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[资源] Springer Tracts in Modern Physics Vol.211--Evaluating Feynman integrals 2004

Preface
The goal of this book is to describe in detail how Feynman integrals1  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. 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 –
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 my previous book.
In this method, it is also necessary to apply some analytical methods of eval-
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.
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
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 B. Feucht, A.G. Grozin and J. Piclum for
careful reading of preliminary versions of the whole book and numerous com-
ments and suggestions; to M. 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. K¨uhn for permanent support.
I am thankful to my family for permanent love, sympathy, patience and
understanding.
Moscow – Hamburg,
V.A. Smirnov
October 2004

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[ Last edited by zhq025 on 2008-1-16 at 10:38 ]

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