Determining the masses of invisible particles: Application to Higgs boson invisible decay
aa r X i v : . [ h e p - ph ] M a y MITP/13-069
Determining the masses of invisible particles: Application to Higgs boson invisibledecay
Jian Wang ∗ PRISMA Cluster of Excellence & Mainz Institute for Theoretical Physics,Johannes Gutenberg University, D-55099 Mainz, Germany (Dated: September 25, 2018)To know the total width of the recently discovered Higgs boson particle, it is important to measurethe invisible decay width of the Higgs boson. However, the signal for this measurement at the LHC,i.e., a charged lepton pair and missing energy in the final state, cannot be definitely understood asthe product of the intermediate produced Z and H bosons due to the possible interaction betweendark matter and a Z boson or quarks, which can be described by representative effective operators.First, we consider the relic abundance, the LUX direct detection experiment and the result ofsearching for Higgs boson invisible decay at the LEP and LHC to find the allowed parameterregion for the effective operators. Then we investigate the transverse momentum distribution of themissing energy and propose two observables that can be used to distinguish the different underlyingprocesses. Moreover, with these two observables, we may be able to determine the masses of invisibleparticles. PACS numbers: 14.80.Bn, 95.35.+d, 12.38.Qk
INTRODUCTION
It is important to precisely measure the properties ofthe Higgs boson after its discovery [1, 2]. The total widthof the Higgs boson is difficult to measure at a hadroncollider due to the unmeasurable partonic center-of-massenergy. As a result, one can only get information onthe total width from the global fit and sum over vari-ous decay channels of the Higgs boson [3] . The largestdecay channel is H → b ¯ b , which suffers from overwhelm-ing QCD backgrounds. Although it becomes possible todetect H → b ¯ b by using a delicate method proposed inRef. [6] and the ATLAS and CMS collaborations havesearched for the b ¯ b decay in the V ( W, Z ) H associatedproductions, the statistical uncertainties of the resultsare still too large [7, 8]. The decay channel H → gg faces the same problem, but there is no b-tagging tech-nology that can be used in H → b ¯ b . The decay width ofthis channel can only be obtained from a global fit of therelevant Higgs couplings.In addition, there are still possibilities for the Higgsboson decaying to invisible particles [1, 2, 9–16]. Tomeasure the invisible decay width of the Higgs boson atcolliders, it is necessary to consider the associated pro-duction of the Higgs boson with some visible particles.The Z boson is a good choice due to its large coupling tothe Higgs boson, large production rate and clear signa-ture at colliders. Because of the limited center-of-massenergy, the experiment at the LEP has only excluded We notice that a constraint is presented on the total width of theHiggs boson, i.e., Γ H <
22 MeV at the 95% confidence level [4],by using its relative on-shell and off-shell production and decayrates to a pair of Z bosons [5]. the Higgs boson mass range below 114.4 GeV via theHiggs-strahlung process e + e − → HZ [17]. Recently, theATLAS collaboration at the LHC carried out a similarsearch in the HZ associated production with H decay-ing invisibly and the Z boson decaying into a chargedlepton pair. The present result shows no deviation fromthe Standard Model (SM) expectation and constrains theinvisible branching fractions to be less than 65% at the95% confidence level [18]. The CMS collaboration hasobtained a similar result [19].We are interested in the question of, if some deviationfrom the SM expectation is observed in the future as theintegrated luminosity of the LHC is increased, if can wedetermine the invisible decay width of the Higgs boson.In fact, the experimentally observed final state is just acharged lepton pair and missing energy. And the anal-ysis is based on the assumption that a resonance and a Z boson have been associated produced with the crosssection predicted by the SM and the resonance totallydecaying invisibly. It is reasonable that the charged lep-ton pair can be attributed to the decay product of theintermediate Z boson if the invariant mass of the chargedlepton pair is around the Z boson mass M Z . However, itis not very convincible to interpret the missing energy asa decay product of the intermediate Higgs boson becausethere are other possible origins of the missing energy. Forexample, it is likely that some dark matter (DM) can in-teract with the Z boson or quarks, which has been alsostudied extensively in the recent years. They will appearat the LHC with the same signature. Thus, it is essentialto extract more information about the missing energy.In this work, we perform such a study toward this di-rection, focusing on the mass of the missing particles. Ifone can determine the mass of the missing particle, onecan be more confident to judge whether the intermedi- Z ∗ ZSSq ¯ q ( a ) ( b ) q ¯ q Z ∗ ZH SSν ¯ νZq ¯ q Z ( c ) q SS ¯ q Z ( d ) FIG. 1: Leading Feynman diagrams for a Z boson and missingenergy associated production. The diagram ( a ) is the signalprocess in the analysis of the ATLAS experiment and thediagram ( c ) is the main irreducible background. The diagrams( b ) and ( d ) are induced by the effective operators in Eq.