# An Earth-mass Planet in a Time of Covid-19: KMT-2020-BLG-0414Lb

Weicheng Zang Department of Astronomy, Tsinghua University, Beijing 100084,
China  Cheongho Han  Department of Physics, Chungbuk National University,
Cheongju 28644, Republic of Korea  Iona Kondo  Department of Earth and Space
Science, Graduate School of Science, Osaka University, Toyonaka, Osaka
560-0043, Japan  Jennifer C. Yee  Center for Astrophysics $|$ Harvard &
Smithsonian, 60 Garden St.,Cambridge, MA 02138, USA  Chung-Uk Lee  Korea
Astronomy and Space Science Institute, Daejon 34055, Republic of Korea
University of Science and Technology, Korea, (UST), 217 Gajeong-ro Yuseong-gu,
Daejeon 34113, Republic of Korea  Andrew Gould  Max-Planck-Institute for
Astronomy, Königstuhl 17, 69117 Heidelberg, Germany  Department of Astronomy,
Ohio State University, 140 W. 18th Ave., Columbus, OH 43210, USA  Shude Mao
Department of Astronomy, Tsinghua University, Beijing 100084, China  National
Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China
Leandro de Almeida  Universidade Federal do Rio Grande do Norte (UFRN),
Departamento de F1́sica, 59078-970, Natal, RN, Brazil  Yossi Shvartzvald
Department of Particle Physics and Astrophysics, Weizmann Institute of
Science, Rehovot 76100, Israel  Xiangyu Zhang  Department of Astronomy,
Tsinghua University, Beijing 100084, China  Michael D. Albrow  University of
Canterbury, Department of Physics and Astronomy, Private Bag 4800,
Christchurch 8020, New Zealand  Sun-Ju Chung  Korea Astronomy and Space
Science Institute, Daejon 34055, Republic of Korea  University of Science and
Technology, Korea, (UST), 217 Gajeong-ro Yuseong-gu, Daejeon 34113, Republic
of Korea  Kyu-Ha Hwang  Korea Astronomy and Space Science Institute, Daejon
34055, Republic of Korea  Youn Kil Jung  Korea Astronomy and Space Science
Institute, Daejon 34055, Republic of Korea  Yoon-Hyun Ryu  Korea Astronomy and
Space Science Institute, Daejon 34055, Republic of Korea  In-Gu Shin  Korea
Astronomy and Space Science Institute, Daejon 34055, Republic of Korea  Sang-
Mok Cha  Korea Astronomy and Space Science Institute, Daejon 34055, Republic
of Korea  School of Space Research, Kyung Hee University, Yongin, Kyeonggi
17104, Republic of Korea  Dong-Jin Kim  Korea Astronomy and Space Science
Institute, Daejon 34055, Republic of Korea  Hyoun-Woo Kim  Korea Astronomy and
Space Science Institute, Daejon 34055, Republic of Korea  Seung-Lee Kim  Korea
Astronomy and Space Science Institute, Daejon 34055, Republic of Korea
University of Science and Technology, Korea, (UST), 217 Gajeong-ro Yuseong-gu,
Daejeon 34113, Republic of Korea  Dong-Joo Lee  Korea Astronomy and Space
Science Institute, Daejon 34055, Republic of Korea  Yongseok Lee  Korea
Astronomy and Space Science Institute, Daejon 34055, Republic of Korea  School
of Space Research, Kyung Hee University, Yongin, Kyeonggi 17104, Republic of
Korea  Byeong-Gon Park  Korea Astronomy and Space Science Institute, Daejon
34055, Republic of Korea  University of Science and Technology, Korea, (UST),
217 Gajeong-ro Yuseong-gu, Daejeon 34113, Republic of Korea  Richard W. Pogge
Department of Astronomy, Ohio State University, 140 W. 18th Ave., Columbus, OH
43210, USA  John Drummond  Possum Observatory, Patutahi, Gisbourne, New
Zealand  Thiam-Guan Tan  Perth Exoplanet Survey Telescope, Perth, Australia
José Dias do Nascimento Júnior  Universidade Federal do Rio Grande do Norte
(UFRN), Departamento de F1́sica, 59078-970, Natal, RN, Brazil  Center for
Astrophysics $|$ Harvard & Smithsonian, 60 Garden St.,Cambridge, MA 02138, USA
Dan Maoz  School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv
6997801, Israel  Matthew T. Penny Department of Physics and Astronomy,
Louisiana State University, Baton Rouge, LA 70803 USA  Wei Zhu  Department of
Astronomy, Tsinghua University, Beijing 100084, China  Canadian Institute for
Theoretical Astrophysics, University of Toronto, 60 St George Street, Toronto,
ON M5S 3H8, Canada  Ian A. Bond  Institute of Natural and Mathematical
Sciences, Massey University, Auckland 0745, New Zealand  Fumio Abe  Institute
for Space-Earth Environmental Research, Nagoya University, Nagoya 464-8601,
Japan  Richard Barry  Code 667, NASA Goddard Space Flight Center, Greenbelt,
MD 20771, USA  David P. Bennett  Code 667, NASA Goddard Space Flight Center,
Greenbelt, MD 20771, USA  Department of Astronomy, University of Maryland,
College Park, MD 20742, USA  Aparna Bhattacharya  Code 667, NASA Goddard Space
Flight Center, Greenbelt, MD 20771, USA  Department of Astronomy, University
of Maryland, College Park, MD 20742, USA  Martin Donachie  Department of
Physics, University of Auckland, Private Bag 92019, Auckland, New Zealand
Hirosane Fujii  Department of Earth and Space Science, Graduate School of
Science, Osaka University, Toyonaka, Osaka 560-0043, Japan  Akihiko Fukui
Department of Earth and Planetary Science, Graduate School of Science, The
University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  Instituto
de Astrofísica de Canarias, Vía Láctea s/n, E-38205 La Laguna, Tenerife, Spain
Yuki Hirao  Department of Earth and Space Science, Graduate School of Science,
Osaka University, Toyonaka, Osaka 560-0043, Japan  Yoshitaka Itow  Institute
for Space-Earth Environmental Research, Nagoya University, Nagoya 464-8601,
Japan  Rintaro Kirikawa  Department of Earth and Space Science, Graduate
School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan  Naoki
Koshimoto  Department of Astronomy, Graduate School of Science, The University
of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  National Astronomical
Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  Man Cheung
Alex Li  Department of Physics, University of Auckland, Private Bag 92019,
Auckland, New Zealand  Yutaka Matsubara  Institute for Space-Earth
Environmental Research, Nagoya University, Nagoya 464-8601, Japan  Yasushi
Muraki  Institute for Space-Earth Environmental Research, Nagoya University,
Nagoya 464-8601, Japan  Shota Miyazaki  Department of Earth and Space Science,
Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
Clément Ranc  Code 667, NASA Goddard Space Flight Center, Greenbelt, MD 20771,
USA  Nicholas J. Rattenbury  Department of Physics, University of Auckland,
Private Bag 92019, Auckland, New Zealand  Yuki Satoh  Department of Earth and
Space Science, Graduate School of Science, Osaka University, Toyonaka, Osaka
560-0043, Japan  Hikaru Shoji  Department of Earth and Space Science, Graduate
School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan  Takahiro
Sumi  Department of Earth and Space Science, Graduate School of Science, Osaka
University, Toyonaka, Osaka 560-0043, Japan  Daisuke Suzuki  Institute of
Space and Astronautical Science, Japan Aerospace Exploration Agency, 3-1-1
Yoshinodai, Chuo, Sagamihara, Kanagawa, 252-5210, Japan  Yuzuru Tanaka
Department of Earth and Space Science, Graduate School of Science, Osaka
University, Toyonaka, Osaka 560-0043, Japan  Paul J. Tristram  University of
Canterbury Mt. John Observatory, P.O. Box 56, Lake Tekapo 8770, New Zealand
Tsubasa Yamawaki  Department of Earth and Space Science, Graduate School of
Science, Osaka University, Toyonaka, Osaka 560-0043, Japan  Atsunori Yonehara
Department of Physics, Faculty of Science, Kyoto Sangyo University, 603-8555
Kyoto, Japan  Andreea Petric  CFHT Corporation, 65-1238 Mamalahoa Hwy,
Kamuela, Hawaii 96743, USA  Space Telescope Science Institute, Baltimore, MD
21211  Todd Burdullis  CFHT Corporation, 65-1238 Mamalahoa Hwy, Kamuela,
Hawaii 96743, USA  Pascal Fouqué  CFHT Corporation, 65-1238 Mamalahoa Hwy,
Kamuela, Hawaii 96743, USA  Université de Toulouse, UPS-OMP, IRAP, Toulouse,
France

###### Abstract

We report the discovery of KMT-2020-BLG-0414Lb, with a planet-to-host mass
ratio $q_{2}=0.9$–$1.2\times 10^{-5}=3$–$4~{}q_{\oplus}$ at $1\sigma$, which
is the lowest mass-ratio microlensing planet to date. Together with two other
recent discoveries ($4\lesssim q/q_{\oplus}\lesssim 6$), it fills out the
previous empty sector at the bottom of the triangular $(\log s,\log q)$
diagram, where $s$ is the planet-host separation in units of the angular
Einstein radius $\theta_{\rm E}$. Hence, these discoveries call into question
the existence, or at least the strength, of the break in the mass-ratio
function that was previously suggested to account for the paucity of very
low-$q$ planets. Due to the extreme magnification of the event, $A_{\rm
max}\sim 1450$ for the underlying single-lens event, its light curve revealed
a second companion with $q_{3}\sim 0.05$ and $|\log s_{3}|\sim 1$, i.e., a
factor $\sim 10$ closer to or farther from the host in projection. The
measurements of the microlens parallax $\bm{\pi}_{\rm E}$ and the angular
Einstein radius $\theta_{\rm E}$ allow estimates of the host, planet, and
second companion masses,
$(M_{1},M_{2},M_{3})\sim(0.3M_{\odot},1.0M_{\oplus},17M_{J})$, the planet and
second companion projected separations,
$(a_{\perp,2},a_{\perp,3})\sim(1.5,0.15~{}{\rm or}~{}15)$ au, and system
distance $D_{\rm L}\sim 1$ kpc. The lens could account for most or all of the
blended light ($I\sim 19.3$) and so can be studied immediately with high-
resolution photometric and spectroscopic observations that can further clarify
the nature of the system. The planet was found as part of a new program of
high-cadence follow-up observations of high-magnification events. The
detection of this planet, despite the considerable difficulties imposed by
Covid-19 (two KMT sites and OGLE were shut down), illustrates the potential
utility of this program.

##  1 Introduction

It has long been known that intensive monitoring of high-magnification
microlensing events is sensitive to planets of one-to-few Earth/Sun mass
ratio, $q_{\oplus}=3\times 10^{-6}$, planets. Dong et al. (2006) showed that
the $A_{\rm max}\sim 3000$ event OGLE-2004-BLG-343 would have had such
sensitivity had it been observed over peak (see their Figure 9). Yee et al.
(2009) showed such sensitivity for the actual data covering the peak of the
$A_{\rm max}\sim 1600$ event OGLE-2008-BLG-279 (see their Figure 7). In both
cases (and, indeed, for high magnification events in general, Gaudi et al.
2002; Gould et al. 2010), the sensitivity diagrams have a triangular
appearance that is symmetric in $\log s$ about the origin. That is, the
contour limits meet at $s=1$, where $s$ is the planet-host separation in units
of the Einstein radius $\theta_{\rm E}$. Hence, the limiting sensitivity in
$q$ is via a so-called “resonant caustic”. For $s\gg 1$, the caustic structure
consists of a small quadrilateral caustic near the host and a larger
quadrilateral caustic near the planet. For $s\ll 1$, it consists of a similar
small quadrilateral caustic near the host and two triangular caustics located
on the opposite side of the planet. For $s\sim 1$, these two sets of caustics
merge into a relatively large 6-sided “resonant” caustic, which is what makes
the detection feasible at the very limit of sensitivity111In fact, Yee et al.
(2021) showed that “semi-resonant” caustics, which have not quite merged but
still have long magnification ridges extending from the central caustic, as
well as exceptionally large planetary caustics, are just as sensitive as
resonant caustics..

