An Interlude — Question 8

Question 8

Equal-energy colliders use more beam energy for new rest mass

A particle of rest energy m=1.00 GeV and total lab energy E=9.00 GeV strikes an identical stationary particle. Use c=1 in the invariant formulas. Compare with two identical counterpropagating beams each of total energy 9.00 GeV.

1. Calculate center-of-mass energy W for both arrangements.

2. For a required final invariant energy W=6.00 GeV, find the fixed-target beam total energy and the energy of each symmetric collider beam.

Question 8 - Solution

Equal-energy colliders use more beam energy for new rest mass

(a) Method and calculation

$$W_{\mathrm{fixed}}^2=2m^2+2mE\Rightarrow\boxed{W_{\mathrm{fixed}}=\sqrt{20}=4.47214\,\mathrm{GeV}}.$$$$\boxed{W_{\mathrm{collider}}=2E=18.0\,\mathrm{GeV}}\quad(\mathbf p_{\mathrm{total}}=0).$$

(b) Interpretation and further calculation

$$\boxed{E_{\mathrm{fixed}}=(W^2-2m^2)/(2m)=17.0\,\mathrm{GeV}}.$$$$\boxed{E_{\mathrm{each,collider}}=W/2=3.00\,\mathrm{GeV}}.$$

Physical check

These are total beam energies including rest energy. Reaction thresholds also require all additive quantum numbers to balance.