\documentclass{sigma}
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\setcounter{MaxMatrixCols}{30}
\begin{document}
\renewcommand{\PaperNumber}{} \FirstPageHeading{}
\ShortArticleName{The form factor program}
\ArticleName{The form factor program: a review and new results
- the nested $\mathbf{SU(N)}$ off-shell Bethe ansatz\footnote
{To appear in the
Proceedings of the O'Raifertaigh Symposium on Non-Perturbative and
Symmetry Methods in Field Theory (June 22-24, 2006; Budapest, Hungary)}
} \Author{Hratchya M.  Babujian~$^\dag$, Angela Foerster~$^\ddag$,
and Michael Karowski~$^\S$}
\AuthorNameForHeading{H. Babujian, A. Foerster and M. Karowski}
\Address{$^\dag$~Yerevan Physics Institute,
Alikhanian Brothers 2, Yerevan, 375036, Armenia}
\EmailD{babujian@physik.fu-berlin.de}
\Address{$^\ddag$~Instituto de F\'{\i}sica da UFRGS,
Av. Bento Gon\c{c}alves 9500, Porto Alegre, RS - Brazil}
\EmailD{angela@if.ufrgs.br}
\Address{$^\S$~Theoretische Physik, Freie Universit\"{a}t Berlin,
Arnimallee 14, 14195 Berlin, Germany} \EmailD{karowski@physik.fu-berlin.de}
\URLaddressD{http://www.physik.fu-berlin.de/\~{}karowski/}
\Abstract {The
purpose of the ``bootstrap program'' for integrable quantum field theories in
1+1 dimensions is to construct explicitly a model in terms of its Wightman
functions. In this article, this program is mainly illustrated in terms of
the sinh-Gordon model and the $SU(N)$ Gross-Neveu model. The nested off-shell
Bethe ansatz for an $SU(N)$ factorizing S-matrix is constructed. We review
some previous results on sinh-Gordon form factors and the quantum operator
field equation. The problem of how to sum over intermediate states is
considered in the short distance limit of the two point Wightman function for
the sinh-Gordon model.}
\Keywords{Integrable quantum field theory; Form factors}
\Classification{81T08; 81T10; 81T40}

\section{Introduction}

The `bootstrap program for integrable quantum field theories in
1+1-dimen\-sions' does \emph{not} start with any classical
Lagrangian. Rather this program classifies integrable quantum field theoretic
models and in addition provides their explicit exact solutions in term of all
Wightman functions. We achieve contact with the classical models only, when
at the end we compare our exact results with Feynman graph (or other)
expansions which are usually based on Lagrangians.

One of the authors (M.K.) et al.~\cite{KTTW} formulated the on-shell program
i.e. the exact determination of the scattering matrix using the Yang-Baxter
equations. The concept of generalized form factors was introduced by one of
the authors (M.K.) et al.~\cite{KW}. In this article consistency equations
were formulated which are expected to be satisfied by these quantities.
Thereafter this approach was developed further and studied in the context of
several explicit models by Smirnov \cite{Sm} who proposed the form factor
equations $(i)-(v)$ (see below) as extensions of similar formulae in the
original article \cite{KW}. These formulae were then proven by two of the
authors et al.~\cite{BFKZ}. In the present article we apply the form factor
program for an $SU(N)$ invariant S-matrix (see \cite{BFK1}).
The procedure is similar to that for the $Z(N)$ and $A(N-1)$ cases
\cite{BK04,BFK} because the bound state fusions are similar in these
three models. However, the algebraic structure of the form factors for the
$SU(N)$ model is more intricate, because the S-matrix also describes backward
scattering. We have to apply the nested off-shell Bethe ansatz.

Finally the Wightman functions are obtained by taking integrals and sums over
intermediate states. The explicit evaluation of all these integrals and sums
remains an open challenge for almost all models, except the Ising model
\cite{Ba,MTW,SMJ,BKW,YZ}. In this article we discuss this problem for the
examples of the sinh-Gordon model (see \cite{BK5}). We investigate the short
distance behavior of the two-point Wightman function of the exponentiated
field.

\section{The ``bootstrap program''}

The `bootstrap program' for integrable quantum field theories in
1+1-dimensions provides the solution of a model in term of all its Wightman
functions. The result is obtained in three steps:

\begin{enumerate}
\item The S-matrix is calculated by means of general properties such as
unitarity and crossing, the Yang-Baxter equations (which are a consequence of
integrability) and the additional assumption of `maximal analyticity'. This
means that the two-particle S-matrix is an analytic function in the physical
plane (of the Mandelstam variable $(p_{1}+p_{2})^{2}$) and possesses only
those poles there which are of physical origin. The only input which depends
on the model is the assumption of a particle spectrum with an underlining
symmetry. Typically there is a correspondence of
fundamental representations with multiplets of particles. A
\emph{classification} of all S-matrices obeying the given properties is
obtained.

\item Generalized form factors which are matrix elements of local operators
\[
^{out}\left\langle \,p_{m}^{\prime},\ldots,p_{1}^{\prime}\left\vert
\mathcal{O}(x)\right\vert p_{1},\ldots,p_{n}\,\right\rangle ^{in}\,
\]
are calculated by means of the S-matrix. More precisely, the equations
$(i)-(v)$ as listed in section \ref{s3} are solved. These equations follow
from LSZ-assumptions and again the additional assumption of `maximal
analyticity' \cite{BFKZ}.

\item The Wightman functions are obtained by inserting a complete set of
intermediate states. In particular the two point function for a hermitian
operator $\mathcal{O}(x)$ reads
\begin{multline*}
\langle\,0\left\vert \mathcal{O}(x)\,\mathcal{O}(0)\right\vert 0\,\rangle
=\sum_{n=0}^{\infty}\frac{1}{n!}\idotsint\frac{dp_{1}\ldots dp_{n}}%
{\,(2\pi)^{n}2\omega_{1}\dots2\omega_{n}}\\
\times\left\vert \left\langle \,0\left\vert \mathcal{O}(0)\right\vert
p_{1},\ldots,p_{n}\,\right\rangle ^{in}\right\vert ^{2}e^{-ix\sum p_{i}}.
\end{multline*}
Up to now a direct proof that these sums converge exists only for the scaling
Ising model \cite{Ba,MTW,SMJ} and the non-unitary `Yang-Lee' model \cite{Sm3}.
Recently, Lechner \cite{Le} has shown that models with factorizing S-matrices
exist in the frame work of algebraic quantum field theory.
\end{enumerate}

\subsection*{Integrability}

Integrability in (quantum) field theories means that there exist infinitely
many local conservation laws
\[
\partial_{\mu}J_{L}^{\mu}(t,x)=0\quad(L=\pm1,\pm3,\dots)\,.
\]
A consequence of such conservation laws in 1+1 dimensions is that there is no
particle production and the n-particle S-matrix is a product of 2-particle
S-matrices
\[
S^{(n)}(p_{1},\dots,p_{n})=\prod_{i<j}S_{ij}(p_{i},p_{j})\,.
\]
If backward scattering occurs the 2-particle S-matrices will not commute and
one has to specify the order. In particular for the 3-particle S-matrix there
are two possibilities
\begin{gather*}
\mathbf{S^{(3)}=S_{12}S_{13}S_{23}=S_{23}S_{13}S_{12}}\\[2mm]%
\begin{array}
[c]{c}%
\unitlength5mm\begin{picture}(17,4)(-6,0) \thicklines
\put(-5.5,.8){\line (1,1){3}}
\put(-4,.8){\line(0,1){3}}
\put(-2.5,.8){\line(-1,1){3}}
\put(-4,2.3){\makebox(0,0){\Large$\bullet$}}
\put(-5.6,0){$1$}
\put (-4.2,0){$2$}
\put(-2.8,0){$3$}
\put(-1.5,2){$=$}
\put(0,1){\line(1,1){3}}
\put(0,3){\line(1,-1){3}}
\put(2,0){\line(0,1){4}}
\put(4.3,2){$=$}
\put(6,0){\line(1,1){3}}
\put(6,4){\line(1,-1){3}}
\put(7,0){\line(0,1){4}}
\put(.2,.5){$1$}
\put(1.3,0){$2$}
\put(3,.2){$3$}
\put(5.5,.2){$1$}
\put(7.3,0){$2$}
\put(8.4,.4){$3$} \end{picture}
\end{array}
\end{gather*}
which yield the \textbf{\textquotedblleft Yang-Baxter
Equation\textquotedblright.}

