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\title{Subprocess Size in Hard Exclusive Scattering}
\author{Paul Hoyer$^1$, Jonathan T.\ Lenaghan$^{2,3}$, Kimmo
Tuominen$^4$ and Carsten Vogt$^4$\\
$^1$Department of Physical Sciences and
Helsinki Institute of Physics\\ \vspace{3mm}
POB 64, FIN-00014 Helsinki University, Finland\\
$^2$Department of Physics, University of Virginia\\\vspace{3mm}
382 McCormick Rd.,Charlottesville, VA 22903, USA\\
$^3$Niels Bohr Institute and $^4$Nordita\\ Blegdamsvej 17, DK-2100
Copenhagen, Denmark}

\preprint{October 8, 2002\\ HIP-2002-47/TH \\ NORDITA-2002-67 HE\\ 
\hepph{0210124}}

\abstract{The interaction region of hard exclusive hadron
scattering can have a large transverse size due to endpoint
contributions, where one parton carries most of the hadron
momentum. The endpoint region is enhanced and can dominate
in processes involving multiple scattering and quark
helicity flip. The endpoint Fock states have perturbatively
short lifetimes, hence are not Sudakov suppressed and scatter softly
in the target. Endpoint contributions can explain the apparent
absence of color transparency in fixed angle exclusive scattering
and the dimensional scaling of transverse $\rho$ photoproduction
at high momentum transfer, which requires quark helicity flip.}

\keywords{Perturbative QCD, Exclusive reactions}

\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The dynamics of endpoint contributions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In the Brodsky-Lepage (BL)
framework of exclusive scattering
\cite{bl:1980,Brodsky:1989pv}, the amplitude for a process $A+B \to C+D$
factorizes at large $t=(p_A-p_C)^2$ into a hard subprocess $a+b
\to c+d$ times distribution amplitudes $\phi_{a/A},\ldots$ for
each external hadron. Here $a$ represents the valence Fock state
of hadron $A$ (\eg, $a = uud$ for the proton). The distribution
amplitude is the valence Fock state amplitude at equal Light-Cone
(LC) time, integrated over the relative transverse momenta of the
partons up to a hard scale of $\morder{\sqrt{-t}}$. Hence all
hadrons involved in the scattering are in transversally compact
configurations. In the subprocess amplitude the momenta of the
partons in each hadron are effectively parallel, their relative
transverse momenta being negligible compared to the hard scale.

This factorization framework can fail due to endpoint
contributions in the integration over the longitudinal momentum
fractions of the quarks \cite{Brodsky:1989pv,ilr}.
The longitudinal momentum of a quark with fraction 
$z \lsim \lqcd/p^+$ of its parent hadron momentum~$p^+$ 
(in a frame where the hadron moves fast along the $z$-axis)
is no larger than its transverse momentum.
Hence the quark is isotropically
distributed in momentum space and outside the light-cone
formed by the fast quarks of the subprocess. The virtualities of
subprocess propagators decrease with $z$, allowing hadron Fock
states of large transverse size to contribute near the endpoints.

The lifetime $\tau$ of a Fock state in a parent hadron of high 
momentum $p^+= E+p^z$ is inversely proportional to the difference 
between its energy and that of the hadron,
\beq \label{lifetime}
\frac{1}{\tau} \simeq \sum_i E_i - E \simeq \sum_i  
\frac{k_{\perp i}^2 + m_i^2}{z_i \, p^+} - \frac{M^2}{p^+} \,,
\eeq
where $M$ is the mass of the parent. The second equality between 
the ordinary and light-cone energy differences is valid when 
$z_i \, p^+=k_i^+ \gg k_{\perp i},m_i$ for all constituents~$i$ and 
is thus {\em not} valid for Fock states in the endpoint region. 
The lifetime of the endpoint Fock states is $\sim 1/\lqcd$, 
which is short compared to the typical lifetimes of 
\order{p^+/\lqcd^2} of Fock states where all constituents have 
comparable momentum fractions~$z_i$.

The nature of endpoint dynamics is illustrated by Deep Inelastic 
Scattering ($e \, p \to e \, X$, DIS). In the aligned jet (parton 
model) regime the virtual photon with $q^+ \simeq 2 \, \nu$ splits 
asymmetrically into a $q\bar q$ pair, such that $z_{\bar q} \sim 
\lqcd^2/Q^2$ and $k_{\perp\bar q} \sim \lqcd$~\cite{bks,bhm}. 
Thus the antiquark momentum $k_{\bar q}^+ = z_{\bar q} \, q^+ 
\simeq \lqcd^2/(m_N x_B)$ stays finite in the Bjorken limit. The 
probability $\propto 1/Q^2$ of the asymmetric photon splitting 
determines the scaling of the DIS cross section, 
$\sigma_{tot}(\gamma^*p) \propto 1/Q^2$. The non-perturbative 
scattering cross section $\sigma[(q\bar q)N] \sim 1/\lqcd^2$ of  
the $q\bar q$ Fock state corresponds to the quark distribution 
$f_{q/N}(x_B)$ in the $q^- \simeq 2 \, \nu$ (or Breit) frame.

