Abelian dominance and quark confinement in Yang-Mills theories

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 $R\ge R_c$. The string tension of mesons is obtained as ${\sigma }_q=\frac{{g_c}^2}{3\pi {R_c}^2}$, where $g_c$ is the effective coupling constant at the critical resolution $R_c$. We estimate ${R_c}^{-1\mathrm{}}$ 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 ${\alpha }_g^{\prime }=\frac{1}{3}{\alpha }^{\prime }$ between Regge slopes of mesons (${\alpha }^{\prime }$) and of gluonia (${\alpha }_g^{\prime }$). This relation agrees with experimental data remarkably well provided that gluonia are identified with states lying on the Pomeranchuk trajectory.