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21r problem analytic continuation analytic properties barycentric system branch cut calculation channel 11 Chew and Mandelstam contribution cross section crossing relation 26a crossing symmetry cut by poles decay rate derive determined by nearby determined from experiment dispersion relations effective-range formula element for photopion energy formula 12 four-vector homogeneous integral equation interaction interchange intermediate isospin-independent function isotopic spin Lawrence Radiation Laboratory left cut left-hand cut Legendre polynomials locate the singularities LOW-ENERGY PHOTOPION PRODUCTION Mandelstam representation matrix element meson method momenta nearby singularities neutral pions NEUTRAL-PION DECAY nucleon obtain once the parameters p-wave amplitude p-wave integral equation partial-wave amplitudes partial—wave decomposition photon physical region pion-pion amplitude polynomial expansion production from pions pseudoscalar real axis region of convergence scalar transition amplitude scattering amplitude sign function spectral function symmetry point two-dimensional representation proposed two-pole formula Two-pole solution unitary condition unknown parameter unphysical vanish variable A2 zero