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 $\chi'(0)$ 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 $\chi'(0)$ (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 $\chi'(0)$ and $\Sigma$ were found. The same value of $\chi'(0)$ 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.