Fundamental Theories of Physics - Strange...

Fundamental Theories of Physics

Volume 162

Series Editors

Philippe Blanchard, Universitat Bielefeld, Bielefeld, GermanyPaul Busch, University of York, Heslington, York, United KingdomBob Coecke, Oxford University Computing Laboratory, Oxford, United KingdomDetlef Duerr, Mathematisches Institut, Munchen, GermanyRoman Frigg, London School of Economics and Political Science, London, United KingdomChristopher A. Fuchs, Perimeter Institute for Theoretical Physics, Waterloo, Ontario, CanadaGiancarlo Ghirardi, Universita di Trieste, Trieste, ItalyDomenico Giulini, Leibnitz Universitat Hannover, Hannover, GermanyGregg Jaeger, Boston University CGS, Boston, USAClaus Kiefer, Universitat Koln, Koln, GermanyKlaas Landsman, Radboud Universiteit Nijmegen, Nijmegen, The NetherlandsChristian Maes, K.U. Leuven, Leuven, BelgiumHermann Nicolai, Max-Planck-Institut fur Gravitationsphysik, Golm, GermanyVesselin Petkov, Concordia University, Montreal, CanadaAlwyn van der Merwe, University of Denver, Denver, USARainer Verch, Universitat Leipzig, Leipzig, GermanyReinhard Werner, Leibnitz Universitat Hannover, Hannover, GermanyChristian Wuthrich, University of California, San Diego, La Jolla, USA

For further volumes:http://www.springer.com/series/6001

Luc Blanchet Alessandro SpallicciBernard WhitingEditors

Mass and Motionin General Relativity

ABC

EditorsLuc BlanchetInstitut dAstrophysique de Paris98 bis Boulevard Arago75014 [email protected]

Alessandro SpallicciUniversite dOrleansObservatoire des Sciences

de lUnivers en region CentreLPC2E-CNRS, 3A Avenue

de la Recherche Scientifique45071 [email protected]

Bernard WhitingUniversity of FloridaDepartment of PhysicsP.O. Box 118440, GainesvilleFL 32611-8440, [email protected]

ISBN 978-90-481-3014-6 e-ISBN 978-90-481-3015-3DOI 10.1007/978-90-481-3015-3Springer Dordrecht Heidelberg London New York

Library of Congress Control Number: 2010938712

c Springer Science+Business Media B.V. 2011No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or byany means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without writtenpermission from the Publisher, with the exception of any material supplied specifically for the purposeof being entered and executed on a computer system, for exclusive use by the purchaser of the work.

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Springer is part of Springer Science+Business Media (www.springer.com)

Mass and Motion in General Relativity

From the infinitesimal scale of particle physics to the cosmic scale of the universe,research is concerned with the nature of mass. While there have been spectacularadvances in physics during the past century, mass still remains a mysterious entityat the forefront of current research. Particle accelerators in the quest for the Higgsboson responsible for the mass of particles, laser interferometers that are sensitiveenough to respond to gravitational waves generated by the motion of astrophysi-cal bodies, equivalence principle tests of the relationship between gravitational andinertial mass are among the most ambitious and expensive experiments that funda-mental physics has ever envisaged.

Our current perspective on gravitation has arisen over millennia, through fallingapples, lift thought experiments and stars spiraling into black holes. In this volume,the worlds leading scientists offer a multifaceted approach to mass by giving a con-cise and introductory presentation into their particular research on gravity. The maintheme is mass and its motion within general relativity and other theories of gravity,particularly for compact bodies. Within this framework, all articles are tied togethercoherently, covering post-Newtonian and related methods applied to in-spiralingcompact binaries, as well as the self-force approach to the analysis of motion.

All contributions reflect the fundamental role of mass in physics, from issuesrelated to Newtons laws, via the effect of self-force and radiation reaction withintheories of gravitation, to the role of the Higgs boson in modern physics. Precisionmeasurements are described in detail; modified theories of gravity reproducing ex-perimental data are investigated as alternatives to dark matter and the fundamentalproblem of reconciling the theory of gravity with the physics of quantum fields isaddressed.

Radiation and motion have been hotly debated within general relativity from theinception of the theory well beyond the theoreticians arena. Mass and motion areintimately intertwined as self-acceleration depends directly on the mass of the body

Lectures from the School on Mass held at Orleans on 2325 June 2008Organised by the Observatoire des Sciences de lUnivers en region Centre OSUC, UniversitedOrleans UO, Centre National de la Recherche Scientifique CNRS

v

vi Mass and Motion in General Relativity

experiencing it. Recent developments have shown that the computation of radiationreaction is unavoidable for determining the gravitational waveforms emitted notonly by large bodies in binary formation but also from sources such as the captureof stellar size objects by super-massive black holes.

The main theme of this volume is indeed mass and its motion within generalrelativity (and other theories of gravity), particularly for compact bodies, to whichmany articles directly refer.

Within this framework, after a presentation of the mass and momentum in gen-eral relativity (Jaramillo and Gourgoulhon), there are chapters on post-Newtonian(Blanchet, Schafer), effective one-body (Damour and Nagar) methods as well as onthe self-force approach to the analysis of motion (Wald with Gralla, Detweiler, Pois-son, Barack, Galtsov). post-Newtonian and self-force methods converge in theircommon domain of applicability (Blanchet, Detweiler, Le Tiec and Whiting). Asnapshot on the state of the art of the self-force (Burko) and the historic devel-opment of the field including future perspectives for the classic free fall problem(Spallicci) conclude this central part.

Auxiliary chapters set the context for these theoretical contributions within awider context. The space mission LISA (Jennrich) has been designed to detect thegravitational waves from EMRI captures. Motion in modern gravitation demandsan account of the relation between vacuum fluctuations and inertia (Jaekel andReynaud). A volume centred on the fundamental role of mass in physics should faceissues related to the basic laws of mechanics proposed by Newton (Lammerzahl)and precision measurements (Davis).

The role of the Higgs boson within physics is to give a mass to elementary parti-cles (Djouadi), by interacting with all particles required to have a mass and therebyexperiencing inertia.

Motion of stars and of galaxies are explicable according to most researchersby only evoking yet undetected matter and energy constituting around 95% of ouruniverse. A proposed alternative to dark matter theories is due to the modified theo-ries of gravity (Esposito-Farese) such as MOND (MOdified Newtonian Dynamics).Even if general relativity does not explain gravity, there still remains the fundamen-tal problem of reconciling any theory of gravity with the physics of quantum fields(Noui), itself so well verified experimentally.

The book is based upon the lectures of the School on Mass held in Orleans,France, in June 2008. The school was funded by CNRS Centre National dela Recherche Scientifique, INSU Institut National des Sciences de lUnivers,UO Universite dOrleans, Region Centre, Conseil Regional du Loiret, Observa-toire de Paris and was organised by OSUC Observatoire des Sciences de lUniversen region Centre and its associated laboratory LPC2E Laboratoire de Physique etChimie de lEnvironnement et de lEspace.

