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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 some calculations, e.g. low-energetic πA or γA collision, it is a good assumption, that the
target nucleus stays very close to its ground state. Henceforth, one keeps as an approximation
the target nucleus constant in time. 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, they are neglected in the calculation of the actual densities. One can simulate in
this fashion the effects of the almost constant target on the outgoing nucleons without modifying
the target. E.g. in πA collisions we initialize all initial state pions as *perturbative* test-particles.
Thus the target stays automatically constant 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 each other or modify the *real*
particles in a reaction, one can also split a *perturbative* particle in \(N_{test}\) 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 ou use the seperation into *real* and *perturbative* particles like this, you want 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 \(d\sigma^j_{tot}=d\sigma^N_{tot}(\vec r_j,\vec p_j)\) 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.