Springer Tracts in Modern Physics Volume 211 ManagingEditor:G.Ho¨hler,Karlsruhe Editors: J.Ku¨hn,Karlsruhe Th.Mu¨ller,Karlsruhe A.Ruckenstein,NewJersey F.Steiner,Ulm J.Tru¨mper,Garching P.Wo¨lfle,Karlsruhe StartingwithVolume165,SpringerTractsinModernPhysicsispartofthe[SpringerLink]service. ForallcustomerswithstandingordersforSpringerTractsinModernPhysicsweofferthefull textinelectronicformvia[SpringerLink]freeofcharge.Pleasecontactyourlibrarianwhocan receiveapasswordforfreeaccesstothefullarticlesbyregistrationat: springerlink.com Ifyoudonothaveastandingorderyoucanneverthelessbrowseonlinethroughthetableof contentsofthevolumesandtheabstractsofeacharticleandperformafulltextsearch. Thereyouwillalsofindmoreinformationabouttheseries. Springer Tracts in Modern Physics SpringerTractsinModernPhysicsprovidescomprehensiveandcriticalreviewsoftopicsofcurrent interest in physics. The following fields are emphasized: elementary particle physics, solid-state physics,complexsystems,andfundamentalastrophysics. Suitablereviewsofotherfieldscanalsobeaccepted.Theeditorsencourageprospectiveauthorsto correspondwiththeminadvanceofsubmittinganarticle.Forreviewsoftopicsbelongingtothe abovementionedfields,theyshouldaddresstheresponsibleeditor,otherwisethemanagingeditor. Seealsospringeronline.com ManagingEditor Solid-StatePhysics,Editors GerhardHo¨hler AndreiRuckenstein EditorforTheAmericas Institutfu¨rTheoretischeTeilchenphysik Universita¨tKarlsruhe DepartmentofPhysicsandAstronomy Postfach6980 Rutgers,TheStateUniversityofNewJersey 76128Karlsruhe,Germany 136FrelinghuysenRoad Phone:+49(721)6083375 Piscataway,NJ08854-8019,USA Fax:+49(721)370726 Phone:+1(732)4454329 Email:[email protected] Fax:+1(732)445-4343 www-ttp.physik.uni-karlsruhe.de/ Email:[email protected] www.physics.rutgers.edu/people/pips/ ElementaryParticlePhysics,Editors Ruckenstein.html JohannH.Ku¨hn PeterWo¨lfle Institutfu¨rTheoretischeTeilchenphysik Institutfu¨rTheoriederKondensiertenMaterie Universita¨tKarlsruhe Universita¨tKarlsruhe Postfach6980 Postfach6980 76128Karlsruhe,Germany 76128Karlsruhe,Germany Phone:+49(721)6083590 Phone:+49(721)6083372 Fax:+49(721)6087779 Fax:+49(721)370726 Email:woelfl[email protected] Email:[email protected] www-ttp.physik.uni-karlsruhe.de/∼jk www-tkm.physik.uni-karlsruhe.de ThomasMu¨ller ComplexSystems,Editor Institutfu¨rExperimentelleKernphysik FrankSteiner Fakulta¨tfu¨rPhysik Universita¨tKarlsruhe AbteilungTheoretischePhysik Postfach6980 Universita¨tUlm 76128Karlsruhe,Germany Albert-Einstein-Allee11 Phone:+49(721)6083524 89069Ulm,Germany Fax:+49(721)6072621 Phone:+49(731)5022910 Email:[email protected] Fax:+49(731)5022924 www-ekp.physik.uni-karlsruhe.de Email:[email protected] www.physik.uni-ulm.de/theo/qc/group.html FundamentalAstrophysics,Editor JoachimTru¨mper Max-Planck-Institutfu¨rExtraterrestrischePhysik Postfach1312 85741Garching,Germany Phone:+49(89)30003559 Fax:+49(89)30003315 Email:[email protected] www.mpe-garching.mpg.de/index.html Vladimir A. Smirnov Evaluating Feynman Integrals With48Figures 123 VladimirA.Smirnov LomonosovMoscowStateUniversity II.Institutfu¨rTheoretischePhysik SkobeltsynInstituteofNuclearPhysics Universita¨tHamburg Moscow119992,Russia LuruperChaussee149 E-mail:[email protected] 22761Hamburg,Germany E-mail:[email protected] LibraryofCongressControlNumber:2004115458 PhysicsandAstronomyClassificationScheme(PACS): 12.38.Bx,12.15.Lk,02.30.Gp ISSNprintedition:0081-3869 ISSNelectronicedition:1615-0430 ISBN3-540-23933-2SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialisconcerned, specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,reproductionon microfilmorinanyotherway,andstorageindatabanks.Duplicationofthispublicationorpartsthereofis permittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,1965,initscurrentversion,and permissionforusemustalwaysbeobtainedfromSpringer.ViolationsareliableforprosecutionundertheGerman CopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springeronline.com ©Springer-VerlagBerlinHeidelberg2004 PrintedinGermany Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply,evenin theabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelawsandregulations andthereforefreeforgeneraluse. Typesetting:bytheauthorandTechBooksusingaSpringerLATEXmacropackage Coverconcept:eStudioCalamarSteinen Coverproduction:design&productionGmbH,Heidelberg Printedonacid-freepaper SPIN:10985380 56/3141/jl 543210 Preface The goal of this book is to describe in detail how Feynman integrals1 can be evaluatedanalytically.TheproblemofevaluatingLorentz-covariantFeynman integrals over loop momenta originated in the early days of perturbative quantumfieldtheory.Overaspanofmorethan fifty years,agreatvarietyof methods for evaluating Feynman integrals has been developed. This book is a first attempt to summarize them. I understand that if another person – in particular one actively involved in developing methods for Feynman integral evaluation – made a similar attempt, he or she would probably concentrate on some other methods and would rank the methods as most important and less important in a different order. I believe, however, that my choice is reasonable. At least I have tried to concentrate on the methods that have been used in the past few years in the most sophisticated calculations, in which world records in the Feynman integral ‘sport’ were achieved. The problem of evaluation