Phenomenological aspects of new
physics at high energy hadron
colliders
Andreas Papaefstathiou
Robinson College
A dissertation submitted to the University of Cambridge
for the degree of Doctor of Philosophy
April 2011
Phenomenological aspects of new physics at high
energy hadron colliders
Andreas Papaefstathiou
Abstract
This thesis contains studies of phenomenological aspects of new physics at hadron
colliders, such as the Large Hadron Collider (LHC). After a general introduction in chap-
ter 1, in chapter 2 we outline the main features of the Standard Model (SM) of particle
physics and the theoretical motivations for going beyond it. We subsequently provide
brief descriptions of a few popular models that aim to solve the issues that arise within
the SM.
In chapter 3 we describe the general Monte Carlo method for evaluating multidimen-
sional integrals and show how it can be used to construct a class of computational tools
called Monte Carlo event generators. We describe the main generic features of event
generators and how these are implemented in the HERWIG++ event generator.
By applying resummation techniques, we provide, in chapter 4, analytical calcula-
tions of two types of hadron collider observables. The first, global inclusive variables,
are observables that make use of all measured particle momenta and can provide useful
information on the scale of new physics. The second observable is the transverse energy
of the QCD initial state radiation (ET ), associated with the either Drell-Yan gauge boson
production or Higgs boson production. In both cases we provide comparisons to results
obtained from Monte Carlo event generators.
In chapter 5 we examine two well-motivated models for new physics: one of new heavy
charged vector bosons (W prime), similar to the SM W gauge bosons, and a model moti-
vated by strong dynamics electroweak symmetry breaking that contains new resonances,
leptoquarks, that couple primarily to quarks and leptons of the third generation. In the
prior model, we improve the current treatment of the W ′ by considering interference ef-
fects with the SM W and construct an event generator accurate to next-to-leading order
which we use to conduct a phenomenological analysis. For the leptoquark model, starting
from an effective Lagrangian for production and decay, we provide an implementation in
the HERWIG++ event generator and use it to form a strategy for mass reconstruction. The
thesis ends with some conclusions and suggestions for extensions of the work presented.
Further details and useful formulæ are given in the appendices.
iii
Declaration
This dissertation is the result of my own work, except where explicit reference is made
to the work of others, and has not been submitted for another qualification to this or any
other university.
The original work in chapter 4 was done in collaboration with Bryan Webber and
Jennifer Smillie and appears in [1–3]. The original work in chapter 5 was done in collab-
oration with Oluseyi Latunde-Dada, published in [4], and with Bryan Webber, Kazuki
Sakurai and Ben Gripaios, published in [5].
This thesis does not exceed the 60,000 word limit prescribed by the Degree Committee
for Physics and Chemistry.
Andreas Papaefstathiou
v
t€ înta Êànai te pˆnta kaÈ mènein oÎdèn.
– <Hrˆkleito
 å >Efèsio
‘All things move and nothing remains still.’
– Heracletus of Ephesus
********
Dedicated to my family, for their love and support.
vii
Acknowledgements
I will begin by thanking my supervisor, Professor Bryan Webber, for his guidance, en-
couragement and support throughout the past four years. He has not only been a brilliant
teacher, but also an inspiring collaborator. Moreover I am grateful for his help, patience
and time spent on preparing this thesis. I have been fortunate to have collaborated with
very energetic and inspiring people who have taught me a lot during our work together.
These include Seyi Latunde-Dada, Kazuki Sakurai, Ben Gripaios and Jennie Smillie dur-
ing my time in Cambridge, as well as Jeff Forshaw and Andy Pilkington for provoking
my interest in particle physics while at Manchester.
In addition, I’ve learned a lot through discussions with members of the Cambridge Su-
persymmetry Group, such as Ben Allanach, Andy Parker, Chris Lester and Are Raklev.
A great amount of knowledge on computational issues has also come through my partic-
ipation in the HERWIG++ phone meetings, and I would like to particularly thank Peter
Richardson and David Grellscheid for providing me with assistance whenever required. It
has been an honour and a privilege to have been part of the Cavendish HEP group and
interacted with lively, interesting and hard-working people. Moreover, as a phenomenolo-
gist, the interaction with the experimentalists has given me invaluable insight to the more
‘practical’ aspects of our field.
My research has been funded by the Science and Technology Facilities Council. I have
also received funding from the European Training Network MCnet and the Cavendish
HEP group which have allowed me to attend conferences and visit collaborators.
On a more personal note, I would like to thank my (near-)officemates, Seyi Latunde-
Dada, Marco Sampaio, Jo Gaunt, Lucian Harland-Lang, Eleni Vryonidou and Edwin
Stansfield, as well as Professor James Stirling and Steve Kom, for creating such an inspir-
ing working environment. I would also like to thank the group’s former members, Deirdre
Black and Are Raklev, who have both provided me with guidance in terms of physics, as
well as career advice.
A great amount of love, support and patience throughout the years has come from my
family, and I would like to thank them all for believing in me always. I would never have
made it to this point without them.
Finally, I will end by thanking Bryan, Lucian, Eleni and Ed for proofreading the thesis.
Any remaining errors are of course my sole responsibility.
ix
Contents
1 Introduction 1
2 The Standard Model and beyond 5
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 Gauge theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.2 Electroweak theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.3 Quantum chromodynamics . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Beyond the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.1 The need for BSM physics . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.2 Survey of BSM theories . . . . . . . . . . . . . . . . . . . . . . . . 25
3 Monte Carlo methods and event generators 31
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Monte Carlo event generators . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.1 Unweighted event generation . . . . . . . . . . . . . . . . . . . . . . 34
3.2.2 General features of event generators . . . . . . . . . . . . . . . . . . 34
3.2.3 The HERWIG++ event generator . . . . . . . . . . . . . . . . . . . . 39
3.3 Next-to-leading order matching . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3.1 MC@NLO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3.2 POWHEG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4 Effects of QCD radiation on hadron collider observables 49
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Effects of QCD radiation on global inclusive variables . . . . . . . . . . . . 52
4.2.1 Global inclusive variables . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.2 The variable sˆmin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2.3 ISR effects without invisible particle emission . . . . . . . . . . . . 54
4.2.4 Mellin transform inversion . . . . . . . . . . . . . . . . . . . . . . . 65
4.2.5 ISR effects including invisible particle emission . . . . . . . . . . . . 68
4.2.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
xi
xii CONTENTS
4.3 Resummation of ET in vector boson and Higgs boson production at hadron
colliders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.3.1 Resummation method . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.3.2 Resummed distributions . . . . . . . . . . . . . . . . . . . . . . . . 96
4.3.3 Matching to fixed order . . . . . . . . . . . . . . . . . . . . . . . . 97
4.3.4 Monte Carlo comparisons . . . . . . . . . . . . . . . . . . . . . . . 101
4.3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5 New physics searches at hadron colliders 109
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.2 NLO production of heavy charged vector bosons . . . . . . . . . . . . . . . 110
5.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.2.2 W ′ at leading order . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.2.3 W −W ′ interference . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.2.4 Extension to NLO . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2.5 Experimental bounds . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.2.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.2.7 Extraction of limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.3 Searching for third-generation composite leptoquarks . . . . . . . . . . . . 132
5.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.3.2 Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.3.3 Reconstruction strategies . . . . . . . . . . . . . . . . . . . . . . . . 143
5.3.4 (tτ)(tτ) decay mode . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.3.5 (qν)(qν) decay modes . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.3.6 (q′τ)(qν) decay modes . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.3.7 (bτ)(bτ) decay mode . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.3.8 Determination of quantum numbers . . . . . . . . . . . . . . . . . . 162
5.3.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6 Conclusions and Outlook 165
A Illustration of a Monte Carlo event 169
B Pair-production cross sections 173
C The Cabibbo-Kobayashi-Maskawa matrix 175
D Supplementary material for ET resummation 177
D.1 Relation of ET resummation to qT resummation . . . . . . . . . . . . . . . 177
D.2 Results for LHC at 7 TeV . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
CONTENTS xiii
E Supplementary material for W ′ study 181
E.1 Model discrimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
E.1.1 Likelihood ratios of probability density functions . . . . . . . . . . . 181
E.1.2 Poisson likelihood ratios . . . . . . . . . . . . . . . . . . . . . . . . 182
E.1.3 Application to a toy model . . . . . . . . . . . . . . . . . . . . . . . 183
E.2 The W ′ Drell-Yan cross section . . . . . . . . . . . . . . . . . . . . . . . . 184
F Supplementary material for leptoquark study 191
F.1 Feynman rules and diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 191
F.2 The effective Lagrangian for derivatively-coupled leptoquarks . . . . . . . . 193
F.3 (tτ)(tτ) reconstruction method . . . . . . . . . . . . . . . . . . . . . . . . 195
F.4 (q′τ)(qν) reconstruction method . . . . . . . . . . . . . . . . . . . . . . . . 196
xiv CONTENTS
Chapter 1
Introduction
The Human kind, by good fortune, has developed the ability to ponder and investigate
natural phenomena. We have been intrinsically acting as scientists for thousands of years.
However, the form of what we call ‘science’, a term originating from the Latin sciencia,
meaning knowledge, has evolved dramatically through the ages. The Greek philosopher
Aristotle, who lived in the 4th century BC, profoundly affected those who followed him
with his views on natural phenomena. His persistent beliefs included that substances
that make up the Earth (‘earth’, ‘air’, ‘water’, ‘fire’) were different than those that made
up the heavens (‘aether’) and that objects moved as long as they were being pushed.
His writings were largely qualitative and, although he had basic ideas regarding a few of
the fundamental concepts of nature, such as speed and temperature, he was lacking the
proper instruments to make quantitative statements about them. It was not until the 16th
century AD, and the ideas of Galileo Galilei, an Italian natural scientist, that things began
to change. Galileo, with the aid of the newly-invented telescope which he had improved,
made detailed astronomical observations that made it more plausible that the heavens
and the Earth were made from the same materials. He also proposed the ‘law of inertia’,
whereby objects tended to maintain their state of motion instead of preferring to be at
rest. Aristotle’s misconceptions originated from his lack of understanding of frictional
forces. Isaac Newton, unarguably one of the most important scientists of recent centuries,
effectively weaved his theories of motion and gravitation based on the groundwork laid
by scientists like Galileo.
Newton’s Universe was like clockwork: mechanical and perfectly deterministic. Space
was absolute: the scene in which the heavenly bodies and the Earth executed their eternal
motion, a rigid grid of three dimensions. Time, according to Newton, was flowing always
1
2 Chapter 1. Introduction
at the same rate (‘equably’), and ‘without regard to anything external’. These concepts
persisted until the advent of further scientific revolutions that occurred in the early 20th
century: Albert Einstein’s theory of relativity, doing away with absolute space and time,
and quantum mechanics, a theory of subatomic particles, whose results for the evolution
of a physical system were of probabilistic nature. The revolutions were either instigated
by experimental facts (e.g. Einstein’s explanation of the photoelectric effect) or guided
by the philosophy that natural laws should be ‘beautiful’ (e.g. the Dirac equation). It is
important to emphasise, however, that Newton’s theories were not discarded completely;
rather, they were shown to be specific limits of the theories that encompass them. In the
words of the mathematician David Hilbert, in a lecture delivered before the International
Congress of Mathematicians at Paris in 1900,
‘History teaches the continuity of the development of science. We know
that every age has its own problems, which the following age either solves or
casts aside as profitless and replaces by new ones.’
Old problems are viewed from different perspectives by scientists of the following gen-
erations, in a different framework of thought, possessing more powerful analytical and
experimental tools. Some are solved, some discarded, and new questions are posed.
Nowadays, the study of the fundamental nature of matter is called ‘particle physics’,
or ‘high energy physics’. Particle physicists are currently faced with a multitude of un-
solved puzzles. The quest to address them may lead to a revolution of our view of the
fundamental principles of equal magnitude as the ones that have occurred before. We
present an overview of the current understanding of the subatomic world in chapter 2.
This framework is called the ‘Standard Model’ of particle physics. We will also examine
the issues that are thought to plague this framework and outline some suggestions that
have been put forward to address some of them.
Science is based on careful observations, known as experiments. Through experiments,
we put our predictions to a test in a controlled environment, in a reproducible way.
In particle physics, the most common form of experiment is rudimentary: we ‘throw’
particles onto one another and study the scattering process. Though basic, the idea is
powerful: very detailed quantitative predictions can be made and theories can be put to
a rigorous test. The Large Hadron Collider (LHC), at the European Organisation for
Nuclear Research (CERN), near Geneva, Switzerland, involves such experiments. It will
primarily1 collide protons to protons at energies we have never examined before, about
1The LHC is also a heavy-ion collider.
314000 times the rest mass energy of the proton (∼ 1 GeV). There, we expect to at least
observe a hypothetical particle that is required for the consistency of the Standard Model,
the Higgs boson.2
 1e-06
 0.001
 1
 1000
 1e+06
 1e+09
 1e+12
 1e+15
 1e+18
M
as
s 
(G
eV
)
τ
µ
e
t
bc
s
d
u
Z
W
Figure 1.1: The Standard Model spectrum of known massive particles. The mass is shown on
the vertical axis in logarithmic scale, in units of GeV, from 10−6 GeV to 1019 GeV.
The photon, gluon and the neutrinos are not shown.
Forgetting for the moment the more concrete theoretical reasons (which we will review
in the following chapter) to expect the observation of new phenomena at the energy scales
of the LHC, we can make a plausibility argument for their existence by a simple observa-
tion. Figure 1.1 demonstrates the spectrum of known fundamental particles present in the
SM that possess mass.3 The vertical axis shows the mass in logarithmic scale, extending
from 10−6 GeV to the scale 1019 GeV, the fundamental scale of gravity, known as the
Planck scale.4 The Planck scale is the highest scale at which our current understanding
of physics makes sense. By examining Fig. 1.1, one may observe something bizarre and
slightly suspicious: many particles have been discovered with masses ranging from frac-
tions of a GeV to fractions of a TeV, and one more (the Higgs boson) is predicted below
2Or, if it is absent, we expect to observe a mechanism that explains that absence.
3The neutrinos also possess mass, albeit a small one. However, only mass differences are known and
we do not show them here.
4The Planck scale is the scale at which the strength of the gravitational interaction between particles
becomes of order one. An estimate is given by MPl = G
−1/2
N ∼ 1019 GeV, where GN is Newton’s
gravitational constant.
4 Chapter 1. Introduction
the TeV scale, but no known particles exist above that scale! The question that arises
is: why should there be such a ‘desert’ of energy (or mass) scales between the TeV scale
and the Planck scale? If we do not accept the existence of such a puzzling absence of
particles, then we can only conclude one of the following:
• We do not currently possess the correct understanding of the fundamental scale of
gravity, the Planck scale. Either there exists some specific mechanism explaining
why such a large hierarchy arises between the TeV scale and the Planck scale, or
the Planck scale is much lower than 1019 GeV.
• There exists a multitude of new particles and new interactions, waiting for potential
discovery by future experiments.
Both of these prospects are very exciting. The LHC will be instrumental in exploring the
TeV scale, perhaps revealing a whole new set of particles or phenomena that will provide
an explanation for the above observation.
This thesis focuses on hadron colliders, of which the LHC is the most ‘powerful’ ever
built. It is an extraordinary experiment, based on a collaboration which transcends bor-
ders, involving tens of thousands of researchers from more than a hundred nations. The
task that particle physicists are facing at the LHC is non-trivial. Certain ingenuity is
required if we wish to maximise the amount of physics results obtained from the experi-
ments. A solid bridge of communication between the theorists and the experimentalists
needs to be established. This is where phenomenology comes into play: it provides an
interface between theory and experiment, for example through powerful computational
tools, Monte Carlo programs, which we will be discussing in detail in chapter 3. These
tools are indispensable to both experimentalists and theorists. In chapter 4, we make
analytical phenomenological predictions of the effects of the theory of the strong force,
quantum chromodynamics (QCD), to observables that experimenters will be using, either
in their searches for new physics, or for better understanding of known SM processes.
In the final chapter, 5, we take two well-motivated new physics scenarios and provide a
phenomenological analysis to act as a guide for experimentalists in their search.
In another part of his 1900 talk in Paris, Hilbert asks the following:
‘Who of us would not be glad to lift the veil behind which the future lies
hidden; to cast a glance at the next advances of our science and at the secrets
of its development during future centuries?’
We hope that soon we will have reached such a point.
Chapter 2
The Standard Model and beyond
2.1 Introduction
In his popular science book ‘QED: The Strange Theory of Light and Matter’ [6], Richard
Feynman writes:
Therefore the possibility exists that the three W ’s and the photon are all dif-
ferent aspects of the same thing. Stephen Weinberg and Abdus Salam tried to
combine quantum electrodynamics with what’s called the ‘weak interactions’
into one quantum theory, and they did it. But if you just look at the results
they get you can see the glue, so to speak. It’s very clear that the photon and
the three W ’s are interconnected somehow, but at the present level of under-
standing, the connection is difficult to see clearly – you can still see the ‘seams’
in the theories; they have not yet been smoothed out so that the connection
becomes more beautiful and, therefore, probably more correct.
Despite Feynman’s observation, the so-called ‘Standard Model’ of particle physics (ab-
breviated as SM) has been tremendously successful at describing experimental data. In
fact, the SM can be arguably considered as the quantitatively most successful theory our
species has ever constructed, with astounding agreement between theory and experiment:
the anomalous magnetic moments of the electron and of the muon are amongst the most
precise measurements in the whole of physics.
The ‘glue’-ing that Feynman refers to is the fact that the SM is a gauge theory of a
product of gauge symmetry groups:
SU(3)c × SU(2)L × U(1)Y , (2.1)
5
6 Chapter 2. The Standard Model and beyond
where the SU(3)c describes quantum chromodynamics (QCD) and the SU(2)L×U(1)Y the
electroweak (EW) theory. The SU(3)c symmetry, known as colour, is explicitly manifest
in Nature, whereas the SU(2)L×U(1)Y symmetry is broken down to U(1)em via the Higgs
Mechanism.
The Higgs Mechanism, introduced to explain the masses of fermions and bosons in the
theory by breaking the electroweak symmetry, is currently the least understood part of
the SM and is thus a source of inspiration to many extensions to the SM. Many of these
extensions attempt to explain the so-called ‘hierarchy problem’, which can be briefly
described as the lack of explanation for the stability of the Higgs boson mass against
radiative corrections. Due to this issue and other open theoretical questions, the SM is
often believed to be an incomplete description of particle physics. Its amazing success
at describing experiments, however, ensures that it will be a subset of a ‘true’ theory,
understood in the framework of effective field theories as capturing the low-energy limit
of some more fundamental, microscopic physics [7].
In section 2.2 we give a general introduction to the SM at its current state, starting
from the general principles for writing down a relativistic quantum field theory. We will be
focusing on the phenomenological aspects of the EW theory and QCD and mathematical
details will be kept to a minimum. In section 2.3 we examine some important open
theoretical and experimental questions that suggest the need for extensions to the SM
and present a brief overview of BSM theories.
The reader is referred to Ref. [8] for further details on the fundamentals of quantum
field theory. For further details on the basic principles of QCD and particularly on QCD
phenomenology, [9,10] are recommended and for a general introduction to gauge theories
and the Standard Model [7, 11, 12].
2.2 The Standard Model
2.2.1 Gauge theories
Symmetries provide good candidates for underlying first principles in Physics. The gauge
principle is an economical guide for using local symmetry to construct renormalisable
quantum field theories. The fact that local invariance is required is motivated by the
relativistic viewpoint: each observer, at every space-time point, possesses some freedom
of convention. This of course may be considered by some as an aesthetical argument;
however, the gauge principle has been proven to produce phenomenologically successful
2.2. The Standard Model 7
field theories and accounts for quantum electrodynamics (QED) as well as EW theory
and QCD.
The gauge principle is most simply illustrated in the case of invariance under the
unitary group U(1), which yields an abelian gauge theory that describes QED. Consider
the classical Lagrangian density1 describing the interaction of a spin 1/2, charged fermion
field ψ with the gauge field Aµ:
Lem = −1
4
FµνF
µν + ψ¯(i /D −m)ψ , (2.2)
where F µν = ∂µAν − ∂νAµ is the field strength tensor and the covariant derivative is
defined by /Dψ ≡ γµDµψ ≡ γµ(∂µ + ieAµ)ψ, where m and e are respectively the mass
and electric charge of ψ and we have also defined the Feynman slash convention. The
Lagrangian is invariant under the simultaneous gauge transformations:
Aµ → A′µ ≡ Aµ − ∂µχ , (2.3)
ψ → ψ′ ≡ eieχψ , (2.4)
where χ = χ(x, t) is an arbitrary function of space-time. Had we not included the field
Aµ, the Lagrangian would not have remained invariant under the local transformation of
the field ψ alone, owing to the derivatives present in the fermion kinetic terms. This can
be seen explicitly if we examine the transformation of the term ψ¯γµ∂µψ under Eq. (2.4):
ψ¯γµ∂µψ → ψ¯e−ieχγµ∂µ(eieχψ) = ψ¯(γµ∂µ + ieγµ(∂µχ))ψ , (2.5)
which is evidently not invariant unless we use the covariant derivative, which introduces
Aµ, to cancel out the extra term:
ψ¯γµDµψ → ψ¯e−ieχγµDµ(eieχψ) = ψ¯(γµ∂µ + ieγµAµ − ieγµ(∂µχ)ψ + ieγµ(∂µχ))ψ
= ψ¯γµDµψ (invariant) . (2.6)
Thus, the requirement that the Lagrangian (and hence the equations of motion) is invari-
ant under local U(1) transformations requires the existence of a gauge field Aµ. This field
corresponds to the electromagnetic field and hence to the photon.
1In what follows, and the rest of this thesis, we will be referring to the ‘Lagrangian density’ simply as
the ‘Lagrangian’.
8 Chapter 2. The Standard Model and beyond
2.2.2 Electroweak theory
As we alluded in the introduction, the electroweak sector of the SM can be described by
a non-abelian gauge theory based on the group SU(2)L × U(1)Y , where the L subscript
indicates the left-handed chiral nature of the coupling of the gauge fields and Y is the
hypercharge, to be distinguished from the electromagnetic charge, Q. The SU(2)L quan-
tum number is referred to as the weak isospin. We can separate the Lagrangian into three
parts as
LEW = Lbosons + LHiggs + Lfermions , (2.7)
where Lbosons, LHiggs and Lfermions correspond to the gauge bosons, the Higgs field and the
fermions respectively.
2.2.2.1 Boson masses
The gauge boson Lagrangian now contains two gauge fields:
Lbosons = −1
4
FAµνF
Aµν − 1
4
BµνB
µν , (2.8)
where the FAµν and Bµν are the field tensors corresponding to the SU(2)L and U(1)Y
symmetries and the index A labels the SU(2)L weak isospin quantum numbers:
FAµν = ∂µWAν − ∂νWAµ − gfABCWBµWCν ,
Bµν = ∂µBν − ∂νBµ , (2.9)
where the fABC are the group structure constants (the alternating tensor in the SU(2)
case) and g is the SU(2)L charge. It is important to note that the non-abelian field
tensor FAµν now contains a self-interaction term ∝WBµWCν , a feature that is even more
significant in SU(3) non-abelian gauge theory, as we shall see.
The SU(2)L × U(1)Y symmetry is not manifested in nature. It is in fact, a sponta-
neously broken symmetry: the W and Z gauge bosons are massive. The minimal way
to break it within the SM and give masses to the gauge bosons, while preserving the
gauge-invariant nature of the theory, is to introduce the complex scalar Higgs field that
2.2. The Standard Model 9
is an SU(2)L doublet and possesses hypercharge Y = 1/2:
2
φ =
(
φ+
φ0
)
,
φ† =
(
φ¯0 φ−
)
,
(2.10)
where the meaning of the labels on the components will become apparent subsequently.
The corresponding Lagrangian, LHiggs is given by
LHiggs = (Dµφ)†(Dµφ)− V (φ†φ) , (2.11)
where the covariant derivative, which introduces the interaction between the Higgs field
and the gauge fields, is defined by
Dµ ≡ ∂µ + ig(T ·W µ) + iY g′Bµ . (2.12)
In the above, we have suppressed weak isospin indices, the T are matrix representations
of the SU(2)L generators and g and g
′ are the SU(2)L and U(1)Y gauge charges respec-
tively. The potential term is given a special form containing quadratic and quartic terms,
commonly referred to as the ‘mexican hat’ potential:
V (φ†φ) = −µ2(φ†φ) + λ(φ†φ)2 , (2.13)
and the constants are chosen such that µ2, λ > 0. This potential possesses a minimum
at (φ†φ)min = µ2/2λ ≡ v2, an unstable maximum at the origin and goes off to positive
infinity as (φ†φ)→∞ (hence the resemblance to the mexican hat). Figure 2.1 illustrates
the shape of the potential on the complex φ plane. To break the symmetry, a particular
direction (referred to as a ‘vacuum expectation value’ or VEV) in the SU(2) space for the
minimum of φ is chosen:
〈φ〉 = 1√
2
(
0
v
)
. (2.14)
2The hypercharge for the Higgs field could have been chosen Y = 1. The choice affects the hypercharges
for the rest of the matter content in the theory. The relation between electric charge, hypercharge and
the third component of weak isospin will contain factors of 2 accordingly.
10 Chapter 2. The Standard Model and beyond
Figure 2.1: The ‘mexican hat’ potential given by Eq. (2.13), with µ2, λ > 0.
Notice that, at the minimum, the theory still possesses a residual U(1) symmetry. To
determine the properties of the theory at this minimum, we need to expand the fields
about it, using an SU(2) transformation of the field:
U(ξ) = e−iT ·ξ/v , (2.15)
φ =
(
0
(H + v)/
√
2
)
U(ξ) , (2.16)
where we have three ξ components and one scalar degree of freedom, H . At first order this
is just an expansion of the field about the minimum. Since the Lagrangian of Eq. (2.11)
is gauge-invariant, we should simultaneously perform an SU(2) gauge transformation:
T ·W µ → UT ·W µU−1 + i
g
(∂µU)U−1 . (2.17)
We obtain the following form for the Higgs boson Lagrangian:
L = 1
2
∂µH∂
µH−V
(
(v +H)2
2
)
+
(v +H)2
8
χ†(2gT ·Wµ+g′Bµ)(2gT ·W µ+g′Bµ)χ , (2.18)
where we have used the vector χ = (0, 1), a unit vector along the direction of the Higgs
2.2. The Standard Model 11
boson VEV. Evidently the three degrees of freedom from the ξ field do not appear ex-
plicitly in the Lagrangian. These degrees of freedom have been absorbed by the gauge
bosons and provide the longitudinal degree of freedom: a massive vector boson has three
polarisation states whereas a massless one has two.
We now consider the terms quadratic in the vector boson fields:
LM = v
2
8
[
(gW 3µ − g′Bµ)(gW 3µ − g′Bµ) + 2g2W−µ W+µ
]
. (2.19)
We need to rewrite this in ‘diagonal form’, i.e. in terms of mass eigenstates. We define
two electrically neutral fields Zµ and Aµ:
Zµ = cos θwW
3
µ − sin θwBµ ,
Aµ = sin θwW
3
µ + cos θwBµ , (2.20)
where we have also defined the electroweak mixing angle, sin2 θw = g
′2/(g2+ g′2) ≃ 0.23.3
Using the definitions of Eq. (2.20) we have
LM = g
2v2
4
W+µ W
−µ +
(g2 + g′2)v2
8
ZµZ
µ , (2.21)
from which we can now deduce the vector boson masses:
MA = 0 , MW =
1
2
vg , MZ =
1
2
v
√
g2 + g′2 . (2.22)
Thus, with the particular choice for the Higgs boson representation, masses have been
generated for the weak vector bosons, Z and W±, while one field, A, has remained mass-
less. This corresponds to a U(1)-symmetric force which we identify with QED and a
boson which we identify with the photon, γ. We can thus write the following symmetry
breaking relation:
SU(2)L × U(1)Y → U(1)em , (2.23)
and we can associate the different charges of fields by
Q = T 3 + Y , (2.24)
where Q is the electromagnetic charge, T 3 is the third component of weak isospin and Y
3The current Particle Data Group value is sin2 θˆ(MZ)(MS) = 0.23116(13) [13].
12 Chapter 2. The Standard Model and beyond
Representation T 3 Y Q
e−R 0 -1 -1 νeL
e−L
 +1/2 0- 1/2
−1/2 -1
νeR 0 0 0
d′R 0 −1/3 −1/3 uL
d′L
 +1/2 +2/3+1/6
1/2 −1/3
uR 0 +2/3 +2/3
Table 2.1: The first-generation fermion representations in the electroweak SU(2)L×U(1)Y the-
ory. The third component of weak isospin, T 3, the hypercharge, Y and the resulting
electromagnetic charge Q are given. The primes on the quark sector indicate that
they are not mass eigenstates. The right-handed neutrino is hypothetical, does not
couple to the SM particles and is shown for completeness.
is the hypercharge. Q is essentially the only remaining unbroken generator. We can now
see that the upper component of the Higgs doublet has Q = 1, whereas the lower one has
charge Q = 0, which explains why the VEV was chosen as in Eq. (2.14).
2.2.2.2 Fermion masses and couplings
Introducing fermion masses explicitly into the fermion Lagrangian would break gauge
invariance. The Higgs mechanism in its simplest form is the conventional method to
give masses to the SM fermions without any adverse effects. The choices of fermion
representations in the SM and their charges are given in Table 2.1, for the first generation.
The SM has been observed to contain three ‘copies’ of this structure: three generations.
Notice that the difference between the hyperchages of the singlet (right-handed) and
doublet (left-handed) is ±1/2, which allows us to use the Higgs doublet (with Y = 1/2)
2.2. The Standard Model 13
to form gauge-invariant terms:
L = gee(ν¯eL, e¯L)
(
φ+
φ0
)
eR + gdd(u¯L, d¯
′
L)
(
φ+
φ0
)
d′R
+ gνν(ν¯eL, e¯L)
(
φ¯0
φ−
)
ν¯eR + guu(u¯L, d¯
′
L)
(
φ¯0
φ−
)
uR
+ (other flavours) + h.c. , (2.25)
Thus, when the Higgs field obtains a VEV and the symmetry is broken, we obtain terms
quadratic in the fermion fields: mass terms proportional to the VEV, mff = gffv/
√
2,
and mixing terms between the fermions. To take care of this mixing we conventionally
define  d
′
s′
b′
 = V
 ds
b
 , (2.26)
where V is known as the Cabbibo-Kobayashi-Maskawa (CKM) matrix.4 V is a 3 × 3
unitary matrix, given by
V =
 Vud Vus VubVcd Vcs Vcb
Vtd Vts Vtb
 , (2.27)
and can be parametrized by three mixing angles and a charge-parity (CP) violating phase.
In the SM, this phase provides a source of CP violation, observed experimentally via mix-
ing in the neutral kaon system (K0-K¯0 mixing) [11]. See appendix C for parametrizations
of the CKM matrix and measured values of its matrix elements.
In practical calculations, the CKM matrix elements Vij can be inserted into the am-
plitude of diagrams where a W boson couples to quarks i and j. If we define the right-
and left-handed projection operator for fermions, PR,L =
1
2
(1 ± γ5), we may write down
4Kobayashi and Maskawa were awarded the Nobel Prize in 2008, on work related to the CKM matrix,
and specifically ‘for the discovery of the origin of the broken symmetry which predicts the existence of at
least three families of quarks in nature’ [14].
14 Chapter 2. The Standard Model and beyond
the interaction term between the SM fermions f and f ′ and the W boson:
LWff = g√
2
Vff ′ f¯γµPLf
′W µ + h.c. , (2.28)
where Vff ′ = 1 for leptons (ℓνℓ) and Vff ′ = Vij for quarks i and j.
2.2.3 Quantum chromodynamics
Quantum chromodynamics (QCD) is also formulated in terms of a gauge theory, based
on the non-abelian group SU(3). It possesses several distinct features: it is unbroken
in Nature, contains self-interacting degrees of freedom, the gluons, and exhibits asymp-
totic freedom, which reveals that only in the short-distance limit we can use perturbative
methods legitimately.
We will briefly review the construction of the theory and subsequently present pertur-
bative tools that will be employed in calculations that will follow in this thesis.
2.2.3.1 SU(3) gauge theory
The fermions that carry SU(3) charge, or colour charge, are the quarks. The full quantum
Lagrangian is given by
LQCD = Lclassical + Lgauge−fixing + Lghost . (2.29)
The expression for the classical Lagrangian is similar to what we have written down for
the QED and EW theories:
Lclassical = −1
4
FAµνF
Aµν +
∑
flavours
q¯a(i /D −mq)abqb , (2.30)
where the sum over the index A is over the eight colour degrees of freedom of the gluon
field GAµ , the sum over flavours is over the nf quark flavours and the field strength tensor
FAµν is defined as
FAµν ≡ ∂µGAν − ∂νGAµ − gsfABCGBµGCν , (2.31)
where gs is the strong charge and the f
ABC are the structure constants of SU(3). We
may also define the strong coupling constant αs ≡ g2s/4π. The covariant derivative Dµ is
2.2. The Standard Model 15
defined according to whether it acts on triplet or octet fields:
(Dµ)ab = ∂
µδab + ig(t ·Gµ)ab , (Dµ)AB = ∂µδAB + ig(T ·Gµ)AB , (2.32)
where t and T are generators (matrices) in the fundamental and adjoint representations
of SU(3) respectively. They satisfy the following relations:
[tA, tB] = ifABCtC , [TA, TB] = ifABCTC , (TA)BC = −ifABC . (2.33)
The following identities are true for SU(N) gauge theories:
Tr(tAtB) =
1
2
δAB ≡ TRδAB ,∑
A
tAabt
A
bc =
N2 − 1
2N
δac ≡ CF δac ,
Tr(TCTD) =
∑
A,B
fABCfABD = NδCD ≡ CAδCD , (2.34)
which imply that for QCD, for which N = 3: TR = 1/2, CF = 4/3 and CA = 3.
In practical calculations an explicit representation for the tA is not necessary, and the
identities of Eq. (2.34) are used.
The classical Lagrangian, Eq. (2.30), is invariant under the simultaneous SU(3) trans-
formations:
qa → q′a =
(
eit·θ
)
ab
qb ≡ Uabqb ,
t.Gµ → t.G′µ = Ut ·GµU−1 +
i
gs
(∂µU)U
−1 , (2.35)
where θA = θA(x, t) are eight arbitrary real functions of space-time.
The Lagrangian of Eq. (2.30) cannot be used immediately to calculate Feynman rules
for QCD: in this form, a propagator for the gluon field cannot be defined.5 We can exploit
gauge invariance to add a gauge-fixing term to the QCD Lagrangian which amends this
issue:
Lgauge−fixing = −1
λ
(∂µGAµ )
2 , (2.36)
where λ is an arbitrary parameter. Provided we work in the covariant gauge, that is, a
gauge in which we choose ∂µGAµ = 0, we have not made any changes in the physics and
5The same issue arises when defining the photon propagator in QED.
16 Chapter 2. The Standard Model and beyond
(a) (b) (c) (d)
Figure 2.2: The quark-gluon vertex, (a), and part of the one-loop correction to it, (b), are
shown. We need to decide whether we assign the loop correction as a correction to
the vertex, as in (c), or to the rest of the diagram, as in (d). This is done at some
scale µR, above which the loop is part of the vertex and below which it is part of
the rest of the diagram.
we can now define a propagator for the gluon fields.
Finally, it is necessary to add an extra term to the Lagrangian which is related to the
need for ghost particles, Lghost, whose purpose is to cancel unphysical degrees of freedom
that may arise when renormalising a non-abelian gauge theory. For further details on
non-abelian gauge theory renormalisation and the need for ghost fields, see [8].
2.2.3.2 Renormalisation and the running of αs
Besides the masses of the quarks, the only other parameter which appears in the QCD
Lagrangian is the strong charge,6 gs. One should be cautious however: parameters in a
Lagrangian are not necessarily physical quantities. Physical observables can be calculated
as functions of these parameters, in this case of gs. What we would like to do is reformulate
the theory so we can write a physical observable as a function of another. This process is
called renormalisation.
As an illustrative example, as given in Ref. [9], consider the quark-gluon vertex shown
in Fig. 2.2. Figure 2.2a shows the lowest order of the vertex in perturbation theory, for
which we can define gs to be the strength. If we now start considering higher orders, and
specifically part of the one-loop correction shown in Fig. 2.2b, we must decide whether
we will include this as part of the vertex, as in Fig. 2.2c, or as part of the rest of the
diagram, as in Fig. 2.2d. To do this we choose some scale µR, called the renormalisation
scale, above which the loop is part of the vertex, as in Fig. 2.2c, and below which it is
6In practical calculations one usually employs αs rather than gs.
2.2. The Standard Model 17
part of the rest of the diagram, as in Fig. 2.2d. This scale should not have any physical
significance: it is simply a device we introduce to avoid double-counting. Due to this,
µR should not affect any physical prediction. However, the strength gs now becomes a
function of the scale µR.
We can see the effect of this procedure on the theory by considering a dimensionless
physical observable R that is a function of a physical scale Q2. After applying the above
renormalisation procedure, the observable R can only be a function of Q2, µ2R and αs(µ
2
R).
Dimensional analysis restricts R to depend on Q2 and µ2R only through their ratio, giving
R = R(Q2/µ2R, αs(µ
2
R)) . (2.37)
We can now employ the fact that a physical observable cannot depend on the value of the
renormalisation scale, µR and write, using the chain rule:
µ2R
d
dµ2R
R(Q2/µ2R, αs) = 0 ,
⇒
[
µ2R
∂
∂µ2R
+ β(αs)
∂
∂αs
]
R = 0 , (2.38)
where we have defined
β(αs) ≡ µ2R
∂αs
∂µ2R
. (2.39)
An important observation is that the Q-dependence of the quantity R would not have
come about in a classical theory: it is a feature of the renormalised theory which arises
due to the introduction of the scale µR.
Even though the β-function of QCD is currently known to four-loop accuracy,
β(αs) = −α2s(β0 + β1αs + β2α2s + β3α3s + ...) , (2.40)
only the one-loop result is necessary for qualitative understanding of QCD:
β0 =
11CA − 4TRnf
12π
. (2.41)
For most phenomenology the number of active quark flavours can be taken to be nf = 5
and hence the β-function is negative when αs is small. This implies that the running
18 Chapter 2. The Standard Model and beyond
coupling αs decreases to zero as an inverse power of lnQ
2. To one-loop order, we have
αs(Q
2) =
αs(µ
2
R)
1 + αs(µ2R)β0 ln(Q
2/µ2R)
. (2.42)
Hence QCD interactions become weak at high energy, a feature called asymptotic freedom,
and strong at low energy. This is in contrast to QED, where the observed charge of the
electron is smaller at large distances. This can be thought of as being due to the self-
interactions of the gluons: emission of virtual gluons, which are themselves charged, causes
the colour charge of the source that emitted them to ‘leak out’ to the vacuum. In effect,
this reduces the colour force at short distances since the interaction between distributed
charges that overlap is weaker than that between point charges.
2.2.3.3 Parton evolution
Free quarks or gluons have not been observed in Nature. This has led to the confinement
hypothesis: the only energy eigenstates of QCD that can exist in Nature have to be
colour-neutral (or colour-singlets). There is strong circumstantial evidence in favour of the
hadrons as bound states of quarks and anti-quarks. For example, there exists quantitative
understanding of high energy inelastic scattering of hadrons once they are taken to be
composites of quarks and gluons. In fact, hadron-hadron and hadron-lepton scattering can
be described in terms of perturbation theory and the parton model, a direct consequence
of the property of asymptotic freedom. The basic assumption of the parton model is that
interactions of hadrons are due to interactions of more elementary entities, called partons,
which turn out to be the quarks and gluons of QCD. The number and momenta of partons
are most conveniently described in terms of parton density functions (PDFs). Theoretical
and experimental details on the parton model can be found in [10].
The PDFs are fundamentally non-perturbative and at present cannot be predicted
from first principles. However, the evolution equations for their scale-dependence can be
derived (see, for example, [10]). We denote the fraction of momentum of the proton that a
parton i possesses at scale Q2 as x. The momentum fraction distribution is then denoted
by fi(x,Q
2). The equation describing the evolution of fi(x,Q
2), known as the DGLAP
(Dokshitzer-Gribov-Lipatov-Altarelli-Parisi) equation, takes the following form:
Q2
∂
∂Q2
fi(x,Q
2) =
∑
j
∫ 1
x
dz
z
αs
2π
Pij(z)fj(x/z,Q
2) , (2.43)
2.2. The Standard Model 19
where Pij(z) are the so called (regularised) splitting functions, related to the probability
of finding a parton i in a parton j and the integral is taken over all possible momentum
fractions for the splittings, z. The regularised splitting functions at leading order are
given by
Pqq(z) = CF
[
1 + z2
(1− z)+ +
3
2
δ(1− z)
]
,
Pqg(z) = TR[z
2 + (1− z)2] ,
Pgg(z) = 2CA
[
z
(1− z)+ +
1− z
z
+ z(1− z)
]
+
1
6
(11CA − 4nfTR)δ(1− z) , (2.44)
where we have used the so-called ‘plus’ prescription,∫ 1
0
dx
f(x)
(1− x)+ =
∫ 1
0
dx
f(x)− f(1)
1− x , (2.45)
and the values of the constants CA, CF and TR for QCD have been given in section 2.2.3.1.
2.2.3.4 Parton branching
Perturbative calculations in QCD are hard beyond leading order: the work involved in-
creases roughly factorially with the order. However, there are cases when we cannot
truncate the series to a fixed order since there are higher order terms that are enhanced
in certain regions of phase space. Such a region is collinear parton emission from a parton
involved in a scattering process. This can be either an incoming parton or an outgoing
parton. Branchings of outgoing partons are called ‘time-like’ (Fig. 2.3) and branchings of
incoming partons are called ‘space-like’ (Fig. 2.4).
We consider the kinematics for the time-like branching shown in Fig. 2.3 first. Our aim
is to calculate the modification to the n-body cross section when we have a single branching
in the collinear approximation. We define the opening angle between the outgoing partons
b and c as θ = θb + θc. The collinear approximation implies that θ → 0. If we neglect the
parton masses then p2b ≈ 0 and p2c ≈ 0. If we then define t ≡ p2a, using pa = pb + pc, we
may write
t = 2pb · pc = 2EbEc(1− cos θ) , (2.46)
where Eb and Ec are the corresponding energies of the partons b and c. Defining the
20 Chapter 2. The Standard Model and beyond
Mn a
b
c
θb
θc
Figure 2.3: The kinematics of time-like branching: an outgoing parton, part of an n-body
matrix element Mn, branches into partons c and b at angles θb and θc.
