Laurent Hoffmann, \footnote{email:{hoffmann@physik.uni-kl.de}} Anastasios C. Petkou \footnote{email:{petkou@physik.uni-kl.de}} and Werner R\" uhl \footnote{ email:{ruehl@physik.uni-kl.de}}
\if@twocolumn \section*{Abstract} \else %\begin{center} {Abstract\vspace{-.5em}\vspace{0pt}} \end{center} \quotation \fi
4ptf_letter.tex
\documentclass[11pt]{article}
%\input defs
\input epsf
\usepackage{psfrag}
\usepackage{graphicx}

\def\baselinestretch{1.2}
\parskip 6 pt

\textheight 22.5 cm
\topmargin -.5 cm
\textwidth 16cm
\oddsidemargin 0 in
\evensidemargin 0 in
%

%
% Putting citations together
%
\catcode`\@=11
%------------
%NAME: cite.sty
%
%              Replacement for \@citex
%
%     Allow, but strongly discourage, line breaks within a long
%     series of citations.  Compress lists of successive numbers
%     to one number range, e.g., 5,6,7,8,9 --> 5--9.  Compatible
%     with versions of \@cite that use exponents.
%              -- Donald Arseneau  1989
%
\def\@citex[#1]#2{%
\if@filesw \immediate \write \@auxout {\string \citation {#2}}\fi
\@tempcntb\m@ne \let\@h@ld\relax \def\@citea{}%
\@cite{%
  \@for \@citeb:=#2\do {%
    \@ifundefined {b@\@citeb}%
      {\@h@ld\@citea\@tempcntb\m@ne{\bf ?}%
      \@warning {Citation `\@citeb ' on page \thepage \space undefined}}%
%   else
      {\@tempcnta\@tempcntb \advance\@tempcnta\@ne%
      \@tempcntb\number\csname b@\@citeb \endcsname \relax%
      \ifnum\@tempcnta=\@tempcntb %Number follows previous--hold on to it
        \ifx\@h@ld\relax%
%         % first pair of successive
          \edef \@h@ld{\@citea\csname b@\@citeb\endcsname}%
        \else%
%         % compressible list of successive
          \edef\@h@ld{\ifmmode{-}\else--\fi\csname b@\@citeb\endcsname}%
        \fi%
      \else%   %  non-successor--dump what's held and do this one
        \@h@ld\@citea\csname b@\@citeb \endcsname%
        \let\@h@ld\relax%
      \fi}%
    \def\@citea{,\penalty\@highpenalty\,}%
  }\@h@ld
}{#1}}

%    \@cite macro using []'s:
%
\def\@citeb#1#2{{[#1]\if@tempswa , #2\fi}}
%
%    \@cite macro using exponents:
%
\def\@citeu#1#2{{$^{#1}$\if@tempswa , #2\fi }}
%
%    \@cite macro with nothing:
%
\def\@citep#1#2{{#1\if@tempswa , #2\fi}}

%
%   Style of citations:
%
\def\bcites{         % cite with []'s
        \catcode`\@=11
        \let\@cite=\@citeb
        \catcode`\@=12
}

\def\upcites{         % cite with exponents
        \catcode`\@=11
        \let\@cite=\@citeu
        \catcode`\@=12
}

\def\plaincites{      % cite without brackets
        \catcode`\@=11
        \let\@cite=\@citep
        \catcode`\@=12
}

%
%   Draft stuff
%

\newcount\hour
\newcount\minute
\newtoks\amorpm
\hour=\time\divide\hour by 60
\minute=\time{\multiply\hour by 60 \global\advance\minute by-\hour}
\edef\standardtime{{\ifnum\hour<12 \global\amorpm={am}%
        \else\global\amorpm={pm}\advance\hour by-12 \fi
        \ifnum\hour=0 \hour=12 \fi
        \number\hour:\ifnum\minute<10 0\fi\number\minute\the\amorpm}}
\edef\militarytime{\number\hour:\ifnum\minute<10 0\fi\number\minute}

\def\draftlabel#1{{\@bsphack\if@filesw {\let\thepage\relax
   \xdef\@gtempa{\write\@auxout{\string
      \newlabel{#1}{{\@currentlabel}{\thepage}}}}}\@gtempa
   \if@nobreak \ifvmode\nobreak\fi\fi\fi\@esphack}
        \gdef\@eqnlabel{#1}}
\def\@eqnlabel{}
\def\@vacuum{}
\def\marginnote#1{}
\def\draftmarginnote#1{\marginpar{\raggedright\scriptsize\tt#1}}
\overfullrule=0pt

\def\draft{
        \pagestyle{plain}
        \overfullrule=2pt
        \oddsidemargin -.5truein
        \def\@oddhead{\sl \phantom{\today\quad\militarytime} \hfil
        \smash{\Large\sl DRAFT} \hfil \today\quad\militarytime}
        \let\@evenhead\@oddhead
        \let\label=\draftlabel
        \let\marginnote=\draftmarginnote
        \def\ps@empty{\let\@mkboth\@gobbletwo
        \def\@oddfoot{\hfil \smash{\Large\sl DRAFT} \hfil}
        \let\@evenfoot\@oddhead}
        \def\@eqnnum{(\theequation)\rlap{\kern\marginparsep\tt\@eqnlabel}%
        \global\let\@eqnlabel\@vacuum}  }


%
%               Black stuff
%
\def\blackfonts{
        \font\blackboard=msbm10 scaled\magstep1
        \font\blackboards=msbm8
        \font\blackboardss=msbm6
}

\def\yblack{
        \blackfonts
        \newfam\black
        \textfont\black=\blackboard
        \scriptfont\black=\blackboards
        \scriptscriptfont\black=\blackboardss
        \def\ZZ{{\fam\black\relax Z}}
        \def\NN{{\fam\black\relax  N}}
        \def\CC{{\fam\black\relax  C}}
        \def\RR{{\fam\black\relax  R}}
        \def\QQ{{\fam\black\relax  Q}}
        \def\PP{{\fam\black\relax  P}}
}

\def\nblack{            % For people without blackboard fonts
        \def\ZZ{{Z \n{10} Z}}
        \def\NN{{N \n{14} N}}
        \def\CC{{C \n{11} C}}
        \def\RR{{R \n{11} R}}
        \def\QQ{{Q \n{12} Q}}
        \def\PP{{P \n{11} P}}
}

%
%               Two column
%

\def\prep{         % twocolumn.sty  Changed by Marek and Neil
        \catcode`\@=11
        \input art10.sty
        \catcode`\@=12
        \special{landscape}
        \let\small\null
        \def\blackfonts{
                \font\blackboard=msbm10
                \font\blackboards=msbm7
                \font\blackboardss=msbm5
        }
        \let\sl\it
        \twocolumn
        \sloppy
        \voffset=-2.54truecm
        \hoffset=-2.54truecm
        \flushbottom
        \parindent 1em
        \leftmargini 2em
        \leftmarginv .5em
        \leftmarginvi .5em
        \marginparwidth 48pt
        \marginparsep 10pt
        \setlength{\columnsep}{2truecm}
        \setlength{\textwidth}{25.4truecm}
        \setlength{\textheight}{17truecm}
        \baselineskip=16pt
        \oddsidemargin .18truein
        \evensidemargin .17truein
}

%TeX \eqalign

\def\eqalign#1{\null\,\vcenter{\openup\jot\m@th
  \ialign{\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil
      \crcr#1\crcr}}\,}
\def\eqalignno#1{\displ@y \tabskip\centering
  \halign to\displaywidth{\hfil$\@lign\displaystyle{##}$\tabskip\z@skip
    &$\@lign\displaystyle{{}##}$\hfil\tabskip\centering
    &\llap{$\@lign##$}\tabskip\z@skip\crcr
    #1\crcr}}

%
%    Personal choice here
%
\def\section{\@startsection {section}{1}{\z@}{3.ex plus 1ex minus
 .2ex}{2.ex plus .2ex}{\large\bf}}
\def\subsection{\@startsection{subsection}{2}{\z@}{2.75ex plus 1ex minus
 .2ex}{1.5ex plus .2ex}{\bf}}        
\def\subsect#1{\par\penalty1000{\noindent \bf #1}\par\penalty500}
\def\appendix{{\newpage\section*{Appendix}}\let\appendix\section%
        {\setcounter{section}{0}
        \gdef\thesection{\Alph{section}}}\section}
\def\thefootnote{\arabic{footnote}}
\def\abstract{\if@twocolumn
\section*{Abstract}
\else %\small
\begin{center}
{\bf Abstract\vspace{-.5em}\vspace{0pt}}
\end{center}
\quotation
\fi}

