11institutetext: Instituto Nacional de Astrofísica, Óptica y Electrónica,
Apartado Postal 51 y 216, 72000, Puebla, Mexico. 11email<EMAIL_ADDRESS>22institutetext: Max Planck Institut für Radioastronomie, Auf dem Hügel 69,
D-53121 Bonn, Germany. 33institutetext: Institut de radioastronomie
millimétrique, 300 rue de la piscine, 38406 Saint Martin d’Hères, France.
44institutetext: Instituto de Radioastronomía y Astrofísica, Universidad
Nacional Autónoma de México, Morelia, 58089, Mexico. 55institutetext:
Instituto de Astronomía, Universidad Nacional Autónoma de México, Apdo Postal
70-264, Ciudad de México, Mexico. 66institutetext: MIT Haystack Observatory,
99 Millstone Road, Westford, MA 01886, USA 77institutetext: National Radio
Astronomy Observatory (NRAO), 520 Edgemont Road, Charlottesville, VA 22903,
USA

# The distance to the Serpens South cluster from H2O masers

Gisela N. Ortiz-León 1122 Sergio A. Dzib 33 2 2 Laurent Loinard 4455 Yan Gong
22 Thushara Pillai 66 Adele Plunkett 77

In this Letter we report Very Long Baseline Array observations of 22 GHz water
masers toward the protostar CARMA–6, located at the center of the Serpens
South young cluster. From the astrometric fits to maser spots, we derive a
distance of $440.7\pm$3.5 pc for the protostar ($1\%$ error). This represents
the best direct distance determination obtained so far for an object this
young and deeply embedded in this highly obscured region. Taking depth effects
into account, we obtain a distance to the cluster of $440.7\pm 4.6$ pc. Stars
visible in the optical that have astrometric solutions in the Gaia Data
Release 3 are, on the other hand, all located at the periphery of the cluster.
Their mean distance of $437^{+51}_{-41}$ pc is consistent within $1\sigma$
with the value derived from maser astrometry. As the maser source is at the
center of Serpens South, we finally solve the ambiguity of the distance to
this region that has prevailed over the years.

###### Key Words.:

astrometry – water masers – star forming regions – Serpens South – stars: low-
mass – techniques: interferometric

##  1 Introduction

The Serpens South region hosts a rich protostellar cluster that was discovered
by Gutermuth et al. (2008) from Spitzer imaging. The cluster is embedded in a
massive molecular filament (André et al. 2010; Tanaka et al. 2013) and
contains approximately 400 young members (see the recent work by Sun et al.
2022), of which very few are visible in the optical (Herczeg et al. 2019).
Since its discovery, this protocluster has become a favorite target for
observations of young stars in the earliest phases of their development (e.g.,
Maury et al. 2011) and studies of related phenomena, such as molecular gas
kinematics (e.g., Fernández-López et al. 2014; Friesen et al. 2016; Dhabal et
al. 2018), protostellar outflows (Plunkett et al. 2015), and magnetic fields
(Pillai et al. 2020). Serpens South is the most active region, in terms of
star formation, projected in the direction of a larger complex of molecular
clouds known as the Aquila Rift (or the Serpens-Aquila Rift). In the vicinity
of Serpens South ($\sim 20^{\prime}$ to the east) resides the W40 region
(Westerhout 1958), which hosts an HII region, hundreds of candidate young
stars (e.g., Povich et al. 2013; Sun et al. 2022), and several OB stars
(Shuping et al. 2012). The well-studied Serpens Main molecular cloud is
located about 3 degrees north of Serpens South and is also an active star-
forming region in the Serpens-Aquila Rift (see the review by Eiroa et al.
2008).

The distance to the Serpens South cluster has been debated over the years.
Early works assumed a distance of 260 pc, the estimated distance to the front
clouds in the line of sight toward the Aquila Rift (Straižys et al. 2003, see
also the discussion by Dzib et al. 2010). However, later measurements of the
distance to star-forming regions within the Aquila Rift, including Serpens
South, are in the range $\approx 380-480$ pc (Dzib et al. 2010; Ortiz-León et
al. 2017; Herczeg et al. 2019; Zucker et al. 2019). For instance, Herczeg et
al. (2019) reported distances to subclusters and dark clouds in the Aquila
Rift based on Gaia Data Release 2 (DR2) data and star counts. These authors
found distances of 438$\pm$11, 478$\pm$6, 383$\pm$2, and 407$\pm$16 pc for
Serpens Main, Serpens Northeast, Serpens Far-South, and LDN 673, respectively.
In addition, they reported larger distances of $\sim$500-700 pc for the dark
clouds (which they refer to as the Aquila Rift) located northeast and
southwest of the active star-forming regions.

More recently, Anderson et al. (2022) used Gaia Early Data Release 3 (EDR3)
data from the Serpens South and W40 regions and arrived at a mean distance of
$455\pm 50$ pc for this region. We note that this is currently the most direct
estimation of the distance to the Serpens South cluster, as it is directly
derived from the parallaxes. However, the error bar is large and reflects the
dispersion on the parallax measurements, as these regions are highly obscured
at the Gaia bands. On the other hand, Ortiz-León et al. (2018) measured Very
Long Baseline Array (VLBA) parallaxes of eight radio continuum sources in the
star-forming regions Serpens Main and W40, both within the Aquila Rift,
resulting in a mean distance of 436.0$\pm$9.2 pc.

It is important to note that, using the VLBA, Ortiz-León et al. (2017) and
Ortiz-León et al. (2018) did not detect any radio continuum sources associated
with the Serpens South region. However, based on Gaia DR2 parallaxes to two
stars in the outskirts of the Serpens South region and their similar values to
the VLBA parallaxes of Serpens Main and W40 stars, Ortiz-León et al. (2018)
conclude that the three regions reside at the same distance. On the other
hand, estimations based on studies of the X-ray luminosity function of Serpens
South have placed this region at the significantly closer distance of 260 pc
(Winston et al. 2018). Following their indirect distance estimation, Winston
et al. (2018) suggest that Serpens South is at the front edge of the Aquila
Rift complex, closer than W40 and Serpens Main and not physically associated
with them. Other authors have also assumed a closer distance (e.g., $250$ pc
by Maury et al. 2019 and $350$ pc by Galametz et al. 2019 and Podio et al.
2021). As there are no direct distance determinations to the youngest stars in
this region, these results did not resolve the debate regarding the distance
to the Serpens South cluster. However, using incorrect distances affects the
estimates of envelope and disk masses, as well as of the outflow properties of
their stellar members. For instance, using a distance of 436 pc would increase
internal luminosities and envelope masses reported in Maury et al. (2011) by a
factor of 3.

Recently, Ortiz-León et al. (2021) discovered 22 GHz water maser emission from
two protostars in the Serpens South cluster, CARMA–6 and CARMA–7. These masers
offer an opportunity to attempt an independent and direct measurement of the
distance to Serpens South. In this Letter we present the results from VLBA
follow-up observations of the 22 GHz water masers associated with CARMA–6,
which we use to derive the distance to the Serpens South cluster. Section 2
describes the observations and data calibration. In Sect. 3 we present the
astrometric fits. Finally, our results are discussed in Sect. 4.

##  2 Observations and data reduction

We observed CARMA–6 with the VLBA during a total of 12 epochs between
September 2020 and September 2021 (Table 2). Two epochs were reported in
Ortiz-León et al. (2021), while the rest are analyzed here for the first time.
The phase center was at position $\alpha$(J2000) = 18:30:03.538,
$\delta$(J2000) = –02:03:08.377. The data were taken at 22.2 GHz with four
intermediate frequency bands of 16 MHz bandwidth each. The third band was
centered at the $6_{1,6}-5_{2,3}$ H2O transition (rest frequency 22,235.080
MHz) and correlated with a spectral resolution of 15.625 kHz, corresponding to
a resolution of $\sim$0.2 km s-1 in velocity.

We performed fast-switching observations between our target and the phase
reference calibrator J1824+0119 ($\alpha$(J2000) = 18:24:48.143436,
$\delta$(J2000) = +01:19:34.20183), switching sources every $\approx$30
seconds. Additional 30-minute blocks of calibrators distributed over a wide
range of elevations were observed at 23.7 GHz every $\approx$2 hours.

The data were calibrated with the Astronomical Imaging System (AIPS; Greisen
2003), using the ParselTongue scripting interface (Kettenis et al. 2006) and
following standard procedures for phase-referencing observations (e.g., Reid
et al. 2009). As the masers were variable in both flux density and velocity
(see Sect. 3.1), we opted to derive the fringe-fitting solutions from the
scans on the phase-reference calibrator and then applied the solutions to the
target. The calibrated data were imaged in individual channels using
$4096\times 4096$ pixels and a pixel size of $50\leavevmode\nobreak\ \mu$as.
The sensitivities and beam sizes of the resulting data cubes are given in
Table 2. Spot positions and fluxes were determined by fitting a Gaussian to
the brightness distribution at individual channels using the AIPS task jmfit.
We note that the expected statistical positional errors are on the order of 70
$\mu$as for emission detected at S/N=5.

##  3 Results

###  3.1 Maser spots

Figure 1: Averaged VLBA spectra of the water maser emission observed toward
CARMA–6. They were obtained by integrating over the area that covers all
detected spots, and their epochs are indicated in the legends. The vertical
dashed line at $V_{\rm LSR}$=8 km s-1 marks the systemic velocity of the
cloud.

Spectra of water emission from CARMA–6 are shown in Fig. 1, where the $x$-axis
represents the radial velocity with respect to the local standard of rest
(LSR), $V_{\rm LSR}$. The spectra display three main velocity components, at
$V_{\rm LSR}\approx 8.5$, 11, and 13 km s-1, that change over time and are
redshifted with respect to the systemic velocity of the cloud, $V_{\rm
sys}\approx 8$ km s-1 (Kirk et al. 2013; Plunkett et al. 2015). The 8.5 km s-1
component disappeared after our observation in October 2020, while the other
two components became brighter than before. Overall, the intensity of the
emission starts declining after the maximum reached in September 2021.

Figures 2 and 5 show the distribution of the spot positions measured with
jmfit at individual velocity channels during all observed epochs. At
velocities from $V_{\rm LSR}=9.76$ to 10.81 km s-1, we clearly see that the
spots describe a helical motion due to the combined effect of the parallax and
proper motion. From $V_{\rm LSR}=11.02$ to 13.34 km s-1 the detection of many
spots per epoch hampers the identification of motion of individual spots. It
is not clear why many more maser spots (and, correspondingly, the highest
maser intensity) appear at these velocities, but this suggests that the water
masers could be excited in internal shocks of the redshifted flow emerging
from the protostar. Outside the $9.76-13.97$ km s-1velocity range, the spots
persist over fewer epochs due to variability.

###  3.2 Astrometric fits

Table 1: Astrometric fits.

Velocity channels | Parallax | $\mu_{\alpha}\cos\delta$ | $\mu_{\delta}$ | RA rms | Decl. rms | Distance
---|---|---|---|---|---|---
(km s-1) | (mas) |  (mas yr-1) |  (mas yr-1) | (mas) | (mas) | (pc)
9.76 – 10.81a | $2.280\pm 0.233$ | $-2.815\pm 0.334$ | $-6.788\pm 0.177$ | 0.72 | 0.35 | $439\pm 45$
9.76 – 10.81b | $2.287\pm 0.072$ | $-1.324\pm 0.085$ | $-7.382\pm 0.151$ | 0.11 | 0.21 | $437\pm 14$
10.40 | $2.275\pm 0.038$ | $-1.204\pm 0.039$ | $-7.447\pm 0.128$ | 0.010 | 0.040 | $439.6\pm 7.4$
10.60 | $2.265\pm 0.021$ | $-1.063\pm 0.025$ | $-7.744\pm 0.075$ | 0.006 | 0.019 | $441.5\pm 4.1$
10.81 | $2.332\pm 0.191$ | $-1.321\pm 0.314$ | $-7.580\pm 0.273$ | 0.077 | 0.076 | $429\pm 35$
111 $a$$a$footnotetext: Fit to maser spots detected in epochs 1 to 9.
$b$$b$footnotetext: Fit to maser spots detected in epochs 4 to 9.

To perform the astrometric fittings, following Sanna et al. (2017), we first
selected maser spots that persist for five epochs or more and that can be
distinguished as isolated maser centers. We note that we did not attempt to
fit spots detected at velocities from $V_{\rm LSR}=11.02$ to 13.34 km s-1,
because at these velocities we detect many spots at multiple positions in the
same epoch (see Fig. 2), which hampers the spot identification. The spots that
satisfy the criteria given above and that are outside the 11.02 to 13.34 km
s-1 range are those at $V_{\rm LSR}$ = 9.76, 9.97, 10.18, 10.4, 10.6, and
10.81 km s-1. The positions of these spots are given in Table 3.

