Testing Invisible Momentum Ansatze in Missing Energy Events at the LHC
PPrepared for submission to JHEP
CERN-TH-2017-061
Testing invisible momentum ansatze in missing energyevents at the LHC
Doojin Kim, a Konstantin T. Matchev, b Filip Moortgat, c Luc Pape c a Theory Division, CERN, CH-1211 Geneva 23, Switzerland b Physics Department, University of Florida, Gainesville, FL 32611, USA c CERN, Geneva CH-1211, Switzerland
Abstract:
We consider SUSY-like events with two decay chains, each terminating in aninvisible particle, whose true energy and momentum are not measured in the detector.Nevertheless, a useful educated guess about the invisible momenta can still be obtained byoptimizing a suitable invariant mass function. We review and contrast several proposalsin the literature for such ansatze: four versions of the M T -assisted on-shell reconstruc-tion (MAOS), as well as several variants of the on-shell constrained M variables. Wecompare the performance of these methods with regards to the mass determination of anew particle resonance along the decay chain from the peak of the reconstructed invariantmass distribution. For concreteness, we consider the event topology of dilepton t ¯ t eventsand study each of the three possible subsystems, in both a t ¯ t and a SUSY example. Wefind that the M variables generally provide sharper peaks and therefore better ansatzefor the invisible momenta. We show that the performance can be further improved bypreselecting events near the kinematic endpoint of the corresponding variable from whichthe momentum ansatz originates. a r X i v : . [ h e p - ph ] M a r ontents M T -assisted and M -assisted mass reconstructions of mass peaks 16 M -based methods 184.3 Comparison of M T -assisted and M -assisted reconstruction schemes 21 The bread and butter method for discovering a new particle in high energy physics isthe “bump hunt”: one identifies and measures the momenta and energies of all relevantdecay products, and forms their total invariant mass. Signal events, which are due to theproduction of a new resonance, appear as a localized “bump” feature over the relativelysmooth background continuum. This technique has led to many discoveries in the past,including the most recent discovery of the Standard Model Higgs boson, which was firstobserved as an invariant mass peak in the four-lepton and di-photon channels [1, 2].However, this tried and true method faces a major challenge when one (or more) of thedecay products are neutral, weakly interacting particles, which are invisible in the detector,and as a result their energies and momenta remain unknown. Many well-motivated modelsof new physics Beyond the Standard Model (BSM) contain such particles, as they arethe prototypical dark matter candidates. Consequently, one has to develop alternativemethods for discovery (and mass measurement) which are applicable to the case of suchsemi-invisible resonance decays. As well as measuring its mass and lifetime. The case of a fully invisible decay, i.e., when the new resonance decays to invisible particles only, israther trivial and will not be considered in this paper. – 1 –he situation is further complicated by the fact that most BSM models with darkmatter candidates introduce a conserved discrete symmetry, often a Z parity, in orderto protect the lifetime of the dark matter particle. The new particles which are chargedunder this symmetry are necessarily pair produced; therefore, each event contains not one,but at least two invisible (dark matter) particles whose 4-momenta q µ and q µ are notindividually measured. At hadron colliders, it is only the sum (cid:126)q T + (cid:126)q T of their transversemomenta which can be measured in the form of the missing transverse momentum /(cid:126)P T ofthe event: (cid:126)q T + (cid:126)q T = /(cid:126)P T . (1.1)However, the partitioning of the measured /(cid:126)P T into (cid:126)q T and (cid:126)q T is a priori unknown, andfurthermore, the longitudinal components q z and q z remain arbitrary at this point aswell.Over the last 15-20 years, a large number of methods have been proposed to dealwith measurements in such “SUSY-like” events, i.e., events with two decay chains, eachterminating in an invisible particle (see Ref. [3] for a recent review). One possibility is totry to calculate exactly the unknown individual 4-momenta q µi of the invisible particles,which in turn would allow one to reconstruct an invariant mass peak again. Unfortunately,this idea can only be applied to very specific event topologies, where the decay chains aresufficiently long, yielding enough mass-shell constraints in addition to (1.1) [4–7]. This iswhy the majority of the proposed methods have abandoned the idea of directly measuringa mass peak, and instead focused on measuring a kinematic endpoint for a suitably definedvariable.Now, what constitutes a “good” kinematic variable for a kinematic endpoint measure-ment? The answer to this question in principle depends on several factors, including theassumed event topology, the nature of the visible SM particles in the final state, the preci-sion with which their momenta p j are measured, etc. Roughly speaking, we can divide theset of kinematic variables into two categories: • Variables built only from directly measured quantities, i.e., the momenta p j of thevisible final state particles and the missing transverse momentum /(cid:126)P T . The primaryexample of such a variable is the invariant mass of a collection of visible particles. Thisidea forms the basis of the classic method for mass determination in supersymmetryfrom kinematic endpoints [8–14]. Other variables belonging to this class include thescalar sum H T of the transverse momenta of visible objects (jets or leptons), theeffective mass M eff [8, 15], the contransverse mass variable M CT [16, 17] and itsvariants M CT ⊥ and M CT (cid:107) [18], the ratio of visible transverse energies [19, 20], andthe energy itself [21–26]. The advantage of these variables is their simplicity, sinceone does not have to even face the question about the individual momenta q i ormasses ˜ m i of the invisible particles in the event. In principle, these variables are verygeneral and can be usefully applied in certain situations; however, they also fail to Throughout this paper we shall employ the convention where the letter p ( q ) is used to denote themeasured (unmeasured) momentum of a particle which is visible (invisible) in the detector. In addition,the true (hypothesized) mass of the i -th invisible particle will be denoted with m i ( ˜ m i ). – 2 –ake advantage of the specific characteristics of the event, and become suboptimalfor more complex event topologies. • Variables defined in terms of both the measured momenta p j and the invisible mo-menta q i . Of course, since the individual invisible momenta q i are unknown, thedefinition of any such variable v ≡ v ( p j , q i ) (1.2)must be supplemented with a procedure for fixing the values of the invisible momenta q i through a suitable ansatz. More concretely, the ansatz should allow us to computethe invisible 4-momenta q µi in terms of the measured visible 4-momenta p νj and a setof hypothesized masses ˜ m i for the invisible particles: q µi = q µi ( p νj , ˜ m i ) , (1.3)so that at the end of the day, the kinematic variable (1.2) can be equivalently ex-pressed in terms of visible momenta p j and invisible masses ˜ m i only: v = v ( p j , q i ( p j , ˜ m i )) . (1.4)If one is solely interested in the kinematic variable v itself and its properties (differen-tial distribution, kinematic endpoints, etc.), the intermediate step (1.3) of computingthe individual invisible momenta q i is unimportant and can be regarded simply asa convenient calculational tool. In fact, many of the computer codes on the mar-ket which are used to compute kinematic variables of the type (1.2), by default donot even report the values for the invisible momenta found from the ansatz (1.3).There are also some special cases, e.g., the minimum partonic center-of-mass energy √ ˆ s min [27, 28], the razor variables [29, 30], or the transverse mass M T [31, 32], whereone can solve for the ansatz (1.3) analytically, eliminate the invisible momenta, andderive an exact analytical expression for the variable v in the form of (1.4), which canthen serve as an alternative definition, without reference to any invisible momentaat all.Perhaps the two best known examples of variables of the type (1.2) are the transversemass [31, 32] and the Cambridge M T variable [33, 34]. Recently this set of variableswas expanded significantly and now includes M T ⊥ and M T (cid:107) [35], the asymmetric M T [36, 37], M C [38, 39], M CT [40, 41], M approxT [42], and the constrained M variables [43–47]. As the index “2” suggests, all these variables were designed forthe case of SUSY-like events with two decay chains, and they also carry an implicitdependence on the test masses ˜ m i of the invisible particles, as indicated in (1.4).Despite the large number of such variables on the market, they all share the same At first, the dependence on the unknown masses ˜ m i was considered undesirable, which perhaps pre-vented the more widespread use of variables of the type (1.2). Later on, it was realized that the ˜ m i dependence itself contains a large amount of useful information, e.g., a “kink” develops at the true value m i of the invisible particle mass [36, 48–51] (related techniques for measuring the invisible particle massesby utilizing the ˜ m i dependence are described in [18, 35, 52, 53]). – 3 –ommon idea [54]: choose a suitable target function and minimize it over all possi-ble values of the individual invisible momenta q i which are consistent with the /(cid:126)P T condition (1.1). The variations arise because one faces a menu of choices: – Partitioning of the event.
One groups the final state particles according to theassumed production process — single production, pair production, etc. Ideally,one should also have a separate category for jets which are suspected to comefrom initial state radiation [55–58]. – Choice of target function.
The target function can be a full (3+1)-dimensionalinvariant mass, as in the case of √ ˆ s min [27], M C [38, 39] and M [43, 44]; a(2+1)-dimensional transverse mass, e.g., M T [31, 32] or M T [33], and even a(1+1)-dimensional mass as in the case of M T ⊥ and M T (cid:107) [35]. Note that theprojection to lower dimensions in general does not commute with the partition-ing, so by performing those two operations in different order, one obtains inprinciple different variables [54]. – Imposing additional on-shell constraints.
