1 Introduction

Supersymmetry (SUSY), devised to elegantly solve the gauge hierarchy problem, used to and will provide the major impetus for building high energy colliders, such as the LHC. Guided by the naturalness argument [1,2,3,4], stops, among a bunch of new particles predicted by SUSY, should be light and therefore, along with their color charges, may furnish the first “smoking-gun” signature for SUSY at the LHC.

Nevertheless, confirmatory hints for light stops at the LHC are absent so far. Considering that the LHC is now already running at the CM energy \(\sqrt{s}=13\) TeV, the null results arouse concerns about the existence of low energy SUSY, or more concretely a light stop below the TeV scale. Actually, the current LHC search strategies [5,6,7,8,9,10,11,12] still leave a wide room for a relatively light stop (\({\sim }500\) GeV), provided that a very large missing energy from stop decay is not present, say due to a compressed spectrum. Such a spectrum is characterized by very close mass between stop \(\tilde{t}\) and the lightest sparticle (LSP) [13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38], or more loosely speaking \(m_{\tilde{t}}\sim m_t+m_\mathrm{LSP}\) and \(m_{\tilde{t}}\sim m_b+m_W+m_\mathrm{LSP}\).Footnote 1 Despite allowing for a natural low energy supersymmetry, it is challenging to uncover such a stop at the LHC. Nevertheless, if naturalness is reliable, the heavier stop and the lighter sbottom should not lie far above \(m_{\tilde{t}_1}\), thus being detectable. This motivates the searches for the signal of \(\tilde{t}_2\) pair production with \(\tilde{t}_2 \rightarrow \tilde{t}_1 h/Z \) decay at the LHC run-I [40, 41]. Moreover, through searching for the final state of multi-leptons and/or multi-b-jets [42], the heavier stop/sbottom with masses below \(\sim 1\) TeV decaying into \(\tilde{t}_1\) and heavy bosons are found to be detectable at the 13/14 TeV LHC with an integrated luminosity of \(\mathcal {O}(100)~\text {fb}^{-1}\) [43,44,45,46].

On the other hand, boosted objects such as a boosted top quark, vector bosons and the Higgs boson being new physics signatures have been receiving increasing experimental attention [47,48,49,50], where the new physics scale is pushed into the higher and higher region. The substructures of these boosted objects furnish a powerful tool to distinguish the signatures from the huge QCD backgrounds. Taking into account that they (top etc.) dominantly decay into hadrons, the substructure approach may be more efficient than the searching approach utilizing their leptonic final states. This leads us to reconsider the strategy of searching for the heavier stop/sbottom in the compressed SUSY scenario. If there is a relatively large mass splitting between the heavier stop/sbottom (\(\tilde{t}_2 / \tilde{b}_1\)) and the lighter stop (\(\tilde{t}_1\)), the h / Z / W boson in the decay chain \(\tilde{t}_2 \rightarrow h/Z \tilde{t}_1\) and \(\tilde{b}_1 \rightarrow W \tilde{t}_1\) will be quite energetic. Hence, hunting for \(\tilde{t}_2 / \tilde{b}_1\) by tagging these boosted bosons may be a promising way. It was already tried in an earlier paper [51], which employed the boosted boson tag technique to probe the highly mixed stop sector and obtained a satisfactory sensitivity for \(m_{\tilde{t}_2} \sim 1\) TeV and \(m_{\tilde{t}_1} \sim 400\) GeV. But this study focused on the case of degeneracy between \(\tilde{t}_1\) and the LSP, with \(m_{\tilde{t}_1} - m_{\tilde{\chi }^0_1} \sim \mathcal {O}(10)\) GeV, which requires the flavor-violating decay \(\tilde{t}_1\rightarrow c \tilde{\chi }^0_1\) and renders \(\tilde{t}_1\) invisible. Whereas for the moderately compressed spectrum considered in this paper, additional visible particles from \(\tilde{t}_1\) (flavor conserving) decay are available.

So, in this paper we consider a (simplified) quasi-natural pattern of low energy supersymmetry where the lighter stop of a few hundred GeV is right-handed stop like and lives in the compressed regions, due to its close mass with bino or Higgsinos; whereas states in the doublet \(\tilde{Q}_3\) are around the TeV scale. Thus, the characteristic signatures of this model contain fairly boosted bosons from decays \(\tilde{Q}_3\rightarrow \tilde{t}_1+W/Z/h\). To demonstrate the prospects of those signatures at the LHC, we choose four benchmark points corresponding to four possible decay modes of \(\tilde{t}_1\): (1) \(\tilde{t}_1 \rightarrow b \tilde{\chi }^\pm _1\); (2) \(\tilde{t}_1 \rightarrow b f f \tilde{\chi }^0_1\); (3) \(\tilde{t}_1 \rightarrow b W \tilde{\chi }^0_1\); (4) \(\tilde{t}_1 \rightarrow t \tilde{\chi }^0_1\), which produce extra detectable b-jets and leptons as well as missing transverse energy (MET). Therefore, boosted bosons plus MET, associated with b-jets/leptons constitute the smoking-gun signature for such a compressed SUSY. By adopting the boosted decision tree (BDT) method for signal and background discrimination, we find that the resulting search strategy with boosted bosons can have better sensitivity than those utilizing multi-leptons.

