What is the definition of beam energy in particle physics? For example, the proton beams in the LHC collider have 7 TeV energy. Does this mean that the individual protons in the beam have 7 TeV energy or that the energy of all the protons in the beam add up to 7 TeV?
 A: In the LHC, each individual collision has a center-of-mass energy of roughly 14 TeV. Since the collisions are symmetric (two protons with equal energy, moving in opposite directions, collide), we can say that each individual proton has roughly* 7 TeV of energy.
As you can probably tell, adding up the energy of all of the protons in the ring at any given time gives you a pretty colossal amount of energy.
*In reality, there's a distribution of energies that the protons in the beampipe can have, due to the fact that accelerators can only be so precise.The distribution is centered around 7 TeV, and is sharp enough that we can safely talk about collisions having a pretty uniform center-of-mass energy in most cases.
A: Collision experiments are done to create particles that can not be studied under normal circumstances. Energy and momentum conservation as well as the famous Einstein equation $E=mc^2$ tell us that a heavier particle can not just "pop out of thin air". But if we let two particles with enough energy collide, they can create a new particle, if the incoming energy is high enough. Therefore the individual particles in the collision beam need to have a high energy. 
