Graviphoton mixing with KK modes in the 2D effective theory of charged black holes

We study the effects of massive Kaluza–Klein modes on the two-dimensional low-energy effective theory obtained by the $S^2$ reduction of four-dimensional Einstein–Maxwell theory on a Reissner–Nordström black hole background. We retain the first excited ($l=1$) Maxwell Kaluza–Klein modes in addition to the massless sector. The background electric flux mixes the Maxwell Kaluza–Klein scalars with the graviphotons, and tree-level exchange of the massive scalars generates a correction to the graviphoton kinetic term proportional to the square of the black hole charge. Within the leading local approximation, we also perturbatively evaluate the one-loop correction to the graviphoton kinetic term and find no dependence on the black hole charge. We find that near extremality, the charge-dependent tree-level correction becomes parametrically comparable to the classical graviphoton kinetic term.