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%                         FIGURES 
%
%\label{fig:diag}           diag_es3.eps*  [SAMIM.UA8.SE_PAPER]diag.kumac
%\label{fig:elastic}     elastic_es3.eps*  [SAMIM.UA8.SE_PAPER]factordiag.kumac
%\label{fig:zeus}           zeus_es3.eps*  [SAMIM.UA8.SE_PAPER]plotzeus.kumac
%\label{fig:h1}               h1_es3.eps*  [SAMIM.UA8.SE_PAPER]ploth1.kumac 
%\label{fig:apvseps}     apvseps_es3.eps*  [SAMIM.UA8.SE_PAPER]apvseps.kumac
%\label{fig:epsvsq2}     epsvsq2_es3.eps*  [SAMIM.UA8.SE_PAPER]ploteps.kumac
%\label{fig:ratio}         ratio_es3.eps*  [SAMIM.UA8.SE_PAPER]f2d2f.kumac
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\newcommand{\x}{\cdot}
\newcommand{\ra}{\rightarrow}
\newcommand{\pom}{\mbox{${\rm \cal P}$omeron}}
\newcommand{\flux}{\mbox{$F_{{\cal P}/p}(t, \xi)$}}
\newcommand{\fluxpap}{\mbox{$F_{{\cal P}/p}^{\bar{p}p}(t, \xi)$}}
\newcommand{\fluxgmp}{\mbox{$F_{{\cal P}/p}^{ep}(t, \xi)$}}
\newcommand{\ap}{\mbox{$\bar{p}$}}
\newcommand{\pap}{\mbox{$\bar{p} p$}}
\newcommand{\SPS}{\mbox{S\pap S}}
\newcommand{\xp}{\mbox{$x_{p}$}}
\newcommand{\sumet}{\mbox{$\Sigma E_t$}}
\newcommand{\mpr}{\mbox{${m_p}$}}
\newcommand{\mpi}{\mbox{${m_\pi}$}}
\newcommand{\rs}{\mbox{$\sqrt{s}$}}
\newcommand{\rsp}{\mbox{$\sqrt{s'}$}}
\newcommand{\rsps}{\mbox{$\sqrt{s} = 630 $ GeV}}
\newcommand{\lum}{\mbox{$\int {\cal L} {dt}$}}
\newcommand{\T}{\mbox{$t$}}
\newcommand{\abt}{\mbox{${|t|}$}}
\newcommand{\di}{\mbox{d}}
\newcommand{\HS}{\mbox{$xG(x)=6x(1-x)^1$}}
\newcommand{\sigdifjets}{\mbox{$\sigma_{sd}^{jets}$}}
\newcommand{\sigpomjets}{\mbox{$\sigma_{{\cal P}p}^{jets}$}}
\newcommand{\sigdiftot}{\mbox{$\sigma_{\bar{p} p}^{\rm tot \, diff}$}}
\newcommand{\sigpomtot}{\mbox{$\sigma_{p {\cal P}}^{\rm tot}$}}
\newcommand{\sigpompom}{\mbox{$\sigma_{{\cal P} {\cal P}}^{\rm tot}$}}
\newcommand{\sigpptot}{\mbox{$\sigma_{pp}^{\rm tot}$}}
\newcommand{\sigpomzero}{\mbox{$\sigma_{{\cal P}p}^o$}}
\newcommand{\dsig}{\mbox{${d^2 \sigma }\over{d \xi dt}$}}
\newcommand{\alamb}{\mbox{$\overline{\Lambda^{\circ}}$}}
\newcommand{\lamb}{\mbox{$\Lambda^{\circ}$}} 
\newcommand{\peetee}{\mbox{${ p_t}$}}
\newcommand{\PRET}{\mbox{\Proton-\sumet}}
%
%C new symbols for erhan-schlein II paper -------------------------
%\newcommand{\xpom}       {\mbox{$x_{pom}$}}
\newcommand{\xpom}  {\mbox{$x_{I \! \! P}$}} 
\newcommand{\pmm}         {\mbox{${\cal P}$}}
\newcommand{\gm}         {\mbox{$\gamma^{*}$}}
\newcommand{\gmp}         {\mbox{$\gamma^{*} p$}}
\newcommand{\siggp}      {\mbox{$\sigma_{\gamma^{*} p}^{\rm tot}$}}
\newcommand{\siggpm}     {\mbox{$\sigma_{\gamma^{*} {\cal P}}^{\rm tot}$}}
\newcommand{\FtwoDtwo}       {\mbox{$F_2^{D(2)}$}}
\newcommand{\FtwoDthree}       {\mbox{$F_2^{D(3)}$}}
\newcommand{\xpomFtwo}     {\mbox{$\xi \FtwoDthree$}}
\newcommand{\qsq}        {\mbox{$Q^2$}}
\newcommand{\mx}         {\mbox{$M_X$}}
\newcommand{\mxsq}       {\mbox{$M_X^2$}}
\newcommand{\w}          {\mbox{$W$}}
\newcommand{\wsq}        {\mbox{$W^2$}}
\newcommand{\fluxint}    {\mbox{$f_{{\cal P}/p}(\xi)$}}
\newcommand{\eps}        {\mbox{$\epsilon$}}
\newcommand{\alf}        {\mbox{$\alpha '$}}
%
\begin{titlepage}
\vspace{4cm}
\begin{flushright}  
{31 January, 2003}
\end{flushright}
%
\vspace{4ex}
% 
%
\begin{center}
\LARGE
{\bf {\boldmath A new $\gamma ^{*} p/\pap$ factorization test in}}\\
{\bf {\boldmath diffraction, valid below $\qsq \sim 6$~GeV$^2$}}\\
\normalsize
\vspace{9 ex}
Samim Erhan and Peter Schlein \\
\vspace{3.0mm}
University of California$^{*}$, Los Angeles, California 90095, USA. \\
\end{center}
%
\vspace{11 ex}
% 
\begin{abstract}

