# Chemo-kinematic analysis of metal-poor stars with unsupervised machine
learning

André R. da Silva1, Rodolfo Smiljanic1, and Riano E. Giribaldi1

###### Abstract

Metal-poor stars play an import role in the understanding of Galaxy formation
and evolution. Evidence of the early mergers that built up the Galaxy might
remain in the distributions of abundances, kinematics, and orbital parameters
of the stars. In this work, we report on preliminary results of an on-going
chemo-kinematic analysis of a sample of metal-poor ([Fe/H] $\leq$ -1.0) stars
observed by the GALAH spectroscopic survey. We explored the chemical and
orbital data with unsupervised machine learning (hierarchical clustering,
k-means cluster analysis and correlation matrices). Our final goal is to find
an optimal way to separate different Galactic stellar populations and stellar
groups originating from merging events, such as Gaia-Enceladus and Sequoia.

1Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, ul.
Bartycka 18, 00-716, Warsaw, Poland<EMAIL_ADDRESS>
##  1 Introduction

Galaxies grow through the accretion and merging of smaller stellar systems
(e.g., Bland-Hawthorn & Freeman 2014). It is sometimes possible to identify
signatures of these early mergers as chemo-kinematic stellar substructures. In
many instances, however, the criteria used to separate and study such
substructures is arbitrarily defined by carving boxes out of the parameter
space. In this work, we are investigating ways to improve the selection
criteria using unsupervised machine learning, and trying to account for the
possible distributions of stellar properties of the objects belonging to each
substructure.

The first unequivocal evidence of mergers was the discovery of the Sagittarius
dwarf galaxy by Ibata et al. (1994). Since then, others have discovered more
pieces of the Galactic merger puzzle. Recently, Helmi et al. (2018) and
Belokurov et al. (2018) independently discovered a possible merger that
occurred in the Milky Way about 8-11 Gyr ago. The object that merged with the
Galaxy, often called Gaia-Enceladus or Gaia-Sausage (GES), could be the origin
of the inner halo and the thick disk of the Milky Way (Helmi et al. 2018). The
second data release (DR2) of Gaia (Gaia Collaboration 2018) has helped in the
identification of additional merger candidates such as the Sequoia (Myeong et
al. 2018) and the Thamnos substructures (Koppelman et al. 2019).

##  2 Data and analysis

In this work, we used the proper motions and parallaxes of Gaia DR2. Chemical
parameters were obtained from the DR2 of GALAH (Galactic Archaeology with
Hermes, Buder et al. 2018). GALAH delivered chemical abundances of 26 chemical
elements for $\sim$340 000 stars from the analysis of spectra with R $\sim$ 28
000. We cross-matched metal-poor stars from the GALAH survey (selected with
[Fe/H] $\leq$ $-$1.0, 2.0 $\leq\log g\leq$ 5.0, $T_{\rm eff}$ $\geq$ 4 kK and
with cannon_flag = 0) with Gaia (selecting those with
$\sigma_{\varpi}/\varpi<$ 20%). This resulted in 1 072 stars.

Using Gaia proper motions and GALAH radial velocities, we then integrated the
stellar orbits using galpy (Bovy 2015) for a period of 10 Gyr using the Milky
Way potential from McMillan (2017). From galpy, we extracted the action angles
(Jr, Jϕ,Jz), two integrals of motion (E, Lz), eccentricity, cylindrical
radius, and other orbital parameters. We also performed Monte Carlo
simulations in order to estimate the uncertainties.

t-distributed stochastic neighbor embedding (t-SNE) is a manifold learning
method that uses affinity between the data points as probability. This is
particularly useful for exploring structures from an N dimensional problem in
a 2D map. The location of the points in this map is given by minimizing the
Kullback-Leibler divergence. For the analysis that we report here, we used
t-SNE to identify the stars in our sample that are close together in the
parameter space. The quantities used were: Jr, Jϕ, Jz, and the orbital
eccentricity, together with the chemical parameters [Fe/H] - representing
iron-peak elements, [Mg/Fe] - representing the $\alpha$-elements, [Ba/Fe] -
representing the s-process elements and [Eu/Fe] - representing the r-process
elements.

In order to decide which is the best number of cluster components for these
data, we did a k-means clustering analysis with increasing number of clusters.
The first number of clusters past the mark of 95% of distortion explanation
was considered as the optimal number (in this case nine). We then used
agglomerative clustering using Ward hierarchical method to separate the t-SNE
map into the nine clusters. We used the python module scikit-learn (Pedregosa
et al. 2011) to perform this analysis.

##  3 Discussion

In Fig. 1, the t-SNE map is shown with the color-coded clusters. The
definition of structures associated to mergers have in general been done using
only kinematical parameters (see Helmi et al. 2018; Belokurov et al. 2018;
Myeong et al. 2018; Massari et al. 2019), with a few exceptions like in Naidu
et al. (2020). Massari et al. (2019) and Myeong et al. (2018) disagree on
criteria used to classify the GES and Sequoia stars (see Fig. 2). Figure 3
helps to compare which proposed Galactic substructures found in the literature
is closer to each of the clusters that we found.

Comparing the results of our analysis with those of the literature, we find
that: in gray scale, we have the stellar groups that correspond to what is
generally agreed to be the disk of the Galaxy; in pink scale is the GES close
to the limits defined by Massari et al. (2019); the red points agree with
Myeong et al. (2018) and Naidu et al. (2020) for the Sequoia substructure; the
stars in blue are intermediary between the disk and the halo of the Milky Way;
and the lighter blue cluster is on the same region as Naidu et al. (2020)
defines the Wukong substructure (as a prograde, metal-poor and $\alpha$-rich
substructure). In Fig. 4, it is possible to separate the pink shaded group
from the gray. The dark blue cluster is on the same region of what Naidu et
al. (2020) called a “metal-weak thick disk”. To summarize, we found that
unsupervised machine learning techniques are powerful tools that can help to
recover halo substructures associated to early merging events of the Milky
Way.

Figure 1.: t-SNE projection of the parameter space for Jϕ, Jr, Jz, [Fe/H],
[Mg/Fe], [Ba/Fe], [Eu/Fe], orbital eccentricity. Gray-colored symbols
represent the disk, in pink is the Gaia-Enceladus-Sausage (GES), in red is
Sequoia and in blue the thick disk transition to the halo. Figure 2.: Action
angles by total action. The blue box represents the limits of Sequoia in
Myeong et al. (2018) and the red box the limits of GES in Myeong et al.
(2018). Cluster 5 (in red) agrees with Myeong et al. (2018). Figure 3.: Total
orbital energy by angular momentum in the Galactic pole direction (Lindblad
diagram). Points colored in intermediate pink could be Wukong as described by
Naidu et al. (2020). Figure 4.: Orbital eccentricity by the average
cylindrical radius of the orbit. The GES stars are mostly concentrated in high
eccentricity.

### Acknowledgments

A.R.S, R.S., and R.E.G. acknowledge support by the Polish National Science
Centre through project 2018/31/B/ST9/01469.

## References

  * Belokurov et al. (2018) Belokurov, V., Erkal, D., Evans, N. W., Koposov, S. E., & Deason, A. J. 2018, MNRAS, 478, 611
  * Bland-Hawthorn & Freeman (2014) Bland-Hawthorn, J., & Freeman, K. 2014, Saas-Fee Advanced Course, 37, 1
  * Bovy (2015) Bovy, J. 2015, ApJS, 216, 29
  * Buder et al. (2018) Buder, S., Asplund, M., Duong, L., , & Galah Collaboration 2018, MNRAS, 478, 4513
  * Gaia Collaboration (2018) Gaia Collaboration 2018, A&A, 616, A1
  * Helmi et al. (2018) Helmi, A., Babusiaux, C., Koppelman, H. H., Massari, D., Veljanoski, J., & Brown, A. G. A. 2018, Nat, 563, 85
  * Ibata et al. (1994) Ibata, R. A., Gilmore, G., & Irwin, M. J. 1994, Nat, 370, 194
  * Koppelman et al. (2019) Koppelman, H. H., Helmi, A., Massari, D., Price-Whelan, A. M., & Starkenburg, T. K. 2019, A&A, 631, L9
  * Massari et al. (2019) Massari, D., Koppelman, H. H., & Helmi, A. 2019, A&A, 630, L4
  * McMillan (2017) McMillan, P. J. 2017, MNRAS, 465, 76
  * Myeong et al. (2018) Myeong, G. C., Evans, N. W., Belokurov, V., Sand ers, J. L., & Koposov, S. E. 2018, ApJ, 863, L28
  * Naidu et al. (2020) Naidu, R. P., Conroy, C., Bonaca, A., Johnson, B. D., Ting, Y.-S., Caldwell, N., Zaritsky, D., & Cargile, P. A. 2020, ApJ, 901, 48
  * Pedregosa et al. (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al. 2011, the Journal of machine Learning research, 12, 2825


Quadratic-in-Spin Hamiltonian at $\mathcal{O}(G^{2})$
from Scattering Amplitudes

Dimitrios Kosmopoulos and Andres Luna

Mani L. Bhaumik Institute for Theoretical Physics,
Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095

###### Abstract

We obtain the quadratic-in-spin terms of the conservative Hamiltonian
describing the interactions of a binary of spinning bodies in General
Relativity through $\mathcal{O}(G^{2})$ and to all orders in velocity. Our
calculation extends a recently-introduced framework based on scattering
amplitudes and effective field theory to consider non-minimal coupling of the
spinning objects to gravity. At the order that we consider, we establish the
validity of the formula proposed in [1] that relates the impulse and spin kick
in a scattering event to the eikonal phase.

###### Contents

  1. 1 Introduction
  2. 2 Review of Spin Formalism
  3. 3 Full theory amplitudes
    1. 3.1 Constructing the full-theory amplitudes
    2. 3.2 The amplitudes in the center-of-mass frame
  4. 4 Hamiltonian from effective field theory
    1. 4.1 EFT scattering amplitudes
    2. 4.2 Conservative spin Hamiltonian
    3. 4.3 Comparison to the literature
  5. 5 Observables from the eikonal phase
  6. 6 Conclusions

##  1 Introduction

The detection of gravitational waves by the LIGO and Virgo collaborations [2,
3] promises intriguing new discoveries. The main sources of gravitational
waves are binary systems of compact astrophysical objects. Therefore, the
great experimental advances also press for the development of high-precision
theoretical tools for the modeling of the evolution of such systems. In the
present paper we consider the inspiral phase of the evolution of the binary. A
well-developed theoretical tool to study this phase is the post-Newtonian (PN)
approximation. This approach consists of an expansion in small velocities and
weak gravitaional field. Several methods based on General Relativity (GR) [4,
5] as well as effective field theory (EFT) [6] have been developed in this
direction. We instead choose to use the post-Minkowskian (PM) approximation,
an expansion in Newton’s constant $G$ which yields the exact velocity
dependence. The PM approximation has a long history in GR [7] but has gained
prominence recently (see e.g. [8, 9, 10]) due in part to the successful
adaptation of modern scattering-amplitudes techniques.

The application of quantum-field-theory (QFT) methods to the study of the two-
body problem dates back to the 1970’s [11]. However, it was recently that Ref.
[12] proposed the application of the well-established scattering-amplitudes
toolkit to the derivation of gravitational potentials (see Refs. [13, 14, 15]
for reviews on the modern amplitudes program). Along these lines, Ref. [16]
developed an EFT of non-relativistic scalar fields which allowed the
construction of the 2PM111The $n$PM order corresponds to $\mathcal{O}(G^{n})$.
canonical Hamiltonian from a one-loop scattering amplitude. This Hamiltonian
was equivalent to the one of Ref. [7]. Refs. [17, 18, 19] later implemented
this approach to obtain novel results at 3PM order. Ref. [20] followed up
shortly after to compare these results against numerical relativity in terms
of the energetics of the binary. Very recently, Ref. [21] obtained the
conservative binary potential at 4PM order.

Besides making use of a non-relativistic EFT, various approaches have been
developed to extract the dynamics of compact non-spinning objects from
scattering data. Refs. [22, 23] established a formalism to obtain physical
observables from unitarity cuts. Refs. [24, 25] made use of the Lippman-
Schwinger equation. Refs. [26, 27] developed a boundary-to-bound (B2B)
dictionary, and Refs. [28, 29] implemented a worldline PM EFT. Ref. [21]
discovered an amplitude-action relation that allows the calculation of
physical observables directly from the scattering amplitude.

The techniques mentioned above have been extended in multiple directions in
recent years. Indeed, Refs. [30, 31, 32] applied similar methods to
supergravity. Ref. [33] studied three-body dynamics, while Refs. [34, 35, 36,
37] incorporated the radiation emitted by the binary into their analysis.
Refs. [38, 39, 40, 41, 42, 43, 44] considered tidal deformations of the
astrophysical objects. In the present paper we explore a different direction
and focus on effects due to the spin of the compact objects.

When considering intrinsic angular momentum in the problem of a binary of
compact astrophysical objects, one assumes that the spin of the objects is
subdominant to the angular momentum of the system. In this way, we organize
the effects we consider in a systematic expansion in the spin of the objects.
Along these lines, there has been great progress in incorporating spin effects
in the PN approximation. Refs. [45, 46, 47, 48, 49] approached these effects
with traditional GR techniques. Ref. [50] extended the worldline EFT methods
of Ref. [6] in this direction, and since then there have been substantial
developments [51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65] (see
Refs. [66, 67] for reviews).

The current state-of-the-art results at the 5PN222The $n$PN order corresponds
to $\mathcal{O}(G^{a}v^{2b}S^{c})$ with $a+b+c=n+1$, where $v$ is the relative
velocity of the binary system and $S$ corresponds collectively to the spins of
the objects.  order include the linear-in-spin [68] and quadratic-in-spin [69]
interactions at next-to-next-to-next-to-leading order and the cubic-in-spin
[70] and quartic-in-spin [71] interactions at next-to-leading order. The PM
literature on the other hand is less developed. Refs. [72, 73] recently
obtained results at the 1PM and 2PM orders for effects linear in the spin of
the objects via GR considerations. Ref. [74] treated the black hole (BH) case
at 1PM order and exactly in the spin by matching an effective action to the
linearized Kerr solution. Refs. [75, 76] obtained the 2PM-order scattering
angle in the special kinematic configuration where the spins of the BHs are
aligned to the orbital angular momentum of the binary.

