MODEL-INDEPENDENT MASS AND SPIN DETERMINATION FOR A SEQUENTIALDECAY WITH A JET AND TWO LEPTONS

By

MICHAEL E. BURNS

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2009

1

c© 2009 Michael E. Burns

2

“I pledge allegiance to effective field theory, and to the renormalization for which itstands; one theory under GUT with running parameters for all.”

3

ACKNOWLEDGMENTS

I owe tremendous gratitude to Kyoungchul Kong, a former student of my advisor,

and Myeonghun Park, a fellow student, for countless calculations, simulations, helpful

discussions, and help with mundane type-setting. I am also grateful for having such

a wonderful office-mate and companion, Sung-Soo Kim, whose fresh perspectives and

enthusiasm reminded me everyday that I was actually still interested in physics. I thank

Darlene Latimer, Nathan Williams, and Pam Marlin for their patience and help taking

care of the administrative B.S. (and I don’t mean Bachelor of Science). Of course, I

would never have completed this dissertation without the enduring support of my advisor,

Konstantin Matchev, who somehow managed to put up with me for 4 years.

4

TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.1 The Basic Experimental Situation . . . . . . . . . . . . . . . . . . . . . . . 111.2 Intermediate Resonances of the Standard Model . . . . . . . . . . . . . . . 121.3 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.4 Experimental Challenges Pertaining to our Analysis . . . . . . . . . . . . . 21

2 MASS DETERMINATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.1 The Method of Kinematic Endpoints . . . . . . . . . . . . . . . . . . . . . 232.1.1 Forward Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.1.2 Inversion Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 Duplication Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3 Kinematic Boundary Lines for the (m2

jl(lo),m2jl(hi)) Distribution . . . . . . . 39

2.4 Kinematic Boundary Lines for the (m2ll,m

2jll) Distribution . . . . . . . . . . 49

3 SPIN DETERMINATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.1 Classification of Helicity Combinations . . . . . . . . . . . . . . . . . . . . 613.1.1 Helicity Basis Functions FIJ . . . . . . . . . . . . . . . . . . . . . . 633.1.2 Helicity Coefficients KIJ . . . . . . . . . . . . . . . . . . . . . . . . 66

3.2 Observable Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.2.1 Invariant Mass Distributions in the Helicity Basis {FIJ} . . . . . . . 683.2.2 Invariant Mass Distributions in the Observable Basis {Fα,Fβ,Fγ,Fδ} 71

3.3 Determination of Model Parameters {α, β, γ} . . . . . . . . . . . . . . . . 753.4 Twin Spin Scenarios FSFS/FSFV and FVFS/FVFV . . . . . . . . . . . . 76

4 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.1 Kinematic Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.2 Spin Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.3 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

APPENDIX

A A BSM EXAMPLE: SUSY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5

B EXAMPLE INVERSION FORMULAS FOR Njl = 1, 2, 3 . . . . . . . . . . . . . 87

C HELICITY BASIS FUNCTIONS {FIJ} . . . . . . . . . . . . . . . . . . . . . . 89

D OBSERVABLE SPIN BASIS FUNCTIONS {Fα,Fβ,Fγ,Fδ} . . . . . . . . . . . 95

E SIMPLE SPIN FITTING PROCEDURE . . . . . . . . . . . . . . . . . . . . . . 100

F SPIN DETERMINATION EXAMPLES . . . . . . . . . . . . . . . . . . . . . . 103

F.1 The SPS1a Study Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103F.2 Input from SFSF (S = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107F.3 Input from FSFS (S = 2) and FSFV (S = 3) . . . . . . . . . . . . . . . . . 109F.4 Input from FVFS (S = 4) and FVFV (S = 5) . . . . . . . . . . . . . . . . 111F.5 Input from SFVF (S = 6) . . . . . . . . . . . . . . . . . . . . . . . . . . . 113F.6 Remarks on Spin Determination at the Tevatron . . . . . . . . . . . . . . . 114

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6

LIST OF TABLES

Table page

1-1 Some representative comparisons of our notation, Rij. . . . . . . . . . . . . . . . 17

1-2 The six spin assignments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2-1 Examples of mass ambiguities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2-2 Two examples of exact duplication. . . . . . . . . . . . . . . . . . . . . . . . . . 37

3-1 Shorthand notation for the six different spin assignments. . . . . . . . . . . . . . 57

3-2 Classification of model parameters according to their contribution to KIJ . . . . 64

3-3 Available measurements of the model parameters α, β and γ. . . . . . . . . . . 77

4-1 Expected outcomes from our spin discrimination analysis. . . . . . . . . . . . . 82

C-1 Helicity basis functions for the dilepton. . . . . . . . . . . . . . . . . . . . . . . 94

C-2 Helicity basis functions for j`n. . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

D-1 Observable basis functions for the dilepton. . . . . . . . . . . . . . . . . . . . . 99

D-2 Observable basis functions for j`n. . . . . . . . . . . . . . . . . . . . . . . . . . 99

7

LIST OF FIGURES

Figure page

1-1 Example decays in the SM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1-2 Four basic decay scenarios of the Higgs boson. . . . . . . . . . . . . . . . . . . . 14

1-3 We considered these two “model-independent” decay diagrams. . . . . . . . . . 15

1-4 A representation of the 11 Regions of input mass parameter space. . . . . . . . . 18

1-5 Decision tree for choosing the spin assignments. . . . . . . . . . . . . . . . . . . 21

2-1 An artificial single-variable histogram to demonstrate kinematic endpoints. . . . 26

2-2 The maps T13 : R1 7−→ R3 and T23 : R2 7−→ R3. . . . . . . . . . . . . . . . . . . 32

2-3 The map T21 : R2 7−→ R1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2-4 The minimum value RminCD (RBC , RAB) required for duplication. . . . . . . . . . . 36

2-5 The generic shape ONPF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2-6 Obtaining the shape of the (m2jl(lo),m

2jl(hi)) bivariate distribution by folding. . . . 43

2-7 The generic shape of the bivariate distribution (m2jl(lo),m

2jl(hi)) . . . . . . . . . . 45

2-8 Scatter plots of (m2jl(lo),m

2jl(hi)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2-9 The generic shape OV US. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2-10 Scatter plots of (m2ll,m

2jll). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3-1 The 8 different helicity combinations. . . . . . . . . . . . . . . . . . . . . . . . . 63

3-2 A contour plot of cos ϕ̃c as a function of cosϕc and f . . . . . . . . . . . . . . . 69

A-1 An example of Fig. 1-3A from mSUGRA. . . . . . . . . . . . . . . . . . . . . . 86

E-1 Contour plots of χ2(α, β, γ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

F-1 Dilepton invariant mass distributions, L+−S . . . . . . . . . . . . . . . . . . . . . 104

F-2 Sum of j`+ and j`− invariant mass distributions, S+−. . . . . . . . . . . . . . . 105

F-3 Difference of j`+ and j`− invariant mass distributions, D+−. . . . . . . . . . . . 106

F-4 Decay diagrams for FSFS and FSFV. . . . . . . . . . . . . . . . . . . . . . . . . 110

F-5 Decay diagrams for FVFS and FVFV. . . . . . . . . . . . . . . . . . . . . . . . 112

F-6 Decay diagram for SFVF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

8

Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

MODEL-INDEPENDENT MASS AND SPIN DETERMINATION FOR A SEQUENTIALDECAY WITH A JET AND TWO LEPTONS

By

Michael E. Burns

August 2009

Chair: Konstantin MatchevMajor: Physics

We reconsidered existing techniques for the determination of the masses and spins of

the four narrow resonances, A, B, C, and D, that might occur in a sequential decay, D →

jC → j`±B → j`±`∓A, where j is a quark jet, `±`∓ is a charged lepton (electron or muon)

pair, A is undetectable, and the energy and longitudinal momentum of D are unknown.

We found that these existing techniques rely on new physics model assumptions beyond

the mere existence of the decay chain. We propose model-independent generalizations in

order to improve these mass and spin determination techniques.

We identified a two-fold ambiguity in the method of kinematic endpoints for

mass determination that uses extreme kinematic values of m2ll, m2

jl(lo), m2jl(hi), and

m2jll as observables, namely m2max

ll , m2 maxjl(lo) , m2max

jl(hi) , m2 maxjll , and m2 min

jll(θ>π/2). While

similar ambiguities have already been recognized, these ambiguities have not been

well-understood, and the existing resolutions rely on model-dependent features of the

single-variable invariant mass distributions, such as their shapes. In order to obtain a

model-independent resolution to the two-fold ambiguity, we generalized from kinematic

endpoints of single-variable distributions to kinematic boundary lines of double-variable

distributions. In particular, we demonstrated a resolution of the ambiguity by using the

double-variable invariant mass distributions in (m2ll,m

2jll) and (m2

jl(lo),m2jl(hi)). Furthermore,

our technique automatically determines if mB > mC .

9

For spin determination, we reverted back to the single-variable invariant mass

distributions in m2ll and m2

jl±. Existing spin determination techniques that use these

distributions assume fixed chiral projections for each decay vertex, where these chiral

projections are determined by the assumed new physics model. These fixed chiral

projections can artificially enhance the distinction between two different spin assignments.

In order to isolate the spin determination, we derived basis functions that allow the

spin-dependence of the invariant mass distributions to be separated from the dependence

on the chiral projections of the vertices. Then, we demonstrated our spin determination

technique by fitting simulated invariant mass distributions to our basis functions with

floating chiral projection coefficients. While the floating chiral projections can weaken the

distinction between two different spin assignments, our new basis functions clearly show

that an intermediate vector particle will always result in a characteristic deviation from

pure phase space, regardless of the chiral projections. A fortunate byproduct of our spin

determination technique is a set of constraints on the chiral projection coefficients, as a

result of the fitting procedure, which may further distinguish a particular model of new

physics beyond the distinction of the spin assignments alone.

10

CHAPTER 1INTRODUCTION

1.1 The Basic Experimental Situation

Particle physicists expect that the Higgs field, along with associated phenomena

beyond the Standard Model (BSM), should exhibit invariant mass resonances in the

range from about 100 GeV up to a few TeV [1, 2]. The lower end of this mass range is

just within reach of the Tevatron at Fermilab, and they continue to search for evidence

of the Higgs boson [1, 3, 4]. The former Large Electron-Positron Collider (LEP) at The

European Organization for Nuclear Research (CERN) also probed the lower end of this

mass range [3, 5], and the newly commissioned Large Hadron Collider (LHC) at CERN

is designed to probe even higher masses than Tevatron or LEP [3]. There has been some

collider-based observation of the decay of every unstable fundamental particle in the

Standard Model of particle physics (SM), except the Higgs boson. So, a major goal of the

LHC is, either to produce Higgs bosons and observe their decays, or to strongly rule out

the Higgs mechanism. Both of these scenarios are exciting. Many particle physicists also

want to see experimental evidence for an extension of the SM, even if a Higgs boson is

observed [1, 6]. So, the great expectation is that a signal will be discovered in the LHC

data that cannot be explained by any particle or interaction that has ever been observed.

However, this signal must still be interpreted, which presents several challenges.

In order to understand the data that are obtained from a particle collider, the data

must be analyzed and interpreted on various levels. The raw data are simply the physical

excitations of the particle detector, which is a complex and carefully planned arrangement

of materials. These raw data are filtered, quite nontrivially, into a numerical format, which

is then interpreted into reconstructed collider objects. The possible reconstructed objects

for a hadron collider such as the LHC are [7]:

• photon

• lepton: either electron or muon

11

• jet: either untagged, b-tagged, or tau

• missing transverse momentum

Some model is assumed at each level of data interpretation. Detector excitations

are modeled in terms of the electrostatic and nuclear forces that bind matter [7, 8]. The

reconstruction of collider objects is modeled in terms of the set of electromagnetic and

nuclear properties of each particle that could be produced in a collision. The possibility

and frequency of combinations of species and physical properties (i.e. the “signatures”) of

these reconstructed objects are modeled in terms of the underlying physics. We focused on

a specific signature, namely a jet and two leptons, taking all other levels of interpretation

for granted. The actual appearance of any specific signature depends on the underlying

physics, and our analysis applies only to data in which the signature appears.

1.2 Intermediate Resonances of the Standard Model

We approximated the reconstructed electrons, muons, and jets as massless. In this

context the only essentially massive particles of the SM are [9]:

• the weak bosons, Z and W

• the top quark, t

• the Higgs boson, h

So, we approximate that a SM collision product can be, either an essentially massless

particle that does not decay into any distinct reconstructed objects, or one of the massive

objects listed above.

The two lightest particles in the list that decay to distinct reconstructed objects are

the weak bosons, both of which undergo a simple two-body decay (Fig. 1-1A&B). The

top quark undergoes a two-body decay to a W boson with a bottom quark, and then

the W boson decays to a two-particle state (Fig. 1-1C). This is an example of what we

call a “sequential decay”. A sequential decay is a decay that produces an intermediate

on-shell resonance (a resonance that is kinematically allowed to be on its mass shell),

which then itself decays, either to a lighter intermediate on-shell resonance, or generically

12

W±

f

f ′

Z0

f

f̄

f

f ′

b

t

W

A B C

Figure 1-1. Example decays in the SM. In all three diagrams, {f, f ′} can be, either alepton and neutrino, or a pair of quarks, and {f, f̄} is a genericfermion-antifermion pair. A) and B) depict the generic SM decays of the weakbosons, which are both kinematically simple two-body decays. C) depicts theSM decay of the top quark, which is a two-stage sequential decay.

to some final state. In the case of the top quark decay, there is only one intermediate

on-shell resonance, the W boson, so the top quark exhibits a two-stage sequential decay.

In general, a sequential decay has two or more stages, one more than the number of

intermediate on-shell resonances.

The relevant decay scheme of the Higgs boson depends on its mass.1

A) If the mass of the Higgs boson is at the lower end of its allowed range (around120 GeV), then it primarily undergoes a two-body decay to a bottom-flavoredquark-antiquark pair (Fig. 1-2A) [3].

B) If the mass is in its lower intermediate range (less than about 160 GeV), then itprimarily undergoes a three-body decay to a state containing a single intermediateweak boson (Fig. 1-2B) [3]. This provides another SM example of a two-stagesequential decay, in which the first stage is a three-body decay, and the last stage isa two-body decay.

C) If the mass is in its upper intermediate range (between about 160 GeV and 350GeV), then it primarily undergoes a two-body decay to a weak boson pair [3], eachof which undergoes a two-body decay (Fig. 1-2C). This scenario does not producea sequential decay, because the first stage of the decay produces two intermediateon-shell resonances instead of only one.

1 At the time of this research, the mass of the Higgs boson is unknown, but it isexpected to be around 120 GeV [3].

13

A

h

b

b̄

B

h

f ′′′

f ′′

V

f ′

f

D

h

t

t̄

b

W+

f

f ′

b̄

W−f ′′

f ′′′ C

h

V

V̄

f

f ′

f ′′

f ′′′

Figure 1-2. Four basic decay scenarios of the Higgs boson, depending on its mass [3, 10].A) is perhaps the most expected scenario, since the Higgs boson is expected tobe in the light range. B) is the sequential decay scenario for the Higgs boson,producing a four-body final state, and it is dominant in the lower intermediatemass range. C) is simply related to B) by replacing the off-shell V̄ in B) (notshown) with an on-shell V̄ in C). V can be any of {W±, Z0}, but then V̄ mustbe the corresponding {W∓, Z0}, respectively. D) provides a significant decaychannel in the heavy mass range.

D) If the mass is in its high range (around 500 GeV), then the upper-intermediatedecay scheme (Fig. 1-2C) remains dominant, but a two-body decay to a topquark-antiquark pair (Fig. 1-2D) also contributes significantly to the decay [3],which has a distinct signature. However, this scenario does not result in a sequentialdecay of the Higgs boson, again because the first stage produces two intermediateon-shell resonances.

So, in the SM, there are possibly two types of sequential decay, both of which are

two-stage decays. This implies that a three-stage sequential decay is evidence for

underlying physics BSM.

1.3 Definitions

A given signature has advantages and disadvantages. For example, the choice of

including jets and leptons in a signature demonstrates this trade-off. We assume that a

14

D C B A

j `± `∓

D CA

j `±`∓

A B

Figure 1-3. We considered these two “model-independent” decay diagrams. A, B, C andD represent some heavy resonances with unknown masses and spins. Decayscenario B) is an alternative to decay scenario A) in which mB > mC . Weapplied our analysis to the jet (j), “near” lepton (l±n ), and “far” lepton (l∓f ),and we allowed the vertices to have arbitrary couplings to the differentchiralities of these three final-state particles. We restricted our spin analysis toA) with a spin-1/2 jet.

2-to-2 subprocess of a hadronic collision is most likely to contain QCD-charged objects

in the final state. So a signature that includes jets can provide higher luminosity (i.e.

more statistics) compared to a purely leptonic signature. However, due to the composite

structure of hadrons and the relative strength of the QCD coupling, a significant amount

of hadronic background radiation is expected in a hadron collision, making the separation

between background jets and signature jets ambiguous. Also, the kinematics of the

underlying parton (i.e. quark) that we would like to use in our analysis may be quite

different from the kinematics of the jet that we associate with that parton. So, purely

leptonic signatures are typically more reliable and easier to interpret than signatures

that include jets. We compromised by assuming that the QCD-charge from the hadronic

collision results in a single initial hadronic jet and a QCD-neutral intermediate resonance.

Then, we selected events in which the QCD-neutral state decays to two leptons and some

other QCD-neutral particle that we ignored (e.g. a particle of dark matter, DM). So, in

terms of Fig. 1-3, D is QCD-charged, and C, B, and A are all QCD-neutral.

We imagined the existence of a sequential decay of some heavy particle, D, that

produces a single massless jet and two massless leptons (Fig. 1-3). We considered a jet

from the first decay in the decay chain because we expect that QCD hard processes are

15

dominant at the LHC, so that D carries color. We considered the subsequent decay of

C to two opposite-sign-same-flavor (OSSF) leptons in order to cope with background.

Indeed, our restriction to OSSF was a model assumption. However, this plausible choice

was made only for concreteness of the discussion and does not represent a fundamental

limitation of our methods; the basic idea applies to the general case where the visible

particles are any 3 SM fermions. Of course, the occurrence of such a decay process is not

guaranteed, but we believe that it provides a good combination of simplicity, plausibility,

and distinctiveness.

In general, we limited our analysis to the two independent mass hierarchies shown in

Inequalities 1–1.

mD > mC > mA and mB > mA (1–1)

We always assumed positive masses, and this assumption is implicit in all of our

expressions. In these hierarchies, A is always the lightest resonance in the chain.

Kinematically, these mass hierarchies always allow the decay from D to C, and the

subsequent decay from C to A; however, the intermediate decay from C to B is not

necessarily allowed. We refer to an occurrence of the cascade decay in which the decay

from C to B does occur as a “three-stage” or “on-shell” scenario (Fig. 1-3A). We refer

to an occurrence of the cascade decay in which the decay from C to B does not occur

as a “two-stage” or “off-shell” scenario (Fig. 1-3B). The reference to the “shell” refers to

the mass shell of B: we assumed that mB < mC would allow only a three-stage scenario

(in which B is on-shell) and that mB > mC would allow only a two-stage scenario (in

which B is off-shell). While we recognize that it is more meaningful to reserve the names

“three-stage” and “two-stage” to refer to the different kinds of decay topology and to

reserve the names “on-shell” and “off-shell” to refer to the different kinds of mass spec-

trum, we used the names for both the topology and the spectrum interchangeably, and

we made no attempt at a consistent usage. Indeed, while it is physically meaningful to

consider an “off-shell” spectrum that exhibits a cascade decay with the “three-stage”

16

Table 1-1. Some representative comparisons of our notation, Rij, with two othernotations for squared mass ratios that appear in the literature [11, 12]. We usedthe Rij notation for the mass determination, and we used the x,y,z notation forthe spin analysis.

basicexpression

Rij Ref. [11] Ref. [12]

m2A/m2

B RAB RA zm2

B/m2C RBC RB y

m2C/m2

D RCD RC xmA/mC

√RAC

√RARB

√yz

mB/mD

√RBD

√RBRC

√xy

mA/mD

√RAD

√RARBRC

√xyz

topology (i.e. there exists a Feynman diagram with an off-shell intermediate particle), we

ignored such cases in order to achieve a more model-independent analysis.

We found a convenient mass parametrization (Equation 1–2) for the mass determination

(Chapter 2) part of our analysis, which is essentially a generalization of the notation used

in [11].

Rij ≡m2

i

m2j

where i, j ∈ {A, B, C, D} (1–2)

We adopted a simpler alternative notation (Equation 1–3) for our spin determination

(Chapter 3), as used in [12].

x ≡ m2C

m2D

y ≡ m2B

m2C

z ≡ m2A

m2B

(1–3)

Table 1-1 shows the comparison of our notation to that of [11, 12].

For our mass determination analysis (Chapter 2), we defined 11 Regions of the

input mass parameter space by extending the definitions in [11] to include the off-shell

scenario. A slice through parameter space at a constant mC and mD (Fig. 1-4) shows

these Regions in terms of RAB and RBC (or, equivalently, a slice at constant x shows these

Regions in terms of y and z). The borders of these Regions depend on mC and mD only

through RCD. We name these Regions with an ordered pair of whole numbers, (Njll, Njl),

whose meaning is based on the method of kinematic endpoints. Similarly to Ref. [11],

we assigned these numbers arbitrarily according to algebraic definitions (Equations

17

Figure 1-4. A representation of the 11 Regions of input mass parameter space, (Njll,Njl),at a fixed value of RCD = 0.3. These Regions represent the conditions onEquations 2–7 through 2–11, and they are color-coded to match Figs. 2-2, 2-3,2-5, 2-7, 2-8, and 2-10. The shapes of the four colored Regions, Njl, areindependent of RCD.

1–4 and 1–5), which partition the mass hierarchy (Equation 1–1) into many different

sub-hierarchies. The general hierarchy (Equation 1–1) is implied in these definitions

(Equations 1–4 and 1–5).

Njll =

1 ,√

mAmD > mC > mB

2 , mC

√

mA

mD> mB

3 , mC > mB >√

mAmD

4 , mC >√

mAmD > mB > mC

√

mA

mD

5 , mB > mC > mA >m2

C

mD

6 , mB > mC >√

mAmD

(1–4)

Njl =

1 , m2B − m2

A > m2C − m2

B > 0

2 , m2B − m2

A >m2

A

m2B

(m2C − m2

B) >m2

A

m2B

(m2B − m2

A)

3 ,√

mAmC > mB

4 , mB > mC

(1–5)

18

In terms of the squared mass ratios (Equation 1–2), these conditions can be written as

Equations 1–6 and 1–7.

Njll =

1 , RAB > RAC > RCD

2 ,√

RAD > RBC

3 , RCD > RBD > RAB

4 , RBC >√

RAD ∩ RAB > RBD ∩ RCD > RAC

5 , RBC > 1 > RAC > RCD

6 , RBC > 1 > RCD > RAC

(1–6)

Njl =

1 , 1 > RBC > (2 − RAB)−1 > 0

2 , 1 > (2 − RAB)−1 > RBC > RAB

3 , 1 > RAB > RBC

4 , RBC > 1 > RAC

(1–7)

The on-shell scenario occurs if RBC < 1, and this occurs in the Njl = 1, 2, 3 Regions (the

green, magenta, and cyan Regions in Fig. 1-4, respectively), which are collectively the

same as the Njll = 1, 2, 3, 4 Regions, and these are the Regions considered in [11]. The

off-shell scenario occurs if RBC > 1, and this occurs in the Njl = 4 Region (the yellow

Region in Fig. 1-4), which is collectively the same as the Njll = 5, 6 Regions [13].

Restricting the spin analysis to the three-stage scenario (Fig. 1-3A), we identified the

general form of the two types of BSM vertex functions, shown in Equations 1–8 and 1–9.