One of the key experimental issues in \pom\ physics is the extent 
to which factorization of \pom\ emission and interaction is valid in
diffractive processes.
In the present paper, we present the results of a new test in 
diffractive $\gm p$ and \pap\ interactions, which does not rely on
the assumption of universality of the \pom\ flux factor in the proton.
The test is satisfied 
to within $\sim 20\%$ for $1 < \qsq \sim 6$~GeV$^2$ and $\beta < 0.4$ in the 
$\gm p$ interactions, suggesting that multi--\pom --exchange has a limited 
effect on factorization.
However, a clear breakdown is observed at larger \qsq .
Kharzeev and Levin suggest that this can be 
attributed to the onset of perturbative QCD effects due to the \pom 's 
structure. 
The breakdown occurs in a \qsq\ region which agrees with their estimates of a 
small \pom\ size.

\end{abstract} 
%
\vspace{2 ex}
\begin{center}
submitted to European Physical Journal C\\
\end{center}
\vspace{15 ex}
\rule[.5ex]{16cm}{.02cm}
$^{*}$\ Supported by U.S. National Science Foundation Grant \\
\end{titlepage}
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\section{Introduction}
\label{sect:intro}
\indent

Studies of the inclusive inelastic production of beam-like particles
with momenta within a few percent of the associated incident beam 
momentum, as in:
%
\begin{equation}
\ap \, \,  \, \, + \, \, p_i \, \,  \, \, \ra \, \, \, \, X \, \, 
+ \, \, p_f 
\label{eq:dif}
\end{equation}
%
\begin{equation}
\gamma^{*} \, \, + \, \, p_i \, \,  \, \, \ra \, \, \, \, X \, \, 
+ \, \, p_f 
\label{eq:gams}
\end{equation}
%
have led to the development of a rich Regge 
phenomenology \cite{collins,basicphen,is}
of these ``single-diffraction"
\pom\ exchange processes illustrated in Fig.~\ref{fig:diag}.
The observed final--state proton momentum
reflects the exchanged \pom 's momentum fraction
in the proton\footnote{We use the symbol $\xi$ for this variable in view
of its simplicity and its increasing use in the literature.}, 
$\xi \equiv x_{I \! \! P} = 1 - \xp$, and momentum transfer, \T . 

One of the relatively recent ideas~\cite{is} 
underlying the phenomenology is that, although
the \pom 's existence in the proton is due to non-perturbative
QCD, once the \pom\ exists, perturbative QCD processes can occur
in the proton-\pom\ and $\gamma^{*}$-\pom\ interactions
of Reacts.~\ref{eq:dif} and~\ref{eq:gams}, respectively. 
Ref.~\cite{is} proposed the study of such hard processes 
in Reacts.~\ref{eq:dif} and~\ref{eq:gams}
in order to determine the \pom 's structure.
Hard diffraction scattering was discovered in \pap\ interactions by
the UA8 experiment \cite{ua8} at the \SPS --Collider and in $ep$ 
interactions by the ZEUS \cite{zeusfirst} and H1 \cite{h1first} 
experiments at HERA.

All available inclusive diffractive data from Reacts.~\ref{eq:dif} 
and~\ref{eq:gams} are well described \cite{ua8dif,zeus94,h194} 
by expressing the observed single-diffraction differential cross section as a
product of factors describing \pom\ emission and \pom\ interaction,
for example a \pom\ flux factor in the proton, \flux , 
and a proton--\pom\ or \gm --\pom\ total cross section, respectively.
%
\begin{equation}
{{d^2 \sigma_{\bar{p} p}^{\rm diff}}\over{d \xi dt}} \, \, 
= \, \,  \fluxpap \, \, \x \, \, \sigpomtot(s')
\label{eq:factorhad}
\end{equation}
%
\begin{equation}
{{d^2 \sigma_{\gamma^{*} p}^{\rm diff}}\over{d \xi dt}} \, \, 
= \, \,  \fluxgmp \, \, \x \, \, \siggpm(s', \qsq )
\label{eq:factorgam}
\end{equation}
%
\qsq\ is the momentum transfer
of the \gm\ in React.~\ref{eq:gams}.
$s'$ is the squared invariant mass of the $X$ systems in 
Reacts.~\ref{eq:dif} and~\ref{eq:gams}. To good approximation,
$s' = \xi s$ in React.~\ref{eq:dif} and $s' = \xi W^2 -\qsq$ in
React.~\ref{eq:gams} (see Fig.~\ref{fig:diag}). 

There is, however, one complicating issue in the successful description
of the data by Eqs.~\ref{eq:factorhad} and \ref{eq:factorgam}.
The empirical \pom\ flux factor, \flux , is not universal in the two 
equations. 
More specifically, the efffective \pom\ Regge trajectory in the common
factor, $\flux \sim \xi^{1-2\alpha (t)}$, required to fit the data
is different in React.~\ref{eq:dif} and \ref{eq:gams}.
The ZEUS \cite{zeus94} and H1 \cite{h194} collaborations both demonstrated
that the  effective \pom\ trajectory at low--$|t|$  in 
React.~\ref{eq:gams} lies above the effective trajectory which 
characterizes React.~\ref{eq:dif} (the evidence for this is shown below in 
Sect.~\ref{sect:gamphenom}). 
This remarkable situation
is presumably a consequence of different non-perturbative \pom\
formation processes in Reacts.~\ref{eq:dif} and \ref{eq:gams}. 
Furthermore, the $s$--dependence of multi--\pom --exchange in 
React.~\ref{eq:dif} causes
the empirical effective \pom\ trajectory to decrease \cite{es2} with increasing 
$s$, as predicted by Kaidalov et al.\ \cite{kaidpomt}.
Only below $s \sim 550$~GeV$^2$ is the effective \pom\ trajectory
equal to the trajectory which describes the $s$-dependence of the
$pp$ and $p\ap$ total cross sections \cite{cudell,dino2}. 
. 

