Abstract

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 $\mathrm{Re}l>\ensuremath{\alpha}$ 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 ${t}^{\ensuremath{\alpha}}$ 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 ($\frac{{t}^{\ensuremath{\beta}}}{{s}^{\ensuremath{\gamma}}}, \frac{{s}^{\ensuremath{\beta}}}{{t}^{\ensuremath{\gamma}}}$) if $s$ and $t$ are both sufficiently large with $\ensuremath{\gamma}>0$ and $\ensuremath{\beta}<min(1, \ensuremath{\gamma})$. With the help of the $\frac{N}{D}$ technique it is then possible to continue analytically the partial-wave amplitude up to the line $\mathrm{Re}l=\ensuremath{\beta}$ and show that it is meromorphic in the region $\ensuremath{\beta}<\mathrm{Re}l<~\ensuremath{\alpha}$. 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 ${|l|}^{\ensuremath{-}\frac{1}{2}}$ 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 $\mathrm{Re}l=\ensuremath{-}1$ even if one assumes $\ensuremath{\beta}<\ensuremath{-}1$. It has also been shown that accumulation of poles at $l=\ensuremath{-}\frac{1}{2}$ near threshold, a feature which has been pointed out by several authors, is also manifested in the analytically continued partial-wave amplitude.

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