Feynman Integral Calculus
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1 Feynman Integral Calculus
2 Vladimir A. Smirnov Feynman Integral Calculus ABC
3 Vladimir A. Smirnov Lomonosov Moscow State University Skobeltsyn Institute of Nuclear Physics Moscow , Russia Library of Congress Control Number: ISBN-10 ISBN Springer Berlin Heidelberg New York Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the author and techbooks using a Springer LATEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN: /techbooks
4 Preface This is a textbook version of my previous book [190]. Problems and solutions have been included, Appendix G has been added, more details have been presented, recent publications on evaluating Feynman integrals have been taken into account and the bibliography has been updated. The goal of the book is to describe in detail how Feynman integrals 1 can be evaluated analytically. The problem of evaluating Lorentz-covariant Feynman integrals over loop momenta originated in the early days of perturbative quantum field theory. Over a span of more than fifty years, a great variety of methods for evaluating Feynman integrals has been developed. Most powerful modern methods are described in this book. I understand that if another person in particular one actively involved in developing methods for Feynman integral evaluation wrote a book on this subject, 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 recently 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 one needs not only to organize new experiments but also perform rather nontrivial 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 numerical 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 [186]. In this method, 1 Let us point out from beginning that two kinds of integrals are associated with Feynman: integrals over loop momenta and path integrals. We will deal only with the former case.
5 VI Preface it is also necessary to apply some analytical methods of evaluation which were described there only very briefly. So the present book (and/or its previous version [190]) can be considered as Volume 1 with respect to [186], which might be termed Volume 2, or the sequel. 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 two winter semesters of and at the University of Hamburg (and in at the University of Karlsruhe) as a DFG Mercator professor in Hamburg. It is my pleasure to thank the students, postgraduate students, postdoctoral fellows and professors who attended my lectures for numerous stimulating discussions. I am grateful very much to A.G. Grozin, B. Jantzen and J. Piclum for careful reading of preliminary versions of the book and numerous comments and suggestions; to M. Czakon, M. Kalmykov, P. Mastrolia, J. Piclum, M. Steinhauser and O.L. Veretin for valuable assistance in presenting examples in the book; to C. Anastasiou, K.G. Chetyrkin, A.I. Davydychev and A.V. Smirnov for various instructive discussions; to P.A. Baikov, M. Beneke, Z. Bern, K.G. Chetyrkin, A. Czarnecki, A.I. Davydychev, L. Dixon, A.G. Grozin, G. Heinrich, B. Jantzen, A.A. Penin, A. Signer, A.V. Smirnov, M. Steinhauser and O.L. Veretin for fruitful collaboration on evaluating Feynman integrals; to M. Czakon, A. Czarnecki, T. Gehrmann, V.P. Gerdt, J. Gluza, K. Melnikov, T. Riemann, E. Remiddi, O.V. Tarasov and J.B. Tausk for stimulating competition; to Z. Bern, L. Dixon, C. Greub, G. Heinrich, and S. Moch for various pieces of advice; and to B.A. Kniehl and J.H. Kühn for permanent support. I am thankful to my family for permanent love, sympathy, patience and understanding. Moscow April 2006 V.A. Smirnov
6 Contents 1 Introduction Notation Feynman Integrals: Basic Definitions and Tools Feynman Rules and Feynman Integrals Divergences Alpha Representation Regularization Properties of Dimensionally Regularized FeynmanIntegrals Evaluating by Alpha and Feynman Parameters Simple One- and Two-Loop Formulae Auxiliary Tricks Recursively One-Loop Feynman Integrals Partial Fractions Dealing with Numerators One-Loop Examples FeynmanParameters Two-LoopExamples Problems Evaluating by MB Representation One-Loop Examples Evaluating Multiple MB Integrals More One-Loop Examples Two-LoopMasslessExamples Two-LoopMassiveExamples Three-Loop Examples MoreLoops MB Representation versus Expansion by Regions Conclusion Problems
7 VIII Contents 5 IBP and Reduction to Master Integrals One-Loop Examples Two-LoopExamples Reduction of On-Shell Massless Double Boxes Conclusion Problems Reduction to Master Integrals by Baikov s Method Basic Parametric Representation Constructing Coefficient Functions. SimpleExamples General Recipes. Complicated Examples Two-Loop Feynman Integrals for the Heavy Quark Static Potential Conclusion Problems Evaluation by Differential Equations One-Loop Examples Two-LoopExample Conclusion Problems A Tables A.1 TableofIntegrals A.2 SomeUsefulFormulae B Some Special Functions C Summation Formulae C.1 SomeNumberSeries C.2 Power Series of Levels 3 and 4 in Terms of Polylogarithms C.3 Inverse Binomial Power Series up to Level C.4 PowerSeriesofLevels5and6inTermsofHPL D Table of MB Integrals D.1 MB Integrals with Four Gamma Functions D.2 MB Integrals with Six Gamma Functions D.3 The Gauss Hypergeometric Function andmbintegrals
8 Contents IX E Analysis of Convergence and Sector Decompositions E.1 Analysis of Convergence E.2 Practical Sector Decompositions F A Brief Review of Some Other Methods F.1 DispersionIntegrals F.2 Gegenbauer Polynomial x-space Technique F.3 Gluing F.4 Star-Triangle Relations F.5 IR Rearrangement and R F.6 Difference Equations F.7 Experimental Mathematics and PSLQ G Applying Gröbner Bases to Solve IBP Relations G.1 Gröbner Bases for Ideals of Polynomials G.2 Constructing Gröbner-Type Bases for IBP Relations G.3 Examples G.4 Perspectives Solutions References List of Symbols Index
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