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 $0\le s,t,u\le {\mu }^2$, with the four external mass variables off the mass shell and restricted so that ${q_1}^2=0\mathrm{}$, ${q_2}^2=s$, ${q_3}^2=t$, 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.