Non-Lagrangian Models of Current Algebra

Abstract

An alternative is proposed to specific Lagrangian models of current algebra. In this alternative there are no explicit canonical fields, and operator products at the same point [say, $j_{\mu }\mathrm{}\left(x\right)\mathrm{}{j}_{\mu }\mathrm{}\left(x\right)\mathrm{}$] have no meaning. Instead, it is assumed that scale invariance is a broken symmetry of strong interactions, as proposed by Kastrup and Mack. Also, a generalization of equal-time commutators is assumed: Operator products at short distances have expansions involving local fields multiplying singular functions. It is assumed that the dominant fields are the $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{}$ currents and the $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{}$ multiplet containing the pion field. It is assumed that the pion field scales like a field of dimension $\Delta$, where $\Delta$ is unspecified within the range $1\le \Delta <4\mathrm{}$; the value of $\Delta$ is a consequence of renormalization. These hypotheses imply several qualitative predictions: The second Weinberg sum rule does not hold for the difference of the $K^{*}$ and axial-$K^{*}$ propagators, even for exact $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$; electromagnetic corrections require one subtraction proportional to the $I=1\mathrm{}$, $I_z=0\mathrm{}\sigma$ field; $\eta \to 3\pi$ and ${\pi }_0\to 2\gamma$ are allowed by current algebra. Octet dominance of nonleptonic weak processes can be understood, and a new form of superconvergence relation is deduced as a consequence. A generalization of the Bjorken limit is proposed.