\documentclass{elsart}
\usepackage{epsfig}
\usepackage{amssymb}
\begin{document}
\begin{frontmatter}
\title{
$ t $-channel unitarity and photon cross sections}
\author{J.R. Cudell}
\ead{JR.Cudell@ulg.ac.be}
\author{E. Martynov\corauthref{cor1}}
\corauth[cor1]{on leave from Bogolyubov Institute for Theoretical Physics,
Kiev.}
\ead{E.Martynov@guest.ulg.ac.be}
\author{G. Soyez}
\ead{G.Soyez@ulg.ac.be}
\address{Universit\'e de Li\`ege, B\^at. B5-a, Sart Tilman, B4000 Li\`ege,
Belgium}
\begin{abstract}
We analyze the consequences of $ t $-channel unitarity for photon
cross sections and show what assumptions are necessary to allow the existence
of new
singularities at $ Q^{2}=0 $ for the $ \gamma p $ and $ \gamma \gamma  $
total cross sections. For nonzero $ Q^{2} $, such singularities
can in general be present, but we show that, apart from the perturbative
singularity associated with $ \gamma ^{*}\gamma ^{*}\rightarrow q\bar q $,
no new ingredient is needed to reproduce the data from LEP
and HERA, in the Regge region.
\end{abstract}
\begin{keyword}
Unitarity \sep factorization \sep photon cross sections \sep $S$ matrix \sep
DIS

% PACS codes here, in the form: \PACS code \sep code
\PACS 13.85.Lg \sep 14.70.Bh \sep 13.60.Hb \sep 11.55.Bq \sep 11.55.J
\end{keyword}
\end{frontmatter}
% main text
\section{Introduction }
\label{intro}
The DIS and total cross section data \cite{H1,ZEUS,total} from HERA
have opened
new avenues in our understanding of strong interactions, and models
\cite{DoLa,DM,CS} now exist which provide a unified description
of $ \gamma p $
interactions for  a photon virtuality ranging from
$ Q^{2}=0 $ to $ Q^{2}=30000$~GeV$^{2} $. The theoretical situation
is nevertheless not clear.

Indeed, a wide range of data can be described for $ Q^{2}\geq 2 $ GeV$ ^{2} $
by the DGLAP evolution \cite{DGLAP1,DGLAP2,DGLAP3}. Several theoretical
questions need however to be addressed in this context. Firstly, the evolution
is leading twist, and hence one should remove higher-twist contributions
from the data before one uses the DGLAP equation. Secondly, the evolution
introduces extra singularities in the complex $ j $ plane at $ j=1 $.
These singularities start to appear at the arbitrary factorization
scale $ Q_{0} $, and their resummation leads to an essential singularity.
No trace of it is however present in soft cross sections. Finally,
the DGLAP evolution should be replaced at small $ x $ by the BFKL
resummation. The latter does not lead to an essential singularity
in the complex $ j $ plane, but unfortunately it does
not seem to be stable against next-to-leading order corrections.

Given these problems, Donnachie and Landshoff have proposed to use
the soft pomeron as a higher-twist background to be subtracted from
the evolution, while a new simple pole, the ``hard pomeron''\cite{hardpom},
would reproduce the DIS data. Furthermore, they have shown \cite{DoLa}
that this new singularity evolves according to DGLAP, provided that
one removes the $j$-plane
singularities induced by DGLAP evolution, and keeps only their effect on
the hard
pomeron residue.
Again the question arises whether such a new pole should be present
in total cross sections and whether it is perturbative or not.

Finally, we have shown that in fact no new singularity is needed to
reproduce the DIS data \cite{DM,CS}, provided that one assumes a
logarithmic behaviour of cross sections as functions of $ \nu $.
Double or triple poles at $ j=1 $ provide such a behaviour, and
enable one to reproduce all soft and hard $ \gamma p $ data within
the Regge region.

How to bridge the gap between those models and QCD remains a challenge,
as the description of the proton, being non-perturbative, remains
at best tentative. However, LEP has now provided us with a variety
of measurements of the $ \gamma \gamma  $ total cross sections, for
on-shell photons, and of $ F_{2}^\gamma $ for off-shell ones \cite{L3,OPAL}.
One may hope that this will be a good testing ground for perturbative
QCD \cite{BL}, and that these measurements will provide guidance for
the QCD understanding of existing models. Hence it is important to
build a unified description of all photon processes, and to explore
where perturbative effects may manifest themselves. The natural framework
for such a goal is the ``factorization theorem'' of the analytic
$ S $ matrix, which relates $ \gamma \gamma  $, $ \gamma p $
and $ pp $ amplitudes. This theorem is based on $ t $-channel
unitarity, \emph{i.e}. unitarity in the crossed channel, and in the
case of simple poles one obtains the factorization of the
residues at each pole. For more general analytic structures, one obtains
more complicated relations, which we shall spell out in Section 2.

Furthermore, a relation between $ \gamma \gamma  $ and $ \gamma p $
processes may be of practical use as some of the measurements have
big systematic uncertainties. As it is now well known \cite{MCstudy}, the
LEP measurements are sensitive to the theoretical Monte Carlo used
to unfold the data, leading to rather different conclusions as to
the energy dependence of the data. This problem is manifest in the
case of total cross sections, where the unfolding constitutes the
main uncertainty. In the case of HERA data, the measurement of the
total cross section also seems to be affected by large
uncertainties. Again, a joint study of both processes could help constrain
the possible behaviours of these cross sections.

To decide whether new singularities can appear in $ \gamma p $
and $ \gamma \gamma  $ scattering, one must first recall why singularities
are supposed to be universal in hadronic cross sections. The original
argument \cite{factorisation,multifact} made use of analytic continuation of
amplitudes in the complex $ j $ plane from one side of a 2-particle
threshold to the other, which lead to universal simple poles and factorization
of their residues. We show in section 2 that it is in fact possible
to reproduce these results without analytic continuation to the second
sheet, and that one can obtain a general formula for complex $ j $ plane
amplitudes, which is valid no matter what the singularity is, and
which leads to consequences similar to factorization. We argue in
the third section that such a formula may be applicable to photon
cross sections at $ Q^{2}=0 $, and give its generalization to off-shell
photons. If we assume as in \cite{DM,CS} that no other singularity
is present in DIS, stringent constraints come from the positivity
requirement for $ \gamma \gamma  $ total cross sections and $F_2$. We show
that it is possible to obtain a good fit to all photon
data for $Q^2<150$ GeV$^2$
by using either double or triple-pole parametrisations. For total cross
sections, no extra singularity seems to be needed, suggesting that an $S$
matrix
may be defined for photons. For high $Q^2$ data, it seems
that extra singularities are needed. We conclude this study by
outlining its consequences on the evolution of parton distributions and
on the possibility of observing the BFKL pomeron.
\section{$\bf t$-channel unitarity in the hadronic case}
\label{tcu}
\subsection{General argument}
\label{ga}
We start by giving a proof of the factorization theorem in the hadronic
case for spin-averaged amplitudes. We have extended
the standard proof to the general case of n-particle thresholds, and
this point will be useful in the next section.

We start by considering the amplitudes for three related processes:\\
\begin{center}
\epsfig{file=diags.eps,width=11cm}
\end{center}
We shall refer to the momenta of the incoming particles as $p$ and $q$, and
we use the Mandelstam variables $s=(p+q)^2$ and $t=(p-q)^2$.
In the $ s $ channel, these diagrams describe the processes
$ aa\rightarrow aa $,
$ ab\rightarrow ab $, $ bb\rightarrow bb $, whereas in the $ t $
channel, they describe the processes $ a\overline{a}\rightarrow a\overline{a}
$,
$ a\overline{a}\rightarrow b\overline{b} $, $ b\overline{b}
\rightarrow b\overline{b} $.
Assuming that $ m_{b} $ is the lowest hadronic mass, we know that
the latter processes have thresholds for $ t>4m_{a}^{2}>4m_{b}^{2} $
(for instance, think of $ a=p $, $ b=\pi  $, and the n$ \pi  $
thresholds in $ pp $, $ \pi p $ and $ \pi \pi  $ amplitudes).
In general, if $ t $ is large enough, there are many possible
intermediate states (not only n$ \pi  $, but also n$ K $, etc.)
for each process under consideration, which we must in principle take
into account to write the unitarity relations. These states can be grouped
into subsets which have the same quantum numbers, and for which one can
derive factorization.