Similarly to the non-spinning case, we may use scattering amplitudes to study
the gravitational potential between spinning objects. Indeed, Ref. [77]
calculated a one-loop amplitude using Feynman rules, which allowed them to
obtain a 2PM-order potential by means of a Born iteration. Following the
approach of [12], Ref. [78] reproduced Hamiltonians describing the
interactions between spinning BHs by considering spinning particles minimally
coupled to gravity. Later, Ref. [79] used the generalization of minimal-
coupling amplitudes of [80] and the holomorphic classical limit of [81] to
show that amplitudes encode information about BHs that is exact in spin. Refs.
[82, 83] used the massive spinor-helicity formalism of [80] to study 2PM-order
gravitational scattering from a one-loop amplitude. Furthermore, Ref. [84]
related classical observables of a scattering process between spinning
particles directly to the scattering amplitude, extending the formalism of
[22]. Using this formalism, Refs. [85, 86, 87] obtained a 1PM-order
Hamiltonian that reproduced the result of [74]. Finally, Ref. [1] obtained the
conservative 2PM-order potential that is bilinear in the spin of the objects
and valid for arbitrary spin orientations.

Studies of the classical physics of spinning particles have also revealed
double copy structures. Refs. [88, 89, 90, 91, 92, 93] applied the definition
of minimal coupling of [80] to classical solutions. In this way they made
contact with the classical double copy of Ref. [94] and with an effective
theory of on-shell heavy spinning particles [95]. The latter generalizes the
heavy black hole effective theory of Ref. [96], whose amplitudes are known to
double copy [97].

A suprising structure that emerged from the calculation of Ref. [1] is the
expression of the observables in a scattering event in terms of the eikonal
phase [98]. Similar relations already existed in the non-spinning case [98,
99, 100, 101, 102, 103, 104, 105, 31, 32]. In the spinning case there was
evidence for such a relation in the special kinematic configuration where the
spins of the particles are parallel to the angular momentum of the system [82,
75, 76]. The formula of [1] was the first example of such a relation for
arbitrary orientations of the spins. This striking observation potentially
implies that all physical observables are obtainable via simple manipulations
of the scattering amplitude.

The goal of the present paper is to obtain a 2PM-order Hamiltonian that
describes the dynamics between a binary of generic spinning objects in GR
including effects that are up to quadratic in the spin. We take the masses of
the two objects to be $m_{1}$ and $m_{2}$ and the rest-frame spin three
vectors to be $\bm{S}_{1}$ and $\bm{S}_{2}$. We denote the relative distance
between the objects as $\bm{r}$ and the momentum three vector in the center-
of-mass frame as $\bm{p}$. The Hamiltonian then reads

$\displaystyle H$
$\displaystyle=\hbox{}\sqrt{\bm{p}^{2}+m_{1}^{2}}+\sqrt{\bm{p}^{2}+m_{2}^{2}}+V^{(0)}(\bm{r}^{2},\bm{p}^{2})+V^{(1,1)}(\bm{r}^{2},\bm{p}^{2})\frac{\bm{L}\cdot\bm{S}_{1}}{\bm{r}^{2}}+V^{(1,2)}(\bm{r}^{2},\bm{p}^{2})\frac{\bm{L}\cdot\bm{S}_{2}}{\bm{r}^{2}}$
$\displaystyle+V^{(2,1)}(\bm{r}^{2},\bm{p}^{2})\frac{(\bm{r}\cdot\bm{S}_{1})(\bm{r}\cdot\bm{S}_{2})}{\bm{r}^{4}}+V^{(2,2)}(\bm{r}^{2},\bm{p}^{2})\frac{\bm{S}_{1}\cdot\bm{S}_{2}}{\bm{r}^{2}}+V^{(2,3)}(\bm{r}^{2},\bm{p}^{2})\frac{(\bm{p}\cdot\bm{S}_{1})(\bm{p}\cdot\bm{S}_{2})}{\bm{r}^{2}}$
$\displaystyle+V^{(2,4)}(\bm{r}^{2},\bm{p}^{2})\frac{(\bm{r}\cdot\bm{S}_{1})^{2}}{\bm{r}^{4}}+V^{(2,5)}(\bm{r}^{2},\bm{p}^{2})\frac{\bm{S}_{1}^{2}}{\bm{r}^{2}}+V^{(2,6)}(\bm{r}^{2},\bm{p}^{2})\frac{(\bm{p}\cdot\bm{S}_{1})^{2}}{\bm{r}^{2}}+\dots\,,$
(1.1)

where $\bm{L}=\bm{r}\times\bm{p}$ is the orbital angular momentum, and the
ellipsis stands for terms of higher order in the spin. Note that we omit terms
quadratic in $\bm{S}_{2}$ as they are obtained from the ones quadratic in
$\bm{S}_{1}$ via appropriate relabeling. The terms in Eq. (1.1) take the form

$\displaystyle V^{A}(\bm{r}^{2},\bm{p}^{2})$
$\displaystyle=\frac{G}{|\bm{r}|}c_{1}^{A}(\bm{p}^{2})+\left(\frac{G}{|\bm{r}|}\right)^{2}c_{2}^{A}(\bm{p}^{2})+{\cal
O}(G^{3})\,,$ (1.2)

where the label $A$ takes the values indicated in Eq. (1.1).

Our task is to determine the coefficients $c_{i}^{A}$ appearing in Eq. (1.2).
For simplicity, and since the bilinear-in-spin interactions were given in Ref.
[1], we may consider one of the bodies to be non spinning. This amounts to
formally setting $\bm{S}_{2}=0$ in Eq. (1.1). We have explicitely verified
that the results of this paper do not change if we take into account all the
terms in Eq. (1.1).

Following Refs. [16, 1], we obtain the potential coefficients in question via
a matching calculation. First, we calculate a one-loop scattering amplitude in
our so-called full theory. This is a theory that describes particles of
arbitrary spin coupled to gravity. Specifically, it captures minimal and non-
minimal coupling of the particles to gravity. In terms of our Lagrangian, we
include all possible operators that are up to quadratic in the spin of the
massive particle and up to linear in the curvature. Then, we calculate the
corresponding amplitude in an EFT of spinning particles interacting via the
Hamiltonian of Eq. (1.1). Our EFT generalizes that of Refs. [16, 1] to
consider effects quadratic in the spin of one of the particles.

In obtaining these amplitudes we restrict to the piece that captures the
classical dynamics. We implement the classical limit by rescaling
$q\rightarrow\lambda q,\,S_{1}\rightarrow(1/\lambda)S_{1}$ and expanding in
$\lambda$, where $q$ denotes graviton momenta and $S_{1}$ the covariant spin
of the spinning particle. Finally, we fix the desired coefficients by matching
the two computed amplitudes.

The remainder of this paper is structured as follows: In Sec. 2 we review some
aspects of the spin formalism introduced in [1] that we use throughout the
paper. Namely, we describe our field-theory approach to higher spin and its
classical limit. We compute the necessary full-theory tree and one-loop
amplitudes in Sec. 3. We adopt the method of generalized unitarity [106, 107,
108] to produce the loop-level amplitude, using tree-level amplitudes as
building blocks. We then express the amplitudes in the center-of-mass frame,
which facilitates the matching to the EFT. Sec. 4 contains the setup of the
EFT, along with the computation of the EFT amplitudes. By equating the full-
theory and EFT amplitudes, we obtain the desired two-body Hamiltonian. We
compare our result against PN [109] and test-body [110] Hamiltonians in the
literature. Finally, in Sec. 5 we use the derived Hamiltonian to compute
scattering observables. We then establish that the conjecture of Ref. [1],
which directly relates these observables to the eikonal phase [98], holds
unaltered when we include the quadratic-in-spin effects. We present our
concluding remarks in Sec. 6.

Note added: As this paper was in its latest stages we learned about [111],
which contains overlap with our work. Ref. [111] extended the worldline PM EFT
of [28, 29] to include spin degrees of freedom. We have explicitly verified
that, where overlapping, our results are in agreement with those of [111].

##  2 Review of Spin Formalism

$k_{2}$$k_{1}$$k_{2}$$k_{1}$$k_{3}$$k_{4}$$k_{3}$$(a)$$(b)$

Figure 1: The Feynman vertices used to compute full-theory amplitudes. The
three-particle vertex (a) determines the $\mathcal{O}(G)$ dynamics. The
Compton amplitude, which requires the contact vertex (b), captures the
$\mathcal{O}(G^{2})$ dynamics. The straight lines correspond to the spinning
particle, while the wiggly lines correspond to gravitons.

In this section, we review the aspects of the higher-spin formalism that we
use in the paper. For further details, we refer the reader to Ref. [1].

We identify spinning compact astrophysical objects with higher-spin particles.
We describe these massive particles of integer-spin $s$ by real symmetric
traceless rank-$s$ tensor fields $\phi_{s}$. For brevity, we suppress the
indices of $\phi_{s}$, implying matrix multiplication when necessary.

We use a Lagrangian to organize the interactions of higher-spin fields with
gravity. Ref. [112] obtained such a Lagrangian using auxiliary fields to
eliminate all but the spin-$s$ representation of the $SO(3)$ rotation group.
Here we relax this requirement, and interpret the theory as a relativistic
effective theory that captures all spin-induced multipole moments of spinning
objects coupled to gravity. We write the higher-spin Lagrangian ${\cal L}$ and
action $S$ as

${\cal L}={\cal L}_{\rm min}+{\cal L}_{\rm nonmin}\ ,\qquad S=\int
d^{4}x\sqrt{-g}{\cal L}\ .$ (2.1)

The minimal Lagrangian contains terms with up to two derivatives,

$\displaystyle{\cal L}_{\rm min}$
$\displaystyle=-R(e,\omega)+\frac{1}{2}g^{\mu\nu}\nabla(\omega)_{\mu}\phi_{s}\nabla(\omega)_{\nu}\phi_{s}-\frac{1}{2}m^{2}\phi_{s}\phi_{s}\,.$
(2.2)

The covariant derivative is

$\nabla(\omega)_{\mu}\phi_{s}\equiv\partial_{\mu}\phi_{s}+\frac{i}{2}\omega{}_{\mu}{}_{ef}M^{ef}\phi_{s}\,,$
(2.3)

where $\omega$ is the spin connection, and $M^{ab}$ are the Hermitian Lorentz
generators. The gravitational field is described in the vielbein formulation.
The non-minimal Lagrangian containing all the terms linear in the graviton and
bilinear in the higher-spin field is

$\displaystyle{\cal L}_{\text{non-min}}$
$\displaystyle=\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n}}{\left(2n\right)!}\frac{C_{ES^{2n}}}{m^{2n}}\nabla(\omega)_{f_{2n}}\cdots\nabla(\omega)_{f_{3}}R_{f_{1}af_{2}b}\nabla(\omega)^{a}{\phi}_{s}\,{{\mathbb{S}}}^{(f_{1}}\dots{\mathbb{S}}^{f_{2n})}\nabla(\omega)^{b}\phi_{s}$
(2.4)
$\displaystyle-\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n}}{\left(2n+1\right)!}\frac{C_{BS^{2n}}}{m^{2n+1}}\nabla(\omega)_{f_{2n+1}}\cdots\nabla(\omega)_{f_{3}}\frac{1}{2}\epsilon_{ab(c|f_{1}}R^{ab}{}_{|d)\,f_{2}}\nabla(\omega)^{c}{\phi}_{s}{{\mathbb{S}}}^{(f_{1}}\dots{\mathbb{S}}^{f_{2n+1})}\nabla(\omega)^{d}\phi_{s}\,.$

where we use an off-shell analog of the Pauli-Lubanski vector

${{\mathbb{S}}}^{a}\equiv-\frac{i}{2m}\epsilon^{abcd}M_{cd}\nabla(\omega)_{b}\,.$
(2.5)

The operators in Eq. (2.4) are in direct correspondence to the non-minimal
couplings in the worldline spinning-particle action of Ref. [113]. One could,
in principle, include terms with dependence on higher powers of the curvature,
but we do not attempt to do so in the present paper. Since our objective is to
describe the dynamics up to spin squared, we focus on the first non-minimally
coupled term,

$\displaystyle{\cal L}_{ES^{2}}=$
$\displaystyle-\frac{C_{ES^{2}}}{2m^{2}}R_{f_{1}af_{2}b}\nabla^{a}{\phi}_{s}\,{{\mathbb{S}}}^{(f_{1}}{{\mathbb{S}}}^{f_{2})}\nabla^{b}\phi_{s}.$
(2.6)

Ref. [50] first studied the effects captured by this operator at leading order
in the PN approximation. The extensions to next-to-leading and next-to-next-
to-leading orders were considered in Refs. [53] and [109] respectively, while
Ref. [114] studied its contributions to higher orders in spin. We instead
consider its effects in the PM approximation.

To extract Feynman rules, we define the graviton as the fluctuation of the
metric around Minkowski space. We determine the spin connection $\omega$ as
the solution of the vielbein postulate, $\nabla_{\mu}(\omega)e_{\nu}^{\ a}=0$.
This yields the following expansions for the needed quantities

$\displaystyle g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}\,,\hskip
28.45274pte{}_{\mu}{}^{a}=\delta_{\mu}^{a}+\frac{1}{2}h{}_{\mu}{}^{a}-\frac{1}{8}h_{\mu\rho}h^{a\rho}+{\mathcal{O}}(h^{3})\,,$
$\displaystyle\omega(e){}_{\mu}{}_{cb}=-\partial_{[c}h_{b]\mu}-\frac{1}{4}h{}^{\rho}{}_{[c}\partial_{\mu}h{}_{b]\rho}+\frac{1}{2}h{}^{\rho}{}_{[c}\partial_{\rho}h{}_{b]\mu}-\frac{1}{2}h{}^{\rho}{}_{[c}\partial_{b]}h_{\mu\rho}+{\mathcal{O}}(h^{3})\,.$
(2.7)

After substituting this expansion into the Lagrangian of Eq. (2.6), we follow
a straightforward procedure to obtain the Feynman vertices in Fig. 1. These
are the vertices necessary to determine the dynamics through
$\mathcal{O}(G^{2})$.

