source detections are located along the spikes of the PSF. Those spurious
detections are filtered out by our detection fraction filter with high
efficiency (see Sect. 3.6), since in different scans the spikes are at
different orientations relative to the scene.

####  Iterations

Unlike the nominal IPD, SIF CF IPD has the ability to review the source
detections and PSF fits in the context of the source surroundings. Once all
sources are located and fit for the first time, the PSF fits are repeated,
starting again with the brightest source. In this second fit iteration, the
information about the source surroundings is improved, since now even for the
brightest source there is a full scene available, consisting of modelled
sources and background.

When the second PSF fit is completed for all sources, the source detection is
run again. It now takes into account the knowledge about all already detected
and fitted sources. This allows the pipeline to detect fainter sources and
sources closer to bright ones. The second source detection run is followed by
another two PSF fit iterations for all sources, old and newly detected ones.

Finally, a third block of one source detection and two PSF fit runs is
performed. The combination of three source detection and six PSF fit runs
turned out to be the best trade-off between performance improvement and run
time increase. In every SIF CF image, between 100 to 700 000 source detections
are found; this extreme range is a result of the drastic differences in image
size rather than simply the source density within the images.

When source detection and PSF fit runs are completed, some basic filters are
applied to remove internal duplicates and detections located outside the CCD
detectors. The full list of filters is given in Sect. 3.6.

####

In short, SIF CF IPD iterates source finding and PSF fitting for all sources
in the SIF CF image ordered by source brightness. In doing so, it takes into
account an artificial image constructed from PSF models of already detected
sources and background to create well defined new source detections.

###  3.2 SIF CF XM

The SIF CF Crossmatch (SIF CF XM) module assigns the SIF CF detections to
already known and new sources. During this process, clusters of SIF CF
detections belonging to the same physical light source on the sky are formed,
and a unique source identifier is assigned to each such cluster of detections.

The SIF CF XM code is a slightly altered version of IDU XM, see Torra et al.
(2021) and Section 3.4.13 in Castañeda et al. (2022). The SIF CF version does
not include the first XM stage (the detection classifier), which distinguishes
real from spurious detections. As explained in the previous section, SIF CF
detections are found on-ground by SIF CF IPD from SIF CF images and thus are
not affected by limitations of the Gaia on-board detection software and
processing resources. The on-ground detection performance significantly
reduces the appearance of spurious detections and their patterns do not match
those from nominal data. Subsequent XM stages use the same algorithms as the
IDU XM but with different parameters and source catalogue as described below.

####  Parameters

The main difference between nominal IDU XM and SIF CF XM processing is the
choice of input parameters regarding detection clustering. These parameter
differences are caused by the fact that the default XM parameters are
optimised for typical source densities of the Gaia catalogue, and SIF CF XM
operates in a much denser environment for which a different set of parameters
is needed. To find the best parameter set, SIF CF XM was run with various
parameter settings, improving the clustering performance on the dense regions
of SIF CF data. The results were compared to an HST reference catalogue
(Bellini et al. 2017, see Sect. 4.6). The optimal set of parameters increased
the number of matches to the reference catalogue, and hence decreased the
number of unmatched sources from both, SIF CF and HST.

####  Nominal source input

Instead of providing just the SIF CF detections as input to the SIF CF XM
software, we additionally fed a subset of nominal Gaia sources to it, that is,
exactly those nominal sources that were already published in Gaia DR3 for the
$\omega$ Cen region. SIF CF detections could then be assigned to either a
newly created SIF CF source or to an existing nominal source, and SIF CF XM
returned two easily distinguishable source sets:

  * $\bullet$

Known Gaia DR3 sources together with their matched SIF CF detections. As these
sources are published in Gaia DR3 already, they are excluded from further
processing in the SIF CF pipeline, and not published again in this FPR. They
are used as input for the photometric overlap calibration though, and also for
internal validation.

  * $\bullet$

Newly created SIF CF sources with their matched SIF CF detections. These are
new sources with respect to Gaia DR3. Astrometry and photometry are computed
for them, and these sources are published in this FPR.

Contrary to what is done in the nominal XM processing, the nominal input
sources are not altered by SIF CF XM. Therefore the SIF CF XM configuration is
changed to prevent merging or splitting these sources.

The reason to discard sources that have a nominal match in Gaia DR3 from the
output list and not to publish them is that they are already present in DR3.
These sources are created, used internally for the photometric overlap
calibration, and then discarded. Doing so ensures the FPR data to be a truly
disjunct extension to Gaia DR3.

####  No detection mixing

All SIF CF sources are based on SIF CF detections only, and thus there is no
mixing of SIF CF and nominal Gaia data in any of the sources. Gaia DR3 sources
are based on nominal Gaia SM and AF observations only, and SIF CF sources are
derived from SIF CF observations only.

####  Source identifiers

The SIF CF XM module assigns a unique source identifier, the sourceId, see
Section 20.1.1 in Hambly et al. (2022), to all newly created SIF CF sources.
This sourceId follows the same scheme as for nominal Gaia sources and codes a
spatial index, the source provenance (code of the data processing centre or
system where it was created), a running number, and a component number.111The
SIF CF sourceIds use an as of yet unused provenance code (code = 6), which
allows one to easily distinguish SIF CF sourceIds from nominal Gaia sourceIds
using the following expression:
(sourceId >> 32) & 7 == 6.

####

For a more detailed description of the different stages and algorithms used in
XM, we refer to Torra et al. (2021). The next step in the chain addresses the
calculation of coverage maps for the SIF CF images.

###  3.3 SIF CF Coverage

Figure 1: $\omega$ Cen area coloured by the number of SIF CF scans.

The SIF CF Coverage task filters unreliable sources by calculating the number
of possible detections for each source location n_scans, and comparing it to
the number of actual detections assigned to the source matched_transits. In
order to assess the number of possible detections, which is the number of SIF
CF images covering the precise location within a SIF CF region, the image
boundaries are transferred to sky coordinates. As SIF CF images can be very
long, the image is described as a polygon rather than a rectangle. The
combination of all image descriptions provides the number of scans for each
source location (see Fig. 1).

We define the image boundary as a polygon in the Cartesian space of right
ascension and declination. This is not equivalent to a polygon in spherical
coordinates. Minor attitude inconsistencies, rounding errors etc. add to this
effect. As a consequence, for some sources located at the image borders,
n_scans might be incorrect. In a test dataset, this was true for just one out
of $\sim$ 200 000 sources. Even if errors add up for several images, numbers
are so small that it seems tolerable, and it was decided not to implement
proper spherical polygons in this task.

###  3.4 SIF CF AGIS

The Astrometric Global Iterative Solution (AGIS) software provides precise
astrometry for all nominal Gaia sources. A detailed description of AGIS can be
found in Lindegren et al. (2021) and Section 4.1.1 in Hobbs et al. (2022). The
nominal AGIS software ran unaltered for the SIF CF sources.

The operational SIF CF AGIS run for this FPR took place in March 2022 and
created AGIS solutions for all SIF CF sources published in this FPR. For SIF
CF, in the AGIS run only the astrometry of the sources is updated. This is
known as the AGIS secondary solution. When running AGIS in secondary mode, it
needs a calibrated initial attitude and a geometrical calibration model as
input. These are not updated by the SIF CF AGIS pipeline but supplied from a
previous nominal AGIS run in its primary mode. Not a single source failed to
converge.

The SIF CF AGIS run used a six parameter solution (position, proper motion,
parallax, and pseudo colour) with a pseudo-colour prior222As for faint sources
the colour is less crucial than for bright ones, and SIF CF sources are
generally rather faint, the default nominal pseudo colour was used as colour
prior to optimise software run time. of $1.43\pm 0.1\,\,{\mu}m^{-1}$. The
quality of the pseudo colour measurements obtained without any colour
information, thus purely from astrometry, is rather low. It is not advised to
use these as a proxy for the colour.

Apart from the astrometry for the newly created SIF CF sources, SIF CF AGIS
also produced AGIS solutions for those SIF CF detections assigned to an
already known Gaia source. These SIF CF solutions of already published Gaia
DR3 sources, were used for the photometric calibration of SIF CF sources (see
Sect. 3.5) and for internal validation. They were discarded after use in order
to avoid a duplication of existing Gaia DR3 sources in the FPR catalogue.

###  3.5 SIF CF photometry

In order to ensure consistency between the SIF CF photometry and the most-
recently released Gaia catalogue, the photometric calibration of the SIF CF
data follows a more traditional approach compared to the one for Gaia DR3.
Rather than using the data themselves to define the photometric system, a set
of calibrators was selected among those unpublished SIF CF sources that had
counterparts in Gaia DR3. The Gaia DR3 photometry was then used as a reference
to calibrate the SIF CF observations.333An alternative approach would involve
applying the photometric calibrations derived in the nominal photometric
processing (Riello et al., 2021) for the SM CCDs. However, for this first run
of the SIF CF processing, several inconsistencies in the IPD, most importantly
the use of a different PSF library with respect to the nominal run, made this
option not applicable. This choice will be re-evaluated for future releases.

The time evolution of the photometric calibration was taken into account by
splitting the input data into groups of observations separated by more than 30
satellite revolutions and not longer than six revolutions. For each group, one
calibration per CCD (and thus per FOV) was solved for independently. The
calibration model is described by a quadratic function in AC coordinate and
provides the ratio between observed SIF CF and Gaia DR3 reference flux. No
magnitude term was introduced in this step.

Applying these calibrations to the observed fluxes effectively translates them
to the photometric system of Gaia DR3, removing all differential effects due
to the evolution in time of the instrument, differences between the CCDs, and
other elements in the optical paths. This allows the procedure to accumulate
all single observations of a source to produce a mean source photometry.

Among all sources from Gaia DR3 with observations in SIF CF mode, only those
with gaia_source. phot_variable_flag =’NOT_AVAILABLE’, $15<G<20$ and
$0.7<G_{\rm BP}-G_{\rm BP}<4.0$ were used as calibrators. The magnitude and
colour ranges were defined to avoid strong magnitude and colour terms that
might affect the observed fluxes with respect to the reference photometry.
These are visible in Fig. 2, where the difference between the SIF CF
calibrated magnitude and the reference one are shown versus colour for a range
of magnitudes in the top panel, and versus magnitude for a range of colours in
the bottom panel.

Figure 2: Colour and magnitude terms between the SIF CF calibrated magnitudes
and the reference magnitudes from Gaia DR3. The magnitude terms have been
corrected for in this FPR.

Using the colour information available for the calibrators, an ad hoc
calibration could be derived. However, as no colour information is available
for SIF CF sources, it is not possible to apply such calibration. As a result
of this, the calibrated photometry for new SIF CF sources bluer than 0.7 in
$G_{\rm BP}-G_{\rm RP}$ will suffer from a systematic colour term of magnitude
similar to that shown by the blue line in the top panel of Fig. 2. This
includes sources in the Blue Horizontal Branch. These are visible in the
bottom panel of Fig. 2 as a population of sources that, starting at around $G$
magnitude $15$, show large (of the order of a few percent) differences between
the SIF CF calibrated magnitudes and the reference Gaia DR3 magnitudes.