F

S

f

= gLPL + gRPR (1–8)

F

V

f

= γµ (gLPL + gRPR) (1–9)

f represents a massless SM fermion (jet or lepton), F represents a massive fermion, S

represents a massive scalar boson, and V represents a massive vector boson. PL and PR

19

Table 1-2. The six spin assignments of the heavy particles D, C, B and A in the decaychain of Fig. 1-3A. The last column gives one typical SUSY or UED example.

Spins D C B A Example

SFSF Scalar Fermion Scalar Fermion q̃ → χ̃02 → ˜̀→ χ̃0

1

SFVF Scalar Fermion Vector Fermion ?FSFS Fermion Scalar Fermion Scalar q1 → ZH → `1 → γH

FSFV Fermion Scalar Fermion Vector q1 → ZH → `1 → γ1

FVFS Fermion Vector Fermion Scalar q1 → Z1 → `1 → γH

FVFV Fermion Vector Fermion Vector q1 → Z1 → `1 → γ1

represent projection operators for left-handed and right-handed fermions, respectively,

and they act on the SM fermion wave-function to the right. For example, if f is a Dirac

fermion, then PL = 12(1 − γ5). The vertex functions may be Hermitian-conjugated,

depending on the matrix element of the physical process. gL and gR represent the details

of the vertex factor, and so these “coupling constants” essentially parametrize our

ignorance of the dynamics of the particular physics model. Specifically, we name g → c for

the decay vertex from D to C, g → b for the decay vertex from C to B, and g → a for the

decay vertex from B to A.

Since the SM particles in Fig. 1-3A are all spin-1/2 fermions, the particles A, B, C

and D must alternate between fermions (F ) and bosons (S or V ) in order to maintain

Lorentz invariance. Table 1-2 lists by name the 6 spin assignments that we studied for our

spin determination technique, which were also considered in [14, 15]. Figure 1-5 shows a

decision tree representing the 6 spin assignments that we considered. Our setup follows

closely the conventions of Refs. [12, 14–18].

So, our starting point was to essentially parametrize our ignorance of the matrix

element of the decay chain in terms of four masses, six (generally complex-valued)

coupling constants, and the various ways to assign spins to the heavy resonances. We

emphasize that these parameters represent effective parameters; they do not necessarily

represent parameters that would appear in some BSM Lagrangian.

20

?

D: S F

C: F S V

B: S V F F

A: F F S V S V

Figure 1-5. Decision tree for choosing the spin assignments of particles D, C, B, and A,as labeled in Table 1-2. S implies spin-0, F implies spin-1/2, and V impliesspin-1. The possibilities are determined by the sum rules for angularmomentum, assuming that each of the three final-state SM particles arespin-1/2, and restricting the spin to the three possibilities: S, F , and V . Wedid not consider the possibility of a vector D, which would have added twomore spin assignments, VFSF and VFVF, to Tab. 1-2.

1.4 Experimental Challenges Pertaining to our Analysis

There are a number of experimental challenges to uniquely identify the particles

coming from the cascade of Fig. 1-3.

E1 Jet Combinatorics. Collision events at a hadron collider are expected to contain anumber of “extra” jets. Some of these jets may come from initial state radiation,others may originate from the opposite cascade in the same event, and there mayalso be jets appearing from the decays of heavier particles into particle D. Thisposes a combinatorics problem: which one of the many jets that are observed in theevent corresponds to the jet in Fig. 1-3? The jet may be selected as one of the twohardest jets in the event if D and C have a large mass splitting, but unfortunatelythe jet in question must also be used in order to determine if this is indeed the case.Fortunately, there does exist a technique to address this problem: the mixed eventtechnique. The mixed event technique statistically removes the wrong jets [19] fromthe data. Ref. [20] successfully applied the mixed event technique to demonstrateSUSY mass measurement for the SPS1a study point. A subtraction by a mixedevent technique is particularly well-suited for our purposes, since our analysis relieson only the cumulative distributions, so that the the identification of the correct jetin a single event is not critical.

21

E2 Lepton combinatorics. There may be additional isolated leptons in the event, soone might consider selecting events exclusively with exactly two leptons. However,even then, it is not guaranteed that those two leptons are coming from the samedecay chain. Fortunately, there again exists a technique to address this problem:opposite lepton flavor subtraction (OLFS). This is again a statistical technique,similar in idea to the mixed event technique. For our signature, OLFS solves thelepton combinatorics problem [19] by forming the linear combination of {e+e−} +{µ+µ−} − {e+µ−} − {µ+e−}, in which the effects of the uncorrelated leptons in thesignal (including SM backgrounds involving top quarks, b-jets, W bosons, and taujets, and even uncorrelated BSM backgrounds) cancel out.

E3 Quark-antiquark jet ambiguity. Assuming that A is neutral, the charge of thejet in Fig. 1-3 is fixed by the charge of D. However, since the charge of a jet isdifficult to determine2 , we assumed that we could not determine the charge of D.So our invariant mass distributions actually represent the sum of the individualcontributions from both D’s and anti-D’s.

We assumed that the jet and lepton combinatorics have been appropriately resolved.

We incorporated the “quark-antiquark” ambiguity into our analysis, showing that the

distinction is unnecessary for mass determination and can also be accounted as an

additional model parameter for spin determination.

2 If q is a heavy flavor, then the distinction can be made (statistically). We ignored thispossibility in order to demonstrate that our method works even in the worst case scenarioof jet-charge ambiguity.

22

CHAPTER 2MASS DETERMINATION

2.1 The Method of Kinematic Endpoints

With the imminence of data from the LHC there have been numerous developments

of mass determination techniques [11,13,19,21–61]. Our contribution to mass determination

applies to the two different scenarios shown in Fig. 1-3. Our goal was to improve the

method of kinematic endpoints to determine the unknown masses, either {mD, mC , mB, mA}

in the on-shell scenario, or {mD, mC , mA} in the off-shell scenario. It may be possible to

determine mB even in the off-shell scenario (Fig. 1-3B) [33]. However, a number of

influences, such as the spins of the heavy resonances and the backgrounds, plague this

determination, and we preferred our analysis to be insensitive to these influences by

relying on only absolute kinematic effects.

We applied the method of kinematic endpoints to various representative spectra,

using the invariant masses of the four different combinations of the jet and two leptons

{mll, mjll, mjln, mjlf}. We assumed that two of the invariant masses, mll and mjll, are

unambiguously observable, but the other two combinations are ambiguous because the

decay vertices corresponding to l±n and l∓f cannot be determined on an event-by-event

basis. One possibility is to distinguish the two leptons based on their charges, which

leads to invariant mass distributions that are particularly well-suited to spin studies

[12, 14, 62, 63]. However, for mass determination we used the popular event variables

defined in Equations 2–1 and 2–2 [11, 26].

mjl(lo) ≡ smaller of{

mjln , mjlf

}

for each event (2–1)

mjl(hi) ≡ larger of{

mjln, mjlf

}

for each event (2–2)

The method of kinematic endpoints, when applied to our decay chain, begins with the

set of four observables listed in Expression 2–3.

mmaxll , mmax

jll , mmaxjl(lo), m

maxjl(hi) (2–3)

23

In using the method of kinematic endpoints, one assumes that these four observables

(endpoints) depend on the masses, mA, mB, mC and mD, and then expects that the

expressions for the endpoints in terms of the masses can, in principle, be inverted [64].1

Unfortunately, the inversion of the expressions for the four endpoints alone can be

ambiguous [31, 64–71], and we categorize various reasons for the ambiguity.

1. There may not be four independent expressions for the four measurements(Expression 2–3) in terms of the four masses. Indeed, in Regions (2,3), (3,2),and (3,1) of Fig. 1-4, these expressions exhibit the dependence shown in Equation2–4 [31].

m2 maxjll = m2 max

jl(hi) + m2 maxll (2–4)

In this case, the four measurements (Equation 2–3), when thought of as fourconstraints on the 4-dimensional mass space, all simultaneously intersect along anextended curve rather than at a discrete point solution [37]. So one must supplementthese constraints with an additional measurement constraint that will select aparticular point along this curve. The distribution of mjll subject to the constraintin Equation 2–5 provides a lower kinematic endpoint (i.e. “threshold”), mmin

jll(θ> π2),

that has been suggested for this purpose [26].

mminjll(θ> π

2) ≡ mmin

jll s.t. m2ll >

1

2m2max

ll (2–5)

(θ is the angle between the two leptons in the rest frame of B.) We simply precludedthe ambiguity in Equation 2–4 by extending the original set of 4 measurements(Expression 2–3) to the set of five measurements listed in Expression 2–6.

mmaxll , mmax

jll , mmaxjl(lo), m

maxjl(hi), m

minjll(θ> π

2) (2–6)

2. The five measurements (Expression 2–6) inevitably come with some experimentalerrors, so that within those experimental uncertainties, two or more solutions arepossible [31, 32, 65]. Even with 300 fb−1 of data at the LHC, the experimentaluncertainties (finite detector resolution, statistical and systematic errors, etc.)will still allow two solutions [72]. We show an example of this ambiguity in theSPS1a columns of Tab. 2-1. The experimental resolution should improve withtime so that this ambiguity may eventually be resolved. For example, increasing

1 Incidentally, we also recently developed a mass determination technique that wecall sub-mT2 [59]. The sub-mT2 technique can be used to determine masses in an evenshorter decay chain, under certain assumptions in addition to the ones that we have maderegarding the method of kinematic endpoints.

24

the integrated luminosity reduces the statistical error. It is also worth noting thatRef. [31] conservatively assigned a rather large systematic error for the mmin

jll(θ> π2)

measurement, since the analytical shape of its edge was unknown at the time. Theshape was afterward derived in [36], so that now all five measurements (Expression2–6) can be considered on equal footing.

3. Discrete ambiguities arise due to the very nature of the mathematical problem.The expressions for the endpoints in terms of the masses are piecewise-definedfunctions, and the method of kinematic endpoints provides no criteria to determinewhich definition is the relevant one. The method of kinematic endpoints requiresone to consider all possibilities, obtain each solution, and test for consistencyat the very end. So, even in the ideal case of a perfect experiment, which wouldyield results for all five measurements (Expression 2–6) with vanishing error bars,there may still be multiple solutions to the inversion. We identified the specificcircumstances when this takes place. We emphasize that these discrete ambiguitiesarise due to mathematically identical values for all five observables for more thanone discrete set of input masses. Therefore, no improvements in the experimentalresolution are able to resolve this duplication. Two examples of a similar kind ofambiguity are given in [66], but their analysis uses only four out of the five availablemeasurements (Expression 2–3). Table 2-1 shows that the inclusion of mmin

jll(θ> π2)

resolves the ambiguities presented in [66]. In contrast, we use all five measurements(Expression 2–6) and we still find some parts of the mass parameter space in whichthe ambiguity occurs, namely in Regions (2,3), (3,2), and (3,1) of Fig. 1-4, preciselythe Regions in which mmin

jll(θ> π2) is required to resolve the ambiguity due to Equation

2–4. Numerical examples of a duplication similar to ours have previously beenpresented in [64], and our analytical formulas help to understand the reason for theduplication presented there.

2.1.1 Forward Formulas

For convenience, we use the lowercase letters a, b, c, d, and e to represent the

kinematic endpoint values (not to be confused with the uppercase letters A, B, C and D

that label the unknown resonances). The endpoints are given in terms of the input masses

according to known expressions (Equations 2–7 through 2–11) [13, 19, 22, 24, 29, 31]. Njll

and Njl are Region labels that are determined by the spectrum (Equations 1–4 and 1–5).

a ≡ m2maxll =

m2DRCD(1 − RBC)(1 − RAB) , Njl = 1, 2, 3

m2DRCD(1 −

√RAC)2 , Njl = 4

(2–7)

25

mv

even

tco

unt

per

bin

mv ≡√

E2v − |~pv|2

mv = 0 mminv(c) mmax

v

Figure 2-1. An artificial single-variable histogram to demonstrate the meaning of akinematic endpoint, mmax

v , and threshold, mminv(c), neglecting experimental

ambiguities, such as background, that would obscure the locations of thesepoints. v stands for a generic combination of observable particles, i.e.v ∈ {``, j``, j`(lo), j`(hi)}. (c) stands for a generic restriction on the sample(e.g. Equation 2–5). The method of kinematic endpoints uses the extremevariables mmax

v and mminv(c), ignoring the shape of the distribution at

intermediate values of mv.

b ≡ m2maxjll =

m2D(1 − RCD)(1 − RAC) , Njll = 1, 5

m2D(1 − RBC)(1 − RABRCD) , Njll = 2

m2D(1 − RAB)(1 − RBD) , Njll = 3

m2D

(

1 −√

RAD

)2, Njll = 4, 6

(2–8)

c ≡ m2maxjl(lo) =

m2D(1 − RCD)(1 − RBC) , Njl = 1

m2D(1 − RCD)(1 − RAB)(2 − RAB)−1 , Njl = 2, 3

12m2

D(1 − RCD)(1 − RAC) , Njl = 4

(2–9)

d ≡ m2maxjl(hi) =

m2D(1 − RCD)(1 − RAB) , Njl = 1, 2

m2D(1 − RCD)(1 − RBC) , Njl = 3

m2D(1 − RCD)(1 − RAC) , Njl = 4

(2–10)

26

e ≡ m2minjll(θ> π

2) =

14m2

D

{

(1 − RAB)(1 − RBC)(1 + RCD) + 2(1 − RAC)(1 − RCD)

− (1 − RCD)√

(1 + RAB)2(1 + RBC)2 − 16RAC

}

if Njl = 1, 2, 3

14m2

D(1 −√

RAC)

{

2RCD(1 −√

RAC)

+ (1 − RCD)

(

3 +√

RAC −√

1 + RAC + 6√

RAC

)}

if Njl = 4

(2–11)

Equations 2–7 through 2–11 are piecewise-defined: the expressions for the endpoints

depend on the Region of mass parameter space (Fig. 1-4) in which the spectrum occurs.

Using Equations 2–7, 2–8 and 2–10, it is easy to verify Equation 2–4 (written in

simplified notation as b = a + d) in Regions (3,1), (3,2) and (2,3). Therefore, one must rely

on the additional measurement of e (Equation 2–11) in order to obtain a discrete solution

in these Regions.

Notice the absence of the “B” index in the expressions for the endpoints if Njl = 4

or Njll = 5, 6. This simply implies that, if B is off-shell, then the expressions for the

endpoints are independent of mB.

2.1.2 Inversion Formulas

We derived the exact analytical inversion formulas for all of the relevant parameter

space Regions of Fig. 1-4. Until now, inversion formulae had been derived for only 6 of the

11 regions, namely (1,1), (1,2), (1,3), (4,1), (4,2) and (4,3), and even these did not include

the e measurement [31]. Note that, in general, the system appears to be over-constrained

since there are four unknowns (mA, mB, mC and mD) in terms of five measurements (a, b,

c, d and e).

The approach of Ref. [31] (which considered only the on-shell case in Fig. 1-3A) was

to use only a, b, c, and d. One reason to neglect e was that the “forward” expression for e

27

(Equation 2–11) appears too complicated to use in an analytical inversion. However, there

are two problems with this approach:

• It cannot produce a discrete solution in the three on-shell Regions (3,1), (3,2) and(2,3), due to Equation 2–4.

• It leads to discrete ambiguities even in other Regions, exemplified by SU1 and SU3in Tab. 2-1.

Therefore, in order to obtain inverse relations that are valid for any allowed spectrum

(Equation 1–1), the threshold mminjll(θ> π

2) must be used, and one of a, b or d, can be

eliminated. We dropped the b measurement from our list (Expression 2–6), leaving

the four measurements shown in Expression 2–12.

a ≡ m2maxll , c ≡ m2 max

jl(lo) , d ≡ m2 maxjl(hi) , e ≡ m2 min

jll(θ>π/2) (2–12)

Since we did not rely on the b measurement in our inversion, the linear dependence

between a, b and d (Equation 2–4) was never an issue. The same 4 endpoints (Equation

2–12) can be used in all parameter space Regions (Fig. 1-4). Also, we only needed to

consider the 4 Regions according to Njl (the 4 color-coded Regions of Fig. 1-4) for our

inversion. These regions can be defined in terms of only two mass parameters, RAB and

RBC .

We found a convenient form for the inversion formulae that applies to all 4 Regions.

m2A = GNjl

(

αNjl− 1) (

βNjl− 1) (

γNjl− 1)

(2–13)

m2B = GNjl

(

αNjl− 1) (

βNjl− 1)

γNjl(2–14)

m2C = GNjl

(

αNjl− 1)

βNjlγNjl

(2–15)

m2D = GNjl

αNjlβNjl

γNjl(2–16)

This form provides modularity to the inversion formulae (which is convenient for coding

the inversion formulae into FORTRAN, for example). The quantities GNjl, αNjl

, βNjl,

and γNjlare particular combinations of the measured endpoints given in Equations 2–17

28

through 2–20. They depend on the Region (Njl), just like the “forward” expressions for

the endpoints in terms of the input masses.

G1 ≡g (2d − g) − 2c (d − g)

gα1 ≡

a + G1

G1β1 ≡

d

G1γ1 ≡

c

G1(2–17)

G2 ≡g (2d − g) (d − c)

g (d − c) + 2c (d − g)α2 ≡

a + G2

G2

β2 ≡d

G2

γ2 ≡c

d − c(2–18)

G3 ≡(g (2d − g) − 2c (d − g)) d

gd + 2c (d − g)α3 ≡

a + G3

G3β3 ≡

c (d + G3)

dG3γ3 ≡

d

G3(2–19)

G4 ≡ −d + g +√

(2d − g)g α4 ≡a + G4

G4

β4 ≡ γ4 ≡d + G4

2G4

(2–20)

We simplified the expressions for GNjlby realizing a particular combination of endpoints,

Equation 2–21.

g ≡ 2e − a (2–21)

A word of caution is in order regarding the off-shell scenario (Equation 2–20). In that

case, mB is not a relevant parameter, so there are only really three unknowns: mA, mC

and mD. At the same time, there are one fewer independent constraints, since Equations

2–9 and 2–10 imply Equation 2–22.

c =1

2d (2–22)

For the purpose of inversion, in the off-shell case we chose to omit c and work only with

{a, d, g}, which are the only three endpoints appearing in Equation 2–20. The appearance

of the square root in Equation 2–20 should not be a problem, since Inequality 2–23 is

satisfied in the off-shell scenario.

1 <d

g< 2 +

√2 (2–23)

The set of analytical inversion formulae (Equation 2–13 through 2–16 with 2–17 through

2–21) is our first main result.

2.2 Duplication Analysis

For convenience, we refer to the mass parameter space Region for which Njl = i as

“Ri”. We considered the four color-coded Regions (Fig. 1-4), and asked the question: Is

29

Table 2-1. Examples of mass ambiguities [31, 65, 66]. The “True” column represents thesolution that was found in the same Region of mass parameter space as the“Input” spectrum. The “False” column represents a second solution that wasfound in a different Region. The False solution for SPS1a(α) produces fiveendpoints (Expression 2–6) that are similar to the five endpoints produced bythe True solution, but only within the experimental uncertainties. The Falsesolutions for SU1 and SU3 produce four endpoints (Expression 2–3) thatexactly match the four endpoints produced by the respective True solutions,but they produce conflicting values of mmin

jll(θ> π2).

SPS1a(α) [31, 65] SU1 [66] SU3 [66]Variable Input True False True False True FalseRegion (1,1) (1,1) (1,2) (1,1) (1,3) (1,3) (1,1)mχ̃0

196.1 96.3 85.3 137.0 122.1 118.0 346.8

ml̃R143.0 143.2 130.4 254.0 127.5 155.0 411.1

mχ̃02

176.8 177.0 165.5 264.0 245.9 219.0 451.6

mq̃L537.2 537.5 523.5 760.0 743.6 631.0 899.9

mmaxll 77.0 77.0 77.1 61. 61. 100. 100.

mmaxjl(lo) 298.3 298.3 299.6 194. 194. 322. 322.

mmaxjl(hi) 375.6 375.6 375.7 600. 600. 418. 418.

mmaxjll 425.8 425.8 425.6 609. 609. 499. 499.

mminjll(θ> π

2) 200.6 200.6 205.1 143. 148. 247. 214.

it possible that identically the same values of the endpoints {a, c, d, e} can be obtained

from a mass spectrum belonging to a different parameter space Region Rj 6= Ri? If the

answer was “yes”, we then asked two follow-up questions: first, exactly in which parts of

Ri and Rj does this duplication occur, and second, can the other endpoint measurement,

b, resolve this two-fold ambiguity?

We considered Equations 2–7 and 2–9 through 2–11 as a map Fi of the corresponding

parameter space region Ri onto the space of values of the kinematic endpoints.

RiFi7−→ {a, c, d, e} (2–24)

Similarly, Equations 2–13 through 2–21 provide an inverse map F−1j from the space of

kinematic endpoints back onto the mass parameter space.

{a, c, d, e}F

−1

j7−→ Rj (2–25)

30

The composition of the two maps is a transformation of mass parameter space into itself.

Tij ≡ F−1j · Fi (2–26)

RiTij7−→ Rj (2–27)

For i = j, the composite map is a bijective identity map. For the three on-shell cases,

i = 1, 2, 3, i 6= j, the mapping has the generic form shown in Equations 2–28 through 2–31.

R′AB = fAB(RAB, RBC) (2–28)

R′BC = fBC(RAB, RBC) (2–29)

R′CD = fCD(RAB, RBC , RCD) (2–30)

m′D = mD fD(RAB, RBC , RCD) (2–31)

One important feature of the Tij map (Equation 2–28 through 2–31) is that the two

transformed values, {R′AB, R′

BC}, depend on only the two original values, {RAB, RBC},

regardless of the other two values, {mD, RCD}. Notice that RAB and RBC are precisely

the parameters defining the four regions Ri (Fig. 1-4). Therefore, for the purposes of our

duplication analysis it was sufficient to consider the simpler transformation of Equations

2–28 and 2–29 only, instead of the more general mapping given by all four Equations 2–28

through 2–31.

For a nontrivial mapping, our definitions require that the obtained values, {R′AB, R′

BC},

belong to Rj , which is not automatically guaranteed by Equations 2–7, 2–9 through 2–11

and 2–13 through 2–21. In order to find all such occurrences, we considered all possible

transformations Tij with i 6= j. We began with the on-shell case (i, j = 1, 2, 3), for which

there are 6 possible mappings Tij. For the purpose of finding the duplicated portion of

parameter space, it was sufficient to consider only 3 of them, which we chose as T13, T23

and T21. The corresponding results are shown in Figs. 2-2 and 2-3.

31

Figure 2-2. The maps T13 : R1 7−→ R3 (top two panels) and T23 : R2 7−→ R3 (bottom twopanels). In both cases the target region R3 is shaded in cyan. Under T13, thegreen-shaded region ABD in the top left panel transforms into thegreen-hatched region A′B′D′ of the top right panel. Under T23, themagenta-shaded region BCD in the bottom left panel transforms into themagenta-hatched region B′C ′D′ of the bottom right panel. In both cases, thetransformed (primed) region falls completely within the boundaries of theintended target (R3), implying duplication.

Fig. 2-2 shows the image sets T13(R1) (top two panels) and T23(R2) (bottom two

panels), while Fig. 2-3 shows the image set T21(R2). The color-shaded regions in the left

and right panels exhibit the domains Ri and ranges Rj The image sets occur where the

cross-hatched regions in the right panels overlap the ranges. For example, in Fig. 2-2 T13

maps the whole green-shaded region ABD on the left into the green-hatched region

A′B′D′ on the right, while in Fig. 2-2 (Fig. 2-3) T23 (T21) maps the whole magenta-shaded

region BCD on the left into the magenta-hatched region B′C ′D′ on the right. Duplication

occurs whenever the image set is nonempty. We see that duplication occurs in the case

of T13 and T23, but not for T21. (Although image points of T21(R2) that are on opposite

32

Figure 2-3. The same as Fig. 2-2, but for the map T21 : R2 7−→ R1, where the intendedtarget is the green-shaded region R1. Under T21, the magenta-shaded regionBCD in the left panel transforms into the magenta-hatched region B′C ′D′ ofthe right panel. The image B′C ′D′ has no overlap with its intended target R1,except along the BD = B′D′ boundary, which is invariant under the T12

transformation.

sides, but close to the boundary line BD, give rather similar values for the measured

kinematic endpoints {a, c, d, e}.)