We describe the state of the higher-spin particles by their momentum $p$ and
polarization tensor $\varepsilon(p)$. To take the classical limit of
expectation values, we choose “spin coherent states” [115], whose defining
property is that they minimize the standard deviation of observables.
Following [116, 117], we relate the classical spin tensor and Lorentz
generators via

$\displaystyle\varepsilon(\tilde{p})M^{\mu_{1}\nu_{1}}\varepsilon(p)$
$\displaystyle=S(p)^{\mu_{1}\nu_{1}}\varepsilon(\tilde{p})\cdot\varepsilon(p)+\ldots\,,$
$\displaystyle\varepsilon(\tilde{p})\\{M^{\mu_{1}\nu_{1}},M^{\mu_{2}\nu_{2}}\\}\varepsilon(p)$
$\displaystyle=S(p)^{\mu_{1}\nu_{1}}S(p)^{\mu_{2}\nu_{2}}\varepsilon(\tilde{p})\cdot\varepsilon(p)+\ldots\,,$
(2.8)

where $\\{A,B\\}\equiv\frac{1}{2}(AB+BA)$ and $\tilde{p}\equiv-p-q$ (note that
we use the all-outgoing convention). We can also write analogous expressions
for products with higher powers of the Lorentz generator. Throughout the paper
we omit terms that do not contribute to the classical potential in ellipsis.
These include terms that do not survive in the classical limit and terms that
cancel in the matching between full-theory and EFT amplitudes.

Importantly, one can only interpret the symmetric product of Lorentz
generators as a product of spin tensors. However, it is always possible to
decompose a product of Lorentz generators into a sum of completely symmetric
products by means of the Lorentz algebra,

$[M^{\mu_{1}\nu_{1}},M^{\mu_{2}\nu_{2}}]=i(\eta^{\mu_{3}\mu_{1}}M^{\mu_{4}\mu_{2}}+\eta^{\mu_{2}\mu_{3}}M^{\mu_{1}\mu_{4}}-\eta^{\mu_{4}\mu_{1}}M^{\mu_{3}\mu_{2}}-\eta^{\mu_{2}\mu_{4}}M^{\mu_{1}\mu_{3}})\,.$
(2.9)

We take the spin tensor to obey the so-called covariant spin supplementary
condition,

$p_{\mu}{S}(p)^{\mu\nu}=0\,.$ (2.10)

We define the Pauli-Lubanski spin vector by

$S^{\alpha}(p)=-\frac{1}{2m}\epsilon^{\alpha\beta\gamma\delta}{p}_{\beta}{S}_{\gamma\delta}(p)\,.$
(2.11)

Using the on-shell condition for the spinning particle $p^{2}=m^{2}$ and Eq.
(2.10), we find

$S^{\alpha\beta}(p)=-\frac{1}{m}\epsilon^{\alpha\beta\gamma\delta}{p}_{\gamma}{S}_{\delta}(p)\,.$
(2.12)

For this choice of the spin vector we have

$\displaystyle{S}(p)^{\mu}$
$\displaystyle=\bigg{(}\,\frac{{\bm{p}}\cdot{\bm{S}}}{m},{\bm{S}}+\frac{{\bm{p}}\cdot{\bm{S}}}{m(E+m)}{\bm{p}}\,\bigg{)}\,,$
(2.13)

where ${\bm{S}}$ is the three-dimensional rest-frame spin of the particle and
$p=-(E,{\bm{p}})$. I.e. we obtain the covariant spin vector by boosting its
rest-frame counterpart. Finally, by writing the polarization tensors as boosts
of rest-frame coherent states [115], we have

$\displaystyle\varepsilon(\tilde{p})\cdot\varepsilon(p)=\exp\left[-\frac{\bm{L}_{q}\cdot\bm{S}}{m(E+m)}\right]+\ldots\,,$
(2.14)

where $\bm{L}_{q}\equiv i\bm{p}\times\bm{q}$, and the ellipsis stand for terms
that do not contribute to the classical potential.

Figure 2: The tree-level amplitude that captures the ${\mathcal{O}}(G)$ spin
interactions. The thick (thin) straight line represents the spinning (scalar)
particle, while the wiggly line corresponds to the exchanged graviton.

##  3 Full theory amplitudes

In this section we calculate the scattering amplitudes needed to construct the
desired Hamiltonian. Specifically, we obtain the relevant pieces of the tree-
level and one-loop two-to-two scattering amplitude between a scalar and a
spinning particle. For the tree-level amplitudes we use the Feynman rules
derived in the previous section. We use the generalized unitarity method [106,
107, 108] for the one-loop amplitude. Anticipating the comparison to the EFT
amplitudes, we specialize our results to the center-of-mass frame.

###  3.1 Constructing the full-theory amplitudes

Figure 3: The one-loop scalar box integrals $I_{\Box}$ (a) and $I_{\bowtie}$
(b) and the corresponding triangle integrals $I_{\bigtriangleup}$ (c) and
$I_{\bigtriangledown}$ (d). The bottom (top) solid line corresponds to a
massive propagator of mass $m_{1}$ ($m_{2}$). The dashed lines denote massless
propagators.

The information to determine the $\mathcal{O}(G)$ Hamiltonian is contained in
the tree-level amplitude shown in Fig. 2. We take the incoming momentum of the
spinning (scalar) particle to be $-p_{1}$ ($-p_{2}$) and its outgoing momentum
to be $p_{4}$ ($p_{3}$). Using the Feynman rules obtained above, we find

$\displaystyle\mathcal{M}^{\textrm{tree}}=$ $\displaystyle-\frac{4\pi
G}{q^{2}}\,\varepsilon_{1}\cdot\varepsilon_{4}\,\left(\alpha_{1}^{(0)}+\alpha_{1}^{(1,1)}\mathcal{E}_{1}+\alpha_{1}^{(2,4)}(q\cdot
S_{1})^{2}\right)+\ldots\,,$ (3.1)

where $\mathcal{E}_{1}\equiv
i\epsilon^{\mu\nu\rho\sigma}{p_{1}}_{\mu}{p_{2}}_{\nu}q_{\rho}S_{1}{}_{\sigma}$,
and the labeling scheme for $\alpha_{i}^{A}$ follows that for $c_{i}^{A}$ in
Eq. (1.2). In the ellipsis we omit terms that do not contribute to the
classical limit, along with pieces proportional to $q^{2}$, since they cancel
the propagator and do not yield long-range contributions. The coefficients
$\alpha_{1}^{A}$ take the explicit form

$\displaystyle\alpha_{1}^{(0)}=4m^{4}\nu^{2}(2\sigma^{2}-1)\,,\qquad\alpha_{1}^{(1,1)}=\frac{8m^{2}\nu\sigma}{m_{1}}\,,\qquad\alpha_{1}^{(2,4)}=\frac{2{C_{ES^{2}}}m^{4}\nu^{2}(2\sigma^{2}-1)}{m_{1}^{2}}\,,$
(3.2)

where we use the variables

$\sigma=\frac{p_{1}\cdot p_{2}}{m_{1}m_{2}}\,,\qquad
m=m_{1}+m_{2}\,,\qquad\nu=\frac{m_{1}m_{2}}{m^{2}}\,.$ (3.3)

In order to construct the $\mathcal{O}(G^{2})$ Hamiltonian we further need the
corresponding one-loop amplitude. We may express any one-loop amplitude as a
linear combination of scalar box, triangle, bubble and tadpole integrals
[118]. Refs. [16, 18] showed that the bubble and tadpole integrals do not
contribute to the classical limit. Dropping these pieces we may write

$i\mathcal{M}^{\text{1-loop}}=d_{\Box}\,I_{\Box}+d_{\bowtie}\,I_{\bowtie}+c_{\bigtriangleup}\,I_{\bigtriangleup}+c_{\bigtriangledown}\,I_{\bigtriangledown}\,,$
(3.4)

where the coefficients $d_{\Box}$, $d_{\bowtie}$, $c_{\bigtriangleup}$ and
$c_{\bigtriangledown}$ are rational functions of external momenta and
polarization tensors. The integrals $I_{\Box}$, $I_{\bowtie}$,
$I_{\bigtriangleup}$ and $I_{\bigtriangledown}$ are shown in Fig. 3. The
triangle integrals take the form [16]

$\displaystyle
I_{\bigtriangleup,\bigtriangledown}=-\frac{i}{32m_{1,2}}\frac{1}{\sqrt{-q^{2}}}+\cdots\,.$
(3.5)

The box contributions do not contain any novel $\mathcal{O}(G^{2})$
information. They correspond to infrared-divergent pieces that cancel out when
we equate the full-theory and EFT amplitudes [16, 18]. In this sense, the
explicit values for the box coefficients serve only as a consistency check of
our calculation and we do not show them. Instead, we give the result for

$\displaystyle i\mathcal{M}^{\bigtriangleup+\bigtriangledown}\equiv
c_{\bigtriangleup}\,I_{\bigtriangleup}+c_{\bigtriangledown}\,I_{\bigtriangledown}\,.$
(3.6)

We use the generalized-unitarity method [106, 107, 108, 119] to obtain the
integral coefficients of Eq. (3.4). We start by calculating the gravitational
Compton amplitude for the spinning particle, using the Feynman rules derived
in the previous section. We depict the relevant Feynman diagrams in Fig. 4.
Subsequently, we construct the two-particle cut depicted in Fig. 5(a) by
gluing the Compton amplitude for the spinning particle with that for a scalar.
The latter is a well-known amplitude. The residue of the two-particle cut on
the scalar-matter pole gives the triple cut in Fig. 5(b), while the one on the
spinning-matter pole gives the triple cut in Fig. 5(c). Localizing both matter
poles gives the quadruple cut in Fig. 5(d). Finally, following Refs. [120,
121, 122], we obtain the box and triangle coefficients from the quadruple and
triple cuts respectively. Our result reads

Figure 4: The Compton-amplitude Feynman diagrams. The straight line
corresponds to the spinning particle. The wiggly lines correspond to
gravitons.

56565656

Figure 5: Appropriate residues of the two-particle cut (a) give the triple
cuts (b) and (c), and the quadruple cut (d). The thick straight line
corresponds to the spinning particle, the thin straight line to the scalar,
and the wiggly lines to the exchanged gravitons. All exposed lines are taken
on-shell.

$\displaystyle\mathcal{M}^{\bigtriangleup+\bigtriangledown}=\frac{2\pi^{2}G^{2}\varepsilon_{1}\cdot\varepsilon_{4}}{\sqrt{-q^{2}}}\bigg{[}\alpha_{2}^{(0)}+\alpha_{2}^{(1,1)}\mathcal{E}_{1}+\alpha_{2}^{(2,4)}(q\cdot
S_{1})^{2}+\alpha_{2}^{(2,5)}q^{2}S_{1}^{2}+\alpha_{2}^{(2,6)}q^{2}(p_{2}\cdot
S_{1})^{2}\bigg{]}+\ldots\,,$

where the coefficients are given by

$\displaystyle\alpha_{2}^{(0)}$
$\displaystyle=3m^{5}\nu^{2}(5\sigma^{2}-1)\,,\qquad\alpha_{2}^{(1,1)}=\frac{m^{2}(4m_{1}+3m_{2})(5\sigma^{2}-3)\nu\sigma}{m_{1}(\sigma^{2}-1)}\,,$
$\displaystyle\alpha_{2}^{(2,4)}$
$\displaystyle=-\frac{m_{2}^{2}}{16(\sigma^{2}-1)}\Big{[}-4m_{2}\big{(}-\sigma^{2}+1+{C_{ES^{2}}}{}(30\sigma^{4}-29\sigma^{2}+3)\big{)}$
$\displaystyle\qquad\qquad\qquad\qquad-
m_{1}\big{(}35\sigma^{4}-30\sigma^{2}-5+{C_{ES^{2}}}{}(155\sigma^{4}-174\sigma^{2}+35)\big{)}\Big{]}\,,$
$\displaystyle\alpha_{2}^{(2,5)}$
$\displaystyle=-\frac{m_{2}^{2}}{16(\sigma^{2}-1)}\Big{[}4m_{2}\big{(}15\sigma^{4}-17\sigma^{2}+2+{C_{ES^{2}}}{}(15\sigma^{4}-13\sigma^{2}+2)\big{)}$
$\displaystyle\qquad\qquad\qquad\qquad+m_{1}\big{(}95\sigma^{4}-102\sigma^{2}+7+{C_{ES^{2}}}{}(95\sigma^{4}-102\sigma^{2}+23)\big{)}\Big{]}\,,$
$\displaystyle\alpha_{2}^{(2,6)}$
$\displaystyle=-\frac{1}{8(\sigma^{2}-1)^{2}}\Big{[}2m_{2}\big{(}15\sigma^{4}-14\sigma^{2}-1+{C_{ES^{2}}}{}(15\sigma^{4}-10\sigma^{2}+3)\big{)}$
$\displaystyle\qquad\qquad\qquad\qquad+m_{1}\big{(}65\sigma^{4}-66\sigma^{2}+1+{C_{ES^{2}}}{}(65\sigma^{4}-66\sigma^{2}+17)\big{)}\Big{]}\,.$
(3.7)

We note here that the relation $\alpha_{2}^{(2,4)}=-\alpha_{2}^{(2,5)}$, which
was expected following a pattern observed in Refs. [96, 1], is broken for
generic values of ${C_{ES^{2}}}$. We recover this relation for
${C_{ES^{2}}}=1$, which corresponds to the Kerr black hole [74]. This is in
line with a recent observation in Ref. [39], that this equality fails to hold
in the presence of tidal finite-size effects.