Considering only the velocity range $V_{\rm LSR}=9.76$ to 10.81 km s-1, we see
in Fig. 2 that for epochs 5 to 9 the group of spots detected toward the south
are tracing the same gas cloudlet. These spots have similar positions, as
expected given that the observations were taken $\sim$1 month apart. Thus, we
discarded the spots detected toward the north ($\delta\approx$ –02:03:08.380)
in epochs 6 and 7. On the other hand, for epochs 1, 2, and 3 we used the spots
that are closer in position to the spots detected in the rest of the epochs,
thus discarding those detected at $\delta\approx$ –02:03:08.375 (spots that
are discarded are plotted with empty symbols in Fig. 2, while those used for
the astrometric fits are filled with a cross). Using the least-squares method
(e.g., Loinard et al. 2007), we fit the positions of the remaining spots with
a model that accounts for the effect of parallax ($\varpi$) and proper motions
($\mu_{\alpha},\mu_{\delta}$). The position uncertainties used in the fitting
procedure are inflated with error floors that yield a reduced $\chi^{2}$ of
near unity (e.g., Ortiz-León et al. 2017; Sanna et al. 2017). Then, we used
the covariance matrix to obtain the errors on the astrometric parameters.
Table 1 gives the best-fit astrometric parameters and their associated
uncertainties.

Figure 2: Spatial distribution of the water maser spots. The numbers in the
corners indicate the velocity channel in km s-1. The symbols are color-coded
by epoch (see the figure legend). The spots used for the astrometric fits are
filled with crosses (see Sect. 3.2).

The astrometric fitting to the set of spots selected as described above gives
a best-fit parallax of $2.263\pm 0.231$ mas. Figure 3 shows the best-fit model
and the measured positions of the spots in the plane of the sky (panel a). We
also show the positions of the spots as a function of time after removing
proper motions (panel b) and the residuals from the fit (panel c). The
residuals are large, with post-fit rms values of 0.71 and 0.33 mas in RA and
Decl., respectively, which suggests that we are fitting spots that do not
trace the same gas cloudlet. It is clear that the spots that show the largest
residuals, mainly in the RA direction, are those detected in the first three
epochs. Discarding these spots and fitting only those detected in epochs 4 to
9 gives a parallax of $2.276\pm 0.072$ mas (Fig. A.3). The residuals are
smaller than those obtained in the previous fit, with post-fit rms values of
0.11 and 0.21 mas in RA and Decl., respectively, while the parallax
uncertainty decreases by a factor of $\approx$3\. The rms values are still
high in declination.

(a)

(b)

(c)

Figure 3: Astrometric fits to water maser spots detected in the velocity range
$V_{\rm LSR}=9.76$ to 10.81 km s-1. (a) Measured absolute spot positions,
shown as blue squares. The dashed black line is the fitted model. Expected
positions from the best-fit model are shown as black dots. (b) Measured
positions and best-fit model after proper motions are removed. (c) Residuals
of the astrometric fit in RA (top panel) and Decl. (bottom panel).

Since the spots are not necessarily at the same position in the whole
9.76–10.81 km s-1 velocity range, we fit spots at individual velocity channels
when more than five points were available. Table 1 gives the best-fit
astrometric parameters for velocity channels at 10.4, 10.6, and 10.81 km s-1.
These fits are shown in Figs. A.4, A.5, and A.6. We find that the velocity
channels at 10.4 and 10.6 km s-1 have the smallest post-fit rms values. Thus,
they yield a better fit to the data. We also note that all fits yield similar
solutions for the parallax but different values for the proper motion,
confirming that the spots in individual velocity channels are most likely
tracing different gas regions.

In Fig. 6 we show the distribution of maser spots after removing the parallax
from the fit to the 9.76–10.81 km s-1 velocity range. The dashed red line in
this figure shows the best-fit model with the parallax signature removed, that
is to say, it reflects only the proper motions (see Table 1). The spots in the
velocity range $V_{\rm LSR}$ = 9.76 to 10.81 km s-1 closely follow this line,
that is, they have a linear motion toward the southwest that results from the
effect of a uniform proper motion. We also see that most spots in the velocity
range $V_{\rm LSR}$ = 11.23 to 15.02 km s-1 do not share this motion. Indeed,
spots in the 12.5–14.18 km s-1 range are moving mostly to the south. We can
conclude that these spots are tracing distinct gas cloudlets.

In Appendix B we consider the possibility of an unseen companion in the fits
to the data from the nine epochs and the full 9.76 to 10.81 km s-1velocity
range. When this is compared with the fits presented above, we find that
additional terms in the astrometric model do not improve the fits (Fig. B.1).
This suggests that the large residuals in Fig. 3 are not due to orbital
motion, but due to combining positions from different velocity channels.

##  4 Discussion

###  4.1 The distance to Serpens South

Inverting the weighted mean average of the parallaxes reported in Table 1
gives a distance of $440.7\pm$3.5 pc, where the uncertainty is the standard
error of the weighted mean. This value is consistent within 1$\sigma$ with the
distance reported by Anderson et al. (2022), who used Gaia EDR3 parallaxes in
Serpens South and W40. However, the new distance is an improvement (by an
order of magnitude) over the Gaia EDR3 measurement. In Appendix C we collect
parallaxes of young stellar objects (YSOs) from the most recent data release,
DR3. Although the dispersion of these parallaxes is large (Figs. C.1 and C.2),
we find that their weighted mean is consistent with the maser parallax.

From parallax measurements to eight radio continuum sources in W40 and Serpens
Main , Ortiz-León et al. (2018) obtained a mean distance of $436.0\pm 9.2$ pc
and individual distances consistent with one another within the errors. Our
measurement is consistent with values found previously by Ortiz-León et al.
(2018) using VLBA parallaxes of radio continuum sources in Serpens Main and
W40. Thus, the parallax to one star embedded in the Serpens South cluster
obtained in this work suggests that Serpens South is at the same distance as
Serpens Main and W40, confirming that the three structures are physically
associated.

###  4.2 Proper motions

Figure C.3 shows the proper motions of the maser source and the Gaia sources.
The Gaia sources have a mean value of
$(\overline{\mu}_{\alpha}\cos\delta,\overline{\mu}_{\delta})=(0.37,-6.06)$ mas
yr-1, with a mean error of 0.2 mas yr-1. We also show proper motions of five
radio continuum sources in W40 derived by Ortiz-León et al. (2018). The maser
source has a proper motion different from the motion of the Gaia sources
located in the west of the Serpens South region. This is expected since the
maser emission arises from gas that may be moving along a jet or along a disk
wind (Moscadelli et al. 2022) associated with CARMA–6. We note that the
direction of the maser proper motion is close to perpendicular to the large-
scale molecular outflow, which is oriented in the southeast-northwest
direction (see Fig. 3 in Ortiz-León et al. 2021). This suggests that the
masers are most likely associated with a disk wind. We will investigate this
in a forthcoming paper.

On the other hand, the motions of most Gaia sources are larger in magnitude
than the expected proper motion due to Galactic rotation (see, e.g., Dzib et
al. 2017) for objects at the location of the Serpens South and W40 regions.
The expected proper motion is shown as a black arrow in Fig. C.3 and has
values of $\mu_{\alpha}\cos\delta=0.83$ and $\mu_{\delta}=-5.18$ mas yr-1.

To obtain proper motions of sources in the rest frame of the molecular cloud
complex, we had to remove the expected proper motion from Galactic rotation.
We used a rotation velocity for the LSR of $\Theta_{0}=239$ km s-1, a distance
to the Galactic center of $R_{0}=8.3$ kpc (Brunthaler et al. 2011), and a
Solar motion of $U_{0}=11.10$ km s-1, $V_{0}=12.24$ km s-1, and $W_{0}=7.25$
km s-1 (Schönrich et al. 2010). The resulting stellar motions from this
subtraction are shown in Fig. 4. We note that in this figure sources in the
west of Serpens South have proper motions mostly pointing toward the south,
whereas sources in the east of W40 have RA proper motions mostly pointing
toward the west. This may result from an interaction between W40 and Serpens
South. The proper motions of stars in W40 have magnitudes $\approx 2$ mas
yr-1, which correspond to tangential velocities $\approx 4$ km s-1for a
distance of 440.7 pc.

Shimoikura et al. (2019) suggest that an outer expanding shell created by the
W40 HII region is interacting with the dense gas associated with the Serpens
South young cluster, and that this interaction induced star formation in the
cluster (see also Nakamura et al. 2017). Shimoikura et al. (2019) estimate an
expansion velocity for the shell of $\sim$3 km s-1. This is similar to the
magnitude of the velocity of the Gaia sources in W40 ($\approx 4$ km s-1).
However, we see in Fig. 4 that the Gaia sources at the westernmost side of
Serpens South belong to “filament A” traced by C18O at 6 km s-1(see Fig. 12 of
Shimoikura et al. 2019), and the model shown in their Fig. 18 suggests that
this filament is not interacting with the outer boundary shell of the W40 HII
region. In Appendix C all Gaia stars with $\varpi/\sigma_{\varpi}<5$ were
excluded, a few of which were close to the Serpens South main filament. Thus,
the origin of these motions could be investigated with higher precision
astrometric data.

##  5 Conclusions

We performed VLBA observations of 22 GHz masers arising from the protostar
CARMA–6 in the Serpens South cluster. The high precision astrometry to maser
spots allowed us to derive a distance of $440.7\pm$3.5 pc for this object.
This protostar is at the heart of the cluster, whose young members are
concentrated within molecular filaments (e.g., Sun et al. 2022) that extend
over an area of $\sim 20^{\prime}\times 10^{\prime}$ on the sky ($\sim
2.6\leavevmode\nobreak\ {\rm pc}\times 1.3$ pc at $d=440$ pc). The depth of
the cluster could be $\sim$3 pc, assuming the main filament has a length along
the line of sight similar to its length on the sky. We added in quadrature
this value to the formal uncertainty and conservatively determine the distance
to the young cluster to be $440.7\pm$4.6 pc.

Figure 4: Spatial distribution of Gaia sources (green star symbols) in the
Serpens South and W40 regions. The arrows indicate their proper motions. The
red star and red arrow mark the location of the water maser associated with
CARMA–6 and the proper motion from the astrometric fits. The cyan arrows
indicate proper motions of stars in W40 that have VLBA astrometric solutions
(Ortiz-León et al. 2015). We subtracted the expected proper motion of the
Serpens South and W40 regions due to Galactic rotation from the proper motions
shown here. The background shows the H2 column density map obtained with
Herschel (André et al. 2010).

This result rules out shorter distances suggested in recent works (e.g.,
Winston et al. 2018 and Podio et al. 2021). In addition, it is an improvement
over the Serpens South distance obtained with Gaia DR3 sources in the least
dense areas of the region and an improvement over previous estimates based on
sources in the nearby regions W40 and Serpens Main. Furthermore, our results
combined with previous VLBA parallaxes indicate that Serpens South, W40, and
Serpens Main are all physically associated.

###### Acknowledgements.

G.N.O.L. acknowledges financial support from UNAM-DGAPA postdoctoral
fellowship program. L.L. acknowledges the support of DGAPA PAPIIT grant
IN112820 and CONACYT-CF grant 263356. The National Radio Astronomy Observatory
is a facility of the National Science Foundation operated under a cooperative
agreement by Associated Universities, Inc. 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.

