%% slacpub7204: page file slacpub7204002.tcx.
%% section 2 Semiclassical approach to highenergy scattering [slacpub7204002 in slacpub7204002: slacpub7204003]
%%%% latex2techexplorer block:
%% latex2techexplorer page setup:
\iftechexplorer
\setcounter{section}{1}
\fi
\iftechexplorer
\setcounter{secnumdepth}{2}
\setcounter{tocdepth}{2}
\def\thepart#1{}%
\def\thechapter#1{}%
\newcommand{\partLink}[3]{\docLink{#1.tcx}[part::#2]{#3}\\}
\newcommand{\chapterLink}[3]{\docLink{#1.tcx}[chapter::#2]{#3}\\}
\newcommand{\sectionLink}[3]{\docLink{#1.tcx}[section::#2]{#3}\\}
\newcommand{\subsectionLink}[3]{\docLink{#1.tcx}[subsection::#2]{#3}\\}
\newcommand{\subsubsectionLink}[3]{\docLink{#1.tcx}[subsubsection::#2]{#3}\\}
\newcommand{\paragraphLink}[3]{\docLink{#1.tcx}[paragraph::#2]{#3}\\}
\newcommand{\subparagraphLink}[3]{\docLink{#1.tcx}[subparagraph::#2]{#3}\\}
\newcommand{\partInput}{\partLink}
\newcommand{\chapterInput}{\chapterLink}
\newcommand{\sectionInput}{\sectionLink}
\else
\newcommand{\partInput}[3]{\input{#2.tcx}}
\newcommand{\chapterInput}[3]{\input{#2.tcx}}
\newcommand{\sectionInput}[3]{\input{#2.tcx}}
\fi
\newcommand{\subsectionInput}[3]{\input{#2.tcx}}
\newcommand{\subsubsectionInput}[3]{\input{#2.tcx}}
\newcommand{\paragraphInput}[3]{\input{#2.tcx}}
\newcommand{\subparagraphInput}[3]{\input{#2.tcx}}
\aboveTopic{slacpub7204.tcx}%
\previousTopic{slacpub7204001.tcx}%
\nextTopic{slacpub7204003.tcx}%
\bibfile{slacpub7204005u4.tcx}%
\newmenu{slacpub7204::context::slacpub7204002}{
\docLink{slacpub7204.tcx}[::Top]{Top}%
\sectionLink{slacpub7204001}{slacpub7204001}{Previous: 1. Introduction}%
\sectionLink{slacpub7204003}{slacpub7204003}{Next: 3. Quark pair production in a colour field}%
}
%%%% end of latex2techexplorer block.
%%%% code added by add_nav perl script
\docLink{slacpub7204.tcx}[::Top]{Top of Paper}%

\docLink{pseudo:previousTopic}{Previous Section}%
\bigskip%
%%%% end of code added by add_nav
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%% author definitions added by nc_fix
\def\hbr{\hfil\break} \def\ve{\vfil\eject}
\def\npi{\parindent=0pt } \def\pind{\parindent=6mm }
\def\dn#1{\hfl\hbox{#1}\cr}
\def\0{\over } \def\1{\vec } \def\2{{1\over2}} \def\4{{1\over4}}
\def\5{\bar } \def\6{\partial } \def\7#1{{#1}\llap{/}}
\def\8#1{{\textstyle{#1}}} \def\9#1{{\bf {#1}}}
\def\ul#1{$\underline{\hbox{#1}}$} \def\llp{\hbox to 0pt{\hss /\hskip1.5pt}}
\def\llo{\hbox to 0.2pt{\hss /}} \def\llq{\hbox to 0pt{\hss /\hskip0.5pt}}
\def\so{\supset\hbox to 0pt{\hss $\displaystyle $\hskip1pt}}
\newcommand{\av}[1]{<\!#1\!>}
\def\ds{\displaystyle }
\def\<{\langle } \def\>{\rangle } \def\lb{\left\{} \def\rb{\right\}}
\def\trian{{\scriptscriptstyle \triangle}}
\def\B#1{{\bf #1}} \def\C#1{{\cal #1}} \def\I#1{{\it #1}}
\def\CA{{\cal A}} \def\CD{{\cal D}} \def\CL{{\cal L}}
\def\i{{\rm i}} \def\itrln{\i\tr\log}
\def\bea{\begin{eqnarray}} \def\eea{\end{eqnarray}}
\def\beann{\begin{eqnarray*}} \def\eeann{\end{eqnarray*}}
\def\beq{\begin{equation}} \def\eeq{\end{equation}}
\newcommand{\Dl}{D\!\!\!\!\!\raisebox{1.5ex}{$\leftarrow$}}
\newcommand{\Dls}{D\!\!\!\!\!\raisebox{1.5ex}{$\leftarrow$}\!\!\!\!\!/}
\newcommand{\Gs}{G\!\!\!\!/}
\newcommand{\Ds}{D\!\!\!\!/}
\newcommand{\ls}{l\!\!/}
\newcommand{\ks}{k\!\!\!/}
\newcommand{\bls}{\bar{l}\!\!/}
\newcommand{\bps}{\bar{p}\!\!/}
\newcommand{\psu}{p\!\!/}
\newcommand{\pts}{\tilde{p}\!\!/}
\newcommand{\lts}{\tilde{l}\!\!/}
\newcommand{\qs}{q\!\!\!/}
\newcommand{\epsilons}{\epsilon\!\!/}
\newcommand{\bp}{\bar{p}}
\newcommand{\bl}{\bar{l}}
\newcommand{\dpar}{\Delta_{\}}
\newcommand{\xpar}{x_{\}}
\newcommand{\pdl}{\partial\!\!\!\!\raisebox{1.5ex}{$\leftarrow$}}
\newcommand{\pdls}{\partial\!\!\!\!\raisebox{1.5ex}{$\leftarrow$}\!\!\!\!\!/}
\newcommand{\pds}{\partial\!\!\!/}
\renewcommand{\p}{{\scriptscriptstyle \}}
