Searching for Dark Matter with $t\overline{t}$ Resonance

Many models containing particles which are candidates for dark matter, assume the standard model particles and the dark matter candidates are mediated by a spin-0 particle. At the LHC, one can use these models for dark matter searches. One of the possible approaches for the search of these models is by considering the decay of the spin-0 particle to a pair of $t\bar{t}$, thus modifying the pattern of the top quark pair invariant mass spectrum. This search suggests a good sensitivity in a parameter space different than the more traditional searches. We examine this sensitivity and put limits on two benchmark models containing candidates for dark matter, using previous ATLAS results. It was found that when the mediator mass ($m_{Y_0}$) and the dark matter candidate mass ($m_{\chi}$) have values of $m_{Y_0} \sim 2 \cdot m_{\chi}$, mediator masses in the range of $[400,600]$ GeV are excluded. We compare our results to direct detection experiments and show that we gain sensitivity for new regions which are not covered by other searches.


Introduction
Astrophysical observations support the existence of nonbaryonic component of the universe: Dark Matter (DM) [1][2][3][4]. DM particles have to be stable, massive, and do not participate in the strong and electro-magnetic interactions. There are many searches for DM candidates at the Large Hadron Collider (LHC) experiment that use different approaches to model the signal for DM. One of the most popular candidates is a Weakly Interacting Massive Particle (WIMP) [5]. At the LHC, one can search for DM particles (χ) produced in pp collisions.
Searches for DM using the models with DM and spin-0 mediators as a signal were already presented by ATLAS [6][7][8][9][10][11] and CMS [12][13][14][15][16]  TeV. Those searches focus on production of DM in association with a pair of top or bottoms quarks. In all of those searches, the mediator decays into a pair of DM particles, leaving a signature of high missing transverse momentum in the detector. A complementary search for these models can be achieved if the mediator decays to a pair of top quarks, leaving a more complex signature in the detector. Those searches are challenging since strong interference with the SM tt production is expected [17], leading to a pick-dip shape in the spectrum of the tt invariant mass [18,19]. Good sensitivity is expected in the parameter space where the mediator is heavier than twice the mass of the top quark. Representative Feynman diagrams for leading-order production of a tt pair by a spin-0 mediator (Y 0 ) and by the SM are presented at Figure 1.
There have been a few analyses targeting searches for heavy particles decaying to a pair of top quarks [20][21][22], creating a Breit-Wigner resonance in the m tt spectrum. However, for most of those analyses, spin-0 particles were not taken into account. The recent search published in a similar context, considering interference with the SM for tt production, was done by the ATLAS collaboration [23] using data of pp collisions at a centre-of-mass energy of √ s = 8 TeV and integrated luminosity of 20.3 f b −1 . This analysis used Two-Higgs-Doublet model for the interpretation of the results, using the spin-0 particle mass of 500 − 750 GeV. Here, we show how the search for tt resonance originating from spin-0 particles is important for models consisting of DM and covering parameter space that is not covered by the more traditional searches, especially in a signature with a mediator that decays into a pair of DM particles. This paper is organized as follow: in section 2 we discuss the theoretical framework for models with spin-0 mediators that couples to DM and top quarks; in section 3 we discuss two selected benchmark models; in section 4 we present the results; finally, we present our conclusions at section 5.

Theoretical Framework
In many models consisting of new spin-0 particles, the couplings with the SM fermions are being set as proportional to the SM Yukawa terms, by using the Minimal Flavor Violation assumption [29]. This leads to a motivation for a search in association with heavy flavor quarks. There are many types of these models assuming interactions between DM particles and spin-0 CP-odd or CP-even mediators, see for example [24,27,28].
For a spin-0 particle decays into a pair of Dirac fermions, which can be either DM particles (χχ) or SM fermions ( ff ), the calculation of the mediator width at tree level behaves as the following: Where n = 3 for a scalar (φ ) and n = 1 for a pseudo-scalar (a). Here, χ is a DM particle, f is a SM fermion, m φ /a is the mass of the scalar / pseudo-scalar, m χ is the DM mass, m f and y f are the corresponding mass and Yukawa term for the SM fermion, respectively. The parameters g med−χχ and g med− ff are model dependent couplings. In general, equations 1-2 can be applied to other types of DM particles, however we keep it as a benchmark assumption. In principle, interactions between the dark sector and the SM gauge bosons do exist in part of those models, and are taken into account when analyzing the results. The calculation of the mediator width from equations 1-2 presents an interesting behavior: In the case the mediator is heavy enough to decay into a pair of top quarks (i.e. ff = tt), the partial decay width of (φ /a → χχ) is significantly smaller. This is especially true in the case of high m χ , since the partial decay width of (φ → χχ) becomes even more suppressed. Therefore, the tt resonance search along the search for tt + χχ are complementary: the former gain better sensitivity for low DM masses, while the latter has better sensitivity for high DM masses. The Branching Ratio of this kind of spin-0 decay into a pair of top quarks is presented in Figure 2, assuming couplings only for DM and top quarks, and setting g med−χχ = g med− ff = 1. The behavior discussed above is well observed in those figures.