Neither of the above two events yielded any planet detection, but Udalski et
al. (2018) did find a resonant-caustic planet in OGLE-2017-BLG-1434 ($A_{\rm
max}\sim 23$ for the underlying single-lens event). While its mass ratio was
$q\sim 5.7\times 10^{-5}$ (i.e., $q\sim 19~{}q_{\oplus}$), Udalski et al.
(2018) showed that a planet with exactly the same characteristics, but 30
times less massive, would have been detected (see their Figure 4).

Nevertheless, in spite of the recognized theoretical possibility of such
few-$q_{\oplus}$ planet detections, no planet with mass ratio below that of
Uranus/Sun mass ratio $q\sim 15~{}q_{\oplus}$ was actually detected prior to
2018222Although OGLE-2017-BLG-0173 has a best-fit solution of $q\sim
8~{}q_{\oplus}$, its two degenerate solutions have $q\sim 21~{}q_{\oplus}$ at
$\Delta\chi^{2}=3.5$ (Hwang et al., 2018).. This failure gave rise to several
different suggestions of a paucity of low mass-ratio planets. Suzuki et al.
(2016) argued for a “break” in the mass-ratio function at $q_{\rm br}\sim
57~{}q_{\oplus}$ based on a statistically well-defined sample of planets
detected from MOA-discovered microlensing events. Jung et al. (2019) argued
for a “break”, or possibly a “pile-up” at $q\sim 19~{}q_{\oplus}$ based on the
ensemble of planets with $q<3\times 10^{-4}$. Udalski et al. (2018) used a new
“$V/V_{\rm max}$” method to show that if planets with $q<10^{-4}$ were modeled
as a power-law distribution in $q$, then the distribution was rising toward
higher $q$, seemingly confirming the Suzuki et al. (2016) “break”.

However, in 2018 and 2019, three planets were discovered with $q$ below the
previous record, and hence below the level of the conjectured “break” or
“pile-up”: KMT-2018-BLG-0029Lb ($q\sim 1.8\times 10^{-5}$, Gould et al. 2020),
KMT-2019-BLG-0842Lb ($q\sim 4.1\times 10^{-5}$, Jung et al. 2020), and
OGLE-2019-BLG-0960Lb ($q\sim 1.4\times 10^{-5}$, Yee et al. 2021). All three
were detected via resonant caustics at or near the peak of moderately high
magnification events, with $|s-1|\sim(0.000,0.017,0.003~{}{\rm or}~{}0.029)$.
Moreover, Yee et al. (2021) showed that the recent discoveries of
KMT-2018-BLG-0029Lb and OGLE-2019-BLG-0960Lb populate the previously-vacant
lower region of the $(\log s,\log q)$ diagram. See their Figure 11. It is
therefore clear that the well-established sensitivity to $q\sim q_{\oplus}$
depends primarily on relatively rare (i.e., high-magnification) microlensing
events generated by relatively rare resonant lens configurations. And,
therefore, it is possible that the previous paucity of planets near the
detection limit was more a product of the difficulty of detection than the
intrinsic rarity of the population.

In this context, it is notable that over the last 10 years, microlensing
planet searches have moved away from their previous focus on high-
magnification events, which is one of the two “rare elements” just described
for probing the low-$q$ population. Prior to the inauguration of the wide-
field-camera OGLE-IV survey (Udalski et al., 2015), substantial effort was
expended, particularly by the Microlensing Follow Up Network ($\mu$FUN), to
find high-magnification events, and then to focus intensive observations over
the peak (Gould et al., 2010). As a result, the projected spatial distribution
of planetary events was drawn roughly uniformly from the OGLE-III and MOA-II
surveys. See the blue circles in Figure 8 of Ryu et al. (2020a). That is, the
surveys were able to detect events, more or less regardless of cadence, but
were much less able to detect planets on their own, again regardless of
cadence. Hence, planet-yielding events were unaffected by survey cadence.
However, with the layered approach (higher cadence in more productive fields)
made possible by the introduction of the larger-format OGLE-IV camera, planet
searches came to rely more on survey data, so that planetary discoveries
became more concentrated on high-cadence regions. See green and yellow points
of the same diagram. This remained so with the advent of the still larger-
format KMTNet (Kim et al., 2016) survey (magenta and black points).

Beginning in 2016, KMTNet continuously monitors $\sim 97~{}{\rm deg}^{2}$ area
of sky toward the Galactic bulge field from three 1.6m telescopes equipped
with 4 ${\rm deg}^{2}$ FOV cameras at CTIO in Chile, SAAO in South Africa, and
SSO in Australia. In fact, KMTNet’s three-observatory system is capable of
detecting very low-$q$ planets in its highest cadence, $\Gamma=4\,{\rm
hr}^{-1}$ fields, as was proved by the case of KMT-2019-BLG-0842Lb. The
substantially lower-$q$ planet KMT-2018-BLG-0029Lb was discovered in KMT-only
observations of a $\Gamma=1\,{\rm hr}^{-1}$ field. For the lowest-$q$ planet
OGLE-2019-BLG-0960Lb, although the planetary signal was first recognized by
the $\mu$FUN CT13 data and was most clearly delineated by the $\mu$FUN Kumeu
data, the detection would probably have been regarded as secure using the low-
cadence survey observations ($\Gamma_{\rm KMT}\sim 0.4\,{\rm hr}^{-1}$, 3
sites; $\Gamma_{\rm OGLE}=0.17\,{\rm hr}^{-1}$, 1 site; $\Gamma_{\rm
MOA}=0.6\,{\rm hr}^{-1}$, 1 site).

An additional notable feature of these detections is that the magnification of
the underlying 1L1S event at the time of the planetary anomaly was modest
($A_{\rm anom}\sim 37$ for KMT-2018-BLG-0029Lb, $A_{\rm anom}\sim 22$ for
KMT-2019-BLG-0842Lb and $A_{\rm anom}\sim 45$ for OGLE-2019-BLG-0960Lb),
although $A_{\rm max}\sim 160$ for the latter two cases. The source trajectory
was “oblique”, i.e., close to parallel to the long axis of the resonant
caustic, with $\alpha=8.3^{\circ}$ and $\alpha=15.5^{\circ}$ for
KMT-2019-BLG-0842Lb and OGLE-2019-BLG-0960Lb, respectively. While such oblique
trajectories are relatively rare, they can enhance detection efficiency by
“stretching out” anomalies. Moreover, Yee et al. (2021) found that sensitivity
to very low-$q$ planets can be maximized by intensively monitoring events
whenever they were magnified by a factor $A>10$ (see also Abe et al. 2013). In
order to probe the very low-$q$ planets, KMTNet together with the LCO &
$\mu$FUN Follow-up Team developed a program for focusing on observations and
analysis of $A\gtrsim 20$ events located in KMTNet $\Gamma\leq 1\,{\rm
hr}^{-1}$ fields.

However, with the advent of Covid-19, two of KMT’s three observatories were
shut down, leaving only KMT’s Australia telescope as operational. Hence, the
conditions became much more similar to those of the first decade of this
century, when microlensing alerts came primarily from the OGLE-III survey, and
planets were primarily discovered by follow-up observations of these single-
site alerts, as well as some non-overlapping alerts from MOA. Of course, there
were some differences. In particular, KMT has a much larger format camera than
OGLE-III, and so it operates at substantially higher cadence. However, KMTA
also has the worst conditions of KMT’s three observatories, and so is inferior
to OGLE-III in both weather interruptions and photometric precision.

It was the specific orientation of this program, i.e., follow-up observations
of $A\gtrsim 20$ events that led to the discovery of KMT-2020-BLG-0414Lb, the
lowest mass-ratio microlensing planet discovered to date, $q_{2}\sim 10^{-5}$.
Due to the extreme magnification of the event, $A_{\rm max}\sim 1450$ for the
underlying single-lens event, and high-cadence observations by MOA over the
peak, a second companion was also detected, with $q_{3}\sim 0.05$.

##  2 Observations

KMT-2020-BLG-0414 occurred at equatorial coordinates $(\alpha,\delta)_{\rm
J2000}$ = (18:07:39.60, $-$28:29:06.8), corresponding to Galactic coordinates
$(\ell,b)=(2.82,-3.95)$. It was announced as a “probable microlensing” event
by the KMTNet Alert-Finder system (Kim et al., 2018) on 1 June 2020, about 40
days before peak, when the event was manifested as an $I\sim 18.7$ difference
star. As mentioned in Section 1, by this date, KMTC and KMTS had been closed
down for more than two months due to Covid-19. Hence, only KMTA data
contributed to the alert and to subsequent monitoring of the event. The event
lies in the KMNTet BLG32 field, which has a cadence of $\Gamma=0.4~{}{\rm
hr}^{-1}$, with every tenth $I$-band observation complemented by one in the
$V$ band for the source color measurements333In fact, this $V$-band to
$I$-band ratio applies only to the “normal cycle” of KMT observations. During
the latter part of the season (including the peak of KMT-2020-BLG-0414), these
normal-cycle observations were supplemented by an end-of-night sequence of
Eastern fields, which was purely in the $I$-band. For the low-cadence field
BLG32, these end-of-night observations accounted for $\sim 30\%$ of the total
near the peak of the event..

The event was independently identified by the Microlensing Observations in
Astrophysics (MOA, Sumi et al. 2016) collaboration as MOA-2020-BLG-109 on 20
June 2020 (Bond et al., 2001). MOA observations are carried out with a 1.8m
telescope at the Mt. John University Observatory in New Zealand, which is
equipped with a 2.2 square degree camera. The nominal cadence for this field
was $\Gamma=1.2~{}{\rm hr}^{-1}$ by a MOA-Red filter (which is similar to the
sum of the standard Cousins $R$\- and $I$-band filters), and observations with
the MOA $V$ filter (Bessell V-band) were taken once every clear night. Earlier
in the season, the MOA survey had also been closed for Covid-19 for almost 100
days, but it had re-opened 28 days before peak. Hence, the Covid-19 hiatus had
very little effect on its observations of this event.

The KMTNet Alert-Finder system identified KMT-2020-BLG-0414 with a catalog
star at $(\alpha,\delta)_{\rm J2000}$ = (18:07:39.60,$-28$:29:05.50), which is
about $1.3^{\prime\prime}$ away from the true position of the source. As a
result, the real-time on-line photometry was relatively noisy. Nevertheless,
at UT 17:34 on 7 July 2020 (${\rm HJD}^{\prime}=9038.23,{\rm
HJD}^{\prime}={\rm HJD}-2450000$), the LCO & $\mu$FUN Follow-up Team found
that this event had magnification $A_{\rm now}>10$ based on the two KMTA
points at ${\rm HJD}^{\prime}\sim 9038$ and could peak at a high magnification
2–3 days later444The LCO & $\mu$FUN Follow-up Team recognized the KMTA
photometric centroid shift by its noisy curve and thus started follow-up
observations for security, although $A_{\rm now}$ did not meet the $A\gtrsim
20$ threshold. In fact, the actual magnification at that time was about 35..
Thus, high-cadence follow-up observations were immediately scheduled by Las
Cumbres Observatory (LCO) global network and Observatorio do Pico dos Dias
(OPD) in Brazil (a $\mu$FUN site). The LCO global network conducted
observations from its 1.0m telescopes located at SAAO (LCOS), SSO (LCOA) and
McDonald (LCOM), with the SDSS-$i^{\prime}$ filter. Observations by OPD were
taken from its 0.6m (OPD06) and 1.6m (OPD16) telescopes with the $I$ filter.
At UT 05:58 on 10 July 2020 (${\rm HJD}^{\prime}=9040.75$), the LCO & $\mu$FUN
Follow-up Team identified that this event was currently undergoing an anomaly
and would peak at a very high magnification soon, based on the real-time LCO
and MOA data. Noting that SSO was predicted to be rainy that night, the Team
issued an alert to the MOA collaboration. MOA responded to the alert and
densely observed this event 60 times over the peak. Due to the very high
brightness, the MOA observer decreased the exposure time from 60s to 5.2s over
the peak. We carefully inspected these 60 MOA images and excluded 19 data
points from the analysis due to saturation or bad seeing. At UT 13:53 on 10
July 2020 (${\rm HJD}^{\prime}=9041.08$), the LCO & $\mu$FUN Follow-up Team
also issued an alert to all $\mu$FUN observers. As a result, the 0.3m Perth
Exoplanet Survey Telescope (PEST) in Australia and the 0.4m Possum Observatory
(Possum) at New Zealand responded to the alert and took intensive observations
without a filter. Finally, the event was also observed by the 3.6m Canada-
France-Hawaii Telescope (CFHT) with the SDSS-$i^{\prime}$ filter.