Starting with the unitarity of the $ S $ matrix:
\newcommand{\1}{{1\hspace{-3pt} \rm I}}
\begin{equation}
S^{\dagger }S=SS^{\dagger }=\1
\end{equation}
and setting $ S=\1+iS_{c} $, we obtain
\begin{equation}
\label{unita}
S_{c}-S_{c}^{\dagger }=iS_{c}^{\dagger }S_{c}=iS_{c}S_{c}^{\dagger }.
\end{equation}
One can define the invariant amplitude $ T_{if} $ by the matrix
elements\begin{equation}
<f|S_{c}|i>=(2\pi )^{4}\delta ^{4}(p_{f}-p_{i})T_{if}.\end{equation}
Eq. (\ref{unita}) then becomes the following at the amplitude level:
\begin{equation}
T_{if}-T_{if}^{\dagger }=C_{s}(T,T^{\dagger }).\end{equation}

We have defined the $ C_{s} $ operator as the following convolution:
\begin{equation}
\label{unitarity0}
C_{s}(T^{\dagger },T)=C_{s}(T,T^{\dagger })=2i\sum _{k}\int dPS\:
T_{ik}T^{\dagger }_{kf}
\end{equation}
where $ k $ refers to all possible intermediate on-shell n-particle
states in the $ t $ channel, which can differ by the number and
nature of produced particles, and $ dPS $ represents the differential
n-particle Lorentz-invariant phase space associated with these states.

If the particles are massive, we can enumerate these open channels
and assume that $ k $ runs from 1 to $ N+2 $. In particular, we
shall find in this set of states the $ a\overline{a} $ and $ b\overline{b} $
intermediate states to which we respectively assign the labels $ k=1,\: 2 $.
Note that in general the label $ k $ does not refer to the number
of particles in the intermediate state, and that $ k $ can stand
for particles different from $ a $ and $ b $. So in general
the amplitude $ T_{km} $ represents the following process:
\begin{center}
\epsfig{file=diag2.eps}
\end{center}
Eq. (\ref{unitarity0}) can the be represented by:
\begin{center}
\epsfig{file=diag3.eps,width=9cm}
\end{center}

We can now imagine that we split the amplitude into  charge-parity $+1 $
and  charge-parity $ -1 $ parts, and then perform a Watson-Sommerfeld transform
\begin{equation}
\label{Regge}
T_{ab}^{\pm }(\nu ,t)=16\pi i\int dl\; P_{l}(-\cos (\vartheta _{t}))\;
\frac{2l+1}{2\sin (\pi l)}T_{ab}^{\pm }(l,t)\; \left( 1\pm e^{-i\pi l}\right) .
\end{equation}
with $\nu=p.q$.
(In the following, we shall only consider the charge-parity $+1 $ part of
the amplitudes without carrying the superscript $ + $.) After continuing
this relation to complex $ l\equiv j $, we deform the contour of integration
so that only the singularities of $ T(j,\, t) $ will contribute.
All amplitudes become functions of $ j $, and the operator
$ C_{s} $ changes to $ C $, which has the following properties:
\begin{itemize}
\item It is associative and distributive\begin{equation}
C(\alpha A_{1}+\beta A_{2},B)=\alpha C(A_{1},B)+\beta C(A_{2},B).\end{equation}
\item In the case of 2-particle intermediate states $ k $, the form of
$ C $ is particularly simple:\begin{equation}
C_{2}(T^{\dagger },T)=\rho _{k}T_{ik}T^{\dagger }_{kf}=(TRT^{\dagger
})_{if}\end{equation}
with $ \rho _{k}=2i\sqrt{\frac{t-4m_{k}^{2}}{t}} $, and $ R_{km}=\rho
_{k}\delta _{km} $.
\end{itemize}
To proceed further, we shall represent the $ T $ matrix in the
following form, for $ k\leq N+2 $:
\begin{eqnarray}
T=\left( \begin{array}{cc}
T_{0}(2\times 2) & \cdot\\
\cdot &\cdot
\end{array}\right) &+&\left( \begin{array}{cc}
\cdot & T_{u}(2\times N)\\
\cdot &\cdot
\end{array}\right) \nonumber\\
&+&\left( \begin{array}{cc}
\cdot & \cdot\\
T_{l}(N\times 2) &\cdot
\end{array}\right)
+\left( \begin{array}{cc}
\cdot & \cdot\\
\cdot & T_{r}(N\times N)
\end{array}\right) ,\end{eqnarray}
where we have indicated the dimensions of the sub-matrices in parenthesis.
$ T_{0} $ contains the elastic amplitudes ($ i,\: f $=1, 2),
the upper matrix $ T_{u} $ contains the inelastic amplitudes $ i=1,2\rightarrow
k>2 $,
and the lower matrix $ T_{l} $ the inelastic amplitudes $ k>2\rightarrow i=1,\:
2 $.
$ T_{r} $ stands for the rest of the amplitudes $ k\rightarrow m $,
with $ k $ and $ m $ $ >2 $.

The system (\ref{unitarity0}) can then be written:
\begin{eqnarray}
\label{first}
T_{0}-T_{0}^{\dagger }&=&T_{0}RT_{0}^{\dagger }+C(T_{u},T_{u}^{\dagger }),\\
\label{two}
T_{u}-T_{l}^{\dagger }&=&T_{0}RT_{l}^{\dagger }+C(T_{u},T_{r}^{\dagger }),\\
\label{three}
T_{l}-T_{u}^{\dagger }&=&T_{l}RT_{0}^{\dagger }+C(T_{r},T_{u}^{\dagger }),\\
\label{four}
T_{r}-T_{r}^{\dagger }&=&C(T_{l},T_{l}^{\dagger })+C(T_{r},T_{r}^{\dagger }).
\end{eqnarray}

To derive factorization, it is enough to consider the first two relations
(\ref{first}, \ref{two}). We assume that the second equation can
be solved by a series expansion, yielding
\begin{equation}
T_{u}=M+T_{0}RM
\end{equation}
with $ M $ the solution of $ M=T_{l}^{\dagger }+C(M,T_{r}^{\dagger }) $:
\begin{equation}
M=T_{l}^{\dagger }+C(T_{l}^{\dagger },T_{r}^{\dagger })+C(C(T_{l}^{\dagger
},T_{r}^{\dagger }),T_{r}^{\dagger })+...\end{equation}
We can put this form into Eq.~(\ref{first}), which then gives
\begin{equation}
\label{solved}
T_{0}(\1-RD)=D
\end{equation}
with\begin{equation}
D=\left[ T_{0}^{\dagger }+C(M,T_{u}^{\dagger })\right] .\end{equation}

$T_0$ is then a function of $T^\dagger$ and its singularities
cannot come from singularities in the
right-hand side of Eq.~(\ref{solved}), because they are exactly matched
by corresponding factors in the left-hand side. Hence \emph{ the singularities
of the amplitudes $ T_0^{if} $ are common to all processes} as they
can only come from zeroes $ z_{m} $ of the determinant of the matrix
in bracket in the left-hand side:
\begin{equation}
\Delta =\det \left( \1-RD\right) =0\; {\rm for}\; j=z_{m}.\end{equation}
Near each zero $ z_{m} $, we can write, for $p$ and $q$ equal to $1$ or $2$
\begin{equation}
T_{pq}=\frac{t_{pq}}{\Delta }.\end{equation}
Furthermore, the matrix $D$ is sensitive to the existence of thresholds
associated with bound states, and does not know directly about quarks and
gluons
which do not enter the unitarity equations. Hence \emph{ the zeroes $z_m$
are not calculable perturbatively.}

This is the basis of the complex $ j $-plane factorization of the
amplitudes contained in $ T_{0} $. Indeed, we can write \begin{equation}
\label{factorised}
T_{11}T_{22}-T_{12}T_{21}=\det (T_{0})=\frac{\det (D)}{\Delta }.
\end{equation}
We see that near the zeroes we obtain

\begin{equation}
\label{factoris}
\lim _{j\rightarrow z_{m}}\left[
T_{11}(j)-\frac{T_{12}(j)T_{21}(j)}{T_{22}(j)}\right] =\rm finite\: terms.
\end{equation}

One may note here that one could have a spurious cancellation of the
singularity if $ t_{pq} $ has a zero at $ j=z_{m} $. However,
as both quantities are $ t $-dependent, and as $ t_{pq} $ is
process-dependent, it is unlikely for this cancellation to occur for
all $ t $ or for all processes. It is however possible to {}``hide{}''
a singularity, \emph{e.g.} at $t=0$ for $ pp $ and $ \bar{p}p $
scattering. This might provide an explanation for the absence of
an odderon pole in forward scattering data.