Benchmark Models
In order to emphasize how the behavior described in section 2 affects the sensitivity for models consisting of spin-0 and DM particles, we select two suitable benchmark models.

Benchmark 1: Simplified Models
The first model we consider is a simplified model with a spin-0 mediator couples to both to the SM fermions and to a new dark sector [24][25][26]. We will consider two choices: in the first one the interaction with the SM is mediated by a real scalar, and in the second we consider only a new pseudo-scalar (assuming that the associated scalar is decoupled from the low-energy spectrum). The dark sector, in general, can contain more than a single particle. We assume the dark sector contain only one DM particle which is a Dirac fermion, keeping in mind this assumption effects mostly on the width of the mediator, so the results can be easily converted to more complicated cases.
This model assumes Yukawa-like couplings between the dark sector mediator and the SM fermions. The Lagrangians for the scalar (L φ ) and pseudo-scalar (L a ) are [24]: Here, φ and a are scalar and pseudo-scalar fields (respectively) connect the SM with the dark sector, χ is the DM field, g χ is the DM-mediator coupling, g v is the flavouruniversal SM-mediator coupling and y f are the SM Yukawa couplings for fermions. If g v ∼ g χ and m φ /a > 2m χ , the decay of the mediator to DM is expected to dominate, unless the mediator is heavy enough for the top channel to open. This is true because the Yukawa couplings to light fermions are significantly small comparing to the Yukawa term of the top. The minimal viable value of Γ φ /a can be calculated from the other parameters. The mediator width can have larger values than the minimal one if additional dark sector particles are present. In our interpretation, however, we assume only one type of DM particle. For simplicity, the couplings g χ and g v were set to be equal to each other: g = g χ = g v . This leaves us with only three free parameters: m χ , m φ /a , g.

Benchmark 2: 2HDM+Z
The second model we consider is a type-II two-Higgs-doublet model with an additional U(1) gauge symmetry. This model, introduced in [27], is an extension to the familiar type-II 2HDM model [34], and introduces an extra spin-1 mass eigenstate which is denoted as Z . The pseudo-scalar (A 0 ) is the only one couples to a pair of DM particles, therefore this particle is identified as the mediator between the SM particles and the dark sector. The decoupling limit [35] is assumed. This model has 6 parameters, which is set as following: The mass of the light scalar, m h = 125 GeV, as the measured value at the LHC; the mass of the heavier scalar and charged scalar m H = m H ± = 300 GeV; the mass of the pseudo-scalar, m A 0 = 400 GeV; the mass of Z , m Z = 3 TeV; and the coupling of the new spin-1 boson, g Z = 1. The mass m Z was chosen to be high which makes it effectively decoupled. Therefore, the results are valid also for other values of m Z , as long as it is heavy enough to avoid decays of A 0 into Z with another Higgs boson. In the chosen parameter space the effect of g Z was found to be negligible as well, therefore the value chosen for this parameter was set in an arbitrary way. The ratio between vacuum expectation values of the two Higgs doublets, tan(β ), and the mass of the DM particle, m χ , are free parameters.
In the selected parameter space the masses of the CPeven bosons are set to be lower than the tt decay threshold, and the mass of the CP-odd boson is higher than this threshold. Therefore, the dominant process for tt resonant production is via the pseudo-scalar mediator: A 0 → tt.

