ufdc image array 2 - c 2012 gaurab k....

TOPICS IN BEYOND STANDARD MODEL COLLIDER PHENOMENOLOGY

By

GAURAB K. SARANGI

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

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2012

c© 2012 Gaurab K. Sarangi

2

To my parents

3

ACKNOWLEDGMENTS

I would like to express my gratitude to everyone who made this work and dissertation

possible, in a threefold way.

First and foremost, I would like to thank my first teachers, my parents and my family,

who nurtured and encouraged my curiosity as I was growing up and provided constant

moral and emotional support throughout my career. In particular, I would like to thank

my brother Sampad, who was more than family - a friend who stood by the me my entire

life.

Next, I would like to thank my teachers and mentors who shaped my mind through

various stages of my education. I will forever be indebted to my doctoral advisor,

Konstantin Matchev, a constant source of inspiration and support who guided me not

only in the ways of high energy physics, but also in critical thinking and in expressing my

ideas clearly. I would like to express my gratitude to Asesh K Datta who introduced me

to the world of Feynman diagrams while holding my hands as I took baby steps in the

world of particle physics. I would also like to acknowledge my debt to Prakash Satpathy,

my high school mathematics and physics teacher who introduced me to concepts of

abstraction and beauty in science.

Finally, I would like to acknowledge my gratitude for my friends Manoj, Shawn, MJ,

Mike, Kathleen, Katie, Joe, Joel and Becca, who made my stay in Gainesville fun and

interesting, a must have for a happy and productive research.

4

TABLE OF CONTENTS

page

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

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

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

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.1 Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.1.1 The Success Story . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.1.2 So, are we done then? . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2 Beyond Standard Model Theories . . . . . . . . . . . . . . . . . . . . . . 151.2.1 Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2.2 Extra Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3 Motivation for Our Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.3.1 Large Extra-Dimension (ADD Model) . . . . . . . . . . . . . . . . . 171.3.2 Diquarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.3.3 Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.3.4 Model Independent Studies . . . . . . . . . . . . . . . . . . . . . . 19

2 LARGE EXTRA-DIMENSION SIGNATURE THROUGH Z-PAIR PRODUCTIONAT THE LHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1 Introduction to Extra Dimensions . . . . . . . . . . . . . . . . . . . . . . . 202.2 Large Extra Dimensions: ADD Model . . . . . . . . . . . . . . . . . . . . . 232.3 Phenomenological Signatures . . . . . . . . . . . . . . . . . . . . . . . . . 252.4 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.5 Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.6 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 DIQUARKS IN PYTHIA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1 Diquark Production and Decay . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Implementation in Pythia . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.1 Problem with Implementation . . . . . . . . . . . . . . . . . . . . . 403.2.2 Our Workaround . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2.3 Description of the Implementation . . . . . . . . . . . . . . . . . . . 41

3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5

4 SAME-SIGN DILEPTONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.1 Interpretation of Experimental Results . . . . . . . . . . . . . . . . . . . . 474.2 Calculating Nsig Theoretically . . . . . . . . . . . . . . . . . . . . . . . . . 474.3 Simplified Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.4 Simplified Model: An Illustration . . . . . . . . . . . . . . . . . . . . . . . . 504.5 Model-Independent Procedure . . . . . . . . . . . . . . . . . . . . . . . . 51

4.5.1 Model-Independent Procedure: Fast Simulation . . . . . . . . . . . 524.5.2 Model-Independent Procedure: Emulation . . . . . . . . . . . . . . 61

4.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5 HOW TO LOOK FOR SUPERSYMMETRY UNDER THE LAMPPOST AT THELHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 675.2 Travelling Salesman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.3 8-lepton Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.3.1 Case: A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.3.2 Case: B and C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.4 Grouping Signatures and Hierarchies . . . . . . . . . . . . . . . . . . . . 835.5 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

APPENDIX

A FORTRAN CODE FOR EXTERNAL IMPLEMENTATION OF DIQUARK INPYTHIA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

B PLOTS FROM EMULATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

C HIERARCHY TO SIGNATURES . . . . . . . . . . . . . . . . . . . . . . . . . . 122

D SIGNATURE TO HIERARCHIES . . . . . . . . . . . . . . . . . . . . . . . . . . 168

E W-B TRANSITION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

6

LIST OF TABLES

Table page

2-1 Compactification lengths for different numbers of extra dimensions. . . . . . . . 24

3-1 Quantum numbers for various particles involved in diquark production anddecay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4-1 The seven search regions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5-1 The set of SUSY particles considered in this analysis, shorthand notation foreach multiplet, and the corresponding soft SUSY breaking mass parameter. . . 68

5-2 Construction of a sample hierarchy chain . . . . . . . . . . . . . . . . . . . . . 69

5-3 Input soft SUSY mass parameters (in GeV) for the xxGQWLBEH study points. 78

5-4 Input soft SUSY mass parameters (in GeV) for the xxGUBEWLH study points. 79

5-5 Input soft SUSY mass parameters (in GeV) for the xxGUBEHLW study points. . 80

C-1 Hierarchy to Signatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

D-1 Signature to Hierarchies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

7

LIST OF FIGURES

Figure page

2-1 Representative diagrams for (a) real graviton emission process and (b) virtualgraviton exchange process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2-2 Feynman diagrams for the leading order Z pair production process in SM. . . . 29

2-3 Feynman diagrams for the leading order Z pair production process in ADDmodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2-4 Dependence of total cross-section (σ) for f f̄ → ZZ , on EminT ,Z at a 1 TeV fixedcenter-of-mass energy e+e− collider. . . . . . . . . . . . . . . . . . . . . . . . . 30

2-5 Dependence of total cross-section σ for f f̄ → γγ, on EminT ,γ at a 1TeV fixedcenter-of-mass energy e+e− collider. . . . . . . . . . . . . . . . . . . . . . . . . 31

2-6 Invariant mass distribution of the outgoing Z bosons for pp → ZZ at LHC (14TeV). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2-7 Angular distribution of the outgoing Z bosons for pp → ZZ at LHC (14 TeV). . . 36

2-8 Dependence of the cross-section for pp → ZZ on the string scale at LHC (14TeV). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3-1 Invariant mass distribution of the four outgoing particles with the mass of thelepto-diquark fixed at 100 GeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3-2 Invariant mass distribution of the four outgoing particles with the mass of thelepto-diquark fixed at 300 GeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4-1 The parameter space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4-2 Diagramatic description of our simplified model setup for two same signedlepton signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4-3 PGS: Efficiencies for the seven search regions for M1 = 10 GeV. . . . . . . . . 54

4-4 PGS: Efficiencies for the seven search regions for M1 = 100 GeV. . . . . . . . 55



4-7 PGS: Limits on the x-sections for the seven search regions for M1 = 10 GeV. . 57

4-8 PGS: Limits on the x-sections for the seven search regions for M1 = 100 GeV. 58



8

4-11 PGS: SS2L production x-section for M1 = 10, 100, 200 and 300 GeV. . . . . . 60

4-12 PGS: 95% CL limit on SS2L production x-section for M1 = 10, 100 and 200GeV with a luminosity of 1, 10 and 30 fb−1. . . . . . . . . . . . . . . . . . . . . 63

4-13 Emulation: 95% CL limit on SS2L production x-section for M1 = 10, 100 and200 GeV with a luminosity of 1, 10 and 30 fb−1. . . . . . . . . . . . . . . . . . . 64

4-14 PGS versus the Emulation prescription: Efficiency of acceptance of µ and eas a function of their respective PT ). . . . . . . . . . . . . . . . . . . . . . . . . 65

4-15 PGS versus the Emulation prescription: Number of accepted jets and leptons(µ and e)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4-16 Comparision of HT and 6ET from PGS and the Emulation prescription. . . . . . 66

5-1 Production rate (in percentage) of (a) no LCP, (b) only one LCP and (c) bothLCP in a MQ-MU plain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5-2 Graphical representation of the allowed transitions between the SUSY states. . 72

5-3 Mass spectrum for the hierarchy xxGQWLBEH. . . . . . . . . . . . . . . . . . . 77

5-4 Branching ratio, production cross-section and multi-lepton signatures for thecase of hierarchy xxGQWLBEH . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5-5 Mass spectrum for the hierarchy xxGUBEWLH. . . . . . . . . . . . . . . . . . . 80

5-6 Branching ratio, production cross-section and multi-lepton signatures for thecase of hierarchy xxGUBEWLH . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5-7 Mass spectrum for the hierarchy xxGUBEHLW . . . . . . . . . . . . . . . . . . . 82

5-8 Branching ratio, production cross-section and multi-lepton signatures for thecase of hierarchy xxGUBEHLW . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5-9 X,Y,Color == #leptons,#groups,#channels. . . . . . . . . . . . . . . . . . . . . . 84

5-10 X,Y,Color == #channels,#groups,#leptons. . . . . . . . . . . . . . . . . . . . . . 84

5-11 X,Y,Color == #hierarchies,#groups,#leptons. . . . . . . . . . . . . . . . . . . . . 85

5-12 X,Y,Color == #hierarchies,#groups,#channels. . . . . . . . . . . . . . . . . . . . 86

5-13 The travelling salesman diagram for the case when the electroweak multipletsare considered separately. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

B-1 Emulation: Efficiencies for the seven search regions for M1 = 10 GeV. . . . . . 117

B-2 Emulation: Efficiencies for the seven search regions for M1 = 100 GeV. . . . . 118


9


B-5 Emulation: Limits on the x-sections for the seven search regions for M1 = 10GeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119




E-1 Dependence of w̃ 0 decay width on M2 in e+e−(blue), Z (red) and h (green)production. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

E-2 Dependence of w̃ 0 decay width on additional parameters . . . . . . . . . . . . 207

10

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

TOPICS IN BEYOND STANDARD MODEL COLLIDER PHENOMENOLOGY

By

Gaurab K. Sarangi

May 2012

Chair: Konstantin T. MatchevMajor: Physics

This Dissertation is a comprehensive summary of my work as a doctoral student

at the University of Florida. This document will eloborate our research in the study

of collider phenomenology at the LHC for graviton production inspired by Large

Extra-dimension, diquark production as a supermodel (i.e. models with significant

signals at early LHC), model independent Minimal Supersymmetric Standard Model

(MSSM) and model independent topological analysis of same sign dilepton study by the

CMS collaboration.

