Asymptotic Behavior of Partial-Wave Amplitudes

For infinite energies, we determine the asymptotic behavior of partial-wave amplitudes when the full scattering amplitude satisfies Mandelstam representation and has itself a Regge asymptotic behavior. Particular attention is paid to the behavior of the partial-wave-amplitude discontinuities on their cuts. They are shown to behave as ${\mathrm{}\left|t\right|\mathrm{}}^{\alpha \mathrm{}\left(0\right)-1\mathrm{}}$, where $t$ is the energy squared and $\alpha \mathrm{}\left(0\right)\mathrm{}$ is the leading Regge-pole position at zero energy. This result removes an old-standing difficulty in the Chew-Mandelstam calculation of amplitudes and provides a precise justification of the nearest singularity technique. As an application, we show that no subtraction is necessary in partial-wave-amplitude dispersion relations at physical values of the angular momentum, even for the case of $S$ waves.