Multimeson Resonances and Nucleon-Nucleon Interaction

Relativistic partial-wave dispersion relations are formulated for elastic nucleon-nucleon scattering. These dispersion relations are integral equations with an inhomogeneous term taken from single-particle exchange contributions. The particles under consideration are $\pi (I=1, \mathrm{pseudoscalar})$, $\eta (I=0, \mathrm{pseudoscalar})$, $\rho (I=1,\mathrm{vector})$, $\omega (I=0, \mathrm{vector})$, $\varphi (I=0, \mathrm{vector})$, and $\sigma (I=0, \mathrm{scalar})$. The existence of a $\sigma$ meson is not well established. Two possibilities are considered: (i) The $\sigma$ meson exists, in which case the mass and coupling constants are taken to be two parameters of the present problem. (ii) The $\sigma$ meson does not exist but the $I=0\mathrm{}$, $J=0\mathrm{}$ two-pion continuum is taken into account. This two-pion continuum can be treated as a superposition of scalar particles with a mass spectrum determined by pion-nucleon and pion-pion interactions. Information on the $\pi N$ interaction is obtained from $\pi N$ scattering data, while the $S$-wave $\pi \pi$ interaction is represented with a relativistic scattering-length approximation. In addition to the $\pi \pi$ scattering length, a cutoff on the two-pion spectrum is introduced. Thus two parameters are introduced in either (i) or (ii). Aside from the masses and coupling constants of the particles mentioned, a cutoff parameter is needed for each of the vector mesons $\rho$, $\omega$, and $\varphi$. These are taken to be coefficients in an exponentially decreasing factor suggested by the Regge-pole behavior of composite particles. A total of twelve adjustable parameters is used and a search program is formulated to fit 560 $\mathrm{pp}$ and $\mathrm{np}$ data collected by the Livermore group ranging from 9.68 to 388 MeV. In both cases (i) and (ii), a fit is obtained with a "goodness to fit" value of approximately 8%, meaning that the ${\chi }^2$ is ∼548 if the uncertainty inherent in the theory is assumed to be 8%.