It is also worth mentioning that \emph{each singularity factorizes
separately.} Hence it does not make sense to consider globally factorizing
cross sections or amplitudes in the $ s $, $ t $ representation, unless
of course the amplitude can be reproduced by only one pole.

The relations (\ref{factoris}) lead to a definite prediction
for the residues (or couplings) of the singularities above threshold
$ t>4m_{a}^{2} $. As no singularity occurs when $ t $ is continued
to the physical region for the $ s $ channel processes, these relations
still remain true there.

\subsection{Specific examples}
\label{spec}
Eq. (\ref{factoris}) is usually not mentioned, and only its
consequences for
the residues of simple poles are considered. However, we have shown
that it is true in general, and that it leads leads to specific predictions
for any singularity structure of $ T_{pq}(j) $, \emph{e.g.} for
a given order of the zeroes of $ z_{m} $. We shall give here the
formulae that correspond to simple, double or triple poles, which
seem to be three possibilities emerging from fits to hadronic amplitudes
at $ t=0 $ \cite{compete}. We shall refer to these relations as the $t$-Channel
Unitarity ($t$CU) relations.
The case of cuts will not be explicitly
considered here, although Eq. (\ref{factoris}) holds also in this
case.

For isolated simple poles\begin{equation}
T_{pq}=\sum _{m}\frac{R_{pq}^{m}}{j-z_{m}},\end{equation}
one obtains the usual relations for the residues
\cite{factorisation}\begin{equation}
R_{22}^{m}=\frac{\left( R_{12}^{m}\right) ^{2}}{R_{11}^{m}}.\end{equation}

If $ T_{pq} $ has coinciding simple and double poles\begin{equation}
T_{pq}=\frac{S_{pq}}{j-z}+\frac{D_{pq}}{(j-z)^{2}},\end{equation}
one obtains the new relations
\begin{eqnarray}
D_{11}D_{22}&=&{\left( D_{12}\right) ^{2}},\nonumber\\
\label{double}
D_{11}^2S_{22}&=&{D_{12}(2S_{12}D_{11}-S_{11}D_{12})}.
\end{eqnarray}

In the case of triple poles
\begin{equation}
T_{pq}=\frac{S_{pq}}{j-z}+\frac{D_{pq}}{(j-z)^{2}}
+\frac{F_{pq}}{(j-z)^{3}},\end{equation}
the relations become
\begin{eqnarray}
F_{11}F_{22}&=&{\left( F_{12}\right) ^{2}},\nonumber\\
\label{triple}
F_{11}^2D_{22}&=&{F_{12}(2D_{12}F_{11}-D_{11}F_{12})},\\
F_{11}^3S_{22}&=&F_{11}F_{12}\left( 2S_{12}F_{11}-S_{11}F_{12}\right)
\nonumber\\
&+&D_{12}F_{11}\left( D_{12}F_{11}-2D_{11}F_{12}\right)
+D_{11}^2F_{12}^2.\nonumber
\end{eqnarray}
It is worth pointing out that the double pole relations are not the
limit of the triple pole relations for a vanishing triple pole residue.
Similarly, the simple-pole relations cannot be obtained from the double-pole
ones. The reason for this is that the relations (\ref{factorised})
relate the poles of order $ 2n $ to $ n+1 $, $n$ being the maximal
order of the pole. All these relations give $ 0=0 $ if the leading
pole vanishes, and it is the next relations, which normally would
not give a divergence in Eq. (\ref{factoris}), that now contribute.


\section{The photon case}
\label{photon}
\subsection{On-shell photons}
\label{osp}
The basic problem here concerns the fact that photons are massless.
Because of this, one has perturbatively an infrared singularity
in all amplitudes containing a fixed number of photons. These singularities
are canceled by virtual corrections in inclusive cross sections,
and the standard strategy to solve the problem is to perform a resummation
of soft photons \emph{\`a la} Bloch-Nordsiek \cite{BN}. One then only
considers inclusive quantities which include an infinite resummation
of soft photons. The outcome of this resummation is that the exclusive
amplitudes connecting states with a finite number of photons are identically
zero. This means that the $ S $ matrix is not defined, and that
asymptotic states with a fixed number of particles cannot be used
to build the theory. The formalism that we have developed then breaks
down (or becomes trivial: Eq.~(\ref{factorised}) gives $0=0$),
and one can only use $ S $-matrix theory to treat hadronic
interactions.

If one takes the above point of view, one can salvage part of the $t$CU
relations
if one keeps only the hadronic part of the photon wave function, and
neglects electromagnetic interactions altogether. Assuming that an
$ S $-matrix still describes the interactions of this part of the
wave function, one then keeps a subset of the equations (\ref{factorised}),
effectively removing photon thresholds from the unitarity equations,
and treating photons as external states only. In practice, the equations
(\ref{solved}-\ref{factorised}) remain the same, provided that we write
the threshold matrix $ R $ as
\begin{equation}
R=\left( \begin{array}{cc}
\rho _{p} & 0\\
0 & 0
\end{array}\right) .\end{equation}

This means that $ \Delta  $ will only involve $ D_{pp} $, hence
singularities can now come from other elements of $ D $, and $ \det (D) $
can contain singularities not present in $ \Delta , $ hence breaking
the factorization relation (\ref{factorised}). Namely, we obtain
\begin{eqnarray}
T_{pp}&=&\frac{D_{pp}}{\Delta },\nonumber\\
\label{pessimist}
T_{\gamma p}&=&\frac{D_{\gamma p}}{\Delta },\\
T_{\gamma \gamma }&=&\frac{\rho _{p}D_{\gamma p}^{2}}{\Delta }+
D_{\gamma \gamma }.\nonumber
\end{eqnarray}

Extra singularities can come from $ D_{\gamma p} $ or $ D_{\gamma \gamma } $.
In the first case, the nature of the singularity is different in $ \gamma p $
and in $ \gamma \gamma  $, and the coupling of the singularity,
which contains $ \Delta  $, must be of non-perturbative origin.
On the other hand, singularities in $ D_{\gamma \gamma } $ can
be purely perturbative.

However, this state of affairs is largely unsatisfactory,
as single photons and single electrons do not exist in the theory
anymore - worse, one can show that no pole structure can be associated
to them, hence electrons do not exist as poles in propagators anymore,
but only
as cuts \cite{cuts}. This fundamental question has baffled theorists
for a long time \cite{othersmatrix}. Recently, however, there have been
claims \cite{polemass,dressing}
that an $ S $-matrix formalism could be
developed in QED, and that one construct gauge-invariant free asymptotic
states for QED, provided that one used dressed electrons instead of
the bare ones from the lagrangian. It has further been shown that
these new states correspond to poles in propagators, and hence have
a definite mass.

If this is the case, then the $ S $-matrix formalism
holds, and one-shell photons behave like on-shell pions:
the equations (\ref{factoris}) apply to dressed photon
amplitudes.

The fact that the complex $j$ plane is cut into two parts
which do not communicate ($ T^{\dagger } $ cannot be obtained by
analytic continuation of $ T $) is not important as we did not
rely on the original factorization proof \cite{factorisation,multifact}
which considered continuation around the cut, but only on the existence
of the $ S $ matrix, and on the possibility of inverting Eq.~(\ref{two}).
In the massless limit, the n-particle cuts which we have considered
merge together, but this does not seem to invalidate the above argument
either.

Hence we see that the question of extra singularities has far-reaching
consequences: \emph{the existence of new singularities in total $ \gamma p $
and $ \gamma \gamma  $ cross sections at $Q^2=0$
 would constitute an experimental
proof that an $ S $ matrix formalism cannot be developed for QED.}


\subsection{Off-shell photons}
\label{offsp}
In the DIS case, the situation is very similar whether the QED $S$ matrix is
defined or not. If we exclude the photon thresholds as in Eq.
(\ref{pessimist}),
the equations and the conclusions remain the same. But even if we
consider photon thresholds, we must take into account that the
incoming particles are off shell.
These virtual particles must not be included in the intermediate states
of Eq.~(\ref{unitarity0}). One can still define an $ S $-matrix
in this case, at least in the one-photon approximation, as the electron
contributions can be factored and canceled on each side of the unitarity
equations.

