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Factorization at the LHC: from Pdfs to Initial State Jets (Ph. D. Thesis)

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Factorization at the LHC: from Pdfs to Initial State Jets (Ph. D. Thesis) Factorization at the LHC: From PDFs to Initial State Jets by Wouter J. Waalewijn Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2010 c Massachusetts Institute of Technology 2010. All rights reserved. Author............................................................................ Department of Physics May 12, 2010 Certified by........................................................................ Iain W. Stewart Associate Professor Thesis Supervisor arXiv:1104.5164v1 [hep-ph] 27 Apr 2011 Accepted by....................................................................... Krishna Rajagopal Associate Department Head for Education 2 Factorization at the LHC: From PDFs to Initial State Jets by Wouter J. Waalewijn Submitted to the Department of Physics on May 12, 2010, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Abstract New physics searches at the LHC or Tevatron typically look for a certain number of hard jets, leptons and photons. The constraints on the hadronic final state lead to large logarithms, which need to be summed to obtain reliable theory predictions. These constraints are sensitive to the strong initial-state radiation and resolve the colliding partons inside initial- state jets. We find that the initial state is properly described by \beam functions". We introduce an observable called \beam thrust" τB, which measures the hadronic radiation relative to the beam axis. By requiring τB 1, beam thrust can be used to impose a central jet veto, which is needed to reduce the large background in H WW `ν`ν¯ ! ! from tt¯ W W b¯b. We prove a factorization theorem for \isolated" processes, pp XL ! ! where the hadronic final state X is restricted by τB 1 and L is non-hadronic. This n m factorization theorem enables us to sum large logarithms αs ln τB in the cross section and involves beam functions. The beam thrust spectrum allows us to study initial-state radiation in perturbation theory, which can be compared with experiment and Monte Carlo. We present results for the beam thrust cross section for Drell-Yan and Higgs production through gluon fusion at next-to-next-to-leading-logarithmic order. The beam functions depend on the momentum fraction x and the (transverse) virtuality t of the colliding partons. The t dependence can be calculated in perturbation theory by matching beam functions onto the parton distribution functions (PDFs) at an intermediate scale µB pt. We calculate all the one-loop matching coefficients. Below µB we have the ∼ usual DGLAP evolution for the PDFs. Above µB, the evolution of the beam function is in t and does not change x or the parton type. We introduce the event shape \N-jettiness" τN , which generalizes beam thrust to events with N signal jets. Requiring τN 1 restricts radiation between the signal jets and vetoes additional undesired jets. This yields a factorization formula with inclusive beam and n m jet functions and allows us to sum the large logarithms αs ln τN from the phase space restriction. Thesis Supervisor: Iain W. Stewart Title: Associate Professor Acknowledgments It has been a pleasure to work under the supervision of Prof. Iain Stewart, and I want to thank him for all the advice and time he has given me. The work in this thesis has been done in collaboration with Dr. Frank Tackmann, who I would like to thank for many enjoyable discussions and for sharing some of his physical intuition and passion for physics. I also want to thank Dr. Carola Berger and Claudio Marcantonini with whom I collaborated on some of the research presented in this thesis. Thanks go to my thesis committee members Prof. Robert Jaffe and Prof. Allan Adams. I have enjoyed and benefitted from being part of a research group that has included Dr. Keith Lee, Dr. Chris Arnesen, Dr. Ambar Jain, Dr. Massimiliano Procura, Riccardo Abbate, Teppo Jouttenus and Roberto Sanchis. I am very grateful for the love and support of my parents Caspar and Heleen and my family. I especially want to thank my wife Eileen. Her love, care and encouragement have been a joy and help throughout my PhD. The MIT Graduate Christian Fellowship has been a great place of friendship, fun and growth, in particular the Sidney-Pacific bible study group. I am thankful for the many friends, both in an outside of the Center of Theoretical Physics that have been part of my life during my time at MIT. I realize that all my gifts and talents are not something I deserve, and I am grateful to God for this opportunity as well as the results. This work was supported by the Office of Nuclear Physics of the U.S. Department of Energy, under the grant DE-FG02-94ER40818. 