\catcode`\@=12
%%%%mak.tex%%%%%%%%%definitions%%%%%%%%%%%%%%%%%%%%%%
\def\lin{\bigskip\centerline{\vrule width 5cm height0.4pt}\bigskip}
\def\d{\partial}
\def\dh{\mathop{\vphantom{\odot}\hbox{$\partial$}}}
\def\dl{\dh^\leftrightarrow}
\def\sqr#1#2{{\vcenter{\vbox{\hrule height.#2pt\hbox{\vrule width.#2pt 
height#1pt \kern#1pt \vrule width.#2pt}\hrule height.#2pt}}}}
\def\w{\mathchoice\sqr45\sqr45\sqr{2.1}3\sqr{1.5}3\,} 
\def\fii{\varphi}
\def\eps{\varepsilon}
\def\hq{\hbar}
\def\lo{\cal L_+^\uparrow}
\def\psq{{\overline{\psi}}}
\def\pp{\psi ^\prime}
\def\ppq{\overline{\psi ^\prime}}
\def\sp{\vec\sigma\cdot\vec p}
\def\pdh{(2\pi )^{-3/2}}
\def\ps{\hbox{\rm\rlap/p}}
\def\ec{{e\over c}}
\def\=d{\,{\buildrel\rm def\over =}\,}
\def\iix{\int d^3x\,}
\def\iip{\int d^3p}
\def\inx{\int d^3x_1\ldots d^3x_n}
\def\H{{\cal H}}
\def\F{{\cal F}}
\def\N{\hbox{\bbf N}}
\def\A{\hbox{\bbf A}}
\def\xn{\vec x_1,\ldots ,\vec x_n\,}
\def\vxp{\vec x\,'}
\def\V{\hbox{\bbf V}}
\def\S{\hbox{\bbf S}}
\def\U{{\hbox{\bbf U}}}
\def\HH{\hbox{\bbf H}}
\def\Q{\hbox{\bbf Q}}
\def\i3p{\p32\int d^3p}
\def\psm{\psi ^{(-)}}
\def\psp{\psi ^{(+)}}
\def\px{\vec p\cdot\vec x}
\def\pqp{\overline{\psi}^{(+)}}
\def\pqm{\overline{\psi}^{(-)}}
\def\vq{\overline v}
\def\uq{\overline u}
\def\iep{\int {d^3p\over 2E}}
\def\ipd{\int d^4p\,\delta (p^2-m^2)}
\def\ds{\hbox{\rlap/$\partial$}}
\def\pe{\sqrt{\vec p^2+m^2}}
\def\dsx{\hbox{\rlap/$\partial$}_x}
\def\itn{\int\limits_{-\infty}^{+\infty}dt_1\int\limits_{-\infty}^{t_1}
dt_2\cdots\int\limits_{-\infty}^{t_{n-1}}dt_n}
\def\ipn{\int d^3p_1\cdots\int d^3p_{n-1}\, }
\def\As{A\hbox to 1pt{\hss /}}
\def\np4{\int d^4p_1\cdots d^4p_{n-1}\, }
\def\Sr{S^{\rm ret}}
\def\gp{\vec\gamma\cdot\vec p}
\def\te{\vartheta}
\def\tr{{\rm tr}\, }
\def\Sa{S^{\rm av}}
\def\qs{\hbox{\rlap/q}}
\def\supp{{\rm supp}\, }
\def\Tr{{\rm Tr}\, }
\def\Im{{\rm Im}\, }
\def\sgn{{\rm sgn}\, }
\def\cau{{1\over 2\pi i}}
\def\P{\,{\rm P}}
\def\Re{{\rm Re}\, }
\def\iinf{\int\limits_{-\infty}^{+\infty}}
\def\kx{\vec k\cdot\vec x}
\def\io{\int{d^3k\over\sqrt{2\omega}}\,}
\def\nx4{\int d^4x_1\ldots d^4x_n\, }
\def\xnn{x_1,\ldots ,x_n}
\def\xnm{x_1,\ldots ,x_{n-1},x_n}
\def\Dr{D^{\rm ret}}
\def\Da{D^{\rm av}}
\def\kon#1#2{\vbox{\halign{##&&##\cr
\lower4pt\hbox{$\scriptscriptstyle\vert$}\hrulefill &
\hrulefill\lower4pt\hbox{$\scriptscriptstyle\vert$}\cr $#1$&
$#2$\cr}}}
\def\lra{\longleftrightarrow}
\def\konv#1#2#3{\hbox{\vrule height12pt depth-1pt}
\vbox{\hrule height12pt width#1cm depth-11.6pt}
\hbox{\vrule height6.5pt depth-0.5pt}
\vbox{\hrule height11pt width#2cm depth-10.6pt\kern5pt
      \hrule height6.5pt width#2cm depth-6.1pt}
\hbox{\vrule height12pt depth-1pt}
\vbox{\hrule height6.5pt width#3cm depth-6.1pt}
\hbox{\vrule height6.5pt depth-0.5pt}}
\def\konu#1#2#3{\hbox{\vrule height12pt depth-1pt}
\vbox{\hrule height1pt width#1cm depth-0.6pt}
\hbox{\vrule height12pt depth-6.5pt}
\vbox{\hrule height6pt width#2cm depth-5.6pt\kern5pt
      \hrule height1pt width#2cm depth-0.6pt}
\hbox{\vrule height12pt depth-6.5pt}
\vbox{\hrule height1pt width#3cm depth-0.6pt}
\hbox{\vrule height12pt depth-1pt}}
\def\es{\hbox{\rlap/$\varepsilon$}}
\def\ks{\hbox{\rlap/k}}
\def\konw#1#2#3{\hbox{\vrule height12pt depth-1pt}
\vbox{\hrule height12pt width#1cm depth-11.6pt}
\hbox{\vrule height6.5pt depth-0.5pt}
\vbox{\hrule height12pt width#2cm depth-11.6pt \kern5pt
      \hrule height6.5pt width#2cm depth-6.1pt}
\hbox{\vrule height6.5pt depth-0.5pt}
\vbox{\hrule height12pt width#3cm depth-11.6pt}
\hbox{\vrule height12pt depth-1pt}}
\def\grad{{\rm grad}\, }
\def\diw{{\rm div}\, }
\def\eh{{\scriptstyle{1\over 2}}}
\def\intt{\int dt_1\ldots dt_n\,}
\def\tnn{t_1,\ldots ,t_n}
\def\dett{{\rm det}\,}
\def\lap{\bigtriangleup\,}
\def\HHe{\hbox{\bex H}}
\font\bex=cmbx7
\def\Wiout{W_{{\rm\scriptstyle in}\atop{\rm\scriptstyle out}}}
\def\Win{W_{\rm in}}
\def\Wout{W_{\rm out}}
\def\poi{\cal P _+^\uparrow}
\def\gu{\underline{g}} 
\def\hu{\underline{h}}
\def\i{{\rm int}}
\def\su{\sum_{n=1}^\infty}
\def\suu{\sum_{n=0}^\infty}
\def\c{{\rm cl}}
\def\e{{\rm ext}}
\def\snn{\su {1\over n!}\nx4}
\def\r{{\rm ret}}
\def\a{{\rm av}}
%\def\ra{{{\rm\scriptstyle ret}\atop{\rm\scriptstyle av}}}
\def\dsu{\mathop{\bigoplus}}            
\def\m3{{\mu_1\mu_2\mu_3}}
\def\itt{\int\limits_{t_1}^{t_2}}
\def\gs{\,{\scriptstyle{>\atop <}}\,}
\def\ga{\,{\scriptstyle{>\atop\sim}\,}}
\def\Seh{{\cal S}^\eh_\eh}
\def\SS{{\cal S}({\bf R^4})}
\def\co{{\rm Com}}
\def\ml{{m_1\ldots m_l}}
\def\ik{{i_1\ldots i_k}}
\def\jt{{j_1\ldots j_t}}
\def\is{{i_{k+1}\ldots i_s}}
\def\js{{j_{t+1}\ldots j_s}}
\def\xnr{x_1-x_n,\ldots x_{n-1}-x_n}
\def\qm{q_1,\ldots q_{m-1}}
\def\pn{p_1,\ldots p_{n-1}}
\def\xll{x_l-x_{l+1}}
\def\xmm{x_1,\ldots x_{n-1}}
\def\ph{{\rm phys}}
\def\nab{\bigtriangledown}
\def\p{{(+)}}
\def\ul{\underline}
\def\tu{\tilde u}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\input fon3.tex
%\input mak.tex
%\input definitions


%%%%%%%%%%%%%%%%%%%% defin %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%\newcommand{\myappendix}{\setcounter{equation}{0}\appendix}
%\newcommand{\mysection}{\setcounter{equation}{0}\section}
%\renewcommand{\theequation}{\arabic{equation}}

\def\be{\begin{equation}}       \def\eq{\begin{equation}}
\def\ee{\end{equation}}         \def\eqe{\end{equation}}

\def\bea{\begin{eqnarray}}      \def\eqa{\begin{eqnarray}}
\def\ena{\end{eqnarray}}        \def\eea{\end{eqnarray}}
                                \def\eqae{\end{eqnarray}}

\def\ba{\begin{array}}
\def\ea{\end{array}}
\def\unit{1 \hskip-.3em \raise2pt\hbox{$ \scriptstyle |$ } }
% Accents and foreign (in text):

\let\und=\b % bar-under (but see \un below; it's better)
\let\ced=\c % cedilla
\let\du=\d % dot-under
\let\um=\H % Hungarian umlaut
\let\sll=\l % slashed (suppressed) l (Polish)
\let\Sll=\L % " L
\let\slo=\o % slashed o (Scandinavian)
\let\Slo=\O % " O
\let\tie=\t % tie-after (semicircle connecting two letters)
\let\br=\u % breve
              % Also:      \`       grave
              %     \'       acute
              %     \v       hacek (check)
              %     \^       circumflex (hat)
              %     \~       tilde (squiggle)
              %     \=       macron (bar-over)
              %     \.       dot (over)
              %     \"       umlaut (dieresis)
              %     \aa \AA         A-with-circle (Scandinavian)
              %     \ae \AE         ligature (Latin & Scandinavian)
              %     \oe \OE         " (French)
              %     \ss      es-zet (German sharp s)
              %     \$  \#  \&  \%  \pounds  {\it\&}  \dots

% Abbreviations for Greek letters

\def\a{\alpha}
\def\b{\beta}
\def\c{\gamma} 
\def\d{\delta}
\def\e{\epsilon}           % Also, \varepsilon
\def\f{\phi}               %      \varphi
\def\g{\gamma}
\def\h{\eta}   
\def\i{\iota}
\def\j{\psi}
\def\k{\kappa}                    % Also, \varkappa (see below)
\def\l{\lambda}
\def\m{\mu}
\def\n{\nu}
\def\o{\omega}  \def\w{\omega}
\def\p{\pi}                % Also, \varpi
\def\q{\theta}  \def\th{\theta}                  %     \vartheta
\def\r{\rho}                                     %     \varrho
\def\s{\sigma}                                   %     \varsigma
\def\t{\tau}
\def\u{\upsilon}
\def\x{\xi}
\def\z{\zeta}
\def\D{\Delta}
\def\F{\Phi}
\def\G{\Gamma}
\def\J{\Psi}
\def\L{\Lambda}
\def\O{\Omega}  \def\W{\Omega}
\def\P{\Pi}
\def\Q{\Theta}
\def\S{\Sigma}
\def\U{\Upsilon}
\def\X{\Xi}
\def\del{\partial}              % overwritten by \nabla

% Calligraphic letters

\def\ca{{\cal A}}
\def\cb{{\cal B}}
\def\cc{{\cal C}}
\def\cd{{\cal D}}
\def\ce{{\cal E}}
\def\cf{{\cal F}}
\def\cg{{\cal G}}
\def\ch{{\cal H}}
\def\ci{{\cal I}}
\def\cj{{\cal J}}
\def\ck{{\cal K}}
\def\cl{{\cal L}}
\def\cm{{\cal M}}
\def\cn{{\cal N}}
\def\co{{\cal O}}
\def\cp{{\cal P}}
\def\cq{{\cal Q}}
\def\car{{\cal R}}
\def\cs{{\cal S}}
\def\ct{{\cal T}}
\def\cu{{\cal U}}
\def\cv{{\cal V}}
\def\cw{{\cal W}}
\def\cx{{\cal X}}
\def\cy{{\cal Y}}
\def\cz{{\cal Z}}





% You can never get enough fonts

\def\half{{1 \over 2}}

\def\Bf#1{\mbox{\boldmath $#1$}}       % bold Greek
\def\Sf#1{\hbox{\sf #1}}               % sans serif "

