Two-Nucleon Potential from Pion Field Theory with Pseudoscalar Coupling

The two-nucleon potential is derived from $ps-ps$ pion field theory, using the Tamm-Dancoff method with the subsequent appropriate renormalization of the two-nucleon amplitude, up to orders $g^2{\left(\frac{p}{\kappa }\right)}^2$ and $g^4\left(\frac{p}{\kappa }\right)$, $g^2$ being the equivalent $ps-pv$ coupling constant and $p$, $\kappa$ the nucleon (relative) momentum and rest mass, respectively. Neglected are the mass and charge renormalizations and the pion-pion scattering term. It is shown that the only quadratic term is $-V_2\left(\mathrm{r}\right)\left(\frac{p^2}{2{\kappa }^2}\right)+\mathrm{H}.\mathrm{c}.\mathrm{}$, where $V_2\left(\mathrm{r}\right)$ is the second order static potential. All remaining terms of order $g^2{\left(\frac{p}{\kappa }\right)}^2$ are converted by a canonical transformation to terms of order $g^4\left(\frac{p}{\kappa }\right)$ or a linear combination of a static and an L·S potential. In the case of $ps-ps$ coupling, in particular, no velocity-dependent potential besides the quadratic one mentioned above follows from diagrams of one- and two-pion exchange without nucleon pairs; thus the nucleon-pair diagrams are the only source of an L·S potential, if there is any, up to the orders considered. The nucleon-pair diagrams are also estimated assuming the effective pion-pair interaction Hamiltonian, the static limit of which agrees with the $S$-wave static model of Drell et al. The L·S potential thus obtained has the right sign in the odd state and changes its sign in the even state, while its magnitude seems in both states somewhat too small. As for the static part of our potential, the new correction [$g^4\left(\frac{\mu }{\kappa }\right)$] cancels out the conventional fourth-order term [$g^4$] appreciably; the entire static potential becomes quite close to the second-order static potential down to distances of the order of the pion Compton wavelength, except for the central force in the triplet even state. The details are shown on graphs. The $ps-pv$ coupling case is treated separately in the following paper.