From Boundary Data to Bound States

We introduce a -- somewhat holographic -- dictionary between gravitational observables for scattering processes (measured at"the boundary") and adiabatic invariants for bound orbits (in"the bulk"), to all orders in the Post-Minkowskian (PM) expansion. Our map relies on remarkable connections between the relative momentum of the two-body problem, the classical limit of the scattering amplitude and the deflection angle in hyperbolic motion. These relationships allow us to compute observables for generic closed orbits, such as the periastron advance $\Delta\Phi$, from the unbound case through analytic continuation. A simplified (more geometrical) map can be obtained for circular orbits, enabling us to extract the orbital frequency as a function of the (conserved) binding energy, $\Omega(E)$, from scattering data. As an example, using the results in Bern et al. [1901.04424, 1908.01493], we readily derive $\Omega(E)$ and $\Delta\Phi(J,E)$ to two-loop orders. We also provide a closed-form expression for the orbital frequency and periastron advance at tree-level and one-loop order, respectively, which capture a series of exact terms in the Post-Newtonian expansion. We then perform a partial PM resummation, using a"no-recoil"approximation for the amplitude. This limit is behind the map between the scattering angle for a test-particle and the two-body dynamics to 2PM. We show that it also captures a subset of higher order terms beyond the test-particle limit. While a (rather lengthy) Hamiltonian may be derived as an intermediate step, our map applies directly between gauge invariant quantities. Our findings provide a starting point for an alternative approach to the binary problem. We conclude with future directions and some speculations on the classical double copy.


Introduction
The nascent field of gravitational wave (GW) science and multi-messenger astronomy [1][2][3][4] will be a truly interdisciplinary subject, enriching different branches of physics. Yet, to fully exploit the discovery potential in GW observations, precise theoretical predictions for the binary problem in General Relativity will be mandatory. The computational challenges in precision gravity, however, are enormous [5,6]. Numerical methods cover mostly the late stages and merger regime of the binary's dynamics, where the gravitational interaction becomes strong. However, numerical codes are incapable of solving for the entirety of the observed cycles, which may be of the order of 10 5 with third generation detectors [7] and for several astrophysically motivated sources, such as neutron star binaries [8]. In those cases, the majority of the cycles in the detector's band will occur during the inspiral phase, which is instead described using analytic methods such as the Post-Newtonian (PN) expansion [9][10][11].
The PN regime involves perturbative calculations where ideas from particle physics have already played a prominent role. The Effective Field Theory (EFT) approach put forward in [12] has introduced a series of quantum field theory techniques into classical General Relativity, which have successfully reduced the two-body problem in gravity into a computation of Feynman diagrams [11][12][13][14]. Using the powerful EFT machinery, as well as other methodologies [15][16][17][18], the current level of accuracy in the PN framework has achieved the next-tonext-to-next-to-next-to-leading order (N 4 LO) in the conservative sector, or four loops [19][20][21][22][23], and up to five loops in the static limit [24,25]. The advantage of the PN formalism is the separation of scales in the non-relativistic regime, which in the EFT approach allows the use of the method of regions [26] to compute otherwise intractable integrals. The disadvantage is the non-relativistic truncation (as opposite to a relativistic result), as well as the need of gauge choices and gauge dependent objects (e.g. lengthy potentials) on the way to derive measurable quantities, such as the binding energy for circular orbits. The latter, on the other hand, displays a much simpler analytic form, which suggests that a formalism involving only gauge invariant quantities must exist. We develop such framework in this paper, by relating gravitational observables from scattering processes directly to adiabatic invariants for bound states.
In a parallel development, the field of scattering amplitudes has taken a leading role at the frontier of theoretical physics [27][28][29]. It is then not surprising that, endowed with novel ideas such as the spinor helicity formalism (see e.g. [30,31]), the double copy [32,33], generalized unitarity [34,35], and other tools from quantum field theory, scattering amplitudes have found their way into the (classical) two-body problem in gravity, e.g. . Some of the basic ideas are very simple, going back to the computation by Iwasaki of the 1PN correction using a scattering process [58]. The scattering matrix carries the information about the kick deflecting the particles, which may be extracted iteratively. In its modern incarnation, a judicious classical limit of the quantum amplitudes -taking the large impact parameter and heavy-mass limits -enables the extraction of the gravitational potential. The latter is obtained through a matching computation to an effective (relativistic) theory with only local interactions [47]. In these approaches, the standard perturbative regime of field theory turns into what is known as the Post-Minkowskian (PM) expansion of the two-body problem.
While obtaining physical information out of the scattering amplitude by matching to an effective potential is a valid approach -only specifying a gauge by Fourier transforming in the center of mass frame with respect to the transfer momentum -it still relies on deriving a gauge dependent object as an intermediate step: The Hamiltonian. This is of course useful, and it was used in [36,37] to derive the scattering angle to 3PM. Once applied to bound states, it can also be used to compute the binding energy [59]. However, as we argue, going through the Hamiltonian is unnecessarily complicated (and of increasing complexity, given the value of the coefficients of the PM expanded potential), in comparison with the much simpler form of the scattering amplitude. Moreover, the Hamiltonian also hinders the power of the PM approach versus the PN formalism, mainly because a truncated velocity expansion may be needed to solve for, e.g. the binding energy, in a self-consistent fashion. 1 Motivated by the on-shell philosophy of modern approaches to scattering amplitudes, we show that a map between gauge invariant quantities can be constructed instead, to all orders in velocity. This map neatly illustrates how the physical information is encoded in the amplitude, and how to translate it into observables for bound states.
The connection between scattering data and two-body dynamics has also been explored elsewhere, e.g. via an effective matching to the scattering angle [43,60,61]. An alternative venue was also pursued in [62], where it was shown how the scattering angle to 2PM can be obtained from the test-particle limit, and later used in [63] to construct a (local-in-time) Hamiltonian for circular orbits. In all of these cases, a Hamiltonian (or equivalent) has played a central role; or a resummed version in the effective one-body formalism [64]. As we show here, the gravitational potential can be derived from the scattering data without resorting to a matching computation to an effective description. However, motivated by the simplicity of the physical information encoded in both -the scattering problem and closed orbits -we have instead constructed a map that directly connects the two in a gauge invariant fashion.
Our approach relies on a remarkable connection between the momentum for the (conservative) two-body problem in the center of mass frame, and the (Fourier transform of the) scattering amplitude in the classical limit. This relationship -which we refer to as the impetus formula -allows us, for instance, to relate the amplitude to the deflection angle to all orders in the PM expansion. This is achieved by first inverting the well-known equation that determines the scattering angle from the radial momentum, a problem solved by Firsov many years ago [65]. The direct connection between the scattering matrix and deflection angle illustrates the gauge invariant information encoded in the former. Having the analytic form of the relative momentum, extracted from scattering data, we then move to the case of bound orbits. By analytic continuation in the energy, we construct a radial action from which adiabatic invariants for elliptic motion can be computed. For instance, from the scattering amplitude we can readily obtain the periastron advance. As an example, bypassing the need of an effective Hamiltonian, we obtain the precession of the perihelion from the knowledge of the scattering amplitude to two-loops [36,37,47], which recovers the known result in General Relativity to 2PN order. Moreover, we give a closed-form expression for the precession at one-loop order, which captures a series of exact terms in the PN expansion.
Our framework is universal, and can be used to relate scattering information to observables for bound states in generic configurations (so far ignoring spin effects). For circular orbits, however, a more geometrical program can be pursued. Using an interplay between Firsov's inversion formula and our impetus formula, we extract the orbital elements for hyperbolic motion from the scattering process, to all orders in the PM expansion. Hence, after performing an analytic continuation in the energy (or rapidity) and impact parameter (as well as for the eccentric anomaly), we obtain the orbital elements for elliptic motion. In particular, we determine the eccentricity, which must vanish for circular orbits. This condition allows us to extract the (reduced) angular momentum of the orbit as a function of the binding energy, from which we can derive the orbital frequency using the first law for binary dynamics [66]. As an example, using the results from [36,37], we provide an expression for the orbital frequency as a function of the binding energy valid to 3PM order, and to all orders in velocity. We also provide a closed-form expression for the contribution at tree-level, or 1PM order, which encapsulates an infinite series of exact terms in the PN expansion.
Motivated by the relationship between the test-particle limit and two-body problem to 2PM discovered in [62], we use the impetus formula to perform a resummation of PM effects. This is achieved by implementing the no-recoil approximation for the scattering amplitude. In this limit, the impetus formula allows us to compute the amplitude by boosting to the center of mass frame from the rest frame of one of the particles. In the rest frame, the relative momentum is then given by the expression in Schwarzschild, up to O(ν) corrections. The boost transforms the (reduced) energy of the test-particle into a factor of γ = p 1 · p 2 /(m 1 m 2 ), while the additional overall Γ −1 ≡ (2M )/(2E) discovered in [62] is due to the relativistic normalization ((2E) −1 ) of the amplitude (needed to obtain scalar quantities). As we demonstrate, the no-recoil approximation is exact to 2PM, while recovering the Schwarzschild expression for the scattering angle in the limit Γ → 1. 2 We also show that it captures an infinite subset of terms at higher PM orders. As a whole, however, the no-recoil approximation fails at 3PM. Yet, together with the impetus formula it provides a playground to explore other (potentially more accurate) resummation schemes, and a novel starting point for a gauge invariant approach to the binary inspiral problem. (See [67] for other recent developments.) There are several venues to continue developing our framework. First of all, we have only considered the conservative sector and ignored radiation-reaction effects. These can be incorporated following the analysis in [43], or using the EFT approach for radiation modes [11, 19-21, 23, 68, 69]. We have also concentrated entirely on the non-spinning case. The inclusion of spin, both in the EFT approach [70][71][72][73][74][75][76] and in scattering amplitudes [40,44,48,53,[77][78][79][80], introduces rich new structures. In particular, the map from hyperbolic into elliptic motion, once no longer restricted to a plane, deserves a more careful study. We have focused also on scattering data from the amplitude and deflection angle, yet it would be useful to explore other possibilities to characterize invariant information. In principle, our map applies in the non-perturbative regime. It would be useful to refocus numerical efforts toward scattering processes in General Relativity, to use our dictionary for the -numerically more challenging -binary problem. We will conclude our paper with further discussions on some of these issues. Finally -and in light of the impetus formula -we will also indulge in some speculations about the classical double copy between gauge theory and gravity. This paper is organized as follows. In §2 we review the determination of the scattering angle from the Hamiltonian. In §3 we review Firsov's approach to infer the scattered momentum in the center of mass from the scattering angle, and vice versa. We also describe how to obtain the gravitational potential without resorting to matching to an effective theory. In §4 we demonstrate the impetus formula, which relates the two-body relative momentum to the Fourier transform of the classical scattering amplitude, in the conservative sector. Furthermore, we show how radiation-reaction effects are encoded in terms quadratic in the amplitude. Subsequently, we use the impetus formula to relate the amplitude to the deflection angle, which highlights the gauge invariant information encoded in the scattering matrix. As an example, given the amplitude to one-loop order, we provide exact expressions for the scattering angle. In §5 we develop a map to transform scattering data into adiabatic invariants for bound orbits. We construct the radial action and illustrate the needed steps to compute observables. Using the new results in Bern et. al. [36,37] we then obtain the periastron advance to two-loops, which reproduces the result in General Relativity to 2PN. We also give a closed-form expression at one-loop, which encodes a series of exact PN terms. Afterwards, we show how to obtain the orbital elements for elliptic motion from the hyperbolic case, via analytic continuation in impact parameter and energy. Imposing the vanishing of the eccentricity, we obtain the orbital frequency as a function of the binding energy. The results in [36,37] then allow us derive an expression valid to 3PM, and to all orders in velocity. We also provide an exact result for the orbital frequency to 1PM, which also captures an infinite series of terms in the PN expansion. The no-recoil approximation is described in §6. We show how it exactly recovers the 2PM dynamics, and a subset of higher order terms. We conclude in §7 with a collection of main results, further discussions on open problems and some speculative explorations into the non-perturbative regime and the classical double copy. Appendix A contains a (more direct) proof of the impetus formula restricted to the conservative sector. Throughout the paper we use = c = 1 units, unless otherwise noted.

