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%% subsection 4.3 Longitudinal cross section [slac-pub-7204-0-0-4-3 in slac-pub-7204-0-0-4: slac-pub-7204-0-0-4-4]
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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-7204-0-0-4-3}{Longitudinal cross section}}\label{subsection::slac-pub-7204-0-0-4-3}\label{lcs}
In the phase space region of $q\bar{q}$-jets with high transverse momentum
described at the beginning of the previous subsection the phase space
integration can be given in the form
\bea
d\Phi^{(3)}&=&\frac{2}{(2\pi)^9}\, \frac{dp_+'}{2p_+'}d^2p_\perp'
\frac{dk_+'}{2k_+'}d^2k_\perp'\frac{dl_+'}{2l_+'}d^2l_\perp'\delta(q_+-p_+'
-k_+'-l_+')\label{dphi}\\ \nonumber\\&=&\frac{1}{8(2\pi)^8}\frac{Q^2}{q_+x
\alpha'}d\alpha d\alpha'd\xi d^2k_\perp'd^2\Delta_\perp\, .\nonumber
\eea
One obtains analogous expressions for the regions of high transverse
momentum $qg$-and $\bar{q}g$-jets by replacing $\alpha'$
and $k_\perp'$ by the longitudinal momentum fraction and the
transverse momentum of the soft parton.
Explicit formulae for the partonic part ${\cal M}$ of the amplitude defined
in Sect.~\docLink{slac-pub-7204-0-0-4.tcx}[ampl]{4.1} are given in Appendix C. Treating the colour factors
as described in the previous subsection explicit expressions for the
different contributions to the longitudinal diffractive structure function
are obtained. To stress the common features of these contributions from the
different phase space regions generic kinematical variables are introduced.
The longitudinal momentum fraction and the transverse momentum of one of the
high-$p_\perp$ jets are denoted by $\gamma$ and $r_\perp$ respectively.
The longitudinal momentum fraction of the relatively soft parton
is denoted by $\gamma'$, its transverse
momenta before and after the interaction with the external field are denoted
by $s_\perp$ and $s_\perp'$. For example, in the case of quark and
antiquark jets this notation means that
$\gamma=\alpha\, ,$ $\gamma'=\alpha'\, ,$ $r_\perp=p_\perp\, ,$
$s_\perp=k_\perp\, ,$ and $s_\perp'=k_\perp'\, .$
The different contributions to $F_L^D$ can be given in the form
\beq
F_L^{D,n}(x,Q^2,\xi)=\frac{16\alpha_S}{(2\pi)^5\xi}\ \beta^2(1-\beta)g_L^{(n)}
(\beta)\int_0^1d\gamma\, ,\qquad n=1,2\, .\label{fldi}
\eeq
Here the contribution from the region of high-$p_\perp$
$q\bar{q}$-jets is labeled by $n=1$ and the sum of the contributions from
the regions of high-$p_\perp$ $qg$- and $\bar{q}g$-jets is labeled by
$n=2$. The trivial $\gamma$-integration has been kept
explicitly to allow a more detailed description of the final states.
The $\beta$-spectrum can be different for the two contributions of
Eq.~(\docLink{slac-pub-7204-0-0-4.tcx}[fldi]{53}). Its dependence on the details of the colour field of the
proton is given by the two dimensionless functions
\bea
g_L^{(1)}(\beta)&=&\int_0^\infty\frac{(1+u)^2u\,du}{32(\beta+u)^2}\int
d^2s_\perp'(s_\perp'^2)^2\int_{x_\perp}\left|\int\frac{d^2s_\perp}
{(2\pi)^2}\cdot\frac{\mbox{tr}[\tilde{W}^{\cal A}_{x_\perp}(s_\perp-s_\perp')]
\, t_L^{ij}}{s_\perp'^2(\beta+u)+s_\perp^2(1-\beta)}\right|^2\label{g1}
\\&&\nonumber\\
g_L^{(2)}(\beta)&=&\int_0^\infty\frac{2(1+u)^2du}{9(\beta+u)^2}\int
d^2s_\perp'(s_\perp'^2)\int_{x_\perp}\left|\int\frac{d^2s_\perp}{(2\pi)^2}
\cdot\frac{s_\perp\cdot\mbox{tr}[\tilde{W}_{x_\perp}(s_\perp-s_\perp')]}
{s_\perp'^2(\beta+u)+s_\perp^2(1-\beta)}\right|^2,\label{g2}
\eea
where the tensor $t^{ij}_L$ is given by
\beq
t_L^{ij}=\delta^{ij}+\frac{2s_\perp^is_\perp^j}{s_\perp'^2}\cdot
\frac{1-\beta}{\beta+u}\, ,\qquad i,j=1,2\, .\label{tij}
\eeq
Note, that in the modulus squared in Eq.~(\docLink{slac-pub-7204-0-0-4.tcx}[g1]{54}) the appropriate
contraction of the indices of the two tensors $t_L^{ij}$ is assumed.
