Kinematic Focus Point Method for Particle Mass Measurements in Missing Energy Events

TL;DR

This paper introduces a novel kinematic focus point method for particle mass measurements in missing energy events at hadron colliders, leveraging the properties of extreme events and phase space singularities to improve accuracy.

Contribution

The paper develops a new focus point technique for mass determination that reduces systematic errors and complements existing methods by analyzing the kinematic boundary density.

Findings

Identifies a class of extreme events with true mass points on their kinematic boundary.

Demonstrates the abundance of extreme events in realistic data samples.

Shows the focus point method effectively lifts degeneracies in mass space.

Abstract

We investigate the solvability of the event kinematics in missing energy events at hadron colliders, as a function of the particle mass ansatz. To be specific, we reconstruct the neutrino momenta in dilepton $t\bar{t}$-like events, without assuming any prior knowledge of the mass spectrum. We identify a class of events, which we call extreme events, with the property that the kinematic boundary of their allowed region in mass parameter space passes through the true mass point. We develop techniques for recognizing extreme events in the data and demonstrate that they are abundant in a realistic data sample, due to expected singularities in phase space. We propose a new method for mass measurement whereby we obtain the true values of the mass parameters as the focus point of the kinematic boundaries for all events in the data sample. Since the masses are determined from a relatively sharp peak structure (the density of kinematic boundary curves), the method avoids some of the systematic errors associated with other techniques. We show that this new approach is complementary to previously considered methods in the literature where one studies the solvability of the kinematic constraints throughout the mass parameter space. In particular, we identify a problematic direction in mass space of nearly 100% solvability, and then show that the focus point method is effective in lifting the degeneracy.

Figures

Tables

Table 1. Table 1: Mass spectrum along the flat direction ( 26 ) for the three m ~ C subscript ~ 𝑚 𝐶 \tilde{m}_{\!{C}} values used in the paper.

${\tilde{m}}_C$ (GeV)	${\tilde{m}}_B$ (GeV)	${\tilde{m}}_A$ (GeV)
400	505	694
700	800	1000
1000	1098	1304

Equations

$$t\to b{W}^{+}\to b{\ell }^{+}\nu ,$$

$${\vec{p}}_{\nu T}=/{\vec{P}}_T.$$

$$(p_{{\ell }^{+}}+p_{\nu }{)}^2=m_W^2,$$

$$p_{\nu }^2=m_{\nu }^2,$$

$$(p_{{\ell }^{+}}+p_{\nu }{)}^2$$

$$(p_{{\ell }^{-}}+p_{\bar{\nu }}{)}^2$$

$$p_{\nu }^2$$

$$p_{\bar{\nu }}^2$$

$$(p_b+p_{{\ell }^{+}}+p_{\nu }{)}^2=(p_{\bar{b}}+p_{{\ell }^{-}}+p_{\bar{\nu }}{)}^2,$$

$${\vec{p}}_{\nu T}+{\vec{p}}_{\bar{\nu }T}=/{\vec{P}}_T.$$

$$p_{\nu }^2$$

$$(p_{{\ell }^{+}}+p_{\nu }{)}^2$$

$$pp$$

$$A_i$$

$$q_1^2$$

$$(p_{b_1}+q_1{)}^2$$

$$(p_{a_1}+p_{b_1}+q_1{)}^2$$

$${\vec{q}}_{1T}+{\vec{q}}_{2T}={\vec{\cancel{P}}}_T$$

$$q_1^2$$

$$q_2^2$$

$$(p_{b_1}+q_1{)}^2$$

$$(p_{b_2}+q_2{)}^2$$

$$(p_{a_1}+p_{b_1}+q_1{)}^2$$

$$(p_{a_2}+p_{b_2}+q_2{)}^2$$

$$\left\{M_{T2}^{\max }(ab),M_{T2}^{\max }(a),M_{T2}^{\max }(b),m_{ab}^{\max }\right\}.$$

$${\left[M_{T2}^{\max }(b)\right]}^2=M_{T2}^{\max }(ab)\left[M_{T2}^{\max }(ab)-M_{T2}^{\max }(a)\right],$$

$$\mathcal{E}(p_{a_i},p_{b_i},q_i)\equiv E_{a_1}{E}_{b_1}{E}_{C_1}{E}_{a_2}{E}_{b_2}{E}_{C_2}\left[({\vec{V}}_1\times {\vec{V}}_2)\cdot \hat{z}\right]=0,$$

$${\vec{V}}_i=({\vec{v}}_{a_i}\times {\vec{v}}_{b_i})+({\vec{v}}_{b_i}\times {\vec{v}}_{C_i})+({\vec{v}}_{C_i}\times {\vec{v}}_{a_i}).$$

$${\vec{v}}_{a_i}=\frac{{\vec{p}}_{a_i}}{E_{a_i}};{\vec{v}}_{b_i}=\frac{{\vec{p}}_{b_i}}{E_{b_i}};{\vec{v}}_{C_i}=\frac{{\vec{q}}_i}{E_{C_i}}.$$

$$\min \left(N_{p\cup q}-C_q-C_p,N_p-C_p\right).$$

$${\mathcal{P}}_{\mathrm{vis}}d{V}_{\mathrm{vis}}=\sum _i{\mathcal{P}}_{\mathrm{full},i}d{V}_{\mathrm{full},i},$$

$${\mathcal{P}}_{\mathrm{vis}}=\sum _i{\mathcal{P}}_{\mathrm{full},i}\frac{d{V}_{\mathrm{full},i}}{d{V}_{\mathrm{vis}}}.$$

$${\mathcal{P}}_{\mathrm{mass}}^{db}({\tilde{m}}_A,{\tilde{m}}_B,{\tilde{m}}_C)\sqrt[3]{d^3{V}_{\mathrm{mass}}}=d{V}_1\underset{\begin{array}{c}\mathrm{extreme}\\ \mathrm{events}\end{array}}{\int }{\mathcal{P}}_{\mathrm{vis}}d{V}_{(N_p-C_p-1)},$$

$${\mathcal{P}}_{\mathrm{mass}}^{db}({\tilde{m}}_A,{\tilde{m}}_B,{\tilde{m}}_C)=J\underset{\begin{array}{c}\mathrm{extreme}\\ \mathrm{events}\end{array}}{\int }{\mathcal{P}}_{\mathrm{vis}}d{V}_{(N_p-C_p-1)},$$

$$\frac{d{\tilde{m}}_A^2}{d{\tilde{m}}_C^2}$$

$$\frac{d{\tilde{m}}_B^2}{d{\tilde{m}}_C^2}$$

Full text

1 Introduction

At the Large Hadron Collider (LHC), information about the underlying physics is extracted by studying the kinematic properties of the final state objects (jets, leptons, photons, etc.) which are reconstructed in each collision. Therefore, understanding the unique kinematic features arising from different event topologies is an important step in any LHC data analysis, be it for a new physics search, or a parameter measurement.

For example, the traditional method to discover a new heavy resonance (as well as to measure its mass) is to look for a bump in the invariant mass distribution of its decay products (daughter particles). This procedure is straightforward if the daughter particles are all visible in the detector and their energies and momenta are measured. However, things become more involved if some of the daughter particles are invisible, e.g., they can be neutrinos or new weakly interacting massive particles, as predicted in many models of new physics with dark matter candidates and/or dark sectors Feng:2010gw . A prime example of this situation is provided by semi-leptonic $t\bar{t}$ events, in which one of the top quarks, say $\bar{t}$, decays hadronically, while the other decays as

[TABLE]

where a neutrino $\nu$ goes missing. At this point a natural approach would be to attempt to compute the momentum of the neutrino from the available kinematic information in the event, perhaps supplemented with some theoretical assumptions. For example, assuming that there are no other invisible particles in the event, we can identify the measured missing transverse momentum ${\;/\!\!\!\!\vec{P}_{T}}$ in the event with the transverse component $\vec{p}_{\nu T}$ of the neutrino momentum

[TABLE]

On the other hand, the neutrino arises from the decay of a $W$-boson, which is typically produced on-shell, in turn implying that

[TABLE]

where $p_{\ell^{+}}$ and $p_{\nu}$ are the 4-momenta of the lepton and neutrino, respectively, and $m_{W}$ is the physical mass of the $W$-boson. Finally, the energy $E_{\nu}$ and 3-momentum $\vec{p}_{\nu}$ of the neutrino are related by the mass shell condition

[TABLE]

where $p_{\nu}=(E_{\nu},\vec{p}_{\nu})$. Equations (2-4) provide a total of $2+1+1=4$ constraints on the four components of $p_{\nu}$, allowing to solve for the neutrino 4-momentum, up to discrete ambiguities111The mass constraints (3) and (4) lead to a quadratic equation, which in general has two solutions. Additional discrete combinatorial ambiguities arise if there are more than one lepton and/or more than one $b$-jet present in the event.. In practice, this is typically done through an eventwise kinematic fit which accounts for the detector resolution and the combinatorial uncertainties Abbott:1998dc ; Gripaios:2011jm ; Sirunyan:2018gqx .

Now and for the rest of the paper we shall turn our attention to the much more challenging case of dilepton $t\bar{t}$ events, in which both top quarks decay leptonically as in (1). This event topology is relevant not only for top physics, but also for new physics searches where the decay chains of the newly produced particles terminate in a generic dark matter candidate, i.e., a stable, neutral, weakly interacting particle which is invisible in the detector just like the neutrino. In many such models, e.g., supersymmetry (SUSY) with $R$-parity Martin:1997ns , universal extra dimensions with KK-parity Appelquist:2000nn ; Rizzo:2001sd ; Cheng:2002ab , Little Higgs with $T$-parity ArkaniHamed:2002qy ; Cheng:2004yc ; Schmaltz:2005ky ; Perelstein:2005ka , etc., the lifetime of the dark matter particle is protected by an exact $Z_{2}$ symmetry, which ensures that new particles are pair-produced at the LHC, leading to symmetric events with two decay chains. From that point of view, dilepton $t\bar{t}$ events are an excellent testbed which allows us to test ideas originally developed for the study of new physics signatures, see, e.g., Chatrchyan:2013boa .

