Reconstruction of Inert Doublet Scalars at the International Linear Collider

We study collider signatures for extra scalar bosons in the inert doublet model at the international linear collider (ILC). The inert doublet model is a simple extension of the standard model by introducing an additional isospin-doublet scalar field which is odd under an unbroken Z_2 symmetry. The model predicts four kinds of Z_2-odd scalar bosons, and the lightest of them becomes stable and a candidate of the dark matter as long as it is electrically neutral. Taking into account the constraints from various theoretical and phenomenological conditions, we perform a simulation study for the distinctive signatures of the extra scalars over the standard-model background contributions at the ILC with the center-of-mass energy of sqrt{s} = 250 GeV and 500 GeV. We further discuss observables for determination of the mass of the scalars. We find that the parameter regions which cannot be detected at the large hadron collider can be probed at the ILC.


I. INTRODUCTION
In July 2012, a Higgs-like particle was found at the large hadron collider (LHC) [1,2].
The particle looks like mostly a Higgs boson in the standard model (SM), but the detail properties of the particle and the whole structure of the Higgs sector have not yet been revealed. It is widely believed that the SM has to be extended, since it cannot explain the dark matter, (tiny) neutrino masses, and the baryon asymmetry in the Universe, etc.
Although in the Higgs sector of the SM, only one SU(2) L -doublet scalar field is introduced, there is no theoretical guideline for this choice. Thus, the Higgs sector may be a solid target to probe new physics beyond the SM. The inert doublet model (IDM) is one of the simplest extensions of the SM, where an additional SU(2) L -doublet scalar field is introduced, which is odd under the unbroken Z 2 symmetry [3,4]. As in the case in the general two Higgs doublet model, four kinds of additional scalars appear as physical states, namely neutral CP -even state (H), neutral CP -odd state (A) and charged scalar states (H ± ), all of which are called inert scalars. Due to the Z 2 symmetry, Yukawa interactions of the inert scalars to SM fermions are forbidden, and the possible flavor-changing neutral current is absent at the tree level. The lightest inert particle (LIP) is stable, because of the Z 2 -parity conservation. Therefore, the model provides a dark matter candidate [4][5][6][7][8][9][10][11]. In addition to that, the model has rich phenomenological features such as the electroweak symmetry breaking [12], electroweak phase transition [13][14][15][16], radiatively generating neutrino masses by introducing Z 2 -odd right-handed neutrinos [17], and leptogenesis [18,19], etc.
Collider signatures of the inert scalars in the IDM have been studied in the literature [4,10,11,[20][21][22][23]. In Ref. [21], bounds on the inert scalar masses are obtained by using the experimental results at the LEP II [24][25][26]. Since the inert scalars do not have QCD interactions, it is not suited to search for them at hadron colliders. Even though the parameter regions where the inert scalars could be discovered at the LHC are pointed out [10,22,23], detailed analysis on these scalars such as the precise determination of these masses and quantum numbers would be difficult.
In this letter, we study collider phenomenology for the inert scalars at the international linear collider (ILC). As it is a machinery for precision measurements of Higgs boson properties, the extended Higgs sector can also be investigated in details. We study the characteristic signatures, corresponding backgrounds and the kinematical observables in the processes of HA associated production as well as H + H − pair production. Earlier studies can be found e.g. in Refs. [27,28] where the charged scalar pair production process is studied in detail, and in Ref. [29] where HA associated production is also studied. On the other hand, our study includes all the available processes and decay modes, and the simulation analysis for the signal and the background contributions with appropriate kinematical cuts. Furthermore, we also discuss the procedure for the mass determination of the inert scalars which can be performed at an early stage of the experiment, and introduce a new observable which could be useful to determine the inert scalar masses more precisely than the variables used in Refs. [28,29].
The rest of the letter is organized as follows. In Sec. II, we review the inert doublet model and introduce the benchmark points used in our simulation study. Then we present the simulation studies for observing the inert scalars and determining the masses of them in Sec. III. Sec. IV and Sec. V are devoted to discussions and conclusions, respectively. In Appendix, we evaluate a new observable at e + e − colliders which is used in our analysis for the mass determination.