From Dynamics to Scattering Angles
We start by reviewing the standard procedure to compute the scattering angle given a Hamiltonian.

Hamiltonian Approach
In classical physics we can compute the scattering angle in the center of mass frame from the radial momentum, where J is the conserved angular momentum. Using the Hamiltonian, to solve for p 2 (r, H = E), the scattering angle is then given by (see e.g. [37]) The closest distance between the particles r min (E, J) is obtained by imposing p r (r min ) = 0. Introducing p 2 ∞ = p 2 (r → ∞) and the impact parameter b = J/p ∞ , we can re-write the above expression as: with p = p/p ∞ . Notice that, assuming the interaction turns off at infinity V (r, (2.5) The scattering angle can be computed following these simple steps. For example, in Newtonian mechanics, H N = p 2 2µ − GM µ r , we have (see e.g. [60]) with (µ, M ) the reduced and total mass and ν = µ/M the symmetric mass ratio.

Post-Minkowskian Expansion
In General Relativity, on the other hand, the computation is much more challenging due to the non-linearities involved. For large impact parameter, b GM , the scattering angle can be computed as a series in GM/b, or 1/j, what is known as the Post-Minkowskian (PM) expansion: andp ∞ = p ∞ /µ. While it is straightforward to read off from (2.6) the χ (n) b 's for the Newtonian case [60], the scattering angle up to second order [81], was the state-of-the-art in General Relativity for quite some time. In the above expressions we introduced Γ ≡ E/M = 1 + 2ν(γ − 1) = 1 + νE . (2.12) In terms of these variables, we havep The scattering angle to 3PM order was derived more recently, using novel tools from the theory of scattering amplitudes [27-29, 32, 33] via the 3PM Hamiltonian [36,37]. The result reads: We will discuss the use of scattering amplitudes, and other approaches, which borrow from particle physics later on in §4, when we introduce a derivation of the scattering angle which does not require computing an explicit Hamiltonian. 3 It is somewhat convenient to introduce also the concept of rapidity β ≡ arcosh γ , (2.15) which will be useful later on when we study the map to bound states (for which γ < 1). For example, The expression of the scattering angle as a function of energy and impact parameter will be the starting point to obtain adiabatic invariants for a two-body system with bound orbits.

From Scattering Angles to Dynamics
In this section we proceed in reverse order, and obtain the Hamiltonian of the system in terms of the scattering angle.

Inversion Formula
The inverse problem was solved some time ago by Firsov, who obtained the following formula [65,82] for the relative momentum. Inserting the PM expansion of the scattering angle (2.8) into Firsov's formula and expanding the exponential in a Taylor series, we obtain 2) The f i 's are given in terms of the χ (i) b 's as: For notational convenience we have introduced 4 4 This is simply motivated by the integral appearing in Firsov's formula: with the overall factor of 2 from the definition in (2.8).
In the expression in (3.3), P(i) is the set of all integer partitions of i. Each partition is described by i = σ j σ j (implicit summation) with mutually different σ j 's. In other words, the number σ j appears σ j times in the partition. The coefficients in the above expansion are given by 5 We can also invert (3.3), obtaining for the coefficients of the scattering angle in impact parameter space: These relationships suggest that both the scattering angle and the coefficients of the PM expansion of the momentum carry the relevant boundary data. As we show here, this is indeed the case, and moreover it can be used to compute observables for binary systems in closed orbits.
Of course, the expressions in (3.3) and (3.7) can also be obtained following the derivation in §2.1, order by order, by finding the value of the χ  [37] to 4PM.) It is then possible, if so desired, to construct a Hamiltonian. As we show next, this can be accomplished without matching to a (local) effective theory.