(2). ate particle is the Higgs boson or not. The candidatefor the Higgs boson decay products should have a masslower than one-half of the Higgs boson mass. This studyis closely related to searching for new physics. In the R -parity violating supersymmetry model, some sparticleswould decay into SM particles and a neutralino, whichbecomes invisible if it is lighter than the other supersym-metry particles and stable enough [20]. In the large extradimension model, the Kaluza-Klein graviton can escapefrom the detection at colliders, manifesting itself as miss-ing energy [21]. Cosmology observation has confirmedthe existence of DM in our Universe [22]. It can be pro-duced at colliders if it is light and has interactions withSM particles. Because it is stable, it is not detectable atcolliders [23]. Searching for all these kinds of new physicsrequires analyzing the events with missing energy and re-coiling particles, e.g., a photon [24–28], a lepton [29], ajet [30, 31], a W/Z boson [32–34], or a top quark [35, 36].Therefore, it would be helpful to distinguish them if wecan know more about the missing energy. In this paper,we focus on the case of missing energy and a Z bosonassociated production, which is important to determinethe invisible decay width of the Higgs boson. The methodemployed here can be generalized to other cases. EFFECTIVE OPERATORS
The Higgs boson is special in the SM. Its mass termis not fixed by the gauge invariance and provides an op-portunity for this particle to couple with some SM gaugesinglets still with renormalizable interactions [16]. Thesimplest case is adding to the SM a new real scalar S with the Lagrangian given by [37–40] L min = L SM + ( ∂ µ S ) − m S − λS | H | − λ S S . (1) The new real scalar can account for the observed DMdensity if a global Z symmetry is imposed, under whichall the SM particles are singlets while the new scalar cantransform nontrivially, i.e., S → − S . To maintain this Z symmetry, the new scalar should have no vacuum expec-tation value (VEV), h S i = 0. Thus, its mass m S is givenby m S = m + λv , where (0 , v ) T / √ m S is smaller than the Higgs bo-son mass m H , the Higgs boson can decay into an S pair,namely, Higgs boson invisible decay. This would modifythe total width of Higgs boson and therefore affect thecross section of Higgs boson production and decay intoother final states. Through the mixing with the Higgsboson, the S pair can annihilate to SM particle pairs, forwhich the cross section is constrained by the DM relicdensity. With the same interaction as in annihilation, S can scatter with nucleons, which can be measured by DMdirect detection experiments. Given that the Higgs bo-son mass is around 125 GeV, this model receives strongconstraint after considering the results of searching forHiggs boson at the LHC, cosmological relic density andthe DM direct detection [41–47]. Therefore, more compli-cated models , in which more particles are included, areproposed [48–51]. There are no rules, in principle, thatthese additional particles should be very light [48]. Andthey are perhaps heavy since a lighter particle is oftenmuch easier to find either in the decay product or directproduction at colliders. In this case, the role played bythese particles can be described by effective operators, O Z = m Z Z Z µ Z µ S , O q = m q q ¯ qqS . (2)They can induce the production processes described bythe Feynman diagrams ( b ) and ( d ) in Fig.1, generatingthe same signature at hadron colliders as the process as-sumed in the experimental analysis, e.g., the Feynmandiagrams ( a ) and ( c ). Note that O Z and O q are notgauge invariant. In fact, one can write down gauge invari-ant operators with higher dimensions and get the abovetwo operators after symmetry breaking. For example,the first operator O Z can be generated from the gaugeinvariant dimension-6 effective operator: O (6) Z = κ Λ ( D µ H ) † ( D µ H ) S , (3)where D µ is the usual covariant derivative, and H de-notes the SM Higgs field. After the Higgs field gets anonvanishing vacuum expected value, O (6) H would deduceto O Z and the associate similar operator O W = M W W W + µ W − µ S (4)because the Higgs field couples with W and Z bosons si-multaneously. The second operator O q can be generatedfrom the gauge invariant operator of dimension 6 O (6) q = κλ q Λ ¯ Q L Hq R S + H.c., (5)where λ q is the Higgs-quark-quark Yukawa couplings inthe SM.Here we choose these two effective operators in Eq.