Mn
a
c
θb
θc
b(z)
Figure 2.4: The kinematics of space-like branching: an incoming parton a branches into par-
tons b and c. Parton b carries a momentum fraction z of parton a. Parton b then
takes part in an interaction in some n-body matrix element Mn.
energy fraction of the splitting, z, by
z ≡ Eb
Ea
= 1− Ec
Ea
, (2.47)
t can then be approximated for small θ as
t ≈ z(1− z)E2aθ2 . (2.48)
Due to the propagator factor, the (n + 1)-body matrix element squared, |Mn+1|2, is
proportional to 1/t. In fact, it can be shown that it can be written in terms of the n-body
matrix element squared as
|Mn+1|2 = 8παs
t
Pˆba(z) |Mn|2 . (2.49)
2.2. The Standard Model 21
The splitting function in this case, Pˆba, is ‘unregularised’, i.e. it may diverge as z → 0
or z → 1. It will be shown later how to obtain a regularised splitting function when
including the virtual corrections to the differential cross section, for the case of space-like
branching (section 4.2). The origin of the collinear enhancement is now obvious: it is due
to the 1/t factor which diverges as t→ 0, i.e. when the internal parton line is on-shell.
We can use Eq. (2.49) to compute the cross section for one branching in terms of that
for no branchings, using the following relations:
dσn+1 =
8παs
t
Pˆba(z)F |Mn|2 dΦn+1 ,
dσn = F |Mn|2 dΦn , (2.50)
where F is the flux factor for the initial state and Φm is the relevant m-body phase space.
It can be shown that
dΦn+1 = dΦn
1
4(2π)3
dtdzdφ , (2.51)
where φ is the azimuthal angle related to the branching. Hence, in the collinear ap-
proximation, when all angles are small, we have derived the relation between the n-body
differential cross section and the differential cross section for one time-like emission,
dσn+1 = dσn
dt
t
dz
αs
2π
Pˆba(z) , (2.52)
where an average/sum over initial/final spins has been taken. We can repeat the same
procedure for space-like branchings (Fig. 2.4) for which we define |p2b | ≡ t. We then have
t = EaEbθ
2
c and the phase space factor becomes
dΦn+1 = dΦn
1
4(2π)3
dt
dz
z
dφ , (2.53)
where z = Eb/Ea = 1 − Ec/Ea. But in this case the initial-state flux factor, F , changes
by a factor of z because of the change of energy of the incoming parton from Eb to Ea.
Thus, it turns out that the expression for space-like branching is identical to the one for
time-like branching, Eq. (2.52).
22 Chapter 2. The Standard Model and beyond
t
W, Z, γ H
H H
H H H H
Figure 2.5: The three most ‘dangerous’ quadratically divergent one-loop corrections to the
Higgs boson mass in the Standard Model. From left to right: the top contribution,
the gauge boson contribution and the Higgs boson contribution.
2.3 Beyond the Standard Model
2.3.1 The need for BSM physics
The standard model is phenomenally successful at describing strong, weak and electro-
magnetic interactions, with precision results up to O(100 GeV). Given its success, why
don’t we just look for the ‘missing’ Higgs particle and declare the end of particle physics
once we discover it? The reason is that there are conceptual and phenomenological hints
that the SM is incomplete. The conceptual issues include the multitude of unexplained
parameters, family replication and flavour hierarchies, the inability of the SM to incor-
porate gravity and the hierarchy problem. Phenomenological hints are neutrino masses,
Dark Matter, the cosmological vacuum energy (also known as Dark Energy) and the quest
for Grand Unification and coupling constant merging. We discuss a few of these issues
here.
2.3.1.1 The hierarchy problem and new physics
An important conceptual issue, known as the hierarchy problem, concerns the Higgs field.
Essentially the problem arises since, in its usual Standard Model incarnation, the Higgs
boson is considered to be a fundamental scalar. Fundamental scalars suffer from radia-
tive instability in their masses due to radiative corrections. The three most ‘dangerous’
contributions from radiative corrections to the Higgs boson mass in the SM come from
one-loop diagrams with top quarks, gauge bosons and the Higgs boson itself, as shown in
Fig 2.5. The contributions to the Higgs boson mass squared are proportional to a cut-off
scale squared, Λ2, with the top contribution being negative and the gauge and Higgs bo-
son contributions being positive. If Λ is set to be the Planck scale, MP l ∼ 1019 GeV, then
2.3. Beyond the Standard Model 23
the Higgs boson mass ought naturally to be of that order, barring any unnatural cancella-
tions between the positive and negative contributions. On the other hand, experimental
evidence shows that the Higgs boson should be light: its mass should be . 245 GeV,
assuming no new physics [13]. To achieve this, the cut-off scale Λ should be much lower
than the Planck scale and new physics, possibly in the form of new resonances, is expected
to appear at the electroweak scale ΛEW ∼ O(1 TeV).
2.3.1.2 The little hierarchy problem
If new particles are indeed responsible for cancelling the quadratic divergences to the
square of the Higgs boson mass, their masses should be O(1 TeV) by naturalness. How-
ever, current experimental data already set some constraints on possible new physics at
the TeV scale. For example, absence of nucleon decays and strong bounds on flavour-
changing neutral currents indicate that these effects cannot receive any significant contri-
butions from TeV-scale physics. Precision electroweak measurements put constraints on
operators arising by exchanging new heavy particles and scales which suppress them are
required to be larger than 2−7 TeV [15]. Therefore, there exists an issue of compatibility
between the experimental data and the expectation of the masses of the new particles re-
quired in order to satisfy the naturalness of the low Higgs boson mass. This is sometimes
referred to as the ‘little hierarchy problem’.
A possible ‘solution’ to this problem relies on the fact that the quadratic sensitivity to
the high energy physics of the Higgs boson mass is a result of loop contributions: to cancel
the quadratic divergences, the new TeV-scale particles only need to appear in interaction
vertices in pairs or more. Therefore, we can suppress the tree-level contributions, while
maintaining the cancellation of the loop contributions, by introducing a new symmetry
acting on the new particles, under which all the SM fields are neutral. The simplest, and
most common, choice is a Z2 symmetry, or parity.
2.3.1.3 Dark Matter
The nature of Dark Matter (DM) has been an open question in astrophysics since the
initial observations by Fritz Zwicky in 1933 which required the existence of non-luminous,
weakly interacting matter to explain the orbital velocities of galaxies in clusters. Following
the initial discovery, observations have indicated that galaxy rotation curves do not fall
off with radial distance from the galactic centre and evidence from the cosmic microwave
24 Chapter 2. The Standard Model and beyond
background (CMB) indicates that DM makes up about 25% of the energy density of the
Universe [16].
There are two basic types of candidates for DM. The first is Massive Compact Halo
Objects (MACHOs). MACHOs are objects of baryonic origin such as black holes, brown
dwarf stars and giant planets. It has been shown, however that MACHOS cannot account
for more than 20% of DM [17]. The second type of candidates are weakly interacting
particles. The only candidates within the SM, the neutrinos, do not possess the necessary
density to compose the Dark Matter. In addition, the existence of hot (i.e. relativistic)
DM is not consistent with observations and hence any candidates should be relatively
heavy. Thus DM is thought to consist of cold (i.e. non-relativistic), stable, or at least
meta-stable [18, 19], massive particles which appear in theories beyond the SM. These
are usually called Weakly Interacting Massive Particles, or WIMPs. There are good
arguments that these particles might appear at the TeV scale [20].
2.3.1.4 Dark Energy
Strong evidence for the existence of Dark Energy comes from distant supernovae observa-
tions, indicating that our Universe is currently undergoing an accelerated expansion. The
question of the nature of Dark Energy is considered by theorists to be even more severe
than the Dark Matter problem, not only because Dark Energy is thought to contribute
about 70% of the energy density of the Universe, but also because we currently have no
strong theoretical explanations for it: it is totally mysterious. It could possibly be the
vacuum energy, in the form of a cosmological constant. The issue is much worse than the
previously mentioned hierarchy problem: calculations of the energy density of vacuum in
quantum field theory, with a cut-off at 1014 GeV,7 give a result for a cosmological con-
stant of the order of 1054 GeV4, whereas the measured value of the dark-energy density is
10−47 GeV4 [21]. It is likely that solving the problem requires a complete quantum theory
of gravity [22].
2.3.1.5 GUTs and coupling constant merging
A troubling feature of the Standard Model is the fact that it contains almost 20 param-
eters. It is thought that a fundamental theory of nature should be able to explain the
origin of their values through some underlying principle. We would also need to address
7This scale could be the Grand Unification scale, for example.
2.3. Beyond the Standard Model 25
the issue Feynman wasn’t comfortable with at the conception of the Standard Model: the
fact that the different components ‘have not yet been smoothed out’. It basically boils
down to the fact that we have three gauge couplings of different magnitudes. In mathe-
matical terms, the question is why is the gauge group of the Standard-Model semi-simple
and not simple?
A resolution to this issue is to assume that the underlying gauge symmetry of nature
is in fact a simple group, but it is broken at some high energy scale down to the gauge
group of the SM. For example, the underlying gauge group could be the SU(5) group,
which can be shown to contain an SU(3) × SU(2) × U(1) subgroup. The SU(5) group
can then be easily broken down to the SM group via the introduction of a set of Higgs
fields [21]. Although SU(5) can successfully incorporate the SM, it predicts proton decay
with a lifetime that has already been excluded [23]. Nevertheless, it is a simple example
of a Grand Unified Theory (GUT).
A GUT can possibly solve the problem of seemingly separate gauge couplings. As
we have already demonstrated in the case of the strong coupling constant, αs, the gauge
couplings are scale-dependent quantities. If the hypothesis of unification is to hold, they
must all equal each other at some scale, the grand unification scale. Below this scale, the
simple group (e.g. SU(5)) is broken down to the SM group and the separate couplings
have different behaviour. In fact, below the grand unification scale, the couplings for each
group SU(3), SU(2) and U(1) evolve according to the respective β-functions with no
memory of the simple group they originated from. If the exercise is performed, evolving
from low to high scale, however, it is found that the unification of the couplings in the
Standard Model is a ‘near-miss’, as can be seen by the dashed lines in Fig. 2.6.
2.3.2 Survey of BSM theories
2.3.2.1 Supersymmetry
Supersymmetry is one of the most popular extensions to the Standard Model, and not
without good reasons. Conceptually it is very appealing: it is the only possible extension
to the spatial symmetries of the theory in flat, four-dimensional space. It appears to be
able to accommodate a solution to the hierarchy problem, contains natural candidates
for Dark Matter and solves the ‘near-miss’ of the gauge-coupling unification in the SM.
It is also a powerful tool for understanding quantum field theories, especially in the non-
perturbative regime [25].
26 Chapter 2. The Standard Model and beyond
2 4 6 8 10 12 14 16 18
Log10(Q/1 GeV)
0
10
20
30
40
50
60
α
−1
 
α1
−1
α2
−1
α3
−1
Figure 2.6: Renormalisation group evolution of the inverse gauge couplings for SU(3), SU(2)
and U(1) (α−13,2,1(Q) respectively) in the Standard Model (dashed lines) and the
Minimal Supersymmetric Standard Model (solid lines). The super-particle mass
thresholds are varied between 250 GeV and 1 TeV, α3(mZ) between 0.113 and
0.123 and two-loop effects are included [24].
Supersymmetry introduces fermionic operators Qα and Q¯α˙ to the Poincare´ generators
P µ (corresponding to translations) and Mµν (corresponding to rotations and Lorentz
boosts).8 The new operators satisfy the following anti-commutation relation:
{Qα, Q¯β˙} = 2(σµ)αβ˙Pµ , (2.54)
where σµ is the 4-vector of Pauli matrices. The anti-commutation relation implies that
two symmetry transformations QαQ¯β˙ have the effect of a translation. This is to be
expected since the operators themselves carry spin angular momentum, so it is clear that
supersymmetry must be a space-time symmetry.
Let |B〉 be a bosonic state and |F 〉 a fermionic state. Then we have, schematically,
Qα |F 〉 = |B〉 ,
Q¯β˙ |B〉 = |F 〉 , (2.55)
8Hence, the Poincare´ group corresponds to the basic symmetries of special relativity.
2.3. Beyond the Standard Model 27
H H
t t˜
H H
Figure 2.7: Cancellation between the quadratically divergent contributions to the Higgs boson
mass squared from top and stop loops in the MSSM render its mass finite.
and, as a result,
QQ¯ : |B〉 → |B(translated)〉 . (2.56)
Supersymmetry is a symmetry between bosonic particles and fermionic particles. The
matter fields (in the fundamental representations) of the Standard Model, fit into chiral
multiplets, while the gauge fields fit into vector multiplets. In supersymmetry, a chiral
multiplet contains a fermion field and a scalar field, related by a supersymmetry transfor-
mation such as the one described by Eqs. (2.55) and a vector multiplet contains a vector
field and a fermion field also related by the same transformation. If supersymmetry is
exact (i.e. unbroken) the particles related to fields in the same multiplet should possess
the same mass.
The minimal additional content of a supersymmetric theory to the matter fields of
the SM consists of a (super-)scalar for each chirality of the SM fermions and a (super-
)fermion for each SM vector boson. The Higgs boson sector for electroweak symmetry
breaking becomes more complicated, requiring the introduction of a second Higgs doublet,
otherwise the electroweak gauge symmetry would suffer a gauge anomaly. The conditions
for cancellation of gauge anomalies are already miraculously satisfied in the SM by the
known quarks and leptons, but in a supersymmetric theory, the multiplet which contains
the SM Higgs boson will now contain a fermionic partner. This would spoil the anomaly
cancellation, but can be avoided if a second doublet which has Y = −1/2 is introduced,
so that the contributions from the two fermionic members of the Higgs multiplets are
cancelled out. Furthermore, the Y = −1/2 doublet is required so that masses can be
given to the −1/3 down-type quarks and charged leptons [24].
The Minimal Supersymmetric Standard Model (MSSM) provides an elegant solution
to the hierarchy problem of the SM: the supersymmetric partner corrections cancel out the
quadratically divergent SM particle corrections to the square of the Higgs boson mass,
as illustrated in Fig. 2.7. Supersymmetry can also accommodate candidates for Dark
28 Chapter 2. The Standard Model and beyond
Matter. As we have mentioned in section 2.3.1.2, to solve the little hierarchy problem
we need to introduce a new discrete Z2 symmetry, or parity. In the MSSM, and in other
supersymmetric models, R-Parity is introduced, under which the super-partners are odd
and the SM particles are even. If R-parity is exact, the lightest supersymmetric particle
(LSP) cannot decay to lighter SM particles and thus may be stable and contribute to
Dark Matter.
Exact supersymmetry is of course not manifested in nature, otherwise we would have
already observed the super-multiplets. A method to break supersymmetry is thus re-
quired; in fact this is where the elegance of the theory ends and the complications begin:
as a result of the breaking, many free parameters are added to the model. To maintain
the successful cure for the hierarchy problem, resulting in the observed W and Z masses,
the masses of super-partners should be around the TeV scale, with the lightest ones at
most about 1 TeV [24].
2.3.2.2 Extra Dimensions
Another possibility of extending the space-time symmetries of Nature is to introduce extra
spatial dimensions. This has been a long-discussed idea and has re-surfaced in different
contexts. There are many flavours of models with extra dimensions, each attempting
to address different issues that plague the SM: e.g. Universal Extra Dimensions [26] or
Randall-Sundrum type scenarios [27].
The Randall-Sundrum scenario is particularly interesting. In its simplest form, it is a
5-dimensional theory. The extra dimension is an interval, with the surfaces (or ‘branes’)
at the end of each interval being (3+1)-dimensional. One surface is at y = 0 and the
other at y = πR, where R is a length related to the extra dimensions. The metric
changes from y = 0 to y = πR as ηµν → e−kπRηµν where k is a constant. This implies
that all the length and energy scales change with y. If the fundamental scale is the
Planck scale, MP l, the y = 0 brane carries physics at MP l, but all energy scales are
‘red-shifted’ by the exponential factor until the other brane is reached, where we would
have an exponentially smaller scale. In particular, this scale can be the electroweak scale:
MEW ≈ MP le−kπR ∼ 1 TeV. In fact this method ‘shifts’ the hierarchy problem to a
problem of finding the proper mechanism to fix the size of the extra dimensions. The
Randall-Sundrum scenario can also potentially incorporate a mechanism that explains
the flavour structure and mass hierarchy of the SM [28].
Very often, extra-dimensional models suffer from a ‘little hierarchy’ problem and a
2.3. Beyond the Standard Model 29
discrete parity needs to be introduced to restrict the production of heavier resonances
to pairs. This is sometimes called ‘Kaluza-Klein’ parity (KK parity). The lightest KK
resonance is sometimes also considered to be a viable Dark Matter candidate.
2.3.2.3 Strong dynamics
Strong coupling dynamics can provide an elegant and natural solution to the hierarchy
problem. It is natural in a literal sense since we have already observed an example in
Nature in which a large hierarchy arises: between the Planck scale and the proton mass
scales (1 GeV). This is a result of the running of the QCD coupling constant and the
strong coupling regime in the infrared.
In technicolour models, as some strong dynamics models are usually called, mass is
given to the electroweak gauge bosons via some new strong dynamics. Extensions to
these models allow mass to be given to the SM fermions as well. In typical types of
extensions we obtain large flavour-changing neutral currents which are highly constrained
by experiment. Technicolour models also have trouble facing the electroweak precision
tests. Minimal Walking Technicolour (MWTC) [29] is a model which has the smallest
deviation from precision data, with the most economical particle content. In MWTC
theories the coupling remains large and nearly constant over a wide range of energy
scales. Technicolour theories have also been combined with supersymmetry to alleviate
some of the issues present [30].
Strong dynamics theories may contain composite fermionic or bosonic resonances or
composite Higgs particles. We will examine a model which contains scalar leptoquark
resonances in section 5.3.
2.3.2.4 Dimensional deconstruction and Little Higgs models
Dimensional deconstruction involves building extra dimensions instead of starting with
them [31]. The idea can be used to construct renormalisable four-dimensional gauge
theories that dynamically generate extra dimensions. In this framework, extra dimensions
can be used purely as an inspiration and may be discarded in the end, together with all
the issues they introduce (for example, without the need of justification of the size of
the extra dimension). As a result, realistic theories of electroweak symmetry breaking in
four-dimensions can be constructed, with the new feature that they are perturbative (as
supersymmetry is) and insensitive to high energy details up to a cut-off scale much larger
than O(TeV).
30 Chapter 2. The Standard Model and beyond
‘Little Higgs’ models are such models, inspired by the method of dimensional decon-
struction [32], although most do not have a simple five-dimensional or ‘theory space’
interpretation. In Little Higgs models a special ‘collective’ pattern is created in which
the gauge and Yukawa couplings break some global symmetries. As a consequence of
this special pattern, the one-loop contribution to the Higgs boson mass is not generated.
This can be viewed as the cancellation between divergences caused by the SM particles
(Fig. 2.5) and new resonances, as in supersymmetry. Unlike supersymmetry, the new
resonances have the same statistics as their SM counterparts. The remaining corrections
to the Higgs boson mass parameter are smaller and no fine tuning is required to keep
the Higgs boson light. At energies of O(10 TeV), the Little Higgs description becomes
strongly-coupled and the model needs to be completed in the ultraviolet regime, for exam-
ple by a QCD-like gauge theory with a confinement scale around 10 TeV [33]. Note that
in phenomenologically viable Little Higgs models, a discrete parity, called T -parity, needs
to be introduced to solve strains that arise from the electroweak precision tests [34, 35].
The phenomenological consequences are similar to those in supersymmetry: new heavy
resonances will be pair-produced and the theory may contain Dark Matter candidates.
2.3.2.5 String theory and all that
String theory [21] was ‘discovered’ by accident in the late 1960s, first proposed as a theory
of strong interactions. It was later dismissed as a valid theory of hadronic physics, but
some theorists considered the fact that it contained spin-2 resonances as an indication
that it could possibly lead to a theory of gravity. A lot of work has been done in the
‘framework’ of string theory and it has since become a popular candidate for a quantum
theory of gravity. However, organising principles for the ‘theory’ do not exist and at
present it seems that the number of possible solutions is practically infinite.
Another candidate for a theory of quantum gravity is ‘loop quantum gravity’. It
is a non-perturbative approach to a quantum theory of gravity, in which no classical
background metric is used. One of the consequences is that quantities such as area and
volume are quantised in units of the Planck length. It has modest aims, not attempting
unification; rather, its goal is to quantise Einstein’s general theory of relativity in four
dimensions [36].
Whatever the theory of quantum gravity looks like, what is certain at this point is
that we need a major revision of our understanding of the nature of space-time to discover
a path towards it.
Chapter 3
Monte Carlo methods and event
generators
3.1 Introduction
Theories of physical systems are formulated in terms of ‘equations of motion’. These are
usually differential equations which describe the rate of change of variables with respect
to system parameters, such as time. For concreteness, let us assume that the rate of
change of a particle’s position in one dimension, X, is given by the following differential
equation:
dX
dt
= f(t) , (3.1)
where f(t) is a given function of time. To calculate the displacement of the system from
t1 to t2, we need to integrate the differential equation:
X =
∫ t2
t1
f(t)dt . (3.2)
In fact, a vast majority of problems in Physics can be reduced to solving particular
integrals. Most of these integrals cannot be solved analytically, forcing us to resort to
numerical techniques to evaluate them.
In one dimension, calculating integrals numerically is just a matter of applying certain
well-known techniques, such as the trapezium rule, Simpson’s rule, gaussian quadrature
and so on. These take the values of f onN points {t1, ..., tN}, at certain fixed intervals, and
31
32 Chapter 3. Monte Carlo methods and event generators
yield an estimate of the integral. These ‘quadrature’ methods are based on approximating
the function f(t) with some polynomial. An alternative technique is based on the fact
that the value of the integral can be recast as the average of the integrand:
X = (t2 − t1) 〈f(t)〉 . (3.3)
We can approximate the average of the integrand by taking N values of t, uniformly
distributed on (t1, t2), and hence obtain a reasonable estimate of X:
X ≈ (t2 − t1) 1
N
N∑
i=1
f(ti) . (3.4)
The order in the sum in Eq. (3.4) is not of any significance so it is possible to draw the
N values of ti randomly. We can then write
X ≈ (t1 − t2) 〈f(ti)〉 ,
ti = (t2 − t1)ρi + t1 , (3.5)
where ρi is a random number
1 on the interval (0, 1). This method of using random
numbers to aid the calculation of integrals is known as ‘Monte Carlo’ integration.2 If we
assume N ≫ 1, we can estimate the accuracy of the method using the Central Limit
Theorem. The distribution of 〈f(ti)〉 will tend to a Gaussian with standard deviation
σ = σi/
√
N , where σi is the standard deviation of the values of f(xi). This implies that
the inaccuracy of our estimate simply decreases as 1/
√
N .
In particle physics the dimensionality of the integrals is usually very large and vari-
able: for an n-particle final state, there are 3n − 4 dimensions, coming from the three
components of momentum and taking into account the total 4-momentum conservation,
plus flavour and spin labels. Table 3.1 shows the rate of convergence of the various in-
tegral approximation techniques in d-dimensions: the ‘quadrature’ rules all suffer from
1A random number is one whose value is unpredictable from any initial information. In practice
pseudo-random numbers are used in Monte Carlo event generators. These are sequences of numbers
which are fully deterministic, but are supposed to be indistinguishable from random numbers. We will
assume that a pseudo-random number generator has been provided and we will not make the distinction
in this thesis.
2The name is, of course, inspired by the ‘Casino de Monte Carlo’ which is a ‘legendary casino, a jewel
of the arts ‘Belle Epoque’, the absolute reference for all players. Its wide range of table games is the
most prestigious and the most complete in Europe’ [37]. The only association of the author of the thesis
to the famous casino appears at http://www.hep.phy.cam.ac.uk/theory/andreas/mc.jpg.
3.2. Monte Carlo event generators 33
Technique Convergence
trapezium 1/N2/d
Simpson’s 1/N4/d
mth-order gaussian quadrature 1/N (2m−1)/d
Monte Carlo 1/
√
N
Table 3.1: The rate of convergence with the number of points N used for each method in
d-dimensions.
the problem that they converge in accordance to N1/d, the number of points along each
axis.3 The convergence of the Monte Carlo technique overtakes each of the methods
mentioned at d = 4, 8 and d = 4m − 2 respectively and hence it is well-suited for the
high-dimensional integrals that appear in particle physics. The Monte Carlo integration
method in fact possesses many advantages over numerical quadrature methods: it con-
verges fast in many dimensions, it can handle complex integration regions, it has a small
‘feasibility limit’ (the number of function evaluations which must be made for the method
to work) and it is easy to estimate the accuracy of the result. Moreover, it is useful in
the study of fundamentally random processes since there can be a direct correspondence
between the parameter space points and actual events being modelled. This is one of the
main reasons why the Monte Carlo method has become an important tool for collider
experiments, through its use in constructing event generators and detector simulators.
3.2 Monte Carlo event generators
An event generator can be defined as any program which aims to simulate individual
events, rather than the bulk properties of a physical process. Using the Monte Carlo
method, an event generator is capable of simulating a wide range of interesting processes
that are expected at hadron colliders such as the LHC.
As we have already noted at the end of the previous section, in particle physics the
Monte Carlo method is closely related to the physical process under study in a way that
allows us to make a direct connection between phase space points and events. The Monte
Carlo method can be used to generate ‘unweighted’ events which can be manipulated like
those obtained by experiment.
3However this is only true for the quadrature methods if the derivatives exist and are bounded.
34 Chapter 3. Monte Carlo methods and event generators
3.2.1 Unweighted event generation
The unweighting of events (or phase space points) is performed by a method called ‘hit-
or-miss’. The method can be briefly described in the following steps:
• We find the maximum of the distribution f(t) in the interval (t1, t2), fmax, during an
initial sampling. This is taken to be the maximum weight in the integration region.
• In a subsequent sampling process, we choose whether we keep (accept) or throw
away (reject) a randomly chosen event with properties given by t, with probability
f(t)/fmax. We thus end up with a sample of events which have been ‘accepted’.
• During the initial sampling, the value of the integral, X, can also be calculated
using Monte Carlo integration as described by Eq. (3.5).
It should be understood that t is now a multidimensional phase space point. The above
steps can be translated to particle physics ‘language’ readily: the point t would be a set of
particle momenta and other quantum numbers, f(t) would be the matrix element squared
for the configuration t and X would be the total cross section within the cuts set by t1
and t2.
Both the convergence of the integral and the event generation efficiency can be im-
proved if importance sampling is employed. The basic idea is to perform a Jacobian
transformation so that the integrand is flatter in the new integration variable, thus reduc-
ing the standard deviation of the weights. Using importance sampling can improve the
efficiency by many orders of magnitude.
3.2.2 General features of event generators
We outline the components of an event generator. For further details the recent review [38]
is useful. We also discuss jets: although not explicitly part of the event generation, they
are important for making the connection between the parton picture, arising from the
theoretical calculations, and the hadrons observed in experiments.
3.2.2.1 Hard subprocesses
A particle physics event generator has at its core the simulation of ‘hard subprocesses’:
particle scattering at large invariant momentum transfer. These processes are calculable
3.2. Monte Carlo event generators 35
in the framework of QCD, as it becomes asymptotically free at high energy and hence
perturbative.
Unweighted hard scattering events are produced using the ‘hit-or-miss’ method as
described in section 3.2.1. In the case of hadron colliders, with incoming hadrons h1 and
h2, partons i and j are ‘extracted’ from each respectively, with corresponding momentum
fractions xi and xj . The probability density functions for the momentum fractions are then
given by the relevant Parton Density Functions (PDFs): fi/h1(xi, µ
2) and fj/h2(xj , µ
2), as
defined in section 2.2.3.3. The cross section for the hard subprocess h1h2 → n + X,
can then be calculated by assuming that the non-perturbative hadron physics and the
short-distance hard physics can be factorized [38]:
σ =
∑
i,j
∫ 1
0
dxidxj
∫
fi/h1(xi, µ
2
F )fj/h2(xj , µ
2
F )dσˆij→n(µF , µR) , (3.6)
where µF and µR are the relevant factorisation [10] and renormalisation scales and the
parton-level differential cross section may be written as
dσˆij→n(µF , µR) =
1
2sˆ
|Mij→n|2 (Φ;µF , µR)dΦn , (3.7)
where Φ is the final-state phase space, sˆ = xixjs is the partonic centre-of-mass energy
squared in terms of the hadronic centre-of-mass energy squared, s, and |Mij→n|2 is the
matrix element squared for the process, averaged over initial spin and colour. The phase
space dΦn over the n final-state particles is given by
dΦn =
n∏
k=1
d3pk
(2π)32Ei
(2π)4δ(pi + pj −
n∑
k=1
pk) , (3.8)
where pi and pj are the initial-state momenta.
3.2.2.2 Parton showers
The particles which participate in the hard subprocesses at hadron colliders are likely to
carry QCD charge. Of course, this is always true for the incoming partons. Inevitably,
colour-charged particles will radiate via the QCD interaction: quarks and gluons can
radiate gluons, gluons can branch out to quark-anti-quark pairs. This is in accordance
with the parton branching picture we presented in section 2.2.3.4, where we wrote down
the expression for the differential cross section for a single branching, Eq. (2.52). This
36 Chapter 3. Monte Carlo methods and event generators
equation contains a 1/t factor, and hence the t→ 0 phase space region is enhanced. This
is in fact a collinear-enhanced region, since, as we have already shown (Eq. 2.48), t ∝ θ2,
where θ is the opening angle for the branching.
The Monte Carlo method is well-suited for describing parton branching. This is done
in terms of the Sudakov form factor, which forms the basis of the ‘parton shower’. The
Sudakov form factor,
∆ba(t0, t) = exp
[
−
∫ t
t0
dt′
t′
∫
dz
αs
2π
Pˆba(z)
]
, (3.9)
is simply the probability of a parton evolving from scale (squared) t0 to t without branch-
ing, for a certain type of evolution ba (a parton a evolving to a parton b). The integral
over z would diverge at z = 0 and z = 1 and so needs to be cut-off at appropriate val-
ues, functions of the cut-off scale t0. The Monte Carlo branching algorithm, details of
which can be found in [10], evolves the parton from squared scale and momentum frac-
tion (t1, x1) to (t2, x2) in steps in (t, x) space. The algorithm takes into account collinear
enhancements to all orders in perturbation theory, a procedure called ‘resummation’ to
which we will return in chapter 4.
In addition to the collinear enhancements, there are enhancements due to the emission
of soft gluons. These are treated in detail in [10]. It turns out that these follow a
remarkable property called angular ordering: an example of a coherence effect common to
all gauge theories. The upshot of this effect is that parton emissions occur at successively
smaller angles as one evolves from the high scale of the hard subprocess to lower scales.
In the case of final-state showers, this means as one evolves ‘forwards’ from the hard
subprocess to the hadronization scale at which the outgoing hadrons are formed. But for
initial-state showers, it means as one evolves ‘backwards’ from the hard subprocess to the
constituent partons of the incoming hadrons. It is convenient to generate the initial-state
showers by backwards evolution because the hard subprocess kinematics must be specified
first and the shower must evolve to satisfy them.
3.2.2.3 Hadronization
The hard subprocess in conjunction with the parton shower will produce a set of partons
(quarks/gluons) from the scattering of the incoming hadrons, possibly associated with
some heavier resonances (heavy quarks, gauge bosons or new heavy particles) which would
then subsequently be decayed. Yet quarks and gluons are never visible in their own right.
3.2. Monte Carlo event generators 37
After the parton shower has terminated, we are left with a set of partons with virtuali-
ties (square of 4-momenta) of the order of the cut-off scale, which lies in the low momentum
transfer, long-distance regime. This regime is dominated by non-perturbative effects, such
as the conversion of the partons into hadrons, or hadronization. As a non-perturbative ef-
fect, hadronization can currently only be described by phenomenological models that are
inspired by QCD. The general approach assumes the hypothesis of local parton-hadron
duality, which associates the flow of momentum and quantum numbers at parton level
with those at hadron level. This hypothesis is important as it allows us to associate the
theoretical parton-level calculations with what is observed at collider experiments, the
hadrons. Here we briefly discuss the main features of two popular hadronization models:
the string model and the cluster model. Further details on these can be found in [10,38].
The string model is based on the assumption of linear confinement at large distances.
For example, for the production of a qq¯ pair, the model assumes the physical picture of a
‘colour flux tube’ being stretched between the q and the q¯. For a uniform tube, this leads
to a confinement picture with a linearly rising potential V (r) = κr, where κ is the string
constant, phenomenologically taken to have the value κ ≈ 0.2 GeV2. As the q and q¯
move apart the string may break, creating a new q′q¯′ pair, and the system breaks into two
colour-singlet systems qq¯′ and q′q¯. The string model offers a very predictive framework and
applies to complicated multiparton distributions but contains many parameters related
to flavour properties which need to be determined by data.
The cluster model is based on the so-called preconfinement property of parton showers:
the colour structure of the shower at any evolution scale is such that the colour-singlet
combinations of partons (i.e. clusters) can be formed with an asymptotically universal (i.e.
dependent only on the evolution scale and the QCD scale) invariant mass distribution. In
this model, cluster hadronization starts with non-perturbative splitting of gluons into qq¯
pairs (or diquark-anti-diquark) and the formation of clusters from colour-connected pairs.
Clusters above a certain (flavour-dependent) maximum mass are first fragmented in a
string-like fashion into lighter clusters. Most clusters then undergo sequential two-body
phase space decays. In the cluster model, heavy flavour, strangeness and baryon produc-
tion are suppressed and transverse momenta are limited by the cluster mass distribution.
The model, in combination with the angular-ordered shower, gives a fairly good overall
description of high-energy collider data, less good than the string model but with fewer
parameters.
38 Chapter 3. Monte Carlo methods and event generators
3.2.2.4 Jets
As we have seen, almost immediately after they are produced, quarks and gluons fragment
and hadronize, leading to sprays of energetic hadrons, which we may call ‘jets’. However,
the definition of a jet is not a simple one, even though we may easily identify the structure
on an event display. Defining a jet is complicated for several reasons:
• Partons have divergent branching probabilities in perturbative QCD.
• A jet may originate from the hadronic decay of a heavy particle or it may be radia-
tive, arising through the emission of a parton via parton branching.
• It is also never possible to identify a jet with a single parton: jets are colour-singlets
whereas partons are not.
To address these issues and create a consistent definition of what a jet is we need to define
a ‘jet algorithm’. These algorithms provide a set of rules for grouping particles into jets.
They involve a parameter that defines a distance measure between the particles, used
as a criterion to judge whether they belong to the same jet or not. A ‘recombination
scheme’ is also required, which indicates what the momentum of the combination of two
particles is when they are grouped together. A jet algorithm together with a recombination
scheme form a ‘jet definition’. A jet definition should possess the following properties, the
‘Snowmass accord’ [39], set in 1990 by a group of influential theorists and experimentalists:
• It has to be simple to implement in an experimental analysis and a theoretical
calculation.
• It has to be defined at any order of perturbation theory and yield a finite cross
section at any order of perturbation theory.
• It has to yield a cross section that is relatively insensitive to hadronization.
Another important and desirable property not explicitly mentioned but implied by the
above list is ‘infrared and collinear safety’ (IRC): if one modifies an event by either adding
a collinear or a soft emission, the hard jets that are found in the event should remain
unchanged. Some modern jet definitions do satisfy the Snowmass criteria and are IRC
safe. Examples are the kt and anti-kt algorithms, the Cambridge/Aachen algorithm and
seedless cone algorithms (e.g. SISCone). For a further, more detailed discussion on
carefully defining and using IRC-safe and Snowmass-accord jet algorithms see Ref. [40].
3.2. Monte Carlo event generators 39
3.2.2.5 The underlying event
Strong evidence for the existence of the ‘underlying event’ has existed since the CERN
ISR experiment, through measurements of momentum imbalance [41]. Experimentally,
what is called the ‘underlying event’ (UE) includes all activity in a hadronic collision
that is not related to the ‘signal’ particles from the hard subprocess (for example, the
leptons in the Drell-Yan process). This definition will obviously include initial- and final-
state radiation described by the parton shower, but for the sake of modelling, these extra
emissions can be attributed to the hard subprocess. It is then thought that the UE
originates from additional soft scatters that occur during a hadron-hadron collision. It is
a significant effect to consider when it comes to measuring jet properties: jet algorithms
will inevitably gather together any energy deposits from the soft particles in the vicinity
when constructing a jet. This will alter the energy and internal structure of the jets
formed by the algorithm.
Currently, description and understanding of the UE comes mainly through the use
of phenomenological models. The simplest model, called the UA5 model [42], treats the
UE as additional soft hadronic activity generated by additional clusters which have been
formed flat in rapidity, with an exponentially falling transverse momentum distribution.
A more recent, and phenomenologically more successful, model treats the UE as a
sequence of more or less independent parton interactions which include full parton show-
ers [43, 44]. This is capable of describing the jet-like structure of the UE. The additional
scatters are always modelled as simple 2 → 2 QCD scatterings as long as the scattering
contains a hard jet of at least a few GeV.
We will be examining the effect of the UE described by a model of the latter philosophy
of multiple parton interactions on certain hadron collider variables in chapter 4.
3.2.3 The HERWIG++ event generator
There is a healthy choice of general-purpose Monte Carlo event generators on the mar-
ket. The latest incarnations of the more popular generators are Pythia 8 [45] and
HERWIG++ [46], and a ‘new-comer’, SHERPA [47]. These differ mainly in physics, philos-
ophy and implementation of the various components described in section 3.2.2, a variety
which allows cross-checking of models and implementations.
The author of the present thesis has been a heavy user of the HERWIG++ event genera-
tor as well as member of the HERWIG++ collaboration and hence this thesis will inevitably
40 Chapter 3. Monte Carlo methods and event generators
be heavily biased towards that event generator. In this section we briefly review the main
features of the HERWIG++ event generator, in reference to the general features we have
presented in section 3.2.2. For further details, one can consult the manual [46] and public
webpages [48] which are updated to keep up with the latest developments.
HERWIG++ is based on the FORTRAN event generator HERWIG (which stands for Hadron
Emission Reactions With Interfering Gluons), first published in 1986 [49]. HERWIG++ is
not just a rewrite of the earlier FORTRAN version in the C++ language, but also introduces
physics improvements whenever necessary and feasible.
3.2.3.1 ThePEG
HERWIG++ is distributed as a comprehensive collection of plugin modules to ThePEG, the
‘Toolkit for High Energy Physics Event Generation’ [50], which provides all the infras-
tructure that is necessary to construct an event generator. It can handle random number
generation, the event record and a mechanism for creating physics implementations for
all steps of event generation. It also provides a reader for the Les Houches Accord event
format [51], a feature we make use of in section 5.2 to perform a phenomenological analysis
of a heavy charged vector boson model.
3.2.3.2 Hard process generation, parton shower and decays
Three main mechanisms for simulation of hard processes are available in HERWIG++:
1. A hand-coded set of matrix elements for common processes in hadron, lepton and
deep inelastic scattering collisions. These are written using a reimplementation of
the HELAS helicity amplitude formalism, which allows the spin correlations to be
carried forward to the remaining event simulation consistently.
2. A generic matrix element calculator for 2 → 2 processes, mainly used for Beyond-
the-Standard Model physics, which automatically determines the permitted dia-
grams for a set of given external legs from a list of active vertices.
3. As already mentioned, it is possible to read Les Houches Accord event format files
at parton level, with any number of legs to be read from external sources.
The parton shower associated with the hard processes is based on a new evolution
variable q˜, motivated from the branching of gluons off heavy quarks [52]. The evolution
in this variable ensures the angular ordering of emissions, to take into account the colour
3.2. Monte Carlo event generators 41
coherence effects. Prior to the parton shower, heavy unstable particles (e.g. Higgs bosons,
W , Z, top quarks and other BSM particles) are decayed. All intermediate coloured lines
are also showered.
3.2.3.3 Hadronization and hadron decays
HERWIG++ uses the cluster hadronization model, described in section 3.2.2.3. The unsta-
ble hadrons that are formed are decayed via the same framework as fundamental unstable
particles: using either a general matrix element based on the spin structure, or a specific
matrix element for the important decay modes.
3.2.3.4 The underlying event
The implementation of the multiparton scattering in HERWIG++ is connected to the parton
shower and hadronization models. Event generation starts with sampling the hard process
according its matrix element and PDFs. The parton shower evolves the final-state partons
from the scale of the hard interaction down to the cut-off scale for the hadronization. The
incoming partons are extracted out of the hadrons and the chosen number of secondary
interactions is sampled according to the QCD 2 → 2 matrix elements and the same
PDFs as for the hard process. The partons involved in the additional hard scatters also
undergo parton showers. Secondary interactions are showered in an identical way to the
hard process. However, backward evolution has to be modified: for example, an additional
scattering may lead to partons with more energy than the remaining energy of the hadron
remnants, and has to be vetoed. A further modification is that any incoming partons are
always evolved back to an initial gluon, with a gluon distribution identical to the one in
the initial hadron. Further details of the implementation can be found in [44].
3.2.3.5 Beyond the Standard Model
Each new physics model in HERWIG++ is implemented in a model class which includes
the model parameters and vertex structure. A model input text file allows for changes
in the parameters and all possible production and decay matrix elements with up to four
external legs are constructed. The BSM models currently available in HERWIG++ (version
2.5.0) are [53]:
• Supersymmetric models: MSSM and NMSSM implemented with flexibility in the
parameters via the Susy Les Houches Accord file reader [54].
42 Chapter 3. Monte Carlo methods and event generators
• A model for universal extra dimensions and an implementation of the Randall-
Sundrum model and ADD-type gravitons.
• A model for transplanckian scattering.
• A model involving non-derivatively and derivatively-coupled leptoquark resonances.
We will study the implementation and phenomenology of the leptoquark model in detail
in section 5.3.
3.3 Next-to-leading order matching
Fixed-order matrix elements are excellent for simulating well separated, hard partons,
but are unable to describe collinear and soft partons, which have logarithmically divergent
probabilities. Parton showers are good in the opposite region: hard, wide-angled emissions
are handled poorly while the enhanced soft and collinear emissions are well-described, even
for multiple emissions. Generic Monte Carlo event generators start the parton shower from
a leading order (LO) distribution of partons to produce a high multiplicity hadronic state
with relatively low transverse momenta. The fact that the shower starts from the LO
distribution implies that the total cross section is also accurate to that order.
However, for many processes there exist next-to-leading order (NLO) perturbative
calculations. These may provide significant corrections to both the total cross section and
the shape of distributions of observables. They are also essential in providing control over
the scale dependence of our calculations, absent from a LO calculation. We would thus like
to combine the NLO matrix elements with the parton shower. The task is non-trivial for
several reasons, which we shall discuss. We present a brief overview of two popular NLO
‘matching’ methods, the ‘Monte Carlo at Next-to-leading Order’ (MC@NLO) [55,56] method
and the ‘Positive Weight Hardest Emission Generation’ (POWHEG) [57, 58] method. The
discussion in this section has been adapted from [38] and [59]. We use these two methods
to simulate the production of heavy charged vector bosons at NLO in section 5.2.
