Charging the Love numbers: Charged scalar response coefficients of Kerr-Newman black holes

We study the electrically-charged massless scalar response to the general Kerr-Newman black hole and obtain the exact (complex) response coefficients, whose real parts are the Love numbers. We find there is a curious discontinuity between the neutral and charged scalar. The Love numbers vanish for the neutral scalar, but are inversely proportional to the electric charge of the scalar in the small charge limit. We study the near-zone of the dynamical equation with non-vanishing frequency ω, and show that the system have an SL(2,R) Love-like symmetry. The symmetry selects a special frequency ωcr, which vanishes in the neutral case, so that the real part of the response coefficient vanishes identically. We also generalize the results into higher dimensions to some extend.