The symmetry group of vector and axial vector currents

Abstract

We review, modify slightly, generalize, and attempt to apply a theory proposed earlier of a higher broken symmetry than the eightfold way. The integrals of the time components of the vector and axial vector current octets are assumed to generate, under equal time commutation, the algebra of $\mathrm{SU}(3)\phantom{\rule{0.25em}{0ex}}\ifmmode\times\else\texttimes\fi{}\phantom{\rule{0.25em}{0ex}}\mathrm{SU}(3)$. The energy density of the strong interactions is assumed to consist of a piece invariant under the algebra, a piece that violates conservation of the axial vector currents only and belongs to the representation $(3,\phantom{\rule{0.25em}{0ex}}3*)$ and $(3*,\phantom{\rule{0.25em}{0ex}}3)$, and a piece that violates the eightfold way and probably belongs to $(1,\phantom{\rule{0.25em}{0ex}}8)$ and $(8,\phantom{\rule{0.25em}{0ex}}1)$. Assuming the algebraic structure is exactly correct, there is still the question of whether one can assign particles approximately to super-supermultiplets. The pseudoscalar meson octet, together with a pseudoscalar singlet, a scalar octet, and a scalar singlet, may belong to $(3,\phantom{\rule{0.25em}{0ex}}3*)$ and $(3*,\phantom{\rule{0.25em}{0ex}}3)$. The vector meson octet, together with an axial vector octet, may belong to $(1,\phantom{\rule{0.25em}{0ex}}8)$ and $(8,\phantom{\rule{0.25em}{0ex}}1)$. The baryon octet with $\mathrm{J}\phantom{\rule{0.15em}{0ex}}=\phantom{\rule{0.15em}{0ex}}1/{2}^{+}$, together with a singlet with $\mathrm{J}\phantom{\rule{0.15em}{0ex}}=\phantom{\rule{0.15em}{0ex}}1/{2}^{\ensuremath{-}}$, may belong to $(3,\phantom{\rule{0.25em}{0ex}}3*)$ and $(3*,\phantom{\rule{0.25em}{0ex}}3)$, as suggested before. Several crude coupling patterns and mass rules emerge, to zeroth or first order in the symmetry violations. Some are roughly in agreement with experiment, but certain predictions, like that of the existence of a scalar octet, have not been verified. Whether or not they are useful as an approximate symmetry, the equal time commutation rules fix the scale of the weak interaction matrix elements. Further rules of this kind are found to hold in certain Lagrangian field theory models and may be true in reality. In particular, we encounter an algebraic system based on $\mathrm{SU}(6)$ that relates quantities with different kinds of behavior under Lorentz transformations.