6 Contents 1 Introduction 15 1.1 Effective Physics . 15 1.2 Predicting Collisions at the LHC . 16 1.3 Weak Interactions and Effective Field Theory . 22 1.4 Soft-Collinear Effective Theory . 28 1.5 Outline . 35 2 Introduction to Beam Functions and Isolated Drell-Yan 37 2.1 Drell-Yan Factorization Theorems . 38 2.2 Generalized Observables . 47 2.3 Relating Beam Functions and PDFs . 50 2.4 Comparison with Initial-State Parton Shower . 53 2.5 Relation to Fixed-Order Calculation . 54 2.6 Renormalization Group Evolution . 57 3 Beam Functions 61 3.1 Definition . 61 3.2 Renormalization and RGE . 67 3.3 Operator Product Expansion . 72 3.4 Tree-level Matching onto PDFs . 74 3.5 Analytic Structure and Time-Ordered Products . 76 4 The Quark Beam Function 81 4.1 Quark PDF with Offshellness IR Regulator . 82 4.2 Quark Beam Function with Offshellness IR Regulator . 88 7 4.3 Renormalization and Matching . 93 5 The Gluon Beam Function 97 5.1 The Gluon PDF at One Loop . 98 5.2 SCET Feynman Rules . 99 5.3 The Gluon Beam Function at One Loop . 100 5.4 Renormalization, Running and Matching . 104 6 Beam Function Plots 107 7 Isolated Factorization Theorem 115 7.1 SCET . 116 7.2 Kinematics . 119 7.3 Derivation of Isolated Factorization Theorem . 123 7.3.1 Cross Section in QCD . 123 7.3.2 Matching QCD onto SCET . 125 7.3.3 Soft-Collinear Factorization . 127 7.3.4 Cancellation of Glauber Gluons . 135 7.4 Final Results for Drell-Yan . 138 7.4.1 The Leptonic Tensor . 138 7.4.2 The Hard Function . 139 7.4.3 The qq¯ Soft Function . 142 7.4.4 Final Cross Section for Beam Thrust . 143 8 Beam Thrust Cross Section for Drell-Yan 145 9 Beam Thrust Cross Section for Higgs Production 151 9.1 Introduction . 151 9.2 Plots . 155 10 N-Jettiness: An Inclusive Event Shape to Veto Jets 159 10.1 Introduction . 159 10.2 Definition . 161 10.2.1 N = 2 for e+e− Jets. 162 ! 10.2.2 N = 0 for Drell-Yan . 163 8 10.2.3 General Case . 164 10.2.4 Generalizations . 165 10.3 Factorization formula . 166 11 Conclusions and Outlook 169 A Notation Cheat Sheet 171 A.1 Lightcone Coordinates . 171 A.2 Symbols . 172 B Plus Distributions and Discontinuities 175 C Renormalization Structure of the Beam Function 177 D Quark Beam Function Matching in Pure Dimensional Regularization 181 E Perturbative Results 185 E.1 Fixed-Order Results for Drell-Yan . 185 E.2 Fixed-Order Results for Higgs Production . 186 E.3 Renormalization Group Evolution . 187 9 10 List of Figures 1-1 Schematic picture of a proton-(anti)proton collision at the LHC or Tevatron. 17 1-2 Examples of events at the LHC. 19 1-3 The tree-level process for c sud¯ in the full and effective theory. 23 ! 1-4 The one-loop QCD corrections to c sud¯ in the full and effective theory. 24 ! 1-5 Schematic diagram of a proton-(anti)proton collision up to the hard interaction. 29 1-6 Feynman rule for the interaction between a collinear quark and a soft gluon. 30 1-7 Schematic diagram of a proton-(anti)proton collision including Glauber gluons. 32 1-8 The matching for Higgs production through gluon fusion. 33 1-9 The decoupling of soft radiation from collinear modes in pictures. 34 2-1 The final-state configuration for inclusive, isolated and threshold Drell-Yan and threshold and isolated dijet production. 39 2-2 Definition of hemispheres and kinematic variables for isolated Drell-Yan. 42 2-3 Generalized definition of hemispheres for a boosted partonic center-of-mass. 49 2-4 Evolution of the initial state. 51 2-5 Factorization for isolated Drell-Yan in pictures. 55 2-6 RGE running for inclusive, threshold and isolated Drell-Yan. 58 3-1 Tree-level diagram for the quark PDF and the quark beam function. 75 3-2 Cuts in the complex b+ plane for the time-ordered product in Eq. (3.52). 78 4-1 One-loop diagrams for the quark PDF. 82 4-2 One-loop diagrams for the quark beam function. 89 5-1 The SCET Feynman diagrams for the gluon field strength. 99 5-2 One-loop diagrams for the gluon beam function. 101 11 6-1 The u,u ¯ and g beam functions at the hard scale. 109 6-2 The u and d beam functions at the beam scale. 111 6-3 Theu ¯ and d¯ beam functions at the beam scale. 112 6-4 The gluon beam function at the beam scale. 112 7-1 Definition of the different collinear momenta related to the incoming beams. 119 7-2 RGE running including potential Glauber modes. 135 8-1 Drell-Yan cross section dσ=dQ with a cut on beam thrust.
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