Complex decay chains of top and bottom squarks

Current searches for the top squark mostly focus on the decay channels of $\tilde{t}_1 \rightarrow t \chi_1^0$ or $\tilde{t}_1 \rightarrow b \chi_1^\pm \rightarrow bW \chi_1^0$, leading to $tt/bbWW+\not\mathrel{E}_T$ final states for top squark pair production at the LHC. In supersymmetric scenarios with light gauginos other than the neutralino lightest supersymmetric particle (LSP), different decay modes of the top squark could be dominant, which significantly weaken the current top squark search limits at the LHC. Additionally, new decay modes offer alternative discovery channels for top squark searches. In this paper, we study the top squark and bottom squark decay in the Bino-like LSP case with light Wino or Higgsino next-to-LSPs (NLSPs), and identify cases in which additional decay modes become dominant. We also perform a collider analysis for top squark pair production with mixed top squark decay final states of $\tilde{t}_1 \to t {\chi}_2^0 \to th {\chi}_1^0$, $\tilde{t}_1 \to b {\chi}_1^\pm \to bW {\chi}_1^0 $, leading to the $bbbbjj\ell+\not\mathrel{E}_T$ collider signature. The branching fraction for such decay varies between 25\% and 50\% for a top squark mass larger than 500 GeV with $M_2=M_1+150$ GeV. At the 14 TeV LHC with 300 ${\rm fb}^{-1}$ integrated luminosity, the top squark can be excluded up to about 1040 GeV at the 95\% C.L., or be discovered up to 940 GeV at 5$\sigma$ significance.


I. INTRODUCTION
The discovery of a 125 GeV Higgs at the Large Hadron Collider (LHC) [1,2] motivates the consideration of new physics beyond the Standard Model (SM). In the SM, the Higgs receives unstable quadratically divergent radiative corrections to its mass from the top quark loop. An unnatural cancellation is needed to recover the light physical Higgs mass, which is the so called "naturalness problem" [3]. Supersymmetry (SUSY) provides a solution to the naturalness problem by introducing superpartners to the SM particles, with interactions following the SUSY relations. The quadratic divergence from the su-  [4]. Given the tight connection between the stop and Higgs sectors, it is important to fully explore the discovery potential of the stop at the LHC.
Most of the current searches for the light stop focus on the decayt → tχ 0 1 ort → bχ ± 1 → bW χ 0 1 , leading to tt + E T or bbW W + E T final states for stop pair production at the LHC. However, due to the large SM background from tt, searches for the stop can be very challenging. The current limits from ATLAS and CMS experiments exclude stops with masses up to about 645 GeV for a light neutralino LSP [5][6][7][8][9][10][11][12][13][14][15]. At energy of 14 TeV with 100 fb −1 of integrated luminosity, the expected discovery sensitivity for stops at the LHC is about 1 TeV [16].
The current stop search limits, however, always assume the dominance of the stop decay channels mentioned above. The current limits could be significantly weakened when other stop decay modes open, which could occur in many regions of MSSM parameter space. On the other hand, the opening of new channels offers alternative discovery potential for stops at the LHC. Therefore, it is timely to study the non-minimal stop decay pattern as well as assess the stop discovery potential at the 14 TeV LHC.
Even under the usual assumption of a Bino-like LSP, the existence of other light neutralino states, for example, Wino-like or Higgsino-like next-to-LSPs (NLSPs) could lead to new decay channels for the stop. For instance,t could decay to tχ 0 2,3 , with χ 0 2,3 further decaying to Zχ 0 1 , hχ 0 1 . Given the relatively large SU(2) L coupling and top Yukawa coupling, compared to the U(1) Y coupling relevant for the Bino-like LSP, decays to tχ 0 2,3 could even be dominant despite the phase space suppression. In this paper, we study the stop decay branching fractions for the Wino-or Higgsino-like NLSP case, considering the minimal mixing and the maximal mixing scenarios in the stop sector, and outline the main search channels for the stops at the LHC.
