Isospin analysis of low-energyπN→ππNdata and chiral-symmetry breaking

The amplitudes for $\pi N$ single-pion production are decomposed into four independent isospin amplitudes in the $\left(\pi \pi \right)N$ representation. Reaction cross sections for the ${\pi }^{-}p$ and ${\pi }^{+}p$ initial states below total energies of 1370 and 1470 MeV, respectively, are simultaneously fitted with near-threshold measurements of the double-differential cross section $\frac{d^2\sigma }{d\Omega \mathrm{dT}}$ for the ${\pi }^{+}$ produced in ${\pi }^{-}p\to {\pi }^{+}{\pi }^{-}n$. The validity of soft-pion calculations for $\pi N\to \pi \pi N$ is demonstrated by showing that the threshold $s$-wave isospin amplitudes can be described consistently by a single value of the chiral-symmetry-breaking parameter.