The editors wish to thank the OSUC director (Elisabeth Verges) for continuoussupport and organisation of the school; the OSUC staff (S. Bouquet, T. Cantalupo,L. Catherine, N. Rolland) who dealt with all issues related to the practical or-ganisation and running of two international events (the School followed up bythe 11th Capra meeting on radiation reaction); the LPC2E director (M. Tagger)

Mass and Motion in General Relativity vii

for suggestions and hosting the Capra workshop; the local CNRS delegation(P. Letourneux) for assistance and support; M. Volkov (Univ. Tours) for suggestionsand all members of the scientific and organisation committees, especially S. Cordier(MAPMO - Univ. Orleans).

Both events are shown on the OSUC web pages: http://www.cnrs-orleans.fr/osuc/conf/

The contributions to this book have been anonymously refereed and revised bythe editors.

Luc BlanchetAlessandro Spallicci

Bernard WhitingEditors

Contents

Mass and Motion in General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

The Higgs Mechanism and the Origin of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Abdelhak Djouadi1 The Standard Model and the Generation of Particle Masses . . . . . . . . . . . . . . . . 1

1.1 The Elementary Particles and Their Interactions . . . . . . . . . . . . . . . . . . . . . 21.2 The Standard Model of Particle Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 The Higgs Mechanism for Mass Generation .. . . . . . . . . . . . . . . . . . . . . . . . . 5

2 The Profile of the Higgs Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1 Characteristics of the Higgs Boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Constraints on the Higgs Boson Mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 The Higgs Decay Modes and Their Rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Higgs Production at the LHC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1 The Large Hadron Collider . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 The Production of the Higgs Boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Detection of the Higgs Boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4 Determination of the Higgs Boson Properties . . . . . . . . . . . . . . . . . . . . . . . . 18

4 The Higgs Beyond the Standard Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Testing Basic Laws of Gravitation Are Our Postulateson Dynamics and Gravitation Supported by ExperimentalEvidence? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Claus Lammerzahl1 Introduction Why Gravity Is So Exceptional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 Key Features of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Standard Tests of the Foundations of Special and General Relativity . . . . . . . 28

3.1 Tests of Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2 Tests of the Universality of Free Fall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3 Tests of the Universality of the Gravitational Redshift . . . . . . . . . . . . . . . 313.4 The Consequence.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

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4 Tests of Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.1 The Gravitational Redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2 Light Deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.3 Perihelion/Periastron Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.4 Gravitational Time Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.5 LenseThirring Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.6 Schiff Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.7 The Strong Equivalence Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5 Why New Tests? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.1 Dark Clouds Problems with GR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.2 The Search for Quantum Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.3 Possible New Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6 How to Search for New Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.1 Better Accuracy and Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.2 Extreme Situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.3 Investigation of Exotic Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

7 Testing Exotic but Fundamental Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507.1 Active and Passive Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507.2 Active and Passive Charge .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517.3 Active and Passive Magnetic Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527.4 Charge Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527.5 Small Accelerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537.6 Test of the Inertial Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547.7 Can Gravity Be Transformed Away? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Mass Metrology and the International System of Units (SI) . . . . . . . . . . . . . . . . . . 67Richard S. Davis1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672 The SI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.1 Base Units/Base Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.2 Gaussian Units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702.3 Planck Units, Natural Units, and Atomic Units . . . . . . . . . . . . . . . . . . . . . . . 71

3 Practical Reasons for Redefining the Kilogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.1 Internal Evidence Among 1 kg Artifact Mass Standards .. . . . . . . . . . . . 713.2 Fundamental Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.3 Electrical Metrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.4 Relative Atomic Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4 Routes to a New Kilogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785 Realizing a New Kilogram Definition in Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.1 Watt Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.2 Silicon X-Ray Crystal Density (XRCD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.3 Experimental Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Contents xi

6 Proposals for a New SI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.1 Consensus Building and Formal Approval .. . . . . . . . . . . . . . . . . . . . . . . . . . . 836.2 An SI Based on Defined Values of a Set of Constants. . . . . . . . . . . . . . . . 84

7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Mass and Angular Momentum in General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . 87Jose Luis Jaramillo and Eric Gourgoulhon1 Issues Around the Notion of Gravitational Energy in General Relativity . . . 88

1.1 EnergyMomentum Density for Matter Fields . . . . . . . . . . . . . . . . . . . . . . . 881.2 Problems when Defining a Gravitational EnergyMomentum .. . . . . . 901.3 Notation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

2 Spacetimes with Killing Vectors: Komar Quantities . . . . . . . . . . . . . . . . . . . . . . . . . 942.1 Komar Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942.2 Komar Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3 Total Mass of Isolated Systems in General Relativity. . . . . . . . . . . . . . . . . . . . . . . . 953.1 Asymptotic Flatness Characterization of Isolated Systems . . . . . . . . . . 953.2 Asymptotic Euclidean Slices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963.3 ADM Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.4 Bondi Energy and Linear Momentum .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105

4 Notions of Mass for Bounded Regions: Quasi-Local Masses . . . . . . . . . . . . . . .1084.1 Ingredients in the Quasi-Local Constructions. . . . . . . . . . . . . . . . . . . . . . . . .1084.2 Some Relevant Quasi-Local Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1094.3 Some Remarks on Quasi-Local Angular Momentum . . . . . . . . . . . . . . . .1144.4 A Study Case: Quasi-Local Mass of Black Hole IHs. . . . . . . . . . . . . . . . .115

5 Global and Quasi-Local Quantities in Black Hole Physics . . . . . . . . . . . . . . . . . .1185.1 Penrose Inequality: a Claim for an Improved Mass

Positivity Result for Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1195.2 Black Hole (Thermo-)dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1195.3 Black Hole Extremality: a MassAngular Momentum Inequality . . .121

6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .121References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .123

Post-Newtonian Theory and the Two-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . .125Luc Blanchet1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1252 Post-Newtonian Formalism.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .128

2.1 Einstein Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1282.2 Post-Newtonian Iteration in the Near Zone . . . . . . . . . . . . . . . . . . . . . . . . . . .1312.3 Post-Newtonian Expansion Calculated by Matching .. . . . . . . . . . . . . . . .1352.4 Multipole Moments of a Post-Newtonian Source . . . . . . . . . . . . . . . . . . . .1392.5 Radiation Field and Polarization Waveforms . . . . . . . . . . . . . . . . . . . . . . . . .1432.6 Radiative Moments Versus Source Moments . . . . . . . . . . . . . . . . . . . . . . . . .145

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3 Inspiralling Compact Binaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1473.1 StressEnergy Tensor of Spinning Particles . . . . . . . . . . . . . . . . . . . . . . . . . .1473.2 Hadamard Regularization .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1503.3 Dimensional Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1533.4 Energy and Flux of Compact Binaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1563.5 Waveform of Compact Binaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1603.6 SpinOrbit Contributions in the Energy and Flux . . . . . . . . . . . . . . . . . . . .162