11

CHAPTER 1INTRODUCTION

At the dawn of the 21st century, we find ourselves standing at the bleeding edge

of high energy physics in terms of both science and technology. Being in the field of

phenomenology has given me the front row seat to see how they come together to

propel forward our knowledge of reality. As we try to probe into the unknown we are

faced with the following challenges: a) new cosmological data asking for a newer theory,

b) newer theories with predictions which can’t be validated with current experimental

setups, asking for better experiments/data, and c) newer theories with predictions that

have a chance of being validated (or rejected) in current experimental setup.

With the mega beast called the Large Hadron Collider at CERN, Geneva colliding

proton beams at a never seen before 7 TeV center of mass energy we now have a great

opportunity in being able to do some serious development especially in the case ‘c’

mentioned above. But having already discovered a good portion of physics with a similar

setup (pp̄ collider @ 2 TeV) at the Fermilab, Batavia, we now have to identify what once

was considered a good signal to be mere background. Thus one of the major challenges

that we face today is to find that signal amidst an overwhelming amount of background,

not very far from trying to find a needle in a haystack, except for the fact that we won’t

even be sure if the needle was produced at all. This calls for superior techniques in data

handling and statistical analysis.

In this chapter I’ll try to shed some light on the current status of our verified

knowledge in high energy physics, the theoretical ideas which just became open for

prying and prodding, and the analytic techniques being employed to dissect the data.

1.1 Standard Model

The Standard Model of particle physics is a Quantum Field Theory that tries to

explain the electro-magnetic, weak and strong interaction between various particles. It

incorporates Quantum Chromo-Dynamics (QCD) by Politzer, Wilczek and Gross, and

12

Electro-Weak Symmetry Breaking (EWSB) as prescribed by Glashow with the Higgs

mechanism incorporated into it by Salam and Weinberg.

1.1.1 The Success Story

The Standard Model successfully explains the interaction between all observed

particles in terms of a fewer number of fundamental particles and interactions, reducing

the number of free parameters in the theory to just 19. It categorizes all particles into

matter particles (fermions) and force carriers (bosons). Furthermore, the matter particles

are divided into 3 generations. In each generation we have a chirally left lepton doublet

(e.g. left handed electron and electron nutrino), a chirally right lepton singlet, a chirally

left quark doublet and 2 chirally right quark singlets. The interaction particles are 8

gluons, 3 weak bosons and a hypercharge boson. Upon EWSB, the neutral weak and

hypercharge gauge bosons mix to give rise to Z 0 and γ.

The Standard Model made predictions about the W and Z bosons, charm, bottom

and top quarks. The W and Z bosons were observed at the Large Electron Positron

collider and their observed masses matched the prediction. The charm quark was

discovered at Brookhaven National Laboratory and Stanford Linear Accelerator Collider

(1974) and the bottom and the top quarks were discovered at Fermilab (bottom in 1977

and top in 1999).

The only thing left so far is the elusive Higgs particle. The Higgs mechanism was

first postulated by Peter Higgs and was later incorporated into Sheldon Glashow’s

Electro-weak theory by Steven Weinberg and Abdus Salam. This introduced the scalar

Higgs boson in the theory. It has been 45 years since it was proposed in 1967 and

remains as the last piece that completes all Standard Model predictions.

1.1.2 So, are we done then?

It looks like once we find Higgs we can happily say Standard Model has explained

everything it set forth to explain. But, that in no way means there aren’t any unanswered

questions. These issues can be grouped in 2 categories.

13

Theoretical Issues: Although the Standard Model is theoretically very well

formulated it doesn’t explain away some of the issues arising from such formulations. At

the same time it also doesn’t attempt to shed light on other physical features of reality.

a) Hierarchy problem: The top loop contribution to the Higgs mass is quadratically

divergent, which means this will lead to Higgs having mass on the order of the cut off

scale, e.g. Planck mass (MPlanck ).

b) Strong CP Problem: In the QCD Lagrangian there appear terms which in

principle can violate CP (charge and parity) symmetry. However, we have not observed

any such case of CP violation in the experiments.

c) 19 parameters!: A huge number of free parameters are required to descibe the

theory completely, which leaves the theory a bit aesthetically challenging.

d) Why three generations: The Standard Model organizes the matter particles

(quarks and leptons) neatly in three different families. But, there is no theoretical

motivation behind it and it doesn’t try to explain as to why that should be the case.

e) What about gravity?: Even though Standard Model explains the electro-magnetic,

weak and strong interactions very well, it is silent when it comes to gravity. It is

incompatible with the General Theory of Relativity, the most accepted and well tested

theory of gravity to date.

Experimental Issues:

a) Neutrino masses: The Standard Model takes the masses of the neutrinos of

the three generations to be exactly zero. But, we have experimentally seen that the

neutrinos have tiny masses. They also oscillate from one flavor to another thus violating

lepton flavor number which is presumed to be conserved in the Standard Model.

b) Dark Matter and Dark Energy: The matter particles included in the Standard

Model constitute all the visible matter in the universe. But from astronomical data we find

that this visible portion of matter only constitutes about 14% of all matter and 4% of all

mass-energy in the universe. From the rotation of the galaxies we now know that there

14

is some other kind of matter in the universe that takes up about 86% of the remaining

matter and 23% of all mass-energy in the universe. This kind of matter has been dubbed

as Dark matter. The rest 74% of all mass-energy content has been named Dark Energy

which accounts for the accelerated expansion of the universe.

c) Matter-Antimatter Asymetry: According to the Standard Model there should be

equal amount of matter and antimatter in the universe. But, what we see is that the

universe is filled with matter.

1.2 Beyond Standard Model Theories

Since, we see that Standard Model is not enough in tackling the challenges

described above we need to explore theories which go beyond that. There have been

many such theories with various motivations. Here, we mention a few.

1.2.1 Supersymmetry

Supersymmetry is a Beyond Standard Model theory primarily motivated by

• Gauge unification, and

• The hierarchy problem,

• Dark matter

To solve the hierarchy problem, it introduces new particles (and thus loop diagrams)

which cancel out the quadratic divergences encountered in the hierarchy problem. For

example, the top loop that contributes to the quadratic divergence for the Higgs mass is

canceled out by another diagram with an ”almost” similar particle in the loop contributing

with an opposite sign. The calculations reveal that this particle has to be exactly like

top quark except for one key difference, it has to be a scalar. Thus the Supersymmetric

lagrangian has a new symmetry called Supersymmetry that relates a particle with its

superpartner which has all of the same quantum numbers as the particle but with a

15

difference of 12

in the spin. Thus, if Q is a supersymmetric transformation,

Q|Boson >= |Fermion >

and

Q|Fermion >= |Boson > .

When we incorporate Supersymmetry into the Standard Model we get twice as many

particles with a new set of superpartners for every particle in the Standard Model.

The simplest possible supersymmetric model consistent with the Standard Model is

called the Minimally Supersymmetric Standard Model (MSSM). Apart from solving

the hierarchy problem it also predicts that the gauge couplings unify at a very high

energy scale (1016GeV ). It also introduces the concept of R-parity, defined as PR =

(−1)2S+3B+L, where, S ,B and L are the particle spin, baryon number and lepton number

respectively. If R-parity is conserved then a supersymmetric particle can’t decay into

only Standard Model decay products. This gives us a way of describing Dark matter

particles. This is because, if R-parity is conserved then the Lightest Supersymmetric

Particle (LSP) can’t decay any further. Such particles in principle can constitute the

observed Dark matter in the universe, if they are electrically and color neutral.

1.2.2 Extra Dimensions

This theory was first postulated as the Kaluza-Klein (KK) theory by Theodor Kaluza

and Oscar Klein in 1926 seeking to unify gravitation and electromagnetism. But, since

then it has developed into a myriad of different approaches and interpretations, and also

addresses problems like Dark matter and hierarchy as well. The basic idea of the theory

is to introduce n more spatial dimensions to the 3 observable spatial dimensions, where,

n can be any positive integer. Depending on the nature of the extra dimension we have

various theories.

