#  OGLE-2016-BLG-1093Lb: A Sub-Jupiter-mass Spitzer Planet Located in Galactic
Bulge

In-Gu Shin  Korea Astronomy and Space Science Institute, Daejon 34055,
Republic of Korea  Jennifer C. Yee  Center for Astrophysics $|$ Harvard &
Smithsonian 60 Garden St., Cambridge, MA 02138, USA  Kyu-Ha Hwang  Korea
Astronomy and Space Science Institute, Daejon 34055, Republic of Korea  Andrew
Gould  Max Planck Institute for Astronomy, Königstuhl 17, D-69117 Heidelberg,
Germany  Department of Astronomy, The Ohio State University, 140 W. 18th Ave.,
Columbus, OH 43210, USA  Andrzej Udalski  Astronomical Observatory, University
of Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa, Poland  Ian A. Bond  Institute
of Information and Mathematical Sciences, Massey University, Private Bag
102-904, North Shore Mail Centre, Auckland, New Zealand  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  Cheongho Han  Department of Physics, Chungbuk National University,
Cheongju 28644, Republic of Korea  Youn Kil Jung  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  Hyoun Woo Kim  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  Yossi Shvartzvald  Department of
Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot
76100, Israel  Weicheng Zang  Department of Astronomy and Tsinghua Centre for
Astrophysics, Tsinghua University, Beijing 100084, China  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  Seung-Lee Kim  Korea Astronomy and Space Science
Institute, Daejon 34055, Republic of Korea  Chung-Uk Lee  Korea Astronomy and
Space Science Institute, Daejon 34055, 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, The Ohio State
University, 140 W. 18th Ave., Columbus, OH 43210, USA  Przemek Mróz
Astronomical Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478
Warszawa, Poland  Michał K. Szymański  Astronomical Observatory, University of
Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa, Poland  Jan Skowron  Astronomical
Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa, Poland
Radek Poleski  Astronomical Observatory, University of Warsaw, Al. Ujazdowskie
4, 00-478 Warszawa, Poland  Igor Soszyński  Astronomical Observatory,
University of Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa, Poland  Paweł
Pietrukowicz  Astronomical Observatory, University of Warsaw, Al. Ujazdowskie
4, 00-478 Warszawa, Poland  Szymon Kozłowski  Astronomical Observatory,
University of Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa, Poland  Krzysztof
Ulaczyk  Department of Physics, University of Warwick, Gibbet Hill Road,
Coventry, CV4 7AL, UK  Charles A. Beichman  IPAC, Mail Code 100-22, Caltech,
1200 E. California Blvd., Pasadena, CA 91125, USA  Geoffery Bryden  Jet
Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove
Drive, Pasadena, CA 91109, USA  Sebastiano Calchi Novati  IPAC, Mail Code
100-22, Caltech, 1200 E. California Blvd., Pasadena, CA 91125, USA  Sean Carey
IPAC, Mail Code 100-22, Caltech, 1200 E. California Blvd., Pasadena, CA 91125,
USA  B. Scott Gaudi  Department of Astronomy, Ohio State University, 140 W.
18th Ave., Columbus, OH 43210, USA  Calen B. Henderson  IPAC, Mail Code
100-22, Caltech, 1200 E. California Blvd., Pasadena, CA 91125, USA  Wei Zhu
Department of Astronomy, Tsinghua University, Beijing 100084, China  Fumio Abe
Institute for Space-Earth Environmental Research, Nagoya University, Nagoya
464-8601, Japan  Richard K. Barry  Astrophysics Science Division, NASA/Goddard
Space Flight Center, Greenbelt, MD20771, USA  David P. Bennett  Laboratory for
Exoplanets and Stellar Astrophysics, NASA / Goddard Space Flight Center,
Greenbelt, MD 20771, USA  Department of Astronomy, University of Maryland,
College Park, MD 20742, USA  Aparna Bhattacharya  Laboratory for Exoplanets
and Stellar Astrophysics, NASA / Goddard Space Flight Center, Greenbelt, MD
20771, USA  Department of Astronomy, University of Maryland, College Park, MD
20742, USA  Hirosane Fujii  Institute for Space-Earth Environmental Research,
Nagoya University, Nagoya 464-8601, 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
Depertment of Earth and Space Science, Graduate School of Science, Osaka
University, 1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japan  Laboratory for
Exoplanets and Stellar Astrophysics, NASA / Goddard Space Flight Center,
Greenbelt, MD 20771, USA  Department of Astronomy, University of Maryland,
College Park, MD 20742, USA  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  Iona Kondo  Depertment of Earth and
Space Science, Graduate School of Science, Osaka University, 1-1
Machikaneyama, Toyonaka, Osaka 560-0043, Japan  Yutaka Matsubara  Institute
for Space-Earth Environmental Research, Nagoya University, Nagoya 464-8601,
Japan  Sho Matsumoto  Department of Earth and Space Science, Graduate School
of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan  Shota Miyazaki
Depertment of Earth and Space Science, Graduate School of Science, Osaka
University, 1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japan  Yasushi Muraki
Institute for Space-Earth Environmental Research, Nagoya University, Nagoya
464-8601, Japan  Greg Olmschenk  Code 667, NASA Goddard Space Flight Center,
Greenbelt, MD 20771, USA  Arisa Okamura  Department of Earth and Space
Science, Graduate School of Science, Osaka University, Toyonaka, Osaka
560-0043, Japan  Clément Ranc  Sorbonne Université, CNRS, UMR 7095, Institut
d’Astrophysique de Paris, 98 bis bd Arago, 75014 Paris, France  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
Stela Ishitani Silva  Department of Physics, The Catholic University of
America, Washington, DC 20064, USA  Code 667, NASA Goddard Space Flight
Center, Greenbelt, MD 20771, USA  Takahiro Sumi  Depertment of Earth and Space
Science, Graduate School of Science, Osaka University, 1-1 Machikaneyama,
Toyonaka, Osaka 560-0043, Japan  Daisuke Suzuki  Depertment of Earth and Space
Science, Graduate School of Science, Osaka University, 1-1 Machikaneyama,
Toyonaka, Osaka 560-0043, Japan  Taiga Toda  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  Aikaterini Vandorou
Code 667, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
Department of Astronomy, University of Maryland, College Park, MD 20742, USA
Hibiki Yama  Department of Earth and Space Science, Graduate School of
Science, Osaka University, Toyonaka, Osaka 560-0043, Japan

###### Abstract

OGLE-2016-BLG-1093 is a planetary microlensing event that is part of the
statistical Spitzer microlens parallax sample. The precise measurement of the
microlens parallax effect for this event, combined with the measurement of
finite source effects, leads to a direct measurement of the lens masses and
system distance: $M_{\rm host}=0.38$–$0.57\,M_{\odot}$,
$m_{p}=0.59$–$0.87\,M_{\rm Jup}$, and the system is located at the Galactic
bulge ($D_{L}\sim 8.1$ kpc). Because this was a high-magnification event, we
are also able to empirically show that the “cheap-space parallax” concept
(Gould & Yee, 2012) produces well-constrained (and consistent) results for
$|\pi_{{\rm E}}|$. This demonstrates that this concept can be extended to many
two-body lenses. Finally, we briefly explore systematics in the Spitzer light
curve in this event and show that their potential impact is strongly mitigated
by the color-constraint.

Gravitational microlensing (672) — Microlensing parallax (2144) —
Gravitational microlensing exoplanet detection (2147)

##  1 Introduction

The effect of the Bulge environment on planet formation has yet to be
determined. A few early studies, such as Gonzalez et al. (2001) and Lineweaver
et al. (2004), investigated how various properties that vary throughout the
Galaxy, such as metallicity and supernova rate, might impact planet formation.
These issues were subsequently revisited in Gowanlock et al. (2011). Later,
Thompson (2013) suggested the ambient temperature of the Bulge could inhibit
the formation of ices, and thus, of giant planets. Because of its ability to
find planets in both the Disk and the Bulge of the Galaxy, microlensing is the
best technique for directly measuring the frequency of Bulge planets.

Two statistical studies have attempted to address the relative frequency of
Disk and Bulge planets. Penny et al. (2016) compared the distances (some
measured and some estimated with a Bayesian analysis) of $31$ known
microlensing planets with the expected distribution from a Galactic model for
a range of relative Disk/Bulge planet frequencies. Their limit on the relative
planet frequency suggests fewer or no planets in the Bulge relative to the
Disk. Then, Koshimoto et al. (2021) published an analysis comparing the
observed lens-source relative proper motion, $\mu_{\rm rel}$, and Einstein
timescale, $t_{\rm E}$, distributions with the expectations from a Galactic
model. They find the distribution is consistent with no dependence on
Galactocentric radius, but with large uncertainties. Part of the reason their
result is so imprecise is that $\mu_{\rm rel}$ and $t_{\rm E}$ only provide a
mass-distance relation for each object in the sample.

By contrast to these previous two studies, the Spitzer microlensing parallax
program was undertaken to directly measure distances to planets in order to
infer the relative frequency of planets in the Disk and the Bulge (Calchi
Novati et al., 2015; Yee et al., 2015; Zhu et al., 2017). In this paper, we
present the analysis of OGLE-2016-BLG-1093, the eighth planet in the
statistical sample of Spitzer planets. We begin with an overview of the
observations in Section 2. The analysis of the ground-based light curve is
presented in Section 3. In Section, 4, we fit the Spitzer data for the
satellite microlens parallax effect. We discuss various tests of the Spitzer
parallax and low-level systematics in the Spitzer light curve in Section 5. We
derive the physical properties of the lens in Section 6 and verify membership
in the Spitzer sample in Section 7. Finally, we give our conclusions in
Section 8.

Figure 1:  Light curve with the best-fit model (see Table 2). The left panels
show the full view of the light curve from both the ground and Spitzer. The
upper-right panels show just the ground-based data for the the anomaly part of
the light curve (caustic crossings at $t_{1}$ and $t_{2}$), which is induced
by the planetary system. For comparison, we present the 1L1S model light curve
(dashed red line) with residuals. The lower-right panel presents the caustic
geometry of the event.

##  2 Observations

The microlensing event occurred on a background star (i.e., source) located at
$(\alpha,\delta)=(17^{h}56^{m}01^{s}.03,\,-32^{\circ}42{{}^{\prime}}48{{}^{\prime\prime}}.5)$
in equatorial coordinates, which corresponds to
$(\ell,b)=(-2^{\circ}.108,\,-3^{\circ}.848)$ in Galactic coordinates. This
event was first announced by the Early Warning System (EWS; Udalski et al.,
1994; Udalski, 2003) of the Optical Gravitational Lensing Experiment survey
(OGLE-IV; Udalski et al., 2015) on 2016 June 11. Thus, this event is
designated OGLE-2016-BLG-1093. The event was observed using the $1.3\,m$
Warsaw telescope ($1.4\,{\rm deg}^{2}$ science camera) located at Las Campanas
Observatory in Chile.

Two other microlensing surveys independently discovered this event. The
Microlensing Observations in Astrophysics (MOA; Bond et al., 2001; Sumi et
al., 2003) detected this event on June $24$, $2016$, using the $1.8\,m$
telescope located at the University of Canterbury Mount John Observatory in
New Zealand. The Korea Microlensing Telescope Network (KMTNet; Kim et al.,
2016) also detected this event using a telescope network consisting of three
identical $1.6\,m$ telescopes ($4\,{\rm deg}^{2}$ science cameras) located at
the Cerro Tololo Inter-American Observatory in Chile (KMTC), the South African
Astronomical Observatory in South Africa (KMTS), and the Siding Spring
Observatory in Australia (KMTA). As a result, the light curve of
OGLE-2016-BLG-1093 is well covered by five ground-based observatories (see
Figure 1).

