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%% section 2 Basic physics of $F_2(x,Q^2)$ at small $x$ [slacpub7096002 in slacpub7096002: slacpub7096003]
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\section{\usemenu{slacpub7096::context::slacpub7096002}{Basic physics of $F_2(x,Q^2)$ at small $x$}}\label{section::slacpub7096002}
The familiar parton picture of deepinelastic scattering is easy to
apply at small $x$, especially in a typical HERA laboratory frame of
reference. A rightmoving electron ``sees" a leftmoving wee parton
in the extremerelativistic leftmoving proton, and Coulombscatters
from it with momentum transfer $Q$. The legoplot picture of the
finalstate particles is sketched in Fig. \docLink{slacpub7096002.tcx}[fig1]{1}a; one sees the
electron and the struckquark jet each with transverse momentum $Q$.
This Coulombscattering picture is accurate in the frame of
reference where $\eta = 0$ is chosen to be the rapidity halfway
between these two features.
Also of interest is the (approximate) location of the initial state
quark before it was struck (the socalled ``hole" fragmentation
region \cite{7}). It is a distance of order $\ell n\, Q$ to the
left of the quark jet, because $\ell n\, \theta/2$ has changed by
approximately that amount because of the Coulomb scattering. It is a
distance $\ell n\, 1/x$ from the leadingproton fragments.
It is also convenient, especially theoretically, to view the same
process in a collinear virtualphoton proton reference frame (cf.
Fig. \docLink{slacpub7096002.tcx}[fig1]{1}b). In such a frame there are generically no
large$p_T$ jets, at least at the level of naive, oldfashioned
parton model. With QCD, there will be extra gluon initialstate and
finalstate radiation. Most of this will look like minijet
production in collinear reference frames, but occasionally there
will be extra genuine gluon jets, especially in the phasespace
region between the hole and the leading quark fragments. Note that
now the amount of phasespace to the right of the ``hole" region is
of order $\ell n\, Q^2$; the extra $\ell n\, Q$ amount of phase space is
in the quark jet in HERA reference frames. The total phase space is
evidently $\ell n\, Q^2 + \ell n\, 1/x = \ell n\, W^2$, as it should be.
\vspace{.5cm}
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsfxsize=5in
\epsfbox{8117A01.eps}
\end{center}
\caption[*]{Legoplot of finalstate hadrons in small$x$ deep
inelastic scattering:
\hfill\break (a) HERA laboratory frame, and (b) collinear
$\gamma^*  p$ reference frames.}
\label{fig1}
\end{figure}
We now are ready to introduce Gribov's paradox. He viewed the same
process in the laboratory frame of the nucleon, but considered for
simplicity replacement of the nucleon by a large, heavy nucleus of
radius $R$. The picture is that first there is the virtual
dissociation of the virtual photon into a hadron system upstream of
the target hadron. For HERA conditions and Ioffe's estimate of
longitudinal distances, this is a distance of hundreds of fermis in
this fixedtarget reference frame. This virtual dissociation process
is followed by just the geometrical absorption of the virtualhadron
system on the nucleus. Gribov used oldfashioned (or in modern terms
lightcone) perturbation theory to make the estimate, which gives a
simple and utterly transparent result:
\begin{equation}
\sigma_T = (1  Z_3) \, \pi R^2 \ .
\end{equation}
Note that the estimate is for $\sigma_T$, not $F_2$, and that
$(1  Z_3)$, with $Z_3$ the charge renormalization of the photon,
is just the probability the photon is hadron, not photon:
\begin{equation}
1  Z_3 = \frac{\alpha}{3\pi} \int \frac{ds~s~R(s)}{(Q^2+s)^2}\approx
\frac{\alpha}{3\pi}~\bar{R}~\ell n\, \frac{1}{x}\ ,
\end{equation}
where $R(s)$ is the sum of squared charges of partons, as used in
describing the $e^+$$e^$ annihilation cross section. So up to
logarithmic factors, the result is that $\sigma_T$, not $F_2$, is
independent of $Q^2$. Since
\begin{equation}
F_2 = \frac{Q^2~\sigma_T}{4\pi^2~\alpha}\ ,
\end{equation}
this means the aforementioned scaling violation by an extra power of
$Q^2$. Gribov's structure function is much too big (at least at
present energies)!!
There are (at least) two ways out of the paradox. One way is, in
modern jargon, ``color transparency". Typically the virtual photon
dissociates into a bare $q \bar{q}$ system which on arrival at the
nucleus is a small color dipole of spatial extent $Q^{1}$. It can
only interact perturbatively with the target via single gluon
exchange. And since the cross section goes as the square of the
dipole moment, one gets $\sigma_T$ proportional to $Q^2$, as is
needed. Note however that the final state morphology is different
from what has been given for the naive parton model; it contains two
leading jets (in the virtual photon direction) and a recoilparton
jet in the proton direction, all typically with a $p_T$ scale $Q$
(this in the collinear $\gamma^*$proton frame; cf. Fig.
\docLink{slacpub7096002.tcx}[fig2]{2}). Also the Adependence for this mechanism is generically
$A^1$.
The second mechanism is associated with more infrequent
configurations, where the $q$ and $\bar{q}$ created by the virtual
photon do not have large $p_T$, but are aligned along the
virtualphoton beam direction. This clearly leads to one particle
(call it the quark $q$) carrying almost all the momentum and the
other (the $\bar{Q}$) carrying much less. When the kinematics is
worked out, one finds that the typical momentum carried by the
``slow" $\bar{Q}$ is of order $x^{1}$ GeV. But do note that this is
still hundreds of GeV for HERA conditions, in Gribov's fixedtarget
reference frame.
\vspace{.5cm}
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsfxsize=5in
\epsfbox{8117A02.eps}
\end{center}
\caption[*]{Legoplot of finalstate hadrons for the
``colortransparency", ``BetheHeitler", or ``noncollinear
photongluon fusion" mechanism at small $x$.}
\label{fig2}
\end{figure}
There is enough time, according to Ioffe's basic estimate, for this
``slow" $\bar{Q}$ to evolve nonperturbatively \cite{8}, and
in particular it will be found at large transverse distances from
the ``fast" quark, of order the hadronic size scale. (For example
there is enough time and space for a nonperturbative color fluxtube
to grow between $q$ and $\bar{Q}$.) So on arrival at the target
hadron the hadronic progeny of the virtual photon look something
like a Bmeson, the fast pointlike $q$ analogous to the pointlike
$b$quark, and the slow structured $\bar{Q}$ looking something like
the light constituent antiquark orbiting the $b$.
It follows that when this configuration evolves, it should be
absorbed geometrically on the target nucleus or nucleon, as assumed
by Gribov. But the probability, per incident virtual photon, that
this configuration actually occurs is easily worked out to be $({\rm
constant})/Q^2$, so the scaling of the structure function $F_2$ is
recovered. Also, the finalstate structure in collinear reference
frames contains no jets, so the partonmodel final state structure
is recovered. The $\bar{Q}$ finalstate fragmentation products are
in fact located in the ``hole" fragmentation region already
described. An additional expectation is that the Adependence at
small $x$ is $A^{2/3}$, even at large $Q^2$.
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