Large Extra Dimension: This model (also called ADD model) was proposed

in 1998[1]. According to this model the extra dimension is compactified and is large

16

compared to the Planck length. In this model the Standard Model particles and

interactions are confined in the 3 spatial dimensions but gravity can propagate through

all spatial dimensions. Thus while the graviton can have an infinite number of KK

excitation modes the Standard Model particles just have the zero excitation (ground

state) mode. We discuss this model in a greater detail in Chapter 2.

Warped Extra Dimension: This model (also called Randall-Sundrum model) was

proposed in 1999 [2, 3] as a solution to the Higgs hierarchy problem. It postulaes a five

dimensional space-time geometry with the fifth dimesnion strongly curved (warped) by

a large cosmological constant. This five dimensional manifold lies in the anti de Sitter

space (AdSn) since the de Sitter space describes a warped geometry with a positive

cosmological constant. In the RS model, the Standard Model interactions and particles

lie in the 3+1 brane while gravity is spread out in the bulk and is thus weak compared to

the SM interactions.

Universal Extra Dimension: In this framework, the metric of the bulk is flat and not

warped. In contrast to the ADD model and the RS model in Universal Extra-dimension

not only gravity but also all the Standard Model fields can propagate in the bulk [4, 5].

Thus, where as in Large Extra-dimension we have KK excitation states for graviton, in

UED we have KK excitation states for all the SM fields as well.

1.3 Motivation for Our Work

Our work spanned across a variety of interesting topics in BSM physics. Here, we

would briefly describe the motivation behind our work in the specific topics.

1.3.1 Large Extra-Dimension (ADD Model)

Since in the ADD model only graviton propagates in the bulk and has higher KK

excitation states, the collider searches consider signatures from processes involving

gravitons. At the time of our work, such searches primarily focused on (jets + 6ET )

and (2γ) signatures. Since, both signatures come with their share of overpowering

17

backgrounds at the LHC, we wanted to consider 2-Z0 bosons produced through an

s-channel graviton as a signature. The main benefits of this signatures are,

a) Very small background at the LHC,

b) The background is from a t-channel process at tree level, hence a very different

η distribution (the 2-Z0’s mostly in the forward/backward direction) compared to the

s-channel graviton process and

c) LHC would be hunting for this signal anyway, since, it is one of the search

channels for a Higgs heavier than the sum of the masses of the two Z0 bosons.

The details of our work are described in Chapter 2.

1.3.2 Diquarks

The motivation for our work with diquarks came from experimental needs. A diquark

being a quark-quark (qq) resonance, its production cross-section at the LHC is huge

compared to the Tevatron (R.I.P.). This can be seen without going into much detail,

simply by looking at the parton distribution functions (PDF ’s) of the colliding particles.

At the Tevatron we have proton-anti-proton colliding with each other. To form a qq

resonance one of the quarks has to be a valence quark of the proton and the other has

to be a sea quark of the anti-proton. In contrast, at the LHC, since both the colliding

particles are protons, both the quarks in the qq resonance are valence quarks. Hence,

with the LHC switching on it becomes important to study this as a supermodel [6], since

for the very first time such a signal will be available even at the early LHC. This brings

us to the next step, i.e. doing the analysis which involves event generation and their

detector simulation. But, at the time of our work there was no existing event generator

to do so. Moreover, the diquark under study is a color sextet and there was no event

generator where such a particle could be introduced. So, we implemented the whole

production process as an external process in Pythia [7]. We chose Pythia because of

our familiarity with it as well as the vast experience and expertise of the community that

uses it. A detail description of our work is presented in Chapter 3.

18

1.3.3 Supersymmetry

There has been more literature written on supersymmetry than one can read in a

lifetime. So, it become crucial for us at the outset of this presentation to explain why our

work was important in the field. As mentioned, being so highly investigated a theory,

it has led to many models. All of them constrain the full parameter space in some

way or another, with mSUGRA being the most constrained and the most investigated.

With different models different signatures become important and different search

strategies are required. Our work was the first in the field to take the model independent

supersymmetric mass spectrum and categorize them based on their signatures. This

is one step towards tackling the inverse problem. A detail description of our work is

presented in Chapter 4.

1.3.4 Model Independent Studies

With so many BSM theories in the market, a rigorous analysis can only be done in

a select few of them. Thus, experimental collaborations pick the models which are the

most reviewed and are more likely to give a definite answer. If such an analysis could be

recycled for other models motivated by a completely different theory then it could save

a lot of time and effort and could be a first check to see the validity of a model. Chapter

5, describes in detail our work in identifying and separating the model free aspect of the

CMS analysis of same-sign dilepton search for supersymmetry in the mSUGRA m0-m 12

plane from the model dependent part.

19

CHAPTER 2LARGE EXTRA-DIMENSION SIGNATURE THROUGH Z-PAIR PRODUCTION AT THE

LHC

2.1 Introduction to Extra Dimensions

The idea of extra dimensions was originally proposed by T. Kaluza and O. Klein

way back in 1926 in an attempt to unify Gravity with Electro-magnetism. In Kaluza-Klein

theories, the extra dimensions are spatial dimensions and they are much different

than our regular three dimensions, in the sense that they are compactified with some

compactification scale R−1c 1 . For example, if we have one extra dimension, it can

be a circle of radius Rc . If we have more than one extra dimension it can be a higher

dimensional sphere or a torus with our regular (3+1)-dimension confined to the 4-d

hypersurface. A Kaluza-Klein (1+3+d)-dimensional space-time will have the geometry

of a direct product M4 × X d , M4 being four-dimensional Minkowski space-time and X d

a compact manifold of the extra spatial dimensions. One of the most exciting theoretical

developments of recent years has been the idea that the observable Universe could be

confined to a four-dimensional hypersurface, called brane, in a higher-dimensional ‘bulk’

spacetime. Thus, only gravity propagates in the bulk and all Standard Model fields are

confined to the brane.

We start by applying the idea of extra dimension to the simplest of cases of a real

scalar field in 5-dimensional (one extra dimension) space-time. The Lagrangian looks

like

L = −12∂MΦ∂

MΦ, M = 0, 1, 2, 3, 4, (2–1)

where, Φ(M) ≡ Φ(xµ, y) ≡ Φ(t,~x , y), xµ being the 4-dimensional co-ordinate with

µ = 0, 1, 2, 3, and y is the coordinate in the extra dimension. As the extra dimension is

1 For simplicity of our phenomenological analysis, we assume torroidalcompactification with all dimensions having the same compactification radius Rc .

20

compactified on a circle of radius Rc ,

Φ(xµ, y) = Φ(xµ, y + 2πRc). (2–2)

Then we expand this field in the harmonics of a circle

Φ(xµ, y) =

+∞∑n=−∞

φn(xµ)einy/Rc . (2–3)

Substituting this expansion of the field in the expression for the Lagrangian we get

L = −12

+∞∑m,n=−∞

(∂µφn∂

µφm −nm

R2cφmφn

)e i(m+n)y/Rc . (2–4)

Hence, the action takes the form

S =

∫d4x

∫ 2πRc0

dyL =∫d4x

(−12∂µφ0∂

µφ0

)−∫d4x

+∞∑n=1

(∂µφn∂

µφ∗n +n2

R2cφnφ

∗n

).(2–5)

Here, we have used the fact that φ∗n = φ−n and absorbed the factor of 2πRc coming from

the integration over y by redefining φn =√2πRcφn

Thus, from this action we can see that we have

• A single real massless scalar field, φ0;

• An infinite number of massive complex scalar fields (φn) with masses inverselyproportional to the compactification radius, mn = nRc .

These states are called the Kaluza-Klein modes, with the massless scalar field being the

zero-mode. At low energy or distances much larger than the compactification scale only

the zero-mode is important, but at higher energies all the modes have to be considered.

Instead of taking a scalar field as done above, if we take an Abelian gauge field,

with the Lagrangian

L = − 14g2FMNF

MN , M,N = 0, 1, 2, 3, 4 (2–6)

and treat it the exact same way as the scalar field, then we get

• A single real massless gauge field A(0)µ with gauge coupling = g2/(2πRc);

21

• An infinite number of massive gauge bosons with masses, mn = nRc .

• A massless scalar field, A(0)5 .

Now, using the same idea for the gravity, the reason why this theory was proposed

in the first place, we take the gravitational action

S =M3Pl∗2

∫d4xdy

√GR5, . (2–7)

where, MPl∗ is the Plank scale in 5-dimension, GMN is the 5-dimensional metric, G = GMM

and R5 is the Ricci scalar in 5-dimensional space-time.

Treating it the same way as the scalar and vector fields, we get

• A single massless spin-2 graviton, g(0)µν .

• An infinite number of massive spin-2 gravitons with masses inversely proportionalto the compactification radius, mn = nRc .

• A massless scalar field, g(0)55 .

• A massless gauge field, g(0)µ5 .

Furthermore comparing this 5-dimensional action to the 4-dimensional action for gravity

we get

M2Pl = M3Pl∗(2πRc). (2–8)

The expression (for generalized d-dimension) above also comes from using Gauss’s law

on gravitational field and comparing the gravitational force between test charges m1 and

m2 separated by a distance of r in 4-dimension and d-dimension.