No attempt was made to calibrate the strong magnitude term for $G<13$ as only
a handful of SIF CF sources exist in that magnitude range. The magnitude term
affecting fainter sources instead was calibrated out and removed from the mean
source photometry.

Once the initial mean photometry was obtained applying the calibrations to all
observations, we evaluated the remaining effects not described by our model
and estimated the remaining magnitude term. It is calibrated empirically as a
simple offset to be applied to the SIF CF mean photometry in order to bring it
into agreement with the corresponding Gaia DR3 values using sources for which
mean photometry is available in both Gaia DR3 and SIF CF in the magnitude
range $[13,20]$ in $G$ and colour $0.7<G_{\rm BP}-G_{\rm RP}<4.0$. When
applying this additional correction to the mean photometry, sources outside
the magnitude range were corrected using the value for the closest magnitude
included in the range. When applying the correction to sources in the covered
magnitude range, interpolation is used between the bin values.

Figure 3: Comparison between SIF CF calibrated magnitudes and reference
magnitudes from Gaia DR3 for the sources in common after the removal of the
magnitude term.

Figure 3 shows the same comparison as the bottom panel of Fig. 2 after
application of the magnitude term calibration. The correction was applied to
the mean source photometry and to all relevant quantities, which are to
median, minimum, maximum, and quartiles. It has no effect on photometric
uncertainties as the error on the magnitude term is insignificant with respect
to the uncertainties of the photometric measurements.

It is probably useful to stress again that the calibrations described in this
section are not published in this FPR, but ensure consistency between the SIF
CF and Gaia DR3 photometry. Systematics affecting the reference photometry,
that is Gaia DR3, will also be present in the SIF CF FPR catalogue. Interested
users are therefore encouraged to refer to the Gaia Early Data Release 3
(Riello et al., 2021; Fabricius et al., 2021) and DR3 (Gaia Collaboration et
al., 2023) papers where the known systematics are discussed.

###  3.6 Filters

Unlike in nominal Gaia data releases, no specific filtering is applied as a
last step of the data processing. Some detections and sources did get
discarded though during operation of the SIF CF pipeline:

####  Filters at detection level

In SIF CF IPD and SIF CF XM we removed the following detections:

  * •

Detections for which the PSF fit did not converge or found a source position
outside the hypothetical observation window. Most likely these sources are
actually just noise.

  * •

Detections with internal duplicates. For the very rare cases, in which after
the status cleaning two sources are located within the same subsample region,
only the brightest source is kept. Most likely the fitting procedure here used
two PSFs to fit one source.

  * •

Detections that are located outside the SM CCD detector.

  * •

Detections that would result outside the other instrument CCD detectors, as
these cannot be processed by the downstream systems.

  * •

Detections with less than 50 counts, which corresponds to a magnitude of 22.5,
because they are most likely spurious.

  * •

Detections with a positional uncertainty of more than 1 pixel and/or AL offset
from PSF fit window centre of more than 1.2 pixels.

####  Filters at source level

In SIF CF XM, SIF CF Coverage, and SIF CF AGIS post processing we removed the
following sources:

  * •

Sources with less than 11 observations in total. This we regarded as the
minimum number of detections required for a reliable source.

  * •

Sources with a detection fraction of less than 50 % (see Sect. 1). Sources
that were observed in less than half of the cases that they could have been
observed are most likely spurious.

  * •

Sources with positional error bars larger than 100 mas (30 sources removed).

  * •

Sources within $0\aas@@fstack{\prime\prime}16$ distance from a brighter star.
This was done irrespective of whether the brighter star was a Gaia DR3 source
or a SIF CF source. We do not trust sources that are too close to brighter
ones, as these can easily result from spike remnants left over when
subtracting the scene containing the PSF model of a bright source from its
observed image before source detection (673 sources removed).

  * •

Non-converged sources (this did not actually happen).

The specification of SIF CF filters follows the nominal processing rules as
closely as possible. The last three filters, applied during AGIS post-
processing, are in fact identical except for the precise distance to a
brighter star. This distance was increased from the nominal value of
$0\aas@@fstack{\prime\prime}12$ to $0\aas@@fstack{\prime\prime}16$ for SIF CF.

After filtering, 526 587 SIF CF sources persisted in $\omega$ Cen. These are
published with this FPR.

##  4 Validation

In order to assure the quality of the SIF CF data product, a scientific
validation has been performed. For this, we defined a fixed colour for each
dataset that we use. Except for the last figure, throughout the whole document
we use yellow for Gaia DR3 data, violet for SIF CF data published in this FPR,
and dark blue for the combination of these two. Additionally, there are
unpublished SIF CF data, which were used for photometric calibration only, in
blue and HST reference data in teal.

###  4.1 Numbers

The number of detections in a single image ranges from 208 to 471 039. In
total, SIF CF IPD produced 66 660 921 detections from 2 329 SIF CF images of
$\omega$ Cen. From these, 45 747 947 detections were matched to a source, and
27 500 167 of these were matched to a new source not present in Gaia DR3.

Originally, SIF CF XM created 848 988 sources, of which 527 290 sources
persisted after filtering for already known Gaia DR3 sources, the detection
fraction, and minimum number of observations (see Sect. 3.6). Filtering for
sources too close to brighter stars and sources with high uncertainties
removed another 703 sources. The remaining 526 587 sources were consolidated
from 27 476 327 detections.

Figure 4: Number of detections and detection fraction. Distribution of sources
with a certain amount of detections (left) and detection fraction (right) for
SIF CF sources.

Figure 4 shows the distribution of the absolute number of detections for the
SIF CF sources (left) and the fraction of the maximum possible observations
(right). Most SIF CF sources have a very good detection fraction; they are
observed in almost every image covering their location. A smaller amount of
sources is observed in 50 to 90 percent of the scans. The absolute number of
detections (left) shows that SIF CF sources are generally well probed, in most
cases with 50 - 60 detections.

###  4.2 Photometry

To evaluate the effect of the photometric calibration on the uncertainties, it
is interesting to see how these change with the application of the
calibration, and how they compare with expectations from Gaia DR3. In the
first instance, this can be done for the set of calibrators, for which the
expected and actual uncertainties from Gaia DR3 are available. Both are
valuable to quantify the effects of crowding on the photometric uncertainties.

Figure 5: Median G magnitude uncertainty versus G magnitude. G magnitude
uncertainty for all calibrators with data density in grey, see the legend and
text for a description of the various overlaid curves. Figure 6: Distribution
of magnitude and magnitude uncertainty for SIF CF sources in $\omega$ Cen.
Top: Magnitude distribution. Bottom: Distribution of the mean G magnitude
uncertainty versus G magnitude with data density in grey, see the legend and
text for a description of the various overlaid curves.

The density plot in Fig. 5 shows the distribution of photometric uncertainties
versus Gaia DR3 magnitude for all the sources selected as calibrators. The
dashed blue line illustrates the median photometric error for the uncalibrated
SIF CF data. In this case, the source photometry is generated by simply taking
a weighted average of the fluxes as measured by SIF CF IDU. The continuous
blue line presents the median uncertainty of the calibrated SIF CF photometry.
The density distribution of these data is provided in grey. The continuous
yellow line provides the median photometric error measured from the Gaia DR3
photometry for the same set of sources. Finally, the dashed yellow line shows
the expected Gaia DR3 photometry.

We provide this last curve to offer an idea of the quality of the photometry
for the sources used as calibrators with respect to the Gaia DR3 photometric
catalogue. It is expected that the Gaia DR3 photometry for sources in a region
centred on $\omega$ Cen will be affected by crowding. All photometric errors
have been scaled to correspond to a number of 50 observations, which is close
to the average number of observations for the SIF CF FPR catalogue.

The G magnitude distribution of all SIF CF sources is presented in the top
panel of Fig. 6. The bottom panel of the same figure shows the distribution of
the mode of the photometric uncertainties for all SIF CF sources. The data
density is given in grey. As for these sources no photometry is available from
Gaia DR3, only the expected nominal Gaia DR3 photometric uncertainties are
provided for comparison (dashed yellow line). The continuous violet line in
this plot depicts the mode rather than the median (as in Fig. 5) given the
rather skewed distribution at the faint end.

The comparison of the blue dashed and continuous lines in Fig. 5 demonstrates
that the applied calibration is efficient in reducing the scatter (as inferred
from the measured uncertainties) between observed fluxes for a given source.
The remaining scatter is larger than for the Gaia DR3 photometry. This is
expected due to the different observing conditions, as only measurements from
SM CCDs contribute to the SIF CF photometry. A larger scatter can also be due
to the fact that all of these sources are observed in dense regions, where a
high fraction of sources is blended and different scanning directions may give
rise to different contamination from nearby objects. This is also apparent
from the larger Gaia DR3 photometric uncertainties characterising the
calibrators in this region with respect to the level expected from the entire
catalogue.

###  4.3 Position

Figure 7: Median positional uncertainties. Uncertainties in right ascension
$\sigma_{\alpha^{*}}$ (dashed lines) and declination $\sigma_{\delta}$ (solid
lines) for SIF CF (violet) and Gaia DR3 (yellow) in $\omega$ Cen.

A direct comparison of SIF CF and Gaia DR3 astrometry is not straightforward.
The Gaia DR3 data have very different input as these sources are created from
unbinned and more AF observations per transit; they are generally brighter,
situated in less dense environments, and their calibration including the PSF
model is better defined. The SIF CF sources instead are based on binned SM
data with just one observation per transit; as all sources with Gaia DR3
counterpart were removed, these are fainter sources situated in denser
environments (see Fig. 14); and as SM data was originally not meant to be used
for scientific exploitation, the calibrations are less accurate. However, a
few important things can be noticed nevertheless.

In Fig. 7 we show the median uncertainties of source positions
$\sigma_{\alpha^{*}}$ (dashed line) and $\sigma_{\delta}$ (solid line) as a
function of G magnitude in $\omega$ Cen for newly created SIF CF sources
(violet) and Gaia DR3 sources (yellow). Overall, the SIF CF data have a median
positional uncertainty of 3.03 mas in right ascension (RA) and 2.69 mas
declination (Dec). The reason for the difference between uncertainties in RA
and Dec is most likely connected to the asymmetry in scan direction
distribution.

The median positional uncertainties (and also uncertainties in parallax and
proper motion, see Fig. 10 and Fig. 12) are substantially larger for SIF CF
data (violet lines) than for Gaia DR3 (yellow lines). This is not at all
unexpected due to the above mentioned limitations.

###  4.4 Proper motion

Figure 8: Distribution of proper motions in $\omega$ Cen coloured by source
density. Mean proper motion of SIF CF sources (violet cross). Proper motions
for Gaia DR3 (left) and SIF CF sources (right).  Figure 9: Distribution of
proper motions in $\omega$ Cen coloured by correlation between proper motion
in RA and proper motion in Dec. Mean proper motion of SIF CF sources (white
cross). Proper motion for Gaia DR3 sources (left) and SIF CF sources (right).

