ABSTRACT
Title of dissertation: NOVEL DARK MATTER
PHENOMENOLOGY AT COLLIDERS
Kyle Wardlow, Doctor of Philosophy, 2015
Dissertation directed by: Professor Kaustubh S. Agashe
Department of Physics
While a suitable candidate particle for dark matter (DM) has yet to be discov-
ered, it is possible one will be found by experiments currently investigating physics
on the weak scale. If discovered on that energy scale, the dark matter will likely be
producible in significant quantities at colliders like the LHC, allowing the proper-
ties of and underlying physical model characterizing the dark matter to be precisely
determined. I assume that the dark matter will be produced as one of the decay
products of a new massive resonance related to physics beyond the Standard Model,
and using the energy distributions of the associated visible decay products, develop
techniques for determining the symmetry protecting these potential dark matter
candidates from decaying into lighter Standard Model (SM) particles and to simul-
taneously measure the masses of both the dark matter candidate and the particle
from which it decays.
NOVEL DARK MATTER
PHENOMENOLOGY AT COLLIDERS
by
Kyle Patrick Wardlow
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2015
Advisory Committee:
Professor Kaustubh Agashe, Chair/Advisor
Professor Zakaria Chacko,
Professor Thomas Cohen,
Professor Nicolas Hadley,
Professor Massimo Ricotti
c© Copyright by
Kyle Patrick Wardlow
2015
Dedication
I would like to dedicate this dissertation to my grandmother,
Ramona Wardlow, for providing constant inspiration and enduring,
unconditional love and support as I grew up. The memory of her
generous spirit, kind heart, and diligent mind continue to provide an
example worth striving toward.
ii
Acknowledgments
The work done in this thesis is the result of the help, support, and collaboration
of so many people. I would foremost like to thank my advisor, Kaustubh Agashe,
for his assistance and patience throughout the years. His help has greatly expanded
my knowledge of particle physics and I owe him much gratitude for all he has done
in the past five years.
I would also like to thank Doojin Kim, for his assistance in all things phe-
nomenology and research. His insight was instrumental in helping me understand
the ins and outs of properly analyzing our data; Roberto Franceschini, for his expert
advice and ability to simplify the seemingly difficult problems encountered in our
research to tractable, manageable hurdles.
This work was supported in part by NSF grants Nos. PHY-0652363, PHY-
1315155, and PHY-0968854 and the Maryland Center for Fundamental Physics. I
would like to thank all the members of the center, and especially Tom Cohen, Zakaria
Chacko, Ted Jacobson, and Raman Sundrum for their insightful and instructive
lectures and discussions over the years.
I would like to express my gratitude to Evan Berkowitz for his immense help
with Mathematica, programming, life, the universe, and everything. I would like to
also express my thanks to all of grad students in the center for their camaraderie
and enthusiasm.
My family deserves no small praise in supporting me and having high expec-
tations that I would succeed and get this far. My mother for more reasons than I
iii
can tell and my father and family for their constant belief in me.
And finally, Karen for her patience, love, support, and so much more.
iv
Table of Contents
List of Tables vii
List of Figures viii
List of Abbreviations ix
1 Introduction 1
1.1 Dark Matter and WIMPs . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Dark matter and Colliders . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Energy Peaks and Stabilization Symmetry 12
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Theoretical observations on kinematics . . . . . . . . . . . . . . . . . 18
2.2.1 The peak of the energy distribution of a visible daughter . . . 19
2.2.1.1 Two-body decay . . . . . . . . . . . . . . . . . . . . 19
2.2.1.2 Three-body decay . . . . . . . . . . . . . . . . . . . 23
2.2.2 The kinematic endpoint of the MT2 distribution . . . . . . . . 27
2.2.2.1 Two-body decay, one visible and one invisible . . . . 28
2.2.2.2 Three-body decay, one visible and two invisibles . . . 29
2.3 General Strategy to distinguish Z2 and Z3 . . . . . . . . . . . . . . . 32
2.4 Application to b quark partner decays . . . . . . . . . . . . . . . . . . 35
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3 Mass extraction 51
3.1 Introduction and general strategy of the mass measurement . . . . . 52
3.2 A template for the energy spectrum of a massive child particle . . . 60
3.3 “Event mixing” to estimate the combinatorial background . . . . . . 65
3.4 Application to the gluino decay . . . . . . . . . . . . . . . . . . . . . 70
3.4.1 Signal process: gluino decay . . . . . . . . . . . . . . . . . . 71
3.4.2 Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.4.2.1 Standard Model backgrounds and event selection . . 74
3.4.2.2 Combinatorial background and mixed event subtrac-
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
v
3.5 Mass measurement results and discussion . . . . . . . . . . . . . . . . 90
3.5.1 Measurement of gluino and neutralino masses . . . . . . . . . 90
3.5.2 Study of systematic effects . . . . . . . . . . . . . . . . . . . 100
3.5.3 Improving the mass measurement using the mbb endpoint . . . 107
3.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 109
4 Conclusions 114
4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
A Event Mixing: Signal and background 117
Bibliography 120
vi
List of Tables
2.1 Cross-sections of Z2 and Z3 signals and dominant BG . . . . . . . . . 41
3.1 Gluino signal and dominant BG cross sections before and after cuts . 73
3.2 Fit results for massless and massive templates for all mass slices . . . 97
3.3 Conditions for over-, under-, and consistent estimation of the masses . 100
3.4 Samples used to isolate the primary source of error in the template
fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
vii
List of Figures
1.1 DM Discovery Channels . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 DM Decay Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Z2 and Z3 signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Z3 separation from Z2 peak . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Effect of ST cuts on the dominant Z2 and Z3 backgrounds . . . . . . 43
2.4 MT2 distributions post-cuts for Z2 and Z3 . . . . . . . . . . . . . . . 45
2.5 Bottom energy distributions for Z2 and Z3 after cuts with BG . . . . 46
3.1 Real three-body decay and effective two-body decay used to form
distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Mixed event and correct gluino signal di-b-jet invariant mass and R
distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.3 Di-b energy distributions and ratio for mbb = 300GeV,700GeV. . . . . 83
3.4 Mixed event signal and background interference . . . . . . . . . . . . 88
3.5 Bin-by-bin ratio of Ebb from mixed event subtraction and the correct
pairings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.6 Goodness of fit comparison for massive and massless template . . . . 91
3.7 Fit results for massless and massive templates for m¯bb = 250 GeV
and m¯bb = 650 GeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.8 Mass extraction using the E∗bb parameter extracted from the massless
and massive template fits . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.9 Contour plot in the plane of the gluino and DM mass around their
best fit mass values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.10 Determination of the source of bias in the fit results . . . . . . . . . . 105
3.11 Contour plot of improved mass paramaters using the invariant mass
endpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
viii
List of Abbreviations
SM Standard Model of particle physics
LHC Large Hadron Collider
DM Dark Matter
Λ− CDM Lambda Cold Dark Matter Cosmological Model
TeV Tera-electronVolt
GeV Giga-electronVolt
MeV Mega-electronVolt
MT2 Stranverse Mass
ISR Initial State Radiation
BSM Beyond Standard Model
Z2 Cyclic group of order two
Z3 Cyclic group of order three
MCFP Maryland Center for Fundamental Physics
NSF National Science Foundation
DOE Department of Energy
ix
Chapter 1: Introduction
While it has been extraordinarily successful in providing an accurate descrip-
tion for an extremely broad range of observed phenomena, the Standard Model does
not constitute a complete accounting of the physics of the universe. Some of the ob-
servational discrepancies and omissions apparent in the model can be addressed by
the well-motivated addition of complementary extensions and modifications, such as
Λ-CDM a Big Bang cosmological model, but problems remain even after taking these
into account - there is no provision for new physics above the electro-weak scale,
leaving the Planck-weak hierarchy problem unresolved, and there are no reasonable
candidate particles for dark matter (DM) [1,2].
The fact that the Standard Model includes no new physics beyond the electro-
weak scale it not necessarily problematic per se - however, the theory seems strangely
fine-tuned without the addition of new particles between the weak and Planck scales.
We know that a new description of physics must take over at the Planck scale,
because it is the scale at which quantum gravitational fluctuations become too large
to neglect, but according to our best knowledge of the universe as it currently stands,
there are no new states with masses between the weak scale (mWeak = 246GeV)
and the Planck scale (MPl ≈ 1018GeV) The mass of the Higgs particle, the scalar
1
boson that breaks the electro-weak symmetry of the Standard Model and gives the
fundamental elementary leptons their masses, is susceptible to physics at higher
energy scales, and without new physics between the Planck and weak scales, would
receive corrections to its mass based on the order of the Planck scale. Because the
Planck scale is so large, the fact that the observed mass of the Higgs is on the order
of weak scale, mH = 126GeV, suggests that there is a very precise cancellation to
these higher order corrections to the Higgs mass. This is highly suggestive of new
physics at or above the weak scale. Since the energies below the weak scale have
been well-studied, we therefore expect that new physics that protects the Higgs mass
will come in at around the TeV scale and that the LHC will be able to probe these
new phenomena. An interesting possibility is that this new physics will also include
a weakly-interacting, massive particle (WIMP) that constitutes a viable candidate
for dark matter.
The existence of dark matter has been solidly established by astrophysical
observation - the rotation curves of objects in many observed galaxies cannot be
explained without either modifying gravity or assuming that there is additional, in-
visible matter keeping the luminous matter in the orbits seen. Modifying gravity is
at the present moment strongly disfavored - observations such as the Bullet Cluster
put strong bounds on the extent to which gravitational interactions can be altered
from what is expected in general relativity - and so ‘dark’ matter is the likely can-
didate for the observed phenomena [1]. Observations also force us to conclude that
the dark matter is not baryonic in nature - it does not interact with electromagnetic
radiation - indeed, it has only been observed gravitationally at present. Cosmic
2
microwave background data also corroborate that dark matter are a significant por-
tion of the mass-energy of the known universe; in fact, it constitutes approximately
26.8% of this mass-energy, making it about 5.5 times more plentiful than ordinary
matter [3].
Dark matter is quite possibly one of most interesting topics in particle physics
at the present time - we have a clear signal of its existence and we have good reason
to believe that the discovery of its underlying physical basis is within our grasp. We
must therefore discuss new, non-gravitational methods of probing the properties of
this dark matter. To that end, I will discuss how dark matter can be seen directly
and indirectly, and potentially produced experimentally. After this, I will elaborate
on why WIMPs are an interesting candidate for this dark matter and will lay out
how this kind of dark matter relates to collider experiments and discuss the ways in
which it can be seen and probed at the current generation of experiments. I will then
quickly outline the remainder of the dissertation by focusing on some of the most
interesting properties of this dark matter and how these specifically can be measured
at the LHC if and when dark matter is produced there. In depth discussion and
results from my phenomenological investigations into the possibility of determining
these properties will then follow in the body of the dissertation, and I will conclude
with a summation of these results and comments on the outlook for future areas
of phenomenological interest. This dissertation essentially follows work done with
Kaustubh Agashe, Roberto Francescini, Doojin Kim, and Sung-Woo Hong, and is
based on that work in chronological order [4, 5].
3
Figure 1.1: The ways in which dark matter can be observed
1.1 Dark Matter and WIMPs
Generally speaking, if dark matter interacts with normal matter at all, it can
in principle be detected via those interactions. Direct detection involves dark matter
scattering off of normal matter, which in the context of direct detection experiment
usually means a large, uniform mass highly shielded from external background radia-
4
tion and sensitive Cherenkov-detecting apparatuses to detect the energy transferred
in this scattering. Indirect detection typically involves the astrophysical observation
of regions of high dark matter density and looking for unexpected lines of radiation;
these unexpected lines would imply the bipartite self-annihilation of the dark mat-
ter into visible particles (typically photons) and would be an easy way to access the
mass of the dark matter particles. Production is the method of observation that we
shall direct our focus toward, and typically involves dark matter particles being pro-
duced at collider experiments as a result of the high energy collisions there. These
dark matter particles can be produced in a variety of ways, but one of the most
interesting is when they are the result of the decay of some heavier, new physics
state produced in the collider experiment and the dark matter must be produced in
association with visible particles in order for the events to be of interest because it
is essentially invisible to the detection equipment at current colliders.
As mentioned above, WIMP dark matter is an especially interesting candidate
for the dark matter observed in the universe; this is due to what is known as the
“WIMP Miracle.” The WIMP Miracle, briefly, is the fact that particles that have
an interaction cross section with normal matter and masses of order the weak scale
give approximately the observed dark matter relic density left over after the Big
Bang and the period known as inflation in the early universe. This is tantalizing;
as discussed above, we have strong reason to suspect that there is new physics just
around the corner at the TeV scale, which consider approximately equivalent to the
weak scale. This means that whatever new physics might be discovered at, say, the
LHC may also naturally bring with it dark matter candidates at no extra theoretical
5
cost.
This raises the exciting possibility that we will see signs of dark matter candi-
dates being produced at the LHC and it is for this reason that I restrict our focus
here on collider dark matter phenomenology. In the investigations undertaken in
the following, a technical distinction must be made: I motivate this research with
the conspicuous coincidence of the WIMP miracle as it relates to dark matter, but
our results are in fact more general and can be applied to other situations where
there are invisible particles. I use the heuristic shortcut of labeling these invisible
dark matter candidates as dark matter, because the hope is that if similar invisi-
ble particles are discovered, they would match the criteria we already know from
cosmological observations and thus earn the appellation “dark matter.”
There are many theoretical extensions of the Standard Model that give rise
to these invisible particles; in fact, most are constructed so that they include this
attractive feature. Supersymmetry (SUSY) is perhaps the most famous, where
the particles known as neutralinos act as excellent WIMP dark matter when they
are the lightest (stable) supersymmetric partner to the Standard Model [6]. In
order to make this class of dark matter stable, it must be imbued with a conserved
charge that prevents it from decaying to lighter SM final states; this is typically
implemented in the form of an R-parity symmetry. But there are yet more interesting
and exotic models that also give rise to dark matter candidates - Extra dimensional
models can easily incorporate dark matter in the form of the Kaluza-Klein modes of
neutral Standard Model particles. If the extra dimensions are warped, one can also
ameliorate the Planck-weak hierarchy problem, which makes this class of models
6
competitive with SUSY [7, 8]. Extra dimensional models are also often distinct
from SUSY in the way that they protect their dark matter candidates (among other
things, such as protons) from decaying to lighter SM particles - typically it is done by
giving particles charge based on their baryon number (gauging and then breaking the
baryon number symmetry in a proscribed way), which results in a different symmetry
of stabilization for these models; we shall return to this later in this chapter and
in Chapter 2 [9]. There are also models that include axion dark matter, which is
related to the resolution of the strong CP problem, and additional generations of
highly decoupled neutrinos, known as sterile neutrinos; however, I do not consider
these here, and my general program will be to consider only WIMP dark matter.
1.2 Dark matter and Colliders
I will now consider some general phenomenological characteristics of the be-
havior of dark matter in collision events. First, in order for the dark matter to be
“seen” in a collider experiment, it must paradoxically escape the detector without
decaying to visible particles. For this reason, I have required that the dark matter
considered in the dissertation to be stable. Of course, we know that astrophysical
dark matter must be stable at least on the order of the lifetime of the universe, so
this is a well-justified assumption.
Now let us turn to what collision events involving dark matter look like: Natu-
rally, in order to make any determination of the properties of these dark matter par-
ticles, they must first be discovered. There are several means of making a definitive
7
discovery of dark matter, both in the context of detection design and construction,
and in what discovery channel, i.e direct vs. indirect detection vs. production as has
already been mentioned; I will assume in the following that a definitive discovery
has already been made and the results that follow are predicated upon that. In
general, the signature of dark matter in a collision event is a large amount of miss-
ing momentum and energy - the dark matter flies through the detector leaving no
direct trace of its presence. Because of this, it must be produced along with visible,
SM particles; otherwise the “event” has no way of registering in the detector, as
described in the following schematic equation:
SM + SM → SM +DM = SM + E/T (1.1)
There are a variety of events that could have this E/T + SM final state; we will
consider only those where the dark matter is produced in pairs so that the charge
that the dark matter has under its stabilization symmetry is conserved. To wit:
SM + SM → SM + DM + ¯DM The dark matter pairs could be the result of the
decay of some heavy intermediate state produced directly by the collision of partons
- this channel would require that it be produced in association with some initial state
radiation in order for it to be seen, and the ISR would give a handle on the missing
energy and momentum. They could also be produced as the subsequent decay
products of heavier new physics states that are also charged under the stabilization
symmetry; this is the case that we take under consideration for the remainder of
this dissertation.
In this category of production, there are further subcategories related to the
8
S˜M
SM
DM
SM
DM
S˜MSM
SM
Figure 1.2: Typical dark matter-inclusive decay topology from a pair-produced new
heavy resonance.
mass and energy of the pair produced intermediate states:
• Some or all of the intermediate states are on-shell, which means that the
precise decay topology plays an important role in the ultimate kinematics of
the decay products.
• The intermediate states are all far off their mass shell(s), leading to contact-
like interactions, meaning that the decay is effectively N-body, where N is the
total number of decay products.
Of these, our focus is directed toward the latter. The former is interesting in its
own right, and the author would direct the interested reader to [10,11].
Now that the production channel of the dark matter has been sufficiently
specified, let us consider which dark matter properties are of interest and accessible
from the data accessible to collider experiments. Of paramount interest are the
masses of not just the dark matter, but of all the particles involved in the interaction
9
leading to the emission of the dark matter particles. Knowing the masses of these
particles would give physicists a much better handle on the parameter space and
scale of the new physics, and would help to better direct theoretical development of
the nascent BSM theory that the discovery of such particles would necessitate. There
are other parameters one would generally like to know, such as the spin, but one
that is often overlooked is the stabilization symmetry. It may seem at first glance
that accessing this information is extraordinarily difficult, but it is in fact quite
simple in some cases to garner insight about this symmetry from simple kinematics.
It is upon these two categories of determination - the stabilization symmetry and
the masses of the dark matter and its originating particle - that will occupy the rest
of time in the following.
Broadly, the stabilization symmetry directly affects how many particles a mas-
sive parent that is also charged under the symmetry will ultimately decay to. For
R parity-like (Z2) symmetries, the decays are typically two-body with one invisible
and one SM particle in the final state, and for baryon number-like symmetries (Z3),
the final state of a massive parent are three-body, two of which are invisible [10].
In the second chapter, I will develop the theory behind how this affects the kine-
matic distributions of the visible particles in the decay, and I will show how the
stabilization symmetry can be determined using event simulated under a largely
model-independent, toy process using the peak location of the energy distribution
of the visible particles. In this chapter, I will assume that the visible particles are
massless, which is critical to the simplicity of the result.
In the third chapter, I will generalize the result in chapter two to massive visible
10
decay products and, considering a different decay topology, discuss how the mass
of both the parent and the dark matter particle can be extracted from the energy
distribution of the decay-wise sum of the energy of the visible decay products. In
order to do this, I simulate events for an example process - pair-produced gluinos
decaying to two SM particles and one invisible each, and fit the data in the pair-
wise energy sum distribution of the visible particles to a function that allows me
to ultimately extract the mass parameters of both the parent and DM particle. In
order to do so, the constructed energy data must first be treated for a combinatoric
ambiguity - there is generally more than one way to combine the particles in the
overall multi-body final state, and without removing the distortion to distribution
that is the inevitable result of this, one cannot extract any useful information.
The critical take-away is the use of the energy distribution in both of these
examples: typically, Lorentz invariants are used to extract information from parti-
cle collision event samples, and it is remarkable (and truly novel) that the energy
distribution proves to be so useful in this regard. I will close with some comments
on this and on the outlook for further uses of novel kinematic observables, such as
the energy distribution, going into the future.
11
Chapter 2: Energy Peaks and Stabilization Symmetry
2.1 Introduction
As mentioned previously, there are many motivations for extensions to the
Standard Model (SM) of particle physics at the TeV scale; perhaps the most im-
portant among these motivations are the necessity of a fundamental mechanism for
electroweak symmetry breaking (EWSB) and a resolution of the related Planck-weak
hierarchy problem. In such extensions of the SM, there generally exists a new par-
ticle at or below the TeV scale which cancels the quadratic divergence of the Higgs
mass from the top quark loop in the SM. Such a particle is typically a color triplet
with a significant coupling to the SM top quark, and has an electric charge of +2/3.
