Compressed Stop Searches with Two Leptons and Two jets
Abstract
In top squark (stop) searches with a compressed spectrum, it is very helpful to consider the stop production recoiling against a hard jet from the initial state radiation to obtain a significant amount of missing transverse energy. In particular, the kinematic variable which measures the ratio of the lightest neutralino mass and the stop mass proved to be crucial in separating the signals from the backgrounds in both the allhadronic decay and the semileptonic decay of the stops. Here we generalize the search method to the dileptonic stop decays. In this case, due to the two missing neutrinos, there are not enough kinematic constraint equations to solve for the variable exactly, but only render an allowed interval consistent with the event. However, we show that the minimum and the maximum values of this allowed interval still provide useful variables in discriminating signals from the backgrounds. Although in the traditional stop decay to a top quark and the lightest neutralino, the dileptonic mode is not as competitive due to its small branching ratio, it becomes the main search mode if the stops decay through the charginos and sleptons with a compressed spectrum. We show that with the new variables, the dileptonic search of the stop can cover regions of the parameter space which have not been constrained before.
I Introduction
Since the discovery of a 125 GeV Higgs boson in 2012 Aad:2012tfa (); Chatrchyan:2012xdj (), the Large Hadron Collider (LHC) has not discovered any other new elementary particle or observed major deviations from the Standard Model (SM). It leaves the naturalness problem of the SM still a mystery, as the large quadratic contribution to the Higgs masssquared parameter from the top quark loop would destabilize the electroweak (EW) scale if it is not canceled. A major theoretical endeavor to address the naturalness problem is to extend the SM by supersymmetry (SUSY), so that the Higgs mass is protected by this additional symmetry from the quadratic divergence. Under SUSY, every SM fermion (boson) has its bosonic (fermionic) partner. The superpartners must receive large enough masses from SUSY breaking effects so that they have escaped the experimental detection so far. However, to make the SM natural, the superpartners of the top quark (top squarks or stops) should have masses close to or below 1 TeV in order to cancel the large top loop contribution. Another benefit of SUSY is that it provides natural candidates for dark matter in the universe if the parity is conserved. The lightest supersymmetric particle (LSP) is stable and can be a weakly interacting massive particle (WIMP) dark matter if it is not charged under or , e.g., the lightest neutralino which is a linear combination of the superpartners of the EW gauge bosons and the Higgs boson. At colliders, superparticles are pairproduced and decay down to the LSPs which escape the detectors, leaving missing energy signals as one of the signatures of SUSY.
There have been extensive searches for the stops at the LHC in various channels. With the new Run 2 results, CMS and ATLAS ATLAS:2017kyf (); ATLAS:2017tmd (); ATLAS:2017msx (); Sirunyan:2017cwe (); Sirunyan:2017cwe (); CMS:2017qjo (); CMS:2017arv (); CMS:2017vbf () have pushed the lower limit of the stop mass to TeV, assuming that the stop decays to the LSP and the top, and that GeV. The limit becomes weaker for a smaller mass difference between the stop and the , or if the stop decays differently in the case that the LSP is the sneutrino Chala:2017jgg (). For the difficult cases of very compressed spectra, , various search channels and kinematic variables have been proposed Hikasa:1987db (); Muhlleitner:2011ww (); Boehm:1999tr (); Das:2001kd (); Drees:2012dd (); Bai:2013ema (); Carena:2008mj (); Hagiwara:2013tva (); An:2015uwa (); Macaluso:2015wja (); Cheng:2016mcw (); Jackson:2016mfb () and new techniques have been adopted in recent experimental searches ATLAS:2017kyf (); ATLAS:2017tmd (); ATLAS:2017msx (); Sirunyan:2017cwe (); Sirunyan:2017cwe (); CMS:2017arv (); CMS:2017vbf ().
