aainstitutetext: Dipartimento di Fisica e Astronomia, Università degli Studi
di Padova, Via Marzolo 8, I-35131 Padova, Italy.bbinstitutetext: INFN, Sezione
di Padova, Via Marzolo 8, I-35131 Padova, Italy.ccinstitutetext: Mani L.
Bhaumik Institute for Theoretical Physics, University of California at Los
Angeles, Los Angeles, CA 90095, USA.ddinstitutetext: Max Planck Institute for
Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, Potsdam
D-14476, Germanyeeinstitutetext: Institut für Physik und IRIS Adlershof,
Humboldt-Universit ät zu Berlin, Zum Großen Windkanal 2, D-12489 Berlin,
Germany

#  Renormalizing Love: tidal effects at the third post-Newtonian order

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

We present the conservative effective two-body Hamiltonian at the third order
in the post-Newtonian expansion with gravitoelectric quadrupolar dynamical
tidal-interactions. Our derivation of the effective two-body Lagrangian is
based on the diagrammatic effective field theory approach and it involves
Feynman integrals up to three loops, which are evaluated within the
dimensional regularization scheme. The elimination of the divergent terms
occurring in the effective Lagrangian requires the addition of counterterms to
ensure finite observables, thereby introducing a renormalization group flow to
the post-adiabatic Love number. As a limiting case of the renormalized
dynamical effective Hamiltonian, we also derive the effective Hamiltonian for
adiabatic tides, and, in this regime, calculate the binding energy for a
circular orbit, and the scattering angle in a hyperbolic scattering.

HU-EP-23/43-RTG

##  1 Introduction

A new era in astronomy and cosmology has begun with the recent gravitational
wave (GW) detections by the LIGO-Virgo-KAGRA collaboration
LIGOScientific:2021djp . 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:2017pwj  continues to grow, and will grant access to an ever broader
frequency band with higher sensitivity.

Among the most significant sources of GWs are the neutron star (NS) binaries
LIGOScientific:2017vwq ; LIGOScientific:2017ync ; LIGOScientific:2020aai ,
which provide insights into the physics of dense nuclear matter within these
stars. In a binary system, a NS develops a quadrupole moment due to the tidal
interaction with its companion Flanagan:2007ix . The imprint of such tidal
interactions was observed in the GW signal GW170817 LIGOScientific:2017vwq
and led to constraints on the underlying NS equation of state (EOS)
LIGOScientific:2018hze ; LIGOScientific:2018cki ; Chatziioannou:2020pqz ;
Pradhan:2022rxs . These tidal interactions also give rise to oscillation modes
of NS McDermott:1983ApJ…268..837M ; McDermott:1988ApJ…325..725M ;
christensen1998lecture ; Kokkotas:1999bd ; Andersson:2021qdq , in particular,
the f-mode dynamical tides Will:1983vlw , which have been argued to be
important in inferring the NS EOS Pratten:2021pro  in upcoming observing runs
of present GW detectors. Neglecting the dynamical tidal effects can introduce
significant biases in the estimation of the tidal deformability, consequently
impacting the accuracy of the EOS inferences. Moreover, the inclusion of
dynamical tides is essential to improve the agreement between GW models and
numerical relativity simulations Steinhoff:2016rfi ; Hinderer:2016eia ;
Andersson:2019dwg ; Schmidt:2019wrl  and it is necessary to fulfill the
scientific objectives of the next generation ground-based GW observatories
Punturo:2010zza ; Reitze:2019iox ; Ackley:2020atn . In this article, we aim to
model the effects of such dynamic oscillations of NS on the dynamics of the
binary system.

We begin by considering the relativistic Lagrangian developed in Ref.
Steinhoff:2016rfi  for the quadrupole $f$-mode oscillation of a NS coupled to
an external tidal field, describing the dynamical tidal (DT) effects,111 This
representation of the Lagrangian is not unique, as it is subject to field
redefinition: for instance, the last term could be equivalently replaced with
a redefined $\lambda$ along with terms involving $(E_{\mu\nu})^{2}$ and $({\rm
d}E_{\mu\nu}/{\rm d}\tau)^{2}$.

$\displaystyle\mathcal{L}_{\rm
DT}=\frac{z}{4\lambda\omega_{f}^{2}}\left[\frac{c^{2}}{z^{2}}\frac{{\rm
d}Q_{\mu\nu}}{{\rm d}\tau}\frac{{\rm d}Q^{\mu\nu}}{{\rm
d}\tau}-\omega_{f}^{2}Q_{\mu\nu}Q^{\mu\nu}\right]-\frac{z}{2}Q^{\mu\nu}E_{\mu\nu}-\kappa_{d}\frac{G_{d}^{2}m^{2}}{c^{6}}\frac{z}{2}Q^{\mu\nu}\frac{{\rm
d}^{2}E_{\mu\nu}}{{\rm d}\tau^{2}}\,,$ (1)

where $\omega_{f}$ is the frequency of the mode, $\lambda$ is the tidal
deformability Flanagan:2007ix , $Q_{\mu\nu}$ is a symmetric trace-free
tensor222We also impose a supplementary condition $Q_{\mu\nu}u^{\mu}=0$ on the
quadrupole to project out the unphysical degrees of freedom. that models the
relativistic quadrupole moment of the star, $u^{\mu}={\rm d}x^{\mu}/{\rm
d}\tau$ is the four-velocity,
$E_{\mu\nu}=-c^{2}R_{\mu\alpha\nu\beta}u^{\alpha}u^{\beta}/z^{2}$ is the
gravitoelectric field and $z=\sqrt{u^{2}}$ is the redshift factor. We work
with the gravitational constant in $(d+1)$ spacetime dimensions written as
$G_{d}=(\sqrt{4\pi\exp(\gamma_{\rm E})}\,R)^{\epsilon}\,G_{N}$. The last term
in the above equation is the first non-minimal coupling that starts
contributing to the conservative sector from 3PN order and
$\kappa_{d}=(\sqrt{4\pi\exp(\gamma_{\rm E})}\,R)^{-2\epsilon}\kappa$ is the
post-adiabatic Love number in $(d+1)$ spacetime dimensions. We express $G_{d}$
and $\kappa_{d}$ in this particular form because later on we will employ the
modified minimal subtraction scheme Collins:1984xc , and hence the appearance
of the $4\pi$, the Euler-Mascheroni constant $\gamma_{\rm E}$, and the
(arbitrary) lenghtscale $R$. In the adiabatic limit Hinderer:2007mb ;
Binnington:2009bb ; Damour:2009vw , where $\omega_{f}\rightarrow\infty$, the
tides do not oscillate independently and are instead locked to the external
tidal field as

$\displaystyle{Q}^{\mu\nu}=-\lambda{E}^{\mu\nu}-\lambda\kappa_{d}\frac{G_{d}^{2}m^{2}}{c^{6}}\frac{{\rm
d}^{2}{E}^{\mu\nu}}{{\rm d}\tau^{2}}\,.$ (2)

The first term is the leading order term in the small frequency expansion of
the conservative response function of the quadrupole Chakrabarti:2013xza ,
when sourced by external tidal field and the second term is the next-to-
leading-order correction of it, known as the post-adiabatic term. Substituting
the above equation in Eq. (1), we obtain the Lagrangian for (post-)adiabatic
tides (AT), which is given by,

$\displaystyle\mathcal{L}_{\rm
AT}=\frac{z\lambda}{4}E_{\mu\nu}E^{\mu\nu}+\lambda\kappa_{d}\frac{G_{d}^{2}m^{2}}{c^{6}}\frac{z}{2}E_{\mu\nu}\frac{{\rm
d}^{2}E^{\mu\nu}}{{\rm d}\tau^{2}}\,.$ (3)

We can also write a similar adiabatic Lagrangian for the higher multipole
moments which are studied in Ref. Bini:2012gu . See also Refs. Thorne:1984mz ;
Zhang:1986cpa ; Damour:1990pi ; Binnington:2009bb ; Damour:2009wj ;
Damour:2009vw ; Henry:2019xhg . In general relativity, in addition to the
relativistic gravitoelectric tides, we also get a new sector of
gravitomagnetic tides Favata:2005da ; Landry:2015cva ; Pani:2018inf ;
Banihashemi:2018xfb ; Poisson:2020eki ; Poisson:2020mdi ; Poisson:2020ify ;
Gupta:2020lnv  that are coupled to the odd-parity normal modes of the NS,
modeled by the current-type multipole moments. For the adiabatic limit of the
gravitomagnetic sector, see Ref. Bini:2012gu . In a systematic EFT
construction of dynamical tides, further couplings should be included, as
outlined in Gupta:2020lnv , and applied to gravitomagnetic tides, which we
leave for future work.

In this article, we quantify the influence of the dynamic tides on the
behavior of a compact binary system. To achieve this goal, we employ effective
field theory (EFT) techniques Goldberger:2004jt  to analyze the inspiral phase
of the binary, which occurs when its components are moving at nonrelativistic
velocities and the orbital separation is large as compared to the length scale
associated to each compact object. In this context, we apply a perturbative
approach that involves a series expansion based on powers of $v/c$, where $v$
represents the orbital velocity of the binary and $c$ is the speed of light.
The virial theorem dictates that, the kinetic energy is ($-1/2$) times the
potential energies of a bound state system. Thus, we set up the _post-
Newtonian_ (PN) analysis, expanding in two perturbative parameters: $v/c$ and
$G_{N}$, where $G_{N}$ denotes Newton’s constant. Here, terms of the form
$(v/c)^{n}$ are referred to as being of $(n/2)$PN order. The PN analysis of
the binary dynamics can be categorized into two sectors: the conservative
sector, where emitted radiation is neglected and the orbital separation
remains constant, and the radiative sector, where radiation carries away
energy and momentum. At higher PN orders, these sectors can interact due to
tail effects, originating from radiation being scattered by the orbital
background curvature and affecting the orbital dynamics (see, e.g., Ref.
Blanchet:2013haa .) By employing the EFT approach, we can determine
observables at any given PN order using diagrammatic methods, first proposed
in Goldberger:2004jt  and modern integration methods Kol:2013ega ;
Foffa:2016rgu , turning the problem into the determination of scattering
amplitudes, which can be systematically obtained through the calculation of
corresponding Feynman diagrams (see Refs. Porto:2016pyg ; Levi:2018nxp ;
Goldberger:2022ebt , for reviews). Following the same computational strategy
as developed in Mandal:2022nty ; Mandal:2022ufb ; Mandal:2023lgy , in order to
compute the effective Lagrangian, we make use of an automated in-house code,
interfaced to QGRAF NOGUEIRA1993279 , for the diagram generation, to xTensor
xAct , for tensor algebra manipulation, and to LiteRed Lee:2013mka , for the
integral decomposition.

The effective Hamiltonian for conservative dynamical gravitoelectric tides was
first computed in Refs. Vines:2010ca ; Steinhoff:2016rfi  up to 1PN and then
was extended up to 2PN in Ref. Mandal:2023lgy . The effects of spin and tides
were analyzed together in Refs. Gupta:2020lnv ; Steinhoff:2021dsn ;
Gupta:2023oyy  for gravitomagnetic tides. In the adiabatic limit, the 2PN
effective Hamiltonian was computed in Refs. Bini:2012gu ; Henry:2019xhg  for
both gravitoelectric and gravitomagnetic tides. Other works in PN theory can
be found in Refs. Vines:2011ud ; Bini:2012gu ; Vines:2010ca ;
Abdelsalhin:2018reg ; Banihashemi:2018xfb ; Landry:2018bil . In the post-
Minkowskian (PM) expansion, where the perturbative series is controlled by
$G_{N}$ alone, the adiabatic tidal corrections were studied to 3PM order in
Refs. Jakobsen:2022psy ; Kalin:2020lmz . See also Refs. Kalin:2020mvi ;
Bini:2020flp ; Cheung:2020sdj ; Haddad:2020que ; Cheung:2020gbf ; Bern:2020uwk
; Mougiakakos:2022sic . Adiabatic tidal effects where also included in
effective-one-body waveform models Buonanno:1998gg ; Buonanno:2000ef  in Refs.
Bini:2012gu ; Damour:2009wj ; Baiotti:2011am ; Bernuzzi:2012ci ; Bini:2014zxa
and in Refs. Hinderer:2016eia ; Steinhoff:2016rfi ; Steinhoff:2021dsn  for the
case of dynamical tides. In this paper, we extend the state-of-the art of the
_analytic calculations of dynamical and adiabatic gravitoelectric tides in the
conservative sector to 3PN order_.

Within the diagrammatic EFT approach, higher-order perturbative corrections to
the scattering amplitudes may contain divergent contributions. By adopting the
dimensional regularization scheme, the divergent terms of multi-loop Feynman
integrals are parameterised by poles in $\epsilon=(d-3)$, where $d$ is the
continuous number of space-time dimension. The so-called ultraviolet (UV)
divergences can be absorbed in the redefinition of the free parameters of the
theory, known as renormalization, thus yielding finite results. The process of
renormalization can be understood as a coarse-graining transformation on the
system PhysRevB.4.3174 . One can understand the effects of renormalization in
classical context using the Kadanoff’s block-spin transformations
Kadanoff:1966wm . See Ref. Solodukhin:1997xn ; Goldberger:2001tn  for
renormalization in classical field theories coupled to external spatial
sources.

Besides the UV divergences, the computation of the conservative potential for
a binary system may also involve infrared (IR) divergences. In the point
particle sector, starting from the 4PN order, we encounter IR divergences in
the conservative sector, which are due the artificial separation of
conservative and dissipative dynamics Foffa:2019yfl . However, in the
computation of 3PN dynamical tides, we encounter only UV divergences. Part of
these divergent terms can be removed by adding counterterms based on the
world-line operators, which eventually can be removed with field-
redefinitions. This procedure is equivalent to finding a suitable canonical
transformation (in the Hamiltonian picture) or adding a total derivative (in
the Lagrangian picture) Foffa:2019yfl . We dub these divergences as spurious
divergences as they do not have any effect on any physical observables.
Additionally, we encounter UV divergences that are not spurious, which can be
eliminated by the renormalization procedure, namely by introducing a new
counterterm in the point particle Lagrangian, eventually yielding finite
results. This counterterm affects the source coupling of the quadrupole moment
with the gravitons. Hence, the renormalization procedure introduces the
running of the post-adiabatic Love number $\kappa$.