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .164

Post-Newtonian Methods: Analytic Results on the BinaryProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .167Gerhard Schafer1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1672 Systems in Newtonian Gravity in Canonical Form.. . . . . . . . . . . . . . . . . . . . . . . . . .1693 Canonical General Relativity and PN Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . .171

3.1 Canonical Variables of the Gravitational Field . . . . . . . . . . . . . . . . . . . . . . .1733.2 BrillLindquist Initial-Value Solution for Binary Black Holes . . . . . .1753.3 Skeleton Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1763.4 Functional Representation of Compact Objects . . . . . . . . . . . . . . . . . . . . . .1793.5 PN Expansion of the Routh Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1853.6 Near-Zone Energy Loss Versus Far-Zone Energy Flux . . . . . . . . . . . . . .185

4 Binary Point Masses to Higher PN Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1874.1 Conservative Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1874.2 Dynamical Invariants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1884.3 ISCO and the PN Framework .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1904.4 PN Dissipative Binary Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .192

5 Toward Binary Spinning Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1925.1 Approximate Hamiltonians for Spinning Binaries. . . . . . . . . . . . . . . . . . . .196

6 Lorentz-Covariant Approach and PN Expansions .. . . . . . . . . . . . . . . . . . . . . . . . . . .2006.1 PM and PN Expansions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2026.2 PN Expansion in the Near Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2036.3 PN Expansion in the Far Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .205

7 Energy Loss and Gravitational Wave Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2067.1 Orbital Decay to 4 PN Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2067.2 Gravitational Waveform to 1.5 PN Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . .207

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .209

The Effective One-Body Description of the Two-Body Problem . . . . . . . . . . . . . .211Thibault Damour and Alessandro Nagar1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2112 Motion and Radiation of Binary Black Holes: PN Expanded Results. . . . . . .2133 Conservative Dynamics of Binary Black Holes: the EOB Approach . . . . . . .2154 Description of RadiationReaction Effects in the EOB Approach . . . . . . . . . .224

4.1 Resummation of OF Taylor Using a One-ParameterFamily of Pade Approximants: Tuning vpole . . . . . . . . . . . . . . . . . . . . . . . . . .227

4.2 Parameter-Free Resummation of Waveform and Energy Flux . . . . . . .230

Contents xiii

5 EOB Dynamics and Waveforms.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2385.1 PostPost-Circular Initial Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2385.2 EOB Waveforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2395.3 EOB Dynamics .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .241

6 EOB and NR Waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2437 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .248References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .249

Introduction to Gravitational Self-Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .253Robert M. Wald1 Motion of Bodies in General Relativity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2532 Point Particles in General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2543 Point Particles in Linearized Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2554 Lorenz Gauge Relaxation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2565 Hadamard Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .256

5.1 Hadamard Expansions for a Point Particle Source . . . . . . . . . . . . . . . . . . .2586 Equations of Motion Including Self-Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .259

6.1 The MiSaTaQuWa Equations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2596.2 The DetweilerWhiting Reformulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .260

7 How Should Gravitational Self-Force Be Derived? . . . . . . . . . . . . . . . . . . . . . . . . . .261References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .262

Derivation of Gravitational Self-Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .263Samuel E. Gralla and Robert M. Wald1 Difficulties with Usual Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2632 Rigorous Derivation Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2643 Limits of Spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2644 Our Basic Assumptions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .265

4.1 Additional Uniformity Requirement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2655 Geodesic Motion .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2666 Corrections to Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .267

6.1 Calculation of the Perturbed Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2687 Interpretation of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2698 Self-Consistent Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2699 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .270References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .270

Elementary Development of the Gravitational Self-Force . . . . . . . . . . . . . . . . . . . . .271Steven Detweiler1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .271

1.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2731.2 Notation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .274

2 Newtonian Examples of Self-Force and Gauge Issues . . . . . . . . . . . . . . . . . . . . . . .2753 Classical Electromagnetic Self-Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .277

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4 A Toy Problem with Two Length Scales That Createsa Challenge for Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2784.1 An Approach Which Avoids the Small Length Scale . . . . . . . . . . . . . . . .2794.2 An Alternative That Resolves Boundary Condition Issues . . . . . . . . . . .281

5 Perturbation Theory.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2825.1 Standard Perturbation Theory in General Relativity. . . . . . . . . . . . . . . . . .2835.2 An Application of Perturbation Theory: Locally

Inertial Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2855.3 Metric Perturbations in the Neighborhood of a Point Mass . . . . . . . . . .2875.4 A Small Object Moving Through Spacetime . . . . . . . . . . . . . . . . . . . . . . . . .289

6 Self-Force from Gravitational Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . .2916.1 Dissipative and Conservative Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2926.2 Gravitational Self-Force Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . .293

7 Perturbative Gauge Transformations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2958 Gauge Confusion and the Gravitational Self-Force . . . . . . . . . . . . . . . . . . . . . . . . . .2979 Steps in the Analysis of the Gravitational Self-Force . . . . . . . . . . . . . . . . . . . . . . . .29810 Applications .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .300

10.1 Gravitational Self-Force Effects on Circular Orbitsof the Schwarzschild Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .300

10.2 Field Regularization Via the Effective Source . . . . . . . . . . . . . . . . . . . . . . . .30111 Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .304References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .306

Constructing the Self-Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .309Eric Poisson1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3092 Geometric Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3113 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3124 Field Equation and Particle Motion .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3165 Retarded Greens Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3166 Alternate Greens Function .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3187 Fields Near the World Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3198 Self-Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3219 Axiomatic Approach .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32210 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .324References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .325

Computational Methods for the Self-Force in Black HoleSpacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .327Leor Barack1 Introduction and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .327

1.1 The MiSaTaQuWa Formula.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3291.2 Gauge Dependence .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3301.3 Implementation Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .331

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2 Mode-Sum Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3352.1 An Elementary Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3362.2 The Mode-Sum Formula .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3382.3 Derivation of the Regularization Parameters . . . . . . . . . . . . . . . . . . . . . . . . . .339

3 Numerical Implementation Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3433.1 Overcoming the Gauge Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3443.2 Numerical Representation of the Point Particle . . . . . . . . . . . . . . . . . . . . . . .347

4 An Example: Gravitational Self-Force in Schwarzschild Via1C1D Evolution in Lorenz Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3524.1 Lorenz-Gauge Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3524.2 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .354

5 Toward Self-Force Calculations in Kerr: the Puncture Methodand m-Mode Regularization .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3565.1 Puncture Method in 2C1D.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3565.2 m-Mode Regularization .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .358

6 Reflections and Prospects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .360References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .364

Radiation Reaction and EnergyMomentum Conservation. . . . . . . . . . . . . . . . . . .367Dmitri Galtsov1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3672 EnergyMomentum Balance Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .369

2.1 Decomposition of the Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3712.2 Bound Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3742.3 The Rest Frame (Nonrelativistic Limit) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .377

3 Flat Dimensions Other than Four . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3784 Local Method for Curved Space-Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .379

4.1 Hadamard Expansion in Any Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . .3804.2 Divergences .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3814.3 Four Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3844.4 Self and Radiative Forces in Curved Space-Time . . . . . . . . . . . . . . . . . . . .386

