Spectral-Function Sum Rules and the Sugawara Model

Abstract

We investigate the implications of Sugawara's theory of currents for the properties of the vector and axialvector spectral functions ${\rho }_V$ and ${\rho }_A$. Using only the requirement of Lorentz invariance of the equal-time commutators of the components of the stress-energy tensor, we show that in order that the theory be nontrivial, the integrals $\int d{m}^2{\rho }_{V,A}\left(m^2\right)$ involved in Weinberg's second sum rule must diverge. This in turn enables us to predict the asymptotic behavior of the total cross section for the process $e^{+}+e^{-}\to \mathrm{hadrons}$, and is a rigorous consequence of the nontrivial Sugawara model. Our arguments, unlike Dashen and Frishman's, apply for the $\mathrm{SU}\left(3\right)\mathrm{}$ version of the Sugawara model as well.