Majorana Collaboration

# Exotic dark matter search with the Majorana Demonstrator

I.J. Arnquist  Pacific Northwest National Laboratory, Richland, WA 99354, USA
F.T. Avignone III  Department of Physics and Astronomy, University of South
Carolina, Columbia, SC 29208, USA  Oak Ridge National Laboratory, Oak Ridge,
TN 37830, USA  A.S. Barabash  National Research Center “Kurchatov Institute”
Institute for Theoretical and Experimental Physics, Moscow, 117218 Russia
C.J. Barton  Department of Physics, University of South Dakota, Vermillion, SD
57069, USA  K.H. Bhimani  Department of Physics and Astronomy, University of
North Carolina, Chapel Hill, NC 27514, USA  Triangle Universities Nuclear
Laboratory, Durham, NC 27708, USA  E. Blalock  Department of Physics, North
Carolina State University, Raleigh, NC 27695, USA  Triangle Universities
Nuclear Laboratory, Durham, NC 27708, USA  B. Bos  Department of Physics and
Astronomy, University of North Carolina, Chapel Hill, NC 27514, USA  Triangle
Universities Nuclear Laboratory, Durham, NC 27708, USA  M. Busch  Department
of Physics, Duke University, Durham, NC 27708, USA  Triangle Universities
Nuclear Laboratory, Durham, NC 27708, USA  M. Buuck  Present address: SLAC
National Accelerator Laboratory, Menlo Park, CA 94025, USA  Center for
Experimental Nuclear Physics and Astrophysics, and Department of Physics,
University of Washington, Seattle, WA 98195, USA  T.S. Caldwell  Department of
Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514,
USA  Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA  Y-D.
Chan  Nuclear Science Division, Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, USA  C.D. Christofferson  South Dakota Mines, Rapid City,
SD 57701, USA  P.-H. Chu  Los Alamos National Laboratory, Los Alamos, NM
87545, USA  M.L. Clark  Department of Physics and Astronomy, University of
North Carolina, Chapel Hill, NC 27514, USA  Triangle Universities Nuclear
Laboratory, Durham, NC 27708, USA  C. Cuesta  Centro de Investigaciones
Energéticas, Medioambientales y Tecnológicas, CIEMAT 28040, Madrid, Spain
J.A. Detwiler  Center for Experimental Nuclear Physics and Astrophysics, and
Department of Physics, University of Washington, Seattle, WA 98195, USA  Yu.
Efremenko  Department of Physics and Astronomy, University of Tennessee,
Knoxville, TN 37916, USA  Oak Ridge National Laboratory, Oak Ridge, TN 37830,
USA  H. Ejiri  Research Center for Nuclear Physics, Osaka University, Ibaraki,
Osaka 567-0047, Japan  S.R. Elliott  Los Alamos National Laboratory, Los
Alamos, NM 87545, USA  G.K. Giovanetti  Physics Department, Williams College,
Williamstown, MA 01267, USA  M.P. Green  Department of Physics, North Carolina
State University, Raleigh, NC 27695, USA  Triangle Universities Nuclear
Laboratory, Durham, NC 27708, USA  Oak Ridge National Laboratory, Oak Ridge,
TN 37830, USA  J. Gruszko  Department of Physics and Astronomy, University of
North Carolina, Chapel Hill, NC 27514, USA  Triangle Universities Nuclear
Laboratory, Durham, NC 27708, USA  I.S. Guinn  Department of Physics and
Astronomy, University of North Carolina, Chapel Hill, NC 27514, USA  Triangle
Universities Nuclear Laboratory, Durham, NC 27708, USA  V.E. Guiseppe  Oak
Ridge National Laboratory, Oak Ridge, TN 37830, USA  C.R. Haufe  Department of
Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514,
USA  Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA  R.
Henning  Department of Physics and Astronomy, University of North Carolina,
Chapel Hill, NC 27514, USA  Triangle Universities Nuclear Laboratory, Durham,
NC 27708, USA  D. Hervas Aguilar  Department of Physics and Astronomy,
University of North Carolina, Chapel Hill, NC 27514, USA  Triangle
Universities Nuclear Laboratory, Durham, NC 27708, USA  E.W. Hoppe  Pacific
Northwest National Laboratory, Richland, WA 99354, USA  A. Hostiuc  Center for
Experimental Nuclear Physics and Astrophysics, and Department of Physics,
University of Washington, Seattle, WA 98195, USA  M.F. Kidd  Tennessee Tech
University, Cookeville, TN 38505, USA  I. Kim<EMAIL_ADDRESS>Los Alamos
National Laboratory, Los Alamos, NM 87545, USA  R.T. Kouzes  Pacific Northwest
National Laboratory, Richland, WA 99354, USA  T.E. Lannen V  Department of
Physics and Astronomy, University of South Carolina, Columbia, SC 29208, USA
A. Li  Department of Physics and Astronomy, University of North Carolina,
Chapel Hill, NC 27514, USA  Triangle Universities Nuclear Laboratory, Durham,
NC 27708, USA  A.M. Lopez  Department of Physics and Astronomy, University of
Tennessee, Knoxville, TN 37916, USA  J.M. López-Castaño  Oak Ridge National
Laboratory, Oak Ridge, TN 37830, USA  E.L. Martin  Present address: Duke
University, Durham, NC 27708  Department of Physics and Astronomy, University
of North Carolina, Chapel Hill, NC 27514, USA  Triangle Universities Nuclear
Laboratory, Durham, NC 27708, USA  R.D. Martin  Department of Physics,
Engineering Physics and Astronomy, Queen’s University, Kingston, ON K7L 3N6,
Canada  R. Massarczyk  Los Alamos National Laboratory, Los Alamos, NM 87545,
USA  S.J. Meijer  Los Alamos National Laboratory, Los Alamos, NM 87545, USA
S. Mertens  Max-Planck-Institut für Physik, München, 80805, Germany  Physik
Department and Excellence Cluster Universe, Technische Universität, München,
85748 Germany  T.K. Oli  Department of Physics, University of South Dakota,
Vermillion, SD 57069, USA  G. Othman  Present address: Universität Hamburg,
Institut für Experimentalphysik, Hamburg, Germany  Department of Physics and
Astronomy, University of North Carolina, Chapel Hill, NC 27514, USA  Triangle
Universities Nuclear Laboratory, Durham, NC 27708, USA  L.S. Paudel
Department of Physics, University of South Dakota, Vermillion, SD 57069, USA
W. Pettus  Department of Physics, Indiana University, Bloomington, IN 47405,
USA  IU Center for Exploration of Energy and Matter, Bloomington, IN 47408,
USA  A.W.P. Poon  Nuclear Science Division, Lawrence Berkeley National
Laboratory, Berkeley, CA 94720, USA  D.C. Radford  Oak Ridge National
Laboratory, Oak Ridge, TN 37830, USA  A.L. Reine  Department of Physics and
Astronomy, University of North Carolina, Chapel Hill, NC 27514, USA  Triangle
Universities Nuclear Laboratory, Durham, NC 27708, USA  K. Rielage  Los Alamos
National Laboratory, Los Alamos, NM 87545, USA  N.W. Ruof  Center for
Experimental Nuclear Physics and Astrophysics, and Department of Physics,
University of Washington, Seattle, WA 98195, USA  D.C. Schaper  Los Alamos
National Laboratory, Los Alamos, NM 87545, USA  D. Tedeschi  Department of
Physics and Astronomy, University of South Carolina, Columbia, SC 29208, USA
R.L. Varner  Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA  S.
Vasilyev  Joint Institute for Nuclear Research, Dubna, 141980 Russia  J.F.
Wilkerson  Department of Physics and Astronomy, University of North Carolina,
Chapel Hill, NC 27514, USA  Triangle Universities Nuclear Laboratory, Durham,
NC 27708, USA  Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA  C.
Wiseman<EMAIL_ADDRESS>Center for Experimental Nuclear Physics and
Astrophysics, and Department of Physics, University of Washington, Seattle, WA
98195, USA  W. Xu  Department of Physics, University of South Dakota,
Vermillion, SD 57069, USA  C.-H. Yu  Oak Ridge National Laboratory, Oak Ridge,
TN 37830, USA  B.X. Zhu  Present address: Jet Propulsion Laboratory,
California Institute of Technology, Pasadena, CA 91109, USA  Los Alamos
National Laboratory, Los Alamos, NM 87545, USA

###### Abstract

With excellent energy resolution and ultra-low level radiogenic backgrounds,
the high-purity germanium detectors in the Majorana Demonstrator enable
searches for several classes of exotic dark matter (DM) models. In this work
we report new experimental limits on keV-scale sterile neutrino DM via the
transition magnetic moment from conversion to active neutrinos,
$\nu_{s}\rightarrow\nu_{a}$. We report new limits on fermionic dark matter
absorption ($\chi+A\rightarrow\nu+A$) and sub-GeV DM-nucleus 3$\rightarrow$2
scattering ($\chi+\chi+A\rightarrow\phi+A$), and new exclusion limits for
bosonic dark matter (axionlike particles and dark photons). These searches
utilize the 1–100 keV low energy region of a 37.5 kg-y exposure collected by
the Demonstrator between May 2016 and Nov. 2019, using a set of 76Ge-enriched
detectors whose surface exposure time was carefully controlled, resulting in
extremely low levels of cosmogenic activation.

††preprint: APS/123-QED

As large-scale dark matter experiments have rejected much of the traditional
WIMP parameter space, interest in alternative dark matter models has
increased. Popular models include light (sub-GeV) dark matter, axionlike
particles (ALP), dark photons, and keV-scale sterile neutrinos. Experimental
searches have previously placed limits on bosonic dark matter in the 1–100 keV
range (scalar or axionlike particles and vector or dark photons) [1, 2], with
the most stringent limits to date on the scalar and vector couplings set by
XENON1T [3]. GERDA has set limits up to 1 MeV [4], while DAMIC has achieved
the lowest electron recoil energy thresholds of 200 eV [5], and nuclear-recoil
detectors such as CRESST have achieved sensitivity to nuclear recoil
ionization to 30 eV [6]. Recently, interest has increased in fermionic dark
matter, which may convert to neutrinos upon interaction with a target [7, 8,
9, 10]. A low-background detector array with excellent energy resolution [11]
is an ideal environment to search for rare peaks from several exotic DM
models.

Figure 1: Energy spectra from natGe and enrGe HPGe detectors, with a 1 keV
lower threshold. The spectrum in the natGe below 20 keV is tritium-dominated.
Lower levels of tritium, 55Fe, 65Zn, 68Ge show that surface exposure control
of the enrGe material significantly reduced cosmogenic activation. The 65Zn
contribution at 9.0 keV is small but nonzero, and can be seen in Fig. 3. The
average enrGe count rate is 0.011 $\pm$ 0.002 counts/(keV kg d) in the 20–40
keV range.

