Gravito-magnetic Polarization of Schwarzschild Black Hole

We determine the gravito-magnetic Love numbers of non-rotating black holes in all spacetime dimensions through a novel and direct derivation. The Ishibashi- Kodama master field and its associated field equation are avoided. The matching to the EFT variables is simple. This method allows us to correct the values in the literature. Moreover, we highlight a parity-based selection rule for nonlinear terms that include both electric-type and magnetic-type gravitational field tensors, enabling us to conclude that many of the nonlinear response coefficients in the Schwarzschild black hole effective action vanish.


Introduction
Love numbers characterize the response of a celestial object to an external gravitational field [1], namely a tidal deformation.More generally, they can refer to the response of any object to any external field.Love numbers of black holes are especially interesting, both for intrinsic theoretical reasons being fundamental properties of black holes and relativistic gravity (aka General Relativity or GR), and for observational reasons through their influence on the motion of inspiraling binaries, see e.g.[2][3][4].
The first black hole Love numbers that were determined were those of the Schwarzschild black hole [5][6][7][8].It appears that this determination requires only the knowledge of the Schwarzschild metric [9], which was available for almost a century earlier.However, the nonlinear nature of GR introduces an issue, namely nonlinear contributions appear to mix with the black hole response and thereby mask it. 1This was overcome by dimensional regularization [8].On the way, the Love numbers of static black holes in all dimensions, known as the Tangherlini solutions [10], were determined.Interestingly, the 4d BH Love numbers were found to vanish for all l, the spherical harmonic index.Another possible reason for the time that it took to determine the Schwarzschild Love numbers, is that perhaps the question was not asked.
Following the first determination of BH Love numbers numerous generalizations suggested themselves.Magnetic Love numbers correspond to the response to a gravitomagnetic external field.In 4d, they were evaluated already in [5,6], while in arbitrary spacetime dimension, their value was stated more recently in [11].Additional generalizations include: non-gravitational Love numbers (the first such study might be that of scalar Love numbers in [8]), nonzero frequency Love numbers, Love numbers of more general black holes such as Kerr (see more below) or in higher dimensions, and nonlinear response coefficients [12,13].Magnetic Love numbers are central to this work and the nonlinear ones will also be studied here.
A second wave of progress on BH Love numbers started with the evaluation of Kerr Love numbers [14,15], following earlier work on slowly rotating black holes [16][17][18].The wave progressed to proposals for a symmetry that is responsible for the vanishing of 4d Love numbers [19][20][21][22], sometimes called Love symmetry.
An Effective Field Theory (EFT) perspective is implicit in the very notion of Love numbers, since by definition they allow to replace an object by a point particle equipped with some response coefficients.In field theory, such a procedure where the short distance scale of the object size is eliminated and replaced by effective couplings with the long distance scale of the external field, is known as an Effective Field Theory.
In the case at hand, the EFT is called the effective point particle description of a black hole.In fact, this EFT is an ingredient of another EFT, the Non-Relativistic GR (NRGR) [23][24][25][26] which presents the post-Newtonian approximation in the EFT framework.
In this paper, we examine two questions.The first question is to derive the magnetic Love numbers through the EFT perspective in all spacetime dimensions.[11] studied these numbers by using the Ishibashi-Kodama equation [27] and stated values for them.As we shall see, it would turn out that not only do we get a novel and somewhat economized derivation for these numbers, but in fact, we correct a prefactor in the literature.The corrected value is now accepted by the authors of [11] -see its revised version.The nature of the economization is that both the Ishibashi-Kodama master field and the associated field equation are avoided, being replaced by a more direct, though not gauge-invariant, derivation, and the matching to the EFT variables becomes simpler.
The second question concerns nonlinear response coefficients.We shall point out a parity-based selection rule for nonlinear terms that include both the electric-type and magnetic-type gravitational field tensors.This means that many of the nonlinear response coefficients vanish, thereby allowing future work to concentrate on a considerably smaller set of such coefficients.
This work is organized as follows.In the Sect.2, we review the non-relativistic decomposition of the Einstein metric field, which is instrumental to this work.Sect. 3 determines the magnetic Love numbers, starting with a GR analysis followed by an EFT matching.Sect. 4 describes and derives the nonlinear selection rule.Finally, a discussion and open questions are presented in Sect. 5. Appendix A provides some details regarding the black hole point-particle effective action, while Appendix B is dedicated to the calculation of a class of Feynman diagrams.

