CERN-PH-TH-2010-298 KA-TP-40-2010 Cavendish-HEP-10/21 DCPT/10/202 MAN/HEP/2010/23 IPPP/10/101 SLAC-PUB-14333 LU TP 10-28 HD-THEP-10-24 MCnet-11-01 General-purpose event generators for LHC physics Andy Buckleya, Jonathan Butterworthb, Stefan Giesekec, David Grellscheidd, Stefan H¨ochee, Hendrik Hoethd, Frank Kraussd, Leif L¨onnbladf,g, Emily Nurseb, Peter Richardsond, Steffen Schumannh, Michael H. Seymouri, Torbj¨ornSj¨ostrandf, Peter Skandsg, Bryan Webberj aPPE Group, School of Physics & Astronomy, University of Edinburgh, EH25 9PN, UK bDepartment of Physics & Astronomy, University College London, WC1E 6BT, UK cInstitute for Theoretical Physics, Karlsruhe Institute of Technology, D-76128 Karlsruhe dInstitute for Particle Physics Phenomenology, Durham University, DH1 3LE, UK eSLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA fDepartment of Astronomy and Theoretical Physics, Lund University, Sweden gPH Department, TH Unit, CERN, CH-1211 Geneva 23, Switzerland hInstitute for Theoretical Physics, University of Heidelberg, 69120 Heidelberg, Germany iSchool of Physics and Astronomy, University of Manchester, M13 9PL, UK jCavendish Laboratory, J.J. Thomson Avenue, Cambridge CB3 0HE, UK Abstract We review the physics basis, main features and use of general-purpose Monte Carlo event generators for the simulation of proton-proton collisions at the Large Hadron Collider. Topics included are: the generation of hard- scattering matrix elements for processes of interest, at both leading and next- to-leading QCD perturbative order; their matching to approximate treat- ments of higher orders based on the showering approximation; the parton and dipole shower formulations; parton distribution functions for event gen- erators; non-perturbative aspects such as soft QCD collisions, the underly- ing event and diffractive processes; the string and cluster models for hadron formation; the treatment of hadron and tau decays; the inclusion of QED radiation and beyond-Standard-Model processes. We describe the principal features of the Ariadne, Herwig++, Pythia 8 and Sherpa generators, to- gether with the Rivet and Professor validation and tuning tools, and discuss the physics philosophy behind the proper use of these generators and tools. This review is aimed at phenomenologists wishing to understand better how parton-level predictions are translated into hadron-level events as well as ex- perimentalists wanting a deeper insight into the tools available for signal and background simulation at the LHC. Preprint submitted to Physics Reports January 14, 2011 Work supported in part by US Department of Energy contract DE-AC02-76SF00515. Keywords: QCD, hadron colliders, Monte Carlo simulation Contents 1 General introduction 6 I Review of physics behind MC event generators 11 2 Structure of an event 11 2.1 Jets and jet algorithms . 13 2.2 The large-Nc limit . 14 3 Hard subprocesses 14 3.1 Factorization formula for QCD cross sections . 15 3.2 Leading-order matrix-element generators . 17 3.3 Choices for renormalization and factorization scales . 17 3.4 Choices for PDFs . 18 3.5 Anatomy of NLO cross section calculations . 18 3.6 Summary . 20 4 Parton showers 21 4.1 Introduction: QED bremsstrahlung in scattering processes . 21 4.2 Collinear final state evolution . 22 4.3 Soft gluon emission . 29 4.4 Initial state evolution . 31 4.5 Connecting parton showers to the hard process . 34 4.6 Quark mass effects . 39 4.7 The dipole approach to parton showering . 41 4.8 Summary . 43 5 ME and NLO matching and merging 44 5.1 Introduction . 44 5.2 Correcting the first emission . 48 5.2.1 The NLO cross section . 48 5.2.2 The first emission in a parton shower . 50 5.2.3 Powheg ......................... 52 5.2.4 MC@NLO . 53 2 5.3 Tree-level multi-jet merging and CKKW . 54 5.3.1 Merging for the first emission . 54 5.3.2 Multi-jet merging . 55 5.4 Multi-jet NLO merging . 57 5.5 Summary . 57 6 PDFs in event generators 58 7 Soft QCD and underlying event physics 60 7.1 Primordial k ........................... 61 ? 7.2 Soft QCD processes . 64 7.3 Models based on multiple parton interactions (MPI) . 68 7.3.1 Basics of MPI . 68 7.3.2 Impact parameter dependence . 72 7.3.3 Perturbative corrections beyond MPI . 75 7.3.4 Non-perturbative aspects . 76 7.4 Colour reconnections . 80 7.5 Diffraction and models based on pomerons . 81 7.6 Summary . 82 8 Hadronization 84 8.1 Definition and early developments . 84 8.2 String model . 86 8.3 Cluster model . 95 8.4 Summary . 100 9 Hadron and tau decays 101 10 QED radiation 106 11 BSM in general-purpose generators 108 II Specific reviews of main generators 111 12 Ariadne 111 12.1 Introduction . 111 12.2 Hadronic collisions . 113 12.3 The Ariadne program and the LHC . 114 3 13 Herwig++ and ThePEG 115 13.1 Introduction . 115 13.2 ThePEG . 116 13.3 Hard processes . 117 13.4 BSM physics . 117 13.5 Parton showering . 118 13.6 Multiple parton interactions and beam remnants . 119 13.7 Hadronization . 