Following the literature, I will generically call such particles “top partners” and
denote them by T ′ 1. These top partners often come along with bottom partners,
which I similarly denote as B′. The typical reason for this is that the left-handed
(LH) top quark is in a doublet of SU(2)L with the LH bottom quark. I then ex-
pect top and bottom quark-rich events from the production and decay of these new
particles at the LHC. Because the aforementioned extensions also generally contain
1In this chapter this name applies as long as the partners have interactions with the relevant
SM particle, even if the partners do not directly cancel the Higgs mass divergence.
12
candidate particle for dark matter [1], in the form of WIMPs or other more ex-
otic candidates [2], and these scenarios will often involve heavier new particles that
are charged under both the symmetry that keeps the DM stable and the SM gauge
group. These new particles should then be copiously produced at the LHC and
must decay into DM particles and SM states, given that the latter are not charged
under the DM stabilization symmetry. Thus I expect this new physics to give rise
to events at the LHC with large missing energy, in association with jets, leptons,
and photons. We therefore explore scenarios which employ extensions that have
the above characteristics; In this case, it is likely that the top and bottom partners
are also charged under the DM stabilization symmetry. These extensions will then
result in top and bottom quark-rich events at the LHC in which the new particles
give rise to missing energy. There are many examples of these extensions [6–8], but
in essence I find that a search for events exhibiting the characteristics of having a
top or bottom partner and missing energy should be a top priority for the LHC.
Once the existence of new physics has been established, the most urgent issue
that will then have to be addressed is the determination of the details of the dynam-
ics underlying this new physics. In particular, it will be crucial to determine the
properties of the top and bottom partners using as model-independent an approach
as possible. This detailed study would also offer major hints regarding the resolu-
tion of the Planck-weak hierarchy problem. For largely model-independent work on
fermionic bottom and top partners’ discovery potential at the LHC see Refs. [12,13]
and for the determination of generic partners’ spin and mass see Refs. [14].
However, I remark that in this literature it has been assumed that the top or
13
bottom partner decays into only one DM particle, which is expected when the DM
is stabilized by a Z2 symmetry. While Z2 is perhaps the simplest DM stabilization
symmetry, it is by no means the only possibility: see references [15,16]. The point,
especially in the case of such non-Z2 symmetries, is that more than one DM can
appear in the decays of top and bottom (and other SM) partners: for example, two
DM are allowed with Z3 as in [15], but not with Z2.
I believe that a truly model-independent approach to the determination of the
top and bottom partners’ properties should include this possibility of multiple DM in
addition to different spins for the top and bottom partners. With this goal in mind,
I aim to devise a strategy that uses experimental data to determine the number of
DM in these decays and accordingly to identify the stabilization symmetry of the
dark matter. Below, I outline a general strategy and then apply it to the specific
case of bottom partner decays.
I concentrate on the distinction between two general decay topologies:
A→ bX and A→ bX Y (2.1)
where b is a (single) SM visible particle, X and Y are two potentially different
invisible particles and A is a heavier particle that belongs to the new physics sector.
In the context of the models that I have discussed, A is the heavy particle charged
under the DM stabilization symmetry and the particles labeled X and Y are the
DM particles. In particular, I focus on scenarios where the two decays are mutually
exclusive, i.e. where the stabilization symmetry and the charges of the involved
particles are such that one decay can happen and not the other. This mutual
14
exclusivity can be the case with both Z2 and Z3 as the stabilization symmetry. To
wit, if the SM particle b is not charged under the stabilization symmetry and all the
new particles A,X, Y are, then the Z2 symmetry allows only for two-body decays
of A. On the other hand, both the two and three-body decays of A are allowed
by the Z3 symmetry by itself. However, I assume that other considerations forbid
(or suppress) the two-body decay in this model. I choose to concentrate on this
realization of the Z3 -symmetric model in part because this is the case that cannot
be resolved using the results of previous work on the DM stabilization symmetry.
This is the case, for instance, in Ref. [10], where purely two-body decays of A could
be distinguished from mixed two- and three-body decays, but not from the purely
three-body decays that I am now taking into consideration.
In this chapter, I develop a method based primarily on the features of the
energy distribution of the visible final state b to differentiate between the cases
of purely two- and three-body decays. I remark that this is the first use of the
energy distribution of the the decay products to study the stabilization symmetry
of the DM. In fact, other work has typically focused on using Lorentz invariant
quantities or quantities that are invariant under boosts along the beam direction of
the collider. This is the case for Refs. [10, 11,17,18]. In particular, Refs. [10, 11,17]
used the endpoints of kinematic distributions to probe the stabilization symmetry
of the DM, whereas our method relies quite directly on peak measurements and
only marginally on endpoint measurements. Additionally, I note that the methods
developed in Refs. [11, 18] apply only to the case where there are more than one
visible particle per decay. Therefore, our result for cases where there is only one
15
visible particle per decay is complementary to the results of the above references.
Our basic strategy is explained in the following. It relies on a new result:
assuming massless visible decay products and the unpolarized production of the
mother particles, I will show that in a three-body decay the peak of the observed
energy of a massless decay product is smaller than its maximum energy in the
rest frame of the mother. This observation can be used in conjunction with a
previously observed kinematic characteristic of the two-body decay to distinguish
the stabilization symmetry of the DM. Specifically, it was shown in Ref. [19, 20]
that for an unpolarized mother particle, the peak of the laboratory frame energy
distribution of a massless daughter from a two-body decay coincides with its (fixed)
energy in the rest-frame of the mother.
Clearly, to make use of these observations in distinguishing two from three-
body decays, I need to measure the “reference” values of the energy that are involved
in these comparisons. Moreover, the procedure that is to be used to obtain this
reference value from the experimental data should be applicable to both two and
three-body decays. To this end, I find that when the mother particles are pair
produced, as happens in hadronic collisions, the MT2 variable can be used. Thus,
these observations make counting the number of invisible decay products possible
by looking only at the properties of the single detectable particle produced in the
decay. However, it is worth noting that our proof of the above assertion regarding
the kinematics of two- and three-body decays is only valid with a massless visible
daughter and an unpolarized mother. Therefore, care must be taken when discussing
cases with a massive daughter or a polarized mother.
16
To illustrate the proposed technique, I will study how to distinguish between
pair-produced bottom partners each decaying into a b quark and one DM from pair-
produced bottom partners each decaying into a b quark and two DM particles at the
LHC 2. As discussed above, a bottom partner appears in many motivated extensions
to the SM, so I posit that this is a relevant example. Furthermore, I remark that the
b quark is relatively light compared to the expected mass of the bottom partner, so
that our theoretical observation for massless visible particles is expected to apply.
Additionally, the production of bottom partners proceeds dominantly via QCD and
is thus unpolarized. In this sense, the example of a bottom partner is well-suited to
illustrate our technique. Finally, it is known that the backgrounds to the production
of bottom partners may be rendered more easily manageable than for those of top
partners [12], which would be a well-motivated alternative example.
Specializing to the example of bottom partners, our goal then is to distinguish
the two processes illustrated in Figure 2.1 at the collider
pp→ B′B¯′ → bb¯χχ for Z2 , (2.2)
pp→ B′B¯′ → bb¯χχχ¯χ¯ for Z3 , (2.3)
where χ is an invisible particle and a bar denote anti-particles. In these processes,
I assume that there are no on-shell intermediate states. I consider the case where
the decay into two χ can happen only if the stabilization symmetry of the DM is
Z3, while the decay into one χ is characteristic of the Z2 case. As said before, I
2To the best of our knowledge, none of the earlier work on distinguishing DM stabilization
symmetries at colliders has studied this specific case.
17
p                  B'                 χ
p                  B'                 χ
b
b
p                  B' 
p                  B'    
b
b
χ
χ
χ
χ
Figure 2.1: The signal processes of interest for Z2 (left panel) and Z3 (right panel)
stabilization symmetry of the dark matter particle χ.
focus on this scenario because it has thus far been left uninvestigated by previous
studies on the experimental determination of the stabilization symmetry of the dark
matter [10,11].
From here, I organize our findings as follows: In Section 2.2, I review the
current theory and I derive new results about the energy spectrum of the decay
products of two- and three-body decays. These are then the foundation of the
general technique presented in Section 2.3 for differentiating decays into one DM
particle from those into two DM particles. In Section 2.4, I apply this technique to
the specific case of bottom partners at the LHC. I conclude in Section 2.5.
2.2 Theoretical observations on kinematics
I begin first by reviewing the relevant theoretical observations about the kine-
matics of two-body and three-body decays. Specifically, I review the remarks on
two-body decays described in [19]. I then generalize this result to three-body decay
18
kinematics and study the features that distinguish it from two-body decay kinemat-
ics. I also briefly review applications of the kinematic variable MT2 to two-body
and three-body decays and discuss the distinct features of the two different decay
processes [10, 21].
For the two-body decay, I assume that a heavy particle A decays into a massless
visible daughter b and another daughter X which can be massive and invisible:
A→ bX. (2.4)
On the other hand, for a three-body decay the heavy particle A decays into particles
b, X and another particle Y
A→ bX Y . (2.5)
Like particle X, particle Y can also be massive and invisible, but it is not necessarily
the same species as particle X.
2.2.1 The peak of the energy distribution of a visible daughter
2.2.1.1 Two-body decay
It is well-known that the energy of particle b in the rest frame of its mother
particle A is fixed, which implies a δ function-like distribution, and the simple ana-
lytic expression for this energy can be written in terms of the two mass parameters
mA and mX :
E∗b =
m2A −m2X
2mA
. (2.6)
19
Typically, the mother particle is produced in the laboratory frame at colliders with
a boost that varies with each event. Since the energy is not an invariant quantity, it
is clear that the δ function-like distribution for the energy as described in the rest
frame of the mother is smeared as I go to the laboratory frame. Thus, naively it
seems that the information encoded in eq. (2.6) might be lost or at least not easily
accessed in the laboratory frame. Nevertheless, it turns out that such information is
retained. I denote the energy of the visible particle b as measured in the laboratory
frame as Eb. Remarkably, the location of the peak of the laboratory frame energy
distribution is the same as the fixed rest-frame energy given in eq. (2.6):
Epeakb = E∗b , (2.7)
as was shown in [19,20].
Let us briefly review the proof of this result while looking ahead to the discus-
sion of the three-body case. As mentioned before, the rest-frame energy of particle
b must be Lorentz-transformed. The energy in the laboratory frame is given by
Eb = E∗bγ(1 + β cos θ∗) = E∗b (γ +
√
γ2 − 1 cos θ∗) , (2.8)
where γ is the Lorentz boost factor of the mother in the laboratory frame and θ∗
defines the angle between the emission direction of the particle b in the rest frame of
the mother and the direction of the boost ~β, and where I have used the relationship
γβ =
√
γ2 − 1. If the mother particle is produced unpolarized, i.e., it is either a
scalar particle or a particle with spin produced with equal likelihood in all possible
polarization states, the probability distribution of cos θ∗ is flat, and thus so is that
20
of Eb. Since cos θ∗ varies between −1 and +1 for any given γ, the shape of the
distribution in Eb is simply given by a rectangle spanning the range
Eb ∈
[
E∗b (γ −
√
γ2 − 1), E∗b (γ +
√
γ2 − 1)
]
. (2.9)
It is crucial to note that the lower and upper bounds of the above-given range are
always smaller and greater, respectively, than Eb = E∗b for any given γ, so that E∗b is
covered by every single rectangle. As long as the distribution of the mother particle
boost is non-vanishing in a small region near γ = 1, E∗ is the only value of Eb to
have this feature. Furthermore, because the energy distribution is flat for any boost
factor γ, no other energy value has a larger contribution to the distribution than
E∗b . Thus, the peak in the energy distribution of particle b is unambiguously located
at Eb = E∗b .
The existence of this peak can be understood formally. From the fact that
the differential decay width in cos θ∗ is constant, I can derive the differential decay
width in Eb for a fixed γ as follows:
1
Γ
dΓ
dEb
∣∣∣∣
fixed γ
=
1
Γ
dΓ
d cos θ∗
d cos θ∗
dEb
∣∣∣∣
fixed γ
=
1
2E∗b
√
γ2 − 1
Θ
[Eb
E∗b
− (γ − βγ)
]
Θ
[
−
Eb
E∗b
+ (γ + βγ)
]
,(2.10)
where the two Θ(Eb) are the usual Heaviside step functions, which here merely de-
fine the range of Eb. To obtain the full expression for any given Eb, one should
integrate over all γ factors contributing to this Eb. Letting g(γ) denote the prob-
ability distribution of the boost factor γ of the mother particles, the normalized
21
energy distribution f2-body(Eb) can be expressed as the following integral
f2-body(Eb) =
∫ ∞
1
2
(
Eb
E∗b
+
E∗b
Eb
) dγ g(γ)
2E∗b
√
γ2 − 1
. (2.11)
The lower limit in the integral can be computed by solving the following equation
for γ:
Eb = E∗b
(
γ ±
√
γ2 − 1
)
(2.12)
with the positive (negative) signature being relevant for Eb ≥ E∗b (Eb < E∗b ). I can
also calculate the first derivative of eq. (2.11) with respect to Eb as follows:
f ′2-body(Eb) = −
1
2E∗bEb
sgn
(Eb
E∗b
−
E∗b
Eb
)
g
(
1
2
(Eb
E∗b
+
E∗b
Eb
))
. (2.13)
The solutions of f ′2-body(Eb) = 0 give the extrema of f2-body(Eb), and given the
expression f ′2-body(Eb) in eq. (2.13), these zeros originate from those of g(γ). For
practical purposes, one can take g(γ) to be non-vanishing for particles produced at
colliders for any finite value of γ greater than 1 3. As far as zeros are concerned, two
possible cases arise for g(1) (corresponding to Eb = E∗b ). If it vanishes, f ′2-body(Eb =
E∗b ) ∝ g(1) = 0, which implies that the distribution has a unique extremum at
Eb = E∗b . If g(1) 6= 0, f ′2-body(Eb) has an overall sign change at Eb = E∗b . As a result,
the distribution has a cusp and is concave-down at Eb = E∗b . Moreover, the function
f2-body(Eb) has to be positive to be physical, and has to vanish as Eb approaches
either 0 or ∞, which is manifest from the fact that in those two limits the definite
3It must be noted that due to the finite energy of the collider, there is a kinematic upper limit
for the boost factor γ of the heavy mother particles. However, this kinematic limit is usually very
large and can effectively be taken as infinite.
22
integral in eq. (2.11) is trivial. Combining all of these considerations, one can easily
see that the point Eb = E∗b is necessarily the peak value of the distribution in both
cases.
2.2.1.2 Three-body decay
I now generalize the above argument to three-body decays. I denote the energy
of the visible particle b measured in the rest frame of the mother particle A as E¯b.
I also denote the normalized rest-frame energy distribution of particle b as h(E¯b).
In the two-body decay, this rest-frame energy is single-valued (see eq. (2.6)), and so
the corresponding distribution h(E¯b) was trivially given by a δ-function. However,
when another decay product is introduced, for instance, particle Y in eq. (2.5), then
the energy of particle b is no longer fixed, even in the mother’s rest frame: h(E¯b) 6=
δ
(
E¯b − E∗b
)
. Although the detailed shape of this rest-frame energy distribution is
model-dependent, the kinematic upper and lower endpoints are model-independent.
Since particle b is assumed massless, the lower endpoint corresponds to the case
where energy-momentum conservation is satisfied by particles X and Y alone. On
the other hand, the upper endpoint is obtained when the invariant mass of X and Y
equals mX +mY , which corresponds to the situation where X and Y are produced
at rest in their overall center-of-mass frame. Thus, I have
E¯minb = 0 , (2.14)
E¯maxb =
m2A − (mX +mY )2
2mA
. (2.15)
For any fixed γ, the differential decay width in the energy of particle b in the
23
laboratory frame is no longer a simple rectangle due to non-trivial h(E¯b). For any
specific laboratory frame energy Eb, contributions should be taken from all relevant
values of E¯b and weighted by h(E¯b). This can be written as
1
Γ
dΓ
dEb
∣∣∣∣
fixed γ
=
∫ E¯>b
E¯<b
dE¯b
h(E¯b)
2E¯b
√
γ2 − 1
, (2.16)
where
E¯<b = max
[
E¯minb ,
Eb
γ +
√
γ2 − 1
]
=
Eb
γ +
√
γ2 − 1
, (2.17)
E¯>b = min
[
E¯maxb ,
Eb
γ −
√
γ2 − 1
]
, (2.18)
with Eb running from 0 to E¯maxb
(
γ +
√
γ2 − 1
)
. Again, since the visible particle
is assumed massless, E¯minb is zero and so the second equality in eq. (2.17) holds
trivially.
Finding an analytic expression for the location of the peak is difficult because
of the model-dependence of h(E¯b), and it follows that the precise location of the peak
is also model-dependent. Nevertheless, I can still obtain a bound on the position
of the peak for fixed γ. Suppose that I am interested in the functional value of the
energy distribution at a certain value of Eb in the laboratory frame; according to
the integral representation given above, the relevant contributions to this Eb come
from a range of center of mass energies which go from E¯ ′b to E¯ ′′b , where these are
defined by
E¯ ′b(γ +
√
γ2 − 1) = Eb , (2.19)
E¯ ′′b (γ −
√
γ2 − 1) = Eb . (2.20)
Each energy contributes with weight described by h(E¯b), as implied by eq. (2.16).
24
Let us assume that E¯ ′′b = E¯maxb and denote the corresponding energy in the
laboratory frame as Elimitb , given by
Elimitb = E¯maxb (γ −
√
γ2 − 1). (2.21)
From these considerations, it follows that all rest-frame energies in the range from
E¯ ′b =
Elimitb
(γ+
√
γ2−1)
to E¯ ′′b = E¯maxb contribute to a chosen energy in the laboratory
frame, Elimitb . On the other hand, any laboratory frame energy greater than Elimitb
has contributions from E¯ ′b >
Elimitb
(γ+
√
γ2−1)
to E¯ ′′b = E¯maxb ; the relevant range of the
rest-frame energy values will shrink so that the peak cannot exceed Elimitb :
Epeakb
∣∣∣
fixed γ
< E¯maxb (γ −
√
γ2 − 1) ≤ E¯maxb for any fixed γ. (2.22)
In order to ensure that the first inequality holds even for γ = 1, I assume in the
last equation that h
(
E¯maxb
)
= 0, which is typically the case for a three-body decay.
In order to obtain the shape of the energy distribution of particle b in the labora-
tory frame, all relevant values of γ should be integrated over as with the two-body
kinematics in the previous section. Hence, the laboratory frame distribution reads
f3-body(Eb) =
1
Γ
dΓ
dEb
=
∫ E¯>b
E¯<b
dE¯b
∫ ∞
γmin(Eb, E¯b)
dγ g(γ)h(E¯b)
2E¯b
√
γ2 − 1
. (2.23)
Since the argument leading to eq. (2.22) holds for every γ, the superposition of
contributions from all relevant boost factors does not alter this observation. There-
fore, I can see that irrespective of g(γ) and h(E¯b), the peak position of the energy
distribution of particle b in the laboratory frame is always less than the maximum
rest-frame energy:
Epeakb < E¯maxb . (2.24)
25
1.00 1.01 1.02 1.03 1.04 1.05
0.75
0.80
0.85
0.90
0.95
1.00
Γ
U
p
p
e
r
b
o
u
n
d
o
n
E
b
p
e
a
k

E
bm
a
x
Figure 2.2: Relative separation of the peak of the laboratory energy distribution from
the maximal energy in the center-of-mass frame of the three-body decay kinematics
as per eq. (2.24). The horizontal red dashed line marks a 10% variation of the peak
energy from the maximal value in the rest frame.
To gain intuition on the magnitude of the typical difference between the peak
of the energy distribution in the laboratory frame and the maximum rest frame
energy, I show the ratio of the two as a function of γ in Fig. 2.2. From the figure, it
is clear that as the typical γ increases beyond γ = 1, i.e., as the system becomes more
boosted, the location of the peak in the energy distribution becomes smaller. An
appreciable shift of order 10% is achieved for a modest boost of order γ− 1 ' 10−2.