In this letter, we consider four benchmark points for the masses of inert scalars listed in Table I, which satisfy all the available theoretical and also phenomenological constraints [32].
The bounds on the masses of inert scalars are briefly summarized as follows. By the constraints from dark matter relic abundance and direct searches, the mass of LIP should be 40 m H 80 GeV [5, 7, 10] 1 . For m H < 80 GeV, the mass of the second neutral scalar has to satisfy |m A − m H | < 8 GeV or m A > 100 GeV, to avoid the bounds from the direct searches at the LEP experiments [21]. The mass differences of inert scalars result into additional contributions [4] to the electroweak S and T parameters. To be consistent with the current experimental data [33], must be satisfied. In Ref. [10] these constraints are more extensively studied.
In the four benchmark points, the mass of H is fixed to 65 GeV, so that it does not induce the invisible decay of the SM Higgs boson. While it could be up to ∼ 80 GeV concerning the dark matter relic abundance analysis [5,[7][8][9][10][11] [5] and also the bounds on the cross section of dark matter direct production. On the other hand, the method to determine λ 2 has not yet been established.
For our four benchmark points, the production cross sections of inert scalars at the ILC are large enough to be tested. In Table I There are two important effects in the property of the SM-like Higgs boson, the invisible decay [20,34] and the charged scalar loop contribution to the two-photon decay amplitude [11,34,35]. The invisible decay mode opens if m h > 2m H . If this is the case, the branching ratio for the invisible decay mode can be typically several tens percent [20,34].
Thus, it could be discovered at the LHC [36]. The two-photon branching ratio of the SM-like Higgs boson can be directly related to the production cross-section of the Higgs-boson at the LHC is not modified in the IDM. It is shown [34] that R γγ can be enhanced with relatively light H ± (m H ± 130 GeV), negative λ 3 and no invisible decays.

III. COLLIDER SIGNATURES IN THE IDM AT THE ILC
In this section, we perform the simulation studies for the detection and the mass determination of the inert scalars in the IDM at the ILC. The kinematical distributions are calculated by using MadGraph [37] at the parton level with basic cuts for event generations. For charged leptons, we set p ℓ T > 1 GeV, |η ℓ | < 2.5 and ∆R ℓℓ > 0.2 for the isolation requirement, where p T is the transverse momentum, η is the pseudo-rapidity, and ∆R(= ∆η 2 + ∆φ 2 ) is the distance of the two particle in the η − φ plane. For jets, we restrict ourselves with p j T > 5 GeV, |η j | < 2.5 and ∆R jj , ∆R ℓj > 0.4. Furthermore, we require the missing transverse energy E / T to be greater than 10 GeV to reduce background events from two photon scattering processes. The two photon scattering processes and QED radiation effects are not estimated in our analysis.
We note that, these cuts may be rather conservative, so that our parton level analysis makes sense. At e + e − colliders, hadronic final-states would be utilized even if they do not form narrow jets. Then, the cuts on p T and the isolation for partons may not be necessary.
We include these cuts so that the number of jets in an event can be more easily accounted the number of outgoing partons in the processes. In case we loose these cuts, the number of available events would increase, but we expect more background contributions from the events with less partons. An isolation requirements for leptons may be weakened as well in the real experiment.
A. e + e − → HA process First, we consider the HA production process followed by A → Z ( * ) H decay. Since H is neutral and stable, it escapes from detection. Thus, it gives the signature with a dilepton (dijet) plus large missing energy. Expected background contributions come from dilepton (dijet) plus two neutrinos production in the SM.
First we study the collider signature of these events at √ s = 250 GeV, for the parameter sets (I, III) and (II) where A decays into H and off-shell Z-boson. To reduce the SM background contributions, we apply following kinematical cuts; the scaled acoplanarityφ acop , which is the acoplanarity 2 multiplied by the sine of the smallest angle between a lepton (jet) and the beam axis, is larger than 100 • ; | cos θ ℓℓ | < 0.8 for dilepton or | cos θ jj | < 0.6 for dijet, where θ ℓℓ(jj) is the polar angle of the dilepton (dijet) 3-momenta.
