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\sectionLink{slac-pub-7204-0-0-4}{slac-pub-7204-0-0-4}{Above: 4. Radiation of an additional gluon}%
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\subsection{\usemenu{slac-pub-7204::context::slac-pub-7204-0-0-4-1}{Amplitude}}\label{subsection::slac-pub-7204-0-0-4-1}\label{ampl}
In this section the process $\gamma^*\to q\bar{q}g$ in an external colour
field is calculated in the kinematical region with two final state partons
having high transverse momentum. This extends the analysis of the previous
section (see also \cite{8}) to the case where an additional fast gluon is
radiated. The interaction of the gluon with the external field is treated
in the high-energy approximation in analogy to the two quarks.
\begin{figure}[h]
\begin{center}
\parbox[b]{10cm}{\psfig{width=10cm,file=fig3.eps}}
\end{center}
\refstepcounter{figure}
\label{dqqg}
{\bf Fig.\docLink{slac-pub-7204-0-0-4.tcx}[dqqg]{3}} Diagrams for the process $\gamma^*\to q\bar{q}g$.
\end{figure}
The amplitude is given by the sum of the two diagrams shown in
Fig.~\docLink{slac-pub-7204-0-0-4.tcx}[dqqg]{3}. Separating explicitly the part which describes the
scattering of the gluon off the external field, the amplitude can be given in
the form
\beq
S^b_{\m}=\int\frac{d^4k}{(2\pi)^4} A^a_{\mu\nu}
\frac{-ig^{\nu\rho}}{k^2}B^{ab}_{\rho\s}\e_{(\la')}^{*\s}(k')\, .\label{sbab}
\eeq
Here $A$ refers to the production of the $q\bar{q}g$-system, including the
interactions of the quarks with the external field, and $B$ describes the
scattering of the gluon. Like the amplitude $S_\m$ defined in Sect.~3,
$S^b_\m$ is a $3\times 3$ colour matrix. The index $b$ denotes the colour of
the outgoing gluon.
Using the approximate $k_-$-independence of $B$, which is analogous to the
quark scattering amplitude of the previous section, the $k_-$-integration
in Eq.~(\docLink{slac-pub-7204-0-0-4.tcx}[sbab]{35}) can be performed in such a way that the gluon
propagator goes on shell. Since $A$ and $B$ are now physical amplitudes and
therefore gauge invariant, only physical polarizations contribute to the
gluon propagator connecting $A$ and $B$,
\beq
S^b_{\m}=\sum_{\lambda=1,2}\int\frac{dk_+d^2k_\perp}{2(2\pi)^3}
A^a_{\mu\nu}\epsilon^{*\nu}_{(\lambda)}(k)\frac{1}
{k_+}\epsilon^\rho_{(\lambda)}(k)B^{ab}_{\rho\s}\e_{(\la')}^{*\s}(k')\, .
\eeq
The expression for the high-energy scattering of the gluon is very similar
to the analogous expression for the quark given in the previous section,
\bea
T^g_{\lambda'\lambda}(k,k')&=&
\epsilon^\rho_{(\lambda)}(k)B_{\rho\s}\e_{(\la')}^{*\s}(k')\nn\\
&=&2\pi i\, 2k_+\delta_{\lambda'\lambda}
\delta(k_+'-k_+)(\tilde{F}^\dagger_{\cal A}(k_\perp'-k_\perp)-
(2\pi)^2\de(k'_\t-k_\t))\, .
\eea
A derivation is sketched in Appendix A. The main difference to the quark
case lies in the eikonal factor, which is now taken in the adjoint
representation,
\beq
F_{\cal A}^{ab}(x_\perp)\equiv{\cal A}(F(x_\perp))^{ab}\, .
\eeq
The quark propagators $i/\psu$ and $-i/\ls$, where $l=q-p-k$, are treated in
the high-energy approximation as explained in Sect.~\docLink{slac-pub-7204-0-0-3.tcx}[qq]{3}. The
$p_-$-integration implicit in $A$ can be performed in such a way that
$p$ goes on shell in the first diagram of Fig.~\docLink{slac-pub-7204-0-0-4.tcx}[dqqg]{3} and $l$ goes on
shell in the second diagram. The $p_+$- and $k_+$-integrations are
performed using two of the three $\delta$-functions from the amplitudes for
the scattering off the external field. As a result of these manipulations
the following expression is obtained,
\beq
S^a_{\m}=eg\, 2\pi\delta(q_+-p_+'-k_+'-l_+')\int\frac{d^2p_\perp}{(2\pi)^2}
\frac{d^2k_\perp}{(2\pi)^2}\frac{2q_+\cdot {\cal M}_{\m}\cdot C^a}{\left(Q^2+
\frac{p_\perp^2}{1-\alpha-\alpha'}+\frac{k_\perp^2}{\alpha'}+
\frac{l_\perp^2}{\alpha}\right)}\, .\label{saab}
\eeq
Here $\alpha=\l_+/q_+\, ,\,\alpha'=k_+/q_+$ and ${\cal M}_\m$ describes the
purely partonic part of the amplitude given by
\beq
{\cal M}_\m=\bar{u}_{s'}(\bar{p})\left[\g_\m\frac{1}{\qs-\bps}
\epsilons_{(\lambda')}(k)-\epsilons_{(\lambda')}(k)\frac{1}{\qs-\bls}
\g_\m\right]v_{r'}(\bar{l})\, .\label{mcal}
\eeq
All the non-abelian eikonal factors are combined in $C^a$,
\beq
C^a=\int_{x_\t,y_\t,z_\t} e^{i[x_\t(p_\t-p'_\t)+y_\t(k_\t-k'_\t)
+z_\t(l_\t-l'_\t)]}F(x_\t,y_\t,z_\t)^a\, ,
\eeq
\beq
F(x_\t,y_\t,z_\t)^a={\cal A}(F^\dagger(y_\t))^{ab}
(F^\dagger(x_\t)T^bF(z_\t)) - T^a\, .\label{fxyz}
\eeq
The last term in
Eq.~(\docLink{slac-pub-7204-0-0-4.tcx}[fxyz]{42}) subtracts the unphysical contribution where none of the
partons is scattered by the external field (cf. Eq.~(\docLink{slac-pub-7204-0-0-3.tcx}[amp3]{20})).
One can think of $x_\perp,y_\perp$ and $z_\perp$ as the transverse
positions at which quark, gluon and antiquark penetrate the proton field,
picking up corresponding non-abelian eikonal factors.