\paragraph{The two particle S-matrix}

is of the form%
\[
S_{\alpha\;\beta}^{\beta^{\prime}\alpha^{\prime}}(\theta_{12})=%
\begin{array}
[c]{c}%
\unitlength2mm\begin{picture}(5,6.3) \thicklines
\put(0,1){\line(1,1){4}}
\put(2,3){\makebox(0,0){$\bullet$}}
\put(4,1){\line(-1,1){4}}
\put (0,-.5){$\alpha$}
\put(3.5,-.5){$\beta$}
\put(3.5,5.8){$\alpha'$}
\put(0,5.8){$\beta'$} \end{picture}
\end{array}
\]
where $\alpha,\beta$ etc denote the type of the particles and the rapidity
difference $\theta_{12}=\theta_{1}-\theta_{2}>0$ is defined by $p_{i}%
=m_{i}(\cosh\theta_{i},\sinh\theta_{i})$. We also use the short hand notation
$S_{12}(\theta_{12})$. It satisfies unitarity
\begin{equation}
S_{21}(\theta_{21})S_{12}(\theta_{12})=1~:~~~~~%
\begin{array}
[c]{c}%
\unitlength3mm\begin{picture}(8,5)
\put(0,1){\line(1,1){2}}
\put(2,1){\line(-1,1){2}}
\put(0,3){\line(1,1){2}}
\put(2,3){\line(-1,1){2}}
\put(6,1){\line(0,1){4}}
\put(8,1){\line(0,1){4}}
\put(3.7,2.7){$=$}
\put(0,-.5){$1$}
\put(1.5,-.5){$2$}
\put(6,-.5){$1$}
\put(7.5,-.5){$2$} \end{picture}
\end{array}
\label{2.2}%
\end{equation}
and crossing
\begin{gather}
S_{12}(\theta_{1}-\theta_{2})=\mathbf{C}^{2\bar{2}}\,S_{\bar{2}1}(\theta
_{2}+i\pi-\theta_{1})\,\mathbf{C}_{\bar{2}2}=\mathbf{C}^{1\bar{1}}%
\,S_{2\bar{1}}(\theta_{2}-(\theta_{1}-i\pi)\,\mathbf{C}^{\bar{1}1}%
\label{2.4}\\%
\begin{array}
[c]{c}%
\unitlength3mm\begin{picture}(4,5)
\put(0,1){\line(1,1){4}}
\put(4,1){\line(-1,1){4}}
\put(0,-.5){$1$}
\put(3.7,-.5){$2$} \end{picture}
\end{array}
~~~~~~=~~~~~~%
\begin{array}
[c]{c}%
\unitlength3mm\begin{picture}(6,5)
\put(1,1){\line(1,1){4}}
\put(4,1){\line(-1,2){2}}
\put(1,5){\oval(2,8)[lb]}
\put(5,1){\oval(2,8)[tr]}
\put(3.5,-.5){$1$}
\put(5.7,-.5){$2$} \end{picture}
\end{array}
~~~~~~=~~~~~~%
\begin{array}
[c]{c}%
\unitlength3mm\begin{picture}(6,5)
\put(2,1){\line(1,2){2}}
\put(5,1){\line(-1,1){4}}
\put(1,1){\oval(2,8)[lt]}
\put(5,5){\oval(2,8)[br]}
\put(0,-.5){$1$}
\put(2,-.5){$2$} \end{picture}
\end{array}
~~~~~\nonumber
\end{gather}
where $\mathbf{C}^{1\bar{1}}$ and $\mathbf{C}_{1\bar{1}}$ are charge
conjugation matrices. We have introduced the following graphical rule, that
a line changing the \textquotedblleft time direction\textquotedblright\ also
interchanges particles and anti-particles and changes the rapidity as
$\theta\rightarrow\theta\pm i\pi$
\begin{equation}
\mathbf{C}_{\alpha\bar{\beta}}=%
\begin{array}
[c]{c}%
\unitlength4mm\begin{picture}(5.5,3)
\put(2,1){\oval(2,4)[t]}
\put(0,1){$\theta$}
\put(3.3,1){$\theta-i\pi$}
\put(.7,0){$\alpha$}
\put(2.7,-.1){$\bar\beta$} \end{picture}
\end{array}
,~~~~\mathbf{C}^{\alpha\bar{\beta}}=%
\begin{array}
[c]{c}%
\unitlength4mm\begin{picture}(5.5,3)
\put(2,2){\oval(2,4)[b]}
\put(0,1){$\theta$}
\put(3.3,1){$\theta+i\pi$}
\put(.7,2.2){$\alpha$}
\put(2.7,2.2){$\bar\beta$} \end{picture}
\end{array}
. \label{2.6}%
\end{equation}


\paragraph{Bound states:}

Let $\gamma$ be a bound state of particles $\alpha$ and $\beta$ with mass
\[
m_{\gamma}=\sqrt{m_{\alpha}^{2}+m_{\beta}^{2}+2m_{\alpha}m_{\beta}\cos\eta
}~,~~~(0<\eta<\pi)\,.
\]
Then the 2 particle S-matrix has a pole such that
\begin{align}
i\operatorname*{Res}_{\theta=i\eta}S_{\alpha\beta}^{\beta^{\prime}%
\alpha^{\prime}}(\theta)  &  =\Gamma_{\gamma}^{\beta^{\prime}\alpha^{\prime}%
}\,\Gamma_{\alpha\beta}^{\gamma}\label{2.8}\\[0.2in]
i\operatorname*{Res}_{\theta=i\eta}%
\begin{array}
[c]{c}%
\unitlength2.7mm\begin{picture}(5,6) \thicklines
\put(0,1){\line(1,1){4}}
\put(2,3){\makebox(0,0){$\bullet$}}
\put(4,1){\line(-1,1){4}}
\put(0,-.3){$\alpha$}
\put(3.5,-.3){$\beta$}
\put(3.5,5.5){$\alpha'$}
\put(0,5.5){$\beta'$} \end{picture}
\end{array}
~~  &  =~~~%
\begin{array}
[c]{c}%
\unitlength2mm\begin{picture}(5,8) \thicklines
\put(0,1){\line(1,1){2}}
\put(2,3){\makebox(0,0){$\bullet$}}
\put(4,1){\line(-1,1){2}}
\put(2,3){\line(0,1){2}}
\put(2,5){\makebox(0,0){$\bullet$}}
\put(2,5){\line(1,1){2}}
\put(2,5){\line(-1,1){2}}
\put(0,-.5){$\alpha$}
\put(3.5,-.5){$\beta$}
\put(2.5,3.7){$\gamma$}
\put(3.5,7.7){$\alpha'$}
\put(0,7.7){$\beta'$} \end{picture}
\end{array}
\nonumber
\end{align}
where $\eta$ is called the fusion angle and $\Gamma_{\alpha\beta}^{\gamma}$ is
the `bound state intertwiner' \cite{K1,BK}. The \textbf{bound state S-matrix,
}that is the scattering matrix of the bound state (12) with a particle 3, is
obtained by the
\textbf{\textquotedblleft bootstrap equation\textquotedblright} \cite{K1}
\begin{align*}
S_{(12)3}(\theta_{(12)3})\,\Gamma_{12}^{(12)}  &  =\Gamma_{12}^{(12)}%
\,S_{13}(\theta_{13})S_{23}(\theta_{23})\,\\%
\begin{array}
[c]{c}%
\unitlength3mm\begin{picture}(5,5) \thicklines
\put(2,2){\line(1,1){2}}
\put(4.5,1.5){\line(-1,1){3}}
\put(2,2){\line(-4,-1){2}}
\put(2,0){\line(0,1){2}}
\put(0,.4){$1$}
\put(1,0){$2$}
\put(3.5,.7){$3$}
\put(4.2,4){$(12)$}
\put(2,2){\makebox(0,0){$\bullet$}} \end{picture}
\end{array}
~~~  &  =~~~%
\begin{array}
[c]{c}%
\unitlength3mm\begin{picture}(5,5) \thicklines
\put(3,3){\line(1,1){1.2}}
\put(4,0){\line(-1,1){4}}
\put(0,2){\line(3,1){3}}
\put(2,0){\line(1,3){1}}
\put(0,.8){$1$}
\put(1,0){$2$}
\put(4,.4){$3$}
\put(4.4,4){$(12)$}
\put(3,3){\makebox(0,0){$\bullet$}} \end{picture}
\end{array}
\end{align*}
where we use the usual short hand notation of matrices acting in the spaces
corresponding to the particles 1, 2, 3 and (12).

