  Infrared excesses around metal polluted white dwarfs have been associated
with the accretion of dusty, planetary material. This work analyses the
available infrared data for an unbiased sample of white dwarfs and demonstrates
that no more than 3.3% can have a wide, flat, opaque dust disc, extending to
the Roche radius, with a temperature at the disc inner edge of $T_{in} =
1,400$K, the standard model for the observed excesses. This is in stark
contrast to the incidence of pollution of about 30%. We present four potential
reasons for the absence of an infrared excess in polluted white dwarfs,
depending on their stellar properties and inferred accretion rates: i) their
dust discs are opaque, but narrow, thus evading detection if more than 85% of
polluted white dwarfs have dust discs narrower than $\delta r< 0.04r$, ii)
their dust discs have been fully consumed, which only works for the oldest
white dwarfs with sinking timescales longer than hundreds of years, iii) their
dust is optically thin, which can supply low accretion rates of $<10^7
$gs$^{-1}$ if dominated by PR-drag, and higher accretion rates, if inwards
transport of material is enhanced, for example due to the presence of gas, iv)
their accretion is supplied by a pure gas disc, which could result from the
sublimation of optically thin dust for T* > 20, 000K. Future observations
sensitive to faint infrared excesses or the presence of gas, can test the
scenarios presented here, thereby better constraining the nature of the
material fuelling accretion in polluted white dwarfs.
