Vanishing of Nonlinear Tidal Love Numbers of Schwarzschild Black Holes

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I. INTRODUCTION
The tidal deformability of a compact object refers to its propensity to respond when acted upon by an external longwavelength gravitational field.It is in general characterized in terms of complex coefficients that capture the conservative and dissipative parts of the response.The coefficients associated to the conservative response of the object are usually referred to as Love numbers-conceptually, one can think of the Love numbers as the analogues of the electric polarizability of a material in electromagnetism.The tidal response coefficients are important because they offer insights into the gravitational behavior and the body's internal structure.In the case of a neutron star, the tidal deformability is tightly related to the physics inside the object and its equation of state [1].In the case of black holes, the tidal response coefficients depend on the physics at the horizon, and can be used to access and test the fundamental properties of gravity in the strong-field regime, including the existence of symmetries of the black hole perturbations [2][3][4][5][6][7][8].
In a binary system of compact objects, the way one body responds to the gravitational perturbation of its companion becomes more relevant in the last stages of the inspiral, influencing the waveform of the emitted gravitational waves.The tidal coefficients can be measured or constrained with gravitational-wave data.They can be used to detect binary neutron star systems [9,10] and have been the subject of recent searches in the LIGO-Virgo data [11].Future observations will achieve much better accuracy and demand highprecision calculations, such as those developed with various schemes in [12][13][14][15][16].The incorporation of tidal effects in these schemes will be crucial, as highlighted, e.g., in [17][18][19][20][21].
It is well known that isolated asymptotically flat black holes in general relativity have exactly vanishing Love numbers [22][23][24][25][26][27][28][29][30][31][32][33].Quite interestingly, this result holds only in four dimensions, while higher-dimensional black holes display in general a non-vanishing conservative response [25,[34][35][36][37].Most of the results in this context have regarded so far linear perturbation theory only.However, nonlinearities are an intrinsic property of general relativity.Nonlinerities have been studied for instance in relation with quasinormal modes (see, e.g., [38][39][40][41][42][43][44][45]), but much less is known regard-ing nonlinear corrections to the tidal response of compact objects. 1 In this work we make progress in this direction and derive quadratic corrections to the static Love numbers of Schwarzschild black holes in general relativity in four spacetime dimensions.Our strategy will be to compute the response of a perturbed Schwarzschild black hole solution to an external gravitational field in the static limit and perform the matching up to quadratic order in the external perturbation with the worldline effective field theory (EFT).The latter provides a robust framework to define the tidal response of compact objects [49][50][51].For simplicity, we will consider an external field with a quadrupolar structure and even under parity transformation.The two main results of our work can be summarized as follows: (i) the vanishing of the linear Love numbers, defined as Wilson couplings of quadratic derivative operators in the worldline EFT, is robust against nonlinear corrections;2 (ii) the quadratic Love number couplings also vanish.
The structure of the paper is as follows.In section II, we introduce the worldline EFT.In section III, we solve the nonlinear Einstein equations up to second order in perturbation theory and in the static limit.For illustrative purposes, we will focus on the even sector only, and assume quadrupolar tidal boundary conditions at large distances for the metric perturbation.In section IV, we perform the matching between the EFT and the full solution in general relativity, up to second order in the external tidal field amplitude.Some details and useful technical results are collected in the appendices.In particular, appendix A provides all the equations necessary for the computation of the metric solution at second order in the Regge-Wheeler gauge, while appendix B summarizes the Feynmann rules for reference.Conventions.We use the mostly-plus signature for the metric, (−, +, +, +), and work in natural units, h = c = 1.We use the notation κ = √ 32πG = 2M −1 Pl and the curvature convention R ρ σ µν = ∂ µ Γ ρ νσ + . . .and R µν = R ρ µρν .We use round brackets to identify a group of totally symmetrized indices, e.g., Our convention for the decomposition in spherical harmonics is For simplicity, we will often omit the arguments on Ψ altogether and drop the tilde, relying on the context to discriminate between the different meanings.

II. WORLDLINE EFFECTIVE THEORY AND LOVE NUMBER COUPLINGS
A robust way of defining the tidal response of a compact object is in terms of the worldline EFT [49][50][51].By taking advantage of the separation of scales in the problem, the worldline EFT implements the idea that any object, when seen from distances much larger than its typical size, appears in first approximation as a point source.Finite-size effects can then be consistently accounted for in terms of higher-dimensional operators localized on the object's worldline.As in any genuine EFT, they are organized as an expansion in the number of derivatives and fields.