At this stage of the analysis, we wondered whether the result (Fig. 2-2) would

be sufficient to prove the existence of duplication. Indeed, Fig. 2-2 told us nothing

about the remaining two parameters RCD and mD and more specifically about their

transformed values R′CD and m′

D under the mappings T13 and T23. However, we found

that all mappings of RCD and mD lead to allowed values of R′CD and m′

D. Therefore, the

duplication examples shown in Fig. 2-2 truly represent a problematic two-fold ambiguity

in the method of kinematic endpoints. We then performed a similar analysis involving the

off-shell region R4 and found no occurrence of duplication, which was not surprising, since

the off-shell case is more restricted, due to Equation 2–22. We summarize our result as

follows:

33

For every point with RAB < RBC < 1 (i.e. R1 or R2) and arbitrary values of

RCD and mD, there exists another parameter space point with RBC < RAB < 1

(i.e. R3) and certain (in general different) values of RCD and mD, which would

result in identical predictions for all four kinematic endpoints {a, c, d, e}.

The inverse is not true: not every point with RBC < RAB < 1 (i.e. R3) is subject to

duplication. Referring to the right panels of Fig. 2-2, only the cross-hatched portions of

R3 (cyan) are duplicated.

Having found duplication examples for the limited set of measurements {a, c, d, e},

we then considered whether the additional measurement of the kinematic endpoint b

would help. We found that, as might have been expected, the b measurement resolves the

duplication whenever it is independent of the other endpoints. Unfortunately, in the three

sub-regions (3,1), (3,2) and (2,3), b is not an independent measurement, and thus the

duplication persists even for the full set of 5 measurements {a, b, c, d, e}. We summarize

the two cases of duplication (Fig. 2-2) in Equations 2–32 and 2–33.

(3, 1)T13−→ (2, 3), (2–32)

(3, 2)T23−→ (2, 3). (2–33)

If the original parameter space point belongs to either Region (3,1) or (3,2), and its image

under Tij belongs to Region (2,3), then the resulting two sets of endpoints (Expression

2–6) are identical.

Not every parameter space point in regions (3,1), (3,2) and (2,3) is duplicated. The

boundaries of the dangerous sub-regions (3,1), (3,2) and (2,3) depend on RCD, thus

the range of RCD values resulting in duplication is restricted. Therefore, we revise our

conclusion as follows:

For every point with RAB < RBC < 1 and any mD, there exists a range of RCD

for which exactly the same values of the five kinematic endpoint measurements

34

{a, b, c, d, e} can also be obtained from a different parameter space point with

RBC < RAB < 1 and some other (generally different) values of RCD and mD.

For any given point in the (RBC , RAB) plane, there may exist a range of values for RCD

which would cause duplication. We denote the minimum and maximum values of that

range by RminCD and Rmax

CD , correspondingly. Both RminCD and Rmax

CD are functions of RBC and

RAB. The duplicated parameter space can be simply described as the set of all points

{RAB, RBC , RCD}, that satisfy Inequality 2–34.

RminCD (RBC , RAB) < RCD < Rmax

CD (RBC , RAB). (2–34)

If, on the other hand, the values of RBC and RAB are such that duplication does not occur

for any value of RCD, we can simply take RminCD = Rmax

CD , resulting in a set of zero measure.

In order to delineate the duplicated parameter space, we only need to supply its

boundaries RminCD (RBC , RAB) and Rmax

CD (RBC , RAB). Within the duplicated region we

always found that RmaxCD (RBC , RAB) = 1, while the function Rmin

CD (RBC , RAB) is plotted in

Fig. 2-4. Duplication does not occur in the uniformly red region in the upper left corner,

so there we chose to plot RminCD = Rmax

CD = 1, in accordance with our convention. Within the

multi-colored region in Fig. 2-4, duplication exists for any value of mD, as long as RCD is

larger than the RminCD value shown in the figure.

We illustrate the duplication with two specific examples of duplicate mass spectra,

one for Equation 2–32 and another for Equation 2–33. Table 2-2 shows the corresponding

input masses and mass ratios, as well as the resulting kinematic endpoints. Since we

know from Fig. 2-4 that every pair of {RAB, RBC} in R1 and R2 is duplicated, it was

convenient to first choose the values of RAB and RBC from one of those Regions, and

then find the duplicated point in R3. We selected nice round numbers like RAB = 0.4

and RBC = 0.8. This choice is indicated in Fig. 2-4 with the white asterisk inside region

R1. Then, RminCD = 0.686, so we chose a somewhat larger value: RCD = 0.7. This choice

of RAB, RBC and RCD already guarantees duplication for any value of mD: we simply

35

Figure 2-4. The minimum value RminCD (RBC , RAB) required for duplication, as a function

of RBC and RAB. The white asterisks (circles) mark the duplicate pair ofpoints P31 and P23 (P32 and P ′

23) in Table 2-2. Duplication does not occur inthe solid red region.

chose mD = 500 GeV (another nice round number). The resulting masses mA, mB and

mC are readily computed in terms of mD and the mass ratios. We named the resulting

spectrum “study point P31”, which is listed in the second column of Table 2-2. Given P31,

we used the transformation T13 to obtain the matching spectrum in R3, which is listed in

the third column of Table 2-2 under the name of “study point P23”. The corresponding

point is marked with the white asterisk in the upper right corner of Fig. 2-4. In the case

of Equation 2–33, we followed a similar procedure, except that we started with a point in

region R2 (indicated with a white circle in Fig. 2-4) and then used the T23 transformation

to obtain the corresponding point in R3 (also indicated with a white circle in Fig. 2-4).

The two resulting mass spectra (called P32 and P ′23) are given in the fourth and fifth

columns of Table 2-2, respectively.

Additional constraints are needed in order to resolve the ambiguity. One option

is to consider a longer decay chain, which would yield several additional endpoint

measurements. For example, the decay chains in Fig. 1-3 may begin with an even heavier

36

Table 2-2. Two examples of exact duplication under Equations 2–32 and 2–33. The pairsof study points P31 and P23, as well as P32 and P ′

23, exhibit identical values forall five kinematic endpoints (Expression 2–6). Point P31 belongs to R1, pointP32 belongs to R2, and points P23 and P ′

23 belong to R3. The last eight rowsgive the endpoint measurements that are available from the two-dimensionaldistributions (m2

jl(lo), m2jl(hi)) (Equations 2–41 through 2–43) and (m2

ll, m2jll)

(Equations 2–62 through 2–65 and 2–67).R1 ↔ R3 R2 ↔ R3

(3,1) (2,3) (3,2) (2,3)Variable P31 P23 P32 P ′

23

mA (GeV) 236.643 915.618 126.491 241.618mB (GeV) 374.166 954.747 282.843 346.073mC (GeV) 418.33 1083.10 447.214 554.133mD (GeV) 500.00 1172.57 500.00 610.443RAB 0.400 0.920 0.200 0.487RBC 0.800 0.777 0.400 0.390RCD 0.700 0.853 0.800 0.824Rmin

CD 0.686 0.845 0.774 0.800mmax

ll (GeV)√

a 145. 145. 310. 310.

mmaxjll (GeV)

√b 257. 257. 369. 369.

mmaxjl(lo) (GeV)

√c 122. 122. 149. 149.

mmaxjl(hi) (GeV)

√d 212. 212. 200. 200.

mminjll(θ> π

2) (GeV)

√e 132. 132. 248. 248.

mmaxjlf

(GeV)√

f 212. 127. 200. 183.

m(p)jlf

(GeV)√

p 190. 112. 126. 115.

mmaxjln (GeV)

√n 122. 212. 173. 200.

mjll(+)(0) (GeV)√

s 226. 240. 214. 230.

mjll(+)(at) (GeV)√

t 263. 257. 374. 369.mjll(+)(aon) (GeV)

√u 257. 257. 369. 369.

mjll(−)(aon) (GeV)√

v 190. 193. 355. 360.mjll(+)(aoff) (GeV)

√w 256. 243. 372. 367.

particle (say, E), at the expense of a single new parameter (the mass of particle E) [32].

Alternatively, one may supplement (or even replace) Expressions 2–6 with measurements

from a future lepton collider [65]. Instead, we concentrated on the question, “What

additional information that is already present in the hadron collider data can be used to

resolve the ambiguity?”

In very general terms, the kinematics of the decay in Fig. 1-3 is governed by some

three-dimensional differential distribution, Expression 2–35, where α, β and γ are some

37

suitably chosen angles specifying the particular decay configuration [12].

d3Γ

dα dβ dγ(2–35)

Through a change of variables, these angles can be traded for three invariant mass

combinations of the observed decay products in Fig. 1-3, e.g. mll, mjl+ , mjl− [15], but any

generic set of masses {m1, m2, m3} is possible. So, one can replace Expression 2–35 with

Expression 2–36.

d3Γ

dm1dm2dm3

(2–36)

We imagine that the distribution (Expression 2–36) is observable as a three-dimensional

histogram. It provides much more information about the decay in Fig. 1-3 compared to

the single variable distributions. One can simply integrate out two of the three masses in

order to obtain the single variable distribution for say, m1 as demonstrated in Equation

2–37, which exhibits a kinematic endpoint, mmax1 .

dΓ

dm1

≡∫

dm2dm3d3Γ

dm1dm2dm3

(2–37)

However, the integrated distribution ignores some of the original information contained in

Expression 2–36. The integration on a single invariant mass produces a two-dimensional

(bivariate) distribution, e.g. (m1, m2) as demonstrated in Equation 2–38, and maintains

some of the information that is integrated out for the corresponding single variable

distributions, i.e. m1 and m2 separately.

d2Γ

dm1dm2≡∫

dm3d3Γ

dm1dm2dm3(2–38)

A two-dimensional distribution exhibits boundary lines (or curves) rather than a single

endpoint. Given that bivariate and trivariate distributions are more informative than the

simple one-dimensional histograms, we find it surprising that they have not been used

more often in the previous analyses of mass determination.

38

We found that the shape of the bivariate distribution (m2jl(lo), m

2jl(hi)) can be used

to qualitatively identify the Region (Fig. 1-4), thus resolving the duplication [67, 69–71].

We also found analytical formulas in terms of the parameters Rij (Definition 1–2) for

the boundaries of the kinematically allowed distributions, which can be used to further

quantitatively improve on the mass determination [71]. We identified some definitive

points on the boundary lines, for which we provide analytic expressions in terms of Rij .

These definitive points are typically hidden as ambiguous features of the one-dimensional

distributions but are easy to see and understand on the bivariate distributions.

2.3 Kinematic Boundary Lines for the (m2jl(lo),m

2jl(hi)) Distribution

We examined the kinematic boundary lines of the two-variable invariant mass

distribution in (m2jl(lo),m

2jl(hi)).

d2Γ

dm2jl(lo) dm2

jl(hi)

(2–39)

We found that the qualitative shape of the boundary lines of this distribution can be used

to resolve the duplication problem, and also that extra measurements can be obtained

from these boundary lines.

The variables mjl(lo) and mjl(hi) (Equations 2–1 and 2–2) deal with the ambiguity in

the identification of the “near” and “far” leptons l±n and l∓f in Fig. 1-3. The shape of the

(mjl(lo),mjl(hi)) distribution is related to the corresponding (m2jln

,m2jlf

) distribution.

d2Γ

dm2jln

dm2jlf

(2–40)

In principle, both distributions (Equation 2–39 and 2–40) depend not only on the mass

spectrum, but also on the spins and on the chiralities of the coupling constants of the

particles A, B, C and D [14, 17, 63]. However, the location and the shape of the boundary

lines are determined by kinematics alone, and do not depend on the spin and type of

couplings. To the extent that we were only interested in these boundary lines, it was

therefore sufficient to consider only pure phase space decays, in which case the analytical

results for the distributions were, in principle, already available [11]. So, for our mass

39

Figure 2-5. The generic shape ONPF of the bivariate distribution in the (m2jln

, m2jlf

)plane.

determination analysis we use the term “shape” to refer to the shape of the boundary

lines, and not the probability distribution.

The shape of the (m2jln

,m2jlf

) distribution is a simple right-angle trapezoid, illustrated

in Fig. 2-5. Point O is simply the origin of the coordinate system. Point N (for “near”)

lies on the m2jln

axis, and its coordinate is the maximum possible value of the jet-near

lepton invariant mass, given in terms of the input spectrum according to Equation 2–41.

n ≡(

mmaxjln

)2= m2

D (1 − RCD) (1 − RBC) (2–41)

Similarly, point F (for “far”) lies on the m2jlf

axis, and its coordinate is the maximum

possible value of the jet-far lepton invariant mass, given in terms of the input spectrum

according to Equation 2–42.

f ≡(

mmaxjlf

)2

= m2D (1 − RCD) (1 − RAB) (2–42)

40

The coordinates of point P are (n, p), where p is given in terms of the input spectrum

according to Equation 2–43.

p ≡(

m(p)jlf

)2

≡ fRBC = m2D (1 − RCD) RBC (1 − RAB) (2–43)

Since N and P share the same m2jln coordinate n, point P always lies directly above point

N . At the same time, the definition of p implies Equation 2–44, so that point P always

lies lower than point F (Fig. 2-5).

p < f (2–44)

In R2 and R3, we identified one more point, Q: it is the point where the FP side of the

trapezoid intersects the line m2jln

= m2jlf

. The 2 coordinates of point Q are equal by

definition, and are given by Equation 2–45.

q ≡(

mmaxjl(eq)

)2= m2

D (1 − RCD)1 − RAB

2 − RAB(2–45)

The 4 quantities n, p, f and q are not all independent, but obey Equation 2–46.

f

q= 1 +

f − p

n(2–46)

The trapezoid ONPF can be equivalently defined through a parametric expression

of the boundary line segment FP . A convenient choice for the line parameter is n̂ ≡

m2jln

/m2 maxjln

∈ [0, 1]. Then the parametric equation of the line FP is given by Equation

2–47.

FP : m2jlf

(n̂) = f − (f − p)n̂ (2–47)

In terms of this parametrization, the three special m2jlf

values introduced in Fig. 2-5 are

given by Equations 2–48 through 2–50.

f = m2jlf

(0) (2–48)

p = m2jlf

(1) (2–49)

q = m2jlf

(q/n) (2–50)

41

The colored regions in Fig. 2-5 show the allowed locations of point P , and are

color-coded to match Fig. 1-4. The two white areas in Fig. 2-5 are not accessible to point

P . The region with m2jlf

> f is forbidden according to Equation 2–44. Similarly, the white

triangular area near the origin, defined by Equation 2–51 is also not allowed according to

Inequality 2–52 following from Equations 2–41 through 2–43.

m2jlf

< f − m2jln (2–51)

p ≥ f − n (2–52)

Therefore, point P must belong to one of the three colored regions in Fig. 2-5. These three

regions are distinguished based on the value of n relative to f and p.

1. n < p < f : Point P then lies somewhere within the green-shaded area in Fig. 2-5.Using Equations 2–41 through 2–43, the conditions n < p < f imply Equation 2–53,which is precisely the defining relation for R1 in Equation 1–5.

n < p < f =⇒ 1

2 − RAB< RBC < 1 (2–53)

Therefore, in Fig. 2-5 we have labeled and color-coded the area with n < p < f tomatch R1 in Fig. 1-4.

2. p < n < f : Point P then belongs to the magenta-shaded triangular area in Fig. 2-5.The conditions p < n < f now imply Equation 2–54, which is the definition of R2 inEquation 1–5.

p < n < f =⇒ RAB < RBC <1

2 − RAB(2–54)

Once again, we labeled and color-coded this region to match Fig. 1-4.

3. p < f < n: Point P then falls somewhere within the cyan-shaded semi-infiniterectangular strip in Fig. 2-5. Using Equations 2–41 through 2–43, the constraintsp < f < n now translate into 2–55, which is the definition of R3 in Equation 1–5,again matching Fig. 2-5 to Fig. 1-4.

p < f < n =⇒ 0 < RBC < RAB (2–55)

The off-shell scenario of Fig. 1-3B should be handled with care, since the “near” and “far”

lepton distinctions are meaningless in that case. Nevertheless, the off-shell scenario can

42

Figure 2-6. Obtaining the shape of the (m2jl(lo),m

2jl(hi)) bivariate distribution by folding the

(m2jln

,m2jlf

) distribution across the line m2jln

= m2jlf

. This particular exampleapplies to R3. For the other three regions, refer to Fig. 2-7.

still be represented in Fig. 2-5, and in fact this representation is unique: there is a single

allowed location for point P at n = f and p = 0. In Fig. 2-5 this unique location is

indicated with a yellow-shaded circle, which corresponds to the whole yellow-shaded region

R4 in Fig. 1-4. In other words, in the off-shell case we can randomly assign “near” and

“far” labels to the two leptons in each event, and then the shape ONF of the resulting

(m2jln

, m2jlf

) distribution will be an isosceles right triangle.

The boundaries of the distribution (Expression 2–40) contain some useful information:

their shape uniquely identifies the parameter space region (Ri) and yields the measurements

{n, f, p, q} given in Equations 2–41 through 2–43 and 2–45. We found that it is actually

possible to preserve and subsequently extract this additional information from the

observable distribution (Expression 2–39), using the simple intuitive understanding of the

shape exhibited in Fig. 2-5. The key was to realize that the reordering of the (m2jln

, m2jlf

)

pair into the (m2jl(lo), m

2jl(hi)) pair (Equation 2–1 and 2–2) simply corresponds to “folding”

the trapezoid ONPF in Fig. 2-5 across the line m2jln

= m2jlf

. This procedure is shown

pictorially in Fig. 2-6, where for illustration we used an example from R3. Panel A

shows the trapezoidal shape of the original (m2jln ,m2

jlf) distribution from Fig. 2-5. The

43

(m2jln

, m2jlf

) distribution can be converted into the (m2jl(lo), m

2jl(hi)) distribution simply by

reinterpreting the m2jln

axis as m2jl(lo) and the m2

jlfaxis as m2

jl(hi). From that point of view,

the trapezoid ONPF in Fig. 2-6 divides into two adjacent regions: OQF (blue-shaded)

and ONPQ (red-shaded). Within the blue-shaded area OQF we have m2jln

< m2jlf

, so

that the coordinate pair (m2jln , m2

jlf) can be directly identified with (m2

jl(lo), m2jl(hi)). Thus

the blue-shaded area OQF in panel A remains unchanged and appears identically in

panel B, where it is marked with a blue cross-hatch. In contrast, within the red-shaded

area ONPQ of panel A, the coordinates m2jln

and m2jlf

are in the wrong order, and need

to be swapped when going to (m2jl(lo), m

2jl(hi)). This reversal corresponds to “folding”

the trapezoid ONPF across the line OQ, as shown in Fig. 2-6. The resulting image

ON ′P ′Q in Fig. 2-6B is then overlaid on the original OQF portion. Any (m2jl(lo), m

2jl(hi))

distribution therefore exhibits two characteristic types of population density. For example,

in the blue-hatched red area of Fig. 2-6B we expect the density to be generally larger than

the other area, since the folded distribution ON ′P ′Q is overlaid on top of the existing

distribution OQF underneath. In Fig. 2-7, we marked such “double-density” areas with

a blue cross-hatch in addition to the solid red shading. In contrast, region FQP ′N ′

in Fig. 2-6B is a “single-density” region, since the folded distribution happened to fall

onto empty space, where originally there were no points to begin with. A single density

region can also be obtained when portions of the original (m2jln

, m2jlf

) scatter plot are not

overlaid in the process of folding. In either case, we denote a single-density region by a

solid (red) color-shading with no cross-hatch. Fig. 2-7 displays our results, where we show

the characteristic shape for each Region. In addition, we show the original location of

the point P in the (m2jln

, m2jlf

) plot. The allowed positions of point P in each case are

color-coded to Figs. 1-4 and 2-5.

A nice feature of all the plots in Fig. 2-7 is that they are composed entirely of straight

lines. This is a consequence of the fact that the original trapezoid in Fig. 2-5 is made

up of straight lines, and then the “folding” of Fig. 2-6 does not curve the boundaries.

44

Figure 2-7. The generic shape of the bivariate distribution (m2jl(lo),m

2jl(hi)) for each of the

four parameter space regions: A) R1, B) R2, C) R3, and D) R4. Each panelshows the typical shape (red-shaded) of the resulting (m2

jl(lo), m2jl(hi))

distribution, after the “folding” in Fig. 2-6. Hatched (unhatched) areascorrespond to double-density (single-density). Each panel also shows theoriginal location of the point P in the (m2

jln, m2

jlf) plot, as well as the allowed

positions of point P , following the color conventions of Figs. 1-4 and 2-5.

45

The shape of the (m2jl(lo), m

2jl(hi)) distribution allows us to uniquely determine the region

of mass parameter space. For example, the typical shape for Region R1, exhibited in

Fig. 2-7A, consists of a right-angle triangular region OO′N ′ of double density and a

right-angle trapezoidal region N ′O′PF of single density. In this case, point P is directly

observable, and its coordinates immediately yield the quantities n and p defined in

Equations 2–41 and 2–43. In addition, one can also measure the location f of point F

along the m2jl(hi) axis, given by Equation 2–42. This gives a total of three independent

measurements: n, p and f , which should be ordered as n < p < f (Equation 2–53). The

one-dimensional distributions are obtained by projecting the (m2jl(lo), m

2jl(hi)) distribution

shown in Fig. 2-7A onto the two axes. In this case the endpoint of the single-variable

m2jl(lo) distribution will be given by c = n, while the endpoint of the signal-variable m2

jl(hi)

distribution will be given by d = f , and neither of those will reveal the quantity p. In

contrast, p can be identified on the two-variable distribution, and provides an additional

independent measurement. Similar analysis applies to Figures 2-7B and 2-7C, with

observation of the additional point, Q, as well.

Fig. 2-7D represents R4. Due to the symmetry between the “near” and “far” leptons

in the off-shell case, the folded region ONQ has an identical triangular shape to the

underlying region OFQ, so that, after folding, the two match perfectly and we obtain

a single triangular region of double density, and no single-density areas. The off-shell

scenario offers only one independent endpoint measurement, which can be taken as f .

The latter appears as the endpoint d in the one-dimensional m2jl(hi) distribution, while

the endpoint c of the other one-dimensional distribution, m2jl(lo), is then simply given as

c = f/2, in agreement with Equation 2–22. The (m2jl(lo), m

2jl(hi)) distribution in R4 can still

be helpful in discriminating a potential regional ambiguity. The triangular double-density

shape of the scatter plot in the off-shell case of Fig. 2-7D can in principle also be obtained

in the on-shell cases of Fig. 2-7B and Fig. 2-7C, provided that the image P ′ of point P

ends up very close to point F . In terms of the (m2jln

, m2jlf

) distribution of Fig. 2-5, this

46

situation corresponds to the on-shell cases of Regions R2 or R3, with point P lying very

close to the yellow dot representing Region R4. In spite of having the same shape of their

boundary lines, the two distributions will be quite different, as they will exhibit a different

point density. In particular, for all three on-shell cases, the pure phase space probability

distribution is given by Equation 2–56.

d2Γ

dm2jln

dm2jlf

=1

n(

m2jlf

(m2jln

)) =

1

fn − (f − p) m2jln

(for RBC < 1) (2–56)

Within the kinematically allowed region, the density is independent of m2jlf

. In the limit

p → 0, Expression 2–56 becomes singular when m2jln

→ n. This singularity is regularized

by the width of particle B and the branching fraction for the C → B decay. In contrast,

the corresponding density in the off-shell case is quite different, and in particular does not

exhibit such singular features. Furthermore, we found that the (m2jll, m

2ll) boundary lines

would appear quite different for such an on-shell point compared to an off-shell point.