5 Gravitational Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3875.1 Bianchi Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3875.2 Vacuum Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3895.3 Gravitational Radiation for Non-Geodesic Motion . . . . . . . . . . . . . . . . . . .390


The State of Current Self-Force Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .395Lior M. Burko1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3952 The Teukolsky Equation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .397

2.1 The Inhomogeneous Teukolsky Equationwith a Distributional Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .397

2.2 Adiabatic Waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .398

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2.3 Numerical Solution of the Teukolsky Equation .. . . . . . . . . . . . . . . . . . . . . .3992.4 The Linearized Einstein Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .400

3 Frequency-Domain Calculations of the Self-Force. . . . . . . . . . . . . . . . . . . . . . . . . . .4013.1 Mode-Sum Regularization .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4013.2 The DetweilerWhiting Regular Part of the Self-Force . . . . . . . . . . . . . .402

4 Time-Domain Calculations of the Self-Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4034.1 1C1D Numerical Simulations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4034.2 2C1D Numerical Simulations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .404

5 Post-adiabatic Self-Force-Driven Orbital Evolution . . . . . . . . . . . . . . . . . . . . . . . . .4085.1 The Importance of Second-Order Self-Forces . . . . . . . . . . . . . . . . . . . . . . . .4085.2 Conservative Self-Force Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .412

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .413

High-Accuracy Comparison Between the Post-Newtonianand Self-Force Dynamics of Black-Hole Binaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .415Luc Blanchet, Steven Detweiler, Alexandre Le Tiec,and Bernard F. Whiting1 Introduction and Motivation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4162 The Gauge-Invariant Redshift Observable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4183 Regularization Issues in the SF and PN Formalisms . . . . . . . . . . . . . . . . . . . . . . . . .4194 Circular Orbits in the Perturbed Schwarzschild Geometry . . . . . . . . . . . . . . . . . .4215 Overview of the 3PN Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .423

5.1 Iterative PN Computation of the Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4235.2 The Example of the Zeroth-Order Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . .426

6 Logarithmic Terms at 4PN and 5PN Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4276.1 Physical Origin of Logarithmic Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4276.2 Expression of the Near-Zone Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .429

7 Post-Newtonian Results for the Redshift Observable . . . . . . . . . . . . . . . . . . . . . . . .4308 Numerical Evaluation of Post-Newtonian Coefficients . . . . . . . . . . . . . . . . . . . . . .433

8.1 Overview.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4348.2 Framework for Evaluating PN Coefficients Numerically . . . . . . . . . . . .4358.3 Consistency Between Analytically and Numerically

Determined PN Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4378.4 Determining Higher Order PN Terms Numerically . . . . . . . . . . . . . . . . . .4388.5 Summary.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .439

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .441

LISA and Capture Sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .443Oliver Jennrich1 LISA A Mission to Detect and Observe Gravitational Waves . . . . . . . . . . . . .443

1.1 Mission Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4441.2 Sensitivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4451.3 Measurement Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .446

2 Capture Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4483 Science Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .449

Contents xvii

4 Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4514.1 Capture Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4514.2 Signal Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4524.3 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .454

5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .456References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .457

Motion in Alternative Theories of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .461Gilles Esposito-Farese1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4612 Modifying the Matter Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4623 Modified Motion in Metric Theories? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4644 Scalar-Tensor Theories of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .468

4.1 Weak-Field Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4694.2 Strong-Field Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4714.3 Binary-Pulsar Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4724.4 Black Holes in Scalar-Tensor Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .476

5 Extended Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4776 Modified Newtonian Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .479

6.1 Mass-Dependent Models? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4806.2 Aquadratic Lagrangians or k-Essence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4816.3 Difficulties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4826.4 Nonminimal Couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .485


Mass, Inertia, and Gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .491Marc-Thierry Jaekel and Serge Reynaud1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4912 Vacuum Fluctuations and Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .494

2.1 Linear Response Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4942.2 Response to Motions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4982.3 Relativity of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5022.4 Inertia of Vacuum Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .504

3 Mass as a Quantum Observable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5083.1 Quantum Fluctuations of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5083.2 Mass and Conformal Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .510

4 Metric Extensions of GR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5144.1 Radiative Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5154.2 Anomalous Curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5184.3 Phenomenology in the Solar System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .520

5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .526References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .527

xviii Contents

Motion in Quantum Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .531Karim Noui1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .531

1.1 The Problem of Defining Motion in Quantum Gravity .. . . . . . . . . . . . . .5311.2 Quantum Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5321.3 Three-Dimensional Quantum Gravity Is a Fruitful Toy Model . . . . . .5341.4 Outline of the Article. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .535

2 Casting an Eye Over Loop Quantum Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5362.1 The Classical Theory: Main Ingredients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5362.2 The Route to the Quantization of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5382.3 Spin-Networks Are States of Quantum Geometry .. . . . . . . . . . . . . . . . . . .5392.4 The Problem of the Hamiltonian Constraint . . . . . . . . . . . . . . . . . . . . . . . . . .541

3 Three-Dimensional Euclidean Quantum Gravity. . . . . . . . . . . . . . . . . . . . . . . . . . . . .5423.1 Construction of the Noncommutative Space . . . . . . . . . . . . . . . . . . . . . . . . . .5433.2 Constructing the Quantum Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5503.3 Particles Evolving in the Fuzzy Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5523.4 Reduction to One Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .553

4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .558References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .559

Free Fall and Self-Force: an Historical Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . .561Alessandro Spallicci1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5622 The Historical Heritage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5633 Uniqueness of Acceleration and the Newtonian Back-Action .. . . . . . . . . . . . . .5654 The Controversy on the Repulsion and on the Particle Velocity

at the Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5695 Black Hole Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5746 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5787 Relativistic Radial Fall Affected by the Falling Mass. . . . . . . . . . . . . . . . . . . . . . . .582

7.1 The Self-Force .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5827.2 The Pragmatic Approach.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .587

8 The State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5908.1 Trajectory .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5918.2 Regularisation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5928.3 Effect of Radiation Reaction on the Waveforms During Plunge .. . . .592

9 Beyond the State of the Art: the Self-Consistent Prescription .. . . . . . . . . . . . . .59310 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .595References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .597

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .605

The Higgs Mechanism and the Origin of Mass

Abdelhak Djouadi

Abstract The Higgs mechanism plays a key role in the physics of elementaryparticles: in the context of the Standard Model, the theory which, describes ina unified framework the electromagnetic, weak, and strong nuclear interactions,it allows for the generation of particle masses while preserving the fundamen-tal symmetries of the theory. This mechanism predicts the existence of a newtype of particle, the scalar Higgs boson, with unique characteristics. The detec-tion of this particle and the study of its fundamental properties is a major goal ofhigh-energy particle colliders, such as the CERN Large Hadron Collider or LHC.