Figure 8 presents the distribution of proper motions for Gaia DR3 sources
(left panel) and SIF CF sources (right panel) in the $\omega$ Cen region. For
DR3 sources, we can clearly distinguish two sub-populations – a dense blob of
$\omega$ Cen cluster stars and a more extended distribution of fore- and
background stars. In the distribution of the SIF CF sources we see, as
expected, primarily cluster members. This is the case, because most SIF CF
sources lie in the dense cluster core, where the fraction of fore- and
background stars is rather small. The mean proper motion of the SIF CF sources
is $\mu_{\alpha*}=-3.21$ mas/yr in RA and $\mu_{\delta}=-6.24$ mas/yr in Dec.
It is marked as violet cross in Fig. 8.

We also notice that the distribution of proper motions for SIF CF sources is
much more extended than for DR3 sources. This is again due to the SIF CF
sources being systematically fainter and detected in a much denser
environment, which results in larger proper motion uncertainties.

Additionally, the proper motions of the SIF CF sources reveal an X-shaped
feature not visible in the DR3 distribution. The X-shape of this feature is
most likely caused by the inhomogeneous distribution of the scanning angles,
which leads to a strong correlation between proper motions in RA and Dec. This
is visualised in Fig. 9, where the colour scale reflects the correlation
coefficient between proper motion in right ascension and proper motion in
declination. This is a dimensionless quantity in the range $[-1,+1]$. The
sources located in the X of the distribution show strong correlations between
proper motions in RA and Dec.

Figure 10: Median proper motion uncertainties as functions of $G$ magnitude.
We show $\sigma_{\mu_{\alpha^{*}}}$ (dashed lines) and $\sigma_{\mu_{\delta}}$
(solid lines) for SIF CF (violet) and Gaia DR3 (yellow) sources in $\omega$
Cen.

The median uncertainties of proper motions $\mu_{\alpha^{*}}$ (dashed lines)
and $\mu_{\delta}$ (solid lines) in $\omega$ Cen for Gaia DR3 (yellow) and SIF
CF (violet) sources are given in Fig. 10. As expected (see Sect. 4.3), the
median proper motion uncertainties are substantially larger for new SIF CF
sources (violet) than for Gaia DR3 sources (yellow). The median proper motion
uncertainty for SIF CF sources in $\omega$ Cen amounts to 2.02 mas/year in RA
and 2.06 mas/year in Dec. This is less than the systematic proper motion of
$\omega$ Cen, but greater than the internal proper motion dispersion for
cluster stars, which is just over 0.5 mas/yr in both RA and Dec. An accuracy
comparable to the internal proper motion dispersion is reached only for the
brighter part of the sample.

Figure 11: Median proper motion uncertainties $\sigma_{\mu_{\delta}}$ as
functions of source density. Proper motion uncertainties for bright
($G<18^{m}$), medium ($18^{m}\leq G<20^{m}$), and faint ($G\geq 20^{m}$)
sources in SIF CF (violet) and Gaia DR3 (yellow).

Figure 11 shows the median uncertainties of proper motions $\mu_{\delta}$ as
functions of the source density in three magnitude ranges. The uncertainties
for the SIF CF sources (violet) are larger than for the Gaia DR3 sources
(yellow) within the same magnitude range and same density. However, the
difference decreases rapidly with increasing density. At the highest densities
for which Gaia DR3 sources of a given magnitude range are still present in the
catalogue, the uncertainty difference between SIF CF and Gaia DR3 sources
almost vanishes.

###  4.5 Parallax

Figure 12: Median parallax uncertainties $\sigma_{\varpi}$ versus $G$
magnitude in $\omega$ Cen. Uncertainties for SIF CF (violet) and Gaia DR3
(yellow). Data density in grey.

The distribution of uncertainties for parallax ($\varpi$) is provided in Fig.
12. The uncertainties in the $\omega$ Cen region are plotted with respect to
the G magnitude and grey shades mark the density of data points. As for
positions and proper motions (compare Sect. 4.3), the median uncertainty of
the parallax for SIF CF sources, shown as a violet line, is substantially
larger than the median uncertainty of the parallax for Gaia DR3 (yellow).
Overall, the median parallax uncertainty for SIF CF sources in $\omega$ Cen
amounts to 3.95 mas.

###  4.6 Reference data

Thanks to its largely stare-mode observing approach, HST data tend to be much
deeper than Gaia DR3 or SIF CF data. They therefore provide a good opportunity
to probe the power of the SIF CF pipeline.

The higher angular resolution of HST images allows us to test SIF CF
deblending abilities. We use a dedicated processing of multiple HST images for
the $\omega$ Cen cluster core, carried out by Bellini et al. (2017). In order
to better match the Gaia data, we applied the following corrections to this
HST sample:

  * •

As G–band proxy, we use the HST F606W filter magnitude with an offset:
$G_{Bellini}=F606W-0^{m}.07$.

  * •

An astrometric correction of $+12.3$ mas in right ascension and $-11.26$ mas
in declination was applied.

  * •

For some HST sources, the photometric uncertainty was not provided. In such
cases, we estimated it by means of an empirical function:
$\sigma_{G}=e^{(F606W-26.59)\cdot 0.675}+0.0105$

Figure 13: Digitized Sky Survey view of $\omega$ Cen. Extent of the SIF CF
region (black circle) and HST reference data (teal square).

As reference for our validation we selected a region in the core of $\omega$
Cen that is slightly smaller than the area covered by Bellini et al. (2017).
The less reliable outer parts were cut, and the area was limited to give best
coverage with SIF CF data. In Fig. 13, it is shown as teal polygon. For this
region, deep HST data are available with which we cannot examine the
astrometry, but we can verify the reliability of faint SIF CF source
detections in the most crowded parts of $\omega$ Cen.

###  4.7 Small-scale completeness

In order to assess the small-scale completeness of the SIF CF catalogue, we
calculate the density of source neighbours as a function of their separation
from the source. In a survey with infinite resolution, this would be a
constant function. In real surveys the density decreases for smaller
separations and goes to zero as the separation goes to zero – it is
increasingly more difficult to resolve sources with smaller separations.

Figure 14: Normalised source density as a function of source separation.
Source density for SIF CF (violet), Bellini HST (teal) and Gaia DR3 data
(yellow).

In Fig. 14 the normalised source density is plotted as a function of the
source separation for Gaia DR3 data (yellow), Gaia SIF CF data (violet), and
the Bellini HST sample (teal, see Sect. 4.6). The HST images have very good
angular resolution, thus it is not surprising that 50 % density is reached at
a separation of $0\aas@@fstack{\prime\prime}06$ already. Gaia DR3, on the
other hand, reaches 50 % density at $0\aas@@fstack{\prime\prime}57$
separation, which roughly corresponds to 10 pixels in AL or 3 pixels in AC.
This is consistent with the small-scale completeness reported for Gaia in
Fabricius et al. (2021). Unlike the nominal Gaia pipeline, the SIF CF pipeline
can be applied to full images and profits from iterative source detection and
parametrisation. The SIF CF dataset therefore reaches 50 % density for a
separation of $0\aas@@fstack{\prime\prime}22$ – this is a factor of 2.6 better
than Gaia DR3. It corresponds to 1.7 SM samples in AL and 0.6 samples in AC.

A more detailed study has shown that the SIF CF pipeline can separate sources
down to $0\aas@@fstack{\prime\prime}2$ distance in case their brightness
differs by less than three magnitudes, which is the case for the biggest part
of the data. If their brightness differs by more than three magnitudes though,
the SIF CF resolving power decreases. The reason for this is that with faint
sources of say $G=21^{m}$ and a magnitude difference of more than three, the
brighter source already has $G=18^{m}$ or brighter, which means that these are
increasingly bright sources already, and around such sources it becomes more
and more complicated to detect $G=21^{m}$ sources.

In order to describe the resolution limit for SIF CF sources and its
dependence on the magnitude, we defined the following approximate relation:

$\displaystyle\mathrm{Sep}<\left\\{\begin{aligned} &0.2\,\arcsec,&\textrm{if
}\Delta m\leq 3^{m}\\\ &0.1\,\arcsec\cdot(\Delta m-1),&\textrm{if }\Delta
m>3^{m}\end{aligned}\right..$ (1)

This limit should be viewed as a soft one though. A considerable number of
sources are detected below that limit, and also a number of sources are missed
above.

###  4.8 Source density and depth

Figure 15: Source density as function of distance from cluster centre in
$\omega$ Cen. Source density for Gaia DR3 sources (yellow), SIF CF sources
(violet), and the combined catalogue (dark blue)

The source density as a function of distance from the $\omega$ Cen cluster
centre in stars per square arcmin is shown in Fig. 15. Gaia DR3 source density
is given in yellow, SIF CF source density in violet, and the combined
catalogue in dark blue. At a radius of about 9′ the same number of sources is
found in SIF CF and Gaia DR3 data. Within this radius, SIF CF sources dominate
the combined sample. In the central parts of the $\omega$ Cen cluster SIF CF
produces about ten times more sources than Gaia DR3.

This is further demonstrated in Fig. 16, where quantiles of the magnitude
distribution are shown as a function of the distance from the cluster centre.
The quantiles are given for Gaia DR3 alone (yellow) and for the combined
catalogue of Gaia DR3 supplemented with the SIF CF sources (dark blue). As
expected, the quantiles for Gaia DR3 sources and the combined catalogue are
relatively close for radii beyond 9′, where the contribution of new SIF CF
sources is small. However, at smaller radii the combined catalogue is
considerably deeper. In the cluster core, the 95th percentile of the G
magnitude distribution amounts to 16.9 for the nominal Gaia catalogue, while
for the combined catalogue it reaches 20.4. In the densest region of $\omega$
Cen the source magnitudes of the combined catalogue go more than three
magnitudes deeper than those of Gaia DR3 alone.

Figure 16: Quantiles of the magnitude distribution versus distance to cluster
centre. 33 % (thin line), 50 %, 67 %, and 95 % (thick line) for Gaia DR3
sources (yellow) and the combined catalogue (dark blue).

###  4.9 Reliability and completeness

Figure 17: Comparison of SIF CF data extended with Gaia DR3 and HST data.
Left: Distribution of the combined Gaia SIF CF and Gaia DR3 sources (solid
dark blue) versus Gaia G magnitude; of these, sources matched (dashed) and not
matched (dash-dotted) to HST; of the latter, not blended sources (dotted).
Centre: Distribution of all HST sources (solid teal) versus Gaia G magnitude;
HST sources matched (dashed) and not matched (dash-dotted) to SIF CF sources;
Unmatched HST sources potentially resolvable by SIF CF (dotted). Right:
Fraction of combined SIF CF and Gaia DR3 sources not matched to HST (dash-
dotted dark blue); of these, not HST blends (dotted dark blue); fraction of
HST sources not matched to SIF CF (dash-dotted teal); fraction of resolvable,
unmatched HST sources (dotted teal) versus (estimated) G magnitude

.

In the core of the $\omega$ Cen cluster, nominal Gaia DR3 sources become less
numerous and less accurate. This is, where SIF CF data are expected to be
deeper than nominal data.

To probe the reliability and completeness of the SIF CF pipeline results, we
selected an area in the $\omega$ Cen cluster core for which data are available
in Bellini et al. (2017). The pre-processing applied to this dedicated dataset
is described in Sect. 4.6. We should bear in mind though, that absolute
astrometry with HST is either poor or tied to Gaia data, and should thus be
used with care in this validation. The HST photometry can be used to some
extent, as the HST F606W magnitudes are with a certain offset close to the
Gaia G–band magnitudes.