In addition, this event was chosen as a target of the Spitzer Microlensing
Campaign (see Yee et al. 2015 for the details of target selection criteria)
and observed using the Spitzer Space Telescope with the $3.6\,{\mu m}$ channel
of the IRAC camera. The event was initially selected as a “potentially good”
target (i.e., as a “subjective, secret” event) on 2016 June 12 so that
observations would be taken during the first week of the Spitzer campaign,
i.e., starting 2016 June 18. We announced the event as a “subjective,
immediate” event on 2016 June 18 at UT 16:34 (HJD${}^{\prime}=7558.19$), just
after the start of the Spitzer observations. We discuss the implications of
the selection in Section 7.

Table 1: Best-fit parameters of ground-based models

| 1L1S | 2L1S | 1L2S
---|---|---|---
Parameter |  | STD | APRX ($u_{0}<0$) | APRX ($u_{0}>0$) | Parameter |
$\chi^{2}_{\rm ground}/{\rm N_{data}}$ | $11151.00/10692$ | $10688.11/10692$ | $10684.919/10692$ | $10684.597/10692$ | $\chi^{2}_{\rm ground}/{\rm N_{data}}$ | $10879.919/10692$
$t_{0}$ [${\rm HJD^{\prime}}$] | $7560.354\pm 0.006$ | $7560.317\pm 0.007$ | $7560.316\pm 0.007$ | $7560.315\pm 0.007$ |  $t_{0,{S_{1}}}$ $[{\rm HJD^{\prime}}]$ | $7560.378\pm 0.006$
$u_{0}$ | $0.022\pm 0.001$ | $0.021\pm 0.001$ | $-0.021\pm 0.001$ | $0.021\pm 0.001$ |  $t_{0,{S_{2}}}$ $[{\rm HJD^{\prime}}]$ | $7556.494\pm 0.012$
$t_{\rm E}$ [days] | $54.563\pm 1.164$ | $55.395\pm 1.214$ | $56.222\pm 1.185$ | $56.832\pm 1.312$ | $u_{0,{S_{1}}}$ | $0.021\pm 0.001$
$s$ |  $\cdots$ | $1.018\pm 0.001$ | $1.017\pm 0.001$ | $1.017\pm 0.001$ |  $u_{0,{S_{2}}}$ ($\times 10^{-3}$) | $0.039\pm 0.376$
$q$ ($\times 10^{-3}$) |  $\cdots$ | $1.536\pm 0.104$ | $1.463\pm 0.104$ | $1.432\pm 0.109$ |  $t_{\rm E}$ [days] | $56.815\pm 1.279$
$\alpha$ [rad] |  $\cdots$ | $2.528\pm 0.026$ | $-2.537\pm 0.027$ | $2.543\pm 0.027$ | $q_{\rm flux}$ | $0.003\pm 0.001$
$\rho_{\ast}$ ($\times 10^{-3}$) |  $\cdots$ | $2.123\pm 0.305$ | $2.170\pm 0.293$ | $1.883\pm 0.297$ | $\rho_{\ast,{S_{1}}}$ |  $\cdots$
$\pi_{{\rm E},N}$ |  $\cdots$ |  $\cdots$ | $-0.104\pm 0.102$ | $-0.131\pm 0.097$ |  $\rho_{\ast,{S_{1}}}$ ($\times 10^{-3}$) | $0.921\pm 0.257$
$\pi_{{\rm E},E}$ |  $\cdots$ |  $\cdots$ | $0.256\pm 0.147$ | $0.271\pm 0.149$ | $\cdots$ |  $\cdots$

Note. — ${\rm HJD^{\prime}=HJD-2450000.0}$. For the 1L2S model, the angular
radius of the first source ($\rho_{\ast,{S_{1}}}$) is not measured.

##  3 Ground-based Light Curve Analysis

The light curve of OGLE-2016-BLG-1093 is shown in Figure 1. The event
increases in brightness by $\sim 3$ magnitudes suggesting a possible high-
magnification event. There is a small perturbation at $t_{\rm pl}=7556.5$. The
best-fit point-source/point-lens (1L1S) model has $t_{0}=7560.3$, $t_{\rm
E}=55$ days, $u_{0}=0.022$, where $t_{0}$ is the time at the peak of the light
curve, $t_{\rm E}$ is the timescale that the source travels the angular
Einstein ring radius ($\theta_{\rm E}$), and $u_{0}$ is the separation in
units of $\theta_{\rm E}$ at $t_{0}$.

To find the best-fit solution of the binary-lens and single-finite-source
(2L1S) model to describe the observed light curve, we first conduct a grid
search over the parameters $s$ and $q$. Here, $s$ is the projected separation
between binary components, and $q\equiv M_{\rm pl}/M_{\rm h}$ is the mass
ratio of binary components, where $M_{\rm pl}$ is the mass of a planet and
$M_{\rm h}$ is the mass of a host star. The parameters $s$ and $q$ are related
to the caustic morphology. Thus, we set them as grid parameters. For five
other parameters of the static 2L1S model (STD), we minimize $\chi^{2}$ using
the Markov Chain Monte Carlo algorithm (Dunkley et al., 2005). These five
parameters are $t_{0}$, $u_{0}$, $t_{\rm E}$, $\alpha$, and $\rho_{\ast}$. The
set of $[t_{0},u_{0},t_{\rm E}]$ are the same as in the 1L1S case, $\alpha$ is
the angle of the source trajectory with respect to the binary axis, while
$\rho_{\ast}$ is the angular source radius ($\theta_{\ast}$) in units of
$\theta_{\rm E}$.

We set a wide range for the $q$ parameter, $\log(q)\in[-5.5,1.0]$. For the $s$
parameter, the grid range is $\log_{10}(s)\in[-1.0,1.0]$. Once we find local
minima from the grid search, we use a finer grid to search for localized
minima. For competitive solutions found from the grid search, we refine models
by allowing all parameters to vary. The best-fit solution has $s\sim 1$ and
$q\sim 1.5\times 10^{-3}$. The detailed parameters of this model are presented
as the 2L1S STD solution in Table 1.

We find that the light curve of OGLE-2016-BLG-1093 is qualitatively similar to
that of KMT-2019-BLG-0842Lb (Jung et al., 2020) or OGLE-2019-BLG-0960Lb (Yee
et al., 2021), both of which have much smaller mass ratios ($q<10^{-4}$) than
our best-fit solution. Following the formalism of Gaudi & Gould (1997), we
estimate the properties of such a solution. Because the planetary perturbation
occurs at $u_{\rm pl}\equiv\sqrt{u_{0}^{2}+\tau_{\rm pl}^{2}}=0.074$, where
$\tau_{\rm pl}\equiv(t_{\rm pl}-t_{0})/t_{\rm E}=0.069$, this line of
reasoning would imply $s=\left[\sqrt{\left(u_{\rm pl}^{2}+4\right)}+u_{\rm
pl}\right]/2=1.04$ and $\alpha=\tan^{-1}(u_{0}/\tau_{\rm pl})=162^{\circ}$
(2.83 radians). Such a trajectory would be very similar to those of
KMT-2019-BLG-0842Lb and OGLE-2019-BLG-0960Lb. Hence, one might expect that
OGLE-2016-BLG-1093 may also contain a planet with a very small mass ratio.
Thus, we explicitly search for small mass-ratio solutions similar to
KMT-2019-BLG-0842Lb and OGLE-2019-BLG-0960Lb, but find that those solutions
are disfavored by $\Delta\chi^{2}\sim 180$ compared to the
$(s,q)\sim(1,1.5\times 10^{-3})$ solution.

###  3.1 Annual Microlens Parallax (APRX)

Figure 2:  The ($\mbox{$\pi_{{\rm E},E}$},\,\mbox{$\pi_{{\rm E},N}$}$)
contours of the APRX models. Each color indicates $\Delta\chi^{2}_{\rm(chain-
best)}\leq n^{2}$, where $n=1$ (red), $2$ (yellow), $3$ (green), $4$ (light
blue), $5$ (blue), and $6$ (purple), respectively.

The ground-based light curve analysis shows that the timescale of this event
is $\sim 55.4$ days. This is long enough ($t_{\rm E}>15$ days) to check the
effect caused by the annual microlens parallax (APRX; Gould, 1992) on the
light curve. Thus, we try to check the APRX effect by introducing the
microlens-parallax parameters: ($\pi_{{\rm E},N}$, $\pi_{{\rm E},E}$), which
indicate the north and east directions of the microlens-parallax vector
($\mbox{\boldmath$\pi$}_{\rm E}$), respectively.

In Table 1, we present the best-fit parameters of APRX models. From the APRX
modeling, we find $\chi^{2}$ improvements of $\Delta\chi^{2}=3.2$ and
$\Delta\chi^{2}=3.5$ for ($u_{0}<0$) and ($u_{0}>0$) cases, respectively.
These improvements are too minor to claim that the APRX measurements can be
used to determine the lens properties.

However, we find that the APRX contours do not represent a complete non-
detection of the APRX effect. The contours are well converged as shown in
Figure 2, which gives strong upper limits on the parallax.

###  3.2 Lens-orbital Effect

Because the orbital motion of the binary components can affect the APRX
(Batista et al., 2011), we also check the lens-orbital (OBT) only model by
introducing the lens-orbital parameters: $(ds/dt,\,d\alpha/dt)$ where $ds/dt$
is the variation of the binary separation as a function of time and
$d\alpha/dt$ is the variation of the source trajectory angle as a function of
time.

We find a negligible $\chi^{2}$ improvement (i.e., $\Delta\chi^{2}_{\rm(STD-
OBT)}=1.67$) when the lens-orbital effect is considered. Also, we find that
this very small improvement comes from outside the region of the caustic-
crossing. In general, the lens-orbital effect is most sensitive to the
caustic-crossing parts of the light curve (that are mostly covered by KMTC and
KMTS). Thus, we can conclude that there is no significant lens-orbital effect
for this event.

###  3.3 2L1S/1L2S Degeneracy

As Gaudi (1998) pointed out, the single-lens/binary-source (1L2S)
interpretation can mimic planetary anomalies. Also, Shin et al. (2019) shows
that this 2L1S/1L2S degeneracy can appear in wider ranges of cases,
especially, the degeneracy appears in cases having non-optimal observational
coverage.

For this event, there is a gap in the observational coverage of the second
bump. Thus, we check the 2L1S/1L2S degeneracy. For the 1L2S model, we adopt
parameters described in Shin et al. (2019) (A-type, see Appendix of the
reference) shown in Table 1. We find that the $\chi^{2}$ difference between
the 1L2S and 2L1S models is $\sim 192$, which is enough to resolve the
degeneracy. In particular, the 1L2S cannot properly describe the caustic-
crossing feature of the first bump. Thus, we conclude that this event does not
suffer from the 2L1S/1L2S degeneracy.