###  3.2 The amplitudes in the center-of-mass frame

In preparation for the matching procedure in the following section, we
specialize our expressions to the center-of-mass frame. In this frame, the
independent four-momenta read

$\displaystyle p_{1}=-(E_{1},\bm{p})\,,\hskip
42.67912ptp_{2}=-(E_{2},-\bm{p})\,,\hskip 42.67912ptq=(0,\bm{q})\,,\hskip
42.67912pt\bm{p}\cdot\bm{q}=\bm{q}^{2}/2\,.$ (3.8)

Using Eq. (2.13), we have

$\displaystyle q\cdot S_{1}=\bm{q}\cdot\bm{S}_{1}$
$\displaystyle-\frac{\bm{q}^{2}\bm{p}\cdot\bm{S}_{1}}{2m_{1}(E_{1}+m_{1})}\,,\qquad
i\epsilon^{\mu\nu\rho\sigma}{p_{1}}_{\mu}{p_{2}}_{\nu}q_{\rho}{S_{1}}_{\sigma}=E\,\bm{L}_{q}\cdot\bm{S}_{1}\,,\qquad
p_{2}\cdot S_{1}=-\frac{E}{m_{1}}\bm{p}\cdot\bm{S}_{1}\,.$ (3.9)

Furthermore, Eq. (2.14) becomes

$\displaystyle\varepsilon_{1}\cdot\varepsilon_{4}=1-\frac{\bm{L}_{q}\cdot\bm{S}_{1}}{m_{1}(E_{1}+m_{1})}+\frac{(\bm{L}_{q}\cdot\bm{S}_{1})^{2}}{2m_{1}^{2}(E_{1}+m_{1})^{2}}+\ldots\,.$
(3.10)

Using the above expressions, our amplitudes take the form

$\displaystyle\frac{\mathcal{M}^{\text{tree}}}{4E_{1}E_{2}}$
$\displaystyle=\frac{4\pi
G}{\bm{q}^{2}}\bigg{[}a^{(0)}_{1}+a^{(1,1)}_{1}\bm{L}_{q}\cdot\bm{S}_{1}+a^{(2,4)}_{1}(\bm{q}\cdot\bm{S}_{1})^{2}\bigg{]}\,,$
(3.11)
$\displaystyle\frac{\mathcal{M}^{\bigtriangleup+\bigtriangledown}}{4E_{1}E_{2}}$
$\displaystyle=\frac{2\pi^{2}G^{2}}{|\bm{q}|}\bigg{[}a^{(0)}_{2}+a^{(1,1)}_{2}\bm{L}_{q}\cdot\bm{S}_{1}+a^{(2,4)}_{2}(\bm{q}\cdot\bm{S}_{1})^{2}+a^{(2,5)}_{2}\bm{q}^{2}\,\bm{S}_{1}^{2}\
+a^{(2,6)}_{2}\bm{q}^{2}\,(\bm{p}\cdot\bm{S}_{1})^{2}\bigg{]}\,.$

The coefficients $a_{i}^{A}$ are given in terms of the $\alpha_{i}^{A}$ of
Eqs. (3.2) and (3.7) by333Note that unlike Ref. [1] we do not introduce the
coefficients $a_{cov}$. This means that factors of the spin in Eq. (3.11)
appear both because we specialize in the center-of-mass frame and due to Eq.
(3.10).

$\displaystyle a^{(0)}_{i}$
$\displaystyle=\frac{\alpha_{i}^{(0)}}{4m^{2}\gamma^{2}\xi}\,,\qquad\
a^{(1,1)}_{i}=\frac{\alpha_{i}^{(1,1)}}{4m\gamma\xi}-\frac{1}{m_{1}^{2}(\gamma_{1}+1)}\frac{\alpha_{i}^{(0)}}{4m^{2}\gamma^{2}\xi}\,,$
$\displaystyle a^{(2,j)}_{i}$
$\displaystyle=\frac{\alpha_{i}^{(2,j)}\tilde{\zeta}^{(j)}}{4m^{2}\gamma^{2}\xi}-\frac{\zeta^{(j)}}{m_{1}^{2}(\gamma_{1}+1)}\frac{\alpha_{i}^{(1,1)}}{4m\gamma\xi}+\frac{\zeta^{(j)}}{m_{1}^{4}(\gamma_{1}+1)^{2}}\frac{\alpha_{i}^{(0)}}{8m^{2}\gamma^{2}\xi}\,,$
(3.12)

where $i=1,2$ and the structure-dependent coefficients are given by

$\displaystyle\zeta^{(4)}=-\zeta^{(5)}=\bm{p}^{2},\qquad\zeta^{(6)}=1,\qquad\tilde{\zeta}^{(4)}=\tilde{\zeta}^{(5)}=1,\qquad\tilde{\zeta}^{(6)}=-\frac{E^{2}}{m_{1}^{2}}.$
(3.13)

In addition to the definitions in Eq. (3.3) we use

$\gamma=\frac{E}{m}\,,\qquad\gamma_{1}=\frac{E_{1}}{m_{1}}\,,\qquad
E=E_{1}+E_{2}\,,\qquad\xi=\frac{E_{1}E_{2}}{E^{2}}\,.$ (3.14)

##  4 Hamiltonian from effective field theory

We now turn our attention to the task of translating the scattering amplitudes
of higher-spin fields to a two-body conservative Hamiltonian. We do this by
matching the scattering amplitude computed in the last section to the two-to-
two amplitude of an EFT of the positive-energy modes of higher-spin fields.
Ref. [16] developed this matching procedure for higher orders in $G$ and all
orders in velocity, while Ref. [1] extended the formalism to include spin
degrees of freedom. We conclude this section by comparing our answer with
previous results in the literature.

###  4.1 EFT scattering amplitudes

The action of the effective field theory for the higher-spin fields $\xi_{1}$
and $\xi_{2}$ is given by

$\displaystyle S=$
$\displaystyle\int_{\bm{k}}\,\sum_{a=1,2}\xi_{a}^{\dagger}(-\bm{k})\left(i\partial_{t}-\sqrt{\bm{k}^{2}+m_{a}^{2}}\right)\xi_{a}(\bm{k})-\int_{\bm{k},\bm{k}^{\prime}}\,\xi_{1}^{\dagger}(\bm{k}^{\prime})\xi_{2}^{\dagger}(-\bm{k}^{\prime})\,V(\bm{k}^{\prime},\bm{k},{\hat{\bm{S}}}_{1})\,\xi_{1}(\bm{k})\xi_{2}(-\bm{k})\,,$
(4.1)

where $\int_{\bm{k}}=\int\frac{d^{D-1}\bm{k}}{(2\pi)^{D-1}}$, and the
interaction potential $V(\bm{k}^{\prime},\bm{k},{\hat{\bm{S}}}_{1})$ is a
function of the incoming and outgoing momenta $\bm{k}$ and $\bm{k}^{\prime}$,
and the spin operator ${\hat{\bm{S}}}_{1}$. We consider kinematics in the
center-of-mass frame. As in the full theory side, we choose the field
$\xi_{2}$ to be a scalar, while the asymptotic states of $\xi_{1}$ are taken
to be spin coherent states. We obtain the classical rest-frame spin vector as
the expectation value of the spin operator with respect to these on-shell
states.

We build the most general potential containing only long-range classical
contributions, up to quadratic order in spin. In momentum space, a minimal
basis of interactions in the on-shell scheme is given by the operators

$\hat{\mathbb{O}}^{(0)}=\mathbb{I}\,,\hskip
14.22636pt\hat{\mathbb{O}}^{(1,1)}=\bm{L}_{\hat{q}}\cdot{\hat{\bm{S}}}_{1}\,,\hskip
14.22636pt\hat{\mathbb{O}}^{(2,4)}=\big{(}\hat{\bm{q}}\cdot\bm{\hat{S}}_{1}\big{)}^{2}\,,\hskip
14.22636pt\hat{\mathbb{O}}^{(2,5)}=\hat{\bm{q}}^{2}\
\bm{\hat{S}}_{1}^{2}\,,\hskip
14.22636pt\hat{\mathbb{O}}^{(2,6)}=\hat{\bm{q}}^{2}\big{(}\bm{k}\cdot\bm{\hat{S}}_{1}\big{)}^{2}\,,$
(4.2)

where444The three-vectors $\hat{\bm{q}}$ and $\hat{\bm{p}}$ are not to be
confused with unit-norm vectors. $\hat{\bm{q}}\equiv\bm{k}-\bm{k}^{\prime}$
and $\bm{L}_{\hat{q}}\equiv i\bm{k}\times\bm{\hat{}}{\bm{q}}$. Their
expectation values with respect to spin coherent states are in one-to-one
correspondence with the monomials in the full theory amplitude, Eq. (3.11).
The labeling scheme for the operators follows the conventions of Eq. (1.1). We
use the following ansatz for the potential operator

$\displaystyle V(\bm{k}^{\prime},\bm{k},{\hat{\bm{S}}}_{1})=$
$\displaystyle\sum_{A}V^{A}(\bm{k}^{\prime},\bm{k})\,\hat{\mathbb{O}}^{A}\,,$
(4.3)

where $A$ runs over the superscripts of the operators in Eq. (4.2).
$V^{A}(\bm{k}^{\prime},\bm{k})$ are free coefficients with the same structure
as the spin-independent potential of Refs. [18, 17],

$\displaystyle V^{A}(\bm{k}^{\prime},\bm{k})=$ $\displaystyle\frac{4\pi
G}{\hat{\bm{q}}^{2}}d^{A}_{1}\left(\hat{\bm{p}}^{2}\right)+\frac{2\pi^{2}G^{2}}{|\hat{\bm{q}}|}d^{A}_{2}\left(\hat{\bm{p}}^{2}\right)+{\mathcal{O}}(G^{3})\,,$
(4.4)

where $\hat{\bm{p}}^{2}\equiv(\bm{k}^{2}+\bm{k}^{\prime 2})/2$. At the
$\mathcal{O}(G)$ level, the operators containing a factor of
$\hat{\bm{q}}^{2}$ can be ignored, as they lead to contact terms. Therefore we
choose

$d^{(2,5)}_{1}=d^{(2,6)}_{1}=0\,.$ (4.5)

However, the factor of $\hat{\bm{q}}^{2}$ does not cancel out with the
$\mathcal{O}(G^{2})$ denominator, so we need to keep $d^{(2,5)}_{2}$ and
$d^{(2,6)}_{2}$.

We now evaluate the EFT two-to-two scattering amplitude. To this end we use
the Feynman rules derived from the EFT action (Eq. (4.1)),

$\includegraphics[scale={.55},trim=0.0pt 5.69046pt 0.0pt
0.0pt,clip]{figs/EFTFeynman_prop-eps-converted-
to.pdf}=\frac{i\,\mathbb{I}}{E-\sqrt{\bm{k}^{2}+m^{2}}+i\epsilon}\,,\qquad\vbox{\hbox{\includegraphics[scale={.55},trim=0.0pt
14.22636pt 28.45274pt 0.0pt,clip]{figs/EFTFeynman_vertex-eps-converted-
to.pdf}}}=-iV(\bm{k}^{\prime},\bm{k},{\hat{\bm{S}}}_{1})\,.$ (4.6)

Using these rules we compute the amplitude up to $\mathcal{O}(G^{2})$ directly
evaluating the relevant Feynman diagrams, omitting terms that do not
contribute to long range interactions. The spin-dependent vertices must be
treated as operators, and thus their ordering is important. After carrying out
the energy integration, we obtain an expression for the amplitude

$\displaystyle\mathcal{M}^{\textrm{EFT}}=$
$\displaystyle-V(\bm{p}^{\prime},\bm{p},\bm{S}_{1})-\int_{\bm{k}}\frac{V(\bm{p}^{\prime},\bm{k},\bm{S}_{1})\,V(\bm{k},\bm{p},\bm{S}_{1})}{E_{1}+E_{2}-\sqrt{\bm{k}^{2}+m_{1}^{2}}-\sqrt{\bm{k}^{2}+m_{2}^{2}}}\
.$ (4.7)

Similarly to the full theory, in order to extract the classical limit, one
needs to first decompose products of the spin vector into irreducible
representations of the rotation group, by repeated use of the $SO(3)$ algebra.

At $\mathcal{O}(G)$ the EFT amplitude receives a contribution only from the
first term of Eq. (4.7),

$\displaystyle\mathcal{M}^{\rm EFT}_{\rm 1PM}=\frac{4\pi
G}{\bm{q}^{2}}\left[a^{(0)}_{1}+a^{(1,1)}_{1}\bm{L}_{q}\cdot\bm{S}_{1}+a^{(2,4)}_{1}\,(\bm{q}\cdot\bm{S}_{1})^{2}\right]\,.$
(4.8)

The $a_{1}^{A}$ are given directly in terms of the momentum-space potential
coefficients,

$a^{A}_{1}=-d^{A}_{1}\,.$ (4.9)

The EFT amplitude at $\mathcal{O}(G^{2})$ receives contributions from both
terms in Eq. (4.7) and can be written as

$\displaystyle\mathcal{M}^{\rm EFT}_{\rm 2PM}$
$\displaystyle=\frac{2\pi^{2}G^{2}}{|\bm{q}|}\bigg{[}a^{(0)}_{2}+a^{(1,1)}_{2}\bm{L}_{q}\cdot\bm{S}_{1}+a^{(2,4)}_{2}(\bm{q}\cdot\bm{S}_{1})^{2}+a^{(2,5)}_{2}\bm{q}^{2}\,\bm{S}_{1}^{2}+a^{(2,6)}_{2}\bm{q}^{2}\,(\bm{p}\cdot\bm{S}_{1})^{2}\bigg{]}$
$\displaystyle\hskip 170.71652pt\hbox{}+(4\pi G)^{2}\,a_{\rm
iter}\int\frac{d^{D-1}\bm{\ell}}{(2\pi)^{D-1}}\frac{2\xi
E}{\bm{\ell}^{2}(\bm{\ell}+\bm{q})^{2}(\bm{\ell}^{2}+2\bm{p}\cdot\bm{\ell})}\,,$
(4.10)

where $\bm{\ell}=\bm{k}-\bm{p}$ and we only keep terms that are relevant in
the classical limit. The above coefficients are given by

$\displaystyle a^{(0)}_{2}$ $\displaystyle=-d^{(0)}_{2}+\frac{1}{2\xi
E}\,\tilde{A}_{0}\left[\left(d^{(0)}_{1}\right)^{2}\right]\,,$ (4.11)
$\displaystyle a^{(1,1)}_{2}$ $\displaystyle=-d^{(1,1)}_{2}+\frac{1}{2\xi
E}\,\tilde{A}_{2}\left[d^{(0)}_{1}d^{(1,1)}_{1}\right]\,,$ $\displaystyle
a_{2}^{(2,4)}$ $\displaystyle=-d_{2}^{(2,4)}+\frac{3}{8\xi
E}\tilde{A}_{4/3}\bigg{[}d_{1}^{(0)}d_{1}^{(2,4)}\bigg{]}+\frac{\bm{p}^{2}}{16\xi
E}\tilde{A}_{4}\left[\left(d_{1}^{(1,1)}\right)^{2}\right]+\frac{\xi
E}{4}d_{1}^{(1,1)}d_{1}^{(2,4)}\,,$ $\displaystyle a_{2}^{(2,5)}$
$\displaystyle=-d_{2}^{(2,5)}-\frac{1}{8\xi
E}\tilde{A}_{4}\bigg{[}d_{1}^{(0)}d_{1}^{(2,4)}\bigg{]}-\frac{\bm{p}^{2}}{8\xi
E}\tilde{A}_{4}\bigg{[}\left(d_{1}^{(1,1)}\right)^{2}\bigg{]}+\frac{\xi
E}{4}d_{1}^{(1,1)}d_{1}^{(2,4)}\,,$ $\displaystyle a_{2}^{(2,6)}$
$\displaystyle=-d_{2}^{(2,6)}+\frac{\xi
E}{\bm{p}^{4}}d_{1}^{(0)}d_{1}^{(2,4)}+\frac{1}{8\xi
E}\tilde{A}_{4}\bigg{[}\left(d_{1}^{(1,1)}\right)^{2}\bigg{]}-\frac{\xi
E}{\bm{p}^{2}}d_{1}^{(1,1)}d_{1}^{(2,4)}\,,$

where we define the function

$\displaystyle\tilde{A}_{j}[X]$
$\displaystyle=\left[(1-3\xi)+\frac{j\xi^{2}E^{2}}{\bm{p}^{2}}+2\xi^{2}E^{2}\partial\right]X,$
(4.12)

and the derivative is taken with respect to the square of the center-of-mass
momentum $\partial=\partial/\partial{\bm{p}^{2}}$. The second term in Eq.
(4.10) is infrared divergent and should cancel out when we equate the full-
theory and EFT amplitudes. We have explicitly verified this cancellation at
leading order in the classical expansion.