NRG decomposition
In this paper, we focus on stationary gravitational fields.Therefore, it is natural to perform a temporal Kaluza-Klein dimensional reduction [8,26] This relation defines a change of variables from g µν to (φ, A I , γ IJ ), where capital Latin indices run over the d − 1 dimensional space, with I, J = 1, 2, . . ., d − 1.The new variables are referred to as "Non-Relativistic Gravity" (NRG) fields.In fact, the parameterization of the metric in terms of NRG fields is general and can also be applied to time-dependent setups [28].However, it is particularly useful in the case of stationary or nearly stationary gravitational systems, see e.g., [29][30][31][32][33][34][35][36]. 2or time-independent fields, the Einstein-Hilbert action takes the form where the spatial metric γ IJ is used to raise and lower the indices, is the Ricci scalar of the metric γ IJ .Note that the integral in the definition of S NRG is (d − 1)-dimensional because time factors out and can be stripped off.In particular, time-independent geometries satisfy the following equations of motion where ∇ I denotes the covariant derivative compatible with γ IJ , and the stress-energy tensors are (2.4) For static geometries, A I vanishes.An example of such a geometry is provided by the Schwarzschild black hole metric in a general number of space-time dimensions, commonly known as the Tangherlini solution [10], where r s is the Schwarzschild radius.We denote the NRG fields that describe the Schwarzschild black hole by φ S , A S I and γ S IJ .Using (2.1), one can read off their explicit expressions Here and in the following sections, lower-case Latin indices run over the (d − 2)dimensional sphere, and Ω ij represents the metric on a unit sphere.In the next section, we employ the NRG decomposition to explore gravito-magnetic perturbations of the Schwarzschild black hole.

Gravito-magnetic perturbations of black hole
In this section, we delve into the linear response of the Tangherlini solution (2.5) to stationary gravito-magnetic perturbations.We determine the corresponding Love numbers without relying on the Regge-Wheeler-Zerilli equation [38][39][40] and its generalization [27,41].The master field governed by the Regge-Wheeler equation is replaced with the NRG vector potential A I .We argue that at leading order, A I decouples from other fields in the action, and its radial profile satisfies a hypergeometric equation.

Full General Relativity analysis
It is convenient to rewrite the equation of motion for A I in (2.3) as follows Imposing the following gauge condition for the perturbation the radial equation (3.2) then takes the form Here, F (Ω) is an arbitrary function on a sphere, representing the residual gauge, subject to the condition S d−2 F (Ω) = 0, and ∇ S d−2 denotes the covariant derivative on a unit sphere.We choose F (Ω) = 0. Hence, our gauge conditions read In this gauge, the equations of motion along the sphere, (3.3), are given by Alternatively, employing the commutator of covariant derivatives in the second term, A k = (d − 3)A j , and using the gauge condition (3.6) leads to where ∆ S d−2 represents the covariant Laplacian of a vector field on a unit sphere.Next, we introduce a dimensionless radial coordinate and separate variables where Y lm i (Ω) is a divergence-free vector spherical harmonic on S d−2 (to satisfy the equation on the right of (3.6)), which is an eigenvector of the vector Laplacian on a unit sphere4 The equations of motion for A j reduce to a single ODE for a radial profile, where l = l/(d − 3), and we used The regular solution at the horizon is given by Expanding it around X = 0 leads to where ellipses encode terms that include either integer or higher-order fractional powers of X, and B(x, y) is the beta function,

