# Searching for $|V_{cd}|$ through the exclusive decay $D_{s}^{+}\to
K^{0}e^{+}\nu_{e}$ within QCD Sum Rules

Hai-Jiang Tian1 Yin-Long Yang1 Dan-Dan Hu2 Hai-Bing Fu1<EMAIL_ADDRESS>Tao
Zhong1<EMAIL_ADDRESS>Xing-Gang Wu2<EMAIL_ADDRESS>1Department of
Physics, Guizhou Minzu University, Guiyang 550025, P.R.China  2Department of
Physics, Chongqing Key Laboratory for Strongly Coupled Physics, Chongqing
University, Chongqing 401331, P.R.China

###### Abstract

In this paper, we carry out an investigation into the semileptonic decays
$D_{s}^{+}\to K^{0}\ell^{+}\nu_{\ell}$ with $\ell=(e,\mu)$ by employing the
QCD light-cone sum rules approach. The vector transition form factor (TFF)
$f_{+}^{D_{s}^{+}K^{0}}(q^{2})$ for $D_{s}^{+}\to K^{0}$ decay is calculated
while considering its next-to-leading order contribution. Subsequently, we
briefly introduce the twist-2, 3 kaon distribution amplitudes, which are
calculated by using QCD sum rules within the framework of the background field
theory. At the large recoil point, the TFF has
$f_{+}^{D_{s}^{+}K^{0}}(0)=0.692_{-0.026}^{+0.027}$. Then, we extrapolate
$f_{+}^{D_{s}^{+}K^{0}}(q^{2})$ to the whole physical $q^{2}$-region via the
simplified $z(q^{2},t)$-series expansion, and the behavior of TFF
$f_{+}^{D_{s}^{+}K^{0}}(q^{2})$ is exhibited in the numerical results part,
including the theoretical and experimental predictions for comparison. In
addition, we compute the differential branching fraction
$\mathcal{B}(D_{s}^{+}\to K^{0}\ell^{+}\nu_{\ell})$ with the electron and muon
channels, which are expected to be $\mathcal{B}(D_{s}^{+}\to
K^{0}e^{+}\nu_{e})=3.379_{-0.275}^{+0.301}\times 10^{-3}$ and
$\mathcal{B}(D_{s}^{+}\to K^{0}\mu^{+}\nu_{\mu})=3.351_{-0.273}^{+0.299}\times
10^{-3}$ as well as contained other results for comparison. Our results show
good agreement with the BESIII measurements and theoretical predictions.
Furthermore, we present our prediction with respect to the CKM matrix element
$|V_{cd}|$ by using the $\mathcal{B}(D_{s}^{+}\to K^{0}e^{+}\nu_{e})$ result
from BESIII Collaboration, yielding $|V_{cd}|=0.221_{-0.010}^{+0.008}$.
Finally, we provide the ratio between $D_{s}^{+}\to K^{0}e^{+}\nu_{e}$ and
$D_{s}^{+}\to\eta e^{+}\nu_{e}$ channels, i.e.
$\mathcal{R}_{K^{0}/\eta}^{e}=0.144_{-0.020}^{+0.028}$.

###### pacs:

13.25.Hw, 11.55.Hx, 12.38.Aw, 14.40.Be

Introduction.–Heavy-to-light meson exclusive decay is one of the significant
processes in the Standard Model (SM), which plays a crucial role in extracting
the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements and in-depth understanding
the non-perturbative properties of quantum chromodynamics (QCD). Accurate
measurement of the CKM matrix elements has been recognized as a central topic,
both theoretically and experimentally. In particular, the Cabibbo-suppressed
(semi)-leptonic charm decays are a valuable application for determining the
CKM matrix elements $|V_{cd}|$ and $|V_{cs}|$, which have been extensively
measured in the BaBar BaBar:2007zgf ; BaBar:2014xzf , Belle Belle:2006idb ,
CLEO CLEO:2008ffk ; CLEO:2009svp , and BESIII BESIII:2018xre ; BESIII:2013iro
; BESIII:2015tql ; BESIII:2017ylw ; BESIII:2018eom ; BESIII:2019vhn ;
Ablikim:2020hsc  Collaborations. The upgraded BESIII has collected an
abundance of data samples at the center-of-mass energy of $\sqrt{s}=3.773$,
4.009, 4.13-4.23 and 4.6-4.7 GeV from 2010 to 2023 years, which reported a
battery of results with respect to $|V_{cd}|$ and $|V_{cs}|$ by using 2.93
fb-1 of data sample adopted at 3.773 GeV. In extracting the CKM matrix
elements, which involve the non-perturbative strong effects appearing in the
transition from the initial state to the final state, those effects are
parameterized by the hadronic invariant form factors. Accordingly, determining
the hadronic form factor with greater precision would significantly improve
accuracy. In 2018, the BESIII Collaboration reported the first measurements of
the form factor for $D_{s}^{+}\to K^{0}e^{+}\nu_{e}$ and the improved
measurements with regard to the absolute branching fraction, which have been
made with 3.19 fb-1 $e^{+}e^{-}$ annihilation data recorded BESIII:2018xre .
In addition, they extracted the CKM $|V_{cd}|=0.217\pm 0.026\pm 0.004$ based
on their acquired data sample. The data sample is the biggest that has ever
been collected for an experiment in a clean, near-threshold setting for
$D_{s}^{+}$ investigations.

In the theoretical aspect, for semileptonic decays that involve non-
perturbative form factors, one should employ a non-perturbative method to
implement the calculations. There are some theoretical predictions concerning
semileptonic decays $D_{s}^{+}\to K^{0}\ell^{+}\nu_{\ell}$ derived from
various techniques, such as the constituent quark model (CQM) Melikhov:2000yu
, QCD light-cone sum rules (LCSRs) Wu:2006rd , light-front quark model (LFQM)
Wang:2008ci ; Verma:2011yw , lattice QCD (LQCD) Koponen:2012di , covariant
confined quark model (CCQM) Soni:2018adu , schwinger function methods (SFMs)
Yao:2021pdy , and 4-flavor holographic QCD (4FhQCD) Ahmed:2023pod . It is
worth noting that, while the application of each individual method is limited,
combining them offers a more comprehensive understanding of the highlighted
physics. Melikhov:2000yu .

As a well-established theory, the LCSRs method incorporates both the hard and
soft contributions in the computation of hadron transitions. This feature
enables the LCSRs method to furnish some insights when tackling hadron physics
problems spanning various energy scales. Therefore, in this manuscript, we
prepare to compute the $D_{s}^{+}\to K^{0}$ transition form factor (TFF)
$f_{+}^{D_{s}^{+}K^{0}}(q^{2})$ based on the QCD LCSRs technique.
Additionally, the meson’s distribution amplitudes (DAs) are regarded as
universal non-perturbative inputs that play a crucial role in the
calculations. Therefore, a detailed investigation of them can enhance the
theoretical predictions of the QCD LCSRS approach. In our previous works
Zhong:2022ecl ; Zhong:2011rg , we carefully computed the kaon twist-2 and
twist-3 DAs by integrating the phenomenological light-cone harmonic oscillator
model with QCD sum rules, all within the framework of background field theory.

Theoretical Framework.–The $D_{s}^{+}\to K^{0}\ell^{+}\nu_{\ell}$ with
$\ell=(e,\mu)$ differential decay width in relation to the mass squared
$q^{2}$ of the $\ell^{+}\nu_{\ell}$ system can be written as:

$\displaystyle\frac{d\Gamma(D_{s}^{+}\to
K^{0}\ell^{+}\nu_{\ell})}{dq^{2}}=\frac{G_{F}^{2}|V_{cd}|^{2}}{24\pi^{3}}|\vec{p}_{K}|^{3}|f_{+}^{D_{s}^{+}K^{0}}(q^{2})|^{2},$
(1)

with $G_{F}=1.166\times 10^{-5}$ GeV-2, $\vec{p}_{K}$ stands for kaon three
momentum in the rest frame of $D_{s}$-meson, and
$f_{+}^{D_{s}^{+}K^{0}}(q^{2})$ refers to the $D_{s}^{+}\to K^{0}$ vector TFF,
respectively. To calculate the $D_{s}^{+}\to K^{0}$ TFF’s analytical
expression within the QCD LCSRs technique, one can start with the following
correlation function (also called correlator):

$\displaystyle\Pi_{\mu}(p,q)$ $\displaystyle=i\int d^{4}e^{iq\cdot x}\langle
K(p)|T\\{j_{\mu}(x),j_{5}^{\dagger}(0)\\}|0\rangle,$ (2)

where $j_{\mu}(x)=\bar{s}\gamma_{\mu}c(x)$ and
$j_{5}^{\dagger}(0)=m_{c}\bar{s}(0)i\gamma_{5}c(0)$. The $q$ and $p$ stand for
the momentum transfer and kaon momentum. The correlator can be separated into
vector and scalar form factors based on the transition matrix element $\langle
K^{0}(p)|\bar{s}\gamma_{\mu}c|D_{s}^{+}(p+q)\rangle=2f_{+}^{D_{s}^{+}K^{0}}(q^{2})p_{\mu}+\tilde{f}^{D_{s}^{+}K^{0}}(q^{2})q_{\mu}$.
Meanwhile, the traditional current correlator can well highlight the
contributions from the kaon twist-2, 3 DAs, thus better ensuring precision
from QCD LCSRs. Since the semileptonic decay amplitudes for $D_{s}^{+}\to
K^{0}\ell^{+}\nu_{\ell}$ with $\ell=(e,\mu)$ are solely dependent on the TFF
$f_{+}^{D_{s}^{+}K^{0}}(q^{2})$, we exclusively present the detailed
calculation methodology pertaining to this vector TFF. Subsequently, we insert
the complete intermediate state, which has the same quantum numbers as the
current operator $(\bar{c}i\gamma_{5}s)$, into the correlation function in the
time-like $q^{2}$-region and by isolating the pole term of the
$D_{s}^{+}$-meson ground-state contributions for the invariant amplitude
$F_{0(1)}=[q^{2},(p+q)^{2}]$, we can then obtain:

$\displaystyle
f_{+}^{D_{s}^{+}K^{0}}(q^{2})=\frac{e^{m_{D_{s}^{+}}^{2}/M^{2}}}{2m_{D_{s}^{+}}^{2}f_{D_{s}^{+}}}$
$\displaystyle\qquad\times\bigg{\\{}F_{0}[q^{2},M^{2},s_{0}]+\frac{\alpha_{s}G_{F}}{4\pi}F_{1}[q^{2},M^{2},s_{0}]\bigg{\\}}.$
(3)

In which the invariant amplitudes $F_{0(1)}[q^{2},M^{2},s_{0}]$ stand for the
leading-order (LO) and next-leading-order (NLO) contribution, respectively.
Those functions have similar expressions as those of the LO TFF of Hu
Hu:2021zmy  and the NLO TFF of Duplancic Duplancic:2008ix . As a cross check,
we have confirmed that we get the same results via proper transformations. So
to short the length of the manuscript, we do not provide the complex
expression of $F_{0(1)}[q^{2},M^{2},s_{0}]$ here.

As a further step, the kaon twist-2,3 DAs are the key parametric inputs in
computation. Since they have already been studied in our past works
Zhong:2022ecl ; Zhong:2011rg , we shall just briefly describe them here. The
basic idea originates from the BHL prescription Huang:1994dy , which can link
the equal-time wavefunction (WF) in the rest frame and the light-cone WF and
then use the acquired WF to construct the twist-2, 3 DAs. The relationship
between the kaon twist-2, 3 DAs and their WFs can be written as
$\phi_{i;K^{0}}(x,\mu)={\cal
N}_{i}\int_{|\mathbf{k}_{\bot}|\leqslant\mu^{2}}\frac{d^{2}\mathbf{k}_{\bot}}{16\pi^{3}}\Psi_{i;K^{0}}(x,\mathbf{k}_{\bot})$,
where the index $i=(2,3)$ stand for twist-2, 3 DAs respectively. The
normalization coefficients are ${\cal N}_{2}=2\sqrt{6}/f_{K}$ and ${\cal
N}_{3}=1$. Then, the twist-2, 3 WFs are expressed as,

$\displaystyle\Psi_{2;K^{0}}(x,\mathbf{k}_{\bot})=\frac{\tilde{m}}{\sqrt{\mathbf{k}_{\bot}^{2}+\tilde{m}^{2}}}A_{2;K^{0}}\varphi_{2;K^{0}}(x)$
$\displaystyle\quad\times\exp{\left[-\frac{1}{8\beta_{2:K^{0}}^{2}}\left(\frac{\mathbf{k}_{\bot}^{2}+\hat{m}_{s}^{2}}{x}+\frac{\mathbf{k}_{\bot}^{2}+\hat{m}_{q}^{2}}{\bar{x}}\right)\right]},$
(4)

with
$\varphi_{2;K^{0}}(x)=\sum_{n=1}^{2}{[x\bar{x}]^{\alpha_{2;K^{0}}}[1+\hat{B}^{n}_{2;K^{0}}C_{n}^{3/2}(\xi)]}$
dominates the WF’s longitudinal distribution, and

$\displaystyle\Psi_{3;K^{0}}^{(p,\sigma)}(x,\mathbf{k}_{\bot})=\frac{A_{3;K^{0}}^{(p,\sigma)}\varphi_{3;K^{0}}^{(p,\sigma)}(x)}{x\bar{x}}$
$\displaystyle\quad\times\exp\left[-\frac{1}{8(\beta_{3;K^{0}}^{(p,\sigma)})^{2}}\left(\frac{\hat{m}_{q}^{2}+\mathbf{k}_{\bot}^{2}}{x}+\frac{\hat{m}_{s}^{2}+\mathbf{k}_{\bot}^{2}}{\bar{x}}\right)\right],$
(5)

with
$\varphi_{3;K^{0}}^{(p,\sigma)}(x)=\sum_{n=1}^{2}{\left[1+\hat{B}_{3;K^{0}}^{(p,\sigma),n}C_{n}^{(1/2,3/2)}(\xi)\right]}$.
The tags that appear above have the following descriptions: $\bar{x}=(1-x)$,
$\xi=(2x-1)$, $\tilde{m}=(\hat{m}_{q}x+\hat{m}_{s}\bar{x})$ with $\hat{m}_{q}$
and $\hat{m}_{s}$ are the constituent quark masses, $\mathbf{k}_{\bot}$ stands
for the transverse momentum, $C_{n}^{(1/2,3/2)}(\xi)$ represent for the
Gegenbauer polynomial, while other marks refer to the WF’s model parameters.
Note that we set $\hat{B}^{1}_{2;K^{0}}=0.4\hat{B}^{2}_{2;K^{0}}$ where the
factor 0.4 comes from the ratio of the first and second Gegenbauer moments.
Those WF’s model parameters can be fitted through the kaon twist-2, 3 DAs
$\langle\xi^{n}\rangle$-moments, which are computed by using QCD sum rules
within the framework of background field theory. Usually, we start with the
related correction function to achieve the final formula undergoing the finest
calculation. Furthermore, for the twist-4 DAs, we chose them in Ref.
Ball:2006wn .