The minimization of (3+1)-dimensionalmass target functions over the invisible momenta can be performed by takinginto account the /(cid:126)P T constraint (1.1) only, or by adding additional kinematicconstraints which are motivated by the assumed event topology [43, 44, 59, 60],a prior kinematic endpoint measurement [38], or by the presence of a known SMparticle in the decay chain (for example, a W boson [59, 61–63] or a τ lepton[59, 64, 65]). The additional on-shell constraints further restrict the alloweddomain of values for the components of the individual invisible momenta q i andin general lead to a different outcome from the minimization procedure.Note that whenever the target function is a transverse mass in (2+1) dimensions, theminimization fixes only the transverse components (cid:126)q iT of the invisible momenta, andfor the longitudinal components one must rely on additional measurements or assump-tions. For example, in the M T -assisted on-shell (MAOS) reconstruction method, oneassumes knowledge of the mass of the mother particle and enforces its on-shell condi-tion, which allows to solve for the longitudinal momenta [66]. The method was thentested in examples where the mothers are known SM particles, e.g. top quarks, W -bosons or τ -leptons [67–72]. Since the on-shell constraints are nonlinear functions, theMAOS approach typically yields multiple solutions for the longitudinal momentumcomponents, so one must also specify a prescription for handling this multiplicity. Incontrast, target functions defined in (3+1) dimensions automatically yield ansatze forthe full energy-momentum 4-vectors q µi , without any need for additional assumptions[54]. Another benefit of the (3+1) formulation is that the obtained solutions for thelongitudinal components q iz are typically unique [44, 47].In this paper, we would like to reemphasize the existence of various ansatze (1.3) forthe individual invisible momenta in missing energy events, and demonstrate their utility inthe context of a mass measurement through a “bump hunt”. Following previous studies,we shall consider the general event topology of dilepton t ¯ t events, which already have– 4 –ery rich kinematics, as one can define and study three different subsystems [73]: oneassociated with the two b -jets, another associated with the two leptons, and a third onereferring to the event as a whole (see Fig. 1 below). After briefly introducing our notationand conventions in Section 2, in the next Section 3 we shall carefully define and contrastthe different ansatze for invisible momenta which follow from some of the most commonlydiscussed in the literature variables of type (1.2): M T , M , and √ ˆ s min . The transversevariable M T is already at the heart of (as well as in the name of) the MAOS method [66].In addition to the traditional MAOS method described earlier, in Section 3.1 we shall alsoconsider two modified MAOS prescriptions [67–70], which avoid using information aboutthe mother particle mass, and instead rely on the calculated value of M T in the event.(There will also be a fourth variant of the MAOS method, which will assume a knownmass for a particle other than the parent.) Then in Section 3.2 we shall consider the caseof (3+1)-dimensional target functions, since it automatically provides an ansatz for thelongitudinal invisible momenta [27, 44].Next we would like to compare the performance of the difference ansatze (1.3). Onepossibility is to compare the momenta predicted by (1.3) to the true invisible momentain the event. However, the ultimate goal of any invisible momentum reconstruction is toperform some kind of physics measurement. In particular, once we have a guess for theinvisible momenta, we can revisit the original idea for a bump hunt, and compare theprecision of mass measurements performed with different ansatze. This will be the subjectof Section 4, in which we shall study the position and the sharpness of the correspondingreconstructed invariant mass peak. Our main result will be that the invisible momentaprovided by M -type variables generally lead to the most accurate mass measurements.In Section 5 we shall generalize our discussion to the case of BSM collider signalsexhibiting the t ¯ t event topology. In particular, we shall explore the general mass parameterspace of the three particles in each decay chain, and analyze the performance of the invisiblemomentum reconstruction from M -type variables as a function of parameter space. Indoing so, we shall identify the parameter space regions where the accuracy is degraded, andthen propose a solution for recovering sensitivity by applying a preselection cut. The sameidea has already been used successfully in the case of MAOS [66] and here we demonstrateits validity in a more general context. Sec. 6 is reserved for our conclusions. In this paper we shall largely follow the notation and terminology of Ref. [44], which webriefly review here for the reader’s convenience.
We focus on the generic event topology which is schematically depicted in Fig. 1. Weassume the pair production of two heavy particles, A and A , whose subsequent decaychains consist of two two-body decays: pp → A A , A i → a i B i , B i → b i C i , ( i = 1 , . (2.1)– 5 – B C A B C a b a b ( a ) ( b )( ab ) Figure 1 . The decay topology under consideration in this paper with the corresponding threesubsystems explicitly delineated. Each parent particle, A i , ( i = 1 ,
2) decays to two visible particles, a i and b i , and an invisible daughter particle, C i , through an intermediate on-shell resonance, B i .The blue dotted, green dot-dashed, and black solid lines indicate the subsystems ( a ), ( b ), and ( ab ),respectively. Here the particles a i and b i are SM particles which are visible in the detector, so thattheir 4-momenta p µa , p µb , p µa , and p µb are measured known quantities. In contrast, theparticles C i are invisible in the detector — they can be dark matter candidates or SMneutrinos — and their 4-momenta, q µi , are a priori unknown, being constrained only bythe /(cid:126)P T measurement (1.1) and our conjectured values ˜ m C and ˜ m C for their masses: q i = ˜ m C i , ( i = 1 , . (2.2)As usual, all visible particles are assumed massless (this is done merely for simplicity). Themasses of the intermediate resonances in Fig. 1 are denoted by m A i and m B i , with m A i >m B i . The process (2.1) depicted in Fig. 1 covers a large class of interesting and motivatedscenarios, including dilepton t ¯ t events in the SM, stop pair production in supersymmetrywith ˜ t → b ˜ χ + , followed by χ + → (cid:96) + ˜ ν (cid:96) , gluino pair production in supersymmetry with˜ g → ¯ q ˜ q , followed by ˜ q → q ˜ χ , and many more. As first discussed in the context of the M T variable [73], within the original event one canconsider several useful subsystems which are delineated by the colored rectangles in Fig. 1.Each subsystem is defined by a choice of parent particles and a choice of daughter particlesamong the set of three particles { A i , B i , C i } . Since the parents must be heavier than thedaughters, there are only three possibilities, and in each case, the remaining third type ofparticles will be referred to as relatives . Following the notation of [44], we shall label eachsubsystem by the set of visible particles on each decay side which are used to construct thekinematic variable: • The ( ab ) subsystem. This system refers to the event as a whole and is indicated bythe solid black box in Fig. 1. Here the A i ’s are the two parent particles and the– 6 – i ’s are the daughter particles, leaving the intermediate resonances B i as the relativeparticles. The visible particles on each side, a i and b i , are combined into a compositevisible particle with 4-momentum p µa i + p µb i . • The ( b ) subsystem. This subsystem is outlined by the green dot-dashed box in Fig. 1.Now the parents are the B i particles, the daughters are the C i particles, and therelatives are the A i particles. The kinematic variables for this subsystem will bedefined in terms of the 4-momenta p µb i of the visible particles b i . • The ( a ) subsystem. This subsystem is depicted by the blue dotted box in Fig. 1.The A i particles are again treated as parents, but the daughters are now the B i particles, while the relatives are the C i particles. The kinematic variables will usethe 4-momenta p µa i of the visible particles a i . We are now in position to define the different kinematic variables of interest, for each of thethree subsystems: ( ab ), ( b ) and ( a ). For each variable (1.2), we first identify a target func-tion, which is then minimized over all possible values of the individual invisible momenta q µi consistent with the missing transverse momentum condition (1.1). This minimizationwill yield the required ansatz for the missing momenta (1.3). In Section 3.1 we begin ourdiscussion with (2+1)-dimensional target functions defined on the transverse plane, wherethe minimization fixes only the transverse components (cid:126)q iT of the invisible momenta. Onethen needs to impose an additional requirement in order to obtain a suitable value for thelongitudinal components, and we shall review the different options discussed in the liter-ature. Then in Section 3.2 we shall proceed to discuss (3+1)-dimensional invariant masstarget functions, where the minimization results in fully specified invisible momenta q µi .In preparation for the numerical comparisons to follow in the next two sections, we shallagain review the different possibilities arising from applying various on-shell constraints onthe parent and/or relative particles. At hadron colliders, where the longitudinal momentum of the initial state is a prioriunknown, transverse variables are attractive since they are invariant under longitudinalboosts. When targeting an event topology with two separate decay chains like that ofFig. 1, one should consider the two parent particles P i and their corresponding decay prod-ucts { a i , b i , C i } . In order to obtain a useful generalization of the canonical transverse massvariable for this case, one follows the prescription behind the Cambridge M T variable [33]— first form the individual transverse masses M T P i of the two parents, then choose thelarger of the two, max ( M T P , M T P ), as our target function, and minimize it with respectto the transverse components of the momenta of the daughter particles, subject to the /(cid:126)P T constraint (1.1). We obtain three different versions of the M T variable, depending onthe subsystem under consideration [73]. For subsystem ( ab ), the parents are A i and the– 7 –aughters are C i , thus M T ( ab ) ≡ min (cid:126)q T ,(cid:126)q T { max [ M T A ( (cid:126)q T , ˜ m C ) , M T A ( (cid:126)q T , ˜ m C )] } . (3.1) (cid:126)q T + (cid:126)q T = /(cid:126)P T In subsystem ( b ), the parents are the B i particles, and one gets M T ( b ) ≡ min (cid:126)q T ,(cid:126)q T { max [ M T B ( (cid:126)q T , ˜ m C ) , M T B ( (cid:126)q T , ˜ m C )] } . (3.2) (cid:126)q T + (cid:126)q T = /(cid:126)P T The case of subsystem ( a ) is somewhat more complicated since the daughters are the B i particles and the minimization is performed in terms of their momenta as opposed to themomenta of the C i particles. If we introduce the 4-momenta of the B i particles, Q i ≡ q i + p b i , (3.3)we can define M T ( a ) ≡ min (cid:126)Q T , (cid:126)Q T (cid:110) max (cid:104) M T A ( (cid:126)Q T , ˜ m B ) , M T A ( (cid:126)Q T , ˜ m B ) (cid:105)(cid:111) , (3.4) (cid:126)Q T + (cid:126)Q T = /(cid:126)P T + (cid:126)p b T + (cid:126)p b T where instead of (2.2) we have Q i = ˜ m B i , ( i = 1 , . (3.5)The three minimizations in (3.1), (3.2) and (3.4) in principle provide three independentansatze for the transverse momenta (cid:126)q iT of the C i particles , as shown in Table 1. As forthe longitudinal components q iz , one has to impose additional constraints and compute q iz independently. There are several different options: • MAOS1: use the known mass of a parent particle.