(2)because they are the only ones in the leading power order,and representative, one of them inducing an s -channeland the other inducing a t -channel process; see Feynmandiagrams ( b ) and ( d ) in Fig.1. The detailed discussionon the possible gauge symmetry breaking mechanism isbeyond the scope of this paper. The above effective op-erators have been discussed in the study of DM [52, 53].The main difference between Feynman diagrams ( a )and ( b ) is whether the missing particles have a fixedinvariant mass. In some phase space points where theinvariant mass of the S pair is around the Higgs bosonmass, Feynman diagram ( b ) would be equivalent to Feyn-man diagram ( a ) up to some constant coefficients. How-ever, the phase space for missing particles cannot be fixedexperimentally and must be integrated in any observable.It is the purpose of this study to find a way to reveal thisdifference. NUMERICAL DISCUSSION
To distinguish the different processes shown in Fig.1and extract precise information on Higgs boson invisi-ble decay, we should understand the properties of thecross sections first. At the moment, we neglect the in-terference among the Feynman diagrams ( a, b, c, d ). Thisis reasonable because the total widths of the Higgs and Z bosons are small enough so that we can use narrowwidth approximation in calculating the Feynman dia-grams ( a, c ) and thus do not need to consider their inter-ference with the Feynman diagrams ( b, d ). In numericaldiscussion, we have taken m H = 125 GeV, M Z = 91 . Z boson and missing energy associatedproduction at the LEP and LHC, and find agreementwith MadGraph5v1.3.3 [54] for processes at the LHC.The CTEQ6L1 parton distribution function (PDF) set[55] is chosen and the renormalization and factorizationscales are set to be 200 GeV, which is about the sum of m H and M Z .Before proceeding, we have to find the allowed param-eter space of (Λ Z/q , m S ). In the following, we will discussthe DM relic abundance and direct detection experimentLUX as well as the Z boson and missing energy associ-ated production at the LEP and LHC. Constraint from relic abundance
The DM relic abundance is a precision observable incosmology and imposes constraint on any DM model. Itis determined by the annihilation cross section of DM toSM particles, given by [56]Ω DM h ≈ . × GeV − x f M Pl g / ∗ ( a + 3 b/x f ) , (6)where Ω DM is the cold DM energy density of the Universenormalized by the critical density, and h = 0 . ± . a and b are the coeffi-cients in the partial wave expansion of the DM annihila-tion cross section, σ an v Møl = a + bv + O ( v ). v Møl is called Møller velocity, defined as [57] v Møl = p | v − v | − | v × v | , (7)where v and v are the velocities of colliding DMs inthe cosmic comoving frame. Note that this velocity isdifferent from that in collisions at colliders, since the col-liding DMs in the Universe are not necessarily movingalong a line. And v Møl cannot be transformed into thecenter-of-mass frame of the two colliding DMs becausethe thermally averaged total annihilation cross section h σ an v Møl i has to be evaluated in a common frame for allcollisions. In general, it is difficult to calculate h σ an v Møl i due to the complex definition of v Møl . Fortunately, it isproved that [57] h σv Møl i = h σv lab i lab = h σv cm i cm , (8)in which v lab = | v , lab − v , lab | is the relative velocity inthe rest frame of one of the incoming particles and v cm isthe velocity in the center-of-mass frame of the two collid-ing DMs. The DM is moving at nonrelativistic velocitieswhen freezing out; thus, v ≪ g ∗ is the number ofrelativistic degrees of freedom available at the freeze-outepoch x f . And x f is evaluated by [56] x f = ln A − . A + ln (cid:18) ba ln A (cid:19) (9)with A = 0 . ag/g / ∗ M Pl m S . g counts the internaldegree of freedom and is equal to 1 for a real scalar inour case.For SS → ZZ or q ¯ q , the annihilation cross section is σ Z an = 132 πs (cid:18) M Z Λ Z (cid:19) p s − M Z p s − m S (cid:18) s M Z − sM Z + 3 (cid:19) ,σ q an = N c π (cid:18) m q Λ q (cid:19) q s − m q p s − m S s − m q s . (10)These results are in agreement with those in Refs.[23, 58,59]. Substituting s ≈ m S + m S v + 3 m S v /
