I am new to QFT and found myself wondering the following. Particle physics experiments usually consist of looking at what happens when we smash two protons together. As such, we look to calculate amplitudes for 2-> n scattering, with n the number of particles that emerge from the other side. Is there ever a case in which we need to think about this for high n? And if not, why need we worry about the divergences n-point Green's functions being cured for lower n, since we're unlikely to have to use them to calculate anyway? Any thoughts most appreciated!