Evaluating Feynman Integrals

250 Pages·2005·1.42 MB·english

by  Vladimir Smirnov (auth.)

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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]
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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