Let us start from the bulk action, which we take to be the standard Einstein-Hilbert term in general relativity: The point-particle action is where M is the mass of the point particle, s is its proper time and τ parametrizes the worldline.
To capture finite-size effects we now include derivative operators attached to the worldline.Neglecting dissipative effects [52,53]-which are absent for the static response of nonrotating Schwarzschild black holes-and focusing for the moment on the lowest order of the derivative expansion, the quadrupolar (ℓ = 2) Love number operators can be written as [49,[54][55][56]] where E µν is the electric (even) component of the Weyl tensor C µρνσ , defined as where u µ ≡ dx µ /ds is the particle's four-velocity, normalized to unity, u µ u µ = −1.Since we will focus only on the even response, in (3) we omitted to write explicitly operators involving the odd part of the Weyl tensor [54][55][56].One can easily extend (3) to higher ℓ by introducing the multi-index operators [56] E µ 1 ...µ ℓ ≡ P ν 3 (µ 3 . . .
where P is the projector on the plane orthogonal to u µ , i.e., In (3), λ are the (quadrupolar) Love number couplings at the n th order in response theory.This provides an unambiguous way of defining the tidal deformability, which is independent of the choice of coordinates and the field parametrization.Putting all together, the EFT for the point-particle is At this level, λ n are generic couplings, which will then be determined after performing the matching with the full theory.

III. NONLINEAR STATIC DEFORMATIONS OF SCHWARZSCHILD BLACK HOLES
In this section we solve the quadratic static equations for the metric perturbations of a Schwarzschild black hole in general relativity, given some suitable tidal boundary conditions at large distances.We will denote here with g Sch µν the Schwarzschild solution for the metric, where r s = 2GM, and with δ g µν = g µν − g Sch µν the metric perturbation.
The static linearized solutions are well studied and lead to the well-known fact that the induced static response of a Schwarzschild black hole is zero, once regularity of the physical solution is imposed at the black hole horizon [22][23][24][25][26][27]34] (see also Appendix A below).Once the linear solution for δ g (1) is known, the source on the right-hand side of (8) becomes fully fixed and the inhomogeneous solution to eq. ( 8) can be derived using standard Green's function methods.We shall stress that there are two expansion parameters in the problem: there is κ ≡ 2/M Pl , which controls the number of graviton field insertions, and there is the amplitude of the external tidal field, which we will denote with E and which controls the nonlinear response.The two should in general be kept separate, as they appertain to different power countings in the EFT (see section IV).
In the following we will compute nonlinear corrections to the Love numbers by explicitly solving the second-order equations (8) in some particular cases.As briefly reviewed in Appendix A, we will parametrize the metric perturbations δ g µν by distinguishing them in even (polar) and odd (axial) components, δ g µν = δ g even µν + δ g odd µν (see eqs. (A1) and (A2) for the explicit expressions).We will assume that the external tidal field is purely even.As such, we can just focus on the even sector and set the odd perturbations δ g odd µν to zero: at quadratic order in perturbation theory, an external even tidal field cannot induce a parity-odd response (see Appendix A for further details).
In full generality, we shall parametrize δ g even µν as in eq.(A1).After choosing the Regge-Wheeler gauge (A3) and solving the nonlinear (tr) constraint equation, as outlined in Appendix A, the expression for δ g even µν takes a simple diagonal form: where we decomposed the field perturbations in spherical harmonics.Plugging (9) into the Einstein equations, one finds the following decoupled equation for H 0 (see also eq. ( A9)): where SH 0 is fully dictated by the known linearized solution for δ g even µν .Note that to write (10) we have projected the equation for H 0 in real space with an (ℓ, m) spherical harmonic.As a result, the right-hand side of ( 10) is proportional to an integral of the product of three spherical harmonics, which enforces the standard angular momentum selection rule ℓ = ℓ 1 ⊗ ℓ 2 .Given the tensor product (ℓ 1 , m 1 ) ⊗ (ℓ 2 , m 2 ) between two different representations of the rotation group, the resulting total angular momentum ℓ satisfies the triangular condition |ℓ 1 − ℓ 2 | ≤ ℓ ≤ ℓ 1 + ℓ 2 , while the total magnetic quantum number is given by the sum m = m 1 + m 2 .For the sake of the presentation, we will focus in the following on the case in which the external tidal field contains only a single quadrupolar harmonic, i.e., ℓ 1 = ℓ 2 = 2.The analysis will be analogous with a more general tidal field and for higher harmonics.