##  4 Spitzer Parallax

###  4.1 Joint Spitzer$~{}+$ Ground

Figure 3:  The ($\pi_{{\rm E},E}$, $\pi_{{\rm E},N}$) contours of SPRX models.
Top panels show SPRX contours of Joint Spitzer$~{}+$ Ground cases. Bottom
panels show SPRX contours of Spitzer-“only” cases. Right and left panels
present ($u_{0}<0$) and ($u_{0}>0$) cases, respectively. The black stars
indicate the best-fit ($\pi_{{\rm E},E}$, $\pi_{{\rm E},N}$) values of Joint
Spitzer$+$ Ground cases. The labeled black $+$’s indicate the ($\pi_{{\rm
E},N}$, $\pi_{{\rm E},N}$) locations of the four minima shown in Figure 4

To measure the microlens parallax effect using Spitzer data, we jointly fit
the Spitzer and ground-based light curves. This fit is constrained by a color-
color relation constructed from nearby stars:

$(I-L)_{\rm pyDIA}=-5.505\pm 0.025.$ (1)

We note that the color constraint for modeling is $(I-L)_{\rm pySIS}=2.218\pm
0.025$ (allowed $2\sigma$ ranges). The value of this color constraint is
different from the value shown in Equation 1. Because the Spitzer zero point
of the color constraint shown in Equation 1 is $25$ magnitude. However, for
our modeling scheme, we use an $18$ magnitude as the zero point. Thus, we
convert the magnitude system from $25$th to $18$th by adding $7$ magnitudes.
In addition, we use the “pySIS” photometry of KMTC for modeling to use the
best quality of photometry. While, we use the “pyDIA” photometry for the
source color estimation (the “pyDIA” photometry is optimized to obtain
$V$-band light curve and the color-magnitude diagrams). These two datasets
have different zero points. For this event, the relation between them is
$I_{\rm pySIS}-I_{\rm pyDIA}=0.723\pm 0.002$. Thus, the final color constraint
for the modeling is that $(I-L)_{\rm pySIS}=(I-L)_{\rm pyDIA}+7+0.723=2.218\pm
0.025$. We apply the color constraint on the models using an additional
$\chi^{2}$ (i.e., $\chi^{2}_{\rm penalty}$) that is weighted by the different
amount of the source color. The details of $\chi^{2}_{\rm penalty}$ are
described in Shin et al. (2017).

The top panels of Figure 3 show that the resulting constraints on the parallax
are very tight for the both $(u_{0}<0)$ and $(u_{0}>0)$ solutions. These
panels show two well-constrained minima at $\mbox{\boldmath$\pi$}_{\rm
E}(N,E)=(-0.044,0.047)$ and $(0.058,0.025)$ for the $(u_{0}<0)$ and
$(u_{0}>0)$ solutions, respectively. This is in contrast to the “usual”
situation for point lens events, which typically show a four-fold degeneracy
(Refsdal, 1966; Gould, 1994) or an arc (often seen for partial light-curves
like this one; Gould, 2019). We revisit this degeneracy in the next section.
These minima imply parallax values of $\pi_{\rm E}=0.064$ and $0.063$,
respectively. They are also consistent with the broad constraints on the
parallax from the annual parallax effect (see Figure 2). The full fits are
given in Table 2.

Figure 4:  Four solutions from Spitzer-“only” fitting. Left panels show the trajectories of the source as seen from the ground (black line with arrow) and Spitzer (red line with arrows) over the full span of the observations from Spitzer (epochs of observations shown as purple circles). The caustics are shown as a black, closed curve. Center panels show a zoom of the caustic region. Right panels show the Spitzer light curve (purple points) with the best-fit model light curve (red line) for each solution. The “a” and “b” solutions are disfavored relative to the “A” and “B” solutions; lettering corresponds to the minima indicated in Figure 3.  Table 2: Best-fit parameters of SPRX Tests with the ($I-L$) color constraint  | SPRX | Cheap-SPRX
---|---|---
Parameter | ($u_{0}<0$) | ($u_{0}>0$) | ($u_{0}<0$) | ($u_{0}>0$)
$\chi^{2}_{\rm Total}$ | $10735.013$ | $10737.220$ | $10687.513$ | $10686.936$
$\chi^{2}_{\rm ground}/{\rm N_{data}}$ | $10686.57/10692$ | $10689.34/10692$ | $10686.55/10692$ | $10686.07/10692$
$\chi^{2}_{\rm Spitzer}/{\rm N_{data}}$ | $48.28/36$ | $47.70/36$ | $0.84/4$ | $0.84/4$
$\chi^{2}_{\rm penalty}$ | $0.164$ | $0.125$ | $0.121$ | $0.030$
$t_{0}$ [${\rm HJD^{\prime}}$] | $7560.318\pm 0.007$ | $7560.319\pm 0.007$ | $7560.315\pm 0.007$ | $7560.318\pm 0.007$
$u_{0}$ | $-0.021\pm 0.001$ | $0.021\pm 0.001$ | $-0.021\pm 0.001$ | $0.021\pm 0.001$
$t_{\rm E}$ [days] | $56.355\pm 1.168$ | $56.311\pm 1.242$ | $55.401\pm 1.241$ | $56.370\pm 1.241$
$s$ | $1.017\pm 0.001$ | $1.017\pm 0.001$ | $1.018\pm 0.001$ | $1.018\pm 0.001$
$q$ ($\times 10^{-3}$) | $1.473\pm 0.099$ | $1.470\pm 0.102$ | $1.523\pm 0.105$ | $1.469\pm 0.104$
$\alpha$ [rad] | $-2.532\pm 0.026$ | $2.535\pm 0.025$ | $-2.532\pm 0.026$ | $2.531\pm 0.026$
$\rho_{\ast}$ ($\times 10^{-3}$) | $2.142\pm 0.303$ | $2.103\pm 0.309$ | $2.216\pm 0.312$ | $2.185\pm 0.312$
$\pi_{{\rm E},N}$ | $-0.044\pm 0.003$ | $0.058\pm 0.002$ |  $\cdots$ |  $\cdots$
$\pi_{{\rm E},E}$ | $0.047\pm 0.006$ | $0.025\pm 0.006$ |  $\cdots$ |  $\cdots$
$|\pi_{{\rm E}}|$ | $0.064\pm 0.003$ | $0.063\pm 0.003$ | $0.066_{-0.010}^{+0.013}$ | $0.088_{-0.027}^{+0.007}$
$f_{\rm S,KMTC\,{\it(I)}}$ | $0.044\pm 0.001$ | $0.044\pm 0.001$ | $0.045\pm 0.001$ | $0.044\pm 0.001$
$f_{\rm S,Spitzer,{\it(L)}}$ | $0.341\pm 0.011$ | $0.341\pm 0.011$ | $0.341\pm 0.011$ | $0.339\pm 0.011$
$(I-L)_{\rm pySIS}$ | $2.223\pm 0.043$ | $2.223\pm 0.043$ | $2.209\pm 0.043$ | $2.223\pm 0.043$

###  4.2 Spitzer-“only”

As a check, we also conduct the Spitzer-“only” test for the full Spitzer
dataset. We conduct this Spitzer-“only” analysis following the formalism laid
out in Gould et al. (2020) but using a full planet model as in Zang et al.
(2020). Specifically, we fix the seven parameters of the model ($t_{0}$,
$u_{0}$, $t_{E}$, $\rho$, $\alpha$, $s$, $q$) to their ground-based values,
vary $\pi_{E,N}$ and $\pi_{E,E}$ on a grid, and fit only the Spitzer data with
the color-constraint applied. The resulting parallax constraints shown in the
bottom panels of Figure 3 are more extended arcs compared to the constraints
from the full, joint fit. In fact, they show four local minima as might be
expected from the four-fold degeneracy discussed above. Nevertheless, the
four-fold degeneracy is broken because the caustic structure induces
additional structure to the light curve (see Figure 4), and the Spitzer-“only”
fits ultimately recover the same global minimum in each case ($u_{0}>0$ and
$u_{0}<0$). Then, comparing the top and bottom panels in Figure 3 shows that
the joint fitting further suppresses these alternate minima leaving a two-fold
degeneracy rather than a four-fold degeneracy.

##  5 Tests of the Spitzer Parallax

Figure 5:  Spitzer light curve in instrumental flux units. The solid vertical
line indicates the peak of the light curve as observed from Earth
($t_{0}\equiv t_{0,\oplus}$) and the dotted vertical line is $+t_{\rm E}$
away.

Koshimoto & Bennett (2020) have suggested that systematics in the Spitzer
light curve may bias the resulting parallax measurements. Such systematics
have been seen at the level of 1–2 instrumental flux units (Gould et al.,
2020; Hirao et al., 2020; Zang et al., 2020), and so are most likely to play a
significant role in events with small changes in flux as measured by Spitzer.
Figure 5 shows the Spitzer light curve of OGLE-2016-BLG-1093 in instrumental
flux units and illustrates that it is in this regime. However, because
OGLE-2016-BLG-1093 is in the high-magnification regime, this permits several
tests that can be used to verify the parallax signal, which we describe below.

###  5.1 Heuristic Analysis

First, overall, the Spitzer light curve appears to decline during the Spitzer
observation window, suggesting that the event peaked earlier as seen from the
ground. Because Spitzer was separated from Earth by $D_{\perp}=1.021$ au (as
projected on the sky), this suggests $\pi_{\rm E}>\Delta\tau\,({\rm
AU}/D_{\perp})$ where $\tau=(t_{0,\oplus}-t_{\rm start,Spz})/t_{\rm E}=0.043$,
so $\pi_{\rm E}>0.042$, which is consistent with the values derived from the
full-fit.

Second, the characteristics of OGLE-2016-BLG-1093 are similar to the criteria
set out by Gould & Yee (2012) for “cheap-space parallaxes”. Specifically, the
event is in the high-magnification regime ($A_{\rm peak}\sim 50$), the Spitzer
observations start before the ground-based peak ($t_{\rm
start,Spz}=7558.0<t_{0,\oplus}=7560.3$), and, as we show below, was observed
by Spitzer close to baseline. The basic logic is that if Spitzer observes at
the peak as seen from the ground, then $u_{\oplus}\sim 0$ and

$\pi_{\rm E}\sim\frac{{\rm AU}}{D_{\perp}}u_{\rm Spz}.$ (2)

In this case, we observe $\Delta f_{\rm Spz}\sim 4$ flux units (between
$t_{0,\oplus}$ and $t_{\rm last,Spz}$), and we can estimate $f_{\rm
source,Spz}$ from the color-color relation in Equation 1. Given $I_{\rm
source}=20.9$, this implies $f_{\rm source,Spz}=0.274$. The last Spitzer
observation is taken at HJD${}^{\prime}=7589.1$ when the event is magnified by
$A_{\oplus}=2.1$ as seen from Earth. Because we know the event peaked earlier
from Spitzer, we know that $A_{\rm last,Spz}<2.1$. Then,

$\begin{split}\Delta f_{\rm Spz}&=f_{\rm Spz}(t_{0,\oplus})-f_{\rm
Spz}(7589.1)\\\ &=f_{\rm source,Spz}[A_{\rm Spz}(t_{0,\oplus})-A_{\rm
last,Spz}],\end{split}$ (3)

Hence, $15.6<A_{\rm Spz}(t_{0,\oplus})<16.7$. This yields $0.060<u_{\rm
Spz}(t_{0,\oplus})<0.064$ and an estimate of the parallax: $0.059<\pi_{\rm
E}<0.063$ (for $D_{\perp}=1.021$ au), although the parallax could be somewhat
larger for $u_{\oplus}>0$.

An additional caveat is that the “cheap-space parallaxes” formalism was
developed for point lenses, whereas OGLE-2016-BLG-1093 contains a large
resonant caustic structure. This caustic structure has the potential to cause
this formalism to break down. At the same time, we can already see that the
measured values of the parallax are in good agreement with this simplified
heuristic analysis.