Gauss’s law for (3+1)-dimension:∮2

F · dA = 1M2Plm1m2, (2–9)

F =1

M2Plm1m2

1

r 2. (2–10)

22

Gauss’s law for D-dimension (D=1+3+d extra dimension):∮2+d

F · dA = 1M2+dPl∗

m1m2, (2–11)

F =(4π)d/2Γ(d/2)

M2+dPl Rdc

m1m21

r 2, (2–12)

M2Pl(4π)d/2Γ(d/2) = M2+dPl∗ R

dc . (2–13)

Where, MPl∗ is the fundamental scale of the theory.

2.2 Large Extra Dimensions: ADD Model

Such ideas of extra dimension proposed by Kaluza and Klein gave rise to

elegant solutions[1, 8, 9] to the well-known gauge hierarchy problem of high energy

physics, which is just the instability against quantum corrections. What is even more

interesting, perhaps, is the suggestion[10, 11] that there could be observable signals

of quantum gravity at current and future accelerator experiments, and this possibility

has spawned a vast and increasing body of work over the past five years. This

relatively new set of ideas, commonly dubbed ‘Brane World Phenomenology’, bases

itself on two main principles: the concept of hidden compact dimensions and the

string-theoretic idea of Dp-branes. The simplest brane-world scenario is the so-called

Arkani-Hamed—Dimopoulos—Dvali (ADD) model[1, 8, 9], in which there are d extra

spatial dimensions, compactified on a d-torus of radius Rc each way. Together with

the four canonical Minkowski dimensions, this constitutes the ‘bulk’ spacetime. In

this scenario the radius Rc of the extra dimensions can be as large as 44 µm[12].

However, the SM fields are confined to a four-dimensional slice of spacetime, with

thickness not more than 10−17 cm, which is called the ‘brane’. If the ADD model is

embedded in a string-theoretic framework, the ‘brane’ is, in fact a D3-brane, i.e. a 3+1

dimensional hypersurface on which the ends of open strings are confined. However, it

is not absolutely essential to embed the model in a string theory, and the word ‘brane’

or ‘wall’ is then used simply to denote the hypersurface (or thin slice) where the SM

23

fields are confined. A crucial feature of this model is that gravity, which is a property of

spacetime itself, is free to propagate in the bulk. As a result

a) Planck’s constant in the bulk MPl∗ is related to Planck’s constant on the brane MPl

(' 1.2× 1019 GeV) by

(MPl∗)2+d = (4π)d/2 Γ(d/2) M2Pl (Rc)

−d . (2–14)

Table 2-1. Compactification lengths for different numbers of extra dimensions.number of extra dimensions Rc (in m)d = 1 ∼ 1012d = 2 ∼ 10−3d = 3 ∼ 10−8

b) At MPl∗ = 1TeV , Table 2-1 gives the compactification lengths for different number

of extra dimensions. This means, for Rc ≤ 44µm [12], it is possible to have MPl∗ as low

as a TeV for d ≥ 3. (The normalization of reference [11] has been adopted here). This

solves the gauge hierarchy problem simply by bringing down the scale of new physics

(i.e. strong gravity in this case) to about a TeV and thereby providing a natural cut-off

to the SM, since the string scale MPl∗ now controls graviton-induced processes on the

brane.

c) There is a huge number of massive Kaluza-Klein excitations of the (bulk) graviton

field on the brane, with masses mn = n/Rc , and these collectively produce gravitational

excitations of electroweak strength, which may be observable at current experiments

and those planned in the near future.

d) As higher energies are reached, the distances probed are smaller, and for

distances comparable to or smaller than the compactification length, the dependence

of gravitational force on distance changes from inverse square to something steeper

depending on the number of extra dimensions.

It is only fair to mention that a major drawback of the ADD model is that it creates

a new hierarchy between the ‘string scale’ MPl∗ ∼ 1 TeV and the size of the extra

24

dimensions R−1c ∼ 1 µeV. In fact, the huge size of the extra dimensions (compared

to the Planck length) is not stable under quantum corrections, which tend to shrink

it down until MPl∗ ∼ R−1c ∼ MPl ∼ 1019 GeV, at which stage the original hierarchy

problem is reinstated. Nevertheless, there are several variants of the ADD model which

address this problem in various ways, and some of these ideas may not be far from the

truth. From a phenomenological point of view, it is, therefore, reasonable to postpone

addressing the stability issue, and proceed to study the minimal ADD model and its

consequences for experiment.

2.3 Phenomenological Signatures

The key features in ADD-phenomenology are,

a) Massive gravitons couple to anything with energy and momentum.

b) Gravitons are color/flavor blind.

c) Graviton coupling increases with energy.

d) Individual gravitons (Gn massive graviton of nth mode in the KK tower) escapes

detection.

e) The collective effect of gravitons in the whole tower has a near-electroweak

interaction strength.

The experimental consequences of ADD gravity have been mainly studied in the

context of

Figure 2-1. Representative diagrams for (a) real graviton emission process and (b)virtual graviton exchange process, producing Standard Model particles at ahadron collider

25

Real Graviton Emission: In this case, the final state gravitons in the ADD model

are ‘invisible’, escaping the detector because of their feeble individual interactions

(∼ M−1Pl ) with matter. Fig. 2-1(a) is a representative diagram of such process, where the

initial state consists of a quark and an anti-quark, annihilating into a gluon and a real

graviton (the signature being a jet and missing energy). The real graviton in fact can be

emitted from either of the two other external legs or the vertex as well. The final state,

involving missing energy due to gravitons, will be built up by making an incoherent sum

over the tower of graviton modes. Thus the cross-section will be,

σ(q + q̄ → g + Gn) =∑n

σ(mn)

=

∫ √s0

dmρ(m)σ(m) (2–15)

where, the density of states, ρ(m) is given by

ρ(m) =2Rncm

n−1

(4π)n/2Γn/2(2–16)

and the integration is cut off at the kinematic limit of√s.

Virtual Graviton Exchange: In this case the graviton is not produced as a final

state particle. Fig. 2-1(b) is a representative diagram of such process, where the initial

state consists of a quark and an anti-quark and the final state consists of an electron

and a positron, with an s-channel graviton propagator. Unlike the case of real graviton,

all final states are exactly same for virtual gravitons from all the KK modes. Thus we do

a coherent sum (at amplitude level) to calculate the cross-section. While calculating the

amplitude for each propagator (corresponding to different KK mode graviton), we get a

factor of 1s−m2n

, where, s is the center of mass energy and mn is the mass of the nth KK

mode graviton. Summing over all the modes is done in [10] as,

1

M̄2Pl

∑n

1

s −m2n=4π

Λ4, (2–17)

26

where Λ is the cut off string scale and M̄Pl is the reduced Planck scale (M̄Pl = MPl4π ). Also,

for some of the processes, we can get the same final state through a Standard Model

process (for example in the case of Fig. 2-1(b) the standard model process can be an

s-channel process with a photon or Z-boson propagator). Thus the standard model

process is also added using the coherent sum. Therefore, when we calculate the total

cross-section, along with a Standard Model and a Beyond Standard Model (BSM) term,

we also get an interference term. Because of Λ8 suppression in pure BSM compared

to Λ4 suppression in the interference term, the interference contribution is a lot more

important than the pure BSM term. In either case, it may be shown[10, 11] that, after

summing, the Planck mass MPl cancels out of the cross-section, leaving an interaction

of near-electroweak strength.

At a hadron collider most frequently, the signal coming from real graviton production

is accompanied by an isolated jet. These real production channels have strong

dependence on the number of extra dimensions. However, one can expect to have

better reach on the string scale, as, virtual graviton exchange processes have very weak

or no dependence on the number of extra dimension. In this paper we study the effect of

graviton propagation on the Z-pair production at the LHC and compare to the dominant

SM background.

2.4 Limits

Below are the limits imposed on the size and number of extra-dimension from

various experiments.

Gravitational Inverse-square law: Experiments were conducted by Kapner et al.

[12], to test the validity of Gravitational Inverse-Square Law below Dark-Energy Length

scale. They concluded that the law holds down to a length scale of 56 µm and that an

extra dimension must have a size smaller than that (Rc ≤ 56µm).

27

Collider search:The LEP collaborations have searched for Real Graviton emission

and L3 has the best limit. Their limits are MPl∗ > 1.5TeV − 0.5TeV for number of extra

dimensions n=2 to 8. CDF puts the limit on MPl∗ > 0.6TeV − 0.55TeV for n=4 to 8.

Cosmological Data: There are also limits on Large Extra Dimensions from

Supernova cooling [13]. For n = 2 they put a limit on MPl∗ > 84TeV and Rc ≤ 90µm. For

n = 3 they put a limit on MPl∗ > 7TeV and Rc ≤ 0.19µm.

2.5 Calculation

We explore the ADD Model for Large Extra Dimension with d extra-dimensions, in

which we consider the Z-pair production by proton-proton collision, mediated by a Virtual

Graviton. In our calculations we include fermion anti-fermion as well as gluon-gluon

initial states. We are interested in this particular process, mainly because, Z-pair

production (which then decays into four leptons) is one of the most important modes

in the search for higgs and experimentalists are definitely going to be looking for four

lepton signals.

For our calculations, we used the Giudice, Rattazi and Wells (GRW) notation [10].

