Remarks on the Analytic Continuation of the Partial-Wave Amplitude

Following Mandelstam, we consider the asymptotic properties of a certain function $F(s, l)$ relevant to the problem of analytic continuation of the partial-wave amplitude $T(s, l)$ by the $\frac{N}{D}$ method. As a function of $s$, $F(s, l)$ is defined to have only the left-hand ($0>~s>~-\infty$) discontinuity of $T(s, l)$. A representation for $F(s, l)$, suitable for discussing its asymptotic properties, is obtained. It is shown that for large $l$, $F(s, l)$ must fall off at least as fast as ${\mathrm{}\left|l\right|\mathrm{}}^{-\frac{1}{2}}$ if $s$ is above threshold. By virtue of crossing symmetry, the $s$-asymptotic behavior of the left-hand discontinuity is related to the behavior of $A_t(s, t)$, the absorptive part of the scattering amplitude in the $t$ channel, as $s$ and/or $t$ tend to infinity. If $A_t(s, t)$ is bounded by $t^{\alpha }$ for fixed $s$, then, under certain assumptions, it is possible to show that $F(s, l)$ is bounded by $s^{\gamma }$ where $\gamma$ is the larger of ($\alpha -l-1\mathrm{}$, -1). A more stringent bound on the asymptotic behavior of $F(s, l)$, although not ruled out, can be established only if one knows the detailed structure of $A_t(s, t)$. It is suggested that in the absence of crossing symmetry, e.g., in potential scattering, the left-hand discontinuity may behave asymptotically as stipulated by Mandelstam so that analytic continuation of $T(s, l)$ by the $\frac{N}{D}$ method would be possible.