Gravitational Potential
Given the relationship between the relative momentum and the scattering angle, it is straightforward to construct the gravitational potential. Using (2.5) and (3.2), we can find the Hamiltonian iteratively in powers of G, by solving the equation: (Notice we use a slightly different normalization for the c i coefficients than the one used in [36,37,47].) Clearly, at zeroth order we have: At higher orders, we find a recursion relation of the type: The B i,k 's are partial Bell polynomials, and G m (p 2 ) is given by 6 with P ( ) n = ∂ ∂E P n (E). By the properties of the Bell polynomial, only factors of c k with k < i appear on the RHS of (3.10). Notice that the pattern discovered in [37] follows directly from the first term in the expansion of the square-root in (3.8).
The above formulae reproduce the expressions given to 4PM order in Eqs. (11.27)-(11.30) of [37], when inverted to solve for c i 's as a function of P i 's (and accounting for the different normalizations). Hence, once the P i 's (or f i 's) are written in terms of the scattering angle through (3.3), the c i 's agree with the explicit expressions as a function of p 2 given in Eq. (10.10) of [37]. For instance, as expected at leading order, Let us emphasize that the coefficients of the expanded Hamiltonian are obtained directly in terms of the coefficients of the PM series for the scattering angle, without performing a matching computation. This suggests that we may bypass the use of a Hamiltonian to compute observable quantities, as we demonstrate here.

From Amplitudes to Scattering Angles
In the context of the two-body problem in gravity, up to now the (classical limit of the) scattering amplitude has been computed as a series expansion in the PM framework and been used to derive the gravitational Hamiltonian. We refer the reader to the comprehensive analysis in [37] for further details. In this section we explore the possibility to use the gauge invariant information encoded in the scattering amplitude to directly compute the scattering angle, without resorting to a gauge dependent Hamiltonian. Along the way we will find a remarkable connection between the amplitude and the relative momentum of the two-body problem, from which we can readily obtain the scattering data. 6 Formally, one can show that the Gm's can be written as Gm(p 2 ) = ∂G(p 2 )/∂G evaluated at G = 0, with a generating function given by Since intermediate IR divergences are expected to cancel out in observables quantities, we will restrict ourselves to IR-safe quantities. Moreover, in the classical limit of the conservative sector we can simply retain the contributions which are non-analytic in q, and real. 7 For the sake of notation, in this and subsequent sections we will simply denote (unless otherwise noted) the relativistically normalized IR finite piece of the classical amplitude in the center of mass frame.
As it was shown in [36,37,47], a non-relativistic EFT can be constructed to read off the gravitational potential through a matching condition: 8 where the M n are obtained as a series expansion in powers of G: In principle, it is not obvious how to obtain the scattering angle from the amplitude without going through a Hamiltonian. A key observation in this direction was made in [37], where it was speculated that the M EFT n 's are proportional to the P n 's (or f n 's) entering in the expansion of the momentum (see §3). 9 In fact, introducing the Fourier transform of the amplitude (up to a relativistic normalization factor), we observe that the proportionality is exactly what is needed to obtain the relation where M n (E) are the coefficients in the expansion of M(r, E) in powers of G/r. Provided Eq. (4.4) holds to all orders in the PM expansion, it implies the following -remarkably simple -formula: relating the (Fourier transform of the) full amplitude and the momentum in the center of mass frame. This expression then allows for the computation of the scattering angle directly from the amplitude. We will refer to it as the impetus formula. 7 There are of course also imaginary parts which account for internal gravitons going on-shell. In what follows we will concentrate on the conservative sector. See §7 for more on radiation-reaction. 8 The relativistic amplitude is related to the non-relativistic version by an overall factor of (4E1E2). 9 This was explicitly checked to 5PM order. (We thank Mikhail Solon for sharing this with us.) Following the steps outlined in [37] (see also [51]), one could in principle demonstrate that (4.4) persists to all PM orders. However, the validity of (4.5) deserves a better understanding than just an accident of the computation in the EFT side. We provide a derivation below which highlights the (physical) origin of (4.5), as well as how to correct it to incorporate radiation-reaction effects. We give an alternative (more explicit) proof in Appendix A.
Let us start by mapping the computation of the scattering amplitude in the relativistic field theory with to that of potential scattering in standard quantum mechanics. To that purpose, we re-write the evolution equation of the relativistic state, in terms of an effective Schrödinger-like equation, The effective potential is constructed by solving iteratively in powers of G, and replacing F (p 2 , r)E|ψ → F (p 2 , r)H PM (p 2 , r)|ψ to the given PM order. (Notice we can choose any ordering, since the commutators do not contribute in the → 0 limit.) As long as we keep all the PM corrections, we expect the solution to the original and effective Hamiltonians to match into each other in the classical limit. 10 We compute the following expectation value: withP the momentum operator, and we have inserted the identity in the form |r r|d 3 r = 1.
The full solution with 'energy' p 2 ∞ of the scattering problem, ψ p (p ∞ ), is obtained in terms of the amplitude for potential scattering, defined as where r|p = φ p = e ip·r √ Vol are the free momentum states (with 'Vol' the usual volume factor). 11 It is clear that the total energy/momentum is conserved. However, we can use the above expression in (4.9) to define a classical, localized and instantaneous (square of the) scattered momentum as follows: 12 1 Vol We can justify this definition by considering the classical limit. Re-installing the 's, the wave-function takes the form ψ e iS cl / / √ Vol, with S cl the classical action. Including the non-interacting piece, (4.11) implies p 2 cl = (∇S cl ) 2 + O( ), as expected. Following [43], we split (4.11) into two contributions: the purely conservative part I (1) which is linear in the amplitude, and radiation-reaction effects I (2) which depend quadratically instead. For the first term, we keep only the real part of the RHS in (4.11) to linear order in ψ p (p ∞ ). (The imaginary parts must cancel out by construction). The remaining term, quadratic in f (p, p ), carries information about radiation effects, which we do not incorporate at this moment (see §7 for further details).
Before we proceed, let us make a few important remarks about intermediate IR divergences. In general, IR poles exponentiate into an overall phase, which cancels out in observables quantities (such as the cross section). By the definition in (4.11), an overall phase would also cancel out in p 2 sc , making it an IR-safe observable. On the other hand, both the I (1) and I (2) terms can in principle be IR divergent. (They may also contain imaginary parts from intermediate soft modes.) Yet the IR poles must cancel out in the sum Therefore, to obtain the physical contribution from I (1) to p 2 sc , we remove its IR divergent piece.
Notice that the expression in (4.13) involves an infinite series of terms, from iterations of the Lippmann-Schwinger equation. However, as we demonstrate below, these terms are precisely the combination which enters in the scattering amplitude. To show this we use the fact that ψ p (r, p ∞ ) obeys the Schrödinger equation in (4.7). Hence, where we inserted the momentum identity |k k|d 3 k = 1 and used (4.10). Gathering the pieces together, and identifying the momentum transfer q = p − k, we find: (4.17) (As we explained above, IR divergences cancel out between the two contributions in (4.12), and therefore we keep only the IR finite part of the scattering amplitude.) To finish the proof, we must identify the effective Schrödinger-like scattering amplitude with the (relativistic) one from the original problem. This can be done in a number of ways, for instance by matching the (gauge-and coordinate-invariant) cross section to all PM orders: From here, taking the classical limit on both sides, 13 we conclude which together with (4.17) leads to the impetus formula in (4.5). See Appendix A for an alternative (and also more explicit) derivation.

. . . → Deflection Angle
Once the relationship between the scattered momenta and amplitude is established, we can then swiftly remove the scaffolding we use in the form of the Hamiltonian evolution. Applying the impetus formula, it is now straightforward to compute the scattering angle by simply using (2.4), 14 where M(r, E) = M(r, E)/p 2 ∞ . The point of closest approach can be obtained from the The above equations provide a non-perturbative map between χ(b, E) and M(p, q). This relationship can also be expressed in terms of Firsov's formula in Eq. (3.1). First we notice that r min is given by where we used p 2 (r min , E) = b 2 /r 2 min . This implies, returning to the scattering amplitude, The reader will immediately notice that this expression closely resembles the eikonal approximation. Indeed, taking r min = b + · · · , we find Expanding the exponential in the PM approximation, we have as expected. Higher orders terms can be determined iteratively.
Alternatively, using together with (3.7), we can solve for the scattering angle as a function of the amplitude. For instance, to 3PM: These equations ultimately illustrate the physical information encoded in the scattering amplitude, with the f n 's representing the (gauge invariant) boundary data that is intimately linked to the scattering angle, and vice versa.
Notice that, in principle, the knowledge of the f i 's allows us to read off an infinite series of PM terms for the deflection angle. For example, for an 'f 1 -theory' -obtained from M 1 at 1PM -we find from (3.7) if n is odd and zero otherwise. This reproduces all the f 1 terms in (4.27), and beyond. Needless to say, this sums into the Newtonian form (see (2.6)), with y ≡ GM f 1 /b. We can also perform similar manipulations for the 'f 1,2 -theory', extracted from the 2PM scattering amplitude. We find from (3.7), , n = 0, 1, . . . , (4.30) and performing the sum we obtain with F 2 ≡ f 2 /f 2 1 (see the next section for more details). In the next section we will discuss how the boundary information from scattering processes can be used to construct adiabatic invariants for closed orbits.