Figure 6:  The ($\pi_{{\rm E},E}$, $\pi_{{\rm E},N}$) contours of cheap–SPRX
models with SPRX contours for comparison. Top panels show cheap–SPRX contours
of Spitzer–“only” cases. Middle panels show cheap–SPRX contours of Joint
Spitzer$~{}+$ Ground cases. The bottom panels show the SPRX contours for
comparison. The right and left panels present the ($u_{0}<0$) and
($u_{0}>0$)cases, respectively. The circles indicate $|\pi_{{\rm E}}|$ values
(solid lines) measured from the cheap–SPRX with their $1\sigma$ errors (dashed
lines). The blue and black star marks indicate the best-fit values of SPRX and
cheap–SPRX models, respectively.

###  5.2 Cheap-Space Parallax Limit

To further explore the application of “cheap-space parallaxes” to this event,
we next fit a subset of the Spitzer data, as might be obtained by such an
observational program. Specifically, we restrict the fitting to the Spitzer
observation taken closest to $t_{0,\oplus}$ and the last three Spitzer
observations (technically, only the last observation is needed, but the
observed scatter in the light curve suggests that, in this case, a single
observation may not accurately reflect the mean magnification). Under the
“cheap-space parallaxes” formalism, for a points lens with $u_{0}=0$, we would
expect this test to produce an annulus on the $\mbox{\boldmath$\pi$}_{\rm E}$
plane centered at the origin (e.g., Shin et al., 2018).

Figure 7:  Constraints on the microlens parallax vector without including the
color constraint. The top panels show Spitzer-“only” cases. The bottom panels
show Spitzer$+$ Ground cases. The star mark indicates the best-fit parallax
with including the color constraint. These differ significantly from the
constraints in Figure 3 because the correlated noise in the Spitzer light
curve can be fit by the planetary model if an arbitrary source flux is
allowed.

####  5.2.1 Spitzer-“only” Cheap-Space Parallaxes

For the first test, we conduct Spitzer-“only” fitting with this restricted
dataset. The full contours are shown in the top panels of Figure 6. They show
that, for this event with its large resonant caustic, the contours are still
similar to a ring shape. That ring is displaced from the origin by $\sim
u_{0,\oplus}$, as would be expected for the 1L1S case with $|u_{0,\oplus}|>0$.
The uncertainty in $\pi_{\rm E}$ due to $|u_{0,\oplus}|>0$ is comparable to
the uncertainty due to 2L1S departures from the ring-shape. This indicates
that the “cheap-space parallaxes” formalism still applies in 2L1S cases with
$q\ll 1$ (i.e., the magnification map is still dominated by the effect of the
primary lens).

Finally, from this fit, we derive $0.06<\mbox{$|\pi_{{\rm E}}|$}<0.10$, which
much stronger than the constraint from APRX alone and constrains the lens to
be in the bulge (when combined with $\theta_{\rm E}$, see Section 6).

####  5.2.2 Spitzer $+$ Ground Cheap-Space Parallaxes

We also conduct a joint fit to the ground-based data and the “cheap-space
parallax” subset of the Spitzer data (and including the color-constraint). The
results of the joint fit are shown in the middle panels of Figure 6 and the
parameters are given in Table 2. These contours show the influence of the APRX
constraint from the ground-based data, which weakly prefer parallaxes in the
middle-left panel. Hence, in the “cheap-space parallax” limit, even weak
constraints or upper limits from APRX can be meaningful.

###  5.3 Test of Spitzer Systematics

####  5.3.1 Role of the Color-constraint

Because the overall change in the Spitzer flux is small relative to the
magnitude of the systematics, the color-constraint plays a very important role
in this event. When we fit the Spitzer data without including the color-
constraint, we find a completely different solution for the parallax. Figure 7
shows the contours for the Spitzer-“only” case without the color-constraint
(the joint fit is similar because of the weakness of the APRX). For the best-
fit solution, $f_{\rm S,Spz}\sim 0.9$, which a factor of $\sim 3$ larger than
the value expected from the color-constraint (Section 5.1). Close inspection
of the Spitzer light curve in Figure 5 shows an apparent “dip” in the light
curve between HJD${}^{\prime}\sim 7561$ and HJD${}^{\prime}\sim 7574$ at the
level of $\sim 1$ flux unit compared to a smooth decline. Thus, in the absence
of the color-constraint, this dip can be fit by the “trough” induced by the
$s<1.02$ planet. This confirms that correlated noise can exist in the Spitzer
light curves at the level of $\sim 1$ flux unit.

Figure 8:  The contours of the $\rho_{\ast}$ parameter. The color scheme is
identical to Figure 2.

####  5.3.2 Potential Impact of Systematics

Systematics at the level of $\sim 1$ flux unit have been seen in several
events (e.g., Poleski et al., 2016; Dang et al., 2020; Gould et al., 2020;
Hirao et al., 2020) and are confirmed in this case. The potential impact of
systematics will be most pronounced in events for which the overall flux
change is small (e.g., for $\Delta f_{\rm Spz}\lesssim 10$ flux units). For
example, such an impact was seen in KB180029 (Gould et al., 2020), for which
the baseline flux differed by $\sim 1$ flux unit between seasons.

Because the overall flux change in OGLE-2016-BLG-1093 is only $\Delta f_{\rm
Spz}\sim 4$ flux units, we briefly consider how such systematics might affect
the parallax. We have already shown that the “cheap-space parallaxes”
formalism is a reasonable approximation in this case and that our simple
estimate of the parallax from Section 5.1 is reasonably accurate. If
systematics affect either (or both) the peak or the baseline of the Spitzer
light curve for OGLE-2016-BLG-1093, then the true change in flux might be as
low as 3 flux units or as high as 5 flux units. Taking into account this range
of fluxes (and the uncertainty in $A_{\rm last}$), the parallax would still be
confined to the range $\pi_{\rm E}=[0.05,0.08]$, yielding a factor of $\sim
1.3$ uncertainty in the lens mass and a $\sim 0.25$ kpc uncertainty in the
distance. Hence, the lens is still constrained to be a low-mass dwarf in the
bulge.

##  6 Source Color and Lens Properties

Figure 9:  Color-magnitude diagrams of the KMTNet.

###  6.1 Finite-source Effect

In addition to measuring $\pi_{\rm E}$, one must also determine the angular
Einstein ring radius ($\theta_{\rm E}$), in order to derive the lens
properties such as mass of the lens system ($M_{\rm L}$ and the distance to
the lens system ($D_{\rm L}$):

$M_{\rm L}=\frac{\theta_{\rm E}}{\kappa\pi_{\rm E}},~{}~{}D_{\rm L}=\frac{\rm
au}{\pi_{\rm E}\theta_{\rm E}+\pi_{\rm S}},$ (4)

where $\kappa=8.144\,{\rm mas}\,M_{\odot}^{-1}$, $\pi_{\rm S}\equiv{\rm
au}/D_{\rm S}$ is the parallax of the source, and $D_{\rm S}$ is a distance to
the source.

The $\theta_{\rm E}$ can be determined using the finite-source effect:
$\rho_{\ast}\equiv\theta_{\ast}/\theta_{\rm E}$ where $\theta_{\ast}$ is the
angular source radius, which is an observable that can be routinely measured
(see Section 6.2). For this event, there are caustic-crossing features that
are well-covered by KMTS and KMTC. Thus, it is possible to measure
$\rho_{\ast}$ from the finite-source effect. In Figure 8, we present the
contours of $\rho_{\ast}$. The contours show that the $\rho_{\ast}$ values are
securely measured.

Table 3: Lens properties determined using SPRX and cheap–SPRX  | SPRX | Cheap-SPRX
---|---|---
Properties | ($u_{0}<0$) | ($u_{0}>0$) | ($u_{0}<0$) | ($u_{0}>0$)
$\theta_{\rm E}$ (mas) | $0.24\pm 0.04$ | $0.25\pm 0.04$ | $0.23\pm 0.04$ | $0.24\pm 0.04$
$M_{\rm host}$ ($M_{\odot}$) | $0.46\pm 0.08$ | $0.48\pm 0.09$ | $[0.31,0.60]$ | $[0.25,0.56]$
$M_{\rm planet}$ ($M_{\rm Jup}$) | $0.71\pm 0.12$ | $0.74\pm 0.13$ | $[0.49,0.96]$ | $[0.39,0.86]$
$D_{\rm S}$ (kpc) | $9.40\pm 0.87$ | $9.39\pm 1.21$ | $9.16\pm 1.08$ | $9.20\pm 1.09$
$D_{\rm L}$ (kpc) | $8.12\pm 0.69$ | $8.06\pm 0.91$ | $7.93\pm 0.84$ | $7.79\pm 0.82$
$D_{\rm LS}$ (kpc) | $1.28\pm 0.27$ | $1.33\pm 0.37$ | $1.23\pm 0.36$ | $1.41\pm 0.43$
$a_{\perp}$ (au) | $2.13\pm 0.33$ | $2.23\pm 0.43$ | $2.01\pm 0.34$ | $2.00\pm 0.34$

###  6.2 Angular Source Radius

The angular source radius can be measured using the conventional method (Yoo
et al., 2004). The method requires multi-band observations. The KMTNet survey
regularly observes $V$-band data. In $2016$, KMTC made one $V$-band
observation for every $10$ $I$-band observations, while KMTS made $V$-band
observations at half this rate. We use the KMTC observations to determine the
color. Using the multi-band observations, we construct the KMTNet color-
magnitude diagrams (CMD) shown in Figure 9. Then, we measure the reddened and
de-reddened colors of the source using Yoo et al.’s method and the intrinsic
color and magnitude of the red giant clump adopted from Bensby et al. (2011)
and Nataf et al. (2013):

$(V-I)=2.102\pm 0.013,$ (5) $(V-I)_{0}=0.842\pm 0.052.$ (6)

We estimate the angular source radius using the color conversion of Bessell &
Brett (1988) and the color/surface-brightness relation of Kervella et al.
(2004):

$\theta_{\ast}=0.52\pm 0.03\,\mu{\rm as}$ (7)

Then, by combining the $\rho_{\ast}$ measurements, we determine the angular
Einstein ring radius of each solution shown in Table 3.

###  6.3 Lens properties

In Table 3, we present lens properties determined using measurements of
$|\pi_{{\rm E}}|$ and $\theta_{\rm E}$ (see Equation 4). The binary lens
system is revealed as a planetary system consisting of a sub-Jupiter-mass
planet ($M_{\rm pl}\sim 0.7\,M_{\rm Jup}$) orbiting an M-dwarf host star
($M_{\rm h}\sim 0.5\,M_{\odot}$) with a projected separation of $\sim 2$ au.

Because the source distance is not precisely known, $\pi_{\rm rel}$ (and
therefore $D_{\rm LS}\equiv D_{\rm S}-D_{\rm L}\simeq\left(\pi_{\rm rel}/{\rm
au}\right)D_{\rm clump}^{2}$) is much better constrained than $D_{\rm L}$.
Here, we adopt $D_{\rm clump}=8.54$ kpc from Nataf et al. (2013). We find

$D_{\rm LS}\simeq\frac{\pi_{\rm rel}}{\rm au}D_{\rm clump}^{2}=1.1\pm
0.2\,{\rm kpc}$ (8)

Because, the source is almost certainly in the bulge, this small value of
$D_{\rm LS}$ provides strong evidence that the lens is in the bulge as well.

Moreover, we estimate distances to lens ($D_{\rm L}$) and source ($D_{\rm S}$)
using the Bayesian analysis with constraints of $t_{\rm E}$, $\theta_{\rm E}$,
and $\mbox{\boldmath$\pi$}_{\rm E}$. The Bayesian formalism is adopted from
Shin et al. (2021) with the mass function of Chabrier (2003). The Bayesian
results indicate that the lens is located at the bulge ($D_{\rm L}\sim 8.1$
kpc). This result directly supports the $D_{\rm LS}$ argument, which shows the
planet is located in the bulge. We present the distances of each case in Table
3.