Black hole effective action
Consider a black hole of size r s moving in a background with a typical length scale L ≫ r s .Given the hierarchy of scales, it is natural to employ the point particle approximation.However, this approximation has limitations, notably in its disregard for most of the physical properties of the object, particularly the finite-size effects.To overcome these limitations, one adopts an effective field theory approach [23].Within this framework, the full GR action gives way to an effective action for the black hole, wherein the finite-size effects and internal degrees of freedom are encapsulated by an infinite tower of generally covariant terms supported at the worldline of a point-like black hole, where dτ is the proper time of the point-like black hole moving along the worldline trajectory x(τ ), and are the gravito-electric and gravito-magnetic components of the Riemann tensor. 5he coefficients C E and C B correspond to the electric-type and magnetic-type quadrupole susceptibilities of the black hole, respectively.They represent the leading finite-size effects of a spinless black hole.Notably, the gravito-electric and gravitomagnetic components of the Riemann tensor are characterized by the angular momentum with l = 2, whereas tensors with higher angular momentum can be constructed by applying covariant derivatives to E µν and Multiplying and contracting these tensors in diverse ways results in higher order finite-size effects, which represent interactions involving multipoles with different values of l.
The numerical values of C E and C B are determined by matching the physical observables calculated within the EFT approach with their counterparts evaluated using the full GR theory.
In Appendix A we show that if the gravitational fields are weak, then the black hole effective action simplifies to 6 where ellipsis encodes terms that are cubic in the weak gravitational fields, and the coefficients C E l and C B l are referred to as Love numbers for the electric-type and magnetictype multipole susceptibilities of the black hole.
For completeness, we include here the values of the electric-type Love numbers, C E l as computed in [4,8,11], where Ω d represents the area of a unit sphere in d-dimensional space, The full EFT action consists of S p.p. and an additional term, denoted as S bulk , which governs the dynamics of the long-wavelength modes of the metric.Due to general covariance, S bulk takes the same form as (2.2), except the fields far away from the black hole are weak.Hence, we can expand around a flat metric, where σ IJ = γ IJ − δ IJ ≪ 1, σ = δ IJ σ IJ , and ellipsis encode self-interactions of the NRG fields. 6The relation between C E , C B in (3.16) and C E l , C B l for l = 2 is given by By construction, the EFT action is gauge invariant, so it needs to be supplemented with the gauge-fixing term S GF .We choose the harmonic gauge,7 The full EFT action is given by In the next subsection, we introduce a weak gravito-magnetic perturbation to the Schwarzschild black hole and use S EFT to perturbatively evaluate the asymptotic value of A I .Matching it to (3.14) allows us to determine C B l in a general number of dimensions.

Matching gravito-magnetic Love
Consider a background metric of the form where symmetric in all the last l indices, totally traceless, and obeys the condition This background not only solves the linearized Einstein equations (2.3) but also satisfies the gauge condition (3.6),8 The total angular momentum carried by A i is characterized by l.Furthermore, by construction C IJ 1 •••J l belongs to an irreducible representation of the permutation group. Assuming we can treat this background as a small perturbation of the Schwarzschild black hole located at the origin.To systematically evaluate perturbative gravito-magnetic corrections to the black hole's metric in the region far away from the event horizon, one needs to redefine the vector potential A i → A I + A I in the effective action S EFT and subsequently calculate Feynman diagrams with one external propagator of A I and an appropriate number of A I insertions.
In particular, the linear response of the Schwarzschild black hole induced by an external perturbation A I is associated with a class of Feynman diagrams featuring a single insertion of A I .This class can be split into two families.The first family is shown in Fig. 1(a Thus, we deduce that the diagrams belonging to the first family recover integer powers of X = rs r d−3 within parentheses in (3.14).In contrast, fractional powers of X in (3.14) indicate the presence of the gravito-magnetic finite-size effect, they correspond to the diagrams in Fig. 1(b).
To determine the value of C B l , it is sufficient to evaluate the diagram shown in Fig. 2 and match it to the leading term of the fractional power series in (3.14).We have, where The propagator of the gravito-magnetic vector potential is essentially the inverse of the quadratic in A I term in the effective action S EFT , Using this propagator, yields (3.30) Hence, the EFT expression for A I is given by where ellipses represent terms involving either integer or higher-order fractional powers of rs This determination of the gravito-magnetic Love numbers of Schwarzschild-Tangherlini black holes in all dimensions is one of the main results of this paper.Comparing with version 3 of [11], one finds that most of the expression matches, including the positions of all poles and zeros.However, there is a remaining discrepancy, a fraction which depends on d and l, and in particular it changes the residues.Following correspondence regarding (3.32), [11] was revised (version 4) to agree with it.
The two real parameters, l± , defined in (3.13), are strictly positive in the region l ≥ 2, d ≥ 4. Consequently, the beta function B( l+ , l− ) in (3.32) is smooth and positive in this range.In contrast, B(− l+ , − l− ) exhibits simple poles when one or both of l± equal an integer and vanishes for non-integer l± if their sum is an integer.This leads to three types of C B l : finite, divergent, or vanishing.The specific type of magnetic Love number is entirely determined by the values of l± .If both parameters, along with their sum, are non-integers, the Love number in question is finite.If both are non-integers, while their sum is a positive integer, then the Love number diverges.Finally, if at least one of the two parameters is a positive integer, the Love number vanishes.
For instance, in d = 6, Love numbers satisfying l mod 3 = 0 diverge, whereas the remaining Love numbers vanish.In d = 5, Love numbers with odd l vanish, and those with even l diverge.Notably, all magnetic Love numbers vanish in d = 4.