Once we consider dilepton $t\bar{t}$ events, we have to face the complication that now there are twice as many unknown momentum components, i.e., the four momenta, $p_{\nu}$ and $p_{\bar{\nu}}$, of the two missing neutrinos. The number of on-shell constraints is also doubled — (3) is replaced by

[TABLE]

while (4) is replaced by

[TABLE]

We also gain a new constraint, that the two parent particles (top quarks) have equal mass:

[TABLE]

but the missing transverse momentum constraint (2) remains a single 2D vector equation:

[TABLE]

As a result, the seven constraints (5-8) are not sufficient to determine the eight unknown components $p_{\nu}$ and $p_{\bar{\nu}}$ uniquely, and therefore, one cannot reconstruct the top as a mass bump. The problem is exacerbated in new physics applications, where a priori one does not know the masses of the new particles playing the roles of the $W$-boson and the neutrino, so that the best one can do is to apply the equal mass constraints

[TABLE]

in place of (5-6), leaving us with only 5 constraints (7-10) for 8 unknowns.

There are several possible approaches to tackle this conundrum Barr:2010zj :

• •

Gain additional constraints. This can be done in several ways. First, one may consider longer decay chains which may occur in new physics scenarios. The classic examples are the three-step squark decay chain and the four-step gluino decay chain in supersymmetry, where the trick of analyzing several events simultaneously could provide the required number of constraints to solve for all unknown momenta, up to discrete ambiguities Nojiri:2003tu ; Kawagoe:2004rz ; Cheng:2007xv ; Cheng:2008mg ; Cheng:2009fw ; Webber:2009vm . One may also impose constraints resulting from existing preliminary measurements of the kinematic endpoints of suitable variables Nojiri:2007pq ; Djilkibaev:2008pj ; Casadei:2010nf . Finally, one could preselect a sub-sample of events with special properties, e.g., events located at the kinematic endpoint of a suitable variable, in which case the kinematics of the decays is further constrained Kersting:2009ne ; Kang:2009sk .

• •

Adopt an ansatz for the undetermined components of the invisible momenta. The idea here is to assign values for the remaining invisible momentum components through a suitable ansatz which preserves some useful property, e.g., that a parent mass bound is not exceeded Barr:2011xt ; Kim:2017awi . Operationally the values are assigned by extremizing some relevant function of the invisible momenta. Often the minimum of the function itself becomes a useful kinematic variable — some well-known examples include the Cambridge $M_{T2}$ variable Lester:1999tx ; Barr:2003rg and its variants Burns:2008va ; Barr:2009jv ; Konar:2009wn ; Konar:2009qr , the $\sqrt{s}_{min}$ variable Konar:2008ei ; Konar:2010ma ; Swain:2014dha , a variety of constrained transverse mass variables Barr:2009mx ; Gross:2009wia ; Barr:2011he ; Barr:2011ux , the $M_{CT2}$ variable Cho:2009ve ; Cho:2010vz , the $M_{2C}$ variable Ross:2007rm ; Barr:2008ba , the MAOS method Cho:2008tj ; Choi:2009hn ; Park:2011uz ; Guadagnoli:2013xia , the $M_{2}$ class of variables Mahbubani:2012kx ; Cho:2014naa ; Cho:2014yma ; Kim:2014ana ; Cho:2015laa ; Konar:2015hea ; Konar:2016wbh ; Goncalves:2018agy , etc. While this approach has useful practical applications, it still only represents an approximate treatment and does not lead to a mass reconstruction through a bump.

• •

Use kinematic endpoints of observable invariant mass distributions. Another logical possibility is to ignore the invisible momenta altogether and design the analysis entirely in terms of the measured momenta of the reconstructed visible objects in the event. Historically this was the original approach which led to the classic analyses of SUSY mass measurements from kinematic endpoints in invariant mass distributions Hinchliffe:1996iu ; Bachacou:1999zb ; Allanach:2000kt ; Gjelsten:2004ki ; Gjelsten:2005aw ; Matchev:2009iw ; Matchev:2019sqa . However, as in the previous case, the method relies on the measurement of kinematic endpoints instead of mass bumps. Endpoint measurements are generally challenging — they may be difficult to extract in the presence of background, their location is rather sensitive to the effects of particle widths and detector resolution, and they require fitting to a suitable profile Agashe:2010tu ; Curtin:2011ng , which introduces additional systematics.

• •

Check the solvability of an enlarged set of kinematic constraints. Here the idea is first to enlarge the existing set of kinematic constraints by assuming test values for the unknown masses, and then instead of focusing on the actual solutions for the invisible momenta, simply ask the question whether such solutions exist or not. If solutions do not exist, the corresponding test ansatz for the particle masses is inconsistent with the kinematics of the particular event and those masses are disfavored Kawagoe:2004rz ; Cheng:2007xv ; Cheng:2008hk . By repeating the procedure over the full event sample, one can gradually restrict the allowed mass parameter space, hopefully until it shrinks to a point.222A variant of this technique assigns weights derived from the parton distribution functions to different points in the allowed region of mass parameter space in order to arrive at the most likely value of the particle masses Anagnostou:2011aa . Indeed, the true masses should be compatible with every signal event, while for any wrong choice of particle masses, one would hope that, given sufficient statistics, there would be at least one signal event in the data which would have incompatible kinematics. We will illustrate these ideas more explicitly later on in Sections 3 and 4. For now we just mention that this method appears to have the desired property that the true particle masses are revealed by a bump in mass parameter space (for example, when plotting the fraction of compatible events). However, as we shall show in Section 4, this expectation is misleading, as there is generally a whole “flat direction” of near 100% solvability in mass parameter space.

In this paper we present a new approach to measuring masses in events with ${\;/\!\!\!\!\vec{P}_{T}}$, by building up on the solvability method discussed in the last bullet above. Our method makes crucial use of the singularities in the phase space density which arise from projecting the full phase space of final state momenta into the visible phase space Kim:2009si . This method, which we shall call the “kinematic focus points method”333Not to be confused with the focus point phenomenon in the RGE running of certain SUSY mass parameters Feng:1999hg ; Feng:1999zg ., has similarities to a traditional mass-bump search in the sense that the density of an appropriately defined quantity will peak at the true values of the unknown masses in the event topology. Previous attempts to design an analysis for measuring the unknown masses in SUSY-like events with missing energy from such bump-like features were limited to the case of a single decay chain Lester:2013aaa ; Agrawal:2013uka ; Debnath:2016gwz ; Altunkaynak:2016bqe ; Debnath:2018azt . Instead, here we will be interested in the more typical situation of two symmetric decay chains. While our method is applicable more generally, we shall introduce it in the context of the dilepton $t\bar{t}$ event topology discussed above.

The paper is organized as follows. In Section 2 we specify the process under study, introduce our conventions and terminology, and provide some details on our simulations. In Section 3 we list the kinematic constraints for our event topology, and briefly review the idea of using their solvability in order to perform mass measurements (or at the very least, to restrict the allowed region of mass parameter space). In Section 4 we discuss the expected outcome from this approach and highlight its advantages as well as potential pitfalls. In particular, we identify a flat direction in mass parameter space of nearly 100% solvability, along which the masses are consistent with the kinematics of nearly all events in the data. We then show that by performing complementary measurements of kinematic endpoints one is able to lift the flat direction and identify the true mass point.444To the careful reader, this should not come as a surprise, since kinematic endpoint measurements by themselves are already able to fully determine the mass spectrum Burns:2008va ; Chatrchyan:2013boa .

In the remaining sections, we define the kinematic focus point method and illustrate its performance. First in Section 5 we discuss the event-wise kinematic boundaries in mass parameter space, and introduce the notion of “extreme” events, i.e., events for which the kinematic boundary passes through the true mass point. We argue that extreme or near-extreme events are quite abundant in a realistic data sample. More importantly, we show that the density of extremeness boundaries in the mass parameter space is singular at the true mass point555This is a purely mathematical statement which is valid in the zero width approximation and in the absence of detector smearing. Both of those effects will lead to some smearing of the singularity, the extent of which is studied later on in Sections 6.4 and 6.5.. Using the results from Section 5, in Section 6 we motivate the kinematic focus points method and proceed to investigate its performance for two cases — a SUSY-like mass spectrum (where top events are part of the SM background) as well as for top pair-production itself, thus allowing for a measurement of the top, $W$-boson and neutrino masses independently666An analogous procedure based on kinematic endpoints was advertised in Burns:2008va and tested in Chatrchyan:2013boa .. In Section 7, we present our conclusions and outlook for future studies. In Appendix A we derive the parametric equation for the flat direction of low mass sensitivity encountered in Sections 4 and 6.

2 Notations and Setup

2.1 Conventions

In this paper, we consider the $t\bar{t}$-like event topology depicted in Fig 1.

A pair of heavy particles, $A_{1}$ and $A_{2}$, is initially produced, and each $A_{i}$ decays via two successive two-body decays:

[TABLE]

to two visible Standard Model particles $a_{i}$ and $b_{i}$ and an invisible particle $C_{i}$. The two branches in Fig. 1 are assumed to be identical and the intermediate resonances $A_{i}$ and $B_{i}$ will be taken to be on-shell. The true masses of particles $A_{i}$, $B_{i}$ and $C_{i}$ are a priori unknown, and will be denoted by $m_{\!{A}}$, $m_{\!{B}}$ and $m_{\!{C}}$, respectively. The goal is to measure $m_{\!{A}}$, $m_{\!{B}}$ and $m_{\!{C}}$ independently from a sample of events with the topology of Fig. 1.

