Quadratic-in-Spin Hamiltonian at \mathcal{O}(G^2) from Scattering Amplitudes
QQuadratic-in-Spin Hamiltonian at O ( G ) from Scattering Amplitudes Dimitrios Kosmopoulos and Andres Luna
Mani L. Bhaumik Institute for Theoretical Physics,Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095
Abstract
We obtain the quadratic-in-spin terms of the conservative Hamiltonian describing the in-teractions of a binary of spinning bodies in General Relativity through O ( G ) and to all ordersin velocity. Our calculation extends a recently-introduced framework based on scattering am-plitudes and effective field theory to consider non-minimal coupling of the spinning objects togravity. At the order that we consider, we establish the validity of the formula proposed in [1]that relates the impulse and spin kick in a scattering event to the eikonal phase. a r X i v : . [ h e p - t h ] F e b ontents Introduction
The detection of gravitational waves by the LIGO and Virgo collaborations [2, 3] promises in-triguing new discoveries. The main sources of gravitational waves are binary systems of compactastrophysical objects. Therefore, the great experimental advances also press for the developmentof high-precision theoretical tools for the modeling of the evolution of such systems. In the presentpaper we consider the inspiral phase of the evolution of the binary. A well-developed theoreticaltool to study this phase is the post-Newtonian (PN) approximation. This approach consists of anexpansion in small velocities and weak gravitaional field. Several methods based on General Rela-tivity (GR) [4, 5] as well as effective field theory (EFT) [6] have been developed in this direction.We instead choose to use the post-Minkowskian (PM) approximation, an expansion in Newton’sconstant G which yields the exact velocity dependence. The PM approximation has a long his-tory in GR [7] but has gained prominence recently (see e.g. [8–10]) due in part to the successfuladaptation of modern scattering-amplitudes techniques.The application of quantum-field-theory (QFT) methods to the study of the two-body problemdates back to the 1970’s [11]. However, it was recently that Ref. [12] proposed the applicationof the well-established scattering-amplitudes toolkit to the derivation of gravitational potentials(see Refs. [13–15] for reviews on the modern amplitudes program). Along these lines, Ref. [16]developed an EFT of non-relativistic scalar fields which allowed the construction of the 2PM canonical Hamiltonian from a one-loop scattering amplitude. This Hamiltonian was equivalent tothe one of Ref. [7]. Refs. [17–19] later implemented this approach to obtain novel results at 3PMorder. Ref. [20] followed up shortly after to compare these results against numerical relativity interms of the energetics of the binary. Very recently, Ref. [21] obtained the conservative binarypotential at 4PM order.Besides making use of a non-relativistic EFT, various approaches have been developed to extractthe dynamics of compact non-spinning objects from scattering data. Refs. [22, 23] establisheda formalism to obtain physical observables from unitarity cuts. Refs. [24, 25] made use of theLippman-Schwinger equation. Refs. [26, 27] developed a boundary-to-bound (B2B) dictionary, andRefs. [28, 29] implemented a worldline PM EFT. Ref. [21] discovered an amplitude-action relationthat allows the calculation of physical observables directly from the scattering amplitude.The techniques mentioned above have been extended in multiple directions in recent years. In-deed, Refs. [30–32] applied similar methods to supergravity. Ref. [33] studied three-body dynamics,while Refs. [34–37] incorporated the radiation emitted by the binary into their analysis. Refs. [38–44] considered tidal deformations of the astrophysical objects. In the present paper we explore adifferent direction and focus on effects due to the spin of the compact objects.When considering intrinsic angular momentum in the problem of a binary of compact astrophys- The n PM order corresponds to O ( G n ). order include the linear-in-spin [68] and quadratic-in-spin [69] interactions at next-to-next-to-next-to-leading order and the cubic-in-spin [70] andquartic-in-spin [71] interactions at next-to-leading order. The PM literature on the other handis less developed. Refs. [72, 73] recently obtained results at the 1PM and 2PM orders for effectslinear in the spin of the objects via GR considerations. Ref. [74] treated the black hole (BH) case at1PM order and exactly in the spin by matching an effective action to the linearized Kerr solution.Refs. [75, 76] obtained the 2PM-order scattering angle in the special kinematic configuration wherethe spins of the BHs are aligned to the orbital angular momentum of the binary.Similarly to the non-spinning case, we may use scattering amplitudes to study the gravitationalpotential between spinning objects. Indeed, Ref. [77] calculated a one-loop amplitude using Feyn-man rules, which allowed them to obtain a 2PM-order potential by means of a Born iteration.Following the approach of [12], Ref. [78] reproduced Hamiltonians describing the interactions be-tween spinning BHs by considering spinning particles minimally coupled to gravity. Later, Ref. [79]used the generalization of minimal-coupling amplitudes of [80] and the holomorphic classical limitof [81] to show that amplitudes encode information about BHs that is exact in spin. Refs. [82, 83]used the massive spinor-helicity formalism of [80] to study 2PM-order gravitational scattering froma one-loop amplitude. Furthermore, Ref. [84] related classical observables of a scattering processbetween spinning particles directly to the scattering amplitude, extending the formalism of [22].Using this formalism, Refs. [85–87] obtained a 1PM-order Hamiltonian that reproduced the resultof [74]. Finally, Ref. [1] obtained the conservative 2PM-order potential that is bilinear in the spinof the objects and valid for arbitrary spin orientations.Studies of the classical physics of spinning particles have also revealed double copy structures.Refs. [88–93] applied the definition of minimal coupling of [80] to classical solutions. In this waythey made contact with the classical double copy of Ref. [94] and with an effective theory ofon-shell heavy spinning particles [95]. The latter generalizes the heavy black hole effective theoryof Ref. [96], whose amplitudes are known to double copy [97].A suprising structure that emerged from the calculation of Ref. [1] is the expression of theobservables in a scattering event in terms of the eikonal phase [98]. Similar relations already existedin the non-spinning case [31, 32, 98–105]. In the spinning case there was evidence for such a relation The n PN order corresponds to O ( G a v b S c ) with a + b + c = n + 1, where v is the relative velocity of the binarysystem and S corresponds collectively to the spins of the objects.
