Probing stops in the coannihilation region at the HL-LHC: a comparative study of different processes

In the minimal supersymmetric model, the coannihilation of the lighter stop $\tilde{t}_1$ and bino-like dark matter $\chi$ provides a feasible way to accommodate the correct dark matter relic abundance. In this scenario, due to the compressed masses, $\tilde{t}_1$ merely appears as missing energy at the LHC and thus the pair production of $\tilde{t}_1$ can only be probed by requiring an associated energetic jet. Meanwhile, since $\tilde{t}_2$ and $\tilde{b}_1$ are correlated in mass and mixing with $\tilde{t}_1$, the production of $\tilde{t}_2\tilde{t}_2^*$ or $\tilde{b}_1\tilde{b}_1^*$, each of which dominantly decays into $\tilde{t}_1$ plus $Z$, $h$ or $W$ boson, may serve as a complementary probe. We examine all these processes at the HL-LHC and find that the $2\sigma$ sensitivity to $\chi$ mass can be as large as about 570 GeV, 600 GeV and 1.1 TeV from the production process of $\tilde{t}_1\tilde{t}_1^*+{\rm jet}$, $\tilde{t}_2\tilde{t}_2^*$ and $\tilde{b}_1\tilde{b}_1^*$, respectively.


I. INTRODUCTION
The nature of dark matter (DM) remains a mystery in particle physics. In minimal supersymmetric standard model (MSSM) with conserved R−parity, the lightest neutralino χ can serve as a DM candidate. However, the null results of DM direct detections [1][2][3] give significant constraints on the neutralino sector in the MSSM. It is notable that the stop-bino coannihilation, in which DM is the bino-like lightest supersymmetric particle (bino-LSP) and the stop (t 1 ) is the next-to-lightest supersymmetric particle (NLSP) and nearly degenerate with the bino-LSP, provides a feasible mechanism to accommodate the DM relic abundance.
Because of the extremely weak interaction between the bino-LSP and nucleons, this scenario can easily evade the DM direct detection constraints [4]. However, the search of stops at the LHC in this scenario is rather challenging 1 . The reason is that due to the compressed masses,t 1 is merely appearing as missing energy and the pair production oft 1 can only be probed by requiring an associated energetic jet.
On the other hand, we should note thatt 2 andb 1 are correlated witht 1 sincet L,R mix into mass eigenstatest 1,2 (see the following section) whileb L (b 1 =b L , neglecting the sbottom mixing) has the same soft mass ast L . Furthermore, to avoid fine-tuning, these particles should not be too heavy 2 because at one-loop level we approximately have [13,14] ∆ ≡ δm 2 where m SUSY = √ mt 1 mt 2 , Λ is the cut-off scale, Q 3 = (t L ,b L ) and U 3 =t R . Therefore, the production oft 2t * 2 orb 1b * 1 , followed by the dominant decays intot 1 plus Z, h or W boson, may serve as a complementary probe of stops in such a stop-bino coannihilation scenario.
In this work we perform a comprehensive study for all these correlated processes at the HL-LHC (14 TeV, 3000 fb −1 ). We will first perform a scan to figure out the stop-bino coannihilation parameter space. Then we display the properties oft 1,2 andb 1 in this stopbino coannihilation parameter space. For thet 1t * 1 + jet production which has been searched at the LHC, we will show its current sensitivity and then extend the coverage to the HL- 1 The search of stops at the LHC has been a hot topic and numerous studies have been performed in various cases, e.g., the large or small stop-top or stop-LSP mass splitting [5][6][7], the single stop production [8], the stop in natural SUSY [9], machine learning in stop production [10] and other miscellaneous cases [11]. 2 Note that the stops cannot be too light in order to give the 125 GeV Higgs mass except a singlet is introduced [12].
LHC. For the productionst 2t * 2 andb 1b * 1 , followed the dominant decayst 2 →t 1 + Z/h and b 1 →t 1 + W , we will examine the HL-LHC sensitivities through Monte Carlo simulations of the signals and backgrounds.
The structure of this paper is organized as follows. In Sec. II, we briefly review stop-bino coannihilation scenario and discuss the details of our scan. In Sec. III, we perform detailed Monte Carlo simulations for the productions oft 1t * 1 + jet,t 2t * 2 andb 1b * 1 at the HL-LHC. Finally, we give our conclusions in Sec. IV.

