Leading Nonlinear Tidal Effects and Scattering Amplitudes
Zvi Bern, Julio Parra-Martinez, Radu Roiban, Eric Sawyer, Chia-Hsien Shen
aa r X i v : . [ h e p - t h ] O c t CALT-TH/2020-041
Leading Nonlinear Tidal Effectsand Scattering Amplitudes
Zvi Bern, a , Julio Parra-Martinez b , Radu Roiban, c ,Eric Sawyer a and Chia-Hsien Shen, a,d a Mani L. Bhaumik Institute for Theoretical Physics,Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095, USA b Walter Burke Institute for Theoretical Physics,California Institute of Technology, Pasadena, USA c Institute for Gravitation and the Cosmos,Pennsylvania State University, University Park, PA 16802, USA d Department of Physics 0319, University of California at San Diego,9500 Gilman Drive, La Jolla, CA 92093, USA
Abstract
We present the two-body Hamiltonian and associated eikonal phase, to leading post-Minkowskian order, for infinitely many tidal deformations described by operators with ar-bitrary powers of the curvature tensor. Scattering amplitudes in momentum and positionspace provide systematic complementary approaches. For the tidal operators quadratic incurvature, which describe the linear response to an external gravitational field, we work outthe leading post-Minkowskian contributions using a basis of operators with arbitrary numbersof derivatives which are in one-to-one correspondence with the worldline multipole operators.Explicit examples are used to show that the same techniques apply to both bodies interactingtidally with a spinning particle, for which we find the leading contributions from quadratic incurvature tidal operators with an arbitrary number of derivatives, and to effective field theoryextensions of general relativity. We also note that the leading post-Minkowskian order contri-butions from higher-dimension operators manifest double-copy relations. Finally, we commenton the structure of higher-order corrections. ontents E and B tidal effects 15 Introduction
The remarkable discovery of gravitational waves by the LIGO and Virgo collaborations [1] hasushered in a new era of exploration that promises major new discoveries on black holes, neutronstars and perhaps even new basic insights into fundamental physics. Theoretical tools of increasedprecision, matching that of gravitational-wave signals not only from current detectors but also fromproposed gravitational-wave observatories [2], are required.The evolution of a compact binary and the ensuing gravitational-wave emission can be divided inthree distinct phases — inspiral, merger and ring down — according to their underlying properties.The inspiral part of binary mergers, which is the subject of this paper, is analyzed through modelssuch as the effective one-body (EOB) formalism [3]. The weak gravitational field during this phasemakes it suitable for a perturbative approach and these models import information from post-Newtonian (PN) gravity [4–7], as well as the self-force framework [8] and numerical relativity [9].More recently, the post-Minkowskian (PM) expansion [10–16] has gained prominence due to itscapture of the complete velocity dependence at fixed order in Newton’s constant. By exposing theanalytic structure of each order, this expansion also offers new insight into features of gravitationalperturbation theory, exposes hereto unexpected structure in certain observables, and may open apath to the resummation of perturbation theory in the classical limit. The PN, PM and self-forceexpansions provide important nontrivial cross checks in their overlapping regions of validity [7, 13,14, 17]. For recent reviews see Refs. [18].Over the years a close link between classical physics and scattering amplitudes has been de-veloped [11–13, 15, 19–21] and led to a robust and powerful means for obtaining two-body Hamil-tonians [11] and observables in the post-Minkowskian expansion. It was obtained by combiningmodern techniques, such as generalized unitarity [22], which emphasize gauge-invariant buildingblocks at all stages and build higher-order contributions from lower-order ones with effective fieldtheory methods. This framework proved its effectiveness through the construction of the soughtafter two-body Hamiltonian at the third order in Newton’s constant [12, 13] and the identification ofsurprising simplicity in physical observables of interacting spinning black holes [23]. The scatteringangle is of particular importance, as it provides a direct link [20] with the EOB framework [3] usedto predict gravitational wave emission from compact binaries.In this paper we investigate the effects of tidal deformations [24] on the conservative two-bodyHamiltonian during the inspiral phase, focusing on their structure in the post-Minkowskian expan-sion. The tidal deformations offer a window into the equation of state of neutrons stars [25] andtest our understanding of black holes [21, 26–30] and of possible exotic physics [31]. While tidaleffects are expected to vanish for black holes in general relativity [32], they are of crucial importancefor understanding the equation of state of neutron stars. These corrections are formally equiva-lent to fifth-order post-Newtonian effects [5], highlighting the importance of precision perturbative3alculations.Properties of extended bodies that relate to their finite size can be encoded in local-operatordeformations of a point-particle theory by integrating out their internal degrees of freedom. The setof all possible tidal operators is constrained only by the symmetry properties of the fundamentaltheory, such as parity. We introduce our organization of tidal operators in close analogy with thecase of electromagnetic susceptibilities. Indeed, not only is there a formal similarity between gaugetheory and gravity, but the integrand of gravitational scattering amplitudes can be obtained directlyfrom gauge theory using the double copy [33, 34]. For the relatively simple case of the leading-PMorder contribution of a given tidal operator to scattering amplitudes, these relations follow from thefactorization of the point-particle energy-momentum tensor and from the fact that the linearizedRiemann tensor is a product of two gauge-theory field strengths. Thus, in analogy with the caseof electromagnetic interactions of extended bodies, tidal operators may contain arbitrarily-highnumber of Riemann curvature tensors with an arbitrary number of derivatives.Curvature-squared tidal operators, describing the linear response of an extended body to anexternal gravitational field, were recently classified in Ref. [30], where an expression for the two-body Hamiltonian and scattering angle at leading post-Minkowskian order was conjectured. Herewe prove the conjecture for a basis of operators whose Wilson coefficients in the four-dimensionalpoint-particle effective action are exactly the same as the worldline electric and magnetic tidalcoefficients, related to the corresponding multipole Love numbers by factors of the typical scale ofthe body, see e.g. Ref. [5, 25, 28]. The lowest-order matrix elements of our tidal operators are, byconstruction, the same as the matrix elements of the worldline tidal operators. To establish themap beyond leading order it is necessary to compare physical quantities. At the next-to-leadingorder the contributions of low-derivative R tidal operators to the two-body Hamiltonian and tothe scattering angle were determined in Refs. [21, 29].We also obtain the leading-order modifications of the two-body Hamiltonian and of the scatteringangle due to tidal operators with arbitrarily-high number of Weyl tensors, which describe thenonlinear response of extended bodies to external gravitational field. As usual we organize theoperators in terms of electric and magnetic-type components, E and B , of the Riemann (or Weyl)tensor. The finite rank of these tensors leads to nontrivial relations between different operators,allowing us to express the contributions of E n and B n -type operators for n ≥ R and R and compare them with existing results. The two-body Hamiltonian and associated observables fora point-particle deformed by tidal operators interacting with a spinning particle can also be derivedthrough similar methods. To leading PM order, only the single-graviton interaction of the spinningparticle is relevant and it is captured by the stress tensors described in [23, 41, 42]. As an example,we find the leading spin-orbit contributions from E -type tidal operators with an arbitrary numberof derivatives interacting with a spinning particle.This paper organized as follows. In Sec. 2 we present a description of the operators encoding tidaldeformations. In Sec. 3 we discuss the leading-order tidal contributions from R -type operators withan arbitrary number of derivatives. This section also demonstrate how to incorporate spin effectsfor the second body. We proceed to derive in Sec. 4 the leading contributions of various infiniteclasses of R n -type tidal operators and also comment on their higher-order contributions. In Sec. 5we discuss the application of our methods to the case of R n extensions of General Relativity. Wepresent our conclusions in Sec. 6. An appendix gives the explicit results for the contributions of acollection of high-order tidal operators to the two-body Hamiltonian and the associate scatteringamplitudes. Note added:
While this project was ongoing we became aware of concurrent work by Cheung,Shah and Solon [43] based on using the geodesic equation and containing some overlap on leadingcontributions to the two-body Hamiltonian from the R n tidal operators. In addition, the methodsdeveloped there determine the two-body Hamiltonian for a tidally-deformed test particle interactingwith a Schwarzschild black hole, to all orders in the Schwarzschild radius of the latter. We aregrateful for interesting and helpful discussions and sharing drafts. In this work we study tidal or finite-size effects in the gravitational interactions of two massiveextended bodies. They are encoded in a classical two-body Hamiltonian of the form H ( p , r ) = q p + m + q p + m + V ( p , r ) , (2.1)5nd is extracted systematically, following the general approach introduced in [11], by matching QFTscattering amplitudes to a non-relativistic EFT. If the size of the two bodies is much smaller thantheir separation, non-analytic/long-distance classical potential has the form V ( p , r ) ∼ c i ( p ) m Gm | r | ! i , (2.2)where m carries unit mass dimension and the momentum transfer q , Fourier-conjugate to r , is muchsmaller than the center of mass momentum p . Such a conservative potential arises from integratingout gravitons with momenta ℓ in the potential region which has the scaling behavior ℓ = ( ℓ , ℓ ) ∼ ( | q || v | , | q | ) , (2.3)where | v | ∼ O ( | p | /m ). Note that Gm is of the order of the effective Schwarzschild radius of theparticles R s , so the classical expansion of the potential is an expansion in R s / | r | . If the separationof the two bodies can be of the same order as their typical size R , then the classical potential takesthe form V ( p , r ) ∼ c i,k ( p ) m Gm | r | ! i R | r | ! k . (2.4)For black holes R ∼ R s so the size of terms with powers of R is comparable to higher PM orders.For other bodies R > R s so the contribution should be bigger. For reference, neutrons stars have R/R s ∼
10, and the sun has
R/R s ∼ . In practice, it is convenient to always use R s /r as theexpansion parameter so that the tidal effects just modify the coefficients in the usual PM potential,i.e. c i,k ∼ ∆ c i + k .From our point of view, the new scale R s is introduced by integrating out the degrees of freedomthat describe the tidal dynamics of an extended body to yield a point-particle effective theory. Insuch an effective theory the finite size effects are encoded as higher-dimension operators O i whichare suppressed by powers of R s | q | . Their Wilson coefficients can be determined either by matchingto the complete theory that includes the tidal degrees of freedom, or by comparing to experiment.A side effect of choosing R s instead of R as the scale characterizing finite-size effects is that for lesscompact bodies the Wilson coefficients are not necessarily O (1). This approach was pioneered inthe context of a worldline PN formalism in Ref. [5], and recently adapted to the PM framework inRef. [29]. In the QFT language this approach has been recently used in Refs. [21, 30]. In sectionwe provide a systematic treatment of such effective actions and write a basis of operators whichsimplifies the translation between QFT and worldline formalisms and makes the relation to familiar in-in observables manifest.The cases that we focus on in this paper correspond to leading contributions from tidal orother operators. Although these operators first contribute to loop amplitudes, the determination The amplitude also contains non-analytic terms which we will not study here, corresponding to quantum contri-butions to the potential of the form ( ℓ p /r ) n , where ℓ p is the Planck length.
6f their leading-order contribution to the two-body potential is straightforward and formally givenby inverting the Born relation between the scattering amplitude and the potential: V O ( p , r ) = − E E Z d D − q (2 π ) D − e − i q · r M O ( p , q ) . (2.5)Here M O is the leading-order four-scalar scattering amplitude with with a single insertion of O ,center of mass momentum p , transferred momentum q . In general the potential is gauge dependentand not unique. In the above equation we choose to expose the on-shell condition on q first suchthat p · q ≃ O ( q ) ∼
0. This naturally gives the potential in the isotropic gauge.Alternatively, the effective two-body Hamiltonian can be constructed by matching its conserva-tive observables — such as the conservative scattering angle, or the impulse and spin kick — or theclosely-related eikonal phase [44], δ O ( p , b ) = 14 m m √ σ − Z d D − q (2 π ) D − e − i b · q M O ( p , q ) , (2.6)with the corresponding quantities in the complete theory. Here we use − p i = − m i u i as the incomingmomenta of particle 1 and 2 and σ ≡ p · p m m = u · u . (2.7)In either case, the matching is carried out order by order in Newton’s constant G , that is orderby order in the post-Minkowskian expansion. The relation between the eikonal and conservativeobservables holds also for the scattering of spinning particles. To leading nontrivial order, the effectof a composite operator O on the impulse and spin kick in the center-of-mass frame is∆ p = −∇ b δ O ( b ) + . . . , ∆ S i = −{ S i , δ O ( b ) } + . . . , (2.8)where the ellipsis stand for higher-order terms that depend on O and {• , •} is the Poisson bracket.We expect that the all-order relation between the eikonal phase and conservative observables putforth in Ref. [23] holds in the presence of deformations by tidal and other composite operators. Atleading order, the semiclassical approximation implies that the eikonal phase coincides with theradial action integrated over the scattering trajectory. We discuss this further in Section 3.3. Thelatter allows us to make contact with Ref. [28] in which tidal effects were computed using a classicalworldline formalism for a subset of tidal operators.Alternatively the matching can be performed by directly computing a physically meaningfulquantity such as the conservative scattering angle, corresponding to the scattering with radiationreaction turned off; or the closely related eikonal phase. In either case matching is performedorder by order in perturbation theory in Newton’s constant, G , that is order by order in the post-Minkowskian expansion. M O ( q ) = | q | A M O , (2.9)7 O ( r ) = − E E A Γ (cid:16) ( D − A ) (cid:17) π ( D − / Γ( − A ) | r | − A − ( D − M O , (2.10) δ O ( b ) = 14 m m √ σ − A Γ (cid:16) ( D − A ) (cid:17) π ( D − / Γ( − A ) | b | − A − ( D − M O , (2.11)where we have used the formula for the Fourier transform of a power Z d D q (2 π ) D e − i x · q | q | A = 2 A Γ (cid:16) ( D + A ) (cid:17) π d/ Γ( − A ) | x | − ( A + D ) . (2.12)Here A is power of the soft q carried by the amplitude. For an operator with n power of Riemann orWeyl tensors with n ∂ derivatives acting on them, the leading contribution to the two-to-two scalaramplitude is A = 3 n + n ∂ − − ǫ ( n − , (2.13)where we use D = 4 − ǫ . For example, for the electric and magnetic operators E and B we willintroduce shortly, n = 2 and n ∂ = 0 so A = 3 − ǫ , and every pair of derivatives acting of theseincreases n ∂ and A by two. We now explain how to parametrize the response of a general body to an external field and howthis can be encoded in an effective action. We will discuss this in detail in the simpler case ofelectromagnetism, which will easily generalize to the gravitational case.
