Pion Scattering Lengths

Abstract

The current commutation ${\mathrm{relations}}^{1}$ and partially conserved axial-vector current (PCAC) ${\mathrm{assumption}}^{2,3}$ allow the calculation of the matrix elements for emission and absorption of any number of soft ${\mathrm{pions}}^{4}$ and, therefore, in particular, determine the scattering length of a pion on any target particle. In this note we give a simple formula for pion scattering on any particle but a ${\mathrm{pion}}^{5}$, and then extend this result to the more difficult case of pion-pion scattering.