Number of Subtractions in Partial-Wave Dispersion Relations

Abstract

Under the usual assumption of unitarity and analyticity for the partial-wave amplitude ${f}_{l}(s)$, it is proved that the dispersion relation for ${f}_{l}(s)$ requires no more than one subtraction for any angular momentum $l$, provided that $|{f}_{l}(s)|\ensuremath{\le}\mathrm{exp}[C{(\mathrm{ln}|s|)}^{2\ensuremath{-}\ensuremath{\epsilon}}]$, $\ensuremath{\epsilon}g0$, holds for $|s|\ensuremath{\rightarrow}\ensuremath{\infty}$, and that the number of times that the sign of the discontinuity $\mathrm{Im}{f}_{l}(s+i0)$ changes in the interval ($s$, 0) does not increase more rapidly than ${C}^{\ensuremath{'}}{(\mathrm{ln}|s|)}^{1\ensuremath{-}\ensuremath{\epsilon}}$ as $s\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$ along the negative real axis.