Since the final state particles $a_{i}$ and $b_{i}$ are visible in the detector, their 4-momenta, $p_{a_{i}}$ and $p_{b_{i}}$, are measured known quantities for each event. In many new physics models with this event topology, $a_{i}$ and $b_{i}$ are bottom quarks and leptons, respectively, just like the case of dilepton $t\bar{t}$ events. Then, the $a$’s can be distinguished from the $b$’s, but there remains a twofold ambiguity in partitioning the two $a$’s and the two $b$’s into the two branches. As there are already several suggestions in the literature on how to address this combinatorial issue Baringer:2011nh ; Choi:2011ys ; Debnath:2017ktz , we shall not dwell on the combinatorial problem any further. Note that our proposed method below will be robust against combinatorial issues, since it will rely on detecting a peak structure which is absent from the combinatorial background. Therefore, for clarity of the presentation, in what follows we shall assume that the visible particles have been properly assigned to the two branches.

The final state particles $C_{i}$ are invisible in the detector ($C$ is typically a dark matter candidate). Consequently, their 4-momenta, $q_{i}$, are a priori unknown. The 4-momenta of $B_{i}$ and $A_{i}$ are denoted by $q_{{\!{B}}_{i}}(\equiv p_{b_{i}}+q_{i})$ and $q_{{\!{A}}_{i}}(\equiv p_{a_{i}}+p_{b_{i}}+q_{i})$, respectively. Note that the notation here follows that of Barr:2011xt , where the letter $p$ is used to denote known momenta, while $q$ is used for a priori unknown momenta. An overhead arrow will be used to denote the spatial component $\vec{p}$ of a 4-momentum $p$. The beam axis (the longitudinal direction) is chosen to be the $z$ axis, while the transverse components of a 4-momentum $p$ will be denoted by $\vec{p}_{T}$. The product of two 4-momenta will refer to their Lorentz inner product.

We shall often consider hypothesized values for the a priori unknown masses, i.e., test or trial masses, which will be denoted by a tilde: $\tilde{m}_{\!{A}}$, $\tilde{m}_{\!{B}}$, and $\tilde{m}_{\!{C}}$. For convenience, the singular form true mass and test mass will sometimes be used to refer to a set of true masses $(m_{\!{A}},m_{\!{B}},m_{\!{C}})$ and a set of test masses $(\tilde{m}_{\!{A}},\tilde{m}_{\!{B}},\tilde{m}_{\!{C}})$. For notational convenience, $q_{i}$, $q_{{\!{B}}_{i}}$ and $q_{{\!{A}}_{i}}$ will be used interchangeably as variables or particular values for these variables, and the meaning will be clear from the context.

2.2 Simulation details

The analyses in this study were performed on events generated as follows.

• •

Obviously, the event topology of Fig. 1 can be applied directly to the study of top quark pair production, in which case the particles $A$, $B$ and $C$ are the top quark, the $W$-boson and the neutrino, respectively. While their masses $m_{t}$, $m_{W}$ and $m_{\nu}$, are already known, the techniques discussed below can still be used to perform more sensitive measurements of those SM parameters. Our primary target, however, is a new physics event topology of the type shown in Fig. 1, where the three masses $m_{\!{A}}$, $m_{\!{B}}$ and $m_{\!{C}}$ are a priori unknown. For example, in SUSY $A$ can be a top squark, $B$ a chargino and $C$ a sneutrino. Thus for our main simulation we choose a study point with a spectrum of $m_{\!{A}}=1000\;\mathrm{GeV}$, $m_{\!{B}}=800\;\mathrm{GeV}$ and $m_{\!{C}}=700\;\mathrm{GeV}$. Note that the mass differences are relatively modest, so that the $p_{T}$ and energy distributions of the jets and leptons in the signal are similar to those of the main background (dilepton $t\bar{t}$ events), so that a template-matching method is unlikely to be very sensitive.

• •

For clarity of the presentation, the resonances $A_{i}$ and $B_{i}$ were kept on-shell. Including small widths for these particles will lead to a slight smearing of the peaks and endpoints discussed in the study Grossman:2011nh ; Kim:2017qdi .

• •

The visible final state particles $a_{i}$ and $b_{i}$ were taken to be massless. This is a very good approximation in the typical scenario where $a_{i}$ and $b_{i}$ are bottom quarks and leptons, respectively. Regardless, the mass measurement methods considered here do not rely on this choice, and this assumption can be easily relaxed if needed.

• •

For definiteness the decay vertices were taken to be non-chiral, i.e., there are no non-trivial spin effects. This assumption also does not impact our results, since the methods studied below are based on purely kinematics arguments and are therefore model independent.

• •

The parent particles $A_{i}$ were pair produced at LHC center of mass energy of $13\;\mathrm{TeV}$. For simplicity, initial state radiation (ISR) was not turned on. Again, this assumption has little bearing on our results — for example the ISR jets are unlikely to be $b$ jets in which case one might worry that they can be confused with particles $a_{i}$ in Fig. 1, complicating the combinatorial issue.

• •

The events were generated at the parton level with MadGraph Alwall:2011uj and resonances were decayed by phase space. Initially, we will show results with no detector simulation applied, before studying the effects of the detector resolution in Sections 6.4 and 6.5.

3 Solvability of Kinematic Constraints

The general problem with missing energy events is that there exist final state momenta ($q_{i}$ in our case) which are not measured. However, if we could somehow calculate the invisible momenta $q_{i}$ (and from there $q_{{\!{B}}_{i}}$ and $q_{{\!{A}}_{i}}$), the unknown masses can be simply obtained by $m_{\!{A}}^{2}=q_{{\!{A}}_{i}}^{2}$, $m_{\!{B}}^{2}=q_{{\!{B}}_{i}}^{2}$ and $m_{\!{C}}^{2}=q_{i}^{2}$. This motivates us to study the kinematic constraints obeyed by the invisible momenta.

With the notation from Section 2, the five kinematic constraints (7-10) for the event topology in Fig. 1 become

[TABLE]

[TABLE]

These five constraints are clearly insufficient to determine all 8 unknown components of $q_{i}$. To this end, we introduce test masses $(\tilde{m}_{\!{A}},\tilde{m}_{\!{B}},\tilde{m}_{\!{C}})$ with $\tilde{m}_{\!{A}}>\tilde{m}_{\!{B}}>\tilde{m}_{\!{C}}$. The constraints in eq. (12) are thus enlarged to

[TABLE]

Together with eq. (13), these six constraints make up a total of 8 kinematic constraints which can be solved by standard means Sonnenschein:2005ed ; Sonnenschein:2006ud ; Betchart:2013nba in order to yield test values for the unknown momenta $q_{i}$ corresponding to the given choice of test masses $(\tilde{m}_{\!{A}},\tilde{m}_{\!{B}},\tilde{m}_{\!{C}})$.

At this point it might seem that we have not made much progress, since we have traded one set of unknowns for another. The key idea now is to focus not on the actual solutions for the invisible momenta, but on the existence (or lack thereof) of viable solutions in the first place Kawagoe:2004rz ; Cheng:2007xv ; Cheng:2008hk . Indeed, not all choices of test masses will lead to real solutions for $q_{i}$. In general, the kinematic constraints (13) and (14) allow four complex solutions. Non-real solutions come in complex conjugate pairs Gripaios:2011jm , and therefore, there can be either zero, two or four real (possibly degenerate) solutions for a given test mass. As a result, each event separates the three-dimensional mass parameter space $(\tilde{m}_{\!{A}},\tilde{m}_{\!{B}},\tilde{m}_{\!{C}})$ into allowed and disallowed regions, depending on whether there exists a real solution for $q_{i}$ or not. The event will be said to be solvable by the test mass if there exists a real solution for $q_{i}$. Note that for signal events the true mass point will always solve the event777Barring the effects from the detector resolution and finite particle widths., and thus can never be ruled out.

Figs. 2-4 illustrate the above discussion for two representative signal events. In the figures, we show “‘solvability plots” constructed as follows. Given that it is difficult to visualize the full three-dimensional mass parameter space, we choose to present slices at fixed values of $\tilde{m}_{\!{C}}$: in Fig. 2 $\tilde{m}_{\!{C}}$ is set to the true value of $m_{\!{C}}=700\;\mathrm{GeV}$, while in Fig. 3 we take $\tilde{m}_{\!{C}}=600\;\mathrm{GeV}$ and in Fig. 4 we have $\tilde{m}_{\!{C}}=800\;\mathrm{GeV}$. In each panel, we color-code the $(\tilde{m}_{\!{B}},\tilde{m}_{\!{A}})$ plane according to the number of real solutions to the kinematic constraints (13) and (14): white (uncolored) areas correspond to no real solutions, green areas have two real solutions and yellow areas allow four real solutions. The cross in Fig. 2 marks the true mass point $m_{\!{B}}=800$ GeV, $m_{\!{A}}=1000$ GeV (the cross is absent in Figs. 3 and 4 since the test mass $\tilde{m}_{\!{C}}$ in those figures is different from the true value $m_{\!{C}}$).

There are several important lessons which can be learned from these figures. First, it is rather remarkable that the large majority of the plotted areas are actually ruled out by this single event — notice how the white color is the dominant one on the plots. We also observe that as the third (fixed) test mass $\tilde{m}_{\!{C}}$ is decreased (increased), the allowed region in the $(\tilde{m}_{\!{B}},\tilde{m}_{\!{A}})$ plane correspondingly shifts towards lower (higher) values of $\tilde{m}_{\!{B}}$ and $\tilde{m}_{\!{A}}$. In turn, the allowed region itself has two sub-regions, with 2 and 4 real solutions, correspondingly. The (yellow-colored) region with 4 real solutions sometimes appears at relatively large masses, and this just happens to be the case with the event used for the left panels in Figs. 2-4. Finally, it is worth noting that for both of these randomly chosen events, the true mass point marked with the cross in Fig. 2 is located rather close to the boundary of the allowed region. At this point, this may seem to be just a coincidence, but a more extensive search throughout the signal event sample reveals that this property is actually quite generic. This observation then brings up the question whether it is possible for the true mass point to be located exactly on the boundary of the allowed region. This is one of the central issues in this paper, which will be addressed below in Section 5. But first we shall present and discuss the mass measurement method based on solvability.

