Nonlinear tails of massive scalar fields around a black hole

Nonlinear effects play a fundamental role in the late-time ringdown of black holes, with direct implications for gravitational-wave observations. For massive fields, these dynamics become richer, yet their nonlinear signatures remain poorly understood. Here, we systematically study nonlinear tails of massive scalar perturbations, from a toy model with ingoing and outgoing sources to a self-interacting scalar model, revealing nonlinear tails and contrasting the results with their linear counterparts. We find that the nonlinear tails of massive scalar fields, opposite to massless ones, decay as the same rate as linear tails in the intermediate time, independent of source parameters or initial conditions. Nevertheless, quadratic quasinormal modes could serve as a probe to the nonlinear effects of massive fields.