Divergent Love numbers
As pointed out in the previous subsection, the magnetic Love numbers (3.32) diverge if l± are non-integers, while their sum is a positive integer.In contrast, physical observables are finite.Therefore, divergent C B l requires interpretation along with careful treatment to isolate a finite number that characterizes the physical polarization of the black hole.In this subsection, we argue that these divergences are classical analogs of the counterterms ubiquitous in the context of quantum field theory.In particular, to remedy this pathology, one follows the standard guidelines of regularization and renormalization.We illustrate that the physical magnetic Love numbers are finite and exhibit classical Renormalization Group (RG) flow when C B l in (3.32) is singular.For non-integer l± such that l+ + l− = n, where n is a positive integer, the two power series in (3.14) merge, resulting in a regular power series augmented by a logarithmic term.The expansion (3.14) is replaced with where H n represents the n-th harmonic number.We display only the asymptotic behaviour and the leading logarithmic fall off, with the remaining terms represented by the ellipses.
To recover (3.33) using the EFT calculation, we must sum the diagram in Fig. 2 with those Feynman graphs in Fig. 1(a) that have n+ 1 mass vertices on the worldline. 10The divergence associated with the singularity in the Love number is canceled by a similar divergence in the diagrams from Fig. 1(a), resulting in a finite expression that matches (3.33).
To regulate the divergence, we analytically continue the calculation to d + ǫ dimensions, where ǫ ≪ 1 is a small dimensional regulator.The structure of diagrams in d + ǫ dimensions that feature n + 1 mass vertices on the worldline is given by, Fig. 1(a where the ellipses represent the contribution of diagrams with powers of mass either higher or lower than n + 1. Notably, the coordinate dependence of the Feynman graphs is entirely determined by dimensional analysis, while the residue of the pole in 1/ǫ is the outcome of the calculation.Here, we fix it based on the requirement that the poles in 1/ǫ of Fig. 1(a) and Fig. 2 cancel, without the need for explicit calculation.Towards the end of this subsection, we evaluate this residue in a specific example, demonstrating its agreement with the anticipated pattern of the form 11 In the ǫ → 0 limit, m R , up to a numerical factor, represents the mass of the black hole in a d-dimensional spacetime. 12 In field theory terminology, C B l in (3.32) is referred to as a bare coupling, and its divergence serves as a counterterm essential for canceling the above 1/ǫ pole to 10 In the special cases that we study in this subsection, the mass dimension of C B l satisfies therefore, the diagrams in Fig. 2 and Fig. 1(a) mix. 11In principle, one could add to this expression a finite term proportional to m n+1 R .However, it can be eliminated by properly rescaling L. 12 The relation between the mass of a black hole and its Schwarzschild radius in a general number of dimensions is given by render the gravito-magnetic amplitude, A I , finite.Specifically, the bare C B l is usually decomposed as follows where L is an arbitrary length scale in the EFT, C R l (L) denotes a finite renormalized coupling representing the physical polarization of the black hole, and the counterterm C C.T. l stands for the leading-order divergent term in the expansion of (3.32) around ǫ = 0. Substituting C B l into (3.31),adding (3.35) to it, and taking the ǫ → 0 limit yields a finite expression The gravito-magnetic vector potential, A EFT i , is independent of an arbitrary scale L. Therefore, the physical Love number, C R l , exhibits a classical RG flow, Solving the RG flow equation fixes C R l up to an additive constant.This constant is defined by matching A EFT i to the full solution (3.33).Identifying L with r, yields To further illustrate the preceding discussion and provide evidence for our claims, let us examine the scenario where d = 7 and l = 2, corresponding to l+ = 3  4 and l− = 1 4 , thereby leading to n = 1, indicating a divergence in the Love number.This case is the simplest, as it is characterized by the lowest possible values of angular momentum and the number of mass insertions on the worldline, n + 1 = 2.The relevant Feynman diagrams featuring two mass insertions on the worldline are depicted in Fig. 3.We present the details of calculating these diagrams in Appendix B. Here, we provide the final answer in general d: .
As expected, these diagrams reveal a simple pole at d = 7.The total residue matches (3.35), and is given by