The search in the compressed region has been challenging because its signature is hard to distinguish from the SM production at the LHC. For which is called the top corridor, the LSP and top are almost static in the rest frame of the stop decay. Therefore, in the lab frame, the top and the LSP would be collinear and that
(1) 
In the stop pair production, the two LSPs tend to travel back to back, resulting in a cancellation of their transverse momenta, thus leaving little trace for s. A way to separate the signals from the backgrounds is to consider the stop pair production together with a hard jet from the initial state radiation (ISR) Carena:2008mj (); Hagiwara:2013tva (); An:2015uwa (); Macaluso:2015wja (); Cheng:2016mcw (); Jackson:2016mfb (). From the conservation of momentum,
(2) 
both LSPs tend to be emitted antiparallel to the ISR jet, resulting in a significant amount of missing transverse momentum (). By studying the fully hadronic decays of such events, it was pointed out that the ratio between and , defined as , can be a useful kinematic variable to differentiate between the stop and top decays An:2015uwa (); Macaluso:2015wja (). Since the missing momentum in stop decays is mainly due to the presence of LSPs, together with equations (1) and (2), the stop pair production tends to have
(3) 
which is expected between zero and one. On the other hand, the for the hadronic decays of is due to the detector smearing effect, so is expected close to zero for the background. Using this variable, with the help of the recursive jigsaw method Jackson:2016mfb () to separate the ISR from the stop system, ATLAS has excluded the stop mass between 235–590 GeV along the top corridor () with 36.1fb of integrated luminosity ATLAS:2017kyf ().
For the semileptonic and dileptonic decays of the stops, becomes less informative if the neutrinos’ contribution to cannot be easily separated from that of neutralinos. In general for the same , the signal events are still expected to have larger missing transverse energies (MET) than those of the backgrounds. This can be used to constrain the compressed stop, though the current bounds are somewhat weaker than the allhadronic channel ATLAS:2017tmd (); ATLAS:2017msx (); CMS:2017arv (). In a previous paper Cheng:2016mcw () we showed that for stop semileptonic decays, the neutrino momentum can be solved (up to a twofold ambiguity) by exploiting the kinematic features of the compressed stop spectrum in the top corridor. Therefore the neutrino momentum can be subtracted from and a modified version of can be defined:
(4) 
where is the neutrino’s contribution to the missing momentum along the ISR direction. The variable can be similarly used in the stop search along the top corridor for signal events with one lepton.
For the dileptonic stop decays, the situation is tougher. The branching fraction is small if the stops decay through the tops. Furthermore, there are more unknowns than the number of kinematic constraints available from the decay chain, hence the two neutrinos’ contribution to cannot be solved exactly and be completely removed from the total . One would think that the final states with may not be a competitive channel for the compressed stop search. However, with a more complicated spectrum, there could be other decay chains of stops which end up with the same dileptonic final states and they can be even dominant in certain cases. For example, this can occur if there are also charginos and sleptons in the spectrum between the stop and the lightest neutralino as shown in Fig. 1. The all hadronic and semileptonic searches may become ineffective if the dileptonic decays become dominant. Therefore, it is still worthwhile to consider the dileptonic search channel for the compressed stop. Another potential merit of the dileptonic channel is that the only significant background to the final states is the dileptonic decays, so one only needs to focus on suppressing this background. In contrast, the all hadronic and the semileptonic signals also suffer from decays with an extra lepton which is not identified, and they are often the major backgrounds in the variable.
Despite the fact that the two neutrinos’ momenta can not be solved exactly for the dileptonic events from the or the stop pair in the top corridor with ISR, the kinematic constraints still strongly limit the ranges of their possible values, which translate to an allowed range of values for each 2jet(s) final state event. For the background, is expected to be close to zero and contain the point zero. For the stop events, due to the additional ’s contribution to , it is expected to shift to larger values. Even for stop events decaying through sleptons as in Fig. 1, which do not have the correct kinematics for the kinematic constraint equations of , one can still obtain the corresponding ranges anyway, and there is no reason that they should be close to zero as the background. As a result, the allowed range, characterized by its minimum and maximum values , , may be used to distinguish the signals from the backgrounds of the 2jet(s) final state, irrespective of the stop decay topologies.