In this case, we want to indicate explicitly whether the external
legs of the $ 2\rightarrow 2 $, $ 2\rightarrow n $ and $ n\rightarrow 2 $
amplitudes are off-shell or not. We introduce the notations $
T_{0}(Q_{in},Q_{out}) $,
$ T_{u}(Q_{in}) $ and $ T_{l}(Q_{out}) $, where $ Q_{in} $
stands for the two virtualities $ (Q_1^{2},Q_2^{2}) $ of the
initial states in the $ t $ channel, and $ Q_{out} $ for the
two virtualities $ (Q_{3}^{2},Q_{4}^{2}) $ of the final states,
and we write $ Q_{in}=0 $ or $ Q_{out}=0 $ in the case of on-shell
states, and the relations (\ref{unitarity0}) can be visualized as
follows:

\vspace{0.375cm}
{\centering \includegraphics{diag4.eps} \par}
\vspace{0.375cm}

The system of equations (\ref{first}-\ref{four}) then becomes:

\begin{eqnarray}
T_{0}(Q_{in},Q_{out})-T_{0}^{\dagger
}(Q_{in},Q_{out})&=&T_{0}(Q_{in},0)RT_{0}^{\dagger }(0,Q_{out})\nonumber\\
&+&C(T_{u}(Q_{in}),T_{u}^{\dagger }(Q_{out})),
\label{First}\\
\label{Two}
T_{u}(Q_{in})-T_{l}^{\dagger }(Q_{in})&=&T_{0}(Q_{in},0)RT_{l}^{\dagger
}(0)+C(T_{u}(Q_{in}),T_{r}^{\dagger }),\\
\label{Three}
T_{l}(Q_{out})-T_{u}^{\dagger }(Q_{out})&=&T_{l}(Q_{out})RT_{0}^{\dagger
}(0,0)+C(T_{r},T_{u}^{\dagger }),\\
\label{Four}
T_{r}-T_{r}^{\dagger }&=&C(T_{l}(0),T_{l}^{\dagger }(0))+C(T_{r},T_{r}^{\dagger
}).
\end{eqnarray}

The resolution of the system proceeds as before with the elimination
of\break $T_{u}(Q_{in}) $:
\begin{equation}
T_{u}(Q_{in})=M(Q_{in})+T_{0}(Q_{in},0)RM(0)\end{equation}
with $ M $ the solution of $ M(Q)=T_{l}^{\dagger }(Q)+C(M(Q),T_{r}^{\dagger })
$:\begin{equation}
M(Q_{in})=T_{l}^{\dagger }(Q_{in})+C(T_{l}^{\dagger }(Q_{in}),T_{r}^{\dagger
})+C(T_{l}^{\dagger }(Q_{in}),T_{r}^{\dagger }),T_{r}^{\dagger
})+...\end{equation}

The first equation however now gives
\begin{equation}
\label{Master}
T_{0}(Q_{in},Q_{out})=D(Q_{in},Q_{out})+T_{0}(Q_{in},0)RD(0,Q_{out})
\end{equation}
with \begin{equation}
D(Q_{in},Q_{out})=T_{0}^{\dagger }(Q_{in},Q_{out})+C(M(Q_{in}),T_{u}^{\dagger
}(Q_{out})).\end{equation}

For DIS, we consider $ Q_{out}=0 $ and $ Q_{1}^{2}=Q_{2}^{2}=Q^{2}\equiv -q^2
$.
(Note that the same kind of relations and conclusions would hold for
off-forward parton distribution functions). This gives us
\begin{equation}
T_{0}(Q_{in},0)(\1-RD(0,0))=D(Q_{in},0).\end{equation}

Hence we see that all the on-shell singularities must be present in
the off-shell case, but we can have new ones coming from the singularities
of $ D(Q_{in},0) $. These singularities can be of perturbative
origin (\emph{e.g.} the singularities generated by the DGLAP evolution)
but their coupling will depend on the threshold matrix $ R $, and
hence they must know about hadronic masses, or in other words they
are not directly accessible by perturbation theory.

In the case of $ \gamma ^{*}\gamma ^{*} $ scattering, we take $Q_1^2=Q_2^2=Q^2$
and $Q_3^2=Q_4^2=P^2=-p^2$, and Eq.~(\ref{Master})
gives
\begin{equation}
\label{Master2}
T_{0}(Q_{in},Q_{out})=D(Q_{in},Q_{out})
+\frac{D(Q_{in},0)RD(0,Q_{out})}{\1-RD(0,0)}.
\end{equation}
This shows that the DIS singularities will again be present, either through
$ \Delta  $, or through extra singularities present in DIS (in
which case their order will be different in $ \gamma \gamma  $
scattering, at least for $Q_{in}=Q_{out}$).

It is also possible to have extra singularities purely from $ D(Q_{in},Q_{out})
$.
\emph{A priori} these could be independent from the threshold matrix,
and hence be of purely perturbative origin
(\emph{e.g.} $ \gamma ^{*}\gamma ^{*}\rightarrow \bar{q}q $
or the BFKL pomeron coupled to photons through a perturbative impact factor).

We also want to point out that the intercepts of these new singularities
can depend on $ Q^{2} $, and as the off-shell states do not enter
unitarity equations, these singularities can be fixed in $ t $.
However, their residues must vanish as $ Q^{2}\rightarrow 0 $.

\section{Test of $t$CU relations}
\label{testtcu}
In order to test the previous equations, and to evaluate the need
for new singularities, we shall use models that reproduce $ pp $, $ \gamma p $
and $ \gamma \gamma  $ cross sections. Previous studies \cite{compete}
have shown that there are at least three broad classes of models that
can reproduce all forward hadron and photon data.

The general form of these parametrisations is given, for total cross
sections of $a$ on $b$, by the generic formula\footnote{The real part of
the amplitudes, when needed to fit the $\rho$ parameter, is obtained from
$s\leftrightarrow u$ crossing symmetry.}
\begin{equation} \sigma^{tot}_{ab}=(R_{ab}+H_{ab})/s \end{equation}where
$R_{ab}$ is the contribution of the highest meson trajectories ($\rho$,
$\omega$, $a$ and $f$)
and the rising term $H_{ab}$ stands for the pomeron. The first term
is parametrised via Regge theory, and we allow the lower trajectories
to be partially non-degenerate, {\it i.e.} we allow one intercept
for the charge-even trajectories, and another one for the charge-odd ones
 \cite{CKK}.
Hence we use
\begin{equation} R_{ab}= Y_{ab}^{+} \left({\tilde s}\right)^{\alpha _{+}} \pm
Y_{ab}^{-} \left({\tilde s}\right)^{\alpha _{-}}\label{lower} \end{equation}
with
$\tilde s=2\nu/(1$ GeV$^2)$.

As for the pomeron term, we consider the following possibilities:
\begin{eqnarray}
H_{ab}&=&X_{ab}\left[ \widetilde{s}\right] ^{\alpha _{\wp }},\\
\label{doubpar}
H_{ab}&=&sD_{ab}\left[ \log \widetilde{s}+\log C_{ab}\right], \\
\label{trippar}
H_{ab}&=&st_{ab}\left[ \left( \log \frac{\widetilde{s}}{d_{ab}}\right)
^{2}+\log \left( c_{ab}\right) \right] .
\end{eqnarray}
These forms come from simple, double or triple poles in Eq. (\ref{Regge}),
in the limit of $ \cos (\vartheta _{t}) $ large, so that the contribution
from the integration contour vanishes, and that one can keep only
the leading meson trajectories and the pomeron contribution.

