Spin Effects in the Effective Field Theory Approach to Post-Minkowskian Conservative Dynamics
PPrepared for submission to JHEP
DESY 21-020
Spin Effects in the Effective Field Theory Approach toPost-Minkowskian Conservative Dynamics
Zhengwen Liu, Rafael A. Porto and Zixin Yang
Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, 22607 Hamburg, Germany
E-mail: [email protected] , [email protected] , [email protected] Abstract:
Building upon the worldline effective field theory (EFT) formalism for spinningbodies developed for the Post-Newtonian regime, we generalize the EFT approach to Post-Minkowskian (PM) dynamics to include rotational degrees of freedom in a manifestly covariantframework. We introduce a systematic procedure to compute the total change in momentumand spin in the gravitational scattering of compact objects. For the special case of spinsaligned with the orbital angular momentum, we show how to construct the radial action forelliptic-like orbits using the Boundary-to-Bound correspondence. As a paradigmatic example,we solve the scattering problem to next-to-leading PM order with linear and bilinear spineffects and arbitrary initial conditions, incorporating for the first time finite-size corrections.We obtain the aligned-spin radial action from the resulting scattering data, and derive theperiastron advance and binding energy for circular orbits. We also provide the (square of the)center-of-mass momentum to O ( G ), which may be used to reconstruct a Hamiltonian. Ourresults are in perfect agreement with the existent literature, while at the same time extendthe knowledge of the PM dynamics of compact binaries at quadratic order in spins. a r X i v : . [ h e p - t h ] F e b ontents A.1 Trajectories 26A.1.1 Velocity & Position 26A.1.2 Spin 28A.2 Scattering Data 28
B One-loop Integration 31 – 1 –
Introduction
The power of gravitational wave (GW) science [1] is predicated on the precise reconstructionof the GW signal as a function of the parameters of the sources, notably binary compact ob-jects [2–4]. As one may anticipate, effects due to spin play a key role in the problem, e.g. [5],particularly due to the expectation that binary black holes in the observable universe maybe rapidly rotating, e.g. [6]. Spinning black holes have also attracted interest in recent yearsdue their ability to harvest clouds of putative ultralight particles [7], which can be fleshedout either through mass/spin distributions, in GW stochastic backgrounds [8–10] or, morepromising, the precise reconstruction of the GW signal emitted from binary systems [11, 12].Rotating black holes have puzzled relativists for decades, taking almost 50 years after thediscovery of Schwarzschild’s solution to arrive at the Kerr metric [13]. Not surprisingly, thesituation does not improve in the two-body problem. Consequently, prior to the developmentof the effective field theory (EFT) approach [14, 15] (see [16–18] for detailed reviews), incor-porating spin effects in the gravitational dynamics of binary systems was a daunting task. This was the case even for conservative contributions in the perturbative Post-Newtonian(PN) regime of small-velocity/weak-gravity, with only spin-orbit results known at the timeto next-to-leading order (NLO) [20]. Presently, spin effects in the PN conservative dynamicsof binary compact systems are known up to N LO [20–29], with partial results also at higherorders, e.g. [30, 31]. Moreover, spin-independent conservative contributions — both frompotential and radiation-reaction effects — are known up to N LO [45–58], with partial resultsknown at higher orders using various methodologies, e.g. [59–62]. Most of these results in thePN regime, notably spin effects, were obtained for the first time following variants of the EFTapproach developed in [14, 15]. The goal of this paper is therefore to repurpose the worldlinetheory for spinning bodies, originally introduced for the PN expansion, to calculate spin ef-fects in the Post-Minkowskian (PM) regime using the EFT approach and boundary-to-bound(B2B) correspondence recently developed in [63–67].One of the main advantages of an EFT framework for rotating bodies — presently widelyadopted [18, 68] — is the introduction of a point-particle effective action to describe compactobjects in gravitational backgrounds [15], in contrast to applying Mathisson-Papapetrou-Dixon (MPD) equations-of-motion (EoM) independently for momentum and spin [69–71].In addition, finite-size effects can be readily incorporated as a series of corrections beyondminimal coupling constrained solely by diffeomorphism invariance [14, 15], without the needof an ansatz for the stress energy tensor. The EFT framework therefore reduces the numberof free parameters in previous (more traditional) approaches, e.g. [26]. There are a few other Needless to say, numerical simulations for spinning black holes in the strongly coupled regime are alsosignificantly more involved than non-rotating counterparts, e.g. [19]. On the other hand, spin-dependent radiation effects are only known to NLO and up to quadratic orderin the spins [32–39]. The radiated power without spin was (re-)obtained in the EFT framework of [14, 40] toN LO in [41]. Absorption effects can also be studied within an EFT worldline theory, see e.g. [42–44]. – 2 –ubtleties when dealing with the spin dynamics of point-like objects. For instance, the gaugeredundancy in describing rotational degrees of freedom in relativistic theories means that
SpinSupplementarity Conditions (SCCs) are often invoked [72]. Moreover, rather than a position the spin angular momentum behaves as a conjugated variable. This naturally leads us to aneffective theory written in terms of a Routhian [18, 22–24, 73]; that is, half as a Lagrangian(for the position/velocity) and half as a Hamiltonian (for the spin variables). The EFTmachinery then sets in, systematically “integrating out” in the saddle point approximationthe potential and radiation modes of the gravitational field, via a series of Feynman diagrams.As it is customary, divergences of the point-particle approximation are then naturally handledvia regularization/renormalization.Up until recently, efforts to solve the conservative binary dynamics of compact objectshave focused on the direct calculation of the Hamiltonian [51] or Lagrangian [52–56] as anintermedia step towards building waveforms. This was no different for spin effects [18, 68].However, building upon novel ideas from scattering amplitudes [74–76], in the last years weexperienced an explosion of work using the classical limit of amplitudes, either to compute theimpulse or to extract an effective Hamiltonian which can then be used to study generic orbits,e.g. [77–116]. These developments, which notably belong to the realm of the PM expansion,have produced the state-of-the-art for the conservative dynamics of non-spinning bodies inthe PM regime at N LO order [78–80], and very recently also partial results (with potentialmodes) at N LO [116]. On the other hand, for spinning bodies, spin-orbit and spin -spin contributions are presently known to NLO in the PM regime [82, 89, 90, 103, 104, 117, 118].(Radiation effects in the PM expansion have also been recently approached in e.g. [119–123].)While the derivation of a (classical) Hamiltonian from a (quantum) scattering amplitudeas an intermedia step is a perfectly viable option (dating back to the work of Iwasaki [124]), oneof the main paradigms in modern approaches is to avoid the introduction of gauge-dependentobjects [74, 75]. For instance, this was adopted in [81, 82] to solve for the (classical) scatteringproblem, although without providing yet the necessary link to bound states. This then becamethe main motivation for the B2B correspondence — to remain entirely within the on-shell philosophy. The B2B dictionary was introduced in [63, 64], mapping scattering data toobservables for elliptic-like orbits via a radial action and the analytic continuation in bindingenergy and angular momentum. This allowed us to directly relate gravitational observableswithout ever invoking a Hamiltonian. In its first incarnation [63], the B2B map relied onthe connection (dubbed ‘impetus formula’) between the center-of-mass momentum and the(infrared-finite) scattering amplitude in the classical limit, which was used to compute theradial action. In the second version, suggested in [63] and elaborated in [64], the dictionarywas entirely constructed from the knowledge of the scattering angle instead. Moreover, in[64] we showed how spin effects are incorporated in the B2B correspondence through theconnection between the periastron advance and scattering angle, albeit for configurations See [116] for further developments in the amplitude-action link motivated by the B2B map [63, 64]. – 3 –here spins are aligned with the orbital angular momentum. Once the B2B correspondenceis written in terms of the deflection angle, bypassing the need to go through the classicallimit of a scattering amplitude, the remaining task is to systematically compute the formerentirely within the classical domain. Following the pioneering work in [14], an EFT formalismwas developed in [65] to solve for the impulse and scattering angle in the PM regime viaFeynman diagrams, originally without spin effects. Shortly after the EFT approach wasintroduced, and benefiting from the simplifications of the classical framework together withpowerful tools for computing ‘loop’ integrals via differential equations [76, 110], the EFTformalism in [65] rapidly achieved the state-of-the-art at 3PM [66], subsequently yieldingalso new results for tidal effects beyond leading order [67]. In this paper, building uponthe EFT in [15, 23, 24, 27], we continue the development of the EFT approach in the PMregime by incorporating spin effects in the scattering problem, insofar for the conservativesector. We then implement the B2B dictionary to derive observables for bound orbits withaligned spins. Mirroring the simplifications already reflected in [65–67], the inclusion of spinin classical scattering and consequently in elliptic-like motion become remarkably simplerthan computing the Hamiltonian. As a result, we will readily achieve the state-of-the-art forspin-orbit and spin -spin effects in the PM regime, and present spin -spin contributions toNLO, including finite-size effects, for the first time.This paper is organized as follows. In § § § § § arXiv submission with more detailed results. Conventions : We use η µν = diag(+ , − , − , − ) for the Minkowski metric. The productof four-vectors is denoted as k · x = η µν k µ x ν , and k · x = δ ij k i x j for the Euclidean case,with boldface letters representing three-vectors. We use the convention (cid:15) = 1 from the xCoba package. We work in dimensional regularization in D = d − (cid:15) dimensions, with d either 4 , (cid:82) k ≡ (cid:82) d D k/ (2 π ) D , as well as ˆ δ ( x ) ≡ πδ ( x ). We use M − ≡ √ πG for the Planck mass, in (cid:126) = c = 1 units, with G Newton’s constant.– 4 –
Spinning bodies in the PM EFT approach
We start by briefly reviewing the worldline effective theory approach for rotating compactbodies [15, 23, 24]. Afterwards we show how to solve for the momentum and spin impulse ingravitational scattering to all orders in G . For more details in the EFT framework see [18]. As it is well-known, e.g. [72], additional constrained variables are needed in order to introducea local (off-shell) effective action describing a spinning body in a relativistic framework.Following the analogy with angular momentum, we use a spin tensor, S αβ , that is subject toa SSC (technically a second class constraint) [15]. In order to preserve covariance (withoutbackground fields) it is customary to resort to a covariant one, S αβ p β = 0 , (2.1)with p µ the particle’s momentum. The preservation of the SSC upon evolution implies [18] p α = 1 √ v (cid:18) mv α + 12 m R βρµν S αβ S µν v ρ + · · · (cid:19) , (2.2)with v µ ≡ dx µ dσ the particle’s velocity and σ and affine parameter. The ellipses account forhigher orders in spin and curvature. To bilinear order in the spin, the SSC in (2.1) becomes S αβ v β = 0 + O ( S ) . (2.3)The mass, m ≡ m ( S ), can be read-off from the on-shell condition, p = m , which alsoserves as a constraint enforcing reparameterization invariance [72].To introduce a worldline action, from which the equations of motion can be derived, itis convenient to use a tetrad field, e Iµ , which co-rotates with the (compact) body [15]. Usinga locally-flat frame, e aµ (with g µν e aµ e bν = η ab ), the co-rotating tetrad can be parameterized interms of an element of the Lorentz algebra, Λ Ia , via e Iµ = Λ Ia e aµ . Using these fields (and timederivatives) as degrees of freedom we can introduce an action such that we obtain the MPDequations of motions, with the spin tensor emerging as a momentum variable conjugate tothe angular velocity of the co-rotating field [15]. Because of this, rather than a Lagrangian(or a Hamiltonian), it turns out to be useful to use a Routhian instead to describe rotatingbodies, with the spin promoted to a lead-actor in the effective theory. Furthermore, it is alsoconvenient to write the worldline theory using the spin variables projected onto the locally-flatframe, S ab ≡ S µν e aµ e bν , (2.4)such that the S ab matrices obey the SO (1 ,