4n the special kinematic configuration where the spins of the particles are parallel to the angularmomentum of the system [75, 76, 82]. The formula of [1] was the first example of such a relation forarbitrary orientations of the spins. This striking observation potentially implies that all physicalobservables are obtainable via simple manipulations of the scattering amplitude.The goal of the present paper is to obtain a 2PM-order Hamiltonian that describes the dynamicsbetween a binary of generic spinning objects in GR including effects that are up to quadratic in thespin. We take the masses of the two objects to be m and m and the rest-frame spin three vectorsto be S and S . We denote the relative distance between the objects as r and the momentumthree vector in the center-of-mass frame as p . The Hamiltonian then reads H = q p + m + q p + m + V (0) ( r , p ) + V (1 , ( r , p ) L · S r + V (1 , ( r , p ) L · S r + V (2 , ( r , p ) ( r · S )( r · S ) r + V (2 , ( r , p ) S · S r + V (2 , ( r , p ) ( p · S )( p · S ) r + V (2 , ( r , p ) ( r · S ) r + V (2 , ( r , p ) S r + V (2 , ( r , p ) ( p · S ) r + . . . , (1.1)where L = r × p is the orbital angular momentum, and the ellipsis stands for terms of higher orderin the spin. Note that we omit terms quadratic in S as they are obtained from the ones quadraticin S via appropriate relabeling. The terms in Eq. (1.1) take the form V A ( r , p ) = G | r | c A ( p ) + G | r | ! c A ( p ) + O ( G ) , (1.2)where the label A takes the values indicated in Eq. (1.1).Our task is to determine the coefficients c Ai appearing in Eq. (1.2). For simplicity, and sincethe bilinear-in-spin interactions were given in Ref. [1], we may consider one of the bodies to be nonspinning. This amounts to formally setting S = 0 in Eq. (1.1). We have explicitely verified thatthe results of this paper do not change if we take into account all the terms in Eq. (1.1).Following Refs. [1, 16], we obtain the potential coefficients in question via a matching calcu-lation. First, we calculate a one-loop scattering amplitude in our so-called full theory. This is atheory that describes particles of arbitrary spin coupled to gravity. Specifically, it captures minimaland non-minimal coupling of the particles to gravity. In terms of our Lagrangian, we include allpossible operators that are up to quadratic in the spin of the massive particle and up to linear inthe curvature. Then, we calculate the corresponding amplitude in an EFT of spinning particlesinteracting via the Hamiltonian of Eq. (1.1). Our EFT generalizes that of Refs. [1, 16] to considereffects quadratic in the spin of one of the particles.In obtaining these amplitudes we restrict to the piece that captures the classical dynamics. Weimplement the classical limit by rescaling q → λq, S → (1 /λ ) S and expanding in λ , where q denotes graviton momenta and S the covariant spin of the spinning particle. Finally, we fix thedesired coefficients by matching the two computed amplitudes.5 k k k k k k ( a ) ( b )Figure 1: The Feynman vertices used to compute full-theory amplitudes. The three-particle vertex (a) de-termines the O ( G ) dynamics. The Compton amplitude, which requires the contact vertex (b), captures the O ( G ) dynamics. The straight lines correspond to the spinning particle, while the wiggly lines correspondto gravitons. The remainder of this paper is structured as follows: In Sec. 2 we review some aspects of thespin formalism introduced in [1] that we use throughout the paper. Namely, we describe our field-theory approach to higher spin and its classical limit. We compute the necessary full-theory treeand one-loop amplitudes in Sec. 3. We adopt the method of generalized unitarity [106–108] toproduce the loop-level amplitude, using tree-level amplitudes as building blocks. We then expressthe amplitudes in the center-of-mass frame, which facilitates the matching to the EFT. Sec. 4contains the setup of the EFT, along with the computation of the EFT amplitudes. By equatingthe full-theory and EFT amplitudes, we obtain the desired two-body Hamiltonian. We compare ourresult against PN [109] and test-body [110] Hamiltonians in the literature. Finally, in Sec. 5 we usethe derived Hamiltonian to compute scattering observables. We then establish that the conjectureof Ref. [1], which directly relates these observables to the eikonal phase [98], holds unaltered whenwe include the quadratic-in-spin effects. We present our concluding remarks in Sec. 6.
Note added : As this paper was in its latest stages we learned about [111], which containsoverlap with our work. Ref. [111] extended the worldline PM EFT of [28, 29] to include spin degreesof freedom. We have explicitly verified that, where overlapping, our results are in agreement withthose of [111].