The goal of this work is to study the usefulness of the variables in the dileptonic decay channels of the compressed stop search and their search reaches. The rest of the paper is organized as follows. In section II, we review for the semileptonic decays and generalize the concept to dileptonic decays. Even though a unique cannot be obtained in the dileptonic case because of insufficient kinematic constraints, the minimum and maximum allowed values from the constraints can still provide useful variables to distinguish signals and backgrounds. In section III, we perform analyses using the , variables on the chosen benchmarks, for both the stopslepton decay and the tradition stop dileptonic decay scenarios. Section IV contains our conclusions. A detailed description on how we calculate , , and a comparison of significances in search analyses with and without using the , variables are collected in appendices.
Ii Kinematics and Variables
Since the assumptions and tools we employ to analyze the dileptonic stop decays share many similarities to the semileptonic stop decays, we begin by reviewing the concept of in semileptonic signals following Section 2 of Ref. Cheng:2016mcw (). For the stop pair production with a hard ISR jet in the top corridor, due to the neutralinos is antiparallel to the of the ISR, because the components perpendicular to ISR from the two neutralinos cancel each other. As a result, the component of perpendicular to the ISR can be attributed to the presence of the neutrino. Once the is identified, is uniquely determined from the experimental measurements. Combining it with the three massshell conditions
(5) 
and the measured momenta of the lepton and the jet (assuming that the jet from the corresponding top decay can be identified), we can solve for the neutrino momentum (up to a twofold ambiguity due to the quadratic massshell equations). After obtaining , we can subtract its contribution from and get the relation Eq.(4):
For the type of dileptonic stop decays depicted in Fig. 2, which will be referred to as the traditional stop decay from now on, there are 6 massshell equations from both top decay chains in an event:
(6) 
There are two more equations from the transverse momentum conservation, from the components perpendicular and parallel to the ISR.
(7) 
where we assumed which is valid in the top corridor.
For such an event we have 8 equations but 9 variables, namely the fourmomenta for the two neutrinos and the transverse momentum due to the neutralinos, . However, we note that for the SM background, where there is no contamination from the neutralinos, these equations are exactly solvable for . Although we cannot reconstruct all particle momenta for this signal topology, we may ask the question: what is the range of that is compatible with all the kinematic constraints from the 8 equations? This allows us to obtain an upper and a lower bounds for the value of , which is now adjusted to take into account of the dileptonic nature of the signal:
(8) 
Because we do not obtain a single value for but just an allowed range, we include a sign in the definition of such that it is positive if the (solved) is antiparallel to and negative if they are parallel. Effectively, one can include a factor
(9) 
in the definition. This is in accordance with the convention that the true value is positive for the stop events.
For the background, the allowed range for should contain the point zero, and the upper and lower bounds of are expected to converge to zero in the highly boosted (large ISR) regime. Similarly, for the compressed stop events which follow the traditional decays, the allowed range is expected to be around its true value and converges to it in the highly boosted case. On the other hand, for the dileptonic stop decays through sleptons as depicted in Fig. 1, which we will refer to as the stopslepton decay, there is no true value for because the kinematic equations do not apply to this topology. Nevertheless, due to the contributions to the missing momentum from the LSPs at the end of the decay chains, in general we do not expect , to be close to zero as in the case of the background. Therefore, they may also be used to suppress the background in this case.
In calculating and for an event with the 2jet(s) final state, there are some practical issues. First, there are two possible combinations of pairing the jet and the lepton on the same decay chain. In addition, because the kinematic equations combine into a quartic equation, there could be two disjoint allowed regions for in the solution, although it does not happen very often. There is no canonical way to deal with the multiple solutions. In Appendix A, we describe the procedure of solving the kinematic equations and how we define and for our benchmark studies in more details. Different strategies of choosing and due to these ambiguities do not have significant effects on the final results.
