Feynman Integral Calculus

The problem of evaluating Feynman integrals over loop momenta has existed from the early days of perturbative quantum field theory. The goal of the book is to summarize those methods for evaluating Feynman integrals that have been developed over a span of more than fifty years. `Feynman Integral Calculus' characterizes the most powerful methods, in particular those used for recent, quite sophisticated calculations, and then illustrates them with numerous examples, starting from very simple ones and progressing to nontrivial examples This book characterizes methods of evaluation of Feynman integrals in a systematic way. It concentrates on the methods that have been employed recently for most sophisticated calculations. Feynman Integral Calculus explains how the problem of evaluation has become ever more important since what could be easily evaluated has already been evaluated years ago. It demonstrates and explains how to perform the newest important calculations, while showing how to choose adequate methods and combine evaluation methods in a non-trivial way. The most powerful methods of calculation of Feynman integrals are characterized and then illustrated through numerous examples, starting from very simple ones and culminating in rather nontrivial examples. This is a textbook version of the previous book (Evaluating Feynman integrals, STMP 211) of the author. 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 summarize those methods for evaluating Feynman integrals that have been developed over a span of more than fifty years The book characterizes the most powerful methods and illustrates them with numerous examples starting from very simple ones and progressing to nontrivial examples The book demonstrates how to choose adequate methods and combine evaluation methods in a non trivial way The most powerful methods are characterized and then illustrated through numerous examples This is an updated textbook version of the previous book Evaluating Feynman integrals STMP 211 of the author The problem of evaluating Feynman integrals over loop momenta has existed from the early days of perturbative quantum field theory The goal of the book is to summarize those methods for evaluating Feynman integrals that have been developed over a span of more than fifty years Feynman Integral Calculus characterizes the most powerful methods in particular those used for recent quite sophisticated calculations and then illustrates them with numerous examples starting from very simple ones and progressing to nontrivial examples This book characterizes methods of evaluation of Feynman integrals in a systematic way It concentrates on the methods that have been employed recently for most sophisticated calculations Feynman Integral Calculus explains how the problem of evaluation has become ever more important since what could be easily evaluated has already been evaluated years ago It demonstrates and explains how to perform the newest important calculations while showing how to choose adequate methods and combine evaluation methods in a non trivial way The most powerful methods of calculation of Feynman integrals are characterized and then illustrated through numerous examples starting from very simple ones and culminating in rather nontrivial examples This is a textbook version of the previous book Evaluating Feynman integrals STMP 211 of th 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. 1 ThegoalofthebookistodescribeindetailhowFeynmanintegrals canbe evaluatedanalytically.TheproblemofevaluatingLorentz-covariantFeynman integrals over loop momenta originated in the early days of perturbative quantum?eld theory. Over a span of more than?fty years, a great variety of methodsforevaluatingFeynmanintegralshasbeendeveloped.Mostpowerful modern methods are described in this book. Iunderstandthatifanotherperson–inparticularoneactivelyinvolvedin 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 di?erent 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. Introduction. Feynman Integrals: Basic Definitions and Tools. Evaluating by Alpha and Feynman Parameters. Evaluating by MB Representation. IBP and Reduction to Master Integrals. Reduction to Master Integrals by Baikov's Method. Evaluating by Differential Equations. Appendix A: Tables. Appendix B: Some Special Functions. Appendix C: Summation Formulae. Appendix D: Table of MB integrals. Appendix E: Analysis of Convergence and Sector Decompositions. Appendix F: A brief Review of some other Methods. Appendix G: Applying Gröbner Bases to Solve IBP Relations.- Solutions. The important mathematical problem of evaluating Feynman integrals arises quite naturally in elementary-particle physics when one treats various quantities in the framework of perturbation theory.