Erratum to: Revisiting Kerr–Newman black hole’s charged scalar cloud: flux balance (The European Physical Journal C, (2025), 85, 12, (1383), 10.1140/epjc/s10052-025-15128-3)

Abstract

In the original manuscript, the discussion of charged scalar cloud formation did not explicitly consider the case of small positive charge configurations. We would like to clarify this point, which slightly modifies the conclusion regarding allowed charge configurations. In Eq. (83) of the original manuscript, for positive charges qrQ>0, the physical bound-state branch requires, (Formula presented.) (Formula presented.) Multiplying both sides by 2, isolating the square root and squaring both sides, we obtain, (Formula presented.) which leads to the following previously missing inequality, (Formula presented.) Therefore, small positive charge configurations satisfying (4) can, in principle, support a scalar cloud. For larger positive charges, no bound-state solution exists. This addendum slightly modifies the conclusion of the original manuscript by identifying a previously overlooked range of small positive charges that can support scalar clouds. Apart from this refinement, all other results and analyses presented in the original work remain valid.