To make a proper comparison, we supplemented SIF CF sources with Gaia DR3
sources, creating a complete Gaia-and-SIF-CF source list. The contribution of
Gaia DR3 sources is small though – there are only around 1200 DR3 sources
fainter than $16^{m}$ available in the area under consideration, which is
about 1.5 % of the SIF CF sources in this area. As the fraction of added Gaia
DR3 sources is really small in this sample, we keep referring to it as SIF CF,
even though strictly speaking it is a combined catalogue.

The HST data cover a fraction of the cluster only, but they do go
substantially deeper and offer a higher resolution. We can thus assume that
the HST data should contain all sources observed in SIF CF. Hence those
sources found by SIF CF but not in the HST data are likely spurious or blends,
while sources present in HST but missed in SIF CF demonstrate what SIF CF
might be missing.

For Fig. 17, we define ’matches’ as pairs for which the separation between HST
and SIF CF sources is less than 200 mas, and the following inequality for
corresponding source magnitudes $m_{\mathrm{HST}}$ and $m_{\mathrm{SIF}}$
holds:

$|m_{\mathrm{HST}}-m_{\mathrm{SIF}}|<3(\sigma_{\mathrm{HST}}+\sigma^{*}_{\mathrm{SIF}}).$
(2)

Here, $\sigma_{\mathrm{HST}}$ is the magnitude uncertainty for the HST source,
and $\sigma^{*}_{\mathrm{SIF}}$ is derived from the SIF CF source magnitude
uncertainty $\sigma_{\mathrm{SIF}}$ as:

$\sigma^{*}_{\mathrm{SIF}}=\sqrt{\sigma^{2}_{\mathrm{SIF}}+\left(0.1+10^{4.407-m_{\mathrm{SIF}}/3}\right)^{2}}.$
(3)

This complex equation is an empirical relation introduced to account for
systematic scatter between SIF CF and HST magnitudes at the bright end due to
differences in Gaia G and HST F606W bands and reduced reliability for HST
magnitudes at the brighter end due to saturation.

With this match definition, we find HST matches for 90.6 % of the 83 245 SIF
CF sources present in the selected area, suggesting that the vast majority of
the SIF CF sources are reliable. This can be seen in Fig. 17 left panel,
comparing those SIF CF sources matched to HST (dashed dark blue) to all SIF CF
sources (solid dark blue line).

In the 9.4 % unmatched SIF CF sources (dash-dotted dark blue), there are 1.8 %
of sources brighter than 16 mag. Actually most of the SIF CF sources brighter
than $16^{m}$ do not have an HST counterpart. The reason for this is simply
that the Bellini data do not completely cover brighter sources. This can be
seen in the abrupt cut-off of HST sources (solid teal) in Fig. 17 middle
panel. It is reflected in the right panel of Fig. 17 as well, in the dash-
dotted dark blue line that shows the fraction of SIF CF sources not matched to
HST sources versus estimated G magnitude.

For SIF CF sources fainter than $16^{m}$, the number of sources without HST
counterpart (dash-dotted dark blue), and thus the number of possibly spurious
sources, amounts to 6 368. Fig. 17 right panel shows, that this is 7.6 %
(dash-dotted dark blue) of all SIF CF sources (solid dark blue, left panel)
only. Some of these are likely blends: We call a SIF CF source blended, if the
criteria from Eq. (2) holds for the combined flux of all HST sources within
$0\aas@@fstack{\prime\prime}2$ from it. This is the case for 2 400 of the
unmatched SIF CF sources. So 38 % of the unmatched SIF CF sources fainter than
$16^{m}$ seem to be blends of several HST sources.

Looking at HST sources that could not be matched to a SIF CF source (dash-
dotted teal), we find 63 706 HST sources brighter than mag 21.5 that SIF CF
missed. These are 46 % of all HST sources in that area. In Fig. 17 middle
panel, we see that the number of unmatched HST sources increases gradually
from 16th magnitude onward. The right panel gives the fraction of unmatched
HST sources (dash-dotted teal) as magnitude distribution. It reveals the
magnitudes for which SIF CF is missing sources. At mag 20.4, half of all
sources present in HST are missed by SIF CF. Looking at the depth of SIF CF
data close to the centre of the cluster in Fig. 16, this is immediately
explained.

However, due to the fact that SIF CF images have a lower resolution than HST,
there is a considerable number of unmatched HST sources that SIF CF is simply
not able to detect. To estimate this number, we consider as unresolvable all
HST sources for which no SIF match exists and a brighter HST source exists
within 200 mas. These 24 019 unresolvable sources constitute 38 % of the
unmatched HST sources. Fig. 17 middle panel shows the resolvable HST sources
(dotted). The right panel provides the fraction of unmatched HST sources (teal
dash-dotted) and resolvable HST sources (teal dotted) from all HST sources
versus estimated G magnitude.

We note that the fact that the fraction of unmatched SIF CF sources that
appear to be blends and the fraction of unmatched HST sources that are
unresolvable both being approximately 38% is merely a coincidence.

We stress here that a 200 mas separation corresponds to 50 % completeness of
SIF catalogue, so there are many more unresolved pairs with larger
separations. Our estimate of 38 % of the unmatched HST sources actually not
being resolvable by SIF CF is therefore a conservative lower limit.

###  4.10 Known issues

Figure 18: Four examples of blends in Gaia DR3 that are de-blended in SIF CF
FPR. SIF CF detections (small dots), colour-coded by sourceId for resolved
(blue, orange) and blended, unresolved (grey) detections connected to their
corresponding SIF CF and rDR3 source (black dots), together with the blended
Gaia DR3 source (black star).

Even though newly created SIF CF sources are removed from the sample when they
match an existing Gaia DR3 source (see Sect. 3.2), a few sources from the SIF
CF FPR catalogue do match a nominal Gaia DR3 source. When using a positional
offset of less than 160 mas as the matching criterion, this is the case for
379 of the 526 587 published sources. A list of these matches is provided with
the paper444The list of remaining duplicates between the SIF CF and Gaia DR3
catalogue is available in electronic form at the CDS via anonymous ftp to
cdsarc.cds.unistra.fr (130.79.128.5) or via https://cdsarc.cds.unistra.fr/cgi-
bin/qcat?J/A+A/ .

At first sight, this might look like an error, but that is not the case. We
first clarify why these sources got created, even though they are located so
close to a matching DR3 source, and then explain why they were not filtered,
even though sources within a 160 mas distance from brighter sources were
removed in the filtering process.

The reason for the creation of these SIF CF sources in the majority of cases
is that SIF CF observed two sources, where in DR3 there was only one: The SIF
CF XM module found two clusters of detections and assigned two sources to them
(see Fig. 18). One of these sources, called rDR3 for recreated DR3, is matched
to the existing Gaia DR3 source and thus removed from the output list. The
other source, called SIF here, is created as a new SIF CF source and published
in this FPR.

In most cases rDR3 is close in magnitude to DR3, while SIF is fainter as
expected. In a handful of cases it is the opposite though, and the kept and
disregarded sources should ideally have been swapped.

The position of the unpublished rDR3 source differs slightly from that of the
published DR3 source. During the filtering process, new SIF CF sources are
removed, in case their distance to a brighter neighbour is less than
$0\aas@@fstack{\prime\prime}16$ (see Sect. 3.6). Here the SIF sources are
compared to other new SIF CF sources and rDR3 sources. As the position of a
removed rDR3 source and a known DR3 source is not the same, new SIF CF sources
can end up within a $0\aas@@fstack{\prime\prime}16$ radius of a DR3 source.
This happened in 347 cases.

In the remaining 32 cases the DR3 sources were of too low quality and
therefore not included in the SIF CF XM input list. This was revealed in the
validation step only and is reported here for clarity.

In conclusion, all matches are well explained. While 32 of them are due to low
quality Gaia DR3 input, the remaining 347 matches are actually resolved
blends. Furthermore, these examples demonstrate that in some cases SIF CF
indeed outperforms the nominal processing in deblending sources.

##  5 Archive table

The new SIF CF sources are provided via the Gaia data archive in an archive
table named crowded_field_source. This new table resembles a shortened version
of the standard gaia_source, featuring the astrometric and photometric data
for new sources found by Gaia’s dedicated SIF CF pipeline in the $\omega$ Cen
region. Those SIF CF sources that were matched to an existing Gaia DR3 source
were excluded from publication.

Apart from the standard fields known from the gaia_source table,
crowded_field_source contains two additional new fields:

  * •

region_name identifies the SIF CF region. In this FPR, all published data stem
from the same SIF CF region, so all sources will have the same value ’Omega
Centauri’.

  * •

n_scans gives the number of scans for the source location on the sky, which is
the number of times that the location of this source was contained in an SIF
CF image.

Apart from n_scans, the number of potential observations, the table also
contains the nominal field matched_transits, which provides the number of
actual detections assigned to this source. The division of these two produces
the observation fraction, that was used as filter criterion (see Sect. 3.6 and
Fig. 4).

For information about all other fields, we refer to the online documentation
of the SIF CF Gaia archive table crowded_field_source as well as the
description of the nominal gaia_source table, see Section 20.1.1 in Hambly et
al. (2022).

The SIF CF FPR data can be downloaded from all known Gaia archive mirror
sites. An example query showing how to obtain both Gaia DR3 and SIF CF sources
in $\omega$ Cen can be found in Appendix B. Along with this paper we also
publish the python code to reproduce most of the plots presented here.555The
python code for most of the plots in this paper is available at
https://gitlab.aip.de/dpac_gaia/fpr_plots/.

##  6 Conclusion

In the $\omega$ Cen region, Gaia’s new SIF CF pipeline produced 66 660 921
source detections, which were matched to 848 988 sources (Sect. 4.1). Of
these, 321 698 sources were already present in Gaia DR3 and 703 sources were
filtered for other reasons (see Sect. 3.6). The remaining 526 587 sources are
published in this Focused Product Release. The SIF CF astrometry has been
calculated using the nominal AGIS software. The photometry is provided in the
photometric system of Gaia DR3.

The comparison to the HST Bellini et al. (2017) data shows that in the dense
cluster centre most of the SIF CF sources, at least 90.6 %, can be matched to
an HST source and can thus be regarded as real (Sect. 4.9, Fig. 17). From the
9.4 % SIF CF sources that do not match an HST source, 1.8 % are brighter than
16th mag and are simply not included in the HST sample. From the remaining 7.6
% unmatched SIF CF sources, 38 % correspond to blends that are resolved by
HST.

Regarding the HST sources, we cannot find a SIF CF match for about 46 % of
them (Sect. 4.9, Fig. 17). Most of these unmatched HST sources are faint
sources. A large fraction (38 %) of the unmatched HST sources lie in the
vicinity of a brighter source. It is therefore unlikely that these sources
could be detected in the SIF CF data.