                    % Also:        \rm    Roman (default for text)
                    %       \bf     boldface
                    %       \it     italic
                    %       \mit    math italic (default for equations)
                    %       \em     emphatic
                    %       \tt     typewriter
                    % (but not \sl slanted, which I hate/redefined)
                    % and sizes:  \tiny
                    %             \scriptsize
                    %             \footnotesize
                    %             \small
                    %             \normalsize
                    %             \large
                    %             \Large
                    %             \LARGE
                    %             \huge
                    %             \Huge         see big.tex
                    %             \HUGE

% Math symbols

\def\bop#1{\setbox0=\hbox{$#1M$}\mkern1.5mu
        \vbox{\hrule height0pt depth.04\ht0
        \hbox{\vrule width.04\ht0 height.9\ht0 \kern.9\ht0
        \vrule width.04\ht0}\hrule height.04\ht0}\mkern1.5mu}
\def\Box{{\mathpalette\bop{}}}                        % box
\def\pa{\partial}                              % curly d
\def\de{\nabla}                                       % del
\def\dell{\bigtriangledown} % hi ho the dairy-o
\def\su{\sum}                                         % summation
\def\pr{\prod}                                        % product
\def\iff{\leftrightarrow}                      % <-->
\def\conj{{\hbox{\large *}}} % complex conjugate
\def\lconj{{\hbox{\footnotesize *}}}          % little "
\def\dg{\sp\dagger} % hermitian conjugate
\def\ddg{\sp\ddagger} % double dagger

\def\>{\rangle} %right angle

\def\<{\langle} %left angle
\def\Dsl{D \hskip-.6em \raise1pt\hbox{$ / $ } }

                    % Also:  \int  \oint   integral, contour
                    %       \hbar                h bar
                    %       \infty               infinity
                    %       \sqrt                square root
                    %       \pm  \mp             plus or minus
                    %       \cdot  \cdots        centered dot(s)
                    %       \oplus  \otimes      group theory
                    %       \equiv               equivalence
                    %       \sim                        ~
                    %       \approx              approximately =
                    %       \propto              funny alpha
                    %       \ne                  not =
                    %       \le \ge              < or = , > or =
                    %       \{  \}               braces
                    %       \to  \gets           -> , <-
                    % and spaces:  \,  \:  \;  \quad  \qquad
                    %             \!             (negative)

% Math stuff with 1 argument (but see >1 below)

\def\sp#1{{}^{#1}}                             % superscript (unaligned)
\def\sb#1{{}_{#1}}                             % sub"
\def\oldsl#1{\rlap/#1}                 % poor slash, except for Roman
\def\sl#1{\rlap{\hbox{$\mskip 1 mu /$}}#1}% good slash for l.c.
\def\Sl#1{\rlap{\hbox{$\mskip 3 mu /$}}#1}     % " upper
\def\SL#1{\rlap{\hbox{$\mskip 4.5 mu /$}}#1}   % " fat stuff (e.g., M)
\def\Tilde#1{\widetilde{#1}}                   % big tilde
\def\Hat#1{\widehat{#1}}                       % big hat
\def\Bar#1{\overline{#1}}                      % big bar
\def\bra#1{\Big\langle #1\Big|}                       % < |
\def\ket#1{\Big| #1\Big\rangle}                       % | >
\def\VEV#1{\Big\langle #1\Big\rangle}                 % < >
\def\abs#1{\Big| #1\Big|}                      % | |
\def\sbra#1{\left\langle #1\right|}            % variable < |
\def\sket#1{\left| #1\right\rangle}            % variable | >
\def\sVEV#1{\left\langle #1\right\rangle}      % variable < >
\def\sabs#1{\left| #1\right|}                  % variable | |
\def\leftrightarrowfill{$\mathsurround=0pt \mathord\leftarrow \mkern-6mu
       \cleaders\hbox{$\mkern-2mu \mathord- \mkern-2mu$}\hfill
       \mkern-6mu \mathord\rightarrow$}
\def\dvec#1{\vbox{\ialign{##\crcr
       \leftrightarrowfill\crcr\noalign{\kern-1pt\nointerlineskip}
       $\hfil\displaystyle{#1}\hfil$\crcr}}}          % <--> accent
\def\hook#1{{\vrule height#1pt width0.4pt depth0pt}}
\def\leftrighthookfill#1{$\mathsurround=0pt \mathord\hook#1
       \hrulefill\mathord\hook#1$}
\def\underhook#1{\vtop{\ialign{##\crcr                 % |_| under
       $\hfil\displaystyle{#1}\hfil$\crcr
       \noalign{\kern-1pt\nointerlineskip\vskip2pt}
       \leftrighthookfill5\crcr}}}
\def\smallunderhook#1{\vtop{\ialign{##\crcr      % " for su'scripts
       $\hfil\scriptstyle{#1}\hfil$\crcr
       \noalign{\kern-1pt\nointerlineskip\vskip2pt}
       \leftrighthookfill3\crcr}}}
\def\der#1{{\pa \over \pa {#1}}}               % partial derivative
\def\fder#1{{\d \over \d {#1}}} % functional derivative
              % Also math accents:      \bar
              %                   \check
              %                   \hat
              %                   \tilde
              %                   \acute
              %                   \grave
              %                   \breve
              %                   \dot  (over)
              %                   \ddot (umlaut)
              %                   \vec  (vector)

% Math stuff with >1 argument

%\def\frac#1#2{{\textstyle{#1\over\vphantom2\smash{\raise.20ex
%       \hbox{$\scriptstyle{#2}$}}}}}                  % fraction
\def\ha{\frac12}                               % 1/2
\def\sfrac#1#2{{\vphantom1\smash{\lower.5ex\hbox{\small$#1$}}\over
       \vphantom1\smash{\raise.4ex\hbox{\small$#2$}}}} % alt. fraction
\def\bfrac#1#2{{\vphantom1\smash{\lower.5ex\hbox{$#1$}}\over
       \vphantom1\smash{\raise.3ex\hbox{$#2$}}}}      % "
\def\afrac#1#2{{\vphantom1\smash{\lower.5ex\hbox{$#1$}}\over#2}}  %"
\def\dder#1#2{{\pa #1\over\pa #2}}        % partial derivative of
\def\secder#1#2#3{{\pa\sp 2 #1\over\pa #2 \pa #3}}          % second "
\def\fdder#1#2{{\d #1\over\d #2}}         % functional derivative of
\def\on#1#2{{\buildrel{\mkern2.5mu#1\mkern-2.5mu}\over{#2}}}%acc.over
\def\On#1#2{\mathop{\null#2}\limits^{\mkern2.5mu#1\mkern-2.5mu}}%alt."
\def\under#1#2{\mathop{\null#2}\limits_{#1}}          % accent under
\def\bvec#1{\on\leftarrow{#1}}                 % backward vector accent
\def\oover#1{\on\circ{#1}}                            % circle accent
\def\dt#1{\on{\hbox{\LARGE .}}{#1}}                   % (big) dot over
\def\dtt#1{\on\bullet{#1}}                      % alternate (bigger) "
\def\ddt#1{\on{\hbox{\LARGE .\kern-2pt.}}#1}             % double dot
\def\tdt#1{\on{\hbox{\LARGE .\kern-2pt.\kern-2pt.}}#1}   % triple dot

% Young tableaux:  \boxup<a>{\boxes<a>...\boxes<b>}

\def\boxes#1{
       \newcount\num
       \num=1
       \newdimen\downsy
       \downsy=-1.5ex
       \mskip-2.8mu
       \bo
       \loop
       \ifnum\num<#1
       \llap{\raise\num\downsy\hbox{$\bo$}}
       \advance\num by1
       \repeat}
\def\boxup#1#2{\newcount\numup
       \numup=#1
       \advance\numup by-1
       \newdimen\upsy
       \upsy=.75ex
       \mskip2.8mu
       \raise\numup\upsy\hbox{$#2$}}

% Aligned equations

\newskip\humongous \humongous=0pt plus 1000pt minus 1000pt
\def\caja{\mathsurround=0pt}
\def\eqalign#1{\,\vcenter{\openup2\jot \caja
       \ialign{\strut \hfil$\displaystyle{##}$&$
       \displaystyle{{}##}$\hfil\crcr#1\crcr}}\,}
\newif\ifdtup
\def\panorama{\global\dtuptrue \openup2\jot \caja
       \everycr{\noalign{\ifdtup \global\dtupfalse
       \vskip-\lineskiplimit \vskip\normallineskiplimit
       \else \penalty\interdisplaylinepenalty \fi}}}
\def\li#1{\panorama \tabskip=\humongous                % eqalignno
       \halign to\displaywidth{\hfil$\displaystyle{##}$
       \tabskip=0pt&$\displaystyle{{}##}$\hfil
       \tabskip=\humongous&\llap{$##$}\tabskip=0pt
       \crcr#1\crcr}}
\def\eqalignnotwo#1{\panorama \tabskip=\humongous
       \halign to\displaywidth{\hfil$\displaystyle{##}$
       \tabskip=0pt&$\displaystyle{{}##}$
       \tabskip=0pt&$\displaystyle{{}##}$\hfil
       \tabskip=\humongous&\llap{$##$}\tabskip=0pt
       \crcr#1\crcr}}

% Tables: filler space for tabular*

\def\phil{@{\extracolsep{\fill}}}
\def\unphil{@{\extracolsep{\tabcolsep}}}

% Journal abbreviations

\def\NP{Nucl. Phys. B}
\def\PL{Phys. Lett. }
\def\PR{Phys. Rev. Lett. }
\def\PRD{Phys. Rev. D}
%\def\ref#1{$\sp{#1]}$}
\def\Ref#1{$\sp{#1)}$}