For the light curve analysis, the KMTNet, MOA, CFHT and LCO data were reduced
using custom implementations of the difference image analysis technique
(Tomaney & Crotts, 1996; Alard & Lupton, 1998): pySIS (Albrow et al., 2009)
for the KMTNet data, Bond et al. (2001) for the MOA data and ISIS (Alard &
Lupton, 1998; Alard, 2000; Zang et al., 2018) for the CFHT and LCO data. The
OPD, PEST and Possum data were reduced using DoPHOT (Schechter et al., 1993).
On the OPD16 images, the target was affected by a bleed trail from a saturated
star, resulting in some systematics. We therefore do not include OPD16 data in
the analysis. The $I$-band magnitude of the KMTA light curve has been
calibrated to the standard $I$-band magnitude. For the source color
measurements, we use the MOA $V$-band data, while KMTA $V$-band data are not
used due to poor seeing. The errors from photometric measurements for each
data set $i$ were renormalized using the formula
$\sigma_{i}^{\prime}=k_{i}\sqrt{\sigma_{i}^{2}+e^{2}_{i,{\rm min}}}$, where
$\sigma_{i}$ and $\sigma_{i}^{\prime}$ are original errors from the photometry
pipelines and renormalized error bars in magnitudes, and $k_{i}$ and
$e_{i,{\rm min}}$ are rescaling factors. We obtained the rescaling factors
using the procedure of Yee et al. (2012), which enables $\chi^{2}/{\rm dof}$
for each data set to become unity. We derived the rescaling factors using the
binary lens (2L1S) and triple lens (3L1S) models, respectively, in order to
understand how the event would have been interpreted in the absence of MOA
data on the peak. The data used in the analysis, together with corresponding
data reduction method and rescaling factors are summarized in Table 1.

##  3 2L1S Analysis

Figure 1 shows the observed light curve of KMT-2020-BLG-0414. Although the
light curve can be regarded as “single peak” in the sense that it
monotonically rises and then falls, its significantly asymmetric shape cannot
be fitted by a single-lens single-source (1L1S) model. A 1L1S model is usually
described by three Paczyński (1986) parameters $(t_{0},u_{0},t_{\rm E})$,
i.e., the time of closest lens-source approach, the impact parameter scaled to
$\theta_{\rm E}$, and the Einstein crossing time,

$t_{\rm E}=\frac{\theta_{\rm E}}{|\bm{\mu}_{\rm rel}|};\qquad\theta_{\rm
E}=\sqrt{\kappa M_{\rm L}\pi_{\rm
rel}};\qquad\kappa\equiv\frac{4G}{c^{2}\mathrm{au}}\simeq 8.144\frac{{\rm
mas}}{M_{\odot}},$ (1)

where $M_{\rm L}$ is the mass of the lens and $(\pi_{\rm rel},\bm{\mu}_{\rm
rel})$ are the lens-source relative (parallax, proper motion). In the present
case, we also consider finite-source effects (Gould, 1994; Witt & Mao, 1994;
Nemiroff & Wickramasinghe, 1994), which occur when the source passes close to
singular structures in the magnification pattern. This requires a fourth
parameter $\rho=\theta_{*}/\theta_{\rm E}$, where $\theta_{*}$ is the angular
radius of the source.

As we will show in Section 4, the full light curve cannot be explained by a
2L1S model, and in fact requires 3L1S. However, if we exclude the MOA data on
the peak ($9040.78<{\rm HJD}^{\prime}<9041.20$, the only data set to cover the
peak), then the remaining data are quite well fit by a 2L1S model. We
therefore begin by analyzing this restricted (non-MOA-peak) data set. We are
motivated by two considerations. First, and most importantly, 3L1S models
often “factor” into two 2L1S models (Han, 2005; Gaudi et al., 2008; Gould et
al., 2014; Han et al., 2013, 2017, 2019a). In particular, the 3L1S caustic is
often very nearly the superposition of the two 2L1S caustics. In such cases,
one can often exclude the data from the neighborhood of the anomaly from the
“second” 2L1S model in order to accurately determine the parameters of the
“first” 2L1S model. In these cases, the “first” 2L1S model provides a powerful
basis for finding the full 3L1S model, by one of several techniques. This
proves to be the case for KMT-2020-BLG-0414. Second, it is of independent
scientific interest to understand how the event would have been interpreted
and check the so-called “higher-order effects” in the absence of MOA data on
the peak. For example, MOA could have been weathered out on the night of the
peak (as was KMTA). The analysis would have led to a report of a single low-
mass-ratio planet and a lens that is much brighter than the blended light. The
comparison of this reconstructed “report” with the full model can inform our
understanding of other 2L1S events with incomplete light-curve coverage.

Therefore, for the remainder of this section, we will exclude the MOA data on
the peak. These data will then be incorporated in Section 4.

###  3.1 Static Binary Lens Model

The 2L1S model requires three additional parameters $(s,q,\alpha)$, which are
respectively the separation of the two lens bodies scaled to $\theta_{\rm E}$,
the mass ratio between these bodies, and the angle of the source trajectory
relative to the binary axis. For modeling, we use the advanced contour
integration code (Bozza, 2010; Bozza et al., 2018),
VBBinaryLensing555http://www.fisica.unisa.it/GravitationAstrophysics/VBBinaryLensing.htm.
We initially carry out a sparse grid searches for the parameters ($\log s,\log
q,\alpha$). The grid consists of 21 values equally spaced between
$-1.0\leq\log s\leq 1.0$, 20 values equally spaced between
$0^{\circ}\leq\alpha<360^{\circ}$, and 61 values equally spaced between
$-6.0\leq\log q\leq 0.0$. For each set of ($\log s,\log q,\alpha$), we find
the minimum $\chi^{2}$ by Markov chain Monte Carlo (MCMC) $\chi^{2}$
minimization using the emcee ensemble sampler (Foreman-Mackey et al., 2013),
with fixed $\log q$, $\log s$ and free $t_{0},u_{0},t_{\rm E},\rho,\alpha$. We
identify one local minimum at $(\log s,\log q)\simeq(0.0,-5.1)$, similar to
the case of Yee et al. (2021). We thus conduct a similar dense grid search as
Yee et al. (2021) that consists of 51 values equally spaced between
$-0.02\leq\log s\leq 0.03$ and 41 values equally spaced between $-6.0\leq\log
q\leq-4.0$. Often, such a grid search yields two local minima at $s>1$ and
$s<1$ (e.g., Jung et al., 2020; Yee et al., 2021), which must then be
individually further explored and compared. However, in the present case,
there is only one local minimum at $s<1$, while the $s>1$ model is disfavored
by $\Delta\chi^{2}>900$. We refine this minimum by allowing all parameters to
vary. The parameters with their $68\%$ uncertainty range from the MCMC are
shown in Table 2, and the fit and residuals are shown in Figure 1. The very
low mass ratio $q\sim 10^{-5}$ indicates that the companion is a very-low-mass
planet.

###  3.2 Microlens Parallax Model

Even without detailed analysis, the results for the static model, that are
listed in Table 2, imply a large (and so potentially measurable) microlens
parallax (Gould, 1992, 2000),

$\bm{\pi}_{\rm E}\equiv\frac{\pi_{\rm rel}}{\theta_{\rm E}}\frac{\bm{\mu}_{\rm
rel}}{\mu_{\rm rel}}.$ (2)

According to $\theta_{*}$ and the blended light in Section 5, $\theta_{\rm
E}=\theta_{*}/\rho\geq 1.68~{}{\rm mas}$ and the lens light $I_{\rm L}\geq
18.9$ at $3\sigma$ level. The two values correspond roughly to an $M\sim
0.5M_{\odot}$ at a distance of 1.2 kpc. Thus, we can expect666One exception to
this limit would be if the lens were a massive remnant.

$\pi_{\rm E}=\frac{\theta_{\rm E}}{\kappa M_{\rm L}}\geq 0.41.$ (3)

Moreover, given the long Einstein timescale, $t_{\rm E}\sim 103$ days, the
projected velocity on the observer plane, $\tilde{v}\equiv{\rm au}/(\pi_{\rm
E}t_{\rm E})\lesssim 42~{}{\rm km~{}s}^{-1}$ is close to the changes of
Earth’s velocity over the course of the event, taking account of which could
impact other parameters as well.

Therefore, it is essential to include microlens-parallax effects in the fit.
We fit the annual microlens-parallax effect by introducing two additional
parameters $\pi_{\rm E,N}$ and $\pi_{\rm E,E}$, the North and East components
of $\pi_{\rm E}$ in equatorial coordinates (Gould, 2004). We also fit
$u_{0}>0$ and $u_{0}<0$ models to consider the “ecliptic degeneracy” (Jiang et
al., 2004; Poindexter et al., 2005). Table 2 shows the results of fitting the
light curve with the microlens parallax effect. Because the annual parallax
effect can be degenerate with the effects of lens orbital motion (Batista et
al., 2011; Skowron et al., 2011), we also introduce two linearized parameters
($ds/dt,d\alpha/dt$), the instantaneous changes in the separation and
orientation of the two lens components defined at $t_{0}$. We restrict the
MCMC trials to $\beta<0.8$, where $\beta$ is the ratio of projected kinetic to
potential energy (Dong et al., 2009)

$\beta\equiv\left|\frac{{\rm KE}_{\perp}}{{\rm
PE}_{\perp}}\right|=\frac{\kappa M_{\odot}{\rm
yr}^{2}}{8\pi^{2}}\frac{\pi_{\rm E}}{\theta_{\rm
E}}\gamma^{2}\left(\frac{s}{\pi_{\rm E}+\pi_{\rm S}/\theta_{\rm
E}}\right)^{3};\qquad\bm{\gamma}\equiv\left(\frac{ds/dt}{s},\frac{d\alpha}{dt}\right),$
(4)

where we adopt the source parallax $\pi_{\rm S}=0.128$ mas based on the mean
distance to clump giant stars in this direction (Nataf et al., 2013). See
Table 2 for the results. We find that the addition of lens orbital motion
effect provides improvements of $\Delta\chi^{2}=2.4~{}{\rm and}~{}5.4$ for the
$u_{0}>0$ and $u_{0}<0$ solution, respectively, and $\bm{\pi}_{\rm E}$ is
basically the same compared to the “parallax” model.

Although the angular Einstein radius $\theta_{\rm E}$ estimated from the
parallax modeling is smaller than the value from the static model, the
resulting parallax for 2L1S is still strongly inconsistent with the constraint
of the blended light at about $3\sigma$. We will further discuss the
implication of the 2L1S results in Section 6.1.

###  3.3 Binary-Source (1L2S) Model

In some cases, planetary (2L1S) light curves can be imitated by binary-source
(1L2S) events (Gaudi, 1998). We do not expect that this will be case for
KMT-2020-BLG-0414 because the planetary anomaly is mainly characterized by
sharp changes in slope, rather than a smooth short-lived bump. Nevertheless,
as a matter of due diligence, we search for such models including both
microlens-parallax and microlens-xarallap effects (Griest & Hu, 1992; Han &
Gould, 1997; Poindexter et al., 2005). We find that while the introduction of
a second source yields a huge improvement with respect to the 1L1S model with
$\Delta\chi^{2}=\chi^{2}({\rm 1L1S})-\chi^{2}({\rm 1L2S})>10000$, the 1L2S
model still does not compete with the 2L1S model with
$\Delta\chi^{2}=\chi^{2}({\rm 1L2S})-\chi^{2}({\rm 2L1S})>400$.

