aainstitutetext: INFN, Sezione di Padova, Via Marzolo 8, I-35131 Padova,
Italy.bbinstitutetext: Dipartimento di Fisica e Astronomia, Università degli
Studi di Padova, Via Marzolo 8, I-35131 Padova, Italy.ccinstitutetext: Max
Planck Institute for Gravitational Physics (Albert Einstein Institute), Am
Mühlenberg 1, Potsdam 14476, Germanyddinstitutetext: Institut für Physik und
IRIS Adlershof, Humboldt-Universit ät zu Berlin, Zum Großen Windkanal 2,
D-12489 Berlin, Germanyeeinstitutetext: Indian Institute of Science Education
and Research Bhopal, Bhopal Bypass Rd, Bhauri, Madhya Pradesh 462066, India.

# Gravitational Spin-Orbit Hamiltonian at NNNLO in the post-Newtonian
framework

Manoj K. Mandal b,a Pierpaolo Mastrolia c,d,e Raj Patil c Jan Steinhoff
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

We present the result of the spin-orbit interaction Hamiltonian for binary
systems of rotating compact objects with generic spins, up to corrections
within the post-Newtonian expansion. The calculation is performed by employing
the effective field theory diagrammatic approach, and it involves Feynman
integrals up to three loops, evaluated within the dimensional regularization
scheme. We apply canonical transformations to eliminate the non-physical
divergences and spurious logarithmic behaviours of the Hamiltonian, and use
the latter to derive the gauge-invariant binding energy and the scattering
angle, in special kinematic regimes.

HU-EP-22/28-RTG

##  1 Introduction

The first gravitational wave (GW), emerging from the collision of two black
holes, which shook the optical systems of the GW observatories
LIGOScientific:2016aoc  can be considered the trigger of a new era in
precision astronomy and cosmology. Since then the LIGO-Virgo-KAGRA
collaboration detected already 90 GW events LIGOScientific:2021djp  and the
worldwide network of ground-based LIGOScientific:2014pky ; VIRGO:2014yos ;
KAGRA:2020agh ; Saleem:2021iwi ; LIGOScientific:2016wof ; Punturo:2010zza  as
well as space based GW detectors LISA  continues to grow, granting access to
an ever broader frequency band and higher sensitivity.

The experimental advances have been boosting the theoretical investigation, by
pushing the application of canonical methods as well as the development of
novel techniques aiming at providing the most accurate description of the
dynamics of coalescing binary systems. Earlier developments dedicated to the
post-Newtonian (PN) Einstein:1938yz ; Einstein:1940mt ; Ohta:1973je ;
Jaranowski:1997ky ; Damour:1999cr ; Blanchet:2000nv ; Damour:2001bu ;
Damour:2014jta ; Jaranowski:2015lha ; Bernard:2016wrg ; Bini:2019nra ;
Bini:2020wpo ; Bini:2020uiq ; Bini:2020nsb ; Bini:2020hmy ; Bini:2020rzn , and
the post-Minkowskian (PM) approximation Bertotti:1956pxu ; Kerr:1959zlt ;
Bertotti:1960wuq ; Portilla:1979xx ; Westpfahl:1979gu ; Portilla:1980uz ;
Bel:1981be ; Westpfahl:1985tsl ; Damour:2016gwp ; Damour:2017zjx , the
gravitational self-force formalism Mino:1996nk ; Quinn:1996am , the
effective–one-body formalism Buonanno:1998gg ; Buonanno:2000ef , as well as
the effective field theory (EFT) approach for the PN formulation of general
relativity Goldberger:2004jt ; Gilmore:2008gq ; Chu:2008xm , have been flanked
by modern Feynman diagrams- Foffa:2011ub ; Foffa:2012rn ; Foffa:2019rdf ;
Foffa:2016rgu ; Foffa:2019yfl ; Blumlein:2020pog ; Foffa:2019ahl ;
Foffa:2019hrb ; Blumlein:2021txe ; Blumlein:2021txj ; Blumlein:2020znm , and
Scattering Amplitudes-based approaches Donoghue:1994dn ; Bjerrum-Bohr:2002gqz
; Neill:2013wsa ; Bjerrum-Bohr:2018xdl ; Cheung:2018wkq ; Kosower:2018adc ;
Bern:2019crd ; Kalin:2019inp ; Kalin:2019rwq ; Kalin:2020fhe ; Kalin:2020mvi ;
Bern:2021dqo ; Bern:2021yeh ; Bjerrum-Bohr:2021din ; Kol:2021jjc ;
Brandhuber:2021eyq ; DiVecchia:2021bdo ; Dlapa:2021npj ; Dlapa:2021vgp , which
turned out to be particularly powerful for the determination of the
perturbative corrections to the classical potential within the EFT approach.

Several studies were also performed to include the spinning motion of the
compact objects in the analysis. Initial studies extending the classical
techniques were done in Tulczyjew:1959 ; Damour:1982 , which was later
extended by Tagoshi:2000zg ; Faye:2006gx ; Damour:2007nc ; Hartung:2011te ;
Hartung:2013dza ; Steinhoff:2008zr ; Steinhoff:2009ei ; Bohe:2012mr ;
Aoude:2022thd ; FebresCordero:2022jts ; Bern:2022kto ; Liu:2021zxr ;
Jakobsen:2022fcj . The spin effects were then also included in the PN
formalism of GR in Porto:2005ac ; Porto:2010tr ; Levi:2010zu ; Levi:2020kvb ;
Porto:2006bt ; Porto:2008tb ; Levi:2008nh ; Porto:2008jj ; Levi:2011eq ;
Levi:2014sba ; Levi:2015ixa ; Kim:2021rfj ; Levi:2020uwu ; Levi:2019kgk ;
Levi:2015msa ; Levi:2014gsa . More recently, quantum scattering amplitudes
involving massive particles of arbitrary spin were used to obtain classical
spin corrections to the two-body effective potential Guevara:2017csg ;
Vines:2018gqi ; Guevara:2018wpp ; Chung:2018kqs ; Guevara:2019fsj ;
Chung:2019duq ; Siemonsen:2019dsu ; Guevara:2020xjx ; Arkani-Hamed:2019ymq .
See Levi:2018nxp ; Porto:2016pyg ; Blanchet:2013haa  for recent reviews and a
more comprehensive reference to literature.

In particular, the PN analysis follows the system during the so-called
inspiral phase, where the components of the binary move at non-relativistic
velocities and their orbital separation is slowly decaying. At this stage, the
non-relativistic regime of the motion allows for a perturbative treatment of
the problem, that can be studied by a series expansion in powers of $v/c$,
where $v$ is the orbital velocity of the compact binary and $c$ is the speed
of light. For bound state systems like binaries of compact objects, in this
phase, the kinetic and potential energy must share the same order of magnitude
according to the virial theorem, namely $v^{2}=G_{N}M/r$, where $G_{N}$ is the
Newton’s constant, $M$ is the typical mass of the system and $r$ is the
separation between the two bodies. From this perspective, the PN approach is
an expansion in two parameters $v$ and $G_{N}$, whose $n^{th}$ order is
referred to as $n$PN (or usually referred to as NnLO), has contributions from
terms proportional to $G_{N}^{l}(v^{2})^{n+1-l}$ where $l=1,2,...,n+1$. The
analysis then is separated in two sectors, the conservative sector, where the
outgoing radiation is ignored and orbits don’t decay, and the radiation
sector, where the emitted radiation is analysed. In the radiative sector, the
emitted radiation also carries energy and angular momentum and, therefore,
back-reacts on the orbital dynamics giving us the so-called tail effects.

The EFT approach is very well suited for this problem because it can be well
separated into three different length scales, namely, the length scale
associated with the compact object $R_{s}$ (Schwarzschild radius), the radius
of the orbit $r$, and the wavelength of the emitted gravitational wave
$\lambda$. Then an EFT at each scale exactly does the job of determining the
essential degrees of freedom. In this method, the determination of any
observable quantity at a given PN is equivalent to the computation of
corresponding scattering amplitudes, which can be systematically computed
through the evaluation of corresponding Feynman diagrams. The leading
contribution to scattering amplitudes is given by tree diagrams, whereas
higher order corrections are determined by diagrams with multiple loops. Tree
diagrams represent rational functions of the kinematic variables (energy,
momenta, and masses of the particles), therefore they are easy to compute.
Loop diagrams, instead, represent very challenging integrals. This means that
computing a scattering amplitude for each new level of precision requires
drawing exponentially more Feynman diagrams and solving a vastly more
complicated mathematical formula.

In this article we focus on the EFT techniques developed in Goldberger:2004jt
; Porto:2005ac  to carry the post-Newtonian analysis in the conservative
sector of the two body problem.

The analysis of compact objects without spin up to the 1PN correction in the
conservative orbital dynamics was reported in Goldberger:2004jt  for the first
time using EFT techniques. Following this, the 2PN was computed in
Gilmore:2008gq ; Chu:2008xm , the 3PN was computed in Foffa:2011ub  and the
4PN was computed in Foffa:2012rn ; Foffa:2019rdf ; Foffa:2016rgu ;
Foffa:2019yfl ; Blumlein:2020pog . The current state of the art for the post-
Newtonian calculation for the conservative potential is the 5PN correction
computed in Foffa:2019hrb ; Blumlein:2021txe . Several partial results of 6PN
Blumlein:2021txj ; Blumlein:2020znm  have been recently added to the
catalogue.

Compact objects are expected to be rotating rapidly and hence the analysis of
the spin becomes important for systems with such objects. The analysis of spin
in the formalism of EFTs was initiated in Porto:2005ac . For systems with
spinning compact objects, the post-Newtonian analysis is usually separated
into different sectors with different powers of the spin variable appearing in
the effective Lagrangian. The spin-orbit sector is analogous to the fine
structure correction obtained to the hydrogen atom, which describe the
interaction between the orbital angular momentum of the binary and the spin of
one of its constituents. Whereas, the quadratic in spin are analogous to the
hyperfine structure corrections to the hydrogen atom and describe the
interaction between each other/self of the spins of the binary constituents.

For the spin-orbit sector, the leading order (LO) effective potential was
first computed in Porto:2005ac . The next-to-leading order (NLO) potential in
Porto:2010tr ; Levi:2010zu , and N2LO in Levi:2015uxa . Partial results for
the N3LO correction were recently presented in Levi:2020kvb . Similarly, for
the quadratic in spin sector, the LO effective potential was also computed in
Porto:2005ac . The NLO in Porto:2006bt ; Porto:2008tb ; Levi:2008nh ;
Porto:2008jj , N2LO in Levi:2011eq ; Levi:2014sba ; Levi:2015ixa  and N3LO was
computed in Kim:2021rfj ; Levi:2020uwu . The computation for cubic and higher
orders in the spin variables can be found in Levi:2019kgk ; Levi:2014gsa ;
Levi:2015msa , whereas the finite size effects are described in detail in
Porto:2008jj ; Levi:2014gsa ; Levi:2015msa . The spin-orbit N3LO was then
completed in Antonelli:2020aeb  using the techniques of first-order self-
force.

In this article, we present the complete conservative N3LO spin-orbit
interaction, using the framework of EFT Goldberger:2004jt ; Porto:2005ac  and
an extension of the diagrammatic approach presented in Foffa:2016rgu . In
particular, we begin with deriving the required Feynman rules and then the
Feynman diagrams:
$1\,(0\text{PN})+4\,(1\text{PN})+21\,(2\text{PN})+130\,(3\text{PN})$ = 156,
for the non-spinning sector;
$2\,(\text{LO})+13\,(\text{NLO})+100\,(\text{N${}^{2}$LO
})+894\,(\text{N${}^{3}$LO })$ = 1009, for the spinning sector, The
corresponding tensor integrals are decomposed in terms of scalar loop
integrals, up to three loops, which are decomposed in terms of a minimal basis
of master integrals (MIs), by means of integration-by-parts (IBP) identities
Chetyrkin:1981qh ; Laporta:2000dsw . The contribution of each diagram to the
scattering amplitude is then obtained after substituting the analytic
expression of the MIs. Finally, the Fourier transform of the amplitude
generates the the effective Lagrangian, later converted into a Hamiltonian
which is the main result of this article. The computational setup has been
fully automated within an in-house Mathematica routine interfaced to QGRAF
NOGUEIRA1993279 , used for the diagram generation, xTensor xAct , for tensor
algebra manipulation, LiteRed Lee:2013mka , for the IBP decomposition,
elaborating on some of the ideas implemented in EFTofPNG Levi:2017kzq .

The effective Lagrangian obtained in this way contain higher order derivatives
of the position and the spin, which are eliminated by employing suitable
coordinate transformations. The EFT Hamiltonian is then obtained by applying
the Legendre transform, and it is found to contains poles in $\epsilon=d-3$
($d$ being the number of the continuous space dimensions), and logarithmic
terms depending on the size of the binary system, which are eliminated by
suitable canonical transformations.