3.3.1 MC@NLO
We begin by describing MC@NLO method for combining the NLO matrix element with the
parton shower. The parton shower and the NLO result contain terms of the same order
3.3. Next-to-leading order matching 43
and hence when defining any matching method we have to take care to avoid double-
counting. The MC@NLO method, as we shall see, exhibits the complication that a small
fraction of the generated events possess negative weights. These are few enough so that
the number of events required for constructing smooth distributions is comparable to that
for an ordinary LO process.
We demonstrate the procedure by applying it to a toy model, by assuming the emission
of only photons for simplicity. Consider a system that radiates particles with energy x,
such that 0 ≤ x ≤ 1. In perturbation theory, the cross section for a process at NLO (one
photon emission), after dimensional regularisation in d = 4− 2ǫ dimensions is given by4
σNLO = lim
ǫ→0
∫ 1
0
dxx−2ǫ
[(
dσ
dx
)
B
+
(
dσ
dx
)
V
+
(
dσ
dx
)
R
]
, (3.10)
where the factor x−2ǫ has been retained from the phase space factor and(
dσ
dx
)
B
= Bδ(x) ,(
dσ
dx
)
V
= a
(
B
2ǫ
+ V
)
δ(x) ,(
dσ
dx
)
R
= a
R(x)
x
, (3.11)
where a is the coupling constant (analogous to αs in QCD), B and V are constant with
respect to x and represent the Born (leading order) and virtual contributions, and R(x)→
B as x→ 0, where R is the real contribution.
Now, at leading order, an infrared-safe observable O(x) has an expectation value given
by
〈O〉LO = BO(0)
= lim
ǫ→0
∫
dxx−2ǫBδ(x)O(x) , (3.12)
where we have written the second line in a way convenient to be used below. We can then
4Dimensional regularisation is a technique used when renormalising a theory: the idea is to compute a
Feynman diagram as an analytic function of the dimensionality of space d. Any loop-momentum integral
will converge for sufficiently large d (in the case of infrared divergences). Hence in this case ǫ < 0 is
required.
44 Chapter 3. Monte Carlo methods and event generators
write down the equivalent NLO prediction for O using Eqs. (3.10) and (3.11):
〈O〉NLO = limǫ→0
∫
dxx−2ǫO(x)
[
Bδ(x) + a
(
B
2ǫ
+ V
)
δ(x) + a
R(x)
x
]
= (B + aV )O(0) + lim
ǫ→0
[
a
B
2ǫ
O(0) +
∫ 1
0
dxx−2ǫO(x)a
R(x)
x
]
. (3.13)
For an observable O(x), the ǫ parameter gives rise to poles of opposite sign in the virtual
and real contributions which cancel to give a finite integral. To show this explicitly we
start by using the fact that
aBO(0)
∫ 1
0
dx
x−2ǫ
x
= −aB
2ǫ
O(0) , (3.14)
to write the expectation value of O(x) as
〈O〉NLO = (B + aV )O(0) + limǫ→0
∫ 1
0
dxx−2ǫ
[
a
O(x)R(x)−BO(0)
x
]
, (3.15)
where Eq. (3.14) has been added and subtracted from Eq. (3.13). The integrand in the
second term does not contain any singularities and hence we can set ǫ = 0 to get:
〈O〉NLO =
∫ 1
0
dx
[
O(0)
(
B + aV − aB
x
)
+O(x)
R(x)
x
]
. (3.16)
This method of making the finiteness of the NLO expression explicit is called ‘subtraction’.
It yields an NLO-accurate expression for an observable. The MC@NLO formalism aims to
match the shower Monte Carlo to the NLO calculation, reproducing the expression of
Eq. (3.16).
An observable at leading order, i.e. with no radiation from the system before being
interfaced to the shower Monte Carlo, is given by the first line in Eq. (3.12). The energy
of the system in this case is xM = 1. A shower MC sums the enhanced higher order terms
to all orders to give the following for the distribution of the observable O:
dσ
dOLO+MC
= BIMC(O, xM) = BIMC(O, 1) , (3.17)
where IMC(O, xM) is the distribution of the observable after an MC shower starting with
energy xM . The above expression implies that the total rate at LO is given by B, which
3.3. Next-to-leading order matching 45
is due to the unitary nature of the parton shower. We can make an attempt to extend
Eq. (3.17) to NLO by simply replacing O(0) and O(x) in Eq. (3.16) with the MC shower
observables IMC(O, 1) and IMC(O, 1− x) respectively:
〈O〉naive =
∫ 1
0
dx
[
IMC(O, 1)
(
B + aV − aB
x
)
+ IMC(O, 1− x)R(x)
x
]
. (3.18)
We have assumed that the energy of the system is xM = 1 − x after the radiation of a
photon of energy x. The na¨ıve expression of Eq. (3.18) implies that we need to generate
two events for a randomly chosen x:
• A ‘no emission + shower MC’-type event with xM = 1 and weight given by the
integrand in the first term, B + aV − aB/x, and
• a ‘one emission + shower MC’-type event with xM = 1− x and weight given by the
integrand in the second term, aR(x)/x.
We have of course called Eq. (3.18) the ‘na¨ıve’ result since it is plagued by two issues.
The first is that as x → 0, the weights for the two types of events diverge even though
the integral is finite. The second issue is that this procedure introduces double-counting:
terms that appear in the NLO emission also appear in the shower MC. For an explicit
demonstration of how double counting occurs see Ref. [59].
Both identified problems of the na¨ıve subtraction can be solved by introducing a
‘modified subtraction’:
〈O〉mod =
∫ 1
0
dx
[
IMC(O, 1)
(
B + aV − aB(1−Q(x))
x
)
+ IMC(O, 1− x)R(x)− aBQ(x)
x
]
,
(3.19)
where we have added to the first term the following:
IMC(O, 1)
aBQ(x)
x
, (3.20)
and subtracted from the second term:
IMC(O, 1− x)aBQ(x)
x
, (3.21)
where aQ(x)/x is analogous to a splitting function αsP (z) in QCD and Q(x) is a mono-
46 Chapter 3. Monte Carlo methods and event generators
tonic function that satisfies
0 ≤ Q(x) ≤ 1 , lim
x→0
Q(x) = 1 , lim
x→1
Q(x) = 0 . (3.22)
The difference between the added and subtracted terms does not contribute to the ob-
servable at O(a) since IMC(O, xM) is independent of xM at O(a0). The function Q(x)
is dependent on the shower MC used, which effectively requires the MC@NLO method to
be customised for each MC. The modified procedure can be shown to coincide with the
NLO result at O(a) and the integrand for the two ‘types’ of events can be shown to be
finite as x→ 0 [59], hence solving both issues. The extension to QCD from the toy model
does not require any significant changes to the procedure we have described. A necessary
modification involves an extra term related to the initial-state collinear divergences.
Notice that the weights in Eq. (3.19) can be either positive or negative. These lead to
‘unphysical’ negative weight events, which can be handled during the unweighting process
by assigning to them a weight -1 instead of +1. This can be easily taken into account
when producing histograms of distributions, by removing an event from the histogram bin
it corresponds to rather than adding it there. The events with negative weights, however,
can be manipulated in the same way as the positively-weighted ones: e.g. cuts can be
applied to them and they can be processed by detector simulation software.
3.3.2 POWHEG
The ‘Positive Weight Hardest Emission Generation’ (POWHEG) method was proposed to
overcome the problem of negatively-weighted events generated by MC@NLO. The goal is to
generate the hardest emission first using the exact NLO matrix element and yield only
positively-weighted events. Another advantage of the POWHEG method is that it does not
depend upon the subsequent shower MC.
In Monte Carlo event generators with angular-ordered showers, such as HERWIG++,
the first emission is not necessarily the hardest one. Hence, implementing the POWHEG
method requires the use of a transverse momentum veto to ensure that any emissions
that follow the first one are softer. Also, a ‘truncated shower’, extra soft radiation, must
be generated to recover the double-logarithm accuracy of the shower.
We first write down the inclusive differential cross section as given by the first emission
3.3. Next-to-leading order matching 47
in a parton shower,
dσPS = dΦ0B
[
∆(Q2, Q20) +
∫
Q20
dq2
q2
∫
dz
αs
2π
P (z)∆(Q2, q2)
]
, (3.23)
where ∆(q21, q
2
2) is the Sudakov form factor (see Eq. (3.9)), Q
2 and Q20 are the starting scale
and cut-off scales for the shower respectively and dΦ0 is the Born phase space. Performing
the integral would give a total cross section equal to the Born cross section, as previously
mentioned.
Note that the real emission matrix element, R, can be split into a singular part and
a non-singular part: R = Rs + Rns. We can replace the splitting function in Eq. (3.23)
with the singular part of the real emission matrix element,
dq2
q2
dz
αs
2π
P (z)→ dΦrR
s
B
, (3.24)
where dΦr is the phase space of the radiation variables. The replacement can be carried
over to the Sudakov form factor by defining [38]
∆¯(Q2, q2) = exp
[
−
∫
dΦrαs
Rs
B
]
. (3.25)
For angular-ordered parton showers, a hard matrix element correction, related to the non-
singular term Rns, is necessary to cover the whole phase space and we get the following
result:
dσPScorr = dΦ0B
[
∆¯(Q2, Q20) +
∫
dΦrαs
Rs
B
∆¯(Q2, q2)
]
+ dΦ0,rαsR
ns . (3.26)
In the POWHEG method, we define Rs = R and hence Rns = 0. We then write
dσPOWHEG = dΦ0B¯
[
∆¯(Q2, Q20) +
∫
dΦrαs
Rs
B
∆¯(Q2, q2)
]
, (3.27)
where we have now used the NLO-weighted Born matrix element, B¯, defined by
B¯ ≡ B + αsV +
∫
(R− C) dΦr , (3.28)
where C are counter-terms chosen to approximate R with the same singularities so that
48 Chapter 3. Monte Carlo methods and event generators
the integral over four dimensions is finite. Parton showering using Eq. (3.27) will give
similar emissions as the first term in Eq. (3.26), but with a global NLO reweighing B¯/B,
which can be considered to be a local k-factor.5 The replacement B → B¯ will result in
the integrated cross section being correct to NLO.
We note that the term αsV +
∫
(R− C), which can be negative, is formally of order
αs and therefore it is always overcome by the positive-definite term B (otherwise the
perturbation expansion would not be valid). This results in positive weights being assigned
to each event, since the NLO-weighted B¯ is positive-definite.
5A k-factor is the ratio between a cross section at LO and one at higher order: k = σh.o./σLO.
Chapter 4
Effects of QCD radiation on hadron
collider observables
The original work in this chapter was done in collaboration with Bryan Webber and
Jennifer Smillie and appears in [1–3].
4.1 Introduction
We have already laid down a few convincing arguments for why we should expect new
particles or phenomena at the TeV scale in section 2.3.1. These include the naturalness
of the expected Higgs mass and the electroweak symmetry breaking scale, and arguments
that dark matter should naturally occur at the TeV scale. The LHC has been designed
to investigate the electroweak scale and hence it is hopeful that it will shed light to
some of the puzzles of the Standard Model within the next few years, by uncovering new
phenomena.
But searching for new physics at a hadron collider such as the LHC is a non-trivial task.
There are two main challenges we need to face. First of all, possible new physics signals can
be quite complex (see Fig. 4.1 for an illustration). New heavy resonances can decay into
multiple jets and/or leptons. If the new dynamics possesses some discrete parity, such as
R-parity in supersymmetry, then the new particles will be pair-produced, resulting in two
identical (or similar) decay chains and introducing a combinatorial problem. Moreover,
the end-point of likely long decay chains can very possibly be a particle which interacts
weakly with ordinary matter, especially since we expect it to be a dark matter candidate
(a WIMP). This would result in missing energy in our detector which would complicate
the issue further, especially in combination with the SM neutrinos which would also escape
49
50 Chapter 4. Effects of QCD radiation on hadron collider observables
χ
χ
p
p
M
Figure 4.1: New physics processes are expected to provide complex signals: multiple jets
and/or leptons resulting in combinatorial issues plus missing energy. The diagram
shows the production of two heavy resonances (blue) via some matrix element
M and their subsequent decay chains into SM particles (black), new resonances
(red) and finally into some weakly interacting particles that escape detection (red
dashes). This could be, for example, the production of a gluino and a squark with
subsequent decay into a χ in supersymmetry.
the detector. Moreover, signals are expected to have low production rates, especially in
comparison against backgrounds usually occurring orders of magnitude more often.
The second set of challenges we need to face is intrinsic to a high energy hadron collider
(see Fig 4.2). These are mainly due to the fact that the incoming hadrons are ‘bags’ of
quarks and gluons and the high energy interacting partons radiate gluons before they
interact, according to the parton branching picture we presented in section 2.2.3.4. Thus,
processes are always associated with copious collinear initial-state radiation (ISR) which
affects the longitudinal momentum components of particles. Moreover, it is not possible
to measure the initial longitudinal (i.e. along the z-axis, defined parallel to the detector’s
beam pipe) components of the interacting partons in hadron colliders, since what is left of
the colliding hadrons (the ‘remnants’) travels down the beam pipe and is not detected.1
But the issues are not only limited to ISR and our inability to measure the z-components
of incoming partons: secondary partonic interactions between the interacting hadrons also
play an important role, contaminating the signal with soft particles. It is also worthwhile
to note that since the LHC is destined to become a high-luminosity machine,2 secondary
1Note, however, that central exclusive production, in which the protons remain intact and forward
detectors can be used to measure the z-component of the central system, has been investigated. See, for
example, [60].
2Here we refer to the instantaneous luminosity of a collider with bunched beams. If two bunches
containing n1 and n2 particles collide with frequency f and the beam profiles are characterised by σx
and σy, the transverse beam profiles in the horizontal and vertical directions respectively, then the
4.1. Introduction 51
χ
χ
p
p
M
(...)
(...)
Figure 4.2: Physics searches at hadron colliders are complex: high energy processes are always
associated with copious initial-state radiation (blue gluon lines), and secondary
interactions are very likely (secondary partons linking to small blue interaction).
We do not show secondary proton-proton interactions (pile-up).
proton-proton interactions (or ‘pile-up’) are also expected to add soft particles to the
events. Of course, one cannot forget the various experimental challenges due to beam
halo, noise and so on.3
Several search variables of varying complexity and assumptions have been constructed
to tackle the aforementioned issues. This is done, for example, either through cleverly
defining variables that have discriminatory power against backgrounds, after cuts have
been imposed to the event sample, or variables that are insensitive to QCD effects in
certain ways.
Kinematic variables can be constructed in order to constrain the momenta of particles
and hence search for new physics. They represent the first step in understanding the
observations at hadron colliders, since they make very few assumptions about the details
of the underlying physical model, such as the gauge groups, spins or couplings, providing
model-independent, robust information [62]. The very simplest class of the kinematic
variables are global inclusive variables, i.e. those that make use of all observed momenta,
without hypothesising any particular structure of the final state. Since the longitudinal
instantaneous luminosity is given by L = (fn1n2)/(4πσxσy). The rate of events for a certain process of
cross section σ is then given by dN/dt = L × σ [13].
3See, for example, Ref. [61] for details of the expected performance of the ATLAS experiment at the
LHC.
52 Chapter 4. Effects of QCD radiation on hadron collider observables
momentum of the hard process is unknown, transverse variables of this class have been
generally investigated so far. Examples are the observed transverse energy ET , the missing
transverse energy, /ET , and their sum, HT = ET+ /ET . The distributions of these quantities
can provide information on the energy scales of new processes such as supersymmetric
particle production [63–65].
Although longitudinal components of final-state momenta are strongly influenced by
ISR, they do contain information about the underlying hard process. Indeed, the amount
of ISR emitted is determined by the energy scale of the subprocess. This has motivated
several studies of global inclusive variables that contain longitudinal components, such as
the sˆ
1/2
min and the total visible mass M variables, which will be investigated in section 4.2.
Initial-state radiation also modifies the distributions of the products of the hard pro-
cess. This effect has been studied in great detail for the processes of electroweak boson
production, with the result that the transverse momentum and rapidity distributions
of W , Z and Higgs bosons at the Tevatron and LHC are predicted with good preci-
sion [66–68]. In contrast, the equivalent ET distributions have received little attention.
In section 4.3 we will be investigating the ET distribution in vector boson production and
Higgs boson production.
4.2 Effects of QCD radiation on global inclusive vari-
ables
4.2.1 Global inclusive variables
Global inclusive variables are easily defined with reference to Fig. 4.3, as found in [69].
We focus on a specific subprocess, formed by the interaction of two incoming partons from
the hadrons (protons or anti-protons). The resulting final-state particles can be either
visible, Xi, or invisible, χi. The visible particles Xi can originate either from the hard
process itself or from ISR and can be jets, electrons, muons and photons. The invisible
particles consist of SM neutrinos, which we take to be massless, and some new massive
BSM particles, not necessarily of the same type. We can define, using the sum of the
4-momenta of all the visible particles, the total 3-momentum P and the total energy E.
The only experimental information we possess about the invisibles is the total missing
transverse momentum 2-vector, /P T . Any global inclusive variable can be defined by using
E, components of P and /P T .
4.2. Effects of QCD radiation on global inclusive variables 53
X1
X2
X3
X4
Xnvis
χninv
χnχ+2
χnχ+1
p(p¯)
p(p¯)
χnχ
χ2
χ1
E,Px, Py, Pz
6~PT
Figure 4.3: Global inclusive variables can be defined with reference to the above figure, taken
from [69]. Black (red) lines correspond to SM (BSM) particles. The solid lines de-
note SM particles Xi, i = 1, 2, . . . , nvis, which are visible in the detector, e.g. jets,
electrons, muons and photons. The SM particles may originate either from initial-
state radiation, or from the hard scattering and subsequent cascade decays (indi-
cated with the green-shaded ellipse). The dashed lines denote neutral stable invisi-
ble particles χi, i = 1, 2, . . . , ninv which consist of some number nχ of BSM particles
(indicated with the red dashed lines), as well as some number nν = ninv − nχ of
SM neutrinos (denoted with the black dashed lines). The global event variables
describing the visible particles are: the total energy E, the transverse components
Px and Py and the longitudinal component Pz of the total visible momentum P.
The only experimentally available information regarding the invisible particles is
the missing transverse 2-momentum /P T .
4.2.2 The variable sˆmin
In Ref. [69] various global variables were investigated, including those that make use
of longitudinal as well as transverse momentum components. The quantities studied
included the total energy E visible in the detector (as defined in section 4.2.1) and the
visible invariant mass M ,
M =
√
E2 − P 2z − /P 2T , (4.1)
where Pz is the visible longitudinal momentum. In addition a new variable was introduced,
defined as
sˆ
1/2
min(Minv) ≡
√
M2 + /P
2
T +
√
M2inv + /P
2
T , (4.2)
54 Chapter 4. Effects of QCD radiation on hadron collider observables
where the parameter Minv is a variable estimating the sum of masses of all invisible
particles in the event:
Minv ≡
ninv∑
i=1
mi . (4.3)
The variable sˆ
1/2
min is simply the minimum value of the parton-level Mandelstam variable sˆ
(the square of the partonic centre-of-mass energy) which is consistent with the observed
values of the total energy E, Pz and /P T in a given event. It was argued that the peak
in the distribution of sˆ
1/2
min is a good indicator of the mass scale of new physics processes
involving heavy particle production in the absence of ISR and multiple parton interactions.
In Ref. [69] specific examples were given in tt¯, gluino pair-production and gluino-LSP
associated production and the dependence on Minv was also investigated.
It was also recognised that the effects of the ISR and the multiple parton interactions
make this measurement impossible, since sˆ
1/2
min would measure the total energy of the
full system, whereas the interest is on the energy of the hard scattering. In Ref. [70]
an alternative approach was proposed, preserving the definition, but instead of using
information from the calorimeters to construct the quantities E, P and /P T , reconstructed
objects were used. That means, objects that have been recognised via some algorithm as
jets, muons, electrons or photons were used. This seems to recover some of the attributes
of sˆ
1/2
min, but the distributions are not calculable analytically and hence we do not pursue
this method here.
4.2.3 ISR effects without invisible particle emission
In the present section we examine the effects of ISR on global inclusive variables, first in
an approximate fixed-order treatment, taking into account collinear-enhanced terms, and
then in an all-orders resummation of such terms. We quantify the way the distributions
of quantities that involve longitudinal momenta depend on the scale of the underlying
hard subprocess and on the properties of the detector, in particular the maximum visible
pseudorapidity ηmax. Initially, we ignore the effect of invisible final-state particles: we
assume that all the final-state particles from the hard subprocess are detected. We will
present the treatment of invisibles in section 4.2.5.
The Monte Carlo results presented in Ref. [69] show that the second term on the
right-hand side of Eq. (4.2) is not strongly affected by ISR. The first term is intended to
add extra longitudinal information about the hard subprocess, allowing a more reliable
4.2. Effects of QCD radiation on global inclusive variables 55
determination of its mass scale. The extra longitudinal information enters through the
visible mass M , and we therefore concentrate on this quantity.
4.2.3.1 Born approximation
In the Born (or leading-order) approximation, assuming that no beam remnants are de-
tected, M yields a perfect estimate of the centre-of-mass energy of the hard subprocess
with no invisibles. For incoming partons with momentum fractions x1,2:
E =
1
2
√
S(x1 + x2) , Pz =
1
2
√
S(x1 − x2) , (4.4)
where
√
S is the hadron-hadron centre-of-mass energy, so that
M2 = x1x2S , Y ≡ 1
2
ln
(
E + Pz
E − Pz
)
=
1
2
ln
x1
x2
, (4.5)
where we have also defined the rapidity of the visible system, Y . The differential cross
section for parton flavours a, b is thus:
dσab
dM2dY
=
∫
dx1 dx2 fa(x1)fb(x2)δ(M
2 − x1x2S)δ
(
Y − 1
2
ln
x1
x2
)
σˆab(x1x2S) , (4.6)
where fa,b are the relevant parton distribution functions for the incoming hadrons and σˆab
is the hard subprocess cross section. Hence, at Born level we find
S
dσab
dM2dY
= fa
(
M√
S
eY
)
fb
(
M√
S
e−Y
)
σˆab(M
2) . (4.7)
The parton distributions are sometimes given as Fi(x) = xfi(x), in terms of which we
have
M2
dσab
dM2dY
= Fa
(
M√
S
eY
)
Fb
(
M√
S
e−Y
)
σˆab(M
2) . (4.8)
If the partonic cross section σˆab has a threshold or peak, indicating that the ab subprocess
has a characteristic scale Q, then this is also manifest in the Born cross section (4.8) at
M ∼ Q, provided the relevant parton distributions are large enough for that subprocess
to contribute significantly.
56 Chapter 4. Effects of QCD radiation on hadron collider observables
4.2.3.2 Quasi-collinear NLO correction
To examine the sensitivity of the above results to ISR, let us first compute the NLO
contribution due to quasi-collinear gluon emission and the associated virtual corrections.
Consider first the emission of a gluon from parton a. If the emission angle θ, defined with
respect to the beam direction in the lab frame, is large enough, say θ > θc, the gluon
enters the detector and contributes toM . In the small-angle approximation we then have
E =
1
2
√
S(x1/z + x2) , Pz =
1
2
√
S(x1/z − x2) , (4.9)
where x1/z is the momentum fraction of parton a before the emission, so that
M2 = x1x2S/z , Y =
1
2
ln
x1
zx2
. (4.10)
The correction associated with a detected emission from parton a is then:
αs
π
∫
θc
dθ
θ
dz
z
dx1 dx2Pˆa(z)fa(x1/z)fb(x2)δ(M
2 − x1x2S/z)δ
(
Y − 1
2
ln
x1
zx2
)
σˆab(x1x2S)
(4.11)
where Pˆa(z) is the unregularised a → ag splitting function and we have neglected the
running of αs for the moment.
On the other hand if the gluon misses the detector (θ < θc), E and Pz are still given
by Eq. (4.4), so the contribution is
αs
π
∫ θc
0
dθ
θ
dz
z
dx1 dx2Pˆa(z)fa(x1/z)fb(x2)δ(M
2 − x1x2S)δ
(
Y − 1
2
ln
x1
x2
)
σˆab(x1x2S) .
(4.12)
Finally the associated virtual correction is the term that regularises the splitting function,
which in this approximation is
−αs
π
∫
dθ
θ
dz dx1 dx2Pˆa(z)fa(x1)fb(x2)δ(M
2 − x1x2S)δ
(
Y − 1
2
ln
x1
x2
)
σˆab(x1x2S) .
(4.13)
Adding everything together gives a correction
δ
(
dσab
dM2dY
)
=
αs
π
∫
dθ
θ
dz dx1 dx2 Pˆa(z)fb(x2)σˆab(x1x2S)
×
[1
z
fa(x1/z)δ
(
Y − 1
2
ln
x1
zx2
)
δ(M2 − x1x2S/z)Θ(θ − θc)
4.2. Effects of QCD radiation on global inclusive variables 57
+
{1
z
fa(x1/z)Θ(θc − θ)− fa(x1)
}
δ
(
Y − 1
2
ln
x1
x2
)
δ(M2 − x1x2S)
]
.
(4.14)
Setting aside for the moment the possibility of splittings other than a→ ag, the DGLAP
evolution equation for fa(x1) is
q
∂
∂q
fa(x1) =
αs
π
∫
dz Pˆa(z)
[
1
z
fa(x1/z)− fa(x1)
]
, (4.15)
where q represents the scale at which the parton distribution is measured. Hence the
correction may be written as
δ
(
dσab
dM2dY
)
=
∫
dθ
θ
dx1 dx2 fb(x2)σˆab(x1x2S)
[
q
∂fa
∂q
δ
(
Y − 1
2
ln
x1
x2
)
δ(M2 − x1x2S)
+
αs
π
∫
dz
z
Pˆa(z) fa(x1/z)
{
δ
(
Y − 1
2
ln
x1
zx2
)
δ(M2 − x1x2S/z)
− δ
(
Y − 1
2
ln
x1
x2
)
δ(M2 − x1x2S)
}
Θ(θ − θc)
]
. (4.16)
Since dθ/θ = dq/q, the first term represents a change of scale in the Born term. It replaces
the reference scale in fa by the scale Q of the hard subprocess. The remaining terms give
a correction
δ
(
dσab
dM2dY
)
=
αs
πS
∫
θc
dθ
θ
∫
dz
z
Pˆa(z)fb
(
M√
S
e−Y
)
×
[
fa
(
M√
S
eY
)
zσˆab(zM
2)− fa
(
M
z
√
S
eY
)
σˆab(M
2)
]
. (4.17)
In leading-log approximation the θ integration just gives a factor of − ln θc. In the same
approximation, we may set − ln θc = ηmax,4 the maximum pseudorapidity seen by the
detector. Note that this is a different quantity from Y , the true rapidity of the visible
system. The correction associated with parton b gives the same expression with a ↔ b
and Y → −Y . Thus, defining
x¯1 =
M√
S
eY , x¯2 =
M√
S
e−Y , (4.18)
4Note that this is an approximation to the conventional definition of the pseudorapidity η =
− ln tan( θ
2
), consistent with the leading-long approximation.
58 Chapter 4. Effects of QCD radiation on hadron collider observables
we have
S
dσab
dM2dY
= fa(x¯1, Q)fb(x¯2, Q)σˆab(M
2)
+ ηmax
αs
π
∫
dz
z
[
z{Pˆa(z) + Pˆb(z)}fa(x¯1, Q)fb(x¯2, Q)σˆab(zM2)
− {Pˆa(z)fa(x¯1/z,Q)fb(x¯2, Q) + Pˆb(z)fa(x¯1, Q)fb(x¯2/z,Q)}σˆab(M2)
]
.
(4.19)
Expressing this in terms of Fi(x) = xfi(x), as in Eq. (4.8),
M2
dσab
dM2dY
= Fa(x¯1, Q)Fb(x¯2, Q)σˆab(M
2)
+ ηmax
αs
π
∫
dz
[
{Pˆa(z) + Pˆb(z)}Fa(x¯1, Q)Fb(x¯2, Q)σˆab(zM2) (4.20)
− {Pˆa(z)Fa(x¯1/z,Q)Fb(x¯2, Q) + Pˆb(z)Fa(x¯1, Q)Fb(x¯2/z,Q)}σˆab(M2)
]
.
Results for tt¯ production at the LHC (pp at
√
S = 14 TeV) with ηmax = 5 and Y = 0
are shown in Fig. 4.4. Leading-order MSTW parton distributions [71] were used. For
simplicity we have taken Q = M . Recall that the simplifying assumption made here is
that all tt¯ decay products are detected, so theM distribution vanishes below tt¯ threshold.
We see that there is a large negative NLO correction near threshold, followed by a broad
positive peak. To understand these qualitative features, consider the case a = b, as in
gg → tt¯, and Y = 0, so that x¯1,2 = M/
√
S ≡ x¯. Then the NLO correction becomes
simply:
δ
(
M2
dσ
dM2dY
)
= 2ηmax
αs
π
F (x¯)
∫
dzPˆa(z)
[
F (x¯)σˆ(zM2)− F (x¯/z)σˆ(M2)] . (4.21)
The first term is positive-definite, contributes only above threshold, and diverges at thresh-
old. It produces the broad positive peak. The second term is negative-definite, contributes
around threshold, and has a divergent coefficient. It provides the sharp negative peak.
4.2.3.3 Resummation
By adding and subtracting the expression
{P˜a(z) + P˜b(z)}Fa(x¯1, Q)Fb(x¯2, Q)σˆab(M2) , (4.22)
4.2. Effects of QCD radiation on global inclusive variables 59
Figure 4.4: Distribution of visible mass M in tt¯ production at LHC for ηmax = 5 and Y = 0:
leading and approximate next-to-leading order.
in the integrand of Eq. (4.20) and comparing with Eq. (4.15), we see that the last line of
that equation corresponds to a change of scale Q→ Qc ∼ θcQ in the parton distributions,
leading to
M2
dσab
dM2dY
= Fa(x¯1, Qc)Fb(x¯2, Qc)Σab(M
2) . (4.23)
The above definition of Σab includes the approximation Qc ≈ θcM in the evaluation of the
PDFs, since they do not vary substantially with scale. To first order we can then write
Σab(M
2) = σˆab(M
2)
+ ηmax
αs
π
∫
dz{Pˆa(z) + Pˆb(z)}{σˆab(zM2)− σˆab(M2)} . (4.24)
The interpretation of this result is simple: undetected ISR at angles less than θc, cor-
responding to scales less than ∼ θcQ, is absorbed into the structure of the incoming
hadrons.
To resum the effects of gluons at angles greater than θc, consider first the real emission
of n such gluons from parton a. In the quasi-collinear approximation these form an
60 Chapter 4. Effects of QCD radiation on hadron collider observables
angular-ordered sequence, giving rise to a contribution to Σab of
δR,n(Σab) =
(αs
π
)n ∫
θc
dθ1
θ1
∫
θ1
dθ2
θ2
. . .
∫
θn−1
dθn
θn
∫ 1
0
dz1 . . .dznPˆa(z1) . . . Pˆa(zn)σˆab(z1 . . . znM
2)
=
1
n!
(
ηmax
αs
π
)n ∫ 1
0
dz1 . . .dznPˆa(z1) . . . Pˆa(zn)σˆab(z1 . . . znM
2) , (4.25)
where again we have made the identification − ln θc = ηmax and we are still neglecting the
running of αs. The multiple convolution of the momentum fractions zi can be transformed
into a product by taking Mellin moments. Defining∫ ∞
0
dM2
(
M2
)−N
σˆab(M
2) ≡ σˆabN , (4.26)
we have(αs
π
)n ∫ ∞
0
dM2
(
M2
)−N ∫
dz1 . . .dznPˆa(z1) . . . Pˆa(zn)σˆ(z1 . . . znM
2) = (γˆaN)
n σˆabN ,
(4.27)
where
γˆaN =
αs
π
∫ 1
0
dz zN−1Pˆa(z) . (4.28)
Therefore defining correspondingly,∫ ∞
0
dM2
(
M2
)−N
Σab(M
2) ≡ ΣabN , (4.29)
the contribution (4.25) to this quantity will be
δR,n(ΣabN ) =
1
n!
(ηmaxγˆ
a
N)
n σˆabN , (4.30)
which summed over n gives∑
n
δR,n(ΣabN ) = exp (ηmaxγˆ
a
N) σˆ
ab
N . (4.31)
The corresponding virtual contributions give a Sudakov-like form factor,
exp
(
−αs
π
∫
θc
dθ
θ
∫ 1
0
dz Pˆa(z)
)
. (4.32)
4.2. Effects of QCD radiation on global inclusive variables 61
and therefore the total contribution from parton a is
exp (ηmaxγ
a
N) σˆ
ab
N , (4.33)
where γaN is the anomalous dimension,
γaN =
αs
π
∫ 1
0
dz
(
zN−1 − 1) Pˆa(z) = αs
π
∫ 1
0
dz zN−1Pa(z) , (4.34)
Pa(z) being the regularised a→ ag splitting function. Note that it is the virtual correction,
introduced via the form factor in Eq. (4.32), that has regularised the splitting function.
Parton b gives a similar factor with γbN in place of γ
a
N , so the result for the quantity (4.29)
is simply
ΣabN = e
ηmax(γaN+γ
b
N )σˆabN . (4.35)
We can see as follows that this result is qualitatively correct. The anomalous dimensions
are positive for small N and negative for large N . Thus, for θc ≪ 1, ΣabN is enhanced
relative to σˆabN at small N and suppressed at large N . Now from the moment definition
(4.26) small N corresponds to large M and vice versa. Hence the distribution of M is
suppressed at small M and enhanced at large M relative to the Born term, as observed
in the Monte Carlo [69] and NLO results.
The emission of partons other than gluons is included by introducing the anomalous
dimension matrix ΓN with elements given by
(ΓN )ba =
αs
π
∫ 1
0
dz zN−1Pba(z) , (4.36)
where Pba(z) is the regularised a→ b splitting function. Then
ΣabN = σˆ
a′b′
N
(
eηmaxΓN
)
a′a
(
eηmaxΓN
)
b′b
. (4.37)
The corresponding generalisation of the evolution equation (4.15) is
q
∂
∂q
fb(x) =
αs
π
∫
dz
z
Pba(z)fa(x/z) . (4.38)
62 Chapter 4. Effects of QCD radiation on hadron collider observables
Defining the moments of the parton distribution functions
faN =
∫ 1
0
dxxN−1fa(x) , (4.39)
we see that
q
∂
∂q
f bN = (ΓN)baf
a
N , (4.40)
with solution
f bN (q) =
(
[q/q0]
ΓN
)
ba
faN(q0) . (4.41)
Hence
f bN (Q) =
(
eηmaxΓN
)
ba
faN(Qc) , (4.42)
where
Qc = θcQ = Qe
−ηmax , (4.43)
showing that the evolution of the visible mass distribution is related to that of the parton
distributions over the same range of scales.
Taking into account the running of the strong coupling αs(q) in the evolution equation
(4.38), Eq. (4.42) becomes
f bN(Q) = K
ba
N f
a
N(Qc) , (4.44)
where
KbaN =
([
αs(Qc)
αs(Q)
]p∆N)
ba
, (4.45)
with the factor p = 6/(11CA−2nf ), coming from the QCD one-loop β-function, Eq. (2.41),
given in section 2.2.3.2, and
(∆N )ba =
π
αs
(ΓN)ba =
∫ 1
0
dz zN−1Pba(z) . (4.46)
The running of αs will affect Eq. (4.37) similarly, giving
ΣabN = σˆ
a′b′
N K
a′a
N K
b′b
N . (4.47)
To invert the above we can write the following double convolution
Σab(M
2) =
∫ 1
0
dz1 dz2 σˆa′b′(z1z2M
2)Ka′a(z1)Kb′b(z2) , (4.48)
4.2. Effects of QCD radiation on global inclusive variables 63
where
Kb′b(z) =
1
2πi
∫
C
dN z−NKb
′b
N . (4.49)
In the above inversion, performed on the complex plane, the contour C is formally defined
to be to the right of all singularities of the integrand and runs parallel to the imaginary
axis from −i∞ to +i∞. We will be discussing the details of the inversion in section 4.2.4.
It then follows from Eq. (4.45) that Kb′b(z) obeys an evolution equation like that of the
parton distributions,
Q
∂
∂Q
Kb′b(z) =
αs(Q)
π
∫
dz′
z′
Pb′a(z
′)Kab(z/z′) . (4.50)
Putting everything together, the visible mass distribution is related to the hard sub-
process cross section (in the absence of invisible final-state particles) as follows:
M2
dσab
dM2dY
=
∫
dz1 dz2 σˆa′b′(z1z2M
2)Ka′a(z1)Fa(x¯1, Qc)Kb′b(z2)Fb(x¯2, Qc) , (4.51)
where the kernel functions Ka′a(z) and Kb′b(z) can be obtained by solving the evolution
equation (4.50) with the initial condition that Kab(z) = δabδ(1 − z) at Q = Qc or by
directly using Eq. (4.49) to invert the Mellin transform. The results shown in this section
use the former method whereas in section 4.2.5 results given by the latter method will
be shown. Note also that the assumption Qc ≈ θcM that we had made in Eq. (4.23) has
been alleviated in Eq. (4.51).
To verify that the integrated cross section is not affected by resummation, define
x1,2 = z1,2x¯1,2 and write Eq. (4.51) as
M2
dσab
dM2dY
=
∫
dx1 dx2 σˆa′b′(x1x2S)Ka′a(x1/x¯1)fa(x¯1, Qc)Kb′b(x2/x¯2)fb(x¯2, Qc) .
(4.52)
Now
dM2
M2
dY =
dx¯1
x¯1
dx¯2
x¯2
, (4.53)
and ∑
a
∫
dx¯1
x¯1
Ka′a(x1/x¯1)fa(x¯1, Qc) = fa′(x1, Q) . (4.54)
64 Chapter 4. Effects of QCD radiation on hadron collider observables
Hence∑
ab
∫
dM2dY
dσab
dM2dY
=
∑
a′b′
∫
dx1 dx2 σˆa′b′(x1x2S)fa′(x1, Q)fb′(x2, Q) , (4.55)
in agreement with Eq. (4.6).
Resummed results corresponding to Fig. 4.4 are shown in Fig. 4.5. We see that the
peak of the distribution has moved to much higher mass, beyond 1 TeV. This is due to
multiple emission of ISR partons in the evolution of the initial state from the detection
scale Qc to the hard subprocess scale Q. As the value of ηmax is reduced, the range of
evolution becomes smaller, less ISR is emitted, and the peak moves closer to the hard
subprocess scale, as illustrated in Fig. 4.6. Results for higher values of the visible rapidity
Y are shown in Fig. 4.7. The peak moves to lower mass as Y increases, as a consequence
of the suppression of high masses by the rapid fall-off of the parton distributions at high x.
All the results shown in the present section have been obtained by solving the evolution
equation (4.50) directly.
Figure 4.5: Resummed distribution of visible mass M in tt¯ production at LHC for ηmax = 5
and Y = 0.
4.2. Effects of QCD radiation on global inclusive variables 65
Figure 4.6: Resummed distribution of visible mass M in tt¯ production at LHC for Y = 0 and
lower values of ηmax: colour scheme as in Fig. 4.5.
Figure 4.7: Resummed distribution of visible mass M in tt¯ production at LHC for ηmax = 5:
results at non-zero visible rapidity Y .
4.2.4 Mellin transform inversion
In the previous section we presented results using the direct integration of Eq. (4.50). In
the following sections we will be using the inverse Mellin transform to calculate the kernel
functions. We outline the procedure in this subsection.
Equation (4.49), sometimes called the Bromwich integral, defines the inverse of a
Mellin transform. Formally, the contour C on the complex N -plane is to the right of all
singularities in the integrand and runs parallel to the imaginary axis from −i∞ to +i∞.
The formal contour C is shown in Fig. 4.8 in blue dashes. However, we often need to
perform the inversion on the complex plane numerically. The formal contour C turns out
66 Chapter 4. Effects of QCD radiation on hadron collider observables
φ
Figure 4.8: Integration contours, C, C0 and C1, for the inverse Mellin transform as given by
the Bromwich integral, Eq. (4.49). The angle φ is used in the parametrization of
C0.
to be inappropriate for a numerical calculation. To see this, consider the parametrization
of the contour C passing through the point a on the real axis: N = a + it, where t is a
parameter which runs from −∞ to +∞. Then the Bromwich integral becomes
Kb′b(z) =
z−a
2π
∫ +∞
−∞
dt e−it ln zKb
′b
N(t) . (4.56)
The above expression will not converge numerically because of the oscillatory nature of
the factor e−it ln z. In PDF evolution this is ameliorated by deforming the contour C to
contour C0, shown in Fig. 4.8 in purple dot-dashes. If there are no singularities in the
integrand in the region C0 − C, we expect the result of the integration not to change by
the deformation. This is indeed the case for all the functions we will be considering. The
contour C0 can be parametrized as N = a + te
iφ where φ is a constant related to the
slope of the straight lines as shown in Fig. 4.8. It is easy to see that if φ > π/2, then a
damping term of the form et cosφ ln(1/z) is introduced in the integrand and hence numerical
convergence can be achieved. The contour C0 is usually used to evolve parton density
functions [72], according to Eq. (4.44).
However, in the case of the evolution kernels Ka′a(z), the linear contour C0 does not
4.2. Effects of QCD radiation on global inclusive variables 67
δC0(z) δC1(z)
Figure 4.9: A comparison of the resulting ‘delta’ functions obtained by using C0 (left) and C1
(right). It is obvious that the function obtained by using C1 behaves much more
like a delta function: going from z = 0.8 to z ≈ 1, δC0 goes from a value of ∼ 1 to
a value of ∼ 10, whereas δC1 goes from ∼ 10−8 to ∼ 10−5. Note the logarithmic
scale on the vertical axis in the right-hand figure.
provide sufficient accuracy to reproduce the function from its transform. This is due
to its inability to accurately invert a constant function fN = c to the correct analytic
result, a delta function. This implies that the inversion does not reproduce the necessary
initial condition, Ka′a(z,Q = Qc) ∝ δ(1 − z). A numerically more accurate contour is
available in the literature, used in the so-called ‘Fixed-Talbot algorithm’. This contour,
C1 is shown in solid red in Fig. 4.8. It has the form Re(N) = ImN cot(ImN/r), where
r is a parameter which we will set to r = 0.4m/ log(1/z) during the computation, m
being the required precision in number of decimal digits, a value derived from numerical
experiments. The contour is related to the ‘steepest descent’ path for a certain class of
functions. For further details on its origin and accuracy see Ref. [73]. A comparison of
the resulting ‘delta’ functions obtained by using C0 and C1 can be seen in Fig. 4.9. It is
obvious that the function obtained by using C1 behaves much more like a delta function.
We first rewrite the evolution kernel in a more convenient basis:
Ka
′a
N =
(
O−1
[
αS(Qc)
αS(Q)
]pdiag(λN,i)
O
)
a′a
, (4.57)
68 Chapter 4. Effects of QCD radiation on hadron collider observables
where O is the matrix of eigenvectors of ∆N and diag(λN , i) is the diagonal matrix of its
eigenvalues. This is equivalent to using, implicitly, the singlet and non-singlet basis [10].