Non-linear polarization
Our previous discussion focused on a subset of terms within S p.p. -specifically those quadratic in B α 1 •••α d−2 and its covariant derivatives.These terms govern the linear response of a black hole to an external gravito-magnetic perturbation.While the linear response is dominant in the presence of a weakly curved background, it provides an incomplete description of the black hole's internal structure due to the inherently nonlinear nature of General Relativity.In addition, our analysis confirmed that C B l = 0 for the Schwarzschild black hole in d = 4 [5,6,11].A similar linear behavior is exhibited by the four-dimensional Kerr black hole [14,15,17,18,[43][44][45]. Consequently, within the framework of Einstein's theory of gravity, upcoming observational data -capturing physics beyond the pointparticle approximation -is anticipated to be shaped by non-linear effects.
To gain a comprehensive understanding of these effects beyond the linear response approximation, it is imperative to explore higher order finite-size coefficients.In this subsection, we take a step in this direction by partially unraveling the higher curvature structure of S p.p. .Recent studies on this subject can be found in [12,13].
To determine the finite-size coefficients governing non-linear effects associated with cubic and higher-order curvature terms in S p.p. , one must resort to a higher-order perturbation theory.However, as we argue below, the NRG decomposition in (2.1) allows us to isolate a symmetry that protects certain coefficients of this type; that is, they vanish in a general number of space-time dimensions, including d = 4.
To this end, we note that for stationary geometries, the NRG action in (2.2) exhibits manifest invariance under A I → −A I .This symmetry is a remnant of the invariance of the full Einstein-Hilbert action under time reversal, T , accompanied by an appropriate transformation of the metric.In terms of NRG fields in (2.1), it boils down to This orientation-reversing diffeomorphism results in a simple transformation of the spatial part of the gravito-electric and gravito-magnetic components of the Riemann tensor: E IJ remains unchanged under time reversal (T -even), while By construction, the effective action respects all the symmetries of Einstein's theory of gravity.Consequently, one might erroneously conclude that worldline terms containing an odd number of B I 1 •••I d−2 's are not allowed in S p.p. due to their T -odd nature.This is not universally true, as the coefficients multiplying curvature invariants in S p.p. are not necessarily T -even.These coefficients encode the internal degrees of freedom of the black hole, and their transformation may compensate for the emergent minus sign.A notable example of this kind is the coupling of a black hole's spin to a weakly curved background [29,[46][47][48] where the antisymmetric tensor S IJ represents the spin variables of a rotating black hole.Both S IJ and F IJ change sign under time reversal, ensuring the invariance of δS p.p. under T .That being said, the Schwarzschild background is static with A S I = 0, rendering the metric invariant under t → −t.Therefore, the finite-size coefficients of the effective action, which encode the internal structure of the Schwarzschild black hole, must be T -even.Thus, the upshot of our discussion is:  This conclusion becomes particularly transparent when analyzing perturbations around the Schwarzschild black hole using NRG decomposition (2.1).Let us expand the NRG fields as follows φ = φ S + φ (1) + φ (2) , where the superscript n denotes the order of the perturbation in the weak background A I .Notably, the equations of motion for φ and γ IJ in (2.3) are quadratic in A I .Thus, A S I = 0 implies that the gravito-magnetic perturbation A I leads to φ (1) = σ (1) IJ = 0. 13 Consequently, the non-trivial corrections to the Schwarzschild fields φ S and γ S IJ begin at the second order in A I , resulting in A (2) I = 0, as the equation defining A I in (2.3) is already linear in the perturbation. 14Extending this discussion to higher orders reveals that the perturbation A I gives rise to non-trivial A I = 0 is equivalent to the absence of terms that are cubic in the gravito-magnetic component of the Riemann tensor.In d = 4, this includes the coefficient of B µν B να B µ α , which vanishes.In terms of NRG fields (2.1), the absence of these terms in the black hole's effective action has a straightforward diagrammatic explanation.As derived from (2.2), all bulk vertices of the effective action are either independent of A I or quadratic in A I .Consequently, any A I -line originating from the external leg (either A I or A I ) continues seamlessly without branching to another external leg or terminates at the worldline vertex representing a finite-size effect.In particular, diagrams of the form shown in Fig. 4 (a) are nonexistent, because the bulk of these diagrams has three external A I -legs.The only non-trivial diagrams that contribute to A Similarly, the vanishing of φ (3) forbids mixing of 3 gravito-magnetic components B α 1 •••α d−2 with a single E µν . 15