Our novel result for the spin-orbit interaction Hamiltonian are used to derive
the binding energy for circular orbits and the scattering angle, and we are
glad to report that both expressions agree with the results given in
Antonelli:2020aeb .

The paper is organised as follows. In section 2, we review the description of
the spinning binaries within the EFT formalism. In section 3, we present the
computation for the N3LO spin-orbit potential employing the Feynman
diagrammatic approach within the EFT framework. Then, in section 4, we
describe the procedure of removing the residual divergences and logarithms to
derive the EFT Hamiltonian. We provide our main result of the spin-orbit
Hamiltonian up to N3LO in section 5. In section 6, we compute two observable
from the EFT Hamiltonian, namely, the binding energy of the binary system in
circular orbits with aligned spin configuration and scattering angle for two
spinning compact objects with aligned spins. We summarize our main results in
section 7. The article contain three appendices: in appendix A, we describe
the notations used in this article; in appendix B, we provide the required
master integrals and their expression; in appendix C, we provide the
Hamiltonians till 3PN in the non-spinning sector and till NNLO in the spin-
orbit sector.

We provide the required EFT Feynman rules in the ancillary file
Feynman_Rules.m and the analytic results of the Hamiltonian till N3LO in the
ancillary file Hamiltonian.m.

##  2 EFT of spinning objects

In this section, we describe the action for the degrees of freedom of the
gravitational field and the degrees of freedom of the compact objects, namely
their center of mass and their spin. Then we describe the techniques of the
Post-Newtonian formulation of GR in detail and briefly outline the procedure
to compute the effective action.

###  2.1 Action

The dynamics of the gravitational field ($g_{\mu\nu}$) is given by the
Einstein-Hilbert action along with a gauge fixing term,

$\displaystyle S_{\text{EH}}=-\frac{c^{4}}{16\pi G_{N}}\int
d^{4}x\sqrt{g}~{}R[g_{\mu\nu}]+\frac{c^{4}}{32\pi G_{N}}\int
d^{4}x\sqrt{g}~{}g_{\mu\nu}\Gamma^{\mu}\Gamma^{\nu}\,,$ (1)

where $\Gamma^{\mu}=\Gamma^{\mu}_{\rho\sigma}g^{\rho\sigma}$ (in the harmonic
gauge $\Gamma^{\mu}=0$), $\Gamma^{\mu}_{\rho\sigma}$ is the Christoffel
symbol, $G_{N}$ is the Newton’s constant, $R$ is the Ricci scalar, and $g$ is
the determinant of the $g_{\mu\nu}$.

To model the dynamics of the compact object, we utilize worldlines
$x_{(a)}^{\mu}(\tau)$ parametrized by an affine parameter $\tau$ and define
tetrads $\Lambda^{\mu}_{(a)A}(\tau)$ along the worldlines which connects the
body-fixed frame (denoted by upper case Latin indices) of the $a^{\text{th}}$
compact object and the general coordinate frame (denoted by Greek indices).
Then the angular velocity tensor of the spinning object can be defined as

$\displaystyle\Omega_{(a)}^{\mu\nu}=\Lambda^{\mu}_{(a)A}\frac{d\Lambda_{(a)}^{A\nu}}{d\tau}\,,$
(2)

and the corresponding conjugate momenta to the $\Lambda^{\mu}_{(a)A}$ is given
by

$\displaystyle S_{(a)\mu\nu}=-2\frac{\partial
L_{\text{pp}}}{\partial\Omega_{(a)}^{\mu\nu}}\,.$ (3)

Then by demanding the reparameterization invariance of the point particle
action, the dynamics for the compact objects is governed by the worldline
point particle action given by Levi:2015msa ,

$\displaystyle S_{\text{pp}}=\sum_{a=1,2}c\int
d\tau\left(-m_{(a)}\sqrt{u_{(a)}^{2}}-\frac{1}{2}S_{(a)\mu\nu}\Omega_{(a)}^{\mu\nu}-\frac{S_{(a)\mu\nu}u_{(a)}^{\nu}}{u_{(a)}^{2}}\frac{du_{(a)}^{\mu}}{d\tau}+\mathcal{L}_{(a)\text{SI}}\right)\,,$
(4)

which, corresponds to Pryce, Newton, and Wigner gauge for spin supplementarity
condition (SSC) given by
$S_{(a)\mu\nu}(u_{(a)}^{\nu}+\sqrt{u_{(a)}^{2}}\delta^{\nu 0})\approx 0$. In
the above action, the center of mass of the compact object is modeled by the
position of the point particle $x_{(a)}^{\mu}$ and the spin of the compact
object is modeled by $S_{(a)\mu\nu}$. The $\mathcal{L}_{(a)\text{SI}}$ is
given by equation (4.16) of Levi:2015msa  which corresponds to the spin-
induced non minimal couplings that does not contribute up to the N3LO spin-
obit sector at 4.5PN. Here $u_{(a)}^{\mu}=dx_{(a)}^{\mu}/d\tau$ is the four-
velocity and $u_{(a)}^{2}=g_{\mu\nu}u_{(a)}^{\mu}u_{(a)}^{\nu}$, whereas the
proper time $\tau$ is related to the coordinate time $t$ by $d\tau=c~{}dt$.

As we will be performing the computation using the techniques of multi-loop
Feynman diagrams, it is necessary to write the gravitational coupling constant
in $d$ dimensions as

$\displaystyle G_{d}=G_{N}\Big{(}\sqrt{4\pi
e^{\gamma_{E}}}R_{0}\Big{)}^{d-3}\,,$ (5)

where, $R_{0}$ is an arbitrary length scale.

###  2.2 Post-Newtonian formulation of General Relativity

In the bound state of two compact objects, we have three length scales, namely
the length scale associated with the compact object $R_{s}$ (Schwarzschild
radius), the radius of the orbit $r$, and the wavelength of the emitted
gravitational wave $\lambda$. We assume the velocities of the particles to be
small as compared to the velocity of light and the particles are far from each
other, hence propagate on a flat background
($g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$), where the gravitational interaction
between the two particles is governed by the gravitons $h_{\mu\nu}$. Then we
have a hierarchy of length scales

$\displaystyle\lambda\gg r\gg R_{s}\,.$ (6)

As we are only interested in the long-distance physics at the scales of
$\lambda$, we first decompose the graviton fields in short distance modes -
potential gravitons $H_{\mu\nu}$ with scaling
$(k_{0},\textbf{k})\sim(v/r,1/r)$ and long-distance modes - radiation
gravitons $\bar{h}_{\mu\nu}$ with scaling $(k_{0},\textbf{k})\sim(v/r,v/r)$
Goldberger:2004jt .

Noting that $v^{2}\sim 1/r$ for bound orbits due to the virial theorem (or the
third Kepler law), the dimensionless expansion parameter can be taken as
$v^{2}/c^{2}\sim G_{N}M/rc^{2}$, which formally scales as $1/c^{2}$. Hence,
following the majority of the PN literature, we equivalently adopt a formal
expansion in $1/c$ with one PN order corresponding to $1/c^{2}$. For the spin
variables, it holds $S_{(a)}=Gm_{(a)}^{2}\chi_{(a)}/c$ where the dimensionless
spins $\chi_{(a)}$ are at most $\mathcal{O}(1)$ for black holes and
(realistic) neutron stars, so that $S_{(a)}\sim 1/c$. Henceforth we rescale
the spins as $S_{(a)}\rightarrow S_{(a)}/c$ in order to make the PN counting
in $1/c$ manifest.

Now to compute the conservative binding potential of the two-body system, we
ignore the radiation modes and decompose the potential modes in the Kaluza-
Klein (KK) parameterization Kol:2007bc ; Kol:2007rx . In this, the different
components of metric $g_{\mu\nu}$ ($=\eta_{\mu\nu}+H_{\mu\nu}$) are encoded in
three fields, a scalar $\bm{\phi}$, a 3-dimensional vector $\bm{A}_{i}$ and a
3-dimensional symmetric rank two tensor $\bm{\sigma}_{ij}$. The decomposition
is given by,

$g_{\mu\nu}=\begin{pmatrix}e^{2\bm{\phi}/c^{2}}\,\,\,&-e^{2\bm{\phi}/c^{2}}\frac{\bm{A}_{j}}{c^{2}}\\\
-e^{2\bm{\phi}/c^{2}}\frac{\bm{A}_{i}}{c^{2}}\,\,\,\,\,\,&-e^{-2\bm{\phi}/c^{2}}\bm{\gamma}_{ij}+e^{2\bm{\phi}/c^{2}}\frac{\bm{A}_{i}}{c^{2}}\frac{\bm{A}_{j}}{c^{2}}\end{pmatrix}\,,$

where, $\bm{\gamma}_{ij}=\bm{\delta}_{ij}+\bm{\sigma}_{ij}/c^{2}$.

The 2-body effective action is then given by integrating out the gravitational
degrees of freedom from the above derived actions as

$\displaystyle\text{exp}\Big{[}{{\bf i}\int
dt~{}\mathcal{L}_{\text{eff}}}\Big{]}=\int
D\bm{\phi}D\bm{A}_{i}D\bm{\sigma}_{ij}~{}e^{{\bf
i}(S_{\text{EH}}+S_{\text{pp}})}\,,$ (7)

where, $\mathcal{L}_{\rm eff}$ is the effective Lagrangian further decomposed
as

$\mathcal{L}_{\rm eff}=\mathcal{K}_{\rm eff}-\mathcal{V}_{\rm eff}\,,$ (8)

where, $\mathcal{K}_{\rm eff}$ is the kinetic term and $\mathcal{V}_{\rm eff}$
is the effective contribution due to gravitational interactions between the
two objects.

The effective potential can be represented in the form of connected,
classical, 1 particle irreducible (1PI) scattering amplitudes as

$\displaystyle\mathcal{V}_{\text{eff}}={\bf i}\lim_{d\rightarrow
3}\int_{\textbf{p}}e^{{\bf
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}\definecolor{pgffillcolor}{rgb}{0.75,0.75,0.75}\pgfsys@moveto{11.3811pt}{-22.7622pt}\pgfsys@moveto{11.3811pt}{-22.7622pt}\pgfsys@lineto{11.3811pt}{0.0pt}\pgfsys@lineto{34.14331pt}{0.0pt}\pgfsys@lineto{34.14331pt}{-22.7622pt}\pgfsys@closepath\pgfsys@moveto{34.14331pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}{}{{}}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{45.52441pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}{}{{}}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-22.7622pt}\pgfsys@lineto{45.52441pt}{-22.7622pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }\,,$ (9)

where p is the momentum transfer between the two particles and the box diagram
in the above equation refers to all possible Feynman diagrams with gravitons
($\bm{\phi}$, $\bm{A}_{i}$, and $\bm{\sigma}_{ij}$) mediating the
gravitational interaction between the two point particle represented by the
two solid black lines.

Our aim in this article is to compute the spin-orbit effective potential up to
N3LO . For this, we further decompose the kinetic and potential terms as
$\mathcal{K}_{\rm eff}=\mathcal{K}_{\rm pp}+\mathcal{K}_{\rm spin}$ and
$\mathcal{V}_{\rm eff}=\mathcal{V}_{\rm pp}+\mathcal{V}_{\rm spin}$ where
$\mathcal{K}_{\rm pp}$ and $\mathcal{V}_{\rm pp}$ represent the kinetic and
potential terms for center of mass degrees of freedom for the point particle
and $\mathcal{K}_{\rm spin}$ and $\mathcal{V}_{\rm spin}$ represent the
kinetic and potential terms for the spin degrees of freedom. The expression
for the kinetic terms are given by