It should be noted that all results here for both two-body and three-body
decays are valid to leading order in perturbation theory. The presence of extra
radiation in the decay will effectively add extra bodies to the relevant kinematics.
Specifically, extra radiation can turn a two-body decay into a three-body one, which
for our investigation would constitute a fake signal of two DM particles being pro-
duced in the decay of a heavy new physics particle. Therefore, I have to remark
26
that in some cases, for instance, when the heavy new physics is typically produced
with very small boost, the differences between the two scenarios of DM stabilization
may be tiny and a study beyond leading order may be necessary. From Fig. 2.2 it
seems, however, that the typical effect of the presence of two dark matter particles
per decay of the heavy new particle is to easily induce an order one effect on the
peak position. Therefore, I anticipate that such an effect would be much larger than
the expected uncertainty from higher order corrections, which I estimate to be of
order 10%.
Before closing this section, I emphasize that I shall use the right-hand sides
of eqs. (2.7) and (2.24) as “reference” values to which the measurements of their
respective left-hand side values (extracted from the energy distribution) are to be
compared. In the next section, I show that such a reference value can, in fact, be
extracted from an analysis of MT2.
2.2.2 The kinematic endpoint of the MT2 distribution
In this section, I review how the MT2 variable is implemented for the two- and
three-body decays of heavy particles produced at a collider. For our MT2 analysis,
I make further assumptions as follow:
1) all massive decay products, i.e., particles X and Y in eqs. (2.4) and (2.5), are
invisible;
2) the mother particles A are produced in pairs;
3) the entire decay process is symmetric in the sense that the mother particles
27
are pair-produced and then decay to the same decay products, that is
pp→ AA , A→ X b or A→ b X Y , (2.25)
for the two-body decay and the three-body decay, respectively.
The last assumption is especially relevant to make contact with the problem of
distinguishing the Z2 and the Z3 dark matter interactions, as detailed in the intro-
duction.
2.2.2.1 Two-body decay, one visible and one invisible
The MT2 variable generalizes the transverse mass to the cases where pair-
produced mother particles each decay into visible particles along with missing par-
ticles (see Ref. [21] and references therein for a detailed review). Specifically, it can
be evaluated for each event by a minimization of the two transverse masses in each
decay chain, under the constraint that the sum of all the transverse momenta of the
visible and invisible particles vanishes.
By construction, each of the transverse masses in both decay chains involve
the mass of the invisible particle(s), and thus so does MT2. Since a priori I am
not aware of the invisible particles’ masses, I am required to introduce a trial mass
parameter into the definition of MT2. I denote this trial mass by m˜. The dependence
of the definition of MT2 on the trial mass makes it a function of m˜. This function
has been shown in Ref. [21] to have a kinematic endpoint
MmaxT2,2−body(m˜) = C2−body +
√
C22−body + m˜2 , (2.26)
28
where the C parameter is given by
C2−body =
m2A −m2X
2mX
. (2.27)
This C parameter can be deduced from eq. (2.26) by substituting the experimental
value of the kinematic endpoint and the chosen trial DM mass.
2.2.2.2 Three-body decay, one visible and two invisibles
As previously mentioned, for three-body decays I assume that the extra par-
ticle Y is also invisible. Therefore, as far as the detectable final state is concerned,
the three-body decay looks like a two-body process. Since I am not a priori aware of
the number of invisible particles involved in the decay process, a natural assumption
is to hypothesize a single invisible particle per decay chain as in a two-body decay.
In this context, I shall refer to this supposition as the “na¨ıve” MT2 method (for
three-body decay) [10].
In each event, this three-body decay can be understood as a two-body decay
process where the two invisible particles X and Y behave like a single invisible
particle with an effective mass equal to the invariant mass of the system formed
by particles X and Y . As is well-known, the invariant mass of the particles X
and Y follows a distribution and ranges from mX + mY to mA. Therefore, the
overall kinematic endpoint in the corresponding MT2 distribution arises when the
invariant mass of the X-Y system is minimized [10]. The theoretical expectation
for MmaxT2,3−body is similar to that of the two-body decay:
MmaxT2,3−body(m˜) = C3−body +
√
C23−body + m˜2 , (2.28)
29
where the C parameter is given by
C3−body =
m2A − (mX +mY )2
2mA
. (2.29)
When comparing to the two-body case, two different features should be noted.
First, given the same mother particle, visible state, and trial DM mass, the kinematic
endpoint of the MT2 distribution for the three-body process is expected to be smaller
than that of the two-body process. This is because for the three-body decay, one
more invisible particle, Y , is involved (see and compare eqs. (2.27) and (2.29), i.e.,
mX +mY ≥ mX). Second, the fall-off of the distribution of the three-body process
at the endpoint is faster than in the two-body process. This is because in the three-
body case more kinematic constraints need to be satisfied to reach the kinematic
endpoint [10,17].
Before closing the Section, a further critical observation is in order. According
to eqs. (2.26) and (2.28), I see that the observed values of MmaxT2 as a function of the
various chosen trial DM masses (m˜) can be fitted with the same equation in both
the two- and three-body cases:
MmaxT2,obs. = C +
√
C + m˜2 , (2.30)
where the parameter C can be extracted from the fit. This will be used in the
following to extract the C parameter without making any assumption on the number
of invisible products in the decay.
The fact that the MT2 endpoint can be described with the same parametriza-
tion in terms of a generic C parameter, as in eq. (2.30), is not surprising. In fact,
for the two-body case in events near the endpoint each mother needs to have its
30
decay products (b and X) emitted at the same rapidity (although the two moth-
ers A can be at different rapidities) [21]. Analogously for the three-body case, the
two invisible decay products (X and Y ) and the particle b produced at the same
interaction vertex all need to share the same rapidity. In such a situation, the two
invisible particles are kinematically equivalent to a single invisible particle, and so
the decay can still be effectively reduced to a two-body decay. In this sense, MmaxT2
for the three-body case corresponds to the same kinematic configuration that gives
the endpoint for the two-body case. However, it must be noted that the C param-
eter actually provides different information in the two cases. For two-body decays,
the C parameter in eq. (2.27) is the same as the rest-frame energy of particle b in
eq. (2.6), whereas for three-body decays, the C parameter in eq. (2.29) is the same
as the maximum energy of particle b in the rest frame in eq. (2.15) 4:
C =



E∗b for two-body decays
E¯maxb for three-body decays.
(2.31)
This observation puts us in the position to extract the C parameter from the
MT2 distribution and compare it with the peak value in the energy distribution of
the visible particle so as to test the nature of the decay.
4Alternatively one can interpret the C parameter of the three-body decay as the analogy of the
two-body case where the mass of the single DM particle is replaced by the mass of the effective
single body made of the two DM, i.e. the sum of the mass of the two DM particles, as apparent
from the comparison of eqs. (2.27) and (2.29).
31
2.3 General Strategy to distinguish Z2 and Z3
I now apply the above theoretical observation to the determination of the
underlying DM stabilization symmetry. To pinpoint this stabilization symmetry, I
study the energy distribution of the particle b from the process defined in eq. (2.25).
In particular, I exploit relation between this energy distribution and the distribution
of the MT2 variable in the same process. As will be clear from the following analysis,
the correlation between features of the distribution of these two observables will
allow us to make a much firmer statement than merely utilizing one of them.
In point of fact, the MT2 distribution of the process eq. (2.25) could itself in
principle be a good discriminator between Z2 and Z3 models. Indeed, as discussed
in Section 2.2.2.2, the kinematic endpoint in the MT2 distribution of the visible
particles from a duplicate three-body decay, which is realized under Z3 symmetry,
develops a longer tail than that of two-body decays, the latter being realized under
Z2 symmetry. Therefore, a less sharp fall-off near the endpoint could be a sign of
more than one invisible particle in the decay [10, 17]. However, shape analyses of
the tail of the MT2 distribution are rather delicate, especially in the presence of
a background. Besides the issues raised by the backgrounds, there are also some
inherent complications in using only the shape of the MT2 distribution to determine
the underlying stabilization symmetry. For example, the effects of spin correlation
could change the shape of the MT2 distribution, particularly the behavior near the
upper endpoint of the distribution. In other words, a certain “choice” of spin corre-
lation could alter the sharp edge of the MT2 distribution in Z2 models, mimicking
32
the typical distribution shape characteristic of Z3 models, and vice versa.
Alternatively, one could try to use the energy distribution of the b particles in
events from the process eq. (2.25). Recall that the distribution of the visible particle
energy in their mother particle’s rest frame is δ function-like in Z2 models, whereas
the distribution in Z3 models is non-trivial. Therefore, once the decay products are
boosted to the laboratory frame from their mother particle’s rest frame, the energy
distribution for Z3 physics is expected to be relatively broader for a given mother
particle. However, it is very hard to quantify the width of the resulting energy
distributions in both Z2 and Z3 models because it is strongly model-dependent. In
particular, the shape of the energy distribution in the laboratory frame is governed
by the boost distributions of the mother particles, which are subject to uncertainties.
Such uncertainties come from the fact that I am not a priori aware of the underlying
dynamics governing the new physics involved in the process eq. (2.25), which affects,
for instance, the production mechanism of the mother particles.
In order to overcome the difficulties described above, I propose here a combined
analysis of the two distributions. The goal is to obtain a more robust technique
that is sensitive to the differences between the Z2 and the Z3 models but largely
independent of the other details of the models. Also, I aim at formulating a method
that is less demanding from an experimental standpoint and more stable against the
inclusion of experimental errors. The analysis proceeds in two steps as explained in
the following.
From the data, one first produces the MT2 distribution using a trial DM mass
and extracts the kinematic endpoint MmaxT2,obs.. Then, by substituting the measured
33
endpoint into the function given in eq. (2.30), one obtains the C parameter. As illus-
trated in eq. (2.31), the C parameter has different physical implications depending
on the stabilization symmetry of the DM. For the Z2case, it is the energy of the
visible particle in the rest frame of its mother particle, and by virtue of [19, 20], it
is expected to be the value of the peak of the energy distribution in the laboratory
frame. Alternatively, for a Z3 model the C parameter is an upper bound to the
peak of the energy distribution in the laboratory frame. Therefore, the comparison
between the extracted C parameter and the peak position in the b particle energy
distribution enables us to determine whether the relevant physics is Z2 or Z3. This
observation can be summarized as follows:
Epeakb,obs. = Cobs. =
m2B′ −m2χ
2mB′
for Z2
Epeakb,obs. < Cobs. =
m2B′ − 4m2χ
2mB′
for Z3. (2.32)
Some remarks must be made about our proposal. First, the use of the dis-
tribution of MT2 is needed only to the extent that this is useful to extract the C
parameter. In fact, in order to find the reference value needed for the comparison
of eq. (2.32), any other observable that is sensitive to the relevant combination of
masses could be used. Second, spin correlation effects do not change the location of
the peak in the energy distribution of the b particle as long as the bottom partners
are produced unpolarized, as discussed earlier. Additionally, although the overall
shape near the endpoint of the MT2 distribution could be affected by non-trivial
spin correlation effects, the endpoint value is not. Furthermore, substantial errors
in the determination of the MT2 endpoint can be tolerated. In fact, as shown in
34
Fig. 2.2, the difference between the reference value and the typical peak of the energy
distribution in a three-body decay is quite large.
For the above reasons, I believe that compared with other methods which
utilize only MT2, the method presented here is more general and more robust in
highlighting the different kinematic behavior inherent to the two different stabiliza-
tion symmetries.
In order to demonstrate the feasibility of the proposed analysis, I work out in
detail an application of our method to the case of pair production of partners of
the b quark that decay into a b quark and one or two invisible particles in the next
section.
2.4 Application to b quark partner decays
In this Section, I study in detail the production of b quark partners, B′, and
their subsequent decay into b quarks and one or two DM particles. As mentioned in
the introduction, b quark partners occur in many well-motivated extensions to the
SM. In the following, I apply the results of Sections 2.2 and 2.3 with the underlying
goal of “counting” the number of DM particles in the above decay process. Although
I employ DM and a b quark partner with specific spin for the purpose of illustrating
our technique, I emphasize that our method can be applied for any appropriate
choice of spins for the involved particles. In fact, the choice of spins does not alter
our results so long as the mother particles are produced unpolarized.
Because the b quark partners are charged under QCD, the dominant pro-
35
duction channel at hadron colliders would be via color gauge interactions, which
guarantee that the b quark partners would be produced unpolarized and in pairs.
Due to the fact that these particles are produced in pairs, the above results given
for MT2 are in force. Furthermore, the unpolarized production guarantees that the
results of Section 2.2 can be applied to the energy distribution.
In what follows, I consider the QCD pair production of heavy b quark partners
at the LHC running at a center-of-mass energy
√s = 14 TeV, and I take as signal
processes:
pp→ B′B¯′ → bb¯χχ for Z2 , (2.33)
pp→ B′B¯′ → bb¯χχχ¯χ¯ for Z3 , (2.34)
where χ is the DM particle. Once produced, I assume that each B′ decays into a
b quark and either one or two stable neutral weakly-interacting particles (see also
Fig. 2.1). These processes will appear in the detector as jets from the two b quarks
and missing transverse energy
pp→ bb¯+ E/T for both Z2 and Z3. (2.35)
Note that our program is meant to be carried out only after the discovery of
heavy b quark partner. In fact, our focus is not on discovery, but on determining
what type of symmetry governs the associated decays of such a particle once the
discovery is made, specifically in the bb¯+E/T channel. In order to achieve this goal,
a high integrated luminosity would be required to make a definitive determination
of the underlying symmetry. Likewise, compared with the criteria necessary to
36
claim the discovery of such a resonance, a different set of event selection conditions
would be likely have to be used in order to make a definitive determination of the
underlying stabilization symmetry.
For our proof-of-concept example, I take mB′ = 800 GeV and mχ = 100 GeV
while noting that searches for scalar b quark partners such as Ref. [22, 23] are in
principle sensitive to our final state. Unfortunately, there is no available interpreta-
tion of this search in terms of a fermionic partner; a naive rescaling of the current
limits on a scalar partner with mass of about 650 GeV shows that our choice of
mass parameters might be on the verge of exclusion. However, I remark that our
choice is only for the purpose of illustrating our technique, and can just as easily be
applied to a heavier B′.
There are several SM backgrounds that are also able to give the same detector
signature as our signal. Since I require a double b-tagging, the main backgrounds
to our signal are the following three processes: i) Z + bb¯, where Z decays into two
neutrinos, ii) W± + bb¯, where the W decay products are not detected, and iii) tt¯
where again the two W ’s from the top decay go undetected 5. The first background
is irreducible, while the latter two are reducible.
To reduce these backgrounds to a level that allows clear extraction of the
features of the b-jet energy and MT2 distribution, I put constraints on the following
observables:
• pT, j1 is the transverse momentum of the hardest jet in the event,
5By undetected I mean that the decay products do not pass our selection criteria or are legiti-
mately undetected.
37
• E/T = |−
∑
i ~pT, i| is the missing transverse energy of the event and is computed
summing over all reconstructed objects,
• ST = 2λ2λ1+λ2 is the transverse sphericity of the event. Due to the tendency of
QCD to produce strongly directional events, the background processes typi-
cally have small sphericity, while decay products of a heavy B′ are expected
to be significantly more isotropic and hence will preferentially have a larger
sphericity [24].
In general, the mismeasurement of the momenta of the observable objects used
to compute E/T can produce an instrumental source of E/T , as opposed to a “physical”
source of E/T which originates from invisible particles carrying away momentum. The
mismeasurement of E/T can grow as objects of larger pT are found in an event, and
it is therefore useful to compare the measured missing transverse energy with some
measure of the global transverse momentum of the event. For this reason, I introduce
the quantity 6
f = E/T/Meff where Meff ≡ E/T + |pT j1 |+ |pT j2| ,
which is expected to be small for events where the E/T comes from mismeasurements,
but should be large for events where invisible particles carry away momentum. Fur-
thermore, when the instrumental E/T originates mostly from the mismeasurement of
a single object, the E/T is expected to point approximately in the direction of one of
the visible momenta. Therefore, the events where the E/T is purely instrumental are
6Sometimes a slightly different quantity f ′ = E/T /
∑
i |pT,i| is used in the same context of our
f . The two variables have the same meaning and give similar results.
38
expected to have a small
∆φ(E/T , jets),
which is the angle between the direction of the missing transverse momentum and
any ~pT j.
To select signal events and reject background events, I choose the following
set of cuts:
0 leptons with |ηl| < 2.5 and pT l > 20 GeV for l = e, µ, τ , (2.36a)
2 b-tagged jets with |ηb| < 2.5 and pT b1 > 100 GeV, pT b2 > 40 GeV, (2.36b)
E/T > 300 GeV , (2.36c)
ST > 0.4 , (2.36d)
f > 0.3 , (2.36e)
∆φmin(E/T , bi) > 0.2 rad for all the selected b-jets bi . (2.36f)
Note that the our cuts are of the same sort used in experimental searches for new
physics in final states with large E/T , 0 leptons and jets including 1 or more b-jets (see,
for instance, [25]). However, notice that in our analysis, I privilege the strength of
the signal over the statistical significance of the observation. As already mentioned,
I imagine this investigation being carried out after the initial discovery of a B′
has taken place. Hence, I favor enhancing the signal to better study the detailed
properties of the interaction(s) of B′. For this reason, I cut more aggressively on
E/T and ST than in experimental searches and other phenomenological literature
focusing on the discovery of B′s (see, for example, [12]).
39
I consider quarks separated by ∆R > 0.7 as jets. With this as our con-
dition on jet reconstruction, the cuts of eq. (2.36) can be readily applied to the
signals and to the Z + bb¯ background; the resulting cross-sections are shown in Ta-
ble 3.1. These cross-sections are computed from samples of events obtained using
the Monte Carlo event generator MadGraph5 v1.4.7 [26] and parton distribution func-
tions CTEQ6L1 [27]. For the sake of completeness, I specify that in generating these
event samples I assumed a fermionic B′ and a weakly interacting scalar χ. However,
as already stressed, I anticipate that different choices of spin for these particles will
not significantly affect our final result because the production via QCD gives rise to
an effectively unpolarized sample of b quark partners.
The estimate of the reducible backgrounds requires more work, as it is partic-
ularly important to accurately model the possible causes that make
pp→ tt¯→ bb¯+X and pp→ W± + bb¯
a background to our 2b+E/T signal. In fact, these processes have larger cross sections
than Z + bb¯. However, they also typically give rise to extra leptons or extra jets
with respect to our selection criteria in eq. (2.36). Therefore, in order for us to
consider them as background events, it is necessary for the extra leptons or jets to
fail our selection criteria. Accordingly, the relevant cross-section for these processes
is significantly reduced compared to the total. In fact, I find that tt¯ and W±bb¯ are
subdominant background sources compared to Z + bb¯. In what follows, I describe
how I estimated the background rate from tt¯ and W±bb¯.
An accurate determination of the proportion of tt¯ andW±bb¯ background events
40
Cut Z2 (B → bχ) Z3 (B → bχχ¯) Z + bb¯ (Z → νν¯)
No cuts 159.75 159.75 –
Precuts 139.89 136.73 2927
pj1T > 100 GeV, pj2T > 40 GeV 139.64 133.76 971.9
E/T > 300 GeV 101.73 69.01 19.93
f > 0.3 89.66 65.21 19.40
∆φmin > 0.2 88.95 64.31 18.81
ST > 0.4 30.03 16.07 1.96
2 b-tagged jets 13.29 7.18 0.87
Table 2.1: Cross-sections in fb of the signals and the dominant background Z + bb¯
after the cuts of eqs. (2.36). The mass spectrum for the signals is mB′ = 800 GeV
and mχ = 100 GeV. The line “No cuts” is for the inclusive cross-section of the
signal. The line “precuts” gives the cross-section after the cuts E/T > 60 GeV, pT,b >
30 GeV, ηb < 2.5,∆Rbb > 0.7 that are imposed solely to avoid a divergence in the
leading order computation of the background. In the last line, the rate of tagging b
quarks is assumed 66% [28].
that pass the cuts in eq. (2.36) depends on the finer details of the detector used to
observe these events. However, the most important causes for the extra jets and
leptons in the reducible backgrounds to fail our jet and lepton identification criteria
can be understood at the matrix element level. I estimate the rate of the reducible
backgrounds by requiring that at the matrix element level, a suitable number of
final states from the tt¯ and W + bb¯ production fail the selections of eq. (2.36) for
41
one of the following reasons:
• the lepton or quark is too soft, i.e., pT,l < 20 GeV, pT,j < 30 GeV
• or the lepton or quark is not central, i.e. |ηl,j| > 2.5 .
