Limit of Scattering Angle Theta in Lab Frame

The quantity is known as the

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

In the Lab Frame

[math]\displaystyle{ u={\mathbf P_1^{2}}+ {\mathbf P_2^{'2}}-2 {\mathbf P_1} {\mathbf P_2^{'}}={\mathbf P_2^{2}}+ {\mathbf P_1^{'2}}-2 {\mathbf P_2} {\mathbf P_1^{'}} }[/math]

[math]\displaystyle{ u=2m^2-2E_1E_2^{'}+2 \vec p_1 \vec p_2^{'}=2m^2-2E_2E_1^{'}+2 p_2 p_1^{'} }[/math]

with [math]\displaystyle{ p_2=0 }[/math]

and [math]\displaystyle{ E_2=m }[/math]

[math]\displaystyle{ u=2m^2-2E_1E_2^{'}+2 \vec p_1 \vec p_2^{'}=2m^2-2mE_1^{'} }[/math]

[math]\displaystyle{ u=2m^2-2E_1E_2^{'}+2 \left | p_1 \right | \left | p_2^{'} \right | \cos \theta_{1\ 2^{'}}=2m^2-2mE_1^{'} }[/math]