Similarly, the current sbottom searches focus onb → bχ 0 1 , with bb + E T being the dominant search channel. Given data collected at the LHC 7/8 TeV, sbottoms with masses up to 700 GeV are excluded [17]. Even in parameter space with highly degenerate sbottom and LSP masses, the sbottom is excluded with mass up to about 400 GeV [18]. The lefthanded sbottom mass is related to the left-handed stop mass since they are controlled by the same soft SUSY breaking mass parameter. In this paper, we also study the left-handed sbottom decay pattern for various scenarios, as well as its collider signatures.
Given the discovery of the SM-like Higgs boson at the LHC, we can now use final states with a Higgs boson to search for new physics beyond the SM. To explore the 14 TeV LHC reach for the exotic stop decay channels, we performed a detailed collider analysis with a Higgs in the final state: pp →t 1t1 * with one stop decaying viat 1 → tχ 0 2 → thχ 0 1 , and other stop decaying viat 1 → bχ ± 1 → bW χ 0 1 , leading to bbbbjjℓ + E T signature. By designing cuts to identify the signal while suppressing SM backgrounds, we obtained the 95% C.L. exclusion limit as well as the 5σ discovery reach at the 14 TeV LHC with 300 fb −1 integrated luminosity. Final states with χ 0 2 → Zχ 0 1 are left for future studies. The rest of the paper is organized as the following. In Sec. II, we present the third generation squark sector in the MSSM and discuss its connection to the Higgs sector. In Sec. III, we discuss the stop and sbottom decays for various scenarios, as well as the collider signatures for stop/sbottom pair production. In Sec. IV, we summarize the current LHC stop and sbottom search results from both ATLAS and CMS. In Sec. V, we investigate the 14 TeV reach of the stop via final states with a Higgs. In Sec. VI, we conclude.

II. MSSM STOP SECTOR
In this study, we work in the MSSM and focus primarily on the third generation squark sector. We decouple other SUSY particles: the gluino, sleptons, and the first and second generation squarks. We also decouple the non-SM Higgs particles by setting M A large. The remaining SUSY particles in the model are the third generation squarks, the neutralinos and charginos.
The gauge eigenstates for the superpartners of the top and bottom quarks are (t L ,b L ),t R andb R , with the left-handed states grouped as an SU(2) L doublet and the right-handed states as singlets. The mass matrix for the stop sector is with M 2 3SQ and M 2 3SU representing the soft SUSY breaking masses fort L andt R , m 2 t term coming from the F-term contribution in the SUSY Lagrangian and the ∆ terms coming from the D-term contribution. The off-diagonal termÃ t is given by: for A t representing the trilinear coupling and µ representing the supersymmetric bilinear mass term in the Higgs sector.
The stop mass matrix can be diagonalized with a stop mixing angle θ t : with mass eigenstatest 1 ,t 2 : mt 1 < mt 2 . For M 3SQ < (>)M 3SU ,t 1 is mostly left-handed (right-handed), while for M 2 3SQ ∼ M 2 3SU ,t 1,2 could be a mixture of the left-and righthanded states.
Given the large top Yukawa coupling, the stop sector provides the dominant contribution to the radiative corrections of the SM-like Higgs mass in the MSSM. For M 3SQ = M 3SU = M SU SY , the correction to the SM-like Higgs mass squared is [19]: In the minimal mixing case withÃ t = 0, a large M SU SY is needed to provide a SM-like Higgs mass of 125 GeV. In the maximal mixing case withÃ t = √ 6M SU SY , a relatively small M SU SY can be accommodated given the additional contribution from theÃ t term.
In the general MSSM when M 2 3SQ = M 2 3SU , to provide a SM-like Higgs mass of 125 GeV, the light stopt 1 can still be as light as 200 GeV. A large mass splitting between the stop mass eigenstates (and a largeÃ t term), however, is typically needed, resulting in mt 2 > ∼ 500 GeV in general [20,21].