In the top left (right) panel in Fig. 1, we see the dilepton (dijet) energy distributions for the signal and background processes at √ s = 250 GeV with the integrated luminosity of L int = 250 fb −1 . After the cuts described above, background events are well reduced. The difference of the number of the signal events in dilepton and dijet signature comes from the branching ratio of the Z-boson, and that of the background events comes from the absence of W W production in the dijet case.
The endpoints of the E ℓℓ(jj) distribution are related with the masses of H and A as where λ(a, b, c) = a 2 + b 2 + c 2 − 2(ab + bc + ca). Thus, the masses can be determined by measuring these endpoints. Since the distribution is quite steep around the maximum value, 2 The acoplanarity angle φ acop is defined as the supplement of the difference of azimuthal angles of the leptons (jets). E max ℓℓ(jj) should be a good observable to be measured precisely. It becomes 24 GeV even for the cases with small mass splitting (I, III), and 80 GeV for the case (II) at √ s = 250 GeV.
On the other hand, E min ℓℓ(jj) measurement may be difficult since the distribution is gradual around the minimum, and E min ℓℓ(jj) is too small for the cases (I, III) [E min ℓℓ(jj) = 2.4 GeV at √ s = 250 GeV].
In the bottom left (right) panel in Fig. 1, we show the E ℓℓ(jj) distributions for the pa-  For the case with the off-shell Z-boson, from the maximum of the dilepton (dijet) invariant-mass distribution, the difference of the two scalar masses can be determined as We note that, for the case with small mass splitting, the measurement would be affected by QED radiation backgrounds and acceptance cuts.
B. e + e − → H + H − process We here turn to the H + H − pair production, where H ± predominantly decays into HW ± , and W ± further into ℓ ± ν or qq ′ . We study the semi-leptonic and all-hadronic decay modes as successful signatures.
First, we study the semi-leptonic decay mode, where the signature is a charged lepton plus dijet plus large missing energy. The expected leading background process is τ ± νjj production followed by the leptonic decay of τ . The ℓ ± νjj background process can be Here, we note that the invariant mass of all hadrons vanishes at the endpoints. Therefore, the hadronic system would be actually observed as one jet near the endpoints. When we apply a cut on the smallest of the dijet invariant-mass at M cut , the endpoints of the energy distribution would be replaced by with . Thus, the mass information can be still obtained. Furthermore, the maximum value of the invariant mass of all hadrons is just the difference between m H ± and m H , In the right panel in Fig. 2, the E had distribution in the semi-leptonic decay modes are plotted by using the parameter sets (II) and (IV) at the ILC with √ s = 500 GeV and In the middle panel of Fig. 3, the same distributions are plotted but for the signal processes using parameter sets (II) and (IV) at √ s = 500 GeV with L int = 500 fb −1 . By the kinematical cut of | cos θ miss | < 0.8, the SM background is sufficiently reduced except at M rec ≃ m Z . As it is shown in Appendix, the peak of the signal distribution is given by It is the advantage of this observable that this relation holds even when the W -boson in H ± → W ± H is off-shell. Thus, the ratio of m H and m H ± can be determined.
In the right panel of Fig. 3, the M vis distributions are plotted for the signal processes using parameter sets (II) and (IV) at √ s = 500 GeV with L int = 500 fb −1 . In addition to the kinematical cut applied in the previous panel, the cut of M rec > 110 GeV is applied to reduce the SM background which has a missing energy from Z → νν. After these cuts, the SM background is sufficiently reduced except at M vis ≃ m Z . The signal distribution has a peak at when the W -boson in H ± → W ± H is on-shell [the case (IV)]. When the W -boson is off-shell, the relation on the peak position no more holds.

C. Mass Determination
Here, we summarize the observables for determining the masses of inert scalars. in the threshold behavior of the production cross sections, the scattering angular dependence, and the decay angular dependence. It has been discussed that these measurements are possible for the processses we considered at the ILC [28,38,39]. On the contrary, those measurements at the LHC are difficult, because the center-of-mass energy in the partonic scattering is not fixed and also the center-of-mass system of the partonic scattering cannot be reconstructed.