\paragraph{\textbf{Examples }}

of integrable models in 1+1-dimensions (which we will consider in this review)
are the \textbf{sinh-Gordon} model defined by the classical field equation
\[
\ddot{\varphi}(t,x)-\varphi^{\prime\prime}(t,x)+\frac{\alpha}{\beta}\sinh
\beta\varphi(t,x)=0
\]
and the $\mathbf{SU(N)}$ \textbf{Gross-Neveu} model described by the
Lagrangian
\[
\mathcal{L}=\bar{\psi}\,i\gamma\partial\,\psi+\frac{g^{2}}{2}\left(
(\bar{\psi}\psi)^{2}-(\bar{\psi}\gamma^{5}\psi)^{2}\right)  \,,
\]
where the Fermi fields form an $SU(N)$ multiplet.

Further integrable quantum field theories are: scaling $Z_{N}$-Ising models,
nonlinear $\sigma$-models, $O(N) $Gross-Neveu models, Toda models etc.

\subsection*{The S-matrix}

\paragraph{The sinh-Gordon S-matrix}

is given by the analytic continuation $\beta\rightarrow i\beta$ of the
sin-Gordon breather S-matrix \cite{KT}
\begin{equation}
S(\theta)=\frac{\sinh\theta+i\sin\pi\nu}{\sinh\theta-i\sin\pi\nu}%
\quad\text{with }-1\leq\nu=\frac{-\beta^{2}}{8\pi+\beta^{2}}\leq0\,.
\label{2.10}%
\end{equation}
The model has the self-dual point at%
\[
\nu=-\,\frac{1}{2}~~\mathrm{or}~~\beta^{2}=4\pi\,.
\]


\paragraph{The $SU(N)$ S-matrix:}

All solutions of $U(N)$-invariant S-matrix satisfying unitarity, crossing and
the Yang-Baxter equation have been obtained in \cite{BKKW}. Following
\cite{KuS,KS}, we adopt the view that in the $SU(N)$ Gross-Neveu model, the
anti-particles are bound states of $N-1$ particles. This implies that there is
no particle anti-particle backward scattering and that the $SU(N)$ S-matrix
should be given by solution II of \cite{BKKW} . The scattering of the
fundamental particles which form a multiplet corresponding to the vector
representation of $SU(N)$ is (see \cite{KKS,BW,ABW} and \cite{Be} for $N=2$)
\begin{equation}
S_{\alpha\beta}^{\delta\gamma}(\theta)=%
\begin{array}
[c]{c}%
\unitlength3mm\begin{picture}(6,6)
\put(1,1){\line(1,1){4}}
\put(1,1){\vector(1,1){1.2}}
\put(3,3){\vector(1,1){1.2}}
\put(5,1){\line(-1,1){4}}
\put(5,1){\vector(-1,1){1.2}}
\put(3,3){\vector(-1,1){1.2}}
\put(.5,.1){$\scriptstyle\alpha$}
\put(5,.1){$\scriptstyle\beta$}
\put(5,5.3){$\scriptstyle\gamma$}
\put(.5,5.3){$\scriptstyle\delta$}
\put(1.8,1){$\scriptstyle p_1$}
\put(3.5,1){$\scriptstyle p_2$}
\put(3.5,4.7){$\scriptstyle p_3$}
\put(1.8,4.7){$\scriptstyle p_4$} \end{picture}
\end{array}
=\delta_{\alpha\gamma}\delta_{\beta\delta}\,b(\theta)+\delta_{\alpha\delta
}\delta_{\beta\gamma}\,c(\theta) \label{2.12}%
\end{equation}
where due to Yang-Baxter $c(\theta)=-\frac{2\pi i}{N\theta}\,b(\theta)$
holds and the highest weight amplitude is given as
\begin{equation}
a(\theta)=b(\theta)+c(\theta)=-\frac{\Gamma\left(  1-\frac{\theta}{2\pi
i}\right)  \Gamma\left(  1-\frac{1}{N}+\frac{\theta}{2\pi i}\right)  }%
{\Gamma\left(  1+\frac{\theta}{2\pi i}\right)  \Gamma\left(  1-\frac{1}%
{N}-\frac{\theta}{2\pi i}\right)  }\,. \label{2.14}%
\end{equation}
There is a bound state pole at $\theta=i\eta=2\pi i/N$ in the antisymmetric
tensor sector which agrees with Swieca's \cite{KuS} picture that the bound
state of $N-1$ particles is to be identified with the anti-particle. Similar
as in the scaling $Z(N)$-Ising and $A(N-1)$-Toda models \cite{BK04,BFK} this
will be used to construct the form factors in $SU(N)$ model \cite{BFK1}.

\section{Form factors\label{s3}}

For a local operator $\mathcal{O}(x)$ the generalized form factors \cite{KW}
are defined as
\[
\,F_{\alpha_{1}\dots\alpha_{n}}^{\mathcal{O}}\left(  \theta_{1},\dots
,\theta_{n}\right)  =\langle\,0\,|\,\mathcal{O}(0)\,|\,p_{1},\dots
,p_{n}\,\rangle_{\alpha_{1}\dots\alpha_{n}}^{in}%
\]
for $\theta_{1}>\dots>\theta_{n}$. For other orders of the rapidities they are
defined by analytic continuation. The index $\alpha_{i}$\thinspace denotes the
type of the particle with momentum $p_{i}$. We also use the short notations
$F_{\underline{\alpha}}^{\mathcal{O}}(\underline{\theta})$ or $F_{1\dots
n}^{\mathcal{O}}(\underline{\theta})$\footnote{The later means the co-vector
in a tensor product space with the components $F_{\underline{\alpha}%
}^{\mathcal{O}}$.}.

For the $SU(N)$ Gross-Neveu model $\alpha$ denotes the types of particles
belonging to all fundamental representations of $SU(N)$ with dimension
$\binom{N}{r},~r=1,\dots,N-1$. In most formulae we restrict $\alpha=1,\dots,N$
to the multiplet of the vector representation. Similar as for the S-matrix,
`maximal analyticity' for generalized form factors means again that they are
meromorphic and all poles in the `physical strips' $0\leq\operatorname{Im}%
\theta_{i}\leq\pi$ have a physical interpretation. Together with the usual
LSZ-assumptions \cite{LSZ} of local quantum field theory the following form
factor equations can be derived:

\begin{itemize}
\item[$(i)$] The Watson's equations describe the symmetry property under the
permutation of both, the variables $\theta_{i},\theta_{j}$ and the spaces
$i,j=i+1$ at the same time
\begin{equation}
F_{\dots ij\dots}^{\mathcal{O}}(\dots,\theta_{i},\theta_{j},\dots)=F_{\dots
ji\dots}^{\mathcal{O}}(\dots,\theta_{j},\theta_{i},\dots)\,S_{ij}(\theta_{ij})
\label{3.2}%
\end{equation}
for all possible arrangements of the $\theta$'s.

\item[$(ii)$] The crossing relation which implies a periodicity property under
the cyclic permutation of the rapidity variables and spaces
\begin{multline}
^{~\text{out,}\bar{1}}\langle\,p_{1}\,|\,\mathcal{O}(0)\,|\,p_{2},\dots
,p_{n}\,\rangle_{2\dots n}^{\text{in,conn.}}\\
=\mathbf{C}^{\bar{1}1}\sigma_{1}^{\mathcal{O}}F_{1\ldots n}^{\mathcal{O}%
}(\theta_{1}+i\pi,\theta_{2},\dots,\theta_{n})=F_{2\ldots n1}^{\mathcal{O}%
}(\theta_{2},\dots,\theta_{n},\theta_{1}-i\pi)\mathbf{C}^{1\bar{1}}
\label{3.4}%
\end{multline}
where $\sigma_{\alpha}^{\mathcal{O}}$ takes into account the statistics of the
particle $\alpha$ with respect to $\mathcal{O}$ (e.g. . $\sigma_{\alpha
}^{\mathcal{O}}=-1$ if $\alpha$ and $\mathcal{O}$ are both fermionic, these
numbers can be more general for anyonic or order and disorder fields, see
\cite{BFK}). For the charge conjugation matrix $\mathbf{C}^{\bar{1}1}$ we
refer to (\ref{2.6}).

