ARCHMM!
Exclusive Cone Jet Algorithms
MASSACHUSETTS INSTITUTE
OF rECHNOLOLGY
for High Energy Particle Colliders
by
AUG 10 2015
LIBRARIES
Thomas Frederick Wilkason, Jr.
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
Bachelor of Science in Physics
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2015
@
Massachusetts Institute of Technology 2015. All rights reserved.
A uthor ............................
Signature redacted
Department of Physics
May 7, 2015
C ertified by ..........................
Signature redacted
6'
Jesse Thaler
Assistant Professor of Physics
Thesis Supervisor
Signature redacted
I
-
Accepted by........
'Nergis Mavalvala
Senior Thesis Coordinator, Department of Physics
2
Exclusive Cone Jet Algorithms
for High Energy Particle Colliders
by
Thomas Frederick Wilkason, Jr.
Submitted to the Department of Physics
on May 7, 2015, in partial fulfillment of the
requirements for the degree of
Bachelor of Science in Physics
Abstract
In this thesis, I develop an exclusive cone jet algorithm based on the principles of jet
substructure and demonstrate its use for physics analyses at the Large Hadron Collider. Based on the event shape N-jettiness, this algorithm, called "XCone," partitions
the event into a fixed number of conical jets of size RO in the rapidity-azimuth plane.
This algorithm is designed to locate substructure independent of momentum, allowing accurate resolution of jets at both low and high energy scales. I present three
potential analyses using XCone to search for heavy resonances, Higgs bosons, and
top quarks at various momenta and show that it reconstructs these particles with
efficiencies between 60% and 80% without any additional substructure techniques,
and maintains this efficiency over a wide kinematic range. This algorithm provides
many key advantages over traditional jet algorithms that make it appealing for use
at the LHC and other high energy particle colliders.
Thesis Supervisor: Jesse Thaler
Title: Assistant Professor of Physics
3
4
Acknowledgments
First and foremost, I'd like to thank my advisor Jesse Thaler for his incredible mentorship over the past two years. Working on this and other projects under his guidance
has taught me an enormous amount about how to conduct physics research, and I
will be forever grateful for his efforts in teaching me how to become a better scientist.
I will always admire his excitement and passion for the field, and I sincerely hope
that we will have the opportunity to work more in the future.
I'd also like to thank my first advisor Christoph Paus for offering me a UROP
position in the MIT CMS group during my freshman year and for taking me to CERN
during the summer of 2013. The opportunity to learn about real particle physics
research at such an early stage in my career helped solidify my own excitement and
passion for the field.
I'd like to thank my friends in the physics department, in East Campus, and all
over MIT for making my time here so enjoyable. I'd also like to thank my family for
always supporting me in my life decisions, however strange they may seem.
This work is based on two upcoming publications. Chapter 2 is based primarily
on "XCone: N-jettiness as an Exclusive Cone Jet Algorithm" by lain Stewart, Frank
Tackmann, Jesse Thaler, Christopher Vermilion, and myself. Chapter 3 is essentially
identical to "Resolving Boosted Jets with XCone" by Jesse Thaler and myself. I would
like to thank my collaborators Jesse, lain, Frank, and Chris for allowing me to use this
material for my thesis. I would also like to thank the MIT Undergraduate Research
Opportunities Program for providing funding for this research through the Paul E.
Gray Endowed Fund.
5
6
Contents
1
13
Introduction and Background
1.1
Physics of the Large Hadron Collider . . . . . . . . . . . . . . . . . .
13
1.2
Jet Definitions and Jet Algorithms
. . . . . . . . . . . . . . . . . . .
15
19
2 Introducing the XCone Jet Algorithm
2.1
N-jettiness as a Jet Algorithm . . . . . . . . . . . . . . . . . . . . . .
20
2.2
Choice of Measure
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.2.1
The Conical Measure . . . . . . . . . . . . . . . . . . . . . . .
22
2.2.2
The Geometric Measure
. . . . . . . . . . . . . . . . . . . . .
25
2.2.3
The Conical Geometric Measure . . . . . . . . . . . . . . . . .
26
2.2.4
The XCone Default Measure . . . . . . . . . . . . . . . . . . .
27
. . . . . . . . . . . . . . . . . . . . . . .
28
2.3.1
One-Pass Minimization . . . . . . . . . . . . . . . . . . . . . .
28
2.3.2
Update Step for General Measures
. . . . . . . . . . . . . . .
29
2.3.3
Seed Axes for Minimization
. . . . . . . . . . . . . . . . . . .
31
2.3
3
Minimization of N-jettiness
35
Resolving Boosted Jets with XCone
3.1
3.2
. . . . . . . . . . . . .
36
3.1.1
N = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
3.1.2
N = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
3.1.3
N = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
Boosted Higgs Bosons and Dynamic Split/Merge . . . . . . . . . . . .
41
N =1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
Dijet Resonances and Comparison to Anti-kT
3.2.1
7
3.3
4
3.2.2
N = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
3.2.3
N = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
Boosted Top Quarks and High-Multiplicity Final States . . . . . . . .
48
3.3.1
N = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
3.3.2
N = 2 x 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
57
Conclusion
8
List of Figures
2-1
Event displays and jet regions with various N-jettiness measures for
N = 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2-2
Event displays and jet regions with XCone measures for N = 2.
24
2-3
Density plots showing alignment of global TN minimized axes and gen-
. . .
eralized kT axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-1
Single Jet kinematics of N = 1 XCone versus hardest anti-kT jet,
measured on the dijet resonance sample.
. . . . . . . . . . . . . . . .
37
. . . . .
38
3-2
Same N = 1 comparison as Fig. 3-1 with area distributions.
3-3
Dijet kinematics of N = 2 XCone versus the two hardest anti-kT jets,
measured on the dijet resonance sample.
3-4
. . . . . . . . . . . . . . . .
Comparing the third hardest jet kinematics between XCone N
anti-kT, measured on the dijet resonance sample.
=
. . . . . . . . . . .
Same N = 3 comparison as Fig. 3-4 for area distributions.
3-6
Comparing mass and area of fat jet Higgs reconstruction between
XCone N= 1 and hardest anti-kT jet.
. . . . . . . . . . . . . . . . .
41
43
with the mass window m, C (100, 150) GeV. . . . . . . . . . . . .
44
Comparing mass and area of dijet Higgs reconstruction between XCone
N
3-9
. . . . . .
40
Efficiency for N = 1 fat jet Higgs reconstruction as a function of Higgs
PT,
3-8
39
3 and
3-5
3-7
33
=
2 and the two hardest anti-kT jets . . . . . . . . . . . . . . . . .
46
Efficiency for N = 2 resolved dijet Higgs reconstruction with and without IRS tagging as a function of Higgs PT. . . . . . . . . . . ... . . . .
9
47
3-10 Comparing mass and area of three-jet top reconstruction between XCone
with N = 6 and the six hardest anti-kT jets. . . . . . . . . . . . . . .
50
3-11 Comparing boosted top signal efficiency, background efficiency, and
signal significance for XCone with N = 6 as a function of top PT.
. .
51
3-12 Comparing mass of three-jet top reconstruction between XCone with
N
=
2 x 3 and the boosted and resolved anti-kT strategies. . . . . . .
53
3-13 Comparing boosted top signal efficiency, background efficiency, and
signal significance for XCone with N = 2 x 3 as a function of top PT.
10
53
List of Tables
2.1
N-jettiness measures studied in this thesis. . . . . . . . . . . . . . . .
2.2
Fraction of jets and events found with the heuristic minimum that are
aligned with the "true minimum". . . . . . . . . . . . . . . . . . . . .
3.1
21
34
Fraction of events where the nth hardest XCone jet is aligned with the
nth hardest anti-kT jet. . . . . . . . . . . . . . . . . . . . . . . . . . .
11
41
12
Chapter 1
Introduction and Background
The high energy collisions of the Large Hadron Collider (LHC) create an environment that is well-suited for the discovery of new physics. This environment is also
dominated by the emission of jets, collimated collections of hadrons that result from
the production of quarks or gluons. As a result, our ability to correctly understand
and interpret signals of new physics relies heavily on effective and powerful tools for
analyzing jets. In this thesis, I will present the idea of an exclusive cone jet algorithm,
a new jet algorithm based on the principles of jet substructure that is designed to improve jet finding and reconstruction. I will also demonstrate the use of this algorithm
in several key LHC channels to show its advantages.
1.1
Physics of the Large Hadron Collider
The Large Hadron Collider is a 27-km particle accelerator located outside of Geneva,
Switzerland that is capable of producing proton-proton collisions at a center of mass
energy of V
= 14 TeV. The LHC is designed to probe the energy frontier of particle
physics and search for new particles both in and beyond the Standard Model. With
the discovery of the Higgs boson in 2012, the LHC has effectively confirmed the
validity of Standard Model in describing particle physics at low energies. However,
the Standard Model has a number of theoretical failings, including but not limited to
the hierarchy problem and the nature of dark matter. It is expected that new physics
13
should appear at the TeV scale to answer some of these questions, and so new results
from the LHC are expected to help explain what lies beyond the Standard Model.
Since the LHC is designed to search for incredibly rare and short-lived particles,
many of the events in the LHC are produced as a result of common, Standard Model
processes. Some of the most prevalent and important types of events in the LHC
are those containing jets. Due to the property of asymptotic freedom in the strong
interaction, these high energy collisions are capable of producing individual partons
(i.e. quarks and gluons) in the collision. However, due to color confinement, these
partons cannot be individually measured.
After the partons are produced in the
collision, they quickly begin radiating off additional quarks and gluons, creating a
showering effect known as "fragmentation." At the same time, the partons excite the
vacuum into creating quark/anti-quark pairs that then couple to the original partons
to create colorless bound states in a process called "hadronization." The combination
of these two processes leads to the creation of jets in these collisions.
Jets serve a dual purpose at the LHC, as they can result both from interesting
signal processes and from expected background processes. Partons are often the final
product of the decays from both Standard Model and beyond the Standard Model
particles, and as such, jets are a key component in the discovery of new physics.
For example, supersymmetric models often include long decay chains down into a
large multiplicity of Standard Model partons, leading to a large number of jets in
the central region of the detector [14]. Indeed, one of the key search channels for the
Higgs boson is the decay into two bottom quarks, resulting in two b-jets [23, 41. The
ability to accurately tag and resolve each of these different jets is thus required to
accurately detect and measure the properties of these new particles.
In addition, due to the fact that the LHC is a hadron collider, there are a large
number of soft interactions between the two protons that lead to a huge number
of background jets. These include the 1) underlying event due to the unscattered
partons in the proton and soft interactions between the two colliding protons that
produce a large number of low-energy jets, and 2) pile-up, which occurs due to the
large number of simultaneous proton-proton collisions that further produces a large
14
number of low-energy jets. As a result, the key to accurately resolving important,
signal jets lies in the ability to discriminate signal jets from the background jets and
the ability to mitigate these backgrounds so they have minimal effect on the final
data.