The Majorana Demonstrator conducted a search for neutrinoless double-beta
decay ($\beta\beta(0\nu)$) from 2015 to 2019 with a 29.7-kg set of 76Ge-
enriched high-purity germanium (HPGe) p-type point contact (PPC) detectors
manufactured by AMETEK-ORTEC [12, 13], at the 4850’ level of the Sanford
Underground Research Facility in Lead, South Dakota [14, 15]. In 2020 a set of
inverted-coaxial PPC detectors were swapped in for evaluation prior to use in
the LEGEND-200 experiment [16]. A final exposure of $\sim$65 kg-y was
collected before all enrGe detectors were removed in 2021 for installation in
the LEGEND-200 experiment. The Demonstrator continues to operate with 14.3 kg
of natural-abundance Ge detectors for background studies and other rare-event
searches.

From the primary data taking period, 37.5 kg-y enrGe and 15.1 kg-y natGe
exposure were selected for an exotic dark matter analysis in the lowest-energy
region of the spectrum, 1–100 keV. The energy estimator, time-dependent
detector selection, and muon veto cut are computed in the $\beta\beta(0\nu)$
analysis, and additional low energy pulse shape parameters are computed for
data cleaning and surface event rejection. An event multiplicity cut is
applied to select only single-detector events. The DM-matter interaction
cross-section for all models considered here is known to be extremely small,
and the probability of DM multiple-scattering in the detector array is
negligible for this exposure. Multi-detector events from lightly ionizing
particles (LIPs) were searched for in Ref. [17]. A technical paper with
additional details on the low energy data cleaning and efficiency
determination is in preparation.

Figure 2: Total efficiency and uncertainty for enrGe and natGe detectors. A
fit to a cumulative Weibull distribution (red) gives the central value, while
the gray region gives the data-driven $1\sigma$ uncertainty, with
contributions from changing energy thresholds and pulse shape analysis cuts.

The surface exposure of the enrGe detectors was carefully limited during
fabrication and storage, resulting in the lowest cosmogenic activation of any
Ge experiment to date and increased sensitivity to low-statistics rare events.
Since the surface exposure time of the natGe detectors was not limited, they
show a strong tritium feature and associated cosmogenic lines, while the enrGe
detectors show significantly reduced 68Ge, 65Zn, and tritium. Fig 1 gives the
energy spectrum of the enrGe and natGe detectors used in this analysis after
data cleaning.

Energy-degraded events near the millimeter-thick Ge(Li) n+ detector surfaces
are known to produce pulses with a slower rising edge that may be rejected by
pulse shape analysis [18]. Our algorithm fits each wavelet-denoised waveform
to a sigmoid function, giving pulse shape rejection capability to 1 keV,
similar to Ref. [19]. The turnover of the natGe spectrum at energies below 5
keV shows the successful rejection of slow pulse signals by the data cleaning
algorithm. The enrGe spectrum shows a rising spectral shape below 10 keV which
persists after aggressive slow pulse cuts, indicating the excess signal is
dominated by fast events. These may originate from ionization in the main
fiducial (bulk) volume, events near the p+ contact, or from the micrometer-
thick amorphous Ge passivation layer for the PPC detector geometry [20].
Despite detector storage in nitrogen environments, residual contamination of
the detector component surfaces by long-lived Rn progeny including 210Pb,
210Bi, and 210Po plausibly explain the signal. We observe the 46.5 keV peak
from 210Pb at the same intensity in both sets of detectors, indicating roughly
uniform distribution of the source in the array, but notably do not observe
the associated 10.8 keV line, which may not penetrate the passivated surface
region. Low energy $\beta$ emission from this decay chain is also expected,
which may penetrate the region and cause the rise.

The signal acceptance efficiency of the pulse shape analysis cuts is computed
by considering the time-dependent detector energy thresholds, and the
detector-specific efficiency of the n+ slow pulse cut. This is determined from
weekly 228Th calibration data using the 238.6 keV line from 212Pb, which is
not emitted in coincidence with other gamma rays. Paired detector events with
sum energy $\sim$239 keV are likely to include a small-angle Compton scatter
in the first detector, which can deposit as little as 1 keV with a mean free
path of 14 mm, well into the bulk region. The combined efficiency for enrGe
and natGe detectors is given in Fig. 2, and each is fit to the cumulative
distribution of a Weibull function, a heuristic choice of an asymmetric
function [21]. The fit constants (shape, location, scale, amplitude) are
(0.4633, 0.9866, 0.4960, 0.9283) for the enrGe and (0.5922, 0.9880, 1.9911,
0.9394) for the natGe detectors. The efficiency in the enrGe is 15.9% at 1
keV, rising to 79.1% at 3 keV.

In this Letter, we give exclusion limits for several classes of DM models
using the 37.5 kg-y exposure obtained from the enrGe detectors in the
Demonstrator. The models considered here are expected to produce a sharp peak
with a width close to the expected energy resolution, $\sigma_{D}(E)$ [15]. We
scan for a peak at energies $E_{i}$ between 1–100 keV, sampling at intervals
proportional to half the expected detector energy resolution, ranging from
0.15 keV FWHM at 1 keV to 0.23 keV at 100 keV, taking a 30% resolution
uncertainty based on studies of cosmogenic and 228Th calibration peak widths.
At each energy we evaluate the maximum possible counts attributable to a rare
signal, $N_{U}$. To set an upper limit on the various couplings of interest at
each $E_{i}$, we perform an unbinned extended maximum likelihood fit in a
$\pm$(7$\sigma_{D}$ \+ 1) keV window, weighting the counts by the analytic cut
efficiency and its uncertainty shown in Fig. 2. We profile the likelihood
function to a 90% confidence level (CL) to identify any excess in the number
of counts at each peak location, and we find no significant peak-like excess
in the 1–100 keV region that do not originate from known backgrounds. The
total probability density function $\mathcal{P}$ used for each local spectrum
model at $E_{i}$ is given by

$\mathcal{P}(E_{i})=n_{0}\mathcal{P}_{\mathrm{pol2}}+\sum^{n_{\mathrm{pks}}}n_{i}\mathcal{P}_{G,i}+N_{U}\mathcal{P}_{\mathrm{rare}}$
(1)

where $\mathcal{P}_{\mathrm{rare}}$ are the Gaussian-distributed rare signal
peaks at varying energies, and $\mathcal{P}_{G,i}$ are the Gaussian PDFs for
known cosmogenic peaks. The yields (counts) $n_{i}$ are convolved with the cut
efficiency and its uncertainty at each energy $E_{i}$ as Gaussian constraints
in each unbinned fit. When the rare peak $\mathcal{P}_{\mathrm{rare}}$ is
degenerate with a known cosmogenic line, $N_{U}$ is constrained to include the
full yield $n_{i}$ at the peak. $\mathcal{P}_{\mathrm{poly}}$ is a second-
order Chebyshev polynomial estimation for the continuum shape in each fit
window. This heuristic choice assigns the least strength to background counts
in the window, and requires only an initial guess for the yields $n_{i}$. The
expected number of detected counts at energy $E$ for a mass $m_{\chi}$ is
given by

$\frac{dN}{dE}=\Phi_{\mathrm{DM}}(m_{\chi})\ \sigma(m_{\chi})\ \eta(E)\
\mathcal{P}_{\mathrm{rare}}(E)\ MT\ \frac{m_{\mathrm{enr}}}{m_{\mathrm{nat}}}$
(2)

Here, $\Phi_{\mathrm{DM}}$ is the dark matter flux, the exposure $MT$ is in
kg-y, and the detection efficiency $\eta(E)$ is convolved with the rare signal
peak $\mathcal{P}_{\mathrm{rare}}$. The rate is weighted by the relative
molecular weights of the enrGe detectors (75.668 $\pm$ 0.010 g/mol) to the
standard weight of natGe, 72.63 g/mol. The interaction cross-section $\sigma$
varies with the DM model of interest. An example calculation of $N_{U}$ at
10.8 keV is shown in Fig. 3.

Figure 3: An example fit to the energy spectrum that provides a calculation of
the 90% UL for a rare peak at 10.8 keV, where a 210Pb line is possible but not
seen. Initial guesses for the cosmogenic yields are taken from a preliminary
fit without the signal peak. When the signal peak has any overlap with a
cosmogenic line, the cosmogenic heights are minimized to maximize $N_{U}$.

Sterile neutrino transition magnetic moment.– Sterile neutrinos have been
proposed as a cold (nonrelativistic) DM candidate [22, 23, 24, 25, 26, 27],
capable of producing a detectable upscattering event in a DM detector. The
nonzero mass of the neutrino allows radiative decay between states [28],
including transitions of heavy right-handed sterile neutrinos $\nu_{s}$ with
keV-MeV scale masses into active $\nu_{a}=\nu_{e,\mu,\tau}$. Radiative decay
of a 7.1 keV DM $\nu_{s}$ may explain anomalous X-ray emission lines at
$\sim$3.5 keV observed by the XMM-Newton space observatory [29, 30], though
the explanation is contested [31].

Current best limits on any neutrino transition magnetic moment (TMM) from
active-to-sterile conversion come from solar neutrino-electron scattering in
Borexino ($\nu_{\mu}\rightarrow\nu_{s}$), with the best limit $\mu_{\mu
s}<7\times 10^{-11}\ \mu_{B}$ for sterile neutrino masses 10–100 keV [32].
Flavor-dependent couplings have been proposed to avoid constraints from
supernova SN1987A and the CMB [33, 34]. An active-to-sterile transition
($\nu_{a}\rightarrow\nu_{s}$) from incident solar neutrinos via TMM may also
account for the excess of events below 5 keV observed by XENON1T [35, 36].
While this interpretation does not depend on any DM assumption on $\nu_{s}$, a
limit on $\mu_{sa}$ can impact both the XENON1T excess interpretation and the
dark matter sterile neutrino hypothesis.