Using the asymptotic expansion of the Legendre polynomials $ P_{l} $
\begin{equation}
\label{Legendre}
P_{l}(-\cos (\vartheta _{t}))\rightarrow \frac{\Gamma (2l+1)}{(\Gamma
(l+1))^{2}2^{l}}\left( \frac{\nu }{m_{p}^{2}}\right) ^{l},
\end{equation}
 we obtain, by the residue theorem, from Eq. (\ref{Regge}) the following
contributions to the total cross section for simple, double, and triple
poles:
\begin{eqnarray}
\label{SWsimp}
T(j,0)&=&\frac{g}{j-\alpha }\rightarrow \sigma_{tot}=g\left( \frac{\nu
}{m_{p}^{2}}\right) ^{\alpha }\frac{(2\alpha +1)\Gamma (2\alpha +1)}{(\Gamma
(\alpha +1))^{2}2^{\alpha }},\\
\label{SWdoub}
T(j,0)&=&\frac{g}{(j-\alpha )^{2}}\rightarrow \sigma _{tot}=g\left( \frac{\nu
}{m_{p}^{2}}\right) ^{\alpha }\log \left( \frac{\nu }{m_{p}^{2}}\right)
\frac{(2\alpha +1)\Gamma (2\alpha +1)}{(\Gamma (\alpha +1))^{2}2^{\alpha }},\\
\label{SWtrip}
T(j,0)&=&\frac{g}{(j-\alpha )^{3}}\rightarrow \sigma _{tot}=g\left( \frac{\nu
}{m_{p}^{2}}\right) ^{\alpha }\log \left( \frac{\nu }{m_{p}^{2}}\right)
^{2}\frac{(2\alpha +1)\Gamma (2\alpha +1)}{(\Gamma (\alpha +1))^{2}2^{\alpha
+1}}.
\end{eqnarray}

In the photon case, things are a little different. Looking first at
the $ \gamma p $ amplitude with off-shell photons, we have \begin{equation}
\label{cosg}
|\cos (\vartheta _{t})|=\frac{\nu }{m_{p}\sqrt{Q^{2}}}.
\end{equation}

In the on-shell limit $ Q^{2}\rightarrow 0 $, the Legendre polynomial
of Eq. (\ref{Regge}) becomes infinite, hence one must assume that
the amplitude goes to zero in a way that will make the limit finite.
One can take for instance\begin{equation}
\label{aigpl}
T^{\gamma p}=\tilde{T}^{\gamma p}\left( \frac{\sqrt{Q^{2}}}{q_{\gamma
}(Q^{2})}\right) ^{j}
\end{equation}
with $ q_{\gamma }(0) $ finite. Such a choice introduces a new
scale that effectively replaces $ \sqrt{Q^{2}} $
with $ q_{\gamma }(Q^{2}) $ in $ \cos (\vartheta _{t}) $, and
$ T $ with $ \tilde{T} $. In
the $ \gamma \gamma  $ case, in order to keep the unitarity
relations(\ref{factorised})
for the amplitude $ \tilde{T} $ instead of $ T $, one needs to
assume that \begin{equation}
\label{aigpl2}
T^{\gamma \gamma }=\tilde{T}^{\gamma \gamma }\left(
\frac{\sqrt{Q^{2}P^2}}{q_{\gamma }(Q^{2})q_{\gamma }(P^{2})}\right) ^{j}
\end{equation}
and the scales $q_{\gamma }(Q^{2}) $  and $ q_{\gamma }(P^2)$ replace $ m_{p}$
in Eqs. (\ref{SWsimp}-\ref{SWtrip}).

\subsection{Regge region}
\label{reggereg}
One can think of translating the minimum $ \sqrt{s} $ of
the $ pp $ case into a bound for $ {\nu }/{m_{p}^{2}} $,
$ {\nu }/({m_{p}q_{\gamma }(Q^{2})}) $ and $ {\nu }/{q_{\gamma }^{2}(Q^{2})} $,
and use the same bound in the three processes. Unfortunately, the
situation is really more complicated because one cannot extract $q_\gamma(Q^2)$
from the data as the
$\log\nu$ terms come from  a combination of simple, double (and triple) poles
at $j=1$, which can always be reshuffled among themselves.

In the following, we shall use a cut on $2\nu$, and a cut on
$\cos(\vartheta_t)$. We find that data are well reproduced in the region
\begin{eqnarray}
\cos(\vartheta_t) & \ge & \frac{49}{2m_p^2},\\
\sqrt{2\nu}  & \ge & 7\ \rm GeV.
\end{eqnarray}
For the $\gamma \gamma$ and the $\gamma p$ total
cross sections, as well as for the photon structure function
where $P^2 \rightarrow 0$,
$\cos(\vartheta_t)\rightarrow \infty$, and only the cut on $2\nu$ constrains
the Regge region.

Furthermore, in the case of one virtual photon, experimentalists measure
the $ ep $ or the $ e\gamma  $ cross sections. From these, one
can extract a cross section for $ \gamma ^{*}p $ or $ \gamma ^{*}\gamma ^{*} $
scattering, provided one factors out a flux factor. As is well known,
the latter is univoquely defined only for on-shell particles:\begin{equation}
\label{sigf2}
\sigma_{tot} =\lim_{Q^2\rightarrow 0}\frac{4\pi ^{2}\alpha }{Q^{2}}F_{2}.
\end{equation}
The flux factor can then be modified arbitrarily, provided that the
modifications vanish as $ Q^{2}\rightarrow 0 $. This means, for
instance, that we can always multiply the left-hand side of (\ref{sigf2})
by an arbitrary power of $ (1-x) $. Hence one should in principle
limit oneself to small values of $ x $ only.
We find that we can obtain good fits in the region
\begin{equation}
x\leq 0.3.
\end{equation}
Note that in the case of two
off-shell photons, experimentalists measure $ \sigma _{TT}+\sigma _{TL}+\sigma
_{LT}+\sigma _{LL} $,
which is precisely the quantity entering the factorization theorem.
Hence no flux factor is necessary here.

Finally, all the residues are expected to be functions of $Q^2$. These form
factors are unknown, and are expected to contain higher twists.
In order to check factorization, we do not want to be too dependent on these
guesses, hence we choose a modest region of
\begin{equation}
Q^2\leq 150 \ {\rm GeV}^2.
\end{equation}

We shall consider in the next section possible extensions to a wider region.

\subsection{Factorizing $ t $CU relations }
\label{factorising}
As explained above, the simple-pole singularities will factor in the
usual way. Note that there is no charge-odd singularity in the photon
case, hence only the $ a/f $ lower trajectory will enter the relations.
One then gets \begin{equation}
\label{lowerf}
Y_{pp}Y_{\gamma \gamma }(P^{2},Q^{2})=Y_{\gamma p}(P^{2})Y_{\gamma p}(Q^{2}).
\end{equation}

In the case of a soft-pomeron pole, one obtains similarly\begin{equation}
\label{softpom}
X_{pp}X_{\gamma \gamma }(P^{2},Q^{2})=X_{\gamma p}(P^{2})X_{\gamma p}(Q^{2}).
\end{equation}

The case of multiple poles is given by Eqs. (\ref{double}, \ref{triple}),
and can be made more transparent by using the forms (\ref{doubpar},
\ref{trippar}) which give factorization-looking relations for the
constants (but not for all the residues $-$ see Eqs. (\ref{double})
and~(\ref{triple})~$-$!):
\begin{equation}
f_{pp}f_{\gamma \gamma }(P^{2},Q^{2})=f_{\gamma p}(P^{2})f_{\gamma
p}(Q^{2})\end{equation}
with $f=D$, $C$, $t$, $d$ or $c$.

\subsection{Dataset}
\label{dataset}
For the total cross sections, we have used the updated COMPETE dataset
\cite{PRL}, which is the same as that of \cite{RPP} except for
the inclusion of the latest ZEUS results on $\gamma p$ cross section
\cite{ZEUS}
and for the inclusion of cosmic-ray data.

For $\gamma p$ scattering, we have used the full set of available data
\cite{H1,ZEUS,others}.

For the $\gamma\gamma$ measurements of $F_2^\gamma$, we have used the data
of \cite{L3,OPAL}, whenever these included
the joint $x$ and $Q^2$ (and $P^2$) dependence. We have not included other
data as they do not have points in the Regge region.
Note that we have not taken the
uncertainties in $x$ into account, hence the $\chi^2$ values are really
upper bounds in the $\gamma\gamma$ case.

\subsection{Previous parametrisations}
\label{previous}
We have first considered the results using previous studies \cite{DM,CS}
of $ \gamma ^{(*)}p $ and $ pp $ scattering. Making use of the $t$CU
relations (\ref{double}) and (\ref{triple}), we have obtained
reasonably good predictions for $ \sigma _{\gamma \gamma } $ and
$ F_{2}^{\gamma } $. However, the formalism breaks
down in the case of $ \gamma ^{*}\gamma ^{*} $ scattering, because
the form factors that we used do not guarantee the positivity of the
charge-even part of the
cross sections. Refitting them enables one to get closer to the data,
but the problem of negativity remains in some part of the physical
region. Hence, at this point, the factorization relations have one major
consequence: the parametrisations of \cite{DM,CS} are ruled out.

