Some Consequences of a Piecewise-Analytic Scattering Amplitude

Abstract

In the indefinite-metric quantum field theory formulated by one of the authors and in related work the scattering amplitude does not satisfy the usual Mandelstam analyticity. It becomes piecewise analytic at the threshold for the production of at least one of the negative-metric particles of the theory, i.e., the amplitude above the threshold is not the analytic continuation of the amplitude. Some consequences of such a piecewise-analytic scattering amplitude are investigated. The physical two-body scattering amplitude $G(s, t)$ is taken to be of the form $G(s, t)=F(s, t)+{h}_{1}(s, t)\ensuremath{\theta}(s\ensuremath{-}{s}_{0})+{h}_{2}(s, t)\ensuremath{\theta}(t\ensuremath{-}{t}_{0})+{h}_{3}(s, t)\ensuremath{\theta}(u\ensuremath{-}{u}_{0})$, where $F(s, t)$ and the ${h}_{i}(s, t)$ are analytic functions of $s$ and $t$ with the negative-metric thresholds occurring at $s={s}_{0}$, $t={t}_{0}$, and $u={u}_{0}$ in the $s$, $t$, and $u$ channels, respectively. The modified forms of the Pomeranchuk theorem, dispersion relations, and finite-energy sum rules due to this general form of piecewise analyticity are derived and the interpretation of experimental results in terms of them are discussed. In particular, the modified forward dispersion relations for ${\ensuremath{\pi}}^{+}p$ and ${\ensuremath{\pi}}^{\ensuremath{-}}p$ scattering differ from the normal forms by a function $\ensuremath{\xi}(\ensuremath{\nu})$ which depends on the piecewise-analytic contributions ${h}_{i}(s, t)$ above, where $\ensuremath{\nu}$ is the laboratory momentum of the pion. The forward dispersion relations for the symmetric and antisymmetric combinations of the real part of the ${\ensuremath{\pi}}^{+}p$ and ${\ensuremath{\pi}}^{\ensuremath{-}}p$ scattering amplitudes ${D}^{+}(\ensuremath{\nu})$ and ${D}^{\ensuremath{-}}(\ensuremath{\nu})$, respectively, are tested. The best fits to the latest total cross-section data for ${\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}p$ scattering from 8 to 65 GeV/c which do not satisfy the Pomeranchuk theorem are used. No test for ${D}^{\ensuremath{-}}(\ensuremath{\nu})$ which must be twice subtracted is possible since it depends strongly on the $\ensuremath{\pi}N$ coupling constant ${f}^{2}$ which is itself determined by dispersion relations. The result for ${D}^{+}(\ensuremath{\nu})$ allows for a violation, but the evidence is not compelling.