Measurement of charge symmetry breaking by the comparison of ${\pi }^{+}\overrightarrow{d}\mathrm{pp}\eta$ with ${\pi }^{-}\overrightarrow{d}\mathrm{nn}\eta$

We report a measurement of charge symmetry breaking in the $\mathrm{NN}\eta$ system. We have measured the ratio of the differential cross sections of the charge-symmetric reactions ${\pi }^{+}\overrightarrow{d}\mathrm{pp}\eta$ and ${\pi }^{-}\overrightarrow{d}\mathrm{nn}\eta$ in the energy region of the $\eta$ threshold. Our result is $R\equiv d\sigma \left({\pi }^{+}\overrightarrow{d}\mathrm{pp}\eta \right)/d\sigma \left({\pi }^{-}\overrightarrow{d}\mathrm{nn}\eta \right)=0.938\mathrm{}\pm 0.009$ after a phase-space correction is made for the difference in the threshold energies of the two reactions. The deviation of R from unity is an indication of charge symmetry breaking, which is mostly due to ${\pi }^0-\eta$ mixing. A theoretical model for $\eta$ production which includes ${\pi }^0-\eta$ mixing was used to fit the data and yields a mixing angle of $(1.5\mathrm{}\pm 0.4)°.$ Our result is consistent with the mixing angle determined in particle decay and isospin-forbidden processes as well as predictions by several theoretical analyses which yield $\approx 1°$ and another which yields $\approx 2°.$