1.2
Jet Definitions and Jet Algorithms
The dual processes of fragmentation and hadronization leave the definition of jets
fundamentally ambiguous, as the evolution from a single parton into a final-state
jet cannot be fully predicted or understood using conventional methods of quantum
field theory. As a result, the analysis of jets requires the use of jet algorithms that
define the jet based on characteristic properties such as direction, momentum, and
size. These algorithms all must fulfill basic physical requirements, such as infrared
and collinear (IRC) safety to guarantee the cancellation of divergences at each order
in perturbative QCD, but take different approaches to defining and classifying jets
[31, 46]. As a result, different algorithms will produce different jets for the same
event, each with different locations, momenta, and shapes, and each with their own
key analysis advantages and disadvantages.
Although there a large number of existing jet algorithms, the most popular at the
LHC are the kT-like sequential clustering algorithms [18, 20]. These algorithms create
jets by taking each pairwise combination of particles and calculating a jet metric (dij)
and a beam metric (diB)- If dij <diB, then the two particles are recombined according
to a pre-defined recombination scheme. If diB < dij, then the ith particle is declared
a separate jet and removed the list. This process is then iterated until either there
are no more particles (the "inclusive" variant), or until the predefined number of jets
has been reached (the "exclusive" variant) [21, 33].
The original use of this metric was the kT algorithm, which clusters the particles
from soft to hard, mimicking the original showering process by which the particles
were created [21, 33]. Another variant is the Cambridge-Aachen algorithm, which
clusters particles only based on their relative geometry with no regard to momentum
15
[29, 57, 56].
The above two approaches have the advantage of creating jets that
retain characteristic information about the distribution of energy within the jet, but
have the disadvantage that they are oddly shaped and difficult to calibrate in the
experiment. The most commonly used algorithm at the LHC is the anti-kT algorithm,
which clusters particles from hard to soft 118]. This opposes the original intuition of
the showering process, but it was found that this algorithm accurately finds the hard
centers of jets with ease. Further, for well-separated jets, the anti-kT algorithm creates
nearly conical boundaries, making jets easier to calibrate in the detector and calculate
using theoretical tools.
The anti-kT algorithm, while useful for finding large jets at the LHC, has certain
limitations that make some LHC analyses more difficult. In order to maintain sensible
results with this algorithm, jets should be found only using the inclusive version [181,
which means that the same type of event can lead to a different number of jets.
Also, if two jets are closer than the characteristic radius, the anti-kT algorithm will
merge them into a single jet. Further, because the particles are clustered from hard
to soft, much of the structure inside the jet, which can often give interesting and
useful information, is lost. This becomes a significant problem in the highly boosted
regime (i.e. for large jet PT), as many of the interesting decay products become highly
collimated and lost inside the same anti-kT jet. As a result, jets resulting from heavy
particles such as W/Z bosons and top quarks, which typically have a multi-prong
structure inside the jet, become virtually indistinguishable from background QCD
jets [25, 27, 2].
This problem has led to the development of the field of jet substructure. Jet substructure is a recent development in jet physics that utilizes the internal information
of the jet, such as the relative distribution of energy, in order to find out more about
the event. Substructure techniques, such as jet groomers [38, 341, as well as substructure observables, such as N-subjettiness [53, 54], both modify the jets produced from
anti-kT and other algorithms to find interesting structure within the jet to improve
mass resolution and background discrimination.
These techniques have also been
useful in improving jet resolution in the face of the effects of pile-up and the under16
lying event [50, 171. However, all analyses that utilize these techniques first require
jet finding with a standard algorithm and then performing additional techniques to
extract this information.
This motivates the development of a new jet algorithm derived from the principles
of jet substructure. By treating each of the important decay products of the original
particle as separate, independent jets that can be found independent of
PT
scale, the
original particle can be reconstructed more accurately. Further, the principles behind
substructure techniques, such as recoil insensitivity and background supression, can
be built into the algorithm itself so that the jets remain insensitive to effects from
pileup and other soft interactions. In this way, similar types of events can be analyzed as a group, which gives a stronger means of identifying events with new and
interesting physics. This also allows identification of jets at different energy scales,
reducing the necessity of tuning jet radii for different energy scales. In particular,
these algorithms allow improved resolution on the substructure of the jet to capture
more information about all the separate decay products contained in the jet, resulting in a more accurate measurement. These algorithms would thus simplify analyses
by accurately reconstructing the original particles without requiring any additional
machinery.
The primary novelty of this thesis is the development of an exclusive cone jet
algorithm based on the principles described above.
The exclusive cone algorithm
has a number of novel features that give it great power over existing algorithms. In
channels where the number of jets is known in advance, such as Higgs -+ bb, the
exclusive cone algorithm allows the expected number jets to be found regardless of
the
PT
scale of the event. As a result, even for large boosts, this algorithm can locate
the hard decay products of the event, where they would otherwise be lost in an antikT
algorithm. Further, these jets remain conical regardless of their separation, and
dynamically partition via Voronoi tessellation at the scale where the jets begin to
overlap. This allows smooth interpolation between the highly resolved regime at low
PT
and the highly boosted regime at high
PT,
where currently different algorithms
and different techniques have to be used [24, 28]. It also has the flexibility to be used
17
in cases where the number of jets in the final state differs from the expected value,
such as in the case of initial state radiation or final state radiation.
The remainder of this thesis is organized as follows. In chapter 2, we present a
concrete implementation of an exclusive cone jet algorithm, called "XCone", based on
the principle of minimization of the N-jettiness event shape [52J. We start by briefly
reviewing the N-jettiness shape and its use as a jet finding technique. We will discuss
the different possible measures that can be used to define N-jettiness, and outline a
new measure that utilizes insights from previous uses of N-jettiness. We will then
discuss the implementation of the minimization routine and how we can approximate
the minimum of N-jettiness with a sequential clustering algorithm.
In Chapter 3, we outline three potential analyses for searches at the LHC made
possible with the XCone algorithm. We will first study the dijet decays of heavy
bosons to show XCone's similarity to standard algorithms in the case of resolved
jets. We will then show how XCone's unique properties allows jets to be accurately
resolved both at low PT and high PT. We will demonstrate this both with a Higgs -+ bb
decay and a boosted top pair to demonstrate XCone's usage both in low and high
multiplicity final states. These three analyses will demonstrate how XCone can be
successfully used in a wide variety of analyses across different kinematic regimes with
high performance.
18
Chapter 2
Introducing the XCone Jet Algorithm
In this chapter, we present a new jet algorithm based on the event shape N-jettiness
[521, using insights from the jet shape N-subjettiness [53, 54]. The key novelty is that
N-jettiness defines an exclusive cone jet algorithm (or "XCone"). Like the exclusive
kT algorithm
[211, the XCone algorithm returns a fixed number of jets, relevant for
physics applications where the number of jets is known in advance. Like anti-kT jets
[181, XCone jets are nearly conical for well-separated partons, such that they have
a fixed active jet area [17, 19]. Unlike previous jet algorithms, XCone can smoothly
interpolate between widely-separated conical jets and merged jets with substructure
[5, 8, 7, 6], a possibility that will be explored in Chapter 3.
As we will see, there is considerable flexibility in how one defines N-(sub)jettiness.
Because N-jettiness and N-subjettiness had different original intended applications,
they ended up having different default definitions. Here, as the XCone default, we
propose a "conical geometric" measure that incorporates lessons from previous work
[52, 53]. This measure is based on the dot product between particles and light-like
axes as in Ref. [52] but incorporates an angular exponent
#
as in Ref. [54], as well as
a beam exponent -y for additional flexibility. Crucially for the purposes of jet finding
at the LHC, this measure yields (nearly) conical jets over a wide rapidity range, and
the user can choose the desired jet radius RO.
For the physics applications, we propose a default setting of / = 2 and -y
=
1,
which acts similarly to existing cone algorithms (see e.g. [51, 11, 32, 47]) in that
19
the resulting jet regions are (approximately) stable cones where the jet momentum
and the jet axis align. The key difference with algorithms like SISCone [471 is that
XCone does not require a split/merge step.
In particular, typical inclusive cone
algorithms have an overlap parameter which determines whether two abutting stable
cones should be joined or remain separate. However, XCone has the advantage that
the split/merge decision is determined dynamically through N-jettiness minimization.
In the following chapter, we will show examples of quasi-boosted kinematics that
capitalize on this exclusive approach to cone jet finding.
Our practical XCone implementation will use recursive clustering algorithms [21,
33, 29, 57, 56] to approximate N-jettiness minima.
Roughly speaking, we run a
generalized kT clustering algorithm to determine the jet axes, and then use N-jettiness
to define the jet regions. Separating jet axes finding from jet region finding appeared
previously in the context of recoil-free jets 140, 421, where a fixed radius cone was
centered on winner-take-all axes
[10,
40, 45j or broadening axes [54, 40j. XCone
allows us to extend this strategy to N jet events, with / = 1 yielding recoil-free jets
and
#
= 2 yielding traditional cones where the jet axes and jet momenta are (nearly)
aligned.
The remainder of this chapter is organized as follows. In Sec. 2.1, we review how to
define an exclusive jet algorithm via minimizing N-jettiness. We then discuss a variety
of N-jettiness measures in Sec. 2.2, including the conical geometric measure that is the
basis for XCone. In Sec. 2.3, we discuss some details of our XCone implementation,
most especially the choice of seed axes for finding a (local) N-jettiness minimum. The
XCone algorithm will be available through the NSUBJETTINESS FAsTJET CONTRIB
[20, 1].
2.1
N-jettiness as a Jet Algorithm
Given a set of normalized light-like axes nA -
{ 1, nA},
a generic definition of N-
jettiness is
TN
min Ipjet (p,i
), . , Pjet (Pi, nN), Pbeam(Pi),
20
(2.1)
Name
Conical [54|
General Conical
piARiA
pTi/A R" f (pi)
Geometric [36]
Modified Geometric
nA Pi
nA P
pjet (pi, nA)
Conical Geometric
XCone Default (/ = 2, y
Recoil-Free Default
(#
PTi
_
(2osy4Y
=
2
Pbeam(Pi)
PTiRO
pTiRo f(pi)
mTR min{eyi, e yi}
mTiR'(2 cosh yi)-
A-Pi
fTA PTi
2nA P/nTA
1)
1, y = 1)
PTi
(2coshyi)
-
0
PTiR
/2nA APiP /nA PriRo
Table 2.1: N-jettiness measures studied in this thesis. The conical geometric measure
with
#
= 2 and -y = 1 is the suggested XCone default, giving stable cone jets (like the
conical measure) through dot-product distances (like the geometric measure). The
recoil-free variant with # = 1 centers the jet around its hardest cluster, making the
jet regions less sensitive to soft contamination.
where the sum runs over particles i in the event with four-vector pi, Pjet (pi, nA) is a
distance measure to the A-th jet direction, and Pbeam(Pi) is a distance measure to the
beam.
