MODEL INDEPENDENT PARTICLE MASS MEASUREMENTS IN MISSING ENERGY EVENTS AT HADRON COLLIDERS

Transcription

1 MODEL INDEPENDENT PARTICLE MASS MEASUREMENTS IN MISSING ENERGY EVENTS AT HADRON COLLIDERS By MYEONGHUN PARK A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 011

2 011 Myeonghun Park

3 I dediate this to my grand mother SunAe Seo and my wife Kelly Chung. 3

4 ACKNOWLEDGMENTS First and foremost, I am deeply indebted to my advisor, Prof. Konstantin Mathev, for his time, patiene, enouragement, muh stimulating advie. To me, he is a father who keeps trying to teah his son about everything that a son needs to know to be prepared for a world in front of him. In addition that, I ve always been happy to play games with his first son, Anton. I ve learned from my advisor about many things beyond physis, for example I learned how I an be a nie husband even though I need to spend lots of time in my offie. I am also extremely grateful to my ollaborator Dr. K.C. Kong for advie and useful disussions. I would like to thank Dr. Partha Konar for interesting disussions and ollaboration. All of them are really nie like as elder brothers. Like as I love my family, I love my group members. During my journey to physis, it was a great pleasure to attend Prof. Pierre Ramond s lasses. Beause of him, I opened my eyes to the beauty of symmetries and realized how symmetry an guide us to deep questions and handle problems. I also want to thank Prof. Rik Field for supporting me to have various experienes. I would like to thank UF CMS group for allowing me to join CMS ollaboration. It is really fantasti to handle real data from the LHC. I am grateful to Dr. Joseph Lykken and Dr. Stephen Mrenna. To have opportunities to work with them was really interesting to me. There are lots of interesting tasks that we need to do, and this is always my pleasure. I should mention about senior physiists who have helped me a lot and gave advies too. Visiting Pusan and Tokyo lead me into world wide ollaborations. All of these ould not be done without supports from Professor Konstantin Mathev, Professor Deog-Ki Hong and Professor Mihoko M. Nojiri. I would like to thank Dr. Christopher Lester, Professor Mihael Peskin and Dr. Jay Waker who gave me useful omments and advie, among many other physiists that 4

5 I have met at workshops and seminar visits. I would like to show my gratitude to Prof. Paul W. Chun s advies and great are on UF Korean students. I do not know how I an show my gratitude to my parents and family. As the first son, I know how muh they miss me. In fat I really miss them too. During my study abroad, the very reason that I have not suffered from any home sikness is beause I have a family here too. I really love my parents in law and appreiate their endless support. Now, I will leave here and go to another plae. I do not worry about my future sine I am with my wife. My wife is the best friend all the time. I really appreiate my wife s understanding so that I ould spend lots of time in other plaes for professional visits. All of my works here were made possible beause of my beloved fathers, my father in Korea, my father in law, my advisor and my Lord. 5

6 TABLE OF CONTENTS page ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES ABSTRACT CHAPTER 1 INTRODUCTION GENERAL ANALYSIS WITHOUT ANY ASSUMPTIONS The Need for a Universal, Global and Inlusive Mass Variable Definition of s min s min and the Underlying Event Problem Definition of the RECO level Variable s (reo) min Definition of the Subsystem Variable s (sub) min SM example: Dilepton Events from t t prodution Event Simulation Details s (reo) min Variable s (sub) min Variable An Exlusive SUSY Example: Multijet Events From Gluino Prodution An Inlusive SUSY Example: GMSB Study Point GM1b Comparison to Other Inlusive Collider Variables INVARIANT MASS ENDPOINTS METHOD Three Generi Problems in Invariant-mass Endpoint Methods Near-far Lepton Ambiguity Insuffiient Number of Measurements Parameter Spae Region Ambiguity Posing The Problem New Variables The Union mjl n mjl f The Produt m jln m jlf The Sums mjl α n + mjl α f Kinemati Endpoints of mjl(s) (α) with α Kinemati Endpoints of mjl(s) (α) with α < 1 and α The Differene mjl n mjl f Theoretial Analysis Our Method And The Solution For The Mass Spetrum Disambiguation Of The Two Solutions For m B

7 Invariant mass endpoint method Invariant mass orrelations Numerial Examples Mass Measurements At Study Points LM1 and LM Eliminating The Fake Solution for m B SUBSYSTEM M T METHOD A Short Deay Chain X X 1 X The Subsystem Variable M (1,1,0) T The Subsystem Variable M (,,1) T The Subsystem Variable M (,,0) T The Subsystem Variable M (,1,0) T M T -based Mass Measurement methods Pure M T Endpoint Method M T Endpoint Shapes And Kinks Hybrid Method: M T Endpoints Plus An Invariant Mass Endpoint ONE DIMENSIONAL PROJECTION METHOD Detailed Study On M T s Charateristis Using the P T of the Upstream Jet with M T Method Using Full Phase Spae Information With M CT ASYMMETRIC EVENT TOPOLOGY Generalizing M T To Asymmetri Event Topologies The Conventional Symmetri M T Definition Computation Properties Property I: Knowledge Of M p As A Funtion of M Property II: Kink In M T (max) At The True M Property III: P UTM Invariane Of M T (max) At The True M The Generalized Asymmetri M T Definition Computation Properties Property I: Knowledge Of M p As A Funtion Of M (a) And M (b) Property II: Ridge In M T (max) Through The True M (a) And M (b) Property III: P UTM Invariane Of M T (max) At The True M (a) And M (b) Examples Combinatorial Issues

8 6.4 The Simplest Event Topology: One Standard Model Partile On Eah Side Asymmetri Case Symmetri Case Mixed Case A More Complex Event Topology: Two Visible Partiles On Eah Side Off-shell Intermediate Partile On-shell Intermediate Partile Appliation To More General Cases CONCLUSIONS s min Invariant Mass Endpoint Method Subsystem M T Method One Dimensional Projetion Method Asymmetri Event Topology APPENDIX A ANALYTICAL EXPRESSIONS FOR THE SHAPES OF THE INVARIANT MASS DISTRIBUTIONS A.1 Dilepton Mass Distribution mll A. Combined Jet-lepton Mass Distribution mjl(u) A.3 Distribution of the sum mjl(s) (α = 1) A.4 Distribution Of The Differene mjl(d) (α = 1) A.5 Distribution Of The Produt mjl(p) B ANALYTICAL EXPRESSIONS FOR M (n,p,) T,max ( M, p T ) B.1 The Subsystem Variable M (n,n,n 1) T,max ( M n 1, p T ) B. The Subsystem Variable M (n,n,n ) T,max ( M n, p T ) B.3 The Subsystem Variable M (n,n 1,n ) T,max ( M n, p T ) C THE SYMMETRIC M T IN THE LIMIT OF INFINITE P UTM REFERENCES BIOGRAPHICAL SKETCH

9 Table LIST OF TABLES page -1 Seleted s quantities for events in Figure -4, -5 and Masses (in GeV) of the SUSY partiles at the GM1b study point Cross-setions (in pb) and parent mass thresholds (in GeV) for the dominant prodution proesses at the GM1b study point The relevant part of the SUSY mass spetrum for the LM1 and LM6 study points Seleted spartile masses (in GeV) at point LM Mass spetra for the two examples studied in Setions and

10 Figure LIST OF FIGURES page 1-1 A shemati desription of the mass-determination methods desribed in this dissertation The generi event topology used to define the s min Event topology of a subsystem Distributions of various s min for the dilepton t t PGS alorimeter map of the energy deposits for a dilepton t t event with only two reonstruted jets PGS alorimeter map of the energy deposits for a dilepton t t event with more than two reonstruted jets Distribution of various s min for W + W subsystem of t t with two reonstruted leptons Unit-normalized distribution of jet multipliity in dilepton t t events Distributions of various s min for the dilepton t t sample for subsystems Unit-normalized distribution of jet multipliity in gluino pair prodution events Distribution of various s min with a SUSY example of gluino pair prodution, with eah gluino deaying to four jets and a χ Distribution of various s min with a SUSY example of gluino pair prodution, with eah gluino deaying to two jets and a χ 0 1 LSP as in ( 44) PGS alorimeter map of the energy deposit for a SUSY event of gluino pair prodution, with eah gluino fored to deay to 4 jets and the LSP Distribution of the s (al) min and s (reo) min variables in inlusive SUSY prodution for the GMSB GM1a benhmark study point Distributions of various s min for the GMSB SUSY example Comparison various s min with other transverse variables for t t prodution Comparison various s min with other transverse variables for the gluino pair prodution, with eah gluino deaying to 4 jets Comparison various s min with other transverse variables for gluino pair prodution with eah gluino deaying to jets The typial asade deay hain of a BSM partile, N asade =

11 3- Comparison of the preditions for the kinemati endpoints mjl(s) max (α) of the real and fake solutions Predited satter plots of m jl(u) versus m ll, for the ase of the real and fake solutions for eah of the two study points LM1 and LM One-dimensional invariant mass distributions for the ase of LM1 and LM1 spetra One-dimensional invariant mass distributions for the LM6 mass spetrum and the LM6 mass spetrum Some other one-dimensional invariant mass distributions of interest for the ase of the LM1 mass spetrum and LM Some other one-dimensional invariant mass distributions of interest for the LM6 mass spetrum and the LM6 mass spetrum Illustration of the subsystem M (n,p,) T variable The subsystem M (n,p,) T variables whih are available for n = 1 and n = events Dependene of the M (1,1,0) T,max upper kinemati endpoint on the value of p T Dependene of the M (,,0) T,max and M(,1,0) T,max upper kinemati endpoints on the value of the test mass M The amount of kink as a funtion of the mass ratios y and z Unit-normalized distributions of M (n,p,) T variables in dilepton events from W + W pair prodution and t t pair prodution Unit-normalized m bl invariant mass-squared distributions in dilepton t t events The typial SUSY event topology produing two isolated same-sign leptons at point LM The two speial momentum onfigurations defined in Equations (5 1,5 ) M max T versus the test mass M Saling fators relating the error δ M p in the extration of the M T endpoint The generi event topology for one-dimensional projetion Deomposition of the observed transverse momentum vetors The unit-normalized M T distribution Observable M T distribution after hard uts for 100 fb 1 of LHC data

12 5-9 The funtion ˆN( M ) Zero-bin subtrated M CT distribution after uts, for t t dilepton events Satter plots of M CT versus M CT and M CT versus M CT D CT distributions for four different values of M p Fitted values of D min CT as a funtion of M p The generi event topology with different missing partiles M T (max) ( M, P UTM ) and M T (max) ( M, P UTM ) for several fixed values of P UTM The three different event-topologies for various deaying types Unit-normalized M T distributions for the event topology of Figure 6-3(b) M T (max) as a funtion of the two test hildren masses, M (a) and M (b), for the event topology of Figure 6-3(a) M T (max) for the event topology of Figure 6-3(a) with fixed upstream momentum of P UTM = 1 TeV M T ridge struture when two missing partiles are different M T (max) for the event topology of Figure 6-3(a) with the symmetri mass spetrum II from Table 6-1 without upstream momentum M T (max) for the event topology of Figure 6-3(a) with the symmetri mass spetrum II from Table 6-1 with fixed upstream momentum M T ridge struture when two missing partiles have same mass spetrum Unit-normalized, zero-bin subtrated M T distribution for the full mixed event sample The four regions in the ( M (a), M (b) ) parameter plane leading to the four different types of solutions for the M T endpoint for the offshell senario illustrated in Figure 6-3(b) M T (max) as a funtion of the two test hildren masses, M (a) and M (b), for the event topology of Figure 6-3(b) The four regions in the ( M (a), M (b) ) parameter plane leading to the four different types of solutions for the M T endpoint for the onshell senario illustrated in Figure 6-3() [M T (max) as a funtion of the two test hildren masses, M (a) and M (b), for the event topology of Figure 6-3()

13 6-16 Event topology for the effetively different missing parties A-1 Comparison of the numerially obtained differential invariant mass distributions for study point LM1 with the analytial results C-1 The parameter plane of test hildren masses squared, divided into the four different regions R i used to define the M T endpoint funtion C- M T (max) for the event topology of Figure 6-3(a) with fixed upstream momentum of P UTM = C-3 M T (max) for the event topology of Figure 6-3(a) with the symmetri mass spetrum II from Table 6-1 with upstream momentum P UTM C-4 A study of the sharpness of the M T ridge for the ase when missing partiles are different C-5 A study of the sharpness of the M T ridge for the ase when missing partiles are the same

14 Abstrat of Dissertation Presented to the Graduate Shool of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Dotor of Philosophy MODEL INDEPENDENT PARTICLE MASS MEASUREMENTS IN MISSING ENERGY EVENTS AT HADRON COLLIDERS Chair: Konstantin T. Mathev Major: Physis By Myeonghun Park May 011 This dissertation desribes several new kinemati methods to measure the masses of new partiles in events with missing transverse energy at hadron olliders. Eah method relies on the measurement of some feature (a peak or an endpoint) in the distribution of a suitable kinemati variable. The first method makes use of the Gator variable s min, whose peak provides a global and fully inlusive measure of the prodution sale of the new partiles. In the early stage of the LHC, this variable an be used both as an estimator and a disriminator for new physis over the standard model bakgrounds. The next method studies the invariant mass distributions of the visible deay produts from a asade deay hain and the shapes and endpoints of those distributions. Given a suffiient number of endpoint measurements, one ould in priniple attempt to invert and solve for the mass spetrum. However, the non-linear harater of the relevant oupled quadrati equations often leads to multiple solutions. In addition, there is a ombinatorial ambiguity related to the ordering of the deay produts from the asade deay hain. We propose a new set of invariant mass variables whih are less sensitive to these problems. We demonstrate how the new partile mass spetrum an be extrated from the measurement of their kinemati endpoints. The remaining methods desribed in the dissertation are based on transverse invariant mass variables like the Cambridge transverse mass M T, the Sheffield 14

15 ontrasverse mass M CT and their orresponding one-dimensional projetions M T, M T, M CT, and M CT with respet to the upstream transverse momentum U T. The main advantage of all those methods is that they an be applied to very short (single-stage) deay topologies, as well as to a subsystem of the observed event. The methods an also be generalized to the ase of non-idential missing partiles, as demonstrated in Chapter 7. A omplete set of analytial results for the alulation of the relevant variables in eah event, as well as the dependene of their endpoints on the underlying mass spetrum is given for eah ase. In some irumstanes, the whole shape of the differential distribution an be theoretially predited as well. The methods are illustrated with examples from supersymmetry and from top quark prodution in the standard model. 15

16 CHAPTER 1 INTRODUCTION The Large Hadron Collider (LHC) at CERN has begun its long awaited exploration of the TeV sale and reahed integrated luminosity of L 40pb 1 at 7TeV. Starting with the hunt for standard model Higgs partile, we expet to see Beyond the Standard Model (BSM) phenomena at the LHC whih may hold the key to our understanding of some very basi questions about our universe: What is the dark matter? What are the fundamental symmetries of Nature? Are there any hidden dimensions of spae? A potential disovery of a missing energy signal at the LHC may relate to all three of these questions. Perhaps the most ompelling phenomenologial evidene for BSM partiles and interations at the TeV sale is provided by the dark matter problem [1]. It is a tantalizing oinidene that a neutral, weakly interating massive partile (WIMP) in the TeV range an explain all of the observed dark matter in the Universe. A typial WIMP does not interat in the detetor and an only manifest itself as missing energy. The WIMP idea therefore greatly motivates the study of missing energy signatures at the Tevatron and the LHC []. The long lifetime of the dark matter WIMPs is typially ensured by some new exat symmetry, e.g. R-parity in supersymmetry [3], KK parity in models with extra dimensions [4], T -parity in Little Higgs models [5, 6] et. The partiles of the Standard Model (SM) are not harged under this new symmetry, but the new partiles are, and the lightest among them is the dark matter WIMP. This setup guarantees that the WIMP annot deay, and more importantly, that WIMPs are always pair-produed at olliders. The ross-setions for diret prodution of WIMPs (tagged with a jet or a photon from initial state radiation) at hadron olliders are typially too small to allow observation above the SM bakgrounds [7]. Therefore one typially onentrates on the pair prodution of the other, heavier partiles (e.g. superpartners, KK-partners, or T -partners), whih also arry nontrivial new quantum numbers just like the WIMPs. One produed, those 16

17 heavier partners will asade deay down, emitting SM partiles whih are in priniple observable in the detetor. However, eah suh asade also inevitably ends up with an invisible WIMP, whose energy and momentum are unknown. Sine the heavy partners are being pair-produed, there are two suh asades in eah event, and therefore, two unknown WIMP momenta. In addition, at hadron olliders the total parton level energy and momentum in the enter of mass frame are also unknown, and thus the exat reonstrution of the deay hains on an event by event basis is a very hallenging task 1. In this dissertation, we present how we an resolve these diffiulties more systematially. Our approah is desribed shematially in the Figure 1-1. Systemati Approahes for Mass Determinations. The very first question that we may want to ask would be what is the sale of the new physis. To answer this question, it would be best to avoid any assumptions about the event-topology. We studied variables whih mimi true prodution energy ŝ of new partiles. The referene [50] provides s min that is a minimization of ŝ with only one ondition from missing energy onstraint. It turned out that the peak of s min provides a global and fully inlusive measure of the prodution sale of the new partiles. In the early stage of the LHC, this variable an be used both as an estimator and a disriminator for new physis over the standard model bakgrounds. More detailed studies on s min will be provided in Chapter. Sine we have not speified on the event-topology, s min itself is not preise enough to determine the full mass spetrum of the new partiles. In addition to s min one an also study invariant mass of various visible partiles. As we will point out, it is possible to reonstrut the intermediate partiles masses when the asade deay hain is long enough. The problem of using invariant-mass is that we need 1 See Dark matter and ollider phenomenology of universal extra dimensions (Phys. Rept. 453, 9 (007)) [8] for a reent review. 17

18 to speify whih visible partile omes from whih intermediate partile, namely we suffer from ombinatorial problems, thus if N asade (the length of deaying hain) is large, it beomes more diffiult to hoose the right set of visible partiles to form the invariant mass. On top of this problem, when we invert the invariant mass endpoints to solve for the mass spetra, there will be multiple solutions oming from the non-linear harateristis of oupled quadrati equations [9]. Those diffiulties initiate our projets on invariant mass methods, and detailed studies will be presented in Chapter 3. While invariant mass methods rely on a single asade deay mode of a new partile, usually new physis will ome with pair produed partiles due to the new exat symmetry. Thus if we use this extra ondition about produed partiles, we an reonstrut new partiles mass even for short deaying modes like as N asade = whih is not possible with the invariant mass endpoints method. We developed a subsystem onept whih we an apply to speifi deay modes. We applied the subsystem to a Cambridge transverse variable alled M T [10] in Chapter 4. While a subsystem M T in priniple works for N asade 1, when N asade = 1 the resolution of the mass determination is not so good. This is beause the mass determination depends on the hardness of the upstream momentum, whih is not big enough when upstream objets ome from the initial state radiation. We proposed orthogonal deompositions of known kinemati variables suh as M CT and M T onto that speial transverse diretion [11, 1]. We realized that the doubly transverse quantities like M CT and M T are partiularly useful, sine their kinemati endpoints are independent of U T and make it possible to use the whole struture of phase spae so that we an determine the mass spetra of related partiles in a very short deay hain. We provide these methods in Chapter 5. s min does not suffer from this kind of ombinatorial problem. 18

19 Given our utter ignorane about the struture of the dark matter setor, we set out to develop the neessary formalism for arrying out missing energy studies at hadron olliders in a very general and model-independent way, without relying on any assumptions about the nature of the missing partiles. In partiular, we did not assume that the two missing partiles in eah event are the same. We generalized the M T idea to asymmetri events with different missing partiles in Chapter 6. 19

20 Chapter 3 (1 N asades ) Invariant Mass endpoints (3 N asades < ) Chapter 6 Chapter smin Asymmetri M T (1 N asades < ) Assumptions on event topology? Yes No Are pair produed partiles same? Yes No Are missing partiles same? Yes No Effets from upstream partiles? Chapter 5 1D projeted variablem CT No Chapter 4 1D projeted variable M T (1 N asades < ) Yes SubsystemM T (1 N asades < ) Figure 1-1. A shemati desription of the mass-determination methods desribed in this dissertation. 0

21 CHAPTER GENERAL ANALYSIS WITHOUT ANY ASSUMPTIONS.1 The Need for a Universal, Global and Inlusive Mass Variable Most methods are model-dependent in the sense that eah method ruially relies on the assumption of a very speifi event topology. One ommon flaw of all methods on the market is that they usually do not allow any SM neutrinos to enter the targeted event topology, and the missing energy is typially assumed to arise only as a result of the prodution of (two) new dark matter partiles. Furthermore, eah method has its own limitations. For example, the traditional invariant mass endpoint methods [9, 13 ] require the identifiation of a suffiiently long asade deay hain, with at least three suessive two-body deays [3]. The polynomial methods [4 31] also require suh long deay hains and furthermore, the events must be symmetri, i.e. must have two idential deay hains per event, or else the deay hain must be even longer [3]. The reently popular M T methods [10, 1, 3 39] do not require long deay hains [3], but typially assume that the parent partiles are the same and deay to two idential invisible partiles 1. The limitations of the M CT methods [11, 4, 43] are rather similar. The kinemati usp method [44] is limited to the so alled antler event topology, whih ontains two symmetri one-step deay hains originating from a single s-hannel resonane. In light of all these various assumptions, it is ertainly desirable to have a universal method whih an be applied to any event topology. The s min variable is defined in terms of the total energy E and 3-momentum P observed in the event, and thus does not make any referene to the atual event topology. It is ompletely general, universal and fully inlusive, and to the fullest extent makes use of the available experimental information. 1 See Dark Matter Partile Spetrosopy at the LHC: Generalizing MT to Asymmetri Event Topologies (JHEP 1004, 086 (010)) [41] for a more general approah whih avoids this assumption. 1

22 Figure -1. The generi event topology used to define the s min variable in s min : A Global inlusive variable for determining the mass sale of new physis in events with missing energy at hadron olliders, (JHEP 0903, 085 (009))[50]. Blak (red) lines orrespond to SM (BSM) partiles. The solid lines denote SM partiles X i, i = 1,,..., n vis, whih are visible in the detetor, e.g. jets, eletrons, muons and photons. The SM partiles may originate either from initial state radiation (ISR), or from the hard sattering and subsequent asade deays (indiated with the green-shaded ellipse). The dashed lines denote neutral stable partiles χ i, i = 1,,..., n inv, whih are invisible in the detetor. In general, the set of invisible partiles onsists of some number n χ of BSM partiles (indiated with the red dashed lines), as well as some number n ν = n inv n χ of SM neutrinos (denoted with the blak dashed lines). The identities and the masses m i of the BSM invisible partiles χ i, (i = 1,,..., n χ ) do not neessarily have to be all the same, i.e. we allow for the simultaneous prodution of several different speies of dark matter partiles. The global event variables desribing the visible partiles are: the total energy E, the transverse omponents P x and P y and the longitudinal omponent P z of the total visible momentum P. The only experimentally available information regarding the invisible partiles is the missing transverse momentum /P T.

23 .1.1 Definition of s min Consider the most generi missing energy event topology shown in Figure -1. In defining s min, one imagines a ompletely general setup eah event ontains some number n vis of Standard Model (SM) partiles X i, i = 1,,..., n vis, whih are visible in the detetor, i.e. their energies and momenta are in priniple measured. Examples of suh visible SM partiles are the basi reonstruted objets, e.g. jets, photons, eletrons and muons. The visible partiles X i are denoted in Figure -1 with solid blak lines and may originate either from ISR, or from the hard sattering and subsequent asade deays (indiated with the green-shaded ellipse). In turn, the missing transverse momentum /PT arises from a ertain number n inv of stable neutral partiles χ i, i = 1,,..., n inv, whih are invisible in the detetor. In general, the set of invisible partiles onsists of some number n χ of BSM partiles (indiated with the red dashed lines), as well as some number n ν = n inv n χ of SM neutrinos (denoted with the blak dashed lines). As already mentioned earlier, the /P T measurement alone does not reveal the number n inv of missing partiles, nor how many of them are neutrinos and how many are BSM (dark matter) partiles. This general setup also allows the identities and the masses m i of the BSM invisible partiles χ i, (i = 1,,..., n χ ) in priniple to be different, as in models with several different speies of dark matter partiles [45 49]. Of ourse, the neutrino masses an be safely taken to be zero m i = 0, for i = n χ + 1, n χ +,..., n inv. ( 1) Given this very general setup, if we try to minimize the parton-level Mandelstam invariant mass variable s whih is onsistent with the observed visible 4-momentum vetor P µ (E, P), we will get the following minimum of s as smin ( /M) E Pz + /M + /P T, ( ) 3

24 where the mass parameter /M is nothing but the total mass of all invisible partiles in the event: n inv /M m i = n χ i=1 i=1 m i, ( 3) and the seond equality follows from the assumption of vanishing neutrino masses ( 1). The result ( ) an be equivalently rewritten in a more symmetri form smin ( /M) = M + P T + /M + /P T ( 4) in terms of the total visible invariant mass M defined as M E P x P y P z E P T P z. ( 5) Notie that in spite of the omplete arbitrariness of the invisible partile setor at this point, the definition of s min depends on a single unknown parameter /M - the sum of all the masses of the invisible partiles in the event. For future referene, one should keep in mind that transverse momentum onservation at this point implies that P T + /P T = 0. ( 6) The main result from s min : A Global inlusive variable for determining the mass sale of new physis in events with missing energy at hadron olliders, (JHEP 0903, 085 (009))[50] was that in the absene of ISR and MPI, the peak in the s min distribution niely orrelates with the mass threshold of the newly produed partiles. This observation provides one generi relation between the total mass of the produed partiles and the total mass /M of the invisible partiles..1. s min and the Underlying Event Problem At the same time, it was also reognized that effets from the underlying event (UE), most notably ISR and MPI, severely jeopardize this measurement. The problem is that in the presene of the UE, the s min variable would be measuring the total energy of the full system shown in Figure -1, while for studying any new physis we are mostly 4

25 interested in the energy of the hard sattering, as represented by the green-shaded ellipse in Figure -1. The inlusion of the UE auses a drasti shift of the peak of the smin distribution to higher values, often by as muh as a few TeV [50 5]. As a result, it appeared that unless effets from the underlying event ould somehow be ompensated for, the proposed measurement of the s min peak would be of no pratial value. The main purpose of this hapter is to propose two fresh new approahes to dealing with the underlying event problem whih has plagued the s min variable and prevented its more widespread use in hadron ollider physis appliations. We propose two new variants of the s min variable, whih we label s (reo) min and s (sub) min and define in Setions. and.3, orrespondingly. We illustrate the properties of these two variables with several examples in Setions These examples will show that both s (reo) min and s (sub) min are unharmed by the effets from the underlying event, thus resurreting the original idea of s min proposed in Referene [50] to use the peak in the s min distribution as a first, quik, model-independent estimate of the new physis mass sale. In Setion.7 we ompare the performane of s min against some other inlusive variables whih are ommonly used in hadron ollider physis for the purpose of estimating the new physis mass sale.. Definition of the RECO level Variable s (reo) min In the first approah, we shall not modify the original definition of s min and will ontinue to use the Equation ( ) (or its equivalent Equation ( 4)), preserving the desired universal, global and inlusive harater of the s min variable. Then we shall onentrate on the question, how should one alulate the observable quantities E, P and /P T entering the defining Equations ( ) and ( 4). The previous s min studies [50 5] used alorimeter-based measurements of the total visible energy E and momentum P as follows. The total visible energy in the 5

26 alorimeter E (al) is simply a salar sum over all alorimeter deposits E (al) α E α, ( 7) where the index α labels the alorimeter towers, and E α is the energy deposit in the α tower. As usual, sine muons do not deposit signifiantly in the alorimeters, the measured E α should first be orreted for the energy of any muons whih might be present in the event and happen to pass through the orresponding tower α. The three omponents of the total visible momentum P were also measured from the alorimeters as P x(al) = α E α sin θ α os φ α, ( 8) P y(al) = α E α sin θ α sin φ α, ( 9) P z(al) = α E α os θ α, ( 10) where θ α and φ α are orrespondingly the polar and azimuthal angular oordinates of the α alorimeter tower. The missing transverse momentum an similarly be measured from the alorimeter as (Equation ( 6)) P T (al) P T (al). ( 11) Using these alorimeter-based measurements ( 7-11), one an make the identifiation E E (al), ( 1) P P (al), ( 13) /PT P T (al) ( 14) 6

27 in the definition ( ) and onstrut the orresponding alorimeter-based s min variable as (al) s min ( /M) E (al) Pz(al) + /M + PT (al). ( 15) This was preisely the quantity whih was studied in [50 5] and shown to exhibit extreme sensitivity to the physis of the underlying event. Here we propose to evaluate the visible quantities E and P at the RECO level, i.e. in terms of the reonstruted objets, namely jets, muons, eletrons and photons. To be preise, let there be N obj reonstruted objets in the event, with energies E i and 3-momenta P i, i = 1,,..., N obj, orrespondingly. Then in plae of Equations ( 1-14), let us instead identify N obj E E (reo) E i, ( 16) P P (reo) i=1 N obj i=1 P i, ( 17) /PT P T (reo) = P T (reo), ( 18) and orrespondingly define a RECO-level s min variable as (reo) s min ( /M) E (reo) Pz(reo) + /M + PT (reo), ( 19) whih an also be rewritten in analogy to Equation ( 4) as (reo) s min ( /M) M (reo) + PT (reo) + /M + PT (reo), ( 0) where P T (reo) and P T (reo) are related as in Equation ( 18) and the RECO-level total visible mass M (reo) is defined by M (reo) E (reo) P (reo). ( 1) 7

28 What are the benefits from the new RECO-level s min defined as in Equations ( 19, 0) in omparison to the old alorimeter-based s min definition in an Equation ( 15)? In order to understand the basi idea, it is worth omparing the alorimeter-based missing transverse momentum /P T (whih in the literature is ommonly referred to as missing transverse energy /E T ) and the analogous RECO-level variable /H T, the missing H T. The /H T vetor is defined as the negative of the vetor sum of the transverse momenta of all reonstruted objets in the event: N obj /HT P Ti. ( ) Then it is lear that in terms of our notation here, /H T is nothing but P T (reo). It is known that /H T performs better than /E T [53]. First, /H T is less affeted by a number of adverse instrumental fators suh as: eletroni noise, faulty alorimeter ells, pile-up, et. These effets tend to populate the alorimeter uniformly with unlustered energy, whih will later fail the basi quality uts during objet reonstrution. In ontrast, the true missing momentum is dominated by lustered energy, whih will be suessfully aptured during reonstrution. Another advantage of /H T is that one an easily apply the known jet energy orretions to aount for the nonlinear detetor response. For both of these reasons, CMS is now using /H T at both the trigger level and offline [53]. Now realize that s (al) min i=1 is analogous to the alorimeter-based /E T, while our new variable s (reo) min is analogous to the RECO-level /H T. Thus we may already expet that (reo) s min will inherit the advantages of /H T and will be better suited for determining the new physis mass sale than the alorimeter-based quantity s (al) min. This expetation is onfirmed in the expliit examples studied below in Setions.4 and.5. Apart from the already mentioned instrumental issues, the most important advantage of s (reo) min from the physis point of view is that it is muh less sensitive to the effets from the underlying event, whih had doomed its alorimeter-based s (al) min ousin. 8

29 Stritly speaking, the idea of s (reo) min does not solve the underlying event problem ompletely and as a matter of priniple. Every now and then the underlying event will still produe a well-defined jet, whih will have to be inluded in the alulation of (reo) s min. Beause of this effet, we annot any more guarantee that s (reo) min provides a lower bound on the true value s true of the enter-of-mass energy of the hard sattering the additional jets formed out of ISR, pile-up, and so on, will sometimes ause (reo) s min to exeed s true. Nevertheless we find that this effet modifies only the shape of the s (reo) min distribution, but leaves the loation of its peak largely intat. To the extent that one is mostly interested in the peak loation, s (reo) min enough for all pratial purposes. should already be good.3 Definition of the Subsystem Variable s (sub) min In this setion we propose an alternative modifiation of the original s min variable, whih solves the underlying event problem ompletely and as a matter of priniple. The downside of this approah is that it is not as general and universal as the one disussed in the previous setion, and an be applied only in ases where one an unambiguously identify a subsystem of the original event topology whih is untouhed by the underlying event. The basi idea is shematially illustrated in Figure -, whih is nothing but a slight rearrangement of Figure -1 exhibiting a well defined subsystem (delineated by the blak retangle). The original n vis visible partile X i from Figure -1 have now been divided into two groups as follows: 1. There are n sub visible partiles X 1,..., X nsub originating from within the subsystem. Their total energy and total momentum are denoted by E (sub) and P (sub). The subsystem partiles are hosen so that to guarantee that they ould not have ome from the underlying event.. The remaining n vis n sub visible partiles X nsub +1,..., X nvis are reated upstream (outside the subsystem) and have total energy E (up) and total momentum P (up). The upstream partiles may originate from the underlying event or from deays of heavier partiles upstream this distintion is inonsequential at this point. 9

30 Figure -. A rearrangement of Figure -1 into an event topology exhibiting a well defined subsystem (delineated by the blak retangle) with total invariant mass s (sub). There are n sub visible partiles X i, i = 1,,..., n sub, originating from within the subsystem, while the remaining n vis n sub visible partiles X nsub +1,..., X nvis are reated upstream, outside the subsystem. The subsystem results from the prodution and deays of a ertain number of parent partiles P j, j = 1,,..., n p, (some of) whih may deay semi-invisibly. All invisible partiles χ 1,..., χ ninv are then assumed to originate from within the subsystem. We also assume that all invisible partiles χ 1,..., χ ninv originate from within the subsystem, i.e. that no invisible partiles are reated upstream. In effet, all we have done in Figure - is to partition the original measured values of the total visible energy E and 3-momentum P from Figure -1 into two separate omponents as E = E (up) + E (sub), ( 3) P = P (up) + P (sub). ( 4) 30

31 Notie that now the missing transverse momentum is defined as /PT P T (up) P T (sub), ( 5) while the total visible invariant mass M (sub) of the subsystem is given by M (sub) = E (sub) P (sub). ( 6) There would be ambiguities in ategorizing a given visible partile X i as a subsystem or an upstream partile. Sine our goal is to identify a subsystem whih is shielded from the effets of the underlying event, the safest way to do the partition of the visible partiles is to require that all QCD jets belong to the upstream partiles, while the subsystem partiles onsist of objets whih are unlikely to ome from the underlying event, suh as isolated eletrons, photons and muons (and possibly identified τ-jets and, to a lesser extent, tagged b-jets). With those preliminaries, we are now ready to ask the usual s min question: Given the measured values of E (up), E (sub), P (up) and P (sub), what is the minimum value s (sub) min of the subsystem Mandelstam invariant mass variable s (sub), whih is onsistent with those measurements? Proeeding as in [50], one again we find a very simple universal answer, whih, with the help of Equations ( 5) and ( 6), an be equivalently written in several different ways as follows: s (sub) min ( /M) = = { ( ) } 1 E(sub) P z(sub) + /M + /P T PT (up) { ( ) } 1 M(sub) + P T (sub) + /M + /P T PT (up) ( 7) ( 8) = { ( ) } 1 M(sub) + P T (sub) + /M + /P T ( P T (sub) + /P T ) ( 9) = p T (sub) + p T, ( 30) 31

32 where in the last line we have introdued the Lorentz 1+ vetors p T (sub) p T ( M (sub) + PT (sub), P ) T (sub) ; ( 31) ( ) /M + /P T, /P T. ( 3) As usual, the length of a 1+ vetor is omputed as p = p p = p 0 p 1 p. Before we proeed to the examples of the next few setions, as a sanity hek of the obtained result it is useful to onsider some limiting ases. First, by taking the upstream visible partiles to be an empty set, i.e. P T (up) 0, we reover the usual expression for s min given in Equations (, 4). Next, onsider a ase with no invisible partiles, i.e. /M = 0 and orrespondingly, /P T = 0. In that ase we obtain that (sub) s min = M (sub), whih is of ourse the orret result. Finally, suppose that there are no visible subsystem partiles, i.e. E (sub) = P (sub) = M (sub) = 0. In that ase we obtain (sub) s min = /M, whih is also the orret answer. As we shall see, the subsystem onept of Figure - will be most useful when the subsystem results from the prodution and deays of a ertain number n p of parent partiles P j with masses M Pj, j = 1,,..., n p, orrespondingly. Then the total ombined mass of all parent partiles is given by n p M p M Pj. ( 33) By the onjeture of Referene [50], the loation of the peak of the s (sub) min ( /M) distribution will provide an approximate measurement of M p as a funtion of the unknown parameter /M. By onstrution, the obtained relationship M p ( /M) will then be ompletely insensitive to the effets from the underlying event. At this point it may seem that by exluding all QCD jets from the subsystem, we have signifiantly narrowed down the number of potential appliations of the (sub) s min variable. Furthermore, we have apparently reintrodued a ertain amount of j=1 3

33 model-dependene whih the original s min approah was trying so hard to avoid. Those are in priniple valid objetions, whih an be overome by using the s (reo) min variable introdued in the previous setion. Nevertheless, we feel that the s (sub) min variable an prove to be useful in its own right, and in a wide variety of ontexts. To see this, note that the typial hadron ollider signatures of the most popular new physis models (supersymmetry, extra dimensions, Little Higgs, et.) are preisely of the form exhibited in Figure -. One typially onsiders prodution of olored partiles (squarks, gluinos, KK-quarks, et.) whose ross-setions dominate. In turn, these olored partiles shed their olor harge by emitting jets and deaying to lighter, unolored partiles in an eletroweak setor. The deays of the latter often involve eletromagneti objets, whih ould be targeted for seletion in the subsystem. The s (sub) min variable would then be the perfet tool for studying the mass sales in the eletroweak setor (in the ontext of supersymmetry, for example, the eletroweak setor is omposed of the harginos, neutralinos and sleptons). Before we move on to some speifi examples illustrating these ideas, one last omment is in order. One may wonder whether the s (sub) min variable should be omputed at the RECO-level or from the alorimeter. Sine the subsystem will usually be defined in terms of reonstruted objets, the more logial option is to alulate s (sub) min RECO-level and label it as s (sub,reo) min. However, to streamline our notation, in what at the follows we shall always omit the reo part of the supersript and will always impliitly assume that s (sub) min s (reo) min is omputed at RECO-level..4 SM example: Dilepton Events from t t prodution In this and the next two setions we illustrate the properties of the new variables and s (sub) min with some speifi examples. In this setion we disuss an example taken from the Standard Model, whih is guaranteed to be available for early studies at the LHC. We onsider dilepton events from t t pair prodution, where both W s deay leptonially. In this event topology, there are two missing partiles (two neutrinos). 33

34 Therefore, these events very losely resemble the typial SUSY-like events, in whih there are two missing dark matter partiles. In the next two setions, we shall also onsider some SUSY examples. In all ases, we perform detailed event simulation, inluding the effets from the underlying event and detetor resolution..4.1 Event Simulation Details Events are generated with PYTHIA [54] (using its default model of the underlying event) at an LHC of 14 TeV, and then reonstruted with the PGS detetor simulation pakage [55]. We have made ertain modifiations in the publily available version of PGS to better math it to the CMS detetor. For example, we take the hadroni alorimeter resolution to be [56] while the eletromagneti alorimeter resolution is [56] ( σ E σ E = 10% E, ( 34) ) = ( S E ) + ( ) N + C, ( 35) E where the energy E is measured in GeV, S = 3.63% is the stohasti term, N = 0.14 is the noise and C = 0.6% is the onstant term. Muons are reonstruted within η <.4, and we use the muon global reonstrution effiieny quoted in [56]. We use default p T uts on the reonstruted objets as follows: 3 GeV for muons, 10 GeV for eletrons and photons, and 15 GeV for jets. For the t t example presented in this setion, we use the approximate next-to-next-to-leading order t t ross-setion of σ t t = 894 ± pb at a top mass of m t = 175 GeV [57]. For the SUSY examples in the next two setions we use leading order ross-setions..4. s (reo) min Variable We first onsider SUSY-like missing energy events arising from t t prodution, where eah W -boson is fored to deay leptonially (to an eletron or a muon). 34

35 Figure -3. Distributions of various s min quantities disussed in the text, for the dilepton t t sample at the LHC with 14 TeV CM energy and 0.5 fb 1 of data. The dotted (yellow-shaded) histogram gives the true s distribution of the t t pair. The blue histogram is the distribution of the alorimeter-based s (al) min variable in the ideal ase when all effets from the underlying event are turned off. The red histogram shows the orresponding result for s (al) min in the presene of the underlying event. The blak histogram is the distribution of the s (reo) min shown for /M = 0. variable introdued in Setion.. All s min distributions are We do not impose any trigger or offline requirements, and simply plot diretly the output from PGS. We show various s quantities of interest in Figure -3, setting /M = 0, sine in this ase the missing partiles are neutrinos and are massless. The dotted (yellow-shaded) histogram represents the true s distribution of the t t pair. It quikly rises at the t t mass threshold M p m t = 350 GeV ( 36) Therefore, our plots in this subsetion are normalized to a total number of events equal to σ t t BR(W e, µ). 35