##  7 Membership in the Spitzer sample

According to the protocols of Yee et al. (2015), because the first Spitzer
observation (at HJD${}^{\prime}=7557.96$) was taken before the event was
announced as a Spitzer target, this point must be excluded from the evaluation
of the Spitzer parallax for the purposes of evaluating whether or not the
event is part of the statistical Spitzer planet sample111However, if the event
is found to be in the sample based on the limited dataset, the full dataset
can be used to characterize the parallax.. Therefore, we repeat the joint
ground+Spitzer fitting but without the first Spitzer observation. We find
$\mbox{$|\pi_{{\rm E}}|$}=0.062\pm 0.003$ and $\mbox{$|\pi_{{\rm
E}}|$}=0.063\pm 0.003$ for the $(u_{0}<0)$ and $(u_{0}>0)$ solutions,
respectively.

In terms of calculating membership in the Spitzer sample, we need to calculate
$D_{8.3}$ and its uncertainty (i.e., the lens distance for a source at 8.3
kpc, see Zhu et al. 2017 for details). We find $D_{8.3}=7.3$ kpc and
$\sigma_{D_{8.3}}=0.2$ kpc. According to Zhu et al. (2017), events with
$\sigma_{D_{8.3}}<1.4$ kpc can be included in the statistical Spitzer sample.
Hence, OGLE-2016-BLG-1093 meets this criterion.

Then, we should consider whether or not the planet in OGLE-2016-BLG-1093 can
be included in the sample. Yee et al. (2015) have specified that only planets
(and planet sensitivity) from after the selection can be included in the
statistical analysis. In this case, the event was not selected until
HJD${}^{\prime}=7558.19$, which is after the planet perturbation at $\sim
7556.5$. However, the last datum that was available to the Spitzer team when
it made its decision was at $7555.63$, i.e., before the anomaly. That is,
although at the time of the decision, OGLE had taken one additional
observation (at $7557.57$), this was not posted to the OGLE web page until
$7558.29$, i.e., after the decision. Moreover, MOA did not issue its alert
until $7564.05$, and KMT did not reduce its data until after the season. The
anomaly was first recognized by KMTNet member K.-H. Hwang in May 2019. Hence,
no information about the planet (or possible planet) was available to the team
at the time of the decision and all of the KMT and MOA data can be included in
the statistical analysis.

##  8 Conclusions

Using Spitzer parallax combined with finite source effects, we find that
OGLE-2016-BLG-1093Lb is a sub-Jupiter-mass planet with the mass of
$0.59$–$0.87\,M_{\rm Jup}$. This planet lies beyond the snow line
($a_{\perp}\sim 2$ au) of an M-dwarf host with the mass of
$0.38$–$0.57\,M_{\odot}$ ($a_{\rm snow}=1.0$–$1.5$ au: Kennedy & Kenyon,
2008). This planet is part of the statistical sample of Spitzer microlensing
planets with measured distances. Although the planet perturbation occurred
prior to the selection of the event for Spitzer observations, no anomalous
data points were available to the Spitzer Team until after it made its
selection, and so it meets the criteria of Yee et al. (2015) for inclusion in
the sample. Including, OGLE-2016-BLG-1093Lb, the total number of planets in
the sample is now eight (two-thirds of the expected number for the program).
Moreover, the recent discovery of the planet in OGLE-2019-BLG-1053 (Zang et
al., 2021) suggests additional planets remain undiscovered in the Spitzer
sample.

Because this is a high magnification event ($A_{\rm max}\sim 50$), we are able
to conduct a number of tests of the SPRX measurement. First, we conducted a
test by adopting the cheap–SPRX idea from Gould & Yee (2012), in which we use
only the Spitzer data taken closest to the ground-based peak of the event and
the least magnified points. From this test, we found that the cheap–SPRX
measurement is consistent with the full SPRX measurement at the $1\sigma$
level. This test demonstrates that the cheap space-parallax method can be
applied to 2-body lenses in addition to point lenses, at least in cases for
which the 2-body perturbation is small relative to the overall magnification
effect.

We also perform the Spitzer-“only” test from Gould et al. (2020) and
investigate systematics in the Spitzer light curve. Because the annual
parallax signal is very weak, the Spitzer-“only” test is very similar to the
joint Spitzer\+ ground fitting for the full dataset; i.e., the parallax
measurement is dominated by the SPRX. However, it has some influence on the
“cheap–SPRX” fits because relatively little Spitzer data are used. Finally, we
confirm that there are systematics in the Spitzer light curve of
OGLE-2016-BLG-1093 at the $\sim 1$ flux unit level. In a completely free fit,
these systematics may be fit by features in the planetary magnification
pattern. However, such models are ruled out once the constraint on the source
flux is included.

In addition to OGLE-2016-BLG-1093Lb, there are several other cases of giant
planets with distances measured to be in or very near the bulge. Based on
measurements of the lens flux resolved from the source, Vandorou et al. (2020)
find $m_{p}=2.74\,M_{\rm Jup}$, $D_{L}=6.72$ kpc for MOA-2013-BLG-220Lb, and
Bhattacharya et al. (2021) find $m_{p}=1.71\,M_{\rm Jup}$, $D_{L}=6.89$ kpc
for MOA-2007-BLG-400Lb. Batista et al. (2014) measured a flux excess for
MOA-2011-BLG-293 and inferred $m_{p}=4.8\,M_{\rm Jup}$, $D_{L}=7.72$ kpc
assuming this excess is due to the lens. However, Koshimoto et al. (2020) have
suggested this excess is due to a source companion rather than lens;
nevertheless, they still infer a giant planet in or near the bulge. In
addition, the planet in OGLE-2016-BLG-1093 is similar to OGLE-2017-BLG-1140Lb,
which is $1.6\,M_{\rm Jup}$ planet orbiting a $0.2\,M_{\odot}$ host with
$D_{8.3}=7.4\,$ kpc (see Zhu et al., 2017, for the definition of $D_{8.3}$);
although, in that case, the measured proper motions suggested that the lens
was a member of the disk population. Regardless, the growing number of
detected giant planets at bulge distances would seem to contradict the
expectation from Thompson (2013) that giant planets cannot form in the bulge.
OGLE-2016-BLG-1093 also suggests a definitive answer will be possible once
analysis of the full Spitzer sample is complete.

Acknowledgments. 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. Work by C.H. was supported by grants of the National Research
Foundation of Korea (2017R1A4A1015178 and 2019R1A2C2085965). J.C.Y.
acknowledges support from N.S.F Grant No. AST–2108414 and JPL grant 1571564.
The MOA project is supported by JSPS KAK–ENHI Grant Number JSPS24253004,
JSPS26247023, JSPS23340064, JSPS15H00781, JP16H06287,17H02871 and 19KK0082.

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11institutetext: INFN, Sezione di Trieste, SISSA, Via Bonomea 265, 34136,
Trieste, Italy22institutetext: Department of Physics, University of
Cincinnati, Cincinnati, Ohio 45221, USA33institutetext: INFN Sezione di
Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino, Italy

# Direct CKM determination from W decays at future lepton colliders

David Marzocca 2 , Manuel Szewc 3 , Michele Tammaro<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We project the reach of future lepton colliders for measuring CKM elements
from direct observations of $W$ decays. We focus our attention to $|V_{cs}|$
and $|V_{cb}|$ determinations, using FCC-ee as case study. We employ state-of-
the-art jet flavor taggers to obtain the projected sensitivity, and scan over
tagger performances to show their effect. We conclude that future lepton
collider can sizeably improve the sensitivity on $|V_{cs}|$ and $|V_{cb}|$,
albeit the achievable reach will strongly depend on the level of systematic
uncertainties on tagger parameters.

##  1 Introduction

The Cabibbo-Kobayashi-Maskawa (CKM) matrix describes transitions between up
and down quark mass eigenstates in weak interactions. The theoretical and
experimental establishment of quark mixings marked one of the milestones of
the Standard Model (SM) of particle physics. Five decades of efforts lead to
pinning down the size and phases of CKM elements with a precision ranging from
sub per-mille to few percent ParticleDataGroup:2022pth . The knowledge of the
absolute size of CKM matrix elements allows to test the unitarity of the CKM
matrix, a crucial prediction of the SM. These values also appear as inputs in
rare flavour-changing processes and can be the limiting factor in the
sensitivity to possible physics beyond the SM. Currently, CKM matrix elements
are extracted from measurements of leptonic and semileptonic decays of
hadrons, hadronic $\tau$ decays, or studies of meson mixing. In all these
cases a precise knowledge of non-perturbative hadronic matrix elements is
required, which is often the limiting factor for the achievable precision.

A complementary measurement of CKM matrix elements, free from hadronic matrix
elements uncertainties, can be performed at future lepton colliders, such as
FCC-ee, ILC, and CepC, from the direct on-shell production of $W$ bosons and
subsequent decay in two quark jets Charles:2020dfl ; deBlas:2024bmz . The
decay width of the process $W^{+}\to u_{i}\bar{d}_{j}$, as well as its
conjugate, is directly proportional to the CKM element $|V_{ij}|^{2}$.
Colliding electrons and positrons at center of mass energies close to the $WW$
production threshold, these colliders will provide a large sample of events in
an exquisitely clean environment. Additionally, state-of-the-art jet flavor
taggers, calibrated precisely thanks to the $\approx 10^{12}$ $Z$ bosons
produced in the $Z$-pole run, allow to reconstruct the initial quark flavors
of an event with high efficiency and relatively low mistag probabilities. For
the remainder of this study we focus on the projected performances of jet-
flavor taggers at the FCC-ee future collider, obtained with the IDEA detector
concept Bedeschi:2022rnj ; Gouskos:taggers ; Selvaggi:taggers , and perform
our analysis based on these values. However, we expect similar results to
apply also for CepC CEPCPhysicsStudyGroup:2022uwl ; Liang:2023yyi . At the
ILC, while a similar number of $W$ bosons can be expected, the lack of a
Tera-$Z$ run does not allow to reach the same level of precision in the jet
tagging performance ILDConceptGroup:2020sfq . A measurement of this kind for
$|V_{cs}|$ was performed at LEP2 DELPHI:1998hlc , reaching a precision of
$\sim 16\%$.

In this work we project the sensitivity of FCC-ee to $|V_{cs}|$ and
$|V_{cb}|$, which are the CKM elements for which a substantial improvement,
compared to the current precision, can be expected thanks to the $b$, $c$, and
$s$ taggers. Currently, the measurements of these two matrix elements arise
from the study of leptonic and semileptonic decays of $D$ and $B$ mesons. The
most precise measurement of $|V_{cs}|$ stems from $D\to K\ell\nu$ decays,
where the dominating uncertainty comes from the lattice QCD calculation of the
hadronic form factors. The $|V_{cb}|$ matrix element can be extracted either
from inclusive $B\to X_{c}\ell\nu$ or exclusive $B\to D^{(*)}\ell\nu$
semileptonic decays. These two determinations currently have similar precision
but are in tension with each other. The uncertainty on $|V_{cb}|$ is already
systematically limited and cannot be reduced in the future unless the Belle-II
detector performance is better understood Belle-II:2018jsg . Regarding
$|V_{cs}|$, a precision of approximately $0.2\%$ is expected to be reached at
the STCF Liu:2021qio . The current PDG ParticleDataGroup:2022pth  averages for
$|V_{cs}|$ and $|V_{cb}|$ are reported in the first column of Table 1.

