RR Lyrae mid-infrared Period-Luminosity-Metallicity and Period-Wesenheit-Metallicity relations based on Gaia DR3 parallaxes

We present new empirical infrared Period-Luminosity-Metallicity (PLZ) and Period-Wesenheit-Metallicity (PWZ) relations for RR Lyrae based on the latest Gaia EDR3 parallaxes. The relations are provided in the WISE $W1$ and $W2$ bands, as well as in the $W(W1, V - W1)$ and $W(W2, V - W2)$ Wesenheit magnitudes. The relations are calibrated using a very large sample of Galactic halo field RR Lyrae stars with homogeneous spectroscopic [Fe/H] abundances (over 1,000 stars in the $W1$ band), covering a broad range of metallicities ($-2.5 \lesssim \textrm{[Fe/H]} \lesssim 0.0$). We test the performance of our PLZ and PWZ relations by determining the distance moduli of both galactic and extragalactic stellar associations: the Sculptor dwarf spheroidal galaxy in the Local Group (finding $\bar{\mu}_{0}=19.47 \pm 0.06$), the Galactic globular clusters M4 ($\bar{\mu}_{0}=11.16 \pm 0.05$) and the Reticulum globular cluster in the Large Magellanic Cloud ($\bar{\mu}_{0}=18.23 \pm 0.06$). The distance moduli determined through all our relations are internally self-consistent (within $\lesssim$ 0.05 mag) but are systematically smaller (by $\sim$ 2-3$\sigma$) than previous literature measurements taken from a variety of methods/anchors. However, a comparison with similar recent RR Lyrae empirical relations anchored with EDR3 likewise shows to varying extents a systematically smaller distance modulus for PLZ/PWZ RR Lyrae relations.