  The study of the planet-debris disk connection can shed light on the
formation and evolution of planetary systems, and may help predict the presence
of planets around stars with certain disk characteristics. In preliminary
analyses of the Herschel DEBRIS and DUNES surveys, Wyatt et al. (2012) and
Marshall et al. (2014) identified a tentative correlation between debris and
low-mass planets. Here we use the cleanest possible sample out these surveys to
assess the presence of such a correlation, discarding stars without known ages,
with ages < 1 Gyr and with binary companions <100 AU, to rule out possible
correlations due to effects other than planet presence. In our sample of 204
FGK stars, we do not find evidence that debris disks are more common or more
dusty around stars harboring high-mass or low-mass planets compared to a
control sample without identified planets, nor that debris disks are more or
less common (or more or less dusty) around stars harboring multiple planets
compared to single-planet systems. Diverse dynamical histories may account for
the lack of correlations. The data show the correlation between the presence of
high-mass planets and stellar metallicity, but no correlation between the
presence of low-mass planets or debris and stellar metallicity. Comparing the
observed cumulative distribution of fractional luminosity to those expected
from a Gaussian distribution, we find that a distribution centered on the Solar
system's value fits well the data, while one centered at 10 times this value
can be rejected. This is of interest in the context of future terrestrial
planet characterization because it indicates that there are good prospects for
finding a large number of debris disk systems (i.e. with evidence of harboring
the building blocks of planets) with exozodiacal emission low enough to be
appropriate targets for an ATLAST-type mission to search for biosignatures.