\item[$(iii)$] There are poles determined by one-particle states in each
sub-channel. In particular the function $F_{\underline{\alpha}}^{\mathcal{O}%
}({\underline{\theta}})$ has a pole at $\theta_{12}=i\pi$ such that
\begin{equation}
\operatorname*{Res}_{\theta_{12}=i\pi}F_{1\dots n}^{\mathcal{O}}(\theta
_{1},\dots,\theta_{n})=2i\,\mathbf{C}_{12}\,F_{3\dots n}^{\mathcal{O}}%
(\theta_{3},\dots,\theta_{n})\left(  \mathbf{1}-\sigma_{2}^{\mathcal{O}}%
S_{2n}\dots S_{23}\right)  \,. \label{3.6}%
\end{equation}


\item[$(iv)$] If there are also bound states in the model the function
$F_{\underline{\alpha}}^{\mathcal{O}}({\underline{\theta}})$ has additional
poles. If for instance the particles 1 and 2 form a bound state (12), there is
a pole at $\theta_{12}=i\eta~(0<\eta<\pi)$ such that
\begin{equation}
\operatorname*{Res}_{\theta_{12}=\eta}F_{12\dots n}^{\mathcal{O}}(\theta
_{1},\theta_{2},\dots,\theta_{n})\,=F_{(12)\dots n}^{\mathcal{O}}%
(\theta_{(12)},\dots,\theta_{n})\,\sqrt{2}\Gamma_{12}^{(12)} \label{3.8}%
\end{equation}
where the bound state intertwiner $\Gamma_{12}^{(12)}$ is defined by
(\ref{2.8}) (see \cite{K1,BK}).

\item[$(v)$] Naturally, since we are dealing with relativistic quantum field
theories we finally have
\begin{equation}
F_{1\dots n}^{\mathcal{O}}(\theta_{1}+\mu,\dots,\theta_{n}+\mu)=e^{s\mu
}\,F_{1\dots n}^{\mathcal{O}}(\theta_{1},\dots,\theta_{n}) \label{3.10}%
\end{equation}
if the local operator transforms under Lorentz transformations as
$F^{\mathcal{O}}\rightarrow e^{s\mu}F^{\mathcal{O}}$ where $s$ is the
\textquotedblleft spin\textquotedblright\ of $\mathcal{O}$.
\end{itemize}

The properties (i) - (iv) may be depicted as
\[%
\begin{array}
[c]{rrcl}%
(i) &
\begin{array}
[c]{c}%
\unitlength3.2mm\begin{picture}(7,3)
\put(3.5,2){\oval(7,2)}
\put(3.5,2){\makebox(0,0){${\cal O}$}}
\put(1,0){\line(0,1){1}}
\put(3,0){\line(0,1){1}}
\put(4,0){\line(0,1){1}}
\put(6,0){\line(0,1){1}}
\put(1.4,.5){$\dots$}
\put(4.4,.5){$\dots$} \end{picture}
\end{array}
& = &
\begin{array}
[c]{c}%
\unitlength3.2mm\begin{picture}(7,4)
\put(3.5,3){\oval(7,2)}
\put(3.5,3){\makebox(0,0){${\cal O}$}}
\put(1,0){\line(0,1){2}}
\put(3,0){\line(1,2){1}}
\put(4,0){\line(-1,2){1}}
\put(6,0){\line(0,1){2}}
\put(1.4,1){$\dots$}
\put(4.4,1){$\dots$} \end{picture}
\end{array}
\\
(ii) & ~~~~~~%
\begin{array}
[c]{c}%
\unitlength3.2mm\begin{picture}(5,4)(2,0)
\put(2,0){\line(0,1){1}}
\put(3.5,3){\line(0,1){1}}
\put(3.5,2){\oval(5,2)}
\put(3.5,2){\makebox(0,0){${\cal O}$}}
\put(5,0){\line(0,1){1}}
\put(6,3){\scriptsize conn.}
\put(2.8,.3){$\dots$} \end{picture}
\end{array}
~~=~~\sigma^{\mathcal{O}}%
\begin{array}
[c]{c}%
\unitlength3.2mm\begin{picture}(6,4)
\put(0,1){\line(0,1){3}}
\put (1,1){\oval(2,2)[b]}
\put(3.5,2){\oval(5,2)}
\put(3.5,2){\makebox(0,0){${\cal O}$}}
\put(5,0){\line(0,1){1}}
\put(2.6,.3){$\dots$} \end{picture}
\end{array}
~~ & = &
\begin{array}
[c]{c}%
\unitlength3.2mm\begin{picture}(7,4)
\put(7,1){\line(0,1){3}}
\put (6,1){\oval(2,2)[b]}
\put(3.5,2){\oval(5,2)}
\put(3.5,2){\makebox(0,0){${\cal O}$}}
\put(2,0){\line(0,1){1}}
\put(2.6,.3){$\dots$} \end{picture}
\end{array}
\\
(iii) & \dfrac{1}{2i}\,\operatorname*{Res}\limits_{\theta_{12}=i\pi}~~~%
\begin{array}
[c]{c}%
\unitlength3.2mm\begin{picture}(6,4)
\put(3,2){\oval(6,2)}
\put(3,2){\makebox(0,0){${\cal O}$}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(3,0){\line(0,1){1}}
\put(5,0){\line(0,1){1}}
\put(3.4,.5){$\dots$} \end{picture}
\end{array}
& = &
\begin{array}
[c]{c}%
\unitlength3.2mm\begin{picture}(5,4)
\put(.5,0){\oval(1,2)[t]}
\put(3,2){\oval(4,2)}
\put(3,2){\makebox(0,0){${\cal O}$}}
\put(2,0){\line(0,1){1}}
\put(4,0){\line(0,1){1}}
\put(2.4,.5){$\dots$} \end{picture}
\end{array}
~~-\sigma^{\mathcal{O}}~~%
\begin{array}
[c]{c}%
\unitlength3.2mm\begin{picture}(6,5)
\put(0,0){\line(0,1){3}}
\put(3,3){\oval(6,4)[t]}
\put(3,3){\oval(6,4)[br]}
\put(3,0){\oval(4,2)[tl]}
\put(3,3){\oval(4,2)}
\put(3,3){\makebox(0,0){${\cal O}$}}
\put(2,0){\line(0,1){2}}
\put(4,0){\line(0,1){2}}
\put(2.4,1.5){$\dots$} \end{picture}
\end{array}
\\
(iv) & \dfrac{1}{\sqrt{2}}\,\operatorname*{Res}\limits_{\theta_{12}=\eta}~%
\begin{array}
[c]{c}%
\unitlength3.2mm\begin{picture}(5,3)
\put(2.5,2){\oval(5,2)}
\put(2.5,2){\makebox(0,0){${\cal O}$}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(4,0){\line(0,1){1}}
\put(2.4,.5){$\dots$} \end{picture}
\end{array}
& = &
\begin{array}
[c]{c}%
\unitlength3.2mm\begin{picture}(5,4)
\put(2.5,3){\oval(5,2)}
\put(2.5,3){\makebox(0,0){${\cal O}$}}
\put(1.5,0){\oval(1,2)[t]}
\put(1.5,1){\line(0,1){1}}
\put(4,0){\line(0,1){2}}
\put(2.4,1){$\dots$} \end{picture}
\end{array}
\end{array}
\]


These equations have been proposed by Smirnov \cite{Sm} as generalizations of
equations derived in the original articles \cite{KW,BKW,K2}. They have been
proven \cite{BFKZ} by means of the LSZ-assumptions and `maximal analyticity'.

We will now provide a constructive and systematic way of how to solve the
equations (i) -- (v) for the co-vector valued function $F_{1\dots
n}^{\mathcal{O}}$ once the scattering matrix is given.

\subsection{Two-particle form factors}

For the two-particle form factors the form factor equations are easily
understood. The usual assumptions of local quantum field theory yield
\[
\langle\,0\,|\,\mathcal{O}(0)\,|\,p_{1},p_{2}\rangle^{in/out}=F\left(
(p_{1}+p_{2})^{2}\pm i\varepsilon\right)  =\,F\left(  \pm\theta_{12}\right)
\]
where the rapidity difference is defined by $p_{1}p_{2}=m^{2}\cosh\theta_{12}%
$. For integrable theories one has particle number conservation which implies
(for any eigenstate of the two-particle S-matrix)
\[
\langle\,0\,|\,\mathcal{O}(0)\,|\,p_{1},p_{2}\rangle^{in}=\langle
\,0\,|\,\mathcal{O}(0)\,|\,p_{2},p_{1}\rangle^{out}\,S\left(  \theta
_{12}\right)  \,.
\]
Crossing (\ref{3.4}) means
\[
\langle\,p_{1}\,|\,\mathcal{O}(0)\,|\,p_{2}\rangle=F\left(  i\pi-\theta
_{12}\right)
\]
where for one-particle states the in- and out-states coincide.
Therefore Watson's equations follow
\begin{equation}%
\begin{array}
[c]{l}%
F\left(  \theta\right)  =F\left(  -\theta\right)  S\left(  \theta\right)
\vspace{5pt}\\
F\left(  i\pi-\theta\right)  =F\left(  i\pi+\theta\right)  \,.
\end{array}
\label{3.12}%
\end{equation}
For general theories Watson's \cite{Wa} equations only hold below the particle
production thresholds. However, for integrable theories there is no particle
production and therefore they hold for all complex values of $\theta$. It has
been shown \cite{KW} that these equations together with \textquotedblleft
maximal analyticity\textquotedblright\ have a unique solution.

