Analyticity of the Positions and Residues of Regge Poles

The amplitude $a(l, k)$ for scattering by a superposition of Yukawa potentials is considered as a meromorphic function of angular momentum $l$ and linear momentum $k$. An $\frac{N}{D}$ representation with the usual properties is explicitly given, and is then used to show that the position, $\alpha \mathrm{}\left(k\right)\mathrm{}$, and the residue, $\beta \mathrm{}\left(k\right)\mathrm{}$, of its poles in $l$—the so-called Regge poles—are holomorphic functions of $k$, under a certain assumption. Further considered as functions of energy, $\alpha$ and $\frac{\beta }{k^{2\alpha }}$ are shown to be real-analytic with no left-hand cut.