Low Energy Model

In the low energy parameterized model the mean number of hadrons produced in a hadron-nucleus collision is given by

$$N_m = C(s) A^{1/3} N_{ic}$$ (20.1)

where $A$ is the atomic mass, $C(s)$ is a function only of the center of mass energy $s$, and $N_{ic}$ is approximately the number of hadrons generated in the initial collision. Assuming that the collision occurs at the center of the nucleus, each of these hadrons must traverse a distance roughly equal to the nuclear radius. They may therefore potentially interact with a number of nucleons proportional to $A^{1/3}$. If the energy-dependent cross section for interaction in the nuclear medium is included in $C$ then Eq. 20.1 can be interpreted as the number of target nucleons excited by the initial collision. Some of these nucleons are added to the intra-nuclear cascade. The rest, especially at higher momenta where nucleon production is suppressed, are replaced by pions and kaons.

Once the mean number of hadrons, $N_m$ is calculated, the total number of hadrons in the intra-nuclear cascade is sampled from a Poisson distribution about the mean. Sampling from additional distribution functions provides

• the combined multiplicity $w(\vec{a},n_{i})$ for all particles $i$, $i = \pi^{+}, \pi^{0}, \pi^{-}, p, n, .....$, including the correlations between them,
• the additive quantum numbers $E$ (energy), $Q$ (charge), $S$ (strangeness) and $B$ (baryon number) in the entire phase space region, and
• the reaction products from nuclear fission and evaporation.

A universal function $f(\vec{b},x/p_{T},m_{T})$ is used for the distribution of the additive quantum numbers, where $x$ is the Feynman variable, $p_T$ is the transverse momentum and $m_T$ is the transverse mass. $\vec{a}$ and $\vec{b}$ are parameter vectors, which depend on the particle type of the incoming beam and the atomic number $A$ of the target. Any dependence on the beam energy is completely restricted to the multiplicity distribution and the available phase space.

The low energy model can be applied to the $\pi^+$, $\pi^-$, $K^+$, $K^-$, $K^0$ and $\overline{K^0}$ mesons. It can also be applied to the baryons $p$, $n$, $\Lambda$, $\Sigma^+$, $\Sigma^-$, $\Xi^0$, $\Xi^-$, $\Omega^-$, and their anti-particles, as well as the light nuclei, $d$, $t$ and $\alpha$. The model can in principal be applied down to zero projectile energy, but the assumptions used to develop it begin to break down in the sub-GeV region.