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Electronic Theses, Treatises and Dissertations The Graduate School

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Direct Photon Production In Association With AHeavy QuarkTzvetalina P. StavrevaFlorida State University

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Recommended CitationStavreva, Tzvetalina P., "Direct Photon Production In Association With A Heavy Quark" (2009). Electronic Theses, Treatises andDissertations. Paper 1572.

FLORIDA STATE UNIVERSITY

COLLEGE OF ARTS AND SCIENCES

DIRECT PHOTON PRODUCTION IN ASSOCIATION WITH A HEAVY

QUARK

By

TZVETALINA P. STAVREVA

A Dissertation submitted to theDepartment of Physics

in partial fulfillment of therequirements for the degree of

Doctor of Philosophy

Degree Awarded:Spring Semester, 2009

The members of the Committee approve the Dissertation of Tzvetalina P. Stavreva defended

on April 3, 2009.

Joseph F. Owens IIIProfessor Directing Dissertation

William DewarOutside Committee Member

Laura ReinaCommittee Member

Horst WahlCommittee Member

Marcia FenleyCommittee Member

The Graduate School has verified and approved the above named committee members.

ii

To mum and dad

iii

ACKNOWLEDGEMENTS

I would like to thank my parents for showing me how captivating science can be at an

early age, and also for giving me all the support and encouragement that I needed.

I would also like to thank my friends, for being there for me, whenever I needed a helping

hand, and for creating a home away from home. I would especially like to thank Cecily

Oakley for being a wonderful friend, and of course for her amazing cupcakes that would

provide a good start to the working day, Christelle Castet for being a great roommate and

friend, and someone who was always there to have an adventure with. As well as that I

am thankful to my fellow high energy graduate students, Jose Lazoflores, Edgar Carrera,

Benjamin Thayer, Paulo Rottmann, and Dan Duggan (especially for providing the so needed

experimental results) with all of whom it was great to share an office, as well as many

meaningful (and some not so meaningful, but nonetheless fun) discussions all these years,

also I am indebted to Fernando Febres Cordero and Bryan Field for sharing their knowledge

with me and helping me come to grasps with the subtleties of QCD. And a very special thank

you to Shibi Raj, for being helpful and supportive, and for putting things in perspective, at

times when I needed inspiration.

I want to thank the High Energy Physics Group, Faculty and Staff, for creating a very

pleasant environment in which to work, and also my committee members, with a special

thanks to Laura Reina and Horst Wahl. I am particularly grateful to my advisor Jeff Owens,

first for giving me the opportunity to come back to physics and fulfill one of my goals, to

study high energy physics, and of course for leading me through the intriguing terrain of

next-to-leading-order QCD calculations.

— Tzvetalina

iv

TABLE OF CONTENTS

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. NLO CALCULATION OF THE DIRECT PHOTON AND HEAVY QUARKCROSS SECTION AT HADRON COLLIDERS . . . . . . . . . . . . . . . . . 82.1 Basic Overview of a Hadronic Cross Section Calculation . . . . . . . . . 92.2 Numerical Methods - Phase Space Slicing Method . . . . . . . . . . . . . 122.3 γ+Q calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.1 LO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.2 NLO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Stability of the Calculation . . . . . . . . . . . . . . . . . . . . . . . . . 32

3. RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.1 Tevatron Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.1.1 Photon Isolation and NLO Fragmentation . . . . . . . . . . . . . . 463.1.2 Intrinsic Charm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2 LHC Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4. COMPARISON TO DATA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5. MASSIVE VERSUS MASSLESS COMPARISON . . . . . . . . . . . . . . . . 715.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

A. PARTON DISTRIBUTION FUNCTIONS . . . . . . . . . . . . . . . . . . . . 80

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

v

B. PHASE SPACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

C. 2 → 3 NLO MASSLESS MATRIX ELEMENTS . . . . . . . . . . . . . . . . 85

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

vi

LIST OF TABLES

1.1 A list of the particles comprising the SM. . . . . . . . . . . . . . . . . . . . . . 2

2.1 A list of all 2 → 2 LO hard-scattering subprocesses with at least one heavy quark,Q, in the final state (where Q, can be either a heavy quark or an antiquark, whereappropriate) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 a list of all 2 → 3 NLO hard-scattering subprocesses . . . . . . . . . . . . . . . . 21

2.3 A list of all 2 → 3 NLO hard-scattering subprocesses, O(α3s), with at least one

heavy quark, Q, in the final state (where Q, can be either a heavy quark or anantiquark, where appropriate) . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.1 Subprocesses that contribute to the LO massive calculation . . . . . . . . . . . 72

A.1 A summary of the power of logs resummed, dependent on the order of the splittingfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

vii

LIST OF FIGURES

2.1 Self Energy of the Photon - an example of a loop diagram . . . . . . . . . . . . . 12

2.2 Compton Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 An example of Leading Order Fragmentation Contributions 1) gg → QQ, with thephoton fragmenting from one of the final state heavy quarks 2) gQ → gQ, with thephoton fragmenting off from the final state gluon . . . . . . . . . . . . . . . . . 18

2.4 Virtual Feynman diagrams for the Compton subprocess . . . . . . . . . . . . . . 19

2.5 Example Feynman diagrams for all the possible real subprocesses . . . . . . . . . 21

2.6 Initial collinear singularities for gQ → γgQ, the final state gluon is collinear to theinitial state heavy quark, or the final state gluon is collinear to the initial state gluon. 23

2.7 Feynman diagrams for the subprocess qQ → γQq. . . . . . . . . . . . . . . . . . 27

2.8 Feynman diagrams for the subprocess qq → γQQ, diagrams 1) and 2) are s-channeldiagrams, and diagrams 3) and 4) are t-channel diagrams. . . . . . . . . . . . . 29

2.9 Feynman diagram for the subprocess gg → QQg shown in diagram 1). A photonbeing produced by fragmentation from one of the final states for this subprocess,diagram 2) from the final state quark, diagram 3) from the final state gluon, anddiagram 4) from the final state anti-quark . . . . . . . . . . . . . . . . . . . . . 32

2.10 δs dependence of σNLO(pp → γbX), at√

S = 1.96 TeV, for δc = 5 × 10−5 whenδs > 0.001 , and δc = 1 × 10−5, when δs < 0.001 . The upper plot shows thedependence on δs of the two-body (dashed line), and three-body (dot dashed line)contributions, and also the sum of the two (solid line). The lower plot showsthe dependence on the full NLO cross section, with the statistical error included,µ = µr = µf = µF , µ = pTγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.11 δc dependence of σNLO(pp → γbX), at√

S = 1.96 TeV, for δs = 0.01. The upperplot shows the dependence on δc of the two-body (dashed line), and three-body(dot dashed line) contributions, and also the sum of the two (solid line). The lowerplot shows the dependence on the full NLO cross section, with the statistical errorincluded, µ = µr = µf = µF , µ = pTγ . . . . . . . . . . . . . . . . . . . . . . . . 34

viii

3.1 The LO differential cross section at the Tevatron, dσ/dpTγ for the production ofa direct photon and a bottom quark as a function of pTγ for

√S = 1.96 TeV at

LO, with the scale dependence shown, where the three different scales have beenset to be equal µ = µr = µf = µF , µ = pTγ (solid line), µ = pTγ/2 (dashed line),µ = 2pTγ (dotted line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 The differential cross section at the Tevatron, dσ/dpTγ for the production of a directphoton and a bottom quark as a function of pTγ for

√S = 1.96 TeV, at NLO (solid

line), and at LO (dashed line), µ = pTγ . . . . . . . . . . . . . . . . . . . . . . 40

3.3 The differential cross section at the Tevatron, dσ/dpTγ for the production of a directphoton and a charm quark as a function of pTγ for

√S = 1.96 TeV, at NLO (solid

line), and at LO (dashed line), µ = pTγ . . . . . . . . . . . . . . . . . . . . . . 41

3.4 Contributions of the different subprocesses to the differential cross section, NLO(solid line), annihilation qq → γQQ (dashed line), Qq → γqQ, and Qq →γqQ(dotted line), gQ → γgQ (dot dashed line), gg → γQQ+LO (dash dot dottedline), QQ → γQQ, and QQ → γQQ (dot dash dotted line), µ = pTγ . . . . . . . . 42

3.5 A comparison between the differential cross sections, dσ/dpTγ for the productionof a direct photon and a bottom quark, and that of a direct photon plus a charmquark at NLO and LO, charm at NLO (solid line), bottom at NLO (dashed line),charm at LO (dot dashed line), bottom at LO (dotted line), µ = pTγ . . . . . . . . 43

3.6 The ratio of the charm and bottom differential cross sections versus pTγ , at NLO(solid line) and at LO (dashed line), µ = pTγ . . . . . . . . . . . . . . . . . . . . 44

3.7 Scale dependence of the NLO differential cross section, dσ/dpTγ for the productionof a direct photon and a bottom quark, where the three different scales have beenset to be equal µ = µr = µf = µF , µ = pTγ (solid line), µ = pTγ/2 (dashed line),µ = 2pTγ (dotted line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.8 Comparison between the differential cross section, dσ/dpTγ without isolation re-quirements and with them, no isolation (solid line) , isolation (dashed line), µ = pTγ . 47

3.9 Ratio between the differential cross section dσ/dpTγ , with NLO fragmentationcontribution included and the differential cross section with just LO fragmentationincluded, no isolation required (solid line), isolation (dashed line), µ = pTγ . . . . . 48

3.10 Comparison between the three different charm PDFs at scale Q = 40 GeV,CTEQ6.6M (solid line), BHPS or CTEQ6.6C2 (dashed line), sea-like or CTEQ6.6C4(dotted line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.11 The differential cross section, dσ/dpTγ , for the production of a direct photon anda charm quark for the three different PDF cases, CTEQ6.6M (solid line), BHPS orCTEQ6.6C2 (dashed line), sea-like or CTEQ6.6C4 (dotted line), µ = pTγ . . . . . 51

ix

3.12 The differential cross section versus the transverse momentum of the photondσ/dpTγ for the production of a direct photon and a bottom quark at LHC centerof mass energies,

√S = 14 TeV, NLO (solid line), LO (dashed line), µ = pTγ . . . . 52

3.13 K factor, or the ratio of the NLO to the LO differential cross section forpp → bγX at

√S = 14 TeV, µ = pTγ. . . . . . . . . . . . . . . . . . . . . . . 53

3.14 Contributions of the different subprocesses to the differential cross section, dσ/dpTγ

for pp → bγX at√

S = 14 TeV, NLO (solid line), annihilation qq → γQQ (dashedline), Qq → γqQ, and Qq → γqQ (dotted line), gQ → γgQ (dot dashed line),gg → γQQ+LO (dash dot dotted line), QQ → γQQ, and QQ → γQQ (dot dashdotted line), µ = pTγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.15 Scale dependence of the differential cross section, dσ/dpTγ for the production of adirect photon and a bottom quark at the LHC, where the three different scales havebeen set to be equal µ = µr = µf = µF ,for the NLO cross section µ = pTγ (solidline), µ = 2pTγ (dashed line), µ = pTγ/2 (dotted line) and for the LO cross sectionµ = pTγ (dot dashed line), µ = 2pTγ (dashed dot dot line), µ = pTγ/2 (dasheddashed dot line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1 The differential cross section, dσ/dpTγ , for the production of a direct photon and ab quark, γ + b. In the graph to the left, the solid line is the NLO theory curve andthe circular dots are the data points measured by the DØ collaboration, for regionone. In the graph to the right, the dashed line is the NLO theory curve and thesquares are the data points measured by the DØ collaboration, for region one. . . 58

4.2 The differential cross section, dσ/dpTγ , for the production of a direct photon and ab quark, γ + c. In the graph to the left, the solid line is the NLO theory curve andthe circular dots are the data points measured by the DØ collaboration, for regionone. In the graph to the right, the dashed line is the NLO theory curve and thesquares are the data points measured by the DØ collaboration, for region one. . . 59

4.3 Scale dependence of the differential cross section, dσ/dpTγ , for the production of adirect photon and a c quark, γ +c, compared to data (circular dots), for region one.The three different scales have been set to be equal µ = µr = µf = µF , µ = pTγ

(solid line), µ = pTγ/2 (dashed line), µ = 2pTγ (dotted line). . . . . . . . . . . . 61

4.4 Scale dependence of the differential cross section, dσ/dpTγ , for the production of adirect photon and a c quark, γ +c, compared to data (circular dots), for region two.The three different scales have been set to be equal µ = µr = µf = µF , µ = pTγ

(solid line), µ = pTγ/2 (dashed line), µ = 2pTγ (dotted line). . . . . . . . . . . . 62

4.5 The NLO differential cross section, dσ/dpTγ , for the production of a direct photonand a c quark, γ +c, with the use of CTEQ6.6M PDFs (solid line), using the BHPSintrinsic charm PDF, (dashed line) and with the use of the sea-like model intrinsiccharm PDF (dotted line), compared to data (circular dots), region one. . . . . . . 63

x

4.6 The NLO differential cross section, dσ/dpTγ , for the production of a direct photonand a c quark, γ +c, with the use of CTEQ6.6M PDFs (solid line), using the BHPSintrinsic charm PDF, (dashed line) and with the use of the sea-like model intrinsiccharm PDF (dotted line), compared to data (circular dots), region two. . . . . . . 64

4.7 Scale dependence of the NLO differential cross section, dσ/dpTγ , for the productionof a direct photon and a c quark, γ + c, with the use of the BHPS PDFs, µ = pTγ

(dashed line), µ = pTγ/2 (dotted line), compared to data (squares), region two. . . 65

4.8 Scale dependence of the differential cross section, dσ/dpTγ , for the production of adirect photon and a b quark, γ+b, compared to data (circular dots), for region one.The three different scales have been set to be equal µ = µr = µf = µF , µ = pTγ

(solid line), µ = pTγ/2 (dashed line), µ = 2pTγ (dotted line). . . . . . . . . . . . 67

4.9 Scale dependence of the differential cross section, dσ/dpTγ , for the production of adirect photon and a b quark, γ+b, compared to data (circular dots), for region two.The three different scales have been set to be equal µ = µr = µf = µF , µ = pTγ

(solid line), µ = pTγ/2 (dashed line), µ = 2pTγ (dotted line). . . . . . . . . . . . 68

4.10 The NLO differential cross section, dσ/dpTγ , for the production of a direct photonand a b quark, γ + b, with the use of CTEQ6.6M PDFs (solid line), using theBHPS intrinsic charm PDF, (dashed line) and with the use of the sea-like modelintrinsic charm PDF (dotted line), compared to data (circular dots), region one,µ = µr = µf = µF , µ = pTγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.11 The NLO differential cross section, dσ/dpTγ , for the production of a direct photonand a b quark, γ + b, with the use of CTEQ6.6M PDFs (solid line), using theBHPS intrinsic charm PDF, (dashed line) and with the use of the sea-like modelintrinsic charm PDF (dotted line), compared to data (circular dots), region two,µ = µr = µf = µF , µ = pTγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.1 Feynman diagrams for the gg → γQQ subprocess. In diagram 1) the two initialgluons go into a pair of heavy quarks and the photon is emitted for the final stateheavy quark. In diagram 2) one of the initial state gluons splits into a heavy quarkanti quark pair. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2 The gg → γQQ subprocess, in its collinear limit can be written as the Comptonsubprocess times a splitting function. . . . . . . . . . . . . . . . . . . . . . . . 74

5.3 The differential cross section, dσ/dpTγ for the production of a direct photon and abottom quark as a function of pTγ for

√S = 1.96 TeV, at massless NLO (solid line),

at massless LO (dashed line), and at massive LO (dotted line), µ = µr = µf = µF ,µ = pTγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

xi

5.4 The differential cross section, dσ/dpTγ for the production of a direct photon and abottom quark as a function of pTγ for

√S = 14 TeV, at massless NLO (solid line),

at massless LO (dashed line), and at massive LO (dotted line), µ = µr = µf = µF ,µ = pTγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

xii

ABSTRACT

In this Thesis we present the Next-To-Leading-Order calculation, O(αα2s), of the inclusive

cross section for a photon and a heavy quark (charm or bottom), pp/pp → γ + Q + X,

(Q = c, b) at hadron colliders. We include fragmentation effects through the Next-

To-Leading-Order. This calculation is performed with the use of a phase space slicing

technique so that the effects of experimental cuts can be easily included. We study in

detail the characteristics of this process at both the Tevatron and the LHC. Results for

the ratios of the charm and bottom cross sections are presented and the systematics of the

various subprocesses are compared and contrasted. The theory predictions are compared to

experimental measurements from the DØ collaboration at Fermilab. A brief overview of the

LO massive calculation is also presented, and compared to the NLO massless case.

We predict that the investigation of this process and our results will be relevant in the

study of heavy quark PDFs at the LHC.

xiii

CHAPTER 1

INTRODUCTION

The idea that the world is composed of indivisible building blocks has been around since

antiquity. The ancient Greeks used the term atom, which means uncuttable, to refer to

these building blocks. This word ended up as somewhat of a misnomer, but the idea stood

the tests of time and experimental observation. And today we still believe that the universe

comprises indivisible building blocks, which we refer to as elementary particles. To describe

the fundamental interactions of these particles, a theory known as the Standard Model (SM)

was developed [1, 2, 3, 4]. It is a quantum field theory (QFT) that incorporates three of the

four known forces: the strong, the weak and the electromagnetic ones. The SM comprises

the particles shown in Table 1.1. They can be divided into two types of particles, fermions

which are spin half particles, and bosons which have integral spin. The fermions, i.e., the

quarks and leptons, form three generations, each generation differing from the previous only

by the mass of the particles in it. The fermions interact with each other via the gauge

bosons which are the mediators of the three forces. They include the eight gluons for the

strong force, the photon for the electromagnetic force and the Z and the W± bosons for the

weak force. There is one more particle predicted by the SM known as the Higgs boson. It

is necessary in order for the massive particles in the SM, to acquire their mass. It is the

only particle in the SM that has eluded experimental observation as of yet. Hopes for its

discovery are resting on the new particle collider at CERN, Geneva, [5] known as the Large

Hadron Collider (LHC), which is expected to begin operation in late 2009.

