source: trunk/documents/UserDoc/UsersGuides/PhysicsReferenceManual/latex/hadronic/theory_driven/BinaryCascade/angularDistributions.tex

%\subsection{Angular distributions \editor{Johannes Peter}}

Angular distributions for final states other than 
nucleon elastic scattering are
calculated analytically, derived from the collision term of the
in-medium relativistic Boltzmann-Uehling-Uhlenbeck equation,
absed on the nucleon nucleon elastic scattering cross-sections:

$$
\sigma_{NN\rightarrow NN}(s,t) = 
{1\over(2\pi)^2s}\left(D(s,t)+E(s,t) + (inverted t,u)\right)
$$

Here $s$, $t$, $u$ are the Mandelstamm variables, 
$D(s,t)$ is the direct term, and $E(s,t)$ is the exchange
term, with
\begin{eqnarray}
D(s,t) = &{(g_{NN}^\sigma)^4(t-4m^{*}2)^2 \over2(t-m_\sigma^2 )^2} +
          {(g_{NN}^\omega)^4(2s^2+2st+t^2-8m^{*2}s+8m^{*4}) \over (t-m_\omega^2)^2} + \nonumber\\
         & {24(g_{NN}^\pi)^4m^{*2}t^2\over (t-m_\pi^2)^2} -  
           {4(g_{NN}^\sigma g_{NN}^\omega)^2(2s+t-4m^{*2})m^{*2}\over
          (t-m_\sigma^2)(t-m_\omega^2)}, \nonumber
\end{eqnarray}
and
\begin{eqnarray}
E(s,t) = &{(g_{NN}^\sigma)^4\left( t(t+s)+4m^{*2}(s-t)\right)\over
          8(t-m_\sigma^2)(u-m_\sigma^2) }+
         {(g_{NN}^\omega)^4(s-2m^{*2})(s-6m^{*2}))\over 2(t-m_\omega^2)(u-m_\omega^2) } - \nonumber\\
         &{6(g_{NN}^\pi)^4(4m^{*2}-s-t)m^{*4}t\over (t-m_\pi^2)(u=m_pi^2) }+
         {3(g_{NN}^\sigma g_{NN}^\pi)^2 
           m^{*2} (4m^{*2}-s-t)(4m^{*2}-t) \over (t-m_\sigma^2)(u-m_\pi^2) } + \nonumber\\
         &{3(g_{NN}^\sigma g_{NN}^\pi)^2 
           t(t+s)m^{*2}\over 2(t-m_\pi^2)(u-m_\sigma^2) } + 
         {(g_{NN}^\sigma g_{NN}^\omega)^2
           t^2-4m^{*2}s-10m^{*2}t+24m^{*4}\over4(t-m_\sigma^2)(u-m_\omega^2) } + \nonumber\\
         &{(g_{NN}^\sigma g_{NN}^\omega)^2
           (t+s)^2-2m^{*2}s+2m^{*2}t\over 4(t-m_\omega^2)(u-m_\sigma^2)} + 
         {3(g_{NN}^\omega g_{NN}^\pi)^2
           (t+s-4m^{*2})(t+s-2m^{*2})\over (t-m_\omega^2)(u-m_\pi^2)} + \nonumber\\
         &{3(g_{NN}^\omega g_{NN}^\pi)^2
           m^{*2} (t^2-2m^{*2}t)\over (t-m_\pi^2)(u-m_\omega^2)}. \\
\end{eqnarray} 
Here, in this first release, the in-medium mass was set to the free mass, and the nucleon
nucleon coupling constants used were 1.434 for the $\pi$, 7.54 for the $\omega$, and 6.9 for
the $\sigma$. This formula was used for elementary hadron-nucleon differential cross-sections 
by scaling teh center of mass energy squared accordingly.

Finite size effects were taken into account at the meson nucleon vertex,
using a phenomenological form factor (cut-off) at each vertex.