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new physics beyond the standard model ( sm ) has drawn extensive attention since the discovery of the sm higgs boson @xcite . while a few problems such as how to stabilize the higgs mass against ultraviolet radiative corrections are tied to new physics of high mass scale , in this paper
we instead focus on dark matter with a mass near the weak scale .
in contrast to new physics which appears at a rather high mass scale , such a dark matter model has promising prospect for discovery at both astrophysical and particle collider experiments .
in particular , we are interested in the simplest higgs - portal dark matter model , in which the dark matter communicates with sm particles via the higgs scalar . unlike the fermion dark matter setting , a scalar dark matter in the so - called higgs - portal scalar dark matter model ( hsdm ) @xcite still survives the latest data of direct detections at xenon100 @xcite and lux @xcite , indirect detections at fermi - lat @xcite , and higgs invisible decay at the lhc run 1 @xcite .
detailed discussions about this model have been given in the literature ( @xcite-@xcite ) .
fitting the experimental data indicates that the dark matter mass is either near the resonant mass region between @xmath3 gev and @xmath4 gev or in a large - mass region above @xmath5 gev .
while the large - mass region between 185 gev and 3 tev can be probed by the future xenon1 t @xcite , most of the resonant mass region is beyond the reach of this facility . in this paper
, we discuss the collider signatures of the scalar dark matter in the hsdm model with a mass between @xmath3 gev and @xmath4 gev at the 14 tev lhc and the future 100 tev proton collider ( fcc ) .
we will show that similar to circular electron positron collider ( cepc ) @xcite , fcc will be a useful machine for searching dark matter in this narrow mass region .
we will show that for fcc with a luminosity of 10 @xmath6 the exclusion and discovery sensitivities reach to 57 gev and 56 gev respectively through the vector boson fusion ( vbf ) channel , and 54.8 gev and 53.9 gev respectively via the mono-@xmath7 channel .
it indicates that fcc with 10 @xmath6 is a competitive facility in comparison with cepc or xenon1 t .
the remaining parts of the paper are organized as follows . in sec .
ii , we briefly discuss the direct and indirect detection constraints on the hsdm . in sec.iii
we address the collider phenomenologies for the hsdm with dark matter mass in the narrow resonant mass region at the 14 tev lhc and the 100 tev fcc , where we focus on both the vbf channel and mono-@xmath7 channel .
our main results are presented in sec .
iv , where we show the luminosities required for the @xmath1 exclusion and @xmath2 discovery .
finally we conclude in sec . v.
in the simplest hsdm model , the dark matter @xmath8 communicates with the sm particles through the sm higgs scalar .
the lagrangian for this mode reads as @xmath9 where @xmath10 , @xmath11 and @xmath12 are the singlet scalar bare mass , the self - interaction coupling constant , and the coupling constant between dark matter and sm higgs , respectively . a @xmath13 parity , under which @xmath8 is odd and other fields are even , is imposed to make the dm stable , which reduces the number of model parameters .
after the electroweak symmetry breaking one can obtain @xmath14 where @xmath15 is the physical mass of the singlet scalar , and @xmath16 , @xmath17 and @xmath18 gev .    among the three model parameters , the self - interaction coupling @xmath19 does not directly affect the dm relic abundance , the dm - nucleon scattering cross section and dm production cross section at colliders , we simply decouple this parameter from the dm phenomenology discussed below .
it turns out that the remaining two parameters are strongly constrained .
assume that the cold dark matter is saturated by the singlet scalar @xmath8 , @xmath8 should account for the dm relic density measured by the planck and wmap @xcite , @xmath20 from which one infers the correlation between @xmath21 and @xmath12 as shown in fig.[figic ] .
besides the relic abundance in eq.([relic ] ) , there are other indirect constraints , including the higgs invisible decay @xmath22 in the mass region @xmath23 and the @xmath24-ray limits from the fermi - lat @xcite . for the higgs invisible decay , fig.[figic ]
shows the latest limits at the 8 tev lhc @xcite , hl - lhc and cepc @xcite , which indicates that @xmath21 below 52 gev is excluded by the data @xmath25 , while the hl - lhc and cepc can reach 54 gev and 57 gev , respectively .     from the dark matter relic abundance and higgs
invisible decays at the lhc run 1 , hl - lhc and cepc.,scaledwidth=70.0% ]      the direct detection at lux and xenon1 t can further constrain the parameter space , according to the spin - independent dm - nucleon scattering cross section given by @xmath26 where @xmath27 is the nucleon mass , @xmath28 is the dm - nucleon reduced mass , and @xmath29 is the hadron matrix element @xcite .