Similarly, the mass matrix for the sbottom is given as: Given the suppression of the off-diagonal terms by the small bottom mass, large mixing among the sbottom mass eigenstates is less common.
Since the stop sector provides the dominant contribution to the Higgs mass corrections, we decouple the right-handed sbottom in our analysis. The left-handed sbottom mass, however, is determined by M 3SQ and could be relatively light. Given m bÃb , M 2 3SQ << M 2 3SD , the light sbottom mass eigenstate is mostly left-handed:b 1 ∼b L . Although the sbottom corrections to the Higgs mass are small compared to the stop corrections, there can be significant modifications to the Higgs couplings, especially the bottom Yukawa coupling [22].
• Case IB, Bino-like LSP with Higgsino-like NLSPs: The decays of the light stop or sbottom highly depend on the low-lying neutralino/chargino spectrum, as well as the composition of the light stop and sbottom.
In each scenario, we consider two limiting cases with different stop left-right mixing. In the minimal mixing case,Ã t = A t −µ cot β = 0, the lightest stop mass eigenstatet 1 is either

A. Case I: Bino-like LSP with decoupled Wino and Higgsino
The simplest case has a mass spectrum with stop(s), left-handed sbottom, and only the low-lying neutralino being the Bino-like LSP.
In the minimal mixing case with the light stopt 1 as a pure left-or right-handed state, t 1 either directly decays to tχ 0 1 when it is kinematically accessible or through bW * χ 0 1 with 100% branching fraction. Similarly, in the case of small M 3SQ ,b 1 decays directly through bχ 0 1 with 100% branching fraction.
In the maximal mixing case,t 1 ,t 2 , andb 1 appear in the spectrum, with a typical mass order mt 1 < mb 1 < mt 2 with relatively large mass splittings of 150 GeV or larger. While the decay oft 1 is straightforward (100% into bW ( * ) χ 0 1 ), the decays ofb 1 andt 2 could have multiple competing channels, as shown in Fig conventional channels wheret → t/bW χ 0 1 andb → bχ 0 1 dominate. σ ×BR is the same as the production cross sections for the stop pair and sbottom pair. The middle panel of Fig. 3 shows the σ×BR forb 1b1 pair production in the maximal mixing scenario. The conventional channel bb+ E T is highly suppressed, while bbW W W W + E T becomes dominant. The right panel of Fig. 3 shows the σ × BR fort 2t2 pair production in the maximal mixing scenario.  Fig. 1). Also note that all the cross sections shown in the plots are leading order only. The next leading order K-factor for stop and sbottom pair production process is about 1.33 at the 14 TeV LHC [24,25], which has been taken into account in our collider analysis below in Sec. V. The low lying neutralino/chargino spectrum in Case IA comprises of a Bino-like LSP, as well as a pair of Wino-like states: χ 0 2 and χ ± 1 with nearly degenerate masses. In the minimal mixing scenario, the decay branching fractions are shown in Fig. 4 for left-handed t 1 (left),b 1 (middle), and right-handedt 1 (right). For the left-handedt 1 , decays to bχ ± 1 (∼ 70% for large mt 1 ) and tχ 0 2 (∼ 30% for large mt 1 ) dominate over tχ 0 1 once kinematically accessible, due to the stronger SU(2) L coupling compared to the relatively weaker U(1) Y coupling. Similarly,b 1 → tχ ± 1 (∼ 65%) andb 1 → bχ 0 2 (∼ 30%) dominate over the conventional channel of bχ 0 1 for sbottom. Given the dominant decay channels of the Wino-like neutralino/chargino 1 : Whent 1 is mostly right-handed, it decays to tχ 0 1 almost 100%, since its couplings to the Wino-like neutralino/charginos are highly suppressed.
