Some bounds on pion-scattering amplitudes and their applications

If one uses the method introduced by Meinam and further developed by Martin, together with unitarity crossing symmetry and analyticity, upper and lower bounds to the πdeg πged scattering amplitudes inside the Mandelstam triangle are obtained in terms of the scattering lengths. These bounds are used to set a lower bound to the renormalized pion coupling constant lambda. In terms of the D-wave scattering length a2, the results are a0 greater than or equal to -23.18 square root a2 + 129.07a2 and -7.71 square root a2 less than or equal to lambda less than or equal to 7.71 square root a2+2.5a2. For a2=7.10-4 they yield a2 greater than or equal to -0.52 and lambda less than or equal to 0.20 where the pion, mass is set equal to unity