Resonant Higgs pair production as a probe of stop at the LHC

Searching for top squark (stop) is a crucial task of the LHC. When the flavor conserving two body decays of the stop are kinematically forbidden, the stops produced near the threshold will live long enough to form bound states which subsequently decay through annihilation into the Standard Model (SM) final states. In the region of stop mixing angle $\theta_{\tilde{t}} \to 0$ or $\pi/2$, we note that the LHC-13 TeV diphoton resonance data can give a strong bound on the spin-$0$ stoponium ($\eta_{\tilde{t}}$) and exclude the constituent stop mass $m_{\tilde{t}}$ up to about 290 GeV. While in the large stop mixing region, the stoponium will dominantly decay to the Higgs pair. By analyzing the process $pp \to \eta_{\tilde{t}} \to h (\to b\bar{b} )h (\to \tau^+\tau^-)$, we find that a large portion of the parameter space on the $m_{\tilde{t}_1}$ - $\theta_{\tilde{t}}$ plane can be probed at $2\sigma$ significance level at the LHC with the luminosity ${\cal L}=3000$ fb$^{-1}$.


Introduction
Since the discovery of the Higgs boson at the Run 1 of the Large Hadron Collider (LHC) in 2012 [1,2], the persuit of physics beyond the SM (BSM) becomes the primary goal in particle physics community. One of the most important guidelines in this endeavor is the famous naturalness principle which states that the physics at weak scale should be insensitive to quantum effects from much higher scales. Among all the proposed scenarios, supersymmetry (SUSY) remains as one of the most popular models, in which the quantum correction to the Higgs mass from the top quark is canceled by that from the stop. In this regard, the search for stop [3][4][5][6][7][8][9][10][11][12][13][14][15][16] is an important direction of testing SUSY naturalness at the LHC .
Till now, numerous efforts have been dedicated to the searching for stop in the LHC experiments. The experimental signatures of stop pair production depend on the stop-LSP mass splitting which leads to different decay modes. For instance, when mt 1 > m t + mχ0 1 andt 1 mainly decays to tχ 0 1 , the top quark from stop decay can be quite energetic and a stop mass up to 940 GeV for a massless lightest neutralino has been excluded by the very recent LHC run-2 data [17]. When the flavor-conserving two body decays channels liket 1 → tχ 0 1 andt 1 → bχ + 1 are kinematically forbidden, the primary decay channels of the light stop would be the three-body decayt 1 → W + bχ 0 1 , the two-body flavor-changing decayt 1 → cχ 0 1 or the four-body decayt 1 → bf fχ 0 1 [18][19][20][21]. The current null results of LHC searches for these decay channels have correspondingly excluded the stop mass up to ∼ 500 GeV, 310 GeV and 370 GeV for certain mass splitting between the stop and the LSP [17].
It should be mentioned that such a light stop usually has very small decay width [22] compared to the typical binding energy oft 1t stop. Therefore, it is expected that the search of stoponium can provide a complementary probe to the direct stop pair production at the LHC.
The phenomenologies of the stoponium have been studied at colliders [22][23][24][25][26][27][28][29][30]. In particular, the diphoton channel was studied and found to be a promising way to observe stoponium at the LHC in Refs. [22][23][24]. The diboson decay of stoponium with W W and ZZ final states were also examined in [28,29]. In [31], the authors investigated the di-Higgs decay of stoponium with bbγγ final states and found it to be a viable channel at the LHC. But the loop induced diphoton decay of the Higgs boson can be sizably affected by other sparticles, such as the light stau in the MSSM [32].
In this paper, we first confront the stoponium with the recent data of searching for high mass resonances at 13 TeV LHC. Then we explore the potential of probing the stop in Higgs pair production with bbτ + τ − final states at high-luminosity LHC (HL-LHC). As a comparison with bbγγ channel, although the bbτ + τ − channel suffers from relatively complicated backgrounds, it has a larger branching ratio. Besides, it is expected that the reconstruction efficiency of τ can reach ∼ 80% with the likelihood τ taggers in the future LHC experiment [33,34]. This will make bbτ + τ − channel become another promising way of discovering, or confirming the stoponium at the LHC. The paper is organized as follows. In Sec. 2, we introduce productions and decays of the stoponium and display the limits on stoponium mass from the LHC-13 TeV data. In Sec. 3, we investigate the observability of the di-Higgs decay of the stoponium with bbτ − τ + final states at the LHC. Finally, we draw our conclusions in Sec. 4.

Diphoton resonance constraint on the stoponium
In the gauge-eigenstate basis, the stop mass matrix is given by where mQ 3L and mŨ 3R denote the soft-breaking mass parameters of the third generation left-handed squark doubletQ 3L and the right-handed stopŨ 3R , respectively. A t is the soft-breaking trilinear parameter. We neglect the generation mixing in our study. The hermitian matrix Eq. (2.1) can be diagonalized by a unitary transformation: where θt ∈ [0, π) is the mixing angle between left-handed (t L ) and right-handed (t R ) stops. A very narrow decay width of stop 1 can naturally appear in the compressed region, in which the decay width of stop is suppressed either by phase space or loop factor. If the Γt 1 is much smaller than binding energy, stop pair produced near the threshold could form a bound state due to the strong attractive force mediated by gluons. Then, these bound states will proceed annihilation decay rather than the prompt decay of the constituent stop.
