The MUSE Ultra Deep Field (MUDF). V. Characterizing the Mass-Metallicity Relation for Low Mass Galaxies at $z\sim 1$-$2$

Using more than 100 galaxies in the MUSE Ultra Deep Field with spectroscopy from the Hubble Space Telescope's Wide Field Camera 3 and the Very Large Telescope's Multi Unit Spectroscopic Explorer, we extend the gas-phase mass-metallicity relation (MZR) at $z\approx\,$1$\,$-$\,$2 down to stellar masses of M$_{\star}$ $\approx$ 10$^{7.5}$ M$_{\odot}$. The sample reaches six times lower in stellar mass and star formation rate (SFR) than previous HST studies at these redshifts, and we find that galaxy metallicities decrease to log(O/H) + 12 $\approx$ 7.8 $\pm$ 0.1 (15% solar) at log(M$_{\star}$/M$_{\odot}$) $\approx$ 7.5, without evidence of a turnover in the shape of the MZR at low masses. We validate our strong-line metallicities using the direct method for sources with [O III] $\lambda$4363 and [O III] $\lambda$1666 detections, and find excellent agreement between the techniques. The [O III] $\lambda$1666-based metallicities double existing measurements with S/N $\geq$ 5 for unlensed sources at $z~>$ 1, validating the strong-line calibrations up to $z \sim$2.5. We confirm that the MZR resides $\sim$0.3 dex lower in metallicity than local galaxies and is consistent with the fundamental metallicity relation (FMR) if the low mass slope varies with SFR. At lower redshifts ($z\sim$0.5) our sample reaches $\sim$0.5 dex lower in SFR than current calibrations and we find enhanced metallicities that are consistent with extrapolating the MZR to lower SFRs. Finally, we detect only a $\sim$0.1 dex difference in the metallicities of galaxies in groups versus isolated environments. These results are based on robust calibrations and reach the lowest masses and SFRs that are accessible with HST, providing a critical foundation for studies with the Webb and Roman Space Telescopes.