Furthermore, we note that the same counterterm can also remove the UV
divergences arising from the radiative sector Blanchet:1997jj ;
Goldberger:2009qd . The appearance of an identical divergent term in both the
conservative and radiative sector was observed in Ref. Goldberger:2004jt  just
for the case of spurious divergences, which can be removed through field
redefinition completely.

Analogously, in the recent computation of 4PM observables due to scalar
interactions Barack:2023oqp , the renormalization of one of the scalar tidal
Love numbers was necessary to obtain finite results. Renormalization and
running for conservative and dissipative Love numbers was also seen in Ref.
Saketh:2023bul  by determining the counter terms in the tidal response using
BHPT (black-hole perturbation theory) for the case of Kerr black holes. In
this article, _we present the renormalization of post-adiabatic Love number as
a consequence of genuine divergences appearing in the conservative sector of
the gravitational two-body system._

The paper is organized as follows. In Section 2, we review the description of
tidally-interacting binaries in the EFT formalism. In Section 3, we present
the algorithm used to compute the 3PN dynamic tidal potential. In Section 4,
we present the procedure of renormalization required for the post-adiabatic
Love number to get a finite interaction Hamiltonian. Our main result, the
effective dynamical tidal Hamiltonian (38), is presented in Section 5. In
Section 6, we consider the adiabatic limit, and derive an effective adiabatic
tidal Hamiltonian. We compute two gauge-independent observables: (i) the
binding energy of a circular binary and (ii) the scattering angle for the
hyperbolic encounter of two stars. Finally, we present our conclusions and
avenues for future work in Section 7. This work is supplemented with three
ancillary files: Hamiltonian-DT.m, containing the analytic expression of the
Hamiltonian for the dynamic tides, Hamiltonian-AT.m, containing the analytic
expression of the Hamiltonian for the adiabatic tides and Poincare_Algebra.m,
containing the result for the center of mass of the system which completes the
Poincaré algebra and, hence, validates our results.

Notation – The mostly negative signature for the metric is employed. Bold-face
characters are used for three-dimensional variables, and normal-face font, for
four-dimensional variables. The subscript ${(a)}$ labels the binary components
on all the corresponding variables, like their position $\bm{x}_{(a)}$ and
quadrupole moment $\bm{Q}_{(a)}$. An overdot indicates the time derivative,
e.g., $\bm{v}_{(a)}=\dot{\bm{x}}_{(a)}$ is the velocity,
$\bm{a}_{(a)}=\ddot{\bm{x}}_{(a)}$ the acceleration and $\dot{{\bm{Q}}}={\rm
d}{\bm{Q}}/{\rm d}t$. The separation between two objects is denoted by
$\bm{r}=\bm{x}_{(1)}-\bm{x}_{(2)}$, with absolute value $r=|\bm{r}|$ and the
unit vector along the separation is $\bm{n}=\bm{r}/r$.

##  2 An EFT description of tides

In this section, we review the formalism developed to model the dynamic tidal
oscillations of compact objects in Ref. Steinhoff:2016rfi . The Lagrangian
given in equation (1) can be written as

$\displaystyle\mathcal{L}_{{\rm DT}(a)}$
$\displaystyle=cP_{(a)\mu\nu}\frac{{\rm d}Q_{(a)}^{\mu\nu}}{{\rm
d}\tau}-z_{(a)}\left[\lambda_{(a)}\omega_{f(a)}^{2}P_{(a)}^{\mu\nu}P_{(a)\mu\nu}+\frac{1}{4\lambda}Q_{(a)}^{\mu\nu}Q_{(a)\mu\nu}\right]$
$\displaystyle\quad-\frac{z_{(a)}}{2}Q_{(a)}^{\mu\nu}E_{\mu\nu}-\kappa_{d(a)}\frac{G_{d}^{2}m_{(a)}^{2}}{c^{6}}\frac{z_{(a)}}{2}Q_{(a)}^{\mu\nu}\frac{d^{2}E_{\mu\nu}}{d\tau^{2}}\,,$
(4)

where the conjugate momenta $P_{\mu\nu}$ with respect to the quadrupole moment
are defined as,

$\displaystyle
P_{(a)\mu\nu}=\frac{1}{c}\frac{\partial\mathcal{L}}{\partial({{\rm
d}Q_{(a)}^{\mu\nu}}/{{\rm
d}\tau})}=\frac{c}{2\lambda\omega_{f}^{2}z_{(a)}}\frac{{\rm
d}Q_{(a)\mu\nu}}{{\rm d}\tau}\,.$ (5)

The action for dynamical tides, written explicitly in terms of the physical
333The supplementary condition for the dynamical degrees of freedom in the
rest frame of the star becomes: $Q_{(a)}^{A0}=0\,,\quad\text{and}\quad
P_{(a)}^{A0}=0$ where, we now explicitly see that $Q^{AB}_{(a)}$ and
$P^{AB}_{(a)}$ are spatial tensors that encode only the physical degrees of
freedom. Thus, we define the spatial tensor
$Q_{(a)}^{AB}\delta_{A}^{i}\delta_{B}^{j}=\bm{Q}_{(a)}^{ij}$ and
$P_{(a)}^{AB}\delta_{A}^{i}\delta_{B}^{j}=\bm{P}_{(a)}^{ij}$.  degrees of
freedom, $\bm{Q}_{(a)}^{ij}$ and $\bm{P}_{(a)}^{ij}$, can be obtained by
expressing the dynamical variables in the rest frame of each body 444
Different reference frames: (i) the general coordinate frame (denoted by Greek
indices), (ii) the local Lorentz frame (denoted by small Latin indices), (iii)
and the rest frame of the compact objects (denoted by capital Latin indices),
and the Lorentz transformation, which boosts between the local Lorentz frame
and the rest frame of the body is given by  $\displaystyle
B^{a}_{(a)A}=\eta^{a}_{~{}A}+2\frac{u_{(a)}^{a}\delta_{A}^{0}}{z_{(a)}}-\frac{(u_{(a)}^{a}+z_{(a)}\delta^{a}_{0})(u_{(a)A}+z_{(a)}\delta_{A}^{0})}{z_{(a)}(z_{(a)}+u^{a}\delta_{a}^{0})}\,.$
. This gives us the effective point-particle (“pp”) action

$\displaystyle S_{\text{pp}}=\sum_{a=1,2}\int\frac{{\rm
d}\tau}{c}\left[-m_{(a)}z_{(a)}c^{2}+{\cal L}_{{\rm FD}(a)}+{\cal L}_{{\rm
MQ}(a)}+{\cal L}_{\rm{EQ}(a)}+\mathcal{L}_{{\rm\ddot{E}Q}(a)}\right]\,.$ (6)

The first term is simply the action for a point particle, while the remaining
terms originate from the Lagrangian (4) as follows. The first term in Eq. (4)
gives rise to,

$\displaystyle\mathcal{L}_{{\rm FD}(a)}$
$\displaystyle=\bm{P}_{(a)}^{ij}\dot{\bm{Q}}_{(a)}^{ij}+c\left[-u_{(a)}^{\mu}\omega_{\mu}^{ij}\left(\frac{\bm{S}_{Q(a)}^{ij}}{2}-\frac{\bm{S}_{Q(a)}^{ik}u^{k}_{(a)}u_{(a)}^{j}}{z_{(a)}(z_{(a)}+u_{(a)}^{a}\delta_{a}^{0})}\right)-u_{(a)}^{\mu}\omega_{\mu}^{ai}\delta_{a}^{0}\bm{S}_{Q(a)}^{ij}\frac{u^{j}_{(a)}}{z_{(a)}}\right.$
$\displaystyle\qquad\qquad\qquad\quad\,\,\left.+\frac{\bm{S}_{Q(a)}^{ij}u^{i}_{(a)}}{z_{(a)}(z_{(a)}+u_{(a)}^{a}\delta_{a}^{0})}\frac{{\rm
d}u^{j}_{(a)}}{{\rm d}\tau}\right]\,,$ (7)

which describes frame-dragging (“FD”) effects on the quadrupole moment of each
binary component, where, the “tidal spin” tensor
$\bm{S}_{Q(a)}^{ij}=2\,(\bm{Q}_{(a)}^{ki}\bm{P}_{(a)}^{jk}-\bm{Q}_{(a)}^{kj}\bm{P}_{(a)}^{ik})$
, which describes the angular momentum of the dynamical quadrupole moment. The
second term in Eq. (4) yields,

$\displaystyle\mathcal{L}_{{\rm
MQ}(a)}=-z_{(a)}\left[\lambda_{(a)}\omega_{f(a)}^{2}\bm{P}_{(a)}^{ij}\bm{P}_{(a)}^{ij}+\frac{1}{4\lambda_{(a)}}\bm{Q}_{(a)}^{ij}\bm{Q}_{(a)}^{ij}\right]=-z_{(a)}\bm{M}_{Q(a)}\,.$
(8)

This term governs the dynamics of the quadrupole moment, which, by the second
equality, can be described as a time-dependent effective mass term for
quadrupole moment (“MQ”). Finally, the last two terms in Eq. (4) results in,

$\displaystyle\mathcal{L}_{{\rm
EQ}(a)}=-\frac{z_{(a)}}{2}\bm{Q}_{(a)}^{ij}\bm{E}_{(a)}^{ij}\,\quad\quad\text{and}\quad\quad\mathcal{L}_{{\rm\ddot{E}Q}(a)}=-\kappa_{d(a)}\frac{G_{d}^{2}m_{(a)}^{2}}{c^{6}}\frac{z_{(a)}}{2}\bm{Q}_{(a)}^{ij}\ddot{\bm{E}}_{(a)}^{ij}\,.$
(9)

These terms act as a driving source for the quadrupole moment’s dynamics and
are induced on each of the binary components by the gravitoelectric tidal
field
$\bm{E}_{(a)}^{ij}=B^{a}_{(a)i}B^{b}_{(a)j}e^{\mu}_{~{}a}e^{\nu}_{~{}b}E_{\mu\nu}$
of its companion.

Here the Poisson bracket of dynamical quantities i.e. the dynamic quadrupole
$\textbf{Q}^{ij}$ and its conjugate momenta $\bm{P}^{ij}$ is given as

$\displaystyle\\{\bm{Q}^{ij},\bm{P}^{kl}\\}$
$\displaystyle=\frac{1}{2}\left(\delta^{ik}\delta^{jl}+\delta^{il}\delta^{jk}\right)-\frac{1}{3}\delta^{ij}\delta^{kl}\,,$
(10)

which then implies a SO(3) angular momentum algebra for the tidal spin tensor
$\bm{S}_{Q}^{ij}$.

##  3 Computational algorithm

###  3.1 The effective Lagrangian

In this section, we present the used computational algorithm. The resulting
potential will then be used in the next section to obtain the effective two-
body Hamiltonian. The dynamics of the gravitational field $g_{\mu\nu}$ is
given by the Einstein-Hilbert action along with a harmonic gauge fixing term
in $d+1$ spacetime dimensions,

$\displaystyle S_{\text{EH}}=-\frac{c^{4}}{16\pi G_{d}}\int{\rm
d}^{d+1}x~{}\sqrt{g}~{}\mathcal{R}+\frac{c^{4}}{32\pi G_{d}}\int{\rm
d}^{d+1}x~{}\sqrt{g}~{}g_{\mu\nu}~{}\Gamma^{\mu}\Gamma^{\nu}\,,$ (11)

where $\Gamma^{\mu}=\Gamma^{\mu}_{\rho\sigma}g^{\rho\sigma}$,
$\Gamma^{\mu}_{\rho\sigma}$ is the Christoffel symbol, $\mathcal{R}$ is the
Ricci scalar, and $g$ is the metric determinant.