As an example we write the sinh-Gordon \cite{KW} and the(highest weight)
$SU(N)$ form factor function \cite{BFK1}
\begin{align}
F^{SHG}(\theta)  &  =\exp\int_{0}^{\infty}\frac{dt}{t\sinh t}\,\left(
\frac{\cosh(\frac{1}{2}+\nu)t}{\cosh\frac{1}{2}t}-1\right)  \cosh t\left(
1-\frac{\theta}{i\pi}\right) \label{3.14}\\
F^{SU(N)}\left(  \theta\right)   &  =c\exp\int\limits_{0}^{\infty}\frac
{dt}{t\sinh^{2}t}e^{\frac{t}{N}}\sinh t\left(  1-\frac{1}{N}\right)  \left(
1-\cosh t\left(  1-\frac{\theta}{i\pi}\right)  \right)
\end{align}
which are the minimal solution of (\ref{3.12}) with $S^{SHG}\left(
\theta\right)  $ as given by (\ref{2.10}) and $S^{SU(N)}\left(  \theta\right)
=a\left(  \theta\right)  $ as given by (\ref{2.14}), respectively.

\subsection{The general form factor formula}

As usual \cite{KW} we split off the minimal part and write the form factor for
n particles as%
\begin{equation}
F_{\alpha_{1}\dots\alpha_{n}}^{\mathcal{O}}(\theta_{1},\dots,\theta
_{n})=K_{\alpha_{1}\dots\alpha_{n}}^{\mathcal{O}}(\underline{\theta}%
)\prod\limits_{1\leq i<j\leq n}F(\theta_{ij})\,.\label{3.16}%
\end{equation}
By means of the following {\textquotedblleft off-shell Bethe ansatz"} for the
(co-vector valued) K-function%
\begin{equation}
\fbox{$\rule[-0.18in]{0in}{0.45in}~K_{\alpha_{1}\dots\alpha_{n}}^{\mathcal{O}%
}(\underline{\theta})=\displaystyle\int_{\mathcal{C}_{\underline{\theta}}%
}dz_{1}\cdots\displaystyle\int_{\mathcal{C}_{\underline{\theta}}}%
dz_{m}\,h(\underline{\theta},\underline{z})\,p^{\mathcal{O}}(\underline
{\theta},\underline{z})\,\Psi_{\alpha_{1}\dots\alpha_{n}}(\underline{\theta
},\underline{z})~$}\label{3.18}%
\end{equation}
we transform the complicated form factor equations $(i)-(v)$ into simple ones
for the p-functions which are scalar and polynomials in $e^{\pm z_{i}}$. The
\textbf{\textquotedblleft off-shell Bethe ansatz"}{ state }$\Psi_{\alpha
_{1}\dots\alpha_{n}}(\underline{\theta},\underline{z})$ is obtained as a
product of S-matrix elements and the integration contour $\mathcal{C}%
_{\underline{\theta}}$ depends on the model (see below for the $SU(N)$-model).
The scalar functions%
\begin{align}
h(\underline{\theta},\underline{z}) &  =\prod_{i=1}^{n}\prod_{j=1}^{m}%
\phi(\theta_{i}-z_{j})\prod_{1\leq i<j\leq m}\tau(z_{i}-z_{j})\,,\label{3.20}%
\\
\tau(z) &  =\frac{1}{\phi(z)\phi(-z)}\nonumber
\end{align}
depend on $F(\theta)$ only (see (\ref{3.21}) below), i.e.~on the S-matrix,
whereas the p-function $p^{\mathcal{O}}(\underline{\theta},\underline{z})$
depends on the operator.

\subsubsection*{The $SU(N)$ form factors}

The form factors for n fundamental particles (of the vector representation of
$SU(N)$) are given by (\ref{3.16}) - (\ref{3.20}) where the
\textbf{\textquotedblleft nested Bethe ansatz"} is needed. This means that
$\Psi${ is of the form}%
\begin{equation}
\Psi_{\alpha_{1}\dots\alpha_{n}}(\underline{\theta},\underline{z}%
)=L_{\beta_{1}\dots\beta_{m}}(\underline{z})\Phi_{\alpha_{1}\dots\alpha_{n}%
}^{\beta_{1}\dots\beta_{m}}(\underline{\theta},\underline{z}) \label{3.29}%
\end{equation}
where the indices $\alpha_{i}$ take the values $\alpha=1,\dots,N$ and the
summations run over $\beta_{i}=2,\dots,N$. The \textbf{\textquotedblleft
off-shell Bethe ansatz"}{ state }$\Phi_{\alpha_{1}\dots\alpha_{n}}^{\beta
_{1}\dots\beta_{m}}(\underline{\theta},\underline{z})$ is obtained using the
techniques of the algebraic Bethe ansatz as follows.

