Two-Nucleon Potential from Pion Field Theory with Pseudovector Coupling

Abstract

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