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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:

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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 ?

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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.

The computation of the luminosity of real-world collider setups is non-trivial. For an introduction see e.g. "Concept of luminosity" [pdf link] by Herr and Muratori.

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