The SM background events where leptons arise from decays, such as or , tend to have small azimuthal angle separation between the and the light leptons. This can be explored to further suppress the SM dileptonic backgrounds. For single leptonic final states, removing such backgrounds can be done with the transverse mass , which has a sharp dropoff around for the SM backgrounds. For the dileptonic events, there are two leptons in the final states. A variable called leverage inspired by the transverse mass was defined in Ref. Cheng:2016npb () to further reduce the SM backgrounds:
(10) 
where is the number of isolated leptons. By definition, if a lepton has a low azimuthal angle separation from the direction, its contribution to is small. As the stop signal events tend to have larger MET and larger lepton azimuthal separations due to the ’s in the final states, a minimum cut on can effectively suppress the SM backgrounds and enhance the signal significance.
Iii Compressed Stop in Dileptonic Searches
In this section, we perform some more detailed collider studies for both the stopslepton decay case and the traditional stop decay case with compressed specta. We use MadGraph 5 Alwall:2011uj () and Pythia 6 Sjostrand:2006za () to generate both the background and the signal events. MLM matching scheme Mangano:2002ea () is applied for both the background and the SUSY signal production in order to prevent doublecounting between the matrix elements and the parton shower. The detector simulation is performed by Delphes 3 deFavereau:2013fsa (). For the signals, the production cross section is normalized to 13 TeV NLO+NLL results Borschensky:2014cia (). The tagging efficiency is taken to be a universal with an overall lightflavor mistag rate of 1.5.
We expect that the SM production to be the dominant background. In this work, we have simulated +jet events with an amount equivalent to an integrated luminosity of 235 at LHC 13 TeV. Besides , the backgrounds from and production were simulated with a luminosity greater than 300 . Other SM backgrounds, such as diboson production or jets, have small cross sections or low signal efficiencies. Consequently we ignore these background processes for the rest of our discussion.
For our benchmark studies, all the events must satisfy the preliminary selections as described below. All the events are required to have two tagged jets with GeV, two light leptons with GeV. Since our analysis relies on a hard ISR jet, we require that the hardest nontagged jet has GeV. To take into account the cases where there are more than one ISR jets, we define to be the vector sum of the three leading nontagged jets. We also require that GeV, since the signal is expected to have a substantial amount of missing transverse momentum. Furthermore, to reduce and related backgrounds, we veto all events with a tagged jet or oppositesign sameflavor lepton pairs with .
iii.1 StopCharginoSlepton Scenario
For the stop chargino slepton type of decay, we take a simplified model approach. Assuming that and are degenerate, so are and . The simplified model is therefore characterized by four masses: , , and . To suppress the direct decay, the gap is chosen to be smaller than . We assume that all stops decay through the chargino and the (first two generation) sleptons with a 50% branching ratio through either the charged sleptons or the sneutrinos. Since we are interested in the “compressed” spectrum, and are chosen to be small, so that the searches based on the type of variables CMS:2017qjo () are ineffective. This allows us to explore the usefulness of the type of variables. In this subsection we show detailed analyses on two signal benchmarks: BMP1 GeV, GeV, GeV, GeV) and BMP2 GeV, GeV, GeV, GeV). An expected search reach for GeV in the – plane is presented later.
In Fig. 3, we plot a series of twodimensional distributions for the benchmark BMP1 signals and the background after the preliminary selections. It can be clearly seen that the signal events tend to have a larger MET, especially for a larger . We also notice that in the vs. plane, signals have a different feature of the distribution compared to the backgrounds. As discussed in Sec. II, the SM events are expected to have close to 0. This can be recognized in the distribution, as shown in the bottom panel of Fig. 3. The distribution for the background is smaller than that of the signal, especially when MET increases.
To obtain a better search reach, one can divide the events into many signal regions based on these kinematic variables and perform a multivariate analysis. However, to get a good intuition on how the kinematic variables discriminate the signal and the backgrounds, we perform a simpler cutandcount analysis to select a few signal regions in this theoretical study. First, we require GeV and GeV. Also, since the benchmark spectra focused in this study are compressed, we impose a cut on the leading lepton GeV for all events. Then we apply cuts on the preselected events and divide them into the following three exclusive signal regions based on their MET:

for events with MET 550 GeV, is required (SRH);

for events with MET GeV, and Leverage 80 GeV are required (SRM);

for events with MET GeV, , Leverage 80 GeV, , and are required (SRL).
