Final Lab Frame Moller Electron 4-momentum components in XZ Plane

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Final Lab Frame Moller Electron 4-momentum components in XZ Plane

Figure 3: Definition of Moller electron variables in the Lab Frame in the x-z plane. Using [math]\theta '_2=\arccos \left(\frac{p^'_{2(z)}}{p^'_{2}}\right)[/math]

[math]\Longrightarrow {p^'_{2(z)}=p^'_{2}\cos(\theta '_2)}[/math]

Checking on the sign resulting from the cosine function, we are limited to:

[math]0^\circ \le \theta '_2 \le 90^\circ \equiv 0 \le \theta '_2 \le \pi \ Radians[/math]

Since,

[math]\frac{p^'_{2(z)}}{p^'_{2}}=cos(\theta '_2)[/math]

[math]\Longrightarrow p^'_{2(z)}\ should\ always\ be\ positive with respect to the total momentum. This is effectively a ratio of momentum in the z direction versus total momentum. [/math]