Chirality Conservation and Soft Pion Production

A formally ${\gamma }_5$-invariant system consisting of a Dirac field and a massless pseudoscalar field allows chirality conservation in the sense that its expectation value is a constant of motion. This leads to the consequence that in any reaction a change in the fermion chirality (∼helicity×velocity) is compensated for by the emission of a massless boson at zero energy, which can be expressed by a simple formula relating the radiative amplitude to the elastic amplitude. Assuming the pion-nucleon system to be ${\gamma }_5$-invariant when the pion mass can be neglected, the formula is applied to the processes $N+\pi \to N+\pi \mathrm{and} N+2\mathrm{}\pi$. A reasonable agreement with experiment is obtained in a case dominated by the 3-3 resonance.