Results
Limits on spin-0 mediator models were already set at 95% Confidence Level (CL) at [17], using the latest ATLAS ttresonance search with an available data [21]. Limits were also put on spin-0 mediators at [23]. In both of the cases interference effects with the SM were considered, and the signal was generated at Next to Leading Order (NLO) and Next to Next to Leading Order (NNLO) in QCD corrections, respectively. For those limits, however, no interaction with DM particles was taken into account, as we do in this section. The results of the former was found to be more efficient for mediator mass which is lower than 500 GeV, while the latter is more sensitive for mediator masses which is higher than 500 GeV. This is expected since the corresponding AT-LAS analysis used spin-0 particle masses which are higher than 500 GeV.
The experimental resolution on the tt invariant mass, m tt , was calculated to be 8% at both of the analyses. Since the width of the mediator has a strong effect on the shape of the pure signal and interference distributions, an upper limit Γ total m φ /a < 8% was set, where Γ total is the total decay width of the mediator. In the case that the mediator decays into a pair of DM particles, Γ total m φ /a < 40% was used to keep the narrow width approximation valid. Results with higher total widths were discarded. Figure 3 presents upper limits at 95% CL on the coupling g. The figure presents the lowest coupling excluded for the model. Both scalar and pseudo-scalar mediator cases are considered. The best limits obtained from [10] searching for tt + χχ processes are presented as well for comparison. The exclusion contour is more stringent for the tt resonance searches when m φ /a ≥ 400 GeV, especially when the DM mass is high. The limits obtained from the tt resonance are stronger for the scalar case since the width calculation (see equations 1, 2) allows higher values for the pseudo-scalar case with similar parameters, leading to higher total widths which we discard.

=8 TeV, L = 20.3 fb s
Limits at 95% CL   Fig. 3 Exclusion contour for DM simplified models in the (m φ /a , m χ ) plane for the scalar (upper) and pseudo-scalar (bottom) scenarios. The contour corresponds to the lowest value of the coupling g = g χ = g v allowed. Results from both tt resonance and tt + chiχ signatures are presented for comparison. Figure 4 presents exclusion contour at 95% CL in the plane of (m χ ,tanβ ). The rest of the parameters of the model are fixed as discussed at section 3. The best limits obtained from [10] searching for tt + χχ processes are not presented since they do not provide any constraint in the considered parameter space. In this model the coupling between the top quarks and A 0 is proportional to (tanβ ) −1 . Therefore, the lower part of the exclusion contour is due to mediator width values which are higher than 8% (see equation 2 with g med− ff = (tanβ ) −1 , while the upper part is due to lower cross sections excluded by the analysis.

Comparison to Direct Detection Experiments
Results from LHC analyses with DM simplified models can be compared to direct detection experiments, for both spinindependent and spin-dependent cases. For this purpose, we use the prescription described at [36], setting limits on DMnucleon interaction.

Spin-Independent
For scalar simplified models, we set an upper limit on the DM-nucleon cross section (σ SI ) as follows [36]: σ SI 6.9 · 10 −43 · cm 2 · (g ν · g χ ) 2 125GeV m φ 4 µ nχ 1GeV 2 (5) Where we introduce the nucleon mass, m n 0.939 GeV, and the reduced DM-nucleon mass, m n · m χ /(m n + m χ ). In the results presented at 4.1, we find that scalar masses in the range of m φ ∈ [400, 600] GeV are excluded, with the lowest couplings in the range of g ∈ [1.1 − 2.0] and m χ ≥ 160GeV . In order to use the prescription above the couplings should be fixed, therefore we choose g χ = g v = 1.5. The results obtained in this paper are presented along with direct detection experiments [37][38][39][40][41] and with the contour calculated at [10] at Figure 5. There are two caveats for this comparison, however: first, the results we state are at 95% CL, while the results of the other experiments are at 90% CL; second, one has to keep in mind that the comparison is model dependent.

Spin-Dependent
For the pseudo-scalar scenario, a velocity suppression term in the non-relativistic limit creates large difference by several orders of magnitude in favor of LHC results [10, 36], and therefore it is not presented. However, this actually makes the motivation to use tt resonance interpretation for DM models even stronger: it covers regions with higher DM masses, for which both tt + χχ and direct detection searches are insensitive to.
Dirac, g Fig. 5 Upper limit on spin-independent DM-nucleon cross section (σ SI ) as a function of the DM mass. The exclusion limit of the tt resonance obtained in this paper at 95% CL is compared with limits from direct detection experiments and from the latest tt + χχ by ATLAS at 90% CL.

Conclusion
The necessity of tt resonance search as a complementary measurement for models consisting of DM has been emphasized for two different models. Despite this signature is challenging from the experimental point of view, it provides unique access to regions in the parameter space with high DM masses, where other signatures quite often do not have the sensitivity to search. Examining those searches using centre of mass energy of √ s = 13 TeV and with higher luminosity now recorded by ATLAS and CMS collaborations allows to expand even more the parameter space covered by this signature.