We consider the atomic ionization process by incoming sterile neutrinos
$\nu_{s}+A\rightarrow\nu_{a}+A^{+}+e^{-}$. In this process, the limit on the
4-momentum transfer $q^{2}\rightarrow 0$ is kinematically accessible due to
the two-body atomic final state of the positive ion plus the ionization
electron [37]. Near this limit, the atomic ionization process can be viewed as
a two-step process, where the exchange photon emitted from the incoming
$\nu_{s}$ interacts coherently with an atom in the detector target, producing
ionization. The singularity due to the real photon pole in the interaction
cross section is accessed, and the cross section is enhanced by orders of
magnitude at the energy transfer $T=m_{s}/2$, half the mass of the sterile
neutrino [37]. This would produce a peak-like signature in the energy
spectrum. Considering this resonance in presence of a rich $\nu_{s}$ source
allows more stringent limits to be set on $\mu_{sa}$ than previous limits from
solar neutrino-electron scattering [37]. In this analysis, we assume that the
local DM halo consists of $\nu_{s}$, for comparison with Ref. [37] 111The
original $\mu_{sa}$ limit using the same postulate in Ref. [37] was found to
be erroneous by a factor 14, which has been taken into account in Fig. 4. An
erratum is in preparation by the authors [38]..

Near the resonant energy $E=m_{s}/2$, the equivalent photon approximation
(EPA) becomes valid [37]. Within the interval $E=m_{s}/2\ \pm\
|\vec{k}_{s}|/2$, where $\vec{k}_{s}$ is the sterile neutrino momentum vector,
the amplified differential cross section has the form

$\frac{d\sigma(m_{s},v)}{dE}\approx\Big{(}\frac{\mu_{sa}}{2m_{e}}\Big{)}^{2}\frac{\alpha}{2n_{A}}\frac{m_{s}^{2}}{|v|^{2}}~{},$
(3)

where $\alpha$ is the electromagnetic fine structure constant, $n_{A}$ is the
number density of Ge atoms and $v$ is the velocity of the incoming sterile
neutrino. As a consequence, a peaked signature at $E=m_{s}/2$ is expected.
With Eq. 3, we can set a limit on $\mu_{sa}$ in terms of the incoming sterile
neutrino flux at any value of $m_{s}$. For any dark matter model, the DM
interaction rate per unit mass in a terrestrial detector with isotopic mass
$m_{A}$ can be calculated by

$\frac{dR}{dE}=\frac{\rho_{\chi}}{m_{\chi}m_{A}}\int^{\infty}_{u_{\mathrm{min}}}\Big{[}\frac{d\sigma(m_{s},\vec{u})}{dE}uf(\vec{u})\Big{]}d^{3}u$
(4)

where $\vec{u}=\vec{v}+\vec{v_{e}}$ is the velocity of the dark matter in the
Earth’s reference frame, $\vec{v}$ is the velocity of the dark matter in the
Galactic rest frame, $\vec{v_{e}}=\vec{v_{e}}(t)$ is the circular motion of
the Earth in the Galactic frame, $u_{\mathrm{min}}$ is the minimum speed of
the DM to produce detectable recoil and $f(\vec{u})$ is the DM velocity
distribution in the Earth’s reference frame. If the sterile neutrinos are
taken as a DM candidate, $m_{\chi}=m_{s}$, and the standard local DM density
$\rho_{\chi}=0.4$ GeV/cm3.

We use a property of the EPA cross section, which has a plateau at $E=m_{e}/2$
with $u^{-2}$-dependent height and $u$-dependent width. All $u$-dependence in
the integrand of Eq. 4 is cancelled except $f(\vec{u})$, which integrates to
unity by definition. The expression for the event rate $R$ is independent of
the specific DM velocity distribution.

With the EPA differential cross section at the resonant energy and setting
$u_{\mathrm{min}}=0$ for inelastic scattering, Eq. 4 can be integrated to get
an expression of $R$, and set equal to the experimental upper limit:

$R=\frac{\rho_{\chi}\ \mu_{sa}^{2}\ \alpha\ m_{s}^{2}}{m_{A}\ 4m_{e}^{2}\
2n_{A}}=\frac{N_{U}}{MT}$ (5)

Our limit on $\mu_{sa}$ is compared to the $\mu_{sa}$ limits from TEXONO and
Borexino neutrino experiments in Fig. 4. The parameter spaces expected to
cause the XENON1T excess are also illustrated. Note that other active-to-
sterile transition searches do not necessarily make any assumption of sterile
neutrino dark matter. Our result implies that if the DM halo is assumed to
consist of keV-scale sterile neutrinos, the value of $\mu_{sa}$ is too low to
produce the excess of events observed by XENON1T.

Figure 4: The 90% limit on the sterile neutrino transition magnetic moment via
atomic ionization from Majorana (red), under the assumption of the DM sterile
neutrinos. Our result implies that the sterile neutrino TMM interpretation of
the XENON1T excess is inconsistent with the DM sterile neutrino hypothesis.

Fermionic DM.– Recent papers have discussed fermionic DM-neutrino
interactions, modeled as a Yukawa-like interaction with a bosonic mediator [8,
7, 39, 9]. In the presence of this mediator, fermionic DM can scatter off a
nucleus $A$ and convert to standard model neutrinos via the 2$\rightarrow$2
neutral-current (NC) interaction $\chi+A\rightarrow\nu+A$. If the incoming DM
is non-relativistic, the signature of this inelastic scattering is a
monoenergetic peak at the nuclear recoil energy $E_{R}\simeq
m^{2}_{\chi}/2M_{T}$, where $M_{T}$ is the isotopic mass of the target. The
total absorption rate is given by

$\frac{N_{U}}{MT}=\frac{\rho_{\chi}}{m_{\chi}}\sigma_{\mathrm{NC}}\sum_{j}N_{Aj}A^{2}_{j}F_{j}(m_{\chi})^{2}\Theta(E_{R,j}-E_{\mathrm{th}})~{}.$
(6)

The main quantity of interest is the DM-nucleon scattering cross section
$\sigma_{\mathrm{NC}}$. The local DM density is taken to be $\rho_{DM}=0.4$
GeV/cm3 as in Ref. [40]. The sum over target nuclei $j$ is performed over the
five most abundant Ge isotopes in the enrGe detectors. $N_{Aj}$ is the number
of nuclei for each isotope with mass number $A_{j}$, and $F_{j}(m_{\chi})$ is
the normalized Helm form factor [41, 42] evaluated at momentum transfer
$q=m_{\chi}$ for non-relativistic incoming DM. The step function $\Theta$ is
the experimental energy threshold $E_{th}$ and represents the nuclear recoil
energy threshold for detectable signals.

A quenching factor for germanium must also be applied to convert the observed
electron equivalent ionization (keVee) to the energy of the nuclear recoil
$E_{R}$ (keVnr) [43, 44, 45]. To describe the stopping of few-keV ions in Ge
at $\sim$77 K, we select the quenching factor
$Q=0.2(E_{\mathrm{nr}}/\mathrm{keVnr})^{1.12}$ suggested by CoGeNT [18], as
most applicable to the 1–100 keV energy range [46, 47, 43]. For energies above
1 keVee, there is little dependence on the choice of quenching factor, and the
Lindhard model is adequate. At 1 keVee, our nuclear recoil threshold is
$E_{\mathrm{nr}}=4.5$ keVnr, corresponding to a fermionic DM mass of
$m_{\chi}=$25.5 MeV.

Figure 5: 90% exclusion limit for the $\chi+A\rightarrow\nu+A$ fermionic DM
absorption process. Previous constraints from LHC $Z_{0}$ monojet searches are
shown in gray. The Helm form factor for Ge determines the overall shape of the
curve, and contains a pole which produces the peak-like structure at 174 MeV
[41, 42].

Only two other searches for fermionic dark matter have been performed to date.
$Z_{0}$ monojet searches at the LHC provided indirect constraints [48], and
very recently PandaX released a search using 0.63 ton-yr of exposure from the
PandaX-4T detector [10]. Our search for fermionic dark matter, shown in Fig. 5
is the first done with a Ge array, surpassing the $Z_{0}$ monojet constraints
by roughly five orders of magnitude, but is surpassed by the PandaX-4T result
due to the significantly larger exposure. Our result is the first to set
experimental bounds above masses of 120 MeV. A larger-scale future Ge array
with backgrounds comparable to Majorana could be competitive with the Xe
results. We find a best 90% upper limit on the DM-nucleon coupling
$\sigma_{\mathrm{NC}}<1.620\times 10^{-48}$ cm2 at a DM mass of 48.07 MeV.

Sub-GeV DM-nucleus 3$\rightarrow$2 scattering.– Recent work has shown that the
detection probability of observing DM-nucleus interactions in Ge would be
significantly enhanced for sub-GeV DM if the 3$\rightarrow$2 process
$\chi+\chi+A\rightarrow\phi+A$ is considered [49]. The signature of this
process is an absorption peak at the nuclear recoil energy
$E_{R}\simeq(4-\xi^{2})m_{\chi}^{2}/2M_{T}$, where $\xi$ is the mass ratio of
the final and initial dark matter states $\phi$ and $\chi$, and $M_{T}$ is the
isotopic mass of the target. The value of $\xi$ is model-dependent, and is 0
for a massless (dark photon) final state. For a bound DM final state, it is
obtained by $\xi=(2m_{\chi}+\epsilon_{1})/m_{\chi}$, where the binding energy
is $\epsilon_{1}=-(g_{D}^{4}m_{\chi})/(64\pi^{2})$ and $g_{D}$ is the new
gauge coupling [50]. Setting the gauge coupling $|g_{D}|=3$ for the bound
state DM as in Ref. [49], we obtain $\xi=1.87$ for the bound final state.

The total rate of nuclear recoil events has a similar form to the fermionic DM
absorption (Eq. 6):

$\displaystyle\begin{split}\frac{N_{U}}{MT}&=\Big{(}\frac{\rho_{\chi}}{m_{\chi}}\Big{)}^{2}\langle\sigma_{\mathrm{NC}}^{3\rightarrow
2}v^{2}\rangle\\\ &\times\sum_{j}N_{Aj}\ A^{2}_{j}\ F_{j}(q)^{2}\
\Theta(E_{R,j}-E_{\mathrm{th}})\end{split}$ (7)

where $\langle\sigma_{\mathrm{NC}}^{3\rightarrow 2}v\rangle$ is the average
three-body inelastic cross section per nucleon with the initial DM velocity
$v$.

Figure 6: 90% exclusion limit for the DM-nucleus 3-2 inelastic scattering
process, $\chi+\chi+A\rightarrow\phi+A$. $\xi=0$ is a massless dark photon
final state and $\xi=1.87$ is a bound final state.