In this section, we review the aspects of the higher-spin formalism that we use in the paper. Forfurther details, we refer the reader to Ref. [1].We identify spinning compact astrophysical objects with higher-spin particles. We describe thesemassive particles of integer-spin s by real symmetric traceless rank- s tensor fields φ s . For brevity,we suppress the indices of φ s , implying matrix multiplication when necessary.We use a Lagrangian to organize the interactions of higher-spin fields with gravity. Ref. [112]obtained such a Lagrangian using auxiliary fields to eliminate all but the spin- s representation ofthe SO (3) rotation group. Here we relax this requirement, and interpret the theory as a relativistic6ffective theory that captures all spin-induced multipole moments of spinning objects coupled togravity. We write the higher-spin Lagrangian L and action S as L = L min + L nonmin , S = Z d x √− g L . (2.1)The minimal Lagrangian contains terms with up to two derivatives, L min = − R ( e, ω ) + 12 g µν ∇ ( ω ) µ φ s ∇ ( ω ) ν φ s − m φ s φ s . (2.2)The covariant derivative is ∇ ( ω ) µ φ s ≡ ∂ µ φ s + i ω µef M ef φ s , (2.3)where ω is the spin connection, and M ab are the Hermitian Lorentz generators. The gravitationalfield is described in the vielbein formulation. The non-minimal Lagrangian containing all the termslinear in the graviton and bilinear in the higher-spin field is L non-min = ∞ X n =1 ( − n (2 n )! C ES n m n ∇ ( ω ) f n · · · ∇ ( ω ) f R f af b ∇ ( ω ) a φ s S ( f . . . S f n ) ∇ ( ω ) b φ s (2.4) − ∞ X n =1 ( − n (2 n + 1)! C BS n m n +1 ∇ ( ω ) f n +1 · · · ∇ ( ω ) f (cid:15) ab ( c | f R ab | d ) f ∇ ( ω ) c φ s S ( f . . . S f n +1 ) ∇ ( ω ) d φ s . where we use an off-shell analog of the Pauli-Lubanski vector S a ≡ − i m (cid:15) abcd M cd ∇ ( ω ) b . (2.5)The operators in Eq. (2.4) are in direct correspondence to the non-minimal couplings in the worldlinespinning-particle action of Ref. [113]. One could, in principle, include terms with dependence onhigher powers of the curvature, but we do not attempt to do so in the present paper. Since ourobjective is to describe the dynamics up to spin squared, we focus on the first non-minimally coupledterm, L ES = − C ES m R f af b ∇ a φ s S ( f S f ) ∇ b φ s . (2.6)Ref. [50] first studied the effects captured by this operator at leading order in the PN approximation.The extensions to next-to-leading and next-to-next-to-leading orders were considered in Refs. [53]and [109] respectively, while Ref. [114] studied its contributions to higher orders in spin. Weinstead consider its effects in the PM approximation.To extract Feynman rules, we define the graviton as the fluctuation of the metric aroundMinkowski space. We determine the spin connection ω as the solution of the vielbein postulate, ∇ µ ( ω ) e aν = 0. This yields the following expansions for the needed quantities g µν = η µν + h µν , e µa = δ aµ + 12 h µa − h µρ h aρ + O ( h ) , ( e ) µcb = − ∂ [ c h b ] µ − h ρ [ c ∂ µ h b ] ρ + 12 h ρ [ c ∂ ρ h b ] µ − h ρ [ c ∂ b ] h µρ + O ( h ) . (2.7)After substituting this expansion into the Lagrangian of Eq. (2.6), we follow a straightforwardprocedure to obtain the Feynman vertices in Fig. 1. These are the vertices necessary to determinethe dynamics through O ( G ).We describe the state of the higher-spin particles by their momentum p and polarization tensor ε ( p ). To take the classical limit of expectation values, we choose “spin coherent states” [115], whosedefining property is that they minimize the standard deviation of observables. Following [116, 117],we relate the classical spin tensor and Lorentz generators via ε (˜ p ) M µ ν ε ( p ) = S ( p ) µ ν ε (˜ p ) · ε ( p ) + . . . ,ε (˜ p ) { M µ ν , M µ ν } ε ( p ) = S ( p ) µ ν S ( p ) µ ν ε (˜ p ) · ε ( p ) + . . . , (2.8)where { A, B } ≡ ( AB + BA ) and ˜ p ≡ − p − q (note that we use the all-outgoing convention).We can also write analogous expressions for products with higher powers of the Lorentz generator.Throughout the paper we omit terms that do not contribute to the classical potential in ellipsis.These include terms that do not survive in the classical limit and terms that cancel in the matchingbetween full-theory and EFT amplitudes.Importantly, one can only interpret the symmetric product of Lorentz generators as a productof spin tensors. However, it is always possible to decompose a product of Lorentz generators into asum of completely symmetric products by means of the Lorentz algebra,[ M µ ν , M µ ν ] = i ( η µ µ M µ µ + η µ µ M µ µ − η µ µ M µ µ − η µ µ M µ µ ) . (2.9)We take the spin tensor to obey the so-called covariant spin supplementary condition, p µ S ( p ) µν = 0 . (2.10)We define the Pauli-Lubanski spin vector by S α ( p ) = − m (cid:15) αβγδ p β S γδ ( p ) . (2.11)Using the on-shell condition for the spinning particle p = m and Eq. (2.10), we find S αβ ( p ) = − m (cid:15) αβγδ p γ S δ ( p ) . (2.12)For this choice of the spin vector we have S ( p ) µ = p · S m , S + p · S m ( E + m ) p ! , (2.13)8 Figure 2:
The tree-level amplitude that captures the O ( G ) spin interactions. The thick (thin) straightline represents the spinning (scalar) particle, while the wiggly line corresponds to the exchanged graviton.
