[Abridged] Debris discs around main-sequence stars indicate the presence of larger rocky bodies. The components of the nearby binary aCentauri have higher than solar metallicities, which is thought to promote giant planet formation. We aim to determine the level of emission from debris in the aCen system. Having already detected the temperature minimum, Tmin, of aCenA, we here attempt to do so also for the companion aCenB. Using the aCen stars as templates, we study possible effects Tmin may have on the detectability of unresolved dust discs around other stars. We use Herschel and APEX photometry to determine the stellar spectral energy distributions. In addition, we use APEX for spectral line mapping to study the complex background around aCen seen in the photometric images. Models of stellar atmospheres and discs are used to estimate the amount of debris around these stars. For solar-type stars, a fractional dust luminosity fd 2e-7 could account for SEDs that do not exhibit the Tmin-effect. Slight excesses at the 2.5 sigma level are observed at 24 mu for both stars, which, if interpreted to be due to dust, would correspond to fd (1-3)e-5. Dynamical disc modelling leads to rough mass estimates of the putative Zodi belts around the aCen stars, viz. <~4e-6 MMoon of 4 to 1000 mu size grains, distributed according to n a^-3.5. Similarly, for filled-in Tmin emission, corresponding EKBs could account for ~1e-3 MMoon of dust. Light scattered and/or thermally emitted by exo-Zodi discs will have profound implications for future spectroscopic missions designed to search for biomarkers in the atmospheres of Earth-like planets. The F-IR SED of aCenB is marginally consistent with the presence of a minimum temperature region in the upper atmosphere. We also show that an aCenA-like temperature minimum may result in an erroneous apprehension about the presence of dust around other stars.
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