Numerical Analysis.–In processing the numerical analysis, we adopt the
following basic input parameters from Particle Data Group (PDG)
ParticleDataGroup:2022pth : $m_{D_{s}^{+}}=1968.35\pm 0.07$ MeV,
$m_{K^{0}}=497.611\pm 0.013$ MeV, the $d$ and $s$-quark masses are
$m_{d}=4.67_{-0.17}^{+0.48}$ MeV and $m_{s}=93_{-3.4}^{+8.6}$ MeV, while the
charm quark mass is $\hat{m}_{c}=1.27\pm 0.02$ GeV. The $D_{s}^{+}$ and $K$
meson decay constants are taken as $f_{D_{s}^{+}}=251.1\pm 2.4\pm 3.0$ MeV
BESIII:2021bdp  and $f_{K}/f_{\pi}=1.1932$
FlavourLatticeAveragingGroup:2019iem  with $f_{\pi}=130.2(1.2)$ MeV
ParticleDataGroup:2022pth .

In this part, we directly exhibit the numerical values of the twist-2, 3 DAs
$\left<\xi^{n}\right>$-moments in Table 1, whose specific calculation process
can be seen in Refs. Zhong:2022ecl ; Zhong:2011rg . By means of those
numerical results, one can fit the model parameters of the Eqs. (Searching for
$|V_{cd}|$ through the exclusive decay $D_{s}^{+}\to K^{0}e^{+}\nu_{e}$ within
QCD Sum Rules) and (Searching for $|V_{cd}|$ through the exclusive decay
$D_{s}^{+}\to K^{0}e^{+}\nu_{e}$ within QCD Sum Rules) so as to proceed with
the next calculation. Subsequently, we give the model parameters’s results
corresponding to the applicable energy scale
$(m_{D_{s}}^{2}-m_{c}^{2})^{1/2}\approx 1.5$ GeV for TFF
$f_{+}^{D_{s}^{+}K^{0}}(q^{2})$,

$\displaystyle A_{2;K^{0}}=15.93,$ $\displaystyle\alpha_{2;K^{0}}=0.402,$
$\displaystyle\hat{B}^{2}_{2;K^{0}}=0.007,$
$\displaystyle\beta_{2;K^{0}}=2.660,$ (6)

for twist-2 kaon DA. Meanwhile, the parameters for kaon twist-3 DAs are

$\displaystyle A_{3;K^{0}}^{p}=133.6,$
$\displaystyle\hat{B}_{3;K^{0}}^{p,1}=-0.002,$
$\displaystyle\hat{B}_{3;K^{0}}^{p,2}=1.602,$
$\displaystyle\beta_{3;K^{0}}^{p}=0.547,$ $\displaystyle
A_{3;K^{0}}^{\sigma}=153.1,$
$\displaystyle\hat{B}_{3;K^{0}}^{\sigma,1}=0.015,$
$\displaystyle\hat{B}_{3;K^{0}}^{\sigma,2}=0.223,$
$\displaystyle\beta_{3;K^{0}}^{\sigma}=0.471.$ (7)

Table 1: Results of the kaon twist-2,3 DAs $\langle\xi^{n}\rangle$-moments at the scale $\mu_{0}=1$ with $n=(1,...,10)$ for twist-2 and $n=(1,2)$ for twist-3. $\hskip 45.52458pt\mathrm{Odd}$ | $\hskip 45.52458pt\mathrm{Even}$
---|---
$\langle\xi^{1}\rangle_{2;K^{0}}|_{\mu_{0}}=-0.0438_{-0.0075}^{+0.0053}$ | $\langle\xi^{2}\rangle_{2;K^{0}}|_{\mu_{0}}=+0.262_{-0.010}^{+0.010}$
$\langle\xi^{3}\rangle_{2;K^{0}}|_{\mu_{0}}=-0.0210_{-0.0035}^{+0.0024}$ | $\langle\xi^{4}\rangle_{2;K^{0}}|_{\mu_{0}}=+0.132_{-0.006}^{+0.006}$
$\langle\xi^{5}\rangle_{2;K^{0}}|_{\mu_{0}}=-0.0134_{-0.0021}^{+0.0014}$ | $\langle\xi^{6}\rangle_{2;K^{0}}|_{\mu_{0}}=+0.082_{-0.005}^{+0.005}$
$\langle\xi^{7}\rangle_{2;K^{0}}|_{\mu_{0}}=-0.0087_{-0.0014}^{+0.0009}$ | $\langle\xi^{8}\rangle_{2;K^{0}}|_{\mu_{0}}=+0.058_{-0.004}^{+0.004}$
$\langle\xi^{9}\rangle_{2;K^{0}}|_{\mu_{0}}=-0.0058_{-0.0010}^{+0.0007}$ | $\langle\xi^{10}\rangle_{2;K^{0}}|_{\mu_{0}}=+0.044_{-0.004}^{+0.004}$
$\langle\xi_{p}^{1}\rangle_{3;K^{0}}|_{\mu_{0}}=-0.126_{-0.050}^{+0.034}$ | $\langle\xi_{p}^{2}\rangle_{3;K^{0}}|_{\mu_{0}}=+0.425_{-0.064}^{+0.063}$
$\langle\xi_{\sigma}^{1}\rangle_{3;K^{0}}|_{\mu_{0}}=-0.094_{-0.044}^{+0.030}$ | $\langle\xi_{\sigma}^{2}\rangle_{3;K^{0}}|_{\mu_{0}}=+0.329_{-0.052}^{+0.057}$
Table 2: Comparison of $f_{+}^{D_{s}^{+}K^{0}}(0)$ for $D_{s}^{+}\to K^{0}$ transition. | $f_{+}^{D_{s}^{+}K^{0}}(0)$
---|---
This Work | $0.692_{-0.026}^{+0.027}$
BESIII’18-I BESIII:2018xre  | $0.720\pm 0.084\pm 0.013$
CQM’00 Melikhov:2000yu  | $0.72\pm 0.20$
LCSR’06 Wu:2006rd  | $0.820_{-0.071}^{+0.080}$
LFQM’08 Wang:2008ci  | $0.67$
LFQM’12 Verma:2011yw  | $0.66$
CCQM’18 Soni:2018adu  | $0.60\pm 0.09$
SFMs’21 Yao:2021pdy  | $0.681(10)$
4FhQCD’23 Ahmed:2023pod  | $0.57$

Furthermore, to obtain the numerical results for $D_{s}^{+}\to K^{0}$ TFF, one
need to determine two parameters, $\it i.e.,$ the continuum threshold $s_{0}$
and Borel parameters $M^{2}$. We can determine them based on the basic
criteria of the sum rules approach Tian:2023vbh : the continuum threshold
$s_{0}=6.7(0.5)$ GeV2 is taken to be close to the squared mass of the
$D_{s}^{+}$-meson excited state, $D_{s0}(2590)$, and the suitable Borel window
is found to be $M^{2}=14(1)$. According to those determined parameters, we
present the $D_{s}^{+}\to K^{0}$ TFF at the large recoil point
$f_{+}^{D_{s}^{+}K^{0}}(0)$, including other results for comparison, $\it
e.g.,$ the BESIII’18-I BESIII:2018xre , CQM’00 Melikhov:2000yu , LCSR’06
Wu:2006rd , LFQM’08(12) Wang:2008ci ; Verma:2011yw , CCQM’18 Soni:2018adu ,
SFMs’21 Yao:2021pdy , and 4FhQCD’23 Ahmed:2023pod , which are listed in Table
2. Upon examining the Table 2, it is evident that the predictions fall within
the range of $0.5\sim 0.9$, and our results lie in the upper-middle section,
demonstrating a good degree of agreement. In addition, the contribution
proportion of the NLO is approximately $12\%$ in our total results.

Figure 1: The curves of $D_{s}^{+}\to K^{0}$ TFF
$f_{+}^{D_{s}^{+}K^{0}}(q^{2})$ along with associated uncertainties, is done
throughout the whole $q^{2}$-region. Compared with various results from the
theoretical and experimental, $\it i.e.,$ the BESIII’18-I BESIII:2018xre ,
CQM’00 Melikhov:2000yu , LCSR’06 Wu:2006rd , LFQM’12 Verma:2011yw , LQCD’12
Koponen:2012di , SFMs’21Yao:2021pdy , and 4FhQCD’23 Ahmed:2023pod . The
results of 4FhQCD’23 cited here were obtained from
$f_{+}^{D_{s}^{+}K^{0}}(q^{2})/f_{+}^{D_{s}^{+}K^{0}}(0)$ multiplying
$f_{+}^{D_{s}^{+}K^{0}}(0)$ Ahmed:2023pod . Figure 2: Differential decay rates
for $D_{s}^{+}\to K^{0}e^{+}\nu_{e}$ as a function of the electron invariant
mass squared($q^{2}$) with the shaded bands stand for its uncertainties,
including the experimental results for comparison from the BESIII’18-I
BESIII:2018xre .

The physical allowable region for the $D_{s}^{+}\to K^{0}$ TFF is
approximately $q^{2}\leqslant(m_{D_{s}^{+}}-m_{K^{0}})^{2}\approx 2.16$ GeV2.
Then, according to the applicable $q^{2}$-region of the LCSRs approach, $\it
i.e.,$ the low and intermediate region, we take $q^{2}\in[0,0.8]$ GeV2, while
the high $q^{2}$-region can be extrapolated through $z(q^{2},t)$ converging
the simplified series expansion (SSE). The $D_{s}^{+}\to K^{0}$ TFF can expand
as:

$\displaystyle
f_{+}^{D_{s}^{+}K^{0}}\left(q^{2}\right)=\frac{1}{1-q^{2}/m_{H}^{2}}\sum_{k=0,1,2}{\beta_{k}z^{k}\left(q^{2},t\right)},$
(8)

in which $H$ is $D_{s}^{+}$-meson, $\beta_{k}$ refers to the fitting
parameters, while the function
$z(q^{2},t)=(\sqrt{t_{+}-q^{2}}-\sqrt{t_{+}-t_{0}})/(\sqrt{t_{+}-q^{2}}+\sqrt{t_{+}-t_{0}})$
with $t_{\pm}=(m_{D_{s}^{+}}\pm m_{K^{0}})^{2}$ and
$t_{0}=t_{\pm}(1-\sqrt{1-t_{-}/t_{+}})$. The SSE method can remain the
analytic structure correct in the complex plane and assure the suitable
scaling, $f_{+}^{D_{s}^{+}K^{0}}(q^{2})\sim 1/q^{2}$.

After extrapolating the $D_{s}^{+}\to K^{0}$ TFF to the complete physical
$q^{2}$-region, one can obtain the entire behavior of
$f_{+}^{D_{s}^{+}K^{0}}(q^{2})$, which is exhibited in Fig. 1. In which, the
solid line is the central behavior of $f_{+}^{D_{s}^{+}K^{0}}(q^{2})$, the
darker band stands for the LCSRs prediction, and the lighter band refers to
the SSE result. Meanwhile, we also present the theoretical and experimental
results for comparison from the CQM’00 Melikhov:2000yu , LCSR’06 Wu:2006rd ,
LFQM’12 Verma:2011yw , LQCD’12 Koponen:2012di , BESIII’18-I BESIII:2018xre ,
SFMs’21 Yao:2021pdy , and 4FhQCD’23 Ahmed:2023pod .