We have also considered the hard pomeron fit of \cite{DoLa}
where the charge-parity $+1$
rising term contains two different simple poles: the soft and the hard pomeron.
In this case, the soft pomeron residues factorize. The hard pomeron, with
intercept $\alpha_h$
not present in $pp$ cross sections, then comes in as a double
pole in $\gamma\gamma$ cross sections, see Eq. (\ref{pessimist}), and
produces a cross section proportional to $\nu^{\alpha_h}\log\nu$.
Its residue will then depend on the value of $\Delta(\alpha_h)$,
which is unknown. This means that factorization does not say much about
the hard pomeron contribution, which can always be arbitrarily re-scaled.
It is possible to get good fits using these forms, but as they do not
test factorization, we shall not present these results here.
\subsection{New parametrisation: triple pole}
\label{new}
In the triple-pole case, the problem of negativity
can be cured through the introduction
of another functional form for the form factors. To convince ourselves
that this is possible, we have fitted $ F_{2} $ in several $ Q^{2} $
bins to\begin{equation}
F_{2}^{p}(Q^{2})=a(\log \nu +b)^{2}+c\nu ^{-0.47}.\end{equation}
{}From the values of $ a, $ $ b $ and $ c $, and the $t$CU relations,
one can then predict
the symmetric $ F_{2}^{\gamma }(Q^{2},Q^{2}) $. The result
of this exercise is shown in Fig. \ref{2branch}.
\begin{figure}
\centering{\epsfig{file=facto3.eps}}
\caption{Prediction from $t$CU relations for the $\gamma^*\gamma^*$ cross
section, including the box diagram of Fig.~\ref{box}.}
\label{2branch}
\end{figure}
One clearly sees that there are two branches in the fit to HERA data:
one with positive $ b $, and another one with negative $ b $.
Both have comparable $ \chi ^{2} $, but one produces positive $ \gamma \gamma
$
cross sections, whereas the other one does not. Armed with this information,
we found that the resulting form factors could be well approximated
by the following forms: \begin{eqnarray}
t_{\gamma p}(Q^{2})&=&t_{1}\left( \frac{1}{1+\frac{Q^{2}}{Q_{t}^{2}}}\right)
^{\epsilon _{t}},\nonumber\\
\label{formfact}
Y^{+}_{\gamma p}(Q^{2})&=&Y_{1}\left(
\frac{1}{1+\frac{Q^{2}}{Q_{y}^{2}}}\right) ^{\epsilon _{y}},\\
\log d_{\gamma p}(Q^{2})&=&d_{1}\left( \frac{Q^{2}}{Q^{2}+Q_{d}^{2}}\right)
^{\epsilon _{d}},\nonumber\\
\log c_{\gamma p}(Q^{2})&=&c_{0}+c_{1}\left(
\frac{1}{1+\frac{Q^{2}}{Q_{c}^{2}}}\right) ^{\epsilon
_{c}}.\nonumber\end{eqnarray}
With the form factors obtained from our fit, we have then checked
that the $ \gamma ^{*}\gamma ^{*} $ cross section remains positive
everywhere.

\subsection{New parametrisation: double pole}
\label{newdp}
In the case of a double pole, Fig. \ref{2branch} shows that the situation
is more difficult, as one cannot guarantee positivity. We have tried
several possibilities, among which a further splitting of leading
meson trajectories along the lines of \cite{DM2}, but found that
positivity is still not guaranteed.

However, it is possible to obtain a good fit, positive everywhere,
if one assumes a slightly modified version
of the double pole \cite{DM3}.

Instead of taking an $\tilde s D\log \tilde s  $ term in $H_{ab}$ as
in Eq.~(\ref{doubpar}),
 one can consider
\begin{equation}
H_{ab}=\tilde s D_{ab}\ [L_{ab} +\log C_{ab}]
\end{equation}
with
\begin{equation}  L_{ab}(\tilde
s)=\frac{1}{2}\Re e[\log(1+\Lambda_{ab}\tilde
s\,^{\delta})+\log(1+\Lambda_{ab}(-\tilde s)\,^{\delta})].
\end{equation}
Asymptotically, this gives the same form as a double pole. Furthermore,
one can rewrite
$\log(1+\Lambda_{ab}(\tilde s)\,^{\delta})=\delta \log(\tilde s)+
\log(\Lambda_{ab}+1/(\tilde s)\,^{\delta})$. The first term comes
from a double pole at $j=1$, whereas the Taylor expansion of the remaining
term comes from a series of simple poles. Hence $D_{ab}$ and $\Lambda_{ab}$
factorizes according to
\begin{eqnarray}
D_{\gamma\gamma}(P^2,Q^2)D_{pp}&=&D_{\gamma p}(P^2) D_{\gamma
p}(Q^2),\nonumber\\
\Lambda_{\gamma\gamma}(P^2,Q^2)\Lambda_{pp}&=&\Lambda_{\gamma p}(P^2)
\Lambda_{\gamma p}(Q^2).
\end{eqnarray}
We found good fits using the following form factors:
\begin{eqnarray}
D_{\gamma p}&=&D_1\left (\frac{1}{1+\frac{Q^{2}}{Q_{d}^{2}}} \right
)^{\epsilon_{d}},\nonumber\\
\label{formfacd}
D_{\gamma p}\log C_{\gamma p}&=&C_1\left
(\frac{1}{1+\frac{Q^{2}}{Q_{c}^{2}}} \right )^{\epsilon_{c}},\\
\Lambda_{\gamma p}&=&\Lambda_1\left
(\frac{1}{1+\frac{Q^{2}}{Q_{\lambda}^{2}}} \right
)^{\epsilon_{\lambda}}.\nonumber
\end{eqnarray}
\begin{table}[ht]
\small
\centering{\begin{tabular}{|l|c|c|c|c|c|}
\hline
 & & \multicolumn{2}{c|}{double} & \multicolumn{2}{c|}{triple} \\
\hline
Quantity & Nb Data & $\chi^2$ & $\chi^2/pt$ & $\chi^2$ & $\chi^2/pt$ \\
\hline
$F_2^p$        &  821 &  789.624 & 0.962 &  870.599 & 1.060 \\
$F_2^\gamma$        &   65 &   57.686 & 0.887 &   59.963 & 0.923 \\
$\sigma_{\gamma\gamma}$  &   32 &   19.325 & 0.604 &   15.568 & 0.487 \\
$\sigma_{\gamma p}$  &   30 &   17.546 & 0.585 &   21.560 & 0.719 \\
$\sigma_{pp}$  &   90 &  100.373 & 1.115 &   82.849 & 0.921 \\
$\sigma_{p\bar p}$ &   49 &   55.240 & 1.127 &   58.900 & 1.202 \\
$\rho_{pp}$    &   67 &   93.948 & 1.402 &   98.545 & 1.471 \\
$\rho_{p\bar p}$   &   11 &   16.758 & 1.523 &    4.662 & 0.424 \\
\hline
Total          & 1165 & 1150.500 & 0.988 & 1212.645 & 1.041 \\
\hline
\end{tabular}}
\caption{Results of fits to a generalized double pole model and to a triple
pole model, using the form factors of Eqs.~(\ref{formfact}) and
(\ref{formfacd}).}
\label{tab1}
\end{table}
\subsection{The box diagram}
\label{boxdi}
One new singularity may be present in $\gamma\gamma$ scattering:
it is the box diagram,
shown in Fig. \ref{box}, which couples directly two photons to quarks.
This diagram must be present when the photons are far
off-shell and
pQCD applies. As we have explained above, it is not at all obvious that
it is present in the case of total cross sections, and
in fact we get better fits if we include it only for off-shell photons.
Hence it seems that it appears as an extra perturbative singularity
in Eq. (\ref{Master2}).
\begin{figure}
\centering{\epsfig{file=figbox.eps,width=9cm}}
\caption{The box diagram contribution.}
\label{box}
\end{figure}

We have re-calculated it and confirm the results of \cite{Budnev}\footnote{
 We want
to point out that one need to calculate $\sigma_{LL}$, $\sigma_{TL}$,
$\sigma_{LT}$
and $\sigma_{TT}$ separately and sum them to obtain the off-shell cross
section. A contraction of $g_{\mu\nu}$ does not resum the helicities
properly \cite{peskin} which probably explains the discrepancies between
\cite{DoDo} and \cite{Budnev}.}.