The full non-linear response of a body to an external electric field E i is described by the inducedelectric dipole moment density D i . In the rest frame of the body, it has a formal expansion inpowers of the electric field [45]: D i ( t, x ) = χ (1) i i ( t, x ) E i ( t, x ) + χ (2) i i i ( t, x ) E i ( t, x ) E i ( t, x ) + · · · . (2.14)The first term is the familiar linear response function; the subsequent terms encode the propertiesof the body in the susceptibility tensors, χ ( n ) , which are symmetric in their indices. Similarly,in the presence of a magnetic field B i , one can write magnetic susceptibilities, as well as generalsusceptibilities capturing the response under a general electromagnetic field.It is convenient to transform Eq. (2.14) to Fourier space, where it takes the form D i ( − ω , − q ) = χ (1) i i ( ω , q ; ω , q ) E i ( ω , q )+ χ (2) i i i ( ω , q ; ω , q ; ω , q ) E i ( ω , q ) E i ( ω , q ) + · · · . (2.15)8ere we have adopted a generalized summation convention where repeated frequencies and momentaare integrated over, and the Fourier susceptibilities include energy-momentum-conservation deltafunctions χ ( n − i ··· i n = δ X i ω i ! δ X i q i ! ˜ χ ( n − i ··· i n , (2.16)which account for the fact that the position-space product in Eq. (2.14) becomes a Fourier spaceconvolution in Eq. (2.15).The dipole density can be related to a generating function — or effective action — S ( E ), viathe usual response formula D i ( − ω , − q ) = ∂S ( E ) ∂E i ( ω , q ) . (2.17)The effective action, following from formally integrating Eq. (2.14), is given by S ( E ) = 12 χ (1) i i ( ω , q ; ω , q ) E i ( ω , q ) E i ( ω , q )+ 13 χ (2) i i i ( ω , q ; ω , q ; ω , q ) E i ( ω , q ) E i ( ω , q ) E i ( ω , q ) + · · · . (2.18)This makes clear that the momentum space susceptibilities are completely symmetric tensors, aswell as symmetric functions of all their arguments. S ( E ) could be put in a form closer to an actionby series expanding the susceptibilities and rewriting the powers of frequency and three-momentaas derivatives. For instance one can rewrite some terms in the expansion as follows ∂χ (1) i i ∂ω ∂ q j (0) ω q j E i ( ω , q ) E i ( ω , q ) ∼ ∂χ (1) i i ∂ω ∂ q j (0) ∂ t E i ( t, x ) ∇ j x E i ( t, x ) . (2.19)Note that the expansion in the three momenta here simply corresponds to a multipole expansionof the electric fields.So far we have been working in the rest frame of the object. The choice of a frame breaksmanifest Lorentz invariance down to the rotations around the position of the object. We would liketo covariantize the expressions above so that is they are valid in an arbitrary reference frame, inwhich the body moves with velocity v . This can be done by considering the four-velocity of theobject u µ = γ (1 , v ), where γ is the Lorentz factor. As is well known the electric field and magneticfields in the rest frame of the body can be covariantly written as E µ = F µν u ν , B µ = ∗ F µν u ν , (2.20)where F µν is the electromagnetic field strength, and ∗ F µν its dual. Similarly, it is clear that anyfrequency and spatial momenta can be written as ω i → u · q ≡ u µ q µ , q i → ( q ⊥ ) µ ≡ P µν q ν , (2.21)9here we have introduced the four momentum of the field, q µi and a projector, P µν = η µν − u µ u ν , (2.22)which makes indices purely spatial in the rest frame of the object. Naively this covariantizationrequires adding components to the polarizabilities so that χ ( n − i ··· i n → χ ( n − µ ··· µ n , and we can write S ( E ) = χ (1) µ µ ( u · q , q ⊥ ; u · q , q ⊥ ) E µ ( q ) E µ ( q )+ χ (2) µ µ µ ( u · q , q ⊥ ; u · q , q ⊥ , u · q , q ⊥ ) E µ ( q ) E µ ( q ) E µ ( q ) + · · · , (2.23)due to the fact that u µ E µ = u µ B µ = 0, which follows from the antisymmetry of the field strength.The generating function written above describes the non-linear response of an arbitrary material,including those that violate rotational and Lorentz invariance. In the following we will be onlyinterested in Lorentz-preserving effects, which impose addition constraints on the susceptibilitytensors. Firstly, Lorentz invariance constrains the index structure of the susceptibility, which canonly be carried by Lorentz-covariant tensors. If we impose parity, the only such tensors are themetric itself and the graviton momenta, so the tensor susceptibility must decompose in a set ofscalar susceptibilities as follows χ (1) µ µ = χ (1)0 g µ µ + χ (1)1 q ⊥ µ q ⊥ µ (2.24) χ (2) µ µ µ = χ (2)0 ( g µ µ q ⊥ µ + g µ µ q ⊥ µ + g µ µ q ⊥ µ ) , (2.25) χ (3) µ µ µ µ = χ (3)0 g ( µ µ g µ µ ) + χ (3)1 ( g µ µ q ⊥ µ q ⊥ µ + perms) + χ (3)2 q ⊥ µ q ⊥ µ q ⊥ µ q ⊥ µ , (2.26)where in general each tensor structure must be summed over permutations which respect the symme-try ( µ i ↔ µ j ) while simultaneously swapping q ⊥ i ↔ q ⊥ j . Another consequence of Lorentz invarianceis that the scalar susceptibilities only depend on Lorentz invariant combinations of momenta, sothat χ ( n − a ( u · q i ; q ⊥ i ) → χ ( n − a ( u · q i ; q ⊥ i · q ⊥ j ) . (2.27)Note that in the rest frame q ⊥ i · q ⊥ j = q i · q j . It is now easy to generalize the tidal response for electromagnetism to its gravitational analog. Inthis case we start from the induced quadrupole moment, written in terms of the gravito-electricfield Q i j ( t, x ) = χ (1) i j i j ( t, x ) E i j ( t, x ) + χ (2) i j i j i j ( t, x ) E i j ( t, x ) E i j ( t, x ) + · · · , (2.28)where now the gravitational susceptibilities are more general tensors symmetric in each pair of i and j indices χ ··· ij ··· = χ ··· ji ··· , χ ··· i a j a ··· i b j b ··· = χ ··· i b j b ··· i a j a ··· . (2.29)10n the rest frame of the object the electric field is related to the Weyl tensor as E ij = C i j . Similarexpressions can be written for the response to a gravito-magnetic or to a mixed field.All of these quantities can be covariantized by introducing E µν ≡ C µανβ u α u β , B µν ≡ ( ∗ C ) µανβ u γ u δ ≡ ǫ αβγµ C αβδν u γ u δ , (2.30)where all indices are curved and the Levi-Civita tensor is defined as ǫ = +1. As in the electro-magnetic case the following relations hold E µν u ν = 0 , B µν u ν = 0 , (2.31)as well as E µµ = 0 , B µµ = 0 , (2.32)where the first equality is a consequence of the tracelessness of the Weyl tensor. The correspondinggenerating function for tidal response is then simply S grav ( E ) = χ (1) µ ν µ ν ( u · q , P q ; u · q , P q ) φ ( p ′ ) E µ ν ( q ) E µ ν ( q ) φ ( p ) (2.33)+ χ (2) µ ν µ ν µ ν ( u · q , P q ; u · q , P q , u · q , P q ) φ ( p ′ ) E µ ν ( q ) E µ ν ( q ) E µ ν ( q ) φ ( p ) + · · · where, as above, a convolution over all momenta is implicit, and the covariant susceptibilities aretraceless in each pair of µ, ν indices η µν χ ··· µν ··· = 0. Once again, Lorentz invariance will furtherconstraint the form of the susceptibility tensors in a way analogous to Eqs. (2.24)-(2.26). We now proceed to connect our discussion to a QFT effective action, focusing on the case of gravity;the electromagnetic case is completely analogous.The connection can be easily made by interpreting the generating function, S grav ( E ) as theexpectation value in a background field of an operator in a one-particle state | p i with four momentum p = mu , and zero spin. In second-quantized language the one-particle state is created by a scalarfield, φ , at infinity and S tidal = χ (1) µ µ ( u · q , q ⊥ ; u · q , q ⊥ ) φ ( p ) E µ ( q ) E µ ( q ) φ ( p ′ )+ χ (2) µ µ µ ( u · q , q ⊥ ; u · q , q ⊥ , u · q , q ⊥ ) φ ( p ) E µ ( q ) E µ ( q ) E µ ( q ) φ ( p ′ ) + · · · , (2.34)can be identified as the momentum-space effective action that encodes the response to the back-ground field. Note that, in order to enforce momentum conservation, the Fourier-transformedsusceptibilities must satisfy χ ( n − µ ··· µ n = δ X i q i − q ! ˜ χ ( n − µ ··· µ n , (2.35)11here q = − ( p + p ′ ). Note that the susceptibilities are initially only defined for q = 0, so theircovariantization requires an extension to q = 0. This does not affect the classical limit. As above,each term in the expansion of susceptibilities is encoded by a higher-dimension operator in theeffective action, where now the factors of four-velocity u can be identified with derivatives actingon the scalar field. For instance, ∂ nω χ (1) µ ν µ ν (0 , u · q ) n + ( u · q ) n ] φ ( p ′ ) E µ ν ( q ) E µ ν ( q ) φ ( p ) , ↔ ∂ nω χ (1) µ ν µ ν (0 , Z d x √− g m n φE µ ν ∇ ( ρ ··· ρ n ) E µ ν ∇ ( ρ ··· ρ n ) φ . (2.36)where the classical limit is implicit on the left-hand side. To write a generic operator appearing inthis expansion it is convenient to introduce the combinations,ˆ E µ µ ...µ n = i m Sym µ ...µ n [ ∇ ν n . . . ∇ ν C µ αµ β ˆ P ν n µ n . . . ˆ P ν µ ∇ α ∇ β ] , ˆ B µ µ ...µ n = i m Sym µ ...µ n [ ∇ ν n . . . ∇ ν ( ∗ C ) µ αµ β ˆ P ν n µ n . . . ˆ P ν µ ∇ α ∇ β ] , ˆ E ( l ) µ µ ...µ n = i m +2 m m +2 Sym µ ...µ n [ ∇ ν n . . . ∇ ν ∇ ρ . . . ∇ ρ l C µ αµ β ˆ P ν n µ n . . . ˆ P ν µ ∇ ( ρ . . . ∇ ρ l ) ∇ α ∇ β ] , ˆ B ( l ) µ µ ...µ n = i m +2 m m +2 Sym µ ...µ n [ ∇ ν n . . . ∇ ν ∇ ρ . . . ∇ ρ l ( ∗ C ) µ αµ β ˆ P ν n µ n . . . ˆ P ν µ ∇ ( ρ . . . ∇ ρ l ) ∇ α ∇ β ] , (2.37)where all the derivatives on the right of the Weyl tensor act on the scalar field, and the position-spaceprojector is ˆ P νµ = 1 m ( ∂ µ ∂ ν − δ νµ ∂ ) . (2.38)The terms in the expansion that encode the most general linear response are then S QFTtidal (cid:12)(cid:12)(cid:12) linear = m Z d x √− g ∞ X n =2 ∞ X l =0 ( µ ( n,l ) φ ˆ E ( l ) µ ··· µ n ˆ E ( l ) µ ··· µ n φ + σ ( n,l ) φ ˆ B ( l ) µ ··· µ n ˆ B ( l ) µ ··· µ n φ ) (2.39)where the coefficients are related to the susceptibility as µ ( n,l ) ∼ ( ∂ ω ) l ( ∂ q · q ) l χ (1)0 (0; 0), and themagnetic susceptibilities are related to σ ( n,l ) in a similar way. Operators like φE ( l ) µ µ E ( l ) µ µ φ with l = l are related to operators with l = l by integration by parts and use of scalar field equationsof motion. We therefore can ignore them at this order. Similarly, the effective action S QFTtidal (cid:12)(cid:12)(cid:12) non-linear = m Z d x √− g ∞ X n =2 ( ρ ( n ) e φ ˆ E µ µ ˆ E µ µ · · · ˆ E µ n µ φ + ρ ( n ) m φ ˆ B µ µ ˆ B µ µ · · · ˆ B µ n µ φ ) + · · · (2.40)encodes part of the lowest-multipole time-independent non-linear response,It is not difficult to translate the different terms in the response functions into a first quantizedframework. This leads to a one-to-one relation between the higher-dimension operators in the QFTeffective action and worldline operators. The factors of u are identified with the four-velocity of12he worldline u µ = dx µ /dτ and the factors of ( u · ∇ ) simply become derivatives with respect to theproper time τ . Thus, the analog of the operators in the effective worldline action are E µ µ ...µ n = Sym µ µ ...µ n h P ν µ . . . P ν n µ n ∇ ν . . . ∇ ν n C µ αµ β i u α u β ,B µ µ ...µ n = Sym µ µ ...µ n h P ν µ . . . P ν n µ n ∇ ν . . . ∇ ν n ( ∗ C ) µ αµ β i u α u β ,E ( m ) µ ...µ n = ( u α ∇ α ) m E µ ...µ n = ( ∂ τ ) m E µ ...µ n ,B ( m ) µ ...µ n = ( u α ∇ α ) m B µ ...µ n = ( ∂ τ ) m B µ ...