## References

  * Anderson et al. (2022) Anderson, A. R., Williams, J. P., van der Marel, N., et al. 2022, ApJ, 938, 55
  * André et al. (2010) André, P., Men’shchikov, A., Bontemps, S., et al. 2010, A&A, 518, L102
  * Brunthaler et al. (2011) Brunthaler, A., Reid, M. J., Menten, K. M., et al. 2011, Astronomische Nachrichten, 332, 461
  * Curiel et al. (2020) Curiel, S., Ortiz-León, G. N., Mioduszewski, A. J., & Torres, R. M. 2020, AJ, 160, 97
  * Dhabal et al. (2018) Dhabal, A., Mundy, L. G., Rizzo, M. J., Storm, S., & Teuben, P. 2018, ApJ, 853, 169
  * Dunham et al. (2015) Dunham, M. M., Allen, L. E., Evans, II, N. J., et al. 2015, ApJS, 220, 11
  * Dzib et al. (2010) Dzib, S., Loinard, L., Mioduszewski, A. J., et al. 2010, ApJ, 718, 610
  * Dzib et al. (2017) Dzib, S. A., Loinard, L., Rodríguez, L. F., et al. 2017, ApJ, 834, 139
  * Eiroa et al. (2008) Eiroa, C., Djupvik, A. A., & Casali, M. M. 2008, in Handbook of Star Forming Regions (The Southern Sky Vol. 2), ed. B. Reipurth (San Francisco, CA: ASP), 693
  * Fernández-López et al. (2014) Fernández-López, M., Arce, H. G., Looney, L., et al. 2014, ApJ, 790, L19
  * Friesen et al. (2016) Friesen, R. K., Bourke, T. L., Di Francesco, J., Gutermuth, R., & Myers, P. C. 2016, ApJ, 833, 204
  * Gaia Collaboration et al. (2022) Gaia Collaboration, Vallenari, A., Brown, A. G. A., et al. 2022, arXiv e-prints, arXiv:2208.00211
  * Galametz et al. (2019) Galametz, M., Maury, A. J., Valdivia, V., et al. 2019, A&A, 632, A5
  * Greisen (2003) Greisen, E. W. 2003, ASSL, 285, 109
  * Gutermuth et al. (2008) Gutermuth, R. A., Bourke, T. L., Allen, L. E., et al. 2008, ApJ, 673, L151
  * Herczeg et al. (2019) Herczeg, G. J., Kuhn, M. A., Zhou, X., et al. 2019, ApJ, 878, 111
  * Kettenis et al. (2006) Kettenis, M., van Langevelde, H. J., Reynolds, C., & Cotton, B. 2006, in Astronomical Society of the Pacific Conference Series, Vol. 351, Astronomical Data Analysis Software and Systems XV, ed. C. Gabriel, C. Arviset, D. Ponz, & S. Enrique, 497
  * Kirk et al. (2013) Kirk, H., Myers, P. C., Bourke, T. L., et al. 2013, ApJ, 766, 115
  * Lindegren et al. (2018) Lindegren, L. et al. 2018, Gaia Technical Note: GAIA-C3-TN-LU-LL-124-01
  * Loinard et al. (2007) Loinard, L., Torres, R. M., Mioduszewski, A. J., et al. 2007, ApJ, 671, 546
  * Maury et al. (2011) Maury, A. J., André, P., Men’shchikov, A., Könyves, V., & Bontemps, S. 2011, A&A, 535, A77
  * Maury et al. (2019) Maury, A. J., André, P., Testi, L., et al. 2019, A&A, 621, A76
  * Moscadelli et al. (2022) Moscadelli, L., Sanna, A., Beuther, H., Oliva, A., & Kuiper, R. 2022, Nature Astronomy, 6, 1068
  * Nakamura et al. (2017) Nakamura, F., Dobashi, K., Shimoikura, T., Tanaka, T., & Onishi, T. 2017, ApJ, 837, 154
  * Ortiz-León et al. (2017) Ortiz-León, G. N., Dzib, S. A., Kounkel, M. A., et al. 2017, ApJ, 834, 143
  * Ortiz-León et al. (2018) Ortiz-León, G. N., Loinard, L., Dzib, S. A., et al. 2018, ApJ, 869, L33
  * Ortiz-León et al. (2015) Ortiz-León, G. N., Loinard, L., Mioduszewski, A. J., et al. 2015, ApJ, 805, 9
  * Ortiz-León et al. (2021) Ortiz-León, G. N., Plunkett, A. L., Loinard, L., et al. 2021, AJ, 162, 68
  * Pillai et al. (2020) Pillai, T. G. S., Clemens, D. P., Reissl, S., et al. 2020, Nature Astronomy, 4, 1195
  * Plunkett et al. (2015) Plunkett, A. L., Arce, H. G., Corder, S. A., et al. 2015, ApJ, 803, 22
  * Podio et al. (2021) Podio, L., Tabone, B., Codella, C., et al. 2021, A&A, 648, A45
  * Povich et al. (2013) Povich, M. S., Kuhn, M. A., Getman, K. V., et al. 2013, ApJS, 209, 31
  * Reid et al. (2009) Reid, M. J., Menten, K. M., Brunthaler, A., et al. 2009, The Astrophysical Journal, 693, 397
  * Sanna et al. (2017) Sanna, A., Reid, M. J., Dame, T. M., Menten, K. M., & Brunthaler, A. 2017, Science, 358, 227
  * Schönrich et al. (2010) Schönrich, R., Binney, J., & Dehnen, W. 2010, MNRAS, 403, 1829
  * Shimoikura et al. (2019) Shimoikura, T., Dobashi, K., Nakamura, F., Shimajiri, Y., & Sugitani, K. 2019, PASJ, 71, S4
  * Shuping et al. (2012) Shuping, R. Y., Vacca, W. D., Kassis, M., & Yu, K. C. 2012, AJ, 144, 116
  * Straižys et al. (2003) Straižys, V., Černis, K., & Bartašiūtė, S. 2003, A&A, 405, 585
  * Sun et al. (2022) Sun, J., Gutermuth, R. A., Wang, H., et al. 2022, MNRAS, 516, 5244
  * Tanaka et al. (2013) Tanaka, T., Nakamura, F., Awazu, Y., et al. 2013, ApJ, 778, 34
  * Taylor (2005) Taylor, M. B. 2005, in Astronomical Society of the Pacific Conference Series, Vol. 347, Astronomical Data Analysis Software and Systems XIV, ed. P. Shopbell, M. Britton, & R. Ebert, 29
  * Westerhout (1958) Westerhout, G. 1958, Bull. Astron. Inst. Netherlands, 14, 215
  * Winston et al. (2018) Winston, E., Wolk, S. J., Gutermuth, R., & Bourke, T. L. 2018, AJ, 155, 241
  * Zucker et al. (2019) Zucker, C., Speagle, J. S., Schlafly, E. F., et al. 2019, ApJ, 879, 125

##  Appendix A Supplementary figures and tables

Table 2 lists the details of the observations performed with the VLBA. Table 3
gives the positions of maser spots obtained from Gaussian fits to the
brightness distribution and using the AIPS task jmfit.

Figures 5 and 6 show the distribution of the detected water maser spots with
and without the parallax signature subtracted, respectively. Figure A.3 shows
the astrometric fit to maser spots detected in epochs 4 to 9 and the velocity
range 9.76–10.81 km s-1. Figures A.4, A.5, and A.6 show the astrometric fits
for velocity channels 10.4, 10.6, and 10.81 km s-1, respectively.

Table 2: VLBA observed epochs. # | ID | Observation | Beam | P.A. | Channel
---|---|---|---|---|---
|  | Date | Size |  | rms
|  |  |  (mas$\times$mas) |  (o) |  (mJy beam-1)
1 | BO061A3 | 2020 Sep 21 |  1.6$\times$0.3 | $-17$ | 9
2 | BO061A4 | 2020 Oct 25 |  1.3$\times$0.4 | $-17$ | 8
3 | BO061A5 | 2021 Apr 02 |  1.4$\times$0.4 | $-17$ | 9
4 | BO061A6 | 2021 Sep 04 |  1.6$\times$0.3 | $-18$ | 10
5 | BO068A | 2022 Mar 14 |  1.5$\times$0.4 | $-18$ | 7
6 | BO068B | 2022 Apr 11 |  1.4$\times$0.6 | $-7$ | 10
7 | BO068C | 2022 May 06 |  1.3$\times$0.4 | $-15$ | 10
8 | BO068D | 2022 May 30 |  1.5$\times$0.4 | $-18$ | 9
9 | BO068E | 2022 Jun 08 |  1.4$\times$0.3 | $-17$ | 12
10 | BO068F | 2022 Jul 07 |  1.8$\times$0.3 | $-18$ | 13
11 | BO068G | 2022 Aug 27 |  2.9$\times$0.3 | $-19$ | 15
12 | BO068H | 2022 Sep 08 |  1.2$\times$0.3 | $-17$ | 12
Table 3: Maser spot positions. Velocity channel | Julian Day |  $\alpha$(J2000) | $\sigma_{\alpha}$ |  $\delta$(J2000) | $\sigma_{\delta}$ |
---|---|---|---|---|---|---
(km s-1) |  | (h:m:s) | (s) |  (o:′:′′) |  (′′) |
9.76 | 2459114.56829 | 18:30:03.53822820 | 0.00000185 | -02:03:08.376282 | 0.000078 |
| 2459148.47313 | 18:30:03.53818104 | 0.00000234 | -02:03:08.377222 | 0.000092 |
| 2459307.04326 | 18:30:03.53832871 | 0.00000200 | -02:03:08.379177 | 0.000083 |
| 2459462.61304 | 18:30:03.53791920 | 0.00000014 | -02:03:08.382839 | 0.000004 |
| 2459462.61304 | 18:30:03.53788515 | 0.00000020 | -02:03:08.381776 | 0.000007 |
| 2459653.08302 | 18:30:03.53813337 | 0.00000295 | -02:03:08.386172 | 0.000100 |
| 2459705.93832 | 18:30:03.53809439 | 0.00000217 | -02:03:08.386643 | 0.000073 |
9.97 | 2459114.56829 | 18:30:03.53822891 | 0.00000341 | -02:03:08.376468 | 0.000098 |
| 2459307.04326 | 18:30:03.53833487 | 0.00000359 | -02:03:08.379005 | 0.000122 |
| 2459462.61304 | 18:30:03.53790362 | 0.00000071 | -02:03:08.382417 | 0.000026 |
| 2459462.61304 | 18:30:03.53791405 | 0.00000014 | -02:03:08.382718 | 0.000004 |
| 2459653.08302 | 18:30:03.53813780 | 0.00000130 | -02:03:08.386316 | 0.000047 |
| 2459681.00657 | 18:30:03.53813554 | 0.00000222 | -02:03:08.386480 | 0.000074 |
10.18 | 2459462.61304 | 18:30:03.53788712 | 0.00000010 | -02:03:08.381886 | 0.000004 |
| 2459653.08302 | 18:30:03.53813892 | 0.00000066 | -02:03:08.386292 | 0.000023 |
| 2459681.00657 | 18:30:03.53813117 | 0.00000132 | -02:03:08.386483 | 0.000043 |
| 2459705.93832 | 18:30:03.53809611 | 0.00000124 | -02:03:08.386650 | 0.000049 |
10.40 | 2459307.04326 | 18:30:03.53822214 | 0.00000171 | -02:03:08.380405 | 0.000067 |
| 2459462.61304 | 18:30:03.53789370 | 0.00000053 | -02:03:08.382123 | 0.000022 |
| 2459653.08302 | 18:30:03.53813846 | 0.00000039 | -02:03:08.386357 | 0.000014 |
| 2459681.00657 | 18:30:03.53813106 | 0.00000050 | -02:03:08.386486 | 0.000015 |
| 2459705.93832 | 18:30:03.53809904 | 0.00000055 | -02:03:08.386761 | 0.000019 |
| 2459729.87277 | 18:30:03.53804777 | 0.00000303 | -02:03:08.386955 | 0.000088 |
10.60 | 2459307.04326 | 18:30:03.53822791 | 0.00000132 | -02:03:08.380376 | 0.000050 |
| 2459462.61304 | 18:30:03.53788743 | 0.00000008 | -02:03:08.381954 | 0.000003 |
| 2459653.08302 | 18:30:03.53813450 | 0.00000034 | -02:03:08.386322 | 0.000011 |
| 2459681.00657 | 18:30:03.53813058 | 0.00000026 | -02:03:08.386534 | 0.000009 |
| 2459705.93832 | 18:30:03.53809796 | 0.00000026 | -02:03:08.386766 | 0.000009 |
| 2459729.87277 | 18:30:03.53804620 | 0.00000087 | -02:03:08.387044 | 0.000028 |
10.81 | 2459307.04326 | 18:30:03.53823056 | 0.00000110 | -02:03:08.380358 | 0.000041 |
| 2459462.61304 | 18:30:03.53788776 | 0.00000009 | -02:03:08.381989 | 0.000003 |
| 2459653.08302 | 18:30:03.53812683 | 0.00000025 | -02:03:08.386320 | 0.000008 |
| 2459681.00657 | 18:30:03.53812948 | 0.00000021 | -02:03:08.386523 | 0.000007 |
| 2459705.93832 | 18:30:03.53809741 | 0.00000023 | -02:03:08.386759 | 0.000008 |
| 2459729.87277 | 18:30:03.53804580 | 0.00000041 | -02:03:08.387017 | 0.000013 |
| 2459738.84548 | 18:30:03.53800918 | 0.00000296 | -02:03:08.386988 | 0.000079 |
Figure 5: Similar to Fig. 2 but for the velocity range 11.23 to 15.02 km s-1.
Figure 6:  Spatial distribution of the water maser spots. The parallax
signature has been subtracted from all positions presented here. The numbers
in the corners indicate the velocity channel in km s-1. The symbols are color-
coded by epoch (see the figure legend).  Figure A.2: continued.

(a)

(b)

(c)

Figure A.3: Same as Fig. 3 but after removing spots from epochs 1 to 3.

(a)

(b)

(c)

Figure A.4: Same as Fig. A.3 but for the 10.40 km s-1 channel.

(a)

(b)

(c)

Figure A.5: Same as Fig. A.3 but for the 10.60 km s-1 channel.

(a)

(b)

(c)

Figure A.6: Same as Fig. A.3 but for the 10.81 km s-1 channel.

##  Appendix B Astrometric fits taking an unseen companion into account

It is possible that the large residuals seen in Fig. 3 are due to an
unresolved companion that is associated with CARMA–6 and perturbs its motion.
To test this possibility, specifically of an unseen companion in a long-period
orbit ($\gg$1 yr), we fit the maser data with a model that includes
acceleration terms. The results are presented in Fig. B.1. The parallax and
distance resulting from this fit are $2.16\pm 0.10$ mas and $464\pm 22$ pc,
respectively. The rms of the residuals (0.34 mas in both RA and Decl.) are
smaller than the residuals from the fit that does not include acceleration by
an unseen companion (Fig. 3 and Table 1). However, except for the first two
epochs, we see that the residuals show a similar temporal trend as those shown
Fig. 3. When including acceleration terms, the data from the third and fourth
epochs still show large deviations from the best fit (see Fig. B.1). Thus, a
model that takes an unseen companion in a long-period orbit into account does
not produce a better fit to the full data set (9.76–10.81 km s-1 velocity
range and nine epochs).

