The t quantity is known as the square of the 4-momentum transfer
[math]t \equiv \left({\mathbf P_1^*}- {\mathbf P_1^{'*}}\right)^2=\left({\mathbf P_2^{*}}- {\mathbf P_2^{'*}}\right)^2[/math]
In the CM Frame
[math]{\mathbf P_1^{*}}=-{\mathbf P_2^{*}}[/math]
[math]{\mathbf P_1^{'*}}=-{\mathbf P_2^{'*}}[/math]
[math]E_1^{*}=E_1^{'*}=E_2^{*}=E_2^{'*}[/math]
[math]\left | \vec p_1^* \right |=\left | \vec p_1^{'*} \right |=\left | \vec p_2^* \right |=\left | \vec p_2^{'*} \right |[/math]
[math]t \equiv \left({\mathbf P_1^*}- {\mathbf P_1^{'*}}\right)^2=\left({\mathbf P_2^{*}}- {\mathbf P_2^{'*}}\right)^2[/math]
[math]{\mathbf P_1^{*2}}+ {\mathbf P_1^{'*2}}-2 {\mathbf P_1^*} {\mathbf P_1^{'*}}[/math]</center
In the Lab Frame
[math]{\mathbf P_1^{2}}+ {\mathbf P_1^{'2}}-2 {\mathbf P_1} {\mathbf P_1^{'}}[/math]
[math]=2m^2-2E_1E_1^'+2 p_1 p_1^'=2m^2-2E_2E_2^'+2 p_2 p_2^'[/math]
with [math]p_2=0[/math]
and [math]E_2=m[/math]
[math]t=2m^2-2mE_2^'=2(m^2-E_2^'m)[/math]
[math]\textbf{\underline{Navigation}}[/math]
[math]\vartriangleleft [/math]
[math]\triangle [/math]
[math]\vartriangleright [/math]