We consider a state with $n$ particles and define the monodromy matrix
\begin{equation}
T_{1\dots n,0}(\underline{\theta},\theta_{0})=S_{10}(\theta_{10})\,\cdots
S_{n0}(\theta_{n0})=%
\begin{array}
[c]{c}%
\unitlength3mm\begin{picture}(9,4.5)
\put(0,2){\line(1,0){9}}
\put(2,0){\line(0,1){4}}
\put(7,0){\line(0,1){4}}
\put(1,0){$1$}
\put(5.8,0){$ n$}
\put(8.2,.7){$ 0$}
\put(3.6,3){$\dots$} \end{picture}
\end{array}
\nonumber
\end{equation}
as a matrix acting in the tensor product of the \textquotedblleft quantum
space\textquotedblright, a space of $n$ particles (with respect to their
quantum numbers) $V^{1\dots n}=V^{1}\otimes\cdots\otimes V^{n}$ and the
\textquotedblleft auxiliary space\textquotedblright\ $V^{0}$. The $V^{i}%
\cong\mathbb{C}^{N}$ denote the spaces of all fundamental particles. The
Yang-Baxter algebra relation for the S-matrix yields
\begin{gather}
T_{1\dots n,a}(\underline{\theta},\theta_{a})\,T_{1\dots n,b}(\underline
{\theta},\theta_{b})\,S_{ab}(\theta_{a}-\theta_{b})=S_{ab}(\theta_{a}%
-\theta_{b})\,T_{1\dots n,b}(\underline{\theta},\theta_{b})\,T_{1\dots
n,a}(\underline{\theta},\theta_{a})\label{1.32}\\
\unitlength4mm\begin{picture}(19,5)
\put(.3,0){$1$}
\put(4.3,0){$n$}
\put(0,2.3){$b$}
\put(0,4.3){$a$}
\put(7.5,2.3){$b$}
\put(7.5,.5){$a$}
\put(2.5,3){$\dots$}
\put(1,0){\line(0,1){5}}
\put(0,2){\line(1,0){8}}
\put(5,0){\line(0,1){5}}
\put(0,2){\oval(14,4)[rt]}
\put(8,2){\oval(2,4)[lb]}
\put(9.3,2.3){$=$}
\put(13.3,0){$1$}
\put(17.3,0){$n$}
\put(11,2){$b$}
\put(11,4.2){$a$}
\put(18.5,3.4){$b$}
\put(18.5,1.4){$a$}
\put(15.5,2){$\dots$}
\put(14,0){\line(0,1){5}}
\put(18,0){\line(0,1){5}}
\put(11,3){\line(1,0){8}}
\put(19,3){\oval(14,4)[lb]}
\put(11,3){\oval(2,4)[rt]} \end{picture}\nonumber
\end{gather}
which in turn implies the basic algebraic properties of the sub-matrices
$A,B,C,D$ with respect to the auxiliary space defined by
\begin{equation}
T_{1\dots n,0}(\underline{\theta},z)\equiv\left(
\begin{array}
[c]{cc}%
A_{1\dots n}(\underline{\theta},z) & B_{1\dots n,\beta}(\underline{\theta
},z)\\
C_{1\dots n}^{\beta}(\underline{\theta},z) & D_{1\dots n,\beta}^{\beta
^{\prime}}(\underline{\theta},z)
\end{array}
\,\right)  ,~~~~2\leq\beta,\beta^{\prime}\leq N\,. \label{1.34}%
\end{equation}
The basic Bethe ansatz co-vectors $\Phi$ of equation (\ref{3.29})
are obtained by an application of the operators $C$ to a \textquotedblleft
pseudo-vacuum\textquotedblright\ state%
\begin{equation}
\Phi_{1\dots n}^{\underline{\beta}}(\underline{\theta},\underline{z}%
)=\Omega_{1\dots n}C_{1\dots n}^{\beta_{m}}(\underline{\theta},z_{m})\cdots
C_{1\dots n}^{\beta_{1}}(\underline{\theta},z_{1}) \label{1.38}%
\end{equation}
This may be depicted as%
\[%
\begin{array}
[c]{r}%
\Phi_{\underline{\alpha}}^{\underline{\beta}}(\underline{\theta},\underline
{z})=%
\begin{array}
[c]{c}%
\unitlength4mm\begin{picture}(12,8)
\put(9,5){\oval(14,2)[lb]}
\put(9,5){\oval(18,6)[lb]}
\put(4,1){\line(0,1){4}}
\put(8,1){\line(0,1){4}}
\put(3.7,0){$\alpha_1$}
\put(7.8,0){$\alpha_n$}
\put(-.2,5.4){$\beta_1$}
\put(1.8,5.4){$\beta_m$}
\put(3.8,5.4){$1$}
\put(7.8,5.4){$1$}
\put(9.2,1.8){$1$}
\put(9.2,3.8){$1$}
\put(4.3,1){$\theta_1$}
\put(8.3,1){$\theta_{n}$}
\put(5.5,2.342){$z_1$}
\put(5.5,4.3){$z_m$}
\put(.6,4.3){$\dots$}
\put(5.8,1.3){$\dots$}
\put(8.6,2.6){$\vdots$} \end{picture}
\end{array}
\text{with }%
\begin{array}
[c]{l}%
2\leq\beta_{i}\leq N\\
1\leq\alpha_{i}\leq N\,.
\end{array}
\end{array}
\]
It means that $\Phi_{\underline{\alpha}}^{\underline{\beta}}(\underline
{\theta},\underline{z})$ is a product of S-matrix elements as given by the
picture where at all crossing points of lines there is an S-matrix
(\ref{2.12}) and the sum over all indices of internal lines is to be taken.
The \textquotedblleft pseudo-vacuum\textquotedblright\ is the highest weight
co-vector (with weight $w=(n,0,\dots,0)$)
\[
\Omega_{1\dots n}=e(1)\otimes\cdots\otimes e(1)
\]
where the unit vectors $e(\alpha)~(\alpha=1,\dots,N)$ correspond to the
particle of type $\alpha$ which belong to the vector representation of
$SU(N)$. The pseudo-vacuum\ basic vector satisfies
\begin{equation}%
\begin{array}
[c]{rcl}%
\Omega_{1\dots n}\,B_{1\dots n}^{\beta}(\underline{\theta},z) & = & 0\\
\Omega_{1\dots n}\,A_{1\dots n}(\underline{\theta},z) & = & \prod
\limits_{i=1}^{n}a(\theta_{i}-z)\Omega_{1\dots n}\\
\Omega_{1\dots n}\,D_{1\dots n,\beta}^{\beta^{\prime}}(\underline{\theta},z) &
= & \delta_{\beta}^{\beta^{\prime}}\prod\limits_{i=1}^{n}b(\theta_{i}%
-z)\Omega_{1\dots n}\,.
\end{array}
\label{1.40}%
\end{equation}
The amplitudes of the scattering matrices are given by eq. (\ref{2.12}). The
technique of the nested Bethe ansatz means that for the co-vector valued
function $L_{\beta_{1}\dots\beta_{m}}(\underline{z})$ in (\ref{3.29}) one
makes the second level Bethe ansatz.
This ansatz is of the same form as (\ref{3.18}) only that
the range of the indices is reduced by 1. Iterating this nesting procedure one
finally arrives at a scalar function.
The integration contour $\mathcal{C}_{\underline{\theta}}$ depends on the
$\underline{\theta}$ and is depicted in figure \ref{f2}.
\begin{figure}[tbh]
\begin{center}
\unitlength4mm\begin{picture}(27,13)
\thicklines
\put(1,0){
\put(0,0){$\bullet~\theta_n-2\pi i$}
\put(.19,5){\circle{.3}~$\theta_n-2\pi i\frac1N$}
\put(0,6){$\bullet~~\theta_n$}
\put(.2,6.2){\oval(1,1)}
\put(.5,6.68){\vector(1,0){0}}
\put(0,11){$\bullet~\theta_n+2\pi i(1-\frac1N)$}
}
\put(8,6){\dots}
\put(12,0){
\put(0,0){$\bullet~\theta_2-2\pi i$}
\put(.19,5){\circle{.3}~$\theta_2-2\pi i\frac1N$}
\put(0,6){$\bullet~~\theta_2$}
\put(.2,6.2){\oval(1,1)}
\put(.5,6.68){\vector(1,0){0}}
\put(0,11){$\bullet~\theta_2+2\pi i(1-\frac1N)$}
}
\put(20,1){
\put(0,0){$\bullet~\theta_1-2\pi i$}
\put(.19,5){\circle{.3}~$\theta_1-2\pi i\frac1N$}
\put(0,6){$\bullet~~\theta_1$}
\put(.2,6.2){\oval(1,1)}
\put(.5,6.68){\vector(1,0){0}}
\put(0,11){$\bullet~\theta_1+2\pi i(1-\frac1N)$}
}
\put(9,2.5){\vector(1,0){0}}
\put(0,3){\oval(34,1)[br]}
\put(27,3){\oval(20,1)[tl]}
\end{picture}
\end{center}
\caption{\emph{The integration contour }$C_{\underline{\theta}}$\emph{. The
bullets belong to poles of the integrand resulting from }$a(\theta_{i}%
-z_{j})\,\phi(\theta_{i}-z_{j})$\emph{ and the small open circles belong to
poles originating from }$b(\theta_{i}-z_{j})$\emph{ and }$c(\theta_{i}-z_{j}%
)$\emph{. }}%
\label{f2}%
\end{figure}

Swieca's picture is that the bound state of $N-1$ fundamental particles is
to be identified with the anti-particle and lead together with the form factor
recursion relations $(iii)+(iv)$ to the equation for the
function $\phi(z)$ (see \cite{BFK,BFK1})
\begin{equation}
\prod_{k=0}^{N-2}\phi\left(  z+ki\eta\right)  \prod_{k=0}^{N-1}F\left(
z+ki\eta\right)  =1~,~~~\eta=\frac{2\pi}{N} \label{3.21}%
\end{equation}
with the solution
\[
\fbox{ $\phi(\theta)=\Gamma\left(  \dfrac{\theta}{2\pi i}\right)
\Gamma\left(  1-\dfrac{1}{N}-\dfrac{\theta}{2\pi i}\right).$}
\]
In \cite{BFK1} is shown that the form factors given by (\ref{3.16}) and
(\ref{3.18}) satisfy the form factor equations $(i)-(v)$ if some simple
equations for the p-function are satisfied. We note that form factors of
this model were also calculated in \cite{Sm,jap} using other techniques.

\subsubsection*{Locality}

Let us discuss a further property of the form factors which is directly
related to the nature of the operators, that is \textit{locality}. In fact,
this property touches the very heart and most central concepts of relativistic
quantum field theory, like Einstein causality and Poincar\'{e} covariance,
which are captured in local field equations and commutation relations.

So far we have presented a formulation of a quantum field
theory, which starts from a particle picture. It is basically possible to
obtain the particle picture from the field formulation by means of the
LSZ-reduction formalism. The reverse problem, namely of how to
reconstruct the entire field content, or at least part of it, from the
scattering theory is in general still an outstanding challenge. Besides this
classification issue there remains also the general question if the operators
which are related to the solutions of $(i)-(v)$ are genuinely local, meaning
that they (anti)-commute for space-like separations with themselves. It has
been shown that $(i)-(v)$ (see e.g. \cite{Sm,BFK}) imply
\begin{equation}
^{in}\langle\,\phi\,|\,\left[  \mathcal{O}(x),\mathcal{O}(y)\right]
\,|\,\psi\,\rangle^{in}=0\quad\text{for}\quad(x-y)^{2}<0 \label{3.22}%
\end{equation}
for all matrix elements. Note that commutation rules such as (\ref{3.22})
hold if
the statistics factors in $(i)-(v)$ are trivial, $\sigma^{\mathcal{O}}=1$.
However, for more general statistics factors there are also fermionic, anyonic
and even order-disorder commutation rules as for example for the scaling
$Z_{N}$-Ising model (see \cite{BFK}).