###  4.2 Conservative spin Hamiltonian

As mentioned in Sec. 1, our final result is the position-space Hamiltonian,

$\displaystyle H=\hbox{}$
$\displaystyle\sqrt{\bm{p}^{2}+m_{1}^{2}}+\sqrt{\bm{p}^{2}+m_{2}^{2}}+V^{(0)}(\bm{r}^{2},\bm{p}^{2})+V^{(1,1)}(\bm{r}^{2},\bm{p}^{2})\frac{\bm{L}\cdot\bm{S}_{1}}{\bm{r}^{2}}$
$\displaystyle+V^{(2,4)}(\bm{r}^{2},\bm{p}^{2})\frac{(\bm{r}\cdot\bm{S}_{1})^{2}}{\bm{r}^{4}}+V^{(2,5)}(\bm{r}^{2},\bm{p}^{2})\frac{\bm{S}_{1}^{2}}{\bm{r}^{2}}+V^{(2,6)}(\bm{r}^{2},\bm{p}^{2})\frac{(\bm{p}\cdot\bm{S}_{1})^{2}}{\bm{r}^{2}}+\dots\,.$
(4.13)

The potentials take the form

$\displaystyle V^{A}(\bm{r}^{2},\bm{p}^{2})$
$\displaystyle=\frac{G}{|\bm{r}|}c_{1}^{A}(\bm{p}^{2})+\left(\frac{G}{|\bm{r}|}\right)^{2}c_{2}^{A}(\bm{p}^{2})+{\cal
O}(G^{3})\,.$ (4.14)

We obtain the position-space Hamiltonian by taking the Fourier transform of
the momentum-space Hamiltonian with respect to the momentum transfer $\bm{q}$,
which is the conjugate of the separation between the particles $\bm{r}$. In
this way, we express the position-space coefficients $c_{i}^{A}$ in terms of
the momentum-space coefficients $d_{i}^{A}$ via linear relations dictated by
the $\bm{q}$-dependence of the spin operators,

$\displaystyle c_{1}^{(0)}$ $\displaystyle=d_{1}^{(0)}\,,\quad$ $\displaystyle
c_{1}^{(1,1)}$ $\displaystyle=-d_{1}^{(1,1)}\,,\quad$ $\displaystyle
c_{1}^{(2,4)}$ $\displaystyle=-3d_{1}^{(2,4)}\,,\quad$ $\displaystyle
c_{1}^{(2,5)}$ $\displaystyle=d_{1}^{(2,4)}\,,\quad$ $\displaystyle
c_{1}^{(2,6)}$ $\displaystyle=0\,,$ (4.15) $\displaystyle c_{2}^{(0)}$
$\displaystyle=d_{2}^{(0)}\,,\quad$ $\displaystyle c_{2}^{(1,1)}$
$\displaystyle=-2d_{2}^{(1,1)}\,,\quad$ $\displaystyle c_{2}^{(2,4)}$
$\displaystyle=-8d_{2}^{(2,4)}\,,\quad$ $\displaystyle c_{2}^{(2,5)}$
$\displaystyle=2d_{2}^{(2,4)}-2d_{2}^{(2,5)}\,,\quad$ $\displaystyle
c_{2}^{(2,6)}$ $\displaystyle=-2d_{2}^{(2,6)}\,.$

We determine the momentum-space coefficients $d_{i}^{A}$ in terms of the
amplitudes coefficients $a_{i}^{A}$ by the relations in Eqs. (4.9) and (4.11).
We may now obtain $a_{i}^{A}$ by demanding that the EFT amplitude matches the
full-theory one,

$\displaystyle{\cal M}^{\text{EFT}}_{\text{1PM}}=\frac{{\cal
M}^{\text{tree}}}{4E_{1}E_{2}}\ ,\qquad{\cal
M}^{\text{EFT}}_{\text{2PM}}=\frac{{\cal M}^{\text{1-loop}}}{4E_{1}E_{2}}\ ,$
(4.16)

where the factors of the energy account for the non-relativistic normalization
of the EFT amplitude. Using Eq. (3.12) we relate $a_{i}^{A}$ to
$\alpha_{i}^{A}$, which are explicitly shown in Eqs. (3.2) and (3.7). Putting
everything together, we obtain novel expressions for the position-space
coefficients $c_{i}^{A}$ which are lengthy, and so we only provide them in the
ancillary file coefficients.m.

###  4.3 Comparison to the literature

In order to ensure the validity of our result, we compare it with existing
Hamiltonians in the General Relativity literature. Specifically, we compare
with overlapping results in Ref. [109], which obtained the next-to-next-to-
leading order post-Newtonian Hamiltonian, and in Ref. [110], which calculated
the test-body Hamiltonian. Both references included interactions of up to
quadratic order in the spins.

One way to establish the equivalence of two Hamiltonians is to construct a
canonical transformation that extrapolates between them. Alternatively, we may
compare the gauge invariant scattering amplitudes calculated from the two
Hamiltonians by means of the EFT. We take the latter approach here. To do so,
we promote the spin vector in the classical Hamiltonians to the spin operator,
and we account for the non-isotropic terms according to the conventions of
[16, 17, 18].

In this way we obtain EFT amplitudes in the form of Eqs. (4.8) and (4.10). The
relevant coefficients for our purposes obtained using the Hamiltonian of Ref.
[109] read

$\displaystyle a_{1}^{(2,4)}$
$\displaystyle=\frac{m_{2}{C_{ES^{2}}}}{2m_{1}}-\frac{8m_{1}m_{2}+7m_{2}^{2}-{C_{ES^{2}}}\left(6m_{1}^{2}+16m_{1}m_{2}+6m_{2}^{2}\right)}{8m_{1}^{3}m_{2}}{\bm{p}}^{2}$
(4.17)
$\displaystyle\qquad\qquad-\frac{3m_{2}^{2}(7m_{1}^{2}+4m_{1}m_{2}-2m_{2}^{2})+{C_{ES^{2}}}(5m_{1}^{4}-18m_{1}^{2}m_{2}^{2}+5m_{2}^{4})}{16m_{1}^{5}m_{2}^{3}}{\bm{p}}^{4}+\ldots\,,$

and

$\displaystyle a_{2}^{(2,4)}$
$\displaystyle=\frac{m_{1}m_{2}^{3}{C_{ES^{2}}}}{4(m_{1}+m_{2}){\bm{p}^{2}}}+\frac{m_{2}\big{(}10m_{1}^{2}-7m_{1}m_{2}-13m_{2}^{2}+{C_{ES^{2}}}(32m_{1}^{2}+61m_{1}m_{2}+29m_{2}^{2})\big{)}}{16m_{1}(m_{1}+m_{2})}$
$\displaystyle\qquad\qquad\qquad+\frac{15m_{1}^{4}-73m_{1}^{3}m_{2}-361m_{1}^{2}m_{2}^{2}-343m_{1}m_{2}^{3}-82m_{2}^{4}}{64m_{1}^{3}m_{2}(m_{1}+m_{2})}{\bm{p}^{2}}$
$\displaystyle\qquad\qquad\qquad+\frac{{C_{ES^{2}}}(93m_{1}^{4}+467m_{1}^{3}m_{2}+707m_{1}^{2}m_{2}^{2}+397m_{1}m_{2}^{3}+64m_{2}^{4})}{64m_{1}^{3}m_{2}(m_{1}+m_{2})}{\bm{p}^{2}}+\ldots\,,$
$\displaystyle a_{2}^{(2,5)}$
$\displaystyle=-\frac{m_{1}m_{2}^{3}{C_{ES^{2}}}}{4(m_{1}+m_{2}){\bm{p}^{2}}}-\frac{m_{2}\big{(}22m_{1}^{2}+19m_{1}m_{2}+m_{2}^{2}+{C_{ES^{2}}}(20m_{1}^{2}+35m_{1}m_{2}+15m_{2}^{2})\big{)}}{16m_{1}(m_{1}+m_{2})}$
(4.18)
$\displaystyle\qquad\qquad\qquad-\frac{51m_{1}^{4}+115m_{1}^{3}m_{2}-53m_{1}^{2}m_{2}^{2}-155m_{1}m_{2}^{3}-50m_{2}^{4}}{64m_{1}^{3}m_{2}(m_{1}+m_{2})}{\bm{p}^{2}}$
$\displaystyle\qquad\qquad\qquad-\frac{{C_{ES^{2}}}(57m_{1}^{4}+279m_{1}^{3}m_{2}+399m_{1}^{2}m_{2}^{2}+209m_{1}m_{2}^{3}+32m_{2}^{4})}{64m_{1}^{3}m_{2}(m_{1}+m_{2})}{\bm{p}^{2}}+\ldots\,,$
$\displaystyle a_{2}^{(2,6)}$
$\displaystyle=\frac{m_{1}m_{2}^{3}{C_{ES^{2}}}}{2(m_{1}+m_{2})\bm{p}^{4}}+\frac{m_{2}\big{(}8m_{1}^{2}+8m_{1}m_{2}+m_{2}^{2}+{C_{ES^{2}}}(7m_{1}^{2}+11m_{1}m_{2}+4m_{2}^{2})\big{)}}{4m_{1}(m_{1}+m_{2}){\bm{p}^{2}}}$
$\displaystyle\qquad\qquad\qquad+\frac{33m_{1}^{4}+97m_{1}^{3}m_{2}+13m_{1}^{2}m_{2}^{2}-73m_{1}m_{2}^{3}-28m_{2}^{4}}{32m_{1}^{3}m_{2}(m_{1}+m_{2})}$
$\displaystyle\qquad\qquad\qquad+\frac{{C_{ES^{2}}}(39m_{1}^{4}+185m_{1}^{3}m_{2}+245m_{1}^{2}m_{2}^{2}+115m_{1}m_{2}^{3}+16m_{2}^{4})}{32m_{1}^{3}m_{2}(m_{1}+m_{2})}+\ldots\,,$

where the ellipsis stands for higher orders in $\bm{p}$. These coefficients
are in complete agreement with the velocity expansion of our amplitudes. The
Hamiltonian of Ref. [110] produces the coefficients

$\displaystyle a_{1}^{(2,4)}$
$\displaystyle=\frac{{m_{2}}\left({{C_{ES^{2}}}}\left(2{\gamma_{1}}^{3}+2{\gamma_{1}}^{2}-{\gamma_{1}}-1\right)-2{\gamma_{1}}^{3}-2{\gamma_{1}}^{2}+3{\gamma_{1}}+1\right)}{2{\gamma_{1}}({\gamma_{1}}+1){m_{1}}}\,,$
(4.19)

and

$\displaystyle a_{2}^{(2,4)}$
$\displaystyle=\frac{{m_{2}}^{2}\left({{C_{ES^{2}}}}\left(30{\gamma_{1}}^{4}-29{\gamma_{1}}^{2}+3\right)-30{\gamma_{1}}^{4}+59{\gamma_{1}}^{2}-24{\gamma_{1}}-5\right)}{16{\gamma_{1}}\left({\gamma_{1}}^{2}-1\right){m_{1}}}\,,$
$\displaystyle a_{2}^{(2,5)}$
$\displaystyle=-\frac{{m_{2}}^{2}\left({{C_{ES^{2}}}}\left(15{\gamma_{1}}^{4}-13{\gamma_{1}}^{2}+2\right)-15{\gamma_{1}}^{4}+43{\gamma_{1}}^{2}-24{\gamma_{1}}-4\right)}{16{\gamma_{1}}\left({\gamma_{1}}^{2}-1\right){m_{1}}}\,,$
$\displaystyle a_{2}^{(2,6)}$
$\displaystyle=\frac{{m_{2}}^{2}\left({{C_{ES^{2}}}}\left(15{\gamma_{1}}^{4}-10{\gamma_{1}}^{2}+3\right)-15{\gamma_{1}}^{4}+46{\gamma_{1}}^{2}-24{\gamma_{1}}-7\right)}{16{\gamma_{1}}\left({\gamma_{1}}^{2}-1\right)^{2}{m_{1}}^{3}}\,,$
(4.20) $\displaystyle a_{2}^{(2,\tilde{4})}$
$\displaystyle=\frac{\left(95{\gamma_{1}}^{4}-102{\gamma_{1}}^{2}+15\right){m_{1}}}{32{\gamma_{1}}\left({\gamma_{1}}^{2}-1\right)}\,,\qquad
a_{2}^{(2,\tilde{5})}=\frac{95{\gamma_{1}}^{4}{m_{1}}-102{\gamma_{1}}^{2}{m_{1}}+15{m_{1}}}{32{\gamma_{1}}-32{\gamma_{1}}^{3}}\,,$
$\displaystyle a_{2}^{(2,\tilde{6})}$
$\displaystyle=\frac{65{\gamma_{1}}^{4}-66{\gamma_{1}}^{2}+9}{16{\gamma_{1}}\left({\gamma_{1}}^{2}-1\right)^{2}{m_{1}}}\,,$

where the coefficients $a_{2}^{(2,\jmath)}$ correspond to the spinning
particle as the test body, while $a_{2}^{(2,\tilde{\jmath})}$ correspond to
the scalar particle as the test body. These coefficients exactly reproduce the
test body expansion of our amplitudes.