4 Solvability as a Mass Measurement Method

4.1 Superposition of individual events

In the previous section, we saw that by requiring solvability, a single event can already rule out a sizable chunk of the three-dimensional mass parameter space $(\tilde{m}_{\!{A}},\tilde{m}_{\!{B}},\tilde{m}_{\!{C}})$. Repeating the same analysis with a different event, we would expect a slightly different portion of the mass space to be disfavored. Then, by taking the intersection of the two regions allowed by the individual events, we will further constrain the mass parameter space Kawagoe:2004rz ; Cheng:2007xv ; Cheng:2008hk ; Lester:2013aaa .

This procedure is pictorially illustrated in Fig. 5, where we use the same two events as in Figs. 2-4 above.

In the top row, we replot the result from Fig. 2 for $\tilde{m}_{\!{C}}=m_{\!{C}}=700\;\mathrm{GeV}$, only now we remove the distinction between the green and yellow regions (with 2 and 4 real solutions, respectively) and uniformly shade the allowed region in light blue. Then in the plot shown in the bottom row, we superimpose these two allowed regions, thus obtaining a new partition of the $(\tilde{m}_{\!{B}},\tilde{m}_{\!{A}})$ plane into three possible regions: where neither event is solvable (white areas), where only one of the events is solvable but not the other (light blue areas) and where both events are solvable (dark blue areas). By construction, the dark blue region where both events are solvable is smaller than each of the individual allowed regions seen in the plots in the top row. This demonstrates the benefit from adding more events to the discussion, i.e., increasing the statistics. At the same time, a pessimist might point out that a) the benefit does not seem to be that great, since the improvement (the light blue areas in the bottom panel of Fig. 5) is relatively minor when compared to the overall size of the remaining allowed region (the dark blue area); and b) that the remaining dark blue allowed region still seems to extend out to infinity in the “northeast” direction. A crucial question, therefore, becomes how much further the allowed region will shrink once we add all of our remaining events in the data sample. We will answer this question in the next two subsections 4.2 and 4.3, where for clarity of the presentation we split the discussion into two parts: in Sec. 4.2 we focus on the issue of measuring $m_{\!{A}}$ and $m_{\!{B}}$ for a given value of $\tilde{m}_{\!{C}}$ (which throughout that subsection will be fixed to be the true mass $m_{\!{C}}$) and then in Sec. 4.3 we shall tackle the question of measuring $m_{\!{C}}$ itself. The reason for this separation is twofold: first, it is difficult to present and visualize our results in the full three-dimensional mass parameter space $(\tilde{m}_{\!{A}},\tilde{m}_{\!{B}},\tilde{m}_{\!{C}})$, and second, as we shall see below, the three-dimensional mass parameter space exhibits a direction of relatively low sensitivity, which can be parametrized by the value of $\tilde{m}_{\!{C}}$ (see Appendix A for details). As a result, the measurement of $m_{\!{C}}$ will turn out to be much more challenging than the measurements of $m_{\!{A}}$ and $m_{\!{B}}$.

However, before proceeding to show results with the full event sample, it is useful to consider one more simple exercise, illustrated in Fig. 6, where we repeat the superposition of allowed regions done in Fig. 5, still with just a handful of events, in this case three. The important new twist here is that the three events in Fig. 6 were not chosen at random, as was done in Fig. 5, but were more carefully selected. The idea was to pick events which are maximally incompatible with each other, and would therefore rule out the largest amount of mass space by themselves. The top left, top right and bottom left panels in Fig. 6 show the allowed regions for the three selected events. There are three notable features of these individual allowed regions.

• •

For all three events, the cross marking the true mass point $(m_{\!{B}},m_{\!{A}})$ lies on the boundary of the allowed region. This answers the question posed towards the end of the previous section, demonstrating that a kinematic boundary may pass through the true mass point. We shall have a lot more to say about that in the next Section 5.

• •

The shapes of the three allowed regions are very different. For the events in the top row, we see the appearance of two apparently disjoint branches — however, those are actually connected to each other at large values of $\tilde{m}_{\!{B}}$ and $\tilde{m}_{\!{A}}$ beyond the plot range.

• •

More importantly, the locations of the individual allowed regions happen to be different, so that when they are superimposed in the bottom right panel, there is only a very narrow sliver of allowed mass space left (the red-shaded area). Note the different location of the cross (the true mass point) for the three events: in the top left (top right) panel the cross is on the upper (lower) boundary of the allowed region, so that when the two events are superimposed, they will leave only the “southwest-to-northeast” direction as viable. On the other hand, the cross in the lower left panel is right at the tip of the allowed region, thus eliminating the “southwest” portion, and leaving only the red-shaded area extending to the northeast of the true mass point.

The lesson from Figs. 5 and 6 is that some events are better at ruling out mass parameter space than others. Unfortunately, since a priori we do not know the values of the true masses, i.e., the location of the cross in Figs. 5 and 6, we are unable to preselect events which are “good at” eliminating parameter space, and the best we can do is go over the whole sample, thus guaranteeing ourselves that at some point we will eventually hit on some “good” events as well. This is precisely what we intend to do in the next two subsections.

4.2 Measuring $m_{B}$ and $m_{A}$ for a given trial mass $\tilde{m}_{C}$

In this subsection we focus on a slice of the three-dimensional mass parameter space $(\tilde{m}_{\!{A}},\tilde{m}_{\!{B}},\tilde{m}_{\!{C}})$ at a constant $\tilde{m}_{\!{C}}$, i.e., we shall attempt to measure the values of $m_{\!{B}}$ and $m_{\!{A}}$ given a value for $\tilde{m}_{\!{C}}$, which we shall take to be the true mass, $m_{\!{C}}$. However, this choice is inconsequential, and the results will be similar for any other choice of $\tilde{m}_{\!{C}}$ (in the next subsection we shall return to the question of measuring $m_{\!{C}}$ itself).

To this end, we follow the procedure illustrated in Figs. 5 and 6, only this time we use the full event sample. With realistic detector resolutions and finite widths for the resonances, even the true mass point is not expected to solve 100% of the events. Its success rate will be further reduced in the presence of background events. So, instead of successively restricting the allowed mass space with more and more events, which would eventually result in an empty set, a better approach is to look at the fraction of events that are solvable by a given test mass hypothesis, with the hope that the true test mass exhibits the highest such fraction.

The result888From now on such plots will be referred to as “solvability heat maps”. is plotted in Fig. 7, where we use only signal events and for now ignore the effects of detector resolution and finite particle widths. The color represents the fraction of events which are solvable for the given mass hypothesis $(\tilde{m}_{\!{A}},\tilde{m}_{\!{B}})$.

Fig. 7 confirms that, as expected, the true mass point $m_{\!{B}}=800\;\mathrm{GeV}$ and $m_{\!{A}}=1000\;\mathrm{GeV}$ has a 100% solvability rate. However, in addition to the true mass point, there also seems to be a whole line of masses with nearly 100% solvability. This “solvability flat direction” is precisely the northeast-pointing red-shaded region previously encountered in Fig. 6. What we are now finding in Fig. 7 is that this problematic region persists even after considering the full statistics in the sample. This certainly presents a problem — it suggests that in its current form the solvability method cannot uniquely determine the masses, even in this simplified exercise where we only look at the two-dimensional $(\tilde{m}_{\!{B}},\tilde{m}_{\!{A}})$ plane.

One can think of several possible ways out. Perhaps the flat direction seen in Fig. 7 is not exactly flat, but has a gentle slope, which would nevertheless pick out the true masses. To test this, in the right panel of Fig. 7 we change the color scale, zooming in on the events with maximal solvability (above 90%). We observe that the problematic direction is still pretty flat, and any existing gentle slope will be washed out once we add the realistic effects of detector resolution and backgrounds.999The careful reader will note that the flat direction appears to terminate at the true mass point, so that the true mass point is given not by the condition of maximal solvability, but by the sudden drop in the solvability rate along the flat direction. This conjecture is correct Cheng:2008hk and can be further motivated by ideas from Sections 5 and 6 below. We conclude that for any practical purposes the “solvability flat direction” is there, and we need additional information in order to lift this degeneracy.

For example, we can augment the solvability method with constraints on the masses from kinematic endpoint measurements. The available measurements for this topology are Burns:2008va :

[TABLE]

The first three are the upper kinematic endpoints for the three possible subsystem $M_{T2}$ variables in the event topology of Fig. 1, where we have used the notation of Ref. Cho:2014naa . The last one is the upper kinematic endpoint in the distribution of the invariant mass of $a_{i}$ and $b_{i}$. The measurement of any one of these endpoints acts as an additional constraint on the masses. In particular, Fig. 8 shows the effect of each individual kinematic endpoint measurement on the $(\tilde{m}_{\!{B}},\tilde{m}_{\!{A}})$ parameter space from Fig. 7. Since $\tilde{m}_{\!{C}}$ is still fixed at $\tilde{m}_{\!{C}}=m_{\!{C}}=700\;\mathrm{GeV}$, each kinematic endpoint measurement leads to a relation among $\tilde{m}_{\!{B}}$ and $\tilde{m}_{\!{A}}$ as given by the corresponding colored curve: blue for $M^{\mathrm{max}}_{T2}(ab)$, orange for $M^{\mathrm{max}}_{T2}(a)$, green for $M^{\mathrm{max}}_{T2}(b)$, and red for $m^{\mathrm{max}}_{ab}$. We see that all four curves intersect at the true mass point $(m_{\!{B}},m_{\!{A}})$, as they should. More importantly, neither of the four curves is aligned with the solvability flat direction, and consequently, any one of them can be used for lifting the degeneracy. We thus conclude that the solvability method is able to pinpoint the correct mass values in the $(\tilde{m}_{\!{B}},\tilde{m}_{\!{A}})$ plane (at fixed $\tilde{m}_{\!{C}}=m_{\!{C}}$), but only when supplemented with a kinematic endpoint measurement.

