Chiral symmetry and the nucleon-nucleon interaction: Tensor decomposition of Feynman diagrams

Abstract

A class of Lagrangians that describe the interaction of nucleons and pions and that provide a nonlinear representation of chiral symmetry is considered. We simplify the form of these Lagrangians by making an expansion in inverse powers of ${\mathit{f}}_{\mathrm{\ensuremath{\pi}}}$ and calculate the irreducible fermion-fermion scattering amplitude to order ${\mathit{f}}_{\mathrm{\ensuremath{\pi}}}^{\mathrm{\ensuremath{-}}4}$. Some of the integrals encountered in these calculations are divergent and are regulated with a (Euclidean) momentum-space cutoff, \ensuremath{\Lambda}, where \ensuremath{\Lambda}\ensuremath{\simeq}1 GeV. While elements of the S matrix are independent of the form of the Lagrangian used, somewhat different results are obtained for the irreducible amplitudes calculated with Lagrangians that have either pseudoscalar or pseudovector coupling of the pion field to the nucleon. We compare our results for the isoscalar irreducible amplitudes to the potentials used in the one-boson-exchange model of the nucleon-nucleon force. In the case of pseudovector coupling, there is only a single relevant diagram of order ${\mathit{g}}^{4}$, a ``crossed-box'' diagram. We find that this crossed-box diagram is well represented by the exchange of a ``pseudo-eta'' particle, that is, an isoscalar-pseudoscalar meson with an imaginary coupling constant. There is also a relatively small scalar attraction seen, while tensor, vector, and axial vector terms are quite small. We also consider a Lagrangian with pseudoscalar coupling. In this case, there are four diagrams of order ${\mathit{g}}^{4}$ in the irreducible amplitude. Again, we find a significant attractive pseudoscalar exchange term (``pseudo-eta''). Relatively small repulsive interactions of vector and scalar type are also found in this case. Further analysis of the isoscalar amplitude requires that we extend our model to reproduce the dynamics of correlated two-pion exchange.