%%%%%%%% Bas macro's for refs
\def\JSP#1#2#3{J. Stat. Phys. {\bf #1} (#2) #3}
\def\PRL#1#2#3{Phys. Rev. Lett. {\bf#1} (#2) #3}
\def\PR#1#2#3{Phys. Rev. {\bf#1} (#2) #3}
\def\EPL#1#2#3{{\it Europhys. Lett.} {\bf#1} (#2) #3}
\def\NPB#1#2#3{Nucl. Phys. {\bf B#1} (#2) #3}
\def\NPBFS#1#2#3#4{Nucl. Phys. {\bf B#2} [FS#1] (#3) #4}
\def\CMP#1#2#3{{\it Comm. Math. Phys.} {\bf #1} (#2) #3}
\def\CPAM#1#2#3{{\it Comm. Pure Appl. Math.} {\bf #1} (#2) #3}
\def\PRD#1#2#3{ Phys. Rev. {\bf D#1} (#2) #3}
\def\PRB#1#2#3{Phys. Rev. {\bf B#1} (#2) #3}
\def\PLB#1#2#3{Phys. Lett. {\bf #1B} (#2) #3}
\def\PLA#1#2#3{{Phys. Lett.} {\bf #1A} (#2) #3}
\def\JMP#1#2#3{{\it J. Math. Phys.} {\bf #1} (#2) #3}
\def\JMM#1#2#3{{\it J. Math. Mech.} {\bf #1} (#2) #3}
\def\PTP#1#2#3{{\it Prog. Theor. Phys.} {\bf #1} (#2) #3}
\def\SPTP#1#2#3{{\it Suppl. Prog. Theor. Phys.} {\bf #1} (#2) #3}
\def\AoP#1#2#3{{\it Ann. of Phys.} {\bf #1} (#2) #3}
\def\APNY#1#2#3{{\it Ann. Phys. (N.Y.)} {\bf #1} (#2) #3}
\def\PNAS#1#2#3{{\it Proc. Natl. Acad. Sci. USA} {\bf #1} (#2) #3}
\def\RMP#1#2#3{{\it Rev. Mod. Phys.} {\bf #1} (#2) #3}
\def\PRep#1#2#3{{\it Phys. Reports} {\bf #1} (#2) #3}
\def\AoM#1#2#3{{\it Ann. of Math.} {\bf #1} (#2) #3}
\def\UMN#1#2#3{{\it Usp. Mat. Nauk} {\bf #1} (#2) #3}
\def\RMS#1#2#3{{\it Russian Math Surveys} {\bf #1} (#2) #3}
\def\FAP#1#2#3{{\it Funkt. Anal. Prilozheniya} {\bf #1} (#2) #3}
\def\FAaIP#1#2#3{{\it Functional Analysis and Its Application} {\bf #1} (#2)
#3}
\def\BSMF#1#2#3{{\it Bull. Soc. Mat. France} {\bf #1} (#2) #3}
\def\BAMS#1#2#3{{\it Bull. Am. Math. Soc.} {\bf #1} (#2) #3}
\def\TAMS#1#2#3{{\it Trans. Am. Math. Soc.} {\bf #1} (#2) #3}
\def\AIHP#1#2#3{{\it Ann. Inst. Henri Poincar\'e} {\bf #1} (#2) #3}
\def\ANYAS#1#2#3{{\it Ann. New York Acad. Sci.} {\bf #1} (#2) #3}
\def\AIF#1#2#3#4{{\it Ann. Inst. Fourier} {\bf #1,#2} (#3) #4}
\def\PAMS#1#2#3{{\it Proc. Am. Math. Soc.} {\bf #1} (#2) #3}
\def\PRS#1#2#3{{\it Proc. Roy. Soc.} {\bf #1} (#2) #3}
\def\CMJ#1#2#3{{\it Czechosl. Math. J.} {\bf #1} (#2) #3}
\def\CompM#1#2#3{{\it Compositio Math.} {\bf #1} (#2) #3}
\def\Compt#1#2#3{{\it Compt. Rend. Acad. Sci. Paris} {\bf #1} (#2) #3}
\def\Invm#1#2#3{{\it Invent. math.} {\bf #1} (#2) #3}
\def\LMP#1#2#3{{\it Letters in Math. Phys.} {\bf #1} (#2) #3}
\def\IJMPA#1#2#3{{\it Int. J. Mod. Phys.} {\bf A#1} (#2) #3}
\def\IJMPB#1#2#3{{\it Int. J. Mod. Phys.} {\bf B#1} (#2) #3}
\def\AdM#1#2#3{{\it Advances in Math.} {\bf #1} (#2) #3}
\def\AdP#1#2#3{{\it Advances in Phys.} {\bf #1} (#2) #3}
\def\RMaP#1#2#3{{\it Reports on Math. Phys.} {\bf #1} (#2) #3}
\def\IJM#1#2#3{{\it Ill. J. Math.} {\bf #1} (#2) #3}
\def\APP#1#2#3{{\it Acta Phys. Polon.} {\bf #1} (#2) #3}
\def\TMP#1#2#3{{\it Theor. Mat. Phys.} {\bf #1} (#2) #3}
\def\JPA#1#2#3{{\it J. Physics} {\bf A#1} (#2) #3}
\def\JPC#1#2#3{{\it J. Physics} {\bf C#1} (#2) #3}
\def\JPCM#1#2#3{{\it J. Physics Cond. Mat.} {\bf #1} (#2) #3}
\def\JPSJ#1#2#3{{\it J. Phys. Soc. Japan} {\bf #1} (#2) #3}
\def\Phy#1#2#3{{\it Physica} {\bf #1} (#2) #3}
\def\JSM#1#2#3{{\it J. Soviet Math.} {\bf #1} (#2) #3}
\def\MPLA#1#2#3{{\it Mod. Phys. Lett.} {\bf A#1} (#2) #3}
\def\MPLB#1#2#3{{\it Mod. Phys. Lett.} {\bf B#1} (#2) #3}
\def\JETP#1#2#3{{\it Sov. Phys. JETP} {\bf #1} (#2) #3}
\def\JETPL#1#2#3{{\it Sov. Phys. JETP Lett.} {\bf #1} (#2) #3}
\def\SPD#1#2#3{{\it Sov. Phys. Dokl.} {\bf #1} (#2) #3}
\def\CMH#1#2#3{{\it Comment. Math. Helv.} {\bf #1} (#2) #3}
\def\ZP#1#2#3{{\it Z.Phys.} {\bf #1} (#2) #3}
\def\ZPB#1#2#3{{\it Z.Phys.} {\bf B#1} (#2) #3}
\def\sci#1#2#3{{\it Science} {\bf #1} (#2) #3}
\def\LNC#1#2#3{{\it Lett. Nuovo Cimento} {\bf #1} (#2) #3}
\def\HPA#1#2#3{{\it Helv. Phys. Acta} {\bf #1} (#2) #3}
\def\SSC#1#2#3{{\it Solid State Comm.} {\bf #1} (#2) #3}


\def\to{\rightarrow}

%%%%%%%%%%%%%%%%%%%%%%%%%% can %%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\tj{\tilde{\j}}
\def\td{\tilde{D}}
\def\tv{\tilde{\varphi}}
\def\rld{\rlap{\,/}D}
\def\dv{\dot{\varphi}}
\def\dj{\dot{\j}}
\def\bv{\bar{\varphi}}
\def\bj{\bar{\j}}
\def\rld{\rlap{\,/}D}
\def\rla{\rlap{\,A}\bigcirc}
\def\1ov4{{1\over 4}}
\def\bc{\bar{\chi}}
\def\dox{\dot{x}}
\def\dc{\dot{\chi}}
\def\trld{\tilde{\rlap{\,/}D}}

%%%%%%%%%%%%%%%%%%%%% spin 3/2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\vecnab{\vec{\nabla}}
\def\vx{\vec{x}}
\def\vy{\vec{y}}
\def\arrowk{\stackrel{\rightarrow}{k}}
\def\kbar{k\!\!\!^{-}}
\def\karrow{k\!\!\!{\rightarrow}}
\def\arrowl{\stackrel{\rightarrow}{\ell}}


%%%%%%%%%%%%%%%%%%%% trace anomalies %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\SAmpl{ \langle z,\bar\eta | \exp \left( - { \b \over
\hbar } \hat H \right) | y,\chi \rangle }
\def\SAmplb{ \langle z | \exp \left( - { \b \over
\hbar } \hat H \right) | y \rangle }
\def\tr{{\rm tr}}
\def\Tr{{\rm Tr}}


%%%%%  Curci-Ferrari   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\pa{\partial}
\def\xx{\times}
\def\dd{\bar{\d}}

%%%%%%%%%%%%%%%%%%%%%%%%%%   berkovits-string %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




\def\dda{\dot{\alpha}} 
\def\ddb{\dot{\beta}}
\def\ddc{\dot{\chi}}
\def\ddd{\dot{\delta}}
\def\dde{\dot{\epsilon}}
\def\ddf{\dot{\phi}}
\def\ddg{\dot{\gamma}}
\def\ddh{\dot{\h}}
\def\ddi{\dot{\i}}
\def\ddj{\dot{\j}}
\def\ddk{\dot{\k}}
\def\ddl{\dot{\l}}
\def\ddm{\dot{\m}}
\def\ddn{\dot{\n}}
\def\ddo{\dot{\omega}}
\def\ddp{\dot{\p}}
\def\ddq{\dot{\q}}
\def\ddr{\dot{\r}}
\def\dds{\dot{\s}}
\def\ddt{\dot{\t}}
\def\ddu{\dot{\u}}
\def\ddx{\dot{\x}}
\def\ddz{\dot{\z}}

\def\ta{\tilde{\a}} 
\def\tb{\tilde{\b}}
\def\tc{\tilde{\c}}
\def\td{\tilde{\d}}
\def\te{\tilde{\e}}
\def\tf{\tilde{\f}}
\def\tg{\tilde{\g}}
\def\th{\tilde{\h}}
\def\ti{\tilde{\i}}
\def\tj{\tilde{\j}}
\def\tk{\tilde{\k}}
\def\tl{\tilde{\l}}
\def\tm{\tilde{\m}}
\def\tn{\tilde{\n}}
%\def\to{\tilde{\ome}}
\def\tp{\tilde{\p}}
\def\tq{\tilde{\q}}
\def\tr{\tilde{\r}}
\def\ts{\tilde{\s}}
\def\dott{\tilde{\t}}
\def\tu{\tilde{\u}}
\def\tx{\tilde{\x}}
\def\tz{\tilde{\z}}

\def\ome{\omega}

\def\pa{\partial}
\def\del{\nabla}
\def\delbar{\bar{\nabla}}

\def\xx{\times}

\def\ab{\bar{a}}
\def\bb{\bar{b}}
\def\cb{\bar{c}}
\def\db{\bar{d}}
\def\eb{\bar{e}}
\def\fb{\bar{f}}  \def\Fb{\bar{\F}}
\def\kb{\bar{k}}  \def\Kb{\bar{K}}
\def\lb{\bar{l}}  \def\Lb{\bar{L}} 
\def\mb{\bar{m}}  \def\Mb{\bar{M}}
\def\nb{\bar{n}}  \def\Tb{\bar{T}} 
\def\zb{\bar{z}}  \def\Gb{\bar{G}} 
\def\wb{\bar{w}}  \def\Jb{\bar{J}}
\def\pb{\bar{p}}  
\def\Ib{\bar{I}}


\def\Pbb{\bar{\P}} 
\def\jbb{\bar{\j}}
\def\qbb{\bar{\q}}
\def\Sbb{\bar{\S}}
\def\Pbb{\bar{\P}}
\def\dbb{\bar{\delta}}
\def\kbb{\bar{\kappa}} 
\def\ebb{\bar{\epsilon}}