Additionally, when any quark or lepton is too close to a b quark, I consider
them as having been merged by the detector, and the resulting object is counted
as a b quark (i.e., ∆Rbl < 0.7, ∆Rbj < 0.7), or if any light quark or lepton is too
close to a light jet, they are likewise merged, and the resulting object is counted
as a light quark (i.e., ∆Rjl < 0.7, ∆Rjj < 0.7). In the latter case, the light ”jet”
resulting from a merger must then also satisfy the pT and η criteria given above for
going undetected.
Using our method to estimate the results on the backgrounds in Ref. [12],
the analysis of which was carried out with objects reconstructed at the detector
level, I find that our estimates agree with Ref. [12] within a factor of two. Because
I successfully captured the leading effect, I did not feel the necessity of pursuing
detector simulations in our analysis.
Estimating the reducible background after the selections in eq. (2.36), I find
that tt¯ and W + bb¯ are subdominant compared to Z+ bb¯. The suppression of the re-
ducible backgrounds, and in particular, of tt¯, comes especially from the combination
of the ST and E/T cuts. This is shown in Fig. 2.3, where I plot the E/T distributions
of the three backgrounds under different ST cuts: ST > 0, ST > 0.2, and the cut
ST > 0.4, which is used in our final analysis. Clearly, one can see that for a E/T as
large as our requirement in eq. (2.36), the dominant background is Z + bb¯, and that
42
Z + bb
W + bb
tt
0 100 200 300 400 500 600 700
0.01
0.1
1
10
100
MET H GeVL
d
Σ
d
H
M
E
T
L
H
f
b

1
0
G
e
V
L
i
n
L
o
g
s
c
a
l
e
S
T
> 0.0
Z + bb
W + bb
tt
0 100 200 300 400 500 600 700
0.01
0.1
1
10
100
MET H GeVL
d
Σ
d
H
M
E
T
L
H
f
b

1
0
G
e
V
L
i
n
L
o
g
s
c
a
l
e
S
T
> 0.2
Z + bb
W + bb
tt
0 100 200 300 400 500 600 700
0.01
0.1
1
10
100
MET H GeVL
d
Σ
d
H
M
E
T
L
H
f
b

1
0
G
e
V
L
i
n
L
o
g
s
c
a
l
e
S
T
> 0.4
Figure 2.3: E/T distributions for the three backgrounds (Z + bb¯, W± + bb¯, and tt¯)
with ST cuts of increasing magnitude, ST > 0.0, > 0.2, and > 0.4 from the left panel
to the right panel. In each plot, the black solid, blue dotdashed, and red dashed
curves represent Z + bb¯, W± + bb¯, and tt¯, respectively.
in particular, the tt¯ is significantly suppressed by simultaneously requiring a large
E/T and moderate ST cut (rightmost panel in the figure).
As the first step in our analysis, I compute the MT2 distributions expected
at the LHC for our two potential cases of new physics interactions, Z2 and Z3 .
The distributions for the two cases are shown in Fig. 2.4. Since I found that with
selections of eq. (2.36), the Z + bb¯ process is the dominant background, as seen
in the figure, I consider it the only background process. The two distributions
43
have been computed assuming a trial mass m˜ = 0 GeV and have an endpoint at
787.5 GeV and 750 GeV for the Z2 and the Z3 cases, respectively. Interpreting the
distributions under the na¨ıve assumption of one invisible particle per decay of the
B′, I obtain from eq. (2.30) a C parameter that is 383.75 GeV and 375 GeV for Z2
and Z3 , respectively. These are the reference values that I need for the analysis of
the energy distributions 7.
As the final step in our analysis, I need to compare the obtained reference
values with the peaks of the energy distributions. These distributions are shown
in Fig. 2.5. I clearly see that the location of the peak in the energy distribution
the Z2 case coincides with the associated reference value, whereas for the Z3 case
the peak is, as expected, at an energy less than the associated reference value. I
remark that in the Z3 case, the peak of the energy distribution is significantly
displaced with respect to the reference value. Therefore, I expect our test of the Z2
nature of the interactions of the B′ to be quite robust under the inclusion of both
experimental and theoretical uncertainties, such as the smearing of the peak due to
the resolution on the jet energy, the errors on the extraction of the reference value
obtained from the MT2 analysis, and the shift of the peak that is expected due to
radiative corrections to the leading order of the decay of the B′.
7I remark that as apparent from the figure, the signal rate is much larger than that of the
background, and therefore the shape of the distribution expected at the LHC largely reflects the
features of the signal. In this case, it seems particularly straightforward to extract the endpoint
of the distribution. In other cases where the background is larger, the extraction of the endpoint
may require a more elaborate procedure, especially for the Z3 case where the endpoint is much less
sharp (see, for example, [10, 29–31]).
44
Signal+ BG
Signal
BG
0 200 400 600 800
0.0
0.1
0.2
0.3
0.4
M
T2
H GeV L
d
Σ
d
M
T
2
H
f
b

1
0
G
e
V
L
Signal H Z
2
L + BG
Signal+ BG
Signal
BG
0 200 400 600 800
0.0
0.1
0.2
0.3
0.4
0.5
M
T2
H GeV L
d
Σ
d
M
T
2
H
f
b

1
0
G
e
V
L
Signal H Z
3
L + BG
Figure 2.4: MT2 distributions after the cuts of eq. (2.36). The chosen masses for
the new particles are mB′ = 800 GeV and mχ = 100 GeV. The left panel is for the
Z2 signal while the right panel is Z3 (both in blue). In both cases, the background
is Z + bb¯ (red). In both panels, the black line represents the sum of signal and
background. The black vertical dashed lines denote the theoretical prediction for
the endpoints.
45
Signal+ BG
Signal
BG
0 200 400 600 800 1000
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
E
b
H GeV L
d
Σ
d
E
b
H
f
b

1
0
G
e
V
L
Signal H Z
2
L + BG
Signal+ BG
Signal
BG
0 200 400 600 800 1000
0.00
0.05
0.10
0.15
0.20
0.25
E
b
H GeV L
d
Σ
d
E
b
H
f
b

1
0
G
e
V
L
Signal H Z
3
L + BG
Figure 2.5: Energy distributions of the b quarks after the cuts of eqs. (2.36). The
chosen masses for the new particles are mB′ = 800 GeV and mχ = 100 GeV. The
left panel is for the Z2 signal, while the right panel is Z3 (both in blue). In both
cases, the background is Z + bb¯ (red). In both panels, the black line represents the
sum of signal and background. The black vertical dashed lines denote the reference
values extracted from the MT2 distributions of Fig. 2.4 using eq. (2.30).
46
2.5 Conclusions
In this chapter, I studied the problem of the experimental determination of the
general structure of the interactions of an extension to the SM that hosts collider-
stable WIMPs. If these new particles are charged under a new symmetry and the
SM particles are not, then the lightest such WIMP is stable and is concomitantly a
candidate for the DM of the universe. In the context of such DM models, the above
is thus relevant for the determination of the stabilization symmetry of this DM.
In more detail, such models typically have heavier new particles that are charged
under both the SM gauge group and the DM stabilization symmetry. Thus, these
particles can be produced via the collision of SM particles, and will decay into DM
plus SM particles. The number of DM particles in such a decay depends on the DM
stabilization symmetry. Our goal was to devise a strategy to count this number of
DM and thus probe the nature of this symmetry, based only on the visible part of
the decays.
To illustrate the technique, I studied models with fermionic b quark partners,
i.e. colored fermions with electric charge −1/3 with sizable coupling to the b quark.
In our example, I considered the case of b quark partners with mass at or below the
TeV scale. The possibility of such is motivated by extensions to the SM that solve
the Planck-weak hierarchy problem, since they contain top partners and, thus by
SU(2)L symmetry, bottom partners. In the same model, it is also possible to have
a WIMP DM. The b quark partners, as the typical states of the new physics sector,
are charged under this stabilization symmetry and will then decay into a bottom
47
quark, plus DM. Furthermore, thanks to their color gauge interactions, the b quark
partners have a large production cross-section at hadronic colliders. Therefore the
study of b quark partners is very well-suited to illustrate our technique.
The literature on b quark partners thus far has only considered single DM in
each decay chain, as would be the case in models where the DM is stabilized by a
Z2 symmetry. However, in general, there can be more than one DM in this decay
chain; for example, two DM are allowed in the case of a Z3 stabilization symmetry,
albeit not in the case of a Z2 symmetry. So, the question I posed is whether I can
distinguish the hypothesis of one vs. (say) two DM particles appearing in each of
these decay chains. As mentioned above, in this way I can probe the nature of
the DM stabilization symmetry. The question is non-trivial, because in either case
the detectable particles produced are the same, and so is the signal of the b quark
partners’ production, i.e. bb¯+ E/T .
To distinguish between one and two DM in each b quark partner decay chain,
the first result I used is that the measured MT2 endpoints can be fitted by the
formula eq. (2.30) irrespectively of how many DM particles are produced.
The value of the free parameter obtained by fitting eq. (2.30) to the data is
used in the next step of our analysis as follows. The second theoretical observation
is that the peak of the distribution of the b quark energy in the laboratory frame is
the same as the mother rest frame value for the two-body decay, but is smaller than
the maximum value in the mother rest frame for the three-body decay. The crux
is that the rest frame energy that is used as a reference value in this comparison is
precisely the parameter obtained in the above MT2 analysis. Combining the above
48
two facts, I showed that the peak of observed bottom-jet energy being smaller than
(vs. same as) the reference value obtained from the MT2 endpoint provides evidence
for two (vs. one) DM particles in the decay of a b quark partner, and thus a Z3
symmetry can be distinguished from Z2.
I verified our theoretical observations in B′ pair production and decay at the
LHC. To assess the feasibility of the determination of the stabilization symmetry
with our method, I simulated the signal and the dominant SM backgrounds. Using
suitable cuts, I showed that the background in this case is due mostly to Z + bb¯.
I studied in detail the case where the b quark partner has a mass mB′ = 800 GeV
and the invisible particles have a mass mχ = 100 GeV. In this case, the background
can be made small compared to the signal using the cuts of eq. (2.36). In Figures
2.4 and 2.5, I show the resulting MT2 and b quark energy distributions relevant
to our analysis. I observed that the peak in the b quark energy distribution for
Z2 models is consistent with the reference value from the MT2 endpoint, while
that of Z3 models is apparently less than the corresponding reference value. The
determinations of the peak of the energy distribution and of the reference value
needed for our analysis are subject to uncertainties, e.g. those that propagate from
the error in the determination of the MT2 endpoint. However, the evidence for a
Z3 stabilization symmetry comes from a difference between the peak of the energy
distribution and the reference value. The theoretical prediction for this difference is
large enough compared to the relevant uncertainties so that the proposed method
seems to be quite robust, and should allow a clear discrimination of the stabilization
symmetry of the DM.
49
In next chapter I shall extend the theory of Section 2.2 to deal with massive
visible decay products and will tackle the issue of multi-body decay channels where
there are potentially many identical particles in the final state. Having handled
that, I will develop a method of simultaneously determining the masses of the DM
and parent particles.
50
Chapter 3: Mass extraction
In previous works I have demonstrated how the energy distribution of massless
decay products in two body decays can be used to measure the mass of decaying
particles. In this work I show how such results can be generalized to the case of
multi-body decays. The key ideas that allow us to deal with multi-body final states
are an extension of our previous results to the case of massive decay products and
the factorization of the multi-body phase space. The mass measurement strategy
that I propose is distinct from alternative methods because it does not require an
accurate reconstruction of the entire event, as it does not involve, for instance,
the missing transverse momentum, but rather requires measuring only the visible
decay products of the decay of interest. To demonstrate the general strategy, I
study a supersymmetric model wherein pair-produced gluinos each decay to a stable
neutralino and a bottom quark-antiquark pair via an off-shell bottom squark. The
combinatorial background stemming from the indistinguishable visible final states
on both decay sides can be treated by an “event mixing” technique, the performance
of which is discussed in detail. Taking into account dominant backgrounds, I am able
to show that the mass of the gluino and, in favorable cases, that of the neutralino
can be determined by this mass measurement strategy.
51
3.1 Introduction and general strategy of the mass measurement
As previously mentioned, a stable WIMP is a well-motivated candidate for
dark matter. Specifically, many models incorporating this WIMP-type DM contain
particles that are not only heavier than the DM and charged under the DM stabiliza-
tion symmetry, but also that interact via SM gauge bosons. If these heavier particles
(dubbed “parent” particles) do interact via say, QCD, they could be copiously pro-
duced at hadron colliders, which would then be followed by their subsequent decay
into the concomitant DM and SM particles. By design, the DM particle leaves no
visible trace in the particle detector, thus its presence in an event is typically in-
ferred from the missing transverse momentum (
pT ), which can be interpreted as a
loss of specificity in the kinematic information of the event.
The primary goal of this chapter is to devise a strategy for the simultaneous
measurement of the masses of the parent and the DM particles in the associated
processes despite this loss of information. This strategy for the mass measurement
also has further applications beyond the study of DM particles; it can be applied
to any case where a new particle decays to a semi-invisible final state. Again, the
invisible particle neither has to be a DM candidate, nor has to be absolutely stable
– only insofar as its time-of-flight out of the detector is concerned. However, for
notational simplicity I shall still refer to it as “DM”.
A full reconstruction of such a decay chain is typically not possible, given
that it contains an invisible particle. On top of this, due to the DM stabilization
symmetry the parent particles are typically pair-produced, implying that each event
52
comes with two invisible particles. The presence of two invisible particles involves
an even greater loss of kinematic information from each event, and poses a sizable
challenge in the associated mass measurement. Methods using the MT2 variable and
its variations [21,30,32–35] have been proposed as a solution to overcome this chal-
lenge. This class of variables is well-known for its usefulness both in measuring the
masses of particles [36] and in isolating new physics signals from their backgrounds
(e.g. Ref. [37]). Despite their utility, these variables have a possible drawback when
aiming at a precise mass measurement: they all require information about the total
missing momentum. Unfortunately, a precise measurement of the missing momen-
tum is often difficult, for instance due to the relatively poor reconstruction of the
jets that are usually a part of the overall event structure. This is an unpleasant fea-
ture of missing momentum measurements, especially in those cases in which many
of the jets that are involved in the measurement of the missing momentum are actu-
ally not involved in the decay process of interest. Said another way, in general, the
missing transverse momentum is measured as the opposite of the sum of the momen-
tum of all the reconstructed objects (leptons, jets, photons,...) in the event, which
means that the measurement of the missing momentum is an inherently “global”
measurement of said event.
In light of this, I have recently proposed complementary methods for mass
measurements which instead use only the energy of the visible particles. The reason
to pursue this strategy is, of course, that it relies intrinsically on more “local”
information, ideally using only a subset of the particles coming from a given decay
53
chain.1 The main idea behind the method that I propose is to use the energy spectra
of the visible particles. The basic result upon which the method is predicated was
shown in Ref. [19]: namely, for a massless child from the two-body decay of an
unpolarized parent, the peak in the energy spectrum of the child (henceforth denoted
as “energy-peak”) seen in the laboratory frame (henceforth denoted by “laboratory-
frame” energy) is the same as the fixed value of its energy in the rest frame of the
associated parent (henceforth denoted by “rest-frame” energy). The latter value is
given by a simple relation in terms of the masses of the two massive particles (the
parent and the other child) involved in the decay, and hence can give information
about these masses. In a subsequent paper [45], my colleagues then applied this
observation to measuring the unknown masses in the semi-invisible decay of a heavy
new particle involving a multi-step cascade of two-body decays.2
In this chapter, I consider instead a single-step, three-body, semi-invisible decay
of a heavier new particle, which I denote as
B → Aab (3.1)
where a, b are visible SM particles and A, B are massive new particles with A
assumed invisible. In order to deal with this specific decay topology, I need to
extend the result of Ref. [19] to multi-body decays. The key idea is to map a
multi-body final state into a two-body one, by the factorization of phase-space. In
1See also Refs. [18, 38–42] for other recent methods of mass measurement that do not use the
missing transverse momentum and Ref. [43,44] for a general review of mass measurement methods.
2I also showed that our energy-peak result of Ref. [19] can be used for “counting”’ DM particles
in decays [4], which is a powerful probe of the DM stabilization symmetry.
54
carrying out this mapping, I will take particular care in correctly partitioning and
grouping the multi-body decay products and selecting an appropriate region of the
multi-body phase space upon which to apply the above two-body result.
B
A
a
b
mab
B
A
(ab) mab
Figure 3.1: The three-body decay of interest (left panel) and the effective two-body
decay (right panel) with the mass of the visible system being mab. (ab) denotes the
effective visible pseudo-particle formed by the two visible particles a and b.
To reduce the multi-body final state to a two body one, I first form a compound
system made of all of the visible particles, labeled as a and b in eq. (3.1). I denote this
compound system as (ab) and graphically represent the corresponding partitioning
in Figure 3.1. After this partitioning the decay does not yet look like a truly two-
body decay because the compound system (ab) does not have a fixed mass. The
combination of (ab) will have its own phase space in invariant mass. This is apparent
from the well-known [46] recursive formula for the multi-particle phase-space of N
particles of masses m1, ...,mN , which can be thought of as the sum of many two-
body phase-spaces, a single particle on its own and the remaining (N − 1) particles
clustered into a single object whose mass now depends on the momenta and the
angles between the N − 1 clustered particles:
55
φN(m1, ...,mN) =
∫
dµ dφ2 (mN , µ(m1, ...,mN−1) · dφN−1(m1, ...,mN−1)) . (3.2)
Considering each value of the masses that the compound system (ab) can take
separately, I can regard the N body final state as a weighted sum of a collection
of two-body systems, each of which is characterized by the mass of the compound
system denoted by µ and its probability dφN−1. This probability, together with the
actual squared matrix element of the decay, would give the rate of decay in that
particular kinematic configuration. In the following, I do not assume any knowledge
of the matrix element of the decay and I shall make no use of these rates; all I will
need for our strategy to work is the ability to represent the multi-body final state as
the sum of the collection of all possible two-body final states. The fact that I do not
need to know the rate for each possible kinematic configuration of the multi-particle
final states is a remarkable point of strength of our method; it is especially powerful
when applied to newly discovered particles, as their matrix elements are a priori
essentially unknown.
For the case of a three-body decay, the above outlined procedure gives
B → Aab =
∑
mab
(B → A (ab)mab) ,
where the equality should be taken in the sense of an equivalence. I also remark
that for practical reasons the integral for the phase-space factorization formula has
been discretized. In this way, I can form a finite number of compound systems
(ab)mab of mass mab ± δ with δ  mab. This procedure ensures that each of the
compound systems has an approximately fixed invariant mass, and I can think of
56
it as a “pseudo-particle” having a width of order δ. This means that I partition, or
“slice”, the data according to the total invariant mass of a compound particle formed
from the a and b particles and apply the result for two-body decays to each mass
partition as the overall system has been reduced to an effective two-body decay. In
the rest frame of the parent particle, each partition of the pseudo-particle (ab) has
energy:
E∗(ab) =
m2B −m2A +m2ab
2mB
, (3.3)
based simply on two-body kinematics for decay of B into A and (ab). Using the
appropriate extension of the result in Ref. [19] to the case mab 6= 0 (see more on this
point in Section 3.2), I am able to extract E∗(ab) from the laboratory-frame energy
distribution of that particular (ab) compound particle. I then repeat this procedure
for each of the mass partitions in the overall range of mab. When plotted versus m2ab,
the fitted data for E∗(ab) extracted from the energy distributions should lie along a
straight line as per eq. (3.3). It is straightforward to see that mB can be determined
from the slope of this line and that mA can be determined from the intercept on the
vertical axis once mB has been determined. The available information can be fully
utilized in constructing this straight line by analyzing the data in all slices.3
I remark that, although the characteristic signature of the production of an
invisible particle A is missing momentum, our method does not make explicit use this
quantity and yet still offers a way to obtain a measurement of mass of this particle!