36 and then eventually falls off at large s due to the parton density funtion suppression. Beause the top quarks are typially produed with some boost, the s true distribution in Figure -3 peaks a little bit above threshold: ( strue ) peak > M p. ( 37) It is lear that if one ould diretly measure the s true distribution, or at least its onset, the t t mass sale will be easily revealed. Unfortunately, the esaping neutrinos make suh a measurement impossible, unless one is willing to make additional model-dependent assumptions 3. Figure -3 also shows two versions of the alorimeter-based s (al) min variable: the blue (red) histogram is obtained by swithing off (on) the underlying event (ISR and MPI). These urves reveal two very interesting phenomena. First, without the UE, the peak of the s (al) min threshold [50]: distribution (blue histogram) is very lose to the parent mass no UE = ( s ) (al) min peak M p. ( 38) The main observation of Partha et al. [50] was that this orrelation offers an alternative, fully inlusive and model-independent, method of estimating the mass sale M p of the parent partiles, even when some of their deay produts are invisible and not seen in the detetor. 3 For example, one an use the known values of the neutrino, W and top masses to solve for the neutrino kinematis (up to disrete ambiguities). However, this method assumes that the full mass spetrum is already known, and furthermore, uses the knowledge of the top deay topology to perfetly solve the ombinatoris problem disussed in the Introdution. As an example, onsider a ase where the lepton is produed first and the b-quark seond, i.e. when the top first deays to a lepton and a leptoquark, whih in turn deays to a neutrino and a b-quark. The kinemati method would then be using the wrong on-shell onditions. The advantage of the s min approah is that it is fully inlusive and does not make any referene to the atual deay topology. 36

37 s (al) min Unfortunately, the no UE limit of Equation ( 38) is unphysial, and the orresponding distribution (blue histogram in in Figure -3) is unobservable. What is worse, when one tries to measure the s (al) min distribution in the presene of the UE (red histogram in Figure -3), the resulting peak is very far from the physial threshold: with UE = ( s ) (al) min peak M p. ( 39) In the t t example of Figure -3, the shift is on the order of 1 TeV! It appears therefore that in pratie the s (al) min peak would be unorrelated with any physial mass sale, and instead would be ompletely determined by the (uninteresting) physis of the underlying event. One the nie model-independent orrelation of Equation ( 38) is destroyed by the UE, it beomes of only aademi value [8, 50 5, 58]. However, Figure -3 also suggests the solution to this diffiult problem. If we look at the distribution of the s (reo) min returned to the desired value: variable (blak solid histogram), we see that its peak has ( s ) (reo) min peak M p, ( 40) In order to measure physial mass thresholds, one simply needs to investigate the distribution of the inlusive s (reo) min variable, whih is alulated at RECO-level. Eah peak in that distribution signals the opening of a new hannel, and from Equation ( 40) the loation of the peak provides an immediate estimate of the total mass of all partiles involved in the prodution. Our first main result is therefore niely summarized in Figure -3, whih shows a total of 4 distributions, 3 of whih are either unphysial (the blue histogram of s (al) min in the absene of the UE), unobservable (the yellow-shaded histogram of s true ), or useless (the red histogram of s (al) min in the presene of the UE). The only distribution in Figure -3 whih is physial, observable and useful at the same time, is the distribution of s (reo) min (solid blak histogram). 37

38 E al (parton level) (parton level) E T,al q e 30 4 q e 30 φ 3 e q (GeV) E al φ 3 e q (GeV) E T,al η η 0 E al (detetor level) E T,al (detetor level) q e 30 4 q e 30 φ 3 e q (GeV) E al φ 3 e q (GeV) E T,al η η 0 Figure -4. PGS alorimeter map of the energy deposits, as a funtion of pseudorapidity η and azimuthal angle ϕ, for a dilepton t t event with only two reonstruted jets. At the parton level, this partiular event has two b-quarks and two eletrons. The loation of a b-quark (eletron, muon) is marked with the letter q ( e, µ ). A grey irle delineates (the one of) a reonstruted jet, while a green dotted irle denotes a reonstruted lepton. In the upper two plots the alorimeter is filled at the parton level diretly from PYTHIA, while the lower two plots ontain results after PGS simulation. The left plots show absolute energy deposits E α, while in the right plots the energy in eah tower is shown projeted on the transverse plane as E α os θ α. 38

39 E al (parton level) E T,al (parton level) 6 e 40 6 e φ 3 q µ (GeV) E al φ 3 q µ (GeV) E T,al q 5 1 q η η 0 E al (detetor level) E T,al (detetor level) 6 e 40 6 e φ 3 q µ (GeV) E al φ 3 q µ (GeV) E T,al q 5 1 q η η 0 Figure -5. PGS alorimeter map of the energy deposits for a dilepton t t event with more than two reonstruted jets 39

40 Table -1. Seleted s quantities (in GeV) for the events shown in Figures -4, -5 and -1. The seond olumn shows the true invariant mass s true of the parent system: top quark pair in ase of Figures -4 and -5, or gluino pair in ase of Figure -1. The third olumn shows the value of the s (al) min variable ( 15) alulated at the parton level, without any PGS detetor simulation, but with the full detetor aeptane ut of η < 4.1. The fourth olumn lists the value of s (al) min obtained after PGS detetor simulation, while the last olumn shows the value of the s (reo) min variable defined in Equation ( 19). Event type PYTHIA parton level after PGS simulation strue s (al) min s (al) min s (reo) min t t event in Figure t t event in Figure SUSY event in Figure Before onluding this subsetion, we explain the reason for the improved performane of the s (reo) min variable in omparison to the s (al) min version. As already antiipated in Setion., the basi idea is that energy deposits whih are due to hard partiles originating from the hard sattering, tend to be lustered, while the energy deposits due to the UE tend to be more uniformly spread throughout the detetor. In order to see this pitorially, in Figures -4 and -5 we show a series of alorimeter maps of the ombined ECAL+HCAL energy deposits as a funtion of the pseudorapidity η and azimuthal angle ϕ. Sine the alorimeter in PGS is segmented in ells of ( η, ϕ) = (0.1, 0.1), eah alorimeter tower is represented by a square pixel, whih is olor-oded aording to the amount of energy present in the tower. We have hosen the olor sheme so that larger deposits orrespond to darker olors. Eah alorimeter map in Figures -4 and -5 has four panels. In the upper two panels the alorimeter is filled at the parton level diretly from PYTHIA. This orresponds to a perfet detetor, where we ignore any smearing effets due to the finite energy resolution. The lower two plots in Figures -4 and -5 show the orresponding results after PGS simulation. Thus by omparing the plots in the upper row to those in the bottom row, one an see the effet of the detetor resolution. While the finite detetor resolution does play some role, 40

41 we find that it is of no partiular importane for understanding the reason behind the big swings in the s min peaks observed in Figure -3. Let us instead onentrate on omparing the plots in the left olumn versus those in the right olumn. The left plots show the absolute energy deposit E α in the α alorimeter tower, while in the right plots this energy is shown projeted on the transverse plane as E α os θ α. The differene between the left and the right plots is quite striking. The plots on the left exhibit lots of energy, whih is deposited mostly in the forward alorimeter ells (at large η ) [50]. The plots on the right, on the other hand, show only a few lusters of energy, onentrated mostly in the entral part of the detetor. Those energy lusters give rise to the objets (jets, eletrons and photons) whih are reonstruted from the alorimeter. Furthermore, eah energy luster an be easily identified with a parton-level partile in the top deay hain. In order to exhibit this orrelation, in Figures -4 and -5 we use the following notation for the parton-level partiles: a b-quark (eletron, muon) is marked with the letter q ( e, µ ). A grey irle delineates (the one of) a reonstruted jet, while a green dotted irle marks a reonstruted lepton (eletron or muon). The lepton isolation requirement implies that green irles should be void of large energy deposits off-enter, and indeed we observe this to be the ase. In partiular, Figure -4 shows a bare-bone dilepton t t event with just two reonstruted jets and two reonstruted leptons (whih happen to be both eletrons). As seen in the Figure -4, the two jets an be easily traed bak to the two b-quarks at the parton level, and there are no additional reonstruted jets due to the UE ativity. Beause the event is so lean and simple, one might expet to obtain a reasonable value for s min, i.e. lose to the t t threshold. However, this is not the ase, if we use the alorimeter-based measurement s (al) min. As seen in Table -1, the measured value of s (al) min is very far off on the order of 1 TeV, even in the ase of a perfet detetor. 41

42 The reason for this disrepany is now easy to understand from Figure -4. Reall that s (al) min is defined in terms of the total energy E (al) in the alorimeter, whih in turn is dominated by the large deposits in the forward region, whih ame from the underlying event. More importantly, those ontributions are more or less equally spread over the forward and bakward region of the detetor, leading to anellations in the alulation of the orresponding longitudinal P z(al) momentum omponent. As a result, the first term in Equation ( 15) beomes ompletely dominated by the UE ontributions [51]. Let us now see how the alulation of s (reo) min is affeted by the UE. Sine objet reonstrution is done with the help of minimum transverse uts (for lustering and objet id), the relevant alorimeter plots are the maps on the right side in Figure -4. We see that the large forward energy deposits whih were ausing the large shift in s (al) min are not inorporated into any reonstruted objets, and thus do not ontribute to the (reo) s min alulation at all. In effet, the RECO-level presription for alulating s min is leaving out preisely the unwanted ontributions from the UE, while keeping the relevant ontributions from the hard sattering. As seen from Table -1, the alulated value of (reo) s min for that event is 363 GeV, whih is indeed very lose to the t t threshold. It is also smaller than the true s value of 47 GeV in that event, whih is to be expeted, sine by design s min s, and this event does not have any extra ISR jets to spoil this relation. It is instrutive to onsider another, more omplex t t dilepton event, suh as the one shown in Figure -5. The orresponding alulated values for s (al) min and s (reo) min shown in the seond row of Table -1. As seen in Figure -5, this event has additional jets and a lot more UE ativity. As a result, the alulated value of s (al) min are is shifted by almost TeV from the nominal s true value. Nevertheless, the RECO-level presription niely ompensates for this effet, and the alulated s (reo) min value is only 736 GeV, whih is within 100 GeV of the nominal s true = 638 GeV. Notie that in this example 4

43 we end up with a situation where s (reo) min happens quite often the tail of the s (reo) min > s true. Figure -3 indiates that this distribution is more populated than the (yellow-shaded) s true distribution. This should be no ause for onern. First of all, we are only interested in the peak of the s (reo) min any omparisons between s (reo) min distribution, and we do not need to make and s true. Seond, any suh omparison would be meaningless, sine the value of s true is a priori unknown, and unobservable..4.3 s (sub) min Variable Before onluding this setion, we shall use the t t example to also illustrate the idea of the subsystem s (sub) min variable developed in Setion.3. Dilepton t t events are a perfet testing ground for this idea, sine the WW subsystem deays leptonially, without any jet ativity. We therefore define the subsystem as the two hard isolated leptons resulting from the deays of the W -bosons. Correspondingly, we require two reonstruted leptons (eletrons or muons) at the PGS level 4, and plot the distribution of the leptoni subsystem s (sub) min variable in Figure -6. As before, the dotted (yellow-shaded) histogram represents the true s distribution of the W + W pair. As expeted, it quikly rises at the WW threshold (denoted by the vertial arrow), then falls off at large s. Sine the s (WW ) true the best we an do is to study the orresponding s (sub) min distribution is unobservable, distribution shown with the solid blak histogram. In this subsystem example, all UE ativity is lumped together with the upstream b-jets from the top quarks deays, and thus has no bearing on the properties of the leptoni s (sub) min. In partiular, we find that the value of s (sub) min is always smaller than the true s (WW ) true. More importantly, Figure -6 demonstrates that the peak in the (sub) s min distribution is found preisely at the mass threshold of the partiles (in this ase the two W bosons) whih initiated the subsystem. 4 The seletion effiieny for the two leptons is on the order of 60%, whih explains the different normalization of the distributions in Figures -3 and

44 Figure -6. [Distribution of various s min for the dilepton subsystem in dilepton t t events with two reonstruted leptons in PGS. The dotted (yellow-shaded) histogram gives the true s distribution of the W + W pair in those events. The blak histogram shows the distribution of the (leptoni) subsystem variable s (sub) min defined in Setion.3. In this ase, the subsystem is defined by the two isolated leptons, while all jets are treated as upstream partiles. The vertial arrow marks the W + W mass threshold. In analogy to ( 40) we an also write ( s ) (sub) min peak M (sub) p, ( 41) where M (sub) p is the ombined mass of all the parents initiating the subsystem. Figure -6 shows that in the t t example just onsidered, this relation holds to a very high degree of auray. This example should not leave the reader with the impression that hadroni jets are never allowed to be part of the subsystem. On the ontrary the subsystem may very well inlude reonstruted jets as well. The t t ase onsidered here in fat provides a perfet example to illustrate the idea. 44

45 Figure -7. Unit-normalized distribution of jet multipliity in dilepton t t events. Figure -8. Distributions of various s min for the dilepton t t sample, in addition to the two leptons, the subsystem now also inludes: exatly two b-tagged jets (blak histogram); the two highest p T jets (blue histogram); or all jets (red histogram). The dotted (yellow-shaded) histogram gives the true s distribution of the t t pair. 45

46 Let us reonsider the t t dilepton sample, and redefine the subsystem so that we now target the two top quarks as the parents initiating the subsystem. Correspondingly, in addition to the two leptons, let us allow the subsystem to inlude two jets, presumably oming from the two top quark deays. Unfortunately, in doing so, we must fae a variant of the partitioning 5 ombinatorial problem disussed in the introdution: as seen in Figure -7, the typial jet multipliity in the events is relatively high, and we must therefore speify the exat proedure how to selet the two jets whih would enter the subsystem. We shall onsider three different approahes. B-tagging. We an use the fat that the jets from top quark deay are b-jets, while the jets from ISR are typially light flavor jets. Therefore, by requiring exatly two b-tags, and inluding only the two b-tagged jets as part of the subsystem, we an signifiantly inrease the probability of seleting the orret jets. Of ourse, ISR will sometimes also ontribute b-tagged jets from gluon splitting, but that happens rather rarely and the orresponding ontribution an be suppressed by a distribution for the subsystem of leptons and b-tagged jets is shown in Figure -8 with the blak histogram. We see that, as expeted, the distribution peaks at the t t threshold and this time provides a measurement of the top quark mass: further invariant mass ut on the two b-jets. The resulting s (sub) min ( s (sub) min ) peak M (sub) p = m t = 350 GeV. ( 4) The disadvantage of this method is the loss in statistis: ompare the normalization of the blak histogram in Figure -8 after applying the two b-tags, to the dotted (yellow-shaded) distribution of the true t t distribution in the seleted inlusive dilepton sample (without b-tags). Seletion by jet p T. Here one an use the fat that the jets from top deays are on average harder than the jets from ISR. Correspondingly, by hoosing the two highest p T jets (regardless of b-tagging), one also inreases the probability to selet the orret jet pair. The orresponding distribution is shown in Figure -8 with the blue histogram, and is also seen to peak at the t t threshold. An important 5 By onstrution, the s min and s (sub) min ombinatorial problem. variables never have to fae the ordering 46

47 advantage of this method is that one does not have to pay the prie of redued statistis due to the two additional b-tags. No seletion. The most onservative approah would be to apply no seletion riteria on the jets, and inlude all reonstruted jets in the subsystem. Then the subsystem s (sub) min variable essentially reverts bak to the RECO-level inlusive variable s (reo) min already disussed in the previous subsetion. Not surprisingly, we find the peak of its distribution (red histogram in Figure -8) near the t t threshold as well. All three of these examples show that jets an also be usefully inorporated into the subsystem. The only question is whether one an find a reliable way of preferentially seleting jets whih are more likely to originate from within the intended subsystem, as opposed to from the outside. As we see in Figure -8, in the t t ase this is quite possible, although in general it may be diffiult in other settings, like the SUSY examples disussed in the next setion..5 An Exlusive SUSY Example: Multijet Events From Gluino Prodution Sine s min is a fully inlusive variable, arguably its biggest advantage is that it an be applied to purely jetty events with large jet multipliities, where no other method on the market would seem to work. In order to simulate suh a hallenging ase, we onsider gluino pair prodution in supersymmetry, with eah gluino fored to undergo a asade deay hain involving only QCD jets and nothing else. In this setion, two different possibilities for the gluino deays were onsidered: In one senario, the gluino g is fored to undergo a two-stage asade deay to the LSP. In the first stage, the gluino deays to the seond-lightest neutralino χ 0 and two quark jets: g q q χ 0. In turn, χ 0 itself is then fored to deay via a 3-body deay to quark jets and the LSP: χ 0 q q χ 0 1. The resulting gluino signature is 4 jets plus missing energy: g jj χ 0 jjjj χ 0 1. ( 43) Therefore, gluino pair prodution will nominally result in 8 jet events. Of ourse, as shown in Figure -9, the atual number of reonstruted jets in suh events is even higher, due to the effets of ISR, FSR and/or string fragmentation. As in Figure -9, eah suh event has on average 10 jets, presenting a formidable 47

48 ombinatoris problem. We suspet that all 6 mass reonstrution methods on the market are doomed if they were to fae suh a senario. It is therefore of partiular interest to see how well the s min method (whih is advertized as universally appliable) would fare under suh dire irumstanes. In the seond senario, the gluino deays diretly to the LSP via a three-body deay g jj χ 0 1, ( 44) so that gluino pair-prodution events would nominally have 4 jets and missing energy. For onreteness, in eah senario we fix the mass spetrum as was done in [50]: we use the approximate gaugino unifiation relations to relate the gaugino and neutralino masses as m g = 3m χ 0 = 6m χ 0 1. ( 45) We an then vary one of these masses, and hoose the other two in aord with these relations. Sine we assume three-body deays in Equations ( 44) and ( 43), we do not need to speify the SUSY salar mass parameters, whih an be taken to be very large. In addition, as implied by Equation ( 45), we imagine that the lightest two neutralinos are gaugino-like, so that we do not have to speify the higgsino mass parameter either, and it an be taken to be very large as well. After these preliminaries, our results for these two senarios are shown in Figures -10 and -11, orrespondingly. In Figure -10 (Figure -11) we onsider the 8-jet signature arising from ( 43) (the 4-jet signature arising from ( 44)). Panels (a) orrespond to a light mass spetrum m g = 600 GeV, m χ 0 = 00 GeV and m χ 0 1 = 100 GeV; while panels (b) orrespond to a heavy mass spetrum m g = 400 GeV, m χ 0 = 800 GeV and m χ 0 1 = 400 GeV. Eah plot shows the same four distributions 6 With the possible exeption of the M Tgen method of referene, C. Lester and A. Barr, MTGEN : Mass sale measurements in pair-prodution at olliders, (JHEP 071, 10 (007)). [33], see Setion.7 below. 48

49 as in Figure -3. The s min distributions are all plotted for the orret value of the missing mass parameter, namely /M = m χ 0 1. Overall, the results seen in Figures -10 and -11 are not too different from what we already witnessed in Figure -3 for the t t example. The (unobservable) distribution strue shown with the dotted yellow-shaded histogram has a sharp turn-on at the physial mass threshold M p = m g. If the effets of the UE are ignored, the position of this threshold is given rather well by the peak of the s (al) min Unfortunately, the UE shifts the peak in s (al) min the distribution of the RECO-level variable s (reo) min ontamination, and its peak is still in the right plae (near M p ). distribution (blue histogram). by 1- TeV (red histogram). Fortunately, (blak histogram) is stable against UE Having already seen a similar behavior in the t t example of the previous setion, these results may not seem very impressive, until one realizes just how ompliated those events are. For illustration, Figure -1 shows the previously disussed alorimeter maps for one partiular 8 jet event. This event happens to have 11 reonstruted jets, whih is onsistent with the typial jet multipliity seen in Figure -9. The values of the s quantities of interest for this event are listed in Table -1. We see that the RECO presription for alulating s min is able to ompensate for a shift in s of more than 1.5 TeV! A asual look at Figure -1 should be enough to onvine the reader just how daunting the task of mass reonstrution in suh events is. In this sense, the ease with whih the s min method reveals the gluino mass sale in Figures -10 and -11 is quite impressive. 49

50 Figure -9. Unit-normalized distribution of jet multipliity in gluino pair prodution events, with eah gluino deaying to four jets and a χ 0 1 LSP as in ( 43). 50

51 Figure -10. Distribution of various s min with a SUSY example of gluino pair prodution, with eah gluino deaying to four jets and a χ 0 1 LSP as indiated in ( 43). The mass spetrum is hosen as: (a) m g = 600 GeV, m χ 0 = 00 GeV and m χ 0 1 = 100 GeV; or (b) m g = 400 GeV, m χ 0 = 800 GeV and m χ 0 1 = 400 GeV. All three s min distributions are plotted for the orret value of the missing mass parameter, in this ase /M = m χ 0 1. Figure -11. Distribution of various s min with a SUSY example of gluino pair prodution, with eah gluino deaying to two jets and a χ 0 1 LSP as in ( 44). 51

52 E al (parton level) E T,al (parton level) 6 q q 40 6 q q q q q q φ 3 q 5 0 (GeV) E al φ 3 q 5 0 (GeV) E T,al q q q q q 5 1 q η η 0 E al (detetor level) E T,al (detetor level) 6 q q 40 6 q q q q q q φ 3 q 5 0 (GeV) E al φ 3 q 5 0 (GeV) E T,al q q q q q 5 1 q η η 0 Figure -1. PGS alorimeter map of the energy deposit for a SUSY event of gluino pair prodution, with eah gluino fored to deay to 4 jets and the LSP as in ( 43). The SUSY mass spetrum is as in Figures -10(a) and -11(a): m g = 600 GeV, m χ 0 = 00 GeV and m χ 0 1 = 100 GeV. As in Figures -4 and -5, the irles denote jets reonstruted in PGS, and here q marks the loation of a quark from a gluino deay hain. Therefore, a irle without a q inside orresponds to a jet resulting from ISR or FSR, while a letter q without an aompanying irle represents a quark in the gluino deay hain whih was not subsequently reonstruted as a jet. 5

53 Figure -13. Distribution of the s (al) min (dotted red) and s (reo) min (solid blak) variables in inlusive SUSY prodution for the GMSB GM1a benhmark study point with parameters Λ = 80 TeV, M mes = 160 TeV, N mes = 1, tan β = 15 and µ > 0. The dotted yellow-shaded histogram is the true s distribution of the parent pair of SUSY partiles produed at the top of eah deay hain (the identity of the parent partiles varies from event to event). The s min distributions are shown for /M = 0 and are normalized to 1 fb 1 of data. The vertial arrows mark the mass thresholds for a few dominant SUSY pair-prodution proesses. Table -. Masses (in GeV) of the SUSY partiles at the GM1b study point. Here ũ and d ( l and ν l ) stand for either of the first two generations squarks (sleptons). ũ L d L ũ R d R l L ν l l R χ ± χ 0 4 χ 0 3 g t 1 b 1 t b τ ν τ τ 1 χ ± 1 χ 0 χ 0 1 G An Inlusive SUSY Example: GMSB Study Point GM1b In the Introdution we already mentioned that s min is a fully inlusive variable. Here we would like to point out that there are two different aspets of the inlusivity property of s min : Objet-wise inlusivity: s min is inlusive with regards to the type of reonstruted objets. The definition of s (reo) min does not distinguish between the different types 53

54 of reonstruted objets (and s (al) min makes no referene to any reonstruted objets at all). This makes s min a very onvenient variable to use in those ases where the newly produed partiles have many possible deay modes, and restriting oneself to a single exlusive signature would ause loss in statistis. For illustration, onsider the gluino pair prodution example from the previous setion. Even though we are always produing the same type of parent partiles (two gluinos), in general they an have several different deay modes, leading to a very diverse sample of events with varying number of jets and leptons. Nevertheless, the s (reo) min pinpoint the gluino mass sale, as explained in Setion.5. distribution, plotted over this whole signal sample, will still be able to Event-wise inlusivity: s min is inlusive also with regards to the type of events, i.e. the type of new partile prodution. For simpliity, in our previous examples we have been onsidering only one prodution mehanism at a time, but this is not really neessary s min an also be applied in the ase of several simultaneous prodution mehanisms. In order to illustrate the last point, in this setion we shall onsider the simultaneous prodution of the full spetrum of SUSY partiles at a partiular benhmark point. We hose the GM1b CMS study point [59], whih is nothing but a minimal gauge-mediated SUSY-breaking (GMSB) senario on the SPS8 Snowmass slope [81]. The input parameters are Λ=80 TeV, M mes =160 TeV, N mes =1, tan β = 15 and µ > 0. The physial mass spetrum is given in Table -. Point GM1b is haraterized by a neutralino NLSP, whih promptly deays (predominantly) to a photon and a gravitino. Therefore, a typial event has two hard photons and missing energy, whih provide good handles for suppressing the SM bakgrounds. We now onsider inlusive prodution of all SUSY subproesses and plot the s min distributions of interest in Figure -13. As usual, the dotted yellow-shaded histogram is the true s distribution of the parent pair of SUSY partiles produed at the top of eah deay hain. Sine we do not fix the prodution subproess, the identity of the parent partiles varies from event to event. Naturally, the most ommon parent partiles are the ones with the highest prodution ross-setions. For point GM1b, at a 14 TeV LHC, strong SUSY prodution dominates, and is 87% of the total ross-setion. A few of the dominant subproesses and their ross-setions are listed in Table

55 Table -3. Cross-setions (in pb) and parent mass thresholds (in GeV) for the dominant prodution proesses at the GM1b study point. The listed squark ross-setions are summed over the light squark flavors and onjugate states. The total SUSY ross-setion at point GM1b is 9.4 pb. Proess χ ± 1 χ0 χ + 1 χ 1 g g g q R g q L q R q R q L q R q L q L σ (pb) M p (GeV) The true s distribution in Figure -13 exhibits an interesting double-peak struture, whih is easy to understand as follows. As we have seen in the exlusive examples from Setions.4 and.5, at hadron olliders the partiles tend to be produed with s lose to their mass threshold. As seen in Table -, the partile spetrum of the GM1b point an be broadly divided (aording to mass) into two groups of superpartners: eletroweak setor (the lightest hargino χ ± 1, seond-to-lightest neutralino χ0 and sleptons) with a mass sale on the order of 00 GeV and a strong setor (squarks and gluino) with masses of order GeV. The first peak in the true s distribution (near s 500 GeV) arises from the pair prodution of two partiles from the eletroweak setor, while the seond, broader peak in the range of s GeV is due to the pair prodution of two olored superpartners 7. Eah one of those peaks is made up of several ontributions from different individual subproesses, but beause their mass thresholds 8 are so lose, they annot be individually resolved, and appear as a single bump. If one ould somehow diretly observe the true s SUSY distribution (the dotted yellow-shaded histogram in Figure -13), this would lead to some very interesting onlusions. First, from the presene of two separate peaks one would know immediately 7 The attentive reader may also notie two barely visible bumps (near 950 GeV and 1150 GeV) refleting the assoiated prodution of one olored and one unolored partile: g χ ± 1, g χ0 and q χ ± 1, q χ0, orrespondingly. 8 A few individual mass thresholds are indiated by vertial arrows in Figure

56 that there are two widely separated sales in the problem. Seond, the normalization of eah peak would indiate the relative size of the total inlusive ross-setions (in this example, of the partiles in the eletroweak setor versus those in the strong setor). Finally, the broadness of eah peak is indiative of the total number of ontributing subproesses, as well as the typial mass splittings of the partiles within eah setor. It may appear surprising that one is able to draw so many onlusions from a single distribution of an inlusive variable, but this just omes to show the importane of s as one of the fundamental ollider physis variables. Unfortunately, beause of the missing energy due to the esaping invisible partiles, the true s distribution annot be observed, and the best one an do to approximate it is to look at the distributions of our inlusive s min variables disussed in Setion.: the alorimeter-based s (al) min (dotted red histogram in Figure -13) and the RECO-level s (reo) min histogram in Figure -13). First let us onentrate on the alorimeter-based version s (al) min variable variable (solid blak (dotted red histogram). We an immediately see the detrimental effets of the UE: first, the eletroweak prodution peak has been almost ompletely smeared out, while the strong prodution peak has been shifted upwards by more than a TeV! This behavior is not too surprising, sine the same effet was already enountered in our previous examples in Setions.4 and.5. Fortunately, we now also know the solution to this problem: one needs to onsider the RECO-level variable s (reo) min instead, whih traks the true s distribution muh better. We an see evidene of this in Figure -13 as well. In partiular, s (reo) min does show two separate peaks (indiating that SUSY prodution takes plae at two different mass sales), the peaks are in their proper loations (relative to the missing mass sale /M), and have the orret relative width, hinting at the size of the mass splittings in eah setor. We thus onlude that all of the interesting physis onlusions that one would be able to reah from looking at the true s distributions, an still be made based on the inlusive distribution of our RECO-level s (reo) min variable. 56

57 Figure -14. Distributions of various s min for the GMSB SUSY example onsidered in Figure -13. Here the subsystem is defined in terms of the two hard photons resulting from the two χ 0 1 G + γ deays. The vertial arrow marks the onset for inlusive χ 0 1 χ 0 1 prodution. Before onluding this setion, we shall take the opportunity to use the GM1b example to also illustrate the s (sub) min variable proposed in Setion.3. As already mentioned, the GM1b study point orresponds to a GMSB senario with a promptly deaying Bino-like χ 0 1 NLSP. Most events therefore ontain two hard photons from the two χ 0 1 deays to gravitinos. Then it is quite natural to define the exlusive subsystem in Figure - in terms of these two photons. The orresponding s (sub) min distribution is shown in Figure -14 with the blak solid histogram. For ompleteness, we also show the true s distribution of the χ 0 1 pair (dotted yellow-shaded histogram). The vertial arrow marks the loation of the χ 0 1 χ 0 1 mass threshold. We notie that the peak of the (sub) s min distribution niely reveals the loation of the neutralino mass threshold, and from there the neutralino mass itself. We see that the method of s (sub) min provides a very simple way of measuring the NLSP mass in suh GMSB senarios (for an alternative approah based on M T, see [61]). 57

58 .7 Comparison to Other Inlusive Collider Variables Having disussed the newly proposed variables s (reo) min and s (sub) min in various settings in Setions.4-.6, we shall now ompare them to some other global inlusive variables whih have been disussed in the literature in relation to determining a mass sale of the new physis. For simpliity here we shall onentrate only on the most model-independent variables, whih do not suffer from the topologial and ombinatorial ambiguities mentioned in the Introdution. At the moment, there are only a handful of suh variables. Depending on the treatment of the unknown masses of the invisible partiles, they an be lassified into one of the following two ategories: Variables whih do not depend on an unknown invisible mass parameter. The most popular members of this lass are the missing H T variable N obj /H T P Ti, ( 46) whih is simply the magnitude of the /H T vetor from Equation ( ), and the salar H T variable N obj H T /H T + P Ti. ( 47) i=1 Here we follow the notation from Setion., where P Ti is the measured transverse momentum of the i-th reonstruted objet in the event (i = 1,,..., N obj ). The main advantage of /H T and H T is their simpliity: both are very general, and are defined purely in terms of observed quantities, without any unknown mass parameters. The downside of /H T and H T is that they annot be diretly orrelated with any physial mass sale in a model-independent way 9. i=1 9 Some early studies of H T -like variables found interesting linear orrelations between the peak in the H T distribution and a suitably defined SUSY mass sale in the ontext of speifi SUSY models, e.g. minimal supergravity (MSUGRA) [13, 63, 64], minimal GMSB [63], or mixed moduli-mediation [65]. However, any suh orrelations do not survive further srutiny in more generi SUSY senarios, see e.g. [66]. 58

59 Variables whih exhibit dependene on one or more invisible mass parameters. As two representatives from this lass we shall onsider M Tgen from Referene [33] and s (reo) min from Setion. here. We shall not repeat the tehnial definition of M Tgen, and instead refer the uninitiated reader to the original paper [33]. Suffie it to say that the method of M Tgen starts out by assuming exatly two deay hains in eah event. The arising ombinatorial problem is then solved by brute fore by onsidering all possible partitions of the event into two sides, omputing M T for eah suh partition, and taking the minimum value of M T found in the proess. Both M Tgen and s (reo) min introdue a priori unknown parameters related to the mass sale of the missing partiles produed in the event. In the ase of s (reo) min, this is simply the single parameter /M, measuring the total invisible mass (in the sense of a salar sum as defined in Equation ( 3)). The M Tgen variable, on the other hand, must in priniple introdue two separate missing mass parameters /M 1 and /M (one for eah side of the event). However, the existing appliations of M Tgen in the literature have typially made the assumption that /M 1 = /M, although this is not really neessary and one ould just as easily work in terms of two separate inputs /M 1 and /M [40, 41]. The inonveniene of having to deal with unknown mass parameters in the ase of M Tgen and s (reo) min is greatly ompensated by the luxury of being able to relate ertain features of their distributions to a fundamental physial mass sale in a robust, model-independent way. In partiular, the upper endpoint M (max) Tgen of the M Tgen distribution gives the larger of the two parent masses max{m P1, M P } [6]. Therefore, if the two parent masses are the same, i.e. M P1 = M P, then the parent mass threshold M p = M P1 + M P is simply given by M p = M (max) Tgen. ( 48) On the other hand, as we have already seen in Setions.4-.6, the peak of the s (reo) min is similarly orrelated with the parent mass threshold, see Equation ( 40). In priniple, all four 10 of these variables are inlusive both objet-wise and event-wise. It is therefore of interest to ompare them with respet to: 1. The degree of orrelation with the new physis mass sale M p.. Stability of this orrelation against the detrimental effets of the UE. Figures -15, -16 and -17 allow for suh omparisons. 10 We aution the reader that H T is often defined in a more narrow sense than Equation ( 47). For example, sometimes the /H T term is omitted, sometimes the sum in Equation ( 47) is limited to the reonstruted jets only; or to the four highest p T jets only; or to all jets, but starting from the seond-highest p T one. 59

60 Figure -15. Comparison various s min with other transverse variables for t t prodution. In addition to the true s (yellow shaded) and s (reo) min (blak) distribution, we also plot the distributions of M Tgen (red dots), M TTgen (magenta dots), H T (green dots) and /H T (blue dots), all alulated at the RECO-level. All results inlude the full simulation of the underlying event. For plotting onveniene, the /H T distribution is shown saled down by a fator of. The vertial dotted line marks the t t mass threshold M p = m t = 350 GeV. In Figure -15 we first revisit the ase of the dilepton t t sample disussed in Setion.4. In addition to the true s (yellow shaded) and s (reo) min (blak) distribution already appearing in Figure -3, we now also plot the distributions of M Tgen (red dots), H T (green dots) and /H T (blue dots), all alulated at the RECO-level. For ompleteness, in Figure -15 we also show a variant of M Tgen, alled M TTgen (magenta dots), where all visible partile momenta are first projeted on the transverse plane, before omputing M Tgen in the usual way [33]. All results inlude the full simulation of the underlying event. For plotting onveniene, the /H T distribution is shown saled down by a fator of. Based on the results from Figure -15, we an now address the question, whih inlusive distribution shows the best orrelation with the parent mass sale (in this ase the parent mass sale is the t t mass threshold M p = m t = 350 GeV marked 60

61 by the vertial dotted line in Figure -15). Let us begin with the two variables, /H T and H T, whih do not depend on any unknown mass parameters. Figure -15 reveals that the /H T distribution peaks very far from threshold, and therefore does not reveal muh information about the new physis mass sale. Consequently, any attempt at extrating new physis parameters out of the missing energy distribution alone, must make some additional model-dependent assumptions []. On the other hand, the H T distribution appears to orrelate better with M p, sine its peak is relatively lose to the t t threshold. However, this relationship is purely empirial, and it is diffiult to know what is the assoiated systemati error. Moving on to the variables whih arry a dependene on a missing mass parameter, (reo) s min, M Tgen and M TTgen, we see that all three are affeted to some extent by the presene of the UE. In partiular, the distributions of M Tgen and M TTgen are now smeared and extend signifiantly beyond their expeted endpoint ( 48). Not surprisingly, the UE has a larger impat on M Tgen than on M TTgen. In either ase, there is no obvious endpoint. Nevertheless, one ould in priniple try to extrat an endpoint through a straight-line fit, for example, but it is lear that the obtained value will be wrong by a ertain amount (depending on the hosen region for fitting and on the assoiated bakgrounds). All these diffiulties with M Tgen and M TTgen are simply a refletion of the hallenge of measuring a mass sale from an endpoint as in ( 48), instead of from a peak as in ( 40). By omparison, the determination of the new physis mass sale from the s (reo) min distribution is muh more robust. As shown in Figure -15, the s (reo) min peak is barely affeted by the UE, and is still found preisely in the right loation. All of the above disussion an be diretly applied to the SUSY examples onsidered in Setion.5 as well. As an illustration, Figures -16 and -17 revisit two of the gluino examples from Setion.5. We onsider gluino pair-prodution with a light SUSY spetrum (m χ 0 1 = 100 GeV, m χ 0 = 00 GeV and m g = 600 GeV). Then in Figure -16 eah gluino deays to 4 jets as in Equation ( 43), while in Figure

62 eah gluino deays to jets as in Equation ( 44). (Thus Figure -16 is the analogue of Figure -10(a), while Figure -17 is the analogue of Figure -11(a).) The onlusions from Figures -16 and -17 are very similar. These results onfirm that /H T is not very helpful in determining the gluino mass sale M p = m g = 100 GeV (indiated by the vertial dotted line). The H T distribution, on the other hand, has a nie well-defined peak, but the loation of the H T peak always underestimates the gluino mass sale (by about 50 GeV in eah ase). Figures -16 and -17 also onfirm the effet already seen in Figure -15: that the underlying event auses the M Tgen and M TTgen distributions to extend well beyond their upper kinemati endpoint, thus violating ( 48) and making the orresponding extration of M p rather problemati. In fat, just by looking at Figures -16 and -17, one might be tempted to dedue that, if anything, it is the peak in M Tgen that perhaps might indiate the value of the new physis mass sale and not the M Tgen endpoint. Finally, the s (reo) min distribution also feels to some extent the effets from the UE, but always has its peak in the near viinity of M p. Therefore, among the five inlusive variables under onsideration here, s (reo) min appears to provide the best estimate of the new physis mass sale. The orrelation of Equation ( 40) is seen to hold very well in Figure -17 and reasonably well in Figure

63 Figure -16. Comparison various s min with other transverse variables for the gluino pair prodution example from Setion.5, with eah gluino deaying to 4 jets as in ( 43). We use the light SUSY mass spetrum from Figure -10(a). The vertial dotted line now shows the g g mass threshold M p = m g = 100 GeV. Figure -17. Comparison various s min with other transverse variables for gluino pair prodution with eah gluino deaying to jets as in ( 44). Compare to Figure -11(a). 63

64 CHAPTER 3 INVARIANT MASS ENDPOINTS METHOD In this hapter we onentrate on the lassi method of kinematial endpoints [13]. We set out to redesign this standard algorithm for performing these studies, by pursuing two main objetives: Improving on the experimental preision of the SUSY mass determination. For example, we required that our analysis be based exlusively on upper invariant mass endpoints, whih are expeted to be measured with a greater preision than the orresponding lower endpoints (a.k.a. thresholds). Consequently, we did not make use of the threshold measurement mjll(θ> min π ), whih has been an integral part of most SUSY studies sine Referene [16]. In the same vein, we also demanded that we should not rely on any features observed in a twoor a three-dimensional invariant mass distribution suh measurements are expeted to be less preise than the (upper) endpoints extrated from simple one-dimensional histograms. Avoiding any parameter spae region ambiguities. It is well known that some of the invariant mass endpoints used in the onventional analyses are pieewise-defined funtions. This feature may sometimes lead to multiple solutions for the SUSY mass spetrum in the LHC inverse problem [9, 17, 67 69]. In order to safeguard against this possibility, we onservatively demanded from the outset that none of our endpoint measurements be given by pieewise defined funtions. This rather strit requirement rules out three of the standard endpoint measurements m max m max jl(lo), and mmax jl(hi). In order to meet these objetives, in Setion 3. we proposed a set of new invariant mass variables whose upper kinemati endpoints an be alternatively used for SUSY mass reonstrution studies. Then in Setion 3.3 we outlined a simple analysis whih was based on the partiular set of four invariant mass variables (3 41), all of whih satisfy our requirements. In Setion we provided simple analytial formulas for the SUSY mass spetrum in terms of the four measured endpoints in Equation (3 41). Our solutions revealed a surprise: in spite of the two-fold ambiguity as in Equations (3,3 3) in the interpretation of two of our endpoints Mjl(u) max and mmax jl(u), the answer for three (m D, m C and m A ) out of the four SUSY masses is unique! jll, 64

65 Figure 3-1. The typial asade deay hain under onsideration in this hapter. Here D, C, B and A are new BSM partiles, while the orresponding SM deay produts are: a QCD jet j, a near lepton l ± n and a far lepton l f. This hain is quite ommon in SUSY, with the identifiation D = q, C = χ 0, B = l and A = χ 0 1, where q is a squark, l is a slepton, and χ 0 1 ( χ 0 ) is the first (seond) lightest neutralino. In what follows we shall quote our results in terms of the D mass m D and the three dimensionless squared mass ratios R CD, R BC and R AB defined in Equation (3 6). The fourth mass (m B ) is also known, up to the two-fold ambiguity as in Equation (3 55), whih an be easily resolved by a variety of methods disussed and illustrated in Setions 3.3. and In Setion 3.4 we applied our tehnique to two speifi examples the LM1 and LM6 CMS study points. Following the previous SUSY studies, for illustration of our results we shall use the generi deay hain D jc jl ± n B jl ± n l f A shown in Figure 3-1. Here D, C, B and A are new BSM partiles with masses m D, m C, m B and m A. Their orresponding SM deay produts are: a QCD jet j, a near lepton l ± n and a far lepton l f. This deay hain is quite ommon in SUSY, with the identifiation D = q, C = χ 0, B = l and A = χ 0 1, where q is a squark, l is a slepton, and χ 0 1 ( χ 0 ) is the first (seond) lightest neutralino. However, our analysis is not limited to SUSY only, sine the hain in Figure 3-1 also appears in other BSM senarios, e.g. Universal Extra Dimensions [71]. For onreteness, we shall assume that all three deays exhibited in Figure 3-1 are two-body, i.e. we shall onsider the mass hierarhy m D > m C > m B > m A > 0. (3 1) 65