Solving first the homogeneous linearized equation (10) and imposing regularity at the horizon yields the following linear solution for the radial profile of H 0 : where the amplitude E m depends on the magnetic quantum number m.The other components of δ g even µν are obtained from (12) via the constraint equations.The linearized solutions for H 2 and K are Using ( 12) and ( 13), the right-hand side of ( 10) is completely fixed.At second order, a general solution for (10) is given by a superposition of the homogeneous solution and a particular one.The latter can be obtained via standard Green's function methods (see Appendix A 1).One of the two integration constants for the homogeneous solution simply corresponds to a redefinition of the tidal field amplitude in (12) and can be set to zero.The other integration constant is chosen in such a way that the solution at second order preserves regularity at the horizon.Note that, from the standard angular momentum selection rules, an ℓ = 2 can induce at second order in perturbation theory the harmonics ℓ = 0, 2 and 4. In the following, we will focus on the quadrupole, which contributes to the leading order in the derivative expansion (3).We find the following quadratic solution for the (ℓ = 2, m) harmonic of H 0 , up to quadratic order in perturbation theory: Similarly, for H 2 and K, we find Note that the quadratic terms in E m are small corrections as long as E m r 2 ≪ r 2 s .This should not surprise because the tidal field is formally divergent at large distances, and sufficiently far away perturbation theory is expected to break down.However, in physical situations, such as in binary systems, this does not happen, because the external field acts as a growing source only on a finite region, beyond which it decays to zero at infinity.In practice, we will perform the matching with the worldline EFT in the region r s ≪ r ≪ r s / √ E m , which is sufficiently far from the black hole that the object can be treated as a point particle, but still within the range of validity of the perturbative expansion. 3he previous results have been derived under the assumption that the external source is composed by a single quadrupolar harmonic.However, they can be generalized to the case of more general tidal fields, such as a superposition of different harmonics, using the same procedure.
FIG. 1. Feynman diagrams that reconstruct the Schwarzschild metric up to order r 2 s .

IV. MATCHING WITH EFFECTIVE THEORY
Given the results of section III, we now need to perform the matching with the worldline effective theory (7) and derive the Love number couplings in eq. ( 3).We shall see explicitly that the matching with the calculation in general relativity can be performed with just (1) and ( 2), without turning on any of the Love number couplings in (3).
For this computation, it is convenient to use the background field method [57][58][59].We shall then expand the metric in eq. ( 7) around a non-trivial background as follows where the background metric ḡµν represents the external tidal field that satisfies the vacuum Einstein equations, while h µν parametrizes perturbative corrections in G to this tidal field and, possibly, a response.At this point, we can explicitly compute the one-point function of h µν induced by the external tidal field coupled to the point particle by performing a path integral as follows: up to a normalization factor.In the above action we have introduced the usual gauge-fixing term S GF arising from a Faddev-Popov procedure.Since we are ultimately interested in the classical limit of the above equation, following Ref.[49] we shall discard all diagrams with closed graviton loops.Hence, we do not need to add any ghost field.Finally, in order to maintain covariance of the final result with respect to the external metric ḡµν , we work with the following gauge-fixing action, Here, ∇µ is the covariant derivative associated to the metric ḡµν and ḡµν is the inverse of the background metric.At a practical level, we shall expand also the tidal field as The one-point function can then be constructed by considering all Feynman diagrams with one external h µν .We will use the following diagrammatic conventions: Feynman diagrams needed for the computation of h µν .Diagram (a) yields the order-r s correction to the linear tidal field solution.Diagrams (b), (c) and (d) represent instead order-r s corrections to the tidal source at second order in the external field amplitude.
For the comparison with section III, we need to compute the diagrams represented in fig. 1 and 2. Their explicit expressions can be found using the Feynman rules listed in appendix B. We shall compute all diagrams in the rest frame of the point-particle, which means that τ = t and the worldline is given by4 x µ = (t, 0, 0, 0) , where v µ ≡ dx µ dτ .The advantage of working with the background field method is that, as we mentioned, the final result for h µν is covariant under diffeomorphisms of the external metric ḡµν .This means that we can choose the tidal field in any convenient gauge of our choice.Hence, we choose the gauge such that H µν satisfies the vacuum Einstein equation on a flat background consistent with the Regge-Wheeler gauge used in section III.In particular, to compare with the results of that section we focus on a tidal field composed by just the harmonic ℓ = 2. Written in cartesian coordinates, this reads where A µν is a symmetric-trace-free, purely spatial constant tensor (of mass dimension 2), i.e., A µν v µ = 0 and A µν η µν = 0. To be concrete, in spherical coordinates one has where r = x i x j δ i j and we have chosen the amplitude E m of the external tidal field in such a way as to match the notation of section III.
