Pion Production in Nucleon-Nucleon Collisions at Threshold

We present a theory of soft-pion production by nucleons which provides a connection between the production amplitude and nucleon-nucleon scattering. This is done by applying the algebra of currents to the $S$-matrix element for the process $\gamma +2n\to 2N+\pi$. After reducing out both the photon and pion, the hypothesis of partially conserved axial-vector current is used to replace the pion field by the divergence of the axial-vector current. Using the current commutation relations, we then obtain the pion production amplitude in terms of the amplitude for photoproduction of a pion on two nucleons, plus a correction integral which vanishes in the soft-pion limit. Upon taking the soft-photon—soft-pion limit, only certain pole terms survive. In this way, pion production is directly related to off-shell nucleon-nucleon scattering. Results are presented for the reactions $\mathrm{pp}\to \mathrm{np}{\pi }^{+}$, $\mathrm{np}\to \mathrm{pp}{\pi }^{-}$, and $\mathrm{pp}\to \mathrm{pp}{\pi }^0$. The theoretical predictions are found to be in good agreement with the cross-section data near threshold for ${\pi }^{-}$ and ${\pi }^0$ production, and with an extrapolation from higher-energy data for ${\pi }^{+}$ production.