µ n , (2.41)where P µν = g µν − u µ u ν is the u -orthogonal projector on the worldline. The effective action encodingthe linear response are S worldlinetidal | linear = Z dτ ∞ X n =2 ∞ X l =0 µ ( n,l ) ( E ( l ) µ ··· µ n E ( l ) µ ··· µ n + σ ( n,l ) B ( l ) µ ··· µ n B ( l ) µ ··· µ n ) . (2.42)Note that here we use a different normalization than Ref. [28], the relation between our coefficientsis µ ( n,l )BDG = 2 l ! µ ( n,l ) and σ ( n,l )BDG = 2( l + 1)! σ ( n,l ) . The non-linear response is captured by S worldlinetidal (cid:12)(cid:12)(cid:12) non-linear = Z dτ ∞ X n =2 ρ ( n ) e E µ µ E µ µ · · · E µ n µ + ρ ( n ) m B µ µ B µ µ · · · B µ n µ ) + · · · . (2.43)Thus, for a particle of mass m i described by the scalar field φ i , the correspondence betweenworldline operators and QFT Lagrangian operators is Z dτ E ( l ) µ ...µ n E ( l ) µ ...µ n ←→ m i Z d x √− gφ i ˆ E ( l ) µ ...µ n ˆ E ( l ) µ ...µ n φ i , (2.44) Z dτ B ( l ) µ ...µ n B ( l ) µ ...µ n ←→ m i Z d x √− gφ i ˆ B ( l ) µ ...µ n ˆ B ( l ) µ ...µ n φ i . (2.45)The normalization of the QFT operators is fixed such that their four-point matrix elements in theclassical limit reproduce the expectation value of the worldline operators, provided that the normal-ization of the asymptotic states is the same for both of them, i.e. it is a nonrelativistic normalizationfor the QFT states. One may similarly construct a correspondence between worldline and QFToperators with more factors of the Riemann tensor. For more details about the correspondencebetween QFT amplitudes and worldline matrix elements see e.g. Ref. [46]. In any fixed dimension, the operators described above satisfy relations stemming from their finitenumber of components ; thus they give an overcomplete description of the physics of extendedbodies.One class of relations follows from the the electric and magnetic fields being tensors of finiterank. Naively they have rank four, but because E µν u ν = B µν u ν = 0 their rank is lowered to three. In a different context these relations are known as evanescent operators which are operators whose matrixelements vanish in four-dimensions but not in general dimension [47]. E µν and B µν are the covariant versionsof the purely spatial E ij , B ij in the rest frame. The simplest relation following from the finitenessof the ranks of E and B is E [ µ µ E µ µ E µ µ E µ ] µ = 0 , (2.46)which, together with the tracelessness of E , implies that E = 1 / E ) . More generally, relationscan be found which involve mixed powers of the electric and magnetic fields. For operators with noderivatives all such relations can be generated by evaluating the following determinant as a formalpower series det[1 + t ( E + rB )] = ∞ X i =2 i X j =0 R i,j t i r j . (2.47)The rank-three property of an arbitrary combination of E and B implies that R i ≥ ,j = 0. A sampleof such relations is2 R , = ( E ) − E ) = 0 , R , = 2( EB ) + ( B )( E ) − EBEB ) − E B ) = 0 , R , = ( E ) −
56 ( E )( E ) = 0 , R , = 6( E BEB ) + 6( E B ) − ( B )( E ) − E )( EB ) = 0 , R , = 2( EB ) − ( B )( EB ) = 0 , (2.48)as well as the ones that follow by interchanging E and B . Here the round parenthesis denote thematrix trace, ( O ) ≡ Tr[ O ] . (2.49)Recursively solving them implies that any operator of the form ( E n ≥ ) can be written as a polyno-mial in E and E as follows( E n ) = n X p +3 q = n p q Γ( p + q )Γ( p + 1)Γ( q + 1) ( E ) p ( E ) q . (2.50)A similar relation holds for ( B n ), while ( B n +1 ) = 0 in a parity-invariant theory such as GR.Another class of relations follows from the vanishing of the Gram determinants of any five ormore four-momenta. They imply that certain terms in the power series expansion of susceptibilitiesare not linearly independent. For instance,det( v i · v j ) = 0 with v i ⊂ { p , p , q , q , q } . (2.51)A final class of relations, which we will not detail any further, follows from the over-antisymmetrizationof indices of both derivatives and E or B .An exhaustive enumeration of the E - and B -type operators was carried out in Ref. [30], usingHilbert series techniques [48], which automatically eliminate the redundancies described here. Incontrast, we will not make an attempt to eliminate all redundant operators, but rather use theirrelations as a check on our framework and calculations.14 ℓ ℓ Figure 1:
The generalized cut for leading-order contributions to E - or B -type tidal operators. Each blobis an on-shell amplitude, which in this case is local. Each exposed line is taken to be on shell and the blobsrepresent tree amplitudes. The dark blob contains an insertion of an E - or B -type higher-dimensionoperator with an arbitrary number of additional derivatives. The external momenta are all outgoing andthe arrows indicated the direction of graviton momenta. E and B tidal effects In this section we discuss the leading-order contribution of the two-graviton tidal operators con-structed in Section 2. The analysis parallels to some extent that of Ref. [30], with the main differencebeing the choice of operator basis. Our choice aligns with the worldline approach [5, 28] makingit straightforward to compare Love numbers. We also evaluate all integrals providing a proof ofthe results with arbitrary numbers of derivatives. Here we work in an amplitudes-based approachfollowing Refs. [11–13, 21].
The first task is to write down a scattering amplitude from which classical scattering anglesand Hamiltonians can be extracted. To obtain the integrand we use the generalized unitaritymethod [22]. In this method, the integrand is constructed from the generalized unitarity cut whichwe define to be
C ≡ X states M tree(1) M tree(2) M tree(3) · · · M tree( m ) , (3.1)where the M tree( i ) are tree amplitudes, some of which can have operator insertion. As a simpleexample, Fig. 1 displays the unitarity cut containing the leading-order effect of an R tidal operator.In general, the cuts that can contribute to the conservative classical Hamiltonian satisfy somesimple rules. The first is that generalized unitarity cuts must separate the two matter lines toopposite sides of a cut, which follows from the fact we are interested only in long-range interactions.Another general rule is that every independent loop must have at least one cut matter line, so theenergy is restricted to a matter residue. Any contribution with a graviton propagator attached tothe same matter line also does not contribute to the conservative classical part. Further details arefound in Ref. [13].In constructing the amplitude integrand we may immediately expand in soft-graviton momenta,since each power of graviton momentum effectively carries an additional power of ~ and is quantum15uppressed. This expansion can be carries out either on at the level of the input tree amplitudes orafter assembling the cuts. The order to which a give term needs to be expanded is dictated by simplecounting rules. Terms with too high a scaling in the graviton momenta are dropped. For example,at one-loop for the case without tidal or other higher-dimension operators this implies that any termin a diagram numerator with more than a single power of loop momentum in the numerators yieldsonly quantum-mechanical contributions; some terms require fewer loop-momentum factors. In thepresence of higher-dimension operators, the leading classical contributions can have higher powersof loop momentum dictated simply by the number of extra derivatives in the operator comparedto to the usual two derivative minimal coupling; the extra implicit powers of ~ are made up by thecoefficient so the entire expression corresponds to a classical result.In general to sew the trees together into generalized cuts one should use physical-state projectorswhich depend on null reference momenta P µνρσ = X states ε µν ( − p ) ε ρσ ( p ) = 12 (cid:18) P µρ P νσ + P µρ P νσ (cid:19) − D − P µν P ρσ , (3.2)where P µρ = η µρ − ( n µ p ρ + n ρ p µ ) / ( n · p ) and n µ is the null reference momentum. However, thereference momenta will drop out if the seed amplitudes are manifestly transverse. In fact, one canalways arrange for such terms to automatically drop out [49].Alternatively, we can also use four-dimensional helicity states to sew gravitons across unitaritycuts. In general, some caution is required in the presence of infrared or ultraviolet singularites, al-though at least through third post-Minkowskian order helicity methods have been shown to correctlycapture all contributions [13]. For cases without non-trivial infrared or ultraviolet divergences ,we can straightforwardly apply four-dimensional methods. In our cases, the above D -dimensionalsewing is simple enough so we will not use four-dimensional helicities here.Finally, the information from multiple generalized cuts must be merged into a single expression.This can either be accomplished at the level of the integrand or after integration. For leading tidalcoefficients, effectively only a single cut contributes, so merging information from the cuts is trivial. The on-shell amplitudes in the unitarity cut simplifies dramatically if we are only interested atleading classical order. Because there is no enhancement from iteration, any terms beyond theleading order in graviton momenta are quantum mechanical and can thus be ignored. For example,consider a three-point scalar-graviton-scalar amplitude at tree level M ( φ ( p ) , h ( ℓ ) , φ ( p ′ )) = − κp µ p ν ε µν ( ℓ ) , (3.3) There are ultraviolet divergence at even loop orders that local in momentum transfer q , e.g. in the 3PM scat-tering [12, 13]. However, these are irrelevant for long-range dynamics because they can be absorbed by a contactinteraction. κ is related to Newton’s constant by κ = 32 πG . For any of the three-point amplitudesinserted in Fig. 1, we can replace the scalar momenta p by the external momentum p at leadingclassical order. Physically this implies that we ignore all back reaction on the particle 2, so allthree-point amplitudes in Fig. 1 are approximately the same.For the amplitude with higher-dimension operator, it suffices to use linearized version of thecurvature operators. Expanding the metric in the usual way, g µν = η µν + κh µν , we find the Weyltensor to leading order is C µνρσ = − κ∂ [ µ | ∂ [ ρ h σ ] | ν ] + O ( κ , (cid:3) h ) . (3.4)In deriving this expression we have also dropped terms proportional to the equations of motion forthe graviton; this is because they do not contribute to the on-shell matrix elements necessary forthe evaluation of the leading-order amplitude. The linearized Weyl tensor in momentum space thenreads C lin µνρσ ( ℓ ) ≡ κ ℓ µ ℓ ρ ε ( ℓ ) νσ − ℓ ν ℓ ρ ε ( ℓ ) µσ − ℓ µ ℓ σ ε ( ℓ ) νρ + ℓ ν ℓ σ ε ( ℓ ) µρ ] . (3.5)The linearized Weyl tensor can be written a form that manifests the double copy in terms of twogauge-theory field strengths C lin µνρσ ( ℓ ) = κ F lin µν ( ℓ ) F lin ρσ ( ℓ ) , (3.6)where F lin µνρσ ( ℓ ) ≡ ℓ µ ε ( ℓ ) ν − ℓ ν ε ( ℓ ) µ , (3.7)and we identify the graviton polarization tensor as ε ( ℓ ) νσ = ε ( ℓ ) ν ε ( ℓ ) σ . This simple example of adouble-copy relation [33, 34], which is trivial at the linearized level, then implies that the leading-order amplitudes for tidal operators display double-copy relations. The gauge invariance is manifest.To make the gravitational