119 13.8 Hadron decays and QED radiation . 120 13.9 Outlook . 121 14 Pythia 8 121 14.1 Introduction . 121 14.2 Hard processes . 122 14.3 Soft processes . 124 14.4 The perturbative evolution . 125 14.5 Parton showering . 126 14.6 Multiple parton interactions and beam remnants . 127 14.7 Hadronization . 128 14.8 Program structure and usage . 129 14.9 Summary . 129 15 Sherpa 129 15.1 Introduction . 129 15.2 Hard processes . 130 15.3 Parton showering . 133 15.4 Matrix-element parton-shower merging . 134 15.5 Multiple parton interactions and beam remnants . 135 15.6 Hadronization . 136 15.7 Hadron decays and QED radiation . 137 15.8 Interfaces and extensions . 137 15.9 Summary . 140 III The use of generators 141 4 16 Physics philosophy behind phenomenology and generator val- idation 141 16.1 Physical observables and Monte Carlo truth . 141 16.2 Making generator-friendly experimental measurements . 142 16.3 Evaluation of MC-dependent systematic errors . 146 17 Validation and tuning 148 17.1 Generator validation and tuning strategies . 148 17.2 Rivet . 152 17.3 Professor . 154 18 Illustrative results 157 Acknowledgements 170 IV Appendices 170 Appendix A Monte Carlo methods 170 Appendix A.1 Generating distributions . 170 Appendix A.2 Monte Carlo integration and variance reduction . 171 Appendix A.3 Veto method . 173 Appendix B Evaluation of matrix elements 175 Appendix B.1 Matrix element calculation . 175 Appendix B.2 Phase-space integration . 180 Appendix B.3 Interface structures . 183 Appendix C Top quark mass definitions 184 References 192 5 1. General introduction Understanding the final states of high energy particle collisions such as those at the Large Hadron Collider (LHC) is an extremely challenging theo- retical problem. Typically hundreds of particles are produced, and in most processes of interest their momenta range over many orders of magnitude. All the particle species of the Standard Model (SM), and maybe some beyond, are involved. The relevant matrix elements are too laborious to compute beyond the first few orders of perturbation theory, and in the case of QCD processes they involve the intrinsically non-perturbative and unsolved prob- lem of confinement. Once these matrix elements have been computed within some approximation scheme, there remains the problem of dealing with their many divergences and/or near-divergences. Finally they must be integrated over a final-state phase space of huge and variable dimension in order to obtain predictions of experimental observables. Over the past thirty years an armoury of techniques has been developed to tackle these seemingly intractable problems. The crucial tool of factoriza- tion allows us to separate the treatment of many processes of interest into different regimes, according to the scales of momentum transfer involved. At the highest scales, the constituent partons of the incoming beams interact to produce a relatively small number of energetic outgoing partons, leptons or gauge bosons. The matrix elements of these hard subprocesses are per- turbatively computable. At the very lowest scales, of the order of 1 GeV, incoming partons are confined in the beams and outgoing partons interact non-perturbatively to form the observed final-state hadrons. These soft pro- cesses cannot yet be calculated from first principles but have to be modelled. The hard and soft regimes are distinct but connected by an evolutionary process that can be calculated in principle from perturbative QCD. One con- sequence of this scale evolution is the production of many additional partons in the form of initial- and final-state parton showers, which eventually par- ticipate in the low-scale process of hadron formation. All three regimes of this highly successful picture of hard collisions are eminently suited to computer simulation using Monte Carlo techniques. The large and variable dimension of the phase space, 3n 4 dimensions1 plus flavour and spin labels for an n-particle final state, makes− Monte Carlo the 1Three components of momentum per produced particle, minus four constraints of overall energy-momentum conservation. 6 integration method of choice: its accuracy improves inversely as the square root of the number of integration points, irrespective of the dimension. The evolution of scales that leads to parton showering is a Markov process that can be simulated efficiently with Monte Carlo techniques, and the avail- able hadronization models are formulated as Monte Carlo processes from the outset. Furthermore the factorized nature of the problem means that the treatment of each regime can be improved systematically as more precise perturbative calculations or more sophisticated hadronization models become available. Putting all these elements together, one has a Monte Carlo event gen- erator capable of simulating a wide range of the most interesting processes that are expected at the LHC, which can be used for several distinct pur- poses in particle physics experiments.