For a given form of Pjet and Pbeam, the minimum inside of FN partitions the
event into N jet regions and one unclustered beam region. To use N-jettiness as a
jet algorithm, one minimizes FN over all possible light-like axes directions:
TN
Min
n1,n2,---,.N
TN.
(2.2)
The locations of the axes at the minimum define the centers of the jet regions, and
we discuss methods to approximate the rN minimum in Sec. 2.3.
For any choice of measure, N-jettiness minimization defines an exclusive jet algorithm.
In particular, rN always identifies N jet regions (and one beam region),
regardless of how close the axes nA might be to each other. When the axes are well
separated, the boundary of the jet regions is determined through competition between
Pjet and Pbeam. When the axes are close together, the jet regions are determined by
the competition between different Pjet.
To go from an exclusive jet algorithm to an exclusive cone jet algorithm (i.e. XCone),
one wants the jet boundaries to be approximately circles in the rapidity-azimuth
plane, which requires an appropriate choice of jet and beam measures.
21
In Sec. 2.2
we study a variety of measures summarized in Table 2.1, including a general conical
measure (see Eq. (2.4)) that yields exact cones for widely separated jets.
In addition to partitioning the event into regions,
TN
is a quality criteria that
measures how well an event is characterized by N jets. For narrow jets (i.e. small
effective jet radius), rN is typically dominated by the beam measure. Thus, for LHC
applications, one typically wants
Pbeam(pi)
to be proportional to
PTi
(the transverse
momentum of particle i) such that N-jettiness minimization results in the least unclustered
2.2
PT.
Choice of Measure
Every choice of jet and beam measure defines some kind of N-jettiness jet algorithm.
We now review previous measures in the literature en route to explaining the logic behind the new XCone default measure. Example jet regions found from these measures
are shown in Figs. 2-1 and 2-2.
2.2.1
The Conical Measure
The first conical N-jettiness measure was proposed in Ref. [54]:1
Conical Measure
Pbe(A)
PTi/ROA7
(2.3)
,
Pbeam Pi) =pTi R0
where RO is the characteristic jet radius and ,3 is an angular weighting exponent. The
parameter RO acts just like the radius in a cone algorithm, since particle i can only
be clustered into jet A if pjet(pi, nA) <
Pbeam(Pi),
which is equivalent to ARiA < Ro.
Thus, the measure in Eq. (2.3) yields perfect conical jets in the rapidity/azimuth
plane, unless two jet axes are closer than RO, in which case the jet regions are determined by Voronoi partitioning (i.e. nearest neighbor).
This yields "clover jet"
configurations as shown in Fig. 2-la.
'Strictly speaking, the measure in Ref. [54] has an extra rapidity cut and a slightly different
normalization.
22
Conical Measure (B
=
6
Geometric Measure
2-
2
0
-1
1
2
N=6
RO = 0.5
5
4
-2
INNINIL-
6
4
-3
-
2)
N=6
RO =0.5
5
- --
-
-INSEN- -
3
-3
-2
-1
0
1
2
3
y
(b)
(a)
XCone Default (8
6
=
Recoil-free Default (/8 = 1)
2)
6-
N=6
RO =0.5
5
N=6
RO =0.5
5
4
4
3:
2
2
1
1
0
-3
-2
-1
0
1
2
3
-3
-2
-1
0
y
1
2
3
y
(d)
(c)
Figure 2-1: Jet regions found with various N-jettiness measures. This is a boosted
tt event from the Boost 2010 event sample [51, and every measure has N = 6 and
Ro = 0.5. (a) Conical measure with # = 2. (b) Geometric measure. (c) The XCone
default, conical geometric measure with # = 2. (d) The recoil-free default, conical
geometric measure with / = 1. Both the conical and conical geometric measures yield
(approximately) circular jets, with the overlap region partitioned by nearest neighbor.
23
..
......
...
....
....
......
..
Recoil-free Default (0 = 1)
XCone Default (#B = 2)
6
6-
N=2
N=2
Ro =1.0
Ro =1.0
W
4
4
*2
S3
2
w
1
-3 -_
3'
1....
1
-1 '-
3
2
-2
0
-1
1
2
3
y
y
(b)
(a)
Figure 2-2: Same as Figs. 2-1c and 2-1d, but for N = 2 and Ro = 1.0. Here we see
that the XCone default (0 = 2, left) yields jets roughly centered along the total jet
momentum while the recoil-free default (# = 1, right) yields jets centered along the
hardest cluster within the jet. Note that the jets are elongated in azimuth compared
to rapidity, which is consequence of the trigonometric functions in Eq. (2.8).
For small R0 , TN is dominated by the beam measure, which is just the unclustered
in an event (multiplied by Rg). Thus, this measure typically finds the N hardest
jets in an event. By adjusting the exponent 0, the jet axis can be varied to point
PT
along the jet direction (0
=
2) or along the hardest cluster inside a jet (0 = 1) (see
also Refs. [54, 40, 42]).
Naively, the conical measure would seem to be the only measure that would lead
to conical jets. After all, any change to the measure should change the style of event
partitioning since it would affect the competition between Pjet and Pbeam. One can
maintain conical jets, however, if one deforms Eq. (2.3) via
Pjet (pi,nA)=-pri ARf f(pi),
iA
General Conical Measure
(2.4)
Pbeam(Pi) = PTiR'f (Pi),
where f(pi) is any function of the particle four-momentum. This still returns conical
jets since the factor of f(pi) drops out when comparing Pjet to Pbeam, and overlapping
jets still have Voronoi partitioning since f(pi) also drops out when comparing two
24
. ....
. ......
--
-
......
...
..
different pjet. Of course, the f(pi) factor does play a role in determining the minimum
in Eq. (2.2), so the final jets will be different depending on the choice of f(pi). We will
exploit this f(pi) function when defining the conical geometric measure in Sec. 2.2.3.
2.2.2
The Geometric Measure
A variety of N-jettiness measures were proposed and studied in Ref.
[36].
For the
purposes of defining a cone jet algorithm, the most promising choice is the geometric
measure:
Geometric Measure
where nra,b
=
Pjet(Pi, UA)
P
Pbeam(Pi) = R min{na - pi, nb -
(2.5)
}
{1, 0, 0, +1} are the beam directions. Despite the presence of the Lorentz
dot product nA pi, this measure actually behaves quite similarly to the conical measure. To see this, note that the four-vectors for particle i and light-like axis nA can
be expressed as
(2.6)
A = {mTi cosh yi, P-Ti, mTi sinh yi},
nA
where mTi =
i
+ mi, and
nA -*Pi
nTA Pri
{1,
'TA,
InTAI
=-
(2.7)
tanhyA},
=1 / cosh YA. Taking the dot product, we have
'Mri
Pri
coshyiA
R2
-
cosqiA ~
-,
2
(2.8)
where the last approximation holds in the small angle limit for massless particles. 2
Thus, the pjet for the geometric measure acts similarly to the general conical measure
with /3 = 2 and f (pi) = 1/(2 cosh yi).
The presence of the nA - pi dot product is very natural from a theoretical perspective, since soft and collinear modes are usually defined in terms of their light-like
2
This dot product form was also suggested in Ref. [21] in the context of recursive clustering
algorithms, though to our knowledge it has never been used in a physics study.
25
momenta.
But because the geometric measure does not take the precise form of
Eq. (2.4), it yields football-like jets in the central region (y ~ 0, see Fig. 2-1b), which
is rather unnatural from a collider physics perspective. To yield more conical jets,
though, we just need to ensure that this same factor of f(pi) appears in the beam
measure. This motivates a modified geometric measure:
PTiR2
nA) = nA - A
i
Modified Geometric Measure
,
Pjet(pi,
2 cosh yi
piR
MriRS2
Pbem~p) =2 cosh yi
(2.9)
2
2 cosh y,
Here, the approximations are the same as for Eq. (2.8), such that 1/nrTA = cosh YA ~
cosh yi. Since this modified measure is approximately the same as Eq. (2.4) with
/ = 2 and f(pi)
2.2.3
=
1/(2 cosh yi), it yields nearly conical jet regions.3
The Conical Geometric Measure
Combining the lessons of the conical and geometric measures, we now introduce the
"conical geometric" measure.
For a specific choice of parameters, this will be the
recommended XCone default measure.
Like the conical measure, we want a measure that returns (nearly) conical jets, and
we also want a parameter 3 in the jet measure to adjust the jet axes behavior. Like
the geometric measure, we want a measure that depends on the dot products between
light-like axes and particles, since that is the natural distance to use in theoretical
calculations. For additional flexibility, we will introduce a separate -y parameter to
adjust the beam measure. These requirements lead us to
Conical Geometric Measure
Pjet(Pi, nA) =
Pbeam(A) =
PTi
(2 cosh y)7PTi
1
2n_
pA/2
nTA PTZi
_R
(2 cosh y)7-1
(2.10)
In the jet measure, we recognize the last factor in parentheses as the approximate
3
1n the next subsection, we will make the jets even more conical by changing the jet measure as
well.
26
form for RiA in Eq. (2.8), such that 3 acts just like the 0 factor in the conical measure.
We have chosen f(pi) = (2 cosh yi)l-Y, such that we maintain (nearly) conical jets.
For the beam measure, y = 1 corresponds to the original conical measure and y = 2
corresponds to the modified geometric measure.
The XCone Default Measure
2.2.4
For LHC applications, our recommended XCone default is / = 2 and y = 1:
XCone Default Measure (/
=
Pjet (Pi, nA)
2)
=
2
rA
(2.11)
nTA
Pbeam Pi) = pTi RO.
By choosing -y = 1, the beam measure is the same as the conical measure, so minimizing
TN
will tend to minimize the unclustered
PT.
By choosing
#
=
2, the jet axis
(approximately) aligns with the jet momentum, as is typical for traditional stable
cone algorithms.
Alternatively, we use
/
= 1 to handle cases where recoil-sensitivity [22, 30, 9, 41]
is an issue (such as in high pileup environments [421):
Recoil-Free Default Measure (/3
MPet (pi,
1)
Ti
PA)
=TA
(2.12)
Pbeam(Pi) - PTiRo.
Here, the jet center aligns approximately along the broadening axis of the jet [54, 40],
which is the axis that minimizes the summed transverse momentum relative to it.
This is similar to finding the "median" jet energy and nA tends to point along the
hardest cluster within a given jet.
When the number of clusters is the same as the chosen value of N, the difference
between
/
= 2 and
/
= 1 is very small (in the absence of jet contamination), analogous
to the way that the mean and median of a peaked distribution are very similar. This
is demonstrated in Figs. 2-1c and 2-1d for a boosted top event with N
a jet has substructure, the "mean"
(0
=
2) and "median"
27
(0
=
=
6. When
1) axes are offset, as
shown in Figs. 2-2a and 2-2b for a boosted top event with N = 2. These measures
will be the basis for our LHC case studies in the following chapter, where we find
that both
3
= 2 and
3
= 1 give comparable results (again, in the absence of jet
contamination).
