Exploring the Inert Doublet Model through the dijet plus missing transverse energy channel at the LHC

In this study of the Inert Doublet Model (IDM), we propose that the dijet + missing transverse energy channel at the Large Hadron Collider (LHC) will be an effective way of searching for the scalar particles of the IDM. This channel receives contributions from gauge boson fusion, and $t-$channel production, along with contributions from $H^+$ associated production. We perform the analysis including study of the Standard Model (SM) background with assumed systematic uncertainty, and optimise the selection criteria employing suitable cuts on the kinematic variables to maximise the signal significance. We find that with high luminosity option of the LHC, this channel has the potential to probe the IDM in the mass range of up to about 400 GeV, which is not accessible through other leptonic channels. In a scenario with light dark matter of mass about 65 GeV, charged Higgs in the mass range of around 200 GeV provides the best possibility with a signal significance of about $2\sigma$ at an integrated luminosity of about 3000 fb$^{-1}$.


Introduction
The discovery of Higgs boson by the ATLAS and CMS collaborations of the LHC [1,2] has definitely put the spotlight of particle physics research on Higgs phenomenology. While all measurements so far indicate that the new particle is indeed a Higgs boson, compatible with that predicted by the Standard Model (SM) of particle physics, detailed questions about the exact nature of the Higgs potential and the coupling of the Higgs particle to other SM particles need to be investigated. This information, along with the popular reasoning that the SM Higgs mechanism is only an effective description to understand Electroweak Symmetry Breaking (EWSB), of a more fundamental high-energy theory has led researchers to study the implication of many variant models. It is believed that the Standard Higgs mechanism with only one physical scalar is too minimalistic, and in reality, there could be more than one Higgs field sharing the responsibility of EWSB. Such multi-Higgs models are also motivated by other drawbacks of the SM. For example, low-energy supersymmetric models (Minimal Supersymmetric Standard Model -MSSM) proposed as a remedy to the hierarchy problem requires two doublet Higgs fields, resulting in the physical spectrum with five more scalars, three of which are neutral. The two-Higgs Doublet Models (2HDM) without supersymmetry has also been a popular theoretical option beyond the minimal Higgs mechanism proposed by the SM. The issue of dark matter, required by astrophysical observations but for which we lack a suitable candidate in the SM, is another important reason to attempt to go beyond the SM and, in these attempts, it is often the minimalism of the scalar sector of the SM that is sacrificed.
The 2HDM with one of the doublet fields not having any direct (at the level of the Lagrangian) interaction with the SM particles, except the gauge particles, is a promising candidate model in this regard. This is achieved by the imposition of a Z 2 symmetry under which one of the doublets is odd, while all other fields are even. Such an Inert Doublet Model (IDM) [3] would have the Higgs phenomenology, quite different from that of the SM as well as the MSSM or the usual 2HDM scenarios. For example, in the physical Higgs boson sector, all neutral scalars except one are odd under the Z 2 symmetry and are, therefore, always produced in pairs. This also means that the lightest of these cannot decay, and thus could be a candidate for dark matter. Adding a Z 2 -odd right-handed neutrino to this model can also generate small neutrino masses radiatively [4], and to generate leptogenesis [5], ideas which are followed up in further studies [6][7][8][9][10][11][12][13][14]. The model is shown to be helpful in explaining the LEP-paradox [15][16][17], and could also generate EWSB at one-loop level through Coleman-Weinberg mechanism [18].
Most of these studies focus on the pair production of the inert scalars, and consider final states involving leptons and missing energy. The purely hadronic channels are generically marred by the large irreducible background owing to the hadronic environment of the LHC.
A comprehensive report on the IDM search at Run 2 of the LHC is provided by the report of the Dark Matter Forum [44]. For the ILC, the effect of IDM on the triplet Higgs couplings is studied by Ref. [45]. In this work, we consider the dijet along with missing energy as the signature of IDM, and explore the possible parameter reach at the LHC, with moderate to high luminosity. Apart from the pair production and subsequent cascade decays, this channel receives significant contribution from the vector boson fusion (VBF) 5 , t−channel with the invisible Higgs (H) radiating from the mixed propagator, and the s−channel with quartic coupling involving H and W/Z. This paper is organized in the following way. In Section II, we describe the model including the present theoretical and experimental constraints available on the model parameters.
In Section III, we discuss our analysis, and finally, in Section IV we present the summary and conclusions of our study.