As a test of the evolution using Mellin inversion, we used the contour C1 to evolve
u-quark and gluon PDFs from a starting scale of Qc = 10 GeV to Q = 10
4 GeV. The form
of fa(x,Qc) at Qc = 10 GeV was extracted directly from the the leading-order MSTW
parton distributions [71] and the evolved results at Q = 104 GeV were compared to the
actual values given by the MSTW PDFs. The results are shown in Fig. 4.10, exhibiting
good agreement for most of the range of values of x. The discrepancy at high x is due to
the difference in treatment of fa(x) as x→ 1.
 1e−07
 1e−06
 1e−05
 0.0001
 0.001
 0.01
 0.1
 1
 0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1
 1e−12
 1e−10
 1e−08
 1e−06
 0.0001
 0.01
 1
 100
 0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1
fu(x) fg(x)
x x
Figure 4.10: The MSTW PDFs for the u-quark PDF (left) and the gluon PDF (right) evolved
from their given form at Qc = 10 GeV to Q = 10
4 GeV using Mellin inversion on
the complex plane via the C1 contour (red points). The actual values from the
MSTW PDFs are given for comparison (green points).
4.2.5 ISR effects including invisible particle emission
Suppose now that an invisible 4-momentum pµinv is emitted from the hard subprocess. If we
define the total lab frame 4-momentum of the incoming partons a and b as P µ = (E, ~P ),
then
P µ =
1
2
√
S[(x¯1 + x¯2), 0, 0, (x¯1 − x¯2)] , (4.58)
4.2. Effects of QCD radiation on global inclusive variables 69
and the visible 4-momentum will be given by P µ− pµinv. By definition, the visible mass is
then
M2 = (P − pinv)2 = P µPµ + pµinvpinv,µ − 2pµinvPµ . (4.59)
Equation (4.59) demonstrates the interplay between two effects: on one hand ISR in-
creases the ‘true’ scale of the hard process Q, to the ‘apparent’ scale M by contaminating
the detector with extra particles, and on the other hand the invisible particle emission
decreases M by the loss of particles. In the case of gluino pair-production both effects
are equally important, as we will show.
Substituting from Eq. (4.58) in Eq. (4.59) and defining p±inv ≡ p0inv ± p3inv, we obtain
M2 = x¯1x¯2S +m
2
inv −
√
S[x¯1p
−
inv + x¯2p
+
inv] , (4.60)
where minv represents the total invariant mass of the invisibles, m
2
inv = p
µ
invpinv,µ.
The momenta pµinv are defined in the lab frame, relative to which the centre-of-mass
frame of the hard subprocess is boosted by an amount defined by the momentum fractions
x1 and x2 of the partons entering the subprocess. This implies that the p
±
inv transform as
p+inv =
√
x1
x2
q+inv , p
−
inv =
√
x2
x1
q−inv , (4.61)
where q±inv ≡ q0inv±q3inv, defined in terms of the invisible momentum, qµinv, in the centre-of-
mass frame of the hard subprocess. Substituting the expressions of Eq. (4.61) in Eq. (4.60),
we find an expression for the visible invariant mass:
M2 = m2inv + x¯1x¯2S
[
1− z1f+inv − z2f−inv
]
, (4.62)
where we have defined f±inv = q
±
inv/Q and used Q
2 = x¯1x¯2z1z2S. We may now solve
Eq. (4.62) for Q2 to obtain Q2 in terms of M2:
Q2 =
z1z2(M
2 −m2inv)
1− z1f+inv − z2f−inv
. (4.63)
The above expression for the hard subprocess scale now becomes the argument of the
70 Chapter 4. Effects of QCD radiation on hadron collider observables
parton-level cross section, σˆa′b′ in Eq. (4.51):
S
dσab
dM2dY
=
∫
dz1 dz2Ka′a(z1)fa(x¯1, Qc)Kb′b(z2)fb(x¯2, Qc) σˆa′b′
(
z1z2(M
2 −m2inv)
1− z1f+inv − z2f−inv
)
.
(4.64)
The functions f±inv, which are related to the invisible particle 4-momenta, remain to be
determined. The visible system rapidity, Y , is also modified by the presence of invisible
particles as
Y =
1
2
log
(
x¯1(1− z1f+inv)
x¯2(1− z2f−inv)
)
, (4.65)
and therefore Eqs. (4.18) for x¯1,2 become:
x¯1 =
√
(M2 −m2inv)(1− z2f−inv)
S(1− z1f+inv − z2f−inv)(1− z1f+inv)
eY ,
x¯2 =
√
(M2 −m2inv)(1− z1f+inv)
S(1− z1f+inv − z2f−inv)(1− z2f−inv)
e−Y . (4.66)
The kinematic constraints restrict Q2 to be greater than the threshold energy squared for
the process and the true invariant mass, M2true ≡ x¯1x¯2S = Q2/(z1z2), to be greater than
the visible invariant mass, M2. These result in the following two constraints for Q2:
Q2 > Q2threshold , Q
2 > z1z2M
2 . (4.67)
4.2.5.1 Single-invisible decays
The benchmark scenario for a single invisible decay originating from the hard process is
tt¯ production in which one of the two tops decays into bqq′ (hadronic) and the other into
bℓν (semi-leptonic), the neutrino comprising the missing 4-momentum. Excluding the
proton remnants, we assume that all other particles within the pseudorapidity coverage
are detected. We will refer to the neutrino as the invisible particle and the W as the
intermediate particle in the tt¯ case, but the treatment is readily applicable to the gluino
case where the invisible particle is the χ01 and the intermediate particle is a squark (treated
in section 4.2.5.2).
To calculate the functions f±inv and obtain Q
2, we need to calculate the neutrino 4-
momentum in the hard process frame. This is done by choosing the neutrino 4-momentum
in the frame of its parent W and then applying two subsequent Lorentz boosts: one going
4.2. Effects of QCD radiation on global inclusive variables 71
from the W frame to the top frame, and one from the top frame to the hard process
frame. The decay chain is shown in Fig. 4.11. Each of these boosts involves two angular
variables which originate from the ‘decay’ of the parent particle. Hence the 4-momentum
qµinv of the neutrino may be written as
qµinv = Λ
µ
κ
(
Q, θˆ, φˆ
)
Λκλ
(
θ˜, φ˜
)
p¯λν(θ¯, φ¯) , (4.68)
where the Λ’s are Lorentz boost matrices and where quantities with a hat refer to the
hard process frame, quantities with a tilde refer to the top frame and quantities with a bar
refer to the W frame. The angles θ and φ represent the usual polar angles, defined with
respect to the direction of the ‘sister’ particle (see Fig. 4.11). For example, in the case
W+ → ℓ+νℓ, where the W+ was produced from the top decay along with a bottom quark,
the angles (θ¯, φ¯) are defined with respect to the direction of motion of the b quark in the
W+ frame. The two boost vectors have magnitudes given by |~βi| = |~pi|/Ei (i = t,W ),
the ratio of the parent 3-momentum magnitude and its energy. The boosts, as well as
the magnitude of the invisible particle 4-momentum, can be obtained by considering the
kinematics in each frame as:
p¯λν(θ¯, φ¯) =
mW
2
(1, ~¯r) , (4.69)
~βW =
m2t−m2W
m2t+m
2
W
~˜r ,
~βt =
√
1− 4m2t
Q2
~ˆr ,
where ~r = (sin θ cosφ, sin θ sinφ, cosφ) is the unit vector in spherical polar coordinates in
the appropriate frame and mW , mt are the W and top quark masses respectively. The
4-vector fµinv, and hence the functions f
±
inv, are calculated by f
±
inv = q
±
inv/Q. Evidently,
the functions f±inv are functions of Q
2, giving an implicit equation for Q2. To make this
more explicit, we rewrite Eq. (4.63):
Q2 =
z1z2 [M
2 −m2inv(Q2,Ω)]
1− z1f+inv(Q2,Ω)− z2f−inv(Q2,Ω)
, (4.70)
and analogously for Eq. (4.66), where Ω represents the set of all angular variables. In the
present case minv(Q
2,Ω) = mν ≃ 0 but for multiple invisible particles it will also be a
function as indicated.
Equation (4.70) needs to be solved numerically for each set (z1, z2,Ω) in the region
72 Chapter 4. Effects of QCD radiation on hadron collider observables
g˜/t
(θ¯, φ¯)
(θ˜, φ˜)
(θˆ, φˆ)
q˜/W
χ01/ν
q
q/ℓ
Figure 4.11: The sequential two-body decay chain under consideration in the invisible particle
treatment. The relevant production angles in the parent centre-of-mass frame
are also shown in parentheses.
(4m2t/g˜, z1z2S), where S is the square of the proton centre-of-mass energy, along with
the restriction that the visible invariant mass should be lower than the ‘true’ invariant
mass, M ≤Mtrue. The numerical solution was found using the Van Wijngaarden-Dekker-
Brent method [74,75], a bracketing method for finding roots of one-dimensional equations.
Since Q is not uniquely determined for each M , different values of the ‘true’ centre-of-
mass energy Q contribute to the cross section. Note that not all possible configurations
(z1, z2,Ω) are kinematically allowed to contribute to the cross section at M and hence
some configurations do not yield roots of Eq. (4.70). Once Q2 is obtained, the parton-
level cross section for the hard process partons, σˆa′b′(Q
2), is calculated. This result is then
multiplied with the parton density functions for the incoming partons, fa,b(x¯1,2, Qc), and
the kernels for evolution from incoming partons a and b to hard process partons a′ and b′
(Ka′a(z1) and Kb′b(z2)). We then integrate over all possible values of z1 and z2, according
to Eq. (4.64). Finally, to obtain the full resummed result we have to integrate over the
distribution of the angular variables Ω. Notice that the visible invariant mass distribution
becomes non-zero below the threshold for production, M < 2mt/g˜, owing to the loss of
invisible particles.
4.2.5.2 Double-invisible decays
We now turn to the case where both particles produced in the hard process decay invisibly.
For illustration we refer to sequential decays of the gluino: g˜ → q˜q → χ01qq. Although
this decay mode is generally not the dominant one, it is useful for illustration of the
procedure. We extend the treatment given in the semi-leptonic/hadronic top case by
4.2. Effects of QCD radiation on global inclusive variables 73
writing out functions related to the two invisible particle 4-momenta in the decay chain
(which we call χ and χ′):
qµχ = Λ
µ
κ
(
Q, θˆ, φˆ
)
Λκλ
(
θ˜, φ˜
)
p¯λχ(θ¯, φ¯) , (4.71)
qµχ′ = Λ
µ
κ
(
Q, θˆ′, φˆ′
)
Λκλ
(
θ˜′, φ˜′
)
p¯λχ′(θ¯
′, φ¯′) , (4.72)
where the primed quantities now distinguish between the two invisibles. Since both of
these 4-vectors are defined in the hard subprocess frame, we simply have
f±inv =
1
Q
(
q±χ + q
±
χ′
)
. (4.73)
The rest of the treatment is identical to the case for one invisible particle: an implicit
equation has to be solved to obtain Q2 for each (z1, z2, Ω) set and then an integral over
Ω is taken to obtain the resummed result.
4.2.5.3 Angular distributions
The distributions of the angular variables Ω = (θˆ, φˆ, θ˜, φ˜, θ¯, φ¯), appearing in the treatment
of invisibles given in the previous sections, are process-dependent. They represent the an-
gles at which the daughter particle is emitted in the frame of the parent particle. We in-
vestigated the angular distributions using HERWIG++ version 2.4.0 and subsequently used
the results in calculating the f±inv functions. The results for SPS1a gluino pair-production
are shown in Fig. 4.12, where the uniform distributions are shown for comparison (red
horizontal line). Figure 4.13 shows the distributions as obtained for top pair-production.
The neutrino angle in the W frame is also compared to the analytic calculation. As ex-
pected, all the φ angles, in both cases, were found to be uniform (not shown). The form
of all the distributions can be justified using general spin considerations:
θˆi: The angular distribution of the angle θˆi at which the fermions are produced in
the hard process frame is expected to have the form ∼ 1 + β cos2 θˆi, where β is a
process-dependent constant.
θ˜i: The angle θ˜i, is defined between the direction of the daughter boson (W or q˜)
with respect to the direction of polarisation of the parent (t or g˜). The angular
74 Chapter 4. Effects of QCD radiation on hadron collider observables
distribution for a spin-up fermion parent is then given by [76]
1
N↑
dN↑
d cos θ˜i
=
1
2
(1 + Pαi cos θ˜i) , (4.74)
where αi is a constant and P is the modulus of the polarization of the parent.
Since the production processes for both tt¯ and g˜g˜ are parity-conserving, there is
also an equal spin-down (N↓) contribution to the total distribution with the sign of
αi reversed. This results in a uniform distribution for cos θ˜i.
ˆ ˜ ¯
Figure 4.12: Monte Carlo results for the gluino pair-production decay chain angles. From left
to right: the production angle of the gluino in the hard process frame, the angle of
the outgoing squark in the gluino frame and the angle of the outgoing neutralino
in the squark frame. The uniform distributions are shown for comparison.
θ¯i: In gluino pair-production, the decay products of the squark, q˜, which is a scalar,
are uniformly distributed in cos θ¯. In top pair-production, on the other hand, the
decay W → ℓνℓ is parity-violating and the distribution of cos θ¯ is forward-backward
asymmetric in the W frame [10]. The angle θ¯ (sometimes called Ψ, see e.g. [77])
can be used experimentally to infer helicity information on the W . The distribution
may be written as
1
N
dN
d cos θ¯
=
3
2
[
F0
(
sin θ¯√
2
)2
+ FL
(
1− cos θ¯
2
)2
+ FR
(
1 + cos θ¯
2
)2]
,
(4.75)
where FL, FR and F0 are the probabilities for left-handed, right-handed and lon-
gitudinal helicities of the W in top quark decay respectively. The SM predictions,
4.2. Effects of QCD radiation on global inclusive variables 75
(FL, FR, F0) = (0.304, 0.001, 0.695), yield the blue solid curve shown on the right in
Fig. 4.13.
ˆ ˜ ¯
Figure 4.13: Monte Carlo results for the top pair-production decay chain angles. From left to
right: the production angle of the top in the hard process frame, the angle of the
outgoing W boson in the top frame and the angle of the outgoing neutrino in the
W frame. The uniform distributions are shown for comparison. The neutrino
angle in the W frame is also compared to the analytic calculation.
The spins of the two produced fermions (tops or gluinos) are correlated and this may
cause a degree of correlation between the distributions of particles in the decay chains.
We investigated whether these correlations play an important role in the calculation of the
invisible particle effects on the visible mass. By comparing the invariant mass distributions
with and without the spin correlations in the Monte Carlo we concluded that the effect
is small in both top and gluino pair-production and can be safely neglected.
4.2.6 Results
We present the resummed distributions obtained for tt¯ and g˜g˜ production according to
Eq. (4.64). All results are for the LHC at design energy, i.e. pp collisions at
√
s = 14 TeV.
We have integrated over the visible system rapidity, Y , in the range |Y | < 5. We first
compare our results to those obtained using the HERWIG++ event generator at parton
level (i.e. no hadronization or underlying event) and excluding the proton remnants.5 In
sections 4.2.6.3 and 4.2.6.4 we examine the effects of hadronization and the underlying
event. Parton-level top and gluino pair-production cross section formulae are given in
5We verified, using the event generator, that the contribution of the proton remnants to the total
invariant mass in the considered rapidity range is negligible.
76 Chapter 4. Effects of QCD radiation on hadron collider observables
Figure 4.14: The tt¯ visible mass distributions for a pseudorapidity cut ηmax = 5, comparing
hadronic (no invisibles) and semi-leptonic (one invisible) decays. The leading-
order tt¯ invariant mass distribution is shown (red dot-dashes) for comparison.
appendix B. The PDF set used both in the resummation calculation and HERWIG++ is
the MRST LO** (MRSTMCal) set [78, 79].
4.2.6.1 Top quark pair-production
We present resummed results in comparison to Monte Carlo for Standard Model tt¯ pro-
duction, where we include particles with maximum pseudorapidity ηmax = 5. In Figs. 4.14
and 4.15 we show separate results for combinations of hadronic and semi-leptonic decays
of the top, leading to zero, one or two invisible neutrinos from the hard process. The
effect of the invisibles in both the fully semi-leptonic case and the hadronic/semi-leptonic
case are small compared to the effects of hadronization, to be discussed in section 4.2.6.3.
The differences between the Monte Carlo and resummed curves in Figs. 4.14 and 4.15
may be attributed to sensitivity to the behaviour of the PDFs and parton showering at
low scales, since Qc can be as low as 2mt× e−5 ∼ 2 GeV in the case of tt¯ production, and
the precise definition of Qc in terms of ηmax.
4.2. Effects of QCD radiation on global inclusive variables 77
Figure 4.15: The tt¯ visible mass distributions for a pseudorapidity cut ηmax = 5, comparing
hadronic (no invisibles) and fully leptonic (two invisibles) decays. The leading-
order tt¯ invariant mass distribution is shown (red dot-dashes) for comparison.
4.2.6.2 Gluino pair-production
We focus on the SPS1a point [80], which has gluino and lightest neutralino masses mg˜ =
604.5 GeV andmχ01 = 97.0 GeV respectively (and see table 4.1 for the squark masses). For
simplicity we set the squark mass in the invisible particle treatment to 550 GeV. We also
present results for a modified SPS1a point, with mg˜ = 800 GeV. In this process only the
two-invisibles case is realistic, but for comparison we also show results for no invisibles,
i.e. imagining that the two lightest neutralinos are also detected. When ηmax = 5, 3,
Particle Mass (GeV) Particle Mass (GeV)
g˜ 604.5 s˜L 570.7
χ01 97.0 s˜R 547.9
u˜L 562.3 b˜1 515.3
u˜R 548.2 b˜2 547.7
d˜L 570.7 t˜1 400.7
d˜R 547.9 t˜2 586.3
Table 4.1: The relevant particle masses in the supersymmetric model used in the invisible
study, SPS1a. The modified SPS1a point differs in that it has mg˜ = 800 GeV.
78 Chapter 4. Effects of QCD radiation on hadron collider observables
there is fairly good agreement between the Monte Carlo and resummation predictions in
both the two-invisibles and no-invisibles cases, and for both gluino masses, as shown in
Figs. 4.16 and 4.17, where one should compare the dashed histograms (Monte Carlo) to
the solid curves of the same colour (resummation).
The shift in the peak of the visible mass distribution in going from no to two invisibles
is much larger than that in top pair-production, amounting to 600-700 GeV, roughly
independent of ηmax and the gluino mass. This results mainly from the higher masses
of the intermediate particles in the decays (mq˜ ≃ 550 GeV vs. mW = 80 GeV), which
implies a higher energy release, rather than the masses of the invisible particles themselves
(mχ01 = 97 GeV vs. mν = 0).
One of the assumptions of the resummation is that all the visible hard process decay
products are detected, which is not true when the maximum pseudorapidity ηmax is re-
stricted to lower values. When ηmax ∼ 2 in the Monte Carlo analysis, a significant number
of hard process particles begin to be excluded and hence the curves shift to lower values
compared to the resummed predictions. Figure 4.18 shows the rapidity distribution of the
decay products of the gluino at parton level formg˜ = 604.5 GeV. For the case shown, cuts
of ηmax = 5, 3, 2 and 1.4 correspond to exclusion of, respectively, ∼0.002%, 1.1%, 7.5%
and 20.0% of the gluino decay products from the detector. The effect of this appears in
Figs. 4.19 and 4.20, where the Monte Carlo distributions are narrower and peak at lower
masses than the resummed predictions. The variation between the resummed ηmax = 2
and 1.4 curves is smaller than that between ηmax = 5 and 3, since they correspond to
smaller differences in Qc.
The heavy and light gluino scenarios exhibit similar behaviour when varying the pseu-
dorapidity coverage and the number of invisibles, showing the lack of dependence of the
resummation on the mass of the pair-produced particle. The sensitivity to low-scale PDF
behaviour and showering is reduced compared to the tt¯ case since we are considering
higher centre-of-mass energies, with the lowest possible Qc now being of the the order
2mg˜ × e−5 ∼ 8 GeV. The position of the curves is again also sensitive to the precise
definition of Qc in terms of ηmax.
Table 4.2 shows a summary of the peak positions for all cases and different pseudo-
rapidity cuts. For the higher values of ηmax, the agreement between the Monte Carlo
and resummation is satisfactory. There is a large difference in the peak positions for no
invisibles and ηmax = 5, but this is mainly due to the broad shape of the peak in this case,
while the overall distributions agree better. For ηmax ≤ 2 there is a growing discrepancy,
4.2. Effects of QCD radiation on global inclusive variables 79
mg˜ (GeV). ηmax MC (GeV) (0 inv./2 inv.) Resum. (GeV) (0 inv./2 inv.)
604.5 5 2280/1560 1785/1620
604.5 3 1680/1080 1593/1204
604.5 2 1440/840 1497/1204
604.5 1.4 1380/660 1497/1204
800.0 5 2820/2100 2569/1870
800.0 3 2220/1620 2128/1684
800.0 2 1920/1380 1865/1683
800.0 1.4 1740/1140 1865/1683
Table 4.2: Summary of the positions of the peaks of the gluino pair-production visible mass
distributions as given by the Monte Carlo and the resummation, for different values
of the maximum pseudorapidity and for no and two invisibles.
Figure 4.16: The SPS1a gluino pair-production visible mass distributions for pseudorapidity
cuts ηmax = 5 (left) and ηmax = 3 (right). The leading-order distribution is shown
(red dot-dashes) for comparison.
especially for the realistic case of two invisibles, due to the loss of particles coming from
the hard process.
4.2.6.3 Hadronization effects
We have assumed that ISR partons emitted at pseudorapidities above ηmax do not con-
tribute to the visible invariant mass. This would be true if the hadronization process were
perfectly local in angle. However, as a result of hadronization high rapidity ISR partons
can produce lower rapidity hadrons and thus ‘contaminate’ the detector and shift the
visible mass to higher values.
As we have already discussed in section 3.2.3, the hadronization model employed in
80 Chapter 4. Effects of QCD radiation on hadron collider observables
Figure 4.17: The modified SPS1a gluino pair-production (with mg˜ = 800 GeV) results for
pseudorapidity cuts ηmax = 5 (left) and ηmax = 3 (right). The leading-order
distribution is shown (red) for comparison.
Figure 4.18: The SPS1a gluino pair-production pseudorapidity distribution of gluino decay
products, for the case mg˜ = 604.5 GeV.
the HERWIG++ Monte Carlo is a refinement of the cluster model. The model involves
clustering of partons into colour-singlet objects that decay into hadrons, resulting in a
smearing of the pseudorapidity distribution which causes the increase in the visible mass
described above. The effect is shown in Fig. 4.21 for gluino and top pair-production
(excluding the invisible particles from the hard process). The effect was found to be
larger for tt¯ production where the mass distribution is shifted significantly, whereas in
gluino pair-production the shift is negligible.6
6This was found to dependent solely on the mass of the pair-produced particle, with a similar effect
to the gluino case appearing if the top mass is increased to ∼ 605 GeV.
4.2. Effects of QCD radiation on global inclusive variables 81
Figure 4.19: The SPS1a gluino pair-production results for pseudorapidity cuts ηmax = 2 (left)
and ηmax = 1.4 (right). The leading-order distribution is shown (red dot-dashes)
for comparison.
Figure 4.20: The modified SPS1a gluino pair-production (with mg˜ = 800 GeV) results for
pseudorapidity cuts ηmax = 2 (left) and ηmax = 1.4 (right). The leading-order
distribution is shown (red) for comparison.
4.2.6.4 Underlying event
The underlying event, which we have discussed in section 3.2.2.5, is a further source of
non-perturbative contributions to the visible mass. If P µH represents the ‘hard’ visible
4-momentum studied in earlier sections and P µU represents that due to the underlying
event, the total visible mass is given by
M2 = (PH + PU)
2 = M2H +M
2
U + 2(EHEU − PzHPzU)
= M2H +M
2
U + 2MU
√
M2H+ 6E2T cosh(YH − YU) . (4.76)
82 Chapter 4. Effects of QCD radiation on hadron collider observables
Figure 4.21: The tt¯ fully semi-leptonic (left) and SPS1a gluino pair-production (right, with
mg˜ = 604.5 GeV) visible mass distributions for a pseudorapidity cut ηmax = 5
with and without hadronization (black and red respectively).
Figure 4.22: The tt¯ fully hadronic visible mass distributions for pseudorapidity cuts ηmax = 5
(left) and ηmax = 3 (right), with and without multiple parton interactions (black
and red respectively) and the reconstructed curves (blue dot-dashes). The ηmax =
5 curve was reconstructed using the resummed results for the visible mass and
rapidity, whereas the ηmax = 3 curve was reconstructed using the Monte Carlo
visible mass and rapidity.
where we neglect transverse momentum associated with the underlying event. Thus, even
if the visible invariant mass due to the underlying event is small, its effect on the overall
visible mass may be enhanced through the last term on the right-hand side.
As we have already mentioned, the underlying event is simulated in HERWIG++ by a
multiple parton interaction model. In this model, for the rapidity ranges considered here,
the underlying event is approximately process-independent and exhibits little correlation
with the rest of the event. Therefore, to a good approximation, the distributions of the
variables related to the underlying event, YU andMU , can be determined once and for all at
4.2. Effects of QCD radiation on global inclusive variables 83
Figure 4.23: The SPS1a gluino pair-production (with mg˜ = 604.5 GeV) visible mass dis-
tributions for pseudorapidity cuts ηmax = 5 (left) and ηmax = 3 (right), with
and without multiple parton interactions (black and red respectively) and the
reconstructed curves from the Monte Carlo visible masses and rapidities (blue
dot-dashes).
each collider energy. The process-dependence comes primarily through the dependence on
YH and MH , which can be calculated using the resummation formula given in Eq. (4.64).
The overall visible mass distribution can then be obtained by convolution using Eq. (4.76).
The effects of including the underlying event in the visible mass distribution are shown
in Figs. 4.22 and 4.23 for tt¯ and gluino pair-production, respectively. The multiple parton
interactions push the peak value to substantially higher masses. The shift amounts to
about 250 GeV at ηmax = 3 and 1.2 TeV at ηmax = 5, and is roughly process-independent.
However, since the underlying event is approximately uncorrelated with the hard process,
the visible mass distributions can be reconstructed well by the convolution procedure
outlined above, as shown by the blue dot-dashed curves in Figs. 4.22 and 4.23. The
distributions for the underlying event, MU , used to obtain Figs. 4.22 and 4.23, are shown
in Fig. 4.24. These features of the underlying event will need to be validated by LHC
data on a variety of processes. Accurate modelling of the underlying event is important
for practically all aspects of hadron collider physics.
4.2.7 Conclusions
We have presented detailed predictions on the total invariant mass M of the final-state
particles registered in a detector, as a function of its pseudorapidity coverage ηmax by
considering the effects of QCD initial-state radiation, first in the quasi-collinear NLO
approximation and then in an all-orders resummation of the collinear-enhanced terms.
84 Chapter 4. Effects of QCD radiation on hadron collider observables
Figure 4.24: The tt¯ (left) and SPS1a gluino pair-production (right, with mg˜ = 604.5 GeV)
underlying event MU distributions for pseudorapidity cuts ηmax = 5 (red) and
ηmax = 3 (black).
This quantity provides the dominant contribution to many global inclusive observables
such as the new variable sˆ
1/2
min (Eq. (4.2)), which can provide information on the energy
scales of hard processes. We have extended the resummation method presented to include
the effects of invisible particle emission from the hard process. We have considered the
case of one or two invisible particles and presented results for Standard Model top quark
pair-production and SPS1a gluino pair-production, obtained using a numerical Mellin
moment inversion method.
In the case of tt¯ production the invisible particles are neutrinos from W boson decays
and their effect on the visible invariant mass distribution is small, even when both decays
are leptonic. This is mainly a consequence of the small W boson mass compared to
the overall invariant mass, rather than the negligible neutrino mass. For gluino pair-
production the invisibles are a pair of massive LSPs from squark decays. The LSP mass
is again small compared to the overall invariant mass, but the squark masses are not,
leading to a substantial downward shift in the visible mass distribution, of the order of
the squark mass. In both cases the resummed predictions are in fair agreement with Monte
Carlo estimates of the position of the peak in the distribution, provided the pseudorapidity
range covered by the detector is large enough (ηmax >∼ 3). For ηmax ∼ 3, the difference
between the Monte Carlo prediction and resummed predictions is of the order of 100
GeV for both the heavy and light gluino SPS1a points. The agreement becomes worse
when the pseudorapidity range is restricted, due to particle loss from the hard process.
Table 4.2 shows the positions of the peaks of the distributions for the Monte Carlo results
from HERWIG++ and the resummation.
4.3. Resummation of ET in vector boson and Higgs boson production at
hadron colliders 85
These comparisons were made with Monte Carlo visible mass distributions at parton
level. We found that non-perturbative effects, especially the underlying event, tend to shift
the invariant mass distributions to significantly higher values than expected from a purely
perturbative calculation. According to the underlying event model used in HERWIG++,
the shift amounts to about 250 GeV at ηmax = 3 and 1.2 TeV at ηmax = 5. This effect is
also expected in other observables sensitive to longitudinal momentum components, such
as sˆ
1/2
min. However, in the model found in HERWIG++ version 2.4.x, the underlying event is
only weakly correlated with the rest of the event and hence its effects can be determined
once and for all at each collider energy. The modelling of the underlying event is an
important feature of the Monte Carlo programs that needs to be validated by comparison
with experiment. Once this has been done, a wide range of global inclusive observables,
including the visible invariant mass, will be reliably predicted and useful for establishing
the scales of contributing hard subprocesses.
It is important to note that recent UE results from the LHC experiments have shown
that the model present in HERWIG++ 2.4.x does not describe the data adequately [81].
A more recent version of the event generator, 2.5.0 [53], which includes a model for
colour reconnection, an extension of the cluster model of hadronization, achieves major
improvements in the description of the UE LHC data.
4.3 Resummation of ET in vector boson and Higgs
boson production at hadron colliders
QCD radiation from incoming partons changes the distributions of the products of the
hard process. This effect has been studied in great detail for the processes of electroweak
boson production, with the result that the transverse momentum and rapidity distri-
butions of W , Z and Higgs bosons at the Tevatron and LHC are predicted with good
precision.7 The predictions for the transverse momentum (qT ) distributions in particu-
lar include resummation of terms enhanced at small qT to all orders in αs, matched with
fixed-order calculations at higher qT values. The transverse momentum of the boson arises
(neglecting the small intrinsic transverse momenta of the partons in the colliding hadrons)
7See [66–68] and references therein.
86 Chapter 4. Effects of QCD radiation on hadron collider observables
from its recoil against the transverse momenta of the radiated partons: qT = |~qT |, where:
~qT = −
∑
i
~pT i . (4.77)
The resummation of enhanced terms therefore requires a sum over emissions i subject to
the constraint (4.77), which is most conveniently carried out in the transverse space of
the impact parameter ~b Fourier conjugate to ~qT :
δ(~qT +
∑
~pT i) =
1
(2π)2
∫
d2~b eiqT ·b
∏
i
eipTi·b . (4.78)
One then finds that the cumulative distribution in b = |~b| contains terms of the form
αns ln
p(Qb), where Q is the scale of the hard process, set in this case by the mass of
the electroweak boson, and p ≤ 2n. These terms, which spoil the convergence of the
perturbation series at large b, corresponding to small qT , are found to exponentiate [82–88]:
that is, they can be assembled into an exponential function of terms that are limited to
p ≤ n + 1. This resummation procedure improves the convergence of the perturbation
series at large values of b and hence allows one to extend predictions of the qT distribution
to smaller values.
Together with its vector transverse momentum ~pT i, every emission generates a con-
tribution to the total hadronic transverse energy of the final state, ET , which, neglecting
parton masses, is given by
ET =
∑
i
|~pT i| . (4.79)
To first order in αs (0 or 1 emissions) this quantity coincides with qT , but they differ in
higher orders. In particular, at small qT there is the possibility of vectorial cancellation
between the contributions of different emissions, whereas this cannot happen for the scalar
ET . Thus the distribution of ET vanishes faster at the origin, and its peak is pushed to
higher values. To resum these contributions at small ET , one should perform a one-
dimensional Fourier transformation and work in terms of a ‘transverse time’ variable τ
conjugate to ET :
δ(ET −
∑
|~pT i|) = 1
2π
∫
dτ e−iET τ
∏
i
ei|pTi|τ . (4.80)
Since the matrix elements involved are the same, one finds a similar pattern of enhanced
4.3. Resummation of ET in vector boson and Higgs boson production at
hadron colliders 87
terms at large τ as was the case for large b: terms of the form αns ln
p(Qτ) with p ≤ 2n,
which arise from an exponential function of terms with p ≤ n + 1. Evaluation of the
exponent to a certain level of precision (leading-logarithmic, LL, for p = n + 1, next-
to-leading, NLL, for p = n, etc.) resums a corresponding class of enhanced terms and
extends the validity of predictions to lower values of ET .
The resummation of ET in this way has received little attention since the first papers
on this topic, over 20 years ago [89–91]. This is surprising, as most of the effects of QCD
radiation from incoming partons mentioned above depend on this variable rather than qT .
A possible reason is that, unlike qT , ET also receives an important contribution from the
underlying event, which is thought to arise from secondary interactions between spectator
partons, as discussed in section 3.2.2.5. At present this can only be estimated from
Monte Carlo simulations that include multiple parton interactions (MPI). Nevertheless it
is worthwhile to predict as accurately as possible the component coming from the primary
interaction, which carries important information about the hard process. For example, we
expect the ET distributions in Higgs and vector boson production to be different, as they
involve primarily gluon-gluon and quark-anti-quark annihilation, respectively. Accurate
estimates of the primary ET distribution are also important for improving the modelling
of the underlying event.
In the present section we extend the resummation of ET in vector boson production
to next-to-leading order (NLO) in the resummed exponent, parton distributions and co-
efficient functions, and present for the first time the corresponding predictions for Higgs
boson production. In section 4.3.1 the resummation procedure is reviewed and extended
to NLO; results on the resummed component are presented in section 4.3.2. This compo-
nent alone is not expected to describe the region of larger ET values, of the order of the
boson mass; in section 4.3.3 we describe and apply a simple procedure for including the
unresummed component at order αs. Section 4.3.4 presents ET distributions generated
using the parton shower Monte Carlo programs HERWIG version 6.510 [92] and HERWIG++
version 2.4.0 [46], which are compared with the analytical results and used to estimate of
the effects of hadronization and the underlying event. Our conclusions are summarised
in section 4.3.5. Appendix D.1 gives mathematical details of a comparison between the
resummation of the transverse energy ET and transverse momentum qT and appendix D.2
shows results for the LHC at lower centre-of-mass energy (7 TeV).
88 Chapter 4. Effects of QCD radiation on hadron collider observables
4.3.1 Resummation method
4.3.1.1 General procedure
Here we generalise the results of Ref. [90] to NLO resummation. The resummed compo-
nent of the transverse energy distribution in the process h1h2 → FX at scale Q takes the
form:[
dσF
dQ2 dET
]
res.
=
1
2π
∑
a,b
∫ 1
0
dx1
∫ 1
0
dx2
∫ +∞
−∞
dτ e−iτET fa/h1(x1, µ) fb/h2(x2, µ)
× W Fab(x1x2s;Q, τ, µ) , (4.81)
where fa/h(x, µ) is the parton distribution function (PDF) of parton a in hadron h at
factorisation scale µ, taken to be the same as the renormalisation scale here. In what
follows we use the MS renormalisation scheme. As mentioned earlier, to take into account
the constraint that the transverse energies of emitted partons should sum to ET , the
resummation procedure is carried out in the domain that is Fourier conjugate to ET , using
Eq. (4.80). The transverse energy distribution (4.81) is thus obtained by performing the
inverse Fourier transformation with respect to the transverse time, τ . The factor W Fab
is the perturbative and process-dependent partonic cross section that embodies the all-
order resummation of the large logarithms ln(Qτ). Since τ is conjugate to ET , the limit
ET ≪ Q corresponds to Qτ ≫ 1.
As in the case of transverse momentum resummation [93], the resummed partonic
cross section can be written in the following universal form:
W Fab(s;Q, τ, µ) =
∑
c
∫ 1
0
dz1
∫ 1
0
dz2 Cca(αs(µ), z1; τ, µ) Cc¯b(αs(µ), z2; τ, µ) δ(Q
2 − z1z2s)
× σFcc¯(Q,αs(Q)) Sc(Q, τ) . (4.82)
Here σFcc¯ is the cross section for the partonic subprocess c+ c¯→ F , where c, c¯ = q, q¯ (the
quark qf and the anti-quark q¯f ′ can possibly have different flavours f, f
′) or c, c¯ = g, g.
The term Sc(Q, τ) is the quark (c = q) or gluon (c = g) Sudakov form factor. In the case
of ET resummation, this takes the form [90,91]:
Sc(Q, τ) = exp
{
−2
∫ Q
0
dq
q
[
2Ac(αs(q)) ln
Q
q
+Bc(αs(q))
] (
1− eiqτ)} , (4.83)
4.3. Resummation of ET in vector boson and Higgs boson production at
hadron colliders 89
with c = q or g. The functions Ac(αs), Bc(αs), as well as the coefficient functions Cab in
Eq. (4.82), contain no ln(Qτ) terms and are perturbatively computable as power expan-
sions with constant coefficients:
Ac(αs) =
∞∑
n=1
(αs
π
)n
A(n)c , (4.84)
Bc(αs) =
∞∑
n=1
(αs
π
)n
B(n)c , (4.85)
Cab(αs, z) = δab δ(1− z) +
∞∑
n=1
(αs
π
)n
C
(n)
ab (z) . (4.86)
Thus a calculation to NLO in αs involves the coefficients A
(1)
c , A
(2)
c , B
(1)
c , B
(2)
c and C
(1)
ab .
All these quantities are known for both the quark and gluon form factors and associated
coefficient functions. Knowledge of the coefficients A(1) leads to the resummation of the
leading logarithmic (LL) contributions at small ET , which in the differential distribution
are of the form αns ln
p(Q/ET )/ET where p = 2n − 1. The coefficients B(1) give the
next-to-leading logarithmic (NLL) terms with p = 2n − 2, A(2) and C(1) give the next-
to-next-to-leading logarithmic (N2LL) terms with p = 2n − 3, and B(2) gives the N3LL
terms with p = 2n− 4. With knowledge of all these terms, the first term neglected in the
resummed part of the distribution is of order α3s ln(Q/ET )/ET .
In general the coefficient functions in Eq. (4.82) contain logarithms of µτ , which are
eliminated by a suitable choice of factorisation scale. To find the optimal factorisation
scale, we note that, to NLL accuracy,∫ Q
0
dq
q
lnp q
(
1− eiqτ) ≃ ∫ Q
iτ0/τ
dq
q
lnp q , (4.87)
where τ0 = exp(−γE) = 0.56146 . . ., γE being the Euler-Mascheroni constant. See ap-
pendix D.1 for a derivation. The effective lower limit of the soft resummation becomes
iτ0/τ , and the parton distributions and coefficient functions should be evaluated at this
scale. However, evaluation of parton distribution functions at an imaginary scale using
the standard parametrizations is not feasible. We avoid this by noting that
fa/h(x, q
′) =
∑
b
∫ 1
x
dz
z
Kab(z; q
′, q)fb/h(x/z, q) , (4.88)
90 Chapter 4. Effects of QCD radiation on hadron collider observables
whereKab is the DGLAP evolution kernel, also used in section 4.2.3.3 (see, e.g. Eq. (4.54)).
Therefore:
fa/h(x, iµ) =
∫ 1
x
dz
z
Kab(z; iµ, µ)fb/h(x/z, µ) , (4.89)
where the evolution kernel Kab(z; iµ, µ) is given to NLO by
Kab(z; iµ, µ) = δab +
i
2
αs(µ)Pab(z) , (4.90)
where Pab(z) is the leading-order DGLAP splitting function. Similarly, in the coefficient
functions we can write αs(iµ) in terms of αs(µ) using the definition of the running coupling,
given in section 2.2.3.2: ∫ iµ
µ
dαs
β(αs)
= 2
∫ iµ
µ
dq
q
= iπ , (4.91)
where β(αs) = −bα2s +O(α3s), so that
αs(iµ) = αs(µ)− iπb[αs(µ)]2 +O(α3s) . (4.92)
Furthermore, as the expressions (4.81) and (4.82) are convolutions, we can transfer the
extra terms from (4.89) into the coefficient functions to obtain
W Fab(s;Q, τ) =
∑
c
∫ 1
0
dz1
∫ 1
0
dz2 C˜ca(αs(τ0/τ), z1) C˜c¯b(αs(τ0/τ), z2) δ(Q
2 − z1z2s)
× σFcc¯(Q,αs(Q)) Sc(Q, τ) , (4.93)
where
C˜ca(αs(µ), z) =
∑
d
∫ 1
z
dz′
z′
Ccd(αs(iµ), z/z
′)Kda(z′; iµ, µ) . (4.94)
Now the lowest-order coefficient function is of the form:
C˜(0)ca (z) = C
(0)
ca (z) = δcaδ(1− z) , (4.95)
and therefore
C˜(1)ca (z) = C
(1)
ca (z) + i
π
2
Pca(z) . (4.96)
4.3. Resummation of ET in vector boson and Higgs boson production at
hadron colliders 91
Putting everything together, we have[
dσF
dQ2 dET
]
res.
=
1
2πs
∑
c
∫ +∞
−∞
dτ e−iτETSc(Q, τ) Rc(s;Q, τ) σFcc¯(Q,αs(Q)) , (4.97)
where, taking all PDFs and coefficient functions to be evaluated at scale µ = τ0/τ ,
Rc(s;Q, τ) =
∑
a,b
∫ 1
0
dx1
x1
dx2
x2
dz1
z1
fa/h1(x1) fb/h2(x2) C˜ca(z1) C˜c¯b
(
Q2
z1x1x2s
)
. (4.98)
To write (4.97) as an integral over τ > 0 only, we note from (4.89) and (4.90) that when
τ → −τ , to NLO the real parts of fa/h1 and fb/h2 are unchanged but the imaginary parts
change sign. All other changes in (4.98) are beyond NLO. Thus, writing
Rc = R
(R)
c + iR
(I)
c , (4.99)
R
(R)
c is symmetric with respect to τ and R
(I)
c is antisymmetric. Defining
F (R)c (Q, τ) = 2
∫ Q
0
dq
q
[
2Ac(αs(q)) ln
Q
q
+Bc(αs(q))
]
(1− cos qτ) ,
F (I)c (Q, τ) = 2
∫ Q
0
dq
q
[
2Ac(αs(q)) ln
Q
q
+Bc(αs(q))
]
sin qτ , (4.100)
we therefore obtain[
dσF
dQ2 dET
]
res.