We also investigated the possibility that there is a short-period companion
perturbing the motion of CARMA–6. Following Curiel et al. (2020), we fit a
model that includes a Keplerian orbit induced on the star by an unseen
companion. We explored initial orbital periods from 0.2 to 2 yr but did not
find a reasonable fit to the data, as neither the period nor eccentricity
converge.

(a)

(b)

(c)

Figure B.1: Same as Fig. 3 but including acceleration terms in the fitted
model.

##  Appendix C Comparison with Gaia DR3

We followed a similar analysis to that done in Ortiz-León et al. (2018) in
order to compare the distance obtained from the water masers with that derived
from Gaia DR3 (Gaia Collaboration et al. 2022). We used the catalog of YSO
candidates from the Spitzer Core two Disks (c2d) and Gould Belt survey (Dunham
et al. 2015). This catalog contains 1319 sources in the Aquila Rift, including
Serpens South and W40. We cross-matched the positions of the YSO candidates
against Gaia DR3 positions using a matching radius of 1′′ and the Virtual
Observatory tool TOPCAT (Taylor 2005). Then, we selected all sources with the
following criteria: (i) $18^{\rm h}28^{\rm m}00^{\rm s}\leq\alpha\leq 18^{\rm
h}34^{\rm m}00^{\rm s}$, (ii) $-3^{\rm o}\leq\delta\leq-1^{\rm o}$, (iii)
$\varpi/\sigma_{\varpi}\geq 5$, and (iv) renormalized unit weight error (RUWE)
$\leq 1.4$ (Lindegren et al. 2018). This resulted in 21 sources.

We also used the list of 316 Class I to Class III YSOs of Anderson et al.
(2022), and found that 46 of them are in the DR3 catalog and satisfy the
criteria listed above. Combining them with the c2d sample by matching their
Gaia IDs resulted in 35 sources. The distribution of these parallaxes is shown
in Fig. C.1, while Figs. 4 and C.3 show the spatial distribution and proper
motions of the 35 sources. We see that all of them are projected in the least
dense parts of the Serpens South and W40 regions, while the maser source is
located at the center of the Serpens South cluster. The parallaxes show a
Gaussian distribution (Fig. C.1), with a peak and standard deviation of
$2.11\pm 0.02$ mas and 0.38 mas, respectively. We note, however, that there
are a few sources outside the distribution, with parallaxes above 3 mas and
below 1 mas, respectively.

Excluding parallaxes $\varpi>3$ mas and $\varpi<1$ mas, we find that the
weighted mean parallax of the Gaia stars is $\varpi=2.29\pm 0.23$ mas, which
translates to $d=437^{+49}_{-40}$ pc. This distance is consistent within
$1\sigma$ with the water maser distance.

We also used the Gaia DR3 catalog to select all sources with (i) $r\leq
15^{\prime}$, (ii) $1\leq\varpi[{\rm mas}]\leq 5$, (iii)
$\varpi/\sigma_{\varpi}\geq 5$, and (iv) RUWE $\leq 1.4$, where $r$ is the
angular separation from the center of the Serpens South region (at $18^{\rm
h}30^{\rm m}00^{\rm s}$ –$02^{\rm o}00^{\rm m}00^{\rm s}$). The parallax
distribution of these stars is shown in Fig. C.2. As expected, when the sample
is not restricted to known or candidate YSOs, the resulting parallax
distribution is broader than the one shown in the histogram of Fig. C.1. To
estimate the width and peak of this distribution, we fit a Gaussian function.
The peak is at $2.61\pm 0.04$ mas, and the width is 0.7 mas. The parallax peak
corresponds to a distance of $d=382.5$ pc. The weighted mean of the
distribution is $2.84\pm 0.71$ mas ($d=353^{+118}_{-71}$ pc), where the error
bar is the standard deviation.

This broader distribution with a larger parallax peak results from the
contribution of stars that do not belong to the Serpens South cluster.
Therefore, larger parallaxes (or smaller distances) in this sample would be
associated with a group of stars located at the front edge of the Aquila Rift
complex but that are not related to the young cluster, which is embedded in
the Serpens South region.

The value found above of $d=353$ pc is close to the Gaia-based distance used
by Galametz et al. (2019) and Podio et al. (2021). We speculate that Galametz
et al. (2019) identified a jump in extinction around 300–350 pc and assigned
the distance to this extinction layer as the distance to the Serpens South
cloud (the authors do not provide details for their distance derivation).
However, the analysis presented above shows that the YSOs are embedded in a
more distant layer.

Figure C.1: Distribution of Gaia DR3 parallaxes for the sample of YSOs
selected from the catalogs published by Dunham et al. (2015) and Anderson et
al. (2022) and that satisfy the criteria listed in Sect. C. The dashed black
line is a Gaussian fit to the distribution.  Figure C.2: Distribution of Gaia
DR3 parallaxes for stars within $15^{\prime}$ of the Serpens South center and
that satisfy the criteria listed in Sect. C. The vertical line indicates the
peak of the parallax distribution shown in Fig. C.1, which only includes
sources classified as YSOs ($\varpi=2.11$ mas). The dashed black line is a
Gaussian fit to estimate the width and peak of the distribution.  Figure C.3:
Spatial distribution of Gaia sources (green star symbols) in the Serpens
South/W40 region. The arrows indicate their published proper motions. The red
star and red arrow mark the location of the water maser associated with
CARMA–6 and the proper motion from the astrometric fits. The cyan arrows
indicate proper motions of stars in W40 that have VLBA astrometric solutions
(Ortiz-León et al. 2015). The black arrow indicates the expected proper motion
of Serpens South/W40 from Galactic rotation. The background shows the H2
column density map obtained with Herschel (André et al. 2010).


# Correlating the CDF $W$-boson mass shift with the $b\to s\ell^{+}\ell^{-}$
anomalies

Xin-Qiang Li<EMAIL_ADDRESS>Ze-Jun Xie<EMAIL_ADDRESS>Ya-
Dong Yang<EMAIL_ADDRESS>Xing-Bo Yuan<EMAIL_ADDRESS>Institute of
Particle Physics and Key Laboratory of Quark and Lepton Physics (MOE),
Central China Normal University, Wuhan, Hubei 430079, China  School of Physics
and Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001, China

###### Abstract

The recently updated measurement of the $W$-boson mass by the CDF
collaboration exhibits a $7\sigma$ deviation from the SM expectation, which
may imply a sign of new physics beyond the SM. The observed discrepancy could
be explained by new fermions that carry the electroweak gauge charges and
affect the vacuum polarization of gauge bosons. Notably, if the new fermions
also have the same quantum numbers as of the SM quarks, they can mix with the
latter and thus modify the penguin diagrams governing the $b\to
s\ell^{+}\ell^{-}$ transitions. Therefore, the $W$-boson mass shift could be
related to the $b\to s\ell^{+}\ell^{-}$ anomalies observed by the LHCb
collaboration during the past few years. To investigate this possibility, we
consider in this paper a model containing a vector-like top partner gauged
under a new $U(1)^{\prime}$ symmetry. It is found that the latest CDF $m_{W}$
measurement and the $b\to s\ell^{+}\ell^{-}$ anomalies can be simultaneously
accommodated at $2\sigma$ level.

##  1 Introduction

The Standard Model (SM) of particle physics has been remarkably successful in
explaining most phenomena observed in experiments [1]. After the discovery of
the Higgs boson [2, 3], one important goal of the Large Hadron Collider (LHC)
and its high-luminosity upgrade is the direct searches for new phenomena
beyond the SM [4]. While the LHC has not yet discovered any direct evidence of
new physics (NP) beyond the SM, several interesting deviations from the SM
predictions have been emerging from the precision measurements, such as the
anomalies observed in $b\to s\ell^{+}\ell^{-}$ processes [5, 6, 7].

In the field of precision measurements, the electroweak (EW) precision
observables have played an important role in establishing the structure of the
EW sector of the SM and can provide a sensitive probe of NP [8, 9, 10].
Recently, using the complete dataset collected by the CDF II detector at the
Fermilab Tevatron, corresponding to $8.8\,{\,\rm fb}^{-1}$ of integrated
luminosity, the CDF collaboration has reported an updated measurement of the
$W$-boson mass [11]

$\displaystyle m_{W}^{\rm CDF}=80.4335\pm 0.0064_{\rm stat}\pm 0.0069_{\rm
syst}{\,\rm GeV}.$ (1)

Interestingly, such a high precision measurement shows a $7\sigma$ deviation
from the SM expectation obtained by a global EW fit, $m_{W}^{\rm SM}=80.357\pm
0.006{\,\rm GeV}$ [1]. The latest CDF result also shows a significant shift
compared to the average of the previous measurements from LEP, CDF, D0 and
ATLAS, $m_{W}^{\rm PDG}=80.379\pm 0.012{\,\rm GeV}$ [1], as well as the LHCb
measurement, $m_{W}^{\rm LHCb}=80.354\pm 0.031{\,\rm GeV}$ [12]. If confirmed
by future measurements, the discrepancy could imply a sign of NP beyond the SM
and has motivated numerous studies of the phenomenological implications of
such a marvellous measurement (see, e.g., refs. [13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38,
39]).

The latest CDF $W$-boson mass shift could be induced by new fermions, which
carry the EW gauge charges and contribute to the vacuum polarization of gauge
bosons [40, 41, 42, 43, 44, 45]. Notably, if the new fermions also have the
same quantum numbers as of the SM quarks, they can mix with the latter and
thus modify the $\gamma$\- and $Z$-penguin diagrams that govern the flavour-
changing neutral-current (FCNC) $b\to s$ transitions. Therefore, the observed
discrepancy of the latest CDF $m_{W}$ measurement could be related to the
$b\to s\ell^{+}\ell^{-}$ anomalies observed by the LHCb collaboration [46, 47,
48, 49, 50]. To investigate this possibility, we consider in this paper the NP
model introduced in ref. [51]. Specifically, the model is characterized by a
vector-like top partner with an additional $U(1)^{\prime}$ gauge symmetry,
where the top quark and the top partner have the same quantum numbers under
the SM gauge symmetry, while only the latter is charged under the
$U(1)^{\prime}$ symmetry. Similar models have been investigated, e.g., in
refs. [52, 53, 54, 55, 56, 57, 58].

This paper is organized as follows. In section 2, we introduce the NP model.
In section 3, the top-partner contributions to the $W$-boson mass shift and
the oblique parameters $S$, $T$ and $U$ are investigated. In section 4, we
consider and compute the NP contributions to the $b\to s\ell^{+}\ell^{-}$
transitions. In section 5, we first derive the constraints on the model
parameters from the global EW fit in view of the latest CDF $m_{W}$
measurement, and then investigate the possibility of simultaneously explaining
the observed $W$-boson mass shift and the $b\to s\ell^{+}\ell^{-}$ anomalies.
Our conclusion are finally made in section 6. For convenience, the loop
functions involved are relegated in the appendix.

##  2 Model

As mentioned in the last section, to simultaneously accommodate the latest CDF
$W$-boson mass shift and the $b\to s\ell^{+}\ell^{-}$ anomalies, the new
fermions contributing to the vacuum polarization of gauge bosons should have
the same quantum numbers as of the SM quarks. To this end, one simple
realization is the ${Z^{\prime}}$ model introduced in ref. [51]. The model is
characterized by a new $U(1)^{\prime}$ gauge symmetry that is spontaneously
broken by the vacuum expectation value (vev) of a new scalar field $\Phi$,
transforming as $(1,1,0,q_{t})$ under the $SU(3)_{C}\otimes SU(2)_{L}\otimes
U(1)_{Y}\otimes U(1)^{\prime}$ gauge symmetry. While all the SM fields do not
carry $U(1)^{\prime}$ charge, a coloured singlet vector-like top partner
$U_{L,R}^{\prime}$, transforming as $(3,1,2/3,q_{t})$, is introduced in the
model. Specific to the case where only the third-generation SM quarks mix with
the top partner, the general renormalizable interactions take the form [51,
57]

$\displaystyle\mathcal{L}_{\text{int }}=$
$\displaystyle\big{(}\lambda_{H}\bar{Q}_{3L}\tilde{H}u_{3R}+\lambda_{\Phi}\bar{U}_{L}^{\prime}u_{3R}\Phi+\mu\bar{U}_{L}^{\prime}U_{R}^{\prime}+\text{h.c.}\big{)}$
$\displaystyle\qquad+q_{t}g_{t}\left(\bar{U}_{L}^{\prime}\gamma^{\mu}U_{L}^{\prime}+\bar{U}_{R}^{\prime}\gamma^{\mu}U_{R}^{\prime}\right)Z_{\mu}^{\prime},$
(2)

where $q_{t}$ and $g_{t}$ denote the charge and coupling of the
$U(1)^{\prime}$ gauge symmetry, respectively. $Q_{3L}=(u_{3L},d_{3L})^{T}$ and
$u_{3R}$ denote the third-generation left-handed quark doublet and right-
handed quark singlet in the SM, respectively. Matter fields in the above
Lagrangian are all given in the interaction eigenbasis. For simplicity, the
scalar potential and the interaction between ${Z^{\prime}}$ and $\Phi$ are
suppressed here.

