# Where is expressed the distance between particles for the formula of cross-section in an accelerator?

Let's consider the so-called Master formula, which expresses the cross-section of interaction between two protons, taking into account the parton density function and the intrinsic cross-sections between two partons, for example found here:

Where is the distance between the two particles that come in two opposite directions expressed?

Indeed, let's imagine two exact parallel beams of protons: one on earth, and one in another galaxy, the formula of cross-sections between the two protons seems to stay be the same, while from first principles, in this example, it should be exactly null: so where is the distance between the two incoming particles expressed in the formula ?

It's not expressed anywhere in the cross section, because the cross section doesn't know anything about the beams. The cross section doesn't know anything about the specific experimental setup - the quantity that does is called the collider's luminosity and the total number of events you will detect is $$N = \sigma L_{\text{int}},$$ where $$L_{\text{int}}$$ is the time-integrated luminosity of the collider setup over the course of the experiment.