2.3
Minimization of N-jettiness
For a given N-jettiness measure, we need to implement the minimization procedure in
Eq. (2.2). In general, the only guaranteed method to find the global minimum of
TN
is to brute force test all possible partitions of the particles into N jet regions and one
beam region. Since that is computationally prohibitive, we will rely on approximate
methods that are only guaranteed to find local minima of TN.
2.3.1
One-Pass Minimization
Our minimization routine is based on iteratively improving the axes directions nA,
starting from a set of seed axes. In order to be infrared and collinear (IRC) safe,
this procedure must be deterministic, so we always work with one-pass minimization
algorithms where a set of IRC safe seed axes are iteratively improved to a local
minimum of TN, without any stochastic elements. For the conical measure in Eq. (2.3),
Ref. [54] introduced a modification of Lloyd's method [43] that finds a local minimum
of
TN
for 1 </
< 3. We can adopt this same strategy for more general measures.
This iterative procedure has two pieces: an assignment step and an update step.
In the assignment step, particles are assigned to one of the N jet regions or to the
beam region via the rN partitioning in Eq. (2.1). The assignment step can be easily
implemented for any choice of measure since it depend directly on the competition
between the jet measures Pjet(nA, pi) and the beam measure Pbeam(P).
In the update step, the axes nA are improved to minimize the value of TN within
each jet region, keeping the jet constituents fixed. Different update steps are needed
for different measures, since there is no general procedure to find the axes nA that
28
minimize pjet (nA, pi)." Once an appropriate update step is found, the assignment and
update steps can be iterated until the axes converge to within some specified accuracy.
In Sec. 2.3.2, we give a general update step that works well for the measures studied
in this thesis.
As discussed in Ref.
[541,
these one-pass minimization procedures are quite effec-
tive for N-subjettiness, often converging to the global minimum. There are additional
complications, however, for N-jettiness.
The reason is that N-jettiness has a beam
region, and particles in the bulk of the beam region are insensitive to small changes
to the location of the jet axes nA. Even minimization routines that try to go "uphill"
to escape local minima may never find the right jet regions. Keep in mind that rN
corresponds roughly to the unclustered PT in an event (for -y = 1), so failing to find
a decent rN minimum means that one will identify too many soft jets.
Therefore, for XCone to be a practical jet algorithm, one has to find a good set
of seed axes for one-pass minimization. In Sec. 2.3.3, we show how to find such seed
axes through recursive clustering algorithms.
2.3.2
Update Step for General Measures
We now construct a general update step that converges to a local minimum of
Pjet(nA, pi) for a fixed set of jet constituents.
This approach works for a wide va-
riety of jet measures, including the XCone defaults.
To motivate our general procedure, we start with the special case of the modified
geometric measure, where finding a local minimum of Pjet is particularly straightforward. Within a given jet region, we want to find the axis nA that minimizes
ZnA
-Pi = nA *PA,
(2.13)
iEA
where
PA =
EieA pi is the total jet four-momentum. Introducing a Lagrange multi-
4
Even if one does find such a procedure, one has to check on a case-by-case basis whether this
procedure actually converges, and some pathological cases were discussed in Ref. [54].
29
plier A, the quantity
nA -PA + A(A
-- 1)
(2.14)
is minimized for
PPA ,
(2.15)
argmin nA PA
=
IP-AI
nA
such that the jet axis exactly aligns with the jet three-momentum. Thus, minimizing
the modified geometric measure is equivalent to finding N mutually stable (Voronoibounded) cones. In this same spirit, any measure of the form
Pjet(nAPi) = 2 nA - p f(pi)
(2.16)
will be minimized by
nA =
qA = I:
q
Pi f (Pi),
(2.17)
iqA
where qA is an effective four-vector for the f-rescaled jet constituents.
For both
of these cases, one-pass minimization will terminate in a finite number of assignment/update steps. 5
The conical geometric measure does not take the form of Eq. (2.16), but rather
takes the general form
pjet (nA, pi) = 2 nA -pi g(pi, nA),
(2.18)
where the jet measure has non-linear dependence on nA. This means that the jet axis
and the jet three-momentum do not in general align. For the XCone default measure
in particular, the extra factor of 1/nrTA in the jet measure means that there is an
offset between the axis and the momentum proportional to the jet mass. Thus, we
cannot directly use the stable cone finding logic to minimize pjet. Just like in Ref. [54],
5
For practical purposes, it is sometimes necessary to include an "effective mass" term by changing
2nA - A
--+ 2 nA - pi + EE to avoid potential divide by zero errors.
30
though, we can simply define an update step based on the previous nA value:
=
ne
1
,A
pi g (pi, nold).
qA-
(2.19)
i EA
As long as the dependence on nA is mild enough (roughly 1 ; 3 < 3 for the conical
geometric measure), this procedure will converge within a desired accuracy in a reasonable number of iterations. We adopt this one-pass minimization strategy for the
XCone default measures.
2.3.3
Seed Axes for Minimization
Recursive clustering algorithms are particularly effective to find seed axes for one-pass
minimization. When run in exclusive mode, a recursive clustering algorithm returns
exactly N jets which can then be interpreted as N light-like seed axes. In fact, the
axes are often so good in practice that the iterative improvement step is unnecessary.
One could even imagine a more general strategy that separates jet axis finding (here
using recursive clustering) from jet region finding (here using N-jettiness partitions).
Unlike generic cluster optimization, recursive clustering algorithms are computationally efficient, and this efficiency is inherited by our XCone implementation (at the
expense of only guaranteeing a local minimum for
TN).
6
For the conical geometric measures with -y = 1 (including the XCone defaults),
good seed axes can be found by running the generalized kT clustering algorithm with a
generalized Et recombination scheme. The generalized kT clustering measure [18, 201
is parametrized by an exponent p and a jet radius RO:
dij = min (p,p)
AR?.
S,
diB
=p 2
,
(2.20)
where p = 1 is the kT algorithm [21, 33] and p = 0 is the Cambridge/Aachen (C/A)
algorithm [29, 57, 56].
'One can further improve TN minimization by running an exclusive jet algorithm to find N + n
axes and then testing N + n choose N options to find the best minimum. This option is available
in the NSUBJETTINEsS code (though not recommended by default for reasons of speed).
31
The generalized Et recombination scheme is parametrized by an energy weighting
power 6, such that one obtains a massless recombined four-vector with
~PTi + PTj
PTr
where 6
1 is the original Et scheme, 6
is the winner-take-all scheme [10, 40,
451.
=
2 is the El scheme [21, 131, and 6
It is possible to handle the 'y
$
=
oc
1 case by
further modifications of Eq. (2.20) (such as those proposed in Ref. [12]), but we have
not attempted that for the present XCone implementation.
For finding seed axes, the recommended parameters for 0 < #3 < 2 are
1(
1
with matching radius parameter R0 . To understand this heuristic, consider starting
with a final state of N +1
particles and running one iteration of exclusive generalized
kT to find N axes. For this procedure to give good seed axes for
TN
minimization, we
want to choose the values of p and 6 that match the N-jettiness behavior as closely
as possible. Essentially, we want diB to match the beam measure Pheam, di3 to match
the jet measure Pjet, and the recombination scheme to appropriately place the merged
axis in the desired location.
We perform this heuristic analysis for the conical measure, which a bit easier
to understand than the conical geometric measure with ~y
=
1, though the same
conclusions hold. To match the conical beam measure, generalized kT with p ;> 0
already gives the right behavior, since the softest particle further than R 0 from any
other particle is merged with the beam. To match the conical jet measure, we want
di, to depend on the combination PTnRG, which is achieved for
132
(2.23)
Min. Axes for
P = 2 (One pass)
Min. Axes for
0.9
3.
(One pass)
0.9
3.
0.8
0.7
2.
P= I
0.8
0.7
2.
0.6
0.6
0.5
0.5
0.4
1.
0.4
1
0.3
*0.3
0.2
0.
0.2
0.5[
0.1
S0.31
0 0.5
4
1
1.5
2 2.5
6
3 3.5 4
6
(a)
(b)
Figure 2-3: Fraction of events where all generalized kT jets align with the axes from
global TN minimization after one-pass minimization. This is for a boosted top sample,
using the conical geometric measure with N = 6 and Ro = 0.5. (a) The XCone
default (3 = 2). (b) The recoil-free default (,=
1). Here, p and 6 parametrize
the generalized kT metric and recombination scheme, respectively. The boxes here
indicate the preferred values of p and 6 predicted by the heuristic.
To match the conical axis behavior, we have to know which axis minimizes the rN
value for a jet region consisting of two particles. Labeling the particles 1 and 2 and
simplifying to one dimension q without loss of generality, we have
TN D PT1 11 -
where
#A
OA I"
+ PT2|#2 - OA 3,
is the location of the axis. Solving dTN/dA
=
(2.24)
0 to find the location of the
minimum, we have
OA=
PTIO + Pr 2 02
(5
6
PT1 + PT2
6 1
/3-1'
(2.25)
which is exactly the generalized Et recombination scheme. This is the logic behind
the heuristic given in Eq. (2.22).
To validate this heuristic, we consider a sample of boosted top quarks from the
Boost 2010 report [5], using N = 6 and Ro = 0.5. We first brute force determine (as
best we can) the global TN minimum by performing one-pass minimization on a wide
range of seed axes. We then run the generalized kT procedure for a range of p and 6
33
(0 = 2)
No Min
With Min
(0 = 1)
No Min
With Min
Jets
0.96
0.96
Jets
0.93
0.95
Events (>4)
0.99
0.99
Events (>4)
0.97
0.97
Events (>5)
Events (6)
0.93
0.80
0.93
0.83
Events (>5)
Events (6)
0.94
0.71
0.96
0.78
(a)
(b)
Table 2.2: (a) Fraction of jets found with the heuristic minimum that are aligned
with the "true" minimum both before and after one-pass minimization, along with
the fraction of events with 4 or more, 5 or more, and all 6 jets aligned with the true
minimum, for axes found with 3 = 2 minimization. (b) Same as Table (a) with # = 1
minimization.
values and find the fraction of generalized
kT
jets (without any one-pass minimization)
that are within AR < 0.1 of the axes found from global rN minimization. The results
are shown in Fig. 2-3 for the XCone default measures. As shown by the density plots,
the heuristic choice in Eq. (2.22) is satisfied for the two
#
values presented, indicating
that this heuristic provides a simple and accurate means of locating seed axes.
The percentage of aligned jets (with and without one-pass minimization), as well
as the percentage of events with 4 or more, 5 or more, and all 6 jets aligned (with and
without one-pass minimization) are shown in Tables 2.2a and 2.2b. Even without one-
pass minimization, > 95% of the jets are aligned for the heuristic choice in Eq. (2.22)
for both 3 = 1 and 3 = 2, with a range of p and 6 values giving similar results.
This suggests that finding local
TN
minima from generalized
procedure that often results in a global TN minimum.