In the present paper, we explore the extent to which factorization of 
\pom\ formation and interaction in Eqs.~\ref{eq:factorhad} and 
\ref{eq:factorgam} survives the non-universality of $F_{{\cal P}/p}$,
whereby there are different non-perturbative effects of \pom\ formation
in the two reactions.
Such a scenario could place constraints on the types of
multi-\pom -exchange processes that participate in Reacts.~\ref{eq:dif} 
and \ref{eq:gams}.

We propose to test this type of factorization 
by asking if the \pom --exchange components of the
extracted $\gamma ^*$-\pom\ and $p$-\pom\ total cross 
sections satisfy the relationship:
%
\begin{equation}
{{\siggpm}\over{\siggp}} \, \, = \, \, {{\sigpomtot}\over{\sigpptot}},
\label{eq:ratiosig}
\end{equation}
%
where all four cross sections are evaluated at the same value of 
$s$~\cite{ryskin}.
Equation~\ref{eq:ratiosig} is obtained from the optical theorem and the 
ratios of the forward elastic amplitudes shown in Fig.~\ref{fig:elastic}.
In practice, since Eqs.~\ref{eq:factorhad} and~\ref{eq:factorgam} show
that only the product of the \flux\ normalization constant, $K$, and the
cross section are experimentally accessible in each case,
the form of Eq.~\ref{eq:ratiosig} to be compared with data is actually:
%
\begin{equation}
{{K_{ep} \, \siggpm}\over{\siggp}} \, \, = \, \, 
{{K_{pp} \, \sigpomtot}\over{\sigpptot}},
\label{eq:Kratio}
\end{equation}
%
$K$ is labeled differently on the left and right sides of Eq.~\ref{eq:Kratio}, 
to allow for the possibility that they may be different. 
This point is discussed further in Sect.~\ref{sect:factor}. 

In Sect.~\ref{sect:difphenom} we review the existing phenomenological analyses 
of React.~\ref{eq:dif} in terms of Eq.~\ref{eq:factorhad}
and extract the right-hand side of Eq.~\ref{eq:Kratio}.
In Sect.~\ref{sect:gamphenom} the HERA diffractive data
on React.~\ref{eq:gams} is reanalyzed and the left-hand side of 
Eq.~\ref{eq:Kratio} is extracted from the data.
In Sect.~\ref{sect:factor}, the factorization test is carried out 
using Eq.~\ref{eq:Kratio}. 
Our conclusions are given in Sect.~\ref{sect:conclude}.

\section{Analysis of diffractive {\boldmath $\ap p$} and {\boldmath $pp$} data}
\label{sect:difphenom}
\indent

The UA8 collaboration \cite{ua8dif} fit Eq.~\ref{eq:factorhad} to the joint 
$\xi$-$t$ distributions of the available data on React.~\ref{eq:dif} 
at the \SPS\ ($\rs = 630$~GeV and $1.0 < |t| < 2.0$~GeV$^2$) 
and the corresponding $pp$ data at the ISR~\cite{albrowisr} 
($\rs = 23.5, 30.5$~GeV and $|t| < 2.0$~GeV$^2$),
all with $\xi < 0.09$.
They obtained parametrizations of \flux\ and \sigpomtot\ 
which embody some features not previously  known:
 
\begin{equation}
\flux = K \cdot |F_1(t)|^2 \cdot e^{(1.1 \pm 0.2)t} \cdot \xi^{1-2\alpha (t)} 
\label{eq:flux}
\end{equation}

\begin{equation}
\alpha(t) = 1 + \epsilon + \alpha ' t + \alpha '' t^2 = 
1.10 + 0.25 t + (0.079 \pm 0.012) t^2
\label{eq:traj}
\end{equation}

\begin{equation}
K_{pp} \, \sigpomtot (s') \, \, = 
\, \, (0.72 \pm 0.10) \cdot [(s')^{0.10} + 
(4.0 \pm 0.6) (s')^{-0.32}]\,\,\,\,\, {\rm mb \, GeV^{-2}}. 
\label{eq:spomprot}
\end{equation}

