Correlator of topological charge densities at low $Q^2$ in QCD: connection with proton spin problem

Abstract

Vacuum correlator of topological charge densities $\chi (q^2)$ at low $q^2$ in QCD is discussed: its value $\chi (0)$ and first derivative ${\chi }^{\prime }(0)$ at $q^2=0$ and the contributions of pseudoscalar quasi-Goldstone bosons to $\chi (q^2)$ at low $q^2$. The QCD sum rule \cite{1}, giving the connection of ${\chi }^{\prime }(0)$ (for massless quarks) with the part of the proton spin $\Sigma$, carried by $u,d,s$ quarks, is presented. From the requirement of selfconsistancy of the sum rule the values of ${\chi }^{\prime }(0)$ and $\Sigma$ were found. The same value of ${\chi }^{\prime }(0)$ follows also from the experimental data on $\Sigma$. The contributions of $\pi$ and $\eta$ to $\chi (q^2)$ are calculated basing on low energy theorems. In such calculation the $\pi -\eta$ mixing, expressed in terms of quark mass ratios is of importance.

Comments: V. Gribov Memorial Volume, Heavy Physics, Academiai Kiado, Budapest, in press, 20 pages, LaTeX 2e, (file hip-atrc.sty is enclosed), 1 eps figure included using epsf.sty