Crossing Relations and Legendre Expansions in Pion-Pion Scattering

The direct use of crossing relations for pion-pion scattering amplitudes, outside the triangle bordered by the lines $s=0\mathrm{}$, $t=0\mathrm{}$, and $u=0\mathrm{}$ in the Mandelstam diagram, is not generally possible. This is because the regions of convergence of the usual Legendre expansions for the amplitudes are restricted by cross-channel cuts in appropriate $cos\theta$ variables. These convergence difficulties may be relieved by suitably decomposing each amplitude into two terms. One term differs in analytic properties from the actual amplitude in that portions of the cross-channel cuts in $cos\theta$ nearest $cos\theta =\pm 1$ are absent. The Legendre expansion of this term has a larger region of convergence than that for the actual amplitude. The other term in the decomposition is expressed in terms of the Legendre series for physical scattering in the cross channels. The amplitudes so represented may now be continued from one physical region to another, and crossing relations may, in general, be directly applied outside the triangle. As a simple application of the formalism, the existence and approximate mass and width of the $\rho$ meson are found to be simple consequences of analyticity, unitarity, and crossing symmetry.