$\mathcal{K}_{\rm
pp}=\sum_{a=1,2}m_{(a)}\left[\frac{1}{2}\textbf{v}_{a}^{2}+\frac{1}{8}\textbf{v}_{(a)}^{4}\left(\frac{1}{c^{2}}\right)+\frac{1}{16}\textbf{v}_{(a)}^{6}\left(\frac{1}{c^{4}}\right)+\frac{5}{128}\textbf{v}_{(a)}^{8}\left(\frac{1}{c^{6}}\right)\right]+\mathcal{O}\left(\frac{1}{c^{8}}\right)\,,$
(10) $\displaystyle\mathcal{K}_{\rm
spin}=\sum_{a=1,2}\Bigg{\\{}-\frac{1}{2}\textbf{S}_{(a)}^{ij}\bm{\Omega}_{(a)}^{ij}\left(\frac{1}{c}\right)+\textbf{S}_{(a)}^{ij}\textbf{v}_{(a)}^{i}\textbf{a}_{(a)}^{j}\left(\frac{1}{c^{3}}\right)$
$\displaystyle\Bigg{[}\frac{1}{2}+\frac{3}{8}\textbf{v}_{(a)}^{2}\left(\frac{1}{c^{2}}\right)+\frac{5}{16}\textbf{v}_{(a)}^{4}\left(\frac{1}{c^{4}}\right)$
$\displaystyle+\frac{35}{128}\textbf{v}_{(a)}^{6}\left(\frac{1}{c^{6}}\right)\Bigg{]}+\mathcal{O}\left(\frac{1}{c^{11}}\right)\Bigg{\\}}\,,$
(11)

and the decomposition of the potential terms is defined as follows

$\mathcal{V}_{\rm pp}=\mathcal{V}_{\rm
N}+\left(\frac{1}{c^{2}}\right)\mathcal{V}_{\rm
1PN}+\left(\frac{1}{c^{4}}\right)\mathcal{V}_{\rm
2PN}+\left(\frac{1}{c^{6}}\right)\mathcal{V}_{\rm
3PN}+\mathcal{O}\left(\frac{1}{c^{8}}\right)\,,$ (12) $\mathcal{V}_{\rm
spin}=\left(\frac{1}{c^{3}}\right)\left[\mathcal{V}^{SO}_{\rm
LO}+\left(\frac{1}{c^{2}}\right)\mathcal{V}^{SO}_{\rm
NLO}+\left(\frac{1}{c^{4}}\right)\mathcal{V}^{SO}_{\rm
N^{2}LO}+\left(\frac{1}{c^{6}}\right)\mathcal{V}^{SO}_{\rm
N^{3}LO}\right]+\mathcal{O}\left(\frac{1}{c^{11}}\right)\,,$ (13)

where, $\mathcal{V}_{\rm N}$ stands for the Newtonian potential and
$\mathcal{V}_{j}$ with $j={\\{\rm 1PN,\,2PN,\,3PN\\}}$ refers to the
corresponding PN correction for the non-spinning part of the potential. The
$\mathcal{V}^{SO}_{j}$ with $j=\\{\rm LO,\,NLO,\,N^{2}LO,$ $\rm\,N^{3}LO\\}$
refers to the corresponding correction to the spin-orbit coupling of the
binary system. Then our aim in this article would be to compute the
$\mathcal{V}_{j}$ and $\mathcal{V}^{SO}_{j}$ using the techniques of multi-
loop scattering amplitude, as further described in the next sections.

##  3 Computational Algorithm

To obtain the effective potential from the diagrammatic approach as shown in
equation (9), we begin by generating all the relevant generic topologies
contributing at different orders of $G_{N}$. It could be easily seen from the
virial theorem that NnLO has contributions from terms proportional to
$G_{N}^{l}$ where $l=1,2,...,n+1$, and we consider all the topologies at $l-1$
loops contributing to the specific order $l$. So, for the computation of the
N3LO spin-orbit potential, we generate all the topologies till the order
$G_{N}^{4}$ (3-loop) using QGRAF NOGUEIRA1993279 . There is 1 topology at
order $G_{N}$ (tree-level), 2 topologies at order $G_{N}^{2}$ (one-loop), 9
topologies at $G_{N}^{3}$ (two-loop), and 32 topologies at order $G_{N}^{4}$
(three-loop). Then we dress these topologies with the KK field and Feynman
rules derived from the action of PN111The Feynman rules obtained from the
actions in equation (1) and (4) after the KK parameterization are provided in
an ancillary file Feynman_Rules.m with this article. expansion of GR given in
(1) and (4), to obtain all the Feynman diagrams that contribute to the given
order of $G_{N}$ and $v$ depending on the specific perturbation order. The
number of diagrams that contribute at particular order in $1/c$ and of
particular loop topology are given in table 1(a) and 1(b)222While considering
the spin effects, we count only the representative Feynman diagrams, where the
spin can contribute from any of the world-line graviton interaction vertex
present in the diagram. Additionally, the diagrams, which can be obtained from
the change in the label $1\leftrightarrow 2$, are not counted as separate
diagrams..

GravityDiagrams$\longleftrightarrow$$\equiv$Multi-loopDiagrams Figure 1:

Order | Diagrams | Loops | Diagrams
---|---|---|---
0PN | 1 | 0 | 1
1PN | 4 | 1 | 1
0 | 3
2PN | 21 | 2 | 5
1 | 10
0 | 6
3PN | 130 | 3 | 8
2 | 75
1 | 38
0 | 9
(a) Non-spinning sector

Order | Diagrams | Loops | Diagrams
---|---|---|---
LO | 2 | 0 | 2
NLO | 13 | 1 | 8
0 | 5
N2LO | 100 | 2 | 56
1 | 36
0 | 8
N3LO | 894 | 3 | 288
2 | 495
1 | 100
0 | 11
(b) Spin-orbit sector

Table 1: Number of Feynman diagrams contributing different sectors.

Within the EFT framework, the sources remain static and as a result the
generated Feynman diagrams are mapped to two-point multi-loop Feynman diagrams
with mass-less internal lines and an external momentum (the momentum
transferred between two sources) as shown in figure 1. We translate these
Feynman diagrams to their corresponding Feynman amplitudes after performing
the tensor algebra using xTensor xAct . The generic form of the effective
potential corresponding to any $l$-loop Feynman graph $G$ can be expressed as

${\cal V}_{G}^{(l)}=\underbrace{\vphantom{\sum_{\begin{subarray}{c}a\\\
b\end{subarray}}}N^{\mu_{1},\mu_{2},\cdots}_{C~{}~{}~{}\nu_{1},\nu_{2},\cdots}\big{(}x_{(a)},\cdots,S_{(a)},\cdots\big{)}}_{\begin{subarray}{c}\textrm{Coefficient
that depends}\\\ \textrm{on orbital
variables}\end{subarray}}~{}\underbrace{\vphantom{\sum_{\begin{subarray}{c}a\\\
b\end{subarray}}}\int_{p}e^{{\bf
i}p_{\mu}(x_{(1)}-x_{(2)})^{\mu}}N_{F~{}~{}~{}\mu_{1},\mu_{2},\cdots}^{\alpha_{1},\alpha_{2},\cdots}(p)}_{\textrm{Fourier
integral}}~{}\underbrace{\vphantom{\sum_{\begin{subarray}{c}a\\\
b\end{subarray}}}\prod_{i=1}^{l}\int_{k_{i}}\frac{N_{M~{}~{}~{}\alpha_{1},\alpha_{2},\cdots}^{\nu_{1},\nu_{2},\cdots}(k_{i})}{\prod_{\sigma\in
G}D_{\sigma}(p,k_{i})}}_{\textrm{Multi-loop integral}}\,,$ (14)

where,

  1. (i)

$N_{C}$ is a tensor polynomial depending on the world-line coordinates
($x_{(a)}^{\mu}$), the spin tensor ($S_{(a)\mu\nu}$), and their higher-order
time derivatives,

  2. (ii)

$p$ is the momentum transfer between the sources (Fourier momentum),

  3. (iii)

$N_{F}$ is the tensor polynomial built out of momenta $p$,

  4. (iv)

$k_{i}$ are the loop momentums,

  5. (v)

$D_{\sigma}$ denotes the set of denominators corresponding to the internal
lines of $G$,

  6. (vi)

$N_{M}$ stands for a tensor polynomial built out of external momenta $p$ and
loop momenta $k_{i}$.

We perform the reduction of the multi-loop tensor integrals to scalar
integrals by applying a set of projectors exploiting the Lorentz invariance.
We build the projectors depending on the Lorentz invariant external
momentum($p$) and the background metric. After applying the projector the
numerator ($N_{M}$) of the multi-loop integral is transformed as

$N_{M~{}~{}~{}\alpha_{1},\alpha_{2},\cdots}^{\nu_{1},\nu_{2},\cdots}(p,k_{i})\longrightarrow\tilde{N}_{M}(p,k_{i})\tilde{N}_{F~{}~{}~{}\alpha_{1},\alpha_{2},\cdots}^{\nu_{1},\nu_{2},\cdots}(p)\,,$
(15)

where, $\tilde{N}_{M}$ is a polynomial depending on the scalar products
between the external momentum ($p$) and the loop momentums ($k_{i}$). We use
projectors up to rank 6 for the complete evaluation of the N3LO Spin-Orbit
effective potential. The generated scalar integrals are not all independent
and there exist linear relations between these integrals, which are envisaged
via the Integration-By-Parts (IBP) relations. We employ LiteRED Lee:2013mka
to generate these IBP relations to obtain a smaller set of independent scalar
integrals, known as Master integrals (MIs). For the specific computation of
N3LO Spin-Orbit effective potential, we obtain 1 MI at one-loop, 2 MIs at two-
loop, and 3 MIs at three-loop. These MIs are well known and admit closed
analytic expressions in $d$ dimension. We provide the explicit expressions in
B.1 for convenience.

After the evaluation of the multi-loop integral, we perform the Fourier
transform of the tensor polynomials, which have the form

$\int_{p}e^{p_{\mu}(x_{(1)}-x_{(2)})^{\mu}}N_{F~{}~{}~{}\mu_{1},\mu_{2},\cdots}^{\alpha_{1},\alpha_{2},\cdots}(p)\tilde{N}_{F~{}~{}~{}\alpha_{1},\alpha_{2},\cdots}^{\nu_{1},\nu_{2},\cdots}(p)~{}f(p)\,,$
(16)

where, $N_{F}$ and $\tilde{N}_{F}$ are the tensor polynomial built out of
Fourier momenta $p$ and $f(p)$ is a function of the external momentum arising
from the multi-loop integral. The Fourier integrals up to rank 8 are required,
which are obtained by iterative differentiation of the expression given in
section B.2. Then, we obtain the effective potential that only depends on the
orbital variables $\textbf{x}_{(a)}$, $\textbf{S}_{(a)}$, and their higher-
order time derivatives by expanding the complete scattering amplitude as
Laurent series in $\epsilon$ around $d=3$.

Integrand generationMulti-loop methods Topologies Feynman rules Feynman
diagrams Integrands Tensor reduction IBP Reduction to Master integrals Master
integrals Tensor Fourier integrals Effective Lagrangian Laurent expansion
around $d=3$ Figure 2: Flowchart of the computational algorithm

The computation of the effective potential starting from the generation of the
required Feynman diagrams, expressing them in multi-loop integrands,
performing IBP reduction, and then applying the Fourier transformations have
been automated through an in-house code, elaborating on some of the ideas
implemented in EFTofPNG Levi:2017kzq , and using xTensor xAct  for tensor
algebra manipulations as well as successful interface to LiteRed Lee:2013mka ,
Reduze vonManteuffel:2012np , KIRA Maierhofer:2017gsa  for the IBP reduction.
A flow chart for the complete computational algorithm for the effective
potential as implemented in our in-house code is shown in Fig. 2.

##  4 Processing the effective Lagrangian

The effective potential obtained in this way usually contains higher-order
time derivatives of the position ($\textbf{a}_{(a)}$,
$\dot{\textbf{a}}_{(a)}$, $\ddot{\textbf{a}}_{(a)}$,$\cdots$) and the spin
($\dot{\textbf{S}}_{(a)}$, $\ddot{\textbf{S}}_{(a)}$,$\cdots$). In our
computation of the N3LO spin-orbit potential, we have 6th order time
derivative of position and 3rd order of time derivative in spin. We need to
eliminate these higher-order time derivatives from the potential to facilitate
the computation of the Hamiltonian, to obtain gauge-invariant quantities as
well as to pave the way for the implementation within the effective one-body
formalism. Additionally, the potential in the non-spinning sector at 3PN and
the N3LO spin-orbit potential contains poles in the dimensional regularization
parameter $\epsilon$ and logarithmic terms of the form log($r\over R_{0}$). We
need to remove the divergence further to derive the EFT Hamiltonian. Moreover,
as we do not observe any tail effect either at 3PN or at N3LO Spin-orbit
sector, all the logarithms are a remnant of a bad choice of coordinates and
need to be removed by finding a suitable coordinate transformation. First, we
discuss the procedure to eliminate the higher-order time derivatives and then
we show how we remove the divergent pieces and the logarithmic terms to obtain
the EFT Hamiltonian free of any divergences and logarithmic terms.

