Testing Higgs portal dark matter via $Z$ fusion at a linear collider

We investigate the possibility of detecting dark matter at TeV scale linear colliders in the scenario where the dark matter is a massive particle weakly interacting only with the Higgs boson $h$ in the low energy effective theory (the Higgs portal dark matter scenario). The dark matter in this scenario would be difficult to be tested at the CERN Large Hadron Collider when the decay of the Higgs boson into a dark matter pair is not kinematically allowed. We study whether even in such a case the dark matter $D$ can be explored or not via the $Z$ boson fusion process at the International Linear Collider and also at a multi TeV lepton collider. It is found that for the collision energy $\sqrt{S}>1$ TeV with the integrated luminosity 1 ab$^{-1}$, the signal ($e^{\pm}e^-\to e^{\pm}e^-h^\ast \to e^{\pm}e^-DD$) can be seen after appropriate kinematic cuts. In particular, when the dark matter is a fermion or a vector, which is supposed to be singlet under the standard gauge symmetry, the signal with the mass up to 100 GeV can be tested for the Higgs boson mass to be 120 GeV.


Introduction
Dark Matter is one of the biggest mysteries in present physics and astronomy. It has been established that more than one fifth of the energy density in our Universe is occupied by dark matter [1]. If the essence of the dark matter is a kind of particle, it must be electrically neutral and must be weakly interacting. As it has turned out that neutrinos cannot be the candidate, the dark matter should necessarily be a new massive content in physics beyond the standard model (SM). A plausible candidate for the dark matter is therefore a weakly interacting massive particle (WIMP). According to the WMAP experiment [1], the mass of the WIMP dark matter is at the TeV scale or less. Various direct and indirect dark matter search experiments are currently being performed [2]- [6] and planned [7]- [9]. Moreover, we may be able to directly produce the dark matter and to test it at collider experiments such as the CERN Large Hadron Collider (LHC) and future linear colliders.
The fact that the mass scale of the WIMP dark matter is similar to that of the electroweak symmetry breaking would indicate that there is a connection between the Higgs boson and the dark matter. There are many new physics models involving a dark matter candidate. In some of them, it can happen that the dark matter couples only to the Higgs boson in the low energy effective theory, where stability of the dark matter is guaranteed by an unbroken discrete symmetry [10]- [15]. Such a scenario is often called the Higgs portal dark matter scenario [13].
In the scenario of the Higgs portal dark matter, a collider signal at the LHC is expected to come from the W boson fusion process pp → jjW * W * → jjh * → jjDD, where D represents the Higgs portal dark matter whose spin is either 0, 1/2, or 1 [16], while j is a jet originating in an energetic quark. When the mass of D is less than one half of that of h, the invisible decay process h → DD opens, so that the signal would be detectable after appropriate kinematic cuts [17] unless the coupling constant between h and D is too small. On the other hand, if the decay h → DD is not kinematically allowed, the detection of the signal would be hopeless for the dark matter which is consistent with the WMAP and direct detection experiments [16].
In this Letter, we investigate the possibility whether the Higgs portal dark matter can be tested at TeV scale linear colliders such as the International Linear Collider (ILC) [18] and the Compact Linear Collider (CLIC) [19] even in the case where the decay h → DD is not kinematically allowed. In the case of m D < m h /2, the process e + e − → Zh * → ZDD has been studied for the collision with the center of mass energy of √ s = 350 GeV [20]. We here study pair production processes of the dark matter via Z boson fusion from electron-positron (e + e − ) and electron-electron (e − e − ) collisions. It is found that for the collision energy √ s > 1 TeV with the integrated luminosity 1 ab −1 , the signal (e ± e − → e ± e − h * → e ± e − DD) could be seen even for m D > m h /2 after appropriate kinematical cuts, when the mass of D is not much heavier than that of the W boson, especially for the dark matter D being a fermion or a vector.

The model
We here consider the simple model in which a dark matter field is added to the SM.
We impose an unbroken Z 2 parity, under which the dark matter is assigned to be odd while the SM particles are to be even. Stability of the dark matter is guaranteed by the Z 2 parity. We consider three possibilities for the spin of the dark matter; i.e., the real scalar φ, the Majorana fermion χ and the real massive vector V µ .
The Lagrangian for each case of the dark matter is given by where M i (i = S, F and V) are the bare masses of φ, χ and V µ , c i and d i are dimensionless coupling constants, Λ is a dimensionfull parameter, and V µν and B µν are Abelian field strength tensors. The last term in Eq. (2) is expected to be small because this is induced at the one loop level. Hence, we neglect this term in the following analysis. In this case, the dark matter in Eqs. physical mass of each dark matter particle is therefore given by In our analysis, physical masses m i and coupling constants c i are treated as free parameters. Theoretical constraints and experimental bounds from the WMAP data and the direct search results on these models are discussed in Ref. [16].