Yet even with the discovery of the Higgs, a few important questions will remain

unanswered by the SM. These range from the Higgs hierarchy problem, to the incorporation

of the fourth force - gravity - in a QFT framework. Also the SM does not explain why there is

more matter than anti-matter, known as the matter anti-matter asymmetry, in the universe.

1

Table 1.1: A list of the particles comprising the SM.

fermions bosonsI II III vector scalar

quarksu c t

gH

d s b

leptonse µ τ γνe νµ ντ Z,W±

As well as that, the SM does not provide particle candidates to what is mysteriously referred

to as dark matter and dark energy, the need for which comes from the fact that the known

baryonic and radiation matter accounts for only four percent of the mass of the universe,

i.e., its mass is largely unaccounted for. In order to be able to answer these questions, and

create a new theory that incorporates the old one and elucidates these queries, one thing is

clear, a detailed understanding and a thorough check of the SM are necessary. It is the aim

of this dissertation to provide a small part to this understanding.

To test the SM and be able to find out more about the substructure of matter, we

look at collisions characterized by large momentum transfers amongst the particles involved.

Such large momentum transfer collisions probe the short distance structure of the colliding

particles. The investigation of these processes requires effort from both the theoretical and

the experimental communities. Experimentally, these processes are observed at particle

accelerators. The currently highest center of mass energy particle collider is the Tevatron

at Fermilab [6], with a center of mass energy,√

S = 1.96 TeV. At the predecessors of

the Tevatron, at the beginning of the 1950’s and 1960’s, a multitude of new particles were

discovered. In order to classify them and bring some sense of order to what was referred

to as the ‘particle zoo’, a SU(3) flavor symmetry scheme was proposed independently by

Gell-Mann and Ne’eman [7, 8], as a means by which to classify the hadronic states, through

the existence of particles named quarks. At the same time the parton model was developed

by Feynman [9], in order to understand the dynamics of high energy particle scattering

processes, with the partons and quarks turning out to be the same particles. As a way

of developing a gauge theory of the interacting colored quarks, Quantum Chromodynamics

(QCD) was proposed, and the combination of QCD and the Electroweak theory was named

2

the SM. The SM has been deemed a very successful theory, as it predicted the existence of

the gluon, the top and charm quark, the tau neutrino and the W± and Z bosons, before they

were observed. The discoveries of these particles along with a number of high precision data

measurements obtained from the Tevatron and the LEP collider at CERN, have confirmed

the SM. However there have been no indications of signals of new physics, and so the last two

decades have been marked by a slow-down of new breakthrough experimental observations.

The LHC is expected to rectify the lack of data that will drive high energy physics in a

direction beyond the SM. It is going to probe energy scales in the TeV range, with a center

of mass energy seven times higher than the one at the Tevatron.

With these long-awaited measurements expected from the LHC in the near future,

calculations in perturbative QCD (pQCD) will prove to be essential in the discovery of new

physics and the discrimination between new physics signals and already observed background

events. As the number of these background events are expected to be much larger than

the searched for signal, it becomes necessary to know them to a high degree of precision.

Furthermore, since there are many predicted possibilities and also unknown physics that can

be discovered a thorough cross-check with the SM is needed.

In order to make theoretical predictions which can be compared to data, a cross section

is calculated. This cross section gives the probability of a certain type of event to occur,

and is computed with the use of perturbative techniques. Since most of the observed SM

events involve the strong interactions, which are described by QCD, it is very fortunate that

we can use perturbation theory to study them. Perturbative techniques are possible to use

in the study of QCD through the existence of asymptotic freedom. Asymptotic freedom is

the term used to describe the fact that at large momentum transfers or short distances the

strong force becomes weaker and we can view the constituents of the hadrons, the quarks and

the gluons (collectively known as partons), as almost free. In Eq. 1.1 the 1-loop leading-log

expression for the dependence of the strong coupling constant αs on the scale, Q, is shown

[10],

αs(Q) =2π

b0 log(Q/Λ), (1.1)

where b0 = (11 − 2/3nf ), nf is the number of quark flavors, and Λ is the mass scale below

which non-perturbative effects take over. From Eq. 1.1 we can clearly see that as the energy

scale Q increases the size of the strong coupling decreases. This is due to the fact that we

3

are dealing with a non-abelian theory, where the gauge bosons, the gluons, can interact with

each other, and, as such, the constant b0 is positive, as long as the number of flavors remains

less than sixteen, nf ≤ 16. Thus we can calculate the cross section as an infinite series in

the expansion of the strong coupling constant. However each following term in the series

becomes more and more difficult to calculate, and we have to limit ourselves to just the first

few terms. Some time ago only the first term of the series or the leading order (LO) term was

necessary in order to have a somewhat reliable prediction for the experimental observations.

But now as the experimental measurements are becoming more and more precise, so too

need be the theoretical calculations. The expected order is now the next-to-leading (NLO)

order, or the second term in the infinite expansion series. And with the measurements of

the LHC, next-to-next-to-leading order (NNLO) calculations are most likely to be needed

for certain processes.

One aspect that complicates a higher precision QCD calculation is the existence of

confinement. Even though due to asymptotic freedom the quarks and gluons can be treated

as free in the high energy processes, there are no free quarks or gluons in nature; they form

bound states in hadrons. The description of the behavior of quarks and gluons inside these

bound states is done through the parton distribution functions (PDFs). The PDFs tell us

the probability of finding a parton inside a hadron with some momentum fraction of the

parent hadron’s momentum, at a given energy scale. In order to do most QCD calculations,

the PDFs of the partons inside the colliding protons or antiprotons need to be known.

However the PDFs’ dependence on the momentum fraction is not calculable and needs to

be measured experimentally. Since confinement can pose somewhat of a challenge in the

experimental study of PDFs, a way to get information about the structure of the hadrons,

and probe directly what goes on inside the large momentum transfer or hard scattering

subprocess is to look at particles unaffected by confinement or the strong interactions. One

such particle is the photon, or the carrier of the electromagnetic force. As a carrier of the

electromagnetic force it couples to the quark’s charge and is blind to the strong force. Since

we want a photon that comes directly from the hard scattering process, and is not produced

after the partons have hadronized, it should be distinguished from a photon that comes from

the decays of hadrons, e.g. such as π0 → γγ, η → γγ or ω → π0γ. We call a photon produced

at the hard scattering level or at the level before hadronization a direct photon.

The study of direct photons is aimed at providing information about the PDFs [11]

4

and also at testing the adequacy of the perturbative techniques used to calculate the hard

scattering subprocesses [12, 13]. It provides a solid testing ground for QCD interactions as the

QFT of the electromagnetic interactions - Quantum Electrodynamics (QED) has been well

tested and understood. Due to the beneficial aspects it provides, direct photon production

has been studied thoroughly since the construction of the SM [14, 15, 16, 17, 18, 19, 20]. It is

the single photon cross section that provides the basic observable for direct photon studies.

The calculation of this cross section involves integrations over the phase space variables of

the accompanying partons, thereby limiting the information which can be obtained about

the underlying subprocesses. Thus the calculation of the direct photon cross section can be

viewed as the sum of many pieces. It is of interest to take this sum apart and investigate

in detail one particular piece of the direct photon calculation. Here we concentrate on such

a piece, namely the study of direct photon production in association with a heavy quark,

where the heavy quark can be either a charm or a bottom quark.

The charm and the bottom quarks were discovered in 1974 [21, 22] and 1977 [23]

respectively. Their discovery offered yet more proof of the validity of the SM and also new

ways of investigating the strong interactions. The charm and bottom quarks have masses

of 1.16 to 1.34 GeV and 4.13 to 4.37 GeV in the MS scheme, correspondingly [24]. They

are known as heavy quarks due to the fact that their masses satisfy mQ Λ, where Λ is

shown in Eq. 1.1, and has a value of about 200 MeV. As a consequence of the larger energy

required to create the heavy quarks, they are produced in less abundance than the light

quarks, namely the up, down and strange ones, at the Tevatron. However at the LHC due

to the higher center of mass energy of 14 TeV they are expected to play a prominent role.

Generally at colliders it is not just the heavy quark meson that is detected, but it is the

heavy quark jet. Typically, jets are made up of hundreds of particles almost all of which

contain light quarks. A jet containing a heavy quark provides quite a distinctive tag, due to

the longer lifetimes of the charm and bottom quarks. They are distinguished through the

secondary vertex formed as a result of this longer decay time.

The combined study of direct photons and heavy quarks can be used as a further test of

the validity of perturbative QCD and as a way to learn more of the heavy quark’s role in

the nucleon. Furthermore final states involving electroweak gauge bosons and heavy quarks

can be components of signals of new physics [25]. From this study we can test the validity

of the approximations we use in our QCD calculations. We will also be able to learn more

5

about the structure of the proton and antiproton by being able to constrain more precisely

the distribution functions of the heavy quarks. Furthermore, an understanding of photon

plus heavy quark production will help us be prepared to identify possible signatures of new

physics.

In this dissertation we have calculated the inclusive cross section for the production of a

photon in association with a heavy quark, pp/pp → γ + Q + X. There have been previous

studies of this process. In Ref. [26, 27], the NLO cross section is presented for the production

of a direct photon and a charm quark. However photon fragmentation effects, which are

needed for a complete NLO calculation were included at lowest order. The fragmentation

effects occur since a direct photon can be produced not only at the hard scattering level,

but through fragmenting from a quark or a gluon. These fragmentation effects are described

by the photon fragmentation functions (FFs), which give the probability for a photon to

originate from a quark or a gluon, with a certain momentum fraction of the quark’s or

gluon’s momenta. We need to introduce the photon FFs in order to take care of collinear

singularities occurring in the case when a photon is emitted collinearly to a final state quark.

Here we extend these previous efforts by presenting a complete NLO cross section for the

first time.

We also investigate the direct photon plus bottom cross section, and compare it with

the direct photon and charm cross section. Results for this process at the Tevatron and

at the LHC have been obtained. This calculation has been done in the limit of a massless

approximation, i.e., both the charm and bottom quarks are treated as massless. In order to

be able to work in the massless approximation the heavy quarks and photons produced need

to carry a transverse momentum, pT , which is a few times larger than the mass of the heavy

quark mQ, i.e. pT ≥ 10 GeV. Since the lower bounds for the values of the transverse momenta

for direct photons and heavy quarks measurable at both the DØ and CDF collaborations at

Fermilab are above pT ≥ 10 GeV, a comparison with a massless calculation is appropriate.

Another important reason for working in the massless approximation is the fact that currently

the most abundantly used PDFs and FFs both treat the heavy quarks as massless in their

evolution. The calculation was performed with the use of the two cutoff phase space slicing

technique [28], so that the effects of experimental cuts can be included. The inclusion of

isolation requirements in the theoretical calculation is also eased by this method. Isolation

requirements need to be imposed in the experimental observation of a direct photon, in

6

order to acquire a reliable measurement of the photon’s energy, this imposition also helps

ease the differentiation of direct photons from those photons coming from the decays of

hadrons. By isolating the photon we require that it is not surrounded by a large amount

of hadronic energy, thereby reducing the background of secondary photons. This will affect

direct photons produced via fragmentation as they are produced in close proximity to the

partons from which they have been radiated.

The dissertation is ordered as follows: in Chapter 2 a description of the theory and

techniques for the calculation are outlined. In Chapter 3 results for the differential cross

section are shown. Predictions for what process at both the Tevatron and LHC are presented

and a comparison between the differential cross sections is given. The effects of including

the NLO fragmentation terms are shown, as well as the effect of the use of different charm

PDFs on the cross section. In Chapter 4 we present the comparison between the theoretical

calculation and data from DØ for this process. In Chapter 5 we briefly outline the differences

between a massive and massless theoretical calculation, and in Chapter 6 we summarize and

conclude our findings.

7

CHAPTER 2

NLO CALCULATION OF THE DIRECT PHOTON

AND HEAVY QUARK CROSS SECTION AT

HADRON COLLIDERS

In this chapter the basics of a NLO QCD calculation are presented. We give specific details

for the calculation of the inclusive massless cross section for the associated production

of a photon and a heavy quark at NLO, which is O(αα2s) in this case, where α is the

electromagnetic coupling constant. As mentioned in the Introduction, direct photons are

produced alongside many different particles, and here we have concentrated on one particular

combination. The direct photon cross section constitutes a sum over all these different parts.

In order to calculate the cross section for the associated production of a direct photon and

a heavy quark, we can take that sum apart and retain only those parts from the direct

photon cross section that are relevant to this process. Thus, the general outline of the

calculation follows that of the direct photon production computation. We need to remove

the subprocesses that do not contain heavy quarks in their final state, and, as such, contribute

only to the direct photon production case. Also due to the fact that we have two particles

that we are interested in, in the final state we also need to modify how we treat some

singularities with respect to just the direct photon computation. The reasons for this being,

that when collinear singularities arise in the final state they are treated differently depending

on whether the particle is observed or not. Since the calculation becomes quite complex at

NLO, it is done numerically. The numerical method used to perform this calculation is the

two cutoff phase space slicing method (PSS) [28], where the 2 → 3 body phase space is

divided into a soft, a collinear and a hard region, more detail about this method will be

given in Section 2.2.

8

2.1 Basic Overview of a Hadronic Cross SectionCalculation

Any inclusive hadronic cross section σ for the production of a particle C can be written in

the form:

σ(AB → C + X) =

∫dxadxbdzcGa/A(xa, µf )Gb/B(xb, µf )σ(ab → cd..)DC/c(zc, µF ), (2.1)

where A and B are the colliding hadron beams. By inclusive we mean that C can be produced

along side all allowed particles for the process. Since we integrate over their phase space, we

are not concerned with what X is.

There are two different components that go into the calculation of the inclusive hadronic

cross section, in Eq. (2.1). One is calculable via perturbation theory and is referred to

appropriately as short distance physics. The other is concerned with nonperturbative effects,

such as hadronization and is referred to as long distance physics. From the uncertainty

principle, ∆x∆p ≥ ~/2, we can deduce that the short distance physics during the hard

scattering occurs through the exchange of large transverse momentum. The further or

previous emission of collinear particles occurs at lower transverse momentum scales, i.e.,

over longer distances. These long distance processes cannot affect what will happen at the

hard scattering level, and we can freely separate the two regions from each other. This

separation of long distance and short distance physics is proved possible by the factorization

theorems [29, 30].

As is shown in Eq. (2.1) the long distance behavior is separated out into the PDFs:

Ga/A(xa, µf ) and Gb/B(xb, µf ), and the Fragmentation Function (FF): DC/c(zc, µd), both

of which are probability density functions. Thus Ga/A(xa, µf )dxa as mentioned in the

Introduction gives the probability of finding parton a inside hadron A, with a momentum

fraction between xa = pa/PA and xa + dxa of the parent’s momentum PA, where pa is

the parton’s momentum. The FF is similar in nature to the PDF. DC/c(zc, µd)dzc gives

the probability of parton c to fragment (or hadronize) into hadron C, which has a fraction

zc = PC/pc to zc + dzc of the initial parton’s momentum. Both the PDFs and FFs are

universal, i.e., they do not depend on the process at hand. Unfortunately we cannot

compute the PDFs and FFs with the use of perturbation theory, since we need to deal

with energy scales below Λ in order to determine the hadron structure, and at these scales

nonperturbative techniques need to be used. Thus the dependence of the distribution and

9

fragmentation functions on the momentum fraction has to be determined empirically, through

experimental measurements.

Since perturbation theory cannot be used in this energy region, the PDFs dependence

on x and the FFs dependence on z need to be measured experimentally.

In Eq. (2.1), the short distance part of the hadronic cross section is described by the

partonic cross section σ(ab → cd..). It gives the probability for the hard scattering subprocess

to occur and is calculable via perturbation theory, as an infinite series in the strong coupling

constant αs:

σ =∞∑

n=0

αn+is σ(n), (2.2)

where i can be 0, 1, 2, ... depending on the process. At first glance the calculation of σ

in Eq. (2.8) seems, although toilsome, fairly straightforward, especially if we are limiting

ourselves to the calculation of just a few terms (n = 0, 1, 2). However as soon as we go

beyond LO, i.e., n > 0, divergences appear. There are three different types of divergences

that can appear in a higher precision calculation. These are the ultraviolet (UV), the infrared

(IR) and collinear divergences. The possibility that, for example, a boson can split into a

pair of virtual particles, gives rise to the UV divergences in loop diagrams. Since the virtual

particles need not be on shell, the square of their four momentum does not have to equal

their mass squared. Thus, although the energy and momenta at each of the vertices has

to be conserved, the internal loop momenta of the virtual particles pi and pj can take

on any values as long as pi + pj = p, where p is the initial momenta of the boson, Fig.

2.1. This means that the integral over the virtual particles momenta can run to infinity,

which makes it logarithmically divergent. This fact was very puzzling at the time of the

construction of QFTs, as it was not known how to deal with these large quantities. It

slowed down the progress of the development of QFT and it took about 20 years to solve

this conundrum. The solution was the introduction of the concept of renormalization [31]

independently by Feynman [32] via the path integral formulation, and an operator method

developed by Schwinger [33] and Tomonaga [34]. The two methods were proved equivalent

and merged by Dyson [35]. The basic idea behind renormalization is the notion that we can

redefine the fields, and the constants of the theory. We assume that these appear in the

Lagrangian in their bare form, which is divergent, and is different from the experimentally

measured value. As such we can introduce counter terms for the coupling constants of the

10

theory, δZα, and for the wave function of the external fields, δZfield, which will cancel against

the aforementioned UV divergences. In order for this to happen it is convenient to expose

the divergences as poles. This is done with the use of a regulator. Each way of regularizing a

theory should yield the same results, however the length and the ease of the calculation are

different. The current method used in most calculations is called dimensional regularization

(DR) [36]. In DR the number of dimensions is reduced from 4 to d, where d is no longer

an integer, and is usually given by, d = 4− 2ε, with ε being a very small number. Now the

scattering amplitudes are computed in d dimensions where they are no longer divergent, but

where the UV singularities appear as poles in 1/εUV . After this procedure the UV poles are

taken care of with the use of renormalization, and we are free to return back to 4 dimensions.