The endpoint dynamics was also studied for quarkonium 
hadroproduction, $\pi N \to J/\psi + X$ \cite{bhmt}. QCD 
factorization breaks down when the quarkonium carries large 
fractional momentum, $x_F \simeq 1- \lqcd^2/M_{J/\psi}^2$. In  
this regime there is no hard scattering on a target parton (\eg, 
in a subprocess such as $gg \to c\bar c$). Rather, a compact 
Fock state in the pion  projectile fluctuates into an endpoint 
state where nearly all momentum is carried by the heavy quark 
pair. The light valence quarks have transverse momenta of 
\order{\lqcd} and their soft, non-perturbative scattering in the 
target liberates the heavy quarks, which then appear in the final 
state.

According to Eq. \eq{lifetime} all endpoint configurations have 
short lifetimes in spite of their large transverse size. Like 
compact states they thus have a low number of constituents -- in 
particular, comoving fields that have long formation times are 
absent. Hence there is no enhanced forward radiation in 
scattering processes, \ie, endpoint configurations are not 
suppressed by the Sudakov form factor~\cite{bls}.

The large size of endpoint configurations favors multiple 
scattering in the target. This enhances their importance in 
diffractive processes which require color singlet exchange. The 
aligned jet configuration contributes at leading twist to 
diffractive DIS, whereas multiple scattering of the compact, 
symmetric $q\bar q$ configurations is power suppressed. Endpoint 
configurations are also enhanced in scattering on nuclear 
targets due to the increased importance of multiple scattering. 
The nuclear dependence in effect measures the size of the  
contributing Fock states. The failure to observe color 
transparency in large angle $e \, p \to e \, p$ \cite{ctep} and 
$p \, p \to p \, p$ \cite{ctpp} scattering, with the target proton 
embedded in a nucleus, may signal dominant endpoint contribution 
in these processes\footnote{See Refs.~\cite{Brodsky:1987xw,Ralston:rb} 
for alternative explanations.}.

Quark helicity flip in hard photon and gluon interactions is 
suppressed by a factor $m_q/k_\perp$. Helicity is therefore 
conserved at leading twist in BL factorization. On 
the other hand, the low $k_\perp \sim \lqcd$ of endpoint 
constituents implies that quark helicity flip is not suppressed. 
The relative importance of amplitudes with quark helicity flip 
is thus another measure of endpoint contributions.

The LC energy difference in \eq{lifetime} diverges when any 
fractional momentum $z_i \to 0$. This is the reason why 
distribution amplitudes, which are defined at equal LC time, 
vanish at the endpoints \cite{bl:1980,Brodsky:1989pv}. As we 
emphasized above, however, target scattering is soft in the 
endpoint regime, implying a breakdown of LC dominance and of 
factorization into hard subprocess and distribution amplitudes. 
For this dynamics it is more natural to use the difference of
ordinary energies in~\eq{lifetime} which stays finite (albeit large) 
in the $z_i \to 0$ limit. This increases the importance of endpoint 
contributions in convolution integrals.


In this paper we study two processes where data indicates that 
endpoint contributions dominate. The perturbative QCD (PQCD) 
estimate for $d\sigma/dt(\gamma \, p \to \pi^+ n)$, obtained from 
the semi-exclusive process $\gamma \, p \to \pi^+ Y$ 
(Fig.~1)~\cite{BDHP:1998} using Bloom-Gilman duality~\cite{BG, CEBAF}, 
is two orders of magnitude below the data \cite{hoyer:2001}. In 
section~2 we show that the transverse size of the  $\gamma \, u \to 
\pi^+ d$ subprocess is effectively large, and that the color 
transparency assumed in the semi-exclusive process is thus 
likely to be violated.

Recent high energy data on $\rho^0$ and $\phi$ meson 
photoproduction show a dominance of quark helicity flip out to 
large momentum transfer $|t| \lsim 12 \gev^2$~\cite{Chekanov:2002}. 
In section~3 we study the properties of the 
$\gamma \, g \to \rho\, g$ subprocess amplitude. The 
amplitude for longitudinally polarized $\rho$ mesons (which 
conserves quark helicity) vanishes for real external photons. 
The amplitude for transversely polarized
$\rho$'s (which dominates in the data) has strongly enhanced 
endpoint contributions. Due to the factor of $m_q/\sqrt{-t}$ 
resulting from the quark helicity flip only the endpoint 
contributions can potentially explain the observed dimensional 
scaling of the cross section. We show how such asymptotic 
scaling naturally arises.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The size of $\gamma \, u \to \pi^+ d$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\FIGURE[t]{\epsfig{file=Fig1.eps,width=.6\textwidth}
\caption{Semi-exclusive scattering. In the limit (2.1) the cross
section factorizes into a hard subprocess cross section
$\hat\sigma$ times a target parton distribution.}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Our study of the $\gamma \, u \to \pi^+ d$ process is motivated by
its role as a subprocess of semi-exclusive $\gamma \, p \to \pi^+
Y$ scattering. In the kinematic limit
\beq \label{sexlimit}
s \gg -t, M_Y^2 \gg \lqcd^2 \,,
\eeq
where (\cf\ Fig.~1) $s=(q+p)^2 = E_{\rm CM}^2$ and the invariant
momentum transfer $t=(q-P)^2$, the semi-exclusive cross section
reads~\cite{BDHP:1998}
\beq \label{sexsigma}
\frac{d\,\sigma}{dt}(\gamma \, p \to \pi^+ Y) = \sum_{q=u,\bar d}
f_{q/p}(x)\, \frac{d\hat{\sigma}}{dt}(\gamma \, q \to \pi^+ q') \,.
\eeq
The fractional momentum of the struck quark is $x=-t/(M_Y^2-t)$
and the subprocess cross section is given by
($\hat s = xs$)
\begin{eqnarray}\label{subsigma}
      \frac{d\hat{\sigma}}{dt}(\gamma \, u \to \pi^+ d) =
          \frac{256\,\pi^2\,\alpha\,\as^2}{27\,\hat{s}^2\,|t|} \,
          (e_u - e_d)^2 \,
          \left[ \int_0^1 dz \, \frac{\Phi_\pi(z)}{z} \right]^2 \,.
\end{eqnarray}
Here $\Phi_\pi(z)$ is the pion distribution amplitude, \ie, its
Fock state wave function for a $u\bar d$ pair at short
transverse distance $\sim \morder{1/\sqrt{-t}}$, with the
$u$-quark carrying a fraction $z$ of the pion momentum. A color
singlet $u\bar d$ pair of small transverse size does not
rescatter in the proton target, giving the simple expression
\eq{sexsigma} for the semi-exclusive cross section.

