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

\subsection{The longitudinal and kinky strings are produced 
in hadronic collisions.}

\subsection{The number of the cut Pomerons.}

\hspace{1.0em}In the case of a nondiffractive interaction we can
determine $n$ of cut Pomerons or $2n$ produced strings according to
probability \cite{AGK74}
\begin{equation}
\label{SLKS1}
 p^{(n)}_{ij}(\vec{ b}_i-\vec{ b}_j,s)
=c^{-1}\exp{\{-2u(b_{ij}^2,s)\}}
 \frac{[2u(b_{ij}^2,s)]^{n}}{n!}.
\end{equation}

\subsubsection{Separation of the longitudinal soft strings from 
 the kinky hard strings.}
 
\hspace{1.0em}We assume  \cite{ASGABP95}, \cite{WDFHO97} that each cut
Pomeron can be substituted either by the two longitudinal soft strings
or by the two kinky hard strings.

At the moment it is not completely clear how to choose which cut 
pomeron should be 
substituted by longitudinal and which one should be 
substituted by kinky strings.

One recipe is based on the eikonal model \cite{RCT94}, \cite{WDFHO97}
\begin{equation}
\label{SLKS2}u(b_{ij}^2,s)=u_{soft}(b_{ij}^2,s) + u_{hard}(b_{ij}^2,s).
\end{equation}
The soft eikonal part is defined as
\begin{equation}
\label{SLKS3}u_{soft}(b_{ij}^2,s) =
\frac{\gamma _{soft}}{\lambda_{soft} (s)}(s/s_0)^{\Delta_{soft}} 
\exp (-b_{ij}^2/4\lambda_{soft} (s)). 
\end{equation}
The hard part is calculated according to 
\begin{equation}
\label{SLKS4}u_{hard}(b_{ij}^2,s)=
\frac{\sigma_{jet}}{8\pi\lambda_{hard}(s)}
(s/s_0)^{\Delta_{hard}} 
\exp (-b_{ij}^2/4\lambda_{hard} (s)). 
\end{equation} 
The $\sigma_{jet}=0.027$ mbarn and $\Delta_{hard}=0.47$ were found from the fit of the two--jet 
experimental cross section \cite{UA1}. Then from the global fit of 
the total and
elastic cross sections for $pp$ collisions the values of $\gamma_{soft} = 35.5$ mbarn, $\Delta_{soft}=
0.07$ and $R^2_{hard} = R^2_{soft} = 3.56$ GeV$^{-2}$ were
found.

Thus we can examine each cut Pomeron and substitute it by two kinky
strings with probability
\begin{equation}
\label{SLKS5}
P_{hard}(b^2_{ij},s) = \frac{u_{hard}(b_{ij}^2,s)}{u_{soft}(b_{ij}^2,s)+ 
u_{hard}(b_{ij}^2,s)}.
\end{equation}