\def\ua{\underline{a}}
\def\ub{\underline{b}}
\def\uc{\underline{c}}
\def\ud{\underline{d}}
\def\ue{\underline{e}}

\def\uua{\underline{\a}}
\def\uub{\underline{\b}}
\def\uuc{\underline{\c}}
\def\uud{\underline{\d}}
\def\uue{\underline{\e}}
\def\uug{\underline{\g}}


\def\hL{\hat{L}} 
\def\hM{\hat{M}}
\def\hj{\hat{\j}} 
\def\hf{\hat{\f}}
\def\hMb{\bar{\hat{M}}}
\def\hfb{\bar{\hat{\f}}}


\def\dif{\partial}
\def\difb{\bar{\partial}} \def\dbar{\bar{\partial}}
\def\pab{\bar{\pa}}
\def\nonu{\nonumber \\{}}
\def\half{{1 \over 2}}

\def\bfx{{\bf x}}
\def\bfy{{\bf y}}
\def\bfk{{\bf k}}
\def\bfl{{\bf l}}

\font\mybb=msbm10 at 12pt
\def\bb#1{\hbox{\mybb#1}}
\def\bZ {\bb{Z}}
\def\bR {\bb{R}}
\def\bE {\bb{E}}
\def\bT {\bb{T}}
\def\bM {\bb{M}}
%%%%%%%%%%%%%%%%%
\def\II{{\unit}}
\def\cM {{\cal{M}}}







%\draft

\newcommand{\non}{\nonumber}
\renewcommand{\arraystretch}{1}
\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}
\newcommand{\rar}{\rightarrow}
\newcommand{\lar}{\leftrarrow}
\newcommand{\lrar}{\leftrightarrow}
\def\pa{\partial}
\renewcommand{\a}{\alpha}
\renewcommand{\b}{\beta}
\renewcommand{\c}{\gamma}
\renewcommand{\d}{\delta}
%\newcommand{\e}{\epsilon}
\newcommand{\rmd}{{\rm d}}
%\newcommand{\m}{\mu}
%\newcommand{\n}{\nu}
\newcommand{\bpsi}{{\bar{\psi}}}
\newcommand{\bx}{{\bar{x}}}
\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\tD}{{\tilde{\Delta}}}
%\newcommand{\D}}{\Delta}
\def\ba{\begin{eqnarray}}
\def\ea{\end{eqnarray}}
\newcommand{\txt}{\textstyle}

\begin{document}
 


%%%%%%%%%

% Front page here

%\vspace*{1cm}
\null\vskip-24pt
\hfill KL-TH 00/01
\vskip-10pt
\hfill {\tt hep-th/0002025}
\vskip0.3truecm
\begin{center}
\vskip 3truecm
{\Large\bf
%\titleline
A note on the analyticity of AdS scalar exchange graphs in the crossed channel
}\\ 
\vskip 1.5truecm
%\vfill
{\large\bf
Laurent Hoffmann, 
 \footnote{email:{\tt hoffmann@physik.uni-kl.de}} Anastasios C. Petkou
   \footnote{email:{\tt
       petkou@physik.uni-kl.de}}   
and Werner R\" uhl \footnote{
email:{\tt ruehl@physik.uni-kl.de}}
}\\
\vskip 1truecm
%\addresses
{\it Department of Physics, Theoretical Physics\\
University of Kaiserslautern, Postfach 3049 \\
67653 Kaiserslautern, Germany}\\

\end{center}
\vskip 3truecm
\centerline{\bf Abstract}

We discuss the analytic properties of AdS scalar exchange
graphs in the crossed channel. We show that the possible non-analytic 
terms drop out by virtue of 
non-trivial properties of generalized hypergeometric functions. The
absence of non-analytic terms is
a necessary condition for the existence of an operator product
expansion for CFT amplitudes obtained from AdS/CFT correspondence.


\newpage

Scalar exchange conformal graphs are basic ingredients in a
calculation of conformally invariant $n$-point functions for $n\geq 
4$. Such graphs have been extensively studied in earlier works on  
conformal field theory (CFT) in $d>2$ \cite{fradkin,ruehl1,tassos1}.
Consider a conformal four-point function \footnote{For clarity we
  choose  all fields in the four-point function to have equal
  dimensions. The generalization of our results in the case of 
  different dimensions is straightforward.} ${\cal G}(x_1,..,x_4)= \langle{\cal
  O}(x_1)..{\cal O}(x_4)\rangle$ where the scalar field ${\cal O}(x)$
has dimension $\tilde{\Delta}$. One contribution to this
four-point function is given by the skeleton graphs in which a scalar
field $\sigma(x)$  of dimension $\D$ is exchanged between
external legs corresponding to ${\cal O}(x)$. For such skeleton graphs
to exist  it suffices that the three-point
function $\langle {\cal O}(x_1){\cal O}(x_2)\sigma(x_3)\rangle $ is
non-zero. \footnote{Conformal invariance determines the form of
three-point functions. Then, using the D'EPP formula
\cite{depp} for amputation one can obtain the full three-point
vertex function which is used in the skeleton graphs.} 
The relevant graph is depicted in Fig.1. 
Then, bose symmetry requires that ${\cal G}(x_1,..,x_4)$ 
receives contributions from all different graphs which can be obtained
from Fig.1  by suitable relabeling  of the external points.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\centering
\begin{minipage}{14cm}
\centering
\psfrag{x1}{$x_1$}
\psfrag{x2}{$x_2$}
\psfrag{x3}{$x_3$}
\psfrag{x4}{$x_4$}
\psfrag{C}{$B(x_1,x_3;x_2,x_4)\,=$}
\includegraphics[width=7cm]{normal4pt1.eps}
\centering
\caption{{\it \small The standard CFT$_{d}$ scalar exchange graph. The
    solid lines 
  represent the full propagator of ${\cal O}(x)$, the dotted line
  represents the full propagator of $\sigma(x)$ and the dark blobs
  stand for the full vertex functions }\cite{ruehl1,tassos1}.}
\end{minipage}
\end{figure} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

One of the basic implications of conformal invariance is the existence
of an operator product expansion (OPE) \cite{mack}  which is
essentially equivalent to a partial-wave decomposition of conformal
$n$-point functions. Such a property 
finds an  explicit application in the study of the four-point
function ${\cal 
  G}(x_1,..,x_4)$, as it implies that its calculation in terms of
skeleton graphs should be compatible with a conformal OPE.  By
conformal invariance ${\cal G}(x_1,..,x_4)$  is determined up to 
an arbitrary analytic function of the  two biharmonic ratios
\beq
u=\frac{x_{13}^2x_{24}^2}{x_{12}^2 x_{34}^2}\,\,\,\,,\,\,\,\,
v=\frac{x_{14}^2x_{23}^2}{x_{12}^2 x_{34}^2}\,,\,\,\,\, x_{ij}=x_i-x_j
\,.
\eeq
Then, calculating the relevant skeleton graphs obtained
from  Fig.1 one finds a general expansion of the form 
\ba
\label{gen4pt}
{\cal G}^{(1)}(x_1,x_2,x_3,x_4) &=&
\frac{1}{(x_{12}^2x_{34}^2)^{\tilde{\Delta}}}
\sum_{n,m=0}^{\infty}\frac{u^n
  (1-v)^m}{n!m!}\Biggl[ -A_{nm}(\tD,\D) \ln u +B_{nm}(\tD,\D) \nonumber \\
 & & \hspace{2cm}+u^{\frac{1}{2}\Delta
 -\tilde{\Delta}} C_{nm}(\tilde{\Delta},\Delta) 
+u^{\frac{1}{2}(d-\Delta)-\tilde{\Delta}}  
C_{nm}(\tilde{\Delta},d-\Delta) \Biggl] \,,
\ea
where the coefficients $A_{nm}$, $B_{nm}$ and $C_{nm}$ have
been given in a number of works \cite{ruehl1,tassos1}. This expression
is compatible with a conformal OPE when the latter is 
inserted in the {\it direct channel} $x_1\rightarrow x_3$,
$x_2\rightarrow x_4$ or equivalently $u\rightarrow 0$, $v\rightarrow
1$. The first two terms on the r.h.s. of (\ref{gen4pt}) together with
the contributions from the zeroth order terms (disconnected graphs)
\footnote{Here and in the following we set to 1 the normalization of
  two-point functions.}
\beq
\label{disc}
{\cal G}^{(0)}(x_1,x_2,x_3,x_4) =
\frac{1}{(x_{13}^2x_{24}^2)^{\tilde{\Delta}}} +
\frac{1}{(x_{12}^2x_{34}^2)^{\tilde{\Delta}}}
\left[1+v^{-\tilde{\Delta}}\right]\,, 
\eeq
represent 
the contributions of infinitely many conformal tensor fields with
definite dimension and rank. The last two terms on the r.h.s. of
(\ref{disc}) determine the canonical dimensions of these tensor
fields, while the logarithmic terms on the
r.h.s. of (\ref{gen4pt}) determine their anomalous dimension. A general
method to identify and study 
the contributions of conformal tensor fields in four-point functions is
described in \cite{HPR}. 

In order that a four-point function admits a conformal OPE
it is necessary that it does not contain non-analytic
terms e.g. the double summation in 
(\ref{gen4pt}) is restricted over
$n,m\in{\bf N}_0$. The 
analyticity of conformally
invariant amplitudes is a non-trivial property of CFT$_d$ and should
be checked in all explicit calculations. In this letter
we show that general AdS$_{d+1}$ scalar
exchange graphs are analytic in the crossed channel expansion and
therefore 
admit a conformal OPE. More technical details of
our calculation as well as a thorough OPE analysis of AdS$_{d+1}$
scalar exchange graphs are
contained in  \cite{HPR}. 

