Three body final states : in the vacuum

GiBUU

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Three body final states : $\mathbf{X \rightarrow c d e}$ in the vacuum

In analogy to the two-particle final and utilizing equation (1.7)

$\displaystyle \frac{d\sigma_{a b \rightarrow c d e }}{d\mu_c\, d\mu_d\,d\mu_e\ d\vert\vec{p}_{c}\vert\, d\Omega_c \, d\vert\vec{p}_{d}\vert\, d\phi_d}(s)$	$\textstyle =$	$\displaystyle \frac{1}{8\left(2\pi\right)^5} \frac{1}{p_{ab} \sqrt{s}} \frac{\l... ..._{d}\right\vert}{E_c E_d} \mathcal{A}_c(\mu_{c},p_c) \mathcal{A}_d(\mu_{d},p_d)$
		$\displaystyle \times \, \mathcal{A}_e(\mu_{e},p_e)\left\vert\mathcal{M}_{a b \rightarrow c d e} (s)\right\vert^2 \,.$	(2.15)

and the variable transformation

$\begin{displaymath} y_i(\mu_i)=2\,\arctan\left[2\,\frac{\mu_i-M_i^0}{\Gamma^0_i}\right]; i\in\left\lbrace c,d\right\rbrace \end{displaymath}$

we get

$\begin{eqnarray*} \frac{d\sigma_{a b \rightarrow c d e }}{dy_c\, dy_d\,dy_e\ d\v... ... \, \frac{d\mu_c}{dy_c} \frac{d\mu_d}{dy_d} \frac{d\mu_e}{dy_e}. \end{eqnarray*}$

Hence we need to choose $y_c, y_d, y_e, \vert\vec{p}_{c}\vert, \Omega_c$ and $\vert\vec{p}_{d}\vert$ independent of each other. The limits for the

are given by the smallest and largest possible masses. The absolute values of the momenta $\vert\vec{p}_{i}\vert$ are limited by the energy conservation. Evaluated in CM-System

$\begin{displaymath}\sum_i E_{i}=\sum_i \sqrt{m_i^2+p_i^2}=\sqrt{s}\end{displaymath}$

this demand sets the limits to

$\begin{displaymath}\vert\vec{p}_{i}\vert< \sqrt{s} \, . \end{displaymath}$

The value of $\Omega_c$ is choosen by choosing $\cos(\theta)\in[-1,1]$ and $\phi\in[0,2\pi]$ by random. Once we have chosen those parameters, the full kinematics is fixed. Having choosen the parameters according to flat distribution we can define a probability to accept such a configuration by

$\begin{displaymath} p_{accept}= \frac{\frac{\left\vert\vec{p}_{c}\right\vert \le... ...},p_e) \frac{d\mu_d}{dy_d} \frac{d\mu_e}{dy_e}}{\mathcal{MAX}} \end{displaymath}$

where $\mathcal{MAX}$ is chosen such that it is larger than the maximum of the nominator. With a Monte-Carlo decision we now accept or reject this configuration. We evaluate different configurations until we get one, which is accepted.

Next: Medium Corrections Up: Final State Decisions Previous: Two body final states Contents

Oliver Buss 2005-03-16