After the $U(1)^{\prime}$ and the EW symmetry breaking, where the vevs of the
two scalar fields are given respectively by
$\langle\Phi\rangle=v_{\Phi}/\sqrt{2}$ and $\langle
H\rangle=v_{H}/\sqrt{2}\simeq 174{\,\rm GeV}$, both the SM fermions and the
top partner as well as the ${Z^{\prime}}$ boson obtain their masses. By
diagonalizing the mass matrix, the physical top quarks $(t_{L},t_{R})$ and its
partners $(T_{L},T_{R})$ can be expressed in terms of the fermion fields in
eq. (2) through the following rotation matrices [57]:

$\displaystyle\begin{pmatrix}t_{L}\\\ T_{L}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\cos\theta_{L}&-\sin\theta_{L}\\\
\sin\theta_{L}&\hphantom{-}\cos\theta_{L}\end{pmatrix}\begin{pmatrix}u_{3L}\\\
U_{L}^{\prime}\end{pmatrix},$ (3) $\displaystyle\begin{pmatrix}t_{R}\\\
T_{R}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\cos\theta_{R}&-\sin\theta_{R}\\\
\sin\theta_{R}&\hphantom{-}\cos\theta_{R}\end{pmatrix}\begin{pmatrix}u_{3R}\\\
U_{R}^{\prime}\end{pmatrix},$ (4)

where the mixing angles $\theta_{L}$ and $\theta_{R}$ parameterize the
rotation matrices of the left- and right-handed quarks, respectively. In terms
of the physical parameters, the top-quark mass $m_{t}$, the top-partner mass
$m_{T}$, as well as the ${Z^{\prime}}$-boson mass
$m_{{Z^{\prime}}}=g_{t}v_{\Phi}$, the two mixing angles are related to each
other through

$\displaystyle\tan\theta_{L}=\frac{m_{t}}{m_{T}}\tan\theta_{R},$ (5)

with $m_{t}$ and $m_{T}$ determined, respectively, by

$\displaystyle\lambda_{H}$
$\displaystyle=\frac{\cos\theta_{L}}{\cos\theta_{R}}\frac{\sqrt{2}\,m_{t}}{v_{H}},$
$\displaystyle\left|\lambda_{\Phi}\right|$
$\displaystyle=g_{t}\left|\cos\theta_{L}\sin\theta_{R}\right|\frac{\sqrt{2}\,m_{T}}{m_{Z^{\prime}}}\left(1-\frac{m_{t}^{2}}{m_{T}^{2}}\right).$
(6)

Turning to the fermion mass eigenbasis, the explicit expressions of the gauge
interactions involving the top quark and the top partner can be written as
[51, 57]

$\displaystyle\mathcal{L}_{\gamma}$
$\displaystyle=\frac{2}{3}e\bar{t}\not{A}t+\frac{2}{3}e\bar{T}\not{A}T\,,$ (7)
$\displaystyle\mathcal{L}_{W}$
$\displaystyle=\frac{g}{\sqrt{2}}V_{td_{i}}\left(c_{L}\bar{t}\,\not{W}P_{L}d_{i}+s_{L}\bar{T}\,\not{W}P_{L}d_{i}\right)+\text{
h.c.}\,,$ (8) $\displaystyle\mathcal{L}_{Z}$
$\displaystyle=\frac{g}{c_{W}}\left(\bar{t}_{L},\bar{T}_{L}\right)\begin{pmatrix}\frac{1}{2}c_{L}^{2}-\frac{2}{3}s_{W}^{2}&\frac{1}{2}s_{L}c_{L}\\\\[4.26773pt]
\frac{1}{2}s_{L}c_{L}&\frac{1}{2}s_{L}^{2}-\frac{2}{3}s_{W}^{2}\end{pmatrix}\not{Z}\begin{pmatrix}t_{L}\\\\[4.26773pt]
T_{L}\end{pmatrix}$
$\displaystyle+\frac{g}{c_{W}}\left(\bar{t}_{R},\bar{T}_{R}\right)\bigg{(}-\frac{2}{3}s_{W}^{2}\bigg{)}\not{Z}\begin{pmatrix}t_{R}\\\
T_{R}\end{pmatrix}\,,$ (9) $\displaystyle\mathcal{L}_{{Z^{\prime}}}$
$\displaystyle=q_{t}g_{t}\left(\bar{t}_{L},\bar{T}_{L}\right)\begin{pmatrix}s_{L}^{2}&-s_{L}c_{L}\\\\[4.26773pt]
-s_{L}c_{L}&c_{L}^{2}\end{pmatrix}\not{Z}^{\prime}\begin{pmatrix}t_{L}\\\\[4.26773pt]
T_{L}\end{pmatrix}$ $\displaystyle+(L\to R)\,,$ (10)

where $s_{L,R}=\sin\theta_{L,R}$, $c_{L,R}=\cos\theta_{L,R}$, and
$s_{W}=\sin\theta_{W}$ with the Weinberg mixing angle $\theta_{W}$; $d_{i}$
stand for the down-type quarks $d,s,b$, $g$ denotes the SM weak coupling
constant, and $V_{td_{i}}$ are the CKM matrix elements. The Yukawa
interactions of the physical SM Higgs $h$ with the top quark and the top
partner are given by

$\mathcal{L}_{h}=\frac{m_{t}}{v_{H}}\left(\bar{t}_{L},\bar{T}_{L}\right)\left(\begin{array}[]{cc}c_{L}&c_{L}\tan\theta_{R}\\\\[4.26773pt]
s_{L}&s_{L}\tan\theta_{R}\end{array}\right)h\begin{pmatrix}t_{R}\\\\[4.26773pt]
T_{R}\end{pmatrix}+\text{h.c.}$ (11)

Assuming that the effective ${Z^{\prime}}$ couplings to the charged leptons
are flavour diagonal and focusing only on its interaction with the muon, we
can then write the general renormalizable $\mu^{+}\mu^{-}{Z^{\prime}}$ vertex
as [59, 51, 57]

$\displaystyle\mathcal{L}_{\mu}=\bar{\mu}\not{Z}^{\prime}\left(g_{\mu}^{L}P_{L}+g_{\mu}^{R}P_{R}\right)\mu\,,$
(12)

where the effective couplings $g_{\mu}^{L}$ and $g_{\mu}^{R}$ depend on the
specific UV completions. For example, one can introduce vector-like leptonic
partners to generate such effective couplings [60]. In this paper, since we
only concentrate on the possible connection between the latest CDF $W$-boson
mass shift and the $b\to s\mu^{+}\mu^{-}$ anomalies, details of the lepton
sector are not so much relevant. Thus, as in refs. [51, 57, 59], we shall
consider the general effective $\mu^{+}\mu^{-}{Z^{\prime}}$ interaction and
leave the complete UV completions for future work.

##  3 $W$-boson mass shift and oblique parameters

The global fit of the SM to the EW precision data, known as the global EW fit
[8, 9, 10], is a powerful tool to explore the validity of the SM and provides
also a sensitive probe of NP beyond it [61, 62, 63, 64]. As the EW parameters
of the SM are closely related to each other, it is generally expected that
some observables in the global EW fit may be affected once the latest CDF
$m_{W}$ measurement is considered [13, 14, 15, 16, 17]. Most of the NP effects
on the EW sector can be parameterized in terms of the oblique parameters $S$,
$T$ and $U$ [65, 66, 67]. These parameters are in turn related to the NP
contributions to the vacuum polarization of gauge bosons and can be written,
respectively, as [65, 66]

$\displaystyle S$
$\displaystyle=\frac{4s_{W}^{2}c_{W}^{2}}{\alpha}\bigg{[}\frac{\Pi_{ZZ}\left(m_{Z}^{2}\right)-\Pi_{ZZ}(0)}{m_{Z}^{2}}$
$\displaystyle\qquad\qquad\qquad-\frac{c_{W}^{2}-s_{W}^{2}}{s_{W}c_{W}}\Pi_{Z\gamma}^{\prime}(0)-\Pi_{\gamma\gamma}^{\prime}(0)\bigg{]}\,,$
(13) $\displaystyle T$
$\displaystyle=\frac{1}{\alpha}\bigg{[}\frac{\Pi_{WW}(0)}{m_{W}^{2}}-\frac{\Pi_{ZZ}(0)}{m_{Z}^{2}}\bigg{]}\,,$
(14) $\displaystyle U$
$\displaystyle=\frac{4s_{W}^{2}}{\alpha}\bigg{[}\frac{\Pi_{WW}\left(m_{W}^{2}\right)-\Pi_{WW}(0)}{m_{W}^{2}}$
$\displaystyle\qquad\qquad-\frac{c_{W}}{s_{W}}\Pi_{Z\gamma}^{\prime}(0)-\Pi_{\gamma\gamma}^{\prime}(0)\bigg{]}-S\,,$
(15)

where $\Pi_{XY}$ denotes the NP contribution to the vacuum polarization of the
gauge bosons with $X,Y=W,Z,\gamma$, $c_{W}=\cos\theta_{W}$, and $\alpha$ is
the fine structure constant. Then, the $W$-boson mass shift induced by the
oblique corrections can be expressed as [65]

$\displaystyle\Delta m_{W}^{2}=\frac{\alpha
c_{W}^{2}m_{Z}^{2}}{c_{W}^{2}-s_{W}^{2}}\left[-\frac{S}{2}+c_{W}^{2}T+\frac{c_{W}^{2}-s_{W}^{2}}{4s_{W}^{2}}U\right].$
(16)

Thus, to make an explanation of the discrepancy between the latest CDF $m_{W}$
measurement and the SM expectation, one needs a global EW fit to the oblique
parameters $S$, $T$ and $U$, which encode the potential NP effects.

In the model introduced in section 2, extra contributions to the vacuum
polarization of gauge bosons arise from the modified fermion-gauge couplings
that are characterized by the mixing angle $\theta_{L}$, as well as from the
loops involving the top partner (cf. eqs. (7)–(10)). Their contributions to
the oblique parameters can be written as

$\displaystyle S$
$\displaystyle=\frac{s_{L}^{2}}{12\pi}\Big{[}K_{1}(y_{t},y_{T})+3c_{L}^{2}K_{2}(y_{t},y_{T})\Big{]},$
(17) $\displaystyle T$ $\displaystyle=\frac{3s_{L}^{2}}{16\pi
s_{W}^{2}}\bigg{[}x_{T}-x_{t}-c_{L}^{2}\Big{(}x_{T}+x_{t}+\frac{2x_{t}x_{T}}{x_{T}-x_{t}}\ln\frac{x_{t}}{x_{T}}\Big{)}\bigg{]},$
(18) $\displaystyle U$
$\displaystyle=\frac{s_{L}^{2}}{12\pi}\Big{[}K_{3}(x_{t},y_{t})-K_{3}(x_{T},y_{T})\Big{]}-S,$
(19)

with $x_{q}=m_{q}^{2}/m_{W}^{2}$ and $y_{q}=m_{q}^{2}/m_{Z}^{2}$ for $q=t,T$.
Explicit expressions of the loop functions $K_{1,2,3}(x,y)$ are listed in the
appendix. In the NP model considered, the oblique parameters $S$, $T$ and $U$
are solely determined by the two NP parameters $\theta_{L}$ and $m_{T}$.

##  4 $b\to s\ell^{+}\ell^{-}$ transitions

Figure 1: One-loop Feynman diagrams for the $b\to s\mu^{+}\mu^{-}$
transitions, corresponding to the $W$-box, as well as the $\gamma$-, $Z$-, and
${Z^{\prime}}$-penguin contributions, respectively.

The FCNC processes mediated by the quark-level $b\to s\ell^{+}\ell^{-}$
transitions, such as the rare decays $B_{s}\to\ell^{+}\ell^{-}$, $B\to
K^{(*)}\ell^{+}\ell^{-}$ and $B_{s}\to\phi\ell^{+}\ell^{-}$, can also provide
promising probes of potential NP effects [5, 6, 7]. Specific to the model
introduced in section 2, the NP contributions to the $b\to s\ell^{+}\ell^{-}$
transitions start at the one-loop level, with the relevant Feynman diagrams
shown in figure 1. Here the NP effects can arise from the modified SM box and
penguin diagrams that are featured by the mixing angle $\theta_{L}$, as well
as from the penguin diagrams involving the top partner and/or ${Z^{\prime}}$
boson.