##  5 Observables from the eikonal phase

The conservative Hamiltonian we obtained in the previous section enables the
calculation of physical observables for a binary of compact objects
interacting through gravity. On the one hand, one may calculate quantities
that describe bound trajectories of the binary, as the bound-state energy. On
the other hand, observables pertaining to unbound orbits have received a surge
of attention. The main reason for this is that these observables serve as
input to important phenomenological models, as the effective one-body
Hamiltonian [123, 124, 125, 126, 127, 128]. Recently, there has been great
progress in obtaining these observables directly from the scattering amplitude
[22, 84, 9, 25]. Moreover, in the non-spinning case, Refs. [26, 27] developed
a dictionary between observables for unbound and bound orbits.

One prominent connection between physical observables and the scattering
amplitude is made via the eikonal phase [98]. There are several studies of
this connection, especially in the non-spinning case [98, 99, 100, 101, 102,
103, 104, 105, 31, 32]. Refs. [82, 75, 76] verified the applicability of this
approach for spinning particles in the special configuration where the spins
of the particles are orthogonal to the scattering plane. More recently, Ref.
[1] conjectured a formula that expresses physical observables in terms of
derivatives of the eikonal phase for arbitrary orientation of the spin
vectors.

In this section we extend the analysis of Refs. [82, 75, 76] and [1].
Specifically, we start by obtaining the eikonal phase via a fourier transform
of our amplitudes. By restricting to the aligned-spin configuration we obtain
a scattering angle which matches that of Ref. [82, 75, 76] when we specialize
to the black-hole case. Then, we verify the conjecture of Ref. [1] by solving
Hamilton’s equations for the impulse and spin kick, and relating them to
derivatives of the eikonal phase.

The eikonal phase $\chi=\chi_{1}+\chi_{2}+\mathcal{O}(G^{3})$ is given by

$\displaystyle\chi_{1}$
$\displaystyle=\frac{1}{4m_{1}m_{2}\sqrt{\sigma^{2}-1}}\int\frac{d^{2}\bm{q}}{(2\pi)^{2}}e^{-i\bm{q}\cdot\bm{b}}\mathcal{M}^{\rm
tree}(\bm{q})\,,$ $\displaystyle\chi_{2}$
$\displaystyle=\frac{1}{4m_{1}m_{2}\sqrt{\sigma^{2}-1}}\int\frac{d^{2}\bm{q}}{(2\pi)^{2}}e^{-i\bm{q}\cdot\bm{b}}\mathcal{M}^{\bigtriangleup+\bigtriangledown}(\bm{q})\,.$
(5.1)

Using our amplitudes expressed in the center-of-mass frame (see Eq. (3.11)) we
find

$\displaystyle\chi_{1}$ $\displaystyle=\frac{\xi
EG}{|\bm{p}|}\biggl{[}-a^{(0)}_{1}\ln\bm{b}^{2}-\frac{2a^{(1,1)}_{1}}{\bm{b}^{2}}(\bm{p}\times\bm{S}_{1})\cdot\bm{b}+a^{(2,4)}_{1}\left(\frac{2}{\bm{b}^{2}}\bm{S}_{1\perp}^{2}-4\frac{(\bm{S}_{1\perp}\cdot\bm{b})^{2}}{\bm{b}^{4}}\right)\biggr{]}\,,$
$\displaystyle\chi_{2}$ $\displaystyle=\frac{\pi\xi
EG^{2}}{|\bm{p}|}\bigg{[}\,\frac{a^{(0)}_{2}}{|\bm{b}|}-\frac{a^{(1,1)}_{2}}{|\bm{b}|^{3}}(\bm{p}\times\bm{S}_{1})\cdot\bm{b}+a^{(2,4)}_{2}\left(\frac{1}{|\bm{b}|^{3}}\bm{S}_{1\perp}^{2}-3\frac{(\bm{S}_{1\perp}\cdot\bm{b})^{2}}{|\bm{b}|^{5}}\right)$
(5.2) $\displaystyle\hskip
227.62204pt\hbox{}-\left(a^{(2,5)}_{2}\bm{S}_{1}^{2}+a^{(2,6)}_{2}(\bm{p}\cdot\bm{S}_{1})^{2}\right)\,\frac{1}{|\bm{b}|^{3}}\bigg{]}\,,$

where we define $\bm{S}_{\perp
1}\equiv\bm{S}_{1}-\frac{\bm{S}_{1}\cdot\bm{p}}{\bm{p}^{2}}\,\bm{p}$.

We may now use the eikonal phase to obtain certain classical observables. We
start by considering the aligned-spin kinematics of Ref. [82, 75, 76].
Specifically, we take the spin to be parallel to the orbital angular momentum,
and hence orthogonal to the scattering plane. This implies the relations

$\bm{S}_{1}\cdot\bm{b}=\bm{S}_{1}\cdot\bm{p}=0.$ (5.3)

Since the scattering process is confined to a plane, it can be described by
one scattering angle $\theta=\theta_{1}+\theta_{2}+\mathcal{O}(G^{3})$, which
we obtain as a derivative of the eikonal phase [98]

$\displaystyle\theta_{i}=-\frac{E}{m_{1}m_{2}\sqrt{\sigma^{2}-1}}\partial_{b}\chi_{i}\,,\quad
i=1,\,2\,,$ (5.4)

where $b=|\bm{b}|$. The novel piece of the 2PM angle we obtain is quadratic in
spin and given by

$\displaystyle\theta_{2,\,\bm{S}_{1}^{2}}=\frac{3E\pi
G^{2}\bm{S}_{1}^{2}}{32m_{1}^{2}b^{4}(\sigma^{2}-1)^{2}}\Big{\\{}$
$\displaystyle
m_{2}\left(6(5\sigma^{4}-6\sigma^{2}+1)+2{C_{ES^{2}}}(45\sigma^{4}-42\sigma^{2}+5)\right)$
$\displaystyle+$ $\displaystyle
m_{1}\left((65\sigma^{4}-66\sigma^{2}+1)+{C_{ES^{2}}}(125\sigma^{4}-138\sigma^{2}+29)\right)\Big{\\}}\,.$
(5.5)

By specializing to the black-hole case (${C_{ES^{2}}}=1$) we reproduce the
result of Ref. [82].

Ref. [1] conjectured a formula that directly relates observables in a
scattering event with arbitrary spin orientations to the eikonal phase. The
observables in question are the impulse $\Delta\bm{p}$ and spin kick
$\Delta\bm{S}_{1}$, where

$\displaystyle\bm{p}(t=\infty)$ $\displaystyle=\bm{p}+\Delta\bm{p}\,,\hskip
27.88358pt\bm{p}$ $\displaystyle(t=-\infty)=\bm{p}\,,$
$\displaystyle\bm{S}_{1}(t=\infty)$
$\displaystyle=\bm{S}_{1}+\Delta\bm{S}_{1}\,,\quad\bm{S}_{1}$
$\displaystyle(t=-\infty)=\bm{S}_{1}\,.$ (5.6)

Specifically, by obtaining the impulse and spin kick through
$\mathcal{O}(G^{2})$ using Hamilton’s equations, we find that they may be
written as

$\displaystyle\Delta\bm{p}_{\perp}$
$\displaystyle=-\\{\bm{p}_{\perp},\chi\\}-\frac{1}{2}\,\\{\chi,\\{\bm{p}_{\perp},\chi\\}\\}-\mathcal{D}_{SL}\left(\chi,\\{\bm{p}_{\perp},\chi\\}\right)+\frac{1}{2}\,\\{\bm{p}_{\perp},\mathcal{D}_{SL}\left(\chi,\chi\right)\\}\,,$
$\displaystyle\Delta\bm{S}_{1}$
$\displaystyle=-\\{\bm{S}_{1},\chi\\}-\frac{1}{2}\,\\{\chi,\\{\bm{S}_{1},\chi\\}\\}-\mathcal{D}_{SL}\left(\chi,\\{\bm{S}_{1},\chi\\}\right)+\frac{1}{2}\,\\{\bm{S}_{1},\mathcal{D}_{SL}\left(\chi,\chi\right)\\}\,.$
(5.7)

In Eq. (5.7) we use the definitions

$\displaystyle\\{\bm{p}_{\perp},f\\}\equiv-\frac{\partial
f}{\partial\bm{b}}\,,\hskip 28.45274pt\\{\bm{S}_{1},f\\}\equiv\,\frac{\partial
f}{\partial\bm{S}_{1}}\times\bm{S}_{1}\,,\hskip
28.45274pt\mathcal{D}_{SL}\left(f,g\right)\equiv-\,\bm{S}_{1}\,\cdot\left(\frac{\partial
f}{\partial\bm{S}_{1}}\,\times\frac{\partial g}{\partial\bm{L}_{b}}\right)\,,$
(5.8)

where $\bm{L}_{\bm{b}}\equiv\bm{b}\times\bm{p}$. In the above we decompose the
impulse as

$\Delta\bm{p}=\Delta
p_{\parallel}\frac{\bm{p}}{|\bm{p}|}+\Delta\bm{p}_{\perp}\,.$ (5.9)

Eq. (5.7) does not give $\Delta p_{\parallel}$. Instead, we obtain $\Delta
p_{\parallel}$ from the on-shell condition
$(\bm{p}+\Delta\bm{p})^{2}=\bm{p}^{2}$.

Our calculation establishes the conjecture of Ref. [1] at the quadratic-in-
spin level. The fact that the relation holds without modification when we
include these higher-in-spin terms is strong indication for its validity in
general. Our calculation further serves as evidence in favor of the
surprisingly compact all-order formula that relates the scattering observables
to the eikonal phase,

$\Delta\mathcal{O}=ie^{-i\chi\mathcal{D}}\\{\mathcal{O},e^{i\chi\mathcal{D}}\\}\,,$
(5.10)

where for our case $\mathcal{O}=\bm{p}_{\perp}$ or $\bm{S}_{1}$, and
$\chi\mathcal{D}g\equiv\chi g+i\mathcal{D}_{SL}(\chi,g)$.

##  6 Conclusions

In this paper we obtained the 2PM-order Hamiltonian that describes the
conservative dynamics of two spinning compact objects in General Relativity up
to interactions quadratic in the spin of one of the objects. We followed the
approach of Refs. [1, 16] which was based on scattering amplitudes and EFT.
Along with the results of [1] for the bilinear-in-the-spins interactions, this
completes the $\mathcal{O}(G^{2})$ analysis of quadratic-in-spin effects not
including tidal effects.

To construct the Hamiltonian we followed a matching procedure. Ref. [16]
developed this procedure for non-spinning particles, while Ref. [1] extended
it to the spinning case. Specifically, we calculated and matched two
amplitudes, one in our full theory and one in an EFT. Ref. [1] introduced the
full theory to describe the minimal and non-minimal coupling of particles of
arbitrary spin to gravity. The Lagrangian contains operators that are in one-
to-one correspondence with those of the worldline EFT of [113]. The EFT we
used captures the dynamics of non-relativistic spinning particles interacting
via a potential with unfixed coefficients. This EFT extended the one of [1] to
include operators quadratic in the spin of one of the particles. By matching
the amplitudes computed in these two theories, we fixed these coefficients and
hence determined the desired Hamiltonian.

In our calculation we considered effects up to quadratic in the spin of one of
the particles, while we took the other particle to be non-spinning. In terms
of our full theory, we included the first non-minimal-coupling operator along
with the corresponding arbitrary Wilson coefficient ${C_{ES^{2}}}$. Unlike the
linear-in-spin results, the effects of this operator are not universal and
generic bodies are described by different values of ${C_{ES^{2}}}$. As a
specific example, ${C_{ES^{2}}}=1$ describes the Kerr black hole. For
arbitrary values of ${C_{ES^{2}}}$, we found that the amplitude depends on
$q^{2}S_{1}^{2}$ and $\left(q\cdot S_{1}\right)^{2}$ independently, rather
than on the linear combination $q^{2}S_{1}^{2}-\left(q\cdot S_{1}\right)^{2}$.
The latter was expected based on an observation in Refs. [96, 1]. Recently,
Ref. [39] also remarked that finite-size effects spoil the above expectation.
Interestingly, for the Kerr black-hole case (${C_{ES^{2}}}=1$) the amplitude
indeed depends on the linear combination $q^{2}S_{1}^{2}-\left(q\cdot
S_{1}\right)^{2}$.

The produced conservative Hamiltonian enables the calculation of observables
pertaining to binary systems of spinning black holes or neutron stars. For
example, one may study bound states of the binary by choosing suitable initial
conditions. In this paper we chose to compute scattering observables instead,
which may be used in the construction of important phenomenological models as
the effective one-body Hamiltonian [123, 124, 125, 126, 127, 128].
Specifically, by solving Hamilton’s equations we obtained the relevant impulse
and spin kick. In this way we verified the conjecture of Ref. [1], which
expresses these observables in terms of the eikonal phase via the simple
compact formula in Eq. (5.10). The existence of such a formula has intriguing
implications in classical mechanics. Specifically, it hints towards a
formalism that bypasses using Hamilton’s equations, and directly expresses the
observables in terms of derivatives of a single function of the kinematics.

In order to establish the validity of our result for the quadratic-in-spin
two-body Hamiltonian, we performed several checks against the literature. We
did this by comparing at the level of the gauge-invariant amplitudes in the
regime where they overlap. Firstly, we verified that our amplitude expanded in
velocity matches the one calculated using the Hamiltonian of Ref. [109], which
was obtained in the PN approximation. Secondly, by expanding our amplitude in
the test-body limit we found agreement with the amplitude obtained by the
Hamiltonian of Ref. [110]. As a third check, we computed the scattering angle
for the kinematic configuration where the spin vector is aligned with the
orbital angular momentum of the system and confirmed that it reproduces the
one of Ref. [82] for the BH case, ${C_{ES^{2}}}=1$. Finally, we compared the
impulse in Eq. (5.7) with the one given in Ref. [111] in covariant form and
found agreement.

Our calculation serves as evidence that the formalism of Ref. [1] can capture
the effects of non-minimal coupling to gravity. Therefore, an obvious future
direction is to extend this analysis to include more powers of spin. Moreover,
a number of pressing questions remain interesting and unanswered. These
include the extension of these methods to higher PM orders, the proof of the
relation between classical scattering observables and the eikonal phase, along
with potential extensions of this relation to bound-orbit observables.