This search is the first to set an experimental limit for this 3$\rightarrow$2
scattering process. We place our limit on the
$(m_{\chi},\sigma_{\mathrm{NC}}^{3\rightarrow 2}v^{2}n_{\chi})$ parameter as
suggested in Ref. [49], where $n_{\chi}=\rho_{\chi}/m_{\chi}$ is the DM number
density. The 90% exclusion curves are shown in Fig. 6. For the massless final
state, the best limit on the coupling is
$\langle\sigma_{\mathrm{NC}}^{3\rightarrow 2}v^{2}\rangle n_{\chi}<4.570\times
10^{-48}$ cm2, at a mass of 24.04 MeV. For the bound final state, we find a
best limit of $1.297\times 10^{-47}$ cm2 at a DM mass of 68.22 MeV.

Bosonic DM.– Several experiments have searched for both pseudoscalar
(axionlike) and vector (dark photon) bosonic dark matter as an alternative to
WIMP models. These are non-relativistic DM candidates whose mass energy is
absorbed by a target atom through a variation of the photoelectric (or
axioelectric) effect [1, 51, 52], producing a peak at the rest mass energy.
For axionlike particles, we assume a DM density $\rho_{\mathrm{DM}}$ = 0.3 GeV
cm-3, and a DM velocity such that the energy is approximately equal to its
mass, $\beta=v_{\chi}/c=0.001$. This follows standard assumptions in Refs. [1,
4, 2], defining $m_{\chi}$ as the rest mass in keV. The DM flux (cm-2 d-1)
becomes

$\Phi_{\mathrm{DM}}=\frac{\rho_{\mathrm{DM}}\
v_{\chi}}{m_{\chi}}=\frac{7.8\times 10^{-17}}{m_{\chi}}$ (8)

The interaction, known as the axioelectric effect [51, 52] has a cross section
given by

$\sigma_{ae}(E)=g_{ae}^{2}\frac{E^{2}\sigma_{pe}(E)}{\beta}\bigg{(}\frac{3}{16\pi\alpha
m_{e}^{2}}\bigg{)}$ (9)

Here, $m_{e}$ is the electron mass in keV, $\sigma_{pe}$ is the photoelectric
cross section for Ge at $E_{i}$ [53]. The upper limit on the pseudoscalar
coupling $g_{ae}$ can be expressed as follows, factoring it out of
$\sigma_{ae}$ such that $\sigma_{ae}\equiv g_{ae}^{2}\sigma_{ae}^{\prime}$:

$|g_{ae}|\leq\bigg{(}\frac{N_{U}\ m_{\chi}}{MT\ (7.8\times 10^{17})\
\sigma_{ae}^{\prime}(m_{\chi})}\bigg{)}^{1/2}$ (10)

The resulting exclusion plot is shown in Fig. 7.

Recently, Ferreira et. al [54] have argued that the parameter space previously
searched by Majorana and other experiments is already ruled out since the
lifetime of the ALPs would be shorter than the age of the Universe. Our search
reaches below 5 keV, surpassing the limits from x-ray and gamma-ray limits on
lifetime [54]. In the full energy range, we find a best upper limit on the
pseudoscalar axion-electron coupling constant $g_{ae}<1.425\times 10^{-13}$ at
17.28 keV, and below 5 keV, a best upper limit of $g_{ae}<1.542\times
10^{-13}$ at 2.64 keV. The best limits to date are set by large-scale Xe
experiments due to their much larger active mass [3].

Figure 7: Exclusion limits for pseudoscalar (axionlike) bosonic dark matter.
The Majorana curve (black) is the best limit for any Ge experiment, with the
increased exposure and a factor 5 reduction in energy threshold from the
previous analysis. Xe experiments are capable of tighter limits due to their
factor O(10) larger active mass [55, 35].

For vector bosonic dark matter (dark photons), the coupling constant to
electrons $\alpha^{\prime}$ is related to the electromagnetic fine structure
constant $\alpha$, with experimental limits set on the kinetic mixing
$\kappa^{2}=\alpha^{\prime}/\alpha$ or its logarithm. To compute the expected
counts, the product of the DM flux and interaction cross section (kg-1 d-1) is
replaced by [1]:

$\Phi_{DM}\ \sigma_{ve}=\frac{4\times
10^{23}}{m_{\chi}}\bigg{(}\frac{\alpha^{\prime}}{\alpha}\bigg{)}\frac{\sigma_{pe}(m_{\chi})}{A}$
(11)

Searching for a rare peak at each energy $E_{i}$ as before, we obtain a limit
on the coupling as a function of the mass $m_{\chi}$:

$\frac{\alpha^{\prime}}{\alpha}\leq\bigg{(}\frac{N_{U}\ A\ m_{\chi}}{MT\
(4\times 10^{23})\ \sigma_{pe}(m_{v})}\bigg{)}$ (12)

Fig. 8 shows the exclusion plot for the vector DM search. We find a best upper
limit on the coupling at 2.29 keV, $\alpha^{\prime}/\alpha<2.414\times
10^{-30}$.

Figure 8: Exclusion limits for vector bosonic (dark photon) dark matter.

Discussion.– In this Letter we report results of several exotic DM searches
using the Majorana low energy data set, using a 90% CL profile likelihood
method to set an upper limit on counts attributable to rare peaks in the 1–100
keV energy region. Our result indicates that the transition magnetic moment
between sterile and active neutrinos is too low to produce the XENON1T excess,
if keV-scale $\nu_{s}$ DM is assumed. We set new experimental limits on the
($\chi+A\rightarrow\nu+A$) inelastic fermionic DM absorption, and sub-GeV DM-
nucleus 3$\rightarrow$2 scattering for both massless and bound final DM
states. Our limits for fermionic and bosonic dark matter are currently the
best among Ge experiments, while Xe experiments have surpassed this with their
significantly larger active mass. Controlling the surface exposure time of the
detectors in the Demonstrator was key to increase statistical sensitivity in
the low-energy region. The proposed next-generation $\beta\beta(0\nu)$ LEGEND
experiment [16] may be capable of extending the results of the Demonstrator,
with a significantly larger (ton-scale) active mass, lower backgrounds, and
equally low energy thresholds. With its final goal of 10 ton-years exposure,
significant improvements on the limits presented here may be realized.

###### Acknowledgements.

This material is based upon work supported by the U.S. Department of Energy,
Office of Science, Office of Nuclear Physics under contract / award numbers
DE-AC02-05CH11231, DE-AC05-00OR22725, DE-AC05-76RL0130, DE-FG02-97ER41020, DE-
FG02-97ER41033, DE-FG02-97ER41041, DE-SC0012612, DE-SC0014445, DE-SC0018060,
and LANLEM77/LANLEM78. We acknowledge support from the Particle Astrophysics
Program and Nuclear Physics Program of the National Science Foundation through
grant numbers MRI-0923142, PHY-1003399, PHY-1102292, PHY-1206314, PHY-1614611,
PHY-1812409, PHY-1812356, and PHY-2111140. We gratefully acknowledge the
support of the Laboratory Directed Research & Development (LDRD) program at
Lawrence Berkeley National Laboratory for this work. We gratefully acknowledge
the support of the U.S. Department of Energy through the Los Alamos National
Laboratory LDRD Program and through the Pacific Northwest National Laboratory
LDRD Program for this work. We gratefully acknowledge the support of the South
Dakota Board of Regents Competitive Research Grant. We acknowledge the support
of the Natural Sciences and Engineering Research Council of Canada, funding
reference number SAPIN-2017-00023, and from the Canada Foundation for
Innovation John R. Evans Leaders Fund. This research used resources provided
by the Oak Ridge Leadership Computing Facility at Oak Ridge National
Laboratory and by the National Energy Research Scientific Computing Center, a
U.S. Department of Energy Office of Science User Facility located at Lawrence
Berkeley National Laboratory. We thank our hosts and colleagues at the Sanford
Underground Research Facility for their support.

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# Effective metric of spinless binaries with radiation-reaction effect up to
fourth Post-Minkowskian order in effective-one-body theory

Jiliang Jing111<EMAIL_ADDRESS>Department of Physics, Key Laboratory of
Low Dimensional Quantum Structures and Quantum Control of Ministry of
Education, and Synergetic Innovation Center for Quantum Effects and
Applications, Hunan Normal University, Changsha, Hunan 410081, P. R. China
Center for Gravitation and Cosmology, College of Physical Science and
Technology, Yangzhou University, Yangzhou 225009, P. R. China  Weike Deng
Department of Physics, Key Laboratory of Low Dimensional Quantum Structures
and Quantum Control of Ministry of Education, and Synergetic Innovation Center
for Quantum Effects and Applications, Hunan Normal University, Changsha, Hunan
410081, P. R. China  Sheng Long  Department of Physics, Key Laboratory of Low
Dimensional Quantum Structures and Quantum Control of Ministry of Education,
and Synergetic Innovation Center for Quantum Effects and Applications, Hunan
Normal University, Changsha, Hunan 410081, P. R. China  Jieci Wang222
<EMAIL_ADDRESS>Department of Physics, Key Laboratory of Low Dimensional
Quantum Structures and Quantum Control of Ministry of Education, and
Synergetic Innovation Center for Quantum Effects and Applications, Hunan
Normal University, Changsha, Hunan 410081, P. R. China

###### Abstract

By means of the scattering angles, we obtain an effective metric of spinless
binaries with radiation-reaction effects up to fourth post-Minkowskian order,
which is the foundation of the effective-one-body theory. We note that there
are freedoms for the parameters of the effective metric because one equation
corresponds to two parameters for each post-Minkowskian order. Accordingly, in
order to construct a self-consistent effective-one-body theory in which the
Hamiltonian, radiation-reaction forces and waveforms for the “plus” and
“cross” modes of the gravitational wave should be based on the same physical
model, we can fix these freedoms by requiring the null tetrad component of the
gravitationally perturbed Weyl tensor $\Psi_{4}^{B}$ to be decoupled in the
effective spacetime.

post-Minkowskian approximation, effective-one-body theory, effective metric,
energy map, radiation-reaction effect

###### pacs:

04.25.Nx, 04.30.Db, 04.20.Cv

##  I Introduction

The gravitational waveform template plays a very important role in the
detection of gravitational wave events Abbott2016 ; Abbott2016(2) ; Abbott2017
; Abbott2017(2) ; Abbott2017(3) ; Abbott2019 ; Abbott20211 ; Abbott20212 ;
Abbott20213 ; Nitz2021  generated by coalescing binary systems. The basis of
the gravitational waveform template is the theoretical model of gravitational
radiation, in which the key point is the late dynamical evolution of a
coalescing binary system.