For the conservative dynamics of the system, we decompose the metric as
$g_{\mu\nu}=\eta_{\mu\nu}+H_{\mu\nu}$, where $H_{\mu\nu}$ is the potential
graviton. We use the Kaluza-Klein parametrization to decompose the metric,
where the 10 degrees of freedom of $H_{\mu\nu}$ are encoded in three fields: a
scalar $\bm{\phi}$, a three-dimensional vector $\bm{A}$ and a three-
dimensional symmetric rank two tensor $\bm{\sigma}$ Kol:2007bc ; Kol:2007rx .
In this parametrization, we write the metric as

$g_{\mu\nu}=\begin{pmatrix}e^{2\bm{\phi}/c^{2}}\,\,\,&-e^{2\bm{\phi}/c^{2}}{\bm{A}_{j}}/{c^{2}}\\\
-e^{2\bm{\phi}/c^{2}}{\bm{A}_{i}}/{c^{2}}\,\,\,\,\,\,&-e^{-2\bm{\phi}/((d-2)c^{2})}\bm{\gamma}_{ij}+e^{2\bm{\phi}/c^{2}}{\bm{A}_{i}}{\bm{A}_{j}}/{c^{4}}\end{pmatrix}\,,\quad{\rm
with}\quad\bm{\gamma}_{ij}=\bm{\delta}_{ij}+\bm{\sigma}_{ij}/c^{2}\,.$ (12)

We can now obtain the effective action for the binary by integrating out the
gravitational degrees of freedom as follows,

$\displaystyle\exp\left[{{\rm i}\,\int{\rm
d}t~{}\mathcal{L}_{\text{eff}}}\right]=\int{\rm D}\bm{\phi}\,{\rm
D}\bm{A}_{i}\,{\rm D}\bm{\sigma}_{ij}\,\exp[{\rm
i}\,(S_{\text{EH}}+S_{\text{pp}})]\,,$ (13)

where the Einstein-Hilbert action is given by Eq. (11) and the point-particle
action is given by Eq. (6). To perform this integration, we first decompose
the effective Lagrangian ${\cal L}_{\rm eff}$ as

$\mathcal{L}_{\rm eff}=\mathcal{K}_{\rm eff}-\mathcal{V}_{\rm eff}\,,$ (14)

where $\mathcal{K}_{\rm eff}$ is an effective kinetic term, which does not
dependent on any integration of potential graviton (i.e., it is independent of
integration over ${\bm{\phi}}$, ${\bm{A}}$, and ${\bm{\sigma}}$). We can
compute $\mathcal{K}_{\rm eff}$ directly up to the required PN order.
Explicitly, we decompose $\mathcal{K}_{\rm eff}$ in a point-particle, a frame-
dragging, and a “quadrupole mass” contribution, i.e., ${\cal K}_{\rm
eff}={\cal K}_{\rm pp}+{\cal K}_{\rm FD}+{\cal K}_{\rm MQ}$,

$\displaystyle\mathcal{K}_{\rm pp}=$
$\displaystyle\sum_{a=1,2}m_{(a)}\left[\frac{1}{2}\bm{v}_{a}^{2}+\frac{1}{8}\bm{v}_{(a)}^{4}\left(\frac{1}{c^{2}}\right)+\frac{1}{16}\bm{v}_{(a)}^{6}\left(\frac{1}{c^{4}}\right)+\frac{5}{128}\bm{v}_{(a)}^{8}\left(\frac{1}{c^{6}}\right)\right]+\mathcal{O}\left(\frac{1}{c^{8}}\right)\,,$
(15a) $\displaystyle\mathcal{K}_{\rm FD}=$
$\displaystyle\sum_{a=1,2}\left\\{\bm{P}_{(a)}^{ij}\dot{\bm{Q}}_{(a)}^{ij}+\bm{S}_{Q(a)}^{ij}\bm{v}_{(a)}^{i}\bm{a}_{(a)}^{j}\left[\frac{1}{2}\left(\frac{1}{c^{2}}\right)+\frac{3}{8}\bm{v}_{(a)}^{2}\left(\frac{1}{c^{4}}\right)+\frac{5}{16}\bm{v}_{(a)}^{4}\left(\frac{1}{c^{4}}\right)\right]\right\\}+\mathcal{O}\left(\frac{1}{c^{8}}\right)\,,$
(15b) $\displaystyle\mathcal{K}_{\rm MQ}=$
$\displaystyle\sum_{a=1,2}\bm{M}_{(a)}\left[1+\frac{1}{2}\bm{v}_{a}^{2}\left(\frac{1}{c^{2}}\right)+\frac{1}{8}\bm{v}_{(a)}^{4}\left(\frac{1}{c^{4}}\right)+\frac{1}{16}\bm{v}_{(a)}^{6}\left(\frac{1}{c^{6}}\right)\right]+\mathcal{O}\left(\frac{1}{c^{8}}\right)\,.$
(15c)

The terms that are obtained after performing the explicit integral are
collectively denoted by the potential $\mathcal{V}_{\rm eff}$. These terms are
computed by summing over the connected Feynman diagrams without graviton
loops, as shown below,

$\displaystyle\mathcal{V}_{\text{eff}}={\rm i}\,\lim_{d\rightarrow
3}\int\frac{{\rm d}^{d}\bm{p}}{(2\pi)^{d}}~{}e^{{\rm
i}\,\bm{p}\cdot(\bm{x}_{(1)}-\bm{x}_{(2)})}\,\parbox{71.13188pt}{
\leavevmode\hbox to69.58pt{\vbox
to34.43pt{\pgfpicture\makeatletter\hbox{\hskip 7.16602pt\lower-28.59521pt\hbox
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{}{}\pgfsys@beginscope\pgfsys@invoke{
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where $\bm{p}$ is the linear momentum transferred between the two bodies. For
this, we begin with generating all the topologies that correspond to graviton
exchanges between the worldlines of the two compact objects. There is 1
topology at tree-level ($G_{N}$), 2 topologies at one-loop ($G_{N}^{2}$), 9
topologies at two-loop ($G_{N}^{3}$), and 32 topologies at three-loop
($G_{N}^{4}$). We then dress these topologies with the Kaluza-Klein fields
$\bm{\phi}$, $\bm{A}$ and $\bm{\sigma}$. The number of diagrams555The diagrams
which can be obtained from the change in the label $1\leftrightarrow 2$, are
not counted as separate diagrams. appearing in the point-particle sector is
given in Table 1(a), whereas that in the tidal sector are given in Tables
1(b), 1(c), 1(d) and 1(e). We use an in-house code that uses tools from
EFTofPNG Levi:2017kzq  and xTensor xAct , for the tensor algebra manipulation,
and LiteRED Lee:2013mka , for the integration-by-parts reduction, to compute
these Feynman diagrams. This reduction recasts the Feynman diagrams in terms
of two point massless master integrals Foffa:2016rgu  as shown in Fig. 1.

GravityDiagrams$\longleftrightarrow$$\equiv$Multi-loopDiagrams Figure 1: The
diagrammatic correspondence between the four-point EFT-Gravity graphs and the
two-point quantum-field-theory (QFT) graphs.

Once the exact expressions for the master integrals are substituted, we
perform a Fourier transform to obtain the position-space effective potential
${\cal V}_{\rm eff}$. The details of the algorithm and the expressions for the
master integrals up to three loops can be found in Ref. Mandal:2022nty .

After carrying out all these steps, the effective potential can be decomposed
into a point-particle and a dynamical tide contribution, i.e., ${\cal V}_{\rm
eff}={\cal V}_{\rm pp}+{\cal V}_{\rm DT}$, where

$\displaystyle\mathcal{V}_{\rm pp}$ $\displaystyle=\mathcal{V}_{\rm
N}+\left(\frac{1}{c^{2}}\right)\mathcal{V}_{\rm
1PN}+\left(\frac{1}{c^{4}}\right)\mathcal{V}_{\rm
2PN}+\left(\frac{1}{c^{6}}\right)\mathcal{V}_{\rm
3PN}+\mathcal{O}\left(\frac{1}{c^{8}}\right)\,,$ (17a)
$\displaystyle\mathcal{V}_{\rm DT}$
$\displaystyle=\sum_{n=0}^{3}\left(\frac{1}{c^{2}}\right)^{n}\left({\cal
V}^{\rm EQ}_{n{\rm PN}}+{\cal V}^{\rm FD}_{n{\rm PN}}+{\cal V}^{\rm MQ}_{n{\rm
PN}}\right)+\left(\frac{1}{c^{6}}\right)\mathcal{V}^{\rm\ddot{E}Q}_{\rm
3PN}+\mathcal{O}\left(\frac{1}{c^{6}}\right)\,,$ (17b)

and we remark that ${\cal V}_{\rm DT}$ has contributions due to the driving
source, the “quadrupole-mass” and the frame-dragging terms, and the post
adiabatic terms.

Order | Diagrams | Loops | Diagrams
---|---|---|---
3PN | 130 | 0 | 9
1 | 38
2 | 75
3 | 8
(a) Point particle sector

Order | Diagrams | Loops | Diagrams
---|---|---|---
3PN | 168 | 0 | 9
1 | 43
2 | 101
3 | 15
(b) EQ sector

Order | Diagrams | Loops | Diagrams
---|---|---|---
3PN | 100 | 0 | 8
1 | 36
2 | 56
(c) FD sector

Order | Diagrams | Loops | Diagrams
---|---|---|---
3PN | 21 | 0 | 6
1 | 10
2 | 5
(d) MQ sector

Order | Diagrams | Loops | Diagrams
---|---|---|---
3PN | 1 | 0 | 1
(e) ${\rm\ddot{E}Q}$ sector

Table 1: Number of Feynman diagrams contributing different sectors.

###  3.2 Removal of higher order time derivatives

The potential ${\cal V}_{\rm eff}$ computed in the previous section, is a
function of the dynamical variables $\bm{x}_{(a)}$, $\bm{Q}_{(a)}$, and
$\bm{S}_{Q(a)}$ and $\bm{M}_{Q(a)}$, and their higher order time derivatives.
The first and higher-order time derivatives of $\bm{Q}_{(a)}$,
$\bm{S}_{Q(a)}$, and $\bm{M}_{Q(a)}$ are removed using integration by parts,
while second and higher-order time derivatives of $\bm{x}_{(a)}$ are removed
using a coordinate transformation
$\bm{x}_{(a)}\rightarrow\bm{x}_{(a)}+\delta\bm{x}_{(a)}$ Levi:2014sba ;
Schafer:1984mr ; Damour:1990jh ; Damour:1985mt ; Barker:1980spx . This
coordinate transformation changes the Lagrangian as

$\displaystyle\delta\mathcal{L}=\frac{\delta\mathcal{L}}{\delta\bm{x}_{(a)}^{i}}\,\delta\bm{x}_{(a)}^{i}+\mathcal{O}(\delta\bm{x}_{(a)}^{2})\,,$
(18)

where $\delta\bm{x}_{(a)}$ is chosen such that it removes the undesirable
terms from our final Lagrangian. In our case, the process of removing the
higher order time derivatives using a coordinate transformation is equivalent
to the substitution of the equation of motion for the acceleration
$\bm{a}_{(a)}$ and its higher order time derivatives back into the Lagrangian
Damour:1990jh . This procedure converts the Lagrangian $\mathcal{L}_{\rm eff}$
into the Lagrangian $\bar{\mathcal{L}}_{\rm eff}$ which depends only on
$\bm{x}_{(a)}$, $\bm{v}_{(a)}$, and $\bm{Q}_{(a)}$, but still contains
divergent terms.

###  3.3 Removal of spurious divergences

After the removal of the higher order time derivatives, the effective
Lagrangian $\bar{\mathcal{L}}_{\rm eff}$ contains several divergent terms.
These divergent terms are generated by the two topologies as shown in Fig. 2.
Part of these divergent terms can be removed by adding counter-terms based on
the world-line operators, which ultimately can be removed by using field-
redefinitions. So, this procedure becomes equivalent to adding total
derivative terms to the Lagrangian or finding a canonical transformation on
the corresponding Hamiltonian. As a result these divergences do not have any
effect on the physical observable and are dubbed as spurious divergences.
666The spurious divergences appearing in 3PN order can also be removed from
the equation of motion Blanchet:2003gy . For the removal of the 3PN spurious
divergence in the point particle sector, following Foffa:2011ub , we can add a
total derivative term

$\displaystyle\mathcal{L}_{\text{TD1}}=\left(\frac{1}{c^{6}}\right)\frac{{\rm
d}}{{\rm
d}t}\left[\frac{G_{N}^{3}}{r^{2}}\Big{(}c_{1}\left(\bm{v}_{(1)}\cdot\bm{n}\right)+c_{2}\left(\bm{v}_{(2)}\cdot\bm{n}\right)\Big{)}\right]\,,$
(19)

and gaining insights from Levi:2014sba , we choose another total derivative
term

$\displaystyle\mathcal{L}_{\text{TD2}}$
$\displaystyle=\left(\frac{1}{c^{6}}\right)\frac{{\rm d}}{{\rm
d}t}\Bigg{\\{}\frac{G_{N}^{3}}{r^{4}}\left[c_{3}\Big{(}\bm{Q}_{(1)}^{ij}\bm{v}_{(1)}^{i}\bm{n}^{j}\Big{)}+c_{4}\Big{(}\bm{Q}_{(1)}^{ij}\bm{v}_{(2)}^{i}\bm{n}^{j}\Big{)}+c_{5}\Big{(}\bm{Q}_{(2)}^{ij}\bm{v}_{(1)}^{i}\bm{n}^{j}\Big{)}+c_{6}\Big{(}\bm{Q}_{(2)}^{ij}\bm{v}_{(2)}^{i}\bm{n}^{j}\Big{)}\right]$
$\displaystyle+\frac{G_{N}^{3}}{r^{4}}\Bigg{[}\Big{(}\bm{Q}_{(1)}^{ij}\bm{n}^{i}\bm{n}^{j}\Big{)}\Big{(}c_{7}\left(\bm{v}_{(1)}\cdot\bm{n}\right)+c_{8}\left(\bm{v}_{(2)}\cdot\bm{n}\right)\Big{)}+\Big{(}\bm{Q}_{(2)}^{ij}\bm{n}^{i}\bm{n}^{j}\Big{)}\Big{(}c_{9}\left(\bm{v}_{(1)}\cdot\bm{n}\right)+c_{10}\left(\bm{v}_{(2)}\cdot\bm{n}\right)\Big{)}\Bigg{]}\Bigg{\\}}\,,$
(20)

to remove the spurious divergence in the 3PN tidal sector. Here the
coefficients are represented in the notation,

$\displaystyle c_{n}\equiv c_{\epsilon
n}\frac{1}{\epsilon}+c_{Ln}\log\left(\frac{r}{R}\right)\,,\,\,\text{for}\,\,n=1,2,\cdots
10\,,$ (21)

and their particular values are given as,

$\displaystyle c_{\epsilon 1}$
$\displaystyle=\frac{1}{3}\left(4m_{(1)}^{3}m_{(2)}-m_{(1)}m_{(2)}^{3}\right),$
$\displaystyle c_{\epsilon 2}$
$\displaystyle=\frac{1}{3}\left(m_{(1)}^{3}m_{(2)}-4m_{(1)}m_{(2)}^{3}\right)\,,$
$\displaystyle c_{\epsilon 3}$
$\displaystyle=-\frac{107}{35}m_{(1)}^{2}m_{(2)}^{2},$ $\displaystyle
c_{\epsilon 4}$ $\displaystyle=\frac{107}{35}m_{(1)}^{3}m_{(2)}\,,$
$\displaystyle c_{\epsilon 5}$
$\displaystyle=-\frac{107}{35}m_{(1)}m_{(2)}^{3},$ $\displaystyle c_{\epsilon
6}$ $\displaystyle=\frac{107}{35}m_{(1)}^{2}m_{(2)}^{2}\,,$ $\displaystyle
c_{\epsilon 7}$ $\displaystyle=\frac{107}{14}m_{(1)}^{2}m_{(2)}^{2},$
$\displaystyle{\epsilon 8}$
$\displaystyle=-\frac{107}{14}m_{(1)}^{3}m_{(2)}\,,$ $\displaystyle
c_{\epsilon 9}$ $\displaystyle=\frac{107}{14}m_{(1)}m_{(2)}^{3},$
$\displaystyle c_{\epsilon 10}$
$\displaystyle=-\frac{107}{14}m_{(1)}^{2}m_{(2)}^{2}\,,$ (22)

and

$\displaystyle c_{Li}$ $\displaystyle=-3c_{\epsilon i}~{}~{}~{}\text{ for
}i=1,2\,$ $\displaystyle c_{Li}$ $\displaystyle=-2c_{\epsilon
i}~{}~{}~{}\text{ for }i=3,\dots,10\,.$ (23)

Now the modified Lagrangian
$\bar{\mathcal{L}}_{\text{eff}}+\mathcal{L}_{\text{TD1}}+\mathcal{L}_{\text{TD2}}$
is free of all the spurious divergences and still contains divergent terms in
the 3PN tidal sector, which requires renormalization and we analyse it in the
next section.

