Behavior of Current Divergences underSU3×SU3

Abstract

We investigate the behavior under ${\mathrm{SU}}_{3}\ifmmode\times\else\texttimes\fi{}{\mathrm{SU}}_{3}$ of the hadron energy density and the closely related question of how the divergences of the axial-vector currents and the strangeness-changing vector currents transform under ${\mathrm{SU}}_{3}\ifmmode\times\else\texttimes\fi{}{\mathrm{SU}}_{3}$. We assume that two terms in the energy density break ${\mathrm{SU}}_{3}\ifmmode\times\else\texttimes\fi{}{\mathrm{SU}}_{3}$ symmetry; under ${\mathrm{SU}}_{3}$ one transforms as a singlet, the other as the member of an octet. The simplest possible behavior of these terms under chiral transformations is proposed: They are assigned to a single (3,${3}^{*}$)+(${3}^{*}$,3) representation of ${\mathrm{SU}}_{3}\ifmmode\times\else\texttimes\fi{}{\mathrm{SU}}_{3}$ and parity together with the current divergences. The commutators of charges and current divergences are derived in terms of a single constant $c$ that describes the strength of the ${\mathrm{SU}}_{3}$-breaking term relative to the chiral symmetry-breaking term. The constant $c$ is found not to be small, as suggested earlier, but instead close to the value ($\ensuremath{-}\sqrt{2}$) corresponding to an ${\mathrm{SU}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathrm{SU}}_{2}$ symmetry, realized mainly by massless pions rather than parity doubling. Some applications of the proposed commutation relations are given, mainly to the pseudoscalar mesons, and other applications are indicated.