=
1
πs
∑
c
∫ ∞
0
dτ e−F
(R)
c (Q,τ)
[
R(R)c (s;Q, τ) cos{F (I)c (Q, τ)− τET}
−R(I)c (s;Q, τ) sin{F (I)c (Q, τ)− τET}
]
σFcc¯(Q,αs(Q)) , (4.101)
where, inserting (4.95) and (4.96) in (4.98) and defining ξ = Q2/s, we have to NLO,
R(R)c (s;Q, τ) = R
(R)
c (ξ = Q
2/s, τ)
=
∫
dx1
x1
dx2
x2
{
fc/h1(x1)fc¯/h2(x2) +
αs
π
∑
a
[
fa/h1(x1)fc¯/h2(x2)C
(1)
ca
(
ξ
x1x2
)
+fc/h1(x1)fa/h2(x2)C
(1)
c¯a
(
ξ
x1x2
)]}
,
R(I)c (s;Q, τ) = R
(I)
c (ξ = Q
2/s, τ)
92 Chapter 4. Effects of QCD radiation on hadron collider observables
=
αs
2
∑
a
∫
dx1
x1
dx2
x2
[
fa/h1(x1)fc¯/h2(x2)Pca
(
ξ
x1x2
)
+fc/h1(x1)fa/h2(x2)Pc¯a
(
ξ
x1x2
)]
. (4.102)
It will be more useful to write, for example,∫
dx1
x1
dx2
x2
fa/h1(x1)fc¯/h2(x2)Pca
(
ξ
x1x2
)
=
∫
dx1
x1
dx2
x2
dz δ
(
z − ξ
x1x2
)
fa/h1(x1)fc¯/h2(x2)Pca(z)
=
∫
dx1
x1
dz
z
fa/h1(x1)fc¯/h2
(
ξ
zx1
)
Pca(z) . (4.103)
This makes it more straightforward to interpret the plus prescription, which appears in
some splitting functions, as∫
dx1
x1
dz
z
fa/h1(x1)fc¯/h2
(
ξ
zx1
)
P (z)+
=
∫
dx1
x1
fa/h1(x1)
∫ 1
0
dz
[
1
z
fc¯/h2
(
ξ
zx1
)
− fc¯/h2
(
ξ
x1
)]
P (z)
=
∫ 1
ξ
dx1
x1
fa/h1(x1)
∫ 1
ξ/x1
dz
[
1
z
fc¯/h2
(
ξ
zx1
)
− fc¯/h2
(
ξ
x1
)]
P (z)
−
∫ 1
ξ
dx1
x1
fa/h1(x1)fc¯/h2
(
ξ
x1
)∫ ξ/x1
0
dz P (z) . (4.104)
We show in appendix D.1 that the results of resummation of the scalar transverse
energy are identical to those of the more familiar resummation of vector transverse mo-
mentum at order αs, as they should be since at most one parton is emitted at this order.
The transverse energy computed here is the resummed component of hadronic initial-
state radiation integrated over the full range of pseudorapidities η. In Ref. [90] the ET
distribution of radiation emitted in a restricted rapidity range |η| < ηmax was also es-
timated. This was done by replacing the lower limit of integration in Eqs. (4.100) by
Qc = Q exp(−ηmax), i.e. assuming that radiation at q < Qc does not enter the detected
region. This is justified at the leading-logarithmic level, where q/Q ∼ θ ∼ exp(−η) and
the scale dependence of the parton distributions and coefficient functions in Eq. (4.98) can
be neglected. Then when ηmax = 0 the form factor Sc is replaced by unity and Eq. (4.97)
correctly predicts a delta function at ET = 0 times the Born cross section. However, this
4.3. Resummation of ET in vector boson and Higgs boson production at
hadron colliders 93
simple prescription cannot be correct at the NLO level, where the τ -dependence of the
scale must be taken into account. Therefore we do not consider the ET distribution in a
restricted rapidity range here.
It is worth noting at this point, an existing related resummation of a variable called
the ‘beam thrust’, essentially defined as the ET weighted by exp(−η) [94].
4.3.1.2 Vector boson production
One of the best studied examples of resummation is in vector boson production through
the partonic subprocess q + q¯′ → V (V =W or Z):
σFcc¯(Q,αs(Q)) = δcqδc¯q¯′δ(Q
2 −M2V )σVqq′ , (4.105)
where, at lowest order,
σWqq′ =
π
3
√
2GFM
2
W |Vqq′|2 ,
σZqq′ =
π
3
√
2GFM
2
Z(V
2
q + A
2
q)δqq′ , (4.106)
with Vqq′ the appropriate CKM matrix element and Vq, Aq the vector and axial couplings
to the Z0. The coefficients in the quark form factor Sq(Q, τ) are [86, 95]:
A(1)q = CF , A
(2)
q =
1
2
CFK , B
(1)
q = −
3
2
CF ,
B(2)q = C
2
F
(
π2
4
− 3
16
− 3ζ3
)
+ CF CA
(
11
36
π2 − 193
48
+
3
2
ζ3
)
+ CF nf
(
17
24
− π
2
18
)
,
(4.107)
where ζn is the Riemann ζ-function (ζ3 = 1.202 . . .), CF = 4/3, CA = 3, nf is the number
of light flavours, and:
K =
(
67
18
− π
2
6
)
CA − 5
9
nf . (4.108)
The above expression for B
(2)
q is in a scheme where the subprocess cross section is
given by the leading-order expression (4.105). In the same scheme the NLO coefficient
functions are [95, 96]:
Cqq(αs, z) =
{
1 +
αs
4π
CF (π
2 − 8)
}
δ(1− z) + αs
2π
CF (1− z)
94 Chapter 4. Effects of QCD radiation on hadron collider observables
≡
(
1 +
αs
π
c(1)q
)
δ(1− z) + αs
2π
CF (1− z) ,
Cqg(αs, z) =
αs
2π
z(1− z) , (4.109)
where the second line defines c
(1)
q . The corresponding splitting functions are:
Pqq(z) = CF
[
1 + z2
(1− z)+ +
3
2
δ(1− z)
]
,
Pqg(z) =
1
2
[
z2 + (1− z)2] . (4.110)
Equations (4.102)–(4.104) therefore give
R(R)q (ξ, τ) =
∫ 1
ξ
dx1
x1
{
fq/h1(x1)fq¯/h2
(
ξ
x1
)(
1 +
αs
π
2c(1)q
)
+
αs
π
∫ 1
ξ/x1
dz
z
[
fq/h1(x1)fq¯/h2
(
ξ
zx1
)
CF (1− z)
+
{
fg/h1(x1)fq¯/h2
(
ξ
zx1
)
+ fq/h1(x1)fg/h2
(
ξ
zx1
)}
1
2
z(1− z)
]}
,
R(I)q (ξ, τ) =
αs
2
∫ 1
ξ
dx1
x1
∫ 1
0
dz
z
{
2fq/h1(x1)fq¯/h2
(
ξ
zx1
)
Pqq(z) (4.111)
+
[
fg/h1(x1)fq¯/h2
(
ξ
zx1
)
+ fq/h1(x1)fg/h2
(
ξ
zx1
)]
Pqg(z)
}
=
αs
2
∫ 1
ξ
dx1
x1
{
2CFfq/h1(x1)fq¯/h2
(
ξ
x1
)[
2 ln
(
1− ξ
x1
)
+
3
2
]
+
∫ 1
ξ/x1
dlz
z
[
2CFfq/h1(x1)
{
fq¯/h2
(
ξ
zx1
)
1 + z2
1− z − fq¯/h2
(
ξ
x1
)
2z
1− z
}
+
{
fg/h1(x1)fq¯/h2
(
ξ
zx1
)
+ fq/h1(x1)fg/h2
(
ξ
zx1
)}
1
2
{
z2 + (1− z)2}]} .
4.3.1.3 Higgs boson production
In the case of Higgs boson production the corresponding LO partonic subprocess is gluon
fusion, g + g → H , through a massive-quark loop:
σFcc¯(Q,αs(Q)) = δcgδc¯gδ(Q
2 −m2H)σH0 , (4.112)
4.3. Resummation of ET in vector boson and Higgs boson production at
hadron colliders 95
where in the limit of infinite quark mass:
σH0 =
α2s(mH)GFm
2
H
288π
√
2
. (4.113)
The coefficients in the gluon form factor Sg(Q, τ) are [97–99]:
A(1)g = CA , A
(2)
g =
1
2
CAK , B
(1)
g = −
1
6
(11CA − 2nf) ,
B(2)Hg = C
2
A
(
23
24
+
11
18
π2 − 3
2
ζ3
)
+
1
2
CF nf − CA nf
(
1
12
+
π2
9
)
− 11
8
CFCA .
(4.114)
Here again, the above expression for B
(2)
g is in a scheme where the Higgs boson sub-
process cross section is given by the leading-order expression (4.112). In the same scheme
the NLO coefficient functions are [100]:
Cgg(αs, z) =
{
1 +
αs
4π
[
CA
(
2− π
2
3
)
+ 5 + 4π2
]}
δ(1− z)
≡
(
1 +
αs
π
c(1)g
)
δ(1− z) ,
Cgq(αs, z) = Cgq¯(αs, z) =
αs
2π
CF z . (4.115)
The corresponding splitting functions are:
Pgg(z) = 2CA
[
z
(1− z)+ +
1− z
z
+ z(1− z)
]
+
1
6
(11CA − 2nf)δ(1− z) ,
Pgq(z) = Pgq¯(z) = CF
1 + (1− z)2
z
. (4.116)
Equations (4.102)–(4.104) therefore give
R
(R)
g (ξ, τ) =
∫ 1
ξ
dx1
x1
{
fg/h1(x1)fg/h2
(
ξ
x1
)(
1 +
αs
π
2c(1)g
)
+
αs
π
∫ 1
ξ/x1
dz
z
[
fg/h1(x1)fs/h2
(
ξ
zx1
)
+ fs/h1(x1)fg/h2
(
ξ
zx1
)]
1
2
CFz
}
,
R
(I)
g (ξ, τ) =
αs
2
∫ 1
ξ
dx1
x1
∫ 1
0
dz
z
{
2fg/h1(x1)fg/h2
(
ξ
zx1
)
Pgg(z)
+
[
fg/h1(x1)fs/h2
(
ξ
zx1
)
+ fs/h1(x1)fg/h2
(
ξ
zx1
)]
Pgq(z)
}
96 Chapter 4. Effects of QCD radiation on hadron collider observables
=
αs
2
∫ 1
ξ
dx1
x1
{
2fg/h1(x1)fg/h2
(
ξ
x1
)[
2CA ln
(
1− ξ
x1
)
+
1
6
(11CA − 2nf)
]
+
∫ 1
ξ/x1
dz
z
[
4CAfg/h1(x1)
{
fg/h2
(
ξ
zx1
)[
z
1− z +
1− z
z
+ z(1− z)
]
− fg/h2
(
ξ
x1
)
z
1− z
}
+
{
fg/h1(x1)fs/h2
(
ξ
zx1
)
+ fs/h1(x1)fg/h2
(
ξ
zx1
)}
CF
1 + (1− z)2
z
]}
, (4.117)
where fs =
∑
q(fq + fq¯).
4.3.2 Resummed distributions
4.3.2.1 Vector boson production
Figure 4.25: Resummed component of the transverse energy distribution in Z0 boson produc-
tion at the Tevatron and LHC. The curves show the effects of the coefficients in
the quark form factor: black, all coefficients; magenta omitting B
(2)
q ; blue A
(1)
q
and A
(2)
q only; red A
(1)
q only.
Figure 4.25 shows the resummed component of the transverse energy distribution in
Z0 boson production at the Tevatron (pp¯ at
√
s = 1.96 TeV) and LHC (pp at
√
s = 14
TeV).8 For all calculations, we use the MSTW 2008 NLO parton distributions [71]. The
different curves show the effects of the subleading coefficients (4.107) in the quark form
factor. We see that while B
(1)
q has a large effect (the difference between the blue and
magenta curves), the effects of the other subleading coefficients are quite small.
8Results for pp at
√
s = 7 TeV are given in appendix D.2.
4.3. Resummation of ET in vector boson and Higgs boson production at
hadron colliders 97
The peak of the resummed distribution lies at around ET ∼ 10 GeV at the Tevatron,
rising to ∼ 14 GeV at the LHC. This is comfortably belowMZ , justifying the resummation
of logarithms of ET/MZ in the peak region. However, at LHC energy the predicted
distribution has a substantial tail at larger values of ET , indicating that the higher-order
terms generated by the resummation formula remain significant even when the logarithms
are not large. In addition, the LHC prediction does not go to zero as it should at small
ET . However, this region is sensitive to the treatment of non-perturbative effects such as
the behaviour of the strong coupling at low scales (we freeze its value below 1 GeV) and
the upper limit in the integral over transverse time (we set τmax = 1/Λ where Λ is the
two-loop QCD scale parameter, set to 200 MeV here).
The resummed component forW± boson production looks very similar, apart of course
from the overall normalisation, and therefore we do not show it here. Predictions with
matching to fixed order will be presented in section 4.3.3.
4.3.2.2 Higgs boson production
Figure 4.26 shows the resummed component of the transverse energy distribution in Higgs
boson production at the Tevatron and LHC, for a Higgs boson mass of 115 GeV. The
effects of subleading terms in the gluon form factor (4.114) are more marked than those
of the quark form factor discussed above. The distribution peaks at large values of ET ,
around 40 GeV at the Tevatron, rising to ∼ 50 GeV at the LHC. This is due to the
larger colour charge of the gluon. However, together with the large effects of subleading
terms, it does make the reliability of the resummed predictions more questionable. Also
in contrast to the vector boson case, the suppression at low and high ET is if anything too
great, resulting in negative values below 16 GeV and above 120 GeV at Tevatron energy.
We verified, by cutting-off the gluon PDFs at zero, that the negative values are due to
the resummation and not due to the gluon PDF becoming negative at low x values.
4.3.3 Matching to fixed order
The resummed distributions presented above include only terms that are logarithmically
enhanced at small ET . To extend the predictions to larger ET we must match the resum-
mation to fixed-order calculations. To avoid double counting of the resummed terms, the
corresponding contribution must be subtracted from the fixed-order result.
We consider here only matching to first order in αs. To this order the ET distribution
98 Chapter 4. Effects of QCD radiation on hadron collider observables
Figure 4.26: Resummed component of the transverse energy distribution in Higgs boson pro-
duction at the Tevatron and LHC. The curves shows the effects of the coefficients
in the gluon form factor: black, all coefficients; magenta omitting B
(2)
g ; blue A
(1)
g
and A
(2)
g only; red A
(1)
g only.
for ET > 0 has the form:
dσ
dET
=
1
ET
(A lnET +B) + C(ET ) , (4.118)
where A and B are constants (for a given process and collision energy) and the function
C(ET ) is regular at ET = 0. The terms involving A and B are already included in the
resummed prediction, and therefore we have only to add the regular function C to it to
obtain a prediction that is matched to the O(αs) result. This function is determined
by fitting the O(αs) prediction for ET dσ/dET to a linear function of lnET at small
ET , extracting the coefficients A and B, and then subtracting the enhanced terms in
Eq. (4.118).
4.3.3.1 Vector boson production
The above matching procedure is illustrated for Z0 production at the Tevatron in Fig. 4.27.
The fit to the logarithmically enhanced terms gives excellent agreement with the order-αs
result out to around 20 GeV, confirming the dominance of such terms throughout the
region of the peak in Fig. 4.25. The remainder function C(ET ) vanishes at small ET
and rises to around 10 pb/GeV, falling off slowly at large ET . Consequently the matching
correction to the resummed prediction is small and roughly constant throughout the region
4.3. Resummation of ET in vector boson and Higgs boson production at
hadron colliders 99
Figure 4.27: Left: O(αs) ET distribution in Z0 production at the Tevatron; solid, full predic-
tion; dashed, fit to enhanced terms. Right: difference between full prediction and
fit to enhanced terms.
Figure 4.28: Predicted ET distribution in Z
0 production at the Tevatron and LHC. Solid:
resummed prediction matched to O(αs). Dashed: resummed only. Dot-dashed:
O(αs) only.
40–100 GeV, as shown in Fig. 4.28.
As shown on the right in Fig. 4.28, the situation is similar at LHC energy: the matching
correction is small, although in this case it is negative below about 40 GeV. The large tail
at high ET and the bad behaviour at low ET , due to uncompensated higher-order terms
generated by resummation, are not much affected by matching to this order.
The corresponding matched predictions forW± boson production are shown in Fig. 4.29.
100 Chapter 4. Effects of QCD radiation on hadron collider observables
Figure 4.29: Predicted ET distribution in W
++W− production at the Tevatron and LHC.
Solid: resummed prediction matched to O(αs). Dashed: resummed only. Dot-
dashed: O(αs) only.
As remarked earlier, the form of the resummed distribution is very similar to that for Z0
boson production, and again the matching correction is small.
Note that at high ET the O(αs) distributions should approximate the matched distri-
butions, although this is not apparent in the figures.
4.3.3.2 Higgs boson production
Adopting the same matching procedure for Higgs boson production, we find the results
shown in Figs. 4.30 and 4.31. The form of the matching correction is similar to that
for vector bosons, but its effect is rather different. The roughly constant, then slowly
decreasing, correction in the region 20–100 GeV is not small compared to the resummed
result and therefore it raises the whole distribution by a significant amount throughout
this region. This has the beneficial effect of compensating the negative values at low
and high ET at Tevatron energy. However, it further enhances the high ET tail of the
distribution at LHC energy. This, together with the relatively large correction in the peak
region, casts further doubt on the reliability of the predictions in the case of Higgs boson
production.
4.3. Resummation of ET in vector boson and Higgs boson production at
hadron colliders 101
Figure 4.30: Left: O(αs) ET distribution in Higgs boson production at the Tevatron; solid,
full prediction; dashed, fit to enhanced terms. Right: difference between full
prediction and fit to enhanced terms.
Figure 4.31: Predicted ET distribution in Higgs boson production at the Tevatron and LHC.
Solid: resummed prediction matched to O(αs). Dashed: resummed only. Dot-
dashed: O(αs) only.
4.3.4 Monte Carlo comparisons
In this section we compare the resummed and matched distributions obtained above with
the predictions of the parton shower Monte Carlo programs HERWIG [92] and HERWIG++ [46].
Comparisons are performed first at the parton level, that is, after QCD showering
from the incoming and outgoing partons of the hard subprocess. We say ‘incoming and
outgoing’ because both programs apply hard matrix element corrections: in addition to
102 Chapter 4. Effects of QCD radiation on hadron collider observables
the Born process, O(αs) real emission hard subprocesses are included in phase space
regions not covered by showering from the Born process.
After showering, the Monte Carlo programs apply a hadronization model to convert
the partonic final state to a hadronic one. We show the effects of hadronization in the case
of HERWIG only; those in HERWIG++ are broadly similar since both programs use basically
the same cluster hadronization model we described in section 3.2.2.3. The programs also
model the underlying event, which arises from the interactions of spectator partons (see
section 3.2.2.5) and makes a significant contribution to the hadronic transverse energy.
In this case we show only the underlying event prediction of HERWIG++, since the default
model used in HERWIG has been found to give an unsatisfactory description of Tevatron
data. For an improved simulation of the underlying event, HERWIG can be interfaced
to the multiple interaction package JIMMY [43], which is similar to the model built into
HERWIG++.
4.3.4.1 Vector boson production
Figure 4.32 shows the comparisons for Z0 production at the Tevatron and LHC. The
HERWIG predictions are renormalised by a factor of 1.3 to account for the increase in the
cross section from LO to NLO. The HERWIG++ results were not renormalised, because
they were obtained using LO** parton distributions [78], which aim to reproduce the NLO
cross section. We see that the parton-level Monte Carlo predictions of both programs
agree fairly well with the matched resummed results above about 15 GeV, but HERWIG++
generates a substantially higher number of events with low values of ET . A similar
pattern is evident in the results on W± boson production, shown in Fig. 4.33. The effects
of hadronization, shown by the difference between the blue and magenta histograms, are
also similar for both vector bosons. They generate a significant shift in the distribution,
of around 10 GeV at Tevatron energy and 20 GeV at LHC.
4.3.4.2 Higgs boson production
As may be seen from Fig. 4.34, the agreement between the resummed and parton-level
Monte Carlo results is less good in the case of Higgs boson production than it was for vector
bosons. Here we have renormalised the HERWIG predictions by a factor of 2 to allow for the
larger NLO correction to the cross section. Then the Monte Carlo ET distributions agree
quite well with each other but fall well below the matched resummed predictions. Fair
agreement above about 40 GeV can be achieved by adjusting the normalisation, but then
4.3. Resummation of ET in vector boson and Higgs boson production at
hadron colliders 103
Figure 4.32: Predicted ET distribution in Z
0 boson production at the Tevatron and LHC.
Comparison of resummed and Monte Carlo results.
Figure 4.33: Predicted ET distribution in W
++ W− boson production at the Tevatron and
LHC. Comparison of resummed and Monte Carlo results.
the Monte Carlo programs predict more events at lower ET . The effect of hadronization
is similar to that in vector boson production, viz. a shift of about 10 GeV at the Tevatron
rising to 20 GeV at the LHC, which actually brings the HERWIG distribution into somewhat
better agreement with the resummed result.
4.3.4.3 Modelling the underlying event
Figures 4.35 and 4.36 show the parton-level HERWIG++ predictions for the ET distribution
in Z0 and Higgs boson production, respectively, with the contributions from initial-state
104 Chapter 4. Effects of QCD radiation on hadron collider observables
Figure 4.34: Predicted ET distribution in Higgs boson production at the Tevatron and LHC.
Comparison of resummed and Monte Carlo results.
Figure 4.35: Predicted ET distribution in Z
0 boson production at the Tevatron and LHC.
Monte Carlo results including underlying event.
radiation (in red, already shown in Figs. 4.32 and 4.34), the underlying event (blue)
and the combination of the two (green). As we have already seen in section 3.2.2.5, the
underlying event is modelled using multiple parton interactions. Clearly it has a very
significant effect on the ET distribution. However, this effect is substantially independent
of the hard subprocess, as we have already found in section 4.2.6.4 when examining the
total invariant mass, M . This can also be seen from the comparison of the ET of the UE
associated to different subprocesses in Fig. 4.37.
We find that the probability distribution of the ET contribution of the underlying event
4.3. Resummation of ET in vector boson and Higgs boson production at
hadron colliders 105
Figure 4.36: Predicted ET distribution in Higgs boson production at the Tevatron and LHC.
Monte Carlo results including underlying event.
in the HERWIG++ Monte Carlo can be represented quite well by a Fermi distribution:
P (ET ) =
1
N
1
exp
(
ET−µ
T
)
+ 1
, (4.119)
for which the normalisation, N , is given by
N = T ln
[
exp
(µ
T
)
+ 1
]
. (4.120)
The dependence of the ‘chemical potential’ µ and ‘temperature’ T on the hadronic collision
energy is shown in Fig. 4.38. The red curves show fits to the energy dependence of the
form:
µ =
A
√
s
1 +B
√
s
, T = q
(
1− e−r
√
s
)
, (4.121)
where the coefficients in the fits are A = 20(1), B = 0.030(4), q = 36(2), r = 0.28(3).
Example fits for the LHC case,
√
s = 14 TeV, are shown in Fig. 4.37.
4.3.5 Conclusions
We have extended the resummation of the hadronic transverse energy ET in vector boson
production to next-to-leading order (NLO) in the resummed exponent, parton distribu-
tions and coefficient functions, and also presented for the first time the corresponding
predictions for Higgs boson production. We have matched the resummed results to the
corresponding O(αs) predictions, by adding the contributions in that order which are not
106 Chapter 4. Effects of QCD radiation on hadron collider observables
Figure 4.37: Comparison of ET distributions of the HERWIG++ underlying event in different
subprocesses at the LHC. Fits obtained using the Fermi distribution of Eq. 4.119
for the mean values of the parameters given below Eq. 4.121 are shown, as well as
two example fits, obtained by varying the parameters by one standard deviation
in different directions (lowest: A = 21, B = 0.026, q = 34, r = 0.31, highest:
A = 19, B = 0.034, q = 38, r = 0.25).
included in the resummation. In addition we have compared our results to parton shower
Monte Carlo predictions and illustrated the effects of hadronization and the underlying
event.
In the case of vector boson production, the resummation procedure appears stable and
the parton-level results should be quite reliable. The leading-order mechanism of quark-
anti-quark annihilation typically generates a moderate amount of transverse energy in
initial-state QCD radiation. Consequently the effects of subleading resummed terms and
fixed-order matching are small and the peak of the ET distribution lies well below the
boson mass scale, where resummation makes good sense. The comparisons with Monte
Carlo programs reveal some discrepancies but these are at the level of disagreements
between different programs; in this case the resummed predictions should be more reliable
(at parton level) than existing Monte Carlo programs. The programs suggest that the
4.3. Resummation of ET in vector boson and Higgs boson production at
hadron colliders 107
Figure 4.38: Fitted values of the parameters of the HERWIG++ underlying event in Higgs
production in pp collisions at various energies.
non-perturbative effects of hadronization and the underlying event are substantial. These
effects can however be modelled in a process-independent way. We have suggested a
simple parametrization of the contribution of the underlying event through the model
given in the HERWIG++ event generator. We stress once again, however, that recent UE
results from the LHC experiments have shown that this model does not describe the data
adequately [81], and as of version 2.5.0, HERWIG++ includes an improved description [53]
via a colour reconnection model. The effect of the improved model on the ET distributions
remains to be investigated in future work.
The situation in Higgs boson production is not so good. The dominant mechanism
of gluon fusion generates copious ISR and the effects of subleading terms and matching
are large. The resummed ET distribution peaks at a value that is not parametrically
smaller than the Higgs boson mass and the behaviour at low and high ET is unphysi-
cal before matching. The discrepancies between the matched resummed and Monte Carlo
predictions are substantially greater than those between different programs, even allowing
for uncertainties in the overall cross section. All this suggests that there are significant
higher-order corrections that are not taken into account, either further subleading loga-
rithms or unenhanced terms beyond NLO. It would be interesting (but very challenging)
to attempt to extract such terms from the available NNLO calculations of Higgs boson
production.
108 Chapter 4. Effects of QCD radiation on hadron collider observables
Chapter 5
New physics searches at hadron
colliders
The original work in this chapter was done in collaboration with Oluseyi Latunde-
Dada [4], Bryan Webber, Kazuki Sakurai and Ben Gripaios [5].
5.1 Introduction
The challenges present at hadron colliders have been discussed in section 4.1, where we
emphasised that new physics signals can be difficult to observe and interpret. The diffi-
culties arise due to multiple jets and/or leptons, the presence of invisible particles and the
huge backgrounds that may potentially imitate the topology and kinematics of a signal
of relatively low rate. It is thus extremely important to investigate the phenomenology
of concrete models of new physics. At the same time, we need to ensure that we do not
weaken our reach due to biases introduced by focusing on specific models. The strategies
that we develop must allow us to explore the possibilities for generic new physics signa-
tures, motivated by theoretically and experimentally plausible models. Furthermore, it is
important for the observables that we construct to be well-defined and calculable so they
can provide unambiguous information for the discovery of new physics and determination
of its properties.
In this chapter we investigate two different new physics scenarios. In section 5.2
we present an investigation on a generic model that involves the production of a new
heavy charged vector boson, called W ′ (W prime), essentially a heavy version of the
standard model W gauge boson. Such heavy bosons may arise from the breaking of a
large symmetry group to the SM symmetry group, as excitations of the SM W in models
109
110 Chapter 5. New physics searches at hadron colliders
with extra dimensions, or as a composite particle in a strong dynamics theory. We study
the interference effects of a W ′ with the SM W and extend the treatment to next-to-
leading order by using the MC@NLO and POWHEG methods described in section 3.3. We
incorporate these features in a publicly available event generator and use this to examine
the detection reach at the Tevatron and the LHC. In section 5.3 we examine a model
that contains composite leptoquarks, particles that couple to leptons and quarks (as the
name suggests), which may arise in strong dynamics theories. These couple primarily
to the third generation of fermions, a feature motivated by a model of fermion mass
generation which aims to solve the problem of flavour-changing neutral currents in theories
where electroweak symmetry is broken via strong dynamics. We propose a general search
strategy for discovery and mass reconstruction of leptoquarks.1
5.2 NLO production of heavy charged vector bosons
5.2.1 Introduction
There exists a proliferation of theories which contain new heavy, electrically neutral or
charged, gauge bosons referred to as Z ′ and W ′ respectively. Both the Z ′ and W ′ have
been studied extensively and reviews can be found in [13] and [101] respectively. The
study of this section focuses on W ′ bosons.
The new charged vector bosons may or may not have similar properties to the SM
bosons, depending on the underlying theory. In particular they may have right-handed in-
stead of left-handed couplings, may couple to new fermions, or may even be fermiophobic.
Models which predict new charged vector bosons may be based on extensions of the elec-
troweak gauge group, SU(2)×U(1), for example to the gauge group SU(2)1 × SU(2)2 ×
U(1) [102], or groups that contain the electroweak symmetry, such as SU(3) × U(1)
or SU(4) × U(1) [103]. Several models with extra dimensions contain W ′ bosons as
Kaluza-Klein excitations in the bulk. Examples of these models include the Randall-
Sundrum model (section 2.3.2.2) with bulk gauge fields [27] and Universal Extra Dimen-
sions [26,104]. Theories which break the electroweak sector via strong dynamics may also
contain the W ′ as a composite spin-1 particle [105, 106].
1It is interesting to mention that these two scenarios of new physics are not mutually exclusive. Both
leptoquarks and heavy vector bosons may arise in the same model. This could be, for example, a strong
dynamics theory that contains composite scalar particles, acting as leptoquarks, and composite vector
particles, acting as W ′’s. In these models the W ′ bosons would also potentially couple preferentially to
the fermions of the third generation.
5.2. NLO production of heavy charged vector bosons 111
W/W ′
ℓ
νℓ
Figure 5.1: Schematic diagram for the Drell-Yan process pp→W/W ′ → ℓνX.
Current Monte Carlo simulations of Drell-Yan-type W ′ production at hadron collid-
ers rely mainly on leading-order matrix elements and parton showers. There exists no
treatment of next-to-leading (NLO) QCD effects which simultaneously includes the in-
terference effects for the W ′. Here, we present the results of the event generator package
Wpnlo [107] which improves the treatment of Drell-Yan production of heavy charged gauge
bosons. We consider the interference effects with the Standard ModelW , which have been
shown to provide valuable information [108], but have not been considered in experimen-
tal searches. We use the MC@NLO and POWHEG methods, discussed in section 3.3 to match
the NLO QCD calculation to the parton shower, producing fully exclusive events using
the HERWIG++ event generator. Note that a similar implementation for the Z ′ exists
for the NLO MC@NLO event generator, which matches the complete NLO matrix elements
with the parton shower and cluster hadronization model of the Fortran HERWIG event
generator [109].
5.2.2 W ′ at leading order
The W ′ reference model is based on the one which originally appeared in Ref. [110]. In
the model described therein, sometimes referred to as the Sequential Standard Model,
the magnitudes of the W ′ couplings to fermions are directly transcribed from the SM
W , i.e. it is a heavy copy of the SM W . In the present treatment we allow both right-
and left-handed couplings, ∝ (1± γ5) respectively. In the case of right-handed couplings,
112 Chapter 5. New physics searches at hadron colliders
we assume that the right-handed neutrinos are light compared to the W ′, but not light
enough for the Z boson to decay into them. The W ′ and W couplings to fermions are
given by
LWiff ′ =
(
GFM
2
W√
2
)1/2
Vff ′C
ℓ,q
i f¯γµ(ki − hiγ5)f ′W µi + h.c. , i = {W,W ′}, (5.1)
where GF is the Fermi coupling constant,
2 MW is the SM W mass, C
ℓ,q
i are the coupling
strengths of boson i to leptons and quarks respectively, W µ is the massive boson polari-
sation vector, f and f ′ are the Dirac spinors for the fermions and Vff ′ is the unit matrix
when ff ′ are leptons and the CKM matrix, given in appendix C, when ff ′ are quarks.
The ki and hi represent the structure of the vector-axial vector (V-A) coupling of the
bosons, where for the case i = W we have kW = hW = 1, i.e. purely left-handed cou-
pling. Using the above coupling to fermions, we show in appendix E.2 that the hadronic
differential cross section for the process pp→ W+/W ′+ → ℓνX (Fig. 5.1) is given by
dσ
dτdydz
=
G2FM
4
W
192π
∑
qq′
|Vqq′|2
[
SG+qq′(1 + z
2) + 2AG−qq′z
]
, (5.2)
where z = cos θ is defined as the scattering angle between the u-type quark and the
outgoing neutrino (both being fermions) in the centre-of-mass frame, y is the rapidity of
the intermediate boson, τ = sˆ/s is the ratio of the squares of the quark centre-of-mass
energy to the proton centre-of-mass energy. S = S(sˆ) and A = A(sˆ) are functions of the
quark centre-of-mass energy:
S =
∑
i,j
Si,j =
∑
i,j
Pij(CiCj)
ℓ(CiCj)
q(kikj + hihj)
2 , (5.3)
A =
∑
i,j
Ai,j =
∑
i,j
Pij(CiCj)
ℓ(CiCj)
q(kjhi + hjki)
2 , (5.4)
with
Pij = sˆ
(sˆ−M2i )(sˆ−M2j ) + ΓiΓjMiMj
[(sˆ−M2i )2 + Γ2iM2i ][i→ j]
, (5.5)
where i, j can be eitherW orW ′ andMi, Γi are the mass and width of boson i respectively.
The functions G±qq′ that appear in the differential cross section are even or odd products
2The constantGF is related to the weak coupling constant g, which appears in Eq. (2.28), by GF /
√
2 =
g2/(8M2W ).
5.2. NLO production of heavy charged vector bosons 113
of parton density functions for the relevant hadrons, given by
G±qq′ =
[
fq/A(xa, sˆ)fq′/B(xb, sˆ)± fq/B(xb, sˆ)fq′/A(xa, sˆ)
]
, (5.6)
where fq/h(x, sˆ) is the parton density function for a quark q in a hadron h carrying
hadron momentum fraction x, in a collision in which the quark pair centre-of-mass energy
squared is sˆ. The A,B indices represent the type of the ‘left’ (travelling in the positive
z-direction) or ‘right’ (travelling in the negative z-direction) hadrons respectively. This
definition allows for easy modification of the pp → W/W ′ → ℓνX cross section to the
pp¯→W/W ′ → ℓνX, by changing the PDFs accordingly. Analogous expressions can also
be obtained for the case of (W−, W ′−), by appropriately modifying the functions G±qq′ and
taking z → −z. The width can be taken to be a free parameter in the reference model:
the couplings of the W ′ to other gauge bosons or the Higgs boson are model-dependent.3
Here we shall assume for illustration that the fermionic decay width4 scales with the mass
as ΓW ′→ff ′ = (4ΓWMW ′/3MW ) (provided that MW ′ ≫ Mt, the mass of the top quark)5
and that the tri-boson W ′WZ vertex is suppressed by a small mixing angle and can be
neglected in the analysis.
5.2.3 W −W ′ interference
The narrow width approximation (NWA) is often used when discussing the production of
new vector bosons. This approximation is usually claimed to be valid up to O(ΓW ′/MW ′)
corrections. However, W -W ′ interference effects can become important in certain regions
even as the width ΓW ′ → 0, see for example [108], and as we also show below. Use of the
NWA may thus lead to invalid conclusions, as pointed out in [111].
Interference effects arise because the Drell-Yan process pp → W/W ′ → ℓνX can
proceed either via an intermediate SM W or a W ′ in the reference model. The matrix
element squared for the process may be decomposed in the following way:
|M|2 = |MW |2 + |MW ′|2 + 2Re(M∗WMW ′) . (5.7)
The last term, which contains the interference, depends on the functions S(sˆ) and A(sˆ)
3An exception is the photon, for which the coupling is fixed by gauge invariance.
4The fermionic decay width is thus also taken to be the lower bound on the total width.
5The factor of 4/3 comes from the extra decay channel that opens up when MW ′ > Mtop, e.g.
W ′+ → tb¯.
114 Chapter 5. New physics searches at hadron colliders
(Eqs. (5.3) and (5.4)). Here we discuss the function S(sˆ) when studying interference
effects, although the arguments for A(sˆ) are equivalent. The function S(sˆ) can be de-
composed into pieces that arise individually due to the W or the W ′, and an interference
piece:
S = SW,W + SW ′,W ′ + SW,W ′ + SW ′,W = SW,W + SW ′,W ′ + 2Sint , (5.8)
where we have defined the interference term Sint ≡ SW,W ′ = SW ′,W . Explicitly, this term
may be written as
Sint =
[
sˆ
(sˆ−M2W )(sˆ−M2W ′) + ΓWΓW ′MWMW ′
[(sˆ−M2W )2 + Γ2WM2W ][W →W ′]
]
(1 + hWhW ′)
2 , (5.9)
where we have set all the couplings Cℓ,qW/W ′ = 1 and kW = kW ′ = 1. It is evident that
since hW = 1 for the SM W , when we set hW ′ = −1 then the interference term vanishes:
Sint = 0. This implies that there is no interference for the case of the SM W and a
purely right-handed W ′, and the square of the total matrix element for the process can
be written as the sum of the squares of the individual matrix elements for the W and W ′:
|M(hW ′ = −1)|2 = |MW |2 + |MW ′|2 . (5.10)
It is simple to see why this is so: the W ′ and SM W decay to final-state particles of
different helicities, which are distinguishable, and hence the two processes cannot interfere.
However, when hW = 1 and hW ′ = 1, i.e. both bosons possessing left-handed couplings,
we have Sint 6= 0. In fact, by examining of the expression for Si,j (Eq. (5.3)), we observe
that Sint should be of the same order of magnitude as SW ′,W ′ and SW,W . Figure 5.2 shows
the variation of the interference term, as well as the W and W ′ terms SW,W and SW ′,W ′,
for the case MW ′ = 1 TeV. We observe that Sint is negative (green, blue, purple) in
the intermediate mass squared region sˆ ∈ (M2W ,M2W ′) = (∼ 6400 GeV2, 106 GeV2). The
total cross section in this region is less than the sum of the individual W and W ′ cross
sections. We note that the possibility of a reduced cross section is seldom considered
in experimental searches. It is important to realise that the interference term is non-
vanishing and comparable in magnitude to the other terms in S(sˆ) even as ΓW ′ → 0, a
clear indication that the narrow width approximation is not justified in the intermediate
region.
5.2. NLO production of heavy charged vector bosons 115
2#10
5
4#10
5
6#10
5
8#10
5
1#10
6
K0.00010
K0.00005
0
0.00005
0.00010
sˆ (GeV2)
Si,j (GeV
−2)
Figure 5.2: The interference term Sint for hW ′ = 1, MW ′ = 1 TeV, plotted against sˆ, for dif-
ferent widths: ΓW ′ = 1, 35, 100 GeV (green, blue, purple respectively). The terms
SW,W (red) and SW ′,W ′ (black) are shown for comparison. It is evident that Sint is
negative in the intermediate region (M2W ,M
2
W ′) = (∼ 6400 GeV2, 106 GeV2). It is
also clear that the magnitude of the interference term is comparable to SW,W and
SW ′,W ′. As the width decreases the negative peak becomes narrower, but there al-
ways exists a portion of the curve in the intermediate region which is independent
of the width.
5.2.4 Extension to NLO
Next, we extend the simulation to NLO using the MC@NLO and the POWHEG methods. We
briefly discuss their application to W ′ boson production. Full details of the application
of the MC@NLO method to vector boson production can be found in section 6 of Ref. [112].
Details of the application of the POWHEG method can be found in chapter 4 of Ref. [59],
where vector boson production is discussed in detail.
116 Chapter 5. New physics searches at hadron colliders
5.2.4.1 The MC@NLO method
The NLO cross section for the production of W ′ bosons can be written as a sum of two
contributions:
σNLO = σqq¯′ + σ(qq¯′)g , (5.11)
where σqq¯′ is the contribution from qq¯
′ annihilation and σ(qq¯′)g is the contribution from the
Compton subprocesses. In the modified minimal subtraction (MS) factorisation scheme,
these are:
σqq¯′ = σ0
∑
q
∫
dx1dx2
x[Dq(x1)Dq¯′(x2) + q ↔ q¯′]
Dq(xq)Dq¯′(xq¯′)
[
δ(1− x) + αS
2π
CF
{
−21 + x
2
1− x ln x
+ 4(1 + x2)
(
ln(1− x)
1− x
)
+
+
(
−8 + 2
3
π2
)
δ(1− x)
}]
,
σ(qq¯′)g = σ0
∑
q,q¯′
∫
dx1dx2
x[D(q,q¯′)(x1)Dg(x2) + (q, q¯
′)↔ g]
Dq(xq)Dq¯′(xq¯′)
αS
2π
TR
[
1
2
+ 3x− 7
2
x2
+ (x2 + (1 + x2)) ln
(1− x)2
x
]
, (5.12)
where σ0 is the Born differential cross section
d2σ0
dQ2dY
with Q the invariant mass and Y
the rapidity of the vector boson. The x1, x2 are the NLO momentum fractions and xq, xq¯′
are the Born momentum fractions with Q2 = xqxq¯′S, if S denotes the hadronic centre-of-
mass energy. Also, x =
xqxq¯′
x1x2
and Dq(x1) = x1fq(x1) etc., with fq(x1) being the parton
distribution function of parton q.
Focusing on the qq¯′ annihilation process for the moment, if we introduce the variable
y = cos θ , (5.13)
where θ is the scattering angle of the emitted parton in the partonic centre-of-mass frame,
we can rewrite
σqq¯′
σ0
as an integral over x and y:
σqq¯′
σ0
=
∑
q
∫
dxdy
[{
x[Dq(x1)Dq¯′(x2) + q ↔ q¯′]
Dq(xq)Dq¯′(xq¯′)
1
2
(
δ(1− x) + αS
2π
CF
(
−21 + x
2
1− x ln x
+ 4(1 + x2)
(
ln(1− x)
1− x
)
+
+
(
−8 + 2
3
π2
)
δ(1− x)
))
−Mqq¯′(x, y)
}
+Mqq¯′(x, y)
]
,
(5.14)
5.2. NLO production of heavy charged vector bosons 117
where Mqq¯′(x, y) is the real emission matrix element. Since we have subtracted this
contribution from the total cross section, in the curly brackets we are left with the sum
of the Born, virtual and QCD PDF correction contributions. Now we can define an
infrared-safe observable O whose NLO expectation value is given by
〈Oqq¯′〉 =
∑
q
∫
dxdy
[
OW ′
{
x[Dq(x1)Dq¯′(x2) + q ↔ q¯]
Dq(xq)Dq¯′(xq¯′)
1
2
(δ(1− x)
+
αS
2π
CF
(
−21 + x
2
1− x ln x+ 4(1 + x
2)
(
ln(1− x)
1− x
)
+
+
(
−8 + 2
3
π2
)
δ(1− x)
) )
−Mqq¯′(x, y)
}
+OW ′gMqq¯′(x, y)
]
, (5.15)
where OW ′ and OW ′g are observables arising from hadronic final states generated from
q+ q¯ → W ′ and q+ q¯ →W ′ + g starting configurations respectively. This however is not
entirely correct because of double counting in the final states represented by OW ′ which
are already included in the states arising from OW ′g. The solution to this is to subtract
the parton shower contributions, which we denote MCqq¯′ (x, y), from the regions in which
the parton shower contributes (the jet region J) and integrate the full matrix element in
the hard emission region D, left untouched by the shower. This gives for 〈Oqq¯′〉:
〈Oqq¯′〉 =
∑
q
∫
J
[
OW ′
{
x[Dq(x1)Dq¯′(x2) + q ↔ q¯′]
Dq(xq)Dq¯′(xq¯′)
1
2
(
δ(1− x) + αS
2π
CF
(
−21 + x
2
1− x ln x
+ 4(1 + x2)
(
ln(1− x)
1− x
)
+
+
(
−8 + 2
3
π2
)
δ(1− x)
))
−Mqq¯′ +MCqq¯′
}
+ OW ′g
{
Mqq¯′ −MCqq¯′
}]
+
∑
q
∫
D
[
OW ′
{
x[Dq(x1)Dq¯′(x2) + q ↔ q¯′]
Dq(xq)Dq¯′(xq¯′)
1
2
(
δ(1− x) + αS
2π
CF
(
−21 + x
2
1− x ln x
+ 4(1 + x2)
(
ln(1− x)
1− x
)
+
+
(
−8 + 2
3
π2
)
δ(1− x)
))
−Mqq¯′
}
+OW ′gMqq¯′
]
.
