Two-Variable Expansion of the Scattering Amplitude for any Mass and Crossing Symmetry for Partial Waves

A two-variable expansion of the scattering amplitude for the process $a+b\to c+d$ is proposed, where $a$, $b$, $c$, and $d$ are spinless particles of arbitrary mass. It is diagonal in angular momentum, displays the threshold and pseudothreshold behavior of partial waves, and leads to sum rules which contain a finite number of partial waves due to the crossing symmetry of the collision amplitude. The results of our previous work are recovered when the masses are equal. The reaction $\pi +N\to \pi +N$ is treated with the inclusion of nucleon spin. The expansion is valid over the Dalitz plot for a decay amplitude. A simple method to derive sum rules which relate a finite number of partial waves without the use of the two-variable expansion is also outlined.