Limit of Energy in Lab Frame

The t quantity is known as the square of the 4-momentum transfer

$t \equiv \left({\mathbf P_1^*}- {\mathbf P_1^{'*}}\right)^2=\left({\mathbf P_2^{*}}- {\mathbf P_2^{'*}}\right)^2$

In the CM Frame

${\mathbf P_1^{*}}=-{\mathbf P_2^{*}}$

${\mathbf P_1^{'*}}=-{\mathbf P_2^{'*}}$

In the Lab Frame

${\mathbf P_1^{2}}+ {\mathbf P_1^{'2}}-2 {\mathbf P_1} {\mathbf P_1^{'}}$

$=2m^2-2E_1E_1^'+2 p_1 p_1^'=2m^2-2E_2E_2^'+2 p_2 p_2^'$

with $p_2=0$

and $E_2=m$

$t=2m^2-2mE_2^'=2(m^2-E_2^'m)$

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