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%% subsection 4.2 Colour structure [slac-pub-7204-0-0-4-2 in slac-pub-7204-0-0-4: slac-pub-7204-0-0-4-3]
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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-7204-0-0-4-2}{Colour structure}}\label{subsection::slac-pub-7204-0-0-4-2}\label{cols}
In the phase space region with two of the three final state particles having
high $p_\perp$, in the $\gamma^*p$ center-of-mass system,
the form of the three-particle colour
factor $C^a$ simplifies significantly. To see this, the three possible
configurations, i.e. high-$p_\perp$ $q\bar{q}$-jets, high-$p_\perp$
$qg$-jets and high-$p_\perp$ $\bar{q}g$-jets, have to be distinguished.
Consider first the case of high-$p_\perp$ quark and antiquark, i.e.
$p_\perp'^2,\, l_\perp'^2\gg\Lambda^2$. In analogy to the results of
Sect.~\docLink{slac-pub-7204-0-0-3.tcx}[qq]{3} a leading twist contribution to diffraction can only appear
if the gluon is relatively soft, i.e. in the region of small $k_\perp'^2$
and $\alpha'$. Therefore the relations $k_\t'^2 \sim \La^2$ and
$\alpha'\ll 1$ are used in the calculations below. This will also be
justified by the final formulae, which show that these kinematical regions
dominate the integrations.
The assumption of a smooth external field implies small transverse momentum
transfer from the proton, i.e. $|p_\t''| \sim \La$, where
$p_\perp''= p_\perp'-p_\perp$. The $p_\perp$-integration
in Eq.~(\docLink{slac-pub-7204-0-0-4.tcx}[saab]{39}) can be trivially replaced by a $p_\perp''$-integration,
substituting at the same
time
\beq
p_\perp=p_\perp'-p_\perp''\qquad\mbox{and}\qquad l_\perp=-p_\perp'+p_\perp''
-k_\perp\, .\label{pl}
\eeq
Neglecting $p_\perp''$ in ${\cal M}$ and in the energy denominator in
Eq.~(\docLink{slac-pub-7204-0-0-4.tcx}[saab]{39}), which is justified since $|p_\perp''^2|\ll|p_\perp'^2|,\,
|l_\perp'^2|$, the only remaining $p_\perp''$-dependence is in the colour
factor $C^a$. This simplifies the $p_\perp''$-integration to
\beq
\int\frac{d^2p_\perp''}{(2\pi)^2}\ C^a\, .
\eeq
Defining $\Delta\equiv p'+k'+l'-p-k-l$ to be the total momentum
transferred from the proton, $C^a$ can be given in the from
\beq
C^a =\int_{x_\t} e^{-ix_\perp\Delta_\perp}\int_{y_\t,z_\t}
e^{i[y_\perp(k_\t-k'_\t)+z_\perp(l_\t-l'_\t)]}F(x_\perp,x_\perp+y_\perp,
x_\perp+z_\perp)^a\, ,
\eeq
where $l_\perp$ is given by Eq.~(\docLink{slac-pub-7204-0-0-4.tcx}[pl]{43}). The $p_\perp''$-integration
gives a $\delta$-function for the variable $z_\perp$, thus resulting in the
final formula
\beq
\int\frac{d^2p_\perp''}{(2\pi)^2}\ C^a = \int_{x_\t}\ e^{-ix_\perp
\Delta_\perp}\int_{y_\t}\ e^{iy_\perp(k_{\perp}-k_{\perp}^{\prime})}
F(x_\perp,x_\perp+y_\perp,x_\perp)^a\, .
\eeq
This result shows that in the kinematical situation with two high-$p_\perp$
quark jets and a relatively soft gluon the leading twist contribution is not
affected by the transverse separation of the quarks. It is the
transverse separation between quark-pair and gluon which tests large
distances in the proton field and which can lead to non-perturbative effects.
The colour-singlet projection of the colour-tensor $C$ reads
\beq
S(C)=\2\ \mbox{tr}[C^a T^a]\, .
\eeq
Using the identity
\beq
{\cal A}(U)^{ab}=2\ \mbox{tr}[U^{-1}T^aUT^b]\quad,\quad U\in \mbox{SU(3)}\, ,
\eeq
the contribution relevant for diffraction, i.e. the production of a
colour-singlet $q\bar{q}g$-system, becomes
\beq
\int\frac{d^2p_\perp''}{(2\pi)^2}\ S(C)=\int_{x_\t} e^{-ix_\t\Delta_\t}\,
\frac{1}{4}\, \mbox{tr}[\tilde{W}^{\cal A}_{x_\perp}(k_\perp-k_\perp')]\, ,
\label{cf1}
\eeq
\beq
W^{\cal A}_{x_\perp}(y_\perp)={\cal A}(F^\dagger(x_\perp+y_\perp)F(x_\perp))
-1\, .
