What does energy mean in the context of particles accelerators? In the context of particle collisions I keep seeing references to specific quantities of energy being required to generate new particles. However, energy is a frame dependent quantity. The collision could have one quantity of energy in one frame and another quantity in another. So what energy is being referenced in these contexts?
 A: The energy required to create a new particle is the energy in the rest frame of the collision. For instance, in a "threshold reaction" $\rm e^+e^- \to \mu^+\mu^-$ where the muon pair is created at rest, the total energy (including rest masses) on both sides must be twice the muon's rest mass.
In fixed-target experiments, one usually talks about the beam's kinetic energy, because that's what the accelerator folks can measure easily and put on the wall.  If a person needs specifically to discuss the center of mass energy, they'll say so.  In the limit of a relativistic beam, the kinetic energy and the total energy are the same.  For instance, at Jefferson Lab's 12 GeV electron accelerator, the electrons leave the injector at about 50 MeV, which is already $\gamma \approx 100$.  But at the proton accelerators at LANSCE or the Spallation Neutron Source, the 1 GeV proton beam refers just to the proton's kinetic energy, and you have to know things to conclude the protons have $\gamma \approx 2$.
In collider experiments such as at RHIC or CERN, the colliding beams generally have approximately equal but opposite momenta, and so the sum of the beam energies is a good approximation to the center-of-mass energy.  There is usually an intentional nonzero momentum to the collision, so that your collision region can be offset from your detectors by a little bit.  Correcting for this difference is a detail that gets buried in the middle of a paper.