From Scattering Data to Adiabatic Invariants
In this section we transform the information from the scattering process, with E > M , to the case of bound states, with E < M . In principle, this can be done by first deriving a Hamiltonian from the boundary data, as discussed in §3.2, and afterwards searching for closed orbits. As we demonstrate here, we can bypass the use of a Hamiltonian and proceed directly from observable quantities in scattering processes, such as the deflection angle, to adiabatic invariants for elliptic and circular orbits, such as the binding energy. Throughout this section we use the non-relativistic energy, as well as the reduced angular momentum j ≡ J/(GM µ), introduced in (2.7). We will also use the definition = −2E, which is often standard in the PN literature.

Radial Action
Following [83], we can introduce the radial action: from which gravitational observables may be computed. The points r ± are the real positive roots of p r (r) = 0, with 0 < r − < r + . These solutions exist only for bound states, a condition which can be enforced once p 2 (r, E) is known. As we discussed in the previous section, the functional form of p 2 (r, E) may be obtained from the knowledge of the scattering amplitude, through (4.5). By analytic continuation to the region E < 0, we can then compute gravitational observables for bound systems. Therefore, provided we consider classical processes (without anomalous thresholds), the radial action takes the form where the scattering amplitude is analytically continued to E < 0. This equation allows us to compute gravitational observables directly from the knowledge of the amplitude. For instance, the periastron to periastron period as well as the periastron advance: At the end of the day, these expressions will be written in terms of analytic continuations of boundary data, i.e. f n (E) or the scattering angle χ (n) (E). As usual, the precise form of the analytic continuation requires a little work, as well as finding the r ± endpoints for the radial motion. We will give a more concrete procedure as we move along, in particular when we concentrate on circular orbits. In general, since the f i 's are functions of γ, and themselves functions of β through (2.15), we will adopt the prescription β → iβ, such that γ → cos β, with 0 < β < π. For example, at 1PM we have which is negative, allowing for bound orbits.
Before we conclude the general case, let us give a few useful formula to compute the radial action in the PM expansion. As we see below, for some applications the precise knowledge of r ± is not needed.

Post-Minkowskian Expansion
Expanding the radial action in the PM framework using (3.2), which are obtained through the scattering amplitude via (4.4), we encounter expressions of the type: where we introduced the split: with λ a formal small parameter associated with the PM expansion.
In order to obtain the radial action, we follow [83], where it was demonstrated that (5.7) can be computed in terms of a contour integral with residues at 0 and ∞, without the need of the values for the r ± , the turning points. First of all, we expand the action in powers of λ, One can then show that the S (n) r (J, E)'s are given as polynomials of the D i 's, times a series of master integrals, These master integrals can then be evaluated using residues: 15 where 2 F 1 are hypergeometric functions. Notice that the reality condition for the radial action is directly connected with the existence of bound states, for which p 2 ∞ (E) < 0. At the end of the day, the action will take the form:

Periastron Advance to Two-Loops
For example, we can use the radial action to compute the precession of the perihelion. From the scattering amplitude at one-loop order [47], we have In this case the radial action can be computed exactly, and (5.5) automatically yields for the periastron advance to 2PM order, and to all orders in velocity. As expected, using that γ = 1 + O(v 2 ) this result reproduces the leading order value in General Relativity, ∆Φ = 6π/j 2 + · · · , at 1PN order.
The computation at higher orders is more involved. For instance, at two-loops we have D 1 ∝ M 3 , with [36,37] 20) and the radial action involves a series expansion, as shown in (5.17), which can be written as It is straightforward to show that G 3 terms (and generically of the form G 2n+1 ) are not present. Therefore, we need to calculate the G 4 contribution, which includes also the amplitude at three-loops, through M 4 . It is straightforward to expand the radial action up to 4PM, such that (5.17) yields which in principle includes an infinite series of velocity corrections. By restricting to the contribution to two-loops, and performing a PN expansion, we have (5.23) which reproduces the known result to 2PN order in General Relativity, see e.g. Eq. (5.8) in [84], but it includes also a (partial) series of PN corrections. 16 In particular, the contribution from the one-loop amplitude in (5.19)  binding energy. Therefore, the O(E 2 /j 2 ) and O(E 3 /j 2 ) terms at 3PN and 4PN, respectively, are already included in the above result, and we can already predict the 5PN contribution (shown in the last line of (5.23)).

From Hyperbolas to Ellipses . . .
The above procedure is very generic. However, there is also a more geometrical approach to construct adiabatic invariants, which will be useful when we study the circular case. The main observation is that the point of closest approach, r min , is a root of r 2 p 2 = b 2 , namely Our task is to take the boundary data from the scattering problem, encoded in the f i 's, and find the two real (positive) solutions r ± (E, J) for the bound state. Of course, we can find these solutions from (5.24) after analytic continuation, with b = J/p ∞ , as we would do also with the Hamiltonian. However, as we shall see, this can be done as well through an analytic continuation of the impact parameter.
First of all we start with hyperbolic motion (see Fig. 1), described by the equation where u is the eccentric anomaly andã andẽ the orbital elements. These can be written as in terms of the two real solutions of (5.24), one of which is negative. As we discuss momentarily, by performing a series of analytic continuations we can find the two roots, r ± , for bound orbits. After these roots are found, we perform one last analytic continuation in the eccentric anomaly u → iu [85], which transforms the hyperbolic into elliptic motion (see Fig. 2):  Figure 2: Bound elliptic motion in the center of mass frame (gray dot). The black ellipses mark the paths of each individual body. The heavier one lies on the focus of the green dashed ellipse, which describes the worldline of the lighter body in the companion's reference frame. The dotted circle of radius a defines the eccentric anomaly, u, which can be used to paramaterize the orbit. See the text for a description of the orbital elements.
The elliptic orbit can then be obtained as a result of analytic continuations from the two roots of the unbound problem. The form in (5.27) will be helpful later on to compute the binding energy for circular orbits, for which the eccentricity vanishes (e = 0).

Analytic Continuation
As a warm up, let us first consider the scattering problem to 1PM order. The boundary information in this case is encoded in f 1 , obtained from M 1 . In this "f 1 -theory", the condition for r min follows from the second order equation The roots are given byr with the reversed ∓ chosen for later convenience. For the scattering problem we have f 1 > 0 and b 2 > 0, such that we only have one real and positive solution: On the other hand, for a bound orbit we have f 1 < 0. Moreover, b 2 = J 2 /p 2 ∞ takes on negative values. Under these new conditions, it is clear that (5.30) has two positive roots, obeying: 0 < r − < r + . As we show below, the two real roots can also be obtained from the hyperbolic solution via an analytic continuation. Using Firsov's formula, the procedure can be constructed entirely in terms of the scattering angle.
We start with r − , which can be readily obtained from r min , through the following analytic continuation r − (J, E) = r min (ib, iβ) . (5.32) and β defined in (2.15).
To find the other solution, r + , we proceed as follows. (What we describe below can be equally applied to read offr + from the knowledge ofr − ) We first re-write r − in the form To obtain the r + solution we perform the map b → −b, which sends These manipulations can easily be extended to 2PM order. For the "f 1,2 -theory" we have the same type of quadratic equation for the scattering problem: There are, of course, also two solutions and the same steps as above allow us to construct r ∓ for the bound system: At this point the reader may wonder how the procedure will extend to higher orders -in particular, once the equation in (5.24) becomes next-to-impossible to solve in closed analytic form. As we discuss next, Firsov's formula gives us a prescription which can be applied to find both the required r ± solutions, to all PM orders.