For obvious reasons, the SIF CF uncertainties for position, parallax, and
proper motion are significantly higher than Gaia DR3 uncertainties (Sects.
4.3, 4.4, and 4.5). We report a median positional uncertainty of 3.03 mas and
2.69 mas in RA/Dec, a median parallax uncertainty of 3.95 mas, and a median
proper motion uncertainty of 2.02 mas/year in RA and 2.06 mas/year in Dec.
Additionally, a strong correlation is reported between proper motion in RA and
proper motion in Dec (Sect. 4.4).

For stars separated by $\sim$ 200 mas, the published SIF CF FPR data are 50 %
complete (Fig. 14). The HST Bellini et al. (2017) data reveal how many sources
are still missing in Gaia SIF CF: at mag 20.4, half of the HST sources are
missed in SIF CF (Fig. 17, right panel).

Most new SIF CF sources are added in the very dense region of the cluster
core. Starting at a radius of 12′ from the centre of $\omega$ Cen, SIF CF
sources contribute to the combined Gaia DR3 and SIF CF catalogue. Within 9′
from the cluster centre, SIF CF sources dominate the combined catalogue. In
the cluster core itself, SIF CF contributes ten times as many sources to the
combined catalogue as Gaia DR3 (Fig. 15, Sect. 4.8). Here, with respect to
Gaia DR3 alone SIF CF sources increase the completeness of the combined
catalogue by more than three magnitudes (Fig. 16, Sect. 4.8).

###### Acknowledgements.

This work 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. All of the authors are
current or past members of the ESA Gaia mission team and of the Gaia DPAC, and
their work has been supported by the German Aerospace Agency (Deutsches
Zentrum fur Luft- und Raumfahrt e.V., DLR) grant number 50QG1904 and 50QG1404;
the Spanish Ministry of Economy (MINECO/FEDER, UE) through grants
ESP2016-80079-C2-1-R, RTI2018-095076-B-C21 and the Institute of Cosmos
Sciences University of Barcelona (ICCUB, Unidad de Excelencia ‘María de
Maeztu’) through grant MDM-2014-0369 and CEX2019-000918-M; the Tenure Track
Pilot Programme of the Croatian Science Foundation and the Ecole Polytechnique
Fédérale de Lausanne and the Project TTP-2018-07-1171 Mining the variable sky,
with the funds of the Croatian-Swiss Research Programme. The authors
thankfully acknowledge the computer resources from MareNostrum, and the
technical expertise and assistance provided by the Red Española de
Supercomputación at the Barcelona Supercomputing Center, Centro Nacional de
Supercomputación.

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##  Appendix A List of acronyms

Table 2: Acronyms used in this paper Acronym |  Description
---|---
AC |  Across scan direction
ADQL |  Astronomical Data Query Language
AF |  Astrometric Field
AGIS |  Astrometric Global Iterative Solution
AL |  Along scan direction
BP |  Blue Photometer
CCD |  Charge-Coupled Device
CF |  Crowded Field
DAOPHOT |  Code for crowded field stellar photometry
DEC |  Declination
DPAC |  Data Processing and Analysis Consortium
DR3 |  Data Release 3
EDR3 |  Data Release 3 Early
ESA |  European Space Agency
ESAC |  European Space Astronomy Centre
FOV |  Field of View
FPR |  Focused Product Release
HST |  Hubble Space Telescope
IDU |  Intermediate Data Update
IPD |  Image Parameter Determination
LMC |  Large magellanic cloud
NGC |  New General Catalogue
PSF |  Point Spread Function
RA |  Right Ascension
RP |  Red Photometer
RVS |  Radial Velocity Spectrometer
SIF |  Service Interface Function
SM |  Sky Mapper
SMC |  Small magellanic cloud
TDI |  Time Delayed Integration
VBS |  Very Bright Star
XM |  Crossmatch

##  Appendix B Gaia archive query

Example, showing how to retrieve all Gaia SIF CF and Gaia DR3 sources within
0.8 degree radius around $\omega$ Cen cluster centre from the Gaia archive via
Astronomical Data Query Language (ADQL). For DR3 data, the field n_scans does
not exist, so a value of -1 is used there.

    SELECT source_id, ref_epoch, ra, ra_error, dec,
    dec_error, pmra, pmra_error, pmdec, pmdec_error,
    phot_g_mean_mag, l, b, n_scans,
    ’fpr’ as origin
    FROM gaiafpr.crowded_field_source
    UNION
    SELECT source_id, ref_epoch, ra, ra_error, dec,
    dec_error, pmra, pmra_error, pmdec, pmdec_error,
    phot_g_mean_mag, l, b, -1 as nscans,
    ’dr3’ as origin
    FROM gaiadr3.gaia_source
    WHERE
    1=CONTAINS(POINT(’ICRS’, ra, dec),
    CIRCLE(’ICRS’,
    201.69399972775088, -47.484610741298994, 0.8))


# The roles of the $T_{c\bar{s}0}(2900)^{0}$ and $D_{0}^{*}(2300)$ in the
process $B^{-}\to D_{s}^{+}K^{-}\pi^{-}$

Wen-Tao Lyu  School of Physics and Microelectronics, Zhengzhou University,
Zhengzhou, Henan 450001, China  Yun-He Lyu  School of Physics and
Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001, China  Man-Yu
Duan  School of Physics, Southeast University, Nanjing 210094, China  De-Min
Li<EMAIL_ADDRESS>School of Physics and Microelectronics, Zhengzhou
University, Zhengzhou, Henan 450001, China  Dian-Yong Chen<EMAIL_ADDRESS>School of Physics, Southeast University, Nanjing 210094, China  Lanzhou Center
for Theoretical Physics, Lanzhou University, Lanzhou 730000, P. R. China  En
Wang<EMAIL_ADDRESS>School of Physics and Microelectronics, Zhengzhou
University, Zhengzhou, Henan 450001, China

###### Abstract

Motivated by the recent LHCb observations of $T_{c\bar{s}0}(2900)^{0}$ and
$T_{c\bar{s}0}(2900)^{++}$ in the processes
$B^{0}\to\bar{D}^{0}D_{s}^{+}\pi^{-}$ and
$B^{+}\to\bar{D}^{0}D_{s}^{+}\pi^{+}$, we have investigated the decay
$B^{-}\to D_{s}^{+}K^{-}\pi^{-}$ by taking into account the contributions from
the $S$-wave vector-vector interactions, and the $S$-wave $D^{+}_{s}K^{-}$
interactions. Our results show that the $D_{s}^{+}K^{-}$ invariant mass
distribution has an enhancement structure near the threshold, associated to
the $D^{*}_{0}(2300)$, which is in good agreement with the Belle measurements.
We have also predicted the $D^{+}_{s}\pi$ invariant mass distribution and the
Dalitz plot, which show the significant signal of the $T_{c\bar{s}0}(2900)$.
Our results could be tested by the Belle II and LHCb experiments in the
future.

##  I Introduction

Since the charmonium-like state $X(3872)$ was observed in the
$\pi^{+}\pi^{-}J/\psi$ invariant mass distribution of the process $B^{\pm}\to
K^{\pm}\pi^{+}\pi^{-}J/\psi$ by the Belle Collaboration in the year of 2003
[1], many candidates of the exotic states have been reported experimentally,
and called many theoretical attentions, which largely deepens our
understanding of the hadron spectra and hadron-hadron interactions [2, 3, 4,
5].

In the year of 2020, the LHCb Collaboration has observed two states
$X_{0}(2900)$ and $X_{1}(2900)$ with masses about 2900 MeV in the $D^{-}K^{+}$
invariant mass distribution of the process $B^{+}\to D^{+}D^{-}K^{+}$ [6, 7],
in agreement with the predictions of Ref. [8]. As the open-flavor
$\bar{c}\bar{s}ud$ states, $X_{0}(2900)$ and $X_{1}(2900)$ have called many
attentions, and there are several different theoretical interpretations about
their nature, such as the compact tetraquark [9, 10, 11, 12, 13], molecular
structure interpretations [14, 15, 16, 19, 20, 21, 17, 18, 22], or triangle
singularity [23].

Recently, the LHCb Collaboration has reported two new states
$T_{c\bar{s}0}(2900)^{0}$ and $T_{c\bar{s}0}(2900)^{++}$ in the
$D_{s}^{+}\pi^{-}$ and $D_{s}^{+}\pi^{+}$ invariant mass distributions of the
processes $B^{0}\to\bar{D}^{0}D_{s}^{+}\pi^{-}$ and
$B^{+}\to\bar{D}^{0}D_{s}^{+}\pi^{+}$ decays, respectively [25, 24], where the
significance is found to be $8.0\sigma$ ($N_{df}=7.29$) for the
$T_{c\bar{s}0}(2900)^{0}$ state and $6.5\sigma$ ($N_{df}=8.53$) for the
$T_{c\bar{s}0}(2900)^{++}$ state. Their masses and widths are determined as,

$\displaystyle M_{T_{c\bar{s}0}(2900)^{0}}=(2892\pm 14\pm 15)\mathrm{MeV},$
(1) $\displaystyle\Gamma_{T_{c\bar{s}0}(2900)^{0}}=(119\pm 26\pm
12)\mathrm{MeV},$ $\displaystyle M_{T_{c\bar{s}0}(2900)^{++}}=(2921\pm 17\pm
19)\mathrm{MeV},$ $\displaystyle\Gamma_{T_{c\bar{s}0}(2900)^{++}}=(137\pm
32\pm 14)\mathrm{MeV},$

respectively. Their masses and widths are close to each other, which implies
that $T_{c\bar{s}0}(2900)^{0}$ and $T_{c\bar{s}0}(2900)^{++}$ with the flavor
components $c\bar{s}\bar{u}d$ and $c\bar{s}u\bar{d}$ should be two of the
isospin triplet. Some interpretations for their structure are proposed
theoretically [26]. For instance, the authors in Ref. [27] argue that
$T_{c\bar{s}0}(2900)^{++}$ and $T_{c\bar{s}0}(2900)^{0}$ may be modelled as
molecules $D_{s}^{*+}\rho^{+}$ and $D_{s}^{*+}\rho^{-}$, respectively, using
two-point sum rule method. In addition, the $T_{c\bar{s}0}(2900)$ also can be
considered as a virtual state created by the $D_{s}^{*}\rho$ and $D^{*}K^{*}$
interaction in coupled channels [28]. Thus, it is crucial to confirm the
existence of $T_{c\bar{s}0}(2900)$ in other processes, which will provide more
precise properties and shed light on its internal structure.

As we known, many candidates of the exotic states were observed in the decays
of $B$ meson, and Belle and LHCb Collaborations have accumulated many events
of $B$ mesons, which provides an important lab to study the hadron resonances
[3, 29, 30, 32, 31, 33]. For instance, we have proposed to search for the
open-flavor tetraquark $T_{c\bar{s}0}(2900)^{++}$ in the process $B^{+}\to
K^{+}D^{+}D^{-}$ by assuming $T_{c\bar{s}0}(2900)^{++}$ as a $D^{*+}K^{*+}$
molecular state in Ref. [34]. According to the Review of Particle Physics
(RPP) [35], one can find that the branching fraction of the process $B^{+}\to
D^{-}_{s}\pi^{+}K^{+}$ is $(1.80\pm 0.22)\times 10^{-4}$, in the same order of
magnitude with the branching fraction of the process $B^{+}\to
D^{-}D^{+}K^{+}$ [$(2.2\pm 0.7)\times 10^{-4}$], which implies that it is
reasonable to propose to search for the $T_{c\bar{s}0}(2900)$ in the process
$B^{+}\to D^{-}_{s}\pi^{+}K^{+}$.