3) algebra, { S ab , S cd } = η ac S bd + η bd S ac − η ad S bc − η bc S ad . (2.5)– 5 –he SSC is then easily incorporated through Lagrange multipliers which are fixed by thepreservation upon evolution. These extra parameters then yield an additional (curvature-dependent) term in the worldline Routhian (see (2.8) below) [18]. As in the non-spinningcase, the worldline theory can also readily include spin-dependent finite-size effects throughdiffeomorphism invariant contributions beyond minimal coupling [14, 15]. For instance, theself-induced quadrupole moment of a rotating body is described by the coupling, first intro-duced in [15, 23, 24], C ES m (cid:90) E µν √ v e µa e νb S ac S cb dσ , (2.6)where E µν is the electric compoment of the Weyl tensor. The Wilson coefficient , C ES ,parameterizes our ignorance about the internal degrees of freedom of the compact object,either a black hole, neutron star, or any other exotic possibility. For example, for a Kerr blackhole we have C Kerr ES = 1 [24], but (much) larger values may be obtained in other scenarios, forinstance with clouds of ultralight particles surrounding black holes [11, 12].Before we move on, there is yet another important simplification that occurs when study-ing scattering processes. As it was discussed in [65–67], without spin, we can introduce aneinbein, e , and a Polyakov-type action linear in the metric field. We can then choose thegauge e = 1, which coincides with the proper-time for incoming and outgoing states. It isstraightforward to extend the same reasoning to the case of spinning bodies, resulting in apoint-particle wordline action that can be written as S pp ≡ (cid:90) + ∞−∞ dτ R , (2.7)with τ the proper-time (at ±∞ ). The Routhian, R , in the covariant SSC then takes the form R = − (cid:18) m g µν v µ v ν + ω abµ S ab v µ + 1 m R βρµν e αa e βb e µc e νd S ab S cd v ρ v α − C ES m E µν e µa e νb S ac S cb + · · · (cid:19) , (2.8)to linear order in curvature and quadratic order in the spins, with ω abµ the Ricci rotationcoefficients. The last two (curvature-dependent) terms account for the conservation of theSSC as well as finite-size effects to quadratic order in the spins, respectively. The EoM areobtained via [18] δδx µ S pp = 0 , ddτ S ab = { S ab , R} . (2.9)Notice that after expanding in the weak field limit, g µν = η µν + h µν /M Pl , (2.10)the mass coupling remains linear in the metric [65], but that is not the case for the otherterms, which instead yield non-linear gravitational interactions both at linear and bilinearorder in the spin. In what follows we show how to use this formalism to compute the totalmomentum and spin impulses in gravitational encounters.– 6 – .2 Momentum & Spin impulses The computation follows similar steps as described in [65]. We start by ‘integrating out’ thegravitational field in the potential region in a saddle-point approximation ( A = 1 , e iS eff [ x A ,S abA ] = (cid:90) D h µν e iS EH [ h ]+ iS GF [ h ]+ i (cid:82) dτ R [ x A ,S abA ,h ] , (2.11)where S EH and S GF are the Einstein-Hilbert action and gauge-fixing terms, respectively. Asexplained in [65], we adapt S GF (as well as total time-derivatives) to simplify the resultingFeynman rules. The effective Routhian/action then becomes a (local-in-time) function ofthe position and spin of the two-body systems, S eff = (cid:88) n (cid:90) dτ R n [ x ( τ ) , S ( τ ); x ( τ ) , S ( τ )] . (2.12)The R n ’s are the O ( G n ) contribution to the worldline Routhian after evaluating the Feynmanintegrals for generic configurations. From here we can then use (2.9) to obtain the EoM, whichwe can solve iteratively in powers of G , both for the position variables [65], x µA ( τ A ) = b µA + u µa τ a + (cid:88) n δ ( n ) x µA ( τ A ) ,v νA ( τ A ) = u νA + (cid:88) n δ ( n ) v νA ( τ A ) , (2.13)as well as the spin in a locally-flat frame, S abA ( τ A ) = S abA + (cid:88) n δ ( n ) S abA ( τ A ) . (2.14)The initial values, { b µA , u µA , S abA } , are related to the impact parameter, b ≡ b − b , incomingvelocity and spin, respectively. Since the perturbation vanishes at infinity, we have e aµ → δ aµ .Hence, the locally-flat frame and Lorentzian one (where the initial spins and velocities aredefined) coincide. This observation allows us to enforce the SSC in (2.3) via the constraint S µν u ν = 0 , (2.15)on the initial data, which is then preserved by the evolution equations in (2.9). The totalmomentum impulse is obtained as in [65], but with a Routhian rather than a Lagrangian,∆ p µA = − η µν (cid:88) n (cid:90) + ∞−∞ d τ A ∂ R n ∂x νA . (2.16)Similarly to the non-spinning case, the iterations of the EoM on lower order contributions tothe effective action play an important role [65], and we have the same type of decomposition∆ ( n ) p µA = (cid:88) k ≤ n ∆ ( n ) R k p µA , (2.17) We are ignoring here the non-local contributions due to radiation-reaction (tail) effects [48, 50, 54], whichinclude also spin-dependent effects at higher PM orders. – 7 –t n PM order, with∆ ( n ) R k p µA ≡ − η µν (cid:90) + ∞−∞ d τ A (cid:32) ∂∂x νA R k (cid:34) b A ( B ) + u A ( B ) τ A ( B ) + n − k (cid:88) r =0 δ ( r ) x A ( B ) ; S abA ( B ) + n − k (cid:88) r =0 δ ( r ) S abA ( B ) (cid:35)(cid:33) O ( G n ) . (2.18)Likewise, the total change of spin follows from∆ S abA = (cid:88) n (cid:90) + ∞−∞ dτ A (cid:110) S abA , R n (cid:111) , (2.19)which must be evaluated iteratively on solutions to the EoM, yielding the same structure, i.e.∆ ( n ) R k S abA , as in (2.18).It is somewhat convenient to re-write the final covariant expressions, obtained after usingthe EoM though (2.9), in terms of the initial Pauli-Lubanski vector, S µA = m A a µA ≡ (cid:15) µναβ S αβA u νA , (2.20)where we take advantage of the fact that both the incoming velocity and spin tensor live inthe same (inertial) frame. This observation drastically simplifies the handling of the SSC inthe scattering problem. By considering only incoming/outgoing states in Minkowski space,the complexity due to the mismatch between the locally-flat and ‘PN frame’ disappears.Furthermore, for the case of spins aligned with the angular momentum, it is easy to see thatthe motion remains in a plane, and we can compute the standard deflection angle, e.g. [65],2 sin (cid:16) χ (cid:17) = (cid:112) − ∆ p p ∞ , (2.21)where the momentum at infinity, p ∞ , is given by p ∞ = µ (cid:112) γ −
1Γ = µ ˆ p ∞ , (2.22)and γ ≡ u · u , (2.23)Γ ≡ E/M = (cid:112) ν ( γ − , (2.24)with E the total energy in the CoM frame. Throughout the remaining of this paper we usethe notation M = m + m for the total mass, µ = m m /M for the reduced mass, and ν ≡ µ/M for the symmetric mass ratio. We also introduce the (reduced) binding energy, E ,such that E = M (1 + ν E ) . (2.25)– 8 – Aligned-spin Boundary-to-Bound correspondence
In principle, the B2B dictionary with generic spins would require a map for non-planar motion.However, a major simplification arises for aligned-spin configurations, which we have shownin [64] is amenable to the same correspondence between the periastron advanced, ∆Φ, andscattering angle, χ , ∆Φ( J, E )2 π = χ ( J, E ) + χ ( − J, E )2 π , E < , (3.1)albeit with the canonical total angular momentum, J ≡ L + S + S , as opposite to the orbitalangular momentum, L , which enters in the non-spinning case. We review in what followshow to use (3.1) to reconstruct the bound radial action from scattering data for aligned spins.For convenience, we will write various results in terms of the spin vector in (2.20), whichfor aligned spins obeys a µA u Aµ = a µA b µ = 0, and introduce the scalar variables a A ≡ a A · L .Moreover, we often use the spin parameters a ± = a ± a for the two-body state, as well as there-scaled variables ˜ a ± ≡ a ± / ( GM ), (cid:96) ≡ L/GM µ for the spin and orbital angular momentum.
As it was shown in [64], for the case of non-spinning bodies the relationship in (3.1) allowsus to construct the (reduced) radial action for the bound problem, i r ( E , (cid:96) ), in terms of theanalytic continuation to negative binding energy of the PM coefficient of the scattering angle, χ (cid:88) n χ ( n ) b ( E ) (cid:18) GMb (cid:19) n = (cid:88) n χ ( n ) (cid:96) ( E ) (cid:96) n , (3.2)yielding (cid:0) with sg(ˆ p ∞ ) ≡ ˆ p ∞ / (cid:112) − ˆ p ∞ (cid:1) i r ( E , (cid:96) ) = sg(ˆ p ∞ ) χ (1) (cid:96) ( E ) − (cid:96) (cid:32) π (cid:88) n =1 χ (2 n ) (cid:96) ( E )(1 − n ) (cid:96) n (cid:33) (without spin) . (3.3)The expression in (3.3) does not translate directly to the spinning case. For starters, theexpansion in (3.2) gets modified into a two-scale expansion with spin effects, such that inaddition to the standard factors of GM/b we also have an expansion in a ± /b . This can becircumvented by the introduction of the dimensionless variables ˜ a ± = a ± / ( GM ), which allowsus to conveniently keep the same type of expansion as in (3.2) (with χ ( n ) (cid:96) ( E , ˜ a ± ) coefficients)at the expenses of a minor mismatch in the G power-counting. Hence, using the fact that therelationship in (3.1) involving both the orbital and spin angular momentum still applies, wecan once again integrate with respect to L and perform the same manipulations as in [64] toobtain, after some trivial re-arrangement, i r ( E , (cid:96), ˜ a ± ) = sg(ˆ p ∞ ) χ (1) (cid:96) ( E ) + (cid:96) − π ∞ (cid:88) n =1 χ (2 n +1) (cid:96), odd ( E , ˜ a ± )2 n (cid:96) n +1 + χ (2 n ) (cid:96), even ( E , ˜ a ± )(2 n − (cid:96) n . (3.4)– 9 –he χ ( k ) (cid:96), odd(even) ( E , ˜ a ± ) are the odd (and even) contributions in the ˜ a ± spin variables, analyt-ically continued to negative binding energies. The expression in (3.4) plays a similar role as(3.3), with the addition of the odd contributions accounting for spin-orbit corrections. Theeven terms including not only spin-independent factors, but also effects quadratic in spin.Higher orders in spin follow the same pattern.There is still an important caveat in the B2B dictionary for spinning bodies. The solutionto the scattering problem produces results in an expansion in GM/b , with b the covariant impact parameter, as in (3.2). However, for rotating bodies the latter is not directly relatedto (cid:96) , the canonical orbital angular momentum. Instead we have [117, 118] (cid:96) = ˆ p ∞ bGM + Γ − ν (cid:18) ˜ a + − δ Γ ˜ a − (cid:19) , (3.5)where δ ≡ √ − ν ( m − m ) / | m − m | . This introduces an additional expansion in ˜ a ± /(cid:96) once the scattering angle in (3.2) is written in covariant form, mixing the power-counting.For example, it leads to spin-dependent contributions stemming off of the spin-independentdeflection angle in impact-parameter space [64]. As it was demonstrated in [63, 64], the B2B dictionary relies on the connection between theorbital elements for hyperbolic- and elliptic-like motion [63, 64]. The orbital elements areobtained from the roots of the radial momentum, which can be solved as a function of thebinding energy using a gauge where the (canonical) impetus takes the quasi-isotropic form P r = p ∞ (cid:32) ∞ (cid:88) i =1 f i ( E , (cid:96) ˜ a ± , ˜ a ± , · · · ) ( GM ) i r i (cid:33) − L r . (3.6)The expression in (3.6) also allows us to re-write the radial action in terms of the f i ’s, using thesame algebraic relationships uncovered in [63, 64]. For instance, for the case of non-spinningbodies, the coefficients in the PM expansion of (3.2) are related to the CoM momentum in(3.6), via [63] χ ( n ) (cid:96) ( E ) = √ π (cid:18) n + 12 (cid:19) (cid:88) σ ∈P ( n ) ˆ p n ∞ Γ (cid:0) n − Σ k (cid:1) (cid:89) k f σ k σ k ( E ) σ k ! (without spin) , (3.7)which follows from Firsov’s solution to the scattering problem [125]. (See [63] for details onthe combinatorial manipulations involved in (3.7).) Using the expression in (3.3), the relationin (3.7) then leads to an alternative representation for the radial action — so far for the caseof non-spinning bodies. However, as demonstrated in [64] (see its Appendix A), the resultingform in terms of the f i ’s coincides with the PM expansion of the radial action that followsfrom the direct integration of the radial momentum, i.e. i r ( (cid:96), E , ˜ a ± ) = 12 πGM µ (cid:73) P r ( (cid:96), E , ˜ a ± ) dr , (bound) (3.8)– 10 –sing Sommerfeld’s contour in the complex plane (originally performed to all orders in [63]).Hence, after noticing that the spin and angular momentum are simple spectators in all manip-ulations involving integration over the radial coordinate, it is straightforward to conclude thatthe general solution for the radial action in terms of the coefficients of the CoM momentum(in isotropic gauge) carries over unscathed onto the spinning case, obtaining i r ( E , (cid:96), ˜ a ± ) = ˆ p ∞ (cid:112) − ˆ p ∞ f (cid:96) √ π ∞ (cid:88) n =0 (cid:18) ˆ p ∞ (cid:96) (cid:19) n Γ (cid:18) n − (cid:19) (with spin) × (cid:88) σ ∈P (2 n )