By means of the determined $D_{s}^{+}\to K^{0}$ TFF from the LCSRs, one can
further proceed with the calculation with respect to the physical observables
of the semi-leptonic decay $D_{s}^{+}\to K^{0}\ell^{+}\nu_{\ell}$ with
$\ell=(e,\mu)$. Then, we presented the integrated values of differential decay
widths for $D_{s}^{+}\to K^{0}\ell^{+}\nu_{\ell}$ with two channels:

$\displaystyle\mathrm{\Gamma}(D_{s}^{+}\to
K^{0}e^{+}\nu_{e})=4.413_{-0.359}^{+0.393}\times 10^{-15}~{}\mathrm{GeV},$ (9)
$\displaystyle\mathrm{\Gamma}(D_{s}^{+}\to
K^{0}\mu^{+}\nu_{\mu})=4.376_{-0.357}^{+0.390}\times 10^{-15}~{}\mathrm{GeV}.$
(10)

As a further step, we can obtain the branching fraction results through the
utilization of the lifetime of the initial state $D_{s}^{+}$-meson, $\it
i.e.,$ $\tau_{D_{s}^{+}}=0.504\pm 0.004$ ps ParticleDataGroup:2022pth . The
predictions are listed in Table 3, including the theoretical and experimental
results for a comparison, $\it e.g.,$ the BESIII’18-I BESIII:2018xre , LCSR’06
Wu:2006rd , LFQM’08 Wang:2008ci , CCQM’18 Soni:2018adu , PDG’22
ParticleDataGroup:2022pth , HIETALA’15 Hietala:2015jqa , LFQM’17 Cheng:2017pcq
, and CLEO’09-I CLEO:2009dyb . In which, our numerical results can match with
those experimental results within their uncertainties, however, the
statistical uncertainties remain significantly large, and thus it is necessary
to update the more precise experimental results in the near future. In Fig. 2,
we exhibit the behavior for the differential width of $D_{s}^{+}\to
K^{0}e^{+}\nu_{e}$ in whole $q^{2}$-region, including the experimental results
for comparison.

Table 3: The branching fractions (in unit: $10^{-3}$) for $D_{s}^{+}\to K^{0}e^{+}\nu_{e}$ and $D_{s}^{+}\to K^{0}\mu^{+}\nu_{\mu}$ channels. For comparison, the theoretical and experimental results are also presented.  | $\mathcal{B}(D_{s}^{+}\to K^{0}e^{+}\nu_{e})$ | $\mathcal{B}(D_{s}^{+}\to K^{0}\mu^{+}\nu_{\mu})$
---|---|---
This Work | $3.379_{-0.275}^{+0.301}$ | $3.351_{-0.273}^{+0.299}$
BESIII’18-I BESIII:2018xre  | $3.25\pm 0.38\pm 0.16$ | -
LCSR’06 Wu:2006rd  | $3.90_{-0.57}^{+0.74}$ | $3.83_{-0.56}^{+0.72}$
LFQM’08 Wang:2008ci  | $2.9$ | -
CCQM’18 Soni:2018adu  | $2$ | $2$
PDG’22 ParticleDataGroup:2022pth  | $0.34\pm 0.04$ |
HIETALA’15 Hietala:2015jqa  | $3.9\pm 0.8\pm 0.3$ | -
LFQM’17 Cheng:2017pcq  | $0.27\pm 0.02$ | $0.26\pm 0.02$
CLEO’09-I CLEO:2009dyb  | $3.7\pm 1\pm 0.2$ | -
Figure 3: Our prediction of $|V_{cd}|$ obtained from the decay channel
$D_{s}^{+}\to K^{0}e^{+}\nu_{e}$, compared with other theoretical and
experimental results of various channels.

Furthermore, to provide a new result on $|V_{cd}|$ within the QCD LCSRs
approach, we extract the CKM matrix element $|V_{cd}|$ based on the branching
fractions $\mathcal{B}(D_{s}^{+}\to K^{0}e^{+}\nu_{e})$ from the BESIII’18-I
BESIII:2018xre , which is listed as follows:

$\displaystyle|V_{cd}|=0.221_{-0.010}^{+0.008}.$ (11)

In addition, for comparison, we gathered the CKM matrix element $|V_{cd}|$
results from the theoretical and experimental studies, which are presented in
Fig. 3. Those results originate from the CLEO’08(09-II) CLEO:2008ffk ;
CLEO:2009svp , BESIII’(14-20) BESIII:2018xre ; BESIII:2013iro ; BESIII:2015tql
; BESIII:2017ylw ; BESIII:2018eom ; BESIII:2019vhn ; Ablikim:2020hsc , PDG’22
ParticleDataGroup:2022pth , HPQCD’11 Na:2011mc , ETM’17 Lubicz:2017syv , and
HFLAV’21 HFLAV:2022esi . And our theoretical prediction aligns well with the
experimental outcomes.

Lastly, we attempt to give a theoretical ratio $\mathcal{R}_{K^{0}/\eta}^{e}$
with respect to the different decay channels as follows:

$\displaystyle\mathcal{R}_{K^{0}/\eta}^{e}$
$\displaystyle=\frac{\mathcal{B}(D_{s}^{+}\to
K^{0}e^{+}\nu_{e})}{\mathcal{B}(D_{s}^{+}\to\eta e^{+}\nu_{e})}$
$\displaystyle=0.144_{-0.020}^{+0.028}.$ (12)

The result $\mathcal{B}(D_{s}^{+}\to\eta
e^{+}\nu_{e})=2.346_{-0.331}^{+0.418}\times 10^{-2}$ comes from our previous
work Hu:2021zmy . The ratio presented can be viewed as a test for the
$D_{s}^{+}$-meson’s internal structure and the properties of related
pseudoscalar mesons.

Summary.–In this paper, we conducted an investigation into the semileptonic
decays $D_{s}^{+}\to K^{0}\ell^{+}\nu_{\ell}$ with $\ell=(e,\mu)$. Firstly, we
calculated the $D_{s}^{+}\to K^{0}$ vector TFF $f_{+}^{D_{s}^{+}K^{0}}(q^{2})$
by using the QCD LCSRs approach up to the NLO correction and briefly
introduced the kaon twist-2, 3 DAs. Then we plotted the behavior of the
$D_{s}^{+}\to K^{0}$ TFF $f_{+}^{D_{s}^{+}K^{0}}(q^{2})$ in the whole
$q^{2}$-region, which is exhibited in Fig. 1, including the theoretical and
experimental results for comparison. Meanwhile, according to the derived TFF
$f_{+}^{D_{s}^{+}K^{0}}(q^{2})$, we also computed the differential width of
$D_{s}^{+}\to K^{0}e^{+}\nu_{e}$ and presented it in Fig. 2. As the further
step, we calculated the branching fractions of the semileptonic decays
$D_{s}^{+}\to K^{0}\ell^{+}\nu_{\ell}$ with $\ell=(e,\mu)$ by means of the
lifetime of the initial state $D_{s}^{+}$-meson, which are
$\mathcal{B}(D_{s}^{+}\to K^{0}e^{+}\nu_{e})=3.379_{-0.275}^{+0.301}\times
10^{-3}$ and $\mathcal{B}(D_{s}^{+}\to
K^{0}\mu^{+}\nu_{\mu})=3.351_{-0.273}^{+0.299}\times 10^{-3}$. Furthermore, we
extracted the CKM matrix element $|V_{cd}|$ by using the
$\mathcal{B}(D_{s}^{+}\to K^{0}e^{+}\nu_{e})$ from the BESIII Collaboration,
which is expected to be $|V_{cd}|=0.221_{-0.010}^{+0.008}$. Lastly, we
presented a theoretical ratio
$\mathcal{R}_{K^{0}/\eta}^{e}=0.144_{-0.020}^{+0.028}$.

Acknowledgments.–This work was supported in part by the National Natural
Science Foundation of China under Grant No.12265010, No.12265009, No.12175025
and No.12347101, the Project of Guizhou Provincial Department of Science and
Technology under Grant No.ZK[2021]024 and No.ZK[2023]142.

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# Search for the wide-orbit massive companion of XO-7b in the follow-up
radial-velocity and transit-timing data: no significant clues

Z. Garai,1,2,3 T. Pribulla,3 and R. Komžík,3
1HUN-REN-ELTE Exoplanet Research Group, 9700 Szombathely, Szent Imre h. u.
112, Hungary
2ELTE Gothard Astrophysical Observatory, 9700 Szombathely, Szent Imre h. u.
112, Hungary
3Astronomical Institute, Slovak Academy of Sciences, 05960 Tatranská Lomnica,
Slovakia
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Last updated 2020 June 10; in original form 2013 September 5)

###### Abstract

XO-7b is a hot Jupiter transiting a $V=10.52$ mag G0V-type star. The planetary
system is interesting because the linear slope in the discovery radial-
velocity (RV) data indicated a wide-orbit massive companion. In 2020 we
started an RV campaign for the system with the main scientific goal to follow-
up this linear slope, and to put constraints on the orbital period of the
companion. Furthermore, we aimed at refining the system parameters and we
wanted to probe transit timing variations (TTVs) of XO-7b in order to search
for long-term dynamical signs of the companion of XO-7b in the observed-minus-
calculated (O-C) data of mid-transit times. Apart from the discovery RVs, we
obtained and analyzed 20 follow-up RV observations and TESS photometric data.
The previously observed significant linear RV slope was not confirmed with the
follow-up RV data, where we detected only a marginal linear slope with the
opposite trend. If the announced companion really exists, the most convincing
explanation is that both RV datasets were collected near its quadrature
position. Based on the RVs we estimated the minimum orbital period, which is
$P_{\mathrm{orb,min,3}}\gtrsim 7900\pm 1660$ d, and the ’minimum’ minimum mass
of the companion, which is $(M_{3}\sin i)_{\mathrm{min}}=16.7\pm
3.5~{}\mathrm{M_{Jup}}$. We did not find significant evidence of the companion
of XO-7b in the O-C dataset of mid-transit times. We can again conclude that
if the announced companion really exists, this is in agreement with previous
results that distant companions of exoplanets are only known by RV solutions.

###### keywords:

methods: observational – techniques: photometric – techniques: spectroscopic –
planets and satellites: individual: XO-7b

††pubyear: 2020††pagerange: Search for the wide-orbit massive companion of
XO-7b in the follow-up radial-velocity and transit-timing data: no significant
clues–Search for the wide-orbit massive companion of XO-7b in the follow-up
radial-velocity and transit-timing data: no significant clues

##  1 Introduction

Nowadays the Transiting Exoplanet Survey Satellite (TESS) mission (Ricker et
al., 2014, 2015), launched in 2018, is the most active transiting-exoplanet
searching program. During its nominal and extended mission duration, up to
September 19, 2023, discovered 392 confirmed planets and 6788 planet
candidates111This information is based on the NASA Exoplanet Archive database
(see https://exoplanetarchive.ipac.caltech.edu/index.html).. In comparison
with the formerly active Kepler (Borucki et al., 1996, 2010, 2011) and K2
(Howell et al., 2014) projects it has a big advantage, namely that TESS
targeted bright stars, brighter than $T$ (TESS) = 13 mag. Bright stars are
needed to get true masses of the exoplanets via ground-based radial velocity
(RV) measurements (Mayor & Queloz, 1995). Moreover, TESS has one more big
advantage in comparison with Kepler, i.e. that it is performing a near-all-sky
survey in sectors. Every sector has a dimension of $24\times 96$ deg, and the
sectors are monitored for at least 27 days. The photometric precision is
dominated by pointing jitter. A 1-hour integration at a $T=10$ mag star gives
about 230 ppm precision, which is sufficient to detect super-Earths. These
characteristics make this space telescope not only a powerful tool in the
discovery of new exoplanets but also a useful tool for the follow-up of known
exoplanets. Apart from the Characterising Exoplanet Satellite (CHEOPS) space
observatory (Benz et al., 2021), TESS is the most used space-based instrument
for these purposes.

Known exoplanets from ground-based surveys were frequently followed-up using
TESS data. For example, physical and orbital parameters of WASP-18Ab,
WASP-19b, and WASP-77Ab were refined based on the TESS data by Cortés-Zuleta
et al. (2020), and planetary architecture and stellar variability constraints
were improved at WASP-166b by Doyle et al. (2022). Transit time variations
(TTVs) were probed with precise TESS data, e.g., in the cases of WASP-4b
(Bouma et al., 2019; Baluev et al., 2020), XO-6b (Ridden-Harper et al., 2020),
or HATS-18b (Southworth et al., 2022). The decaying orbit of WASP-12b was
confirmed based on TESS observations by Turner et al. (2021) and Wong et al.
(2022). The orbital decay was probed also in the cases of WASP-18b
(Maciejewski et al., 2020) and WASP-43b (Garai et al., 2021; Davoudi et al.,
2021) using TESS data.