4.3 Measuring $m_{C}$

We now turn our attention to determining $m_{\!{C}}$. We begin by recreating the solvability heat maps from Fig. 7 in Sec. 4.2, only this time we choose different values of the trial mass parameter $\tilde{m}_{\!{C}}$, away from the true value $m_{\!{C}}=700\;\mathrm{GeV}$. Results for $\tilde{m}_{\!{C}}=600\;\mathrm{GeV}$ and $\tilde{m}_{\!{C}}=800\;\mathrm{GeV}$ are shown in Figs. 9 and 10, respectively. They confirm the trend observed in Figs. 2-4, namely, that as we increase (decrease) the value of $\tilde{m}_{\!{C}}$, the preferred values of $\tilde{m}_{\!{B}}$ and $\tilde{m}_{\!{A}}$ increase (decrease) as well. Unfortunately, the solvability heat maps in Figs. 9 and 10 look qualitatively very similar to the heat map in Fig. 7 for $\tilde{m}_{\!{C}}=m_{\!{C}}$. Therefore, it appears very difficult to extract the correct value of $m_{\!{C}}$ solely on the basis of these two-dimensional heat maps in the $(\tilde{m}_{\!{B}},\tilde{m}_{\!{A}})$ plane; some additional input is needed.

The measurements of the kinematic endpoints (15) might just be this missing input. Fig. 8 demonstrated that with the correct value of $\tilde{m}_{\!{C}}$, the measurements (15) are consistent among themselves, as well as with the solvability heat map. It is worth checking whether this consistency is retained when we use erroneous values of $\tilde{m}_{\!{C}}$, i.e., values away from the true mass. This test is performed in Fig. 11, which shows analogues of Fig. 8 for $\tilde{m}_{\!{C}}=400\;\mathrm{GeV}$ (upper left panel), $\tilde{m}_{\!{C}}=600\;\mathrm{GeV}$ (upper right panel), $\tilde{m}_{\!{C}}=800\;\mathrm{GeV}$ (lower left panel) and $\tilde{m}_{\!{C}}=1000\;\mathrm{GeV}$ (lower right panel).

There are several interesting features present in Fig. 11. First, we note that the blue, orange and green curves corresponding to the three $M_{T2}$ kinematic endpoints always cross at a single point, and therefore the three $M_{T2}$ measurements are consistent with each other, regardless of the chosen value of $\tilde{m}_{\!{C}}$. This is due to the fact that these measurements are not independent, but obey the relation Burns:2008va

[TABLE]

which is in fact why the fourth measurement of $m^{\mathrm{max}}_{ab}$ needed to be added to the set (15). On the other hand, the red curve corresponding to the invariant mass endpoint $m^{\mathrm{max}}_{ab}$ is not consistent with the others, as it always misses the common crossing point for the three $M_{T2}$ measurements, which is an indication that the chosen value for $\tilde{m}_{\!{C}}$ was incorrect. The mismatch grows as $\tilde{m}_{\!{C}}$ moves further and further away from the true value $m_{\!{C}}$.

Fig. 11 also reveals a discrepancy between the high solvability region, on the one hand, and the various kinematic endpoint measurements, on the other. In particular, at the lower values of $\tilde{m}_{\!{C}}$ below the true mass $m_{\!{C}}$, the red curve for $m^{\mathrm{max}}_{ab}$ tends to miss the solvable region, while for values of $\tilde{m}_{\!{C}}$ which are too high, it is the blue $M^{max}_{T2}(ab)$ curve which fails to come into contact with the high solvability region. Therefore, by plotting the maximum fraction of solvable events along these constraint curves in the $(\tilde{m}_{\!{B}},\tilde{m}_{\!{A}})$ plane, as a function of $\tilde{m}_{\!{C}}$, we can identify the true $m_{\!{C}}$ as the point where the maximum solvable fraction is highest. This is the technique proposed by H.-C. Cheng and Z. Han in Ref. Cheng:2008hk (for an illuminating discussion on the relation between solvability and kinematic endpoints, edges and kinks, see Barr:2009jv ; Lester:2013aaa ).

In conclusion of this section, we have seen that the solvability method, supplemented with one or more kinematic endpoint measurements, is in principle capable of determining all three unknown masses $m_{\!{A}}$, $m_{\!{B}}$ and $m_{\!{C}}$. However, as revealed by the preceding discussion in Sections 4.2 and 4.3, the method is not as sensitive in finding the true value of $m_{\!{C}}$ as it is in finding the true values of $m_{\!{B}}$ and $m_{\!{A}}$, given $m_{\!{C}}$. This is indicative of a “flat direction” of low sensitivity along $\tilde{m}_{\!{C}}$. This is not unexpected — it is known that mass differences can be measured much better than the overall mass scale, as has been demonstrated with specific studies for the case of pair production Cho:2007qv ; Barr:2007hy ; Cho:2007dh ; Matchev:2009fh ; Matchev:2009ad and a single decay chain Gjelsten:2004ki ; Gjelsten:2005aw ; Gripaios:2007is ; Debnath:2016gwz ; Betancur:2017kqe .

5 Extreme Events and Kinematic Boundaries in Mass Space

With the intuition developed in the previous two sections, we are now ready to present our main idea. First we shall introduce some terminology.

5.1 Definitions

As illustrated in the solvability plots in Figs. 2-4, for any given event, the kinematic constraints (13) and (14) partition the three-dimensional parameter space $(\tilde{m}_{\!{A}},\tilde{m}_{\!{B}},\tilde{m}_{\!{C}})$ into regions allowing 0, 2 or 4 real solutions for the invisible momenta. We shall call the two kinematic101010In the sense that they are derived from kinematic constraints. boundary surfaces separating those mass space regions degeneracy boundaries, since on those boundaries there exists a pair of degenerate real solutions. For completeness, we shall also give a special name to the boundary separating the regions with zero and two solutions and call it a solvability boundary, since the event is solvable on one side of the boundary and not solvable on the other. In our event topology, the kinematic boundaries of an event are two dimensional surfaces in the three dimensional mass space $(\tilde{m}_{\!{A}},\tilde{m}_{\!{B}},\tilde{m}_{\!{C}})$. Therefore, when we take a slice of the parameter space at fixed $\tilde{m}_{\!{C}}$ as in Figs. 2-4, we correspondingly obtain degeneracy curves in the $(\tilde{m}_{\!{B}},\tilde{m}_{\!{A}})$ plane, some of which (the ones which delineate the region with zero solutions) are solvability curves.

Fig. 12 illustrates the notion of degeneracy curves, using the event from the right panels in Figs. 2-4. In the left panel of Fig. 12, we reproduce the solvability plot from Fig. 2 which was made for $\tilde{m}_{\!{C}}=m_{\!{C}}=700\;\mathrm{GeV}$. Again we can see the three different regions — with 0, 2 and 4 real solutions (colored white, green and yellow, respectively). Then in the right panel of Fig. 12 we ignore the interiors of those regions and only plot the degeneracy curves for that event. As before, the cross marks the true mass point $(m_{\!{B}},m_{\!{A}})$. We observe that the cross does not lie on any of the degeneracy curves — indeed, there is no reason to think that the true masses will necessarily lead to degenerate real solutions for the momenta. Since the event used in Fig. 12 was chosen at random, its properties are generic and follow this expectation.

At the same time, Fig. 6 also revealed the presence of events in the sample for which the cross will lie on a degeneracy curve. Now is a good time to give a name to such special events, since they will be the main topic of discussion in this section. In general, whenever a given trial mass point $(\tilde{m}_{\!{A}},\tilde{m}_{\!{B}},\tilde{m}_{\!{C}})$ lies on a degeneracy boundary of an event, the event will be said to be an extreme event for that mass point. Extreme events for the true mass point $(m_{\!{A}},m_{\!{B}},m_{\!{C}})$ will be referred to simply as extreme events. Note that a priori we do not know which events in the event sample are extreme in the latter sense, since we do not know the true masses. Therefore, in order to analyze extreme events, we either have to cheat and use Monte Carlo information, or develop some alternative strategies for recognizing such events in the data, see Section 5.2 below.

As demonstrated in Fig. 6, the primary motivation for studying extreme events is that they are very efficient in ruling out incorrect mass hypotheses. This is especially important in the region in the immediate vicinity of the true mass point. Indeed, by definition, any event which is not extreme, will leave the whole neighborhood around the true mass point as viable. In that sense, extreme events are also extremely valuable, as they provide the only chance to probe the mass space close to the true mass point. This point is illustrated in Fig. 13, which repeats the exercise from Fig. 6 with three other, randomly selected, extreme events. Note that while the events were required to be extreme in the sense that the true mass point marked with the cross lies on a degeneracy boundary, they were not required to be complementary to each other, as was previously done in Fig. 6. As a result, their superposition now leaves a much larger red-shaded area in which all three events are solvable (compare to Fig. 6). Even so, notice that the true mass point is on the very tip of the allowed red-shaded region. In other words, three of the four possible directions away from the true mass point are ruled out, leaving only a relatively narrow funnel in the north-northeast direction. This is to be contrasted with the result from Fig. 5, for example, where the immediate vicinity of the true mass point is viable in any direction.

The observant reader would have noticed that each of the three events in Fig. 13 happened to be extreme because the true mass point lies on a solvability boundary as opposed to the other degeneracy boundary (the one between the regions with 2 and 4 real solutions). This is somewhat accidental — in the data sample we do find extreme events of both types. If we were to categorize them, we would find that the extreme events on a solvability boundary outnumber the others by a factor of roughly $5:1$. This is consistent with the outcome in Fig. 13 — given three extreme events, the chances that they all have the true mass point on a solvability boundary are approximately $\left(\frac{5}{6}\right)^{3}\approx 58\%$. One might question the usefulness of the remaining $1/6$ of extreme events which do not involve a solvability boundary. Indeed, they do not help to shave the allowed mass space near the true mass point. Nevertheless, as we shall see in the next section, both types of extreme events will contribute on an equal footing to the mass measurement method proposed in Section 6, therefore from now on we shall stop making the distinction between them.