The fits show that 
the effective \pom\ Regge trajectory flattens in the domain, $1.0 < |t| < 
2.0$~GeV$^2$, as described by
by the quadratic term in Eq.~\ref{eq:traj}, when $\epsilon$ and $\alpha '$
are fixed at 0.10 and 0.25 GeV$^{-2}$, respectively.
In Eq.~\ref{eq:spomprot}, with the 
exponents\footnote{In this formula and others like it, 
``$s'$" stands for ``$s'/s_0"$, where $s_0$ = 1~GeV$^2$.} 
of $s' = \xi s$ fixed at 0.10 and -0.32, respectively,
$K_{pp} \sigpomtot (s')$ requires the presence of  
both \pom -- and Reggeon--exchange terms, as shown. 
Ref.~\cite{ua8dif} confirms the flattening of the effective 
trajectory at larger $|t|$ values, as well as the presence of the 
Reggeon--exchange term in Eq.~\ref{eq:spomprot}, by fitting the 
resolution--smeared $\xi$--dependences at fixed $t$ values 
when\footnote{The sensitivity of fitting at small $\xi$ comes from the fact 
that, at small momentum transfer, the rapid increase of 
$\xi^{1-2\alpha (0)} \sim 1/\xi^{1+2\epsilon}$ dominates the relatively 
weak dependence of \sigpomtot\ on $\xi$ (via $s' = \xi s$).} 
$\xi < 0.03$.
Finally, with  $|F_1(t)|^2$ in Eq.~\ref{eq:flux} set equal to the 
Donnachie-Landshoff \cite{dl1} form factor\footnote{$F_1(t)={{4 m_p ^2 - 2.8t}
\over{4 m_p ^2 - t}}\, \x \, {1\over{(1-t/0.71)^2}}$}, the additional
exponential factor is required.

A description of the phenomenology of React.~\ref{eq:dif} is incomplete without 
inclusion of the explicit effects of multi--\pom --exchange.
It has been widely known for some time that the observed $s$--dependence
of the total single--diffractive cross section, \sigdiftot , is not described by 
the $t$--integrated Eq.~\ref{eq:factorhad} (for $\xi < 0.05$) with a fixed
\pom\ Regge trajectory. Such a calculated \sigdiftot\ rises rapidly with
energy and soon violates unitarity, while the observed \sigdiftot\ tends to
level off or plateau at high energy \cite{dino,es1}. 
Since there is no built-in mechanism in the single-\pom -exchange 
process of Fig.~\ref{fig:diag} to account for the plateauing of \sigdiftot , 
there have been continuing theoretical efforts to satisfy $s$--channel 
unitarity \cite{unitarity};
the flattening effect is attributed to multiple--\pom --exchange and 
is referred to variously in the literature
as damping, screening, shadowing or absorption.
Kaidalov et al.\ \cite{kaidpomt} showed that multi--\pom --exchange diagrams 
lead to an effective \pom\ trajectory whose $t = 0$ intercept decreases with 
increasing energy.

In order to quantitatively assess these effects, Ref.~\cite{es2} performed fits 
of the $t$--integrated Eq.~\ref{eq:factorhad} to the observed $d\sigma /dt$   
with $\xi < 0.05$ of all available ISR \cite{isrdsdt} and \SPS\ 
\cite{ua8dif,ua4dsdt} data.
In fitting to the complete set of ISR $d\sigma/dt$ data
over the energy range, $s$ = 549 to 3840 GeV$^2$, the only free parameters
in Eqs.~\ref{eq:flux}, \ref{eq:traj} and \ref{eq:spomprot} were those 
in the effective \pom\ trajectory, each of which was assumed to have a 
simple $s$--dependence. Their fit results are:
%
\begin{tabbing}
\hspace{3cm}\=$\epsilon  (s)$ \hspace{6mm}\= =\hspace{4mm}\=
$(0.096 \pm 0.004) - (0.019 \pm 0.005) \cdot log (s/549)$.\\ 
\>$\alpha '  (s)$ \> = \> $(0.215 \pm 0.011) - (0.031 \pm 0.012) \cdot 
log (s/549)$.\\ 
\>$\alpha '' (s)$ \> = \> $(0.064 \pm 0.006) - (0.010 \pm 0.006) \cdot 
log (s/549)$.\\
\end{tabbing}
%
At the lowest ISR energy, $s = 549$~GeV$^2$,  $\epsilon = 0.096$, 
$\alpha ' = 0.215$~GeV$^{-2}$ and $\alpha '' = 0.064$~GeV$^{-4}$, while each
of these is seen to decrease with increasing energy.
This is consistent with fixing $\epsilon = 0.10$ and $\alpha ' = 0.25$ 
in the fits of Ref.~\cite{ua8dif}, since
the only low--$|t|$ data in those fits were at the lowest ISR energies.

Ref.~\cite{es2} finds that the effective \pom\ trajectory continues
to decrease at higher energy. 
At the \SPS , (\rs\ = 630~GeV), the effective trajectory is:

\begin{center}
$\alpha(t) \, = \, 1 + \epsilon + \alpha' t + \alpha'' t^2 \, = \, 
1.035 + 0.165 t + 0.059 t^2$
\end{center}
%
This form is also seen \cite{es2} to be consistent with the published functional 
form that is used to describe data at the Tevatron.

Although damping effects appear to be mainly in the low--$\xi$, low--$|t|$ 
region \cite{es2,es1}, where most of the cross section is, there is no 
evidence for damping in the momentum transfer range, $1 < |t| < 2$~GeV$^2$. 
That is, the effective $\alpha (t)$ exhibits no $s$--dependence in this range 
of \T\ and the average trajectory value at $|t| = 1.5$~GeV$^2$ is~\cite{es2} 
$\alpha(t) = 0.92 \pm 0.03$.  
It is therefore interesting to note that both data sets used in the fits of 
Ref.~\cite{ua8dif} are in kinematic regions where damping effects in 
React.~\ref{eq:dif} seem to be smallest.