The left, middle and right panels of Fig. 5 show the σ×BR for pure left-handedt 1t1 ,b 1b1 and pure right-handedt 1t1 pair production, respectively, in the minimal mixing scenario 1 For χ 0 2 , whether it decays preferably to Zχ 0 1 or hχ 0 1 depends on the sign of µ, as explained in detail in Ref. [26].  For the maximally mixed scenario, the decay oft 1 ,b 1 andt 2 are shown in the left, middle and right panels of Fig. 6, respectively. Fort 1 with large mass, the decay to bχ ± 1 , tχ 0 2 still dominates over tχ 0 1 , but the corresponding branching fractions are smaller compared to the pure left-handed case (Fig. 4) due to the decrease of the coupling to the Wino-like state caused by the right-handed stop component. Forb 1 , while tχ ± 1 and bχ 0 2 modes still dominate over bχ 0 1 mode, the new decay channel of Wt 1 opens up and even dominates over most of the mass range. Its branching fraction varies between 100% to about 40% for mb 1 between 400 GeV to 1800 GeV. Fort 2 , in addition to bχ ± 1 and tχ 0 1,2 (about a few percent to 20%), decays to a light stop/sbottom plus a gauge boson become comparable or even dominant: about 50% − 70% for Zt 1 and about 20% − 15% for Wb 1 .
The left, middle and right panels of Fig. 7 show the σ × BR fort 1t1 ,b 1b1 , andt 2t2   in the minimal mixing scenario.

C. Case IB: Bino-LSP with Higgsino-NLSP
The low lying neutralino/chargino spectrum in Case IB comprises of a Bino-like LSP, as well a pair of Higgsino-like neutralino states χ 0 2,3 and chargino states χ ± 1 with nearly degenerate masses. Fig. 8 shows the branching fractions of left-handedt 1 andb 1 and righthandedt 1 in the left, middle and right panels for the minimal mixing scenario. Fort 1 , decays to tχ 0 2,3 dominate over bχ ± 1 and tχ 0 1 since the former ones are controlled by the large top Yukawa coupling, compared to the small bottom Yukawa coupling and U(1) Y couplings for the latter two. However, forb 1 , the decay of tχ ± 1 becomes dominant since thẽ b LtRH + u coupling is proportional to the top Yukawa while its couplings to χ 0 2,3 and χ 0 1 are suppressed by the bottom Yukawa coupling and U(1) Y couplings. For the right-handedt 1 case, it dominantly decays to bχ ± 1 , reaching almost 50%, while decays to tχ 0 2 +tχ 0 3 are about 20%. All channels are controlled by the top Yukawa coupling while the latter ones have extra phase space suppression. Given the near degeneracy of the two Higgsino states χ 0 2,3 , contributions from final states involving χ 0 2,3 are usually summed over in collider analyses. Given the further decays of χ ± 1 → W χ 0 1 , χ 0 2,3 → Zχ 0 1 /hχ 0 1 as discussed in detail in [26], the pair production of stops and sbottoms lead to complicated final states at the collider.
The left, middle and right panels of Fig. 9 show the σ × BR for pure left-handedt 1t1 , b 1b1 and pure right-handedt 1t1 pair production in the minimal mixing scenarios of Case   bχ ± 1 (dominant fort R ) (see the left and right panel of Fig. 8) have roughly the same decay branching fraction, around 30% each. Decay to the conventional state of tχ 0 1 is typically a few percent, unless other decay modes are kinematically unaccessible at small mt 1 .
Forb 1 , the relative strength of tχ ± 1 and bχ 0 2,3 is similar to that of theb 1 in the minimal mixing scenario, but the opening of the Wt 1 mode dominates the decay for most of the mass range, leading to the suppression of the tχ ± 1 and bχ 0 2,3 modes. With increasing mb 1 , tχ ± 1 becomes more and more important, which dominates over Wt 1 when mb 1 > ∼ 1300 GeV. Fort 2 , decay to Zt 1 is dominant, about 60% − 30% for mt 2 in the range of 700 − 1700 GeV. Decays to bχ ± 1 , tχ 0 2,3 are sub-dominant, around 10% − 20% for each channel. t 2 → Wb 1 is typically around 10% to about a few percent, whilet 2 → tχ 0 1 is only at a few percent level.