We should mention that while using DR some dimensionless quantities in 4 dimensions such

as the coupling constant can gain dimension, and g → µεrg, where µr has units of mass.

There is also some ambiguity as to what is absorbed with the use of renormalization once

the poles have been exposed. There are some finite terms that appear alongside the pole in

the calculations. Instead of carrying those terms around we can redefine the constants of

the theory and absorb not only the pole, but also these finite terms. In our calculation the

MS scheme is used, where one absorbs the following combination,

(4π)ε exp(−εγE)1

ε→ 1

ε, (2.3)

with γE denoting the Euler-Mascheroni constant.

DR is very useful as it also exposes the IR and collinear divergences as poles in 1/ε. The

IR divergences occur when the momenta of the virtual particles tend to 0, and also when a

radiated boson’s energy tends to zero. We refer to such bosons as soft. The IR divergences

cancel between these two cases. Collinear divergences arise when two massless particles are

emitted parallel to each other. This means that if we label the four momenta of the two

collinear massless particles as pc1 and pc2, their dot product tends to zero as they become

collinear, since the space angle between them tends to zero:

pc1.pc2 = Ec1Ec2 − ~pc1.~pc2

= Ec1Ec2 − |pc1||pc2|cosθcosθ→1→ Ec1Ec2 − |pc1||pc2| = 0. (2.4)

Once again we can expose the collinear singularity as a pole via DR. Then this singularity

is factored off into the PDF for an initial case collinearity, and into the FF for a final state

11

p p

pi

pj

Figure 2.1: Self Energy of the Photon - an example of a loop diagram

collinearity. However in the collinear case there is one more thing that might upset the

stability of the calculation. This can be due to logarithms of the form log(Q/Λ), which

can occur through the integration over the transverse momenta of the collinear particles.

From Eq. 1.1 we can see that the product of the strong coupling constant and the collinear

logarithm, αslog(Q/Λ), is of order one. In order to take care of these large logarithms, we

can resum them into the PDFs or FFs. This is done with the help of the Dokshitzer-Gribov-

Lipatov-Altarelli-Parisi (DGLAP) [37, 38, 39, 40] evolution equations, the solutions of which

give the scale dependence of the distribution and fragmentation functions.

In the following subsections we will go into detail as to how these procedures apply to

the study of the process at hand.

2.2 Numerical Methods - Phase Space Slicing Method

Due to the complexity of the calculation, it has to be performed numerically. As mentioned in

the Introduction we have used the two cutoff PSS method [28], which is one of the state of the

art methods for performing beyond the LO calculations. Other numerical methods include

the one cutoff PSS [41] and the dipole subtraction method [42]. At LO, we do not encounter

any singularities and so we perform the necessary integrations for the computation of the

cross section with the use of Monte Carlo integration. As we saw in section 2.1, divergences

appear at NLO. At this order there can be 3 particles in the final state. Once we have taken

care of the UV singularities using renormalization, the PSS method provides a way of dealing

with the IR and collinear divergences. To do that in PSS the 2 → 3 body phase space is

12

divided into a soft region, and a hard region:

σReal = σH + σS. (2.5)

The soft region is the region where a massless particle’s energy, such as the gluon tends to

zero, and thus a soft singularity can occur, while we want to keep the hard region free of soft

divergences. The division or slicing of phase space of the two regions is done with the use

of a parameter, called a soft cutoff - δs. In order to separate the different regions from each

other, we need to define the Mandelstam variables [43], which are invariant under Lorentz

transformations and are given by

sij = (pi + pj)2,

tij = (pi − pj)2. (2.6)

The soft region is defined as the region where the gluon’s energy, Eg, is less than the product

of the soft cutoff and the center of mass energy of the hard scattering,√

s12,

0 ≤ Eg ≤ δs

√s12/2. (2.7)

Since there are also collinear singularities that need to be dealt with, the hard region is

further divided into a hard non-collinear and a hard collinear region,

σH = σHNon−Coll + σHColl. (2.8)

Here the collinear region is separated with the use of a collinear cutoff - δc. With the use of

the Mandelstam variables the collinear region is defined as the region of phase space where

the variables are less than the product of the partonic center of mass energy and the collinear

cutoff,

sij, |tij| < δcs12. (2.9)

We can use the numerical PSS method after the UV and IR singularities have been

exposed and canceled in n dimensions, then we can move back to 4 dimensions and integrate

over the finite region of phase space that is left. To perform this integration, we use VEGAS,

which is an adaptive multidimensional integration Monte Carlo algorithm [44].

In the next section we will show how the PSS method applies to our calculation. We

have to keep in mind that since the two cutoffs are arbitrary and are not physical, the final

13

result should be independent of them. Even though all three regions, the soft, collinear, and

hard non-collinear region depend on the cutoffs, once the numerical values for each of those

are added, the result should not depend on them.

2.3 γ+Q calculation

2.3.1 LO

Compton Subprocess

To LO, which is order ααs, there is only one hard scattering subprocess that can give a

final state photon and a heavy quark. This is the Compton subprocess,

g(p1) + Q(p2) → γ(p3) + Q(p4),

where an initial state gluon and an initial state heavy quark scatter into a photon and a

heavy quark, with p1 + p2 = p3 + p4. This differs from the LO for the fully inclusive direct

photon production, where there is a second subprocess that needs to be included, which is

qq → γg. As there are no final state heavy quarks produced by it, this process does not

pertain to our calculation.

There are two diagrams describing the Compton subprocess. These are shown in Fig.

2.2. In order to calculate the cross section we need to write the expression for the scattering

amplitude AComp:

AComp = AsComp +Au

Comp, (2.10)

where AsComp and Au

Comp are the amplitudes for the s and u channel respectively. With the

use of the Feynman rules from Fig. 2.2 we arrive at:

AsComp = ieeQigst

aiju(p4)γ

νi/q

q2γµu(p2)εµ,g(p1)ε

∗ν,γ(p3),

AuComp = ieeQigst

aiju(p4)γ

µi/q′

q′2γνu(p2)εµ,g(p1)ε

∗ν,γ(p3), (2.11)

where q = p1 + p2 and q′ = p1 − p4. Here gs is the strong charge, e is the electromagnetic

charge, and eQ is the fractional charge of the heavy quarks, with, ec = 2/3 and eb = −1/3.

The generators of the SU(3) color group are represented by the matrices taij. Using the

following relations: ∑r

ε∗rµ,p(p3)εrρ,p(p3) → −gµρ, (2.12)

14

g(p1)

Q(p2)

γ(p3)

Q(p4)

g(p1)

Q(p2)

γ(p3)

Q(p4)

Figure 2.2: Compton Scattering

∑s′

us′(p1)u

s′(p1) = /p1

, (2.13)

we get for the matrix element squared, summed over final spins and color and averaged over

initial spins and color, the result:∑|AComp|2 =

(1

4

)(1

3

)(1

8

)g2

se2e2

Q32(s12

t14+

t14s12

). (2.14)

And making the final substitution, αs = g2

4π, α = e2

4π, we get for the squared matrix element:∑

|AComp|2 = 16π2e2Qααs

(1

3

)(s12

t14+

t14s12

). (2.15)

In order to get the hadronic cross section for the Compton subprocess, we convolute the

squared amplitude,∑|AComp|2 with the PDFs and integrate over the phase space of the two

final state particles, dPS2, as shown in Eq. 2.1, and include a flux factor, 12s

,

σComp =

∫dx1dx2Gg(x1, µf )GQ(x2, µf )

1

2s

∑|AComp(x1, x2, µr)|2dPS2 +

∫1 ↔ 2. (2.16)

The phase space for the two final state particles is given by the general formula:

dPS2 =d3p3

(2π)32E3

d3p4

(2π)32E4

(2π)4δ4(p1 + p2 − p3 − p4). (2.17)

Making use of the fact that since the scattering is symmetric about the collision axis the

integration over the azimuthal angle φ yields 2π, and also after some further manipulation

Eq. 2.17 can be written as:

dPS2 =dcosθ

16π. (2.18)

In Eq. 2.18, θ is the scattering angle with respect to the beam axis.

15

LO Fragmentation

As we can see from the above calculation, the Born term for this process is of order ααs.

The photon however can be produced not only in a hard scattering, but by fragmenting off

of a parton. As briefly mentioned in the Introduction, the photon fragmentation function

is introduced in order to take care of final state collinear singularities, where the photon is

emitted collinearly to one of the final state partons. One such example is the case when the

photon may be emitted collinear with the final state q, in the qQ → qQγ subprocess. This

case will give rise to a collinear singularity. This singular contribution can be absorbed into

the photon fragmentation function Dγ/q., as we pointed out in Section 2.1,

The part of the hadronic cross section for the photon to be produced by fragmentation

will look as follows:

σFragm =∑

i

∫dx1idx2idzG(x1i, µf )G(x2i, µf )σ

iFragmDγ/pi

(z, µF ) + (1 ↔ 2). (2.19)

In Eq. 2.19, Dγ/pi(z, µF )dz is the photon fragmentation function density, which gives the

probability for a parton pi = u, d, s, c, b, g to fragment into a photon with fraction z to

z + dz of its momenta, at a fragmentation scale µF . As was pointed out in Section 2.1, after

factorizing the collinear singularity there still remain large logs coming from the integration

over the angle between the photon and the quark to which it is collinear. These logs are

resummed into the photon FFs by solving the DGLAP equations. The photon couples

directly to the quark via the electromagnetic force. Thus due to the nature of the photon’s

coupling to the quark, which is pointlike, the photon FFs can be derived as a solution to the

set of inhomogeneous coupled equations shown in Eq. 2.20, [14].

dDγ/q(z, t)

dt=

α

2πPγq(z) +

αs

2π[Dγ/q ⊗ Pqq + Dγ/g ⊗ Pgq],

dDγ/g(z, t)

dt=

αs

2π[Dγ/q ⊗ Pqg + Dγ/g ⊗ Pgg], (2.20)

where t = ln(Q2/Λ2) and ⊗ denotes a convolution integral as defined in Appendix A. As a

first iteration to solving the above equations, we can write Eq. 2.20, as

dDγ/q(z, t)

dt=

α

2π

1

2[1 + (1− z)2],

dDγ/g(z, t)

dt= 0, (2.21)

16

Table 2.1: A list of all 2 → 2 LO hard-scattering subprocesses with at least one heavy quark, Q,in the final state (where Q, can be either a heavy quark or an antiquark, where appropriate)

LO FragmentationSubprocessesgg → QQgQ → gQqQ → qQqQ → qQqq → QQQQ → QQQQ → QQ

where we have substituted for the value of the splitting function Pγq(z) = 12[1 + (1 − z)2].

Thus we can clearly see that Dγ/q(z, t) is proportional to αt. If we go back to Eq. 1.1,

however we see that αs ∼ 1/ ln(Q/Λ), i.e. αs ∼ 1/t. As such,

Dγ/pi∼ α

αs

(2.22)

Therefore, another class of contributions of order ααs consists of 2 → 2 QCD subprocesses

with at least one heavy quark in the final state convoluted with the appropriate photon FF,

as listed in Eq. 2.19, since α2s ⊗ Dγ/pi

is of order ααs. In Eq. 2.19 i runs from 1 to 7,

since there are seven subprocesses that contribute at this order. In Table 2.1 we list these

appropriate 2 → 2 QCD subprocesses. We have to keep in mind that we have to have at least

one heavy quark in the final state. As well as that, we should note that since we require that

the photon be isolated, i.e., the hadronic energy by which it can be surrounded is limited,

the direct photon and the heavy quark cannot be both experimentally detected if they are

collinear to each other. Due to this fact the photon cannot fragment from the heavy quark in

the case when there is only one heavy quark in the final state, i.e., gQ → γgQ, Qq → γqQ,

Qq → γqQ, since we require that both the photon and heavy quark are experimentally

observed. Thus also in the case when there are two final state heavy quarks, the photon can

originate only from the unobserved heavy quark.

Example Feynman diagrams are shown in Fig. 2.3. In diagram 1) the LO QCD subprocess

gg → QQ is shown, with the photon radiating from the final state antiquark. The photon

can be a fragment of either final state quark. However, the contribution to the cross section

17

Q(p1)

g(p2) Q(p4)

γ(zp3)g(p1)

g(p2) Q(p4)

γ(zp3)

1) 2)

Figure 2.3: An example of Leading Order Fragmentation Contributions 1) gg → QQ, withthe photon fragmenting from one of the final state heavy quarks 2) gQ → gQ, with the photonfragmenting off from the final state gluon

can only come from the heavy quark which does not emit a photon by bremsstrahlung, since

isolation requirements exclude the other case. In diagram 2) the subprocess Qg → gQ is

shown. In this diagram, even though the photon can fragment from either the gluon or the

heavy quark, we do not include the case when it is produced by the heavy quark.

The complete LO hadronic cross section is the sum of the two types of contributions,

σLO = σCompton + σLOFragm. (2.23)

We can see from Eqs. 2.23,2.19,2.16, that σLO, depends on the renormalization scale, µr,

the fragmentation scale, µf , and the factorization scale µF . We want to reduce this scale

dependence and also obtain a more precise value of σ, by calculating the NLO contributions,

the calculation of which we describe in the next section.

2.3.2 NLO

To calculate the NLO terms we need to go up by one order of αs. Thus, for this calculation

we must consider all possible subprocesses that are of order αα2s, and contain a photon and

at least one heavy quark in the final state. At this order there are two types of contributions,

those are the virtual and real ones.

Virtual Corrections

The virtual corrections are calculated as the interference between the LO Born diagrams,

shown in Fig. 2.2 and the virtual diagrams in Fig. 2.4. In the virtual diagrams, just like it

18

Figure 2.4: Virtual Feynman diagrams for the Compton subprocess

is shown for Fig. 2.1, the momenta of the virtual particles are not completely determined

by the external lines. Thus, in this case the virtual momenta can take on values from zero

to infinity. After integration over the virtual momenta, two types of singularities can occur,

ultraviolet and infrared, coming from the upper and lower bound of the integration region

respectively. In order to expose the singularities as poles we evaluate the matrix elements

corresponding to the diagrams in n = 4 − 2ε dimensions. After convoluting them with the

PDFs, the virtual correction can be written as,

σV irt =

∫dx1dx2Gg(x1, µf )GQ(x2, µf )σV irt + (1 ↔ 2), (2.24)

19

where,

σV irt = σsingV irt(1/εUV , 1/εIR) + σfinite

V irt . (2.25)

The virtual corrections have been calculated in Ref. [45]. In Eq. 2.25 the singularities

are contained in σsingV irt and σfinite

V irt is finite. The poles resulting from the UV singularities

are taken care of by renormalizing the coupling constant and fields, and we are left with a

dependence only on the IR singularities, σ′sing

V irt(1/εIR). The IR poles will cancel with soft

singularities coming from the real corrections, while the remaining collinear singularities in

the real corrections part will be absorbed into the distribution and fragmentaion functions.

We proceed to describe the calculation of the real contributions to the NLO cross section.

Real Corrections

The phase space for producing a photon in association with a heavy quark increases now

that we have an extra parton emitted in the final state. And now there are seven possible

hardscattering subprocesses, which are listed in Table 2.2. An example Feynman diagram

of each is listed in Fig. 2.5. The real corrections contribution can be calculated by,

σReal =∑

i

∫dx1idx2iG(x1i, µf )G(x2i, µf )σ

iReal + (1 ↔ 2), (2.26)

where i runs through the seven subprocesses listed in Table 2.2, and

dσReal =1

2s12

|A3|2dPS3. (2.27)

Here the three body phase space dPS3 in n dimensions is given by:

dPS3 =dn−1p3

2p03(2π)n−1

dn−1p4

2p04(2π)n−1

dn−1p5

2p05(2π)n−1

(2π)nδn(p1 + p2 − p3 − p4 − p5). (2.28)

Since we expect soft and collinear singularities to appear at this point in the calculation, we

partition the real phase space into three regions with the use of the PSS method. These three

regions are the soft, the collinear and the finite region. The finite region does not contain

any singularities and we are free to integrate over it with the use of Monte Carlo techniques,

just as we did for the calculation of the LO cross section. It is the soft and collinear regions

that pose a bigger challenge in the real part of the NLO calculation. In what follows we

describe the calculation for these regions.

20

Table 2.2: a list of all 2 → 3 NLO hard-scattering subprocesses

NLO subprocessesgg → γQQgQ → γgQQq → γqQQq → γqQqq → γQQQQ → γQQQQ → γQQ

γ(p3)

Q(p4)Q(p1)

q(p5)/q(p5)

Qq → γQq/Qq → γQq

q(p2)/q(p2)

γ(p3)

Q(p4)

Q(p5)

q(p1)

q(p2)

qq → γQQ

g(p2)

g(p1)

γ(p3)

Q(p4)

Q(p5)

gg → γQQ

γ(p3)

gQ→ γgQ

g(p2)

Q(p1)

Q(p4)

g(p5)

QQ→ γQQ/QQ→ γQQ

γ(p3)

Q(p4)Q(p1)

Q(p5)/Q(p5)Q(p2)/Q(p2)

Figure 2.5: Example Feynman diagrams for all the possible real subprocesses

21

Soft Region

The soft, or infrared, singularities appear when the gluon’s energy tends to zero. There is

only one subprocess that allows such a configuration, this being gQ → γgQ, which is shown

in the second diagram in Fig. 2.5. In this case the final state gluon’s energy tends to zero,

Eg = E5 → 0. This region is separated from the rest of phase space with the use of Eq. 2.7.