This is the idea of the originalMAOS method [66]. If we imagine that the mass of the parent particle is alreadyknown from a prior measurement , we can enforce two mass shell conditions (one In what follows, to simplify the notation we shall not indicate explicitly the parent mass dependenceon the visible momenta p a i and p b i , which should be clear from the chosen subsystem. Strictly speaking, in the case of subsystem (a), we initially obtain an ansatz for the transverse momenta (cid:126)Q iT of the intermediate particles B i , but they can be easily related to (cid:126)q iT with the help of eq. (3.3). It is important to distinguish the two different situations in which we can use such information aboutthe parent mass. First, the parents can be SM particles, which decay semi-invisibly, e.g., top quarks, W -bosons or tau leptons. In this case the parent mass is known exactly. Second, the parents can beBSM particles, whose masses are a priori unknown, but some partial information can be obtained from thestandard kinematic endpoint measurements, which typically establish a relationship between the mass ofthe daughter and the mass of the parent. In this case, the left-hand sides of eqs. (3.6-3.8) should be thoughtof as functions of the test mass of the daughter particle, ˜ m C i or ˜ m B i , depending on the subsystem. In otherwords, in practical applications of the MAOS method to BSM analyses, one first introduces a value for thedaughter test mass, after which the parent mass can be computed from a kinematic endpoint measurementand substituted in (3.6-3.8). – 8 –nsatz for the invisible momentaMethod required inputs longitudinal No of transversecomponents solutions componentsMAOS1(ab) m A , m C eq. (3.6) up to 4 M T ( ab )MAOS2(ab) m C eq. (3.9) up to 2MAOS3(ab) m C eq. (3.12) uniqueMAOS4(ab) m B , m C eq. (3.15) up to 4MAOS1(b) m B , m C eq. (3.7) up to 4 M T ( b )MAOS2(b) m C eq. (3.10) uniqueMAOS3(b) eq. (3.13)MAOS4(b) m A , m C eq. (3.16) up to 4MAOS1(a) m A , m B eq. (3.8) up to 4 M T ( a )MAOS2(a) m B eq. (3.11) uniqueMAOS3(a) eq. (3.14)MAOS4(a) m C , m B eq. (3.17) up to 4 Table 1 . A summary of the different possible MAOS schemes. The transverse invisible momentaare fixed by the M T calculation in one of the three possible subsystems ( ab ), ( a ), and ( b ), whilethe longitudinal invisible momenta can be computed from any one of the four conditions MAOS1,MAOS2, MAOS3 and MAOS4 described in the text. The second column lists the required massinputs for each case. for each parent) in order to determine the longitudinal momentum of the respectiveinvisible particle. Depending on the subsystem under considerations, the MAOS1constraint reads Subsystem (ab) : m A i = ( p a i + p b i + q i ) , (3.6)Subsystem (b) : m B i = ( p b i + q i ) , (3.7)Subsystem (a) : m A i = ( p a i + Q i ) . (3.8)The first two relations will provide an ansatz directly for q iz , while the last one canbe solved for Q iz , after which q iz will be obtained from (3.3). In all cases, we haveto deal with a quadratic equation for each decay chain, thus we may end up with upto four valid solutions, as indicated in Table 1. • MAOS2: use the value of M T calculated in the event. The main disadvantage ofthe original MAOS1 scheme is that one needs precise prior knowledge of the mass ofthe parent particle, which may not be available immediately. In order to circumventthis difficulty, an alternative proposal, which does not require the parent mass as aninput, was suggested in Refs. [67–70]. The idea is to use the numerical value of theevent-wise M T value in place of the parent mass. Depending on the subsystem, we– 9 –ave: Subsystem (ab) : M T ( ab ) = ( p a i + p b i + q i ) , (3.9)Subsystem (b) : M T ( b ) = ( p b i + q i ) , (3.10)Subsystem (a) : M T ( a ) = ( p a i + Q i ) . (3.11)At first glance, these relations may look weird, since the left-hand side is a transversequantity, while the right-hand side is a genuine (3+1)-dimensional invariant mass.This observation is the key to understanding the physical meaning of the ansatz: theinvisible momentum is chosen so that its rapidity is the same as the rapidity of theagglomerated visible decay products, which allows a longitudinal boost to a framewhere the momenta are purely transverse, and the transverse mass becomes the sameas the mass [54]. • MAOS3: use the individual parent transverse masses obtained in the M T calculation. One remaining disadvantage of the MAOS2 method is that the obtained solutionfor the longitudinal momenta may not be unique. This occurs for the so-called“unbalanced” events, where the minimum of the target function is at a point wherethe transverse masses of the two parents are not equal [44]. This motivated anotherchoice, where one makes use of the individual parent transverse masses in each branch[69], namely Subsystem (ab) : M T A i = ( p a i + p b i + q i ) , (3.12)Subsystem (b) : M T B i = ( p b i + q i ) , (3.13)Subsystem (a) : M T A i = ( p a i + Q i ) . (3.14)With this prescription, the obtained values for the longitudinal momenta are unique.As shown in Table 1, the distinction between MAOS2 and MAOS3 only arises in thecase of subsystem ( ab ), since subsystems ( a ) and ( b ) always lead to balanced events,for which MAOS2 and MAOS3 are identical procedures. • MAOS4: use the known fixed mass of a relative particle.
This method is similar inspirit to MAOS1, only this time we use as an input the mass of a relative particle.In analogy to (3.6-3.8), we getSubsystem (ab) : m B i = ( p b i + q i ) , (3.15)Subsystem (b) : m A i = ( p a i + p b i + q i ) , (3.16)Subsystem (a) : m C i = ( Q i − p b i ) . (3.17)As shown in Table 1, three different versions of MAOS4 are possible in dilepton t ¯ t events. For example, MAOS4(ab) requires that the lepton and the neutrino on eachside of the event reconstruct to the true W -boson mass, which makes it suitable forstudying the reconstructed top quark mass. On the other hand, in MAOS4(b) onedemands that the two top quarks have nominal masses, in which case the interestingvariable to study would be the reconstructed W -boson mass.– 10 –n principle, all twelve MAOS methods listed in Table 1 are valid procedures for ob-taining the invisible momenta and they will all be illustrated in Section 4.1 below. Tothe best of our knowledge, only some of the options in Table 1 have been used in theliterature so far. The original proposal [66] focused on MAOS1(ab), while MAOS2(ab)and MAOS3(ab) were introduced later in [67–70]. Ref. [71] made use of MAOS1(ab) andMAOS4(ab) to tackle the two-fold combinatorial ambiguity in dilepton t ¯ t events [74, 75].The possibility to use different subsystems for MAOS reconstruction was pointed out inRef. [72], which performed a comparison of MAOS1(ab), MAOS1(a) and MAOS1(b) usingdilepton t ¯ t events and concluded that the best ansatz for the momenta of the invisible par-ticles is provided by MAOS1(ab), followed by MAOS1(b) and finally, MAOS1(a). Quiterecently, the MAOS1(b) version was used by the CMS collaboration to measure the topmass in dilepton t ¯ t events [76]. Following [43, 44, 54], one could also consider target functions in (3+1)-dimensions. Start-ing with the actual parent masses, M P i , we can schematically define the (3+1)-dimensionalanalogues of (3.1), (3.2) and (3.4) as M ( ˜ m ) ≡ min (cid:126)q ,(cid:126)q { max [ M P ( (cid:126)q , ˜ m ) , M P ( (cid:126)q , ˜ m )] } , (3.18) (cid:126)q T + (cid:126)q T = /(cid:126)P T where ˜ m is the daughter test mass for the corresponding subsystem and the minimizationis performed over all 3-components of the vectors (cid:126)q and (cid:126)q . If (3.18) is left as is, we willobtain nothing new — the result of the minimization will be equal to the correspondingvalue of M T [38, 44, 54], and furthermore, we will derive the same invisible momentaas with the MAOS2 method. This motivates us to modify the naive definition (3.18)appropriately, by taking into account the specific features of the event topology of Fig. 1[44]. For example, in many BSM realizations of Fig. 1, the two decay chains are symmetricin the sense that the original parent particles A i are identical (or at worst a particle-antiparticle pair) and decay in the same fashion. As a result, the corresponding masses onthe two sides of the event are the same: m A = m A ≡ m A , (3.19) m B = m B ≡ m B , (3.20) m C = m C ≡ m C , (3.21)and we can incorporate some number of these constraints into the definition of the kinematicvariable. Note that the first equal sign in eqs. (3.19-3.21) refers to the symmetry of theevent topology, while the second additionally implies knowledge of the actual value of the Recall that in the case of subsystem (a) we are actually using the momenta Q i which are related to q i by eq. (3.3). As in the case of MAOS, for BSM applications the parent mass may only be known as a function ofthe test daughter mass, as the latter is always a necessary input to the analysis. – 11 –ass, m A , m B or m C . Due to the freedom of choosing different sets among the constraints(3.19-3.21), several classes of variables are possible. • Equality of the two parent masses.
