Past exam of the particle physics course of the University of Oxford / 2016 / 5 / 4 / Solution

The isospin coupling $1\otimes 1\to 1$, with third component $-1$, has the normalized stateThe two Clebsch-Gordan coefficients have equal squared magnitude. Neglecting electromagnetic effects and charged-neutral mass differences, the two rho meson charge channels have equal rates. Their subsequent strong decays are ${\rho }^0\to {\pi }^{+}{\pi }^{-}$ and ${\rho }^{-}\to {\pi }^{-}{\pi }^0$, with nearly unit probabilities. Consequentlyin the ideal isospin model.

Each final state contains one pair of identical pions. Summing the alternative resonant-pair assignments coherently, with the corresponding identical-particle phase space factors, does not generate an extra factor of two for just one channel. In the narrow-rho meson approximation the equality is already evident from equal $\rho \pi$ rates; the consistent isospin amplitudes extend it to the integrated three-body distributions.

The tabulated ratio $9.02/9.30=0.970$ is close to this expectation but is not exactly one. Treating the printed errors as independent gives an uncertainty about $0.012$, so a percent-level comparison requires isospin breaking, radiative corrections, realistic resonant dynamics and experimental correlations. The idealized argument predicts the leading pattern rather than an exact identity for the rounded measurements.