##  4 3L1S Analysis

While the 2L1S models described in Section 3 fit the non-MOA-peak data very
well, they completely fail to explain the features of the MOA data in the peak
region. Moreover, a 2L1S grid search that includes all the data fails to
return any model that even approximately traces the data over the peak. See
Figure 1. We therefore conduct a 3L1S grid search. Relative to static 2L1S
models, 3L1S models have three additional parameters, $(s_{3},q_{3},\psi)$.
These are, respectively, the normalized separation of the third body from the
primary, the mass ratio of the third body to the primary, and the angle of the
second from the third body, as seen from the primary. Note that, to avoid
confusion, we rename $(s,q)\rightarrow(s_{2},q_{2})$.

###  4.1 3L1S Static Models

We begin by conducting a grid search for static 3L1S solutions that is
analogous to the one carried out previously for 2L1S solutions, but is
substantially more computationally intensive. In a grid search (whether 2L1S
or 3L1S), the lens geometry is held fixed at each grid point. However, for
2L1S, the geometry is specified by just two parameters, $(s_{2},q_{2})$,
whereas for 3L1S, five geometric parameters are required,
$(s_{2},q_{2},s_{3},q_{3},\psi)$. To reduce the grid of geometries from five
to three dimensions, we consider a $(70\times 70\times 180)$ grid in
$(s_{3},q_{3},\psi)$. We hold $(s_{2},q_{2})$ fixed at the best-fit 2L1S
model. We seed the remaining five parameters, $(t_{0},u_{0},t_{\rm
E},\rho,\alpha)$, at the best-fit 2L1S model, and we then allow these to vary.
We initially consider only close models for the third body, i.e., $s_{3}<1$.
For the grid-search phase, which relies on fixed geometries, we apply the map-
making technique of Dong et al. (2006) to evaluate the magnifications. This
grid search yields only one local region of candidate solutions.

We then seed an additional MCMC with the best grid point from this region and
allow all 10 parameters to vary. Because the geometry now varies with each
step in the MCMC, we use the adaptive-image inverse-ray-shooting technique to
evaluate the magnifications. The resulting parameters are shown in Table 3,
and the model light curve is compared to the data in Figure 1. The
corresponding caustic structure in the upper panel of Figure 2 shows that
KMT-2020-BLG-0414 is a classic case of “caustic factorization”. The caustic is
nearly the superposition of two well-known caustic types: a large resonant
caustic associated with the planet, and a smaller, nearly Chang-Refsdal (Chang
& Refsdal, 1979), caustic associated with the third body. As is often the
case, the two caustic structures “interact” and become intertwined where they
overlap. Because the caustic factors and $q_{3}\ll 1$, it is straightforward
to guess the alternate wide $(s_{3}>1)$ solution according to the prescription
of Griest & Safizadeh (1998): $s_{3}\rightarrow s_{3}^{-1}$. We seed this
guess into an MCMC to yield the alternate wide solution, whose parameters are
given in Table 3 and whose geometry is shown in the lower panel of Figure 2.
The most striking difference between the 3L1S close and wide solutions is that
$s_{2,\rm close}=0.99914\pm 0.00011$, while $s_{2,\rm wide}=0.96882\pm
0.00018$, which appears to be a “$100\sigma$” difference. We address this
issue in Appendix § A.

###  4.2 3L1S Parallax-only Models

As discussed in Section 3.2, the microlens-parallax parameters $\bm{\pi}_{\rm
E}$ (Equation (2)) can be degenerate with the orbital-motion parameters
$\bm{\gamma}$ (Equation (4)). Hence, both should be considered together.
However, as in that section, we proceed step-by-step, in part due to the
increasing computational load as more parameters are introduced, and therefore
the importance of understanding which are really necessary.

When only $\bm{\pi}_{\rm E}$ is added to the 3L1S static model, there are 12
parameters. As was the case for 2L1S, adding parallax to the fit results in
doubling the number of solutions, i.e., there is a $\pm u_{0}$ pair of
solutions for each of the close and wide solutions found in Section 4.1. Hence
there are four solutions altogether. The resulting parameters and $\chi^{2}$
values are given in Table 3. It is found that including parallax significantly
improves the fit by $\Delta\chi^{2}>130$. The wide solutions are disfavored by
$\Delta\chi^{2}>17$. Between the two close solutions, the $u_{0}>0$ solution
is significantly favored by $\Delta\chi^{2}\sim 12$. We note that the
magnitude of $\bm{\pi}_{\rm E}$ for 3L1S is substantially larger than the
value estimated from the 2L1S modeling regardless of the lens-orbital effect.
Figure 3 shows the cumulative distribution of $\Delta\chi^{2}=\chi^{2}(\rm
static)-\chi^{2}(\rm parallax)$ for the four solutions. Overall,
$\Delta\chi^{2}$ grows steadily over time, giving credence to the parallax
measurement. An important feature of this diagram is that the contribution to
$\Delta\chi^{2}$ during the short time interval starting from one day before
the peak and ending two days after the peak is about $40\%$ of the total
$\Delta\chi^{2}$. By contrast, one normally expects the parallax signal to be
dominated by the wings of the light curve. This time interval is essentially
the duration of contact with the planetary caustic, in particular as the
source “rides the caustic” for two days after the peak. This gives a plausible
explanation for the sensitivity of $\bm{\pi}_{\rm E}$ to the near-peak region
of the light curve. Hence, it is essential to include orbital motion.

###  4.3 3L1S Parallax and Planet Orbital Motion

We now include both $\bm{\pi}_{\rm E}$ and $\bm{\gamma}$ for the planet in the
3L1S fit, for a total of 14 chain parameters. We show the parameters of this
fit in the Table 4. It is found that including planet orbital motion
significantly changes $\bm{\pi}_{\rm E}$ in both magnitude and direction, and
$|\pi_{\rm E}|$ of the $u_{0}<0$ solution is $\sim 1.8$ times greater than
that of the $u_{0}>0$ solution. The close $u_{0}<0$ solution has the best fit
to the observed data, while other solutions are only disfavored by
$\Delta\chi^{2}<7$. Thus, we cannot exclude any solution from the light-curve
analysis. The ratio $\beta$ of projected kinetic to potential energy is well
measured, and all of the solutions have $\beta<0.1$ at $3\sigma$. This
relatively low value of the ratio suggests that the planet and the host may be
aligned along the line of sight.

###  4.4 Possible Orbital Motion of the Third Body

The third body (with a brown-dwarf-like mass ratio $q_{3}\sim 0.05$) must also
undergo orbital motion. In the wide solutions, the period would be of order
100 years, implying that orbital motion of the third body would not affect the
lensing light curve. For the close solutions, the period would be of order 50
days, and thus its orbital motion could affect the light curve. Nevertheless,
we do not attempt to model orbital motion of the third body for several
reasons. First, the duration of its pronounced perturbation ($\sim 0.3\,$days
over the peak) is 10 times shorter than for the duration of the planetary
signal, and its impact on the light curve is quadratic in the duration.
Second, if there were clear prospects of a scientifically important result,
such work would be warranted, but there are no such prospects (see Section
5.4). Finally, the results of 3L1S parallax + planet orbital motion required
about two weeks of computations with 400 processors. The already prodigious
use of computer time (which scales $\propto(n/2)!$ where $n$ is the number of
chain parameters) would increase by a factor eight. We therefore decline to
pursue this aspect of the problem.

##  5 Physical Parameters

Normally, if the angular Einstein radius $\theta_{\rm E}$ and the microlens
parallax $\pi_{\rm E}$ are well measured, one can simply determine the lens
total mass $M_{\rm L}$ and the lens distance $D_{\rm L}$ by (Gould, 1992,
2000)

$M_{\rm L}=\frac{\theta_{\rm E}}{{\kappa}\pi_{\rm E}};\qquad D_{\rm
L}=\frac{\mathrm{AU}}{\pi_{\rm E}\theta_{\rm E}+\pi_{\rm S}}.$ (5)

However, in the present case, the lens system is very close ($D_{\rm L}\sim 1$
kpc), and the large symmetric errors of $\pi_{\rm E}$ can lead to an
asymmetric distribution in inferred lens distance. Hence, we conduct a
Bayesian analysis to estimate the lens physical parameters in Section 5.3.
Before doing so, we estimate $\theta_{*}$ by a color-magnitude diagram (CMD)
analysis (Yoo et al., 2004) in Section 5.1, in order to estimate the angular
Einstein radius by $\theta_{\rm E}=\theta_{*}/\rho$. We also study the blended
light in Section 5.2 to obtain constraints on the lens light. Finally in
Section 5.4, we illustrate how future high-resolution photometric and
spectroscopic observations would clarify the nature of the system.

###  5.1 Color-Magnitude Diagram (CMD)

Figure 4 shows the CMD of stars from the OGLE-III catalog (Szymański et al.,
2011) located within a square region with one side length of
$240^{\prime\prime}$ centered at the location of KMT-2020-BLG-0414, together
with the source position (blue) and the centroid of the red giant clump (red).
We measure the centroid of the red giant clump as $(V-I,I)_{\rm cl}=(2.04\pm
0.01,15.55\pm 0.02)$. For the intrinsic centroid of the red giant clump, we
adopt $(V-I,I)_{\rm cl,0}=(1.06,14.35)$ (Bensby et al., 2013; Nataf et al.,
2013). This implies that $A_{I}=1.20$ and $E(V-I)=0.98$ toward this direction.
For the source color, which is independent of any model, we get $(V-I)_{\rm
S}=1.82\pm 0.01$ by regression of MOA $V$ versus $R$ flux as the source
magnification changes and a calibration to the OGLE-III scale using the field-
star photometry from the same reductions. Because the source apparent
brightness slightly depends on the model, for simplicity, we explicitly derive
results for $I_{\rm S}=19.12$ and then present a scaling relation for
different source magnitudes. From this procedure, we obtain the intrinsic
color and brightness of the source as $(V-I,I)_{\rm S,0}=(0.84\pm
0.03,17.92\pm 0.03)$, suggesting that the source is a mid-G type dwarf
(Bessell & Brett, 1988). Using the color/surface-brightness relation of Adams
et al. (2018), we obtain

$\theta_{*}=0.943\pm 0.047~{}\mu{\rm as}.$ (6)

where the 5% error is given by Table 3 of Adams et al. (2018). Then, for any
particular model with source magnitude $I_{\rm S}$, one can infer
$\theta_{*}=0.943\times 10^{-0.2(I_{\rm S}-19.12)}$.

###  5.2 The Blended Light

For KMT-2020-BLG-0414, the baseline object was detected by the OGLE-III
survey, $(V,I)_{\rm base}=(20.70\pm 0.08,18.46\pm 0.03)$. We further consider
the statistical errors due to the mottled background from unresolved stars
(Park et al., 2004). We follow the approach of Ryu et al. (2020b) using the
GalSim package (Rowe et al., 2015) with the readout noise of Udalski et al.
(2015). We find $\sigma_{I}=0.08$ mag and $\sigma_{V}=0.012$ mag, and thus the
baseline object has $(V,I)_{\rm base}=(20.70\pm 0.14,18.46\pm 0.09)$, yielding
the blended light of $(V-I,I)_{\rm B}=(3.1^{+1.0}_{-0.6},19.32\pm 0.20)$. This
value is consistent with the lens properties that are predicted by the
microlensing light-curve and CMD analyses. For example, using the median
$\theta_{\rm E}$ and $\pi_{\rm E}$ values of the close $u_{0}<0$ solutions,
the host mass $M_{1}=0.25~{}M_{\odot}$ and it would have rough intrinsic
brightness and color of $M_{I}\sim 9.8$, $(V-I)_{0}\sim 3.0$. Assuming an
extinction curve with a scale height of 120 pc, it would have
$(A_{I},E(V-I))_{\rm L}=(0.38,0.31)$ at the lens distance $D_{\rm L}=0.74$
kpc. These corresponds to $(V-I,I)_{\rm L}\sim(3.3,19.5)$, which is shown as
the cyan point (“naive lens”) in Figure 4 and is quite consistent with the
blend.