Via the Scattering Angle
The main observation to construct a generalized map is the representation ofr − in terms of the scattering angle in (4.22), which we reproduce here for the reader's convenience: In the PM expansion of the scattering angle, it takes the form: As an example, let us show how this representation reproduces (5.31) for the f 1 -theory. In order to see this we need the f 1 contribution to the scattering angle at all orders, which is given in (4.28). Plugging it into (5.40) and performing the summation, we find after using the identity arcsinh(x) = log x + √ x 2 + 1 . According to our prescription, we obtain r − (b, β) by analytic continuation, It is also clear that r + (b > 0) = r − (−b, β).
The same manipulations can be applied to the f 1,2 -theory. Using (4.30), the representation in (5.40) yields Let us compute one sum at a time. The first one is straightforward: The second sum is slightly more involved, we find which is nothing but the expansion of arcsinh: and we finally get Putting the two sums together, which agrees with (5.37), and can be shown to reproduce the correct r ± via analytic continuation, with and Of course, obtaining the solution to a quadratic equation is significantly simpler than the type of resummations we just performed. However, finding the roots of higher order polynomials is a much more difficult problem, while Firsov's formula provides a compact representation which readily identifies the two roots we need to characterize the elliptic problem (something which is far less transparent in the generic form of the solution to (5.24)). 17 This becomes more relevant for the case of circular orbits, which we study next.

. . . to Circular Orbits
In the limit where we collapse the ellipse into a circle, the eccentricity vanishes. The condition e = 0 implies that the roots are degenerate: r + = r − . Given the representation we have for both roots, and promoting the impact parameter to a complex number z the circularity condition turns into Notice that the LHS is complex for z = ib, such that z 2 < 0 (for b > 0), which is required to find solutions for closed orbits. Once the above condition is met, we can solve for b, and subsequently for the reduced angular momentum j(E) = |p ∞ |b/(GM µ), as a function of the f n 's. Notice that, for circular orbits, the above steps bypass the need to compute the radial action. In this special configuration, we can then read off the orbital frequency as a function of the binding energy from the second law of binary dynamics [66], Since the PM computations depend on γ, it is natural to introduce a new object GM Ω γ ≡ ∂j(γ) ∂γ −1 .

(5.55)
Using that ∂γ/∂E = Γ, we find which characterizes the gauge invariant information for the bound state in a circular orbit. As we shall see, it can be obtained systematically at any PM order, and to all orders in velocity, from the scattering data.

The f 1,2 -Theory
Let us demonstrate the necessary steps for the derivation to 2PM order. We have already computed this sum for the f 1,2 -theory, which now runs over the odd values, χ (2n+1) [f 1,2 ] in (4.30). The result, as expected, is the arcsinh function. The circular orbit condition becomes with obtained directly either from the scattering amplitude or deflection angle (via (3.3)). The ellipsis includes contributions from higher f n 's, which we will discuss momentarily. To find a solution to this equation we must also evaluate the f 1 and f 2 functions with γ < 1, through the analytic continuation β → iβ.
Using the values for f 1,2 in (5.59), together with (5.58), we readily obtain: where we followed the convention in [9] with = −2E. The result in (5.60) is valid at 2PM, to all orders in velocity. However, it does not include all effects needed in the PN framework for the bound system. Nevertheless, it does capture relevant information, while also including a partial resummation of higher order terms. To see this, we perform a PN expansion in powers of ∼ v 2 , resulting inj circ = 1 + 9 + ν 4 + 1 16 (−55 + 48ν + ν 2 ) 2 + · · · . (5.61) We immediately notice that (5.61) reproduces the 1PN term (see p. 140 of [9]). From here we can readily compute the orbital frequency, using the first law of binary dynamics [66]. The full 2PM result is a little messy, however, we can introduce the standard PN parameter, 18 Notice that, in principle, the same expression can be obtained directly from the vanishing of the radial action for circular orbits to 2PM, which becomes (−A)(−C) = B 2 , see §5. In terms of the amplitude, this can be written as which reproduces (5.58), after using Mn = fn/(p 2 ∞ M n ) and noticing p 2 ∞ < 0 for a bound state. While imposing Sr = 0 for circular orbits is straightforward at one-loop, it becomes much more cumbersome at higher orders (see the next subsection).
which is written here in powers of , the binding energy. The above relationship can be inverted, precisely reproducing the value of the binding energy as a function of frequency to 1PN order (see e.g. Eq. (232) in [9]).

Orbital Frequency to 3PM
In order to incorporate the newly obtained 3PM effects to the orbital frequency, we need to include the f 3 (γ) contribution. Using (4.4) (or via the deflection angle through (3.3)), we have where we used the IR finite piece of the M 3 scattering amplitude in Eq. (9.3) of [37] (after proper relativistic normalization). Before we proceed, it is worthwhile noticing how the analytic continuation in rapidity works with the new terms. Implementing β → iβ goes smoothly, except for the arcsinh, which leads to a complex number. However, there is also a complex denominator, such that In principle, the manipulations in (5.53) can be extended to add the f 3 part. However, while numerically straightforward, the resulting equations are somewhat analytically cumbersome. When restricted to circular orbits, it is useful to develop a hybrid approach, where we input the fact that the roots in Frisov's form are also the zeros of the original equation in (5.24). In this case, the existence of a circular orbit requires that all the roots are equal, and therefore the discriminant of the cubic equation must vanish. Furthermore, we will assume that the solution matches the 2PM result for small f 3 , which uniquely fixes the relevant roots. (We discuss the power counting below.) Under these conditions we find (for z = ib, with b > 0) wheref i = (GM/z) i f i (no summation over repeated indices). The factor of w is given by: 3 2 e i arg(w) .

(5.67)
In general, there are three solutions for z 2 in (5.66). As we mentioned, we chose the one that reduces to the known solution for the 2PM theory in the limit f 3 → 0, yielding This expression can be rewritten and solved for j, obtaining: 19 as a series expansion in F 3 ≡ f 3 /f 3 1 ; with j 2 2PM the solution in (5.58). The above result encapsulates the full 3PM information for the bound state system to all orders in velocity (through γ), extracted from the scattering data.
By power counting we observe that F 3 O( 2 ). Therefore, in search of 2PN accuracy, we would only need to keep the first extra term in (5.69), such that with j 2 1PM = |p 2 ∞ |f 2 1 /4. It is straightforward to computej circ , keeping the leading term in (5.64), and we find which reproduces the 2PN result in [9]. Notice that the ν 2 2 term in (5.61) remains the same, such that the 2PM result already carried part of the information for the 2PN dynamics. That is the case because f 3 /f 1 ∼ α + βν 1 at leading order in . (In fact, the (ν ) n terms are entirely driven by the f 1 -theory, see below.) 19 As for the 2PM case (see footnote 18), one can show that the expression in (5.69) implies the vanishing of the radial action, including the D1 term. However, notice that solving for j(E) (or E(j)) to all orders using the radial action, see e.g. (5.21), becomes much more difficult than the procedure we outlined for the specific case of circular orbits.
The orbital frequency is obtained via (5.56). Taking a derivative of (5.69) with respect to γ, we have (5.72) Using the standard PN parameter in (5.62), we find From here we can invert to obtain the binding energy, These results agree with the known value to 2PN (see Eqs. (232) and (349) in [9]), while they include also an infinite series of velocity terms. However, still missing are the PM corrections necessary to complete higher levels of PN accuracy, even to recover the correct test-particle limit at 3PN. (Of course, this is expected since the O(G 4 ) term is needed.) The Exact f 1 -Theory The 2PN order is as far as we can go with the 3PM amplitude/angle. Yet, notice that the expression in (5.74) (or (5.73)) still captures a subset of higher PN corrections. For example, the same pattern we found before reappears. Namely, the terms with the highest powers of ν at a given PN order in (5.74) reproduce the correct answer. For instance the 77 31104 ν 4 x 4 at 4PN order (see e.g. [10]). However, this is not that surprising, and it is entirely driven by the 1PM theory (the f 2 1 term in (5.58)), as it was pointed out in [86]. In the PM framework, this is due to the scaling of the f n terms in powers of and ν, relative to f 1 . This observation allows us to incorporate all of the O (ν n x n ) terms in the binding energy, which are controlled by the 1PM contribution to the orbital frequency, given in closed form by the expression: (xν) n π(n + 1)! + · · · , (5.76) which captures the exact O (ν n x n ) at each PN order, and is not modified by higher PM effects.