##  4 Renormalization

In this section, we first analyze the divergent terms present in the effective
Lagrangian, which cannot be eliminated through the addition of total
derivatives. These divergent terms are generated by the two topologies shown
in Fig. 2. We show that to eliminate these remaining divergences, a
renormalization procedure is necessary, where the divergent contributions are
absorbed into the bare post-adiabatic Love number $\kappa^{\rm B}$. As a
consequence, we obtain a renormalized post-adiabatic Love number $\kappa(R)$,
which exhibits a nontrivial renormalization group flow.

###  4.1 Analysis of divergent terms

First, we conduct a complete validity check of our computation, following the
observation in Ref. Steinhoff:2016rfi . Specifically, the Hamiltonians
$\widetilde{\mathcal{H}}^{\text{EQ}}$ and
$\widetilde{\mathcal{H}}^{\text{FD}}$ exhibit similarities to the spin-induced
quadrupole ($\widetilde{\mathcal{H}}^{\text{ES}^{2}}$) and spin-orbit
Hamiltonians ($\widetilde{\mathcal{H}}^{\text{SO}}$) when certain replacements
are applied. Our analysis starts with
$\bar{\mathcal{L}}_{\text{eff}}+\mathcal{L}_{\text{TD1}}$, from which we
derive the corresponding Hamiltonians
$\widetilde{\mathcal{H}}^{\text{EQ}}_{3{\rm PN}}$ and
$\widetilde{\mathcal{H}}^{\text{FD}}_{3{\rm PN}}$. These Hamiltonians are free
of divergent and logarithmic terms in the point particle sector but contain
such terms in the tidal sectors, respectively. On the other hand, following
Mandal:2022nty ; Mandal:2022ufb , we obtain the
$\widetilde{\mathcal{H}}^{\text{SO}}_{\text{N${}^{2}$LO }}$ and
$\widetilde{\mathcal{H}}^{\text{ES}^{2}}_{\text{N${}^{3}$LO }}$, which are
free of divergent and logarithmic terms in the point particle sector, but
contains such terms in the spinning sector. Further investigations led us to
find that these Hamiltonians are equivalent to each other up to a canonical
transformation, with $\widetilde{\mathcal{H}}^{\text{EQ}}_{3{\rm PN}}$ being
equivalent to $\widetilde{\mathcal{H}}^{\text{ES}^{2}}_{\text{N${}^{3}$LO }}$,
and $\widetilde{\mathcal{H}}^{\text{FD}}_{3{\rm PN}}$ being equivalent to
$\widetilde{\mathcal{H}}^{\text{SO}}_{\text{N${}^{2}$LO }},$ provided certain
replacements are applied. This result serves as a robust consistency check,
confirming the accuracy of our computation of the effective Lagrangian and the
procedure of removing the higher-order time derivatives. However, an important
distinction arises, while eliminating the divergent terms using a canonical
transformation, between the tidal and the spinning sector. Notably, the
kinetic term of $\bm{Q}^{ij}$ begins at 0PN order, whereas the kinetic term of
$\bm{S}^{ij}$ starts at 0.5PN order. This discrepancy has a crucial
consequence while computing the canonical transformation. The contributions
from the Poisson bracket of the spin tensor at N3LO can be ignored (see Ref.
Mandal:2022ufb , Sec. 4.3.2), whereas the equivalent contributions from the
Poisson bracket for the quadrupole tensor (Eq. (10)) at 3PN order cannot be
ignored. This distinction accounts for the origin of the residual divergent
terms present in the tidal sector while all the divergent pieces in the
spinning sector can be removed by finding a suitable canonical transformation.

(a)

(b)

Figure 2: The topologies that give divergent contribution in the tidal sector
at 3PN. Solid lines represent the compact object’s worldlines and the dashed
lines represent the potential gravitons $H_{\mu\nu}$.

After the removal of all the spurious divergent terms, we obtain the following
residual divergent term

$\displaystyle\Big{(}\bar{\mathcal{L}}_{\text{eff}}+\mathcal{L}_{\text{TD1}}+\mathcal{L}_{\text{TD2}}\Big{)}_{1/\epsilon}=\Bigg{(}\frac{107}{105}m_{(1)}^{2}G_{N}^{2}\frac{1}{c^{6}}\frac{1}{\epsilon}\Bigg{)}\Bigg{(}\frac{3}{2}\frac{G_{N}m_{(2)}}{r^{3}}\Big{(}\ddot{\bm{Q}}_{(1)}^{ij}\bm{n}^{i}\bm{n}^{j}\Big{)}\Bigg{)}+(1\leftrightarrow
2)\,,$ (24)

where we perform some algebraic manipulations to present the expression in a
compact form. We observe that the divergent terms in Eq. (24) cannot be
expressed as a total derivative and the tail effects in the case of dynamic
tides start at 4PN order. So, we absorb the divergence in one of the coupling
constants of the Lagrangian given in Eq. (1), by designing a counterterm,
which will eventually lead to finite results. By examining the structure of
terms in Eq. (24), we identify the following counterterm, which we can add to
the point particle action (6).

$\displaystyle{\cal
L}_{\rm{CT}(a)}=-\delta_{\kappa(a)}\frac{G_{N}^{2}m_{(a)}^{2}}{c^{6}}\frac{z_{(a)}}{2}\bm{Q}_{(a)}^{ij}\ddot{\bm{E}}_{(a)}^{ij}\,,$
(25)

where $\delta_{\kappa(a)}$ contains the divergence. The counter term
Lagrangian contributes through a tree level diagram at this order as shown in
Fig. 3, which renders the result finite, eventually. Now, the particular value
of $\delta_{\kappa(a)}$ that cancels the divergent terms presented in the
effective Lagrangian (24) is,

$\displaystyle\delta_{\kappa(a)}=-\frac{107}{105}\frac{1}{\epsilon}\,.$ (26)

We note that the counterterm has the structure similar to ${\cal
L}_{{\rm\ddot{E}Q}(a)}$, while the divergence produced in the Eq. (24) is
sourced by the ${\cal L}_{{\rm EQ}(a)}$. The addition of the counterterm to
the point particle action in Eq. (6), leads to the renormalization of the
post-adiabatic Love number

$\displaystyle\kappa_{d(a)}=\Big{(}\sqrt{4\pi\exp(\gamma_{\rm
E})}\,R\Big{)}^{-2\epsilon}~{}\Big{(}\kappa_{(a)}+\delta_{\kappa(a)}\Big{)}\,,$
(27)

where, the renormalized post-adiabatic Love number $\kappa_{(a)}$ depends on
the external scale $R$, which we analyse in detail in Sec. 4.2. One important
thing to note is that the Lagrangian presented in Eq. (1) can be changed by a
field redefinition, which in turn redefines the different Love numbers. Hence,
the above mentioned procedure of renormalizing the post-adiabatic Love number
$\kappa$ is particular to this representation of the Lagrangian.

Moreover, the counterterm presented in Eq. (25) has an intriguing form, as it
turns out to be identical to the counterterms introduced to eliminate the
divergences in the radiation emitted by each NS, as discussed in
Blanchet:1997jj ; Goldberger:2009qd . In Sec. 4.3, we discuss further into the
connection of this counterterm with radiation. Similar set of counterterms was
also observed in Refs. Barack:2023oqp ; Saketh:2023bul .

Figure 3: Counter-term diagram. The the cross symbol stands for the insertion
of the counterterm constant $\delta_{\kappa(a)}$. Solid lines represent the
compact object’s worldlines and the dashed lines represent the potential
gravitons $H_{\mu\nu}$.

###  4.2 Beta-function for post-adiabatic Love number

Now both the gravitational constant and the post-adiabatic Love number depends
on the scale $R$. However, any physical quantity should be independent of the
arbitrary scale $R$. Indeed by demanding the scale independence of the bare
couplings in the Lagrangian we obtain the corresponding renormalization group
equations.

For the $d$-dimensional gravitational coupling, we obtain,

$\displaystyle 0=\frac{{\rm d}G_{d}}{{\rm d}R}=\frac{{\rm d}}{{\rm
d}R}\left[\left(\sqrt{4\pi e^{\gamma_{E}}}R\right)^{\epsilon}G_{N}\right]\,,$
(28)

which implies a trivial beta function for $G_{N}$ given by,

$\displaystyle\beta(G_{N})\equiv R\,\frac{{\rm d}G_{N}}{{\rm
d}R}=-\epsilon\left(\sqrt{4\pi
e^{\gamma_{E}}}R\right)^{\epsilon}G_{N}\stackrel{{\scriptstyle\epsilon\rightarrow
0}}{{=}}0\,.$ (29)

Similarly, for the post-adiabatic coupling, we obtain

$\displaystyle 0=\frac{{\rm d}\kappa_{d(a)}}{{\rm d}R}$
$\displaystyle=\frac{{\rm d}}{{\rm d}R}\left[\Big{(}\sqrt{4\pi\exp(\gamma_{\rm
E})}\,R\Big{)}^{-2\epsilon}\left(\kappa_{(a)}+\delta_{\kappa_{(a)}}\right)\right]\,,$
(30)

which results in the following beta function, 777Similar beta-function can be
found for the parameter $\lambda_{8}$ of the following scalar theory,
$\displaystyle\mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi)^{2}-\frac{1}{2}m^{2}\phi^{2}-\frac{1}{6!}\lambda_{6}\phi^{6}-\frac{1}{8!}\lambda_{6}^{2}\lambda_{8}\phi^{8}$

$\displaystyle\beta(\kappa_{(a)})\equiv R\,\frac{{\rm d}\kappa_{(a)}}{{\rm
d}R}$ $\displaystyle=-\frac{214}{105}\,.$ (31)

The solution of the above given beta function is

$\displaystyle\kappa_{(a)}(R)=\kappa_{(a)}(R_{0})-\frac{214}{105}\log\left(\frac{R}{R_{0}}\right)\,,$
(32)

where $R_{0}$ in the integration constant. Now the idea for choosing $R$ will
be the following Goldberger:2004jt . In principle, we are allowed to choose
the scale $R$ to whatever value we want. But as shown in Eq. (41), the
$R$-dependence enters the effective Hamiltonian via $\kappa(R)$ and
logarithmic terms of the form $\log\left(r/R\right)$.

When matching the point particle theory to an overarching theory for NSs, we
should choose the scale $R$ somewhere around the cutoff of the point particle
theory, i.e. $R\approx R_{0}\approx R_{\rm NS}$ where $R_{\rm NS}$ is some
characteristic lengthscale of the NS. This will remove all the logarithmic
terms from the matching procedure and we obtain a value for
$\kappa(R_{0}\approx R_{\rm NS})$. Then, when we analyse the two body system,
we have $R_{\rm orb}\approx r$ which is much larger than $R\approx R_{\rm
NS}$, where $R_{\rm orb}$ is the characteristic lengthscale of the binary, and
hence the logarithm in the Lagrangian blows up. So, in this case we will have
to pick $R\approx R_{\rm orb}$, so that the logarithm is small and we get
$\kappa(R_{\rm orb})$ in the effective Lagrangian. Hence the logarithmic terms
plays a role only in flowing the $\kappa$ from $\kappa(R_{\rm NS})$ to
$\kappa(R_{\rm orb})$ using the RG solution in Eq. (32). Something peculiar to
observe is that, $\kappa(R_{\rm orb})\rightarrow-\infty$ when $R_{\rm
orb}\rightarrow\infty$ i.e. when the radius of the binary is large. This is
acceptable because as $R_{\rm orb}\rightarrow\infty$ there are factors of
$1/r^{m}$ with the $\kappa(R_{\rm orb})$ in the effective Lagrangian that
suppress this logarithmic growth.

Using the above procedure, we will see that the observables will be
independent of the arbitrary scale $R$ only after performing the matching to
an overarching UV theory of NSs. An example of this can be seen in Ref.
Blanchet:1997jj , where the authors computed the flux emitted by a binary and
obtained scale dependent logarithmic terms similar to that in Eq. (32). Such
terms drop out when they include terms obtained by matching the quadrupole of
the binary to the dynamics of two point particle constituents of the binary
Blanchet:2013haa  and the total flux is independent of the arbitrary scale.

###  4.3 Radiation from a single neutron star

In this section, we explore the relationship between the renormalization
procedure used in the previous section and the computation of radiative
observables from a single NS with a dynamical quadrupole oscillating at a
frequency $\omega$. This is intriguing as the counter term required to achieve
a finite result in our 3PN conservative computation is the same counter term
that addresses the divergences in the radiative sector.

