Total 4-momentum
As was shown earlier the scalar product of a 4-Momentum vector with itself ,
[math]\displaystyle{ {\mathbf P_1}\cdot {\mathbf P^1}=E_1E_1-\vec p_1\cdot \vec p_1 =m_{1}^2=s }[/math] ,
and the length of a 4-Momentum vector composed of 4-Momentum vectors,
[math]\displaystyle{ {\mathbf P^2}=({\mathbf P_1}+{\mathbf P_2})^2=(E_1+E_2)^2-(\vec p_1 +\vec p_2 )^2=(m_1+m_2)^2=s }[/math],
are invariant quantities.
It was further shown that
[math]\displaystyle{ {\mathbf P^*}^2={\mathbf P}^2 }[/math]
where [math]\displaystyle{ {\mathbf P^*}=({\mathbf P_1^*}+{\mathbf P_2^*})^2 }[/math] represents the 4-Momentum Vector in the CM frame
and [math]\displaystyle{ {\mathbf P}=({\mathbf P_1}+{\mathbf P_2})^2 }[/math] represents the 4-Momentum Vector in the initial Lab frame
which can be expanded to
[math]\displaystyle{ {\mathbf P^*}^2={\mathbf P^{'*}}^2={\mathbf P}^2={\mathbf P^'}^2 }[/math]
where [math]\displaystyle{ {\mathbf P^'}=({\mathbf P_1^'}+{\mathbf P_2^'})^2 }[/math] represents the 4-Momentum Vector in the final Lab frame
and [math]\displaystyle{ {\mathbf P^{'*}}=({\mathbf P_1^{'*}}+{\mathbf P_2^{'*}})^2 }[/math] represents the 4-Momentum Vector in the final CM frame