So, the shape of the kinematic boundary lines of the (m2jl(lo), m

2jl(hi)) distribution

uniquely identifies the region, as shown in Fig. 2-7. Since the duplicate solutions that we

found always appear in two different regions, this is in principle sufficient to eliminate the

wrong solution. Also, the scatter plots offer the possibility of additional measurements,

and at the very least a measurement of the quantity p. As can be seen from Table 2-2, the

value of p is already different for each pair of duplicate spectra, and, provided that it can

be measured with sufficient precision, can also be used to remove the ambiguity.

Our conclusions are confirmed by Fig. 2-8, which shows (m2jl(lo), m

2jl(hi)) scatter

plots for the four study points from Table 2-2. The figure indeed shows that each pair

of duplicate points has identical values for the endpoints of the separate single-variable

distributions m2jl(lo) and m2

jl(hi). However, the shapes of the scatter plots have obvious

differences. We therefore conclude that the two-variable distribution in (m2jl(lo), m

2jl(hi)) in

principle can resolve the two-fold ambiguity.

47

Figure 2-8. Scatter plots of (m2jl(lo),m

2jl(hi)) for the 4 study points from Table 2-2. A

quadratic scale was used on both axes. The theoretical kinematic boundarylines are outlined with the corresponding color for each Region, following thecolor coding conventions of Figs. 1-4 and 2-5. Each plot has 10,000 datapoints. We assumed that all particles A, B, C and D are exactly on-shell, andthat all 4-momenta are perfectly measured.

48

As a final remark, we point out that when the two-dimensional scatter plots like

those in Fig. 2-8 are projected onto the axes to obtain the corresponding one-dimensional

distributions of either m2jl(lo) or m2

jl(hi), the latter often exhibit some peculiar features that

were classified as either “feet” or “drops” in Ref. [11]. The origin of these features is now

easy to understand in terms of the two-variable distribution. For example, the scatter

plots in Figs. 2-8B and 2-8D, when projected onto the m2jl(hi) axis, both exhibit a “drop”

at the m2jl(hi) endpoint, which is simply due to the flat upper boundary P ′N ′ in Fig. 2-7C.

Similarly, the projection of the distribution in Fig. 2-8C onto the m2jl(hi) axis exhibits a

“foot” extending from n to f . The “foot” can be easily understood in terms of the generic

shape of Fig. 2-7B, where it arises from the projection of the single density area N ′HF .

2.4 Kinematic Boundary Lines for the (m2ll,m

2jll) Distribution

In addition to the kinematic boundary lines of the (m2jl(lo),m

2jl(hi)) distribution, we

also examined the kinematic boundary lines of the distribution in Expression 2–57, whose

generic shape OV US is shown in Fig. 2-9.

d2Γ

dm2jll dm2

ll

(2–57)

The red area represents kinematically allowed values for the pair of observables (m2ll,m

2jll).

The kinematic boundary lines of Expression 2–57 generally consist of 4 segments. The

upper (SU) and lower (OV ) curved boundaries are segments of a hyperbolic curve

(OWS), while the left (OS) and right (UV ) boundaries are straight line segments. We

describe the shape of the (m2ll, m

2jll) boundary with parametric equations for the upper

and lower curved boundaries SW and OW , together with the location of the vertical line

UV . We chose m2ll as the line parameter describing the curved boundaries. The upper

boundary line SUW is given by Equation 2–58 while the lower boundary line OV W is

given by Equation 2–59 [13].

m2jll(+)(m

2ll) =

1 + RCD

2

m2ll

RCD

+1

2m2

D(1 − RCD)(1 − RAC)

49

Figure 2-9. The generic shape OV US of the bivariate distribution (Expression 2–57) inthe (m2

ll, m2jll) plane.

+1 − RCD

2

{

[(

m2ll

RCD

)

− m2D(1 + RAC)

]2

− 4m4DRAC

}1

2

(2–58)

m2jll(−)(m

2ll) =

1 + RCD

2

m2ll

RCD

+1

2m2

D(1 − RCD)(1 − RAC)

− 1 − RCD

2

{

[(

m2ll

RCD

)

− m2D(1 + RAC)

]2

− 4m4DRAC

}1

2

(2–59)

The vertical straight line segment UV is in general located at m2ll = a, where a is the value

of the dilepton invariant mass endpoint (Equation 2–7).

Since the expression for a depends on the mass shell condition of B, we introduce

separate notation for the endpoint a in each of these two cases. In the on-shell scenario

(Fig. 1-3A) we use aon, and in the off-shell scenario (Fig. 1-3B) we use aoff . From Equation

2–7 we determine Equation 2–60, which is obvious on Fig. 2-9.

aon ≤ aoff (2–60)

50

Equality in Equation 2–60 is achieved when RAB = RBC , i.e. when the on-shell spectrum

happens to lie exactly on the border between R2 and R3 (Fig. 1-4). In physical terms this

means that mB is equal to the geometric mean of mA and mC , shown in Equation 2–61.

mB =√

mAmC ⇒ aon = aoff (2–61)

This represents another potential source of confusion in extracting the mass spectrum –

the measurement of the dilepton invariant mass endpoint a alone tells us nothing about

whether the intermediate particle B is on-shell or off-shell [33]. In particular, when

Equation 2–61 holds, the entire area inside OWS is kinematically accessible, and the

boundary of the (m2ll,m

2jll) distribution in the on-shell scenario cannot be distinguished

from the boundary of the (m2ll,m

2jll) distribution in the off-shell scenario. However, we

already determined that the shape of the (m2jl(lo), m

2jl(hi)) distribution distinguishes the

off-shell scenario (Fig. 2-7D) from the on-shell scenario (Fig. 2-7A through 2-7C).

We identified several points along the hyperbola OWS in Fig. 2-9. Point O is simply

the origin (0, 0) of the (m2ll, m

2jll) coordinate system. Point W is the tip of the hyperbola

at m2ll = aoff , where the upper branch m2

jll(+)(m2ll) meets the lower branch m2

jll(−)(m2ll). The

m2jll coordinate of W is given by Equation 2–62.

w ≡ m2jll(+)(aoff) ≡ m2

jll(−)(aoff) = m2D

(

1 − RCD

√

RAC

)(

1 −√

RAC

)

(2–62)

Point S is the m2jll-intercept of the upper kinematic boundary line m2

jll(+)(m2ll). The m2

jll

coordinate of point S is given by Equation 2–63.

s ≡ m2jll(+)(0) = m2

D (1 − RCD) (1 − RAC) (2–63)

Points U and V label the intersections of the vertical boundary UV with the upper and

lower hyperbolic branches (Equations 2–58 and 2–59), respectively. They share the same

m2ll coordinate aon, while their m2

jll coordinates are correspondingly given by Equations

51

2–64 and 2–65.

u ≡ m2jll(+)(aon) (2–64)

=1

2m2

D

[

(1 + RCD)(1 − RBC)(1 − RAB) + (1 − RCD)(1 − RAC + |RBC − RAB|)]

v ≡ m2jll(−)(aon) (2–65)

=1

2m2

D

[

(1 + RCD)(1 − RBC)(1 − RAB) + (1 − RCD)(1 − RAC − |RBC − RAB|)]

Point T occurs at the maximum of the upper branch (Equation 2–58). The m2ll coordinate

at of point T is given by Equation 2–66.

at ≡ m2D

(

RCD −√

RAD

)(

1 −√

RAD

)

(2–66)

The m2jll coordinate t of point T can be found by substituting Equation 2–66 into

Equation 2–58, and is given in terms of the input mass spectrum by Equation 2–67.

t ≡ m2jll(+)(at) = m2

D

(

1 −√

RAD

)2

(2–67)

Point W is not part of the actual boundary except in the off-shell scenario, or if

Equation 2–60 holds. Point T may not even be defined if at < 0. We calculate the slope of

the upper branch m2jll(+)(m

2ll) at point S.

(

dm2jll(+)

dm2ll

)

m2ll=0

=RCD − RAC

RCD(1 − RAC)(2–68)

Since the denominator on the R.H.S. of Equation 2–68 is always positive, the sign of

the derivative is determined by the relative size of RCD and RAC . When RCD < RAC ,

the slope is negative, and T is undefined. In that case, the maximum value of m2jll over

the whole scatter plot OV US is obtained exactly at S, and is given by Equation 2–63.

Comparing to Equation 2–8, we see that this happens precisely for Njll = 1 and Njll = 5.

In contrast, for the other four cases Njll = 2, 3, 4, 6, the slope at point S is positive and

point T is well defined. However, this does not mean that point T would then lie on the

52

boundary OV US. In the off-shell case of Njll = 6, point T lies on the boundary, and the

maximum value of m2jll is given by t, in agreement with Equation 2–8. However, in the

remaining three on-shell cases (Njll = 2, 3, 4) point T is included only if it lies to the left of

the UV line (Equation 2–69).

at < aon (2–69)

Using Equations 2–66 and 2–7, Equation 2–69 can be rewritten as Equation 2–70.

(RBC − RABRCD)(RAB − RBD) > 0 (2–70)

Alternatively, point T will not be on the boundary if Equation 2–71 is satisfied.

(RBC − RABRCD)(RAB − RBD) < 0 (2–71)

The two factors entering Equations 2–70 and 2–71 cannot be simultaneously negative: if

that were the case, we would have Equation 2–72, which contradicts our basic assumption

(Equation 1–1).

RBC − RABRCD < 0 ⇒ m2B < mA

mDm2

C

RAB − RBD < 0 ⇒ mA mD < m2B

⇒ mD < mC (2–72)

Therefore, whenever one of the two factors in Equation 2–72 is negative, the other is

guaranteed to be positive. It is also possible that both factors are positive. Altogether,

this leads to three different possibilities, which are related to the Njll = 2, 3, 4 cases of

Equation 2–8.

• Njll = 2: In this case, the first factor in Equation 2–72 is negative.

RBC − RABRCD < 0 ⇒ RAB − RBD > 0

⇒ (RBC − RABRCD)(RAB − RBD) < 0 ⇒ at > aon (2–73)

In this case, T is not part of the boundary. Then, the maximum value of m2jll is

obtained at point U and is given by Equation 2–64. Since in this case RBC <RABRCD < RAB, the absolute value sign in Equation 2–64 can be resolved as|RBC − RAB| = RAB − RBC and then Equation 2–64 simplifies to Equation 2–8 forNjll = 2.

53

• Njll = 3: In this case, the second factor in Equation 2–72 is negative.

RAB − RBD < 0 ⇒ RBC − RABRCD > 0

⇒ (RBC − RABRCD)(RAB − RBD) < 0 ⇒ at > aon (2–74)

Once again, point T does not belong to the boundary, and the maximum value ofm2

jll is obtained at point U and is given by Equation 2–64. This time, however,RAB < RBD = RBCRCD < RBC , and correspondingly, |RBC − RAB| = RBC − RAB.Then, Equation 2–64) simplifies to Equation 2–8 for Njll = 3.

• Njll = 4: This is the case when both factors in Equation 2–72 are positive.

RBC − RABRCD > 0

RAB − RBD > 0

}

⇒ (RBC−RABRCD)(RAB−RBD) > 0 ⇒ at < aon (2–75)

In this case, T belongs to the boundary, and t is the maximum value of the m2jll

distribution, in agreement with Equation 2–8 for Njll = 4.

Fig. 2-9 allows us to understand geometrically the meaning of the threshold e =

(mminjll(θ> π

2))

2 (Equation 2–11). By restricting the (m2ll,m

2jll) distributions to points with

m2ll > 1

2aon, i.e. to the right of the dashed line EE ′, the single-variable m2

jll distribution

will exhibit a lower endpoint, whose value e is given by the m2jll coordinate of point E in

Fig. 2-9. In the on-shell case, e is defined by Equation 2–76, and, in the off-shell case, e is

defined by Equation 2–77.

e ≡ m2jll(−)(aon/2) (2–76)

e ≡ m2jll(−)(aoff/2) (2–77)

In principle, the newly introduced quantities s, t, u, v, and w can be determined

from the observable boundary of the (m2ll,m

2jll) distribution (Fig. 2-9), where t and w may

require extrapolation. Table 2-2 lists their square root values for our four study points

P31, P23, P32 and P ′23. As expected, the value of u is matched identically for each pair.

However, the two other directly observable quantities s and v differ, and in principle

can be used to resolve the duplication. This is illustrated in Fig. 2-10. Unlike Fig. 2-8,

the differences between the (m2ll,m

2jll) scatter plots for each duplicated pair are only

quantitative, and may be difficult to observe in practice. Duplication occurs only in

54

Figure 2-10. The same as Fig. 2-8, but for (m2ll,m

2jll).

regions with Njll = 2 or Njll = 3. In both cases, the shape of the (m2ll, m

2jll) scatter plot is

rather similar: the slope at point S is positive, and the upper boundary SU excludes point

T . Furthermore, the duplication analysis ensures that the rightmost vertical boundary UV

occurs in the same location aon. Anyway, we already determined that the qualitative shape

of the kinematic boundary of the (m2jl(lo), m

2jl(hi)) distribution can resolve the duplication.

55

CHAPTER 3SPIN DETERMINATION

After considering the mass determination of particles A, B, C and D from the

decay process represented in Fig. 1-3A, we also wanted to develop a model-independent

technique to determine the spins of these particles. (We assumed only the three-stage

cascade decay for the spin determination.) As in the mass determination, we assumed that

we could not observe the energy and momentum of A, so that we could not reconstruct

the resonances. For simplicity, we assumed that the masses {mA, mB, mC , mD} had

already been determined.

Recently there has been a lot of effort on developing various techniques for discriminating

among different model scenarios [12, 14–18, 35, 62, 73–86]. Since we assumed that the spin

of A is unknown, this gives rise to several distinct possibilities for the spin assignments of

the heavy resonances. Even if the spin of A were known, this still would not completely fix

the spins of the preceding particles B, C and D.

The last column of Table 1-2 gives some typical examples involving the squarks q̃,

sleptons ˜̀ and neutralinos χ̃0i in supersymmetry, the KK quarks q1, KK leptons `1 and

KK gauge bosons Z1 and γ1 in 5D (or 6D) UED [87], and the spinless gauge bosons γH

and ZH in 6D UED [88]. Case SFVF would require either a scalar leptoquark or a gauge

boson that carries lepton number, so we do not provide an example [63]. Nevertheless, we

included this case in our study in order to compare to the results of [14, 15]. We emphasize

that the supersymmetry and UED examples in Table 1-2 serve only as illustrations, and

we never restricted the analysis to any particular model. In particular, our analysis did

not rely on any features of the mass spectrum nor the couplings that might be expected in

SUSY or UED. It is for this reason that we claim a model-independent analysis.

Our main goal was to assess the possibility of discriminating between the 6 different

alternatives in Table 1-2, using observable invariant mass distributions of the visible

particles (the jet and the two leptons) in Fig. 1-3A. Our method of spin determination

56

Table 3-1. Shorthand notation for the six different spin assignments listed in Table 1-2shorthand: S = 1 2 3 4 5 6spin assignment: SFSF FSFS FSFV FVFS FVFV SFVF

also provides an independent measurement of certain combinations of couplings and

mixing angles of the heavy partners. For convenience, we provide a shorthand notation

for the 6 different spin assignments, identifying each one with an integer from 1 to 6, as

shown in Table 3-1.

Since we do not know whether the jet represents a quark or an antiquark, then we

also do not know whether the cascade was initiated by a particle D or its antiparticle

D̄. At a pp̄ collider such as the Tevatron, we assume that the fraction f of D particles

produced in the data is equal to the fraction f̄ of antiparticles D̄. However, at a pp

collider such as the LHC we expect an excess of particles over antiparticles, but without

knowledge of the precise value. Therefore f , the fraction of particles (as opposed to

antiparticles) produced from the collision, is in principle an unknown parameter, which

significantly affects the observable mjl+ and mjl− distributions. Previous studies of spin

measurements have fixed f to a specific value for some corresponding study point [?].

However, we decided that this was unjustified. The influence of f on the spin extraction

was considered in [77, 79], where f was left as a floating parameter and consequently

the extraction of the spins became much more difficult. We followed a similar approach,

treating f as a floating input parameter.

We assume that the three SM particles in our signature are all spin-1/2 particles,

whose couplings to the heavy partners at each vertex are unknown. The observed

invariant mass distributions depend on the chirality of those couplings, and this

presents an ambiguity in determining the spins. A given set of measured invariant

mass distributions could in principle be explained by more than one spin configuration,

depending on the chiralities for the fermion couplings. We made those couplings

completely arbitrary, and parametrized them in terms of independent chirality coefficients

57

at each vertex (Equations 1–8 and 1–9). In general, there are three different sets of vertex

coefficients {gL, gR}, one at each vertex, denoted as {cL, cR}, {bL, bR} and {aL, aR} from

left to right in Fig. 1-3A. The couplings {cL, cR} are associated with the D-C-j vertex,

the couplings {bL, bR} are associated with the C-B-`n vertex, and the couplings {aL, aR}

are associated with the B-A-`f vertex. Notice that this parametrization allows the same

coefficients to describe either kind of interaction (Equation 1–8 or 1–9) at each vertex

without discrimination. This allowed us to float these parameters independently of the

spin assignment. Since we were not concerned with the actual cross-section, but only

the shape of the distributions, we found it convenient to unit normalize the couplings

(Equations 3–1).

|aL|2 + |aR|2 = 1 |bL|2 + |bR|2 = 1 |cL|2 + |cR|2 = 1 (3–1)

It turns out that only the relative helicities of the final-state SM particles determine

the shapes of the distributions, so we parametrized the relative chirality of each vertex

with a single tangent (Equations 3–2).

tan ϕa =|aR||aL|

tanϕb =|bR||bL|

tanϕc =|cR||cL|

(3–2)

Since gL and gR are in general complex parameters, each of the SM fermion interactions is

parametrized by four real values. The normalization condition (Equation 3–1) eliminates

one degree of freedom, and Equation 3–2 identifies the interesting degree of freedom

explicitly as tan ϕ. The remaining two degrees of freedom, the complex phases of the

coupling constants, remain arbitrary and cannot be measured from the invariant mass

distributions that we considered. Fortunately, these complex phases have no influence on

the spin determination.

We emphasize that the couplings gL and gR in Equations 1–8 and 1–9 are the

couplings of mass eigenstates. Therefore, whenever there is mixing among the heavy

partner states, our couplings gL and gR are in general matrices, which are related

58

to the couplings g(0)L and g

(0)R of the interaction eigenstates through rotations by the

corresponding mixing angles. Due to this mixing, we do not expect the experimentally

measured couplings gL and gR to be purely chiral, even in models with purely chiral

couplings g(0)L and g

(0)R . The effect of heavy fermion mixing in a specific UED model was

previously considered in [80]. We generalized the analysis to include also arbitrary heavy

boson mixing and arbitrary couplings. One of our main results was identifying which

particular combinations of the model parameters can in principle be measured from the

invariant mass distributions of the three SM fermions (j, `n, and `f ), and to propose the

actual method for measuring them. There are three such combinations, which we called α,

β and γ. Each one of them is observable in principle, and represents some combination of

couplings and mixing angles. It is in this sense that our method yields a measurement of

the couplings and mixing angles of the heavy partners.

Our basic assumption is that the shapes of the invariant mass distributions depend on

the spins of the heavy particles along the decay chain. Pure phase space predicts squared

mass distributions that are, either

• flat throughout for the two “near-type” distributions (m2ll and m2

jln), or

• composed of a flat piece in the low range juxtaposed with a logarithmic piece in thehigh range of the m2

jlfdistribution [11].

Deviation from this pure phase space prediction implies some kind of spin correlations

[62]. However, observing distributions which are consistent with the pure phase space

prediction does not necessarily imply that all particles involved in the decay are scalars –

spin correlations may have been present for the individual helicities, but may have been

washed out when the different helicity contributions were combined to form the observable

distributions [80].

The general approach in previous spin studies has been to compare the data from

a given study point within one specific model to the corresponding data obtained from

another model alternative with different choice of spins for the heavy partners. A common

59

flaw in all such studies was that the couplings and particle-antiparticle fraction were fixed

to be identical in the two models, so that any remaining difference can be interpreted as

a manifestation of spins. Such an approach to spin measurement depends on the specific

model that is assumed, and has little use beyond the assumed model. Since the chirality

parameters ϕa, ϕb and ϕc and the particle-antiparticle fraction f are not independently

measured prior to the spin determination, there is no reason to require same values for

these parameters for each of the different spin configurations under study. Therefore, the

proper question to ask instead is:

Does a particular set of spin assignments fit the data for some choice

of the chirality parameters ϕa, ϕb and ϕc, and for some choice of the

particle-antiparticle ratio f?

We developed some mathematical tools to address this question in a model-independent

way. Our tools do not require the value of the fraction f , nor the values of the chirality

tangents. So, we divided the question into two parts: For a given mass spectrum,

• What are the spins?

• What are the unknown model parameters (i.e. the particle-antiparticle fraction andthe chirality tangents)?

Our method allows us to address the spin question independently of the model parameter

question. However, the actual answer to the spin question may not be unique, i.e. two

sets of spins may fit the data. In particular we found that the model pairs {FSFS, FSFV}

and {FVFS, FVFV} can be indistinguishable, and the indistinguishability depends on the

mass spectrum.

Since we separated the spin dependence from the dependence on the particle-antiparticle

fraction, our method is not limited to pp colliders such as LHC, and is equally applicable

to the Tevatron. In contrast, the lepton charge asymmetry proposed in [62] is greatly

affected by the value of f . For example, the charge asymmetry is predicted to be

identically zero at the Tevatron and has no discriminating power there with regard to

60

spins. In contrast, our technique not only addresses the spin question in a model-independent

way, but also provides a measured constraint on f .

3.1 Classification of Helicity Combinations

Spin correlations in invariant mass distributions were already derived [14]; however,

we extended the calculation by including arbitrary chiral projections for both vertices. We

found that the spin correlation depends on only the relative chiral projections of the two

vertices. Our results assume that all helicities have been summed, for three reasons:

• Since QCD does not discriminate left from right, we expect that the two polarizationsof the initial fermion (or antifermion) are equally likely.

• We assume that the polarizations of the final-state particles will not be observed.

• The numerator algebra for polarization sums provides some nice simplifications,especially with the assumption of massless SM fermions.

Since we assumed massless SM fermions, the chiral projections can actually be interpreted

as the helicities of the SM fermions (or opposite the helicities of the antifermions), so that,

if one wishes to consider the effects of polarization, one can simply make the appropriate

assignments to the chiral coupling coefficients, and then the polarization sums can still

be used. In fact, the results of such calculations led us to the helicity basis functions

(Appendix C).

We found that each two-particle invariant mass distribution can be written in the

form of Equation 3–3.

(

dN

dm̂2p

)

S

=

2∑

I=1

2∑

J=1

K(p)IJ (f, ϕa, ϕb, ϕc)F (p)

S;IJ(m̂2p; x, y, z) (3–3)

The sub- and super-script p denotes one of the five possible SM particle pairs, j`−n ,

j`+n , j`−f , j`+

f , or `+`−, and m̂2p is the squared invariant mass of this pair divided by its

kinematic maximum (Equation 3–4).

m̂2p ≡

m2p

m2 maxp

0 ≤ m̂2p ≤ 1 (3–4)

61

We assumed that m2maxp was already measured (e.g. from kinematic endpoints). The

coefficients KIJ depend on only the particle-antiparticle fraction (f) and the chirality

tangents (tan ϕa, tan ϕb, tan ϕc), and their form is independent of the spin assignment.

The basis functions F (p)S;IJ depend on only the masses (mA, mB, mC , mD, and, of course,

mp), but their form depends on the spin assignment (Table 1-2). It is this dependence of

the form of the basis functions on the spin assignments that allows the discrimination of

spin.

