Gluonic Higgs Scalar, Abelianization and Monopoles in QCD -- Similarity and Difference between QCD in the MA Gauge and the NAH Theory

We study the similarity and the difference between QCD in the maximally abelian (MA) gauge and the nonabelian Higgs (NAH) theory by introducing the ``gluonic Higgs scalar field'' $\vec \phi(x)$ corresponding to the ``color-direction'' of the nonabelian gauge connection. The infrared-relevant gluonic mode in QCD can be extracted by the projection along the color-direction $\vec \phi(x)$ like the NAH theory. This projection is manifestly gauge-invariant, and is mathematically equivalent to the ordinary MA projection. Since $\vec \phi(x)$ obeys the adjoint gauge transformation and is diagonalized in the MA gauge, $\vec \phi(x)$ behaves as the Higgs scalar in the NAH theory, and its hedgehog singularity provides the magnetic monopole in the MA gauge like the NAH theory. We observe this direct correspondence between the monopole appearing in the MA gauge and the hedgehog singularity of $\vec \phi(x)$ in lattice QCD, when the gluon field is continuous as in the SU($N_c$) Landau gauge. In spite of several similarities, QCD in the MA gauge largely differs from the NAH theory in the two points: one is infrared monopole condensation, and the other is infrared enhancement of the abelian correlation due to monopole condensation.