Scalar Black-Hole Love Numbers

• Scalar black-hole Love numbers measure the ratio of the decaying tidal response to the growing scalar field source, capturing key deformation characteristics.
• In general relativity, a hidden ladder and SL(2,R) symmetry enforce a vanishing Love number, ensuring the regularity of the horizon-regular solution.
• Nonlinearities, modified gravity, higher dimensions, and quantum corrections can induce nonzero or running Love numbers, potentially affecting gravitational-wave signatures.

A scalar black-hole Love number quantifies the induced multipolar deformation of a black hole (BH) in response to an externally applied, static, long-wavelength scalar field. Scalar Love numbers encode the conservative tidal response of the BH to external static scalar perturbations, and are extracted from the coefficient of the decaying (response) tail in the asymptotic expansion of the field at spatial infinity, relative to the amplitude of the growing tidal source. Their properties are deeply connected to hidden symmetries of the linearized equations governing scalar perturbations in black-hole backgrounds.

1. Definition and Extraction of Scalar Black-Hole Love Numbers

Given a massless scalar field $\phi$ on a Schwarzschild background (horizon radius $r_h = 2GM$), the static, multipole-$\ell$ perturbation is expanded as

$$\phi_{\ell m}(r) = \phi_{\ell m}(x) Y_{\ell m}(\theta, \phi), \qquad x \equiv r/r_h.$$

At large $x$ (spatial infinity), the general solution behaves as

$$\phi_{\ell m}(x) \simeq A_\ell x^\ell + B_\ell x^{-\ell-1} + \ldots$$

The scalar Love number $k_S^{(\ell)}$ is defined as the response-to-source ratio: $k_S^{(\ell)} \equiv \frac{B_\ell}{A_\ell}$ Alternatively, in effective field theory (EFT) language, this is matched to multipolar worldline operators that encode the static tidal response coefficients.

Regularity at the horizon fixes all integration constants such that only solutions nonsingular at $r = r_h$ are allowed. The Love number is determined by extracting the subleading decaying amplitude in the asymptotic expansion of the unique horizon-regular mode (Sharma et al., 2024, Katagiri et al., 2023).

2. Vanishing Love Numbers in General Relativity: Ladder and SL(2,$\mathbb{R}$) Symmetry

In four-dimensional, asymptotically flat, stationary black holes of general relativity (Schwarzschild, Reissner–Nordström, Kerr, and related families including Konoplya–Rezzolla–Zhidenko deformations), all scalar black-hole Love numbers vanish identically: $k_S^{(\ell)} = 0 \qquad \forall \ell$ This result is a consequence of a hidden ladder symmetry of the static scalar perturbation operator. The key technical point is that the radial equation for the scalar perturbation admits algebraic raising and lowering (ladder) operators $D^{\pm}_\ell$, satisfying

$$H_{\ell+1} D^+_\ell = D^+_\ell H_\ell, \qquad H_{\ell-1} D^-_\ell = D^-_\ell H_\ell$$

where $H_\ell$ is the radial "Hamiltonian" associated to the $\ell$-th mode.

The unique solution regular at the horizon is a finite polynomial in $r$, strictly proportional to $r^\ell$ at large $r$: it contains no $r^{-\ell-1}$ tail, enforcing $k_S^{(\ell)} = 0$. Any attempt to include the decaying branch in the solution would violate regularity at the horizon, yielding a logarithmic singularity (Sharma et al., 2024, Hui et al., 2021, Rai et al., 2024).

This structure can be interpreted as the radial equation possessing a representation-theoretic SL(2,$\mathbb{R}$) ("Love symmetry") structure in the near-horizon (near-zone) region (Charalambous, 2024, Charalambous et al., 2023). The highest-weight nature of the horizon-regular solution selects a finite-dimensional SL(2,$\mathbb{R}$) representation, and the absence of an allowed descendant with the required asymptotics suppresses the response tail.

3. Beyond GR: Nonlinearities, Modified Gravity, and Nonzero Scalar Love Numbers

When nonlinear interactions or higher-derivative operators are introduced into the bulk action for the scalar field, the vanishing of the Love numbers generically fails. Consider a bulk scalar theory

$$S_{\rm bulk} = -\frac{1}{2} \int d^4x \sqrt{-g} \left[ (\partial \phi)^2 + \alpha \mathcal{O}(\phi) \right]$$

Possible interactions include power-law potentials, higher-derivative shift-symmetric interactions $(\partial \phi)^4$, or non-linear sigma-model structures.

Onset conditions:

• For $\phi^n$-type potential interactions, nonzero Love numbers emerge at order $\alpha^2$ in perturbation theory, due to log-enhanced terms in the source generated by nonlinearities.
• For derivative interactions such as $(\partial\phi)^4$, nonzero Love numbers appear already at order $\alpha$, with dilogarithms and double logarithmic asymptotic behavior producing the required decaying tail.

In contrast, for a non-linear sigma model (NL$\sigma$M), i.e., a two-derivative scalar theory with a nontrivial target-space metric $G_{IJ}(\phi)$, all scalar Love numbers vanish to all orders. The NL$\sigma$M can always be field-redefined into a free field (at the order relevant for the response calculation), so the underlying ladder symmetry remains unbroken. This mirrors the situation in GR and underlines the underlying symmetry rationale for vanishing Love numbers (Luca et al., 2023).

