source: trunk/documents/UserDoc/DocBookUsersGuides/PhysicsReferenceManual/latex/hadronic/theory_driven/PartonString/KinkyStringExcitation.tex @ 1211

\section {Kinky string excitation.}

\hspace{1.0em}Having sampled the configuration of kinky strings  
we generate outgoing gluon--kink momenta.
 
We assume that kinky strings are produced as result of $gg \rightarrow
gg$ hard interactions. Our generation of the outgoing gluons (kinks)
momenta is based on the two-jets inclusive production cross section:
\begin{equation}
\label{KSE1} \frac{d\sigma_{gg}}{dx_g^{+}dx_g^{-}d\cos{\theta}}=
 f(x_g^{+},Q^2)f(x_g^{-},Q^2)\frac{d\sigma_{gg}(\hat{s})}{d\cos{\theta}},
\end{equation}
where we take $\hat{s}=Q^2=x_g^{+}x_g^{-}s$ and $s$ is the total
colliding system center of mass energy squared, which calculated using
 $x_i$ and $q_{ti}$ and $m^2_i$ string end partons. The value
of $s$ should be large enough to produce gluons with the transverse momentum
cutoff $Q^2_{0} = 2 \ GeV^2$ choosen.  The QCD gluon -- gluon
interaction cross section
\begin{equation}
\label{KSE2}
\frac{d\sigma_{gg}(\hat{s})}{d\cos{\theta}} = \frac{9\pi
\alpha^2_s(Q^2)}{32s}\frac{(3+\cos^2{\theta})^3}{(1-\cos^2{\theta})^2}
\end{equation}
was calculated in the Born approximation \cite{CKR77}.  The $\theta$ is
the scattering angle in the center of mass of the parton-parton system
and $-z_0 < \cos{\theta} < z_0$ with
\begin{equation}
\label{KSE3} z_0 = \sqrt{1 - \frac{4Q^2_0}{sx^{+}x^{-}}}.
\end{equation}  
The QCD running coupling constant 
\begin{equation}
\label{KSE4}\alpha_s(Q^2) = \frac{12 \pi}{25 \ln{(Q^2/\Lambda^2)}},
\end{equation} 
which is corresponding to four flavours and $\Lambda^2 = 0.01$ GeV$^2$
is taken.

$f(x,Q^2)$ is the momentum fraction distribution of gluons in hadron. It
is choosen from\cite{CKMT95}:
\begin{equation}
\label{KSE5}
f(x,Q^2) = A(Q^2)\frac{x^{-\Delta(Q^2)}}{x}(1 - x)^{n(Q^2)+3} 
\end{equation}
with parameters $\Delta(Q^2) = \Delta (1 + \frac{2Q^2}{Q^2 + 1.12})$ and
$n(Q^2) = \frac{3}{2}(1+\frac{Q^2}{Q^2 + 3.55})$ and $A(Q^2)$ is
calculated from energy-momentum conservation sum rule.  At $Q^2_{0} = 2
\ GeV^2$
\begin{equation}
\label{KSE6}
f(x,Q^2_{0}) = 1.71\frac{x^{-0.18}}{x}(1 - x)^{5}. 
\end{equation}

Thus the MC procedure to build the kinky strings can be outlined as 
follows:
%
\begin{itemize}

\item Sample $x_i$, ${\bf q_{it}}$ and $m^2_i$, where $i = 1,2,...,2n$,
 for partons, which will be on the $2n$ string ends for both the soft
 and kinky strings.

\item For each pair of kinky string calculate total center of mass energy 
$s$ and sample $x^{+}_{min} < x^{+}$ and $x^{-}_{min} < x^{-}$, where 
$x_{min} = 2Q_{0}/\sqrt{s}$,  
for ingoing gluon momenta
using gluon distribution function defined by Eq.($\ref{KSE6}$) at
$Q^2_0 = 2$ GeV$^2$.

\item Sample the outgoing gluon center of mass scattering angle $\theta$ 
using Eq. ($\ref{KSE2}$).

\item For each kinky string recalculate parton string end energies and
momenta.

\end{itemize}
This procedure should be improved taking into account initial and final state 
gluon radiation.