II. STOP-BINO COANNIHILATION
In the MSSM, the mass matrix of stop sector in gauge-eigenstate basis (t L ,t R ) is given by where The mixing betweent L andt R is induced by X t = A t − µ cot β, where A t is the stop soft-breaking trilinear coupling. One can diagonalize the mass matrix through a rotation wheret 1 andt 2 are the mass eigenstates of lighter and heavier stops, respectively. The mixing angle θt betweent L andt R is determined by In the early universe, the freeze-out number density for the bino-LSP DM will be overabundant because the annihilation cross section σ in the Boltzmann equation is too small to keep DM thermal equilibrium with SM particles for sufficient time 3 . When the stop (t 1 ) mass is close to bino-LSP mass, the annihilation cross section σ is replaced by the effective cross section [15] where ∆ i = (m i − m χ )/T , m i and g i are the mass and degrees of freedom of the particle i = {χ,t 1 }, and σ ij denotes the cross section of particle i annihilating with particle j. The annihilation modes oft 1 with χ or itself can enhance σ eff ift 1 is nearly degenerate with the bino-LSP. We can also see that σt 1t1 is suppressed by double exponents compared to σ χχ , while σt 1 χ is suppressed by single exponent. Therefore, when the mass splitting ∆t 1 is small, the contribution to relic abundance from thet 1 χ annihilation tends to be more important than that from thet 1t * 1 annihilation, although this also depends on their respective cross section.
In order to obtain the stop-bino coannihilation parameter space, we use SuSpect 2.41 [16] to calcualte the mass spectrum and SDECAY 1.5 [17] to evaluate sparticle decay width and branching ratio. We regard the lighter stop as right-handed dominated. The reason for such assumption is that if mt R = mt L at some high energy scale, mt R tends to be smaller than mt L at the electroweak scale from the renormalization group equations (RGE) evolution [18].
The stop mixing angle cos 2 θt < ∼ 0.5 is required so that the lighter stopt 1 is right-handed [19] is used to compute the DM relic abundance Ω χ h 2 .
In our scan we impose the following constraints 4 : (i) The lighter CP-even Higgs mass is required to be in the range of 125 ± 3 GeV [20,22].
(ii) The DM relic abundance satisfies the observed value Ω χ h 2 = 0.1186 ± 0.0020 within 2σ range [23]. 4 Here we do not require SUSY to explain the muon g-2 anomaly, which requires light sleptons [21] (iii) To avoid the existence of a color or charge breaking vaccum deeper than the electroweak vacuum in the scalar potential, the trilinear coupling A t should not exceed the upper In the left panel of Fig. 1, we display the stop-bino coannihilation parameter space that satisfies the constraints (i)-(iii), where the B physics constraints are ignored because of the decoupled higgsino mass parameter, and the contribution of the stops to h → γγ (and gg) [18] is also negligibly small. We can see that the mass splitting ∆m(t 1 , χ) increases with where is O(10 −4 ) if all soft-breaking parameters have the same order of magnitude. It is clear that the four-body decay width increases more sharply with ∆m(t 1 , χ) than the FCNC two-body decay width, as shown in the right panel of Fig. 1. However, due to the ratio of Γ 4−body /Γ(t 1 → cχ) is suppressed by ∆m(t 1 , χ) 6 /m 2 t m 4 W and the small coefficient, the four-body decay is not competitive with thet 1 → cχ decay.
Because the soft c-jet from thet 1 → cχ decay is hard to detect, the search strategy for this coannihilation scenario is usually to exploit thet 1t * 1 production in association with an energetic jet from the initial state radiation (ISR) which boostst 1t * 1 system and produces large missing energy at the LHC. The parton level events of the signal and backgrounds are generated with MadGraph5 aMC@NLO [30]. Then, the event parton showering and hadronization are performed by Pythia [31]. We use Delphes [32] to implement detector simulations where the anti-k t jet algorithm and ∆R = 0.4 [33] are set for the jet clustering.
To discriminate the signal and backgrounds, we require a leading jet with p T (j 1 ) > 300 GeV, |η| < 2.4 and azimuthal angle ∆φ(j 1 , p miss T ) > 0.4. We veto events with electrons with p T > 20 GeV, |η| < 2.47 or muons with p T > 10 GeV, |η| < 2.5 to reduce the W (→ ν )j and tt backgrounds. Events having more than four jets with p T > 30 and |η| < 2.8 are vetoed. The signal regions are defined with E miss T cuts: 300 GeV, 500 GeV, 700 GeV and 900 GeV. The signal significance is calculated as S/ √ B in which the total background where the systematic error on the backgrounds is set to 1%.
In the left panel of Fig. 1, we display the 2σ exclusion limits at the 13 TeV LHC with L = 36.1 fb −1 (the region on the left side of the curve is excluded) and the sensitivity at the 14 TeV LHC with L = 3000 fb −1 . We can see that the current monojet search gives a loose limit on the bino-LSP DM mass m χ > ∼ 260 GeV and this limit can be raised to 570 GeV at the 14 TeV LHC with L = 3000 fb −1 .