We also check the astrometric alignment between the source and the baseline
object from KMTA imaging. We find that the baseline object lies
$(0.18^{\prime\prime},0.01^{\prime\prime})$ west and south of the source.
Because the source position is derived from difference image analysis on
highly magnified images, the uncertainty in the source position ($\lesssim
0.01^{\prime\prime}$) is negligible relative to the error in the baseline
position. We estimate the error of baseline position by the fractional
astrometric error being equal to the fractional photometric error (Jung et
al., 2020), $\sigma_{\rm ast}=0.39\sigma_{I}{\rm FWHM}$ =
$0.09^{\prime\prime}$. Hence, the baseline object is astrometrically
consistent with the source (and thus lens) at $2\sigma$ level. Thus, it is
plausible that most or all of the blended light is due to the lens.

The alignment between the source and the baseline object can be immediately
checked (i.e., 2021 bulge season) by the Hubble Space Telescope (HST) or by
ground-based adaptive optics (AO) mounted on large ground-based telescopes
(e.g., Keck, Subaru). Even if the alignment was demonstrated, the blended
light could in principle come from a stellar companion to either the source or
the lens. However, if the alignment is $\lesssim 50~{}{\rm mas}$, an
additional stellar companion to the lens would have generated significant
deviations on the peak, so the confirmation that the blend is well aligned to
the lens could probably rule out the lens-companion scenario. Because the
source and the blended light have significantly different colors, the
possibility of the source companion can be checked by a measurement of the
astrometric offset in different bands (Bennett et al., 2006). It is a priori
unlikely that the blended light is primarily due to ambient stars that are
unassociated with the event because of the low surface density of stars
relative to the 180 mas offset. If high-resolution imaging nevertheless showed
that this were the case, it would imply that the parallax was even larger and
the lens was less massive and closer than the best-estimated values 777Or the
host could conceivably be a white dwarf..

###  5.3 Bayesian Analysis

In the Bayesian analysis, we choose the log-normal initial mass function of
Chabrier (2003) as the mass distribution of the lens. For the bulge and disk
stellar number density, we choose the model used by Zhu et al. (2017) and
Bennett et al. (2014), respectively. For the dynamical distribution of the
disk lens, we assume the disk lenses follow a rotation of $240~{}{\rm
km~{}s}^{-1}$ (Reid et al., 2014) with the velocity dispersion used by Han et
al. (2020a). For the source and bulge lens dynamical distributions, we examine
a _Gaia_ CMD (Gaia Collaboration et al., 2016, 2018) using the stars within
$5^{\prime}$ and derive the proper motion of “clump” stars
($16.4<G<17,1.9<B_{p}-R_{p}<2.35$). We obtain (in the heliocentric frame)

$\langle\bm{\mu}_{\rm
bulge}(\ell,b)\rangle=(-6.19,-0.36)\pm(0.15,0.13)~{}\text{masyr}^{-1},$ (7)
$\sigma(\bm{\mu}_{\rm bulge})=(2.78,2.39)\pm(0.11,0.09)~{}\text{masyr}^{-1}.$
(8)

We create a sample of $5\times 10^{9}$ simulated events from the Galactic
model. We weight each simulated event, i, by

$\omega_{{\rm Gal},i}=\Gamma_{i}\mathcal{L}_{i}(t_{\rm
E})\mathcal{L}_{i}(\theta_{\rm E})\mathcal{L}_{i}(\pi_{\rm E,N},\pi_{\rm
E,E}),$ (9)

where $\Gamma_{i}\varpropto\theta_{{\rm E},i}\times\mu_{{\rm rel},i}$ is the
microlensing event rate, $\mathcal{L}_{i}(t_{\rm E})$,
$\mathcal{L}_{i}(\theta_{\rm E})$ and $\mathcal{L}_{i}(\pi_{\rm E,N},\pi_{\rm
E,E})$ are the likelihoods of its inferred parameters given the distributions
of these quantities. In addition, we adopt the $3\sigma$ upper limit of the
blended light, $I_{\rm B,limit}=18.9$, as the upper limit of the lens flux. We
then adopt the mass-luminosity relation of Wang et al. (2018),

$M_{I}=4.4-8.5\log{(\frac{M_{\rm L}}{M_{\odot}})},$ (10)

where $M_{I}$ is the absolute magnitude in the $I$ band, and reject trial
events for which the lens properties obey

$M_{I}+5\log{\frac{D_{\rm L}}{10{\rm pc}}}+A_{I,D_{\rm L}}<I_{\rm B,limit},$
(11)

where $A_{I,D_{\rm L}}$ is the extinction at $D_{\rm L}$, which is derived by
an extinction curve with a scale height of 120 pc.

In Table 5, we list the angular Einstein radius, $\theta_{\rm E}$, the
microlens parallax, $\pi_{\rm E}$, the relative proper motion, $\mu_{\rm
rel}$, the estimated masses of the individual lens components, $M_{1}$,
$M_{2}$ and $M_{3}$, the distance to the lens, $D_{\rm L}$, and the projected
separations to the host, $a_{\perp,2}$ and $a_{\perp,3}$ for all of the four
3L1S solutions from the Bayesian analysis. For both solutions, it is found
that the host is probably an $M_{1}\sim 0.3M_{\odot}$ M-type dwarf, the second
body is an Earth-mass terrestrial planet at a projected separation of
$a_{\perp,2}\sim 1.5$ au, and the third body is an object at the planet/brown-
dwarf boundary at a projected separation of $a_{\perp,3}\sim 0.15$ au for the
close solutions and $a_{\perp,3}\sim 15$ au for the wide solutions. The lens
system is located in the Galactic disk with a distance of $D_{\rm L}\sim 1.0$
kpc.

###  5.4 Characterizing the Lens System by Future Observations

The highly degenerate 3L1S solutions and their very different $\bm{\pi}_{\rm
E}$, with large uncertainties, show that the characteristics of the lens
system cannot be completely determined by our work. The ambiguous elements
mainly include two aspects: the mass and distance of the lens system and the
projected separation of the third body. They can be clarified by future high-
resolution photometric and spectroscopic observations.

For the lens mass and distance, they can be unambiguously determined by
conducting high-resolution AO imaging observations when the source and lens
are resolved. Bhattacharya et al. (2018) resolved the source and lens of
OGLE-2012-BLG-0950 using Keck AO and the $\it HST$ when they were separated by
$\sim$ 34 mas, in a case for which the source and lens had approximately equal
brightness. In the present case, the blended light and source also have
approximately equal brightness, so the lens and source can be resolved in 2024
provided that the lens contributes a significant part of the blended light.
The lens-source resolution can yield the measurement of the lens flux and the
lens-source relative proper motion vector, $\bm{\mu_{\rm rel}}$ (e.g., Alcock
et al., 2001; Kozłowski et al., 2007; Batista et al., 2015). The lens flux
would provide an independent mass-distance relationship, and $\bm{\mu_{\rm
rel}}$ would further constrain $\theta_{\rm E}$ (Equation 1) and $\pi_{\rm E}$
(Equation 2) by its magnitude and direction, respectively. Together, these can
significantly reduce the uncertainties of the physical lens parameters.

For the orbit of the third body, future spectroscopy observations to measure
the radial velocity (RV) can determine whether the third body lies inside or
outside the planetary orbit. Following the procedure of Han et al. (2019b), we
estimate that the close solution has a RV amplitude $v\sin(i)$ of order
$1~{}{\rm km}~{}{\rm s}^{-1}$ with a period of order $0.15$ yr, and the wide
solution has a $v\sin(i)$ of order $100~{}{\rm m}~{}{\rm s}^{-1}$ with a
period of order $100$ yr. These RV amplitudes and differences are big enough
to be distinguished by high-resolution spectrometers on VLT/Espresso or future
30m telescopes. Note that if the lens indeed has comparable brightness to the
source, then these observations can be made immediately, i.e., long before the
lens and source separate on the sky, because the lens and source RVs likely
differ by tens or hundreds of km/s.

##  6 Discussion

###  6.1 Inconsistency of the 2L1S Model

In Section 3, we adopted the perspective that there are no data over peak, and
we, in particular, asked how the event would have been analyzed and reported
in the absence of such data. Of course, one obvious difference is that the
data fit well with the 2L1S model and so there would have been no report of a
“third body”. But here we see that there would also have been a report of a
“major puzzle” about this parallax inconsistency. The derived parallax values
in Table 2 ($\pi_{\rm E}=0.229\pm 0.042$ for the $u_{0}>0$ solution or
$\pi_{\rm E}=0.173\pm 0.040$ for the $u_{0}<0$ solution) are strongly
inconsistent with Equation (3), which was derived from a combination of the
well-measured value of $\theta_{\rm E}$ and the photometric contraints on
blended light. Most likely, this would have been “explained” as due to
“probable low-level systematics in the KMTA and/or MOA data”. Perhaps there
would have been some additional commentary about the possibility that the host
was a white dwarf, which would enable evasion of the photometric constraints.
Then, it is very likely that the final reported parameters would have been
given as the static solution, while the physical parameters would have been
derived from a Bayesian analysis that simply ignored the parallax measurement.
And finally, it would have been pointed out that the apparent discrepancies
could ultimately be resolved by future high-resolution follow-up observations.

While the scenarios laid out in the previous paragraph are necessarily
somewhat speculative, we believe that the great majority of people familiar
with the microlensing-planet literature would broadly agree with this
assessment of what would have happened. We rehearse them here in detail
because all of the “explanations”, “caveats”, etc., in the previous paragraph
are completely wrong. In fact, once the peak data are added back into the
light curve and a third body is included in the modeling, the supposed
“tension” surrounding Equation (3) entirely disappears. We note that the
problem of the impact of undetected “third bodies” on microlensing solutions
was analyzed by Zhu et al. (2014a). While that study focused on the impact on
the measured parameters of the detected planet, the present work shows that
higher-order parameters can be affected as well.

###  6.2 Two Paths for Very Low Mass-Ratio Planets

Over the past two years, the previously empty “tip” of the $(\log s,\log q)$
diagram has been gradually populated by new discoveries. While
KMT-2019-BLG-0842Lb lies just below the “pile up” associated with the Jung et
al. (2019) “break” (see Figure 5), three other recent discoveries
(KMT-2018-BLG-0029Lb, OGLE-2019-BLG-0960Lb, KMT-2020-BLG-0414Lb), with
$q\sim(6.0,4.5,3.6)~{}q_{\oplus}$, all lie well below it. All four of these
recent discoveries were detected via resonant caustics. Moreover, for three of
these (all except KMT-2018-BLG-0029Lb), the source trajectory angle with
respect to the binary axis was oblique. For two of the four
(OGLE-2019-BLG-0960Lb and KMT-2020-BLG-0414Lb), followup observations played a
major or dominant role in the detection and characterization.

Due to inhomogeneous (and difficult to model) selection, it is probably not
possible to draw precise statistical conclusions from this particular sample.
However, the broad features listed above tend to support the idea that there
is not a strong break below Neptune mass ratios, as had been previously
conjectured. Rather, the conditions and lens geometries required for such
detections are rare, including resonant caustic geometries and dense coverage
over/near the peak of high-magnification events, often aided by oblique
trajectories. In two cases, the dense coverage was provided by KMTNet high
$(\Gamma=1\,{\rm hr}^{-1})$, or very high $(\Gamma=4\,{\rm hr}^{-1})$
3-continent monitoring, and in the other two cases by dense 3-continent
follow-up observations.

These characteristics suggest two paths forward for probing the very low-$q$
cold-planet population. First, as discussed by Yee et al. (2021), it should be
possible to construct a rigorous statistical sample by systematic
investigation of all KMTNet high-magnification events, where the threshold for
“high-magnification” would be set at some definite value, like $A_{\rm
max}>20$. Each event with pipeline parameters near or above this boundary
would require tender-loving care (TLC) reductions to determine if it were
truly in the sample, and, if so, it would be subjected to a dense grid search
for planets. Both KMT-2018-BLG-0029Lb and KMT-2019-BLG-0842Lb would definitely
be recovered by such a search, and it is possible that OGLE-2019-BLG-0960Lb
would be recovered as well. These reductions are costly in human effort, but
if restricted to high-magnification events, the project would be feasible.