In this limit the calculation of the matrix element is simplified by the use of the eikonal, or

double pole, approximation [46]. In it we set the gluon’s energy to zero when it appears in

the numerator of the matrix element. We have to retain the terms where Eg appears in the

denominator, as those terms are divergent. As such the matrix element can be written as

the product of the LO matrix element squared and an eikonal factor,

|Asoft(gQ → γgQ)|2 = g2µ2εr |AComp|2Φeikonal. (2.29)

In Eq. 2.29 the eikonal factor Φeikonal in general is calculated as,

Φeikonal =∑i,j

pi.pj

pi.pgpj.pg

Cij. (2.30)

In Eq. 2.30 C is a color factor and the sum runs over the number of partons from which the

gluon can be emitted, which in our case are the initial state gluon and heavy quark, and the

final state heavy quark.

There is another approximation, which we can use in this limit. The three body phase

space in Eq. 2.28 can also be greatly simplified in this region, so that it can be written as a

combination of the two-body phase space multiplied by a soft phase space factor,

dPSsoft3 = dPS2dPSsoft. (2.31)

With the use of Eq. 2.29 and 2.31 we can write the soft contribution to the cross section as

σsoft =[αs

2π

Γ(1− ε)

Γ(1− 2ε)

(4πµ2r

s12

)ε]σComp

∫ΦeikonaldS, (2.32)

with dS given by

dS =1

π

( 4

s12

)−ε∫ δs

√s12/2

0

dEgE1−2εg sin1−2εθ1dθ1sin

−2εθ2dθ2. (2.33)

After the integration we can write the soft cross section as a sum of two parts,

σsoft = σsingsoft(εIR) + σfinite

soft , (2.34)

22

g(p1)

Q(p2)

γ(p3)

Q(p4)Q(p′

2)

g(p5)

Q(p2)

g(p1)

g(p5)

g(p′

1)

Figure 2.6: Initial collinear singularities for gQ → γgQ, the final state gluon is collinear to theinitial state heavy quark, or the final state gluon is collinear to the initial state gluon.

where σsingsoft contains the soft poles and σfinite

soft is finite.

Collinear Region

The last type of singularities are the collinear singularities. These occur in the region

when two massless particles are emitted collinearly to each other. Depending on their nature

they are treated in different ways. The initial state collinear singularities are absorbed in

the PDFs. The final state ones are either absorbed in the photon FF or when summed

over experimentally unobserved states cancel due to the Kinoshita-Lee-Nauenberg theorem

[47, 48, 49]. Let us as an example analyze the collinear structure of gQ → γgQ. We label the

partons as shown in Fig. 2.6. Two initial state collinear singularities can occur, in the case

when the final state gluon is collinear to the initial state heavy quark, or when the final state

gluon is collinear to the initial state gluon. The final state heavy quark cannot be collinear

to the initial gluon in our case, since we require that it be experimentally detectable. In

this case the collinear singularity occurs when either t15 or t25 → 0, and the collinear region

in the PSS method is defined as the region where |t15| or |t25| < δcs12. Let us concentrate

on the case where the initial and the final state gluons are collinear. In this limit we can

approximate the momenta of particles g(p1), g(p′1) and g(p5) as follows:

p1 = (p, 0, 0, p)

p′1 = (zp + p2t /2zp, pt, 0, zp)

p5 = ((1− z)p + p2t /2(1− z)p,−pt, 0, (1− z)p) (2.35)

23

Here z is the fraction of momentum taken by the emitted particle. Using another collinear

approximation the 2 → 3 phase space, in Eq. 2.28 can be approximated to:

dPS3coll =[ dn−1p3

2p03(2π)n−1

dn−1p4

2p04(2π)n−1

(2π)nδn(zp1+p2−p3−p4)] (4π)ε

16π2Γ(1− ε)dzdt15[−(1−z)t15]

−ε,

(2.36)

where the part in the square brackets corresponds to the 2 → 2 phase space of partons

g(p′1)Q(p2) → γ(p3)Q(p4). In this region we can also use another approximation. This is the

leading pole, or collinear, approximation of the matrix element. This approximation comes

about from the use of the collinear kinematics, shown in Eq. 2.35, in the 2 → 3 matrix

element, which written out in n = 4− 2ε dimensions takes the form:∑|A3(g + Q → γ + Q + g)|2 =

∑|A2(gQ → γQ)|2Pgg(z, ε)g

2µ2εr

−2

zt15, (2.37)

where z is the momentum fraction carried by gluon 1′, as shown in Eq. 2.35, and the bar over

the sum denotes summing over final spin and color and averaging over initial spin and color.

Here Pgg(z, ε) is the unregulated splitting function in d dimensions, Pgg(z, ε) = Pgg(z)+εP ′gg.

With Pgg(z) and P ′gg given by:

Pgg(z) = 2N[ z

1− z+

1− z

z+ z(1− z)

],

P ′gg(z) = 0. (2.38)

Putting Eq. 2.37 and Eq. 2.36 together, we get for the collinear part of the partonic cross

section that we are discussing:

dσgQ→γQgcoll,init1 = dσComp(zs12, t, u)Pgg(z, ε)

[αs

2π

(4πµ2r)

ε

Γ(1− ε)

](1− z)−ε

zdz

t−ε15

t15dt15 + (1 → 2). (2.39)

Looking more closely at the integration over t15 in Eq. 2.39, we get:∫ δcs12

0

dt15t−1−ε15 = −1

ε(δcs12)

−ε. (2.40)

By performing the integration in Eq. 2.40 we have made the collinear singularity explicit

as a pole in ε. Using the approximations 1Γ(1−ε)

' Γ(1−ε)Γ(1−2ε)

, and δ−εc ' 1 − ε ln(δc), and also

shifting the momentum p1 → p1/z, we obtain for the hadronic cross section:

dσgQ→γQgcoll,init1 = Gg/A(x1/z)GQ/B(x2)dσComp(s12, t, u)Pgg(z, ε)

[αs

2π

Γ(1− ε)

Γ(1− 2ε)

(4πµ2r

s12

)ε]×

[−1

ε+ ln δc

](1− z

z

)−ε dz

zdx1dx2 + (1 → 2). (2.41)

24

Now that the singularity structure is explicit, we need to take care of it. We do that by

introducing a scale dependent PDF, using the MS convention [50], given by Eq. 2.42.

Gg/A(x, µf ) = Gg/A(x) +(−1

ε

)[αs

2π

Γ(1− ε)

Γ(1− 2ε)

(4πµ2r

µ2f

)ε] ∫ 1

x

dz

zPga(z)Ga/A(x/z), (2.42)

where Gg/A(x) is the bare PDF, present in Eq. 2.44, and Pga(z) are the regulated splitting

functions, with Pgg(z) given by:

Pgg(z) = 2N[ z

1− z+

1− z

z+ z(1− z)

]+

(11

6N − 1

3nf

)δ(1− z). (2.43)

dσComp = Gg/A(x1)GQ/B(x2)dσCompdx1dx2 (2.44)

After substituting Eq. 2.42 in Eq. 2.44 and adding together the αs pieces we get:

dσgQ→γQgcoll,init1 = dσComp

[αs

2π

Γ(1− ε)

Γ(1− 2ε)

(4πµ2r

s12

)ε][(−1

ε+ ln

δc(1− z)

z

)Pgg(z, ε)

+(1

ε+ ln

s12

µ2f

)Pgg(z)

]dzGg/A(x1/z, µf )GQ/B(x2, µf )dx1dx2 + (1 → 2),

(2.45)

where we have used the approximation, −1ε

(µ2

r

µ2f

)ε

= −(

µ2r

s12

)ε(1ε

+ ln s12

µ2f

), and we have also

expanded(

1−zz

)−ε

to order ε. Here we can explicitly see the 1ε

poles canceling. However the

integration limits over z for the two expressions differ. This is due to the fact that we require

the final state gluon to be hard, which means that E5 ≥ δs√

s12

2. Since p′1 + p2 = p3 + p4 and

substituting p′1 ' zp1, we get that zs12 ' s34,

E5 =s12 − s34

2√

s12

=(1− z)

2

√s12 ≥ δs

√s12

2,

z ≤ 1− δs. (2.46)

Due to this mismatch the final expression for the part of the cross section associated with

the initial gg collinearity is given by Eq. 2.47:

dσgQ→γQgcoll,init1 = dσComp

[αs

2π

Γ(1− ε)

Γ(1− 2ε)

(4πµ2r

s12

)ε][G′g/A(x1, µf )

+(Asc

1 (g → gg)

ε+ Asc

0 (g → gg))Gg/A(x1, µf )

]GQ/B(x2, µf )dx1dx2 + (1 → 2).

(2.47)

25

In Eq. 2.47 Asc1 (g → gg) and Asc

0 (g → gg) are given by:

Asc1 (g → gg) = 2N ln δs +

11N − 2nf

6,

Asc0 (g → gg) = Asc

1 (g → gg) lns12

µ2f

. (2.48)

The modified parton distribution function G′g/A(x1, µf ) is given by

G′g/A(x1, µf ) =

∫ 1−δs

x1

dz

zG′g/A(x1/z, µf )Pgg(z), (2.49)

and

Pgg(z) = Pgg(z) ln(δc

1− z

z

s12

µ2f

)− P ′gg(z). (2.50)

For the other case, when partons Q(p2) and g(p5) are collinear, we use exactly the same

procedure, and we end up with the following expression:

dσgQ→γQgcoll,init2 = dσComp

[αs

2π

Γ(1− ε)

Γ(1− 2ε)

(4πµ2r

s12

)ε][G′Q/B(x2, µf )

+(Asc

1 (q → qg)

ε+ Asc

0 (q → qg))GQ/B(x2, µf )

]Gg/A(x1, µf )dx1dx2 + (1 → 2).

(2.51)

Now that we have dealt with the initial collinear singularities let us focus on the possible final

state collinear singularities for this subprocess. Generally if we were to look at the inclusive

cross section for direct photon production, the case when the final state photon is emitted

collinearly from the final state heavy quark would occur. However since we require that both

the photon and heavy quark are experimentally observable we do not observe this singularity

in our calculation. Thus for this subprocess we only have the possibility of the final state

gluon and heavy quark to be collinear to each other. Using the leading pole approximation

for the matrix element as well as the collinear phase space approximation, that we outlined

above, we arrive at the following expression,

dσgQ→γQgcoll,fin = dσComp

[αs

2π

Γ(1− ε)

Γ(1− 2ε)

(4πµ2r

s12

)ε](Aq→qg1

ε+ Aq→qg

0

)+ (1 → 2). (2.52)

One further point to note is the need of a jet definition in this case. Since we have divided

the phase space into a 2 → 2 and a 2 → 3 contribution, when the gluon and the heavy

quark are collinear this contributes to the two-body phase space, however when they are

26

Q(p1)Q(p1)

Q(p1) Q(p1)

q(p2)

q(p2)

q(p2)

q(p2)

γ(p3)

γ(p3) γ(p3)

Q(p4)

Q(p4)

Q(p4)

Q(p4)

q(p5) q(p5)

q(p5)q(p5)

2)1)

γ(p3)

3) 4)

Figure 2.7: Feynman diagrams for the subprocess qQ → γQq.

almost collinear this configuration will contribute to the three-body phase space. In this

case the photon’s transverse momentum will be balanced by the transverse momenta of both

the heavy quark and the gluon. In order to ensure that proper cancelation between the

two and three body phase space occurs a jet definition is introduced, when now the heavy

quark jet momentum is the sum of the momenta of the two partons. And now the photon’s

momentum on one side is balanced by the momentum of the jet which the collinear gluon

and heavy quark have formed.

In the subprocess described above, as explained we do not observe a collinear singularity

involving the photon. However in all the rest of the NLO subprocesses listed in Table 2.2

such a singularity can occur. We also have to keep in mind that in the case when we have two

final state heavy quarks the photon can be collinear only to the unobserved heavy quark. Let

us take a closer look at the case when a photon is emitted collinearly to the final state light

quark in the subprocess, Qq → γQq. There are four Feynman diagrams for this process,

listed in Fig. 2.7. The only collinear singularity involving the photon in our case, is the

27

final state collinear singularity that can occur in diagram 4. In this case the Mandelstam

variable s35 goes to zero. Here following the same procedure as the initial collinear singularity

treatment we reach Eq. 2.53,

dσQq→γQqcoll,fin = σQq→Qq

[αe2q

2π

]Dγ/q(z, µF )dz + (1 → 2). (2.53)

Eq. 2.53 differs from Eq. 2.47, due to the exchange of the strong coupling constant αs for the

electromagnetic one α, and also due to the fact that here there are no soft collinear terms as

we require the photon to be detectable, i.e. it cannot be soft. The modified FF Dγ/q(z, µF ),

is given by the same expression as the modified PDF, in Eq. 2.49. To order α for Dγ/q, we

get

Dγ/q(z, µF ) =

∫ 1

z

dy

yDγ/γ(z/y)Pγq(y), (2.54)

with Dγ/γ(z/y) = δ(1 − z/y). After substituting for Dγ/γ(z/y), we get for the modified

fragmentation function,

Dγ/q(z, µF ) = Pγq(z). (2.55)

Here Pγq(z) follows the form of Eq. 2.50,

Pγq(z) = Pγq(z) ln(δcz(1− z)

s12

µ2F

)− P ′γq(z), (2.56)

with the difference that z is now in the numerator instead of the denominator, which comes

out from the fact that we are dealing with a final collinear singularity, instead of an initial

one. The splitting function is given by

Pγq(z) =1 + (1− z)2

zP ′γq(z) = −z. (2.57)

Substituting Eq. 2.55 and 2.56 into Eq. 2.53, we get,

dσQq→γQqcoll,fin = σQq→Qq

[αeq

2π

][ln

(z(1− z)δc

s12

µ2F

)(1 + (1− z)2

z

)+ z

]dz + (1 → 2). (2.58)

We treat the rest of the final state photon collinear singularities in the same fashion. This

is done by substituting σQq→Qq, and eq in Eq. 2.58 with the appropriate hard scattering

subprocess, and quark charge. We note that there are no photon and gluon collinear

singularities and as such Pγg(z) = 0.

28

q(p2)

q(p1)

γ(p3)

Q(p4)

Q(p5)q(p2)

q(p1)

γ(p3)

Q(p4)

Q(p5)

Q(p4)

Q(p5)

q(p1)

q(p2) γ(p3)

q(p1)

q(p2)

γ(p3)

Q(p4)

Q(p5)3) 4)

2)1)

Figure 2.8: Feynman diagrams for the subprocess qq → γQQ, diagrams 1) and 2) are s-channeldiagrams, and diagrams 3) and 4) are t-channel diagrams.

There is one last final state collinear singularity that needs to be taken care of. It occurs

in the annihilation subprocess, qq → γQQ shown in Fig. 2.8. In the t-channel diagrams 3)

and 4) in Fig. 2.8, the pair of final state heavy quarks, Q(p4) and Q(p5) are produced by

gluon splitting. In this case there is a possibility of a collinear singularity as the spatial angle

between the two quarks goes to zero. If we were to calculate the inclusive cross section for a

production of a direct photon without tagging on the heavy quark, as we do here, this singular

region would be integrated over yielding a two-body contribution dependent on δc. This

two-body contribution would be proportional to the subprocess qq → γg. The contribution

would be added to the one-loop corrections for the qq → γg subprocess, the poles in ε would

cancel and there would be a residual δc contribution to the qq → γg subprocess. This would

cancel against a similar contribution from qq → γQQ once a suitable jet definition has been

implemented in the calculation. However in this calculation the subprocess qq → γg does

not enter at LO as it does not contain any heavy quarks in the final state. Since we are

dealing with heavy quarks, we can address this problem by remembering that the pair of

29

heavy quarks cannot be produced unless their invariant mass is equal to or larger than 4m2Q.

Imposing this constraint on the events generated for qq → γQQ avoids the problem of the

uncanceled δc dependence. And as such the situation where the invariant s45 → 0 is avoided,

by the constraint that s45 ≥ 4m2Q.

After discussing all the possible collinear singularities and their treatment, we can write

the collinear cross section using the following general expression,

σcoll =

∫dx1dx2(Gg(x1, µf )GQ(x2, µf ) + Gg(x1, µf )GQ(x2, µf ))σComp

+∑

i

∫dx1idx2idzG(x1i, µf )G(x2i, µf )σ

iFragmDγ/pi

(z, µF ) + (1 ↔ 2)

+ σsingcoll (1/εIR). (2.59)

Total Cross Section

Now that we have the expressions for the different parts that contribute to the cross

section, we can write the NLO cross section as a sum of the two-body and three-body

components,

σNLO = σ2bodyNLO + σ3body

NLO . (2.60)

In Eq. 2.60 σ3bodyNLO is calculated as shown in Eq. 2.26, with the integration limits imposed

by the PSS requirements Eq. 2.7 and 2.9. The two body cross section is the sum of all the

different pieces that have two particles in the final state,

σ2bodyNLO = σLO + σsoft + σvirt + σcoll. (2.61)

We note that the singular parts once added, cancel,

σsingvirt + σsing

soft + σsingcoll = 0. (2.62)

The cross section in Eq. 2.60, is finite, however it is not the complete NLO cross section,

as there are NLO fragmentation effects that need to be added to it. We have extended

the calculations of Ref. [26] by adding this contribution. We proceed to describe their

calculation.