By an explicit calculation, we find that these NP contributions affect only
the Wilson coefficients ${\mathcal{C}}_{9}$ and ${\mathcal{C}}_{10}$ in the
low-energy effective weak Hamiltonian governing the $b\to s\mu^{+}\mu^{-}$
transitions [68]

$\displaystyle\mathcal{H}_{\rm
eff}\supset-\frac{4G_{F}}{\sqrt{2}}V_{tb}V_{ts}^{*}\frac{\alpha}{4\pi}\left({\mathcal{C}}_{9}^{\mu}\mathcal{O}_{9}^{\mu}+{\mathcal{C}}_{10}^{\mu}\mathcal{O}_{10}^{\mu}\right)+\text{
h.c.}\,,$ (20)

with the four-fermion operators
$\mathcal{O}_{9}=\left(\bar{s}\gamma^{\mu}P_{L}b\right)\left(\bar{\mu}\gamma_{\mu}\mu\right)$
and
$\mathcal{O}_{10}=\left(\bar{s}\gamma^{\mu}P_{L}b\right)\left(\bar{\mu}\gamma_{\mu}\gamma_{5}\mu\right)$.
Explicitly, the NP contributions to ${\mathcal{C}}_{9}$ and
${\mathcal{C}}_{10}$ can be written, respectively, as

$\displaystyle{\mathcal{C}}_{9}^{\rm NP}$
$\displaystyle=s_{L}^{2}I_{1}+s_{L}^{2}\left(1-\frac{1}{4s_{W}^{2}}\right)\left(I_{2}+c_{L}^{2}I_{3}\right)+\Delta{\mathcal{C}}_{+}^{Z^{\prime}}\,,$
(21) $\displaystyle{\mathcal{C}}_{10}^{\rm NP}$
$\displaystyle=\frac{s_{L}^{2}}{4s_{W}^{2}}\left(I_{2}+c_{L}^{2}I_{3}\right)+\Delta{\mathcal{C}}_{-}^{Z^{\prime}}\,,$
(22)

with the ${Z^{\prime}}$-penguin contributions given by

$\displaystyle\Delta{\mathcal{C}}_{\pm}^{Z^{\prime}}=\frac{(g_{L}\pm
g_{R})q_{t}g_{t}}{e^{2}}\frac{m_{W}^{2}}{m_{Z^{\prime}}^{2}}c_{L}^{2}s_{R}^{2}\left(I_{4}-\frac{c_{L}^{2}}{c_{R}^{2}}I_{5}\right),$
(23)

where $e=\sqrt{4\pi\alpha}$ is the electromagnetic coupling constant. The loop
integrals $I_{1-5}$ are all functions of the masses $m_{t}$ and $m_{T}$, whose
explicit expressions can be found in the appendix. In the limit of
$\theta_{L}\to 0$ or $\theta_{R}\to 0$, the NP Wilson coefficients
${\mathcal{C}}_{9}^{\rm NP}$ and ${\mathcal{C}}_{10}^{\rm NP}$ will vanish.

Here, the NP contributions to the Wilson coefficients ${\mathcal{C}}_{9}^{\rm
NP}$ and ${\mathcal{C}}_{10}^{\rm NP}$ are calculated in the unitary gauge. We
implement our computation in two setups by making use of different packages
including FeynRules [69], FeynArts [70], FeynCalc [71, 72, 73], Package-X
[74], as well as some in-house routines. After expanding in terms of the ratio
$m_{W}^{2}/m_{T}^{2}$ and keeping only the leading terms in
$s_{R}=\sin\theta_{R}$, our results are in agreement with those obtained in
ref. [57].

##  5 Numerical analysis

###  5.1 $W$-boson mass and global EW fit

As all the three oblique parameters $S$, $T$ and $U$ could be affected by the
NP effects (cf. eqs. (17)–(19)), a global EW fit is required to explain the
latest CDF $m_{W}$ measurement [11]. To this end, several groups have recently
updated the global EW fit on the $S$, $T$ and $U$ parameters by including the
latest CDF measurement of the $W$-bosn mass [13, 14, 15, 16, 17]. Here, we
adopt the result obtained in ref. [16],111It is found that using the global EW
fit performed in ref. [17] with the package HEPfit [75] does not make
substantial changes to our numerical results. which is based on the package
Gfitter [61, 62, 63]. The resulting values of the $S$, $T$ and $U$ parameters
as well as the correlation matrix read [16]

$\displaystyle\begin{array}[]{l}S=0.06\pm 0.10,\\\\[4.26773pt] T=0.11\pm
0.12,\\\\[4.26773pt] U=0.14\pm 0.09,\end{array}\quad{\rm
corr.=}\begin{bmatrix}1.00&0.90&-0.59\\\\[4.26773pt]
&1.00&-0.85\\\\[4.26773pt] &&\hphantom{-}1.00\end{bmatrix}.$ (27)

As described in section 3, the NP contributions to the $S$, $T$ and $U$
parameters depend only on the top-partner mass $m_{T}$ and the mixing angle
$\theta_{L}$. To avoid large modification of the $\bar{t}bW^{+}$ coupling, we
require $\cos\theta_{L}>0.9$ in the following. This range is also consistent
with that derived from the top-Higgs coupling through global fits to the SM
Higgs properties [52, 57]. After considering the constraints from the global
EW fit, we find that there still exist parameter regions allowed at $2\sigma$
level, which is shown in figure 2. Numerically, $\cos\theta_{L}=0.98\sim 0.99$
is allowed for $500<m_{T}<2000{\,\rm GeV}$.

Figure 2: Allowed parameter space in the $(m_{T},\cos\theta_{L})$ plane at
$2\sigma$ level from the global EW fit with the latest CDF measurement of the
$W$-boson mass.

Such a small mixing angle suppresses the NP contributions from the $W$-box,
$\gamma$\- and $Z$-penguin diagrams, because they are all proportional to
$\sin^{2}\theta_{L}$, as can be seen from eqs. (21) and (22). However, the
${Z^{\prime}}$-penguin diagrams do not suffer from this suppression and may
affect the $b\to s\ell^{+}\ell^{-}$ processes, as will be exploited in the
next subsection.

###  5.2 $b\to s\ell^{+}\ell^{-}$ anomalies

In the model considered, the NP affects only the Wilson coefficients
${\mathcal{C}}_{9}$ and ${\mathcal{C}}_{10}$ and, in order to find the
parameter regions required to explain the $b\to s\ell^{+}\ell^{-}$ anomalies,
we should perform a global fit to the $b\to s\ell^{+}\ell^{-}$ observables,
including, but not limited to, the lepton-flavour-universality-violating
ratios $R_{K^{(*)}}$ [46, 47, 48] and the angular observables in $B\to
K^{*}\mu^{+}\mu^{-}$ decays [49, 50]. When performing the global fit, we
follow the same prescription as in ref. [76] and use the package flavio [77].
More details can be found in ref. [78]. We refer to refs. [79, 80, 81, 82, 83]
for the latest global fit by other groups.

The NP contributions to the $b\to s\ell^{+}\ell^{-}$ transitions depend on the
mixing angles $\theta_{L,R}$, the top-partner mass $m_{T}$, the
$U(1)^{\prime}$ charge $q_{t}$, the ${Z^{\prime}}$ mass $m_{Z^{\prime}}$, the
$\bar{t}t{Z^{\prime}}$ coupling $g_{t}$, as well as the
$\mu^{+}\mu^{-}{Z^{\prime}}$ couplings $g_{\mu}^{L,R}$. From eqs. (21) and
(22), one can see that the ${Z^{\prime}}$-related parameters always appear as
the product $q_{t}g_{t}g_{\mu}^{L,R}/m_{Z^{\prime}}^{2}$. Thus, without loss
of generality, we take in the numerical analysis $q_{t}=1$, $g_{t}=1$ and
$m_{Z^{\prime}}=200{\,\rm GeV}$, while keeping the effective
$\mu^{+}\mu^{-}{Z^{\prime}}$ couplings $g_{\mu}^{L,R}$ as free parameters.
After considering the relation in eq. (5), the independent NP parameters can
be chosen as $(\cos\theta_{L},m_{T},g_{\mu}^{L},g_{\mu}^{R})$.

Figure 3: Constraints on the effective $\mu^{+}\mu^{-}{Z^{\prime}}$ couplings
$(g_{\mu}^{L},g_{\mu}^{R})$ by the $b\to s\ell^{+}\ell^{-}$ processes, in the
$2\sigma$ allowed regions of the $(\cos\theta_{L},m_{T})$ plane obtained from
the global EW fit with the latest CDF $m_{W}$ measurement. The allowed regions
are shown in red ($1\sigma$) and black ($2\sigma$) for $m_{T}=500$, $1000$,
$1500$, $2000{\,\rm GeV}$, respectively. Here, without loss of generality,
$q_{t}=1$, $g_{t}=1$ and $m_{Z^{\prime}}=200{\,\rm GeV}$ have been taken.

With the $2\sigma$ allowed regions of $\cos\theta_{L}$ and $m_{T}$ obtained
from the global EW fit (cf. figure 2), constraints on the NP parameters
$g_{\mu}^{L,R}$ can then be derived from the $b\to s\ell^{+}\ell^{-}$
processes. In figure 3, we show the allowed regions of
$(g_{\mu}^{L},g_{\mu}^{R})$ for $m_{T}=500$, $1000$, $1500$, $2000{\,\rm
GeV}$, respectively. It can be seen that, even after considering the latest
CDF measurement of the $W$-boson mass, the model introduced in section 2 can
still accommodate the $b\to s\ell^{+}\ell^{-}$ anomalies. For a ${Z^{\prime}}$
boson with mass of hundreds of ${\,\rm GeV}$, its couplings to the top and the
muon are both less than ${\mathcal{O}}(1)$, being therefore safely in the
perturbative region. Furthermore, smaller ${Z^{\prime}}$ couplings to the top
and the muon are required in the case of a heavier top partner.

###  5.3 Collider searches

Direct searches for single and pair productions of the vector-like top partner
have been performed at the LHC, and strong bounds on the mass and mixing angle
of the top partner have been derived [84, 85, 86, 87]. However, most of these
searches assume that the top partner decays exclusively into the SM particles,
i.e., $T\to bW/tZ/th$. In our case, however, $T\to t{Z^{\prime}}$ is the
dominant decay channel in the range $m_{Z^{\prime}}<m_{T}-m_{t}$, which could
relax the constraints from these direct searches [88, 89, 90].

In the model considered, the ${Z^{\prime}}$ boson couples exclusively to the
top quark and the top partner. Collider phenomenologies of such a top-philic
${Z^{\prime}}$ have been investigated in refs. [54, 55, 56, 51, 57]. At the
LHC, the dominant production channels include $pp\to t\bar{t}{Z^{\prime}}$ at
the tree and $pp\to j{Z^{\prime}}$ at the one-loop level [57]. The
${Z^{\prime}}\to\mu^{+}\mu^{-}$ decay makes the dimuon resonance searches [91,
92] sensitive to this scenario. In addition, searches for multi-top final
states could also provide important probe of the ${Z^{\prime}}$ boson as well
as the top partner [57]. A complete analysis of the collider searches may
provide interesting constraints, which is however beyond the scope of this
work.

##  6 Conclusion

The latest CDF measurement of the $W$-boson mass shows about $7\sigma$
deviation from the SM expectation obtained by a global EW fit. We have pointed
out that the latest CDF $m_{W}$ measurement and the $b\to s\ell^{+}\ell^{-}$
anomalies could be simultaneously accommodated by introducing additional
fermions, which have the same quantum numbers as of the SM quarks. As a simple
example, the model introduced in ref. [51], which is characterized by a
vector-like top partner with an additional $U(1)^{\prime}$ gauge symmetry, is
considered. We have calculated its contributions to the oblique parameters
$S$, $T$ and $U$ in the global EW fit as well as the Wilson coefficients
${\mathcal{C}}_{9}$ and ${\mathcal{C}}_{10}$ in the $b\to s\ell^{+}\ell^{-}$
transitions. Our numerical results show that the top-partner loops in the
vacuum polarization of gauge bosons can explain the latest CDF $m_{W}$
measurement and, at the same time, the ${Z^{\prime}}$-penguin diagrams
involving the top partner can account for the $b\to s\ell^{+}\ell^{-}$
anomalies. Both the ${Z^{\prime}}$ boson and the top partner can be as light
as of a few hundreds of ${\,\rm GeV}$ and thus may be accessible at the LHC
Run III and its upgrade. Therefore, our work has demonstrated that the latest
CDF $W$-boson mass shift could be related to the $b\to s\ell^{+}\ell^{-}$
anomalies, which is expected to inspire more complete model buildings for
unified theories containing new fermions.

## Note Added

As this paper was being finalized, refs. [40, 41, 42, 43, 44, 45, 93] appeared
that also consider possible explanations of the latest CDF $m_{W}$ measurement
with vector-like top partners.

## Acknowledgements

This work is supported by the National Natural Science Foundation of China
under Grant Nos. 12135006, 12075097, and 11805077, as well as by the self-
determined research funds of CCNU from the colleges’ basic research and
operation of MOE under Grant Nos. CCNU19TD012 and CCNU20TS007. XY is also
supported in part by the Startup Research Funding from CCNU.