###  4.1 Elimination of higher-order time derivatives

We briefly describe the procedure to remove all the higher-order time
derivatives, using coordinate transformations Levi:2014sba ; Schafer:1984mr ;
Damour:1990jh ; Damour:1985mt ; Barker:1980spx . The Lagrangian under a
coordinate transformation
$\textbf{x}_{(a)}\rightarrow\textbf{x}_{(a)}+\delta\textbf{x}_{(a)}$ changes
by

$\displaystyle\delta\mathcal{L}=\left(\frac{\delta\mathcal{L}}{\delta\textbf{x}_{(a)}^{i}}\right)~{}\delta\textbf{x}_{(a)}^{i}+\mathcal{O}\left(\delta\textbf{x}_{(a)}^{2}\right)$
(17)

So, when the equation of motion (EOM) is linear in $\textbf{a}_{(a)}$ at LO,
we can construct the $\delta\textbf{x}_{(a)}$ such that all the terms
depending on $\textbf{a}_{(a)}$ drop out in $\mathcal{L}+\delta\mathcal{L}$.
Similarly, for terms involving higher-order time derivatives of
$\textbf{a}_{(a)}$, we can use the modified equations by taking the total time
derivative on the EOM. Following the same approach, when we apply a small
transformation simultaneously to the rotation matrix
$\bm{\Lambda}_{(a)}^{ij}\rightarrow\bm{\Lambda}_{(a)}^{ij}+\delta\bm{\Lambda}_{(a)}^{ij}$
and the spin
$\bm{S}_{(a)}^{ij}\rightarrow\bm{S}_{(a)}^{ij}+\delta\bm{S}_{(a)}^{ij}$, which
relates the co-rotating frame and the local frame of the spinning object, the
local frame transforms as

$\displaystyle\delta\bm{\Lambda}_{(a)}^{ij}=\bm{\Lambda}_{(a)}^{ik}\bm{\omega}_{(a)}^{kj}+\frac{1}{2}\bm{\Lambda}_{(a)}^{ik}\bm{\omega}_{(a)}^{kl}\bm{\omega}_{(a)}^{lj}+\mathcal{O}\left(\bm{\omega}_{(a)}^{3}\right)\
,$ (18)

where the $\bm{\omega}_{(a)}^{kj}$ is the antisymmetric generator of the
rotation matrix,333A generic rotation can be written as a matrix exponential
$e^{\bm{\omega}_{(a)}}$ such that
$\bm{\Lambda}_{(a)}\rightarrow\bm{\Lambda}_{(a)}e^{\bm{\omega}_{(a)}}$ and
$\bm{S}_{(a)}\rightarrow
e^{-\bm{\omega}_{(a)}}\bm{S}_{(a)}e^{\bm{\omega}_{(a)}}$. and similarly the
spin transforms as

$\displaystyle\delta\bm{S}_{(a)}^{ij}=2\bm{S}_{(a)}^{k[i}\bm{\omega}_{(a)}^{j]k}+\bm{S}_{(a)}^{kl}\bm{\omega}_{(a)}^{ik}\bm{\omega}_{(a)}^{jl}+\bm{S}_{(a)}^{k[i}\bm{\omega}_{(a)}^{j]l}\bm{\omega}_{(a)}^{kl}+\mathcal{O}\left(\bm{\omega}_{(a)}^{3}\right)\
.$ (19)

Due to such transformation, the Lagrangian changes by444Here
$V\equiv-\left(\mathcal{L}-\left(-\frac{1}{2}\textbf{S}_{(a)}^{ij}\bm{\Omega}_{(a)}^{ij}\right)\right)$.

$\displaystyle\delta\mathcal{L}=$
$\displaystyle-\left(\frac{1}{c}\right)\frac{1}{2}\dot{\bm{S}}_{(a)}^{ij}\bm{\omega}_{(a)}^{ij}-\left(\frac{1}{c}\right)\frac{1}{2}\bm{S}_{(a)}^{ij}\dot{\bm{\omega}}_{(a)}^{ik}\bm{\omega}_{(a)}^{kj}-\left(\frac{\delta
V}{\delta\bm{S}_{(a)}^{ij}}\right)\delta\bm{S}_{(a)}^{ij}+\mathcal{O}\left(\bm{\omega}_{(a)}^{3},\delta\bm{S}_{(a)}^{2}\right)$
(20)

So, when the above equation is linear in $\dot{\textbf{S}}_{(a)}$ at LO, we
can construct the $\bm{\omega}_{(a)}^{ij}$ such that all the terms depending
on $\dot{\textbf{S}}_{(a)}$ drops out in $\mathcal{L}+\delta\mathcal{L}$. The
higher-order time derivatives of $\dot{\textbf{S}}_{(a)}$ can also be
eliminated by using the modified equations obtained by taking the total time
derivative on the $\bm{\omega}_{(a)}^{ij}$. We apply this procedure
iteratively to eliminate the higher-order time derivatives from the N3LO spin-
orbit potential. Specifically, we perform 4 iterations, where

  1. 1.

we remove the terms with $\textbf{a}_{(a)}$ and its higher-order time
derivatives from the LO and NLO spin-orbit potentials,

  2. 2.

we remove $\dot{\textbf{S}}_{(a)}$ and its higher-order time derivatives from
the NLO spin-orbit potential,

  3. 3.

we remove the $\textbf{a}_{(a)}$ and its higher-order time derivatives from
non-spinning 2PN and 3PN potentials,

  4. 4.

we remove the $\textbf{a}_{(a)}$, $\dot{\textbf{S}}_{(a)}$ and their higher-
order time derivatives from the N2LO and N3LO spin-orbit potentials.

After each iteration, we obtain a new Lagrangian (with new EOM) to be used in
the next iteration. One can check that at each step contributions quadratic in
$\delta\textbf{x}_{(a)}$ and cubic in $\bm{\omega}_{(a)}$ are negligible
(higher order in spin or the PN approximation). Following these steps, we
obtain the effective Lagrangian, which depends on the position, velocity, and
spin only.

###  4.2 Computation of the EFT Hamiltonians

We apply the Legendre transformation on the effective Lagrangian obtained in
the previous step to derive the EFT Hamiltonian

$\displaystyle\mathcal{H}(\textbf{x},\textbf{p},\textbf{S})=\sum_{a=1,2}\textbf{p}_{(a)}^{i}\dot{\textbf{x}}_{(a)}^{i}-\mathcal{L}(\textbf{x},\dot{\textbf{x}},\textbf{S})\
,$ (21)

where the canonical momenta $\textbf{p}^{i}$ is defined as

$\displaystyle\textbf{p}_{(a)}^{i}=\frac{\partial\mathcal{L}(\textbf{x},\dot{\textbf{x}},\textbf{S})}{\partial\dot{\textbf{x}}_{(a)}^{i}}\,.$
(22)

We invert this relation (order by order in $1/c$) to express
$\dot{\textbf{x}}_{(a)}^{i}$ in terms of $\textbf{p}_{(a)}^{i}$. We use this
relation between $\dot{\textbf{x}}_{(a)}^{i}$ and $\textbf{p}_{(a)}^{i}$ in
the equation (21) to obtain the required Hamiltonian
$\mathcal{H}(\textbf{x},\textbf{p},\textbf{S})$

###  4.3 Removal of the poles and logarithms

The Hamiltonian derived in the previous step, and the Lagrangian both contain
divergent pieces and logarithmic terms, which can be removed by finding
suitable coordinate transformations. While using the Lagrangian description,
we can add total time derivative terms with arbitrary coefficients, and fix
these coefficients to a set of values such that the divergent piece drops out
while removing the higher-order time derivatives as described in section 4.1.

On the other hand, in the Hamiltonian description, we can define a canonical
transformation given by 555where the Poisson bracket is defined as
$\\{A,B\\}=\sum_{i=1}^{N}\left(\frac{\partial A}{\partial r_{i}}\frac{\partial
B}{\partial p_{i}}-\frac{\partial A}{\partial p_{i}}\frac{\partial B}{\partial
r_{i}}\right)$ (23)

$\displaystyle\mathcal{H}^{\prime}=\mathcal{H}+\\{\mathcal{H},\mathcal{G}\\}\
,$ (24)

where $\mathcal{G}$ is the infinitesimal generator of the canonical
transformation. Usually, the ansatz for the terms in the generic total time
derivative to be added to the Lagrangian contains a large number of terms.

Here, we remove the poles and the logarithms in the Hamiltonian description by
defining a suitable canonical transformation. We put up an ansatz for the
infinitesimal generator $\mathcal{G}$ with arbitrary coefficients and build a
system of linear equations in these coefficients by demanding the cancellation
of the divergent pieces in $\mathcal{H}^{\prime}$. We use the solution to fix
the arbitrary coefficients to a set of values, thereby obtaining the final
Hamiltonian $\mathcal{H}^{\prime}$ free of poles and logarithms. In the
following sections, we describe the procedures for the removal of the poles
and logarithms from the 3PN non-spinning sector and the Spin-orbit N3LO
sector.

####  4.3.1 3PN non-spinning sector

The 3PN non-spinning Lagrangian has been studied extensively Foffa:2011ub  and
we follow the steps suggested to remove the divergent pieces. Specifically,
following Foffa:2011ub  we add a total derivative term with the complete 3PN
Lagrangian

$\displaystyle\mathcal{L}_{\text{TD}}=\left(\frac{1}{c^{6}}\right)\frac{1}{\epsilon}\frac{d}{dt}\left[\frac{G_{N}^{3}}{r}\Big{(}c_{1}\left(\textbf{v}_{(1)}\cdot\textbf{n}\right)+c_{2}\left(\textbf{v}_{(2)}\cdot\textbf{n}\right)\Big{)}\right]\,,$
(25)

where,

$\displaystyle
c_{1}=\frac{1}{3}\left(4m_{(1)}^{3}m_{(2)}-m_{(1)}m_{(2)}^{3}\right)\quad\quad,\quad\quad
c_{2}=\frac{1}{3}\left(m_{(1)}^{3}m_{(2)}-4m_{(1)}m_{(2)}^{3}\right)\,.$ (26)

We follow the usual procedure of removing $\textbf{a}^{i}$ and
$\dot{\textbf{S}}^{ij}$ and their higher-order time derivatives as described
in Sec. 4.1 including the total derivative term with the 3PN Lagrangian. It
turns out that the divergent pieces drop out and we obtain a finite Lagrangian
till 3PN. Then we remove the terms involving logarithms by finding a canonical
transformation to the Hamiltonian obtained from the above Lagrangian. For this
purpose, we use the following ansatz for the infinitesimal generator,

$\displaystyle\mathcal{G}_{\text{3PN}}=$
$\displaystyle\left(\frac{1}{c^{6}}\right)\frac{G_{N}^{3}}{r^{2}}\left(g_{1}\frac{1}{m_{(1)}}\left(\textbf{p}_{(1)}\cdot\textbf{n}\right)+g_{2}\frac{1}{m_{(2)}}\left(\textbf{p}_{(2)}\cdot\textbf{n}\right)\right)\,,$
(27)

where,

$\displaystyle g_{n}\equiv
g_{Ln}\log\left(\frac{r}{R_{0}}\right)\,\,,\,\,\text{for}\,\,n=1,2\,.$ (28)

We use this generator to build a canonical transformation following Eq. 24. By
requiring the removal of the logarithm terms in the new Hamiltonian, we build
a system of equations and solving them we obtain

$\displaystyle
g_{L1}=m_{(1)}m_{(2)}^{3}-4m_{(1)}^{3}m_{(2)}\quad\quad,\quad\quad
g_{L2}=4m_{(1)}m_{(2)}^{3}-m_{(1)}^{3}m_{(2)}\,.$ (29)

####  4.3.2 Spin-orbit NNNLO sector

For the Spin-orbit Hamiltonian at N3LO , we remove both the divergent terms as
well as terms involving logarithms by finding a suitable canonical
transformation. Inspired by Levi:2014sba , we build the following ansatz for
the infinitesimal generator

$\displaystyle\mathcal{G}_{\rm SO-N^{3}LO}=\left(\frac{1}{c^{9}}\right)$
$\displaystyle\Bigg{\\{}\frac{G_{N}^{3}}{r^{3}}\left[g_{3}\frac{1}{m_{(1)}m_{(2)}}\Big{(}\textbf{S}_{(1)}\cdot(\textbf{p}_{(1)}\times\textbf{p}_{(2)})\Big{)}+g_{4}\frac{1}{m_{(1)}m_{(2)}}\Big{(}\textbf{S}_{(2)}\cdot(\textbf{p}_{(1)}\times\textbf{p}_{(2)})\Big{)}\right]$
$\displaystyle+\frac{G_{N}^{3}}{r^{3}}\Bigg{[}\frac{1}{m_{(1)}}\Big{(}\textbf{S}_{(1)}\cdot(\textbf{p}_{(1)}\times\textbf{n})\Big{)}\left(g_{5}\frac{1}{m_{(1)}}\left(\textbf{p}_{(1)}\cdot\textbf{n}\right)+g_{6}\frac{1}{m_{(2)}}\left(\textbf{p}_{(2)}\cdot\textbf{n}\right)\right)$
$\displaystyle\quad\quad\quad+\frac{1}{m_{(2)}}\Big{(}\textbf{S}_{(1)}\cdot(\textbf{p}_{(2)}\times\textbf{n})\Big{)}\left(g_{7}\frac{1}{m_{(1)}}\left(\textbf{p}_{(1)}\cdot\textbf{n}\right)+g_{8}\frac{1}{m_{(2)}}\left(\textbf{p}_{(2)}\cdot\textbf{n}\right)\right)$
$\displaystyle\quad\quad\quad+\frac{1}{m_{(1)}}\Big{(}\textbf{S}_{(2)}\cdot(\textbf{p}_{(1)}\times\textbf{n})\Big{)}\left(g_{9}\frac{1}{m_{(1)}}\left(\textbf{p}_{(1)}\cdot\textbf{n}\right)+g_{10}\frac{1}{m_{(2)}}\left(\textbf{p}_{(2)}\cdot\textbf{n}\right)\right)$
$\displaystyle\quad\quad\quad+\frac{1}{m_{(2)}}\Big{(}\textbf{S}_{(2)}\cdot(\textbf{p}_{(2)}\times\textbf{n})\Big{)}\left(g_{11}\frac{1}{m_{(1)}}\left(\textbf{p}_{(1)}\cdot\textbf{n}\right)+g_{12}\frac{1}{m_{(2)}}\left(\textbf{p}_{(2)}\cdot\textbf{n}\right)\right)\Bigg{]}\Bigg{\\}}\,,$
(30)

where the arbitrary coefficients $g_{n}$ are defined as

$\displaystyle g_{n}\equiv g_{\epsilon
n}\frac{1}{\epsilon}+g_{Ln}\log\left(\frac{r}{R_{0}}\right)\,,\,\,\text{for}\,\,n=3,4,\cdots
12\,.$ (31)

