Number of Subtractions in Partial-Wave Dispersion Relations

Abstract

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