As a sanity check, we have verified that the sum of the diagrams in figs. 1 and 2 satisfies the gauge condition Γµ = 0 , (25) and that the diagrams in fig. 1 give the Schwarzschild metric up to order G 2 in the gauge (25). 5This is consistent with the well known result of, e.g., Refs.[49,60] and reproduces the background metric g Sch µν in section III.We can now match the result of the other diagrams to the full-theory solution δ g µν derived in section III.However, while H µν is already in the gauge used in section III, ⟨h µν ⟩ is not.Therefore, to do the comparison we must first transform ⟨h µν ⟩ from the coordinates x µ defined by the gauge condition (25) into the coordinate x µ RW defined by the Regge-Wheeler gauge.The gauge transformation reads where ξ µ = x µ RW − x µ is given below.This allows us to define which is now in the Regge-Wheeler gauge and can be compared to the full-theory solution δ g µν .
For simplicity, we will compare only the (tt) component, the other components of δ g µν being fixed in terms of δ g tt via the Einstein equations.Since ⟨h µν ⟩ is static, then the gauge transformation must be time-independent.Therefore, if we focus on the (tt) component, the gauge transformation simplifies to ⟨h RW tt ⟩ = ⟨h tt ⟩ − ξ i ∂ i ḡtt .The derivative of the background metric is at least of order of the amplitude A of the tidal field, hence, we only need to find ξ i up to order r s A. Explicitly this is given by Moreover, since λ is independent of m, it is enough to perform the matching for any particular configuration of (m, m 1 , m 2 ) in order to compute its value and show that it vanishes.For concreteness, we will consider the case where m 1 = 0 and m 2 = 0, resulting in m = 0 at second order.Performing the coordinate transformation above and projecting the result on the (ℓ = 2, m = 0) harmonic, we find where E ≡ E m=0 .The first term on the right-hand side, proportional to E , results from the diagram 2(a) and reproduces the r s /r correction at leading order in the tidal field [61].The second term on the right-hand side, proportional to E 2 , results from the last three diagrams in fig. 2. In particular, diagram 5 Intermediate divergences coming from diagram 1(b) are handled using dimensional regularization. ( λ (b) is simply the iteration of (a) due to the solution of the tidal field at order E 2 .Instead, diagrams (c) and (d) are a double insertion of the lowest-order tidal field given in eq. ( 23).More details on the calculation of these diagrams will be provided in [62].The sum of all the four diagrams matches the full-theory solution, δ g tt = (1 − r s /r)H 0 , with H 0 given in eq. ( 14) for m = m 1 = m 2 = 0, upon expanding for small r s /r and using π .The matching with the full theory in section III is obtained without the inclusion of any of the higher dimensional operators (3) in the point-particle action.In other words, up to quadratic order in the external source, the couplings associated with the diagrams in fig.3, which capture the induced response of the body, vanish for black holes.Note that, for the ℓ = 2 induced response, this conclusion can be reached directly from a simple dimensional analysis: the higher dimensional operators (3) correspond to a scaling ∼ 1/r ℓ+1 in the one-point function of h µν which is absent in the full solution (14).

V. CONCLUSIONS
In this work, we have derived the static nonlinear response of Schwarzschild black holes in general relativity.We have explicitly solved the nonlinear Einstein equations in the static limit and up to second order in the perturbations of the Schwarzschild metric.We have then performed the matching with the worldline EFT, which provides a robust and unambiguous way of defining the tidal deformability of the object.At given order in powers of the external field amplitude, different types of diagrams contribute in the EFT: there are the diagrams in fig. 3 corresponding to the operators E 2 and E 3 in (3), and there are those in figs. 1 and 2 obtained from the interaction vertices in (1) and ( 2).The former capture the true induced (linear and quadratic, respectively) response of the object, while the latter combine to resum the external source.By comparing the full solution in general relativity with the EFT, we have concluded that λ = 0 in (3) up to quadratic order in perturbation theory.For simplicity, we have focused on the leading order in the derivative expansion in the EFT and considered only parity-even perturbations.Our approach can be employed to study higher multipoles and odd perturbations [62].
To summarize, the vanishing of the nonlinear Love numbers is a consequence of the following results: (i) at quadratic or-der in perturbation theory the inhomogeneous solution is constructed from a source (see eq. (A18)) that is made of only the linear tidal field; (ii) the point-particle EFT can be matched with the full solution without turning on Love number couplings, while nonlinear corrections to the static solution in general relativity can be reconstructed from the EFT, at all orders in r s , via just graviton bulk nonlinearities.
