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Extremal Love: tidal/electromagnetic deformability, logarithmic running and the weak gravity conjecture

Toshifumi Noumi, Sam S. C. Wong

TL;DR

This work shows that extremal charged black holes in Einstein-Maxwell EFT acquire nonzero tidal responses once four-derivative corrections are included, with the vector ($\ell=1$) and parity-odd tensor ($\ell\ge 2$) sectors exhibiting logarithmic running in the Love numbers. The authors derive explicit expressions for electric and magnetic susceptibilities in the $\ell=1$ sector and demonstrate that unitarity and the Weak Gravity Conjecture constrain the signs and RG running of these quantities, linking microscopic UV consistency to macroscopic deformability. They uncover gravito-electromagnetic mixing in the $\ell\ge 2$ sector, producing symmetric cross-log corrections that are naturally captured by a worldline EFT with cross-coupling operators. The results imply that Love numbers are scale-dependent EFT parameters and highlight subtleties in bulk-to-worldline matching, motivating extensions to non-extremal or rotating BHs and potential implications for gravitational-wave phenomenology.

Abstract

In General Relativity, the tidal Love numbers of black holes vanish, implying they are resistant to tidal deformation. This "rigidity" is easily broken in the presence of higher-derivative corrections. Focusing on extremal charged black holes in Einstein-Maxwell EFT, we compute the static linear response for both the vector ($\ell=1$) and parity-odd tensor ($\ell \ge 2$) sectors. We find that the resulting tidal Love numbers are non-zero and exhibit logarithmic running, a hallmark of quantum corrections. Crucially, we show that the sign of these deformations is not arbitrary; the induced electric and magnetic susceptibilities and their log runnings in the $\ell=1$ sector are constrained by unitarity and the Weak Gravity Conjecture. Furthermore, due to gravito-electromagnetic mixing, we find the cross log runnings and show that they are the same, which we explain through the worldline effective field theory.

Extremal Love: tidal/electromagnetic deformability, logarithmic running and the weak gravity conjecture

TL;DR

This work shows that extremal charged black holes in Einstein-Maxwell EFT acquire nonzero tidal responses once four-derivative corrections are included, with the vector () and parity-odd tensor () sectors exhibiting logarithmic running in the Love numbers. The authors derive explicit expressions for electric and magnetic susceptibilities in the sector and demonstrate that unitarity and the Weak Gravity Conjecture constrain the signs and RG running of these quantities, linking microscopic UV consistency to macroscopic deformability. They uncover gravito-electromagnetic mixing in the sector, producing symmetric cross-log corrections that are naturally captured by a worldline EFT with cross-coupling operators. The results imply that Love numbers are scale-dependent EFT parameters and highlight subtleties in bulk-to-worldline matching, motivating extensions to non-extremal or rotating BHs and potential implications for gravitational-wave phenomenology.

Abstract

In General Relativity, the tidal Love numbers of black holes vanish, implying they are resistant to tidal deformation. This "rigidity" is easily broken in the presence of higher-derivative corrections. Focusing on extremal charged black holes in Einstein-Maxwell EFT, we compute the static linear response for both the vector () and parity-odd tensor () sectors. We find that the resulting tidal Love numbers are non-zero and exhibit logarithmic running, a hallmark of quantum corrections. Crucially, we show that the sign of these deformations is not arbitrary; the induced electric and magnetic susceptibilities and their log runnings in the sector are constrained by unitarity and the Weak Gravity Conjecture. Furthermore, due to gravito-electromagnetic mixing, we find the cross log runnings and show that they are the same, which we explain through the worldline effective field theory.
Paper Structure (15 sections, 115 equations, 2 figures)

This paper contains 15 sections, 115 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Einstein-Maxwell EFT up to four derivative operators
  3. Vector Love numbers and susceptibility
  4. Point particle effective action
  5. Causality constraints
  6. Vector ($\ell=1$) Love numbers of extremal black holes
  7. Parity even sector, electric susceptibility of extremal black hole
  8. Parity odd sector, magnetic susceptibility
  9. Magnetic and parity odd tensor tidal responses for $\ell \ge 2$
  10. $\ell =2$ and $\ell =3$ responses
  11. Worldline EFT operators with gravito-EM mixing
  12. Discussion
  13. Electromagnetic polarizability in the background of extremal black holes (ignoring gravitational perturbations)
  14. Parity even sector, electric susceptibility of extremal black hole
  15. Parity odd sector, magnetic susceptibility

Figures (2)

  • Figure 1: Positive electric polarizability is simply the intuition that positive charges are being pushed away by electric field, as depicted by the figure, such that any medium screen the electric field inside.
  • Figure 2: Blue region: the constraint from $72 \alpha_1+45 \alpha_3 \ge 0$. Brown region: $4\alpha_1 -|\alpha_3| \ge 0$.