coupling manifest in all equations, we will extract all factors of κ from the building blocks of amplitudes. The linearized electric and magnetic components of thelinearized Weyl tensor (3.5) follow from Eq. (2.30) E µ µ ( ℓ, p ) = 12 m h ℓ µ ℓ µ ( p · ε ( ℓ ) · p ) − ( p · ℓ ) ( ℓ µ ε ( ℓ ) µ ρ p ρ + ℓ µ ε ( ℓ ) µ ρ p ρ ) + ε ( ℓ ) µ µ ( p · ℓ ) i , (3.8) B µ µ ( ℓ, p ) = 14 m ǫ αβγµ h ( p · ℓ ) ( ℓ α ε ( ℓ ) βµ − ℓ β ε ( ℓ ) αµ ) + ℓ β ℓ µ ( p · ε ( ℓ )) α − ℓ α ℓ µ ( p · ε ( ℓ )) β i , (3.9)where the particle momentum and its four-velocity are related in the usual way, p µ = mu µ . It isthen straightforward to assemble the amplitude with insertions of a higher-dimension operator fromabove formulae.In general to sew trees into generalized cuts one should use physical-state projectors whichdepend on null reference momenta. However, for the leading-order contributions that we will mostlybe studying here, the terms containing dependence on the reference momentum automatically drop17ut because they are contracted into manifestly gauge-invariant (transverse) quantities . Effectively,we can use the numerator of the de Donder gauge propagator, P µνρσ = X states ε µν ( − p ) ε ρσ ( p ) → (cid:18) η µρ η νσ + η µρ η νσ (cid:19) − D − η µν η ρσ , (3.10)to sew gravitons across cuts. Combining the projector with the three-point amplitude in Eq. (3.3)at leading classical order, effectively turns the graviton polarization tensors of the higher-dimensionoperator into ε µν ( ℓ ) → T µν ( p ) = p ,µ p ,ν − m D s − η µν ! . (3.11)Crucially the result is independent of the loop momentum, implying that the sewing automaticallyimposes Bose symmetry for the gravitons of the higher-dimension operator. As we will outlinein Sec. 4, this no longer holds beyond leading order where back-reaction becomes important. Forexample, at next-to-leading order pairs of the stress tensor in Eq. (3.3) can source a single graviton,acting as a sort of “impurity”, which may be interpreted as the first correction to the gravitationalfield of a free particle towards that of a Schwarzschild black hole.The discussion above can be extended to include the leading-order scattering of scalars deformedby higher-dimension operators off higher-spin particles described the Lagrangian in Ref. [23]. Fora generic spinning body the stress tensor is M ( φ s ( p ) , h ( ℓ ) , φ s ( p ′ )) = − κ V µν ( φ s ( p ) , h ( ℓ ) , φ s ( p ′ )) ε µν ( ℓ ) , (3.12) V µν ( φ s ( p ) , h ( ℓ ) , φ s ( p ′ )) = p µ p ν ∞ X n =0 C ES n (2 n )! ℓ · S ( p ) m ! n − iℓ ρ p ( µ S ( p ) ν ) ρ ∞ X n =0 C BS n +1 (2 n + 1)! ℓ · S ( p ) m ! n , where ℓ is the graviton momentum and S ( p ) µ and S ( p ) µν are the covariant spin vector and spintensor, related by S µν ( p ) = − m ǫ µνγδ p γ S δ ( p ) , S µ ( p ) = − m ǫ µβγδ p β S γδ ( p ) , (3.13)and we recall that in the classical limit ℓ · S ( p ) /m = O (1).For the Kerr black hole the stress tensor, originally found in Ref. [42] from different considera-tions, is obtained by setting C ES n = C BS n = 1 and has the closed-form expression M Kerr3 ( φ s ( p ) , h ( ℓ ) , φ s ( p ′ )) = − κ exp( ia ∗ ℓ ) ( µρ p ν ) p ρ ε µν ( ℓ ) , (3.14)where a µ = 12 p ǫ µνρσ p ν S ρσ ( p ) , ( a ∗ ℓ ) µν ≡ ǫ µνρσ a ρ ℓ σ . (3.15) In fact, one can always arrange for such terms to automatically drop out [49]. ε µν ( ℓ ) → T Kerr µν ( ℓ, p ) = exp( ia ∗ ℓ ) ( αρ p β ) p ρ (cid:18) δ µα δ νβ − η αβ η µν D s − (cid:19) . (3.16)We note that only the terms with an even number of spin vectors, in general governed by thecoefficients C ES n , contribute to the trace part of this replacement. To shorten the ensuing equations,in the following we will use the replacement ǫ µν ( ℓ ) → T µν gen ( ℓ, p ) = p µ p ν − m D s − η µν ! A ( ℓ ) − i ℓ ρ ( p µ S νρ ( p ) + p ν S µρ ( p )) B ( ℓ ) , (3.17)where A ( ℓ ) and B ( ℓ ) can be read off Eqs. (3.12) and (3.16). Before discussing the leading-order effects of the most general tidal operators introduced in Sec. 2, wediscuss here the simpler case of operators E ( m ) µ µ , corresponding to the multipoles of the gravitationalfield of the quadrupole operator E µν .The construction of the relevant four-point matrix element of the operator φE ( m ) µ µ E ( m ) µ µ φ ,corresponding to the darker blob in Fig. 1, is straightforward. The matrix element is M E l, ( h ( ℓ ) , h ( ℓ ) , φ ( p ) , φ ( p )) = 2 κ m (cid:16) D E l, ( p , ℓ , p , ℓ ) + D E l, ( p , ℓ , p , ℓ ) (cid:17) ,D E l, ( p , ℓ , p , ℓ ) = (cid:18) im (cid:19) l ( p · ℓ ) l ( p · ℓ ) l E µ µ ( ℓ , p ) E µ µ ( ℓ , p ) . (3.18)As noted earlier, because tidal operators are gauge invariant and constructed out of Weyl tensors,this matrix element obeys the transversality conditions for the two gravitons. Thus, their contribu-tion to generalized unitarity cut in Fig. 1 automatically accounts for the physical-state projection.The sewing is then simply given by the replacement in Eq. (3.11). To leading order in soft expansionwe can also replace all p · ℓ = − p · ℓ + O ( q ).The resulting amplitude is M E l, ( p , q ) = iκ Z d D ℓ (2 π ) D M E l, ( h ( ℓ ) , h ( ℓ ) , φ ( p ) , φ ( p )) (cid:12)(cid:12)(cid:12) ε µν ( ℓ i ) → T µν ( p ) ℓ (( ℓ − p ) − m )( q − ℓ ) = 4 i m κ Z d D ℓ (2 π ) D ( u · ℓ ) l E µ µ ( ℓ , p ) E µ µ ( ℓ , p ) (cid:12)(cid:12)(cid:12) ε µν ( ℓ i ) → T µν ( p ) ℓ (( ℓ − p ) − m )( q − ℓ ) , (3.19)where the numerator is given more explicitly by E µ µ ( ℓ , p ) E µ µ ( ℓ , p ) (cid:12)(cid:12)(cid:12) ε µν ( ℓ i ) → T µν ( p ) (3.20)19 18 m (cid:20) ( u · ℓ ) (( u · ℓ ) + 12 q ) − σ q ( u · ℓ ) + 18 q (1 − σ ) (cid:21) + O ( q ) . Further expanding the amplitude in the soft limit leads to M E l, ( p , q ) = 64 iπ G | q | l m m ((1 − σ ) I l + 4( − σ ) I l ) + 8 I l ) ) , where I l are triangle integrals I l = Z d D ℓ (2 π ) D | q | − l +1 ( ℓ · u ) l ℓ ( − ℓ · u )( ℓ − q ) , (3.21)which must be evaluated in the potential region. The results of these integrals were conjectured inRef. [30]. Here we present the proof, by going to the frame in which particle 2 is at rest u µ = − ( σ, , , √ σ − , u µ = − (1 , , , , q µ = (0 , q ) = (0 , q x , q y , q z ) , (3.22)under which ℓ iµ = ( ℓ i , ℓ i ) = ( ℓ i , ℓ xi , ℓ yi , ℓ zi ). Note that since q z = q · ˆ z = O ( q ) by on shell conditions,we can treat q z ≈ I l = ( σ − l Z d D ℓ (2 π ) D | q | − l +1 ( ℓ z ) l (2 ℓ ) ℓ ( ℓ − q ) = i ( σ − l l +1 (4 π ) ( D − / Z d D − ℓ π ( D − / | q | − l +1 (2 ℓ z ) l ℓ ( ℓ − q ) , (3.23)where in the second equality we have evaluated the residue of the energy pole with a symmetry factor1 / Z d D − ℓ π ( D − / ( q ) a + b + c − ( ℓ − i a [( ℓ − q ) − i b (2 ℓ z − i c (3.24)= e iπc | q | − ǫ Γ (cid:16) c (cid:17) Γ (cid:16) − a − c − ǫ (cid:17) Γ (cid:16) − b − c − ǫ (cid:17) Γ (cid:16) a + b + c + ǫ − (cid:17) a )Γ( b )Γ( c )Γ(3 − a − b − c − ǫ ) , for q · ˆ z = 0 which is valid for leading order in the classical limit. The result is I l = − i ( σ − l l +2 − ǫ (4 π ) / − ǫ | q | − ǫ Γ (cid:16) − ǫ (cid:17) Γ (cid:16) + ǫ (cid:17) Γ (cid:16) − l (cid:17) Γ (1 − ǫ + l ) . (3.25)Using the result for these integrals with ǫ = 0 the amplitude is M E l, ( p , q ) = | q | l M E l, ( p ) , (3.26) M E l, ( p ) = G m m ( − l π / Γ( + l )2 l ) Γ(3 + l ) (3.27) × ( σ − l (11 + 4 l (3 + l ) − l ) σ + (5 + 2 l )(7 + 2 l ) σ ) . The corresponding potential and eikonal phase are V E l, ( p , r ) = − E E | r | l +6 l Γ(3 + l ) π / Γ( − − l ) M E l, ( p ) , (3.28)20 E l, ( p , b ) = 14 m m √ σ − | b | l +5 l Γ( + l ) π Γ( − − l ) M E l, ( p ) . (3.29)It is not difficult to see that, for l = 0 and l = 1, eq. (3.29) reproduces the expectation values ofthe operators E and ( ˙ E ) evaluated in Ref. [28].The calculation above can be easily repeated for the operator B ( l ) µν B µν ( l ) ; it amounts to replacingin Eq. (3.19) E with B given in Eq. (3.9). The resulting amplitude, potential and eikonal phase are: M B l, ( p , q ) = | q | l M B l, ( p ) , (3.30) M B l, ( p ) = G m m ( − l π / Γ( + l )2 l +1) Γ(3 + l ) (5 + 2 l )( σ − l +1 (1 + 2 l + (7 + 2 l ) σ ) , (3.31) V B l, ( p , r ) = − E E | r | l +6 l Γ(3 + l ) π / Γ( − − l ) M B l, ( p ) , (3.32) δ B l, ( p , b ) = 14 m m √ σ − | b | l +5 l Γ( + l ) π Γ( − − l ) M B l, ( p ) . (3.33)Similarly to eq. (3.29), the eikonal phase above evaluated on l = 0 and l = 1 reproduces theexpectation values of the operators B and ( ˙ B ) found in [28]. Alternatively, the calculation can be done in position space, more specifically in the rest frame ofparticle 2 as in Eq. (3.22). This approach will provide a simple way to generalize the analysis beyondone loop. There are two key observations here. First, the amplitude with C operator insertion inEq. (3.18) factorizes into a product of the multipole expansions of electric or magnetic tensors M E l, ( h ( ℓ ) , h ( ℓ ) , φ ( p ) , φ ( p )) = 4 m κ (cid:18) im (cid:19) l (( p · ℓ ) l E µ µ ( ℓ , p ))(( p · ℓ ) l E µ µ ( ℓ , p ))+ O ( q l +4 ) , (3.34)where we have applied the classical limit p = − p + O ( q ) to Eq. (3.18). Second, in the potentialregion, we can integrated out graviton energy component by picking up residue from the matterpropagator [11, 13]. This sets ℓ = ℓ = 0 and implies the graviton momenta ℓ , ℓ are purelyspatial. To exploit the factorization at the integrand level, we further Fourier transform the spatial q in Eq. (3.19) to position space M E l, ( p , r ) ≡ Z d D − q (2 π ) D − e − i r · q f M E l, ( p , q ) (3.35)= κ m Y i =1 Z d D − ℓ i (2 π ) D − e − i r · ℓ i ℓ i M E l, ( h ( ℓ ) , h ( ℓ ) , φ ( p ) , φ ( p )) (cid:12)(cid:12)(cid:12) ε µν ( ℓ i ) → T µν ( p ) . The Fourier transform acts on the amplitude with generic off-shell q , which is three dimensional. We use f M ( p , q )to denote amplitude with off-shell q . ℓ , ℓ factorizes and each of them canbe treated as an independent variable. Together with the factorization in Eq. (3.34), the Fouriertransform acts on individual electric tensor E µ µ ( ℓ i , p ). We define E µν ( r , p ) ≡ Z d D − ℓ i (2 π ) D − e − i r · ℓ i ℓ i E µν ( ℓ i , p ) (cid:12)(cid:12)(cid:12) ε ρσ ( ℓ ) → T ρσ ( p ) = − m π | r | (cid:20) (cid:16) r + 2( σ − z (cid:17) u µ u ν − σ r ( u µ u ν + u µ u ν ) + 2 r u µ u ν + 3(2 σ − r µ r ν − σ √ σ − z ( u µ r ν + u µ r ν ) + 3 √ σ − z ( u µ r ν + u µ r ν )+ ((3 σ − r − σ − z ) η µν (cid:21) , (3.36)where r µ = (0 , r ) = (0 , x, y, z ) in the frame of Eq. (3.22) as the electric field sourced by p inposition space. The Fourier transform of scalar-graviton amplitude (with the graviton propagators)is then M E l, ( h , h , φ ( p ) , φ ( p ) | r ) ≡ Y i =1 Z d D − ℓ i (2Π) D − e − i r · ℓ i ℓ i M E l, ( h ( ℓ ) , h ( ℓ ) , φ ( p ) , φ ( p ))= M E l, ( h ( ℓ ) , h ( ℓ ) , φ ( p ) , φ ( p ) |E µ µ ( ℓ j , p ) → E µ µ ( r j , p ) , ℓ i → i ∇ j ) (cid:12)(cid:12)(cid:12) r j → r , (3.37)where any loop momentum ℓ j is replaced with the gradient on the position r j of the electric field E µ µ ( r j , p ) and all r j are identified with r . The two-scalar scattering amplitude in position spacethen has a simple form M E l, ( p , r ) = κ m M E l, ( h , h , φ ( p ) , φ ( p ) | r )= κ m m ( σ − l h ( ˆ z · ∇ ) l E µ µ ( r , p ) i , (3.38)where in the second line we plug in the result in Eq. (3.34), apply the replacement in Eq. (3.37)and ˆ z is the unit vector along z direction.The position-space result is generally not isotropic; namely, it could depend on ˆ z · r . To make theresult isotropic, we go back to momentum space and impose the on-shell condition ˆ z · q = O ( q ) ≃ M O ( p , q ) = Z d D − r e + i r · q M O ( p , r ) (cid:12)(cid:12)(cid:12) ˆ z · q =0 . (3.39)Since the result only depends on the covariant variables σ and q = − q , it can be promoted to anyother frame. All Fourier-transforms that appear in this calculation are of the form Z d D − r e i r · q ( ˆ z · r ) s r h = ( − s/ π D/ h − s − D +1 | q | h − s − D +1 sin( π ( D − − h )) Γ( (1 + s ))Γ( h )Γ(1 + ( h − s − D + 1)) , (3.40)for some exponents h and integer s . The isotropic potential then follows from Eq. (2.10).22rom the position-space amplitude we can directly obtain the eikonal phase, although it can becalculated easily once we have the amplitude M O ( p , b ). To see this, we simply invert the amplitudein terms of Eq. (2.5) and plug it into Eq. (2.6) δ O ( p , b ) = 14 m m √ σ − Z d D − q (2 π ) D − e − i b · q Z d D − r e i r · q M O ( p , r ) (cid:12)(cid:12)(cid:12)(cid:12) q =( q x ,q y , = 14 m m √ σ − Z ∞−∞ dz M O ( p , r = ( b , z )) , (3.41)where we use b = ( b x , b y ,
