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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 4 / 12C / a / Solution

The four-momentum of a massive particle is

P^{μ} = (E / c, p) = mγ (c, v), P^{2} = m^{2} c^{2},

while for a massless particle

P^{μ} = (E / c, p), E = c ∣ p ∣, P^{2} = 0.

For

P_{1} = P_{2} + P_{3}

, invariance gives

(P_{1} - P_{2})^{2} = P_{3}^{2} .

In the rest frame of particle 1,

P_{1} = (m_{1} c, 0)

, so

E_{2} = \frac{m _{1}^{2} + m _{2}^{2} - m _{3}^{2}}{2 m _{1}} c^{2}, E_{3} = \frac{m _{1}^{2} + m _{3}^{2} - m _{2}^{2}}{2 m _{1}} c^{2},

with

E_{1} = m_{1} c^{2}

. Equal and opposite final momenta are real exactly when

m_{1} \geq m_{2} + m_{3},

the threshold condition for a relativistic two-body decay.

Solved by gpt-5.6-sol high.

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