Describe false-vacuum transitions near larger black holes

Describe the false-vacuum decay transitions near Schwarzschild black holes with radii at or above the termination radius of the computed aspherical critical-bubble branch, where the paper does not identify the relevant transition configuration or mechanism.

Background

The paper studies thermal false-vacuum decay of a Higgs-like scalar field near a Schwarzschild black hole in equilibrium with a Hawking-temperature environment. For black-hole radii slightly above the threshold at which spherical critical bubbles acquire additional negative modes, the authors numerically find aspherical critical bubbles and follow their dominant branch as the black-hole radius increases.

The aspherical branch terminates at a radius denoted by r_s^{(c)}. The authors report that their numerical method fails for r_s ≥ r_s^{(c)}, while the aspherical solutions exist only below this value. Although later sections identify an infinitesimally thin Fubini–Lipatov bounce near the horizon as the dominant configuration at sufficiently large radius, the quoted passage explicitly records the unresolved problem of describing transitions in the regime beyond the computed aspherical-bubble branch.