The cut flow and the number of events passing the cuts for two benchmark points and the SM backgrounds, normalized to an integrated luminosity of 300 fb are shown in Table 1. As expected, the SM is the dominant background for all signal regions.
Initial  Preselection  225  SRL  SRM  SRH  
BMP1  8.9  167  144  144  115  24.5  27.5  10.7 
BMP2  2.6  145  121  116  96.2  4.8  19.6  11.5 
SM  1.46  1.16  8.17  2.60  86.6  89.2  11.4  
SM  8.1  627  459  293  124  1.7  2.6  0.9 
SM  2.8  166  121  78  32  0.7  0.7  1.4 
SM total  1.47  1.17  8.21  2.76  89  92.5  13.7 
To calculate the signal significances for the benchmark models, we use the likelihood method with the assumption that the overall number of background events in each signal region respects the normal distribution with a fractional uncertainty . The likelihood is defined to be
(11) 
where and are corresponding numbers of signal and background events, , and is the normalized normal distribution with the mean and a standard deviation . The final significance from this method is simply given by . For the case with no systematic error, , this equation simply reduces to the standard formula Cowan:2010js ():
(12) 
For BMP1, we get a significance of for 300 with (without) a 10 background uncertainty. For BMP2, we get a () significance. As shown in the cut flows for BMP2, the contribution from the higher MET bins (SRH, SRM) become more important for larger signal masses. Since the number of background events significantly decreases in SRH, the background uncertainty affects the significance far less for BMP2 compared to BMP1.
One could ask how much the new variables really help the stop search in this case, given that the signal and background distributions already look different in the standard variables such as and . In Appendix B we compare the analyses with and without the variables by dividing the variable space into the same number of signal regions and show that the inclusion of variables does substantially improve the signal significance.
To explore the search reach of the stopslepton decay case, we repeat the analysis for a range of different spectra. To simplify the study, we follow the assumptions as adopted in the benchmark study. We take and as the free parameters and fix 30 GeV, and . We then calculate the significances for points in the plane of vs. and obtain the 2 exclusion reach for an integrated luminosity of 300 fb. The result is shown in Fig. 4. The most stringent constraint for this kind of spectrum comes from the production, which gives 3 lepton or samesign dilepton final states. The current limit on the chargino mass can be up to 1150 GeV for a light LSP mass CMS:2017fdz (); ATLAS:2016uwq (). However, the mass reach for a compressed spectrum is limited. The current CMS exclusion limit for is also plotted in Fig. 4. According to the current LHC study, keeping and fixed while changing would not change the exclusion limit drastically.
One can see that the dileptonic stop search in the stopslepton decay case can probe the parameter space which is not excluded by the current experimental limits. Of course one should note that our search is based on the stop pair production while the exclusion limit on Fig. 4 comes from production which is independent of the stop mass, besides the difference in the integrated luminosities. With a relatively light stop, the stop pair production has the advantage of larger production cross section. Although introducing jets in our final state would increase background events, the kinematic variables introduced in this work allow us to better handle the backgrounds. We should also point out that our projected reach is based on the simple cutandcount analysis. It might be further improved with a more sophisticated multivariate analysis.
iii.2 Traditional Stop Decays with a Compressed Spectrum
In this subsection we turn to the compressed stop searches with the traditional decay into the final states. The all hadronic and semileptonic channels have obtained quite strong limits in the compressed region with the help of the hard ISR and the variable ATLAS:2017kyf (); CMS:2017vbf (); CMS:2017arv (); ATLAS:2017msx (). Stop mass up to 590 GeV has been excluded along the top corridor ATLAS:2017kyf (). The dileptonic channel suffers from the small branching ratio. In order not to have too few signal events, we choose the point =600 GeV and =427 GeV just beyond the current limit for our analysis. In contrast to the stopcharginoslepton scenario, also has a physical interpretation for the signals in this case, and would be close to if the signals are very boosted. Therefore, in addition to the preselection criteria as described in the beginning of this section, we trigger on events with GeV instead of GeV for a single jet and .
