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, t, u^2, with the four external mass variables off the mass shell and restricted so that {q₁}^2=0, {q₂}^2=s, {q₃}^2=t, and {q₄}^2=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 {q₈}^2. 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.