Inert Doublet Model
The IDM has one additional scalar doublet (under SU (2) L ), compared to the SM. This additional scalar, denoted by Φ 2 is odd under a discrete Z 2 symmetry imposed, while all the SM fields are even under this new symmetry. This Z 2 symmetry prohibits the Yukawa interactions of Φ 2 with the SM fields. The inert doublet, however, can have direct interaction with the gauge fields, providing the mechanism to generate the corresponding particles. A consequence of the Z 2 symmetry is that the lightest particle state belonging to Φ 2 is stable, and thus providing a candidate dark matter. Denoting the SM scalar doublet as Φ 1 , the scalar potential respecting SU (2) L ⊗ U (1) Y gauge invariance is given by 4 In the present study, bechmark points resulting from the analysis of [27] are used. 5 While this manuscript was being prepared, the study of Ref. [46] on dark matter searches through VBF appeared.
In the CP-conserved version, the parameters µ 2 1 , µ 2 2 , λ 1 , λ 2 , λ 3 , λ 4 , λ 5 are considered to be real. In the version with exact Z 2 symmetry, Φ 2 does not acquire any non-zero vacuum expectation value (VEV), and therefore, only the SM field, Φ 1 takes part in the electroweak symmetry breaking (EWSB). After the EWSB these scalar doublets may be written in the following form in the unitary gauge.
where v = 246 GeV is the vacuum expectation value of Φ 1 . Apart from the SM-like Higgs h, this presents a neutral scalar, H, a neutral pseudoscalar, A, and two charged Higgs bosons H ± , with the other degrees of freedom of Φ 1 becoming part of the massive gauge bosons through the Higgs mechanism. The masses of these physical scalars can be written in terms of parameters of the potential and v as We may note that the parameters are not completely free and independent of each other.
There are theoretical constraints arising from the vacuum stability [30,47], given by and to ensure perturbativity [16,47] we need to keep |λ i | ≤ 8π. Considering the case m H ± > (m H , m A ), Eq. 3 gives λ 5 < 0 for m H < m A , and λ 5 > 0 for m A < m H . Thus, the sign of λ 5 dictates whether H or A is the dark matter candidate. Apart from these theoretical constraints, we have experimental constraints coming from LEP observations [48,49]. From the non-observation of Z and W decays to dark Higgs bosons, we require The oblique parameter, T , receives contributions from the IDM, which could be written in terms of the mass splittings as  [24,42,54]. Also, the future Cherenkov Telescope Array (CTA) may be able to rule out heavier dark matter masses [55].
Previous studies of the LHC phenomenology include Ref. [20,28,29]. Most of these studies In this article, we shall focus on the dijet plus missing energy signal arising in the IDM scenario. As we shall see in the next section, this signal can originate from the production of H ± in association with H, with the subsequent decay of the charged Higgs, as well as from other VBF channels, and s−channels with quartic V V HH couplings, where V = Z, W , and t−channel with mixed propagator, radiating HH.