30

Table 2.3: A list of all 2 → 3 NLO hard-scattering subprocesses, O(α3s), with at least one

heavy quark, Q, in the final state (where Q, can be either a heavy quark or an antiquark, whereappropriate)

NLO FragmentationSubprocessesgg → QQggQ → QgggQ → QqqgQ → QQQgq → qQQQq → QqgQq → Qqgqq → QQgQQ → QQgQQ → QQg

NLO Fragmentation

Just as in the LO case, there are fragmentation contributions that need to be taken

into account at this order, so that we have a complete NLO calculation. At this order

we need to convolute all 2 → 3 QCD subprocesses of order α3s that have at least one

heavy quark in the final state with the photon FF. Thus we get, as in section 2.3.1,

O(α3s) ⊗ Dγ/q,g ∼ α3

sα/αs = αα2s. The subprocesses contributing at this stage are listed

in Table 2.3, the matrix elements for which are computed in Ref. [51]. Let us look as an

example at the first subprocess in Table 2.3, which is gg → QQg. An example Feynman

diagram of this subprocess is shown in diagram 1) of Fig. 2.9. In order to get a final state

that has a photon from this subprocess, the photon can be produced by fragmentation from

either one of the final state partons as is shown in diagrams 2) - 4). As was shown earlier in

this section when we go beyond the LO, divergences start to plague the calculation. Thus

this subprocess can exhibit initial state collinear singularities where either the final state

heavy quark, anti-quark or gluon can be collinear to the initial state gluons. In addition to

that, we can have final state collinear singularities when the final state gluon is collinear to

either the heavy quark or antiquark, and also the two heavy quarks can be collinear to each

other. There is also the possibility of the gluon’s energy to go to zero, i.e for it to become

31

1)g(p1)

g(p2)

Q(p4)

g(p3)

Q(p5)

g(p3)

g(p1)

g(p2) Q(p5)

γ(zp4)2)

g(p1)

g(p2)

Q(p4)

g(p3)

γ(zp5)

4)

γ(zp3)

3)

g(p1)

g(p2)

Q(p4)

Q(p5)

Figure 2.9: Feynman diagram for the subprocess gg → QQg shown in diagram 1). A photonbeing produced by fragmentation from one of the final states for this subprocess, diagram 2) fromthe final state quark, diagram 3) from the final state gluon, and diagram 4) from the final stateanti-quark

soft. All these singularities need to be taken care of. This is done by exactly the same

procedure that was followed for the NLO hard scattering calculation. We remember that

we included the 2 → 2 QCD subprocesses that contribute to the LO fragmentation part.

Thus when we calculate all the virtual corrections to the subprocesses listed in Table 2.1,

the resulting poles cancel with the soft singularities and the final state collinear singularities,

while the initial state collinear singularities are absorbed into the the PDFs.

2.4 Stability of the Calculation

Once the calculation has been performed we need to make sure that the final result is indeed

independent of the cutoffs. In Fig. 2.10, we present the independence of the cross section

on the soft cutoff δs, while we have kept the value of δc = 5 × 10−5 constant. In the upper

window we show the dependence of the two-body and the three-body parts on the cutoff

δs. The two-body provides a negative contribution to the cross section and the three-body

32

-200

-100

0

100

200σ(

pb)

σ2−>2σ2−>3σ

tot

0.0001 0.001 0.01 0.1δ

s

181920

Figure 2.10: δs dependence of σNLO(pp → γbX), at√

S = 1.96 TeV, for δc = 5 × 10−5 whenδs > 0.001 , and δc = 1 × 10−5, when δs < 0.001 . The upper plot shows the dependence on δs ofthe two-body (dashed line), and three-body (dot dashed line) contributions, and also the sum ofthe two (solid line). The lower plot shows the dependence on the full NLO cross section, with thestatistical error included, µ = µr = µf = µF , µ = pTγ .

gives a positive one, with the size of these contributions depending on the magnitude of the

cutoff. When the two contributions are added, their sum gives the total cross section, which

is independent of the cutoff as can be seen by the solid line in Fig. 2.10. We should note that

the two-body contribution contains parts which are not dependent on the cutoffs, such as

the LO cross section, but just give a constant shift to the contribution. In the lower window

we show an enlargement of the cutoff dependence of the total cross section. We can see that

the total is constant, while dropping off for the first and last values of δs. This means that

33

-50

0

50

100

σ(pb

)σ2−>2σ2−>3σ

tot

1e-05 0.0001 0.001δ

c

181920

Figure 2.11: δc dependence of σNLO(pp → γbX), at√

S = 1.96 TeV, for δs = 0.01. The upperplot shows the dependence on δc of the two-body (dashed line), and three-body (dot dashed line)contributions, and also the sum of the two (solid line). The lower plot shows the dependence onthe full NLO cross section, with the statistical error included, µ = µr = µf = µF , µ = pTγ .

we must have a sufficiently small value for the cutoff in order to work with the phase space

slicing method, i.e., δs < 0.1. The kink in the graph comes from the change of the value of

the collinear cutoff δc, from δc = 5 × 10−5 to δc = 1 × 10−5, this was done in order to keep

the ratio of δc

δsfrom becoming large, which is what happens in the last point where again

there is an observed drop off.

In Fig. 2.11, we show the collinear cutoff versus the total cross section, while this time

keeping the soft cutoff constant, δs = 0.01. As in the previous plot, we see that there is no

dependence on the cutoff for sufficiently small values of δc. Again there is a dropoff for the

34

larger values of δc, thus we require δc < 10−3.

We should note that the reason that the two cutoffs need to be kept sufficiently small is

so that the eikonal and the leading-pole approximations remain valid.

At this point since we have a stable cross section we can analyze its behavior. We present

this analysis in the next chapter.

35

CHAPTER 3

RESULTS

In this section we present the results of the NLO calculation from Chapter 2. Firstly we are

going to examine the predictions for this process at the Tevatron σNLO(pp → γQ). There

are two current experiments at the Tevatron that can provide measurements for this process;

CDF and DØ . Each of these two detectors has different capabilities in the kinematic ranges

of the particles it can observe. Here we are going to mainly focus on producing theoretical

predictions for the DØ experiment, as they have recently published measurements for the

inclusive differential cross sections, d3σ/(dpγT dyγdyjet) for both γ+b and γ+c production [52].

As such, the kinematic cuts we have applied reproduce the ones used by the DØ experiment

at the Tevatron in FermiLab. These kinematic cuts are as follows: the lower bound on the

transverse momentum on the photon is pTγ > 30 GeV. The corresponding cut for the heavy

quark is pTQ > 15 GeV. Both particles are limited to be in the central region of the detector,

and the restrictions on their rapidities are, |yγ| < 1, |yb| < 0.8. Here is where the versatility

of using numerical methods such as the PSS is extremely useful, as applying these cuts is

quite simple in the theoretical calculation, as it just requires changes in a few lines of code.

As well as the kinematic cuts listed above, we also need a jet definition. Since partons

hadronize, what will be experimentally measurable is the jet that the heavy quark forms. We

also need the jet definition in order to get a theoretically stable cross section as mentioned

in Section 2.3.2. We use the jet definition of the DØ detector [53]. Thus if two particles are

produced within a cone of radius R = 0.5 of each other they are merged into a single jet.

The radius R is defined as follows, R =√

(ηQ − ηpi)2 + (φQ − φpi

)2, with ηQ, ηpibeing the

rapidities of the heavy quark and the parton that is merged in the jet, and φQ, φpibeing the

azimuthal angles ( in the plane transverse to the proton / antiproton beam direction ) of the

36

two particles. The rapidity of a particle is defined as follows:

η =1

2ln

(E + pz

E − pz

), (3.1)

where pz is the momentum component along the beam axis. If a particle is massless Eq. 3.1

can be reduced to:

η = ln cot(θ/2), (3.2)

where θ is the angle between the particle’s momentum and the beam axis, making the

rapidity very useful variable to measure experimentally.

Another important issue to be addressed is isolation, since in order to be experimentally

detectable, a photon needs to be isolated. Through this requirement charged particles are

prevented from entering the same cell in the detector as the photon, which could interfere

with the measurement of the photon’s energy. In order for a photon to be isolated, it

should not be surrounded by hadronic energy more than a certain fraction of its own energy,

Eh = ε ∗ Eγ in a cone of radius Riso around it [54, 55, 56]. Where the radius of the cone

is defined as Riso =√

(ηγ − ηQ)2 + (φγ − φQ)2. To make sure that the cross section we are

calculating is infrared safe, we also need to impose the isolation requirement on the photons

produced through fragmentation. Thus the momentum fraction z = pTγ/pTparton, defined in

Section 2.1 also needs to be restricted, by making the condition,

pTj < εpTγ, (3.3)

where pTj represents the momenta of the remainder of the particles that come from the

original parton after fragmentation, with pTj = (1− z)pTparton, so that pTγ + pTj = pTparton.

After substituting the expressions for pTγ and pTj in Eq. 3.3, we get,

(1− z)pTparton < εzpTparton

(1− z) < εz (3.4)

such that, from Eq. 3.4, we get the following condition z > 11+ε

. The photon isolation

requirements imposed to model the DØ detection of a direct photon are: R1 < 0.2, ε1 < 0.04

and R2 < 0.4, ε2 < 0.07 [52].

The PDFs we have used for the numerical calculation are the CTEQ6.6M set [57], unless

otherwise stated. And we have also used a 2-loop αs corresponding to αs(MZ) = 0.118. For

37

the photon FFs a set of next to leading log fragmentation functions determined by L. Bourhis,

M. Fontannaz, and J.P. Guillet was used [58]. The values for the charm and bottom quark

masses used are mb = 4.2 GeV and mc = 1.25 GeV [59], and their corresponding charges

are ec = 23|e|, and eb = −1

3|e|, with |e| being the charge of the electron.

After these technical details, we proceed with the numerical results for the Tevatron.

We will investigate how the inclusion of the NLO fragmentation affects the cross section, as

well as the effect of the imposition of isolation. We will also check if the idea that there is

a nonperturbative intrinsic charm component in the nucleon is testable with the currently

used cuts and detectors. In the last section of this chapter we will present results for this

process at the LHC, σNLO(pp → γQ).

3.1 Tevatron Predictions

We start by presenting the LO differential cross section for the production of a direct photon

and a bottom quark, pp → γbX as a function of the transverse momentum of the photon,

at the Tevatron center of mass energy,√

S = 1.96 TeV, Fig. 3.1. There we have presented

the scale dependence of the LO cross section. We have set the three different scales, the

renormalization, µr, the factorization, µf and the fragmentation one µF , to be equal, i.e.

µ = µr = µf = µF and have varied this scale through pTγ/2, to 2pTγ. We hope to decrease

the scale dependence in Fig. 3.1 with the inclusion of the next order in αs. We know from the

renormalization group equation that any physical observable, σ in our case, is independent

of the scale, µ:

µ2 d

dµ2σ(Q2/µ2, αs) ≡ 0. (3.5)

However since σ is only calculable via perturbation theory, there will be a residual scale

dependence coming from the truncation of the perturbation series, but this scale dependence

will decrease with the inclusion of each following term in the series. And in our case part of

the NLO scale dependence will cancel against the LO scale dependence, thereby reducing it.

Next we proceed to compare the NLO and the LO differential cross sections, which are

both shown in Fig. 3.2, solid line and dashed line respectively. The most striking feature

in Fig. 3.2 is the substantial increase in the difference between the LO and NLO curves

with growing pTγ. In order to ensure that this trend is not caused by the bottom quark’s

charge or PDF, we present the corresponding graph for the production of a photon and a

38

0 50 100 150 200

pTγ(GeV)

0.0001

0.01

1

dσ/d

p Tγ(p

b/G

eV)

LO µ=pTγ/2

LO µ=pTγ

LO µ=2pTγ

p+p -> γ+b+X√

S =1.96 TeV

Figure 3.1: The LO differential cross section at the Tevatron, dσ/dpTγ for the production of adirect photon and a bottom quark as a function of pTγ for

√S = 1.96 TeV at LO, with the scale

dependence shown, where the three different scales have been set to be equal µ = µr = µf = µF ,µ = pTγ (solid line), µ = pTγ/2 (dashed line), µ = 2pTγ (dotted line).

charm quark in Fig. 3.3. Where again we see the same trend - the difference between the

NLO curve and LO curve increases as the transverse momentum of the photon increases.

Generally the ratio between the two orders, is expected to stay of the same order. Thus, it

is essential to figure out where this difference is coming from.

In order to understand the origin of this effect it is necessary to take the process apart

and investigate how the different subprocesses listed in Table 2.2 contribute to the total

cross section. To perform this decomposition we have to make sure that all the singularities

described in Section 2.3.2 are canceled. This calculation is eased through the use of the

39

50 100 150 200

pTγ(GeV)

0.0001

0.01

1

dσ/d

p Tγ(p

b/G

eV)

NLOLO

p+p -> γ+b+X√

S =1.96 TeV

Figure 3.2: The differential cross section at the Tevatron, dσ/dpTγ for the production of a directphoton and a bottom quark as a function of pTγ for

√S = 1.96 TeV, at NLO (solid line), and at

LO (dashed line), µ = pTγ .

PSS method, where we have separated the singularities corresponding to each subprocess,

by placing appropriate switches in our code. How each of the subprocesses add to the cross

section is shown in Fig. 3.4.

In Fig. 3.4, the solid curve represents the NLO differential cross section, and the rest

of the curves show the contributions to the total of the different subprocesses. The dashed

curve shows the annihilation subprocess qq → γQQ. It is apparent from Fig. 3.4 that the

effect shown in Fig. 3.2 is driven by it. It overtakes the Compton contribution (dash dot

dotted line, and dot dashed line) and starts dominating the cross section at pTγ ∼ 70 GeV.

40

50 100 150 200

pTγ(GeV)

0.0001

0.01

1

dσ/d

p Tγ(p

b/G

eV)

NLOLO

p+p -> γ+c+X√

S =1.96 TeV

Figure 3.3: The differential cross section at the Tevatron, dσ/dpTγ for the production of a directphoton and a charm quark as a function of pTγ for

√S = 1.96 TeV, at NLO (solid line), and at LO

(dashed line), µ = pTγ .

Here the LO subprocess and gg → γQQ are added together (dash dot dotted line), since the

net contribution of gg → γQQ is negative. This negative contribution is what remains after

the appropriate collinear terms are subtracted, when using the phase space slicing method.

The gQ → γgQ (dot dashed line) and Qq → γqQ, and Qq → γqQ (dotted line) subprocesses

contribute to the cross section about equally, with gQ → γgQ prevailing over Qq → γqQ

/Qq → γqQ at small pTγ, where the gluon PDF is larger than the light quark PDF, and

then at large pTγ, Qq → γqQ, Qq → γqQ takes over when the light quark PDFs become

larger than the gluon PDFs. The QQ → γQQ, and QQ → γQQ are added together (dot

41

50 100 150 200

pTγ(GeV)

1e-08

1e-06

0.0001

0.01

1

dσ/d

p Tγ(p

b/G

eV)

NLOqq->QQγqQ->qQγ ; qQ->qQγgQ->gQγLO+gg->QQγQQ->QQγ ; QQ->QQγ

p+p -> γ+b+X√

S =1.96 TeV

Figure 3.4: Contributions of the different subprocesses to the differential cross section, NLO (solidline), annihilation qq → γQQ (dashed line), Qq → γqQ, and Qq → γqQ(dotted line), gQ → γgQ(dot dashed line), gg → γQQ+LO (dash dot dotted line), QQ → γQQ, and QQ → γQQ (dot dashdotted line), µ = pTγ .

dash dotted line), since their role is almost negligible, as the heavy quark PDFs are much

smaller than the light quark and gluon PDFs.

Now let us turn our attention to the annihilation subprocess, since it drives the cross

sections behavior at high pTγ. We recall from Chapter 2 that there are two channels through

which the annihilation process can be produced, with the corresponding Feynman diagrams

shown in Fig. 2.8. Since the photon couples to the final state heavy quarks in the s-channel,

diagrams 1 and 2, these diagrams are proportional to the heavy quark charge squared e2Q.

Whereas in the t-channel, diagrams 3 and 4, the photon couples to the initial state light

42

50 100 150 200

pTγ(GeV)

0.0001

0.01

1

dσ/d

p Tγ(p

b/G

eV)

charm NLObottom NLOcharm LObottom LO

p+p -> γ+Q+X√

S =1.96 TeV

Figure 3.5: A comparison between the differential cross sections, dσ/dpTγ for the production of adirect photon and a bottom quark, and that of a direct photon plus a charm quark at NLO and LO,charm at NLO (solid line), bottom at NLO (dashed line), charm at LO (dot dashed line), bottomat LO (dotted line), µ = pTγ .

quarks, and thus these diagrams are proportional to the light quark’s charge squared, e2q. The

t-channel diagrams begin to dominate as pTγ grows. In this configuration the photon goes

to one side and the two heavy quarks go to the opposite side, i.e., the photon’s momentum

is balanced by the sum of the two heavy quarks momenta. Thus in this case the photon

is not in the vicinity of the heavy quarks. Through the s-channel production the photon is

radiated by a heavy quark, and in this case the photon is more likely to be in the proximity

of the heavy quark, in this case this production channel is suppressed due to the fact that

we require the photon to be isolated. Thus at high pTγ the cross section is largely driven by

43

50 100 150 200

pTγ(GeV)

0

2

4

6

8

char

m/b

otto

m r

atio

NLOLO

p+p -> γ+Q+X√

S =1.96 TeV

Figure 3.6: The ratio of the charm and bottom differential cross sections versus pTγ , at NLO(solid line) and at LO (dashed line), µ = pTγ .

a process that is independent of the heavy quark’s charge and also their PDF, since we only

have light quarks in the initial state. As this is the driving subprocess for both the bottom

and charm cross sections, we can expect the difference between the two to decrease as pTγ

increases.

This indeed is the case as can be seen from Fig. 3.5. There the NLO differential cross

sections for both the charm quark (solid line) and the bottom quark (dashed line) are shown.