4, we obtain a Z = 132 π (cid:18) M Z Λ Z (cid:19) p m S − M Z (cid:0) m S − m S M Z + 3 M Z (cid:1) m S M Z ,b Z = 132 π (cid:18) M Z Λ Z (cid:19) m S − m S M Z + 5 M Z )16 m S M Z p m S − M Z . (11)and a q = m q DM around the Earth can scatter elastically withatomic nuclei, resulting in recoiling movements of nu-clei. These events, if observed, can be explained by DM-nucleon collisions, which can be divided into spin depen-dent and spin independent according to the DM-quark in-teractions. Generally, the spin independent elastic scat-tering cross section has a large value and thus gets a morestringent constraint from direct detection experiments. allowed 100 150 200 250 300050010001500 m S @ GeV D L Z @ G e V D FIG. 2: The allowed space of (Λ Z , m S ) from relic abundance. The region below the solid blue line is allowed. The grayregion corresponds to Λ Z < m S . allowedLUX excluded 20 40 60 80 100050010001500 m S @ GeV D L q @ G e V D FIG. 3: The allowed space of (Λ q , m S ) from relic abundanceand the direct detection experiment. The region below thesolid blue line is allowed by the relic abundance. The re-gion below the dashed red line is excluded by the LUX directdetection experiment [60]. The gray region corresponds toΛ q < m S . The most stringent limit on the spin independent elasticscattering cross section comes from the LUX experiment[60]. Therefore in this analysis, we only consider the dataof LUX in order to obtain the allowed parameter space.The DM-proton spin independent elastic scatteringcross section is given by [23, 61] σ SISp = m p π ( m S + m p ) [ f ( p ) Sp ] , (14)where f ( p ) Sp = X q = u,d,s f ( p ) T q C Sq m p m q + 227 f ( p ) T g X q = c,b,t C Sq m p m q , (15)with [62] f ( p ) T u ≈ . ± . , f ( p ) T d ≈ . ± . ,f ( p ) T s ≈ . ± . , f ( p ) T g ≈ − f ( p ) T u − f ( p ) T d − f ( p ) T s . (16)In our case, C Sq = m q / Λ q . After comparing with theLUX data [60], we obtain the allowed parameter regionin Fig. 3. It is observed that the combination of therelic abundance and direct detection experiment resultsin very stringent constraints on O q . Only a limited pa-rameter space with m S < . q < 110 GeV isallowed. We notice that there are no such constraints onthe parameters of the operator O Z since a nucleon doesnot contain a Z boson. Therefore, we will discuss onlythe case of O Z in the following part. Constraints from Z boson and missing energyassociated production Apart from DM annihilation and elastic scatteringwith nucleons, DM can be produced at the colliders. Herewe are interested in the process of Z boson and missingenergy associated production at the LEP and LHC, i.e., e + e − ( pp ) → Z ∗ → SSZ . For the LEP experiments, wetake the upper limit on the invisible Higgs boson produc-tion cross section at 206.0 GeV, given in Ref. [17]. Weshow the excluded parameter space in Fig. 4. It is foundthat the region excluded by the current LHC has coveredall that by LEP, indicating the better sensitivity of theLHC to the new physics. As the increasing of the dataaccumulated at the LHC, the excluded region will en-large if no signal of new physics is observed. We show inFig. 4 as well the curves corresponding to the upper limiton the cross section of HZ production at the LHC witha Higgs boson branching fraction of 65% , , and 5%,respectively.On the other hand, if an excess of events is found inthe future, it may be induced by O Z with the parame-ter combination lying on the curve that corresponds to a In practice, the DM-neutron spin independent elastic scatteringcross section is almost identical to the DM-proton one [23]. LEP Excluded LHC Excluded65 % % % 20 40 60 80 100020406080100 m S @ GeV D L Z @ G e V D FIG. 4: The allowed space of (Λ Z , m S ) from Z boson andmissing energy associated production. The shaded region isexcluded by the limit on the missing energy and Z bosonproduction at the LEP [17] and the LHC [18]. The threecurves from the bottom up correspond to the cross sectionof HZ production at the LHC with a Higgs boson branchingfraction of 65% , , and 5%, respectively. fixed Higgs boson branching fraction. Then it is essen-tial to judge that this excess results from Higgs bosoninvisible decay or DM associated production with a Z boson. Missing transverse momentum distribution Now we show the normalized p miss T distributions formissing energy and Z boson production at the 8 TeVLHC in Fig.5 for the process induced by O Z . Wesee that the shape of the process induced by O Z with m S = 30 GeV is very similar to the SM process of H (invisible decay) Z production. This similarity suggeststhat one cannot simply interpret the signal of missing en-ergy and a charged pair production as the associated HZ production with Higgs boson invisible decay.From Fig.5, we find that the peak position for the pro-cess induced by O Z is moving toward the large p miss T re-gion, and the tail of the distribution drops more slowly asthe DM mass increases. To make it explicit, we presentthe fitted formulas describing the peak positions: p peak T, O Z = 15 . 