New Consistency Conditions on Pion-Pion Amplitudes and Their Determination to Fourth Order in External Momenta

We derive a set of new consistency conditions for the pion-pion scattering amplitude. These conditions hold for any s, t, u in the cube 0≤s,t,u≤μ², with the four external mass variables off the mass shell and restricted so that q₁²=0, q₂²=s, q₃²=t, and q₄²=u. Using these consistency conditions, we determine the coefficients of the power-series expansion of the pion-pion amplitude up to and including second-order terms in the variables s, t, u, and qᵢ². We use this expansion to calculate the pion-pion S-wave scattering lengths and thus check the consistency of Weinberg's recent calculation of these numbers to the next higher order. The final result is within 10% of that obtained by Weinberg.