Baryon Currents in the Fermi-Yang Model

Abstract

All possible baryon currents consisting of trilinear local products of anticommuting nucleon fields are constructed and classified with respect to $\mathrm{SU}(2)\ensuremath{\bigotimes}\mathrm{SU}(2)$. In contrast to the quark-model baryon currents investigated by Hwa and Nuyts, the Fermi-Yang model (nucleon field) baryon currents may be isospin $\frac{3}{2}$ as well as \textonehalf{}. Commutators of the baryon currents with axial charges are investigated, as well as anticommutators of baryon currents with the Hermitian conjugates of their own divergences. Although isospin mixing does occur for some of the axial-charge---baryon-current commutators, it is found that those terms in the commutators carrying the appropriate isospin are always proportional to the baryon current being commuted. For an isospin-\textonehalf{} baryon current ${B}^{\ensuremath{\mu}}(x)$, it is found that $\ensuremath{\delta}({x}_{0}\ensuremath{-}{y}_{0}){{B}^{\ensuremath{\mu}}(x),{\ensuremath{\partial}}^{\ensuremath{\nu}}{{B}_{\ensuremath{\nu}}}^{\ifmmode\dagger\else\textdagger\fi{}}(y)}=\ensuremath{\delta}(x\ensuremath{-}y)\mathcal{O}(\ensuremath{\mu},x)+(\mathrm{Schwinger}\mathrm{term})$, where $〈0|\mathcal{O}(\ensuremath{\mu},x)|0〉=0$. This property of $\mathcal{O}(\ensuremath{\mu},x)$ is equivalent to a spectral-function sum rule, the single-particle approximation of which relates the positive- and negative-parity spin-\textonehalf{}, isospin-\textonehalf{} nucleon resonances.