The paper is organized as the following. In Sect. 2 we establish the quasi-natural SUSY which can hide the lighter stop involving the minimal degrees of freedom and demonstrate the distribution of \(\tilde{t}_2\) and \(\tilde{b}_1\) decays in the MSSM. In Sect. 3 we detail the signal and background analysis at the LHC. Discussions and conclusions are presented in the final section.

2 Quasi-natural supersymmetry

In this section we will present the quasi-natural model with minimal field content and analyze the decay modes of the heavier stop and sbottom, in particular the bosonic modes, analytically and numerically. Accordingly, benchmark points are selected.

2.1 A minimal setup

Asides from a light stop sector, naturalness arguments in general favor a weak scale \(\mu \)-term, thus light Higgsinos. On the other hand, considering the SUSY status after the discovery of a relatively heavy SM-like Higgs boson but there being no hints for light stops, we may have to abandon the ideal naturalness criterion and tolerate fine-tuning to some degree, say 1% or even worse [52, 53]. Such a situation inspires us to consider a quasi-natural SUSY involving a minimal set of particles that accommodate a light stop \(\tilde{t}_1\) with or without weak scale Higgsinos; other superpartners, including \(\tilde{b}_R\) and winos, are simply assumed to decouple for simplicity. The resulting Lagrangian most relevant to our discussions derived from the flavor basis is (we just schematically list the terms).Footnote 2

$$\begin{aligned} -\mathcal{L}_{QNS}&=m_{\tilde{Q}_3}^2|\tilde{Q}_3|^2+\frac{m_{\tilde{B}}}{2} \tilde{B}\tilde{B}+m_{\tilde{t}_R}^2|\tilde{t}_R|^2\nonumber \\&\quad +\left( h_tA_t\tilde{Q}_3 H_u t_R^\dagger +|D_\mu \tilde{Q}_3|^2\right) \nonumber \\&\quad +i\sqrt{2}\left( \frac{g_Y}{6}\tilde{Q}_3^\dagger \tilde{B} t_L-\frac{2g_Y}{3}\tilde{t}_R^\dagger \tilde{B} t_R\right) \nonumber \\&\quad +\left( \mu \tilde{H}_u\tilde{H}_d+h_t \tilde{Q}_3 \tilde{H}_u t_R^\dagger +h_b \tilde{Q}_3\tilde{H}_d b_R^\dagger \right) +\mathrm{{h.c.}}, \end{aligned}$$
(2.1)

where \(D_\mu =\partial -i\frac{g_2}{\sqrt{2}}\left( T^+W_\mu ^++T^-W_\mu ^-\right) -i\frac{g_2}{\cos \theta _W}Z_\mu \left( T^3-\sin ^2\theta _WQ\right) +...\) with \(\theta _W\) the Weinberg mixing angle. In the second line, terms in the first and second brackets may be irrelevant if \(\tilde{B}\) and \(\mu \) are much heavier than all other particles therein, respectively. For simplicity, we will consider that either the bino or the Higgsino is light and might provide the LSP. Although a large \(A_t\) is not necessarily required in this setup, we will see that it is crucial viewing things in the perspective of collider searches; besides, recalling the difficulty in achieving a relatively heavy SM-like Higgs boson in natural SUSY, a large \(A_t\), which could really help to radiatively enhance the Higgs boson mass, is well motivated. A good case in point of such a kind of quasi-natural SUSY is the Higgs deflected gauge mediated SUSY-breaking [54].

A compressed superpartner spectrum could make \(\tilde{t}_1\) hard to detect. If the mass degeneracy between \(\tilde{t}_1\) and LSP is mild and \(\tilde{t}_1\rightarrow t+\tilde{\chi }^0_1\) or \(b+\tilde{\chi }_1^\pm \) is kinematically accessible, they will become the main decay modes of \(\tilde{t}_1\). If degeneracy becomes severer, the above channels are closed and \(\tilde{t}_1\) will dominantly have three- (four)-body decays into \(bW^{(*)}\tilde{\chi }^0_1\), assuming that the flavor changing decay \(\tilde{t}_1\rightarrow c \tilde{\chi }^0_1\), which strongly depends on the unknown flavor structure of squarks, is negligible. As a matter of fact, the four-body decay case is particularly well motivated after identifying the bino as the dark matter candidate: the bino is a gauge singlet, so, in order to reduce its relic density during the freeze-out era, usually coannihilation with a nearly degenerate stop is necessary; for a sub TeV bino DM, a fairly small mass difference \(m_{\tilde{t}_1}-m_{\tilde{\chi }_1^0}\sim \) 30 GeV is needed [55].

To hide a light \(\tilde{t}_1\) at the current LHC, it is better to let \(\tilde{t}_1\) dominantly reside in \(\tilde{t}_R\); otherwise, the accompanying \(\tilde{b}_L\), which has close mass with \(\tilde{t}_1\approx \tilde{t}_L\), would have been uncovered via \(\tilde{b}_L\rightarrow \tilde{\chi }^0_1 b\) except for the highly degeneracy between \(\tilde{t}_1\) and \(\tilde{\chi }^0_1\), a case that has been extensively discussed before [45]. Moreover, in this paper we focus on the doublet \(\tilde{Q}_3=(\tilde{b}_L, \tilde{t}_L)\) being considerably heavier than \(\tilde{t}_R\approx \tilde{t}_1\), and therefore, by tagging the boosted bosons from \(\tilde{Q}_3\) decaying into \(\tilde{t}_1\), they may show a more promising prospect at the LHC than \(\tilde{t}_1\), which is somewhat hidden as before. On the contrary, \(\tilde{Q}_3\) having similar mass to \(\tilde{t}_1\) may be hard to discover, because their decay final states typically are soft. In this sense the heavier stop/sbottom may instead provide the smoking gun for (quasi-)natural SUSY.

