Canonical Trace Anomalies

We discuss the anomalies present in broken-scale-invariance trace identities which result from assuming that products of hadronic currents have a canonical singularity structure at short distances. The analysis is performed qualitatively in configuration space and quantitatively in momentum space. Canonical anomalies are found in trace identities involving two electromagnetic currents, or two axial-vector currents or their divergences. There are related canonical anomalies in trace identities involving three or four currents. They can be represented by the anomalous trace equation ${\theta }_{\lambda }^{\lambda \mathrm{anomalous}}\mathrm{}\left(x\right)={\theta }_{\lambda }^{\lambda }\mathrm{}\left(x\right)+\left(\frac{R}{32{\pi }^2}\right){\tilde{F}}_{\mu \nu }^i{\tilde{F}}_i^{\mu \nu }$, where ${\tilde{F}}_{\mu \nu }^i={\partial }_{\mu }{F}_{\nu }^i-{\partial }_{\nu }{F}_{\mu }^i+h_{\mathrm{ijk}}{F}_{\mu }^j{F}_{\nu }^k$, with the $F_{\mu }^i$ external fields coupled to the SU(3) × SU(3) currents, and $h_{\mathrm{ijk}}$ the structure constants of SU(3) × SU(3). The electromagnetic current trace anomaly is related to the high-energy cross section for $e^{-}{e}^{+}\to \gamma \to \mathrm{hadrons}$, and via P0T (partially zero trace) to the coupling of a scalar meson to photons. These are connected by $\left(12\mathrm{}{\pi }^2{F}_{\sigma }\right)g_{\sigma \gamma \gamma }=R=limit\; of\frac{\sigma (e^{-}{e}^{+}\to \gamma \to \mathrm{hadrons})}{\sigma (e^{-}{e}^{+}\to \gamma \to {\mu }^{-}{\mu }^{+})}\text{as}{p}^2\to \infty$, where $F_{\sigma }$ is defined by $〈0\left|{\theta }_{\mu }^{\mu }\right|\sigma 〉=m_{\sigma }^{}{}_{}{}^2{F}_{\sigma }$. The axial-vector current anomalies are related to the high-energy cross sections for $e^{-}{\overline{\nu }}_e\left({\mu }^{-}{\overline{\nu }}_{\mu }\right)\to \mathrm{hadrons}$; they do not affect previous estimates of the $\sigma \pi \pi$ coupling made using broken scale invariance and P0T.