Abelian dominance and quark confinement in Yang-Mills theories

Abstract

We define the Abelian component of the gauge field as a gauge-invariant object in the Yang-Mills theory. Then, by assuming that the Abelian component dominates in the theory at a long-distance scale, we demonstrate that quarks as well as gluons are confined by electric vortices. The vacuum structure is shown to depend on resolution $R$. That is, the vacuum has two phases in $R$ and monopole condensation occurs for $R\ensuremath{\ge}{R}_{c}$. The string tension of mesons is obtained as ${\ensuremath{\sigma}}_{q}=\frac{{{g}_{c}}^{2}}{3\ensuremath{\pi}{{R}_{c}}^{2}}$, where ${g}_{c}$ is the effective coupling constant at the critical resolution ${R}_{c}$. We estimate ${{R}_{c}}^{\ensuremath{-}1}$ to be 0.6 GeV in the presence of static quarks, and the bag constant ${B}^{\frac{1}{4}}$ to be 0.2-0.4 GeV. We also derive a relation ${\ensuremath{\alpha}}_{g}^{\ensuremath{'}}=\frac{1}{3}{\ensuremath{\alpha}}^{\ensuremath{'}}$ between Regge slopes of mesons (${\ensuremath{\alpha}}^{\ensuremath{'}}$) and of gluonia (${\ensuremath{\alpha}}_{g}^{\ensuremath{'}}$). This relation agrees with experimental data remarkably well provided that gluonia are identified with states lying on the Pomeranchuk trajectory.