$|V_{ij}|$ | Current |  | FCC-ee | FCC-ee
---|---|---|---|---
(PDG) |  | ($\delta_{\epsilon}=1\%$) | ($\delta_{\epsilon}=0.1\%$)
$|V_{cs}|$ | $0.975\pm 0.006$ | $(0.6\%)$ | $0.36\%$ | $0.05\%$
$|V_{cb}|$ | $(40.8\pm 1.4)\times 10^{-3}$ | $(3.4\%)$ | $0.52\%$ | $0.16\%$
Table 1: The firs column shows the current values of $|V_{cs}|$ and $|V_{cb}|$
from PDG ParticleDataGroup:2022pth . The second and third columns show the
relative precision projected for FCC-ee by assuming 1% and 0.1% relative
systematic uncertainty on the tagger parameters, respectively.

Early studies Charles:2020dfl  estimated, from the projected statistics, a
relative precision of approximately $0.4\%$ on $|V_{cb}|$, using ILD jet
tagging performances as reference. This was later revised to $0.15\%$ using
FCC-ee performances Monteil:VcbWW2024 . Our work follows the same direction,
improving the previous ones by discussing possible sources of background and
considering the effect of systematic uncertainties on tagger performances. Our
results for the expected sensitivity from $WW$ decays, for two different
values of relative systematic uncertainty $\delta_{\epsilon}=1\%$ and $0.1\%$,
are summarized in Table 1 and compared with current values. Such a measurement
of CKM matrix elements would be completely independent from those obtained
from meson decays. In particular, these will not require any lattice QCD
inputs and could provide a final answer to the longstanding tension between
inclusive and exclusive determinations of $|V_{cb}|$.

The manuscript is organized as follows. In Section 2 we describe the desired
signal and the respective branching fractions, while in Section 3 we study
possible backgrounds. Next, in Section 4 we describe the probabilistic model
to extract the FCC-ee sensitivity, including all systematic uncertainties.
Finally, in Section 5 we present the results and discuss the effect of
different tagger parameters on the projected reach.

##  2 Signal

The measurement of CKM elements at future lepton colliders is based on
counting the number of jets with specific flavors produced by a $W$ decay. At
center of mass $\sqrt{s}\approx 162$ GeV, the FCC-ee total planned luminosity
will amount to $N_{WW}\approx 3\times 10^{8}$ of $W^{+}W^{-}$ pairs. Each $W$
is emitted approximately at rest and decays promptly to two quark jets or to a
charged lepton and a neutrino, with branching ratios ${\rm Br}(W\to{\rm
had})\equiv\mathcal{B}_{\rm had}\simeq 67.4\%$ and ${\rm
Br}(W\to\ell\nu)\simeq 32.6\%$ respectively ParticleDataGroup:2022pth .

The decay width of the process $W^{+}\to u_{i}\bar{d}_{j}$ is

$\Gamma(W^{+}\to
u_{i}\bar{d}_{j})=3|V_{ij}|^{2}\Gamma_{0}\equiv\Gamma^{+}_{ij}\,,$ (1)

where $\Gamma_{0}\approx g_{2}^{2}m_{W}/(48\pi)$, $g_{2}$ is the ${\rm
SU}(2)_{L}$ coupling constant, $m_{W}=80.4$ GeV is the $W$ boson mass, and the
factor of 3 comes from the number of quark colors. The indices run over the
kinematically allowed quark flavors: $i=1,2$ for up-quarks and $j=1,2,3$ for
down quarks. The same expression is valid for $W^{-}\to\bar{u}_{i}d_{j}$.

The total hadronic width follows by summing over all indices $ij$, that is

$\Gamma^{\pm}_{\rm
had}=\sum_{ij}\Gamma^{\pm}_{ij}=3\Gamma_{0}\sum_{ij}|V_{ij}|^{2}={\cal B}_{\rm
had}\Gamma_{\rm tot}\,,$ (2)

where $\Gamma_{\rm tot}$ is the total $W$ boson decay width. Using this, we
can rewrite the branching ratio for the flavour set $ij$ as

${\cal B}_{ij}=\frac{\Gamma_{ij}^{\pm}}{\Gamma_{\rm
tot}}=\frac{|V_{ij}|^{2}}{\sum_{lm}|V_{lm}|^{2}}\mathcal{B}_{\rm had}\,.$ (3)

We consider as signal events the processes

$e^{+}e^{-}\to
W^{+}W^{-}\to\left(u_{i}\bar{d}_{j}\right)\left(d_{k}\bar{u}_{z}\right)\,,$
(4)

that is where both $W$ decay hadronically. The expected fraction of events per
channel is then

$F_{ij}={\cal A}_{W}\times{\cal B}_{ij}\,,$ (5)

where ${\cal A}_{W}$ indicates the total detector acceptance for an hadronic
$W$ decay. Finally, the expected number of events for a flavor combination
$ijkz$ is

$N_{ijkz}=N_{WW}\times F_{ij}\times F_{kz}\,.$ (6)

The CKM element $|V_{ij}|$ can be measured by counting the number of decays
into the final state $\left(u_{i}d_{j}\right)$, where we are considering both
charge combinations. The couple of jets can be produced by either one $W$
boson, while the other produces a different pair of jets, or by both $W$
bosons at the same time. The efficiency of this counting depends on the
performance of the jet flavor taggers, which are discussed in Section 4.

QCD corrections can affect the final $4q$ state in several ways. Corrections
to the $W\to q\bar{q}^{\prime}$ decay cancel, up to negligible kinematical
effects, when we write the branching ratio ${\cal B}_{ij}$ in terms of the
measured total hadronic branching ratio as in Eq. (3). On the other hand, QCD
interactions between the two pairs of quarks can induce colour reconnection
effects, affecting the final hadron distribution Gustafson:1988fs ;
Sjostrand:1993hi ; ALEPH:2013dgf . This phenomenon takes place during the
fragmentation process and can be modelled by modern showering algorithms
Christiansen:2015yca , which is a requirement for a precise measurement of the
$W$ boson mass. Since the goal of this work is to provide an estimate of the
sensitvity reach in the CKM matrix element extractions, in the following we
work at the parton level doing a simple counting analysis and we assume that
any uncertainties due to the colour reconnection modelling can be embedded in
the systematic uncertainties of the jet flavour tagging efficiencies.

Finally, we note that the semileptonic channel, $e^{+}e^{-}\to
WW\to\left(\ell\nu\right)\left(ud\right)$, could also be exploited for the CKM
measurement. The main advantage would be the absence of Drell-Yan backgrounds,
which we discuss in the next Section, and of colour reconnection effects. On
the downside, the smaller branching ratio in leptons leads to a factor of
$\sim 1/2$ in the total number of hadronic events, while the reconstruction
efficiency is reduced due to the invisible neutrino. Nevertheless, the
semileptonic channel can be employed to expand the present study, or to
control the tagger efficiencies in the $WW$ runs. We leave these
considerations for future works and proceed here to consider the purely
hadronic decays.

##  3 Background

$W_{1}$ | $(\bar{d}d)$ | $(\bar{s}s)$ | $(\bar{b}b)$ | $(\bar{u}u)$ | $(\bar{c}c)$ | $(\bar{d}g)$ | $(\bar{s}g)$ | $(\bar{b}g)$ | $(\bar{u}g)$ | $(\bar{c}g)$
---|---|---|---|---|---|---|---|---|---|---
$W_{2}$ | $(gg)$ | $(gg)$ | $(gg)$ | $(gg)$ | $(gg)$ | $(dg)$ | $(sg)$ | $(bg)$ | $(ug)$ | $(cg)$
$\sigma$ [fb] | 14 | 14 | 6.9 | 20 | 22 | 15 | 15 | 8.7 | 19 | 22
$N_{\rm ev}$ [$10^{6}$] | 1.7 | 1.7 | 0.83 | 2.4 | 2.7 | 1.8 | 1.8 | 1.1 | 2.3 | 1.6
Table 2: Detailed composition of the flavour content of background Drell-Yan
events of type $2q2g$, after the $m_{W}$ invariant mass cut and paired
accordingly, with the corresponding cross section.

The main background to the $e^{+}e^{-}\to W^{+}W^{-}\to 4j$ process is the
Drell-Yan induced one, $e^{+}e^{-}\to q\bar{q}+2j$, where the two extra jets
can be either both gluons or a quark anti-quark pair. To estimate its
importance we simulate $10^{6}$ Drell-Yan (DY) induced events at LO with
MadGraph5_aMC Alwall:2011uj , restricting for simplicity to parton level. We
fix the center of mass energy to $\sqrt{s}=162$ GeV and impose the following
cuts: $p_{T}^{j}>5$ GeV, $\eta_{j}<2$, $\Delta R_{jj}>0.1$, yielding the total
cross section $\sigma\approx 22.7$pb.

We apply a simple selection cut to the simulated events, by requiring events
in which there exist two pairs of jets with invariant masses $(m_{12},m_{34})$
within 5 GeV of $m_{W}$, consistent with the signal being four jets coming
from two $W$ decays. We find that $\sim 6\times 10^{3}$ background events pass
this cut, with the resulting cross section $\sigma_{4j-m_{W}}\approx 159$ fb.
The latter can be decomposed in two categories: final state with two quarks
and two gluons, with $\sigma_{2q2g}\approx 158$ fb, and final state with four
quarks, with $\sigma_{4q}\approx 0.95$ fb. The vast majority of the
brackground events thus falls in the first category, while only $\sim 1\%$
goes in the second. The detailed flavour decomposition of these background
events is reported in Table 2. To derive a corresponding number of events, the
reported cross section should be multiplied by the integrated luminosity at
the $WW$ threshold scan $\mathcal{L}_{WW}\approx 120{\rm ab}^{-1}$ FCC:2018evy
. Given the dominance of gluons in the background, we expect an increased
rejection rate due to the low mistag probabilities of b-, c-, and s-flavor
taggers versus light jets, see Table 4.

Parameter |  Value |  Rel. Unc. (%)
---|---|---
$N_{WW}$ | $3\times 10^{8}$ | $0.2$
${\rm Br}(W\to{\rm had})$ | $0.6741$ | $0.4-0.01$
${\rm Br}(W\to\ell\nu)$ | $0.3278$ | $0.9-0.03$
${\cal A}_{W}$ | 0.9 | –
Table 3: Nuisance parameters considered in the fit. Relative uncertainties are
shown in the last column FCC:2018byv . For $W$ decay branching ratios, both
present and FCC-ee projected uncertainties are listed.

##  4 Probabilistic model and systematics

We present the probabilistic model for measuring $|V_{cb}|$ with two
orthogonal taggers in order to introduce the notation in a simple way, then
extend the same to the general $|V_{ij}|$ with two or three orthogonal
taggers.