Second, KMT-2020-BLG-0414Lb shows that follow-up observations of high-
magnification events in low-cadence fields can detect the most extreme
systems. It is challenging, but not impossible, to derive rigorous statistical
conclusions from such a sample. For example, Gould et al. (2010) showed that
their sample of 13 very high-magnification $(A_{\rm max}>200)$ events was
statistically well-grounded because they showed that the $\mu$FUN follow-up
campaign randomly sampled half of the underlying OGLE-III $A_{\rm max}>200$
events during 2004-2008. This achievement required prodigious human effort
because, in the majority of cases, it was not possible to determine which
events would be very high magnification based on (often sparse) OGLE-III data
alone. Hence, it was necessary to “patrol” a much larger set of OGLE events
with 1.3m SMARTS telescope observations to even determine which events should
be densely monitored. And even choosing the events to be patrolled required
several hours of human effort per day. This level of effort, for 1–2 planets
per year was the main reason for abandoning this approach once massive surveys
were underway.

However, with the advent of KMTNet’s 3-continent survey, together with
(beginning in full in 2019) its real-time Alert-Finder system (Kim et al.,
2018), it has become possible to identify rising high-magnification events
with little or no “patrolling”. The LCO network has eight 1m robotic
telescopes at the same sites as KMTNet and a reaction time of about 15
minutes, which allows the rapid necessary coverage for high-magnification
events. $\mu$FUN is mainly composed of amateur volunteers and has different
sites888(see http://www.astronomy.ohio-state.edu/~microfun/microfun.html) from
the current surveys, which can provide targeted coverage for high-
magnification events when the survey sites have poor weather. Indeed, our 2020
follow-up program was meant to be a pilot for such a patrol-free (or patrol-
lite) high-magnification ($A\gtrsim 20$) planet search. This original idea was
undermined by Covid-19, which resulted in the closure of two out of three
KMTNet observatories for the bulk of the 2020 season. Substantial patrolling
was therefore required in this season. For KMT-2020-BLG-0414Lb, the beginning
of follow-up observations at “$A_{\rm now}>10$” was actually the result of
such patrolling. Nevertheless, the detection of KMT-2020-BLG-0414Lb shows that
this approach is viable.

###  6.3 Multiple Low Mass-Ratio Companions

Including KMT-2020-BLG-0414, 15 triple-lens events have been published. Among
these, six lens systems contain two planets orbiting a star and seven consist
of a planet in a binary-star system (for details, see Table 1 of Han et al.
2021). OGLE-2016-BLG-0613L (Han et al., 2017) and KMT-2020-BLG-0414L are
composed of a star, a low-mass brown dwarf and a planet999For
OGLE-2016-BLG-0613L, there are two degenerate solutions (a planet + binary
stars) disfavored by $\Delta\chi^{2}\geq 10$.. Among the eight systems
containing a host and two low mass-ratio ($q<0.1$) companions, five were
detected by a high-magnification event: OGLE-2006-BLG-109 ($u_{0}=0.0035$,
Gaudi et al. 2008; Bennett et al. 2010), OGLE-2012-BLG-0026 ($u_{0}=0.0088$,
Han et al. 2013; Beaulieu et al. 2016; Madsen & Zhu 2019), OGLE-2018-BLG-0532
($u_{0}=0.0079$, Ryu et al. 2020b), KMT-2019-BLG-1953 ($u_{0}=0.0007$, Han et
al. 2020b) and KMT-2020-BLG-0414 ($u_{0}=0.0007$)101010Here we adopt the mean
$u_{0}$ of the underlying 1L1S event for degenerate solutions.. High-
magnification events provide an efficient channel for multiple low mass-ratio
companions because the source trajectory goes very close to the host, where
each planet/BD induces distortions in the magnification profile via their
central or resonant caustics (Gaudi et al., 1998).

There have been only two microlensing studies about the occurrence rate of
systems with multiple low mass-ratio companions. Using the one two-planet
system OGLE-2006-BLG-109L among the 13 $\mu$FUN $A>200$ sample, Gould et al.
(2010) found that the frequency of solar-like systems is 1/6 for all planetary
systems. Using the two multiplanetary systems OGLE-2006-BLG-109L and
OGLE-2014-BLG-1722L, and the detection efficiency of the six-year MOA survey
combined with the four-year $\mu$FUN follow-up observations, Suzuki et al.
(2018) estimated that $6\%\pm 2\%$ of stars host two cold giant planets. The
detection of KMT-2020-BLG-0414L suggests that the new follow-up program for
high-magnification events can also give an estimate of the occurrence rate of
systems with multiple low mass-ratio companions.

In addition, although the new follow-up program aims to observe events located
in KMTNet $\Gamma\leq 1\,{\rm hr}^{-1}$ fields, it is still important to
follow up very high-magnification events located in KMTNet $\Gamma=4\,{\rm
hr}^{-1}$ fields. First, a simulation of Zhu et al. (2014b) shows that even
KMTNet $\Gamma=6\,{\rm hr}^{-1}$ cadence is not intensive enough to capture
the subtle anomalies for very high-magnification events. This has been
demonstrated by the two less secure multiplanetary events OGLE-2018-BLG-0532
and KMT-2019-BLG-1953, for which a cadence of $\Gamma=4\,{\rm hr}^{-1}$ was
only barely adequate to detect the second planet. Second, the survey
observations could suffer from nonlinearity or saturation on the very bright
peak of very high-magnification events. For KMT-2020-BLG-0414, its $I=11.1$
peak is too bright for the current surveys with their normal exposure time
(60s for KMTNet with three 1.6m telescopes, 100-120s for OGLE with a 1.3m
telescope and 60s for MOA with a 1.8m telescope). The LCO network and most of
the $\mu$FUN telescopes have smaller apertures and more flexible exposure
time, so the new follow-up program can provide supplementary coverage for the
very bright peak of very high-magnification events in KMTNet $\Gamma=4\,{\rm
hr}^{-1}$ fields.

W.Z. thank Subo Dong for fruitful discussions. W.Z., S.M. and X.Z. acknowledge
support by the National Science Foundation of China (Grant No. 11821303 and
11761131004). Work by C.H. was supported by the grants of National Research
Foundation of Korea (2019R1A2C2085965 and 2020R1A4A2002885). Work by JCY was
supported by JPL grant 1571564. This research has made use of the KMTNet
system operated by the Korea Astronomy and Space Science Institute (KASI) and
the data were obtained at three host sites of CTIO in Chile, SAAO in South
Africa, and SSO in Australia. This research uses data obtained through the
Telescope Access Program (TAP), which has been funded by the TAP member
institutes. The MOA project is supported by JSPS KAKENHI Grant Number
JSPS24253004, JSPS26247023, JSPS23340064, JSPS15H00781, JP16H06287, and
JP17H02871. Wei Zhu was supported by the Natural Sciences and Engineering
Research Council of Canada (NSERC) under the funding reference #CITA
490888-16.

##  Appendix A The difference in $s_{2}$

For the 3L1S model, in fact, what the light curve is sensitive to is not
directly $s_{2}$, i.e., the angular separation between $M_{2}$ and $M_{1}$,
scaled to $\theta_{\rm E}$. Rather, it is directly sensitive to “$x_{2}$”,
which is the offset between $M_{2}$ and the center of mass of the body(ies)
interior to $s_{2}$, scaled to $\theta_{\rm E,interior}$, i.e., the Einstein
radius associated with the mass interior to $s_{2}$. Hence, we do not expect
the $s_{2}$ from the various solutions (in particular, the wide and close
topologies) to be equal, but rather their associated $x_{2}$.

For example, for the wide 3L1S static solution, $M_{1}$ is the only body
interior to $s_{2}$. Hence, its position is also the center of mass. However,
$\theta_{\rm E,interior}=\theta_{\rm E}/\sqrt{1+q_{3}}$. Hence,

$x_{2,\rm wide}=\sqrt{1+q_{3}}s_{2,\rm wide}=0.99493\pm 0.00011.$ (A1)

For the corresponding close solution, $\theta_{\rm E,interior}=\theta_{\rm
E}$. However, the center of mass of the material interior to $s_{2}$ is offset
from $M_{1}$ by $\Delta x=s_{3}q_{3}/(1+q_{3})$ and thus

$x_{2,\rm close}=\sqrt{s_{2}^{2}-2s_{2}\Delta x\cos{\psi}+\Delta
x^{2}}=0.99522\pm 0.00010.$ (A2)

That is, as expected, $x_{2,\rm close}=x_{2,\rm wide}$ to within $2.8\times
10^{-4}$.

