Our goal is to create and detect positrons, which are the antimatter counterpart of electrons. They are formed in a process called pair production, so named because a positron and electron are simultaneously created as a pair. The mechanism by which pair production would occur from colliding a proton beam with a stationary target is from the decay of a virtual photon that mediates the electromagnetic interaction between the proton and a nucleus. If the photon is the correct energy in the gamma range, it may decay into an electron-positron pair. The likelihood of this reaction occurring is small, but measurable.
In order for pair production to occur from a proton collision, the incident particle must have a certain minimum energy. Contrary to what one may think, the energy that needs to be added to an incident particle in order to create secondary particles is actually much higher than just the rest energy of the secondary particles. This is only true in the center-of-mass frame between the particles and not the lab frame. For example, the rest energy of an electron and positron is 0.511 MeV. To create a pair, the proton must be accelerated to at least 2 x 0.511 MeV more than its rest energy, but this is only within the center-of-mass frame. With respect to the lab frame of reference, the proton actually needs more energy to provide the secondary particles with some initial velocity as well.
To do this, we must consider the particle four-vectors with respect to different reference frames.
In the lab frame, one proton is accelerated and the other, in the target, is stationary. The four-vectors of the two protons are given as:
Where gamma and beta are the relativistic factors for the incident proton in the laboratory frame.
Before the collision, the protons have the following 4-vectors in the center-of-mass frame (in which the protons have equal and opposite velocities pointing toward the center of mass):
In the same frame of reference, after the collision the electron, positron, and protons have the 4-vectors:
The energy conservation between conditions before and after collision in the center-of-mass frame gives:
We can rearrange the Lorentz factor using the frame-invariance of the dot product before and after the collision:
Therefore, the required kinetic energy is:
