The extended ultraviolet (XUV) disk galaxies are one of the most interesting objects studied in the last few years. The UV emission, revealed by GALEX, extends well beyond the optical disk, after the drop of H$alpha$ emission, the usual tracer of star formation. This shows that sporadic star formation can occur in a large fraction of the HI disk, at radii up to 3 or 4 times the optical radius. In most galaxies, these regions are poor in stars and dominated by under-recycled gas, therefore bear some similarity to early stages of spiral galaxies and high-redshift galaxies. One remarkable example is M83, a nearby galaxy with an extended UV disk reaching 2 times the optical radius. It offers the opportunity to search for the molecular gas and characterise the star formation in outer disk regions, traced by the UV emission. We obtained CO(2-1) observations with ALMA of a small region in a 1.5$times$ 3 rectangle located at $r_{gal}=7.85$ over a bright UV region of M83. There is no CO detection, in spite of the abundance of HI gas, and the presence of young stars traced by their HII regions. Our spatial resolution (17pc x 13pc) was perfectly fitted to detect Giant Molecular Clouds (GMC), but none were detected. The corresponding upper limits occur in an SFR region of the Kennicutt-Schmidt diagram where dense molecular clouds are expected. Stacking our data over HI-rich regions, using the observed HI velocity, we obtain a tentative detection, corresponding to an H$_2$-to-HI mass ratio of $<$ 3 $times$ 10$^{-2}$. A possible explanation is that the expected molecular clouds are CO-dark, because of the strong UV radiation field. The latter preferentially dissociates CO with respect to H$_2$, due to the small size of the star forming clumps in the outer regions of galaxies.
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