Some methods to evaluate complicated Feynman integrals
I discuss a progress in calculations of Feynman integrals based on the Gegenbauer Polynomial Technique and the Differential Equation Method. %\vspace{1pc} \vspace{-0.5cm}
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\title{Some methods to evaluate complicated Feynman integrals
}

\author{A.V. Kotikov\address[MCSD]{ BLThPh,
%Bogolubov Laboratoty of Theoretical Physics,
Joint Institute for Nuclear Physics,
141980 Dubna, Russia}}
%        \thanks{Footnotes should appear on the first page only to
%                indicate your present address (if different from your
%                normal address), research grant, sponsoring agency, etc.
%                These are obtained with the {\tt\ttbs thanks} command.},
%        \addressmark\thanks{For following authors with the same
%                address use the {\tt\ttbs addressmark} command.}}

       
\begin{document}

\begin{abstract}
I discuss a progress in calculations of Feynman integrals 
based on the Gegenbauer Polynomial Technique and 
the Differential Equation Method. 
%\vspace{1pc}
\vspace{-0.5cm}
\end{abstract}

\maketitle

%\vspace{-1cm}

%\newpage
\section{ Introduction }


Last years there was an essential progress in calculations
of Feynman integrals. It seems that most important results have
been obtained for two-loop four-point massless Feynman diagrams:
in on-shall case (see \cite{Smirnov,GeRede}) and for a class of off-shall
legs (see \cite{Offshall}). A review of the results can be found in
\cite{GeReconf}. Moreover, very recently results for a class of these
diagrams have been obtained
\cite{mass} in the case when some propagators have a nonzero mass.


In the letter, I review very shortly results obtained with help of 
two methods for calculations of Feynman diagrams (for details,
see \cite{Winter02}): 
%The first one, so-called 
the Gegenbauer Polynomial Technique \cite{2} (see also \cite{3,4}) and
 the Differential Equation Method (DEM) \cite{DEM1}.
The additional information about a modern progress in calculations
of Feynman integrals can be found, for example, also in recent articles 
\cite{Passarino}.




\section{Applications of the Gegenbauer Polynomial Technique}

%{\bf 3.1}~~~
The Gegenbauer Polynomial Technique
%above results have 
has been used for evaluation
%calculations 
of very complicated Feynman diagrams  (see also \cite{4})
which contribute mostly in calculations based on
%in 
various type of $1/N$ expansions: 
%to the following results:
\begin{itemize}
\item
In the calculation (in \cite{QED3}) of the next-to-leading (NLO)
corrections to the value of dynamical mass generation (see \cite{Nash})
in the framework of three-dimensional Quantum Electrodynamics. 
\item
In the evaluation (in \cite{KoKo}) of the correct value of 
of the leading order contribution to the $\beta$-function
of the $\theta$-term in Chern-Simons theory. The $\beta$-function
is zero in the framework of usual perturbation theory but it takes nonzero
values in $1/N$ expansion (see \cite{korea}).
\item
In the evaluation (in \cite{Fadin}) of NLO corrections to the value of
gluon Regge trajectory (see discussions in \cite{Fadin} and references 
therein).
\item
In the calculation (in \cite{KoLi}) of the next-to-leading 
corrections to the BFKL intercept of spin-dependent part of high-energy
asymptotics of hadron-hadron cross-sections. 
\item
In the calculation (in \cite{KoLi,KoLi1}) of the next-to-leading 
corrections to the BFKL equation at arbitrary conformal spin.
\item
In the evaluation (in \cite{BrKo}) of the 
most complicated parts of ${\rm O}(1/N^3)$ contributions to 
critical exponents of $\phi^4$-theory,
for any spacetime dimensionality $D$.
\end{itemize}


\section{The recent progress in calculation of Feynman integrals by
the DEM.} 

\begin{itemize}
\item
%{\bf 1.} 
The articles \cite{FKV1} and \cite{FKV}: 