To prepare for the factorization test of Eq.~\ref{eq:Kratio}, we need to
evaluate its right--hand--side.
Its numerator is given by Eq.~\ref{eq:spomprot},
while its denominator is the $pp$ total cross section,
which we take from the fits of Refs.~\cite{cudell,dino2}:
%
\begin{equation}
\sigpptot =   18 \, s^{0.10} - 27 \, s^{-0.50} 
+ 55 \, s^{-0.32}\,\,\,\,\,{\rm mb}.
\label{eq:Ksigpp}
\end{equation}
%
Since we are interested only in the \pom\ exchange terms in
Eqs.~\ref{eq:spomprot} and~\ref{eq:Ksigpp}, we drop the Reggeon-exchange
terms in both numerator and denominator.
The right-hand-side of Eq.~\ref{eq:Kratio} is then given by:
%
\begin{equation}
{{K_{pp} \, \sigpomtot}\over{\sigpptot}} 
\, \, \, \, = \, \, \, \, 0.041 \pm 0.007 \, \, {\rm GeV}^{-2}.  
\label{eq:ratioa}
\end{equation}

\section{Analysis of diffractive {\boldmath $\gamma ^{*}p$} data}
\label{sect:gamphenom}
\indent

In order to carry out the factorization tests, we first reanalyze
HERA $ep$ diffractive data samples. 
The diffractive structure function, $\xi \FtwoDthree$, for the 
ZEUS 1994 data~\cite{zeus94} is displayed in Fig.~\ref{fig:zeus} 
and for the H1 1994 data~\cite{h194} in Fig.~\ref{fig:h1}.  
In both cases, the recoil proton is not detected and the data are thus
integrated over momentum transfer, \T .

For the \T -integrated data, the diffractive structure 
function \cite{ingelpry}, \FtwoDthree , is defined in terms of
the diffractive $\gamma^{*}$--proton differential
cross section by:
%
\begin{equation}
{{d\sigma_{\gamma^{*}p}^{diff}}\over{d \xi}} \, \, = \, \,
{{4 \pi^2 \alpha}\over{\qsq }} \cdot \FtwoDthree (\beta, \qsq , \xi)
\label{eq:f2d3}
\end{equation}
%
where, as noted earlier, we use the variable, $\xi \equiv \xpom$.
Eq.~\ref{eq:f2d3} is obtained from Eq.~8 of Ref.~\cite{zeus94}.

As in Ref.~\cite{is}, \FtwoDthree\ is written in factorized form, as the 
product of a \pom\ flux factor (in this case, $t$--integrated) 
and a \pom\ structure function, \FtwoDtwo :
%
\begin{equation}
\FtwoDthree(\beta, \qsq , \xi) =  
\int \fluxgmp dt \, \cdot \, \FtwoDtwo (\beta, \qsq ) \, \, \approx
\, \, {{K_{ep}}\over{\xi ^{1 + 2\eps} \cdot (3.9 - 2 \alf \, ln \, \xi)}}
\cdot \FtwoDtwo (\beta, \qsq ).
\label{eq:factorize1}
\end{equation}
%
or:
%
\begin{equation}
\xi \FtwoDthree(\beta, \qsq , \xi) \, \, \,  = \, \, \, 
{{K_{ep} \, \FtwoDtwo (\beta, \qsq )}\over
   {\xi ^{2\eps} \cdot (3.9 - 2 \alf \, ln \, \xi)}}.
\label{eq:factorize2}
\end{equation}
%
This approximate form of the $t$--integrated flux integral arises from assuming
$e^{3.9 t}\xi^{1-2\alpha(t)}$ for the functional form of \flux .
3.9 is the value which makes the integral equal to that of the $|t|$--integral 
of the full \flux\ in Eq.~\ref{eq:flux} when $\alpha ' = 0.25$ is 
used\footnote{This constant decreases to 3.7 and 3.5, for $\alpha$ = 0.15 and 
0.05, respectively.}.

Fig.~\ref{fig:zeus} shows the fits of Eq.~\ref{eq:factorize2} to the ZEUS 
data.
The free parameters are \eps , \alf\ and an independent 
$K \FtwoDtwo$ at each of the twelve \qsq\ and \mx\ combinations. 
These fits, and those made to the H1 data in Fig.~\ref{fig:h1},
confirm factorization of \pom\ production and 
interaction in the diffractive \gmp\ interactions. 
In the domain, $\xi < 0.01$, where Reggeon exchange can be ignored at all 
$\beta$, 
all the observed dependence on $\xi$ is described by the flux factor
in Eq.~\ref{eq:factorize2}.
Although this factorization has been known for some time from the H1 and ZEUS 
experiments, it is perhaps not widely recognized how remarkable it is.

From the fits to the ZEUS data, Fig.~\ref{fig:apvseps} shows the results when we
fix \alf\ at the series of four values shown and determine \eps\ and the twelve
normalization constants in Fig.~\ref{fig:zeus}. 
Fig.~\ref{fig:apvseps} shows a 1$\sigma$ error ``band" of allowed
\alf\ and \eps\ values. 
All points along the valley of the coutour are equally acceptable as 
solutions and there is 
insignificant discrimination between them with the present
data. Although we assume $\alpha ' = +0.25$ for the factorization analysis
in this paper (and for the curves shown in Figs.~\ref{fig:zeus}
and \ref{fig:h1}), 
we note that if the true effective $\alpha '$ were
as small as +0.15, the final ratios used in the factorization analysis 
only change by about 10\% and do not effect our conclusions.

We note in Fig.~\ref{fig:apvseps} that the band of allowed values for the
$\gamma^{*}$--p interactions
is seen to be {\it inconsistent} with the conventional hadronic
``soft-\pom " effective Regge trajectory parameters \cite{cudell,dino2,dl1}, 
\alf\ = 0.25 and \eps\ = 0.10 which characterize React.~\ref{eq:dif}.
This had been noted earlier by both ZEUS \cite{zeus94} and H1 \cite{h194},
but neither pointed out that there is a band of possible solutions
when $t$--integrated data is used. 
Fig.~\ref{fig:epsvsq2} shows the fitted values of $\epsilon$ vs.\ \qsq\ for a 
fixed $\alpha ' = 0.25$. The ZEUS points correspond to simultaneous fits to the 
three distributions at each \qsq\ shown in Fig.~\ref{fig:zeus}. 
The H1 points are from combined fits made to the distributions with $\xi < 10^{-
2}$, at each two neighboring \qsq\ values (4.5 and 7.5, 9 and 12, etc.) shown in 
Fig.~\ref{fig:h1}.