The left, middle and right panel of Fig. 11 show the σ ×BR fort 1t1 ,b 1b1 ,
Searches for the second stop utilize the decay oft 2 →t 1 Z/h, looking for signals including b-jets and large E T with either same flavor leptons reconstruction of the Z boson [8] and/or high p T jet and b-jet multiplicities with additional leptons [13,14]. The second stop mass is excluded up to about 600 GeV. Stop searches in scenarios with a Gravitino LSP are explored in Refs. [8,15]. Stop searches in the R-parity violating MSSM can be found in Ref. [27].
There are other theoretical studies in the literature on the stop searches at the LHC, mostly focusing on the stop decaying to light generation quarks [28,29] or a light stop with little missing energy, which mimics the W W signal at the LHC [30][31][32][33][34].

V. COLLIDER ANALYSIS
In this section, we study the detectability of the light stop in Case IA with a mass hierarchy of M 1 < M 2 < M 3SQ ≪ |µ|, M 3SU , resulting in a mass spectrum including a mostly left-handed stop and mostly left-handed sbottom, Wino-like NLSPs, and a Bino-like LSP.
We choose a benchmark point with the specific set of parameters and the corresponding mass spectrum shown in Table I. The value ofÃ t is chosen such that the SM-like Higgs mass is around 125 GeV. Note that even thoughÃ t is set to a large value, the large mass splitting between M 3SQ and M 3SU results in a mostly left-handedt 1 and mostly righthandedt 2 . Therefore, the decay patterns oft 1 andb 1 follow those of the Case IA: purely left-handed stop/sbottom in the minimal mixing scenario.  The decay channels for the light stop of the benchmark point are shown in Table II.
While the dominant decay channel ist 1 → bχ + 1 with 78% branching fraction, the subdominant channelt 1 → tχ 0 2 is about 20%, providing an interesting signal where χ 0 2 can either decay to a Higgs or a Z boson. For our choice of parameters with µ > 0, χ 0 2 dominantly decays to hχ 0 1 , as shown in Table II. Flipping the sign of µ could lead to another interesting channel of χ 0 2 → Zχ 0 1 , which is left for future study.
Decay Branching Fractioñ  In our analysis, we study the stop pair production with one stop decaying viat 1 → tχ 0 2 → bW hχ 0 1 and the other stop decaying viat 1 → bχ + 1 → bW χ 0 1 . We consider semileptonic decays of the two W s and the Higgs decay to two b-quarks. The signal contains four b-jets, two jets, one isolated lepton and large missing energy. The presence of a single lepton helps to reduce QCD backgrounds without significant branching fraction suppression.
The dominant SM backgrounds are tt and ttbb. While tth is an irreducible background, the production cross section is typically small. Other backgrounds consist of ttW , ttZ and bbW W .
Event samples are generated using Madgraph MG5 aMC V2 2 1 [35], processed through Pythia 6.420 [36] for fragmentation and hadronization and then through Delphes-3.1.2 [37] with the Snowmass combined LHC detector card [38] for detector simulation. Both the SM backgrounds and the stop pair production signal are normalized to theoretical cross sections, calculated including higher-order QCD corrections [24,25,[39][40][41][42][43]. We apply the following event selection cuts: • Events are required to have at least four isolated jets with • Among the jets, at least two are b-tagged jets.
Additional optimization cuts are applied to further enhance the signal and suppress the SM backgrounds: • E T , defined as the magnitude of the missing transpose momentum, p miss T , ranging from 100 to 200 GeV, in increments of 20 GeV.
• H T , defined as the scalar sum of the p T of all surviving jets: H T = p jet T , ranging from 400 to 600 GeV, in increments of 50 GeV.