is very important at the moment. What could be easily evaluated was evaluated many years ago. To perform important calculations at the two-loop level and higher one needs to choose adequate methods and combine them in a non-trivial way. In the present situation – which might be considered boring because the Standard Model works more or less properly and there are no glaring contradictions with experiment – oneneedsnotonlytoorganizenewexperimentsbutalsoperformrathernon- trivial calculations for further crucial high-precision checks. So I hope very much that this book will be used as a textbook in practical calculations. I shall concentrate on analytical methods and only briefly describe nu- merical ones. Some methods are also characterized as semi-analytical, for example, the method based on asymptotic expansions of Feynman integrals in momenta and masses which was described in detail in my previous book. Inthismethod,itisalsonecessarytoapplysomeanalyticalmethodsofeval- uation which were described there only very briefly. So the present book can be considered as Volume 1 with respect to the previous book, which might be termed Volume 2, or the sequel. 1Letuspointoutfrombeginningthattwokindsofintegralsareassociatedwith Feynman: integrals over loop momenta and path integrals. We will deal only with the former case. VI Preface Although all the necessary definitions concerning Feynman integrals are provided in the book, it would be helpful for the reader to know the basics of perturbative quantum field theory, e.g. by following the first few chapters of the well-known textbooks by Bogoliubov and Shirkov and/or Peskin and Schroeder. This book is based on the course of lectures which I gave in the winter semester of 2003–2004 at the Universities of Hamburg and Karlsruhe as a DFGMercatorprofessorinHamburg.Itismypleasuretothankthestudents, postgraduate students, postdoctoral fellows and professors who attended my lectures for numerous stimulating discussions. I am grateful very much to B. Feucht, A.G. Grozin and J. Piclum for carefulreadingofpreliminaryversionsofthewholebookandnumerouscom- mentsandsuggestions;toM.Czakon,M.Kalmykov,P.Mastrolia,J.Piclum, M. Steinhauser and O.L. Veretin for valuable assistance in presenting exam- ples in the book; to C. Anastasiou, K.G. Chetyrkin and A.I. Davydychev for various instructive discussions; to P.A. Baikov, M. Beneke, K.G. Chetyrkin, A.Czarnecki,A.I.Davydychev,B.Feucht,G.Heinrich,A.A.Penin,A.Signer, M. Steinhauser and O.L. Veretin for fruitful collaboration on evaluating Feynman integrals; to M. Czakon, A. Czarnecki, T. Gehrmann, J. Gluza, T. Riemann, K. Melnikov, E. Remiddi and J.B. Tausk for stimulating com- petition; to Z. Bern, L. Dixon, C. Greub and S. Moch for various pieces of advice; and to B.A. Kniehl and J.H. Ku¨hn for permanent support. I am thankful to my family for permanent love, sympathy, patience and understanding. Moscow – Hamburg, V.A. Smirnov October 2004 Contents 1 Introduction.............................................. 1 1.1 Notation .............................................. 8 References ................................................. 9 2 Feynman Integrals: Basic Definitions and Tools ............................... 11 2.1 Feynman Rules and Feynman Integrals.................... 11 2.2 Divergences............................................ 14 2.3 Alpha Representation ................................... 18 2.4 Regularization ......................................... 20 2.5 Properties of Dimensionally Regularized Feynman Integrals...................................... 24 References ................................................. 29 3 Evaluating by Alpha and Feynman Parameters................................. 31 3.1 Simple One- and Two-Loop Formulae ..................... 31 3.2 Auxiliary Tricks........................................ 34 3.2.1 Recursively One-Loop Feynman Integrals............ 34 3.2.2 Partial Fractions ................................. 35 3.2.3 Dealing with Numerators.......................... 36 3.3 One-Loop Examples .................................... 38 3.4 Feynman Parameters ................................... 41 3.5 Two-Loop Examples .................................... 43 References ................................................. 52 4 Evaluating by MB Representation ........................ 55 4.1 One-Loop Examples .................................... 56 4.2 Multiple MB Integrals .................................. 63 4.3 More One-Loop Examples ............................... 65 4.4 Two-Loop Massless Examples ............................ 71 4.5 Two-Loop Massive Examples ............................ 81 4.6 Three-Loop Examples................................... 92 4.7 More Loops............................................ 98 4.8 MB Representation versus Expansion by Regions ........... 102 VIII Contents 4.9 Conclusion ............................................ 105 References ................................................. 106 5 IBP and Reduction to Master Integrals................... 109 5.1 One-Loop Examples .................................... 109 5.2 Two-Loop Examples .................................... 114 5.3 Reduction of On-Shell Massless Double Boxes.............. 120 5.4 Conclusion ............................................ 127 References ................................................. 