Fig. 5 shows the distributions of a couple of kinematic variables that can help to discriminate the signal from the background. Based on the distributions, we require that MET GeV and GeV.
In this case, and are expected to be around the true value which is larger than 0 for the signal, hence we scan the and in the range from 0 to 2. Fig. 6 shows the two dimensional distribution of and for the signal and the background after the MET and the cuts. Clearly, the background events aggregate near the region where . Based on this, we apply the final selection: and . This gives us a significance at 300 fb without taking into account of any background uncertainty.
In Table 2 we show a comparison between the semileptonic (Semi) analysis and the dileptonic (Di) analysis for the chosen benchmark point. The search strategy for the semileptonic stop decays is the same as described in Ref. Cheng:2016mcw (). For the preselections of the Semi analysis, we require at least 4 jets, 1 or 2 tagged jets, exactly one lightflavor lepton, GeV, and that the hardest jet (ISR) has GeV. Then we apply the following cuts: GeV, and GeV, which is grouped under “Other cuts” in Table 2. Finally we require that lies within the range (, 1).
As one can see from Table 2, for the chosen benchmark point, the strengths of the semileptonic and the dileptonic channels based on the simple analyses are not far from each other. The dileptonic channel has fewer signal events but also less background. Since these numbers are for a future integrated luminosity of 300 fb, they are not expected to compete with the projected allhadronic analysis. It is possible that the significances can be improved with more sophisticated analyses, although the improvement may be limited if the signal is statistically limited as in the case of the dileptonic decay. Nevertheless, analyses based on different final states provide complementary probes of various stop decay channels.
Initial  Preselection  Other cuts  cuts  
(Di)  5.4  69  27  15 
SM (Di)  2.4  1.24  382  60 
(Semi)  2.3  597  35.8  24.3 
SM (Semi)  1.2  5.82  327  15.2 
SM (Di)  2.4  1.24  352  103 
Iv Conclusions
The stops are the most relevant particles for the naturalness of the SM if SUSY is the solution to the hierarchy problem. The experimental verification of whether they exist is of undisputed importance in testing SUSY as a possible new symmetry principle of the universe and our understanding of the naturalness in quantum field theories. LHC has put very strong bounds on their masses which could challenge both issues. However, there are still search holes in the lower mass region if the SUSY spectrum is compressed. It is therefore very important to devise new methods in experimental analyses to cover these regions where the stop could still be hiding. A main difficulty in identifying the signal events is the lack of a significant MET in the stop decays if the spectrum is compressed. Recently it has been shown that this could be overcome by requiring the stop production recoiling against a hard ISR jet, which results in a large MET in the opposite direction of the ISR for the stop signal events. The experimental measurement of the variable which corresponds to provides a powerful way to distinguish signals from the backgrounds for both allhadronic and semileptonic decay channels. Consequently, significant regions in previous search gaps of the compressed spectrum have been excluded in the most recent ATLAS and CMS analyses.
In this paper, we extend the study to the dileptonic decay channel of the stop search. With two missing neutrinos, there are not enough kinematic constraints to solve for the to get a unique answer for a given event. However, we can find two new variables and which bound the interval of that is kinematically consistent with that event. In the limit of large ISR, they tend to converge to the true value. We found that these variables provide additional discriminating power, beyond the standard variables and MET, between the signals and backgrounds. For the traditional stop decay to in the top corridor, the dileptonic search is probably not as competitive as the allhadronic or semileptonic channels due to the small decay branching ratios. It is not far behind though so it can still provide a complementary analysis. On the other hand, the dileptonic search mode becomes most useful in the scenario where the stops dominantly decay through the charginos and sleptons with a compressed spectrum. In this case, the 2 can be the dominant final states, therefore the allhadronic and semileptonic searches are not effective. The strong trilepton or samesign dilepton charginoneutralino search constraints also diminish in the compressed region. Even though the signal topology is different from the one where the kinematic constraint equations for are derived, we have shown that the variables are still useful to suppress the dominant background which does have the topology of the constraint equations. It can cover a significant region of the parameter space which has not been experimentally explored by other methods before.