5.2 Characterizing extreme events

As mentioned earlier, the kinematic constraints (13) and (14) allow four (in general complex) solutions for the invisible momenta $q_{i}$ of particles $C_{i}$. The four solutions are continuous functions of $(\tilde{m}_{\!{A}},\tilde{m}_{\!{B}},\tilde{m}_{\!{C}})$ and any non-real solutions among them appear in complex conjugate pairs Gripaios:2011jm . On the degeneracy boundaries, where the number of real solutions changes, the equations have a pair of degenerate real solutions.

We can therefore characterize extreme events (for a given choice of masses $\tilde{m}_{\!{A}}$, $\tilde{m}_{\!{B}}$ and $\tilde{m}_{\!{C}}$) by finding the condition for a particular solution $(q_{1},q_{2})$ to be degenerate. It can be shown that a solution $(q_{1},q_{2})$ is degenerate iff

[TABLE]

where $\hat{z}$ is the unit vector along the $z$-axis, the $E$’s refer to the energy components of the corresponding 4-momenta, and the three-vectors $\vec{V}_{i}$ are defined as

[TABLE]

Here the 3-vectors $\vec{v}$ are the velocities111111Not to be confused with the spatial components of the 4-velocities. of the particles, i.e., $\vec{v}=\vec{p}/E$:

[TABLE]

Eq. (17) is useful in understanding the kinematic configurations of extreme events. As discussed later in Section 5.4, extreme events correspond to kinematic endpoints, therefore eq. (17) (and its analogues for other topologies) can be useful in understanding and deriving various endpoints. Eq. (17) also provides an efficient way of generating a pure sample of extreme events for dedicated studies of such events121212Any simulated event (for which we know the invisible momenta) can be converted into an extreme event by appropriately rotating the decay of $B_{2}$ about its direction of motion to satisfy eq. (17).. Of course, the quantity ${\mathcal{E}}$, being a scalar, conveniently quantifies the level of extremeness of the event: the smaller the magnitude of ${\mathcal{E}}$, the more extreme an event is.

One downside of the parametrization (17) is that it refers to the particular solution $(q_{1},q_{2})$, so that in order to use this criterion in practice, one has to solve the constraint equations first. Since they may lead to up to 4 real solutions, one must test the condition (17) for each real solution and then use the smallest value of ${\mathcal{E}}$ thus obtained in order to judge whether the event is extreme or not.

5.3 Extreme events are more common than you think

Having motivated extreme events through their useful and interesting properties, the crucial question to address next is, how frequent are they in a realistic data sample? The pessimist from Sec. 4.1 could again raise an objection that the volume of a two-dimensional surface in three-dimensional space is zero, and therefore, the chances that an event will land exactly on a degeneracy boundary and thus become extreme are vanishing as well. However, this argument contains a hidden assumption — that the distribution of events throughout the three-dimensional space is a normal, well-behaved function, i.e., the event density has no singularities. In this subsection we shall argue that the hidden assumption is false, and that in fact, in the narrow width approximation, the density of events is expected to be singular precisely on the degeneracy boundaries, thus greatly increasing the odds of encountering extreme events in the sample. The gist of our argument goes back to the algebraic singularity method of I.-W. Kim Kim:2009si . As a specific example, phase space singularities on kinematic surfaces were recently explored in the case of a single SUSY-like decay chain in Agrawal:2013uka ; Debnath:2016gwz ; Debnath:2018azt . However, our main argument here is very general, thus we shall try to keep the discussion independent of the underlying topology.

Consider the space of the $N_{p\,\cup\,q}$ components of the 4-momenta of all final state particles, visible and invisible. Out of these, let $N_{p}$ be the total number of visible momentum components $\{p\}$ and $N_{q}$ be the total number of invisible momentum components $\{q\}$, where trivially $N_{p\,\cup\,q}=N_{{p}}+N_{{q}}$. On-shell events generated from a given set of true masses will obey a certain number, $C_{{q}}$, of kinematic constraints involving the invisible momenta, plus possibly also a certain number of constraints $C_{{p}}$ involving only the visible momenta $\{{p}\}$ and no unknown parameters. As a result of the kinematic constraints, the final state momenta will lie on a hypersurface of dimension $N_{{p}\,\cup\,{q}}-C_{{q}}-C_{{p}}$, which is embedded in the original $(N_{{p}\,\cup\,{q}})$-dimensional space. This hypersurface represents the full phase space for this process, and events will be more or less evenly distributed in the full phase space.

Now the crucial question is, how are events distributed in the allowed $(N_{{p}}-C_{{p}})$-dimensional visible subspace? For this purpose, we need to project the full $(N_{{p}\,\cup\,{q}}-C_{{q}}-C_{{p}})$-dimensional phase space onto the $(N_{{p}}-C_{p})$-dimensional allowed visible subspace Kim:2009si . The dimensionality of the resulting projection is

[TABLE]

In this paper, we shall restrict our attention to the case where $N_{{p}\,\cup\,{q}}\,-\,C_{{q}}\,-\,C_{{p}}=N_{{p}}\,-\,C_{{p}}$, i.e., the dimensionality of the full phase space equals the dimensionality of the allowed visible subspace. Since $N_{{p}\,\cup\,{q}}=N_{{p}}+N_{{q}}$, an equivalent way to state the same condition is simply $N_{{q}}=C_{{q}}$, i.e., there are just enough constraints in order to solve for the invisible momenta $\{{q}\}$. This was precisely the case in Section 3: we were able to solve for the invisible momenta in the event topology of Fig. 1. Let us check: there were 4 visible final state particles ($N_{{p}}=4\times 4=16$), 2 invisible final state particles ($N_{{q}}=2\times 4=8$), two constraints from (13) and six constraints from (14) for a total of $C_{{q}}=2+6=8$. There are also four mass shell constraints on the visible momenta ($C_{{p}}=4$). Thus both the full phase space and the visible phase space are 12-dimensional: $N_{{p}\,\cup\,{q}}-C_{{q}}-C_{{p}}=(16+8)-8-4=12$ and $N_{{p}}-C_{{p}}=16-4=12$.

As already mentioned above, the probability density of events on the full phase space is well behaved, i.e., there are no singularities or discontinuities. However, this is not the case with events in the visible phase space. First we note that the mapping of the full phase space onto the visible phase space is not one-to-one (or invertible) — multiple points in the full phase space can have the same visible phase space projection, as evidenced by the existence of multiple solutions for the invisible momenta. To keep a simple visual analogy in mind, think of the projection of a two-dimensional hollow sphere onto the equatorial plane. Any image point in the equatorial plane has two possible preimages, one in the Northern hemisphere and one in the Southern hemisphere.

Let $\mathcal{P}_{\mathrm{full}}$ be the density of events in the full phase space, and let $\mathcal{P}_{\mathrm{vis}}$ be the density of events in the visible phase space. Consider an infinitesimal volume $dV_{\mathrm{vis}}$ in the visible phase space and the corresponding infinitesimal volumes $dV_{\mathrm{full},i}$ which can be mapped onto it from the full phase space. Here the index $i$ labels the “sheet” on which $dV_{\mathrm{full},i}$ is located131313Since the visible phase space and the full phase space have equal dimensionalities, there can only be a finite number of such “sheets”. — for example, in the hollow sphere example above, the two values of $i=1,2$ would label the Northern and Southern hemispheres. The two event densities are related as

[TABLE]

or equivalently,

[TABLE]

As already mentioned, $\mathcal{P}_{\mathrm{full},i}$ is well behaved. However, at points where $dV_{\mathrm{vis}}/dV_{\mathrm{full},i}$ is vanishing (for one of the $i$ values), $\mathcal{P}_{\mathrm{vis}}$ will be singular Kim:2009si . The vanishing of $dV_{\mathrm{vis}}/dV_{\mathrm{full},i}$ occurs at those locations where the tangent plane of the full phase space is perpendicular to the tangent plane of the visible phase space. (In the hollow sphere example above, this singularity occurs on the equator, where the surface of the sphere is perpendicular to the equatorial plane.) It is also easy to see that those locations are precisely where two different branches in the full phase space (i.e., corresponding to two different values of $i$) meet (in the hollow sphere example, think of the Northern and Southern hemisphere merging at the equator). In other words, the points where $\mathcal{P}_{\mathrm{vis}}$ is singular correspond to extreme events which have degenerate real solutions for the invisible momenta. Consequently, the dimensionality of the subspace spanned by extreme events is $N_{{p}}-C_{{p}}-1$, due to the additional “degeneracy condition”.

5.4 The connection between extreme events and kinematic endpoints

For events on the boundary of the visible phase space, the true mass point will lie on the solvability boundary in mass space. In other words, events on the boundary of the visible phase space lead to a particular class of extreme events, namely the ones involving the solvability boundary. As already discussed at the end of Section 5.1, those are the large majority of extreme events.

The shape of the boundary of the visible phase space is dependent on the underlying mass spectrum and can therefore reveal information about it. A standard approach to extract this information is to measure kinematic endpoints in the distributions of variables suitably constructed from the visible momenta Barr:2010zj ; Matchev:2019sqa . It is clear that the events for which the kinematic variable under consideration acquires its endpoint value, are necessarily events on the boundary of the visible phase space Cheng:2008hk ; Barr:2009jv ; Lester:2013aaa . Since those are extreme events, it follows that extreme events are responsible for the endpoints in the distributions of kinematic variables. The reverse, however, is not necessarily true — not all of the extreme events are necessarily mapped onto the kinematic endpoint value. Therefore, compared to endpoints in kinematic distributions, extreme events present an interesting and more general target for studies of event kinematics.