But, in case of ZZ we noticed that in the GRW notation, there was no Feynman rule

for coupling of Graviton to massive vector boson, for which we used the Feynman

rules in Han, Lykken and Zhang (HLZ) notation [11] (normalized with that of the GRW

notation). After we started our calculations, we saw simillar work done in [14] and

[15], but our result disagreed with their result. Then another paper [16] was published

with calculations at next to leading order and our result matched exactly with theirs at

leading order level. In [14], they take the contribution from qq̄ → ZZ to be zero and their

cross-section blows up for mZ → 0.

For the tensor manipulations while doing amplitude squaring, fermion trace

calculations etc. we used the program FORM (a Symbolic Manipulation System). For

cross-section calculations we used Mathematica and Fortran. We used the sub-routine

28

Figure 2-2. Feynman diagrams for the leading order Z pair production process in SM.

Figure 2-3. Feynman diagrams for the leading order Z pair production process in ADDmodel.

Vegas offered in the package Cuba for Mathematica, for numerical integration. We used

the Parton Distribution Function, CTEQ5L.

We define, x = tŝ, z = (MZ√

ŝ) where, ŝ is the center of mass energy.

For, f f̄ → ZZ

29

0.01

0.1

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

σ (

pb)

EminT,Z (TeV)

Λ = 2.3 TeV

Λ = 3.0 TeV

Λ = 3.8 TeV

SM

Figure 2-4. Dependence of total cross-section (σ) for f f̄ → ZZ , on EminT ,Z at a 1 TeV fixedcenter-of-mass energy e+e− collider. EminT ,Z is the minimum value of ET(Transverse Energy) of outgoing Z-bosons. In the plots the symbolsrepresent results from the event generator Sherpa and the lines representour analytical result using (2–18). The lower line represents thecross-section coming from only Standard Model and the other threerepresent cross-sections including the contribution from extra dimension andcorrespond to three different values of the string scale Λ namely 2.3 TeV, 3.0TeV and 3.8 TeV.

dσf f̄→ZZdx

= − πŝ3

8Λ8[12z8 − 12z6(4x + 1) + z4(72x2 + 48x + 5)

−2z2(24x3 + 30x2 + 11x + 2) + x(12x3 + 24x2 + 17x + 5)]

+2πŝα

Λ4[2z8 − z6(−8x + 1) + z4x(12x2 + 2x − 1)− z2x(8x2 + 7x + 1)

+x(2x3 + 4x2 + 3x + 1)]C1

− 2πα2

ŝ

[4z8 − 4z6(3x + 1) + z4(14x2 + 6x + 1)− 2z2x(2x + 1)2

+x(2x3 + 4x2 + 3x + 1)]C2, (2–18)

30

1

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

σ (

pb)

EminT,γ (TeV)

Λ = 2.3 TeV

Λ = 3.0 TeVΛ = 3.8 TeV

SM

Figure 2-5. Dependence of total cross-section σ for f f̄ → γγ, on EminT ,γ at a 1TeV fixedcenter-of-mass energy e+e− collider. EminT ,γ is the minimum value of ET(Transverse Energy) of outgoing photons. In the plots the symbols representresults from the event generator Sherpa and the lines represent ouranalytical result using (2–19). The lower line represents the cross-sectioncoming from only Standard Model and the other three representcross-sections including the contribution from extra dimension andcorrespond to three different values of the string scale Λ namely 2.3 TeV, 3.0TeV and 3.8 TeV.

where

C1 =

[(g2Af + g

2Vf )

xw(xw − 1)x(−2z2 + x + 1)

],

C2 =

[(g4Af + 6g

2Af g

2Vf + g

4Vf )

x2w(xw − 1)2x2(−2z2 + x + 1)2

],

and gAf and gVf are the axial vector and vector coupling of the incoming quark to Z

boson.

For, f f̄ → γγ

31

dσf f̄→γγdx

= − πŝ3

8Λ8[4x(x + 1)(2x2 + 2x + 1)

]+2πŝα

Λ4[2x2 + 2x + 1

]− 2πα

2

ŝ

[(2x2 + 2x + 1)

x(x + 1)

], (2–19)

which is same as that of GRW [10].

For gg → ZZ ,

dσgg→ZZdx

= − πŝ3

24Λ8[12z8 − 12z6(4x + 1) + z4(72x2 + 48x + 5)

− 2z2(24x3 + 30x2 + 11x + 2) +(12x3 + 24x2 + 17x + 5)], (2–20)

and for, gg → γγ

dσgg→γγdx

= − πŝ3

16Λ8[(2x4 + 4x3 + 6x2 + 4x + 1)

], (2–21)

For comparison with numerical results from event generator Sherpa, we made plots

with the cross-section vs the PT cut of the out going Z boson(and γ). We wanted to

match our analytical result for the case of hadron collider with the numerical result from

the event generator Sherpa. But we couldn’t get Sherpa to calculate the cross-section

for the process gg → ZZ and gg → γγ. We found out that Sherpa couldn’t calculate

the cross-section of a process that doesn’t have a Standard Model counterpart. So,

to cross-check our analytical result with the numerical one from Sherpa we chose

e− + e+ → Z + Z and e− + e+ → γ + γ, where we fixed the center of mass energy to

be s = 1TeV . In Figures (2-5) and (2-4) the dotted lines represent result from Sherpa

and the continuous lines represent our analytical result. The x-axis corresponds to the

EminT (transverse energy cut of the out going Z or γ) in TeV. The y-axis corresponds to

the cross-section in pb. The 4 sets of lines represent SM and 3 ADD results based on 3

values of Λ namely 2.3 TeV, 3.0 TeV and 3.8 TeV. One interesting thing to note is that the

32

percentage change in Standard Model cross-section with addition of ADD contribution

increases with increasing EminT . Moreover as we decrease the cutoff string scale the

contribution from the extra dimension process increases. Also as we go from γγ to ZZ

this difference is even more magnified.

To compare with existing results, we also considered several other processes

namely γγ pair production from gluon-gluon as well as fermion anti-fermion initial state,

such as, an electron and a positron annihilating into a massless fermion-anti-fermion

pair. For the sake of completeness we also included an electron and a positron

annihilating into a pair of top quark and anti-top quark.

For, eē → f f̄ ,

dσeē→f f̄dx

=dσ(eē→f f̄ )SM

dx+dσ(eē→f f̄ )INT

dx+dσ(eē→f f̄ )ADD

dx

=dσ(eē→f f̄ )SM

dx

+Nf πŝ

3

32Λ8[32x4 + 64x3 + 42x2 + 10x + 1

]− Nf απŝ

2Λ4[QeQf (8x

3 + 12x2 + 6x + 1)

+1

xw(1− xw)(1− z2)(gVegVf (8x

3 + 12x2 + 6x + 1)

+ gAegAf (6x2 + 6x + 1)

)]+ δef

[−παŝ2Λ4

[Q2e (x

3 + 11x2 + 24x + 22 +9

x)

+g2Ve + g

2Ae

xw(1− xw)(1− z2)(x3 + 6x2 + 9x + 4)

+1

xw(1− xw)(x − z2)(g2Ve(5x

3 + 15x2 + 18x + 9)

+ g2Ae(5x3 + 15x2 + 10x + 1)

)]+

πŝ3

32Λ8(9x4 + 60x3 + 126x2 + 114x + 40)

]. (2–22)

33

For, f f̄ → tt̄

dσeē→tt̄dx

=dσ(eē→tt̄)SMdx

+dσ(eē→tt̄)INT

dx+dσ(eē→tt̄)ADD

dx

=dσ(eē→tt̄)SMdx

+3π

32ŝΛ8[40m8t − 8m6t ŝ(18x + 5) + 4m4t ŝ2(50x2 + 34x + 5)

− 2m2t ŝ3(64x3 + 80x2 + 26x + 1) + ŝ4(32x4 + 64x3 + 42x2 + 10x + 1)]

− απŝ2Λ4

[−12m6t + 2m4t ŝ(14x + 5)

− 4m2t ŝ2(6x2 + 5x + 1) + (8x3 + 12x2 + 6x + 1)]

+3απ

2ŝ2Λ4xw(1− xw)(1− z2)2×[

gVegVt(−12m6t + 2m4t ŝ(14x + 5)− 4m2t ŝ2(6x2 + 5x + 1)

+ (8x3 + 12x2 + 6x + 1))+ gAegAt

(2m2t + ŝ)(6m

4t − 4m2t ŝ(3x + 1)

+ ŝ2(6x2 + 6x + 1))].

2.6 Analysis

We have all the necessary expressions for the differential cross-section for Z-pair

production as we see in Equations (2–20) and (2–18). So we are in a position to do an

analysis to see how the Z-pair signal stands out in the presence of the Standard Model

background. In Figure (2-6) we have the differential cross-section for pp → ZZ at LHC

(14 TeV). On the x-axis we have the invariant mass of the two out going Z-bosons (mZZ )

in GeV and on the y-axis we have the differential cross-section in pb/GeV. The areas

under the dotted lines represents ADD contributions from uū, dd̄ and gg separately. The

contribution from Standard Model falls sharply with increasing mZZ . This behaviour can

be attributed to the parton distribution function of the quarks and gluons (which behaves

like e−ŝ). But the contribution from extra dimension increases with increasing mZZ . This

is because in the differential cross-section terms coming from extra dimension we have

positive powers of ŝ (ŝ3 in the pure extra dimension term and ŝ in the interference term).