21 43 31 42 321 4 2 31 4 ( a ) ( b ) ( c ) ( d ) Figure 3:
The one-loop scalar box integrals I (cid:3) (a) and I ./ (b) and the corresponding triangle integrals I (c) and I (d). The bottom (top) solid line corresponds to a massive propagator of mass m ( m ). Thedashed lines denote massless propagators. where S is the three-dimensional rest-frame spin of the particle and p = − ( E, p ). I.e. we obtain thecovariant spin vector by boosting its rest-frame counterpart. Finally, by writing the polarizationtensors as boosts of rest-frame coherent states [115], we have ε (˜ p ) · ε ( p ) = exp " − L q · S m ( E + m ) + . . . , (2.14)where L q ≡ i p × q , and the ellipsis stand for terms that do not contribute to the classical potential. In this section we calculate the scattering amplitudes needed to construct the desired Hamiltonian.Specifically, we obtain the relevant pieces of the tree-level and one-loop two-to-two scattering am-plitude between a scalar and a spinning particle. For the tree-level amplitudes we use the Feynmanrules derived in the previous section. We use the generalized unitarity method [106–108] for theone-loop amplitude. Anticipating the comparison to the EFT amplitudes, we specialize our resultsto the center-of-mass frame.
The information to determine the O ( G ) Hamiltonian is contained in the tree-level amplitude shownin Fig. 2. We take the incoming momentum of the spinning (scalar) particle to be − p ( − p ) and9ts outgoing momentum to be p ( p ). Using the Feynman rules obtained above, we find M tree = − πGq ε · ε (cid:16) α (0)1 + α (1 , E + α (2 , ( q · S ) (cid:17) + . . . , (3.1)where E ≡ i(cid:15) µνρσ p µ p ν q ρ S σ , and the labeling scheme for α Ai follows that for c Ai in Eq. (1.2). In theellipsis we omit terms that do not contribute to the classical limit, along with pieces proportionalto q , since they cancel the propagator and do not yield long-range contributions. The coefficients α A take the explicit form α (0)1 = 4 m ν (2 σ − , α (1 , = 8 m νσm , α (2 , = 2 C ES m ν (2 σ − m , (3.2)where we use the variables σ = p · p m m , m = m + m , ν = m m m . (3.3)In order to construct the O ( G ) Hamiltonian we further need the corresponding one-loop am-plitude. We may express any one-loop amplitude as a linear combination of scalar box, triangle,bubble and tadpole integrals [118]. Refs. [16, 18] showed that the bubble and tadpole integrals donot contribute to the classical limit. Dropping these pieces we may write i M = d (cid:3) I (cid:3) + d ./ I ./ + c I + c I , (3.4)where the coefficients d (cid:3) , d ./ , c and c are rational functions of external momenta and polarizationtensors. The integrals I (cid:3) , I ./ , I and I are shown in Fig. 3. The triangle integrals take the form [16] I , = − i m , √− q + · · · . (3.5)The box contributions do not contain any novel O ( G ) information. They correspond to infrared-divergent pieces that cancel out when we equate the full-theory and EFT amplitudes [16, 18]. In thissense, the explicit values for the box coefficients serve only as a consistency check of our calculationand we do not show them. Instead, we give the result for i M + ≡ c I + c I . (3.6)We use the generalized-unitarity method [106–108, 119] to obtain the integral coefficients ofEq. (3.4). We start by calculating the gravitational Compton amplitude for the spinning particle,using the Feynman rules derived in the previous section. We depict the relevant Feynman diagramsin Fig. 4. Subsequently, we construct the two-particle cut depicted in Fig. 5(a) by gluing theCompton amplitude for the spinning particle with that for a scalar. The latter is a well-knownamplitude. The residue of the two-particle cut on the scalar-matter pole gives the triple cut in10 Figure 4:
The Compton-amplitude Feynman diagrams. The straight line corresponds to the spinningparticle. The wiggly lines correspond to gravitons. ( a )1 ( b ) ( c ) ( d )1 1 142 3 4 4 43 3 32 2 2 Appropriate residues of the two-particle cut (a) give the triple cuts (b) and (c), and the quadruplecut (d). The thick straight line corresponds to the spinning particle, the thin straight line to the scalar,and the wiggly lines to the exchanged gravitons. All exposed lines are taken on-shell.
Fig. 5(b), while the one on the spinning-matter pole gives the triple cut in Fig. 5(c). Localizingboth matter poles gives the quadruple cut in Fig. 5(d). Finally, following Refs. [120–122], we obtainthe box and triangle coefficients from the quadruple and triple cuts respectively. Our result reads M + = 2 π G ε · ε √− q " α (0)2 + α (1 , E + α (2 , ( q · S ) + α (2 , q S + α (2 , q ( p · S ) + . . . , where the coefficients are given by α (0)2 = 3 m ν (5 σ − , α (1 , = m (4 m + 3 m )(5 σ − νσm ( σ − ,α (2 , = − m σ − (cid:20) − m (cid:16) − σ + 1 + C ES (30 σ − σ + 3) (cid:17) − m (cid:16) σ − σ − C ES (155 σ − σ + 35) (cid:17)(cid:21) ,α (2 , = − m σ − (cid:20) m (cid:16) σ − σ + 2 + C ES (15 σ − σ + 2) (cid:17) + m (cid:16) σ − σ + 7 + C ES (95 σ − σ + 23) (cid:17)(cid:21) ,α (2 , = − σ − (cid:20) m (cid:16) σ − σ − C ES (15 σ − σ + 3) (cid:17) + m (cid:16) σ − σ + 1 + C ES (65 σ − σ + 17) (cid:17)(cid:21) . (3.7)We note here that the relation α (2 , = − α (2 , , which was expected following a pattern observedin Refs. [1, 96], is broken for