2.2 Bosonic decay modes of \(\tilde{t}_2\) and \(\tilde{b}_1\): roles of a large \(A_t\)

In this section, we examine the bosonic decay modes of \(\tilde{t}_2\) and \(\tilde{b}_1\) and see the conditions which make them the dominant modes. These decays do not depend on the nature of LSP. Concretely, their decay widths are given by [56]

$$\begin{aligned} \Gamma (\tilde{b}_1 \rightarrow \tilde{t}_1 W)&\approx \frac{g_2^2\cos ^2\theta _{\tilde{t}}}{32\pi }\frac{m_{\tilde{b}_1 }^3}{m_W^2}\lambda ^{3/2}(m^2_{\tilde{b}_1},m^2_{\tilde{t}_1},m^2_W),\\\Gamma (\tilde{t}_2\rightarrow \tilde{t}_1Z)&\approx \frac{g_2^2}{\cos ^2\theta _W}\frac{\sin ^22\theta _{\tilde{t}}}{256\pi }\frac{m_{\tilde{t}_2}^3}{m_Z^2}\lambda ^{3/2}(m^2_{\tilde{t}_2}, m^2_{\tilde{t}_1}, m^2_Z),\\\Gamma (\tilde{t}_2\rightarrow \tilde{t}_1h)&\approx \frac{g_2^2\cos ^22\theta _{\tilde{t}}}{64\pi }\frac{m_t^2}{m_W^2}\frac{A_t^2}{m_{\tilde{t}_2}} \lambda ^{1/2}(m^2_{\tilde{t}_2}, m^2_{\tilde{t}_1}, m^2_h),\nonumber \\ \end{aligned}$$
(2.2)

with \(\lambda (a,b,c)=[1-(b+c)/a]^2-4bc/a^2\approx 1\). We have taken \(h\sim \mathrm{Re}(H_u^0)\) to get the last expression. Here \(\tilde{\theta }_t\) is the mixing angle between the left- and right-handed stops, defined through

$$\begin{aligned} \tilde{t}_{L}=\cos {\theta _{\tilde{t}}}\tilde{t}_{1}-\sin \theta _{\tilde{t}}\tilde{t}_2,\quad \tilde{t}_{R}=\sin {\theta _{\tilde{t}}}\tilde{t}_{1}+\cos \theta _{\tilde{t}}\tilde{t}_2, \end{aligned}$$
(2.3)

and \(\tan 2\theta _{\tilde{t}}=2X_t m_t/(m_{\tilde{Q}_3}^2-m^2_{\tilde{t}_R})\) with \(X_t=A_t+\mu /\tan \beta \). If \(\tilde{t}_1\) is very \(\tilde{t}_R\)-like, one will have \(\tilde{\theta }_t\rightarrow \pi /2\) and consequently all the bosonic modes except for \(\tilde{t}_2\rightarrow \tilde{t}_1 h\) will be highly suppressed. A large \(A_t\) is thus indispensable: It does not only generate sizable LR stop mixing but also directly enhances \(\tilde{t}_2\rightarrow \tilde{t}_1h\).Footnote 3 In practice, we do not need a fairly sizable \(\theta _{\tilde{t}}\) because the (longitudinal) W / Z modes are enhanced by a factor like \(\left( m_{\tilde{t}_2}/m_Z\right) ^2\sim \mathcal{O}(10^2)\) for a TeV scale \(m_{\tilde{Q}_3}\), which could easily compensate the mild suppression from the small mixing.

Now we analyze their heavy quark decay modes based on the quasi-natural SUSY, see Eq. (2.1), which are sensitive to the LSP components. In the most general cases, the decay widths take the forms of [56]

$$\begin{aligned}&\Gamma (\tilde{q}_i\rightarrow q\tilde{\chi }^0_k)=\frac{g_2^2}{16\pi m_{\tilde{q}_i}}\nonumber \\&\quad \times \left[ \left( (h^q_{ik})^2+(f^q_{ik})^2\right) \left( m_{\tilde{q}_i}^2-m_q^2-m_{\tilde{\chi }^0_k}^2\right) -4h^q_{ik}f^q_{ik}m_q m_{\tilde{\chi }^0_k} \right] ,\end{aligned}$$
(2.4)
$$\begin{aligned}&\Gamma (\tilde{q}_i\rightarrow q'\tilde{\chi }^\pm _k)=\frac{g_2^2}{16\pi m_{\tilde{q}_i}}\nonumber \\&\quad \times \left[ \left( (l^{\tilde{q}}_{ik})^2+(k^{\tilde{q}}_{ik})^2\right) \left( m_{\tilde{q}_i}^2-m_{q'}^2-m_{\tilde{\chi }^0_k}^2\right) -4l^{\tilde{q}}_{ik}k^{\tilde{q}}_{ik}m_{q'} m_{\tilde{\chi }^\pm _k} \right] , \end{aligned}$$
(2.5)

where \(\tilde{q}_i\) denote \(\tilde{t}_{1, 2}\) and \(\tilde{b}_{1}\). The matrices \(h^q_{ik}\) etc. encode couplings between quark and squark, neutralinos; in the following we will give their concrete expressions in the Higgsino- and bino-LSP limits.