66 This presents the most hallenging ase, in whih one has to determine all four masses m D, m C, m B and m A. The idea of the kinemati endpoint method is very simple. Given the SM deay produts j, l n and l f exhibited in Figure 3-1, form the invariant mass 1 of every possible ombination, m ll, m jln, m jlf, and m jll, plot the resulting distributions and measure the orresponding upper kinemati endpoints [13, 16, 17] (m max ll ) = m D R CD (1 R BC ) (1 R AB ); (3 ) ( m max jl n ) = m D (1 R CD ) (1 R BC ) ; (3 3) ( m max jl f ) = m D (1 R CD ) (1 R AB ) ; (3 4) m D (1 R CD)(1 R AC ), for R CD < R AC, ase (1, ), ( ) m max jll = m D (1 R BC)(1 R AB R CD ), for R BC < R AB R CD, ase (, ), (3 5), m D (1 R AB)(1 R BD ), for R AB < R BD, ase (3, ), m D ( 1 RAD ), otherwise, ase (4, ). Here and below we write all results in terms of an overall mass sale (given by the mass m D of the heaviest BSM partile D) and three dimensionless squared mass ratios R ij m i m j, i, j {A, B, C, D}. (3 6) 1 We shall see below that the formulas simplify onsiderably if we use invariant masses squared instead. This distintion is not entral to our analysis. 66

67 Note that there are only three independent ratios in Equation (3 6). We shall take those to be R AB, R BC, and R CD (see Figure 3-1), and their definition domain will be the interval (0, 1). 3.1 Three Generi Problems in Invariant-mass Endpoint Methods In spite of their transparent theoretial meaning, the set of four endpoints in Equations (3 through 3 5) by themselves have (justifiably) never been used as the sole basis for a SUSY mass determination analysis. This is due to three generi problems, whih are all very well known, and are separately reviewed in the next three subsetions 3.1.1, 3.1. and Our new approah to resolving these three problems, and the outline of the rest of the paper are presented in Setion Near-far Lepton Ambiguity The first problem is that one annot differentiate between the near and far leptons l n and l f on an event-by-event basis. Sine all deays in Figure 3-1 are prompt, both leptons point bak to the primary interation vertex and there is no way to tell whih ame first and whih ame seond. Consequently, one annot separately onstrut the individual m jln and m jlf invariant mass distributions, whose upper endpoints would be given by Equations (3 3) and (3 4). This problem has motivated most of the previous invariant mass studies in the literature, beginning with [16], to introdue an alternative definition of the two jl distributions, simply by ordering the two m jl entries in eah event by invariant mass as follows m jl(lo) min {m jln, m jlf }, (3 7) m jl(hi) max {m jln, m jlf }. (3 8) As seen in Equation (3 5), at times we shall also utilize one or more of the other three ratios, R AC, R AD and R BD, whenever this will lead to a simplifiation of the formulas. Of ourse, the latter three ratios are related to our preferred set {R AB, R BC, R CD } due to the transitivity property R ij R jk = R ik. 67

68 Both of the newly defined quantities m jl(lo) and m jl(hi) also exhibit upper kinemati endpoints (mjl(lo) max and mmax jl(hi), orrespondingly). Sine the individual m jl(lo) and m jl(hi) distributions are observable, their endpoints are experimentally measurable and an be related to the underlying SUSY mass spetrum as follows [16, 17] ( ) m max jl n, for ( RAB ) 1 < R BC < 1, ase (, 1), ( m max ( jl(lo)) = mjl(eq)) max, for RAB < R BC < ( R AB ) 1, ase (, ), ( m max jl(hi)) = ( m max jl(eq)), for 0 < RBC < R AB, ase (, 3); ( m max jl f ), for ( RAB ) 1 < R BC < 1, ase (, 1), ( m max jl f ), for RAB < R BC < ( R AB ) 1, ase (, ), (3 9) (3 10) ( m max jl n ), for 0 < RBC < R AB, ase (, 3); where and m max jl n and m max jl f ( m max jl(eq)) = m D (1 R CD ) (1 R AB ) ( R AB ) 1 (3 11) were already defined in Equations (3 3) and (3 4), orrespondingly. With this approah, the original set of 4 endpoints in Equations (3-3 5) is replaed by m max ll, mjll max, mjl(lo), max mjl(hi). max (3 1) In ontrast to this onventional approah in the literature, we shall adopt a very different attitude towards resolving the problem of the near-far lepton ambiguity. We will do the simplest possible thing, namely, we shall do nothing. We shall never ask the question whih lepton was l n and whih one was l f?. We shall also not use the ordering in Equations (3 7,3 8). Instead, we shall simply take the two m jl entries in eah event, and always treat them in a symmetri fashion. For example, any observable invariant mass distribution that we will build out of the two measured quantities m jln and 68

69 m jlf should be invariant under the symmetry m jln m jlf. (3 13) The advantages of our approah may not be immediately obvious at this point, but will beome lear in the proess of our mass determination analysis in Setion 3.3 below Insuffiient Number of Measurements. The seond problem assoiated with the original set of four measurements in Equations (3-3 5), as well as the alternative set in Equation (3 1), is that the measured endpoints may not all be independent from eah other. Indeed, there are ertain regions of parameter spae where one finds the following orrelation [17] ( ) m max ( ) jll = m max jl(hi) + (m max ll ). (3 14) In this ase, the four measurements in Equation (3 1) are learly insuffiient to pin down all four independent input parameters m D, m C, m B and m A. Therefore, one has to measure an additional independent endpoint. To this end, it has been suggested to onsider the onstrained distribution m jll(θ> π ), whih exhibits a useful lower kinemati endpoint m min jll(θ> π ) [16] { ( ) mjll(θ> min 1 π = ) 4 m D (1 R AB )(1 R BC )(1 + R CD ) (3 15) + (1 R AC )(1 R CD ) (1 R CD ) (1 + R AB ) (1 + R BC ) 16R AC }. The distribution m jll(θ> π ) is nothing but the usual m jll distribution over a subset of the original events, subjet to the additional dilepton mass onstraint m max ll < m ll < m max ll. (3 16) 69

70 In the rest frame of partile B, this ut implies the following restrition on the opening angle θ between the two leptons [70] θ > π, (3 17) thus justifying the notation for m jll(θ> π ). The advantage of the threshold endpoint measurement (3 15) is that it is always independent of the other four measurements in (3 1). As a result, it would appear that the enlarged set of five kinemati endpoint measurements m max ll, mjll max, mjl(lo), max mjl(hi), max mjll(θ> min π ) (3 18) should be in priniple suffiient to determine all four unknown masses (see, however Setion below). Unfortunately, the threshold (3 15) also suffers from ertain disadvantages, whih are mostly of experimental nature. It is generally expeted that the experimental preision on the determination of the lower kinemati endpoint (3 15) will be rather inferior ompared to the preision on the other four upper kinemati endpoints (3 1) [17]. There are several generi reasons for suh a pessimisti attitude. First, the region in the m jll(θ> π ) distribution near its lower endpoint (3 15) is rather sparsely populated, resulting in a shallow edge and sizable statistial errors. To make matters worse, the m jll(θ> π ) distribution near its lower edge is a onvex funtion [7], whih makes it even more diffiult to tell where the signal ends and the tails from various soures begin [17]. Finally, the low mass region of almost any invariant mass distribution in SUSY is generally assoiated with larger SM (as well as SUSY ombinatorial) bakgrounds ompared to its high mass ounterpart. Overall we find all these disadvantages suffiiently onvining so that we will drop the measurement in Equation (3 15) altogether and will never use it in the ourse of our analysis in Setion 3.3 below. We will be justified in doing so, sine the linear 70

71 dependene problem (3 14), whih has plagued previous studies and was the prime motivation for introduing the mjll(θ> min π ) measurement in the first plae, will have no effet on our analysis. In fat, we will not be using the endpoint measurement mjl(hi) max (for the reasons given in the previous subsetion 3.1.1) and we will not be using the endpoint measurement m max jll (for the reasons given in the following subsetion 3.1.3). One these two problemati measurements are removed from onsideration, the linear dependene problem (3 14) does not arise, and the threshold measurement (3 15) is not entral to the analysis any more Parameter Spae Region Ambiguity The third problem with the onventional set of measurements (3 18) is immediately obvious from the defining Equations (3 5), (3 9) and (3 10) for the kinemati endpoints mjll max, mmax jl(lo), and mmax jl(hi), orrespondingly. One an see that the relevant expressions are pieewise-defined funtions, i.e. they depend on the values of the independent variables m A, m B, m C and m D. For example, there are four different ases for m max jll, and three different ases for the pair of (mjl(lo) max, mmax jl(hi) ). Altogether, these give rise to 9 different ases 3 whih must be separately onsidered [9, 17]. Of ourse, this represents a problem, sine the masses are a priori unknown, and it is not lear whih ase is the relevant one. Barring any model-dependent assumptions, one is fored to onsider all possibilities, obtain a solution for the spetrum, and only at the very end, test whether the solution falls within the parameter spae appliable for the ase at hand. This proedure may often result in several alternative solutions [9, 17, 67 69, 73, 74]. In fat, we reently proved that there exists a sizable parameter spae region in whih even the full set of measurements (3 18) would always yield two alternative solutions, even under ideal experimental onditions[9]. The problem is further exaerbated by the inevitable 3 The remaining 3 ases are always unphysial [17]. 71

72 experimental errors on the measurements (3 18), whih would allow for an even larger number of fake or dupliate solutions [9, 68, 69]. Having identified the root of the dupliation problem as the pieewise definition of the mathematial formulas in (3 5,3 9,3 10), our solution to the problem will be again very simple and onservative. We will simply avoid using any kinemati endpoints whih are given in terms of pieewise-defined expressions. This requirement automatially eliminates from onsideration the three onventional endpoints mjll max, mmax jl(lo), and mmax jl(hi). Sine we already gave up on mjll(θ> min π ) in the previous subsetion, this leaves mmax ll as the only measurement out of the onventional set (3 18) that we shall use in our analysis. This is perhaps the most drasti differene between our approah and all previous studies in the literature Posing The Problem In the previous three subsetions we disussed eah of the three generi theoretial 4 problems with the previous appliations of the kinemati endpoint method for mass determination. We are now ready to expliitly formulate our main goal in this paper. We aim to design a method for measuring the masses of the partiles in the deay hain of Figure 3-1, whih is based on kinemati endpoint information, and satisfies the following requirements: 4 In addition, there are problems whih are of experimental nature, e.g. identifying the orret jet and the orret lepton pair resulting from the deay hain in Figure 3-1. There exists a set of standard experimental tehniques whih are aimed at overoming these problems, e.g. the opposite flavor subtration for the two leptons and the mixed event subtration for the jet [75]. Wrong ll and jl pairings an also be identified and a posteriori removed whenever an invariant mass entry for m ll, m jl or m jll exeeds the orresponding kinemati endpoint mll max, mjl(hi) max or mmax jll. In what follows we shall assume that those preliminary steps have already been done and the samples we are dealing with have already been appropriately subtrated to remove the ombinatorial bakground. 7

73 It does not make use of any kinemati endpoints whose interpretation is ambiguous, i.e. whose expressions in terms of the physial masses are pieewise-defined funtions. It does not make use of any lower kinemati endpoints suh as the threshold mjll(θ> min π ), due to the experimental hallenges with suh measurements. It relies solely on 1-dimensional distributions, unlike the methods reently advertised in [9, 1, 73, 74, 76], whih utilize -dimensional orrelation plots. While the latter do provide a wealth of valuable information, they also typially require more data in order to obtain good enough statistis for drawing any robust onlusions from them. In ontrast, the one-dimensional distributions should be available rather early on, and with suffiient statistis for endpoint measurements. As already alluded to in the previous subsetions, the first two requirements already eliminate four out of the five onventional inputs (3 18). Obviously, we will need to find a way to replae those with an alternative set of kinemati endpoint measurements whih nevertheless satisfy the above requirements. In Setion 3. we introdue and investigate a new set of invariant mass variables whose upper endpoints an be useful for our analysis. Then in Setion 3.3 we outline our basi method, whih makes use of some of these new variables. We illustrate our disussion in Setion 3.4 with two numerial examples: the LM1 and LM6 CMS study points. In Appendix A we supply the analyti expressions for the shapes of the 1-dimensional invariant mass distributions used in our main analysis in Setion Those results an be useful in improving the preision on the extration of the kinematial endpoints. 3. New Variables In this setion we propose a new set of invariant mass (squared) variables. As already explained in the Introdution, our variables should be omposed of m jl n and m jl f in a symmetri way, in aordane with (3 13). Consequently, any plotting manipulations or mathematial operations involving m jl n and m jl f should obey the symmetry implied by Equation (3 13). 73

74 3..1 The Union mjl n mjl f We begin with the simplest ase, where we postpone applying any mathematial operations to mjl n and mjl f, and instead simply plot them. The requirement of Equation (3 13) implies that the only possibility is to plae both of them together on the same plot, in essene forming the union mjl(u) mjl n mjl f (3 19) of the individual m jl n and m jl f distributions. Sine eah individual distribution is smooth and has a kinemati endpoint, the same two kinemati endpoints should be visible on the ombined distribution m jl(u) as well5. We shall denote the larger of the two endpoints with ( ) { M max (m ) jl(u) max max ( ) } jl n, m max jl f (3 0) and the smaller of the two endpoints with ( m max jl(u) ) min { (m max jl n ), ( m max jl f ) }. (3 1) The newly introdued quantities Mjl(u) max and mmax jl(u) are nothing but the usual kinemati endpoints m max jl n and mjl max f, given by (3 3) and (3 4), orrespondingly. Of ourse, at this point we do not know whih is whih, and we have an apparent two-fold ambiguity: we an have either M max jl(u) = m max jl n, m max jl(u) = m max jl f, if R AB R BC, (3 ) or M max jl(u) = m max jl f, m max jl(u) = m max jl n, if R AB R BC. (3 3) Notie that both (3 0) and (3 1) are offiially upper kinemati endpoints, and thus satisfy our basi requirements. 5 For speifi numerial examples, refer to Setion

75 The benefits of our alternative treatment (3 19) in response to the near-far lepton ambiguity problem of Setion 3.1.1, are now starting to emerge. With the onventional ordering (3 7,3 8) one has to deal with a three-fold ambiguity in the interpretation of the endpoints mjl(lo) max and mmax jl(hi), as seen in Equations (3 9,3 10). Instead, the simple union (3 19) leads only to the two-fold ambiguity of Equations (3,3 3). More importantly, the analysis of Setion below will reveal that in spite of the remaining two-fold ambiguity in Equations (3,3 3), one an nevertheless uniquely determine all three of the masses m D, m C and m A! We onsider this to be one of the important results of this paper. 3.. The Produt m jln m jlf In the remainder of this setion, we shall onstrut new invariant mass squared variables out of the two entries m jl n and m jl f, simply by applying various mathematial operations on them in a symmetri fashion. We begin with the produt m jl(p) m jln m jlf (3 4) whose endpoint is given by ( ) m max jl(p) 1 m D (1 R CD) 1 R AB, for R BC 0.5, m D (1 R CD) R BC (1 R BC )(1 R AB ), for R BC 0.5. (3 5) Unfortunately, this endpoint also turns out to be pieewise-defined, thus failing one of our basi requirements from the Introdution. Therefore we shall not use this endpoint in the ourse of our analysis The Sums m α jl n + m α jl f Another possibility is to onsider various sums, for example m jl n + m jl f or (m jln + m jlf ), as originally proposed in [76]. Here we generalize the disussion in [76] and introdue a whole set of new variables, mjl(s) (α), labelled by the ontinuous parameter 75

76 α, whih are defined as mjl(s)(α) ( ) mjl α n + mjl α 1 α f. (3 6) Sine α is a ontinuous parameter, in priniple there are infinitely many m jl(s) variables! Notie that the onventional variables mjl(lo) and m jl(hi) from (3 7) and (3 8) are also inluded in our set, and are simply given by m jl(lo) m jl(s)( ), (3 7) m jl(hi) m jl(s)( ). (3 8) We see that our new set (3 6) is a very broad generalization of the onventional definitions (3 7) and (3 8), whih just orrespond to the two extreme ases α = ±. Of ourse, the user is free to hoose α at will, and any finite value of α will lead to a new variable m jl(s) (α). In order to make the new variables mjl(s) (α) useful for mass spetrum studies, we need to provide the formulas for their kinemati endpoints (m max jl(s) (α)). These formulas are easy to derive, using the results from [9], and we present them in the next two subsetions, where it is onvenient to onsider separately the following two ases: α 1 (in Setion ) and α < 1, but α 0 (in Setion 3..3.) Kinemati Endpoints of mjl(s) (α) with α 1 When one hooses a value of α 1, the mjl(s) (α) endpoint is given by the following expression ( m max jl(s)(α 1) ) where m max jl f α-dependent quantity ( m max jl f ), RAB 1 (1 R BC ) (1 R α BC) 1 α, ( m max jl (α) ), RAB 1 (1 R BC ) (1 R α BC) 1 α, (3 9) was already defined in Equation (3 4), and mjl max (α) is a newly defined, ( m max jl (α) ) m D (1 R CD ) [R αbc(1 ] 1 R AB ) α + (1 R BC ) α α. (3 30) 76

77 As a ross-hek, one an verify that in the limit α the expression in Equation (3 9) redues to Equation (3 10), in agreement with Equation (3 8). In that ase, the upper line in Equation (3 9) orresponds to options (, 1) and (, ) in Equation (3 10), where m max jl(hi) = mmax jl f, while the lower line in Equation (3 9) orresponds to option (, 3) in Equation (3 10), where m max jl(hi) = mmax jl n. Unfortunately, just like the produt endpoint of Equation (3 5), the endpoint of Equation (3 9) is in general pieewise-defined, and does not meet our riteria. However, there is one important exeption, namely the ase of α = 1, in whih we do get a singly defined funtion. Aording to the general definition in Equation (3 6), mjl(s) (α = 1) is simply the sum of the two m jl entries in eah event: m jl(s)(α = 1) m jl n + m jl f. (3 31) Using the identity Equation (3 31) an be equivalently rewritten as m jll = m jl n + m jl f + m ll, (3 3) m jl(s)(α = 1) m jll m ll. (3 33) To find the expression for its endpoint, one an set α = 1 in Equation(3 9), and then realize that the logial ondition for exeuting the upper line beomes R AB 0, whih is impossible, sine the mass ratios R ij in Equation (3 6) are always positive definite. Therefore, the endpoint mjl(s) max (α = 1) is always alulated aording to the lower line in Equation (3 9), whih results in [76] ( m max jl(s)(1) ) m D (1 R CD )(1 R AC ). (3 34) Note that this endpoint is perfet for our purposes sine the Equation (3 34) is always unique, i.e. it is independent of the parameter spae region. The variable mjl(s) (α = 1) will thus play a ruial role in our analysis below. 77

78 3..3. Kinemati Endpoints of mjl(s) (α) with α < 1 and α 0 Finally, in the ase when α < 1, but α 0, the mjl(s) (α) endpoint is given by the following expression ( m max jl(s)(α < 1) ) ( m max jl (α) ) ], RBC [1 + (1 R AB ) α 1 α 1, ] md (1 R CD) [1 + (1 R AB ) α 1 α ] α 1 α, R BC [1 + (1 R AB ) α 1 α 1, where mjl max (α) was already defined in Equation (3 30). Again as a ross-hek, one an verify that in the limit α the expression in Equation (3 35) redues to (3 35) Equation (3 9), in agreement with Equation (3 7). In the α ase, the upper line in Equation (3 35) orresponds to option (, 1) in Equation (3 9), where m max jl(lo) = mmax jl n, while the lower line in Equation (3 35) orresponds to options (, ) and (, 3) in Equation (3 9), where mjl(lo) max = mmax jl(eq). Unfortunately, the endpoint funtion in Equation (3 35) is again pieewise-defined, and does not meet one of our basi riteria spelled out in the introdution. In passing, we note that the speial ase of α = 1, whih involves the linear sum of the two masses m jl(s)(α = 1 ) (m jl n + m jlf ), (3 36) was previously explored in [76, 77]. In that ase, from Equation (3 35) we find for its endpoint ( mjl(s)( max 1 ) ) m D (1 R CD) ( RBC (1 R AB ) + 1 R BC ), RBC 1 R AB R AB, m D (1 R CD)( R AB ), 3..4 The Differene m jl n m jl f R BC 1 R AB R AB. (3 37) Finally, one an also onsider a set of variables whih involve the absolute value of differenes between m jl n and m jl f. In analogy with Equation (3 6), we an define 78

79 another infinite set of variables mjl(d)(α) m α jln mjl α 1 α f. (3 38) One again, the user is free to onsider arbitrary values of α. However, this freedom is redundant, when it omes to the issue of the kinemati endpoints of the variables in Equation (3 38). It is not diffiult to see that the endpoints of mjl(d) (α) are always given by ( m max jl(d)(α) ) ( M max jl(u)) (3 39) and are in fat independent of α! Therefore, for the purposes of our disussion, it is suffiient to onsider just one partiular value of α. In the following we shall only use α = 1: m jl(d)(α = 1) m jln m jl f, (3 40) whih is the analogue of mjl(s) (α = 1) defined in Equation (3 31). The result of Equation (3 39) implies that the endpoint in Equation (3 40) does not ontain any new amount of information, whih was not already present in the two kinemati endpoints Mjl(u) max and mmax jl(u) disussed in Setion Nevertheless, the independent measurement of (m max jl(d) (1)) an still be very useful, sine it will mark the loation of (Mjl(u) max) on the mjl(u) distribution. Then one will be looking for the seond endpoint (m max jl(u) ) to the left, i.e. in the region of smaller m jl(u) values. This ompletes our disussion of the new invariant mass variables and their kinemati endpoints. For our basi proof-of-priniple measurement tehnique presented in the next Setion 3.3.1, we shall use only three of them, namely M max jl(u), mmax jl(u), and mjl(s) max (α = 1). However, the remaining variables are in priniple just as good, their only disadvantage being that they failed our arbitrarily imposed ondition at the beginning that the endpoint funtions should all be region independent. Of ourse, one ould, and in fat should, use all of the available kinemati endpoint information, whih in a 79

80 global fit analysis an only inrease the experimental preision of the spartile mass determination. 3.3 Theoretial Analysis Our Method And The Solution For The Mass Spetrum Our starting point is the set of four measurements mll max, Mjl(u), max mjl(u), max mjl(s)(α max = 1) (3 41) in plae of the onventional set in Equation (3 18). It is easy to verify that the measurements in Equation (3 41) are always independent of eah other, and thus never suffer from the linear dependene problem disussed in Setion Given the set of four measurements in Equation (3 41), it is easy to solve for the mass spetrum. To simplify the notation, we introdue the following shorthand notation for the endpoints of the mass squared distributions L (mll max ), M ( ) Mjl(u) max ( ), m m max ( jl(u), S m max jl(s)(α = 1) ) (3 4) The solution for the mass spetrum is then given by md = mc = mb = ma = Mm(L + M + m S) (M + m S) ; (3 43) MmL (M + m S) ; (3 44) ML(S M), if R (M+m S) AB R BC, (3 45) ml(s m) (M+m S), if R AB R BC ; L(S m)(s M) (M + m S). (3 46) It is easy to verify that the right-hand side expressions in these equations are always positive definite, so that one an safely take the square root and ompute the linear masses m D, m C, m B and m A. Notie that in spite of the two-fold ambiguity in Equations 80

81 (3,3 3), the solution for m D, m C and m A is unique! Indeed, the expressions for m D, m C and m A are symmetri under the interhange M m. The remaining two-fold ambiguity for m B is preisely the result of the ambiguous interpretation in Equations (3 and 3 3) of the two mjl(u) endpoints, and is related to the symmetry under Equation (3 13), or equivalently, under the interhange R AB R BC. (3 47) In the next subsetion we disuss several ways in whih one an lift the remaining two-fold degeneray for m B whih is due to Equation (3 47). Notie the great simpliity of this method. The expressions in Equations (3 43), (3 44) and (3 46) are region independent and therefore one does not have to go through the standard trial and error proedure involving the 9 parameter spae regions (N jll, N jl ) [9, 17] assoiated with the various interpretations of the endpoints mjll max, mmax jl(lo) and m max jl(hi) Disambiguation Of The Two Solutions For m B The method outlined in Setion allowed us to find the true masses of partiles A, C and D, but yields two separate possible solutions for the mass m B of partile B. We shall now disuss several ways of lifting the remaining two-fold degeneray for m B Invariant mass endpoint method One possibility is to use an additional measurement of an invariant mass endpoint. Indeed, as shown in Setion 3., there are still quite a few one-dimensional invariant mass distributions at our disposal, whih we have not used so far. Those inlude the onventional distributions of mjll, m jl(lo) and m jl(hi), as well as the new distributions mjl(p), m jl(s) (α) and m jl(d) (1) whih we introdued in Setion 3.. Whih of them an be used for our purposes? Note that the dupliation in Equation (3 45) arose due to the symmetry in Equation (3 47), so that any kinemati endpoint whih violates this symmetry will be able to distinguish between the two solutions. 81

82 Figure 3-. Comparison of the preditions for the kinemati endpoints mjl(s) max (α) of the real and fake solutions, as a funtion of ϕ artan α (in units of π), for the two examples disussed in detail in Setion 3.4: (a) the LM1 CMS study point and (b) the LM6 CMS study point. In eah panel, the predition of the real (fake) solution is plotted in red (blue). The vertial dotted line indiates the ase of ϕ = π (α = 1), for whih the two solutions give an idential 4 answer, marked with a green dot. The horizontal dotted lines show the orresponding asymptoti values mjl(hi) max and mmax jl(lo), obtained at α ± (ϕ ± π). Let us begin with the onventional distributions m jll, m jl(lo), m jl(hi) and m jll(θ> π ), whose endpoints we did not use in our analysis so far. It is easy to hek that m max jll, mjl(hi) max and mmin jll(θ> π ) are invariant under the interhange as in Equation (3 47) and annot be used for disrimination. However, mjl(lo) max is not symmetri under Equation (3 47) and an do the job. In fat, one an show that the two dupliate solutions for m B always 6 give different preditions for m max jl(lo). More importantly, many of our new variables from Setion 3. an provide an independent ross-hek on the orret hoie for the solution. For example, the kinemati endpoint in Equation (3 5) of the produt variable mjl(p), also violates the symmetry of Equation (3 47) and distinguishes among the two solutions. The infinite set of variables mjl(s) (α) an also be used, and for almost the whole range of α < 1. To see 6 The only exeption is the trivial ase of R AB = R BC, but then the two solutions for m B oinide, and m B is again uniquely determined. 8

83 this, in Figure 3- we ompare the preditions for the kinemati endpoints mjl(s) max (α) of the real and fake solutions, for the two examples disussed in detail in Setion 3.4: (a) the LM1 CMS study point and (b) the LM6 CMS study point. The orresponding mass spetra are listed in Table 3-1 below. For onveniene, we plot versus the parameter ϕ artan α, (3 48) whih allows us to map the whole definition domain (, ) for α into the finite region ( π, π ) for ϕ. Figure 3- shows that for most of the allowed ϕ range, the two solutions predit different values for the kinemati endpoints mjl(s) max (α). In fat, for ϕ < π, the two 4 preditions are always different, apart from the trivial ase of ϕ = 0 (α = 0). Even for ϕ > π, there still exists a range of ϕ, for whih, at least theoretially, a disrimination an 4 be made. The preditions are guaranteed to oinide only for ϕ = π 4 (α = 1) (as they should, see Equation (3 41)), and for a ertain range of the largest possible values of ϕ Invariant mass orrelations Another way to resolve the twofold ambiguity in our solution in Equation (3 45) is to simply go bak to the original measurements of Mjl(u) max and mmax jl(u) and already at that point try to deide whih of the two measured m jl(u) endpoints is m max jl n and whih one is mjl max f. As already disussed in [1, 76], this identifiation is in priniple possible, if one onsiders the orrelations whih are present in the two-dimensional distribution mjl(u) versus m ll. The basi idea is illustrated in Figure 3-3, where we show satter plots of m jl(u) versus m ll, for the two examples used in Figure 3- and disussed in detail later in Setion 3.4. Figure 3-3(a) (Figure 3-3(b)) shows the result for the real (fake) solution orresponding to the LM1 study point, while Figures 3-3() and 3-3(d) show the analogous results for the LM6 study point. In eah plot we used 10,000 entries, whih roughly orresponds to 0 fb 1 (00 fb 1 ) of data for the atual LM1 (LM6) SUSY study point. Here and below we show the ideal ase where we neglet smearing effets due to the finite detetor resolution, finite partile widths and ombinatorial bakgrounds. 83

84 Figure 3-3. Predited satter plots of m jl(u) versus m ll, for the ase of the real and fake solutions for eah of the two study points LM1 and LM6: (a) the real solution LM1; (b) the fake solution LM1 ; () the real solution LM6; and (d) the fake solution LM6. The red solid horizontal (blue dashed inlined) line indiates the onditional maximum m max jl n (m ll ) (m max jl f (m ll )) given by Equation (3 49) (Equation (3 50)). Eah panel ontains 10,000 entries. The results shown here are idealized in the sense that we neglet smearing effets due to the finite detetor resolution, finite partile widths and ombinatorial bakgrounds. Notie the use of quadrati power sale on the two axes, whih preserves the simple shapes of the satter plots, even when plotted versus the linear masses m jl(u) and m ll. All of our plots are at the parton level (using our own Monte-Carlo phase spae generator) and without any uts. Notie that in order to avoid dealing with the large numerial values of the squared masses, we use a quadrati power sale on both axes, whih allows us to preserve the simple shapes of the satter plots when plotting versus the linear masses themselves. 84

85 Figure 3-3 shows that the ombined distribution mjl(u) is simply omposed of the two separate distributions m jl n and m jl f, but they are orrelated differently with the dilepton distribution mll. In partiular, let us onentrate on the onditional maxima mmax jl n (m ll ) and mjl max f (m ll ), i.e. the maximum allowed values of m jln and m jlf, respetively, for a given fixed value of m ll [1, 76]. A lose inspetion of Figure 3-3 shows that the values of m jl n and m ll are unorrelated, and as a result, the onditional maximum mmax jl n (m ll ) does not depend on m ll. In turn, this implies that the endpoint value (mjl max n ) given in Equation (3 3) an be obtained for any m ll : n ( ) mjl max [ n = m max jl n (m ll ) ] = m D (1 R CD ) (1 R BC ), m ll [ 0, mll max ]. (3 49) Beause of Equation (3 49), the shape of the mjl n versus mll satter plot is a simple retangle [1, 76]. This is onfirmed by the plots in Figure 3-3, where the (red) horizontal solid line indiates the onstant value as in Equation (3 49) for the onditional maximum m max jl n (m ll ). In ontrast, the values of m jl f and mll are orrelated. The onditional maximum mjl max f (m ll ) does depend on the value of m ll as follows: where we introdue the shorthand notation used in [9] ( m max jl f (m ll ) ) f p = p + L m ll, (3 50) f ( m max jl f ) = m D (1 R CD ) (1 R AB ), (3 51) p R BC f = m D (1 R CD ) R BC (1 R AB ). (3 5) The absolute maximum of m jl f, whih is given by Equation (3 4) and denoted here by f, an only be obtained when mll itself is at a maximum [1, 76]: f [ mjl max f (mll max ) ]. (3 53) 85

86 On the other hand, the onditional maximum mjl max f (m ll ) obtains its minimum value at m ll = 0 and orresponds to [1, 76] p [ m max jl f (0) ] f. (3 54) Equations (3 53 and 3 54) imply that the shape of the m jl f versus mll satter plot is a right-angle trapezoid. This is onfirmed by the plots in Figure 3-3, where we mark with a (blue) dashed line the onditional maximum in Equation (3 50). With suffiient statistis, this differene in the kinemati boundaries may be observable, and would reveal the identity of m max jl n and m max jl f [1, 76]. One the individual m max jl n and m max jl f are known, the solution for the mass spetrum is unique see e.g. Appendix A in [9]. Of ourse, in ases where p f, namely R BC 1, it may be diffiult in pratie to tell whih of the two boundaries in the satter plot is inlined and whih one is horizontal 7. One example of this sort is offered by point LM6, whih has R BC = 0.91 and leads to a rather flat mjl max f (m ll ) funtion, as seen in Figure 3-3(). An alternative and somewhat related method will be to investigate the shapes of the one-dimensional distributions themselves [78]. In Appendix A we provide the analytial expressions for the shapes of the four invariant mass distributions m ll, m jl(u), mjl(s) (1) and m jl(d) (1) used in our basi analysis from Setion Given what we have already seen in Figure 3-3, it is not surprising that the true and the fake solutions predit different shapes for the one-dimensional distributions as well. In the LM1 and LM6 examples onsidered below in Setion 3.4, this differene is partiularly notieable for the mjl(u) and m jl(d) (1) distributions (see Figures 3-4(b), 3-4(d), 3-5(b) and 3-5(d)), and an be tested experimentally. 7 A separate problem, whih arises in the ase of p f, will be disussed below in Setion

87 Table 3-1. The relevant part of the SUSY mass spetrum for the LM1 and LM6 study points. The orresponding dupliated solutions LM1 and LM6 are obtained by interhanging R BC R AB as in Equation (3 47). In the table we also list the orresponding values for various invariant mass endpoints. The first four of those represent our basi set of measurements of Equation (3 41) disussed in detail in Setion 3.4.1, while the last two (mjl max n and mjl max f ) are not diretly observable. The remaining invariant mass endpoints are onsidered in Setion In the ase of mjl(s) max (α), we show several representative values for α. For the omplete α variation, refer to Figure 3-. Reall that (+ ) = mmax ( ) = mmax m max jl(s) jl(hi) and mmax jl(s) jl(lo). Variable LM1 LM1 LM6 LM6 m A (GeV) m B (GeV) m C (GeV) m D (GeV) R AB R BC R CD m max ll (GeV) (GeV) M max jl(u) m max jl(u) m max jl(s) (GeV) (α = 1) (GeV) m max jll (GeV) m min jll(θ> π m max jl(hi) m max jl(s) m max jl(s) m max jl(s) m max jl(s) m max jl(s) m max jl(lo) m max jl(p) ) (GeV) (GeV) (α = ) (GeV) (α = 1.5) (GeV) (α = 0.5) (GeV) (α = 0.5) (GeV) (α = 1) (GeV) (GeV) (GeV) mjl max n (GeV) mjl max f (GeV)

88 3.4 Numerial Examples We shall now illustrate the ideas of the previous setion with two speifi numerial examples: the LM1 and LM6 SUSY study points in CMS [75]. The mass spetra at LM1 and LM6 are listed in Table 3-1. Point LM1 is similar to benhmark point A (A ) in Referene [79] (Referene [80]) and to benhmark point SPS1a in Referene [81]. Point LM6 is similar to benhmark point C (C ) in Referene [79] (Referene [80]). The Table 3-1 also lists the orresponding dupliate solutions LM1 and LM6, whih are obtained by interhanging R BC R AB, or equivalently, by replaing the mass of B via m B m B = m Am C m B. (3 55) It is interesting to note that LM1 and LM6 represent both sides of the ambiguity in Equation (3 47): at LM1, we have R AB > R BC and orrespondingly, m max jl n and Equation (3 ) applies. On the other hand, at LM6 we have R AB m max jl n > m max jl f < R BC and < mjl max f, so that Equation (3 3) applies. Another interesting differene is that at LM1 partile B is the right-handed slepton l R, while at LM6 the role of partile B is played 8 by the left-handed slepton l L. To the extent that we are interested in kinematial features, this differene is not relevant Mass Measurements At Study Points LM1 and LM6 Given the mass spetra in Table 3-1, it is straightforward to onstrut and investigate the relevant invariant mass distributions. For the purposes of illustration, we shall ignore spin orrelations, referring the readers interested in those effets to Referene [8 84]. We are justified to do so for several reasons. First, our method relies on the measurement of kinemati endpoints, whose loation is unaffeted by the presene of spin orrelations. Seond, in the ase of supersymmetry (whih is really what we have 8 Although the right-handed slepton l R is also kinematially aessible at point LM6, the wino-like neutralino χ 0 deays muh more often to l L as opposed to l R. 88

89 in mind here), partile B is a salar, whih automatially washes out any spin effets in the m ll and m jl f distributions. Furthermore, if partiles D and their antipartiles D are produed in equal numbers, as would be the ase if the dominant prodution is from gg and/or q q initial state, any spin orrelations in the m jl n distribution are also washed out. Under those irumstanes, therefore, the pure phase spae distributions shown here are in fat the orret answer. We begin our disussion with the four invariant mass distributions m ll, m jl(u), mjl(s) (α = 1) and m jl(d) (α = 1), whih form the basis of our method outlined in Setion Figure 3-4 (Figure 3-5) shows those four distributions for the ase of study point LM1 (LM6). In eah panel, the red (solid) histogram orresponds to the nominal spetrum (LM1 or LM6), while the blue (dotted) histogram orresponds to the fake solution (LM1 or LM6 ), whih is obtained through the replaement (3 55). For all figures in this setion, we use the same 4 samples of 10,000 events eah, whih were already used to make Figure 3-3. Notie our somewhat unonventional way of filling and then plotting the histograms in this setion. First, we show differential distributions in the orresponding mass squared, i.e. dn/dm. This is done in order to preserve the onnetion to the analytial results in Appendix A, whih are written the same way. More importantly, the shapes of the one-dimensional histograms are muh simpler in the ase of dn/dm as opposed to dn/dm [8 84]. In the next step, however, we hoose to plot the thus obtained histogram versus the mass itself rather than the mass squared. This allows one to read off immediately the orresponding endpoint and ompare diretly to the values listed in Table 3-1. It also keeps the x-axis range within a manageable range. However, sine the histograms were binned on a mass squared sale, if we were to use a linear sale on the x-axis, we would get bins with varying size. This would be rather inonvenient and more importantly, would distort the nie simple shapes of the dn/dm distributions. Therefore, we use a quadrati sale on the x-axis, whih preserves the nie shapes and leads to a onstant bin size on eah plot. 89

90 Figure 3-4. One-dimensional invariant mass distributions for the ase of LM1 (red solid lines) and LM1 (blue dotted lines) spetra. The kinemati endpoints in Equation (3 41) used in our analysis in Setion an be observed from these distributions as follows: mll max is the upper kinemati endpoint of the m ll distribution in panel (a); Mjl(u) max is the absolute upper kinemati endpoint seen in both the ombined m jl(u) distribution in panel (b), or the differene distribution m jl(d) (1) in panel (d); mjl(u) max is the intermediate kinemati endpoint seen in panel (b); and mjl(s) max (α = 1) is the upper kinemati endpoint of the m jl(s) (α = 1) distribution in panel (). 90

91 Figure 3-5. One-dimensional invariant mass distributions for the LM6 mass spetrum (red solid lines) and the LM6 mass spetrum (blue dotted lines). 91

92 Figures 3-4 and 3-5 illustrate how eah one of the measurements in Equation (3 41) an be obtained. For example, m max ll is the lassi upper kinemati endpoint of the m ll distributions in Figures 3-4(a) and 3-5(a). This endpoint is very sharp and should be easily observable. Mjl(u) max is the absolute upper kinemati endpoint seen in the ombined m jl(u) distribution in Figures 3-4(b) and 3-5(b). Notie that the same endpoint an independently also be observed as the absolute upper kinemati limit of the differene distributions m jl(d) (1) shown in Figures 3-4(d) and 3-5(d). The fat that there are two independent ways of getting to the endpoint Mjl(u) max should allow for a reasonable auray of its measurement. Upon loser inspetion of the ombined m jl(u) distribution in Figures 3-4(b) and 3-5(b), we also notie the intermediate kinemati endpoint m max jl(u) seen around 30 GeV in Figure 3-4(b) and around 40 GeV in Figure 3-5(b). Finally, m max jl(s) (α = 1) is the upper kinemati endpoint of the m jl(s)(α = 1) distribution shown in Figures 3-4() and 3-5(). It is also rather well defined, and should be well measured in the real data. At this point we would like to omment on one potential problem whih is not immediately obvious, but nevertheless has been enountered in pratial appliations of the invariant mass tehnique for SUSY mass determinations [78]. It has been noted that in the ase of p f (Equations (3 51,3 5)), the numerial fit for the mass spetrum beomes rather unstable. Given our analytial results in Setion 3.3.1, we are now able to trae the root of the problem. Notie that p f implies that R BC 1. In this limit, from Equations (3 ), (3 3), (3 4) and (3 34) we find lim (L) = 0, lim R BC 1 (n) = 0, lim R BC 1 (M + m S) = 0. (3 56) R BC 1 This means that the funtions in Equations (3 43 through 3 46) giving the solution for the mass spetrum will all behave as 0 0, and, given the statistial flutuations in an atual analysis, will have very poor onvergene properties. We note that this problem is not limited to our preferred set of measurements (3 41) and is rather generi, but has been missed in most previous studies simply beause the ase of R BC 1 was 9