Following Eqn. 24, we define a canonical transformation using the above ansatz
for the infinitesimal generator and build a system of linear equations by
requiring the removal of the divergent as well as the logarithmic terms. We
solve these set of equations and find the arbitrary coefficients as

$\displaystyle g_{\epsilon
3}=\frac{7\left(51m_{(1)}^{2}m_{(2)}-25m_{(2)}^{3}\right)}{30}\,,$
$\displaystyle g_{\epsilon
4}=\frac{7\left(25m_{(1)}^{3}-51m_{(1)}m_{(2)}^{2}\right)}{30}\,,$
$\displaystyle g_{\epsilon 5}=-\frac{63m_{(1)}^{2}m_{(2)}}{10}\,,$
$\displaystyle g_{\epsilon
6}=-\frac{20m_{(2)}^{3}-47m_{(1)}^{2}m_{(2)}}{2}\,,$ $\displaystyle
g_{\epsilon 7}=-\frac{122m_{(1)}^{2}m_{(2)}-75m_{(2)}^{3}}{10}\,,$
$\displaystyle g_{\epsilon
8}=-\frac{10m_{(1)}^{2}m_{(2)}-77m_{(2)}^{3}}{10}\,,$ $\displaystyle
g_{\epsilon 9}=-\frac{10m_{(1)}m_{(2)}^{2}-77m_{(1)}^{3}}{10}\,,$
$\displaystyle g_{\epsilon
10}=-\frac{122m_{(1)}m_{(2)}^{2}-75m_{(1)}^{3}}{10}\,,$ $\displaystyle
g_{\epsilon 11}=-\frac{20m_{(1)}^{3}-47m_{(1)}m_{(2)}^{2}}{2}\,,$
$\displaystyle g_{\epsilon 12}=-\frac{63m_{(1)}m_{(2)}^{2}}{10}\,,$
$\displaystyle
g_{L3}=\frac{1}{30}\left(525m_{(2)}^{3}-1126m_{(1)}^{2}m_{(2)}\right)\,,$
$\displaystyle
g_{L4}=\frac{1}{30}\left(1126m_{(1)}m_{(2)}^{2}-525m_{(1)}^{3}\right)\,,$
$\displaystyle g_{L5}=\frac{122}{5}m_{(1)}^{2}m_{(2)}\,,$ $\displaystyle
g_{L6}=-2\left(38m_{(1)}^{2}m_{(2)}-15m_{(2)}^{3}\right)\,,$ $\displaystyle
g_{L7}=\frac{3}{10}\left(122m_{(1)}^{2}m_{(2)}-75m_{(2)}^{3}\right)\,,$
$\displaystyle
g_{L8}=\frac{3}{10}\left(10m_{(1)}^{2}m_{(2)}-77m_{(2)}^{3}\right)\,,$
$\displaystyle
g_{L9}=\frac{1}{10}(-3)\left(77m_{(1)}^{3}-10m_{(1)}m_{(2)}^{2}\right)\,,$
$\displaystyle
g_{L10}=\frac{1}{10}(-3)\left(75m_{(1)}^{3}-122m_{(1)}m_{(2)}^{2}\right)\,,$
$\displaystyle g_{L11}=2\left(15m_{(1)}^{3}-38m_{(1)}m_{(2)}^{2}\right)\,,$
$\displaystyle g_{L12}=\frac{122}{5}m_{(1)}m_{(2)}^{2}\,.$ (32)

Following these steps, we ultimately obtain the full effective Hamiltonian
free of divergences and logarithms for the non-spinning sector till 3PN and
for the spin-orbit sector till N3LO .

##  5 Results

To begin, we first define a set of dimensionless variables to express the
Hamiltonian in a compact form. We introduce the total mass
$M=m_{(1)}+m_{(2)}$, the reduced mass of the two body system
$\mu=m_{(1)}m_{(2)}/M$, the mass ratio $q=m_{(1)}/m_{(2)}$, and the symmetric
mass ratio $\nu=\mu/M$. These are related as follows

$\displaystyle\nu=\frac{m_{(1)}m_{(2)}}{M^{2}}=\frac{\mu}{M}=\frac{q}{(1+q)^{2}}\,.$
(33)

Additionally, we express the results in the center of mass (COM) frame of
reference and define the momentum in COM frame as
$\textbf{p}\equiv\textbf{p}_{(1)}=-\textbf{p}_{(2)}$. In the COM, the orbital
angular momentum is defined as $\textbf{L}=(\textbf{r}\times\textbf{p})$. So,
we can write $p^{2}=p_{r}^{2}+L^{2}/r^{2}$, where
$p_{r}=\textbf{p}\cdot\textbf{n}$, $p\equiv|\textbf{p}|$ and
$L\equiv|\textbf{L}|$. All the variables are rescaled to write the expressions
in the form of dimensionless parameters

$\displaystyle\widetilde{\textbf{p}}=\frac{\textbf{p}}{\mu
c}\quad\,,\quad\quad\widetilde{\textbf{r}}=\frac{\textbf{r}~{}c^{2}}{G_{N}M}\quad\,,\quad\quad\widetilde{\textbf{L}}=\frac{\textbf{L}~{}c}{G_{N}M\mu}\quad\,,\quad\quad\widetilde{\textbf{S}}_{(a)}=\frac{\textbf{S}_{(a)}c}{G_{N}M\mu}\quad\,,\quad\quad\widetilde{\mathcal{H}}=\frac{\mathcal{H}}{\mu
c^{2}}\,.$ (34)

The total EFT Hamiltonian can be written as

$\displaystyle\widetilde{\mathcal{H}}=\widetilde{\mathcal{H}}_{\rm
pp}+\widetilde{\mathcal{H}}_{\rm SO}\ ,$ (35)

where,

$\displaystyle\widetilde{\mathcal{H}}_{\rm pp}$
$\displaystyle=\widetilde{\mathcal{H}}_{\text{0PN}}+\left(\frac{1}{c^{2}}\right)\widetilde{\mathcal{H}}_{\text{1PN}}+\left(\frac{1}{c^{4}}\right)\widetilde{\mathcal{H}}_{\text{2PN}}+\left(\frac{1}{c^{6}}\right)\widetilde{\mathcal{H}}_{\text{3PN}}+\mathcal{O}\left(\frac{1}{c^{8}}\right)\
,$ (36) $\displaystyle\widetilde{\mathcal{H}}_{\rm SO}$
$\displaystyle=\left(\frac{1}{c^{3}}\right)\widetilde{\mathcal{H}}_{\text{LO}}+\left(\frac{1}{c^{5}}\right)\widetilde{\mathcal{H}}_{\text{NLO}}+\left(\frac{1}{c^{7}}\right)\widetilde{\mathcal{H}}_{\rm
N^{2}LO}+\left(\frac{1}{c^{9}}\right)\widetilde{\mathcal{H}}_{\rm
N^{3}LO}+\mathcal{O}\left(\frac{1}{c^{11}}\right)\,.$ (37)

In the non-spinning part, the Hamiltonian till 3PN is known in the literature
Foffa:2011ub  and in the spinning sector the Hamiltonian is known till NNLO
Levi:2015uxa . The novel result of the $\tilde{\mathcal{H}}_{\rm N^{3}LO}$ in
the EFT approach in the COM frame is presented here

$\displaystyle\widetilde{\mathcal{H}}_{\rm N^{3}LO}$
$\displaystyle=\frac{\left(\widetilde{\textbf{S}}_{(1)}\cdot\widetilde{\textbf{L}}\right)}{q}\Bigg{\\{}\frac{\left(10864+2025\pi^{2}\right)\nu^{3}}{800\widetilde{r}^{6}}+\frac{\left(17702+2895\pi^{2}\right)\nu^{2}}{960\widetilde{r}^{6}}-\frac{181\nu}{8\widetilde{r}^{6}}$
$\displaystyle\quad\quad\quad\quad\quad+\widetilde{p}_{r}^{2}\left(\frac{\left(53153+4050\pi^{2}\right)\nu^{3}}{400\widetilde{r}^{5}}+\frac{\left(60989+225\pi^{2}\right)\nu^{2}}{480r^{5}}-\frac{10\nu}{\widetilde{r}^{5}}\right)$
$\displaystyle\quad\quad\quad\quad\quad+\widetilde{p}_{r}^{4}\left(-\frac{263\nu^{4}}{32\widetilde{r}^{4}}-\frac{29\nu^{3}}{\widetilde{r}^{4}}+\frac{741\nu^{2}}{32\widetilde{r}^{4}}-\frac{13\nu}{4\widetilde{r}^{4}}\right)+\widetilde{p}_{r}^{6}\left(\frac{425\nu^{4}}{32\widetilde{r}^{3}}-\frac{745\nu^{3}}{64\widetilde{r}^{3}}+\frac{223\nu^{2}}{64\widetilde{r}^{3}}-\frac{45\nu}{128\widetilde{r}^{3}}\right)$
$\displaystyle\quad\quad\quad\quad\quad+\widetilde{L}^{2}\bigg{(}-\frac{3\left(-412+675\pi^{2}\right)\nu^{3}}{800\widetilde{r}^{7}}+\frac{\left(59116-225\pi^{2}\right)\nu^{2}}{1920\widetilde{r}^{7}}-\frac{85\nu}{8\widetilde{r}^{7}}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+\widetilde{p}_{r}^{2}\left(-\frac{17\nu^{4}}{4\widetilde{r}^{6}}-\frac{4257\nu^{3}}{64\widetilde{r}^{6}}+\frac{2253\nu^{2}}{32\widetilde{r}^{6}}-\frac{13\nu}{2\widetilde{r}^{6}}\right)$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+\widetilde{p}_{r}^{4}\left(\frac{339\nu^{4}}{32\widetilde{r}^{5}}-\frac{1479\nu^{3}}{64\widetilde{r}^{5}}+\frac{597\nu^{2}}{64\widetilde{r}^{5}}-\frac{135\nu}{128\widetilde{r}^{5}}\right)\bigg{)}$
$\displaystyle\quad\quad\quad\quad\quad+\widetilde{L}^{4}\bigg{(}-\frac{29\nu^{4}}{32\widetilde{r}^{8}}-\frac{789\nu^{3}}{64\widetilde{r}^{8}}+\frac{163\nu^{2}}{16\widetilde{r}^{8}}-\frac{13\nu}{4\widetilde{r}^{8}}+\widetilde{p}_{r}^{2}\left(\frac{153\nu^{4}}{32\widetilde{r}^{7}}-\frac{1053\nu^{3}}{64\widetilde{r}^{7}}+\frac{525\nu^{2}}{64\widetilde{r}^{7}}-\frac{135\nu}{128\widetilde{r}^{7}}\right)\bigg{)}$
$\displaystyle\quad\quad\quad\quad\quad+\widetilde{L}^{6}\left(\frac{29\nu^{4}}{32\widetilde{r}^{9}}-\frac{249\nu^{3}}{64\widetilde{r}^{9}}+\frac{151\nu^{2}}{64\widetilde{r}^{9}}-\frac{45\nu}{128\widetilde{r}^{9}}\right)\Bigg{\\}}$
$\displaystyle+\left(\widetilde{\textbf{S}}_{(1)}\cdot\widetilde{\textbf{L}}\right)\Bigg{\\{}\frac{\left(10864+2025\pi^{2}\right)\nu^{3}}{800\widetilde{r}^{6}}+\frac{\left(-85957+8400\pi^{2}\right)\nu^{2}}{3600\widetilde{r}^{6}}-\frac{34\nu}{\widetilde{r}^{6}}$
$\displaystyle\quad\quad\quad\quad\quad+\widetilde{p}_{r}^{2}\left(\frac{81\left(1451+100\pi^{2}\right)\nu^{3}}{800\widetilde{r}^{5}}-\frac{\left(52076+5175\pi^{2}\right)\nu^{2}}{1200\widetilde{r}^{5}}-\frac{37\nu}{\widetilde{r}^{5}}\right)$
$\displaystyle\quad\quad\quad\quad\quad+\widetilde{p}_{r}^{4}\left(-\frac{263\nu^{4}}{32\widetilde{r}^{4}}-\frac{599\nu^{3}}{16\widetilde{r}^{4}}+\frac{561\nu^{2}}{32\widetilde{r}^{4}}-\frac{3\nu}{\widetilde{r}^{4}}\right)+\widetilde{p}_{r}^{6}\left(\frac{2025\nu^{4}}{128\widetilde{r}^{3}}-\frac{325\nu^{3}}{32\widetilde{r}^{3}}+\frac{153\nu^{2}}{128\widetilde{r}^{3}}\right)$
$\displaystyle\quad\quad\quad\quad\quad+\widetilde{L}^{2}\bigg{(}+\frac{\left(5461-2025\pi^{2}\right)\nu^{3}}{800\widetilde{r}^{7}}+\frac{\left(490976+5175\pi^{2}\right)\nu^{2}}{4800\widetilde{r}^{7}}+\frac{11\nu}{\widetilde{r}^{7}}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+\widetilde{p}_{r}^{2}\left(-\frac{17\nu^{4}}{4\widetilde{r}^{6}}-\frac{2677\nu^{3}}{32\widetilde{r}^{6}}+\frac{2115\nu^{2}}{32\widetilde{r}^{6}}-\frac{2\nu}{\widetilde{r}^{6}}\right)+\widetilde{p}_{r}^{4}\left(\frac{1671\nu^{4}}{128\widetilde{r}^{5}}-\frac{471\nu^{3}}{32\widetilde{r}^{5}}+\frac{375\nu^{2}}{128\widetilde{r}^{5}}\right)\bigg{)}$
$\displaystyle\quad\quad\quad\quad\quad+\widetilde{L}^{4}\left(-\frac{29\nu^{4}}{32\widetilde{r}^{8}}-\frac{327\nu^{3}}{16\widetilde{r}^{8}}-\frac{71\nu^{2}}{32\widetilde{r}^{8}}+\frac{\nu}{\widetilde{r}^{8}}+\widetilde{p}_{r}^{2}\left(\frac{777\nu^{4}}{128\widetilde{r}^{7}}-\frac{387\nu^{3}}{32\widetilde{r}^{7}}+\frac{471\nu^{2}}{128\widetilde{r}^{7}}\right)\right)$
$\displaystyle\quad\quad\quad\quad\quad+\widetilde{L}^{6}\left(\frac{151\nu^{4}}{128\widetilde{r}^{9}}-\frac{101\nu^{3}}{32\widetilde{r}^{9}}+\frac{109\nu^{2}}{128\widetilde{r}^{9}}\right)\Bigg{\\}}$
$\displaystyle+(1\leftrightarrow 2)\,.$ (38)