Fig. 1
figure 1

\(\tilde{t}_2\) (left panels) and \(\tilde{b}_1\) (right panels) decay branching ratios for bino LSP (upper panel) and Higgsino LSP (lower panels)

Full size image

We first consider the Higgsinos to be light, while the bino can be dropped; moreover, we will use the strip \(m_b+|\mu |< m_{\tilde{t}_1}\lesssim m_t+|\mu |\) to hide \(\tilde{t}_1\). In the limit of a left-handed sbottom, while we have the right-handed light stop, namely \(\theta _{\tilde{t}}\rightarrow \pi /2\), and a Higgsino LSP (actually two with almost degenerate masses involved), one obtains

$$\begin{aligned} h_{21}^t&\approx h_{22}^t\approx 0, \quad |f_{21}^t|\approx |f_{22}^t|\approx \frac{m_t}{2m_W}\sin \theta _{\tilde{t}},\nonumber \\ |h_{11}^b|&\approx |h_{12}^b|\approx \frac{m_b}{2m_W}\tan \beta , \quad f_{11}^b\approx f_{12}^b\approx 0,\end{aligned}$$
(2.6)
$$\begin{aligned} l_{21}^{\tilde{t}}&\approx 0 ,\quad |k_{21}^{\tilde{t}}| \approx \frac{m_b}{\sqrt{2}m_W\cos \beta }\sin \theta _{\tilde{t}}, \nonumber \\ l_{11}^{\tilde{b}}&\approx 0 ,\quad |k_{11}^{\tilde{b}}| \approx \frac{m_t}{\sqrt{2}m_W\sin \beta }, \end{aligned}$$
(2.7)

where a relatively large \(\tan \beta \) at least is assumed. Next we move to the other case where the Higgsinos are decoupled and the bino is the LSP. In this case a lighter \(\tilde{t}_1\) is allowed if its mass does not significantly exceed \(m_t+m_{\tilde{\chi }_1^0}\). Now the couplings are reduced to

$$\begin{aligned} |h_{21}^t|&\approx \frac{2\sqrt{2}}{3}\sin \theta _W\sin \theta _{\tilde{t}}, \quad |f_{11}^b|\approx \frac{\sqrt{2}}{3}\sin \theta _W\sin \theta _{\tilde{t}}. \end{aligned}$$
(2.8)

All others are suppressed by small mixing angles, and they thus are of no importance. Moreover, since the charginos are decoupled, here we do not need to consider \(l^{\tilde{b}}_{1j}\), etc.

We would like to stress that, in the bino-LSP, case the decay modes of \(\tilde{t}_2\) and \(\tilde{b}_1\) into the heavy flavors, such as \(\tilde{t}_2\rightarrow t+\tilde{\chi }_1^0\) and \(\tilde{b}_1\rightarrow b+\tilde{\chi }_1^0\), are substantially suppressed, because now they come from hypercharge gauge interactions rather than the \(y_t\)-Yukawa interaction as in the light Higgsino case. One can clearly see this situation from Fig. 1, which shows that those branching ratios typically are below \(\mathcal{O}(1\%)\) in the bino LSP scenario. Such a situation makes good for the more boosted bosons from \(\tilde{t}_2/\tilde{b}_1\) decay. But even in the Higgsino LSP case, for a heavier \(m_{\tilde{Q}_3}\) with a large \(A_t\) coupling, these bosonic modes generically have quite sizable branching ratios, \({\gtrsim }\mathcal{O}(10\%)\), by virtue of the significant Goldstone enhancement factor stressed before. In particular, \(\tilde{b}_1\), which has less decay modes than \(\tilde{t}_2\), almost dominantly decays into W plus \(\tilde{t}_1\) in both cases. It can be understood from the estimation (in the Higgsino-LSP limit):

$$\begin{aligned} \frac{\Gamma (\tilde{b}_1\rightarrow W\tilde{\chi }^\pm _1)}{\Gamma (\tilde{b}_1\rightarrow b \tilde{\chi }^0_1)}\simeq \left( \frac{m_{\tilde{b}_1}}{m_t}\right) ^2\cos ^2\theta _{\tilde{t}}\gtrsim \frac{A_t^2}{m_{\tilde{Q}_3}^2}. \end{aligned}$$
(2.9)
Table 1 Benchmark points for different decay modes of the right-handed dominant \(\tilde{t}_1\). Br\((\tilde{t}_1 \rightarrow bW^{(*)} \tilde{\chi }^0_1)\) of T14B (1000) is slightly smaller than one because the flavor changing decay \(\tilde{t}_1 \rightarrow c \tilde{\chi }^0_1\) is also important here
Full size table

In the next section, we will choose several benchmarks points to embody the above possible scenarios for quasi-natural SUSY.

