Non-Lagrangian Models of Current Algebra

TLDR: In this article, an alternative to specific Lagrangian models of current algebra is proposed, in which scale invariance is a broken symmetry of strong interactions, as proposed by Kastrup and Mack.

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, ${j}_{\ensuremath{\mu}}(x){j}_{\ensuremath{\mu}}(x)$] 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 $\mathrm{SU}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(3)$ currents and the $\mathrm{SU}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(3)$ multiplet containing the pion field. It is assumed that the pion field scales like a field of dimension $\ensuremath{\Delta}$, where $\ensuremath{\Delta}$ is unspecified within the range $1\ensuremath{\le}\ensuremath{\Delta}l4$; the value of $\ensuremath{\Delta}$ 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 $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$; electromagnetic corrections require one subtraction proportional to the $I=1$, ${I}_{z}=0\ensuremath{\sigma}$ field; $\ensuremath{\eta}\ensuremath{\rightarrow}3\ensuremath{\pi}$ and ${\ensuremath{\pi}}_{0}\ensuremath{\rightarrow}2\ensuremath{\gamma}$ 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.