generic values of C ES . We recover this relation for C ES = 1, which11orresponds to the Kerr black hole [74]. This is in line with a recent observation in Ref. [39], thatthis equality fails to hold in the presence of tidal finite-size effects. In preparation for the matching procedure in the following section, we specialize our expressions tothe center-of-mass frame. In this frame, the independent four-momenta read p = − ( E , p ) , p = − ( E , − p ) , q = (0 , q ) , p · q = q / . (3.8)Using Eq. (2.13), we have q · S = q · S − q p · S m ( E + m ) , i(cid:15) µνρσ p µ p ν q ρ S σ = E L q · S , p · S = − Em p · S . (3.9)Furthermore, Eq. (2.14) becomes ε · ε = 1 − L q · S m ( E + m ) + ( L q · S ) m ( E + m ) + . . . . (3.10)Using the above expressions, our amplitudes take the form M tree E E = 4 πG q " a (0)1 + a (1 , L q · S + a (2 , ( q · S ) , (3.11) M + E E = 2 π G | q | " a (0)2 + a (1 , L q · S + a (2 , ( q · S ) + a (2 , q S + a (2 , q ( p · S ) . The coefficients a Ai are given in terms of the α Ai of Eqs. (3.2) and (3.7) by a (0) i = α (0) i m γ ξ , a (1 , i = α (1 , i mγξ − m ( γ + 1) α (0) i m γ ξ ,a (2 ,j ) i = α (2 ,j ) i ˜ ζ ( j ) m γ ξ − ζ ( j ) m ( γ + 1) α (1 , i mγξ + ζ ( j ) m ( γ + 1) α (0) i m γ ξ , (3.12)where i = 1 , ζ (4) = − ζ (5) = p , ζ (6) = 1 , ˜ ζ (4) = ˜ ζ (5) = 1 , ˜ ζ (6) = − E m . (3.13)In addition to the definitions in Eq. (3.3) we use γ = Em , γ = E m , E = E + E , ξ = E E E . (3.14) Note that unlike Ref. [1] we do not introduce the coefficients a cov . This means that factors of the spin inEq. (3.11) appear both because we specialize in the center-of-mass frame and due to Eq. (3.10). Hamiltonian from effective field theory
We now turn our attention to the task of translating the scattering amplitudes of higher-spin fields toa two-body conservative Hamiltonian. We do this by matching the scattering amplitude computedin the last section to the two-to-two amplitude of an EFT of the positive-energy modes of higher-spin fields. Ref. [16] developed this matching procedure for higher orders in G and all orders invelocity, while Ref. [1] extended the formalism to include spin degrees of freedom. We conclude thissection by comparing our answer with previous results in the literature. The action of the effective field theory for the higher-spin fields ξ and ξ is given by S = Z k X a =1 , ξ † a ( − k ) (cid:18) i∂ t − q k + m a (cid:19) ξ a ( k ) − Z k , k ξ † ( k ) ξ † ( − k ) V ( k , k , ˆ S ) ξ ( k ) ξ ( − k ) , (4.1)where R k = R d D − k (2 π ) D − , and the interaction potential V ( k , k , ˆ S ) is a function of the incoming andoutgoing momenta k and k , and the spin operator ˆ S . We consider kinematics in the center-of-mass frame. As in the full theory side, we choose the field ξ to be a scalar, while the asymptoticstates of ξ are taken to be spin coherent states. We obtain the classical rest-frame spin vector asthe expectation value of the spin operator with respect to these on-shell states.We build the most general potential containing only long-range classical contributions, up toquadratic order in spin. In momentum space, a minimal basis of interactions in the on-shell schemeis given by the operatorsˆ O (0) = I , ˆ O (1 , = L ˆ q · ˆ S , ˆ O (2 , = (cid:16) ˆ q · ˆ S (cid:17) , ˆ O (2 , = ˆ q ˆ S , ˆ O (2 , = ˆ q (cid:16) k · ˆ S (cid:17) , (4.2)where ˆ q ≡ k − k and L ˆ q ≡ i k × ˆ q . Their expectation values with respect to spin coherent statesare in one-to-one correspondence with the monomials in the full theory amplitude, Eq. (3.11). Thelabeling scheme for the operators follows the conventions of Eq. (1.1). We use the following ansatzfor the potential operator V ( k , k , ˆ S ) = X A V A ( k , k ) ˆ O A , (4.3)where A runs over the superscripts of the operators in Eq. (4.2). V A ( k , k ) are free coefficients withthe same structure as the spin-independent potential of Refs. [17, 18], V A ( k , k ) = 4 πG ˆ q d A (cid:16) ˆ p (cid:17) + 2 π G | ˆ q | d A (cid:16) ˆ p (cid:17) + O ( G ) , (4.4) The three-vectors ˆ q and ˆ p are not to be confused with unit-norm vectors. p ≡ ( k + k ) /
2. At the O ( G ) level, the operators containing a factor of ˆ q can be ignored,as they lead to contact terms. Therefore we choose d (2 , = d (2 , = 0 . (4.5)However, the factor of ˆ q does not cancel out with the O ( G ) denominator, so we need to keep d (2 , and d (2 , .We now evaluate the EFT two-to-two scattering amplitude. To this end we use the Feynmanrules derived from the EFT action (Eq. (4.1)), ( E, k ) = i I E − √ k + m + i(cid:15) , − k ′ k − k k ′ = − iV ( k , k , ˆ S ) . (4.6)Using these rules we compute the amplitude up to O ( G ) directly evaluating the relevant Feynmandiagrams, omitting terms that do not contribute to long range interactions. The spin-dependentvertices must be treated as operators, and thus their ordering is important. After carrying out theenergy integration, we obtain an expression for the amplitude M EFT = − V ( p , p , S ) − Z k V ( p , k , S ) V ( k , p , S ) E + E − q k + m − q k + m . (4.7)Similarly to the full theory, in order to extract the classical limit, one needs to first decomposeproducts of the spin vector into irreducible representations of the rotation group, by repeated useof the SO (3) algebra.At O ( G ) the EFT amplitude receives