In the absence of any knowledge of the actualmasses of the parent particles, the best one can do is to apply the constraint ofidentical parents M P = M P . (3.22)Following the notation of [44], variables for which this condition is enforced, will carrya first index C for “constrained”. • Equality of the two relative masses . In analogy to (3.22), we can demand that thetwo relative particles in each decay chain are the same: M R = M R . (3.23)Following the notation of [44], variables for which this condition is enforced, will carrya second index C . • Fixed mass for the two relatives.
An even stronger constraint arises if we enforce therelative mass to be equal to some fixed value M R (compare to the MAOS4 methodintroduced above in Section 3.1): M R = M R ≡ M R . (3.24)Further expanding upon the notation of [44], variables for which this condition isenforced, will carry a second index R indicating the relative particle whose mass isknown. For example, in the special case of the event topology of Fig. 1 applied todilepton t ¯ t events, the index R can take the values R = t in subsystem (b), R = W in subsystem (ab), and R = ν in subsystem (a).In summary, the M class of variables will be labelled by two subscripts. The firstrefers to the parent hypothesis and takes a value C if (3.22) is applied, and X otherwise.The second subscript refers to the relative hypothesis and takes a value C if (3.23) isapplied, a value R if (3.24) is applied, and X otherwise. Altogether, we have six possiblevariables: M XX , M XC , M XR , M CX , M CC , and M CR .In Table 2 we collect the full set of 6 × M . The tableis organized as follows. We group the variables by subsystem — first (ab), then (b),and finally, subsystem (a). Within each subsystem, we order the variables according tothe amount of theoretical input — variables with fewer (more) constraints appear earlier For simplicity, in this paper we shall always assume the masses of the two daughter particles in agiven subsystem to be the same, otherwise we would need a third index for the daughter particles. Thisassumption is done only for simplicity and can be easily relaxed, see, e.g., [36, 37]. Additional variables can be obtained if we make further assumptions about the event topology. Forexample, if we assume an “antler” topology, where the two parents A i arise from the decay of a heavyresonance G with a known mass m G , one can further impose the constraint ( (cid:80) i ( p a i + p b i + q i )) = m G [60, 65]. – 12 –ubsystem Mass Applied constraints forVariable type inputs parents relatives M XX ( ab ) (ab) m C — — M XC ( ab ) (ab) m C — m B = m B M CX ( ab ) (ab) m C m A = m A — M CC ( ab ) (ab) m C m A = m A m B = m B M XR ( ab ) (ab) m B , m C — m B = m B = m B M CR ( ab ) (ab) m B , m C m A = m A m B = m B = m B M XX ( b ) (b) m C — — M XC ( b ) (b) m C — m A = m A M CX ( b ) (b) m C m B = m B — M CC ( b ) (b) m C m B = m B m A = m A M XR ( b ) (b) m A , m C — m A = m A = m A M CR ( b ) (b) m A , m C m B = m B m A = m A = m A M XX ( a ) (a) m B — — M XC ( a ) (a) m B — m C = m C M CX ( a ) (a) m B m A = m A — M CC ( a ) (a) m B m A = m A m C = m C M XR ( a ) (a) m B , m C — m C = m C = m C M CR ( a ) (a) m B , m C m A = m A m C = m C = m C Table 2 . A summary of the 6 × M defined in the text. For each of thethree subsystems ( ab ), ( a ), and ( b ), one may choose to apply (or not) the parent constraint (3.22),and then choose to apply (or not) one of the relative constraints (3.23) or (3.24). (later) in the list. As indicated by the entries in the third column of Table 2, four ofthe variables within each subsystem require a single input mass parameter, namely thehypothesized mass of the daughter particle for this subsystem. These 12 variables, oftype M XX , M CX , M XC , and M CC , are precisely the on-shell constrained M variablesdiscussed in [44]. The remaining 6 variables in Table 2 require an additional mass input— the mass of the relative particle. In this sense, they are the analogues of the MAOS1 orMAOS4 schemes for invisible momentum reconstruction, which also required an additionalmass input, see Table 1.The pros and cons of the different types of M variables from Table 2 will be discussedin our numerical examples below (see Section 4.2). The exact definition for each variableshould be clear from our earlier discussion (see also [44]), but at this point it may still beinstructive to give a few specific examples, particularly for the newly introduced variables M XR and M CR which employ the stricter constraint (3.24).For concreteness, let us consider the dilepton t ¯ t realization of the event topology ofFig. 1, in which the visible particles are: a pair of b-quarks ( a = b , a = ¯ b ) and a pairof leptons ( b = (cid:96) + , b = (cid:96) − ). One could imagine that the leptons are still the result ofleptonic decays of SM W -bosons to neutrinos, so that m B i = m W and m C i = 0, while the– 13 –arents A i are some new particles, e.g., 4th generation up-type quarks. Then, the physicsprocess under consideration (2.1) becomes pp → t (cid:48) ¯ t (cid:48) , t (cid:48) → bW + , W + → (cid:96) + ν (cid:96) . (3.25)In this case, it makes sense to consider the variable M CW ( b(cid:96) ) defined as M CW ( b(cid:96) ) ≡ min (cid:126)q ,(cid:126)q (cid:8) max (cid:2) ( p b + p (cid:96) + + q ) , ( p ¯ b + p (cid:96) − + q ) (cid:3)(cid:9) , (3.26) q = 0 q = 0 (cid:126)q T + (cid:126)q T = /(cid:126)P T ( p (cid:96) + + q ) = m W ( p (cid:96) − + q ) = m W ( p b + p (cid:96) + + q ) = ( p ¯ b + p (cid:96) − + q ) whose upper kinematic endpoint would be the mass of the top partner t (cid:48) .Another possibility is to consider stop production in SUSY, followed by sequentialdecays to charginos and sneutrinos: pp → ˜ t ˜ t ∗ , ˜ t → b ˜ χ + , ˜ χ + → (cid:96) + ˜ ν (cid:96) . (3.27)In this case, a prior measurement of the M T ( (cid:96) ) kinematic endpoint could provide knowl-edge of the chargino mass as a function of the sneutrino mass, m ˜ χ ± ( m ˜ ν (cid:96) ), which wouldallow us to consider the maximally constrained kinematic variable M C ˜ χ ± ( b(cid:96) ) defined as M C ˜ χ ± ( b(cid:96) ) ≡ min (cid:126)q ,(cid:126)q (cid:8) max (cid:2) ( p b + p (cid:96) + + q ) , ( p ¯ b + p (cid:96) − + q ) (cid:3)(cid:9) , (3.28) q = m ν (cid:96) q = m ν (cid:96) (cid:126)q T + (cid:126)q T = /(cid:126)P T ( p (cid:96) + + q ) = m χ ± ( m ˜ ν (cid:96) )( p (cid:96) − + q ) = m χ ± ( m ˜ ν (cid:96) )( p b + p (cid:96) + + q ) = ( p ¯ b + p (cid:96) − + q ) The minimizations in (3.26) and (3.28) are essentially one-dimensional minimizations, sincethey involve a total of seven constraints for the eight unknown components q µ and q µ .One could also consider situations where the masses for the A i particles are knowninstead. If we stick to the case where A i is the SM top quark, we can imagine that theparticles B i are not W bosons, but some other charged scalars H ± . Then the process underconsideration becomes pp → t ¯ t, t → bH + , H + → (cid:96) + ν (cid:96) . (3.29)– 14 –he relevant variable now is M Ct ( (cid:96) ) ≡ min (cid:126)q ,(cid:126)q (cid:8) max (cid:2) ( p (cid:96) + + q ) , ( p (cid:96) − + q ) (cid:3)(cid:9) , (3.30) q = 0 q = 0 (cid:126)q T + (cid:126)q T = /(cid:126)P T ( p (cid:96) + + q ) = ( p (cid:96) − + q ) ( p b + p (cid:96) + + q ) = m t ( p ¯ b + p (cid:96) − + q ) = m t whose upper kinematic endpoint is the mass of the charged boson H ± . Note that the first M subscript “ C ” in (3.30) refers to the presence of the parent mass constraint (3.22) forthe H ± particles in the leptonic subsystem, while the second subscript “ t ” identifies therelative particles A i as top quarks.In conclusion of this section, we also mention the possibility to define a class of vari-ables, M , where one minimizes a target mass function without any partitioning of theevent [54]. If this minimization is performed in the absence of any additional kinematicconstraints besides (1.1), one obtains the usual √ s min variable [27, 58]. In the example ofthe t ¯ t event topology we have s min ( b(cid:96) ) ≡ M XX ( b(cid:96) ) ≡ min (cid:126)q ,(cid:126)q (cid:8) ( p b + p (cid:96) + + q + p ¯ b + p (cid:96) − + q ) (cid:9) , (3.31) q = 0 q = 0 (cid:126)q T + (cid:126)q T = /(cid:126)P T where we have assumed zero test masses for the two invisible particles. However, onemay also choose to partition the event post factum in order to define a suitable kinematicconstraint of the type (3.22), (3.23) or (3.24). Consider, for example, the single productionof a heavy Higgs boson, H , subsequently decaying to two on-shell W -bosons, which inturn decay leptonically: pp → H , H → W + W − , W + → (cid:96) + ν (cid:96) , W − → (cid:96) − ¯ ν (cid:96) . (3.32)The relevant variable to consider in this case would be M W ( (cid:96) ) ≡ min (cid:126)q ,(cid:126)q (cid:8) ( p (cid:96) + + q + p (cid:96) − + q ) (cid:9) , (3.33) q = 0 q = 0 (cid:126)q T + (cid:126)q T = /(cid:126)P T ( p (cid:96) + + q ) = m W ( p (cid:96) − + q ) = m W – 15 –hich was called m boundT in [62] and ˆ s consmin in [59]. In all those cases, the minimization againresults in an ansatz for the invisible 3-momenta (cid:126)q and (cid:126)q , so that the M class of variablescan in principle also be used for fixing the momenta of the invisible particles. M T -assisted and M -assisted mass reconstructions of mass peaks In the previous section, we identified a number of different ways in which one can obtain anansatz for the unknown momenta of the invisible particles in the event. The main purposeof this section is to compare the usefulness of these different ansatze with regards to massmeasurements through bump hunting. To be specific, we shall focus on the dilepton t ¯ t event topology from Fig. 1 and we shall consider the three subsystems, ( ab ), ( a ) and ( b ).In subsection 4.1 we shall first discuss the twelve versions of the traditional MAOS methodwhich are listed in Table 1, while in subsection 4.2 we shall compare the different typesof M -based reconstructions from Table 2. Depending on the procedure, one expects toobtain an invariant mass bump for one of the three particles involved — the top quark, the W -boson or the neutrino, as the case may be. The sensitivity of the mass measurementwill be judged by the width of the obtained invariant mass distribution — a narrow (broad)peak will indicate high (reduced) sensitivity. Finally, in subsection 4.3 we shall contrastthe MAOS methods from Sec. 4.1 to the M -based methods from Sec. 4.2. First we compare the performance of the twelve different MAOS schemes introduced inSec. 