4. Extensions: Higher Dimensions, Charges, and Quantum Corrections

Higher dimensions:

In $D>4$, the structure of scalar Love numbers is more intricate. The dimensionless index $\hat\ell = \ell/(D-3)$ controls the asymptotics. For general (noninteger) $\hat\ell$, static scalar Love numbers are nonzero, taking values such as

$$k_\ell^{\rm (scalar)} = \frac{\Gamma(-2\hat\ell-1)}{\Gamma(2\hat\ell+1)} \frac{\Gamma(\hat\ell+1)^2}{\Gamma(-\hat\ell)^2} \frac{\sin^2 \pi \hat\ell}{\sin 2\pi \hat\ell}$$

Vanishing ("magic zero") occurs when $\hat\ell \in \mathbb{N}$; otherwise, the deformability is real and finite. For half-integer values of $\hat\ell$, running Love numbers with logarithmic scale dependence emerge, reflecting degenerate indicial exponents and an associated "classical RG flow" in the effective theory (Charalambous, 2024, Xia et al., 12 Nov 2025).

Charges and nonminimal couplings:

For neutral scalars in charged (Reissner–Nordström) or rotating (Kerr, Kerr-Newman) backgrounds in $D=4$, the vanishing result persists (Rai et al., 2024, Ivanov et al., 2022, Ma et al., 2024). However, if the perturbing scalar is itself electrically charged, the degeneracy of growing/decaying modes is lifted due to a shift in the effective multipole index $L$. The consequence is a nonzero, genuinely conservative Love number even for classical, non-extremal black holes. As the scalar's charge $q_e\to0$, the Love number diverges $\sim1/q_e$, indicating a discontinuity in the response of the system and subtle limits in defining the neutral case (Ma et al., 2024, Pereñiguez et al., 15 Sep 2025).

Magnetically-charged black holes (magnetic Reissner–Nordström) similarly acquire nonzero scalar Love numbers when probed by electrically-charged scalars, and the decaying/growing tail distinction becomes meaningful without ambiguities (Pereñiguez et al., 15 Sep 2025).

Quantum gravity modifications:

In loop-quantized black holes (Ashtekar–Olmedo–Singh model), small departures from the classical Schwarzschild solution generate nonzero, negative, mass-dependent scalar Love numbers: $\kappa_\ell^0 \propto -M^{-2/3}$ This arises from the "polymerization" parameter breaking the integer spacing of the characteristic exponents and removing the magic zero structure, with quantum deformability vanishing in the classical large-mass limit (Motaharfar et al., 15 Jan 2025).

5. Running, Renormalization, and Dynamical Scalar Love Numbers

Deviations from the classic two-derivative theory, e.g., by including environmental or effective-field-theory corrections proportional to $(r_H/r)^j$, yield a systematic hierarchy of Love number types:

Correction type	Condition	Scalar Love number behavior
Non-running (constant)	$j \ge 2\ell + 4$	Constant ($k_\ell \neq 0$, $K_\ell=0$)
Running (logarithmic)	$3 \le j \le 2\ell+3$	$k_\ell = 0$, non-zero $K_\ell$ (log scaling)
Vanishing (classical GR, 2-deriv)	$3 \le j \le 2|s|+2$ (classical range)	Zero

Logarithmic corrections reflect the appearance of scale dependence (classical RG flow) in the multipolar coefficients (Katagiri et al., 2023, Xia et al., 12 Nov 2025). For modified gravity or singular core models (e.g., Hayward black holes in scalar–tensor EFTs), model-dependent running and nonzero Love numbers are generically produced (Barura et al., 2024).

For dynamical scalar Love numbers probed by time-dependent fields or considered in the framework of EFT (e.g., "shell" EFT approach), the full frequency-dependent response can be computed to high order in $G$, and exhibits remarkable structure in terms of the Riemann zeta function for the leading transcendental part of the response coefficients (Kosmopoulos et al., 3 Dec 2025).

6. Physical Observables and Gravitational-Wave Signatures

Nonzero scalar Love numbers modify the multipole structure of the black-hole exterior, imprinting additional tidal phasing in gravitational-wave observables from binary coalescences. The scalar Love number of multipole $\ell$ enters the waveform at post-Newtonian order $2\ell+10$, making quadrupolar ($\ell=2$) effects 5PN beyond leading order. The vanishing of these parameters in GR is a sharp prediction: any observational detection of nonzero (or "running") scalar Love numbers would unambiguously reveal physics beyond GR or the influence of quantum/modified gravity effects (Katagiri et al., 2023, Motaharfar et al., 15 Jan 2025).

In summary, scalar black-hole Love numbers are highly nontrivial probes of both the local symmetries of the black-hole perturbation equations and the validity of fundamental gravitational dynamics. Their observed vanishing in GR is enforced by a robust ladder/SL(2,$\mathbb{R}$) symmetry structure, but this tuning is destabilized by nonlinearities, modified-gravity corrections, scalar charges, higher dimensions, or quantum effects, leading to generally nonzero, and sometimes scale-dependent, tidal deformabilities in those scenarios.