\eeq
This is analogous to the quark-pair production of the previous section (cf.
Eq.~(\docLink{slac-pub-7204-0-0-3.tcx}[amp3]{20})). However, now the two lines probing the field
at positions $x_\perp$ and $x_\perp+y_\perp$ correspond to matrices in the
adjoint representation. An intuitive explanation of this result is that
the two high-$p_\perp$ quarks are close together and are rotated in colour
space like a vector in the octet representation. This situation is
illustrated in Fig.~\docLink{slac-pub-7204-0-0-4.tcx}[qqj]{4}.
\begin{figure}[h]
\begin{center}
\parbox[b]{10cm}{\psfig{width=10cm,file=fig4.eps}}\\
\end{center}
\refstepcounter{figure}
\label{qqj}
{\bf Fig.\docLink{slac-pub-7204-0-0-4.tcx}[qqj]{4}} Space-time picture in the case of fast, high-$p_\perp$
quark and antiquark, passing the proton at small transverse separation
with a relatively soft gluon further away.
\end{figure}
To make this last statement more precise, recall that an upper bound for the
Ioffe-time of the fluctuation with two high-$p_\perp$ quarks is given by
$q_0/p_\perp^2$. This means that the distance between the point
where the virtual photon splits into the $q\bar{q}$-pair and the proton can
not be larger than $q_0/p_\perp^2$. As long as the pair shares the
longitudinal momentum of the photon approximately equally, i.e.
$\a (1-\a) = {\cal O}(1)$, the opening angle is
$\sim p_\perp/q_0$. Therefore, the transverse distance between quark and
antiquark is $\sim 1/p_\perp\ll 1/\Lambda$ when they hit the proton.
In the case where quark and gluon have high transverse momentum
and the relatively soft, low-$p_\perp$ antiquark is responsible for the
non-perturbative interaction, we have
$|p_\perp'^2|\simeq|k_\perp'^2|\gg|l_\perp'^2| \sim \La^2$
and $\alpha''\equiv l_+/q_+\ll 1$.
The calculation proceeds along the lines of the soft gluon case described
previously. It is convenient to make the integration over the soft
transverse momentum explicit by substituting $d^2l_\perp$ for $d^2k_\perp$
in Eq.~(\docLink{slac-pub-7204-0-0-4.tcx}[saab]{39}). The $p_\perp$-integration is replaced by a
$p_\perp''$-integration and the $p_\perp''$-dependence entering ${\cal M}$
and the energy-denominator via the relations $p_\perp=p_\perp'-p_\perp''$
and $k_\perp=-p_\perp'+p_\perp''-l_\perp$ is neglected. The
$p_\perp''$-integration gives a $\delta$-function for the variable $y_\perp$,
leading to the following result for the singlet projection of the colour
tensor $C$,
\beq
\int\frac{d^2p_\perp''}{(2\pi)^2}\ S(C)=\int_{x_\t}\ e^{-ix_\perp
\Delta_\perp}\,\frac{2}{3}\,\mbox{tr}[\tilde{W}_{x_\perp}(l_\perp-l_\perp')]
\, .
\eeq
The function $W_{x_\perp}$ has been defined in Eq.~(\docLink{slac-pub-7204-0-0-3.tcx}[wdef]{22}).
Now quark and gluon, having high transverse momentum,
are close together and are colour rotated like a vector in the fundamental
representation (see Fig.~\docLink{slac-pub-7204-0-0-4.tcx}[qgj]{5}). Therefore, the colour structure of the
amplitude is the same as for the quark-pair production of Sect.~3.
\begin{figure}[h]
\begin{center}
\parbox[b]{10cm}{\psfig{width=10cm,file=fig5.eps}}\\
\end{center}
\refstepcounter{figure}
\label{qgj}
{\bf Fig.\docLink{slac-pub-7204-0-0-4.tcx}[qgj]{5}} Space-time picture in the case of fast, high-$p_\perp$
quark and gluon, passing the proton at small transverse separation
with a relatively soft antiquark further away.
\end{figure}
The case of high-$p_\perp$ $\bar{q}g$-jets is completely analogous to the
case of high-$p_\perp$ $qg$-jets and will not be discussed separately.