Here, $(1\leftrightarrow 2)$ implies changing the label $1$ and $2$ on the
spin variables along with $q\leftrightarrow 1/q$.

The computation of the previously known Hamiltonians, together with the novel
computation of the $\tilde{\mathcal{H}}_{\rm N^{3}LO}$, allows us to obtain
the complete expression for the spin-orbit N3LO Hamiltonian within the EFT
framework. We provide the known Hamiltonians in appendix C. The Hamiltonian in
the generic frame is also provided in the ancillary file Hamiltonian.m with
this article.

##  6 Computation of observables with spin

The obtained Hamiltonian is still gauge dependent as it depends on the radial
coordinate, we compute gauge invariant observable to compare with the
literature Antonelli:2020aeb . In this section, we describe the procedure for
the computation of two gauge invariant observable: the binding energy for
circular orbit with aligned spin configuration and the scattering angle with
aligned spin configuration.

For this, we work in the COM defined by $\textbf{p}_{(1)}+\textbf{p}_{(2)}=0$
as given in section 5. With this, we assume that the spins are aligned to the
direction of the orbital angular momentum of the binary system. Such aligned
spin configuration is given by

$\displaystyle\textbf{S}_{(a)}\cdot\textbf{r}=\textbf{S}_{(a)}\cdot\textbf{p}=0\implies\textbf{S}_{(a)}\cdot(\textbf{r}\times\textbf{p})=S_{(a)}L\,,$
(39)

where, $L=|\textbf{L}|$ and $S_{(a)}=|\textbf{S}_{(a)}|$.

###  6.1 Binding energy for circular orbits with aligned spins

We compute the gauge invariant relation between the binding energy and the
orbital frequency for circular orbits by removing the dependence on the radial
coordinate. For circular orbits, we know

$\displaystyle
p_{r}=0\quad\quad\quad\text{and}\quad\quad\quad\frac{dp_{r}}{dt}=0\,,$ (40)

which implies

$\frac{\partial\widetilde{\mathcal{H}}(\widetilde{r},\widetilde{L},\widetilde{S}_{(a)})}{\partial\widetilde{r}}=0\,.$
(41)

We invert the above relation to express $\widetilde{r}$ as a function of
$\widetilde{L}$ and we define the orbital frequency as

$\displaystyle\widetilde{\omega}=\frac{\partial\widetilde{\mathcal{H}}(\widetilde{L},\widetilde{S}_{(a)})}{\partial\widetilde{L}}\,.$
(42)

We invert the above relation again to express $\widetilde{L}$ as a function of
$\widetilde{\omega}$. Additionally, we define a gauge invariant PN parameter
$x=\widetilde{\omega}^{2/3}$ . Using the above two equations, we express
$\tilde{L}$ as a function of $x$, and substituting this in the Hamiltonian, we
obtain the gauge invariant binding energy of the circular orbits in the
aligned spin configuration $\tilde{E}(x,\tilde{S}_{(a)})$. Following the above
procedure with the Hamiltonian given in section 5 we obtain

$\displaystyle
E(x,\widetilde{S}_{(a)})=E_{\text{pp}}(x)+E_{\text{SO}}(x,\widetilde{S}_{(a)})\,,$
(43)

where,

$\displaystyle E_{\text{pp}}(x)$
$\displaystyle=-x\frac{1}{2}+x^{2}\left\\{\frac{3}{8}+\frac{\nu}{24}\right\\}+x^{3}\left\\{\frac{27}{16}-\frac{19}{16}\nu+\frac{1}{48}\nu^{2}\right\\}$
$\displaystyle+x^{4}\left\\{\frac{675}{128}+\left(-\frac{34445}{1152}+\frac{205\pi^{2}}{192}\right)\nu+\frac{155}{192}\nu^{2}+\frac{35}{10368}\nu^{3}\right\\}\,,$
(44)

and

$\displaystyle E_{\text{SO}}(x,\widetilde{S})$
$\displaystyle=x^{5/2}\Bigg{\\{}S^{*}\left(-\nu\right)+S\left(-\frac{4}{3}\nu\right)\Bigg{\\}}$
$\displaystyle+x^{7/2}\Bigg{\\{}S^{*}\left(-\frac{3}{2}\nu+\frac{5}{3}\nu^{2}\right)+S\left(-4\nu+\frac{31}{18}\nu^{2}\right)\Bigg{\\}}$
$\displaystyle+x^{9/2}\Bigg{\\{}S^{*}\left(-\frac{27}{8}\nu+\frac{39}{2}\nu^{2}-\frac{5}{8}\nu^{3}\right)+S\left(-\frac{27}{2}\nu+\frac{211}{8}\nu^{2}-\frac{7}{12}\nu^{3}\right)\Bigg{\\}}$
$\displaystyle+x^{11/2}\Bigg{\\{}S^{*}\left(-\frac{135}{16}\nu+\frac{565}{8}\nu^{2}-\frac{1109}{24}\nu^{3}-\frac{25}{324}\nu^{4}\right)$
$\displaystyle\quad\quad\quad+S\left(-45\nu+\left(\frac{19679}{144}+\frac{29\pi^{2}}{24}\right)\nu^{2}-\frac{1979}{36}\nu^{3}-\frac{265}{3888}\nu^{4}\right)\Bigg{\\}}\,,$
(45)

$S=\widetilde{S}_{(1)}+\widetilde{S}_{(2)}$, and
$S^{\star}=\widetilde{S}_{(1)}/q+\widetilde{S}_{(2)}q$. Our result given in
equation (6.1) agrees with the result given in equation (10) of
Antonelli:2020aeb .

###  6.2 Scattering angle with aligned spins

In this section, we study the scattering angle considering an aligned spin
binary system following Vines:2018gqi . First, we re-scale the spin variables
as $a_{(a)}=S_{(a)}/(m_{(a)}c)$. In the COM, the Hamiltonian $\mathcal{H}$ is
expressed as a function of $p_{r}$, $L$, $r$, and $S{(a)}$ and inverting that
we obtain

$\displaystyle p_{r}=p_{r}(\mathcal{H},L,r,S_{(a)})\,.$ (46)

Then the scattering angle $\chi$ is given by

$\displaystyle\chi(\mathcal{H},L,S_{(a)})=-\int dr\frac{\partial
p_{r}(\mathcal{H},L,r,S_{(a)})}{\partial L}-\pi\,.$ (47)

Now, the total center of mass energy ($E$) and the total energy per total rest
mass ($\Gamma$) is given by

$\displaystyle\mathcal{H}=E$ $\displaystyle=Mc^{2}\sqrt{1+2\nu(\gamma-1)}\,,$
(48) $\displaystyle\Gamma$
$\displaystyle=\frac{\mathcal{H}}{Mc^{2}}=\sqrt{1+2\nu(\gamma-1)}\,,$ (49)

where $\gamma$ is the Lorentz factor. We invert the above relation to express
the Lorentz factor $\gamma$ in terms of $\Gamma$, and we obtain

$\displaystyle\gamma=\frac{1}{\sqrt{1-v^{2}/c^{2}}}=1+\frac{\Gamma^{2}-1}{2\nu}\,,$
(50)

where, $v\equiv|\dot{\textbf{r}}|$ is the relative velocity of the compact
objects. Moreover, the total angular momentum $L$ can be expressed in terms of
the impact parameter $b$ and the aligned spin configurations $a_{(+)}$ and
$a_{(-)}$ as

$\displaystyle L=\frac{\mu\gamma
vb}{\Gamma}+Mc\left(\frac{\Gamma-1}{2}\right)\left(a_{+}-\frac{\delta}{\Gamma}a_{-}\right)\,,$
(51)

where, $\delta=(m_{(1)}-m_{(2)})/M$, $a_{(+)}=a_{(1)}+a_{(2)}$ and
$a_{(-)}=a_{(1)}-a_{(2)}$. We use equation (48) and (50) to trade $H$ for $v$
and using equation (51) we trade $L$ for $b$. This allows us to express the
scattering angle as

$\displaystyle\chi(v,b,S_{(a)})=-\frac{\gamma}{\mu\gamma v}\int
dr\frac{\partial p_{r}(v,b,r,S_{(a)})}{\partial b}-\pi\,.$ (52)

Now, applying the above procedure with the Hamiltonian given in section 5, we
obtain the scattering angle computed in the COM for aligned spins, which can
be expressed as