3In practice, one could end up not using some of the slices if treating them becomes too prob-
lematic, e.g. because of backgrounds or sensitivity to the cuts. The fraction of unused slices will
in any case be kept to a minimum.
57
In other words, any specific property of the invisible particle is almost completely
irrelevant to our method, except for the assumption that there is at least one invisible
particle per decay chain. I again emphasize that this achievement is remarkable,
especially in comparison with other mass measurement methods involving 
pT such
as MT2 and its variants.
Despite the simplicity of the general idea, there are still some potential is-
sues that would need to be properly dealt with in order to successfully execute the
strategy outlined above. First, the compound particle (ab), the visible child particle
from the effective two-body decay, has a non-negligible mass; thus, it is essential to
generalize the result in Ref. [19] on the energy-peak to the case for a massive visible
child particle. I refer to work done by my collaborators to this end [47], which is
devoted to studying how to deal with these massive child particles in more detail.
In this dissertation, I shall merely report the final result of their work and use this
result for our present investigation. Nevertheless, the discussion presented here is
largely independent of the derivation of this result.
As mentioned earlier, the DM model under consideration has the parent par-
ticles being produced in pairs. If both of them decay to the same final state, a
combinatorial ambiguity arises in attempting to correctly partition and group those
particles originating from the same parent; multiple pairs can be formed from the
final state as seen in the detector, but it is not known a priori which is the correct
pairing - that is, that the particles in the pairing originated from the same decay.
This partitioning is a crucial necessity in forming the (ab) compound system that
plays the role of the child pseudo-particle. For this reason, I allot Section 3.3 to the
58
thorough discussion of the treatment of this combinatorial ambiguity. In particular
I propose an “event mixing” technique [48–51] as a way to remove the combinatorial
background.
To illustrate our method I discuss in detail its application to a specific pro-
cess. As a concrete example, I choose the pair production of gluinos in an R-parity
conserving supersymmetric model. Here the gluinos are assumed to subsequently
decay into bb¯ and an invisible light stable neutralino via an off-shell bottom squark:
pp→ g˜g˜ → bb¯bb¯χχ. (3.4)
This scenario is chosen primarly because it has been thoroughly studied in the
literature, and thus should be familiar and interesting to a large audience. Indeed,
this process has also been investigated at the LHC [37,52–56]. In order to provide a
fully realistic example, and to demonstrate some of the issues that arise in using our
method, I shall incorporate the relevant Standard Model backgrounds in our analysis
as well. I take particular care in devising cuts for background rejection so that these
selections do not affect the shape of the energy spectrum near the peak, which is the
critical region of interest for our energy peak method. Obviously, the optimization of
these cuts is a process-dependent issue, and so must be evaluated on a case-by-case
basis. The goal of our discussion is to present potential systematic uncertainties
and biases arising from the specific details of our method, such as those induced by
phase space slicing, event mixing, imperfect knowledge of the background, and overly
restrictive event selection criteria. I also present other complementary observables
that enhance the findings obtained using the energy peak method, one of which is
59
the kinematic endpoint of the di-jet invariant mass distribution.
The rest of the chapter is organized as follows. I continue in the next section
with a discussion of a template function used to describe the energy spectrum of
a massive child particle. Section 3.3 is devoted to dealing with the combinatorial
ambiguity inherent in our chosen decay topology. Then in Section 3.4, I detail our
selected signal process and the relevant backgrounds, both those from SM processes
and the “event mixing” scheme for signal combinatorics. Section 3.5 contains the
main results for the mass measurement of the aforementioned example process to-
gether with a discussion of several opportunities for improvement to the method. In
Section 3.6 I present our conclusions and outlook.
3.2 A template for the energy spectrum of a massive child particle
As outlined in the previous section, the essence of our mass measurement
technique is to fit the data to get the value of E∗ for each of the fixed masses of the
compound system (ab) and fit them onto the straight line in eq. (3.3). The mass
of the compound system (ab), being a system of two particles, is not fixed and in
general spans a range fixed by the masses of all the particles in the decay. Since I am
a priori unaware of the masses of the parent and invisible child particles, it is not
possible to know whether or not a given value of mab is small enough in comparison
to those unknown masses to justifiably trust the validity of our previous results
for effectively massless child particles [19], and thus I am motivated to extend the
finding to massive child particles.
60
The primary difficulty in generalizing to a massive child is the potential loss
of correlation between the peak location in the laboratory-frame energy distribution
of the massive child and its energy in the rest frame of the parent. This can readily
be seen when considering the decay of a massive particle B → X ψ at the kinematic
end point of the phase-space, i.e., mB = mX + mψ. For any value of mX , ψ will
be at rest in the B rest frame, hence E∗ψ = mψ. If each particular event is boosted
to the laboratory frame, the energy of ψ becomes simply γBmψ, with γB being
the boost of particle B relative to the laboratory. This direct linear relationship
between Eψ and γB implies that the shape of the energy distribution of particle ψ in
the laboratory frame should simply be that of the boost distribution of particle B.
In this case, it is clear that the peak of the energy distribution of the massive child
ψ carries essentially no information about the masses; rather, it carries information
on the most probable boost of particle B. This is contrast to the “invariance” that
holds for a massless child: the energy-peak in the laboratory frame is the same as
the rest-frame energy value irrespective of the details in the boost distribution of
particle B.
For a more formal understanding of this problem, it is instructive to analyze
the Lorentz transformation of a massive child particle from the rest frame of its
parent particle, where it has energy-momentum (E∗, p∗), to the laboratory frame.
Given the boost factor γ of the parent particle and the emission angle of the child θ∗
relative to the boost direction, I find the energy of the child particle in the laboratory
61
frame (denoted by E) to be
E = E∗γ (1 + β∗β cos θ∗) , (3.5)
where I have used p∗ = β∗E∗. I observe that the laboratory-frame energy E becomes
equal to the rest-frame energy E∗ only if
cos θ∗ = − 1β∗β
(γ − 1
γ
)
. (3.6)
Denoting this θ∗ as the “reference” angle, I see that any value of cos θ∗ smaller
(larger) than eq. (3.6) gives rise to a laboratory-frame energy value E smaller (larger)
than E∗.
To ensure the existence of the reference angle, the boost factor should satisfy
the following relation:
γ < 1 + (β
∗)2
1− (β∗)2 = 2(γ
∗)2 − 1 ≡ γcr . (3.7)
When this condition is satisfied, the energy distribution in the laboratory frame
is non-zero at E = E∗, which is a necessary, but not sufficient, condition to have
a maximum at E∗. Obviously, if for some γ the condition set by eq.(3.7) is not
satisfied, then E > E∗ for all cos θ∗, which potentially invalidates the statement
that the peak in the laboratory-frame energy distribution appears at E∗. Actually,
for the typical boost distributions of parent particles produced at hadron colliders,
one can see that if any of the boosts of the parent particle(s) lie outside of the
range given by (3.7), it is then guaranteed that the peak of the energy distribution
in the laboratory frame will not be located at the rest-frame energy value 4. A
4The displacement of the maximum with respect to E∗ may still be small, but strictly speaking,
the “invariance” that I demonstrated in Ref. [19] for the massless particle will be broken.
62
more rigorous method of characterizing the energy distribution of a massive child is
presented in [47]. I hope that the argument above, while not as rigorous as the above
reference, will convince the reader that the maximum boost of the parent particle
is the key parameter that controls the position of the peak in the laboratory-frame
energy distribution of a massive child particle.
On top of affecting the peak position, the overall shape of the energy distribu-
tion for the massive child is expected to differ from that for the massless child. This
means that the function used to fit the massless child energy spectra in previous
works cannot be used in the present work. In order to obtain a suitable description
of the massive child energy spectrum, I revisit the corresponding discussion for the
case with a massless child particle. The value of the energy distribution at a given
laboratory-frame energy E is given by a Lebesque-type integral within the range of
γ values contributing to the E together with the associated weight for the γ [19].
For the case at hand, this range is found by solving eq. (3.5) for γ with cos θ∗ = ±1,
which gives:
γ+(E) ≡ γ∗2
(√
1−
1
γ∗2
√
E2
E∗2 −
1
γ∗2 +
E
E∗
)
, (3.8)
γ−(E) ≡ γ∗2
E
E∗
(
1−
√
1−
1
γ∗2
√
1−
E∗2
γ∗2E2
)
. (3.9)
I see that for a massless child particle, i.e., γ∗ →∞, γ+(E) diverges, whereas in the
same limit γ−(E) converges to a finite value:
lim
γ∗→∞
γ−(E) =
E
E∗ +
E∗
E ≡ γ
(∞)
− (E) . (3.10)
In light of eq. (3.10), I can express the energy spectrum for a massless child that I
63
used in previous works [19] as
exp
(
−w · γ(∞)− (E)
)
. (3.11)
Motivated by the success of this exponential form in the massless case [19,45]
and exploiting the identification in the massless limit eq. (3.11), I propose the
following ansatz on the shape of the laboratory frame energy spectrum of a massive
child
f(E) = N (exp[−w · γ−(E)]− exp[−w · γ+(E)]) , (3.12)
whereN and w are a normalization factor and the width of the function, respectively.
A complete evaluation of the accuracy with which this function describes the energy
spectrum in the laboratory frame of a massive child is presented by colleagues in
the companion to the work presented here [47]. For the purposes of this chapter, it
will be sufficient to know that this function reproduces our ansatz for massless child
particles for γ∗ →∞. In any case, I will explicitly show that this function provides
a good description of simulation data for our example process below.
A comment on the location of the maximum of this function is in order. The
maximum of this function coincides with E∗ only in the limit w → ∞, in which
the function becomes a δ function. For all finite values of w, the actual location
of the maximum of the function is slightly larger than E∗. However, I empirically
observe that for parent particles that would typically be produced at colliders, and
for γ∗ somewhat larger than 1, the typical value of w is large enough that this effect
is negligible. Therefore, I expect that eq. (3.12) properly describes a large class
of energy spectra. Because the peak location and the E∗ parameter are no longer
64
necessarily the same 5, the determinations of the best fit values of E∗ and w are
interrelated when fitting the data to the massive template function eq.(3.12). The
fact that the maximum of the spectrum is a function of both E∗ and w is an inherent
feature of our ansatz for the massive child energy spectrum, which did not exist in
the massless case of Ref. [19]. In this chapter I will study the possible effects that
arise in our explicit example due to this feature of eq.(3.12). For a fully general
investigation of this issue I refer to the companion to this chapter [47]. In Sec. 3.4
I shall fit the energy distribution of each mass partition of the pseudo-particle (ab)
both with the new template for massive children eq. (3.12), and with its massless
limit (i.e γ∗ → ∞), the latter of which was the template employed previously for
massless child particles. The comparison of the results from these two templates
will allow us to demonstrate the necessity of using eq. (3.12) and generalizing what
I had used in our previous work [19,45].
3.3 “Event mixing” to estimate the combinatorial background
As explained in the Introduction and depicted in Figure 3.1, the success of
our strategy hinges on correctly identifying the pairs of particles coming from the
same decay side. If this identification is done correctly, the idea of phase-space
factorization can safely be applied to reduce the multi-body final state to a two-body
final state. Generally speaking, the identification of the correct pairs of particles to
be grouped together is a tremendously difficult task, as I have no systematic way
5Again, for the case with massless children, E∗ conforms to the peak irrespective of w.
65
of knowing which particles are related to the same decay, and can thus be correctly
paired in the analysis.
One approach to surmounting this challenge is to attempt to correctly identify
the pairs of particles that come from the same decay by exploiting some preferential
kinematic correlation between particles that originate from the same decay. This
correlation could then be translated into a selection criterion that would correctly
pair the appropriate particles with relatively high accuracy. Of course, this selection
process would not be 100% effective and would fail to pick the correct pairings for
some fraction of events. As a result, I would be left with a certain amount of com-
binatoric pollution from pairings whose constituent particles did not come from the
same decay. Several event-by-event strategies have been developed to identify which
pairs of particles come from the same decay (see, for example, Refs. [29,57–60]). It
is however generally true that, in order to maximize the chance of pairing particles
correctly, the kinematic selection criteria which form the basis of each method must
be rather restrictive, so as to guarantee a sufficient rejection of unwanted pairings.
Events that pass these highly constrained kinematic criteria will be preferentially
selected from isolated regions of the kinematic phase space of the scattering, and
therefore, the kinematic distributions of the final state will be significantly altered
by the imposition of these criteria. Our method of mass extraction relies critically
on the fidelity of the energy distribution around the location of the peak; without
this, the templates I use to extract the masses return biased information. Thus,
because the above set of procedures for selecting correct decay siblings greatly dis-
turbs the resultant kinematic distributions, they are not suitable for use alongside
66
our method. For this reason, I do not even attempt to identify the correct pairings
in each event; I instead try to obtain the energy distribution of the correct pairs
without knowing which are the correct pairs event-by-event.
In order to determine the distribution of a given observable as it would arise
from picking just the correct pairs, I use an event mixing technique whose basic idea
is as follows. I consider a scattering
pp→ B1B2 → (A1a1b1)(A2a2b2), (3.13)
where two heavy particles B1 and B2 are produced and decay to final states of the
same kind, B → Aab, which are labeled to correspond with their respective parent.6
In the analysis of events of this nature, I follow the procedure laid out in Section 3.1
for making pseudo-particles out of the a and b particles for all possible equivalent
pairings (e.g. a1 with b1, but also a1 with b2 and so on.) From these pairs, I obtain
a fully inclusive distribution of the observable of interest, which in our case here is
the total energy of the pair,
dσ
dEab
(all pairs). (3.14)
In order to obtain the distribution stemming only from the pairs of particles
coming from the same decay, it is sufficient to come up with an estimate of the
distribution that stems from the pairs that I would like to discard and subtract it
from the fully inclusive distribution:
dσ
dEab
(same-decay pairs) =
dσ
dEab
(all pairs)−
dσ
dEab
(different-decay pairs). (3.15)
6Strictly speaking, Ai’s need not be invisible as long as they are distinguishable from ai and bi,
which are assumed indistinguishable from each other.
67
This equality might seem trivial, but is in fact very powerful. To wit, the object
that I must have for our method to be successful - the distribution of pairings from
the same decay - which is quite difficult to obtain in itself, is now expressed in
terms of two objects that are much simpler to obtain. The first piece is obviously
attainable, as it is simply the distribution from all the pairs that can be formed in
each event. The second piece, the distribution from pairs not coming from the same
decay, can be estimated by the “event mixing” technique, which is based on making
a distribution from pairs of jets that come from different events. The intuition
that justifies the usefulness of the event mixing technique originates from the fact
that pairs (a1b2) and (a2b1) from the same event are made of particles which are
produced with almost no kinematic correlation. Therefore, it seems reasonable to
mimic the effect of these “incoherent” pairs with the pairs of particles taken from
different events, which intuitively have no correlation. More precisely, I can see that
the phase-space point from which a1 and b1 originate in the decay of B1 is very close
to being uncorrelated with the phase-space point from which a2 and b2 originate in
the decay of B2. This approximately vanishing correlation between the products of
different decays implies that pairs formed by particle a1 taken from one event, and
particle b2 taken from another event are expected to be statistically equivalent to
pairs (a1b2), where both particles are taken from the same event. This means that
the distributions of a quantity over a given sample of events obtained from either
the (a1b2)-type incorrect pairings within the same event or from pairing a1 in one
event and b2 in another event are equivalent. This implies that I can estimate the
68
distribution stemming from pairs of particles from the same decay as
dσ
dEab
(same-decay pairs) '
dσ
dEab
(all pairs)−
dσ
dEab
(different-events pairs) . (3.16)
This observation is at the center of the event mixing idea, and I henceforth de-
note the procedure described by the right-hand side of eq. (3.16) as “mixed event
subtraction”.
The plausibility of the distribution obtained by pairing particles from differ-
ent decays within the same event being equivalent to the distribution arising from
pairings between different events can be seen intuitively in certain simple cases. To
illustrate, imagine having an ensemble E of pairs of particles (B1, B2),
E =
{
(B(1)1 , B
(1)
2 ), (B
(2)
1 , B
(2)
2 ), (B
(3)
1 , B
(3)
2 ), ...
}
, (3.17)
where the superscripts denote the associated event number. For simplicity, I take
particles B to be scalars sitting at rest in the laboratory, where they decay B → abA.
It is obvious that the distributions made from pairs coming from different B particles
in the same event of the ensemble are the same as the distribution made from pairs
coming from B particles from different events in the ensemble. In this example, I
am simply sampling the phase space of the B decay in two different ways, in one
case taking kinematic information from instances of the decay that happen at the
same time and in the other taking that information from instances of the decay that
are separated in time.
However, one must note that the situation described above may not corre-
spond to reality. As a concrete example, I take the pair of (B(i)1 , B
(i)
2 ) and assume
that the system of the two B particles has a center of mass energy
√
sˆi(> 2mB). In
69
the laboratory frame, the phase-space accessible to the decay products of the two
B particles depends on
√
sˆi. Let us denote such a phase-space as Φ(sˆi). If I then
consider the jth event, with the intention of mixing particles from the ith and jth
events, I am forced to confront the fact that the center of mass energy in the jth
event,
√
sˆj, and thus the phase space accessible to the jth event’s particles, is differ-
ent than that of ith event. The mismatch in the phase-space accessible to particles
in events at different
√
sˆ is clearly a potential source of error in the identification of
the “different-decay” and “different-event” distributions, which naturally poses an
a threat to the successful application of the event mixing technique.
It is quite difficult to estimate the size of this inaccuracy, though I expect that
for typical situations at hadron colliders the event mixing technique works quite
well. One reason for this is because of the small variance of the boosts of the B
particles produced in typical collisions at hadron colliders. In addition to this effect,
however, there may be other potential sources of error in the event mixing technique,
and a case-by-case study is needed to check the performance of the method. Because
of this, I take a pragmatic approach in the following and apply the event mixing
technique to our example while explicitly checking the performance of this method
for our example process.
3.4 Application to the gluino decay
I now demonstrate how the general strategy detailed above is realized by tak-
ing as an example a particular gluino decay channel. I first illustrate the signal
70
process and its particular characteristics, and then move to discussing the possible
backgrounds of this signal process. The discussion of these backgrounds is sepa-
rated into two categories: 1) the real background from SM processes, and 2) the
systematic background from incorrect pairings of the final state particles used in
forming the invariant mass and energy distributions. As I detail the various pro-
cesses involved, I shall utilize Monte Carlo simulation to generate the relevant event
samples, construct and analyze the appropriate kinematic data from these samples,
and end with a discussion of the effectiveness of our technique based on the results
of this analysis.
3.4.1 Signal process: gluino decay
I apply the general idea developed in the previous sections for the case of pair-
produced SUSY gluinos and their subsequent decay into two bottom quarks and a
neutralino via a three-body decay:
pp→ g˜g˜ → b¯bb¯b+ χχ (3.18)
at the 14 TeV LHC. In terms of the notation used in Section3.1, the gluino and the
neutralino correspond to particles B and A, respectively, and two visible particles
a and b are the bottom quark and anti-quark in a decay chain. In reality, the
particle detector cannot reliably discriminate between bottom and anti-bottom, thus
particles a and b in this example are considered indistinguishable.