93 rarely onsidered. Figures 3-4 and 3-5 reveal that, as expeted, the real (red solid lines) and fake (blue dotted lines) solutions always give idential results for our basi set of four endpoint measurements in Equation (3 41). This is by design, and in order to disriminate among the real and the fake solution, we need additional experimental input, as disussed in Setion Before we proeed with the disambiguation analysis in the next subsetion, we should stress one again that the real and fake solutions agree on 75% of the relevant mass spetrum, i.e. they give the same values for the masses of partiles D, C and A (Table 3-1). The only question mark at this point is, what is the mass of partile B. This issue is addressed in the following subsetion Eliminating The Fake Solution for m B As already disussed in Setion 3.3., there are several handles whih ould disriminate among the two alternative values of m B in the real and the fake solution. One possibility is to use additional independent measurements of M T kinemati endpoints. Another possibility, disussed in Setion and demonstrated expliitly with Figure 3-3, is to use the different orrelations in the -dimensional invariant mass distributions (m ll, m jl n ) and (m ll, m jl f ). The near-far lepton ambiguity is avoided by studying the satter plot of (mll, m jl(u) ), shown in Figure 3-3, whih should be in priniple suffiient to disriminate among the two alternatives. In keeping with the main theme of this paper, in this subsetion we shall onentrate on the third possibility, already suggested in Setion We shall simply explore additional invariant mass endpoint measurements, whih would hopefully disriminate among the two solutions for m B. Figures 3-6 and 3-7 show several invariant mass distributions whih have already been mentioned at one point or another in the ourse of our previous disussion. Figure 3-6 shows the following 6 distributions: (a) m jll ; (b) mjl(hi) ; () m jl(p) ; (d) m jl(lo) ; (e) m jl(s) (α = 1) and (f) m jl(s) (α = 1 ), for the LM1 mass spetrum (red solid lines) and its LM1 ounterpart (blue dotted lines). Figure 3-7 shows the same 6 distributions, but for the LM6 and LM6 mass spetra. In Figures 3-6 and 3-7, 93

94 we follow the same plotting onventions as in Figures 3-4 and 3-5: we form the mass squared distribution dn/dm, and then plot versus the orresponding linear mass m using a quadrati sale on the x-axis. Notie that the sum of the mjl(hi) distribution in Figure 3-6(b) (Figure 3-7(b)) and the mjl(lo) distribution in Figure 3-6(d) (Figure 3-7(d)) preisely equals the ombined distribution mjl(u) in Figure 3-4(b) (Figure 3-5(b)). In order to be able to see this by the naked eye, we have kept the same x and y ranges on the orresponding plots. As seen in Figures 3-6 and 3-7, not all of the remaining invariant mass distributions are able to disriminate among the two m B solutions. As explained in Setion , the suitable distributions are those whose endpoints violate the symmetry in Equation (3 47), whih aused the m B ambiguity in the first plae. For example, Figures 3-6(a) and 3-7(a) show that the endpoint of the mjll distribution is the same for the real and the fake solution. This is to be expeted, sine the defining Equation (3 5) for m max jll symmetri under Equation (3 47). Figures 3-6(a) and 3-7(a) also show that even the shapes of the mjll distributions for the real and fake solution are very similar. In spite of this, the observation of the mjll endpoint an still be very useful, e.g. in reduing the experimental error on the mass determination. Similar omments apply to the mjl(hi) distributions shown in Figures 3-6(b) and 3-7(b). Here again the endpoint is a symmetri funtion of R AB and R BC, and the real and fake solutions predit idential endpoints. However, while the endpoints are the same, this time the shapes are not. The shape differene is more pronouned in the ase of LM1 shown in Figure 3-6(b), and less visible in the ase of LM6 shown in Figure 3-7(b). The remaining four distributions shown in Figures 3-6(-f) and 3-7(-f) already have different endpoints and an thus be used for disrimination among the real and fake solution for m B. All of the endpoints in Figures 3-6(-f) and 3-7(-f) are relatively sharp and should be measured rather well. One should not forget that in Figures 3-6 and 3-7 is 94

95 we show mjl(s) (α) distributions for only three representative values of α: α = in panels (d), α = 1 in panels (e), and α = 0.5 in panels (f). As seen in Figure 3-, there are infinitely many other hoies for α, whih would still exhibit different endpoints for the real and fake m B solutions. Our onlusion is that through a suitable ombination of additional endpoint measurements one would be able to tell apart the real solution for m B from its fake ousin. 95

96 Figure 3-6. Some other one-dimensional invariant mass distributions of interest, for the ase of the LM1 mass spetrum (red solid lines) and LM1 mass spetrum (blue dotted lines): (a) mjll distribution; (b) m jl(hi) distribution; () m jl(p) distribution; (d) mjl(lo) distribution; (e) m jl(s) (α = 1) distribution; (f) mjl(s) (α = 1 ) distribution. All distributions are then plotted versus the orresponding mass, on a quadrati sale for the x-axis. 96

97 Figure 3-7. Some other one-dimensional invariant mass distributions of interest for the LM6 mass spetrum (red solid lines) and the LM6 mass spetrum (blue dotted lines). 97

98 CHAPTER 4 SUBSYSTEM M T METHOD The idea for a subsystem M T was first disussed in [85] and applied in [86] for a speifi supersymmetry example (assoiated squark-gluino prodution and deay). Here we shall generalize that onept for a ompletely general deay hain. The subsystem M T variable will be defined for the subhain inside the blue (yellow-shaded) box in Figure 4-1. Before we give a formal definition of the subsystem M T variables, let us first introdue some terminology for the BSM partiles appearing in the deay hain. We shall find it onvenient to distinguish the following types of BSM partiles: Grandparents. Those are the two BSM partiles X n at the very top of the deay hains in Figure 4-1. Sine we have assumed symmetri events, the two grandparents in eah event are idential, and arry the same index n. Of ourse, one may relax this assumption, and onsider asymmetri events, as was done in [86, 87]. Then, the two grandparents will be different, and one would simply need to keep trak of two separate grandparent indies n (1) and n (). Parents. Those are the two BSM partiles X p at the top of the subhain used to define the subsystem M T variable. In Figure 4-1 this subhain is identified by the blue (yellow-shaded) retangular box. The idea behind the subsystem M T is simply to apply the usual M T definition for the subhain inside this box. Notie that the M T onept usually requires the parents to be idential, therefore here we will haraterize them by a single parent index p. Children. Those are the two BSM partiles X at the very end of the subhain used to define the subsystem M T variable, as indiated by the blue (yellow-shaded) retangular box in Figure 4-1. The hildren are also haraterized by a single index. In general, the true mass M of the two hildren is unknown. As usual, when alulating the value of the M T variable, one needs to hoose a hild test mass, whih we shall denote with a tilde, M, in order to distinguish it from the true mass M of X. Dark matter andidates. Those are the two stable neutral partiles X 0 appearing at the very end of the asade hain. We see that while those are the partiles responsible for the measured missing momentum in the event, they are relevant for M T only in the speial ase of = 0. 98

99 ISR x n x p x + 1 x 1 p( p) X n X n 1 X p X p 1 X + 1 X X 1 X 0 p( p) X n X n 1 X p X p 1 X + 1 X X 1 X 0 ISR x n x p x + 1 x 1 Figure 4-1. Illustration of the subsystem M (n,p,) T variable defined in Equation (4 3). With those definitions, we are now ready to generalize the onventional M T definition [10, 3]. From Figure 4-1 we see that any subhain is speified by the parent index p and the hild index, while the total length of the whole hain (and thus the type of event) is given by the grandparent index n. Therefore, the subsystem M T variable will have to arry those three indies as well, and we shall use the notation M (n,p,) T. In the following we shall refer to this generalized quantity as either subsystem or subhain M T. It is lear that the set of three indies (n, p, ) must be ordered as follows: n p > 0. (4 1) We shall now give a formal definition of the quantity M (n,p,) T, generalizing the original idea of M T [10, 3]. The parent and hild indies p and uniquely define a subhain, within whih one an form the transverse masses M (1) T and M() T of the two parents: M (k) T (p(k) p, p (k) p 1,..., p(k) +1, P (k) T ; M ), k = 1,. (4 ) Here p (k) i, + 1 i p, are the measured 4-momenta of the SM partiles within the subhain, P (k) T are the unknown transverse momenta of the hildren, while M is their unknown (test) mass. Then, the subsystem M (n,p,) T is defined by minimizing the larger of the two transverse masses as in Equation (4 ) over the allowed values of the hildren s 99

100 transverse momenta P (k) T : M (n,p,) T ( M ) = min k=1 P (k) T = k=1 n j=+1 (k) p jt p T { { max M (1) T, M() T }}, (4 3) where p T indiates any additional transverse momentum due to initial state radiation (ISR) (Figures 4-1). Notie that in this definition, the dependene on the grandparent index n enters only through the restrition on the hildren s transverse momenta P (k) T. Using momentum onservation in the transverse plane k=1 P (k) 0T + n k=1 j=1 p (k) jt + p T = 0, (4 4) Furthermore, the measurement of the missing transverse momentum p T,miss provides two additional onstraints on the unknown transverse momentum omponents P (k) 0T P (1) 0T + P () 0T = p T,miss (4 5) Now we an rewrite the restrition on the hildren s transverse momenta P (k) T k=1 P (k) T = k=1 P (k) 0T + k=1 j=1 p (k) jt = p T,miss + k=1 j=1 as p (k) jt, (4 6) where in the last step we used Equation (4 5). Equation (4 6) allows us to rewrite the subsystem M (n,p,) T depend on the grandparent index n: M (n,p,) definition of Equation (4 3) in a form whih does not manifestly T ( M ) = min k=1 P (k) T = p T,miss+ k=1 (k) j=1 p jt { { max M (1) T, M() T }}. (4 7) However, the grandparent index n is still impliitly present through the global quantity p T,miss, whih knows about the whole event. We shall see below that the interpretation of the experimentally observable endpoints, kinks, et., for the so defined subsystem M (n,p,) T quantity, does depend on the grandparent index n, whih justifies our notation. 100

101 We are now in a position to ompare our subsystem M (n,p,) T quantity to the onventional M T variable. The latter is nothing but the speial ase of n = p and = 0: M T M (n,n,0) T, (4 8) i.e. the onventional M T is simply haraterized by a single integer n, whih indiates the length of the deay hain. We see that we are generalizing the onventional M T variable in two different aspets: first, we are allowing the parents X p to be different from the partiles X n originally produed in the event (the grandparents), and seond, we are allowing the hildren X to be different from the dark matter partiles X 0 appearing at the end of the asade hain and responsible for the missing energy. The benefits of this generalization will beome apparent in the next setion, where we shall disuss the available measurements from the different subsystem M (n,p,) T 4.1 A Short Deay Chain X X 1 X 0 variables. A a relatively long (n 3) new physis deay hain an be handled by a variety of mass measurement methods, and in priniple a omplete determination of the mass spetrum in that ase is possible at a hadron ollider. We also showed that a relatively short (n = 1 or n = ) deay hain would present a major hallenge, and a omplete mass determination might be possible only through M T methods. From now on we shall therefore onentrate only on this most problemati ase of n. First let us summarize what types of subsystem M (n,p,) T measurements are available in the ase of n. As illustrated in Figure 4-, there exist a total of 4 different M (n,p,) T Eah M (n,p,) T quantities. distribution would exhibit an upper endpoint M (n,p,) T,max, whose measurement would provide one onstraint on the physial masses. In order to be able to invert and solve for the masses of the new partiles in terms of the measured endpoints, we need to know the analytial expressions relating the endpoints M (n,p,) T,max to the physial masses M i. 101

102 Figure 4-. The subsystem M (n,p,) T variables whih are available for n = 1 and n = events. In this setion we summarize those relations for eah M (n,p,) T quantity with n. Some of these results (e.g. portions of Setions and Setions 4.1.3) have already appeared in the literature, and we inlude them here for ompleteness. The disussion in Setions 4.1. and Setions 4.1.4, on the other hand, is new. In all ases, we shall allow for the presene of an arbitrary transverse momentum p T due to ISR. This represents a generalization of all existing results in the literature, whih have been derived in the two speial ases p T = 0 [37] or p T = [36]. We shall find it onvenient to write the formulas for the endpoints M (n,p,) T,max terms of the atual masses, but in terms of the mass parameters µ (n,p,) M n ( ) 1 M Mp not in. (4 9) The advantage of using this shorthand notation will beome apparent very shortly The Subsystem Variable M (1,1,0) T We start with the simplest ase of n = 1 shown in Figure 4-(a). Here M (1,1,0) T the only possibility, and it oinides with the onventional M T variable, as indiated by Equation (4 8). Therefore, the previous results in the literature whih have been derived is 10

103 for the onventional M T variable in Equation (4 8), would still apply. In partiular, in the limit of p T = 0, the upper endpoint M (1,1,0) T,max depends on the test mass M 0 as follows [37] M (1,1,0) T,max ( M 0, p T = 0) = µ (1,1,0) + µ (1,1,0) + M 0, (4 10) where the parameter µ (1,1,0) is defined in terms of the physial masses M 1 and M 0 aording to Equation (4 9): µ (1,1,0) M 1 ( ) 1 M 0 = M 1 M0. (4 11) M1 M 1 As usual, the endpoint in Equation (4 10) an be interpreted as the mass M 1 of the parent partile X 1, so that Equation (4 10) provides a relation between the masses of X 0 and X 1. In the early literature on M T, this relation had to be derived numerially, by building the M T distributions for different values of the test mass M 0, and reading off their endpoints. Nowadays, with the work of Cho et al. [ Measuring superpartile masses at hadron ollider using the transverse mass kink, JHEP 080, 035 (008)][37], the relation is known analytially, and, as seen from Equation (4 10), is parameterized by a single parameter µ (1,1,0). Therefore, in order to extrat the value of this parameter, we only need to perform a single measurement, i.e. we only need to study the M T distribution for one partiular hoie of the test mass M 0. We shall find it onvenient to hoose M 0 = 0, in whih ase Equations (4 10) and (4 11) give M (1,1,0) T,max ( M 0 = 0, p T = 0) = µ (1,1,0) = M 1 M 0 M 1, (4 1) providing the required measurement of the parameter µ (1,1,0) demonstrates the usefulness of the M T onept just a single measurement of the endpoint of the M T distribution for a single fixed value of the test mass M 0 is suffiient to provide us with one onstraint among the unknown masses (M 1 and M 0 in this ase). Unfortunately, one single measurement in Equation (4 1) is not enough to pin down two different masses. In order to measure both M 0 and M 1, without any 103

104 theoretial assumptions or prejudie, we obviously need additional experimental input. From the general expression as Equation (4 10) it is lear that measuring other M (1,1,0) T,max endpoints, for different values of the test mass M 0, will not help, sine we will simply be measuring the same ombination of masses µ (1,1,0) over and over again, obtaining no new information. Another possibility might be to onsider events with the next longest deay hain (n = ), whih, as advertised in the Introdution and shown below in Setion 4., will be able to provide enough information for a omplete mass determination of all partiles X 0, X 1 and X. However, the existene and the observation of the n = deay hain is ertainly not guaranteed to begin with, the partiles X may not exist, or they may have too low ross-setions. It is therefore of partiular importane to ask the question whether the n = 1 proess in Figure 4-(a) alone an allow a determination of both M 0 and M 1. The answer to this question, at least in priniple, is Yes [36], and what is more, one an ahieve this using the very same M T variable M (1,1,0) T. The key is to realize that in reality at any ollider, and espeially at hadron olliders like the Tevatron and the LHC, there will be sizable ontributions from initial state radiation (ISR) with nonzero p T, where one or more jets are radiated off the initial state, before the hard sattering interation. (In Figures 4-1 and 4- the green ellipse represents the hard sattering, while ISR stands for a generi ISR jet.). This effet leads to a drasti hange in the behavior of the M (1,1,0) T,max ( M 0, p T ) funtion, whih starts to exhibit a kink at the true loation of the hild mass M 0 = M 0 : ( M (1,1,0) T,max ( ) ( M 0, p T ) M (1,1,0) T,max ( ) M 0, p T ) M 0 M 0 M 0 =M 0 ϵ and furthermore, the value of M (1,1,0) T,max well: M 0 =M 0 +ϵ, (4 13) at that point reveals the true mass of the parent as M (1,1,0) T,max ( M 0 = M 0, p T ) = M 1. (4 14) 104

105 This kink feature in Equations (4 13,4 14) was observed and illustrated in A. J. Barr et al. [Weighing Wimps with Kinks at Colliders: Invisible Partile Mass Measurements from Endpoints, JHEP 080, 014 (008)] Setion 4.4 [36]. We find that it an also be understood analytially, by generalizing the result of Equation (4 10) to aount for the additional ISR transverse momentum p T. Reall that Equation (4 10) was derived in Ref. [37] under the assumption that the missing transverse momentum due to the two esaping partiles X 0 is exatly balaned by the transverse momenta of the two visible partiles x 1 used to form M (1,1,0) T : P (1) 0T + P () (1) () 0T + p 1T + p 1T = 0. (4 15) We may sometimes refer to this situation as a balaned momentum onfiguration 1. In the presene of ISR with some non-zero transverse momentum p T, Equation (4 15) in general eases to be valid, and is modified to P (1) 0T + P () (1) () 0T + p 1T + p 1T = p T, (4 16) in aordane with (4 4). Inluding the ISR effets, we find that the expression (4 10) for the M (1,1,0) T,max endpoint splits into two branhes M (1,1,0) T,max ( M 0, p T ) = F (1,1,0) L ( M 0, p T ), if M 0 M 0, F (1,1,0) R ( M 0, p T ), if M 0 M 0, (4 17) 1 This should not be onfused with the term balaned used for the analyti M T solutions disussed in [33, 37]. 105

106 where F (1,1,0) L ( M 0, p T ) = F (1,1,0) R ( M 0, p T ) = [ µ (1,1,0) (p T ) + [ µ (1,1,0) ( p T ) + ( µ (1,1,0)(p T ) + p T ( µ (1,1,0)( p T ) p T ) + M 0 ) + M 0 ] p T 4 1 ] p T 4, (4 18) 1 (4 19), and the p T -dependent parameter µ (1,1,0) (p T ) is defined as ( ) µ (1,1,0) (p T ) = µ (1,1,0) pt 1 + p T. (4 0) M 1 M 1 Both branhes orrespond to extreme momentum onfigurations in whih all three transverse vetors p (1) () 1T, p 1T F (1,1,0) L F (1,1,0) R related as and p T are ollinear. The differene is that the left branh ( ) orresponds to the onfiguration p (1) () 1T p 1T p T, while the right branh ( ) orresponds to p (1) () 1T p 1T p T. Therefore, the two branhes are simply F (1,1,0) R ( M 0, p T ) = F (1,1,0) L ( M 0, p T ). (4 1) It is easy to verify that in the absene of ISR, (i.e. for p T = 0) our general result (4 17) redues to the previous formula (4 10). Our result (4 17) for the M (1,1,0) T,max upper kinemati endpoint as a funtion of the test mass M 0 is illustrated in Figure 4-3(a). We onsider a single ISR jet and show results for several different values of its transverse momentum p T, starting from p T = 0 (the green solid line) and inreasing the value of p T in inrements of p T = 100 GeV. The uppermost solid line orresponds to the limiting ase P T. The true value of the parent (hild) mass is marked by the horizontal (vertial) dotted line. The red (blue) lines orrespond to the funtion F (1,1,0) to the true M (1,1,0) T,max L (F (1,1,0) R ). The solid portions of those lines orrespond endpoint, while the dashed segments are simply the extension of F (1,1,0) L and F (1,1,0) R into the wrong region for M 0, giving a false endpoint. 106

107 400 µ 350 Ô Ì ½ ½ ½ ¼µ Ì ¾ Ñ Ü Î µ Å Å ½ ¼¼ Î Å ¼ ½¼¼ Î Å ¼ Î µ Figure 4-3. (a) Dependene of the M (1,1,0) T,max upper kinemati endpoint (solid lines) on the value of the test mass M 0, for M 1 = 300 GeV, and M 0 = 100 GeV, and for different values of the transverse momentum p T of the ISR jet, starting from p T = 0 (green line), and inreasing up to p T = 3 TeV in inrements of p T = 100 GeV, from bottom to top. The uppermost line orresponds to the limiting ase p T. The horizontal (vertial) dotted line denotes the true value of the parent (hild) mass. Solid (dashed) lines indiate true (false) endpoints. The red lines orrespond to the funtion F (1,1,0) L defined in Equation (4 18), while the blue lines orrespond to the funtion F (1,1,0) R defined in Equation (4 19). (b) The value of the kink Θ (1,1,0) defined in (4 8), as a funtion of the dimensionless ratios p T M 1 and M 0 M 1. Figure 4-3(a) reveals that the two branhes in Equations (4 18) and (4 19) always ross at the point (M 0, M 1 ), in agreement with Equation (4 14). Interestingly, the sharpness of the resulting kink at M 0 = M 0 depends on the hardness of the ISR jet, as an be seen diretly from (4 17). For small p T, the kink is barely visible, and in the limit p T 0 we obtain the old result (4 10) for the balaned momentum onfiguration, shown with the green solid line, whih does not exhibit any kink. In the other extreme, at very large p T, we see a pronouned kink, whih has a well-defined limit as p T. The M (1,1,0) T,max kink exhibited in Equation (4 17) and in Figure 4-3(a) is our first, but not last, enounter with a kink feature in an M (n,p,) T variable. Below we shall see that the M T kinks are rather ommon phenomena, and we shall enounter at least two other kink types by the end of Setion 4.1. Therefore, we find it onvenient to quantify the 107

108 sharpness of any suh kink as follows. Consider a generi subsystem M (n,p,) T variable whose endpoint M (n,p,) T,max ( M, p T ) exhibits a kink: M (n,p,) T,max ( M, p T ) = F (n,p,) L ( M, p T ), if M M, F (n,p,) R ( M, p T ), if M M. The kink appears beause M (n,p,) T,max ( M, p T ) is not given by a single funtion, but has two separate branhes. The first ( low ) branh applies for M M, and is given by some funtion F (n,p,) L ( M, p T ), while the seond ( high ) branh is valid for M M, (4 ) and is given by a different funtion, F (n,p,) R ( M, p T ). The funtion M (n,p,) T,max ( M, p T ) itself is ontinuous and the two branhes oinide at M = M : F (n,p,) L (M, p T ) = F (n,p,) R (M, p T ), (4 3) but their derivatives do not math: ( ) F (n,p,) L M M =M ( ) F (n,p,) R M M =M, (4 4) leading to the appearane of the kink. Let us define the left and right slope of the M (n,p,) T,max ( M, p T ) funtion at M orrespondingly: tan Θ (n,p,) L tan Θ (n,p,) R = M in terms of two angles Θ (n,p,) L and Θ (n,p,) R, ( ) F (n,p,) L ( M ) M M =M, (4 5) ( ) F (n,p,) R ( M ) M M =M. (4 6) Now we shall define the amount of kink as the angular differene Θ (n,p,) between the two branhes: Θ (n,p,) Θ (n,p,) R Θ (n,p,) L = artan ( tan Θ (n,p,) R 1 + tan Θ (n,p,) R tan Θ (n,p,) L tan Θ (n,p,) L ). (4 7) 108

109 A large value of Θ (n,p,) implies that the relative angle between the low and high branhes at the point of their juntion M = M is also large, and in that sense the kink would be more pronouned and relatively easier to see. This definition an be immediately applied to the M (1,1,0) T,max kink that we just disussed. Substituting the formulas (4 18) and (4 19) for the two branhes F (1,1,0) L and F (1,1,0) R into the definitions (4 5,4 6) and subsequently into (4 7), we obtain an expression for the size Θ (1,1,0) of the M (1,1,0) T,max kink: ( ) Θ (1,1,0) M 0 (M1 M = artan 0) p T 4M 1 + pt M 1 (M1 M 0 ) + M0 M 1 (4M1 + p T ). (4 8) The result (4 8) is illustrated numerially in Figure 4-3(b). As an be seen from (4 8), Θ (1,1,0) depends on the two masses M 0 and M 1, as well as the size of the ISR p T. However, sine Θ (1,1,0) is a dimensionless quantity, its dependene on those three parameters an be simply illustrated in terms of the dimensionless ratios p T M 1 and M 0 M 1. This is why in Figure 4-3(b) we plot Θ (1,1,0) (in degrees) as a funtion of p T M 1 and M 0 M 1. Figure 4-3(b) onfirms that the kink develops at large p T, and is ompletely absent at p T = 0, a result whih may have already been antiipated on the basis of Figure 4-3(a). For any given mass ratio M 0 M 1, the kink is largest for the hardest possible p T. In the limit p T we obtain ( ) M lim p T Θ(1,1,0) = artan 1 M0 M 0 M 1. (4 9) From Figure 4-3(b) one an see that at suffiiently large p T, the Θ (1,1,0) ontours beome almost horizontal, i.e. the size of the kink Θ (1,1,0) beomes very weakly dependent on p T. A areful examination of Figure 4-3(b) reveals that the asymptoti behavior at p T is in agreement with the analytial result (4 9). Notie that the maximum possible value of any kink of the type (4 ) is Θ (n,p,) max = 90. Aording to Figure 4-3(b) and Equation (4 9), in the ase of Θ (1,1,0) the absolute maximum an be obtained only in the p T and M 0 0 limit. The former ondition will never be 109

110 realized in a realisti experiment, while the latter ondition makes the observation of the kink rather problemati, sine the low branh F L of the M (1,1,0) T,max ( M 0, p T ) funtion is too short to be observed experimentally. Therefore, under realisti irumstanes, we would expet the size of the kink Θ (1,1,0) to be only on the order of a few tens of degrees, whih are the more typial values seen in Figure 4-3(b). Aording to Figure 4-3(b), for a given fixed p T, the sharpness of the Θ (1,1,0) kink depends on the mass hierarhy of the partiles X 1 and X 0. When they are relatively degenerate, i.e. their mass ratio M 0 M 1 is large, the kink is relatively small. Conversely, when X 0 is muh lighter than X 1, the kink is more pronouned. The optimum mass ratio M 0 M 1 for p T whih maximizes the kink for a given p T, is rather weakly dependent on the p T, and eventually goes to zero, in agreement with Equation (4 9). However, for more reasonable values of p T as the ones shown on the left half of the plot, the optimal ratio M 0 M 1 M 0 M 1 = 1 3 varies between 0.3 (at p T 0) to 0.1 (at p T 5M 1 ). In this sense, the value of whih was hosen for the illustration in Figure 4-3(a). In onlusion of this subsetion, it is worth summarizing the main points from it. The good news is that the Θ (1,1,0) kink in priniple offers a seond, independent piee of information about the masses of the partiles X 0 and X 1. When taken together with the M (1,1,0) T,max endpoint measurement (4 1), it will allow us to determine both masses M 0 and M 1, in a ompletely model-independent way. Our analytial results regarding the Θ (1,1,0) kink omplement the study of A. J. Barr et al. [ Weighing Wimps with Kinks at Colliders: Invisible Partile Mass Measurements from Endpoints, JHEP 080, 014 (008)]. [36], where this kink was first disovered. However, on the down side, we should mention that muh of our disussion regarding the Θ (1,1,0) kink may be of limited pratial interest, for several reasons. First, as seen in Figure 4-3, the kink beomes visible only for suffiiently large values of the p T. Sine the ISR p T spetrum is falling rather steeply, one would need to ollet relatively large amounts of data, in order to guarantee the presene of events with suffiiently hard ISR jets. Even then, 110

111 the olleted events may not ontain the momentum onfiguration required to give the maximum value of M (1,1,0) T. An alternative approah to make use of the kink struture would be to measure the endpoint funtion M (1,1,0) T,max ( M 0, p T ) for several different p T ranges, and then fit it to the analytial formula (4 17). Whether and how well this an work in pratie, remains to be seen, but the results of [36] from a toy exerise in the absene of any bakgrounds and detetor resolution effets do not appear very enouraging. Nevertheless, while the kink struture Θ (1,1,0) may be diffiult to observe, the measurement (4 1) of the endpoint M (1,1,0) T,max ( M 0 = 0, p T = 0) should be relatively straightforward. In Setions 4..1 and 4..3 we shall see that the additional M T information from events with n = deay hains will eventually allow us to determine all the unknown masses The Subsystem Variable M (,,1) T The subsystem variable M (,,1) T subhain within the smaller retangle on the left. M (,,1) T is illustrated in Figure 4-(b), where we use the is a genuine subhain variable in the sense that we only use the SM deay produts x, and ignore any remaining objets arising from the two x 1 s. In the absene of ISR (p T = 0) one an adapt the results from [37] and show that the formula for the M (,,1) T endpoint is M (,,1) T,max ( M 1, p T = 0) = µ (,,1) + µ (,,1) + M 1, (4 30) where the parameter µ (,,1) was defined in Equation (4 9): µ (,,1) M ( ) 1 M 1 = M M1. (4 31) M M Almost all of our disussion from the previous Setion an be diretly applied here as well. For example, in order to measure the parameter µ (,,1), we only need to extrat the endpoint of a single distribution, for a single fixed value of the test mass M 1. As 111

112 before, we hoose to use M 1 = 0. The resulting endpoint measurement M (,,1) T,max ( M 1 = 0, p T = 0) = µ (,,1) = M M 1 M (4 3) provides the required measurement of the parameter µ (,,1) appearing in Equation (4 30), as well as one onstraint on the masses M 1 and M involved in the problem. More importantly, the new onstraint in Equation (4 3) is independent of the relation in Equation (4 1) found previously in Setion The new variable M (,,1) T will also exhibit a kink in the plot of its endpoint M (,,1) T,max as a funtion of the test mass M 1. This is the same type of kink as the one disussed in the previous subsetion, therefore all of our previous results would apply here as well. In partiular, the analytial expression for the kink is given by M (,,1) T,max ( M 1, p T ) = F (,,1) L ( M 1, p T ), if M 1 M 1, F (,,1) R ( M 1, p T ), if M 1 M 1, (4 33) where F (,,1) L ( M 1, p T ) = F (,,1) R ( M 1, p T ) = [ µ (,,1) (p T ) + [ µ (,,1) ( p T ) + ( µ (,,1)(p T ) + p T ( µ (,,1)( p T ) p T ) + M 1 ) + M 1 ] p T 4 1 ] p T 4, (4 34) 1 (4 35), and the p T -dependent parameter µ (,,1) (p T ) is defined in analogy to Equation (4 0) ( ) µ (,,1) (p T ) = µ (,,1) pt 1 + p T. (4 36) M M The size of the new kink Θ (,,1) an be easily read off from Equation (4 8), where one should make the obvious replaements M 0 M 1 and M 1 M. 11

113 We an now generalize the two examples disussed so far (M (1,1,0) T and M (,,1) T ) to the ase of an arbitrary grandparent index n, with p = n and = n 1. We get where M (n,n,n 1) T,max ( M n 1, p T ) = F (n,n,n 1) L ( M n 1, p T ) = F (n,n,n 1) R ( M n 1, p T ) = [ µ (n,n,n 1) (p T ) + [ µ (n,n,n 1) ( p T ) + F (n,n,n 1) L ( M n 1, p T ), if M n 1 M n 1, F (n,n,n 1) R ( M n 1, p T ), if M n 1 M n 1, ( µ (n,n,n 1)(p T ) + p T ( µ (n,n,n 1)( p T ) p T ) + M n 1 ) + M n 1 ] p T 4 (4 37) ] p T 4 1, (4 38) 1, (4 39) and the p T -dependent parameter µ (n,n,n 1) (p T ) is simply the generalization of Equations (4 0) and (4 36): µ (n,n,n 1) (p T ) = µ (n,n,n 1) 1 + ) p T. (4 40) M n M n ( pt For n = 1 or n =, the general formula of Equation (4 37) reprodues our previous results in Eauations (4 17) and (4 33), orrespondingly The Subsystem Variable M (,,0) T The variable M (,,0) T is illustrated in Figure 4-(b), where we use the whole hain within the larger retangle. As long as we ignore the effets of any ISR, we have a balaned momentum onfiguration and the analytial results from Ref. [37] would apply. In partiular, the endpoint M (,,0) T,max ( M 0, p T = 0) is given by [37] M (,,0) T,max ( M 0, p T = 0) = F (,,0) L ( M 0, p T = 0), if M 0 M 0, F (,,0) R ( M 0, p T = 0), if M 0 M 0, (4 41) In the sense of Equation (4 15). See the disussion following Equation (4 15). 113

114 where F (,,0) L ( M 0, p T = 0) = µ (,,0) + µ (,,0) + M 0, (4 4) F (,,0) R ( M 0, p T = 0) = µ (,,1) + µ (,1,0) + (µ(,,1) µ (,1,0) ) + M 0, (4 43) and the various parameters µ (n,p,) are defined in (4 9). Notie that these expressions are valid only for p T = 0. We have also derived the orresponding generalized expression for M (,,0) T,max ( M 0, p T ) for arbitrary values of p T, whih we list in Appendix B. The most striking feature of the endpoint funtion (4 41) is that it will also exhibit a kink Θ (,,0) at the true value of the test mass M 0 = M 0. However, as emphasized in [36], the physial origin of this kink is different from the kinks Θ (1,1,0) and Θ (,,1) whih we enountered previously in Setions and This is easy to understand in Setions and 4.1. we saw that the kinks Θ (1,1,0) and Θ (,,1) arise due to ISR effets, while Equation (4 41) holds in the absene of any ISR. The explanation for the Θ (,,0) kink has atually already been provided in [37]. In essene, one an treat the SM deay produts x 1 and x in eah hain as a omposite partile of variable mass, and the two branhes F (,,0) L and F (,,0) R orrespond to the two extreme values for the mass of this omposite partile. In spite of its different origin, the kink in the funtion (4 41) shares many of the same properties. Let us use a speifi example as an illustration. Consider a popular example from supersymmetry, suh as gluino pair-prodution, followed by sequential two-body deays to squarks and the lightest neutralinos. This is preisely a asade of the type n =, in whih X is the gluino g, X 1 is a squark q, and X 0 is the lightest neutralino χ 0 1. Let us hoose the superpartner masses aording to the SPS1a mass spetrum, whih was also used in [37]: M = 613 GeV, M 1 = 55 GeV, M 0 = 99 GeV. (4 44) 114

115 Figure 4-4. Dependene of the M (,,0) T,max and M(,1,0) T,max upper kinemati endpoints on the value of the test mass M 0, for (a) the SPS1a parameter point in MSUGRA: M = 613 GeV, M 1 = 55 GeV, and M 0 = 99 GeV; or (b) a split spetrum M = 000 GeV, M 1 = 00 GeV, and M 0 = 100 GeV. The horizontal (vertial) dotted lines denote the true value of the parent (hild) mass for eah ase. Solid (dashed) lines indiate true (false) endpoints, while red (blue) lines orrespond to F (n,p,) L (F (n,p,) R ) branhes. The resulting funtion M (,,0) T,max ( M 0, p T = 0) is plotted in Figure 4-4(a) with the upper set of lines. There are several noteworthy features of M (,,0) T,max ( M 0, p T = 0) whih are evident from Figure 4-4(a). First, when the test mass M 0 is equal to the true hild mass M 0, the M T endpoint yields the true parent mass, in this ase M : M (,,0) T,max ( M 0 = M 0, p T = 0) = M. (4 45) This property of M T is true by design, and is onfirmed by the dotted lines in Figure 4-4(a). Seond, as seen from Equation (4 41), M (,,0) T,max ( M 0, p T = 0) is not given by a single funtion, but has two separate branhes. The first ( low ) branh F (,,0) L M 0 applies for M 0, and is shown in Figure 4-4(a) with red lines. The seond ( high ) branh F (,,0) R is valid for M 0 M 0 and is shown in blue in Figure 4-4(a). While the two branhes oinide at M 0 = M 0 : F (,,0) L (M 0, p T = 0) = F (,,0) R (M 0, p T = 0). (4 46) 115

116 Figure 4-5. The amount of kink: (a) Θ (,,0) and (b) Θ (,1,0) in degrees, as a funtion of the mass ratios y and z. The white dot and the white asterisk denote the loations in this ( y, z) parameter spae of the two sample spetra (4 44) and (4 6) used for Figures 4-4(a) and 4-4(b), orrespondingly. Their derivatives do not math: ( ) F (,,0) L M 0 M 0 =M 0 leading to a kink Θ (,,0) in the funtion M (,,0) T,max ( M 0, p T ( ) F (,,0) R M 0 M 0 =M 0, (4 47) general definition (4 7), we obtain the size of this kink quantitatively, ( ) (1 y)(1 z) yz Θ (,,0) = artan (y + z)(1 + yz) + 4yz where we have defined the squared mass ratios = 0) [34 37]. Applying the, (4 48) y M 1 M, z M 0 M 1. (4 49) The result (4 48) is plotted in Figure 4-5(a) as a funtion of the mass ratios y and z. Figure 4-5(a) demonstrates that as both y and z beome small, the kink Θ (,,0) gets more pronouned. Figure 4-5(a) also shows that the kink Θ (,,0) is a symmetri funtion of y and z, as an also be seen diretly from Equation (4 48). Therefore, the kink Θ (,,0) will be best observable in those ases where y and z are both small, and in addition, the mass spetrum happens to obey the relation y = z, i.e. M 1 = 116

117 M0 M. For this speial value of M 1 = M 0 M, the upper endpoint of the invariant mass distribution M x1 x is the same as in the ase when the intermediate partile X 1 is off-shell, i.e. when M 1 > M. Then we find that the M (,,0) T,max formulas and orresponding kink strutures are idential in the on-shell and off-shell ases. We provide more details in Appendix B. Unfortunately, the SPS1a study point is rather far from this ategory the spetrum (4 44) orresponds to the values y = and z = 0.189, whih are indiated in Figure 4-5(a) by a white dot. This onlusion is also supported by Figure 4-4(a), whih shows a rather mild kink in the SPS1a ase. We shall be rather ambivalent in our attitude toward the Θ (,,0) kink as well. While the interpretation of the kink is straightforward, its observation in the atual experiment is again an open issue. On the one hand, the experimental preision would depend on the partiular signature, i.e. the type of the SM partiles x 1 and x. If those are leptons, their 4-momenta p (k) 1 and p (k) will be measured relatively well and the kink might be observable. However, when x 1 and x are jets, the experimental resolution may not be suffiient. Seondly, as seen in Figure 4-4(a), the kink itself may not be very pronouned, and its observability will in fat depend on the partiular mass spetrum. The main lesson from the above disussion is that while the existene of the kink is without a doubt, its atual observation is by no means guaranteed. Therefore, our main mass measurement method, desribed later in Setion 4..1, will not use any information related to the kink. In fat in Setion 4..1 we shall show that one an ompletely reonstrut the mass spetrum of the new partiles, using just measurements of M T endpoints, eah done at a single fixed value of the orresponding test mass. It is worth noting that, in general, an endpoint in a spetrum is a sharper feature than a kink of the type (4 7). Therefore, we would expet that the experimental preision on the extrated endpoints will be muh better than the orresponding preision on the kink loation. The kink will also not play any role in our hybrid method, 117

118 desribed in Setion Only for the method desribed in Setion 4.., we shall try to make use of the kink information. Let us now return to our original disussion of the M (,,0) T endpoint (4 41). Following our previous approah from Setions and 4.1., we would hoose a fixed value of the test mass M 0 and measure the orresponding M T endpoint. However, the presene of two branhes in Equations (4 4) and (4 43) leads to a slight ompliation: for a randomly hosen value of M 0, we will not know whether we should use Equations (4 4) or (4 43) when interpreting the endpoint measurement. This requires us to make very speial hoies for the fixed value of M 0, whih would remove this ambiguity. It is easy to see that by hoosing M 0 = 0, we an ensure that the endpoint is always desribed by the low branh in Equation (4 4), and the M (,,0) T,max measurement an then be uniquely interpreted as M (,,0) T,max ( M 0 = 0, p T = 0) = µ (,,0) = M M 0 M. (4 50) However, we ould also design a speial hoie of M 0, whih would selet the high branh in Equation (4 4) and again uniquely remove the branh ambiguity. For this purpose, we must hoose a value for the test mass M 0 whih is suffiiently large, in order to safely guarantee that it is well beyond the true mass M 0. Sine the true mass M 0 an never exeed the beam energy E b, one obvious safe and rather onservative hoie for M 0 ould be M 0 = E b, in whih ase from (4 41) we get M (,,0) T,max ( (µ(,,1) ) M 0 = E b, p T = 0) = µ (,,1) + µ (,1,0) + µ (,1,0) + E b (4 51) = M M ( ) ( ) M 1 + M 0 M M + 1 M 0 + E M M1 4 M M1 b. Notie that the high branh funtion F (,,0) R in Equation (4 43) is rather unique in one very important aspet: it depends not just on one, but on two mass parameters, namely the ombinations µ (,,1) + µ (,1,0) and µ (,,1) µ (,1,0). In ontrast, the low branh F (,,0) L, as well as the previously disussed endpoint funtions M (n,n,n 1) T,max ( M 0, p T = 0), 118

119 eah ontained a single µ parameter. As a result, in those ases we did not benefit from any extra measurements for different values of the test mass M 0 had we done that, we would have been measuring the same µ parameter over and over again. However, the situation with F (,,0) R is different, and here we will benefit from an additional measurement for a different value of M 0. For example, let us hoose M 0 = Ẽ b, with Ẽ b > E b, whih will still keep us on the high branh. We obtain another onstraint M (,,0) T,max ( (µ(,,1) ) M 0 = Ẽ b, p T = 0) = µ (,,1) + µ (,1,0) + µ (,1,0) + Ẽ b (4 5) = M M ( ) ( ) M 1 + M 0 M M + 1 M 0 + Ẽ M M1 4 M M1 b. It is easy to hek that the onstraints in Equations (4 50 through 4 5) are all independent, thus providing three independent equations 3 for the three unknown masses M 0, M 1 and M. These three Equations (4 50 through 4 5) an be solved rather easily 4, and one obtains the proper solution for the masses M 0, M 1 and M, up to a two-fold ambiguity: M M, M 1 M 0 M 1 M, M 0 M 0, (4 53) whih is nothing but the interhange y z at a fixed M. The ambiguity arises beause the expression (4 41) for the endpoint M (,,0) T,max (and onsequently, the set of onstraints 3 In pratie, instead of relying on individual endpoint measurements for three different values of M 0, one may prefer to use the experimental information for the whole funtion M (,,0) T,max ( M 0, p T = 0) and simply fit to it the analytial expression (4 41) for the three floating parameters M 0, M 1 and M, as was done [37]. As we shall see shortly, this method does not lead to any new information, and may only improve the statistial error on the mass determination. Therefore, to keep our disussion as simple as possible, we prefer to talk about the three individual measurements in Equations (4 50 through 4 5) as opposed to fitting the whole distribution (4 41). 4 The general solution for M, M 1 and M 0 in terms of the measured endpoints ( ) is rather messy and not very illuminating, therefore we do not list it here. 119