There is as yet no data on $\gamma \, p \to \pi^+ Y$ in the
kinematic region~\eq{sexlimit}.
Assuming that semi-exclusive processes obey Bloom-Gilman duality,
one may relate the $\gamma \, p \to \pi^+ Y$ cross section
to the one for $\gamma \, p \to \pi^+ n$.
However, the measured $\gamma \, p \to \pi^+ n$ cross section
is so large that Bloom-Gilman duality would have to fail by two
orders of magnitude for the prediction of the semi-exclusive
cross section to be correct~\cite{hoyer:2001}.
A more likely explanation is that~\eq{sexsigma} is an
underestimate of the true $\gamma \, p \to
\pi^+ Y$ cross section due to a lack of color transparency.

We shall use the photon virtuality as a probe of the transverse
size of the subprocess \eq{subsigma}. The cross section is 
independent of $Q^2$ when the size of the scattering region is
small compared to $1/Q^2$. We employ the asymptotic distribution
amplitude
\beq \label{asdist}
\Phi_\pi(z) = \frac{\sqrt{6}}{2}f_\pi\, z(1-z)
\eeq
($f_\pi\simeq 130$ MeV) and neglect quark masses ($m_q=0$). In
Fig.~2 we show the differential cross section
$d\sigma/dt(\gamma_\mrm{\, T}^*(Q^2) \, u \to \pi^+ d)$ (solid line)
for a transversely polarized virtual photon as a function the
dimensionless ratio $Q^2/|t|$. While the real photon cross
section given by~\eq{subsigma} is finite, its {\em slope} at
$Q^2=0$ is (as we shall see, logarithmically) infinite. Thus,
however big the momentum transfer $|t|$ is, the transverse size
of the photon scattering region remains large. For comparison we
also show (dashed line in Fig.~2) that the Compton scattering
$\gamma^* e \to \gamma \, e$ cross section is independent of $Q^2$
in the limit $s \gg |t|$, as expected due to the pointlike nature
of the photon.

The divergent slope of the meson photoproduction cross section
has already been noted by the authors of~\cite{ginz:1996}. They
concluded that the onset of the perturbative regime for the
production of light vector mesons is at rather large momentum
transfers, well beyond $30 \gev^2$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\FIGURE[h]{\epsfig{file=Fig2.eps,width=0.6\textwidth}
\caption{$d\sigma/dt(\gamma_\mrm{\, T}^*(Q^2)+u \to \pi^+ +d)$
for a transversely polarized photon as a function of $Q^2/|t|$
(solid line). For comparison, we show the corresponding plot for
Compton scattering, $\gamma^*(Q^2)+e \to \gamma +e$ (dashed line).
The normalization is arbitrary.}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


