s Channel
The s quantity is known as the square of the center of mass energy (invariant mass)
[math]\displaystyle{ s \equiv \left({\mathbf P_1^*}+ {\mathbf P_2^{*}}\right)^2=\left({\mathbf P_1^{'*}}+ {\mathbf P_2^{'*}}\right)^2 }[/math]
[math]\displaystyle{ s \equiv \left({\mathbf P_1^*}+ {\mathbf P_2^{*}}\right)^2 }[/math]
[math]\displaystyle{ s \equiv \mathbf P_1^{*2}+2 \mathbf P_1^* \mathbf P_2^*+ \mathbf P_2^{*2} }[/math]
As shown earlier, the square of a 4-momentum is
[math]\displaystyle{ \mathbf P^{2} \equiv m^2 }[/math]
[math]\displaystyle{ s \equiv m^{2}+2 \mathbf P_1^* \mathbf P_2^*+ m_2^{2} }[/math]
This gives
[math]\displaystyle{ s \equiv 2m^{2}+2 \mathbf P_1^* \mathbf P_2^* }[/math]
Similarly, the scalar product of two 4-momentums
[math]\displaystyle{ s \equiv 2m^2+2(E_1^*E_2^*-\vec p_1^* \vec p_2^*) }[/math]
In the center of mass frame of reference,
[math]\displaystyle{ E_1^*=E_2^* \quad and \quad \vec p_1^*=-\vec p_2^* }[/math]
[math]\displaystyle{ s \equiv 2m^2+2(E_1^{*2}+\vec p_1^{*2} ) }[/math]
Using the relativistic energy equation
[math]\displaystyle{ E^2 \equiv p^2+m^2 }[/math]
[math]\displaystyle{ s \equiv 2m^2+2((m^2+p_1^{*2})+p_1^{*2}) }[/math]
[math]\displaystyle{ s=4(m_{CM}^2+p_{CM}^2) }[/math]