$\displaystyle\chi(v,b,S_{(a)})=\chi_{\text{pp}}(v,b)+\chi_{\text{SO}}(v,b,S_{(a)})\,,$
(53)

where,

$\displaystyle\frac{\chi_{\text{pp}}}{\Gamma}$
$\displaystyle=\left(\frac{G_{N}M}{v^{2}b}\right)\left\\{2+2\left(\frac{v^{2}}{c^{2}}\right)+\mathcal{O}\left(\frac{v^{8}}{c^{8}}\right)\right\\}$
$\displaystyle+\pi\left(\frac{G_{N}M}{v^{2}b}\right)^{2}\left\\{3\left(\frac{v^{2}}{c^{2}}\right)+\frac{3}{4}\left(\frac{v^{4}}{c^{4}}\right)+\mathcal{O}\left(\frac{v^{8}}{c^{8}}\right)\right\\}$
$\displaystyle+\left(\frac{G_{N}M}{v^{2}b}\right)^{3}\left\\{-\frac{2}{3}+2\frac{15-\nu}{3}\left(\frac{v^{2}}{c^{2}}\right)+\frac{60-13\nu}{2}\left(\frac{v^{4}}{c^{4}}\right)+\frac{40-277\nu}{12}\left(\frac{v^{6}}{c^{6}}\right)+\mathcal{O}\left(\frac{v^{8}}{c^{8}}\right)\right\\}$
$\displaystyle+\pi\left(\frac{G_{N}M}{v^{2}b}\right)^{4}\left\\{15\frac{7-2\nu}{3}\left(\frac{v^{4}}{c^{4}}\right)+\left(\frac{105}{4}-\frac{437}{8}\nu+\frac{123}{128}\pi^{2}\nu\right)\left(\frac{v^{6}}{c^{6}}\right)+\mathcal{O}\left(\frac{v^{8}}{c^{8}}\right)\right\\}$
$\displaystyle+\mathcal{O}(G_{N}^{5})\,,$ (54)

and

$\displaystyle\frac{\chi_{\text{SO}}}{\Gamma}=\frac{v}{bc}\begin{bmatrix}a_{(+)}&~{}\delta
a_{(-)}\end{bmatrix}\cdot$
$\displaystyle\Bigg{(}\left(\frac{G_{N}M}{v^{2}b}\right)\left\\{\begin{bmatrix}-4\\\
0\end{bmatrix}+\mathcal{O}\left(\frac{v^{8}}{c^{8}}\right)\right\\}$
$\displaystyle+\pi\left(\frac{G_{N}M}{v^{2}b}\right)^{2}\left\\{-\frac{1}{2}\begin{bmatrix}7\\\
1\end{bmatrix}-\frac{3}{4}\begin{bmatrix}7\\\
1\end{bmatrix}\left(\frac{v^{2}}{c^{2}}\right)+\mathcal{O}\left(\frac{v^{8}}{c^{8}}\right)\right\\}$
$\displaystyle+\left(\frac{G_{N}M}{v^{2}b}\right)^{3}\bigg{\\{}-2\begin{bmatrix}5\\\
1\end{bmatrix}-20\begin{bmatrix}5-\nu/2\\\
1\end{bmatrix}\left(\frac{v^{2}}{c^{2}}\right)-10\begin{bmatrix}5-77\nu/20\\\
1\end{bmatrix}\left(\frac{v^{4}}{c^{4}}\right)$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+\frac{1}{4}\begin{bmatrix}177\nu\\\
0\end{bmatrix}\left(\frac{v^{6}}{c^{6}}\right)+\mathcal{O}\left(\frac{v^{8}}{c^{8}}\right)\bigg{\\}}$
$\displaystyle+\pi\left(\frac{G_{N}M}{v^{2}b}\right)^{4}\bigg{\\{}\frac{3}{4}\begin{bmatrix}-91+13\nu\\\
-21+\nu\end{bmatrix}\left(\frac{v^{2}}{c^{2}}\right)-\frac{1}{8}\begin{bmatrix}1365-777\nu\\\
315-45\nu\end{bmatrix}\left(\frac{v^{4}}{c^{4}}\right)$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad-\frac{1}{32}\begin{bmatrix}1365-\left(\frac{23717}{3}-\frac{733}{8}\pi^{2}\right)\nu\\\
315-\left(\frac{257}{3}+\frac{251}{8}\pi^{2}\right)\nu\end{bmatrix}\left(\frac{v^{6}}{c^{6}}\right)+\mathcal{O}\left(\frac{v^{8}}{c^{8}}\right)\bigg{\\}}\Bigg{)}$
$\displaystyle+\mathcal{O}(G_{N}^{5})\,.$ (55)

Our result given in equation (6.2) agrees with the result given in equation
(7) of Antonelli:2020aeb . We note that the N3LO spin-orbit scattering angle
contains all gauge-invariant information of the corresponding Lagrangian or
Hamiltonian Antonelli:2020ybz , albeit restricted to the center-of-mass frame.

##  7 Conclusion

In this work, we presented the complete evaluation of N3LO Post-Newtonian (PN)
correction to the the spin-orbit Hamiltonian for rapidly rotating compact
objects, within the effective field theory diagrammatic approach of General
Relativity.

The required Feynman diagrams in momentum space were automatically generated
using EFT Feynman rules for the interaction of spinning compact objects. They
were algebraically decomposed in terms of master integrals, by means of
integration by parts identities for dimensionally regulated integrals, whose
expressions were available in the literature. The contribution of each diagram
to the Lagrangian, namely to the effective potential, was found after taking
the Fourier transform to position space, and series expanding around
$d=3+\epsilon$ space dimension. The Hamiltonian is finally found by means of
Legendre transform, followed by suitable canonical transformations, required
for eliminating residual non-physical divergences and spurious logarithmic
behaviours.

We computed the gauge invariant relation between binding energy and the
orbital frequency in the circular orbit with aligned spins as well as the the
scattering angle with aligned spins and they were found in agreement with the
results available in the literature, previously obtained using the self-force
formalism.

Our results complete the description of coalescing rapidly rotating binary
systems up to 4.5 PN order within the spin-orbit sector.

The techniques and the in-house code developed for the current project are
very flexible, and can be applied to other interesting problems, at higher
order in the PN expansion, for both non-spinning and spinning compact objects.

##### Note added

During the completion of this work, a similar study has appeared in
Kim:2022pou .

### Acknowledgements

We would like to thank Jonathan Ronca and William J. Torres Bobadilla for
partial checks on the non-spinning sector and interesting discussions
regarding the reduction methods for Feynman integrals. The work of M.K.M is
supported by Fellini - Fellowship for Innovation at INFN funded by the
European Union’s Horizon 2020 research and innovation programme under the
Marie Skłodowska-Curie grant agreement No 754496. RP is grateful to IISER
Bhopal for the fellowship. RP’s research is funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation), Projektnummer
417533893/GRK2575 “Rethinking Quantum Field Theory”.

##  Appendix A Notation and convention

Spacetime metric $\displaystyle\eta_{\mu\nu}=(1,-1,-1,-1)$ (56a) 4 dimensional
indices $\displaystyle\mu,\nu$ (56b) 3 dimensional indices $\displaystyle i,j$
(56c) Compact object label ${}_{(a)}\quad\textrm{where }a=\\{1,2\\}$ (56d)
Time derivative $\displaystyle\dot{}$ (56e) Position of $a^{\text{th}}$ object
$\displaystyle\textbf{x}_{(a)}$ (56f) Velocity of $a^{\text{th}}$ object
$\displaystyle\textbf{v}_{(a)}\equiv\dot{\textbf{x}}_{(a)}$ (56g) Acceleration
of $a^{\text{th}}$ object
$\displaystyle\textbf{a}_{(a)}\equiv\ddot{\textbf{x}}_{(a)}$ (56h) Separation
vector for binary
$\displaystyle\textbf{r}\equiv\textbf{x}_{(a)}-\textbf{x}_{(a)}$ (56i)
Separation distance for binary $\displaystyle r\equiv|\textbf{r}|$ (56j)
Separation unit vector for binary
$\displaystyle\textbf{n}\equiv\frac{\textbf{r}}{r}$ (56k) Angular momentum of
the binary $\displaystyle\textbf{L}\equiv(\textbf{r}\times\textbf{p})$ (56l)
Spin vector of $a^{\text{th}}$ object
$\displaystyle\textbf{S}_{(a)}^{i}\equiv\bm{\epsilon}^{ijk}\textbf{S}_{(a)}^{jk}$
(56m) $\displaystyle\int_{p}\quad\quad$
$\displaystyle\int\frac{d^{d}p}{(2\pi)^{d}}$ (56n)
$\displaystyle\int_{\textbf{p}}\quad\quad$
$\displaystyle\int\frac{d^{3}p}{(2\pi)^{3}}$ (56o) Center of mass coordinates
$\displaystyle\textbf{p}_{(1)}+\textbf{p}_{(2)}=0$ (56p) Circular orbits
$\displaystyle
p_{r}\equiv\textbf{p}\cdot\textbf{n}=0\quad\quad\textrm{and}\quad\quad\dot{p}_{r}=0$
(56q) Aligned spins
$\displaystyle\textbf{S}_{(a)}\cdot\textbf{r}=\textbf{S}_{(a)}\cdot\textbf{p}=0$
(56r)

##  Appendix B Relevant Integrals

In this appendix, we present all the integrals used in the computation of
results given in section 5. We first give the master integrals at different
loops that are used in the multi-loop methods. Then we give the Fourier
integral used to obtain the effective potential as described computational
algorithm described in figure 2.

###  B.1 Master Integrals

(a) $M_{1,1}$ Figure 3: One loop master integrals

$\displaystyle M_{1,1}$
$\displaystyle=\int_{k_{1}}\frac{1}{k_{1}^{2}}\frac{1}{(k_{1}-p)^{2}}$ (57)
$\displaystyle=(4\pi)^{-\frac{d}{2}}p^{d-4}\frac{\Gamma\left(2-\frac{d}{2}\right)\Gamma\left(\frac{d}{2}-1\right)^{2}}{\Gamma(d-2)}$
(58)

(a) $M_{2,1}$

(b) $M_{2,2}$

Figure 4: Two loop master integrals

$\displaystyle M_{2,1}$
$\displaystyle=\int_{k_{1},k_{2}}\frac{1}{k_{1}^{2}}\frac{1}{(k_{1}-p)^{2}}\frac{1}{(k_{1}+k_{2})^{2}}\frac{1}{(k_{1}+k_{2}-p)^{2}}$
(59)
$\displaystyle=4\,(4\pi)^{-d}p^{2d-8}\frac{1}{(d-4)^{2}(d-3)^{2}}\frac{\Gamma\left(3-\frac{d}{2}\right)^{2}\Gamma\left(\frac{d}{2}-1\right)^{4}}{\Gamma(d-3)^{2}}$
(60) $\displaystyle M_{2,2}$
$\displaystyle=\int_{k_{1},k_{2}}\frac{1}{k_{1}^{2}}\frac{1}{k_{2}^{2}}\frac{1}{(k_{1}+k_{2}-p)^{2}}$
(61)
$\displaystyle=4\,(4\pi)^{-d}p^{2d-6}\frac{1}{(d-4)(d-3)(3d-10)(3d-8)}\frac{\Gamma(5-d)\Gamma\left(\frac{d}{2}-1\right)^{3}}{\Gamma\left(\frac{3d}{2}-5\right)}$
(62)

(a) $M_{3,1}$

(b) $M_{3,2}$

(c) $M_{3,3}$

Figure 5: Three loop master integrals

$\displaystyle M_{3,1}$
$\displaystyle=\int_{k_{1},k_{2},k_{3}}\frac{1}{k_{1}^{2}}\frac{1}{k_{2}^{2}}\frac{1}{k_{3}^{2}}\frac{1}{(k_{1}+k_{2}+k_{3}-p)^{2}}$
(63)
$\displaystyle=(4\pi)^{-\frac{3d}{2}}p^{3d-8}\frac{\Gamma\left(4-\frac{3d}{2}\right)\Gamma\left(\frac{d}{2}-1\right)^{4}}{\Gamma(2d-4)}$
(64) $\displaystyle M_{3,2}$
$\displaystyle=\int_{k_{1},k_{2},k_{3}}\frac{1}{k_{1}^{2}}\frac{1}{k_{2}^{2}}\frac{1}{(k_{1}-k_{3})^{2}}\frac{1}{(k_{1}+k_{2}-p)^{2}}\frac{1}{(k_{1}-k_{3}-p)^{2}}$
(65)
$\displaystyle=(4\pi)^{-\frac{3d}{2}}p^{3d-10}\frac{\Gamma(3-d)\Gamma\left(2-\frac{d}{2}\right)\Gamma\left(\frac{d}{2}-1\right)^{5}}{\Gamma(d-2)\Gamma\left(\frac{3d}{2}-3\right)}$
(66) $\displaystyle M_{3,3}$
$\displaystyle=\int_{k_{1},k_{2},k_{3}}\frac{1}{k_{1}^{2}}\frac{1}{k_{2}^{2}}\frac{1}{k_{3}^{2}}\frac{1}{(k_{1}+k_{2}+k_{3})^{2}}\frac{1}{(k_{1}+k_{3}-p)^{2}}$
(67)
$\displaystyle=(4\pi)^{-\frac{3d}{2}}p^{3d-10}\frac{\Gamma\left(5-\frac{3d}{2}\right)\Gamma\left(2-\frac{d}{2}\right)^{2}\Gamma\left(\frac{d}{2}-1\right)^{5}\Gamma\left(\frac{3d}{2}-4\right)}{\Gamma(4-d)\Gamma(d-2)^{2}\Gamma(2d-5)}$
(68)

###  B.2 Fourier Integrals

The expression for the scalar Fourier integral is given by

$\displaystyle\int_{k}e^{{\bf
i}k^{\mu}x_{\mu}}\left(k^{\mu}k_{\mu}\right)^{-\alpha}=\frac{2^{-2\alpha}\pi^{-\frac{d}{2}}\Gamma\left(\frac{d}{2}-\alpha\right)\left(x^{\mu}x_{\mu}\right)^{\alpha-\frac{d}{2}}}{\Gamma(\alpha)}\,.$
(69)

The required tensor Fourier integrals are then obtained by taking derivatives
on both sides with respect to $x^{\mu}$.

##  Appendix C Lower-order Hamiltonians

In this appendix we give the results for all the Hamiltonians given in eq.(36)
and eq.(37).