Once the spectrum is measured, the functions F (p)S;IJ only depend on m̂ and provide a

unique basis which can be fit to the data for the mass distribution of each final-state SM

two-particle combination. Since these functions do not depend on the model-dependent

parameters f , ϕa, ϕb and ϕc, this fit can be done in a completely model-independent way,

without any prior knowledge of the couplings. The values of the fit coefficients (KIJ)

represent constraints (i.e. measurements) on the couplings and mixing angles of the heavy

partners.

Figure 3-1 represents the 8 distinct helicity combinations of the 3 final-state SM

particles. Table 3-2 further distinguishes the final-state combinations in terms of the

distinctions between fermion and antifermion, listing 32 different coefficients contributing

to the process of Fig. 1-3A. (We count 32 rather than 64 because we assume that

the leptons are always oppositely charged, thus removing one choice of fermion vs.

antifermion.) The 8 blue-colored entries in Tab. 3-2 were considered in [12, 14, 15]. The

remaining 24 red-colored entries represent our contribution to the analysis.

We identified four types of processes (labeled with IJ) based on relative helicity,

where the values of I, J ∈ {1, 2} refer to the relative helicities of the final-state SM

particles (Fig. 3-1).

I = 1: The helicities of the jet and near lepton are the same. The four processes of “Type1” in [12, 14, 15] fall under this category.

I = 2: The helicities of the jet and near lepton are opposite. The four processes of “Type2” in [12, 14, 15] fall under this category.

62

A B

j

`n

`f

j

`n

`f

j

`n

`f

j

`n

`f

j

`n

`f

j

`n

`f

j

`n

`f

j

`n

`f

C D

Figure 3-1. The 8 different helicity combinations of the final-state SM particles, groupedaccording to relative helicities. For brevity, we use the terms “lepton” and“jet” to refer to fermions or antifermions indiscriminately, and “helicity” refersto physical helicity. The distinction between fermion and antifermion is madeexplicit in Tab. 1-2. A) I = 1, J = 2 ⇒ all three helicities are the same. B)I = 2, J = 2 ⇒ the leptons have the same helicity, but they have oppositehelicity to the jet. C) I = 1, J = 1 ⇒ the leptons have opposite helicity, andthe near lepton has the same helicity as the jet. D) I = 2, J = 1 ⇒ the leptonshave opposite helicity, and the near lepton has opposite helicity to the jet.

J = 1: The helicities of the two leptons are opposite. The eight processes treated in[12, 14, 15] are restricted to this type.

J = 2: The helicities of the two leptons are the same. Processes of this type werecompletely neglected in [12, 14, 15].

All contributions from Tab. 3-2 for a given value of IJ lead to the same shape for a given

invariant mass distribution. The new types of processes (J = 2) give a qualitatively new

functional dependence of the dilepton and j`f invariant mass distributions that was not

exhibited in [12, 14, 15].

3.1.1 Helicity Basis Functions FIJ

We formed 9 invariant mass distributions, shown in Equations 3–5 through 3–9, where

we introduced the factor of 12

on the right hand side for convenience.

(

dN

dm̂2q`±n

)

S

=1

2

2∑

I=1

2∑

J=1

K(q`±n )IJ F (j`n)

S;IJ (m̂2q`±n

) (3–5)

63

Table 3-2. Classification of model parameters according to their contribution to KIJ inEquation 3–3. The combinations shown in blue have been previously consideredin [12, 14, 15]. The combinations shown in red are new contributions from ouranalysis.

I = 1, J = 1 I = 1, J = 2{qL`−L`+

L} {q̄L`+L`−L} {qL`−L`+

R} {q̄L`+L`−R}

f |cL|2|bL|2|aL|2 f̄ |cL|2|bL|2|aL|2 f |cL|2|bL|2|aR|2 f̄ |cL|2|bL|2|aR|2{q̄L`−R`+

R} {qL`+R`−R} {q̄L`−R`+

L} {qL`+R`−L}

f̄ |cL|2|bR|2|aR|2 f |cL|2|bR|2|aR|2 f̄ |cL|2|bR|2|aL|2 f |cL|2|bR|2|aL|2{qR`−R`+

R} {q̄R`+R`−R} {qR`−R`+

L} {q̄R`+R`−L}

f |cR|2|bR|2|aR|2 f̄ |cR|2|bR|2|aR|2 f |cR|2|bR|2|aL|2 f̄ |cR|2|bR|2|aL|2{q̄R`−L`+

L} {qR`+L`−L} {q̄R`−L`+

R} {qR, `+L , `−R}

f̄ |cR|2|bL|2|aL|2 f |cR|2|bL|2|aL|2 f̄ |cR|2|bL|2|aR|2 f |cR|2|bL|2|aR|2I = 2, J = 1 I = 2, J = 2{q̄L, `−L , `+

L} {qL, `+L , `−L} {q̄L, `−L , `+

R} {qL, `+L , `−R}

f̄ |cL|2|bL|2|aL|2 f |cL|2|bL|2|aL|2 f̄ |cL|2|bL|2|aR|2 f |cL|2|bL|2|aR|2{qL, `−R, `+

R} {q̄L, `+R, `−R} {qL, `−R, `+

L} {q̄L, `+R, `−L}

f |cL|2|bR|2|aR|2 f̄ |cL|2|bR|2|aR|2 f |cL|2|bR|2|aL|2 f̄ |cL|2|bR|2|aL|2{q̄R, `−R, `+

R} {qR, `+R, `−R} {q̄R, `−R, `+

L} {qR, `+R, `−L}

f̄ |cR|2|bR|2|aR|2 f |cR|2|bR|2|aR|2 f̄ |cR|2|bR|2|aL|2 f |cR|2|bR|2|aL|2{qR, `−L , `+

L} {q̄R, `+L , `−L} {qR, `−L , `+

R} {q̄R, `+L , `−R}

f |cR|2|bL|2|aL|2 f̄ |cR|2|bL|2|aL|2 f |cR|2|bL|2|aR|2 f̄ |cR|2|bL|2|aR|2

(

dN

dm̂2q̄`±n

)

S

=1

2

2∑

I=1

2∑

J=1

K(q̄`±n )IJ F (j`n)

S;IJ (m̂2q̄`±n

) (3–6)

dN

dm̂2q`±

f

S

=1

2

2∑

I=1

2∑

J=1

K(q`±

f)

IJ F (j`f )S;IJ (m̂2

q`±f

) (3–7)

dN

dm̂2q̄`±

f

S

=1

2

2∑

I=1

2∑

J=1

K(q̄`±

f)

IJ F (j`f )S;IJ (m̂2

q̄`±f

) (3–8)

(

dN

dm̂2``

)

S

=1

2

2∑

I=1

2∑

J=1

K(``)IJ F (``)

S;IJ(m̂2``) (3–9)

The same set of functions F (j`n)S;IJ enter both the {q`n} and {q̄`n} distributions, and

similarly, the same set of functions F (j`f )S;IJ enter both the {q`f} and {q̄`f} distributions.

F (q`n)S;IJ (m̂2) = F (q̄`n)

S;IJ (m̂2) ≡ F (j`n)S;IJ (m̂2) (3–10)

F (q`f )S;IJ (m̂2) = F (q̄`f )

S;IJ (m̂2) ≡ F (j`f )S;IJ (m̂2) (3–11)

64

Equations 3–5 through 3–9 show that all invariant mass distributions can be written

in terms of three sets of basis functions: F (j`n)S;IJ (m̂2), F (j`f )

S;IJ (m̂2) and F (``)S;IJ(m̂2). We define

the basis functions to be unit normalized.

∫ ∞

0

F (j`n)S;IJ (m̂2)dm̂2 = 1 (3–12)

∫ ∞

0

F (j`f )S;IJ (m̂2)dm̂2 = 1 (3–13)

∫ ∞

0

F (``)S;IJ(m̂2)dm̂2 = 1 (3–14)

Half of the processes of type J = 1 have been previously considered in [12, 14, 15], so

that the functions F (p)S,11 and F (p)

S,21 in principle already appear there. We found agreement

with [12, 14, 15] for the case of F (p)S,11 and F (p)

S,21, and we supplemented those results with

the remaining two types of functions F (p)S,12 and F (p)

S,22. Not surprisingly, the basis functions

for the m2jln

distribution that differ only in the subscript J are identical, and similarly,

the basis functions for the m2ll distribution that differ only in the subscript I are identical,

since, in both cases the difference is in the relative helicity of a particle that is not used to

form the invariant mass distribution (Tables C-2 and C-1).

Refs. [12, 14, 15] missed the functions F (``)S,12 and F (``)

S,22 as a consequence of their

underlying model assumption: their studies assumed very specific fixed values of the

chirality coefficients (namely, cL = 1, cR = 0, bL = 0, bR = 1, aL = 0, aR = 1 for

the supersymmetry example and cL = 1, cR = 0, bL = 1, bR = 0, aL = 1, aR = 0 for

the UED example) and therefore their results, while correct, are only valid within this

limited model-dependent context. In contrast, the complete set of functions F (``)S,IJ for all

4 helicity combinations allows us to address the spin question in a model-independent

fashion. Similar remarks hold for the F (j`f )S;IJ functions. Again, the functions F (j`f )

S;11 and

F (j`f )S;21 agree1 with the results of [14], while the functions F (j`f )

S;12 and F (j`f )S;22 are new.

1 with the exception of a typo in [14] that was verified by the authors

65

However, whether (and what type of) relations exist between the four functions F (j`f )S;IJ

varies from case to case (i.e. the value of the spin configuration index S). In the three

cases (SFSF, FSFS and FSFV) where there is an intermediate heavy scalar between the

emitted jet and far lepton, the F (j`f )S;IJ set is again reduced to only two unique functions;

however, symmetry under either I or J still depends on the specific spin assignments. This

is again easy to understand. For example, when the two leptons are connected by a scalar

propagator, then the distribution should be insensitive to the relative helicities of the two

leptons (the value of J), which accounts for the equivalence of the spin basis functions in

Equations C–1 and C–2. Similar remarks hold for Equations C–4 through C–8 with the

replacements `f → j and J → I. It is not surprising that all 4 spin basis functions F (j`f )S;IJ

are independent when both pairs of adjacent final-state SM particles are connected by a

particle with nontrivial spin (Equations C–10 through C–23).

3.1.2 Helicity Coefficients KIJ

Using the factors from Table 3-2, for the Kq`11 and K q̄`

11 coefficients we obtain

Equations 3–15 through 3–18.

K(q`−n )11 = K

(q`+f

)

11 = f |cL|2|bL|2|aL|2 + f |cR|2|bR|2|aR|2 (3–15)

K(q̄`−n )11 = K

(q̄`+f

)

11 = f̄ |cL|2|bR|2|aR|2 + f̄ |cR|2|bL|2|aL|2 (3–16)

K(q`+n )11 = K

(q`−f

)

11 = f |cL|2|bR|2|aR|2 + f |cR|2|bL|2|aL|2 (3–17)

K(q̄`+n )11 = K

(q̄`−f

)

11 = f̄ |cL|2|bL|2|aL|2 + f̄ |cR|2|bR|2|aR|2 (3–18)

The Kq`12 and K q̄`

12 coefficients can be obtained from Equations 3–15 through 3–18 by

substituting aL ↔ aR.

K(q`−n )12 = K

(q`+f

)

12 = f |cL|2|bL|2|aR|2 + f |cR|2|bR|2|aL|2 (3–19)

K(q̄`−n )12 = K

(q̄`+f

)

12 = f̄ |cL|2|bR|2|aL|2 + f̄ |cR|2|bL|2|aR|2 (3–20)

K(q`+n )12 = K

(q`−f

)

12 = f |cL|2|bR|2|aL|2 + f |cR|2|bL|2|aR|2 (3–21)

K(q̄`+n )12 = K

(q̄`−f

)

12 = f̄ |cL|2|bL|2|aR|2 + f̄ |cR|2|bR|2|aL|2 (3–22)

66

Replacing f ↔ f̄ and q ↔ q̄ in Equations 3–15 through 3–18 gives the Kq`21 and K q̄`

21

coefficients.

K(q̄`−n )21 = K

(q̄`+f

)

21 = f̄ |cL|2|bL|2|aL|2 + f̄ |cR|2|bR|2|aR|2 (3–23)

K(q`−n )21 = K

(q`+f

)

21 = f |cL|2|bR|2|aR|2 + f |cR|2|bL|2|aL|2 (3–24)

K(q̄`+n )21 = K

(q̄`−f

)

21 = f̄ |cL|2|bR|2|aR|2 + f̄ |cR|2|bL|2|aL|2 (3–25)

K(q`+n )21 = K

(q`−f

)

21 = f |cL|2|bL|2|aL|2 + f |cR|2|bR|2|aR|2 (3–26)

Replacing aL ↔ aR in Equations 3–23 through 3–26 yields the Kq`22 and K q̄`

22 coefficients.

K(q̄`−n )22 = K

(q̄`+f

)

22 = f̄ |cL|2|bL|2|aR|2 + f̄ |cR|2|bR|2|aL|2 (3–27)

K(q`−n )22 = K

(q`+f

)

22 = f |cL|2|bR|2|aL|2 + f |cR|2|bL|2|aR|2 (3–28)

K(q̄`+n )22 = K

(q̄`−f

)

22 = f̄ |cL|2|bR|2|aL|2 + f̄ |cR|2|bL|2|aR|2 (3–29)

K(q`+n )22 = K

(q`−f

)

22 = f |cL|2|bL|2|aR|2 + f |cR|2|bR|2|aL|2 (3–30)

The coefficients K(``)IJ for the dilepton distributions can be expressed in various ways,

for example in terms of the coefficients involving the near lepton `n (Equation 3–31),

in terms of the coefficients involving the far lepton `f (Equation 3–32), in terms of the

coefficients involving the positively charged lepton `+ (Equation 3–33), or in terms of the

coefficients involving the negatively charged lepton `− (Equation 3–34).

K(``)IJ = K

(q`−n )IJ + K

(q̄`−n )IJ + K

(q`+n )IJ + K

(q̄`+n )IJ (3–31)

= K(q`−

f)

IJ + K(q̄`−

f)

IJ + K(q`+

f)

IJ + K(q̄`+

f)

IJ (3–32)

= K(q`+n )IJ + K

(q̄`+n )IJ + K

(q`+f

)

IJ + K(q̄`+

f)

IJ (3–33)

= K(q`−n )IJ + K

(q̄`−n )IJ + K

(q`−f

)

IJ + K(q̄`−

f)

IJ (3–34)

67

3.2 Observable Distributions

3.2.1 Invariant Mass Distributions in the Helicity Basis {FIJ}

We made the conservative assumption (which also happens to be true in many

models) that q is a light flavor quark, so that the experimental distinction between a q and

q̄ cannot be made. So, the observable distributions must be the sum of the distributions

that differ only in q vs. q̄ (Equation 3–35 and 3–36).

(

dN

dm̂2j`±n

)

S

=

(

dN

dm̂2q`±n

)

S

+

(

dN

dm̂2q̄`±n

)

S

≡ 1

2

2∑

I=1

2∑

J=1

K(j`±n )IJ F (j`n)

S;IJ (m̂2j`±n

) (3–35)

dN

dm̂2j`±

f

S

=

dN

dm̂2q`±

f

S

+

dN

dm̂2q̄`±

f

S

≡ 1

2

2∑

I=1

2∑

J=1

K(j`±

f)

IJ F (j`f )S;IJ (m̂2

j`±f

) (3–36)

Since the spin basis functions F (p)S;IJ do not depend on the q-q̄ ambiguity, the new set of

coefficients K(j`±n )IJ and K

(j`±f

)

IJ are simply related to the old ones by Equation 3–37 and

3–38.

K(j`±n )IJ = K

(q`±n )IJ + K

(q̄`±n )IJ (3–37)

K(j`±

f)

IJ = K(q`±

f)

IJ + K(q̄`±

f)

IJ (3–38)

Substituting Equations 3–15 through 3–30 into Equations 3–37 and 3–38 gives Equations

3–39 through 3–42.

K(j`−n )11 = (f |cL|2 + f̄ |cR|2)|bL|2|aL|2 + (f̄ |cL|2 + f |cR|2)|bR|2|aR|2 (3–39)

K(j`−n )12 = (f |cL|2 + f̄ |cR|2)|bL|2|aR|2 + (f̄ |cL|2 + f |cR|2)|bR|2|aL|2 (3–40)

K(j`−n )21 = (f̄ |cL|2 + f |cR|2)|bL|2|aL|2 + (f |cL|2 + f̄ |cR|2)|bR|2|aR|2 (3–41)

K(j`−n )22 = (f̄ |cL|2 + f |cR|2)|bL|2|aR|2 + (f |cL|2 + f̄ |cR|2)|bR|2|aL|2 (3–42)

The remaining K(j`)IJ coefficients are related to these by Equations 3–43 through 3–46.

K(j`−n )11 = K

(j`+f

)

11 = K(j`+n )21 = K

(j`−f

)

21 (3–43)

K(j`−n )12 = K

(j`+f

)

12 = K(j`+n )22 = K

(j`−f

)

22 (3–44)

68

Figure 3-2. A contour plot of cos ϕ̃c as a function of cosϕc and f .

K(j`−n )21 = K

(j`+f

)

21 = K(j`+n )11 = K

(j`−f

)

11 (3–45)

K(j`−n )22 = K

(j`+f

)

22 = K(j`+n )12 = K

(j`−f

)

12 (3–46)

The dependence of these coefficients on f and ϕc always appears through the combinations

f |cL|2+ f̄ |cR|2 = f cos2 ϕc+ f̄ sin2 ϕc and f̄ |cL|2+f |cR|2 = f̄ cos2 ϕc+f sin2 ϕc. We therefore

introduced an alternative model parameter ϕ̃c defined by Equation 3–47, 3–48, or 3–49.

cos2 ϕ̃c = f cos2 ϕc + f̄ sin2 ϕc (3–47)

sin2 ϕ̃c = f̄ cos2 ϕc + f sin2 ϕc (3–48)

cos 2ϕ̃c = (f − f̄) cos 2ϕc (3–49)

Fig. 3-2 shows the relationship between the old parameters, {f, ϕc} and the newly

introduced parameter ϕ̃c.

The K(j`)IJ coefficients are unit-normalized (Equations 3–50 and 3–51).

2∑

I=1

2∑

J=1

K(j`±n )IJ = 1 (3–50)

69

2∑

I=1

2∑

J=1

K(j`±

f)

IJ = 1 (3–51)

Given Equations 3–12 through 3–14 of our basis functions F (p)S;IJ , Equations 3–50 and 3–51

imply that the {j`±n } and {j`±f } distributions (Equations 3–35 and 3–36) are half-unit

normalized (Equations 3–52 and 3–53).

∫ ∞

0

(

dN

dm̂2j`±n

)

S

dm̂2j`±n

=1

2, (3–52)

∫ ∞

0

dN

dm̂2j`±

f

S

dm̂2j`±

f

=1

2. (3–53)

The last (and by far most complicated) step in deriving the observable distributions is

to form the jet-lepton distributions which are based on definite lepton charge. This is done

by adding together the contributions from the m2jln

and m2jlf

for each charge according to

Equation 3–54.(

dN

dm2j`±

)

S

≡(

dN

dm2j`±n

)

S

+

dN

dm2j`±

f

S

(3–54)

Notice that Equation 3–54 is expressed in terms of a dimensionful invariant mass, m2jl,

rather than a unit-normalized mass parameter, m̂2jl (e.g. Equations 3–52 and 3–53).

However, we preferred our distributions to be given in terms of the unit-normalized

mass parameter m̂. To this end, we normalized the generic jet-lepton mass to the m2maxjl(hi)

endpoint (Equation 2–2) according to Equation 3–55.

m̂2jl± ≡

m2jl±

m2 maxjl(hi)

(3–55)

We also used the ratios in Equations 3–56 and 3–57.

rn ≡mmax

j`

mmaxj`n

(3–56)

rf ≡mmax

j`

mmaxj`f

(3–57)

70

Equation 3–58 shows the combined jet-lepton distributions for each lepton charge in terms

of the unit-normalized squared mass variable.

(

dN

dm̂2j`±

)

S

=1

2

2∑

I=1

2∑

J=1

(

K(j`±n )IJ r2

nF(j`n)S;IJ (r2

nm̂2j`±) + K

(j`±f

)

IJ r2fF

(j`f )S;IJ (r2

fm̂2j`±)

)

(3–58)

Whenever the two endpoints mmaxj`n

and mmaxj`f

are different, one of the two ratios rn and rf

is guaranteed to exceed 1, so that there is a range of masses for which the corresponding

argument (r2nm̂

2j` or r2

fm̂2j`) in the corresponding spin basis function (F (j`n)

S;IJ or F (j`f )S;IJ )

exceeds 1. In this mass range, only the other spin basis function (F (j`f)S;IJ or F (j`n)

S;IJ )

contributes. Both of the observable distributions (Equation 3–58) are unit normalized.

∫ ∞

0

(

dN

dm̂2j`±

)

S

dm̂2j`± = 1 (3–59)

We obtained analytical expressions for the three observable mass distributions, the

dilepton distribution (Equation 3–9) and the 2 jet-lepton distributions (Equation 3–58),

that exhibit spin correlations. All three of our formulas are unit normalized and can be

readily rescaled for the actual observed number of events (which is the same for each of

the three distributions). Our formulas are written in terms of a set of known functions

F (p)S;IJ , which are explicitly defined in Appendix C. The coefficients K

(p)IJ appearing in our

formulas depend on three model-dependent parameters: ϕa, ϕb and ϕ̃c (Equations 3–2

and 3–47 through 3–49), but they do not depend on the spin assignment, S. These results

could in principle be used for spin determination, but we chose an alternative approach.

3.2.2 Invariant Mass Distributions in the Observable Basis {Fα,Fβ,Fγ,Fδ}

We actually found a much more convenient set of spin basis functions that are simple

linear combinations of the original helicity basis functions F (p)S;IJ given by Equations 3–60

through 3–63.

F (p)S;α =

1

4

{

F (p)S;11 −F (p)

S;12 + F (p)S;21 −F (p)

S;22

}

(3–60)

F (p)S;β =

1

4

{

F (p)S;11 + F (p)

S;12 − F (p)S;21 −F (p)

S;22

}

(3–61)

71

F (p)S;γ =

1

4

{

F (p)S;11 −F (p)

S;12 − F (p)S;21 + F (p)

S;22

}

(3–62)

F (p)S;δ =

1

4

{

F (p)S;11 + F (p)

S;12 + F (p)S;21 + F (p)

S;22

}

(3–63)

Whereas the original helicity basis functions are distinguished directly in terms of the

4 different relative helicity combinations IJ (which we assume are unobservable), the

new basis functions (Equations 3–60 through 3–63) are distinguished in terms of the

model-dependence of the observable distributions. Thus, the new basis emphasizes what

part of the spin correlation is model-dependent, and in what way. Furthermore, this

new basis (Equations 3–60 through 3–63) reveals that there can indeed be a nontrivial

model-independent spin correlation. The explicit form of these basis functions is given in

Appendix D.

The advantage of the new set of basis functions becomes apparent if we rewrite our

results for the different invariant mass distributions according to Equations 3–64 through

3–66.

(

dN

dm̂2``

)

S

≡ L+−S = F (``)

S;δ (m̂2``) + αF (``)

S;α (m̂2``) (3–64)

(

dN

dm̂2j`±n

)

S

=1

2

{

F (j`n)S;δ (m̂2

j`n) ∓ βF (j`n)

S;β (m̂2j`n

)

}

(3–65)

dN

dm̂2j`±

f

S

=1

2

{

F (j`f )S;δ (m̂2

j`f) + αF (j`f)

S;α (m̂2j`f

)

±βF (j`f )

S;β (m̂2j`f

) ± γF (j`f )S;γ (m̂2

j`f)

}

(3–66)

The three new parameters {α, β, γ} are observable model parameters that play an

analogous role to the coefficients KIJ of the old basis. They are related to the chiral

couplings according to Equations 3–67 through 3–69.