34
kT
seed axes is a robust
Chapter 3
Resolving Boosted Jets with XCone
In the previous chapter, we introduced a new jet algorithm called XCone that blurs
the boundary between resolved and boosted kinematics. As an exclusive cone algorith,
XCone always returns a fixed number of jets N. When jets are well-separated, XCone
yields nearly conical jet regions with radius RO. When jets are overlapping, XCone
dynamically splits the jet regions into nearest neighbor partitions.
Thus, XCone
smoothly interpolates between isolated conical jets and merged jets with substructure,
making it ideally suited for studying the boosted and quasi-boosted regimes.
In this chapter, we present three applications of XCone which are relevant for
LHC physics in and beyond the standard model. In Sec. 3.1, we study high-mass
dijet resonances with isolated final state jets, showing that XCone has nearly identical
performance to the popular anti-kT algorithm [18]. In Sec. 3.2, we study associated
Higgs boson production, showing that XCone can resolve h -÷ bb decays, even when
the bb splitting angle Rbb is less than the radius parameter RO. In Sec. 3.3, we study
the classical example of boosted top quarks, showing how XCone can simultaneously
identify jets and subjets in a high multiplicity final state. These three case studies
highlight the versatility of the XCone algorithm across a wide kinematic range.
35
3.1
Dijet Resonances and Comparison to Anti-kT
For our first case study, we compare the performance of XCone to anti-kT in the
resolved regime of well-separated jets. Inclusive jet algorithms like anti-kT identify a
variable number of jets above some PT threshold, which is useful for classifying events
into different jet multiplicity bins [18, 21, 331. Exclusive jet algorithms like XCone
always return a fixed number of jets N, which is useful if the number of desired jets
is known in advance [21, 33]. For widely separated cone jets, however, the distinction
between inclusive and exclusive cone jet algorithms is rather mild, since for typical
Ro values, exclusive cone jet algorithm will just return the N hardest jets from an
inclusive cone jet algorithm. Since anti-kT acts like an idealized cone algorithm for
well-separated jets [18], XCone jets should be quite similar to the hardest N anti-kT
jets. When we study overlapping jets in Secs. 3.2 and 3.3, the inclusive/exclusive
distinction will become quite important.
A good setting to study the resolved regime is a heavy resonance decay to dijets,
where the two resulting jets are back-to-back and isolated. Here, we consider the
scenario
pp
-
Z' -
(3.1)
qq,
where Z' is a heavy boson with mass mz, and q is a u, d, or s quark. We start
with N = 1 and show that XCone typically matches the hardest anti-kT jet, up to
an expected two-fold ambiguity when the jets are nearly degenerate in PT. Going to
N = 2, both XCone and anti-kT can successfully reconstruct the dijet resonance peak.
Even at N = 3, the found jets are quite similar for typical choices of jet parameters,
though XCone will identify final state jets with substructure.
Overall, XCone has
comparable performance to anti-kT in this basic jet reconstruction scenario.
In the following study, we use Pythia 8.176 [49, 48] to simulate Z' events at the
14 TeV LHC. We take mz, = 1 TeV and assume equal couplings to the three light
quarks. All of the final-state particles (except neutrinos) with Ir/ < 3.0 are considered
for analysis. Anti-kT jets are found using FASTJET 3.0.4 [201. XCone jets are found
36
Single Jet p1
0.1 mz,=1TeV
-13=2
9
-1=10.08131
U0.06-
ak
R= 0.5
--
2
~ -1=
-~5
R= 0.5
0
j2
R= 0.5
51
1.5-
--
U.W4
C
Single Jet A pT vs. A <
~3
-P=2
-P =22
9
0.04
Single Jet pTDiff.
1
g 10-31
0.02
0.5
200
400
600
800
1000
-1
-50
p1 (GeV)
0
PT
100
50
-
(a)
PT, XCone
150
200
-0
(GeV)
(b)
0
0
p
!
-
0
6.
pT,XCone
(GeV)
(c)
Figure 3-1: Single jet kinematics of N = 1 XCone versus the hardest anti-kT jet,
measured on the dijet resonance sample. Shown are both the XCone default (0 = 2)
as well as the recoil-free variant (/ = 1). (a) Single jet PT spectrum. (b) Jet PT
difference between XCone and anti-kT jet, showing that anti-kT jets are slightly harder
on average. (c) Jet PT difference versus azimuth difference, showing the expected two-
fold
#
++ #$+ 7r ambiguity for dijets of comparable PT.
using the NSUBJETTINESS 2.2.01 contrib [53, 54, 1J , using the XCone default measure
with
#
3.1.1
=
2 and 3 = 1. For all algorithms, the jet radius parameter is Ro = 0.5.
N = 1
For N = 1, the XCone jet will tend to align with the hardest anti-kT jet in the
event. The reason is that the XCone measure in Eq. (2.10) penalizes unclustered PT
by design. In Fig. 3-la, we see that anti-kT and XCone yield nearly identical single
jet PT spectra, with the expected structure at mz'/2 from a dijet resonance decay. In
Fig. 3-1b, we compare the found jet PT on an event-by-event basis, and find a sharp
peak at APT
= panti-kr
_
Cone = 0.
On a logarithmic scale, one can see a small tail
extending to 9(50 GeV) for / = 2, with larger deviations possible in the / = 1 case.
As shown in Fig. 3-1c, there is a two-fold
#
ambiguity in the found jets, as expected
from dijet events where both jets have similar
PT
values. Note that the box sizes are
logarithmic in bin counts, and in the majority of cases, XCone and anti-kT find very
similar jet regions.
'At the time of this thesis, this version of
NSUBJETTINESS is not yet
be available by the time this work is officially published.
37
publicly available. It will
Single Jet Area Diff.
Single Jet Area
=2
-@=2
20.35
0.3
10
=1
-
00.25-
T
R= 0.5
0.2
> 1O
0.15
0.05-
..
...I
-1~~~~
84 0'5 06. 0.7 0.8 0.9
1
-0.5-0.40.3-0.2-0.1
1.1 1.2
Area
O 0.10.20.30.40.5
As - Axcone
(b)
(a)
Figure 3-2: Same N = 1 comparison as Fig. 3-1. (a) Single jet areas, showing the
expected peak at irR2. (b) Jet area difference, showing that anti-kT jets occasionally
have a higher area, explaining the PT asymmetry seen in Fig. 3-1b.
It is interesting that the APT distribution is not symmetric, such that anti-kT
jets tend to have a larger PT than XCone jets. In Fig. 3-2a, we plot the active jet
area [17, 19],2 which is quite similar between the two algorithms and peaked at the
expected value of 7rR2. On an event-by-event basis, though, there is a population of
anti-kT jets that are systematically larger than XCone jets, as shown in Fig. 3-2b.
This occurs because anti-kT clustering can yield jets that extend beyond the conical
boundary
[181. The APT asymmetry then arises because these slightly bigger anti-kT
jets contain more particles.
Comparing the performance of different / values, / = 2 jets are more similar to
anti-kT jets since both methods align the jet axis with the jet momentum. The 3 = 1
jets are slightly softer, since they do not recoil away from the hard jet center to absorb
soft radiation. In the absence of pileup, however, the
#
2 and /
1 performance
is quite similar on single jet reconstruction.
3.1.2
N = 2
For dijet resonance reconstruction, N = 2 is the most natural choice for running an
exclusive cone jet algorithm. As shown in Figs. 3-3a and 3-3b, both anti-kT and XCone
2
We use the built-in FASTJET area determination routines using active ghosts. For the general
conical measure introduced in Ref. [53, 541, the active jet area can be determined analytically, though
this is not as straightforward for the conical geometric measure used here.
38
.. ....
. ........
Dijet Inv. Mass
2
0.16 0.14
U0.12
2-Jet Mass Diff.
-@=2
mz,=1 TeV
1
-1=
--ak
R=.5
=0.515
T.
U0.1 R=0.5
R
10-2-
0.08
90.06
10-
0.04
0.02
104
-300 -200 -100
200 400 600 800 100012001400
m,
(GeV)
0
100
200
300
Ma - mxcone (GeV)
(b)
(a)
Figure 3-3: Dijet kinematics of N = 2 XCone versus the two hardest anti-kT jets,
measured on the dijet resonance sample. (a) Dijet mass, showing the expected peak
1 TeV. (b) Dijet mass difference between XCone and anti-kT jet, showing
at mz,
comparable reconstruction.
give a good reconstruction of the resonance peak, and they largely agree on the
in
value on an event-by-event basis, without much of asymmetry in the m.anti-kT r
ne
__XCone
3
distribution. XCone can therefore act as a replacement for anti-kT for dijet resonance
reconstruction, with comparable performance. 3
3.1.3
N = 3
Thus far, XCone and anti-kT have exhibited very similar behavior, but differences
start to appear when considering N
=
3. There are two main ways to achieve three
jet configurations: either there is sufficient initial state radiation (ISR) to form an
additional widely-separated jet, or there is sufficient final state radiation (FSR) to
give one of primary jets some two-prong substructure. In the ISR case, anti-kT and
XCone should still give very similar results since the jets are non-overlapping. In the
FSR case, we expect XCone will often identify two separate prongs inside a fat clover
jet, whereas anti-kT can only identify FSR if it is further away than RO from the hard
jet core.
From the PT spectra in Fig. 3-4a, we see that the third jet is often softer in the
anti-kT case, as expected if anti-kT tends to identify ISR jets. This is highlighted
3
1t is known that the RO = 0.5 cone size is typically too small to capture all of the dijet decay
products [16]. While we did find that better performance could be obtained with somewhat larger
RO, we wanted all of the plots in this chapter to have a common cone size for ease of comparison.
39
..........
Third Jet p Diff.
Third Jet A p vs. A
.
1-
-P=2
80.16 m,=1 TeV -(3=2
U
-
.k......
R =0.5
0.1'.-0 0 R= 0.5
10-
1'
>o.1: 0
p
. -P-=2
gi2
0.5
:R
'
00 124.
.
-
Third Jet p
0 0.18
-0.08
1-
~0.06
0.04
0.02
1
'-0
...
100
200
300
400
500
-200.150-100 -50
pT (GeV)
(a)
P
0
50 100 150 200
- pT, xcone
(GeV)
(b)
-9
150100 50
--
0
pTa
50 100 150 200
-
PT,XCone
(c)
Figure 3-4: Comparing the third hardest jet kinematics between XCone N = 3 and
anti-kT, measured on the dijet resonance sample. (a) Third jet PT spectrum.(b) Third
jet PT difference between XCone and anti-kT, showing that XCone has a somewhat
harder spectrum due to its ability to identify FSR subjets. (c) Third jet PT difference versus azimuth difference, showing a population of events where the third jet
kinematics are completely different between the algorithms.
in Figs. 3-4b and 3-4c, which shows that the third anti-kT jet can have completely
uncorrelated kinematics from the third XCone jet. We can gain further insight in
Fig. 3-5a, which shows the distance between the third jet and the closest harder jet.
