Two-Nucleon Potential from Pion Field Theory with Pseudovector Coupling

The two-nucleon potential is derived from $ps-pv$ pion field theory up to orders $g^2{\left(\frac{p}{\kappa }\right)}^2$ and $g^4\left(\frac{p}{\kappa }\right)$, using the method outlined in the preceding paper, where it was applied to $ps-ps$ theory. It is shown that the only quadratic term is $-V_2\left(\mathrm{r}\right)\left(\frac{p^2}{2{\kappa }^2}\right)+\mathrm{H}.\mathrm{c}.\mathrm{}$ [$V_2\left(\mathrm{r}\right)$ is the second-order static potential], just as in the $ps-ps$ case. The static part is almost the same as the $ps-ps$ potential. However, a big difference appears in the L·S potential; because of the difference in kinematical corrections from $ps-ps$ and $ps-pv$ vertices, large L·S potentials result from both one-pion and two-pion exchange diagrams (with no nucleon pairs), though no L·S potential follows from such diagrams in case of $ps-ps$ theory. The entire L·S potential has the right sign in the odd state and is of the same sign and of larger magnitude [by a factor of two or three] in the even state. We show that this isospin dependence of the L·S potential is not appreciably modified even if we add, besides the $ps-pv$ coupling term, two pion-pair terms which are fitted to low-energy $S$-wave pion-nucleon scattering. This big difference in the L·S potential could eventually be used to discriminate between $ps-ps$ and $ps-pv$ theories. Various L·S potentials, theoretical and phenomenological, are shown on graphs for comparison.