a contribution only from the first term of Eq. (4.7), M EFT1PM = 4 πG q h a (0)1 + a (1 , L q · S + a (2 , ( q · S ) i . (4.8)The a A are given directly in terms of the momentum-space potential coefficients, a A = − d A . (4.9)The EFT amplitude at O ( G ) receives contributions from both terms in Eq. (4.7) and can bewritten as M EFT2PM = 2 π G | q | " a (0)2 + a (1 , L q · S + a (2 , ( q · S ) + a (2 , q S + a (2 , q ( p · S ) + (4 πG ) a iter Z d D − ‘ (2 π ) D − ξE ‘ ( ‘ + q ) ( ‘ + 2 p · ‘ ) , (4.10)14here ‘ = k − p and we only keep terms that are relevant in the classical limit. The abovecoefficients are given by a (0)2 = − d (0)2 + 12 ξE ˜ A (cid:20)(cid:16) d (0)1 (cid:17) (cid:21) , (4.11) a (1 , = − d (1 , + 12 ξE ˜ A h d (0)1 d (1 , i ,a (2 , = − d (2 , + 38 ξE ˜ A / " d (0)1 d (2 , + p ξE ˜ A (cid:20)(cid:16) d (1 , (cid:17) (cid:21) + ξE d (1 , d (2 , ,a (2 , = − d (2 , − ξE ˜ A " d (0)1 d (2 , − p ξE ˜ A " (cid:16) d (1 , (cid:17) + ξE d (1 , d (2 , ,a (2 , = − d (2 , + ξE p d (0)1 d (2 , + 18 ξE ˜ A " (cid:16) d (1 , (cid:17) − ξE p d (1 , d (2 , , where we define the function ˜ A j [ X ] = " (1 − ξ ) + jξ E p + 2 ξ E ∂ X, (4.12)and the derivative is taken with respect to the square of the center-of-mass momentum ∂ = ∂/∂ p .The second term in Eq. (4.10) is infrared divergent and should cancel out when we equate thefull-theory and EFT amplitudes. We have explicitly verified this cancellation at leading order inthe classical expansion. As mentioned in Sec. 1, our final result is the position-space Hamiltonian, H = q p + m + q p + m + V (0) ( r , p ) + V (1 , ( r , p ) L · S r + V (2 , ( r , p ) ( r · S ) r + V (2 , ( r , p ) S r + V (2 , ( r , p ) ( p · S ) r + . . . . (4.13)The potentials take the form V A ( r , p ) = G | r | c A ( p ) + G | r | ! c A ( p ) + O ( G ) . (4.14)We obtain the position-space Hamiltonian by taking the Fourier transform of the momentum-spaceHamiltonian with respect to the momentum transfer q , which is the conjugate of the separationbetween the particles r . In this way, we express the position-space coefficients c Ai in terms ofthe momentum-space coefficients d Ai via linear relations dictated by the q -dependence of the spinoperators, c (0)1 = d (0)1 , c (1 , = − d (1 , , c (2 , = − d (2 , , c (2 , = d (2 , , c (2 , = 0 , (4.15)15 (0)2 = d (0)2 , c (1 , = − d (1 , , c (2 , = − d (2 , , c (2 , = 2 d (2 , − d (2 , , c (2 , = − d (2 , . We determine the momentum-space coefficients d Ai in terms of the amplitudes coefficients a Ai by therelations in Eqs. (4.9) and (4.11). We may now obtain a Ai by demanding that the EFT amplitudematches the full-theory one, M EFT1PM = M tree E E , M EFT2PM = M E E , (4.16)where the factors of the energy account for the non-relativistic normalization of the EFT amplitude.Using Eq. (3.12) we relate a Ai to α Ai , which are explicitly shown in Eqs. (3.2) and (3.7). Puttingeverything together, we obtain novel expressions for the position-space coefficients c Ai which arelengthy, and so we only provide them in the ancillary file coefficients.m. In order to ensure the validity of our result, we compare it with existing Hamiltonians in the GeneralRelativity literature. Specifically, we compare with overlapping results in Ref. [109], which obtainedthe next-to-next-to-leading order post-Newtonian Hamiltonian, and in Ref. [110], which calculatedthe test-body Hamiltonian. Both references included interactions of up to quadratic order in thespins.One way to establish the equivalence of two Hamiltonians is to construct a canonical trans-formation that extrapolates between them. Alternatively, we may compare the gauge invariantscattering amplitudes calculated from the two Hamiltonians by means of the EFT. We take thelatter approach here. To do so, we promote the spin vector in the classical Hamiltonians to the spinoperator, and we account for the non-isotropic terms according to the conventions of [16–18].In this way we obtain EFT amplitudes in the form of Eqs. (4.8) and (4.10). The relevantcoefficients for our purposes obtained using the Hamiltonian of Ref. [109] read a (2 , = m C ES m − m m + 7 m − C ES (6 m + 16 m m + 6 m )8 m m p (4.17) − m (7 m + 4 m m − m ) + C ES (5 m − m m + 5 m )16 m m p + . . . , and a (2 , = m m C ES m + m ) p + m (cid:16) m − m m − m + C ES (32 m + 61 m m + 29 m ) (cid:17) m ( m + m )+ 15 m − m m − m m − m m − m m m ( m + m ) p + C ES (93 m + 467 m m + 707 m m + 397 m m + 64 m )64 m m ( m + m ) p + . . . , (2 , = − m m C ES m + m ) p − m (cid:16) m + 19 m m + m + C ES (20 m + 35 m m + 15 m ) (cid:17) m ( m + m ) (4.18) − m + 115 m m − m m − m m − m m m ( m + m ) p − C ES (57 m + 279 m m + 399 m m + 209 m m + 32 m )64 m m ( m + m ) p + . . . ,a (2 , = m m C ES m + m ) p + m (cid:16) m + 8 m m + m + C ES (7 m + 11 m m + 4 m ) (cid:17) m ( m + m ) p + 33 m + 97 m m + 13 m m − m m − m m m ( m + m )+ C ES (39 m + 185 m m + 245 m m + 115 m m + 16 m )32 m m ( m + m ) + . . . , where the ellipsis stands for higher orders in p . These coefficients are in complete agreement withthe velocity expansion of our amplitudes. The Hamiltonian of Ref. [110] produces the coefficients a (2 , = m ( C ES (2 γ + 2 γ − γ − − γ − γ + 3 γ + 1)2 γ ( γ + 1) m , (4.19)and a (2 , = m ( C ES (30 γ − γ + 3) − γ + 59 γ − γ − γ ( γ − m ,a (2 , = − m ( C ES (15 γ − γ + 2) − γ + 43 γ − γ − γ ( γ − m ,a (2 , = m ( C ES (15 γ − γ + 3) − γ + 46 γ − γ − γ ( γ − m , (4.20) a (2 , ˜4)2 = (95 γ − γ + 15) m γ ( γ − , a (2 , ˜5)2 = 95 γ m − γ m + 15 m γ − γ ,a (2 , ˜6)2 = 65 γ − γ + 916 γ ( γ − m , where the coefficients a (2 , )2 correspond to the spinning particle as the test body, while a (2 , ˜ )2 corre-spond to the scalar particle as the test body. These coefficients exactly reproduce the test bodyexpansion of our amplitudes. The conservative Hamiltonian we obtained in the previous section enables the calculation of physicalobservables for a binary of compact objects interacting through gravity. On the one hand, onemay calculate quantities that describe bound trajectories of the binary, as the bound-state energy.On the other hand, observables pertaining to unbound orbits have received a surge of attention.17he main reason for this is that these observables serve as input to important phenomenologicalmodels, as the effective one-body Hamiltonian [123–128]. Recently, there has been great progressin obtaining these observables directly from the scattering amplitude [9, 22, 25, 84]. Moreover, inthe non-spinning case, Refs. [26, 27] developed a dictionary between observables for unbound andbound orbits.One prominent connection between physical observables and the scattering amplitude is madevia the eikonal phase [98]. There are several studies of this connection, especially in the non-spinning case [31, 32, 98–105]. Refs. [75, 76, 82] verified the applicability of this approach forspinning particles in the special configuration where the spins of the particles are orthogonal to thescattering plane. More recently, Ref. [1] conjectured a formula that expresses physical observablesin terms of derivatives of the eikonal phase for arbitrary orientation of the spin vectors.In this section we extend the analysis of Refs. [75, 76, 82] and [1]. Specifically, we start byobtaining the eikonal phase via a fourier transform of our amplitudes. By restricting to the aligned-spin configuration we obtain a scattering angle which matches that of Ref. [75, 76, 82] when wespecialize to the black-hole case. Then, we verify the conjecture of Ref. [1] by solving Hamilton’sequations for the impulse and spin kick, and relating them to derivatives of the eikonal phase.The eikonal phase χ = χ + χ + O ( G ) is given by χ = 14 m m √ σ − Z d q (2 π ) e − i q · b M tree ( q ) ,χ = 14 m m √ σ − Z d q (2 π ) e − i q · b M + ( q ) . (5.1)Using our amplitudes expressed in the center-of-mass frame (see Eq. (3.11)) we find χ = ξEG | p | " − a (0)1 ln b − a (1 , b ( p × S ) · b + a (2 , b S ⊥ − S ⊥ · b ) b ! ,χ = πξEG | p | " a (0)2 | b | − a (1 , | b | ( p × S ) · b + a (2 , | b | S ⊥ − S ⊥ · b ) | b | ! (5.2) − (cid:16) a (2 , S + a (2 , ( p · S ) (cid:17) | b | , where we define S ⊥ ≡ S − S · pp p .We may now use the eikonal phase to obtain certain classical observables. We start by consid-ering the aligned-spin kinematics of Ref. [75, 76, 82]. Specifically, we take the spin to be parallelto the orbital angular momentum, and hence orthogonal to the scattering plane. This implies therelations S · b = S · p = 0 . (5.3)Since the scattering process is confined to a plane, it can be described by one scattering angle18 = θ + θ + O ( G ), which we obtain as a derivative of the eikonal phase [98] θ i = − Em m √ σ − ∂ b χ i , i = 1 , , (5.4)where b = | b | . The novel piece of the 2PM angle we obtain is quadratic in spin and given by θ , S = 3 EπG S m b ( σ − (cid:26) m (cid:16) σ − σ + 1) + 2 C ES (45 σ − σ + 5) (cid:17) + m (cid:16) (65 σ − σ + 1) + C ES (125 σ − σ + 29) (cid:17) (cid:27) . (5.5)By specializing to the black-hole case ( C ES = 1) we reproduce the result of Ref. [82].Ref. [1] conjectured a formula that directly relates observables in a scattering event with arbitraryspin orientations to the eikonal phase. The observables in question are the impulse ∆ p and spinkick ∆ S , where p ( t = ∞ ) = p + ∆ p , p ( t = −∞ ) = p , S ( t = ∞ ) = S + ∆ S , S ( t = −∞ ) = S . (5.6)Specifically, by obtaining the impulse and spin kick through O ( G ) using Hamilton’s equations, wefind that they may be written as∆ p ⊥ = −{ p ⊥ , χ } − { χ, { p ⊥ , χ }} − D SL ( χ, { p ⊥ , χ } ) + 12 { p ⊥ , D SL ( χ, χ ) } , ∆ S = −{ S , χ } − { χ, { S , χ }} − D SL ( χ, { S , χ } ) + 12 { S , D SL ( χ, χ ) } . (5.7)In Eq. (5.7) we use the definitions { p ⊥ , f } ≡ − ∂f∂ b , { S , f } ≡ ∂f∂ S × S , D SL ( f, g ) ≡ − S · ∂f∂ S × ∂g∂ L b ! , (5.8)where L b ≡ b × p . In the above we decompose the impulse as∆ p = ∆ p k p | p | + ∆ p ⊥ . (5.9)Eq. (5.7) does not give ∆ p k . Instead, we obtain ∆ p k from the on-shell condition ( p + ∆ p ) = p .Our calculation establishes the conjecture of Ref. [1] at the quadratic-in-spin level. The factthat the relation holds without modification when we include these higher-in-spin terms is strongindication for its validity in general. Our calculation further serves as evidence in favor of thesurprisingly compact all-order formula that relates the scattering observables to the eikonal phase,∆ O = ie − iχ D {O , e iχ D } , (5.10)where for our case O = p ⊥ or S , and χ D g ≡ χg + i D SL ( χ, g ).19 Conclusions