Though I am using the specific decay above as a concrete example, I empha-
size that the decay mode and underlying model at hand are chosen only to enable
71
us to demonstrate the proposed technique and that the general idea can be applied
to multi-body decay processes in other models. I also point out that the applica-
bility of the method is not affected in any way by the strengthening of bounds on
supersymmetric particles, because the method is applicable for parent and invisible
(child) particles of any mass. To illustrate our technique, I choose the masses of
these particles to be
mg˜ = 1.2 TeV, mχ = 100 GeV
with a decoupled bottom squark, and assume that the only decay mode of the
gluino is a three-body decay in the form of bb¯χ 7. The Monte Carlo signal for our
study is simulated using MadGraph5 v1.4.8 [26] and the structure of the proton is
parametrized by the parton distribution functions (PDFs) CTEQ6l1 [27], evaluated
with the default renormalization and factorization scale settings of MadGraph5. The
production cross section of the paired gluinos is computed with MadGraph5 and is
reported in the first column of Table 3.1. Since I assume that all produced gluinos
decay into bb¯χ as described above, σ(pp→ g˜g˜) is equal to σ(pp→ g˜g˜ → bb¯bb¯χχ).
The neutralinos in the final state of our signal do not interact with the detectors
of the LHC, resulting in a missing transverse momentum. The four bottom quarks
give rise to jets of hadrons - particularly, B-hadrons. The particular characterisitcs
7During the completion of this work the limit on the gluino mass given by the LHC experiments
has risen to about 1.4 TeV for light χ [55, 56]. Despite these new limits ruling out the spectrum I
consider at a 95% confidence level, this spectrum still serves its purpose as an illustration of the
technique. It should be remarked that I do not expect qualitative differences in the application of
our strategy to the mass measurement of a heavier but not yet experimentally excluded gluino.
72
4b+ 
pT 4b+ 
pT 4b+ Z(→ νν¯) tt¯bb¯
(before cuts) (after cuts) (after cuts) (after cuts)
σ [fb] 54.74 36.53 0.48 0.15
Table 3.1: The cross sections for the signal process (before and after a set of cuts
are imposed) and the (main) background processes (only after cuts are imposed).
The cuts are described in eqs. (3.19)-(3.21), and also include all the identification
and the isolation criteria explained in the text. The effect of the b-tagging efficiency
is not taken into account by the numbers in this table.
of the B-hadrons in the jets allows us to distinguish this type of jet from other jets
that do not originate from the bottom quark, and it is possible to see the traces
of b-quark-initiated jets and tag them in a large fraction of the events. With the
requirement that four of the reconstructed jets in the final state have this tag, the
signal will feature four bottom jets plus missing transverse momentum, 4b+ 
pT .
Before closing this section, I remark that the chosen example process poses
an extra challenge in the application of our method. In fact all visible final state
particles are indistinguishable, hence there are three different ways to form pseudo-
particles from these b-tagged final state particles that must be all be considered in
the analysis. Together with the SM backgrounds, this can be interpreted as another
background, as explained in the next section.
73
3.4.2 Backgrounds
In this section, I discuss the backgrounds relevant to the signal process defined
in the preceding section. As mentioned earlier, there are two types of backgrounds:
those coming from Standard Model processes which give rise to the same signature as
our supersymmetric process, and that which comes from taking the wrong pairings of
two visible particles when I evaluate both the energy sum and the pair-wise invariant
mass. I start by discussing the “real” background from the Standard Model and
then I discuss the combinatorial background.
3.4.2.1 Standard Model backgrounds and event selection
For our collider signature 4b + 
pT , the following two processes in the SM are
identified as the major backgrounds:
pp→ bbb¯b¯+ Z → bbb¯b¯+ νν¯ and pp→ tt¯bb¯ .
The Monte Carlo generation of background events is done using the same event
generator and input PDFs as those for the signal events. Since the detector signature
from these interactions is exactly the same as the one used in our earlier work
Ref. [45], I adopt a similar strategy for handling these backgrounds with only slight
modifications. The Z boson background is irreducible, whereas the tt¯bb¯ is reducible
and can be reduced so that it becomes sub-dominant with respect to the Z boson
background. The tt¯bb¯ background might seem different from the signal process in
terms of its partonic final state, but it can mimic our signal, and thus become a
74
relevant background, by “losing” some of the final state partons in the detectors. In
order to match with the signal’s detector signature, the two W bosons originating
from the decay of the two top quarks must go undetected, which makes those W
bosons the main source of 
pT in the tt¯bb¯ background. Although the rate of the
detector missing the W bosons is expected to be small, the sizable production rate
of pp→ tt¯bb¯ can compensate for this, thus making the pp→ tt¯bb¯ process a possibly
important background.
A W boson will go unseen in the detector for primarily two reasons: 1) when its
decay products are not within the experimental acceptance region of the detector
due to having insufficient pT , supernumerary η, or both, and 2) when its decay
products are not adequately isolated from other particles, i.e., they are merged with
other particles in the reconstruction of a given event. For the first case, I define as
missed any object that satisfies the following criteria:
• for jets, pT,j < 30 GeV or |ηj| > 5,
• for leptons, pT,l < 10 GeV or |ηl| > 3 with l = e, µ, τ .
In the second case, the following rules determine when a particle is missed:
• for merging jets, ∆Rj1j2 < 0.4 with j1 and j2 denoting any jet pairs including
b-jets,
• for merging leptons, ∆Rjl < 0.3 with j and l denoting a jet and a lepton,
respectively.
With the acceptance and isolation requirements listed above, I observe that most of
75
the background events are from either the fully leptonic or the semi-leptonic decay
channels of top quark pairs because these channels require that fewer erstwhile
visible partons be missed in comparison to the fully hadronic top decay channel.
To devise an adequate strategy for rejecting a large number of these back-
ground events while preferentially keeping the signal events, I must adopt event
selection criteria that incorporate the kinematic differences between signal and back-
ground event. I observe first that, thanks to the heaviness of the parent particles,
the signal events will be composed of jets that typically have a larger transverse mo-
mentum than those found in background events. This is a strong hint as to which
cuts will be suitable in rejecting the background. However, one must be especially
cautious in selecting these cuts because the method proposed here is based on ex-
tracting E∗bb, which relies in part on the shape analysis of the energy distributions
for each invariant mass slice. Therefore, cuts should be chosen such that they do not
considerably distort the energy distributions. For this reason, I prefer using softer
cuts than in most searches performed at the LHC, and I choose as our baseline
selection criteria
pT,b > 30 GeV, |ηb| < 5, ∆Rbb > 0.4, (3.19)
for identifying the bottom jets in all events that I analyze.
In order to further suppress the backgrounds from Standard Model processes,
I consider requiring that events have a large missing transverse momentum. In
signal events, the missing transverse momentum is expected to be determined by
some combination of the new particle masses, and thus will be large. On the other
76
hand, the missing transverse momentum in background events is determined by the
larger of the total hardness of the event, the mass of the Z boson, or the mass of
the top quark. Therefore, a large 
pT cut allows us to efficiently discriminate the
signal events from the background. However, in our case special care is needed in
deciding the scale of this cut, as there is the risk of this cut introducing unwanted
bias in the energy distributions. In particular, the missing transverse momentum
can be interpreted as the recoil of the invisible particles against the visible, which
seems to imply that a large 
pT cut is likely to select only events with very hard
visible particles and correspondingly induce some bias in the b-jet energy spectrum
toward higher energies. As mentioned before, this could lead to a misidentification
the value of E∗bb, and as a consequence, an innacurate measurement of the associated
masses. Fortunately, the relatively large mass hierarchy between the gluino and the
neutralino in our signal process ensures multiple hard b-jets on average and thus
a sizable recoil for the invisible neutralinos. I therefore anticipate that the Ebb
distribution will only be mildly affected, even with a fairly hard 
pT cut. For our
signal and backgrounds, I impose

pT > 200 GeV , (3.20)
which strongly suppresses the backgrounds with negligible deformation of the Ebb
distributions.
In addition to the 
pT cut, I introduce another cut that requires each b-jet ~pT
to have some minimum angular separation from the 
pT vector. This enables us to
avoid events where the measured missing energy is caused by the mismeasurement
77
of jets. For our analysis I require
∆φ(

~pT , ~pbT ) > 0.2, (3.21)
which has negligible effect on the shape of the Ebb distributions.
In Table 3.1 I show the cross sections for both signal and background events
after applying the set of cuts listed above. I clearly see that the tt¯bb¯ background
is sub-dominant with respect to the Z + bbbb background. I also remark that the
expected signal-to-background ratio (S/B) is large, which is certainly favorable for
extracting E∗bb from the Ebb distribution. Indeed, if new physics particles are dis-
covered in the forthcoming runs of the LHC, it would then be natural to discuss
measuring their masses in the channels where there is a clean signal, and hence a
large S/B. In this sense, the context in which I present our mass measurement
technique is expected to be typical when attempting a mass measurement beyond
the precision of the order of magnitude, as I do here.
Other than the above-mentioned backgrounds, QCD multi-jet production pp→
bbb¯b¯ is another possible source of background events from the SM, in which the miss-
ing transverse momentum typically arises from imperfectly measuring the energy of
jets. Unsurprisingly, an accurate estimation of this background is quite challenging
because it involves detector effects. I expect that a great deal of the QCD multi-jet
background would largely be suppressed by the cuts eqs. (3.19), (3.20) and (3.21)
to the point that it becomes sub-dominant, and in the following I do not taking this
background into account (see, for example, Ref. [45]).
78
3.4.2.2 Combinatorial background and mixed event subtraction
As mentioned earlier, the procedure of phase space slicing inevitably requires
the formation of the invariant mass of two visible objects. Since the process eq. (3.18)
under consideration has pair-produced parent particles which both decay to indistinguishable
visible children, grouping the four b-jets into two pairs gives the correct choice in
only one case out of the three possible combinations. Following the strategy outlined
in Section 3.3, I form all possible pairs and obtain an inclusive energy distribution. I
then subtract the contributions originating from the wrong combinations by estimat-
ing the corresponding distribution through the event mixing technique as described
before. In order to validate the performance of this mixed event subtraction scheme
in our example, I first study a large number of pure signal events where no selec-
tion cuts are imposed and without including backgrounds. I then discuss how the
inclusion of backgrounds complicates the estimate of the combinatorial background
when using the mixed event subtraction.
i) Pure signal: For our study it is necessary to check that b-jet pairs’ energy and
invariant mass distributions are well reproduced by the mixed event subtraction
scheme. In order to apply it as per eq. (3.16), I need to obtain the distributions of
observables given by forming pairs of b-jets belonging to different events as explained
in Section 3.3. In principle, there are several options for choosing the two events
that one can use to compute these observables. For example, one can compute the
observables using all possible pairs of events, meaning that each event is reused
79
many times, or one can use a procedure such that each event is used only a few
times. The detailed way in which the event mixing is done can, in principle, affect
the result. However, in most cases, the alternative methods give rise to only minor
differences in the relevant result 8. Among those possibilities, I show results for
which the events were mixed as follows: given a sample of N events,
(1) I first randomly shuffle and reorder the events to remove any potential correla-
tion between events arising from the way in which the events were generated,
(2) compute the different-event observables by taking b-jets in the ith and (i+1)th
events,9 so that each event is used twice
(3) finally renormalize these distributions to weigh as much as the contribution
from the incorrect pairings in the signal sample that I intend to remove, i.e.,
two thirds of the total number of events in the signal sample.
I label the inclusive invariant mass distribution formed from all the pairs in the
same event as FSE(mbb) and I denote as FDE(mbb) the distribution obtained from
pairs in different events. Then, given an invariant mass value, I take from the same-
event sample all the pairs whose invariant mass lies within the range of interest and
plot the spectrum of the energy of the sum of the two b-jets, Ebb. I repeat the same
8One can also form “events” out of randomly selected sets of four particles from the entire event
sample and compute the observables by forming pairs from the particles now constituting these
new “events.” I evaluated the efficacy of constructing the distribution to be subtracted using this
alternative method and found little difference between the end results, both in the distributions
and in the energy and mass values extracted.
9The last event is mixed with the first one.
80
operation in the different-event sample, and I obtain the Ebb spectrum for the fixed
ranges of mbb. Denoting the spectrum from the same-event pairs as fSE(Ebb), the
spectrum from different-event pairs as fDE(Ebb), and the resultant spectrum from
the mixed event subtraction as fS(Ebb), our estimate of the energy distribution from
the correct pairs is:
fS(Ebb) = fSE(Ebb)− fDE(Ebb) (3.22)
from which I will ultimately extract the rest-frame energy value E∗bb for each fixed
mbb.
Correct pairings only
All pairings
Mixed events subtracted
0 200 400 600 800 1000 1200 1400
0
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m
bb
HGeV L
R
i
Figure 3.2: The left panel shows the di-b-jet invariant mass distributions that are
normalized to an integrated luminosity of 3 ab−1. The right panel shows the Ri
distribution over mbb.
Similarly to the subtraction for the Ebb spectrum, I obtain an estimate of the
overall invariant mass distribution for the correct pairings using the mixed event
technique, which is
FS(mbb) = FSE(mbb)− FDE(mbb) . (3.23)
81
Note that this distribution is not used to extract the masses, but serves as an
example of the effectiveness of the mixed event subtraction scheme. To quantify
the fidelity of the distribution obtained from the mixed event subtraction scheme, I
define a ratio
Ri ≡
Ni,S
Ni,C
, (3.24)
where Ni,S is the i-th bin-count of the “subtracted” distribution FS(mbb) and Ni,C
is the corresponding bin-count in the distribution obtained considering only the cor-
rect pairs. The latter is obtained by exploiting the fact that in simulated events all
of the history of the particles is available. The left panel in Figure 3.2 compares
the invariant mass distribution from the correct pairs, shown as the blue dot-dashed
histogram, and the distribution obtained by mixed event subtraction, shown as the
red solid histogram. To show how much the original invariant mass distribution
is contaminated by the combinatorial background, the distribution before the sub-
traction procedure is also plotted as the green dashed histogram. Each bin count
is normalized to an integrated luminosity of 3 ab−1. I observe that the distribution
obtained from the mixed event subtraction is very close to that obtained from the
correct pairs. A more quantitative comparison is also provided in the right panel of
Figure 3.2, showing the bin-by-bin ratio Ri for the distribution of the events against
mbb. All of the bin counts are quite close to their associated theoretical values. In
fact, Ri ∼ 1 in all the range of mbb. I do not show the ratio Ri in the vicinity
of both kinematic endpoints because the significantly smaller Ni,C at the endpoints
leads to unreliable Ri values. Besides visualizing the effectiveness of the mixed event
82
subtraction, this check enables us to confirm that for a given invariant mass slice,
a similar amount of data remain available after the mixed event subtraction com-
pared to what would have been available was I able to completely eliminate the
combinatorial background.
Correct parings only
Mixed events subtracted
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Figure 3.3: The two plots in the left panel show the di-b-jet energy distributions for
300 GeV (top) and 700 GeV (bottom) nominal mass slices. They are normalized to
an integrated luminosity of 3/ab. The color codes in the plots in the left panel are
the same as those in Figure 3.2. The two plots in the right panel show the respective
R distributions over Ebb. For computing 〈R〉, only the data within the two black
vertical dashed lines is taken into account.
83
In Figure 3.3 I compare the energy distribution from the correct pairs and
that obtained from the mixed event subtraction. The two distributions are shown
for a mass slice 275 GeV ≤ mbb ≤ 325 GeV in the upper panel of the figure and for
another mass slice 675 GeV ≤ mbb ≤ 725 GeV in the bottom panel. The ratio of
the correct pairs and the subtracted histograms is provided for each choice of mbb
as well. Since the fit to extract E∗bb from the energy spectra will be performed using
only the data around the peak, I show the bin-by-bin ratio only for the energy range
that corresponds to the full width at half maximum (FWHM), which is indicated
by black dashed lines in each plot. In this case, I see that the energy spectrum
processed with the mixed event subtraction (blue histogram) is also quite close
to the associated theory expectation (red histogram). To be more quantitative, I
compute the average of Ri in the FWHM range. This average is denoted as 〈R〉
and is close to 1, which suggests that the mixed event subtraction scheme works
quite well, i.e., the shape of the energy spectrum is reasonably preserved. From this
exploratory analysis, I expect that the extraction of E∗bb from the energy distribution
obtained by the mixed event subtraction is unlikely to have major bias due to the
subtraction.
ii) Background and “signal-background interference”: Once the SM backgrounds
come into play, there is a non-trivial complication that is introduced by the event
mixing. Since I am not aware a priori whether a given event is from the signal or
the background, it is not possible to perform the event mixing using only the signal
events. Therefore, the distribution returned by the whole operation of the mixed
84
event subtraction scheme contains “signal-background interference”, i.e., picking one
particle from a signal event and the other from a background event. In principle, the
overall kinematic characteristics of the background events differ from those of the
signal events, and therefore these interference pairs will make the overall distribution
deviate from that of the pure signal or pure background combinatorially-generated
background.10 As a consequence, a naively subtracted distribution would be dis-
torted with respect to the distribution of a pure signal sample.
To understand the quantitative impact of the inclusion of physical backgrounds,
I first need to assess the hierarchy of the effects that arise from the simple addi-
tion of these backgrounds and from the event mixing. I focus on the situations
where nev events have been collected after the application of selection cuts, e.g.,
eqs. (3.19)-(3.21). These events come both from the signal process and from back-
ground processes. In general, I have ns signal events and nb = nev − ns background
events.11 However, in a situation in which a mass measurement is attempted, I
expect that the signal will dominate the backgrounds, ns  nb. Under this assump-
tion, I can quantify how likely the event mixing procedure is to form pairs where
both particles come from the signal process, both particles come from the back-
ground processes, or one particle comes from signal and the other from background.
10For the dominant background in our case (i.e., Z + 4b), the pure background combinatorial
distribution is somewhat tricky; the distinction between correct and incorrect pairings is meaning-
less because the associated event topology is ill-defined. However, in the interest of generality, I
imagine that our background can also give rise to fictitious correct and wrong combinations.
11Since different types of backgrounds, in principle, will form different distributions, I here
assume only a single type of background to avoid any potential complication.
85
Clearly, the probabilities to select a particle from a signal or a background event in
the sample are
ps =
ns
ns + nb
' 1−
nb
ns
, pb =
nb
ns + nb
'
nb
ns
, (3.25)
for a signal and a background event, respectively. Therefore, most of the pairs
formed in the event mixing procedure are made with two particles from the signal
process. Pairs made of two particles from the background are much less abundant,
and in fact arise only in a small fraction, n2b/n2s, of the cases. Strikingly, pairs made
with one particle coming from the background and one particle from the signal are
much more abundant than pure background pairs, as their probability is 2× nb/ns.
The effect of the pairs involving both backgrounds and signal is not predictable
unless one specifies the signal. However, some general features of this “interference”
contribution to the event mixing estimate of the distribution for the correct signal
pairings can be easily guessed. First, the “interference” distribution tends to produce
an underestimation of the bin-counts in the estimate eq. (3.16) of the distribution
consisting of pairs of b-jets originating from the same decay. The reason is that in
the inclusive distribution from all pairs in the same event, the first term in eq. (3.16),
there are contributions stemming from pairs of events coming both from signal or
both from background, but there is no way to construct a hybrid of the two:
dσ
dEbb
(all-pairs) =
dσS
dEbb
+
dσB
dEbb
, (3.26)
where by σS and σB I mean the signal and background contributions, respectively.
On the other hand, in mixed events, I have
dσ
dEbb
(different-events pairs) =
dσSS
dEbb
+
dσSB
dEbb
+
dσBB
dEbb
. (3.27)
86
As suggested by Figure 3.3, the contribution from pairs where both events come
from the signal, dσSSdEbb , does a good job of estimating the effect of the pairs of b-
jets not coming from the the same decay in the signal. Similarly, the contribution
from pairs of events where both events come from the background, dσBBdEbb , is a good
estimate of the combinatorial background generated from the background itself.
Therefore, the contribution dσSBdEbb is the piece that typically ruins the result because
it gives rise to an excessive subtraction in eq. (3.16). Obviously, this phenomenon
cannot be avoided since I am unable to distinguish signal and background events
with absolute certainty. The presence of this type of “interference” background is
inherent to the event mixing technique and dealing with it requires special care. A
more quantitative argument about the interference is available in App. A for more
interested readers.