120 ( )) is invariant under the transformation (4 53). Beause of this ambiguity, in addition to the original SPS1a input values (4 44) for the mass spetrum, we obtain a seond solution M = 613 GeV, M 1 = GeV, M 0 = 99 GeV. (4 54) This seond solution was missed in the analysis of Ref. [37]. It is easy to hek that the alternative mass spetrum (4 54) gives an idential M (,,0) T,max ( M 0, p T = 0) distribution as the one shown in Figure 4-4(a), so that it is impossible to rule it out on the basis of M (,,0) T,max measurements alone. The previous disussion reveals an important and somewhat overlooked benefit from the existene of the kink one an make not one, not two, but three independent endpoint measurements from a single M (n,p,) T distribution! In fat, we shall argue that the three measurements in Equations (4 50 through 4 5) are muh more robust than the kink measurement (4 7). For example, when the hild mass is relatively small, the lower branh F (,,0) L is relatively short and the kink will be diffiult to see, even under ideal experimental onditions. An extreme example of this sort is presented in Setion 4., where we disuss top quark events, in whih the hild (neutrino) mass M 0 is pratially zero and the kink annot be seen at all. However, even under those irumstanes, the endpoint measurements in Equations (4 50 through 4 5) are still available. More importantly, the onstraints in Equations (4 50 through 4 5) are independent of the previously found relations in Equations (4 1) and (4 3), so that the latter an be used to resolve the two-fold ambiguity in Equation (4 53). Before we move on to a disussion of the last remaining subsystem M T quantity in the next Setion 4.1.4, let us reap our main result derived in this subsetion. We showed that the M (,,0) T variable yields three independent endpoint measurements in Equations (4 50 through 4 5), and possibly a kink measurement in Equation (4 7). The M (,,0) T endpoint measurements by themselves are suffiient to determine all three 10

121 masses M 0, M 1 and M, up to the two-fold ambiguity in Equation (4 53). This represents a pure M T -based mass measurement method, whih does not use any any kink or invariant mass information The Subsystem Variable M (,1,0) T The variable M (,1,0) T is illustrated in Figure 4-(b), where we use the subhain within the smaller retangle on the right. This is another genuine subsystem quantity, sine we only use the SM deay produts x 1 and ignore the upstream objets x. However, the upstream objets x are important in the sense that they have some non-zero transverse momentum, and as a result, the sum of the transverse momenta P (k) 0T of the hildren X 0 is not balaned by the sum of the transverse momenta of the SM objets x 1 used in the M T alulation: P (1) 0T + P () (1) () (1) () 0T + p 1T + p 1T = p T p T p T 0. (4 55) Notie that even in the absene of any ISR p T, this is still an unbalaned onfiguration, due to the transverse momenta p (1) T and p () T of the upstream objets x. Therefore, we annot use the existing analytial results on M T, sine previous studies always assumed that the right-hand side of Equation (4 55) is exatly zero, due to the lak of any partiles upstream. We therefore need to generalize the previous treatments of M T and obtain the orresponding endpoint formulas for our new subsystem M (,1,0) T variable. In partiular, in the absene of any intrinsi ISR (i.e., for p T = 0), we find that the endpoint of the M (,1,0) T M (,1,0) T,max ( M 0, p T = 0) = distribution is given by F (,1,0) L ( M 0, p T = 0), if M 0 M 0, F (,1,0) R ( M 0, p T = 0), if M 0 M 0, (4 56) 11

122 where F (,1,0) L ( M 0, p T = 0) = F (,1,0) R ( M 0, p T = 0) = { [ µ (,,0) µ (,,1) + { [ µ (,1,0) + ] } 1 µ (,,0) + M 0 µ (,,1) ) (µ(,,1) µ (,1,0) + M 0 ] µ (,,1), (4 57) } 1 (4 58), and the various parameters µ (n,p,) are defined in (4 9). The orresponding expressions for general p T (i.e., arbitrary intrinsi ISR) are listed in Appendix B. From Equation (4 56) we see that, one again, the endpoint funtion M (,1,0) T,max ( M 0, p T = 0) would exhibit a kink Θ (,1,0) at the true value of the test mass M 0 = M 0 : ( ) ( ) F (,1,0) L F (,1,0) R. (4 59) M 0 M 0 M 0 =M 0 M 0 =M 0 The existene of this kink should ome as no surprise Ref. [36] showed (in the p T limit) that any type of upstream momentum will generate a kink in an otherwise smooth M T,max funtion. As before, the value of the M T endpoint M (,1,0) T,max loation reveals the true mass of the parent: at the kink M (,1,0) T,max ( M 0 = M 0, p T = 0) = M 1. (4 60) At the same time, the physial origin of this kink is different from either of the two kink types ( Θ (1,1,0) and Θ (,,0) ) disussed earlier. Clearly, the new kink is different from Θ (,,0), whih was due to the varying invariant mass of the {x 1, x } system. Here we are using a single SM partile x 1 whose mass is onstant. Furthermore, the new kink Θ (,1,0) annot be due to any ISR like in the ase of Θ (1,1,0), sine Equation (4 56) does not aount for any ISR effets. The real reason for this new Θ (,1,0) kink is a third one, namely, the kinematial restritions plaed by the deays of the upstream partiles (in this ase, the grandparents X ). We now proeed to investigate the new kink Θ (,1,0) quantitatively. Using the same example of gluino pair-prodution for the SPS1a mass spetrum (4 44), we plot 1

123 the funtion (4 56) in Figure 4-4(a). Comparing the lower and the upper set of lines in Figure 4-4(a), we notie that the M (,1,0) T,max and M(,,0) T,max variables share several ommon harateristis. They both exhibit a kink at the true loation of the hild mass M 0 = M 0, while their values at that point reveal the true parent mass in eah ase: M 1 for M (,1,0) T,max and M for M (,,0) T,max. Using the definition (4 7), we find that the size of the Θ(,1,0) kink is given by ( Θ (,1,0) (1 y )(1 z) ) z = artan, (4 61) z(1 + y ) + y(1 + z ) + yz where the parameters y and z were already defined in (4 49). The kink Θ (,1,0) is plotted in Figure 4-5(b) as a funtion of y and z. We notie that the kink struture beomes more pronouned for relatively small y and z. Comparing Figures 4-5(a) and 4-5(b), we see that for any given set of values for y and z, the Θ (,1,0) kink disussed here is more pronouned than the Θ (,,0) kink from the previous subsetion 5. The differene is partiularly notieable in the region of y 0 and z 0.. The SPS1a mass spetrum (4 44) in our previous example was rather far away from this region, as indiated by the white dots in Figure 4-5. Now let us hoose a different mass spetrum, whih is loser to the region where the differene between the two kinks beomes more notieable, for example M = 000 GeV, M 1 = 00 GeV, M 0 = 100 GeV, (4 6) orresponding to the point marked with the white asterisk in Figures 4-5(a) and 4-5(b). The resulting endpoint funtions M (,,0) T,max and M(,1,0) T,max Indeed we see that with this new spetrum the kink in the M (,1,0) T,max more notieable than the kink in the M (,,0) T,max regarding the M (,1,0) T are plotted in Figure 4-4(b). funtion is muh funtion. Therefore, our first onlusion variable is that its kink is in general sharper and appears to 5 This statement an also be verified using the analytial formulas (4 48) and (4 61). 13

124 be more promising than the previously disussed kink in the M (,,0) T variable from Setion Following our previous strategy, we shall not dwell too long on the kink, but instead we shall disuss the available endpoint measurements for various values of M 0. Again, the presene of two branhes in Equation (4 56) an be used to our advantage. As in Setion 4.1.3, we first hoose a test mass value M 0 = 0, whih would selet the low branh (4 57) and result in an endpoint measurement M (,1,0) T,max ( M 0 = 0, p T = 0) = µ (,,0) (µ (,,0) µ (,,1) ). (4 63) Just as before, we ould also hoose a rather large value for M 0 = E b, whih would selet the high branh (4 58) and result in the measurement M (,1,0) T,max ( M 0 = E b, p T = 0) = { [ µ (,1,0) + (µ(,,1) µ (,1,0) ) + E b ] µ (,,1) A third hoie, M 0 = Ẽ b, with Ẽ b > E b, would yield yet another endpoint measurement { [ ] M (,1,0) T,max ( (µ(,,1) ) M 0 = Ẽ b, p T = 0) = µ (,1,0) + µ (,1,0) + Ẽ b µ (,,1) Again we obtained three Equations (4 63 through 4 65) for the three unknown µ-parameters µ (,,0), µ (,,1) and µ (,1,0), or equivalently, for the three unknown masses M 0, M 1 and M. Equations (4 63 through 4 65) are all independent and an be easily solved, giving a total of four solutions. However, three of the solutions are always unphysial, so that we end up with a single unique solution. This represents an important advantage of the M (,1,0) T,max disussed in Setion There we found that M (,,0) T,max } 1 } 1 variable in omparison with the M(,,0) T,max variable ambiguity in the mass spetrum, while now we see that M (,1,0) T,max. (4 64). (4 65) always gives rise two a two-fold does not suffer from this problem and already by itself allows for a omplete and unambiguous determination of the mass spetrum. 14

125 4. M T -based Mass Measurement methods In this setion we use the analytial results derived in the previous setion to propose three different strategies for determining the masses in n deay hains. We shall illustrate eah of our methods with a speifi example, for whih we hoose to onsider the dilepton samples from W + W and t t events. The former is an example of the n = 1 deay hain exhibited in Figure 4-(a), while the latter is an example of the n = deay hain in Figure 4-(b). Most importantly, these samples already exist in the Tevatron data and will also be among the first to be studied at the LHC. Correspondingly, throughout this setion we shall use the following mass spetrum M = m t = 173 GeV, M 1 = m W = 80 GeV, (4 66) M 0 = m ν = 0 GeV. Before we begin, let us review the four different M (n,p,) T variables whih are in priniple available in that ase. Eah one of them is plotted in Figure 4-6 for five different values of the orresponding test mass (0, 100, 00, 300 and 400 GeV). In Figure 4-6(a) we show the M (1,1,0) T variable from W + W pair prodution events, while in Figures 4-6(b-d) we orrespondingly show the M (,,1) T, M (,,0) T and M (,1,0) T variables from t t events. We used PYTHIA [54] for event generation and did not impose any seletion uts, sine they will not affet the loation of the M T endpoint 6. The plots are made for the Tevatron (a p p ollider with a TeV enter-of-mass energy), where the relevant data is already available. The orresponding analysis for the LHC is very similar. All of our plots in this Setion have the full ISR effets. 6 The uts would have an impat on the overall aeptane and effiieny. This effet is not relevant here, sine we are showing unit-normalized distributions. The uts may also distort the shape of eah distribution, but should preserve the loation of the upper endpoint. 15

126 Figure 4-6. Unit-normalized distributions of M (n,p,) T variables in dilepton events from (a) W + W pair prodution and (b-d) t t pair prodution. Eah panel shows results for five different values (0, 100, 00, 300 and 400 GeV) of the orresponding test mass. The methods of Setions 4..1 and 4..3 only make use of the M T endpoint at zero test mass, M (n,p,) T,max ( M = 0), whih is indiated by the vertial red arrow. In panel (), the two dotted line M (,,0) T distributions orrespond to the orret and the wrong pairing of the two b-jets with the leptons, while the solid line distribution is the average of these two. As disussed in Setion 4.1, the presene of ISR with nonzero p T will inrease the nominal M (n,p,) T endpoints: M (n,p,) T,max ( M, p T ) M (n,p,) T,max ( M, 0), (4 67) where the equality is obtained only when M = M. ISR will therefore introdue some systemati error when one is trying to measure M (n,p,) T,max ( M, 0). The size of this 16

127 error depends on the ISR p T spetrum, whih in turn depends on the type of ollider (Tevatron or LHC). At the Tevatron, this will not be suh a serious issue, as evidened from Figure 4-6, where the observed endpoints in the presene of ISR math pretty well with their expeted values for the p T = 0 ase. On the other hand, at the LHC this may beome a problem, whih an be handled in one of two ways. First, depending on the partiular signature, one may be able to selet a sample with p T 0 (at a ertain ost in statistis), by imposing a suitably designed jet veto to remove jets from ISR. Alternatively, one an use the full event sample (whih would inlude ISR jets), and make use of our general formulas in Appendix B, whih ontain the expliit p T dependene of M (n,p,) T,max. In the previous Setion 4.1 we derived that in the ase of n asades, there are 8 different M T endpoint measurements: one for M (1,1,0) T,max and Setion 4.1.1), one for M (,,1) T,max M (,,0) T,max (see Equation (4 1) (see Equation (4 3) and Setion 4.1.), three for (see Equations (4 50 through 4 5) and Setion 4.1.3), and three for M(,1,0) T,max (see Equations (4 63 through 4 65) and Setion 4.1.4). Given that we are trying to determine only three masses M 0, M 1 and M, it is lear that these 8 measurements should be suffiient to ompletely determine the spetrum. The number of available measurements is in fat muh larger than the number of M (n,p,) T,max variables. Indeed, as shown in Setions and 4.1.4, there are ases where we might be able to obtain more than one mass onstraint from a given M (n,p,) T,max variable. Of ourse, the 8 measurements annot all be independent among themselves, as they only depend on three parameters. Our three methods below will be distinguished based on whih subset of these measurements we are using Pure M T Endpoint Method With this method, we use M T endpoint measurements E np at a single fixed value of the test mass, whih for onveniene we take to be M = 0: E np M (n,p,) T,max ( M = 0, p T = 0). (4 68) 17

128 The orresponding formulas interpreting those measurements in terms of the physial masses M 0, M 1 and M were derived in Setion 4.1: E 110 M (1,1,0) T,max (0, 0) = M 1 M 0 M 1 = M y (1 z), (4 69) E 1 M (,,1) T,max (0, 0) = M M 1 M = M (1 y), (4 70) E 0 M (,,0) T,max (0, 0) = M M 0 M = M (1 yz), (4 71) E 10 M (,1,0) T,max (0, 0) = 1 (M M M 0 )(M 1 M 0 ) = M y(1 z)(1 yz) (4 7). Using the mass spetrum (4 66), the predited loations of these four M T endpoints are E 110 = 80 GeV, (4 73) E 1 = 136 GeV, (4 74) E 0 = 173 GeV, (4 75) E 10 = 80 GeV, (4 76) whih are marked with the vertial red arrows in Figure 4-6. Given that we have four measurements (4 69 through 4 7) for only three parameters M 0, M 1 and M, one should be able to uniquely determine all three of the unknown parameters. Naively, it seems that using just three of the measurements (4 69 through 4 7) should be suffiient for this purpose, and furthermore, that any three of the measurements (4 69 through 4 7) will do the job. However, one should exerise aution, sine not all four measurements (4 69 through 4 7) are independent. It is easy to hek that E 1, E 0 and E 10 obey the following relation E 10 = E 0 (E 0 E 1 ). (4 77) This means that in order to be able to solve for the masses from Equations (4 69 through 4 7), we must always make use of the E 110 measurement in Equation (4 69), 18

129 and then we have the freedom to hoose any two out of the remaining three measurements (4 70 through 4 7). For example, using the set of three measurements {E 110, E 1, E 0 } (i.e. Equations (4 69 through 4 71)), the masses are uniquely determined as M 0 = E { 110 E1 (E 0 E 1 ) [ ]} E 0 (E 0 E 1 ) E110 1 E110 (E, (4 78) 0 E 1 ) M 1 = E 110 E 1 (E 0 E 1 ) E 110 (E 0 E 1 ), (4 79) M = E 110 E 1 E 110 (E 0 E 1 ). (4 80) Similarly, one an solve for M 0, M 1 and M using the set of measurements {E 110, E 0, E 10 }, or alternatively, the set of measurements {E 110, E 1, E 10 }. In eah ase, the remaining fourth unused measurement provides a useful onsisteny hek on the mass determination. 4.. M T Endpoint Shapes And Kinks The method proposed in Setion 4..1 uses the measured endpoints from several different M (n,p,) T single M (n,p,) T variables. Now we disuss an alternative method whih makes use of a variable. Let us begin with the simplest ase of n = 1 as shown in Figure 4-(a). In that ase, we have only one M T variable at our disposal, namely M (1,1,0) T. Its properties were disussed in Setion 4.1.1, where we showed that its endpoint M (1,1,0) T,max an allow the determination of both masses M 0 and M 1, at least as a matter of priniple. Indeed, the endpoint measurement (4 69) at zero test mass provides one relation among M 0 and M 1. The key observation in Setion (whih was first done in [36]) was that with the inlusion of ISR effets, the endpoint funtion M (1,1,0) T,max ( M 0, p T ) exhibits a kink at M 0 = M 0, whih an then be used to determine both masses M 0 and M 1. The method an be readily applied to the existing dilepton event sample from W + W pair prodution, whih will allow an independent measurement of the W mass m W and the neutrino mass m ν. While the preision of this measurement will not be ompetitive 19

130 with existing W and neutrino mass determinations, it is nevertheless useful to test the viability of this approah with real data. Now let us disuss the more ompliated ase of n =, whih in our example orresponds to t t pair prodution with both tops deaying leptonially. As disussed in Setions 4.1., and 4.1.4, here we have a hoie of three different M T variables: M (,,1) T, M (,,0) T, and M (,1,0) T. Beause of the larger t t ross-setion, we expet that the statistial preision on eah one of those three variables will be better than the M (1,1,0) T variable of the n = 1 ase. As shown in Setions and 4.1.4, eah of the two variables M (,,0) T and M (,1,0) T exhibits a kink in its endpoint M (n,p,) T,max when onsidered as a funtion of the test mass M 0, even when the transverse momentum of the intrinsi ISR in the event is zero, p T = 0. Then, whih of these two variables is better suited for a mass determination? The ase of M (,,0) T,max ( M 0, p T = 0) was already disussed in [34, 36, 37]. Here we would like to propose the alternative measurement of M (,1,0) T,max ( M 0, p T M (,1,0) T,max ( M 0, p T ase of M (,,0) T,max ( M 0, p T = 0): = 0). What is more, we would like to emphasize that our funtion = 0) offers several unique advantages over the previously onsidered 1. The subsystem variable M (,1,0) T does not suffer from the ombinatoris problem whih is present for M (,,0) T. Indeed, when onstruting the M (,,0) T distribution, one has to deide how to pair up the b-jets with the two leptons. Beause it is diffiult to distinguish between a b and a b, there is a two-fold ambiguity whih is quite diffiult to resolve by other means. In ontrast, our subsystem variable M (,1,0) T does not make diret use of the b-jets, and is therefore free of suh ombinatoris issues.. As we already saw in Setion 4.1.3, even under perfet experimental onditions, the fit to the M (,,0) T,max endpoint results in two separate solutions for the mass spetrum: one solution (Equation (4 44)) is given by the true values of the input masses, while the seond solution (Equation (4 54)) is obtained by the transformation in Equation (4 53). Using M (,,0) T,max alone, there is no way to tell the differene between these two mass spetra. In ontrast, our variable M (,1,0) T does not suffer from this ambiguity, and aording to our results from Setion the solution is always unique. 130

131 3. The third advantage of the subsystem variable M (,1,0) T is related to the expeted preision on the determination of the masses. As we pointed out in Setion and illustrated expliitly in Figure 4-5, the kink Θ (,1,0) in the M (,1,0) T,max ( M 0, p T = 0) funtion is muh sharper than the orresponding kink Θ (,,0) in the M (,,0) T,max ( M 0, p T = 0) funtion. This an also be seen expliitly from the two examples shown in Figure 4-4. As a result, we expet that the kink struture an be better identified in the ase of M (,1,0) T,max, whih would lead to smaller errors on the mass determination. Of ourse, one ould (and in fat should) use the experimental information from both M (,,0) T,max and M(,1,0) T,max, if available. Our main goal here is simply to point out the obvious advantages of the subsystem variable M (,1,0) T, whih so far has not been used in the literature Hybrid Method: M T Endpoints Plus An Invariant Mass Endpoint Any asade with n will provide a ertain number of measurements like as invariant mass endpoints in addition to the M T measurements disussed so far. In partiular, for the n = example onsidered here, there will be one measurement of the endpoint of the M x1 x invariant mass distribution. The formula for the endpoint M x1 x,max in terms of the unknown physial masses M 0, M 1 and M is in general given by E im M x1 x,max = 1 M 1 (M M 1) (M 1 M 0) = M (1 y)(1 z). (4 81) In the ase of t t events onsidered here, this is simply the endpoint of the invariant mass distribution m bl of eah lepton and its orresponding b-jet. This distribution (unit-normalized) is shown in Figure 4-7. Unfortunately, here one is faing the same ombinatorial problem as with the M (,,0) T variable we annot easily tell the harge of the b-jet, therefore a priori it is not lear whih b-jet goes with whih lepton. Fortunately, there are only two possibilities: the result from the orret (wrong) pairing is shown in Figure 4-7 with the green (blue) dotted line. We see that the green histogram with the orret pairing has an endpoint at the expeted loation (mt mw E im = ) (m W m ν) = GeV, (4 8) mw 131

132 with a relatively small tail due to the finite width effets. More importantly, the (blue) distribution from the wrong pairings is relatively smooth, and as a result the endpoint as in Equation (4 8) is preserved in the experimentally observable (red) distribution, whih inludes all possible bl pairings. Now we an add the new measurement (4 81) to the previously disussed set of measurements in Equations (4 69 through 4 7). We obtain a total of five measurements for the three underlying parameters M 0, M 1 and M, therefore there exist two relations among the measurements. The first relation is already given by Equation (4 77) and does not involve the invariant mass endpoint in Equation (4 81). The seond relation is given by E im = E 1E 110 E 0 E 1. (4 83) We an now onsider a hybrid method, whih would make use of the invariant mass endpoint (4 81), plus any two of the M T measurements in Equations (4 69 through 4 7). In priniple, one again needs to be areful and make sure that the three used measurements are independent. Fortunately, as seen from Equations (4 77,4 83), the invariant mass endpoint E im is independent from any pair of M T measurements. There are 6 possible pairs among the M T measurements (4 69 through 4 7), and in priniple eah one an be used in ombination with the invariant mass endpoint (4 81). What is the best hoie? We find that in all 6 of those ases one obtains a unique solution for the masses M 0, M 1 and M. Therefore, the optimal hoie is ditated by the experimental preision on eah of the measurements (4 69 through 4 7). We expet that the measurement (4 10) of M (1,1,0) T,max ross-setion for W + W prodution. Similarly, M (,,0) T,max will be less preise due to the smaller suffers from the ombinatorial problem already mentioned earlier. Therefore for our illustration of the hybrid method we hoose to use the M (,,1) T endpoint (4 70), the M (,1,0) T endpoint (4 7), and the invariant mass endpoint (4 81). 13

133 Figure 4-7. Unit-normalized m bl invariant mass-squared distributions in dilepton t t events. The green (blue) dotted line orresponds to the orret (wrong) pairing of the leptons and the b-jets, while the red solid line is the average of those two distributions. The endpoint (4 81) of the m bl distribution is marked by the vertial red arrow. The solution for the masses in terms of those three measurements is given by M 0 = M 1 = M = ( ) 1 E1 E im E 1 E10 + E 1 Eim E im E 1 + 4E10 E 1 + E im E 1 E 1 + 4E 10 E1 E im ( E 1 E 1 + 4E 10 E 1 E 1 + E im E 1 E 1 + 4E 10 E 1 Eim E1 + E im E 1 E 1 + 4E10 ) 1, (4 84), (4 85). (4 86) It is easy to hek that substituting the measured values of the endpoints E 1, E 10 and E im from Equations (4 74), (4 76) and (4 8), into Equations (4 86) above yields the values for the neutrino, W and top quark mass, orrespondingly. 133

134 CHAPTER 5 ONE DIMENSIONAL PROJECTION METHOD Most of mass reonstrution methods rely on exlusive hannels where a suffiiently long deay hain an be properly identified. Unfortunately, this almost inevitably requires the use of hadroni jets in some form in the analysis in most SUSY models, the main LHC signal is due to the strong prodution of olored superpartners, whose asade deays to the neutral LSP neessarily involve hadroni jets. For many reasons, jets are notoriously diffiult to deal with, espeially in a hadron ollider environment. Beause of the high jet multipliity in SUSY signal events, any jet-based analysis is bound to fae a severe ombinatorial problem and is unlikely to ahieve any good preision. Thus it is imperative to have alternative methods whih avoid the diret use of jets and instead rely only on the well measured momenta of any (isolated) leptons in the event. In this hapter, we desribe methods, whih are free of the jet ombinatorial problem. In the first two setion, we will use M T and the last setion, we will illustrate how to use full phase spae information with ontravse variable M CT as an example. 5.1 Detailed Study On M T s Charateristis For illustration in two setions, we shall use the standard example of R-parity onserving supersymmetry with a χ 0 1 LSP. Its ollider signatures have been extensively studied, and typially involve jets, leptons and missing transverse energy [89]. Among those, the inlusive same-sign dilepton hannel has already been identified as a unique opportunity for an early SUSY disovery at the LHC [75, 90]. The two leptons of the same harge an be easily triggered on, and provide a good handle for suppressing the SM bakground. In our analysis we use the LM6 CMS study point [75], whose relevant mass spetrum is given in Table 5-1. At point LM6, signal events with two isolated same-sign leptons typially arise from the SUSY event topology in Figure

135 Table 5-1. Seleted spartile masses (in GeV) at point LM6. We list the average q L mass M ql = 1 (M ũ L + M d L ). M g M ql + M χ 1 M l L M νl M χ P T d/ū l + ν p ũ/ d χ + 1 ν l χ 0 1 M p M p ũ/ d χ + 1 ν l χ 0 1 d/ū M T l + ν Figure 5-1. The typial SUSY event topology produing two isolated same-sign leptons at point LM6 (see text for details). The diagram for a pair of negatively harged leptons l l is analogous. Consider the inlusive prodution of same-sign harginos, whih deay leptonially as shown in the yellow-shaded box in Figure 5-1. The resulting sneutrino ( ν l ) ould be the LSP itself, or, as in the ase of LM6, may further deay invisibly to a neutrino ν and the true LSP χ 0 1. Suh same-sign hargino pairs typially result from squark deays, as indiated in Figure 5-1. In turn, the squarks may be produed diretly through a t-hannel gluino exhange, or indiretly in gluino deays. Note that the two same-sign leptons in Figure 5-1 are aompanied by a number of upstream objets (typially jets) whih may originate from various soures, e.g. initial state radiation, squark deays, or deays of even heavier partiles up the deay hain. In order to stay lear of jet ombinatorial issues, we shall adopt a fully inlusive approah to the same-sign dilepton signature, by treating all the upstream objets within the blak retangular frame in Figure 5-1 as a single entity of total transverse momentum P T. Given this very general setup, we now pose the following question: assuming that a SUSY disovery is made in the inlusive same-sign dilepton hannel, is it possible to measure the individual spartile masses M p and M involved in the leptoni deays 135

136 of Figure 5-1, using only the transverse momenta of the two leptons p (1) () lt and p lt, and the total upstream transverse momentum P T? Although it may appear that those three vetors do not provide a lot of information to go on, we shall show that this is possible. We disuss three different approahes. Method I. Let us onentrate diretly on the observed lepton momenta p (i) lt. Consider the two ollinear momentum onfigurations illustrated in Figure 5- and defined as follows. In eah onfiguration, the lepton momenta are the same: p (1) lt and then they an be either parallel or anti-parallel to the measured upstream P T : = p () lt ; s = +1 p (1) lt s = 1 p (1) lt = p () lt P T ; (5 1) = p () lt P T. (5 ) In what follows we shall use the integer s = +1 (s = 1) to refer to the parallel (anti-parallel) onfiguration: s os( p (1) lt, P T ) = os( p () lt, P T ). Now let us measure the maximum lepton momentum in eah onfiguration: p lt (sp T ) p (1) lt () = p lt max os( p (1) lt, P T )=s { p (i) lt }. (5 3) Observe that both p lt (+P T ) and p lt ( P T ) an be diretly measured from the lepton p T distributions. For example, onstrut a D satter plot {x, y} of with the ut p (1) lt x = os( p (1) lt () + p lt, P T ), y = p (1) lt () p lt < ϵ ( 0), and take the limit () + p lt, (5 4) ( y ) p lt (sp T ) = lim. (5 5) x s Armed with the two measurements p lt (+P T ) and p lt ( P T ), we an now diretly solve for the masses M p and M. The formula for p lt (sp T ) is p lt (sp T ) = M p M 4M p ) ( 4M p + (sp T ) sp T. (5 6) 136

137 p (1) lt P T p (1) lt PT p () lt p () lt s = +1 s = 1 Figure 5-. The two speial momentum onfigurations defined in Equations (5 1,5 ). Inverting Equation (5 6), we get M p = plt ( P T ) p lt (+P T ) p lt ( P T ) p lt (+P T ) P T, (5 7) thus fixing the absolute mass sale in the problem. One the parent mass M p is known, the hild mass M is M = M p 1 p lt ( P T ) p lt (+P T ) P T. (5 8) Thus we found the true spartile masses M p and M diretly in terms of the measured lepton momenta p lt (±P T ) and upstream momentum P T. Note that the hoie of the value for P T in Equations (5 7) and (5 8) is arbitrary, whih an be used to our advantage, e.g. to selet the most populated P T bin, minimizing the statistial error. Method II. In our previous method, the lepton momenta p lt (±P T ) were measured diretly from the data as implied by Equation (5 5). Alternatively, we an obtain them indiretly from the endpoint of the Cambridge M T variable. To be more preise, we apply the subsystem M T variable introdued in [3] to the purely leptoni subsystem in the yellow-shaded box of Figure 5-1. Following the generi notation of Referene [3], we denote the input (test) mass of the sneutrino hild as M. The subsystem M T variable is now defined as follows. First form the transverse mass M T for eah (hargino) parent M (i) T ( ) M + p (i) lt M + p (i) T p (i) (i) lt p T 137

138 in terms of the assumed test mass M and transverse momentum p (i) T for eah (sneutrino) hild. Just like the traditional M T, the leptoni subsystem M T variable [3] is defined through a minimization proedure over all possible partitions of the unknown hildren momenta p (k) T, onsistent with transverse momentum onservation (k) k ( p T (k) + p lt ) + P T = 0 M T ( M, P { T, p (i) lt {max ) min M (1) T, M() T }}. (5 9) The M T distribution has an upper kinemati endpoint { MT max ( M, P T ) max M T ( M, } P T, p (i) lt ), (5 10) all events whih an be experimentally measured and subsequently interpreted as the orresponding parent mass M p Mp( M, PT ) M max T ( M, P T ), (5 11) providing one funtional relationship among M p and M, but leaving the individual masses still to be determined. For us the importane of the M T variable (5 9) is that the momentum onfigurations in Figure 5- are preisely the ones whih determine its endpoint MT max. The omplete analytial dependene of the M T endpoint M p ( M, P T ) on both of its arguments M and P T is now known [3]: M p ( M, P T ) = M p ( M, +P T ), if M M, M p ( M, P T ), if M M, (5 1) where M p ( M, sp T ) = {[ ( p lt (sp T ) + plt (spt ) + sp T ) + M ] (sp T ) 4 } 1. (5 13) Thus we an alternatively obtain the spartile masses by measuring just two M T kinemati endpoints, with arbitrary hoies for the test mass M and the upstream 138

139 P T. For onreteness, let us pik some fixed M and P T, form the orresponding M T distribution (5 9) and measure its endpoint M p, also making a note of the onfiguration s : { } { } M, P T measure M p, s. (5 14) Now perform a seond suh measurement { } { } M, P T measure M p, s. (5 15) By inverting Equation (5 13), these two measurements allow the experimental determination of p lt (s P T ) = M p M ( ) 4 M 4 M p p + (s P T ) s P T (5 16) and similarly for p lt (s P T ). Now taking the ratio r p lt (s P T ) p lt (s P T ) = 4M p + (s P T ) s P T 4M p + (s P T ) s P T, (5 17) where in the seond step we used Equation (5 6), we an solve Equation (5 17) for the true parent mass M p in terms of measured quantities: M p = { rs P T s P T (1 r ) ( r s P T s P T and then find the true hild mass M from Equation (5 6) as ) )} (r s P T 1, (5 18) s P T ( M = M p 1 1 ) M 4 M p M p + (s P T ) s P T 4M p + (s P T ) s P T 1 (5 19) with M p already given by Equation (5 18). Note than in this method, the values of M, M, P T M. and P T an be hosen at will, allowing for repeated measurements of M p and Method III. The third and final method for extrating the two masses M p and M will make use of the elebrated kink in the M T endpoint funtion (5 1). 139

140 Figure 5-3. MT max versus the test mass M, as obtained in our simulations (data points) from a sample with P T = 40 ± 50 GeV, or theoretially from Equation (5 1) (blue solid line), as well as their differene (lower panel). Sine M p ( M, +P T ) and M p ( M, P T ) have different slopes at the rossover point M = M, the funtion M p ( M, P T ) has a slope disontinuity preisely at the orret value M of the hild mass, providing an alternative measurement of the absolute mass sale. The proedure is illustrated in Figure 5-3 for the LM6 study point of Table 5-1. The blue solid line shows the theoretially expeted shape from Equation (5 1), for P T = 40 GeV, whih is roughly the mean of the P T distribution at point LM6. In the LM6 ase the kink is very mild, only 3.3 [3]. In order to test the preision of the three methods, we perform event simulations using the PYTHIA event generator [54] and PGS detetor simulation [55]. We onsider the LHC at its nominal energy of 14 TeV and 100 fb 1 of data. To ensure disovery, we use standard CMS uts as follows [75, 91]: exatly two isolated leptons with p T > 10 GeV, at least three jets with p T > (175, 130, 55) GeV, /E T > 00 GeV and a veto on tau jets. With those uts, in the dimuon hannel alone, the remaining SM bakground ross-setion is rather negligible (0.15 fb), while the SUSY signal is 14 fb, already 140

141 leading to a σ disovery with just 10 fb 1 of data [75, 91]. In order to ompare to the theoretial result in Figure 5-3, we selet a ±50 GeV P T bin around P T = 40 GeV and onstrut a series of M T distributions, for different input values of M. For eah ase, we inlude all SM and SUSY ombinatorial bakgrounds, and extrat the M max T endpoint by a linear unbinned maximum likelihood fit, obtaining the data points shown in Figure 5-3. We see that the M T endpoint an be determined rather well (δ M p < 3 GeV), but only on the right branh M M. In ontrast, the M T endpoints on the left branh M M are onsiderably underestimated, washing out the expeted kink. There are two separate reasons behind this effet. Reall that the M T endpoint on the left branh is obtained in the onfiguration s = +1 of Figure 5-, whih requires the lepton to be emitted in the bakward diretion. As a result, the parent boost favors onfigurations with s 1 over s +1. Another onsequene is that leptons with s +1 are softer and more easily rejeted by the offline p T uts. We onlude that M max T measurements on the left branh are in general not very reliable, and tend to jeopardize the traditional kink method. For example, using Method III to fit the data in Figure 5-3 (green dotted line), we find best fit values of only M p(fit) = 1 GeV and M (fit) = 188 GeV. Method I has a similar problem, sine p lt (+P T ) is measured from events in the s = +1 onfiguration. Using the M p measurements from Figure 5-3 at M = 0 and M = 1 TeV, we find from Equation (5 16) that p lt (+40 GeV) = 8.8 GeV and p lt ( 40 GeV) = 50.6 GeV (ompare to the nominal values of 14.8 GeV and 53.6 GeV, orrespondingly). The resulting mass determination via Equations (5 7,5 8) is M p(fit) = 1 GeV and M (fit) = 190 GeV. We see that in both Method I and Method III, the masses are underestimated due to the systemati underestimation of the left M max T branh in Figure 5-3. It is therefore of great interest to have an alternative method, whih relies on the right M max T branh alone. 141

142 Figure 5-4. Saling fators relating the error δ M p in the extration of the M T endpoint to the resulting unertainties δm p and δm on the parent and hild masses alulated from (5 18) and (5 19), as a funtion of the true input masses M and M p. This is where the available freedom in Method II omes into play, sine both test masses M and M an be hosen on the right branh. Taking P T = 350 ± 50 GeV and P T = 500 ± 50 GeV and repeating our earlier analysis, we find that δ M p on the right branh is still on the order of 3 GeV, as in Figure 5-3. The resulting error δm p (δm ) on the measured parent (hild) mass an be easily propagated from Equations (5 18,5 19). The two ratios δm p /δ M p and δm /δ M p are shown in Figure 5-4, where for onreteness we have taken M = M = 1000 GeV. Figure 5-4 reveals that the LM6 input values of M and M p are rather unluky, sine the error δ M p on the M T endpoint is then amplified by a fator of almost 70. However, if M and M p happened to be different, with the rest of the spetrum the same, the preision quikly improves. For example, with δ M p = ±3 GeV, the masses an be determined to within ±30 GeV (±75 GeV) within the yellow (orange) region. One should keep in mind that the dominant 14

143 unertainty on δ M p is due to the SUSY ombinatorial bakground. We have verified that in the absene of suh ombinatorial bakground, δ M p < 1 GeV and the typial preision on M p and M from Figure 5-4 is then at the level of 10%. 5. Using the P T of the Upstream Jet with M T Method Unfortunately, in order to apply previous method, one must work with a subset of events within a relatively narrow fixed P T range of upstream objets (inluding jets), inurring some loss in statistis. To be more general, we treat all upstream partiles into a single setor as upstream objet U, and denote its total transverse momentum U T as in Figure 5-5. In this setion, we propose a new method whih uses the full data set. As in the previous setion, we define the endpoint M max T mass M p with the trial hild mass M for a given U T, : of M T distribution as the parent M p ( M, U T ) M max T ( M, U T ). (5 0) Here we propose to obtain a relation by using the property that the funtion M p ( M, U T ) is independent of U T at the true hild mass M : M p (M, U T + U T ) M p (M, U T ) = 0, U T, (5 1) whih we an rewrite more informatively as M p ( M, U T ) M p ( M, 0) 0, (5 ) with equality being ahieved only for M = M. Equation (5 ) implies that, for any given M, there will always be a ertain number of events whose M T values will exeed the referene value M p ( M, 0), unless the trial mass M happens to oinide with the true hild mass M. In order to quantify this effet, we define the funtion N( M ) all events H ( ) M T M p ( M, 0). (5 3) 143

144 U : U T V 1 : (m 1, p 1T ) p( p) p( p) P M p P C M C V : (m, p T ) Figure 5-5. The generi event topology under onsideration. All partiles visible in the detetor are lustered into three groups: upstream objets U with total transverse momentum U T, and two omposite visible partiles V i (i = 1, ), eah with invariant mass m i and total transverse momentum p it. H(x) is the Heaviside step funtion. From the definition of N( M ) it is lear that it is minimized at M = M, where in theory we would expet N min min{n( M )} = N(M ) = 0. (5 4) In reality, the value of N min will be lifted from 0, due to finite partile width effets, detetor resolution, et. Nevertheless we expet that the loation of the N( M ) minimum will still be at M M : = M, allowing a diret measurement of the hild mass M = { M N( M ) = N min }, (5 5) whih is our first main result. One the hild mass M is found from Equation (5 5), the true parent mass M p is obtained as usual from Equation (5 0) as M p = M p (M, U T ). At this point it is not lear whether we have gained anything statistis-wise, sine the referene quantity M p ( M, 0) appearing in the definition of Equation (5 3) has to be measured at a fixed U T = 0 anyway. Our seond main result is that M p ( M, 0) an in fat be measured from the full data set with no loss in statistis as follows. 144

145 p 1T T p 1T p 1T T p T U T p T p T /P T = p 1T p T U T Figure 5-6. Deomposition of the observed transverse momentum vetors from Figure 5-5 in the transverse plane. Step I. Orthogonal deomposition of the observed transverse momenta with respet to the U T diretion. The Tevatron and LHC ollaborations urrently use fixed axes oordinate systems to desribe their data. Instead, we propose to rotate the oordinate system from one event to another, so that the transverse axes are always aligned with the diretion T seleted by the measured upstream transverse momentum vetor U T and the diretion T orthogonal to it (Figure 5-6). The visible transverse momentum vetors from Figure 5-5 are then deomposed as p it 1 U T ( p it U T ) UT, (5 6) p it p it p it = 1 U UT T ( p it U T ). (5 7) Step II. Construting the transverse and longitudinal ontransverse masses M T and M T. Now we define 1D M T deompositions in omplete analogy with the standard M T definition of Equation (5 9): { }} M T min max {M p 1T + p T = 1T, M T, (5 8) /P T M T min p 1T + p T = /P T {max {M 1T, M T }}. (5 9) 145

