Analytic Continuation of the Froissart-Gribov Partial-Wave Amplitude to the Left-HalflPlane

Abstract

A continuation of the Froissart-Gribov definition of the partial-wave amplitude to the left of the poles in the $l$ plane is obtained under the assumption of power-law behaviors of the Mandelstam weight functions at high energy. Discrete and continuous powers in these weight functions are seen to yield, respectively, poles and cuts in the continued partial-wave amplitude. This continuation is then used to prove that in the presence of cuts a generalized form of the Mandelstam symmetry relation for the partial-wave amplitudes about $l=\ensuremath{-}\frac{1}{2}$ for the half-odd-integral values of $l$ holds at energies where there are no Regge poles passing through half-odd integers. The discontinuity across the cut at a half-odd integer is always equal to discontinuity across the cut at the half odd integer obtained by reflection about $l=\ensuremath{-}\frac{1}{2}$. The case of Regge poles passing through half-odd integers is considered in detail, and the results derived by Mandelstam for potential scattering are shown to follow from our continuation in a straightforward manner. The continued partial-wave amplitude has the desirable feature that every term in it has the correct threshold behavior, ${({q}^{2})}^{l}$.