Pion Scattering Lengths

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.