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:

[math]\displaystyle{ s+t+u=(4(m^2+ p \ ^{*2}))+(-2 p \ ^{*2}(1-cos\ \theta))+(-2 p \ ^{*2}(1+cos\ \theta)) }[/math]

[math]\displaystyle{ s+t+u \equiv 4m^2 }[/math]

Since

[math]\displaystyle{ s \equiv 4(m^2+\vec p \ ^{*2}) }[/math]

This implies

[math]\displaystyle{ s \ge 4m^2 }[/math]

In turn, this implies

[math]\displaystyle{ t \le 0 \qquad u \le 0 }[/math]