In QED we like to define the (differential) cross section for a scattering process as follows:
$$d\sigma \ \dot= \ \frac{w_{fi}dN_f}{|j_{inc}|}\tag{1}$$
where $w_fi$ is the probability of transition between the initial and the final state per unit time (so the probability per unit time that a scattering event will happen I think), $dN_F$ is the number of final states, and $j_{inc}$ is the flux of particles hitting the target. We can check that the result has indeed the dimensions of an area. My question regards the sensibleness of this definition.
For starters I think that $(1)$ defines $d\sigma$ and not $\sigma$ because the definition has a dependency on the number of final states $dN_f$, that fixes a determined momentum, or determined momenta (but I am not sure).
But more crucially if we go back to the standard idea behind the general definition of the cross section $\sigma$ we can say that $\sigma$ must be something like:
$$\sigma \ \dot= \ \frac{numbers \ of \ scattering \ events \ per \ unit \ time}{flux \ of \ particles \ hitting \ the \ target} \tag{2}$$
where the flux in $(2)$ is conceptually the same as the flux in (1), and it is defined as the number of particles that hit the target per unit time and per unit area. This implies that $$w_{fi}dN_f$$ must be the number of scattering events per unit time, i.e. the number of diffused particles per unit time $dN_d/dt$, so it must be true that:
$$\frac{dN_d}{dt}=w_{fi}dN_f \tag{3}$$
I have a problem with $(3)$: I don't understand why it is true. More precisely I would say that the number of scattered particles per unit time $dN_d/dt$ should be equal to the transtion probability per unit time times the number of particles that arrive to the target in an unit time, and not times the number of final states like it is written in $(3)$.
I am sure that I am missing something here, or that maybe I don't properly understand one of the definitions of the quantities involved. In any case $(3)$ also appears in other context so it must be true in some way, but I don't know why or how. How do we justify equation $(3)$?