Our main motivation for studying the analyticity properties of AdS$_{d+1}$
scalar exchange graphs comes from similar studies of 
conformal graphs such as the one depicted in Fig.1. Using the D'EPP
formula and Symanzik's technique \cite{symanzik} the exchange graph
in Fig.1 has been calculated in the direct channel as
\ba
\label{dchann}
B(x_1,x_3;x_2,x_4) & =
&\frac{\pi^du^{-\tD}}{(x_{12}^2x_{34}^2)^{\tD}}\sum_{n,m=0}^{\infty}
\frac{u^n(1-v)^m}{n!m!}\Bigl[u^{\frac{1}{2}\D}c_{nm}(d,\D)
+u^{\frac{1}{2}d-\frac 
  {1}{2}\D} c_{nm}(d,d-\D)\Bigl]\!,  \\
c_{nm}(d,\D)&= & \a(\D)\a^2(\sfrac12 d- \sfrac12\D)
\frac{(\frac{1}{2}\D)_n^2
  (\frac{1}{2}\D)_{n+m}^2}{(\D)_{2n+m}(\D-\frac{1}{2}d+1)_n}\,,\\
\a(b) & = &\frac{\Gamma(\frac12 d-b)}{\Gamma(b)} \,,\,\,\,\, (b)_n
\,\, = \,\, \frac{\Gamma(b+n)}{\Gamma(b)}\,.
\ea
This result is explicitly compatible with a
conformally invariant OPE when the latter is inserted in the direct
channel. Namely, the first term on the r.h.s. of  (\ref{dchann}) is
the full contribution 
of the scalar field $\sigma(x)$ in the four-point function, while the second
term on the r.h.s. represents the so-called {\it shadow symmetric}
  singularities of the first series. \footnote{The term {\it shadow
    symmetry} was 
  introduced for the first time in \cite{parisi}. It corresponds to
  an intertwiner \cite{koller} of the conformal group in $d>2$ that
  maps the equivalent representations with dimensions $\eta$ and
  $d-\eta$ onto each other. Shadow symmetric singularities
  may correspond to physical {\it shadow fields} if the dimensions of
  the latter satisfy
  the unitarity bound e.g. $d-\eta\geq d/2-1$. See 
  \cite{ruehl1,tassos1} and also \cite{klebanov}.} From (\ref{dchann})
we can obtain the 
results for both the  {\it crossed channels}. For clarity we consider
here one of two crossed channels, namely when we set
$x_3\leftrightarrow x_4$ in Fig.1. This is obtained from
(\ref{dchann}) by the interchange  $u\leftrightarrow v$. However,
since we are interested in comparing the result with the OPE
inserted in the direct channel, the above crossing transformation
requires an analytic continuation of the result
in (\ref{dchann}). Indeed, the hypergeometric $m$-sum on the r.h.s. of
(\ref{dchann}) gives, by virtue of the crossing transformation, a
hypergeometric series in the variable $(1-u)$. In order to compare this
with the OPE inserted in the direct channel we use a degenerate Kummer
transformation \cite{grad} to obtain a series in the variable
$u$. Nevertheless, the remaining series is now a series in the
variable $v$ and it is not suitable for a comparison with the direct
channel OPE which requires $v\rightarrow 1$. Eventually, we need to
transform this series into a power series in $(1-v)$ and this step
requires an analytic continuation which may not always be
possible. For the explicit case of 
(\ref{dchann}) 
we obtain after some algebra,
\ba
\label{cchann}
B(x_1,x_4;x_2,x_3) & = &K\frac{1}{(x_{12}^2x_{34}^2)^{\tD}}
\sum_{n,m=0}^{\infty} \frac{u^m}{(m!)^2}{\cal
  D}_m(\frac{\partial}{\partial\xi})\Bigl[(1-v)^{s-2t} {\cal A}_{m}(\xi) +
{\cal B}_{m}(\xi)\Bigl]_{\xi =0}\,,\\
{\cal A}_{m}(\xi) & = & v^{\frac{1}{2}\D}\Gamma(2t-s)
\,{}_2F_1\Bigl(s-t,s-t; 1+s-2t;1-v \Bigl)
  \nonumber \\
 & & -v^{\frac12 d-\frac12 \D}\Gamma(2t-s)
\,{}_2F_1\Bigl(1-t,1-t;1+s-2t;1-v) \label{nan}\,,\\
{\cal B}_m(\xi) & = & v^{\frac12\D}\,\Gamma(s-2t)
\,_2F_1\Bigl(t,t;2t-s+1;1-v\Bigl)\left[\frac{\Gamma^2( t)}{\Gamma^2(s-t)}
  -\frac{\Gamma^2( 1+t-s)}{\Gamma^2(1-t)}\right]\label{an}\!\!,
\ea
where 
\ba
s & = & \D-\sfrac12 d+1 \,,\, t\,= \,\sfrac12 \D +m+\xi\,,\\
{\cal D}_m(\frac{\partial}{\partial\xi}) & = & -\ln u +2\psi(m+1)
  -\sfrac{\partial}{\partial \xi}\,,  
\ea
and $K$ is an overall constant whose explicit value is not important
here \cite{HPR}.
Notice that the second term on the r.h.s. of (\ref {cchann}) is
analytic, while the first is not. Nevertheless, the non-analytic
term is exactly zero as can be easily seen using Euler's identity
\cite{grad} 
for either of the two  hypergeometric functions in (\ref{nan}). The
remaining analytic term can then be easily computed from (\ref{cchann})
and (\ref{an}) and the result was given in \cite{ruehl1,tassos1}. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\centering
\begin{minipage}{14cm}
\centering
\psfrag{x1}{$x_1$}
\psfrag{x2}{$x_2$}
\psfrag{x3}{$x_3$}
\psfrag{x4}{$x_4$}
\psfrag{C}{$C(x_1,x_3;x_2,x_4)=$}
\includegraphics[width=7cm]{ads4pt1.eps}
\centering
\caption{{\it \small The AdS$_{d+1}$ scalar exchange graph. The solid
    lines represent  
  the ``bulk-to-boundary'' propagator of ${\cal O}(x)$ and the dotted
  line represents the standard ``bulk-to-bulk'' propagator of
  $\sigma(x)$ }\cite{freedman1}.}  
\end{minipage}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In the above calculation we observe that the existence of the shadow
singularities in the direct 
channel 
(\ref{dchann}) was essential for the cancellation of the non-analytic
terms in the crossed channel.  Let us turn now to the calculation of
the standard AdS$_{d+1}$ scalar exchange graph \cite{freedman1,liu}
depicted in Fig.2. 
This, 
together with the two crossed symmetric graphs  and the
three disconnected terms (\ref{disc}) contribute to the four-point
function of ${\cal O}(x)$ in the boundary CFT$_d$ in the context of AdS/CFT
correspondence \cite{maldacena}. Our starting point is the following
Mellin-Barnes representation for the graph in Fig.2 which is obtained
using either Symanzik's method \cite{HPR} or by other techniques \cite{liu}
\ba
\label{mb}
C(x_1,x_3;x_2,x_4) & = & \frac{k}{(x_{12}^2x_{34}^2)^{\tD}}
\int_{{\cal C}}\frac{\rmd s}{2\pi{\rm i}} \Gamma^2(-s) \Biggl[
\frac{\Gamma^4(\tD+s) \Gamma^2(\frac12 \D+\tD-\frac12 d)
  \Gamma(\frac12 \D-\tD-s)}{ \Gamma(2\tD +2s)\Gamma(\D-\frac12
  d+1)\Gamma(\frac12 \D+\tD-\frac12 d-s)} \nonumber \\
&& \hspace{-2.5cm} \times _3F_2\left( \sfrac12 \D+\tD-\sfrac12
  d,\sfrac12 \D+\tD-\sfrac12 
  d,\sfrac12 \D-\tD-s; \D-\sfrac12 d +1,\tD+\sfrac12\D-\sfrac12
  d-s;1\right) \nonumber \\
& & \hspace{3cm}\times u^s \,_2F_1\left(\tD +s,\tD +s;2\tD +2s;
  1-v\right)\Biggl]\,, 
\ea  
where $k$ is an unimportant for what follows constant
\cite{HPR}.  The contour ${\cal C}$ is chosen such that it separates
the ``right'' 
from the ``left'' poles of the integrand \cite{liu,titsch}.  The existence of
such a contour is 
necessary for the convergence of (\ref{mb}) which can then be shown to give
a well-defined expansion for the scalar exchange graph of Fig.2 in the
direct channel. This expansion can
be matched with a conformally invariant OPE
\cite{dhoker,HPR}. However, one finds that in the explicit result for
(\ref{mb}) terms similar to the
second term on the r.h.s. of (\ref{dchann}) are missing
\cite{freedman1,liu,dhoker,HPR}. 
In view of the role played by such terms (shadow
singularities), for the analyticity of the standard CFT$_d$ exchange
graphs it is important to check the analyticity of the AdS$_{d+1}$
graph  in Fig.2 in the crossed channel. We choose for clarity to study
the crossed channel which is obtained from (\ref{mb}) by
the interchange $u\leftrightarrow v$. In this case, the argument of
the $_2F_1$ hypergeometric function becomes $(1-u)$ and we again need to use
degenerate Kummer's relations to analytically continue it into a
series in the variable $u$. Furthermore, the generalized
hypergeometric $_3F_2$ function 
in (\ref{mb}) is {\it Saalschutzian} \cite{slater} and we can use the
non-terminating form of Saalschutz's theorem (Eq. 4.3.4.2 of \cite{slater})
to write it as a sum of a ratio of $\Gamma$-functions and another $_3F_2$
function. Then, after some algebra which involves calculating the Mellin-Barnes
integral we obtain
\ba
\label{adscr}
C(x_1,x_4;x_2,x_3) & = &
\frac{k_1}{(x_{12}^2x_{34}^2)^{\tD}} \Biggl[  
  \sum_{m=0}^{\infty} \frac{u^m}{(m!)^2}{\cal
    D}_m(\frac{\partial}{\partial\xi}) \Bigl( v^{\frac12 \D-\tD}
  f_1(v,\xi) -f_2(v,\xi) -f_3(v,\xi)\Bigl)\Biggl]_{\xi=0}\\
f_1(v,\xi) &=& \frac{\Gamma^2(\frac12 \D+m+\xi)}{\Gamma(
  \D-\frac12 d+1)} 
  {}_2F_1(\sfrac12 \D+m+\xi,\sfrac12 \D+m+\xi; \D-\sfrac12 d+1;v
  ) \label{f1}\,,\\
f_2(v,\xi) & = & \frac{\Gamma^2(\tD+m+\xi)}{\Gamma(\frac12
  \D+\tD-\frac12 d+1)\Gamma(\tD-\frac12 \D+1)}\nonumber \\
 & & \times {}_3F_2\left( \tD+m+\xi,
  \tD+m+\xi, 1;\sfrac12 \D+\tD-\sfrac12 d+1,\tD-\sfrac12
  \D+1;v\right),\label{f2}\\
f_3(v,\xi) &= & \frac{\Gamma(2\tD-\frac12 d)}{\Gamma^2( \tD-\frac12\D)
  \Gamma^2(\tD-\frac12 \D-\frac12 d)} \int_{{\cal C}}\frac{\rmd s}{2\pi{\rm i}}
\frac{\Gamma^2(-s)\Gamma^2(\tD+m+s+\xi)}{ (s+1)(\tD+\frac12 \D-\frac12
  d)}v^s  \nonumber \\ 
& & \hspace{-1cm} \times{}_3F_2\left(\sfrac12 \D-\tD+1,\sfrac12
  \D+\tD-\sfrac12 
  d+1+s,1; s+2,\sfrac12 \D+\tD-\sfrac12 d+1;1\right), \label{f3}
\ea
where again the constant $k_1$ is unimportant here. First we show that
$f_3(v,\xi)$ does 
not contain non-analytic terms as
$v\rightarrow 1$. The argument, whose details can be found in
\cite{HPR}, is as follows. The  $f_{3}(v,\xi)$ term has
the form
\beq
\label{genf3}
f_3(v,\xi) = \int_{{\cal C}}\frac{\rmd s}{2\pi{\rm i}}
\,\Gamma^2(-s)\,g(s,\xi)\,v^s\,,
\eeq
for some function $g(s,\xi)$. The analyticity of $f_3(v,\xi)$  at
$v=0$ allows us 
to write the result of the Mellin-Barnes integration in (\ref{genf3}) as 
\beq
\label{f3epsilon}
f_{3}(v,\xi) =
\frac{\partial}{\partial\e}\left[\sum_{n=0}^{\infty}\frac{v^{n-\e}
    g(n-\e,\xi)}{\Gamma^2(n+1-\e)} \right]_{\e=0}\,.
\eeq
One way to obtain (\ref{f3epsilon}) is to shift by an infinitesimal
parameter $\epsilon$ one of the two $\Gamma$-functions in
(\ref{genf3}) in order to regularize the double poles. The possible
non-analytic terms as $v\rightarrow 1$ in 
(\ref{f3epsilon})  are determined by the large-$n$
asymptotics of the ratio $g(n-\e,\xi)/\Gamma^2(n+1-\e)$. This in turn
can be found using, for example, the Stirling formula
(\ref{stirl}) below and an asymptotic expansion for $_3F_2$ obtained
from Eq. 4.3.4.2 of \cite{slater} by setting (in Slater's notation)
$c=\frac12 \D+\tD-\frac12 d+1+s$, $e=s+2$. In this way we obtain
\beq
\label{goverG}
\left. \frac{g(n-\e,\xi)}{\Gamma^2(n+1-\e)}\right|_{n\rightarrow
  \infty} \approx \sum_{i=1}^2 \sum_{\lambda}^{\infty} 
\tilde{\sigma}_{\lambda ,i}
\frac{\Gamma(A_i+n+1-\e-\lambda)}{ \Gamma(n+1-\e)}\,,
\eeq
for some parameters $A_i$, which depend among others on $m$ and
$\xi$. The coefficients $\tilde{\sigma}_{\lambda,i}$ can in principle be
explicitly determined in terms of the Bernoulli numbers (see below),
but in this case we only require their existence. Then, by
virtue of (\ref{goverG}) we obtain the non-analytic terms
\ba
\label{f3final}
 \left. \sum_{n=0}^{\infty}\frac{v^{n-\e}
    g(n-\e,\xi)}{\Gamma^2(n+1-\e)}\right|_{n.a.} &
\approx & \left. \sum_{i=1}^2\sum_{\lambda}^{\infty} \tilde{\sigma}_{\lambda,i}
\,v^{-\e}\frac{\Gamma(A
  +1-\e-\lambda)}{\Gamma(1-\e)}{}_2F_1(A_i+1-\e-\lambda,1;1-\e;v)
\right|_{n.a.} 
\nonumber \\
& \approx & \sum_{i=0}^2\sum_{\lambda}^{\infty} \tilde{\sigma}_{\lambda,i}
\,\Gamma(A_i+1-\lambda) (1-v)^{-A_i-1+\lambda} \,,
\ea  
where to get the second line of (\ref{f3final}) we used a Kummer
transformation \cite{grad}. The crucial point is now that the 
non-analytic terms are independent of the parameter $\e$ and
therefore  drop out  
when we substitute (\ref{f3final}) into (\ref{f3epsilon}).
 
