Constraining the Compressed Top Squark and Chargino along the W Corridor

Studying superpartner production together with a hard initial state radiation (ISR) jet has been a useful strategy for searches of supersymmetry with a compressed spectrum at the Large Hadron Collider (LHC). In the case of the top squark (stop), the ratio of the missing transverse momentum from the lightest neutralinos and the ISR momentum, defined as $\bar{R}_M$, turns out to be an effective variable to distinguish the signal from the backgrounds. It has helped to exclude the stop mass below 590 GeV along the top corridor where $m_{\tilde{t}} - m_{\tilde{\chi}_1^0} \approx m_t$. On the other hand, the current experimental limit is still rather weak in the $W$ corridor where $m_{\tilde{t}} - m_{\tilde{\chi}_1^0} \approx m_W +m_b$. In this work we extend this strategy to the parameter region around the $W$ corridor by considering the one lepton final state. In this case the kinematic constraints are insufficient to completely determine the neutrino momentum which is required to calculate $\bar{R}_M$. However, the minimum value of $\bar{R}_M$ consistent with the kinematic constraints still provides a useful discriminating variable, allowing the exclusion reach of the stop mass to be extended to $\sim 550$ GeV based on the current 36 fb$^{-1}$ LHC data. The same method can also be applied to the chargino search with $m_{\tilde{\chi}_1^\pm} -m_{\tilde{\chi}_1^0} \approx m_W$ because the analysis does not rely on $b$ jets. If no excess is present in the current data, a chargino mass of 300 GeV along the $W$ corridor can be excluded, beyond the limit obtained from the multilepton search.


I. INTRODUCTION
Weak-scale supersymmetry (SUSY) has long been considered as the leading candidate for the new physics beyond the Standard Model (SM). Supersymmetric particles have been searched for at colliders for decades but unfortunately none of them has been found yet. The strong limits from the searches at the Large Hadron Collider (LHC) have raised concerns if SUSY can provide the solution to the hierarchy problem of the Higgs mass scale in SM, as the LHC probes the energy scale into the TeV range. For the hierarchy problem, the most relevant particles are the superpartners of the top quark, top squarks (or simply stops), because the top quark has the largest coupling to the Higgs field and hence gives the largest quadratic correction to the Higgs mass-squared parameter. The stops are needed to be near the weak scale to cut off this contribution in order for the theory to be natural. The current lower bound on the stop has reached beyond 1 TeV in typical search channels at the LHC [1][2][3][4][5][6][7][8][9], which would imply quite severe fine-tuning already.
Of course, there are cases where the stop mass limit is not as strong yet. In particular, the limit degrades if the masses difference between the stop and the lightest supersymmetric particle (LSP) that it decays to become smaller, i.e., they have a compressed spectrum. In this case, the visible SM particles from the stop decay will not carry a large amount of energy.
The missing transverse momentum will also be suppressed because it is just the opposite of the sum of the visible transverse momentum. These signal events are more difficult to be distinguished from the SM backgrounds. The search strategies and search limits depend on how compressed the spectrum is and stop decay chains. For example, for highly compressed spectra (mt − mχ0 1 < m W ), the searches rely on 4-body decays (t → bχ 0 1 f f ) [2,3,[10][11][12][13][14][15] or flavor-changing decays (t → cχ 0 1 W ) [15][16][17][18][19][20], or even monojet [21]. Before the LHC Run 2, the most difficult case used to be the top corridor, where mt ≈ m t + mχ0 1 . This is because the top quark and the neutralino from the stop decay carry little momenta in the stop rest frame and are boosted with the same velocity as the original stop particle. The stop pair is produced back-to-back in the transverse plane, resulting in the cancellation of the two neutralinos' transverse momenta, leaving no trace of their existence. Then the events look exactly like the large SM tt background. The analyses of the Run 1 data provided essentially no constraint along the top corridor.