0) and r = ( x, y, z ). Since we are only interested in the leading order,the particle trajectory can be treated as a straight line. In the frame where particle 2 is rest at theorigin, the position of particle 1 is x µ = ( t, r ) = b µ + u µ τ = τ ( σ, b x , b y , √ σ − δ O ( p , b ) = 14 m m Z ∞−∞ dτ M O ( p , r ( τ )) . (3.42)So the eikonal phase can be obtained straightforwardly from M O ( p , r ( τ )). This is expected becausethe eikonal phase is proportional to the worldline action integrated over a straight line. Our approachhere offers a derivation from purely scattering-amplitudes perspective.The advantage of position-space approach is that it is very general. The discussion above appliesto contribution of any tidal operator at its leading classical order. The only integrals needed, toany loop order, are in Eq. (3.40). We will discuss and illustrate this point in more detail in Sec. 4.The discussion above can be generalized easily to the case with magnetic operators. The position-space magnetic component of the linearized Weyl tensor, contracted with a point-particle stresstensor, is B µν ( r , p ) ≡ Z d D − ℓ i (2 π ) D − e − i r · ℓ i ℓ i B µν ( ℓ i , p ) (cid:12)(cid:12)(cid:12) ε ρσ ( ℓ ) → T ρσ ( p ) . (3.43)We have the scalar-graviton amplitude in position space M B l, ( h ( ℓ ) , h ( ℓ ) , φ ( p ) , φ ( p ) | r ) ≡ Y i =1 Z d D − ℓ i (2Π) D − e − i r · ℓ i ℓ i M B l, ( h ( ℓ ) , h ( ℓ ) , φ ( p ) , φ ( p ))= M B l, ( h ( ℓ ) , h ( ℓ ) , φ ( p ) , φ ( p ) |B µ µ ( ℓ j , p ) → B µ µ ( r j , p ) , ℓ i → i ∇ j ) (cid:12)(cid:12)(cid:12) r j → r . (3.44)Again we identify all r j in the end with r . The position-space amplitude is then M B l, ( r ) = 1 m (cid:18) κ (cid:19) M B l, ( h , h , φ ( p ) , φ ( p ) | r ) . (3.45)Let us comment on an interesting relation between electric and magnetic operators. In positionspace we find E µν ( r , p ) E µν ( r , p ) = 3 m π | r | h σ − r − z )( σ r − ( σ − z ) + r i , (3.46)23 µν ( r , p ) B µν ( r , p ) = 9 m π | r | ( σ − r − z )( σ r − ( σ − z ) . (3.47)The two operators are almost identical. The difference between the two is independent of σ which issub-sub-leading in the high-energy limit σ ≫
1. As explained in Ref. [28], this is expected becausethe difference is proportional to Weyl tensor squared which is independent of σ . This behavior hasalso been observed at the next-to-leading order in Ref. [29]. Following the example discussed in detail in the previous sections, we proceed to evaluate theamplitudes and the corresponding eikonal phases with one insertion of the generic tidal operators φE ( l ) µ ...µ n E ( l ) µ ...µ n φ and φB ( l ) µ ...µ n B ( l ) µ ...µ n φ . As already mentioned for operators with n = 2, wemay choose without loss of generality, the two E and B factors to have equal upper index.The calculations for the two operators are parallel. For this reason, in the common part we willcollectively denote E or B by X , and specialize at them at the end. Thus, to leading order in κ ,the momentum space expressions of ˆ E ( l ) and ˆ B ( l ) defined in Eq. (2.37) are X ( l ) µ µ ...µ n = i l +( n − (cid:18) im (cid:19) l ( p · ℓ ) l Sym µ ...µ n [ P ν µ ( p ) ℓ ν . . . P ν n µ n ( p ) ℓ ν n X ( ℓ, p ) µ µ ] + O ( κ ) , (3.48)where P ν i µ i are the momentum space form of the projectors in Eq. (2.38) and X µ µ ( ℓ, p ) being givenby E µ µ and B µ µ in Eqs. (3.8)-(3.9) for the two operators, respectively. The symmetrization overthe indices µ , . . . , µ n includes division by the number of terms. In the expression above ℓ is thegraviton momentum, p is the scalar momentum and ε ( ℓ ) in the explicit expressions of E µ µ and B µ µ is the graviton polarization tensor.The product of two linearized X ( l ) µ ...µ n with different graviton momenta ℓ and ℓ , and contractedas in Eqs. (2.44) and (2.45), contains three different structures: (1) all projectors are contracted witheach other, (2) all but one projector are contracted with each other and (3) all but two projectorsare contracted with each other. The four-point matrix element of the operator φX ( l ) µ ...µ n X ( l ) µ ...µ n φ needed for the construction of the four-scalar amplitude is M X l,n ( h ( ℓ ) , h ( ℓ ) , φ ( p ) , φ ( p )) = 2 κ m (cid:16) D X l,n ( p , ℓ , p , ℓ ) + D X l,n ( p , ℓ , p , ℓ ) (cid:17) , (3.49)where D X l,n ( p , ℓ , p , ℓ ) = i n − i l ( − l n − n ! ( u · ℓ ) l (cid:20) ( ℓ · P ( p ) · P ( p ) · ℓ ) n − Π X ( p , ℓ , p , ℓ )+ 2( n − ℓ · P ( p ) · P ( p ) · ℓ ) n − Π X ( p , ℓ , p , ℓ ) (3.50)+ 12 ( n − n − ℓ · P ( p ) · P ( p ) · ℓ ) n − Π X ( p , ℓ , p , ℓ ) (cid:21) . The three factors Π X ( p , ℓ , p , ℓ ) are given byΠ X ( p , ℓ , p , ℓ ) = X µ µ ( ℓ , p ) X µ µ ( ℓ , p ) , (3.51)24 X ( p , ℓ , p , ℓ ) = ℓ · P ( p ) · X ( ℓ , p ) · X ( ℓ , p ) · P ( p ) · ℓ , Π X ( p , ℓ , p , ℓ ) = ℓ · P ( p ) · X ( ℓ , p ) · P ( p ) · ℓ ℓ · P ( p ) · X ( ℓ , p ) · P ( p ) · ℓ . To the order we are interested in we may freely replace p → − p , since the difference is of subleadingorder in the expansion in small transferred momentum. For n = 2, the second and third line vanishand, for X ≡ E , we recover the four-point matrix element of the operator φE ( l ) µ µ E ( l ) µ µ φ given inEqs. (3.18).Sewing this matrix element with two three-point scalar-graviton amplitudes in Eq. (3.3) usingthe rule (3.11) leads to M X l,n ( p , q ) = 8(8 πG ) i n − m m n − n ! × (cid:20) M ( l ) n (Π X ) + 2( n − M ( l ) n (Π X ) + 12 ( n − n − M ( l ) n (Π X ) (cid:21) , (3.52) M l,n (Π Xk ) = Z d D ℓ (2 π ) D ( u · ℓ ) l (( u · ℓ ) + q ) n − ℓ (( ℓ − p ) − m )( ℓ − q ) q ( u · ℓ ) + q ! k − M (Π Xk ) , (3.53)where k = 1 , , M (Π E k ) and M (Π B k ) have the same general structure: M (Π Xi ) = A Xi ( u · ℓ ) (( u · ℓ ) + q ) + B Xi q ( u · ℓ ) + C Xi q (( u · ℓ ) + q ) + D Xi q (1 − σ ) . (3.54)The coefficients A, . . . , D for the amplitude with an insertion of an electric-type operator are givenby A E = 1 , B E = − σ , C E = 0 , D E = 18 ,A E = 12 , B E = 18 (1 − σ ) , C E = 0 , D E = 116 ,A E = 12 , B E = − σ , C E = 0 D E = 132 , (3.55)while those for the amplitude with an insertion of the “magnetic” operator are A B = 4 , B B = (1 − σ ) , C B = − , D B = 12 ,A B = 2 , B B = − σ , C B = − , D B = 14 ,A B = 0 B B = 14 (1 − σ ) , C B = − , D B = 18 . (3.56)In the soft limit, all integrals in the amplitude (3.52) are of the type I n, l = Z d D ℓ (2 π ) D | q | − n + l ) ( u · ℓ ) l (( u · ℓ ) + q ) n ℓ ( − u · ℓ )( ℓ − q ) ; (3.57)25hey can be evaluated in terms of the triangle integrals (3.21) found in Sec. 3.2: I n, l = n X u =1 C un (cid:18) − (cid:19) n − u I l + u ) = − i
32 ( − ) n + l l + n Γ( l + ) √ π Γ( l + 1) ( σ − m F (cid:16) + l, − n, l, (1 − σ ) (cid:17) , (3.58)where C un are binomial coefficients. In terms of these integrals, the three terms M ( l ) n (Π Xk ) makingup the complete amplitude are M l,n (Π Xk ) = A Xk I n +1 − k, l +1) + B Xk I n − k, l +1) + q C Xk I n +1 − k, l + (1 − σ ) D Xk I n − k, l ) , (3.59)with coefficients A, . . . , D given in (3.55) and (3.56). Using these building blocks it is then straight-forward to assemble the amplitudes M E l,n ( p , q ) and M B l,n ( p , q ) in Eq. (3.52). The eikonal phasesfollows by Fourier-transforming them to impact parameter space and including the appropriate fac-tors as in Eq. (3.29). Choosing n = 2 we recover the amplitudes in Eqs. (3.26) and (3.30). Last, thetwo-body potential and the eikonal phase are related to the leading-order amplitude in the usualway as in Eqs. (2.5) and (2.6).The position-space analysis also works in this case. In fact. for this approach it is convenientto sidestep the encoding of the tidal effects in a particular basis of higher-dimensions operators andwork directly with the susceptibility χ . From this perspective the matrix element of an arbitrarytidal operator quadratic in the electric field is M χ EE ( h ( ℓ ) , h ( ℓ ) , φ ( p ) , φ ( p )) = 2 m κ χ µ ν µ ν ( u · ℓ , ˆ ℓ ; u · ℓ , ˆ ℓ ) E µ ν ( ℓ , p ) E µ ν ( ℓ , p )+ ( p ↔ p , u ↔ u ) . (3.60)Bose symmetry guarantees that this is symmetric in the two gravitons, so the manipulations inthe previous section can be repeated here. The Fourier transform of the one-loop integrand, aftersewing the unitarity cut and evaluating the energy integral, is M χ EE ( p , r ) = κ m Z d D − ℓ (2 π ) D − e − i r · ℓ ℓ Z d D − ℓ (2 π ) D − e − i r · ℓ ℓ M χ EE ( h ( ℓ ) , h ( ℓ ) , φ ( p ) , φ ( p )) (cid:12)(cid:12)(cid:12) ε µν ( ℓ ) → T µν ( p ) (3.61)= m κ m h χ µ ν µ ν ( v ˆ z · i ∇ , ∇ ⊥ ; v ˆ z · i ∇ , ∇ ⊥ ) E µ ν ( r , p ) E µ ν ( r , p ) i r = r = r , where v = √ σ − ∇ ⊥ = ∇ − v ˆ z ( ˆ z · ∇ ), and we have introduced different positions, r i , for allthe gravitons. They are to be set equal after the derivatives are evaluated. As before, we can obtainthe isotropic potential by first generating the on-shell amplitude through Eq. (3.39) and Fouriertransforming back to the position space. The eikonal phase can either be obtained from Eq. (2.6)or directly from M χ EE ( p , r ) via Eq. (3.42). 