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 non-
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 [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.
