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We know that inside particle accelerators we collide ionized particles. But we also know that in the microworld all collisions are elastic (due only to the electrostatic force?) between the particles. So how exactly two, let's say, protons collide?
One answer that came into my mind, is that they don't collide, but the enormous forces from the electric fields break them apart... Or maybe with the energy we give them they come so close that the strong force is... doing something, I guess?
Your question seems to look for a fundamental picture for particles.
We have been colliding particles since long time, actually trying to find what is there more fundamental to protons. Since we cannot know in detail what hapens in the short time and space where they occur, what we can do is measure everything that comes out and try to understand from this "debris" what happened.
These experiments led to discovering many other particles which were not known, but classifying them and studying them, some regularities were found, which allowed us to say that all of them are composed of smaller elements called quarks whose interactions make up all of them. Later experiments have been consistent with this view and have even confirmed some predictions which were based on this view.
So it seems that when two protons collide, the interaction between these smaller constituents produces other particles. But our evidence for the existence of these quarks and their interactions will always be inferred, because they seem to not exist isolated, but rather in compounds forming protons, neutrons, and all kinds of mesons and barions.
Finally, our current theories cannot provide detailed descriptions of reactions like that between protons. They usually provide interaction and transformation probabilities between the fundamental particles, and other parameters like space distributions and other conserved quantities, but the details of the reactions is out of the current possibilities.
In other words, the answer to "what happens when protons collide?" if we are really honest about our knowledge, looks more like "protons, neutrons mesons, kaons, W, etc., come out with these energies, momenta and correlations". At the moment we cannot give an answer describing internally and by time sequence what happens, since even our theoretical views of particles and their composition cannot define them more precisely than "qantum mechanical bodies which cannot have simultaneous determined position and momentum, which exhibit particle and wave behaviour".