###  C.1 Non-spinning sector up to 3PN

$\displaystyle\widetilde{\mathcal{H}}_{\text{0PN}}=-\frac{1}{\widetilde{r}}+\frac{1}{2}\widetilde{p}_{r}^{2}+\widetilde{L}^{2}\left(\frac{1}{2\widetilde{r}^{2}}\right)$
(70)

$\displaystyle\widetilde{\mathcal{H}}_{\text{1PN}}$
$\displaystyle=\frac{1}{2\widetilde{r}^{2}}+\widetilde{p}_{r}^{2}\left(-\frac{\nu}{\widetilde{r}}-\frac{3}{2\widetilde{r}}\right)+\widetilde{p}_{r}^{4}\left(\frac{3\nu}{8}-\frac{1}{8}\right)$
$\displaystyle+\widetilde{L}^{2}\left(-\frac{\nu}{2\widetilde{r}^{3}}-\frac{3}{2\widetilde{r}^{3}}+\widetilde{p}_{r}^{2}\left(\frac{3\nu}{4\widetilde{r}^{2}}-\frac{1}{4\widetilde{r}^{2}}\right)\right)+\widetilde{L}^{4}\left(\frac{3\nu}{8\widetilde{r}^{4}}-\frac{1}{8\widetilde{r}^{4}}\right)$
(71)

$\displaystyle\widetilde{\mathcal{H}}_{\text{2PN}}$
$\displaystyle=-\frac{\nu}{4\widetilde{r}^{3}}-\frac{1}{2\widetilde{r}^{3}}+\widetilde{p}_{r}^{2}\left(\frac{9\nu}{2\widetilde{r}^{2}}+\frac{9}{4\widetilde{r}^{2}}\right)+\widetilde{p}_{r}^{4}\left(-\frac{\nu^{2}}{\widetilde{r}}-\frac{5\nu}{2\widetilde{r}}+\frac{5}{8\widetilde{r}}\right)+\widetilde{p}_{r}^{6}\left(\frac{5\nu^{2}}{16}-\frac{5\nu}{16}+\frac{1}{16}\right)$
$\displaystyle+\widetilde{L}^{2}\left(\frac{3\nu}{\widetilde{r}^{4}}+\frac{11}{4\widetilde{r}^{4}}+\widetilde{p}_{r}^{2}\left(-\frac{\nu^{2}}{\widetilde{r}^{3}}-\frac{9\nu}{2\widetilde{r}^{3}}+\frac{5}{4\widetilde{r}^{3}}\right)+\widetilde{p}_{r}^{4}\left(\frac{15\nu^{2}}{16\widetilde{r}^{2}}-\frac{15\nu}{16\widetilde{r}^{2}}+\frac{3}{16\widetilde{r}^{2}}\right)\right)$
$\displaystyle+\widetilde{L}^{4}\left(-\frac{3\nu^{2}}{8\widetilde{r}^{5}}-\frac{2\nu}{\widetilde{r}^{5}}+\frac{5}{8\widetilde{r}^{5}}+\widetilde{p}_{r}^{2}\left(\frac{15\nu^{2}}{16\widetilde{r}^{4}}-\frac{15\nu}{16\widetilde{r}^{4}}+\frac{3}{16\widetilde{r}^{4}}\right)\right)+\widetilde{L}^{6}\left(\frac{5\nu^{2}}{16\widetilde{r}^{6}}-\frac{5\nu}{16\widetilde{r}^{6}}+\frac{1}{16\widetilde{r}^{6}}\right)$
(72)

$\displaystyle\widetilde{\mathcal{H}}_{\text{3PN}}$
$\displaystyle=\frac{\left(269-72\pi^{2}\right)\nu}{72\widetilde{r}^{4}}+\frac{11}{8\widetilde{r}^{4}}+\widetilde{p}_{r}^{2}\left(-\frac{5\nu^{2}}{2\widetilde{r}^{3}}-\frac{\left(6368+207\pi^{2}\right)\nu}{288\widetilde{r}^{3}}-\frac{3}{4\widetilde{r}^{3}}\right)+\widetilde{p}_{r}^{4}\left(\frac{475\nu^{2}}{48\widetilde{r}^{2}}+\frac{383\nu}{48\widetilde{r}^{2}}-\frac{25}{16\widetilde{r}^{2}}\right)$
$\displaystyle+\widetilde{p}_{r}^{6}\left(-\frac{\nu^{3}}{\widetilde{r}}-\frac{3\nu^{2}}{\widetilde{r}}+\frac{21\nu}{8\widetilde{r}}-\frac{7}{16\widetilde{r}}\right)+\widetilde{p}_{r}^{8}\left(\frac{35\nu^{3}}{128}-\frac{35\nu^{2}}{64}+\frac{35\nu}{128}-\frac{5}{128}\right)$
$\displaystyle+\widetilde{L}^{2}\Bigg{(}-\frac{5\nu^{2}}{4\widetilde{r}^{5}}+\frac{\left(207\pi^{2}-112\right)\nu}{576\widetilde{r}^{5}}-\frac{21}{4\widetilde{r}^{5}}+\widetilde{p}_{r}^{2}\left(\frac{97\nu^{2}}{8\widetilde{r}^{4}}+\frac{119\nu}{8\widetilde{r}^{4}}-\frac{27}{8\widetilde{r}^{4}}\right)$
$\displaystyle\quad\quad\quad\quad+\widetilde{p}_{r}^{4}\left(-\frac{3\nu^{3}}{2\widetilde{r}^{3}}-\frac{8\nu^{2}}{\widetilde{r}^{3}}+\frac{59\nu}{8\widetilde{r}^{3}}-\frac{21}{16\widetilde{r}^{3}}\right)+\widetilde{p}_{r}^{6}\left(\frac{35\nu^{3}}{32\widetilde{r}^{2}}-\frac{35\nu^{2}}{16\widetilde{r}^{2}}+\frac{35\nu}{32\widetilde{r}^{2}}-\frac{5}{32\widetilde{r}^{2}}\right)\Bigg{)}$
$\displaystyle+\widetilde{L}^{4}\Bigg{(}+\frac{65\nu^{2}}{16\widetilde{r}^{6}}+\frac{121\nu}{16\widetilde{r}^{6}}-\frac{29}{16\widetilde{r}^{6}}+\widetilde{p}_{r}^{2}\left(-\frac{9\nu^{3}}{8\widetilde{r}^{5}}-\frac{115\nu^{2}}{16\widetilde{r}^{5}}+\frac{55\nu}{8\widetilde{r}^{5}}-\frac{21}{16\widetilde{r}^{5}}\right)$
$\displaystyle\quad\quad\quad\quad+\widetilde{p}_{r}^{4}\left(\frac{105\nu^{3}}{64\widetilde{r}^{4}}-\frac{105\nu^{2}}{32\widetilde{r}^{4}}+\frac{105\nu}{64\widetilde{r}^{4}}-\frac{15}{64\widetilde{r}^{4}}\right)\Bigg{)}$
$\displaystyle+\widetilde{L}^{6}\left(-\frac{5\nu^{3}}{16\widetilde{r}^{7}}-\frac{35\nu^{2}}{16\widetilde{r}^{7}}+\frac{17\nu}{8\widetilde{r}^{7}}-\frac{7}{16\widetilde{r}^{7}}+\widetilde{p}_{r}^{2}\left(\frac{35\nu^{3}}{32\widetilde{r}^{6}}-\frac{35\nu^{2}}{16\widetilde{r}^{6}}+\frac{35\nu}{32\widetilde{r}^{6}}-\frac{5}{32\widetilde{r}^{6}}\right)\right)$
$\displaystyle+\widetilde{L}^{8}\left(\frac{35\nu^{3}}{128\widetilde{r}^{8}}-\frac{35\nu^{2}}{64\widetilde{r}^{8}}+\frac{35\nu}{128\widetilde{r}^{8}}-\frac{5}{128\widetilde{r}^{8}}\right)$
(73)

###  C.2 Spin-orbit sector up to N2LO

$\displaystyle\widetilde{\mathcal{H}}_{\text{LO}}=\frac{\left(\widetilde{\textbf{S}}_{(1)}\cdot\widetilde{\textbf{L}}\right)}{q}\left\\{\frac{3\nu}{2\widetilde{r}^{3}}\right\\}+\left(\widetilde{\textbf{S}}_{(1)}\cdot\widetilde{\textbf{L}}\right)\left\\{\frac{2\nu}{\widetilde{r}^{3}}\right\\}+(1\leftrightarrow
2)$ (74)

$\displaystyle\widetilde{\mathcal{H}}_{\text{NLO}}$
$\displaystyle=\frac{\left(\widetilde{\textbf{S}}_{(1)}\cdot\widetilde{\textbf{L}}\right)}{q}\left\\{-\frac{5\nu^{2}}{4\widetilde{r}^{4}}-\frac{5\nu}{\widetilde{r}^{4}}+\widetilde{p}_{r}^{2}\left(\frac{17\nu^{2}}{4\widetilde{r}^{3}}-\frac{5\nu}{8\widetilde{r}^{3}}\right)+\widetilde{L}^{2}\left(\frac{5\nu^{2}}{4\widetilde{r}^{5}}-\frac{5\nu}{8\widetilde{r}^{5}}\right)\right\\}$
$\displaystyle+\left(\widetilde{\textbf{S}}_{(1)}\cdot\widetilde{\textbf{L}}\right)\left\\{-\frac{5\nu^{2}}{4\widetilde{r}^{4}}-\frac{6\nu}{\widetilde{r}^{4}}+\widetilde{p}_{r}^{2}\left(\frac{43\nu^{2}}{8\widetilde{r}^{3}}\right)+\widetilde{L}^{2}\left(\frac{13\nu^{2}}{8\widetilde{r}^{5}}\right)\right\\}$
$\displaystyle+(1\leftrightarrow 2)$ (75)

$\displaystyle\widetilde{\mathcal{H}}_{\rm N^{2}LO}$
$\displaystyle=\frac{\left(\widetilde{\textbf{S}}_{(1)}\cdot\widetilde{\textbf{L}}\right)}{q}\Bigg{\\{}\frac{41\nu^{2}}{8\widetilde{r}^{5}}+\frac{21\nu}{2\widetilde{r}^{5}}+\widetilde{p}_{r}^{2}\left(-\frac{63\nu^{3}}{16\widetilde{r}^{4}}-\frac{399\nu^{2}}{16\widetilde{r}^{4}}+\frac{27\nu}{8\widetilde{r}^{4}}\right)+\widetilde{p}_{r}^{4}\left(\frac{131\nu^{3}}{16\widetilde{r}^{3}}-\frac{7\nu^{2}}{2\widetilde{r}^{3}}+\frac{7\nu}{16\widetilde{r}^{3}}\right)$
$\displaystyle\quad\quad\quad\quad\quad+\widetilde{L}^{2}\left(\widetilde{p}_{r}^{2}\left(\frac{73\nu^{3}}{16\widetilde{r}^{5}}-\frac{11\nu^{2}}{2\widetilde{r}^{5}}+\frac{7\nu}{8\widetilde{r}^{5}}\right)-\frac{17\nu^{3}}{16\widetilde{r}^{6}}-\frac{133\nu^{2}}{16\widetilde{r}^{6}}+\frac{27\nu}{8\widetilde{r}^{6}}\right)+\widetilde{L}^{4}\left(\frac{17\nu^{3}}{16\widetilde{r}^{7}}-\frac{2\nu^{2}}{\widetilde{r}^{7}}+\frac{7\nu}{16\widetilde{r}^{7}}\right)\Bigg{\\}}$
$\displaystyle+\left(\widetilde{\textbf{S}}_{(1)}\cdot\widetilde{\textbf{L}}\right)\Bigg{\\{}\frac{25\nu^{2}}{4\widetilde{r}^{5}}+\frac{13\nu}{\widetilde{r}^{5}}+\widetilde{p}_{r}^{2}\left(-\frac{63\nu^{3}}{16\widetilde{r}^{4}}-\frac{123\nu^{2}}{4\widetilde{r}^{4}}+\frac{3\nu}{\widetilde{r}^{4}}\right)+\widetilde{p}_{r}^{4}\left(\frac{10\nu^{3}}{\widetilde{r}^{3}}-\frac{17\nu^{2}}{8\widetilde{r}^{3}}\right)$
$\displaystyle\quad\quad\quad\quad\quad+\widetilde{L}^{2}\left(\widetilde{p}_{r}^{2}\left(\frac{23\nu^{3}}{4\widetilde{r}^{5}}-\frac{19\nu^{2}}{8\widetilde{r}^{5}}\right)-\frac{17\nu^{3}}{16\widetilde{r}^{6}}-\frac{197\nu^{2}}{16\widetilde{r}^{6}}-\frac{\nu}{\widetilde{r}^{6}}\right)+\widetilde{L}^{4}\left(\frac{11\nu^{3}}{8\widetilde{r}^{7}}-\frac{19\nu^{2}}{16\widetilde{r}^{7}}\right)\Bigg{\\}}$
$\displaystyle+(1\leftrightarrow 2)$ (76)

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