What is the purpose of using different particle collisions in different colliders? eg: LHC using proton and antiproton of same energy. but BELLE uses electron and positron with different energies. Why is it so?
Electrons are elementary particles. So you know that the center of mass energy which goes in has to equal the energy coming out. So they have been the preferred particles for discovery machines, like the LEP at CERN. However electrons are quite light. In order to have many collisions in a collider and thus see signatures with low cross sections which do not happen often, one has to recycle the electrons which have not collided. Thus most colliders have a ring-like shape. Whenever the electron have to travel in a curve, they radiate synchroton radiation. This means that they loose energy. Thus the highest energies the electrons can have is limited. Belle looks into B-physics, which can be done at lower energies. Thus they are o.k. with electrons and positirons. To go to higher energies, one has to use heavier particles. The next best particles to create and accelerate are protons. The protons however are built up from quarks, sea-quarks and gluons. thus the center of mass energy is conserved, but not on the level of the single quark and gluons. However for discovery machines like the LHC this is the way to go.
By the way, circular electron colliders can be used as a source of synchrotron radiation, so taking profit (material structure studies) of this physics phenomenon which limits their rise in energy .
Asymmetric collisions is an unneccessary complication. We had such a situation in fact in case of p-Pb collisions, as the two colliding objects have different mass to charge ratio which defines energy per nucleon alias velocity. Due to this there is a small, but significant difference between the centre of mass and the lab system which necessitates additional Lorentz transformation when analysing collected events This extra CMS boost has also some consequences in the detector acceptance which has to be taken into account. And last but not least, there is a loss of the energy available for production of particles. This was the main reason to go from the fixed target to the collider configuration.