2.3 Decay patterns in quasi-natural SUSY: scanning results and benchmark points

For concreteness, we implement quasi-natural SUSY in the minimal supersymmetric SM (MSSM). There are totally five parameters of interests in each scenario with either decoupled bino or Higgsino. We use Suspect2 [58] and SUSY-HIT [59] to calculate the mass spectrum and the decay branching ratios of stops and sbottom. The parameter scan is performed in the following range:

$$\begin{aligned}&m_{\tilde{Q}_3} \in [700, 1200] ~\text {GeV}, \quad m_{\tilde{U}_3} \in [500,700] ~\text {GeV}, \nonumber \\&|A_t| \in [1,3] ~\text {TeV}, \quad \tan \beta \in [3,50], \nonumber \\&\mu \in [300,700] ~\text {GeV} ~(M_1=2~\text {TeV}) \text {~or~}\nonumber \\&M_1 \in [300, 700]~\text {GeV} ~(\mu =2~\text {TeV}). \end{aligned}$$
(2.10)

The rest of the soft mass parameters of the MSSM are set to 2 TeV, so those sparticles are decoupled from the mass spectrum. The choice of the above parameter pattern is motivated by the non-detection of any stop/sbottom signals at the current stage of LHC [5,6,7,8,9,10,11,12]; the resulting spectrum still allows a light stop with mass \({\sim }500\) GeV if the LSP is relatively heavy (\(m_\mathrm{{LSP}} \gtrsim 300\) GeV). Moreover, we require the mass of the heavier stop and the sbottom to be around the TeV scale to produce relatively boosted bosons in their decay, while still having sizable production rates for discovery in the near future. We note that the searches for a heavier stop at the LHC run-I [40, 41] are only able to exclude models with \(m_{\tilde{t}_2} \lesssim 600\) GeV. As we have discussed in Sect. 2.2, a sizable \(|A_t|\) is needed to enhance \(\mathrm{{Br}}(\tilde{t}_2 \rightarrow h\tilde{t}_1)\) and \(\mathrm{Br}(\tilde{t}_2 \rightarrow Z\tilde{t}_1)\), so a lower limit of \(|A_t|\) is set to improve the scanning efficiency. Since we are expecting new contributions other from the stop in MSSM to the Higgs boson mass, the lighter CP-even Higgs boson (\(H_1 \equiv h\)) mass is set to 125 GeV manually when calculating the decay branching ratios. The heavier CP-even Higgs (\(H_2\)) is decoupled by setting \(m_A=2\) TeV.

In Fig. 1, we plot the decay branching ratios of the heavier stop (\(\tilde{t}_2\)) and the lighter sbottom (\(\tilde{b}_1\)) for either bino LSP or Higgsino LSP. In the upper panels where the bino is the LSP, we can see that the bosonic modes dominate the stop/sbttom decay in the full parameter space, while the branching fractions of the \(\tilde{t}_2 \rightarrow t \tilde{\chi }^0_1\)/\(\tilde{b}_1 \rightarrow b \tilde{\chi }^0_1\) modes typically are two orders of magnitude smaller. The situation changes when Higgsino is the LSP. In the lower panels, decay widths of \(\tilde{t}_2 \rightarrow t \tilde{\chi }^0_1\)/\(\tilde{b}_1 \rightarrow b \tilde{\chi }^0_1\), which are enhanced by the larger top quark Yukawa coupling, become comparable with that of the bosonic modes. Moreover, there will be new decay modes opening due to the charged Higgsino in the final state, i.e., \(\tilde{t}_2 \rightarrow b \tilde{\chi }^\pm _1\) and \(\tilde{b}_1 \rightarrow t \tilde{\chi }^\pm _1\), whose branching fractions are also sizable. Nevertheless, we can observe that the bosonic mode is still one of the dominant decay modes for both \(\tilde{t}_2\) and \(\tilde{b}_1\).

In terms of the scanning results, eight benchmark points are chosen to illustrate the model details in Table 1, which are featured by the different decay modes of the lighter stop, as well as two choices of the \(\tilde{t}_2/ \tilde{b}_1\) masses, that is, \(m_{\tilde{t}_2/ \tilde{b}_1} \sim 800\) GeV and \(m_{\tilde{t}_2/ \tilde{b}_1} \sim 1000\) GeV, respectively. These differences will be used to label each benchmark point in the following discussions, e.g., T1BC (800) corresponding to the one which has \(m_{\tilde{t}_2/\tilde{b}_1}\sim 800\) GeV along with a lighter stop mainly decaying into \(b+\tilde{\chi }_1^0\).

Fig. 2
figure 2

The number of b-jets \(N_b\) and the number of leptons \(N_{\ell }\) distributions for our benchmark points with \(m_{\tilde{t}_L}\)=1 TeV

Full size image

3 Collider phenomenology

3.1 Search strategy

For all benchmark points, the left-handed-like \(\tilde{t}_2\) and \(\tilde{b}_1\) have similar masses, both dominantly decaying into a gauge boson/Higgs boson plus the lighter stop \(\tilde{t}_1\). The high multiplicity of gauge bosons in the final state will lead to multiple leptons events. Moreover, one would expect two bottom quark jets in the final state if the flavor changing decay of \(\tilde{t}_1 \rightarrow c \tilde{\chi }^0_1\) is suppressed. Studies [40, 41, 43, 44, 46] have shown that searching for final states with leptons and b-jets provides as a good probe to study the light stop sector, especially when \(m_{\tilde{t}_1} \sim m_t + m_{\tilde{\chi }^0_1}\). However, for some of our benchmark points, e.g. T14B, the small mass difference between \(m_{\tilde{t}_1}\) and \(m_{\tilde{\chi }^0_1}\) may render the b-jets/leptons undetectable.

To see the point more clearly, we generate the parton level events for our benchmark points with MadGraph5 [60], which are passed to Pythia6 [61] for particle decay, parton showering and hadronization. The Delphes3 [62] with input of the default ATLAS detector card is used for simulating detector effects. In this work, we take the b-jet tagging efficiency as 70%, with the other light quark and gluon mis-tagging probability 1% [63].