\subsubsection*{Examples of operators and their p-functions for the
sinh-Gordon model}

Since there is no backward scattering the Bethe ansatz in (\ref{3.18}) is
trivial, the integrations can be performed \cite{BK2} and the n-particle
K-function may be written as
\begin{equation}
K_{n}^{\mathcal{O}}(\underline{\theta})=\sum_{l_{1}=0}^{1}\dots\sum_{l_{n}%
=0}^{1}(-1)^{\sum l_{i}}\prod_{i<j}\!\left(  1+(l_{i}-l_{j})\frac{i\sin\pi\nu
}{\sinh\theta_{ij}}\right)  \!p_{n}^{\mathcal{O}}(\underline{\theta
},\underline{l})\,. \label{3.24}%
\end{equation}


For several cases the correspondence between local operators and their
p-functions have been proposed in \cite{BK2}\footnote{Sinh-Gordon form
factors have been presented before in a different form in \cite{FMS,KM}}.
Here we provide three examples:

\begin{enumerate}
\item The {normal ordered exponential of the field (see also \cite{BL})}
\begin{equation}
\mathcal{O}(x)=\,:\!e^{\gamma\varphi(x)}\!:\quad\leftrightarrow p(\underline
{\theta},\underline{l})=\left(  \frac{2}{F(i\pi)\sin\pi\nu}\right)  ^{\frac
{n}{2}}\prod\limits_{i=1}^{n}e^{\pi\nu\frac{\gamma}{\beta}(-1)^{l_{i}}}~,
\label{3.26}%
\end{equation}


\item Expanding the last relation with respect to $\gamma$ one obtains the
p-func\-tions for {normal ordered powers }$:\!\varphi(x)^{N}\!:$ in particular
for $N=1$%
\begin{equation}
\,:\!\varphi(x)\!:\quad\leftrightarrow p(\underline{\theta},\underline
{l})=\frac{\pi\nu}{\beta}\left(  \frac{2}{F(i\pi)\sin\pi\nu}\right)
^{\frac{n}{2}}\sum_{i=1}^{n}(-1)^{l_{i}}~, \label{3.28}%
\end{equation}
which yields (for $n=1$) the `wave function renormalization constant'
\[
Z^{\varphi}=\langle\,0\,|\,\varphi(0)\,|p\,\rangle^{2}=\frac{8\pi^{2}\nu^{2}%
}{-F^{SHG}(i\pi)\beta^{2}\sin\pi\nu}%
\]
(see also \cite{KW}).

\item The higher conserved currents $J_{L}^{\mu}(x)$ which are typical for
integrable quantum field theories
\[
J_{L}^{\pm}(x)\leftrightarrow\pm N_{n}^{(J_{L})}\sum_{i=1}^{n}e^{\pm\theta
_{i}}\sum_{i=1}^{n}e^{L\left(  \theta_{i}-\frac{i\pi}{2}(1-(-1)^{l_{i}}%
\nu)\right)  }\,.
\]

\end{enumerate}

\subsubsection*{Quantum sinh-Gordon field operator equation}

Using (\ref{3.26}) and (\ref{3.28}) one finds \cite{BK1,BK,BK2} that the
quantum sinh-Gordon field equation
\begin{equation}
\square\varphi(x)+\frac{\alpha}{\beta}:\!\sinh\beta\varphi\!:(x)=0
\label{3.30}%
\end{equation}
holds for all matrix elements, if the
\textquotedblleft bare\textquotedblright\ mass $\sqrt{\alpha}$
is related to the renormalized mass by
\begin{equation}
\alpha=m^{2}\frac{\pi\nu}{\sin\pi\nu} \label{3.32}%
\end{equation}
where $m$ is the physical mass of the fundamental boson. The result may be
checked in perturbation theory by Feynman graph expansions. In particular in
lowest order the relation between the bare and the renormalized mass
(\ref{3.32}) had already been calculated in the original article \cite{KW}.
The result is
\[
m^{2}=\alpha\left(  1-\frac{1}{6}\left(  \frac{\beta^{2}}{8}\right)
^{2}+O(\beta^{6})\right)
\]
which agrees with the exact formula above. The factor $\frac{\pi\nu}{\sin
\pi\nu}$ in (\ref{3.32}) modifies the classical equation and has to be
considered as a quantum correction. The proof of the quantum field equation
(\ref{3.30}) can be found in \cite{BK2}.

\section{Wightman functions}

As the simplest case we consider the two-point function of two local scalar
operators $\mathcal{O}(x)$ and $\mathcal{O}^{\prime}(x)$%
\[
w(x)=\langle\,0\,|\,\mathcal{O}(x)\,\mathcal{O}^{\prime}(0)\,|\,0\,\rangle.
\]


\subsubsection*{Summation over all intermediate states}

Inserting a complete set of in-states we may write%
\begin{align}
w(x)  &  =\sum_{n=0}^{\infty}\frac{1}{n!}\int\frac{dp_{1}}{2\pi2\omega_{1}%
}\dots\int\frac{dp_{n}}{2\pi2\omega_{n}}e^{-ix(p_{1}+\dots+p_{n})}\nonumber\\
&  \times\langle\,0\,|\,\mathcal{O}(0)\,|p_{1},\ldots,p_{n}\,\rangle
^{in}\,^{in}\langle p_{n},\ldots,p_{1}\,|\,\mathcal{O}^{\prime}%
(0)\,|\,0\,\rangle\nonumber\\
&  =\sum_{n=0}^{\infty}\frac{1}{n!}\int d\theta_{1}\dots\int d\theta
_{n}e^{-ix\sum p_{i}}g_{n}(\underline{\theta})\,. \label{4.2}%
\end{align}
We have introduced the functions%
\begin{align*}
g_{n}(\underline{\theta})  &  =\frac{1}{\left(  4\pi\right)  ^{n}}%
\langle\,0\,|\,\mathcal{O}(0)\,|p_{1},\ldots,p_{n}\,\rangle^{in}\,^{in}\langle
p_{n},\ldots,p_{1}\,|\,\mathcal{O}^{\prime}(0)\,|\,0\,\rangle\\
&  =\frac{1}{\left(  4\pi\right)  ^{n}}\mathcal{O}(\theta_{1},\ldots
,\theta_{n}\,)\,\mathcal{O}^{\prime}(\theta_{n}+i\pi,\ldots,\theta_{1}%
+i\pi)\,.
\end{align*}
where crossing has been used. In particular we consider exponentials of a
scalar bose field
\[
\mathcal{O}^{(\prime)}(x)=\,:\!e^{i\gamma^{(\prime)}\varphi(x)}\!:
\]
where $:\dots:$ means normal ordering with respect to the physical vacuum
and which amounts to the condition
\[
\,\langle\,0\,|:\!e^{i\gamma^{(\prime)}\varphi(x)}\!:|\,0\,\rangle=1
\]
and therefore $g_{0}=1$ holds.

\paragraph{The Log of the two-point function}

For specific operators like exponentials of bose fields it might be convenient
(see below) to consider a resummation of the sum in (\ref{4.2}). For $g_{0}=1$
we may write (see also \cite{Sm2})%
\begin{align*}
w(x)  &  =1+\sum_{n=1}^{\infty}\frac{1}{n!}\int d\theta_{1}\dots\int
d\theta_{n}e^{-ix\sum p_{i}}g_{n}(\underline{\theta})\\
&  =\exp\sum_{n=1}^{\infty}\frac{1}{n!}\int d\theta_{1}\dots\int d\theta
_{n}e^{-ix\sum p_{i}}h_{n}(\underline{\theta})\,
\end{align*}
It is well known that the functions $g_{n}$ and $h_{n}$ are related by the
cummulant formula
\[
g_{I}=\sum_{I_{1}\cup\dots\cup I_{k}=I}h_{I_{1}}\dots h_{I_{k}}\,
\]
where we use the short hand notation $g_{I}=g_{n}(\theta_{1},\dots,\theta
_{n})$ with $I=\left\{  1,\dots,n\right\}  $. The relations for the $g$'s and
the $h$'s may be depicted with ~ $g=\begin{picture}(22,12)
\put(10,5){%
\oval(20,10)}\end{picture}$ ~and~ $h=\begin{picture}(22,12)
\put(0,0){%
\framebox(20,10){}}\end{picture}$ ~as
\[
\unitlength.45mm\begin{picture}(240,32)(0,-2) \footnotesize
\put(20,20){\oval (40,20)[]}
\put(10,5){\line(0,1){5}}
\put(8,-2){$1$}
\put(20,5){\makebox (0,0){$\dots$}}
\put(30,5){\line(0,1){5}}
\put(28,-2){$n$}
\put(47,15){=}
\put(60,10){\framebox(30,20){}}
\put(65,5){\line(0,1){5}}
\put(63,-2){$1$}
\put(75,5){\makebox(0,0){$\dots$}}
\put(85,5){\line(0,1){5}}
\put(83,-2){$n$}
\put(96,15){$+~\displaystyle \sum_{i=1}^n$}
\put(120,10){\framebox(20,15){}}
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\put(127.5,4){.\,.}
\put(135,0){\line(0,1){10}}
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\put(155,5){\line(0,1){5}}
\put(153,-2){$i$}
\put(170,14){$+~\cdots$}
\put(200,10){\framebox(10,10){}}
\put(205,5){\line(0,1){5}}
\put(203,-2){$1$}
\put(220,15){\makebox (0,0){$\dots$}}
\put(230,10){\framebox(10,10){}}
\put(235,5){\line(0,1){5}}
\put(233,-2){$n$} \end{picture}~.
\]
Thus as examples
\begin{align*}
g_{1}  &  =h_{1}\\
g_{12}  &  =h_{12}+h_{1}h_{2}\\
g_{123}  &  =h_{123}+h_{12}h_{3}+h_{13}h_{2}+h_{23}h_{1}+h_{1}h_{2}h_{3}\\
&  \dots
\end{align*}