The production of stoponium is mainly from the gluon fusion at the LHC. In narrowwidth approximation, the leading order (LO) cross section of stoponium is be given by [22] whereŝ is squared center-of-mass energy at the parton level and is taken asŝ = m 2 ηt in our calculation. Γηt →gg is the width of stoponium decay to di-gluon. The next-to-leading order QCD radiative corrections to stoponium production have been calculated in [36]. We include these effects by using the values of K-factor given in [37].
It should be noted that there are two main uncertainties in the computation of stoponium production rate. One of them lies in the parametrization of the wavefunction, which depends on the choice of QCD scale parameter Λ [38]. Larger value of Λ leads to greater coupling and hence stronger binding between the constituent stops. We adopt Λ = 300 MeV by following [37]. The other uncertainty comes from the contributions of excited bound states, such as nS(n ≥ 2) and 1P states. In particular, the effects of higher S-wave states are compared in [37]. The excited states can contribute by either first decaying into the lowest stoponium state (1S) or decaying directly into SM final states. For instance, the non-annihilation decay of the 2S state could go entirely to the 1S state and the signal could be merged with that of the ground state due to the detector energy resolution [22]. In general, states with different angular momentum could have very distinct decay modes. Without thorough knowledge of the decay modes, we will take a conservative approach and focus on the 1S state.
In Fig. 1, we display the decay branching ratios of the stoponium with respect to the mixing angle θt, where we assume tan β = 10, mt 1 = 0.2 TeV and mt 2 = 2 TeV.  Figure 1: The decay branching ratios of the stoponium with respect to the mixing angle θt. Here we take tan β = 10, mt 1 = 0.2 TeV and mt 2 = 2 TeV for example. Note that the branching ratios are symmetric about θt = π/2, we plot only the region θt ∈ [0, π/2] here and also in Fig. 5.
It can be seen that the stoponium dominantly decays to di-gluon when the mixing angle θt approaches 0 or π/2. While ift L andt R have a sizable mixing, the stoponium will dominantly decay to a pair of Higgs bosons because of the enhancement induced by the Higgs-stop coupling λ ht 1t1 2 . We also checked and found that branching ratios of the stoponium have a weak dependence of tan β. So we will assume tan β = 10 in our following calculations. Due to the distinctive signature of two photon final states, the stoponium decay to diphoton offers a very sensitive way to observing stoponium at hadron colliders. The bound on stoponium from 8 TeV run at the LHC is given in [40]. In Fig. 2, we update the result with the LHC-13 TeV diphoton resonance data [41]. We can see that the stoponium mass can be excluded up to about 580 GeV for the mixing angles θt = π/2, which is stronger than that from LHC-13 TeV direct searches for the four-body decaỹ t 1 → bf fχ 0 1 with pure bino LSP in the region of mt 1 − mχ0 1 < 15 GeV [17]. However, due to the branching ratio suppression effect, there is still no constraint on the stoponium from the diphoton data for the mixing angles θt = π/8, π/4. We also checked the bounds on the stoponium from current null results of LHC searches for Zγ and diboson resonances and found that they can not give stronger limits than the diphoton data. 2 The trilinear coupling between the SM Higgs and stop quarkt1 takes the form [40]: ATLAS, s =13 TeV, 15.4 fb -1 Figure 2: Constraint on the stoponium from the LHC-13 TeV diphoton resonance data. The 2σ experimental upper limit (yellow band) is taken from [41]. Here we also assumed tan β = 10 and mt 2 = 2 TeV.
3 Di-Higgs decay of stoponium with bbτ + τ − final states at the LHC Given that the stoponium can have a large branching fraction into the two Higgs bosons, we will investigate its observability through the resonant Higgs pair production with bbτ + τ − final states at the 14 TeV LHC, where one tau lepton decays hadronically (τ had ) and the other decays leptonically. τ had is reconstructed using clusters in the electromagnetic and hadronic calorimeters with medium criterion [42]. We generate parton-level events of the stoponium production and subsequent decay into Higgs pair using the code for resonant Higgs pair production [43] within MG5 aMC@NLO [44], in which τ lepton decays are modeled by TAUOLA [45]. Then we perform parton shower and hadronization with PYTHIA [46]. The fast detector simulation is implemented with Delphes [47]. We use the b-jet tagging efficiency parametrization as 80% [48] and set the misidentification 10% and 1% for c-jets and light jets, respectively. We also assume the τ tagging efficiency is 40%. We set the renormalization scale µ R and factorization scale µ F as the default event-by-event value. We cluster the jets by choosing the anti-k t algorithm with a cone radius ∆R = 0.4 [49]. The major backgrounds come from events with a jet misidentified as τ had , including tt, Z(→ τ + τ − )bb and Z(→ τ + τ − )jj processes.
