Verification of Relativistic Components

Verifying that the kinematics are valid, the Moller momentum from GEANT simulations are extracted using

awk '{print $16, $17, $18}' MollerScattering_NH3_Large.dat > Just_Mol_Mom.dat

Then using the kinematics equations displayed above, used to reverse calculate the scattered electron's momentum and angles. The root histograms of an incoming 11GeV electron (in blue) are overlaid with the relativistic calculations starting with the Moller electron (Reverse Moller in red) and show good agreement.

The differences in these plots come from fact that the "Forward" plots come from a GEANT simulation in which a small amount of energy was lost traveling through the target material.

[math]\left( \begin{matrix} E^{'*}_{1}+E^{'*}_{2} \\ p^{'*}_{1(x)}+p^{'*}_{2(x)} \\ p^{'*}_{1(y)}+p^{'*}_{2(y)} \\ p^{'*}_{1(z)}+p^{'*}_{2(z)}\end{matrix} \right)=\left(\begin{matrix}\gamma^* & 0 & 0 & -\beta^* \gamma^*\\0 & 1 & 0 & 0 \\ 0 & 0 & 1 &0 \\ -\beta^* \gamma^* & 0 & 0 & \gamma^* \end{matrix} \right) . \left( \begin{matrix}E_{1}^'+E_{2}^'\\ p_{1(x)}^'+p_{2(x)}^' \\ p_{1(y)}^'+p_{2(y)}^' \\ p_{1(z)}^'+p_{2(z)}^'\end{matrix} \right)[/math]

[math]\Longrightarrow[/math]This allows momentum in all 3 cartesian coordinates.