\renewcommand{\p}{{\scriptscriptstyle \}}
%\newcommand{\t}{\perp}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%% end of definitions added by nc_fix
\section{\usemenu{slacpub7204::context::slacpub7204002}{Semiclassical approach to highenergy scattering}}\label{section::slacpub7204002}\label{sc}
The subject of the present paper is the analysis of diffractive deep
inelastic scattering in terms of the production of quarkantiquark and
quarkantiquarkgluon final states. These processes are considered in the
small$x$ or highenergy limit, where a considerable part of the total
$\gamma^*p$ cross section is diffractive. The gluon densities are known
to grow rapidly in this small$x$ region. Therefore, as already discussed
in \cite{8}, a description of the proton in terms of a classical colour
field should be adequate. This is closely related to the description of
high energy processes in terms of Wilson lines, which has been discussed
by several authors \cite{11,12}. In this section the basis of
the semiclassical approach shall be discussed in more detail. It is similar
to the method developed by Balitsky \cite{13} for small$x$ deep inelastic
scattering. However, in the following more emphasis shall be placed on the
connection with the proton wave functional in the Schr\"odinger picture.
To keep the discussion as simple as possible, consider first the elastic
scattering of a quark off a proton. Although this process is unphysical
since quarks are confined, it can serve to illustrate the method of
calculation. Therefore, in the following confinement is ignored and
quarks are treated as asymptotic states. The generalization to the
physical case of diffractive electroproduction is straightforward and will
be discussed subsequently.
The scattering of a pointlike quark with initial momentum $q$ off a
relativistic bound state with initial momentum $p$ is illustrated in
Fig.~\docLink{slacpub7204002.tcx}[qp]{1}. Let $m_p$ be the proton mass, and $s$ and $t$ the usual
Mandelstam variables for a $2\to 2$ process. In the highenergy limit,
\hspace{.2cm}$s\gg t,\, m_p^2$,\hspace{.2cm} the
contribution from the annihilation of the incoming quark with a constituent
quark of the proton is negligible. The amplitude is dominated by diagrams
with a fermion line going directly from the initial to the final quark
state. Therefore, the proton can be described by a Schr\"odinger wave
functional $\Phi_p[A]$ (cf.~\cite{14}) depending on the gluon field only.
Quarks are integrated out, yielding a modification of the gluonic action.
\begin{figure}[h]
\begin{center}
\parbox[b]{8cm}{\psfig{width=7cm,file=fig1.eps}}\\
\end{center}
\refstepcounter{figure}
\label{qp}
{\bf Fig.\docLink{slacpub7204002.tcx}[qp]{1}} Scattering of a pointlike quark off the proton bound
state.
\end{figure}
For a scattering process the amplitude can be written in the proton rest
frame as
\beq \label{qamp}
<\!q'p'qp\!>=\lim_{T\to\infty}\int DA_T DA_{T} \Phi_{p'}^*[A_T]
\Phi_p[A_{T}]\int_{A_{T}}^{A_T}DA\, e^{iS[A]}<\!q'q\!>_A\, .
\eeq
Here the fields $A_{T}$ and $A_T$ are defined on threedimensional surfaces
at constant times $T$ and $T$, and $A$ is defined in the fourdimensional
region bounded by these surfaces. The field $A$ has to coincide with
$A_{T}$ and $A_T$ at the boundaries and the action $S$ is defined by an
integration over the domain of $A$. The amplitude $<\!q'q\!>_A$
describes the scattering of a quark by the given external field $A$.
The initial state proton, having well defined momentum $\vec{p}$,
is not well localized in space. However, the dominant field configurations
in the proton wave functional are localized on a scale $\La \sim \La_{QCD}$.
Also the field configurations $A(\vec{x},t)$, which interpolate between
initial and final proton state, are localized in space at each time $t$.