34

Figure 2-6. Differential cross-section for pp → ZZ at LHC (14 TeV). On the x-axis wehave the invariant mass of the two out going Z-bosons (mZZ ) in GeV and onthe y-axis we have the differential cross-section in pb/GeV. The areas underthe dotted lines represents ADD contributions from uū, dd̄ and gg separately.The contribution from Standard Model falls sharply with increasing mZZ . Thisbehaviour can be attributed to the parton distribution function of the quarksand gluons (which behaves like e−ŝ). But the contribution from extradimension increases with increasing mZZ . This is because in the differentialcross-section terms coming from extra dimension we have positive powers ofŝ (ŝ3 in the pure extra dimension term and ŝ in the interference term).

In Figure (2-7) we have the normalized differential cross-section for pp → ZZ at

LHC (14 TeV). The x-axis corresponds to Cosine of the angle (θ) between the beam

direction and the out going Z-boson and the y-axis has the differential cross-section. For

figure (a) θ is measured in center of mass frame and in figure (b) θ is measured in the

lab frame. In both the cases the red (dashed) line corresponds to the result from only

Standard Model and the blue (solid) line corresponds to the result including contributions

extra dimension. The string cutoff scale Λ is set at 1.5 TeV. In Figure (2-8) we have the

dependence of cross-section on the string scale for pp → ZZ at LHC (14 TeV). The

35

Figure 2-7. Angular distribution of the outgoing Z bosons for pp → ZZ at LHC (14 TeV).The x-axis corresponds to Cosine of the angle (θ) between the beamdirection and the out going Z-boson and the y-axis has the differentialcross-section. For figure (a) θ is measured in center of mass frame and infigure (b) θ is measured in the lab frame. In both cases the red (dashed) linecorresponds to the result from only Standard Model and the blue (solid) linecorresponds to the result including contributions extra dimension. The stringcutoff scale Λ is set at 1.5 TeV.

dotted lines represent ADD contributions from uū, dd̄ and gg separately. The x-axis

represents the string cutoff scale Λ and the y-axis represents the cross-section. As the

36

10

1.6 1.8 2 2.2 2.4 2.6 2.8 3

σ (

pb)

Λ (TeV)

pp-> ZZ @ LHC (14TeV)

SMTotal ADD + SM

SM + ADD due to dDSM + ADD due to uUSM + ADD due to GG

Figure 2-8. Dependence of the cross-section for pp → ZZ on the string scale at LHC (14TeV). The dotted lines represents add contributions from uū, dd̄ and ggseparately. The x-axis represents the string cutoff scale Λ and the y-axisrepresents the cross-section. As the string cutoff scale increases, thecontribution from extra dimension decreases rapidly. This is due to the factthat the pure extra dimension contribution term has a factor of Λ−8 and theinterference term has a factor of Λ−4. It can be seen from the plot that theextra dimension contribution from uū is twice that of dd̄ which can beattributed to the parton distribution function, apart from which the amplitudeand phase space are exactly same for both processes.

string cutoff scale increases the contribution from extra dimension decreases rapidly.

This is due to the fact that the pure extra dimension contribution term has a factor of Λ−8

and the interference term has a factor of Λ−4. It can be seen from the plot that the extra

dimension contribution from uū is twice that of dd̄ which can be attributed to the parton

distribution function, apart from which the amplitude and phase space are exactly same

for both the processes.

37

2.7 Results

In scenarios with quantum gravity propagating in large extra dimensions, we

computed the effects of the virtual KK graviton-exchange amplitude. We noticed that

not only the interference term (ADD and SM) contributes to the signal but the pure ADD

amplitude is also quite significant. As mentioned earlier, we were unable to numerically

compute such pure ADD contribution using the event generator Sherpa for processes

with out a Standard Model counterpart. In recent literature these calculations have been

done for pp → ZZ . In this work we were able to cross-check their results and identify the

correct ones from the wrong ones.

38

CHAPTER 3DIQUARKS IN PYTHIA

3.1 Diquark Production and Decay

A diquark is qq resonance with spin zero or one, baryon number 23, and electric

charge −43, 13

or −23. It can transform either as a 6 or 3̄ under SU(3)[6]. For concreteness,

the diquark is considered to be a spin-0 sextet (transforms under SU(3) as a 6).

Table 3-1. Quantum numbers for various particles involved in diquark production anddecay.

Field Spin SU(3)C SU(2)L U(1)Yec 1/2 − − 1uc 1/2 3 − 2/3D 0 6 − 4/3L 1/2 6 − 7/3Lc 1/2 6 − −7/3

It has couplings to SU(2) singlet and SU(3) triplet up-type quarks( and is symmetric

in flavor indices). So, this diquark (D) can thus be produced as a result of collision

between two up-type quarks (uc ) in terms of the production operator as shown below.

OD =κD2D uc uc , (3–1)

where, κD is the coupling constant whose normalization depends on the normalization of

the color matrices Ra with which D is expanded such that D = DaRa. With Tr(RaRb) =

δab, the partonic cross-section for the production of the diquark is,

σ(uu → D) = π6κ2D δ(ŝ −m2D) . (3–2)

Since, it doesn’t have any other coupling available it would decay back into 2 up quarks

with a decay width given by

Γ(D → uu) = 116π

κ2DmD . (3–3)

Another resonance called Lepto − diquark has been introduced into this picture, which

is a vector-like fermion that transforms under SU(3) the same way as the diquark and

39

has the same baryon number, but differs in terms of other quantum numbers. It also has

to be lighter than the diquark such that the decay of diquark into lepto-diquark can be

allowed by phase space. The decay operator will be,

κ̄DDLcec , (3–4)

and the partical decay width will be,

Γ(D → Le) = 116π

κ̄D2mD . (3–5)

This lepto-diquark will further decay into 2 quarks and a lepton via the two operators

above in this section,

Lc → ēcucuc (3–6)

through an off-shell diquark. Thus, the final state has 2 jets and a pair of opposite sign

dileptons.

3.2 Implementation in Pythia

3.2.1 Problem with Implementation

To be able to analyse this model we need to be able to generate such events in

one of the many widely available monte-carlo event generators. Since, this model was

not yet implemented in any of the available event generators this needed to be done

to do any possible analysis. For this purpose we chose Pythia [7] , since, it has a very

neat feature with instruction for incorporating external processes. Also, Pythia comes

with the flexibility for defining a new particle that can be incorporated into the external

process. However, we faced a unique problem in doing so. A particle in Pythia can have

up to one color label and one anti-color label. This would pose no issue if our diquark

was an anti-triplet (we could define it to have no color label and an anti-color label). But,

since the diquark we chose is a sextet it means we need two color labels for it which is

not-trivially possible to do in Pythia.

40

3.2.2 Our Workaround

We defined neither the diquark nor the lepto-diquark as particles in Pythia’s decay

table. We implemented the external process only in terms of the two incoming up quarks

and out going same sign dileptons and the two outgoing up-quarks, thus eliminating the

need to have particles with two color labels. Color stretching was done only with respect

to the incoming and outgoing up quarks such that it was conserved in the process.

3.2.3 Description of the Implementation

The Pythia 6.4 manual has specified a prescription in section 9.9 to include an

external process in Pythia which we followed for our implementation. According to the

prescription, we need to write our own subroutines.

• UPINIT: This subroutine initializes the center of mass energy, the incoming beamsand the external processes.

• UPEVNT: This subroutine samples the phase space, evaluates the processcross-section and sets the color topology and parton shower scales.

These subroutines being already present in Pythia as dummy subroutines we need

to remove them from the Pythia source code and compile it to make it compatible for

use with our external process code. Since the actual physics is implemented in the

subroutine UPEVNT we’ll elaborate it in detailed steps.

Sampling of the Phase Space: The very first thing to do is, decide how the phase

space should be sampled. The total cross section for a 2→ 1 process can be written as

σ =

∫dx1

∫dx2 f1(x1,Q

2) f2(x2,Q2) σ̂(ŝ), (3–7)

or, in terms of τ ≡ x1x2 and y ≡ 12 lnx1x2

, as

σ =

∫dτ

τ

∫dy x1f1(x1,Q

2) x2f2(x2,Q2) σ̂(ŝ). (3–8)

So, a simple recipe would be to pick τ and y uniformly. However, the resonant

cross-section and parton distributions are peaked, so this would be inefficient. In

41

particlar, the cross-section is peaked at ŝ . So, following the instructions from section 7.4

in the Pythia user manual, the integral is rewritten as

σ =π

S

∫hτ(τ)dτ

∫hy(y)dy

x1f1(x1,Q2)x2f2(x2,Q

2)

τ 2hτ(τ)hy(y)

ŝ

πσ̂(ŝ), (3–9)

where, τ and y are generated according to the distributions hτ(τ) and hy(y) given as,

hτ(τ) =e2cmmDΓDτB − τA

1

(ŝ −mD)2 +m2DΓ2Dwhere,

τA = arctan

[ŝmin −m2DmDΓD

]τB = arctan

[ŝmax −m2DmDΓD

]and

hy(y) =1

ymax − yminwhere,

ymax = −1

2ln(τ)

ymin = +1

2ln(τ)

hτ(τ) is defined so, because the process being only in the s-channel τ = ŝS behaves

like a Breit-Wigner. The dependence on y being uniform, hy(y) is defined as a constant

function appropriately normalized.

Montecarlo: Next step is to generate random points for the Montecarlo integration.