Given its peculiar origin, it may be desirable in some cases to remove the
“signal-background interference” contribution. In order to do so, one must discuss
the shape of this distribution, which in general depends on the signal and there-
fore is a priori unknown. However, some general features of the “signal-background
interference” distribution can be predicted using the following argument. The dis-
tribution that arises from pairs made of one background and one signal particle
feels in part the kinematics of the signal events and in part that of the background
events. The background is typically expected to have softer particles than the sig-
nal, and therefore, the “interference” b-jet pair energy distribution is expected to be
skewed towards energies that are somewhat larger than the characteristic values of
the distribution for pairs made out of two background particles. For our example
87
Interference
Background only
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Figure 3.4: The di-b-jet energy distributions of the true background and the in-
terference for 300 GeV (left panel) and 700 GeV (right panel) nominal mass slices.
They are normalized to an integrated luminosity of 3 ab−1. The black distribution
is obtained by subtracting the blue one by the red one. The vertical black dashed
lines denote the associated fitting range for each slice. The bottom panel shows the
performance of the proposed fitting template for the effective backgrounds.
process, I display the distribution from pairs of signal particles in Figure 3.3, while
the distribution from pairs of background particles and the interference distribution
are shown in Figure 3.4. Comparison of these distributions confirms our intuition
from the argument above.
88
From Figure 3.4 I see that the interference effect dominates the pure back-
ground effect essentially everywhere in the distribution. As discussed above, this
is due to the fact that I envision a situation where signal events are much more
abundant than background events. The shape of the interference contribution is
also quite different from that of the pure background contribution.
When discussing results in a later section, I will study the effect of the removal
of background contributions to our results. With this goal in mind, I study what
functional form describes the total effect of backgrounds; that is, the distribution
from the pure background pairs plus that from the “signal-background interference”
pairs. The combination is shown in Figure 3.4; in the lower panel, I show a possible
fit of this distribution. Due to the importance of the signal in determining the
shape of the “signal-background interference” distribution, I decided to model the
total effect of backgrounds with a function of the family eq. (3.12). The fit result in
Figure 3.4 is rather good, but I do not attach any special significance to this finding.
In fact, a better description for this background may exist and might be preferred.
More generally, I stress that the “signal-background interference” distribution is not
universal, and our choice could be unreliable for other signals. In our application
to the gluino decay process, the fairly good description provided by eq. (3.12) and
shown in Figure 3.4 is satisfactory for our current purposes.
89
ŸŸ
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Ÿ
200 400 600 800 1000
0.98
1.00
1.02
1.04
1.06
1.08
1.10
m
bb
H GeV L
<
R
>
Figure 3.5: Average in the FWHM range of the bin-by-bin ratio of the energy
distributions from the event mixing and from just the correct pairs of b-jets. 〈R〉 = 1
implies a good match.
3.5 Mass measurement results and discussion
In this section, I demonstrate the application of the proposed technique to the
gluino decay. Results on the mass measurement from fitting the energy spectra for
the compound system of two b-jets are presented in the following subsections along
with the possible issues and limitations of our method. In the final subsection, I
discuss possible improvements of the mass measurement with the aid of the di-jet
invariant mass endpoint.
3.5.1 Measurement of gluino and neutralino masses
Following the strategy outlined in the previous sections, I present results for the
determination of the masses of the gluino and the neutralino from the b-jet energies.
90
The energy spectra that I study are obtained from simulated event samples generated
as described in Sec. 3.4.1 for both signal and dominant background processes at the
14 TeV LHC. I also recall that the relevant channel is characterized by a large
missing transverse momentum and four bottom-tagged jets which are selected as
per eqs. (3.19)-(3.21). Since the primary interest of this chapter is to study the
theoretical aspects of energy peaks in a multi-body decay, rather than data analysis
under realistic statistics, I take a sufficiently large number of events to minimize
potential statistical fluctuation within the data sample, which is then normalized to
an integrated luminosity of 3 ab−1.
ò
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
d
.
o
.
f
.
Figure 3.6: Comparison of the goodness of the fit between massive template
eq. (3.12) (blue) and the massless template eq. (3.11) (red) as fitter for the en-
ergy spectrum of the b-jet pair system for various values of the invariant mass of
b-jet system m¯bb.
Note that I study the distribution of the sum of the energy of b-jet pairs (say,
b1 and b2), i.e., Ebb = Eb1 + Eb2 , for which the associated invariant mass values
91
belong to a narrow range such that
mbb ∈ [m¯bb −∆mbb/2, m¯bb + ∆mbb/2] , (3.28)
where ∆mbb denotes the width of the mass window. I henceforth identify every
individual mass window by its central value m¯bb. Due to the indistinguishable nature
of the final state particles, I form the Ebb distribution from all possible pairs of b-jets
in an event, and subsequently apply the mixed event subtraction technique described
in Sec. 3.4.2.2 to eliminate the contamination from the pairs of b-jets not coming from
the same gluino. In Figure 3.5, I present the average of the bin-by-bin ratio of the
distributions from only correct pairs and from the mixed event subtraction technique
at various values of m¯bb. I remark that for each point in the figure, the average is
limited to the energy range defined by the full width at half maximum (FWHM)
of the distribution. The figure suggests that the average deviation between the
two distributions is, at most, about 8%, meaning that the mixed event subtraction
scheme reproduces the original distribution fairly well. Although I do not show it
here, I also remark that for a given m¯bb the standard deviation in the bin-by-bin ratio
is small enough that the distribution from the mixed event subtraction consistently
tracks the corresponding distribution from the correct pairs.
For each m¯bb, the rest-frame energy of the b-jet pair system (i.e., E∗bb) is ex-
tracted from the energy distribution by fitting the data to a template function. I
have two possible template functions, given in eqs. (3.11) and (3.12), and I use
both of them to see which one better suits the data and to see if there are sig-
nificant differences between the rest-frame energy and E∗ value found by the two
92
functions. Since eq. (3.11) is suitable only for the case where the relevant m¯bb is ef-
fectively negligible, I expect an increasing discrepancy between the results obtained
by eqs. (3.11) and (3.12) as m¯bb grows. The reduced χ2 values for each fit to the
energy spectrum are shown in Figure 3.6. This χ2 is a measure of how well the
template function describes the data globally, but I do not necessarily attach any
statistical meaning to it. I instead use it as a measure for the distance between
the two template functions. Looking at the figure, I observe that for all m¯bb, the
massive template describes the data as well as or better than the massless template.
As expected, the performance of the massless template becomes progressively worse
as m¯bb increases. The massless template also seems inferior to the massive template
from another aspect as it typically returns an estimate of the rest-frame energy that
is larger than the expected value. On the other hand, the massive template does
not introduce such a pronounced bias.
To demonstrate the difference between fits made with the two templates, I
show in Figure 3.7 sample fit results for two different nominal m¯bb values, 250 GeV
and 650 GeV. In the left panels, I provide results from applying the massive template,
while in the right panels I provide results from applying the massless template.
The errors quoted for the extracted E∗bb were estimated at 95% confidence interval
(C.I.) from the variation of the χ2 of the fit. For both of the m¯bb values, I see
that the massless template estimates E∗bb as being slightly larger than the estimate
given by the massive template, and the discrepancy is larger as I go to larger m¯bb.
Although the discrepancy is within the 95% confidence interval, I feel that this is
an important characteristic of the fit results. In fact, the massless template has a
93
systematic tendency to return larger E∗bb in general, which implies the introduction
of a possible bias to the mass measurement. The results of the fits for all of the
values of m¯bb are reported in Table 3.2, from which I see that massless template
consistently overshoots the estimate of E∗bb obtained from the massive template, and
that it also overestimates the theory values of E∗bb.
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E
bb
*
= 627.2
-19.5
+ 18.7
GeV
Χ
2
‘ d.o. f . = 0.039
0 500 1000 1500 2000 2500
0
10
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30
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bb
HGeV L
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E
bb
*
= 634.8
-19.5
+ 18.7
GeV
Χ
2
‘ d.o. f . = 0.035
0 500 1000 1500 2000 2500
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HGeV L
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E
bb
*
= 802.2
-36.9
+ 21.
GeV
Χ
2
‘ d.o. f . = 0.03
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E
bb
*
= 810.
-40.3
+ 26.8
GeV
Χ
2
‘ d.o. f . = 0.063
0 500 1000 1500 2000 2500
0
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Figure 3.7: Sample fit results for extracting E∗bb using the massless template (right
panels) and the massive template (left panels). The chosen invariant mass slices are
mbb ∈ [225, 275] GeV (top panels) and mbb ∈ [625, 675] GeV (bottom panels). I re-
port only statistical errors. Each fit range is chosen such that it roughly corresponds
to the relevant FWHM.
94
ƒƒ
ƒ
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´
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ƒ
Theory: m
g
Ž
= 1200 GeV, m
Χ
Ž
= 100 GeV
Fit with massive data Hm
bb
Î @200,650DL
: m
g
Ž
= 1042± 65 GeV, m
Χ
Ž
2
= - 159000± 59000 GeV
2
Fit with massless data Hm
bb
Î @200,600DL
: m
g
Ž
= 964± 56 GeV, m
Χ
Ž
2
= - 240000± 41000 GeV
2
E
bb
*
with massive template
E
bb
*
with massless template
200
2
400
2
600
2
800
2
600
700
800
900
1000
1100
1200
1300
m
bb
2
HGeV
2
L
E
b
b
*
H
G
e
V
L
Figure 3.8: The fit of the data points (m2bb, E∗bb) with eq. (3.29). The theoretical
expectation for the given mass spectrum is represented by the solid black line. The
data points obtained by fitting the Ebb distributions with massless and massive
templates are marked by “cross” and “square” symbols, respectively. The mass
measurement done with cross symbols is represented by the red dashed line. For
the blue dot-dashed line, the measurement is done for the data points for which the
massless template work reasonably well.
Finally, I take all the values of E∗bb obtained from the fitting the energy spec-
trum for each m¯bb and fit them to the line given by eq. (3.3), taking into account
and displaying the associated errors from the fit procedure that I used to extract
the values of E∗bb. The expression in eq. (3.3) can be adjusted to match our example
95
òàà
à
ò
68% C.L.
95% C.L.
Best-fit
Theory
0.00042 0.00044 0.00046 0.00048 0.00050
594
595
596
597
598
599
600
12m
g
Ž
H GeV
-1
L
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Ž
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Χ
Ž
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L

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Ž
H
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e
V
L
Massive template
ò
àà
à
ò
68 % C.L.
95 % C.L.
Best-fit
Theory
0.00042 0.00044 0.00046 0.00048 0.00050 0.00052 0.00054
596
598
600
602
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608
12m
g
Ž
H GeV
-1
L
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Ž
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-
m
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Ž
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L

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Ž
H
G
e
V
L
Massless template
Figure 3.9: Contour plots in the plane of s (= 1/2mg˜) vs. y (= (m2g˜ −m2χ˜)/2mg˜)
around the best fit values for the fit results with massive (left panel) and massless
(right panel) templates.
96
m¯bb Fit range Theory Massive [χ2/d.o.f] Massless [χ2/d.o.f]
200 [400, 1000] 612.5 614.8+21.4−24.0 [0.064] 619.1+21.4−24.2 [0.054]
250 [400, 1000] 621.9 627.2+18.7−19.5 [0.039] 634.8+18.7−19.5 [0.035]
300 [400, 1000] 633.3 640.9+15.5−16.0 [0.089] 654.3+15.4−16.0 [0.057]
350 [440, 1000] 646.9 659.2+16.2−16.2 [0.074] 673.1+15.9−16.2 [0.063]
400 [440, 1040] 662.5 670.7+14.3−14.0 [0.074] 693.7+13.6−13.5 [0.061]
450 [500, 1040] 680.2 694.8+14.9−15.5 [0.041] 715.9+14.8−15.6 [0.037]
500 [540, 1040] 700.0 716.3+14.8−15.9 [0.033] 738.8+14.8−16.2 [0.050]
550 [600, 1100] 721.9 742.1+16.7−21.1 [0.038] 760.0+18.5−23.8 [0.064]
600 [640, 1100] 745.8 768.9+17.6−27.2 [0.026] 787.9+19.3−26.3 [0.074]
650 [700, 1200] 771.9 802.2+21.0−36.9 [0.030] 810.0+29.0−40.3 [0.063]
700 [740, 1240] 800.0 832.7+23.4−132.7 [0.011] 840.9+21.4−45.0 [0.060]
750 [800, 1300] 830.2 871.4+28.4−121.5 [0.017] 865.3+39.3−68.7 [0.060]
800 [840, 1340] 862.5 910.5+28.3−110.6 [0.024] 908.3+37.8−64.3 [0.067]
850 [880, 1340] 896.9 952.8+29.4−102.9 [0.019] 961.1+34.4−60.7 [0.12]
900 [920, 1400] 933.3 998.0+29.7−98.0 [0.040] 1015.2+31.6−52.6 [0.21]
Table 3.2: The fit results for fifteen invariant mass slices. For each fit, a mass
slice of 50 GeV was chosen, for example, for m¯bb = 200, Ebb is selected such that
the corresponding mbb is between 175 and 225 GeV. The bin size for all energy
distributions is 20 GeV. The error estimation for each fit parameter is performed by
95% confidence interval. All values but χ2 are given in GeV.
97
process as follows:
E∗bb =
m2g˜ −m2χ + m¯2bb
2mg˜
. (3.29)
I perform the fit of eq. (3.29) on both of the results from the massive and massless
templates. For the fits using the massive template, I use only the results obtained
for m¯bb in the range from 200 GeV to 650 GeV, in which the errors from the fit
of the energy spectra are quite small. Other choices of the m¯bb range give similar
mass measurements, but I simply make a conservative choice for the range of m¯bb
included so to avoid the values of m¯bb where fewer events are expected. From the
results extracted using the massless template, I choose to fit eq. (3.29) only for m¯bb
in the range from 200 GeV to 600 GeV, where it is more reasonably accurate to
treat the b-jet pair system as massless. The fit parameters are the slope of eq. (3.29)
and its vertical intercept (that is E∗bb of m¯bb = 0) and I denote them as
s ≡
1
2mg˜
and y ≡
m2g˜ −m2χ˜
2mg˜
. (3.30)
For our spectrum, the theory values are
s = 4.2× 10−4 GeV−1, y = 595 GeV . (3.31)
Fitting the line eq. (3.29) on the results obtained from the energy spectra, I obtain
the best-fit lines shown in Figure 3.8, which correspond to
s = (4.8± 0.3)× 10−4 GeV−1, y = 597± 5 GeV , (3.32)
for the massive template and
s = (5.2± 0.3)× 10−4 GeV−1, y = 606± 6 GeV , (3.33)
98
for the massless template. Not surprisingly, the values extracted using the massive
template are closer to the input values than those from the massless template,
although even the massless template was able to get a rough estimate of the rest
frame energy values of the bb system for the chosen values of m¯bb. In Figure 3.9, I
also show the 68% and 95% confidence level contours obtained from the χ2 variation
of the fit of the slope and intercept parameters; the result with the massive template
are in the left panel and that with the massless template in the right panel. One
can clearly see that the distance between the theory values and the best-fit values
for the case of the massive template (left panel) is smaller than that for the case of
the massless template (right panel).
As mentioned before, s and y can be easily converted into the masses of gluino
and neutralino. Based on eqs. (3.32) and (3.33), I obtain the following measurements
of the two masses:
Massive template : mg˜ = 1042± 65 GeV, m2χ = −159000± 59000 GeV2 ,(3.34)
Massless template : mg˜ = 964± 56 GeV, m2χ = −240000± 41000 GeV2 .(3.35)
I remark that the gluino mass, while quite precisely determined, is underestimated
by about 20%, with the value from the massive template being closer to the true
value than the value from the massless template. The neutralino mass is poorly
determined using both the massive and the massless template. Possible causes of
this poor estimation will be discussed in the next subsection.
99
3.5.2 Study of systematic effects
The results in the previous subsection on the measurement of the gluino and
neutralino masses are fairly good, considering the challenging circumstances of the
mass measurement, particularly the fully indistinguishable character of final state
particles in our chosen signal process. Despite an adequate result for the gluino mass
measurement, the neutralino mass measurement is very poor; the only conclusion
that one is able to draw is that the neutralino in our example process is consistent
with being massless.
slope
steeper consistent shallow
y-intercept
larger
mg˜,ext < mg˜,in mg˜,ext ≈ mg˜,in mg˜,ext > mg˜,in
m2χ˜01,ext  m
2
χ˜01,in
m2χ˜01,ext < m
2
χ˜01,in
m2χ˜01,ext ≈ m
2
χ˜01,in
consistent
mg˜,ext < mg˜,in mg˜,ext ≈ mg˜,in mg˜,ext > mg˜,in
m2χ˜01,ext < m
2
χ˜01,in
m2χ˜01,ext ≈ m
2
χ˜01,in
m2χ˜01,ext > m
2
χ˜01,in
smaller
mg˜,ext < mg˜,in mg˜,ext ≈ mg˜,in mg˜,ext > mg˜,in
m2χ˜01,ext ≈ m
2
χ˜01,in
m2χ˜01,ext > m
2
χ˜01,in
m2χ˜01,ext  m
2
χ˜01,in
Table 3.3: Comparisons of extracted mass parameters with corresponding input
values for the nine possible combinations of over-, under- or consistent estimation of
the slope and the intercept of the straight line eq. (3.29) fitted on the (mbb, E∗bb) data.
The orange table cell corresponds to the result of the fit of the data in Sec. 3.5.1.
As noted already, the measurement of E∗bb for each m¯bb is statistically compat-
100
ible with the theory value, but still results in a mass measurement that is system-
atically overestimated. From the fit of the data in Figure 3.8, the mismeasurement
of E∗bb primarily implies a slope larger than that predicted by theory, which conse-
quently implies that the extracted gluino mass is biased towards values smaller than
the true mass. This bias is not particularly worrisome per se, as it is about 10%.
However, given the relation between the masses and the observables in eq. (3.30), it
turns out that this underestimation of the gluino mass severely affects the neutralino
mass determination. More generally, there are nine possible cases based on under-,
over-, or consistent estimations of the slope and intercept of the straight line in
eq. (3.29). The implication of each case in terms of the extracted mass parameters
is summarized in Table 3.3.
It is interesting to examine possible causes of this bias in the best-fit line of
Figure 3.8, which also serves as a basis for possible improvements of our method.
In order to clarify the origin of the incorrect estimation of E∗, I study the following
potential sources of inaccuracy in our fits of the energy spectra:
i) an imperfect fit of the data with the massive template eq. (3.12);
ii) contamination due to the background;
iii) biases introduced by the event mixing subtraction;
iv) finite size of the mbb range used to discretize the multi-body phase-space;
v) biases due to events selection.
For the first potential source, I recall the discussion in Section 3.2 where the
101
massive template function was introduced; this template had a maximum at E∗bb
only when w → ∞, which corresponds to producing the gluino(s) at rest. For
practical cases, w is finite and the maximum of the function appears at a somewhat
larger value than E∗bb. On the other hand, physical energy distributions can have
the maximum at E∗bb; in particular, for cases where mbb can be treated as effectively
massless. Therefore, the relevant fit could result in a value that does not match the
corresponding expectation. Fortunately for the case at hand, I find that w is large
enough to cause only a negligible shift in the peak position, i.e., such a potential
mismatch is very tiny. Consequently, I do not ascribe the systematic overestimate
E∗bb to the inaccuracy of the template function eq. (3.12).
In order to see the effect of the other four potential sources of bias on the final
result, I conduct a dedicated analysis for each. In each analysis, I repeat the same
procedure as described in the previous section, that is to say I extract the values
of E∗bb from an event sample that incorporates the effect under study. The event
samples for the study of these possible effects are denoted as “Check Sample” (CS).
I then compare the results obtained from those Check Samples to those obtained
from the Original Sample (OS). The attributes of the check samples that I have
considered are summarized in Table 3.4 and are also described in the following.
