will be probed by XENON100-upgrade and XENON1T. The typical value for the scalar σ SI is a few times 10 −9 pb, whereas σ SI for vectors is larger by a factor of 3 which accounts for the number of degrees of freedom.

DARK MATTER PRODUCTION AT COLLIDERS
The next issue to discuss is how to observe directly the Higgs-portal DM particles at high energy colliders. There are essentially two ways, depending on the Higgs versus DM particle masses. If the DM particles are light enough for the invisible Higgs decay to occur, M DM < ∼ 1 2 m h , the Higgs cross sections times the branching ratios for the visible decays will be altered, providing indirect evidence for the invisible decay channel. In the case of the LHC, a detailed analysis of this issue has been performed in Ref. [7] for instance and we have little to add to it. Nevertheless, if the invisible Higgs branching ratio is smaller than ≈ 10%, its observation would be extremely difficult in view of the large QCD uncertainties that affect the Higgs production cross sections, in particular in the main production channel, the gluon fusion mechanism gg → h  (2) to vector DM and the lower one (1) to scalar DM. The solid, dashed and dotted lines represent XENON100, XENON100 upgrade and XENON1T sensitivities, respectively. [20]. In fact, the chances of observing indirectly the invisible Higgs decays are much better at a future e + e − collider. Indeed, it has been shown that, at √ s ≈ 500 GeV collider with 100 fb −1 data, the Higgs production cross sections times the visible decay branching fractions can be determined at the percent level [21,22].
The DM particles could be observed directly by studying associated Higgs production with a vector boson and Higgs production in vector boson fusion with the Higgs particle decaying invisibly. At the LHC, parton level analyses have shown that, although extremely difficult, this channel can be probed at the 14 TeV upgrade with a sufficiently large amount of data [23] if the fraction of invisible decays is significant. A more sophisticated AT-LAS analysis has shown that only for branching ratios above 30% that a signal can be observed at √ s = 14 TeV and 10 fb −1 data in the mass range m h = 100-250 GeV [24]. Again, at a 500 GeV e + e − collider, invisible decays at the level of a few percent can be observed in the process e + e − → hZ by simply analyzing the recoil of the leptonically decaying Z boson [21,22].
If the DM particles are heavy, M DM > ∼ 1 2 m h , the situation becomes much more difficult and the only possibility to observe them would be via their pair production in the continuum through the s-channel exchange of the Higgs boson. At the LHC, taking the example of the scalar DM particle S, three main processes can be used: a) double production with Higgs-strahlung from either a W or a Z boson, qq → V * → V SS with V = W or Z, b) the W W/ZZ fusion processes which lead to two jets and missing energy qq → V * V * qq → SSqq and c) the gluon-gluon fusion mechanism which is mainly mediated by loops of the heavy top quark that couples strongly to the Higgs boson, gg → h * → SS.
The third process, gg → SS, leads to only invisible particles in the final state, unless some additional jets from higher order contributions are present and reduce the cross section [25] and we will ignore it here. For the two first processes, following Ref. [26] in which double Higgs production in the SM and its minimal supersymmetric extension has been analyzed, we have calculated the production cross sections. The exact matrix elements have been used in the qq → ZSS, W SS processes while in vector boson fusion, we have used the longitudinal vector boson approximations and specialized to the W L W L + Z L Z L → SS case which is expected to provide larger rates at the highest energy available at the LHC i.e. √ s = 14 TeV (the result obtained in this way is expected to approximate the exact result within about a factor of two for low scalar masses and very high energies); the analytical expressions are given in the Appendix.
As can be seen from Fig. ?? where the cross sections are shown as a function of M DM for λ hSS = 1, the rates at √ s = 14 TeV are at the level of 10 fb in the W W + ZZ → SS process for M h < ∼ 120 GeV and one order of magnitude smaller for associated production with W and Z bosons. Thus, for both processes, even before selection cuts are applied to suppress the backgrounds, the rates are small for DM masses of order 100 GeV and will require extremely high luminosities to be observed.