{\bf a)} The set of two-point two-loop Feynman diagrams 
with one- and two-mass
thresholds has been evaluated by DEM (see Fig.1).
The results are presented  in Ref \cite{FKV1,FKV} and in the review 
\cite{Winter02}.
%on pages 2 and 3 and of 
Some of them have been
known before (see 
%discussions in 
\cite{FKV1}). The check of the results 
has been
done by Veretin programs (see discussions in 
\cite{FKV1} and references therein).
%%%  FIGURE  ===  %%%%%%%%%%%%%%%%
   \begin{figure}[tb]
%\label{fig-2}
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%   \epsfig{file=tolya.ps,width=100mm,height=150mm,angle=-90}%
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%}
%\end{picture}
%\vskip -0.5cm
\vskip -0.7cm
 \caption{
%\sl  
Two-loop self-energy diagrams.
Solid lines denote propagators with the mass $m$; dashed lines
denote massless propagators.
%\vspace{-0.7cm}
\vskip -0.8cm}
%\vspace{-0.7cm}
 \end{figure}

{\bf b)} The set of three-point two-loop Feynman integrals
 with one- and two-mass
thresholds has been evaluated (the results of some of them has been
known before (see \cite{FKV1})) by a combination of DEM and Veretin programs
for calculation of first terms in small-moment expansion of Feynman diagrams
(see discussions in \cite{FKV1} and references therein).
\item
%{\bf 2.} 
The article \cite{FKK}: 
The full set of two-point two-loop on-shell master diagrams 
has been evaluated by DEM. The check of the results has been
done by Kalmykov programs (see 
%page 5 and 
discussions in 
\cite{FKK,FKK1} and references therein). 
\item
%{\bf 3.} 
The article \cite{GeRede}: 
The set of three-point and four-point two-loop massless Feynman diagrams 
has been evaluated.
\end{itemize}

\vspace{-0.2cm}

%  {\large {\bf Acknowledgments.}}
{\bf Acknowledgments.}~~Author
 would like to express his sincerely thanks to the Organizing
  Committee of ACAT 2002
%the VIIIth International Workshop on advanced computing
%and analysis techniques in physics research (ACAT 2002) 
for the kind invitation.
% the financial support at  such remarkable Conferences, and 
% for fruitful discussions.
He was supported in part by 
%Alexander von Humboldt fellowship and 
INTAS  grant N366.

