Analytic Continuation of Partial-Wave Amplitude in the Complex Angular-Momentum Plane

An attempt is made to continue analytically the partial-wave amplitude for the scattering of two identical spinless particles in the complex $l$ plane, exploiting unitarity and analyticity properties in $s$. The Froissart-Gribov representation for the partial-wave amplitude is known to be holomorphic in the region $\mathrm{Re}l>\mathrm{}\alpha$ of the complex $l$ plane provided the absorptive part $A_t(s, t)$ of $A(s, t)$, the scattering amplitude in the $t$ channel, is bounded by $t^{\alpha }$ for any fixed $s$. Apart from the above assumptions, two crucial hypotheses on which the present analysis is based are (i) the possibility of extending unitarity in the inelastic region to complex values of $l$, and (ii) the boundedness condition, viz., that both $A_t(s, t)$ and $A(s, t)$ are asymptotically bounded by the maximum of ($\frac{t^{\beta }}{s^{\gamma }}, \frac{s^{\beta }}{t^{\gamma }}$) if $s$ and $t$ are both sufficiently large with $\gamma >0\mathrm{}$ and $\beta <min(1, \gamma )$. With the help of the $\frac{N}{D}$ technique it is then possible to continue analytically the partial-wave amplitude up to the line $\mathrm{Re}l=\beta$ and show that it is meromorphic in the region $\beta <\mathrm{Re}l<~\alpha$. The domain of meromorphy of the partial-wave amplitude obtained by the method of analytic completion is smaller than the preceding one. The analytically continued partial-wave amplitude is bounded by ${\mathrm{}\left|l\right|\mathrm{}}^{-\frac{1}{2}}$ for large values of $\mathrm{Im}l$, so that a Regge representation for $A(s, t)$ can be obtained. The $\frac{N}{D}$ method of analytic continuation does not work beyond the line $\mathrm{Re}l=-1\mathrm{}$ even if one assumes $\beta <-1\mathrm{}$. It has also been shown that accumulation of poles at $l=-\frac{1}{2}$ near threshold, a feature which has been pointed out by several authors, is also manifested in the analytically continued partial-wave amplitude.