1Γ (1 + n − Σ k ) (cid:89) k f σ k σ k ( E , (cid:96), ˜ a ± ) σ k ! , (3.9)in terms of the coefficients in (3.6). For instance, we have [63, 64] i r ( E , (cid:96), ˜ a ± ) = − (cid:96) + ˆ p ∞ (cid:112) − ˆ p ∞ f p ∞ (cid:96) f + ˆ p ∞ (cid:96) (cid:18) f f f + f (cid:19) + ˆ p ∞ (cid:96) (cid:16) f + 6( f f + f ) f + 3( f f + 2 f f + f + f ) (cid:17) + 5ˆ p ∞ (cid:96) (cid:0) · · · + 6 f f + · · · (cid:1) + · · · , (3.10)where we kept only the one piece in the final term which will be needed later on. (The readershould keep in mind that the f i ’s themselves may also depend on (cid:96) .)Notice that, similarly to the non-spinning case, the f k ’s contribute also at n PM order(for n ≥ k ). This will allow us to perform a consistent PN-truncation, as discussed in [63–67].We will return to this point in § It is useful to relate the PM coefficients in (3.6) to the deflection angle in (3.2), also with spineffects. This will allow us to perform the analytic continuation of the CoM momentum tonegative binding energies, and also find a Hamiltonian if so desired. The main observation isthe same we used to arrive at the equivalent representation for the radial action in (3.9). Thatis, spin and angular momentum are going for the ride when the radial action is constructedvia the integral of the radial momentum, regardless of whether we consider bound or unboundorbits. Hence, the result of the integral i r ( E , (cid:96), ˜ a ± ) = 12 πGM µ (cid:90) ∞−∞ P r ( E , (cid:96), ˜ a ± ) dr (unbound) , (3.11)remains also the same, with the f i ’s in (3.6) including spin-dependent parts. Moreover, sincethe scattering angle obeys − ∂∂(cid:96) i r ( E , ˜ a ± ) = 12 + χ ( (cid:96), E , ˜ a ± )2 π (unbound) , (3.12)– 11 –e can solve for the unbound radial action, which can then be written as i r ( (cid:96), E , ˜ a ± ) = − (cid:96) − χ (1) (cid:96) ( E ) log (cid:96)π − (cid:96)π (cid:88) n ≥ χ ( n ) (cid:96) [ f i ](1 − n ) (cid:96) n f i → f i ( E ,(cid:96), ˜ a ± ) (unbound) , (3.13)with the functional form of χ ( n ) (cid:96) [ f i ] given exactly by the expression in (3.7), to all PM orders.Let us stress two related important points regarding (3.13). First of all, there could bea constant of integration (depending only on the binding energy) as in the bound case [63].Moreover, the n = 1 term ( ∝ log (cid:96) ) is a bit subtle when considering the analytic continuation.The constant of integration may be fixed by using the expression in (3.9) and imposing i (bound) r ( E < , (cid:96), ˜ a ± ) = i (unbound) r ( E < , (cid:96), ˜ a ± ) − i (unbound) r ( E < , − (cid:96), − ˜ a ± ) , (3.14)which follows directly from the B2B relation in (3.1). However, this requires a choice forthe branch of the logarithm, when performing the analytical continuation to negative orbitalangular momentum. We find the choice log( (cid:96) ) /π − log( − (cid:96) ) /π → ∓ i , in combination withˆ p ∞ → ± i ˆ p ∞ for the analytic continuation in the binding energy, leads to − χ (1) (cid:96) ( E ) (cid:18) log (cid:96)π − log( − (cid:96) ) π (cid:19) → sg(ˆ p ∞ ) χ (1) (cid:96) ( E ) , (3.15)uniquely fixing the unbound radial action. For convenience, since we work here to quadratic order in the spins, in what follows wedecompose the coefficients of the CoM momentum as f i ( E , (cid:96), ˜ a ± ) = f i ( E ) + (cid:96) (cid:88) A = ± ˜ a A f Ai ( E ) + (cid:88) { A,B } = ± ˜ a A ˜ a B f ABi ( E ) + · · · , (3.16)with f i ( E ) the spin-independent part, and { f Ai ( E ) , f ABi ( E ) } (dimensionless) functions of thebinding energy (and masses). Likewise for the coefficients in (3.2), χ ( n ) (cid:96) ( E , ˜ a ± ) = χ ( n )0 ( E ) + (cid:88) A = ± ˜ a A χ ( n ) A ( E ) + (cid:88) { A,B } = ± ˜ a A ˜ a B χ ( n ) AB ( E ) + · · · (3.17)(we suppress the (cid:96) -subscript on the RHS for notational convenience). Hence, applying (3.12)to (3.13), while keeping track of all the (cid:96) ’s inside the f i ’s in (3.16), we can derive the scatteringangle in terms of the CoM momentum including spin effects to all PM orders. As we willsee momentarily, terms linear and quadratic in the spin first show up at n = 2 and n = 3 This analytic continuation is behind the relationship between the total radiated energy for unbound orbitsand the energy emitted over a period, discussed in [126]. This relationship follows immediately from the B2Bmap applied to the (local part of the) conservative tail effect [48, 57, 62]. The choice must be uniformly adopted to be consistent with the analytic continuation in the (covariant)impact parameter, which was used in [64] to connect the orbital elements. – 12 –n (3.17), respectively. This is intuitively simple to understand, and it follows directly fromthe expansion in impact-parameter of the deflection angle yielding extra factors of a ± /b oncespin is included. As a consequence, f A , ( E ) = f AB , ( E ) = 0 . (3.18)For the remaining coefficients, we find χ (2) A = ˆ p ∞ f A , (3.19) χ (3) A = π ˆ p ∞ (cid:0) f f A + f A (cid:1) , (3.20) χ (3) AB = ˆ p ∞ f ( AB )3 , (3.21) χ (4) AB = 3 π ˆ p ∞ (cid:18) f (0)1 f AB + f AB + 3ˆ p ∞ f A f B (cid:19) , (3.22)where (recall ˜ a ± = a ± / ( GM )) we kept only terms which contribute to O ( G ). Incidentally,notice these values are consistent with the equivalence between the two representations ofthe radial action, in (3.4) and (3.9)-(3.10). It is straightforward to invert these equations toobtain the value of the CoM impetus. See § In this section we apply the EFT formalism for spinning bodies to compute the total mo-mentum and spin impulses to 2PM and quadratic order in the spins. The needed topologiesare shown in Fig. 1 to 2PM order. The vertices at the worldline may include mass andspin couplings, both linear (from the Ricci-rotation coefficients) and bilinear (from the SSCand finite-size terms) in the spins. Because the coupling to the mass is linear in the met-ric perturbation in our (Polyakov-type) gauge [65], the diagram in Fig. 1b only contributesspin-dependent effects. The ‘tree-level’ diagram in Fig. 1a contributes both at 1PM and 2PMorder, the latter through the iteration of the EoM described in § S µA ≡ m A (cid:15) µναβ S αβA p νA , (4.1)which coincides with the value in (2.20) at early times. We obtain the spin impulse in termsof the total change in the spin tensor, using the spin algebra in (2.5) on the Routhian/action,in combination with the momentum impulse. As a non-trivial check, the results below can Notice the leading linear and quadratic terms, scaling as (
GM/b )( a ± /b ) and ( GM/b )( a ± /b ) , have thewrong parity through the B2B map and therefore do not contribute to (3.4). – 13 – a) (b) (c) Figure 1 : Feynman topologies needed to 2PM order. (See text.)be shown to be consistent with the preservation of the SSC, S µ p µ = 0, the on-shell condition, p = m , and the constancy of the magnitude of the spin, S µ S µ = a , to 2PM order.We illustrate the basic ideas and quote the results in what follows, with supplementalmaterial in appendix A, and a few comments on the integration procedure in appendix B.Throughout this section we use the notation | b | ≡ (cid:112) − b µ b µ and ˆ b µ ≡ b µ / | b | . Moreover, wealso use κ ± = C (1) ES ± C (2) ES , and the tensorial structure [82]Π µν ≡ (cid:15) µραβ (cid:15) νργδ u α u β u γ u δ γ − ,T µνρ ≡ ˆ b ρ Π µν + ˆ b ν Π ρµ + ˆ b µ Π ρν ,u µ ⊥ ≡ u µ − γu µ . (4.2) The derivation of the tree-level Routhian/action is straightforward. Following the same stepsas in [65] and evaluating on the unperturbed solutions in (2.13) and (2.14), we obtain∆ (1) a p µ = νGM | b | γ (cid:112) γ − (cid:15) αρβσ a ρ u β u σ (cid:16) Π µα + 2ˆ b µ ˆ b α (cid:17) − (1 ↔ , (4.3)for the spin-orbit contributions, whereas at quadratic order∆ (1) a a p µ = νGM | b | (cid:0) γ − (cid:1)(cid:112) γ − a α a β (cid:16) T αβµ + 4ˆ b α ˆ b β ˆ b µ (cid:17) − (1 ↔ , (4.4)and ∆ (1) a p µ = νGM | b | C (1) ES (cid:0) γ − (cid:1)(cid:112) γ − a α a β (cid:16) T αβµ + 4ˆ b α ˆ b β ˆ b µ (cid:17) − (1 ↔ , (4.5)the latter including the insertion of the finite-size term in (2.8). In all of these expressions wehave (anti-)symmetrized the result (under which b µ → − b µ and δ → − δ ). It is straightforwardto show that all of these 1PM values coincide with the results reported in [82, 117, 118] forthe case of Kerr black holes (with C ES = 1).– 14 – .1.2 Next-to-leading order For the NLO results we must evaluate the one-loop diagrams in Figs. 1b & 1c on the unper-turbed solution, and use the trajectories (shown in Appendix A) to compute the iterationwith the tree-level diagram in Fig 1a. The results are:∆ (2) a p µ = νG M | b | (cid:104) D (cid:15) αρβσ a ρ u β u σ (cid:16) Π µα + 3ˆ b α ˆ b µ (cid:17) + D (cid:15) µαρβ a ρ u β ˆ b α + (cid:16) a ρ u β u σ ˆ b α (cid:15) αρβσ (cid:17) ( D u µ + D u µ ) (cid:105) − (1 ↔ , (4.6)∆ (2) a p µ = νG M | b | (cid:104) D a α a β (cid:16) T αβµ + 5ˆ b α ˆ b β ˆ b µ (cid:17) + D a α ( a · u ) (cid:16) Π αµ + 4ˆ b α ˆ b µ (cid:17) + a α a β ( D u µ − D u µ ) (cid:16) Π αβ + 4ˆ b α ˆ b β (cid:17) + ˆ b µ (cid:0) D a + D ( a · u ) (cid:1) +2 D a µ ( a · u ) + ( a · u ) ( D u µ + D u µ ) − a ( D u µ − D u µ ) (cid:3) − (1 ↔ , (4.7)∆ (2) a a p µ = νG M | b | (cid:20) D a α a β (cid:16) T αβµ + 5ˆ b α ˆ b β ˆ b µ (cid:17) + D a α ( a · u ) (cid:16) Π αµ + 4ˆ b α ˆ b µ (cid:17) + D a α a β u µ (cid:16) Π αβ + 4ˆ b α ˆ b β (cid:17) + 12ˆ b µ ( D ( a · a ) + D ( a · u )( a · u ))+2 D a µ ( a · u ) + D ( a · u )( a · u ) u µ + D ( a · a ) u µ ] − (1 ↔ . (4.8)The D i coefficients are displayed in Appendix A. We now move to the computation of the spin dynamics. As we discussed earlier, we quotethe result in terms of the spin vector in (4.1). We find,∆ (1) a S µ = − νGM | b | (cid:112) γ − (cid:16) (ˆ b · a )( u µ − γu µ ) + 2 γ ˆ b µ ( a · u ) (cid:17) (4.9)at linear order in the spins, while at quadratic order we arrive at∆ (1) a a S µ = νGM | b | (cid:112) γ − (cid:15) µβσρ (cid:16) Π αβ + 2ˆ b α ˆ b β (cid:17) (cid:16) u ρ u σ ( a α ( a · u ) − γa α ( a · u ))+ γ a ρ a α u σ − a α a ρ (cid:0) ( γ − u σ + 2 γu σ ⊥ (cid:1) (cid:17) (4.10)∆ (1) a S µ = − νGM | b | (cid:112) γ − (cid:16)(cid:0) γ − (cid:1) C (1) ES u ρ + 2 γu ρ ⊥ (cid:17) (cid:15) µβσρ a α a σ (cid:16) Π αβ + 2ˆ b α ˆ b β (cid:17) , (4.11)These results are, once again, in agreement with the 1PM variation obtained in [82] for thecase of Kerr black holes. – 15 – .2.2 Next-to-leading order The total change of spin at NLO is significantly more cumbersome. While, based on variousarguments, we suspect an underlying structure that extends the compact expressions at 1PMorder [117], we have not been able to uncover it so far. Yet, we believe these (manifestlycovariant) expressions are perhaps the best hope to unravel a deeper (spacetime) structure.See 6 for more on this point. The results are:∆ (2) a S µ = νG M | b | (cid:104) D a α (Π µα + 2ˆ b α ˆ b µ ) − D a µ + D (cid:16) (ˆ b · a ) u µ ⊥ − ˆ b µ ( a · u ) (cid:17) + (cid:16) D u µ + D ˆ b µ (cid:17) (ˆ b · a ) + (cid:0) D u µ + D u µ ⊥ (cid:1) ( a · u ) (cid:105) (4.12)∆ (2) a a S µ = νG M | b | (cid:15) µναβ (cid:104) − D u ν u β a σ a ρ (cid:16) T αρσ + 4ˆ b α ˆ b ρ ˆ b σ (cid:17) + (cid:16) Π ασ + 3ˆ b α ˆ b σ (cid:17) (cid:16) D (cid:16) u β u ν ( a · u ) + a ν u β ⊥ (cid:17) + (cid:16) D a β a σ u ν + D u β u ν ( a σ ( a · u ) + γa σ ( a · u )) − D a σ a β u ν (cid:17)(cid:17) + (cid:16) Π σν + 2ˆ b σ ˆ b ν (cid:17) (cid:16) − D ˆ b σ a α ( a · u ) (cid:16) u β − γu β (cid:17) − D ˆ b α u β ⊥ a σ ( a · u )+ ˆ b σ u β u α ( D ( a · a ) − γD ( a · u )( a · u )) − D u α u β a σ (ˆ b · a )+ ( γ − D a σ a α ˆ b β + 12 γD a α a σ ˆ b β + D γ ( a · u )ˆ b α a σ ( γu β + u β ) (cid:19) − D (cid:16) Π σρ + 2ˆ b ρ ˆ b σ (cid:17) a σ a ρ ˆ b α u β u ν + D a α u β u ν ( a · ˆ b ) − D a α u β u ν ( a · ˆ b )+ a α ˆ b β ( a · u ) ( D u ν + D u ν ) + ˆ b α u β u ν ( D ( a · a ) + D ( a · u )( a · u ))+ a α ˆ b β ( a · u ) ( D u ν + D u ν ) + D a α u β u ν ( a · u ) + D a α u β u ν ( a · u )+ γD a α a β ˆ b ν − D a α a β u ν (cid:105) , (4.13)∆ (2) a S µ = νG M | b | (cid:15) µναβ (cid:104) D a ρ a σ u β u ν (cid:16) T αρσ + 4ˆ b α ˆ b ρ ˆ b σ (cid:17) + a ν a σ (cid:18) D u β + D u β ⊥ (cid:19) (cid:16) Π ασ + 3ˆ b α ˆ b σ (cid:17) + (cid:16) Π νσ + 2ˆ b σ ˆ b ν (cid:17) (cid:16) D u α u β (cid:16) a ˆ b σ − a σ (ˆ b · a ) (cid:17) + 2 γD ˆ b α a σ ( a · u ) (cid:16) u β ⊥ + (cid:0) γ − (cid:1) u β (cid:17) − D ( a · u ) a α ˆ b σ ( u β − γu β )+ D ( a · u ) a σ ˆ b α u β + D ˆ b β a α a σ (cid:17) + D ˆ b α u ν u β a ρ a σ (cid:16) Π σρ + 2ˆ b ρ ˆ b σ (cid:17) +ˆ b ν u α u β (cid:0) D a + D ( a · u ) (cid:1) + D a ν u α u β ( a · ˆ b )+ a ν ˆ b α ( a · u ) (cid:16) D u β + D u β (cid:17) − D a α u β u ν ( a · u ) (cid:21) , (4.14)with the remaining D i ’s also collected in Appendix A.– 16 – .3 Canonical variables In order to apply the B2B dictionary we must also understand the map to canonical variables,in particular for the orbital angular momentum. This will be useful also to compare our resultswith the derivations in [104], obtained directly in terms of canonical spins. Here we followclosely the analysis put forward in [117, 118] (see also [27]), which we recommend for furtherdetails, while warning the reader to pay attention to the different conventions.The canonical (or Newton-Wigner) spin constraints may be written with the aid of abackground time-like four-vector, U µ , such that the SSC becomes, in contrast to (2.15), S µν can ( U ν + u ν ) = 0 , (4.15)for the (initial) spin and velocities. One can then search a transformation S µν can = S µν + m u [ µ δx ν ] , (4.16)between covariant and canonical variables, with x µ can = x µ + δx µ (obeying δx · u = 0). Weproceed as follows. Firstly, we split the velocity as u µ = ˆ E U µ + u µ ⊥ , (4.17)with ˆ E = E/m ≡ u · U , the body’s (reduced) energy in the U -frame. Hence, introducing thecanonical spin vector as a µ can ≡ m (cid:15) µναβ U ν S αβ can , (4.18)we find a µ can = a µ + u ⊥ · a ˆ E (cid:18) U µ + u µ ⊥ ˆ E + 1 (cid:19) , (4.19)for the relationship to the covariant spin four-vector, and δx µ = − E + 1 S µα can u ⊥ α . (4.20)We now move to the two-body problem and the CoM frame, and choose the backgroundfour-vector as U ν = δ ν . Hence, using the SSC for the covariant spin, we can re-write (4.19) a A, can = 0 , a A, can = a A − u A, ⊥ · a A ˆ E A ( ˆ E A + 1) u A, ⊥ , (4.21)for each particle. For the sake of comparison, it is also convenient to invert the relationship, a A = u A, ⊥ · a A ˆ E A = u A, ⊥ · a A, can , a A = a A, can + u A, ⊥ · a A, can ( ˆ E A + 1) u A, ⊥ . (4.22)where the velocity is given by u A, ⊥ = ( − A +1 p /m A in the CoM frame.– 17 –inally, using that u ⊥ · U = 0, we also find δ x A = u A, ⊥ × a can ( ˆ E A + 1) → b = b can − p × Ξ can ( ˆ E A + 1) , (4.23)for the change of impact parameter between covariant and canonical coordinates, where weintroduced the three-vector Ξ ≡ (cid:88) A a A, can m A ( ˆ E A + 1) . (4.24)From (4.21)-(4.22) it follows that the spin variables remain invariant for aligned spins, forwhich p · a = 0, and moreover do not evolve with time. In addition, the shift in (4.23) yieldsthe relation between covariant and canonical orbital angular momentum in (3.5), which isneeded for the B2B map, we implement momentarily. These transformations also allow us tocompare the results reported in this paper and those in [104]. After applying (4.22)-(4.23) toour results (see also Eq. (2.17) in [104]) we find full agreement for the NLO spin-orbit andspin -spin momentum impulse and spin kick. For the case of spins aligned with the orbital angular momentum, the motion remains in theplane, and the following applies (cid:15) µνασ ˆ b µ u ν u α a σA = (cid:112) γ − a A , (4.25)with the sign of a A (= a A · L ) determined by the direction of the spin w.r.t. the orbitalangular momentum. The scattering angle then follows from the total change of momentumin the CoM, see (2.21). The result, to O ( G ) and quadratic order in the spins, reads∆ ( a,a ) χ Γ = − GM | b | (cid:32) γ (cid:112) γ − a + | b | − γ − γ −
1) ( κ + + 2) a + ( κ + − a − + 2 κ − a − a + | b | (cid:33) (4.26) − π (cid:18) GM | b | (cid:19) (cid:32) γ (5 γ − γ − / a + + δ a − | b | − γ − λ ++ a + λ −− a − + 2 λ + − a + a − | b | (cid:33) including finite-size effects, with λ ++ = 830 γ − γ + 110 + (35 γ − γ + 19) δ κ − + (215 γ − γ + 39) κ + ,λ −− = − γ + 468 γ −
82 + (35 γ − γ + 19) δ κ − + (215 γ − γ + 39) κ + ,λ + − = (215 γ − γ + 39) κ − + ( γ − (cid:0) γ + 10 + (35 γ − δ κ + (cid:1) . (4.27)It is straightforward to show that, for Kerr black holes, we have∆ (1)Kerr χ Γ = − GM | b | (cid:32) γ (cid:112) γ − a + | b | − (cid:0) γ − (cid:1) ( γ − a | b | (cid:33) + O ( a ) , (4.28) Notice that the condition for the spin tensor/vector in Eq. (2.18) of [104] has an overall minus sign,however, they also use the (opposite) convention (cid:15) = − – 18 – (2)Kerr χ Γ = − π G M | b | (cid:18) γ (5 γ − γ − / (cid:18) δ a − | b | + 7 a + | b | (cid:19) − γ − (cid:20) δ (cid:0) γ − γ + 1 (cid:1) a + a − | b | − (cid:0) γ − γ + 1 (cid:1) a − | b | + (cid:0) γ − γ + 47 (cid:1) a | b | (cid:21) (cid:33) + O ( a ) , (4.29)such that our result is consistent with [117] and the conjecture in [118], respectively.In order to apply the B2B map, we must re-write the expression in (4.26) as an expansionin (cid:96) and ˜ a ± , to read off the relevant PM coefficients of the scattering angle. Using thedecomposition in (3.17), we find the following values χ (2)+ = (2 γ − (cid:112) γ − γ −
1Γ + 1 − γ (cid:112) γ − , (4.30) χ (2) − = − (2 γ − (cid:112) γ − γ − δ Γ(Γ + 1) , (4.31) χ (3)+ = π (cid:18) γ − γ − − γ (5 γ − (cid:19) , (4.32) χ (3) − = − πδ (cid:18) γ − γ − (Γ + 1) + (5 γ − γ (cid:19) , (4.33) χ (3)++ = 18Γ (cid:32) γ − (2 γ −
1) Γ (cid:112) γ − − γ ( γ − (cid:112) γ −
1Γ + 1 + 2 (cid:112) γ − γ − κ + + 2) (cid:33) ,χ (3) −− = 18Γ (cid:18) γ − (2 γ − δ (cid:112) γ − + 2 (cid:112) γ − γ − κ + − (cid:19) , (4.34) χ (3) − + = χ (3)+ − = 1Γ (cid:18) (2 γ − γ − δ (cid:112) γ − + 16(2 γ +1)( γ − δ (cid:112) γ − (cid:112) γ − γ − κ − (cid:19) , (4.35) χ (4)++ = 3 π (cid:18) ( γ − (5 γ − − γ ( γ − γ − γ − γ + 110+ δ κ − (35 γ − γ + 19) + κ + (215 γ − γ + 39) (cid:19) , (4.36) χ (4) −− = 3 π (cid:18) δ ( γ − (5 γ − + 64 δ γ ( γ − γ − − γ + 468 γ − δ κ − (35 γ − γ + 19) + κ + (215 γ − γ + 39) (cid:19) , (4.37) χ (4) − + = χ (4)+ − = 3 π (cid:18) γ − γ − δ (Γ + 1) + 64(5 γ + 10 γ − γ + 6 γ + 3) δ Γ + 1 (4.38) − γ − γ − γ − γ − δ + (215 γ − γ +39) κ − + ( γ − γ − δ κ + (cid:19) . – 19 – Bound states to 2PM: linear and bilinear (aligned) spin effects
Once the scattering angle is computed the radial action for the bound problem follows, viaanalytic continuation. Crucially, the relationship between b and (cid:96) in (3.5) also introduces spineffects. This forces us to keep not only spin-dependent terms but also the spin-independentcorrections, computed in [65]. From the radial action it is straightforward to derive thegravitational observables through differentiation. In what follows we illustrate the procedureto 2PM order. We also discuss how to incorporate the extra terms needed to complete theNLO contributions to the binding energy linear and bilinear in spin to 3PN order. Finally,we provide the coefficients in the CoM momentum to O ( G ), and all orders in velocity. From the terms in (4.30)-(4.38), only those which are even in the total angular momentumsurvive the B2B map, yielding for the (bound) radial action to 2PM order: i r ( E , (cid:96), ˜ a ± ) = − (cid:96) + 2 γ − (cid:112) − γ + 34 (cid:96) γ −
1Γ + 1 π (cid:88) A = ± χ (3) A ( γ ) ˜ a A (cid:96) + 23 π (cid:88) { A,B } = ± χ (4) AB ( γ ) ˜ a A ˜ a B (cid:96) , (5.1)after adding the results in [65] for the spin-independent terms. The reader will immediatelynotice that the analytic continuation to negative binding energies ( γ <
1) follows smoothly.While the expression in (5.1) allows us to compute observables, incorporating an infiniteseries of velocity corrections at a given order in 1 /(cid:96) , the fact that we truncate the radialaction in the PM expansion prevents us from having direct access to the information neededto consistently obtain the PN effects associated with our PM results, for example the bindingenergy. This, however, is easily remediated by using the f n ’s in (3.6) and the representationin (3.9), yielding a consistent PN-truncation in higher orders terms. As in the non-spinningcase, we found in § i / r = i r + ∆ i r , (5.2)with∆ i r = π (cid:88) A = ± χ (5) A ( γ ) ˜ a A (cid:96) + 25 π (cid:88) { A,B } = ± χ (6) AB ( γ ) ˜ a A ˜ a B (cid:96) γ → (5.3)= p ∞ (cid:96) (cid:88) A = ± ˜ a A (2 f f f A + ( f ) f A ) + 3ˆ p ∞ (cid:96) (cid:88) { A,B } = ± ˜ a A ˜ a B (cid:0) f AB ( f ) + 2 f AB f f (cid:1) + (cid:88) { A,B } = ± p ∞ (cid:96) ( f ) f A f B ˜ a A ˜ a B (cid:33) γ → – 20 –here we only keep the leading PN corrections. As discussed in [63–67], the remaining termsin (3.10) have fewer factors of f i ’s, thus scaling with additional powers of ˆ p ∞ ∼ E relative tothe ones displayed, and therefore contributing at higher PN order. (This is a consequence ofthe fact that the impetus in the CoM has well-defined static limit, so that f i ∼ / ˆ p ∞ .) The periastron advance follows from the radial action via∆Φ2 π = − ∂∂(cid:96) ( i r + (cid:96) ) . (5.4)However, this is obviously equivalent to the condition in (3.1), which we used to build theradial action. Using (4.26), translated to orbital angular momentum space, we find to O ( G ),∆Φ2 π = 3(5 γ − (cid:96) + (cid:20)
6Γ + 1 (5 γ − γ − a + − δ ˜ a − ) − γ (5 γ − δ ˜ a − + 7˜ a + ) (cid:21) (cid:96) + (cid:20) − γ (5 γ − γ − δ ˜ a − + 7˜ a + )(Γ˜ a + − δ ˜ a − ) (5.5)+ 192(5 γ − γ − (Γ + 1) (Γ˜ a + − δ ˜ a − ) + λ −− ˜ a − + 2 λ + − ˜ a + ˜ a − + λ ++ ˜ a (cid:21) (cid:96) + · · · , where the λ AB coefficients are given in (4.27). In order to compare with the PN literature,we expand it in powers of (cid:15) ≡ − E , yielding∆Φ( (cid:96), a, (cid:15) )2 π = (cid:34) ν − (cid:15) + 3 (cid:0) − ν + 4 ν (cid:1) (cid:15) (cid:35) (cid:96) (5.6)+ (cid:34) − a + + δ ˜ a − − ( ν − δ ˜ a − + (7 ν − a + (cid:15) − (cid:0)(cid:0) − ν + 2 ν (cid:1) δ ˜ a − + (cid:0) − ν + 14 ν (cid:1) ˜ a + (cid:1) (cid:15) (cid:35) (cid:96) + (cid:34) (cid:16) ˜ a − ( κ + −
2) + 2˜ a + ˜ a − κ − + ˜ a ( κ + + 2) (cid:17) − (cid:15) (cid:16) ˜ a − ( δκ − + κ + (13 − ν ) − ν −
25) + 2˜ a + ˜ a − ( κ − (13 − ν ) + δ ( κ + + 11))+ ˜ a ( δκ − + κ + (13 − ν ) − ν + 35) (cid:17) + · · · (cid:35) (cid:96) + · · · . The comparison with the PN result given in Eq. (33) of [127] thus gives perfect agreement,including finite-size effects. Yet, the expression in (5.5) contains all orders in (cid:15) , at O ( a/(cid:96) )and O ( a /(cid:96) ), extending the results in [64] at quadratic order in spins. The error in [127], first pointed out in [64], turns out to be a mere factor of 2, when looking at their resultwith C Q (cid:54) = 1. The correct result is given by replacing (7 − ν + 3 ν ) → (7 − ν + 3 ν ) in their Eq. (33). – 21 – .2.2 Binding Energy The binding energy for circular orbits can be computed in different ways. One option is toget the value of the orbital angular momentum as a function of the energy, (cid:96) c ( E c , a ± ), fromthe condition i r ( (cid:96) c ( E c , a ± ) , E c , a ± ) = 0. (Alternatively we can use the condition r + = r − forthe roots of (3.6) in a circular orbits [63].) From the orbital angular momentum we obtainthe orbital frequency, Ω c , via the first-law [128] x ≡ ( GM Ω c ) / = (cid:18) d(cid:96) c d E c (cid:19) − / , (5.7)from which we obtain the relationship E c ( x, a ± ) for circular orbits. Using the expressionin (5.2), which includes a few terms at higher orders in 1 /(cid:96) needed to account for all thecontributions to NLO in the PN expansion, we find (recall (cid:15) = − E ) (cid:15) c = x − x
