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Two-dimensional gluon propagators in maximally Abelian gauge in SU(2) Lattice QCD

Using SU(2) lattice QCD in two dimensions, we study diagonal and off-diagonal gluon propagators in the maximally Abelian gauge (MAG) with U(1)$_3$ Landau gauge fixing. These propagators are investigated both in momentum space and coordinate space. The Monte Carlo simulation is performed at $β=7.99, 30.5,$ and $120$ on $62^2, 128^2,$ and $256^2$ at the quenched level. In the momentum space, the transverse component of the diagonal gluon propagator shows suppression with increasing $β$ in the infrared region and the dressing function at $β=120$ has a maximum at $p^2 \simeq 4$GeV, while the transverse component of the off-diagonal gluon propagator does not show the $β$-dependence and the dressing function does not have a maximum. This behavior indicates that the effect of the Gribov copies is found for the diagonal gluon, consistent with the result obtained by the Gribov-Zwanziger action in the MAG. In addition, this behavior supports that the Abelian dominance is not found in two dimensions. In the coordinate space, the diagonal gluon propagator decreases as $β$ increases at long distance. In particular, at $β=120$ the diagonal propagator decreases more rapidly with increasing distance than the off-diagonal propagator. These behaviors also indicate the presence of Gribov copies and no Abelian dominance in two dimensions. Furthermore, we also study these propagators at zero-spatial-momentum. The result suggests that all of the spectral functions of diagonal and off-diagonal gluons would have negative regions and thus they show the violation of the Kallen-Lehmann representation.