Discussion
Discovery of the charged Higgs boson will provide a smoking gun signature of the multi-Higgs models. Compatibility of such a scenario, and further identification of the couplings would be one of the first steps in establishing a specific multi-Higgs model. Benchmark Points used in the Table: BP1       and 2.24, respectively, at 3000 fb −1 luminosity. The ratio of the signal to background events is now a somewhat better 2.32%, 1.97% and 1.41, for the respective cases. The other BP with m H + = 500 GeV (BP6) does not spare that well with these selection criteria. The number of signal events at 1000 fb −1 corresponding to this BP is 70, giving a significance of 0.76, which is improved to 1.31 at 3000 fb −1 . The signal to background ratio is now 0.82%, which is less than the expected systematic uncertainty. We have summarised the above results in Table 3. We have employed a uniform selection criteria for all the BP's considered, keeping in mind that such analysis will be easier from the point of view of data analysis. We understand that, the systematic uncertainties could play a critical role while looking for BSM effects with such large SM background events expected. While we do not attempt an involved analysis including the effects of the systematic uncertainties, we have looked at the effects on the significance with an assumed uncertainty of 1% on the background, and 10% systematic uncertainty on the signal events. The resulting significance computed using the formula S √ B+(0.01×B) 2 +(0.1×S) 2 is presented in the Table 3. Clearly, the BP3 leaves a significance of about 2, which is sufficient to give a clear hint of a possible BSM signal. The significance corresponding to BP4 and BP5 lie between 1 and 2, while the other two BP's (BP2 and BP6) provide significance less than one.   Significance with assumed systematic uncertainties are given in the last two columns.
Please note that the above analysis is performed, keeping in mind a generic set of selection criteria that could be employed while searching for signals of the BSM scenarios, the presence of IDM in the present case. We conclude that, in contrast to the phenomenological studies involving leptonic final states, our analysis present a way to probe the large m H + regions up to a value of around 300 -400 GeV with high, but achievable, luminosity at the LHC through the dijet + M ET channel. Beyond these masses, establishing signals above background is somewhat difficult. However, upto even 500 GeV mass ranges, it is possible to probe the model with somewhat smaller significance.

Conclusion
The inert doublet model presents an interesting scenario within the multi-Higgs models, with a candidate dark matter, resulting in distinct phenomenology compared to other models like the 2HDM and MSSM. The model is compatible with all the experimental constraints arising from dark matter searches, as well as from collider experiments including the recent LHC measurements. In a specific scenario, we have considered the mass hierarchy of m H + > m A > m H , so that the neutral scalar is the dark matter candidate. We have considered the possibility to probe the model through 2j + M ET signal at the LHC with high luminosity.
This signal arises in IDM through the cascade decay of pair production of Higgs bosons of the dark sector, along with other production mechanism like VBF, s−channel with quartic Higgsgauge couplings, t−channel with two H radiating from the gauge-Higgs mixed propagator.
Contributions of cascade alone are significantly reduced at larger m H + values, whereas the contributions from other channels are somewhat independent of the Higgs mass, and remains at a few fb level throughout. This provides a promising possibility to probe scenarios with m H + > 150 GeV, which is almost impossible with other channels studied in the literature [20,28,29].
We have specifically considered a few benchmark points with m H + ranging from 80 GeV to 500 GeV. The effect of systematic effects are included through an assumed 1% and 10% uncertainties on the background and signal events. The best case scenarios are the cases with m H + around 200 -400 GeV, which could be probed at the LHC with about 3000 fb −1 integrated luminosity with a signal significance of about 2 for m H + = 200 GeV, and slightly lower, but still better than one for the larger mass regions. For higher mass case of m H + = 500 GeV, the significance is smaller than one, and cases with m H + beyond this range are harder to probe even at such high luminosity. The low mass scenarios with m H + = 80 GeV is also very difficult, mainly owing to the fact that the jets arising from these are too soft, and hard to isolate from the QCD background.
In summary, it is clear that probing 2j + M ET provides good handle on the search for IDM at LHC, and complements search through other leptonic channels. For scenarios like intermediate range of charged Higgs mass, this channel adds to other searches through leptonic and semi-leptonic channels. For larger mass range, where the leptonic channels become inefficient, the dijet plus missing energy channel discussed here proves to be an effective probe mechanism, albeit with the need of large luminosity of the order of 3000 fb −1 .