An alternative subtraction scheme using Nagy-Soper dipoles [Elektronische Ressource] / Cheng Chung

Sujets

• Physik

Informations

• Publié par: rheinisch-westfalischen_technischen_hochschule_-rwth-_aachen
• Publié le: 01 janvier 2011
• Nombre de lectures: 36
• Langue: English

Extrait

An alternative subtraction scheme using
Nagy-Soper dipoles
Von der Fakulta¨t fu¨r Mathematik, Informatik und Naturwissenschaften der
Rheinisch-Westfa¨lischen Technischen Hochschule Aachen
zur Erlangung des akademischen Grades eines Doktors der
Naturwissenschaften genehmigte Dissertation
vorgelegt von
M.Sc. in Physics
Cheng Han Chung
aus Zhubei, Taiwan
Referent: Universita¨tsprofessor Michael Kra¨mer
Korreferent: Universita¨tsprofessor Michal Czakon
Tag der mu¨ndlichen Pru¨fung: 21 Juni 2011
Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek
online verfu¨gbar.Abstract
In this thesis we address an alternative subtraction scheme in high energy colliders at Next-
to-Leading Order (NLO) QCD calculations. In particular, we focus on the treatment of real
radiation contributions in the soft and collinear limits. After UV-renormalization, the remaining
infrared singularities appearing both in the real radiation and in the virtual contributions can be
regularized using dipole subtraction method. In this scheme, dipoles are based on the momentum
mapping and on the splitting functions derived from an improved parton shower formulation
with quantum interference effects. In our new scheme, we employ a slightly altered momentum
mapping such that the number of subtraction terms is greatly reduced in comparison with the
standard Catani-Seymour scheme. In addition, the new scheme also facilitates the matching
of NLO calculations with parton showers using the same splitting functions. We also achieve
the complete integrations of the splitting functions over an unresolved one parton phase space,
obtaining the correct soft and collinear singularity structures that are necessary to cancel the soft
divergences in the virtual contributions. We discuss the general framework setup of the scheme
as well as some scattering processes at colliders; we find complete agreement with the results in
the widely used Catani-Seymour dipole subtraction scheme.Contents
1 A brief review of QCD 6
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 ColourSU(3) and quark confinement . . . . . . . . . . . . . . . . . . . . . . 8C
1.3 QCD Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Local gauge invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6 Parton branching at Next-to-Leading Order . . . . . . . . . . . . . . . . . . . . 18
2 Nagy-Soper dipoles 22
2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 General structure of the NLO cross section and subtraction procedure . . . . . . 25
2.2.1 The general subtraction procedure . . . . . . . . . . . . . . . . . . . . . 25
2.2.2 Generalization to hadron collisions . . . . . . . . . . . . . . . . . . . . 28
2.2.3 Observable-dependent formulation of the subtraction method . . . . . . . 30
2.3 Scheme setup and momentum mapping . . . . . . . . . . . . . . . . . . . . . . 31
2.3.1 Splitting a final state parton . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.2 Splitting an initial state parton . . . . . . . . . . . . . . . . . . . . . . . 36
2.4 Splitting functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4.1 Definition of the splitting amplitudes . . . . . . . . . . . . . . . . . . . 404 CONTENTS
2.4.2 Eikonal factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4.3 Collinear splitting functions: final state splittings . . . . . . . . . . . . . 47
2.4.4 Collinear splitting functions: initial state splittings . . . . . . . . . . . . 52
2.4.5 Soft splitting functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.5 Integrated splitting functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.5.1 Collinear integrals: final state splittings . . . . . . . . . . . . . . . . . . 60
2.5.2 Collinear integrals: initial state splittings . . . . . . . . . . . . . . . . . 61
2.5.3 Interference between initial states . . . . . . . . . . . . . . . . . . . . . 64
2.5.4 Interference between initial and final states . . . . . . . . . . . . . . . . 65
2.5.5 Interference between final (final and initial) states . . . . . . . . . . . . . 66
3 Applications 71
3.1 SingleW production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.1.1 Tree level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.1.2 Real emission, virtual correction and dipole subtraction . . . . . . . . . . 72