The strong $Q^2$ dependence of the $\gamma^* u \to \pi^+ d$
cross section is due to endpoint contributions, even though the
$z$-integral in \eq{subsigma} is not enhanced near $z=0,1$. This
may be seen from the expression for the scattering amplitude,
which is a convolution of the pion distribution amplitude
\eq{asdist} with the $\gamma^* u \to (u\bar d) + d$ subamplitude
${\cal H}_{\mu, \lambda\lambda'}^{\nu\nu'}$ (\cf\ Fig.~3a),
\beq \label{convamp}
{\cal M}_{\mu,\lambda\lambda'}^{AB}(\gamma^* + u \to \pi^+ + d) =
      \delta_{AB} \int_0^1 dz \,
       \frac{1}{\sqrt{2}} \, \Big[ {\cal H}_{\mu,\lambda\lambda'}^{+-}(z)
       - {\cal H}_{\mu,\lambda\lambda'}^{-+}(z) \Big] \, 
\Phi_{\pi}(z) \,.
\eeq
Here $A,B$ are the color indices of the incoming $u$- and outgoing
$d$-quark, $\mu$ is the photon helicity, $\lambda\ (\lambda')$
is the incoming $u$-quark (outgoing $d$-quark) helicity and
$\nu\ (\nu')$ is the $u\ (\bar d)$ quark helicity in the pair
forming the $\pi^+$. For $m_u = m_d =0$ quark helicity is
conserved\footnote{Expressions for the helicity amplitudes for
$Q^2,m_q \neq 0$ are given in the Appendix.} and we shall only
consider the $\mu=+1, \lambda=\lambda'=-\half$ amplitude. In the
semi-exclusive limit~\eq{sexlimit}
($s \gg -t,Q^2$),
\beq \label{subamp}
      {\cal H}^{-+}_{+,--} =
          - \frac{2 \sqrt{2} \, e \, (4 \pi \as) \, C_F}{\sqrt{-t}} \,
           \Bigg[ \frac{e_u}{ z - \bar{z} \, Q^2/t }
          - \frac{ e_d \, \zb }{z \, (\zb - z \, Q^2/t) } \Bigg] \,,
\eeq
where $C_F=(N_c^2-1)/2N_c$ is
the color factor and $\zb=1-z$.
We make the following observations:
\begin{itemize}
\item[(a)] At $Q^2=0$ the amplitude is $\propto (e_u-e_d)/z$.
This endpoint behavior therefore arises both from the photon coupling
to the `slow' $u$-quark and to the `fast' $d$-quark. On the
other hand, the amplitude is finite for $z \to 1$ since the
$u$-quark helicity $\nu = -\half$ is opposite to that of the photon
helicity $\mu = +1$.
Thus the helicity flip between the projectile and
fast outgoing particles is minimized.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\FIGURE[h]{\epsfig{file=Fig3.eps,width=.8\textwidth}
\caption{(a) A diagram contributing to $\gamma+u \to u\bar d +
d$. (b) A diagram contributing to $\gamma+g \to q\bar q + g$.}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item[(b)] $d{\cal H}/dQ^2 \propto e_u/z^2$ for $z \to 0$ at
$Q^2=0$. The $1/z^2$ behavior gives a logarithmic singularity in
the convolution \eq{convamp} when $\Phi_{\pi}(z) \propto z$.
This is the origin of the infinite slope in Fig.~2. We also note
that the singular contribution arises from the photon coupling
to the {\em slow} quark.
\end{itemize}

The $Q^2$-sensitivity is due to the large transverse size of the
$\gamma \, u \to \pi^+ d$ process, as can be seen explicitly from
the kinematics (\cf\ Fig.~3a). We use the notation
$v=[ \, v^+,v^-,\mbf{v}_\perp \, ]$ where $v^\pm = v^0 \pm v^3$
for light-cone coordinates and take the photon momentum in the 
negative $z$-direction so that\footnote{For notational convenience 
we denote the (target) $u$-quark momentum by $p$ and the subprocess 
energy by $s$ rather than $\hat s$.}
\beqa \label{momdef}
p&=&\sqrt{s} \, [ \, 1 \, , \, 0 \, , \, \mbf{0}_\perp \, ] \,,\nn\\
q&=&\sqrt{s} \, [ \, 0 \, , \, 1 \, , \, \mbf{0}_\perp \, ] \,,\nn\\
\\
k_u&=&
\left[\frac{(z \, \mbf{\pit}_\perp+\mbf{\qt}_\perp)^2}{z
\sqrt{s}} \, , \,
z \sqrt{s} \, , \, z \, \mbf{\pit}_\perp +
\mbf{\qt}_\perp\right] \,, \nn\\
k_{\bar d}&=& \left[\frac{(\bar
z \, \mbf{\pit}_\perp-\mbf{\qt}_\perp)^2}{\bar z \sqrt{s}} \, , \, \bar
z \sqrt{s} \, , \, \bar z \, \mbf{\pit}_\perp -
\mbf{\qt}_\perp\right] \nn
\eeqa
for on-shell massless quarks.
$\mbf{\pit}_\perp$ denotes the pion's transverse momentum relative
to the collision axis and $\mbf{\qt}_\perp$ is the relative
transverse momentum of its $u\bar d$ constituents.
The pion momentum $P = k_u+k_{\bar d}$
satisfies\footnote{If the $u$ and $\bar d$ quarks are not
on-shell (as would be the case in the pion) we have
$P^2=(\qt_\perp^2+z k_u^2+\zb k_{\bar d}^2)/z\zb$. Hence an
off-shellness of \order{\qt_\perp^2} is sufficient to keep the
pion on-shell even for $z\to 0,1$.} $P^2= \qt_\perp^2/z\bar z$,
where $\qt_\perp = $ \order{\lqcd},
and the momentum transfer $t=(q-P)^2 = -\pit_\perp^2$.

