Stellar mass is not the best predictor of galaxy metallicity. The gravitational potential-metallicity relation $Φ\rm ZR$

Interpreting the scaling relations followed by galaxies is a fundamental tool for assessing how well we understand galaxy formation and evolution. Several scaling relations involving the galaxy metallicity have been discovered through the years, the foremost of which is the scaling with stellar mass. This so-called mass-metallicity relation is thought to be fundamental and has been subject to many studies in the literature. We study the dependence of the gas-phase metallicity on many different galaxy properties to assess which of them determines the metallicity of a galaxy. We applied a random forest regressor algorithm on a sample of more than 3000 nearby galaxies from the SDSS-IV MaNGA survey. Using this machine-learning technique, we explored the effect of 148 parameters on the global oxygen abundance as an indicator of the gas metallicity. $M_{\rm \star}$/$R_e$, as a proxy for the baryonic gravitational potential of the galaxy, is found to be the primary factor determining the average gas-phase metallicity of the galaxy ($Z_g$). It outweighs stellar mass. A subsequent analysis provides the strongest dependence of $Z_g$ on $M_\star / R_e^{\,0.6}$. We argue that this parameter traces the total gravitational potential, and the exponent $α\simeq 0.6$ accounts for the inclusion of the dark matter component. Our results reveal the importance of the relation between the total gravitational potential of the galaxy and the gas metallicity. This relation is tighter and likely more primordial than the widely known mass-metallicity relation.