source: trunk/documents/UserDoc/DocBookUsersGuides/PhysicsReferenceManual/latex/hadronic/theory_driven/BinaryCascade/inclusive.tex @ 1229

%\subsection{Inclusive cross-sections \editor{Johannes Peter}}
Experimental data are used in the calculation of the total, inelastic and
elastic cross-section wherever available. 
\subsubsection{hadron-nucleon scattering}
For the case of proton-proton(pp) and proton-neutron(pn) collisions, as well as
$\pi^=$ and $\pi^-$ nucleon collisions, experimental
data are readily available as collected by the Particle Data Group (PDG) for both
elastic and inelastic collisions. We use a tabulation based on a sub-set of
these data for $\sqrt{S}$ below 3~GeV. For higher energies, parametrizations from
the CERN-HERA collection are included. 

% As an example, FixME: An example plot.
%Figure \ref{np} shows a
%comparison of the {\sc Geant4} scattering cross-sections with data from the
%PDG\cite{PDG2002}.
\subsection{Channel cross-sections}
A large fraction of the cross-section in individual channels involving meson
nucleon scattering can be modeled as resonance excitation in the s-channel.
This kind of interactions show a resonance structure in the energy dependency of
the cross-section, and can be modeled using the Breit-Wigner function

$$\sigma_{res}(\sqrt{s}) = \sum_{FS}{}{2J+1\over (2S_1+1)(2S_2+1)}
{\pi\over k^2}{\Gamma_IS\Gamma_FS\over (\sqrt{s}-M_R)^2+\Gamma/4},$$

Where $S1$ and $S2$ are the spins of the two fusing particles, $J$ is the spin
of the resonance, $\sqrt(s)$ the energy in the center of mass system, $k$ the
momentum of the fusing particles in the center of mass system, $\Gamma_IS$ and
$\Gamma)FS$ the partial width of the resonance for the initial and final state
respectively. $M_R$ is the nominal mass of the resonance.

The initial states included in the model are pion and kaon nucleon
scattering. The product
resonances taken into account are the Delta resonances with masses 1232, 1600,
1620, 1700, 1900, 1905, 1910, 1920, 1930, and 1950~MeV, the excited nucleons with
masses of 1440, 1520, 1535, 1650, 1675, 1680, 1700, 1710, 1720, 1900, 1990,
2090, 2190, 2220, and 2250~MeV, the Lambda, and its excited states at 1520,
1600, 1670, 1690, 1800, 1810, 1820, 1830, 1890, 2100, and 2110~MeV, and the
Sigma and its excited states at 1660, 1670, 1750, 1775, 1915, 1940, and
2030~MeV. 

% FixME: An example plot.

\subsection{Mass dependent resonance width and partial width}
During the cascading, the resonances produced are assigned reall masses, with
values distributed according to the production cross-section described above.
The concrete (rather than nominal) masses of these resonances may be small
compared to the PDG value, and this implies that some channels may not be open 
for decay. In general it means, that the partial and total width will depend 
on the concrete mass of the resonance.  We are using the 
UrQMD\cite{UrQMD1.BC}\cite{SoH92} approach for calculating these actual width,
\begin{equation}
\Gamma_{R\rightarrow 12}(M) = 
(1+r){\Gamma_{R\rightarrow 12}(M_R)\over p(M_R)^{(2l+1)}}{M_R\over M}
{p(M)^{(2l+1)}\over 1+r(p(M)/p(M_R))^{2l}}.
\label{width}
\end{equation}
Here $M_R$ is the nominal mass of the resonance, $M$ the actual mass, $p$ is
the momentum in the center of mass system of the particles, $L$ the angular
momentum of the final state, and r=0.2.

\subsection{Resonance production cross-section in the t-channel}
In resonance production in the t-channel, single and double resonance 
excitation in nucleon-nucleon collisions are taken into account.  The 
resonance production cross-sections are as much as possible based on 
parametrizations of experimental data\cite{res.BC} for proton proton
scattering.  The basic formula used is motivated from the form of the 
exclusive production cross-section of the $\Delta_{1232}$ in proton proton 
collisions:

$$\sigma_{AB} = 2\alpha_{AB}\beta_{AB}{\sqrt{s}-\sqrt{s_0}
\over (\sqrt{s}-\sqrt{s_0})^2+\beta_{AB}^2}
\left ( {\sqrt{s0}+\beta_{AB}\over \sqrt{s}}\right )^{\gamma_{AB}} $$

The parameters of the description for the various channels 
are given in table\ref{Parameters}. For all other channels, the 
parametrizations were derived from these by adjusting the threshold
behavior.
\begin{table*}[hbt]
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|}\hline
Reaction & $\alpha$ & $\beta$ & $\gamma$ \\ \hline \hline
$\rm pp\rightarrow p\Delta_{1232}$ & 25~mbarn & 0.4~GeV & 3  \\ \hline 
$\rm pp\rightarrow \Delta_{1232}\Delta_{1232}$ & 1.5~mbarn & 1~GeV & 1  \\ \hline 
$\rm pp\rightarrow pp^{*}$    & 0.55~mbarn & 1~GeV & 1 \\ \hline 
$\rm pp\rightarrow p\Delta_{*}   $ & 0.4~mbarn &  1~GeV & 1  \\ \hline 
$\rm pp\rightarrow \Delta_{1232}\Delta^{*}$  & 0.35~mbarn & 1~GeV & 1 \\ \hline
$\rm pp\rightarrow \Delta_{1232}N^{*}$  & 0.55~mbarn & 1~GeV & 1 \\ \hline
\end{tabular}
\caption{\label{Parameters}
Values of the parameters of the cross-section formula for 
the individual channels.
}
\end{center}
\end{table*}

% FiXME: The $\Delta_{1232}$ and omega production as examples.

The reminder of the cross-section are
derived from these, applying detailed balance. Iso-spin invariance
is assumed. The formalism used to apply detailed balance is
\begin{equation}
\sigma(cd\rightarrow ab) = 
\sum_{J,M}{}{\left < j_cm_cj_dm_d\parallel JM\right >^2
\over \left < j_am_aj_bm_b\parallel JM\right >^2}  \\
 {(2S_a+1)(2S_b+1)\over (2S_c+1)(2S_d+1)}\\
 {\left< p_{ab}^2\right> \over \left< p_{cd}^2\right> }\sigma(ab\rightarrow cd)
\end{equation}