In this paper we obtained the 2PM-order Hamiltonian that describes the conservative dynamics oftwo spinning compact objects in General Relativity up to interactions quadratic in the spin of oneof the objects. We followed the approach of Refs. [1, 16] which was based on scattering amplitudesand EFT. Along with the results of [1] for the bilinear-in-the-spins interactions, this completes the O ( G ) analysis of quadratic-in-spin effects not including tidal effects.To construct the Hamiltonian we followed a matching procedure. Ref. [16] developed this pro-cedure for non-spinning particles, while Ref. [1] extended it to the spinning case. Specifically, wecalculated and matched two amplitudes, one in our full theory and one in an EFT. Ref. [1] intro-duced the full theory to describe the minimal and non-minimal coupling of particles of arbitraryspin to gravity. The Lagrangian contains operators that are in one-to-one correspondence withthose of the worldline EFT of [113]. The EFT we used captures the dynamics of non-relativisticspinning particles interacting via a potential with unfixed coefficients. This EFT extended the oneof [1] to include operators quadratic in the spin of one of the particles. By matching the ampli-tudes computed in these two theories, we fixed these coefficients and hence determined the desiredHamiltonian.In our calculation we considered effects up to quadratic in the spin of one of the particles,while we took the other particle to be non-spinning. In terms of our full theory, we included thefirst non-minimal-coupling operator along with the corresponding arbitrary Wilson coefficient C ES .Unlike the linear-in-spin results, the effects of this operator are not universal and generic bodiesare described by different values of C ES . As a specific example, C ES = 1 describes the Kerr blackhole. For arbitrary values of C ES , we found that the amplitude depends on q S and ( q · S ) independently, rather than on the linear combination q S − ( q · S ) . The latter was expectedbased on an observation in Refs. [1, 96]. Recently, Ref. [39] also remarked that finite-size effectsspoil the above expectation. Interestingly, for the Kerr black-hole case ( C ES = 1) the amplitudeindeed depends on the linear combination q S − ( q · S ) .The produced conservative Hamiltonian enables the calculation of observables pertaining tobinary systems of spinning black holes or neutron stars. For example, one may study bound statesof the binary by choosing suitable initial conditions. In this paper we chose to compute scatteringobservables instead, which may be used in the construction of important phenomenological modelsas the effective one-body Hamiltonian [123–128]. Specifically, by solving Hamilton’s equations weobtained the relevant impulse and spin kick. In this way we verified the conjecture of Ref. [1],which expresses these observables in terms of the eikonal phase via the simple compact formulain Eq. (5.10). The existence of such a formula has intriguing implications in classical mechanics.Specifically, it hints towards a formalism that bypasses using Hamilton’s equations, and directlyexpresses the observables in terms of derivatives of a single function of the kinematics.20n order to establish the validity of our result for the quadratic-in-spin two-body Hamiltonian,we performed several checks against the literature. We did this by comparing at the level of thegauge-invariant amplitudes in the regime where they overlap. Firstly, we verified that our amplitudeexpanded in velocity matches the one calculated using the Hamiltonian of Ref. [109], which wasobtained in the PN approximation. Secondly, by expanding our amplitude in the test-body limit wefound agreement with the amplitude obtained by the Hamiltonian of Ref. [110]. As a third check,we computed the scattering angle for the kinematic configuration where the spin vector is alignedwith the orbital angular momentum of the system and confirmed that it reproduces the one of Ref.[82] for the BH case, C ES = 1. Finally, we compared the impulse in Eq. (5.7) with the one givenin Ref. [111] in covariant form and found agreement.Our calculation serves as evidence that the formalism of Ref. [1] can capture the effects of non-minimal coupling to gravity. Therefore, an obvious future direction is to extend this analysis toinclude more powers of spin. Moreover, a number of pressing questions remain interesting andunanswered. These include the extension of these methods to higher PM orders, the proof ofthe relation between classical scattering observables and the eikonal phase, along with potentialextensions of this relation to bound-orbit observables. Acknowledgments:
We thank Zvi Bern, Radu Roiban, Chia-Hsien Shen and Fei Teng for collaboration in related topics,and Donal O’Connell for discussions. We thank Jan Steinhoff and Justin Vines for several helpfuldiscussions, and for supplying ancillary files for the post-Newtonian Hamiltonian of Ref. [113], aswell as a simpler form of the test-mass Hamiltonian based on Ref. [110]. We also thank the authorsof [111] for sharing their results for the linear impulse in advance of its publication. DK and ALare supported by the U.S. Department of Energy (DOE) under award number DE-SC0009937, andby the Mani L. Bhaumik Institute for Theoretical Physics.
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