The situation has completely changed in Run 2 with new techniques being employed to attack this parameter space region. An important observation is that if the stop pair is produced with a hard initial state radiation (ISR) jet, the two neutralinos will be boosted in the opposite direction to the ISR jet, giving missing transverse energy (MET) anti-parallel to the ISR jet [22][23][24][25]. A variable R M which measures the ratio the two-neutralino transverse momentum and the ISR transverse momentum provides a powerful discriminator between the signal and the backgrounds, as it should be equal to mχ0 1 /mt for the stop events while close to zero for the SM tt backgrounds. With a sophisticated method to determine the ISR system [26], the Run 2 analysis has been able to exclude the stop mass below 590 GeV along the top corridor with 36 fb −1 of data [1], assuming 100% branching fraction to tχ 0 1 . This is quite impressive, and the limit is even stronger than the nearby parameter region where the mass difference betweent andχ 0 1 is somewhat off the top quark mass. The analysis with the R M variable is based on the fully hadronic channel where there is no additional MET other than that carried by the two neutralinos. In this case R M is simply given by In semileptonic or dileptonic decays, however, the neutrino(s) coming from the W decays give an additional contribution to MET, which ruin the relation in Eq. (1). One way to deal with this is to try to reconstruct the neutrino momentum so that it can be subtracted from the total MET to obtain the MET due to the neutralinos only. Then a modifiedR M variable related to mχ0 1 /mt can be defined analogously as a discriminating variable. For the semileptonic decay, it was shown [27] that from the 3 mass shell conditions (m t , m W , m ν ) together with the assumption that the perpendicular component of the / p T relative to p T (J ISR ) is entirely due to the neutrino, the neutrino momentum can be solved with a two-fold ambiguity. After subtracting the solved neutrino momentum, theR M variable also provides a strong discriminator for the stop events in the semileptonic decay channel and makes it competitive with the fully hadronic result.
For the dileptonic channel with two final state neutrinos, there is one more unknown than the number of kinematic constraint equations, so we can not completely reconstruct the neutrino momenta. Instead, for each event we can only obtain a finite range ofR M which can be consistent with that event. Nevertheless, the upper and lower limits of the allowedR M , denoted byR max andR min could provide potential variables for discriminating signals from backgrounds. In Ref. [28], it was found indeed thatR max andR min provide more discriminating power than just using p T (J ISR ) and MET. Although, for the case of stop decaying to top plus LSP, the dileptonic channel is not expected to compete with the all-hadronic or semileptonic channels due to the small branching ratios, the dileptonic search can be useful if the SUSY spectrum is such that the stop decays mainly through the chargino and the slepton decays to the LSP, in which case the dileptonic final states can be dominant [28,29].
After the progress in stop search coverage along the top corridor, the W corridor where mt ≈ m W + m b + mχ0 1 remains relatively weakly constrained. In this case, the bottom quark, W andχ 0 1 from the stop decay are also static in the stop rest frame. The missing p T from the twoχ 0 1 's again cancels from the back-to-back boost of the stop-pair in the transverse plane. Such events are difficult to be distinguished from the SM backgrounds, resulting a poor reach in current LHC searches and the stop could still be as light as ∼ 360 GeV around that region [2,3]. A natural thought is again to consider events with an ISR jet to boost the stop-pair in the opposite direction so that theχ 0 1 's will produce some MET. Then one can use the similar R M variables to distinguish signals from backgrounds. A main goal of this study is to explore whether this technique can help to improve the stop mass bound around the W corridor.
We will focus on the semileptonic events where one W decays leptonically and the other W decays hadronically. The b-jets from the stop decays will be soft so they will not be useful due to low tagging efficiencies and large hadronic backgrounds. Compared with the semileptonic stop events along the top corridor, we lose a top quark mass shell constraint because the decay does not go through an on-shell top quark. Therefore the neutrino momentum can not be completely reconstructed and a uniqueR M value can not be obtained. Nevertheless, the kinematic constraints still limitR M into a finite range. We can define theR max andR min variables just as for the case of the dileptonic stop events in the top corridor to examine whether they are useful in suppressing backgrounds. We will find out thatR min does provide a useful discriminating variable in this case.