3.1. Generally, we will be reconstructing the mass of the relative particle — the W boson mass ˜ M W in subsystem ( ab ), the top quark mass ˜ M t in subsystem ( b ) and theneutrino mass ˜ M ν in subsystem ( a ). However, in the case of MAOS4, the result wouldbe trivial since the mass of the relative particle itself is used as one of the constraints.This is why in the case of MAOS4 only we shall instead plot the mass of the parentparticle, i.e., ˜ M W for MAOS4(b) and ˜ M t for MAOS4(ab) and MAOS4(a). Our results arepresented in Figs. 2-4, where events were generated with Madgraph [77] for the LHCwith energy 14 TeV. Since it is difficult to distinguish a b -jet from a ¯ b -jet in practice, thereis a two-fold combinatorial ambiguity which may occur at different stages — in formingthe M T variable, in using the top mass to solve for q iz , or in forming ˜ M t . Either way,this combinatorial ambiguity inevitably affects the results, which is why in the figures weshow separately results for the correct lepton-jet pairing (left panels), the wrong lepton-jetpairing (middle panels) and combining both pairings (right panels).Fig. 2 shows results from reconstructing the top quark mass ˜ M t with the five rele-vant MAOS methods: MAOS1(b) (red solid lines), MAOS2(b) (green dot-dashed lines),MAOS3(b) (blue dotted lines), MAOS4(ab) (orange dashed lines), and MAOS4(a) (cyansolid lines). In all cases, we use the correct test mass when calculating M T : the trueneutrino mass m ν = 0 in subsystems (ab) and (b), and the true W -boson mass m W = 80GeV for subsystem (a). In the case of MAOS2(b) and MAOS3(b), this is the only massinput needed to reconstruct ˜ M t , see Table 1. Unfortunately, this theoretical advantageseems to be offset by the inferior performance of these two methods: even for the correct– 16 – igure 2 . Comparison of the performance of different MAOS schemes in reconstructing the topmass in dilepton t ¯ t events. Distributions of the reconstructed top mass ˜ M t are shown for the caseof the correct lepton-jet pairing (left panel), the wrong lepton-jet pairing (middle panel) and bothpairings (right panel). The top quark is treated as a relative particle in the case of MAOS1(b) (redsolid line), MAOS2(b) (green dot-dashed line) and MAOS3(b) (blue dotted line), and as a parentparticle in the case of MAOS4(ab) (orange dashed line) and MAOS4(a) (cyan solid line). lepton-jet combination, the MAOS2(b) and MAOS3(b) distributions in the left panel inFig. 2 peak below the true top mass m t , so that a bump hunt will systematically under-estimate the value of m t . The remaining three MAOS methods illustrated in the figure,MAOS1(b), MAOS4(ab) and MAOS4(a), use an additional mass input, and are thus ex-pected to perform better. This is confirmed by Fig. 2, which suggests that MAOS4(ab)slightly outperforms the other other two methods, MAOS4(a) and MAOS1(b), which areutilizing the smaller individual subsystems (a) and (b). There are two effects which con-tribute to this. First, for the correct lepton-jet combination (the left panel in Fig. 2) thedistributions for all three methods, MAOS1(b), MAOS4(ab) and MAOS4(a), have theirpeaks very close to the true mass m t , but the peak for MAOS4(ab) is more narrow thanthe other two. Second, for the wrong lepton-jet combination (the middle panel in Fig. 2),the MAOS4(ab) distribution is relatively broad, but happens to peak right around the topquark mass again, while the distributions for MAOS4(a) and MAOS1(b) peak at slightlylower values. If one does not attempt to resolve the combinatorics [71, 74, 75] and insteaddoes the simplest thing, namely, combine the two distributions from the left and middlepanels of Fig. 2, one would obtain the combined distributions shown in the right panelof Fig. 2. We see that among the methods using two mass inputs, MAOS4(ab) appearsto be the best, followed by MAOS4(a) and MAOS1(b). The remaining two procedures,MAOS2(b) and MAOS3(b), rely on a single mass input, and give identical answers, inaccordance with our expectations for subsystem ( b ).Fig. 3 shows the analogous results for the reconstruction of the mass ˜ M W of the W -boson, using MAOS1(ab) (red solid line), MAOS2(ab) (green dot-dashed line), MAOS3(ab) In MAOS1 and MAOS4, the additional mass input is used to solve for the longitudinal momenta. Sincethe relevant equations are non-linear, one may end up with multiple solutions. In such cases, we plotthe result for each solution with a corresponding weight factor so that each event has weight 1. Similarcomments apply to the case of MAOS2, where for unbalanced events one may find two solutions for q iz . – 17 – igure 3 . Comparison of the performance of different MAOS schemes in reconstructing the W -boson mass in dilepton t ¯ t events. Distributions of the reconstructed W -boson mass ˜ M W are shownfor the case of the correct lepton-jet pairing (left panel), the wrong lepton-jet pairing (middlepanel) and both pairings (right panel). The W -boson is treated as a relative particle in the caseof MAOS1(ab) (red solid line), MAOS2(ab) (green dot-dashed line) and MAOS3(ab) (blue dottedline), and as a parent particle in the case of MAOS4(b) (orange dashed line). (blue dotted line), and MAOS4(b) (orange dashed line). In all cases we use the correct testmass as an input to the M T calculation, and then the correct value of the additional massinput required for MAOS1(ab) and MAOS4(b). The left panel of Fig. 3 clearly demon-strates the benefit of the additional mass input, as MAOS1(ab) and MAOS4(b) greatlyoutperform MAOS2(ab) and MAOS3(ab). Since the corresponding wrong-combinationdistributions in the middle panel have similar shapes, this advantage is preserved in thecombined distributions shown in the right panel. Upon closer inspection, MAOS4(b) (or-ange dashed line) appears slightly better than MAOS1(ab) (red solid line). However, in newphysics applications of the MAOS methods, the knowledge of the additional mass input isnot always guaranteed, and one would have to do with MAOS2(ab) or MAOS3(ab), whichperform very similarly. Among the two, MAOS3(ab) has a slight theoretical advantage inthe sense that its invisible momentum ansatz is always unique and well-defined.Our third and final mass reconstruction for the dilepton t ¯ t topology is shown in Fig. 4,where we plot in analogous fashion the reconstructed neutrino mass-squared ˜ M ν whenthe neutrino is treated as a relative particle in subsystem (a): MAOS1(a) (red solid line),MAOS2(a) (green dot-dashed line) and MAOS3(a) (blue dotted line). This time the benefitof the additional mass input m t in the case of MAOS1(a) is not so clear — all threedistributions have similar shapes (the distributions for MAOS2(a) and MAOS3(a) are infact identical, since subsystem ( a ) has only balanced events) and peak near the origin. M -based methods We shall now use the dilepton t ¯ t example to test the accuracy of the invisible momentumreconstruction from the different M -based methods listed in Table 2. We shall not considerall 18 possibilities in Table 2, since some are closely related. For example, it is known thatfor any subsystem, the M XX and M CX variables are identical, and furthermore, equal to– 18 – igure 4 . Comparison of the performance of different MAOS schemes in reconstructing theneutrino mass-squared in dilepton t ¯ t events. Distributions of the reconstructed neutrino mass-squared ˜ M ν are shown for the case of the correct lepton-jet pairing (left panel), the wrong lepton-jetpairing (middle panel) and both pairings (right panel). Here the neutrino is always treated as arelative particle in the case of MAOS1(a) (red solid line), MAOS2(a) (green dot-dashed line) andMAOS3(a) (blue dotted line). the value of the Cambridge transverse mass variable M T [44]: M XX = M CX = M T . (4.1)In spite of this relation, the corresponding three ansatze for the invisible momenta arenot necessarily the same. First of all, M T is a transverse variable and it only fixes thetransverse components (cid:126)q T and (cid:126)q T , while M XX and M CX in addition provide values forthe longitudinal components q z and q z . In the case of balanced events, those predictionsare unique and the same for M XX and M CX , while for unbalanced events, there is atwo-fold ambiguity for q z and q z in the case of M CX and a flat direction in the case of M XX [44]. In what follows, we shall therefore prefer to consider the invisible momentumreconstruction from the M CX variable instead of M XX .Similar considerations apply in the case of the pair of variables M XC and M CC , aswell as for M XR and M CR . In each case, the variables are equal for balanced eventsand only differ for unbalanced events, where this time the obtained invisible momentumconfigurations are unique. This is why we shall also not consider M XC and M XR , andinstead focus on M CC and M CR , respectively.Fig. 5 shows distributions of the reconstructed top quark mass ˜ M t with the four relevant M methods from Table 2: M CR ( ab ) (red solid line), M CX ( b ) (green dot-dashed