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

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

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

$E_1^{*}=E_1^{'*}=E_2^{*}=E_2^{'*}$

$\left | \vec p_1^* \right |=\left | \vec p_1^{'*} \right |=\left | \vec p_2^* \right |=\left | \vec p_2^{'*} \right |$

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

$t={\mathbf P_1^{*2}}+ {\mathbf P_1^{'*2}}-2 {\mathbf P_1^*} {\mathbf P_1^{'*}}={\mathbf P_2^{*2}}+ {\mathbf P_2^{'*2}}-2 {\mathbf P_2^*} {\mathbf P_2^{'*}}$

$t=2m^2-2E_1^*E_1^{'*}+2 \vec p_1^* \vec p_1^{'*}=2m^2-2E_2^*E_2^{'*}+2 p_2^* p_2^{'*}$

$t=2m^2-2E_1^{*2}+2 \left | p_1^{*2}\right | \cos \theta_{1^*\ 1^{'*}}=2m^2-2E_2^{*2}+2 \left | p_2^{*2}\right | \cos \theta_{2^*\ 2^{'*}}$

where $\theta_{1^*\ 1^{'*}}$ and $\theta_{2^*\ 2^{'*}}$is the angle between the before and after momentum in the CM frame

Using the relativistic relation $E^2=m^2+p^2$ this reduces to

$t=-2p_1^{*2}+2 \left | p_1^{*2}\right | \cos \theta_{1^*\ 1^{'*}}=-2p_2^{*2}+2 \left | p_2^{*2}\right | \cos \theta_{2^*\ 2^{'*}}$

$t=-2p_1^{*2}(1- \cos \theta_{1^*\ 1^{'*}})=-2p_2^{*2}(1-\cos \theta_{2^*\ 2^{'*}})$

The maximum momentum is transfered at 90 degrees, i.e. $\cos 90^{\circ}=0$

$t=-2p_1^{*2}$

This can be rewritten again using the relativistic energy relation $E^2=m^2+p^2$

$t=2(m^{2}-E_1^{*2})=2(m^{2}-E_2^{*2})$

In the Lab Frame

$t={\mathbf P_1^{2}}+ {\mathbf P_1^{'2}}-2 {\mathbf P_1} {\mathbf P_1^{'}}={\mathbf P_2^{2}}+ {\mathbf P_2^{'2}}-2 {\mathbf P_2} {\mathbf P_2^{'}}$

$t=2m^2-2E_1E_1^{'}+2 \vec p_1 \vec p_1^{'}=2m^2-2E_2E_2^{'}+2 p_2 p_2^{'}$

with $p_2=0$

and $E_2=m$

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

Maximum Moller Energy in Lab Frame

Since t is invariant between frames

$t=2(m^2-E_2^{'}m)=2(m^2-E_2^{*2})$

$\rightarrow E_2^{'}=\frac{E_1^{*2}}{m}$

with$E_2^{*} \approx 53\ MeV$ for $E_1=11000\ MeV$

The Moller electron has a maximum energy possible of:

$E_2^{'}=5500\ MeV$

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