fig.[figdc ] shows the predicted values of the spin - independent dm - nucleon scattering cross section , together with direct detection limits at xenon100 @xcite and lux @xcite experiments .
the limits at xenon1 t @xcite have been also shown .
it indicates that the dark matter mass is restricted to a narrow resonant region between 53 gev and 63 gev .
once we employ the latest fermi - lat limits @xcite , this narrow mass region is further reduced to a narrow range between 53 gev and 62.5 gev .     from lux and xenon experiments .
the red curve represents the dark matter relic abundance constraint.,scaledwidth=70.0% ]
in this section we study the collider signatures of the scalar dark matter at the 14 tev lhc and 100 tev fcc .
we will explore the sensitivities at these two colliders for the dark matter mass in the narrow resonant region between 53 gev and 62.5 gev .
we consider the dominant vbf channel as well as the sub - leading but relatively clean mon-@xmath7 channel .
we use feynrules @xcite to generate model files prepared for madgraph5 @xcite , which includes pythia 6 @xcite for parton showering and hadronazition , and the package delphes 3 @xcite for fast detector simulation .
in particular , the default cms detector card and the snowmass detector card are used for the 14 tev and 100 tev @xmath30 collider , respectively .
events are generated by using madgraph5 at the leading order with the 125 gev higgs and fixed value @xmath31 for different dark matter masses .
cross sections are reproduced by rescaling @xmath32 which corresponds to @xmath33 .      in the vbf channel ,
the dark matter pairs are produced through the higgs scalar @xmath34 where the higgs @xmath35 should be on - shell in our case .
the primary sm backgrounds to this process include @xmath36jets , @xmath37jets , @xmath38jets and qcd multi - jets . for simplicity
we consider the main contributions arising from @xmath39 and @xmath37jets channels , and adopt the cuts used by the cms vbf analysis @xcite for event selection : @xmath40 where @xmath41 and @xmath42 are the transverse momentum and pseudo - rapidity of the first ( second ) leading jet , respectively .
@xmath43 , @xmath44 and @xmath45 are the rapidity difference , azimuthal - angel difference and invariant mass of the two leading jets , respectively .
any event with an additional jet with @xmath46 gev and pseudo - rapidity between those of the two tagged jets is rejected .
we firstly apply the criteria in eq.([selection1 ] ) to the 8 tev lhc with data of 19.5 @xmath47 .
the number of events for the sm background is 134 in the @xmath36jets channels and 145 in the @xmath37jets channels , respectively .
compared with the reported number of events ( 99 in the @xmath36 jets channels and 183 in the @xmath37 jets channels ) by the cms collaboration @xcite , they are consistent with each other .    the criteria in eq.([selection1 ] )
will be also applied to both the 14 tev lhc and 100 tev fcc simulations for conservation .
it is based on the following facts .
first , there is little difference between the 8 tev lhc and 14 tev lhc except the collision energy , which means the cut on the pseudo - rapidity of the first two leading jets should remain unchanged .
second , the kinetic distribution of the signal events though on - shell higgs decay and the main backgrounds z@xmath48jets and w@xmath48jets have similar changing trends when one modifies the cuts in eq.([selection1 ] ) , as the mass difference between the higgs and @xmath49 boson can be omitted compared to the variation of collision energy .
it turns out that for a benchmark dm mass the effects on the ratio of signal over background events @xmath50 are less than two times due to variations on the cuts in eq.(6 ) such as @xmath51 gev , @xmath52 , @xmath53 gev , @xmath54 and @xmath55 gev .
see table below for details .