1 The Standard Model and the Generationof Particle Masses

The end of the last millennium witnessed the triumph of the Standard Modelof elementary particles, the quantum and relativistic theory which describes in aunified framework three of the four fundamental forces in Nature: the electromag-netic, weak, and strong nuclear interactions. In particular, major progress has beenachieved in the last decade as, on the one hand, the discovery of the top quark hasfinally allowed to fully reconstruct the puzzle formed by matter elementary particlesand, on the other hand, very high precision experiments have asserted the validityof the model for describing the three particle interactions with an unprecedentlyhigh degree of accuracy. Nevertheless, one cornerstone of the theory still remains tobe tested: the mechanism by which the particles acquire mass while preserving thefundamental symmetries of the theory. This mechanism predicts the existence of anew type of particle, the scalar Higgs boson, which is expected to be produced andstudied at the CERN Large Hadron Collider (LHC), which will soon start operation.

In this mini-review, I present a pedagogical introduction to the Standard Modeland the Higgs mechanism for mass generation. I then briefly describe the basic

A. Djouadi ()Laboratoire de Physique Theorique, CNRS et Universite Paris-Sud, F-91405 Orsay, Francee-mail: [email protected].

L. Blanchet, A. Spallicci, and B. Whiting (eds.), Mass and Motion in General Relativity,Fundamental Theories of Physics 162, DOI 10.1007/978-90-481-3015-3 1,c Springer Science+Business Media B.V. 2011

1

2 A. Djouadi

properties of the Higgs particle and discuss the prospects for producing it at theLHC and for studying its basic properties. The Higgs sectors of some scenarios fornew physics beyond the Standard Model will be briefly commented upon.

1.1 The Elementary Particles and Their Interactions

Let us start by briefly summarizing the particle content of the Standard Model andthe basic interactions to which it is subject [1], also sketched in Figs. 1 and 2.

Particles of: matter (s= 12 force (s=1)3 families of fermions gauge bosons Higgs

Qm

quark up

u+2/3 5 MeV

quark charm

c+2/31.6 GeV

quark top

t+2/3172 GeV

gluon

g00

quark down

d1/3

5 MeV

quark strange

s1/30.2 GeV

quark bottom

b1/34.9 GeV

photon

00

neutrino e

Ve0

0

neutrino m

Vm Vt0

0

t neutrino

0 0

boson Z

Z0091.2 GeV

Higgs

H0> 114 GeV

electron

e10.5 MeV

muon

m10.1 GeV

tau

t11.7 GeV

bosons W

W 180.4 GeV

) mass (s=0)

Fig. 1 The elementary particles of the Standard Model, their spin, electric charges, and theirmasses in Giga-Electron Volts (GeV) and in units where the speed of light c is equal to unity

ee

g

a b c

mn

W ene

u g

u

d g

m

Fig. 2 Diagrams (called Feynman diagrams) illustrating the three fundamental interactions of theStandard Model. (a) The electromagnetic interaction, where an electron emits a photon, continuingwith altered momentum; (b) the weak interaction responsible for the decay of a muon, via theexchange of a W boson, into an electron and muonic and electronic antineutrinos; (c) the stronginteraction where the u,u,d quarks constituting the proton interact by exchanging or emitting gluons

The Higgs Mechanism and the Origin of Mass 3

The particles that constitute the building blocks of matter have intrinsic magneticmoment or spin equal to s D 1=2 and are called fermions as they obey to FermiDirac statistics.1 They appear in three families; see Fig. 1. The first family formsordinary matter: it consists of the electron and it associated neutrino, which arecalled leptons, as well as the up and down quarks with fractional electric charges,and which form nuclear matter, that is, the protons and neutrons. The two otherfamilies are perfect replica of the former: the leptons and quarks that constitutethem have exactly the same quantum numbers but larger masses. They decay into thefermions e, e, and u of the first family which, in contrast, are absolutely stable. Notethat the top quark, discovered in 1995, is 330,000 times heavier than the electron,observed by Thomson a century earlier. The latter is far heavier than the neutrinos,which have very small masses that can be safely neglected in the present discussion.

To be complete, one should note that for each particle is associated an antiparticlethat has the same properties but opposite electric charge; these are usually noted witha bar, Nf for the antifermion of the fermion f.

Besides, one has the force particles that mediate the fundamental interactionsbetween the various fermions. They have a spin equal to unity, s D 1, and are calledbosons as they obey to BoseEinstein statistics.2 The photon, denoted , is the mes-senger of the electromagnetic interaction to which are subject charged particles, thatis, all fermions except neutrinos. The WC;W, and Z0 bosons mediate the weak nu-clear interaction responsible for the radioactive decay of heavy particles and which,in principle, concerns all fermions. Finally, eight gluons are the messengers of thestrong nuclear force that binds the atomic nuclei, and which concerns only quarks.

Note that there is a fourth fundamental force in Nature, the gravitationalinteraction for which the messenger is the hypothetical graviton of spin 2. It has amagnitude that is far too weak to play a role at the energies that are being probed inlaboratory experiments. It is thus neglected, except in some cases discussed later.

1.2 The Standard Model of Particle Physics

The quantum and relativistic theory that describes in a unified framework the elec-tromagnetic, weak, and strong forces of elementary particles is called the StandardModel [2, 3]. It is based on a very powerful principle, local or gauge symmetry:the fields corresponding to the particles,3 as well as the particle interactions, areinvariant with respect to local transformations (i.e., for any spacetime point)of a given internal symmetry group. The model is a generalization of Quantum

1 The exclusion principle, put forward by Wolfgang Pauli in 1925, forbids to two fermions to be inthe same quantum configuration.2 In contrast to fermions, several bosons can occupy the same quantum configuration and, thus, canaggregate.3 In a quantum theory, to each particle is associated a field that has a given number of degreesof freedom. For instance, the fields associated to a fermion or to the (massless) photon have twodegrees of freedom, while a real scalar field has a single degree of freedom.

4 A. Djouadi

Electro-Dynamics (QED) [4], the quantum and relativistic theory of electromag-netism which describes the interaction of electrically charged particles through theexchange of photons. The latter is invariant under local phase transformations4 de-scribed by a symmetry group noted U.1/Q and conserves the quantum number thatis the electric charge Q.

The symmetry group of the Standard Model is slightly more complicated and isdenoted by SU.3/C SU.2/L U.1/Y. For the strong interaction [3], based on the symmetry group SU.3/C, the quarks

appear in three different states differentiated by a quantum number called color(which has nothing to do with the usual color) that they exchange via eight inter-mediate massless gluons.5

The electromagnetic and weak interactions are combined to form the electroweakinteraction [2], which is based on the symmetry group SU.2/L U.1/Y. Thefermions appear in two quantum configurations called left- and right-handedchiralities corresponding, for massless fermions, to the two possibilities for theprojection of spin onto the direction of motion (s = 1

2). The fermions with left-

handed chiralities of each family are assembled in a doublet of weak isospin,while the fermions with a right-handed chirality are in singlets of weak isospin.In the case of first-family leptons, for instance, the left-handed electron and itsassociated neutrino always appear in the form of a doublet .eLeL / of isospin (eLhas isospin C1

2while eL has isospin 12 ), while the right-handed electron ap-

pears in a singlet eR (with isospin equal to 0); there is no right-handed neutrinoeR. The same holds for quarks: the left-handed quarks form a doublet .