Finally, we comment on the total comparison for the IDM studies at the LHC and the ILC. It is shown that the LHC has discovery potential for the inert scalars by using the signatures with multileptons plus large missing transverse momentum [10,22,23], if the mass difference is sufficiently large. Thus, in some preferable situations, the evidence of the inert scalars would show up at the LHC before the ILC experiment starts. On the other hand, as we have shown, the ILC can probe the case with smaller mass difference still in a good accuracy, as long as the production processes are kinematically accessible, i.e. √ s > 2m H ± and m H + m A . Thus, there is a parameter region where any evidence can not be observed at the LHC, but the ILC could find it. In any case, at the ILC, the masses of the inert scalars can be reconstructed in good accuracy, and furthermore, the spin of the particles can be measured in principal. Therefore, the final discrimination of the model beyond the SM would be performed at the ILC.

V. CONCLUSIONS
To conclude, we have studied the collider phenomenology of the inert scalar particles at the ILC in the IDM. The model contains a scalar dark matter candidate which is stable due to the unbroken Z 2 symmetry, in addition to another neutral scalar and charged scalar bosons. At collider experiments, because the dark matter would escape from the detector, the signatures always include large missing energy. We have studied the collider signatures of the pair production of the charged scalars and the associated production of the two neutral scalars with appropriate kinematical cuts to reduce the SM backgrounds. We have also investigated the observables to determine the masses of the scalars in these processes. We have shown that distinctive signatures can be observed even if the cases with small mass difference by applying simple kinematical cuts, and that by combining these observables the inert scalar masses can be reconstructed in a good accuracy at the ILC. Here, we consider the kinematics of a general process X is a certain initial-state with the fixed collision energy, A i are scalars with the same mass m A , e.g. the identical particles or the charge-conjugate particles. B i and C i are any particles with their masses m B and m C , respectively, which are produced from the isotropic decay of A i . For the meantime, we consider the case where all the particles are on-shell, i.e. √ s ≥ 2m A ≥ 2m B + 2m C . We consider the invariant-mass distribution of the B 1 B 2 pair: where dΦ n is the n-body phase-space volume element and Γ A is the total decay width of A.
Using the narrow width approximation for A, it is calculated as where (θ 1 , φ) and θ 2 are the decay angles in the rest-frame of A 1 and A 2 , respectively. The azimuthal angle in the decay of A 2 is fixed to be zero, and the scattering angles in the process X → A 1 A 2 , which are irrelevant to M BB , are integrated out. By using the above integration variables, (p B 1 + p B 2 ) 2 is expressed as Here, s = p 2 X , β A = 1 − 4m 2 A /s, ǫ = 1 + m 2 B /m 2 A − m 2 C /m 2 A , β = λ(1, m 2 B /m 2 A , m 2 C /m 2 A ) with λ(a, b, c) = a 2 + b 2 + c 2 − 2(ab + bc + ca), c i = cos θ i for i = 1, 2 and c φ = cos φ.
Although the numerical integration in Eq. (A2) is straightforward, we find that, unless √ s ≃ 2m A , it is a good approximation to neglect the last term in Eq. (A3). In that case, the M BB distribution can be analytically expressed as for M min ≤ M BB ≤ m B √ s/m A , and for m B √ s/m A < M BB ≤ M max , where M max/min = √ s/2 · (ǫ ± β A β). Thus, we find that the distribution has a peak at m B √ s/m A , which therefore depends only on the ratio m B /m A but not on m C .
Here we give some comments. In case with m A ≤ m B + m C , if the particles C i are off- At e + e − colliders where the total 4-momenta of the collision can be fixed, it is possible to assess the invariant mass of the all missing particles (the recoil mass) by M 2 rec = (p in − p vis ) 2 , where p in is the total 4-momenta of the initial e + e − system. Therefore, the invariant mass of the pair of two missing particles can be reconstructed, if these are the only missing particles in the event.