12 ( ν + 9) + x / (cid:18)
13 ( δ ˜ a − + 7˜ a + ) + x (cid:104) (99 − ν )˜ a + − ( ν − δ ˜ a − (cid:105)(cid:19) (5.8)+ 16 x (cid:104) − ( κ + + 2)˜ a − ( κ + − a − − κ − ˜ a − ˜ a + (cid:105) + 572 x (cid:104)(cid:0) ν − κ − − κ + + 5) δ (cid:1) ˜ a − ˜ a + + (cid:0) − δκ − + 10 ν + 3( ν − κ + (cid:1) ˜ a − + (cid:0) − δκ − + 6 ν + 3( ν − κ + (cid:1) ˜ a (cid:105) + · · · , to 3PN order and quadratic in spins. After translating between various conventions, thisagrees with the known value in the literature [20–27, 35, 68]. The PM coefficients of the impetus in (3.6), written using in the decomposition in (3.16),follow by inverting the relations in § f A = 2 χ (2) A ˆ p ∞ , f AB = χ (3) AB ˆ p ∞ (5.9) f A = 4ˆ p ∞ (cid:32) χ (3) A π − χ (1)0 χ (2) A (cid:33) , f AB = 2ˆ p ∞ (cid:32) χ (4) AB π − χ (1)0 χ (3) AB − χ (2) A χ (2) B (cid:33) , (5.10)where we used f = p ∞ χ (1)0 [63]. Then, from the data collected in (4.30)-(4.38), we find f (+)3 = − γ Γ γ − γ − (cid:0) γ − (cid:1) Γ ( γ − (Γ + 1) , (5.11) f ( − )3 = − (cid:0) γ − (cid:1) Γ δ ( γ − γ + 1) (Γ + 1) , (5.12) f (++)3 = (cid:0) γ − (cid:1) Γ ( γ + 1) (Γ + 1) − γ Γ ( γ + 1)(Γ + 1) + (cid:0) γ − (cid:1) Γ ( κ + + 2)4( γ − , (5.13)– 22 – ( −− )3 = (cid:0) γ − (cid:1) Γ δ ( γ + 1) (Γ + 1) + (cid:0) γ − (cid:1) Γ ( κ + − γ −
1) (5.14) f (+ − )3 = f ( − +)3 = 2 γ Γ δ ( γ + 1)(Γ + 1) − (cid:0) γ − (cid:1) Γ δ ( γ + 1) (Γ + 1) + (cid:0) γ − (cid:1) Γ κ − γ − , (5.15) f (+)4 = 8 γ (cid:0) γ − (cid:1) Γ ( γ − − (cid:0) γ − (cid:1) Γ ( γ − ( γ + 1) (Γ + 1) − γ (cid:0) γ − (cid:1) Γ γ − (5.16)+ 3 (cid:0) γ − (cid:1) Γ ( γ − γ + 1) (Γ + 1) ,f ( − )4 = 4 (cid:0) γ − (cid:1) Γ δ ( γ − ( γ + 1) (Γ + 1) − γ (cid:0) γ − (cid:1) Γ δ γ − − (cid:0) γ − (cid:1) Γ δ ( γ − γ + 1) (Γ + 1) , (5.17) f (++)4 = − (cid:0) γ − (cid:1) Γ ( γ − γ + 1) (Γ + 1) + 20 γ (cid:0) γ − (cid:1) Γ ( γ − γ + 1) (Γ + 1) + 3 (cid:0) γ − (cid:1) Γ ( γ + 1) (Γ + 1) − (cid:0) γ − (cid:1) Γ ( κ + + 2)2 ( γ − − γ Γ γ − − γ (cid:0) γ − (cid:1) Γ ( γ − γ + 1) (Γ + 1) (5.18)+ (cid:0) γ − (cid:1) Γ δκ −
64 ( γ −
1) + (cid:0) γ − γ + 39 (cid:1) Γ κ +
64 ( γ − + (cid:0) γ − γ + 55 (cid:1) Γ32 ( γ − ,f ( −− )4 = − (cid:0) γ − (cid:1) Γ δ ( γ − γ + 1) (Γ + 1) − (cid:0) γ − (cid:1) Γ ( κ + − γ − + γ (cid:0) γ − (cid:1) Γ δ ( γ − γ + 1) (Γ + 1)+ 3 (cid:0) γ − (cid:1) Γ δ ( γ + 1) (Γ + 1) + (cid:0) γ − (cid:1) Γ δκ −
64 ( γ −
1) + (cid:0) γ − γ + 39 (cid:1) Γ κ +
64 ( γ − (5.19) − (cid:0) γ − γ + 41 (cid:1) Γ32 ( γ − ,f (+ − )4 = f ( − +)4 = 5 (cid:0) γ − (cid:1) Γ δ ( γ − γ + 1) (Γ + 1) − γ (cid:0) γ − (cid:1) Γ δ ( γ − γ + 1) (Γ + 1) − (cid:0) γ − (cid:1) Γ δ ( γ + 1) (Γ + 1) − (cid:0) γ − (cid:1) Γ κ − γ − + (cid:0) γ − (cid:1) Γ δκ +
64 ( γ − − γ (cid:0) γ − (cid:1) (Γ − δ γ − γ + 1) (Γ + 1) (5.20)+ 5Γ (cid:0) γ + 1 (cid:1) δ
32 ( γ −
1) + (cid:0) γ − γ + 39 (cid:1) Γ κ −
64 ( γ − . The expansion of the CoM momentum can then be analytically continued to negative bindingenergies ( γ < f n ’s are themost natural variables. Therefore, they are preferable to the — much more cumbersome —PM coefficients of the Hamiltonian, which we refrain from displaying here.– 23 – Discussion & Outlook
Building on the EFT approach developed in [14, 15, 23, 24, 27], in this paper we have extendedthe PM framework introduced in [65] to incorporate spin effects. We then used the formalismto compute the NLO momentum and spin impulses with generic initial conditions and toquadratic order in the spins, including for the first time finite-size effects beyond leading order.Afterwards we considered aligned-spin configurations and derived the scattering angle, whichwe used to construct the bound radial action via the B2B correspondence. The latter allowsus to compute all the gravitational observables for elliptic-like orbits. As a notable example,we obtained the periastron advance to O ( G ) and all orders in velocity. We also computedthe linear and bilinear in spin contributions to the binding energy for circular orbits to 3PNorder. In addition, we derived the CoM momentum (or impetus) in a quasi-isotropic gauge,from which one can readily obtain the EoM (or the Hamiltonian) to 2PM order, if so desired.Our results are in perfect agreement with the known literature, notably spin-orbit and spin -spin effects to NLO in the PM expansion in [104], while the spin -spin contributionsfor generic compact bodies are computed here for the first time. We also find agreement withthe conjectured value for the 2PM aligned-spin scattering angle of Kerr black holes [118].In order to perform the comparison with the findings using scattering amplitudes in [104],we have translated our covariant results into canonical variables. The latter have the advan-tage of furnishing a canonical algebra involving only a spin three-vector, yet the Lorentzcovariance of the results is hidden in somewhat cumbersome vectorial expressions. In con-trast, in the former the results are not only manifestly covariant, by construction, but alsoremarkably compact when written in terms of four-dimensional vectors, as displayed here.In fact, due to the conservation of the SSC ( S µ p µ = 0) upon evolution, both the spin andmomentum rotate in spacetime in the same fashion [117]. This implies the simple structure∆ p µ = m δ Λ µα u α , ∆ S µ = m δ Λ µα a α , (6.1)with δ Λ µν = − δ Λ νµ , must hold for both impulses. As shown in [117], the form of the δ Λ µν matrix can be easily found at 1PM order in terms of a four-vector, Z µ , and the velocity u µ , obeying the condition Z · u = 0. This representation is the basis for the “ b → b + ia ”shift which lies at the heart of the (complex) transformation introduced in [129] connectingSchwarzschild and Kerr solutions, (see also [94]). At NLO, however, the construction of the δ Λ µν matrix is less straightforward, mainly due to the new directions in which a non-zeroimpulse appears. For the case of non-spinning bodies, the task is relatively simple and theconcurrent rotation can be easily written down incorporating the impulse in the u µ direction.However, when spin effects are included, the form of the transformation turns out to bemuch more involved, begging instead for a more convenient basis to decompose the impulses. This transformation (applied to perturbations of the background) may also play a key role in understandingthe vanishing of (static) tidal response for rotating black holes [44, 130–132] following the pattern observed inSchwarzschild [133–135] ( ). – 24 –ithout spin, such basis exists, by combining the b µ and u µ impulses into a space-like vector,and it is directly connected to the eikonal representation. It would be interesting to performthe same manipulations for the case of spinning bodies, in particular in light of the remarkablealgebraic structure involving the eikonal phase recently discovered in [104].Another interesting area for further study is the possibility, for the special case of Kerrblack holes, to promote the worldline effective theory into a worldsheet, as advocated in [115].The motivation is also built on the “ x + ia ” shift [129], which suggests extending the worldlineaction into one more dimension. This is not entirely surprising, after all the covariant SSCimplies a non-commutative (Dirac) algebra for the position, { x i , x j } DB (cid:39) (cid:15) ijk a k /m , e.g. [15,72], hinting at the extendedness of spinning particles. The worldsheet idea is also rooted onthe fact that, not only the quadrupole [23], but all of the worldline (Wilson) coefficients obey C ES n = 1 (and similarly with the magnetic terms) when matching to the multipole momentsof a Kerr black hole [115]. This observation allows one to resume all the derivatively coupledhigher-derivative terms in the effective action, which then exponentiate into a translationoperation, e ia · ∂ , that is directly linked to a complex coordinate shift. However, because of theequivalence principle, finite-size effects start with two derivatives (with the 1 and a∂/ e x − /x = (cid:82) dλe λx ,which naturally allows one to introduce a two-dimensional integral for the effective action.Furthermore, it turns out to be natural to introduce a spinor-helicity representation [115](see [136] for other possible routes). Hence, armed with an action incorporating all of the(self-induced) finite-size effects at once, we could then set up the EFT formalism described inthis paper, uplifted to a worldsheet Routhian, to compute the momentum and spin impulsesfor Kerr black holes without having to introduce the curvature terms representing finite-sizeeffects. This possibility is currently under investigation.Yet one more aspect of the framework which deserves more attention is the generalizationof the B2B correspondence to the case of non-aligned spins. As shown in [64], the planar B2Bmap can be extended to spinning bodies by performing an analytic continuation in the total(canonical) angular momentum. However, when ˙ L = − ˙ S (cid:54) = 0, the precession of the planecomplicates matters. Moreover, only canonical variables may be associated through the B2Bdictionary, which requires also transforming the spin variables when we allow for non-planardynamics. In principle, we could consider the periastron shift in the instantaneous planeor, more likely, orbital averages over a period. For instance, because the orbital elementsare related via analytic continuation — with aligned or zero spins — the B2B map yieldsa relationship between the total radiated energy in a scattering process and the integratedpower over an orbit, via analytic continuation (see footnote 5). Hence, we may expect a similarsituation once we include the precession of the orbital plane, thus retaining a link betweenthe total impulse in momentum and an averaged periastron advance. There is as well thetotal spin kick to be considered, and the associated change in orbital angular momentum.Provided a relationship between the orbital elements still holds, we may expect an analogousconnection to the integrated change over an orbit for the bound case. Another interesting– 25 –enue is to explore the modification of the first-law to the case of spinning bodies, e.g. [137],which would allow us to compute the precession frequency once the non-planar B2B map isobtained. We will return to these issues in future work. Finally, since the master integralsto N LO are known [66], the formalism is ready to march forward in the PM expansion.The computation of spin effects to O ( G ) is currently underway. Note added:
While the results of this project were prepared for submission we learnedof the concurrent work of [138], which also computed finite-size contributions to the NLOimpulses via the methods discussed in [104]. After transforming between the different vari-ables (as discussed in § Acknowledgements . We thank Gregor K¨alin, and Justin Vines for helpful discussions.We also thank Andres Luna and Chia-Hsien Shen for useful exchanges on the results in [104].This work is supported by the ERC Consolidator Grant “Precision Gravity: From the LHC toLISA,” provided by the European Research Council (ERC) under the European Union’sH2020 research and innovation programme, grant No. 817791. Our work is also supported bythe DFG under Germany’s Excellence Strategy ‘Quantum Universe’ (No. 390833306). We ac-knowledge extensive use of the xAct packages ( ). A Supplemental Material
A.1 Trajectories
An important element in the computation of the NLO impulses is iteration of the 1PMEoM into the tree-level Routhian. For completeness, we provide here the trajectories for theposition and spin to 1PM order. To simplify the notation we always refer to the dynamics ofparticle 1 and do not include the mirror images. The contribution from the latter, as well asthe corresponding deflection for the second particle, can be derived as explained in [65].