uLdL/ and

the right-handed ones uR; dR are singlets. For a given particle, the quantum number of hypercharge Y is given by the elec-

tric charge and the isospin, Y D 2Q 2I.The electroweak interaction is mediated by the exchange of the gauge bosons6

W;Z0 and the photon . While the photon, the messenger of the long range

4 In QED, the Lagrangian density that describes the theory is invariant under phase transformationson the charged fermionic fields collectively denoted by , .x/ ! eiQ .x/;where xD .x; t /is the spacetime four-vector and Q the electric charge of the fermion. These transformations arecalled gauge or local transformations since the parameter depends on the spacetime four-vector.The photon field mediating the interaction and described by the four-vector A D .A; A0/, trans-forms as: A.x/ ! A.x/ 1Q @.x/; where @ is the derivative with respect to x. In fact,the interaction of fermions via the exchange of photons can be induced in a minimal way in theLagrangian density of the free fermion and photon systems, by substituting the usual derivative @by what is called the covariant derivative: D @ iQA: The gauge transformation group isnoted U.1/Q for the group of unitary matrices of dimension one.5 The transformations of the SU.3/C symmetry group of the strong interaction, called QuantumChromo-Dynamics or QCD, are generated by eight 3 3 unitary matrices with determinant equalto unity. The quarks are triplets of the group (they appear in three colors) while the gluons cor-respond to the eight generators of the group (there are n2 1 generators for SU(n)) and arenon-massive.6 The three generators of the SU.2/L group [n21D3 for nD2], which can be identified with thethree 2 2 Pauli matrices that generate spatial rotations, correspond to the three-vector bosons


electromagnetic force, has zero-mass, the W;Z0 gauge bosons should be massivesince they mediate the weak force that has a short range.

The Standard Model combines esthetics, since gauge invariance provides a sym-metry and is related to a geometrical principle, economy as the number of gaugebosons is fixed and their interactions uniquely determined in a minimal way oncethe symmetry group is chosen, mathematical coherence and, thus, the possibility ofpredicting any phenomenon with infinite precision in principle. Last but not least, ithad a blatant experimental success as some of its predictions have been confirmedat the permille level of accuracy [1, 5]. This makes the Standard Model one of themost successful and most precisely verified theories in Physics.

1.3 The Higgs Mechanism for Mass Generation

A cornerstone of the Standard Model is the mechanism that generates the particlemasses while preserving the gauge invariance of the theory. Indeed, the direct in-troduction of masses for the fermions and for the gauge bosons that mediate theweak interaction violates the invariance with respect to the transformations of theelectroweak symmetry group. In principle, gauge bosons should remain massless topreserve a local symmetry.7 This is for instance the case of the photon for which thezero-mass ensures the invariance of electromagnetism with respect to phase trans-formations. On the other hand, the fact that the left- and right-handed fermions donot have the same isospin quantum numbers prevents them from acquiring a massin a gauge invariant way under isospin symmetry.8

It is the HiggsBroutEnglert mechanism [6, 7], commonly called the Higgsmechanism, which allows the generation of particle masses while preserving thegauge symmetry of electroweak interactions.

The Higgs mechanism postulates the existence of a doublet (under isospin) ofcomplex scalar fields,

D

ReCCi ImCRe0Ci Im0

; (1)

W 1 ;W2 ;W

3 , the messengers of the interaction. The gauge boson associated with the unique

generator of the U.1/Y group is noted B. The four gauge bosons of the electroweak group,W 1 ;W

2 ;W

3 , and B are not the physical ones; the latter are linear combinations of the former:

W D 1p2.W 1 iW 2 /; Z0 D cos WW 3 C sin W B; A D sin WW 3 C cos W B;

where W is the electroweak mixing angle.7 A mass term for the photon and thus a term that is bilinear in the fields, M2AAA

(with thenotation AA DP AA DA20 A A/, will violate the invariance with the transformationsunder the group U.1/Q since one would have: M2AAA

! M2A.A 1Q @/.A 1Q@/ M2AAA

.8 In the case of the first family of leptons for instance, since .eLeL / forms an isospin doublet whileeR forms an isospin singlet, one cannot form a mass term for the electron (which is bilinear in theelectron field), me NeLeR , as this term violates SU.2/L symmetry.

6 A. Djouadi

to which one associates a potential that is invariant under the transformations of theSU.2/L U.1/Y electroweak symmetry group,

V./ D 2 C ./2 : (2)

In this equation, 2 stands for the mass term of the field and the (positive)coupling constant of its self-interaction. For positive values of 2, the potentialV./ has the usual form of an inverted bell in which the minimum of the field ,corresponding to the state of vacuum which should be stable, has zero value. In thiscase, we simply have four additional scalar fields corresponding to four new degreesof freedom or scalar particles, which does not help much toward the solution of themass generation problem. The situation becomes much more interesting if the masssquared term 2 is negative. In this case, the potential V./ has the shape of abottle of Champagne bottom as shown in Fig. 3. The minimum of the potential isnot reached for a zero value of the field (or, rather, for its neutral component0)as usual, but at the nonzero value v Dp2= that is called the nonzero vacuumexpectation value of the field .

When interpreting the field content of the theory starting from this nonsymmetric(the small ball of Fig. 3 having chosen a given minimum) but physical vacuum, onerealizes that three degrees of freedom, among the four degrees of freedom of thecomplex doublet field , have disappeared from the spectrum: they have been ab-sorbed by three gauge bosons of the electroweak interaction. These spin-1 fields,initially massless and with two components or degrees of freedom called transversecomponents, will acquire an additional degree of freedom corresponding to theirlongitudinal component, a characteristic signature of massive spin-1 fields.

The SU.2/L U.1/Y symmetry is then still present but, since the vacuum is notsymmetric, it is not apparent: it is said to be spontaneously broken. Thus, it is thespontaneous breaking of the electroweak symmetry that generates the masses of thevector bosons W and Z0 in a gauge invariant way. The photon remains massless asit should be to explicitly preserve the gauge invariance of electromagnetism.9

Fig. 3 The potential of thescalar field with itsminimum at the value v of thefield Re()

Im()

V()

9 Some technical details of this mechanism are as follows. One first imposes to the Lagrangian den-

sity of the field DC

0

to be invariant under the local transformations of the SU.2/L U.1/Y

symmetry group. The most general form of the Lagrangian is given by:


Using the same scalar field , the masses of the Standard Model fermions canalso be generated in a gauge invariant manner by introducing, for each fermionicfield with left- and right-handed chiralities (and thus, not for the neutrinos whichhave only left-handed chiralities) interaction terms with the scalar field. After spon-taneous symmetry breaking, one identifies the magnitude of the various interactionterms with the experimentally measured values of the fermion masses.10

L D .D/.D/V ./; V ./D2 C ./2where 2 is the mass term and the self-coupling constant. The covariant derivative D inducesthe interactions of the field with the gauge boson fields W a ; B:

D D @ ig2 a2 W a ig1 Y2 Bwith 1

2a and Y

2the generators of the SU.2/L and U.1/Y groups with coupling constants g2 and g1.