###  Acknowledgments:

We thank Zvi Bern, Radu Roiban, Chia-Hsien Shen and Fei Teng for collaboration
in related topics, and Donal O’Connell for discussions. We thank Jan Steinhoff
and Justin Vines for several helpful discussions, and for supplying ancillary
files for the post-Newtonian Hamiltonian of Ref. [113], as well as a simpler
form of the test-mass Hamiltonian based on Ref. [110]. We also thank the
authors of [111] for sharing their results for the linear impulse in advance
of its publication. DK and AL are supported by the U.S. Department of Energy
(DOE) under award number DE-SC0009937, and by the Mani L. Bhaumik Institute
for Theoretical Physics.

## References

  * [1] Z. Bern, A. Luna, R. Roiban, C.-H. Shen, and M. Zeng, “Spinning Black Hole Binary Dynamics, Scattering Amplitudes and Effective Field Theory,” arXiv:2005.03071 [hep-th].
  * [2] LIGO Scientific, Virgo Collaboration, B. P. Abbott et al., “Observation of Gravitational Waves from a Binary Black Hole Merger,” Phys. Rev. Lett. 116 no. 6, (2016) 061102, arXiv:1602.03837 [gr-qc].
  * [3] LIGO Scientific, Virgo Collaboration, B. P. Abbott et al., “GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral,” Phys. Rev. Lett. 119 no. 16, (2017) 161101, arXiv:1710.05832 [gr-qc].
  * [4] L. Blanchet, “Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries,” Living Rev. Rel. 17 (2014) 2, arXiv:1310.1528 [gr-qc].
  * [5] G. Schäfer and P. Jaranowski, “Hamiltonian formulation of general relativity and post-Newtonian dynamics of compact binaries,” Living Rev. Rel. 21 no. 1, (2018) 7, arXiv:1805.07240 [gr-qc].
  * [6] W. D. Goldberger and I. Z. Rothstein, “An Effective field theory of gravity for extended objects,” Phys. Rev. D 73 (2006) 104029, arXiv:hep-th/0409156.
  * [7] K. Westpfahl, “High-Speed Scattering of Charged and Uncharged Particles in General Relativity,” Fortsch. Phys. 33 no. 8, (1985) 417–493.
  * [8] T. Damour, “Gravitational scattering, post-Minkowskian approximation and Effective One-Body theory,” Phys. Rev. D 94 no. 10, (2016) 104015, arXiv:1609.00354 [gr-qc].
  * [9] T. Damour, “High-energy gravitational scattering and the general relativistic two-body problem,” Phys. Rev. D 97 no. 4, (2018) 044038, arXiv:1710.10599 [gr-qc].
  * [10] T. Damour, “Classical and quantum scattering in post-Minkowskian gravity,” Phys. Rev. D 102 no. 2, (2020) 024060, arXiv:1912.02139 [gr-qc].
  * [11] Y. Iwasaki, “Quantum theory of gravitation vs. classical theory. - fourth-order potential,” Prog. Theor. Phys. 46 (1971) 1587–1609.
  * [12] D. Neill and I. Z. Rothstein, “Classical Space-Times from the S Matrix,” Nucl. Phys. B 877 (2013) 177–189, arXiv:1304.7263 [hep-th].
  * [13] L. J. Dixon, “Calculating scattering amplitudes efficiently,” in Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 95): QCD and Beyond, pp. 539–584.  1, 1996.  arXiv:hep-ph/9601359.
  * [14] H. Elvang and Y.-t. Huang, “Scattering Amplitudes,” arXiv:1308.1697 [hep-th].
  * [15] Z. Bern, J. J. Carrasco, M. Chiodaroli, H. Johansson, and R. Roiban, “The Duality Between Color and Kinematics and its Applications,” arXiv:1909.01358 [hep-th].
  * [16] C. Cheung, I. Z. Rothstein, and M. P. Solon, “From Scattering Amplitudes to Classical Potentials in the Post-Minkowskian Expansion,” Phys. Rev. Lett. 121 no. 25, (2018) 251101, arXiv:1808.02489 [hep-th].
  * [17] Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M. P. Solon, and M. Zeng, “Scattering Amplitudes and the Conservative Hamiltonian for Binary Systems at Third Post-Minkowskian Order,” Phys. Rev. Lett. 122 no. 20, (2019) 201603, arXiv:1901.04424 [hep-th].
  * [18] Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M. P. Solon, and M. Zeng, “Black Hole Binary Dynamics from the Double Copy and Effective Theory,” JHEP 10 (2019) 206, arXiv:1908.01493 [hep-th].
  * [19] C. Cheung and M. P. Solon, “Classical gravitational scattering at $\mathcal{O}$(G3) from Feynman diagrams,” JHEP 06 (2020) 144, arXiv:2003.08351 [hep-th].
  * [20] A. Antonelli, A. Buonanno, J. Steinhoff, M. van de Meent, and J. Vines, “Energetics of two-body Hamiltonians in post-Minkowskian gravity,” Phys. Rev. D 99 no. 10, (2019) 104004, arXiv:1901.07102 [gr-qc].
  * [21] Z. Bern, J. Parra-Martinez, R. Roiban, M. S. Ruf, C.-H. Shen, M. P. Solon, and M. Zeng, “Scattering Amplitudes and Conservative Binary Dynamics at ${\cal O}(G^{4})$,” arXiv:2101.07254 [hep-th].
  * [22] D. A. Kosower, B. Maybee, and D. O’Connell, “Amplitudes, Observables, and Classical Scattering,” JHEP 02 (2019) 137, arXiv:1811.10950 [hep-th].
  * [23] L. de la Cruz, B. Maybee, D. O’Connell, and A. Ross, “Classical Yang-Mills observables from amplitudes,” JHEP 12 (2020) 076, arXiv:2009.03842 [hep-th].
  * [24] A. Cristofoli, N. E. J. Bjerrum-Bohr, P. H. Damgaard, and P. Vanhove, “Post-Minkowskian Hamiltonians in general relativity,” Phys. Rev. D 100 no. 8, (2019) 084040, arXiv:1906.01579 [hep-th].
  * [25] N. E. J. Bjerrum-Bohr, A. Cristofoli, and P. H. Damgaard, “Post-Minkowskian Scattering Angle in Einstein Gravity,” JHEP 08 (2020) 038, arXiv:1910.09366 [hep-th].
  * [26] G. Kälin and R. A. Porto, “From Boundary Data to Bound States,” JHEP 01 (2020) 072, arXiv:1910.03008 [hep-th].
  * [27] G. Kälin and R. A. Porto, “From boundary data to bound states. Part II. Scattering angle to dynamical invariants (with twist),” JHEP 02 (2020) 120, arXiv:1911.09130 [hep-th].
  * [28] G. Kälin and R. A. Porto, “Post-Minkowskian Effective Field Theory for Conservative Binary Dynamics,” JHEP 11 (2020) 106, arXiv:2006.01184 [hep-th].
  * [29] G. Kälin, Z. Liu, and R. A. Porto, “Conservative Dynamics of Binary Systems to Third Post-Minkowskian Order from the Effective Field Theory Approach,” Phys. Rev. Lett. 125 no. 26, (2020) 261103, arXiv:2007.04977 [hep-th].
  * [30] S. Caron-Huot and Z. Zahraee, “Integrability of Black Hole Orbits in Maximal Supergravity,” JHEP 07 (2019) 179, arXiv:1810.04694 [hep-th].
  * [31] Z. Bern, H. Ita, J. Parra-Martinez, and M. S. Ruf, “Universality in the classical limit of massless gravitational scattering,” Phys. Rev. Lett. 125 no. 3, (2020) 031601, arXiv:2002.02459 [hep-th].
  * [32] J. Parra-Martinez, M. S. Ruf, and M. Zeng, “Extremal black hole scattering at $\mathcal{O}(G^{3})$: graviton dominance, eikonal exponentiation, and differential equations,” JHEP 11 (2020) 023, arXiv:2005.04236 [hep-th].
  * [33] F. Loebbert, J. Plefka, C. Shi, and T. Wang, “Three-Body Effective Potential in General Relativity at 2PM and Resulting PN Contributions,” arXiv:2012.14224 [hep-th].
  * [34] P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, “Universality of ultra-relativistic gravitational scattering,” Phys. Lett. B 811 (2020) 135924, arXiv:2008.12743 [hep-th].
  * [35] P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, “Radiation Reaction from Soft Theorems,” arXiv:2101.05772 [hep-th].
  * [36] E. Herrmann, J. Parra-Martinez, M. S. Ruf, and M. Zeng, “Gravitational Bremsstrahlung from Reverse Unitarity,” arXiv:2101.07255 [hep-th].
  * [37] T. Damour, “Radiative contribution to classical gravitational scattering at the third order in $G$,” Phys. Rev. D 102 no. 12, (2020) 124008, arXiv:2010.01641 [gr-qc].
  * [38] K. Haddad and A. Helset, “Tidal effects in quantum field theory,” JHEP 12 (2020) 024, arXiv:2008.04920 [hep-th].
  * [39] R. Aoude, K. Haddad, and A. Helset, “Tidal effects for spinning particles,” arXiv:2012.05256 [hep-th].
  * [40] M. Accettulli Huber, A. Brandhuber, S. De Angelis, and G. Travaglini, “From amplitudes to gravitational radiation with cubic interactions and tidal effects,” arXiv:2012.06548 [hep-th].
  * [41] G. Kälin, Z. Liu, and R. A. Porto, “Conservative Tidal Effects in Compact Binary Systems to Next-to-Leading Post-Minkowskian Order,” Phys. Rev. D 102 (2020) 124025, arXiv:2008.06047 [hep-th].
  * [42] C. Cheung and M. P. Solon, “Tidal Effects in the Post-Minkowskian Expansion,” Phys. Rev. Lett. 125 no. 19, (2020) 191601, arXiv:2006.06665 [hep-th].
  * [43] C. Cheung, N. Shah, and M. P. Solon, “Mining the Geodesic Equation for Scattering Data,” Phys. Rev. D 103 no. 2, (2021) 024030, arXiv:2010.08568 [hep-th].
  * [44] Z. Bern, J. Parra-Martinez, R. Roiban, E. Sawyer, and C.-H. Shen, “Leading Nonlinear Tidal Effects and Scattering Amplitudes,” arXiv:2010.08559 [hep-th].
  * [45] G. Faye, L. Blanchet, and A. Buonanno, “Higher-order spin effects in the dynamics of compact binaries. I. Equations of motion,” Phys. Rev. D 74 (2006) 104033, arXiv:gr-qc/0605139.
  * [46] L. Blanchet, A. Buonanno, and G. Faye, “Higher-order spin effects in the dynamics of compact binaries. II. Radiation field,” Phys. Rev. D 74 (2006) 104034, arXiv:gr-qc/0605140. [Erratum: Phys.Rev.D 75, 049903 (2007), Erratum: Phys.Rev.D 81, 089901 (2010)].
  * [47] T. Damour, P. Jaranowski, and G. Schaefer, “Hamiltonian of two spinning compact bodies with next-to-leading order gravitational spin-orbit coupling,” Phys. Rev. D 77 (2008) 064032, arXiv:0711.1048 [gr-qc].
  * [48] J. Steinhoff, S. Hergt, and G. Schaefer, “On the next-to-leading order gravitational spin(1)-spin(2) dynamics,” Phys. Rev. D 77 (2008) 081501, arXiv:0712.1716 [gr-qc].
  * [49] J. Steinhoff, G. Schaefer, and S. Hergt, “ADM canonical formalism for gravitating spinning objects,” Phys. Rev. D 77 (2008) 104018, arXiv:0805.3136 [gr-qc].
  * [50] R. A. Porto, “Post-Newtonian corrections to the motion of spinning bodies in NRGR,” Phys. Rev. D 73 (2006) 104031, arXiv:gr-qc/0511061.
  * [51] R. A. Porto and I. Z. Rothstein, “The Hyperfine Einstein-Infeld-Hoffmann potential,” Phys. Rev. Lett. 97 (2006) 021101, arXiv:gr-qc/0604099.
  * [52] R. A. Porto, “Absorption effects due to spin in the worldline approach to black hole dynamics,” Phys. Rev. D 77 (2008) 064026, arXiv:0710.5150 [hep-th].
  * [53] R. A. Porto and I. Z. Rothstein, “Next to Leading Order Spin(1)Spin(1) Effects in the Motion of Inspiralling Compact Binaries,” Phys. Rev. D 78 (2008) 044013, arXiv:0804.0260 [gr-qc]. [Erratum: Phys.Rev.D 81, 029905 (2010)].
  * [54] M. Levi, “Next to Leading Order gravitational Spin1-Spin2 coupling with Kaluza-Klein reduction,” Phys. Rev. D 82 (2010) 064029, arXiv:0802.1508 [gr-qc].
  * [55] R. A. Porto and I. Z. Rothstein, “Spin(1)Spin(2) Effects in the Motion of Inspiralling Compact Binaries at Third Order in the Post-Newtonian Expansion,” Phys. Rev. D 78 (2008) 044012, arXiv:0802.0720 [gr-qc]. [Erratum: Phys.Rev.D 81, 029904 (2010)].
  * [56] R. A. Porto, “Next to leading order spin-orbit effects in the motion of inspiralling compact binaries,” Class. Quant. Grav. 27 (2010) 205001, arXiv:1005.5730 [gr-qc].
  * [57] R. A. Porto, A. Ross, and I. Z. Rothstein, “Spin induced multipole moments for the gravitational wave flux from binary inspirals to third Post-Newtonian order,” JCAP 03 (2011) 009, arXiv:1007.1312 [gr-qc].
  * [58] M. Levi, “Next to Leading Order gravitational Spin-Orbit coupling in an Effective Field Theory approach,” Phys. Rev. D 82 (2010) 104004, arXiv:1006.4139 [gr-qc].
  * [59] M. Levi, “Binary dynamics from spin1-spin2 coupling at fourth post-Newtonian order,” Phys. Rev. D 85 (2012) 064043, arXiv:1107.4322 [gr-qc].
  * [60] M. Levi and J. Steinhoff, “Equivalence of ADM Hamiltonian and Effective Field Theory approaches at next-to-next-to-leading order spin1-spin2 coupling of binary inspirals,” JCAP 12 (2014) 003, arXiv:1408.5762 [gr-qc].