α ≡(

|aL|2 − |aR|2) (

|bL|2 − |bR|2)

(3–67)

β ≡ (2f − 1)(

|bL|2 − |bR|2) (

|cL|2 − |cR|2)

(3–68)

72

γ ≡ (2f − 1)(

|aL|2 − |aR|2) (

|cL|2 − |cR|2)

(3–69)

Each one of the α, β and γ parameters can take values in the interval [−1, 1]. However, α,

β and γ are not completely unrelated. Given their definitions, they must satisfy certain

relations among themselves, and those are listed in Appendix E.

The F (p)S;δ term appears without any model coefficient. For many spin assignments

and final-state particle combinations, F (p)S;δ simply gives the invariant mass distribution as

predicted by pure phase space, i.e. without spin correlations. This is true whenever the

intermediate heavy partners are scalars or fermions. However, if a heavy vector appears

among the intermediate heavy partners, then the F (p)S;δ function always deviates from

pure phase space. Furthermore, this deviation leads to a deviation of the entire invariant

mass distribution that cannot be canceled by any choice of α, β and γ. Therefore, one

of our general conclusions is that an intermediate heavy vector boson always leads to

deviations from pure phase space and conversely, whenever a pure phase space distribution

is observed, a heavy vector boson is ruled out.

Again the {j`n} and {j`f} distributions (Equations 3–65 and 3–66) are not separately

observable, and instead must be combined in order to form the observable {j`+} and

{j`−} distributions. The {j`f} distribution depends on all three parameters α, β and γ.

Combining Equations 3–65 and 3–66 gives the observable jet-lepton distributions that are

distinguished by the charge of the lepton (Equation 3–70). Both of these distributions

depend on all three model parameters.

(

dN

dm̂2j`±

)

S

=1

2

{

r2nF

(j`n)S;δ (r2

nm̂2j`±) + r2

fF(j`f)S;δ (r2

fm̂2j`±) + αr2

fF(j`f )S;α (r2

fm̂2j`±) (3–70)

±γr2fF

(j`f )S;γ (r2

fm̂2j`±) ± βr2

fF(j`f )

S;β (r2fm̂

2j`±) ∓ βr2

nF(j`n)S;β (r2

nm̂2j`±)

}

However, the same β and γ terms in Equation 3–70 appear with opposite signs in the

{j`+} and the {j`−} distribution. This suggests that, instead of the two individual

distributions(Equation 3–70) we should consider their sum (Equation 3–71) and difference

73

(Equation 3–72).

S+−S ≡

(

dN

dm̂2j`+

)

S

+

(

dN

dm̂2j`−

)

S

= r2nF

(j`n)S;δ (r2

nm̂2j`) + r2

fF(j`f )S;δ (r2

fm̂2j`) + αr2

fF(j`f )S;α (r2

fm̂2j`) (3–71)

D+−S ≡

(

dN

dm̂2j`+

)

S

−(

dN

dm̂2j`−

)

S

= γr2fF

(j`f )S;γ (r2

fm̂2j`) + βr2

fF(j`f )S;β (r2

fm̂2j`) − βr2

nF(j`n)S;β (r2

nm̂2j`) (3–72)

The normalization conditions for the newly defined quantities S+−S and D+−

S are given in

Equations 3–73 and 3–74.

∫ ∞

0

S+−S dm̂2

j` = 2 (3–73)

∫ ∞

0

D+−S dm̂2

j` = 0 (3–74)

Equations 3–71 and 3–72 reveal one of our most important results – that the sum

of the two jet-lepton distributions depends on a single model-dependent parameter, and

this is the same parameter α that determines the dilepton distribution. Therefore, once

α is measured from the relatively clean dilepton data, the observable S+−S distribution is

completely specified. The D+−S distribution can provide a measurement of the other two

model-dependent parameters β and γ. However, the γ parameter cannot be determined

if S = 1, 2, 3, since the basis function F (j`)S;γ vanishes identically so that D+−

S becomes

γ-independent (Appendix D). Similarly, the parameter β cannot be determined if S = 2, 3.

We believe that our basis functions provide a superior alternative to the lepton charge

asymmetry (Equation 3–75) that has been suggested in previous studies [62].

A+−S ≡ D+−

S

S+−S

(3–75)

In general, A+−S is more model-dependent than either S+−

S or D+−S . While S+−

S depends

on the single model parameter α, and D+−S depends on the two other model parameters β

74

and γ, A+−S depends on all three of these model parameters α, β and γ. More importantly,

A+−S is a single distribution, derived from S+−

S and D+−S , therefore it is bound to contain

less information than the two separate distributions S+−S and D+−

S . The information gain

from using the separate S+−S and D+−

S is most striking for the case of a pp̄ collider like the

Tevatron (Appendix F.6).

3.3 Determination of Model Parameters {α, β, γ}

Our spin determination assumes the general form of the observable invariant mass

distributions given in Equations 3–64, 3–71, and 3–72. The functions F (p)S;α, F (p)

S;β, F(p)S;γ

and F (p)S;δ are given in Appendix D, while the model parameters α, β and γ are related

to the chiral couplings and particle-antiparticle fraction (Equations 3–67 through

3–69). Given the data for the three observable distributions (Equations 3–64, 3–71,

and 3–72), our technique is to fit for the unknown model parameters, α, β and γ,

(Appendix E) considering each of the 6 different spin assignments (Tab. 1-2) one at a

time. Our technique results in 6 different sets of “best fit” values for the model parameters

(Appendix F), and an accompanying measure for the goodness of fit in each case. The

goodness of fit for each spin assignment is an indication of the consistency of the data

with that spin assignment, and one can also assign confidence level probabilities to those

statements. This procedure is model-independent, and in fact provides independent

measurements of the model parameters (Table 3-3). For example, when all three model

parameters α, β and γ are measured and found to be non-zero, the relative chiralities of

the three vertices (Fig. 1-3A) are determined according to Equations 3–76 through 3–78.

(

|aL|2 − |aR|2)2

=αγ

β(3–76)

(

|bL|2 − |bR|2)2

=αβ

γ(3–77)

(

|cL|2 − |cR|2)2

=βγ

(2f − 1)2α(3–78)

75

Since the spin correlation depends on only relative chiralities, we cannot determine the

actual handedness of the vertices (as indicated by the squaring in Equations 3–76 through

3–78), only how strongly chiral they are.

While in general α, β and γ can be either positive or negative, Equations 3–67

through 3–69 imply that αβγ

, αγβ

, and βγα

are always non-negative. Furthermore, it follows

that |αβ| ≤ |γ|, |βγ| ≤ |α| and |γα| ≤ |β|. We see that for any given measurement of

α, β and γ, the chirality of the lepton vertices can be uniquely determined, up to the

ambiguity between left-handed and right-handed. In other words, the particle-antiparticle

ambiguity only affects the determination of the chirality of the jet vertex, which is not

at all surprising. The chirality of the jet vertex is not uniquely determined, and instead

is parametrized as a function of f (Equation 3–78). Although f cannot be determined,

consistency (Equations 3–67 through 3–69 and 3–76 through 3–78) restricts the allowed

values of f to be in the range given in Equation 3–79.

0 ≤ f ≤ 1

2

(

1 −√

βγ

α

)

or1

2

(

1 +

√

βγ

α

)

≤ f ≤ 1 (3–79)

At a pp collider like the LHC, in general we expect f > 12, so we would select the higher

f range, while the lower f range would be relevant for a hypothetical p̄p̄ collider. While

Equation 3–79 is only a restriction on the value of f at the LHC, if the measured values

of α, β and γ happen to be such that |βγ| ≈ |α|, then f is restricted to be very close to

unity, and the jet vertex must be strongly chiral.

3.4 Twin Spin Scenarios FSFS/FSFV and FVFS/FVFV

Consulting Appendix D, one can see that the spin basis functions for FSFS (S = 2)

and FSFV (S = 3) generally satisfy the relationships in Equations 3–80 through 3–83.

F (p)3;α = F (p)

2;α

1 − 2z

1 + 2z(3–80)

F (p)3;β = F (p)

2;β = 0 (3–81)

F (p)3;γ = F (p)

2;γ = 0 (3–82)

76

Table 3-3. Available measurements of the model parameters α, β and γ for each of the 6different spin assignments. For SFSF, only β can be determined. For FSFS andFSFV, only α can be determined. For the other three spin assignments, allthree model parameters can be determined.

Spin Parameters measured from distribution:chain L+− S+− D+− L+− ⊕ S+− ⊕ D+−

SFSF − − β βFSFS α α − αFSFV α α − αFVFS α α β, γ α, β, γFVFV α α β, γ α, β, γSFVF α α β, γ α, β, γ

F (p)3;δ = F (p)

2;δ (3–83)

Therefore Equation 3–84 is sufficient to guarantee that all three invariant mass distributions

(Equations 3–85 through 3–87), in the case of FSFS (S = 2) and FSFV (S = 3), are

exactly the same.

α2 = α31 − 2z

1 + 2z(3–84)

L+−2

(

m̂2``; x, y, z, α3

1 − 2z

1 + 2z

)

= L+−3

(

m̂2``; x, y, z, α3

)

(3–85)

S+−2

(

m̂2j`; x, y, z, α3

1 − 2z

1 + 2z

)

= S+−3

(

m̂2j`; x, y, z, α3

)

(3–86)

D+−2

(

m̂2j`; x, y, z, β2, γ2

)

= D+−3

(

m̂2j`; x, y, z, β3, γ3

)

. (3–87)

Equations 3–85 through 3–87 hold identically for any values of the five parameters α3, β3,

γ3, β2 and γ2. The ability to discriminate depends on the values of α2 and z (Equation

3–84). α is defined in the range [−1, 1], while z is defined in (0, 1). Then, for any given

value of α3 ∈ [−1, 1], α2 falls into its allowed range, and an exact duplication is inevitable.

However, not every value of α2 leads to a valid solution for α3, since for large enough

values of |α2|, the value of |α3| would exceed 1, which is not allowed. The two models will

always be confused with each other if nature happens to choose FSFV (S = 3), whereas

if nature chooses FSFS (S = 2), then the confusion arises only if nature also satisfies

77

Equation 3–88.

|α2| ≤∣

∣

∣

∣

1 − 2z

1 + 2z

∣

∣

∣

∣

(3–88)

A similar ambiguity also exists between the FVFS (S = 4) and FVFV (S = 5) spin

assignments. The spin basis functions exhibit the relationships shown in Equations 3–89

through 3–92.

F (p)5;α = F (p)

4;α

1 − 2z

1 + 2z(3–89)

F (p)5;β = F (p)

4;β (3–90)

F (p)5;γ = F (p)

4;γ

1 − 2z

1 + 2z(3–91)

F (p)5;δ = F (p)

4;δ (3–92)

So, if Equations 3–93 through 3–95 can be satisfied by the model parameters, then these

two spin assignments cannot be distinguished.

α4 = α51 − 2z

1 + 2z(3–93)

β4 = β5 (3–94)

γ4 = γ51 − 2z

1 + 2z(3–95)

If nature happens to choose FVFV, then the model will always be confused with FVFS.

However, if nature chooses FVFS, the confusion arises only if nature also satisfies

Equations 3–96 and 3–97.

|α4| ≤∣

∣

∣

∣

1 − 2z

1 + 2z

∣

∣

∣

∣

(3–96)

|γ4| ≤∣

∣

∣

∣

1 − 2z

1 + 2z

∣

∣

∣

∣

(3–97)

However, the ambiguity is resolved unless the values of α4, β4 and γ4 also satisfy the

domain constraints (Equations E–2 through E–5).

78

CHAPTER 4CONCLUSIONS

4.1 Kinematic Boundaries

Once we ignored the endpoint mmaxjll we only needed to consider 4 different cases,

Ri, i = 1, 2, 3, 4, as illustrated with the color-coded regions in Fig. 1-4. In contrast,

previous studies that used the mmaxjll endpoint [11, 31, 32] were forced to consider all

11 different possibilities (Njll, Njl) shown in Fig. 1-4. We provide analytical inversion

formulas (Equations 2–13 through 2–16 with 2–17 through 2–21) that allow the immediate

calculation of the mass spectrum mA, mB, mC and mD in terms of a set of four measured

invariant mass endpoints {a, c, d, e} (Equations 2–7 through 2–11), eliminating the need

for numerical scanning of solution space. Our formulas are valid in all parameter space

regions, since we do not use the endpoint b = mmaxjll , which is problematic in regions (3,1),

(3,2) and (2,3) (Equation 2–4).

We investigated the possibility of finding multiple solutions for the mass spectrum,

in the case of a perfect experiment that measures the values of all five invariant mass

endpoints {a, b, c, d, e} (Equations 2–7 through 2–11) with zero error bars. We found

that there is a certain portion of parameter space (Fig. 2-4) where exact duplication

occurs, i.e. two very different mass spectra yield identical values for all five measurements

{a, b, c, d, e}. The situation only worsens if the inevitable experimental errors on the

endpoint measurements are included in the analysis.

We advertise a new approach to the study of the usual single-variable invariant mass

distributions. In particular, we point out that the multivariate invariant mass distributions

contain more useful information than the individual single-variable histograms that

are usually considered. As two illustrative examples, we considered the two-variable

(m2jl(lo), m

2jl(hi)) and (m2

ll, m2jll) distributions (Figs. 2-7 and 2-9). These two choices

are actually quite natural, since m2jl(lo) and m2

jl(hi) are already related to each other

through their definitions in terms of m2jln

and m2jlf

(Equations 2–1 and 2–2), and since

79

the (m2ll, m

2jll) distribution provides a convenient way to see the threshold e (Fig. 2-9).

The (m2jl(lo), m

2jl(hi)) distribution is always bounded by straight lines (Fig. 2-7), while the

(m2ll, m

2jll) distribution is bounded by a hyperbola (Equations 2–58 and 2–59), and (in the

on-shell case) by the straight line UV (Fig. 2-9). One could also consider other choices of

two-dimensional distributions, for example {m2ll, m

2jl}, {m2

jl, m2jll} [70], or {m2

ll, m2jl(lo)},

{m2ll, m

2jl(hi)} [71]. Those distributions also provide discrimination between the “near” and

“far” lepton endpoints, and will contribute even more data points.

The boundary lines exhibit two useful features. First, the shapes of the kinematic

boundaries are characteristic of the corresponding parameter space region Ri (Fig. 2-7).

This observation can be used to identify the relevant parameter space region, and resolve

potential ambiguities in the extraction of the mass spectrum. Second, the boundary

lines exhibit a number of special points, whose coordinates can in principle be measured,

providing additional experimental information about the mass spectrum. Of course, the

locations of the special points are not independent from each other, since they are all

given in terms of only 4 input masses. Nevertheless, it is certainly preferable to have as

many measurements as possible. The inversion formulas may simplify considerably if we

replace e, whose analytical expression (Equation 2–11) is rather complicated, with some of

the other measurements that can be made from the two-variable boundary lines. One such

example is shown in Appendix B.

An important advantage of the two-variable approach is that one can readily resolve

the ambiguity between the endpoints m2maxjlf

and m2maxjln

. Indeed, the unobservable m2 maxjln

and m2maxjlf

endpoints are Region-independent (among the three on-shell Regions), and

can be directly observed from the boundary lines. Another advantage is that one can

perform a fit to the boundary lines of the scatter plot instead of a fit to the endpoints in

the one-dimensional distributions. This improves the precision of the mass determination,

as demonstrated in [71] using the SPS1a SUSY benchmark example.

80

Perhaps the most pressing consideration is how well the method proposed here

will survive the experimental complications of a full-blown analysis, including detector

simulation, backgrounds (both SM and BSM), finite widths of the particles B, C and D,

varying population density of the scatter plots, etc.. The CMS SUSY working group is

currently investigating these issues, and a future publication is planned to display the

results.

4.2 Spin Correlations

We found analytical expressions for the observable two-particle invariant mass

distributions in m2jl+, m2

jl− and m2ll exhibiting spin correlations between the two particles.

We presented our expressions in the model-independent context of arbitrary couplings and

arbitrary particle-antiparticle fraction, for 6 possible spin assignments (Table 1-2). Our

results generalize those of Refs. [11, 12, 14, 15]. We also derived the exact combinations of

couplings that can be determined as a byproduct of the spin measurement.

We demonstrated our method using the SPS1a mass spectrum, couplings, and

particle-antiparticle fraction to generate the simulated invariant mass distributions. By

assuming that the distributions come from each one of our 6 spin scenarios (Table 1-2)

in turn, we then determined if the other 5 spin assignments could lead to identical

invariant mass distributions for some (possibly different) choice of couplings and

particle-antiparticle fraction. We also proved the general existence of ambiguity in the

determination of the {FSFS, FSFV} and {FVFS, FVFV} model pairs (Table 4-1) by

deriving the relation between the couplings and mixing angles within each pair of models

that would result in identical observable mass distributions for those model pairs. This

ambiguity is exhibited in our case studies for the FSFV and FVFV spin assignments

(Appendices F.3 and F.4).

We considered the example of a quark jet followed by two leptons, which is commonly

encountered in models of supersymmetry or extra dimensions. The three observable

invariant mass distributions for each pair of well-defined objects (in our case {`+`−},

81

Table 4-1. Expected outcomes from our spin discrimination analysis, barring numericalaccidents due to very special mass spectra. The 4 problem entries are coloredred for emphasis. The two cases labeled “maybe” correspond to the potentialconfusion of an FSFS (FVFS) chain with an FSFV (FVFV) chain that iscertain to occur for only a restricted range of the model parameters (Equation3–88 or Equations 3–96 and 3–97).

Data Is this model certain to fit the data?from SFSF FSFS FSFV FVFS FVFV SFVFSFSF of course no no no no noFSFS no of course maybe no no noFSFV no yes of course no no noFVFS no no no of course maybe noFVFV no no no yes of course noSFVF no no no no no of course

{j`+} and {j`−}) were derived. In order to remove the combinatorial ambiguities, one can

perform flavor subtraction on the leptons and mixed-event subtraction on the jet. One

may also apply cuts in order to suppress any SM and new physics backgrounds. The end

product from this step is the 3 distributions L+−, S+− and D+− (Equations 3–64, 3–71,

and 3–72).

No single method is universally applicable, therefore the availability of different and

complementary techniques is important. The success of any given method depends on the

specific new physics scenario. We identify some features of our method that are likely to

make it relevant and successful, if a missing energy signal of new physics is seen at the

LHC and/or the Tevatron.

• Many of the existing techniques for spin determinations (see, for example, [73, 74, 84,85]) have been originally developed in the context of lepton colliders, where the totalcenter of mass energy in each event is known. Consequently, at hadron colliders,those methods are applicable only if the events can be fully reconstructed. In newphysics scenarios with dark matter WIMPs, this appears to be rather challenging,since there are two invisible WIMP particles escaping the detector. In some specialcircumstances, where two sufficiently long decay chains can be identified in theevent, full reconstruction might be possible [39, 46, 51], but in any case, this appearsto require very large data samples. In contrast, our method relies on invariant massdistributions, which are frame-independent, and we do not require full reconstructionof the events. Furthermore, our method does not require the presence of two

82

separate decay chains in the event, and can be in principle also applied to theassociated production of a WIMP with only one other heavy partner.

• One major advantage of our method in comparison to various event countingtechniques [35, 76, 83, 86] is that we do not need to know anything about theproduction cross-sections for the different parton-level initial states, the branchingfractions, the experimental efficiencies, etc.. Our method relies on only the shapes ofthe distributions from the sequential decay (Fig. 1-3A), and so it is insensitive to themodel-dependent cross-section and branching ratio.

• In comparison to studies that have also used invariant mass distributions [12, 14,15, 17, 18, 62, 75, 77–82], the main advantage of our approach is that we make noassumptions about the type of couplings in each vertex of Fig. 1-3A, or about theparticle-antiparticle fraction f . As a result, we even devised measurements of certaincombinations of the couplings and the f parameter.

4.3 General Remarks

Our analysis contains two general themes: model-independence and ambiguities.

Our basic conclusion regarding these themes is that, for a fixed set of observables,

determination of new physics becomes more ambiguous as the method used to make

the determination becomes more model-independent. However, we realized a fortunate

side-effect of model-independence. By forcing ourselves to approach the mass and spin

determination in a model-independent way, we encountered alternative measurements

that can be used to resolve some ambiguity in the determination. Furthermore, whereas

previous techniques require artificial model assumptions, our technique automatically

restricts the possible models of new physics (e.g. discrimination between on-shell and

off-shell, and determination of α, β, and γ).

83

APPENDIX AA BSM EXAMPLE: SUSY

SUSY is a symmetry of the finite representations of the Lorentz Group that may

be imposed on the particle physics Lagrangian. It requires the action functional of

particle physics to be invariant under continuous exchanges of finite Lorentz Group

representations that differ by half-integer spin [1]. However, the generators of SUSY

are also required to commute with the generators of the internal symmetry group of

the SM [1], SU(3)C × SU(2)W × U(1)Y .1 So, lepton and quark fields are each in

distinct representations of SUSY,the Higgs field is in a distinct representation of SUSY,

and the gauge field is in a distinct representation of SUSY. (In fact, left-handed and

right-handed fields are each in distinct representations of SUSY, because they are distinct

representations of SU(2)W .) Thus, there are no fields in the SM that can be exchanged

with each other under SUSY, which implies that the number of fields required for SUSY

must be at least twice the number of fields required for the SM [90]. In other words, SUSY

requires every SM field that is unique under the internal group of the SM to have a unique

“partner” field that does not exist in the SM, commonly referred to as a “superpartner”.

A quantum of a superpartner is commonly referred to as a “superparticle” or “sparticle”,

although the term “superpartner” is also commonly reused for this purpose as well. A

“supermultiplet” (or sometimes “superfield”) is an irreducible representation of SUSY,

which is some linear combination of SM fields and superpartners. [1]

One may wonder what is the experimental evidence for SUSY. The answer is “none”

– so far [91]. Apparently, SUSY does not represent nature – not exactly. One way to

maintain consistency between SUSY and the experimental evidence against it is to

1 I use a subscript “W” (for “weak”) rather than a subscript “L” (for “left”) to denotethe gauge group of weak isospin. I believe that the use of the subscript “L” is confusingfor 2 reasons: 1) there is another SU(2)L in the SM for the left-chiral block of the spinorrepresentation of the Lorentz group, and 2) actually, right-handed rather than left-handed(helicity) antifermions are in the fundamental representation of SU(2)W [89].

84

assume that SUSY is only approximate (i.e broken). One important phenomenological

manifestation of SUSY violation is that a superparticle can have a different mass from its

SM counterpart [1].

Another, more subtle issue is the stability of the proton. SUSY allows for terms in

the Lagrangian that can lead to prompt decay of the proton [1]. Assuming that protons

are, as they seem to be, stable with a half-life greater than the age of the universe [9],

this presents a potential problem for SUSY. A popular resolution to this problem is to

introduce yet another symmetry: R-parity conservation. The fields of the model represent

R-parity, similarly to spatial parity, as either a trivial or a fundamental representation of

some Z2. All SM fields are assigned even R-parity (i.e. they are trivial representations

of this Z2). So, R-parity has no influence on phenomena that are restricted to the SM.

All superpartners are assigned odd R-parity (i.e. they are fundamental representations

of this Z2). So, an important phenomenological consequence of R-parity is that the

number of superparticles must change by a multiple of two. This implies that the lightest

superparticle (LSP) is stable [1]. A neutralino LSP makes a popular DM candidate [92].

The cascade q̃L → χ̃02 → ˜̀

R → χ̃01 of minimal Supergravity (mSUGRA), depicted in

Fig. A-1, provides a popular example of the decay topology and final-state signature that

we examined, with the identifications given in Expression A–1.

q̃L → D χ̃02 → C ˜̀

R → B χ̃01 → A q → j (A–1)

In fact, since mSUGRA is already coded into PYTHIA [93], a popular high-energy

collision simulator, this cascade decay made a convenient starting point for our analysis.

We assumed that the second lightest neutralino has a large wino content and that the

lightest neutralino (and LSP) has a large bino content. We initiated the cascade with a

left-squark instead of a right-squark because the decay of the right-squark to the wino is

prohibited (just as the W boson does not couple to right-handed fermions), which, by our

assumption, greatly suppresses the decay.