Again, the chances of observing DM pair production in the continuum might be higher in the cleaner environment of e + e − collisions. The two most important production processes in this context, taking again the example of a scalar DM particle, are e + e − → ZSS that dominates at relatively low energies and e + e − → Z * Z * e + e − → e + e − SS which becomes important at high energies. The rate for W W fusion is one order of magnitude larger but it leads to a fully invisible signal, e + e − → W * W * νν → ννSS. Following again Ref. [26], we have evaluated the cross sections for e + e − → ZSS at √ s = 500 GeV (the energy range relevant for the ILC) and for Z L Z L → SS at √ s = 3 TeV (relevant for the CERN CLIC) and the results are shown in Fig.6 as a function of the mass M S for λ hSS = 1. One observes that the maximal rate that one can obtain is about 10 fb near the Higgs pole in ZSS production and which drops quickly with increasing M S . The process ZZ → SS becomes dominant for M S > ∼ 100 GeV, but the rates are extremely low, below ≈ 0.1 fb.
The situation should be similar in the case of vector and fermion DM and we refrain from discussing it here. CONCLUSION We have analyzed the implications of the recent LHC Higgs results for generic Higgs-portal models of scalar, vector and fermionic dark matter particles. Requiring the branching ratio for invisible Higgs decay to be less than 10%, we find that the DM-nucleon cross section for electroweak-size DM masses is predicted to be in the range 10 −9 −10 −8 pb in almost all of the parameter space. Thus, the entire class of Higgs-portal DM models will be probed by the XENON100-upgrade and XENON1T direct detection experiments, which will also be able to discriminate between the vector and scalar cases. The fermion DM is essentially ruled out by the current data, most notably by XENON100. Furthermore, we find that light Higgs-portal DM M DM < ∼ 60 GeV is excluded independently of its nature since it predicts a large invisible Higgs decay branching ratio, which should be incompatible with the production of an SM-like Higgs boson at the LHC. Finally, it will be difficult to observe the DM effects by studying Higgs physics at the LHC. Such studies can be best performed in Higgs decays at the planned e + e − colliders. However, the DM particles have pair production cross sections that are too low to be observed at the LHC and eventually also at future e + e − colliders unless very high luminosities are made available.
The differential cross section for the pair production of two scalar particles in association with a Z boson, e + e − → ZSS, after the angular dependence is integrated out, can be cast into the form (v = 174 GeV): where the electron-Z couplings are defined asâ e = −1 andv e = −1 + 4 sin 2 θ W , x 1,2 = 2E 1,2 / √ s are the scaled energies of the two scalar particles, x 3 = 2 − x 1 − x 2 is the scaled energy of the Z boson; the scaled masses are denoted by µ i = M 2 i /s. In terms of these variables, the coefficient Z may be written as (A.2) The differential cross section has to be integrated over the allowed range of the x 1 , x 2 variables; the boundary condition is For the cross section at hadron colliders, i.e. for the process qq → ZSS one has to divide the amplitude squared given above by a factor 3 to take into account color sum/averaging, replace e by q (with a q = 2I 3 q , v q = 2I 3 q − 4e q sin 2 θ W with I 3 q and e q for isospin and electric charge) and the c.m. energy s by the partonic oneŝ; one has then to fold the obtained partonic cross section with the quark/antiquark luminosities. The extension to the qq → W SS case (with a q = v q = √ 2) is straightforward.
For the vector boson fusion processes, one calculates the cross sections for the 2 → 2 processes V L V L → SS in the equivalent longitudinal vector boson approximation and then fold with the V L spectra to obtain the cross section the entire processes e + e − → SSℓℓ and qq → qqSS; see Ref. [26] for details. Taking into account only the dominant longitudinal vector boson contribution, denoting by β V,S the V, S velocities in the c.m. frame,ŝ 1/2 the invariant energy of the V V pair, the corresponding cross section of the subprocess V L V L → SS readŝ (A.4)