146 These deompositions are extremely useful. For one, the 1D variables in Equations (5 8,5 9) an be alulated via simple analyti expressions as shown below. In ontrast, a general formula for the original M T variable in Equation (5 9) in the presene of arbitrary U T is unknown and one still has to ompute M T numerially [38]. More importantly, M T allows us to measure the referene quantity M p ( M, 0) in Equation (5 3) from the full data set, using events with any value of U T. To understand the basi idea, it is suffiient to onsider the simplest, yet most hallenging ase of a single step deay hain. Let V i be a single, (approximately) massless SM partile: m 1 = m = 0. (The disussion for the massive ase proeeds analogously.) In what follows, for illustration we shall use the same-sign dilepton hannel in supersymmetry, where eah V i is a lepton resulting from a hargino deay to a sneutrino [39]. The harginos themselves are produed indiretly in the deays of squarks and gluinos. For onreteness we shall use a SUSY spetrum given by the LM6 CMS study point as in Table 5-1. In our simulations we use the PYTHIA event generator [54] and the PGS detetor simulation program [55]. The variable M T has several unique properties. Eventwise, it an be alulated analytially as M T = A T + A T + M, (5 30) A T 1 ( p 1T p T + p 1T p T ). The endpoint of the M T distribution is given by MT max ( M ) = µ + µ + M, (5 31) in terms of the parameter µ introdued in [3] µ M p ( ) 1 M. (5 3) Mp 146

147 Figure 5-7. The unit-normalized M T distribution (5 53) for the same-sign dilepton hannel in a SUSY model with LM6 CMS mass spetrum and a hoie of test mass M = 100 GeV. The yellow shaded distribution shows the theoretially predited shape (5 53), mathing very well the parton level result from PYTHIA with no uts (red histogram). The green (blue) histogram is the orresponding result after PGS detetor simulation with mild (hard) uts as explained in the text. The endpoint expeted from Equation (5 31) is 13.1 GeV and is marked with the vertial arrow. Equation (5 31) reveals perhaps the most important feature of the M T variable: its endpoint is independent of the upstream P T and an thus be measured with the whole data sample. We an even predit analytially the shape of the (unit-normalized) differential M T distribution dn d N = N 0 δ(m T M ) + (1 N 0 ), (5 33) dm T dm T where N 0 is the fration of events in the lowest M bin M T = M, while the shape of the remaining (unit-normalized) M T distribution is given by (Figure 5-7) ( ) d N = M4 T M 4 µ M T ln. (5 34) dm T µ MT 3 MT M 147

148 Figure 5-8. Observable M T distribution after hard uts for 100 fb 1 of LHC data. The total staked distribution onsists of the SUSY signal (red) and the SM bakground (blue). The solid line is the result of a simple linear fit, revealing endpoints at GeV and 17.4 GeV. Notie that this shape does not depend on any unknown kinemati parameters, suh as the unknown enter-of-mass energy or longitudinal momentum of the initial hard sattering. It is also insensitive to spin orrelation effets, whenever the upstream momentum results from prodution and/or deay proesses involving salar partiles (e.g. squarks) or vetorlike ouplings (e.g. the QCD gauge oupling). It is even independent of the atual value of the upstream momentum P T. Thus we are not restrited to a partiular P T range and an use the whole event sample in the M T analysis. For any hoie of M (in Figure 5-7 we used M = 100 GeV), Equation (5 53) is a one-parameter urve whih an be fitted to the data to obtain the parameter µ and from there the M T endpoint (5 31). As always, there are pratial limitations to the use of suh shape fitting. First, the shape in Equation (5 53) is modified in the presene of mild uts, whih are required for lepton identifiation in PGS (green histogram in Figure 5-7), and more importantly, 148

149 for the disovery of the same-sign dilepton SUSY signal over the SM bakgrounds. To ensure disovery, we use hard uts as follows [75, 91]: exatly two isolated leptons with p T > 10 GeV, at least three jets with p T > (175, 130, 55) GeV, /P T > 00 GeV and a veto on tau jets. With those uts, in the dimuon hannel alone, the remaining SM bakground ross-setion is dominated by t t and is just 0.15 fb, while the SUSY signal is 14 fb, leading to a σ disovery with just 10 fb 1 of data [75, 91]. The distortion of the M T shape with these hard offline uts is illustrated by the blue (rightmost) histogram in Figure 5-7. The atual M T distribution whih we expet to observe with 100 fb 1 of data, is shown in Figure 5-8 and is omprised of a relatively small SM bakground omponent (blue) and a dominant SUSY signal omponent (red). In spite of the presene of a sizable SUSY ombinatorial bakground, the M T endpoint expeted from Figure 5-7 is learly visible and its loation from a simple linear fit is obtained as GeV, whih is very lose to the nominal value of 13.1 GeV. (Interestingly, the data reveals a seond endpoint at 17.4 GeV, whih is due to events in whih one hargino deays through a harged slepton: χ ± 1 l ± L χ0 1 [39]. Its nominal value is 169. GeV.) Our final key observation is that M p ( M, 0) = M max T ( M, 0) = M max T ( M ), (5 35) whih allows to rewrite the funtion N( M ) of Equation (5 3) as N( M ) all events H ( ) M T MT max ( M ). (5 36) The M T analysis just desribed allows a very preise measurement of the benhmark quantity M max T ( M ) appearing in Equation (5 36), so that the funtion N( M ) itself an be reliably reonstruted, using the whole event sample all the way throughout the analysis, without any loss in statistis. 149

150 Figure 5-9. The funtion ˆN( M ) defined in Equation (5 37). The blue (red) set of measurements are with (without) SUSY ombinatorial bakground. The error bars shown are purely statistial. We show our result in Figure 5-9, where for onveniene we unit-normalize the funtion N( M ) as ˆN( M ) = N( M )/ N( M ), (5 37) where the averaging is performed over the plotted range of M. As expeted, the funtion ˆN( M ) exhibits a minimum in the viinity of the true sneutrino mass M = M = 75.7 GeV. Ignoring the SUSY ombinatorial bakground, this measurement (red data points) is quite preise, at the level of a few perent. In order to redue the ombinatorial bakground, we selet events with M < M T < M max T and veto very hard 1 leptons with p T > 60 GeV. The resulting M measurement (blue data points) is at the level of 10%. This preision is learly suffiient to exlude SM neutrinos as the soure of the missing energy, hinting at a potential dark matter disovery at the LHC. Note that the traditional 1 The measured value of M max T in Figure 5-8 already implies that the mass splitting M p M is on the order of 30 GeV, resulting in a rather soft lepton p T spetrum. 150

151 kink for the LM6 study point is only a few degrees and appears diffiult to observe experimentally [39], unlike the result in Figure 5-9. For ompleteness, we now disuss the properties of the variable M T. In any given event, it an be alulated as follows. If p 1T p T 0, then simply M T = M, and the event falls in the lowest M bin. In the alternative ase of p 1T p T > 0, M T an be found from M T = ( ) p 1T + M + p (1) T p 1T + p (1) T, where the hild test momentum is p (1) T = 1 α [ UT + (1 α) p + (1 + α) p 1T T ], M p 1T p T + U T + p 1T + p T p 1T + p T. The endpoint M max T of the M T distribution is idential to the endpoint (5 0) of the M T distribution itself: M max T and is thus known as a funtion of U T [3] {[ MT max = µ(su T ) + = M max T ( M, U T ) = M p ( M, U T ), (5 38) ( µ(sut ) + su T ) + M ] (su T ) 4 } 1, (5 39) where ( ) µ(su T ) µ sut 1 + su T, M p M p and s = ±1 is an integer defined as s sgn(m M ). Unfortunately, the shape of the M T distribution annot be predited in a model-independent way, sine it is sensitive to the underlying ŝ, as well as to the measured U T. Nevertheless, one an imagine several useful appliations of M T. For example, one an study the boundary of the two-dimensional {M T, M T } distribution as we will do with one dimensional projeted variables of M CT. One ould also onsider an analogue of (5 36), defined in terms of 151

152 M T : N ( M ) all events H ( ) M T MT max ( M ), (5 40) whose minimum will also mark the loation of the true hild mass M. 5.3 Using Full Phase Spae Information With M CT In this setion, we show how to get more information from the boundary of phase spae, with 1D-deompsed M CT variable. Compared to M T approahes, M CT method has two advantages. First, it is simpler it uses only the observed objets U, V 1 and V in the event and makes no referene to the missing partile kinematis (or mass). Seond, it is more preise, sine it utilizes the whole kinemati boundary of the relevant two-dimensional distribution and not just the kinemati endpoint of its one-dimensional projetion. Step I. Construting the transverse and longitudinal ontransverse masses M CT and M CT. Our starting point is the original ontransverse mass variable [4] M CT = m 1 + m + (e 1T e T + p 1T p T ), (5 41) where e it is the transverse energy of V i e it = m i + p it. (5 4) For events with U T = 0, M CT has an upper endpoint whih is insensitive to the unknown ŝ, providing one relation among Mp and M [4, 43] MCT max (U T = 0) = m1 + m + m 1m osh (ζ 1 + ζ ), (5 43) where sinh ζ i λ 1 (M p, M, m i ) M p m i, (5 44) λ(x, y, z) x + y + z xy xz yz. (5 45) 15

153 Unfortunately, the U T = 0 limit is not partiularly interesting at hadron olliders (espeially for inlusive studies), sine a signifiant amount of upstream U T is typially generated by ISR (and other) jets. One possible fix is to use all events, but modify the definition (5 41) to approximately ompensate for the transverse U T boost [43]. One then reovers a distribution whose endpoint is still given by Equation (5 43). Alternatively, one ould stik to the original M CT variable, and simply aount for the U T dependene of its endpoint as MCT max (U T ) = m1 + m + m 1m osh (η + ζ 1 + ζ ) (5 46) where ζ i were already defined in Equation (5 44), and sinh η U T M p, osh η 1 + U T 4M p. (5 47) Our approah here is to utilize the one-dimensional projetions from Equations (5 6,5 7) and onstrut one-dimensional analogues of the M CT variable M CT M CT m1 + m + (e 1T e T + p 1T p T ), (5 48) ) m (e 1 + m + 1T e T + p 1T p T, (5 49) where the orresponding transverse energies are e it m i + p it, e it m i + p it. (5 50) The benefit of the deomposition as Equations (5 48,5 49) is that one gets two for the prie of one, i.e. two independent and omplementary variables instead of the single variable in Equation (5 41). The variable M CT in partiular is very useful for our purposes. To illustrate the basi idea, it is suffiient to onsider the most ommon ase, where V i is approximately massless (m i = 0), as the leptons in our t t example. A ruial property of M CT is that 153

154 its endpoint is independent of U T : In fat the whole M CT distribution is insensitive to U T : M max CT = M p M M p, U T. (5 51) dn d N = N 0 δ(m CT ) + (N tot N 0 ), (5 5) dm CT dm CT where N 0 is the number of events in the zero bin M CT = 0. Using phase spae kinematis, we find that the shape of the remaining (unit-normalized) zero-bin-subtrated distribution is simply given by in terms of the unit-normalized M CT variable d N d ˆM CT 4 ˆM CT ln ˆM CT (5 53) ˆM CT M CT M max CT. (5 54) The observable M CT distribution for our t t example is shown in Figure 5-10, for 10 fb 1 of LHC data at 7 TeV. Events were generated with PYTHIA [54] and proessed with the PGS detetor simulator [55]. We apply standard bakground rejetion uts as follows [9]: we require two isolated, opposite sign leptons with p it > 0 GeV, m l + l > 1 GeV, and passing a Z-veto m l + l M Z > 15 GeV; at least two entral jets with p T > 30 GeV and η <.4; and a /E T ut of /E T > 30 GeV ( /E T > 0 GeV) for events with same flavor (opposite flavor) leptons. We also demand at least two b-tagged jets, assuming a flat b-tagging effiieny of 60%. With those uts, the SM bakground from other proesses is negligible [9]. Figure 5-10 demonstrates that the M CT endpoint an be measured quite well. Sine the theoretially predited shape of Equation (5 53) is distorted by the uts, we use a linear slope with Gaussian smearing, and fit for the endpoint and the resolution parameter. 154

155 Figure Zero-bin subtrated M CT distribution after uts, for t t dilepton events. The yellow (lower) portion is our signal, while the blue (upper) portion shows t t ombinatorial bakground with isolated leptons arising from τ or b deays. We find M max CT = 80.9 GeV (ompare to the true value M max CT = 80.4 GeV), whih gives one onstraint (5 51) among M p and M. At this point, a seond, independent onstraint an in priniple be obtained from an analogous measurement of the M max CT endpoint in Equation (5 46) at a fixed value of U T (resulting in loss in statistis), after whih the two masses an be found from M p = M = U T MCT max(u T ) MCT max (MCT max(u T )) (MCT max ), (5 55) ( ) M p Mp MCT max. (5 56) However, the orthogonal deomposition as Equations (5 48,5 49) offers another approah, whih we pursue in the last step. Step II. Fitting to kinemati boundary lines. It is known that two-dimensional orrelation plots reveal a lot more information than one-dimensional projeted histograms []. To this end, onsider the satter plot of M CT vs M CT in Figure 5-11(a), where for illustration we used 10,000 events at the parton level. 155

156 Figure Satter plots of (a) M CT versus M CT and (b) M CT versus M CT, for a fixed representative value U T = 75 GeV. The solid lines show the orresponding boundaries defined in (5 58) and (5 61), for the orret value of M max CT and several different values of M p as shown. For a given value of M CT, the allowed values of M CT are bounded by where M (lo) CT (M CT ) = 0 and M (lo) CT (M CT ) M CT M (hi) CT (M CT ), (5 57) ) M (hi) CT (M CT ) = MCT max ( 1 ˆM CT osh η + sinh η. (5 58) Figure 5-11(a) reveals that the endpoint M max CT is obtained at M CT = 0 of the one-dimensional M CT distribution M max CT = M (hi) CT (0) = MCT max (osh η + sinh η) = 1 ( ) ( ) 1 M 4M M p p + UT + U T. (5 59) Notie that events in the zero bins M CT = 0 and M CT = 0 fall on one of the axes and annot be distinguished on the plot. 156

157 Now onsider the satter plot of M CT vs M CT shown in Figure 5-11(b). M CT is similarly bounded by where this time M (lo) CT (M CT ) = M CT and M (lo) CT (M CT ) M CT M (hi) CT (M CT ), (5 60) ) M (hi) CT (M CT ) = MCT max (osh η + 1 ˆM CT sinh η. (5 61) We see that the endpoint M max CT for M CT = 0: of the one-dimensional M CT distribution is also obtained M max CT = M (hi) CT (0) = Mmax CT (osh η + sinh η) = MCT max. (5 6) Figure 5-11 reveals a oneptual problem with one-dimensional projetions. While all points in the viinity of the boundary lines in Equations (5 58) and (5 61) are sensitive to the masses, the M max CT endpoint is extrated mostly from events with M CT M max CT, while the M max CT and M max CT endpoints are extrated mostly from the events with M CT 0. The events near the boundary, but with intermediate values of M CT, will not enter effiiently either one of these endpoint determinations. So how an one do better, given the knowledge of the boundary line of Equation (5 61)? In the spirit of [93], we define the signed distane to the orresponding boundary, e.g. D CT (M p, M ) M (hi) CT (M CT, U T, M p, M ) M CT and similarly for D CT. The key property of this variable is that for the orret values of M p and M, its lower endpoint D min CT is exatly zero (see Figure 5-1(b)): D min CT (M p, M ) = 0. (5 63) In that ase the boundary line provides a perfetly snug fit to the satter plot notie the green boundary line marked 80 in Figure 5-11(b). 157

158 Figure 5-1. D CT distributions for four different values of M p (and M given from (5 56)). The yellow (light shaded) histograms use only events in the zero bin M CT = 0. The red solid lines show linear binned maximum likelihood fits. While in general Equation (5 63) represents a two-dimensional fit to M p and M, in pratie one an already use the M max CT measurement to redue the problem to a single degree of freedom, e.g. the parent mass M p, as presented in Figures 5-11 and 5-1. We see that the orret parent mass M p = 80 GeV provides a perfet envelope, for whih D min CT = 0. If, on the other hand, M p is too low, a gap develops between the outlying points in the satter plot and their expeted boundary, whih results in D min CT >

159 10 (GeV) min D CT M p = M w min D CT = M p (GeV) Figure Fitted values of D min CT as a funtion of M p. Conversely, if M p is too high, some of the outlying points from the satter plot fall outside the boundary and have D CT Figure 5-1(,d). The resulting fit for D min CT < 0, leading to D min CT < 0, as seen in as a funtion of M p from our PGS data sample is shown in Figure 5-13, whih suggests that a W mass measurement at the level of a few perent might be viable. 159

160 CHAPTER 6 ASYMMETRIC EVENT TOPOLOGY In this hapter, we study asymmetri events based on following reasons. Single dark matter omponent. A ommon assumption throughout the ollider phenomenology literature is that olliders are probing only one dark matter speies at a time, i.e. that the missing energy signal at olliders is due to the prodution of one and only one type of dark matter partiles. Of ourse, there is no astrophysial evidene that the dark matter is made up of a single partile speies: it may very well be that the dark matter world has a rih struture, just like ours [95]. Consequently, if there exist several types of dark matter partiles, eah ontributing some fration to the total reli density, a priori there is no reason why they annot all be produed in high energy ollisions. Theoretial models with multiple dark matter andidates have also been proposed [45 47, ]. Idential missing partiles in eah event. A separate assumption, ommon to most previous studies, is that the two missing partiles in eah event are idential. This assumption ould in priniple be violated as well, even if the single dark matter omponent hypothesis is true. The point is that one of the missing partiles in the event may not be a dark matter partile, but simply some heavier ousin whih deays invisibly. An invisibly deaying heavy neutralino ( χ 0 i ν ν χ 0 1 with i > 1) and an invisibly deaying sneutrino ( ν ν χ 0 1) are two suh examples from supersymmetry. As far as the event kinematis is onerned, the mass of the heavier ousin is a relevant parameter and approximating it with the mass of the dark matter partile will simply give nonsensial results. Another relevant example is provided by models in whih the SUSY asade may terminate in any one of several light neutral partiles [10]. Given our utter ignorane about the struture of the dark matter setor, in this hapter we set out to develop the neessary formalism for arrying out missing energy studies at hadron olliders in a very general and model-independent way, without relying on any assumptions about the nature of the missing partiles. In partiular, we shall not assume that the two missing partiles in eah event are the same. We shall also allow for the simultaneous prodution of several dark matter speies, or alternatively, for the prodution of a dark matter andidate in assoiation with a heavier, invisibly deaying partile. Under these very general irumstanes, we shall try to develop a method for measuring the individual masses of all relevant partiles - the various 160

161 missing partiles whih are responsible for the missing energy, as well as their parents whih were originally produed in the event. 6.1 Generalizing M T To Asymmetri Event Topologies In general, by now there is a wide variety of tehniques available for mass measurements in SUSY-like missing energy events. Suh events are haraterized by the pair prodution of two new partiles, eah of whih undergoes a sequene of asade deays ending up in a partile whih is invisible in the detetor. Eah tehnique has its own advantages and disadvantages 1. For our purposes, we hose to revamp the method of the Cambridge M T variable [10] and adapt it to the more general ase of an asymmetri event topology shown in Figure 6-1. Consider the inlusive prodution of two idential parents of mass M p as shown in Figure 6-1. The parent partiles may be aompanied by any number of upstream objets, suh as jets from initial state radiation [35, 36, 103], or visible deay produts of even heavier (grandparent) partiles [3]. The exat origin and nature of the upstream objets will be of no partiular importane to us, and the only information about them that we shall use will be their total transverse momentum P UTM. In turn, eah parent partile initiates a deay hain (shown in red) whih produes a ertain number n (λ) of Standard Model (SM) partiles (shown in gray) and an intermediate hild partile of mass M (λ). Throughout this hapter we shall use the index λ to lassify various objets as belonging to the upper (λ = a) or lower (λ = b) branh in Figure 6-1. The hild partile may or may not be a dark matter andidate: in general, it may deay further as shown by the dashed lines in Figure For a omparative review of the three main tehniques,[3]. In priniple, the assumption of idential parents an also be relaxed, by a suitable generalization of the M T variable, in whih the mass ratio of the two parents is treated as an additional input parameter [40]. 161

162 Figure 6-1. The generi event topology under onsideration in this hapter. We onsider the inlusive pair-prodution of two parent partiles with idential masses M p. The parents may be aompanied by upstream objets, e.g. jets from initial state radiation, visible deay produts of even heavier partiles, et. The transverse momentum of all upstream objets is measured and denoted by P UTM. In turn, eah parent partile initiates a deay hain (shown in red) whih produes a ertain number n (λ) of SM partiles (shown in gray) and an intermediate hild partile of mass M (λ), where λ = a (λ = b) for the branh above (below). In general, the hild partile does not have to be the dark matter andidate, and may deay further as shown by the dashed lines. The M T variable is defined for the subsystem inside the blue box and is defined in terms of two arbitrary hildren test masses M (a) and M (b) partiles from eah branh form a omposite partile of transverse. The n (λ) SM momentum p (λ) T and invariant mass m (λ), orrespondingly. The trial transverse momenta q (λ) T of the hildren obey the transverse momentum onservation relation shown inside the green box. In general, the number n (λ), as well as the type of SM deay produts in eah branh do not have to be the same. 16

163 We shall apply the subsystem M T onept [3, 85] to the subsystem within the blue retangular frame. The SM partiles from eah branh within the subsystem form a omposite partile of known 3 Sine the hildren masses M (a) transverse momentum p (λ) T and invariant mass m (λ). and M (b) be defined in terms of two test masses M (a) are a priori unknown, the subsystem M T will and M (b). In Figure 6-1, q (λ) T are the trial transverse momenta of the two hildren. The individual momenta q (λ) T unknown, but they are onstrained by transverse momentum onservation: are also a priori q (a) T (b) + q T Q tot = ( p (a) T (b) + p T + P UTM ). (6 1) Given this very general setup, in Setion 6.3 we shall onsider a generalization of the usual M T variable whih an apply to the asymmetri event topology of Figure 6-1. There will be two different aspets of the asymmetry: First and foremost, we shall avoid the ommon assumption that the two hildren have the same mass. This will be important for two reasons. On the one hand, it will allow us to study events in whih there are indeed two different types of missing partiles. We shall give several suh examples in the subsequent setions. More importantly, the endpoint of the asymmetri M T variable will allow us to measure the two hildren masses separately. Therefore, even when the events ontain idential missing partiles, as is usually assumed throughout the literature, one would be able to establish this fat experimentally from the data, instead of relying on an ad ho theoretial assumption. As an be seen from Figure 6-1, in general, the number as well as the types of SM deay produts in eah branh may be different as well. One we allow for the hildren to be different, and given the fat that we start from idential parents, the two branhes of the subsystem will naturally involve different sets of SM partiles. In what follows, when referring to the more general M T variable defined in Setion 6.3, we shall interhangeably use the terms asymmetri or generalized M T. In ontrast, 3 We assume that there are no neutrinos among the SM deay produts in eah branh. 163

164 we shall use the term symmetri when referring to the more onventional M T definition with idential hildren. The traditional M T approah assumes that the hildren have a ommon test mass M M (a) = M (b) and then proeeds to find one funtional relation between the true hild mass M and the true parent mass M p as follows [10]. Construt several M T distributions for different input values of the test hildren mass M and then read off their upper kinemati endpoints M T (max) ( M ). These endpoint measurements are then interpreted as an output parent mass M p, whih is a funtion of the input test mass M : M p ( M ) M T (max) ( M ). (6 ) The importane of this funtional relation is that it is automatially satisfied for the true values M p and M of the parent and hild masses: M p = M T (max) (M ). (6 3) In other words, if we ould somehow guess the orret value M of the hild mass, the funtion (6 ) will provide the orret value M p of the parent mass. However, sine the true hild mass M is a priori unknown, the individual masses M p and M still remain undetermined and must be extrated by some other means. At this point, it may seem that by onsidering the asymmetri M T variable with non-idential hildren partiles, we have regressed to some extent. Indeed, we are introduing an additional degree of freedom in Equation (6 ), whih now reads M p ( M (a), M (b) ) M T (max) ( M (a), M (b) ). (6 4) The standard M T endpoint method will still allow us to find the parent mass M p, but now it is a funtion of two input parameters M (a) and M (b) whih are ompletely unknown. Of ourse, if one knew the orret values of the two hildren masses M (a) and M (b) entering Equation (6 4), the true parent mass M p will be given in a manner analogous to 164

165 Equation (6 3): M p = M T (max) (M (a), M (b) ). (6 5) Our main result is that in spite of the apparent remaining arbitrariness in Equation (6 4), one an nevertheless uniquely determine all three masses M p, M (a) studying the behavior of the measured funtion M p ( M (a) and M (b), just by, M (b) ). More importantly, this determination an atually be done in two different ways! Our first method is simply a generalization of the observation made in Referenes. [3, 34 37] that under ertain irumstanes (varying m (λ) or nonvanishing upstream momentum P UTM ), the funtion (6 ) develops a kink preisely at the orret value M of the hild mass: ( ) ( ) M p ( M ) M p ( M ) = 0, if M M, M M 0, if M = M. M +ϵ M ϵ (6 6) In other words, the funtion (6 ) is ontinuous, but not differentiable at the point M = M. In the asymmetri M T ase, we find that the funtion (6 4) is similarly non-differentiable at a set of points {( M (a), M (b) )}, so that the kink of Equation (6 6) is generalized to a ridge on the -dimensional hypersurfae defined by Equation (6 4) in the three-dimensional parameter spae of { M (a), M (b), M p }. 4 Interestingly enough, the ridge often (albeit not always) exhibits a speial point whih marks the exat loation of the true values (M (a), M (b) ). Our seond method for determining the two hildren masses M (a) and M (b) even more general and is appliable under any irumstanes. The main starting point is that just like the endpoint of the symmetri M T, the endpoint of the asymmetri M T also depends on the value of the upstream transverse momentum P UTM, so that is 4 Referene [40] studied the orthogonal senario of different parents (M p (a) M (b) and idential hildren (M (a) = M (b) ) and found a similar non-differentiable feature, alled a rease, on the orresponding two-dimensional hypersurfae within the three-dimensional parameter spae { M, M p (a), M p (b) }. p ) 165

166 Equation (6 4) is more properly written as M p ( M (a), M (b), P UTM ) = M T (max) ( M (a), M (b), P UTM ). (6 7) Now we an explore the P UTM dependene in Equation (6 7) and note that it is absent for preisely the right values of M (a) M T (max) ( M (a) P UTM and M (b) :, M (b), P UTM ) M (a) =M (a) = 0. (6 8), M (b) =M (b) While this property has been known, it was rarely used in the ase of the symmetri M T, sine it offers redundant information: one the orret hild mass M is found through the M T kink as Equation (6 6), the parent mass M p is given by Equation (6 ) and there are no remaining unknowns, thus there is no need to further investigate the P UTM dependene. In the ase of the asymmetri M T, however, we start with one additional unknown parameter, whih annot always be determined from the ridge information alone. Therefore, in order to pin down the omplete spetrum, we are fored to make use of Equation (6 8). The nie feature of the P UTM method is that it always allows us to determine both hildren masses M (a) and M (b), without relying on the ridge information at all. In this sense, our two methods are omplementary and eah an be used to ross-hek the results obtained by the other Definition 6. The Conventional Symmetri M T We begin our disussion by revisiting the onventional definition of the symmetri M T variable with idential daughters, following the general notation introdued in Figure 6-1. Let us onsider the inlusive prodution of two parent partiles with ommon mass M p. Eah parent initiates a deay hain produing a ertain number n (λ) of SM partiles. In this setion we assume that the two hains terminate in hildren partiles of the same mass: M (a) = M (b) = M. (From Setion 6.3 on we shall remove this assumption.) In most appliations of M T in the literature, the hildren partiles 166

167 are identified with the very last partiles in the deay hains, i.e. the dark matter andidates. However, the symmetri M T an also be usefully applied to a subsystem of the original event topology, where the hildren are some other pair of (idential) partiles appearing further up the deay hain [3, 85]. The M T variable is defined in terms of the measured invariant mass m (λ) and transverse momentum p (λ) T of the visible partiles on eah side (Figure 6-1). With the assumption of idential hildren, the transverse mass of eah parent is M (λ) T ( p (λ) (λ) T ; q T ; m ) (λ); M = m(λ) + M + ( e (λ) ẽ (λ) p (λ) T ) q (λ) T, (6 9) where M is the ommon test mass for the hildren, whih is an input to the M T alulation, while q (λ) T is the unknown transverse momentum of the hild partile in the λ-th hain. In Equation (6 9) we have also introdued shorthand notation for the transverse energy of the omposite partile made from the visible SM partiles in the λ-th hain e (λ) = m(λ) (λ) + p T p (λ) T (6 10) and for the transverse energy of the orresponding hild partile in the λ-th hain ẽ (λ) = M + q (λ) T q (λ) T. (6 11) Then the event-by-event symmetri M T variable is defined through a minimization proedure over all possible partitions of the two hildren momenta q (λ) T [10] M T ( p (a) T q (a) T min, p (b) + q (b) T = Q tot ) T ; m (a),m (b) ; M, P UTM = [ { ( max ; q (a) M (a) T p (a) T T ) ; m (a); M, M (b)( T p (b) (b) T ; q T ; m (b); M ) } ], (6 1) onsistent with the momentum onservation onstraint (6 1) in the transverse plane. 6.. Computation The standard definition as in Equation (6 1) of the M T variable is suffiient to ompute the value of M T numerially, given a set of input values for its arguments. 167

168 The right-hand side of Equation (6 1) represents a simple minimization problem in two variables, whih an be easily handled by a omputer. In fat, there are publily available omputer odes for omputing M T [104, 105]. The publi odes have even been optimized for speed [38] and give results onsistent with eah other (as well as with our own ode) 5. Nevertheless, it is useful to have an analytial formula for alulating the event-by-event M T for several reasons. First, an analytial formula is extremely valuable when it omes to understanding the properties and behavior of omplex mathematial funtions like in Equation (6 1). Seond, omputing M T from a formula will be faster than any numerial sanning algorithm. The omputing speed beomes an issue espeially when one onsiders variations of M T like M T gen, where in addition one needs to san over all possible partitions of the visible objets into two deay hains [33]. Therefore we shall pay speial attention to the availability of analytial formulas and we shall quote suh formulas whenever they are available. In the symmetri ase with idential hildren, an analytial formula for the event-by-event M T exists only in the speial ase P UTM = 0. It was derived in [33] and we provide it here for ompleteness. (In the next setion we shall present its generalization for the asymmetri ase of different hildren.) The symmetri M T is known to have two types of solutions: balaned and unbalaned [3, 33]. The balaned solution is ahieved when the minimization proedure in Equation (6 1) selets a momentum onfiguration for q (λ) T parents are the same: M (a) T = M(b) T in whih the transverse masses of the two. In that ase, typially neither M(a) T nor M(b) T is at its global (unonstrained) minimum. In what follows, we shall use a supersript B to refer to suh balaned-type solutions. The formula for the balaned solution MT B of the 5 Unfortunately, the assumption of idential hildren is hardwired in the publi odes and they annot be used to alulate the asymmetri M T introdued below in Setion 6.3 without additional haking. We shall return to this point in Setion

169 symmetri M T variable is given by [33, 37] [ ( ) ] ( ) MT B p (a) (b) T, p T ; m (a),m (b) ; M = M + A T + 4 M 1 + ( ) A T m(a) A m T m (a) m (b), (b) (6 13) where A T is a onvenient shorthand notation introdued in [37] A T = e (a) e (b) + p (a) T and e (λ) was already defined in Equation (6 10). (b) p T (6 14) On the other hand, unbalaned solutions arise when one of the two parent transverse masses (M (a) T or M(b) T, as the ase may be) is at its global (unonstrained) minimum. Denoting the two unbalaned solutions with a supersript Uλ, we have [3] ( MT Ua ( MT Ub p (a) T p (a) T, p (b) T ; m (a),m (b) ; M ) = m(a) + M, (6 15), p (b) T ; m (a),m (b) ; M ) = m(b) + M. (6 16) Given the three possible options for M T, Equations (6 13), (6 15) and (6 16), it remains to speify whih one atually takes plae for a given set of values for p (a) (b), p T m (a), m (b), M and P UTM = 0 in the event 6. The balaned solution (6 13) applies when the following two onditions are simultaneously satisfied: where M (b) ( T M (a) ( T p (b) (b) (a) T ; q T = q p (a) T ; q (a) (b) T = q ) T (0) + /P T ; m (b) ; M M (a)( ) T (0) + /P T ; m (a) ; M M (b)( q (λ) T (0) = T T p (a) (a) (a) T ; q T = q p (b) (b) (b) T ; q T = q T, T (0) ; m (a), M ) = m(a) + M, (6 17) T (0) ; m (b); M ) = m(b) + M, (6 18) M p (λ) T, (λ = a, b), (6 19) m (λ) 6 Reall that Equation (6 13) only applies for P UTM =

170 gives the global (unonstrained) minimum of the orresponding parent transverse mass M (λ) T. The unbalaned solution MUa applies when the ondition (6 17) is false and T ondition in Equation (6 18) is true, while the unbalaned solution MT Ub applies when the ondition (6 17) is true and ondition in Equation (6 18) is false. It is easy to see that onditions in Equations (6 17) and (6 18) annot be simultaneously violated, so these three ases exhaust all possibilities Properties Given its definition (6 1), one an readily form and study the differential M T distribution. Although its shape in general does arry some information about the underlying proess, it has beome ustomary to fous on the upper endpoint M T (max), whih is simply the maximum value of M T found in the event sample: M T (max) ( M, P UTM ) = ( max [M T all events p (a) T, p (b) T ; m (a),m (b) ; M ) ]. (6 0) Notie that in the proess of maximizing over all events, the dependene on p (a) (b), p m (a) and m (b) disappears, and M T (max) depends only on two input parameters: M T and P UTM, the latter entering through Q tot in the momentum onservation onstraint of Equation (6 1). The measured funtion in Equation (6 0) is the starting point of any M T -based mass determination analysis. We shall now review its three basi properties whih make it suitable for suh studies [1] Property I: Knowledge Of M p As A Funtion of M This property was already identified in the original papers and served as the main motivation for introduing the M T variable in the first plae [10, 3]. Mathematially it an be expressed as T, M p ( M, P UTM ) M T (max) ( M, P UTM ). (6 1) This is the same as Equation (6 ), but now we have been areful to inlude the expliit dependene on P UTM, whih will be important in our subsequent disussion. 170

171 Figure 6-. Plots of (a) the M T endpoint M T (max) ( M, P UTM ) defined in Equation (6 0), and (b) the funtion M T (max) ( M, P UTM ) defined in Equation (6 4) as a funtion of the test hild mass M, for several fixed values of P UTM : P UTM = 0 GeV (solid, green), P UTM = 500 GeV (dot-dashed, blak), P UTM = 1 TeV (dashed, red), and P UTM = TeV (dotted, blue). The proess under onsideration is pair prodution of sleptons of mass M p = 300 GeV, whih deay diretly to the lightest neutralino χ 0 1 of mass M = 100 GeV. As indiated in Equation (6 1), the funtion M p ( M, P UTM ) an be experimentally measured from the M T endpoint of Equation (6 0). The ruial point now is that the relation in Equation (6 1) is satisfied by the true values M p and M of the parent and hild mass, orrespondingly: M p = M T (max) (M, P UTM ). (6 ) Notie that Equation (6 ) holds for any value of P UTM, so in pratial appliations of this method one ould hoose the most populated P UTM bin to redue the statistial error. On the other hand, sine a priori we do not know the true mass M of the missing partile, Equation (6 ) gives only one relation between the masses of the mother and the hild. This is illustrated in Figure 6-(a), where we onsider the simple example of 171

172 diret slepton pair prodution 7, where eah slepton ( l) deays to the lightest neutralino ( χ 0 1) by emitting a single lepton l: l l + χ 0 1. Here the slepton is the parent and the neutralino is the hild. Their masses were hosen to be M p = 300 GeV and M = 100 GeV, orrespondingly, as indiated with the blak dotted lines in Figure 6-(a). In this example, the upstream transverse momentum P UTM is provided by jets from initial state radiation. In Figure 6-(a) we plot the funtion in Equation (6 1) versus M, for several fixed values of P UTM. The green solid line represents the ase of no upstream momentum P UTM = 0. In agreement with Equation (6 ), this line passes through the point (M, M p ) orresponding to the true values of the mass parameters. Notie that the property of Equation (6 ) ontinues to hold for other values of P UTM. Figure 6-(a) shows three more ases: P UTM = 500 GeV (dotdashed blak line), P UTM = 1 TeV (dashed red line) and P UTM = TeV (dotted blue line). All those urves still pass through the point (M, M p ) with the orret values of the masses, illustrating the robustness of the property of Equation (6 ) with respet to variations in P UTM Property II: Kink In M T (max) At The True M The seond important property of the M T variable was identified rather reently [3, 34 37]. Interestingly, the M T endpoint M T (max), when onsidered as a funtion of the unknown input test mass M, often develops a kink as Equation (6 6) at preisely the orret value M = M of the hild mass. The appearane of the kink is a rather general phenomenon and ours under various irumstanes. It was originally notied in event topologies with omposite visible partiles, whose invariant mass m (λ) is a variable parameter [34, 37]. Later it was realised that a kink also ours in the presene of non-zero upstream momentum P UTM [3, 35, 36], as in the example of Figure 6-(a), where P UTM arises due to initial state radiation. As an be seen in Figure 6-(a), the 7 The orresponding event topology is shown in Figure 6-3(a) below with M (a) M (b) = M. = 17

173 kink is absent for P UTM = 0, but as soon as there is some non-vanishing P UTM, the kink beomes readily apparent. As expeted, the kink loation (marked by the vertial dotted line) is at the true hild mass (M = 100 GeV), where the orresponding value of M T (max) (marked by the horizontal dotted line) is at the true parent mass (M p = 300 GeV). Figure 6-(a) also demonstrates that with the inrease in P UTM, the kink beomes more pronouned, thus the most favorable situations for the observation of the kink are ases with large P UTM, e.g. when the upstream momentum is due to the deays of heavier (grandparent) partiles [3]. In Setion we shall see how the kink feature (6 6) of the symmetri M T endpoint M p ( M ) defined by Equation (6 ) is generalized to a ridge feature on the asymmetri M T endpoint M p ( M (a), M (b) ) defined in Equation (6 4) Property III: P UTM Invariane Of M T (max) At The True M This property is the one whih has been least emphasized in the literature. Notie that the M T endpoint funtion of Equation (6 1) in general depends on the value of P UTM. However, the first property of Equation (6 ) implies that the P UTM dependene disappears at the orret value M of the hild mass: M T (max) ( M, P UTM ) P UTM M =M = 0. (6 3) In order to quantify this feature, let us define the funtion M T (max) ( M, P UTM ) M T (max) ( M, P UTM ) M T (max) ( M, 0), (6 4) whih measures the shift of the M T endpoint due to variations in P UTM. The funtion M T (max) ( M, P UTM ) an be measured experimentally: the first term on the right-hand side of Equation (6 4) is simply the M T endpoint observed in a subsample of events with a given (preferably the most ommon) value of P UTM, while the seond term on the right-hand side of Equation (6 4) ontains the endpoint M (max) T of the 1-dimensional 173

174 M T variable introdued in [1]: M T (max) ( M, 0) = M (max) T ( M ). (6 5) Given the definition (6 4), the third property of Equation (6 3) an be rewritten as M T (max) ( M, P UTM ) 0, (6 6) where the equality holds only for M = M : M T (max) (M, P UTM ) = 0, P UTM. (6 7) Equations (6 6) and (6 7) provide an alternative way to determine the true hild mass M : simply find the value of M whih minimizes the funtion M T (max) ( M, P UTM ). This proedure is illustrated in Figure 6-(b), where we revisit the slepton pair prodution example of Figure 6-(a) and plot the funtion M T (max) ( M, P UTM ) defined in Equation (6 4) versus the test mass M, for the same set of (fixed) values of P UTM. Clearly, the zero of the funtion as Equation (6 4) ours at the true hild mass M = M = 100 GeV, in agreement with Equation (6 7). In our studies of the asymmetri M T ase in the next setions, we shall find that the third property in Equation (6 7) is extremely important, sine it will always allow us the omplete determination of the mass spetrum, inluding both hildren masses M (a) 6.3 The Generalized Asymmetri M T and M (b). After this short review of the basi properties of the onventional symmetri M T variable in Equation (6 1), we now turn our attention to the less trivial ase of M (a) M (b). Following the logi of Setion 6., in Setion we first introdue the asymmetri M T variable and then in Setions 6.3. and we disuss its omputation and mathematial properties, orrespondingly. 174

175 6.3.1 Definition The generalization of the usual definition (6 1) to the asymmetri ase of M (a) M (b) is straightforward [40]. We ontinue to follow the onventions and notation of Figure 6-1, but now we simply avoid the assumption that the hildren masses are equal, and we let eah one be an independent input parameter M (λ). Without loss of generality, in what follows we assume M (b) M (a). The transverse mass of eah parent in Equation (6 9) is now a funtion of the orresponding hild mass M (λ) : M (λ) T ( p (λ) (λ) ; q ; m ) ( ) ( (λ); M (λ) = m(λ) + M (λ) + T T e (λ) ẽ (λ) p (λ) T ) q (λ) T, (6 8) where the transverse energy e (λ) of the omposite SM partile on the λ-th side of the event was already defined in Equation (6 10), while the transverse energy ẽ (λ) of the hild is now generalized from Equation (6 11) to ẽ (λ) = ( ) M (λ) + q (λ) T q (λ) T. (6 9) The event-by-event asymmetri M T variable is defined in analogy to Equation (6 1) and is given by [40] M T ( p (a) T q (a) T min + q (b) T = /P T (b), p T ;m (a),m (b) ; M (a), M (b) [ { max M (a) T (,P UTM ) = p (a) T (a) ; q T ; m (a); M (a) ), M (b)( T p (b) T whih is now a funtion of two input test hildren masses M (a) ase of M (a) (b) ; q T ) } ] ; m (b); M (b), (6 30) and M (b). In the speial = M (b) M, the asymmetri M T variable defined in Equation (6 30) redues to the onventional symmetri M T variable (6 1) Computation In this subsetion we generalize the disussion in Setion 6.. and present an analytial formula for omputing the event-by-event asymmetri M T variable (6 30). Just like the formula (6 13) for the symmetri ase, our formula will hold only in the speial ase of P UTM = 0. As before, the asymmetri M T variable has two types of 175