85

q̃L χ̃02

˜̀R χ̃0

1

q `± `∓

Figure A-1. An example of Fig. 1-3A from mSUGRA in which a squark decays to aneutralino, which then decays to a slepton, which then finally decays toanother neutralino (DM candidate). This specific cascade decay is actuallyquite popular [11–14,26, 31, 36, 46].

We show case studies for a popular SUSY model point (SPS1a) in Appendix F.

In particular, the decay chain in Fig. A-1 is an example of the SFSF spin assignment.

However, the other five case studies are not examples of SUSY.

86

APPENDIX BEXAMPLE INVERSION FORMULAS FOR NJL = 1, 2, 3

We discovered that the shape of the kinematic boundaries of the (m2jl(lo), m

2jl(hi))

distribution reveals some new measurements. We provide an example of simple inversion

formulas in terms of these new measurements (Equations B–1 through B–3), assuming

that the shape also indicates one of the three on-shell Regions of mass parameter space.

f ≡ (mmaxjlf

)2 = m2D(1 − RCD)(1 − RAB) (B–1)

p ≡(

m(p)jlf

)2

= m2DRBC(1 − RCD)(1 − RAB) = fRBC (B–2)

n ≡ (mmaxjln )2 = m2

D(1 − RCD)(1 − RBC) (B–3)

In order to determine the 4 unknowns (mA, mB, mC and mD), we need one more

independent measurement.

Fortunately, the dilepton mass edge measurement is simple, robust, and independent

of the on-shell mass Region (Equation B–4), so we chose to use it as our fourth independent

measurement.

a ≡ (mmaxll )2 = m2

DRCD(1 − RBC)(1 − RAB) (B–4)

Equations B–5 through B–8 show the inversion for the squared mass ratios, and Equations

B–9 through B–12 show the inversion for the squared masses themselves.

RAB = 1 − f − p

n(B–5)

RBC =p

f(B–6)

RCD =

(

1 +f − p

a

)−1

(B–7)

m2D =

a f n

(f − p)2

(

1 +f − p

a

)

(B–8)

m2A =

a n p

(f − p)2

(

1 − f − p

n

)

(B–9)

m2B =

a n p

(f − p)2(B–10)

87

m2C =

a n f

(f − p)2(B–11)

m2D =

a n f

(f − p)2

(

1 +f − p

a

)

(B–12)

These inversion formulas are much simpler than Equations 2–13 through 2–16 with 2–17

through 2–21. The simplification is due to the fact that we are not using the threshold

measurement (Equation 2–11) whose analytical expression is somewhat complicated, and

that we (implicitly) used the shape of the (m2jl(lo),m

2jl(hi)) boundary lines to determine the

on-shell scenario (Fig. 2-7).

Equations B–9 through B–12 are only one example of the virtually limitless

possibilities provided by the virtually limitless choices for observable points on the

kinematic boundaries of the two-variable distributions. Furthermore, any point on a

kinematics boundary can now be unambiguously expressed in terms of the masses of the

heavy partners, thanks to the characteristic shape of the boundary lines. As a word of

caution, though, one must take care to select independent observables. Ultimately, the

best procedure is probably to fit the entire kinematics boundary at many different points,

thus over-constraining the mass determination and incorporating redundancy to improve

the confidence of the measurement. However, we emphasize that, in principle, we have

provided two alternative sets of observables that can be immediately inverted for the

masses of the heavy partners in the decay chain, without any need for numerical fitting,

but at the expense of some amount of additional uncertainty compared to the numerical

fit.

88

APPENDIX CHELICITY BASIS FUNCTIONS {FIJ}

The dilepton helicity basis functions F (``)S;IJ(m̂2; x, y, z) are given in Table C-1. The

j`n helicity basis functions F (j`n)S;IJ (m̂2; x, y, z) are given in Table C-2 The j`f helicity basis

functions F (j`f)S;IJ (m̂2; x, y, z) are given in Equations C–1 through C–23.

The parameters x, y, and z are the squared mass ratios defined by Equation 1–3. In

order to simplify the expressions, we defined normalization constants, N IJS , for each spin

assignment, S.

SFSF (S = 1)

F (j`f )1;11 (m̂2) = F (j`f )

1;12 (m̂2) = N IJ1

−(1 − y) − log y if m̂2 ≤ y

−1 + m̂2 − log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(C–1)

F (j`f )1;21 (m̂2) = F (j`f)

1;22 (m̂2) = N IJ1

(1 − y) + y log y if m̂2 ≤ y

1 − m̂2 + y log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(C–2)

N IJ1 =

2

(1 − y)2(C–3)

FSFS (S = 2)

F (j`f )2;11 (m̂2) = F (j`f )

2;21 (m̂2) = N IJ2

−(1 − y) − log y if m̂2 ≤ y

−1 + m̂2 − log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(C–4)

F (j`f )2;12 (m̂2) = F (j`f)

2;22 (m̂2) = N IJ2

(1 − y) + y log y if m̂2 ≤ y

1 − m̂2 + y log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(C–5)

N IJ2 = N IJ

1 =2

(1 − y)2(C–6)

89

FSFV (S = 3)

F (j`f )3;11 (m̂2) = F (j`f )

3;21 (m̂2) = N IJ3

−(1 − y)(1 − 2z) − (1 − 2yz) log y if m̂2 ≤ y

−(1 − m̂2)(1 − 2z) − (1 − 2yz) log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(C–7)

F (j`f )3;12 (m̂2) = F (j`f )

3;22 (m̂2) = N IJ3

(1 − y)(1 − 2z) + (y − 2z) log y if m̂2 ≤ y

(1 − m̂2)(1 − 2z) + (y − 2z) log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(C–8)

N IJ3 =

N IJ2

1 + 2z=

2

(1 − y)2(1 + 2z)(C–9)

FVFS (S = 4)

F (j`f )4;11 (m̂2) = N IJ

4

(1 − y)[4x − y − 4m̂2(2 − 3x)]

+[(−1 + 4x)y + 4m̂2{1 − (2 + y)(1 − x)}] log y if m̂2 ≤ y

(1 − m̂2)[4x(2y + 1) − 5y − 4m̂2(1 − x)]

+[(−1 + 4x)y + 4m̂2{1 − (2 + y)(1 − x)}] log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(C–10)

F (j`f)4;12 (m̂2) = N IJ

4

(1 − y)[2 + 3y − 2x(5 + y) + 4m̂2(2 − 3x)]

+[y(4 + y) − 4x(1 + 2y)

−4m̂2{1 − (2 + y)(1 − x)}] log y if m̂2 ≤ y

(1 − m̂2)[2 + 9y − 2x(5 + 6y) + 2m̂2(1 − x)]

+[y(4 + y) − 4x(1 + 2y)

−4m̂2{1 − (2 + y)(1 − x)}] log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(C–11)

90

F (j`f )4;21 (m̂2) = N IJ

4

(1 − y)[−y − 4m̂2(2 − x)]

−[y + 4m̂2{1 + y(1 − x)}] log y if m̂2 ≤ y

(1 − m̂2)[−5y − 4m̂2(1 − x)]

−[y + 4m̂2{1 + y(1 − x)}] log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(C–12)

F (j`f)4;22 (m̂2) = N IJ

4

(1 − y)[2 + 3y + 2x(1 − y) + 4m̂2(2 − x)]

+[y(4 + y) + 4m̂2{1 + y(1 − x)}] log y if m̂2 ≤ y

(1 − m̂2)[2 + 9y + 2x(1 − 2y) + 2m̂2(1 − x)]

+[y(4 + y) + 4m̂2{1 + y(1 − x)}] log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(C–13)

N IJ4 =

3N IJ1

(1 + 2x)(2 + y)=

6

(1 + 2x)(2 + y)(1 − y)2(C–14)

FVFV (S = 5)

F (j`f )5;11 (m̂2) = N IJ

5

[4x − y + 2z{2 + 3y − 2x(5 + y)}

−4m̂2(2 − 3x)(1 − 2z)](1 − y)

−[y − 2yz(4 + y) + 4x{2z − y(1 − 4z)}

+4m̂2{1 + y − x(2 + y)}(1 − 2z)] log y if m̂2 ≤ y

[4x{1 − 5z + 2y(1 − 3z)} − 5y + 2z(2 + 9y)

−4m̂2(1 − x)(1 − z)](1 − m̂2)

−[y − 2yz(4 + y) + 4x{2z − y(1 − 4z)}

+4m̂2{1 + y − x(2 + y)}(1 − 2z)] log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(C–15)

91

F (j`f )5;12 (m̂2) = N IJ

5

[2 + 3y − 2x(5 + y) + 2(4x − y)z

+4m̂2(2 − 3x)(1 − 2z)](1 − y)

−[4x{1 + 2y(1 − z)} − y(4 + y − 2z)

−4m̂2{1 + y − x(2 + y)}(1 − 2z)] log y if m̂2 ≤ y

[2 − 2x{5 − 4z + 2y(3− 4z)} + y(9 − 10z)

+2m̂2(1 − x)(1 − 4z)](1 − m̂2)

−[4x{1 + 2y(1 − z)} − y(4 + y − 2z)

−4m̂2{1 + y − x(2 + y)}(1 − 2z)] log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(C–16)

F (j`f )5;21 (m̂2) = N IJ

5

[−y + 2{2 + 2x(1 − y) + 3y}z

−4m̂2(2 − x)(1 − 2z)](1 − y)

−[y{1 − 2(4 + y)z} + 4m̂2(1 + y − xy)(1 − 2z)] log y if m̂2 ≤ y

[4(1 + x)z − y{5 − 2(9 − 4x)z}

−4m̂2(1 − x)(1 − z)](1 − m̂2)

−[y{1 − 2(4 + y)z} + 4m̂2(1 + y − xy)(1 − 2z)] log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(C–17)

F (j`f )5;22 (m̂2) = N IJ

5

[2 + 2x(1 − y) + y(3 − 2z)

+4m̂2(2 − x)(1 − 2z)](1 − y)

+[y(4 + y − 2z) + 4m̂2(1 + y − xy)(1 − 2z)] log y if m̂2 ≤ y

[2 + 2x(1 − 2y) + y(9 − 10z)

+2m̂2(1 − x)(1 − 4z)](1 − m̂2)

+[y(4 + y − 2z) + 4m̂2(1 + y − xy)(1 − 2z)] log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(C–18)

N IJ5 =

N IJ4

1 + 2z=

6

(1 + 2x)(2 + y)(1 − y)2(1 + 2z)(C–19)

92

SFVF (S = 6)

F (j`f )6;11 (m̂2) = N IJ

6

(1 − y)[2 − 3z − 2y(1 + z) + 4m̂2(1 − 2z)]

−[z(1 + 4y) − 4m̂2(1 − z − yz)] log y if m̂2 ≤ y

(1 − m̂2)[2 − 3z − 8yz + 2m̂2(1 − z)]

−[z(1 + 4y) − 4m̂2(1 − z − yz)] log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(C–20)

F (j`f )6;12 (m̂2) = N IJ

6

(1 − y)[2 − 3z + 2y(5 − z) + 4m̂2(3 − 2z)]

−[z(1 + 4y) − 4y(2 + y) − 4m̂2(1 + 2y − z − yz)] log y if m̂2 ≤ y

(1 − m̂2)[2 − 3z + 4y(5 − 2z) + 2m̂2(1 − z)]

−[z(1 + 4y) − 4y(2 + y) − 4m̂2(1 + 2y − z − yz)] log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(C–21)

F (j`f )6;21 (m̂2) = N IJ

6

(1 − y)[z − 4m̂2(1 − 2z)]

+[yz − 4m̂2(1 − z − yz)] log y if m̂2 ≤ y

(1 − m̂2)[z(1 + 4y) − 4m̂2(1 − z)]

+[yz − 4m̂2(1 − z − yz)] log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(C–22)

F (j`f )6;22 (m̂2) = N IJ

6

(1 − y)[z − 4y − 4m̂2(3 − 2z)]

−[y(4 − z) + 4m̂2(1 + 2y − z − yz)] log y if m̂2 ≤ y

(1 − m̂2)[z − 4y(3 − z) − 4m̂2(1 − z)]

−[y(4 − z) + 4m̂2(1 + 2y − z − yz)] log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(C–23)

N IJ6 =

3N IJ1

(1 + 2y)(2 + z)=

6

(1 − y)2(1 + 2y)(2 + z)(C–24)

93

Table C-1. Helicity basis functions for the dilepton invariant mass distribution. The normalization constants, N IJS , are given

in Equations C–3, C–6, C–9, and C–24.

S Spins F (``)S;11(m̂

2; x, y, z) = F (``)S;21(m̂

2; x, y, z) F (``)S;12(m̂

2; x, y, z) = F (``)S;22(m̂

2; x, y, z)

1 SFSF 1 12 FSFS 2(1 − m̂2) 2m̂2

3 FSFV2NIJ

3

NIJ2

{1 − (1 − 2z)m̂2} 2NIJ3

NIJ2

{2z + (1 − 2z)m̂2}4 FV FS 2

2+y{y + (2 − y)m̂2} 2

2+y{2 − (2 − y)m̂2}

5 FV FV2NIJ

3

NIJ2

(2+y){y + 4z + (2 − y)(1 − 2z)m̂2} 2NIJ

3

NIJ2

(2+y){2 + 2yz − (2 − y)(1 − 2z)m̂2}

6 SFV FNIJ

6

NIJ1

{

4y + z + 4(1 − 2y − z + yz)m̂2

−4(1 − y)(1 − z)m̂4

}

NIJ6

NIJ1

{

z + 4(1 − z + yz)m̂2

−4(1 − y)(1 − z)m̂4

}

Table C-2. Helicity basis functions for the j`n invariant mass distribution. The normalization constants, N IJS , are given in

Equations C–3 and C–14.

S Spins F (j`n)S;11 (m̂2; x, y, z) = F (j`n)

S;12 (m̂2; x, y, z) F (j`n)S;21 (m̂2; x, y, z) = F (j`n)

S;22 (m̂2; x, y, z)

1 SFSF 2m̂2 2(1 − m̂2)2 FSFS 1 13 FSFV 1 1

4 FV FSNIJ

4

NIJ1

{

y + 4(1 − y + xy)m̂2

−4(1 − x)(1 − y)m̂4

}

NIJ4

NIJ1

{

4x + y + 4(1 − 2x − y + xy)m̂2

−4(1 − x)(1 − y)m̂4

}

5 FV FVNIJ

4

NIJ1

{

y + 4(1 − y + xy)m̂2

−4(1 − x)(1 − y)m̂4

}

NIJ4

NIJ1

{

4x + y + 4(1 − 2x − y + xy)m̂2

−4(1 − x)(1 − y)m̂4

}

6 SFV F 21+2y

{2y + (1 − 2y)m̂2} 21+2y

{1 − (1 − 2y)m̂2}

94

APPENDIX DOBSERVABLE SPIN BASIS FUNCTIONS {Fα,Fβ,Fγ,Fδ}

The dilepton observable basis functions F (``)S;α , F (``)

S;β , F (``)S;γ and F (``)

S;δ are given in

Table D-1. The j`n observable basis functions F (j`n)S;α , F (j`n)

S;β , F (j`n)S;γ and F (j`n)

S;δ are given in

Table D-2. The j`f observable basis functions F (j`f )S;α , F (j`f )

S;β , F (j`f )S;γ and F (j`f )

S;δ are given in

Equations D–1 through D–26.

The parameters x, y, and z are the squared mass ratios defined by Equation 1–3. In

order to simplify the expressions, we defined normalization constants, N δS, for each spin

assignment, S.

SFSF (S = 1)

F (j`f )1;α (m̂2) = F (j`f )

1;γ (m̂2) = 0 (D–1)

F (j`f )1;β (m̂2) = N δ

1

−2(1 − y) − (1 + y) log y if m̂2 ≤ y

−2(1 − m̂2) − (1 + y) log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(D–2)

F (j`f )1;δ (m̂2) =

− log y1−y

if m̂2 ≤ y

− log m̂2

1−yif y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(D–3)

N δ1 =

1

(1 − y)2(D–4)

FSFS (S = 2)

F (j`f )2;α (m̂2) = F (j`f )

1;β (m̂2) = N δ2

−2(1 − y) − (1 + y) log y if m̂2 ≤ y

−2(1 − m̂2) − (1 + y) log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(D–5)

F (j`f )2;β (m̂2) = F (j`f )

2;γ (m̂2) = 0 (D–6)

95

F (j`f )

2;δ (m̂2) = F (j`f )

1;δ (m̂2) =

− log y1−y

if m̂2 ≤ y

− log m̂2

1−yif y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(D–7)

N δ2 = N δ

1 =1

(1 − y)2(D–8)

FSFV (S = 3)

F (j`f )3;α (m̂2) = N δ

3

−2(1 − y) − (1 + y) log y if m̂2 ≤ y

−2(1 − m̂2) − (1 + y) log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(D–9)

F (j`f )

3;β (m̂2) = F (j`f )3;γ (m̂2) = 0 (D–10)

F (j`f )3;δ (m̂2) = F (j`f )

2;δ (m̂2) = F (j`f )1;δ (m̂2) =

− log y1−y

if m̂2 ≤ y

− log m̂2

1−yif y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(D–11)

N δ3 = N δ

2

1 − 2z

1 + 2z=

1 − 2z

(1 − y)2(1 + 2z)(D–12)

FVFS (S = 4)

F (j`f )4;α (m̂2) = N δ

4

(1 − y)[−2(1 + 2y) + 2x(3 + y) − 16m̂2(1 − x)]

−[y(5 + y) − 2x(1 + 3y) + 8m̂2(1 − x)(1 + y)] log y if m̂2 ≤ y

(1 − m̂2)[−2(1 + 7y) + 6x(1 + 2y) − 6m̂2(1 − x)]

−[y(5 + y) − 2x(1 + 3y) + 8m̂2(1 − x)(1 + y)] log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(D–13)

F (j`f)4;β (m̂2) = 2xN δ

4

−2(1 − y) − (1 + y) log y if m̂2 ≤ y

−2(1 − m̂2) − (1 + y) log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(D–14)

96

F (j`f )4;γ (m̂2) = 2xN δ

4

4(1 − y)(1 + m̂2) + [(1 + 3y) + 4m̂2] log y if m̂2 ≤ y

4(1 − m̂2)(1 + y) + [(1 + 3y) + 4m̂2] log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(D–15)

F (j`f )

4;δ (m̂2) = N δ4

2(1 − y)(1 + y)(1 − x)

+[−2x(1 + y) + y(3 + y)] log y if m̂2 ≤ y

2(1 − m̂2)(1 − x){(1 + 2y) − m̂2}

+[−2x(1 + y) + y(3 + y)] log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(D–16)

N δ4 =

3N δ1

(1 + 2x)(2 + y)=

3

(1 + 2x)(2 + y)(1 − y)2(D–17)

FVFV (S = 5)

F (j`f )5;α (m̂2) = N δ

5

(1 − y)[−2(1 + 2y) + 2x(3 + y) − 16m̂2(1 − x)]

−[y(5 + y) − 2x(1 + 3y)

+8m̂2(1 − x)(1 + y)] log y if m̂2 ≤ y

(1 − m̂2)[−2(1 + 7y) + 6x(1 + 2y) − 6m̂2(1 − x)]

−[y(5 + y) − 2x(1 + 3y)

+8m̂2(1 − x)(1 + y)] log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(D–18)

F (j`f)

5;β (m̂2) = 2xN δ4

−2(1 − y) − (1 + y) log y if m̂2 ≤ y

−2(1 − m̂2) − (1 + y) log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(D–19)

F (j`f )5;γ (m̂2) = 2xN δ

5

4(1 − y)(1 + m̂2)

+[(1 + 3y) + 4m̂2] log y if m̂2 ≤ y

4(1 − m̂2)(1 + y)

+[(1 + 3y) + 4m̂2] log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(D–20)

97

F (j`f )

5;δ (m̂2) = N δ4

2(1 − y)(1 + y)(1 − x)

+[−2x(1 + y) + y(3 + y)] log y if m̂2 ≤ y

2(1 − m̂2)(1 − x){(1 + 2y) − m̂2}

+[−2x(1 + y) + y(3 + y)] log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(D–21)

N δ5 = N δ

4

1 − 2z

1 + 2z=

3(1 − 2z)

(1 + 2x)(2 + y)(1 − y)2(1 + 2z)(D–22)

SFVF (S = 6)

F (j`f )6;α (m̂2) = 2yN δ

6

−2(1 − y) − (1 + y) log y if m̂2 ≤ y

−2(1 − m̂2) − (1 + y) log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(D–23)

F (j`f)6;β (m̂2) = N δ

6

(1 − y)[2(1 + 3y) − 2(2 + y)z + 16m̂2(1 − z)]

+[2y(3 + y) − (1 + 5y)z + 8m̂2(1 + y)(1 − z)] log y if m̂2 ≤ y

(1 − m̂2)[2(1 + 8y) − 4(1 + 3y)z + 6m̂2(1 − z)]

+[2y(3 + y) − (1 + 5y)z + 8m̂2(1 + y)(1 − z)] log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(D–24)

F (j`f )6;γ (m̂2) = 2N δ

6

−4(1 − y)(y + m̂2) − [y(3 + y) + 4ym̂2] log y if m̂2 ≤ y

−8y(1 − m̂2) − [y(3 + y) + 4ym̂2] log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(D–25)

F (j`f )6;δ (m̂2) = N δ

6

2(1 − y)(1 + y)(1 − z)

+[−(1 − y)(1 + 2y) + (1 + 3y)(1 − z)] log y if m̂2 ≤ y

2(1 − m̂2)(1 − z){(1 + 2y) − m̂2}

+[−(1 − y)(1 + 2y) + (1 + 3y)(1 − z)] log m̂2 if y ≤ m̂2 ≤ 1

0 if m̂2 ≥ 1

(D–26)

N δ6 =

3N δ1

(1 + 2y)(2 + z)=

3

(1 − y)2(1 + 2y)(2 + z)(D–27)

98

Table D-1. Observable basis functions for the dilepton invariant mass distribution. The normalization constants, N δS, are

given in Equations D–4, D–8, D–12, D–17, D–22, and D–27.

S Spins F (``)S;δ (m̂2; x, y, z) F (``)

S;α (m̂2; x, y, z) F (``)S;β (m̂2; x, y, z) F (``)

S;γ (m̂2; x, y, z)

1 SFSF 1 0 0 02 FSFS 1 1 − 2m̂2 0 0

3 FSFV 1Nδ

3

Nδ2

{1 − 2m̂2} 0 0

4 FV FS 1 −2−y2+y

{1 − 2m̂2} 0 0

5 FV FV 1 − (2−y)Nδ5

(2+y)Nδ4

{1 − 2m̂2} 0 0

6 SFV FNδ

6

Nδ1

{

2y + z + 4(1 − y)(1 − z)m̂2

−4(1 − y)(1 − z)m̂4

}

2yNδ6

Nδ1

{1 − 2m̂2} 0 0

Table D-2. Observable basis functions for the j`n invariant mass distribution. The normalization constants, N δS, are given in

Equations D–4 and D–17.

S Spins F (j`n)S;δ (m̂2; x, y, z) F (j`n)

S;α (m̂2; x, y, z) F (j`n)S;β (m̂2; x, y, z) F (j`n)

S;γ (m̂2; x, y, z)

1 SFSF 1 0 −{1 − 2m̂2} 02 FSFS 1 0 0 03 FSFV 1 0 0 0

4 FV FSNδ

4

Nδ1

{

2x + y + 4(1 − x)(1 − y)(m̂2

−m̂4

}

0 −2xNδ4

Nδ1

{1 − 2m̂2} 0

5 FV FVNδ

4

Nδ1

{

2x + y + 4(1 − x)(1 − y)(m̂2

−m̂4

}

0 −2xNδ4

Nδ1

{1 − 2m̂2} 0

6 SFV F 1 0 −1−2y1+2y

{1 − 2m̂2} 0

99

APPENDIX ESIMPLE SPIN FITTING PROCEDURE

With vanishing error bars, we use the rather naive matching criterion given by

Equation E–1.