Since the b-jets are too soft to be useful in the W corridor of the stop, the signal events look the same as the chargino pair production in the W corridor (mχ± 1 ≈ m W + mχ0 1 ) if one ignores the b-jets. The same analysis can be applied to the chargino search around the W corridor. In SUSY, the chargino is usually accompanied by one or two neutralinos with similar masses, depending on whether it is wino-like or higgsino-like. Hence, one should consider chargino and neutralino pair productions altogether. Under the assumption that the LSP is bino-like and the chargino (neutralino) decays to the LSP plus an (one-shell or offshell) W (Z), the current strongest constraints come from tri-lepton searches [30][31][32][33][34], where the production cross section is taken to be coming from the winos. However, for the same reason due to the large SM W Z background, there is a search gap around mχ0 2 − mχ0 1 ≈ m Z , where the exclusion reach for mχ0 2 is only about 225 GeV [31]. We find that the search usinḡ R M variable with an ISR jet could compete with the tri-lepton search around that region. This paper is organized as follows. In section II, we discuss the kinematic constraints for the stop pair production in the W corridor with an ISR jet in the semileptonc channel. We defineR M for this case and describe how to obtain the minimum and maximum allowed R M values from the constraint equations as the kinematic variables for the stop searches. In section III, we investigate the usefulness of theR min ,R max variables for the stop searches in the W corridor. A more detailed description of the analyses is given for a chosen benchmark stop mass at 450 GeV and it is shown thatR min is quite useful in suppressing certain SM backgrounds. We then perform the study for a series of points along the W corridor to obtain the signal significances in comparison with current search limits. In section IV we apply the similar analysis to the chargino (and second neutralino) pair productions and compare with the tri-lepton search limits. Section V contains our conclusions. A detailed description of the solutions forR min andR max from the kinematic constraints is presented in appendix A. In appendix B we compare the analyses with and without using theR min andR max variables and show that they indeed can improve the signal significances.

II. KINEMATICS AND VARIABLES
For the stop pair production together with and ISR jet, the momentum conservation tells us that In the W corridor where mt ≈ m W + m b + mχ0 1 , the W , b andχ 0 1 from thet decay will simply be co-moving with the same velocity as their mother particlet. Therefore we have Together with Eq. (2) we obtain which is the kinematic variable that we would like to use for discriminating stop signals from backgrounds. However, for semileptonic decays, this quantity is not directly measurable, because the neutrino from the W decay also contributes to the missing transverse momentum / p T besides the twoχ 0 1 's. To obtainR M we need to know the neutrino momentum so that it can be subtracted from / p T to get the total p T of the twoχ 0 1 's. The neutrino momentum satisfies the two mass shell conditions: In addition, the sum of the transverse momenta of the twoχ 0 1 's, p Tχ 0 1 ,1 + p Tχ 0 1 ,2 , should be antiparallel to the ISR jet. If we decompose / p T into components parallel and perpendicular to the p T (J ISR ) direction, the perpendicular component should be attribute to the neutrino: which gives us one more constraint on the neutrino momentum once / p ⊥ T is determined. On the other hand, the parallel component receives contributions from both theχ 0 1 's and the neutrino: Using Eq. (4), we can write where the quantities include signs which represent being parallel or antiparallel to the ISR.
We can see that there is one more unknown than the number of kinematic constraint equations, so we can not completely solve the constraint equations to obtain a unique or discrete solution. 1 However, the kinematic constraints still limit the solutions to a finite range. As in Ref. [28], the minimum and the maximum values of the allowed range ofR M , 1 In contrast, in the top corridor there is an additional mass shell condition of the top quark mass, (p + so the constraint equations can be solved to yield discrete solutions [27].
R min andR max , provide potential variables for the signal and background discrimination.
The detailed computation ofR min andR max is presented in Appendix A. The combination of the kinematic constraints gives a quadratic equation for one neutrino component. To have real solutions, the discriminant of the quadratic equation is required to be 0 . The discriminant is also quadratic inR M , so the two solutions of discriminant = 0 giveR min and R max .
In the above discussion, the information of the b-jets from the stop decays is not used at all. In the W corridor, the b-jets are typically too soft to be identified or to be useful. As a result, the same analysis also applies to the chargino pair production with the chargino decays to W +χ 0 1 and mχ± 1 ≈ m W + mχ0 1 . In SUSY, there is usually at least one neutralino (χ 0 2 ) having a similar mass as that of the chargino, so the neutralino-chargino pair production is also important at the same time. Ifχ 0 2 decays to Z ( * ) (h ( * ) ) +χ 0 1 with Z ( * ) (h ( * ) ) decays hadronically and the W from the chargino decays leptonically, our analysis can also apply. For the chargino-neutralino production, the trilepton search traditionally provides the strongest constraint. We will perform a study based on our method in Sec. IV to compare it with the current trilepton search bound.

A. Signal and Background Generations
We use MadGraph 5 [35] and Pythia 6 [36] to generate both the background and the signal events. MLM matching scheme [37] is applied for both the SM background and the SUSY signal production in order to prevent double-counting between the matrix elements and the parton shower. The detector simulation is performed by Delphes 3 [38], using the anti-k t jet algorithm [39] with the parameter R = 0.5. For the signals, the production cross sections are normalized to 13 TeV NLO+NLL results [40]. The b-jet tagging efficiency is taken to be the same as one of the benchmark operating points shown in [41], with the maximum efficiency ≈ 77%.