## References

  * Abe et al. (2013) Abe, F., Airey, C., Barnard, E., et al. 2013, MNRAS, 431, 2975
  * Adams et al. (2018) Adams, A. D., Boyajian, T. S., & von Braun, K. 2018, MNRAS, 473, 3608
  * Alard (2000) Alard, C. 2000, A&AS, 144, 363
  * Alard & Lupton (1998) Alard, C., & Lupton, R. H. 1998, ApJ, 503, 325
  * Albrow et al. (2009) Albrow, M. D., Horne, K., Bramich, D. M., et al. 2009, MNRAS, 397, 2099
  * Alcock et al. (2001) Alcock, C., Allsman, R. A., Alves, D. R., et al. 2001, Nature, 414, 617
  * Batista et al. (2015) Batista, V., Beaulieu, J.-P., Bennett, D. P., et al. 2015, ApJ, 808, 170
  * Batista et al. (2011) Batista, V., Gould, A., Dieters, S., et al. 2011, A&A, 529, A102
  * Beaulieu et al. (2016) Beaulieu, J. P., Bennett, D. P., Batista, V., et al. 2016, ApJ, 824, 83
  * Bennett et al. (2006) Bennett, D. P., Anderson, J., Bond, I. A., Udalski, A., & Gould, A. 2006, ApJ, 647, L171
  * Bennett et al. (2010) Bennett, D. P., Rhie, S. H., Nikolaev, S., et al. 2010, ApJ, 713, 837
  * Bennett et al. (2014) Bennett, D. P., Batista, V., Bond, I. A., et al. 2014, ApJ, 785, 155
  * Bensby et al. (2013) Bensby, T., Yee, J. C., Feltzing, S., et al. 2013, A&A, 549, A147
  * Bessell & Brett (1988) Bessell, M. S., & Brett, J. M. 1988, PASP, 100, 1134
  * Bhattacharya et al. (2018) Bhattacharya, A., Beaulieu, J. P., Bennett, D. P., et al. 2018, AJ, 156, 289
  * Bond et al. (2001) Bond, I. A., Abe, F., Dodd, R. J., et al. 2001, MNRAS, 327, 868
  * Bozza (2010) Bozza, V. 2010, MNRAS, 408, 2188
  * Bozza et al. (2018) Bozza, V., Bachelet, E., Bartolić, F., et al. 2018, MNRAS, 479, 5157
  * Chabrier (2003) Chabrier, G. 2003, PASP, 115, 763
  * Chang & Refsdal (1979) Chang, K., & Refsdal, S. 1979, Nature, 282, 561
  * Dominik (1999) Dominik, M. 1999, A&A, 349, 108
  * Dong et al. (2006) Dong, S., DePoy, D. L., Gaudi, B. S., et al. 2006, ApJ, 642, 842
  * Dong et al. (2009) Dong, S., Gould, A., Udalski, A., et al. 2009, ApJ, 695, 970
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306
  * Gaia Collaboration et al. (2016) Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1
  * Gaia Collaboration et al. (2018) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, 616, A1
  * Gaudi (1998) Gaudi, B. S. 1998, ApJ, 506, 533
  * Gaudi et al. (1998) Gaudi, B. S., Naber, R. M., & Sackett, P. D. 1998, ApJ, 502, L33
  * Gaudi et al. (2002) Gaudi, B. S., Albrow, M. D., An, J., et al. 2002, ApJ, 566, 463
  * Gaudi et al. (2008) Gaudi, B. S., Bennett, D. P., Udalski, A., et al. 2008, Science, 319, 927
  * Gould (1992) Gould, A. 1992, ApJ, 392, 442
  * Gould (1994) —. 1994, ApJ, 421, L75
  * Gould (2000) —. 2000, ApJ, 542, 785
  * Gould (2004) —. 2004, ApJ, 606, 319
  * Gould et al. (2010) Gould, A., Dong, S., Gaudi, B. S., et al. 2010, ApJ, 720, 1073
  * Gould et al. (2014) Gould, A., Udalski, A., Shin, I. G., et al. 2014, Science, 345, 46
  * Gould et al. (2020) Gould, A., Ryu, Y.-H., Calchi Novati, S., et al. 2020, Journal of Korean Astronomical Society, 53, 9
  * Griest & Hu (1992) Griest, K., & Hu, W. 1992, ApJ, 397, 362
  * Griest & Safizadeh (1998) Griest, K., & Safizadeh, N. 1998, ApJ, 500, 37
  * Han (2005) Han, C. 2005, ApJ, 629, 1102
  * Han & Gould (1997) Han, C., & Gould, A. 1997, ApJ, 480, 196
  * Han et al. (2013) Han, C., Udalski, A., Choi, J. Y., et al. 2013, ApJ, 762, L28
  * Han et al. (2017) Han, C., Udalski, A., Gould, A., et al. 2017, AJ, 154, 223
  * Han et al. (2019a) Han, C., Bennett, D. P., Udalski, A., et al. 2019a, AJ, 158, 114
  * Han et al. (2019b) Han, C., Yee, J. C., Udalski, A., et al. 2019b, AJ, 158, 102
  * Han et al. (2020a) Han, C., Shin, I.-G., Jung, Y. K., et al. 2020a, A&A, 641, A105
  * Han et al. (2020b) Han, C., Kim, D., Jung, Y. K., et al. 2020b, AJ, 160, 17
  * Han et al. (2021) Han, C., Lee, C.-U., Ryu, Y.-H., et al. 2021, arXiv e-prints, arXiv:2102.01806
  * Hwang et al. (2018) Hwang, K.-H., Udalski, A., Shvartzvald, Y., et al. 2018, AJ, 155, 20
  * Jiang et al. (2004) Jiang, G., DePoy, D. L., Gal-Yam, A., et al. 2004, ApJ, 617, 1307
  * Jung et al. (2019) Jung, Y. K., Gould, A., Zang, W., et al. 2019, AJ, 157, 72
  * Jung et al. (2020) Jung, Y. K., Udalski, A., Zang, W., et al. 2020, AJ, 160, 255
  * Kim et al. (2018) Kim, H.-W., Hwang, K.-H., Shvartzvald, Y., et al. 2018, arXiv e-prints, arXiv:1806.07545
  * Kim et al. (2016) Kim, S.-L., Lee, C.-U., Park, B.-G., et al. 2016, Journal of Korean Astronomical Society, 49, 37
  * Kozłowski et al. (2007) Kozłowski, S., Woźniak, P. R., Mao, S., & Wood, A. 2007, ApJ, 671, 420
  * Madsen & Zhu (2019) Madsen, S., & Zhu, W. 2019, AJ, 878, L29
  * Nataf et al. (2013) Nataf, D. M., Gould, A., Fouqué, P., et al. 2013, ApJ, 769, 88
  * Nemiroff & Wickramasinghe (1994) Nemiroff, R. J., & Wickramasinghe, W. A. D. T. 1994, ApJ, 424, L21
  * Paczyński (1986) Paczyński, B. 1986, ApJ, 304, 1
  * Park et al. (2004) Park, B. G., DePoy, D. L., Gaudi, B. S., et al. 2004, ApJ, 609, 166
  * Poindexter et al. (2005) Poindexter, S., Afonso, C., Bennett, D. P., et al. 2005, ApJ, 633, 914
  * Reid et al. (2014) Reid, M. J., Menten, K. M., Brunthaler, A., et al. 2014, ApJ, 783, 130
  * Rowe et al. (2015) Rowe, B. T. P., Jarvis, M., Mandelbaum, R., et al. 2015, Astronomy and Computing, 10, 121
  * Ryu et al. (2020a) Ryu, Y.-H., Navarro, M. G., Gould, A., et al. 2020a, AJ, 159, 58
  * Ryu et al. (2020b) Ryu, Y.-H., Udalski, A., Yee, J. C., et al. 2020b, AJ, 160, 183
  * Schechter et al. (1993) Schechter, P. L., Mateo, M., & Saha, A. 1993, PASP, 105, 1342
  * Skowron et al. (2011) Skowron, J., Udalski, A., Gould, A., et al. 2011, ApJ, 738, 87
  * Sumi et al. (2016) Sumi, T., Udalski, A., Bennett, D. P., et al. 2016, ApJ, 825, 112
  * Suzuki et al. (2016) Suzuki, D., Bennett, D. P., Sumi, T., et al. 2016, ApJ, 833, 145
  * Suzuki et al. (2018) Suzuki, D., Bennett, D. P., Udalski, A., et al. 2018, AJ, 155, 263
  * Szymański et al. (2011) Szymański, M. K., Udalski, A., Soszyński, I., et al. 2011, Acta Astron., 61, 83
  * Tomaney & Crotts (1996) Tomaney, A. B., & Crotts, A. P. S. 1996, AJ, 112, 2872
  * Udalski et al. (2015) Udalski, A., Szymański, M. K., & Szymański, G. 2015, Acta Astron., 65, 1
  * Udalski et al. (2018) Udalski, A., Ryu, Y.-H., Sajadian, S., et al. 2018, Acta Astron., 68, 1
  * Wang et al. (2018) Wang, T., Calchi Novati, S., Udalski, A., et al. 2018, ApJ, 860, 25
  * Witt & Mao (1994) Witt, H. J., & Mao, S. 1994, ApJ, 430, 505
  * Yee et al. (2009) Yee, J. C., Udalski, A., Sumi, T., et al. 2009, ApJ, 703, 2082
  * Yee et al. (2012) Yee, J. C., Shvartzvald, Y., Gal-Yam, A., et al. 2012, ApJ, 755, 102
  * Yee et al. (2021) Yee, J. C., Zang, W., Udalski, A., et al. 2021, arXiv e-prints, arXiv:2101.04696
  * Yoo et al. (2004) Yoo, J., DePoy, D. L., Gal-Yam, A., et al. 2004, ApJ, 603, 139
  * Zang et al. (2018) Zang, W., Penny, M. T., Zhu, W., et al. 2018, PASP, 130, 104401
  * Zhu et al. (2014a) Zhu, W., Gould, A., Penny, M., Mao, S., & Gendron, R. 2014a, ApJ, 794, 53
  * Zhu et al. (2014b) Zhu, W., Penny, M., Mao, S., Gould, A., & Gendron, R. 2014b, ApJ, 788, 73
  * Zhu et al. (2017) Zhu, W., Udalski, A., Calchi Novati, S., et al. 2017, AJ, 154, 210