In 1999, Buonanno and Damour Damour1999  introduced a novel approach for
studying the gravitational radiation generated from a coalescing compact
object binary system by mapping the two-body problem onto an effective-one-
body (EOB) problem. Based on the EOB theory with the post-Newtonian (PN)
approximation, Damour et al. provided an estimate of the gravitational
waveforms emitted throughout the inspiral, plunge and coalescence phases
Damour2000 ; Damour2000(2) . The study was then generalized to the case of
spinning black holes Damour2001 ; Damour2006 . Later, the EOB waveforms were
improved by calibrating the model to progressively more accurate numerical
relativity simulations, and spanning larger regions of the parameter space
Cook ; Pan ; Pan1 ; Damour2008 ; Damour20082 ; Boyle ; Damour20083 ; Pan2 ;
Pan3 ; Damour2009 ; Pan4 ; Bar ; Pan5 ; Damour2015 , which plays a vital role
in the analysis of the gravitational wave signals Pan2014 ; Cao2017 ; Cai2017
.

To release the assumption that $v/c$ should be a small quantity in the PN
approximation, in 2016, Damour Damour2016 ; Da1  presented another theoretical
model by combining EOB theory with the post-Minkowskian (PM) approximation
instead of PN approximation. Khalil et al. Khalil  found that the 4PM dynamics
gives better agreement with numerical relativity (NR) than the 3PM dynamics.
Damour and Rettegno Damour2022  compared NR data on the scattering of equal-
mass binary black holes to analytical prediction based on 4PM dynamics of
inspiralling binaries Dlapa1 ; 2210.05541 ; Dlapa , and pointed out that a
reformulation of PM information in terms of EOB radial potentials leads to
remarkable agreement with NR data, especially when using the radiation-reacted
4PM information. Therefore, this new model may lead to a theoretically
improved version of the EOB conservative dynamics, and may be useful in the
upcoming era of high signal-to-noise-ratio gravitational-wave observations.

The investigation of the gravitational waveform template using EOB theory
consists of two main parts: the dynamical evolution of a coalescing binary
system, and the waveforms of the gravitational wave. The dynamical evolution
of a coalescing binary system for a spinless EOB theory can be described by
the Hamilton equation Damour2000

$\displaystyle\frac{dR}{dt}-\frac{\partial
H[g_{\mu\nu}^{\text{eff}}]}{\partial P_{R}}=0,$ (1a)
$\displaystyle\frac{d\varphi}{dt}-\frac{\partial
H[g_{\mu\nu}^{\text{eff}}]}{\partial P_{\varphi}}=0,$ (1b)
$\displaystyle\frac{dP_{R}}{dt}+\frac{\partial
H[g_{\mu\nu}^{\text{eff}}]}{\partial R}={\cal
F}_{R}[g_{\mu\nu}^{\text{eff}}],$ (1c)
$\displaystyle\frac{dP_{\varphi}}{dt}={\cal
F}_{\varphi}[g_{\mu\nu}^{\text{eff}}],$ (1d)

where ($R,\varphi$) and ($P_{R},P_{\varphi}$) are the polar coordinates and
the corresponding conjugate momenta, respectively,
$H[g_{\mu\nu}^{\text{eff}}]$ is the Hamiltonian, and ${\cal
F}_{R}[g_{\mu\nu}^{\text{eff}}]$ and ${\cal
F}_{\varphi}[g_{\mu\nu}^{\text{eff}}]$ are the radiation-reaction forces. The
Hamilton equation (1) shows that, for a self-consistent EOB theory, the
Hamiltonian, radiation-reaction forces and waveforms for the “plus” and
“cross” modes of the gravitational wave should be based on the same effective
spacetime.

To get the Hamiltonian $H[g_{\mu\nu}^{\text{eff}}]$, we should first know the
effective metric and the relationship between the energy $\mathcal{E}$ of a
real two-body system and that $\mathcal{E}_{0}$ of a EOB system. The
radiation-reaction forces are related to the rate of the energy loss of the
gravitational radiation, which in polar coordinates is described by
$\frac{dE}{dt}=\dot{R}\,{\cal
F}_{R}[g_{\mu\nu}^{\text{eff}}]+\dot{\varphi}\,{\cal
F}_{\varphi}[g_{\mu\nu}^{\text{eff}}]$, with $\frac{dE}{dt}=\frac{1}{4\pi
G\omega^{2}}\int|\psi^{B}_{4}|^{2}r^{2}d\Omega$ Ref:poisson ; TagoshiSasaki745
. That is to say, to get the radiation-reaction forces for the “plus” and
“cross” modes of the gravitational wave, we shall find the energy-loss rate
for these modes, in which the key step is to look for the decoupled equation
for the null tetrad component of the gravitationally perturbed Weyl tensor
$\psi^{B}_{4}=\frac{1}{2}(\ddot{h}_{+}-i\ddot{h}_{\times})$ in the effective
spacetime. On the other hand, it is well known that the waveforms for the
“plus” and “cross” modes of the gravitational wave are also related to
$\psi^{B}_{4}$. These discussions show that we have to know $\psi^{B}_{4}$ in
order to set up a self-consistent EOB theory.

However, for a general effective spacetime, say the effective metric for the
EOB theory based on the PN approximation Damour2000(2) , we note that there
are three non-vanishing null tetrad components of the trace-free Ricci tensor:
$\phi_{00},$ $\phi_{11}$ and $\phi_{22}$. We cannot get the decoupled equation
for $\psi^{B}_{4}$ because too many Newman-Penrose quantities are coupled with
each other. Therefore, in previous studies on waveform template based on the
EOB theory, the Hamiltonian in Eq. (1) is based on the effective metric up to
very high PN order, but the radiation reaction forces are not based on this
effective metric.

On the other hand, Damour Damour  showed that the classical radiation reaction
to the emission of gravitational radiation during the large-impact-parameter
scattering of two classical point masses modifies the conservative scattering
angle by an additional radiation-reaction contribution, which can yield a
finite high-energy 3PM-accurate scattering angle. Dlapa et al. 2210.05541
presented the 4PM scattering angle with radiation-reaction effect. Therefore,
we will also study how the radiation reaction affects the effective spacetime
in the following studies.

Recently, we have wanted to do our best to set up a self-consistent EOB theory
based on the PM approximation. The foundation of the EOB theory is the
effective metric. In this paper, based on Bern’s expression of the
conservative Hamiltonian of a relativistic massive spinless two-body system,
we obtain an effective metric with radiation-reaction effects up to the 4PM
order.

The rest of the paper is organized as follows. We present the scattering
angles for the real two-body and EOB systems up to the 4PM order in Sec. II.
Then in Sec. III, the mapping relationship between the real relativistic
energy $\mathcal{E}$ of the real two-body system and the effective
relativistic energy $\mathcal{E}_{0}$ of the EOB system is investigated, and
the effective metric with radiation-reaction effects up to the 4PM order in
the Schwarzschild-like coordinates is obtained. Finally, conclusions and
discussions are presented in the last section.

In this paper, we take the geometric units with only $c=1$, which is suitable
for the calculations in the PM framework.

##  II Scattering angles for real two-body and EOB systems

The first step to build an EOB theory is to obtain an effective metric. In
principle, we can find the effective metric by using one of the action
variables, the precession and the scattering angles. However, explicit
calculations have shown that the scattering angle contains more information
than the others, and the parameters of the effective metric are independent of
the specific process. Therefore, we will use the scattering angle to find the
effective metric.

###  II.1 Scattering angle with radiation-reaction effect for real two-body
system

The Hamiltonian of a massive spinless binary system Da2 ; Bern20192  is given
by

$\displaystyle
H(\vec{p},\vec{r})=\sqrt{|\vec{p}|^{\;2}+m_{1}^{2}}+\sqrt{|\vec{p}|^{\;2}+m_{2}^{2}}+\sum_{i=1}^{\infty}c_{i}\big{(}\frac{G}{|\vec{r}|}\big{)}^{i},$
(2)

where $m_{1}$ and $m_{2}$ represent the masses of the two particles,
$|\vec{p}|^{2}$ denotes $\vec{p}^{\;2}$, $\vec{r}$ is the distance vector
between particles, and the explicit expressions of $c_{i}$ can be found in
Refs. Da2 ; Bern20192 ; Dlapa 333We should note that $c_{i}$ in the Ref. Da2
is equal to $c_{i}/i!$ in the Ref. Dlapa . In the following we will use

$\displaystyle
m=m_{1}+m_{2}\;,\;\;\;\;\;\;\mu=\frac{m_{1}m_{2}}{(m_{1}+m_{2})}\;,\ \ \ \ \
\nu=\frac{m_{1}m_{2}}{(m_{1}+m_{2})^{2}}\;,$ $\displaystyle
E_{1}=\sqrt{\vec{p}^{\;2}+m_{1}^{2}}\;,\
E_{2}=\sqrt{\vec{p}^{\;2}+m_{2}^{2}}\;,\ \ \ \;E=E_{1}+E_{2}.$ (3)

In spherical coordinates $\\{t,r,\theta,\phi\\}$, we have
$\vec{p}^{\;2}=p_{r}^{2}+\frac{p_{\theta}^{2}}{r^{2}}+\frac{p_{\phi}^{2}}{r^{2}\sin^{2}\theta},$
$\vec{r}\cdot\vec{p}=rp_{r},$ and
$\vec{p}^{\;2}\vec{r}^{\;2}-(\vec{r}\cdot\vec{p})^{2}=p^{2}_{\theta}+\frac{p_{\phi}^{2}}{\sin^{2}\theta}.$

There are two conservative quantities, the energy $\mathcal{E}=-p_{t}$ and the
angular momentum $J=p_{\phi}$ of the real two-body system, associated with the
Hamiltonian (2). Without loss of generality, we set $\theta=\frac{\pi}{2}$ to
denote the plane in which two particles reside. Then, we can express the
reduced action as

$\displaystyle S=-\mathcal{E}t+J\phi+S_{r}(r,\mathcal{E},J)\;.$ (4)

Using the Hamilton-Jacobi equation $\frac{\partial S}{\partial
t}+H(q,\frac{\partial S}{\partial q})=0$, we can obtain the radial momentum
$p_{r}^{2}=(\frac{dS_{r}}{dr})^{2}$ as a function of the radial coordinate
$r$, which up to the 4PM order is given by