We consider the signals of both \(\tilde{t}_2\) and \(\tilde{b}_1\) pair production with subsequent decays for benchmark points with \(m_{\tilde{t}_L}\)=1 TeV. The corresponding \(N_b\) versus \(N_{\ell }\) distributions are given in Fig. 2. It can be seen that even for the benchmark point T1TN, in which \(\tilde{t}_1\) dominantly decays into \(t\tilde{\chi }^0_1\), only around 20% of the total events contain at least one b-jet and one lepton. The fraction becomes even smaller for other benchmark points owing to the heavier \(\tilde{\chi }^0_1\). Events with b-jet multiplicity higher than 2 originate from \(h_{\text {SM}} \rightarrow b\bar{b}\). In all cases, we find that the fraction of events with \(N_{l} \ge 2\) is at the percent level. Consequently, despite relatively low backgrounds, searching for final states with multiple leptons is suffering from serious branching ratio suppressions in the signal processes.

On the other hand, signals with hadronic decaying bosons from \(\tilde{t}_2/\tilde{b}_1\) decay have much larger production rates. Moreover, some recent developments in the jet substructure analysis [47, 50, 64, 65] are found to be very useful in suppressing hadronic SM backgrounds in the boosted region. Because of the relatively large mass splitting between \(\tilde{t}_2/\tilde{b}_1\) and \(\tilde{t}_1\), the h / Z / W bosons from the heavier squarks decay usually are well boosted. Considering the \(\tilde{t}_2 \rightarrow Z \tilde{t}_1\) process as an example and taking \(m_{\tilde{t}_1}=500\) GeV, we plot parton level distributions of the transverse momentum of the Z boson and the angular distance between two fermions from Z decay in Fig. 3. We can see from the figure that the typical transverse momentum of the Z boson exceeds \(\sim 150 ~(200)\) GeV, while the angular distance between the Z boson decay products \(\Delta R(f,f)\) which is roughly proportional to \({2 m_Z}/{p_T(Z)}\), typically is less than \(\sim \)1.5 (1.0) for \(m_{\tilde{t}_2}=800 ~(1000)\) GeV.

Fig. 3
figure 3

Left the transverse momentum distribution for the Z boson in \(\tilde{t}_2 \rightarrow Z \tilde{t}_1\), taking \(m_{\tilde{t}_1}=500\) GeV and \(m_{\tilde{t}_2}=800/1000\) GeV. Right the angular distance \(\Delta R= \sqrt{(\Delta \phi )^2 + (\Delta \eta )^2}\) between the two fermions from the Z boson decay

Full size image

The closeness of the Z boson decay products indicates that they can be reconstructed as a whole, i.e., as a boson jet. A boson jet which has high invariant mass and appropriate substructure can be distinguished from QCD jet, thus providing a most important handle for searching our benchmark points. Besides, there will be extra activities from the subsequent lighter stop \(\tilde{t}_1\) decay, such as leptons and b-jets. In the following, we propose a search for the final state with two boson jets alongside with extra leptons/b-jets.

3.2 Signal and background analysis

The signal processes that we are aiming to search for are

$$\begin{aligned} p ~p&\rightarrow \tilde{t}_2 ~\bar{\tilde{t}}_2, \quad \tilde{t}_2 \rightarrow h/Z ~\tilde{t_1}, \end{aligned}$$
(3.1)
$$\begin{aligned} p~ p&\rightarrow \tilde{b}_1~ \bar{\tilde{b}}_1, \quad \tilde{b}_1 \rightarrow W ~\tilde{t_1}, \end{aligned}$$
(3.2)

with decay branching fractions of \(\tilde{t}_2\), \(\tilde{b}_1\) and \(\tilde{t_1}\) given in Table 1. At the LHC, the signal events can be trigged by requiring a large missing transverse momentum in the final state,   GeV. As for event reconstruction, we first identify isolated electrons and muons with \(p_T(e,\mu )>10\) GeV and \(|\eta (e,\mu )|<2.5\), where the isolation means that the scalar sum of transverse momenta of all particles with \(p_{T}>0.5\) GeV that lie within a cone of radius \(R=0.5\) around the \(e(\mu )\) is less than 12%(25%) of the transverse momentum of \(e(\mu )\). Next, tracks that do not belong to isolated leptons as well as neutral particles are used for jet clustering with fastjet [66]. We adopt the BDRS method [47] for tagging boosted boson jets: (1) reconstructing the boson jet candidates (fat jet) using C/A algorithm [67] with radius \(R=1.2\) and \(p_T >150\) GeV; (2) breaking each fat jet by undoing the clustering procedure. The two boson jets (\(V_1, ~V_2\)) are taken as the two leading fat jets with highest transverse momenta that have large mass dropping to \(\mu <0.67\) and we have a not too asymmetric mass splitting \(y>0.09\) at any step during the declustering; (3) filtering each of the boson jets neighborhood by rerunning the C/A algorithm with a finer angle \(R_\mathrm{{filt}} = \min (0.3,R_{j_1,j_2}/2)\) and taking the three hardest subjets; (4) applying a b-tag on the two leading subjets, where we have followed the b-tagging method that is used in Delphes: identifying the hadronic jet as the generated quark with largest PDG number that lies within the distance of \(\Delta R < R_\mathrm{{filt}}\) of the jet axis. The probabilities of b-tagging a b-jet, c-jet and light flavor jet are taken as 0.7, 0.2 and 0.005 respectively [63]. Finally, for an event that contains two boson jet candidates, we proceed with the reconstruction of narrow jets. The constituents of the two boson jet candidates are removed from particle-flow objects of Delphes output. The remnants are clustered using the anti-\(k_T\) jet clustering algorithm [68] with a jet cone radius of R = 0.4 and \(p_T(j) > 20\) GeV to form narrow jets. The b-tagging is applied to each of the narrow jets with \(|\eta (j)|<2.5\). During the reconstruction, the signal events are required to pass two more preselection cuts: the transverse momenta of two boson jets \(p_T(V_1), p_T(V_2) > 200\) GeV and two boson jets should contain either no b-tagged subjet or exactly two b-tagged subjets.