In the anti-kT case, the third jet is forced to be further away than RO from the jet
core, whereas in the XCone case, the third jet can go nearly to AR = 0, corresponding
to XCone finding substructure within a fat clover jet instead finding a separate ISR
jet.
The same effects are visible in the area distributions in Figs. 3-5b and 3-5c. While
the overall area distributions are not so dissimilar (particularly in the /3 = 2 case),
on an event-by-event basis, there is a population of events where the third XCone jet
has substantially smaller jet area, indicative of jet overlap. This is the flip side of the
area distributions for N = 1 in Fig. 3-2b, where anti-kT jets could grow larger in size
by incorporating a neighboring subjet. In the XCone case, that subjet is separately
identified as its own jet for N = 3.
Despite these differences, the overall jet reconstruction is still rather similar be-
tween XCone and anti-kT. In Tables 3.1b and 3.1a, we show the fraction of events for
which the nth hardest XCone jet is within RO/2 = 0.25 of the nth hardest anti-kT jet.
40
Third Jet/Closest Jet A R
R=0.5
0.2
0.18-
- -
- -
-0=2
-$=1
0.2
... akT
-0=2
n
00.15
~0.08:
~0.06
-
05
1
1.5
2
2.5
R=0.5
0.
't
1073
8.4 0.50.6 07 0.8 0.9
3
U
C4)
90 10-2
R =0.5
0.05
0.04
0.02
-1=2
10-1
-p3=1
0
.
00.147
u0.12
0 .10-
Third Jet Area Diff.
Third Jet Area
00.25
1
1.1 1.2
-0.4 -0.2
Area
AR
(a)
(b)
0
0.2 0.4 0.6 0.8 1
Aa - Axcone
(c)
Figure 3-5: Same N = 3 comparison as Fig. 3-4. (a) Angle of the third hardest jet to
the nearest harder jet, showing that XCone jets can be located as close as AR = 0
whereas anti-kT jets are forced to have AR > Ro. (b) Third jet area distribution,
showing the expected peak at 7rR2, but larger tails than in the N = 1 case in Fig. 3-2a.
(c) Third jet area difference, showing a population of XCone jets with much smaller
areas due to the lack of jet merging.
(0= 2)
AKT 1
AKT 2
AKT 3
AKT 4
XC 1
0.918
0.082
0.000
0.000
XC 2
0.084
0.909
0.002
0.001
XC 3
0.037
0.037
0.805
0.038
(#= 1)
XC 1
XC 2
XC 3
AKT
AKT
AKT
AKT
0.882
0.118
0.000
0.000
0.123
0.864
0.012
0.000
0.078
0.083
0.705
0.051
1
2
3
4
(b)
(a)
Table 3.1: Comparing XCone with N = 3 to anti-kT. Shown is the fraction of events
where the nth hardest XCone jet is within RO/2 = 0.25 of the nth hardest anti-kT
jet. (a) The # = 2 default which behaves most similarly to anti-kT. (b) The #3 1
recoil-free variant where larger differences are possible.
For
#
=
2, the three hardest jets are well aligned 80% to 90% of the time. For /3
=
1,
there are larger deviations, though often this is just because the first and second jets
are reversed in PT ordering. Thus, the choice of using anti-kT or XCone for N = 3
depends on the intended application.
3.2
Boosted Higgs Bosons and Dynamic Split/Merge
To highlight the distinct advantages of XCone, we now consider physics situations
where resolving substructure is a key element of the analysis. Because XCone always
41
identifies N jets, it is well-suited to physics applications with a fixed number of
expected (sub)jets. This is particularly interesting for cases involving jet substructure,
where traditional jet algorithms yield merged fat jets, but XCone can identify jets
and subjets simultaneously.
As one key example at the LHC [23, 41, consider associated Higgs boson production
where the Higgs decays to bottom pairs:
pp -+ HZ
bbvfl.
-
(3.2)
Apart from possible ISR, the final state consists only of two b-jets. To fight QCD
backgrounds, the PT of the Higgs boson must be reasonably large [15]. However,
in this regime, the Higgs decay products are more collimated, often resulting in jet
merging. Roughly speaking, the two b-jets will be merged into a single fat jet when
the Higgs boson is at the scale
merge =
2
mH
(33)
In order to counteract this effect, either RO can be decreased until it is small enough
to resolve two separate b-jets, or jet substructure techniques can be used [15].
This (quasi-)boosted Higgs analysis is well-suited for XCone. At minimum, N = 1
can identify a single fat jet, after which existing jet substructure techniques can be
applied. More intuitively, N = 2 can be used to identity the two hard b-jets in the
event at all PT scales. Here, we describe an analysis in each of these cases to show
the advantages of XCone.
Like the previous dijet study, we use Pythia 8.176 [49, 48] to simulate pp -+ HZ
at the V
=
14 TeV LHC. For simplicity, we force the decays Z -+ vO and H
+
bb.
In order to analyze the properties of the algorithm in different PT regimes, we place
generator-level PT cuts on the Higgs boson between 200 and 1000 GeV. We use the
same RO
=
0.5 jet radius for all analyses in this section, such that
is in the middle of our studied PT range.
42
pmerg,
- 500 GeV
0.3
00.25
-
-$=2
0.2
-
P
R.akT
0~
Jet Mass (p > 800)
Jet Mass (p > 500)
Jet Mass (p > 200)
S0.3
c
-p=2
805
*0.25-
>0.15
R 0.5
Q 0 0.2
0
.>0.15-
.>0.15
R =0.5
0.1-
50
0
200
150
100
AT
R = 0.5
0.05
50
250
- P=1
0. 1
0.051
n
2
-p=
0.25
U0.2-
c
0.05
0.3
100
250
200
150
n
0
50
100
200
150
250
mj (GeV)
mj (GeV)
mj (GeV)
(a)
(b)
(c)
Area of Jet (p > 200)
Area of Jet (p > 500)
Area of Jet (p > 800)
0.35
0.3
=
-P
-
00.25
U 0.2
R
,
.5
0.1
0.05
8. 4 0, 5 0.
07
.8 0.9
1
-p =2
r0.35-
0.3
=1
03
akT
0.25
A T
1 0.15
50.35
.1 1.2
> 0.215
R
0.5
AT
0.1
0.1
0.05
0.6 0.7 0.8 0.9
1
1.1 1.2
O.
0.5 0.6 0.7 0.8 0.9
1
1.1 1.2
Area
Area
(e)
0.5
R
50.15
0.05:
Area
(d)
-$=
00.25
> 0.2
Mo.
8.4
-$= 2
(f)
Figure 3-6: Top row: Comparing fat jet Higgs reconstruction between XCone with
N = 1 and the hardest anti-kT jet. As the Higgs PT increases from (a) 200 GeV to (b)
500 GeV to (c) 800 GeV, both methods captured the merged Higgs decay products,
yielding a growing mass peak at mH = 125 GeV. Bottom row: Comparing area of
Higgs fat jet between XCone with N = 1 and the hardest anti-kT jet. All distributions
show a large peak at A =ir(0.5) 2 , as expected, with anti-kT jets showing a slighly
larger area. This also helps explain the slight mass offset seen in the first row of
figures.
3.2.1
N = 1
Since the pioneering work in Ref. [15], the boosted Higgs channel has often been
been analyzed by finding one fat jet with a large radius parameter, and then using
substructure techniques to analyze its properties [23]. The original BDRS paper used
the C/A algorithm to find this fat jet, and XCone with N = 1 can serve a similar
purpose. 4 Here, we compare XCone to anti-kT with the same jet radius.
4
We do not explicitly compare the Higgs 1-jet channel to the BDRS method since the philosophy
of the two approaches is very different. However, one could in principle use XCone N = 1 to find a
jet and then use the Cambridge/Aachen algorithm to create a clustering sequence from which the
BDRS method could be used.
43
..........
.....
.. ..
......
.
..
..........
...
.....
............
_- ... ...........
........
.......
....
-
- :.....................
:........
.....
Higgs Eff. for 1-jet
0.9 -p=2
R= 0.5
0.8:_ak
0.1--
-
0.70.2-
C 2 0 0 500 40
500 600 700 800 900 1000
(GeV)
p
Figure 3-7: Efficiency for N = 1 fat jet Higgs reconstruction as a function of Higgs
PT, with the mass window mj E (100, 150) GeV. The efficiency grows when the PT
is above the merging scale 2mH/RO, and XCone
/
= 2 outperforms
#
= 1 in the
transition region since the former centers the jet along the Higgs momentum.
Unlike in the dijet resonance study, the difference between
/
=
1 and
/
= 2 is
rather important for quasi-boosted Higgs bosons. As described in Ref. [54],
/
minimization aligns the jet axis with the hardest cluster within a jet, whereas
/
=
1
= 2
minimization places the jet axis approximately in the direction of the jet momentum.
Roughly speaking,
/
= 1 finds the "median" jet axis direction whereas
the "mean" jet axis direction. For the boosted Higgs case, the
/
/
= 2 finds
= 1 jet is more likely
to point in the direction of one of the two b-jets, while the 3 = 2 jet is more likely to
lie in between the two b-jets. Anti-kT acts like
/
=
2 since it also aligns the jet axis
with the jet momentum.
In Figures 3-6a, 3-6b, and 3-6c we show the single jet invariant mass as the
minimum Higgs PT (at generator level) is adjusted from 200 GeV to 500 GeV to
800 GeV. With increasing PT, more of the Higgs decay products are contained inside
a single jet and the peak at mj = 125 GeV grows.
2 distributions are quite similar, whereas the
/
By eye, the anti-kT and
/
=
= 1 case has a somewhat worse
performance since the jet axis is misaligned from the Higgs boson momentum.
We quantify the Higgs reconstruction efficiency in Fig. 3-7, which shows the fraction of jets in the Higgs mass window mj E (100, 150) GeV. At very high PT values,
all algorithms have very similar performance, but 3 = 2 does better in the vicinity
of prmerge. This is because the
/
= 2 jet axis is more likely to lie in between the two
44
...'
, ---
1-1 --. 1 -
,-
41kik"-.
--
- ---
,, ..
'1-1-1-ANA11"- -,
.,-
- .,
-1-1-11 11-1-
.
. -.--
Allidiblilk-l-, ..". -- --
-
,
-. "', -, -1-1.1-
b-jets, so the jet is more likely to capture the full Higgs decay products. As expected,
anti-kT and 0 = 2 are nearly identical.
3.2.2
N = 2
For this associated Higgs scenario, the real power of XCone comes from using N
2.
In the unboosted regime, the standard analysis strategy is to find two b-jets,
reconstruct their invariant mass, and look for a peak at the known Higgs mass. In the
boosted regime with jet merging, though, algorithms like anti-kT are likely to find one
fat Higgs jet and one ISR jet elsewhere in the event, so a dijet reconstruction strategy
is no longer effective. By contrast, since XCone is an exclusive cone algorithm, it will
always identify two jets regardless of the Higgs PT. To find boosted Higgs bosons with
XCone, we can simply run with N = 2 and perform a standard resolved jet analysis.