$\displaystyle p_{r}^{2}$ $\displaystyle=$
$\displaystyle\frac{P_{0}r^{2}-J^{2}}{r^{2}}+P_{1}\Big{(}\frac{G}{r}\Big{)}+P_{2}\Big{(}\frac{G}{r}\Big{)}^{2}+P_{3}\Big{(}\frac{G}{r}\Big{)}^{3}+P_{4}\Big{(}\frac{G}{r}\Big{)}^{4}\;,$
(5)

with

$\displaystyle P_{0}$ $\displaystyle=$
$\displaystyle\Big{(}\frac{\mu}{\Gamma}\Big{)}^{2}(\gamma^{2}-1)\;,$
$\displaystyle P_{1}$ $\displaystyle=$ $\displaystyle
2m\mu^{2}\Big{(}\frac{2\gamma^{2}-1}{\Gamma}\Big{)}\;,$ $\displaystyle P_{2}$
$\displaystyle=$
$\displaystyle\frac{3m^{2}\mu^{2}}{2}\Big{(}\frac{5\gamma^{2}-1}{\Gamma}\Big{)}\;,$
$\displaystyle P_{3}$ $\displaystyle=$ $\displaystyle\Gamma
m^{3}\mu^{2}P_{30},$ $\displaystyle P_{4}$ $\displaystyle=$ $\displaystyle
2m_{1}^{4}m_{2}^{4}f_{4}\,,$ (6)

where
$\gamma=\frac{1}{2}\frac{\mathcal{E}^{2}-m_{1}^{2}-m_{2}^{2}}{m_{1}m_{2}}$,
$\Gamma=\frac{\mathcal{E}}{m_{1}+m_{2}}$,
$P_{30}=\frac{18\gamma^{2}-1}{2\,\Gamma^{2}}+\frac{8\,\nu\,(3+12\gamma^{2}-4\gamma^{4})}{\Gamma^{2}\,\sqrt{\gamma^{2}-1}}\mbox{arcsinh}\sqrt{\frac{\gamma-1}{2}}+\frac{\nu}{\Gamma^{2}}\big{(}1-\frac{103}{3}\gamma-48\gamma^{2}-\frac{2}{3}\gamma^{3}+\frac{3\,\Gamma\,(1-2\gamma^{2})(1-5\gamma^{2})}{(1+\Gamma)(1+\gamma)}\big{)}$,
and the explicit form of $f_{4}$ (which is too long, so we do not present it
here) can be found in the ancillary file of Ref. 2210.05541 .

Then, by using the definition

$\displaystyle\chi^{\text{Nor}}=-\pi+2J\int_{r_{\rm
min}}^{\infty}\frac{dr}{r^{2}\sqrt{p_{r}^{2}}}\;,$ (7)

where the minimum distance $r_{\text{min}}$ is determined by
$p_{r}(r_{min})=0$, we can obtain the scattering angle up to 4PM order without
the radiation-reaction effect, which is described as

$\displaystyle\chi^{\text{Nor}}=\chi^{\text{Nor}}_{1}\frac{G}{J}+\chi^{\text{Nor}}_{2}\Big{(}\frac{G}{J}\Big{)}^{2}+\chi^{\text{Nor}}_{3}\Big{(}\frac{G}{J}\Big{)}^{3}+\chi^{\text{Nor}}_{4}\Big{(}\frac{G}{J}\Big{)}^{4}\;,$
(8)

with

$\displaystyle\chi^{\text{Nor}}_{1}=2\,m_{1}\,m_{2}\,\frac{2\gamma^{2}-1}{\sqrt{\gamma^{2}-1}},\
$
$\displaystyle\chi^{\text{Nor}}_{2}=\frac{3\,\pi\,m_{1}^{2}\,m_{2}^{2}}{4}\,\frac{5\gamma^{2}-1}{\Gamma},\
$
$\displaystyle\chi^{\text{Nor}}_{3}=2\,m_{1}^{3}\,m_{2}^{3}\,\sqrt{\gamma^{2}-1}\,P_{30}+\frac{2}{\pi}\chi^{\text{Nor}}_{1}\,\chi^{\text{Nor}}_{2}-\frac{(\chi^{\text{Nor}}_{1})^{3}}{12},$
$\displaystyle\chi^{\text{Nor}}_{4}=\frac{3\pi}{4}m_{1}^{4}m_{2}^{4}f_{4}+\frac{3\pi}{8}\chi^{\text{Nor}}_{1}\chi^{\text{Nor}}_{3}+\frac{3}{2\pi}(\chi^{\text{Nor}}_{2})^{2}-\frac{3}{4}(\chi^{\text{Nor}}_{1})^{2}\chi^{\text{Nor}}_{2}+\frac{\pi}{32}(\chi^{\text{Nor}}_{1})^{4}.$
(9)

On the other hand, Damour Damour  presented the following 3PM radiation-
reaction effect

$\displaystyle\chi_{3}^{rr}$ $\displaystyle=$
$\displaystyle-\frac{2\nu}{\Gamma^{2}}\frac{\gamma(2\gamma^{2}-1)^{3/2}}{3(\gamma^{2}-1)^{2}}\Big{[}\frac{(5\gamma^{2}-8)\sqrt{\gamma^{2}-1}}{\gamma}+2(9-6\gamma^{2})\mbox{arcsinh}\sqrt{\frac{\gamma-1}{2}}\Big{]},$
(10)

and Dlapa et al. 2210.05541  obtained the 4PM radiation-reaction effect
$\chi^{\text{rr}}_{4}$, which is too long, so we do not list it here. The
explicit form of $\chi^{\text{rr}}_{4}$ can be found in the ancillary file of
Ref. 2210.05541 .

It is worth noting that the relation between our results
$\chi^{\text{Nor}}_{i}$ and the $\chi_{j}^{(i)}$ of Refs. Kalin ; Kalin1 ;
Dlapa  is given by $\chi^{\text{Nor}}_{i}=2(m_{1}m_{2})^{i}\chi_{j}^{(i)}$,
since the PM coefficients of the scattering angle are defined by
$\frac{\chi}{2}=\sum_{i=1}\chi_{j}^{(i)}/j^{i}$ with $j=J/(Gm_{1}m_{2})$ in
Refs. Kalin ; Kalin1 ; Dlapa  (we do not replace $J$ with $j$ because we will
find the relation between $J$ and $J_{0}$, where $J_{0}$ is the angular
momentum of the effective system). Accordingly, the coefficients of the total
scattering angle with radiation-reaction effects up to 4PM order for the real
two-body system are

$\displaystyle\chi^{\text{rel}}_{1}$ $\displaystyle=$ $\displaystyle
2\,m_{1}\,m_{2}\,\frac{2\gamma^{2}-1}{\sqrt{\gamma^{2}-1}},\ $
$\displaystyle\chi^{\text{rel}}_{2}$ $\displaystyle=$
$\displaystyle\frac{3\,\pi\,m_{1}^{2}\,m_{2}^{2}}{4}\,\frac{5\gamma^{2}-1}{\Gamma},\
$ $\displaystyle\chi^{\text{rel}}_{3}$ $\displaystyle=$ $\displaystyle
2\,m_{1}^{3}\,m_{2}^{3}\,\big{(}\sqrt{\gamma^{2}-1}\,P_{30}+\chi_{3}^{rr}\big{)}+\frac{2}{\pi}\chi^{\text{rel}}_{1}\,\chi^{\text{rel}}_{2}-\frac{(\chi^{\text{rel}}_{1})^{3}}{12},$
$\displaystyle\chi^{\text{rel}}_{4}$ $\displaystyle=$ $\displaystyle
2m_{1}^{4}m_{2}^{4}\chi^{\text{rr}}_{4}-\frac{3\pi}{4}m_{1}^{3}m_{2}^{3}\chi^{\text{rel}}_{1}\chi^{\text{rr}}_{3}+\frac{3\pi}{4}m_{1}^{4}m_{2}^{4}f_{4}+\frac{3\pi}{8}\chi^{\text{rel}}_{1}\chi^{\text{rel}}_{3}+\frac{3}{2\pi}(\chi^{\text{rel}}_{2})^{2}$
(11) $\displaystyle-$
$\displaystyle\frac{3}{4}(\chi^{\text{rel}}_{1})^{2}\chi^{\text{rel}}_{2}+\frac{\pi}{32}(\chi^{\text{rel}}_{1})^{4}.$

###  II.2 Scattering angle for EOB system in Schwarzschild-like coordinates

The scattering angle $\chi^{\text{eff}}$ for an EOB system can be found from
the dynamics of a test particle scattered by a black hole described by the
effective metric $g_{\mu\nu}^{\rm eff}$ which can be expressed as

$\displaystyle ds_{\rm
eff}^{2}=Adt^{2}-\frac{D^{2}}{A}dR^{2}-R^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (12)

with

$\displaystyle
A=1+\sum_{i=1}^{\infty}a_{i}\Big{(}\frac{GM_{0}}{R}\Big{)}^{i},$
$\displaystyle
D=1+\sum_{i=1}^{\infty}d_{i}\Big{(}\frac{GM_{0}}{R}\Big{)}^{i},$ (13)

where $M_{0}$ is the mass of the black hole, and $a_{i}$ and $d_{i}$
($i=1,2,3,4$) are dimensionless parameters which will be found in the
following equations.

In the effective spacetime, the effective Hamilton-Jacobi equation reads

$\displaystyle g^{\mu\nu}_{\rm eff}\frac{\partial S_{\rm eff}}{\partial
x^{\mu}}\frac{\partial S_{\rm eff}}{\partial x^{\nu}}+m_{0}^{2}=0\;.$ (14)

Since the effective spacetime described by the line element (12) has spherical
symmetry, without loss of generality, in the equatorial plane
($\theta=\frac{\pi}{2}$), $S_{\rm eff}$ reduces to

$\displaystyle S_{\rm
eff}=-\mathcal{E}_{0}t+J_{0}\Phi+S_{R}^{0}(R,\mathcal{E}_{0},J_{0})\;,$ (15)

where $\mathcal{E}_{0}$ and $J_{0}$ are the effective energy and angular
momentum, respectively. Substituting Eq. (15) into Eq. (14), we obtain

$\displaystyle-\frac{\mathcal{E}_{0}^{2}}{A}+\frac{A}{D^{2}}\Big{(}\frac{dS_{R}^{0}(R,\mathcal{E}_{0},J_{0})}{dR}\Big{)}^{2}+\frac{J_{0}^{2}}{R^{2}}+m_{0}^{2}=0\;,$
(16)

and which leads to

$\displaystyle\frac{dS_{R}^{0}(R,\mathcal{E}_{0},J_{0})}{dR}=\sqrt{\mathcal{R}_{0}(R,\mathcal{E}_{0},J_{0})}\;,$
(17)

with

$\displaystyle\mathcal{R}_{0}(R,\mathcal{E}_{0},J_{0})=\frac{D^{2}}{A^{2}}\mathcal{E}_{0}^{2}-\frac{D^{2}}{A}\bigg{[}m_{0}^{2}+\frac{J_{0}^{2}}{R^{2}}\bigg{]}\;.$
(18)