Table 1: Data used in the analysis with corresponding data reduction method and rescaling factors Collaboration | Site | Filter | Coverage (${\rm HJD}^{\prime}$) | $N_{\rm data}$ | Reduction Method |  $(k,e_{\rm min})$ for 2L1S |  $(k,e_{\rm min})$ for 3L1S
---|---|---|---|---|---|---|---
KMTNet | SSO | $I$ | 8900.3 – 9139.0 | 139 | pySIS1 | (1.23, 0.010) | (1.11, 0.020)
KMTNet | SSO | $V$ | 8931.2 – 9130.9 | 21 | pyDIA2 |  |
MOA |  | Red | 9024.2 – 9145.9 | 240 | Bond et al. (2001) | (1.74, 0.006) | (1.35, 0.020)
MOA |  | $V$ | 9022.0 – 9112.9 | 22 | Bond et al. (2001) |  |
LCO | SSO | $i$ | 9038.9 – 9047.2 | 53 | ISIS2 | (1.08, 0.002) | (0.35, 0.010)
LCO | SAAO | $i$ | 9038.3 – 9055.5 | 107 | ISIS | (1.31, 0.001) | (1.05, 0.005)
LCO | McDonald | $i$ | 9040.7 – 9046.7 | 62 | ISIS | (1.03, 0.003) | (1.10, 0.005)
CFHT |  | $i$ | 9041.9 – 9051.9 | 23 | ISIS | (3.25, 0.000) | (1.10, 0.020)
$\mu$FUN | OPD06 | $I$ | 9040.4 – 9045.7 | 105 | DoPHOT3 | (1.15, 0.002) | (1.33, 0.000)
$\mu$FUN | OPD16 | $I$ | 9038.5 – 9049.6 | 861 | DoPHOT |  |
$\mu$FUN | PEST | unfiltered | 9042.2 – 9051.0 | 98 | DoPHOT | (0.82, 0.016) | (0.71, 0.020)
$\mu$FUN | Possum | unfiltered | 9041.8 – 9052.0 | 143 | DoPHOT | (0.81, 0.027) | (0.87. 0.020)
Table 2: Parameters for 2L1S Model Parameters | Static | Parallax | Parallax + Orbital Motion
---|---|---|---
|  | $u_{0}>0$ | $u_{0}<0$ | $u_{0}>0$ | $u_{0}<0$
$\chi^{2}/dof$ | $948.6/904$ | $902.8/902$ | $921.0/902$ | $900.4/900$ | $917.4/900$
$t_{0}$ (${\rm HJD}^{\prime}$) | 9041.0590 | 9041.0596 | 9041.0561 | 9041.0594 | 9041.0562
| 0.0009 | 0.0010 | 0.0011 | 0.0011 | 0.0012
$u_{0}(10^{-3})$ | 0.125 | 0.153 |  $-$0.156 | 0.160 |  $-$0.170
| 0.014 | 0.015 | 0.016 | 0.018 | 0.019
$t_{\rm E}$ (days) | 102.9 | 93.5 | 91.6 | 95.9 | 95.7
| 6.3 | 4.9 | 4.9 | 6.2 | 6.1
$s$ | 0.99591 | 0.99542 | 0.99521 | 0.99504 | 0.99454
| 0.00025 | 0.00025 | 0.00028 | 0.00052 | 0.00058
$q$ ($10^{-5}$) | 0.69 | 0.95 | 1.01 | 1.07 | 1.24
| 0.13 | 0.15 | 0.17 | 0.22 | 0.25
$\alpha$ (rad) | 0.0177 | 0.0199 |  $-$0.0205 | 0.0214 |  $-$0.0232
| 0.0011 | 0.0011 | 0.0012 | 0.0022 | 0.0024
$\rho$ ($10^{-4}$) | 4.10 | 5.10 | 5.39 | 5.29 | 5.80
| 0.53 | 0.56 | 0.62 | 0.63 | 0.66
$\pi_{\rm E,N}$ | … | 0.035 | 0.080 | 0.035 | 0.076
| … | 0.016 | 0.022 | 0.016 | 0.022
$\pi_{\rm E,E}$ | … | 0.226 | 0.153 | 0.225 | 0.147
| … | 0.041 | 0.038 | 0.041 | 0.038
$ds/dt({\rm yr}^{-1})$ | … | … | … | 0.196 | 0.345
| … | … | … | 0.251 | 0.279
$d\alpha/dt({\rm yr}^{-1})$ | … | …. | … | 0.017 |  $-$0.021
| … | …. | … | 0.059 |  $-$0.061
$I_{\rm S}$ | 19.209 | 19.104 | 19.075 | 19.132 | 19.126
| 0.070 | 0.059 | 0.060 | 0.072 | 0.072
Table 3: Parameters for 3L1S Static and Parallax Model Parameters | Static | Parallax
---|---|---
| Close | Wide | Close $u_{0}>0$ | Close $u_{0}<0$ | Wide $u_{0}>0$ | Wide $u_{0}<0$
$\chi^{2}/dof$ | $1093.8/942$ | $1094.7/942$ | $944.6/940$ | $956.2/940$ | $964.8/940$ | $962.1/940$
$t_{0}$ (${\rm HJD}^{\prime}$) | 9041.0375 | 9041.0362 | 9041.0360 | 9041.0359 | 9041.0365 | 9041.0357
| 0.0009 | 0.0006 | 0.0012 | 0.0009 | 0.0010 | 0.0010
$u_{0}(10^{-3})$ | 0.719 | 0.719 | 0.688 |  $-$0.684 | 0.747 |  $-$0.706
| 0.015 | 0.007 | 0.019 | 0.006 | 0.013 | 0.008
$t_{\rm E}$ (days) | 91.3 | 90.3 | 94.6 | 95.3 | 88.3 | 93.0
| 1.1 | 1.1 | 2.1 | 0.7 | 1.2 | 0.9
$s_{2}$ | 0.99914 | 0.96882 | 0.99827 | 0.99832 | 0.96849 | 0.97315
| 0.00011 | 0.00018 | 0.00019 | 0.00014 | 0.00053 | 0.00023
$x_{2}$ | 0.99522 | 0.99494 | 0.99511 | 0.99521 | 0.99453 | 0.99495
| 0.00012 | 0.00012 | 0.00019 | 0.00011 | 0.00018 | 0.00014
$q_{2}(10^{-5})$ | 1.20 | 1.26 | 1.13 | 1.12 | 1.41 | 1.23
| 0.05 | 0.04 | 0.08 | 0.02 | 0.06 | 0.03
$\alpha$ (rad) | 0.0257 | 0.0218 | 0.0251 |  $-$0.0253 | 0.0230 |  $-$0.0224
| 0.0005 | 0.0002 | 0.0006 | 0.0002 | 0.0004 | 0.0003
$s_{3}$ | 0.0955 | 10.4633 | 0.1096 | 0.1108 | 10.3204 | 9.6413
| 0.0001 | 0.0002 | 0.0034 | 0.0026 | 0.0457 | 0.0366
$q_{3}$ | 0.0584 | 0.0546 | 0.0404 | 0.0395 | 0.0545 | 0.0453
| 0.0011 | 0.0002 | 0.0029 | 0.0021 | 0.0011 | 0.0005
$\psi$ (rad) |  $-$0.729 |  $-$0.746 |  $-$0.730 | 0.733 |  $-$0.730 | 0.741
| 0.008 | 0.005 | 0.014 | 0.010 | 0.013 | 0.011
$\rho(10^{-4})$ | 5.60 | 5.86 | 5.78 | 5.72 | 6.69 | 6.10
| 0.15 | 0.13 | 0.27 | 0.08 | 0.18 | 0.12
$\pi_{\rm E,N}$ | … | … | 0.057 | 0.321 |  $-$0.011 | 0.343
| … | … | 0.042 | 0.035 | 0.035 | 0.045
$\pi_{\rm E,E}$ | … | … | 0.467 | 0.196 | 0.366 | 0.222
| … | … | 0.073 | 0.059 | 0.056 | 0.067
$I_{\rm S}$ | 19.082 | 19.040 | 19.123 | 19.125 | 19.010 | 19.073
| 0.014 | 0.014 | 0.025 | 0.007 | 0.015 | 0.009
Table 4: Parameters for 3L1S Model with Parallax and Planet Orbital Motion Parameters | Close $u_{0}>0$ | Close $u_{0}<0$ | Wide $u_{0}>0$ | Wide $u_{0}<0$
---|---|---|---|---
$\chi^{2}/dof$ | $939.7/938$ | $938.0/938$ | $944.8/938$ | $939.5/938$
$t_{0}$ (${\rm HJD}^{\prime}$) | 9041.0369 | 9041.0362 | 9041.0357 | 9041.0355
| 0.0010 | 0.0012 | 0.0012 | 0.0011
$u_{0}(10^{-3})$ | 0.652 |  $-$0.688 | 0.693 |  $-$0.720
| 0.020 | 0.011 | 0.017 | 0.012
$t_{\rm E}$ (days) | 99.4 | 94.4 | 93.8 | 90.4
| 2.4 | 1.1 | 2.0 | 1.5
$s_{2}$ | 0.99925 | 0.99896 | 0.97202 | 0.96835
| 0.00026 | 0.00035 | 0.00173 | 0.00060
$x_{2}$ | 0.99537 | 0.99521 | 0.99503 | 0.99496
| 0.00023 | 0.00011 | 0.00026 | 0.00027
$q_{2}(10^{-5})$ | 1.01 | 1.13 | 1.14 | 1.21
| 0.11 | 0.06 | 0.07 | 0.06
$\alpha$ (rad) | 0.0255 | -0.0269 | 0.0222 |  $-$0.0233
| 0.0010 | 0.0005 | 0.0006 | 0.0005
$s_{3}$ | 0.0895 | 0.0940 | 9.9713 | 10.5031
| 0.0040 | 0.0051 | 0.4197 | 0.0607
$q_{3}$ | 0.0616 | 0.0578 | 0.0476 | 0.0557
| 0.0038 | 0.0071 | 0.0038 | 0.0013
$\psi$ (rad) |  $-$0.726 | 0.731 |  $-$0.737 | 0.740
| 0.013 | 0.013 | 0.015 | 0.013
$\rho(10^{-4})$ | 5.33 | 5.78 | 5.92 | 6.20
| 0.30 | 0.16 | 0.23 | 0.17
$\pi_{\rm E,N}$ | 0.389 |  $-$0.345 | 0.480 |  $-$0.552
| 0.154 | 0.120 | 0.152 | 0.203
$\pi_{\rm E,E}$ | 0.173 | 0.669 | 0.220 | 0.809
| 0.135 | 0.105 | 0.121 | 0.140
$ds/dt({\rm yr}^{-1})$ | 0.008 |  $-$0.022 | -0.034 |  $-$0.091
| 0.054 | 0.055 | 0.062 | 0.054
$d\alpha/dt({\rm yr}^{-1})$ | 0.466 |  $-$0.848 | 0.496 |  $-$1.054
| 0.169 | 0.137 | 0.181 | 0.212
$\beta$ | 0.042 | 0.058 | 0.035 | 0.055
| 0.018 | 0.006 | 0.014 | 0.005
$I_{\rm S}$ | 19.173 | 19.121 | 19.084 | 19.042
| 0.026 | 0.013 | 0.021 | 0.017
Table 5: Physical Parameters Solutions | Close $u_{0}>0$ | Close $u_{0}<0$ | Wide $u_{0}>0$ | Wide $u_{0}<0$
---|---|---|---|---
$\theta_{\rm E}$ [mas] | $1.475_{-0.091}^{+0.087}$ | $1.586_{-0.074}^{+0.074}$ | $1.489_{-0.083}^{+0.077}$ | $1.574_{-0.060}^{+0.057}$
$\pi_{\rm E}$ | $0.470_{-0.096}^{+0.106}$ | $0.715_{-0.105}^{+0.096}$ | $0.513_{-0.097}^{+0.114}$ | $0.840_{-0.142}^{+0.158}$
$M_{1}$ [$M_{\odot}$] | $0.364_{-0.057}^{+0.072}$ | $0.259_{-0.034}^{+0.041}$ | $0.340_{-0.053}^{+0.063}$ | $0.217_{-0.037}^{+0.045}$
$M_{2}$ [$M_{\oplus}$] | $1.21_{-0.22}^{+0.28}$ | $0.96_{-0.13}^{+0.15}$ | $1.28_{-0.21}^{+0.25}$ | $0.86_{-0.13}^{+0.19}$
$M_{3}$ [$M_{J}$] | $23.3_{-3.9}^{+4.7}$ | $15.4_{-2.5}^{+3.3}$ | $16.9_{-3.0}^{+3.4}$ | $12.5_{-2.1}^{+2.6}$
$D_{\rm L}$ [kpc] | $1.22^{+0.32}_{-0.23}$ | $0.80^{+0.12}_{-0.10}$ | $1.12^{+0.27}_{-0.20}$ | $0.69^{+0.13}_{-0.11}$
$a_{\perp,2}$ [au] | $1.79_{-0.28}^{+0.36}$ | $1.26^{+0.19}_{-0.14}$ | $1.62_{-0.26}^{+0.30}$ | $1.04^{+0.19}_{-0.16}$
$a_{\perp,3}$ [au] | $0.16^{+0.03}_{-0.03}$ | $0.12^{+0.02}_{-0.02}$ | $16.6^{+3.2}_{-2.7}$ | $11.4^{+2.1}_{-1.8}$
$\mu_{\rm hel,N}$ [${\rm mas\,yr^{-1}}$] | $4.92_{-0.81}^{+0.51}$ | $-2.82_{-0.82}^{+0.94}$ | $5.22_{-0.76}^{+0.49}$ | $-3.58_{-0.94}^{+1.41}$
$\mu_{\rm hel,E}$ [${\rm mas\,yr^{-1}}$] | $6.12_{-1.92}^{+1.76}$ | $12.07_{-1.21}^{+1.38}$ | $6.91_{-1.72}^{+1.58}$ | $13.13_{-1.30}^{+1.67}$

Figure 1: Light curve of KMT-2020-BLG-0414 with lensing models. The circles
with different colors are the observed data points for different data sets.
The black solid line is the best-fit 3L1S model using all the data, the cyan
solid line is the best-fit 2L1S model excluding the MOA data on the peak, and
the black dashed line is the 1L1S model derived using the same
($t_{0},u_{0},t_{\rm E},\rho$) as the best-fit 3L1S model. The middle and
bottom panels show a close-up of the main perturbations from the $q\sim
10^{-5}$ planet and the third body ($q\sim 0.05$), respectively. Figure 2:
Geometries of the 3L1S Close (upper panel) and Wide (lower panel) models. In
each panel, the red dashed line represents the caustic structure, the black
solid line is the trajectory of the source, and the arrow indicates the
direction of the source motion. The 3L1S caustic is nearly the superposition
of a large 6-sided “resonant” caustic associated with the $q\sim 10^{-5}$
planet and a small quadrilateral caustic associated with the third body
($q\sim 0.05$). Figure 3: Cumulative distribution of
$\Delta\chi^{2}=\chi^{2}_{\rm static}-\chi^{2}_{\rm parallax}$ between the
four 3L1S parallax solutions and the two 3L1S static solutions. Overall,
$\Delta\chi^{2}$ grows steadily over time and does so steeply from one day
before peak until two days after peak, when the source touches the planetary
caustic. The upper panel shows the best-fit 3L1S parallax model. Figure 4:
Color-magnitude diagram (CMD) for field stars in a $240^{\prime\prime}$ square
centered on KMT-2020-BLG-0414 using the OGLE-III star catalog (Szymański et
al., 2011). The red asterisk, blue dot, magenta dot and green dot represent
the positions of the centroid of the red giant clump, the microlens source,
the baseline object and the blended light, respectively. The cyan dot shows
the position of a naive lens host with $M_{1}=0.25M_{\odot}$ and $D_{\rm
L}=0.74$ kpc, estimated using the median $\theta_{\rm E}$ and $\pi_{\rm E}$
values of the close $u_{0}<0$ solution.

Figure 5: Microlensing parameters $(\log s,\log q)$ for planetary events
(Left: all of the planets; Right: a close-up of $q<10^{-3}$ planet), adapted
from Figure 11 of Yee et al. (2021), but with the addition of
KMT-2020-BLG-0414Lb (red asterisk, average of the median value of the four
3L1S solutions with parallax + planet orbital motion effects). The power-law
“breaks” proposed by Suzuki et al. (2016) and Jung et al. (2019) are indicated
with the blue lines. Subsequently, there were three planetary events from
2018-2020 that lie well below the cluster of planets near the Jung et al.
(2019) “break” (KMT-2018-BLG-0029Lb, OGLE-2019-BLG-0960Lb, and
KMT-2020-BLG-0414Lb), which are marked with text in the right panel. Together,
these three recent discoveries “fill out” the previously empty region at the
bottom of the triangular distribution. Solutions are considered to be “unique”
(black points) if there are no competing solutions within $\Delta\chi^{2}<10$.
Otherwise, they are shown by pairs of green open circles linked by a line
segment. However, eight such pairs for which $q$ differs by more than a factor
of two are excluded on the grounds that $q$ is not accurately measured. In the
right panel, the two magenta dashed lines represent the boundaries between
resonant and non-resonant caustics using the Equation (59) of Dominik (1999).