The virtualities of the internal quark and gluon lines in Fig.~3a are
\beqa \label{virtualities}
\ell_u^2 &=& (q-k_u)^2 = -z \pit_\perp^2- 2 \,
\mbf{\pit}_\perp\cdot\mbf{\qt}_\perp-\qt_\perp^2/z \,,\\
\ell_g^2 &=& (p+\ell_u)^2 = \bar zs +\ell_u^2 \,.
\eeqa
In the semi-exclusive limit \eq{sexlimit} we see that $\ell_u^2$
becomes sensitive to $\mbf{\qt}_\perp$ for
\beq \label{softz}
z \lsim \frac{\qt_\perp}{\pit_\perp} \sim \frac{\lqcd}{\sqrt{-t}} \,.
\eeq
Hence the subprocess is not transversally compact: the distance
between the photon absorption and gluon emission vertices in Fig.~3a
is given by the inverse of $\ell_{u\perp} =|z \,
\mbf{\pit}_\perp+\mbf{\qt}_\perp| = \morder{\lqcd}$ in the region
\eq{softz}. Moreover, BL factorization fails in this
endpoint region since the hard subamplitude depends on the relative
momentum of the quarks in the pion. These consequences of the
kinematics imply the $Q^2$-sensitivity of the process.

We can also see why the large transverse size, \ie,
the $Q^2$-sensitivity, arises only from the photon coupling to the slow
quark. When the $u$-quark is fast, \ie, for $z \simeq 1-
\lqcd/\sqrt{-t}$, the transverse distance $1/\ell_{u\perp} =
\morder{1/\sqrt{|t|}}$ while $\ell_g^2 \propto s\lqcd/\sqrt{-t}$
remains large since $s \gg -t$. Hence the distances between all
interaction vertices in Fig.~3a are short for $z \to 1$.

The fact that the $Q^2$ derivative of the subprocess amplitude 
\eq{subamp} is more endpoint sensitive than the the amplitude 
itself shows that a size measurement introduces inverse factors 
of $z$ and $1-z$. Our result does not change the fact that the 
$z$-integral of the leading twist cross section is flat (for the 
asymptotic distribution amplitude \eq{asdist}), and thus is 
dominated by compact configurations. However, rescattering in 
the target will introduce dipole factors proportional to the 
transverse size and cause the convolution integral to be 
endpoint dominated. A failure of color transparency is thus a 
likely reason for the large discrepancy with data found in 
Ref.~\cite{hoyer:2001} for the $\gamma \, p \to \pi^+ n$ cross 
section.

An analogous sensitivity to small photon virtualities can be
observed for the photon-pion transition form factor~\cite{DKV1},
$\gamma^*(Q^2) \, \gamma^{(*)}(Q'^2) \to \pi^0\,$: its rate of
change is logarithmically divergent as the ratio $Q'^2/Q^2 \to 0$.
At $Q'^2 = 0$ the pion transition form factor is given by the
same integral over the distribution amplitude as appears in the
cross section~\eq{subsigma}. The fact that the
$\pi^0$ is produced in isolation and color transparency thus is
not an issue may explain the phenomenological success
(see~\cite{Gronberg:1997fj} and references therein)
of the PQCD prediction in this case.

As we noted in the previous section, there are no compelling 
reasons to impose a vanishing of the effective distribution 
amplitude in the endpoint region where LC dominance fails. 
Relaxing this suppression makes the convolution integral 
logarithmic near $z=0,1$, implying a leading twist contribution 
from the endpoint region.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Dimensional scaling with quark helicity flip in $\gamma
+p \to \rho+Y$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{The experimental evidence}

The ZEUS collaboration recently published~\cite{Chekanov:2002} 
data on $\rho$ photoproduction, $\gamma +p \to \rho+Y$, in the 
semi-exclusive kinematics specified by Eq.~\eq{sexlimit}. The 
data cover $80 < \sqrt{s} < 120$ GeV, $1.1 < |t| < 12\ \gev^2$ 
and is integrated over $x=-t/(M_Y^2-t) \gsim 0.01$. 
The scattering is believed to be dominantly diffractive and, 
due to the high value of $|t|$, to provide a testing ground for 
the BFKL exchange mechanism~\cite{bfkl}. The hard
subprocess is then $\gamma + g \to \rho+g$, with the t-channel
containing a two-gluon ladder and the $\rho$ emerging via its
distribution amplitude according to BL factorization.

However, the data \cite{Chekanov:2002} pose a serious challenge
to this picture. Dimensional scaling predicts
\beq \label{rhoscaling}
\frac{d\sigma(\gamma \, g \to \rho \, g)}{dt}
\propto \frac{I_\rho^2}{|t|^n}
\eeq
with  $n=3$. Here $I_\rho$ is an integral over the $\rho$
distribution amplitude with dimension GeV. In contrast to the
quark exchange cross section \eq{subsigma} there is no factor
$s$ in the denominator of this gluon exchange cross section. The
data agree with dimensional scaling, giving $n=3.21\pm 0.04 \pm
0.15$ (in $\phi$ production the corresponding power is measured
to be $n=2.7 \pm 0.1 \pm 0.2$). Together with the fact that the
$\phi/\rho$ cross section ratio is consistent with the ratio 2/9 
of the charge factors for $|t| \gsim 4\ \gev^2$, this suggests 
that the $\gamma + g \to \rho+g$ process is hard and perturbative.