Substituting Eq. (II.2) into Eq. (18), up to the 4PM order, we find
$\mathcal{R}_{0}(R,\mathcal{E}_{0},J_{0})$ can be expressed as

$\displaystyle\mathcal{R}_{0}(R,\mathcal{E}_{0},J_{0})$ $\displaystyle=$
$\displaystyle
R_{00}-\frac{J_{0}^{2}}{R^{2}}+\frac{R_{11}G}{R}+\frac{R_{22}G^{2}}{R^{2}}-\frac{R_{31}J_{0}^{2}G-R_{33}G^{3}}{R^{3}}$
(19) $\displaystyle-$
$\displaystyle\frac{R_{42}J_{0}^{2}G^{2}-R_{44}G^{4}}{R^{4}}-\frac{R_{53}J_{0}^{2}G^{3}}{R^{5}}-\frac{R_{64}J_{0}^{2}G^{4}}{R^{6}}\;,$

with

$\displaystyle R_{00}$ $\displaystyle=$
$\displaystyle\mathcal{E}_{0}^{2}-m_{0}^{2}\;,$ $\displaystyle R_{11}$
$\displaystyle=$ $\displaystyle
2M_{0}\left[2\mathcal{E}_{0}^{2}-m_{0}^{2}\right]\;,$ $\displaystyle R_{22}$
$\displaystyle=$ $\displaystyle
M_{0}^{2}\left[(a_{2}-2(2+d_{2}))m_{0}^{2}+2\mathcal{E}_{0}^{2}\left(6-a_{2}+d_{2}\right)\right]\;,$
$\displaystyle R_{31}$ $\displaystyle=$ $\displaystyle 2M_{0}\>,$
$\displaystyle R_{33}$ $\displaystyle=$ $\displaystyle
M_{0}^{3}\left[(4a_{2}+a_{3}-2(4+2d_{2}+d_{3}))m_{0}^{2}+2\mathcal{E}_{0}^{2}\left(16-6a_{2}-a_{3}+4d_{2}+d_{3}\right)\right]\;,$
$\displaystyle R_{42}$ $\displaystyle=$
$\displaystyle(4-a_{2}+2d_{2})M_{0}^{2}\;,$ $\displaystyle R_{44}$
$\displaystyle=$ $\displaystyle
M_{0}^{4}\Big{\\{}-(16-12a_{2}+a_{2}^{2}-4a_{3}-a_{4}+8d_{2}-2a_{2}d_{2}+d_{2}^{2}+4d_{3}+2d_{4})m_{0}^{2}$
$\displaystyle+$
$\displaystyle\mathcal{E}_{0}^{2}\Big{[}80+3a_{2}^{2}-12a_{3}-2a_{4}+24d_{2}+d_{2}^{2}-4a_{2}(12+d_{2})+8d_{3}+2d_{4}\Big{]}\Big{\\}}\;,$
$\displaystyle R_{53}$ $\displaystyle=$
$\displaystyle(8-4a_{2}-a_{3}+4d_{2}+2d_{3})M_{0}^{3}\;,$ $\displaystyle
R_{64}$ $\displaystyle=$
$\displaystyle(16-12a_{2}+a_{2}^{2}-4a_{3}-a_{4}+8d_{2}-2a_{2}d_{2}+d_{2}^{2}+4d_{3}+2d_{4})M_{0}^{4}\;,$
(20)

where we have taken $a_{1}=-2$ and $d_{1}=0$ by requiring that the effective
metric at the 1PM order coincides with the Schwarzschild metric for the sake
of concise expressions.

For the EOB system, the scattering angle can be expressed as

$\displaystyle\chi^{\text{eff}}$ $\displaystyle=$
$\displaystyle-\pi-2\int_{R_{\rm
min}}^{\infty}\frac{\partial\sqrt{\mathcal{R}_{s}(R,\mathcal{E}_{0},J_{0})}}{\partial
J_{0}}dR.$ (21)

where $R_{\text{min}}$ is the minimum distance determined by setting the
vanishing of $\mathcal{R}_{s}(R,\mathcal{E}_{0},J_{0})$. Substituting Eq. (19)
into Eq. (21) and working out the integration up to the 4PM order, we get

$\displaystyle\chi^{\text{eff}}$ $\displaystyle=$
$\displaystyle\frac{R_{11}}{\sqrt{R_{00}}}\frac{G}{J_{0}}+\frac{\pi\big{[}2R_{22}-R_{11}R_{31}+R_{00}(\frac{3}{4}R_{31}^{2}-R_{42})\big{]}}{4}\Big{(}\frac{G}{J_{0}}\Big{)}^{2}$
(22) $\displaystyle+$
$\displaystyle\frac{1}{12R_{00}^{3/2}}\Big{\\{}-R_{11}^{3}-6R_{00}R_{11}^{2}R_{31}+12R_{00}R_{11}\Big{[}R_{22}$
$\displaystyle+$ $\displaystyle
2R_{00}(R_{31}^{2}-R_{42})\Big{]}-8R_{00}^{2}\Big{[}3R_{22}R_{31}-3R_{33}$
$\displaystyle+$ $\displaystyle
2R_{00}(R_{31}^{3}-2R_{31}R_{42}+R_{53})\Big{]}\Big{\\}}\Big{(}\frac{G}{J_{0}}\Big{)}^{3}$
$\displaystyle+$
$\displaystyle\frac{3\pi}{1024}\Big{\\{}128R_{22}^{2}+48R_{11}^{2}\left(5R_{31}^{2}-4R_{42}\right)$
$\displaystyle-$ $\displaystyle
96R_{22}\left(4R_{11}R_{31}-5R_{00}R_{31}^{2}+4R_{00}R_{42}\right)$
$\displaystyle+$ $\displaystyle
16R_{11}\left(16R_{33}+R_{00}\left(-35R_{31}^{3}+60R_{31}R_{42}-24R_{53}\right)\right)$
$\displaystyle+$ $\displaystyle
R_{00}\Big{[}-384R_{31}R_{33}+256R_{44}+3R_{00}\Big{(}105R_{31}^{4}$
$\displaystyle-$ $\displaystyle
280R_{31}^{2}R_{42}+80R_{42}^{2}+160R_{31}R_{53}-64R_{64}\Big{)}\Big{]}\Big{\\}}\Big{(}\frac{G}{J_{0}}\Big{)}^{4}.$

Then, using Eqs. (II.2) and (22) we have

$\displaystyle\chi^{\text{eff}}=\chi^{\text{eff}}_{1}\frac{G}{J_{0}}+\chi^{\text{eff}}_{2}\Big{(}\frac{G}{J_{0}}\Big{)}^{2}+\chi^{\text{eff}}_{3}\Big{(}\frac{G}{J_{0}}\Big{)}^{3}+\chi^{\text{eff}}_{4}\Big{(}\frac{G}{J_{0}}\Big{)}^{4},$
(23)

with

$\displaystyle\chi^{\text{eff}}_{1}$
$\displaystyle=\frac{2M_{0}\left[2\mathcal{E}_{0}^{2}-m_{0}^{2}\right]}{\sqrt{\mathcal{E}_{0}^{2}-m_{0}^{2}}},$
$\displaystyle\chi^{\text{eff}}_{2}$
$\displaystyle=\frac{M_{0}^{2}\pi}{4}\bigg{[}\mathcal{E}_{0}^{2}(15-3a_{2}+2d_{2})+m_{0}^{2}(a_{2}-3-2d_{2})\bigg{]},$
$\displaystyle\chi^{\text{eff}}_{3}$
$\displaystyle=\frac{2\chi^{\text{\text{\text{eff}}}}_{1}\chi^{\text{eff}}_{2}}{\pi}-\frac{\chi^{\text{eff\,3}}_{1}}{12}-\frac{M_{0}^{3}\sqrt{\mathcal{E}_{0}^{2}-m_{0}^{2}}}{3}[(3-3a_{2}-2a_{3}$
$\displaystyle-2d_{2}+4d_{3})m_{0}^{2}+2\mathcal{E}_{0}^{2}(15a_{2}+2(2a_{3}-d_{2}-d_{3})-27)],$
$\displaystyle\chi^{\text{eff}}_{4}$
$\displaystyle=\frac{3\pi}{8}\chi^{\text{eff}}_{1}\chi^{\text{eff}}_{3}+\frac{3}{2\pi}(\chi^{\text{eff}}_{2})^{2}-\frac{3}{4}(\chi^{\text{eff}}_{1})^{2}\chi^{\text{eff}}_{2}+\frac{\pi}{32}(\chi^{\text{eff}}_{1})^{4}+\frac{\pi
M_{0}^{4}(\mathcal{E}_{0}^{2}-m_{0}^{2})}{64}\Big{[}(387-390a_{2}$
$\displaystyle+51a_{2}^{2}-164a_{3}-60a_{4}+12a_{2}d_{2}-4d_{2}(17+6d_{2})-8d_{3}+24d_{4})\mathcal{E}_{0}^{2}-(3+3(a_{2}-2)a_{2}$
$\displaystyle-4a_{3}-12a_{4}+12a_{2}d_{2}-4d_{2}(1+6d_{2})-40d_{3}+24d_{4})m_{0}^{2}\Big{]}.$
(24)

##  III Energy map and effective metric with radiation-reaction for EOB theory

In the EOB theory Damour1999 , the main idea is to map the two-body problem
onto an EOB problem, i.e., a test particle orbits around a massive black hole
described by effective metric. This map can be realized by identifying the
scattering angles for the two systems order by order, i.e., by taking
$\chi^{\text{real}}_{i}=\chi^{\text{eff}}_{i}\;,$ for
$i=1,~{}2,~{}3,~{}4,\cdot\cdot\cdot$. Now, we will find the energy map and the
effective metric with radiation-reaction effects for EOB theory.

###  III.1 Energy map for EOB theory

Following Buonanno and Damour Damour1999 ; Damour2016 , we take

$\displaystyle m_{0}=\frac{m_{1}m_{2}}{m_{1}+m_{2}},$ $\displaystyle
M_{0}=m_{1}+m_{2},$ $\displaystyle J_{0}=J.$ (25)

Then, by requiring that the effective metric at the 1PM order coincides with
the Schwarzschild metric, and taking

$\displaystyle\chi^{\text{real}}_{1}=\chi^{\text{eff}}_{1}\;,$ (26)

we find

$\displaystyle\mathcal{E}_{0}$
$\displaystyle=\frac{\mathcal{E}^{2}-m_{1}^{2}-m_{2}^{2}}{2(m_{1}+m_{2})}\;,$
(27)

which is the energy map between the relativistic energy $\mathcal{E}$ of the
real two-body system and the relativistic energy $\mathcal{E}_{0}$ of the EOB
system.

