# Difference between revisions of "Limits based on Mandelstam Variables"

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# Limits based on Mandelstam Variables

Since the Mandelstam variables are the scalar product of 4-momenta, which are invariants, they are invariants as well. The sum of these invariant variables must also be invariant as well. Find the sum of the 3 Mandelstam variables when the two particles have equal mass in the center of mass frame gives:

Since

This implies

In turn, this implies

At the condition both t and u are equal to zero, we find

Holding u constant at zero we can find the minimum of t

The maximum transfer of momentum would be

The domain of the arccos function is from −1 to +1 inclusive and the range is from 0 to π radians inclusive (or from 0° to 180°). We find as expected for u=0 at

However, from the definition of u being invariant between frames of reference

When u=0, this implies 4 different scenarios

This is the simple solution which would imply no collision.

These two cases show a stationary particle receiving all the momentum of an incident particle. This is not possible for equal mass particles.