Here, first of all we calculate various limits based on the mass of the lepto-diquark (L)

randomly generated in the event according to the Breit-Wigner distribution with the

42

lepto-diquark width.

ŝmin =(pTmin +

√m2L + p

2Tmin

)2τmin =

ŝminE 2cm

ŝmax = E2cm − ŝmax

τmax =ŝmaxE 2cm

Then we randomly generate the required variables (τ and y ) while obeying the limits

above.

τ =1

E 2cm

[m2D +mDΓD tan (τA + (τB − τA)r)

]y = ymin + (ymax − ymin)r ,

where, ‘r’ is a random real number sampled uniformly from 0 to 1. We reject any event

falling outside the phase space region. For each such point in phase space we calculate

the cross-section,

σ = d σ̂ × PSvol ×PDF × CONV , (3–10)

where, from the matrix element we have,

d σ̂ =κ2D6

mDΓD(ŝ −m2D)2 + (mDΓD)2

,

the PDF is supplied by Pythia (with scale Q2 = m2D) as

PDF = x1f1(x1,Q2)x2f2(x2,Q2),

PSvol is the volume of the selected phase space,

PSvol =ŝ

E 2cmτ2hτ(τ)hy(y)

and CONV is conversion factor from GeV−2 to pb.

43

Setting up the particles involved in the partonic process: Here, we set up the

number of incoming and outgoing particles, the identity of the particles, their status

codes, their mother particle codes (0 for incoming particles) and their color topology.

We also initialize the momenta of the outgoing particles to zero and that of the incoming

particles is taken from the variables from montecarlo in the previous section (i.e. x1 and

x2 from the PDFs’).

Event Generation: At this point, we have created an object (as a 2 → 1 process

in the montecarlo section) that has the 4-momentum configuration of the diquark and

other partonic objects with correct quantum numbers and color connections. Now, we

need to generate the 4-momentum configuration of the outgoing particles. This we do

by ”decaying” the ”diquark object” into the 4 outgoing particles (2 up-quarks and a pair

of opposite sign dileptons). This we do in three steps. First we decay it into a lepton and

a ”lepto-diquark object” and then the lepto-diquark object into a lepton and composite

object which finally decays into 2 up-quarks. In the first step we boost to the rest frame

of the diquark and then assign the momenta of the two decay products with one random

number determining the azimuthal angle φ, which has a uniform flat distribution in

{0, 2π}. Then, we boost back to the lab frame and get the momenta for the first outgoing

lepton. Then, we boost to the rest frame of the lepto-diquark object and assign the

momenta of the two decay products with two random numbers (one φ as before and one

for the invariant mass of the composite object with two up quarks randomly sampled

from 0 to mL). Then, we boost back to the lab frame to get the momenta for the second

outgoing lepton. Finally, we boost to the rest frame of the compisite object with two up

quarks and with one random number for φ we assign the momenta for the two outgoing

up quarks. Again, we boost back to the lab frame and get the momenta of the two up

quarks. Thus, at the end we have the momenta configuration for the four outgoing

particles in the lab frame.

44

3.3 Results

To test the code, we generated plots for the invariant mass distribution of the

four outgoing particles for various diquark and lepto-diquark masses, which shows a

Breit-Wigner shape with the correct width. These plots are shown in Figure 3-1 and 3-2.

Figure 3-1 corrspond to a lepto-diquark of mass 100 GeV and figure 3-2 corresponds to

a lepto-diquark of mass 300 GeV.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 100 200 300 400 500 600 700 800 900 1000

(1/σ

)dσ/

dMllq

q

Mllqq (GeV)

MDiquark=350 GeVMDiquark=400 GeVMDiquark=500 GeVMDiquark=600 GeVMDiquark=700 GeVMDiquark=800 GeV

Figure 3-1. Invariant mass distribution of the four outgoing particles with the mass of thelepto-diquark fixed at 100 GeV.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 100 200 300 400 500 600 700 800 900 1000

(1/σ

)dσ/

dMllq

q

Mllqq (GeV)

MDiquark=350 GeVMDiquark=400 GeVMDiquark=500 GeVMDiquark=600 GeVMDiquark=700 GeVMDiquark=800 GeV

Figure 3-2. Invariant mass distribution of the four outgoing particles with the mass of thelepto-diquark fixed at 300 GeV.

45

With the distribution for our Monte-Carlo sampling optimized for the given process,

the efficiency turned out to be 0.3, which means to generate 1000 events the subroutine

has to try around merely 3400 times, which can save a lot of computing time.

46

CHAPTER 4SAME-SIGN DILEPTONS

4.1 Interpretation of Experimental Results

When experimental collaborations do an analysis, they publish the results of their

search i.e. whether the theoretical model they consider for their analysis can be ruled

out or not. To make such a claim they need to estimate the number of background

events that is predicted by existing verified theories and compare it to the predicted

number of signal events from the theoretical model that pass their analysis cuts. This

limit on the maximum number of signal events (Nsig) can be used for other models

having similar signature as the one considered by the collaboration. A theorist can

compute the number of signal events from his/her theoretical model passing the same

cuts as in the experiment and if the number of such events is larger than the limit Nsig

given by the experiment then the model can be ruled out.

4.2 Calculating Nsig Theoretically

The number of signal events depend on the cross-section of the particles produced

at parton level collision, their branching ratios into the final products, the integrated

luminosity and the efficiency with which the events can pass the cuts is given in the

following relationship.

Nsig = σ(Pi)× BR(Pi)× �(Pi)× L, (4–1)

where, σ(Pi) is the parton level production cross-section, BR(Pi) is the branching

ratio of the particles produced at parton level into the final decay products, �(Pi) is

the efficiency for the event to pass the cuts, L is the integrated luminosity and Pi ’s are

the various model parameters (e.g. m0, m 12, tan(β) etc. for msugra). Once a model

is chosen we can calculate the masses of all the particles involved in the process

(Mi ’s) and express Nsig in terms of those relevant masses. L is the same luminosity

that the experimentalists used for their analysis. σ(Mi) and BR(Mi) can be calculated

47

either analytically from the lagrangian or using an event generator like CalcHEP or

Pythia. �(Mi) is harder to compute analytically, since, 1) there is no analytical formula

for Monte-Carlo integration required to generate events and 2) detector simulation of

the final particles into reconstucted objects involves convoluted integrations. When we

generate events using a Monte-Carlo event generator, we get the objects with their true

momentum (ptrue and true forward/backward angle θtrue). But we are interested in finding

their observed values at the detector (pobs and θobs). Thus we are looking for transfer

functions P(pobs |ptrue) and P(θobs |θtrue) such that for a particle with true momentum ptrue ,

P(pobs |ptrue) is the probability distribution of observing pobs in the detector, such that,∫ ∞0

dptrueP(pobs |ptrue) = 1 (4–2)∫ ∞0

dθtrueP(θobs |θtrue) = 1. (4–3)

Hence, the observed matrix element for the process relates to the true matrix element,

as,

|M(pobs , θobs)|2 =∫ ∞0

dptrue

∫ ∞0

dθtrue |M(ptrue, θtrue)|2P(pobs |ptrue)P(θobs |θtrue). (4–4)

Thus, with the relevant cuts on θ and pT , we’ll have to solve the following convoluted

integral.

Nsig = L ×∫ |θcut |−|θcut |

dpobs

∫ ∞pT√1−cos2θ

dθobs |M(pobs , θobs)|2. (4–5)

As, we don’t have analytical formulae for the transfer functions, the above integral has to

be computed numerically using Monte-Carlo which consumes a lot of CPU time.

4.3 Simplified Models

A simplified model is defined by an effective Lagrangian describing the interactions

of a small number of new particles [17]. We use a method such that we can divide

our analysis into model dependent and model independent components. The model

dependent part governs the production cross-section of the Beyond Standard Model

48

Figure 4-1. The parameter space

particles and the branching fractions for their decay into stable particles. The model

independent part depends on the topology and governs the efficiency of such an event

passing the analysis cuts. We start by describing the steps of our analysis, begining with

the model dependent part. For our analysis we chose (details in next sub-section):

• search channel: SS2L analysis by CMS

• production channel: pp → g̃g̃ → χ̃+1 χ̃+1 (χ̃−1 χ̃−1 )4j → l+l+(l−l−)4j2ν2χ̃01

Because of such choices, our bounds are conservative, since there usually are, extra

production subprocesses and extra decay channels, both of which will result in an

increase in signal. Next, we acquire the cuts and the number of background events for

the search channel from the experimental paper, which in our case is [18]. Then we

run the event generator and detector simulation in the chosen parameter space. We

chose our parameter space to be (M1,M2,M3) 4-1, where M1, M2 and M3 are the bino,

wino and gluino mass parameters. For M1 = 10 GeV, 100 GeV, 200 GeV, 300 GeV and

400 GeV, we varied M2 and M3 such that M1 < M2 < M3

(� = NcutNtot). Finally we calculate and plot the reach using the following expression

(σ.BR)max (Mi) =95% CL UL yield

L.�(Mi), (4–6)

as a function of the gaugino mass parameters Mi (i =1, 2, 3), where, 95% CL UL yield

represents observed upper limit on event yields from new physics (from the experimental

analysis paper) and L is the integrated luminosity. In our case, L was taken to be 1fb−1

and 95% CL UL yield was taken from the CMS analysis notes [18].