83 GeV + 6 . 67 GeV m S 10 GeV . (17)We notice that this formula is fitted from the partonicsimulation results, not including the parton shower andhadronization effects, high-order QCD corrections, etc.The exact peak positions may be different after taking [GeV] missT p N o r m a li ze d s p ec t r u m SM H(inv)Z SM Z(inv)Z =10 GeV S m =20 GeV S m =30 GeV S m =40 GeV S m =50 GeV S m FIG. 5: The p miss T distribution for missing energy and Z boson production at the 8 TeV LHC induced by O Z . Wealso show the SM processes of H (invisible decay) Z and Z (invisible decay) Z production. into account all these effects. However, we still use thisformula as long as the difference between the fitted valueand the exact one is not very large. For comparisons, wealso list the values of the peak positions for the processesin the SM: p peak T,HZ = 50 . , (18) p peak T,ZZ = 32 . . (19)At the low integrated luminosity of the LHC, the ob-served events may not be enough to provide a full de-scription of the p miss T distribution. For this reason, wepropose two observables that can be used to distinguishthe underlying processes at the early stage of the LHC,defined as R ≡ σ ( p miss T < p peak T ) σ ( p miss T > p peak T ) , (20) R ≡ σ ( p miss T < p cut T ) σ ( p miss T > p cut T ) . (21)Here p peak T is defined theoretically as the p T value aroundwhich the distribution is the largest (the experimentalvalue of p peak T is discussed in the following). It may takedifferent values for the signal and background events.And the default value of p cut T is chosen to be 150 GeV. R describes the profile of the peak region while R in-corporates the information on the tail region. It is moreconvenient to adopt these two variables rather than thefull distributions to understand the underlying processes.Moreover, because of the ratios in R and R , the coef-ficients of the operators are canceled. As a consequence, R and R are functions of only the DM mass, provid- ing a handle to the masses of invisible particles. In par-ticular, many effects that may change the leading-orderprediction, such as the factorization and renormalizationscales, PDF sets, parton shower, and higher order correc-tions are supposed to be canceled substantially in theseobservables as well.We emphasize again that we are interested in how todistinguish the signal processes between Higgs invisibledecay and DM associated production after the discoveryof the signal. In this case, we can divide the total eventsobserved experimentally to the background (mainly ZZ production) and signals (possible HZ production or SSZ production). And therefore, the two observables, R and R , can be measured separately for the background andsignals.The theoretical prediction for the dependence of R and R on the DM mass is shown in Fig.6. In the plot of R , the curves are obtained by using Eq. (20) with thecorresponding p peak T given in Eqs.(17 ∼ R is insensitive to the DM mass for the processesinduced by O Z and takes discrepant values between theSM processes and processes induced by O Z . On the otherhand, the experiment would give p T distributions in binsfor both the background and signal, from which we canobtain a rough estimate of p peak T . But because of thelimited signal events, the bin is perhaps too wide (forexample 10 GeV) to precisely determine p peak T . In thiscase, we choose p peak T to be the left (right) edge of themaximum bin if the left (right) neighbor of the maximumbin is larger than the right (left) one. Then we countthe left and right bins, obtaining the experimental valuefor R . Given that the definitions of p peak T are differentfrom the theory and experiment sides, it is possible thatthey are not equal to each other. This uncertainty woulddilute the precision in determining the value of DM mass.From Eq.