Discussion
In this paper, we computed the gravito-magnetic Love numbers (3.32) for any spacetime dimension.To facilitate the derivation, we decomposed the metric in terms of 13 This no longer holds in the case of the Kerr black hole. 14Alternatively, the equation for A I , indicating a trivial redefinition of the linear order amplitude. 15Vanishing of φ (1) excludes a term of the form B µν E µν in d = 4, which is a quite trivial result, because linear order perturbation theory does not mix odd and even sectors.
NRG fields (2.1) and calculated the Love numbers using the equation of motion for the vector field A I .At linear order A I decouples from other perturbations thus providing a direct and concise way of determining the Love numbers, avoiding the use of the Ishibashi-Kodama master field.Notably, the revised version of [11] agrees with the final result.
The second thrust of our paper revolves around nonlinear response coefficients.We have identified a parity-based selection rule for nonlinear terms encompassing both electric-type and magnetic-type gravitational field tensors.This implies the vanishing of numerous nonlinear Love numbers, streamlining future research efforts to concentrate on a considerably smaller set of response coefficients.
Moreover, it follows from (2.3) and the analysis in Section 4 that considering perturbations beyond the linear response theory results in the mixing between the gravitomagnetic and gravito-electric sectors.Therefore, to capture the nonlinear effects of the polarization of the Schwarzschild black hole, it is necessary to incorporate mixed-type higher-order curvature terms in the effective action (3.16), involving both the gravitoelectric and gravito-magnetic components of the Riemann tensor.For instance, the full list of mixed-type terms relevant for the second-order effects associated with perturbations characterized by l = 2 in d = 4 consits of two terms As explained in Section 4, the coefficient C EEB vanishes since this term does not respect parity in (4.1).Consequently, only the coefficient C BBE needs to be determined to capture second-order effects induced by perturbations with l = 2. 16 In our future work, we aim to calculate this coefficient [49].
To derive the worldline term in the second line of (3.19), one needs to apply l − 2 derivatives to the gravito-magnetic component and square it.This involves contracting the 2(d + l − 4) free indices in all possible inequivalent ways.Due to the antisymmetry of the Levi-Civita symbol, only two candidates exist for non-trivial (and inequivalent) way of contracting the indices: 1.All free spatial indices of the Levi-Civita symbol in one gravito-magnetic component are contracted with the free indices of the Levi-Civita symbol in the other gravitomagnetic component.The remaining l−1 free indices of derivatives in both components are contracted among themselves.
2. One index of the Levi-Civita symbol is contracted with one of the derivatives present in the other gravito-magnetic component.However, using the following identities, it is straightforward to demonstrate that both alternatives of contracting the indices lead to the same term displayed in the second line of (3.19).17 B Evaluation of diagrams in Fig. 3 In this appendix, we provide a detailed explanation of how to evaluate the diagrams in Fig. 3.We begin by supplementing the effective action (3.22) with interaction terms that are cubic and quartic in the weak NRG fields.Expanding the NRG action (2.2) results in where the ellipsis encode interaction terms that are irrelevant for our purpose.The above terms give rise to the following Feynman rules in position space:18  The other two diagrams in Fig. 3 can be evaluated by following similar steps.Instead of presenting explicit computations, we list three additional master integrals essential for evaluating the two remaining diagrams.Their derivation is straightforward, relying entirely on the Feynman parameterization trick.In all the expressions below, we continue using a shorthand notation where x 12 = x 1 − x 2 .

