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Reactions of the type $\mathbf{a\ b \rightarrow c_1, c_2, c_3, \ldots}$

The general definition of the cross section is given by (e. g. [H⁺02])

$\displaystyle d\sigma_{a\ b \rightarrow f_1, f_2, f_3, \ldots,f_n}=\left( 2 \pi... ...nal} \frac{\mathcal{M}_{a\ b \rightarrow f_1, f_2, f_3, \ldots,f_n}}{j} d\Phi_n$

(1.5)

where denotes the n-particle phase space of the final-state particles, $\mathcal{S}_{final}$ stands for the symmetry factor of the final state and

$\begin{displaymath} j=4\sqrt{\left(p_a p_b\right)^2-m_a^2 m_b^2} \end{displaymath}$

represents the flux factor of the particles

and

. This flux factor can be expressed in the center of mass system by

$\begin{displaymath} j=4 p_{cm} \sqrt{s} \ \end{displaymath}$

with the CM-momentum $p_{cm}$ of the particles

and

. In [Leh03]^1.2 it was shown that one can express the cross section for the production of unstable particles

and

in the final state by

$\displaystyle \frac{d\sigma_{a b \rightarrow c d}}{d\mu_c\, d\mu_d\, d\Omega}(s... ..._d(\mu_d,\Omega)) \left\vert\mathcal{M}_{a b \rightarrow c d } (s)\right\vert^2$

(1.6)

with $p_{ab}$ and $p_{cd}$ denoting the CM momenta of the $a\,b$ and the $c\,d$ -system. Here one needs to assume that the Matrix element is only dependend on

. The spectral functions depend only in the medium, which explicitly breaks Lorentz-invariance, on the four-momenta of the particles. Considering the vacuum-case they will only depend on the squares $\mu=p^{\nu} p_{\nu}$ .

For a three-particle final state one gets a more complicated result due to a rising number of degrees of freedom in the final state

$\displaystyle \frac{d\sigma_{a b \rightarrow c d e }}{d\mu_c\, d\mu_d\,d\mu_e\ d\vert\vec{p}_{c}\vert\, d\Omega_c \, d\vert\vec{p}_{d}\vert\, d\phi_d}(s)$	$\textstyle =$	$\displaystyle \frac{1}{8\left(2\pi\right)^5} \frac{1}{p_{ab} \sqrt{s}} \frac{\l... ..._{d}\right\vert}{E_c E_d} \mathcal{A}_c(\mu_{c},p_c) \mathcal{A}_d(\mu_{d},p_d)$
		$\displaystyle \times \, \mathcal{A}_e(\mu_{e},p_e)\left\vert\mathcal{M}_{a b \rightarrow c d e} (s)\right\vert^2 \,.$	(1.7)

Here $\vec{p}_{c,d}$ denote the CM-momenta of the particles c and d. The CM-momentum of

is given by total momentum conservation

$\begin{displaymath} \vec{p}_c+\vec{p}_d+\vec{p}_e=0 \ . \end{displaymath}$

Subsections

Medium modifications

Next: Medium modifications Up: Reactions: Theoretical framework Previous: The cross section for Contents

Oliver Buss 2005-03-16