By taking the reduced energies as
$\hat{\mathcal{E}_{0}}=\mathcal{E}_{0}/m_{0}$ and
$\hat{\mathcal{E}}=\mathcal{E}/m_{0}$, Eq. (27) can be rewritten as

$\displaystyle\hat{\mathcal{E}_{0}}=\frac{\nu}{2}\hat{\mathcal{E}}^{2}-\frac{1}{2\nu}+1,$
(28)

which is useful in the related calculations.

###  III.2 Effective metric with radiation-reaction effect in EOB theory

We note that there are freedoms for the parameters of the effective metric
because one equation corresponds to two parameters ($a_{i}$, $d_{i}$) for each
post-Minkowskian order $i$. The freedoms can be fixed in the following
discussions.

Substituting Eqs. (11) and (II.2) into

$\displaystyle\chi^{\text{real}}_{i}=\chi^{\text{eff}}_{i}\;,\ \ \ \
\text{(i=2, 3, 4)},$ (29)

we find that the parameters $a_{i}$ in the effective metric are given by

$\displaystyle a_{2}$ $\displaystyle=$
$\displaystyle\frac{3(1-\,\Gamma)(1-5\,\gamma^{2})}{\Gamma\,(3\,\gamma^{2}-1)}\,,$
$\displaystyle a_{3}$ $\displaystyle=$
$\displaystyle\frac{3}{2(4\gamma^{2}-1)}\Big{[}\frac{3-2\,\Gamma-3(15-8\,\Gamma)\gamma^{2}+6(25-16\,\Gamma)\gamma^{4}}{\Gamma\,(3\,\gamma^{2}-1)}-2P_{30}-\frac{2\chi_{3}^{rr}}{\sqrt{\gamma^{2}-1}}\Big{]}$
$\displaystyle-
d_{2}\frac{4-34\gamma^{2}+24\gamma^{4}}{1-7\gamma^{2}+12\gamma^{4}}+d_{3}\frac{2(1-\gamma^{2})}{1-4\gamma^{2}},$
$\displaystyle a_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{(\gamma^{2}-1)(5\,\gamma^{2}-1)}\Big{\\{}-4f_{4}-\frac{32\chi_{4}^{rr}}{3\pi}+\frac{8(2\gamma^{2}-1)\chi_{3}^{rr}}{\sqrt{\gamma^{2}-1}}+\frac{1}{12}\Big{[}(\gamma^{2}-1)\Big{(}-3+387\gamma^{2}$
(30)
$\displaystyle+6a_{2}(1-65\gamma^{2})+3a_{2}^{2}(17\gamma^{2}-1)+4a_{3}(1-41\gamma^{2})\Big{)}\Big{]}\Big{\\}}+\frac{1}{3(5\gamma^{2}-1)}\Big{[}d_{2}-3a_{2}d_{2}$
$\displaystyle+6d_{2}^{2}+10d_{3}-6d_{4}-((17-3a_{2}+6d_{2})d_{2}+2d_{3}-6d_{4})\gamma^{2}\Big{]},$

where the parameters $a_{3}$ and $a_{4}$ include the terms $\chi^{rr}_{3}$ and
$\chi^{rr}_{4}$, which represent the 3PM and 4PM radiation-reaction effects,
respectively. That is to say, the radiation reaction will affect the structure
of the effective spacetime.

How to fix $d_{i}$? Before fixing this issue, let us determine how we can fix
the parameters of the effective metric in the EOB theory based on the PN
approximation. It was shown that the number of equations is less than the
number of parameters of the effective metric at the 3PN order, then some
parameters were chosen artificially Damour2000(2) . It is well known that the
binding energy and the gravitational wave energy flux are two central
ingredients that enter the computation of gravitational waveforms. Therefore,
we have thought that the parameters $d_{i}$ can be fixed by comparing the EOB
predictions for the binding energy of a two-body system on a quasicircular
inspiraling orbit against the results of numerical relativity simulations.
However, we have found that, along this way, we will lose the self-consistency
for an EOB theory, for the following reasons: for the general effective metric
(12) with the parameters (III.2) there are three non-vanishing null tetrad
components of the tracefree Ricci tensor: $\phi_{00},$ $\phi_{11}$ and
$\phi_{22}$. We cannot obtain a decoupled equation for $\psi^{B}_{4}$ in the
effective spacetime because too many Newman-Penrose quantities are coupled
with each other. That is to say, we can use the effective metric to construct
the Hamiltonian, but we cannot obtain the radiation reaction forces and the
waveforms for the “plus” and “cross” modes based on the effective metric.

However, if we take $D=1$ (i.e., $d_{i}=0$ for $i=2,3,4$) which fixed the
freedoms of the parameters of the effective metric, we have

$\displaystyle a_{2}$ $\displaystyle=$
$\displaystyle\frac{3(1-\,\Gamma)(1-5\,\gamma^{2})}{\Gamma\,(3\,\gamma^{2}-1)}\,,$
$\displaystyle a_{3}$ $\displaystyle=$
$\displaystyle\frac{3}{2(4\gamma^{2}-1)}\Big{[}\frac{3-2\,\Gamma-3(15-8\,\Gamma)\gamma^{2}+6(25-16\,\Gamma)\gamma^{4}}{\Gamma\,(3\,\gamma^{2}-1)}-2P_{30}-\frac{2\chi_{3}^{rr}}{\sqrt{\gamma^{2}-1}}\Big{]},$
$\displaystyle a_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{(\gamma^{2}-1)(5\,\gamma^{2}-1)}\Big{\\{}-4f_{4}-\frac{32\chi_{4}^{rr}}{3\pi}+\frac{8(2\gamma^{2}-1)\chi_{3}^{rr}}{\sqrt{\gamma^{2}-1}}+\frac{1}{12}\Big{[}(\gamma^{2}-1)\Big{(}-3+387\gamma^{2}$
(31)
$\displaystyle+6a_{2}(1-65\gamma^{2})+3a_{2}^{2}(17\gamma^{2}-1)+4a_{3}(1-41\gamma^{2})\Big{)}\Big{]}\Big{\\}}.$

Then we have $\phi_{00}=\phi_{22}=0$, and we can obtain the decoupled equation
for the null tetrad component of the gravitationally perturbed Weyl tensor
$\Psi_{4}^{B}$ by means of the Refs. Jing ; Jing1 ; Jing2 , in which we have
shown that the decoupled equation for $\Psi_{4}^{B}$ can be obtained if the
metric takes the form $B=D^{2}/A=1/A$ up to any PM order. In this way, we can
set up a self-consistent EOB theory in which the Hamiltonian, radiation-
reaction forces and waveforms for the “plus” and “cross” modes are based on
the same effective spacetime.

##  IV Conclusions and discussions

With the help of the scattering angles, we obtained the effective metric for
spinless binaries with radiation-reaction effects up to the 4PM order in the
EOB theory. We can show that these results are consistent by means of the
action variables, the precession and scattering angles, which implies that
parameters of the effective metric are independent of any specific process.

Specifically, we can summarize our results as follows. The relations for the
masses and angular momenta between the EOB and the real two-body systems are
taken as $m_{0}=\frac{m_{1}m_{2}}{m_{1}+m_{2}},$ $M_{0}=m_{1}+m_{2},$ and
$J_{0}=J.$ Then, by requiring that the effective metric at the 1PM order
coincide with the Schwarzschild metric, and taking
$\chi^{\text{real}}_{1}=\chi^{\text{eff}}_{1}$, we find that the mapping
relationship between the relativistic energies of the real two-body and the
EOB systems is described by
$\mathcal{E}_{0}=\frac{\mathcal{E}^{2}-m_{1}^{2}-m_{2}^{2}}{2(m_{1}+m_{2})},$
which holds for the 4PM order and we conjecture that it keeps the same form
for any PM order. If we take the reduced energies as
$\hat{\mathcal{E}_{0}}=\mathcal{E}_{0}/m_{0}$ and
$\hat{\mathcal{E}}=\mathcal{E}/m_{0}$, the energy map can be recast as
$\hat{\mathcal{E}_{0}}=\frac{\nu}{2}\hat{\mathcal{E}}^{2}-\frac{1}{2\nu}+1$,
which is useful in the related calculations.

The parameters ($a_{i},\ d_{i}$) for $i=2,~{}3,~{}4$ appearing in the
effective metric (12) are found by identifying the scattering angles for the
two systems order by order, i.e., by taking
$\chi^{\text{real}}_{i}=\chi^{\text{eff}}_{i}\;,$ for $i=2,~{}3,~{}4$.
However, we note that one equation corresponds to two unknowns $a_{i}$ and
$d_{i}$ for each order $i$. Thus, one of them, say $d_{i}$, can be considered
as an undetermined parameter which should be fixed in the specific
application. It should be noted that the effective metrics at the 3PM and 4PM
orders, described by Eq. (12) in the Schwarzschild-like coordinates, include
the terms $\chi^{rr}_{3}$ and $\chi^{rr}_{4}$, which represent the 3PM and 4PM
radiation-reaction effects, respectively. In other words, the structure of the
effective spacetime will be affected by the radiation-reaction effect.

For the general effective metric (12) with the parameters (III.2), there are
three non-vanishing null tetrad components of the tracefree Ricci tensor:
$\phi_{00},$ $\phi_{11}$ and $\phi_{22}$. We cannot isolate the decoupled
equation for the null tetrad component of gravitationally perturbed Weyl
tensor $\Psi_{4}^{B}$ in the effective spacetime because too many Newman-
Penrose equations are coupled with each other. However, if we take $D=1$
(i.e., $d_{i}=0$ for $i=2,3,4$) which fixed the freedoms of the parameters of
the effective metric, we have $\phi_{00}=\phi_{22}=0$ and we can obtain the
decoupled equation for the null tetrad component of the gravitationally
perturbed Weyl tensor $\Psi_{4}^{B}$ by means of the Refs. Jing ; Jing1 ;
Jing2  which showed that the decoupled equation for $\Psi_{4}^{B}$ can be
obtained for the spacetime with the parameter $D=1$ up to any PM order. Then,
based on the results of this paper, we can set up a self-consistent EOB theory
in which the Hamiltonian, radiation-reaction forces and waveforms for the
“plus” and “cross” modes of the gravitational wave are based on the same
effective spacetime.

###### Acknowledgements.

Acknowledgments: We would like to thank professors S. Chen and Q. Pan for
useful discussions on the manuscript. This work was supported by the Grant of
NSFC Nos. 12035005 and 12122504, and National Key Research and Development
Program of China No. 2020YFC2201400.

## References

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