To express the numerator on the left-hand-side of Eq.~\ref{eq:Kratio} 
in terms of the measured structure function, a comparison
of Eqs.~\ref{eq:f2d3} 
and \ref{eq:factorize1} with Eq.~\ref{eq:factorgam} yields:
%
\begin{equation}
K_{ep} \, \siggpm (s', \qsq ) \, \, = \, \, 
{{4 \pi^2 \alpha}\over{\qsq }} \cdot K_{ep} \, \FtwoDtwo (\beta, \qsq )
\label{eq:numer}
\end{equation}
%
where:
%
\begin{equation}
s' = \qsq (1-\beta)/\beta
\label{eq:sp}
\end{equation}

The denominator on the left-hand-side of Eq.~\ref{eq:Kratio}, \siggp , is given
in terms of the $F_2$ structure function by essentially
the same relation, but with an additional factor containing a large--$x$ 
correction \cite{allm}. 
%
\begin{equation}
\siggp (W^2, \qsq ) \, \, = \, \, {{4 \pi^2 \alpha}\over{\qsq }} \cdot
 {{\qsq + 4 m_p^2 x^2}\over{\qsq (1 - x)}} \cdot F_2 (W^2, \qsq )
\label{eq:denom}
\end{equation}
%
where, because both \siggpm\ and \siggp\ are evaluated at the same CM 
energy \cite{ryskin}, $F_2$ uses 
$x = \qsq /(W^2 + \qsq - m_p^2)$ with $W^2 = s'$.
Then, combining Eqs.~\ref{eq:numer} and~\ref{eq:denom},
we have:
%
\begin{equation}
Ratio \, \, \equiv \, \, {{K_{ep} \, \siggpm}\over{\siggp}} \, \, =
\, \, {{\qsq (1-x)}\over{\qsq + 4 m_p^2 x^2}}
\, \, \cdot \, \, {{K_{ep} \, \FtwoDtwo(\beta, \qsq )}\over{F_2 (x, \qsq )}} 
\label{eq:ratiof}
\end{equation}

\section{Factorization test}
\label{sect:factor}
\indent

Fig.~\ref{fig:ratio} shows the $Ratio$ defined in Eq.~\ref{eq:ratiof}
evaluated vs.\ \qsq\ for three different H1 data samples at $\beta$~=~0.04, 
0.10, 0.20 and 0.40. 
For all points, the denominator, $F_2$, in Eq.~\ref{eq:ratiof}
was calculated using parameterizations obtained by fitting real $F_2$ data;
Ref.~\cite{dl_F2} for $x < 0.07$ and Ref.~\cite{h1_F2} for $x > 0.07$.

The solid-square points with $4.5 < \qsq < 28$~GeV$^2$ are calculated using the 
1994 H1 data \cite{h194}.
We see in the figure that these points are in reasonable agreement 
with the factorization prediction in Eq.~\ref{eq:ratioa}
at the lower end of their \qsq\ range (4.5 and 7.5 GeV$^2$) and $\beta < 0.4$,
while for \qsq\ larger than 6 or 7 GeV$^2$, there is a clear 
divergence from agreement with the prediction.
The points plotted in Fig.~\ref{fig:ratio} with solid triangles at lower \qsq\ 
values, $0.8 < \qsq < 5$~GeV$^2$, are preliminary results from the 1995 H1 
data \cite{h1vanc}. 
Those plotted with open circles in the range, $2.5 < \qsq < 12$~GeV$^2$,
are preliminary results from the 1999 H1 data \cite{h1amst}. 
The points are seen to be 
in reasonable agreement in the \qsq\ domains where they overlap.
Eq.~\ref{eq:Kratio}, seems to be satisfied to within 20\% below
$\qsq \sim 6$ or 7~GeV$^2$ and $\beta < 0.4$.

As discussed in the following section, the breakdown of factorization
for \qsq\ above about 6 GeV$^2$ can be attributed to the onset of perturbative 
QCD efects on a small \pom . This is agreement with the magnitude calculated by 
Kharzeev and Levin \cite{kharzeev}.
However, several caveats concerning the results in Fig.~\ref{fig:ratio}
should be noted.
%
\begin{enumerate}
\item The factorization prediction of Eq.~\ref{eq:Kratio}
is only valid for its \pom -exchange components. 
Although we have these for the right-hand-side of the equation as shown
in Eq.~\ref{eq:ratioa} and the bands in Fig.~\ref{fig:ratio}, we are
presently unable to know the \pom -exchange components in the left-hand-side
of Eq.~\ref{eq:Kratio}, or the data points in Fig.~\ref{fig:ratio}.
Thus, agreement between bands and data points in Fig.~\ref{fig:ratio}
implies that \pom -exchange is the same fraction of both numerator
and denominator of $Ratio$.
% 
\item In calculating the points in Fig.~\ref{fig:ratio}, 
\eps\ = 0.15 and \alf\ = 0.25 are assumed. 
Since, with the present data samples, there is no way to know what values to 
use, and how they may depend on \qsq , it is relevant to point out that the 
character of Fig.~\ref{fig:ratio} is not very sensitive to these uncertainties.
For example, as noted in the previous section, if $\alpha ' = +0.15$ is used to 
calculate the ratios in the figure, their values change by only $\sim 10\%$ and 
the conclusions do not change.
%
\item We pointed out above that the $K$ factor in \flux\ might be different in 
Reacts.~\ref{eq:dif} and \ref{eq:gams} and we therefore labeled them 
differently.
However, the approximate factorization agreement that we find at the lower \qsq\ 
values implies that the two $K$ values are probably not very different.
%
\end{enumerate}