Power-Counting
At higher PM orders we expect the above power-counting to be generic, namely the n-th order contribution will scale as powers of F n ≡ f n /f n 1 (n > 1) relative to the leading order. The reason is that, for a bound state we have GM/r ∼ p 2 ∞ ∼ v 2 , and therefore the PM corrections are naturally down by powers of , such that the scaling of the (reduced) angular momentum obeys: (5.77) Each PM order includes a series of terms, each of which can be PN expanded. At the end of the day the result takes the form: to any desired PN order.

No-Recoil Resummation
In this section we perform a partial resummation of PM contributions to the impetus formula, by applying a no-recoil approximation.
The impetus formula provides a unique opportunity to explore the resummation of PM effects. The natural candidate is the no-recoil (or test particle) limit, at leading order in ν (the symmetric mass ratio). Since the amplitude in momentum space is a (dimensionless) relativistic scalar, we can compute it in any frame. For instance, in the rest frame of one of the particle, in which we assume m 1 m 2 , and therefore we can ignore self-force effects. In this limit we have m 2 → µ and m 1 → M , as well as E = M (1 + O(ν)). The (Fourier transform of the) amplitude can then be read off directly from the impetus formula, multiplying (4. and after identifying p 2 ∞ → µ 2 (E 2 0 − 1). The value of the momentum in a Schwarzschild background is given byp where we introducedp = p/µ, E 0 = E 0 /µ, and E 0 is the energy of the heavier particle of mass m 1 . By expanding in powers of G we obtain the "Schwarzschild amplitude", which reproduces M Sch 1,2,3 in Eq. (11.12) of [37] to 3PM (after inserting the non-relativistic normalization factor). Notice that the above formula, in combination with Firsov's (3.1), also neatly reproduces the scattering angle in Schwarzschild. For instance, to 2PM order:

. . . to Two-Body Dynamics
We now move to the center of mass frame of the two-body problem. While the amplitude is invariant, it is useful to write it in a covariant fashion, which we can then evaluate in any frame. This is straightforward, by simply transforming the expression in (6.1) into a relativistic invariant, noticing Hence, in the center of mass frame we have in the no-recoil approximation, where Notice the factor of 1/Γ follows from the relativistic normalization.
Clearly, the no-recoil approximation does not incorporate all of the PM effects. However, it is straightforward to show that it reproduces the 2PM dynamics. This can be seen directly from (6.7), by computing the f no-rec n (E)'s. From the definition in (3.2), we find and so on and so forth. Using (3.3), the knowledge of the f n 's leads directly to the scattering angle. Given that χ (1,2) b ∝ f 1,2 , it is straightforward to show the value of χ for the two-body problem is recovered to 2PM, see (2.10). Since the scattering angle (and/or the f n 's) encode all the required information, we conclude that the no-recoil approximation reproduces the two-body dynamics to 2PM order. The impetus formula thus provide the backbone for the relationship between test-particle and two-body problem discovered in [62].
At the next order, however, the no-recoil approximation fails due to self-force effects. Nevertheless, the expression in (6.9) captures one of the contributions from the full f 3 . The no-recoil approximation amounts to the ν-independent factor inside the parenthesis of (5.64). The series in principle continues ad infinitum. It is easy to see that in the limit Γ = 1 we recover the exact Schwarzschild prediction for the scattering angle. For example, using (4.27), we have Sch (E 0 → γ) , (6.10) in impact parameter space, by construction. However, when Γ = 1, the approximation does not provide a reliable proxy for the O(ν) terms, as it does at lower PM orders. This is related to the peculiar factors of Γ that appear in the f n 's at higher orders. It is though possible to envision a resummation which also includes the factors of Γ, by finding the correct variable.
We will return to this point in future work.
Notice that the connection between two-body dynamics and the test-particle limit to 2PM order not only holds for the scattering angle, it also translates to physical observables for bound orbits. This is expected, since there is a one-to-one map between the scattering angle and the binary dynamics. The connection is in fact remarkably simple, also at the level of the orbital frequency for circular orbits. Notice, that the overall factor of Γ cancels out in (5.58) to 1PM order (but remains at 2PM). This means j 2 Sch (E 0 → γ) is exact to 1PM, after taking M = m 1 + m 2 for the mass. This is yet another indication that General Relativity at 1PM order is recovered by the test-particle limit.