The power of the kinematic variables and techniques studied in this paper and previous works comes from utilizing our maximal knowledge of the kinematic information of the signal and background events. It is conceivable that similar techniques can be used to extend the coverage of other compressed region where the constraint is still weak, like the corridor () in the stop search, or other SUSY searches with compressed spectra. These directions will be explored in the future.
Acknowledgments
We thank Zhangqier Wang for useful discussion about the tradition stop analysis. This work is supported in part by the US Department of Energy grant DESC000999. H.C. C. is also supported by The Ambrose Monell Foundation at the Institute for Advanced Study, Princeton, and thanks HKUST Jockey Club Institute for Advanced Study for hospitality where part of this work was done.
Appendix A Solving for and
In this Appendix, we explain how we determine if Eqs. (6), (7) have real solutions and find the extremum values of that allow real solutions. We first guess a value for . Given a trial , we can reduce Eqs. (6), (7) down to a quartic equation. To determine whether it yields real solutions, one can follow the same procedure as in an efficient method of calculating Cheng:2008hk (). We first compute the Sturm Sequence for the quartic equation and compare the number of the sign changes at the positive and negative infinities. If the number of sign changes at two infinities are different, the quartic equation yields real solutions. The point is that we can determine whether a trial can solve the equations without actually solving them.
Since the background is more likely to have close to zero, the region covered between and ideally should contain zero. We choose the range of the search of between some negative value and some positive value. In the example of the stopslepton decay, the search range of is chosen to be . We first find an value which allow real solutions using the bisection method, then look for () from that value by decreasing (increasing) the value with steps of . Because the final states contain two leptons and two jets, there are two possible ways of grouping the pairs into decay products of the two top quarks. The trial point that solves at least one combination of is considered a viable point. We scan the following the steps until reaching the point where no solutions can be found for either combination of . The last viable point in this scan is designated as ().
In rare cases, there are disjoint intervals of having real solutions. We choose the and of the interval that contains the point 0 or is closest to 0. Specifically, if the initial and the determined in this way do not cover a region which contains zero, we start a new search following the same steps as described above, but with a range , where is the smaller number between the and found in the last search. The search continues recursively until we reach the point where either the and found covers the point 0 or the range itself becomes smaller than the precision .
Appendix B Validating the Usefulness of the Variables
To check the usefulness of as a new dimension for the dileptonic stop search, we perform a study on the stopcharginoslepton scenario and compare the significances of an analysis mainly using ISR and MET, and an analysis including the variables. We use the benchmark BMP1 for the numerical study and focus on the dominant background.
Both signal and background events need to satisfy preliminary selection rules as mentioned in Sec. III, also the leading lepton less than 100 GeV. For the “control” study without variables, we divide the ISRMET plane into 24 nonoverlapping signal regions with 4 ISR bins (200300,300400,400500,500) and 6 MET bins (200250,250300,300350,350400,400500,500), all in units of GeV. For each signal region the significance is calculated with a 10% independent background systematic uncertainty. Then we compare it with an analysis which also include the variable . The signal regions are divided into small boxes in the 3dimensional parameter space. To have a fair comparison of the two analyses, for the 3variable case we make a coarser grid in the ISRMET plane, so that the total number of signal regions is also 24. Specifically, the ISR variable is divided into 2 bins (200400,400), MET variable is divided into 4 bins (200250,250350,350500,500) and the variable is divided into 3 bins (01,12.5,2.5).
The overall combined significance for the analysis including the variable with background is , while for the “control” study without variables but a finer MET/ISR binning, the overall significance is . This corresponds to a improvement. The qualitative conclusion that the new variables help to improve the analysis holds for different choices of signal regions, whereas the extent of improvement depends on the details of signal region selections.
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