In this follow-up project, we selected the transiting exoplanet XO-7b,
discovered by Crouzet et al. (2020) within the framework of the XO project
(McCullough et al., 2005). The XO project discovered seven transiting
exoplanets to date (McCullough et al., 2006, 2008; Burke et al., 2007, 2008;
Johns-Krull et al., 2008; Crouzet et al., 2017, 2020). Although the number of
discoveries is relatively small, XO planets are interesting in many aspects.
XO-7b is also interesting, because its large atmospheric scale height makes it
well suited to atmospheric characterization, and the linear slope in the
discovery RV data indicates a wide-orbit companion with a minimum mass of
$4~{}\mathrm{M_{Jup}}$, which could be a planet, a brown dwarf, or a low-mass
star (Crouzet et al., 2020). In some cases, the outer companions are known to
exhibit their signal in TTVs and RV observations, as well, distant companions
of exoplanets are only known by RV solutions, see e.g., Maciejewski et al.
(2013); Neveu-VanMalle et al. (2016); Dawson et al. (2012); Knutson et al.
(2014), or Hartman et al. (2014).

Motivated by the announced wide-orbit massive companion of XO-7b, in 2020 we
started RV observations of the XO-7 system with the main scientific goal to
follow-up the linear RV slope discovered by Crouzet et al. (2020), and to put
constraints on the orbital period of the companion. Furthermore, we aimed at
combining published RVs and the RV data from this work with TESS photometric
data to derive precise orbital and planetary parameters of the XO-7 system.
Basic stellar parameters of XO-7 were obtained from spectra, which were later
used to extract the RV data. Finally, based on TESS data we probed the TTVs of
XO-7b. Although results on TTVs based on TESS data were recently published by
Maciejewski (2022), who found no additional planet in the XO-7 system down to
sub-Neptune-sized planets, we repeated the analysis with a much longer
timebase TESS data in order to search for long-term dynamical signs of the
announced companion of XO-7b in the TTV data.

The paper is organized as follows. In Sect. 2 a brief description of
observations, the applied spectroscopic instrumentation, and data reduction is
given. We summarise the spectral analysis and the derived stellar parameters
in Sect. 3. Data analysis, model fitting, and the results are detailed in Sect
4. The most interesting result is described and discussed in Sect. 5. We
conclude with the results in Sect. 6.

##  2 Observations and data reduction

###  2.1 Photometric observations

We used 2-min integrated TESS data of XO-7, downloaded from the Mikulski
Archive for Space Telescopes222See
https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html. (MAST). Up to
September 30, 2023, XO-7 was observed with TESS during 11 sectors, Nos. 18,
19, 20, 25, 26, 40, 47, 52, 53, 59, and 60. In the first three sectors,
however, no 2-min integrations were produced, therefore, we downloaded only
the last 8 sectors’ data in the form of Pre-search Data Conditioning Simple
Aperture Photometry (PDCSAP) flux, which was initially smoothed by the PDCSAP
pipeline. This light-curve type is subject to more treatment than the Simple
Aperture Photometry (SAP) flux and is specifically intended for detecting
exoplanets. The pipeline attempts to remove systematic artifacts while keeping
planetary transits intact. This light-curve type is also corrected by the
dilution factor. From MAST we obtained $134230$ PDCSAP datapoints with 2-min
integration (for more details see Tab. 1). These TESS data were first
normalized to unity. Then, since TESS telescope uses as time-stamps
Barycentric TESS Julian Date (i.e. $\mathrm{BJD_{TDB}}-2~{}457~{}000.0$), we
converted all TESS time-stamps to $\mathrm{BJD_{TDB}}$.

Table 1: Log of 2-min integrated TESS PDCSAP photometric observations of XO-7 used in this work. The table shows the time interval of observations, the number of observed transits, and the number of data points obtained from MAST, sorted by the TESS sectors. Sector | Time interval | Number | Data
---|---|---|---
No. | of observation | of transits | points
25 | 2020-05-13 – 2020-06-08 | 9 | 17 246
26 | 2020-06-08 – 2020-07-04 | 8 | 16 942
40 | 2021-06-24 – 2021-07-23 | 10 | 19 611
47 | 2021-12-30 – 2022-01-28 | 8 | 17 311
52 | 2022-05-18 – 2022-06-13 | 8 | 16 749
53 | 2022-06-13 – 2022-07-09 | 8 | 17 305
59 | 2022-11-26 – 2022-12-23 | 9 | 16 476
60 | 2022-12-23 – 2023-01-18 | 7 | 12 590
Total | – | 67 | 134230

###  2.2 Spectroscopic observations and RV data

Table 2: Log of spectroscopic observations of XO-7, obtained at the Skalnaté Pleso Observatory. The table shows the time of the observation, BF RVs and CCF RVs with $\pm 1\sigma$ uncertainties, systematic errors caused by the spectrograph instability ($\sigma_{\mathrm{syst}}$), the signal-to-noise ratio of the combined spectra at 5500 Å, calculated as $S/N=\sqrt{(S/N)_{1}{{}^{2}}+(S/N)_{2}{{}^{2}}+(S/N)_{3}{{}^{2}}}$, where $(S/N)_{\mathrm{n}}$ is the signal-to-noise ratio of individual spectra, and the orbital phase of observation, calculated using the mid-transit time of $T_{\mathrm{c}}=2~{}458~{}779.58040$ $\mathrm{BJD_{TDB}}$ and the orbital period of $P_{\mathrm{orb}}=2.86413296$ d (see Tab. 4). Time [$\mathrm{BJD_{TDB}}$] | RV (BF) [$\mathrm{km~{}s}^{-1}$] | RV (CCF) [$\mathrm{km~{}s}^{-1}$] |  $\sigma_{\mathrm{syst}}$ [$\mathrm{km~{}s}^{-1}$] | $S/N$ | Orbital phase
---|---|---|---|---|---
2458928.385194 | $-13.750\pm 0.047$ | $-13.083\pm 0.073$ | 0.30 | 21.0 | 0.95456
2459108.342633 | $-13.680\pm 0.032$ | $-13.177\pm 0.070$ | 0.09 | 30.5 | 0.78595
2459163.384992 | $-13.790\pm 0.035$ | $-13.259\pm 0.152$ | 0.17 | 27.9 | 0.00376
2459164.350772 | $-13.840\pm 0.031$ | $-13.330\pm 0.177$ | 0.07 | 32.2 | 0.34096
2459196.222682 | $-13.820\pm 0.050$ | $-13.188\pm 0.103$ | 0.35 | 19.9 | 0.46890
2459197.676202 | $-13.850\pm 0.056$ | $-13.198\pm 0.154$ | 0.05 | 17.7 | 0.97639
2459226.470352 | $-13.760\pm 0.036$ | $-13.121\pm 0.116$ | 0.26 | 27.7 | 0.02975
2459624.500127 | $-13.690\pm 0.047$ | $-13.124\pm 0.107$ | 0.03 | 20.9 | 0.00017
2459650.456236 | $-13.810\pm 0.040$ | $-13.210\pm 0.067$ | 0.14 | 24.6 | 0.06264
2459667.415096 | $-13.750\pm 0.063$ | $-13.046\pm 0.209$ | 0.31 | 15.7 | 0.98375
2459699.515815 | $-13.900\pm 0.061$ | $-13.390\pm 0.130$ | 0.10 | 16.2 | 0.19158
2459712.420535 | $-13.720\pm 0.052$ | $-13.137\pm 0.072$ | 0.50 | 19.1 | 0.69721
2459718.393255 | $-13.660\pm 0.050$ | $-13.223\pm 0.068$ | 0.40 | 19.8 | 0.78256
2460065.433550 | $-13.629\pm 0.048$ | $-13.062\pm 0.106$ | 0.10 | 20.8 | 0.95022
2460084.480879 | $-13.607\pm 0.040$ | $-13.167\pm 0.099$ | 0.00 | 25.1 | 0.60052
2460099.463299 | $-13.766\pm 0.044$ | $-12.971\pm 0.069$ | 0.00 | 22.7 | 0.83157
2460114.483999 | $-13.797\pm 0.056$ | $-13.161\pm 0.270$ | 0.10 | 17.6 | 0.07598
2460115.453659 | $-13.813\pm 0.050$ | $-13.194\pm 0.065$ | 0.10 | 19.7 | 0.41454
2460142.428958 | $-13.745\pm 0.049$ | $-13.061\pm 0.177$ | 0.20 | 20.4 | 0.83285
2460153.455588 | $-13.652\pm 0.050$ | $-13.088\pm 0.097$ | 0.00 | 19.9 | 0.68275

Several spectroscopic observations were performed at the Skalnaté Pleso
Observatory in the High Tatras (Slovak Republic), using the 1.3 m f/8.36
Astelco Alt-azimuthal Nasmyth-Cassegrain reflecting telescope, equipped with a
fiber-fed echelle spectrograph of MUSICOS design (Baudrand & Bohm, 1992). Its
fiber injection and guiding unit (FIGU) is mounted in the Nasmyth focus of the
telescope. The FIGU is connected to the calibration unit (ThAr hollow cathode
lamp, tungsten lamp, blue LED) in the control room and to the echelle
spectrograph itself in the room below the dome, where the temperature is
stable. The spectra were recorded by an Andor iKon-936 DZH $2048\times 2048$
pixels CCD camera. The spectral range of the instrument is 4250 – 7375 Å in 56
echelle orders. The maximum resolution of the spectrograph reaches $R\approx
38~{}000$ around 6000 Å. The exposure time was 900 s in all cases. Three raw
spectra were obtained consecutively during an observing night and,
subsequently, combined via IRAF task combine to increase the signal-to-noise
ratio ($S/N$) with the assumption that there is no substantial difference
(Doppler shift) between them.

The raw spectra were reduced using IRAF package tasks, Linux shell scripts,
and FORTRAN programs similarly, as it was described in Pribulla et al. (2015)
and in Garai et al. (2017). In the first step, master dark frames were
produced. In the second step, the photometric calibration of the frames was
done using dark and flat-field frames. Bad pixels were cleaned using a bad
pixel mask, and cosmic hits were removed using the program of Pych (2004).
Order positions were defined by fitting Chebyshev polynomials to the tungsten
lamp and blue LED spectrum. In the following step, scattered light was modeled
and subtracted. Aperture spectra were then extracted for both object and ThAr
frames, and then the resulting 2D spectra were dispersion-solved. Two-
dimensional spectra were finally combined to 1D spectra rebinned to 4250 –
7375 Å wavelength range with a 0.05 Å step, i.e. about 2 – 4 times the
spectral resolution.

Spectra were first analyzed using the broadening function (BF) technique,
developed by Rucinski (1992), to get RVs. As a template star HD 222368
($\iota$ Psc, F7V-type) was used with RV = $5.4~{}\mathrm{km~{}s}^{-1}$
(Nordström et al., 2004). The RVs of XO-7 were measured by fitting a Gaussian
function to the extracted BFs. Because the formal RV uncertainties found using
the BF approach are hard to quantify and depend on BF smoothing, they were
determined from $S/N$ as follows. We first determined RV uncertainties as
1/($S/N$), see Hatzes et al. (2010). Subsequently, we fitted the RV
observations with initial uncertainties using the RMF code (Szabó et al.,
2020; Garai et al., 2021, 2022), and from the best fit we obtained reduced
$\chi^{2}$ ($\chi^{2}_{\mathrm{red}}$). In the next step, we rescaled all
uncertainties to get $\chi^{2}_{\mathrm{red}}=1$. $S/N$ of the spectra can be
obtained from the $1\sigma$ uncertainties as $C/\sigma$, where the scaling
constant $C$ was found to be about 1.0 km s-1. Spectra were also analyzed
using the cross-correlation function (CCF) technique, see Baranne et al.
(1996). The data were cross-correlated with a K0V stellar mask, with an
effective temperature of $T_{\mathrm{eff}}=5250$ K, a surface gravity of $\log
g=4.5$ [cgs], and solar metallicity [Fe/H] = 0.0 dex. To get RVs we used only
a spectral region around the magnesium triplet, i.e., between the wavelengths
of 4900 and 5400 Å, where no telluric lines, nor hydrogen Balmer lines are
present. The advantage of the BF solution is generally smaller uncertainty of
individual RVs, while the CCF technique is more consistent with the discovery
RVs (see Sect. 4.1). An overview of the obtained dataset is shown in Tab. 2,
where we present BF RVs, as well as CCF RVs. The first observation was taken
at $\mathrm{BJD_{TDB}}=2~{}458~{}928.385194$, which corresponds to 21:13:34.28
UT on March 19, 2020, and the last observation
($\mathrm{BJD_{TDB}}=2~{}460~{}153.455588$) is from 22:57:04.96 UT on July 27,
2023. This means that the timebase of the RVs is about 1225 days, fully
covering and exceeding the TESS observations window (compare with Tab. 1).
Unfortunately, we had to discard a couple of spectroscopic observations from
the analysis due to problems with the stability of the spectrograph or due to
outlier data points.

