Narrow loophole for H2-dominated atmospheres on habitable rocky planets around M dwarfs

Habitable rocky planets around M dwarfs that have \ce{H2}-dominated atmospheres, if they exist, would permit characterizing habitable exoplanets with detailed spectroscopy using JWST, owing to their extended atmospheres and small stars. However, the \ce{H2}-dominated atmospheres that are consistent with habitable conditions cannot be too massive, and a moderate-size \ce{H2}-dominated atmosphere will lose mass to irradiation-driven atmospheric escape on rocky planets around M dwarfs. We evaluate volcanic outgassing and serpentinization as two potential ways to supply \ce{H2} and form a steady-state \ce{H2}-dominated atmosphere. For rocky planets of $1-7\ M_{\oplus}$ and early, mid, and late M dwarfs, the expected volcanic outgassing rates from a reduced mantle fall short of the escape rates by $>\sim1$ order of magnitude, and a generous upper limit of the serpentinization rate is still less than the escape rate by a factor of a few. Special mechanisms that may sustain the steady-state \ce{H2}-dominated atmosphere include direct interaction between liquid water and mantle, heat-pipe volcanism from a reduced mantle, and hydrodynamic escape slowed down by efficient upper-atmospheric cooling. It is thus unlikely to find moderate-size, \ce{H2}-dominated atmospheres on rocky planets of M dwarfs that would support habitable environments.