Rigorous sum rules and bounds on pion-pion scattering amplitudes with positivity

Abstract

We derive the complete set of crossing symmetric forward pion-pion amplitudes with non-negative absorptive parts obeying twice-subtracted dispersion relations. We give a method to eliminate subtraction constants for these amplitudes by making a simultaneous use of subtracted dispersion relations for the amplitude and its inverse, and unitarity. We thus deduce from axiomatic field theory (i) lower bounds on forward real parts in terms of physical forward amplitudes, (ii) lower bounds on S- and P-wave scattering lengths in terms of forward amplitudes in any part of the physical region, (iii) sum rules on scattering lengths in terms of physical forward amplitudes which provide a complete test of forward dispersion relations, (iv) upper bounds on scattering lengths in terms of total cross sections and (v) absolute sum rule inequalities on total and forward differential cross sections in terms of the pion mass alone which provide new tests of dispersion relations not involving scattering lengths.