In Fig. 3, we present distributions of the di-tau invariant mass m τ τ , two b-jets invariant mass m bb , the transverse mass of the lepton plus missing energy system m ν T and the di-tau transverse momentum p τ τ T . The simple transverse mass method is used to reconstruct m τ τ from the observed lepton, τ had and E miss T . One can see that m τ τ distribution shows a Figure 3: Distributions of the di-tau invariant mass m τ τ , two b-jets invariant mass m bb , the transverse mass of the lepton plus missing energy system m ν T and the di-tau transverse momentum p τ τ T . The stoponium mass is taken as m ηt = 500 GeV.
relatively broad peak around the Higgs boson mass with a long tail 3 , as a comparison with m bb distribution. Another variable m ν T can effectively reduce tt background since the lepton in signal is not from W boson decay. The variable p τ τ T is used to select the events with the boosted Higgs boson candidate on the transverse plane. For such events, m τ τ resolution is improved and a better separation between the signal ηt → τ τ and the background Z → τ τ is achieved. This selection also has the advantage of reducing the QCD multijet background.
In our analysis, we select events that satisfy the following criteria: • We require exactly one lepton (e or µ) with p T ( ) > 26 GeV , |η e | < 2.47 or |η µ | < 2.5. We further require the presence of a hadronically decayed tau τ h carrying opposite electric charge with p T (τ h ) > 20 GeV and |η τ h | < 2.5.
• We require at least two jets with p T (j) > 30 GeV and |η j | < 2.5 and two of them are b tagged.
• We require 80 GeV < m bb < 150 GeV, 80 GeV < m τ τ < 150 GeV, m ν T < 50 GeV, p τ τ T > 120 GeV and |m bbτ τ − m ηt | < 0.08m ηt . In Table 1, we present a cut flow of cross sections for the signal and backgrounds at 14 TeV LHC. After the di-b jets and di-tau invariant mass cuts, we find that the cut m ν T < 50 GeV can reduce the tt background by about half. The cut p τ τ T > 120 GeV can suppress Z(→ τ τ )jj and Z(→ τ τ )bb backgrounds by an extra factor of six. The total invariant mass cut |m bbτ τ − m ηt | < 0.08m ηt can further hurt tt background by about O(10 2 ) and Z(→ τ τ )jj and Z(→ τ τ )bb by about O(10). In Fig. 4, we plot the cross sections of the process pp → ηt → hh with bbτ + τ − /bbγγ final states needed for the signal significance S/ √ B = 5σ at the HL-LHC. It can be seen that the cross section of the process pp → ηt → hh → bbτ + τ − /bbγγ should be about 800 fb/100 fb to reach 5σ significance at m ηt = 400 GeV. When the stoponium is heavier than about 700 GeV, the required cross section of bbτ + τ − channel for a tau tagging efficiency τ = 40%   Figure 5: 2σ exclusion limits from the di-Higgs decay channel ηt → hh → bbτ + τ − and the di-photon decay channel ηt → γγ for mt 2 = 1 TeV and 2 TeV on the plane of mt 1 versus stop mixing angle θt at the HL-LHC. The result of di-photon decay channel is taken from Ref. [40].
can be comparable with that of bbγγ channel studied in [31]. If τ tagging efficiency can be improved to ∼ 80% estimated in [33,34], the sensitivity of bbτ + τ − channel is expected to become better than that of bbγγ channel for m ηt 570 GeV.
In Fig. 5, we show 2σ exclusion limits from the di-Higgs decay channel ηt → hh → bbτ + τ − and the di-photon decay channel ηt → γγ for mt 2 = 1 TeV and 2 TeV on the plane of mt 1 versus stop mixing angle θt at the HL-LHC. We can see that the stop mass mt 1 can be excluded up to ∼ 380(450) GeV in the large stop mixing region π/7 θt π/3 by the di-Higgs decay channel ηt → hh → bbτ + τ − , since the branching ratio of ηt → hh depends on the Higgs-stop coupling λ ht 1t * 1 . For a given mixing angle θt, a larger mt 2 sets a stronger bound on mt 1 because the Higgs-stop coupling λ ht 1t * 1 is proportional to the mass difference m 2 The di-photon decay channel ηt → γγ mainly excludes small stop mixing region, such as θt π/7 or θt π/3, which is complementary to the di-Higgs decay channel.

Conclusions
In this paper, we confront the stoponium with the recent data of searching for high mass resonances at 13 TeV LHC, and explore the potential of probing the stoponium in resonant Higgs pair production with bbτ + τ − final states at the LHC. We note that the LHC-13 TeV diphoton resonance data can give a strong bound on the spin-0 stoponium (ηt) and exclude the constituent stop mass mt 1 up to about 290 GeV in the small stop mixing region. While in the large stop mixing region, the stoponium will dominantly decay to the Higgs pair. By analyzing the process pp → ηt → h(→ bb)h(→ τ + τ − ), we find that the stop mass mt 1 can be excluded up to ∼ 380 (450) GeV at the LHC with the luminosity L = 3000 fb −1 .