Table 2 Cross sections of benchmark points before and after preselections at 14 TeV LHC
Full size table

The cross sections of the benchmark points at 14 TeV LHC before and after the preselection are given in Table 2. The Next-to-Leading-Order (NLO) production cross sections of \(\tilde{t}_2\bar{\tilde{t}}_2\) plus \(\tilde{b}_1\bar{\tilde{b}}_1\) are calculated by Prospino2 [69]. It can be seen that the signal rate decreases dramatically on increasing the particle mass. The preselection efficiencies are around 10% for benchmark points with \(m_{Q_3}=800\) GeV and become twice larger when \(m_{Q_3} = 1\) TeV.

We list all possible SM backgrounds for our signal in Table 3, as well as their higher order production cross section at the LHC. After the preselection, the dominant backgrounds are \(t\bar{t}\), diboson \(+\) jets and tW processes, in which either an energetic top quark or a QCD jet will be mis-tagged as a boson jet in our analysis, and the large missing transverse momentum is mainly due to the existence of a neutrino in the final state.

Table 3 Cross sections of backgrounds before and after preselections at 14 TeV LHC
Full size table

Comparing Tables 2 and  3, we find that the production rates of our signals are around two orders of magnitude smaller than that of backgrounds after the preselection cuts. Even at the 14 TeV LHC with integrated luminosity of 100 fb\(^{-1}\), the signal significances are only around two. Moreover, because of the smallness of the signal-to-background ratios, the results are quite vulnerable to the systematic uncertainty. We need to apply more refined cuts to obtain a higher signal significance as well as signal-to-background ratio.

Fig. 4
figure 4

Kinematic distributions for signals and backgrounds after preselection

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First of all, the invariant masses of two boson jets should be close to either of the W / Z / h masses in signal processes. In the top panels of Fig. 5, we plot the distributions of the invariant mass of boson jets (\(m_{V_1}\), \(m_{V_2}\)) after pruning [77].Footnote 4 In the figure, all backgrounds have been stacked up with a contribution of each process indicated by different colors and the distributions have been normalized to their production cross sections at 14 TeV LHC. We can see that most of signal events have \(m_{V_1}\) and \(m_{V_2}\) falling between [60, 100] GeV, since the branching ratio to h is suppressed. Meanwhile the backgrounds have relatively flat distribution between [20, 200] GeV, especially for \(m_{V_1}\). This is because of the mis-tagging of the top quark that enhanced the background rate at \(m_{V_1}\sim m_t\). It has to be noted that for the benchmark point T1BC Br\((\tilde{t}_2 \rightarrow t \tilde{\chi }^0_i)\) and Br\((\tilde{b}_1 \rightarrow t \tilde{\chi }^\pm )\) are also sizable. This leads to an enhanced event rate at \(m_{V_1} \sim m_t\) as well.

The effective mass for our signal processes,

(3.3)

which is correlated with \(m_{\tilde{t}_2 / \tilde{b}_1}\), should be higher than that for background processes. As shown in the lower-left panel of Fig. 5, the preselection renders the \(m_{\text {eff}}\) distribution of the background peaks in a wide range between [1000,1200] GeV, while there a large fraction of signal events have \(m_{\text {eff}} >1200\) GeV.

Another useful discriminator that is used frequently in searching supersymmetry is the stransverse mass \(M_{T_2}\) [79, 80], which could reflect the mass difference between the squark and neutralino in the squark pair production channel with subsequent two body decay \(\tilde{q} \rightarrow q \tilde{\chi }^0\). By drawing an analogy between our signal process \(\tilde{t}_2 / \tilde{b} \rightarrow V \tilde{t}_1\) and \(\tilde{q} \rightarrow q \tilde{\chi }^0\), we can define the modified stransverse mass as

(3.4)

where \(\sum \mathbf {p}_T(\ell )\) and \(\sum \mathbf {p}_T(j)\) are vector sums of the transverse momenta of isolated leptons and narrow jets. The \(M_{T_2} (V_1, V_2)\) distribution for signals and backgrounds are presented in the low-right panel of Fig. 4. We can see the signal events are associated to larger values of \(M_{T_2} (V_1, V_2)\) than backgrounds events.

In order to obtain better signal and background discrimination, we employ the BDT method that takes into account the distribution profiles of the following variables:

(3.5)

Furthermore, the information from the decay products of the light top squark may help to improve our signal identification. So we consider three more variables in the BDT analysis:

$$\begin{aligned} n_{\ell }, ~n_b, ~p_T(\ell _1), \end{aligned}$$
(3.6)

where \(p_T(\ell _1)\) is the transverse momentum of the leading lepton if it exists.