The upper part of the subprocess is shown in Fig.~3b. The quark
pair produced at the photon vertex scatters off the two gluons
and forms the vector meson via its distribution amplitude. The
vector meson is expected to be longitudinally polarized since
its quark and antiquark constituents have opposite helicities
due to helicity conservation at the photon and gluon vertices.
However, the ZEUS data show that the $\rho$ meson inherits (to a
good approximation and in the full $t$-range) the transverse
polarization of the incoming photon. In the BL factorization 
framework this implies a quark helicity flip, incurring an
$m_q^2/|t|$ suppression factor in the cross section
\eq{rhoscaling}, which is then expected to scale with a power
$n=4$.

Thus, we are faced with a dilemma. The data obeys simple
dimensional scaling ($n=3$), is consistent with the quark
production process being hard ($\phi/\rho$ flavor symmetry
indicates insensitivity to $m_q$) and the semi-exclusively
produced $\rho$ meson carries both the momentum and the helicity
of the projectile. But these attractive features are mutually
inconsistent within the standard factorization
framework~\cite{bl:1980} of exclusive processes.

The authors of~\cite{ivanov:2000} consider the possibility
that the production of transverse vector mesons is due to a
non-perturbative, chiral-odd wave function of the photon, which
is proportional to the quark condensate. This contribution is 
subleading at asymptotically large momentum transfers. In such 
an approach the dimensional scaling observed~\cite{Chekanov:2002} 
for $|t| \lsim 12 \gev^2$ would be accidental.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The subprocess amplitudes}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

To resolve the dilemma let us consider the structure of the
factorized $\gamma + g \to \rho+g$ amplitude. It is a convolution 
of the quark pair production amplitude 
${\cal G}(\gamma + g \to q\bar q+g)$ and
the $\rho$ distribution amplitude $\Phi_\rho^{\mu'}$.
For example, for transversely polarized $(\mu'= +1)\ \rho$ mesons,
\beq \label{convampg}
{\cal M}_{\mu,\lambda\lambda'}^{ab}(\gamma + g \to \rho+g) =
       \delta_{ab}\int_0^1 dz \, {\cal
G}_{\mu,\lambda\lambda'}^{++}(z) \, \Phi_{\rho}^+(z) \,.
\eeq
Here $a,b$ are gluon color indices, the
$\mu,\lambda,\lambda'$ indices of ${\cal G}$ are the helicities
of the photon, incoming and outgoing gluon, respectively, and
the upper indices refer to the $q$ and $\bar q$ helicities.

For simplicity we consider only the lowest order contribution to
${\cal G}(\gamma + g \to q\bar q+g)$ (\cf\ Fig.~3b). Higher
order diagrams build the gluon ladder and are important for
describing the $s$-dependence, but should not affect the
helicity structure of the upper vertex, which is our present
concern.

The transverse photon ($\mu=+1$) amplitude with
$\lambda=\lambda'= +1$ and no quark helicity flip is
\beqa \label{g-trans-long}
{\cal G}^{+-}_{+,++}(\gamma + g \to q\bar q+g) &=&
    -\frac{\sqrt{2} \, e e_q \, (4\pi\as)}{\sqrt{N_c}\, \sqrt{-t}}
    \frac{Q^2}{t} \,\nn\\
    &\times& \frac{2 \, z - 1}{\bar{z} \, ( z - \bar{z} \, 
Q^2/t - m_q^2/t)( \bar{z} - z \, Q^2/t - m_q^2/t ) } \,\,.
\eeqa
This leading twist amplitude {\em vanishes} in
photoproduction ($Q^2=0$). The quark helicity flip amplitude
contributing to transverse $\rho$ production in \eq{convampg} is
at\footnote{Expressions for $Q^2\neq 0$
are given in the Appendix.} $Q^2=0$
\beqa
\label{g-trans-trans}
    {\cal G}^{++}_{+,++}(\gamma + g \to q\bar q+g) &=&
    - \frac{\sqrt{2}\, e e_q \, (4\pi\as)}{\sqrt{N_c}\, \sqrt{-t}} \,
      \frac{\sqrt{m_q^2/(-t)}}{z \bar{z} \, ( z -  m_q^2/t )
      ( \bar{z} - m_q^2/t)} \nn\\ &&\\
   &=& - \frac{\sqrt{2}\, e e_q \, (4\pi\as)}{\sqrt{N_c}\, \sqrt{-t}}
      \, \frac{\sqrt{m_q^2/(-t)}}{(z \bar{z})^2}\left[1+
      \morder{\frac{m_q^2}{t}}\right] \,. \nn
\eeqa
The factor $(z\zb)^2$ in the denominator enhances the endpoint
regions $z=0,1$ in the convolution \eq{convampg} causing a
(logarithmic) singularity in the $z$-integral for distribution
amplitudes which vanish linearly at the endpoints. This implies
a breakdown of factorization in semi-exclusive $\rho$
photoproduction.

\subsection{Endpoint behavior of the distribution amplitude}

Even though factorization fails due to the strong enhancement of
the subprocess amplitude in the region where one of the produced
quarks carries nearly all the momentum, we may still see
qualitatively how the experimentally observed dimensional
scaling of the cross section can arise. As we emphasized above,
this must be due to the endpoint contributions, since for finite
values of $z$ the transverse $\rho$ cross section is suppressed
by $m_q^2/|t|$, giving $n=4$ in Eq. \eq{rhoscaling}.