As predicted the two curves tend to come closer to each other as the value of the transverse

momentum increases. In Fig. 3.5 we have also shown the curves for the LO differential

cross section for charm (dot-dashed) and for bottom (dotted). The difference between these

44

50 100 150 200

pTγ(GeV)

0.0001

0.01

1

dσ/d

p Tγ(p

b/G

eV)

µ=pTγ/2

µ=pTγ

µ=2*pTγ

p+p -> γ+b+X√

S =1.96 TeV

Figure 3.7: Scale dependence of the NLO differential cross section, dσ/dpTγ for the productionof a direct photon and a bottom quark, where the three different scales have been set to be equalµ = µr = µf = µF , µ = pTγ (solid line), µ = pTγ/2 (dashed line), µ = 2pTγ (dotted line).

curves however, stays about constant. In Fig. 3.6, we also present the ratio of the NLO

charm and bottom differential cross sections (solid line), i.e. σNLO(pp→γcX)σNLO(pp→γbX)

, as well as the

corresponding ratio for the LO (dashed line), σLO(pp→γcX)σLO(pp→γbX)

. The ratio of the LO cross sections

stays almost constant since the main contribution to the LO cross section comes from the

Compton subprocess. The difference between the charm and bottom LO cross sections arises

from the difference in the charges of the charm and bottom quarks and the relative sizes of

the heavy quark PDFs. The ratio of the two LO cross sections depends on the ratio of the

charm and bottom quark charges squared which is e2c/e

2b = 4, and is driven up from that

45

value to about ∼ 7 due to the fact that the charm PDF is larger than the bottom PDF.

From Fig. 3.6 we can also see the NLO ratio decrease and tend to one as pTγ increases, as

expected.

Another consequence of the large dependence of the cross section on the annihilation

subprocess is its scale dependence. Since there is no Born term which involves a qq initial

state, the contributions from the annihilation subprocess start at O(αα2s). As such, the

typical scale compensation between the LO and NLO contributions for this subprocess is

missing and the annihilation subprocess can be thought of as a leading order contribution.

In Fig. 3.7 we show the scale dependence of the cross section, where just like in Fig. 3.1 we

have varied the scale from pTγ/2 to 2pTγ. And we can see that unfortunately the dependence

on the scale increases at large pTγ, where the annihilation subprocess starts to dominate.

The scale dependence increases due to the fact that there are no higher order corrections for

the annihilation process, whose scale dependence will cancel against the annihilation’s scale

dependence. Thus in order for the scale dependence to decrease we need to add corrections

of order αα3s to the qq → γQQ subprocess.

3.1.1 Photon Isolation and NLO Fragmentation

As we have seen in Chapter 2, the NLO fragmentation effects need to be included for a

complete NLO cross section. In this section we are going to investigate what effect their

inclusion will have upon the cross section, and also how the imposition of isolation affects

this contribution. As mentioned previously a photon needs to be isolated in order to give a

clear signal at a detector.

In Fig. 3.8 we show the comparison between the differential cross section with the

inclusion of isolation (dashed line) and without it (solid line). As expected without the

inclusion of isolation, there are more events included in the cross section, and so the solid

line lies higher than the dashed line. However with increasing pTγ, the two curves come close

to each other. This is due to the fact that in that region it is the annihilation subprocess

that dominates the cross section. It is the t-channel of qq → γQQ that contributes the most.

And since through this channel the photon is produced such that its momentum balances the

momenta of the pair of heavy quarks, i.e., the photon is not in the vicinity of any partons,

isolation does not affect this subprocess.

Let us take a look at what effect the NLO fragmentation contributions have upon the

46

50 100 150 200

pTγ(GeV)

0.0001

0.01

1

dσ/d

p Tγ(p

b/G

eV)

NLO no isolationNLO isolation

p+p -> γ+b+X√

S =1.96 TeV

Figure 3.8: Comparison between the differential cross section, dσ/dpTγ without isolationrequirements and with them, no isolation (solid line) , isolation (dashed line), µ = pTγ .

cross section, with the inclusion of isolation and without it. Fig. 3.9 shows the ratio between

the full NLO calculation and the cross section with only LO fragmentation included. The

solid line shows this ratio without the restriction of isolation. We can see that the NLO

fragmentation inclusion increases the cross section by up to ∼ 30%, at low pTγ. As the

transverse momentum grows the NLO fragmentation processes are overshadowed by the

annihilation subprocess and the effects of their inclusion are no longer that noticeable. The

dashed line in Fig. 3.9, shows the above ratio with the inclusion of isolation. We can see that

the NLO fragmentation contribution has now decreased to a few per cent. This is due to the

fact that the isolation requirements affect the photon which is produced by fragmentation

47

0 50 100 150 200

pTγ(GeV)

0

0.1

0.2

0.3

0.4

σ(N

LO

fra

g)/σ

(LO

Fra

g) -

1 no isolationisolation

p+p -> γ+b+X√

S =1.96 TeV

Figure 3.9: Ratio between the differential cross section dσ/dpTγ , with NLO fragmentationcontribution included and the differential cross section with just LO fragmentation included, noisolation required (solid line), isolation (dashed line), µ = pTγ .

the strongest, as it is emitted in close proximity to the parton from which it is fragmenting.

3.1.2 Intrinsic Charm

As mentioned in Chapter 2 the PDFs’ dependence on x is measured experimentally, and their

dependence on the scale Q is obtained from solutions of the DGLAP evolution equations.

Yet at the moment we do not have a clear phenomenological insight about the structure of

the heavy quark’s PDFs. In the CTEQ6.6M set of PDFs used in the previous section both

the charm and bottom PDFs are generated radiatively. What this means is that we assume

48

that at scales below the heavy quark’s mass, mQ, there is no heavy quark content in the

nucleon i.e. the heavy quark PDF, GQ(x, µ) = 0, when µ < mQ. Thus this is our initial

value for the PDF which we input in the DGLAP evolution equation,

dGQ/p(x, t)

dt=

αs

2π[Pqq ⊗GQ/p(x, t) + Pqg ⊗Gg/p(x, t)], (3.6)

i.e. at threshold Eq. 3.6 takes the form,

dGQ/p(x, t)

dt=

αs

2πPqg ⊗Gg/p(x, t). (3.7)

From Eq. 3.7 we can see that the heavy quark PDF depends solely on the gluon’s PDF. This

however does not need to be the case, and there is a possibility that there is an intrinsic non-

perturbative heavy quark component in the nucleon. The models that study this possibility

[60] concentrate on the charm component of the nucleon, as this component is expected to

be much larger than that for the bottom quark.

The PDFs modeling this possibility are the CTEQ6.6C set of PDFs [60]. There are two

models studied in Ref. [60], these are the BHPS model and a sea-like model. The BHPS

model is a light-cone model [61, 62]. In these types of models the intrinsic charm appears

at large x. Since x ∼ pT√S, we expect that the use of this PDF will affect the cross section at

larger momenta. The other model assumes that the shape of the charm distribution follows

that of the light sea quarks, i.e. Gc(x) = Gc(x) ∼ Gu(x) + Gd(x).

The difference between these three cases is shown in Fig. 3.10, where the solid curve

shows the CTEQ6.6M or radiatively generated charm scenario, the dashed curve is the

CTEQ6.6C2, or BHPS model, and the dotted curve is CTEQ6.6C4 PDF or the sea-like

model. As mentioned the difference between the BHPS distribution and the radiatively

generated case are most noticeable at large x, whereas the sea-like model is about equally

larger than the CTEQ6.6M PDF at all values of x. The effect of these different PDFs on the

cross section can be seen from Fig. 3.11. The dotted curve shows the cross section generated

with the use of the sea-like intrinsic charm PDF, and it is larger than the solid curve or the

one generated with the use of the radiatively generated charm by about the same amount

at all values of pTγ. The difference between the radiatively generated charm cross section

and the BHPS charm however is not great at small transverse momentum, but it increases

at large pTγ, as is expected given the differences between the CTEQ6.6M and CTEQ6.6C2

PDFs at large x.

49

0.001 0.01 0.1 1x

1e-08

1e-06

0.0001

0.01

1

xc(x

,Q)

CTEQ6.6MCTEQ6.6C2CTEQ6.6C4

Charm PDFsQ=40 GeV

Figure 3.10: Comparison between the three different charm PDFs at scale Q = 40 GeV,CTEQ6.6M (solid line), BHPS or CTEQ6.6C2 (dashed line), sea-like or CTEQ6.6C4 (dotted line).

3.2 LHC Predictions

At this time the future direction of high energy physics will be determined from the

observations that will come from the LHC, at CERN. As such it is imperative to have a

clear understanding of what the Standard Model (SM) processes observed at the Tevatron

are going to look like at the LHC center of mass energies. These very processes will provide

important means of calibrating and understanding the detectors, and ultimately, are likely

to provide significant backgrounds to new physics signals.

Here we present the differential cross section versus the transverse momentum of the

photon at the LHC, and observe how the seven fold increase in c.m energy and also the

substitution of the anti-proton collision beam by a proton beam affects the cross section.

50

50 100 150 200

pTγ(GeV)

0.0001

0.01

1

100

dσ/d

p Tγ(p

b/G

eV)

CTEQ6.6MCTEQ6.6C2 BHPSCTEQ6.6C4 Sea-like model

p+p -> γ+c+X√

S =1.96 TeV

Figure 3.11: The differential cross section, dσ/dpTγ , for the production of a direct photon anda charm quark for the three different PDF cases, CTEQ6.6M (solid line), BHPS or CTEQ6.6C2(dashed line), sea-like or CTEQ6.6C4 (dotted line), µ = pTγ .

Since it is unlikely that we could apprehend the kinematic cuts that will be applied at the

LHC, we have kept the kinematic cuts that we have used for the DØ case. The NLO and LO

cross sections are both shown in Fig. 3.12. It is apparent from Fig. 3.12 that the increase of

the difference between the NLO (solid line) and the LO (dashed line) grows much less rapidly

with increasing pTγ than was the case for the Tevatron. We also show the K-factor for this

case, in Fig. 3.13, which is defined as the ratio between the NLO and LO cross section,σNLO(pp→γbX)σLO(pp→γbX)

. From Fig. 3.13 we can see that the K-factor stays stable and is around 2.

In order to understand the difference between the LHC and the Tevatron curves, we will

51

50 100 150

pTγ(GeV)

0.01

0.1

1

10

dσ/d

p Tγ(p

b/G

eV)

NLOLO

p+p -> γ+b+X√

S =14 TeV

Figure 3.12: The differential cross section versus the transverse momentum of the photon dσ/dpTγ

for the production of a direct photon and a bottom quark at LHC center of mass energies,√S = 14 TeV, NLO (solid line), LO (dashed line), µ = pTγ .

again look at how the different subprocesses affect the cross section. The contributions of

the different parts contributing to the LHC cross section are shown in Fig. 3.14. From Fig.

3.14 it can be seen that the annihilation subprocess no longer drives the cross section at high

pTγ, and now it is the LO, and the gQ → γgQ subprocesses that are the most prominent.

These differences come about for two reasons. As the LHC collides two beams of protons,

instead of the proton and antiproton beams at Fermilab, there are no longer any valence

light antiquarks present. Hence, the relative contribution of the annihilation subprocess

is decreased. Also, because the LHC will ultimately operate at a center of mass energy

which is about seven times larger than that of the Tevatron, lower values of x ∼ pT /√

s are

52

50 100 150

pTγ(GeV)

1

1.5

2

2.5

3

K-f

acto

rp+p -> γ+b+X

√ S =14 TeV

Figure 3.13: K factor, or the ratio of the NLO to the LO differential cross section forpp → bγX at

√S = 14 TeV, µ = pTγ.

probed at the LHC. For the kinematic region shown in Fig. 3.14 the gluon PDF is dominant,

accounting for the continued importance of the gQ initiated subprocesses.

An interesting consequence of this pattern of subprocess contributions is that the

dominant parts are all proportional to the heavy quark PDFs. Such was not the case for the

Tevatron curves, except for the low end of the pTγ range. Accordingly, heavy quark + photon

measurements at the LHC will have the potential to provide important cross checks on the

perturbatively calculated heavy quark PDFs, as discussed in section 3.1.2. These PDFs are

likely to provide large contributions to other physics signals, either standard model or new

physics, and such checks will be an important part of the search for new physics.

53

50 100 150

pTγ(GeV)

0.0001

0.01

1

100

dσ/d

p Tγ(p

b/G

eV)

NLOqq->QQγqQ->qQγgQ->gQγLO+gg->QQγQQ->QQγ

p+p -> γ+b+X√

S =14 TeV

Figure 3.14: Contributions of the different subprocesses to the differential cross section, dσ/dpTγ

for pp → bγX at√

S = 14 TeV, NLO (solid line), annihilation qq → γQQ (dashed line), Qq → γqQ,and Qq → γqQ (dotted line), gQ → γgQ (dot dashed line), gg → γQQ+LO (dash dot dotted line),QQ → γQQ, and QQ → γQQ (dot dash dotted line), µ = pTγ .

In Fig. 3.15 we show the scale dependence for both the NLO and LO differential cross

sections. In this case unlike the case for the Tevatron, Fig. 3.7, the NLO scale dependence

decreases and the three curves representing the NLO cross section, the solid, the dashed and

the dotted are almost exactly the same.

54

0 50 100 150 200

pTγ(GeV)

0.01

1

100

dσ/d

p Tγ(p

b/G

eV)

NLO µ=pTγ

NLO µ=2pTγ

NLO µ=pTγ/2

LO µ=pTγ

LO µ=2pTγ

LO µ=pTγ/2

p+p -> γ+b+X√

S =14 TeV

Figure 3.15: Scale dependence of the differential cross section, dσ/dpTγ for the production of adirect photon and a bottom quark at the LHC, where the three different scales have been set to beequal µ = µr = µf = µF ,for the NLO cross section µ = pTγ (solid line), µ = 2pTγ (dashed line),µ = pTγ/2 (dotted line) and for the LO cross section µ = pTγ (dot dashed line), µ = 2pTγ (dasheddot dot line), µ = pTγ/2 (dashed dashed dot line).

55

CHAPTER 4

COMPARISON TO DATA

Surmising how the world works can start with a thought and lead to complex and intricate

theories. In order to differentiate those theories that are just conjectures from those that

describe reality the constraints of experimental data are needed. From this comparison we

can also further investigate the viable theories and be able to make future predictions for

them. In this chapter we are going to present one such comparison between data and theory

for this process, and see what can be learned from it.

Until recently there was no available data for the production of a direct photon and

a heavy quark. In 2005 a measurement with a few points was presented by the CDF

collaboration [63]. However these data points were not enough for an in depth comparison,

as their error bars are quite large, and also their integrated luminosity is only 67 pb−1. The

integrated luminosity over time characterizes the amount of accumulated data in that time

period. The luminosity is given by Eq. 4.1:

L =1

σ

dN

dt, (4.1)

where N is the number of events, and σ is the measured cross section. As such the unit for

luminosity is pb−1, with 1fb−1 = 103pb−1. Recently as mentioned in Chapter 3 measurements

from the DØ collaboration have become available [52]. These measurements are based on an

integrated luminosity of 1 fb−1, and the kinematical cuts applied to them are those presented

in Chapter 3.

In Ref. [52] the differential cross section with respect to the transverse momentum of the

photon has been measured for direct photon and bottom production, and for direct photon

and charm production. The measurements in Ref. [52] are separated out into two rapidity

regions. These are region one, where the product of the rapidities is positive, ηγηQ > 0,

56

i.e., the two rapidities are simultaneously either positive or negative, ηγ > 0 and ηQ > 0, or

ηγ < 0 and ηQ < 0, and region two where the product of the rapidities is negative ηγηQ < 0,

i.e., one rapidity is positive and the other negative, ηγ > 0 and ηQ < 0 or ηγ < 0 and

ηQ > 0. This division of the rapidity intervals is done in order to probe a different range

in the parton’s momentum fraction x [64]. For example if we only have two back to back

particles produced in an event, for the two momentum fractions x1 and x2, we have,

x1 =pTγ√

S(eηγ + eηQ),

x2 =pTγ√

S(e−ηγ + e−ηQ). (4.2)

If we pick the value of the transverse momentum to be pTγ = 40 GeV, then in region one,

ηγηQ > 0 we are probing the following values of x, 0.04 < x1 < 0.1 and 0.017 < x2 < 0.04,

and in region two, ηγηQ < 0 the ranges of x is 0.03 < x1 < 0.076 and 0.028 < x2 < 0.066.

In Fig. 4.1 we present the comparison between data obtained from the DØ Run II, for

the production of a direct photon and a b quark, γ + b, and the theoretical curves we have

calculated and presented in Chapter 3, Fig. 3.2. In the graph to the left the results for the

rapidity region one are presented. The solid curve is the NLO theory prediction and the

circular dots are the data points. The graph to the right shows the comparison between

theory and data for rapidity region two, where the dashed line is the NLO theory differential

cross section, and the squares represent the data points. Again as in Section 3.1, we have

set the three different scales to be equal to one another, µ = µr = µf = µF , with µ = pTγ.

And the PDFs used for the curves are the CTEQ6.6M set.

We can see that there is really good agreement in Fig. 4.1 between the experimental data

and the theoretical predictions for γ + b production. Let us now turn our attention to the

direct photon and charm, γ+c, cross section. In Fig. 4.2 we present the comparison between

theory and data for this process, where the labels on the curves and points, the scale and

PDF set are the same as in Fig. 4.1. In this case we can see that there is still really good

agreement between theory and experiment for the data points that lie below pTγ = 70 GeV.

However unlike the photon and b quark production in this case we do not observe the good

agreement for all the data points, and the last two experimental points in Fig. 4.2 lie above

the theory curves for both rapidity regions.

In order to figure out where this discrepancy comes from we have to bear in mind that

while the theory predictions for the charm cross section do not entirely fit the data the

57

0 50 100 150 200

pTγ(GeV)

0.0001

0.01

1

dσ/d

p Tγ(p

b/G

eV)

ηγηQ>0

p+p -> γ+b+X√

S =1.96 TeV

0 50 100 150 200

ηγηQ<0

Figure 4.1: The differential cross section, dσ/dpTγ , for the production of a direct photon and a bquark, γ + b. In the graph to the left, the solid line is the NLO theory curve and the circular dotsare the data points measured by the DØ collaboration, for region one. In the graph to the right,the dashed line is the NLO theory curve and the squares are the data points measured by the DØcollaboration, for region one.