The BDT method uses a 100 tree ensemble that requires a minimum training events in each leaf node of 2.5% and a maximum tree depth of two. It is trained on the half of the preselected signal and backgrounds events and is tested on the rest of the events. We also require that the Kolmogorov–Smirnov test of the BDT analysis should be greater than 0.01 to avoid overtraining.

Having the BDT response distributions for both signal and background, we can impose a cut on the BDT responses to improve the signal significance. Figure 5 shows the signal-to-background ratios (left panel) and the signal significances with 100 fb\(^{-1}\) data sample (right) for all benchmark points. The signal significance is calculated by

$$\begin{aligned} \mathcal {S} = \sqrt{2\left( (s+b) \ln \left( 1+\frac{s}{b}\right) -s\right) }. \end{aligned}$$
(3.7)

We can see that a cut of BDT \({\gtrsim }0.3\) will maximize the signal significance and keep the signal-to-background ratio at \(\mathcal {O}(10)\)% level.

Fig. 5
figure 5

Left the signal-background ratio for varying BDT cut. Right the signal significance at integrated luminosity of 100 fb\(^{-1}\) for varying BDT cut

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In Fig. 6, we plot the signal significances for all benchmark points with different integrated luminosity, where we have chosen the cut BDT \({\ge }0.3\). A heavier stop sector of \({\sim }1\) TeV can be excluded at 95% C.L. at very early stage of the LHC run-II. Since the lighter stops \(\tilde{t}_1\) of the benchmark points are far beyond the reach of the LHC search at 13 TeV 13.3 fb\(^{-1}\), we conclude that the heavier stop provides a better chance for searching supersymmetry. Moreover, comparing to the method in Ref. [44, 46] which utilizes the leptons and b-jets in the final state, our search strategy can achieve a few times larger signal significance because of the higher signal rate.

Fig. 6
figure 6

The signal significance at different integrated luminosities, where the BDT cut is chosen as BDT \({\ge }0.3\). The horizontal line corresponds to the 95% C.L. exclusion limit

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4 Conclusion

A quasi-natural pattern of low energy supersymmetry is considered in this work. The lighter stop is considered to have a mass around a few 100 GeV and to be close to the LSP mass, while the heavier stop and the lighter sbottom have masses around TeV. In this scenario, due to the compressed mass between the lighter stop and the LSP, the lighter stop decay can only produce soft leptons/jets in the final state; thus it evades all current LHC searches and is difficult to probe in future experiments. The heavier stop \(\tilde{t}_2\) and lighter sbottom \(\tilde{b}_1\), in contrast, may provide a better handle for searching the compressed SUSY.

In the framework of MSSM, considering either the bino or the Higgsino as the LSP, we find that the bosonic modes \(h/Z \tilde{t}_1\) (\(W \tilde{t}_1\)) dominate the \(\tilde{t}_2\) (\(\tilde{b}_1\)) decay in the parameter space with relatively large left–right stop mixing as well as large trilinear coupling \(A_t\). With a moderately large mass gap between the heavier members and lightest stop, the bosons in the decay chain are generically quite energetic. This allows us to employ the jet substructure technique for discriminating the natural SUSY signals in searches at the LHC.

We consider the discovery prospects of eight benchmark points at the LHC-14, in terms of four possible decay modes of the lighter stop \(\tilde{t}_1\): (1) \(\tilde{t}_1 \rightarrow b \tilde{\chi }^\pm _1\); (2) \(\tilde{t}_1 \rightarrow b f f \tilde{\chi }^0_1\); (3) \(\tilde{t}_1 \rightarrow b W \tilde{\chi }^0_1\); (4) \(\tilde{t}_1 \rightarrow t \tilde{\chi }^0_1\) as well as two different masses of \(\tilde{t}_2/ \tilde{b}_1\): (a) \(m_{\tilde{t}_2/ \tilde{b}_1} \sim 800\) GeV; (b) \(m_{\tilde{t}_2/ \tilde{b}_1} \sim 1000\) GeV. We search for \(\tilde{t}_2 \bar{\tilde{t}}_2\) and \(\tilde{b}_1 \bar{\tilde{b}}_1\) production in the final state with two boosted boson jets that have substructures and high invariant masses, leptons/b-jets and MET. After considering background contamination and adopting the BDT method for signal discrimination, we find that a heavier stop and lighter sbottom with masses \(\sim 1\) TeV can be excluded at 95% C.L. with integrated luminosity of 10–30 fb\(^{-1}\). Among the four decay modes of \(\tilde{t}_1\), the search sensitivities decrease from \(t\tilde{\chi }^0_1\) to \(bW \tilde{\chi }^0_1\) to \(b ff \tilde{\chi }^0_1\), as the mass difference between \(\tilde{t}_1\) and \(\tilde{\chi }^0_1\) is successively smaller. The \(b\tilde{\chi }^\pm _1\) mode has the least search sensitivity. This is because this mode is possible only when Higgsino is the LSP. Then the decay branching ratios of \(\tilde{t}_2 \rightarrow t \tilde{\chi }^0\) / \(\tilde{b}_1 \rightarrow t \tilde{\chi }^\pm \) become competitive to that of the bosonic decay of \(\tilde{t}_2\) / \(\tilde{b}_1\). Finally, we note that with the aid of the jet substructure and BDT analysis, our search strategy can achieve a few times larger signal significance than the searches proposed in Refs. [44, 46], which utilize the multiple leptons and b-jets in the final state.