The particle antiparticle pair (e⁺ and e^-) collide and annihilate to produce a virtual photon with zero momentum and total energy equal to the sum of the e⁺ and e^- total energies. The 'virtual photon' then materialises into a new e⁺ and e^- pair or any other heavier particle antiparticle pair, such as μ⁺ and μ^-, provided sufficient energy is available. If a μ⁺ and μ^- pair is created they will move off with equal and opposite momenta (remember that momentum is conserved and the total momentum before or after is zero). We assume that the pair move at 90^o to the e⁺ and e^- directions (this simplification does not affect our final conclusions) see figure above.

Let the total energy of each positron and electron be E_e, then the total energy before the collision is 2E_e. After the collision the total energies of each μ particle are equal since they have equal and opposite momenta. Conservation of total energy in the process then requires:

(1)

But we have the relativistic relation connecting energy and momentum for each particle:

(2)

substituting

(3)

and rearranging gives:

(4)

The expression inside the square root must be positive to give a physically possible momentum. Hence:

(5)

  
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