The result λ = 0 was previously obtained in [46] using a different approach, which relies on harmonic coordinates and the framework of post-Newtonian theory.In contrast, our approach is not bound to the post-Newtonian expansion and is manifestly gauge invariant.Our methodology can be applied to prove the vanishing of other types of nonlinear Love numbers, such as those involving couplings with the gravitomagnetic field, or to compute dynamical nonlinear Love numbers beyond the static approximation; see e.g.[46,63].Furthermore, defining the Love numbers as Wilson coefficients of a worldline effective field theory makes it more transparent that their vanishing necessitates the existence of a nonlinear symmetry in general relativity, such as those proposed for linear fields in [3,4] (see also [5][6][7]).It would be interesting to understand to what extent such symmetries can be extended to higher orders in perturbation theory.In addition, it will be interesting to see how our conclusions change for rotating Kerr black holes, black hole solutions in higher dimensions and different spins [32,34,36,37,64].We leave all these aspects for future investigations.sin 2 θ ∂ 2 φ ).Each component of δ g even µν and δ g odd µν can be further decomposed in spherical harmonics as, for instance, Since δ g even µν and δ g odd µν have opposite transformation rules under a parity transforma-tion, the spherical symmetry of the background ḡµν ensures that δ g even µν and δ g odd µν do not couple at the level of the linearized equations of motion.Mixing will appear starting from quadratic order.

Quadratic solution from even tidal field
In this section, we derive the even quadratic equations for the metric perturbations in the static regime and solve them under the assumption of a purely even tidal field at large distances.Without an odd tidal field, since the linear odd static solution is divergent at the horizon, 7 we can set δ g odd µν to zero altogether and just focus on the even perturbations δ g even µν .Gauge fixing.At each order in perturbation theory, we choose to fix the Regge-Wheeler gauge as follows [39,65,67]: Since this is a complete gauge fixing, it can be performed directly in the action without losing any constraints [68].Constraint equation.With the gauge choice (A3), the only off-diagonal metric component in δ g even µν is thus H 1 , which is a constrained variable.It is not hard to see that, in the static limit and in the absence of odd perturbations, at each order in perturbation theory.This follows from solving G tr = 0, where G µν = R µν − 1 2 g µν R is the Einstein tensor.In fact, by construction, G tr is at each order proportional to (derivatives of) H 1 , i.e., it vanishes when H 1 van-ishes.As a result, with the gauge choice (A3) and the nonlinear solution (A4) in the static limit, the metric perturbation δ g even µν boils down to the diagonal form: δ g even µν = diag (1 − r s r )H 0 , H 2 , r 2 K, r 2 sin 2 θ K [69].Next, we plug this into the Einstein-Hilbert action, which we expand up to cubic order in the perturbations H 0 , H 2 and K. Taking then the variation with respect to each of the three metric components, we can write down the quadratic equations for H 0 , H 2 and K. Two of them will lead to constraints, while only one will give the static equation for the physical (even) degree of freedom.To make this manifest, it is first convenient to perform the following field redefinition, where φ is the spherical Laplacian on the 2-sphere, defined with line element dΩ 2 ≡ dθ 2 + sin 2 θ dφ 2 and satisfying ∆ S 2 K = −ℓ(ℓ + 1)K.The resulting equations are where S 1 , S 2 and S 3 are source terms, quadratic in the fieldswhich we will not write explicitly.The goal is to solve (A6)-(A8) in perturbation theory.After straightforward manipulations, one finds that the field components H 2 and K can be solved algebraically for in terms of H 0 and derivatives thereof.
Hence, the problem reduces to solving the H 0 's equation of motion, which, after some massaging of (A6)-(A8), is found to be where SH 0 is a linear combination of (derivatives of) the source terms in (A6)-(A8).
The homogeneous part of (A9) is a (degenerate) hypergeometric equation, which can thus be solved in closed form.To bring it in standard hypergeometric form, it is convenient to perform the following field redefinition, , (B2) α 2 β 2 α 3 β 3 , (B3) where the cubic vertex V 3 is obtained from expanding the Einstein-Hilbert action up to cubic order in h µν .The cubic and quartic vertices V 3 and V 4 are obtained by expanding eqs.( 1) and ( 19) up to quadratic order in both h µν and H µν .Their tensorial structures are handled using the xAct package for Mathematica [71].
FIG. 3. Feynman diagrams for the (a) linear and (b) nonlinear tidal deformation.Note that these diagrams contribute at the same order in powers of the external field amplitude as the diagrams in figs.2(a) and 2(b)-(d), respectively.