χ2(α, β, γ) ≡∫ 1

0

(

f0(m̂2, α0, β0, γ0) − f(m̂2, α, β, γ)

)2

dm̂2 (E–1)

The function f0(m̂2, α0, β0, γ0) represents the observed distribution to be fitted, and

f(m̂2, α, β, γ) is the theoretical test distribution. We minimize the χ2(α, β, γ) function for

α, β and/or γ, as appropriate. α0, β0 and γ0 are fixed constant values that we selected for

the given study point (Equations F–4 and F–9). We simply interpreted any nonzero value

of χ2 as a mismatch. One thing that we found interesting, then, was that we still were

able to fit some “wrong” spin assignments (Appendix F).

The fit to the L+− or S+− distribution is a simple one-parameter fit for α, while the

fit to the D+− distribution is a two-parameter fit for β and γ. Fig. E-1 shows sample

results from our D+− fits for β and γ (Appendix F). In each plot in Fig. E-1, the

f0(m̂2, α0, β0, γ0) distribution comes from the first spin chain (red) written at the top

of each plot, which is then fit to the f(m̂2, α, β, γ) distribution predicted by the second

spin chain (blue) written at the top of each plot. The contour lines represent constant

values of χ2(α, β, γ), where α has already been fixed by fitting to L+−. The blue dot

corresponds to the absolute minimum of χ2, ignoring any restrictions on α, β and γ.

The parameters α, β and γ are not completely independent from each other. For any

given α, the physically allowed region in the (β, γ) parameter space is described by an

envelope (Equations E–2 through E–5).

αβ ≤ γ, βγ ≤ α, γα ≤ β if α > 0, β > 0 and γ > 0 (E–2)

αβ ≥ γ, βγ ≤ α, γα ≥ β if α > 0, β < 0 and γ < 0 (E–3)

αβ ≥ γ, βγ ≥ α, γα ≤ β if α < 0, β > 0 and γ < 0 (E–4)

αβ ≤ γ, βγ ≥ α, γα ≥ β if α < 0, β < 0 and γ > 0 (E–5)

100

Figure E-1. Contour plots of χ2(α, β, γ) as a function of β and γ, with α already fixed bythe fit to the L+− data. The region satisfying the constraints (Equations E–2through E–5) is shaded in white. The blue dot denotes the global χ2

minimum, which does not necessarily represent a consistent set of modelparameters. The green triangle denotes the location of the χ2 minimumwithin the consistent region. The f0(m̂

2, α0, β0, γ0) distribution comes fromthe first spin assignment (shown in red) at the top of each plot, which is thenfitted to the f(m̂2, α, β, γ) distribution (shown in blue).

101

This envelope is represented by the white region in Fig. E-1. The green triangle

corresponds to the minimum of the χ2 function within this restricted parameter space.

The green triangle solution for β and γ was then used for our plots in Fig. F-3. For

the two FVFV (S=5) cases, the global minimum happens to lie within the region of

consistency (white), and so the blue dot and the green triangle coincide. For the extreme

values of α, the region of consistency (white) collapses to one or two lines, as indicated by

Equations E–6 and E–7.

β = 0 or γ = 0 if α = 0 (E–6)

γ = ±β if α = ±1 (E–7)

102

APPENDIX FSPIN DETERMINATION EXAMPLES

F.1 The SPS1a Study Point

We provide an explicit demonstration of our method. We first simulated observable

distributions for each of the 6 spin configurations that are listed in Table 1-2. Then, for

each case we used the simple fitting procedure in Appendix E to determine whether the

observable distributions were consistent with each of the remaining 5 alternatives.

Since we do not yet have real data available, we artificially chose input parameter

values (but only for the sake of demonstration). In order to facilitate comparisons to

previous studies, our input parameters (Equations F–1 through F–9) correspond to the

SPS1a study point in supersymmetry. However, the spin assignments only correspond to

this study point in the case of SFSF. We used the values from Refs. [12, 14].

mA = 96 GeV mB = 143 GeV mC = 177 GeV mD = 537 GeV (F–1)

x = 0.109 y = 0.653 z = 0.451 (F–2)

aL = 0 aR = 1 bL = 0 bR = 1 cL = 1 cR = 0 (F–3)

f = 0.7 f̄ = 0.3 (F–4)

mmax`` = mD

√

x(1 − y)(1 − z) = 77.31 GeV (F–5)

mmaxj`n

= mD

√

(1 − x)(1 − y) = 298.77 GeV (F–6)

mmaxj`f

= mD

√

(1 − x)(1 − z) = 375.76 GeV (F–7)

mmaxj`` = mD

√

(1 − x)(1 − yz) = 425.94 GeV (F–8)

α = 1 β = −0.4 γ = −0.4 (F–9)

Note that α = 1 necessarily implies that β = γ (Equations 3–67 through 3–69). Equation

F–9 defines the input values of the model parameters used in our case study.

103

Figure F-1. Dilepton invariant mass distributions, L+−S . The solid (magenta) line in each

plot represents the simulated observed dilepton distribution for each of the 6spin assignments: A) SFSF; B) FSFS; C) FSFV; D) FVFS; E) FVFV; F)SFVF. The other (dotted or dashed) lines represent our best fits to this inputdistribution, for each of the remaining 5 spin configurations. A dashed (green)line represents an exact fit, and a dotted line represents an inexact fit (coloredother than green or magenta). The best fit value of α for each case is alsoshown, except for cases where it is left undetermined (NA).

104

Figure F-2. Sum of j`+ and j`− invariant mass distributions, S+−. The different linestyles basically represent the same conditions as in Fig. F-1, except that no fitwas performed, and instead the distributions were plotted with the specificvalue of α that was obtained from the fit to L+−.

105

Figure F-3. The same as in Fig. F-1 but for D+− instead of L+−, and a fit to β and γinstead of α.

106

Although the fit can be done simultaneously for all three parameters α, β and γ,

we performed it sequentially, using the fact that the L+−S and S+−

S distributions depend

only on α and not on β and γ. Therefore, we began with the L+−S distribution and first

determined the value of α, which we then used to compare the S+−S test distribution to

the S+−S observable distribution. Already at this stage it may be possible to rule out

all but the correct spin configuration, and we encountered such examples. Sometimes,

however, there may still be several alternatives left, in which case the D+− distribution

must also be considered, where we fit for the values of the coefficients β and γ. At any

rate, there is no harm in using all three distributions, L+−S , S+−

S , and D+−S .

Our results are summarized in Figures F-1 through F-3, which show the L+−S ,

S+−S and D+−

S distributions for each of the 6 input spin assignments, as well as the 5

test distributions for each case. In these figures the solid (magenta) lines represent the

observed distributions, and the other (dotted or dashed) lines represent our best fits

of the test distributions to the observed distributions, for each of the remaining 5 spin

configurations (Table 1-2). A dashed (green) line represents an exact fit to the observed

distribution. The (color-coded) dotted lines represent distributions that could not be fit

exactly to the input distribution. The best fit values of α, β and γ for each case are also

shown, except for those cases (labeled by “NA”) where they are left undetermined by the

fit.

F.2 Input from SFSF (S = 1)

We began the SFSF case study with the observable distributions in Equations

F–10 through F–12. Fig. A-1 gives an example of a possible decay diagram for this spin

assignment.

L+−1 = 1 (F–10)

S+−1 =

2.810 m̂2j` ≤ 0.632

1.228 0.632 ≤ m̂2j` ≤ 0.653

−2.880 log m̂2j` 0.653 ≤ m̂2

j`

(F–11)

107

D+−1 =

−0.668 + 2.002 m̂2j` m̂2

j` ≤ 0.632

−0.035 0.632 ≤ m̂2j` ≤ 0.653

6.633 − 6.633 m̂2j` + 5.481 log m̂2

j` 0.653 ≤ m̂2j`

(F–12)

These distributions are shown with solid (magenta) lines in Figs. F-1A, F-2A and

F-3A, respectively. We first fit the dilepton data in Fig. F-1A. Due to the presence of

an intermediate scalar particle B, the L+− distribution for the SFSF chain (S=1), is

completely flat. However, that does not imply that the spin of particle B is determined

to be zero. In fact, as seen from Fig. F-1A, all other spin configurations except SFVF

can also fit this flat distribution, simply by choosing a vanishing α parameter. Even the

case of S = 6 (SFVF), whose “best fit” prediction is different from the input data, may

still be difficult to discriminate in practice, once experimental errors are introduced. The

bad news, therefore, is that we cannot immediately determine the spins from the L+−

distribution alone, but the good news is that, as anticipated, we obtained a measurement

of the α parameter (i.e. α = 0), which represents some combination of heavy particle

couplings and mixing angles (Equation 3–67).

We compare our Fig. F-1A to Fig. 2a in Ref. [14], where a very similar exercise was

performed. The two results are quite different. For example we found that 4 out of the 5

“wrong” models can perfectly fit the dilepton distribution, while in Ref. [14] all 6 models

give distinct dilepton shapes. The difference arises due to our different philosophy. In

Ref. [14] the parameters corresponding to α, β and γ in our notation were all kept fixed to

the SPS1a values (Equation F–9), whereas our technique allows them to float, since they

would not have been measured in advance. As a result, once we factor in the experimental

realism, we determined that the actual spin measurements are more challenging than

previously anticipated.

S+− depends on exactly the same model parameter α as the dilepton distribution

L+−. Since we measured α by fitting to the L+− distribution, there were no free

parameters left in the S+− distribution, and it was uniquely predicted for each spin

108

assignment. Fig. F-2A shows the resulting predictions for the 6 spin models, using in each

case the corresponding value of α from the L+− fit. In this case, the S+− distribution can

differentiate some of the spin assignments, e.g. it can rule out (in principle) the FVFS and

FVFV spin assignments. Interestingly, SFVF gives a perfect match, but fortunately, it was

already eliminated from consideration by the L+− fit. Unfortunately, the “wrong” spin

scenarios FSFS and FSFV give a perfect match for both L+− and S+−, leaving 3 distinct

possibilities for the spins of the heavy partners.

We were therefore forced to consider our third piece of data, the D+− distribution

(Equation 3–72). This distribution does not depend on the previously fitted parameter

α, and instead must be fitted with the other two model parameters, β and γ. Even

though D+− itself does not explicitly depend on α, the fit is nevertheless impacted by

the measured value of α, as α restricts the allowed range of values for β and γ (Appendix

E). The results from our fitting exercise for D+− are shown in Fig. F-3A. We see that

D+− can now eliminate the remaining two “wrong” spin scenarios FSFS and FSFV,

and, as a result of all three types of fits, we were able to determine uniquely the spin

assignment as SFSF (from the ideal distributions). In addition, we were also able to

obtain a measurement of the parameter β, which carries information about the couplings

and mixing angles of the heavy partners D, C and B. Unfortunately, the parameters α

and γ are not experimentally accessible for SFSF, since their corresponding basis functions

F (p)1;α and F (p)

1;γ are identically zero (Appendix D).

F.3 Input from FSFS (S = 2) and FSFV (S = 3)

We began the FSFS case study with the observable distributions in Equations F–13

through F–15, and we began the FSFV case study with the observable distributions in

Equations F–16 through F–18. Fig. F-4 shows the diagrams for these spin assignments.

L+−2 = 2 − 2m̂2

`` (F–13)

109

D C B A

q `± `∓

D C B A

q `± `∓

A B

Figure F-4. Decay diagrams for A) FSFS and B) FSFV. The only difference betweenthese spin assignments is the spin of the final-state particle A, which can beeither a scalar or a vector.

S+−2 =

2.898 m̂2j` ≤ 0.632

1.316 0.632 ≤ m̂2j` ≤ 0.653

−16.583 + 16.583 m̂2j` − 16.583 log m̂2

j` 0.653 ≤ m̂2j`

(F–14)

D+−2 = 0 (F–15)

L+−3 = 1.052 − 0.104 m̂2

`` (F–16)

S+−3 =

2.815 m̂2j` ≤ 0.632

1.233 0.632 ≤ m̂2j` ≤ 0.653

−0.860 + 0.860 m̂2j` − 3.590 log m̂2

j` 0.653 ≤ m̂2j`

(F–17)

D+−3 = 0 (F–18)

Perhaps the most striking feature is that the D+− distribution, and consequently the

lepton charge asymmetry A+−, are both identically zero. Therefore, they do not convey

any information about the spins, since any spin configuration can fit those distributions

with the proper choice of parameters as shown in Figs. F-3B and F-3C. This being the

case, we concentrated on the L+− and S+− distributions.

We first considered the FSFS spin assignment. Again, we began our analysis with

L+−, which in this case shows very good discrimination, and can already rule out all of

the “wrong” spin combinations. And, while FSFS distributions can sometimes be faked

by FSFV distributions, this could only happen if the input α parameter satisfies Equation

110

3–88. Since for SPS1a α = 1 (Equation F–9), this condition was not satisfied in our

specific case study, and the FSFV model did not fake the FSFS distribution. This is

confirmed by our result in Fig. F-1B.

Since the L+− distribution alone already singles out the correct spin configuration,

we did not even need to consider the S+− distribution. It is worth pointing out, however,

that S+− in this ideal case also can rule out all “wrong” spin models (although the

differences are not so pronounced as for L+− even using these ideal distributions). We also

obtained a measurement of the parameter α. However, the parameters β and γ remain

undetermined, since their corresponding basis functions F (p)2;β and F (p)

2;γ are identically zero

for any p ∈ {``, j`n, j`f} (Appendix D).

We next considered the FSFV spin assignment, which provides an example of an

inconclusive spin determination. Of course, this result was already anticipated, since we

already determined that FSFS can always provide a perfect fit to the FSFV distributions.

The value of α2 that is measured for the fake FSFS model is predicted by Equation F–19.

α2 = α31 − 2z

1 + 2z≈ 0.05 (F–19)

Our numerical study explicitly confirms this general expectation as shown in Figs. F-1C,

F-2C and F-3C. In addition, we verified that the mj`` distributions for these two “twin”

spin models are also identical.

F.4 Input from FVFS (S = 4) and FVFV (S = 5)

We began the FVFS case study with the observable distributions in Equations F–20

through F–22, and we began the FVFV case study with the observable distributions in

Equations F–23 through F–25. Fig. F-5 shows the diagrams for these spin assignments.

L+−4 = 0.492 + 1.016 m̂2

`` (F–20)

111

D C B A

q `± `∓

D C B A

q `± `∓

A B

Figure F-5. Decay diagrams for A) FVFS and B) FVFV. The only difference betweenthese spin assignments is the spin of the final-state particle A, which can beeither a scalar or a vector. These diagrams are related to those in Fig. F-4 bychanging particle C from a scalar to a vector.

S+−4 =

2.307 + 3.455 m̂2j` − 4.553 m̂4

j` m̂2j` ≤ 0.632

1.028 + 0.577 m̂2j` 0.632 ≤ m̂2

j` ≤ 0.653

−42.563 − 12.368 m̂2j` + 54.931 m̂4

j`

−(

7.871 + 90.785 m̂2j`

)

log m̂2j` 0.653 ≤ m̂2

j`

(F–21)

D+−4 =

−0.22 + 0.616 m̂2j` m̂2

j` ≤ 0.632

−0.092 + 0.212 m̂2j` 0.632 ≤ m̂2

j` ≤ 0.653

−3.087 + 3.087 m̂2j`

−(

0.874 + 2.678 m̂2j`

)

log m̂2j` 0.653 ≤ m̂2

j`

(F–22)

L+−5 = 0.974 + 0.053 m̂2

`` (F–23)

S+−5 =

2.496 + 2.908 m̂2j` − 4.553 m̂4

j` m̂2j` ≤ 0.632

1.217 + 0.030 m̂2j` 0.632 ≤ m̂2

j` ≤ 0.653

27.809 − 43.679 m̂2j` + 15.870 m̂4

j`

+(

14.382 − 4.710 m̂2j`

)

log m̂2j` 0.653 ≤ m̂2

j`

(F–24)

D+−5 =

−0.139 + 0.415 m̂2j` m̂2

j` ≤ 0.632

−0.011 + 0.011 m̂2j` 0.632 ≤ m̂2

j` ≤ 0.653

1.109 − 1.109 m̂2j`

+(

1.004 − 0.139 m̂2j`

)

log m̂2j` 0.653 ≤ m̂2

j` ,

(F–25)

112

D C B A

q `± `∓

Figure F-6. Decay diagram for SFVF.

The end result of these two case studies is similar to what we obtained for the other

“twin” model pair FSFS and FSFV, which was also already anticipated, since we already

determined that FVFS can always provide a perfect fit to the FVFV distributions. When

beginning with FVFS and fitting the other 5 spin assignments, we did not encounter any

spin ambiguities. As we already determined for FSFS, this is a numerical accident due to

the particular choice of the SPS1a study point (namely α = 1), which does not satisfy

the necessary condition (Equation 3–96) for the FVFV spin assignment to fake the FVFS

distributions. As a result, the two L+− and S+− distributions were sufficient to determine

the FVFS scenario, and the D+− distribution could then be used as a cross-check, and to

measure the β and γ parameters.

However, when starting with FVFV and fitting the other 5 spin assignments, we

did encounter a spin ambiguity. Again, the reason for this was already anticipated. In

agreement with our analytical results, Figs. F-1E, F-2E and F-3E show that all 3 of the

FVFS distributions provide identical matches to the corresponding FVFV distributions.

However, while we encountered a two-fold ambiguity with respect to the spin of particle A,

for each spin assignment the parameters α, β and γ were all measured.

F.5 Input from SFVF (S = 6)

We began the SFVF case study with the observable distributions in Equations F–26

through F–28. Fig. F-6 shows the decay diagram for this spin assignments.

L+−6 = 1.626 − 0.981m̂2

`` − 0.405m̂4`` (F–26)

113

S+−6 =

2.87 m̂2j` ≤ 0.632

1.288 0.632 ≤ m̂2j` ≤ 0.653

−0.344 − 4.493m̂2j` + 4.837m̂4

j`

− 5.870 log m̂2j` 0.653 ≤ m̂2

j`

(F–27)

D+−6 =

−0.322 + 0.786 m̂2j` m̂2

j` ≤ 0.632

−0.406 + 1.051 m̂2j` 0.632 ≤ m̂2

j` ≤ 0.653

5.870 − 11.674 m̂2j` + 5.804 m̂4

j`

+(

3.384 − 3.595 m̂2j`

)

log m̂2j` 0.653 ≤ m̂2

j`

(F–28)

The SFVF spin assignment is special, since in this case the dilepton invariant mass

distribution exhibits a characteristic m̂4 term which is not present for any of the other 5

spin configurations. Note that the existence of an m̂4 term in the dilepton distribution

does not require any special values of the model parameters. Actually, the m̂4 term cannot

even be removed by any choice of model parameter values. More generally, the dilepton

invariant mass distribution is in general given by some polynomial in terms of m̂2, whose

order is equal to twice the spin of the intermediate particle B. Only in the SFVF case

is there a spin 1 intermediate particle which introduces the m̂4 term in L+−. Therefore,

observation of the m̂4 dependence in the dilepton distribution unambiguously selects

SFVF out of our 6 possible spin assignments. The size of the coefficient of the m̂4 term

depends on the mass spectrum in the model, but it cannot be vanishingly small – this

would require either z = 1 or y = 1, which would correspondingly close off the B → A` or

the C → B` decay, and then the whole decay chain would be unobservable. Our numerical

results in Fig. F-1F confirm this conclusion – we saw that none of the other five models

could reproduce the SFVF dilepton distribution, due to the presence of the m̂4 term.

F.6 Remarks on Spin Determination at the Tevatron

At a pp̄ collider such as the Tevatron we expect the particle-antiparticle fraction to

be f = 12. On the surface, it may appear that this constraint eliminates only one out

114

of the four model-dependent degrees of freedom (f , ϕa, ϕb and ϕc) that we originally

started with. However, as can be deduced from Equations 3–47 and 3–48, this assumption

completely fixes the ϕ̃c parameter (Equation F–29), and as a result both β and γ vanish

identically.

ϕ̃c =π

4(F–29)

In this case, both our D+−S distribution and the lepton charge asymmetry distribution

(A+−) also vanish identically, so that these distributions are useless for spin determination

at a pp̄ collider such as the Tevatron.

However, our results for L+−S and S+−

S still hold, and contain non-trivial spin

information, so that the spin analysis following our method can still be performed.

In fact, our method can already be tested in the top quark semileptonic and dilepton

samples at the Tevatron by looking at the invariant mass distribution of the b-jet and the

lepton [63, 69]. Indeed, our decay chain from Fig. 1-3 can be applied to top quark decays,

for example by identifying C = t, B = W+ and A = ν`, and reinterpreting `n as the

b-jet and `f as the lepton coming from W decay. In that case, the mb` distribution should

be described by Equation 3–64 for L+−6 . Alternatively, one can identify the particles in

Fig. 1-3 as D = t, C = W+, B = ν`, q = b and `n = `. In this case, the mb` distribution

should be described by Equation 3–65 for S = 4 or S = 5. In either case, the characteristic

m̂4 term signals that the W is spin 1 and the top quark and the neutrino are both spin

1/2 (the FVF part of the spin assignment).

115

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BIOGRAPHICAL SKETCH

Michael Burns was born in Lubbock, Texas on 1978 October 8. His family moved

him to Lake Jackson, Texas when he was still an infant. He resided his entire childhood

in the small coastal town south of Houston. He enjoyed many outdoor activities with the

menfolk of his family, and he spent much of his free time hunting, fishing, camping, hiking,

and swimming in the Texas and Colorado wilderness.

Michael attended Brazosport Christian School, a small private religious school, from 4

years of age until he was 14 years of age. He attended the only nearby public high school,

Brazoswood High school (“B-wood”), in the adjacent town of Clute, Texas. During his

junior and senior years of high school, he took several courses at the local community

college, Brazosport College, and also enrolled in an automotive technology program at a

different high school, Brazosport High school.

Due to a religious disagreement with his father, Michael moved out of his father’s

house to support himself before the end of his senior year in high school. He did so rashly,

and was not prepared for the financial responsibility. After several months he reconciled

with his father, who agreed to pay his tuition and housing costs at a popular state school,

Texas A&M University (TAMU), from where his father had also received a degree.

There, Michael chose to major in electrical engineering due to a fascination with complex

numbers. Feeling somewhat trapped by the arrangement with his father, Michael took

heavy course loads and summer school, and managed to graduate within 3 years in 2000

May.

After his graduation, Michael was hired by a global telecommunications firm, Marconi

Communications (Marconi), In Irving, Texas. He soon realized that he did not enjoy

engineering, and requested dismissal from the company. The director of his department

expressed to Michael anger at the request, and suggested that Michael should at least

give another year of service to Marconi. However, within months, the telecommunications

company hit an economic downturn, and Michael was laid-off (perhaps to fulfill his

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request), along with hundreds of other employees from Marconi, and thousands of

employees in the Dallas area.

Facing the end of his unemployment benefits, and stretching his savings thin, Michael

decided to pursue his passion for physics by enrolling in a master’s program at the nearby

University of North Texas (UNT), who offered him a teaching assistantship. He finally

found enjoyment in his work at UNT, especially teaching, where his responsibilities

included tutoring, lab managing, and even lecturing physics courses during his last

summer at UNT. His academic focus at UNT was General Relativity. He received a

master’s in physics from the University of North Texas in 2004 May. Eventually, he

decided that he wanted to learn about high energy physics and the standard model of

particle physics, but he could not find an appropriate advisor for this at UNT. So, after a

year of continuing his assistantship working on a laser experiment at UNT, he transferred

to the University of Florida (UF) in 2005 August, where he received a 4-year fellowship.

Michael’s focus at UF was High Energy Phenomenology. He graduated with a

doctorate in physics in 2009 August, after 4 years at UF. While at UF, he coauthored

three papers with his advisor and other collaborators.

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