(17), ∆ m S ≈ . p peak T,Z . Thus, an uncertaintyof 5 GeV in p peak T,Z would result in an uncertainty of about7.5 GeV in m S . So we will not use p peak T,Z to probe thevalue of m S . However, we can still use R to separatethe HZ production with Higgs boson invisible decay and Z boson with DM associated production processes. InFig.6, we show the uncertainty of R by changing the p peak T by ± R forthe DM associated production and Higgs invisible decayprocesses do not overlap, except for a very small region. R is very sensitive to the DM mass for the process in-duced by O Z , especially in the DM mass range m S < R inter-sects with the curve corresponding to the process inducedby O Z , we can determine the mass of DM. The uncer-tainties arising from the variation of scales and PDF setsare also calculated, explicitly shown in Fig.6, which turnout to be small, as expected. After taking into accountthe scale uncertainties, which are much larger than thosefrom the PDF sets, the accuracy of the determined DM SM H H inv L ZSM Z H inv L Zinduced by O Z 20 40 60 80 1000.00.20.40.60.8 m S @ GeV D R SM H H inv L ZSM Z H inv L Z (cid:144) O Z p T cut = 100 GeV p T cut = 200 GeV D m S D R 20 40 60 80 10002468101214 m S @ GeV D R FIG. 6: The dependence of R and R on the DM mass. Thenarrow bands indicate the scale uncertainties by varying thedefault scale by a factor of 2. The wide bands in the upperplot are obtained by changing the p peak T by ± R value for Z (invisible decay) Z production has been divided bya factor of 3 in the bottom plot. In the example illustratingthe estimation of ∆ m S in the bottom plot, ∆ R is set to be0 . R . mass is estimated to be ∆ m S = 3 . ∼ . R changes from 10 to 2 (corresponding to m S from 15.4 to65.0 GeV); see Fig.7. This estimation is obtained withoutconsidering the effects of parton shower, hadronizationand detector simulation but under the assumption thatthe measured R is accompanied with an uncertainty of0 . R . As the integrated luminosity is increased, the un- p T cut = 150 GeVLHC 8 TeV0 2 4 6 8 10 1220406080100 R m S @ G e V D FIG. 7: The determined m S as a function of R in the caseof p cut T = 150 GeV. The error of R is set to be 0 . R and0 . R for the large and small error bars respectively, whilethe error of m S is derived as illustrated in Fig.6. p T cut = 100 GeVLHC 8 TeV0 1 2 3 4 5 620406080100 R m S @ G e V D p T cut = 200 GeVLHC 8 TeV2 4 6 8 10 12 1420406080100 R m S @ G e V D FIG. 8: The determined m S as a function of R in the caseof p cut T = 100 and 200 GeV. The error of R is set to be 0 . R while the error of m S is derived as illustrated in Fig.6. certainty of R would be reduced. We also show the sit-uation in which the uncertainty of R is 0 . R in Fig.7.Then, the accuracy would be ∆ m S = 2 . ∼ . R changes from 10 to 2.Then we discuss the impact from the choice of p cut T inthe definition of R on the determination of the DM mass.We have chosen p cut T = 150 GeV in the above numericalresults. But it is possible to choose a different value of p cut T as long as it is in the tail region (much greater than p peak T ). The two cases of p cut T = 100 and 200 GeV arealso shown in Figs.6 and 8. We see that for larger p cut T ,the scale uncertainty of R at fixed m S is larger. But R drops faster as m S increase at the same time. Thenet effect results in a similar accuracy in estimating m S .We also notice that too large p cut T would induce a largestatistical uncertainty, and too small p cut T would reducethe sensitivity to m S . As a consequence, p cut T = 150GeV is a good choice. CONCLUSION To know the total width of the recently discoveredHiggs boson particle, it is important to measure the invis-ible decay width of the Higgs boson. However, the signalfor this measurement at the LHC, i.e., a charged lep-ton pair and missing energy in the final state, cannot bedefinitely understood as the product of the intermediateproduced Z and H bosons due to the possible interac-tion between DM and Z boson or quarks, which can bedescribed by representative effective operators. First, weconsider the relic abundance, the LUX direct detectionexperiment and the result of searching for Higgs bosoninvisible decay at the LEP and LHC in order to find theallowed parameter region space for the effective opera-tors. We discover that the interaction between DM andquarks is stringently constrained. Then we investigatethe transverse momentum distribution of the missing en-ergy and propose two observables that can be used todistinguish the different underlying processes. Moreover,with these two observables, we may be able to determinethe mass of invisible particles.In this paper, we only consider the Higgs boson invis-ibly decaying into scalar DM, inspired by the possiblerenormalizable extension of the SM. And we also assumethe Z boson and quarks also interact with this kind ofDM. Given that the DM takes more parts of the energyin the Universe than ordinary matter, it is likely thatthere are many kinds of DM and the kind coupling withthe Z boson and quarks differs from that with the Higgsboson. For example, the Z boson and quarks connectwith fermionic DM, while the Higgs boson decays intoscalar DM. We will explore these scenarios in the future. ACKNOWLEDGEMENT This work was supported by the Cluster of Excellence Precision Physics, Fundamental Interactions and Struc-ture of Matter (Grant No. PRISMA-EXC 1098). This sensitivity can be understood as dR /dm S . ∗ Electronic address: [email protected][1] Georges Aad et al. Observation of a new particle in thesearch for the Standard Model Higgs boson with the AT-LAS detector at the LHC. Phys.Lett. , B716:1–29, 2012.