130 6 Reduction to Master Integrals by Baikov’s Method ...................................... 133 6.1 Basic Parametric Representation ......................... 133 6.2 Constructing Coefficient Functions. Simple Examples ....................................... 138 6.3 General Recipes. Complicated Examples................... 146 6.4 Two-Loop Feynman Integrals for the Heavy Quark Potential ........................... 152 6.5 Conclusion ............................................ 162 References ................................................. 163 7 Evaluation by Differential Equations...................... 165 7.1 One-Loop Examples .................................... 165 7.2 Two-Loop Example..................................... 170 7.3 Conclusion ............................................ 173 References ................................................. 176 A Tables .................................................... 179 A.1 Table of Integrals....................................... 179 A.2 Some Useful Formulae .................................. 185 B Some Special Functions................................... 187 References ................................................. 189 C Summation Formulae ..................................... 191 C.1 Some Number Series .................................... 192 C.2 Power Series of Levels 3 and 4 in Terms of Polylogarithms .............................. 197 C.3 Inverse Binomial Power Series up to Level 4 ............... 198 C.4 Power Series of Levels 5 and 6 in Terms of HPL ............ 200 References ................................................. 204 D Table of MB Integrals .................................... 207 D.1 MB Integrals with Four Gamma Functions................. 207 D.2 MB Integrals with Six Gamma Functions.................. 214 Contents IX E Analysis of Convergence and Sector Decompositions ............................... 221 E.1 Analysis of Convergence................................. 221 E.2 Practical Sector Decompositions.......................... 229 References ................................................. 232 F A Brief Review of Some Other Methods.................. 233 F.1 Dispersion Integrals..................................... 233 F.2 Gegenbauer Polynomial x-Space Technique ................ 234 F.3 Gluing ................................................ 235 F.4 Star-Triangle Relations.................................. 236 F.5 IR Rearrangement and R∗ ............................... 237 F.6 Difference Equations.................................... 240 F.7 Experimental Mathematics and PSLQ .................... 241 References ................................................. 243 List of Symbols ............................................... 245 Index......................................................... 247 1 Introduction TheimportantmathematicalproblemofevaluatingFeynmanintegralsarises quitenaturallyinelementary-particlephysicswhenonetreatsvariousquanti- tiesintheframeworkofperturbationtheory.Usually,itturnsoutthatagiven quantum-field amplitude that describes a process where particles participate cannot be completely treated in the perturbative way. However it also often turns out that the amplitude can be factorized in such a way that different factors are responsible for contributions of different scales. According to a factorization procedure a given amplitude can be represented as a product of factors some of which can be treated only non-perturbatively while others canbeindeedevaluatedwithinperturbationtheory,i.e.expressedintermsof Feynmanintegralsoverloopmomenta.Ausefulwaytoperformthefactoriza- tionprocedureisprovidedbysolvingtheproblemofasymptoticexpansionof Feynman integrals in the corresponding limit of momenta and masses that is determined by the given kinematical situation. A universal way to solve this problem is based on the so-called strategy of expansion by regions [3, 10]. This strategy can be itself regarded as a (semianalytical) method of eval- uation of Feynman integrals according to which a given Feynman integral depending on several scales can be approximated, with increasing accuracy, by a finite sum of first terms of the corresponding expansion, where each term is written as a product of factors depending on different scales. A lot of details concerning expansions of Feynman integrals in various limits of mo- menta and/or masses can be found in my previous book [10]. In this book, however, we shall mainly deal with purely analytical methods. One needs to take into account various graphs that contribute to a given process. The number of graphs greatly increases when the number of loops getslarge.Foragivengraph,thecorrespondingFeynmanamplitudeisrepre- sentedasaFeynmanintegraloverloopmomenta,duetosomeFeynmanrules. The Feynman integral, generally, has several Lorentz indices. The standard waytohandletensorquantitiesistoperformatensorreductionthatenables us to write the given quantity as a linear combination of tensor monomials with scalar coefficients. Therefore we shall imply that we deal with scalar Feynman integrals and consider only them in examples. A given Feynman graph therefore generates various scalar Feynman inte- gralsthathavethesamestructureoftheintegrandwithvariousdistributions V.A.Smirnov:EvaluatingFeynmanIntegrals STMP211,1–9(2004) (cid:1)c Springer-VerlagBerlinHeidelberg2004