For 2 < 0, the minimum of the potential is at the (vacuum expectation) value v D p2=. Toobtain the physical fields, one must describe the Lagrangian L with the true and stable vacuumand the various steps to follow are:

Write the doublet field in terms of four real fields 1;2;3.x/ and H.x/ and use the expansionseries of an exponential, which to first order gives:

.x/ D

2Ci11p2.vCH/i3

' eia.x/a.x/=v 1p

2

0

vCH.x/

Use the freedom to perform a gauge transformation on to eliminate the three fields 1;2;3

.x/ ! eia.x/a .x/ .x/ D 1p2. 0vCH.x//

Develop the kinetic term jD j2 of L which, gives12.@H/

2C g228.vCH/2jW 1CiW 2 j2C 18 .v CH/2jg2W 3g1Bj2:

After having defined the new fields W ; Z0, and A, with sin W D g1=

qg21 C g22 , one

identifies the terms that are bilinear in these fields with the masses of the associated particles. Onerealizes then that the 1;2;3 degrees of freedom have been absorbed by the fields W and Z0 toform their longitudinal components and thus their masses are given by:

MW D 12 vg2 ; MZ D 12 vqg22 C g21 ; MA D 0:

With the value v given by the Fermi constant of weak interaction, v D 1=.p2GF /1=2 D 246GeV, and the experimentally measured values of the coupling constant g2 and g1, one recovers thecorrect masses for the W and Z0 bosons. The photon, instead, remains massless, MA D 0, as itshould. For more details on the Higgs mechanism, see for instance the detailed review of Ref. [7].10 To generate the fermion masses, one introduces a Lagrangian density describing the fermionHiggs interactions in an SU.2/L U.1/Y gauge invariant way. In the case of the electron andthe neutrino for instance, and taking into account that the leptons with left-handed chirality are inSU.2/L doublets while the electron with right-handed chirality is in a singlet, one would have:

LfDfe.Ne; N/LeR ! fe.Nve ; NeL/ 1p2 .0vCH /eR .One then identifies the masses of the electron and the neutrino with the bilinear terms in the fields:

meDfev=p2 and m D0.

8 A. Djouadi

Finally, among the four initial degrees of freedom of the field and after threehave been absorbed by the W and Z0 gauge bosons to acquire their masses, onedegree of freedom will be left over. This residual degree of freedom corresponds toa physical particle,11 the Higgs boson H, the Grail of particle physics.

The Standard Model, despite all of its brilliant successes, will only be completeand validated once this particle has been observed and its fundamental propertiesdetermined. This is the major goal of high-energy colliders and, in particular, of theCERN LHC which has recently started operation.

2 The Profile of the Higgs Particle

2.1 Characteristics of the Higgs Boson

The Higgs boson has remarkable characteristics, which means that it has a uniquestatus in the table of elementary particles given in Fig. 1.

First of all, in contrast to matter particles with spin 1/2 and to gauge particleswith spin 1, it has spin zero.12 It is therefore a boson, as it has integer spin, but itdoes not mediate gauge interactions.

Another unique property of the Higgs particle is that it interacts with or couples toelementary particles proportionally to their masses: the more massive is the particle,the stronger is its interaction with the Higgs boson.13 Thus, the Higgs particle willcouple more strongly to the messengers of the weak interactions, the W and Z0bosons, the masses of which are of the order of hundred GeV. It couples also morestrongly to the top quark, the heaviest particle in the Standard Model, and, to alesser extent, the bottom quark and the leptons, than to the fermions of the firstand second generations which have much smaller masses. Furthermore, it does notcouple to the neutrinos, which are considered as being massless.

The Higgs boson does not couple directly to photons and gluons as the lat-ter have no mass (in the case of gluons, a direct coupling is also absent because

11 The mass of the Higgs boson can be simply deduced from the scalar Higgs potential by isolatingthe terms that are bilinear in the H fields, 1

2M2HH

H , and one obtains

MH Dp2v2:

12 Scalar or spin-zero particles also exist in Nature, but not at the fundamental level: the mesons,for instance, are spin-zero particles but they are bound states of spin 1/2 quarks.13 The interactions of the Higgs boson with the other particles, that is, the terms in the densityLagrangian involving the fields H and two fermionic fields or gauge bosonic fields are describedby the same terms giving the masses, since H always appears in the combination H C v. Theinteraction of the Higgs boson to the particles is thus proportional to their masses:

LHff / mf =v; LHWCW / g2MW ; LHZZ / g2MZ= cos W :


the Higgs boson does not carry color quantum numbers). However, couplings canbe induced in an indirect way through quantum fluctuations. Indeed, according toHeisenbergs uncertainty principle of Quantum Mechanics, the Higgs boson canemit pairs of very heavy particles (such as top quarks for instance) and immediatelyabsorb them; but these virtual particles can, in the meantime, emit photons or gluons.Higgsphotonphoton and Higgsgluongluon couplings are then generated. How-ever, they are expected to be rather small, as they imply intermediate interactions ofthe virtual particles to photons and gluons, which have a small intensity.

Finally, the Higgs boson has also self-interactions, residual of those of the orig-inal scalar field shown in the Higgs potential of Eq. 2; the magnitude of thesetriple and quartic self-interactions are also proportional to the Higgs boson mass(in fact, Higgs mass squared).14

2.2 Constraints on the Higgs Boson Mass

The Higgs boson mass MH is the only free and unknown parameter of the Stan-dard Model, since the coupling constants of the three fundamental interactions thatit describes as well as the masses of the fermion and the gauge boson have beenexperimentally determined. Once this parameter is fixed, the entire profile of theStandard Model Higgs boson is uniquely determined. In particular, its couplings tothe other particles, its production and decay rates can be calculated.

Nonetheless, MH is not a completely free parameter as it is subject to someexperimental and theoretical constraints [8] that we briefly summarize.

The experimental constraints come mainly from the ancestor of the LHC, theLarge Electron Positron collider LEP, an electronpositron collider that has operatedin the 1990s with an energy ranging approximately from 90 to 210 GeV. Importantconstraints come also from the Tevatron, the protonantiproton collider of Fermilabnear Chicago with an energy of 2 TeV. The LEP experiment has first allowed acomprehensive direct search for the Higgs boson and the absence of any signalat LEP led to the lower bound of 114 GeV on its mass. Current results from theTevatron indicate that a Higgs particle with a mass comprised between 160 and 170GeV is probably excluded. In addition, the high-precision measurement of someelectroweak observables of the order of one permille for some of them led toindirect constraints on MH. Indeed, even if it is too heavy to be produced directly incollider experiments, the Higgs boson appears in the small but measurable quantumfluctuations of, for instance, the masses of the Z0 and W bosons, which have been

14 The self interactions between three or four Higgs bosons can be readily obtained from the scalarHiggs potential after electroweak symmetry breaking and read:

LHHH / 3M2H=v; LHHHH / 3M2H=v2:The triple and quartic Higgs couplings are thus proportional to the Higgs mass squared.

