threshold energy

If you want to mint a new, heavy coin in a collision, you must pay for it — and the currency is energy, by way of E equals m c squared. The threshold energy is the minimum collision energy needed to bring a particular new particle, or set of particles, into existence. Below the threshold the reaction simply cannot happen, no matter how many times you try; above it, the reaction becomes possible and grows more likely as you supply more energy.

The subtlety is that not all of the energy you put in is available to pay for new mass. Momentum must be conserved, so the products of a collision usually have to keep moving, and the kinetic energy tied up in that forced motion cannot be spent on creating mass. The cleanest way to see what is available is the center-of-mass frame, where total momentum is zero: there, the entire collision energy can in principle go into rest mass. The threshold is reached when the available center-of-mass energy, sqrt(s), just equals the total rest energy of the particles you want to create.

This single idea drives the design of accelerators. To discover a heavy particle you must build a machine whose collisions exceed its production threshold — which is why ever-heavier discoveries demanded ever-higher energies, from the W and Z bosons to the top quark to the Higgs. It also explains the great advantage of colliders over fixed-target machines: in a fixed-target collision much of the energy is wasted keeping the debris moving forward, so the threshold is reached far more cheaply in a head-on collider where the lab frame is already the center-of-mass frame.