Behavior of Regge Poles in a Potential at Large Energy

Abstract

Following Mandelstam's suggestion, we consider a potential which can be expanded in a power series in $r$, beginning with $\frac{1}{r}$. Then at large energy (positive or negative) there will be Regge poles at all negative integral values of $l$. If the $\frac{1}{r}$ term is absent in the expansion of the potential, the pole at $l=\ensuremath{-}1$ will be absent. Generally, if ${r}^{2n\ensuremath{-}1}$ is the first nonvanishing odd power term in the potential expansion, the poles at $\ensuremath{-}1, \ensuremath{-}2, \ensuremath{\cdots}, \ensuremath{-}n$ are absent. If the potential expansion contains only even non-negative powers of $r$, there will be no Regge poles at finite negative $l$ in the limit of large energy. If the potential behaves as $\ensuremath{-}\frac{a}{{r}^{2}}$ at small $r$, the scattering amplitude has a cut in the $l$ plane from $\ensuremath{-}\frac{1}{2}\ensuremath{-}{a}^{\frac{1}{2}}$ to $\ensuremath{-}\frac{1}{2}+{a}^{\frac{1}{2}}$, and the Regge poles are no longer located at negative integers at large energy. For finite energy, it is shown that a pole at a positive integer or half-integer $l$ implies another pole at $\ensuremath{-}l\ensuremath{-}1$ but that the residues at these poles are, in general, different from each other.