Limits based on Mandelstam Variables

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Limits based on Mandelstam Variables

s Channel

[math]s \equiv \left({\mathbf P_1^*}+ {\mathbf P_2^{*}}\right)^2=\left({\mathbf P_1^{'*}}+ {\mathbf P_2^{'*}}\right)^2[/math]



[math]s \equiv \left({\mathbf P_1^*}+ {\mathbf P_2^{*}}\right)^2[/math]


[math]s \equiv \mathbf P_1^{*2}+2 \mathbf P_1^* \mathbf P_2^*+ \mathbf P_2^{*2}[/math]


As shown earlier, for identical masses in the center of mass frame, and the mass is invariant between frames

[math]P^{*2}=m^2[/math]


[math]s \equiv m^{2}+2 \mathbf P_1^* \mathbf P_2^*+ m_2^{2}[/math]


[math]s \equiv 2m^{2}+2 \mathbf P_1^* \mathbf P_2^*[/math]


[math]s=4(m_{CM}^2+p_{CM}^2)[/math]

t Channel

[math]t \equiv \left({\mathbf P_1^*}- {\mathbf P_1^{'*}}\right)^2=\left({\mathbf P_2^{*}}+ {\mathbf P_2^{'*}}\right)^2[/math]
400px-CMcopy.png


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


[math]t \equiv P_1^{*2}-2P_1^*P_1^{'*}+P_1^{'*2}[/math]


[math]t \equiv 2m_1^2-2E_1^*E_1^{'*}+2p_1^*p_1^{'*}[/math]


[math]t \equiv 2m_1^*-2E_1^{*2}+2p_1^{*2}cos\ \theta[/math]


[math]t \equiv -2p_1^{*2}(1-cos\ \theta)[/math]

u Channel