26 .5 Adding spin It is not difficult to formally the calculation in the previous sections to include spin degrees offreedom for the particle with momentum p . It amounts to changing T µν ( p ) in Eqs. (3.19), (3.35),(3.52) and (3.60) with T Kerr µν ( p , l i ) in Eq. (3.16) or its general form defined from Eq. (3.12) andparametrized as in Eq. (3.17) and multiplying the resulting amplitude by the product of spin- S polarization tensors.With this replacement, the contraction of two electric-type tensors E µ µ ( ℓ i , p ) is E µ µ ( ℓ , p ) E µ µ ( ℓ , p ) (cid:12)(cid:12)(cid:12) ε µν ( ℓ i ) → T gen µν ( p ) (3.62)= 18 m A ( ℓ ) A ( ℓ )(8( ℓ · u ) + 4( ℓ · u ) q (1 − σ ) + q (1 − σ ) ) − i m A ( ℓ ) B ( ℓ ) q σ ( − ℓ · u ) + q ( − σ )) S [ u , q ]+ i m ( A ( ℓ ) B ( ℓ ) + A ( ℓ ) B ( ℓ )) ℓ · u σ (4( ℓ · u ) + q (1 − σ )) S [ ℓ, q ] − i m ( A ( ℓ ) B ( ℓ ) − A ( ℓ ) B ( ℓ )) q σ (4( ℓ · u ) + q (1 − σ )) S [ ℓ, u ]+ 12 m B ( ℓ ) B ( ℓ )( ℓ · u ) ( − ℓ · u ) + q σ ) S [ e µ , q ] S [ e µ , ℓ ]+ 12 m B ( ℓ ) B ( ℓ )( ℓ · u ) (2( ℓ · u ) − q σ ) S [ e µ , ℓ ] S [ e µ , ℓ ]+ m B ( ℓ ) B ( ℓ ) ℓ · u (( ℓ · u ) − q σ ) S [ ℓ, q ] S [ u , q ] − m B ( ℓ ) B ( ℓ ) q (( ℓ · u ) − q σ ) S [ ℓ, p ] S [ u , q ] − m B ( ℓ ) B ( ℓ )( ℓ · u ) σ S [ ℓ, q ] + 12 m B ( ℓ ) B ( ℓ ) q ( − ( ℓ · u ) + q σ ) S [ ℓ, u ] + O ( q ) , where ℓ = ℓ , ℓ = q − ℓ and S [ a, b ] ≡ S ( p ) µν a µ b ν , S [ e µ , a ] S [ e µ , b ] ≡ η µν S ( p ) µρ a ρ S ( p ) νσ b σ . (3.63)For vanishing spin, A ( ℓ i ) = 1 and B ( ℓ i ) = 0, only the first line of Eq. (3.62) survives and we recoverEq. (3.19). One may expand Eq. (3.62) to arbitrary order in spin. For example, to first nontrivialorder, which corresponds to inclusion of the spin-orbit interaction for particle 2, we find E µ µ ( ℓ , p ) E µ µ ( ℓ , p ) (cid:12)(cid:12)(cid:12) ε µν ( ℓ i ) → T gen µν ( p ) = E µ µ ( ℓ , p ) E µ µ ( ℓ , p ) (cid:12)(cid:12)(cid:12) ε µν ( ℓ i ) → T µν ( p ) (3.64)+ i C BS m σ (4 ℓ · u S [ ℓ, q ] + q S [ u , q ])(4( ℓ · u ) + q (1 − σ )) + O (( q · S ) ) , where the first term on the right-hand side is given by Eq. (3.20).It is straightforward, albeit tedious, to write out explicitly an integral representation of theamplitude by plugging in Eq. (3.62) in Eq. (3.19). We will refrain however from doing so, and27ather only comment on its structure. In addition to the integrals in Eq. (3.21), the spin dependenceintroduces also tensor integrals: I µ ...µ s l = Z d D ℓ (2 π ) D | q | − l − s +1 ℓ µ . . . ℓ µ s ( ℓ · u ) l ℓ ( − ℓ · u )( ℓ − q ) ; (3.65)they may be parametrized as a scalar integral I l [ w, s ] by contracting the free indices with an arbitraryvector w , from which the desired tensor integral is extracted by taking s derivatives. Note that,unlike the triangle integrals in Eq. (3.21), here the exponent l is not constrained to be even. Toleading order in spin only the vector integral is relevant. To this order, Eq. (3.64) becomes: M E l, ,S ( p ) ( p , q ) = ε · ε M E l, ( p , q ) (3.66)+ 128( − l C BS G π σ | q | l +3 (cid:18) S [ u , q ] (cid:16) ( − σ ) I l + 4 I l (cid:17) + 4 S [ e µ , q ] (cid:16) (1 − σ ) I µ l − I µ l (cid:17)(cid:19) m m ε · ε + O (( q · S ) ) . It is not difficult to evaluate in the usual way the vector integrals, by writing them as a linearcombination of u , u and q and solving for the coefficients in terms of the scalar triangle integralsin Eq. (3.21). Alternatively, one may re-evaluate the integrals in Eq. (3.21) by treating u , u and q as uncorrelated vectors, differentiate s times with respect to u and then impose u i = 1 , u i · q = 0.For the vector integrals we find I µ l +1 = − u µ − u µ yy − I l +2 . (3.67)Thus, the amplitude with the first spin-dependent term for particle 2 is M E l, ,S ( p ) ( φ ( p ) , φ ( p ) , φ ( p ) , φ ( p )) = ε · ε M E l, ( φ ( p ) , φ ( p ) , φ ( p ) , φ ( p )) (3.68) − C BS G π / Γ( + l )2 l − Γ(3 + l ) m σ ( − σ ) l ( − l ) σ ) | q | l S [ p , ( iq )] ε · ε + O (( q · S ) ) . To extract the two-body potential in terms of the rest-frame spin it is necessary to expand theproduct of polarization tensors to leading order in spin, as discussed in Ref. [23]. Using the relations ε · ε = − i ǫ rsk p r p s S k m ( m + E ( p )) + O (S q ) ! + O ( q ) ,ǫ µνρσ p µ p ν q ρ S iσ = ( E + E ) ( p × q ) · S i , (3.69)the amplitude becomes M E l, ,S ( p ) ( φ ( p ) , φ ( p ) , φ ( p ) , φ ( p )) (3.70)= M E l, ( p ) | q | l +3 + M E l, ( p ) m ( E + m ) + ( E + E ) M E l, , ( p ) | q | l +3 i ( p × q ) · S + O (( q · S ) ) , M E l, , is the coefficient of S [ p , ( iq )] in Eq. (3.68) and, as before, the bar indicates that all q dependence has been extracted. The two-body potential and the eikonal phase are then extracted bythree-dimensional and two-dimensional Fourier-transforms, in terms of their spinless counterpartsand the coefficient of the spin-dependent structure in the amplitude: V E l, ,S ( p , r ) = V E l, ( p , r ) − ( p × r ) · S E E | r | l +8 l Γ(4 + l ) π / Γ( − − l ) × M E l, ( p ) m ( E + m ) + ( E + E ) M E l, , ( p ) + O (( rS ) ) , (3.71) δ E l, ,S ( p , b ) = δ E l, ( p , b ) + 14 m m √ σ − p × b ) · S | b | l +7 l Γ( + l ) π Γ( − − l ) × M E l, ( p ) m ( E + m ) + ( E + E ) M E l, , ( p ) + O (( rS ) ) . (3.72)The position-space analysis extended to include spin degrees of freedom is equally straight-forward. It amounts to substituting in Eqs. (3.36) and (3.61) the stress tensor T µν ( p ) by thegeneral spin-dependent one in Eq. (3.17) or, for the scattering off a Kerr black hole, with T Kerr µν ( p )in Eq. (3.16). As already emphasized, T gen µν ( ℓ i , p ) depends on the graviton momentum ℓ i whichnow makes a leading-order contribution because of the spin dependence. Nevertheless, the con-tribution of T gen µν ( ℓ i , p ) can be organized as a differential operator acting on the position-spacethree-dimensional scalar propagator: E µ µ ( r , p ) = E µ µ ( i ∇ , p ) (cid:12)(cid:12)(cid:12) ε µν ( ℓ i ) → T gen µν ( i ∇ ,p ) Z d D − ℓ i (2 π ) D − e − i r · ℓ i ℓ i . (3.73)The structure of the stress tensor (3.16) implies that, for scattering off a Kerr black hole, thecomplete spin dependence is governed by the non-Abelian Fourier transform Z d D − ℓ i (2 π ) D − e − i (ˆ r − ˆ a ) · ℓ i ℓ i , (3.74)where ˆ r = r and ˆ a is a vector of matrices, (ˆ a σ ) µν = ǫ µνρσ a ρ , with a defined in Eq. (3.15). Onemay evaluate it by formally expanding the integrand in ˆ a .On general grounds, as discussed in Ref. [23], the impulse and spin kick is computed fromthe eikonal phase (3.72) through the relations (2.8) agree with those computed from Hamilton’sequations of motion based on the two-body potential (3.71). The same holds for the magneticanalog of Eqs. (3.72) and (3.71). The amplitude with nonlinear tidal effect, i.e. the scattering with an X n operator insertion, where X stands for E or B , can be constructed from the unitarity cut in Fig. 2. We will mostly focus on29 ℓ ℓ n Figure 2:
The generalized cut for leading order contributions to nonlinear tidal operators. Each blobis simply a (local) on-shell amplitude. The dark blob contains the X n tidal operator. The direction ofgraviton momentum flow is indicated by the arrows. leading contribution for such an operator in this section. In this case, the simplifications describedin Section 3.1.1 are all applicable. Namely, the amplitude with X n tidal operator is still comprisedof linearized electric and magnetic Weyl tensor in Eqs. (3.8) and (3.9); and the sewing of three-point amplitudes with the amplitude with X n tidal operator is effectively replacing the polarization ε µν ( ℓ i ) → T µν ( p ) for each graviton. Start from the the unitarity cut in Fig. 2. After sewing we find M X n ( p , q ) = κ n m n − Z M X n ( h ( ℓ ) , . . . , h ( ℓ n ) , φ ( p ) , φ ( p )) (cid:12)(cid:12)(cid:12) ε µν ( ℓ i ) → T µν ( p ) × ℓ ℓ · · · ℓ n " i ( − u · ℓ ) i ( − u · ℓ ) . . . i ( − u · P n − j =1 ℓ j ) , (4.1)where we integrate over ℓ i with i = 1 , . . . , n − P ni =1 l i = q .As discussed in the previous section, we can include spin degrees of freedom for the field withoutthe tidal deformation by simply replacing in Eq (4.1) the point-particle stress tensor T µν with thatof the general spinning particle T gen µν , cf. Eq. (3.17), or with that of a Kerr black hole, cf. (3.16).The calculations from position space and momentum space also follow similarly as before. Wediscuss them in turn. Start with Eq. (4.1). Again we consider the rest frame of particle 2 in which we have Eq. (3.22).The first step is to integrate out energy in potential region. Using the identity [51] δ n X i =1 ℓ i ! " i ( − u · ℓ + i i ( − u · ℓ + i . . . i ( − u · P n − j =1 ℓ j + i
0) + perm = π n − n Y i =1 δ ( ℓ i ) , (4.2)where perm is the rest of n ! permutations of ℓ ,...,n . Since the integrand is invariant under permu-tations, this localizes all ℓ i = 0 with a 1 /n ! prefactor M X n ( p , q ) = ( − κ ) n (2 m ) n − n ! Z " n Y i =1 d D − ℓ i (2 π ) D − ℓ i δ q − n X i =1 ℓ i ! × M X n ( h ( ℓ ) , . . . , h ( ℓ n ) , φ ( p ) , φ ( p )) (cid:12)(cid:12)(cid:12) ε µν ( ℓ i ) → T µν ( p ) , (4.3)30o evaluate this integral, we use the same manipulations as at one loop. First consider theFourier transform to position space M X n ( p , r ) = Z d D − q (2 π ) D − e − i r · q f M X n ( p , q )= ( − κ ) n (2 m ) n − n ! n Y i =1 Z d D − ℓ i (2 π ) D − e − i r · ℓ i ℓ i M X n ( h ( ℓ ) , . . . , h ( ℓ n ) , φ ( p ) , φ ( p )) (cid:12)(cid:12)(cid:12) ε µν ( ℓ i ) → T µν ( p ) = ( − κ ) n (2 m ) n − n ! M X n ( h , . . . , h n , φ ( p ) , φ ( p ) | r ) , (4.4)where we use Eq. (3.36) to define M X n ( h , . . . , h n , φ ( p ) , φ ( p ) | r ) (4.5) ≡ n Y i =1 Z d D − ℓ i (2 π ) D − e − i r · ℓ i ℓ i M X n ( h ( ℓ ) , . . . , h ( ℓ n ) , φ ( p ) , φ ( p )) (cid:12)(cid:12)(cid:12) ε µν ( ℓ i ) → T µν ( p ) = M X n ( h ( ℓ ) , . . . , h ( ℓ n ) , φ ( p ) , φ ( p ) | X µ µ ( ℓ j , p ) → X µ µ ( r j , p ) , ℓ j → i ∇ j ) (cid:12)(cid:12)(cid:12)(cid:12) r j → r . As before all the coordinates r j are identified as r in the end. The above formula is very general andapplies to higher multipole operators or general susceptibilities similar to Eq. (3.61). Recall that M X n ( h ( ℓ ) , . . . , h ( ℓ n ) , φ ( p ) , φ ( p )) is only a function of E µ µ ( ℓ i , p ), B µ µ ( ℓ i , p ), and Mandelstaminvariants. The Fourier transform simply replaces them with their corresponding in position-spaceexpressions defined in Eqs. (3.36) and (3.43). As before, the result of M X n ( p , r ) is generally notisotropic, because any u · ℓ in momentum space generates dependence on ˆ z · ℓ . To bring it into theisotropic form, we Fourier transform back to momentum space, as in Eq. (3.39).A simple example is the operator E µν E νρ E ρ µ , denoted as ( E ). With the contraction of three E tensors (3.36) given by E µν ( r , p ) E νρ ( r , p ) E ρ µ ( r , p ) = 3 m π | r | h σ − r − z )( σ r − ( σ − z ) + 2 r i , (4.6)the graviton-scalar amplitude is M (E ) ( h , h , h , φ ( p ) , φ ( p ) | r ) = 12 κ m E µν ( r , p ) E νρ ( r , p ) E ρ µ ( r , p ) . (4.7)Plugging into Eq. (4.4) then yields the four-scalar amplitude in position space M (E ) ( p , r ) = − κ m m E µν ( r , p ) E νρ ( r , p ) E ρµ ( r , p ) . (4.8)Using the Fourier transform formula in Eq. (3.40), we arrive the final result M (E ) ( p , q ) = −| q | − ǫ ǫ M (E ) ( p ) = 1811!! G m m π (cid:18) − σ + 10 σ (cid:19) | q | − ǫ ǫ . (4.9)31n important feature of the position-space scalar-graviton amplitude (4.5), which we alreadyencountered in the one-loop analysis in Sec. 3.3, is that it factorizes into a product of position-space E tensor, defined in Eq. (3.36) and its magnetic counterpart, perhaps with additional derivatives.As explained in Sec. 2, the fact that these position-space tensors have rank 3 implies that such aproduct can be further expressed as a sum of products of traces of at most three factors. For example,Eq. (2.50) gives the decomposition of any power of a rank-3 matrix in terms of in terms of tracesof two and three such matrices. It applies directly to the four-scalar amplitude with an insertionof ( E n ) and expresses it as a sum of four-scalar amplitudes with an insertion of ( E ) n ( E ) n with n = 2 n + 3 n . It also applies directly to amplitudes with an insertion of ( B n ). While the resultingamplitude vanishes of n is odd, it also further simplifies if n is even. The parity-odd nature of B µ,ν ( r , p ) and position-space factorization imply that, to leading order, ( B ) = 0 because there areinsufficient vectors to saturate the Levi-Civita tensor. Therefore, to leading order, the analog ofEq. (2.50) for the magnetic operators reduces to( B n =2 k ) = 12 k − ( B ) k . (4.10)The amplitudes collected in the Appendix A verify these formulas for up to n = 8.The momentum-space four-scalar amplitude is related to the position-space four-scalar ampli-tude by single ( D − E ), ( B )or ( E ) have a similar structure, we will discuss them simultaneously, referring to these operatorsas ( O ). They have the form, f M ( O ) = N ( O ) r h (cid:18) a ( O ) + b ( O ) ( r · u ) r + c ( O ) ( r · u ) r (cid:19) , (4.11)where N ( O ) is an operator-dependent normalization factor. For the three operators it is, N (E ) = N (B ) = 2 G π m m , N (E ) = 2 G π m m , (4.12)and the coefficients are a (E ) = 3(1 − σ + 3 σ )2 π , b (E ) = 9(1 − σ )2 π , c (E ) = 92 π ,a (B ) = 9 σ ( σ − π , b (B ) = 9(1 − σ )2 π , c (B ) = 92 π ,a (E ) = − − σ + 9 σ )8 π , b (E ) = − − σ )8 π , c (E ) = − π . (4.13)The exponent of the overall r factor is h = 6 for ( O ) = ( E ) and ( O ) = ( B ) and h = 9 for( O ) = ( E ). 