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 $0\ensuremath{\le}s,t,u\ensuremath{\le}{\ensuremath{\mu}}^{2}$, with the four external mass variables off the mass shell and restricted so that ${{q}_{1}}^{2}=0$, ${{q}_{2}}^{2}=s$, ${{q}_{3}}^{2}=t$, and ${{q}_{4}}^{2}=u$. 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 ${{q}_{i}}^{2}$. 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.