These can be recast in the following form, which may be more transparent
in the present context:

We use $x_1=P^2/(2\nu)$ and $x_2=Q^2/(2\nu)$, with $\nu=p.q$, which give
\begin{eqnarray}
P^2&=&{x_1 w^2\over 1-x_1-x_2},\\
Q^2&=&{x_2 w^2\over 1-x_1-x_2}\end{eqnarray}
with $w^2=s$.
We set
\begin{eqnarray} \mu&=&{m^2\over\nu}={2 m^2(1-x_1-x_2)\over w^2},\\
{\tau}&=&{1-4 x_1 x_2},\\
\delta&=&{-x_1-x_2+1}.\end{eqnarray}
The cross sections then take the form
\begin{equation}\sigma_i={12\alpha^2 \pi\delta\over
w^2}\left[{\sqrt{\delta_\mu}\over \sqrt{\delta \tau} (2\delta x_1
x_2+\tau \mu) \tau^2 }{\Sigma}_i+{\Lambda_i\over\tau^3}
 \log(\rho)\right]\end{equation}
which gives
\begin{equation}\rho={\sqrt{\delta\delta_\mu\tau}-\delta_\mu \tau
\over \sqrt{\delta\delta_\mu\tau}+\delta_\mu \tau
}.\end{equation}

The cross sections then are built from:
\begin{eqnarray}{\Sigma}_{TT}&=&
4 \delta x_1 x_2 [2 x_1 x_2 (x_1^2+x_2^2-1+2 x_1+2x_2)
\nonumber\\
&-&12 x_1^2 x_2^2
-2 (x_1^2+x_2^2)+2 (x_2+x_1)
-1] \nonumber\\
&-&\tau\mu (2 x_1-1)^2 (2 x_2-1)^2  -2 \delta\mu^2 \tau^2, \nonumber\\
\Lambda_{TT}&=&2 \delta \mu \tau-2 \mu^2 \tau^2\nonumber\\
&+&[8 x_1^2x_2^2(x_1^2+x_2^2)+16 x_1^3 x_2^3
-16 x_1^2 x_2^2(x_1+x_2)\nonumber\\
&-&4 x_1x_2(x1^2+x_2^2)
+16 x_1^2 x_2^2+
-2 (x_1+x_2)+2 (x_1^2+x_2^2)+1],\nonumber\\
{\Sigma}_{TL}&=&\mu \tau \delta x_2 [(6 x_1^2+1+2 x_2x_1-6 x_1) \nonumber\\
&+&2 \delta x_1 ((2 x_1^2+1) x_2+(2 x_2^2
+1) x_1-6x_1x_2) ], \nonumber\\
\Lambda_{TL}&=&- x_2[2\delta x_1 (2 x_2-1-2 x_1^2 x_2-2x_1x_2^2-2x_1x_2+2x_1)
\nonumber\\
&+&\mu \tau(2 x_1^2+1-2 x_2x_1+x_1)],\nonumber\\
{\Sigma}_{LT}&=&{\Sigma}_{TL}(x_1\leftrightarrow x_2),\nonumber\\
\Lambda_{LT}&=&\Lambda_{TL}(x_1\leftrightarrow x_2),\nonumber\\
{\Sigma}_{LL}&=&-2 \delta^2 x_1 x_2(3 \delta x_1 x_2+\mu \tau), \nonumber\\
\Lambda_{LL}&=&- \delta^2 x_1 x_2 (2 x_1 x_2+1).\nonumber
\end{eqnarray}
In the following, we shall fix the quark masses at
\begin{eqnarray}
m_u=m_d&=&0.3 {\ \rm GeV},\nonumber\\
m_s&=&0.5 {\ \rm GeV},\\
m_c&=&1.5 {\ \rm GeV},\nonumber\\
m_b&=&4.5 {\ \rm GeV}.\nonumber
\end{eqnarray}
and the quarks are included only above threshold $s=2\nu-P^2-Q^2>4m_q^2$.

\label{results}
\begin{figure}[ht]
\centering{\epsfig{file=stot.eps,width=6cm}
\epsfig{file=rho.eps,width=6cm}}
\caption{Fits to the total cross sections and to the $\rho$ parameter.}
\label{stot}
\end{figure}
\subsection{Results}
As we want to be able to vary the minimum value of $2\nu$,
and as the fits of \cite{compete} neither include the generalized dipole
nor use $2\nu$ as the energy variable,
we have refitted the $pp$ and $\bar pp$
cross sections and $\rho$ parameter together with those for $\gamma^{(*)} p$
and $\gamma^{(*)}\gamma^{(*)}$, and imposed factorization of the residues.
\begin{table}[ht]
\centering{\begin{tabular}{|l|c|l|c|}
\hline
\multicolumn{2}{|c|}{triple} & \multicolumn{2}{c|}{double} \\
\hline
Parameter & Value  & Parameter & Value  \\
\hline
\hline
$t_{pp}$           &0.6264 $\pm$0.0055&$\Lambda_{pp}$  &$1.36 \pm 0.15 $ \\
$log(d_{pp})$      &0.534  $\pm$0.044 &$D_{pp}$     & $40.3 \pm 1.4 $ \\
$t_{pp}\log c_{pp}$&65.86  $\pm$0.48  &$D_{pp}\log C_{pp}$&$-32.7\pm 5.3$\\
$Y^+_{pp}$  & 122.0  $\pm$  1.5       &$Y^+_{pp}$      &  $231.1 \pm 4.7 $ \\
$\alpha_+$         &0.6905$\pm$0.0023&$\alpha_+$   & $0.7263 \pm 0.0010$  \\
 $Y^-_{pp}$       &84.6 $\pm$4.1   &$Y^-_{pp}$   & $97.6 \pm 4.6 $ \\
$\alpha_-$        &0.4596$\pm$0.0010&$\alpha_-$   &  $0.505 \pm 0.015$  \\
\cline{1-1} \cline{2-2}
$c_0$ & $-613.93 \pm  0.91$  & $\delta$ &$0.3313 \pm 0.0092$  \\
\cline{3-3} \cline{4-4}
$c_1$ & 740.8 $\pm$1.2      &$C_1$        &  $-0.105 \pm 0.016$  \\
$Q^2_c$  & 0.1557    $\pm$0.0030 & $Q^{2}_{c}$    &   $0.0219 \pm 0.0076$  \\
$\epsilon_c$ & 0.11619   $\pm$ 0.00047 & $\epsilon_c$ &   $0.553 \pm 0.025$  \\
\hline
$t_1$ &0.001667$\pm$0.000011& $\Lambda_1$       &   $1.49     \pm 0.23  $ \\
$Q^2_t$   & 0.964 $\pm$  0.016 &$Q^2_{\lambda}$ &   $0.111    \pm  0.032$  \\
$\epsilon_t$ & 0.8237$\pm$  0.0034 &$\epsilon_{\lambda}$ &  $0.658 \pm 0.019$\\
\hline
$d_1$  & -8.067$\pm$0.033 & $D_1$        &   $0.1305 \pm 0.0062$  \\
$Q^2_d$         & 7.56   $\pm$0.25  & $Q^2_d$    &   $0.379     \pm 0.061$  \\
$\epsilon_d$ & 0.3081 $\pm$  0.0059 &$\epsilon_d$ & $0.434  \pm 0.021$  \\
\hline
$Y_1$  & 0.1961  $\pm$0.0031 & $Y_{1}$   & $0.515 \pm 0.017$  \\
$Q^2_y$ & 2.056  $\pm$  0.067 & $Q^{2}_{y}$  & $0.158  \pm 0.016$  \\
$\epsilon_y$ & 0.5448    $\pm$  0.0049 &$\epsilon_y$ & $0.709  \pm 0.016$  \\
\hline
\end{tabular}\\~\\}
\caption{Parameters (in natural units) of the global fits.}
\label{tab2}
\end{table}
\begin{figure}[ht]
{\epsfig{file=ft.eps,width=6cm}}{\epsfig{file=fd.eps,width=6cm}}
\caption{Form factors of the triple pole (left) and double pole (right)
 parametrisations.}
\label{tform}
\end{figure}

We show in Table~\ref{tab1} the $\chi^2/dof$ and number of points for each
process.
We see that one obtains a very good global $\chi^2$ for both models. It
is well known \cite{compete} that the partial $\chi^2$ for $\sigma_{p\bar p}$
and $\rho_{pp}$ never reach low values, presumably because of the presence
of contradictory data. We show the corresponding curves in Fig.~\ref{stot}.