\vspace{-0.3cm}

\begin{thebibliography}{99}
\bibitem{Smirnov} V.A. Smirnov, Phys.Lett. B460 (1999) 397;
V.A. Smirnov and O.L. Veretin, Nucl.Phys. B566 (2000) 469; 
J.B. Tausk, Phys.Lett. B469 (1999) 225; C. Anastasiou, et al.,
Nucl.Phys. B (Proc.Suppl.) 89 (2000) 262.
%
\bibitem{GeRede} T. Gehrmann and E. Remiddi, 
Nucl.Phys. B580 (2000) 485; hep-ph/0101147
%
\bibitem{Offshall} V.A. Smirnov, Phys.Lett. B491 (2000) 130; Phys.Lett.
 B500 (2001) 3;\\ C. Anastasiou, et al.,
Nucl.Phys. B565 (2000) 445; T. Gehrmann and E. Remiddi, 
Nucl.Phys. B601 (2001) 248, 
%Nucl.Phys. B601 (2001) 
287.
%
\bibitem{GeReconf} V.A. Smirnov, hep-ph/0209177.
%, hep-ph/0209295. 
%
\bibitem{mass} V.A. Smirnov, 
%DESY preprint DESY 01-190 (hep-ph/0111160).
Phys.Lett. B524 (2002) 129.
%\vspace{-2.5mm}
%
\bibitem{Winter02} A.V. Kotikov, hep-ph/0112347.
%%CITATION = HEP-PH 0112347;%%
%
\bibitem{2} K.G. Chetyrkin, A.L. Kataev and F.V. Tkachov,
Nucl.Phys. B174 (1980) 345.
%%CITATION = NUPHA,B174,345;%%
%
\bibitem{3} D.I. Kazakov and A.V. Kotikov,
Theor. Math.Phys. 73 (1987) 1264.
%%CITATION = TMPHA,73,1264;%%
%\vspace{-2.5mm}
%\bibitem{3.1}  D.I. Kazakov and A.V. Kotikov,
Nucl.Phys. B307 (1988) 721; 
%%CITATION = NUPHA,B307,721;%%
B345 (1990) 299(E);
%%CITATION = NUPHA,B345,299;%%
%\vspace{-2.5mm}
Phys. Lett. B291 (1992) 171.
%%CITATION = PHLTA,B291,171;%%
%\vspace{-2.5mm}
%\bibitem{3.2}
 A.V.Kotikov, 
Theor.Math.Phys. 78 (1989) 134.
%%CITATION = TMPHA,78,134;%%
%\vspace{-2.5mm}
%
\bibitem{4} A.V. Kotikov, 
Phys.Lett. B375 (1996) 240; 
%%CITATION = HEP-PH 9512270;%%
hep-ph/0102177.
%%CITATION = HEP-PH 0102177;%%
%in: 
%{\it Proceedings of the XVth International Workshop 
%``High Energy Physics and Quantum Field Theory``}~ (hep-ph/0102177).
%
\bibitem{DEM1} A.V. Kotikov, 
Phys.Lett. B254 (1991) 158; 
%%CITATION = PHLTA,B254,158;%%
Phys.Lett. B259 (1991) 314;
%%CITATION = PHLTA,B259,314;%%
Phys.Lett. B267 (1991) 123; 
%%CITATION = PHLTA,B267,123;%%
hep-ph/0102178.
%%CITATION = HEP-PH 0102178;%%
%
\bibitem{Passarino} G. Passarino, hep-ph/0101299); 
G. Passarino and S. Uccirati, hep-ph/0112004.
%
\bibitem{QED3} A.V. Kotikov, JETP Lett. 58 (1993) 731.
%%CITATION = JTPLA,58,731;%%
%
\bibitem{Nash} T. Appelquist et al., Phys.Rev. D33 (1986) 3774;
Phys.Rev.Lett. 60 (1988) 2575;
%%CITATION = PRLTA,60,2575;%% 
D. Nash, Phys.Rev. Lett. 62 (1989) 3024.
%%CITATION = PRLTA,62,3024;%%
%
\bibitem{KoKo} I.N. Kondrashuk and A.V. Kotikov, 
Phys. Rev. D53 (1996) 2260.
%%CITATION = HEP-TH 9603076;%%
%
\bibitem{korea} S.H. Park, 
Phys.Rev. D45 (1992) 3332.
%%CITATION = PHRVA,D45,3332;%%
%
%
\bibitem{Fadin}
V.S. Fadin, R. Fiore and M.I. Kotsky,
Phys. Lett. B387 (1996) 593.
%
\bibitem{KoLi}
A.V. Kotikov and L.N. Lipatov, 
Nucl.Phys. B582 (2000) 19. 
%%CITATION = HEP-PH 0004008;%%
%\vspace{-2.5mm}
%
%
\bibitem{KoLi1}
A.V. Kotikov and L.N. Lipatov, hep-ph/0112346, 
%%CITATION = HEP-PH 0112346;%%
hep-ph/0208220. 
%%CITATION = HEP-PH 0208220;%%
%
\bibitem{BrKo}
D.J. Broadhurst and A.V. Kotikov,
Phys. Lett. B441 (1998) 345.
%%CITATION = HEP-TH 9612013;%%
%
\bibitem{FKV1} J. Fleischer, A.V. Kotikov, and O.L. Veretin, 
Nucl.Phys. B547 (1999) 343. 
%%CITATION = HEP-PH 9808242;%%
%
\bibitem{FKV} J. Fleischer, A.V. Kotikov, and O.L. Veretin, 
Phys.Lett. B417 (1998) 163;
%%CITATION = HEP-PH 9707492;%%
%
%\bibitem{FKV2} J. Fleischer, A.V. Kotikov, and O.L. Veretin, 
Acta Phys.Polon. B29 (1998) 2611.
%, hep-ph/9808243.
%%CITATION = HEP-PH 9808243;%% 
%
%
\bibitem{FKK} J. Fleischer, M.Yu. Kalmykov, and A.V. Kotikov, 
Phys.Lett. B462 (1999) 169; B467 (1999) 310(E).
%%CITATION = HEP-PH 9905249;%%
%
\bibitem{FKK1} J. Fleischer, M.Yu. Kalmykov, and A.V. Kotikov, 
hep-ph/9905379;
%%CITATION = HEP-PH 9905379;%%
 J. Fleischer and M.Yu. Kalmykov,
Comput.Phys.Commun. 128 (2000) 531; 
%%CITATION = HEP-PH 9907431;%%
Phys.Lett. B470 (1999) 168.
%A.I. Davydychev, Phys.Rev. D61 (2000) 087701;
%A.I. Davydychev and M.Yu. Kalmykov,
%Nucl.Phys. B605 (2001) 266.
%
\end{thebibliography}
%