## Appendix. Loop functions

The loop functions $K_{1,2,3}(x,y)$ in the oblique parameters $S$, $T$ and $U$
in eqs. (17)–(19) are listed below

$\displaystyle K_{1}(x,y)$ $\displaystyle=-44x+2\ln x+f_{-9}(x,x)-(x\to y),$
(A.28) $\displaystyle K_{2}(x,y)$
$\displaystyle=(x-y)^{2}-\left[(x-y)^{3}-3(x-y)-\frac{6xy}{x-y}\right]\ln x$
$\displaystyle-3x-f_{3}(x,x)+f_{3}(x,y)+(x\leftrightarrow y),$ (A.29)
$\displaystyle K_{3}(x,y)$ $\displaystyle=-3x-6x^{2}-32y+(8-18x+6x^{3})\ln x$
$\displaystyle-(12-18x+6x^{3})\ln(x-1)+4f_{0}(y,y),$ (A.30)

with the function

$\displaystyle f_{n}(x,y)$
$\displaystyle=-\left[(x+y-1)^{2}-4xy-3+n(x+y)\right]$
$\displaystyle\times\sqrt{4xy-(x+y-1)^{2}}\cos^{-1}\left(\frac{x+y-1}{\sqrt{4xy}}\right).$
(A.31)

Explicit expressions of the loop functions in the Wilson coefficients
${\mathcal{C}}_{9}^{\rm NP}$ and ${\mathcal{C}}_{10}^{\rm NP}$ in eqs. (21)
and (22) are given, respectively, by

$\displaystyle I_{1}$
$\displaystyle=f_{t}\left(-\frac{17}{9},-\frac{7}{36},\frac{1}{6}\right)$
$\displaystyle+\left[-\frac{1}{6}+f_{t}\left(2,2,\frac{1}{9},-\frac{1}{6}\right)\right]\ln
x_{t}-(t\to T),$ (A.32) $\displaystyle I_{2}$
$\displaystyle=\frac{x_{t}}{2}-f_{t}\left(\frac{3}{2}\right)+\left[\frac{3}{2}+f_{t}\left(3,\frac{3}{2}\right)\right]\ln
x_{t}-(t\to T),$ (A.33) $\displaystyle I_{3}$
$\displaystyle=\frac{x_{t}+x_{T}}{2}+\frac{x_{t}x_{T}}{x_{T}-x_{t}}\ln\frac{x_{t}}{x_{T}},$
(A.34) $\displaystyle I_{4}$
$\displaystyle=\frac{3x_{t}}{4}\left[\frac{1}{3}-f_{t}(1)+2g(x_{T},x_{t})\ln
x_{t}+(t\leftrightarrow T)\right],$ (A.35) $\displaystyle I_{5}$
$\displaystyle=\frac{x_{t}^{2}}{2}\left[\frac{1}{x_{t}}+\frac{2\ln
x_{t}}{x_{T}-x_{t}}+(t\leftrightarrow T)\right],$ (A.36)

with $x_{t}=m_{t}^{2}/m_{W}^{2}$, $x_{T}=m_{T}^{2}/m_{W}^{2}$, and the
functions

$\displaystyle g(x,y)=\frac{y(4-8y+y^{2})+x(4-2y+y^{2})}{6(x-y)(y-1)^{2}},$
(A.37) $\displaystyle
f_{q}(a_{1},a_{2},a_{3},\dotsc,a_{n})=\sum_{i=1}^{n}\frac{a_{i}}{(x_{q}-1)^{i}}.$
(A.38)