A.1.1 Velocity & Position
The velocity correction at linear order in spin is given by, δ (1) S v µ ( τ ) = − im M m (cid:90) k ˆ δ ( k · u ) k ( k · u − i(cid:15) ) e ik · b e i ( k · u − i(cid:15) ) τ × k α (cid:16) k ν u α (cid:16) S µ ν + 2 u β u µ S νβ (cid:17) − γk µ u ν S αν (cid:17) , (A.1)and one time integration yields the position correction δ (1) S x µ ( τ ) = − m M m (cid:90) k ˆ δ ( k · u ) k ( k · u − i(cid:15) ) e ik · b e i ( k · u − i(cid:15) ) τ × k α (cid:16) k ν u α (cid:16) S µ ν + 2 u β u µ S νβ (cid:17) − γk µ u ν S αν (cid:17) . (A.2)– 26 –t quadratic order in the spin we have several contributions. For the term proportionalto the SSC in (2.8) we find δ (1) RS S v µ ( τ ) = − m M m (cid:90) k ˆ δ ( k · u ) k e ik · b e i ( k · u − i(cid:15) ) τ (cid:104) k ν ( k · u ) S µβ S νβ +2 γk ν S µ ν ( k α u β S αβ ) − u ν ( k · u ) S µ ν ( k α u β S αβ ) (cid:105) , (A.3)and δ (1) RS S x µ ( τ ) = im M m (cid:90) k ˆ δ ( k · u ) k ( k · u − i(cid:15) ) e ik · b e i ( k · u − i(cid:15) ) τ (cid:104) k ν ( k · u ) S µβ S νβ +2 γk ν S µ ν ( k α u β S αβ ) − u ν ( k · u ) S µ ν ( k α u β S αβ ) (cid:105) , (A.4)whereas from the finite-size effects we arrive at δ (1) ES v µ ( τ ) = m M m (cid:90) k ˆ δ ( k · u ) k α k ν k ( k · u − i(cid:15) ) e ik · b e i ( k · u − i(cid:15) ) τ (A.5) × (cid:104) k µ (cid:0) γ − (cid:1) S β α S νβ + k µ u α u ν ( S βρ S βρ − u β u ρ S γ β S ργ )+2 k β u α ( S βρ S ρ ν ( u µ − γu µ ) + u ν (2 u µ u ρ S γ β S ργ − S βρ S µρ )) (cid:105) so that δ (1) ES x µ ( τ ) = − im M m (cid:90) k ˆ δ ( k · u ) k α k ν k ( k · u − i(cid:15) ) e ik · b e i ( k · u − i(cid:15) ) τ (A.6) × (cid:104) k µ (cid:0) γ − (cid:1) S β α S νβ + k µ u α u ν ( S βρ S βρ − u β u ρ S γ β S ργ )+2 k β u α ( S βρ S ρ ν ( u µ − γu µ ) + u ν (2 u µ u ρ S γ β S ργ − S βρ S µρ )) (cid:105) . Finally, we also have spin -spin corrections to the trajectories given by δ (1) S S v µ ( τ ) = 18 M m (cid:90) k ˆ δ ( k · u ) k α k ν k ( k · u − i(cid:15) ) e ik · b e i ( k · u − i(cid:15) ) τ (A.7) × (cid:104) k µ S νβ (cid:16) γ S β α + u β u ρ S αρ (cid:17) + k β u α (cid:16) u ρ S νρ S µ β − u µ S ρ ν S βρ (cid:17)(cid:105) , and δ (1) S S x µ ( τ ) = − i M m (cid:90) k ˆ δ ( k · u ) k α k ν k ( k · u − i(cid:15) ) e ik · b e i ( k · u − i(cid:15) ) τ (A.8) × (cid:104) k µ S νβ (cid:16) γ S β α + u β u ρ S αρ (cid:17) + k β u α (cid:16) u ρ S νρ S µ β − u µ S ρ ν S βρ (cid:17)(cid:105) . – 27 – .1.2 Spin Integrating the spin equation in (2.9) we find (with the notation k [ α q β ] = k α q β − k β q α ) δ (1) S S αβ ( τ ) = − m M (cid:90) k ˆ δ ( k · u ) k ( k · u − i(cid:15) ) e ik · b e i ( k · u − i(cid:15) ) τ × (cid:16) γu ρ k [ α S β ]1 ρ − k ρ S [ α ρ ( u − γu ) β ] (cid:17) , (A.9)for the linear term, whereas at quadratic order δ (1) S S S αβ ( τ ) = − i M (cid:90) k ˆ δ ( k · u ) k ρ k ( k · u − i(cid:15) ) e ik · b e i ( k · u − i(cid:15) ) τ × (cid:16) k σ u ν u [ α S β ]1 ρ S σν + γk σ S [ α ρ S β ]2 σ −S ρσ (cid:16) γk [ α S β ] σ + u σ u ν k [ α S β ]1 ν (cid:17)(cid:17) , (A.10) δ (1) ES S αβ ( τ ) = − im M m S ρσ (cid:90) k ˆ δ ( k · u ) k ( k · u − i(cid:15) ) e ik · b e i ( k · u − i(cid:15) ) τ (A.11) × (cid:16) ( k · u ) u σ u [ α S β ] ρ + k ρ ( k · u ) S [ ασ ( u − γu ) β ] − k [ α S β ] σ (cid:0)(cid:0) γ − (cid:1) k ρ − γu ρ ( k · u ) (cid:1)(cid:17) δ (1) RS S S αβ ( τ ) = im M m (cid:90) k ˆ δ ( k · u ) k ρ k ( k · u − i(cid:15) ) e ik · b e i ( k · u − i(cid:15) ) τ (cid:16) γk σ u ν S σν u [ α S β ]1 ρ +( k · u ) (cid:16) u [ α S β ] σ S ρσ − u ν u σ S ρν u [ α S β ]1 σ (cid:17)(cid:17) . (A.12) A.2 Scattering Data
The value of the D i coefficients for the total momentum and spin impulses are given by: D = πγ (cid:0) γ − (cid:1) ( δ + 7)8 ( γ − / (A.13) D = 1 γ − (cid:2)(cid:0) γ + 4 γ − γ − (cid:1) + (cid:0) γ − γ − γ + 1 (cid:1) δ (cid:3) (A.14) D = γ ( γ − (cid:2) (2 γ + 1) (cid:0) γ + 2 γ − (cid:1) + (2 γ − (cid:0) γ − γ − (cid:1) δ (cid:3) (A.15) D = 1( γ − (cid:2)(cid:0) − γ − γ + 8 γ + 1 (cid:1) + (cid:0) γ − γ + 8 γ − (cid:1) δ (cid:3) (A.16) D = 3 π (cid:112) γ − (cid:0) γ ( δ + 1) + 9 δ + 1 (cid:1) (A.17)+ 3 πC (1) ES
128 ( γ − / (cid:0) γ (7 δ + 55) − γ ( δ + 5) + 23 δ + 47 (cid:1) (A.18)– 28 – = − γγ − γ + 1) + (2 γ − δ ] (A.19) − (cid:0) γ − (cid:1) ( γ − C (1) ES (cid:2)(cid:0) γ + γ − (cid:1) + (cid:0) γ − γ − (cid:1) δ (cid:3) D = (cid:0) γ − (cid:1) ( γ − C (1) ES [( γ + 1) + ( γ − δ ] , D = D ( δ ↔ − δ ) (A.20) D = 3 π (cid:112) γ − (cid:0) γ (7 δ + 31) + δ − (cid:1) (A.21)+ 3 πC (1) ES
128 ( γ − / (cid:0) γ (7 δ + 31) − γ (25 δ + 77) + 15 δ + 31 (cid:1) D = 3 π
64 ( γ − / (cid:0) γ (7 δ + 19) − δ + 1 (cid:1) (A.22)+ 3 πC (1) ES
64 ( γ − / (cid:0) γ (7 δ + 19) − γ (23 δ + 43) + 11 δ + 23 (cid:1) D = 4 γ ( γ − [(3 γ + 2) + (3 γ − δ ] (A.23)+ (cid:0) γ − (cid:1) ( γ − C (1) ES (cid:2) ( γ + 1) (cid:0) γ − γ − (cid:1) + ( γ − (cid:0) γ + 2 γ − (cid:1) δ (cid:3) D = 4 γ ( γ − (cid:2)(cid:0) − γ − γ − (cid:1) + (cid:0) γ − γ + 1 (cid:1) δ (cid:3) (A.24)+ (cid:0) γ − (cid:1) ( γ − C (1) ES (cid:2) ( γ − (cid:0) γ − γ − (cid:1) + ( γ − (cid:0) γ − γ − (cid:1) δ (cid:3) D = 4 γ γ − γ + 1) + ( γ − δ ] + D , D = D ( δ ↔ − δ ) (A.25) D = 3 π (cid:0) γ − γ + 3 (cid:1) γ − / (A.26) D = 1( γ − (cid:2) ( γ + 1) (cid:0) γ − γ − γ + 1 (cid:1) − ( γ − (cid:0) γ + 4 γ − γ − (cid:1) δ (cid:3) (A.27) D =2 (cid:0) γ − (cid:1) ( γ − [( γ + 1) + ( γ − δ ] (A.28) D = 3 πγ (cid:0) − γ (cid:1) ( γ − / (A.29) D = − γ − (cid:2) ( γ + 1) (cid:0) γ + 8 γ − γ − γ − γ + 1 (cid:1) (A.30)+( γ − (cid:0) γ − γ − γ + 8 γ − γ − (cid:1) δ (cid:3) D = 2( γ − (cid:2) ( γ + 1) (cid:0) γ − γ + 1 (cid:1) + ( γ − (cid:0) γ − γ + 1 (cid:1) δ (cid:3) (A.31)– 29 – = − (cid:0) γ − γ + 1 (cid:1) ( δ − γ −
1) (A.32) D = 3 π (cid:0) γ − (cid:1) (cid:112) γ − D =4 (cid:0) − γ (cid:1) D (A.34) D = (cid:0) γ − (cid:1) (cid:0) γ − γ + 1 (cid:1) ( γ − ( γ + 1) − (cid:0) γ − (cid:1) (cid:0) γ + 4 γ + 1 (cid:1) δ ( γ − γ + 1) (A.35) D = δ −
12 ( γ − (A.36) D = − γ (cid:0) γ − (cid:1) ( δ − C (1) ES + 1) γ − C (1) ES + 1)2 D (A.37) D = − γ ( δ − γ − D =(2 γ − D (A.39) D = − (cid:0) − γ (cid:1) ( δ − C (1) ES γ − D = − γ (cid:0) γ − (cid:1) ( C (1) ES − ν ( γ −
1) ( δ + 1) (A.41) D = 8 γ (cid:0) γ − (cid:1) ( C (1) ES − ν ( γ − ( δ + 1) (A.42) D = − γ ( γ −
1) ( δ + 1) (cid:0) γ ν + γ ( δ − ν + 1) − ν (cid:1) + 2 (cid:0) γ − (cid:1) C (1) ES ( γ − ( δ + 1) (cid:0) γ ν + γ (2 δ − ν + 2) − γν − δ + 2 ν − (cid:1) (A.43) D = − γ ( γ − ( δ + 1) (cid:0) γ ν + γ (12 ν − δ + 1)) − γ ν + γ (2 δ − ν + 2) + 7 ν (cid:1) + 2 (cid:0) γ − (cid:1) C (1) ES ( γ − ( δ + 1) (cid:0) γ ν + γ ( − δ + 4 ν − − γν + δ − ν + 1 (cid:1) (A.44) D = 4( γ − ( δ + 1) (cid:0) − γ ν + γ ( − δ + 16 ν −
8) + γ (4 δ − ν + 4) + ν (cid:1) (A.45) D = πγ (cid:0) γ − (cid:1) (7 − δ )8 ( γ − / = 7 − δδ + 7 D (A.46) D = πγ (cid:0) − γ (cid:1) γ − / = − γδ + 7 D (A.47) D = − D γ − = 2 (cid:0) γ + 1 (cid:1) ( δ − γ − D = γ (1 − δ ) + δ + 1( γ − (A.50) D = (cid:0) γ − γ − γ + 1 (cid:1) δγ − (cid:0) − γ − γ + 4 γ + 1 (cid:1) γ − D = − γ (2 γ + 1) (cid:0) γ + 2 γ − (cid:1) ( γ − + γ (2 γ − (cid:0) γ − γ − (cid:1) δ ( γ − (A.52) D = γ (cid:0) − γ + 12 γ − γ − γ + 4 (cid:1) δ ( γ − + γ (cid:0) γ + 12 γ + 4 γ − γ − (cid:1) ( γ − (A.53) D = πγ (cid:0) γ + 2 γ − (cid:1) γ − / = γ + 1 γ (1 − γ ) D (A.54) B One-loop Integration
At 2PM order, all expressions for momentum and spin impulses from Feynman diagramscontain one-loop tensor integrals of the form: I µ ··· µ m ( a ,a ,a ) = (cid:90) k ˆ δ ( k · u j ) k µ · · · k µ m [ k ] a [( k − q ) ] a ( ± k · u /j − i(cid:15) ) a , (B.1)with q · u = q · u = 0 and we use the convention { / , / } , introduced in [66]. The linearpropagators appear due to the iterations where we input the trajectories shown in Appendix Ain the tree-level Routhian/action. In non-spinning cases [65–67], all integrals have at mostone Lorentz index in the numerator. The situation changes when spin is included, and wefind various tensor integrals of rank m ∈ { , , , } . Following the standard method firstproposed by Passarino and Veltman [139],we reduce all the one-loop tensor integral to alinear combination of scalar integrals. The idea is simple, Lorentz covariance implies thatthe tensor structure in the final results can be constructed in terms of only the external data { q µ , u µ , u µ } and the metric tensor g µν . Let us consider for example the integral, which weencountered in the non-spinning case [65], I µ ( a ,a ,a ) = (cid:90) k ˆ δ ( k · u ) k µ [ k ] a [( k − q ) ] a ( ± k · u − i(cid:15) ) a , (B.2)with a >
0. Hence, the tensor decomposition yields I µ ( a ,a ,a ) = q µ I q + u µ I u + u µ I u , (B.3)with the scalar integrals I q , I u and I u often denoted as ‘form factors’ in the literature. Wecan now solve the form factors by performing Lorentz contractions on both sides, with q , u and u . In particular, by contracting with u we immediately find I u = − γ I u (reflecting the– 31 –act that the integral in (B.2) must be perpendicular to u ). As a result, we must computeonly the scalar integrals I q = 1 q (cid:90) k ˆ δ ( k · u ) k · q [ k ] a [( k − q ) ] a ( ± k · u − i(cid:15) ) a , (B.4) I u = − γ − (cid:90) k ˆ δ ( k · u )[ k ] a [( k − q ) ] a ( ± k · u − i(cid:15) ) a − . (B.5)At the end of the day, going to the rest frame of particle 2 to resolve the delta function, allof these integrals belong to the following family: (cid:90) k k ) a [( k − q ) ] a ( ± k · u − i(cid:15) ) a , (B.6)with q · u = 0 and u = γ −
1, in D = 3 − (cid:15) dimensions. While analytical expressions forany { a , a , a } are known, e.g. [140], it is often convenient to use integration-by-parts (IBP)relations to reduce the integrals in (B.6) to a combination of the following masters (cid:90) d D kπ D/ k ( k − q ) = 1( q ) − D/ Γ(2 − D/
2) Γ ( D/ − D − , (B.7) (cid:90) d D kπ D/ k ( k − q ) ( ± k · u − i(cid:15) ) = i √ π ( q ) (5 − D ) / (cid:112) γ − − D ) /
2) Γ (( D − / (cid:112) γ − D − , (B.8)with the factor of π D/ introduced to comply with the present literature. Similar considera-tions apply to higher-ranked tensor decompositions.Finally, we must perform the Fourier transform to impact parameter space, which canbe written in terms of derivatives w.r.t. to b µ , (cid:90) q e iq · b ˆ δ ( q · u )ˆ δ ( q · u ) q µ · · · q µ m ( − q ) n = (cid:0) − i∂ µ b (cid:1) · · · (cid:0) − i∂ µ m b (cid:1) (cid:90) q e iq · b ˆ δ ( q · u )ˆ δ ( q · u )( − q ) n . (B.9)We first notice that the results must lie in the plane orthogonal to both u and u . We canthen construct a projected metric [82, 117] ∂∂b µ b ν = Π µν = η µν + u µ ( u ν − γu ν ) + u µ ( u ν − γu ν ) γ − , (B.10)which we can use to reduce into scalar integrals. Using (B.9), together with (B.10), and themaster integral (cid:90) q e iq · b ˆ δ ( q · u )ˆ δ ( q · u )( − q ) n = 4 − n π (2 − D ) / (cid:112) γ − | b | D − − n Γ( D − − n )Γ( n ) , (B.11) Alternatively, as shown in [65], we can also perform the k integral in the rest frame of particle 1 and pickup the (conservative) pole from the linear propagator only. – 32 –t is straightforward to generate the Fourier integrals of any rank, e.g. (cid:90) q e iq · b ˆ δ ( q · u )ˆ δ ( q · u ) q µ q ν ( − q ) n = − − n π (2 − D ) / (cid:112) γ − | b | D +2 − n Γ( D/ − n )Γ( n ) (cid:16) ( D − n ) b µ b ν + | b | Π µν (cid:17) . (B.12) References [1]
LIGO Scientific, Virgo collaboration,
Open data from the first and second observing runsof Advanced LIGO and Advanced Virgo , .[2] A. Buonanno and B. Sathyaprakash, Sources of Gravitational Waves: Theory andObservations , .[3] R. A. Porto, The Tune of Love and the Nature(ness) of Spacetime , Fortsch. Phys. (2016)723 [ ].[4] R. A. Porto, The Music of the Spheres: The Dawn of Gravitational Wave Science , .[5] S. Vitale, R. Lynch, J. Veitch, V. Raymond and R. Sturani, Measuring the spin of black holesin binary systems using gravitational waves , Phys. Rev. Lett. (2014) 251101 [ ].[6] B. Zackay, T. Venumadhav, L. Dai, J. Roulet and M. Zaldarriaga,
Highly spinning and alignedbinary black hole merger in the Advanced LIGO first observing run , Phys. Rev.