  * [61] M. Levi and J. Steinhoff, “Next-to-next-to-leading order gravitational spin-orbit coupling via the effective field theory for spinning objects in the post-Newtonian scheme,” JCAP 01 (2016) 011, arXiv:1506.05056 [gr-qc].
  * [62] M. Levi and J. Steinhoff, “Next-to-next-to-leading order gravitational spin-squared potential via the effective field theory for spinning objects in the post-Newtonian scheme,” JCAP 01 (2016) 008, arXiv:1506.05794 [gr-qc].
  * [63] N. T. Maia, C. R. Galley, A. K. Leibovich, and R. A. Porto, “Radiation reaction for spinning bodies in effective field theory I: Spin-orbit effects,” Phys. Rev. D 96 no. 8, (2017) 084064, arXiv:1705.07934 [gr-qc].
  * [64] N. T. Maia, C. R. Galley, A. K. Leibovich, and R. A. Porto, “Radiation reaction for spinning bodies in effective field theory II: Spin-spin effects,” Phys. Rev. D 96 no. 8, (2017) 084065, arXiv:1705.07938 [gr-qc].
  * [65] W. D. Goldberger, J. Li, and I. Z. Rothstein, “Non-conservative effects on Spinning Black Holes from World-Line Effective Field Theory,” arXiv:2012.14869 [hep-th].
  * [66] R. A. Porto, “The effective field theorist’s approach to gravitational dynamics,” Phys. Rept. 633 (2016) 1–104, arXiv:1601.04914 [hep-th].
  * [67] M. Levi, “Effective Field Theories of Post-Newtonian Gravity: A comprehensive review,” Rept. Prog. Phys. 83 no. 7, (2020) 075901, arXiv:1807.01699 [hep-th].
  * [68] M. Levi, A. J. Mcleod, and M. Von Hippel, “N3LO gravitational spin-orbit coupling at order $G^{4}$,” arXiv:2003.02827 [hep-th].
  * [69] M. Levi, A. J. Mcleod, and M. Von Hippel, “NNNLO gravitational quadratic-in-spin interactions at the quartic order in G,” arXiv:2003.07890 [hep-th].
  * [70] M. Levi, S. Mougiakakos, and M. Vieira, “Gravitational cubic-in-spin interaction at the next-to-leading post-Newtonian order,” JHEP 01 (2021) 036, arXiv:1912.06276 [hep-th].
  * [71] M. Levi and F. Teng, “NLO gravitational quartic-in-spin interaction,” JHEP 01 (2021) 066, arXiv:2008.12280 [hep-th].
  * [72] D. Bini and T. Damour, “Gravitational spin-orbit coupling in binary systems, post-Minkowskian approximation and effective one-body theory,” Phys. Rev. D 96 no. 10, (2017) 104038, arXiv:1709.00590 [gr-qc].
  * [73] D. Bini, T. Damour, and A. Geralico, “Spin-orbit precession along eccentric orbits: improving the knowledge of self-force corrections and of their effective-one-body counterparts,” Phys. Rev. D 97 no. 10, (2018) 104046, arXiv:1801.03704 [gr-qc].
  * [74] J. Vines, “Scattering of two spinning black holes in post-Minkowskian gravity, to all orders in spin, and effective-one-body mappings,” Class. Quant. Grav. 35 no. 8, (2018) 084002, arXiv:1709.06016 [gr-qc].
  * [75] J. Vines, J. Steinhoff, and A. Buonanno, “Spinning-black-hole scattering and the test-black-hole limit at second post-Minkowskian order,” Phys. Rev. D 99 no. 6, (2019) 064054, arXiv:1812.00956 [gr-qc].
  * [76] N. Siemonsen and J. Vines, “Test black holes, scattering amplitudes and perturbations of Kerr spacetime,” Phys. Rev. D 101 no. 6, (2020) 064066, arXiv:1909.07361 [gr-qc].
  * [77] B. R. Holstein and A. Ross, “Spin Effects in Long Range Gravitational Scattering,” arXiv:0802.0716 [hep-ph].
  * [78] V. Vaidya, “Gravitational spin Hamiltonians from the S matrix,” Phys. Rev. D 91 no. 2, (2015) 024017, arXiv:1410.5348 [hep-th].
  * [79] A. Guevara, “Holomorphic Classical Limit for Spin Effects in Gravitational and Electromagnetic Scattering,” JHEP 04 (2019) 033, arXiv:1706.02314 [hep-th].
  * [80] N. Arkani-Hamed, T.-C. Huang, and Y.-t. Huang, “Scattering Amplitudes For All Masses and Spins,” arXiv:1709.04891 [hep-th].
  * [81] F. Cachazo and A. Guevara, “Leading Singularities and Classical Gravitational Scattering,” JHEP 02 (2020) 181, arXiv:1705.10262 [hep-th].
  * [82] A. Guevara, A. Ochirov, and J. Vines, “Scattering of Spinning Black Holes from Exponentiated Soft Factors,” JHEP 09 (2019) 056, arXiv:1812.06895 [hep-th].
  * [83] M.-Z. Chung, Y.-T. Huang, J.-W. Kim, and S. Lee, “The simplest massive S-matrix: from minimal coupling to Black Holes,” JHEP 04 (2019) 156, arXiv:1812.08752 [hep-th].
  * [84] B. Maybee, D. O’Connell, and J. Vines, “Observables and amplitudes for spinning particles and black holes,” JHEP 12 (2019) 156, arXiv:1906.09260 [hep-th].
  * [85] M.-Z. Chung, Y.-T. Huang, and J.-W. Kim, “Classical potential for general spinning bodies,” JHEP 09 (2020) 074, arXiv:1908.08463 [hep-th].
  * [86] M.-Z. Chung, Y.-t. Huang, J.-W. Kim, and S. Lee, “Complete Hamiltonian for spinning binary systems at first post-Minkowskian order,” JHEP 05 (2020) 105, arXiv:2003.06600 [hep-th].
  * [87] A. Guevara, A. Ochirov, and J. Vines, “Black-hole scattering with general spin directions from minimal-coupling amplitudes,” Phys. Rev. D 100 no. 10, (2019) 104024, arXiv:1906.10071 [hep-th].
  * [88] N. Arkani-Hamed, Y.-t. Huang, and D. O’Connell, “Kerr black holes as elementary particles,” JHEP 01 (2020) 046, arXiv:1906.10100 [hep-th].
  * [89] Y. F. Bautista and A. Guevara, “From Scattering Amplitudes to Classical Physics: Universality, Double Copy and Soft Theorems,” arXiv:1903.12419 [hep-th].
  * [90] M.-Z. Chung, Y.-T. Huang, and J.-W. Kim, “Kerr-Newman stress-tensor from minimal coupling,” JHEP 12 (2020) 103, arXiv:1911.12775 [hep-th].
  * [91] A. Guevara, B. Maybee, A. Ochirov, D. O’Connell, and J. Vines, “A worldsheet for Kerr,” arXiv:2012.11570 [hep-th].
  * [92] W. T. Emond, Y.-T. Huang, U. Kol, N. Moynihan, and D. O’Connell, “Amplitudes from Coulomb to Kerr-Taub-NUT,” arXiv:2010.07861 [hep-th].
  * [93] R. Aoude, M.-Z. Chung, Y.-t. Huang, C. S. Machado, and M.-K. Tam, “Silence of Binary Kerr Black Holes,” Phys. Rev. Lett. 125 no. 18, (2020) 181602, arXiv:2007.09486 [hep-th].
  * [94] R. Monteiro, D. O’Connell, and C. D. White, “Black holes and the double copy,” JHEP 12 (2014) 056, arXiv:1410.0239 [hep-th].
  * [95] R. Aoude, K. Haddad, and A. Helset, “On-shell heavy particle effective theories,” JHEP 05 (2020) 051, arXiv:2001.09164 [hep-th].
  * [96] P. H. Damgaard, K. Haddad, and A. Helset, “Heavy Black Hole Effective Theory,” JHEP 11 (2019) 070, arXiv:1908.10308 [hep-ph].
  * [97] K. Haddad and A. Helset, “The double copy for heavy particles,” Phys. Rev. Lett. 125 (2020) 181603, arXiv:2005.13897 [hep-th].
  * [98] D. Amati, M. Ciafaloni, and G. Veneziano, “Higher Order Gravitational Deflection and Soft Bremsstrahlung in Planckian Energy Superstring Collisions,” Nucl. Phys. B 347 (1990) 550–580.
  * [99] S. Melville, S. G. Naculich, H. J. Schnitzer, and C. D. White, “Wilson line approach to gravity in the high energy limit,” Phys. Rev. D 89 no. 2, (2014) 025009, arXiv:1306.6019 [hep-th].
  * [100] A. Luna, S. Melville, S. G. Naculich, and C. D. White, “Next-to-soft corrections to high energy scattering in QCD and gravity,” JHEP 01 (2017) 052, arXiv:1611.02172 [hep-th].
  * [101] R. Akhoury, R. Saotome, and G. Sterman, “High Energy Scattering in Perturbative Quantum Gravity at Next to Leading Power,” arXiv:1308.5204 [hep-th].
  * [102] A. Koemans Collado, P. Di Vecchia, and R. Russo, “Revisiting the second post-Minkowskian eikonal and the dynamics of binary black holes,” Phys. Rev. D 100 no. 6, (2019) 066028, arXiv:1904.02667 [hep-th].
  * [103] A. Cristofoli, P. H. Damgaard, P. Di Vecchia, and C. Heissenberg, “Second-order Post-Minkowskian scattering in arbitrary dimensions,” JHEP 07 (2020) 122, arXiv:2003.10274 [hep-th].
  * [104] P. Di Vecchia, A. Luna, S. G. Naculich, R. Russo, G. Veneziano, and C. D. White, “A tale of two exponentiations in ${\cal N}=8$ supergravity,” Phys. Lett. B 798 (2019) 134927, arXiv:1908.05603 [hep-th].
  * [105] P. Di Vecchia, S. G. Naculich, R. Russo, G. Veneziano, and C. D. White, “A tale of two exponentiations in $\mathcal{N}$ = 8 supergravity at subleading level,” JHEP 03 (2020) 173, arXiv:1911.11716 [hep-th].
  * [106] Z. Bern, L. J. Dixon, and D. A. Kosower, “Progress in one loop QCD computations,” Ann. Rev. Nucl. Part. Sci. 46 (1996) 109–148, arXiv:hep-ph/9602280.
  * [107] Z. Bern and Y.-t. Huang, “Basics of Generalized Unitarity,” J. Phys. A 44 (2011) 454003, arXiv:1103.1869 [hep-th].
  * [108] R. Britto, F. Cachazo, and B. Feng, “Generalized unitarity and one-loop amplitudes in N=4 super-Yang-Mills,” Nucl. Phys. B 725 (2005) 275–305, arXiv:hep-th/0412103.
  * [109] M. Levi and J. Steinhoff, “Complete conservative dynamics for inspiralling compact binaries with spins at fourth post-Newtonian order,” arXiv:1607.04252 [gr-qc].
  * [110] J. Vines, D. Kunst, J. Steinhoff, and T. Hinderer, “Canonical Hamiltonian for an extended test body in curved spacetime: To quadratic order in spin,” Phys. Rev. D 93 no. 10, (2016) 103008, arXiv:1601.07529 [gr-qc].
  * [111] Z. Liu, R. A. Porto, and Z. Yang, “To appear,”.
  * [112] L. Singh and C. Hagen, “Lagrangian formulation for arbitrary spin. 1. The boson case,” Phys. Rev. D 9 (1974) 898–909.
  * [113] M. Levi and J. Steinhoff, “Spinning gravitating objects in the effective field theory in the post-Newtonian scheme,” JHEP 09 (2015) 219, arXiv:1501.04956 [gr-qc].
  * [114] M. Levi and J. Steinhoff, “Leading order finite size effects with spins for inspiralling compact binaries,” JHEP 06 (2015) 059, arXiv:1410.2601 [gr-qc].
  * [115] J. Klauder and B. Skagerstam, Coherent States: Applications in Physics and Mathematical Physics.  World Scientific, Singapore, 1985.
  * [116] V. Berestetskii, E. Lifshitz, and L. Pitaevskii, QUANTUM ELECTRODYNAMICS, vol. 4 of Course of Theoretical Physics.  Pergamon Press, Oxford, 1982.
  * [117] I. Khriplovich and A. Pomeransky, “Equations of motion of spinning relativistic particle in external fields,” J. Exp. Theor. Phys. 86 (1998) 839–849, arXiv:gr-qc/9710098.
  * [118] G. Passarino and M. Veltman, “One Loop Corrections for e+ e- Annihilation Into mu+ mu- in the Weinberg Model,” Nucl. Phys. B 160 (1979) 151–207.
  * [119] G. Ossola, C. G. Papadopoulos, and R. Pittau, “Reducing full one-loop amplitudes to scalar integrals at the integrand level,” Nucl. Phys. B 763 (2007) 147–169, arXiv:hep-ph/0609007.
  * [120] D. Forde, “Direct extraction of one-loop integral coefficients,” Phys. Rev. D 75 (2007) 125019, arXiv:0704.1835 [hep-ph].
  * [121] W. B. Kilgore, “One-loop Integral Coefficients from Generalized Unitarity,” arXiv:0711.5015 [hep-ph].
  * [122] S. D. Badger, “Direct Extraction Of One Loop Rational Terms,” JHEP 01 (2009) 049, arXiv:0806.4600 [hep-ph].
  * [123] A. Buonanno and T. Damour, “Effective one-body approach to general relativistic two-body dynamics,” Phys. Rev. D 59 (1999) 084006, arXiv:gr-qc/9811091.
  * [124] A. Buonanno and T. Damour, “Transition from inspiral to plunge in binary black hole coalescences,” Phys. Rev. D 62 (2000) 064015, arXiv:gr-qc/0001013.
  * [125] T. Damour, “Coalescence of two spinning black holes: an effective one-body approach,” Phys. Rev. D 64 (2001) 124013, arXiv:gr-qc/0103018.
  * [126] T. Damour, P. Jaranowski, and G. Schaefer, “Effective one body approach to the dynamics of two spinning black holes with next-to-leading order spin-orbit coupling,” Phys. Rev. D 78 (2008) 024009, arXiv:0803.0915 [gr-qc].
  * [127] E. Barausse and A. Buonanno, “An Improved effective-one-body Hamiltonian for spinning black-hole binaries,” Phys. Rev. D 81 (2010) 084024, arXiv:0912.3517 [gr-qc].
  * [128] M. Khalil, J. Steinhoff, J. Vines, and A. Buonanno, “Fourth post-Newtonian effective-one-body Hamiltonians with generic spins,” Phys. Rev. D 101 no. 10, (2020) 104034, arXiv:2003.04469 [gr-qc].