\section{Conclusions}
\label{sect:conclude}
\indent

We have summarized the phenomenology of inclusive single diffractive 
interactions in $pp$ interactions in which all available data are well 
described by a product of two functions which describe \pom\ formation and 
\pom\ interactions (cross sections). The formation factor, often referred to 
as the \pom\ flux factor has the characteristic Regge form, except that the 
empirical \pom\ Regge trajectory is an effective one whose $t=0$ intercept 
and slope decrease with increasing energy. This presumably reflects damping 
or multi--\pom --exchange processes which grow with energy, although it seems 
surprising that the factorized formula continues to describe the data as well 
as it does under these circumstances.

The data on diffractive $\gamma ^{*}p$ interactions in HERA $ep$ collisions 
are also well described by a product of a \pom\ flux factor and a cross 
section factor. However, the effective \pom\ trajectory is distinctly 
different from what is found in the corresponding hadronic interactions referred 
to in the previous paragraph.  A likely interpretation for this fact is that 
multi--\pom --exchange effects are different in $pp$ and $ep$ collisions.  
 
From fits to the two diffractive data sets, we have extracted values for the
$\gamma^{*}$--\pom\  and $p$--\pom\ total cross sections (in each case 
multiplied by the normalization constant of the respective \pom\ flux factor).
We then combined these cross sections with the known total $\gamma^{*}p$ and 
$pp$ total cross sections to test a simple factorization relation between 
their \pom\ exchange components due to the optical theorem.  

The factorization test is reasonably well satisfied, to within about 20\%, 
in the range, $1 < \qsq < \sim$6~GeV$^2$. However, at higher \qsq\ values, 
a clear breakdown in the factorization test is observed.
 
In view of the pronounced and different multi--\pom --exchange or damping 
effects which are observed in the two classes of reactions, the first of 
these two observations is very surprising. It seems to be telling us that the 
dominant damping effects are of such a nature that factorization of \pom\ 
formation and interaction is not too much disturbed.  

According to Kharzeev and Levin \cite{kharzeev}, our second observation 
concerning the breakdown of factorization observed at larger \qsq\ can be
understood in terms of the onset of perturbative scattering in the \pom\
and a small \pom\ size (see also Ref.~\cite{kopeliovich}).  
Arguing that the properties of the soft \pom\ are linked to the scale 
anomaly of QCD, they calculate that the scale, 
$M^2_0  \sim 4  \div 6$~GeV$^2$ is the largest non--perturbative scale in QCD.
This corresponds to $\qsq \sim 1/R^2_{\cal P} \sim M^2_0$, where $R_{\cal P}$
is a typical size of the \pom .

The \qsq\ value at which our observed breakdown occurs gives a measure of 
the size of the \pom :  
$R^2_{\cal P} \sim 1/\qsq = (0.39$~GeV$^2$~mb)~/~(6~GeV$^2$) = 0.065~mb.
The area, $\pi R^2_{\cal P} = 0.20$~mb, agrees well with the recent 
UA8 measurement \cite{pompom} of the \pom --\pom\ total cross section, 
\sigpompom\ = 0.2 mb. In this connection, it is also interesting to note 
that UA8 also obtained a statistically modest, but significant, test of 
factorization in double-\pom --exchange: 
%
 \begin{equation}
\bar{p} \, p  \,\, \rightarrow  \, \, \bar{p} \, X \, p
\label{eq:pompom}
\end{equation}
%
using the relation:
%
\begin{equation}
{{K^2 \, \sigpompom}\over{K \, \sigma^{tot}_{p{\cal P}}}} 
\, \, = \, \, {{K \, \sigma^{tot}_{p{\cal P}}}\over{\sigma^{tot}_{pp}}}
.\label{eq:sratio5}
\end{equation}
 
As pointed out in Sect.~\ref{sect:factor}, there are limitations to the
factorization analysis presented in this paper.
These can be addressed by obtaining larger and improved event samples.
For example, in the case of React.~\ref{eq:gams}, more detailed $t$-dependent 
measurements will allow an unambiguous determination of the effective \pom\ 
trajectory, espectially as a function of \qsq .
The left--hand side in Eq.~\ref{eq:Kratio} should be determined for the
\pom --exchange component alone. 
In the case of React.~\ref{eq:dif}, there is a great need for new and more 
complete data samples over a wider range of $t$ and $s$. 
This will be necessary in order to understand how the validity of our 
factorization test depends on the degree of damping in the reactions. 
This may be a very important issue, which we were not able to address in this 
paper because of a lack of the necessary data. 
To pursue this topic in the future, it will be necessary to have detailed 
studies of React.~\ref{eq:dif} at the Tevatron and a possible new experiment at
RHIC, which can cover the energy range between ISR and \SPS .

In summary, we have shown that, despite the non-universality of the \pom\ 
flux factor in Reacts.~\ref{eq:dif} and \ref{eq:gams}, the differential cross 
sections for these reactions can, to good approximation, still be written as a 
product of \pom\ formation and interaction factors. 
The fact that the \pom\ flux factor is not universal is equivalent to the 
factorization breakdowns reported in Refs.~\cite{whit,cdffact,covolan}.
We hope that our present paper has somewhat clarified these issues.
 