line), M CC ( b ) (blue dotted line), and M CR ( a ) (orange dashed line). In analogy to Fig. 2, weshow separately the distributions obtained for the correct lepton-jet pairing (left panel), thewrong lepton-jet pairing (middle panel), and both pairings (right panel). Note that somedistributions have fewer events, since the constraints cannot be simultaneously satisfied.This is most notable for the case of M CR ( a ), and is typically due to events in which anintermediate resonance (a top quark or a W -boson) is rather off-shell (we expect this effectto be further amplified once we account for the finite detector resolution). Also note that– 19 – igure 5 . The same as Fig. 2, but using the appropriate M variables for fixing the invisiblemomenta. Distributions of ˜ M t are shown for the case of M CR ( ab ) (red solid line), M CX ( b ) (greendot-dashed line), M CC ( b ) (blue dotted line), and M CR ( a ) (orange dashed line). in subsystems ( ab ) and ( a ) the top quark is a parent particle, while in subsystem ( b ) it isa relative particle. This distinction is indicated in the legend of Fig. 5 with a superscript P or R , respectively.Fig. 5 confirms that the more constrained variables generally provide better guesses forthe invisible momenta, as measured by the location and width of the reconstructed masspeak in ˜ M t . The most constrained version of the M variable is M CR , which has one parentconstraint and two relative constraints, leaving a single momentum degree of freedom to beminimized over. In Fig. 5, both M CR ( ab ) and M CR ( a ) seem to work very well — for thecorrect lepton-jet pairing, the reconstructed top mass peak is very well defined and locatedat the correct position (marked with the vertical dashed line). However, the disadvantageof M CR ( ab ) and M CR ( a ) is that one uses both the W -boson mass and the neutrino massas inputs to the calculation, which restricts their applicability to BSM scenarios. Underthose circumstances, the single-input variables M CX ( b ) and M CC ( b ) will be more usefulfor momentum reconstruction — in Fig. 5 the corresponding ˜ M t distributions are shownwith the green dot-dashed and the blue dotted line, respectively. We see that even withthe lack of knowledge of the precise value of the W -boson mass, the M CC ( b ) variablestill provides a good momentum ansatz, as judged by the location of the peak of its ˜ M t distribution.In Figs. 6 and 7 we similarly show distributions of the reconstructed W -boson mass˜ M W and the reconstructed neutrino mass squared ˜ M ν , respectively. (These two figuresare the analogues of Figs. 3 and 4 for the MAOS case.) The ˜ M W distributions in Fig. 6use invisible momentum reconstruction from M CX ( ab ) (green dot-dashed line), M CC ( ab )(blue dotted line), and M CR ( b ) (red solid line), while the ˜ M ν distributions in Fig. 7 usethe invisible momenta obtained by M CX ( a ) (red solid line) and M CC ( a ) (blue dashedline). We again observe that the maximally constrained variable, M CR ( b ), which uses asinputs the neutrino and top quark masses, is able to provide us with a very good ansatzfor the invisible momenta, and the ˜ M W distribution in Fig. 6 exhibits a very narrow peakat the proper location (80 GeV). The remaining four distributions in Figs. 6 and 7 are– 20 – igure 6 . The same as Fig. 3, but using the appropriate M variables for fixing the invisiblemomenta. Distributions of ˜ M W are shown for the case of M CX ( ab ) (green dot-dashed line), M CC ( ab ) (blue dotted line), and M CR ( b ) (red solid line). Figure 7 . The same as Fig. 4, but using the appropriate M variables for fixing the invisiblemomenta. Distributions of ˜ M ν are shown for the case of M CX ( a ) (red solid line) and M CC ( a )(blue dashed line). derived from single-input variables, where we again observe that M CC performs slightlybetter than M CX .In the above discussion of Figs. 5 and 6 we have been focusing on measuring thetop quark mass and the W -boson mass from the peaks of the respective ˜ M t and ˜ M W distributions. However, one should keep in mind that whenever we reconstruct a parent mass, we always have the option of measuring it from a kinematic endpoint as well. Thisis clearly evident in the left panel of Fig. 5 for the case of M CR ( ab ) and M CR ( a ), andin the left panel of Fig. 6 for the case of M CR ( b ). Even with the pollution from thewrong combinatorics in the middle panels of Figs. 5 and 6, the endpoint structures are stillpreserved in the corresponding combined distributions shown in the right panels. M T -assisted and M -assisted reconstruction schemes Having discussed the different versions of the more traditional MAOS method in Sec. 4.1and the different options for M -assisted invisible momentum reconstruction in Sec. 4.2, we– 21 – igure 8 . Comparison of the MAOS and M -assisted methods for top mass reconstruction fromFigs. 2 and 5. The left panel shows distributions of the reconstructed top mass ˜ M t with methodswhich use two mass inputs (the W -boson mass and the neutrino mass): the three MAOS methodsfrom Fig. 2, MAOS4(ab) (blue solid line), MAOS1(b) (green dot-dashed line), and MAOS4(a) (cyandotted line), and the two M -based methods from Fig. 5, M CR ( ab ) (red solid line) and M CR ( a )(orange dashed line). The right panel shows distributions of the reconstructed top mass ˜ M t withmethods which use a single mass input (the neutrino mass): MAOS2(b) (blue dotted line) andMAOS3(b) (green dot-dashed line) from Fig. 2 and M CX ( b ) (orange dashed line) and M CC ( b )(red solid line) from Fig. 5. are now ready to contrast the two methods to each other. For this purpose, we reassemblethe results from the previous two subsections in Figs. 8-10, so that only methods usingthe same number of theoretical mass inputs are compared on each plot: the distributionsshown on the left panels of these figures require two mass inputs, while the distributionsin the right panels need only one. Since we already showed the effects of combinatoricsin the previous two subsections (compare the left and middle panels of Figs. 2-7), herefor simplicity we plot only the combined distributions, which include both the correct andthe wrong lepton-jet assignment. Naturally, the use of the extra mass input should allowfor a better measurement, thus one should expect the distributions in the left panels ofFigs. 8-10 to be more sharply peaked than those in the corresponding right panels.Fig. 8 summarizes our previous results from Figs. 2 and 5 for the reconstruction of thetop mass ˜ M t . The distributions shown in the left panel require prior knowledge of boththe W -boson mass m W and the neutrino mass m ν , while for the distributions shown in theright panel one only needs to know m ν . The left panel of Fig. 8 demonstrates that the two M methods, M CR ( ab ) (the red solid line) and M CR ( a ) (the orange dashed line) clearlyoutperform their MAOS counterparts, MAOS4(ab) (blue solid line), MAOS1(b) (greendot-dashed line) and MAOS4(a) (cyan dotted line) — the peaks reconstructed by means of M are significantly more narrow, which should lead to a more precise mass measurement.Regardless of this width difference, in all five cases the peak of the distribution is correctlycentered on the true top mass used in the simulations (indicated by the vertical dashed– 22 – igure 9 . The same as Fig. 8, but for the reconstructed mass ˜ M W of the W -boson. The leftpanel shows distributions of the reconstructed W -boson mass ˜ M W with methods which use two massinputs (the top mass and the neutrino mass): MAOS1(ab) (green dot-dashed line) and MAOS4(b)(blue dotted line) from Fig. 3 and M CR ( b ) (red solid line) from Fig. 6. The right panel showsdistributions of the reconstructed W -boson mass ˜ M W with methods which use a single mass input(the neutrino mass): MAOS2(ab) (blue dotted line) and MAOS3(ab) (green dot-dashed line) fromFig. 3 and M CX ( ab ) (orange dashed line) and M CC ( ab ) (red solid line) from Fig. 6. line). The right panel of Fig. 8 leads to a very similar conclusion for the set of methodswhich rely on a single mass parameter input — here the method of M CC ( b ) (red solidline) is clearly the best, while the other three, MAOS2(b) (blue dotted line), MAOS3(b)(green dot-dashed line), and M CX ( b ) (orange dashed line), are in a perfect tie, whichis not a numerical coincidence, but rather expected theoretically. First, the proceduresof MAOS2 and MAOS3 differ only for unbalanced events, of which there are none insubsystem ( b ). Furthermore, it is known that the variables M T and M CX are identicalin any subsystem [44], see eq. (4.1). Therefore they would lead to the same invisiblemomentum reconstruction, which is indeed confirmed by the right panel in Fig. 8.Fig. 9 reassembles our previous results from Figs. 3 and 6 for the reconstructed W -boson mass ˜ M W . The left panel shows the distributions which need two mass inputs, thetop mass m t and the neutrino mass m ν . All three distributions peak at the correct valueof the W mass indicated with the vertical dashed line. However, the distribution obtainedwith M CR ( b ) (the red solid line) is slightly more narrow than the other two, correspondingto MAOS1(ab) (the green dot-dashed line) and MAOS4(b) (the blue dotted line). The rightpanel in Fig. 9 collects the distributions from Figs. 3 and 6 which require only the neutrinomass as an input. Here we notice that MAOS2(ab) (the blue dotted line) and MAOS3(ab)(the green dot-dashed line) give slightly different results, due to the presence of unbalancedevents in subsystem ( ab ). Once again, the theorem from [44] ensures that the distributionsfor MAOS2(ab) (the blue dotted line) and M CX ( ab ) (the orange dashed line) are the– 23 – igure 10 . The same as Fig. 8, but for the reconstructed mass-squared ˜ M ν of the neutrino. Theleft panel shows the distribution obtained with MAOS1(a) from Fig. 4, which uses two mass inputs:the mass of the top quark and the mass of the W -boson. The right panel shows distributions ofthe reconstructed neutrino mass squared ˜ M ν with methods which use a single mass input (the W -boson mass): MAOS2(a) (blue dotted line), and MAOS3(a) (green dot-dashed line) from Fig. 4and M CX ( a ) (orange dashed line) and M CC ( a ) (red solid line) from Fig. 7. same . Just like we saw in the right panel of Fig. 8, the distribution obtained from the M CC -type variable, in this case M CC ( ab ) (the red solid line), has the best properties: itspeak is relatively narrow and appears closest to the true W -boson mass.Finally, in Fig. 10 we revisit our results from Figs. 4 and 7 for the reconstructedneutrino mass-squared ˜ M ν . This time there is only a single method, MAOS1(a), whichuses two mass inputs, and correspondingly, it is depicted in the left panel. The remainingfour methods use a single input, the W -boson mass, and are shown in the right panel. Onceagain, in accordance with the theorem from [44], the distributions for M CX ( a ) (the orangedashed line) and MAOS2(a) (the blue dotted line) are identical. The lack of unbalancedevents in subsystem ( a ) implies that the distributions corresponding to MAOS2(a) andMAOS3(a) are also the same. The remaining fourth distribution, based on M CC ( a ) (thered solid line) is different, however, and appears to be the most promising for the purposes The careful reader might notice that in the right panel of Fig. 9, the blue dotted line for MAOS2(ab)and the orange dashed line for M CX ( ab ) are slightly different, in apparent violation of the theorem from[44]. The reason for this is somewhat technical and has to do with the different way in which we produce theplots for MAOS2(ab) and M CX ( ab ). We have verified that for balanced events, the results are identical, asexpected. However, for unbalanced events, the MAOS2(ab) prescription yields two possible values for thelongitudinal momenta, both of which are available to us as the solutions to a simple quadratic equation.Then, when we produce plots for MAOS2, we enter both solutions in the histogram, each with a weight1/2. These two solutions correspond to the two equally deep global minima of the target function usedto compute M CX ( ab ) [44]. Since the M CX ( ab ) minimization is done numerically via Optimass [47], itsnumerical algorithm will randomly pick and converge to one of these minima, giving us only one of the twosolutions, which we then plot with weight 1. – 24 –f a mass measurement of ˜ M ν .In conclusion of this section, let us summarize our main result. We contrasted theMAOS and M methods for invisible momentum reconstruction by examining their poten-tial for a mass measurement of an unknown particle through a bump hunt. We analyzedeach of the three subsystems in the event topology of Fig. 1 and found that the invisi-ble momentum reconstruction offered by the M class of variables is generally superior toMAOS — the reconstructed invariant mass peaks are more narrow and better localized.An additional theoretical advantage of the M approach is that it is less ambiguous, as italways provides a unique ansatz for balanced events. In the next section we shall continueto investigate the M approach in the most general case of the event topology from Fig. 1,where A , B and C are arbitrary new physics particles. Our discussion in the previous section was limited to the SM t ¯ t dilepton event topology. Thedilepton t ¯ t example is appealing to an experimentalist mainly because we know it is presentin the data and can be used as a toy playground for new physics searches [76, 78, 79]. Giventhat the ultimate goal of the LHC is to discover new physics and measure the new particlemass spectrum, in this section we shall abandon the t ¯ t example and instead consider themost general case of the event topology of Fig. 1, where the mass spectrum ( m A , m B , m C )is completely arbitrary, and not ( m t , m W , m ν ), as in the previous section. A concreterealization in SUSY is provided by the process (3.27) of stop production, in which themasses of the top squark, chargino and sneutrino are a priori unknown.The main goal of this section will be to revisit the bump hunting mass measurementtechnique discussed previously and investigate how well it does in the general mass pa-rameter space ( m A , m B , m C ), away from our previous “study point” ( m t , m W , m ν ). Sincethe exact nature of the new physics particles A , B and C is unknown, in the simulationsof this section we shall decay particles A and B by pure phase space. For concreteness,we shall continue to assume that particles A are colored fermions produced similarly totop quarks. For fairness in comparing the sensitivity at different points in mass parameterspace, it would be nice to fix the overall signal rate. An easy way to do this is to fix themass (and hence the cross-section) of the heaviest particle A . In what follows we shallchoose m A = 500 GeV; this has the additional benefit of reducing the dimensionality ofthe relevant mass parameter space to two. The masses of the two remaining particles willbe varied as m B ∈ (0 , m A ) and m C ∈ (0 , m B ). We shall then investigate the sensitivity ofthe method as a function of m B and m C .In Fig. 11 we revisit the main study point considered in [44], namely m A = 500GeV, m B = 300 GeV, m C = 200 GeV. In the left panels we show the distributions of M CC ( ab ) (upper row) and M CC ( b ) (lower row), both made with the correct choice of m C . We observe that both distributions peak very nicely at their kinematic endpoint,allowing a measurement of the corresponding mass ( m A for the case of M CC ( ab ) and m B For concreteness, all plots in this section will be made with the correct lepton-jet pairing, thus avoidingthe combinatorial issue. – 25 – igure 11 . An example of stop production (3.27) which works rather well. The mass spectrum is m A = 500 GeV, m B = 300 GeV, m C = 200 GeV. The top row shows the M CC ( ab ) distribution (leftpanel), the corresponding reconstruction of the relative particle B in the (ab) subsystem (middlepanel), and their correlation (right panel). The bottom row shows the M CC ( b ) distribution (leftpanel), the corresponding reconstruction of the relative particle A in the (b) subsystem (middlepanel), and their correlation (right panel). The dashed lines mark the true values of the particlemasses in each case. for the case of M CC ( b )) from either the peak of the distribution , or the location ofthe kinematic endpoint. For our purposes, however, we are mostly interested in using theinvisible momentum ansatz for reconstructing the mass of the relative particle, namely m B for the case of M CC ( ab ) and m A for the case of M CC ( b ). This reconstruction is shown inthe two middle panels of Fig. 11. We see that the reconstructed relative mass distributionshave very sharp, well-defined peaks positioned very close to the true values of the masses,which are indicated by the vertical dashed lines. We conclude that for the particular studypoint shown in Fig. 11, the invisible momentum reconstruction is quite successful and thebump hunt measurement is very promising. For future reference, the two right panels inFig. 11 then show the correlations between the two variables plotted in the left and middlepanels of each row.Fig. 12 presents the same results, but for a different study point, m A = 500 GeV, m B =450 GeV, and m C = 50 GeV. The mass spectrum was judiciously chosen so that the shapesof the relevant kinematic distributions are adversely affected. For example, as shown in the Note the importance of adding the relative constraint (3.23). Without it, the distribution of M CX ( b )does not peak at the kinematic endpoint, but at lower values [44]. – 26 – igure 12 . The same as Fig. 11, but for a study point which does not work as well: m A = 500GeV, m B = 450 GeV, m C = 50 GeV. upper left panel, the peak in the M CC ( ab ) distribution is now much broader, extendingsignificantly to the left (compare to the upper left panel in Fig. 11). Reconstructing themasses of the relative particles now appears to be a bit more problematic, as illustratedby the middle panels in Fig. 12 — in the upper row, the peak in the distribution of m B reconstructed with the invisible momenta from M CC ( ab ), is very asymmetric. The lowermiddle panel shows an even worse situation: the distribution of m C reconstructed withthe invisible momenta from M CC ( b ) appears flat from 300 GeV all the way to 500 GeV,making the corresponding mass determination quite uncertain.Fortunately, there exists a way to recover sensitivity. The basic idea can be understoodfrom the correlation plots in the right panels of Fig. 12. Notice that the most populatedbins are situated very close to the true values of the masses, m B = 450 GeV and m A = 500GeV. The problem arises because of the appearance of the tail extending towards lowervalues of the reconstructed relative mass M R , so that when we project this two-dimensionalplot on the y -axis, the obtained distribution is skewed towards lower values of M R as well.This basic observation suggests the two possible solutions to the problem. First, instead ofbump hunting on a one-dimensional histogram, one may target directly the most populatedbins in the two-dimensional correlation plots shown in the right panels of Fig. 12 (note thatthis method would have also worked on our previous example shown in Fig. 11). The useof such two-dimensional correlation plots was previously suggested in order to detect thekinematic boundaries of the available phase space [13, 80–84], while here we propose to usethem in order to find the location of the highest density.