For the purpose of spectroscopic analysis of the planet’s host star the
obtained 1D spectra were combined into one spectrum to increase the signal-to-
noise ratio using iSpec333See https://www.blancocuaresma.com/s/iSpec. (Blanco-
Cuaresma et al., 2014; Blanco-Cuaresma, 2019). We shifted all of the 20
obtained spectra into the rest frame and combined the barycentric correction
and the intrinsic RV correction into one step applying the iSpec cross-
correlation routine. We cross-correlated the spectra with a template from the
Munari et al. (2005) synthetic spectrum library. According to the literature
values for the stellar parameters, i.e., $T_{\mathrm{eff}}=6250\pm 100$ K,
$\log g=4.246\pm 0.023$ [cgs], and [Fe/H] = $0.432\pm 0.057$ dex (Crouzet et
al., 2020), we selected a template file from the spectrum library, which
corresponds to $T_{\mathrm{eff}}=6250$ K, $\log g=4.5$ [cgs] and [Fe/H] =
$+0.5$ dex. After shifting the spectra into the rest frame we averaged them
via median into a final spectrum by setting the resolution to $R=35~{}000$ and
setting the sampling to 0.05 Å. We then slightly shifted the overall continuum
level to set it to 1.0 as much as possible. In this way, we obtained the final
averaged spectrum for the planet’s host star XO-7, which we further analyzed
in order to obtain fundamental stellar parameters (see Sect. 3).

##  3 The planet’s host star

Table 3: An overview of fundamental facts about the planet’s host star. The table includes stellar parameters obtained in this work from the spectra of XO-7, as well. Notes: ⋆Derived from the redder region of the final spectrum, see the text in Sect. 3. ⋆⋆Derived from the bluer region of the final spectrum, see the text in Sect. 3. C2020 = Crouzet et al. (2020), A1903 = Argelander (1903), G2023 = Gaia Collaboration et al. (2023), G2021 = Gaia Collaboration et al. (2021), B2018 = Bailer-Jones et al. (2018), H2000 = Høg et al. (2000), C2003 = Cutri et al. (2003). Parameter [unit] | Value | Source
---|---|---
Name | XO-7 | C2020
Catalog Name | BD+85 317 | A1903
Gaia DR3 ID | 2303332931542914048 | G2023
RA (J2000.0) | 18:29:54.9 | G2021
Dec (J2000.0) | +85:13:59 | G2021
$p$ [mas] | $4.3216\pm 0.0132$ | G2023
$\mu_{\alpha}$ [$\mathrm{mas~{}yr^{-1}}$] | $-15.208\pm 0.015$ | G2023
$\mu_{\delta}$ [$\mathrm{mas~{}yr^{-1}}$] | $24.396\pm 0.017$ | G2023
$D$ [pc] | $234.1\pm 1.2$ | B2018
$B$ [mag] | $11.23\pm 0.06$ | H2000
$V$ [mag] | $10.52\pm 0.04$ | H2000
$G$ (Gaia) [mag] | $10.459\pm 0.002$ | G2021
$J$ [mag] | $9.557\pm 0.024$ | C2003
$H$ [mag] | $9.308\pm 0.030$ | C2003
$K$ [mag] | $9.241\pm 0.024$ | C2003
Spectral type | G0V | C2020
$T_{\mathrm{eff}}$ [K] | $6250\pm 100$ | C2020
$T_{\mathrm{eff}}$ [K] |  $5957\pm 388^{\star}$/$6092\pm 164^{\star\star}$ | This work
$\log g$ [cgs] | $4.246\pm 0.023$ | C2020
$\log g$ [cgs] |  $4.06\pm 0.67^{\star}$/$3.76\pm 0.39^{\star\star}$ | This work
Fe/H [dex] | $0.432\pm 0.057$ | C2020
Fe/H [dex] |  $0.39\pm 0.22^{\star}$/$0.42\pm 0.13^{\star\star}$ | This work
$v\sin i$ [$\mathrm{km~{}s}^{-1}$] | $6.00\pm 1.00$ | C2020
$v\sin i$ [$\mathrm{km~{}s}^{-1}$] |  $9.75\pm 1.45^{\star}$/$12.04\pm 1.07^{\star\star}$ | This work
$M_{\mathrm{s}}$ [$M_{\odot}$] | $1.405\pm 0.059$ | C2020
$R_{\mathrm{s}}$ [$R_{\odot}$] | $1.480\pm 0.022$ | C2020
Age [Gyr] | $1.18^{+0.98}_{-0.71}$ | C2020

The final averaged spectrum of XO-7 was fitted using the iSpec fitting
routine, which performs spectral synthesis. We selected two spectral regions
for fitting: as the first (redder) region we selected a wavelength range from
4895 to 5553 Å. As the second (bluer) region with hydrogen Balmer lines from
4220 to 5042 Å was selected and fitted. During the fitting procedure, we fixed
the limb-darkening coefficient $u$ to 0.5, the stellar microturbulent velocity
$v_{\mathrm{mic}}$ to 1.0 $\mathrm{km~{}s}^{-1}$, and the stellar
macroturbulent velocity $v_{\mathrm{mac}}$ to zero, assuming no motion in
larger atmospheric cells. We freely adjusted the following fundamental stellar
parameters: $T_{\mathrm{eff}}$, $\log g$, [Fe/H], and the projected rotational
velocity $v\sin i$.

The obtained results are included in Tab. 3. The observed and averaged stellar
spectrum, overplotted with the synthetic spectrum in the redder region is
depicted, as an illustrative example, in Fig. 1. Although we did not improve
these parameters compared to the discovery values, the final spectrum of XO-7
reveals well the fundamental characteristics of the planet’s host star. Based
on the data listed in Tab. 3 we can see that the best-fitting values obtained
from the bluer region are more precise. The parameter values are in a
$3\sigma$ agreement with the corresponding values found by Crouzet et al.
(2020), except for the $v\sin i$ parameter value derived from the bluer region
of the final spectrum, which is $12.04\pm 1.07$ $\mathrm{km~{}s}^{-1}$, while
the discoverers presented a value of $v\sin i=6.0\pm 1.0$
$\mathrm{km~{}s}^{-1}$. The difference, in this case, is about $5.6\sigma$,
which we can explain with different spectral resolutions of the spectrographs
(see Sections 2.2 and 4.1).

Figure 1: The final averaged spectrum of the host star XO-7, overplotted with the synthetic spectrum in the redder region. Table 4: An overview of the Allesfitter best-fitting parameters of the exoplanet system XO-7 obtained from the TESS 2-min PDCSAP photometry and RV observations. Notes: 1Taken from, or calculated based on Crouzet et al. (2020). 2Based on the stellar parameters of $T_{\mathrm{eff}}=6250$ K and $\log g$ = 4.246 [cgs] (Crouzet et al., 2020). 3Shifted automatically by the Allesfitter software package. Parameter [unit] | Description | Prior | Value
---|---|---|---
$T_{\mathrm{c}}$ [$\mathrm{BJD}_{\mathrm{TDB}}$] | Reference mid-transit time |  $\mathcal{N}$(2457917.47503, 0.00045)1 | $2458779.58040\pm 0.00016^{3}$
$P_{\mathrm{orb}}$ [d] | Orbital period |  $\mathcal{N}$(2.8641424, 0.0000043)1 | $2.86413296\pm 0.00000055$
$R_{\mathrm{p}}/R_{\mathrm{s}}$ | Planet-to-star radius ratio |  $\mathcal{N}$(0.09532, 0.00093)1 | $0.09344\pm 0.00028$
$(R_{\mathrm{p}}+R_{\mathrm{s}})/a$ | Scaled sum of fractional radii |  $\mathcal{N}$(0.1703, 0.0037)1 | $0.1739\pm 0.0013$
$\cos i$ | Cosine of the orbit inclination angle |  $\mathcal{N}$(0.1140, 0.0050)1 | $0.1179\pm 0.0017$
$K$ [$\mathrm{km~{}s^{-1}}$] | RV semi-amplitude |  $\mathcal{N}$(0.0805, 0.0032)1 | $0.0805\pm 0.0021$
$q_{1}$ | Limb darkening coefficient |  $\mathcal{U}$(0.2132, 0.4132)2 | $0.291_{-0.026}^{+0.028}$
$q_{2}$ | Limb darkening coefficient |  $\mathcal{U}$(0.2098, 0.4098)2 | $0.302_{-0.062}^{+0.069}$
$\log\sigma_{\mathrm{TESS}}$ [$\log$ relative flux] | Instrumental noise, TESS data |  $\mathcal{U}$(-15.0, 0.0) | $-6.8592\pm 0.0051$
$\log\sigma_{\mathrm{SOPHIE}}$ [$\log\mathrm{km~{}s^{-1}}$] | Instrumental noise, SOPHIE data |  $\mathcal{U}$(-15.0, 0.0) | $-9.2_{-3.8}^{+3.5}$
$\log\sigma_{\mathrm{MUSICOS}}$ [$\log\mathrm{km~{}s^{-1}}$] | Instrumental noise, MUSICOS data |  $\mathcal{U}$(-15.0, 0.0) | $-8.6\pm 4.3$
$\log S_{\mathrm{0,TESS}}$ | Scaled power, TESS SHO GP kernel |  $\mathcal{U}$(-30.0, 0.0) | $-16.08\pm 0.30$
$\log\omega_{\mathrm{0,TESS}}$ [$\log$ d-1] | Frequency, TESS SHO GP kernel |  $\mathcal{U}$(-15.0, 15.0) | $0.97_{-0.19}^{+0.21}$
$O_{\mathrm{SOPHIE}}$ [$\mathrm{km~{}s^{-1}}$] | Offset, SOPHIE data |  $\mathcal{U}$(-13.15, -12.80) | $-12.9308\pm 0.0038$
$S_{\mathrm{SOPHIE}}$ [$\mathrm{km~{}s^{-1}~{}T_{obs}^{-1}}$] | Linear slope, SOPHIE data |  $\mathcal{U}$(-0.2, 0.0) | $-0.1040\pm 0.0065$
$O_{\mathrm{MUSICOS}}$ [$\mathrm{km~{}s^{-1}}$] | Offset, MUSICOS data |  $\mathcal{U}$(-14.1, -13.4) | $-13.782\pm 0.018$
$S_{\mathrm{MUSICOS}}$ [$\mathrm{km~{}s^{-1}~{}T_{obs}^{-1}}$] | Linear slope, MUSICOS data |  $\mathcal{U}$(-0.2, 0.2) | $0.040\pm 0.029$
Table 5: An overview of the Allesfitter derived parameters of the exoplanet system XO-7 obtained from the TESS 2-min PDCSAP photometry and RV observations. Notes: †Mass ratio. ⋆Total transit duration between the 1st and the 4th contact. ⋆⋆Full transit duration between the 2nd and the 3rd contact. ⋄Assuming $T_{\mathrm{eff}}=6250\pm 100~{}\mathrm{K}$, $R_{\mathrm{s}}=1.480\pm 0.022~{}\mathrm{R}_{\odot}$ (Crouzet et al., 2020), $a=9.31\pm 0.16~{}\mathrm{R}_{\odot}$, the albedo of 0.3, and uniform heat redistribution. ‡Transit depth. Parameter [unit] | Value
---|---
$R_{\mathrm{s}}/a$ | $0.1590\pm 0.0012$
$a/R_{\mathrm{s}}$ | $6.289\pm 0.046$
$R_{\mathrm{p}}/a$ | $0.01486\pm 0.00014$
$R_{\mathrm{p}}$ [$\mathrm{R_{\oplus}}$] | $15.08\pm 0.23$
$R_{\mathrm{p}}$ [$\mathrm{R_{Jup}}$] | $1.346\pm 0.020$
$a$ [$\mathrm{R_{\odot}}$] | $9.31\pm 0.16$
$a$ [au] | $0.04329\pm 0.00072$
$i$ [deg] | $83.228\pm 0.098$
$q_{\mathrm{m}}^{\dagger}$ | $0.000493\pm 0.000015$
$M_{\mathrm{p}}$ [$\mathrm{M_{\oplus}}$] | $231\pm 12$
$M_{\mathrm{p}}$ [$\mathrm{M_{Jup}}$] | $0.726\pm 0.038$
$M_{\mathrm{p}}$ [$\mathrm{M_{\odot}}$] | $0.000693\pm 0.000036$
$b$ | $0.7416\pm 0.0056$
$T_{\mathrm{14}}^{\star}$ [h] | $2.8232\pm 0.0078$
$T_{\mathrm{23}}^{\star\star}$ [h] | $1.829\pm 0.016$
$\rho_{\mathrm{p}}$ [g cm-3] | $0.369_{-0.028}^{+0.030}$
$g_{\mathrm{p}}$ [m s-2] | $9.32\pm 0.29$
$T_{\mathrm{eq}}^{\diamond}$ [K] | $1612\pm 27$
$T_{\mathrm{d}}^{\ddagger}$ [relative flux] | $0.008810_{-0.000028}^{+0.000026}$
$u_{\mathrm{1}}$ | $0.326_{-0.059}^{+0.063}$
$u_{\mathrm{2}}$ | $0.213\pm 0.079$
$\rho_{\mathrm{s}}$ [g cm-3] | $0.574\pm 0.013$

##  4 Data analysis and results

###  4.1 Refined system parameters

Orbital and planetary parameters of the XO-7 system were obtained so far by
Crouzet et al. (2020) from ground-based multicolor photometric data and RV
observations, and by Maciejewski (2022) from three sectors TESS photometric
data alone. In order to derive refined orbital and planetary parameters of the
system, we combined all 2-min integrated TESS PDCSAP photometric data of XO-7
and RV observations. In this case, we used the 20 BF RV measurements described
in Sect. 2.2 and listed in Tab. 2 (these data have smaller uncertainties), and
the RV observations published by Crouzet et al. (2020). These RVs represent 49
observations obtained between $\mathrm{BJD_{TDB}}=2~{}457~{}593.3798$, which
corresponds to 21:08:02.39 UT on July 23, 2016, and
$\mathrm{BJD_{TDB}}=2~{}458~{}303.5649$, which is 01:34:55.27 UT on July 04,
2018, with the SOPHIE spectrograph (Perruchot et al., 2008) installed on the
1.93 m telescope of Observatoire de Haute-Provence (OHP) in France. The
timebase of these observations is about 710 days. The authors used its High-
Resolution (HR) mode, which gives a resolving power of $R=75~{}000$. The
exposure times were around 13 min allowing $S/N$ of around 27 per pixel at
5500 Å. The RV data were determined using the CCF technique. The authors opted
for a K0V stellar mask to cross-correlate the data. The list of the RV
observations with $\pm 1\sigma$ uncertainties can be found in the appendix of
Crouzet et al. (2020), in Tab. 5.