Dark Matter signals at the e + e − collider
We consider the possibility to detect the dark matter at next generation of electronpositron linear colliders such as the ILC and the CLIC. In particular, we are interested in the case of m h < 2m D , where the Higgs boson cannot decay into a pair of dark matters. We concentrate on the Z boson fusion process e + e − → e + e − Z * Z * → e + e − h * → e + e − DD depicted in Fig. 1. This process can, in principle, be used to detect the dark matter by measuring the outgoing electron and positron in the final state and by using the energy momentum conservation.
We impose the polarization for both incident electron and positron beams [18]; where N e −  The cross section of the signal process is the larger as the collision energy √ s increases, and its behavior is ln s as can be seen in Fig. 2, so that the higher collision energy may be more huseful to detect the signal. However, for √ s = 1-5 TeV, the outgoing electron and positron tend to be emitted to forward and backward directions, and the detectability of the leptons near the beam line is therefore essentially important. In this paper, we assume the detectable area as [21] | cos θ| < 0.9999416, where θ is the scattering angle. Assuming the situation that the Higgs boson mass is already known, we impose the condition for the missing invariant mass M inv as in order to discuss the detection of the dark matter in the case m D > m h /2. The production cross sections of the signal process for D = φ, χ and V at the center of mass energy 1 TeV and 5 TeV are shown for m h = 120 GeV in Table 1.

Parton level signal and background
Backgrounds against the signal process are all the process with the final state of e + e − with a missing momentum. The main background processes are those with the final state e + e − ν e ν e , e + e − ν µ ν µ and e + e − ν τ ν τ . After the basic cuts given in Eqs. (8) and (9), the cross sections for e + e − → e + e − ν e ν e and e + e − → e + e − ν i ν i (i = µ or τ ) are evaluated as 1.15 × 10 −1 pb and 8.87 × 10 −4 pb at √ s = 1 TeV, while they are 1.48 × 10 −1 pb and 3.74 × 10 −4 pb at √ s = 5 TeV, respectively: see Table 1. The signal to background ratio amounts to 10 −3 -10 −1 for the coupling constants being taken as c S = c V = 1 and c F /Λ = 0.1 GeV −1 . In order to gain the signal significance we impose kinematical cuts as follows.
First, as seen in the upper panel of Fig. 3, the signal events tend to be located with lower values of the missing energy E inv , while the backgrounds are distributed with larger values. We therefore impose the cut on E inv as By using this cut, the backgrounds from e + e − ν e ν e can be reduced.
Second, as seen in the middle panel of Fig. 3 Finally, the distributions of the azimuthal angle φ ee between outgoing electron and positron are shown for the signal and the background processes in the lower panel of Fig. 3. As can be seen in the figure, the signal is insensitive to the azimuthal angle, while most of the background events are located in the region with relatively large values of the azimuthal angle. We therefore impose the cut on φ ee as by which a considerable amount of the backgrounds can be eliminated.
In Table 1, the event numbers are shown for both the signal and the backgrounds after imposing the above kinematical cuts in Eqs.    result, the significance to detect the signal, which is defined by  5 We are assuming that the scattering cross section is determined only by the diagram in which the Higgs boson is exchanged, Namely, it is not interfered by other diagrams. Furthermore, the detection rate at the direct search has some ambiguities from the hadron matrix element, the dark matter density in the solar system, and the velocity distribution of the dark matter in our galaxy.
Those curves should be therefore regarded as a reference.
First, in Fig. 4(a), the results for the scalar dark matter are shown. There is no overlap between the region of N S / √ N S + N B > 3 and that satisfying the WMAP data even at √ s = 5 TeV. Second, in Fig. 4

Conclusions
We have investigated the possibility of detecting dark matter at TeV scale linear colliders in the Higgs portal dark matter scenario with the scalar, fermion or vector dark matter, via Z boson fusion processes at electron-positron and electron-electron collisions. We have found that a multi-TeV collider can be more useful to explore the dark matter in these models than the 1 TeV collider when the invisible decay of the Higgs boson into a pair of dark matters is kinematically forbidden.