Running Love Numbers of Charged Black Holes

Loops of virtual particles from the vacuum of quantum field theory (QFT) render black holes tidally deformable. We compute the static tidal response of unspinning charged black holes at arbitrary radius, using the perturbative formalism developed in 2501.18684. Since the gravitational and electromagnetic tidal responses mix, we generalize the notion of Love numbers to Love matrices. We derive the coupled equations of motion for the metric and electromagnetic fluctuations around purely electric and magnetic backgrounds. For large charged black holes, which are described by the Effective Field Theory (EFT) of gravity, we compute the full set of Love matrices induced by an arbitrary tower of $F^{2n}$ operators. We find that, although quantum corrections break electromagnetic duality, the Love matrices in electric and magnetic backgrounds are related by a $Z_2$ symmetry under electric-magnetic exchange. Going beyond EFT, we compute the Love matrices of small magnetic black holes. We show that the running of the Love matrices is governed by the running of the $U(1)$ gauge coupling, and we derive the correspondence between Love and $U(1)$ beta functions for arbitrary harmonics. The overall picture that emerges is that the QFT-induced tidal response of magnetic black holes saturates in the strong-field regime. These results imply that nearly-extremal magnetic black holes charged under an Abelian dark sector could be probed by gravitational-wave observations.

💡 Research Summary

The paper investigates how quantum vacuum fluctuations—virtual particle loops in quantum field theory (QFT)—induce a static tidal response in charged, non‑spinning black holes. In classical general relativity, neutral or charged black holes are “rigid”: their Love numbers, which quantify the linear response to external tidal fields, vanish. However, when the vacuum is populated by QFT loops, the black hole experiences an effective non‑empty environment, and non‑zero tidal deformabilities appear.

Because electric and gravitational perturbations couple in a charged background, the authors generalize the scalar Love numbers to Love matrices. For a set of N independent external fields (e.g., metric perturbations, electromagnetic potentials), the response is encoded in an N × N symmetric matrix Kℓ for each spherical harmonic ℓ. The matrix elements describe pure‑gravitational, pure‑electromagnetic, and mixed gravito‑electromagnetic responses.

The analysis proceeds in two regimes:

1. Large black holes (EFT regime).
   When the horizon radius r_h is much larger than the Compton wavelength of the virtual particles, the quantum effects can be captured by a long‑distance effective field theory (EFT) of gravity coupled to a U(1) gauge field. The EFT contains an arbitrary tower of higher‑dimensional operators of the form F^{2n} (pure electromagnetic) and mixed curvature–field operators such as R F F. Treating these operators perturbatively, the authors solve the coupled linearized Einstein–Maxwell equations on the Reissner‑Nordström background, imposing regularity at the horizon and matching to a prescribed tidal field at infinity. They obtain closed‑form expressions for the full Love matrices Kℓ in terms of the EFT coefficients. A striking result is a Z₂ symmetry: the Love matrix in a purely electric background is related to that in a purely magnetic background by an electric‑magnetic exchange, despite quantum corrections breaking electromagnetic duality at the level of the action.

2. Small magnetic black holes (strong‑field, non‑EFT regime).
   For sufficiently small black holes, the EFT expansion breaks down and the dynamics are dominated by the gauge sector. Electric black holes quickly discharge via Schwinger pair production, making a static response ill‑defined. Magnetic black holes, however, can only evaporate into magnetic monopoles of Planckian mass, so they remain stable enough for a static analysis. In this regime the authors incorporate the full Euler‑Heisenberg‑type effective action for the gauge field and retain the quantum loop contributions. They demonstrate that the running of the Love matrices with the renormalization scale μ is governed entirely by the running of the U(1) gauge coupling e(μ). Explicitly, for each harmonic ℓ they derive a relation of the form

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