The radiative sector from the ${\cal L}_{{\rm EQ}(a)}$ term in the Lagrangian
was computed in Refs. Blanchet:1997jj ; Goldberger:2009qd . The radiative
observable was expressed in terms of a matrix element of a single graviton
emitted from the NS. The expression of these amplitudes upto two loops are
given in Eq. (38) of Goldberger:2009qd , where they found that these
amplitudes contain UV divergent terms and they can be removed by modifying the
dynamics of the quadrupole as

$\displaystyle\bm{Q}_{ij}^{B}(\omega)=\bm{Q}_{ij}(\omega,\mu)\Bigg{[}1+\frac{107}{105}\left(\frac{G_{N}m_{(a)}\omega}{c^{3}}\right)^{2}\frac{1}{\epsilon}\Bigg{]}\,.$
(33)

If we only consider the ${\cal L}_{{\rm EQ}(a)}$ term, the above is equivalent
to adding counter terms of the form shown in Eq. (25). This can be seen easily
as follows,

$\displaystyle-\frac{z_{(a)}}{2}\bm{E}_{(a)}^{ij}\bm{Q}_{(a)}^{{\rm B}ij}$
$\displaystyle=-\frac{z_{(a)}}{2}\bm{Q}_{(a)}^{ij}\Bigg{[}1+\frac{107}{105}\left(\frac{G_{N}m_{(a)}}{c^{3}}\right)^{2}\left(-\frac{{\rm
d}^{2}}{{\rm d}t^{2}}\right)\frac{1}{\epsilon}\Bigg{]}\bm{E}_{(a)}^{ij}$
$\displaystyle=-\frac{z_{(a)}}{2}\bm{Q}_{(a)}^{ij}\bm{E}_{(a)}^{ij}-\delta_{\kappa(a)}\frac{G_{N}^{2}m_{(a)}^{2}}{c^{6}}\frac{z_{(a)}}{2}\bm{Q}_{(a)}^{ij}\ddot{\bm{E}}_{(a)}^{ij}\,.\par$
(34)

In our case, since we have a definite dynamics assigned to the quadrupole by
the ${\cal L}_{\rm{FD}(a)}$ and ${\cal L}_{\rm{MQ}(a)}$ terms, the
renormalization of the quadrupole as done in Refs. Blanchet:1997jj ;
Goldberger:2009qd  introduces extra divergent terms. So, rather than
performing the renormalization of the quadrupole, we renormalize the post-
adiabatic Love number by adding the counter term given in Eq. (25), which can
also remove the divergence as obtained in Eq. (38) of Goldberger:2009qd  in
the radiative sector .

Additionally, in Ref. Goldberger:2004jt , it was observed that when starting
with the point particle action (non-spinning and without tides), the stress-
energy tensor and, consequently, the metric produced by a single compact
object contain divergent terms. When we extend this analysis to a binary
system comprising two such point particles, the divergent metric generated by
the first object affects the second object, leading to the emergence of
divergences in the two-body interaction potential. The divergent terms from
both the metric and the interaction potential were removed with the
introduction of a counter term in the point particle action and ultimately
these counter terms vanished by applying a field redefinition in the
Lagrangian. We see an analogous case for the 3PN tidal interaction potential.
The radiative sector from the ${\cal L}_{{\rm EQ}(a)}$ term in the Lagrangian
contains divergent pieces. If one now computes the metric from this, it should
also be divergent. Hence we should expect the same divergence to also show up
in the two body tidal interaction potential. The only difference being, the
counter term that removes the tidal divergence, cannot be removed by a field
redefinition as opposed to the same in the point particle sector.

##  5 Dynamical tides

In this section, we present the result of the effective two-body Hamiltonian
with dynamical gravitoelectric tides. We compute the Hamiltonian $\mathcal{H}$
from the Lagrangian obtained in the previous section using a Legendre
transformation

$\displaystyle\mathcal{H}(\bm{x},\bm{p},\bm{Q})=\sum_{a=1,2}\left(\,\bm{p}_{(a)}^{i}{\bm{v}}_{(a)}^{i}+\bm{P}_{(a)}^{ij}\dot{\bm{Q}}_{(a)}^{ij}\,\right)-\mathcal{L}(\bm{x},{\bm{v}},\bm{Q})\
.$ (35)

We express this Hamiltonian in terms of the total mass of the binary, denoted
by $M=m_{(1)}+m_{(2)}$, the reduced mass by $\mu=m_{(1)}m_{(2)}/M$, the mass
ratio by $q=m_{(1)}/m_{(2)}$, the symmetric mass ratio $\nu=\mu/M$, and the
antisymmetric mass ratio $\delta=(m_{(1)}-m_{(2)})/M$, which are related to
each other by,

$\displaystyle\nu=\frac{m_{(1)}m_{(2)}}{M^{2}}=\frac{\mu}{M}=\frac{q}{(1+q)^{2}}=\frac{(1-\delta^{2})}{4}\,.$
(36)

We express the results in the center-of-mass (COM) frame of reference and
define the momentum in the COM frame as
$\bm{p}\equiv\bm{p}_{(1)}=-\bm{p}_{(2)}$. In the COM frame, the orbital
angular momentum is defined as $\bm{L}=\bm{r}\times\bm{p}$. Hence, we can
write $p^{2}=p_{r}^{2}+L^{2}/r^{2}$, where $p_{r}=\bm{p}\cdot\bm{n}$,
$p=|\bm{p}|$ and $L=|\bm{L}|$. We write the Hamiltonian in terms of
dimensionless quantities, which are obtained by rescaling all the variables as
follows

$\displaystyle\widetilde{\bm{p}}=\frac{1}{c}\frac{\bm{p}}{\mu}\,,\quad\widetilde{\bm{r}}=\frac{c^{2}}{G_{N}}\frac{\bm{r}}{M}\,,\quad\widetilde{\bm{L}}=\frac{c}{G_{N}}\frac{\bm{L}}{M\mu}\,,\quad\widetilde{\mathcal{H}}=\frac{1}{c^{2}}\frac{\mathcal{H}}{\mu}\,,\quad\widetilde{\lambda}=\frac{c^{10}}{G_{N}^{4}}\frac{\lambda}{M^{5}}\,,$
$\displaystyle\widetilde{\bm{Q}}_{(a)}=\frac{c^{4}}{G_{N}^{2}}\frac{\bm{Q}_{(a)}}{M^{2}\mu}\,,\quad\widetilde{\bm{S}}_{Q(a)}=\frac{c}{G_{N}}\frac{\bm{S}_{Q(a)}}{M\mu}\,,\quad\text{and}\quad\widetilde{\bm{M}}_{Q(a)}=\frac{1}{c^{2}}\frac{\bm{M}_{Q(a)}}{\mu}\,.$
(37)

The total EFT Hamiltonian in the dimensionless parameters is given by

$\displaystyle\widetilde{\mathcal{H}}=\widetilde{\mathcal{H}}_{\rm
pp}+\widetilde{\mathcal{H}}_{\rm DT}\ ,$ (38)

where

$\displaystyle\widetilde{\mathcal{H}}_{\rm pp}$
$\displaystyle=\widetilde{\mathcal{H}}_{\text{0PN}}+\left(\frac{1}{c^{2}}\right)\widetilde{\mathcal{H}}_{\text{1PN}}+\left(\frac{1}{c^{4}}\right)\widetilde{\mathcal{H}}_{\text{2PN}}+\left(\frac{1}{c^{6}}\right)\widetilde{\mathcal{H}}_{\text{3PN}}+\mathcal{O}\left(\frac{1}{c^{8}}\right)\,,$
(39a) $\displaystyle\widetilde{{\cal H}}_{\rm DT}$
$\displaystyle=\sum_{n=0}^{3}\left(\frac{1}{c^{2}}\right)^{n}\,\left(\widetilde{{\cal
H}}^{\rm EQ}_{n{\rm PN}}+\widetilde{{\cal H}}^{\rm FD}_{n{\rm
PN}}+\widetilde{{\cal H}}^{\rm MQ}_{n{\rm
PN}}\right)+\left(\frac{1}{c^{6}}\right)\widetilde{\mathcal{H}}^{\rm\ddot{E}Q}_{\rm
3PN}+\mathcal{O}\left(\frac{1}{c^{8}}\right)\,.$ (39b)

The point particle Hamiltonian till 3PN is presented in the same gauge in
Appendix C.1 of Ref. Mandal:2022nty , the tidal Hamiltonian up to 2PN is
presented in Ref. Mandal:2023lgy  and the novel result of 3PN Hamiltonian is
given as,

