At a collider, two high-energy particles A and B with energies EA and EB, which are much greater than either of the masses, collide at a crossing angle θ. Show that the total centre-of-mass energy ECM is given by
E2CM = 2EAEB (1 + cos θ)
If the rest masses are neglected. At the HERA collider (which was operational until 2007) in the DESY laboratory in Hamburg, a 30 GeV electron beam collided with an 820 GeV proton beam at zero crossing angle. Evaluate the total centre-of-mass energy and show that a fixed-target electron accelerator would require a beam energy of approximately 5 × 104 GeV to achieve the same total centre-of-mass energy.
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