Since we require exactly one lepton, large MET and hard extra jets, a number of SM processes can be responsible for such a final state. According to similar/related collider studies [2,42], we expect SM tt, tW , W + jets and di-boson events to be our main back-grounds, since they can naturally provide a lepton and MET with large production rates.
Other backgrounds, such as tt+W/Z, Z+ jets and tb events either suffer from low production cross sections or low signal efficiencies. All SM backgrounds mentioned above are generated by the method aforementioned. Besides the SM backgrounds, the dileptonic decay oftt * can be an irreducible background to the signal. However, this process has a much smaller cross section compared to the SM backgrounds and can be ignored for the rest of our discussion.

B. Event Selection
For our benchmark studies, all events must satisfy the preliminary selection as described below. Each event is required to have at least 2 jets, 0 tau-tagged jet and exactly 1 isolated lepton with 20 GeV < p T < 100 GeV and |η| < 2.5. The upper limit of the transverse momentum is imposed because our signal comes from a compressed spectrum and the W bosons are not very boosted. Since we need a hard ISR jet, the leading jet is required to satisfy p j 1 T > 150 GeV while the rest of the jets must have p j 2 ,j 3 ... In our signal events, the neutralinos are recoiled against the ISR. To make sure that the leading jet is antiparallel to the sum of neutralinos' momenta and is not from the decay of the stops, we require that |φ j 1 − φ MET | ≥ 2 and ∆R j 1 , ≥ 1.5. To take into account the cases where there is more than one ISR jet, we define p T (J ISR ) to be the vector sum of all jets' p T that are inside the ∆R ≤ 2 cone with the leading jet j 1 and outside ∆R ≥ 1 with the lepton.

C. Stop Benchmark Study
For an illustration, we describe in detail the analysis for a benchmark point "Stop-450," mt = 450 GeV, mχ0 1 = 363 GeV along the W corridor in the parameter space. For simplicity, we assume all other supersymmetric particles are decoupled and the stop's decay branching ratio to b + W +χ through an off-shell top is 100%. Since the spectrum of interest is very compressed, the searches based on the M T 2 types of variables are ineffective, which motivates us to explore the usefulness of theR M type of variables.
In Fig. 1 we plot the two-dimensionalR min −R max distributions for the signal benchmark A closer look at the data shows that backgrounds with one single neurtrino as the source of their missing energy tend to give smallR M . These "mono-neutrino" backgrounds include W + jets and semileptonic tt production, where there is no other invisible particle except one neutrino. In principle, these backgrounds should allowR M = 0 as a solution from the However, if the measured / p T purely comes from p Tν , the corresponding transverse mass M T is bounded by the W mass, which would have been removed by the M T ≥ 100 GeV cut. The events that passed the cut must have some additional / p T due to mismeasurements or lost particles, which generally renders a positiveR M as shown in the figure. We also see thatR min is a more useful variable to suppress these backgrounds thanR max which has a wider distribution.
On the other hand, those backgrounds which can produce more than one neutrino such as

required (SRL);
• for events with MET 300 GeV,R min 0.5 and M T > 120 GeV are required (SRH).
For smaller MET where the backgrounds are large, we impose harder cuts onR min and M T to reduce the background events. For large MET,R min and M T cuts can be relaxed a bit to allow more signal events to pass them.
The numbers of events passing the above cuts for the benchmark signal and SM backgrounds, normalized to an integrated luminosity of 36 fb −1 are shown in Table I where S and B are corresponding numbers of signal and background events, L(x, µ) = µ x e −µ x! , and P (B) is the normalized normal distribution with the mean B and a standard deviation σ B . The final significance from this method is simply given by 2 log(Q). For the case with no systematic error, σ B = 0, this equation simply reduces to the standard formula [43]: For the Stop-450 benchmark, we get a significance of 4.3 σ (6.3 σ) for 36 fb −1 with (without) a 10% background uncertainty. For current LHC SUSY searches also using one-lepton final states [2], the systematic uncertainties for different signal regions vary from ∼ 10% to ∼ 30%, which mainly comes from uncertainties in modeling the SM backgrounds and MC simulations rather than the experimental uncertainties. In the future when the integrated luminosity increases from 36 fb −1 to 300 fb −1 , we expect that the systematic uncertainties will further decrease but the actual numbers are hard to predict. Here we use a 10% background systematic uncertainty to demonstrate its impact on the signal significances.
