Soft-Pion Emission in Proton-Antiproton Annihilation

We calculate the amplitude for the annihilation process $\overline{p}p\to K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$, and the differential rate for $\overline{p}p\to K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$, normalized to the $\overline{p}p\to K^{+}{K}^{-}$ rate, in the soft-pion limit, for annihilation in flight at vanishing momentum. The computation uses the formalism of current algebra and PCAC (partially conserved axial-vector current); of particular interest is the use of Low's soft-photon theorem to evaluate the isovector photon term. The final result is expressed as the $\overline{p}p\to K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ rate, differential in all five independent kinematic variables, chosen for their symmetry and their utility in comparing with experimentally determined spectra. The integral of this expression yields the branching ratio $\frac{\overline{p}p\to K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}}{\overline{p}p\to K^{+}{K}^{-}}=2.25\mathrm{}$, in reasonable agreement with experiment.