5.5 The distribution pattern of kinematic boundaries in mass space

In Section 5.3 we argued that the density of extreme events is singular. Here we shall use that result to show that if we consider the degeneracy boundaries in the mass space for a sample of events, the density of these boundaries will be singular at the true mass point.

We shall first give a heuristic justification before building it into a more rigorous argument. Note that any event is extreme for some set of points in mass space, namely, the ones on the degeneracy boundaries of that event. We do not know a priori whether any of those mass points happen to be the true masses or not. So let us then try to figure this out on a statistical basis. Let each event give a vote to all the mass points on its degeneracy boundaries, i.e., all masses for which it is extreme. We saw in Section 5.3 that events which are extreme for the true mass have an enhanced (singular) probability density. On the other hand, events which are extreme for an incorrect test mass point do not necessarily have the same enhancement. By this logic, we can expect the true mass point to pick up more votes in this procedure than an incorrect test mass choice. “More votes” here corresponds to a higher density of degeneracy boundaries.

Let us rephrase this heuristic argument in a slightly more rigorous form. We shall do this by mapping the density of extreme events in the visible phase space to the density of degeneracy boundaries in the mass space.

In our event topology of Fig. 1 the mass space is 3 dimensional and the degeneracy boundaries are 2 dimensional surfaces. Their density, $\mathcal{P}^{db}_{\mathrm{mass}}(\tilde{m}_{\!{A}},\tilde{m}_{\!{B}},\tilde{m}_{\!{C}})$, will be defined as the fraction of degeneracy boundaries per unit $\sqrt[3]{\mathrm{volume}}$ in the mass space.141414This is because the number of randomly distributed $d$-dimensional objects in a $D$-dimensional volume element scales as $\left(\mathrm{volume}\right)^{(1-d/D)}$. In our case $d=2$ dimensional surfaces are distributed in a $D=3$ dimensional volume element and their number scales as $\left({\mathrm{volume}}\right)^{1/3}$. It is given by

[TABLE]

where the integral is over the $(N_{{p}}-C_{{p}}-1)$-dimensional subspace of extreme events for the given mass point $(\tilde{m}_{\!{A}},\tilde{m}_{\!{B}},\tilde{m}_{\!{C}})$. $dV_{1}$ represents a differential element in the visible phase space perpendicular to the space of extreme events. It corresponds to the change in the space of extreme events when the mass point varies within $d^{3}V_{\mathrm{mass}}$. It helps to think of the space of extreme events of a test mass as an $(N_{{p}}-C_{{p}}-1)$-dimensional curve which changes smoothly when the test mass changes. Eq. (23) can be rewritten as

[TABLE]

where $J$ is a Jacobian factor. We can see that $\mathcal{P}^{db}_{\mathrm{mass}}$ will be singular iff the integral is divergent.

Note that the Jacobian factor and the integration volume for a given test mass are both independent of the true mass from which the event sample is generated. So, except possibly for some special test masses (special overall, not in relation to the true mass), we can expect their contribution to the variation of $\mathcal{P}^{db}_{\mathrm{mass}}$ to be well-behaved. Any local singularity in $\mathcal{P}^{db}_{\mathrm{mass}}$ can therefore only come from $\mathcal{P}_{\mathrm{vis}}$. As mentioned in Section 5.3, $\mathcal{P}_{\mathrm{vis}}$ is singular for extreme events of the true mass and for those events only. Thus the density of degeneracy boundaries in the mass space is singular at the true mass151515The density is also singular at masses whose extreme events have an intersection of measure greater than 0 with the true extreme events..

6 The Kinematic Focus Point Method

6.1 The basic idea

Before proceeding further, let us condense the content of Section 5.5 into the following elevator pitch:

Draw the degeneracy boundaries (in the mass space) for all events in the data sample. As a reminder, the degeneracy boundary of an event contains masses for which the given event has degenerate real solutions for the invisible momenta. 2. 2.

The density of these boundaries should peak at the true mass point.

This lays out a straightforward mass determination method pictorially illustrated in Figs. 14 and 15.

We simply plot the degeneracy surfaces for all events in our sample, and look for a focus point where a maximal number of them either intersect or pass close by. Since it is difficult to visualize this in the full three dimensional mass space $(\tilde{m}_{\!{A}},\tilde{m}_{\!{B}},\tilde{m}_{\!{C}})$, in Figs. 14 and 15 we show results in the $(\tilde{m}_{\!{A}},\tilde{m}_{\!{B}})$ plane for fixed $\tilde{m}_{\!{C}}=m_{\!{C}}=700\;\mathrm{GeV}$. In Fig. 14 we use the same plot range as in all of our previous figures, and gradually increase the number of events used for the plot from 5 (top left panel) to 50 (bottom right panel). We observe that as we add more and more events, a focus point indeed gradually emerges near the true mass values of $m_{\!{B}}=800$ GeV and $m_{\!{A}}=1000$ GeV. Our method derives it name from this feature, since the degeneracy curves tend to “focus” at the true mass point. The focus point is very easy to identify by eye, and in the next subsection we shall develop a procedure for quantifying the observed amount of “focusing”. Clearly, if we add even more events, the focus point should become even more pronounced, but unfortunately, the plot would become too cluttered. This is why when we further increase the number of events to 100 in Fig. 15, we zoom in on the region in the vicinity of the focus point. The left (right) panel of Fig. 15 uses a zoom factor of 5 (10) relative to the plot range in Fig. 14. We see that the focus point is very clear, and is in perfect agreement with the true values $m_{\!{B}}=800$ GeV and $m_{\!{A}}=1000$ GeV. For example, if one were to estimate the values of $m_{\!{B}}$ and $m_{\!{A}}$ by simply eyeballing the plots in Fig. 15, the error would not be more than a couple of GeV! One should also keep in mind that the plots in Fig. 15, which were made only for illustration, use only 100 events, while the actual LHC statistics is likely to be much higher.

6.2 Measuring $m_{B}$ and $m_{A}$ for a given trial mass $\tilde{m}_{C}$

Hopefully by now the reader has been convinced by the money plots in Figs. 14 and 15 that the kinematic focus point method outlined in the previous Section 6.1 is a promising new technique for mass measurements. In the remainder of this section, we shall discuss how to implement it in practice, as well as some practical challenges and how they might impact the sensitivity of the method.

Since the kinematic focus point method relies on the high density of degeneracy curves near the true mass point, we need to design a procedure for quantifying this effect. Following the outline of Section 4, let us first concentrate on the task of measuring $m_{\!{B}}$ and $m_{\!{A}}$ at a fixed value of $\tilde{m}_{\!{C}}$, postponing the task of measuring $m_{\!{C}}$ until the next Section 6.3. For this purpose, we need to plot one-dimensional degeneracy curves in the two-dimensional $(\tilde{m}_{\!{B}},\tilde{m}_{\!{A}})$ plane, as was done in Figs. 14 and 15. The discussion from Sec. 5.5 showed that the relevant quantity in that case is the number of degeneracy curves per unit $\sqrt{\mathrm{area}}$, which is what we shall use in our analysis. The area elements in the $(\tilde{m}_{\!{B}},\tilde{m}_{\!{A}})$ plane will be chosen to be $10\;\mathrm{GeV}\times 10\;\mathrm{GeV}$ squares, and so the resulting unit for the density of degeneracy curves will be $(10\;\mathrm{GeV})^{-1}$. For the remainder of this subsection, the test mass $\tilde{m}_{\!{C}}$ will be set to the true value of $700\;\mathrm{GeV}$.

We are now ready to study the density of degeneracy curves in the $(\tilde{m}_{\!{B}},\tilde{m}_{\!{A}})$ plane. The result is shown as a heat map in Fig. 16 and as 3D histograms from different viewpoints in Fig. 17. The density of degeneracy curves is displayed in units of fraction of events per $10\;\mathrm{GeV}$, i.e., instead of plotting the total number of curves passing through a given $10\;\mathrm{GeV}\times 10\;\mathrm{GeV}$ square, we plot the fraction of events in the sample which have a degeneracy curve passing through that square. By normalizing to a fraction, our results become insensitive to the statistics used to make the plots (in this case, Figs. 16 and 17 were made with 1000 events). The figures show that the density has a sharp peak at the true mass ($m_{\!{B}}=800\;\mathrm{GeV},m_{\!{A}}=1000\;\mathrm{GeV}$). The peak is actually very well pronounced: note that over $60\%$ of the events in the sample have one of their degeneracy curves passing through a $10\;\mathrm{GeV}\times 10\;\mathrm{GeV}$ square around the true mass point. This confirms that our method is really effective in finding $m_{\!{B}}$ and $m_{\!{A}}$, given $\tilde{m}_{\!{C}}$. Note the advantages of our method over the solvability method from Sec. 4 — first, we do not encounter the flat direction of $100\%$ solvability seen in Figs. 7, 9 and 10, and as a result we do not need any additional experimental information (from kinematic endpoints or otherwise); and second, we are measuring a sharp peak structure which is centered on the true values of the particle masses.

6.3 Measuring $m_{C}$

As in Sec. 4.3, we now turn to the more difficult task of measuring $m_{\!{C}}$. For this purpose, we repeat the exercise from the previous subsection, only this time with different values of $\tilde{m}_{\!{C}}$. Heat maps analogous to the one in Fig. 16 are shown in the left and right panel of Fig. 18 for two representative values, $\tilde{m}_{\!{C}}=400\;\mathrm{GeV}$ and $\tilde{m}_{\!{C}}=1000\;\mathrm{GeV}$, respectively.

We observe that each of the two plots in Fig. 18 exhibits its own density peak in the $(\tilde{m}_{\!{B}},\tilde{m}_{\!{A}})$ plane. As expected, the location of the peak tracks the value of $\tilde{m}_{\!{C}}$ — for lower (higher) values of $\tilde{m}_{\!{C}}$, the peak is located at lower (higher) values of $\tilde{m}_{\!{B}}$ and $\tilde{m}_{\!{A}}$. From that point of view, there is no qualitative difference in the heat maps at different values of $\tilde{m}_{\!{C}}$. However, there is a quantitative difference: the height of this peak, i.e, the maximum density in the $(\tilde{m}_{\!{B}},\tilde{m}_{\!{A}})$ plane for a given value of $\tilde{m}_{\!{C}}$, is different for different values of $\tilde{m}_{\!{C}}$. The maximum height is expected to occur at the true value of $\tilde{m}_{\!{C}}=m_{\!{C}}$.