Next, by yet another Kummer transformation
it is easy to see that the
$f_1(v,\xi)$ term gives the following non-analytic contribution to
(\ref{adscr})
\beq
\label{naf1}
v^{\frac12 \D-\tD}(1-v)^{1-\frac12 d-2m-2\xi}\Gamma(\sfrac12
d-1+2m+2\xi) \sum_{n=0}^{\infty}\frac{
  (1-v)^n}{n!}\frac{(\frac12 \D-\frac12 d+1-m-\xi)_{n}^2}{(2-\frac12
  d-2m-2\xi)_{n}}\,, 
\eeq 
which is similar to the one found in (\ref{cchann}).
It therefore remains
to see whether the $f_2(v,\xi)$ term involves a non-analytic part
which cancels the above contribution. To this end, we would like to
exploit some kind of ``Kummer-like'' transformations for the generalized
hypergeometric function $_3F_2$ in (\ref{f2}).  However, it seems that such
transformations do not exist in general. For this reason we use again
an asymptotic expansion argument as follows. The relevant term in
(\ref{f2}) reads 
\beq
\label{naf2}
\sum_{n=0}^{\infty} \frac{v^n}{n!}\frac{\Gamma^2(\tD+m+\xi+n)
  \Gamma(n+1)}{ \Gamma(\frac12 \D+\tD-\frac12 d+1+n)\Gamma(
  \tD-\frac12 \D +1+n)}\,,
\eeq
and we have extracted an inessential common constant factor from
(\ref{naf1}) and (\ref{naf2}). In order to study the existence of 
non-analytic terms as $v\rightarrow 1$ in (\ref{naf2}) we can use the
identity
\beq
\label{bin}
\sum_{n=1}^{\infty}\frac{\Gamma(a+n+1)}{\Gamma(n+1)}\,v^n =
\Gamma(a+1)\, (1-v)^{-a-1}\,.
\eeq
To do this, we first need to transform (\ref{naf2}) into the form
(\ref{bin}). This can be accomplished with the help of the Stirling
formula \cite{grad} which gives the asymptotics of the ratio
\ba
\label{stirl}
\left.\frac{\Gamma(a+r+1)}{\Gamma(r+1)}\right|_{r\rightarrow \infty}
&\approx& \exp\left[a\ln(r+1)
    +\sum_{k=1}^{\infty}\frac{{\cal P}_{k+1}(a)}{(r+1)^k}\right]\,,
\\
{\cal P}_{k+1}(a) & = & \frac{(-1)^{k+1}}{k}\sum_{l=0}^{a-1}{l^k}\,. 
\label{Pi}
\ea
Using (\ref{stirl}) we can then write
\beq
\label{stappl}
\left. \frac{\Gamma^2(\tD+m+\xi+n)
  }{ \Gamma(\frac12 \D+\tD-\frac12 d+1+n)\Gamma(
  \tD-\frac12 \D +1+n)}\right|_{n\rightarrow \infty} \approx
\sum_{\lambda=0}^{\infty} \sigma_{\lambda}
\frac{\Gamma(\beta-\lambda+n+1)}{ \Gamma(n+1)}\,,
\eeq
where the coefficients $\sigma_{\lambda}$ are recursively obtained
from
\ba
\label{sigmas}
& & \sum_{\lambda=0}^{\infty}\frac{\sigma_{\lambda}}{(n+1)^{\lambda}}
  \exp\left[\sum_{k=1}^{\infty} \frac{{\cal
      P}_{k+1}(\beta-\lambda) }{(n+1)^{k}}\right] =
\exp\left[\sum_{k=1}^{\infty} 
    \frac{2{\cal P}_{k+1}(t_1)-{\cal P}_{k+1}(t_2)
      -{\cal P}_{k+1}(t_3)}{(n+1)^{k}} \right],\\
& & t_1 =  \tD+m+\xi-1\,,\,\,\,t_2= \tD-\sfrac12
\D\,,\,\,\,t_3= \sfrac12 \D +\tD-\sfrac12 d \,,\\
& & \beta  =  2t_1-t_2-t_3 = \sfrac12 d-2+2m+2\xi\,,
\ea 
by matching the powers of $1/(n+1)$ on both sides of
(\ref{sigmas}). Then, using (\ref{stappl}) and (\ref{sigmas}) we are
able to extract the possible non-analytic behavior of (\ref{naf2}),
which is essentially dictated by its  large-$n$ behavior,
to obtain 
\ba 
\label{naf20}
\left. \sum_{n=0}^{\infty} \frac{v^n}{n!}\frac{\Gamma^2(\tD+m+\xi+n)
  \Gamma(n+1)}{ \Gamma(\frac12 \D+\tD-\frac12 d+1+n)\Gamma(
  \tD-\frac12 \D +1+n)}\right|_{n.a.} & \approx &
\sum_{\lambda}^{\infty} \sigma_{\lambda} 
\left[ \sum_{n=0}^{\infty} \frac{\Gamma(\beta
    -\lambda+n+1)}{\Gamma(n+1)} v^n \right]\nonumber \\
 &  & \hspace{-5cm}\approx\sum_{\lambda=0}^{\infty} \sigma_{\lambda}
 \,\Gamma(\sfrac12 d -1 
 +2m+2\xi-\lambda)(1-v)^{1-\frac12 d-2m-2\xi +\lambda}\,.
\ea
Notice that our approach does not determine the analytic part of
(\ref{naf2}). Therefore, from (\ref{adscr}), (\ref{naf1}) and
(\ref{naf20}) the cancellation of non-analytic terms requires that the
equal powers of $1-v$ coincide to all orders in both sides of the
following identity 
\beq
\label{final}
\sum_{\lambda=0}^{\infty} \sigma_{\lambda}
\frac{(-1)^{\lambda}(1-v)^{\lambda}}{ (2-\frac12 d-2m-2\xi)_{\lambda}} =
\sum_{k,l=0}^{\infty} \frac{(1-v)^{k+l}}{k!l!}\frac{(\tD-\frac12
  \D)_{l}(\frac12 \D-\frac12 d+1-m-\xi)_k^2}{ (2-\frac12 d-2m-2\xi)_k}\,.
\eeq
This highly non-trivial identity, which involves on the l.h.s. among
others the Bernoulli numbers through the polynomials (\ref{Pi}),
is analytically proven in \cite{HPR} and it corresponds to
``Kummer-like'' identities for the generalized hypergeometric
$_3F_2$ function. Its first few lower orders can be also shown to be
satisfied 
with the help of e.g. a simple MAPLE algorithm \cite{HPR}.

Concluding, we outlined our proof that the
possible non-analytic terms in AdS$_{d+1}$
scalar exchange graphs drop out by virtue of some highly non-trivial
identities. This is in contrast with the case of scalar exchange
graphs in standard CFT where the cancellation of the non-analytic
terms was due to the appearance of the shadow singularities in the
direct channel. Our result lends support to the idea that the
amplitudes obtained from AdS/CFT correspondence can be analyzed using
conformal OPE techniques \cite{dhoker}. Such an analysis, which also
includes more technical details of our proof here, is presented in
\cite{HPR}.    


