The Distribution and Chemistry of H$_2$CO in the DM Tau Protoplanetary Disk

H$_2$CO ice on dust grains is an important precursor of complex organic molecules (COMs). H$_2$CO gas can be readily observed in protoplanetary disks and may be used to trace COM chemistry. However, its utility as a COM probe is currently limited by a lack of constraints on the relative contributions of two different formation pathways: on icy grain-surfaces and in the gas-phase. We use archival ALMA observations of the resolved distribution of H$_2$CO emission in the disk around the young low-mass star DM Tau to assess the relative importance of these formation routes. The observed H$_2$CO emission has a centrally peaked and radially broad brightness profile (extending out to 500 AU). We compare these observations with disk chemistry models with and without grain-surface formation reactions, and find that both gas and grain-surface chemistry are necessary to explain the spatial distribution of the emission. Gas-phase H$_2$CO production is responsible for the observed central peak, while grain-surface chemistry is required to reproduce the emission exterior to the CO snowline (where H$_2$CO mainly forms through the hydrogenation of CO ice before being non-thermally desorbed). These observations demonstrate that both gas and grain-surface pathways contribute to the observed H$_2$CO in disks, and that their relative contributions depend strongly on distance from the host star.