[2] Serguei Chatrchyan et al. Observation of a new bosonat a mass of 125 GeV with the CMS experiment at theLHC. Phys.Lett. , B716:30–61, 2012.[3] Vernon Barger, Muneyuki Ishida, and Wai-Yee Keung.Total Width of 125 GeV Higgs Boson. Phys.Rev.Lett. ,108:261801, 2012.[4] Vardan Khachatryan et al. Constraints on the Higgs bo-son width from off-shell production and decay to Z-bosonpairs. 2014.[5] Fabrizio Caola and Kirill Melnikov. Constraining theHiggs boson width with ZZ production at the LHC. Phys.Rev. , D88:054024, 2013.[6] Jonathan M. Butterworth, Adam R. Davison, Math-ieu Rubin, and Gavin P. Salam. Jet substructure as anew Higgs search channel at the LHC. Phys.Rev.Lett. ,100:242001, 2008.[7] Search for the bb decay of the standard model higgs bo-son in associated w/zh production with the atlas detec-tor. Technical Report ATLAS-CONF-2013-079, CERN,Geneva, Jul 2013.[8] Serguei Chatrchyan et al. Search for the standard modelHiggs boson produced in association with a W or aZ boson and decaying to bottom quarks. Phys.Rev. ,D89:012003, 2014.[9] Robert E. Shrock and Mahiko Suzuki. Invisible Decaysof Higgs Bosons. Phys.Lett. , B110:250, 1982.[10] Debajyoti Choudhury and D.P. Roy. Signatures of aninvisibly decaying Higgs particle at LHC. Phys.Lett. ,B322:368–373, 1994.[11] J.F. Gunion. Detecting an invisibly decaying Higgs bosonat a hadron supercollider. Phys.Rev.Lett. , 72:199–202,1994.[12] Oscar J.P. Eboli and D. Zeppenfeld. Observing an invis-ible Higgs boson. Phys.Lett. , B495:147–154, 2000.[13] K. Belotsky, Daniele Fargion, M. Khlopov, R. Konoplich,and K. Shibaev. Invisible Higgs boson decay into massiveneutrinos of fourth generation. Phys.Rev. , D68:054027,2003.[14] R.M. Godbole, M. Guchait, K. Mazumdar, S. Moretti,and D.P. Roy. Search for ‘invisible’ Higgs signals at LHCvia associated production with gauge bosons. Phys.Lett. ,B571:184–192, 2003.[15] Hooman Davoudiasl, Tao Han, and Heather E. Logan.Discovering an invisibly decaying Higgs at hadron collid-ers. Phys.Rev. , D71:115007, 2005.[16] Brian Patt and Frank Wilczek. Higgs-field portal intohidden sectors. 2006.[17] Searches for invisible Higgs bosons: Preliminary com-bined results using LEP data collected at energies up to209-GeV. 2001.[18] Search for invisible decays of a higgs boson produced inassociation with a z boson in atlas. Technical ReportATLAS-CONF-2013-011, CERN, Geneva, Mar 2013.[19] Search for invisible Higgs produced in association witha Z boson. Technical Report CMS-PAS-HIG-13-018,CERN, Geneva, 2013.[20] R. Barbier, C. Berat, M. Besancon, M. Chemtob, A. Deandrea, et al. R-parity violating supersymmetry. Phys.Rept. , 420:1–202, 2005.[21] Gian F. Giudice, Riccardo Rattazzi, and James D. Wells.Quantum gravity and extra dimensions at high-energycolliders. Nucl.Phys. , B544:3–38, 1999.[22] G. Hinshaw et al. Nine-Year Wilkinson MicrowaveAnisotropy Probe (WMAP) Observations: CosmologicalParameter Results. Astrophys.J.Suppl. , 208:19, 2013.[23] Qing-Hong Cao, Chuan-Ren Chen, Chong Sheng Li, andHao Zhang. Effective Dark Matter Model: Relic density,CDMS II, Fermi LAT and LHC. JHEP , 1108:018, 2011.[24] Xiangdong Gao, Chong Sheng Li, Jun Gao, Jian Wang,and Robert J. Oakes. Next-to-leading order QCD pre-dictions for graviton and photon associated productionin the Large Extra Dimensions model at the LHC. Phys.Rev. , D81:036008, 2010.[25] Jian Wang and Chong Sheng Li. Updated predictions forgraviton and photon associated production at the LHC. Phys.Rev. , D86:116008, 2012.[26] Fa Peng Huang, Chong Sheng Li, Jian Wang, andDing Yu Shao. Searching for the signal of dark matterand photon associated production at the LHC beyondleading order. Phys.Rev. , D87:094018, 2013.[27] Serguei Chatrchyan et al. Search for Dark Matter andLarge Extra Dimensions in pp Collisions Yielding a Pho-ton and Missing Transverse Energy. Phys.Rev.Lett. ,108:261803, 2012.[28] Georges Aad et al. Search for dark matter candidatesand large extra dimensions in events with a photon andmissing transverse momentum in pp collision data at √ s = 7 TeV with the ATLAS detector. Phys.Rev.Lett. ,110:011802, 2013.[29] Yang Bai and Tim M.P. Tait. Searches with Mono-Leptons. Phys.Lett. , B723:384–387, 2013.[30] Stefan Karg, Michael Kramer, Qiang Li, and Dieter Zep-penfeld. NLO QCD corrections to graviton productionat hadron colliders. Phys.Rev. , D81:094036, 2010.