Abelian dominance and quark confinement in Yang-Mills theories. II. Oblique confinement andη′mass

Making a working hypothesis of Abelian dominance at a long-distance scale, we analyze the U(1) problem as well as the confinement problem in the $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ Yang-Mills theory with the vacuum angle $\theta$. We show that quarks are confined only when $\frac{\theta }{2\pi }$ is a rational number such that $\frac{\theta }{2\pi }\ne \frac{\mathrm{}(1+nN)\mathrm{}}{\mathrm{mN}}$, $n$ and $m$ being integers. We also calculate a correlation function of the topological charge density $Q\left(x\right)={\left(16\mathrm{}{\pi }^2\right)}^{-1\mathrm{}}\mathrm{Tr}{F}_{\mu \nu }{F}_{\mu \nu }^{*}\mathrm{}\left(x\right)\mathrm{}$ at $\theta =0\mathrm{}$. We derive that $\int d^{4}x〈T\left\{Q\right(x\left)Q\right(0\left)\right\}〉=\frac{N^2}{128{\pi }^4\mathrm{}(N-1\mathrm{})\mathrm{}{\left({\alpha }^{\prime }\right)}^2}$, where ${\alpha }^{\prime }$ denotes the Regge slope of mesons. This formula yields ∼${\mathrm{}\left(150\mathrm{M}\mathrm{e}\mathrm{V}\right)\mathrm{}}^4$ in case of SU(3), which gives rise to a mass ∼ 550 MeV to the ${\eta }^{\prime }$ meson through a chiral anomaly in QCD with massless quarks. This numerical result would explain the ${\eta }^{\prime }$ mass reasonably well in the approximation where pions are massless Goldstone bosons.