Due to Lorentz invariance it is sufficient to consider the value
$x=(-i\tau,0)$. Let $\mathcal{O}(x)$ and $\,\mathcal{O}^{\prime}(x)$ be
scalar operators. Then the functions $h_{n}(\underline{\theta})$ depend
only on the rapidity differences.
We use the formula for the modified Bessel function of the third kind
\[
i\Delta_{+}(x)=\langle\,0\,|\,\varphi(x)\,\varphi(0)\,|\,0\,\rangle=\frac
{1}{4\pi}\int d\theta e^{-\tau m\cosh\theta}=\frac{1}{2\pi}K_{0}(m\tau)
\]
to perform one integration
\begin{align*}
\ln w(x)  &  =\sum_{n=1}^{\infty}\frac{1}{n!}\int d\theta_{1}\dots\int
d\theta_{n}e^{-\tau m\sum\cosh\theta_{i}}h_{n}(\underline{\theta})\,\\
&  =2\sum_{n=1}^{\infty}\frac{1}{n!}\int d\theta_{1}\dots\int d\theta
_{n-1}h_{n}(\theta_{1},\dots,\theta_{n-1},0)K_{0}(m\tau\xi)
\end{align*}
with
\[
\xi^{2}=\left(  \sum_{i=1}^{n-1}\cosh\theta_{i}+1\right)  ^{2}-\left(
\sum_{i=1}^{n-1}\sinh\theta_{i}\right)  ^{2}.
\]


\subsubsection*{Short distance behavior $x\rightarrow0$}

In order to perform the conformal limit of massive models one investigates the
short distance behavior (see e.g.~\cite{Za1,Ca,Sm2}). For small $\tau$ we use
the expansion of the modified Bessel function of the third kind and obtain%
\begin{align*}
\ln w(x)  &  =-2\sum_{n=1}^{\infty}\frac{1}{n!}\int d\theta_{1}\dots\int
d\theta_{n-1}h_{n}(\theta_{1},\dots,\theta_{n-1},0)\\
&  \times\left(  \ln m\tau+\ln\xi+\gamma_{E}-\ln2+O\left(  \tau^{2}\ln
\tau\right)  \right)
\end{align*}
where $\gamma_{E}=0.5772\dots$ is Euler's or Mascheroni's constant. Therefore
the two-point Wightman function has power a like behavior for short distances
\[
w(x)\approx C\left(  m\tau\right)  ^{-4\Delta}\quad\text{for }\tau
\rightarrow0
\]
where the dimension is given by%
\[
\Delta=\frac{1}{2}\sum_{n=1}^{\infty}\frac{1}{n!}\int d\theta_{1}\dots\int
d\theta_{n-1}h_{n}(\theta_{1},\dots,\theta_{n-1,}0)
\]
in case the integrals exist. This is true for the exponentials of bose
fields$\,\mathcal{O}=:\!e^{\gamma\varphi(x)}\!:$ due to the asymptotic
behavior for $\operatorname{Re}\theta_{1}\rightarrow\infty$%
\begin{align*}
F_{n}^{\mathcal{O}}(\theta_{1,}\theta_{2,}\dots)  &  =F_{1}^{\mathcal{O}%
}(\theta_{1})F_{n-1}^{\mathcal{O}}(\theta_{2,}\dots)+O(e^{-\theta_{1}})\\
g_{n}(\theta_{1},\theta_{2},\dots\theta_{n})  &  =g_{1}g_{n-1}(\theta
_{2},\dots\theta_{n})+O(e^{-\left\vert \theta_{1}\right\vert })
\end{align*}
(see e.g. \cite{BK2}). Therefore the functions $h_{n}$ satisfy%
\[
h_{n}(\underline{\theta})=O(e^{-\left\vert \theta_{i}\right\vert }%
)\quad\text{for }\operatorname{Re}\theta_{i}\rightarrow\pm\infty\,.
\]
This follows when we distinguish the variable $\theta_{1}$ in the relation
of the $g$'s and the $h$'s above and reorganize the terms on the right hand
side as follows
\[
g_{I}=\sum_{1\in J\subseteq I}h_{J}g_{I\setminus J}\,.
\]
The constant $C$ is obtained as
\[
C=\exp\left(  -2\sum_{n=1}^{\infty}\frac{1}{n!}\int d\theta_{1}\dots\int
d\theta_{n-1}h_{n}(\theta_{1},\dots,\theta_{n-1},0)\left(  \ln\tfrac{1}{2}%
\xi+\gamma_{E}\right)  \right)
\]
and it should be related to the vacuum expectation value $G=\,\langle
\,0\,|\mathcal{O}(x)|\,0\,\rangle_{C}$ in the `conformal normalization'
\[
C=m^{4\Delta}G^{-2}\,.
\]
Such vacuum a expectation value was calculated in \cite{LZ} for the
sine-Gordon model.

\subsubsection*{Example: the sinh-Gordon model}

The dimension of the exponential of the field $\mathcal{O}(x)=\,:\!e^{\beta
\varphi(x)}\!:$ for the sinh-Gordon model may be calculated in the 1- and
1+2-particle intermediate state approximation (see Fig.~\ref{f1}) as
\begin{align*}
\Delta_{1+2}  &  =\frac{1}{2}\left(  h_{1}+\frac{1}{2!}\int d\theta
\,h_{2}(\theta,0)+\ldots\right) \\
&  =-\frac{\sin\pi\nu}{\pi F(i\pi)}+\left(  \frac{\sin\pi\nu}{\pi F(i\pi
)}\right)  ^{2}\int_{-\infty}^{\infty}d\theta\,\left(  F(\theta)F(-\theta
)-1\right)  +\ldots
\end{align*}
The integral may be calculated exactly with the result \cite{BK4}%
\[
-\frac{\pi}{2}\sin\pi\nu F^{2}(i\pi)-\pi\frac{\cos\pi\nu-1}{\sin\pi\nu
}+2\left(  1-\frac{\pi\nu\cos\pi\nu}{\sin\pi\nu}\right)  \,.
\]
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\put(170,550){$\Delta(\nu)$}
\end{picture}
\end{center}
\caption{\emph{Dimension of an exponential of the field for the sinh-Gordon
model: 1- and 1+2-particle intermediate state contributions}}%
\label{f1}%
\end{figure}
In principle the higher particle intermediate state integrals may also be
calculated, however, up to now we were not able to derive a general formula.
For the scaling Ising model, however, this is possible. The constant $C$ in
the approximation of 1-intermediate states is given as%
\[
C_{1}=\exp\left(  -2h_{1}\left(  \gamma_{E}-\ln2\right)  \right)  =\exp\left(
4\frac{\sin\pi\nu}{\pi F(i\pi)}\left(  \gamma_{E}-\ln2\right)  \right)  \,.
\]
We have not calculated the integrals appearing in higher-particle
intermediate state contributions.

\subsection*{Acknowledgments}

H.B. was supported partially by the grant  Volkswagenstiftung within
in the project "Nonperturbative aspects of quantum field theory in
various space-time dimensions". H.B. also acknowledges to ICTP
condensed matter group for hospitality, where part of this work was
done. A.F. acknowledges support from PRONEX under contract CNPq
66.2002/1998-99 and CNPq (Conselho Nacional de Desenvolvimento
Cient\'{\i}fico e Tecnol\'{o}gico). This work is also supported by
the EU network EUCLID, 'Integrable models and applications: from
strings to condensed matter', HPRN-CT-2002-00325.

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\LastPageEnding


\end{document}
