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We study the massive scalar field equation $\Box_g \phi = m^2 \phi$ on astationary and spherically symmetric black hole $g$ (including in particularthe Schwarzschild and Reissner--Nordstr\"om black holes in the fullsub-extremal range) for solutions $\phi$ projected on a fixed sphericalharmonic. Our problem involves the scattering of an attractive long-rangepotential (Coulomb-like) and thus cannot be treated perturbatively.  We prove precise (point-wise) asymptotic tails of the form $t^{-5/6} f(t)+O(t^{-1+\delta})$, where $f(t)$ is an explicit oscillating profile. Ourasymptotics appear to be the first rigorous decay result for a massive scalarfield on a black hole. Establishing these asymptotics is also an important stepin retrieving the assumptions used in work of the third author regarding theinterior of dynamical black holes and Strong Cosmic Censorship.

The asymptotics of massive fields on stationary spherically symmetric black holes for all angular momenta