production
From the naturalness argument in Sec. I,t 2 can not be too heavy and thet 2t * 2 prodcution can be sizable at the LHC. Since the LSP is bino-like in our scenario, thet 2 decay modes are mainlyt 2 →t 1 Z andt 2 →t 1 h. The corresponding decay widths are given by [34] where λ(a, b, c) = [1 − (b + c)/a] 2 − 4bc/a 2 is the kinematic factor. In the limits m 2 t 2 , m 2 t 1 m 2 Z,h , the factor λ(m 2 t 2 , m 2 t 1 , m 2 Z,h ) approximately equals to (1 − m 2 t 1 /m 2 t 2 ) 2 and then Γ(t 2 →t 1 h) It should be noted that the decay width Γ(t 2 →t 1 Z) is always larger than Γ(t 2 →t 1 h) even though the small loop corrections are taken into account [35,36]. In Fig.2, we plot the branching ratio oft 2 →t 1 Z andt 2 →t 1 h. It is clear that Br(t 2 →t 1 h) is lower than Br(t 2 →t 1 Z) and their difference decreases with the mass splitting ∆m(t 2 ,t 1 ) between heavier and lighter stops. Since the masses oft 1 and the bino-LSP are nearly degenerate, thet 1 will appear as missing energy and the signal oft 2t * 2 production at the LHC is where we neglect the hh+E miss T channel because its production rate is smaller than the above channels. Here we investigate the 2 2b final states, in which leptons come from Z decay and GeV < m < 100 GeV to reconstruct Z bosons.
(ii) Jets must have p T (j) > 30 GeV and |η j | < 2.5. We require two b-jets and the b-jet tagging efficiency is set to be 80%.
(iii) From the right panel of Fig. 3, the signal regions are designed according to E miss  In Table I shown in Fig. 3. After imposing all these cuts, the significance S/ √ B for the benchmark point is about 5.32σ.
In Fig. 4 we present the observability for thet 2t * 2 production. The points to the left of the blue curve have a sensitivity above 2σ level and the colormap shows the change in mb 1 . We can see that this stop pair production can cover m χ < ∼ 600 GeV for mt 2 < ∼ 1100 GeV at 2σ level. This result is not sensitive to the mass splitting ∆m(t 1 , χ).
The sbottomb 1 is lighter than the stopt 2 because of the mixing between left and right handed stops. Sinceb 1 is left-handed in our scenario, it could decay to the longitudal component of W boson in association witht 1 . The branching ratio ofb 1 →t 1 W is depicted in Fig. 2. As we see,b 1 dominantly decays to W boson plust 1 . Then, the signal ofb 1b * 1 production at the LHC is   We display the observability of theb 1b * 1 production in Fig. 6. It can been seen that such sbottom pair production can cover m χ < ∼ 1.1 TeV for mb 1 < ∼ 1375 GeV at 2σ level.
Correspondingly, the lower bound of mt 2 can be pushed up to around 1.4 TeV. Therefore, this result is obviously better than thet 2t *

IV. CONCLUSIONS
We have studied the stop-bino coannihilation region, in which the observed dark matter relic abundance can be reproduced. To test the scenario, we have examined three correlated production processest 1t * 1 + jet (thet 1 's being invisible),t 2t * 2 andb 1b * 1 , followed by the decays t 2 →t 1 + Z/h andb 1 →t 1 + W , at the HL-LHC. Through Monte Carlo simulations for the signals and backgrounds, we found that the 2σ sensitivity to the bino-like LSP can reach about 570 GeV, 600 GeV and 1.1 TeV from the production process oft 1t * 1 + jet,t 2t * 2 and b 1b * 1 , respectively. These three channels should be jointly considered at the future HL-LHC experiment.