The values of the parameters are given in Table \ref{tab2} for the triple-pole
and the double-pole cases, and the
form factors are plotted in Fig. \ref{tform}.

We see that the intercepts of the leading meson trajectories are close,
in fact closer than those of \cite{compete}. This is dues to the smaller
energy region, and to the much larger influence of photon data on
$\alpha_+$.

It may also be
noted, in the double-pole case, that the parameter $\delta$ is close to
the hard pomeron intercept of \cite{DoLa}.
At high $Q^2$, because the form
factor $\Lambda$ falls off, the logarithm starts looking like a power of
$2\nu$, and somehow mimics a simple pole. It may thus be thought of as a
unitarized version of the hard pomeron, which would in fact apply to
hard and soft scatterings.

In the triple-pole case, this is accomplished by a different mechanism: the
scale of the logarithm is a rapidly falling function of $Q^2$, and hence the
$\log^2$ term becomes relatively more important at high $Q^2$.
Interestingly, when one writes the triple-pole parametrisation as a function
of $x$ and $Q^2$, one obtains only very small powers (of the order of
0.1) of $Q^2$, which do not contain any higher twists, contrarily to
the soft pomeron of \cite{DoLa}.

\subsection{Total $\gamma p$ and $\gamma\gamma$ cross sections}
\label{total}
We see from Table \ref{tab1} that one obtains an excellent
$ \chi ^{2} $ for $ \sqrt{2\nu }>7 $ GeV, for a total of 62 points.
The curves are shown
in Fig. \ref{stot}. The fit can in fact be continued
to $ \sqrt{2\nu }=2 $ GeV, with a $ \chi ^{2} $/point of 0.74
for 219 points.

We have checked that adding the box diagram leads to a slight degradation of
the
fit, whether one fits the total cross sections alone or with all other data.
As the contribution of the box is calculated perturbatively,
one might object that one cannot use the result down to $ Q^{2}=0 $, and that
only the $\nu$ dependence should be kept.
Hence we have also tried to add an extra term,
proportional to $ \log \nu /\nu  $
in the total cross section, but found that the fit
prefers to set the proportionality constant to zero. Hence it seems
that this singularity is not needed at $P^2=Q^2=0$.
However, because of large uncertainties
in the data, it is not possible to rule it out altogether.

Similarly,
we do not find the need to introduce any new rising contribution.
However, it is clear in view of the large uncertainties that it is
not possible to rule out completely such a possibility. In fact, our
fit prefers the $\gamma\gamma$ data unfolded with PHOJET \cite{PHOJET},
which rise more slowly than those unfolded with PYTHIA \cite{PYTHIA}.
Interestingly, as we reproduce both HERA and LEP data, for $Q^2$
nonzero, it is not true that an extrapolation of the nonzero $Q^2$ data
leads to a higher estimate of the $\gamma p$ and $\gamma\gamma$
cross sections. Our fit
can be considered as an explicit example for which such an
extrapolation leads to a cross section on the lower side of the
experimental errors.
\begin{figure}[ht]
\epsfig{file=lowq2.eps,width=13cm}
\caption{Fits to $F_2^p$ in the low $Q^2$ region. We show only
graphs for which there are more than 6 experimental points, as well as the
lowest $Q^2$ ones. The curves are as in Fig.~\ref{stot} and the data as
in Fig.~\ref{highq2}.}
\label{lowq2}
\end{figure}
\begin{figure}[ht]
\epsfig{file=highq2.eps,width=13cm}
\caption{Fits to $F_2^p$ in the high  $Q^2$ region. We show only
graphs for which there are more than 6 experimental points.
The curves are as in Fig.~\ref{stot}.
}
\label{highq2}
\end{figure}

\subsection{$F_2^p$}
\label{f2p}
The fit to $F_2$ has quite a good $\chi^2$ as well. We have checked that one
can easily extend it to $Q^2\approx 400$ GeV$^2$ for the triple pole, and to
$Q^2\approx 800$ GeV$^2$ in the double-pole case. It is interesting
that one cannot go as high as in ref. \cite{CS}.
This can be attributed either to too simple a choice for
the form factors, or more probably to the onset of perturbative evolution.

Figs. \ref{lowq2} and \ref{highq2} show the $F_2^p$ fit for
the most populated $Q^2$ bins. As pointed out before, we see that
our fits do reproduce the low-$Q^2$ region quite well, but predict
total cross sections on the lower side of the error bands. Hence
the extrapolation to $Q^2=0$ of DIS data does not require a hard pomeron.


\subsubsection{Fits to $F_2^\gamma$}
\begin{figure}[ht]
\epsfig{file=gamma.eps,width=13cm}
\caption{Fits to $F_2^\gamma$. The curves are as in Fig.~\ref{stot}.
The data are from \cite{L3,OPAL}.}
\label{gamma}
\end{figure}
\begin{figure}
\centering{\epsfig{file=gsame.eps}}
\caption{Fits to $F_2^\gamma$ for $P^2=Q^2$. The curves are as in Fig.~\ref{stot}.
The data are from \cite{L3}.}
\label{gsame}
\end{figure}
\begin{figure}
\centering{\epsfig{file=g37120.eps}}
\caption{Fits to $F_2^\gamma$ for nonzero asymmetric values of $P^2$ and
$Q^2$. The curves are as in Fig.~\ref{stot}.
The data are from \cite{L3}.}
\label{g37120}
\end{figure}
\label{f2g}
As the number of data points is dominated by $pp$ and $\gamma p$ data, the
fit to $\gamma\gamma$ data is really a test of the $t$CU relations. As we
explained
above, the strongest constraint comes from the positivity of the $\gamma^*
\gamma^*$ cross section, which is not guaranteed by the $t$CU relations in the
case
of multiple poles. As Fig. \ref{gamma}, \ref{gsame} and \ref{g37120} show, one
obtains a good description of the points within the Regge region.

Here, we have observed that the quality of the fit improves if
we add the box diagram for nonzero $Q^2$ and $P^2$.
There is no need however to include
other singularities, such as a hard pomeron or a perturbative one.

For $Q^2\neq 0$ and $P^2=0$, the box diagram makes little difference in
the double-pole case, but does reduce the $\chi^2$ appreciably in the
triple-pole case. We have included it in both cases. If one believes
in the existence of the $S$ matrix for QED, this means that it should
enter Eq.~(\ref{Master2}) as $D(Q_{in},0)$, and hence be present in $F_2^p$
as well. However, its contribution there is suppressed by the electromagnetic
coupling and by the fact that it falls with $\nu$.
It is at present undetectable, and the question
of the existence of the $t$CU relations (\ref{Master2}) remains open.

\section{Conclusion}
\label{conclu}
We have shown in this paper that $t$-channel unitarity can be used to
map the regions where new singularities, be they of perturbative
or non-perturbative origin, can occur. Indeed, we have seen that
although hadronic singularities must be universal, this is certainly
not the case for $F_2^p$ and $F_2^\gamma$, as DIS involves off-shell particles.

We have shown however that up to\footnote{
The region we have considered excludes the highest-$Q^2$ $\gamma\gamma^*$
points from OPAL. For the point which falls in the Regge region,
at $P^2=0$, $Q^2=780$ GeV$^2$and $x=0.275$,
the experimental value is $0.93\pm 0.16$,
the extrapolation of the double-pole fit predicts $0.71$,
while the triple pole prediction is $0.74$.
} $Q^2=150$ GeV$^2$, the data do not call for the existence of new
singularities, except perhaps
the box diagram. In the case of total cross sections, this
suggests that it is indeed possible to define an $S$ matrix for QED.

For
off-shell photons, our fits are rather surprising as the standard claim is that
the perturbative evolution sets in quite early.
This evolution is indeed allowed by $t$-channel unitarity constraints: it is
possible to have extra singularities in off-shell photon cross sections, which
are built on top of the non-perturbative singularities. But it seems that Regge
parametrisations can be extended quite
high in $Q^2$ without the need for these new singularities.

Finally, the BFKL singularity can be purely perturbative: the position of
the singularity and the form factor come from pQCD. As such, it can
manifest itself only in $\gamma^*\gamma^*$, but we have seen that there
is no definite need for such a singularity in present data.

\section[*]{Acknowledgments} J.R.C. acknowledges the contribution
of P.V. Landshoff
who initiated this research by suggesting the possibility of extra
singularities from $t$-channel unitarity, E.M. and G.S. are
supported by the Fonds National pour la Recherche Scientifique, Belgium

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