[31] Patrick J. Fox and Ciaran Williams. Next-to-LeadingOrder Predictions for Dark Matter Production at HadronColliders. Phys.Rev. , D87:054030, 2013.[32] M.C. Kumar, Prakash Mathews, V. Ravindran, andSatyajit Seth. Graviton plus vector boson productionto NLO in QCD at the LHC. Nucl.Phys. , B847:54–92,2011.[33] Nicole F. Bell, James B. Dent, Ahmad J. Galea,Thomas D. Jacques, Lawrence M. Krauss, et al. Search-ing for Dark Matter at the LHC with a Mono-Z. Phys.Rev. , D86:096011, 2012.[34] Linda M. Carpenter, Andrew Nelson, Chase Shimmin,Tim M.P. Tait, and Daniel Whiteson. Collider searchesfor dark matter in events with a Z boson and missingenergy. Phys.Rev. , D87:074005, 2012.[35] J. Andrea, B. Fuks, and F. Maltoni. Monotops at theLHC. Phys.Rev. , D84:074025, 2011.[36] Jian Wang, Chong Sheng Li, Ding Yu Shao, and HaoZhang. Search for the signal of monotop production atthe early LHC. Phys.Rev. , D86:034008, 2012.[37] Vanda Silveira and A. Zee. SCALAR PHANTOMS. Phys.Lett. , B161:136, 1985.[38] John McDonald. Gauge singlet scalars as cold dark mat-ter. Phys.Rev. , D50:3637–3649, 1994.[39] C.P. Burgess, Maxim Pospelov, and Tonnis ter Veldhuis.The Minimal model of nonbaryonic dark matter: A Sin-glet scalar. Nucl.Phys. , B619:709–728, 2001. [40] Vernon Barger, Paul Langacker, Mathew McCaskey,Michael J. Ramsey-Musolf, and Gabe Shaughnessy. LHCPhenomenology of an Extended Standard Model with aReal Scalar Singlet. Phys.Rev. , D77:035005, 2008.[41] Martti Raidal and Alessandro Strumia. Hints for anon-standard Higgs boson from the LHC. Phys.Rev. ,D84:077701, 2011.[42] Y. Mambrini. Higgs searches and singlet scalar dark mat-ter: Combined constraints from XENON 100 and theLHC. Phys.Rev. , D84:115017, 2011.[43] Xiao-Gang He and Jusak Tandean. Hidden Higgs Bosonat the LHC and Light Dark Matter Searches. Phys.Rev. ,D84:075018, 2011.[44] Patrick J. Fox, Roni Harnik, Joachim Kopp, and YuhsinTsai. Missing Energy Signatures of Dark Matter at theLHC. Phys.Rev. , D85:056011, 2012.[45] Ian Low, Pedro Schwaller, Gabe Shaughnessy, and Car-los E.M. Wagner. The dark side of the Higgs boson. Phys.Rev. , D85:015009, 2012.[46] Abdelhak Djouadi, Oleg Lebedev, Yann Mambrini, andJeremie Quevillon. Implications of LHC searches forHiggs–portal dark matter. Phys.Lett. , B709:65–69, 2012.[47] Kingman Cheung, Yue-Lin S. Tsai, Po-Yan Tseng, Tzu-Chiang Yuan, and A. Zee. Global Study of the SimplestScalar Phantom Dark Matter Model. JCAP , 1210:042,2012.[48] Abdessamad Abada, Djamal Ghaffor, and Salah Nasri.A Two-Singlet Model for Light Cold Dark Matter. Phys.Rev. , D83:095021, 2011.[49] Maxim Pospelov and Adam Ritz. Higgs decays todark matter: beyond the minimal model. Phys.Rev. ,D84:113001, 2011.[50] Abdessamad Abada and Salah Nasri. RGE of a ColdDark Matter Two-Singlet Model. Phys.Rev. , D88:016006,2013.[51] Admir Greljo, J. Julio, Jernej F. Kamenik, ChristopherSmith, and Jure Zupan. Constraining Higgs mediateddark matter interactions. JHEP , 1311:190, 2013.[52] Eugenio Del Nobile and Francesco Sannino. Dark MatterEffective Theory. Int.J.Mod.Phys. , A27:1250065, 2012.[53] Jessica Goodman, Masahiro Ibe, Arvind Rajaraman,William Shepherd, Tim M.P. Tait, et al. Constraintson Dark Matter from Colliders. Phys.Rev. , D82:116010,2010.[54] Johan Alwall, Michel Herquet, Fabio Maltoni, OlivierMattelaer, and Tim Stelzer. MadGraph 5 : Going Be-yond. JHEP , 1106:128, 2011.[55] J. Pumplin, D.R. Stump, J. Huston, H.L. Lai, Pavel M.Nadolsky, et al. New generation of parton distributionswith uncertainties from global QCD analysis. JHEP ,0207:012, 2002.[56] Edward W. Kolb and Michael S. Turner. The Early Uni-verse. Front.Phys. , 69:1–547, 1990.[57] Paolo Gondolo and Graciela Gelmini. Cosmic abun-dances of stable particles: Improved analysis. Nucl.Phys. ,B360:145–179, 1991.[58] Maria Beltran, Dan Hooper, Edward W. Kolb, andZosia C. Krusberg. Deducing the nature of dark mat-ter from direct and indirect detection experiments in theabsence of collider signatures of new physics. Phys.Rev. ,D80:043509, 2009.[59] Zhao-Huan Yu, Jia-Ming Zheng, Xiao-Jun Bi, Zhibing Li,Dao-Xin Yao, et al. Constraining the interaction strengthbetween dark matter and visible matter: II. scalar, vector and spin-3/2 dark matter. Nucl.Phys. , B860:115–151,2012.[60] D.S. Akerib et al. First results from the LUX dark matterexperiment at the Sanford Underground Research Facil-ity. Phys.Rev.Lett. , 112:091303, 2014.[61] G. Belanger, F. Boudjema, A. Pukhov, and A. Semenov.Dark matter direct detection rate in a generic model with micrOMEGAs 2.2. Comput.Phys.Commun. , 180:747–767, 2009.[62] John R. Ellis, Andrew Ferstl, and Keith A. Olive. Reeval-uation of the elastic scattering of supersymmetric darkmatter.