The distribution amplitude $\Phi_{\rho}^{\mu'}(z)$ appearing in
the convolution \eq{convampg} is the valence Fock state
amplitude {\em at equal light-cone (LC) time}. It has the
general form (see, \eg, section 3.3 of Ref.
\cite{Brodsky:1989pv})
\beq \label{distamp}
\Phi_\rho(z) \sim \int d^2\mbf{k}_{q\perp} \,
\frac{1}{m_\rho^2-\frac{k_{q\perp}^2+m_q^2}{z(1-z)}} \, V \, \Psi \,,
\eeq
where $\Psi$ is the full LC wave function and $V$ is an interaction
potential. Thus $\Phi_\rho(z)$ vanishes at $z=0,1$ because the
LC energy of a parton with momentum fraction $z$
tends to infinity as $z \to 0$. However, as we saw in the
previous section the subprocess is soft and thus not light-cone
dominated for $z \lsim z_s \simeq \lqcd/\sqrt{-t}$. The LC
energy is not relevant in this region -- while the
ordinary energy difference \eq{lifetime} obviously remains 
finite as the longitudinal
momentum of a parton vanishes.

Hence, we should consider the possibility that
$\Phi_{\rho}^{\mu'}(z)$ (effectively) does not vanish at the
endpoints. The $z$-integral is then linearly divergent at
$z=0,1$. Truncating the integration region
where the subprocess becomes soft we get
\beq
\int_{z_s}^{1-z_s} \frac{\Phi_{\rho}^{\mu'}(z)}{z^2\zb^2}
\propto \frac{\sqrt{-t}}{m_q}\, \Phi_{\rho}^{\mu'}(0) \,.
\eeq
Thus we {\em gain} a factor $\sqrt{-t}$ in the amplitude, which
compensates the suppression due to the helicity flip and
restores dimensional scaling ($n=3$ in Eq.~\eq{rhoscaling}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this paper we discussed meson production at large momentum
transfer $|t|$ by real, transversely polarized photons. Based on
features of both PQCD and the data we saw that the scattering is
likely to be endpoint dominated and thus involve $q\bar q$ pairs
of large transverse size. Hence it appears that highly
asymmetric Fock states, where one quark carries nearly all the
momentum, do have a significant overlap with hadron wave
functions. This would explain the absence of color transparency 
in $e \, A \to e \, p \, (A-1)$ \cite{ctep} and $p \, A \to p \, 
p \, (A-1)$~\cite{ctpp}.

Endpoint contributions are enhanced in photoproduction since the 
wave function of transverse photons does not vanish at $z=0,1$. 
This also prevents factorization of the transverse photon 
amplitude in deeply virtual meson production at high $Q^2$ and 
low $|t|$ \cite{cfs}. The longitudinal photon wave function 
$\Psi_\gamma^{\mu=0}(z) \propto z(1-z)$ favors contributions 
from $q\bar q$ pairs of small transverse size $\sim 1/Q$. The 
color transparency observed in $\gamma^*(Q^2)+N \to \rho 
+N$~\cite{Adams:1994bw,Arneodo:1994id,Ackerstaff:1998wt} agrees 
with this.

Due to the short lifetime of the endpoint states their dynamics 
has many of the attributes of hard scattering, despite their 
large transverse size and soft scattering in the target. Thus we 
saw that they can explain the dimensional scaling of the ZEUS 
data. However, the endpoint states are not color transparent, 
nor do they preserve quark helicity. These features make it 
possible to identify their contribution to exclusive processes. 
Since there is no Sudakov suppression \cite{bls} of the 
short-lived endpoint states the dynamics we have discussed 
applies at arbitrarily high momentum transfers.


\acknowledgments

It is a pleasure to thank Stan Brodsky for many helpful 
discussions and  comments on the manuscript.
This work was begun while PH was employed by Nordita, and has
also been supported by the European Commission under contract
HPRN-CT-2000-00130.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\appendix