Conclusions
We have introduced a dictionary relating gravitational scattering data to observables for bound states in generic configurations. Our map can be described (schematically) as follows: Our dictionary relies on the remarkable connection we have shown exists between the relativistic scattering amplitude in the classical limit and the relative momentum of the twobody system: The R.R. stands for radiation-reaction terms, quadratic in the amplitude (see (4.12)), which may also contribute to the conservative sector (see below) [19][20][21]. 20 Together with Firsov's equation [65], relating the momentum to the scattering angle, the above impetus formula allowed us to relate the amplitude directly to the scattering angle: In the PM framework, we were then able to relate the coefficients in the expansion of the amplitude, to the coefficients for the deflection angle in impact parameter space, and vice versa (see (3.3) and (3.7)). For instance, introducing F n ≡ f n /f n 1 , for n > 1, and we find to 3PM order: 21 which includes a resummation of 1PM and 2PM terms. The relationship between χ and M, through the F n 's, suggests that the scattering amplitude by itself carries gauge invariant information in the form of asymptotic charges. It would be useful to properly classify the scattering data, and perhaps recast (3.3) (or (3.7)) and the above expressions in a geometrical (or algebraic) fashion. It would be also interesting to understand the connection with the eikonal phase [56]. 20 Let us emphasize that, by construction, the RHS of the impetus formula is IR finite, such that intermedia IR divergences cancel out between the two terms, see §4.1 and Appendix A. 21 Notice that the PM framework naturally introduces an expansion in GM f1/b. Moreover, in the limit F2,3 = 0, (7.6) generalizes the Newtonian result in (2.6), away from ν = 0 and to all orders in velocity.
By the construction of a radial action, together with a series of analytic continuations in the impact parameter and rapidity (γ = cosh β), of the form b → ib, β → iβ, 22 we are able to relate the scattering information to the computation of adiabatic invariants for bound orbits.
As an example, we can readily obtain the periastron advance to 4PM order, without the need of a Hamiltonian. The scattering amplitude is encoded in the coefficients, M n ≡ M n /2E, of the PM expansion of the RHS of the impetus formula (7.1). The above expression, up to two-loops [36,37], matches the precession of the perihelion in General Relativity to 2PN order, while including also a series of velocity corrections, see §5.2. 23 We also found that the contribution from the one-loop amplitude is exact, and takes the form 24 Given the structure of the PM expansion, it is easy to see the leading PM term in (7.8) captures the O(1/j 2 ) contributions to the periastron advance to all orders in the velocity expansion, see §5.2.
The derivation of adiabatic invariants simplifies for the case of circular orbits, allowing us to solve for the (reduced) angular momentum of the bound state in terms of scattering data, from the condition of a vanishing eccentricity. We found to 3PM order, where j 1PM = (2 cos 2 β − 1) sin β , (7.10) 22 This transformation is intimately linked to the one relating hyperbolic to elliptic motion, via analytic continuation in the eccentric anomaly, u → iu [85]. 23 Notice that the agreement with the known value for the periastron advance provides -yet anotherconfirmation of the validity of the result presented in [36,37] to two-loops. Moreover, from Eq. (5.8) of [84] to 4PN, the expression in (7.7) can be used as a cross-check for the (instantaneous) value of the scattering amplitude to O(G 4 v 2 ). In principle, at 4PM order radiation-reaction terms will also contribute (see below). 24 The expression in (7.8) agrees with the result found in [49] to leading order, after taking into account the proper normalization factors and transforming the amplitude from momentum into real space. Our approach, however, is more generic and it applies to all loop orders via the radial action, without the introduction of a Hamiltonian, see §5. Since only the triangle integral contributes to the IR finite part of the scattering amplitude at one-loop in the classical limit [47], it is also evident that its vanishing would lead to no precession at leading order, as it was found in [49] for the case of N = 8 supergravity.
is the 1PM result. 25 The orbital frequency can be derived from the second law of binary dynamics: = Ω γ /Γ , (7.11) which agrees with the known result to 2PN order. It also includes a resummation of all velocity terms scaling as O(G 3 v n ), see §5.5. Our dictionary thus allows us to obtain the orbital frequency as a function of the binding energy, directly using the gauge invariant information from the scattering amplitude, and without resorting to the (lengthier and gauge dependent) Hamiltonian. Following the steps outlined in this paper, we can translate the scattering data -either from the amplitude or deflection angle -directly to observables, at any order in the PM expansion and to all orders in velocity.
We have also performed a partial resummation of PM terms, using a no-recoil approximation for the scattering amplitude. Using the impetus formula, we have for the scattered momentum of the two-body problem in terms of the (boosted) scattered momentum in Schwarzschild. The factor of 1/Γ arises from the relativistic normalization in (7.1). This (PM-resummed) approximation exactly reproduces the 2PM dynamics, and also provides partial information for the higher order terms. There is, of course, a lot of room for improvement. Even though the no-recoil approximation carries some of the higher order terms (see (6.9)), while encoding at the same time the exact result in Schwarzschild at leading order in ν (see (6.10)), it fails to provide a good proxy for the exact result at 3PM. For instance, while the first term in (2.14) mimics the Schwarzschild result, the overall factor of Γ 3 does not come out of the above approximation, which instead mixes different powers of Γ through the different f n 's. Given the nature of the computation, it appears that the structure Γ k χ (n) j (E 0 → γ) for one of the terms at nPM order is quite generic [60,61]. This suggests that we should be able incorporate the factors of Γ inside the map between E 0 and γ, and to resum also all of these terms. (For instance, using Γ = 1/Γ + O(ν), it is easy to manipulate the expression in (6.9) to recover the first term in (2.14).) We explore this possibility in future work.
The 3PM state-of-the-art for the scattering amplitude/angle leads to the 2PN orbitalfrequency/binding-energy. At this stage, even though a resummation of velocity corrections is included, this is not yet a very accurate result if compared with the present state-of-theart in PN computations [23]. However, a series of improvements can be easily incorporated. In principle we can fix the full O(ν 0 ) contribution using the no-recoil approximation, which includes the Schwarzschild limit. We can also include all of the O (ν n x n ) terms, which are 25 Notice (7.10) agrees with the value in Schwarzschild after replacing E0 → γ.
controlled by the f 1 -theory. For the latter we found the exact result to 1PM (withν = ν −4), This close expression can be inverted, to capture the following series of corrections to the binding energy, = x ∞ n=0 cos (n + 1)π 3 Γ 5 + 2n 6 Γ 1 + 4n 6 (xν) n π(n + 1)! + · · · , (7.14) in the PN expansion, which are solely determined by the 1PM theory. (In some sense, these terms are 'tree-level exact' and do not get renormalized by higher PM orders.) These additional contributions will help in the construction of more accurate waveform models.
In conclusion, the dictionary we introduced in this paper provides a natural way to translate the gauge invariant information in scattering processes into observables for bound states in gravity, without the need of gauge dependent objects. The novel tools in the study of scattering amplitudes can potentially provide the necessary high-precision scattering data, bypassing the combinatorial hurdles of the Feynman technology, and in a manifestly relativistic framework. In principle, relativistic integration with internal massive particleswhich is needed to benefit from the powerful on-shell techniques -makes the problem of computing the relativistic amplitude significantly more laborious than the PN counter-part. That is the reason the 3PM order, or two loops, has been tackled only very recently in [36,37], while the PN expansion is progressing towards the 5PN (or five loops) level of accuracy [23]. Nevertheless, we expect further improvements will streamline the derivation of the required scattering amplitude (and/or independent derivations of the deflection angle) such thatin conjunction with our novel framework and various other techniques -the necessary high level of precision for the future GW astronomy can be achieved.
In what follows we provide a brief discussion over a few directions in which our formalism can be explored further. We restrict ourselves, as for the entire present work, to the case of non-rotating bodies. We will explore the spinning scenario elsewhere.

Radiation-Reaction
The impetus formula in (4.12), also receives corrections due to radiation effects, encoded in the second term, I (2) , which is quadratic in the amplitude. In principle, following the analysis in [43], we can systematize the necessary steps to add radiation-reaction. The latter includes so-called tail effects [19], which in the scattering amplitude can be incorporated through radiation modes, starting at 4PM order [37]. The advantage in the PM formalism is the lack of intermediate (spurious) IR divergences which pollute the PN computations due to the split into regions -which would not be present in the PM framework. Since back-reaction terms involve long-distance modes, it is also possible to compute them using the EFT approach [11], in the form of a radiationreaction force [19,68,69]. The latter also includes a conservative piece, which shifts the value of the binding energy [11,[19][20][21]23]. We can then re-do our analysis in the conservative sector by including the contribution to the binding energy due to the tail, e.g. E → E + E tail , which may be obtained independently (and resummed using the renormalization group evolution discussed in [19]). We will study radiation-reaction in more detail, as well as absorption effects [87,88], in future work.

Scattering Angle From the EFT Approach
Using the EFT approach, in principle we can also derive the scattering angle in the PM framework (see also [60,61]). The idea is simple. From the effective Lagrangian we can read off the total change in momentum, from which we obtain the scattering angle in the center of mass: with |∆p 1 | the total momentum change for particle 1, and the same for the companion. The Lagrangian L can be computed in the EFT approach to any order in the PM expansion. By construction, the computation involves classical (point-like) sources, which simplifies the quantum problem from the onset. However, we still rely on iterated Green's functions in the form of Feynman diagrams (see Fig. 3) and, for the PM calculation, a series of relativistic integrals. Yet, once the deflection angle is known, we can plug it into our machinery to derive the f n 's, thus reconstructing the scattering amplitude. We can also perform the manipulations we describe in this paper to derive gauge invariant observables for closed orbits. We will present the explicit derivation of the scattering angle in the EFT approach for relativistic sources in future work. We will also further explore the connection between the EFT derivation and the one from the scattering amplitude elsewhere.
Let us add a comment on the map between test-particle and two-body dynamics. At 2PM, the computation of the deflection angle involves only the one-loop diagrams shown in (a)-(c) of Fig. 3. In the language of EFT, this is equivalent to computing the one-point function produced by e.g. particle 1, and evaluating it on the worldline effective action of particle 2, plus mirror image. For instance, the diagram in (b) comes from the expansion of the square-root in the point-particle action. This one-point function can also be computed in the rest frame of particle 1, and then boosted to the center of mass. (Alternatively, one can start with the boosted Schwarzschild solution.) The Lorentz transformation will naturally map E 0 → γ, as in (6.5), while the mirror image will take care of the symmetrization. It is Figure 3: Sample of Feynman diagrams needed for the computation of the scattering angle in the EFT approach to 3PM order. Only the diagrams to one-loop order in (a)-(c) are needed to 2PM.
clear then that the momentum change will be the same, obtained directly from the action, and the only difference in the scattering angle is the factor of 1/p ∞ in (7.17). Therefore, where χ 1pt is the scattering angle obtained to one loop order from the one-point function, which is exact to 2PM (see Figs. 3(a)-(c) ), and we used (Notice this resembles the factors entering in the normalization of the amplitude in the norecoil approximation.) This implies, which leads to the map found in [62]. The relationship clearly fails at 3PM, when terms such as the H-diagram (shown in Fig. 3(f)) start to contribute. The no-recoil approximation, however, retains all of the higher order terms which involve the computation of the one-point function in the EFT approach, shown in diagrams (d) and (e) of Fig. 3 at two loops.