It should be pointed out that the processes $B^{+}\to D_{s}^{-}\pi^{+}K^{+}$
and $B^{-}\to D_{s}^{+}\pi^{-}K^{-}$ have been measured by the BABAR [36] and
Belle Collaborations [37], and the $D_{s}^{\pm}K^{\mp}$ invariant mass
distributions exhibit strong enhancements near the threshold. The same
threshold enhancements have also appeared in the $D_{s}K$ invariant mass
distribution in the processes $B^{0}\to D_{s}^{-}K^{0}_{S}\pi^{+}$ and
$B^{+}\to D_{s}^{-}K^{+}K^{+}$ [38], which could be due to the contribution
from the high pole of the $D^{*}_{0}(2300)$ with two-pole structure in the
unitarized chiral perturbation theory [39, 40]111The state $D^{*}_{0}(2300)$
was denoted as $D_{0}^{*}(2400)$ in previous versions of RPP, and more
discussions about this state can be found in the review ‘Heavy Non-$q\bar{q}$
mesons’ of RPP [35]..

Thus, in this work we would investigate the process $B^{-}\to
D_{s}^{+}K^{-}\pi^{-}$ by taking into account the $S$-wave
$D_{s}^{*+}\rho^{-}$ and $D^{*0}K^{*0}$ interactions, which will generate the
resonance $T_{c\bar{s}0}(2900)$. In addition, we will also consider the
contribution from the $S$-wave pseudoscalar meson-pseudoscalar meson
interactions within the unitary chiral approach, which will dynamically
generate the resonance $D^{*}_{0}(2300)$.

This paper is organized as follows. In Sec. II, we present the theoretical
formalism of the process $B^{-}\to D_{s}^{+}K^{-}\pi^{-}$. Numerical results
and discussion are shown in Sec. III. Finally, we give a short summary in the
last section.

##  II Formalism

In this section, we will present the theoretical formalism of the process
$B^{-}\to D_{s}^{+}K^{-}\pi^{-}$. The reaction mechanism via the intermediate
state $T_{c\bar{s}0}(2900)$ is given in Sec. II.1, while the mechanism via the
intermediate state $D^{*}_{0}(2300)$ is given in Sec. II.2. Finally, we give
the formalism of the invariant mass distributions of the process $B^{-}\to
D_{s}^{+}K^{-}\pi^{-}$ in Sec. II.3.

###  II.1 The $T_{c\bar{s}0}(2900)$ role in $B^{-}\to D_{s}^{+}K^{-}\pi^{-}$

Figure 1: (a) Quark level diagrams for the process
$B^{-}\to\rho^{-}D_{s}^{*+}K^{-}$ via the $W^{-}$ external emission, and (b)
the pocess $B^{-}\to K^{*0}D^{*0}K^{-}$ via the $W^{-}$ external emission.

Figure 2: (a) Quark level diagrams for the process
$B^{-}\to\rho^{-}D_{s}^{*+}K^{-}$ process via the $W^{-}$ internal emission,
and (b) the process $B^{-}\to K^{*0}D^{*0}K^{-}$ via the $W^{-}$ internal
emission.

Taking into account that $T_{c\bar{s}0}(2900)$ could be explained as the
molecular state of the $D^{*}_{s}\rho$ and $D^{*}K^{*}$ interaction [28], we
first need to produce the states $D^{*+}_{s}\rho^{-}K^{-}$ and
$D^{*0}K^{*0}K^{-}$ via the external $W^{-}$ emission mechanism and the
internal $W^{-}$ emission mechanism, as depicted in Figs. 1 and 2,
respectively.

In analogy to Refs. [29, 41, 42, 43, 44, 45], as depicted in Fig. 1(a), the
$b$ quark of the initial $B^{-}$ meson weakly decays into a $c$ quark and a
$W^{-}$ boson, then the $W^{-}$ boson decays into $\bar{u}d$ quarks. The
$\bar{u}d$ pair from the $W^{-}$ boson will hadronize into $\rho^{-}$, while
the $\bar{u}$ quark of the initial $B^{-}$ meson and the $c$ quark, together
with the $\bar{s}s$ created from vacuum, hadronize into $K^{-}$ and
$D_{s}^{*+}$. On the other hand, as shown in Fig. 1(b), the $\bar{u}d$ quarks
from the $W^{-}$ boson, together with the $s\bar{s}$ created from vacuum,
hadronize into $K^{-}$ and $K^{*0}$, while the $c$ quark and the $\bar{u}$
from the initial $B^{-}$ meson could hadronize into the $D^{*0}$.

In Fig. 2(a), the $c$ quark from the $B^{-}$ and the $\bar{u}$ from the
$W^{-}$ boson, together with the $\bar{s}s$ created from vacuum, hadronize
into $D_{s}^{*+}$ and $K^{-}$, while the $d$ quark from the $W^{-}$ boson and
the $\bar{u}$ quark of the initial $B^{-}$ meson, hadronize into vector meson
$\rho^{-}$. On the other hand, the $c$ quark from the $B^{-}$ and the
$\bar{u}$ from the $W^{-}$ boson could also hadronize into a $D^{*0}$, while
the $d$ quark from the $W^{-}$ boson and the $\bar{u}$ quark of the initial
$B^{-}$ meson, together with the $\bar{s}s$ created from vacuum, hadronize
into mesons $K^{*0}K^{-}$, as shown in Fig. 2(b). It should be pointed out
that the mechanisms of Figs. 2(a) and 2(b) are $1/N_{c}$ suppressed with
respect to the ones of Fig. 1 [32].

Figure 3: The mechanisms of the process $B^{-}\to D_{s}^{+}K^{-}\pi^{-}$ via
the $S$-wave $\rho^{-}D_{s}^{*+}$ and $D^{*0}K^{*0}$ interactions.

Then, the $S$-wave interactions of $\rho^{-}D_{s}^{*+}$ or $D^{*0}K^{*0}$ will
give rise to the $T_{c\bar{s}0}(2900)^{0}$ state, which could decay into the
final state $D_{s}^{+}\pi^{-}$, as depicted in Fig. 3. The transition
amplitude for the process $B^{-}\to V_{1}V_{2}K^{-}$
($V_{1}V_{2}=\rho^{-}D_{s}^{*+}$, or $D^{*0}K^{*0}$) could be written as,

$\tilde{\mathcal{T}}=\mathcal{Q}(C+1)\vec{\epsilon}(V_{1})\cdot\vec{\epsilon}(V_{2}),$
(2)

where $\vec{\epsilon}(V_{1})$ and $\vec{\epsilon}(V_{2})$ are the polarization
vectors of $\rho^{-}D_{s}^{*+}$ (or $D^{*0}K^{*0}$), and we have the relation
of $\sum_{\rm pol}\epsilon_{i}(V)\epsilon_{j}(V)=\delta_{ij}$. The constant
$\mathcal{Q}$ includes all the dynamical factors of the weakly decay of Fig.
2, while the factor $C=3$ corresponds to the relative weight of the $W^{-}$
external emission mechanism (Fig. 1) with respect to the $W^{-}$ internal
emission mechanism (Fig. 2) [46, 47, 48, 49]. Thus, one could easily obtain
the expression of the $\mathcal{Q}$ as follows,

$\mathcal{Q}^{2}=\frac{\Gamma_{B^{-}}{\mathcal{B}}(B^{-}\to
D^{*0}K^{*0}K^{-})}{\int\frac{3}{(2\pi)^{3}}\frac{(C+1)^{2}}{4M_{B^{-}}^{2}}p_{K^{*}}\tilde{p}_{K}{dM_{\rm
inv}(D^{*0}K^{*0})}}.$ (3)

Since the branching fraction is measured to be $\mathcal{B}(B^{-}\to
D^{*0}K^{*0}K^{-})=(1.5\pm 0.4)\times 10^{-3}$ [35], we could roughly estimate
$\mathcal{Q}^{2}=1.71\times 10^{-13}$ neglecting the contributions from the
possible intermediate resonances.

By taking into account the contributions from the $S$-wave
$\rho^{-}D_{s}^{*+}$ and $D^{*0}K^{*0}$ interactions of Fig. 3, the amplitude
could be expressed as,

$\displaystyle\mathcal{T}^{T_{c\bar{s}0}^{0}}$ $\displaystyle=$
$\displaystyle\mathcal{Q}\vec{\epsilon}(V_{1})\cdot\vec{\epsilon}(V_{2})$ (4)
$\displaystyle\times(C+1)\left[G_{\rho^{-}D_{s}^{*+}}t_{\rho^{-}D_{s}^{*+}\to
D_{s}^{+}\pi^{-}}\right.$
$\displaystyle\left.+G_{D^{*0}K^{*0}}t_{D^{*0}K^{*0}\to
D_{s}^{+}\pi^{-}}\right],$

where $G_{\rho^{-}D_{s}^{*+}}$ and $G_{D^{*0}K^{*0}}$ are the loop functions
of the coupled channels $\rho^{-}D_{s}^{*+}$ and $D^{*0}K^{*0}$, respectively,
and $t_{\rho^{-}D_{s}^{*+}\to D_{s}^{+}\pi^{-}}$ and $t_{D^{*0}K^{*0}\to
D_{s}^{+}\pi^{-}}$ are the transition amplitudes of $\rho^{-}D_{s}^{*+}\to
D_{s}^{+}\pi^{-}$ and $D^{*0}K^{*0}\to D_{s}^{+}\pi^{-}$, respectively. Both
of loop functions $G_{i}$ and transition amplitudes $t_{ij}$ are the functions
of the $D_{s}^{+}\pi^{-}$ invariant mass $M_{D_{s}^{+}\pi^{-}}$. The two-meson
loop function is given by,

$G_{i}=i\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{q^{2}-m_{1}^{2}+i\epsilon}\frac{1}{(p-q)^{2}-m_{2}^{2}+i\epsilon},$
(5)

where $m_{1}$ and $m_{2}$ are the mesons masses of the $i$-th coupled channel.
$q$ is the four-momentum of the meson 1 in the center of mass frame, and $p$
is the four-momentum of the meson-meson system. In the present work, we use
the dimensional regularization method as indicated in Refs. [46, 34], and in
this scheme, the two-meson loop function $G_{i}$ can be expressed as,

$\displaystyle G_{i}=$
$\displaystyle\frac{1}{16\pi^{2}}\left\\{a(\mu)+\ln\frac{m_{1}^{2}}{\mu^{2}}+\frac{s+m_{2}^{2}-m_{1}^{2}}{2s}\ln\frac{m_{2}^{2}}{m_{1}^{2}}\right.$
(6)
$\displaystyle+\frac{|\vec{q}\,|}{\sqrt{s}}\left[\ln\left(s-\left(m_{2}^{2}-m_{1}^{2}\right)+2|\vec{q}\,|\sqrt{s}\right)\right.$
$\displaystyle+\ln\left(s+\left(m_{2}^{2}-m_{1}^{2}\right)+2|\vec{q}\,|\sqrt{s}\right)$
$\displaystyle-\ln\left(-s+\left(m_{2}^{2}-m_{1}^{2}\right)+2|\vec{q}\,|\sqrt{s}\right)$
$\displaystyle\left.\left.-\ln\left(-s-\left(m_{2}^{2}-m_{1}^{2}\right)+2|\vec{q}\,|\sqrt{s}\right)\right]\right\\},$

where $s=p^{2}=M_{\rm inv}^{2}(D_{s}^{+}\pi^{-})$, and $\vec{q}$ is the three-
momentum of the meson in the centre of mass frame, which reads,

