Phase space Limiting Particles

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4.1.2 Phase space Limiting Particles

Since the angle phi has been constrained to remain constant, the x and y components of the momentum will increase in the positive first quadrant. This implies that the z component of the momentum must decrease by the relation:


In the Center of Mass frame, this becomes:

[math]p_x^{*2}+p_y^{*2} = p^{*2}-p_z^{*2}[/math]

Since the momentum in the CM frame is a constant, this implies that pz must decrease. We can use the variable rapidity:

[math]y \equiv \frac {1}{2} \ln \left(\frac{E+p_z}{E-p_z}\right)[/math]

this implies that as

[math]p_z \rightarrow 0 \Rrightarrow \frac{E+p_z}{E-p_z} \rightarrow 1 \Rrightarrow \ln 1 \rightarrow 0 \Rrightarrow y=0[/math]

For forward travel in the light cone:

[math]p_z \rightarrow E \Rrightarrow \frac{E+p_z}{E-p_z} \rightarrow \infin \Rrightarrow \ln \infin \rightarrow \infin \Rrightarrow y \rightarrow \infin [/math]

This corresponds to the scattered electron proven earlier.

For backward travel in the light cone:

[math]p_z \rightarrow -E \Rrightarrow \frac{E+p_z}{E-p_z} \rightarrow 0 \Rrightarrow \ln 0 \rightarrow -\infin \Rrightarrow y \rightarrow -\infin [/math]

Similarly, this corresponds to the Moller electron.

For a particle that transforms from the Lab frame to the CM frame where the particle is not within the light cone:

[math]p_x^2+p_y^2=52.589054^2+9.272868^2=53.400MeV \gt 53.015 MeV (E) \therefore p_z \rightarrow imaginary[/math]

These particles are outside the light cone and are more timelike, thus not visible in normal space. This will reduce the range in theta that Moller electrons will be detected.


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