The first check sample enables us to find the effect of the background on the
extraction of E∗bb. I study first the pure background energy distributions in order
to calibrate the template function describing them in the fit. This calibration is
done for each of the m¯bb slices. Although there are two types of backgrounds, the
dominant SM background and the interference from the event mixing, I employ a
102
∆mbb Event mixing Background Background fit Cuts
OS 50 GeV Yes Included No Yes
CS I 50 GeV Yes Included Yes Yes
CS II 50 GeV No Included No Yes
CS III 50 GeV No Not Included - Yes
CS IV 2 GeV No Not Included - Yes
CS V 2 GeV No Not Included - No
Table 3.4: Description of the original sample (OS) and several selected check
samples (CS). The width of the ranges of mbb for the discretization of the multi-
body phase space is reported in the first column. Samples marked as event mixed are
those in which the mixed event subtraction has been carried out. In those marked as
“no”, the correct pairs are identified in the event record and so eliminate the effect
of combinatorial backgrounds. Samples where the background has been completely
neglected are marked in the third column. For the samples where the background
has been added, I report in the fourth column if I have added a template to fit
the background events to the overall fit of the data . Finally, in the fifth and final
column I report if selection cuts eqs. (3.19) through (3.21) have been applied to the
events or not.
single template in eq. (3.12) to describe both of them collectively. I then repeat
the fit of the energy spectra for each m¯bb, including the template function for the
backgrounds as well. The results in the determination of E∗bb for each m¯bb are labeled
as “CS I” an plotted as red open circles in Figure 3.10. In this figure, the left panel
103
shows the absolute shift of the measured E∗bb from the corresponding theory value
for each sample. The right panel shows the ratio of the values of E∗bb from the
check samples and the corresponding value in the original sample. I observe that
the effect of background modeling is negligible for all m¯bb, and from the right panel
of Figure 3.10, I can in fact see that this engenders less than 1% of the shift in E∗bb.
The next potential source of bias that I study is the event mixing, which is
studied by the second and third check samples, denoted by CS II and CS III in the
following. In these samples I use the event record to identify the correct pairs of
jets coming from the same gluino, and therefore I obtain the correct energy spectra
without applying the mixed event subtraction. The two samples CS II and CS III
differ by the inclusion of the SM background. The results for the determination
of E∗bb are reported in Figure 3.10 by blue filled triangles and blue open triangles,
respectively. From the figure I see that the determination of E∗bb is significantly
improved. In fact, in the left panel of Figure 3.10 I can see that a mild positive
shift of the determined E∗bb values still exists, but is greatly reduced compared to
what I had with the original sample. I remark further that only minor differences
are found between the results obtained from the check samples CS II and CS III,
which can be taken as another way of confirming that the effect from background
events is negligible. Therefore, I conclude that the effect of event mixing is a major
cause of the shift that I observe in the gluino mass determination.
Next, I study the effect of the discretization of the mbb spectrum by taking
smaller ranges for the m¯bb window. This is performed on the check sample denoted
as CS IV. This narrow mbb range analysis is intended to provide an improvement to
104
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í
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ç
ò
ó
ì
í
OS
CS I
CS II
CS III
CS IV
CS V
200
2
400
2
600
2
800
2
- 20
0
20
40
60
80
m
bb
2
HGeV
2
L
D
E
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*
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CS I
CS II
CS III
CS IV
CS V
200
2
400
2
600
2
800
2
0.95
1.00
1.05
m
bb
2
HGeV
2
L
E
b
b
*
H
C
S
L

E
b
b
*
H
O
S
L
Figure 3.10: Comparisons of fit results from the five samples used to assess the
effects of several potential sources of bias in the gluino mass determination as de-
scribed in Table 3.4.
the previous two analyses where the combinatorial issues were artificially resolved
using the information in the event record. In fact, the check sample CS IV for this
analysis is similar to CS III except for the ∆mbb. The results of this analysis are
reported in Figure 3.10 by black filled rhombuses, which suggests that the effect of
105
discretization of the mbb spectrum is negligible. This is not surprising in light of
the following observation: one can easily figure out that, for a given nominal value
of m¯bb the the E∗bb from eq. (3.29) for (m¯bb + 25) GeV (m¯bb − 25) GeV is at most
about 15 GeV larger (smaller) than the E∗bb for the nominal m¯bb. The absolute size
of this shift is already quite small and is further reduced by the fact that for each
mbb range I observe the sum of all the contributions below and above m¯bb. In all,
I expect a very small net effect, making the discretization of the mbb spectrum in
increments of 50 GeV suitable for the precision sought.
Finally, I study the bias induced by the selection cuts. To assess their effect,
I produce a sample along the line of CS IV, but being fully inclusive in the signal
phase-space. The result of fits performed on the energy spectra from this sample
are reported in Figure 3.10 by black open rhombuses. The use of a fully inclusive
sample gives E∗bb from the fits that agree with the theory predictions within few
percents up to m¯bb = 650 GeV.
For the check samples II, III, IV and V I remark that the agreement of the fit
results with the theory value deteriorates as one gets closer to the endpoint of the
range covered by mbb. I observe that good agreement is retained up to m¯bb = 650
GeV, which is in the falling tail of the mbb distribution, as apparent from Figure 3.2.
I suspect that the mismatch of the fitted E∗bb and the theory values is connected to the
massive template becoming less accurate in fitting to the data. Indeed, in Figure 3.8
I can see that the error estimation on the fitted E∗bb becomes larger for m¯bb ≥ 650
GeV. In a realistic application of our mass measurement method, I would not know
up to what precise value of mbb the massive template can be trusted. However, it
106
is clear that the values of mbb for which the extracted E∗bb from the fit comes with
a large error should be avoided. I remark that in Figure 3.8, all the fit results for
mbb ≥ 650 GeV have a significantly large error, thus clearly signaling a transition
to a region of mbb where the fit template eq. (3.12) can no longer be trusted. A
more detailed investigation of this transition boundary is beyond the scope of this
chapter and I instead refer to Ref. [47] for a more systematic study of it.
3.5.3 Improving the mass measurement using the mbb endpoint
In this section, I discuss a possible improvement of the mass measurement
2.0 TeV
1.6 TeV
1.2 TeV
0.8 TeV
0.4 TeV
200
2
400
2
600
2
800
2
400
500
600
700
800
900
m
bb
2
HGeV
2
L
E
b
b
*
H
G
e
V
L
Figure 3.11: The functional dependence of eq. (3.37) for various gluino masses.
The mbb endpoint is set to be 1.1 TeV. The black solid line denotes the case where
mg˜ is identical to that of our study point.
with the aid of the kinematic endpoint of the dijet invariant mass distribution.
The point is that without prior knowledge of the masses, it is not possible to say
107
whether the measurement eq. (3.34) has a bias. However, I can devise a check and
an improvement of the obtained measurement using an independent observable. To
this end, I study a possible combination of the result of fits to energy spectra with
the measurement of the endpoint of the invariant mass distribution of the pairs of
b-jet, which I denote as mmaxbb . This observable is a simple function of the gluino and
neutralino masses:
mmaxbb = mg˜ −mχ , (3.36)
and is expected to be very useful in combination with the results of fits to energy
spectra. In fact, eq. (3.29), in light of eq. (3.36), can be rewritten as
E∗bb = mmaxbb −
(mmaxbb )2 − m¯2bb
2mg˜
, (3.37)
which is a straight line in the plane (E∗bb, m¯2bb) described by just one free parame-
ter. This should be compared to the previous equation I used to find the masses,
eq. (3.29), where there are two independent parameters: the slope and the constant
term of the straight line.
If the mbb endpoint is assumed to be well-measured, I can use eq. (3.37) to
more accurately fit our results to a line in the plane (m2bb, E∗bb). This relation is
shown in Figure 3.11 for mmaxbb = 1.1 TeV. Using this relation as a template for the
fit of the (m2bb, E∗bb) data points in Figure 3.8 along with eq. (3.36), I obtain a mass
measurements:
mg˜ = 1236± 31 GeV, mχ = 134± 31 GeV . (3.38)
In this case, as well as for the analysis in the previous sections, the fit is performed
108
between m¯bb = 200 and m¯bb = 650 GeV, with the data points obtained from the
massive template. This result is more accurate and in better agreement with the
expected values than what I obtained in eq. (3.34) using only energy distributions.
Therefore, I conclude that, depending on the accuracy with which the mbb endpoint
can be experimentally determined, the addition of information from thembb endpoint
can grant a very significant improvement to our results obtained from only the energy
spectra.
3.6 Summary and Conclusions
In this chapter, I have discussed how to use the energy spectra of visible decay
products for the measurement of masses of “parent” particles in semi-invisible multi
(i.e., more than 2)-body decays. The results are an extension of previous results
regarding the properties of the energy distribution of the massless decay products
in a two-body decay. In particular, I extended the results for two-body decays by
discretizing the multi-body phase-space and considering it as the combination of
multiple two-body decays, as suggested by the recursive factorization formula for
the multi-body phase space.
The fictitious two-body systems that are involved in the recursive factorization
of phase-space are necessarily massive. Therefore, I utilized the results of [47] on the
description of energy spectra of massive decay products in two-body decays. Armed
with these results, I should be able to fit the energy spectra of the visible part of
the fictitious two-body decay and extract an estimate of the masses involved in the
109
process using the results of these fits.
A particularly challenging aspect of our analysis had to do with the fact that
the decaying particle (the particle of interest) is typically produced in association
with other particles. In this situation, it is possible that some of the “child” par-
ticles from the decay of the parent are identical to those contained in the rest of
the event, which means that in the process of reducing the multi-body final state
of the parent decay to one with fewer bodies, the particle pairings that I perform
may unintentionally include particles which have nothing to do with the mass mea-
surement at hand. In particular, the parent particles are often produced in pairs:
if the two parents in each event undergo the same decay process, then it is clear
that this combinatorial background is inevitable. These particles extraneous to the
decay potentially hamper the mass measurement of the parent, and thus the con-
tamination they add must be addressed. Most of the general discussion above can
be succinctly illustrated by the consideration of a suitable example. Furthermore,
tackling a concrete example enables us to quantify the quality of the mass measure-
ment that one can achieve using our method. With these goals in mind, I studied in
detail the production of a pair of gluinos in a supersymmetric model, where R-parity
is conserved, and in which the gluinos directly decay to bb¯χ˜01 (a three-body decay)
via an off-shell bottom squark. To this end, I simulated events for the process
pp→ g˜g˜ → bbbb+ 
pT ,
including the relevant SM contribution. I identified selection cuts to remove the SM
backgrounds to a level that further clears a path toward the successful determination
110
of the masses of the new physics states. Simultaneously, I attempted to minimize the
changes in the shape of the energy spectra caused by these event selection criteria.
Our actual analysis then starts with forming all possible pairs of bottom quarks
in each event that passed the selection in eqs. (3.19)-(3.21), from which I then
obtained a distribution of the energy of the b quark pairs, Ebb. This distribution
includes the contribution from pairs formed by bottom quarks not originating from
the same decay, i.e., the wrong combinations. Note that the final state of each decay
in our chosen process is made of two (visible) indistinguishable particles, and so
the pollution from the combination of particles not coming from the same decay
is even more severe; in particular, each event gives 6 combinations of two bottom
quarks, out of which only 2 are correct, cf. the case of distinct particles a and b
from each decay, which would give 2 correct ones out of 4 combinations. To remove
this adulteration of the event sample, I subtracted an estimate that I obtained using
the event mixing technique described in Sec. 3.3. This estimate was obtained from
pairs of b quarks taken from different events, and I showed that the contribution
from pairs of b quarks not coming from the same gluino can thus be effectively
removed, as seen in Figure 3.3. In other words, this method has a natural tendency
to maintain the shapes of the distributions that I need to analyze to carry out our
measurement.
The energy spectra, once effectively rid of the contribution from pairs of b
quarks coming from different gluinos, were fitted in a region around their peak
with the function eq. (3.12), which is taken from Ref. [47] and briefly described in
Sec. 3.2. For comparison, these energy spectra were also fitted with the function
111
eq. (3.11), which was used in our original paper on the peak of energy spectra of
massless particles. The comparison of the results from fits with the two functions
highlights the improvement achieved by our new result for massive decay products.
The better description of the energy spectra with the function eq. (3.12) can be seen
in the comparison of the χ2 for the various fits performed, as reported in Figure 3.6.
The result of the fits of the energy spectra is the extraction of the function
parameter E∗bb, which is exactly the energy of the system of the two b quarks in the
rest frame of the gluino that generated the two b quarks. The determination of E∗bb
is the core of our procedure as this value is connected to the masses of the gluino
and the neutralino via eq. (3.29):
E∗bb =
m2g˜ −m2χ + m¯2bb
2mg˜
for a pair of b quarks of mass m¯bb. The results of the extraction of E∗bb for several
choices of the mass of the two b quark system were shown in Figure 3.8. The
determination of E∗bb for each mbb was fitted using the straight line eq. (3.29) given
above. This fit is essentially our mass measurement, as the gluino mass corresponds
to the slope of the line and the neutralino mass to the constant term of the straight
line. The resulting mass measurement was given in eq. (3.34), which was found to
be within 20% of the gluino mass and a rather poor determination of the neutralino
mass. This inaccuracy of the mass measurement is mainly due to the fact that in
each fit of energy spectra, I tend to overestimate the energy E∗bb. In Sec. 3.5.2, I
studied several possible causes for this error and (in the end) identified the modest
shape changes due to the mixed event subtraction as the primary source.
112
In order to improve the mass measurement, I then studied how including
information about the endpoint location in the mbb distribution altered the quality
of the mass measurements. If well-measured, this quantity should correspond to
the mass difference between the gluino and the neutralino, and hence is expected
to aid in the measurement of these two masses. Indeed, a much more accurate
measurement was obtained using information from this observable, as shown in
eq. (3.38).
113
Chapter 4: Conclusions
4.1 Summary
We know from astrophysical observations that dark matter exists in abundance
throughout the known universe; we also know that to date it has stubbornly eluded
non- gravitational human experiment and observation. However, there is good rea-
son to suspect that the particle or particles that make up this dark matter are
described in electro-weak energy scale extensions to the Standard Model and that
we are on the cusp of discovering these weakly-interacting massive particles. While
there are potentially many means of discovering these presumptive dark matter par-
ticles, the one upon which we focus our interest here is that via direct production
at man-made particle colliders. For the sake of argument, we assume that the dis-
covery of these invisible particles has already been made and fixed our attention on
determining various properties of these dark matter candidates. Naturally, there
are many quantities that characterize the various particles that at a fundamental
level comprise the natural world; these dark matter candidates would be no differ-
ent. Among their measurable properties would be such quantities as mass, spin,
and couplings to various other particles, including both those already adequately
described by the Standard Model and those that may lie beyond the scope of our
114
current knowledge. The investigations above were undertaken primarily in order to
develop and expand data analysis techniques that will consistently and accurately
reveal the nature of these beguiling particles once they have been discovered. Focus
was placed on using novel kinematic distributions, i.e. the energy (or energy sum)
and MT2, in order to uncover the properties of interest. This opens up the possi-
bility that there are further kinematic distributions that have been overlooked or
underused, which could allow us to glean better knowledge from the collision data
available to use currently and in the future.
I first focused on determining broad, model-level characteristics of dark matter-
like invisible particles. Typically, there is a stabilization symmetry underlying the
extensions to the Standard Model that provide the framework from which these
invisible particles arise that protects them from decaying into standard model par-
ticles. Naturally, knowing this property would immensely help the larger particle
physics community in determining where to turn their attention in developing new
physics models and in looking for new physics in the data. It was demonstrated that
it easily possible to determine this symmetry, with only a few modest assumptions.
I next turned my attention to the determination of the masses of an invisible
new physics state and its parent. I demonstrated that the use of energy distributions,
and in particular, the region close to their peaks, is an effective way to measure the
masses of new physics particles involved in a single-step multi-body decay. Taking
the example of gluino production and decay in a R-parity conserving supersymmetric
model, we have found that using only visible decay products of the gluino decay
g˜ → bbχ, it is possible to measure with good accuracy the masses of both the
115
gluino and neutralino, with the best results being obtained when both the energy
and the invariant mass distributions of the pairs of b quarks are used. Rather
strikingly, the mass measurement technique that we discussed does not actually use
information about the missing momentum, except only for event selection purposes.
The example that we considered also required a proper removal of the effect from
pairs of b quarks not coming form the same gluino. To this end, we have shown that
the event mixing technique is especially well-suited.
We anticipate that the general methodology in Chapter 3 can be applied to
mass measurements of other processes and emphasize that the technique to reduce
the combinotorial background is a tool that every phenomenologist should have in
their repertoire.
116
Appendix A: Event Mixing: Signal and background
We begin with the total number of pairings of b-jets (denoted by NNC) and
with the data sample consisting of ns signal events and nb background events. Re-
membering the fact that there are six possible pairings out of four b-jets in each
event, we have
NNC = 2ns + 2nb + 4ns + 4nb (A.1)
where the last two terms represent the total number of wrong pairings of b-jets
denoted by
NWC = 4(ns + nb) .
As mentioned before, our unawareness of which are signal and which are back-
ground events precludes us from having the event mixing (symbolized by ⊗) only
between signal events or background events. Therefore, if we perform the event
mixing procedure on the total (ns + nb) events, we then have signal-signal mixing,
signal-background mixing, and background-background mixing. Since there are four
b-jets in each event, a single event mixing enables us to have up to 4 = 16 b-jet pair-
ings. Of course, for practical purposes, one could use a subset of these 16 pairs,
say m pairs. Now that m is given by a common prefactor for every single event
mixing, we then need to know the number of event mixings. The total number of
117
signal-signal mixing is evaluated by the two-combination in the binomial coefficients:
ns ⊗ ns =
(ns
2
)
=
ns(ns − 1)
2
, (A.2)
and so is that of background-background mixing:
nb ⊗ nb =
(nb
2
)
=
nb(nb − 1)
2
. (A.3)
Likewise, the number of signal-background mixing, i.e., interference, is expressed as
follows:
ns ⊗ nb =
(ns
1
)
×
(nb
1
)
= nsnb. (A.4)
Suppose that we use m mixed pairings out of 16 mixed pairings in each event
mixing. Denoting by NMC the sum of eqs. (A.2), (A.3), and (A.4), we eventually use
m ·NMC b-jet pairs to estimate the distribution from the NWC wrong pairs formed
in the same event distribution.1 Since in general NWC 6= NNC , each mixed pair
should be re-weighted in making the “different events” distributions, so to match
the contribution of NWC wrong pairs. Keeping this issue of the normalization in
mind, let us first see how the wrong b-jet pairings in the signal events can be treated
by the mixed event subtraction even in presence of a small background. Since the
total number of mixed pairings is normalized toNWC , the presence of the background
affects the weight of the “different event” distributions made by just signal events.
In fact, in the “different event” distribution, once normalized so as to match the
contribution of the NWC wrong pairs, the fraction of b-jet pairs stemming from
the signal-signal mixing is given by eq. (A.2) times a rescaling factor to match the
1Note that the only thing that we know is ns + nb, neither ns nor nb separately.
118
normalization to NWC , that is
ns(ns − 1)
2
×
4(ns + nb)
ns(ns − 1)/2 + nsnb + nb(nb − 1)/2
≈ 4ns
(
1−
nb
ns
)
(A.5)
where the common prefactor m is omitted for simplicity and the approximation is
done with the assumptions of ns  nb and ns  1. The implication of this result is
that the combinatorial background (caused by the signal itself) can be almost com-
pletely eliminated with the event mixing technique thanks to the dominance of the
signal assumed throughout in this paper. In the total “different event” distribution
reweighed as to match the expected number of wrong pairing NWC we find that the
fraction of b-jet pairs from the background-background mixing is
nb(nb − 1)
2
×
4(ns + nb)
ns(ns − 1)/2 + nsnb + nb(nb − 1)/2
≈ 4nb
(nb
ns
−
1
ns
)
. (A.6)
In the same distribution we find that the fraction of b-jet pairs from the signal-
background mixing is
nsnb ×
4(ns + nb)
ns(ns − 1)/2 + nsnb + nb(nb − 1)/2
≈ 8nb
(
1−
nb
ns
+
1
ns
)
. (A.7)
Comparing eq. (A.6) and eq. (A.7), we conclude that the effect of “interfer-
ence” is, in general, more important than the contribution from the background-
background mixing, as also argued in the main text. Besides corroborating the
argument given in the main text, these equations quantify more precisely the effect
of background, which becomes more important when the mass measurement strat-
egy described in this paper is applied to situations where S/B is less favorable than
that in our example process.
119
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