\section*{Acknowledgments}
The work of A. C. P. is supported by an Alexander von Humboldt
  Fellowship. 





\begin{thebibliography}{99}

\bibitem{fradkin}
E. S. Fradkin and M. Ya. Palchik, Phys. Rep. {\bf 44} (1978) 249.

\bibitem{ruehl1}
K. Lang and W. R\"uhl, Nucl. Phys. {\bf B377} (1992) 371.


\bibitem{tassos1}
A. C. Petkou,  ``Conserved currents, consistency relations
and operator product expansions in the conformally invariant $O(N)$
vector model for $2<d<4$'', Ann. Phys. {\bf 249} (1996) 180,
hep-th/9410093; ``Operator product expansions and consistency
relations in a $O(N)$ invariant fermionic CFT for $2<d<4$'',
Phys. Lett. {\bf  B389} (1996) 18, hep-th/9602054. 

\bibitem{depp}
M. D'Eramo, G. Parisi and L. Peliti, Lett. Nuovo Cim. {\bf 2} (1971), 878.

\bibitem{mack}
G. Mack and I. T. Todorov, Phys. Rev. {\bf D8} (1973) 1764;\\
V. K. Dobrev, V. B. Petkova, S. G. Petrova and I. T. Todorov,
Phys. Rev. {\bf D13} (1976) 887.

\bibitem{HPR}
L. Hoffmann, A. C. Petkou and W. R\"uhl, ``{\it Aspects of the conformal
operator product expansion in AdS/CFT correspondence}'', to appear. 

\bibitem{symanzik}
K. Symanzik, Lett. Nuovo Cim. {\bf 3} (1972), 734.

\bibitem{parisi}
S. Ferrara, A. F. Grillo and G. Parisi, Lett. Nuovo Cim. {\bf 5}
(1972) 147; 
S. Ferrara, A. F. Grillo, R. Gatto and G. Parisi, Lett. Nuovo
Cim. {\bf 4} (1972) 115;
S. Ferrara and G. Parisi, Nucl. Phys. {\bf B42} (1972) 281.

\bibitem{koller}
K. Koller, Commun. Math. Phys. {\bf 40} (1975) 15.


\bibitem{grad}
I. S. Gradsteyn and I. M. Ryshik, {\it Table of Integrals, Series and
  Products}, 5th Ed. Academic Press (1994).

\bibitem{klebanov}
I. Klebanov and E. Witten, ``AdS/CFT correspondence and symmetry breaking'',
Nucl. Phys. {\bf B556} (1999) 89, hep-th/9905104.

\bibitem{freedman1}
D.Z.~Freedman, S.D.~Mathur, A.~Matusis and L.~Rastelli,
``Comments on 4-point functions in the CFT/AdS correspondence,"
Phys. Lett. {\bf B452} (1999) 61, hep-th/9808006;
E. D'Hoker and D. Z. Freedman, `General scalar
exchange in $AdS_{d+1}$', Nucl. Phys. {\bf B550} (1999) 261, hep--th/9811257;
E.~D'Hoker, D.Z. Freedman, S.D. Mathur, A. Matusis, L. Rastelli,
``Graviton and gauge boson propagators in $AdS_{d+1}$,"
Nucl. Phys. {\bf B562} (1999) 330,
hep-th/9902042;
E.~D'Hoker, D.Z.~Freedman, S.D.~Mathur, A.~Matusis and L.~Rastelli,
``Graviton exchange and complete 4-point functions in the AdS/CFT
correspondence,'' Nucl. Phys. {\bf B562} (1999) 353,
hep-th/9903196;
E.~D'Hoker, D.Z.~Freedman and L.~Rastelli,
``AdS/CFT 4-point functions: How to succeed at z-integrals without
really trying,'' Nucl. Phys. {\bf B562} (1999) 395,
hep-th/9905049.

\bibitem{liu}
H. Liu, `Scattering in Anti--de Sitter space and operator
product expansion', Phys. Rev. {\bf D60} 106005 (1999), hep--th/9811152.

\bibitem{maldacena}
J. Maldacena, ``The large N limit of superconformal field 
theories and supergravity'', Adv. Theor. Math. Phys. {\bf 2} (1998) 
231, hep-th/9711200; 
S.S. Gubser, I.R. Klebanov and A.M. 
Polyakov, ``Gauge theory correlators from noncritical string 
theory'', Phys. Lett. {B428} (1998) 105, hep-th/9802109; 
E. Witten, 
``Anti-de  Sitter space and holography'', Adv. Theor. Math. 
Phys. {\bf 2} (1998) 253, hep-th/9802150;
O.~Aharony, S.S.~Gubser, J.~Maldacena, H.~Ooguri and Y.~Oz,
``Large N field theories, string theory and gravity'',
hep-th/9905111.

\bibitem{titsch}
E. C. Titchmarsh, {\it The theory of Functions}, 2nd Ed. Oxford
University Press (1939). (Reprinted 1975).

\bibitem{slater}
L. J. Slater, {\it Generalized Hypergeometric Functions}, Cambridge
University Press (1966).

\bibitem{dhoker}
E.~D'Hoker, S.D.~Mathur, A.~Matusis and L.~Rastelli, ``The operator
product expansion of ${\cal N}$=4 SYM and the 4-point functions of
supergravity'', hep-th/9911222.


\end{thebibliography}





\end{document}

hep-th0002025.extracted_bib
\begin{thebibliography}{99}

\bibitem{fradkin}
E. S. Fradkin and M. Ya. Palchik, Phys. Rep. {\bf 44} (1978) 249.

\bibitem{ruehl1}
K. Lang and W. R\"uhl, Nucl. Phys. {\bf B377} (1992) 371.


\bibitem{tassos1}
A. C. Petkou,  ``Conserved currents, consistency relations
and operator product expansions in the conformally invariant $O(N)$
vector model for $2<d<4$'', Ann. Phys. {\bf 249} (1996) 180,
hep-th/9410093; ``Operator product expansions and consistency
relations in a $O(N)$ invariant fermionic CFT for $2<d<4$'',
Phys. Lett. {\bf  B389} (1996) 18, hep-th/9602054. 

\bibitem{depp}
M. D'Eramo, G. Parisi and L. Peliti, Lett. Nuovo Cim. {\bf 2} (1971), 878.

\bibitem{mack}
G. Mack and I. T. Todorov, Phys. Rev. {\bf D8} (1973) 1764;\\
V. K. Dobrev, V. B. Petkova, S. G. Petrova and I. T. Todorov,
Phys. Rev. {\bf D13} (1976) 887.

\bibitem{HPR}
L. Hoffmann, A. C. Petkou and W. R\"uhl, ``{\it Aspects of the conformal
operator product expansion in AdS/CFT correspondence}'', to appear. 

\bibitem{symanzik}
K. Symanzik, Lett. Nuovo Cim. {\bf 3} (1972), 734.

\bibitem{parisi}
S. Ferrara, A. F. Grillo and G. Parisi, Lett. Nuovo Cim. {\bf 5}
(1972) 147; 
S. Ferrara, A. F. Grillo, R. Gatto and G. Parisi, Lett. Nuovo
Cim. {\bf 4} (1972) 115;
S. Ferrara and G. Parisi, Nucl. Phys. {\bf B42} (1972) 281.

\bibitem{koller}
K. Koller, Commun. Math. Phys. {\bf 40} (1975) 15.


\bibitem{grad}
I. S. Gradsteyn and I. M. Ryshik, {\it Table of Integrals, Series and
  Products}, 5th Ed. Academic Press (1994).

\bibitem{klebanov}
I. Klebanov and E. Witten, ``AdS/CFT correspondence and symmetry breaking'',
Nucl. Phys. {\bf B556} (1999) 89, hep-th/9905104.

\bibitem{freedman1}
D.Z.~Freedman, S.D.~Mathur, A.~Matusis and L.~Rastelli,
``Comments on 4-point functions in the CFT/AdS correspondence,"
Phys. Lett. {\bf B452} (1999) 61, hep-th/9808006;
E. D'Hoker and D. Z. Freedman, `General scalar
exchange in $AdS_{d+1}$', Nucl. Phys. {\bf B550} (1999) 261, hep--th/9811257;
E.~D'Hoker, D.Z. Freedman, S.D. Mathur, A. Matusis, L. Rastelli,
``Graviton and gauge boson propagators in $AdS_{d+1}$,"
Nucl. Phys. {\bf B562} (1999) 330,
hep-th/9902042;
E.~D'Hoker, D.Z.~Freedman, S.D.~Mathur, A.~Matusis and L.~Rastelli,
``Graviton exchange and complete 4-point functions in the AdS/CFT
correspondence,'' Nucl. Phys. {\bf B562} (1999) 353,
hep-th/9903196;
E.~D'Hoker, D.Z.~Freedman and L.~Rastelli,
``AdS/CFT 4-point functions: How to succeed at z-integrals without
really trying,'' Nucl. Phys. {\bf B562} (1999) 395,
hep-th/9905049.

\bibitem{liu}
H. Liu, `Scattering in Anti--de Sitter space and operator
product expansion', Phys. Rev. {\bf D60} 106005 (1999), hep--th/9811152.

\bibitem{maldacena}
J. Maldacena, ``The large N limit of superconformal field 
theories and supergravity'', Adv. Theor. Math. Phys. {\bf 2} (1998) 
231, hep-th/9711200; 
S.S. Gubser, I.R. Klebanov and A.M. 
Polyakov, ``Gauge theory correlators from noncritical string 
theory'', Phys. Lett. {B428} (1998) 105, hep-th/9802109; 
E. Witten, 
``Anti-de  Sitter space and holography'', Adv. Theor. Math. 
Phys. {\bf 2} (1998) 253, hep-th/9802150;
O.~Aharony, S.S.~Gubser, J.~Maldacena, H.~Ooguri and Y.~Oz,
``Large N field theories, string theory and gravity'',
hep-th/9905111.

\bibitem{titsch}
E. C. Titchmarsh, {\it The theory of Functions}, 2nd Ed. Oxford
University Press (1939). (Reprinted 1975).

\bibitem{slater}
L. J. Slater, {\it Generalized Hypergeometric Functions}, Cambridge
University Press (1966).

\bibitem{dhoker}
E.~D'Hoker, S.D.~Mathur, A.~Matusis and L.~Rastelli, ``The operator
product expansion of ${\cal N}$=4 SYM and the 4-point functions of
supergravity'', hep-th/9911222.


\end{thebibliography}