Particle creation energy in physics
In particle physics, the threshold energy for production of a particle is the minimum kinetic energy that must be imparted to one of a pair of particles in order for their collision to produce a given result. If the desired result is to produce a third particle then the threshold energy is greater than or equal to the rest energy of the desired particle. In most cases, since momentum is also conserved, the threshold energy is significantly greater than the rest energy of the desired particle.
The threshold energy should not be confused with the threshold displacement energy, which is the minimum energy needed to permanently displace an atom in a crystal to produce a crystal defect in radiation material science.
Example of pion creation.
Consider the collision of a mobile proton with a stationary proton so that a formula_1 meson is produced: formula_2
We can calculate the minimum energy that the moving proton must have in order to create a pion.
Transforming into the ZMF (Zero Momentum Frame or Center of Mass Frame) and assuming the outgoing particles have no KE (kinetic energy) when viewed in the ZMF, the conservation of energy equation is:
formula_3
Rearranged to 
formula_4
By assuming that the outgoing particles have no KE in the ZMF, we have effectively considered an inelastic collision in which the product particles move with a combined momentum equal to that of the incoming proton in the Lab Frame.
Our formula_5 terms in our expression will cancel, leaving us with:
formula_6
formula_7
Using relativistic velocity additions:
formula_8
We know that formula_9 is equal to the speed of one proton as viewed in the ZMF, so we can re-write with formula_10:
formula_11
So the energy of the proton must be formula_12 MeV.
Therefore, the minimum kinetic energy for the proton must be formula_13 MeV.
Example of antiproton creation.
At higher energy, the same collision can produce an antiproton:
formula_14
If one of the two initial protons is stationary, we find that the impinging proton must be given at least formula_15 of energy, that is, 5.63 GeV. On the other hand, if both protons are accelerated one towards the other (in a collider) with equal energies, then each needs to be given only formula_16 of energy.
A more general example.
Consider the case where a particle 1 with lab energy formula_17 (momentum formula_18)
and mass formula_19 impinges on a
target particle 2 at rest in the lab, i.e. with lab energy formula_20 and mass formula_21.
The threshold energy formula_22 to produce three particles of masses formula_23, formula_24,
formula_25, i.e.
formula_26
is then found by assuming that these three particles are at rest in the center of mass frame (symbols with
hat indicate quantities in the center of mass frame):
formula_27
Here formula_28 is the total energy available in the center of mass frame.
Using formula_29, formula_30 and 
formula_31 one derives that
formula_32 
References.
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