10 A. Djouadi

0

1

2

3

4

5

6

100 00303

2

had =

Excluded Preliminary

0.02758 0.000350.027490.00012incl. low Q2 data

Theory uncertainty

July 2008 mLimit = 154 GeV

mH GeV

(5)

Fig. 4 Left: the preferred values of the Higgs boson mass in the Standard Model (the minimaof the curves) after a global fit of all electroweak precision data. The full curve in black repre-sents the 68% confidence level result which leads to MH D 84C3626 GeV, the blue band includesthe theoretical uncertainties and the dotted curves are for the results when some experimental in-puts are slightly changed. The domain in yellow represents the excluded region, MH 114 GeV,from direct Higgs searches at LEP; from Ref. [5]. Right: the triviality bound from the finitenessof the Higgs self-coupling (upper curved in red) and the vacuum stability bound from the require-ment of the positivity of the self-coupling (lower curved in green) on the Higgs boson mass as afunction of the new physics or cut-off scale ; the allowed region lies between the bands and thecolored/shaded bands illustrate the impact of various uncertainties; from Ref. [9]

determined with high accuracy. A global analysis of all electroweak high-precisiondata available today [5] allows the imposition of an upper bound, MH


among three or four Higgs particles). This evolution is rather strong and at somestage, the coupling becomes extremely large and the theory completely looses itspredictability.16 If the energy scale up to which the coupling /M 2H remains smallenough, and the Standard Model effectively valid, is of the order of the Higgs massitself, MH should be less than approximately 1 TeV.17 On the other hand, for smallvalues of the self-coupling, and hence of the Higgs boson mass, the quantum fluctu-ations tend to drive the coupling to negative values and, thus, completely destabilizethe scalar Higgs potential to the point where the minimum is not stable anymore (thescalar potential of Fig. 3 is inverted and the minimum is reached when the field is at1). Requiring that the self-coupling stays positive and the minimum stable up toenergies of about 1 TeV implies that the Higgs boson should have a mass above ap-proximately 70 GeV. However, if the Standard Model is to be extended to ultimatescales, such as for instance the Planck scale MP 1018 GeV, these requirementson the self-coupling from finiteness and positivity become much more constrainingand the Higgs mass should lie in the range 120 GeV

12 A. Djouadi

f=t,b,c,t

f

H

V=W,Z

V

H

f

f

V=W,Z

V

H

g /

g /t

H

Fig. 5 The dominant decay processes of the Higgs boson including the loop and three-body decays(left) and the rates or branching ratios of these Higgs decays as a function of MH (center), togetherwith the total Higgs decay width as a function of MH (right); from Ref. [10]

Nevertheless, some decay channels that should normally not appear can be in-duced by quantum fluctuations. This is for instance the case of Higgs decays intotwo gluons, two photons, or a photon plus a Z0 boson. In particular, the decay rateinto two gluons, induced by a loop involving a virtual top quark which couplesstrongly to the Higgs boson, can be comparable to the decay rates into charm quarksand leptons. Instead, the decay mode into two photons and into a photon plus a Z0

boson (which are induced mainly by top quark and W boson loops) are very rare, aconsequence of the fact that the electromagnetic coupling is much smaller than thestrong interaction coupling. For Higgs masses below 130 GeV, the probability forthese two decay modes to occur is at the few permille level.

In addition, the Higgs boson can decay into two rather massive particles, withthe sum of their masses larger than MH, but one of which is virtual and decaysinto two real particles with smaller masses. This is the case, for example, of theHiggs decay into two W (or Z0/ bosons for masses below 2MW.2MZ/ and thus,one of the W .Z0/ bosons must be virtual and decays into a pair of rather lightfermions. In fact, for values of the Higgs mass above 130 GeV (and below 2MW),the rate for the three-body Higgs decay into a real and a virtual W boson becomescomparable and even larger than the otherwise dominating two-body decay into apair of bottomantibottom quarks. This is due to the fact that the virtuality of thegauge boson is partially compensated by the stronger coupling of the Higgs bosonto the W bosons compared to the Higgs coupling to bottom quarks.

For Higgs boson masses of the order of 180 GeV and beyond, the Higgs decaysinto two pairs of real WCW and Z0Z0 bosons largely dominate with branchingfractions of two to one in favor of the former channel. Even for a Higgs boson witha mass larger than 350 GeV, for which the decay channel into pairs of top quarksbecomes kinematically open, these two channels remain dominant thanks to the lon-gitudinal components of the W, Z0 bosons which significantly enhance the rates.


Finally, one should note that the total decay width of the Higgs boson, the inverseof its lifetime, is of the order of only a few MeV for Higgs masses close to 100GeV but it considerably increases with the Higgs mass to reach the GeV range forMH / 180 GeV and becomes of the same order of MH when the latter approaches1 TeV; see the right-hand side of Fig. 5. Thus, the Higgs boson is a very narrowresonance for small masses but the resonance becomes very wide for a very heavyHiggs particle.

3 Higgs Production at the LHC

3.1 The Large Hadron Collider

The LHC, located at CERN near Geneva, forms a circular ring of length 27 kmburied 100 meters underground; see Fig. 6. It is the largest scientific instrument everbuilt and its construction represented a major technological challenge. The LHC is aprotonproton collider operating at an energy of 14 TeV in the center of mass. Sincethe protons are formed by three quarks, the effective energy, that is, the energy inthe quark center of mass, is of the order of 5 TeV. This energy is largely sufficientto probe in depth the TeV scale.

An important characteristic is the luminosity delivered by the machine, corre-sponding to its ability to produce particle collisions. Integrated over time, it has theinverse unit of a cross section, that is the probability of an interaction during the col-lision; the product of the two quantities gives the expected number of events. Theusual unit of a cross section is cm2, but since the events are extremely rare the com-monly used unit is the picobarn (pb), 1 pbD1036 cm2, or femtobarn, 1 pb D 103 fb.The luminosity expected at the LHC is of the order of 10 fb1 per year in the earlyoperating stage and should increase to 100 fb1 per year in the following years.

Fig. 6 The LHC tunnel andthe various associatedexperiments including themultipurpose experimentsATLAS and CMS (courtesyfrom CERN)

CMS

ES40-V10/09/97

CMS

LHC - B

TI 8 TI 2

ATLAS

ATLASCERN

Point 1

Point 8

Point 5

Point 2

LHC - B

Overall view of the LHC experiments.

ALICE

ALICE

SPS

LEP/LHC

14 A. Djouadi

The two multipurpose detectors ATLAS and CMS [11] (there are also otherdetectors dedicated to, for instance, the physics of the bottom quark and that ofheavy ions) have been devised to deliver the maximal amount of information on theinteractions that occur in their inner part and to cover a large spectrum of possiblesignatures from known and new physical phenomena. Their potential has particu-larly been optimized to detect the Higgs particle for masses comprised between 100G

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