The integration variable $u$ can be related to the old kinematic variables
by
\beq
\gamma'=\frac{s_\perp'^2}{M^2}(1+u)\, .\label{udef}
\eeq
The functions $g^i_L$ have been given in terms of an integral in $u$ to make
it obvious that they do not depend on any kinematical variable other than
$\beta$.
For the transverse momentum $r_\perp$ of the two hard jets the relation
\beq
r_\perp^2=\gamma(1-\gamma)\left(M^2-\frac{s_\perp'^2}{\gamma'}\right)=
\gamma(1-\gamma)M^2\frac{u}{1+u}\label{rp}
\eeq
can be derived. Using this relation and the $\gamma$- and
$u$-distributions given by Eq.~(\docLink{slac-pub-7204-0-0-4.tcx}[fldi]{53}) and Eqs.~(\docLink{slac-pub-7204-0-0-4.tcx}[g1]{54}),(\docLink{slac-pub-7204-0-0-4.tcx}[g2]{55})
the $r_\perp$-distribution of the jets in the final state can be easily
recovered.
The above results show that within our model $F_L^D$ has a leading twist
contribution, suppressed by one power of $\alpha_S$. For $\be$ not too close
to 0 and 1 the integrals in $\gamma$ and $u$ are finite and dominated
by the region where both $\gamma$ and $u$ are ${\cal O}(1)$. This justifies
the assumption that $\gamma'\ll 1$ and $s_\perp'^2\ll r_\perp^2$ made at the
beginning of this section. Eq.~(\docLink{slac-pub-7204-0-0-4.tcx}[rp]{58}) also shows that jets with
transverse momentum of order $M$ dominate $F_L^D$. This has to be contrasted
with the case of transverse photon polarization, where the leading
contribution comes from the production of a $q\bar{q}$-pair with small
transverse momenta as discussed in Sect.~\docLink{slac-pub-7204-0-0-3.tcx}[qq]{3}.
The details of the $\beta$-spectrum and the $r_\perp$-distribution depend
on the average over the proton field, which enters via non-abelian eikonal
factors in the adjoint and fundamental representation (see
Eqs.~(\docLink{slac-pub-7204-0-0-4.tcx}[g1]{54}),(\docLink{slac-pub-7204-0-0-4.tcx}[g2]{55})). This is a truly non-perturbative effect
which, in our model, is responsible for leading twist diffraction. It
is realized by one of the three produced partons, which is slower by a
factor $\sim \Lambda^2/M^2$ and can develop a large transverse separation
from the two other partons.
Let us finally consider the behaviour of the diffractive structure function
at small $\beta$, i.e. at large invariant masses. The $u$-integration in
$g^{(1)}_L(\beta)$ is divergent for $\beta=0$. The behaviour of $g^{(1)}_L
(\beta)$ at small $\beta$ is determined by the integration over small values
of $u$. This behaviour can be obtained assuming $\beta\ll 1$, $u\ll 1$ and
$u+\beta\ll 1$ in Eq.~(\docLink{slac-pub-7204-0-0-4.tcx}[g1]{54}). In this region the second contribution of
the tensor $t^{ij}_L$ in Eq.~(\docLink{slac-pub-7204-0-0-4.tcx}[tij]{56}) dominates and the following
expression for $g^{(1)}_L$ is obtained,
\bea
g_L^{(1)}(\beta\to 0)&=&\int_0^\infty\frac{u\,du}{8(\beta+u)^4}\int
d^2s_\perp'\int_{x_\perp}\left|\int\frac{d^2s_\perp}{(2\pi)^2}\cdot
\frac{\mbox{tr}[\tilde{W}^{\cal A}_{x_\perp}(s_\perp-s_\perp')]\ s_\perp^i
s_\perp^j}{s_\perp^2}\right|^2
\\&&\nonumber\\
&=&\frac{1}{48\beta^2}\int d^2s_\perp'\int_{x_\perp}\left|\int
\frac{d^2s_\perp}{(2\pi)^2}\cdot\frac{\mbox{tr}[\tilde{W}^{\cal A}_{x_\perp}
(s_\perp-s_\perp')]\ s_\perp^is_\perp^j}{s_\perp^2}\right|^2\, .
\eea
This means that $F_L^D(x,Q^2,\xi)$ approaches a constant value at fixed
$\xi$ and $\beta\to 0$. For the inclusive structure function, where
one has to integrate over $\be$, this implies a growth $\sim \ln{(1/x)}$.
Since $g^{(2)}_L$ is less singular than $g^{(1)}_L$ at small $\beta$, the
region of large diffractive masses is dominated by the configuration with
high-$p_\perp$ $q\bar{q}$-jets and a relatively soft gluon.