58

0 50 100 150 200

pTγ(GeV)

0.0001

0.01

1

dσ/d

p Tγ(p

b/G

eV)

ηγηQ>0

p+p -> γ+c+X√

S =1.96 TeV

0 50 100 150 200

ηγηQ<0

Figure 4.2: The differential cross section, dσ/dpTγ , for the production of a direct photon and a bquark, γ + c. In the graph to the left, the solid line is the NLO theory curve and the circular dotsare the data points measured by the DØ collaboration, for region one. In the graph to the right,the dashed line is the NLO theory curve and the squares are the data points measured by the DØcollaboration, for region one.

59

bottom ones do. What this signifies is that if something should be adjusted in the theoretical

predictions it cannot depend on any part of the calculation that is the same for both the

charm and the bottom cross sections. Since if we have to adjust it for the charm cross

section we will have to adjust it for the bottom one as well, and as such we will lose the good

agreement that we observe for the γ+b cross section. Owing to this we can conclude that the

difference observed in Fig. 4.2 is not due to any part of the cross section that is dependent

on the charge of the heavy quarks, i.e., we can assume that the discrepancy does not arise

in the calculation of the partonic cross section, where the only difference in the computation

of the charm and for the bottom cross section, comes from the exchange of the magnitude of

the two quark charges. Thus the explanation is likely to come from the treatment of the long

distance physics, i.e., through differences in the distribution and fragmentation functions for

the two cases.

Before we look for an explanation of the observed discrepancy we should check to see

how large the theoretical error is. Thus in Fig. 4.3 we have shown the scale dependence of

the cross section, shown by the solid curve, µ = pTγ, the dashed line for µ = pTγ/2, and

the dotted line for µ = 2pTγ. The scale dependence can be considered as the error due to

theory, and the first three data points agree with the theory within this error. However the

last two points again lie well above these error lines. The shaded region around the curve

for µ = pTγ comes from the PDF uncertainty. The CTEQ6.6M PDF errors are given by

the average of forty PDFs, representing the allowed variations coming from the global fits.

Thus the shaded region incorporates the values of the cross section with the use of each of

these PDFs. As we can see the dependence on this uncertainty is roughly the same as the

dependence on different values of the scale.

In Fig. 4.4 we show the scale dependence, for region two as was done in Fig. 4.3 for

region one. Again the last two data points continue to lie above the data. Thus we need to

consider an alternate explanation to what causes this discrepancy.

In Section 3.1.2 we described the possibility of the existence of an intrinsic heavy quark

component to the nucleon. We concentrated on the intrinsic charm component. We

mentioned that there can be an intrinsic bottom contribution, but that contribution is

negligible. This is due to the fact that we start the evolution of the bottom quarks at

a scale equal to the mass of the bottom quark, which is over three times larger than the

mass of the charm quark. This larger scale does not allow the non-perturbative bottom

60

0 50 100 150 200

pTγ(GeV)

0.0001

0.01

1

dσ/d

p Tγ(p

b/G

eV)

ηγηQ>0

µ=pTγ

µ=pTγ/2

µ=2pTγ

p+p -> γ+c+X√

S =1.96 TeV

Figure 4.3: Scale dependence of the differential cross section, dσ/dpTγ , for the production of adirect photon and a c quark, γ + c, compared to data (circular dots), for region one. The threedifferent scales have been set to be equal µ = µr = µf = µF , µ = pTγ (solid line), µ = pTγ/2(dashed line), µ = 2pTγ (dotted line).

contribution to become comparable to the one coming from the gluon splitting part of the

DGLAP evolutions. And so the intrinsic bottom component, if it exists, remains insignificant

[65]. As such this is a good possibility to investigate.

In Fig. 4.5 we present the comparison between data and theory with the use of the

regular PDF set, and the ones using intrinsic charm, for the γ + c production in the rapidity

region one. The solid line is the NLO differential cross section acquired with the CTEQ6.6M

PDF set, the dashed line is the differential cross section calculated with the use of the BHPS

61

0 50 100 150 200

pTγ(GeV)

0.0001

0.01

1

dσ/d

p Tγ(p

b/G

eV)

ηγηQ<0

µ=pTγ

µ=pTγ/2

µ=2pTγ

p+p -> γ+c+X√

S =1.96 TeV

Figure 4.4: Scale dependence of the differential cross section, dσ/dpTγ , for the production of adirect photon and a c quark, γ + c, compared to data (circular dots), for region two. The threedifferent scales have been set to be equal µ = µr = µf = µF , µ = pTγ (solid line), µ = pTγ/2(dashed line), µ = 2pTγ (dotted line).

IC PDFs, and the dotted one used the sea-like IC model PDFs. As we saw in Fig. 3.10 and

3.11 the increase in the cross section with the use of the sea-like PDF is about constant with

respect to the transverse momenta of the photon, and this is also shown in Fig. 4.5, where

the dotted curve is almost the same distance from the solid one. Even though this line goes

through one of the last two data points, it does not go through the first three data points,

and so it does not describe the data well at all. In Fig. 3.10 and 3.11 we also investigated the

dependence of the cross section on the other intrinsic charm model, which is the BHPS one.

62

0 50 100 150 200

pTγ(GeV)

0.0001

0.01

1

dσ/d

p Tγ(p

b/G

eV)

ηγηQ>0

CTEQ 6.6MBHPS IC modelSea-like IC model

p+p -> γ+c+X√

S =1.96 TeV

Figure 4.5: The NLO differential cross section, dσ/dpTγ , for the production of a direct photonand a c quark, γ + c, with the use of CTEQ6.6M PDFs (solid line), using the BHPS intrinsiccharm PDF, (dashed line) and with the use of the sea-like model intrinsic charm PDF (dottedline), compared to data (circular dots), region one.

We saw that the difference in the cross sections is not great at small pTγ, but increases with

larger transverse momentum. This is what we observe in Fig. 4.5 as well, where the dashed

curve moves away from the solid curve as the momentum increases. As such it follows the

data trend, but still remains substantially below the experimental measurements. And as

in Fig. 4.2 the last two data points lie above that value. Thus we can conclude that the

observed difference between the charm data and the theory can not be explained solely by

the existence of the intrinsic charm component used in the BHPS and sea-like PDFs. The

63

0 50 100 150 200

pTγ(GeV)

0.0001

0.01

1

dσ/d

p Tγ(p

b/G

eV)

ηγηQ<0

CTEQ 6.6MBHPS IC modelSea-like IC model

p+p -> γ+c+X√

S =1.96 TeV

Figure 4.6: The NLO differential cross section, dσ/dpTγ , for the production of a direct photonand a c quark, γ + c, with the use of CTEQ6.6M PDFs (solid line), using the BHPS intrinsiccharm PDF, (dashed line) and with the use of the sea-like model intrinsic charm PDF (dottedline), compared to data (circular dots), region two.

cross section can be further increased with the use of the intrinsic charm BHPS model PDF if

the non perturbative value used for the initial conditions of the DGLAP equations is larger.

However an intrinsic charm component large enough to increase the theory curves by the

amount required is unlikely. Since the intrinsic heavy quark PDFs used are trying to model

the non-perturbative charm component, it is possible that there are different ways to model

this idea [66], which will be able to account for the observed data and theory difference.

In Fig. 4.6 we show the comparison between the data and the differential cross section

64

0 50 100 150 200

pTγ(GeV)

0.0001

0.01

1

dσ/d

p Tγ(p

b/G

eV)

ηγηQ<0

CTEQ 6.6MBHPS IC model µ=p

TγBHPS IC model µ=p

Tγ/2

p+p -> γ+c+X√

S =1.96 TeV

Figure 4.7: Scale dependence of the NLO differential cross section, dσ/dpTγ , for the productionof a direct photon and a c quark, γ + c, with the use of the BHPS PDFs, µ = pTγ (dashed line),µ = pTγ/2 (dotted line), compared to data (squares), region two.

with the use of the intrinsic charm PDFs, for region two, just as was done in Fig. 4.5. As in

region one we observe the same trend as we did in region two.

In view of the above results we cannot deduce that solely the existence of intrinsic

charm is responsible for the observed discrepancy. Presently we cannot conclude if the

disagreement can be fixed by an adjustment in the theoretical calculation or the experimental

measurement. However it might turn out to be a combination of different factors. In Fig.

4.7, we show the scale dependence of the cross section with the use of the BHPS model

65

PDFs, we see that as the scale decreases to µ = pTγ/2, the cross section increases and starts

to resemble the data points more, even though it does not pass through all of them. Thus a

combination of varying the scale and using intrinsic charm PDFs seems to describe the data

better, than just one of them.

In Section 3.1 we investigated how the different subprocesses contribute to the cross

section. For Tevatron energies we found that it is the annihilation subprocess, qq → γQQ

that drives the cross section. It happens to overtake the cross section at pTγ ∼ 70 GeV,

around the value where the difference between data and theory starts. We found out

that this subprocess at high momentum looks the same for both the charm and bottom

cross sections, as it does not depend on either the charm or bottom charge or PDF and

as a result expected the ratio between the two cross sections to decrease, however this

decrease is not experimentally observed. Going back to the calculation of this process we

described the final state collinear singularity that occurs by gluon splitting in 2.3.2. This

singularity was regulated by the requirement that the heavy quarks are massive and there is

a threshold energy needed for them to be produced. By doing so we avoided the use of heavy

quark fragmentation functions. These heavy quark FFs would describe the probability of a

heavy quark to hadronize into a heavy meson. They would also obey the DGLAP evolution

equations just like the photon FF or the PDFs. The collinear singularity described can also

be taken care of by being absorbed into the heavy quark FF. In this case, if any logarithms

of the form ln(pT /mQ) occur they will be resummed by solving the evolution equations.

How this will affect the two cross sections and if it will consolidate the discrepancy will be

investigated in the near future.

In Fig. 4.8, 4.9, 4.10 and 4.11 we show for completeness the same graphs as described

above for the photon and bottom cross section, γ + b. For this case there is for both regions

really good agreement between the data and the theory. And all data points lie within the

PDF uncertainty and the scale dependence band. One thing to point out here is that the two

curves that represent the intrinsic charm PDFs cross section are very close to one another

and to the curve calculated with the regular CTEQ6.6M PDFs. This is due to the fact that

there are very few subprocesess that have an initial charm quark in this case. And as such

the dependence on the charm PDFs is minimal, therefore the use of different PDFs affects

the photon and b quark cross section almost inconceivably.

66

0 50 100 150 200

pTγ(GeV)

0.0001

0.01

1

dσ/d

p Tγ(p

b/G

eV)

ηγηQ>0

µ=pTγ

µ=pTγ/2

µ=2pTγ

p+p -> γ+b+X√

S =1.96 TeV

Figure 4.8: Scale dependence of the differential cross section, dσ/dpTγ , for the production of adirect photon and a b quark, γ + b, compared to data (circular dots), for region one. The threedifferent scales have been set to be equal µ = µr = µf = µF , µ = pTγ (solid line), µ = pTγ/2(dashed line), µ = 2pTγ (dotted line).

67

0 50 100 150 200

pTγ(GeV)

0.0001

0.01

1

dσ/d

p Tγ(p

b/G

eV)

ηγηQ<0

µ=pTγ

µ=pTγ/2

µ=2pTγ

p+p -> γ+b+X√

S =1.96 TeV

Figure 4.9: Scale dependence of the differential cross section, dσ/dpTγ , for the production of adirect photon and a b quark, γ + b, compared to data (circular dots), for region two. The threedifferent scales have been set to be equal µ = µr = µf = µF , µ = pTγ (solid line), µ = pTγ/2(dashed line), µ = 2pTγ (dotted line).

68

0 50 100 150 200

pTγ(GeV)

0.0001

0.01

1

dσ/d

p Tγ(p

b/G

eV)

ηγηQ>0

CTEQ 6.6MBHPS IC modelSea-like IC model

p+p -> γ+b+X√

S =1.96 TeV

Figure 4.10: The NLO differential cross section, dσ/dpTγ , for the production of a direct photonand a b quark, γ + b, with the use of CTEQ6.6M PDFs (solid line), using the BHPS intrinsiccharm PDF, (dashed line) and with the use of the sea-like model intrinsic charm PDF (dottedline), compared to data (circular dots), region one, µ = µr = µf = µF , µ = pTγ .

69

0 50 100 150 200

pTγ(GeV)

0.0001

0.01

1

dσ/d

p Tγ(p

b/G

eV)

ηγηQ<0

CTEQ 6.6MBHPS IC modelSea-like IC model

p+p -> γ+b+X√

S =1.96 TeV

Figure 4.11: The NLO differential cross section, dσ/dpTγ , for the production of a direct photonand a b quark, γ + b, with the use of CTEQ6.6M PDFs (solid line), using the BHPS intrinsiccharm PDF, (dashed line) and with the use of the sea-like model intrinsic charm PDF (dottedline), compared to data (circular dots), region two, µ = µr = µf = µF , µ = pTγ .

70

CHAPTER 5

MASSIVE VERSUS MASSLESS COMPARISON

In Chapter 2 we described the calculation of the NLO massless cross section for the

production of a direct photon and a heavy quark. What is meant by massless is that we set

the quark masses to zero anywhere they would appear in the calculation. We retained them

only in one case, in order to prevent a collinear singularity from occurring. As an example,

the mathematical expression corresponding to the Dirac propagator is:

i(/p + m)

p2 −m2 + iε, (5.1)

which in the massless approximation after setting the mass to be zero it becomes,

i/p

p2 + iε. (5.2)

Combining this along with all the other expressions where the mass has been set to zero

greatly simplifies the calculation. As such the terms that contribute to the matrix elements

in a massless calculation are significantly reduced.

In order to be able to work in the massless approximation the heavy quarks and photons

produced need to carry a transverse momentum, pT , which is few times larger than the mass

of the heavy quark mQ, i.e. pT ≥ 10 GeV. Since the lower bounds for the values of the

transverse momenta for direct photons and heavy quarks measurable at both the DØ and

CDF collaborations are above pT ≥ 10 GeV, the comparison with the massless calculation

that we showed in Chapters 3 and 4 is appropriate.

Here for completeness we are going to review the differences between the two approaches

and give an outline of the massive calculation, and then present a comparison between the

LO massless and the LO massive cross section. Since at the Tevatron it is the annihilation

subprocess that dominates at the NLO, and it also contributes to the LO massive cross

71

Table 5.1: Subprocesses that contribute to the LO massive calculation

LO MassiveSubprocesses

gg → γQQqq → γQQ

section, a comparison between the NLO massless and LO massive cross section will also be

shown.

5.1 Theory

The first step to the massless calculation was to recount the subprocesses that appear at

each order we computed. We saw that at the leading massless order, which was O(ααs)

there was only one hardscattering subprocess that contributed to the cross section. In the

massive calculation the LO subprocesses are different, and in fact the order in αs is also

different. In the massive case at LO there are two subprocesses that contribute, these are

the gg → γQQ and qq → γQQ, Table 5.1. These processes contributed at the NLO in the

massless calculation, and are of order αα2s, which means that the LO in this case is one order

higher in αs than in the massless case. Thus the massive LO is αα2s.

Even though they might look very dissimilar the massive and massless calculations are

just different ways in which we can describe the same process. If we disregard the notion that

there might be any non perturbative heavy quark components to the nucleon, we know that

the only way they come into existence in the proton or neutron, is through gluon splittings.

This is why we can assume that there are no subprocesses that have a heavy quark in the

initial state. And as such the lowest order possible subprocesses that are first to produce a

heavy quark and a photon in the final state, are the ones listed in Table 5.1.

Let us look at the gg → γQQ subprocess in Fig. 5.1. In diagram 1) the two initial state

gluons interact and go into a pair of heavy quarks, with the photon being emitted from the

heavy quark. We can also draw the subprocess as in diagram 2), where one of the initial

state gluons splits into a heavy quark anti quark pair, and then the heavy quark and the

72

g(p2)

g(p1)

Q(p5)g(p2)

g(p1)

γ(p3)

Q(p4)

Q(p5)

1) 2)

Figure 5.1: Feynman diagrams for the gg → γQQ subprocess. In diagram 1) the two initialgluons go into a pair of heavy quarks and the photon is emitted for the final state heavy quark. Indiagram 2) one of the initial state gluons splits into a heavy quark anti quark pair.

other initial state gluon go into a final state heavy quark and photon. We know that in the

case where the angle between the pair of heavy quarks tends to zero a collinear singularity

can occur. In this case however since we are retaining the masses of the heavy quarks this

collinear singularity is regulated and the Mandelstam variable, t25 cannot go to zero.

Nevertheless since the production of heavy quarks at high-pT has the potential to generate

logarithms of the form ln(pT /mQ) as a result of the collinear configuration involving g → QQ,

we need a way to resum these logs. As was shown in Section 2.3.2 with the use of the leading

pole approximation, in the case of a collinear singularity we can approximate the matrix

element squared for the gg → γQQ subprocess as:

|A(gg → γQQ)|2 ∼ αsPqg|A(gQ → γQ)|2, (5.3)

as shown graphically in Fig. 5.2. Since we have to convolute the matrix element in Eq. 5.3,

with the initial state gluon PDFs, we come across the following configuration, αsPqgGg. This

configuration as shown in Appendix A, is a part of the DGLAP evolution equations. Thus

we can invoke a massless heavy quark PDF, GQ, which will be a solution to the DGLAP

equations through means of which the possible large logarithms will be resummed. With the

existence of this PDF we can carry through the massless calculation, starting with the LO

Compton subprocess, Section 2.3.1.

Thus we have shown that the massless and massive calculations are basically two different

approaches to describing the same process. In the literature the massive approach is known

73

g(p2)

g(p1) Q(p4)

Q(p5)

γ(p3)

∼ αsPqg⊗

g(p1)

Q(zp2)

Figure 5.2: The gg → γQQ subprocess, in its collinear limit can be written as the Comptonsubprocess times a splitting function.

as the fixed flavor number scheme (FFN). The name refers to the fact that we are keeping

the number of quark flavors fixed irrespective of the center of mass energy reached at the

hard scattering. Appropriately, the massless case can be referred to as the variable flavor

number scheme (VFN). In the VFN we describe the proton as composed of flavors whose

number depends on the hard scattering energy scale. If the energy is above the threshold

for production of a given quark, i.e. s ∼ 4m2Q, then the quark is taken to be massless and as

a part of the proton. Thus the massless scheme allows us to use the conventional massless

distribution and fragmentation functions [67]. Similarly in Eq. 1.1 describing the evolution

of αs the number of flavors, nf changes by one as soon as we pass the energy threshold for

a flavor production.

