Relativistic field-theoretical approach to the inverse scattering problem

Abstract

The inverse scattering problem for the relativistic three-dimensional equation $(2E_{{\mathbf{p}}^{\prime }}-2E_{\mathbf{p}})<{\mathbf{p}}^{\prime }|{\Psi }_{\mathbf{p}}>=-\int V(t)d^3{\mathbf{p}}^{\prime \prime }<{\mathbf{p}}^{\prime \prime }|{\Psi }_{\mathbf{p}}>$ with $E_{\mathbf{p}}=\sqrt{m^2+{\mathbf{p}}^2}$ and $t=(E_{{\mathbf{p}}^{\prime }}-E_{{\mathbf{p}}^{\prime \prime }}{)}^2-({\mathbf{p}}^{\prime }-{\mathbf{p}}^{\prime \prime }{)}^2$ is considered. The field-theoretical potential $V(t)$ of this equation is constructed in the framework of the old perturbation theory. It contains all contributions of diagrams with intermediate off-mass shell particles. In particular, this potential reproduces the OBE Bonn model of the $NN$ potential exactly.For the $\pi N$ scattering it is generated by $\sigma ,\rho ,\omega$-meson exchange diagrams. The inverse scattering problem is solved by reduction of these relativistic equations to the standard Schrödinger equations $({\Delta }_{\mathbf{r}}+{\mathbf{k}}^2)<\mathbf{r}|{\varphi }_{\mathbf{p}}>=-v(\mathbf{r})<\mathbf{r}|{\varphi }_{\mathbf{k}}>$ with $E_{\mathbf{p}}={\mathbf{k}}^2/2m+m$. The relation between the relativistic potential $V(t)$ and its nonrelativistic representation $v(\mathbf{r})$ is obtained.

Comments: LATEX, 10 pages, 2 figures