The results with different background uncertainties can also be obtained easily from the numbers in Table I Fig. 5. One can see that there is some separation between signal and backgrounds, but compared to Fig. 1 it does not seem to be as good. In Appendix B, we perform an analysis without using theR M variables and find that indeed the signal significance is substantially inferior to the result obtained here with theR M variables.

D. Stop Results at LHC 13TeV
The discussion of the last subsection demonstrated that the analysis based on theR M variables with a hard ISR can yield a large signal significance for a 450 GeV stop in the W corridor with 36 fb −1 integrated luminosity. To study the reach of this method, we perform the same analysis for a series of points along the W corridor. The optimal signal regions may depend on the mass points, but for simplicity and easy comparison we use the same signal regions defined in the previous subsection. The results are shown in Table II and 1 , but leaves a gap below that where the reach degraded to ∼ 360 GeV [3].
They are far beneath the potential reach of the new approach studied here.  We are also interested in the mass parameter region slightly away from the mt − mχ0 1 = m W + m b line to see the coverage of our method. We performed the same analysis for points along the lines of mt − mχ0 The results are also shown in Table II. Away from the mt − mχ0 1 = m W + m b line, some of the kinematic assumptions used in Sec. II are no longer valid. For instance, when the mass gap betweent andχ 0 1 is larger than m W + m b , the W bosons are still on-shell, so Eq. (6) still holds. However, the neutralinos would no longer be static in the rest frame of the stops and consequently the sum of their momentum may no longer be strictly antiparallel to the ISR. Thus, our assumption that the neutrino is solely responsible for / p ⊥ T is no longer justified. This could further smear theR M distribution for the signal, hence reducing its discriminating power.
We see that the significances for the points on the mt − mχ0 1 = m W + m b + 30 GeV line are generally somewhat worse than those on the mt − mχ0 1 = m W + m b line for the same stop mass. On the other hand, for a stop lighter than mχ0 1 + m W + m b , the stop goes through the 4-body decay and the mass shell condition Eq. (6) is no longer valid. However, the mass ratio mχ0 1 /mt is larger for the same mt. The distribution ofR M for the signal also shifts to larger values, resulting in better separation from the backgrounds. The signal significances along the mt − mχ0 1 = m W + m b − 30 GeV line are still comparable to the points along the mt − mχ0 1 = m W + m b line. From these results, we conclude that this new approach can apply to a quite wide region around the W corridor and will extend the coverage on the search gap present in the current experimental analyses.

IV. CHARGINO AND NEUTRALINO SEARCHES ALONG THE W CORRIDOR
The compressed chargino/neutralino that decay viaχ ± 1 → W ± +χ 0 1 orχ 0 2 → Z +χ 0 1 can also give the + jets + MET final state without b jets, thus similar analysis can also be applied to chargino searches along the W corridor. Due to the smaller production cross section, the experimental exclusion limit on mχ± 1 and mχ0 2 is weaker compared to stop searches. The current (36 fb −1 ) reach ofχ ± 1 andχ 0 2 is less than 250 GeV in the compressed region from CMS in the 3 channel, assumingχ ± 1 andχ 0 2 are wino-like and decay to W and Z plusχ 0 1 respectively [30,31]. (The higgsino-likeχ ± 1 ,χ 0 2 can also be constrained as long as they have the same spectrum and decay final states, but their production rate will be a few times smaller.) There is a gap around mχ0 2 − mχ0 1 ≈ m Z where the limit further degrades to ∼ 225 GeV. The limit from the current ATLAS analysis is even weaker [34].
There is no chargino/neutralino search using the 1 channel from either ATLAS or CMS in the compressed region, which motivates us to explore the usefulness of the approach usinḡ R M in the chargino search.
We start with a benchmark (C1N2-300) of 300 GeV degenerateχ ± 1 ,χ 0 2 , with the production rate taken to be wino-like. The LSPχ 0 1 is assumed to be bino-like and has a mass 215 GeV.χ ± 1 ,χ 0 2 are produced through the electroweak process, then decay to W ± (Z) +χ 0 1 with 100% branching ratio. The preliminary selection rules are mostly the same as ones applied in the stop study of the previous section, except for the MET requirement. Because the reach in the mass spectrum will be weaker than the stop case due to the smaller production cross section, for the same ISR momentum the recoiled momentum carried by the twoχ 0 1 's will be lower for smaller mχ0 1 . With that observation, we lower the MET requirement to 180 GeV.