$|\vec{q}\,|=\dfrac{\sqrt{[s-(m_{1}+m_{2})^{2}][s-(m_{1}-m_{2})^{2}]}}{2\sqrt{s}},$
(7)

here we take $\mu_{\rho^{-}D_{s}^{*+}}=1200$ MeV, $\mu_{D^{*0}K^{*0}}=1300$
MeV, $a_{\rho^{-}D_{s}^{*+}}=-1.6$, and $a_{D^{*0}K^{*0}}=-1.474$, which are
the same as those used in the study of the $\rho^{-}D_{s}^{*+}$ interaction
[8, 28] and the $D^{*0}K^{*0}$ interaction [50, 28, 34]. And the transition
amplitudes are given by,

$t_{\rho^{-}D_{s}^{*+}\to
D_{s}^{+}\pi^{-}}=\dfrac{g_{T_{c\bar{s}0}^{0},\rho^{-}D_{s}^{*+}}g_{T_{c\bar{s}0}^{0},D_{s}^{+}\pi^{-}}}{M_{D_{s}^{+}\pi^{-}}^{2}-m_{T_{c\bar{s}0}^{0}}^{2}+im_{T_{c\bar{s}0}^{0}}\Gamma_{T_{c\bar{s}0}^{0}}},$
(8) $t_{D^{*0}K^{*0}\to
D_{s}^{+}\pi^{-}}=\dfrac{g_{T_{c\bar{s}0}^{0},D^{*0}K^{*0}}g_{T_{c\bar{s}0}^{0},D_{s}^{+}\pi^{-}}}{M_{D_{s}^{+}\pi^{-}}^{2}-m_{T_{c\bar{s}0}^{0}}^{2}+im_{T_{c\bar{s}0}^{0}}\Gamma_{T_{c\bar{s}0}^{0}}},$
(9)

where the $m_{T_{c\bar{s}0}^{0}}$ and $\Gamma_{T_{c\bar{s}0}^{0}}$ are given
by Ref. [24, 25]. The constant $g_{T_{c\bar{s}0}^{0},D^{*0}K^{*0}}$
corresponds to the coupling between $T_{c\bar{s}0}(2900)^{0}$ and its
components $D^{*0}K^{*0}$, which could be related to the binding energy by
[51, 52, 53, 34],

$g_{T_{c\bar{s}0}^{0},D^{*}K^{*}}^{2}=16\pi(m_{D^{*}}+m_{K^{*}})^{2}\tilde{\lambda}^{2}\sqrt{\frac{2\Delta
E}{\mu}},$ (10)

where $\tilde{\lambda}=1$ gives the probability to find the molecular
component in the physical states, $\Delta
E=m_{D^{*}}+m_{K^{*}}-m_{T_{c\bar{s}0}}$ denotes the binding energy, and
$\mu=m_{D^{*}}m_{K^{*}}/(m_{D^{*}}+m_{K^{*}})$ is the reduced mass.

The value of the coupling constant $g_{T_{c\bar{s}0}^{0},\rho^{-}D_{s}^{*+}}$
could be obtained from the partial width of
$T_{c\bar{s}0}(2900)^{0}\to\rho^{-}D_{s}^{*+}$, which could be expressed as
follows,

$\Gamma_{T_{c\bar{s}0}^{0}\to\rho^{-}D_{s}^{*+}}=\frac{3}{8\pi}\frac{1}{m_{T_{c\bar{s}0}^{0}}^{2}}|g_{T_{c\bar{s}0}^{0},\rho^{-}D_{s}^{*+}}|^{2}|\vec{q}_{\rho}|,$
(11)

where $\vec{q}_{\rho}$ is the three-momentum of $\rho^{-}$ in the
$T_{c\bar{s}0}(2900)^{0}$ rest frame,

$|\vec{q}_{\rho}|=\dfrac{\lambda^{1/2}(m_{T_{c\bar{s}0}^{0}}^{2},m_{D_{s}^{*+}}^{2},m_{\rho^{-}}^{2})}{2m_{T_{c\bar{s}0}^{0}}}$
(12)

with the K$\ddot{a}$llen function
$\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2yz-2zx$. And similarly we can get the
$g_{T_{c\bar{s}0}^{0},D_{s}^{+}\pi^{-}}$ coupling from the partial width of
$T_{c\bar{s}0}(2900)^{0}\to D_{s}^{+}\pi^{-}$,

$\Gamma_{T_{c\bar{s}0}^{0}\to
D_{s}^{+}\pi^{-}}=\frac{1}{8\pi}\frac{1}{m_{T_{c\bar{s}0}^{0}}^{2}}|g_{T_{c\bar{s}0}^{0},D_{s}^{+}\pi^{-}}|^{2}|\vec{q}_{\pi^{-}}|$
(13)

with

$|\vec{q}_{\pi^{-}}|=\dfrac{\lambda^{1/2}(m_{T_{c\bar{s}0}^{0}}^{2},m_{D_{s}^{+}}^{2},m_{\pi^{-}}^{2})}{2m_{T_{c\bar{s}0}^{0}}}.$
(14)

In Ref. [54], the partial widths of decay modes $\rho^{-}D_{s}^{*+}$ and
$D_{s}^{+}\pi^{-}$ were estimated to be ($2.96\sim 5.3$) MeV and ($0.55\sim
8.35$) MeV, respectively. In the present work, we take the center values of
the decay widths $\Gamma_{T_{c\bar{s}0}^{0}\to\rho^{-}D_{s}^{*+}}=4.13$ MeV
and $\Gamma_{T_{c\bar{s}0}^{0}\to D_{s}^{+}\pi^{-}}=4.45$ MeV [54], from which
we can obtain the coupling constants
$g_{T_{c\bar{s}0}^{0},\rho^{-}D_{s}^{*+}}=2007$ MeV,
$g_{T_{c\bar{s}0}^{0},D_{s}^{+}\pi^{-}}=1104$ MeV, and
$g_{T_{c\bar{s}0}^{0},D^{*0}K^{*0}}=8809$ MeV, respectively.

###  II.2 The mechanism of the $S$-wave meson-meson interaction

Figure 4: Quark level diagrams for the
$B^{-}\to\pi^{-}c(\bar{u}u+\bar{d}d+\bar{s}s)\bar{u}$ process. (a) The $W^{-}$
external emission, (b) the $W^{-}$ internal emission.

In this subsection, we will consider the contribution from the $S$-wave
$D_{s}^{+}K^{-}$ final state interaction. As shown in Fig. 4, the $b$ quark of
the initial $B^{-}$ weakly decays into a $c$ quark and a $W^{-}$ boson, then
the $W^{-}$ boson subsequently decays into a $\bar{u}d$ quark pair, which will
hadronize into a $\pi^{-}$ meson. The $c$ quark from the $B^{-}$ decay and the
$\bar{u}$ quark of the initial $B^{-}$, together with the quark pair
$\bar{q}q=\bar{u}u+\bar{d}d+\bar{s}s$ created from the vacuum with the quantum
numbers $J^{PC}=0^{++}$, hadronize into hadrons pairs, as follows,

$\sum_{i}c(\bar{u}u+\bar{d}d+\bar{s}s)\bar{u}=\sum_{i}P_{4i}P_{i1}=(P^{2})_{41},$
(15)

where $i=1,2,3$ correspond to the $u$, $d$, and $s$ quarks, respectively, and
$P$ is the SU(4) matrix of the pseudoscalar mesons,

$P=\left(\begin{array}[]{cccc}\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta&\pi^{+}&K^{+}&\bar{D}^{0}\\\
\pi^{-}&-\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta&K^{0}&D^{-}\\\
K^{-}&\bar{K}^{0}&-\frac{2}{\sqrt{6}}\eta&D_{s}^{-}\\\
D^{0}&D^{+}&D_{s}^{+}&\eta_{c}\end{array}\right).$ (16)

Then, we could have all the possible pseudoscalar-pseudoscalar pairs after the
hadronization,

$H=\pi^{-}\left(\dfrac{1}{\sqrt{2}}D^{0}\pi^{0}+\dfrac{1}{\sqrt{6}}D^{0}\eta+D^{+}\pi^{-}+D_{s}^{+}K^{-}\right).$
(17)

With the isospin multiplets of ($D^{+},-D^{0}$), ($\bar{D}^{0},D^{-}$), and
($-\pi^{+},\pi^{0},\pi^{-}$), we have,

$\displaystyle D^{0}\pi^{0}$
$\displaystyle=-\left|\frac{1}{2},-\frac{1}{2}\right>\left|1,0\right>$ (18)
$\displaystyle=-\sqrt{\frac{2}{3}}\left|\frac{3}{2},-\frac{1}{2}\right>+\sqrt{\frac{1}{3}}\left|\frac{1}{2},-\frac{1}{2}\right>,$
$\displaystyle D^{+}\pi^{-}$
$\displaystyle=\left|\frac{1}{2},\frac{1}{2}\right>\left|1,-1\right>$ (19)
$\displaystyle=\sqrt{\frac{1}{3}}\left|\frac{3}{2},-\frac{1}{2}\right>+\sqrt{\frac{2}{3}}\left|\frac{1}{2},-\frac{1}{2}\right>,$
$\displaystyle\dfrac{1}{\sqrt{2}}D^{0}\pi^{0}+D^{+}\pi^{-}$
$\displaystyle=\left(-\sqrt{\frac{2}{3}}\cdot\sqrt{\frac{1}{2}}+\sqrt{\frac{1}{3}}\right)|D\pi>^{I=\frac{3}{2}}$
(20)
$\displaystyle+\left(\sqrt{\frac{1}{3}}\cdot\sqrt{\frac{1}{2}}+\sqrt{\frac{2}{3}}\right)|D\pi>^{I=\frac{1}{2}},$
$\displaystyle=\sqrt{\frac{3}{2}}|D\pi>^{I=\frac{1}{2}}.$

Figure 5: The mechanisms of the $B^{-}\to D_{s}^{+}K^{-}\pi^{-}$ decay. (a)
Tree diagram, (b) the mechanism of the $S$-wave meson-meson interactions.