Assume that the incoming quark wave packet is localized such that it passes
the origin $\vec{x}=0$ at time $t=0$. At this instant the field configuration
$A(\vec{x},t)$ is centered at
\beq
\vec{x}[A]\equiv\int d^3\vec{x} E_A(\vec{x})\cdot\vec{x}\Bigg/
\int d^3\vec{x} E_A(\vec{x})\, ,
\eeq
where $E_A(\vec{x})$ is the energy density of the field $A(\vec{x},t)$ at
$t=0$. The amplitude (\docLink{slacpub7204002.tcx}[qamp]{1}) can now be written as
\beq
<\!q'p'qp\!>=\lim_{T\to\infty}\int d^3\vec{x}\int DA_T DA_{T}
\Phi_{p'}^*\Phi_p\int_{A_{T}}^{A_T}DA\, e^{iS}
\delta^3(\vec{x}[A]\vec{x})<\!q'q\!>_A\, .
\eeq
Using the transformation properties under translations,
\beq
<\!q'q\!>_{L_{\vec{x}}A}=e^{i(\vec{q}\vec{q}\,')\vec{x}}<\!q'q\!>_A
\quad,\quad\Phi_{p'}^*[L_{\vec{x}}A]\Phi_p[L_{\vec{x}}A]=
e^{i(\vec{p}\vec{p}\,')\vec{x}}\Phi_{p'}^*[A]\Phi_p[A]\,\, ,
\eeq
where
\beq
L_{\vec{x}}A(\vec{y})\equiv A(\vec{y}\vec{x})\, ,
\eeq
one obtains,
\beq
<\!q'p'qp\!>=2m_p(2\pi)^3\delta^3(\vec{p}\,'+\vec{q}\,'\vec{p}\vec{q})
\, \int_A <\!q'q\!>_A\, .
\eeq
Here $\int_A$ denotes the operation of averaging over all field
configurations contributing to the proton state which are localized at
$\vec{x}=0$ at time $t=0$. It is defined by
\beq
\int_A F\equiv\frac{1}{2m_p}\lim_{T\to\infty}
\int DA_T DA_{T} \Phi_{p'}^*\Phi_p\int_{A_{T}}^{A_T}DA\, e^{iS}
\delta^3(\vec{x}[A])F[A]
\eeq
for any functional $F$. The normalization $\int_A\ 1=1$ follows from
\beq
<\!p'p\!>=2p_0(2\pi)^3\delta^3(\vec{p}\,'\vec{p}\,)\ .
\eeq
More complicated processes can be treated in complete analogy as long as the
proton scatters elastically. In particular, the above arguments apply to
the creation of colour singlet quark antiquark pairs \cite{8}
\beq
<\!q\bar{q}p'\gamma^*p\!>=2m_p(2\pi)^3\delta^3(\vec{k}_f\vec{k}_i)\
\int_A <\!q\bar{q}\gamma^*\!>_A\, ,\label{qqa}
\eeq
where $\vec{k}_i$ and $\vec{k}_f$ are the sums of the momenta in the initial
and final states respectively.
The generalization of this simplest diffractive process to
a process with an additional fast final state gluon, $\gamma^*\to q\bar{q}
g$, will be given in Sect.~\docLink{slacpub7204004.tcx}[qqg]{4}. In contrast to the quarkproton
scattering discussed above, here a colour neutral state is scattered off the
proton. Therefore no immediate contradiction with colour confinement arises.
However, it has to be assumed that the hadronization of the produced
partonic state takes place after the interaction with the proton,
which is described in terms of fast moving partons.
When calculating the cross section from Eq.~(\docLink{slacpub7204002.tcx}[qqa]{9}) the square of the
momentum conserving $\delta$function translates into one momentum
conserving $\delta$function using Fermi's trick. This $\delta$function
disappears after the momentum integration for the final state proton,
resulting in a cross section formula identical to scattering by an external
field.
{}From the above discussion a simple recipe for the calculation of
diffractive processes at high energy follows:
The partonic process is calculated in a given external colour field,
localized at $\vec{x}=0$ at time $t=0$. The weighted average over all
colour fields contributing to the proton state is taken on the amplitude
level. We assume that the typical contributing field is smooth on a
scale $\Lambda$ and is localized in space on the same scale $\Lambda$.
Finally, the cross section is calculated using standard formulae for
the scattering off an external field.
For the comparison of diffractive and inclusive structure functions it is
important to know whether the above semiclassical picture also applies
to nonsinglet partonic final states, e.g. to the production of a
colouroctet $q\bar{q}$pair. In this situation the definition of some
analog of the `averaging'operator in terms of a path integral and the
proton wave functional is not obvious. The problem is that the fragmentation
will in general involve both the diffractively produced final state and the
proton remnant. Below we shall assume that such processes can still
be described by calculating the partonic amplitude in a smooth localized
colour field and by taking an appropriate average over different field
configurations at the end.
%%%% code added by add_nav perl script
\bigskip%
\docLink{pseudo:nextTopic}{Next Section}%

\docLink{slacpub7204005u4.tcx}[::Bottom]{Bottom of Paper}%
%%%% end of code added by add_nav