+ −3.2 Dijet production ine e annihilation . . . . . . . . . . . . . . . . . . . . . . . 78
3.3 Higgs production in gluon-gluon fusion:gg→H . . . . . . . . . . . . . . . . 85
3.3.1 The subprocessqq¯→gH . . . . . . . . . . . . . . . . . . . . . . . . 89
3.3.2 The subprocessqg→qH . . . . . . . . . . . . . . . . . . . . . . . . 91
3.3.3 The subprocessgg→gH . . . . . . . . . . . . . . . . . . . . . . . . 93
3.3.4 One-loop virtual corrections . . . . . . . . . . . . . . . . . . . . . . . . 95
3.3.5 Proof: Nagy-Soper scheme and Catani-Seymour scheme . . . . . . . . . 98
3.3.6 Proof: Catani-Seymour scheme and literature results . . . . . . . . . . . 100
3.4 Higgs decay:H→gg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.4.1 The subprocessH→gqq¯ . . . . . . . . . . . . . . . . . . . . . . . . 102
3.4.2 The subprocessH→ggg . . . . . . . . . . . . . . . . . . . . . . . . 103CONTENTS 5
4 Conclusions 108
5 Appendix 111
5.1 Useful mathematical formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.1.1 Gamma function, Beta function and Hypergeometric function . . . . . . 111
5.1.2 Dilogarithm function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.1.3 The volume element ind dimensions . . . . . . . . . . . . . . . . . . . 113
5.2 Integration measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.2.1 Final state splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.2.2 Initial state splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.2.3 Master integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.2.4 Pole extractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.3 Colour algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.4 Phase space integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.5 Soft photon radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.6 Collinear photon radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.6.1 Final state radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.6.2 Initial state radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.7 One-loop calculations: examples . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.8 A review of the Standard Model (SM) . . . . . . . . . . . . . . . . . . . . . . . 140
5.8.1 Abelian gauge theory: Quantum electrodynamics (QED) . . . . . . . . . 140
5.8.2 Non-abelian gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.8.3 Electroweak theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.8.4 The Standard Model of particle physics . . . . . . . . . . . . . . . . . . 156Chapter 1
A brief review of QCD
1.1 Introduction
In elementary particle physics, the fundamental theory of the strong interaction is described by
Quantum Chromodynamics (QCD). It describes the interactions between quarks and gluons, and
in particular how they bind together to form hadrons (e.g. the proton and the neutron). QCD is a
quantum field theory within a special class described by non-abelian gauge theory [1] (or some-
times called Yang-Mills gauge theory). It is based on theSU(N) gauge group. Understanding
how to use non-abelian gauge theory, combined with the parton model, led to the development
of QCD and it is now a very well established theory in the sense that QCD predictions have
successfully accounted for all the strong interaction experiments observed at colliders, in par-
+ −ticular the phenomena of hadronic jet structure ine e annihilation, the Drell-Yan process and
heavy quark production [100]. QCD has two peculiar properties, which differ from Quantum
Electrodynamics (QED)/electroweak interactions and which also reveal its uniqueness.
• Asymptotic freedom: this means that at very high energies, the strong force (also called
the colour force) of quarks and gluons is so weak that they behave almost as free parti-
cles when the quarks or gluons are really close to each other. This phenomenon is called
asymptotic freedom and it is due to the fact that the strong running coupling constant
2 2 1α (Q ) depends on the energy scaleQ;α (Q ) becomes weaker as the scaleQ increases .s s
To check this, one must determine the running of the coupling constantα , which is gov-s
erned by the renormalization group equation,
∂αs2β(α ) = Qs 2∂Q
1In contrast to QED where the couplingα becomes strong at high energies.1.1 Introduction 7
ifβ < 0, the theory is asymptotically free. The asymptotic freedom of QCD also explains
why we can apply a perturbative approach to explore the structure