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# 'perturbative' and 'real' particles; the perturbative weigth

## 'perturbative' and 'real' particles

(following text is taken - slightly modified - from: O.Buss, PhD thesis, pdf, Appendix B.1)

For reactions which are not violent enough to disrupt the whole target nucleus, e.g. low-energy πA, γA or A collision at not too high energies, the target nucleus stays very close to its ground state. Henceforth, one keeps as an approximation
the target nucleus constant in time ('frozen approximation'). This basically means that the phase space density of
the target is not allowed to change during the run. The test-particles which represent this
constant target nucleus are called *real* test-particles. However, one also wants to consider the
final state particles. Thus one defines another type of test-particles which are called *perturbative*.
The *perturbative* test-particles are propagated and may collide with *real* ones, the products are
*perturbative* particles again. However, *perturbative* particles may not scatter among each other.
Furthermore, their feedback on the actual densities is neglected. One can simulate in
this fashion the effects of the almost constant target on the outgoing pparticles without modifying
the target. E.g. in πA collisions we initialize all initial state pions as *perturbative* test-particles.
Thus the target automatically remains frozen and all products of the collisions of pions and
target nucleons are assigned to the *perturbative* regime.

Furthermore, since the *perturbative* particles do not react among themselves or modify the *real*
particles in a reaction, one can also split a *perturbative* particle into \(N_{test}\) pieces (several *perturbative*
particles) during a run. Each piece is given a corresponding weight \(1/N_{test}\) and one simulates like
this \(N_{test}\) possible final state scenarios of the same *perturbative* particle during one run.

## The perturbative weigth 'perWeight'

Usually, in the cases mentioned above, where one uses the seperation into *real* and *perturbative* particles, one wants to calculate some final quantity like \(d\sigma^A_{tot}=\int_{nucleus}d^3r\int \frac{d^3p}{(2\pi)^3} d\sigma^N_{tot}\,\times\,\dots \). Here we are hiding all medium modifications, as e.g. Pauli blocking, flux corrections or medium modifications of the cross section in the part "\(\,\times\,\dots \)". Now, solving this via the testparticle ansatz (with \(N_{test}\) being the number of test particles), this quantity is calulated as \(d\sigma^A_{tot}=\frac{1}{N_{test}}\sum_{j=1}^{N_{test}\cdot A}d\sigma^j_{tot}\,\times\,\dots \), with \(d\sigma^j_{tot}\) standing for the cross section of the \(j\)-th test-particle.

The internal implementation of calculations like this in GiBUU is, that a loop runs over all \(N_{test}\cdot A\) target nucleons and creates some event. Thus all these events have the same probability. But since they should be weighted according \(d\sigma^j_{tot}\), this is corrected by giving all (final state) particles coming out of event \(j\) the weight \(d\sigma^j_{tot}\).

This information is stored the variable `perWeight`

in the definition of the particle type.

Thus, in order to get the correct final cross section, one has to **sum the perWeight**, and not the particles.

As an example: if you want to calculate the inclusive pion production cross section, you have to loop over all particles and sum the perWeights of all pions. Simply taking the number of all pions would give false results.