(5.16)
A similar functional 〈O(qq¯′)g〉 can be generated for the Compton subprocesses. Events can
then be generated in the different regions of phase space according to their contributions
to the above integrals. These events are then interfaced with HERWIG++ and showered.
Full details of the algorithm for event generation can be found in [112].
118 Chapter 5. New physics searches at hadron colliders
5.2.4.2 The POWHEG method
The POWHEG method, as described in section 3.3, involves the generation of the hardest
radiation from the parton shower according to the real emission matrix element and
independently of the shower Monte Carlo generator used. If we introduce
Rv,r = Mqq¯′ +M(qq¯′)g , (5.17)
where Mqq¯ and M(qq¯′)g are real emission matrix elements for qq¯
′ annihilation and the
Compton subprocesses respectively, we can write the cross section for the hardest gluon
emission event as
dσ =
∑
q
B¯qvdΦv [∆
q(0) + ∆q(pT)Rv,rdΦr] . (5.18)
The index q runs over all quarks and anti-quarks. The subscript v represents the Born
variables, which in this case are the invariant mass Q and the rapidity Y of the boson, r
represents the radiation variables x, y and dΦv, dΦr are the Born and real emission phase
spaces respectively.
∆q(pT) is the modified Sudakov form factor for the hardest emission with transverse
momentum pT, as indicated by the Heaviside function in the exponent of Eq. (5.19):
∆q(pT) = exp
[
−
∫
dΦrRv,rΘ(kT(v, r)− pT)
]
, (5.19)
where kT is the transverse momentum of the hardest emission relative to the splitting
axis and in this case is given by
kT(x, y) =
√
Q2
4x
(1− x)2(1− y2) . (5.20)
Furthermore:
B¯qv = B
q
v + V
q
v +
∫
(Rv,r − Cv,r)dΦr . (5.21)
B¯qv is the sum of the Born (B
q
v), virtual (V
q
v ) and real (Rv,r) terms (with some counter-
terms, Cv,r). The Born variables are generated with distribution B¯
q
v , with the radiation
variables of the first emission generated according to [∆q(0) + ∆q(pT)Rv,rdΦr].
In the MS factorisation scheme, the contribution to the order-αS cross section for W
′
production is given by Eqs. (5.11) and (5.12). The function B¯q in Eq. (5.21) can then
be written as a sum of finite terms using the subtraction method. Here we borrow the
5.2. NLO production of heavy charged vector bosons 119
MC@NLO subtraction formula introduced in Eq. (5.14) and write the function B˜q(Q2, Y ) as
B˜q(Q2, Y ) =
∑
q
∫
dxdydQ2dY
d2σ0
dQ2dY
[
x[Dq(x1)Dq¯′(x2) + q ↔ q¯]
Dq(xq)Dq¯′(xq¯)
1
2
[δ(1− x)
+
αS
2π
CF
{
−21 + x
2
1− x ln x +4(1 + x
2)
(
ln(1− x)
1− x
)
+
+
(
−8 + 2
3
π2
)
δ(1− x)
}]
−Mqq¯ +MCqq¯′ +
{
Mqq¯′ −MCqq¯′
}
+
x[D(q,q¯′)(x1)Dg(x2) + (q, q¯
′)↔ g]
Dq(xq)Dq¯(xq¯)
αS
2π
TF
1
2
[
1
2
+ 3x− 7
2
x2
+ (x2 + (1 + x2)) ln
(1− x)2
x
]
−M(q,q¯′)g +MC(q,q¯′)g +
{
M(q,q¯′)g −MC(q,q¯′)g
}]
,
(5.22)
where we have written the virtual and PDF corrections in terms of the real emission
matrix elements and MC are the subtracted parton shower approximation terms in the
HERWIG++ jet regions. Note that the above prescription does not imply that the POWHEG
method depends on the shower MC used. We have simply used the shower approximation
terms to define a subtraction scheme for the definition of the NLO cross section.
The flavour of the event, the Born variables Q2 and Y , as well as the the radiation
variables x and y are then generated according to the integrand in Eq. (5.22). The
radiation variables are ignored, which amounts to integrating away these variables, leaving
the Born variables distributed according to B¯q(Q2, Y ). The radiation variables x, y are
generated according to
∆q(pT)R(x, y)dxdy . (5.23)
Details of the algorithm used can be found in Ref. [59].
5.2.5 Experimental bounds
We provide a brief overview of direct and indirect searches for W ′ before presenting our
results.
At the Fermilab Tevatron6 both the D0 and CDF collaborations have studied the
eν [113, 114] and tb¯ [115–117] channels. The W ′ was assumed to have narrow width and
SM-like couplings to fermions. In the eν channel the signal consists of a high-energy
6The Tevatron is a proton-antiproton collider located at Fermilab, USA, with a hadronic centre-of-
mass energy of 1.96 TeV.
120 Chapter 5. New physics searches at hadron colliders
electron and missing transverse energy, with an edge in the transverse mass distribution
at MW ′. In the tb¯ channel the signal consists of a W boson decaying leptonically and
two b-jets [13]. The limits obtained from the eν searches, corresponding to 1 fb−1 of data
for D0 and 205 pb−1 for CDF, are MW ′ > 1 TeV and MW ′ > 788 GeV respectively.
The D0 Collaboration has published results using 0.9 fb−1 of data in the tb¯ channel, in
which the limit for left-handed W ′ masses is MW ′ > 731 GeV. Furthermore, for right-
handed W ′ bosons, the limit is MW ′ > 739 GeV assuming the W ′ boson decays to all
fermions, and MW ′ > 768 GeV if it decays only to quarks. CDF has set the limits, using
1.9 fb−1, MW ′ > 800 GeV when MW ′ > MνR, the mass of the right-handed neutrino,
and MW ′ > 825 GeV when MW ′ < MνR . The limits given are at 95% confidence level
(C.L.). A recent review on Tevatron searches is contained in Ref. [118]. More recently,
using 36 pb−1 of LHC data at 7 TeV, the CMS collaboration has published limits for the
sequential SM-like W ′ using the µν channel of MW ′ > 1.40 TeV. This limit was stated to
increase to 1.58 TeV if the analysis was combined with the eν analysis. The equivalent
limit from the ATLAS collaboration, extracted using an equivalent amount of data, is
1.47 TeV [119].
Low energy constraints on W ′’s are strongly model-dependent. If the W ′ couples to
quarks, then box diagrams involving a SM W and a W ′ contribute to meson mixing, for
example to KL −KS mixing. Then the limit arising for the left-right symmetric model,
based on the symmetry group SU(2)L×SU(2)R×U(1)B−L, isMW ′ > 2.5 TeV [120]. This
can be relaxed if we assume no correlation between the right-handed quark and lepton
couplings [121]. Limits also arise from possible contributions ofW ′ bosons in neutrinoless
double-beta decay and right-handed neutrino emissions from supernovae [13].
5.2.6 Results
We present a sample of distributions of variables obtained for∼ 105 events using the Wpnlo
event generator, both at leading and next-to-leading order, using the MC@NLO and POWHEG
methods. The parton-level Wpnlo output was interfaced through the Les Houches interface
to the general purpose event generator HERWIG++, used for showering and hadronization.
The k-factor (where k = σNLO/σLO), for the considered invariant mass range and for fac-
torisation/renormalisation scales set to the default NLO scale µ0 =
√
k2T +Q
2 (where kT
and Q are the dilepton transverse momentum and invariant mass respectively), was found
to be k ≈ 1.3, in all studied cases. The plots have been normalised to unity (with the
exception of Fig. 5.10) to emphasise the differences in the shape of the distributions. For
5.2. NLO production of heavy charged vector bosons 121
Figure 5.3: Transverse momentum distribution at the Tevatron obtained for MC@NLO
with HERWIG++ in the DIS and MS factorisation schemes (PDFs: cteq5d
and cteq5m [122] respectively), POWHEG MS (cteq5m) and LO (PDF:
MRST2001LO [123]), in the mass range (70− 90) GeV.
validation purposes, Fig. 5.3 presents a comparison of theW boson transverse momentum
distribution, (assuming no W ′) between Tevatron data (taken from Ref. [124]) and the
three possible methods: leading order, MC@NLO and POWHEG. The plots include events in the
invariant mass range (70− 90) GeV. The MC@NLO and POWHEG distributions are in agree-
ment with the data within the statistical Monte Carlo and experimental uncertainties.
The leading-order pT distribution is cut off at the W mass since this provides the only
relevant scale in the parton shower, whereas the MC@NLO and POWHEG distributions extend
to higher transverse momentum. The different factorisation schemes given, DIS and MS,
arise due to the arbitrariness of the prescription in how the finite contributions to the
PDFs are treated when factorising the logarithmic singularities. The subsequent figures
in this section represent simulations made for the LHC running at 14 TeV proton-proton
centre-of-mass energy.
Figure 5.4 shows the variation of the NLO cross section for a 1 TeV left-handed
W ′ in the invariant mass range [400, 5000] GeV with factorisation scale, µF , for a fixed
renormalisation scale using the MS scheme. The LO variation with PDF scale is also
shown in an equivalent range. The values have been normalised to the cross sections
at the default scales µ0 =
√
k2T +Q
2 (default NLO) and µ0 = Q (default LO). In the
122 Chapter 5. New physics searches at hadron colliders
NLO case the renormalisation scale was held fixed at MW ′. The NLO cross section
calculation appears to be slightly more stable than the LO calculation. The k-factor at
µ0 was found to be k = 1.288 and the LO cross section at µ0 = Q was found to be
σLO = (2.99 ± 0.07) pb. Figures 5.5 and 5.6 show the transverse mass distributions (in
Figure 5.4: The normalised variation with scale of the cross section calculations at NLO (red
crosses) and LO (black diamonds) are shown for a proton-proton collider at 14
TeV, MW ′ = 1 TeV, ΓW ′ = 36 GeV and left-handed chirality in the invariant
mass range [400, 5000] GeV.
this case we show a ‘theoretical’ transverse mass, defined by MT =
√
M2 + P 2T ) at LO
and NLO for a W ′ at masses and widths of [1 TeV, 36 GeV] and [2 TeV, 72 GeV], for
purely left-handed (hW ′ = 1) and purely right-handed (hW ′ = −1) couplings to fermions
respectively. Figures 5.7 and 5.8 show the corresponding W/W ′ transverse momentum
distributions. In this case the LO distribution cuts off at the W ′ mass. The effect is less
visible for higherW ′ masses. Figure 5.9 shows a comparison between the different methods
of the W/W ′ rapidity, longitudinal momentum and mass distributions for a right-handed
W ′ of mass 2 TeV and width 72 GeV at the LHC.
Finally, Fig. 5.10 shows a comparison between the left- and right-handed W ′ ‘exper-
imental’ transverse mass distributions at NLO (defined by M2T = 2 /ETETℓ − 2~/pT · ~pTℓ,
5.2. NLO production of heavy charged vector bosons 123
using the missing transverse energy and momentum, and the lepton transverse energy
and momentum), using the MC@NLO method. The importance of the interference between
the SM W and the W ′ can be clearly observed: the differential cross section in the region
below MT = MW ′ in the purely left-handed (hW ′ = 1) case is reduced in comparison to
the purely right-handed (hW ′ = −1) case. For a transverse mass greater than the on-shell
mass of the W ′, the interference term becomes positive for the left-handed case, although
this effect is not significant. The SM contribution, in the absence of a W ′ boson, is given
for comparison in both figures. In the right-handed case the contribution of the W ′ is
simply additive to the SM contribution.
Figure 5.5: Transverse mass distributions at the LHC obtained using the MC@NLO and POWHEG
methods (cteq5m/MS) and LO (MRST2001LO) for a purely left-handed W ′.
The plots correspond to masses/widths equal to [1 TeV, 36 GeV] (left) and
[2 TeV, 72 GeV] (right). The invariant mass range was taken to be (0.4−3.0) TeV
for the 1 TeV case and (0.4 − 5.0) TeV for the 2 TeV case. The effect of the
destructive interference can be observed in both cases. Note that the plots have
been normalised to unity.
5.2.7 Extraction of limits
In appendix E.1 we provide a general method for discriminating between two models
given a set of events. Here we apply this method to extract observation limits on the W ′
mass and width at LO. A stand-alone program was written to calculate the quantity R
124 Chapter 5. New physics searches at hadron colliders
Figure 5.6: Transverse mass distributions at the LHC obtained using the MC@NLO and POWHEG
methods (cteq5m/MS) and LO (MRST2001LO) for a purely right-handed W ′.
The invariant mass range, the W ′ mass and widths are identical to those in the
previous figure.
Figure 5.7: Transverse momentum distributions at the LHC obtained using the MC@NLO and
POWHEG methods (cteq5m/MS) and LO (MRST2001LO) for a purely left-handed
W ′. The invariant mass range, the W ′ mass and widths are identical to those in
the previous figures.
at matrix element level, given by Eq. (E.5):
R
p(S)
p(T )
= exp
(
N∑
i=0
log
(
p(MT,i|T )
p(MT,i|S)
))
×
(
N¯T
N¯S
)N
e−(N¯T−N¯S) . (5.24)
5.2. NLO production of heavy charged vector bosons 125
Figure 5.8: Transverse momentum distributions at the LHC obtained using the MC@NLO and
POWHEG methods (cteq5m/MS) and LO (MRST2001LO) for a purely right-handed
W ′. The invariant mass range, the W ′ mass and widths are identical to those in
the previous figures.
The ‘true’ underlying theory, labelled T, was assumed to contain a W ′ at a predefined
mass and theory S was taken to be the SM. Some comments are appropriate:
• Although the total W ′ width was being varied, the decay width to fermions was
always assumed to be ΓW ′→ff ′ = (4ΓWMW ′/3MW ).
• In a real experiment the W ′ mass would be unknown and maximum likelihood
methods should be used to fit the parameters if significant deviation from the SM
is found.
• The R parameter can become very large if a small number of unlikely events occur,
which favour one theory over the other. Experimentally this is unrealistic since
‘unlikely’ events could be the effect of background or detector effects. To take these
into account, one has to introduce nuisance parameters whose behaviour, at this
level of analysis, have to be chosen arbitrarily. Here we avoid the introduction of
such arbitrary parameters.
• The detection curves were drawn for specific data distributions and fluctuations are
expected. In other words, the plots correspond to a specific ‘experimental’ data set.
• The ratio of the prior probability distributions, p(S)/p(T ), was taken to be equal to
unity throughout this analysis: i.e. we assume both models are equally likely prior
to the ‘experiment’.
126 Chapter 5. New physics searches at hadron colliders
Figure 5.9: W/W ′ rapidity (top left), longitudinal momentum (top right) and mass (bot-
tom) distributions at the LHC obtained using the MC@NLO and POWHEG methods
(cteq5m/MS) and LO (MRST2001LO) for a purely right-handedW ′ of mass 2 TeV
and width 72 GeV. The invariant mass range, theW ′ mass and widths are identical
to those in the previous figures.
• A rapidity cut on the leptons corresponding to ycut = 2.5 for the LHC and ycut = 1.3
for the Tevatron was applied to take into account the acceptance regions of the
detectors.
• The distributions p(MT |S) and p(MT |T ) were calculated using the Monte Carlo
event generator itself at higher statistics (∼ 105) than the required number of events
to reduce the required computer time. The sum over i in Eq. (E.5) was taken over
the bins of these distributions and not individual events.
5.2. NLO production of heavy charged vector bosons 127
Figure 5.10: Transverse mass distributions at the LHC obtained using the MC@NLO method
(cteq5m/MS), POWHEG (cteq5m) for purely left- and right-handed W ′’s. The
invariant mass range was taken to be (0.4 − 3.0) TeV. The plots correspond to
masses/widths equal to [1 TeV, 36 GeV] and [2 TeV, 72 GeV]. The significance
of the destructive interference can be observed in the left-handed case; in the
right-handed case the distribution is just the sum of the standard model W and
right-handed W ′ contributions. Note that the plots are normalised to the NLO
cross section for each process.
The limits were drawn on a width-mass plane as logR = C detection curves, where C
is a constant. The variable R can be interpreted as a probability ratio and a discovery
curve logR = C can be interpreted as the limit where the existence of a W ′ is discovered
with certainty 1 − e−C . For example if C = 10, then the detection curve represents
the ∼ 99.9996% confidence level. The LO detection curves can be seen, for different
integrated luminosities at the LHC (14 TeV), in Fig 5.11 for a right-handed W ′ and
Fig. 5.12 for a left-handed W ′. The curves correspond to a single data sample at each
[MW ′,ΓW ′] point, and therefore there are large statistical fluctuations, particularly in the
low-luminosity curves. A comparison between the curves for a left- and right-handed W ′
is shown in Fig. 5.13. It can be observed that a left-handed W ′ has a slightly higher
detection reach, especially at larger widths. By examining Figs. 5.11 and 5.12, we deduce
that the maximum detection reach at the LHC, for example using an integrated luminosity
128 Chapter 5. New physics searches at hadron colliders
of 100 fb−1, for a W ′ decaying primarily to fermions (ΓW ′ ≈ ΓW ′→ff ′), is ∼ 4 TeV. We
 2000  2500  3000  3500  4000  4500  5000
 50
 100
 150
 200
 250
 300
 350
 400
logR = 10, RH
1 fb-1
10 fb-1
100 fb-1
M (GeV)
 2000  2500  3000  3500  4000  4500  5000
 50
 100
 150
 200
 250
 300
 350
 400
logR = 100, RH
1 fb-1
10 fb-1
100 fb-1
M (GeV)
Γ (GeV)
Figure 5.11: The detection reach at the LHC for logR = 10 (left) and logR = 100 (right) at
different integrated luminosities for the right-handed case. The colour scheme is:
green, blue, black corresponding to the luminosities 1, 10, 100 fb−1.
 2000  2500  3000  3500  4000  4500  5000  5500
 50
 100
 150
 200
 250
 300
 350
 400
logR = 10, LH
1 fb-1 10 fb
-1
100 fb-1
M (GeV)
 2000  2500  3000  3500  4000  4500  5000  5500
 50
 100
 150
 200
 250
 300
 350
 400
logR = 100, LH
1 fb-1
10 fb-1
100 fb-1
M (GeV)
Γ (GeV)
Figure 5.12: The detection reach at the LHC for logR = 10 (left) and logR = 100 (right) at
different integrated luminosities for the left-handed case. The colour scheme is
identical to the previous figure. In the logR = 10 and 100 fb−1 case all points
below the contour have logR < 10.
also show the expected limit at the Tevatron (1.96 TeV) in Fig. 5.14 with an integrated
luminosity of 2 fb−1, both at leading and next-to-leading (see below) orders. When theW ′
is only allowed to decay to fermions, i.e. has width ΓW ′ ≈ 36 GeV, the predicted detection
limit for logR ∼ 10 is MW ′ ≈ 1.1 TeV. This is expected to be reduced by experimental
effects. Since the available centre-of-mass energy at the Tevatron is 1.96 TeV, we expect
the saturation of the detection reach to come at aboutMW ′ ∼ 1 TeV without interference
and slightly higher in the left-handed case when interference effects are included. The
5.2. NLO production of heavy charged vector bosons 129
 2000  2500  3000  3500  4000  4500  5000  5500
 50
 100
 150
 200
 250
 300
 350
 400
logR = 10
1 fb-1
10 fb-1
100 fb-1
M (GeV)
 2000  2500  3000  3500  4000  4500  5000  5500
 50
 100
 150
 200
 250
 300
 350
 400
logR = 100
1 fb-1
10 fb-1
100 fb-1
M (GeV)
Γ (GeV)
Figure 5.13: The detection reach at the LHC for logR = 10 (left) and logR = 100 (right) at
different integrated luminosities for the left- and right-handed cases. The colour
scheme is for 1, 10, 100 fb−1 is: left-handed: green, blue, black and right-handed:
orange, light blue, pink.
Tevatron NLO case does not exhibit any substantial difference from the LO case. We
 300  400  500  600  700  800  900  1000  1100  1200
 40
 60
 80
 100
 120
 140
 160
 180
logR = 10
M (GeV)
 300  400  500  600  700  800  900  1000  1100  1200
 40
 60
 80
 100
 120
 140
 160
 180
logR = 100
M (GeV)
Γ (GeV)
Figure 5.14: The detection reach at the Tevatron for logR = 10 (left) and logR = 100 (right)
at 2 fb−1, for the left- and right-handed cases, at LO and NLO. The colour scheme
is for right-handed and left-handed correspondingly, at LO: green, blue and NLO:
light blue, pink.
have performed an equivalent analysis using the NLO method POWHEG at matrix element
level to improve computational time. Working at matrix element level with the POWHEG
method is justified since the transverse mass distribution is not significantly altered after
shower and hadronization and no difficulties arise due to negatively-weighted events, as
would be the case in the MC@NLO case. The comments given at the beginning of the section
for the LO analysis also apply to the NLO analysis. The results are shown in Figs. 5.15
and 5.16 in comparison to the LO results. In the right-handed chirality scenario, NLO
130 Chapter 5. New physics searches at hadron colliders
implies a lower detection reach than indicated at LO. The situation is more complicated in
the left-handed case where the NLO case implies a slightly higher reach for larger widths.
To investigate the dependence of the NLO results on the factorisation scale µF we
 2000  2500  3000  3500  4000  4500  5000  5500
 50
 100
 150
 200
 250
 300
 350
 400
logR = 10
M (GeV)
 2000  2500  3000  3500  4000  4500  5000  5500
 50
 100
 150
 200
 250
 300
 350
 400
logR = 100
1 fb-1
10 fb-1
100 fb-1
M (GeV)
Γ(GeV)
Figure 5.15: The detection reach at the LHC for logR = 10 (left) and logR = 100 (right) at
different integrated luminosities for the right-handed case compared at LO and
NLO. The colour scheme for 1, 10, 100 fb−1 is: LO: green, blue, black and NLO:
orange, light blue, pink.
 2000  2500  3000  3500  4000  4500  5000  5500
 50
 100
 150
 200
 250
 300
 350
 400
 450
logR = 10
M (GeV)
 2000  2500  3000  3500  4000  4500  5000  5500
 50
 100
 150
 200
 250
 300
 350
 400
logR = 100
1 fb-1
10 fb-1
100 fb-1
M (GeV)
Γ(GeV)
Figure 5.16: The detection reach at the LHC for logR = 10 (left) and logR = 100 (right)
at different integrated luminosities for the left-handed case compared at LO and
NLO. The colour scheme is identical to the previous figure.
have reproduced the logR contours for the right-handed W ′ LHC case with an integrated
luminosity of 10 fb−1 at different values of µF while keeping the renormalisation scale
fixed, using the MS scheme. The results are shown in Fig. 5.17. The curves show that
the factorisation scale does not affect the detection reach substantially, for example only
shifting the logR = 10 contour at a width of ΓW ′ ∼ 200 GeV from MW ′ ∼ 3500 GeV to
5.2. NLO production of heavy charged vector bosons 131
MW ′ ∼ 3750 GeV going from µF = 0.5µ0 to µF = 4µ0.
 2000  2500  3000  3500  4000
 50
 100
 150
 200
 250
 300
 350
logR = 10
M (GeV)
 2000  2500  3000  3500  4000
 50
 100
 150
 200
 250
 300
 350
logR = 100
M (GeV)
Γ (GeV)
Figure 5.17: The NLO detection reach at the LHC for logR = 10 (left) and logR = 100 (right)
for an integrated luminosity of 10 fb−1 at different factorisation scale µF . The
colour scheme for µF = 0.5µ0, µ0, 2µ0 and 4µ0 is: green, blue, pink and light
blue.
5.2.8 Conclusions
We have presented a Monte Carlo implementation of the Drell-Yan production of new
charged heavy vector bosons. We have considered the interference effects with the Stan-
dard Model W boson, allowing arbitrary chiral couplings to the leptons and quarks.
Moreover, the implementation is correct up to next-to-leading order in QCD, via the
MC@NLO and POWHEG methods using the HERWIG++ event generator. We have presented
a sample of results at both leading and next-to-leading orders. As expected, the LO
and NLO boson transverse momentum distributions were found to differ significantly, the
NLO extending to higher pT . The dilepton transverse mass, invariant mass, rapidity and
z-momentum distributions were found not to be significantly altered by the NLO treat-
ment. The total cross section was found to increase in the NLO case by a factor of ∼ 1.3
in the region of interest.
Subsequently we applied a theoretical discrimination method to the W ′ reference
model to obtain mass-width observation curves for left- and right-handed chiralities of
the W ′ both at LO and NLO (POWHEG). The NLO curves were shown not to vary signifi-
cantly with factorisation scale. The event generator used throughout this analysis, Wpnlo,
is fully customisable and publicly available [107].
132 Chapter 5. New physics searches at hadron colliders
5.3 Searching for third-generation composite lepto-
quarks
5.3.1 Introduction
The riddle of electroweak symmetry breaking (EWSB), and specifically the hierarchy
problem we have described in section 2.3.1.1, can be addressed in natural way via the
introduction of new physics in the form of a new strong force. An example of new strong
dynamics is ‘technicolour’ - a theory in which EWSB can be explained without adding
any fundamental scalars and the Higgs boson arises as a composite particle of some new
fermion states. This solution is natural in a literal sense, meaning that we have already
observed an example present in Nature: the hierarchy between the Planck scale (1019 GeV)
and the proton mass (∼ 1 GeV) is a result of the logarithmic running of the QCD coupling
constant and the onset of the strong coupling in the infrared.
The Higgs mechanism must explain not only how the masses of the gauge bosons arise,
but also the origin of the fermion masses. We have already described the conventional
mechanism of fermion mass generation in the SM in section 2.2.2.2. This is done con-
ventionally via the so-called ‘Yukawa’ interaction. Before EWSB we have, e.g. for the
fermions of the third generation, couplings of the form:
LYuk.SM = λT T¯OHt+ h.c. , (5.25)
where λT is the top quark Yukawa coupling, T is the third generation fermion SU(2)L
doublet, OH is the fundamental scalar Higgs operator and t is the third generation fermion
SU(2)L singlet. In theories of strong dynamics EWSB it is usual to assume that masses
arise via a similar Yukawa-type interaction:
LYuk.strong = λTs
T¯OHt
Λd−1f
+ h.c. , (5.26)
where we have included a suppression scale Λf , the scale at which the flavour physics
arises, d = [Oh] is the mass dimension of the scalar Higgs operator, now taken to be d ≥ 1
and λTs is a ‘strong’ Yukawa coupling (which is dimensionless like λ
T ). However, in the
framework of an effective theory, other terms will necessarily need to be added to the full
5.3. Searching for third-generation composite leptoquarks 133
Lagrangian. One of these terms has the form:
L ⊃ q¯iqj q¯kql
Λ2f
. (5.27)
This four-fermion operator is potentially catastrophic since it can induce flavour-changing
neutral currents (FCNCs) that can contribute to meson mixing. For example, an operator
involving two strange quarks and two down quarks, of the form ∼ s¯dd¯s, would contribute
to K0 − K¯0 mixing. This type of FCNCs have been well-measured experimentally and
to satisfy the existing constraints, one needs to choose the suppression scale to be Λf &
103 GeV. In the SM we can set Λf as large as we wish and the FCNCs decouple from
the theory since the SM Yukawa interaction term of Eq. (5.25) is not suppressed by any
power of Λf . In theories of strong dynamics, d > 1 in general and hence we cannot set the
scale Λf to be arbitrarily high, otherwise there is a risk of decoupling the Yukawa term of
Eq. (5.26) as well, rendering the theory incapable of producing fermion masses naturally.
This is particularly true for the large top quark mass.
A solution to this issue, that was proposed long ago [125], introduces composite
fermions which arise due to the strong dynamics. The elementary fermions do not couple
directly to the scalar Higgs operator; instead they mix with the composite fermions via
bilinear interactions. Schematically, the Lagrangian terms relevant to the generation of
mass for the third generation is given by
Lmix ∝ mρ
[
yT
gρ
T¯OT + y
t
gρ
t¯Ot + O¯TOT + O¯tOt
]
+ gρO¯TOHOt + h.c. , (5.28)
wheremρ and gρ are the strong coupling mass scale and coupling respectively, y
T and yt are
the mixing parameters of the theory, corresponding to the left- and right-handed top quark
multiplets, OT and Ot are composite left- and right-handed fermions respectively, and OH
is the composite scalar Higgs operator. Note that there are two mixing parameters for each
fermion, one for each chirality. As an example, the fermions OT,t could be technibaryons,
composites of some new ‘technifermions’ in a technicolour theory, just as the protons are
composites of quarks in QCD. The Lagrangian Lmix produces an effective Yukawa term
for the elementary fermions. By examining Fig. 5.18:
LYuk.eff ∝ y
Tyt
gρ
T¯OHt ≡ λT T¯OHt+ h.c. , (5.29)
134 Chapter 5. New physics searches at hadron colliders
where we have used the mixing parameters to define the top Yukawa coupling:
λT ≡ y
Tyt
gρ
. (5.30)
The mechanism is not only capable of producing fermion masses, but also offers the hope
T OT Ot t
OH
Figure 5.18: A schematic diagram demonstrating how the effective T¯OHt vertex is formed
using the mixing terms and the Higgs boson coupling to the composite fermions
in the Lagrangian of Eq. (5.28).
that the observed hierarchies of masses and mixings of the SM fermions may be related to
the electroweak hierarchy via strong-coupling effects. This is because the third generation
can be considered to be the ‘most composite’ and hence should have masses closest to the
strong dynamics scale [125].
It is obvious now that the new strong sector must know about SU(3) colour and must
contain, at the very least, colour-triplet fermionic resonances that mix with the elementary
colour-triplets and make the observed fermions. It is reasonable to also expect the strong
sector to contain other coloured resonances. It may contain bosonic resonances that,
depending on their gauge charges, may be able to couple to a lepton and a quark, playing
the role of the leptoquark states. For example, in Ref. [125], in which the fermionic
resonances were the technibaryons of SU(3), the model would also contain technimesons
that would be able to act as leptoquarks.
It is easy to make an estimate of the magnitude of the leptoquark couplings to fermions
in such models. These arise in much the same way as the effective Higgs boson couplings,
through the following schematic Lagrangian:
LLQ ∝ mρ
[
yQ
gρ
Q¯OQ + y
L
gρ
L¯OL + O¯QOQ + O¯LOL
]
+ gρO¯QOLQOL + h.c. , (5.31)
where yQ is one of the quark mixing parameters, yL is one of the lepton mixing parameters
5.3. Searching for third-generation composite leptoquarks 135
and OLQ is a leptoquark operator with the correct gauge charges to couple to a composite
quark and a composite lepton. It is important to note that the mixing parameters that
appear in Eq. (5.31) are the same as those that appear in Lmix (Eq. (5.28)). The lepto-
quark effective coupling can then be calculated by considering the schematic diagram in
Fig. 5.19. We can deduce that the form of the coupling is
LLQff ∝ y
QyL
gρ
Q¯OLQL ≡ λLQQ¯OLQL+ h.c. , (5.32)
where we have defined the leptoquark coupling to quarks and leptons, λLQ ≡ yQyL/gρ.
We can make estimates of the magnitude of the coupling using the measured Yukawa
couplings if we restrict the mixing parameters to be equal for the quarks and leptons of
each generation:
yQ ∼ yq ∼ yr ,
yL ∼ yν ∼ yℓ , (5.33)
where Q ∈ {U,C, T} and L ∈ {L1, L2, L3} (left-handed doublets), q ∈ {u, c, t} (right-
handed up-type singlets), r ∈ {d, s, b} (right-handed down-type singlets), ν ∈ {νeR, νµR, ντR}
(hypothetical right-handed neutrinos) and ℓ ∈ {eR, µR, τR}. We can then use the measured
Yukawa couplings and estimate each of the mixing parameters that appear in Eq. (5.30)
for the quarks and leptons:
yQ ∼
√
λQgρ , y
L ∼
√
λLgρ . (5.34)
We can substitute these estimates into the leptoquark coupling to fermions defined in
Eq. (5.32), to obtain an estimate in terms of the measured Yukawa couplings:
λLQ ∼
√
λLλQ . (5.35)
From the above equation it is easy to see that couplings to quarks of the third generation
will dominate in this type of models. The SM fermion Yukawa couplings (taken from
Ref. [126]) are given in Table 5.1 and the resulting estimates of the leptoquark couplings
are given in Table 5.2. These estimates evade the constraints coming from flavour experi-
ments, for leptoquark masses even down to ∼ 200 GeV, which may arise if the leptoquarks
appear as pseudo-Nambu-Goldstone bosons [126].
136 Chapter 5. New physics searches at hadron colliders
Q OQ OL L
OLQ
Figure 5.19: A schematic diagram demonstrating how the effective Q¯OLQL vertex is formed
using the mixing terms and the leptoquark coupling to the composite quarks and
leptons in the Lagrangian of Eq. (5.31).
Fermion Yukawa, λF
e 2.87× 10−6
µ 6.09× 10−4
τ 1.02× 10−2
d 2.30× 10−5
s 3.39× 10−4
b 1.40× 10−2
u 2.87× 10−6
c 6.09× 10−4
t 1.02× 10−2
Table 5.1: The Standard Model fermion Yukawa couplings [126].
Lepton Quark 1 2 3
1 8.2× 10−6 1.0× 10−4 1.5× 10−3
2 1.2× 10−4 1.5× 10−3 2.2× 10−2
3 4.9× 10−4 6.0× 10−3 9.0× 10−2
Table 5.2: Estimates of the leptoquark couplings to fermions [126], using the measured Yukawa
couplings that appear in Table 5.1 and the assumption that there is only one mixing
parameter for the quarks and leptons of each generation.
Since the leptoquark couplings to light fermions are highly suppressed, the only rele-
vant couplings for direct collider production and detection are those to third-generation
fermions.7 As a result, the leptoquark states will decay exclusively to third-generation
fermions, that is to tτ or tντ or bτ or bντ . Na¨ıvely, since the leptoquark couplings scale
roughly with the Yukawa couplings, and since the bounds preclude a leptoquark mass
7For an alternative scenario with leptoquarks of this type, see [127].
5.3. Searching for third-generation composite leptoquarks 137
below mt,
8 one might conclude that decays involving the top must dominate. However,
we shall see later that the gauge quantum numbers sometimes preclude couplings to top
quarks and, of course, unknown global symmetries may also preclude one or more cou-
plings. Thus we consider all four possible couplings.
Since leptoquarks couple dominantly to third-generation quarks and leptons, pair-
production through colour gauge interactions will overwhelmingly dominate single-production
at the LHC. The channels of interest therefore involve pair-wise combinations of tτ or tντ
or bτ or bντ .
9 The 2b2τ and 2b+ /ET channels already have been the subject of searches at
the Tevatron [128,129], and can be adapted easily for the LHC. The use of novel kinematic
variables such asMT2 in this 2b+ /ET channel may well improve the prospects for discovery
and mass measurement. The two channels involving the top require more ingenuity, but
merit investigation.
In the present section we perform the first detailed phenomenological study of the pos-
sible production of such states at the LHC. In section 5.3.2 we briefly review their quantum
numbers, couplings and decay modes, which we have implemented in the HERWIG++ event
generator.10 This allows us to propose and investigate strategies for reconstructing third-
generation leptoquark masses from their decay products, including those that involve top
quarks, in the following sections. Our conclusions are presented in section 5.3.9.
Although we focus here on direct searches at the LHC, there are also promising chan-
nels for indirect searches, namely in Bd → Kµ¯µ and Bs → µµ at LHCb, in µ → eγ and
τ → µγ, in µ− e conversion in nuclei, and in τ → ηµ at future B factories [126].
5.3.2 Phenomenology
5.3.2.1 Scalar leptoquark pair-production
We focus on scalar leptoquarks in the present study since their bosonic couplings are
determined completely by QCD and hence their production cross sections only depend
on their masses. Moreover, the lightest (and most easily accessible) leptoquarks in these
scenarios arise as scalar pseudo-Nambu-Goldstone bosons. The type of leptoquarks we are
considering are predominantly pair-produced via gluon-gluon fusion or quark-anti-quark
8Searches at D0 for third-generation scalar leptoquarks decaying exclusively to bτ or bντ yield bounds
of 210 GeV [128] and 229 GeV [129] respectively.
9Note that third-generation lepton-quark couplings are also possible in R-parity-violating supersym-
metric theories.
10PYTHIA [45, 130,131] contains an implementation of a single scalar leptoquark of arbitrary flavour.
138 Chapter 5. New physics searches at hadron colliders
annihilation, due to the fact that they couple to the third-generation quarks and leptons.
Only charge-conjugate leptoquarks can be produced in this way: associated production of
different leptoquarks is forbidden since it would not conserve the Standard Model gauge
quantum numbers. Single-production in association with a lepton is allowed but at a
14 TeV LHC it becomes dominant at leptoquark masses of about 2.2 TeV, at which point
the total cross section, σ ∼ 10−2 fb, is already too low for discovery.
5.3.2.2 Effective Lagrangian for interactions with gluons
The effective Lagrangian describing the interaction of the scalar leptoquarks with gluons
is [132]
LgS =
(
DµijΦ
j
)†
(Dikµ Φk)−M2LQΦi†Φi , (5.36)
where Φ is a scalar leptoquark, i, j, k are colour indices, the field strength tensor of the
gluon field is given by
Gaµν = ∂µAaν − ∂νAaµ + gsfabcAµbAνc , (5.37)
and the covariant derivative is
Dijµ = ∂µδ
ij − igstijaAaµ . (5.38)
The Feynman rules that result from this Lagrangian and the diagrams that contribute to
pair-production of scalar leptoquarks are given in appendix F.1. Expressions for the cross
sections are given in appendix B.
5.3.2.3 Non-derivative fermion couplings
The effective Lagrangian that describes the possible non-derivative couplings of the scalar
leptoquarks to third-generation quarks and leptons is given by [133]
Lnd = (g0Lq¯cLiτ2ℓL + g0Rt¯cRτR)S0
+ g˜0Rb¯
c
RτRS˜0 + g1Lq¯
c
Liτ2τaℓLS
a
1
+ (h1Lt¯RℓL + h1Rq¯Liτ2τR)S1/2 + h2Lb¯RℓLS˜1/2 + h.c. , (5.39)
where the τa are the Pauli matrices, qL and ℓL are SU(2)L quark and lepton doublets
respectively and tR, bR and τR are the corresponding singlet fields. We denote charge-
5.3. Searching for third-generation composite leptoquarks 139
Name SU(3)c T
3 Y Qem Decay mode HERWIG++ id
S0 3¯ 0 1/3 1/3 τ¯Rt¯R, τ¯Lt¯L, ν¯τ,Lb¯L -9911561
S˜0 3¯ 0 4/3 4/3 τ¯Rb¯R -9921551
S
(+)
1 3¯ +1 1/3 4/3 τ¯Lb¯L -9931551
S
(0)
1 3¯ 0 1/3 1/3 τ¯Lt¯L, ν¯τ,Lb¯L -9931561
S
(−)
1 3¯ -1 1/3 -2/3 ν¯τ,Lt¯L -9931661
S
(+)
1/2 3 +1/2 7/6 5/3 tRτ¯L, tLτ¯R 9941561
S
(−)
1/2 3 -1/2 7/6 2/3 bLτ¯R, tRν¯τ,L 9941551
S˜
(+)
1/2 3 +1/2 1/6 2/3 bRτ¯L 9951551
S˜
(−)
1/2 3 -1/2 1/6 -1/3 bRν¯τ,L 9951651
Table 5.3: Numbering scheme, charges and possible decay modes for the non-derivatively cou-
pled scalar leptoquarks. Y represents the U(1)Y charge and T
3 is the third compo-
nent of the SU(2)L charge. Since S1 is an SU(2)L triplet, it contains three complex
scalars. The S1/2 and S˜1/2 are SU(2)L doublets. The naming convention is ex-
plained in the text. The minus sign in the ids of some of the leptoquarks indicates
the fact that they are anti-triplets of SU(3)c.
conjugate fields by f cR,L = (PR,Lf)
c, where the superscript c implies charge conjugation.
In Table 5.3 we give the quantum numbers for the five types of non-derivatively coupled
scalar leptoquarks: the SU(2)L-singlet complex scalars S0 and S˜0, the SU(2)L-triplet
complex scalar S1 and the SU(2)L-doublets S1/2 and S˜1/2.
The numbering scheme used in our implementation of scalar leptoquarks in HERWIG++
is also given in Table 5.3. The particles are numbered as 99NDDDJ , where N distin-
guishes the representation of the standard model gauge group, DDD is the lowest possible
number chosen to relate the leptoquark to the Particle Data Group (PDG) codes of de-
caying fermions, and J = 2S +1, where S is the particle spin. The sign of the PDG code
is negative for colour anti-triplets and positive for colour triplets. Hence, −9911561 is the
‘first’ type of leptoquark, S0, and can decay to particles with codes 15 (τ) and 6 (t).
Notice that the first three kinds of leptoquarks, the S0, S˜0 and the S1 triplet are colour
anti-triplets and the particles (as opposed to the anti-particles) decay into an anti-lepton
and an anti-quark. This is contrast to the S1/2 and S˜1/2 doublets, which are colour-triplets
and decay into quarks and anti-leptons.