\end{document}


hep-ph0303059.extracted_bib
\begin{thebibliography}{99}
\bibitem{Smirnov} V.A. Smirnov, Phys.Lett. B460 (1999) 397;
V.A. Smirnov and O.L. Veretin, Nucl.Phys. B566 (2000) 469; 
J.B. Tausk, Phys.Lett. B469 (1999) 225; C. Anastasiou, et al.,
Nucl.Phys. B (Proc.Suppl.) 89 (2000) 262.
%
\bibitem{GeRede} T. Gehrmann and E. Remiddi, 
Nucl.Phys. B580 (2000) 485; hep-ph/0101147
%
\bibitem{Offshall} V.A. Smirnov, Phys.Lett. B491 (2000) 130; Phys.Lett.
 B500 (2001) 3;\\ C. Anastasiou, et al.,
Nucl.Phys. B565 (2000) 445; T. Gehrmann and E. Remiddi, 
Nucl.Phys. B601 (2001) 248, 
%Nucl.Phys. B601 (2001) 
287.
%
\bibitem{GeReconf} V.A. Smirnov, hep-ph/0209177.
%, hep-ph/0209295. 
%
\bibitem{mass} V.A. Smirnov, 
%DESY preprint DESY 01-190 (hep-ph/0111160).
Phys.Lett. B524 (2002) 129.
%\vspace{-2.5mm}
%
\bibitem{Winter02} A.V. Kotikov, hep-ph/0112347.
%%CITATION = HEP-PH 0112347;%%
%
\bibitem{2} K.G. Chetyrkin, A.L. Kataev and F.V. Tkachov,
Nucl.Phys. B174 (1980) 345.
%%CITATION = NUPHA,B174,345;%%
%
\bibitem{3} D.I. Kazakov and A.V. Kotikov,
Theor. Math.Phys. 73 (1987) 1264.
%%CITATION = TMPHA,73,1264;%%
%\vspace{-2.5mm}
%\bibitem{3.1}  D.I. Kazakov and A.V. Kotikov,
Nucl.Phys. B307 (1988) 721; 
%%CITATION = NUPHA,B307,721;%%
B345 (1990) 299(E);
%%CITATION = NUPHA,B345,299;%%
%\vspace{-2.5mm}
Phys. Lett. B291 (1992) 171.
%%CITATION = PHLTA,B291,171;%%
%\vspace{-2.5mm}
%\bibitem{3.2}
 A.V.Kotikov, 
Theor.Math.Phys. 78 (1989) 134.
%%CITATION = TMPHA,78,134;%%
%\vspace{-2.5mm}
%
\bibitem{4} A.V. Kotikov, 
Phys.Lett. B375 (1996) 240; 
%%CITATION = HEP-PH 9512270;%%
hep-ph/0102177.
%%CITATION = HEP-PH 0102177;%%
%in: 
%{\it Proceedings of the XVth International Workshop 
%``High Energy Physics and Quantum Field Theory``}~ (hep-ph/0102177).
%
\bibitem{DEM1} A.V. Kotikov, 
Phys.Lett. B254 (1991) 158; 
%%CITATION = PHLTA,B254,158;%%
Phys.Lett. B259 (1991) 314;
%%CITATION = PHLTA,B259,314;%%
Phys.Lett. B267 (1991) 123; 
%%CITATION = PHLTA,B267,123;%%
hep-ph/0102178.