Figure 2: SOPHIE (top panels) and MUSICOS (bottom panels) RV observations of
XO-7, overplotted with the best-fitting Allesfitter model (20 curves from
random posterior samples). In the case of MUSICOS, we used BF RVs. Without
phase-folding (left-hand panels) to see the linear slope in the RV data and
with phase-folding (right-hand panels) to see the RV amplitude. Residuals are
also shown. During this modeling procedure, all available 2-min integrated
TESS PDCSAP data and the RV observations were fitted simultaneously. Figure 3:
Stacked and binned TESS transit light curve of XO-7b, overplotted with the
best-fitting Allesfitter model (20 curves from random posterior samples).
Residuals are also shown. During this modeling procedure, all available 2-min
integrated TESS PDCSAP data and the RV observations were fitted
simultaneously.

To derive new orbital and planetary parameters we employed the
Allesfitter444See https://www.allesfitter.com. software package (Günther &
Daylan, 2019, 2021). It is a public software for modeling photometric and RV
data. It can accommodate multiple exoplanets, multi-star systems, starspots,
stellar flares, TTVs, and various noise models. Allesfitter automatically runs
a nested sampling or Markov Chain Monte Carlo (MCMC) fit. For all this, it
constructs an inference framework that unites the versatile packages ellc
(Maxted, 2016), aflare (Davenport et al., 2014), dynesty (Speagle, 2020),
emcee (Foreman-Mackey et al., 2013) and celerite (Kallinger et al., 2014;
Foreman-Mackey et al., 2017; Barros et al., 2020). In order to construct the
combined TESS and RV model we opted for the nested sampling fit option with
initial settings. To speed up the computation process we cut the TESS light
curves and selected only the regions with a data window width of 0.4 d,
centered on the mid-transit points. Several parameters were optimized during
the fitting procedure, including the reference mid-transit time
$T_{\mathrm{c}}$, the orbital period $P_{\mathrm{orb}}$, the planet-to-star
radius ratio $R_{\mathrm{p}}/R_{\mathrm{s}}$, the scaled sum of fractional
radii $(R_{\mathrm{p}}+R_{\mathrm{s}})/a$, the cosine of the orbit inclination
angle ($\cos i$), and the RV semi-amplitude $K$. The quadratic limb darkening
(LD) law was applied during the fitting procedure. The $u_{1}$ and $u_{2}$ LD
coefficients were first linearly interpolated based on the stellar parameters
of $T_{\mathrm{eff}}=6250$ K and $\log g$ = 4.246 [cgs] (Crouzet et al., 2020)
from the tables of coefficients calculated for the TESS passband using the
PHOENIX-COND models by Claret (2018). We then converted these LD coefficients
to $q_{1}$ and $q_{2}$ (Kipping, 2013) and during the optimization procedure,
we applied $\pm 0.1$-wide uniform priors centered on these values to avoid the
non-physical solution of the light curve. Since trends due to stellar and
instrumental noise were still present in the PDCSAP photometry, in order to
model the TESS flux baseline we applied a Gaussian Process (GP) regression
method using the SHOTerm (Simple Harmonic Oscillator – SHO) plus JitterTerm
kernel, implemented in the celerite555See
https://celerite.readthedocs.io/en/stable/. package, with a fixed quality
factor of $Q_{0}=1/\sqrt{2}$, as it is common for quasi-periodic stellar
variability. The regression is done by using $\log\sigma$ (free), $\log Q_{0}$
(fixed), $\log\omega_{0}$ (free), and $\log S_{0}$ (free) hyperparameters with
bounds on the values of these parameters to be inputted by the user. The RV
baseline was modeled with a linear function motivated by the discoverers, who
reported on a linear slope in the discovery RV data with a value of
$S_{\mathrm{SOPHIE}}=-0.1480\pm 0.0010~{}\mathrm{m~{}s^{-1}~{}d^{-1}}$. The
instrumental noise in the RV data was sampled with the JitterTerm kernel,
using the $\log\sigma$ (free) hyperparameter. We assumed a circular orbit of
XO-7b. The list of fitted parameters is presented in Tab. 4 and the derived
parameters are listed in Tab. 5. The fitted RVs are depicted in Fig. 2 and the
stacked and binned TESS PDCSAP transit light curve of XO-7b, overplotted with
the best-fitting Allesfitter model is shown in Fig. 3. We can clearly see that
the previously observed significant linear slope in the SOPHIE RVs was not
confirmed with the follow-up RV data. The linear slope in the case of the
MUSICOS BF RV observations is only marginal. We discuss this interesting
result in Sect. 5.

XO-7b is a hot Jupiter on a close-in orbit with a semi-major axis of
$a=0.04329\pm 0.00072$ au and with a short orbital period below 10 days. We
see its orbit nearly edge-on with an inclination angle of $i=83.228\pm 0.098$
deg, which corresponds to an impact parameter of $b=0.7416\pm 0.0056$. The
total transit duration between the 1st and the 4th contact is
$T_{\mathrm{14}}=2.8232\pm 0.0078$ h. Using all available 2-min integrated
TESS PDCSAP data we significantly improved the linear ephemeris of the planet
compared to the previous values presented by Crouzet et al. (2020) and
Maciejewski (2022). The new linear ephemeris, obtained from the Allesfitter
combined model, is:

$\begin{split}T_{0}=T_{\mathrm{c}}+P_{\mathrm{orb}}\times E=\\\
&=2~{}458~{}779.58040\pm 0.00016~{}\mathrm{BJD_{TDB}}+\\\ &+2.86413296\pm
0.00000055~{}\mathrm{d}\times E,\end{split}$ (1)

where $T_{0}$ is the mid-transit time of an arbitrary transit and $E$ is the
epoch of observation, i.e., the number of the orbital cycle calculated from
the reference mid-transit time $T_{\mathrm{c}}$. XO-7b is an inflated planet
because it has a radius larger than the radius of Jupiter, but its mass is
less than the mass of Jupiter. We obtained a planet-to-star radius ratio,
which is $R_{\mathrm{p}}/R_{\mathrm{s}}=0.09344\pm 0.00028$. Using a stellar
radius of $R_{\mathrm{s}}=1.480\pm 0.022~{}\mathrm{R}_{\odot}$ (Crouzet et
al., 2020), we can get an absolute planet radius of $R_{\mathrm{p}}=1.346\pm
0.020~{}\mathrm{R_{Jup}}$. The mass of the planet was derived based on an RV
semi-amplitude, which is $K=0.0805\pm 0.0021~{}\mathrm{km~{}s^{-1}}$. This
gives $M_{\mathrm{p}}=0.726\pm 0.038~{}\mathrm{M_{Jup}}$. Using a stellar mass
of $M_{\mathrm{s}}=1.405\pm 0.059~{}\mathrm{M}_{\odot}$ (Crouzet et al., 2020)
we can get a mass ratio of $q_{\mathrm{m}}=0.000493\pm 0.000015$ and applying
the obtained planet mass and radius parameter values, we can get a planet
density of $\rho_{\mathrm{p}}=0.369_{-0.028}^{+0.030}$ g cm-3. This is only
about 28.5% of Jupiter’s density. Based on its mass and radius a very similar
planet is HAT-P-30b/WASP-51b (Johnson et al., 2011; Enoch et al., 2011).

###  4.2 TTV analysis

Table 6: The list of the O-C values of mid-transit times of XO-7b derived from the 2-min integrated TESS PDCSAP data using the Allesfitter combined model parameter values (see Tab. 4). The epoch $E$ was counted from $T_{\mathrm{c}}=2~{}458~{}779.58040~{}\mathrm{BJD_{TDB}}$. Transit | $E$ | O-C [d] |  $\pm 1\sigma$ [d]
---|---|---|---
No. 1 | 72 | $-0.00016$ | 0.00062
No. 2 | 73 | $+0.00042$ | 0.00058
No. 3 | 74 | $-0.00003$ | 0.00063
No. 4 | 75 | $+0.00027$ | 0.00062
No. 5 | 76 | $+0.00007$ | 0.00058
No. 6 | 77 | $+0.00026$ | 0.00064
No. 7 | 78 | $-0.00046$ | 0.00059
No. 8 | 79 | $+0.00023$ | 0.00061
No. 9 | 80 | $+0.00044$ | 0.00060
No. 10 | 81 | $-0.00129$ | 0.00060
No. 11 | 82 | $-0.00017$ | 0.00064
No. 12 | 83 | $-0.00044$ | 0.00060
No. 13 | 84 | $+0.00047$ | 0.00061
No. 14 | 86 | $-0.00094$ | 0.00063
No. 15 | 87 | $-0.00039$ | 0.00062
No. 16 | 88 | $-0.00110$ | 0.00067
No. 17 | 89 | $-0.00010$ | 0.00059
No. 18 | 214 | $+0.00004$ | 0.00049
No. 19 | 215 | $+0.00022$ | 0.00049
No. 20 | 216 | $+0.00156$ | 0.00048
No. 21 | 217 | $-0.00077$ | 0.00048
No. 22 | 218 | $+0.00029$ | 0.00048
No. 23 | 219 | $+0.00060$ | 0.00050
No. 24 | 220 | $-0.00080$ | 0.00050
No. 25 | 221 | $+0.00112$ | 0.00048
No. 26 | 222 | $+0.00052$ | 0.00050
No. 27 | 223 | $-0.00051$ | 0.00047
No. 28 | 280 | $-0.00018$ | 0.00049
No. 29 | 281 | $-0.00043$ | 0.00048
No. 30 | 282 | $-0.00021$ | 0.00047
No. 31 | 283 | $-0.00070$ | 0.00048
No. 32 | 285 | $-0.00034$ | 0.00049
No. 33 | 286 | $+0.00015$ | 0.00046
No. 34 | 287 | $-0.00029$ | 0.00049
No. 35 | 288 | $+0.00050$ | 0.00046
No. 36 | 328 | $-0.00024$ | 0.00049
No. 37 | 329 | $+0.00027$ | 0.00050
No. 38 | 330 | $+0.00035$ | 0.00051
No. 39 | 331 | $-0.00024$ | 0.00050
No. 40 | 333 | $-0.00011$ | 0.00049
No. 41 | 334 | $-0.00018$ | 0.00053
No. 42 | 335 | $-0.00074$ | 0.00051
No. 43 | 336 | $-0.00048$ | 0.00048
No. 44 | 337 | $+0.00024$ | 0.00053
No. 45 | 338 | $+0.00006$ | 0.00053
No. 46 | 339 | $-0.00131$ | 0.00058
No. 47 | 340 | $+0.00040$ | 0.00048
No. 48 | 342 | $+0.00082$ | 0.00055
No. 49 | 343 | $-0.00004$ | 0.00050
No. 50 | 344 | $+0.00071$ | 0.00054
No. 51 | 345 | $+0.00005$ | 0.00049
No. 52 | 396 | $-0.00124$ | 0.00048
No. 53 | 397 | $+0.00101$ | 0.00049
No. 54 | 398 | $-0.00059$ | 0.00045
No. 55 | 399 | $+0.00014$ | 0.00047
No. 56 | 401 | $-0.00066$ | 0.00049
No. 57 | 402 | $-0.00026$ | 0.00049
No. 58 | 403 | $-0.00091$ | 0.00048
No. 59 | 405 | $+0.00017$ | 0.00052
No. 60 | 406 | $-0.00031$ | 0.00050
No. 61 | 407 | $-0.00049$ | 0.00051
No. 62 | 408 | $-0.00032$ | 0.00049
No. 63 | 411 | $+0.00101$ | 0.00046
No. 64 | 412 | $+0.00031$ | 0.00050
No. 65 | 413 | $+0.00020$ | 0.00048
Figure 4: Observed-minus-calculated (O-C) diagram of XO-7b mid-transit times,
obtained based on 2-min integrated TESS PDCSAP data, overplotted with the
best-fitting OCFIT models. For more details see Sect. 4.2.