$\displaystyle\widetilde{\mathcal{H}}^{\rm EQ+\ddot{E}Q}_{\rm 3PN}=$
$\displaystyle\left(\widetilde{\bm{Q}}^{ij}_{(1)}\widetilde{\bm{r}}^{\,i}\widetilde{\bm{r}}^{\,j}\right)\Bigg{\\{}\widetilde{L}^{6}\left(-\frac{15\nu^{4}}{32\widetilde{r}^{11}}-\frac{135\nu^{3}}{32\widetilde{r}^{11}}+\frac{45\nu^{2}}{32\widetilde{r}^{11}}\right)+\widetilde{L}^{4}\left(\widetilde{p}_{r}^{2}\left(-\frac{15\nu^{4}}{8\widetilde{r}^{9}}-\frac{585\nu^{3}}{32\widetilde{r}^{9}}+\frac{327\nu^{2}}{32\widetilde{r}^{9}}\right)\right.$
$\displaystyle\left.+\frac{75\nu^{3}}{4\widetilde{r}^{10}}+\frac{137\nu^{2}}{4\widetilde{r}^{10}}-\frac{101\nu}{32\widetilde{r}^{10}}\right)+\widetilde{L}^{2}\left[\widetilde{p}_{r}^{4}\left(-\frac{51\nu^{4}}{16\widetilde{r}^{7}}-\frac{225\nu^{3}}{8\widetilde{r}^{7}}+\frac{1527\nu^{2}}{32\widetilde{r}^{7}}-\frac{105\nu}{8\widetilde{r}^{7}}\right)\right.$
$\displaystyle\left.+\widetilde{p}_{r}^{2}\left(\frac{1977\nu^{3}}{32\widetilde{r}^{8}}-\frac{383\nu^{2}}{16\widetilde{r}^{8}}-\frac{79\nu}{32\widetilde{r}^{8}}\right)+\frac{\left(-3763200~{}\bar{\kappa}_{(1)}+77175\pi^{2}-35396224\right)\nu^{2}}{501760\widetilde{r}^{9}}\right.$
$\displaystyle\left.+\frac{(3675~{}\bar{\kappa}_{(1)}+16651)\nu}{490\widetilde{r}^{9}}-\frac{1269\nu^{3}}{112\widetilde{r}^{9}}\right]+\widetilde{p}_{r}^{2}\left(\frac{3\left(3763200~{}\bar{\kappa}_{(1)}-77175\pi^{2}+11490272\right)\nu^{2}}{627200\widetilde{r}^{7}}\right.$
$\displaystyle\left.-\frac{3(117600~{}\bar{\kappa}_{(1)}+679671)\nu}{19600\widetilde{r}^{7}}-\frac{153\nu^{3}}{7\widetilde{r}^{7}}\right)+\widetilde{p}_{r}^{4}\left(\frac{477\nu^{3}}{32\widetilde{r}^{6}}-\frac{291\nu^{2}}{8\widetilde{r}^{6}}+\frac{327\nu}{32\widetilde{r}^{6}}\right)$
$\displaystyle+\widetilde{p}_{r}^{6}\left(-\frac{6\nu^{4}}{\widetilde{r}^{5}}+\frac{45\nu^{3}}{2\widetilde{r}^{5}}-\frac{135\nu^{2}}{8\widetilde{r}^{5}}+\frac{105\nu}{32\widetilde{r}^{5}}\right)+\frac{\left(705600~{}\bar{\kappa}_{(1)}-1157625\pi^{2}+14115256\right)\nu^{2}}{156800\widetilde{r}^{8}}$
$\displaystyle-\frac{3(14700~{}\bar{\kappa}_{(1)}+44197)\nu}{9800\widetilde{r}^{8}}+\frac{1}{q}\Bigg{[}\widetilde{L}^{6}\left(-\frac{15\nu^{4}}{32\widetilde{r}^{11}}-\frac{45\nu^{3}}{16\widetilde{r}^{11}}+\frac{105\nu^{2}}{32\widetilde{r}^{11}}-\frac{21\nu}{32\widetilde{r}^{11}}\right)$
$\displaystyle+\widetilde{L}^{4}\left(\widetilde{p}_{r}^{2}\left(-\frac{15\nu^{4}}{8\widetilde{r}^{9}}-\frac{399\nu^{3}}{32\widetilde{r}^{9}}+\frac{159\nu^{2}}{32\widetilde{r}^{9}}-\frac{39\nu}{32\widetilde{r}^{9}}\right)+\frac{591\nu^{3}}{32\widetilde{r}^{10}}+\frac{973\nu^{2}}{32\widetilde{r}^{10}}-\frac{15\nu}{2\widetilde{r}^{10}}\right)$
$\displaystyle+\widetilde{L}^{2}\left(\widetilde{p}_{r}^{4}\left(-\frac{51\nu^{4}}{16\widetilde{r}^{7}}-\frac{321\nu^{3}}{16\widetilde{r}^{7}}+\frac{399\nu^{2}}{32\widetilde{r}^{7}}-\frac{3\nu}{32\widetilde{r}^{7}}\right)+\widetilde{p}_{r}^{2}\left(\frac{1977\nu^{3}}{32\widetilde{r}^{8}}+\frac{933\nu^{2}}{32\widetilde{r}^{8}}-\frac{6\nu}{\widetilde{r}^{8}}\right)\right.$
$\displaystyle\left.+\frac{\left(77175\pi^{2}-1536(2450~{}\bar{\kappa}_{(1)}+28169)\right)\nu^{2}}{501760\widetilde{r}^{9}}-\frac{1269\nu^{3}}{112\widetilde{r}^{9}}-\frac{369\nu}{8\widetilde{r}^{9}}\right)$
$\displaystyle\left.+\widetilde{p}_{r}^{2}\left(-\frac{3\left(-3763200~{}\bar{\kappa}_{(1)}+77175\pi^{2}-659872\right)\nu^{2}}{627200\widetilde{r}^{7}}-\frac{153\nu^{3}}{7\widetilde{r}^{7}}+\frac{117\nu}{8\widetilde{r}^{7}}\right)\right.$
$\displaystyle\left.+\widetilde{p}_{r}^{4}\left(\frac{219\nu^{3}}{8\widetilde{r}^{6}}-\frac{735\nu^{2}}{32\widetilde{r}^{6}}+\frac{9\nu}{2\widetilde{r}^{6}}\right)+\widetilde{p}_{r}^{6}\left(-\frac{6\nu^{4}}{\widetilde{r}^{5}}+\frac{27\nu^{3}}{2\widetilde{r}^{5}}-\frac{45\nu^{2}}{8\widetilde{r}^{5}}+\frac{15\nu}{32\widetilde{r}^{5}}\right)\right.$
$\displaystyle\left.+\frac{\left(705600~{}\bar{\kappa}_{(1)}-1157625\pi^{2}+15841456\right)\nu^{2}}{156800\widetilde{r}^{8}}+\frac{63\nu}{2\widetilde{r}^{8}}\Bigg{]}\right\\}$
$\displaystyle+\left(\widetilde{\bm{Q}}^{ij}_{(1)}\widetilde{\bm{L}}^{i}\widetilde{\bm{L}}^{j}\right)\left\\{\widetilde{p}_{r}^{4}\left(-\frac{21\nu^{4}}{16\widetilde{r}^{5}}-\frac{45\nu^{3}}{4\widetilde{r}^{5}}+\frac{24\nu^{2}}{\widetilde{r}^{5}}-\frac{105\nu}{16\widetilde{r}^{5}}\right)+\widetilde{p}_{r}^{2}\left(\frac{225\nu^{3}}{16\widetilde{r}^{6}}+\frac{577\nu^{2}}{8\widetilde{r}^{6}}+\frac{77\nu}{4\widetilde{r}^{6}}\right)\right.$
$\displaystyle\left.+\widetilde{L}^{4}\left(\frac{15\nu^{2}}{16\widetilde{r}^{9}}-\frac{45\nu^{3}}{16\widetilde{r}^{9}}\right)+\widetilde{L}^{2}\left(\widetilde{p}_{r}^{2}\left(-\frac{3\nu^{4}}{8\widetilde{r}^{7}}-\frac{45\nu^{3}}{4\widetilde{r}^{7}}+\frac{21\nu^{2}}{4\widetilde{r}^{7}}\right)+\frac{9\nu^{3}}{2\widetilde{r}^{8}}+\frac{403\nu^{2}}{16\widetilde{r}^{8}}-\frac{209\nu}{16\widetilde{r}^{8}}\right)\right.$
$\displaystyle\left.+\frac{\left(3763200~{}\bar{\kappa}_{(1)}-77175\pi^{2}-25954688\right)\nu^{2}}{1254400\widetilde{r}^{7}}-\frac{(3675~{}\bar{\kappa}_{(1)}+52288)\nu}{1225\widetilde{r}^{7}}-\frac{3\nu^{3}}{2\widetilde{r}^{7}}\right.$
$\displaystyle\left.+\frac{1}{q}\left[\widetilde{L}^{4}\left(-\frac{45\nu^{3}}{16\widetilde{r}^{9}}+\frac{45\nu^{2}}{16\widetilde{r}^{9}}-\frac{9\nu}{16\widetilde{r}^{9}}\right)+\widetilde{L}^{2}\left(\widetilde{p}_{r}^{2}\left(-\frac{3\nu^{4}}{8\widetilde{r}^{7}}-\frac{12\nu^{3}}{\widetilde{r}^{7}}+\frac{39\nu^{2}}{8\widetilde{r}^{7}}-\frac{3\nu}{4\widetilde{r}^{7}}\right)\right.\right.\right.$
$\displaystyle\left.+\frac{9\nu^{3}}{2\widetilde{r}^{8}}+\frac{325\nu^{2}}{16\widetilde{r}^{8}}-\frac{6\nu}{\widetilde{r}^{8}}\right)+\widetilde{p}_{r}^{4}\left(-\frac{21\nu^{4}}{16\widetilde{r}^{5}}-\frac{237\nu^{3}}{16\widetilde{r}^{5}}+\frac{69\nu^{2}}{8\widetilde{r}^{5}}-\frac{3\nu}{16\widetilde{r}^{5}}\right)$
$\displaystyle\left.\left.+\widetilde{p}_{r}^{2}\left(\frac{15\nu^{3}}{\widetilde{r}^{6}}+\frac{1065\nu^{2}}{16\widetilde{r}^{6}}-\frac{3\nu}{\widetilde{r}^{6}}\right)+\frac{\left(384(9800~{}\bar{\kappa}_{(1)}-56157)-77175\pi^{2}\right)\nu^{2}}{1254400\widetilde{r}^{7}}-\frac{3\nu^{3}}{2\widetilde{r}^{7}}-\frac{123\nu}{4\widetilde{r}^{7}}\right]\right\\}$
$\displaystyle+\left(\widetilde{\bm{Q}}^{ij}_{(1)}\widetilde{\bm{r}}^{\,i}\widetilde{\bm{L}}^{j}\right)\widetilde{p}_{r}\Bigg{\\{}\widetilde{L}^{4}\left(\frac{15\nu^{4}}{16\widetilde{r}^{9}}+\frac{45\nu^{3}}{16\widetilde{r}^{9}}-\frac{9\nu^{2}}{16\widetilde{r}^{9}}\right)+\widetilde{L}^{2}\left(\widetilde{p}_{r}^{2}\left(\frac{63\nu^{4}}{16\widetilde{r}^{7}}+\frac{45\nu^{3}}{8\widetilde{r}^{7}}-\frac{45\nu^{2}}{8\widetilde{r}^{7}}\right)\right.$
$\displaystyle\left.-\frac{105\nu^{3}}{8\widetilde{r}^{8}}+\frac{5\nu^{2}}{2\widetilde{r}^{8}}-\frac{\nu}{2\widetilde{r}^{8}}\right)+\widetilde{p}_{r}^{4}\left(\frac{123\nu^{4}}{16\widetilde{r}^{5}}-\frac{45\nu^{3}}{2\widetilde{r}^{5}}+\frac{129\nu^{2}}{16\widetilde{r}^{5}}\right)+\widetilde{p}_{r}^{2}\left(-\frac{81\nu^{3}}{8\widetilde{r}^{6}}-\frac{229\nu^{2}}{2\widetilde{r}^{6}}+\frac{453\nu}{8\widetilde{r}^{6}}\right)$
$\displaystyle\left.+\frac{\left(77175\pi^{2}-384(9800~{}\bar{\kappa}_{(1)}+8103)\right)\nu^{2}}{156800\widetilde{r}^{7}}+\frac{(235200~{}\bar{\kappa}_{(1)}+2662147)\nu}{9800\widetilde{r}^{7}}+\frac{93\nu^{3}}{7\widetilde{r}^{7}}\right.$
$\displaystyle+\frac{1}{q}\left[\widetilde{L}^{4}\left(\frac{15\nu^{4}}{16\widetilde{r}^{9}}+\frac{3\nu^{3}}{\widetilde{r}^{9}}-\frac{33\nu^{2}}{16\widetilde{r}^{9}}+\frac{3\nu}{16\widetilde{r}^{9}}\right)+\widetilde{L}^{2}\left(\widetilde{p}_{r}^{2}\left(\frac{63\nu^{4}}{16\widetilde{r}^{7}}+\frac{147\nu^{3}}{16\widetilde{r}^{7}}-\frac{45\nu^{2}}{8\widetilde{r}^{7}}+\frac{9\nu}{8\widetilde{r}^{7}}\right)\right.\right.$
$\displaystyle\left.\left.-\frac{15\nu^{3}}{\widetilde{r}^{8}}+\frac{3\nu^{2}}{8\widetilde{r}^{8}}+\frac{3\nu}{\widetilde{r}^{8}}\right)+\widetilde{p}_{r}^{4}\left(\frac{123\nu^{4}}{16\widetilde{r}^{5}}-\frac{81\nu^{3}}{16\widetilde{r}^{5}}-\frac{57\nu^{2}}{16\widetilde{r}^{5}}+\frac{15\nu}{16\widetilde{r}^{5}}\right)+\widetilde{p}_{r}^{2}\left(-\frac{171\nu^{3}}{8\widetilde{r}^{6}}-\frac{555\nu^{2}}{8\widetilde{r}^{6}}+\frac{9\nu}{\widetilde{r}^{6}}\right)\right.$
$\displaystyle\left.\left.+\frac{\left(-3763200~{}\bar{\kappa}_{(1)}+77175\pi^{2}-5256352\right)\nu^{2}}{156800\widetilde{r}^{7}}+\frac{93\nu^{3}}{7\widetilde{r}^{7}}+\frac{30\nu}{\widetilde{r}^{7}}\right]\right\\}$
$\displaystyle+(1\leftrightarrow 2)\,,$ (40a)
$\displaystyle\widetilde{\mathcal{H}}^{\rm FD}_{\rm 3PN}=$
$\displaystyle\left(\widetilde{\bm{S}}_{Q(1)}\cdot\widetilde{\bm{L}}\right)\Bigg{\\{}\widetilde{L}^{4}\left(\frac{11\nu^{3}}{8\widetilde{r}^{7}}+\frac{9\nu^{2}}{16\widetilde{r}^{7}}-\frac{\nu}{4\widetilde{r}^{7}}\right)+\widetilde{L}^{2}\left(\widetilde{p}_{r}^{2}\left(\frac{23\nu^{3}}{4\widetilde{r}^{5}}+\frac{21\nu^{2}}{8\widetilde{r}^{5}}-\frac{2\nu}{\widetilde{r}^{5}}\right)-\frac{17\nu^{3}}{16\widetilde{r}^{6}}\right.$
$\displaystyle\left.-\frac{275\nu^{2}}{16\widetilde{r}^{6}}-\frac{17\nu}{4\widetilde{r}^{6}}\right)+\widetilde{p}_{r}^{2}\left(-\frac{63\nu^{3}}{16\widetilde{r}^{4}}-\frac{207\nu^{2}}{8\widetilde{r}^{4}}+\frac{9\nu}{2\widetilde{r}^{4}}\right)+\widetilde{p}_{r}^{4}\left(\frac{10\nu^{3}}{\widetilde{r}^{3}}-\frac{81\nu^{2}}{8\widetilde{r}^{3}}+\frac{2\nu}{\widetilde{r}^{3}}\right)+\frac{75\nu^{2}}{8\widetilde{r}^{5}}+\frac{25\nu}{2\widetilde{r}^{5}}$
$\displaystyle\left.+\frac{1}{q}\left[\widetilde{L}^{4}\left(\frac{17\nu^{3}}{16\widetilde{r}^{7}}-\frac{9\nu^{2}}{4\widetilde{r}^{7}}+\frac{7\nu}{16\widetilde{r}^{7}}\right)+\widetilde{L}^{2}\left(\widetilde{p}_{r}^{2}\left(\frac{73\nu^{3}}{16\widetilde{r}^{5}}-\frac{3\nu^{2}}{\widetilde{r}^{5}}+\frac{7\nu}{8\widetilde{r}^{5}}\right)-\frac{17\nu^{3}}{16\widetilde{r}^{6}}-\frac{133\nu^{2}}{16\widetilde{r}^{6}}+\frac{27\nu}{8\widetilde{r}^{6}}\right)\right.\right.$
$\displaystyle\left.+\widetilde{p}_{r}^{2}\left(-\frac{63\nu^{3}}{16\widetilde{r}^{4}}-\frac{411\nu^{2}}{16\widetilde{r}^{4}}+\frac{27\nu}{8\widetilde{r}^{4}}\right)+\widetilde{p}_{r}^{4}\left(\frac{131\nu^{3}}{16\widetilde{r}^{3}}-\frac{9\nu^{2}}{2\widetilde{r}^{3}}+\frac{7\nu}{16\widetilde{r}^{3}}\right)+\frac{43\nu^{2}}{8\widetilde{r}^{5}}+\frac{21\nu}{2\widetilde{r}^{5}}\right]\Bigg{\\}}$
$\displaystyle+(1\leftrightarrow 2)\,,$ (40b)
$\displaystyle\widetilde{\mathcal{H}}^{\rm MQ}_{\text{3PN}}=$
$\displaystyle\widetilde{\bm{M}}_{Q(1)}\left\\{\widetilde{L}^{6}\left(\frac{5\nu^{2}}{16\widetilde{r}^{6}}-\frac{5\nu^{3}}{8\widetilde{r}^{6}}\right)+\widetilde{L}^{4}\left(\widetilde{p}_{r}^{2}\left(\frac{15\nu^{2}}{16\widetilde{r}^{4}}-\frac{15\nu^{3}}{8\widetilde{r}^{4}}\right)+\frac{3\nu^{3}}{8\widetilde{r}^{5}}+\frac{\nu^{2}}{\widetilde{r}^{5}}\right)\right.$
$\displaystyle\left.+\widetilde{L}^{2}\left(\widetilde{p}_{r}^{2}\left(\frac{\nu^{3}}{\widetilde{r}^{3}}+\frac{\nu^{2}}{4\widetilde{r}^{3}}+\frac{\nu}{\widetilde{r}^{3}}\right)+\widetilde{p}_{r}^{4}\left(\frac{15\nu^{2}}{16\widetilde{r}^{2}}-\frac{15\nu^{3}}{8\widetilde{r}^{2}}\right)+\frac{3\nu^{2}}{4\widetilde{r}^{4}}+\frac{49\nu}{8\widetilde{r}^{4}}\right)+\widetilde{p}_{r}^{2}\left(\frac{7\nu^{2}}{4\widetilde{r}^{2}}+\frac{10\nu}{\widetilde{r}^{2}}\right)\right.$
$\displaystyle\left.+\widetilde{p}_{r}^{4}\left(\frac{\nu^{3}}{\widetilde{r}}+\frac{5\nu}{8\widetilde{r}}\right)+\left(\frac{5\nu^{2}}{16}-\frac{5\nu^{3}}{8}\right)\widetilde{p}_{r}^{6}+\frac{3\nu^{2}}{4\widetilde{r}^{3}}+\frac{\nu}{2\widetilde{r}^{3}}+\frac{1}{q}\left[\widetilde{L}^{6}\left(-\frac{15\nu^{3}}{16\widetilde{r}^{6}}+\frac{5\nu^{2}}{4\widetilde{r}^{6}}-\frac{5\nu}{16\widetilde{r}^{6}}\right)\right.\right.$
$\displaystyle\left.\left.+\widetilde{L}^{4}\left(\widetilde{p}_{r}^{2}\left(-\frac{45\nu^{3}}{16\widetilde{r}^{4}}+\frac{15\nu^{2}}{4\widetilde{r}^{4}}-\frac{15\nu}{16\widetilde{r}^{4}}\right)+\frac{3\nu^{3}}{8\widetilde{r}^{5}}+\frac{45\nu^{2}}{8\widetilde{r}^{5}}-\frac{15\nu}{8\widetilde{r}^{5}}\right)+\widetilde{L}^{2}\left(\widetilde{p}_{r}^{2}\left(\frac{\nu^{3}}{\widetilde{r}^{3}}+\frac{41\nu^{2}}{4\widetilde{r}^{3}}-\frac{15\nu}{4\widetilde{r}^{3}}\right)\right.\right.\right.$
$\displaystyle\left.\left.\left.+\widetilde{p}_{r}^{4}\left(-\frac{45\nu^{3}}{16\widetilde{r}^{2}}+\frac{15\nu^{2}}{4\widetilde{r}^{2}}-\frac{15\nu}{16\widetilde{r}^{2}}\right)-\frac{23\nu^{2}}{8\widetilde{r}^{4}}-\frac{11\nu}{4\widetilde{r}^{4}}\right)+\widetilde{p}_{r}^{2}\left(-\frac{13\nu^{2}}{4\widetilde{r}^{2}}-\frac{9\nu}{4\widetilde{r}^{2}}\right)\right.\right.$
$\displaystyle\left.\left.+\widetilde{p}_{r}^{6}\left(-\frac{15\nu^{3}}{16}+\frac{5\nu^{2}}{4}-\frac{5\nu}{16}\right)+\widetilde{p}_{r}^{4}\left(\frac{\nu^{3}}{\widetilde{r}}+\frac{5\nu^{2}}{\widetilde{r}}-\frac{15\nu}{8\widetilde{r}}\right)+\frac{3\nu^{2}}{4\widetilde{r}^{3}}-\frac{\nu}{2\widetilde{r}^{3}}\right]\right\\}$
$\displaystyle+(1\leftrightarrow 2)\,.$ (40c)

