Phase space Limiting Particles

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4.1.2 Phase space Limiting Particles

Since the angle phi has been constrained to remain constant, the x and y components of the momentum will increase in the positive first quadrant. This implies that the z component of the momentum must decrease by the relation:

p2=p2x+p2y+p2z

In the Center of Mass frame, this becomes:

p2x+p2y=p2p2z


Since the momentum in the CM frame is a constant, this implies that pz must decrease. We can use the variable rapidity:

y12ln(E+pzEpz)

where

P+E+pz
PEpz

this implies that as

pz0


For forward travel in the light cone:

This corresponds to the scattered electron proven earlier.

For backward travel in the light cone:


Similarly, this corresponds to the Moller electron.


For a particle that transforms from the Lab frame to the CM frame where the particle is not within the light cone:

These particles are outside the light cone and are more timelike, thus not visible in normal space. This will reduce the number of particles that will be detected.