D100 (2019)023007 [ ].[7] A. Arvanitaki and S. Dubovsky,
Exploring the String Axiverse with Precision Black HolePhysics , Phys. Rev. D (2011) 044026 [ ].[8] A. Arvanitaki, M. Baryakhtar, S. Dimopoulos, S. Dubovsky and R. Lasenby, Black HoleMergers and the QCD Axion at Advanced LIGO , Phys. Rev. D (2017) 043001[ ].[9] R. Brito, S. Ghosh, E. Barausse, E. Berti, V. Cardoso, I. Dvorkin et al., Stochastic andresolvable gravitational waves from ultralight bosons , Phys. Rev. Lett. (2017) 131101[ ].[10] K. K. Ng, M. Isi, C.-J. Haster and S. Vitale,
Multiband gravitational-wave searches forultralight bosons , Phys. Rev. D (2020) 083020 [ ].[11] D. Baumann, H. S. Chia and R. A. Porto,
Probing Ultralight Bosons with Binary Black Holes , Phys. Rev. D (2019) 044001 [ ].[12] D. Baumann, H. S. Chia, R. A. Porto and J. Stout, Gravitational Collider Physics , Phys. Rev.D (2020) 083019 [ ].[13] R. P. Kerr,
Rotating black holes and the Kerr metric , AIP Conf. Proc. (2008) 9.[14] W. D. Goldberger and I. Z. Rothstein,
An Effective field theory of gravity for extended objects , Phys. Rev.
D73 (2006) 104029 [ hep-th/0409156 ].[15] R. A. Porto,
Post-Newtonian Corrections to the Motion of Spinning Bodies in NRGR , Phys.Rev. D (2006) 104031 [ gr-qc/0511061 ]. – 33 –
16] W. D. Goldberger,
Les Houches lectures on effective field theories and gravitational radiation ,in
Les Houches Summer School - Session 86 , 1, 2007, hep-ph/0701129 .[17] S. Foffa and R. Sturani,
Effective Field Theory Methods to Model Compact Binaries , Class.Quant. Grav. (2014) 043001 [ ].[18] R. A. Porto, The effective field theorist’s approach to gravitational dynamics , Phys. Rept. (2016) 1 [ ].[19] I. Hinder, S. Ossokine, H. P. Pfeiffer and A. Buonanno,
Gravitational waveforms for high spinand high mass-ratio binary black holes: A synergistic use of numerical-relativity codes , Phys.Rev. D (2019) 061501 [ ].[20] G. Faye, L. Blanchet and A. Buonanno, Higher-order spin effects in the dynamics of compactbinaries. I. Equations of motion , Phys. Rev. D (2006) 104033 [ gr-qc/0605139 ].[21] R. A. Porto and I. Rothstein, The Hyperfine Einstein-Infeld-Hoffmann Potential , Phys. Rev.Lett. (2006) 021101 [ gr-qc/0604099 ].[22] R. A. Porto, New results at 3PN via an effective field theory of gravity , in , pp. 2493–2496, 1, 2007, gr-qc/0701106 .[23] R. A. Porto and I. Z. Rothstein,
Spin(1)Spin(2) Effects in the Motion of Inspiralling CompactBinaries at Third Order in the Post-Newtonian Expansion , Phys.Rev.
D78 (2008) 044012[ ].[24] R. A. Porto and I. Z. Rothstein,
Next to Leading Order Spin(1)Spin(1) Effects in the Motionof Inspiralling Compact Binaries , Phys.Rev.
D78 (2008) 044013 [ ].[25] J. Steinhoff, S. Hergt and G. Schaefer,
On the next-to-leading order gravitationalspin(1)-spin(2) dynamics , Phys. Rev. D (2008) 081501 [ ].[26] J. Steinhoff, S. Hergt and G. Schaefer, Spin-squared Hamiltonian of next-to-leading ordergravitational interaction , Phys. Rev. D (2008) 101503 [ ].[27] R. A. Porto, Next-to-Leading Order Spin-Orbit Effects in the Motion of Inspiralling CompactBinaries , Class. Quant. Grav. (2010) 205001 [ ].[28] M. Levi and J. Steinhoff, Next-to-next-to-leading order gravitational spin-orbit coupling via theeffective field theory for spinning objects in the post-Newtonian scheme , JCAP (2016) 011[ ].[29] M. Levi and J. Steinhoff, Complete conservative dynamics for inspiralling compact binarieswith spins at fourth post-Newtonian order , .[30] M. Levi, A. J. Mcleod and M. Von Hippel, NNNLO gravitational quadratic-in-spininteractions at the quartic order in G , .[31] M. Levi, A. J. Mcleod and M. Von Hippel, N LO gravitational spin-orbit coupling at order G , .[32] L. Blanchet, A. Buonanno and G. Faye, Higher-order spin effects in the dynamics of compactbinaries. II. Radiation field , Phys. Rev. D (2006) 104034 [ gr-qc/0605140 ].[33] R. A. Porto, A. Ross and I. Z. Rothstein, Spin induced multipole moments for the gravitationalwave flux from binary inspirals to third Post-Newtonian order , JCAP (2011) 009[ ]. – 34 –
34] R. A. Porto, A. Ross and I. Z. Rothstein,
Spin induced multipole moments for the gravitationalwave amplitude from binary inspirals to 2.5 Post-Newtonian order , JCAP (2012) 028[ ].[35] A. Boh´e, G. Faye, S. Marsat and E. K. Porter,
Quadratic-in-spin effects in the orbitaldynamics and gravitational-wave energy flux of compact binaries at the 3PN order , Class.Quant. Grav. (2015) 195010 [ ].[36] N. T. Maia, C. R. Galley, A. K. Leibovich and R. A. Porto, Radiation reaction for spinningbodies in effective field theory I: Spin-orbit effects , Phys. Rev.
D96 (2017) 084064[ ].[37] N. T. Maia, C. R. Galley, A. K. Leibovich and R. A. Porto,
Radiation reaction for spinningbodies in effective field theory II: Spin-spin effects , Phys. Rev.
D96 (2017) 084065[ ].[38] Z. Yang and A. K. Leibovich,
Analytic Solutions to Compact Binary Inspirals With LeadingOrder Spin-Orbit Contribution Using The Dynamical Renormalization Group , Phys. Rev. D (2019) 084021 [ ].[39] B. A. Pardo and N. T. Maia,
Next-to-leading order spin-orbit effects in the equations ofmotion, energy loss and phase evolution of binaries of compact bodies in the effective fieldtheory approach , Phys. Rev. D (2020) 124020 [ ].[40] W. D. Goldberger and A. Ross,
Gravitational radiative corrections from effective field theory , Phys. Rev.
D81 (2010) 124015 [ ].[41] A. K. Leibovich, N. T. Maia, I. Z. Rothstein and Z. Yang,
Second post-Newtonian orderradiative dynamics of inspiralling compact binaries in the Effective Field Theory approach , Phys. Rev. D (2020) 084058 [ ].[42] W. Goldberger and I. Rothstein,
Dissipative Effects in the Worldline Approach to Black HoleDynamics , Phys. Rev. D (2006) 104030 [ hep-th/0511133 ].[43] R. A. Porto, Absorption Effects due to Spin in the Worldline Approach to Black HoleDynamics , Phys. Rev. D (2008) 064026 [ ].[44] W. D. Goldberger, J. Li and I. Z. Rothstein, Non-conservative effects on Spinning Black Holesfrom World-Line Effective Field Theory , .[45] L. Blanchet, T. Damour and G. Esposito-Farese, Dimensional regularization of the thirdpostNewtonian dynamics of point particles in harmonic coordinates , Phys. Rev. D (2004)124007 [ gr-qc/0311052 ].[46] S. Foffa and R. Sturani, Effective field theory calculation of conservative binary dynamics atthird post-Newtonian order , Phys. Rev. D (2011) 044031 [ ].[47] S. Foffa and R. Sturani, Dynamics of the gravitational two-body problem at fourthpost-Newtonian order and at quadratic order in the Newton constant , Phys. Rev. D (2013)064011 [ ].[48] C. Galley, A. Leibovich, R. A. Porto and A. Ross, Tail Effect in Gravitational RadiationReaction: Time Nonlocality and Renormalization Group Evolution , Phys. Rev. D (2016)124010 [ ]. – 35 –
49] S. Foffa, P. Mastrolia, R. Sturani and C. Sturm,
Effective field theory approach to thegravitational two-body dynamics, at fourth post-Newtonian order and quintic in the Newtonconstant , Phys. Rev. D (2017) 104009 [ ].[50] R. A. Porto and I. Rothstein, Apparent Ambiguities in the Post-Newtonian Expansion forBinary Systems , Phys. Rev. D (2017) 024062 [ ].[51] T. Damour, P. Jaranowski and G. Sch¨afer, Nonlocal-In-Time Action for the FourthPost-Newtonian Conservative Dynamics of Two-Body Systems , Phys. Rev. D (2014)064058 [ ].[52] T. Marchand, L. Bernard, L. Blanchet and G. Faye, Ambiguity-Free Completion of theEquations of Motion of Compact Binary Systems at the Fourth Post-Newtonian Order , Phys.Rev. D (2018) 044023 [ ].[53] S. Foffa and R. Sturani, Conservative dynamics of binary systems to fourth Post-Newtonianorder in the EFT approach I: Regularized Lagrangian , Phys. Rev. D (2019) 024047[ ].[54] S. Foffa, R. A. Porto, I. Rothstein and R. Sturani,
Conservative dynamics of binary systems tofourth Post-Newtonian order in the EFT approach II: Renormalized Lagrangian , Phys. Rev.