While this expectation is consistent with the results in Figs. 16 and 18, we quantify it further in Fig. 19, which shows a plot of the maximum density of degeneracy curves in the $(\tilde{m}_{\!{B}},\tilde{m}_{\!{A}})$ plane (after fixing the value of $\tilde{m}_{\!{C}}$), as a function of $\tilde{m}_{\!{C}}$. As before, the density is plotted in units of fraction/($10\;\mathrm{GeV}$). Fig. 19 nicely confirms that the density peaks at the true value of $m_{\!{C}}=700$ GeV (marked with the vertical red line). As already discussed in Sec. 4.3, the $m_{\!{C}}$ measurement is rather challenging — in our case this is evidenced by the fact that the peak in Fig. 19 is not as sharp as the peak in Figs. 16 and 18. Nevertheless, Fig. 19 confirms that in principle our method can be used to find $m_{\!{C}}$ as well.

6.4 The impact of the detector resolution

Until now, all of our studies were done at the parton level and in the narrow width approximation, without smearing from either detector resolution or particle width effects. We shall now investigate the size of those effects. In this subsection we shall first add the effect of the detector resolution. We shall consider the typical scenario where the particles $a_{i}$ and $b_{i}$ in the event topology of Fig. 1 are bottom quarks and leptons, respectively. Correspondingly, we shall apply smearing to the energies of the two $b$-jets according to typical LHC resolutions Khachatryan:2016kdb : for jet $p_{T}$ up to $\{10,20,30,50,100,400,1000\}$ GeV, the $p_{T}$-dependent energy resolution parameter is $40\%,28\%,19\%,13\%,10\%,6\%,5\%$.

Figs. 20 and 21 show the results analogous to Figs. 16, 18 and 19, only now with the detector resolution included. As expected, the detector effects tend to wash out the peak structures a little bit. Nevertheless, the peaks in Fig. 20 (which shows measurements of $m_{A}$ and $m_{B}$ for fixed $\tilde{m}_{C}$) are still pronounced, and more importantly, the peak found for $\tilde{m}_{C}=m_{C}$ is still located at the proper values of the masses. Fig. 21 shows the measurement of $m_{C}$ (in the presence of detector effects), which was already a difficult task even at the gen-level, see Fig. 19.

6.5 A Standard Model example: dilepton $t\bar{t}$ events

Most of the mass measurement methods applicable to LHC events with missing transverse energy were developed with some kind of new physics model in mind, e.g., supersymmetry Matchev:2019sqa . However, experimenters who wish to test the kinematic focus point method in real data do not have to wait for the discovery of any new physics — the event topology of Fig. 1 is already present in the LHC data in the form of Standard Model dilepton $t\bar{t}$ events, for which the available statistics is enormous. Therefore, it is worth supplementing our previous results with a study showcasing the method for the dilepton $t\bar{t}$ channel, and this is what we shall do in the current subsection.

We generate Standard Model $t\bar{t}$ events with Madgraph Alwall:2011uj , using the proper widths for the top quark and the $W$ boson. We select dilepton events in which both top quarks have decayed leptonically, and repeat the kinematic focus point analysis from Section 6.2, setting the trial neutrino mass $\tilde{m}_{\nu}$ to zero. We then consider the plane spanned by the trial masses for the $W$ boson ($\tilde{m}_{W}$) and for the top quark ($\tilde{m}_{t}$) and compute the density of the degeneracy curves. The heat map analogous to Fig. 16 is shown in Fig. 22, where the left panel depicts the parton level result, while in the right panel we include the effects from the detector smearing as in Section 6.4.

As anticipated, Fig. 22 reveals a peak structure near the true masses $m_{W}=80\;\mathrm{GeV}$ and $m_{t}=173\;\mathrm{GeV}$. The peak is especially pronounced at the parton level (left panel) — over $40\%$ of the events have a degeneracy curve passing through a $5\;\mathrm{GeV}\times 5\;\mathrm{GeV}$ square nearby. The detector resolution does tend to wash out the peak structure in the right panel, but a maximum is still visible. This suggests that the kinematic focus point method could allow the simultaneous independent measurement of the top and $W$-boson masses in the spirit of Refs. Burns:2008va ; Chatrchyan:2013boa .

7 Conclusions and Outlook

In this paper we propose a new approach to mass measurements in events with missing energy, called the kinematic focus points method. The method derives from the study of the solvability of the kinematic constraints which are present in a given event topology Nojiri:2003tu ; Kawagoe:2004rz ; Cheng:2007xv ; Cheng:2008mg ; Cheng:2008hk ; Cheng:2009fw ; Webber:2009vm ; Barr:2009jv . Using the dilepton $t\bar{t}$ topology as our example, we first critically examined the solvability method for measuring masses, and its relation to the measurements of kinematic endpoints. We then showed that any given event divides the relevant three-dimensional mass parameter space into regions with a certain number of (pairs of) real solutions for the unknown invisible momenta. The surface boundaries between those regions are where the solutions become degenerate, and we called those surfaces “degeneracy boundaries”. We illustrated and studied the shape of the degeneracy boundaries, as well as their distribution throughout the mass parameter space. In particular, we singled out a special class of events, extreme events, for which the true mass point lies on a degeneracy boundary.

Our main results were:

• •

Extreme events are very efficient in restricting the allowed mass parameter space.

• •

Extreme events are abundant in realistic data samples, due to singularities in the phase space distribution.

• •

This abundance of extreme events can be harnessed by drawing the degeneracy boundaries in the mass parameter space over the full event sample. The boundaries tend to focus at the true mass point and this property can be used to identify the true masses.

• •

our study of the solvability mass measurement method revealed a flat direction of nearly 100% solvability, which can be lifted only by adding additional information.

The kinematic focus points method has similarities to traditional mass-bump methods, thus allowing for data driven estimation of the relevant backgrounds.

The method also represents an improvement over the traditional kinematic endpoint techniques.

The main advantage stems from the fact that we are able to extract useful information from all events and not just those which populate the vicinity of a kinematic endpoint. In order to illustrate this idea, we can look at the performance of the focus point method on a subset of events which would not assist in the standard extraction of the kinematic endpoints of the signal through a fit. The mass spectrum for our study point ($m_{\!{A}}=1000\;\mathrm{GeV}$, $m_{\!{B}}=800\;\mathrm{GeV}$ and $m_{\!{C}}=700\;\mathrm{GeV}$) happens to be such that all four kinematic endpoints (15) for the signal are higher than the respective endpoints for the $t\bar{t}$ background. Therefore, by restricting ourselves to signal events which satisfy all four background endpoints, we can select signal events far away from the signal endpoints; such events are also hard to distinguish kinematically from the background.

In Fig. 23 we contrast the heat maps (created as in Fig. 16) for Standard Model dilepton $t\bar{t}$ background events (left panel) and so preselected signal events (right panel), both for $\tilde{m}_{C}=700$ GeV. The white $\times$ symbol marks the location $(m_{B},m_{A})$ of the true masses in the signal topology. The signal events entering the plot on the right have been preselected to obey all four kinematic endpoints (15) predicted for the Standard Model dilepton $t\bar{t}$ background. Therefore, as far as kinematic endpoints are concerned, these signal events are very hard to distinguish from the SM background, and in particular, cannot be used for discovery Cho:2014yma . Yet the heat map for these “low utility” signal events in the plot on the right still shows a distinct feature through the correct masses, which is not present for the SM case shown in the left panel.161616The absolute maximum in the right plot of Fig. 23 is shifted away from the white $\times$ symbol due to the bias introduced by the applied preselection. Once the whole signal sample is considered, the peak will shift back to the correct position, as shown in Fig. 16, and will include a substantial contribution from the low utility signal events from Fig. 23.

The focus point method is based purely on event kinematics, and is therefore quite model-independent. Note that in addition to measuring the masses of particles, this technique can also be used as a new physics search technique by designing an analysis which would look for a bump feature in the density of the kinematic boundaries.

Acknowledgements.

DK was supported in part by the Korean Research Foundation (KRF) through the CERN-Korea Fellowship program, and is presently supported by the Department of Energy under Grant No. DE-FG02-13ER41976/DE-SC0009913. The work of KM and PS is supported in part by the United States Department of Energy under Grant No. DE-SC0010296.

Appendix A The Flat Direction of Low Sensitivity in Mass Space

The numerical results in Sections 4.3 and 6.3 suggested the presence of a region (“flat direction”) in mass space which has relatively low sensitivity. In other words, the analysis is least sensitive to the degree of freedom (in mass parameter space) parametrizing the flat direction. In this Appendix we identify and parametrize the flat direction.

One of the key ideas in deriving the flat direction is that the kinematic constraints (13) and (14) are a) Lorentz invariant and b) invariant under separate longitudinal boosts of the two branches in Fig. 1. Therefore, without loss of generality we can restrict our attention to events in which both $A_{1}$ and $A_{2}$ are produced at rest in the longitudinal direction, and have equal and opposite transverse momenta. The other key idea is that the flat direction arises because there is an additional hidden constraint which is approximately satisfied by events in our sample, but is unaccounted for in our analysis. This constraint arises due to the fact that at LHC energies, the transverse momenta of the parent particles are small relative to their masses. This is a good approximation at hadron colliders like the LHC, provided that the $A_{i}$ particles are not too light. In this case, one can show that there exists a flat direction near the true mass point which satisfies

[TABLE]

These equations can be solved to give

[TABLE]

Table 1 lists the mass spectrum predicted by (26) for the three representative values for $\tilde{m}_{\!{C}}$ used on the plots.