• Transverse mass m T , defined as the invariant mass of the lepton and the missing transpose momentum: ranging from 100 to 200 GeV, in increments of 20 GeV. • N j , the number of all surviving jets satisfying p j T > 25 GeV and |η j | < 2.5, to be at least 4, 5, or 6.
• N bj , the number of all tagged b-jets, to be at least 2, 3, or 4.
The distributions of E T and m T for both the signal and the SM backgrounds are shown in Fig. 12. In the E T distribution, the E T for all the SM backgrounds comes only from the neutrino, which is typically smaller than that of the signal with additional E T contribution from the LSP. The transverse mass for the signal process extends beyond the SM threshold of the W boson mass. The H T distribution of the signal is maximum at a higher value compared to the SM backgrounds.
In Table III, we present the cumulative cut efficiencies for the signal and dominant SM backgrounds with optimized cuts. By utilizing strong E T , H T and m T cuts, we significantly reduce the SM backgrounds. The stop signal process typically generates multiple hard jets in our specified decay. The N bj cut further plays an important role in cutting tt, ttW , ttZ and bbW W backgrounds. tt is the dominant background given its large cross section. ttbb is the second dominant background given its relatively large cross section and similar final states to the signal process. tth, ttZ, ttW and bbW W can be sufficiently suppressed due to low cross sections. We optimize the significance S/ √ B for all the combinations of the advanced cuts. We impose a constraint on the number of signal events, N s ≥ 3 for 300 fb −1 in order to obtain sufficient statistics.

VI. SUMMARY AND CONCLUSION
Most of the current stop and sbottom searches at the LHC have been performed considering the channels of tt + E T , bbW W + E T for stop and bb + E T for sbottom, assuming the stop and sbottom decay 100% into these channels. However, in many regions of MSSM parameter space, these decay channels are subdominant, resulting in relaxed bounds from current LHC searches. In this work, we studied decays of the stop and sbottom in the cases of a Bino-like LSP with either Wino-like or Higgsino-like NLSPs in the low energy spectrum, for the left-and right-handed stops and left-handed sbottom in the minimal mixing scenario, andt 1,2 ,b 1 in the maximal mixing scenario. We found that new decay channels oft 1 → tχ 0 2,3 ,b 1 → bχ 0 2,3 , tχ ± 1 , Wt 1 open up, which could even dominate over conventional channels. For the heavier stop state,t 2 , a new channel oft 2 → Wb 1 appears in addition tõ t 2 → Zt 1 in the maximal mixing scenario. Given the further decays of χ 0 2,3 and χ ± 1 , pair production of stops and sbottoms at the LHC typically leads to bb plus multiple gauge bosons plus E T final states. Conventional search channels of bbW W + E T and bb + E T could be highly suppressed.
We performed a detailed collider analysis for the reach of the stop at the 14 TeV LHC with 300 fb −1 integrated luminosity for one particularly interesting channel in the Bino-like LSP with Wino-like NLSP case. We considered left-handed stop pair production with one stop decaying viat 1 → tχ 0 2 → bW hχ 0 1 and the other stop decaying viat 1 → bχ ± 1 → bW χ 0 1 , leading to bbbbjjℓ + E T final states. Our results show that for a LSP mass of 250 GeV, the 95% C.L. exclusion reach is about 1040 GeV for the stop and the 5σ reach is about 940 GeV. The reach decreases with smaller LSP mass.
Considering different low-lying neutralino/chargino spectra provides several promising channels for the stop and sbottom study. In this paper we focused on final states with a Higgs boson. Decays of χ 0 2 to Zχ 0 1 could be dominant with a different choice of sign(µ). Furthermore, a different mass spectrum of neutralino/chargino with LSP being either Winolike or Higgsino-like might give rise to more interesting final states. It is important to identify the leading decay channels in various regions of parameter space to fully explore the reach of the LHC for the third generation squarks, which has important implications for the stabilization of the electroweak scale in supersymmetric models. The strategy developed in our analysis can be applied to the study of top partners in other new physics scenarios as well.