D100 (2019) 024048 [ ].[55] S. Foffa, P. Mastrolia, R. Sturani, C. Sturm and W. J. Torres Bobadilla,
Static two-bodypotential at fifth post-Newtonian order , Phys. Rev. Lett. (2019) 241605 [ ].[56] J. Bl¨umlein, A. Maier and P. Marquard,
Five-Loop Static Contribution to the GravitationalInteraction Potential of Two Point Masses , Phys. Lett. B (2020) 135100 [ ].[57] S. Foffa and R. Sturani,
Hereditary terms at next-to-leading order in two-body gravitationaldynamics , Phys. Rev. D (2020) 064033 [ ].[58] J. Bl¨umlein, A. Maier, P. Marquard and G. Sch¨afer,
The fifth-order post-NewtonianHamiltonian dynamics of two-body systems from an effective field theory approach: potentialcontributions , .[59] J. Bl¨umlein, A. Maier, P. Marquard and G. Sch¨afer, Testing binary dynamics in gravity at thesixth post-Newtonian level , Phys. Lett. B (2020) 135496 [ ].[60] J. Bl¨umlein, A. Maier, P. Marquard and G. Sch¨afer,
The 6th Post-Newtonian Potential Termsat O ( G N ), .[61] D. Bini, T. Damour and A. Geralico, Sixth post-Newtonian local-in-time dynamics of binarysystems , .[62] L. Blanchet, S. Foffa, F. Larrouturou and R. Sturani, Logarithmic tail contributions to theenergy function of circular compact binaries , Phys. Rev. D (2020) 084045 [ ].[63] G. K¨alin and R. A. Porto,
From Boundary Data to Bound States , JHEP (2020) 072[ ].[64] G. K¨alin and R. A. Porto, From boundary data to bound states. Part II. Scattering angle todynamical invariants (with twist) , JHEP (2020) 120 [ ].[65] G. K¨alin and R. A. Porto, Post-Minkowskian Effective Field Theory for Conservative BinaryDynamics , JHEP (2020) 106 [ ]. – 36 –
66] G. K¨alin, Z. Liu and R. A. Porto,
Conservative Dynamics of Binary Systems to ThirdPost-Minkowskian Order from the Effective Field Theory Approach , Phys. Rev. Lett. (2020) 261103 [ ].[67] G. K¨alin, Z. Liu and R. A. Porto,
Conservative Tidal Effects in Compact Binary Systems toNext-to-Leading Post-Minkowskian Order , Phys. Rev. D (2020) 124025 [ ].[68] L. Blanchet,
Gravitational Radiation from Post-Newtonian Sources and Inspiralling CompactBinaries , Living Reviews in Relativity (2014) 2.[69] M. Mathisson, Neue mechanik materieller systemes , Acta Phys. Polon. (1937) 163.[70] A. Papapetrou, Spinning test particles in general relativity. 1. , Proc. Roy. Soc. Lond. A (1951) 248.[71] W. Dixon,
Dynamics of extended bodies in general relativity. I. Momentum and angularmomentum , Proc. Roy. Soc. Lond. A (1970) 499.[72] A. J. Hanson and T. Regge,
The Relativistic Spherical Top , Annals Phys. (1974) 498.[73] K. Yee and M. Bander, Equations of motion for spinning particles in external electromagneticand gravitational fields , Phys. Rev. D (1993) 2797 [ hep-th/9302117 ].[74] H. Elvang and Y.-t. Huang, Scattering Amplitudes in Gauge Theory and Gravity . CambridgeUniversity Press, 4, 2015.[75] Z. Bern, J. J. Carrasco, M. Chiodaroli, H. Johansson and R. Roiban,
The Duality BetweenColor and Kinematics and its Applications , .[76] J. M. Henn, Lectures on differential equations for Feynman integrals , J. Phys. A (2015)153001 [ ].[77] D. Neill and I. Z. Rothstein, Classical Space-Times from the S Matrix , Nucl. Phys.
B877 (2013) 177 [ ].[78] C. Cheung, I. Z. Rothstein and M. P. Solon,
From Scattering Amplitudes to ClassicalPotentials in the Post-Minkowskian Expansion , Phys. Rev. Lett. (2018) 251101[ ].[79] Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M. P. Solon and M. Zeng,
Scattering Amplitudesand the Conservative Hamiltonian for Binary Systems at Third Post-Minkowskian Order , Phys. Rev. Lett. (2019) 201603 [ ].[80] Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M. P. Solon and M. Zeng,
Black Hole BinaryDynamics from the Double Copy and Effective Theory , JHEP (2019) 206 [ ].[81] D. A. Kosower, B. Maybee and D. O’Connell, Amplitudes, Observables, and ClassicalScattering , JHEP (2019) 137 [ ].[82] B. Maybee, D. O’Connell and J. Vines, Observables and amplitudes for spinning particles andblack holes , JHEP (2019) 156 [ ].[83] C. Galley and R. A. Porto, Gravitational Self-Force in the Ultra-Relativistic Limit: the“Large- N ” Expansion , JHEP (2013) 096 [ ].[84] B. R. Holstein and A. Ross, Spin Effects in Long Range Gravitational Scattering , . – 37 –
85] N. Bjerrum-Bohr, J. F. Donoghue and P. Vanhove,
On-shell Techniques and Universal Resultsin Quantum Gravity , JHEP (2014) 111 [ ].[86] V. Vaidya, Gravitational spin Hamiltonians from the S matrix , Phys. Rev.
D91 (2015) 024017[ ].[87] A. Guevara,
Holomorphic Classical Limit for Spin Effects in Gravitational andElectromagnetic Scattering , JHEP (2019) 033 [ ].[88] M.-Z. Chung, Y.-T. Huang, J.-W. Kim and S. Lee, The simplest massive S-matrix: fromminimal coupling to Black Holes , JHEP (2019) 156 [ ].[89] A. Guevara, A. Ochirov and J. Vines, Scattering of Spinning Black Holes from ExponentiatedSoft Factors , JHEP (2019) 056 [ ].[90] A. Guevara, A. Ochirov and J. Vines, Black-hole scattering with general spin directions fromminimal-coupling amplitudes , Phys. Rev. D (2019) 104024 [ ].[91] N. E. J. Bjerrum-Bohr, P. H. Damgaard, G. Festuccia, L. Plante and P. Vanhove,
GeneralRelativity from Scattering Amplitudes , Phys. Rev. Lett. (2018) 171601 [ ].[92] A. Cristofoli, N. E. J. Bjerrum-Bohr, P. H. Damgaard and P. Vanhove,
On Post-MinkowskianHamiltonians in General Relativity , .[93] S. Caron-Huot and Z. Zahraee, Integrability of Black Hole Orbits in Maximal Supergravity , JHEP (2019) 179 [ ].[94] N. Arkani-Hamed, Y.-t. Huang and D. O’Connell, Kerr black holes as elementary particles , JHEP (2020) 046 [ ].[95] N. E. J. Bjerrum-Bohr, A. Cristofoli and P. H. Damgaard, Post-Minkowskian Scattering Anglein Einstein Gravity , .[96] M.-Z. Chung, Y.-T. Huang and J.-W. Kim, From quantized spins to rotating black holes , .[97] Y. F. Bautista and A. Guevara, From Scattering Amplitudes to Classical Physics:Universality, Double Copy and Soft Theorems , .[98] Y. F. Bautista and A. Guevara, On the Double Copy for Spinning Matter , .[99] A. Koemans Collado, P. Di Vecchia and R. Russo, Revisiting the second post-Minkowskianeikonal and the dynamics of binary black holes , Phys. Rev.
D100 (2019) 066028 [ ].[100] H. Johansson and A. Ochirov,
Double copy for massive quantum particles with spin , JHEP (2019) 040 [ ].[101] R. Aoude, K. Haddad and A. Helset, On-shell heavy particle effective theories , .[102] A. Cristofoli, P. H. Damgaard, P. Di Vecchia and C. Heissenberg, Second-orderPost-Minkowskian scattering in arbitrary dimensions , .[103] M.-Z. Chung, Y.-t. Huang, J.-W. Kim and S. Lee, Complete Hamiltonian for spinning binarysystems at first post-Minkowskian order , .[104] Z. Bern, A. Luna, R. Roiban, C.-H. Shen and M. Zeng, Spinning Black Hole BinaryDynamics, Scattering Amplitudes and Effective Field Theory , . – 38 – Universality in the classical limit ofmassless gravitational scattering , .[106] P. Di Vecchia, A. Luna, S. G. Naculich, R. Russo, G. Veneziano and C. D. White, A tale oftwo exponentiations in N = 8 supergravity , Phys. Lett. B (2019) 134927 [ ].[107] A. Antonelli, A. Buonanno, J. Steinhoff, M. van de Meent and J. Vines,
Energetics of two-bodyHamiltonians in post-Minkowskian gravity , Phys. Rev.
D99 (2019) 104004 [ ].[108] A. Brandhuber and G. Travaglini,
On higher-derivative effects on the gravitational potentialand particle bending , JHEP (2020) 010 [ ].[109] C. Cheung and M. P. Solon, Classical gravitational scattering at O (G ) from Feynmandiagrams , JHEP (2020) 144 [ ].[110] J. Parra-Martinez, M. S. Ruf and M. Zeng, Extremal black hole scattering at O ( G ) : gravitondominance, eikonal exponentiation, and differential equations , .[111] C. Cheung and M. P. Solon, Tidal Effects in the Post-Minkowskian Expansion , .[112] M. Accettulli Huber, A. Brandhuber, S. De Angelis and G. Travaglini, Eikonal phase matrix,deflection angle and time delay in effective field theories of gravity , Phys. Rev. D (2020)046014 [ ].[113] Z. Bern, J. Parra-Martinez, R. Roiban, E. Sawyer and C.-H. Shen,
Leading Nonlinear TidalEffects and Scattering Amplitudes , .[114] C. Cheung, N. Shah and M. P. Solon, Mining the Geodesic Equation for Scattering Data , .[115] A. Guevara, B. Maybee, A. Ochirov, D. O’Connell and J. Vines, A worldsheet for Kerr , .[116] Z. Bern, J. Parra-Martinez, R. Roiban, M. S. Ruf, C.-H. Shen, M. P. Solon et al., ScatteringAmplitudes and Conservative Binary Dynamics at O ( G ), .[117] J. Vines, Scattering of two spinning black holes in post-Minkowskian gravity, to all orders inspin, and effective-one-body mappings , Class. Quant. Grav. (2018) 084002 [ ].[118] J. Vines, J. Steinhoff and A. Buonanno, Spinning-black-hole scattering and the test-black-holelimit at second post-Minkowskian order , Phys. Rev. D (2019) 064054 [ ].[119] E. Herrmann, J. Parra-Martinez, M. S. Ruf and M. Zeng, Gravitational Bremsstrahlung fromReverse Unitarity , .[120] P. Di Vecchia, C. Heissenberg, R. Russo and G. Veneziano, Radiation Reaction from SoftTheorems , .[121] G. Mogull, J. Plefka and J. Steinhoff, Classical black hole scattering from a worldline quantumfield theory , .[122] G. U. Jakobsen, G. Mogull, J. Plefka and J. Steinhoff, Classical Gravitational Bremsstrahlungfrom a Worldline Quantum Field Theory , .[123] S. Mougiakakos, M. M. Riva and F. Vernizzi, Gravitational Bremsstrahlung in thePost-Minkowskian Effective Field Theory , . – 39 – Quantum theory of gravitation vs. classical theory. - fourth-order potential , Prog.Theor. Phys. (1971) 1587.[125] O. B. Firsov, Determination of the forces acting between atoms using the differential effectivecross-section for elastic scattering , ZhETP (1953) 279.[126] D. Bini, T. Damour and A. Geralico, Sixth post-Newtonian nonlocal-in-time dynamics ofbinary systems , Phys. Rev. D (2020) 084047 [ ].[127] M. Tessmer, J. Hartung and G. Schafer,
Aligned Spins: Orbital Elements, Decaying Orbits,and Last Stable Circular Orbit to high post-Newtonian Orders , Class. Quant. Grav. (2013)015007 [ ].[128] A. Le Tiec, L. Blanchet and B. F. Whiting, The First Law of Binary Black Hole Mechanics inGeneral Relativity and Post-Newtonian Theory , Phys. Rev.
D85 (2012) 064039 [ ].[129] E. T. Newman and A. I. Janis,
Note on the Kerr spinning particle metric , J. Math. Phys. (1965) 915.[130] H. S. Chia, Probing Particle Physics with Gravitational Waves , Ph.D. thesis, Amsterdam U.,2020. .[131] A. Le Tiec, M. Casals and E. Franzin,
Tidal Love Numbers of Kerr Black Holes , .[132] P. Charalambous, S. Dubovsky and M. M. Ivanov, On the Vanishing of Love Numbers forKerr Black Holes , .[133] T. Binnington and E. Poisson, Relativistic theory of tidal Love numbers , Phys. Rev. D (2009) 084018 [ ].[134] T. Damour and A. Nagar, Relativistic tidal properties of neutron stars , Phys. Rev. D (2009) 084035 [ ].[135] L. Hui, A. Joyce, R. Penco, L. Santoni and A. R. Solomon, Static response and Love numbersof Schwarzschild black holes , .[136] F. A. Berezin and M. S. Marinov, Particle Spin Dynamics as the Grassmann Variant ofClassical Mechanics , Annals Phys. (1977) 336.[137] L. Blanchet, A. Buonanno and A. Le Tiec,
First law of mechanics for black hole binaries withspins , Phys. Rev. D (2013) 024030 [ ].[138] A. Luna and D. Kosmopoulos, to appear , .[139] G. Passarino and M. J. G. Veltman, One Loop Corrections for e + e − Annihilation Into µ + µ − in the Weinberg Model , Nucl. Phys. B (1979) 151.[140] V. A. Smirnov,
Analytic tools for Feynman integrals . Springer, 2012,10.1007/978-3-642-34886-0.. Springer, 2012,10.1007/978-3-642-34886-0.