140 Chapter 5. New physics searches at hadron colliders
Name SU(3)c T
3 Y Qem Decay mode HERWIG++ id
S ′0 3 0 2/3 2/3 tRν¯τ,L, bRτ¯L, bLτ¯R 9961551
S˜ ′0 3 0 5/3 5/3 tRτ¯L, tLτ¯R 9971561
S
′(+)
1 3 +1 2/3 5/3 tRτ¯L, tLτ¯R 9981561
S
′(0)
1 3 0 2/3 2/3 tRν¯τ,L, bLτ¯R, bRτ¯L 9981551
S
′(−)
1 3 -1 2/3 -1/3 bRν¯L 9981651
S
′(+)
1/2 3¯ +1/2 5/6 4/3 b¯Lτ¯L, b¯Rτ¯R -9991551
S
′(−)
1/2 3¯ -1/2 5/6 1/3 b¯Lν¯τ,L, t¯Rτ¯R, t¯Lτ¯L -9991561
S˜
′(+)
1/2 3¯ +1/2 -1/6 1/3 t¯Lτ¯L, t¯Rτ¯R -9901561
S˜
′(−)
1/2 3¯ -1/2 -1/6 -2/3 t¯Lν¯τ,L -9901661
Table 5.4: Numbering scheme, charges and possible decay modes for the derivatively-coupled
scalar leptoquarks. The details are as in Table 5.3.
5.3.2.4 Derivative fermion couplings
We also consider leptoquarks that couple derivatively to the quarks and leptons. The
couplings of the leptoquarks to fermions involve three fields, and hence two independent
positions for the derivative to act, modulo integration by parts. Here, we choose to place
the derivative on either the quark or the lepton, such that the Lagrangian is given by
Ld = −i√
2f
(g′0L,iq¯Lp
µ,iγµℓL + g
′
0R,ib¯Rp
µ,iγµτR)S
′
0
+
−i√
2f
g˜′0R,it¯Rp
µ,iγµτRS˜
′
0 +
−i√
2f
g′1L,iq¯Lp
µ,iγµτaℓLS
′a
1
+
−i√
2f
(h′1L,ib¯
c
Rp
µ,iγµℓL + h
′
1R,iq¯
c
Lp
µ,iγµτR)S
′
1/2 +
−i√
2f
h′2L,it¯
c
Rp
µ,iγµℓLS˜
′
1/2 + h.c. ,
(5.40)
where the index a ∈ {1, 2, 3} and pµ,i, i ∈ {l, q}, denotes the momentum of the lepton or
quark.
The charges of the primed scalar states appear in Table 5.4; they correspond, of course,
to those of vector leptoquarks. Notice that whereas the S0 is a colour anti-triplet, S
′
0 is a
colour triplet and so on.
5.3. Searching for third-generation composite leptoquarks 141
Consider a leptoquark S ′0 that couples derivatively to fermions in the following way:
L ∼ 1√
2f
(
g′0L,it¯L/p
iS ′0νL + g
′
0L,ib¯L/p
iS ′0τL + g
′
0R,ib¯R/p
iS ′0τR
)
+ h.c. , (5.41)
where the f is the sigma model scale for the strong dynamics. Consider the decay of the
S ′0 to on-shell fermions via the coupling g
′
0L,ib¯Lp
µ,iγµtL. We then have
g′0L,ib¯L/p
iτLS
′
0 = g
′
0L,q b¯L/p
qS ′0ℓL + g
′
0L,ℓb¯L/p
ℓτLS
′
0
= g′0L,qmbb¯RS
′
0ℓL + g
′
0L,ℓmτ b¯LS
′
0τR . (5.42)
Note that the chirality of one decay product is reversed in each term by the mass insertion,
which breaks the gauge symmetry. An equivalent manipulation is given in appendix F.2
for terms that contain conjugate fields. For simplicity, we choose to set the quark and
lepton primed couplings for each term equal, g′ℓ = g
′
q = g
′, where g′ represents g′0, g
′
1 or
h′1. As a result of the above manipulation, an effective Lagrangian for the on-shell decay
of a scalar leptoquark S ′0 may be written as
Leff. ∼ 1√
2f
(
g′0Lmtt¯RS
′
0ντ,L + g
′
0Lmbb¯RS
′
0τL + g
′
0Lmτ b¯LS
′
0τR
+ g′0Rb¯LmbS
′
0τR + g
′
0Rb¯RmτS
′
0τL
)
+ h.c. (5.43)
⇒ Leff. ∼
[
1√
2f
(g′0Lmb + g
′
0Rmτ )
]
b¯RS
′
0τL
+
[
1√
2f
(g′0Lmτ + g
′
0Rmb)
]
b¯LS
′
0τR
+
[
1√
2f
(g′0Lmt)
]
t¯RS
′
0ντ,L + h.c. , (5.44)
converting all the derivative couplings to ones that look like those for the unprimed lepto-
quarks, with the lepton or fermion masses appearing in the coupling. See appendix F.2 for
the full effective Lagrangian. Since the scale f is typically a few hundred GeV, couplings
proportional to the top quark mass are expected to dominate when the corresponding
decays are kinematically allowed. The on-shell fermion assumption is realistic since the
widths of the fermions are small in comparison to their masses and hence off-shell effects
are negligible.
142 Chapter 5. New physics searches at hadron colliders
Name λL(ℓq) λR(ℓq) λL(νq)
S0 g0L g0R −g0L
S˜0 0 g˜0R 0
S
(+)
1
√
2g1L 0 0
S
(0)
1 −g1L 0 −g1L
S
(−)
1 0 0
√
2g1L
S
(+)
1/2 h1L h1R 0
S
(−)
1/2 0 −h1R h1L
S˜
(+)
1/2 h2L 0 0
S˜
(−)
1/2 0 0 h2L
Table 5.5: The λi couplings of the non-derivatively-coupled scalar leptoquarks to the different
quark-lepton combinations, as they appear in the Lagrangian.
5.3.2.5 Decay widths
The decay width of non-derivatively coupled scalar leptoquarks in the limit of massless
quarks and leptons is given by [133]
Γ =
MLQ
16π
(
λ2L(ℓq) + λ
2
L(νq) + λ
2
R(νq)
)
, (5.45)
where the couplings λL,R(ℓq) for the types of leptoquarks we are considering are given in
Table 5.5 in terms of the couplings that appear in the Lagrangian. The couplings are taken
to be real. The expression gives, for quark-lepton couplings g ∼ 0.1 and leptoquark mass
of ∼ 400 GeV, a width of ∼ 0.1 GeV. The decay width to massive qℓ is further suppressed
by a phase-space factor compared to the massless quark and lepton width [129]:
F ∼ (1− rq − rℓ)
√
1 + (rq − rℓ)2 − 2rq − 2rl , (5.46)
where rq,ℓ are the squared ratios m
2
q,ℓ/M
2
LQ respectively.
Table 5.6 shows the couplings for the primed, derivatively-coupled, scalar leptoquarks.
The expression for the width given in Eq. (5.45) remains unchanged in the case of the
primed leptoquarks, with the couplings λi taking the appropriate values. Tables 5.7
and 5.8 show example decay widths and branching ratios for scalar leptoquarks of mass
MLQ = 400 GeV and couplings g = 0.1. In the case of derivatively coupled leptoquarks
we choose a suppression scale f = 800 GeV.
5.3. Searching for third-generation composite leptoquarks 143
Name λL(ℓq)×
√
2f λR(ℓq)×
√
2f λL(νq)×
√
2f
S ′0 g
′
0L,qmb + g
′
0R,ℓmτ g
′
0R,qmb + g
′
0L,ℓmτ g
′
0L,qmt
S˜ ′0 g˜
′
0R,ℓmτ g˜
′
0R,qmt 0
S
′(+)
1
√
2g′1L,qmt
√
2g′1L,ℓmτ 0
S
′(0)
1 −g′1L,qmb −g′1L,ℓmτ g′1L,qmt
S
′(−)
1 0 0
√
2g′1L,qmb
S
′(+)
1/2 h
′
1L,qmb + h
′
1R,ℓmτ h
′
1R,qmb + h
′
1L,ℓmτ 0
S
′(−)
1/2 h
′
1R,ℓmτ h
′
1R,qmt h
′
1L,qmb
S˜
′(+)
1/2 h
′
2L,ℓmτ h
′
2L,qmt 0
S˜
′(−)
1/2 0 0 h
′
2L,ℓmt
Table 5.6: The λi couplings of the derivatively-coupled (primed) scalar leptoquarks to the
different quark-lepton combinations, as they appear in the Lagrangian. In our
analysis, we have set the quark and lepton couplings equal for simplicity.
5.3.3 Reconstruction strategies
Table 5.9 provides an overview of our suggested reconstruction strategies for the different
types of leptoquarks. The ‘stransverse’ mass variable,MT2, which appears in the table has
been defined previously in Ref. [134], for the case of identical semi-invisible pair decays
as
MT2 ≡ min
/cT+/c
′
T= /pT
{max (MT ,M ′T )} , (5.47)
where the minimisation is taken over /cT and /c
′
T , the transverse momenta of the invisi-
ble particles, with the constraint that their sum equals /pT, the total missing transverse
momentum, and MT and M
′
T are the transverse masses calculated for the two decay
chains. We assume that the invisible particles are massless and use the jet masses in our
definitions of MT2. The new variables M
bal
min and Mmin will be defined in section 5.3.6.1.
We present our analysis of the mass reconstruction techniques for each pair-production
decay mode separately, initially at parton level and then at detector level, including
discussion of the relevant backgrounds. We focus on the S0 singlet, S1 triplet and S1/2
doublet and outline how to generalise the strategy to all the leptoquark multiplets.
It is evident from Tables 5.7 and 5.8 that the leptoquark decay widths are generally
much smaller than the resolution of the detector components, and hence our analysis is
not sensitive to the decay widths. Throughout the following we have set the leptoquark
144 Chapter 5. New physics searches at hadron colliders
Decay mode Decay width (GeV) BR
S¯0 → τ−t 0.1040 0.5666
S¯0 → ντb 0.07956 0.4334
¯˜S0 → τ−b 0.07956 1
S¯
(+)
1 → τ−b 0.1591 1
S¯
(0)
1 → τ−t 0.05225 0.3964
S¯
(0)
1 → ντ b 0.07956 0.6036
S¯
(−)
1 → ντ t 0.1045 1
S
(+)
1/2 → τ+t 0.1040 1
S
(−)
1/2 → τ+b 0.07956 0.6036
S
(−)
1/2 → ν¯τ t 0.05225 0.3964
S˜
(+)
1/2 → τ+b 0.07956 1
S˜
(−)
1/2 → ν¯τ b 0.07956 1
Table 5.7: Decay widths for non-derivatively-coupled scalar leptoquarks of mass MLQ =
400 GeV and couplings g = 0.1.
couplings to fermions to the value g = 0.1. This value is close to the estimate of the
leptoquark coupling to third-generation quarks and leptons (≈ 0.09) derived using the
measured fermion Yukawa couplings and the assumptions given towards the end of Sec-
tion 5.3.1 (see Tables 5.1 and 5.2). The resulting width-to-mass ratio for the leptoquarks
corresponding to this coupling, according to Eq. (5.45), is O(10−4).
We use the HERWIG++ event generator to generate a number of events corresponding
to an integrated luminosity of 10 fb−1 of the relevant signal and tt¯ background samples.
Subsequently we use the Delphes framework [135] to simulate the detector effects and
assess the feasibility of reconstruction in an experimental situation.11 Delphes includes
the most crucial experimental features: the geometry of the central detector, the effect
of the magnetic field on the tracks, reconstruction of photons, leptons, b-jets, τ -jets and
missing transverse energy. It contains simplifications such as idealised geometry, no cracks
and no dead material. We use the default parameter settings in the Delphes package that
correspond to the ATLAS detector. Crucial features of our analysis are both b- and τ -
tagging of jets and we caution the reader to take into consideration that the relevant
efficiencies will contain a degree of uncertainty at the early stages of the LHC experiment.
11Delphes is a framework for fast simulation of a general-purpose collider experiment.
5.3. Searching for third-generation composite leptoquarks 145
Decay mode Decay width (GeV) BR
S ′0 → τ−b 4.440× 10−6 0.0036
S ′0 → ντ t 0.001239 0.9964
S˜ ′0 → τ−t 0.001239 1
S
′(+)
1 → τ−t 0.002478 1
S
′(0)
1 → τ−b 1.292× 10−6 0.0010
S
′(0)
1 → ντ t 0.001239 0.9990
S
′(−)
1 → ντb 2.193× 10−6 1
S¯
′(+)
1/2 → τ−b 4.440× 10−6 1
S¯
′(−)
1/2 → τ−t 0.001239 0.9991
S¯
′(−)
1/2 → ντb 1.098× 10−6 0.0009
¯˜S
′(+)
1/2 → τ−t 0.001234 1
¯˜S
′(−)
1/2 → ντ t 0.001239 1
Table 5.8: Decay widths for derivatively-coupled (primed) scalar leptoquarks of mass MLQ =
400 GeV, couplings g′ = 0.1 and suppression scale f = 800 GeV.
The b-tagging present in the Delphes framework assumes an efficiency of 40% if the jet
has a parent b-quark, 10% if the jet has a parent c-quark and 1% if the jet is light (i.e.
originating from u, d, s or g). The identification of hadronic τ -jets is consistent with the
one applied in a full detector simulation. The resulting efficiencies for hadronic τ -jets are
in satisfactory agreement with those assumed by ATLAS and CMS. See [135] for further
details.
Throughout the analysis we apply transverse momentum cuts of at least 30 GeV. Since
we are always working with high-transverse momentum objects, we can assume that pile-
up arising due to secondary proton-proton collisions is under experimental control. See,
for example, the ATLAS tt¯H(→ bb¯) study in Ref. [77].
5.3.4 (tτ )(tτ ) decay mode
We examine the possibility of full reconstruction of the topology shown in Fig. 5.20, where
we have, for example, S0(S¯0)→ bjjj1ν1 and S0(S¯0)→ bℓν3j2ν2, where ν1 and ν2 represent
one or more neutrinos coming from the τ decays and ℓ can be either a muon or an electron.
We can assume that the neutrinos ν1,2 associated with the decays of the τs are collinear
with the direction of the jets j1,2 associated with them. The validity of this assumption
146 Chapter 5. New physics searches at hadron colliders
modes types technique
(tτ)(bν) S0, S
(0)
1 jτ ‖ ντ , mass constraints
⇒ edge reconstruction (Mbalmin, Mmin, MT2)
(tτ)(tτ) S0, S
(0)
1 , two jτ ‖ ντ , mass constraints
S
(+)
1/2 , S˜
′
0 ⇒ full reconstruction
(bν)(bν) S0, S
(0)
1 , MT2
S˜
(−)
1/2 , S
′(−)
1
(bτ)(bτ) S
(+)
1 , S˜
(−)
1/2 two jτ ‖ ντ , mass constraints
S˜0, S
′(+)
1/2 , ⇒ full reconstruction
S
′(0)
1
(tν)(tν) S
(−)
1 , S
(−)
1/2
S ′0, S
′(0)
1 , MT2
S˜
′(−)
1/2
(tν)(bτ) S
(−)
1/2 , S
′
0 jτ ‖ ντ , mass constraints
S
′(0)
1 ⇒ edge reconstruction (Mbalmin, Mmin, MT2)
Table 5.9: The table outlines the general reconstruction strategy for leptoquark pair-
production for the different types of leptoquarks. For variable definitions and further
details see the respective sections.
has been confirmed using HERWIG++, for leptoquarks of masses 1, 0.4 and 0.25 TeV, as
may be seen in Fig. 5.21, which shows the distribution of δR =
√
δη2 + δφ2 between
the momenta of the τ jet partons and the τ invisibles. The assumption is employed
in our reconstruction of any leptoquark decay mode containing a τ -jet. The top quark
branching ratios are ∼ 0.216 for the semi-leptonic e, µ modes and ∼ 0.676 for the hadronic
top modes. These appear twice since we have either the t or t¯, resulting in an overall
∼ 0.292 factor for the top decay modes. The branching ratios and cross sections for S0S¯0
production depend on the leptoquark mass and coupling and are shown in Table 5.10,
where the last column is the resulting cross section for the topology under study. We
focus on 400 GeV leptoquarks since these are clearly not excluded by direct searches at
present and still provide a sufficient number of events to be potentially discovered at a
reasonable luminosity (10 fb−1) at 14 TeV.
5.3. Searching for third-generation composite leptoquarks 147
j1
ν1 j
j
b
t
b
ℓ
ν3
j2
ν2
S¯0
S0
t¯
τ¯
τ
Figure 5.20: Pair-production of S0 leptoquarks with decay to (tτ)(tτ), followed by one
hadronic and one semi-leptonic top decay.
MS0 (GeV) σ(pp→ S0S¯0) (pb) BR(tτ) σ(tτ t¯τ¯ → bb¯jjℓ(= e, µ)ντ τ¯ ) (pb)
174.2 (= mtop) 141(1) 0. 0.
250 24.3(3) 0.34 0.729
400 2.000(7) 0.567 0.188
500 0.561(6) 0.606 0.06
1000 5.94(7) ×10−3 0.65 7.3× 10−4
Mtop (GeV) σ(pp→ tt¯) (pb) - σ(tt¯→ bb¯jjℓ(= e, µ)ν) (pb)
174.2 834(1) - 242
Table 5.10: S0S¯0 total cross section at the LHC at 14 TeV pp centre-of-mass energy, branching
ratio to tτ and remaining cross section taking into account the top branching ratios.
The corresponding tt¯ values are given for comparison.
5.3.4.1 Kinematic reconstruction
The final states of S0S¯0 → t¯τ+tτ− processes contain many decay products including
neutrinos. If the system has a large enough number of kinematical constraints, such as
mass-shell conditions and the balance of the total transverse momentum, we can com-
pletely reconstruct the kinematics of the system. The numbers of unknown variables and
constraints are summarised in Table 5.11 for each decay pattern of the tops: (1) both
tops decay hadronically, (2) one top decays semi-leptonically and another hadronically
148 Chapter 5. New physics searches at hadron colliders
Figure 5.21: The distribution of the distance in R-space (δR =
√
δη2 + δφ2) between the
momenta of the τ jet and the τ invisibles in S0 pair-production for MLQ =
1, 0.4, 0.25 TeV.
Decay type # of unknowns # of constraints
(1) had,had 1 + (0 + 2)N (2 + 2)N
(2) had,lep 1 + (4 + 2)N (5 + 2)N
(3) lep,lep 1 + (8 + 2)N (8 + 2)N
Table 5.11: The numbers of unknown variables ([mLQ], [ν from top], [energy fraction of tau])
and constraints ([mass-shell conditions], [balance of missing momentum]) in N
events of each decay type. The mass-shell conditions that constrain the unknown
variables are counted here, i.e. the mass-shell conditions on S0, leptonic top, W
and ν from leptonic top decay.
and (3) both tops decay semi-leptonically. As mentioned above, we assume τ -neutrinos
are collinear to the τ -jets, leaving two unknown parameters associated with the taus,
namely the energy ratios zi (i = 1, 2, zi ≥ 1) which are defined (neglecting masses) by
pτi = zipji ,
pνi = pτi − pji = (zi − 1)pji , (5.48)
5.3. Searching for third-generation composite leptoquarks 149
where pτi , pji and pνi are the four-momenta of the τ leptons, τ -jets and τ -neutrinos,
respectively. Under this assumption, the unknown variables in Table 5.11 are the mass
of the leptoquark, the 4-momenta of neutrinos from leptonic top decays and the energy
fractions associated with the neutrinos from the tau decays. The mass-shell conditions
that could constrain the unknown variables are counted in Table 5.11, i.e. the mass-shell
conditions on S0, leptonic top, W and ν from leptonic top decay.
It is only possible to wholly reconstruct the kinematics of a single event in decay types
(1) and (2). In decays of type (1), it would be difficult to reconstruct both hadronic tops
because of the large combinatorial background. Thus, we focus on decay type (2) and
attempt to determine the leptoquark mass. As we show in appendix F.3, in this case
one obtains a quartic equation for the energy ratio z2, and hence in general up to four
solutions for the leptoquark mass, at least one of which should be close to the true value
if the visible momenta and missing transverse momenta are well-measured.
5.3.4.2 Parton-level reconstruction
We first perform the (tτ)(tτ) analysis of the hard process (no initial- or final-state ra-
diation, no underlying event) at parton level without considering experimental or com-
binatoric effects, to examine its feasibility. For the majority of cases there are only two
physical, approximately degenerate, solutions, which are close to the true leptoquark mass.
The numerical solution of the quartic equation sometimes fails to yield real roots. The
results for true leptoquark massesMS0 = (0.25, 0.4, 1.0) TeV are shown in Fig. 5.22, which
includes histograms of the solutions obtained for 103 events. The histogram includes a
bin at 0 where the events without real solution are placed. These amount to about 10%
of the total events. At this level the reconstruction technique provides a good estimate
of the leptoquark mass for all the trial true masses, lying within a few GeV of the true
mass.
5.3.4.3 Experimental reconstruction
We consider an S0 leptoquark with mass MS0 = 400 GeV, for which the cross section
for production and decay into the topology of Fig. 5.20, S0S¯0 → tτ t¯τ¯ → bb¯jjℓντ τ¯ , is
σ = 0.187 pb. The most significant background in this scenario is tt¯ production, with
two extra jets misidentified as τs and subsequent decay of the tops into bb¯jjℓν. The cross
section for this process is 242.4 pb, overwhelming to begin with. There is also potentially
an irreducible Htt¯→ τ τ¯ tt¯ background which, for a Higgs boson of mass MH = 115 GeV,
150 Chapter 5. New physics searches at hadron colliders
Figure 5.22: Histograms of the solutions obtained at parton level for S0S¯0 and decay to a
semi-leptonic top, a hadronic top and two tau leptons, for MS0 = (0.25, 0.4, 1.0)
TeV (from left to right respectively). The first bin (0 GeV) contains the events
for which no real solution has been found.
has a cross section of approximately 65 fb. Since one of the main rejection mechanisms
is the reconstruction through the solution of the kinematic equations, we do not expect
this background to contribute significantly in the signal region.
We simulate the events with QCD initial-state radiation (ISR), final-state radiation
(FSR) and underlying event (UE). We use the default jet algorithm provided by the
Delphes package for the ATLAS configuration, the anti-kt with the parameter set to
R = 0.7. We then demand a set of relatively loose cuts on the full tt¯ and S0S¯0 samples,
since in a real experiment we would not be able to separate the different decay modes of
the top quark or S0 leptoquark. The cuts applied are the following:
• The existence of a lepton in the event, being either a muon or electron, with pT,ℓ >
30 GeV.
• A minimum of 6 jets.
• The missing transverse momentum in the event, /ET > 20 GeV.
• Two τ -tagged jets, with the extra requirement that they both have pT,τ > 30 GeV.
• No jets tagged as both b- and τ -jets simultaneously.
We also require that the highest-pT lepton is at a distance δR > 0.1 from the τ -tagged
jets, since electrons may create a candidate in the jet collection as well as the lepton
5.3. Searching for third-generation composite leptoquarks 151
collection. The analysis then breaks up into different branches according to the number
of b-tagged jets in an event:
• Two b-tagged jets: we look for one or two further jets (with pT > 30 GeV) that form
an invariant mass close to the top mass, within 20 GeV. One b-jet is then associated
with the semi-leptonic top decay and the other with the hadronic top decay.
• One or no b-tagged jets: when there is one b-tagged jet, we check whether it will
satisfy the top mass conditions with any other (one or two) remaining jets, otherwise
we associate it with the semi-leptonic top. If so, we look for any two or three jets
that satisfy the top mass conditions, and form the hadronic top within a 20 GeV
mass window. For the remaining b-jets (or if there are no b-jets) we look for the
remaining highest-pT jets. Any jets that are found in this way and called b-jets are
required to have a pT,b > 30 GeV.
No solutions are found in the sample of 70 signal events passing the cuts, if we require the
ratios z2 to be purely real. Hence, the solutions to the quartic equation for the momentum
ratios z2, described in appendix F.3, are now allowed to be complex in order to provide
some signal. This is valid since even true leptoquark events are smeared and distorted
by detector and QCD effects. We use the real part of z2 as an input to the calculation of
the rest of the kinematic variables. This is reasonable since the experimental effects are
expected to ‘smear’ the position of the true value of z2 in all directions in the complex
plane. The effect is shown in Fig. 5.23, where we plot the real and imaginary parts of z2
for the events that have passed the kinematic cuts. Evidently, there is a concentration
of solutions around the positive real axis, an effect exemplified by Fig. 5.24, where we
show the ratio of the real part of z2 and its modulus. We have further demanded that the
resulting momentum fractions are physical: R(z1,2) > 1, resulting in only real solutions for
MS0 . Figure 5.25 shows a reconstruction plot for leptoquarks of mass MS0 = 400 GeV.
Note that each event was given weight 1, distributed evenly amongst the solutions it
yields. In the case of complex z2, we assume there is one solution corresponding to the
complex conjugate pair.
Although the cuts applied are relatively weak, most of the background rejection comes
from the requirement of two τ -tagged jets. The background does not produce solutions
in the physical region often enough to be significant.
152 Chapter 5. New physics searches at hadron colliders
-20
-15
-10
-5
 0
 5
 10
 15
 20
-10 -5  0  5  10
Im
(z)
Re(z)
Figure 5.23: The plot shows the complex values of the solutions for z2 after solving the quartic
equation for the events that have passed the experimental cuts. There exists a
higher concentration of events about the positive real axis. The number of entries
is 280 (4 solutions included for each of the 70 events).
Figure 5.24: The plot shows the ratio of the real part of z2 and its modulus. The peak close
to 1 demonstrates the clustering of the real positive solutions about the real axis
and justifies the use of the real part as an input to the rest of the calculation.
The number of entries is 280 (4 solutions included for each of the 70 events).
5.3. Searching for third-generation composite leptoquarks 153
Figure 5.25: Experimental reconstruction of the S0S¯0 → tτ t¯τ¯ → bb¯jjℓ(= e, µ)ντ τ¯ mode using
the method described in the text. Note that each event has weight 1, distributed
evenly amongst the solutions it yields. The signal is shown in red (35 entries)
and the tt¯ background in blue dashes (3 entries).
5.3.5 (qν)(qν) decay modes
We can obtain the mass of the leptoquarks when both of them decay into bν or tν using
the MT2 variable (Eq. (5.3.3)). Examples of these decay mode are S0S¯0 → b¯ν¯bν and
S¯
(−)
1/2S
(−)
1/2 → t¯νtν¯.
5.3.5.1 Parton-level reconstruction
At parton level, the (tν)(tν) and (bν)(bν) decay modes are similar and hence we consider
only the latter here. We first construct the MT2 variable using the parton-level b-quark
4-momenta, in the absence of any experimental effects, ISR, FSR or UE. The result is
shown in Fig. 5.26 for MLQ = (0.25, 0.4, 1) TeV, confirming the expected sharp edge in
these idealised conditions.
5.3.5.2 Experimental reconstruction
As before, we use the Delphes framework to simulate the detector effects, with the settings
stated in section 5.3.3. We demand two b-tagged jets in both the q = b and q = t cases.
154 Chapter 5. New physics searches at hadron colliders
Figure 5.26: The parton-level MT2 distribution constructed for the S0S¯0 → b¯ν¯bν using the
true b-quark momenta, for MLQ = 1, 0.4, 0.25 TeV.
In the latter we search for combinations of 1 or 2 jets with the b-tagged jets which form
the top mass within a window of 10 GeV. We require the following cuts for the (bν)(bν)
case, on the full S0S¯0 sample:
• Two b-tagged jets with pT,b > 120 GeV each.
• No electrons or muons in the event.
• Missing transverse energy /ET > 250 GeV.
For the (tν)(tν) case we require the following cuts on the S¯
(−)
1/2S
(−)
1/2 sample:
• Two b-tagged jets with pT,b > 80 GeV each.
• No electrons or muons in the event.
• Missing transverse energy /ET > 260 GeV.
The resulting MT2 distributions for the signal (blue) and tt¯ background (red) can be seen
in Fig. 5.27. The (tν)(tν) mode appears to be more challenging to reconstruct than the
5.3. Searching for third-generation composite leptoquarks 155
Figure 5.27: Experimental reconstruction of the S0S¯0 → bντ b¯ν¯τ mode (left, 121 background
events, 125 signal events) and S¯−1/2S
−
1/2 → t¯νtν¯ mode (right, 39 background
events, 48 signal events) using MT2. ISR, FSR and the underlying event have
been included in the simulation. The signal is given in red and the tt¯ background
in blue dashes.
(bν)(bν) mode. This is due to the fact that the tt¯ background is very similar to the signal
and the difficulties that are present in reconstructing hadronic tops. Nevertheless, as the
results show, it may be possible to observe an excess over the MT2 distribution of the
background and provide an estimate of the mass.
5.3.6 (q′τ )(qν) decay modes
One possible event topology for the S0S0 → b¯ν¯tτ− processes is shown in Fig. 5.28. When
the top decays hadronically the system has two neutrinos, one from an S0 decay and
another from a τ decay. If we can reconstruct the hadronic top correctly, we can simply use
MT2 to obtain the mass of the leptoquarks. Similar topology is present in the S¯
(−)
1/2S
(−)
1/2 →
(bτ¯ )(t¯ν) decay mode.
It is known that the information from MT2 is the same as that from the ‘minimal
kinetic constraints’, in events where two identical particles decay to missing particles
with the same mass [136–138]. As discussed in section 5.3.4, in this type of event, we can
take advantage of the fact that, to a good approximation, the neutrino from a τ decay is
travelling almost collinearly to the τ -jet in the lab frame. By including this constraint,
we can define kinematical variables, Mmin and M
bal
min, which perform better than MT2 at
parton level, as we will show in the following subsections.
156 Chapter 5. New physics searches at hadron colliders
t
S¯0
S0
τ
ν¯
b¯
j
ν j
j
b
Figure 5.28: Pair production of S0 leptoquarks with decay to (tτ)(bν), followed by hadronic
top decay.
5.3.6.1 Kinematic reconstruction
In the τ collinearity approximation, neglecting masses, we can write
pντ = wpj , w > 0 . (5.49)
The second neutrino comes directly from the S0 decay associated with a b-jet. The
transverse components of the momentum of this neutrino are constrained by
pν = pmiss − wpj . (5.50)
There are two unknown parameters left, w and pzν . In terms of these, we define two
invariant mass variables:
m2tτ (w) = (pt + (1 + w)pj)
2 = m2t + 2(1 + w)pt · pj (5.51)
and
m2bν(w, p
z
ν) = (pb + pν)
2
= 2Eb
√
(pmiss − wpj)2 + (pzν)2 − 2pb · (pmiss − wpj)− 2pzbpzν .
(5.52)
5.3. Searching for third-generation composite leptoquarks 157
Figure 5.29: Parton-level distributions of Mbalmin (left), Mmin (centre) and MT2 (right) for
(bν)(tτ) (solid curve) and (tν)(bτ) (dashed curve).
Note that mtτ does not depend on p
z
ν and is a monotonically increasing function of w
because pt · pj > 0. We can now define two MT2-like variables:
Mmin = min[max{mtτ , mbν}] ≥MT2 , (5.53)
and
Mbalmin = min
mtτ=mbν
[mbν ] , (5.54)
where minimisation is taken for all possible (w, pzν). By construction, both these quantities
have an upper bound equal to the leptoquark mass:
MS0 ≥Mmin, MS0 ≥Mbalmin . (5.55)
Furthermore, we show in appendix F.4 that
Mbalmin ≥Mmin . (5.56)
5.3.6.2 Parton-level reconstruction
Figure 5.29 shows the parton-level distributions of Mbalmin, Mmin and MT2 for 1000 events.
We took only the true combination of the jet assignment. As can be seen, all the distri-
butions have clear edge structures at the input leptoquark mass of 400GeV.
In order to compare these variables we took their differences, shown in Fig. 5.30. The
158 Chapter 5. New physics searches at hadron colliders
Figure 5.30: Parton-level distributions of Mbalmin − Mmin (left) and Mmin − MT2 (right) for
(bν)(tτ) (solid curve) and (tν)(bτ) (dashed curve)
relationMbalmin ≥Mmin ≥MT2 is seen to hold on an event-by-event basis. This implies that
Mbalmin and Mmin are more powerful than MT2 for determining the mass of the leptoquark,
at least at parton level.
5.3.6.3 Experimental reconstruction
The settings for experimental reconstruction used for the Delphes fast simulation remain
unaltered in this analysis (see section 5.3.3). We apply the following event selection cuts
to the full S0S¯0 signal and the tt¯ background:
• At least four jets found in each event.
• Exactly one τ -tagged jet with pT > 120 GeV.
• No, one or two b-tagged jets with pT > 60 GeV.
• Missing transverse energy, /ET > 200 GeV.
For the b-jet originating from the leptoquark decay, we choose the highest-pT b-tagged jet
when there are two b-tagged jets and the highest-pT jet (excluding the τ -tagged jet) when
there are no b-tagged jets. We use all the remaining jets with pT > 30 GeV, (not identified
as the b-jet from the leptoquark) to search for one, two or three jets that form an invariant
mass close to the top mass, within a 20 GeV window. We apply the additional constraint
that the difference between the pT of the τ -tagged jet and the pT of the b-tagged jet,
5.3. Searching for third-generation composite leptoquarks 159
Figure 5.31: Reconstructed distributions Mbalmin (left), Mmin (centre) and MT2 (right) for
(bν)(tτ) signal (red) and tt¯ background events (blue dashes) including ISR, FSR
and the underlying event. There are 37 signal events and 19 background events
in all plots.
pT,τ − pT,b > −10 GeV. This eliminates a high fraction of the tt¯ background since the τs
in that sample originate from the W decay and are expected to have lower pT on average
than the bs that originate directly from the top. On the contrary, in the leptoquark signal
the τ and b transverse momenta are expected to be of the same magnitude on average.
The resulting distributions are shown in Fig. 5.31. Due to the low number of events
passing the selection cuts, it is not obvious whether the Mbalmin observable performs better
than Mmin and MT2. However, we checked that the three distributions satisfy the same
inequalities presented in Fig. 5.30 for the parton-level reconstruction.
The reconstruction strategy for the (tν)(bτ) mode follows the technique described in
section 5.3.6.1 for the (bν)(tτ) case, with the simple replacement b↔ t. The assignment
of b-jets and top-jets is performed in the same way as in the (bν)(tτ) analysis, with the
following cuts applied to the full S¯
(−)
1/2S
(−)
1/2 sample:
• At least four jets found in each event.
• Exactly one τ -tagged jet with pT > 190 GeV.
• No, one or two b-tagged jets with pT > 40 GeV.
• Missing transverse energy, /ET > 120 GeV.
There is also a cut on the reconstructed hadronic top jet, of pT > 120 GeV and that its
invariant mass lies within 20 GeV of the top mass. The results are shown in Fig. 5.32.
160 Chapter 5. New physics searches at hadron colliders
Figure 5.32: Reconstructed distributions Mbalmin (left), Mmin (centre) and MT2 (right) for the
(tν)(bτ) signal (red) and tt¯ background events (blue dashes) including ISR, FSR
and the underlying event. There are 68, 72 and 72 signal events (left to right)
and 22, 23, 23 background events (left to right).
Note that the background that would be present due to the S
(+)
1/2 leptoquark has not been
included.
Although at parton level, the variable Mbalmin performs better than Mmin and MT2, it
seems to become unstable after including experimental effects, with some events failing
to produce a value within the range of the plots shown in Fig. 5.32. The origin of the
instability is the additional assumption of the leptoquark masses being equal, which is
satisfied at parton level (up to small width effects) but does not hold exactly after detector
simulation. For the events for which no solution is found, we assign Mbalmin = Mmin. Even
after this readjustment, there are a few events for which a solution for Mbalmin is found
and lies outside the region shown. Therefore, MT2 and Mmin appear to be preferable as
experimental observables.
5.3.7 (bτ)(bτ) decay mode
5.3.7.1 Kinematic reconstruction
The (bτ)(bτ) mode can be fully reconstructed if one again assumes collinearity of the
τ -jets and τ -neutrinos: pτ,i = zipj,i (i = 1, 2, zi > 1). This implies that the missing
momentum from each τ can be written as /pi = (zi − 1)pj,i. Hence, we may write the
5.3. Searching for third-generation composite leptoquarks 161
following equalities for the components of the measured missing transverse momentum:
pxmiss = p
x
j1(z1 − 1) + pxj2(z2 − 1) ,
pymiss = p
y
j1(z1 − 1) + pyj2(z2 − 1) . (5.57)
The above equations may be written in matrix form and inverted to give
z1 = 1 +
pyj2p
x
miss − pxj2pymiss
pxj1p
y
j2 − pyj2pxj2
,
z2 = 1−
pyj1p
x
miss − pxj1pymiss
pxj1p
y
j2 − pyj2pxj2
. (5.58)
Now the invariant mass of each of the two leptoquarks may be written as m2S = (pb+pτ)
2,
resulting in the following expression:
m2S = 2zipbi · pji , (5.59)
where we have neglected the τ and b-quark mass terms. Using Eqs. (5.58), we obtain two
values of mS per event. At parton level, with the correct jet assignments, these solutions
approximate the leptoquark mass very closely, up to the collinearity approximation.
5.3.7.2 Experimental reconstruction
The Delphes framework has been used with identical settings as in the previous sections.
The following cuts have been applied to the S
(+)
1 S¯
(+)
1 → (b¯τ¯)(bτ) mode:
• At least 4 jets present in the event.
• Two τ -tagged jets with pT > 140 GeV.
• Missing transverse energy /ET > 140 GeV.
We accept events with no, one or two b-tagged jets. If there are less than two b-jets,
we search for the highest-pT non-tagged jet(s) to obtain two b-jets. We apply a cut
of pT > 50 GeV on these. There are two possible assignments of the bτ combination,
resulting in a total of four solutions. The resulting distribution for the mass solutions, as
described in section 5.3.7.1, is show in Fig. 5.33. The tt¯ background appears to be under
control, with 1.75 entries in the mass histogram.
162 Chapter 5. New physics searches at hadron colliders
Figure 5.33: Experimental reconstruction of the (bτ)(bτ) mode using the method described in
the main text. ISR, FSR and the underlying event have been included in the
simulation. Note that each solution has weight 0.25. The signal is shown in red
(18.75 entries) and the tt¯ background in blue dashes (1.75 entries).
We also considered the bb¯jj background, for which we generated events using Alpgen
v2.13 [139], applying the pT cuts on the four parton-level objects. We concluded that we
can safely ignore this background since the expected number of events with two τ -tagged
jets was O(1), before applying any restrictions on the missing transverse energy, /ET . Note
that the backgrounds to this decay channel originating from the other members of the
leptoquark multiplet have not been included.
5.3.8 Determination of quantum numbers
In the ideal scenario where all of the decay modes of a leptoquark multiplet are seen, the
quantum numbers can be deduced without ambiguity. For example if we only observe
combinations of (tτ) and (bν) decay modes, then the only likely candidate is an S0 sin-
glet. However, if in conjunction with these decay modes we observe (bτ) and (tν) decay
modes, with corresponding total rates, then we might guess that we have observed the S1
multiplet.
The issue is more complicated if some decay modes are missed. For example if only
5.3. Searching for third-generation composite leptoquarks 163
the (tτ)(tτ) decay mode has been seen, we might assume that we have observed the pair
production of an S˜ ′0 leptoquark. However, we might have observed the (tτ)(tτ) decay
of an S˜
′(+)
1/2 leptoquark pair and missed the more challenging (tν)(tν) mode of the S˜
′(−)
1/2
leptoquarks. In this case we would need to examine the helicities and charges of the
decay products: the S˜
′(+)
1/2 decays to t¯Lτ¯L and t¯Rτ¯R whereas the S˜
′
0 decays to tRτ¯L and
tLτ¯R. Since we can reconstruct all decay products of the top and τ without combinatorial
ambiguity, using the measured leptoquark mass as an input, there is hope that we could
measure top [140–142] and τ [143,144] polarisations simultaneously. This would allow us
to distinguish these two cases. We leave investigation of the feasibility of this to future
work.
5.3.9 Conclusions
If strongly-coupled dynamics solves the hierarchy problem of electroweak symmetry break-
ing, the question arises of how best to discover it at the LHC. Existing constraints coming
from electroweak precision tests tell us that, at least at low energies, any model of strong
dynamics must be a lot like the Standard Model (with perhaps a faint hope of observable
deviations in the Higgs sector [145,146]). Existing constraints coming from flavour physics
are somewhat different in that, while the data are certainly consistent with the Standard
Model, naturalness arguments suggest that strongly-coupled theories should differ from
the Standard Model in the flavour sector. Indeed, fermion masses should arise via mixing
between elementary and composite fermions of the strongly-coupled sector.
If that is so, then composite leptoquarks (or diquarks) may also appear, coupled pre-
dominantly to third-generation fermions. These would provide a spectacular signature at
the LHC. Their Standard Model quantum numbers imply that they would be produced
strongly as conjugate particle-anti-particle pairs, decaying into third-generation quarks
and leptons in the combinations summarised in Table 5.9. We have proposed a number
of new experimental search strategies adapted to these characteristic final states, also
summarised in Table 5.9, and implemented the relevant processes in the HERWIG++ event
generator version 2.5.0 [53] in order to study their effectiveness in the presence of QCD
radiation, backgrounds and the underlying event. We used the Delphes detector simu-
lation to assess the effects of b- and τ -tagging efficiencies and detector resolution. For
definiteness we assumed a leptoquark mass of 400 GeV and an integrated pp luminosity
of 10 fb−1 at 14 TeV.
In the case of decays of leptoquark pairs to (qτ)(qτ) where q = t or b, the approximate
164 Chapter 5. New physics searches at hadron colliders
collinearity of the missing neutrinos and jets from the tau decays allows full reconstruction
of the leptoquark mass, even when one top decay is semi-leptonic. In the former case
there is a quartic ambiguity in the resulting mass, although not all of the solutions are
real. After detector resolution smearing, the correct solutions for the momentum fraction
z2 may be complex, but we found that using the real parts provides a fair estimate of
the mass, with resolution of the order of ±150 GeV. For (bτ)(bτ) the only ambiguity is
combinatoric but the mass resolution is similar. In both cases the expected background
from qq¯jj is small after cuts and reconstruction.
For decays to (tτ)(bν) or (tν)(bτ), we have proposed an edge reconstruction strategy
similar to those developed for supersymmetric models, but using mass variablesMbalmin and
Mmin that are in principle superior to the classic ‘stransverse mass’ MT2. However, given
the limited statistics expected, the difference in performance between these variables was
not obvious. We found cuts to reduce the background from tt¯ to manageable levels, but
the edge reconstruction remains challenging without higher statistics. For (qν)(qν) the
story is similar for edge reconstruction in MT2, the case of q = t being the more difficult
owing to the similarity of the distributions of the signal and tt¯ background. But even in
that case a clear excess over background should be visible and would give a rough estimate
of the leptoquark mass.
In the event that a discovery is made, one might ask to what extent this provides proof
that electroweak symmetry breaking is driven by strongly-coupled, composite dynamics.