The first TTV analysis of XO-7b was performed by the discoverers. Crouzet et
al. (2020) found that TTVs of XO-7b should be lower than 5 min at $1\sigma$
and 15 min at $3\sigma$ over the two years of discovery observations. Based on
these photometric measurements Crouzet et al. (2020) could rule out the
presence of companions massive and close enough to induce significant TTVs on
shorter timescales. Later, Maciejewski (2022) analyzed TESS data from three
sectors, Nos. 25, 26, and 40, and found no additional planet in the XO-7
system down to sub-Neptune-sized planets. Since this analysis new TESS data
were obtained, we decided to repeat the TTV analysis with a much longer
timebase, which enabled us to focus on long-term dynamical signs of the
announced companion of XO-7b. This could be a sinusoidal modulation of the TTV
data, or rather a fraction of this sinusoid, which could be detectable as a
quadratic trend in the TTV data. For this purpose, we used the TESS data
described in Sect. 2.1. To perform the analysis we again employed the
Allesfitter software package described in Sect. 4.1. We used its TTV function
to fit the individual TESS transits of XO-7b. In this case, the transit
ephemeris is fixed. In addition, we keep fixed the transit shape and the noise
model. It means that during this procedure we fixed every parameter to its
best value from the Allesfitter combined model (see Tab. 4). Only the
observed-minus-calculated (O-C) parameters for individual mid-transit times
were fitted. We applied uniform priors on the O-C values,
$\mathcal{U}$(-0.025, 0.025) d, and again opted for the nested sampling fit
option with initial settings. To speed up the computation process we cut the
TESS light curves and selected only the regions with a data window width of
0.4 d, centered on the mid-transit points. We excluded partial transits from
the dataset. It means that we used 65 TESS transits of XO-7b in total,
covering a timebase of about 977 days. The calculated mid-transit time of the
transit No. 1 in our dataset is $\mathrm{BJD_{TDB}}=2~{}458~{}985.79797$, and
the last transit No. 65 has the calculated mid-transit time of
$\mathrm{BJD_{TDB}}=2~{}459~{}962.46731$. The list of the fitted O-C values of
mid-transit times obtained from this modeling procedure is presented in Tab.
6.

Figure 5: SOPHIE RV observations and MUSICOS BF RVs of XO-7, fitted jointly
with a linear model (left-hand panel). SOPHIE RV observations and MUSICOS CCF
RVs of XO-7, fitted jointly with a quadratic model (right-hand panel). For the
corresponding discussions see the text in Sect. 5.

The O-C (TTV) data were subsequently inputted in the OCFIT666See
https://github.com/pavolgaj/OCFit. code, version 0.2.1 (Gajdoš & Parimucha,
2019), and tested for any quadratic trends in the dataset, which could be an
indicator of the announced massive wide-orbit companion. We first fitted the
O-C data with a linear function using the OCFIT package FitLinear. The free
parameters of the linear model are the reference mid-transit time
$T_{\mathrm{c}}$ and the orbital period $P_{\mathrm{orb}}$. Subsequently, the
O-C data were fitted with a quadratic function, within the OCFIT package
FitQuad. The free parameters of the quadratic model are the reference mid-
transit time $T_{\mathrm{c}}$, the orbital period $P_{\mathrm{orb}}$, and the
quadratic coefficient $Q$, which follows from the quadratic ephemeris formula
of:

$T_{0}=T_{\mathrm{c}}+P_{\mathrm{orb}}\times E+Q\times E^{2},$ (2)

where the quadratic coefficient can be expressed as:

$Q=\frac{1}{2}P_{\mathrm{orb}}\times\dot{P},$ (3)

where $\dot{P}$ means the orbital period change with time $t$, i.e. this is
the so-called orbital period change rate: $\dot{P}=\mathrm{d}P/\mathrm{d}t$.
$\dot{P}$ is a dimensionless quantity, but it can be expressed, e.g., in d
yr-1. During the OCFIT fitting procedure we opted for the robust regression
method, performing 100 000 iteration steps. The uncertainties in the fitted
parameters of $P_{\mathrm{orb}}$, $T_{\mathrm{c}}$, and $Q$ were derived
within the OCFIT packages FitLinear and FitQuad, applying the covariance
matrix method. We measured the quality of the fit using the Bayesian
Information Criterion ($BIC$), calculated automatically by the software. The
O-C values of XO-7b mid-transit times, overplotted with the best-fitting OCFIT
models are depicted in Fig. 4.

We can see that the O-C data of mid-transit times do not show any periodic
features. To check the possible long-term trends, i.e., a fraction of a
sinusoid in the O-C data, we evaluated the OCFIT models as follows. Based on
the linear fit, we obtained a linear ephemeris of:

$\begin{split}T_{0}=T_{\mathrm{c}}+P_{\mathrm{orb}}\times E=\\\
&=2~{}458~{}779.58034\pm 0.00022~{}\mathrm{BJD_{TDB}}+\\\ &+2.86413292\pm
0.00000075~{}\mathrm{d}\times E,\end{split}$ (4)

and a $BIC$ value of 83.6. The quadratic fit resulted in a quadratic ephemeris
of:

$\begin{split}T_{0}=T_{\mathrm{c}}+P_{\mathrm{orb}}\times E+Q\times E^{2}=\\\
&=2~{}458~{}779.58011\pm 0.00040~{}\mathrm{BJD_{TDB}}+\\\ &+2.8641354\pm
0.0000037~{}\mathrm{d}\times E+Q\times E^{2},\end{split}$ (5)

where the quadratic coefficient is $Q=(-5.29\pm 7.68)\times 10^{-9}$ d. We
obtained a dimensionless value of $\dot{P}=(-3.69\pm 5.36)\times 10^{-9}$,
which is equal to $\dot{P}=(-1.34\pm 1.96)\times
10^{-6}~{}\mathrm{d~{}yr^{-1}}$. Since this result is not significant, the
quadratic ephemeris, expressed in Eq. 5, is not justified. This conclusion is
also confirmed by a $BIC$ value of 95.04, which is about 13% larger than in
the case of the linear fit. In summary, we did not find any convincing long-
term dynamical sign of the announced wide-orbit massive companion of XO-7b in
the O-C data. We did not improve the linear ephemeris by this procedure in Eq.
4, therefore we can adopt the linear ephemeris from the Allesfitter combined
model, expressed in Eq. 1, as the final ephemeris solution.

##  5 Discussion

The fitted and derived Allesfitter combined model parameter values, presented
in Tabs. 4 and 5, respectively, are in general in a $3\sigma$ agreement to
those of presented by the discoverers. Several parameter values were also
improved in comparison with these authors. However, very interesting is the
result, which we obtained in the case of the RV slope, announced by Crouzet et
al. (2020). In agreement with the discovery paper, in the case of the SOPHIE
RV data, we obtained a very significant ($\sim 16\sigma$) linear slope, which
is quantified by the parameter of $S_{\mathrm{SOPHIE}}=-0.1040\pm
0.0065~{}\mathrm{km~{}s^{-1}~{}T_{obs}^{-1}}$, which means that the change in
the RVs is expressed during the timebase of observations. We can easily
convert this parameter to the value of $S_{\mathrm{SOPHIE}}=-0.1464\pm
0.0092~{}\mathrm{m~{}s^{-1}~{}d^{-1}}$ if we consider that the time difference
between the first and the last SOPHIE RV observation is about 710 days. This
parameter value is in perfect agreement with the value presented by Crouzet et
al. (2020), which is $S_{\mathrm{SOPHIE}}=-0.1480\pm
0.0010~{}\mathrm{m~{}s^{-1}~{}d^{-1}}$. Following this discovery value and
calculating with a MUSICOS RVs timebase of 1225 days, we expected a linear
slope of
$S_{\mathrm{MUSICOS}}\approx-0.1813~{}\mathrm{km~{}s^{-1}~{}T_{obs}^{-1}}$.
However, in the case of the MUSICOS BF RVs, we could not confirm this
expectation. In this case, we obtained a linear slope, which is expressed with
a value of $S_{\mathrm{MUSICOS}}=0.040\pm
0.029~{}\mathrm{km~{}s^{-1}~{}T_{obs}^{-1}}$, which is equal to
$S_{\mathrm{MUSICOS}}=0.032\pm 0.024~{}\mathrm{m~{}s^{-1}~{}d^{-1}}$. This
value is far from the expected, therefore we can say that the previously
observed significant linear slope in the SOPHIE RVs was not confirmed with the
follow-up RV data. We detected only a marginal ($\sim 1.3\sigma$) linear slope
in the MUSICOS BF RV data. Moreover, this slope has the opposite trend
compared to the SOPHIE RV slope. Based on this result the following
explanations are plausible.

The first possibility is that there is no linear slope in the RVs. The
significant linear slope, observed in the SOPHIE RVs, was caused by a
systematic error. This produced an artificial slope in the SOPHIE RV
observations. Although technically it is possible, we do not strongly believe
that this is the true reason for the marginal detection in the MUSICOS BF RV
data. Secondly, we also should take into consideration that the uncertainties
of the MUSICOS BF RV data are in general larger compared to the SOPHIE RV
observations. On the other hand, the linear RV slope should be increased
linearly with time, so this disadvantage is compensated with a longer timebase
of MUSICOS RV data. The expected linear slope based on the SOPHIE RV data was
$S_{\mathrm{MUSICOS}}\approx-0.1813~{}\mathrm{km~{}s^{-1}~{}T_{obs}^{-1}}$, as
we presented above. This slope should be detectable with MUSICOS BF RV data,
despite the larger uncertainties. Finally, we should also consider the fact
that there is a velocity offset between the SOPHIE and MUSICOS BF RV
observations, which we can express using the absolute difference in the offset
parameter values, i.e.,
$|O_{\mathrm{SOPHIE}}-O_{\mathrm{MUSICOS}}|\approx|12.9-13.7|\approx
0.8~{}\mathrm{km~{}s^{-1}}$. At first glance, this velocity offset could
indicate, for example, that the linear slope is still continuing downward, as
it was presented by the discoverers (see Fig. 5, left-hand panel). However,
the velocity offset is only artificial due to the different data reduction
procedures. The SOPHIE RVs were extracted using a K0V stellar mask via the CCF
method. The MUSICOS BF RVs were obtained using an F7V-type template star. The
offset can result from an incorrect RV of the template star, HD 222368. In
fact, the Simbad database777See https://simbad.u-strasbg.fr/simbad/. gives its
RV from 5.00 to 5.98 km s-1 from measurements since 2002 (we used a value of
5.4 km s-1). Therefore, in this case, we can tentatively use the MUSICOS CCF
RV data (see Sect. 2.2 and Tab. 2), which were extracted using a K0V stellar
mask (with laboratory wavelength), as well (these data have larger
uncertainties). This situation is depicted in Fig. 5 (right-hand panel). In
this case, there is a possibility that both RV datasets were collected near
the quadrature position of the announced wide-orbit massive companion, where
the RVs are nearly stable. This explanation seems to be the most convincing
for us. It is also in agreement with the marginal linear slope detected in the
MUSICOS BF RVs.

The quadratic model depicted in Fig. 5 (right-hand panel) may be used to infer
and discuss some preliminary physical parameters of the announced companion of
XO-7b. To simplify the situation, we will assume that the companion has a
circular orbit. The quadratic model $RV_{\mathrm{quad}}(t)$ in time $t$ can be
described based on Kipping et al. (2011) as:

$\begin{split}RV_{\mathrm{quad}}(t)=O_{\mathrm{SOPHIE}}+d(t-\bar{t})+\frac{1}{2}q(t-\bar{t})^{2},\\\
&RV_{\mathrm{quad}}(t)=O_{\mathrm{MUSICOS}}+d(t-\bar{t})+\frac{1}{2}q(t-\bar{t})^{2},\end{split}$
(6)

where $O_{\mathrm{SOPHIE}}$ is the offset of the SOPHIE RV data,
$O_{\mathrm{MUSICOS}}$ is the offset of the MUSICOS CCF RV data, $d$ is the
linear acceleration coefficient, $q$ is the quadratic acceleration coefficient
(these coefficients are shared in the model between the two datasets), and
$\bar{t}$ is the mean epoch of the merged timebase of SOPHIE and MUSICOS RV
data, which is $\bar{t}=2~{}458~{}441.372~{}\mathrm{BJD_{TDB}}$. The fitted
parameters of the model, as well as the corresponding asymptotic standard
errors, are presented in Tab. 7. Furthermore, we calculated the minimum RV
semi-amplitude $K_{\mathrm{min,3}}$ induced by the companion, as the absolute
difference between the minimum and maximum RV values of the quadratic-fit
curve during the merged timebase of RV observations divided by 2, and we
obtained a value of
$K_{\mathrm{min,3}}=(|RV_{\mathrm{min}}-RV_{\mathrm{max}}|)/2=(|13.1610-12.8880|)/2=0.2730/2=0.1365~{}\mathrm{km~{}s^{-1}}$.
The precision of this measurement is $\pm 0.0005~{}\mathrm{km~{}s^{-1}}$. In
the next step, the minimum orbital period $P_{\mathrm{orb,min,3}}$ of the
announced companion of XO-7b was calculated following Kipping et al. (2011),
and Pinamonti et al. (2022) as a function of the quadratic acceleration
coefficient $q$:

$P_{\mathrm{orb,min,3}}\gtrsim 2\pi\sqrt{\frac{2K_{\mathrm{min,3}}}{q}},$ (7)

and we obtained $7900\pm 1660$ days, where the uncertainty follows mainly from
the uncertainty of $q$. This result seems to be realistic if we take a look at
Fig. 5 (right-hand panel), and if we consider that the time difference between
the first SOPHIE RV observation and the last MUSICOS RV observation is about
2560 days. Last, but not least, from $P_{\mathrm{orb,min,3}}$ and
$K_{\mathrm{min,3}}$, the corresponding ’minimum’ minimum mass $(M_{3}\sin
i)_{\mathrm{min}}$ of the companion can be derived, e.g., by modifying the
formula presented by Torres et al. (2008):

$(M_{3}\sin i)_{\mathrm{min}}=4.919\times
10^{-3}~{}K_{\mathrm{min,3}}~{}(P_{\mathrm{orb,min,3}})^{1/3}~{}(M_{\mathrm{s}})^{2/3},$
(8)

where $K_{\mathrm{min,3}}$ is in $\mathrm{m~{}s^{-1}}$,
$P_{\mathrm{orb,min,3}}$ is in days, $M_{\mathrm{s}}$ is in
$\mathrm{M_{\odot}}$, and the result is expressed in $\mathrm{M_{Jup}}$. In
Eq. 8 we assumed that $M_{3}\ll M_{\mathrm{s}}$ and that the announced
companion has a circular orbit. Using the appropriate parameter values from
Tabs. 3 and 7, we obtained a ’minimum’ minimum mass of $(M_{3}\sin
i)_{\mathrm{min}}=16.7\pm 3.5~{}\mathrm{M_{Jup}}$. Here, the uncertainty is
propagated mainly from the uncertainty of $P_{\mathrm{orb,min,3}}$. This
result means that the companion, if really exists, could be a brown dwarf or a
low-mass star. Compared to the minimum-mass parameter value of
$4~{}\mathrm{M_{Jup}}$, derived by Crouzet et al. (2020), the new value from
this work is higher, and within the $1\sigma$ uncertainty, it excludes the
possibility that the announced companion could have a planetary nature, see
for example Rebolo et al. (1996) and Burrows et al. (1997).

Table 7: An overview of the fitted and derived parameters of the announced companion of XO-7b obtained from the quadratic RV model depicted in Fig. 5 (right-hand panel). For more details see the text in Sect. 5. Parameter [unit] | Value | Comments
---|---|---
$O_{\mathrm{SOPHIE}}$ [$\mathrm{km~{}s^{-1}}$] | $-13.0823\pm 0.0089$ | Fitted value
$O_{\mathrm{MUSICOS}}$ [$\mathrm{km~{}s^{-1}}$] | $-13.096\pm 0.020$ | Fitted value
$d$ [$\mathrm{km~{}s^{-1}~{}d^{-1}}$] | $-0.000161\pm 0.000018$ | Fitted value
$q$ [$\mathrm{km~{}s^{-1}~{}d^{-2}}$] | $0.000000172\pm 0.000000036$ | Fitted value
$K_{\mathrm{min,3}}$ [$\mathrm{km~{}s^{-1}}$] | $0.1365\pm 0.0005$ | Derived value
$P_{\mathrm{orb,min,3}}$ [d] | $7900\pm 1660$ | Derived value
$(M_{3}\sin i)_{\mathrm{min}}$ [$\mathrm{M_{Jup}}$] | $16.7\pm 3.5$ | Derived value

Finally, we discuss the Gaia Data Release No. 3 (DR3) Renormalized Unit Weight
Error (RUWE, $\rho_{\mathrm{R}}$) value for XO-7 in terms of viability of the
announced massive companion. The RUWE value allows distinguishing between
’good’ and ’bad’ astrometric solutions, where $\rho_{\mathrm{R}}\sim 1.4$ is
the threshold between these solutions, and $\rho_{\mathrm{R}}\sim 1.0$
characterizes well-behaved single-star solutions (Lindegren et al., 2018,
2021). The Gaia DR3 RUWE value for XO-7 is $\rho_{\mathrm{R}}=1.116$ (Gaia
Collaboration et al., 2023), which is relatively low and rather corresponds to
a single-star solution. However, Belokurov et al. (2020) conducted a detailed
study of RUWE relation to unresolved binary systems in Gaia Data Release No. 2
(DR2), selected a sample of 411 Jupiter hosts, and found that the peak of the
RUWE distribution for the high-mass ($M_{\mathrm{p}}>1~{}\mathrm{M_{Jup}}$)
hot Jupiters ($P_{\mathrm{orb}}<15$ d) is shifted to a value higher than
$\rho_{\mathrm{R}}=1$. In this group of Jupiters, the peak of the RUWE
distribution is at $\rho_{\mathrm{R}}\sim 1.07$, and the authors concluded
that this deviation is not induced by the known hot Jupiters, but the simplest
interpretation is that these systems harbour an additional distant stellar or
substellar component, which causes the photocentre to wobble. Although XO-7b
does not fit $M_{\mathrm{p}}>1~{}\mathrm{M_{Jup}}$, from this viewpoint, the
Gaia DR3 RUWE value for XO-7 seems to be not so low, and
$\rho_{\mathrm{R}}=1.116$ could indicate a distant companion present in the
system. In order to shed more light on this, we converted the XO-7 RUWE value
to the amplitude of the angular perturbation $\delta\theta$ (Belokurov et al.,
2020):

$\delta\theta\approx\sigma_{\mathrm{AL}}(G)~{}\sqrt{\rho_{\mathrm{R}}^{2}-1},$
(9)

where the $\sigma_{\mathrm{AL}}$ as a function of source magnitude $G$ is
presented in Lindegren et al. (2018) as blue curve in their Fig. 9. Using
$\sigma_{\mathrm{AL}}=0.4$ mas in Eq. 9, we obtained $\delta\theta\approx
0.198$ mas. Taking the distance dependence of the centroid wobble into
account, the corresponding physical displacement in au is (Belokurov et al.,
2020):

$\delta a=\delta\theta~{}D,$ (10)

where $D$ is the distance to the source in kpc calculated as the inverse of
parallax $p$ in mas. Using $p=4.321\pm 0.013$ mas for XO-7 from the Gaia DR3
database (Gaia Collaboration et al., 2023), thus $D=0.2314\pm 0.0007$ kpc in
Eq. 10, we got $\delta a\approx 0.0458$ au. If we take into account that
$P_{\mathrm{orb,min,3}}\gtrsim 7900\pm 1660$ d and $(M_{3}\sin
i)_{\mathrm{min}}=16.7\pm 3.5~{}M_{\mathrm{Jup}}$, we can calculate the
minimum semi-amplitude $a_{\mathrm{min}}$ of the star – companion subsystem,
based on Kepler’s third law, as:

$a_{\mathrm{min}}=[P_{\mathrm{orb,min,3}}^{2}(M_{\mathrm{s}}+(M_{3}\sin
i)_{\mathrm{min}})]^{\frac{1}{3}}.$ (11)

In Eq. 11 the orbital period is in years, the masses are in
$\mathrm{M_{\odot}}$, $a_{\mathrm{min}}$ is expressed in au, we neglected
XO-7b, and assumed a circular orbit. The result of this calculation, together
with the uncertainty dictated mainly by the uncertainty of
$P_{\mathrm{orb,min,3}}$, is $a_{\mathrm{min}}=8.7\pm 1.8$ au. If the
companion is a brown dwarf or a low-mass star, we can assume that it is
significantly fainter compared to XO-7, and therefore the perturbation should
be due to the host star alone, and the photocenter should be the star itself.
In this case, the minimum semi-amplitude of the photocentre perturbation
$a_{\mathrm{min,1}}$ is:

$a_{\mathrm{min,1}}=\frac{a_{\mathrm{min}}(M_{3}\sin
i)_{\mathrm{min}}}{M_{\mathrm{s}}+(M_{3}\sin i)_{\mathrm{min}}}.$ (12)

Based on the Eq. 12 we obtained $a_{\mathrm{min,1}}=0.102\pm 0.021$ au. This
means that the amplitude of the photocentre perturbation should be $\delta
a\gtrsim 0.204$ au, and not $\delta a\approx 0.0458$ au, as we calculated
above. There are several possible explanations for this discrepancy. The first
explanation is that there is no additional companion present in the system and
the RUWE value ($\rho_{\mathrm{R}}=1.116$) corresponds to a single-star
solution. The second possibility is that the announced companion is a more
massive stellar object and the system does not show any extra perturbation
because the photocentre coincides with the centre of mass. However, this
possibility is less probable, because a massive star should be visible in the
spectra of XO-7. Finally, there is one more possibility, which we can mention
in agreement with Belokurov et al. (2020), namely that the minimum orbital
period of the companion ($P_{\mathrm{orb,min,3}}\gtrsim 7900\pm 1660$ d) is
much longer than the timebase of Gaia individual data releases (e.g., 1038
days in the case of Early DR3, see Lindegren et al. (2021)) and the
photocentre perturbation will become quasi-linear and thus will be absorbed
into the proper motion.

##  6 Conclusions

We analyzed the XO-7 exoplanet system with the main scientific goal to follow-
up the linear RV slope reported by the discoverers, and to put constraints on
the orbital period of the announced wide-orbit massive companion of XO-7b.
Furthermore, we aimed at refining the system parameters and we wanted to probe
the TTVs of XO-7b in order to search for long-term dynamical signs of the
companion of XO-7b in the O-C data of mid-transit times. To fulfill these aims
we analyzed the 2-min integrated TESS PDCSAP data from 8 sectors and performed
a long-term RV monitoring of the planetary system. Moreover, in our analysis,
we used the discovery RVs, as well.

Our most interesting result is that the previously observed significant linear
slope, $S_{\mathrm{SOPHIE}}=-0.1040\pm
0.0065~{}\mathrm{km~{}s^{-1}~{}T_{obs}^{-1}}$, in the RVs was not confirmed
with the follow-up RV data. We detected only a marginal linear slope,
$S_{\mathrm{MUSICOS}}=0.040\pm 0.029~{}\mathrm{km~{}s^{-1}~{}T_{obs}^{-1}}$,
in the newly obtained RV data, which has the opposite trend compared to the
linear slope reported by the discoverers. It means that we did not find
significant clues about the companion of XO-7b in the new RVs. We discussed
three possibilities, what could be causing this discrepancy. We can conclude
that if the announced companion really exists, the most convincing explanation
is that both RV datasets were collected near its quadrature position. In this
case, more RVs are needed to put better constraints on the orbital period of
the companion. Based on the SOPHIE and MUSICOS RV observations of XO-7 we were
able to estimate its lower limit, which is $P_{\mathrm{orb,min,3}}\gtrsim
7900\pm 1660$ d. We did not find significant evidence of the companion of
XO-7b in the O-C dataset of mid-transit times. We can again conclude that if
the announced companion really exists, this is in agreement with previous
results that distant companions of exoplanets are only known by RV solutions.

Furthermore, we used the quadratic RV solution of the merged SOPHIE and
MUSICOS data to infer and discuss some preliminary physical parameters of the
announced companion of XO-7b, mainly its mass. From this viewpoint, we can
conclude that if the announced companion really exists, it could be a brown
dwarf or a low-mass star with a ’minimum’ minimum mass of $(M_{3}\sin
i)_{\mathrm{min}}=16.7\pm 3.5~{}\mathrm{M_{Jup}}$. We also discussed the Gaia
DR3 RUWE value for XO-7 in terms of viability of the companion. On the one
hand, we can conclude that the RUWE value, $\rho_{\mathrm{R}}=1.116$, rather
corresponds to a single-star solution. On the other hand, there is a
possibility that the astrometric perturbation effect of the companion is
absorbed into the proper motion of the host star due to the orbital period,
which is much longer than the timebase of Gaia individual data releases.

Finally, we obtained precise orbital and planetary parameters of the system
via a combination of the TESS data and RV observations of XO-7. Precise
parameters of XO-7b, mainly the planet’s mass and radius, will be important
for further analyses. Mass determination can help constrain the internal
structure of the planet, and break degeneracies in atmospheric
characterization follow-up studies. Precise planet radius allows the planetary
composition to be estimated. We also derived the fundamental parameters of the
planet’s host star. These parameter values are in a $3\sigma$ agreement to
those presented by the discoverers, except for the projected rotational
velocity, which we can explain with different spectral resolutions of the
spectrographs.

## Acknowledgements

The authors thank P. Sivanič for the observations and technical assistance.
This work was supported by the Hungarian National Research, Development and
Innovation Office (NKFIH) grant K-125015, the PRODEX Experiment Agreement No.
4000137122 between the ELTE University and the European Space Agency
(ESA-D/SCI-LE-2021-0025), the City of Szombathely under agreement No.
67.177-21/2016, and by the VEGA grant of the Slovak Academy of Sciences No.
2/0031/22. TP acknowledges support from the Slovak Research and Development
Agency – contract No. APVV-20-0148. This paper includes data collected with
the TESS mission, obtained from the MAST data archive at the Space Telescope
Science Institute (STScI). Funding for the TESS mission is provided by the
NASA Explorer Program. STScI is operated by the Association of Universities
for Research in Astronomy, Inc., under NASA contract NAS 5-26555. This work
has made use of data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular, the institutions
participating in the Gaia Multilateral Agreement.

## Data availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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