\section{Appendix}

\subsection{Quark production amplitudes}

We present the helicity amplitudes $\cal{H}^{\nu\nu'}_{\mu,\lambda
\lambda'}$ for the subprocess $\gamma^*(\mu) \, u(\lambda)
\to u(\nu) \bar{d}(\nu') +
d(\lambda')$ for
$Q^2, \, m_q \neq 0$. For transversely polarized photons we find
\begin{eqnarray}
\label{qsubamp_t-l}
   {\cal H}^{+-}_{+,++} = - {\cal H}^{-+}_{-,--} &=&
   \frac{2 \, \sqrt{2} \, e \, (4 \pi \as) \, C_F}{\sqrt{-t}} \,
   \nn \\ &\times& \Bigg[ \frac{e_u \,z}{\bar{z} \,
   ( z - \bar{z}\,Q^2/t - m_q^2/t ) }
   - \frac{ e_d }{ ( \bar{z} - z\,Q^2/t - m_q^2/t ) } \Bigg] \,.
\end{eqnarray}
The amplitudes ${\cal H}^{+-}_{-,++}$ and ${\cal H}^{-+}_{+,--}$ are
obtained from ${\cal H}^{+-}_{+,++}$ and ${\cal H}^{-+}_{-,--}$,
respectively, by exchanging $z \to \bar{z}, \, e_u \leftrightarrow e_d$
and reversing the overall sign. For longitudinally polarized photons
we get
\begin{eqnarray}
\label{qsubamp_l-l}
   {\cal H}^{+-}_{0,++} = {\cal H}^{-+}_{0,--} =
   \frac{4 \, e \, (4 \pi \as) \, C_F}{\sqrt{-t}} \, \left[
   \frac{ e_u \, \sqrt{Q^2/(-t)} }{ z - \bar{z} \, Q^2/t - m_q^2/t}
   +\frac{ e_d \, \sqrt{Q^2/(-t)} }{ \bar{z} - z \, Q^2/t - m_q^2/t}
    \right] \,. \quad
\end{eqnarray}
The quark helicity flip amplitudes read
\begin{eqnarray}
\label{qsubamp_t-t}
   {\cal H}^{++}_{+,--} = {\cal H}^{--}_{-,++} =
   -\frac{2 \, \sqrt{2} \, e \, (4 \pi \as) \, C_F}{\sqrt{-t}} \,
   \left[\frac{e_u \, \sqrt{m_q^2/(-t)}}{z \, \bar{z} \,
   ( z - \bar{z}\,Q^2/t - m_q^2/t ) } \right] \,.
\end{eqnarray}
For ${\cal H}^{++}_{+,++}$ and ${\cal H}^{--}_{-,--}$ we have
to make the replacements $z \to \bar{z}, \, e_u \to e_d$ in
the above respective amplitudes. All other helicity amplitudes
vanish in the kinematical limit~\eq{sexlimit}. The helicity 
non-flip amplitudes agree with those given in Ref.~\cite{huang:2000}
in the limit of large c.m. energies and when appropriate replacements 
for the charge factors are made.\footnote{In Ref.~\cite{huang:2000}
the production of flavor-neutral mesons is considered. Also note 
that the normalization of the subprocess amplitudes is different 
than in our case.}
We note that in the helicity flip amplitudes~(\ref{qsubamp_t-t})
the photon couples only to the quark whose helicity is flipped.
According to \eq{virtualities} the gluon virtuality in Fig. 3a
is of order $s$. Hence to leading order in the limit \eq{sexlimit}
the quark helicity can flip only at the photon vertex.

\subsection{Gluon production amplitudes}

The helicity amplitudes for the subprocess
$\gamma^*(\mu) + g(\lambda) \to q(\nu) \, \bar{q}(\nu') + g(\lambda')$
are denoted by $\cal{G}^{\nu\nu'}_{\mu,\lambda\lambda'}$.
For transversely polarized photons we find
\begin{eqnarray}
\label{gsubamp_-t-l}
   {\cal G}^{+-}_{+,++} = - {\cal G}^{-+}_{-,++} &=&
   -\frac{\sqrt{2} \, e e_q \, (4 \pi \as) }
   {\sqrt{N_{\rm c}} \, \sqrt{-t}} \, \frac{Q^2}{t} \nn \\
   &\times& \frac{2\,z - 1}{ \bar{z} \, ( z - \bar{z} \, Q^2/t - 
   m_q^2/t )\,  ( \bar{z} - z \, Q^2/t - m_q^2/t ) } \,,
\end{eqnarray}
and for longitudinally polarized photons
\begin{eqnarray}
\label{gsubamp_l-l}
   && {\cal G}^{+-}_{0,++} = {\cal G}^{-+}_{0,++} =
     \frac{2\, e e_q \, (4 \pi \as) }{\sqrt{N_{\rm c}} \, \sqrt{-t}} \,
     \frac{\sqrt{Q^2/(-t)}\, (1 - Q^2/t)}
     { (z - \bar{z} \, Q^2/t - m_q^2/t)\,
     ( \bar{z} - z \, Q^2 - m_q^2/t)} \,.
\end{eqnarray}
The amplitudes
${\cal G}^{-+}_{+,++}$ and ${\cal G}^{+-}_{-,++}$ are obtained
from ${\cal G}^{+-}_{+,++}$ and ${\cal G}^{-+}_{-,++}$
respectively by replacing $z \to 1-z$.
For the quark helicity flip amplitudes we obtain
\begin{eqnarray}
\label{gsubamp_t-t}
   {\cal G}^{++}_{+,++} = {\cal G}^{--}_{-,++} &=&
   -\frac{\sqrt{2} \, e e_q \, (4 \pi \as)}{\sqrt{N_{\rm c}} \, 
\sqrt{-t}}
   \,\frac{\sqrt{m_q^2/(-t)}}{z\, \bar{z}} \nn \\
   &\times& \frac{1 - Q^2/t}{ ( z - \bar{z} \, Q^2/t - m_q^2/t ) \,
   ( \bar{z} - z \, Q^2/t - m_q^2/t)} \,.
\end{eqnarray}
For the above combination of quark and photon helicities, the set of
amplitudes with negative gluon helicities is identical, \ie,
\begin{equation}
   {\cal G}^{\nu\nu'}_{\mu,--}={\cal G}^{\nu\nu'}_{\mu,++} \,.
\end{equation}
All other helicity amplitudes vanish in the limit~\eq{sexlimit}.
Again, we find agreement with the results given in
Ref.~\cite{huang:2000}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\end{document}