– 27 – igure 13 . The effect of a preselection cut on the reconstructed relative mass distributions shownin the middle panel plots from Figs. 11, 12 and 15. The preselection cut is applied on the variable M CC ( ab ) (upper row) or M CC ( b ) (lower row). The left, middle and right plots correspond to thestudy points from Fig. 11, Fig. 12 and Fig. 15, respectively. Events are selected in the top 5% (red),top 10% (orange), top 20% (green) and top 50% of the allowed range for the M CC variable. An alternative approach is based on the following observation. The right panels inFigs. 11 and 12 show that the correct value of the relative mass M R is obtained for eventswith extreme values of the M CC kinematic variable plotted on the x -axis. In other words,the ansatz for the invisible momenta tends to work best for events near the M CC kinematicendpoint (this was first pointed out in the context of MAOS reconstruction [66], andfollows from the general principle that kinematic endpoints are attained at very specialextreme momentum configurations [8, 52, 85]). Thus the precision of the one-dimensionalbump hunting method will be recovered, if we simply apply a preselection cut on M CC toeliminate the effect of the tail. Ideally, one would like to select only events which sit rightat the M CC kinematic endpoint, but such a severe cut may cause too large of a loss ofstatistics. This trade-off is illustrated in Fig. 13, which shows the effect of the preselectioncut on the reconstructed relative mass distributions shown in the middle panel plots fromFig. 11 and 12. The preselection cut is applied on the corresponding M CC variable fromthe x -axis of the scatter plots in the right panels of Figs. 11 and 12, M CC ( ab ) (plots inthe upper row) or M CC ( b ) (plots in the lower row). The left (middle) plots correspondto the study point from Fig. 11 (Fig. 12). The middle plots in Fig. 13 nicely illustrate thebenefit from the preselection cut — the unwanted events from the tail are removed andthe mass bump is rendered more symmetric, and is now centered on the correct mass value– 28 – igure 14 . The fraction of events falling within the rightmost 5% of the allowed range for M CC ( ab ) (left panel), M CC ( b ) (middle panel) and M CC ( a ) (right panel). The mass of A is fixedat m A = 500 GeV, while m B and m C are varied as m B ∈ (0 , m A ) and m C ∈ (0 , m B ). for the relative particle. However, those benefits do come at a cost - the number of eventsin the mass bump is correspondingly reduced. (Note, however, the upper middle panel ofFig. 13, where the cut seems to cause no appreciable loss in statistics.) On the other hand,the left plots in Fig. 13 show that for our first study point from Fig. 11, the cut does notlead to a big improvement in the shape of the distribution, but this is because the shapewas already very good to begin with, and thus a preselection cut would be unnecessary.The loss of statistics observed in Fig. 13 as a result of the preselection cut suggeststhat a crucial issue for us to consider is the population of the M CC bins near the upperkinematic endpoint. This is investigated in Fig. 14 as a function of the general massparameter space ( m B , m C ) for a fixed m A = 500 GeV. The figure shows results for each ofthe three subsystems: ( ab ) in the left panel, ( b ) in the middle panel, and ( a ) in the rightpanel. For any given choice of m B and m C , we take the allowed range for the corresponding M CC variable, i.e., the difference between its upper and lower kinematic endpoints, anddivide it into 20 equal size bins. Then the rainbow scale in Fig. 14 indicates the fractionof events which fell into the very last bin, i.e. in the upper 5% of the M CC range, close tothe upper kinematic endpoint.Fig. 14 reveals that throughout the whole mass parameter space, the rightmost bin isvery well populated in the case of the ( ab ) subsystem, and less so in the case of the ( b )and ( a ) subsystems. Given that invisible momentum reconstruction works best for eventsnear the last bin, this suggests that variables based on the ( ab ) subsystem have a certainadvantage in terms of statistics and accuracy. Upon closer inspection of the left panel inFig. 14, we find that the last bin is maximally populated if the mass spectrum satisfies therelation m C = m B m A , (5.1)whose physical meaning is the following — in the rest frame of particle A i , particle C i remains at rest, while the visible particles a i and b i are back-to-back. The relation (5.1)was approximately satisfied for the study point in Fig. 11, where we had m C = 200 GeVand m B /m A = 180 GeV. On the other hand, the study point in Fig. 12 was characterizedby m C = 50 GeV and m B /m A = 405 GeV, which significantly violated (5.1) by m C – 29 – igure 15 . The same as Fig. 12, but for an example with a relatively high value of m C : m A = 500GeV, m B = 300 GeV, m C = 275 GeV. being too low. Note that the relation (5.1) is scale invariant, i.e., the result does notchange if we inflate all masses by the same constant factor. We checked this with explicitsimulations, and verified that much heavier spectra which satisfy (5.1), continue to exhibitnice reconstructed peaks and vice versa.In conclusion of this section, we shall test the prediction (5.1) by choosing a pointfor which m C is too high. Let us again take m A = 500 GeV and m B = 300 GeV, as inFig. 11, only now increase the value of m C to 275 GeV, well above the prediction from(5.1). The result is shown in Fig. 15. The upper left panel shows that, as designed, theevents near the M CC ( ab ) kinematic endpoint are depleted, and the peak of the M CC ( ab )distribution has now moved to lower values, away from the kinematic endpoint. The rightpanels again exhibit tails, only this time the tails curl up towards higher values of thereconstructed relative mass, leading to an overestimate of the mass — the bulk of the M R distributions in the middle panels extend above the nominal mass of the respectiveparent particle. However, these problems can be again overcome by the two techniquesconsidered earlier — applying a preselection cut on events near the M CC ( ab ) kinematicendpoint (see the right panels in Fig. 13), or directly targeting the most populated bins inthe two-dimensional correlation plots in the right panels of Fig. 15.– 30 – Conclusions and outlook
Understanding the kinematics of events with missing transverse momentum at hadroncolliders like the LHC is an important task, since many new physics models have collidersignatures with dark matter particles and/or neutrinos, whose individual momenta andenergies are not measured in the detector. The two traditional approaches for analyzingsuch events are 1) where available, use a sufficient number of on-shell constraints to solvefor the invisible momenta exactly; and 2) use variables which do not require any actualknowledge of the individual invisible momenta. However, recently several prescriptions forassigning approximate values to the individual invisible momenta have emerged. The maingoal of this paper was to advertise the existence of a large number of such ansatze (1.3)and demonstrate their usefulness for the purposes of a mass measurement through a bumphunt. Our specific points are the following. • Many different ansatze are possible.
Quite often, any given prescription for assigninginvisible momenta has many different variations, as we demonstrated in Sections 3.1and 3.2, where we defined 12 versions of the MAOS method and 18 versions of the M method, respectively, in the case of the dilepton t ¯ t event topology. • The M class of variables automatically provides ansatze for the longitudinal com-ponents of the invisible momenta. As discussed in Sections 3 and 4, the importantadvantage of the M class of (3+1)-dimensional invariant mass variables is that theyautomatically provide values for the longitudinal components of the invisible mo-menta, without any need for additional mass inputs. In that sense, they are on thesame theoretical footing as the MAOS2 and MAOS3 versions of the MAOS method,but significantly outperform them in the presence of the relative mass constraint(3.23). • The M -based reconstruction of invisible momenta is superior to the MAOS schemes. In this paper, we compared the performance of the two methods using the exampleof a bump hunt mass measurement. Our results in Sec. 4.3 showed that the invisiblemomenta found by the M class of variables generally lead to a better determinationof the new particle masses. • Software support.
With the release of the public code
Optimass [47] which is capableof computing the on-shell constrained M variables for general event topologies, thecorresponding ansatze for the invisible momenta are also readily available and can beused for phenomenological studies similar to the one in this paper. For example, onecould imagine spin measurements as in Refs. [66, 72, 86], or designing procedures forreducing the combinatorial background [71, 74, 75, 87]. • Sensitivity study throughout the mass parameter space.
In Section 5 we investigatedthe precision of the invisible momentum reconstruction throughout the full massparameter space, and identified the regions where sensitivity can be lost. We proposedto mitigate the problem by either studying the 2D correlations of the reconstructed– 31 –ass and the M variable, or by applying a preselection cut on the M variable inorder to only select events near its kinematic endpoint. Acknowledgments
We would like to thank Won Sang Cho and Sung Hak Lim for useful discussions. This workis supported in part by a US Department of Energy grant DE-SC0010296. DK is supportedby the Korean Research Foundation (KRF) through the CERN-Korea Fellowship program.
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