The Parameter of Interest (POI) in our projection is $\mu=|V_{cb}|$. As the
true value we take the PDG ParticleDataGroup:2022pth  central value: $\mu_{\rm
true}=|V_{cb}|_{\rm PDG}$. Following the method detailed in Refs.
ATLAS:2022ers ; Faroughy:2022dyq ; CMS:2020vac , we apply $b$\- and
$c$-taggers111We are assuming here that the two taggers are orthogonal to
avoid double-countings, e.g. see Refs. Faroughy:2022dyq ; Kamenik:2023hvi  to
the final state jets and separate each couple in tag bins
$(n_{b},n_{c})=\\{(0,0),(1,0),(0,1),(2,0),(0,2),(1,1)\\}$. Here $n_{b}$ and
$n_{c}$ indicate the number of jets tagged as $b$ and $c$ respectively; that
is, we are counting not only events where the jets are tagged correctly,
falling into bin $(1,1)$, but also those where one or both jets are mistagged.
With fully hadronic $W_{1}W_{2}$ events, the expected number of events in the
two pairs of bins, $B_{bc;1}\equiv(n_{b;1},n_{c;1})$ and
$B_{bc;2}\equiv(n_{b;2},n_{c;2})$, is then

$N_{B_{bc;1},B_{bc;2}}(\mu,\nu)=N_{WW}\prod_{i=1,2}\sum_{f}p(n_{b;i},n_{c;i}|f,\nu)F_{f}(\mu,\nu)\,,$
(7)

where the product goes over the two bosons, while the sum spans over all the
possible final states for a $W$, $f=\\{ud,us,ub,cd,cs,cb\\}$. Since we assume
that the taggers cannot distinguish quarks from antiquarks,
$N_{B_{bc;1},B_{bc;2}}$ is a $6\times 6$ symmetric matrix. The off-diagonal
elements can be merged, providing 21 different event categories, however for
simplicity we keep them separate in the discussion since this does not affect
the final result. The expected fraction of events for $f$, $F_{f}$ (see eq.
(5)), depends on the POI and on a set of nuisance parameters $\nu$, described
below. The function $p(n_{b;i},n_{c;i}|f,\nu)$ is the probability distribution
of an event with final state $f$ to fall into the $(n_{b;i},n_{c;i})$ bin. We
provide more details on this function in Appendix A. Note again that this
function depends on both the final state and the nuisance parameters.

|  b |  s |  c |  u |  d |  g
---|---|---|---|---|---|---
$\epsilon_{\beta}^{b}$ | 0.8 | 0.0001 | 0.003 | 0.0005 | 0.0005 | 0.007
$\epsilon_{\beta}^{c}$ | 0.02 | 0.008 | 0.8 | 0.01 | 0.01 | 0.01
$\epsilon_{\beta}^{s}$ | 0.01 | 0.9 | 0.1 | 0.3 | 0.3 | 0.2
Table 4: Jet-flavor taggers Working Points.

The nuisance parameters $\nu$ include physics and detector inputs, like
branching ratios and acceptances, as well as tagger performance parameters. We
list the former in Table 3, showing their numerical values and relative
uncertainties FCC:2018byv . The latter are shown in Table 4: for the
$q$-tagger, we denote as $\epsilon_{\beta}^{q}$ the probability to tag a
$\beta$-jet as a $q$-jet, where $\beta=\\{b,s,c,u,d,g\\}$. Currently, the
level of systematic uncertainty on these parameters,
$\delta_{\epsilon}\equiv\delta\epsilon/\epsilon$, is of the order of few
percent ATLAS:2019bwq ; CMS:2017wtu , while future colliders will be able to
reduce it to ${\cal O}(1)\%$. Recent studies on $b$-taggers at FCC-ee
Selvaggi:Rb  show that a systematic of ${\cal O}(0.1)\%$ is achievable, in
particular thanks to the dedicated calibration during the running at the
$Z$-pole. In order to show the effect of the systematic uncertainties on
$\epsilon_{\beta}^{q}$ in our results, we fix three benchmarks for our
numerical analysis: $\delta_{\epsilon}=0,~{}0.1,~{}1$ %. In each of these
cases, we assume for simplicity that $\delta_{\epsilon}$ is the same for all
tagger parameters.

Given the discussion in Section 3, we are also assuming that the DY background
can be effectively reduced to a negligible fraction, thus not entering in the
definition of $N_{B_{bc;1},B_{bc;2}}(\mu,\nu)$.222We checked that including DY
events in the case $\delta_{\epsilon}=0$ has a negligible impact in the
sensitivity to $|V_{cs}|$ and $|V_{cs}|$. An even smaller effect is expected
when systematic uncertainties are included. As a further simplification in our
analysis, we exploit the large statistics expected in each tag bin $B_{bc;i}$
to bypass the need of detailed MonteCarlo signal simulations, and instead work
in the Asimov approximation Cowan:2010js . In the Asimov approximation, one
sets the observed values of $N_{B_{bc;i}}$ to
$N^{A}_{B_{bc;i}}=N_{B_{bc;i}}(\mu=\mu_{\rm true},\nu=\nu_{0})$, where
$\mu_{\rm true}$ is the input value of $\mu$ (the PDG central value in this
work) and $\nu_{0}$ is the nominal value of each nuisance parameter.

With the above ingredients, we can build the likelihood of the observed number
of events, $N^{A}_{B_{bc;1},B_{bc;2}}$, given the expected number of events
$N$, see Eq. (6), as

${\cal L}(\mu,\nu)={\cal
P}(N^{A}_{B_{bc;1},B_{bc;2}}|N_{B_{bc;1},B_{bc;2}}(\mu,\nu))p(\nu)\,,$ (8)

where $\mathcal{P}$ is the Poisson likelihood

${\cal P}(k|x)=\frac{x^{k}e^{-x}}{k!}\,,$ (9)

and $p(\nu)$ is the appropriate distribution for the nuisance parameters,
which we assume to be a normal distribution, centered on the nominal value and
with the given uncertainty as the standard deviation. We follow Ref.
Cowan:2010js  and define the profile likelihood ratio

$\lambda(\mu)=\frac{{\cal L}(\mu,\hat{\hat{\nu}}(\mu))}{{\cal
L}(\hat{\mu},\hat{\nu})}\,,$ (10)

with the associated test statistic

$t_{\mu}=-2\text{ Ln }\lambda(\mu)\,.$ (11)

Here, $\hat{\hat{\nu}}(\mu)$ are the maximum likelihood estimates (MLE) of the
nuisance parameters, obtained by maximizing $\mathcal{L}({\mu},{\nu})$,
varying $\nu$, but keeping $\mu$ fixed. The MLE $\hat{\mu}$ and $\hat{\nu}$
are instead obtained by finding the global maximum of $\mathcal{L}(\mu,\nu)$,
varying both $\nu$ and $\mu$. In the Asimov approximation, the MLE are by
definition $\hat{\mu}=\mu_{\rm true}$ and $\hat{\nu}=\nu_{0}$.

Finally, we can extract the confidence intervals on $\mu$. In particular, the
$68\%$ confidence interval $[\mu_{\rm low},\mu_{\rm up}]$ is obtained by
solving for $t_{\mu}=1$; given the large number of events, we can practically
approximate the latter by a $\chi^{2}$ distribution. We report our results as
$\delta|V_{cb}|/|V_{cb}|$, that is the relative uncertainty on the
determination of the CKM matrix element.

The model described in this Section can be trivially generalized for all CKM
elements, or extended to fit simultaneously more than one variable. In the
case of interest here, we can perform a combined fit of $|V_{cs}|$ and
$|V_{cb}|$ by defining two POI, $\mu_{cs}$ and $\mu_{cb}$, and extend the
probability distribution function to be defined over 3-dimensional bins.
Namely, we separate events in the tag bins $(n_{b},n_{c},n_{s})$ by means of
the function $p(n_{b},n_{c},n_{s}|f,\nu)F_{f}(\mu_{cs},\mu_{cb},\nu)$ and
compute the corresponding $68\%$ CL contours by solving for
$t_{\mu_{cs},\mu_{cb}}=2.28$. We provide details on
$p(n_{b},n_{c},n_{s}|f,\nu)$ in Appendix A.

##  5 Results

Figure 1: 68% CL contours in the $(|V_{cb}|,|V_{cs}|)$ plane. The magenta
contour corresponds to the present precision ParticleDataGroup:2022pth , while
orange, green, and black contours show the expected sensitivity from $W$
decays assuming 1%, 0.1% and 0 systematic uncertainties on the tagging
efficiencies, respectively.

Here we apply our method to study the cases of $|V_{cb}|$ and $|V_{cs}|$. The
main results are summarized in the last two columns of Table 1 and are
obtained by considering one matrix element at a time and only the two more
relevant taggers, whose efficiencies are set to the values shown in Table 4.
The two columns correspond to two benchmarks for systematic uncertainties,
$\delta_{\epsilon}=1\%,~{}0.1\%$. In all cases, FCC-ee is able to improve on
the current levels of precision. Here and in the following, we only consider
systematic uncertainties for the tagger efficiencies. We have found that the
other systematic uncertainties are subleading and do not qualitatively alter
our results.

In Fig. 1 we show the 68% CL contours in the plane of $|V_{cb}|$ and
$|V_{cs}|$, by performing a combined analysis with the three taggers for
different values of $\delta_{\epsilon}$ (black, green and orange lines),
compared to the present constraints obtained from PDG by neglecting possible
correlations (magenta line). The combined analysis needs to assume that the
three taggers are orthogonal, which may be unjustified for current state-of-
the-art taggers. However, ongoing efforts using multi-class algorithms will
produce such a set of orthogonal taggers with high performance, see e.g. Ref.
Bedeschi:2022rnj ; Gouskos:taggers ; Selvaggi:taggers ; Liang:2023yyi .

Figure 2: Sensitivity reach on $|V_{cs}|$ (blue) and $|V_{cb}|$ (orange) as
function of the systematic uncertainty on the $\epsilon$ parameters.

Since Fig. 1 indicates that there is only a mild correlation between the two
CKM elements (with a correlation coefficient of $\approx 0.5$ for
$\delta_{\epsilon}=1\%$), we can analyze $|V_{cb}|$ and $|V_{cs}|$ separately
using two taggers at a time and directly compare the one-dimensional
confidence intervals with those from PDG. In Fig. 2 we show the projected
relative sensitivity on the two CKM matrix elements as a function of the
systematic uncertainty. The role of systematics is immediately clear in both
cases, albeit with different behaviours. For $|V_{cb}|$ (orange line), the
precision degrades by a factor of 3 if we switch from the best scenario to the
1% systematic level case. Nevertheless, even with the latter, a precision on
$|V_{cb}|$ of about $0.5\%$ is achievable, which is already an order of
magnitude smaller than current results, see Table 1. However, at systematics
of ${\cal O}(0.1\%)$, the result saturates to a constant and stops improving
with even lower uncertainties. This is due to the relatively small number of
events in the $cb$ decay channel. That is, statistical uncertainties on
$|V_{cb}|^{2}$ start to dominate the error budget, and further improvements on
the tagger side do not help. Differently, for $|V_{cs}|$ (blue line) the
result does not flatten at the 0.1% level but keeps decreasing. Unlike the
previous case, the very large statistics in the $cs$ channel renders the
measurement systematic-dominated for any realistic value of systematic
uncertainties.

We now move to study the dependence of our results on the specific working
point for the tagging efficiencies, for the three benchmarks of the systematic
uncertainties. In Fig. 3 (left column) we show isocontours of
$\delta|V_{cb}|/|V_{cb}|$ by varying different combinations of tagger
parameters. In both plots, we show with black, red and green lines the results
with the three different uncertainty assumptions, statistics, 1% and 0.1%
systematics respectively, while the red dot indicates the tagger working point
values. In the top plot, we scan the $\\{\epsilon_{b}^{b},\epsilon_{c}^{c}\\}$
plane. The results present only a mild dependence on these parameters: to drop
the precision by a factor of 2 for small systematics, the efficiencies of both
taggers need to be halved. The same corresponds to a ${\cal O}(10)\%$ drop for
the 1% benchmark.

