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There is a homework question from experimental hadron physics that I am stuck with:

Calculate the cross-section for the reaction $e^- e^+ \to \pi^+ \pi^- \pi^0$ at the peak of the $\omega$ resonance (the branching ratio $\Gamma_i/\Gamma$ for $\omega \to e^- e^+$ is $7.2 \times 10^{-5}$).

The cross-section is to be computed at the peak of the $\omega$ resonance, which will be at Mandelstam $\sqrt s = m_\omega$. It probably also means that we only have to consider the reaction $e^- e^+ \to \omega \to \pi^+ \pi^- \pi^0$ and can ignore all other intermediate states. The intermediate $\omega$ will be at rest when it is created such that we can just treat this as a simple decay into three particles.

All the concrete partial decay widths and partial cross sections are probably to be taken from the Particle Data Booklet. The actual task at hand seems to be the combination of two partial cross sections into a a single one for the whole reaction.

On the level of the Feynman diagrams, one could just use the crossing symmetry and make this a decay of one particle into the leptons and an $\omega$ which then decays into the pions. The kinematics would be different, though, so I do not think that this is the correct way to go about it.

Judging from the dimensions of the cross sections, I need to multiply the two cross sections and then get rid of two energy dimensions. The most promising candidate for this would be $$ \delta^4(P_\mathrm i - P_\mathrm f) \frac{\mathrm d^3 p}{2 E} \,.$$ I am just not sure which momenta to integrate and whether there are some more dimensionless factors that would be still missing.

How would one combine the cross sections of $\omega \to e^+ e^-$ and $\omega \to \pi^+ \pi^- \pi^0$ to give $e^+ e^- \to \omega \to \pi^+ \pi^- \pi^0$?

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