  We employ Pad\'e approximants in the study of the analytic structure of the
four-dimensional $SU(2)$ Landau-gauge gluon and ghost propagators in the
infrared regime. The approximants, which are model independent, serve as
fitting functions for the lattice data. We carefully propagate the
uncertainties due to the fitting procedure, taking into account all possible
correlations. For the gluon-propagator data, we confirm the presence of a pair
of complex poles at $p_{\rm pole}^2 = \left[(-0.37 \,\pm\, 0.05_{\rm
stat}\,\pm\, 0.08_{\rm sys}) \pm i\,(0.66\, \pm\, 0.03_{\rm stat}\, \pm\,
0.02_{\rm sys})\right]\, \mathrm{GeV}^2$, where the first error is statistical
and the second systematic. The existence of this pair of complex poles, already
hinted upon in previous works, is thus put onto a firmer basis, thanks to the
model independence and to the careful error propagation of our analysis. For
the ghost propagator, the Pad\'es indicate the existence of a single pole at
$p^2 = 0$, as expected. In this case, our results also show evidence of a
branch cut along the negative real axis of $p^2$. This is corroborated with
another type of approximant, the D-Log Pad\'es, which are better suited to
studying functions with a branch cut and are applied here for the first time in
this context. Due to particular features and limited statistics of the
gluon-propagator data, our analysis is inconclusive regarding the presence of a
branch cut in the gluon case.
