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Different particle accelerators use different types of collisions. For instance at the LHC they investigated p Pb collisions while its predecessor (LEP), used to collide electrons with proton and at Fermilab it's protons with antiprotons that are colliding.

What is the reason behind these choices? It's clear that energy is a crucial point here but why for example, electron - positron collision process is less interesting at the LHC?

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That is a big topic, and I won't even try to cover it all, but here are a few bits that go into the decision.

  • Electrons give you a clean probe (the lepton--photon or lepton-weak-boson vertex is very well understood.

    Electrons on nuclei (as at JLAB) gives you a precision probe of nucleon or nuclei parameters at moderate energies (in the transition region between nucleon--meson degrees of freedom and quark gluon degrees of freedom.

    Electron-positron scattering gives you access to very pure and very well known initial states. Very useful for precision measurements of fundamental vertexes quantities.

  • Protons can be given more energy for a given bending magnet strength than electrons. Most useful for maximum energy at the primary vertex, but also gives you access to quark--anti-quark interactions (through the nucleon sea). Anti-protons (as at the Tevatron) add some nice flourishes, but it is hard to get a lot of luminosity with them.

  • Muons would combine the pure vertex advantage of electrons with some of the energy advantage of protons. Very interesting, but also pricey. A muon collider always seems to be the project that we will pursue after we do the next generation of [other machines].

  • Heavy ions give you access to non-trivial volumes of dense, high-energy matter during collisions (i.e. quark-gloun plasma). There is a lot to be learned about the rules that govern the behavior of nuclei that are hard to access at low energies. RHIC was a ground-breaker on this.

In recent years accelerator design has become more flexible in terms of being able to handle protons or heavy ions in one machines, so the LHC was designed to use either in either beam allowing three different types of investigations with a single collider.

Different types of machines do different things, and they are all interesting. It just depends on what you can do better than it was done before or what you can do that is new and always, always, always about how much it will cost.

An interesting thing people are looking at for the next generation is a electron--ion collider to do JLAB-like work at one to two orders of magnitude higher CoM energy.

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"""".....What is the reason behind these choices? It's clear that energy is a crucial point here but why for example, electron - positron collision process is less interesting at the LHC?.....""""

This part of your question is very easy to answer. Electrons, and positrons are very low mass (510keV) so they accelerate very quickly under even modest electric field, and as a consequence, they lose energy by EM radiation if you try to accelerate them to high energies in a circular loop accelerator.

Consequently, linear accelerators are much preferred for electron accelerators; which is the reason for the two mile long, Stanford Linear accelerator in Palo Alto California.

Protons are about 1837 times the massive as electrons, and radiate less than electrons at comparable energies.

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