Figure 1 .
Figure 1.A class of Feynman diagrams illustrating the linear response of the Schwarzschild black hole to a weak gravito-magnetic perturbation, A I (denoted by a cross in the diagrams).Solid and dashed lines represent the propagators of φ and A I , respectively, while the black hole's worldline is indicated by a double line.The mass vertex is denoted by a solid dot on the worldline, and the finite-size effect associated with C B l is represented by a solid box.Panel (a) displays a family of diagrams with only mass vertices on the worldline, and panel (b) shows a family of Feynman diagrams with a single insertion of the magnetic-type Love number C B l .
).The diagrams within this family carry only mass vertices on the worldline, and by dimensional analysis, they are proportional to mr d−3 n A I ∼ rs r n(d−3) A I , wheren is an integer.The second family is depicted in Fig.1(b), containing, in addition to the mass terms on the worldline, a single insertion of the finite-size effect associated with C B l .Consequently, the diagrams of this family are proportional to

Figure 2 .
Figure 2. A Feynman diagram illustrating the black hole's response to a gravito-magnetic perturbation, A I (represented by a cross).The black hole's worldline is denoted by a double line, while C B l is symbolized by a solid box.The dashed line represents the propagator of A I .

Figure 3 .
Figure 3. Feynman diagrams that mix with the graph in Fig.2in the case of d = 7 and l = 2.The gravito-magnetic perturbation, A I , is denoted by a cross.Solid, dashed and wiggly lines represent the propagators of φ, A I and σ IJ , respectively.The black hole's worldline is indicated by a double line, while the mass vertex is denoted by a solid dot on the worldline.

Figure 4 .
Figure 4.A class of Feynman diagrams contributing to A (2) I .A solid box represents the non-linear finite-size effect cubic in the gravito-magnetic component of the Riemann tensor, while the remaining Feynman rules are the same as in Fig.1.Panel (a) displays a family of diagrams with only mass vertices on the worldline, and panel (b) shows a family of Feynman diagrams with a single insertion of the non-linear response coefficient.
φ (2n−1) = 0 , σ (2n−1) IJ = 0 , for n ∈ N .(4.5)These constraints on the perturbative expansion (4.4) have significant implications for the structure of S p.p. : consistent with our earlier conclusion, terms with an odd number of B I 1 •••I d−2 are protected, meaning that the corresponding finite-size coefficients vanish.For instance, the condition A

( 2 )
I necessarily include a worldline vertex representing the finite-size effect, as illustrated in Fig.4 (b).Matching the leading-order diagram on the right-hand side of Fig.4 (b) with A (2) I = 0 leads to the vanishing of the finite-size coefficient associated with the term cubic in the gravito-magnetic component of the Riemann tensor.