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

Abstract

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