\section*{Acknowledgements}
\indent

We thank John Dainton and Guenter Wolf for extensive, invaluable 
discussions about diffractive $\gamma^{*}p$ interactions
and providing us with the 1994 H1 and ZEUS diffractive
data sets. 
We are greatly indebted to Dima Kharzeev for pointing out the significance 
of our results in terms of his work with Genya Levin.
We also thank 
Alexei Kaidalov, Boris Kopeliovich, Uri Maor and Misha Ryskin
for helpful discussions.
We are grateful to CERN, where most of this work was done,
for their continued hospitality.

\pagebreak

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\clearpage
 
\begin{figure}
\begin{center}
\mbox{\epsfig{file= diag_es3.eps,width=16cm}}
\end{center}
\caption[]{
Upper: The diffractive $\gamma^{*}$-proton process. 
The total squared energy in 
the interaction is $W^2$. The 4-vector length squared of the $\gamma^{*}$
is \qsq ; Lower: The diffractive \pap\ process. 
In each process, the exchanged \pom\ has a momentum transfer, \T , and 
momentum fraction, $\xi \equiv \xpom = 1 - \xp$, of the
incident proton.  
}
\label{fig:diag}
\end{figure}



\clearpage

\begin{figure}
\begin{center}
\mbox{\epsfig{file= elastic_es3.eps,width=16cm}}
\end{center}
\caption[]{
Ratios of the \gm\ and hadronic forward elastic amplitudes referred to 
in the text. In all cases, the dashed lines are \pom s.
On both left and right sides, the upper vertices cancel, showing that
each is the ratio of \pom -\pom\ to \pom -proton vertices.
Hence, the left and right sides should be equal.
}
\label{fig:elastic}
\end{figure}



\clearpage

\begin{figure}
\begin{center}
\mbox{\epsfig{file= zeus_es3.eps,width=14cm}}
\end{center}
\caption[]{
The ZEUS 1994 data\protect ~\cite{zeus94}: $\xi \FtwoDthree  $, 
vs.\ $ \xi$ ($\xi \equiv \xpom$) for 12 sets of (\mx , \qsq ) values.
At fixed \mx\ and \qsq , $\xi$ and \wsq\ are uniquely 
related ($\xi = (\mxsq + \qsq )/\wsq $). Thus, each set of points
displays the \wsq\ dependence of React.~\ref{eq:gams} at fixed
\mx\ and \qsq .  The curves are the results of fitting 
Eq.~\protect\ref{eq:factorize2}
to the points shown, as discussed in the text.
}
\label{fig:zeus}
\end{figure}

\clearpage

\begin{figure}
\begin{center}
\mbox{\epsfig{file= h1_es3.eps,width=14cm}}
\end{center}
\caption[]{
The H1 1994 data\protect ~\cite{h194}: $\xi \FtwoDthree  $, 
vs.\ $ \xi$ ($\xi \equiv \xpom$) in bins of $\beta$ and \qsq .
At each \qsq , $s' = \qsq \cdot (1-\beta)/\beta$.
The curves are the results of fitting 
Eq.~\protect\ref{eq:factorize2} to the points with $\xi < 10^{-2}$,
as discussed in the text.
}
\label{fig:h1}
\end{figure}

\clearpage
 
\begin{figure}
\begin{center}
\mbox{\epsfig{file= apvseps_es3.eps,width=16cm}}
\end{center}
\caption[]{
Fitted \eps\ vs.\ fixed $\alpha '$ from fits to the ZEUS 1994 data shown
in Fig.~\protect\ref{fig:zeus}. 
The solid circles are from fits to all four \qsq\ ZEUS data sets;
the solid squares are from fits to only the two lowest \qsq\ data sets.
The shaded band represents the 
$\pm \sigma$ fit contour in the first set. 
The open circle shows the ``soft" \pom\ trajectory parameters, obtained 
from fitting the $s$--dependence of total $pp$ and \pap\ cross sections. 
}
\label{fig:apvseps}
\end{figure}

\clearpage

\begin{figure}
\begin{center}
\mbox{\epsfig{file= epsvsq2_es3.eps,width=16cm}}
\end{center}
\caption[]{
Fitted values of \eps\ vs.\ \qsq\ with \alf\ fixed at 0.25~GeV$^{-2}$, as
explained in the text. 
The solid and dashed curves are, respectively, linear and quadratic fits
to the points shown.
}
\label{fig:epsvsq2}
\end{figure}

\clearpage
 
\begin{figure}
\begin{center}
\mbox{\epsfig{file=ratio_es3.eps,width=16cm}}
\end{center}
\caption[]{
$Ratio$ defined in Eq.~\ref{eq:ratiof} vs.\ \qsq\ for $\beta$ = 0.04, 0.10, 0.20 
and 0.4.
The solid-square points are calculated using the 1994 H1 data \cite{h194}
and are the ratios of $K_{ep} \, \siggpm$ to \siggp\ as explained in the text.
The solid triangles and open circle points use the preliminary lower--\qsq\ 
1995 \cite{h1vanc} and 1999 \cite{h1amst} H1 data as measured from figures in
their conference papers.  
The shaded band is the ratio of $K_{pp} \, \sigpomtot$ to 
\sigpptot ; see Eq.~\ref{eq:ratioa}.
In the extraction of the $K_{ep}\siggpm$ values, \eps\ = 0.15 and \alf\ = 0.25
are used.
}
\label{fig:ratio}
\end{figure}

\end{document} 

 