.effects on the ratio @xmath50 due to variations on the cuts in eq.([selection1 ] ) for benchmark dm mass @xmath56 gev at 100 tev fcc . [ cols="^,^,^",options="header " , ]      in the mono-@xmath7 channel the dark matter pairs are produced via the process @xmath57 compared with the vbf channel , the mono-@xmath7 channel is sub - leading but relatively cleaner . for event selection in this channel
we adopt the following cuts as suggested by the cms leptonic mode analysis @xcite : @xmath58 where @xmath59 is the dilepton transverse momentum and @xmath60 is defined as the component of @xmath61 parallel to the direction of @xmath62 .
events are rejected if an additional electron or muon is reconstructed with @xmath63 gev , and any event having two or more jets with @xmath64 gev .
similar to the discussions in the preceding section , the criteria in eq.([selection2 ] ) are examined via the 8 tev lhc simulation with data of 19.7 @xmath47 .
the number of events for the sm background is 88 in the mono-@xmath7 channel .
compared with the reported number of events 138 in the same channel by the cms collaboration @xcite , they are consistent with each other . unlike the cms result , the next - leading order ( nlo ) effects have been neglected in our simulation .
the two numbers of events likely match better if the nlo effects are included in our analysis . following the similar facts mentioned in the preceding section , the criteria in eq.([selection2 ] ) will be directly applied to both the 14 tev lhc and 100 tev fcc simulations .
we present our main results in fig.[fig14 ] and fig.[fig100 ] , which show the integrated luminosity @xmath65 needed for exclusion and discovery at the 14 tev lhc and 100 tev fcc , respectively . here , we take the following definition about significance @xmath66 systematic uncertainties are neglected in both the signal and the background simulations .     and 5@xmath67 discovery determined by @xmath68 in vbf and mono-@xmath7 channel at the 14 tev lhc , respectively .
the solid dark line represents the 10 ab@xmath69 integrated luminosity.,scaledwidth=70.0% ]    , but for the 100 tev fcc.,scaledwidth=70.0% ]    in fig.[fig14 ] one observes that for @xmath70 @xmath47 the scalar dark matter mass up to 53.5 gev can be excluded , and for @xmath71 @xmath47 the exclusion and discovery limits via vbf channel will reach to 54.6 gev and 54 gev , respectively .
in contrast , fig.[fig100 ] shows that for @xmath70 @xmath47 the scalar dark matter mass up to 54.5 gev can be excluded , and for @xmath71 @xmath47 the exclusion and discovery limits at the fcc via vbf channel approaches to 55.8 gev and 55 gev , respectively .
the exclusion limits via the mono-@xmath7 channel are obviously weaker in comparison with the vbf channel .    taking an integrated luminosity @xmath72 @xmath6 at the 100 tev fcc
, one finds that the exclusion and discovery limits approach to 57 gev and 56 gev in the vbf channel , respectively .
compared with either cepc or xenon1 t , where the exclusion limits approach to 56.5 gev and 56 gev , respectively , the fcc with @xmath72 @xmath6 is a competitive facility .
fig.[fig14 ] and fig.[fig100 ] also illustrate that it is unlikely to detect the scalar dark matter in the mass range between 57 gev and 62.5 gev in hsdm model at any present and future facilities mentioned in this paper .
in this paper , we explored the collider signatures of the scalar dark matter in the hsdm model .
our study shows that for the 100 tev fcc with an integrated luminosity of 10 @xmath6 , the exclusion and discovery sensitivities reach to 57 gev and 56 gev respectively through the vbf channel , and 54.8 gev and 53.9 gev respectively via the mono-@xmath7 channel .
compared with either cepc or xenon1 t , where the exclusion limits approach to 56.5 gev and 56 gev , respectively , fcc is a competitive facility .
unfortunately , the scalar dark matter in the mass range between 56.5 gev and 62.5 gev is unlikely to be either directly or indirectly detected at any present and future facility discussed in this paper .
in the simplest higgs - portal scalar dark matter model , the dark matter mass has been restricted to be either near the resonant mass ( @xmath0 ) or in a large - mass region by the direct detection at lhc run 1 and lux . while the large - mass region below roughly 3 tev can be probed by the future xenon1 t experiment , 
 most of the resonant mass region is beyond the scope of xenon1 t . in this paper 
 , we study the direct detection of such scalar dark matter in the narrow resonant mass region at the 14 tev lhc and the future 100 tev hadron collider . 
 we show the luminosities required for the @xmath1 exclusion and @xmath2 discovery .