In Figures 3-8a, 3-8b, and 3-8c we show the reconstructed dijet invariant mass
comparing XCone with anti-kT. For low Higgs PT, all algorithms find the Higgs peak
with roughly the same line shape.5 As the Higgs PT increases, the anti-kT distributions
move to higher dijet masses because of jet mergers, whereas XCone maintains good
performance regardless of PT.
We show the Higgs reconstruction efficiency in Fig. 3-9a as a function of the Higgs
PT.
Anti-kT jets start to merge around PT
significantly.
300 GeV and the Higgs efficiency drops
XCone has nearly flat efficiency as a function of Higgs PT, even as
the PT crosses beyond the
pere
scale. At higher PT values, the
performance degradation, since the
angles. The
=
#
/
#
= 2 jets see a
2 jets are susceptible to radiation at a wide
= 1 jets are able to maintain their performance since the jet axes tend
to always align with the momentum of the Higgs decay products. Overall, the XCone
reconstruction efficiency is around 65% for Ro = 0.5.
5
The low mass tail in each of the plots can be explained by neutrinos from B meson decays within
the b-jet.
45
. 1--
--
-
-
-
-
_:
-
-
Dijet Mass (pT > 200)
,
-
-
-
-
-P
- 0=2
~A
=2
.-
-
~0.2 -j=
.
1
0.15
0.1
T
R = 0.5
R =0.5
0 .1
so
100
150
200
R =0.5
0.05
100
50
250
T
0.1,
0.05-
0.0 5
150
200
50
250
100
150
250
200
mj (GeV)
mj (GeV)
m. (GeV)
(a)
(b)
(c)
Area of Jet (pT > 200)
Area of Jet (pT > 500)
Area of Jet (p T> 800)
0.2 5
00.
-ANL-
Dijet Mass (pT > 800)
0..2
00.1 5-
10
----
S0.25
0. 2
0.
-
Dijet Mass (p > 500)
vO .2 5
Q
-
2
ak
-P3=2
0.2
-P=1
0.5
R =0.5
0.1
.1
0.05
0.0 '5
0.4 0.5 0.6 0.7 0.8 0.9
1
1.1 1.2
NL-
0.25-
-ak
R = 0.5
0.1
0.05
c.4 0.5 0.6 0.7 0.8 0.9
1
1.1 1.2
8.4
0.5 0.6 0.7 0.8
0.9
1
1.1 1.2
Area
Area
Area
(d)
p= 1
90.25
0
uO.15
>0.1 5-
R
0
0.25
(e)
(f)
Figure 3-8: Top row: Comparing resolved dijet Higgs reconstruction between XCone
with N = 2 and the two hardest anti-kT jets. As the Higgs PT increases from (a)
200 GeV to (b) 500 GeV to (c) 800 GeV, anti-kT suffers from jet mergers, whereas
XCone yields a dijet Higgs peak across the entire PT spectrum. Bottom row: Bottom
row: Comparing area between XCone with N = 2 and the two hardest anti-kT jets.
At low PT, both distributions show a large peak at A =7r(0.5) 2 , as expected. As PT
increases, the XCone area falls to roughly half its original value, indicating the jets
become overlapping, while anti-kT jets remain at their original value.
46
. ...........
-
-
-
-
-
-
-
Higgs Eff. for 2-jet + ISR tag
Higgs Eff. for 2-jet
1
1
R= 0.5
0.9
R= 0.5
0.9:-
0.8-OK
0.5
0.5
-0.4
-
S0.4
0.0.14 ---- a =2b
0.2-1
0.=.
0.2--@=1
0.14--ak
03
----
akT........
0.1-..akT
203040506070809010
200 300 400 500 600 700 800 900 1000
pTmin (GeV)
pTmin
(a)
(GeV)
(b)
Figure 3-9: (a) Efficiency for N = 2 resolved dijet Higgs reconstruction as a function
of Higgs PT, with the mass window mj E (100, 150) GeV. We see a degradation in the
efficiency of the anti-kT spectrum at higher PT due to jet merging issue, while XCone
produces constant efficiency across the spectrum at around 65%. Here, # = 1 outperforms 3 = 2, since the former is less susceptible to wide-angle jet contamination.
(b) Same as Fig. 3-9a, but now allowing the Higgs to be reconstructed with either
N = 2 or N = 3 (with the minimum pairwise mass). The N = 3 method allows ISR
jets to be vetoed, improving the performance of all methods, such that XCone /3 1
and / = 2 have comparable performance.
3.2.3
N = 3
To improve the XCone performance, we have to account for ISR, which the leading
cause of misreconstruction. In the presence of hard ISR, XCone can identify an ISR
jet instead of finding one of the two b jets. To address this issue, we can explicitly
identify the ISR jet using N = 3 and find the best reconstruction among the N = 2
and N = 3 options.
We first run XCone with N = 2 and check whether the dijet mass is in the
mnj E (100, 150) GeV window. If not, we run XCone with N = 3 and apply the
Higgs mass test on the pair of jets with the smallest invariant mass, as these are
kinematically the most likely candidates to be the Higgs decay products. Allowing
two pathways for Higgs reconstruction gives improved signal efficiency, and in Fig. 39b, we see that both 3 = 1 and / = 2 now have efficiencies around 75%. Applying the
same 2- and 3-jet technique to anti-kT does improve the signal efficiency somewhat,
though XCone still has better performance for PT > 300 GeV.
47
We conclude that XCone is highly efficient in reconstructing Higgs bosons across
a range of kinematics, from the resolved to quasi-boosted to boosted regimes. This
method could be further improved by using b-tagging to help identify and veto ISR
jets. Using b-tagging is also important for mitigating backgrounds, since a potential
downside of using the combined N = 2,3 method is that background events also have
two pathways to land in the signal window. But the main take away from this study
is that XCone allows traditional resolved analyses to be extended into the boosted
regime, providing a PT-independent method for Higgs reconstruction.
3.3
Boosted Top Quarks and High-Multiplicity Final
States
Given the success of XCone in reconstructing boosted Higgs bosons, we now test
whether XCone can handle the increasingly complex final states possible at LHC
collision energies. An important process at the LHC is pair production of top quarks
with fully hadronic decays:
pp -+ tt,
t -+ Wb
-4
qq'b.
(3.4)
At low mtf, the final state consists of six resolved jets. At high mtf, the jets are
arranged into two fat jets with three-prong substructure, and a variety of substructure
techniques have been developed to tag these boosted tops [37, 55, 44, 54, 39]. Here,
we show that XCone with N = 6 can identify each of the six individual (sub)jets,
regardless of the mtf value, allowing the same analysis strategy to be effective in both
the resolved and boosted regimes. We also show a more efficient N = 2 x 3 method
where the event is first partitioned into hemispheres using N = 2 and then separated
into subjets by applying N = 3 in each hemisphere.
Our study is based on the BOOST 2010 events samples
collisions. 6
6
For the boosted top signal, we use the Herwig tE
[51
-4
for fs
= 7 TeV
hadrons samples
While the BOOST 2011 samples would be more realistic for our studies, they unfortunately
48
where the generator-level top PT ranges from 200-800 GeV in bins of 100 GeV. We
also apply XCone to the the Herwig QCD dijet background sample with the same
PT
bins. As in the boosted Higgs study, we take Ro
=
0.5. When comparing to
traditional fat jet studies, we use anti-kT jets with Ro= 1.0 as recommended in the
BOOST 2010 report [5]. For brevity, we do not include a straight N
=
2 fat jet study
for XCone, since the results are similar to those found in Sec. 3.1.
At the outset, we want to emphasize that XCone is able to handle partially overlapping jets, as expected in the boosted and quasi-boosted regimes. In the highly
boosted limit, however, the subjets are fully overlapping, so substructure methods
based on fat jets are typically more effective at signal/background separation. While
it is possible to combine XCone with jet shapes like N-subjettiness for improved performance in the highly boosted limit, we find in preliminary studies that there is no
real advantage to using N = 6 over a more traditional fat jet analysis with N = 2.
The key advantage of XCone is that it yields relatively uniform performance over a
broad
PT
range, and while specialized techniques can achieve better performance at
extreme kinematics, XCone allows resolved techniques to be applied even when jets
are overlapping.
3.3.1
N = 6
The most straightforward application of XCone for hadronic tops is using N = 6
to resolve six jets.
Like in Sec. 3.2.3, one can try to improve the performance by
explicitly identifying ISR jets using N = 7, but we find that the N = 2 x 3 method
shown below is more effective at dealing with the combinatorial complexity of this
final state. For these studies, we have not incorporated b-tagging information, though
it would be straightforward to use XCone in conjunction with recent subjet b-tagging
methods [26, 3].
After running XCone with N = 6, we want to partition the jets into two top
candidates in a way that is PT-independent. We do this by finding all 6 C 3
=
20 ways
of partitioning the jets into two three-jet clusters, and then finding the configuration
seem to be lost forever.
49
Top Area (400 < p < 500)
Top Mass (400 < pT < 500)
0.35
-P=2
50.16
-P=2
(0-3
-p3=1
(0.14
-p=1
o0.25
...
R =0.5
R = 0.5
0.2
0.08
70.15
0.06
0.1-
0.05
ak
00.12.
AT
-
0.04
-..
0.02-
100
200
30
400
500
04 05 06 07 08 0.9
1
m (GeV)
(b)
(a)
QCD Area (400 <p < 500)
QCD Mass (400 < p < 500)
-0.04=
13=
Q0. 25
-P3=2
90035
-$=2
-P=1
S0.2
0.025-
:.
D0.02-
T
R=0.5
>O.5
R=0.5
0.1-
0015-
0.05
-
0.01
1.1 1.2
Area
0.005:-
100
200
00
400
.4
500
0.5 06 07 08 09
m (GeV)
1
1.1 1.2
Area
(d)
(c)
Figure 3-10: Top row: Comparing resolved three-jet top reconstruction between
XCone with N = 6 and the six hardest anti-kT jets, in the PT E (400,500) GeV bin.
Here, top candidates are identified by minimizing the sum of the three-jet masses. (a)
Candidate top mass distributions, showing that XCone does not have as pronounced
of a high mass tail due to ISR. (b) Area of all six jets, showing a peak at (2/3)7rR2
for XCone expected of clover jet configuration compared to a peak at 7rR2 for anti-kT
expected of separated jets. Bottom row: Same for the QCD background.
. .
...
............
. ......
..
.
50
S 0.7- -0.6
R=0.5
1
F 0.8 -=
Signal Signif. (6-jets)
QCD Eff. (6-jets)
Top Eff. (6-jets)
0--10
R= 0.5
R= 0.5
0. -f=2
=1
---.
0.7
akT
O.60
..
-
7 --- akT
ak
6
o.5
0.5
5
cA 0.40.3
0.2
0.10 ,I
200
0 .4:
4 .
0.3
0.2
0-.1-.
3.
2
1-
. .
. . .
300
.