Non-Perturbative Dictionary
The dictionary described in this paper is valid to all orders in the PM expansion. Therefore, knowledge of higher order terms in the scattering problem can be readily used to derive high-precision invariants for closed orbits. While the steps we described were implemented in the context of the PM framework, in principle Firsov's formula in (3.1), which we can be parameterized as r(λ, E) = λ e −A(λ,E) , with also holds in the non-perturbative regime. This means that, having a solution for the scattering angle as a function of the impact parameter and energy, for instance from numerical simulations, would translate into a solution for the dynamics of the two-body system in elliptic motion. Provided an ansatz for the dependence on the energy and impact parameter can be derived, one could perform the integral and analytically continue in the energy to construct the radial action, from which we can derive adiabatic invariants. As we mentioned earlier, the computation of the action does not require the precise knowledge of the r ± endpoints, and therefore it would only be a matter of finding a suitable representation, in energy and impact parameter space.
We could also use an ansatz for the non-perturbative scattering data to compute the minimum distance, see (7.3), and analytically continue in impact parameter (as well as in the energy) to derive the two roots needed to characterize elliptic motion. Then, the condition for a circular orbit becomes This expression must be understood as an analytic continuation in impact parameter space of the integral, evaluated at b → ±z, as described in §5. The (complex) solutions of the form z = ib (with b > 0) allow us to derive the reduced angular momentum, and subsequently the orbital frequency. Given that numerical simulations for the binary problem are time consuming, while scattering data appears significantly easier to collect, we think our formalism naturally opens up a new venue to explore the non-perturbative regime of the two-body problem in gravity, which deserves further exploration.

Classical Double Copy
It was discovered in [41] that classical spacetimes, such as Schwarzschild or Kerr black holes, can be shown to be double copies of gauge theory configurations. This means that test-particle (geodesic) motion in these background geometries can, in principle, also be mapped into each other. In light of these developments, the impetus formula invites itself to speculations on its connection to the non-perturbative form of the double copy. We will briefly comment on a few directions in what follows and return to this fascinating subject elsewhere.
In the no-recoil approximation, in the rest-frame of the heavy object, the impetus formula relates the scattering amplitude to motion in Schwarzschild's spacetime (see (6.1) and (6.2)), In turn, this can be written as geodesic motion, with g µν Sch the Schwarzschild metric in isotropic coordinates, andp 0 = E 0 . Therefore, M no-rec (r, E) = g µν Sch − η µν p µ p ν . (7.26) Since the equation in (7.25) is manifestly covariant, we can transform now to a different coordinate system. We can then choose the Kerr-Schild coordinates that made the double copy manifest in [41]. In terms of the scattering amplitude, this would correspond to a Fourier transform with respect to a shifted momentum. 26 In these coordinates we have with c a the color charge, the map found in [41] translates into a double copy relationship between scattering amplitudes in gravity and Yang-Mills, in the more traditional sense [29].
In fact, we can also look at this relationship in the other direction. In other words, taking the gauge theory amplitude and postulating the existence of a double copy map to classical gravity would imply the existence of the (linear in G!) Kerr-Schild solution for Schwarzschild. Moreover, following the approach in [89], one could also use the double copy to find other solutions of Einstein's equations.
Notice that the impetus formula also applies to gauge theory amplitudes. Namely, for the relative momentum we have: p 2 (r, E) = p 2 ∞ (E) + 1 2E d 3 r A(p, q)e iq·r + R.R. , (7.30) where A(p, q) is the classical Yang-Mills amplitude. In principle, one can use the traditional double copy relating M to A to find the connection between the classical motion in both theories. By mapping to the Yang-Mills case, this can potentially simplify the derivation of adiabatic invariants for the two body problem in gravity, including the possibility to relate also the strongly coupled regime.
Let us finish with yet another speculative idea. In the derivation of the impetus formula in §4.1 (see also the appendix A) we map the problem into a Lippmann-Schwinger evolution equation for the case of potential scattering, resembling the Schrödinger problem. As such, after analytic continuation to negative binding energies, the levels of the effective Hamiltonian in the Schrödinger-like equation correspond to the energies for elliptic orbits, through the identification of the adiabatic invariants. (A similar idea was the spirit of the original effective one body map in [64].) One can then imagine, following Dirac's steps, taking the square-root of the effective quantum problem. Provided the double copy relation between gravity and gauge theory amplitude holds (schematically) √ M ∼ A, one could imagine then mapping the binding energies of the Dirac problem for Yang Mills to the binding energy for the two-body problem in gravity. The precise form of this dictionary depends on the exact implementation of the double copy at the level of the classical amplitudes. We leave this as a spare time exercise for the reader.

A The Impetus Formula
There is another, more direct, way to prove the impetus formula in (4.5). Let us follow the same steps as in §4, but take instead for the effective potential the expansion of p 2 as a function of the energy: Once again, we expect the solution to the full and effective problem to match to all orders in the PM expansion. Following the same steps as before, we have for the scattering amplitude of this (equivalent) Schrödinger problem: 4π Volf (p 2 , q) = − p + q|V eff |p + · · · = 1 Vol i d 3 r P i (E) G i r i e iq·r + · · · . (A.3) From the relationship in (4.19), adapted to (A.1), we observe that the Born approximation already contains the information encoded in (4.4). Therefore, the impetus formula would hold, provided the additional iterations are composed of (super-classical) IR divergent terms, which do not contribute to the classical limit (since they cancel out between the two contributions described in §4.1). We show that is the case below.
The proof goes as follows. 27 Since the P i (E)'s are only a function of the energy, namely independent of the momentum, the potential in the effective theory of (A.1) can be written as (in d = 3 + dimensions): Moreover, the Green's function of the Schrödinger-like problem is simply given by Following the analysis in [37], we find that the -iteration of the Lippmann-Schwinger equation, encoded in the ellipses in (A.3), can be written as where k 0 = p and k +1 = p . The numerator, N ( ) eff , comes from expanding the potential in the classical limit as described in [37]. For instance, for the first iteration we have a constant N with q = p − p. Notice it is also a super-classical contribution, since it scales with an extra power of |q| −1 with respect to the classical term at this order.
We can now proceed by induction in the power counting. We will show the rather intuitive fact that these super-classical terms cannot generate classical contributions through mixing. Let us return to the Lippmann-Schwinger equation in (A.3). For the n-th order scattering amplitude in the effective Schrödinger problem, we have the recursion formula: 27 For the sake of notation, in what follows we omit the volume factor and the overall 4π in the amplitude.
which we can rewrite as .
Let us assume now that the lower amplitude is also super-classical, and it takes the form We have shown this is the case for the lowest iteration, scaling as 1/|q| 2 rather than 1/|q| 1 , and we are allowing for more generic super-classical terms. Let us concentrate first on the second term, which scales as (up to numerical factors) with ρ = d − m, σ = d − n + + j and γ = 1, after using the form of the potential in (A.4) together with (A.10). We now expand this integral in the classical limit, keep the scaling k 2 − p 2 ∼ q 2 , to transform it into the form in (A.6). Through a change of variables, l = p − k, we can map these integral into sums over integrals like Eq. (7.8) of [37] (with w = p − p = q and z = p in their notation), d d l f (αβγ) (l, p, q) |l| α |l + q| β (2l · p + l 2 ) γ , (A. 12) with α > 0, β > 0, γ = 1, and f (αβγ) (l, p, q) a polynomial. As it was shown in [37], the condition γ = 1 leads to super-classical contributions (which moreover is also IR divergent). The remaining contribution from (1) in (A.10) also takes on the same form, with j = 0. Similarly to the first iteration of the Coulomb potential, and by the same token, the first term in (A.9) also produces IR divergent super-classical contributions. Since these terms cancel out in the classical limit, this completes the proof of the impetus formula. Notice, as a byproduct of the above result, we have shown that there are no correction beyond the Born approximation for the classical scattering in 1/r n potentials, generalizing the Coulomb case. 28