32he position- space amplitude with an insertion of an operator made up of n such traces issimply given by raising (4.11) to the n th power and adjusting the normalization factor, f M ( O ) n = N ( O ) n (cid:20) r h a ( O ) + b ( O ) ( r · u ) r + c ( O ) ( r · u ) r (cid:19) n . (4.14)The change in normalization factor is related to the normalization of the tree-level amplitude withone insertion of the composite operator. We find N (E ) n = N (B ) n = 2 n +2 G n π n m m n +12 , N (E ) n = 2 n +2 G n π n m m n +12 . (4.15)To obtained the momentum-space scattering amplitude with an insertion of an arbitrary operator( O ) n we first use twice the binomial expansion and put the position-space amplitude in the form f M ( O ) n = N ( O ) n r nh n X k =0 k X l =0 nk ! kl ! a n − k O b l O c k − l O ( r · u ) r ! k − l . (4.16)Using then the general tensor Fourier-transform relation (3.40) which enforces q · u = q / → M ( O ) n ( p , q ) = N ( O ) n | q | D − nh − n X k =0 k X l =0 nk ! kl ! a n − k O b l O c k − l O (4.17) × D − hn − π D/ ( σ − k − l Γ( + 2 k − l )sin( π ( D − hn − (3 + hn − D ))Γ(2 k − l + hn ) , where D = 4 − ǫ . The two-body potential and the eikonal phase follow then straightforwardly viaEqs. (2.9)-(2.12): V ( O ) n ( p , r ) = − N ( O ) n E E | r | nh n X k =0 k X l =0 nk ! kl ! a n − k O b l O c k − l O ( σ − k − l Γ( + 2 k − l )Γ( hn ) √ π Γ(2 k − l + hn ) , (4.18) δ ( O ) n ( p , b ) = N ( O ) n m m | b | nh − n X k =0 k X l =0 nk ! kl ! a n − k O b l O c k − l O ( σ − k − l − / Γ( + 2 k − l )Γ( ( hn − k − l + hn ) . As discussed earlier, parity and factorization of the position-space amplitude implies that, toleading order in the classical limit, amplitudes with an insertion of an operator which has at leastone parity-odd factor vanish identically even if the operator is overall parity-even. Thus, Eq. (2.50)with E → B implies that the approach described here yields the two-body potential for all nonlineartidal operators of the type ( B n ).The discussion above can be easily extended to cover amplitudes with one insertion of ( E n ).Eq. (2.50) expresses it as a linear combination of amplitudes with one insertion of ( E ) n ( E ) n with2 n + 3 n = n . The position space form of the latter involves a product of two factors analogousto the right-hand side of Eq. (4.14). Each of them can be binomially expanded (with a slightsimplification based on the equality b ( E ) /b ( E ) = c ( E ) /c ( E ) visible in Eq. (4.13)) and put in a form33nalogous to the right-hand side of Eq. (4.16). Fourier-transforming using Eq. (3.40) and puttingtogether all terms leads to the momentum-space amplitude with one insertion of ( E n ).The general formulas above show explicitly that the difference E n − B n is subleading in thehigh-energy limit. This extends the observations of Refs. [28, 29] beyond the linear order. The above position-space evaluation is a very effective means for evaluating leading contributions toany given tidal operator. Momentum-space methods for evaluating the loop integrals instead offera straightforward way to systematically extend the results to higher orders following the methodspresented in Refs. [11–13]. Indeed following these methods, next to leading order contributions to E and B tidal operators were evaluated in Ref. [21]. A related approach for tidal operators basedon world lines has been recently given in Ref. [29] where additional E operators were evaluated.Here we first re-evaluate the amplitudes in momentum space through C and then discuss theextension to higher orders. The starting point is again the generalized cut shown in Fig. 2. Weevaluate the expressions in D -dimensions. Here we do not make use of the special real-spacefactorization of the integrals discussed in the previous section, but rather simply carry out theevaluation of the cut and then reduce the result to a basis of independent momentum products. Wecan simplify the resulting expressions considerably by applying the cut conditions and expandingin small momentum transfer q . Specifically, we can choose a basis of momentum invariants whichdoes not contain any of the products ( p · ℓ k ), since the cut conditions give − p + k X i =1 ℓ i ! − m = 0 → ( p · ℓ k ) = k X i =2 i − X j =1 ( ℓ i · ℓ j ) − k − X i =1 ( p · ℓ i ) , (4.19)where the final term can be eliminated inductively starting with p · ℓ = 0. Products of the form( p · ℓ k ) can then be eliminated using momentum conservation p = − p − q = − p − P ℓ i . Since thecut graviton momenta scale as O ( q ), the cut conditions thus ensure that the scaling of ( p · ℓ k ) or( p · ℓ k ), which naively would be O ( q ), instead scale as O ( q ). This greatly aids in the simplificationof the integrand after expanding in small q .Unlike in the position-space analysis, the integrals do not decouple into a product, and ingeneral, the momentum-space integrals can be challenging to evaluate. To do so, we use FIRE6 [52]which uses integration by parts methods [35] to reduce the integrals a single master integral, whichcan then be evaluated either by direct integration or by differential equations [53]. Evaluating theintegrals is the most significant bottleneck for this method, but the task is significantly aided bythe use of special variables as described in [36], p = − ( ¯ p − q/ , p = ¯ p + q/ , p = − (¯ p + q/ , p = ¯ p − q/ . (4.20)34
12 34 ℓ L +1 Figure 3:
The L -loop fan integral. The ¯ p i are orthogonal to q by construction: ¯ p i · q = 0 . As described in more detail in Ref. [36], withthese variables the matter propagators reduce to p + ℓ ··· i ) − m = 12¯ p · ℓ ··· i + O ( q ) , (4.21)so the matter propagators are linear in the loop momenta. In addition, we can define normalizedexternal momenta, ¯ u µi = ¯ p µi / q m i − q / , such that ¯ u i = 1 The net effect is that the q dependenceis scaled out of the integral so that it is only a function of a single-scale ¯ u · ¯ u = σ + O ( q ) . Usingthese variables integral encountered at any order of perturbation theory can then be converted toa single scale integral. Such integrals are quite amenable to integration-by-parts methods, greatlyspeeding the evaluation.The restriction to the potential region precludes pinching any propagators and the existence ofirreducible scalar products. Thus, the result of IBP reduction is a single master integral, with acoefficient given by powers of q dictated by dimensional analysis, as well as a polynomial in σ . Themaster integral is the scalar fan integral in Fig. 3, which can be easily evaluated by factorizing theloops by going to position space and Fourier transforming back, with the result I ( L )fan = Z L +1 Y i =1 d D ℓ (2 π ) D ℓ i ! | q | − L δ ( P i ℓ i − q )( − u · ℓ + i − u · ℓ + i · · · ( − u · ℓ ··· n − + i i L +2 L (4 − ǫ ) π L ( − ǫ ) Γ (cid:16) − ǫ (cid:17) L +1 Γ (cid:16) ( ǫ − ) L + 1 (cid:17) Γ( L + 2)Γ (cid:16) ( − ǫ )( L + 1) (cid:17) | q | − ǫL . (4.22)At one loop this agrees with Eq. (3.25) with l = 0 , and at two and three loops it yields I (2)fan = 1768 π ( q ) − ǫ ǫ + O ( ǫ ) , I (3)fan = − i π + O ( ǫ ) . (4.23)The results of the IBP reduction at two loops gives the amplitudes with a single insertion of thetidal operators in terms of a single master integral: M (E ) = 1024385 π G m m | q | (cid:18) − σ + 10 σ (cid:19) I (2)fan , M (EB ) = 10241155 π G m m | q | (cid:16) σ − (cid:17) (cid:16) σ (cid:17) I (2)fan , M (B ) = M E B = 0 . (4.24)35 ℓ n ℓ n +1 ℓ ℓ n +1 Figure 4:
The generalized cuts that need to be evaluated at next to leading order for an R n type tidaloperator. As expected, the parity odd operators E B and B operator do not contribute.At three loops, by reducing the integrand to the sole master integral we find the following forthe amplitudes with an insertion of the single trace operators, M (E ) = − i π G m m | q | (1231 − σ + 18590 σ − σ + 12155 σ ) I (3)fan , M (B ) = − i π G m m | q | ( σ − (1 + 10 σ + 85 σ ) I (3)fan , M (EEBB) = − i π G m m | q | ( σ − σ − σ + 3315 σ ) I (3)fan , M (EBEB) = i π G m m | q | ( σ − σ − σ + 3315 σ ) I (3)fan . (4.25)Similarly, the amplitudes with double trace insertions evaluate to, M (E ) = − i π G m m | q | (1231 − σ + 18590 σ − σ + 12155 σ ) I (3)fan , M (B ) = − i π G m m | q | ( σ − (1 + 10 σ + 85 σ ) I (3)fan , M (E )(B ) = − i π G m m | q | ( σ − σ − σ + 1105 σ ) I (3)fan , M (EB) = 0 , (4.26)It is not difficult to check that these results satisfy the four-dimensional relations described in Sec. 2.In addition, they agree with the results obtained in the previous section for tidal operators witharbitrary numbers of E s and B s and collected in the Appendix for a variety of operators up to E and B .An important aspect of the momentum-space approach is that it gives a systematic means forobtaining corrections higher order in Newton’s constant for any operator insertion. For exampleFig. 4 shows the generalized cuts that would need to be evaluated to obtain the next-to-leading ordercorrections from an C tidal operator. In the first of these cuts the four-point amplitude can appearat any location on the top matter line. The mapping of the integrands resulting from these cutsonto a integral basis generates a number of diagrams. For example, in Fig. 5 we show a sample ofthe diagrams that that are quite easy to evaluate for an R tidal operator, as we can again evaluatethe integral using the real-space technique presented in the previous section. More complicated36 Figure 5:
Sample diagrams for next-to-leading-order contributions for the R tidal operators which aresimple to evaluate.
