The correlations between the modulus of the Polyakov loop, its phase $theta$ and the Landau gauge gluon propagator at finite temperature are investigated in connection with the center symmetry for pure Yang-Mills SU(3) theory. In the deconfined phase, where the center symmetry is spontaneously broken, the phase of the Polyakov loop per configuration is close to $theta = 0$, $pm , 2 pi /3$. We find that the gluon propagator form factors associated with $theta approx 0$ differs quantitatively and qualitatively from those associated to $theta approx pm , 2 pi /3$. This difference between the form factors is a property of the deconfined phase and a sign of the spontaneous breaking of the center symmetry. Furthermore, given that this difference vanishes in the confined phase, it can be used as an order parameter associated to the deconfinement transition. For simulations near the critical temperature $T_c$, the difference between the propagators associated to $theta approx 0$ and $theta approx pm , 2 pi /3$ allows to classify the configurations as belonging to the confined or deconfined phase. This establishes a selection procedure which has a measurable impact in the gluon form factors. Our results also show that the absence of the selection procedure can be erroneously taken as lattice artifacts.
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