Final CM Frame Moller Electron 4-momentum components

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Final CM Frame Moller Electron 4-momentum components

Relativistically, the x and y components remain the same in the conversion from the Lab frame to the Center of Mass frame, since the direction of motion is only in the z direction.

[math]p^*_{2(x)}\Leftrightarrow p_{2(x)}'[/math]

[math]p^*_{2(y)}\Leftrightarrow p_{2(y)}'[/math]

[math]p^*_{2(z)}=-\sqrt {(p^*_2)^2-(p^*_{2(x)})^2-(p^*_{2(y)})^2}[/math]

We choose negative, since the incoming electron in the lab frame is traveling in the positive direction, and the Moller electron is initially at rest, which translates to negative motion in the CM frame.

Redefining the components in simpler terms, we use the fact that

[math]E^*\equiv E^*_1+E^*_2[/math]

[math]s \equiv (P_1^*+P_2^*)^2=(E^*_1+E^*_2)^2=(2E^*_2)^2=(P_1+P_2)^2=2m^2+2E_1E_2-2 \vec p_1 \cdot \vec p_2[/math]

Initially [math]\vec p_2=0 \quad \Rightarrow E^2=p^2+m^2=m^2[/math]

[math]\sqrt{2}E^*_2=\sqrt{2m(m+E_1)}[/math]

[math]E^*_2=\sqrt{\frac{m(m+E_1)}{2}}[/math]

[math] p^*_{2}=\sqrt{E_{2}^{*2}-m^2}[/math]

Initially, before the collision in the CM frame, p₂ was in the negative z direction. After the collision, the direction should reverse to the positive z direction. This same switching of the momentum direction alters p₁ as well.

Using [math]\theta '_2=\arccos \left(\frac{p^'_{2(z)}}{p^'_{2}}\right)[/math]

[math]\Longrightarrow \theta ^*_2=\arccos \left(\frac{p^*_{2(z)}}{p^*_{2}}\right)[/math]