12 34 2 31 4 2 341
Figure 6:
Sample diagrams next-to-leading order contributions for the R tidal operators that involveiteration contributions or nontrivial integrals. diagrams that involve iteration contributions or non-trivial integrations are shown in Fig. 6. Inthese cases, the integrals do not factorize, but the momentum-space approach of evaluating cutsand reducing to a basis of master integrals will still be quite feasible. As noted in Refs. [15, 21] theprobe limit simplifies the evaluation of the contributions. In any case, it is clear that amplitudemethods can be applied beyond leading order to understand the systematics of higher-dimensionoperators. We leave this to future studies. The same methods apply just as well to any operator, not just the tidal ones. For example, we canconsider the R n operators arising from unknown short distance physics. Here we will not classifysuch operators, but pick illustrative examples. The effect of operators up to R has already beendiscussed in some detail in Refs. [38–40]. In order to be concrete here we discuss an effective actionof the form S = 116 πG Z d D x √− g ( − R + c K K µ ...ρ n R µ ν σ ρ R µ ν σ ρ · · · R µ n ν n σ n ρ n ) , (5.1)where the first term is the usual Einstein-Hilbert action, and K µ ...ρ n merely gives the contrac-tion between the Riemann tensors. Each independent contraction carries an independent Wilsoncoefficient c K .We construct the integrands for pure R n modifications of gravity in a similar manner as for thoseof the tidal operators. The leading contribution to the potential due to R n operators is captured by37 ℓ j ℓ j +1 ℓ n
12 34
Figure 7: Cut for a general R n type operator. In the case j = 1 , it is convenient to take the singlegraviton attaching to the bottom matter line as off shell and part of a tree amplitude includingthe lower massive scalar line. All other gravitons and exposed matter lines are taken on shell. Thedirection of graviton momentum flow is indicated by the arrows.the cuts in Fig. 7. The diagrams in general are a product of two fan diagrams, where all gravitonlegs, as well as the matter lines between the three point vertices, are on shell, the exception beingthe case where ( n − on-shell gravitons attach to one of the matter lines, while one graviton whichwe take to be off shell attaches to the other matter line. In this case, it is convenient to include thematter line to which the single graviton propagator is attached as part of a single tree amplitude.To evaluate the cuts in Fig. 7 we use the replacement derived above (see Eq. (3.11)). Thissimplifies the form of the Riemann tensor: R µνρσ ( ℓ i ) (cid:12)(cid:12)(cid:12) ε µν ( ℓ i ) → T µν ( p a ) = − (cid:18) ℓ µi ℓ ρi (cid:18) p νa p σa − η νσ m a (cid:19) − ( σ ↔ ρ ) (cid:19) + (( µ, ρ ) ↔ ( ν, σ )) + O ( q ) , (5.2)where p a and m a are the momentum and mass of the matter line the graviton attaches to. Whencontracted in sequence with other gravitons attaching to the same matter line, products involvingthe matter momenta in the above expression must reduce to p a · p a = m a , or the q scaling willbecome sub-leading, as shown in the previous section.The cut corresponding to Fig. 7 is simply a product of two fans, C R n = K µ ...ρ n O µ ...ρ j ( ℓ , ..., ℓ j ; p ) O µ j +1 ...ρ n ( ℓ j +1 , ..., ℓ n ; p ) , (5.3)where, for instance, O µ ...ρ j ( ℓ , ..., ℓ j ; p ) = R µ ν σ ρ · · · R µ j ν j σ j ρ j j (cid:12)(cid:12)(cid:12) ε µν ( ℓ i ) → T µν ( p ) . (5.4)As in previous sections, the integrands obtained after restoring the cut propagators are also wellsuited for applying position-space techniques. In this case, we must introduce a fictitious momentumtransfer q ′ such that the integrand decouples in two parts, corresponding to the two terms inEq. (5.3) decouple, and the corresponding propagators attached to one matter line or the other.38he energy integrations can be carried out as in the previous sections with the result M R n ( p , q ) = K µ ...ρ n Z d D − q ′ δ ( q + q ′ ) Z j Y a =1 d D − ℓ a (2 π ) D − δ ( P ja =1 ℓ a + q ′ ) O µ ...ρ j ( ℓ , ..., ℓ j ; p ) ℓ · · · ℓ j × Z n Y a = j +1 d D − ℓ a (2 π ) D − δ ( P na = j +1 ℓ a − q ) O µ j +1 ...ρ n ( ℓ j +1 , ..., ℓ n ; p ) ℓ j +1 · · · ℓ n . (5.5)Writing δ ( q + q ′ ) = Z d D − x (2 π ) D − e i ( q + q ′ ) · x (5.6)and taking the Fourier transform of the amplitude we find M R n ( p , r ) = Z d D − q (2 π ) D − e − i q · r M R n ( p , q )= K µ ...ρ n Z d D − x Z j Y a =1 d D − ℓ a (2 π ) D − e − i ℓ a · x ℓ a O µ ...ρ j ( ℓ , ..., ℓ j ; p ) × Z n Y a = j +1 d D − ℓ a (2 π ) D − e − i ℓ a · ( r − x ) ℓ a O µ j +1 ...ρ n ( ℓ j +1 , ..., ℓ n ; p )= K µ ...ρ n Z d D − x O µ ...ρ j ( x ; p ) O µ j +1 ...ρ n ( r − x ; p ) . (5.7)The product in momentum space has become a convolution in position space over x , which canbe viewed as the position in the bulk, i.e. away from the massive particle trajectories, at whichthe R n operator is inserted. Note however that this formula does not have a natural interpretationin position space, given that the energy integrals in each factor were performed by going to therest frame of different particles. In practice, as in previous sections, this formula can be used bytransforming one last time to momentum space, so that the convolution is trivialized and eachfactor can be written in isotropic coordinates.The inclusion of derivatives, ∇ m R n , or of spin on the matter lines poses no obstruction toapplying this method. In the former case one must organize the additional powers of loop momentumin the integrand into either factor in analogy with Eq. (5.3). The factorization argument carriesover and the additional loop momenta become derivatives in position space acting on either factorof Eq. (5.7). For the case of spin, the only difference is that the Fourier transforms in Eq. (5.7)become non-Abelian Fourier transforms defined in Eq. (3.74).As simple examples, consider the cases of O R = R µ ν µ ν R µ ν µ ν R µ ν µ ν and O ( R ) =( R µ ν µ ν R µ ν µ ν ) . The contributing generalized unitarity cut for the R operator are shown inFig. 8(a) while the two potentially contributing cuts for the the R operator are shown in Fig. 8(b,c).In the diagrams the double-line notation indicates that we have not used on-shell conditions on thatline, but consider the two connected blobs as part of a single tree amplitude. Whether on-shell conditions are used on the intermediate leg corresponds to shifting the coefficient of φR n φ operators. c)(a) (b)12 34 12 3412 3 4 Figure 8:
The corrections from (a) R and (b,c) R operators that appear in EFT extensions of GR. Thedouble-line notation indicates that we have not used on-shell conditions on that line. After carrying out the integration, the R and R amplitudes are M R = − c R G π m m ( m + m ) | q | ( σ − , M ( R ) = − c ( R ) G πm m ( m + m ) ( q ) − ǫ ǫ (3 σ − , (5.8)where we took the operators to have coefficient c R and c ( R ) respectively. Taking the Fouriertransform (2.12) to position space gives the potentials V R = 18 E E c R G m m ( m + m )( σ −
1) 1 r ,V ( R ) = 2 E E c ( R ) G m m ( m + m )(3 σ −
1) 1 r . (5.9)The O R amplitude and potential was obtained previously in Refs. [39, 40] and we find agreement.In Ref. [39] the authors also evaluate the effect of an additional R operator, G = O R − R µναβ R βγνσ R σµγα ; (5.10)this is related to tidal operators via a field redefinition up to operators that vanish in four dimensions.This can be seen by evaluating its four-dimensional four-point amplitude, which feeds into the two-graviton cut, using spinor-helicity methods [39]: M G ( φ ( p ) , h ++ ( k ) , h ++ ( k ) , φ ( p )) ∝ [23] ( − q + 2 m ) . (5.11)Since this is a local contribution, it is already captured by tidal operators of the form E , B .Interestingly, though, if this operator were present with a sufficiently large coefficient, it wouldproduce a result equivalent to the leading tidal Love numbers, even if these are set identically tozero for black holes in Einstein gravity [32].The leading PN contribution from the R operator ( O ( R ) ) was calculated in Ref. [38], withwhich we find agreement. We can also easily determine that the other operators consideredin Ref. [38] give no contribution to the leading conservative potential. The contribution from O ( R )( R ˜ R ) = ( R µ ν αβ ǫ αβ µ ν R µ ν µ ν )( R µ ν µ ν R µ ν µ ν ) is zero simply because it is parity-odd.40he operator O ( R ˜ R ) = ( R µ ν αβ ǫ αβµ ν R µ ν µ ν ) , while being parity even, contributes zero at lead-ing order, in analogy to the tidal operator O ( EB ) . In both cases, the factorization of the integrandin real space forces the separate parity-odd factors to evaluate to zero, as discussed in Section 4.1Here we refrain from evaluating the amplitudes for the R and higher operators. However,in these cases, there is an additional link between the R n extensions of Einstein gravity and thetidal operators. After carrying out the soft expansion of the integrand for the R n operators, oneencounters ultraviolet divergences that renormalize tidal operators [5]. For example, in principlethe R operator, which produces a diagram with three gravitons attached to one matter line andtwo attached to the other, could produce a UV subdivergence and thereby renormalize E or B tidal operators (with additional derivatives). It would be an interesting problem to systematicallystudy this interplay for infinite sequences of R n operators. In this paper we evaluated the leading-PM order contributions to the two-body Hamiltonian frominfinite classes of tidal operators using momentum space and position space scattering amplitude andeffective field theory methods. The same principles yield leading-PM order Hamiltonian terms fromtidal deformations probed by a spinning particle and also from effective field theory modifications ofgeneral relativity. Our results offer a new perspective on the general structure of linear and nonlineartidal effects in the relativistic two-body problem while also being of potential phenomenologicalinterest.Our analysis of E and B tidal operators arbitrary number of derivatives is similar to that ofRef. [30], except that we use a basis of operators which aligns with the more standard worldline tidaloperators [5, 28]. Their Wilson coefficients are the same (up to an overall normalization that weprovide) with the worldline electric and magnetic tidal coefficients which in turn are proportional tothe corresponding multipole Love numbers. By directly evaluating all relevant integrals we obtainexplicit expressions for the two-body Hamiltonian and the amplitude’s eikonal phase, from whichboth scattering and closed-orbit observables can be found straightforwardly. We illustrated theinclusion of spin by working out the leading-order tidal contributions from E -type operators witharbitrary number of derivatives for one object interacting with the spin of the other.For tidal operators with arbitrary numbers of electric or magnetic components of the Weyl tensor,the integrand for the leading-order contributions are not difficult to construct because their buildingblocks are tree-level leading order on-shell matrix elements of the point-particle energy-momentumtensor and of the tidal operator. The simple loop-momentum dependence and the permutationsymmetry of the three-point amplitude factors makes the integrals simple to evaluate. Indeed,Fourier-transforming all graviton propagators decouples all integrals from each other, making itstraightforward to write down explicit results for infinite classes of tidal operators. We have verified41hat the results obtained this way thought direct momentum space integration. While positionspace methods make leading-order calculations straightforward, momentum-space methods can beapplied systematically, to arbitrary PM order.An interesting feature of gravitational tidal operators, which we exploited in their descrip-tion, is their close similarity with gauge theory operators describing the interaction of extendedcharge distributions with electromagnetic fields. This formal connection extends to dynamical leveldouble-copy relations. For leading-order contributions this is a straightforward consequence of thefactorization of the linearized Riemann tensor into two gauge-theory field strengths and of thefactorization of the energy-momentum tensor into two gauge theory currents. Such double-copyfactorizations also hold for the energy-momentum tensor [23]. It would be very interesting toinvestigate double-copy relations beyond the leading PM order.In summary, in this paper we took some steps towards systematically evaluating contributionsto the two-body Hamiltonian from infinite families of tidal operators. The leading order in G resultsare remarkably simple, suggesting that much more progress will be forthcoming. Acknowledgments:
We are especially grateful for discussions with Clifford Cheung, Nabha Shah, and Mikhail Solonfor discussions and sharing a draft of their article with us. We also thank Dimitrios Kosmopoulos,Andreas Helset and Andrés Luna for discussions. Z.B. and E.S. are supported by the U.S. De-partment of Energy (DOE) under award number DE-SC0009937. J.P.-M. is supported by the U.S.Department of Energy (DOE) under award number DE-SC0011632. R.R. is supported by the U.S.Department of Energy (DOE) under grant number DE-SC0013699. C.-H.S. is grateful for supportby the Mani L. Bhaumik Institute for Theoretical Physics and by the U.S. Department of Energy(DOE) under award number DE-SC0009919. 42
Appendix: Summary of Explicit Results
In this appendix we collect explicit results for scattering amplitudes with a tidal operator insertion.Using Eq. (2.5), this immediately gives us the potential. Here we consider the amplitudes withoperator insertions of the type E n − m B m . We express the amplitude in terms of the variable σ = p · p /m m . The general formulae for ( E ) n , ( B ) n and ( E ) n are given from Eq. (4.16)to Eq. (4.18) with the coefficients in Eq. (4.13). Here we give explicit results corresponding upto 7 loops in the amplitudes approach. As noted in the text, the amplitudes with an odd B -fieldinsertions vanish by parity so we do not include those. We also do not explicitly list cases where atrace contains an odd number of B s since these also vanish.To list the amplitudes we scale out the powers of | q | from the scattering amplitudes, followingEq. (2.9), M X n = | q | n − M X n = | q | n − C X n , (A.1)for a tidal operator which we build from a total of n E s or B s, independent of the trace structure.For operators where total number of E s and B is odd the rescaling is bit difference because of theappearance of a divergence M X n +1 = | q | n − nǫ M X = − n ǫ | q | n − nǫ C X , (A.2)The long-range classical contribution comes from the log q term that arises from expanding in ǫ .As discussed in Sec. 2, the potential is given in the two-body Hamiltonian is given by a Fouriertransform (2.5) and the eikonal phase is also given by Eq. (2.6). Carrying out the Fourier transformwe have from Eq. (2.10) and Eq. (2.11) V X n = − E E n − Γ(3 n ) π / Γ( − n ) C X n | r | n , (A.3) δ X n = 14 m m √ σ − n − Γ(3 n − ) π Γ( − n ) C X n | b | n − , (A.4)where we only keep the finite term in ǫ . Similarly, for the odd powers V X n +1 = 14 E E ( − n Γ(6 n + 2)2 π C X n +1 | r | n +3 , (A.5) δ X n +1 = 14 m m √ σ − − n − n Γ(3 n + 1) π C X n +1 | b | n +2 . (A.6)For X we have, C (E ) = 52 G m m π (cid:18) − σ + 7 σ (cid:19) , C (B ) = 52 G m m π ( σ − (cid:18) σ (cid:19) , (A.7)43here the parenthesis on the operator denote the matrix trace, as defined in Eq. (2.49)For X : C (E ) = − G m m π (cid:18) − σ + 10 σ (cid:19) , C (EB ) = − G m m π ( σ − (cid:18) σ (cid:19) . (A.8)For X : C (E ) = − · (7!!) G m m π (cid:18) − σ + 130 σ − σ + 85 σ (cid:19) , C (B ) = − · (7!!) G m m π ( σ − (cid:18) σ + 85 σ (cid:19) , C (EEBB) = − · (7!!) G m m π ( σ − (cid:18) σ − σ + 85 σ (cid:19) , C (EBEB) = 11 · (7!!) G m m π ( σ − (cid:18) σ − σ + 85 σ (cid:19) , C (E ) = 2 C (E ) , C (B ) = 2 C (B ) , C (E )(B ) = − · (7!!) G m m π ( σ − (cid:18) σ − σ + 85 σ (cid:19) . (A.9)For X : C (E ) = 12 (19!!) G m m π (cid:18) − σ + 24608 σ − σ + 17280 σ (cid:19) , C (E B ) = 12 G m m π ( σ − (cid:18)
499 + 10144 σ − σ + 51840 σ (cid:19) , C (EBEBE) = − G m m π ( σ − (cid:18)
61 + 1336 σ − σ + 8640 σ (cid:19) , C (EB ) = 3 G m m π ( σ − (cid:18) σ + 120 σ (cid:19) , C (E )(B ) = 32 G m m π ( σ − (cid:18)
61 + 1336 σ − σ + 8640 σ (cid:19) , C (E )(EB ) = 32 G m m π ( σ − (cid:18)
85 + 1600 σ − σ + 5760 σ (cid:19) , C (B )(EB ) = 2 C EB . (A.10)For X , X , X :C (E ) = 17 · · (13!!) G m m π (cid:18) − σ + 60930581 σ − σ + 1834259 σ − σ + 5175 σ (cid:19) , (B ) = 17 · · (13!!) G m m π ( σ − (cid:18) σ + 575 σ + 5175 σ (cid:19) , C (E ) = − (31!!) G m m π (cid:18) − σ + 71940660 σ − σ + 253373120 σ − σ + 69189120 σ (cid:19) , (A.11)C (E ) = − · · · (19!!) G m m π (cid:18) − σ + 3231939466063423 σ − σ + 8252083027 σ − σ + 3294060 σ − σ + 441595 σ (cid:19) , C (B ) = − · · · (19!!) G m m π ( σ − (cid:18)
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