Limits based on Mandelstam Variables

Limits based on Mandelstam Variables

$\Longrightarrow \left({\mathbf P_1^*}+ {\mathbf P_2^{*}}\right)^2=\left({\mathbf P_1^{'*}}+ {\mathbf P_2^{'*}}\right)^2\equiv s$

In the center of mass frame, the momentum of the particles interacting are equal and opposite, i.e. $p_1=-p_2$. However, the 4-momentum still retains an energy component, which as a scalar quantity, can not be countered by another particle's direction of motion.

${\mathbf P_1^*}\equiv \left(\begin{matrix} E_1\\ p_{x_1} \\ p_{y_1} \\ p_{z_1} \end{matrix} \right) \ \ \ \ {\mathbf P_2^*}\equiv \left(\begin{matrix} E_2\\ p_{x_2} \\ p_{y_2} \\ p_{z_2} \end{matrix} \right)$