5.2 Numerical Results

The massive matrix elements for the subprocesses listed in Table 5.1 are obtained by crossing

the matrix elements listed in Ref. [68]. Since there are no divergences at this order, the cross

section is straightforwardly obtained by Monte Carlo integration.

In Fig. 5.3 we present the differential cross section versus the transverse momentum of

the photon, dσ/dpTγ for the production of a direct photon and a bottom quark as a function

of pTγ for√

S = 1.96 TeV, using the DØ kinematic cuts, just as was done in Chapter 3.

The dotted line represents the LO massive cross section, the dashed line which is the LO

massless differential cross section, and the solid line which is the NLO one, are as shown

in Fig. 3.2. We note that in Fig. 5.3 the dotted line, starts close to the LO massless

cross section, but as the transverse momentum grows it increases and tends towards the

74

0 50 100 150 200

pTγ(GeV)

0.0001

0.01

1

dσ/d

p Tγ(p

b/G

eV)

NLO masslessLO masslessLO massive

p+p -> γ+b+X√

S =1.96 TeV

Figure 5.3: The differential cross section, dσ/dpTγ for the production of a direct photon and abottom quark as a function of pTγ for

√S = 1.96 TeV, at massless NLO (solid line), at massless

LO (dashed line), and at massive LO (dotted line), µ = µr = µf = µF , µ = pTγ .

NLO massless curve. This observation is not unexpected, as in Fig. 3.4 of Chapter 3, we

showed that the subprocess that dominates the NLO massless cross section at high pTγ, is the

annihilation subprocess, qq → γQQ. This subprocess is one of the two subprocesses present

at the leading massless order. This being the case, having the same subprocess drive both

the NLO massless and LO massive calculation we see, as expected, that the cross sections

become similar at large values of the transfer momentum. It would also be interesting to

have a comparison between the NLO massive and massless calculations, however there is no

information available on the NLO massive cross section as of yet.

75

50 100 150

pTγ(GeV)

0.01

0.1

1

10

100

dσ/d

p Tγ(p

b/G

eV)

NLO masslessLO masslessLO massive

p+p -> γ+b+X√

S =14 TeV

Figure 5.4: The differential cross section, dσ/dpTγ for the production of a direct photon and abottom quark as a function of pTγ for

√S = 14 TeV, at massless NLO (solid line), at massless LO

(dashed line), and at massive LO (dotted line), µ = µr = µf = µF , µ = pTγ .

In Ref [69] a comparison between the LO massive and LO massless differential cross

sections for the production of a charm quark and a photon at the Tevatron is presented,

in the region below pTγ < 50 GeV, and there the agreement between the LO massless and

massive cases in this energy range is also observed.

In Fig. 5.4 we investigate the LO massive cross section at LHC energies. There we

present the differential cross section versus the transverse momentum of the photon, dσ/dpTγ

for the production of a direct photon and a bottom quark as a function of pTγ, this time

at√

S = 14 TeV. The dotted line represents the LO massive cross section, the dashed line

76

is the LO massless differential cross section, and the solid line is the NLO one, just as in

Fig. 5.3. Here we observe that the LO massive curve is no longer is similar to the NLO

massless curve, but in fact is quite comparable to the LO massless differential cross section.

As described in Section 3.2, due to the higher energies and the exchange of an antiproton

beam for a proton beam, the annihilation subprocess is not dominant in this case, and the

NLO massless cross section is driven by subprocesses containing initial state gluons and

heavy quarks. The dominant subprocess in the LO massive calculation in this case is the

gg → γQQ, and it is comparable to the the massless LO Compton subprocess at the LHC

energies.

In summary we can conclude that the massless treatment is completely sanctioned, and

it is not necessary to retain the heavy quark masses in the entire calculation.

77

CHAPTER 6

CONCLUSION

In this thesis we have presented the NLO calculation and analysis of the associated

production of a direct photon and a heavy quark. Measurements of this process are available

from the DØ collaboration and should be available soon from the CDF experiment at

Fermilab. Comparing these measurements with theory provided yet another check and

confirmation of the standard model. However one of the most important reasons for

performing this calculation is the possibility of learning more about the heavy quarks PDFs.

At this stage we do not have a certain comprehension of the heavy quark structure inside the

nucleon. The radiatively generated heavy quark PDFs that we currently use have worked

well in our presently accessible energy range. It would be enlightning to have conclusive

evidence if there is a non-perturbative heavy quark component to the nucleon, not only for

purely understanding the nucleon better, but also for producing reliable predictions for the

LHC.

At the Tevatron, due to the center of mass energies and the ready availability of both

valence quarks and antiquarks we showed that the annihilation subprocess, qq → γQQ,

dominates the cross section. As such unfortunately we might not be able to extract much

information about the heavy quark PDFs from this comparison, since the annihilation

subprocess does not provide any dependance on the initial state heavy quark behaviour.

After comparing the data provided by the DØ measurements to the NLO theory, we saw

a very good agreement for the γ+b cross section. The γ+c comparison was not as favorable,

and although there was agreement between the first few data points and theory, the last two

lie above it. The possible explanation which we investigated, that the discrepency is caused

by intrinsic charm, did not increase the theory cross section enough to fit the data. One

further possibility we need to check is if the use of heavy quark fragmentation functions will

78

help reduce the observed difference.

However things look very optimistic for the LHC. Since the colliding beams consist only

of protons, there are no longer any interacting valence antiquarks. Thus the annihilation

subprocess does not play the leading role as it does at the Tevatron. The higher center of

mass energies also increase the contributions from processes that involve initial state gluons

and heavy quarks. Once measurements are available from the LHC we can expect to learn

if our radiatively generated heavy quark scenario is indeed correct or if there is an intrinsic

charm and bottom content inside the protons and antiprotons.

79

APPENDIX A

PARTON DISTRIBUTION FUNCTIONS

The scale dependence of the parton distribution functions is given by the DGLAP equations,

which have the following form:

dGqi/p(x, t)

dt=

αs

2π

∫ 1

x

dy

y[Pq←q(y, αs)Gqi/p(x/y, t) + Pqi←g(y, αs)Gg/p(x/y, t)]

dGg/p(x, t)

dt=

αs

2π

∫ 1

x

dy

y[∑

i

Pg←q(y, αs)Gqi/p(x/y, t) + Pg←g(y, αs)Gg/p(x/y, t)].(A.1)

Eq. A.1 is generally written in the shorthand notation:

dGqi/p(x, t)

dt=

αs

2π[Pqq ⊗Gqi/p + Pqig ⊗Gg/p]

dGg/p(x, t)

dt=

αs

2π[Pgq ⊗Gqi/p + Pgg ⊗Gg/p], (A.2)

where ⊗ denotes an integral convolution,

⊗ ≡∫ 1

0

dudvf(u)g(v)δ(uv − z). (A.3)

In Eq. A.1 Gqi,g/p(x, t) are the PDFs and Ppipj(z, αs) are the Altarelli-Parisi splitting

functions. The splitting functions can be written as a power series in the strong coupling

constant.

Ppipj(z, αs) = PLO

pipj(z) + αsP

NLOpipj

(z) + α2sP

NNLOpipj

(z) + ... (A.4)

The LO splitting functions, [37] are listed in Eq. A.5,

PLOqq (z) = CF

[ 1 + z2

(1− z)+

+3

2δ(1− z)

]PLO

qg (z) =1

2

[z2 + (1− z)2

]PLO

gq (z) = CF

[1 + (1− z)2

z

]PLO

gg (z) = 2N[ z

(1− z)+

+1− z

(z+ z(1− z) + δ(1− z)

11N − 2nf

6

](A.5)

80

Table A.1: A summary of the power of logs resummed, dependent on the order of the splittingfunctions

Splitting Functions logs resummed PDFsPLO

pipj(αs ln Q2)n LL

PNLOpipj

αs(αs ln Q2)n−1 NLL

PNNLOpipj

α2s(αs ln Q2)n−2 NLLL

If we plug in the LO splitting functions in the DGLAP equations, A.1, we sum up logs of

the form (αs ln Q2)n, known as the leading logs (LL), where n runs from 1 to infinity. The

solutions obtained are the LL PDFs. With the use of the NLO splitting functions [70, 71] the

subleading or the Next to Leading Logs (NLL) are resummed, and with the NNLO splitting

functions, the NNLL [72, 73] are resummed, Table A.1.

81

APPENDIX B

PHASE SPACE

2 body Phase Space

The two body phase space is given by the following general equation

dPS2 =d3~p1

(2π)32E1

d3~p2

(2π)32E2

(2π)4δ4(p12 − p1 − p2). (B.1)

Using,

δ4(p12 − p1 − p2) = δ3(~p12 − ~p1 − ~p2)δ(E12 − E1 − E2), (B.2)

and

d3~p1 = |~p21|d|p1|dΩ, (B.3)

we get for the 2 body phase space:

dPS2 =p2

1dp1dΩ

16π2E1E2

δ(E12 − E1 − E2). (B.4)

Let us make the following substitution,

w = E1 + E2, (B.5)

working in the center of mass frame of particles one and two, where ~p1 = ~p2, and also

substituting E1 =√

~p21 + m2

1, and E2 =√

~p22 + m2

2, we get

w =√

~p21 + m2

1 +√

~p21 + m2

2, (B.6)

and

dw = p1dp1E1 + E2

E1E2

. (B.7)

82

Putting Eq. B.5 and B.7 into Eq. B.4, we get

dPS2 =dΩ

16π2

p1dw

wδ(E12 − w) (B.8)

=dΩ

16π2

p1

E1 + E2

.

If we work with massless particles then E1 = ~p1, and since E1 + E2 = Ecm, |~p1| = Ecm2

, and

using that

dΩ = dcosθdφ, (B.9)

we get for the massless 2 body phase space

dPSmassless2 =

dcosθdφ

16π2

Ecm/2

Ecm

(B.10)

=dcosθ

16π.

We can substitute cosθ for the variable v which runs from 0 to 1 and as such is convenient

for use in Monte Carlo integrations,

cosθ = −1 + 2v (B.11)

3 body Phase Space

The three body phase space is given by the following general equation

dPS3 =d3~p3

2E3(2π)3

d3~p4

2E4(2π)3

d3~p5

2E5(2π)3(2π)4δ4(p1 + p2 − p3 − p4 − p5), (B.12)

which can be simplified to,

dPS3 =1

23(2π)5

d3~p3

E3

d3~p4

E4E5

δ(E1 + E2 − E3 − E4 − E5). (B.13)

Moving to the particle 4 and 5 center of mass frame, we have

√s45 = E4 + E5, (B.14)

after some manipulation,

d√

s45 =E4 + E5

E4E5

|~p4|dp4. (B.15)

83

Thus,d√

s45√s45

=|~p4|

E4E5

dp4. (B.16)

In the massless case |~p4| = 12

√s45, and so for the three body massless phase space we get,

dPSmassless3 =

1

23(2π)5

d3~p3

E3

1

2

√s45

d√

s45√s45

dΩ45δ(E1 + E2 − E3 −√

s45) (B.17)

=1

24(2π)5

d3~p3

E3

dΩ45

=1

28π4|~p3|dp3dcosθdΩ45

Making the subsitution from p3 and cosθ to the dimensionless variables v and w, which range

from 0 to 1 and are given by,

v = 1 +t13s12

(B.18)

w = − t23s12 + t13

we get for the 3 body phase space,

dPSmassless3 =

1

29π4s12vdvdwdΩ45 (B.19)

84

APPENDIX C

2 → 3 NLO MASSLESS MATRIX ELEMENTS

• g(p1)Q(p2) → γ(p3)Q(p4)g(p5)

|A(gQ → γgQ)|2 = 64π3α2sαe2

Q

(− CF

8

)[(CF −

N

2

)t24 + 2N

(t14t25 + s12s45

t15

)]×( s2

34 + t223t14s45s12t25

+t214 + s2

12

s34s45t23t25+

s245 + t225

s34t14t23s12

)(C.1)

• g(p1)g(p2) → γ(p3)Q(p4)Q(p5)

The matrix element for g(p1)g(p2) → γ(p3)Q(p4)Q(p5) can be obtained by crossing the

one for g(p1)Q(p2) → γ(p3)Q(p4)g(p5) and is,

|A(gg → γQQ)|2 = 64π3α2sαe2

Q

(3CF

64

)[(CF −

N

2

)s45 + 2N

(t14t25 + t15t24s12

)]×( s2

34 + s235

t14t24t15t25+

t214 + t215s34t24s35t25

+t224 + t225

s34t14s35t15

)(C.2)

• Q(p1)Q(p2) → γ(p3)Q(p4)Q(p5)

|A(QQ → γQQ)|2 = 64π3α2sαe2

Q

1

2

CF

N2(−s12

t13t23+

t24t23s34

+t25

t23s35

+t15

t13s35

− s45

s35s34

+

+t14

t13s34

)(s212 + t224 + t215 + s2

45

t14t25+

s212 + t214 + t225 + s2

45

t24t15−

− 1

N

(s212 + s2

45)(s12s45 − t24t15 − t25t14)

t24t25t14t15

)(C.3)

85

• Q(p1)Q(p2) → γ(p3)Q(p4)Q(p5)

The matrix element for Q(p1)Q(p2) → γ(p3)Q(p4)Q(p5) can be obtained by crossing the

one for Q(p1)Q(p2) → γ(p3)Q(p4)Q(p5) and multiplying it by two, and is,

|A(QQ → γQQ)|2 = 64π3α2sαe2

Q

CF

N2( −t15

t13s35

+s45

s35s34

+t25

t23s35

+s12

t13t23− t24

t23s34

+

+t14

t13s34

)(s212 + t224 + t215 + s2

45

t14t25+

t215 + t214 + t225 + t224s12s45

−

− 1

N

(t215 + t224)(−s12s45 + t24t15 − t25t14)

s12s45t14t25

)(C.4)

• Q(p1)q(p2) → γ(p3)Q(p4)q(p5)

|A(Qq → γQq)|2 = 64π3α2sα

2CF

N

(s212 + t224 + t215 + s2

45

t25t14

)[eQeq

(−s12

t13t23+

t24t23s34

+

+t15

s35t13− s45

s35s34

)+ e2

Q

( t14t13s34

)+ e2

q

( t25t23s35

)](C.5)

• q(p1)q(p2) → γ(p3)Q(p4)Q(p5)

The matrix element for q(p1)q(p2) → γ(p3)Q(p4)Q(p5) can be obtained by crossing the

one for Q(p1)q(p2) → γ(p3)Q(p4)q(p5) and is,

|A(qq → γQQ)|2 = 64π3α2sα

2CF

N

(t215 + t225 + t214 + t224s12s45

)[eQeq

( −t15s35t13

+t25

s35t23+

+t14

s34t13− t24

s34t23

)+ e2

q

( s12

t13t23

)+ e2

Q

( s45

s34s35

)](C.6)

86

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BIOGRAPHICAL SKETCH

Tzvetalina P. Stavreva

B.S. in Physics (with honors)

October 2002

Sofia University, St.Kliment Ohridski – Sofia, Bulgaria.

PROFESSIONAL EXPERIENCE

Graduate Research Assistant

01/2005 – present

High Energy Physics Group, Florida State University – Tallahassee, Florida.

Teaching Assistant

08/2004 – 12/2004

Department of Chemistry, Florida State University – Tallahassee, Florida.

Graduate Research Assistant

08/2003 – 12/2004

Institute of Molecular Biophysics, Florida State University – Tallahassee, Florida.

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PUBLICATIONS

• T. Stavreva, J.F. Owens, Direct Photon Production in Association With A Heavy Quark

At Hadron Colliders, arXiv:0901.3791 [hep-ph], http://arxiv.org/abs/0901.3791, Phys.

Rev. D 79, 054017 (2009)

PRESENTATIONS

• Direct Photon Production in Association with a Heavy Quark - Overview

and Comparison with Data, High Energy Seminar, Oklahoma State University, 15

January 2009

• Direct Photon Production in Association with a Heavy Quark - Comparison

with Data, The Coordinated Theoretical-Experimental Project on QCD (CTEQ)

Meeting, Argonne National Laboratory, 05 December 2008

• Direct Photon Production in Association with a Heavy Quark At Hadron

Colliders, DOE visit, Florida State University, 06 October 2008

• Direct Photon Production in Association with a Heavy Quark, American

Physical Society April Meeting, St.Louis, Missouri, 12 April 2008

• Direct Photon Production in Association with Heavy Quarks, Prospectus

Defense, Florida State University, 20 October 2006

CONFERENCES AND SCHOOLS

• The Coordinated Theoretical-Experimental Project on QCD (CTEQ) Meet-

ing, Argonne National Laboratory, 5-7 December 2008

• American Physical Society April Meeting, St.Louis, Missouri, 11-15 April 2008

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• Second Annual Dirac Lectures - Cosmology: From Inflation to the Cosmic

Microwave Background, Florida State University, March 26 - 28 2008

• First Annual Dirac Lectures - Twistors and Twistor Methods in Higher

Order Loop Amplitudes, Florida State University March 14 - 16 2007

• The Coordinated Theoretical-Experimental Project on QCD (CTEQ) Sum-

mer School, Rhodes, Greece, 1-9 July 2006

• American Physical Society April Meeting, Tampa, Florida, 16-19 April 2005

• Electroparamagnetic Resonance Workshop, Cornell University, Ithaca, New

York, 6-8 August, 2004

• Biophysical Society 48th Annual Meeting, Baltimore, Maryland, 14-18 February

2004

• Southeastern Magnetic Resonance Conference, Tallahassee, Florida, 17-19 Oc-

tober 2003

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