 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/ib/paper-4/17b/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

Let the photon momenta be

p_{1}, p_{2}

, with

∣ p_{i} ∣ = E_{i} / c

E_{1} + E_{2}

and total momentum

p_{1} + p_{2}

for the pion. Its mass shell relation therefore gives

m_{π}^{2} c^{4} = (E_{1} + E_{2})^{2} - c^{2} ∣ p_{1} + p_{2} ∣^{2} = 2 E_{1} E_{2} (1 - cos θ) = 4 E_{1} E_{2} sin^{2} (θ /2) .

Since

0 \leq θ \leq π

, the sine is nonnegative, and taking the positive square root yields

sin \frac{θ}{2} = \frac{m _{π} c ^{2}}{2 E _{1} E _{2}} .

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