where,

$\displaystyle\bar{\kappa}_{(1)}(R)=\kappa_{(1)}(R)-\frac{214}{105}\log\left(\frac{r}{R}\right)\,,$
(41)

that combines the terms that depend on the external length scale $R$. The
effective Hamiltonian in a general reference frame is provided in the
ancillary file Hamiltonian-DT.m.

##  6 Adiabatic tides

In this section, we present our results for the adiabatic tides, that is, we
take the $\omega_{f}\rightarrow\infty$ limit of the Hamiltonian (38). This
eliminates the dependence of Hamiltonian on the variables $\bm{Q}_{(a)}$,
$\bm{S}_{Q(a)}$ and $\bm{M}_{Q(a)}$, and hence simplifies our further
calculations. We then compute the binding energy and scattering angle using
the adiabatic Hamiltonian and compare these against known results in the
literature.

The adiabatic limit physically refers to the quadrupole mode being locked to
the external tidal field induced by the binary companion. In this case, the
equation of motion for the $\bm{Q}_{(a)}^{ij}$ is given by

$\displaystyle\bm{Q}^{ij}_{(a)}=-\lambda_{(a)}\bm{E}^{ij}_{(a)}-\lambda_{(a)}\kappa_{d(a)}\frac{G_{d}^{2}m_{(a)}^{2}}{c^{6}}\ddot{\bm{E}}^{ij}_{(a)}\,.$
(42)

We can then substitute Eqs. (42) in the Hamiltonian (38) to obtain the
effective Hamiltonian for adiabatic tides.

Computationally it is efficient to compute dynamic tides and then take the
adiabatic limit, than directly computing the adiabatic tides from the
Lagrangian (3). This is because all the Feynman diagrams generated by (3) will
be factorizable due to $E^{2}$ terms. Hence in this case we can compute 4-loop
($G_{N}^{5}$) terms in the adiabatic observables by doing a 3-loop computation
with the dynamic tides. The counter term for an adiabatic calculation starting
from Lagrangian given in Eq. (3), can be obtained by using Eq. (27) in Eq.
(3).

###  6.1 Effective Hamiltonian

The effective adiabatic Hamiltonian is given by888At leading order, adiabatic
tides contributes at 5PN which can be easily seen writing the Hamiltonian in
the form of the dimensional Love number as shown in Eq. (37). We show here the
relative scaling with respect to the 5PN order contribution.,

$\displaystyle\widetilde{\mathcal{H}}=\widetilde{\mathcal{H}}_{\rm
pp}+\widetilde{\mathcal{H}}_{\rm AT}\ ,$ (43)

where

$\displaystyle\widetilde{\mathcal{H}}_{\rm pp}$
$\displaystyle=\widetilde{\mathcal{H}}_{\text{0PN}}+\left(\frac{1}{c^{2}}\right)\widetilde{\mathcal{H}}_{\text{1PN}}+\left(\frac{1}{c^{4}}\right)\widetilde{\mathcal{H}}_{\text{2PN}}++\left(\frac{1}{c^{6}}\right)\widetilde{\mathcal{H}}_{\text{3PN}}+\mathcal{O}\left(\frac{1}{c^{8}}\right)\
,$ (44a) $\displaystyle\widetilde{\mathcal{H}}_{\rm AT}$
$\displaystyle=\widetilde{\mathcal{H}}^{\text{AT}}_{\text{0PN}}+\left(\frac{1}{c^{2}}\right)\widetilde{\mathcal{H}}^{\text{AT}}_{\text{1PN}}+\left(\frac{1}{c^{4}}\right)\widetilde{\mathcal{H}}^{\text{AT}}_{\text{2PN}}+\left(\frac{1}{c^{6}}\right)\widetilde{\mathcal{H}}^{\text{AT}}_{\text{3PN}}+\mathcal{O}\left(\frac{1}{c^{8}}\right)\
.$ (44b)

The adiabatic Hamiltonian up to 2PN is presented in Mandal:2023lgy  and the
novel result of 3PN Hamiltonian is given as,

$\displaystyle\widetilde{{\cal H}}^{\rm AT}_{3{\rm PN}}=$
$\displaystyle\widetilde{\lambda}_{(1)}\left\\{\widetilde{L}^{6}\left(-\frac{45\nu^{3}}{32\widetilde{r}^{12}}-\frac{15\nu^{2}}{4\widetilde{r}^{12}}-\frac{33\nu}{32\widetilde{r}^{12}}\right)+\widetilde{L}^{4}\left(\widetilde{p}_{r}^{2}\left(-\frac{189\nu^{3}}{32\widetilde{r}^{10}}+\frac{99\nu^{2}}{8\widetilde{r}^{10}}+\frac{711\nu}{32\widetilde{r}^{10}}\right)-\frac{9\nu^{2}}{\widetilde{r}^{11}}+\frac{15\nu}{4\widetilde{r}^{11}}\right)\right.$
$\displaystyle+\widetilde{L}^{2}\left[\widetilde{p}_{r}^{2}\left(-\frac{9\nu^{3}}{2\widetilde{r}^{9}}-\frac{2775\nu^{2}}{16\widetilde{r}^{9}}-\frac{72\nu}{\widetilde{r}^{9}}\right)+\widetilde{p}_{r}^{4}\left(-\frac{99\nu^{3}}{32\widetilde{r}^{8}}+\frac{108\nu^{2}}{\widetilde{r}^{8}}-\frac{1431\nu}{32\widetilde{r}^{8}}\right)\right.$
$\displaystyle\left.+\frac{3(117600~{}\bar{\kappa}_{(1)}+1584771)\nu}{19600\widetilde{r}^{10}}+\frac{93\nu^{2}}{2\widetilde{r}^{10}}\right]+\widetilde{p}_{r}^{2}\left(\frac{9993\nu^{2}}{56\widetilde{r}^{8}}-\frac{3(117600~{}\bar{\kappa}_{(1)}+626121)\nu}{9800\widetilde{r}^{8}}\right)$
$\displaystyle+\widetilde{p}_{r}^{4}\left(-\frac{45\nu^{3}}{\widetilde{r}^{7}}-\frac{729\nu^{2}}{8\widetilde{r}^{7}}+\frac{777\nu}{16\widetilde{r}^{7}}\right)+\widetilde{p}_{r}^{6}\left(\frac{1485\nu^{3}}{32\widetilde{r}^{6}}-\frac{465\nu^{2}}{8\widetilde{r}^{6}}+\frac{465\nu}{32\widetilde{r}^{6}}\right)-\frac{3(29400~{}\bar{\kappa}_{(1)}+429119)\nu}{9800\widetilde{r}^{9}}$
$\displaystyle+\frac{1}{q}\Bigg{[}\widetilde{L}^{6}\left(-\frac{15\nu^{3}}{8\widetilde{r}^{12}}-\frac{75\nu^{2}}{8\widetilde{r}^{12}}-\frac{99\nu}{16\widetilde{r}^{12}}+\frac{33}{32\widetilde{r}^{12}}\right)+\widetilde{L}^{4}\left(\widetilde{p}_{r}^{2}\left(-\frac{9\nu^{3}}{\widetilde{r}^{10}}-\frac{171\nu^{2}}{8\widetilde{r}^{10}}-\frac{27\nu}{16\widetilde{r}^{10}}+\frac{9}{32\widetilde{r}^{10}}\right)\right.$
$\displaystyle\left.+\frac{273\nu^{2}}{16\widetilde{r}^{11}}+\frac{1545\nu}{16\widetilde{r}^{11}}+\frac{495}{16\widetilde{r}^{11}}\right)+\widetilde{L}^{2}\left(\widetilde{p}_{r}^{4}\left(-\frac{99\nu^{3}}{8\widetilde{r}^{8}}+\frac{315\nu^{2}}{8\widetilde{r}^{8}}+\frac{27\nu}{16\widetilde{r}^{8}}-\frac{9}{32\widetilde{r}^{8}}\right)\right.$
$\displaystyle\left.+\widetilde{p}_{r}^{2}\left(-\frac{9\nu^{3}}{2\widetilde{r}^{9}}-\frac{831\nu^{2}}{16\widetilde{r}^{9}}+\frac{567\nu}{8\widetilde{r}^{9}}-\frac{99}{8\widetilde{r}^{9}}\right)+\frac{867\nu^{2}}{28\widetilde{r}^{10}}+\frac{3\left(8384+63\pi^{2}\right)\nu}{512\widetilde{r}^{10}}-\frac{1335}{8\widetilde{r}^{10}}\right)$
$\displaystyle\left.+\widetilde{p}_{r}^{2}\left(\frac{2097\nu^{2}}{16\widetilde{r}^{8}}-\frac{3\left(20576+63\pi^{2}\right)\nu}{256\widetilde{r}^{8}}+\frac{261}{8\widetilde{r}^{8}}\right)+\widetilde{p}_{r}^{4}\left(-\frac{45\nu^{3}}{\widetilde{r}^{7}}-\frac{105\nu^{2}}{8\widetilde{r}^{7}}-\frac{327\nu}{16\widetilde{r}^{7}}+\frac{99}{16\widetilde{r}^{7}}\right)\right.$
$\displaystyle\left.+\widetilde{p}_{r}^{6}\left(\frac{99\nu^{3}}{4\widetilde{r}^{6}}-\frac{69\nu^{2}}{8\widetilde{r}^{6}}-\frac{45\nu}{16\widetilde{r}^{6}}+\frac{15}{32\widetilde{r}^{6}}\right)-\frac{1599\nu^{2}}{56\widetilde{r}^{9}}+\frac{\left(6376-945\pi^{2}\right)\nu}{64\widetilde{r}^{9}}+\frac{519}{4\widetilde{r}^{9}}\Bigg{]}\right\\}$
$\displaystyle+(1\leftrightarrow 2)\,.$ (45)

This Hamiltonian in a general reference frame is provided by us in the
ancillary file Hamiltonian-AT.m. Following the procedure described in Sec. 5.2
of Ref. Mandal:2023lgy , we have validated the above result by computing the
complete Poincare algebra Damour:2000kk ; Levi:2016ofk  using the above
result. The expression of the center of mass $\bm{G}^{i}$ is given in the
ancilary file Poincare_Algebra.m.