,
I , -__0_.......____
500
400
600
700
'0
0
p T, min (GeV)
(a)
Figure 3-11:
0
0
0
0
p T, mm. (GeV)
- -- ~
---
0
0
0
(b)
0
0
0
pT,min. (GeV)
(c)
Comparing boosted top and XCone performance with N = 6 as a
function of top PT for (a) signal efficiency, (b) background mistag, and (c) signal
significance gain. XCone surpasses the performance of anti-kT across the entire PT
range.
that minimizes the scalar sum of the three-jet masses.
Much like in the boosted
Higgs case, we expect that minimizing the mass is most likely to yield the correct
top candidates.
In a more sophisticated analysis, one could use a
x2 -minimization
approach to find the best top candidates, also incorporating W-mass and b-tagging
information. For an apples-to-apples comparison below, we apply the same analysis
strategy on the six hardest anti-kT jets.
In Fig. 3-10a, we show the reconstructed top jet mass in the PT C (400, 500) GeV
bin, comparing XCone and anti-kT with N = 6. The XCone distributions show a
much more resolved top peak, which is more symmetric around the top mass and
has a substantially reduced high-mass tail. In the area distributions in Fig. 3-10b,
we see that XCone jets are peaked at roughly (2/3)irR , where the factor of 2/3 is
expected since the jets are arranged in a clover configuration around the boosted top
direction. The anti-kT jets are peaked around 7rR2, as expected since anti-kT jets do
not typically overlap. Because of subjet mergers, the six found anti-kT jets are not
all associated with the top quarks, leading to large invariant mass values from ISR
jets.
The equivalent dijet background distributions are shown in Figs. 3-10c and 310d. In the absence of genuine three-prong substructure, the XCone jets tend to be
scattered throughout the event, leading to large reconstructed invariant masses and
51
.
.............
7rR' areas. The effect is even stronger in the anti-kT jets, since they avoid overlaps.
To define the top signal region, we take a top mass window of mjjj C (150,200) GeV.7
We also apply a W-tagging cut as described in the CMS analysis [27], by analyzing
each pairwise combination of the three subjects and requiring a minimum pairwise
invariant mass cut of
mjjmin >
50 GeV.
In Figs. 3-11a and 3-11b we show the efficiency and mistag rates for the top and
dijet samples as a function of (generator-level) PT. XCone has a signal efficiency of
around 60% across the entire PT scale, showing the desired scale invariance. While
anti-kT starts with the same 60% efficiency in the resolved regime, the efficiency drops
considerably with PT due to jet mergers, analogous to what was found in Sec. 3.2.2.
Both methods have around a 10% background mistag rate, which is relatively stable
as a function of PT.
The improvement in signal significance (S2 /B) is shown in Fig. 3-11c, where we see
that the performance exceeds that of anti-kT across the entire PT range, and remains
relatively flat. Note that we have not included b-tagging information nor additional
jet shape information, so background rejection factors can be much larger in practice.
From this simple study, we see that a straightforward application of XCone allows for
a PT-independent analysis strategy even for complicated final states.
3.3.2
N = 2x 3
To further improve on the performance of XCone, we can take into account the event
topology.
Even with a moderate boost, the top decay products tend to arrange
themselves into two hemispheres, a feature that is exploited, for example, in the
HEPTogTagger [441. Thus, we can use XCone in multiple stages, first dividing the
event into separate top candidate regions with N = 2 and Ro -* oc, and then finding
jets in each of those regions using N = 3 and Ro = 0.5.
There are two advantages of this N = 2 x 3 approach over the N
=
6 approach.
First, it reduces combinatorial confusion and increases computational efficiency. Sec7
Since the peak from XCone is more narrow and symmetric around the top mass, we can use a
smaller and more symmetric mass window without losing much signal efficiency.
52
QCD Mass (400 < pT < 500)
UO12-
r0.3 5
0.
akT (Bst)
- akT (Res)
3
C) 0.2
0. 2
0.
0.08-
R =0.5
0.0 5
0.02100
200
300
-AT (Bst)
akT (Res)
R = 0.5
~0.04-
0
-
-0.06
0.1 5L
400
500
100
1
200
300
m (GeV)
400
500
m (GeV)
(a)
(b)
Figure 3-12: Reconstructed mass distributions with the N = 2 x 3 strategy, for (a)
top signal events and (b) QCD background events. Here we compare the XCone
N = 2 x 3 method to two traditional methods: a boosted strategy where two anti-kT
RO = 1.0 jets have three exclusive kT subjets, and a resolved strategy where two
exclusive kT hemispheres have three anti-kT RO = 0.5 jets.
QCD Eff. (2x3-jets)
Top Eff. (2x3-jets)
- 2
R = 0.5
U
0o. 5
0.7
0.4
0..-
akT (Bst)
-- a(Res)
Signal Signif. (2x3-jets)
R =0.5
2
9
0.6' -akT
i- 0.64
0.3
0.2
0.
"200
R 0.
10
---akT (Bst)
(Res)
0.7
7-
---- akT (Bst)
6
-
akT (Res)
5-
-.
....-.
.....
.
I2R
=
=0.5
Top Mass (400 < pT < 500)
CU0.4
S0. 3
0.2:-
2-
0.1
300
400
500
600
mp
T, min
(a)
700
.....----------
200
300
400
(GeV)
500
600
700
0200
306
40
pm
T,
min (GeV)
(b)
o500
600
pTmin
700
(GeV)
(c)
Figure 3-13: Comparing boosted top and XCone performance with N = 2 x 3 as
a function of top pT for (a) signal efficiency, (b) background mistag, and (c) signal
significance gain. XCone interpolates between the traditional resolved strategy at low
PT and the traditional boosted strategy at high PT.
53
ond, it ensures that each top candidate has a chance to get three jets. Even without
ISR, the N = 6 method can yield, for example, one four-leaf top clover and one twoleaf top clover, something that is avoided with N
=
2 x 3. While it is possible to get
even higher signal efficiencies by applying N = 2 x 4 and vetoing ISR jets (analogous
to Sec. 3.2.3), such an approach tends to also increase the background mistag rate,
so we will not show N = 2 x 4 results here.
We can compare XCone to traditional top reconstruction methods in two different
ways. For a boosted strategy, we can run anti-kT with Ro = 1.0 to find two fat jets
and then run exclusive kT with N = 3 and Ro = 0.5 on the fat jet constituents
to identify subjets. We denote this as anti-kT (Bst) in the figures. For a resolved
strategy, we can run exclusive kT with N = 2 and Ro -+ oc to find hemisphere regions,
and then run anti-kT with Ro = 0.5 to find the three hardest jets in each hemisphere.
We denote this as anti-kT (Res) in the figures. As we will see, XCone N = 2 x 3
effectively interpolates between these behaviors as a function of PT, reproducing the
best performance of the traditional strategies in their respective domains.
The resulting top mass distributions are shown in Fig. 3-12a, again in the PT E
(400, 500) GeV bin. The top mass distribution for XCone N = 2 x 3 jets is similar to
the N = 6 case, continuing to maintain a peaked, symmetric shape around the top
mass. Crucially, N = 2 x 3 reduces the high-mass tail since there are more correctly
reconstructed top quarks. The anti-kT boosted strategy results in a W shelf caused
by the Ro = 1.0 jet radius not containing the full top decay products, while the antikT resolved strategy has a high-side mass tail from the inclusion of ISR jets. XCone
avoids both of these pitfalls, giving an excellent overall reconstruction.
Similarly, as shown in Fig. 3-12b, the background mass distribution for XCone
falls in between the anti-kT boosted and resolved strategies.
Like in the N = 6
case, both XCone jets and anti-kT N = 6 jets tend to be scattered throughout the
hemisphere, leading to large invariant masses. However, the additional constraint of
the hemisphere helps to control this effect, giving smaller masses on average than
N =6.
The signal efficiency and background mistag rates are given in Figs. 3-13a and
54
3-13b, again for the mjjj c (160, 240) top mass window and mjj,min > 50 GeV W
mass cut. For the signal efficiency, it is clear the XCone interpolates between the
good anti-kT resolved performance at low PT and the good anti-kT boosted performance at high PT, yielding approximately a 60% reconstruction efficiency throughout
the PT range. The improvement over anti-kT is likely due to the inclusion of soft
subjets that fall outside of the normal anti-kT clustering radius, as described in [35].
For the background mistag rate, XCone holds steady at 10%, where again, further
improvements are possible using b-tagging or substructure information.
The improvement in signal significance (S 2 /B) is shown in Fig. 3-13c, where we
see that the performance exceeds both the boosted and resolved strategy of anti-kT
across the entire PT range. Further, the performance is increasing as a function of PT,
indicating a very strong performance of this algorithm in the highly boosted regime.
We conclude that XCone gives a powerful way to extend conventional resolved analysis
strategies into the boosted regime.
55
56
Chapter 4
Conclusion
In this thesis, we have presented the concept of the exclusive cone algorithm designed
for improved analysis of jets at the Large Hadron Collider, and demonstrated its
use in several key LHC analyses. In Chapter 2, we introduced a new jet algorithm
"XCone" based on the N-jettiness event shape that is capable capable of resolving
highly boosted jets by dynamically splitting the jets into equal partitions while still
maintaining the conical boundaries. We also defined a new conical geometric measure
based on previous uses of N-jettiness that has many key theoretical and experimental
advantages. We also outlined the implementation of a simple minimization routine
that approximates stable cone finding, and discussed a straightforward and accurate
way of finding seed axes through sequential clustering algorithms.
In Chapter 3, we presented three case studies to show how XCone can be used in
a variety of LHC analyses, producing comparable or better results than conventional
methods. Remarkably, a single benchmark cone size of RO = 0.5 was able to successfully reconstruct heavy Z' resonances, boosted Higgs bosons, and boosted top quarks
to a high degree of accuracy across a wide range of jet PT. This shows that XCone is
well-equipped to handle a variety of search channels at the LHC over the kinematic
regime probed in current colliders. Although we did not test it in this thesis, we
expect that the use of XCone in conjunction with other, existing, jet substructure
techniques will allow for even greater improvement in signal reconstruction.
There are many future directions that one could take in exploring the wide param-
57
eter space that XCone offers. In particular, the freedom offered by both the choice of
measure and the axis finding and minimization routine allows future implementations
of XCone that may be helpful in vetoing ISR jets, mitigating the QCD background,
and more. Possible important extensions include a study of the role of -Y in the definition of the conical geometric measure, or a new implementation where the strategies
for finding the axes and the jet regions are completely separated. XCone can also be
extended to many other channels beyond those shown in the thesis, including complex
multi-jet searches for supersymmetry and other beyond the Standard Model physics.
At the moment, the XCone code is currently limited to the options presented in this
thesis, but these other options are still left open for future consideration.
Given both the experimental and theoretical advantages that XCone provides, as
well as the promising results of this thesis, we hope that XCone and other exclusive
cone jet algorithms will continue to be researched and implemented for physics studies at the LHC and other colliders. We look forward to potential results with this
algorithm extremely soon as the LHC begins ramping up for Run 2.
58
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