%%CITATION = HEP-PH 0102178;%%
%
\bibitem{Passarino} G. Passarino, hep-ph/0101299); 
G. Passarino and S. Uccirati, hep-ph/0112004.
%
\bibitem{QED3} A.V. Kotikov, JETP Lett. 58 (1993) 731.
%%CITATION = JTPLA,58,731;%%
%
\bibitem{Nash} T. Appelquist et al., Phys.Rev. D33 (1986) 3774;
Phys.Rev.Lett. 60 (1988) 2575;
%%CITATION = PRLTA,60,2575;%% 
D. Nash, Phys.Rev. Lett. 62 (1989) 3024.
%%CITATION = PRLTA,62,3024;%%
%
\bibitem{KoKo} I.N. Kondrashuk and A.V. Kotikov, 
Phys. Rev. D53 (1996) 2260.
%%CITATION = HEP-TH 9603076;%%
%
\bibitem{korea} S.H. Park, 
Phys.Rev. D45 (1992) 3332.
%%CITATION = PHRVA,D45,3332;%%
%
%
\bibitem{Fadin}
V.S. Fadin, R. Fiore and M.I. Kotsky,
Phys. Lett. B387 (1996) 593.
%
\bibitem{KoLi}
A.V. Kotikov and L.N. Lipatov, 
Nucl.Phys. B582 (2000) 19. 
%%CITATION = HEP-PH 0004008;%%
%\vspace{-2.5mm}
%
%
\bibitem{KoLi1}
A.V. Kotikov and L.N. Lipatov, hep-ph/0112346, 
%%CITATION = HEP-PH 0112346;%%
hep-ph/0208220. 
%%CITATION = HEP-PH 0208220;%%
%
\bibitem{BrKo}
D.J. Broadhurst and A.V. Kotikov,
Phys. Lett. B441 (1998) 345.
%%CITATION = HEP-TH 9612013;%%
%
\bibitem{FKV1} J. Fleischer, A.V. Kotikov, and O.L. Veretin, 
Nucl.Phys. B547 (1999) 343. 
%%CITATION = HEP-PH 9808242;%%
%
\bibitem{FKV} J. Fleischer, A.V. Kotikov, and O.L. Veretin, 
Phys.Lett. B417 (1998) 163;
%%CITATION = HEP-PH 9707492;%%
%
%\bibitem{FKV2} J. Fleischer, A.V. Kotikov, and O.L. Veretin, 
Acta Phys.Polon. B29 (1998) 2611.
%, hep-ph/9808243.
%%CITATION = HEP-PH 9808243;%% 
%
%
\bibitem{FKK} J. Fleischer, M.Yu. Kalmykov, and A.V. Kotikov, 
Phys.Lett. B462 (1999) 169; B467 (1999) 310(E).
%%CITATION = HEP-PH 9905249;%%
%
\bibitem{FKK1} J. Fleischer, M.Yu. Kalmykov, and A.V. Kotikov, 
hep-ph/9905379;
%%CITATION = HEP-PH 9905379;%%
 J. Fleischer and M.Yu. Kalmykov,
Comput.Phys.Commun. 128 (2000) 531; 
%%CITATION = HEP-PH 9907431;%%
Phys.Lett. B470 (1999) 168.
%A.I. Davydychev, Phys.Rev. D61 (2000) 087701;
%A.I. Davydychev and M.Yu. Kalmykov,
%Nucl.Phys. B605 (2001) 266.
%
\end{thebibliography}