4.4 Simplified Model: An Illustration

For our illustration, we choose to study two same-sign isolated leptons (2SSL)

search. Within a simplified model approach, we choose three relevant particles, G̃ , W̃±,

and Ñ0. G̃ is a SU(3) gauge partner. For example, G̃ can be a gluino in the MSSM,

or excited KK-gluon in MUED. W̃± is the partner of SU(2) and Ñ0 is of U(1). This is

a minimal set up preserving the gauge structure of the standard model. Thus G̃ is a

new colored particles of SU(3) octet, has a transition to W̃± with two jets. Compared to

SU(3) triplet pair production, this octet pair production is usually preferable due to the

gluion fusions compared to single diagram from flavor preserving interaction of triplet

pair production. A transition of G̃ → W̃± will be a three body decay process.

Transition between W̃± and Ñ0 will give a charged lepton and neutrino if none of

W̃± and Ñ0 will carry a lepton number which is common in most BSMs. This transition

will have two modes depending on the avaiable phase space. For example, if the mass

difference between W̃± and Ñ0 is smaller than the mass of the standard modelW±

gauge boson, then charged leptons will be produced through the three-body decay. On

the other hand, when the phase space is avaiable for an onshellW± boson, then the

W̃± will decay intoW± and Ñ0, followed by a subsequent decay of theW± boson. As

for the branching ratio ofW± into one lepton and neutrino from these two processes, it

will be model dependent. In short, our setup is following:

50

Figure 4-2. Diagramatic description of our simplified model setup for two same signedlepton signal.

G̃two jets−−−−→ W̃± : Three body, (4–7)

W̃±l , ν−→ Ñ0 :

Three body if ∆M < MW±,

ThroughW± if ∆M > MW±,(4–8)

where,

∆M = MW̃± −MÑ0. (4–9)

This procedure is diagrammatically expressed in FIG 4-2. Due to the nature of

“simplified model” of this approach, it would be implemented easily in various new

physics models. In the next sub-section, we breifly explain how this set up can be

derived from the MSSM.

4.5 Model-Independent Procedure

We have applied CMS same-sign dilepton analysis (2SSL) [18] to the simplified

model, which in turn can be applicable to any new physics which has 2SSL signatures at

the LHC. From the various cuts experimetalists apply on their analysis, we can get the

efficiencies of new physics with respect to the relevant parameters. This procedure will

depend on the event topology which resembles of feynman diagram. CMS analyisis for

2SSL.

51

The muon and electron candidates must have pT > 5 and 10 GeV respectively, and

|η| < 2.4. Tau candidates are excluded. Events with 2 or more jets with pT > 40 GeV are

selected. The preliminary cuts for object acceptance were defined as follows,

• |η|µ/e/j < 2.4,

• pT µ > 5 GeV,

• pT e > 10 GeV,

• pT j > 40 GeV.

The choice of the selection cuts need the following considerations.

• The efficiency of event selection depends on the cuts employed.

• The amount of background supression also depends on the cuts.

• A set of cuts might be better in one region in the parameter space and another setof cuts might be better elsewhere.

Thus, it’s important to do an analysis with a variety of cuts and pick different cuts for

exploring different regions in the parameter space. Seven selection criteria (called the

seven search regions) are employed by the CMS collaboration. Since we were following

their analysis (with their numbers for background estimation) we had to stick to their

search regions. The two baseline selections used are:

• inclusive dileptons: events with a pair of same sign dilepton candidate and HT >200 GeV;

• high − pT dileptons: events with a pair of same sign dilepton candidate with boththe leptons having pT > 10 GeV, and at least one lepton having pT > 20 GeV.

Further division into seven search regions and the corresponding cuts are listed in Table

4-1.

4.5.1 Model-Independent Procedure: Fast Simulation

We used Pythia [7] for event generation and PGS [19] for detector simulation. Since,

PGS simulates only a portion of the detector behavior it is not a full simulation. At the

same time, as it neglects some of the other real detector effects (like the magnetic fields)

52

Table 4-1. The seven search regions.Baseline Region Our notation min. HT min. EmissT 95% CL UL

HT EmissT (GeV) (GeV) (events)

high high I/400/120 400 120 3.7Inclusive high low I/400/ 50 400 50 8.9

medium high I/200/120 200 120 7.3high high H/400/120 400 120 3.0

high pT high low H/400/ 50 400 50 7.5medium high H/200/120 200 120 5.2

low low H/ 80/120 80 120 6.0

that demand a lot of computing power, it is much faster than a full detector simulation.

For this reason it is called “Fast Sim”.

The variable model parameters of interest were M1, M2 and M3. For 5 separate

values of M1, namely, 10, 100, 200, 300 and 400 GeV, we evenly sampled M2 and M3

in increments of 10 GeV with M1 > M2 > M3. For each such point we generated

and simulated 10000 events. As we were only interested in decays from gluinos, only

sub-processes 243 and 244 (gluino production) were turned on. The fixed model

parameters were. µ=700 GeV, tan β=10, msleptons ,msquarks=700 GeV and trilinear coupling

parameters RMSS(15)=800, RMSS(16)=800 and RMSS(17)=0. IMSS(3) was set to

1, to make gluino pole mass to be same as M3. ISR, FSR, multiple interactions, and

fragmentation and decay were turned on (default configuration). We were interested

in the decay from gluino to chargino. So, only decays of gluinos into positively charged

charginos were allowed. Each of the chargino was then allowed to decay either into

a W boson and the neutralino-LSP (2 body) or a lepton (e, µ), a neutrino and the

neutralino-LSP (3-body). If it decays to give a W-boson, we allow the W boson to decay

in to a lepton (e, µ) and a neutrino. Then the branching ratio for 2 gluinos to decay into 2

same-sign dileptons (e, µ),

BR(g̃g̃→ss2l) = 2×

[BR(g̃→χ̃+1 +2j) ×

(BR(χ̃+1→χ̃01+e+/µ++νe/µ)

+BR(χ̃+1→χ̃01+W+) × BR(W+→e+/µ++νe/µ)

)]2. (4–10)

53

In the expression above, the prefactor 2 comes from the fact that we need to account for

the negatively charged charginos that we had neglected. The squaring is done to take

into account the decays from both the gluinos produced at parton level.

0

0.05

0.1

0.15

0.2

0.25

0.3

2M50 100 150 200 250 300 350 400 450 500

3M

50

100

150

200

250

300

350

400

450

500

∈

Region=L/400/120M1=10 GeV

∈

0

0.05

0.1

0.15

0.2

0.25

0.3

2M50 100 150 200 250 300 350 400 450 500

3M

50

100

150

200

250

300

350

400

450

500

∈


∈

0

0.05

0.1

0.15

0.2

0.25

0.3

2M50 100 150 200 250 300 350 400 450 500

3M

50

100

150

200

250

300

350

400

450

500

∈


∈

0

0.05

0.1

0.15

0.2

0.25

0.3

2M50 100 150 200 250 300 350 400 450 500

3M

50

100

150

200

250

300

350

400

450

500

∈

Region=H/400/120M1=10 GeV

∈

0

0.05

0.1

0.15

0.2

0.25

0.3

2M50 100 150 200 250 300 350 400 450 500

3M

50

100

150

200

250

300

350

400

450

500

∈


∈

0

0.05

0.1

0.15

0.2

0.25

0.3

2M50 100 150 200 250 300 350 400 450 500

3M

50

100

150

200

250

300

350

400

450

500

∈


∈

0

0.05

0.1

0.15

0.2

0.25

0.3

2M50 100 150 200 250 300 350 400 450 500

3M

50

100

150

200

250

300

350

400

450

500

∈


∈

Figure 4-3. PGS: Efficiencies for the seven search regions for M1 = 10 GeV.

In Figures (4-3,4-4,4-5,4-6, B-1,B-2,B-3,B-4), the seven plots correspond to the

seven search regions. The x-axis represents the chargino mass, the y-axis represents

the gluino mass and the z-axis (color coded) represents the efficiency of an event

passing the cuts for a given x and y value. A higher value for efficiency is desirable.

In each of the plots, we can categorize the parameter space into regions in two ways,

namely,

Low vs high gluino mass (M3) for a fixed chargino mass (M2): As gluino mass

increases, the splitting between the gluino and chargino mass increases. Since, in our

process of interest, the gluino decays into a chargino and a quark. If the mass splitting

is big then the quarks are more likely to have a higher pT , and thus pass the cuts on jet

pT and HT . This we can see in the plots by looking at a fixed value of M2 on the x-axis

54

0

0.05

0.1

0.15

0.2

0.25

0.3

2M50 100 150 200 250 300 350 400 450 500

3M

50

100

150

200

250

300

350

400

450

500

∈


∈

0

0.05

0.1

0.15

0.2

0.25

0.3

2M50 100 150 200 250 300 350 400 450 500

3M

50

100

150

200

250

300

350

400

450

500

∈


∈

0

0.05

0.1

0.15

0.2

0.25

0.3

2M50 100 150 200 250 300 350 400 450 500

3M

50

100

150

200

250

300

350

400

450

500

∈


∈

0

0.05

0.1

0.15

0.2

0.25

ufdc image array 2 - c 2012 gaurab k....

Documents