Massive Scalar Clouds in the Dadhich-Maartens-Papadopoulos-Rezani Braneworld Spacetime

Massive-scalar quasi-bound states are studied around the static braneworld black hole whose exterior has the Reissner–Nordstr¨om form but contains a gravitational tidal charge rather than an electromagnetic charge. The tidal parameter is allowed to be negative, as favored by the original braneworld construction, as well as positive for comparison. The spectrum is obtained with a Frobenius–Leaver continued fraction and verified by independent integration from the horizon and infinity. Negative tidal charge enlarges the event horizon, reduces the field-mass range supporting an exterior potential well, and accelerates absorption. Positive tidal charge has the opposite effect and produces increasingly long-lived clouds. The real energies remain close to their hydrogen-like values, whereas the decay widths respond by factors that grow strongly with angular momentum. The same ordering persists along the first radial excitations, although the fractional response weakens slightly as the clouds extend farther from the horizon. The Schwarzschild spectrum, the hydrogenic limit, and the leading weak-binding tidal shift are all recovered. These results isolate the effect of a bulk-induced tidal field on gravitational atoms without rotation, electric coupling, or superradiant amplification.