Analytic backreaction of a scalar wig on a Schwarzschild black hole

We analytically determine the leading backreaction of a spherically symmetric massive complex scalar quasi-bound state (with mass $\mu$) on a Schwarzschild black hole with (initial) gravitational radius $r_0$. Working in the small-coupling regime, $r_0 \mu \ll 1$, we evaluate the stress-energy tensor of the fundamental scalar $s$-wave and solve the Einstein equations through quadratic order in its amplitude in ingoing Eddington-Finkelstein coordinates. We also determine the small-mass quasi-resonant frequency of the fundamental $s$-wave analytically by matched asymptotic expansions and validate it numerically using Leaver's method. Unlike steady-state treatments, the calculation retains the exponential decay of the quasi-bound state. We obtain explicit expressions for the metric perturbations and Misner-Sharp mass and derive the evolution of the future outer trapping horizon. The black-hole mass grows monotonically with the decaying horizon flux and saturates when the finite scalar cloud has been absorbed, with the decrease of the cloud mass exactly balancing the horizon growth at the perturbative order considered. We also determine the domain in which the scalar small-coupling approximation and the gravitational perturbative expansion are simultaneously valid.