## References

  * [1] P. Zyla, et al., Review of Particle Physics, PTEP 2020 (8) (2020) 083C01, and 2021 update.  doi:10.1093/ptep/ptaa104.
  * [2] G. Aad, et al., Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B 716 (2012) 1–29.  arXiv:1207.7214, doi:10.1016/j.physletb.2012.08.020.
  * [3] S. Chatrchyan, et al., Observation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC, Phys. Lett. B 716 (2012) 30–61.  arXiv:1207.7235, doi:10.1016/j.physletb.2012.08.021.
  * [4] I. Zurbano Fernandez, et al., High-Luminosity Large Hadron Collider (HL-LHC): Technical design report 10/2020 (12 2020).  doi:10.23731/CYRM-2020-0010.
  * [5] S. Bifani, S. Descotes-Genon, A. Romero Vidal, M.-H. Schune, Review of Lepton Universality tests in $B$ decays, J. Phys. G 46 (2) (2019) 023001.  arXiv:1809.06229, doi:10.1088/1361-6471/aaf5de.
  * [6] J. Albrecht, D. van Dyk, C. Langenbruch, Flavour anomalies in heavy quark decays, Prog. Part. Nucl. Phys. 120 (2021) 103885.  arXiv:2107.04822, doi:10.1016/j.ppnp.2021.103885.
  * [7] D. London, J. Matias, $B$ Flavour Anomalies: 2021 Theoretical Status Report (10 2021).  arXiv:2110.13270, doi:10.1146/annurev-nucl-102020-090209.
  * [8] W. F. L. Hollik, Radiative Corrections in the Standard Model and their Role for Precision Tests of the Electroweak Theory, Fortsch. Phys. 38 (1990) 165–260.  doi:10.1002/prop.2190380302.
  * [9] P. Langacker, M.-x. Luo, A. K. Mann, High precision electroweak experiments: A Global search for new physics beyond the standard model, Rev. Mod. Phys. 64 (1992) 87–192.  doi:10.1103/RevModPhys.64.87.
  * [10] J. Erler, M. Schott, Electroweak Precision Tests of the Standard Model after the Discovery of the Higgs Boson, Prog. Part. Nucl. Phys. 106 (2019) 68–119.  arXiv:1902.05142, doi:10.1016/j.ppnp.2019.02.007.
  * [11] T. Aaltonen, et al., High-precision measurement of the W boson mass with the CDF II detector, Science 376 (6589) (2022) 170–176.  doi:10.1126/science.abk1781.
  * [12] R. Aaij, et al., Measurement of the W boson mass, JHEP 01 (2022) 036.  arXiv:2109.01113, doi:10.1007/JHEP01(2022)036.
  * [13] A. Strumia, Interpreting electroweak precision data including the $W$-mass CDF anomaly (4 2022).  arXiv:2204.04191.
  * [14] P. Asadi, C. Cesarotti, K. Fraser, S. Homiller, A. Parikh, Oblique Lessons from the $W$ Mass Measurement at CDF II (4 2022).  arXiv:2204.05283.
  * [15] J. Gu, Z. Liu, T. Ma, J. Shu, Speculations on the W-Mass Measurement at CDF (4 2022).  arXiv:2204.05296.
  * [16] C.-T. Lu, L. Wu, Y. Wu, B. Zhu, Electroweak Precision Fit and New Physics in light of $W$ Boson Mass (4 2022).  arXiv:2204.03796.
  * [17] J. de Blas, M. Pierini, L. Reina, L. Silvestrini, Impact of the recent measurements of the top-quark and W-boson masses on electroweak precision fits (4 2022).  arXiv:2204.04204.
  * [18] J. Fan, L. Li, T. Liu, K.-F. Lyu, $W$-Boson Mass, Electroweak Precision Tests and SMEFT (4 2022).  arXiv:2204.04805.
  * [19] X. Liu, S.-Y. Guo, B. Zhu, Y. Li, Unifying gravitational waves with $W$ boson, FIMP dark matter, and Majorana Seesaw mechanism (4 2022).  arXiv:2204.04834.
  * [20] E. Bagnaschi, J. Ellis, M. Madigan, K. Mimasu, V. Sanz, T. You, SMEFT Analysis of $m_{W}$ (4 2022).  arXiv:2204.05260.
  * [21] A. Paul, M. Valli, Violation of custodial symmetry from W-boson mass measurements (4 2022).  arXiv:2204.05267.
  * [22] Y. Heo, D.-W. Jung, J. S. Lee, Impact of the CDF $W$-mass anomaly on two Higgs doublet model (4 2022).  arXiv:2204.05728.
  * [23] M.-D. Zheng, F.-Z. Chen, H.-H. Zhang, The $W\ell\nu$-vertex corrections to W-boson mass in the R-parity violating MSSM (4 2022).  arXiv:2204.06541.
  * [24] E. d. S. Almeida, A. Alves, O. J. P. Eboli, M. C. Gonzalez-Garcia, Impact of CDF-II measurement of $M_{W}$ on the electroweak legacy of the LHC Run II (4 2022).  arXiv:2204.10130.
  * [25] Y. Cheng, X.-G. He, F. Huang, J. Sun, Z.-P. Xing, Dark photon kinetic mixing effects for CDF W mass excess (4 2022).  arXiv:2204.10156.
  * [26] T.-K. Chen, C.-W. Chiang, K. Yagyu, Explanation of the $W$ mass shift at CDF II in the Georgi-Machacek Model (4 2022).  arXiv:2204.12898.
  * [27] M. Algueró, A. Crivellin, C. A. Manzari, J. Matias, Importance of $Z-Z^{\prime}$ Mixing in $b\to s\ell^{+}\ell^{-}$ and the $W$ mass (1 2022).  arXiv:2201.08170.
  * [28] Q. Zhou, X.-F. Han, The CDF W-mass, muon g-2, and dark matter in a $U(1)_{L_{\mu}-L_{\tau}}$ model with vector-like leptons (4 2022).  arXiv:2204.13027.
  * [29] O. Popov, R. Srivastava, The Triplet Dirac Seesaw in the View of the Recent CDF-II W Mass Anomaly (4 2022).  arXiv:2204.08568.
  * [30] K. Ghorbani, P. Ghorbani, $W$-Boson Mass Anomaly from Scale Invariant 2HDM (4 2022).  arXiv:2204.09001.
  * [31] A. Bhaskar, A. A. Madathil, T. Mandal, S. Mitra, Combined explanation of $W$-mass, muon $g-2$, $R_{K^{(*)}}$ and $R_{D^{(*)}}$ anomalies in a singlet-triplet scalar leptoquark model (4 2022).  arXiv:2204.09031.
  * [32] J. J. Heckman, Extra $W$-Boson Mass from a D3-Brane (4 2022).  arXiv:2204.05302.
  * [33] L. Di Luzio, R. Gröber, P. Paradisi, Higgs physics confronts the $M_{W}$ anomaly (4 2022).  arXiv:2204.05284.
  * [34] Y.-Z. Fan, T.-P. Tang, Y.-L. S. Tsai, L. Wu, Inert Higgs Dark Matter for New CDF W-boson Mass and Detection Prospects (4 2022).  arXiv:2204.03693.
  * [35] T.-P. Tang, M. Abdughani, L. Feng, Y.-L. S. Tsai, Y.-Z. Fan, NMSSM neutralino dark matter for $W$-boson mass and muon $g-2$ and the promising prospect of direct detection (4 2022).  arXiv:2204.04356.
  * [36] D. Borah, S. Mahapatra, D. Nanda, N. Sahu, Type II Dirac Seesaw with Observable $\Delta N_{eff}$ in the light of W-mass Anomaly (4 2022).  arXiv:2204.08266.
  * [37] K. Sakurai, F. Takahashi, W. Yin, Singlet extensions and W boson mass in the light of the CDF II result (4 2022).  arXiv:2204.04770.
  * [38] P. Athron, A. Fowlie, C.-T. Lu, L. Wu, Y. Wu, B. Zhu, The $W$ boson Mass and Muon $g-2$: Hadronic Uncertainties or New Physics? (4 2022).  arXiv:2204.03996.
  * [39] P. Athron, M. Bach, D. H. J. Jacob, W. Kotlarski, D. Stöckinger, A. Voigt, Precise calculation of the W boson pole mass beyond the Standard Model with FlexibleSUSY (4 2022).  arXiv:2204.05285.
  * [40] H. M. Lee, K. Yamashita, A Model of Vector-like Leptons for the Muon $g-2$ and the $W$ Boson Mass (4 2022).  arXiv:2204.05024.
  * [41] A. Crivellin, M. Kirk, T. Kitahara, F. Mescia, Correlating $t\to cZ$ to the $W$ Mass and $B$ Physics with Vector-Like Quarks (4 2022).  arXiv:2204.05962.
  * [42] M. Endo, S. Mishima, New physics interpretation of $W$-boson mass anomaly (4 2022).  arXiv:2204.05965.
  * [43] R. Balkin, E. Madge, T. Menzo, G. Perez, Y. Soreq, J. Zupan, On the implications of positive W mass shift (4 2022).  arXiv:2204.05992.
  * [44] J. Cao, L. Meng, L. Shang, S. Wang, B. Yang, Interpreting the $W$ mass anomaly in the vectorlike quark models (4 2022).  arXiv:2204.09477.
  * [45] S. Baek, Implications of CDF $W$-mass and $(g-2)_{\mu}$ on $U(1)_{L_{\mu}-L_{\tau}}$ model (4 2022).  arXiv:2204.09585.
  * [46] R. Aaij, et al., Test of lepton universality with $B^{0}\rightarrow K^{*0}\ell^{+}\ell^{-}$ decays, JHEP 08 (2017) 055.  arXiv:1705.05802, doi:10.1007/JHEP08(2017)055.
  * [47] R. Aaij, et al., Test of lepton universality in beauty-quark decays, Nature Phys. 18 (3) (2022) 277–282.  arXiv:2103.11769, doi:10.1038/s41567-021-01478-8.
  * [48] R. Aaij, et al., Tests of lepton universality using $B^{0}\to K^{0}_{S}\ell^{+}\ell^{-}$ and $B^{+}\to K^{*+}\ell^{+}\ell^{-}$ decays (10 2021).  arXiv:2110.09501.
  * [49] R. Aaij, et al., Measurement of $CP$-Averaged Observables in the $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ Decay, Phys. Rev. Lett. 125 (1) (2020) 011802.  arXiv:2003.04831, doi:10.1103/PhysRevLett.125.011802.
  * [50] R. Aaij, et al., Angular Analysis of the $B^{+}\rightarrow K^{\ast+}\mu^{+}\mu^{-}$ Decay, Phys. Rev. Lett. 126 (16) (2021) 161802.  arXiv:2012.13241, doi:10.1103/PhysRevLett.126.161802.
  * [51] J. F. Kamenik, Y. Soreq, J. Zupan, Lepton flavor universality violation without new sources of quark flavor violation, Phys. Rev. D 97 (3) (2018) 035002\.  arXiv:1704.06005, doi:10.1103/PhysRevD.97.035002.
  * [52] P. J. Fox, J. Liu, D. Tucker-Smith, N. Weiner, An Effective Z’, Phys. Rev. D 84 (2011) 115006.  arXiv:1104.4127, doi:10.1103/PhysRevD.84.115006.
  * [53] J. Berger, J. Hubisz, M. Perelstein, A Fermionic Top Partner: Naturalness and the LHC, JHEP 07 (2012) 016.  arXiv:1205.0013, doi:10.1007/JHEP07(2012)016.
  * [54] N. Greiner, K. Kong, J.-C. Park, S. C. Park, J.-C. Winter, Model-Independent Production of a Top-Philic Resonance at the LHC, JHEP 04 (2015) 029.  arXiv:1410.6099, doi:10.1007/JHEP04(2015)029.
  * [55] P. Cox, A. D. Medina, T. S. Ray, A. Spray, Novel collider and dark matter phenomenology of a top-philic $Z^{\prime}$, JHEP 06 (2016) 110.  arXiv:1512.00471, doi:10.1007/JHEP06(2016)110.
  * [56] J. H. Kim, K. Kong, S. J. Lee, G. Mohlabeng, Probing TeV scale Top-Philic Resonances with Boosted Top-Tagging at the High Luminosity LHC, Phys. Rev. D 94 (3) (2016) 035023.  arXiv:1604.07421, doi:10.1103/PhysRevD.94.035023.
  * [57] P. J. Fox, I. Low, Y. Zhang, Top-philic $Z^{\prime}$ forces at the LHC, JHEP 03 (2018) 074.  arXiv:1801.03505, doi:10.1007/JHEP03(2018)074.
  * [58] A. Crivellin, C. A. Manzari, M. Alguero, J. Matias, Combined Explanation of the $Z\to b\bar{b}$ Forward-Backward Asymmetry, the Cabibbo Angle Anomaly, $\tau\to\nu\nu$ and $b\to s\ell^{+}\ell^{-}$ Data, Phys. Rev. Lett. 127 (1) (2021) 011801.  arXiv:2010.14504, doi:10.1103/PhysRevLett.127.011801.
  * [59] A. Crivellin, L. Hofer, J. Matias, U. Nierste, S. Pokorski, J. Rosiek, Lepton-flavour violating $B$ decays in generic $Z^{\prime}$ models, Phys. Rev. D 92 (5) (2015) 054013.  arXiv:1504.07928, doi:10.1103/PhysRevD.92.054013.
  * [60] D. Aristizabal Sierra, F. Staub, A. Vicente, Shedding light on the $b\to s$ anomalies with a dark sector, Phys. Rev. D 92 (1) (2015) 015001.  arXiv:1503.06077, doi:10.1103/PhysRevD.92.015001.
  * [61] H. Flacher, M. Goebel, J. Haller, A. Hocker, K. Monig, J. Stelzer, Revisiting the Global Electroweak Fit of the Standard Model and Beyond with Gfitter, Eur. Phys. J. C 60 (2009) 543–583, [Erratum: Eur.Phys.J.C 71, 1718 (2011)].  arXiv:0811.0009, doi:10.1140/epjc/s10052-009-0966-6.
  * [62] M. Baak, J. Cúth, J. Haller, A. Hoecker, R. Kogler, K. Mönig, M. Schott, J. Stelzer, The global electroweak fit at NNLO and prospects for the LHC and ILC, Eur. Phys. J. C 74 (2014) 3046.  arXiv:1407.3792, doi:10.1140/epjc/s10052-014-3046-5.
  * [63] J. Haller, A. Hoecker, R. Kogler, K. Mönig, T. Peiffer, J. Stelzer, Update of the global electroweak fit and constraints on two-Higgs-doublet models, Eur. Phys. J. C 78 (8) (2018) 675.  arXiv:1803.01853, doi:10.1140/epjc/s10052-018-6131-3.
  * [64] J. de Blas, M. Ciuchini, E. Franco, A. Goncalves, S. Mishima, M. Pierini, L. Reina, L. Silvestrini, Global analysis of electroweak data in the Standard Model (12 2021).  arXiv:2112.07274.
  * [65] M. E. Peskin, T. Takeuchi, Estimation of oblique electroweak corrections, Phys. Rev. D 46 (1992) 381–409.  doi:10.1103/PhysRevD.46.381.
  * [66] M. E. Peskin, T. Takeuchi, A New constraint on a strongly interacting Higgs sector, Phys. Rev. Lett. 65 (1990) 964–967.  doi:10.1103/PhysRevLett.65.964.
  * [67] I. Maksymyk, C. P. Burgess, D. London, Beyond S, T and U, Phys. Rev. D 50 (1994) 529–535.  arXiv:hep-ph/9306267, doi:10.1103/PhysRevD.50.529.
  * [68] G. Buchalla, A. J. Buras, M. E. Lautenbacher, Weak decays beyond leading logarithms, Rev. Mod. Phys. 68 (1996) 1125–1144.  arXiv:hep-ph/9512380, doi:10.1103/RevModPhys.68.1125.
  * [69] A. Alloul, N. D. Christensen, C. Degrande, C. Duhr, B. Fuks, FeynRules 2.0 - A complete toolbox for tree-level phenomenology, Comput. Phys. Commun. 185 (2014) 2250–2300.  arXiv:1310.1921, doi:10.1016/j.cpc.2014.04.012.
  * [70] T. Hahn, Generating Feynman diagrams and amplitudes with FeynArts 3, Comput. Phys. Commun. 140 (2001) 418–431.  arXiv:hep-ph/0012260, doi:10.1016/S0010-4655(01)00290-9.
  * [71] R. Mertig, M. Bohm, A. Denner, FEYN CALC: Computer algebraic calculation of Feynman amplitudes, Comput. Phys. Commun. 64 (1991) 345–359.  doi:10.1016/0010-4655(91)90130-D.
  * [72] V. Shtabovenko, R. Mertig, F. Orellana, New Developments in FeynCalc 9.0, Comput. Phys. Commun. 207 (2016) 432–444.  arXiv:1601.01167, doi:10.1016/j.cpc.2016.06.008.
  * [73] V. Shtabovenko, R. Mertig, F. Orellana, FeynCalc 9.3: New features and improvements, Comput. Phys. Commun. 256 (2020) 107478.  arXiv:2001.04407, doi:10.1016/j.cpc.2020.107478.
  * [74] H. H. Patel, Package-X 2.0: A Mathematica package for the analytic calculation of one-loop integrals, Comput. Phys. Commun. 218 (2017) 66–70.  arXiv:1612.00009, doi:10.1016/j.cpc.2017.04.015.
  * [75] J. De Blas, et al., HEPfit: a code for the combination of indirect and direct constraints on high energy physics models, Eur. Phys. J. C 80 (5) (2020) 456.  arXiv:1910.14012, doi:10.1140/epjc/s10052-020-7904-z.
  * [76] W. Altmannshofer, D. M. Straub, New physics in $b\rightarrow s$ transitions after LHC run 1, Eur. Phys. J. C 75 (8) (2015) 382.  arXiv:1411.3161, doi:10.1140/epjc/s10052-015-3602-7.
  * [77] D. M. Straub, flavio: a Python package for flavour and precision phenomenology in the Standard Model and beyond (10 2018).  arXiv:1810.08132.
  * [78] X.-Q. Li, M. Shen, D.-Y. Wang, Y.-D. Yang, X.-B. Yuan, Explaining the $b\to s\ell^{+}\ell^{-}$ anomalies in $Z^{\prime}$ scenarios with top-FCNC couplings (12 2021).  arXiv:2112.14215.
  * [79] L.-S. Geng, B. Grinstein, S. Jäger, S.-Y. Li, J. Martin Camalich, R.-X. Shi, Implications of new evidence for lepton-universality violation in $b\to s\ell^{+}\ell^{-}$ decays, Phys. Rev. D 104 (3) (2021) 035029.  arXiv:2103.12738, doi:10.1103/PhysRevD.104.035029.
  * [80] W. Altmannshofer, P. Stangl, New physics in rare B decays after Moriond 2021, Eur. Phys. J. C 81 (10) (2021) 952.  arXiv:2103.13370, doi:10.1140/epjc/s10052-021-09725-1.
  * [81] C. Cornella, D. A. Faroughy, J. Fuentes-Martin, G. Isidori, M. Neubert, Reading the footprints of the B-meson flavor anomalies, JHEP 08 (2021) 050.  arXiv:2103.16558, doi:10.1007/JHEP08(2021)050.
  * [82] M. Algueró, B. Capdevila, S. Descotes-Genon, J. Matias, M. Novoa-Brunet, $b\to s\ell\ell$ Global Fits after $R_{K_{S}}$ and $R_{K^{*+}}$ (4 2021).  arXiv:2104.08921.
  * [83] T. Hurth, F. Mahmoudi, D. M. Santos, S. Neshatpour, More indications for lepton nonuniversality in $b\to s\ell^{+}\ell^{-}$, Phys. Lett. B 824 (2022) 136838\.  arXiv:2104.10058, doi:10.1016/j.physletb.2021.136838.
  * [84] M. Aaboud, et al., Combination of the searches for pair-produced vector-like partners of the third-generation quarks at $\sqrt{s}=$ 13 TeV with the ATLAS detector, Phys. Rev. Lett. 121 (21) (2018) 211801.  arXiv:1808.02343, doi:10.1103/PhysRevLett.121.211801.
  * [85] A. Tumasyan, et al., Search for single production of a vector-like T quark decaying to a top quark and a Z boson in the final state with jets and missing transverse momentum at $\sqrt{s}$ = 13 TeV (1 2022).  arXiv:2201.02227.
  * [86] J. A. Aguilar-Saavedra, R. Benbrik, S. Heinemeyer, M. Pérez-Victoria, Handbook of vectorlike quarks: Mixing and single production, Phys. Rev. D 88 (9) (2013) 094010.  arXiv:1306.0572, doi:10.1103/PhysRevD.88.094010.
  * [87] H. de la TorreTrisha Farooque, T. Farooque, Looking beyond the Standard Model with Third Generation Quarks at the LHC, Symmetry 14 (3) (2022) 444.  doi:10.3390/sym14030444.
  * [88] J. Serra, Beyond the Minimal Top Partner Decay, JHEP 09 (2015) 176.  arXiv:1506.05110, doi:10.1007/JHEP09(2015)176.
  * [89] A. Anandakrishnan, J. H. Collins, M. Farina, E. Kuflik, M. Perelstein, Odd Top Partners at the LHC, Phys. Rev. D 93 (7) (2016) 075009.  arXiv:1506.05130, doi:10.1103/PhysRevD.93.075009.
  * [90] N. Bizot, G. Cacciapaglia, T. Flacke, Common exotic decays of top partners, JHEP 06 (2018) 065.  arXiv:1803.00021, doi:10.1007/JHEP06(2018)065.
  * [91] G. Aad, et al., Search for high-mass dilepton resonances using 139 fb-1 of $pp$ collision data collected at $\sqrt{s}=$13 TeV with the ATLAS detector, Phys. Lett. B 796 (2019) 68–87.  arXiv:1903.06248, doi:10.1016/j.physletb.2019.07.016.
  * [92] A. M. Sirunyan, et al., Search for resonant and nonresonant new phenomena in high-mass dilepton final states at $\sqrt{s}$ = 13 TeV, JHEP 07 (2021) 208\.  arXiv:2103.02708, doi:10.1007/JHEP07(2021)208.
  * [93] D. Borah, S. Mahapatra, N. Sahu, Singlet-Doublet Fermion Origin of Dark Matter, Neutrino Mass and W-Mass Anomaly (4 2022).  arXiv:2204.09671.