176 solutions balaned and unbalaned. The balaned solution ours when the following two onditions are simultaneously satisfied (ompare to the analogous onditions (Equations 6 17) and (6 18) for the symmetri ase) M (b) ( T M (a) ( T p (b) T ; q (b) (a) T = q T (0) + /P T ;m (b) ; M (b) p (a) (a) (b) T ; q T = q T (0) + /P T ;m (a) ; M (a) where, in analogy to Equation (6 19), q (λ) T (0) ) M (a)( T ) M (b)( T (λ) M = p (a) T ; q (a) (a) T = q T (0) ;m (a), M (a) p (b) (b) (b) T ; q T = q T (0) ;m (b); M (b) ) = m(a) + M (a), (6 31) ) = m(b) + M (b), (6 3) p (λ) T, (λ = a, b), (6 33) m (λ) is the test hild momentum at the global unonstrained minimum of M (λ). The balaned T solution for M T is now given by [ ( ) ] ( MT B m p (a) (b) T, p T ; m (a),m (b) ; M (a), M (b) = M + (b) + A T + ) m (a) A T m(a) M m (b) ( ) ± 4 M M A T m(a) + m (b) A T m(a) A m T m (a) m (b),(6 34) (b) where A T was defined in Equation (6 14). For onveniene, in Equation (6 34) we have introdued two alternative mass parameters M + 1 { ( M (b) M 1 { ( M (b) in plae of the original trial masses M (a) ) ( ) } + M (a), (6 35) ) ( ) } M (a), (6 36) and M (b). The new parameters M + and M are simply a different parametrization of the two degrees of freedom orresponding to the unknown hild masses M (a) and M (b) entering the definition of the asymmetri M T. The parameters M + and M allow us to write formula (6 34) in a more ompat form. More importantly, they also allow to make easy ontat with the known results from 176

177 Setion 6. by taking the symmetri limit M (a) = M (b) M as M + M, M 0. (6 37) It is easy to see that in the symmetri limit as Equation (6 37) our balaned solution in Equation (6 34) for the asymmetri M T redues to the known result of Equation (6 13) for the symmetri M T. An interesting feature of the asymmetri balaned solution is the appearane of a ± sign on the seond line of Equation (6 34). In priniple, this sign ambiguity is present in the symmetri ase as well, but there the minus sign always turns out to be unphysial and the sign issue does not arise [33]. However, in the asymmetri ase, both signs an be physial sometimes and one must make the proper sign hoie in Equation (6 34) as follows. For the given set of test masses ( M (a) transverse enter-of-mass energy ŝ ± T = e (a) + e (b) + (e(b) e (a) ) M A T m(a) m (b) 4 M A T m(a) m (b), M (b) ), alulate the ± (e(b) + e (a) )A T (e (b) m(a) + e(a) m(b) ) A T m (a) m (b) + ( M A T m (a) m (b) ), (6 38) orresponding to eah sign hoie in Equation (6 34), and ompare the result to the minimum allowed value of ŝ T ŝ T(min) = e (a) + e (b) + Q tot + ( M (a) ) + M (b). (6 39) The minus sign in Equation (6 34) takes preedene and applies whenever it is physial, i.e. whenever ŝ T > ŝ T(min). In the remaining ases when ŝ T < ŝ T(min) and the minus sign is unphysial, the plus sign in Equation (6 34) applies. If one of the onditions in Equations (6 31), (6 3) is not satisfied, the asymmetri M T is given by 177

178 an unbalaned solution, in analogy to Equations (6 15) and (6 16): ( MT Ua ( MT Ub p (a) T p (a) T, p (b) T ; m (a),m (b) ; M (a), M (b), p (b) T ; m (a),m (b) ; M (a), M (b) ) = m(a) + M (a), (6 40) ) = m(b) + M (b). (6 41) The unbalaned solution MT Ua of Equation (6 40) applies when the ondition (6 31) is false and ondition (6 3) is true, while the unbalaned solution MT Ub of Equation (6 41) applies when the ondition of Equation (6 31) is true and ondition of Equation (6 3) is false. Equations (6 34), (6 40) and (6 41) represent one of our main results. They generalize the analytial results of Referenes. [33, 37] and allow the diret omputation of the asymmetri M T variable without the need for sanning and numerial minimizations. This is an important benefit, sine the existing publi odes for M T [104, 105] only apply in the symmetri ase M (a) Properties = M (b). All three properties of the symmetri M T disussed in Setion 6..3 readily generalize to the asymmetri ase Property I: Knowledge Of M p As A Funtion Of M (a) And M (b) In the asymmetri ase, the endpoint M T (max) of the M T distribution still gives the mass of the parent, only this time it is a funtion of two input test masses for the hildren: M p ( M (a), M (b), P UTM ) = M T (max) ( M (a), M (b), P UTM ). (6 4) The important property is that this relation is satisfied by the true values of the hildren and parent masses: M p = M T (max) (M (a), M (b), P UTM ). (6 43) Thus the true parent mass M p will be known one we determine the two hildren masses M (a) and M (b). 178

179 Property II: Ridge In M T (max) Through The True M (a) And M (b) In the symmetri M T ase, the endpoint funtion in Equation (6 1) is not ontinuously differentiable and has a kink at the true hild mass M = M. In the asymmetri M T ase, the endpoint funtion in Equation (6 4) is similarly non-differentiable at a set of points {( M (a) (θ), M (b) )} (θ) (6 44) parametrized by a single ontinuous parameter θ. The gradient of the endpoint funtion in Equation (6 4) suffers a disontinuity as we ross the urve defined by Equation (6 44). Sine Equation (6 4) represents a hypersurfae in the three-dimensional parameter spae of { M (a), M (b), M p }, the gradient disontinuity will appear as a ridge (sometimes also referred to as a rease [40]) on our three-dimensional plots below. The important property of the ridge is that it passes through the orret values for the hildren masses, even when they are different: M (a) M (b) = M (a) (θ 0 ), (6 45) = M (b) (θ 0 ), (6 46) for some θ 0. Thus the ridge information provides a relation among the two hildren masses and leaves us with just a single unknown degree of freedom the parameter θ in Equation (6 44). Interestingly, the shape of the ridge provides a quik test whether the two missing partiles are idential or not 8. If the shape of the ridge in the ( M (a) symmetri with respet to the interhange M (a) with respet to the 45 line M (a), M (b) ) plane is M (b), i.e. under a mirror refletion = M (b), then the two missing partiles are the same. 8 To be more preise, the ridge shape tests whether the two missing partiles have the same mass or not. 179

180 Conversely, when the shape of the ridge is not symmetri under M (a) M (b), the missing partiles are in general expeted to have different masses Property III: P UTM Invariane Of M T (max) At The True M (a) And M (b) The third M T property, whih was disussed in Setion , is readily generalized to the asymmetri ase as well. Note that Equation (6 43) implies that the P UTM dependene of the asymmetri M T endpoint in Equation (6 4) disappears at the true values of the hildren masses: M T (max) ( M (a) P UTM, M (b), P UTM ) M (a) =M (a) = 0. (6 47), M (b) =M (b) This equation is the asymmetri analogue of Equation (6 3). Proeeding as in Setion , let us define the funtion M T (max) ( M (a), M (b), P UTM ) M T (max) ( M (a), M (b), P UTM ) M T (max) ( M (a), M (b), 0), whih quantifies the shift of the asymmetri M T endpoint as Equation (6 4) in the presene of P UTM. By definition, (6 48) M T (max) ( M (a), M (b), P UTM ) 0, (6 49) with equality being ahieved only for the orret values of the hildren masses: M T (max) (M (a), M (b), P UTM ) = 0, P UTM. (6 50) The Equation (6 50) reveals the power of the P UTM invariane method. Unlike the kink method disussed in Setion , whih was only able to find a relation between the two hildren masses M (a) and M (b), the P UTM invariane implied by Equation (6 50) allows us to determine eah individual hildren mass, without any theoretial assumptions, and even in the ase when the two hildren masses happen to be different (M (a) M (b) ). 180

181 Figure 6-3. The three different event-topologies under onsideration in this hapter. In eah ase, two parents with mass M p are produed onshell and deay into two daughters of (generally different) masses M (a) and M (b). Case (a), whih is the subjet of Setion 6.4, has a single massless visible SM partile in eah leg and some arbitrary upstream transverse momentum P UTM. In the remaining two ases (b) and (), whih are disussed in Setion 6.5, there are two massless visible partiles in eah leg, whih form a omposite visible partile with varying invariant mass m (λ). The intermediate partile of mass M (λ) i is (b) heavy and off-shell (M (λ) i > M p ), or () on-shell (M p > M (λ) i > M (λ) ). For simpliity, we do not onsider any upstream momentum in ases (b) and () Examples In the next two setions we shall illustrate the three properties disussed so far in Setion with some onrete examples. Instead of the most general event topology depited Figure 6-1, here we limit ourselves to the three simple examples shown in Figure 6-3. The simplest possible ase is when n (λ) = 1, i.e. when eah asade deay ontains a single SM partile, as in Figure 6-3(a). In this example, m (λ) is onstant. For simpliity, we shall take m (λ) 0, whih is the ase for a lepton or a light flavor jet. If the SM partile is a Z-boson or a top quark, its mass annot be negleted, and one must keep the proper value of m (λ). This, however, is only a tehnial detail, whih does not affet our main onlusions below. In spite of its simpliity, the topology of Figure 6-3(a) is atually the most hallenging ase, due to the limited number of available measurements [3]. In order to be able to determine all individual masses 181

182 in that ase, one must onsider events with upstream momentum P UTM, as illustrated in Figure 6-3(a). This is not a partiularly restritive assumption, sine there is always a ertain amount of P UTM in the event (at the very least, from initial state radiation). In Setion 6.4 the topology of Figure 6-3(a) will be extensively studied - first for the asymmetri ase of M (a) M (a) M (b) = M (b) in Setion in Setion 6.4.1, and then for the symmetri ase of Another simple situation arises when there are two massless visible SM partiles in eah leg, as illustrated in Figures 6-3(b) and 6-3(). In either ase, the invariant mass m (λ) is not onstant any more, but varies within a ertain range m min (λ) where m min (λ) = 0, while the value of mmax (λ) depends on the mass M (λ) i intermediate partile. In Figure 6-3(b) we assume M (λ) i partile is off-shell and m (λ) m max (λ), of the orresponding > M p, so that the intermediate m max (λ) = M p M (λ). (6 51) The off-shell ase of Figure 6-3(b) will be disussed in Setion In ontrast, in Figure 6-3() we take M p > M (λ) i > M (λ), in whih ase the intermediate partile is on-shell and the range for m (λ) is now limited from above by m max (λ) ( M = M p 1 (λ) i M p ) 1 We shall disuss the on-shell ase of Figure 6-3() in Setion ( ) M (λ). (6 5) M (λ) i In the event topologies of Figures 6-3(b) and 6-3(), the mass m (λ) is varying and the ridge of Equation (6 44) will appear even if there were no upstream transverse momentum in the event. Therefore, in our disussion of Figures 6-3(b) and 6-3() in Setion 6.5 below we shall assume P UTM = 0 for simpliity. The presene of non-zero P UTM will only additionally enhane the ridge feature. 18

183 6.3.5 Combinatorial Issues Before going on to the atual examples in the next two setions, we need to disuss one minor ompliation, whih is unique to the asymmetri M T variable and was not present in the ase of the symmetri M T variable. The question is, how does one assoiate the visible deay produts observed in the detetor with a partiular deay hain λ = a or λ = b. This is the usual ombinatoris problem, whih now has two different aspets: The first issue is also present in the symmetri ase, where one has to deide how to partition the SM partiles observed in the detetor into two disjoint sets, one for eah asade. In the traditional approah, where the hildren partiles are assumed to be idential, the two sets are indistinguishable and it does not matter whih one is first and whih one is seond. This partiular aspet of the ombinatorial problem will also be present in the asymmetri ase. In the asymmetri ase, however, there is an additional aspet to the ombinatorial problem: now the two asades are distinguishable (by the masses of the hild partiles), so even if we orretly divide the visible objets into the proper subsets, we still do not know whih subset goes together with M (a) and thus gets a label λ = a, and whih goes together with M (b) and gets labelled by λ = b. This leads to an additional ombinatorial fator of whih is absent in the symmetri ase with idential hildren. The severity of these two ombinatorial problems depends on the event topology, as well as the type of signature objets. For example, there are ases where the first ombinatorial problem is easily resolved, or even absent altogether. Consider the event topology of Figure 6-3(a) with a lepton as the SM partile on eah side. In this ase, the partition is unique, and the upstream objets are jets, whih an be easily identified [39]. Now onsider the event topologies of Figures 6-3(b) and 6-3(), with two opposite sign, same flavor leptons on eah side. Suh events result from inlusive pair prodution of heavier neutralinos in supersymmetry. By seleting events with different lepton flavors: e + e µ + µ, we an overome the first ombinatorial problem above and uniquely assoiate the e + e pair with one asade and the µ + µ pair with the other. However, the seond ombinatorial problem remains, as we still have to deide whih of 183

184 the two lepton pairs to assoiate with λ = a and whih to assoiate with λ = b. Reall that the labels λ = a and λ = b are already attahed to the hild partiles, whih are distinguishable in the asymmetri ase. We use the onvention that λ = a is attahed to the lighter hild partile: M (a) M (b), (6 53) whih also ensures that the M parameter defined in Equation (6 36) is real. We an put this disussion in more formal terms as follows. The orret assoiation of the visible partiles with the orresponding hildren will yield M T ( p (a) (b) T, p T ;m (a),m (b) ; M (a), M (b) ), (6 54) while the other, wrong assoiation will give simply M T ( p (a) (b) T, p T ;m (a),m (b) ; M (b) ), M (a). (6 55) Both of these two M T values an be omputed from the data, but a priori we do not know whih one orresponds to the orret assoiation. The solution to this problem is however already known [3, 33]: one an onservatively use the smaller of the two M (<) T min { M T ( p (a) (b) T, p T ;m (a),m (b) ; M (a), M (b) ), MT ( p (a) (b) T, p T ;m (a),m (b) ; M (b), M (a) )} (6 56) in order to preserve the loation of the upper M T endpoint. This is illustrated in Figure 6-4, where we show results for the event topology of Figure 6-3(b) with a mass spetrum as follows: M (a) = 100 GeV, M (b) = 00 GeV and M p = 600 GeV. The test hildren masses are taken to be the true masses: M (a) = M (a) and M (b) = M (b). The dotted blak distribution is the unit-normalized true M T distribution, where one ignores the ombinatorial problem and uses the Monte Carlo information to make the orret assoiation. The red histogram shows the unit-normalized distribution of the M (<) T variable defined in Equation (6 56). 184

185 Figure 6-4. Unit-normalized M T distributions for the event topology of Figure 6-3(b). The mass spetrum is hosen as M (a) = 100 GeV, M (b) = 00 GeV and M p = 600 GeV. The test hildren masses are taken to be the true masses: M (a) = M (a) and M (b) = M (b). The dotted blak distribution is the true M T distribution, ignoring the ombinatorial problem. The red histogram shows the distribution of the M (<) T histogram shows the distribution of the M (>) T Equation (6 58). variable defined in Equation (6 56) while the blue variable defined in We see that the definition in Equation (6 56) preserves the orresponding endpoint: M (<) T (max) = M T (max). (6 57) Of ourse, we an also onsider the alternative ombination M (>) T max { M T ( p (a) (b) T, p T ;m (a),m (b) ; M (a) ) (, M (b), MT p (a) T (b), p T ;m (a),m (b) ; M (b), M (a) )}, (6 58) whose unit-normalized distribution is shown in Figure 6-4 with the blue histogram. One an see that some of the wrong ombination entries in the M (>) T original endpoint M T (max), yet there is still a well defined M (>) T histogram violate the endpoint M (>) T (max) M T (max). (6 59) 185

186 Table 6-1. Mass spetra for the two examples studied in Setions and All masses are given in GeV. Spetrum Case M (a) M (b) M p I Different hildren II Idential hildren M (<) T Stritly speaking, in our analysis in the next setions, we only need to study the endpoint in Equation (6 57), whih ontains the relevant information about the physial M T endpoint. At the same time, with our onvention as Equation (6 53) for the hildren masses, we only need to onentrate on the upper half M (b) the ( M (a) M (>) T the ( M (a) M (a), M (b) ) plane. However, for ompleteness we shall also present results for the endpoint in Equation (6 59), and we shall use the lower ( M (b) of < M (a) ) half of, M (b) ) plane to show those. Thus the M T endpoint shown in our plots below should be interpreted as follows M T (max) = M (<) T (max), if M (a) M (b), M (>) T (max), if M (a) > M (b). (6 60) 6.4 The Simplest Event Topology: One Standard Model Partile On Eah Side In this setion, we onsider the simplest topology with a single visible partile on eah side of the event. We already introdued this example in Setion 6.3.4, along with its event topology in Figure 6-3(a). In Setion below we first disuss an asymmetri ase with different hildren. Later in Setion 6.4. we onsider a symmetri situation with idential hildren masses. The mass spetra for these two study points are listed in Table Asymmetri Case Before we present our numerial results, it will be useful to derive an analytial expression for the asymmetri M T endpoint in Equation (6 4) in terms of the orresponding physial spetrum of Table 6-1 and the two test hildren masses M (a) and M (b). Our result will generalize the orresponding formula derived in [37] for the 186

187 symmetri ase of M (a) = M (b) M and no upstream momentum (P UTM = 0). For the event topology of Figure 6-3(a) the M T endpoint is always obtained from the balaned solution and is given by [37] M T (max) ( M, P UTM = 0) = µ pp + µ pp + M. (6 61) Here we made use of the onvenient shorthand notation introdued in [3] for the relevant ombination of physial masses µ np M n { 1 ( M M p ) }. (6 6) The µ parameter defined in Equation (6 6) is simply the transverse momentum of the (massless) visible partile in those events whih give the maximum value of M T [39]. Squaring Equation (6 61), we an equivalently rewrite it as M T (max)( M, P UTM = 0) = µ pp + M + 4 µ pp(µ pp + M ). (6 63) Now let us derive the analogous expressions for the asymmetri ase M (a) M (b). Just like the symmetri ase, the asymmetri endpoint M T (max) also omes from a balaned solution and is given by MT (max)( M (a), M (b), P UTM = 0) = µ pp + M µ pp( µ pp + M +) + M 4, (6 64) where the parameters M + and M were already defined in Equation (6 35) and (6 36), while µ pp is now the geometri average of the orresponding individual µ pp parameters µ pp µ ppa µ ppb (M p ( M (a) It is easy to hek that in the symmetri limit ) )(M p ( M (b) ) ). (6 65) 4M p M (b) M (a) = µ pp µ pp, M + M, M 0, (6 66) Equation (6 64) redues to its symmetri ounterpart of Equation (6 63), as it should. 187

188 Figure 6-5. M T (max) as a funtion of the two test hildren masses, M (a) and M (b), for the event topology of Figure 6-3(a) with no upstream momentum (P UTM = 0), and the asymmetri mass spetrum I from Table 6-1: (M (a), M (b), M p ) = (50, 500, 600) GeV. We show (a) a three dimensional view and (b) ontour plot projetion on the ( M (a), M (b) ) plane (red ontour lines). The green dot marks the true values of the hildren masses. Panel (b) also shows a gradient plot, where longer (shorter) arrows imply steeper (gentler) slope. A kink struture is absent in this ase. The symmetri endpoint M T (max) ( M ) of Equation (6 61) an be obtained by going along the diagonal orange line M (b) = M (a). We are now ready to present our numerial results for the event topology of Figure 6-3(a). We first take the asymmetri mass spetrum I from Table 6-1 and onsider the ase with no upstream momentum, when formula (6 64) applies. Figure 6-5 shows the orresponding M T endpoint as a funtion of the two test hildren masses M (a) and M (b). In panel (a) we present a three dimensional view, while in panel (b) we show a ontour plot projetion on the ( M (a), M (b) ) plane (red ontour lines). On either panel, the green dot marks the true values of the hildren masses, M (a) M (b). Panel (b) also shows a gradient plot, where longer (shorter) arrows imply steeper (gentler) slope. The symmetri endpoint M T (max) ( M, P UTM = 0) of Equation (6 61) an and be obtained by going along the diagonal orange line M (b) = M (a) in Figure 6-5(b). We remind the reader that the endpoint M T (max) plotted in Figure 6-5 should be interpreted as in Equation (6 60). 188

189 Figure 6-5 illustrates the first basi property of the asymmetri M T variable, whih was disussed in Setion The M T endpoint allows us to find one relation between the two hildren masses M (a) and M (b) and the parent mass M p = M T (max), and in order to do so, we do not have to assume equality of the hildren masses, as is always done in the literature. The ruial advantage of our approah, in whih we allow the two hildren masses to be arbitrary, is its generality and model-independene. It allows us to extrat the basi information ontained in the M T endpoint, without muddling it up with additional theoretial (and unproven) assumptions. Unfortunately, to go any further and determine eah individual mass, we must make use of the additional properties disussed in Setions and In the ase of the simplest event topology of Figure 6-3(a) onsidered here, they both require the presene of some upstream momentum [3, 36]. As a proof of onept, we now reonsider the same type of events, but with a fixed upstream momentum of P UTM = 1 TeV. (The upstream momentum may be due to initial state radiation, or deays of heavier partiles upstream.) The orresponding results are shown in Figure 6-6. Figure 6-6 demonstrates the seond basi property of the asymmetri M T variable disussed in Setion Unlike the result shown in Figure 6-5(a), whih was perfetly smooth, this time the M T (max) funtion in Figure 6-6(a) shows a ridge, orresponding to the slope disontinuity marked with the blak solid line in Figure 6-6(b). The most important feature of the ridge is the fat that it passes through the green dot marking the true values of the hildren masses. Notie that applying the traditional symmetri M T approah in this ase will give a ompletely wrong result. If we were to assume equal hildren masses from the very beginning, we will be onstrained to the diagonal orange line in Figure 6-6(b). The M T endpoint will then still exhibit a kink, but the kink will be in the wrong loation. In the example shown in Figure 6-6(b), we will underestimate the parent mass, while for the hild mass we will find a value whih is somewhere in between the two true masses M (a) and M (b). 189

190 Figure 6-6. M T (max) for the event topology of Figure 6-3(a) with fixed upstream momentum of P UTM = 1 TeV. The ridge struture (shown as the blak solid line) is revealed by the sudden inrease in the slope (gradient) in panel (b). Notie that the ridge goes through the true values of the hildren masses marked by the green dot. Using the ridge information, we now know an additional relation among the hildren masses, whih allows us to express all three masses in terms of a single unknown parameter θ, as illustrated in Figure 6-7(a). Let us hoose to parametrize the ridge by the polar angle in the ( M (a), M (b) ) plane: ( ) M (b) θ = tan 1 M (a). (6 67) Using the ridge information from Figure 6-6, we an then find all three masses as a funtion of θ. The result is shown in Figure 6-7(a). The mass M (a) plotted in red, the mass M (b) of the lighter hild is of the heavier hild is plotted in blue, while the parent mass M p is plotted in blak. With our onvention (6 53) for the hildren masses, only values of θ 45 are physial, and the orresponding masses are shown with solid lines. The dotted lines in Figure 6-7(a) show the extrapolation into the unphysial region θ < 45. Figure 6-7(a) has some important and far reahing impliations. For example, one may now start asking the question: Are there really any massive invisible partiles in those events, or is the missing energy simply due to neutrino prodution [94]? 190

191 Figure 6-7. (a) Partile masses obtained along the M T (max) ridge seen in Figure 6-6. The ridge is parametrized by the angle θ defined in Equation (6 67). The two hildren masses M (a) (θ) (in red) and M (b) (θ) (in blue) as well as the parent mass M p (in blak) are then plotted as a funtion of θ. In our onvention (6 53) only values of θ 45 are physial, and the orresponding masses are shown with solid lines. Dotted lines show the extrapolation for θ < 45. (b) Contour plot of the quantity M T (max) ( M (a), M (b), P UTM = 1 TeV) defined in Equation (6 48), in the ( M (a), M (b) ) plane. This plot is obtained simply by taking the differene between Figure 6-6(a) and Figure 6-5(a). The solid blak urve indiates the loation of the M T (max) ridge. Only the point orresponding to the true hildren masses (the green dot) satisfies the P UTM invariane ondition M T (max) = 0 from Equation (6 50). The ridge results shown in Figure 6-7(a) begin to provide the answer to that quite fundamental question. Aording to Figure 6-7(a), for any value of the (still unknown) parameter θ, the two hildren partiles annot be simultaneously massless. This means that the missing energy annot be simply due to neutrinos, i.e. there is at least one new, massive invisible partile produed in the missing energy events. At this point, we annot be ertain that this is a dark matter partile, but establishing the prodution of a WIMP andidate at a ollider is by itself a tremendously important result. Notie that while we annot be sure about the masses of the hildren, the parent mass M p is 191

192 determined with a very good preision from Figure 6-7(a): the funtion M p (θ) is almost flat and rather insensitive to the partiular value of θ 9. One we have proved that some kind of WIMP prodution is going on, the next immediate question is: how many suh WIMP partiles are present in the data one or two? Unfortunately, the ridge analysis of Figure 6-7(a) alone annot provide the answer to this question, sine the value of θ is still undetermined. If θ = 90, one of the missing partiles is massless, whih is onsistent with a SM neutrino. Therefore, if θ were indeed 90, the most plausible explanation of this senario would be that only one of the missing partiles is a genuine WIMP, while the other is a SM neutrino. On the other hand, almost any other value of θ < 90 would guarantee that there are two WIMP andidates in eah event. In that ase, the next immediate question is: are they the same or are they different? Fortunately, our asymmetri approah will allow answering this question in a model-independent way. If θ is determined to be 45, the two WIMP partiles are the same, i.e. we are produing a single speies of dark matter. On the other hand, if 45 < θ < 90, then we an be ertain that there are not one, but two different WIMP partiles being produed. We see that in order to ompletely understand the physis behind the missing energy signal, we must determine the value of θ, i.e. we must find the exat loation of the true hildren masses along the ridge. One of our main results in this hapter is that this an be done by using the third M T property disussed in Setion The idea is illustrated in Figure 6-7(b), where we show a ontour plot in the ( M (a), M (b) ) plane of the quantity M T (max) ( M (a), M (b), P UTM ) defined in Equation (6 48), for a fixed P UTM = 1 TeV. This plot is obtained simply by taking the differene between 9 Interestingly, for the example in Figure 6-7(a), the maximum value of M p (θ) happens to give the true parent mass M p, but we have heked that this is a oinidene and does not hold in general for other examples whih we have studied. 19

193 Figure 6-6(a) and Figure 6-5(a). (A more pratial method for obtaining this information was proposed in [1].) Reall that the funtion M T (max) was introdued in order to quantify the P UTM invariane of the M T endpoint, and it is expeted that M T (max) vanishes at the orret values of the hildren masses (see Equation (6 50)). This expetation is onfirmed in Figure 6-7(b), where we find the minimum (zero) of the M T (max) funtion exatly at the right spot (marked with the green dot) along the M T (max) ridge. Thus the M T (max) funtion in Figure 6-7(b) ompletely pins down the spetrum, and in this ase would reveal the presene of two different WIMP partiles, with unequal masses M (a) M (b). Our analysis thus shows that olliders an not only produe a WIMP dark matter andidate and measure its mass, as disussed in the existing literature, but they an do a muh more elaborate dark matter partile spetrosopy, as advertized in the title. In partiular, they an probe the number and type of missing partiles, inluding partiles from subdominant dark matter speies, whih are otherwise unlikely to be disovered experimentally in the usual dark matter searhes Symmetri Case While in our approah the two hildren masses M (a) and M (b) are treated as independent inputs, this, of ourse, does not mean that the approah is only valid in ases when the hildren masses are different to begin with. The tehniques disussed in the previous subsetion remain appliable also in the more onventional ase when the hildren are idential, i.e. when olliders produe a single dark matter omponent. In order to illustrate how our method works in that ase, we shall now work out an example with equal hildren masses. We still onsider the simplest event topology of Figure 6-3(a), but with the symmetri mass spetrum II from Table 6-1. We then repeat the analysis done in Figures 6-5, 6-6, and 6-7 and show the orresponding results in Figures 6-8, 6-9 and

194 Figure 6-8. M T (max) for the event topology of Figure 6-3(a) with no upstream momentum. Partiles have the symmetri mass spetrum II from Table 6-1, i.e. (M (a), M (b), M p ) = (100, 100, 300) GeV. Figure 6-9. M T (max) for the event topology of Figure 6-3(a) with fixed upstream momentum P UTM = 1TeV. Partiles have the symmetri mass spetrum II from Table 6-1, i.e. (M (a), M (b), M p ) = (100, 100, 300) GeV. The onlusions from this exerise are very similar to what we found earlier in Setion for the asymmetri ase. The M T endpoint still provides one relation among the two hildren masses M (a) and M (b) and the parent mass M p = M T (max). This relation is shown in Figure 6-8 (Figure 6-9) for the ase without (with) upstream momentum P UTM. As seen in Figure 6-8, in the absene of any upstream P UTM, the funtion M p ( M (a), M (b) ) is smooth and reveals nothing about the hildren masses. 194

195 Figure The same as in Figure 6-7 but for the symmetri mass spetrum II from Table 6-1, i.e. (M (a), M (b), M p ) = (100, 100, 300) GeV. Notie that, in ontrast to Figure 6-7, the minimum of the M T (max) funtion is now obtained at M (a) = M (b), implying that the two missing partiles are the same. However, the presene of upstream momentum signifiantly hanges the piture and the funtion M p ( M (a), M (b) ) again develops a ridge, whih is learly visible 10 in both the three-dimensional view of Figure 6-9(a), as well as the gradient plot in Figure 6-9(b). The ridge information now further onstrains the hildren masses to the blak solid line in Figure 6-9(b), leaving only one unknown degree of freedom. Parametrizing it with the polar angle θ as in (6 67), we obtain the spetrum as a funtion of θ, as shown in Figure 6-10(a). One again we find the fortuitous result that in spite of the remaining arbitrariness in the value of θ, the parent mass M p is very well determined, sine M p (θ) is a very weakly varying funtion of θ. Furthermore, both Figure 6-9(a) and Figure 6-9(b) 10 We aution the reader that here we are presenting only a proof of onept. In the atual analysis the ridge may be rather diffiult to see, for a variety of reasons - detetor resolution, finite statistis, ombinatorial and SM bakgrounds, et. Nevertheless, we expet that the ridge will be just as easily observable as the traditional kink in the symmetri M T endpoint. If the kink an be seen in the data, the ridge an be seen too, and there is no reason to make the assumption of equal hildren masses. Conversely, if the kink is too diffiult to see, the ridge will remain hidden as well. 195

196 exhibit a high degree of symmetry under M (a) M (b), whih is a good hint that the hildren are in fat idential. This suspiion is onfirmed in Figure 6-10(b), where we find that the P UTM dependene disappears at the symmetri point M (a) revealing the true masses of the two hildren. = M (b) = 100 GeV, In the two examples onsidered so far in Setions and 6.4., we used a fixed finite value of the upstream transverse momentum P UTM = 1 TeV, whih is probably rather extreme in realisti models, one might expet typial values of P UTM on the order of several hundred GeV. However, things begin to get muh more interesting if one were to onsider even larger values of P UTM. On the one hand, the ridge feature beomes sharper and easier to observe [3]. More importantly, the ridge struture itself is modified, and a seond set of ridgelines appears 11 at suffiiently large P UTM. All ridgelines interset preisely at the point marking the true values of the hildren masses, thus allowing the omplete determination of the mass spetrum by the ridge method alone. This proedure was demonstrated expliitly in Referene. [40], whih investigated the extreme ase of P UTM = for a study point with different parents and idential hildren. The assumption of P UTM = justified the use of a deoupling argument, in whih the two branhes λ = a and λ = b are treated independently, allowing the derivation of simple analytial expressions for the M T endpoint [40]. In Appendix C we reprodue the analogous analytial results at P UTM for the ase of interest here (idential parents and different hildren) and study in detail the P UTM dependene of the ridgelines. Unfortunately, we find that the values of P UTM neessary to reveal the additional ridge struture, are too large to be of any interest experimentally. On the positive side, the P UTM invariane method disussed in Setion does not require 11 A keen observer may have already notied a hint of those in Figures 6-7(b) and 6-10(b). 196

197 suh extremely large values of P UTM and an in priniple be tested in more realisti experimental onditions Mixed Case For simpliity, so far in our disussion we have been studying only one type of missing energy events at a time. In reality, the missing energy sample may ontain several different types of events, and the orresponding M T measurements will first need to be disentangled from eah other. For onreteness, onsider the inlusive pair prodution of some parent partile χ p, whih an deay either to a hild partile χ a of mass M (a), or a different hild partile χ b of mass M (b). Let the orresponding branhing frations be B a and B b, i.e. B a B(χ p χ a ) and B b B(χ p χ b ). Furthermore, let χ b deay invisibly 1 to χ a. Suh a situation an be easily realized in supersymmetry, for example, with the parent being a squark, a slepton, or a gluino, the heavier hild χ b being a Wino-like neutralino χ 0 and the lighter hild χ a being a Bino-like neutralino χ 0 1. The heavier neutralino has a large invisible deay mode χ 0 χ 0 1ν ν, if its mass happens to fall between the sneutrino mass and the left-handed slepton mass: M ν < M χ 0 < M l L. Let us start with a ertain total number of events N pp in whih two parent partiles χ p have been produed. Then the missing energy sample will ontain N bb = N pp B b symmetri events where the two hildren are χ b and χ b, N aa = N pp Ba symmetri events where the two hildren are χ a and χ a, and N ab = N pp B a B b asymmetri events where the two hildren are χ a and χ b. How an one analyze suh a mixed event sample with a single M T variable? The blak histogram in Figure 6-11 shows the unit-normalized M T distribution for the whole (mixed) event sample (for onveniene, we do not show the zero bin [1]). 1 If χ b deays visibly, then the respetive types of events an in priniple be sorted by their signature. 197

198 Figure Unit-normalized, zero-bin subtrated M T distribution (blak histogram) for the full mixed event sample, as well as the individual omponents χ a χ a (red), χ a χ b (blue) and χ b χ b (green). We took zero test masses for the hildren M (a) = M (b) = 0 and equal branhing fration for the parents B a = B b = 50%. The mass spetrum is taken from the asymmetri study point I in Table 6-1 with M (a) = 50 GeV, M (b) = 500 GeV and M p = 600 GeV. The three arrows indiate the expeted endpoints for eah individual omponent in the sample. For this plot, we used the asymmetri mass spetrum I from Table 6-1: M (a) = 50 GeV, M (b) = 500 GeV and M p = 600 GeV, and hose zero test masses for the hildren M (a) = M (b) = 0. For definiteness, we fixed equal branhing frations B a = B b = 50%, so that the relative normalization of the three individual samples is N aa : N bb : N ab = 1 : 1 :. Figure 6-11 shows that the observable M T distribution is simply a superposition of the M T distributions of the three individual samples χ a χ a, χ a χ b and χ b χ b, whih are shown with the red, blue and green histograms, orrespondingly. Eah individual sample exhibits its own M T endpoint, marked with a vertial arrow, whih an also be seen in the ombined M T distribution. Using Equation (6 64), the three endpoints are found to 198

199 be χ a χ a M (aa) T (max) (0, 0, 0) = M p χ a χ b M (ab) T (max) (0, 0, 0) = M p 1 ( ( 1 ) M (a) = 496 GeV, (6 68) M p M (a) M p ) 1 ( ) M (b) = 301 GeV,(6 69) M p χ b χ b M (bb) T (max) (0, 0, 0) = M p 1 ( ) M (b) = 183 GeV. (6 70) M p Now suppose that all three endpoints as in Equations (6 68 through 6 70) are seen in the data. Their interpretation is far from obvious, and in fat, there will be different ompeting explanations. If one insists on the single missing partile hypothesis, there an be only one type of hild partile, and the only way to get three different endpoints in Figure 6-11 is to have prodution of three different pairs of parent partiles, eah of whih deays in exatly the same way. Sine the three parent masses are a priori unrelated, one does not expet any partiular orrelation among the three observed endpoints in Equations (6 68 through 6 70). Now onsider an alternative explanation where we produe a single type of parents, but have two different hildren types. This situation also gives rise to three different event topologies, with three different M T endpoints, as we just disussed. However, now there is a predited relation among the three M T endpoints, whih follows simply from Equations (6 68 through 6 70): M (ab) T (max) (0, 0, 0) = M (aa) T (max) (0, 0, 0) M(bb) T (max) (0, 0, 0). (6 71) If the parents are the same and the hildren are different, this relation must be satisfied. If the parents are different and the hildren are the same, a priori there is no reason why Equation (6 71) should hold, and if it does, it must be by pure oinidene. The predition of Equation (6 71) therefore is a diret test of the number of hildren 199

200 partiles. Another test an be performed if we ould estimate the individual event ounts N aa, N ab and N bb, although this appears rather diffiult, due to the unknown shape of the M T distributions in Figure In the asymmetri example disussed here, we have another predition, namely N ab = N aa N bb, (6 7) whih is another test of the different hildren hypothesis. Notie that Equation (6 7) holds regardless of the branhing frations B a and B b, although if one of them dominates, the two endpoints whih require the other (rare) deay may be too diffiult to observe. Of ourse, the ultimate test of the single missing partile hypothesis is the behavior of the intermediate M T endpoint in Figure 6-11 orresponding to the asymmetri events of type χ a χ b. Applying either one of the two mass determination methods disussed earlier in Figures 6-7 and 6-10, we should find that M (ab) T (max) is a result of asymmetri events, indiating the simultaneous presene of two different invisible partiles in the data. 6.5 A More Complex Event Topology: Two Visible Partiles On Eah Side In this setion, we onsider two more examples: the off-shell event topology of Figure 6-3(b) is disussed in Setion 6.5.1, while the on-shell event topology of Figure 6-3() is disussed in Setion (For simpliity, we do not onsider any P UTM in this setion.) Now there are two visible partiles in eah leg, whih form a omposite visible partile of varying mass m (λ). In general, by studying the invariant mass distribution of m (λ), one should be able to observe two different invariant mass endpoints, suggesting some type of an asymmetri senario Off-shell Intermediate Partile Here we onentrate on the example of Figure 6-3(b). Sine the intermediate partile is offshell, the maximum kinematially allowed value for m (λ) is given by Equation (6 51). 00

201 Figure 6-1. The four regions in the ( M (a), M (b) ) parameter plane leading to the four different types of solutions for the M T endpoint, for the off-shell event topology of Figure 6-3(b). The green dot marks the true loation of the two hildren masses. Within eah region, we indiate the relevant momentum onfiguration for the visible partiles (red arrows) and the hildren partiles (blue arrows) in eah leg (a or b). The momenta are quoted in the bak-to-bak boosted frame [37], in whih the two parents are at rest. A blue dot implies that the orresponding daughter is at rest and therefore the two visible partiles are emitted bak-to-bak. The two balaned solutions are denoted as B and B, while the two unbalaned solutions are Ua and Ub. The blak solid lines represent phase hanges between different solution types and delineate the expeted loations of the ridges in the M T (max) funtion shown in Figure Reall that for the simple topology of Figure 6-3(a) disussed in the previous setion, the M T endpoint (6 64) always orresponded to a balaned solution. More preisely, the M T variable was maximized for a momentum onfiguration p (λ) T in whih M T was given by the balaned solution (6 34). However, in this setion we shall find that for the more omplex topologies of Figures 6-3(b) and 6-3(), the M T endpoint may result from one of four different ases altogether: two different balaned solutions, whih we shall label as B and B, or the unbalaned solutions Ua and Ub disussed in Setion 01

202 6.3.. Depending on the type of solution giving the endpoint M T (max), the ( M (a), M (b) ) parameter plane divides into the three regions 13 shown in Figure 6-1. The green dot in Figure 6-1 denotes the true hildren masses in this parameter spae. Within eah region, we show the relevant momentum onfiguration for the visible partiles (red arrows) and the hildren partiles (blue arrows) in eah leg (a or b). The momenta are quoted in the bak-to-bak boosted frame [37], in whih the two parents are at rest. The length of an arrow is indiative of the magnitude of the momentum. A blue dot implies that the orresponding daughter is at rest and therefore the two visible partiles are emitted bak-to-bak. The two balaned solutions are denoted as B and B, while the two unbalaned solutions are Ua and Ub. The blak solid lines represent phase hanges between different solution types and delineate the expeted loations of the ridges in the M T (max) funtion shown in Figure 6-13 below. Perhaps the most striking feature of Figure 6-1 is that the three (in fat, all four) regions ome together preisely at the green dot marking the true values of the two hildren masses. The boundaries of the regions shown in Figure 6-1 will manifest themselves as the loations of the ridges (i.e. gradient disontinuities) in the M T (max) funtion. Therefore, we expet that by studying the ridge struture and finding its triple point, one will be able to ompletely determine the mass spetrum. We shall now give analytial formulas for the M T endpoint in eah of the four regions of Figure 6-1. We begin with the two balaned solutions B and B, for whih the event-by-event balaned solution for M T is given by Equation (6 34). In the parameter spae region of Figure 6-1 whih is adjaent to the origin, we find the balaned onfiguration B, in whih all visible partiles have the same diretion in the 13 The fourth ase of the B balaned solution happens to oinide with the two unbalaned solutions along the boundary between Ua and Ub. 0

