Froissart bounds with no arbitrary constants

Summary  A method is described for calculating rigorous upper bounds to averaged total cross-sections $$\overline {\sigma _{tot} (s)} $$ for ππ scattering defined by $$\overline {\sigma _{tot} (s)} = ({{(m + 1)} \mathord{\left/ {\vphantom {{(m + 1)} {(s - 4\mu ^2 }}} \right. \kern-\nulldelimiterspace} {(s - 4\mu ^2 }})^{m + 1} )\int\limits_{4\mu ^2 }^s {(s' - 4\mu ^2 )^m \sigma _{tot} (s')ds'} $$ for all energies and allm≥1, from theD-wave scattering length α 2 t in thet-channel or alternatively from a rigorous bound for theD-wave amplitudea 2(t 0) at one value oft 0 between 0 and 4μ2. It is shown that ass→∞, the bound to $$\overline {\sigma _{tot} (s)} $$ is equivalent to the Froissart bound $$\overline {\sigma _{tot} (s)} \lesssim ({{4\pi (m + 1)} \mathord{\left/ {\vphantom {{4\pi (m + 1)} {t_0 m}}} \right. \kern-\nulldelimiterspace} {t_0 m}})[\log ({s \mathord{\left/ {\vphantom {s {s_0 }}} \right. \kern-\nulldelimiterspace} {s_0 }})]^2 $$ , where the scaling factors 0 can be calculated from the given information andt 0=4μ2 when the scattering length is used. By lettingm→∞ the usual bound is obtained. Similar results are derived for the fixed-angle, fixed-t and forward ππ scattering amplitudes, and again the bounds are asymptotically equivalent to the corresponding Froissart bounds and contain no unknown constants.