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

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

Since

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

This implies

[math]s \ge 4m^2[/math]

In turn, this implies

[math] t \le 0 \qquad u \le 0[/math]

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

[math] t = 0 \qquad u = 0[/math]

[math]-2 p \ ^{*2}(1-cos\ \theta) = 0 \qquad -2 p \ ^{*2}(1+cos\ \theta) = 0[/math]

[math](-2 p \ ^{*2}+2 p \ ^{*2}cos\ \theta) = 0 \qquad (-2 p \ ^{*2}-2 p \ ^{*2}cos\ \theta) = 0[/math]

[math]2 p \ ^{*2}cos\ \theta = 2 p \ ^{*2} \qquad -2 p \ ^{*2}cos\ \theta = 2 p \ ^{*2}[/math]

[math]\cos\ \theta = 1 \qquad \cos\ \theta = -1[/math]

[math]\Rightarrow \theta_{t=0} = \arccos \ 1=0^{\circ} \qquad \theta_{u=0} = \arccos \ -1=180^{\circ}[/math]

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

[math]s+t_{max} \equiv 4m^2[/math]

[math]\Rightarrow t_{max}=4m^2-s[/math]

[math]t_{max}=4m^2-4m^2- 4p \ ^{*2}[/math]

The maximum transfer of momentum would be

[math]t_{max}=-4p \ ^{*2}[/math]

[math]-2 p \ ^{*2}(1-cos\ \theta_{t=max})=-4p \ ^{*2}[/math]

[math](1-cos\ \theta_{t=max})=2[/math]

[math]-cos\ \theta_{t=max}=1[/math]

[math] \theta_{t=max} \equiv \arccos -1[/math]

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 [math]\theta=180^{\circ}[/math]

[math]\theta_{t=max}=180^{\circ}[/math]

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

[math]u \equiv \overbrace{\left({\mathbf P_1^*}- {\mathbf P_2^{'*}}\right)^2=\left({\mathbf P_2^*}- {\mathbf P_1^{'*}}\right)^2}^{CM\ FRAME}=\overbrace{\left({\mathbf P_1}- {\mathbf P_2^{'}}\right)^2 = \left({\mathbf P_2}- {\mathbf P_1^{'}}\right)^2}^{LAB\ FRAME}[/math]

When u=0, this implies 4 different scenarios

[math]\mathbf P_1^*= \mathbf P_2^{'*}=\mathbf P_2^*= \mathbf P_1^{'*}=0[/math]

This is the simple solution which would imply no collision.

[math]\mathbf P_1^*= \mathbf P_2^{'*}; \mathbf P_2^*= \mathbf P_1^{'*}=0 [/math]

[math]\mathbf P_1^*= \mathbf P_2^{'*}=0;\mathbf P_2^*= \mathbf P_1^{'*}[/math]

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

[math]\mathbf P_1^*= \mathbf P_2^{'*}=\mathbf P_2^*= \mathbf P_1^{'*}[/math]

[math]t \equiv \left({\mathbf P_1^*}- {\mathbf P_1^{'*}}\right)^2=\left({\mathbf P_2^{*}}- {\mathbf P_2^{'*}}\right)^2[/math]