203 bak-to-bak boosted frame. As a result, we have m (a) = m (b) = 0 (6 73) and A T = (M p ( M (a) ) )(M p ( M (b) ) ). (6 74) M p Substituting Equations (6 73) and (6 74) in the balaned M T solution (6 34), where we should take the plus sign, we obtain [ M B T (max)( M (a), M (b) ) ] = µ pp + M µ pp( µ pp + M +) + M 4, (6 75) whih we reognize as the balaned solution (6 64) found for the deay topology of Figure 6-3(a). Moving away from the origin in Figure 6-1, we find a seond balaned solution B along the boundary of the unbalaned regions Ua and Ub. In this ase the visible partiles are bak-to-bak, and their invariant mass is maximized: m (λ) = M p M (λ), (6 76) and orrespondingly A T = ( M p M (a) ) ( ) Mp M (b). (6 77) Substituting Equations (6 76) and (6 77) in the balaned M T solution (6 34), we obtain the B -type M T endpoint as [ M B T (max)( M (a), M (b) ) ] = ( Mp M (a) ) ( Mp M (b) ) + M + + M p M (a) M (b) M (b) M (a) The orresponding formulas for the unbalaned ases Ua and Ub are obtained by taking the maximum value for the invariant mass of the visible partiles in the M. (6 78) 03

204 orresponding deay hain: m (a) = m max (a) m (b) = m max (b) = M p M (a) for region (Ua), (6 79) = M p M (b) for region (Ub). (6 80) The orresponding formula for M T (max) is then given by MT Ua (max)( M (a) MT Ub (max)( M (b) ) = M p M (a) ) = M p M (b) + M (a), (6 81) + M (b). (6 8) One an now use the analytial results of Equations (6 75), (6 78), (6 81) and (6 8) to understand the ridge struture shown in Figure 6-1. For example, the boundary between the B and Ua regions is parametrially given by the ondition M B T (max)( M (a), M (b) ) = M Ua T (max)( M (a) ), (6 83) while the boundary between the B and Ub regions is parametrially given by M B T (max)( M (a), M (b) ) = M Ub T (max)( M (b) ). (6 84) On the other hand, the boundary MT Ua (max)( M (a) ) = M Ub T (max)( M (b) ) (6 85) between the two unbalaned regions Ua and Ub is quite interesting. The parametri equation (6 85) is nothing but a straight line in the ( M (a), M (b) ) plane: M (b) = M (b) M (a) + M (a), (6 86) as seen in Figure

205 Figure M T (max) as a funtion of the two test hildren masses, M (a) and M (b), for the off-shell event topology of Figure 6-3(b). We use the mass spetrum from the example in Figure 6-4: M (a) = 100 GeV, M (b) = 00 GeV and M p = 600 GeV and for simpliity onsider only events with P UTM = 0. It is now easy to understand the triple point struture in Figure 6-1. The triple point is obtained by the merging of all three boundaries as in Equations (6 83), (6 84) and (6 85), i.e. when MT B (max)( M (a), M (b) ) = M B T (max)( M (a), M (b) ) = M Ua T (max)( M (a) ) = M Ub T (max)( M (b) ). (6 87) It is easy to hek that M (a) = M (a) and M (b) = M (b) identially satisfy these equations, thereby proving that the triple intersetion of the boundaries seen in Figure 6-1 indeed takes plae at the true values of the hildren masses. These results are onfirmed in our numerial simulations. In Figure 6-13 we present (a) a three dimensional view and (b) a gradient plot of the ridge struture found in events with the off-shell topology of Figure 6-3(b). The mass spetrum for this study point was fixed as in Figure 6-4, namely M (a) = 100 GeV, M (b) = 00 GeV and M p = 600 GeV. Sine the ridge struture for this topology does not require the presene of upstream momentum, for simpliity we onsider only events with P UTM = 0. The ridge pattern is learly evident in Figure 6-13(a), whih shows a three-dimensional view of the M T endpoint funtion M T (max) ( M (a), M (b) ). It is even more apparent in Figure 6-13(b), where one an see a 05

206 sharp gradient hange along the ridge lines: in regions Ua and Ub, the orresponding gradient vetors point in trivial diretions (either horizontally or vertially), in aord with Equations (6 81) and (6 8). On the other hand, the gradient in region B is very small, and the M T endpoint funtion is rather flat. The green dot marks the loation of the true hildren masses (M (a) = 100 GeV, M (b) = 00 GeV) and is indeed the intersetion point of the three ridgelines. As expeted, the orresponding M T (max) at that point is the true parent partile mass M p = 600 GeV. At this point, it is interesting to ask the question, what would be the outome of this exerise if one were to make the usual assumption of idential hildren, and apply the traditional symmetri M T to this situation. The answer an be dedued from Figure 6-13(b), where the diagonal orange dotdashed line orresponds to the usual assumption of M (a) = M (b). In that ase, one still finds a kink, but at the wrong loation: in Figure 6-13(b) the intersetion of the diagonal orange line and the solid blak ridgeline ours at M (a) = M (b) = 65.3 GeV and the orresponding parent mass is M p = GeV. Therefore, the traditional kink method an easily lead to a wrong mass measurement. Then the only way to know that there was something wrong with the measurement would be to study the effet of the upstream momentum and see that the observed kink is not invariant under P UTM. We should note that, depending on the atual mass spetrum, the two-dimensional ridge pattern seen in Figures 6-1 and 6-13(b) may look very differently. For example, the balaned region B may or may not inlude the origin. One an show that if ( M p < M(b) M (b) 4M (a) + 8 ( ) M (a) ) ( + M (b)), (6 88) the boundary between B and Ua does not ross the M (a) axis. In this ase the diagonal line in Figure 6-13(b) does not ross any ridgelines and the traditional M T approah will not produe any kink struture, in ontradition with one s expetations. 06

207 Figure The four regions in the ( M (a), M (b) ) parameter plane leading to the four different types of solutions for the M T endpoint for the onshell senario illustrated in Figure 6-3(). This exerise teahes us that the failsafe approah to measuring the masses in missing energy events is to apply from the very beginning the asymmetri M T onept On-shell Intermediate Partile Our final example is the on-shell event topology illustrated in Figure 6-3(). Now there is an additional parameter whih enters the game the mass M (λ) i the intermediate partile in the λ-th deay hain. As a result, the allowed range of invariant masses for the visible partile pair on eah side is limited from above by Equation (6 5). In this ase we find that the M T endpoint exhibits a similar phase struture as the one shown in Figure 6-1. One partiular pattern is illustrated in Figure 6-14, whih exhibits the same four regions B, B, Ua and Ub seen in Figure 6-1. The differene now is that region B is onsiderably expanded, and as a result, region B does not have a ommon border with regions Ua and Ub any more. The triple point of Figure 6-1 has now disappeared and the orret values of the hildren masses now lie somewhere on of 07

208 the border between regions B and B, but their exat loation along this ridgeline is at this point unknown. Just like we did for the off-shell ase in Setion 6.5.1, we shall now present analytial formulas for the M T endpoint in eah region of Figure In the balaned region B, we find the same results as Equations (6 73 through 6 75) as in the off-shell ase onsidered in the previous Setion The other balaned region B is haraterized by where m max (λ) is given by Equation (6 5), and A T = M p 4 + M p 4 m (λ) = m max (λ), (6 89) ( ) M (a) ( ) i M (a) ( M p M (a) i ( ) M (a) ( ) M (a) ( ) i M (b) ( M (a) M i p M (b) i ) M (b) ( i M p M (b) i M p M (b) M (b) i ) ). (6 90) The formula for the endpoint M B T (max) in region B is then simply obtained by substituting Equations (6 89) and (6 90) into the balaned solution (6 34). Finally, the M T endpoint in the unbalaned regions Ua and Ub is given by MT Ua (max)( M (a) MT Ub (max)( M (b) ) = m max (a) ) = m max (b) + M (a), (6 91) + M (b), (6 9) where m max (a) and m max (b) are given by Equation (6 5). In Figure 6-15 we present our numerial results in this on-shell senario. The mass spetrum is fixed as: M (a) = 100 GeV, M (b) = 00 GeV, M (a) i = M (b) i = 550 GeV and M p = 1 TeV, and we still do not inlude the effets of any upstream momentum. Figure 6-15(a) shows the three-dimensional view of the M T endpoint funtion M T (max) ( M (a), M (b) ), whih exhibits three different sets of ridges, whih are more easily seen in the gradient plot of Figure 6-15(b). 08

209 Figure M T (max) as a funtion of the two test hildren masses, M (a) the event topology of Figure 6-3(), with a mass spetrum M (a) M (b) = 00 GeV, M (a) i = M (b) i = 550 GeV and M p = 1 TeV. and M (b), for = 100 GeV, As usual, the green dot marks the true hildren masses. Figure 6-15(b) shows that the ridgeline separating the two balaned regions B and B does go through the green dot and thus reveals a relationship between the two hildren masses, leaving the ridgeline parameter θ as the only remaining unknown degree of freedom. However, unlike the off-shell ase of Setion 6.5.1, now there is no speial point on this ridgeline, and we annot ompletely pin down the masses by the ridge method. Thus, in order to determine all masses in the problem, one must use an additional piee of information, for example the visible invariant mass endpoint (6 5) or the P UTM invariane method suggested in Setion Appliation To More General Cases In this setion we disuss a few other possible appliations of the asymmetri M T idea, besides the examples already onsidered. 1. Invisible deays of the next-to-lightest partile. Most new physis models introdue some new massive and neutral partile whih plays the role of a dark matter andidate. Often the very same models also ontain other, heavier partiles, whih for ollider purposes behave just like a dark matter andidate: they deay invisibly and result in missing energy in the detetor. For example, in supersymmetry one may find an invisibly deaying sneutrino ν l ν l χ 0 1, in UED one finds an invisibly 09

210 Figure Event topology for the effetively different missing parties. The blak solid lines represent SM partiles whih are visible in the detetor while red solid lines represent partiles at intermediate sages. The missing partiles are denoted by dotted lines. (a) Squark pair prodution with deay hains terminating in two different invisible partiles ( χ 0 1 and ν l, orrespondingly). In this ase ν l deays invisibly. (b) The subsystem M T variable applied to t t events. The W -boson in the lower leg is treated as a hild partile and an deay either hadronially or leptonially. deaying KK neutrino ν 1 νγ 1, et. These senarios an easily generate an asymmetri event topology. For example, onsider the strong prodution of a squark ( q) pair, as illustrated in Figure 6-16(a). One of the squarks subsequently deays to the seond lightest neutralino χ 0, whih in turn deays to the lightest neutralino χ 0 1 by emitting two SM fermions χ 0 l + l χ 0 1 (or χ 0 jj χ 0 1). The other squark deays to a hargino χ ± 1, whih then deays to a sneutrino as χ± 1 l± ν l. Sine ν l an only deay invisibly, we obtain the asymmetri event topology outlined with the blue box in Figure 6-16(a). The two squarks are the parents, the lightest neutralino χ 0 1 is the first hild, and the sneutrino ν l is the seond hild.. Applying M T to an asymmetri subsystem. One an also apply the M T idea even to events in whih there is only one (or even no) missing partiles to begin with. Suh an example is shown in Figure 6-16(b), where we onsider t t prodution in the dilepton or semi-leptoni hannel. In the first leg we an take bl as our visible system and the neutrino ν l as the invisible partile, while in the other leg we an treat the b-jet as the visible system and the W -boson as the hild partile. In this ase, there still should be a ridge struture revealing the true t, W and ν masses. 3. Multi-omponent dark matter. Of ourse, the model may ontain two (or more) different genuine dark matter partiles [45 47, ], whose prodution in various ombinations will inevitably lead at times to asymmetri event topologies. 10

211 CHAPTER 7 CONCLUSIONS We have proposed methods for mass measurements in missing energy events at hadron olliders. In this hapter, we will summarize all proposed variables. We proposed s (reo) min 7.1 s min and s (sub), whih have a lear physial meaning: the min minimum CM energy in the (sub)system, whih is required in order to explain the observed signal in the detetor. The first variant, the RECO-level variable s (reo) min is basially a modifiation of the presription for omputing the original s min variable: instead of using (muon-orreted) alorimeter deposits, as was done in [50, 51], one ould instead alulate s min with the help of the reonstruted objets (jets and isolated photons, eletrons and muons). Our examples in Setions.4,.5 and.6 showed that this proedure tends to automatially subtrat out the bulk of the UE ontributions, rendering the s (reo) min variable safe. Our seond suggestion was to apply s min to a subsystem of the observed event, whih is suitably defined so that it does not inlude the ontributions from the underlying event. The easiest way to do this is to veto jets from entering the definition of the subsystem. In this ase, the subsystem variable s (sub) min is ompletely unaffeted by the underlying event. However, depending on the partiular senario, in priniple one ould also allow (ertain kinds of) jets to enter the subsystem. As long as there is an effiient method (through uts) of seleting jets whih (most likely) did not originate from the UE, this should work as well, as demonstrated in Fig. -6 with our t t example. (reo) s min (and to some extent s (sub) min ) is a general, global, and inlusive variable, whih an be applied to any type of events, regardless of the event topology, number or type of reonstruted objets, number or type of missing partiles, et. For example, all of the arbitrariness assoiated with the number and type of missing partiles is enoded by a single parameter /M. 11

212 The most important property of both s (reo) min and s (sub) min is that they exhibit a peak in their distributions, whih diretly orrelates with the mass sale M p of the parent partiles. Compared to a kinemati endpoint, a peak is a feature whih is muh easier to observe and subsequently measure preisely over the SM bakgrounds. 7. Invariant Mass Endpoint Method With our new proposed sets of invariant mass endpoint, the preision of the BSM mass determination is expeted to improve. We provide the analytial expressions for all differential invariant mass distributions used in our basi analysis : m ll, m jl(u) and mjl(s) (1). We also provide the orresponding expression for the m jl(d) (1) distribution, whose upper endpoint offers an independent measurement of Mjl(u) max Finally, we also list the formula for the differential distribution of mjl(p), whose endpoint an be used for seleting the orret m B solution. The knowledge of the shape of the whole distribution is indispensable and greatly improves the auray of the endpoint extration. In the absene of any analytial results like those in Appendix A, one would be fored to use simple linear extrapolations, whih would lead to a signifiant systemati error. Clearly, not all invariant mass variables will have their endpoints measured with exatly the same preision some endpoints will be measured better than others. This differene an be due to many fators, e.g. the slope of the distribution near the endpoint, the shape (onvex versus onave) of the distribution near the endpoint, the atual loation of the endpoint, the level of SM and SUSY ombinatorial bakground near the endpoint, et. We provide a number of available measurements tremendously exeeds the number of unknown mass parameters. Thus, we an hoose the best invariant mass endpoint variables for speifi appliation. All of the new variables exhibit milder sensitivity to the parameter spae region, in omparison to the onventional endpoint mjll max. The endpoint for eah of our variables is given by at most two different expressions, as opposed to four in the ase of mjll max. A notable exeption is the variable m jl(s) (1), whose endpoint is atually uniquely predited, 1

213 and is independent of the parameter spae region. We therefore strongly enourage the use of m jl(s) (1) in future analyses of SUSY mass determinations. We an already uniquely determine three out of the four masses involved in the problem. Then, the addition of a fifth measurement, as disussed in Setions and 3.4., is suffiient to pin down all four of the BSM masses. In ontrast, with the onventional approah, one also starts with four measurements as in (3 1), but in the worst ase senario this results in infinitely many solutions, due to the linear dependene problem (3 14) disussed in Setion Adding a fifth measurement as in (3 18) helps, but one again, the worst ase senario leads to two alternative solutions [9]. In order to resolve the remaining dupliation, and thus guarantee uniqueness of the solution under any irumstanes, one needs at least 6 measurements. 7.3 Subsystem M T Method We showed that the M T method by itself is suffiient for a omplete mass spetrum determination, even in the problemati ases of N asade = 1 or N aseade =. We baked our laim with two expliit examples: W + W pair prodution, whih is an example of an n = 1 hain, and t t pair prodution, whih is an example of an n = hain. We showed that the M T method in priniple provides more than enough measurements for the unambiguous determination of the omplete mass spetrum. When applying the M T method, we generalized the onept of M T by introduing various subsystem (or subhain) M (n,p,) T variables. The latter are defined similarly to the onventional M T variable, but are labelled by three integers n, p, and, whose meaning is as follows. The integer n labels the grandparent partile originally produed in the hard sattering and initiating the deay hain. We then apply the usual M T onept to the subhain starting at the parent partile labelled by p and terminating at the hild partile labelled by. In general, the hild partile does not have to be the very last (i.e. the missing) partile in the deay hain, just like the parent partile does not have to be the very first partile produed in the event. The introdution of the M (n,p,) T 13

214 subhain variables greatly proliferates the number of available M T -type measurements, and allows us to make full use of the power of the M T onept. 7.4 One Dimensional Projetion Method We proposed one dimensional projetions of the kinemati variables M T and M CT with respet to P T of upstream objets. By doing this deomposition, we an measure BSM partiles mass spetra in a very short deay where pair produed partiles in a hard ollision are deaying into missing partiles and one visible partile (N asade = 1). To the extent that the definition of M T relies only on the diretion and not the magnitude of the upstream P T, our method is insensitive to the jet energy sale error [11]. We have also provided exat analytial formulas for the omputation of the 1D deomposed M T, M T and the shape of the M T distribution. We show how the perpendiular and the parallel projeted variables are related with eah other. By studying the maximum allowed boundary of this o-relation, we an inrease statistis and orrespondingly get more preise measurements. 7.5 Asymmetri Event Topology The dark matter signatures at olliders always involve missing transverse energy. Suh events will be quite hallenging to fully reonstrut and/or interpret. All previous studies have made (either expliitly or impliitly) the assumption that eah event has two idential missing partiles. Our main point is that this assumption is unneessary, and by suitable modifiations of the existing analysis tehniques one an in priniple test both the number and the type of missing partiles in the data. Our proposal here was to modify the Cambridge M T variable [10] by treating eah hildren mass as an independent input parameter. In this approah, one obtains the M T endpoint M T (max) as a funtion of the two hildren masses M (a) properties. The funtion M T (max) ( M (a) and M (b), and proeeds to study its, M (b) ) exhibits a ridge struture (i.e. a gradient disontinuity). The point orresponding to the orret hildren masses always lies 14

215 on a ridgeline, thus the ridgelines provide a model-independent onstraint among the hildren masses, just like the M T endpoint provides a model-independent onstraint on the masses of the hild(ren) and the parent. In general, the M T endpoint funtion also depends on the value of the upstream transverse momentum in the event. M T (max) ( M (a), M (b), P UTM ). However, the P UTM dependene disappears ompletely for preisely the right values of the hildren masses. This provides a seond, quite general and model-independent, method for measuring the individual partile masses in suh missing energy events. 15

216 APPENDIX A ANALYTICAL EXPRESSIONS FOR THE SHAPES OF THE INVARIANT MASS DISTRIBUTIONS This appendix A provides the analytial expressions for the shapes of the invariant mass distributions m ll, m jl(u) m jl n m jl f, m jl(s) (1) m jl n +m jl f, m jl(d) (1) m jl n m jl f, and mjl(p). To simplify the expressions, we introdue the shorthand notation for the orresponding endpoints, whih was already introdued in Equations (3 4), (3 49), (3 51) and (3 5): L (m max ll ) = m D R CD (1 R BC ) (1 R AB ), (A 1) n ( m max jl n ) = m D (1 R CD ) (1 R BC ), (A ) f ( m max jl f ) = m D (1 R CD ) (1 R AB ), (A 3) p R BC f = m D (1 R CD ) R BC (1 R AB ). (A 4) In this appendix, we shall ignore spin orrelations and onsider only pure phase spae deays. General results inluding spin orrelations for m ll, m jl n and m jl f exist and an be found in [84]. We shall unit-normalize the mll, m jl(s), m jl(d) and m jl(p) distributions, to whih eah event ontributes a single entry. In ontrast, the union distribution mjl(u) has two entries per event, so it will be normalized to instead. It is also onvenient to write the distributions in terms of masses squared instead of linear masses. Of ourse, the two are trivially related by dn dm = m dn dm. (A 5) A.1 Dilepton Mass Distribution m ll The differential dilepton invariant mass distribution is given by dn dm ll = 1 L, (A 6) 16

217 whih is unit-normalized: L dmll 0 ( dn dm ll ) = 1. (A 7) A. Combined Jet-lepton Mass Distribution m jl(u) The differential distribution for u mjl(u) is given by dn du = θ (n u) θ (u) 1 ln(f /p) ln(f /u) + θ (p u) θ (u) + θ (f u) θ (u p) n f p f p, where θ(x) is the usual Heaviside step funtion 1, x 0, θ(x) 0, x < 0. (A 8) (A 9) It is easy to verify the normalization ondition M where M (M max jl(u) ) was already defined in Equation. (3 4). 0 du ( ) dn =, (A 10) du In Figure A-1(a) we ross-hek the predition of Equation. (A 8) (blue dashed line) with the numerially obtained mjl(u) distribution in Figure 3-4(b) (red solid line), for the ase of study point LM1. We see that within the statistial errors, our formula is in perfet agreement with the numerial result. A.3 Distribution of the sum mjl(s) (α = 1) The differential distribution for σ mjl(s) (α = 1) is given by dn dσ = 1 f p { ( ) fn θ(m σ) θ(σ) ln fn σ(f p) ( ) M + θ(m σ) θ(σ m) ln M (f p) ( ) } fn σ(f p) + θ(n + p σ) θ(σ M) ln, (A 11) p(n + p f ) 17

218 Figure A-1. Comparison of the numerially obtained differential invariant mass distributions for study point LM1 (red solid lines) with the analytial results presented in this appendix (blue dashed lines): (a) the distribution of the ombined jet-lepton mass u mjl(u) from Figure 3-4(b) versus the analytial predition of Equation. (A 8); (b) the distribution of the sum σ mjl(s) (α = 1) from Figure 3-4() versus the analytial predition of Equation. (A 11); () the distribution of the differene mjl(d) (α = 1) from Figure 3-4(d) versus the analytial predition of Equations (A 15 through A 19); (d) the distribution of the produt ρ mjl(p) from Figure 3-6() versus the analytial predition of Equations (A through A 3). where m (m max jl(u) ) was defined in (3 4), and n, f and p were defined in Equations (A through A 4). The normalization ondition for Equation. (A 11) reads S where S is defined in Equation. (3 4). 0 dσ ( ) dn = 1, (A 1) dσ As a ross-hek, Figure A-1(b) shows that our analytial formula in Equation. (A 11) agrees with the numerial result from Figure 3-4() for the LM1 study point. 18

219 A.4 Distribution Of The Differene mjl(d) (α = 1) The differential distribution for the differene mjl(d) (α = 1) depends on the values of R BC and R AB. To simplify the notation, we define an antisymmetri funtion L(x, y) = L(y, x) ln ( ) nf + x(f p), (A 13) nf + y(f p) whih we heavily use in writing down the result for the differential distribution. Notie that there are various equivalent ways to write down these formulas, due to the transitivity property L(x, y) + L(y, z) = L(x, z). (A 14) If 3 R AB For mjl(d) (α = 1) one needs to onsider five separate ases: R BC < 1, then dn d = 1 f p { [ ] θ(n ) θ( ) L(0, n) + L(, n) + θ(p n ) θ( n) L(0, n) } + θ(f ) θ( (p n)) L(f, ). (A 15) If 1 R AB R BC < 3 R AB, then dn d = 1 f p { [ ] θ(p n ) θ( ) L(0, n) + L(, n) [ ] + θ(n ) θ( (p n)) L(f, ) + L(, n) + θ(f ) θ( n) L(f, ) }. (A 16) 19

220 If R AB R BC < 1 R AB, then dn d = 1 f p { [ ] θ(n p ) θ( ) L(f, ) + L(f, 0) [ ] + θ(n ) θ( (n p)) L(f, ) + L(, n) + θ(f ) θ( n) L(f, ) }. (A 17) If R AB R AB R BC < R AB, then dn d = 1 f p { [ ] θ(n p ) θ( ) L(f, ) + L(f, 0) [ ] + θ(f ) θ( (n p)) L(f, ) + L(, n) + θ(n ) θ( f ) L(, n) }. (A 18) If 0 R BC < R AB R AB, then dn d = 1 f p { [ ] θ(f ) θ( ) L(f, ) + L(f, 0) + θ(n p ) θ( f ) L(f, 0) + θ(n ) θ( (n p)) L(, n) }. (A 19) The normalization ondition now reads M 0 d ( ) dn = 1. (A 0) d As before, in Figure A-1() we ompare the predition of our analytial formula in Equations (A 15 through A 19) to the numerial result obtained earlier in Figure 3-4(d) for the LM1 study point, and we find very good agreement. 0

221 A.5 Distribution Of The Produt m jl(p) Finally, for ompleteness we also list the differential distribution for the produt variable in Equation. (3 4), for whih here we shall use the shorthand notation ρ mjl(p). To further simplify the notation, we define the funtion X ± (ρ) n ( nf ) ± f n + 4(p f )ρ (f p), (A 1) where n, f and p are defined as before in Equations (A through A 4). There are two separate ases: If R BC 0.5, the ρ distribution is made up of two branhes joining at ρ = n p (see, for example the LM1 distribution in Figure 3-6() and the LM6 distribution in Figure 3-7()) { dn dρ = ρ θ ( [ ( ) ( ) n ρ ] n p ρ) θ(ρ) ln + ln n f p X (ρ) ( ) f n + θ f p ρ θ(ρ ( ) } X+ (ρ) n p) ln. (A ) X (ρ) If R BC 0.5, there is a single branh, as illustrated by the LM1 distribution in Figure 3-6() and the LM6 distribution in Figure 3-7(): { dn dρ = ρ ( ) ( ) } n ρ n f θ( n p ρ) θ(ρ) ln + ln. (A 3) p X (ρ) In both of those ases, the normalization ondition is ρ max 0 dρ ( ) dn = 1, (A 4) dρ where ρ max is the orresponding mjl(p) endpoint defined in Equation. (3 5). Figure A-1(d) demonstrates that our analytial result of Equation. (A ) agrees well with the numerially derived mjl(p) distribution in Figure 3-6() for the LM1 study point. 1

222 APPENDIX B ANALYTICAL EXPRESSIONS FOR M (N,P,C) T,MAX ( M C, P T ) The purpose of this Appendix B is to ollet in one plae all relevant formulas for the various subsystem M T endpoints M (n,p,) T,max ( M, p T ) in the presene of initial state radiation (ISR) with arbitrary transverse momentum p T. In all ases, we will find that M (n,p,) T,max ( M, p T ) is given by two branhes: M (n,p,) T,max ( M, p T ) = F (n,p,) L ( M, p T ), if M M, F (n,p,) R ( M, p T ), if M M. (B 1) In what follows we shall list the analyti expressions for eah branh F (n,p,) L and F (n,p,) R, for all possible (n, p, ) ases with n. The grandparents X n, the parents X p and the hildren X are always assumed to be on-shell. However, any intermediate partiles X m with n > m > p or p > m > may or may not be on-shell, and the two ases will have to be treated differently. Suh an example is provided by the endpoint funtion M (n,n,n ) T,max ( M n, p T ) disussed below in Setion B.. For onveniene, our results will be written in terms of the mass parameters µ (n,p,) defined in Equation (4 9) µ (n,p,) M n ( ) 1 M Mp. (B ) These parameters represent ertain ombinations of the masses of the grandparents (M n ), parents (M p ) and hildren (M ), and do not ontain any dependene on the ISR transverse momentum p T. As we disussed in Setions 4.1 and 4., these are generally the quantities whih are diretly measured by experiment. Therefore, with the M T method, the goal of any experiment would be to perform a suffiient number of µ-parameter measurements and then from those to determine the partile masses themselves.

223 In some speial ases, namely n = p, we shall also define p T -dependent µ parameters, where the p T dependene is expliitly shown as an argument: ( ) µ (n,n,) (p T ) = µ (n,n,) pt 1 + p T. M n M n (B 3) When p T = 0, the p T -dependent parameters (B 3) simply redue to the p T -independent ones (B ): µ (n,n,) (p T = 0) = µ (n,n,). (B 4) We also remind the reader that test masses for the hildren are denoted with a tilde: M, while the true mass of any partile does not arry a tilde sign. B.1 The Subsystem Variable M (n,n,n 1) T,max ( M n 1, p T ) The orresponding expressions were already given in Equations (4 38) and (4 39) and we list them here for ompleteness: F (n,n,n 1) L ( M n 1, p T ) = [ ( = µ (n,n,n 1) (p T ) + µ (n,n,n 1)(p T ) + p T ) + M n 1 ] p T 4 1, (B 5) F (n,n,n 1) R ( M n 1, p T ) = [ ( = µ (n,n,n 1) ( p T ) + µ (n,n,n 1)( p T ) p T ) + M n 1 ] p T 4 1, (B 6) where the p T -dependent parameter µ (n,n,n 1) (p T ) was already defined in (B 3): ( ) µ (n,n,n 1) (p T ) = µ (n,n,n 1) pt 1 + p T. M n M n (B 7) As already mentioned in Setion 4.1.1, the left branh F (n,n,n 1) L orresponds to ( ) the momentum onfiguration ( ) p (1) () nt p nt p T, while the right branh F (n,n,n 1) R orresponds to p (1) () nt p nt p T. 3

224 B. The Subsystem Variable M (n,n,n ) T,max ( M n, p T ) In this ase there is an intermediate partile X n 1 between the parent X n and the hild X n (Figures 4-1). Our formulas below are written in suh a way that they an be applied both in the ase when the intermediate partile X n 1 is on shell (M n > M n 1 ) and in the ase when X n 1 is off-shell (M n 1 M n ). In both ases (off-shell or on-shell) we find that the left branh of M (n,n,n ) T,max ( M n, p T ) is given by F (n,n,n ) L ( M n, p T ) = [ µ (n,n,n ) (p T ) + ( µ (n,n,n )(p T ) + p T ) + M n ] p T 4 1, (B 8) where the p T -dependent parameter µ (n,n,n ) (p T ) was already defined in (B 3): ( ) µ (n,n,n ) (p T ) = µ (n,n,n ) pt 1 + p T. M n M n (B 9) The right branh F (n,n,n ) R mass spetrum and the size of the ISR p T : is given by three different expressions, depending on the F (n,n,n ) R ( M n, p T ) = (B 10) F (n,n,n ) L ( M n, p T ), if p T > M n Mn M n, = F (n,n,n ) R,off ( M n, p T ), if p T M n Mn M n and M n,n (p T ) M xn 1 x n,max, F (n,n,n ) R,on ( M n, p T ), if p T M n M n M n and M n,n (p T ) M xn 1 x n,max. Here M n,n (p T ) is a p T -dependent mass parameter defined as whih in the limit p T 0 redues to [ ] M n,n (p T ) Mn + p T 4 M n p T 4 1, (B 11) M n,n (p T = 0) = M n M n, (B 1) 4

225 justifying its notation. Notie that M n,n (p T ) is always well-defined, sine it is only used when the ondition p T (M n M n )/M n is satisfied and the expression under the square root in Equation (B 11) is nonnegative. The other mass parameter appearing in Equation (B 10), M xn 1 x n,max, is the familiar endpoint of the invariant mass distribution of the {x n 1, x n } SM partile pair: 1 M n 1 (M n Mn 1 )(M n 1 M n ), if M n 1 < M n, M xn 1 x n,max M n M n, if M n 1 M n. (B 13) For example, in the speial ase of n = and the intermediate partile X 1 on-shell, Equation (B 13) redues to Equation (4 81). The two expressions F (n,n,n ) R,off F (n,n,n ) R,on appearing in Equation (B 10) are given by F (n,n,n ) R,off ( M n, p T ) = F (n,n,n ) R,on ( M n, p T ) = [ M n + M n,n (p T ) + p T 4 [ Mx n 1 x n,max + pvis (p T ) + ] p T 4 1 M n + (p vis (p T ) p T where M n,n (p T ) and M xn 1 x n,max were already defined in (B 11) and (B 13), and, (B 14) ) ] p T 4 (B 15) orrespondingly. The subsripts off and on in Equations (B 14) and (B 15) an be understood as follows. When the intermediate partile X n 1 is off-shell and M n 1 M n, from Equations (B 11) and (B 13) we get 1, M n,n (p T ) = M n + M n M n M n 1 + p T 4M n (M n M n ) = M x n 1 x n,max. (B 16) Now returning to the logi of Equation (B 10), we see that in the off-shell ase at low p T one would always use the expression F (n,n,n ) R,off ( M n, p T ) defined in Equation (B 14), and never its alternative F (n,n,n ) R,on ( M n, p T ) from Equation (B 15). To put it another way, the expression F (n,n,n ) R,on ( M n, p T ) in Equation (B 15) is only relevant when the intermediate partile X n 1 is on-shell. 5

226 Finally, the quantity p vis (p T ) appearing in Equation (B 15) is a shorthand notation for the total transverse momentum of the visible partiles x n and x n 1 in eah leg: p vis p (k) nt (k) + p (n 1)T. In the ase relevant for F (n,n,n ) R,on, the value of p vis is given by p vis (p T ) (µ (n,n,n 1) + µ (n,n 1,n ) ) p T M n + µ (n,n,n 1) µ (n,n 1,n ) 1 + p T 4M n. (B 17) It is easy to hek that in the limit of p T 0 our Equations (B 8) and (B 10) redue to the known results for the ase of no ISR (Equations (70) and (74) in W. S. Cho et al. [ Measuring superpartile masses at hadron ollider using the transverse mass kink, JHEP 080, 035 (008)] [37]). The left branh F (n,n,n ) L onfiguration while the right branh F (n,n,n ) R In the latter ase, F (n,n,n ) R,off in Equation (B 8) orresponds to the momentum ( ) p (k) (k) nt + p (n 1)T p T, in Equation (B 10) orresponds to ( ) p (k) (k) nt + p (n 1)T p T. is obtained when X n is at rest: P (k) (n )T orresponds to the ase when P (k) (n )T = 1 p T p vis (p T ). = 0, while F (n,n,n ) R,on 6

227 B.3 The Subsystem Variable M (n,n 1,n ) T,max ( M n, p T ) Here we generalize our p T = 0 result of Equation (4 56) from Setion to the ase of arbitrary ISR p T : F (n,n 1,n ) L ( M n, p T ) = ( = µ (n 1,n 1,n ) (ˆp T ) + µ (n 1,n 1,n ) (ˆp T ) + ˆp ) T + M n ˆp T 4 1, (B 18) F (n,n 1,n ) R ( M n, p T ) = ( = µ (n 1,n 1,n ) ( ˆp T ) + µ (n 1,n 1,n ) ( ˆp T ) ˆp ) T + M n ˆp T 4 1 (B 19), where we have introdued the shorthand notation ˆp T p T + µ (n,n,n 1) (p T ). (B 0) Notie that the seond term on the right-hand side ontains the p T -dependent µ parameter defined in Equation (B 7). The left branh F (n,n 1,n ) L onfiguration while the right branh F (n,n 1,n ) R in Equation (B 18) orresponds to the momentum ( ) p (k) (n 1)T p (k) nt p T in Equation (B 19) orresponds to ( ) p (k) (n 1)T p (k) nt p T,. It is worth heking that our general p T -dependent results in Equations (B 18) and (B 19) redue to our previous formulas in Equations (4 57) and (4 58) in the p T 0 limit and in the speial ase of n =. First taking the limit p T 0 from Equations (B 0) and (B 7) we get lim ˆp T = µ (n,n,n 1), p T 0 (B 1) 7

228 lim µ (n 1,n 1,n )( pˆ T ) = µ (n 1,n 1,n ) (µ (n,n,n 1) ) = µ (n,n,n ) µ (n,n,n 1), p T 0 lim µ (n 1,n 1,n )( pˆ T ) = µ (n 1,n 1,n ) ( µ (n,n,n 1) ) = µ (n,n 1,n ). p T 0 (B ) (B 3) Substituting Equations (B 1 through B 3) into Equations (B 18) and (B 19), we get F (n,n 1,n ) L ( M n, p T = 0) = = { [ µ (n,n,n ) µ (n,n,n 1) + ] µ (n,n,n ) + M n µ (n,n,n 1) } 1, (B 4) F (n,n 1,n ) R ( M n, p T = 0) = = { [ µ (n,n 1,n ) + (µ(n,n,n 1) µ (n,n 1,n ) ) + M n ] µ (n,n,n 1) } 1, (B 5) whih are nothing but the generalizations of Equations (4 57) and (4 58) for arbitrary n. 8

229 APPENDIX C THE SYMMETRIC M T IN THE LIMIT OF INFINITE P UTM In this Appendix C, we revisit our previous two examples from Setions and 6.4., this time onsidering the infinitely large P UTM limit [40]. While this situation is impossible to ahieve in a real experiment, its advantage is that it an be treated by analytial means. In the P UTM limit, the deoupling argument holds [40], and one finds the following analytial expression for the M T endpoint as a funtion of the two test hildren masses M (a) M T (max) ( M (a), M (b), ) = and M (b) : Mp (M (a) ) + ( M (a) ), if ( M (a), M (b) ) R 1, M p (M (b) M (b) M M (b) p, ) + ( M (b) ), if ( M (a), M (b) ) R, if ( M (a), M (b) ) R 3, (C 1) M (a) M M (a) p, if ( M (a), M (b) ) R 4, where the four defining regions R i, (i = 1,..., 4) are shown in Figure C-1 and are defined as follows: R 1 : M (b) R : R 3 : M (b) R 4 : M (a) < (M (b) (M (b) ) (M (a) ) + ( M (a) ) M (a) ) (M (a) ) + ( M (a) ) < M (b) < M (b) < M (a) M (a) < M (b) < ( ( M (a) M (b) M (b) M (a) ) ) < M (a), (C ) < M (b), (C 3) M (b), (C 4) M (a). (C 5) Sine the funtional expression for M T (max) within eah region R i is different, there is in general a gradient disontinuity when rossing from one region into the next. Therefore, the ridges on the M T (max) hypersurfae will appear along the ommon boundaries of the four regions R i. Let us denote by L ij the boundary between regions R i and R j. 9

230 Figure C-1. The parameter plane of test hildren masses squared, divided into the four different regions R i used to define the M T endpoint funtion (C 1). Their ommon boundaries L ij are parametrially defined in Equations (C 6-C 9). The blak dot orresponds to the true values of the hildren masses. As indiated in Figure C-1, eah L ij is a straight line in the parameter spae of the hildren test masses squared and is given by L 1 : ( M (b) L 3 : M (b) L 34 : M (b) L 14 : M (a) ) = (M (b) ) (M (a) ) + ( M (a) ), M (a) M (a) ; (C 6) = M (b), M (a) M (a) ; (C 7) = M(b) M (a) M (a), M (a) M (a) ; (C 8) = M (a), M (b) M (b). (C 9) As seen in Figure C-1, all four lines L ij meet at the true hildren mass point M (a) M (b) = M (a), = M (b), where in turn the M T endpoint M T (max) gives the true parent mass M p, in aordane with Equation (6 43). With those preliminaries, we are now in a position to revisit our two examples from Setions and Figures C- and C-3 are the orresponding analogues of Figures 6-6 and 6-9 in the ase of infinite P UTM. Comparing with our earlier results, we notie both quantitative and qualitative hanges in the ridge struture. First, the smooth 30

231 ridge in Figure 6-6(b) (Figure 6-9(b)) has now been deformed into two straight line segments, one horizontal (L 3 ) and the other vertial (L 14 ), whih meet at an angle of 90 preisely at the true values of the hildren masses. More importantly, Figures C- and C-3 now exhibit another pair of ridges L 1 and L 34 (plotted in red in Figures C-(b) and C-3(b)), whih were absent from the earlier figures in Setion 6.4. The system of four ridges seen in Figures C-(a) and C-3(a) is very similar to the rease struture observed in A. J. Barr et al. [ Transverse masses and kinemati onstraints: from the boundary to the rease, JHEP 0911, 096 (009)] [40]. We thus onfirm the result of Referene. [40] that in the infinite P UTM limit there exist four different ridges, whose ommon intersetion point reveals the true masses of the parent and hildren partiles. At this point it is instrutive to ontrast the two sets of ridgelines: L 3 and L 14 (shown in Figures C-(b) and C-3(b) in blak) versus L 1 and L 34 (shown in Figures C-(b) and C-3(b) in red). The boundaries L 3 and L 14 separate the union of regions R 1 and R from the union of regions R 3 and R 4. Along those boundaries, we observe a transition in the onfiguration of visible momenta whih yields the maximum possible value of M T. More preisely, in regions R 1 and R we find that the visible momenta p (λ) T for M T (max) are parallel to the diretion of the upstream momentum P UTM, while in regions R 3 and R 4 we find that p (λ) T are anti-parallel to P UTM. This fat remains true even at finite values of P UTM, whih is why the ridgelines L 3 and L 14 ould also be seen in the earlier plots from Setion 6.4 at finite P UTM = 1 TeV. On the other hand, the ridgelines L 1 and L 34 shown in red in Figures C-(b) and C-3(b) are due to the deoupling argument [40], whih is stritly valid only in the infinite P UTM limit. This is why these ridges beome apparent only at very large values of P UTM, and are gradually smeared out at smaller P UTM. 31

232 Figure C-. M T (max) for the event topology of Figure 6-3(a) with fixed upstream momentum of P UTM =. Figure C-3. M T (max) for the event topology of Figure 6-3(a) with the symmetri mass spetrum II from Table 6-1 with upstream momentum P UTM. 3