In Fig. 7 we plot the two-dimensionalR max vs.R min distributions of 300 GeVχ Due to the smallerR min values for the signal, one may want to lower theR min cut in the signal region selection. However, the backgrounds is large at lowerR min values so a lower R min cut would need to be accompanied by a harder cut on other variables such as MET.
The MET and M T vs.R min distributions for the signal and backgrounds are shown in Fig. 8.
We modify the two signal regions as follows: • for events with 180 MET < 350 GeV, M T 150 GeV andR min 0.7 are required (SRL); • for events with MET 350 GeV, M T 120 GeV andR min 0.4 are required (SRH).
The numbers of events passing the above cuts are listed in Table III. In the table we [30,31,34].
To explore the potential power of this channel in the future, we project the analyses for several mass points along the W corridor to a higher integrated luminosity of L = 300 fb −1 .
Numerical results are presented in Table IV, also with their overall signal efficiency. It is clear that as the chargino becomes heavier, the expectation ofR M also increases, rendering a larger MET andR min on average. As the result, the overall signal efficiency grows and partially compensates the reduction of the production cross section. From the results we can see that for 300 fb −1 13 TeV LHC the reach in the chargino mass around the W corridor could go beyond 400 GeV in the one lepton channel based on this method.

Initial Prelim SRL(R) SRH(R) SRL(C) SRH(C)
χ ±χ∓ 300 6.84 × 10 3 5.  can significantly extend the exclusion reach beyond the limits obtained from the current experimental analyses. The same analysis also applies to chargino/neutralino search in the W corridor. The search reach of the chargino mass is not as good as the stop mass due to the smaller production cross section, but can still surpass the limits set by the current multilepton searches.
A lesson from these studies is that by fully utilizing the kinematic features and constraints of the signal and background events, one can construct discriminating variables that more effectively separate them, and therefore improve the search coverage. This is important for In order for the event process to be physical, the neutrino's momentum must be real which means coefficients a, b, c satisfy the following inequality with the coefficients A, B, C given by A = 4p 2 z (p x p jx + p y p jy ) 2 , −4(E 2 − p 2 z )[(E 2 − p 2 x )p 2 jx + (E 2 − p 2 y )p 2 jy − 2p x p y p jx p jy ] B = 4p 2 z (p x p jx + p y p jy )(2p x/ p x + 2p y/ p y + m 2 W ) + 4(E 2 − p 2 z )(m 2 W p x p jx + m 2 W p y p jy ) − 4(E 2 − p 2 z )[2(E 2 − p 2 x )p jx/ p x + 2(E 2 − p 2 y )p jy/ p y − 2p x p y (p jx/ p y + p jy/ p x )], One can show that the coefficient A is negative as long as p x p jy = p y p jx by the following equivalent relations: (p 2 x + p 2 y )[(p 2 y + p 2 z )p 2 jx + (p 2 x + p 2 z )p 2 jy ] > p 2 z (p 2 x p 2 jx + p 2 y p 2 jy + 2p x p y p jx p jy ) +2(p 2 x + p 2 y )p x p y p jx p jy ⇓ p 2 x p 2 y p 2 jx + p 2 x (p 2 x + p 2 z )p 2 jy + p 2 y (p 2 y + p 2 z )p 2 jx + p 2 x p 2 y p 2 jy > 2(p 2 x + p 2 y + p 2 z )p x p y p jx p jy ⇓ (p 2 x + p 2 y + p 2 z )(p 2 x p 2 jy + p 2 y p 2 jx ) > 2(p 2 x + p 2 y + p 2 z )p x p y p jx p jy (A8) The coefficients A, B, C can be calculated from the experimentally measured lepton momentum p , transverse momentum of the ISR, p T (J ISR ) = (p jx , p jy ), and the missing transverse momentum / p T for each event. After calculating A, B, C, we can obtain the allowed R M range which must satisfy inequality (A6): ]; • if B 2 − 4AC < 0,R M has no real solutions.
The second case may be caused by experimental smearing effects, a wrongly identified ISR system, or even wrong topologies in the case of backgrounds. Instead of simply discarding these events, in our study we also perform an analysis of these events by taking the real part of the solution and defineR M (=R min =R max ) ≡ − B 2A in case that the no real solution result is due to the experimental smearing and the real part may be close to the trueR M value.