###  6.2 Binding energy

In this section, we compute the binding energy in the COM frame for circular
orbits. The gauge invariant relation between the binding energy and the
orbital frequency for circular orbits is obtained by eliminating the
dependence on the radial coordinate. For circular orbits we have
$\partial\widetilde{\mathcal{H}}(\widetilde{r},\widetilde{L})/\partial\widetilde{r}=0\,.$
We invert this relation to express $\widetilde{r}$ as a function of
$\widetilde{L}$. Then, we substitute $\widetilde{L}$, written as a function of
the orbital frequency
$\widetilde{\omega}=\partial\widetilde{\mathcal{H}}(\widetilde{L})/\partial\widetilde{L}$,
in the Hamiltonian (43). Following this procedure we obtain the binding energy
$E$ as,

$\displaystyle E_{\text{AT}}=E^{\rm 0PN}_{\text{AT}}+E^{\rm
1PN}_{\text{AT}}+E^{\rm 2PN}_{\text{AT}}+E^{\rm 3PN}_{\text{AT}}\,,$ (46)

where the energy up to 2PN is presented in Mandal:2023lgy  and the 3PN result
is given as,

$\displaystyle E^{\rm 3PN}_{\text{AT}}$
$\displaystyle=x^{9}\left\\{\left[-\frac{45}{32}\nu^{3}+\frac{74495}{448}\nu^{2}+\left(-\frac{823243}{784}+\frac{8895\pi^{2}}{512}\right)\nu+\frac{894721}{3136}+\frac{321}{7}\left(2\nu-1\right)\log\left(x\widetilde{R}_{\rm
orb}\right)\right]\widetilde{\lambda}_{(+)}\right.$
$\displaystyle\quad\quad\quad\quad+\left.\left[\frac{825}{64}\nu^{2}-\frac{42225}{224}\nu+\frac{378751}{3136}-\frac{321}{7}\log\left(x\widetilde{R}_{\rm
orb}\right)\right]\delta\widetilde{\lambda}_{(-)}-45(\widetilde{\lambda}\kappa)_{(+)}\right\\}\,,$
(47)

where $x=\widetilde{\omega}^{2/3}$, $\widetilde{R}_{\rm orb}=c^{2}R_{\rm
orb}/(G_{N}M)$ and the combined dimensionless Love numbers are denoted by

$\displaystyle\widetilde{\lambda}_{(\pm)}$
$\displaystyle=\frac{m_{(2)}}{m_{(1)}}\widetilde{\lambda}_{(1)}\,\pm\,\frac{m_{(1)}}{m_{(2)}}\widetilde{\lambda}_{(2)}\,,$
(48) $\displaystyle(\widetilde{\lambda}\kappa)_{(\pm)}$
$\displaystyle=\frac{m_{(2)}}{m_{(1)}}\widetilde{\lambda}_{(1)}\kappa_{(1)}(R_{\rm
orb})\,\pm\,\frac{m_{(1)}}{m_{(2)}}\widetilde{\lambda}_{(2)}\kappa_{(2)}(R_{\rm
orb})\,.$ (49)

Here, $\kappa_{(a)}(R_{\rm orb})$ is evaluated at the orbital length scale
$R_{\rm orb}$, after evolving it using the RG Eq. (32) from the matching scale
$R_{\rm NS}$.

###  6.3 Scattering angle

Here we present the scattering angle $\chi$ in the COM frame for the
hyperbolic encounter of two stars. We begin by expressing the Hamiltonian
$\mathcal{H}$ (which is a function of $p_{r}$, $L$ and $r$) to obtain
$p_{r}=p_{r}(\mathcal{H},L,r)$. Then, we use relation between the Lorentz
factor $\gamma$ and the total energy per total rest mass
$\Gamma=\mathcal{H}/(Mc^{2})$ given by

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

where $v\equiv|\dot{\bm{r}}|$ is the relative velocity of the compact objects,
and the total angular momentum $L$ and the impact parameter $b$ are related by
$L=(\mu\gamma vb)/\Gamma$. This allows us to exchange $\mathcal{H}$ for $v$
and $L$ for $b$. With this, we can then write the scattering angle as

$\displaystyle\chi(v,b)=-\frac{\gamma}{\mu\gamma v}\,\int{\rm
d}r\,\frac{\partial p_{r}(v,b,r)}{\partial b}-\pi\,.$ (51)

Performing this procedure with the Hamiltonian (43) yields the scattering
angle computed in the COM frame, which we write as

$\displaystyle\chi_{\rm AT}=\chi^{\rm 0PN}_{\rm AT}+\chi^{\rm 1PN}_{\rm
AT}+\chi^{\rm 2PN}_{\rm AT}+\chi^{\rm 3PN}_{\rm AT}$ (52)

where the scattering angle up to 2PN is presented in Mandal:2023lgy  and the
3PN result is given as 999For computing the logarithmic terms in the
scattering angle, check Ref. Bini:2017wfr .

$\displaystyle\frac{\chi^{\rm 3PN}_{\rm AT}}{\Gamma}$
$\displaystyle=\frac{1}{Mb^{4}}\begin{bmatrix}\lambda_{\rm(+)}&~{}\delta\lambda_{\rm(-)}\end{bmatrix}\cdot\left\\{\pi\left(\frac{G_{N}M}{v^{2}b}\right)^{2}\frac{1575}{256}\begin{bmatrix}1\\\
0\end{bmatrix}\left(\frac{v^{6}}{c^{6}}\right)+\left(\frac{G_{N}M}{v^{2}b}\right)^{3}\frac{1}{70}\begin{bmatrix}18073\\\
2713\end{bmatrix}\left(\frac{v^{6}}{c^{6}}\right)\right.$
$\displaystyle\quad\left.+\,\pi\,\left(\frac{G_{N}M}{v^{2}b}\right)^{4}\left(\frac{1}{458752}\begin{bmatrix}304535296-\left(46848512+231525\pi^{2}\right)\nu\\\
51930496\end{bmatrix}+\frac{1605}{64}\log\left(\frac{2b}{R_{\rm
sc}}\right)\begin{bmatrix}1-2\nu\\\
1\end{bmatrix}\right)\left(\frac{v^{6}}{c^{6}}\right)\right.$
$\displaystyle\quad\left.+\,\left(\frac{G_{N}M}{v^{2}b}\right)^{5}\left\\{192\begin{bmatrix}1\\\
0\end{bmatrix}+48\begin{bmatrix}53-4\nu\\\
5\end{bmatrix}\left(\frac{v^{2}}{c^{2}}\right)+\frac{12}{35}\begin{bmatrix}28753-3480\nu\\\
7473-140\nu\end{bmatrix}\left(\frac{v^{4}}{c^{4}}\right)\right.\right.$
$\displaystyle\quad\quad\quad\quad\left.+\left(\frac{1}{8575}\begin{bmatrix}\left(584325\pi^{2}-68190952\right)\nu+128915306\\\
33894506-2486400\nu\end{bmatrix}+\frac{903936}{1225}\log\left(\frac{b}{2R_{\rm
sc}}\right)\begin{bmatrix}1-2\nu\\\
1\end{bmatrix}\right)\left(\frac{v^{6}}{c^{6}}\right)\right\\}\Bigg{\\}}$
$\displaystyle\quad+\frac{1}{Mb^{4}}\begin{bmatrix}(\lambda\kappa)_{\rm(+)}&~{}\delta(\lambda\kappa)_{\rm(-)}\end{bmatrix}\cdot\left\\{\pi\,\left(\frac{G_{N}M}{v^{2}b}\right)^{4}\frac{1575}{64}\begin{bmatrix}-\nu\\\
0\end{bmatrix}\left(\frac{v^{6}}{c^{6}}\right)+\,\left(\frac{G_{N}M}{v^{2}b}\right)^{5}\frac{25344}{35}\begin{bmatrix}-\nu\\\
0\end{bmatrix}\left(\frac{v^{6}}{c^{6}}\right)\right\\}$
$\displaystyle\quad+\mathcal{O}\left(G_{N}^{6},\frac{v^{8}}{c^{8}}\right)\,,$
(53)

where combined Love numbers are denoted by,

$\displaystyle{\lambda}_{(\pm)}$
$\displaystyle=\frac{m_{(2)}}{m_{(1)}}{\lambda}_{(1)}\,\pm\,\frac{m_{(1)}}{m_{(2)}}{\lambda}_{(2)}\,,$
(54) $\displaystyle({\lambda}\kappa)_{(\pm)}$
$\displaystyle=\frac{m_{(2)}}{m_{(1)}}{\lambda}_{(1)}\kappa_{(1)}(R_{\rm
sc})\,\pm\,\frac{m_{(1)}}{m_{(2)}}{\lambda}_{(2)}\kappa_{(2)}(R_{\rm sc})\,.$
(55)

Here, $\kappa_{(a)}(R_{\rm sc})$ is evaluated at the typical length scale
$R_{\rm sc}$ of the scattering system, after evolving it using the RG Eq. (32)
from the matching scale $R_{\rm NS}$.

##  7 Conclusions

We calculated the conservative two-body effective Hamiltonian, taking into
account the dynamical tidal interaction up to 3PN order. Our approach involves
Feynman diagrams up to three loops, which are evaluated using the dimensional
regularization scheme. We also studied the adiabatic limit and obtained the
adiabatic tidal Hamiltonian up to 3PN order. The fulfillment of Poincaré
algebra constituted an important validation of the Hamiltonian. Additionally,
we presented the analytic expressions of two gauge-invariant observables,
namely the binding energy for bound orbits and the scattering angle for
hyperbolic encounters.

In the considered action, we included the contribution from the non-minimal
coupling (last term of Eq. (1)), which turned out to be crucial for ensuring
the finite observables at the 3PN order. Interestingly, we found that not all
the divergent pieces can be removed by adding a total derivative to the
Lagrangian (canonical transformations on the Hamiltonian). Therefore, we
constructed a counter-term to reabsorb those divergent pieces, yielding the
renormalization of the the post-adiabatic Love number $\kappa_{(a)}$.
Consequently, the latter experiences a renormalization group running, as shown
in Sec. 4.2. According to the corresponding beta function, we observed that
the value of $\kappa_{(a)}$ increases, when reducing the external length scale
$R$.

In the adiabatic limit, our results depend on the two Love numbers (Wilson
coefficients within the context of point particle EFT), namely, $\lambda$ (the
tidal Love number) and $\kappa$ (the post-adiabatic Love number). The
determination of their specific values in the case of BHs requires a matching
computation of an observable between the EFT and BHPT, as described in
Kol:2011vg ; Chia:2020yla ; Hui:2020xxx ; Charalambous:2021mea ; Hui:2021vcv ;
Ivanov:2022qqt ; Ivanov:2022hlo . Notably, the tidal Love number $\lambda$ is
found to be zero for BHs Damour:1982wm ; damour_1984_love_number ; Kol:2011vg
; Damour:2009vw ; Binnington:2009bb . However, the specific value of the post-
adiabatic Love number $\kappa$ remains unknown and requires further
investigation. In the case of NSs, Love numbers are typically expressed in
terms of their EOS. For $\lambda$, this was accomplished in Refs.
Hinderer:2007mb ; Flanagan:2007ix , and $\kappa$ in Ref. Chakrabarti:2013lua .
It will be interesting to compute the particular values of $\kappa$ for
different compact objects Cardoso:2017cfl ; Brustein:2020tpg ; Raposo:2018rjn
. The implications of the flow of the post-adiabatic Love number on the EOS of
NS are very intriguing and warrant further investigation.

The adiabatic Love number $\lambda$ was found to have a non-zero value for
Schwarzschild BHs in spacetime dimensions other than four, as shown in Ref.
Kol:2011vg . Consequently, when using dimensional regularization, the Love
number $\lambda$ yields a finite contribution at $\mathcal{O}(\epsilon=d-3)$,
which, in combination with the divergent terms generated by the
$\mathcal{L}_{\rm EQ}$, results in a finite $\mathcal{O}(\epsilon^{0})$
contribution. However, such contributions to observables could be absorbed
into the contributions from the counterterm, so that the number of degrees of
freedom to be matched does not increase. Furthermore, matching in arbitrary
dimension $d$ or in $d=3$ will in general result in different values for
$\lambda$ and $\kappa$, but observables are the same in the limit
$\epsilon\rightarrow 0$. In this context, it is also very interesting to
explore the role of evanescent operators Collins:1984xc , which are non-zero
only in spacetime dimensions other than four. It was observed in Dugan:1990df
; Buras:1989xd ; Herrlich:1994kh , that the scattering amplitudes from such
operators can affect the matching equation determining the Wilson
coefficients. Also the appearance of such operators in counterterms
originating from physical operators leads to the mixing of physical and
evanescent operators during the RG evolution Dugan:1990df .

In this work, we focused on the dynamical gravitoelectric quadrupolar tides,
but our framework and our automatic computational techniques are general
enough to be extended to obtain further higher-order corrections and also
other types of tidal effects, like the dynamical gravitomagnetic tides
Gupta:2020lnv ; Gupta:2023oyy , as well as to incorporate higher order
multipolar tides. The coupling of the oscillation modes of the NS with other
degrees of freedom, such as its spin Ho:1998hq ; Steinhoff:2021dsn ;
Ma:2020rak ; Bhatt:2023zsy , or other oscillation modes 2012ApJ…751..136W ;
2013ApJ…769..121W ; Landry:2015snx ; Xu:2017hqo ; Essick:2016tkn ; Yu:2022fzw
could also be incorporated. Finally, following Refs. Hinderer:2016eia ;
Steinhoff:2016rfi , our 3PN Hamiltonians can be added into the time-domain
effective-one-body waveform models to improve their agreement with numerical
relativity simulations.

### Acknowledgements

We thank Sumanta Chakraborty, Thibault Damour, Quentin Henry, Mikhail Ivanov,
Gustav Jakobsen, Jung-Wook Kim, Oliver Long, Elisa Maggio, Saketh MVS and Ira
Rothstein for insightful comments and discussions. 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. H.O.S acknowledges funding
from the Deutsche Forschungsgemeinschaft (DFG) - Project No. 386119226. R.P.’s
research is funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation), Projektnummer 417533893/GRK2575 “Rethinking Quantum
Field Theory”.

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