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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:
[math]p^2=p_x^2+p_y^2+p_z^2[/math]
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]
where
[math] P^+ \equiv E+p_z[/math]
[math] P^- \equiv E-p_z[/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 number of particles that will be detected.
[math]\textbf{\underline{Navigation}}[/math]
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