After all, one can easily imagine weakly-coupled theories with such states, for example,
third-generation squarks in R-parity-violating supersymmetric models. Ultimately, TeV-
scale compositeness can only be revealed by experiments probing significantly higher
scales; for that, we shall have to wait some time. In the meantime, the discovery of
leptoquarks coupled to third-generation fermions and their de facto consistency with the
multitude of existing flavour experiments would imply very strong bounds on the couplings
to other fermions. The scenario in which the observed fermions are partially elementary
and partially composite provides, as far as we know, the only mechanism in which the
required suppression can be automatically achieved. Moreover, it gives a prediction for
the size of the other couplings, some of which are not far from current bounds, which
may then be targeted in ongoing flavour experiments. Though circumstantial, this would
seem to be the best possible evidence for compositeness that one might hope for in the
LHC era.
Chapter 6
Conclusions and Outlook
At the time of writing of this thesis, the framework of the Standard Model (SM) is a well-
established set of gauge theories, described by the product group SU(3)c×SU(2)L×U(1)Y
and complemented by the Higgs mechanism, responsible for the breaking of the SU(2)L×
U(1)Y down to U(1)em. It is well-established in the sense that it currently provides
excellent agreement with experimental data. However, as we have discussed in chapter 2,
there remain serious open questions that cast doubt on the status of the SM as a ‘final’
theory of Nature. These include the stability of the scalar Higgs mass against radiative
corrections, known as the hierarchy problem, the ‘near-miss’ of the unification of the
gauge couplings in the SM, and questions originating from astrophysical and cosmological
measurements such as the existence of non-luminous gravitating matter (dark matter)
and a mysterious form of energy that causes the Universe’s expansion rate to accelerate
(dark energy). There is also a strong feeling that the absence of a quantum description of
gravitational interactions in the same framework as the other forces is an indication that
we are far from a complete theoretical description of Nature.
Several solutions, of varying degrees of ambition, have been proposed to address the
problems that plague SM. These range from extensions of space-time symmetries, by
adding extra dimensions or supersymmetries to the Poincare´ symmetry, models with
larger gauge groups with intricate symmetry breaking patterns resulting in interesting
effective theories (e.g. Little Higgs models) or the addition of new strong forces (e.g.
technicolour). High energy particle colliders allow us to explore the high energy realm,
and determine which of the proposed theories, if any, is related to Nature. In these exper-
iments, collisions of particles are performed in a controlled environment and deductions
are made by examining the products of the scattering reactions. The latest and greatest
165
166 Chapter 6. Conclusions and Outlook
experiment is the Large Hadron Collider (LHC), at CERN, near Geneva, Switzerland.
The LHC is a machine that collides protons head-on, with a design nominal energy of
14 TeV.
Theoretical predictions are necessary if we wish to squeeze out every drop of physics
from the LHC experiment. The Monte Carlo method that we described in chapter 3,
provides powerful tools that enable us to make phenomenological predictions, incorpo-
rating perturbative quantum chromodynamics (QCD) and other models inspired by it.
Monte Carlo methods are conceptually easy to associate with experiments. They provide
simulations of particle collisions, starting from the parton-level theoretical predictions,
e.g. derived from a set of Feynman rules, to a full simulation of the effects of interactions
of the resulting particles with the components of the detector. Such simulations also in-
clude perturbative treatment of initial- and final-state QCD showers, as well as models of
secondary partonic interactions, which form the underlying event, a phenomenon under
intense theoretical and experimental investigation. The matrix elements and the show-
ers can nowadays be provided at next-to-leading order (NLO) accuracy, providing more
reliable predictions and better agreement with experiment via the POWHEG and MC@NLO
methods we discussed in section 3.3.
The exploration of physics at high energy hadron colliders is a non-trivial task. We
considered, in the introduction of chapter 4, the complications that we have to face, both
due to the complexity of the possible new physics signals and the difficulties that arise due
to the fact that the colliding particles are hadrons. These are illustrated schematically
in Figs. 4.1 and 4.2. In chapter 4 we constructed analytical predictions of the effects of
QCD phenomena on certain hadron collider variables: on a class of variables called ‘global
inclusive variables’, which make use of all observed momenta, and on the transverse energy
of initial-state radiation in Higgs and Drell-Yan gauge boson production. We compared
the analytical predictions to results obtained from Monte Carlo event generators. In the
future, the calculations of QCD effects on global inclusive variables could be extended
to make use NLO cross sections and splitting functions. Furthermore, the validity of the
calculation of the distribution of transverse energy associated with Higgs production is
uncertain, even after matching to NLO. This could be investigated further with matching
to the full NNLO result.
To be able to cope with the complexity of new physics signals and set the best possible
bounds on model parameters, or improve the potential of discovering new physics, we have
to be adequately prepared. This involves constructing robust general search strategies and
167
predictions. In chapter 5 we first improved the treatment of Drell-Yan production of heavy
charged vector bosons, called W ′ (W prime) by using the POWHEG and MC@NLO methods to
generate fully exclusive events at NLO. We also considered the effects of interference of a
potential W ′ with the SM W . Interference may provide extra information to the nature
of the new particle or improve the detection reach. We also investigated an interesting
model of scalar leptoquarks which couple to third-generation quarks and leptons, inspired
by a theory of strong dynamics electroweak symmetry breaking, although not limited to
it. The signals are challenging at the LHC due to their complexity and potentially huge
backgrounds, but we provided a complete strategy for reconstruction of all the possible
decays in pair-production of charge-conjugate leptoquark states. Future extensions to
this work may involve using NLO matrix elements for the production of the leptoquarks
or investigating the existence of new diquark states that might exist in the same models.
Strategies may be developed for determining the quantum numbers of the leptoquarks,
using methods that measure the helicity of the fermions that originate from their decays,
as we pointed out at the end of section 5.3.
We are at the beginning of an exciting time for particle physics, and science in general.
The quest for understanding the underlying structure of Nature that started thousands of
years ago will surely enter a new chapter during the Large Hadron Collider era. Our duty
is to be prepared, guided by our intuition and the analytical and computational tools that
we have created, so that we will be able to comprehend the new results that will be faced
with. The methods and ideas studied in this thesis could contribute in that direction.
168 Chapter 6. Conclusions and Outlook
Appendix A
Illustration of a Monte Carlo event
In this appendix we illustrate, with the help of schematic diagrams, the set of steps per-
formed by a generic Monte Carlo event generator when producing a full event simulation.
Figures A.1 to A.5 demonstrate the various steps. In each step, the newly added features
are shown in red colour.
1. Hard process generation, Figure A.1: The hard process is generated by choos-
ing a point on the phase space according to the ‘hit-or-miss’ method.
2. Heavy resonance decay, Figure A.2: Heavy resonances with narrow widths are
decayed before the parton shower. In this example the heavy resonance could be a
top quark, decaying to a ℓνℓ and a b-quark.
3. Parton showers, Figure A.3: The incoming partons are showered by evolving
backwards to the incoming hadrons, producing initial-state radiation. Any final-
state particles that are colour-charged also radiate, producing final-state radiation.
4. Multiple parton interactions, Figure A.4: Secondary interactions between
partons within the colliding hadrons, modelled as QCD 2 → 2 interactions, are
generated. The secondary partons are showered and are always evolved backwards
to gluons in the underlying event model present in HERWIG++.
5. Hadronization and hadron decays, Figure A.5: In the cluster model, clusters
are formed and hadrons are produced. Unstable hadrons are subsequently decayed.
169
170 Chapter A. Illustration of a Monte Carlo event
Figure A.1: STEP 1: Generation of the hard process.
Figure A.2: STEP 2: Decay of heavy resonances.
h h
Figure A.3: STEP 3: Parton showers.
171
h h
Figure A.4: STEP 4: Multiple parton interactions.
h h
Figure A.5: STEP 4: Hadronization and hadron decays.
172 Chapter A. Illustration of a Monte Carlo event
Appendix B
Pair-production cross sections
The leading order parton-level cross section for QCD pair-production of particles of mass
mp may be written in terms of scaling functions fij as
σˆij(Q
2) =
α2S(Q
2)
m2p
fij . (B.1)
For heavy quark pair-production, the functions for gluon-gluon and quark-anti-quark
initial states are given by [10]
fgg =
πβρ
192
{
1
β
(ρ2 + 16ρ+ 16) log
∣∣∣∣1 + β1− β
∣∣∣∣− 28− 31ρ} , (B.2)
fqq¯ =
πβρ
27
(2 + ρ) . (B.3)
where ρ = 4m2q/Q
2 and β =
√
1− ρ.
For the case of gluino pair-production, the equivalent functions fij are given by [147]
fgg =
πm2g˜
Q2
{[
9
4
+
9m2g˜
Q2
− 9m
4
g˜
Q4
]
log
∣∣∣∣1 + β1− β
∣∣∣∣− 3β − 51βm2g˜4Q2
}
, (B.4)
fqq¯ =
πm2g˜
Q2
{
β
[
20
27
+
16m2g˜
9Q2
− 8m
2
−
3Q2
+
32m4−
27(m4− +m2q˜Q2)
]
(B.5)
+
[
64m2q˜
27Q2
+
8m4−
3Q4
− 16m
2
g˜m
2
−
27Q2(Q2 − 2m2−)
]
log
(
1− β − 2m2−/Q2
1 + β − 2m2−/Q2
)}
,
where now β =
√
1− 4m2g˜/Q2 and m2− represents the mass-squared difference between
the gluino and the t-channel squark, m2− = m
2
g˜ −m2q˜ .
173
174 Chapter B. Pair-production cross sections
For the case of scalar leptoquark pair-production, the scaling functions are given by
fgg =
πM2LQ
96sˆ
{
β(41− 31β2)− (17− 18β2 + β4) log
∣∣∣∣1 + β1− β
∣∣∣∣} ,
fqq¯ =
2πM2LQ
27sˆ
β3 , (B.6)
where β =
√
1− 4M2LQ/Q2. The differential cross sections with respect to the leptoquark
scattering angle in the partonic centre-of-mass frame, θ, are given by
dσˆgg
SS¯
d cos θ
=
πα2s
6sˆ
β
{
1
32
[25 + 9β2 cos2 θ − 18β2]
− 1
16
(25− 34β2 + 9β4)
1− β2 cos2 θ +
(1− β2)2
(1− β2 cos2 θ)2
}
,
dσˆqq¯
SS¯
d cos θ
=
πα2s
18sˆ
β3 sin2 θ . (B.7)
Appendix C
The Cabibbo-Kobayashi-Maskawa
matrix
The Cabibbo-Kobayashi-Maskawa (CKM) matrix (Eq. (2.27)) is a 3 × 3 unitary matrix
that can be parametrized by three mixing angles θij (i, j ∈ {1, 2, 3}, i 6= j) and a charge-
parity (CP) violating phase, δ. A common choice is
V =
 c12c13 s12c13 s13e
−iδ
−s12c23 − c12s23s13eiδ c12c23 − s12s23s13eiδ s23c13
s12s23 − c12c23s13eiδ −c12s23 − s12c23s13eiδ c23c13

(C.1)
where we have used the shorthand notation cij = cos θij and sij = sin θij . The angles
θij have been chosen to lie in the first quadrant. Experimentally it has been observed
that s13 ≪ s23 ≪ s12 ≪ 1, so it is convenient to demonstrate the hierarchy using the
Wolfenstein parametrisation:
s12 = λ =
|Vus|√|Vud|2 + |Vus|2 , s23 = Aλ2 = λ
∣∣∣∣VcbVus
∣∣∣∣ ,
s13e
iδ = V ∗ub = Aλ
3(ρ+ iη) =
Aλ3(ρ¯+ iη¯)
√
1− A2λ4√
1− λ2 [1−A
2λ(ρ¯+ iη¯)] . (C.2)
175
176 Chapter C. The Cabibbo-Kobayashi-Maskawa matrix
We then have
V =
 1− λ
2/2 λ Aλ3(ρ− iη)
−λ 1− λ2/2 Aλ2
Aλ3(1− ρ− iη) −Aλ2 1
+O(λ4) . (C.3)
The fit for the Wolfenstein parameters defined above gives
λ = 0.2253± 0.0007 , A = 0.808+0.022−0.015 ,
ρ¯ = 0.132+0.022−0.014 , η¯ = 0.341± 0.013 . (C.4)
We can use this to estimate the CP violating phase, δ to be ≈ 68.8◦.
The Particle Data Group world average values, including errors, for the absolute values
of the matrix elements of the CKM matrix are [13]:
V =
 0.97425± 0.00022 0.2252± 0.0009 (3.89± 0.44)× 10
−3
0.230± 0.011 1.023± 0.036 (40.6± 1.3)× 10−3
(8.4± 0.6)× 10−3 (38.7± 2.1)× 10−3 0.88± 0.07
 . (C.5)
Appendix D
Supplementary material for ET
resummation
D.1 Relation of ET resummation to qT resummation
Here we demonstrate the equivalence of transverse energy and transverse momentum
resummation at O(αs). Expanding Eq. (4.83) to this order, using (4.87) and substituting
into (4.81) and (4.82), we find terms involving the integrals
Ip(Q,ET ) = 1
2π
∫ +∞
−∞
dτ e−iτET lnp
(
Qτ
iτ0
)
, (D.1)
with p = 1, 2. At this order, evaluating the PDFs at the scale iτ0/τ leads to single-
logarithmic terms of the same form when we use (4.89) to write
fa/h(x, iτ0/τ)) = fa/h(x,Q)− αs
π
ln
(
Qτ
iτ0
)∑
b
∫ 1
x
dz
z
Pab(z)fb/h(x/z,Q) . (D.2)
The integral (D.1) may be evaluated from
Ip(Q,ET ) = d
p
dup
I(Q,ET ; u)|u=0 , (D.3)
where
I(Q,ET ; u) = 1
2π
∫ +∞
−∞
dτ e−iτET
(
Qτ
iτ0
)u
. (D.4)
177
178 Chapter D. Supplementary material for ET resummation
Writing τ = iz/ET , we have
I(Q,ET ; u) = − i
2πET
(
Q
ET τ0
)u ∫ +i∞
−i∞
dz zu ez . (D.5)
We can safely deform the integration contour around the branch cut along the negative
real axis to obtain
I(Q,ET ; u) = − 1
πET
(
Q
ET τ0
)u
sin(πu) Γ(1 + u) , (D.6)
which, recalling that ln τ0 = −γE = Γ′(1), gives
I1(Q,ET ) = − 1
ET
, I2(Q,ET ) = − 2
ET
ln
(
Q
ET
)
. (D.7)
The resummed component of the transverse momentum (qT ) distribution takes the
form:[
dσF
dQ2 dqT
]
res.
= qT
∑
a,b
∫ 1
0
dx1
∫ 1
0
dx2
∫ ∞
0
db b J0(bqT ) fa/h1(x1, b0/b) fb/h2(x2, b0/b)
× WFab(x1x2s;Q, b) , (D.8)
where b0 = 2 exp(−γE),
W
F
ab(s;Q, b) =
∑
c
∫ 1
0
dz1
∫ 1
0
dz2 Cca(αs(b0/b), z1) Cc¯b(αs(b0/b), z2) δ(Q
2 − z1z2s)
× σFcc¯(Q,αs(Q)) Sc(Q, b) , (D.9)
and
Sc(Q, b) = exp
{
−2
∫ Q
b0/b
dq
q
[
2Ac(αs(q)) ln
Q
q
+Bc(αs(q))
]}
. (D.10)
Expanding to O(αs), we find the same terms as in the ET resummation except that (D.1)
is replaced by
Ip(Q, qT ) = qT
∫ ∞
0
db b J0(bqT ) ln
p(Qb/b0) . (D.11)
It therefore suffices to show that
Ip(Q, qT ) = Ip(Q,ET = qT ) , p = 1, 2 . (D.12)
D.2. Results for LHC at 7 TeV 179
Now corresponding to (D.4) we have
I(Q, qT ; u) = qT
∫ ∞
0
db b J0(bqT )
(
Qb
b0
)u
. (D.13)
Using the result: ∫ ∞
0
dt tµ−1 J0(t) =
2µ
2π
sin
(πµ
2
)
Γ2
(µ
2
)
, (D.14)
gives
I(Q, qT ; u) = − 2
πqT
(
2Q
qT b0
)u
sin
(πu
2
)
Γ2
(
1 +
u
2
)
, (D.15)
and hence
I1(Q, qT ) = − 1
qT
, I2(Q, qT ) = − 2
qT
ln
(
Q
qT
)
, (D.16)
in agreement with (D.7) and (D.12). Notice, however, that the higher (p > 2) derivatives
of I and I differ, corresponding to the difference between ET and qT resummation beyond
O(αs).
D.2 Results for LHC at 7 TeV
Figure D.1: Predicted ET distributions in Z
0 production in pp collisions at
√
s = 7 TeV .
We show here results for the LHC operating at a centre-of-mass energy of 7 TeV,
corresponding to those shown earlier for 14 TeV. Apart from the normalisation, the pre-
dictions for the two energies are very similar, with only a slight downward shift in the
180 Chapter D. Supplementary material for ET resummation
Figure D.2: Predicted ET distributions in W
± production in pp collisions at
√
s = 7 TeV .
Figure D.3: Predicted ET distributions in Higgs boson production in pp collisions at
√
s = 7
TeV .
position of the peak in the ET distribution at the lower energy.
Appendix E
Supplementary material for W ′
study
E.1 Model discrimination
The search for new physics often involves the task of discriminating between two models:
one with new physics, the other without. The actual task of discovering new physics
though is laborious: one has to understand the detector well enough and has to be able
to obtain sufficient statistics to say with certainty that something new has been observed.
Here we adopt a rather theoretical approach: we describe a purely statistical method for
discriminating between models [148, 149]. This will essentially yield an upper bound on
the detection reach of a heavy particle: detector effects and backgrounds will result in a
reduced detection limit. It is useful, however, to be aware of the theoretical possibilities
for discovery.
E.1.1 Likelihood ratios of probability density functions
Consider N data points, of a mass variable measurement, {mi}. Based on these data
points, a theoretical model T is R times more likely than another theoretical model S, if
R =
p(T |{mi})
p(S|{mi}) , (E.1)
181
182 Chapter E. Supplementary material for W ′ study
where p(X|{mi}) is the probability of model X being true given the data set {mi}. We
may use Bayes’ Theorem to rewrite R as
R =
p({mi}|T )p(T )
p({mi}|S)p(S) , (E.2)
where p(T ) and p(S) are the probabilities that S and T are true respectively, called prior
probabilities since they represent any previous knowledge we may possess on the theories.
In the study performed here, we assume that these quantities are equal: there is no strong
evidence for either model. We may simplify Eq. (E.2) further:
R
p(S)
p(T )
=
ΠNi=0p(mi|T )
ΠNj=0p(mj|S)
= ΠNi=0
p(mi|T )
p(mi|S)
⇒ Rp(S)
p(T )
= exp
N∑
i=0
log
(
p(mi|T )
p(mi|S)
)
, (E.3)
where we have assumed that events in the data set {mi} are independent. Eq. (E.3) is
a discrete version of the Kullback-Leibler distance [150], a useful quantity for comparing
the relative likelihood of two theories according to a given data sample. It is important to
note that the distributions p(mi|T ) and p(mi|S) are normalised to unity. This means that
any difference in the number of events predicted by the two theories will not be taken into
account. This will obviously underestimate the significance of a difference in the number
of events as predicted by the two models, for example a substantial excess of events in an
invariant mass peak that may be present. We describe a method which takes this factor
into account in the next section.
E.1.2 Poisson likelihood ratios
In this modification to the method described in the previous section, we simply multiply
the variable R defined in Eq. (E.1) by a ratio of Poisson distributions for the total number
of events:
R =
p(T |{mi})
p(S|{mi})
(
N¯T
N¯S
)N
e−(N¯T−N¯S) , (E.4)
where N¯X = σX .L is the expectation value of the number of events according to theory
X, given by the product of the cross section, σX , and the integrated luminosity, L. This
E.1. Model discrimination 183
expression can be manipulated in a similar manner to Eq. (E.3) to obtain
R
p(S)
p(T )
= exp
(
N∑
i=0
log
(
p(mi|T )
p(mi|S)
))
×
(
N¯T
N¯S
)N
e−(N¯T−N¯S) , (E.5)
For convenience we may define the ‘shape’ and ‘Poisson’ factors respectively:
RS = exp
N∑
i=0
log
(
p(mi|T )
p(mi|S)
)
,
RP =
(
N¯T
N¯S
)N
e−(N¯T−N¯S) , (E.6)
This method takes into account the difference in the total number of events expected
according to each theory at the given integrated luminosity. This is accomplished by
reweighing the ‘shape’ factor RS by a factor RP which gives the ratio of probabilities to
obtain the observed number of events.
E.1.3 Application to a toy model
 0
 1
 2
 3
 4
 5
 1  2  3  4  5  6  7  8  9  10
dσ
dm
m
Figure E.1: The differential cross sections dσdm according to two ‘toy’ theories T and S are
shown. Theory T possesses a Gaussian ‘bump’, at m = 5 whereas S is just a
falling distribution, 1/m. m is in arbitrary mass units and σ in equivalent inverse
area squared units.
184 Chapter E. Supplementary material for W ′ study
Before applying the method to the full W ′ model, it is instructive to present its
application to a simple model involving two analytical ‘toy’ distributions. Events for
the two distributions have been generated by the general Monte Carlo event generation
method. The ‘differential cross sections’ for the two ‘theories’ T and S with respect to a
variable m in arbitrary units, defined in the range m ∈ [0.1, 10], are given by
dσT
dm
=
1
m
+ 0.3e−(m−5)
2
,
dσS
dm
=
1
m
. (E.7)
Theory T has a Gaussian peak at m = 5 on top of a background falling as ∼ 1/m and
theory S falls as ∼ 1/m. The situation is shown in Fig. E.1. This is qualitatively similar to
the SM tail (theory S) and the SM plus a heavy particle (theory T). The ‘cross sections’ in
the rangem = [0.1, 10] are σT = 5.14 and σS = 4.60, in arbitrary area units. Assuming an
integrated ‘luminosity’ of L = 30 (in equivalent arbitrary inverse area units), we have an
expected number of events N¯T = 154 and N¯S = 138. Initially, we assume that theory T is
the correct underlying theory: we produce events that are actually distributed according
to it. The result for the variable R if theory T was ‘true’ was then found to be R = 62.
This implies that theory T is 62 times more likely than theory S given this specific data
set. If, however, the underlying theory is chosen to be S, then we get R = 0.23. This
implies that in the case that theory T is ‘true’, it is easier to exclude theory S than to
exclude theory T in the case that theory S is ‘true’. In other words, it is easier to make
a discovery of a new resonance if it is there than to exclude it if it’s not.
E.2 The W ′ Drell-Yan cross section
In the present section we give details of the derivation of the leading order Drell-Yan
cross section for pp(p¯) → W/W ′ → ℓνX, given in Eq. (5.2). We reproduce the W ′ and
W couplings to fermions given in Eq. (5.1) (with kW = kW ′ = 1):
LWiff ′ =
(
GFM
2
W√
2
)1/2
Vff ′C
ℓ,q
i f¯γµ(1− hiγ5)f ′W µi + h.c. , (E.8)
E.2. The W ′ Drell-Yan cross section 185
Wi
q(p, s)
q¯′(p′, s′)
f(k, r)
f ′(k′, r′)
Figure E.2: Feynman diagram for qq¯′ → Wi → f f¯ ′. The quantities in the parentheses repre-
sent the 4-momentum and spin of the particle respectively.
The propagator for a massive vector boson i is given by
W µνi (q) =
−i
q2 −M2i + iMiΓi
[
gµν − q
µqν
q2 − ζM2i
(1− ζ)
]
, (E.9)
where ζ is the gauge fixing parameter. Two possible gauges are ζ = 1, the Feynman gauge,
and ζ = 0, the Landau gauge. Any observable quantity calculated should be independent
of the gauge fixing parameter. Here we derive the differential cross section using arbitrary
ζ to show this fact explicitly. The invariant matrix element for the parton-level process
qq¯′ →Wi → f f¯ ′ (Fig. E.2) is given by
Mi = GFM
2
W√
2
Vff ′Vqq′C
ℓ
iC
q
i
[
f¯γµ(1− hiγ5)f ′
]
W µνi [q¯
′γν(1− hiγ5)q] , (E.10)
As stated in section 5.2.2, Vff ′ is the unit matrix when ff
′ are leptons, ℓℓ′, so we set
Vℓℓ′ = 1 for the allowed lepton combinations. Vff ′ is a CKM matrix element when ff
′ =
qq′ (see appendix C for the CKM matrix element values). We now form:
MiM∗j = Ωqq′
[
f¯γµ(1− hjγ5)f ′
]
W µνi [q¯
′γν(1− hjγ5)q]
× [f¯ ′γλ(1− hiγ5)f]W λκ∗j [q¯γκ(1− hiγ5)q′] , (E.11)
where we have defined the constant Ωqq′ =
G2FM
4
W
2
|Vqq′|2 (CiCj)ℓ(CiCj)q. We take the sum
over the fermion spins (s, s′, r, r′) and use the identity
∑
s f
s(k)f¯ s(k) = /k±m, where f s(k)
are spinors representing particles (or antiparticles) of mass m, spin s and 4-momentum
k. We have ∑
spins
[
f¯ ′γλ(1− hiγ5)f
] [
f¯γµ(1− hjγ5)f ′
]
186 Chapter E. Supplementary material for W ′ study
=
∑
spins
f¯ ′aγ
ab
λ (1− hiγ5)bcfcf¯dγdeµ (1− hjγ5)eff ′f
= /k′faγabλ (1− hiγ5)bc/kcdγdeµ (1− hjγ5)ef
= Tr
[
/k′γλ(1− hiγ5)/kγµ(1− hjγ5)
]
, (E.12)
and similarly ∑
spins
[q¯γκ(1− hjγ5)q′] [q¯′γν(1− hjγ5)q]
= Tr
[
/pγκ(1− hiγ5)/p′γν(1− hjγ5)
]
. (E.13)
In the above, we have neglected all fermion masses. Putting everything together we obtain∑
spins
MiM∗j = Ωqq′Tr
[
/k′γλ(1− hiγ5)/kγµ(1− hjγ5)
]
W λκ∗j
× Tr [/pγκ(1− hiγ5)/p′γν(1− hjγ5)]W µνi . (E.14)
We can simplify Eq. (E.14) by performing the traces and contracting, using the mathe-
matical package FORM [151]. We obtain
∑
spins
MiM∗j = Ωqq′
(sˆ−M2i )(sˆ−M2j ) +MiMjΓiΓj
[(sˆ−M2i )2 +M2i Γ2i ] [i→ j]
× [8(1 + hihj)2[(tˆ+ sˆ)2 + tˆ2] + 8sˆ(hi + hj)2(2tˆ+ sˆ)] , (E.15)
where sˆ = (p + p′)2 is the square of the quark centre-of-mass energy and tˆ = (p − k)2.
It is reassuring that the gauge-fixing parameter ζ does not appear in Eq. (E.15), as it
should not have any physical significance. We now consider the kinematics, a schematic
diagram of which is shown in Fig. E.3, where pµ = (|~p| , |~p| , 0, 0) for the u-type quark,
kµ = (|~k|, |~k| cos θ, |~k| sin θ, 0) for the outgoing neutrino, ν. The angle θ is defined between
the u-type quark and the neutrino. We have sˆ = (p+p′)2 = 2p·p′ and tˆ = (p−k)2 = −2p·k.
Since we have neglected fermion masses |~p| = |~k| = √sˆ/2 and hence tˆ = −2|~k| |~p| (1 −
cos θ) = − sˆ
2
(1− cos θ). Finally, we obtain
tˆ2 + (tˆ+ sˆ)2 =
sˆ2
2
(1 + cos2 θ) ,
sˆ(2tˆ+ sˆ) = sˆ2 cos θ . (E.16)
E.2. The W ′ Drell-Yan cross section 187
With these relations at hand and by using z = cos θ, we can rewrite
∑
spinsMiM∗j as
∑
spins
MiM∗j =
∑
qq′
Ωqq′
3
(sˆ−M2i )(sˆ−M2j ) +MiMjΓiΓj
[(sˆ−M2i )2 +M2i Γ2i ] [i→ j]
× sˆ2 [(1 + hihj)2(1 + z2) + 2(hi + hj)2z] . (E.17)
We have averaged over initial spins by multiplying by 1/4 and divided by 3 to account
for the fact that the fusing quark and anti-quark must form a colour singlet. We have
also summed over all possible quark flavour combinations, qq′. Using Eq. (E.17) we may
now write the total matrix element squared as |M|2 = |MW |2+M∗WMW ′ +M∗W ′MW +
|MW ′|2. It is not difficult to see that we may group the z-even and z-odd terms into
the functions S(sˆ) and A(sˆ), defined in Eqs. (5.3) and (5.4) respectively. If the collision
u(p, E1)
ν(k, E3)
ℓ¯(k′, E4)
d¯(p′, E2)θ
Figure E.3: Diagram showing the kinematics for the specific case of ud¯ → W+i → ℓ¯ν in the
centre-of-mass frame. The angle θ is defined to be the scattering angle in the
centre-of-mass between the ν and the u, both being fermions.
had involved only quarks of constant centre-of-mass energy then we would simply plug
the matrix element squared at parton level into the expression for the 2 → 2 scattering
differential cross section:
dσ
dzdφ
=
1
64π2sˆ
|M|2 . (E.18)
Integrating over φ would give
dσ
dz
=
1
32πsˆ
|M|2 . (E.19)
In a collision which involves a quark q and an anti-quark q¯′, each can come from either
of the two hadrons. Let us identify the two hadrons to ‘left’ (hadron A) when moving in
the positive z-direction and ‘right’ (hadron B) when moving in the negative z-direction.
Since we do not have any information about which quark came from which hadron, we
188 Chapter E. Supplementary material for W ′ study
have to include both possibilities in the calculation. If the quark q comes from hadron A,
and the anti-quark q¯′ comes from hadron B then the definition of the angle θ, and hence z,
remains unchanged. Contrariwise, if q comes from A and q¯′ comes from B, we have to take
z → −z in our expressions to take into account the fact that the z-axis definition would
change. A schematic diagram can be seen in Fig. E.4. Consider the prior case first. Note
BA
xbxa θ
Figure E.4: Schematic diagram showing the quark momenta fractions, xa corresponding to the
quark coming from the ‘left’ proton, A, and xb corresponding to the quark coming
from the ‘right’ proton, B.
that if we consider the lab frame collision of the quarks, where pµq,lab =
√
s
2
(xa, 0, 0, xa) and
p′µq¯′,lab =
√
s
2
(xb, 0, 0,−xb), we may write sˆ as sˆ = (p + p′)2 = s4 [(xa + xb)2 − (xa − xb)2] =
xaxbs, where s is the hadron centre-of-mass collision energy. The quarks are of course
confined in the hadrons and possess a distribution of momenta, distinct for each quark
flavour. We denote the differential cross section
∑
qq′ qq¯
′ → W/W ′ → f f¯ ′ by dσˆ and
the full hadronic differential cross section, including the PDFs, by dσ. To obtain the
full hadronic differential cross section we multiply by the parton density functions, and
integrate over the momentum fractions xa and xb, using a delta function which ensures
that the quarks have the correct centre-of-mass energy, δ(sˆ − xaxbs). For the case of q
coming A and q¯′ from B:
dσLR
dzdsˆ
=
∫ 1
0
dxadxbδ(sˆ− xaxbs)fq/A(xa, sˆ)fq′/B(xb, sˆ) dσˆ
dzdsˆ
=
∑
qq′
Ωqq′
96
∫ 1
0
dxadxbδ(sˆ− xaxbs)fq/A(xa, sˆ)fq′/B(xb, sˆ)
[
S(sˆ)(1 + z2) + 2A(sˆ)z
]
.
(E.20)
E.2. The W ′ Drell-Yan cross section 189
If we now consider the case when the q comes from B and q¯′ from A, we have to take
z → −z. This will not change the z-even factor, but will change the z-odd factor:
dσRL
dzdsˆ
=
∑
qq′
Ωqq′
96
∫ 1
0
dxadxbδ(sˆ− xaxbs)fq′/A(xa, sˆ)fq/B(xb, sˆ)
[
S(sˆ)(1 + z2)− 2A(sˆ)z] .
(E.21)
To combine the two results into the full differential cross section, we combine the PDFs
into an even and an odd function respectively:
G±qq′ =
[
fq/A(xa, sˆ)fq′/B(xb, sˆ)± fq/B(xb, sˆ)fq′/A(xa, sˆ)
]
. (E.22)
This results in the following expression:
dσ
dzdsˆ
=
∑
qq′
Ωqq′
96
∫ 1
0
dxadxbδ(sˆ− xaxbs)
[
S(sˆ)(1 + z2)G+qq′ + 2A(sˆ)zG
−
qq′
]
.
(E.23)
Since sˆ = xaxbs, we have τ = xaxb and by defining the boson (or dilepton) rapidity
y ≡ 1
2
log(E+pz
E−pz ) =
1
2
log(xa/xb), we finally arrive at the full hadronic differential cross
section:
dσ
dτdydz
=
G2FM
4
W
192π
∑
qq′
|Vqq′|2
[
SG+qq′(1 + z
2) + 2AG−qq′z
]
, (E.24)
where we have set Cℓ,qi = 1. This is exactly what was given in section 5.2.2, Eq. (5.2).
190 Chapter E. Supplementary material for W ′ study
Appendix F
Supplementary material for
leptoquark study
F.1 Feynman rules and diagrams
The Feynman rules [132] relevant to the leptoquark pair-production diagrams are given
in Figs. F.1 and F.2. The relevant parton-level Feynman diagrams are shown in Figs. F.3
and F.4 for gluon-gluon and quark-anti-quark initial states respectively.
k3
k1
k2
V S¯S,aijµ = gs(t
a)ij(k2 − k1)µ .
Figure F.1: Feynman rule for the vertex scalar leptoquark-scalar anti-leptoquark-gluon. All
momenta are incoming and arrows indicate colour flow.
Since the couplings to light generations are suppressed in the kind of models we are
considering, leptoquark single-production in hadron colliders can proceed only via b-quark
gluon fusion, as shown in Fig. F.5. However this is also heavily suppressed due to the low
b-quark PDF and the small couplings to fermions, and can be neglected.
191
192 Chapter F. Supplementary material for leptoquark study
p3
p4
p1
p2
W S¯Sgg,ija1a2(p1, p2, p3, p4) = g
2
s(t
a1ta2 + ta2ta1)ij
× gµ1µ2 .
Figure F.2: Feynman rule for the vertex scalar leptoquark-scalar anti-leptoquark-gluon-gluon.
All momenta are incoming.
Figure F.3: Feynman diagrams relevant to scalar leptoquark pair-production with gluon-gluon
initial states.
Figure F.4: Feynman diagram relevant to scalar leptoquark pair-production with guark-anti-
quark initial states.
F.2. The effective Lagrangian for derivatively-coupled leptoquarks 193
Figure F.5: Feynman diagrams relevant to scalar leptoquark single production. Solid lines
with an arrow indicate quarks, lines without an arrow indicate leptons.
F.2 The effective Lagrangian for derivatively-coupled
leptoquarks
The Lagrangian for derivatively-coupled conjugate fields, which appears in Eq. (5.40),
also contains terms involving the conjugate fields, such as
LS˜′
1/2
∼ t¯cRγµτLpµ,qS˜ ′(+)1/2 . (F.1)
To manipulate the above expression for the case of on-shell S˜ ′1/2 decays as we did in
Eq. (5.42), we need to show that
Ψ¯CR,L/p = mΨ¯
C
L,R , (F.2)
where Ψ is a 4-component spinor and ΨCL,R = (PL,RΨ)
C . This can be demonstrated by
using the following identities [152]:
Ψ¯C = −ΨTC−1 ,
C−1γµ = −γTµC−1 , (F.3)
and hence
Ψ¯CR,L = − [PR,LΨ]T C−1 . (F.4)
So the necessary effective Lagrangian for the decay is given by
Leff ∼ mtt¯cLτLS˜ ′(+)1/2 . (F.5)
194 Chapter F. Supplementary material for leptoquark study
The full list of effective Lagrangians for the primed leptoquarks, from which the decay
modes and couplings in Tables 5.4 and 5.6 can be derived, is given by
LS′0 =
[ −i√
2f
(g′0Lmb + g
′
0Rmτ )
]
b¯RS
′
0τL
+
[ −i√
2f
(g′0Lmτ + g
′
0Rmb)
]
b¯LS
′
0τR
+
[ −i√
2f
(g′0Lmt)
]
t¯RS
′
0ντ,L , (F.6)
LS˜′0 =
[ −i√
2f
(g˜′0Rmtt¯LτR + g˜
′
0Rmτ t¯RτL)S˜
′
0
]
, (F.7)
LS′1 =
[ −i√
2f
√
2g′1L(mtt¯RτL +mτ t¯LτR)S
′(+)
1
+
−i√
2f
√
2g′1Lmbb¯RνLS
′(−)
1
+
−i√
2f
(g′1Lmtt¯RνL − g′1Lmbb¯RτL − g′1Lmτ b¯LτR)S ′(0)1
]
, (F.8)
LS′
1/2
=
[ −i√
2f
(h′1Lmbb¯
c
LνL + h
′
1Rmtt¯
c
RτR + h
′
1Rmτ t¯
c
LτL)S
′(−)
1/2
+
−i√
2f
(h′1Lmb + h
′
1Rmτ )b¯
c
LτLS
′(+)
1/2
+
−i√
2f
(h′1Lmτ + h
′
1Rmb)b¯
c
RτRS
′(+)
1/2
]
, (F.9)
LS˜′
1/2
=
[ −i√
2f
h′2Lmtt¯
c
LνLS˜
′(−)
1/2
+
−i√
2f
(h′2Lmtt¯
c
LτL + h
′
2Lmτ t¯
c
RτR)S˜
′(+)
1/2
]
, (F.10)
where we have defined: S
′(±)
1 ≡ (S ′(1)1 ∓ iS ′(2)1 )/
√
2 (and equivalent definitions for S˜
′(±)
1/2 )
and S
′(0)
1 ≡ S ′(3)1 . We have also used the fact that the doublet leptoquarks may be written
as a vector S ′1/2 = (S
′(−)
1/2 , S
′(+)
1/2 ). We have set the quark and lepton couplings to equal,
gq = gℓ,1 however these can be reinstated trivially by replacing g → gq where a quark
mass term appears and g → gℓ where a lepton mass term appears.
Note that terms appearing in this Lagrangian are no longer SU(2)L × U(1)Y gauge-
invariant. This is consistent since these terms would appear in the Lagrangian after
electroweak symmetry breaking and vanish as the fermion masses tend to zero. The
1The implementation in HERWIG++ version 2.5.0 also includes this simplification.
F.3. (tτ)(tτ) reconstruction method 195
Lagrangian is, of course, U(1)em gauge-invariant.
F.3 (tτ )(tτ ) reconstruction method
In terms of the momentum ratios zi defined in Eq. (5.48), the conditions for balancing
the total missing transverse momentum can be written as
z1 = (p
x
miss − (z2 − 1)pxj2 − pxνl)/pxj1 + 1, (F.11)
pyj1p
x
νl
− pxj1pyνl = pxmiss pyj1 − pymiss pxj1 + (z2 − 1)(pxj1pyj2 − pyj1pxj2). (F.12)
The mass-shell conditions, except for p2νl = 0, can be written as
m2W = (pl + pνl)
2 = 2pl · pνl , (F.13)
m2t = (pb + pl + pνl)
2 = m2W +m
2
b + 2pb · pl + 2pb · pνl , (F.14)
m2S0 = (pt + pτ1)
2 = m˜2t + 2z1pt · pj1 , (F.15)
m2S0 = (pb + pl + pνl + pτ2)
2 = m2t + 2z2(pb + pl) · pj2 + 2z2pj2 · pνl , (F.16)
where m˜t is the reconstructed mass of the hadronic top and mt is the assumed mass of
the semi-leptonic top. By eliminating z1 and mS0 from Eqs. (F.11), (F.15) and (F.16),
one obtains
z2pj2 · pνl +
pt · pj1
pxj1
pxνl = t3 + u3z2, (F.17)
where
t3 =
m˜2t −m2t
2
+
pxmiss + p
x
j1 + p
x
j2
pxj1
pt · pj1 , (F.18)
u3 = −(pb + pl) · pj2 −
pxj2
pxj1
pt · pj1 . (F.19)
Using a vector pνl = (Eνl, p
x
νl
, pyνl, p
z
νl
), Eqs. (F.12), (F.13), (F.14) and (F.17) can be
recasted as
APνl = S (F.20)
196 Chapter F. Supplementary material for leptoquark study
where
A =

El −pxl −pyl −pzl
Eb −pxb −pyb −pzb
z2Ej2 −z2pxj2 + (pt · pj1)/pxj1 −z2pyj1 −z2pzj2
0 pyj1 −pxj1 0
 , (F.21)
and
S =
(
m2W
2
,
m2t−m2b−m2W
2
− pb · pl, t3 + u3z2, t4 + u4z2
)
. (F.22)
t4 and u4 are defined as
t4 = (p
x
miss + p
x
j2
)pyj1 − (pymiss + pyj2)pxj1, (F.23)
u4 = p
x
j1
pyj2 − pyj1pxj2. (F.24)
From Eq. (F.20), we can determine pνl as a function of z2. Finally, z2 can be determined
from the mass-shell condition:
p2νl = (A
−1S)2 = 0. (F.25)
This provides a quartic equation for z2, and we can find up to four real solutions in the
physical range z2 ≥ 1. We can then obtain mS0 by substituting z2 into Eq. (F.16).
F.4 (q′τ )(qν) reconstruction method
Given w in Eq. (5.52), mbν(w, p
z
ν) can be minimised in terms of p
z
ν . The result is
[mminbν (w)]
2 = m2bν(w, p˜
z
ν)
= 2|pb||pmiss − wpj| − 2pb · (pmiss − wpj)
= [mbνT (w)]
2, (F.26)
where
p˜zν ≡
|pmiss − wpj|
|pb| p
z
b (F.27)
F.4. (q′τ)(qν) reconstruction method 197
and mbνT (w) is the transverse mass of the bν system. This allows us to calculate Mmin by
one-parameter minimisation:
Mmin = min
w
[max{mtτ (w), mbνT (w)}]. (F.28)
Since mtτ (w) is a monotonically increasing function of w, if mtτ (0) ≥ mbνT (0):
Mmin = mtτ (0) . (F.29)
Furthermore, since there exists a value pˆzν which fulfils mtτ (0) = mbν(0, pˆ
z
ν), we find
Mbalmin = mtτ (0) . (F.30)
If mbνT (0) > mtτ (0), we have to search for other values of w. For the true w and p
z
ν , say
w∗ and pz∗ν , we have
mbνT (w
∗) < mbν(w∗, pz∗ν ) = mtτ (w
∗) . (F.31)
This assures existence of wˆ which satisfies the relation mbνT (wˆ) = mtτ (wˆ). By scanning w
from 0 to wˆ, one finds:
Mbalmin = m
bν
T (wˆ) , (F.32)
and
Mmin = min
w∈[0−wˆ]
[mbνT (w)] . (F.33)
Hence we have
Mbalmin ≥Mmin. (F.34)
198 Chapter F. Supplementary material for leptoquark study
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