Given the result discussed above and the relatively small statistics of $cb$
final states, the main effect on the precision will come from mistagging a
light jet as a $c$\- or a $b$-jet. Thus, in the bottom plot we turn our
attention to the $\\{\epsilon^{b}_{udsg},\epsilon^{c}_{udsg}\\}$ plane. It is
evident in this scan how a powerful rejection of light jets from the
$b$-tagger, together with low systematics, are necessary to obtain the desired
sensitivity. While projected performances and uncertainties are promising, it
is worth noting that $\epsilon_{udsg}^{b}=0.005$ and
$\delta\epsilon/\epsilon=1\%$ will bring the FCC-ee sensitivity on $|V_{cb}|$
down to the same order as the current levels. On the other hand, the
requirements on the $c$-tagger performance can remain relatively loose, given
the very large number of $c$ quark produced.

Figure 3: Precision contours on when scanning over tagger parameters. The
black, red and green lines indicate the results with statistics, 1% and 0.1%
systematics uncertainties respectively, while the red dot corresponds to the
values of the working point, see Table 4. Left column: precision contours on
$|V_{cb}|$ in the $\\{\epsilon_{b}^{b},\epsilon_{c}^{c}\\}$ (top) and
$\\{\epsilon_{b}^{udsg},\epsilon_{c}^{udsg}\\}$ (bottom) planes. Right column:
precision contours on $|V_{cs}|$ in the
$\\{\epsilon_{c}^{c},\epsilon_{s}^{s}\\}$ (top) and
$\\{\epsilon_{c}^{c},\epsilon_{s}^{udg}\\}$ (bottom) planes.

Regarding $|V_{cs}|$, the large effect from systematic uncertainties on
$\epsilon$, observed in Fig. 2, is evident in Fig. 3 (right column), where we
show isocontours of $\delta|V_{cs}|/|V_{cs}|$ by again varying different
combinations of tagger parameters. In the top plot, we show the percentual
precision as a function of the two relevant true positive rates,
$\\{\epsilon_{c}^{c},\epsilon_{s}^{s}\\}$. For statistics only, the behaviour
is fairly straightforward. Increasing either efficiency increases the
precision, and the isocontours are obtained by decreasing $\epsilon_{s}^{s}$
as $\epsilon_{c}^{c}$ increases. The introduction of systematic uncertainties
for the tagger efficiencies not only degrades the overall performance but also
changes the qualitative features of the scans: the isocontours are almost
independent on $\epsilon_{c}^{c}$. This indicates a relatively large lack of
sensitivity to changes in $\epsilon_{c}^{c}$, with $\epsilon_{s}^{s}$ driving
the precision. A similar behavior is observed in the bottom plot, which shows
the scan on the $\\{\epsilon_{c}^{c},\epsilon_{udg}^{s}\\}$ space.

To understand the flattening of performance curves in the $\epsilon_{c}^{c}$
direction, we need to look at the nuisance parameters of the fit. By adding
systematic uncertainties, we need to take into account the fact that the
nuisance parameters can become constrained by the measurements, at the
expenses of precision on $|V_{cs}|$, and will have a particular correlation
structure, which in turn will affect how each decay channel populates
different bins. On the $c$-tagger side, the mistag parameter
$\epsilon_{udg}^{c}$ is almost negligible, as well as backgrounds with
possible false $c$-tags originating from $b$-jets, given the hierarchical
structure of the CKM matrix
($|V_{ub}|,|V_{cb}|\ll|V_{us}|,|V_{cd}|\ll|V_{ud}|,|V_{cs}|$). On the
$s$-tagger side, the nuisance parameters will impact the fit mainly by
modifying the different sources of $s$-tags. These nuisance parameters become
correlated by the statistical fit and are constrained mainly by the low signal
bins with well-known backgrounds, which we are keeping fixed here. A change in
one of the $\epsilon_{\beta}^{s}$ efficiencies will modify the signal-to-
background ratio for all bins and affect the constraints on the nuisance
parameters, impacting the $|V_{cs}|$ precision, while a change in
$\epsilon_{c}^{c}$ can be almost compensated by a correlated change in all
$\epsilon_{\beta}^{s}$. Additionally, because $\epsilon_{udg}^{s}$ is
particularly large compared to other mistags, its absolute uncertainty becomes
larger.

##  6 Conclusions

In this work we showed the potential of FCC-ee to measure $|V_{cs}|$ and
$|V_{cb}|$ from hadronic $W$ decays, with similar results expected at other
future lepton colliders. Our findings are summarized in Table 1. In both
cases, the relative uncertainty will largely surpass present levels. Assuming
per-mille level systematics on the tagger parameters, one can expect to
increase the precision on CKM elements by one order of magnitude. More
conservative assumptions for the tagger performances will lead to smaller
improvement, especially in the systematic dominated case of $|V_{cs}|$.
Together with the systematic level, final sensitivities strongly depend on the
rejection capability of the taggers, that is on the mistag probabilities.
Projections on tagger working points at FCC-ee indicate that such low mistag
rates can be achieved; particular attention will be needed for $s$-tagger,
where large $s$-to-light jet mistagging can sizeably degrade the results.

The knowledge of CKM matrix elements is often the limiting factor in the
sensitivity to physics beyond the SM from rare decays and meson mixing, which
would therefore benefit from such high precision Charles:2020dfl . This
measurement will also mark the first _lattice-free_ determination of
$|V_{cb}|$, which will be able to resolve the tension between inclusive and
exclusive results from meson decays. Furthermore, such a precise measurement
of CKM elements would allow to use them as inputs in the prediction of hadron
decays. Assuming absence of new physics in leptonic or semi-leptonic charged-
current decays this could facilitate the direct measurement of meson decay
constants and hadronic form factors, providing a benchmark for comparisons
with lattice QCD results.

Additional advancements in tagger performances can open the way to measure
other relevant quantities; e.g., employing a $d$-tagger Selvaggi:taggers  as
an anti-light jet tagger, it would be possible to measure $|V_{cd}|$ with the
same procedure described in this paper, achieving a precision comparable or
superior to present results. We leave these considerations for future work.

## Acknowledgments

The authors would like to thank Jernej Kamenik, Stephane Monteil, Marie-Hélène
Schune, Michele Selvaggi and Jure Zupan for useful discussions. MS
acknowledges support in part by the DOE grant de-sc0011784 and NSF
OAC-2103889. DM acknowledges support by the MUR grant PRIN 20224JR28W.

##  Appendix A Details on the probability function

Here we provide further details on the probability function introduced in
Section 4.

The probability distribution function $p(n_{b},n_{c}|f,\nu)$ is the
probability for a particular decay, with true parton flavor configuration
$f=u_{i},d_{j}$, to end up in the bin tagged as $(n_{b},n_{c})$, with
$n_{b}+n_{c}\leq 2$. These probabilities are built in terms of the jet tagging
efficiencies: $\epsilon^{q}_{\beta}$ is the probability that a jet of true
flavour $\beta$ is tagged as a $q$-jet, where $q$ can be $b$ or $c$. The
probability of a $\beta$-jet to not be tagged as neither of the two is
therefore $(1-\epsilon^{c}_{\beta}-\epsilon^{b}_{\beta})$.

From these, we define the probabilities to have $W$ events in each
$(n_{b},n_{c})$ bin, summing over the hadronic final states $f$ as in Eq. (7):

$P_{W}(n_{b},n_{c})\equiv\sum_{f}p(n_{b},n_{c}|f,\nu)F_{f}(\mu,\nu).$ (12)

Explicitly, the 6 probabilities are given by

$\begin{split}P_{W}(1,1)&=\sum_{u_{i}=u,c}\sum_{d_{j}=d,s,b}(\epsilon^{c}_{u_{i}}\epsilon^{b}_{d_{j}}+\epsilon^{b}_{u_{i}}\epsilon^{c}_{d_{j}}){\cal
B}_{ij}{\cal A}_{W}~{},\\\
P_{W}(0,2)&=\sum_{u_{i}=u,c}\sum_{d_{j}=d,s,b}\epsilon^{c}_{u_{i}}\epsilon^{c}_{d_{j}}{\cal
B}_{ij}{\cal A}_{W}~{},\\\
P_{W}(2,0)&=\sum_{u_{i}=u,c}\sum_{d_{j}=d,s,b}\epsilon^{b}_{u_{i}}\epsilon^{b}_{d_{j}}{\cal
B}_{ij}{\cal A}_{W}~{},\\\
P_{W}(0,1)&=\sum_{u_{i}=u,c}\sum_{d_{j}=d,s,b}(\epsilon^{c}_{u_{i}}(1-\epsilon^{c}_{d_{j}}-\epsilon^{b}_{d_{j}})+\epsilon^{c}_{d_{j}}(1-\epsilon^{c}_{u_{i}}-\epsilon^{b}_{u_{i}})){\cal
B}_{ij}{\cal A}_{W}~{},\\\
P_{W}(1,0)&=\sum_{u_{i}=u,c}\sum_{d_{j}=d,s,b}(\epsilon^{b}_{u_{i}}(1-\epsilon^{c}_{d_{j}}-\epsilon^{b}_{d_{j}})+\epsilon^{b}_{d_{j}}(1-\epsilon^{c}_{u_{i}}-\epsilon^{b}_{u_{i}})){\cal
B}_{ij}{\cal A}_{W}~{},\\\
P_{W}(0,0)&=\sum_{u_{i}=u,c}\sum_{d_{j}=d,s,b}(1-\epsilon^{c}_{u_{i}}-\epsilon^{b}_{u_{i}})(1-\epsilon^{c}_{d_{j}}-\epsilon^{b}_{d_{j}}){\cal
B}_{ij}{\cal A}_{W}~{},\end{split}$ (13)

where ${\cal B}_{ij}$ are the $W$ branching ratios in each channel defined in
Eq. (3) and ${\cal A}_{W}$ is the acceptance. In can be explicitly checked
that the sum of these 6 probabilities gives precisely ${\cal B}_{\rm had}{\cal
A}_{W}$, as expected. Finally, the expected number of events
$N_{B_{bc;1},B_{bc;2}}$ is obtained as

$N_{B_{bc;1},B_{bc;2}}=N_{WW}\,P_{W}(B_{bc;1})\,P_{W}(B_{bc;2})~{}.$ (14)

Also this model can be easily extended to the three jet tagging categories by
modifying $P_{W}(n_{b;i},n_{c;i})\to P_{W}(n_{b;i},n_{c;i},n_{s;i})$, implying
that there would be 10 categories for each $W$ decay, rather than 6. Also, the
probability of a jet not being tagged should be modified to
$(1-\epsilon^{c}_{\beta}-\epsilon^{b}_{\beta}-\epsilon^{s}_{\beta})$.

Given the large number of expected events in all bins, consequence of
$N_{WW}\approx 10^{8}$, the central limit can be applied to obtain accurate
results. In the Gaussian approximation the likelihood of Eq. (8) can be
written as

$-2\ln{\cal
L}=\sum_{B_{bc;1},B_{bc;2}}\frac{\left(N_{B_{bc;1},B_{bc;2}}-N^{A}_{B_{bc;1},B_{bc;2}}\right)^{2}}{N_{B_{bc;1},B_{bc;2}}}+\chi^{2}_{\rm
tag}~{},$ (15)

where as nuisance parameters we consider only the tagging efficiencies with

$\chi^{2}_{\rm
tag}=\sum_{q=b,c}\sum_{\beta=b,s,c,u,d,g}\left(\frac{\epsilon^{q}_{\beta}-\hat{\epsilon}^{q}_{\beta}}{\delta_{\epsilon}\hat{\epsilon}^{q}_{\beta}}\right)^{2}~{}.$
(16)

Here $\hat{\epsilon}^{q}_{\beta}$ are the nominal values of the efficiencies
as in Table 4 and $\delta_{\epsilon}$ is the relative uncertainty, that we
assume to be the same for all taggers.

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