In the isospin basis, we can obtain the $D\pi(I=1/2)$, $D\eta(I=1/2)$, and
$D_{s}\bar{K}(I=1/2)$ channels

$H=\pi^{-}\left({\sqrt{\frac{3}{2}}}D\pi+\dfrac{1}{\sqrt{6}}D^{0}\eta+D_{s}^{+}K^{-}\right).$
(21)

Then, the process $B^{-}\to D_{s}^{+}K^{-}\pi^{-}$ decay could happen via the
tree diagram of Fig. 5(a), and the $S$-wave meson-meson interaction of Fig.
5(b), and the amplitude could be expressed as,

$\displaystyle\mathcal{T}^{D_{0}^{*}(2300)}$
$\displaystyle=\mathcal{Q}^{\prime}(C+1)(h_{D_{s}\bar{K}}+\sum_{i}h_{i}G_{i}t_{i\to
D_{s}\bar{K}})$ (22) $\displaystyle=\mathcal{T}^{tree}+\mathcal{T}^{S},$

where the constant $\mathcal{Q}^{\prime}$ includes all the dynamical factors
of the weakly decay, and $i=1,2,3$ correspond to the $D\pi$, $D\eta$,
$D_{s}\bar{K}$, respectively,

$\begin{array}[]{ll}h_{D\pi}=\sqrt{\frac{3}{2}},\quad
h_{D\eta}=\sqrt{\frac{1}{6}},\quad h_{D_{s}\bar{K}}=1.\end{array}$ (23)

The factor $C=3$ corresponds to the relative weight of the $W^{-}$ external
emission mechanism [Fig. 4(a)] with respect to the $W^{-}$ internal emission
mechanism [Fig. 4(b)]. With the amplitude of Eq. (22), the
$\mathcal{Q}^{\prime}$ is given by,

$\displaystyle\Gamma_{B^{-}}\mathcal{B}(B^{-}\to D_{s}^{+}K^{-}\pi^{-})$ (24)
$\displaystyle=\mathcal{Q}^{\prime
2}\int\frac{1}{(2\pi)^{3}}\frac{(C+1)^{2}}{4M_{B^{-}}^{2}}p_{\pi}\tilde{p}_{K}$
$\displaystyle\times|(h_{D_{s}\bar{K}}+\sum_{i}h_{i}G_{i}t_{i\to
D_{s}\bar{K}})|^{2}{dM_{\rm inv}(D_{s}^{+}K^{-})}.$

According to the experimental measurements of $\mathcal{B}(B^{-}\to
D_{s}^{+}K^{-}\pi^{-})=(1.80\pm 0.22)\times 10^{-4}$ [35], we could roughly
estimate $\mathcal{Q}^{\prime 2}=3.12\times 10^{-12}$.

The $G_{i}$ in Eq. (22) is the loop function of the meson-meson system, and
$t_{i\to D_{s}\bar{K}}$ are the scattering matrices of the coupled channels.
The transition amplitude of $t_{i\to D_{s}\bar{K}}$ is obtained by solving the
Bethe-Salpeter equation,

$T=[1-VG]^{-1}V.$ (25)

The transition potential $V_{ij}$ is taken from Refs. [56, 55],

$\displaystyle V_{ij}$ $\displaystyle=\frac{1}{4f_{\pi}^{2}}[C_{ij}(s-u)]$
(26)
$\displaystyle=\frac{1}{4f_{\pi}^{2}}[C_{ij}(2s-m_{2}^{2}-m_{4}^{2}-2E_{1}E_{3})],$

where the coefficients are given as,

$C_{ij}=\left(\begin{array}[]{ccc}-2&0&-\sqrt{\frac{3}{2}}\\\
0&0&-\sqrt{\frac{3}{2}}\\\
-\sqrt{\frac{3}{2}}&-\sqrt{\frac{3}{2}}&-1\end{array}\right)$ (27)

with $f_{\pi}=93$ MeV. The loop function $G$ in Eq. (25) is given by the
dimensional regularization method, as shown by Eq. (5), and we take $\mu=1000$
MeV and $a=-1.88$ [56].

###  II.3 Invariant Mass Distribution

With above the formalism, one can write down the invariant mass distributions
for the $B^{-}\to D_{s}^{+}K^{-}\pi^{-}$ can be written as,

$\dfrac{d^{2}\Gamma}{dM_{D_{s}^{+}K^{-}}dM_{D_{s}^{+}\pi^{-}}}=\frac{1}{(2\pi)^{3}}\dfrac{2M_{D_{s}^{+}K^{-}}2M_{D_{s}^{+}\pi^{-}}}{32M_{B^{-}}^{3}}|\mathcal{T}^{\text{total
}}|^{2}$ (28)

with the modulus squared of the total amplitude,

$\left|\mathcal{T}^{\text{total
}}\right|^{2}=\left|\mathcal{T}^{T_{c\bar{s}0}^{0}}\right|^{2}+\left|\mathcal{T}^{D_{0}^{*}(2300)}\right|^{2}.$
(29)

For a given value of $M_{12}$, the range of $M_{23}$ is determined by [35],

$\displaystyle\left(m_{23}^{2}\right)_{\min}=\left(E_{2}^{*}+E_{3}^{*}\right)^{2}-\left(\sqrt{E_{2}^{*2}-m_{2}^{2}}+\sqrt{E_{3}^{*2}-m_{3}^{2}}\right)^{2},$
$\displaystyle\left(m_{23}^{2}\right)_{\max}=\left(E_{2}^{*}+E_{3}^{*}\right)^{2}-\left(\sqrt{E_{2}^{*2}-m_{2}^{2}}-\sqrt{E_{3}^{*2}-m_{3}^{2}}\right)^{2},$
(30)

where $E_{2}^{*}$ and $E_{3}^{*}$ are the energies of particles 2 and 3 in the
$M_{12}$ rest frame. $E_{2}^{*}$ and $E_{3}^{*}$ are written as,

$\displaystyle E_{2}^{*}=\dfrac{M_{12}^{2}-m_{1}^{2}+m_{2}^{2}}{2M_{12}},$
$\displaystyle
E_{3}^{*}=\dfrac{M_{B^{-}}^{2}-M_{12}^{2}+m_{3}^{2}}{2M_{B^{-}}},$ (31)

where $m_{1}$, $m_{2}$, and $m_{3}$ are the masses of particles 1, 2, and 3,
respectively. All the masses and widths of the particles are taken from the
RPP [35].

##  III  Numerical results

Figure 6: Modulus squared of the transition amplitudes $t_{i\to D_{s}\bar{K}}$
in $S$-wave.

We first show the transition amplitude $t_{i\to D_{s}\bar{K}}$ of Eq. (25) in
Fig. 6. The red-dotted curve shows the modulus squared of the transition
amplitude $t_{D_{s}\bar{K}\to D_{s}\bar{K}}$, the blue-dashed curve shows the
modulus squared of the transition amplitude $t_{D\eta\to D_{s}\bar{K}}$, and
the black-solid curve shows the modulus squared of the transition amplitude
$t_{D\pi\to D_{s}\bar{K}}$. One can find that the modulus squared of the
transition amplitude $|t_{D\pi\to D_{s}\bar{K}}|^{2}$ has two peaks around
2100 MeV and 2450 MeV, respectively, in consistent with the conclusion of Ref.
[55]. Since the lower pole is far from the $D_{s}\bar{K}$ threshold, the
enhancement structure near the $D_{s}\bar{K}$ threshold of the process
$B^{-}\to D_{s}^{+}K^{-}\pi^{-}$ is mainly due to the contribution from the
high pole.

Figure 7: The $D_{s}^{+}K^{-}$ invariant mass distribution of the process
$B^{-}\to D_{s}^{+}K^{-}\pi^{-}$. The red-dotted curve shows the contribution
from the $T_{c\bar{s}0}(2900)^{0}$, the blue-dashed curve shows the
contribution from the $D_{0}^{*}(2300)$, and the black-solid curve shows the
total contributions. The Belle data are taken from Ref. [37], and have been
rescaled for comparison. Figure 8: The $D_{s}^{+}\pi^{-}$ invariant mass
distribution of the process $B^{-}\to D_{s}^{+}K^{-}\pi^{-}$. The explanations
of the curves are same as those of Fig. 7. Figure 9: The Dalitz plot of
“$M_{D_{s}^{+}\pi^{-}}$” vs. “$M_{D_{s}^{+}K^{-}}$” for the process $B^{-}\to
D_{s}^{+}K^{-}\pi^{-}$.

Figure 10: The $D_{s}^{+}K^{-}$ (a) and $D_{s}^{+}\pi^{-}$ (b) invariant mass
distributions of the process $B^{-}\to D_{s}^{+}K^{-}\pi^{-}$ with
$\Gamma_{T_{c\bar{s}0}^{0}\to D_{s}^{+}\pi^{-}}=12.84$ MeV.

Next, we have shown our results of the $D_{s}^{+}K^{-}$ and $D_{s}^{+}\pi^{-}$
invariant mass distributions of the process $B^{-}\to D_{s}^{+}K^{-}\pi^{-}$
in Fig. 7 and Fig. 8, respectively. The red-dotted curves show the
contribution from the $T_{c\bar{s}0}(2900)^{0}$, the blue-dashed curves show
the contribution from the $D_{0}^{*}(2300)$, and the black-solid curves show
the total contributions. We also show the Belle measurements on the
$D_{s}^{+}K^{-}$ invariant mass distribution of the $B^{-}\to
D_{s}^{+}K^{-}\pi^{-}$ events in Fig. 7, where the Belle data have been
rescaled for comparison [37]222The first three data of Belle are lower than
our predictions, which may be due to the lower detection efficiencies [37],
and we do not show them here.. In Fig. 7, one can find that our results are in
good agreement with the Belle measurements in the region $2700\sim 4000$ MeV,
which implies that the near-threshold structure mainly due to contribution
from the high pole of the $D_{0}^{*}(2300)$. In Fig. 8, one can find a clear
peak around 2900 MeV in the $D_{s}^{+}\pi^{-}$ invariant mass distribution,
which should be associated to the state $T_{c\bar{s}0}(2900)^{0}$.

We have also predicted the Dalitz plot of “$M_{D_{s}^{+}\pi^{-}}$” vs.
“$M_{D_{s}^{+}K^{-}}$” for the process $B^{-}\to D_{s}^{+}K^{-}\pi^{-}$ in
Fig. 9, and one can find that the the $T_{c\bar{s}0}(2900)^{0}$ mainly
contributes to the high energy region of the $D_{s}^{+}K^{-}$ invariant mass
distribution. Indeed, in this region, our results are smaller than the Belle
measurements, as shown by Fig. 7, which implies that the contribution from the
$T_{c\bar{s}0}(2900)^{0}$ is underestimated.

Thus, we take the decay width $\Gamma_{T_{c\bar{s}0}^{0}\to D_{s}^{+}\pi^{-}}$
to be free parameter, and fit it to the $D_{s}^{+}K^{-}$ invariant mass
distribution of the Belle measurements [37], which gives
$\Gamma_{T_{c\bar{s}0}^{0}\to D_{s}^{+}\pi^{-}}=12.84$ MeV, larger than the
prediction of Ref. [54] . With this value, we have renewed the
$D_{s}^{+}K^{-}$ and $D_{s}^{+}\pi^{-}$ invariant mass distributions, as shown
in Figs. 10(a) and 10(b), respectively. One can find that our results of the
$D_{s}^{+}K^{-}$ invariant mass distribution are in good agreement with the
Belle measurements in the region $2600\sim 4700$ MeV, and the peak of the
$T_{c\bar{s}0}(2900)^{0}$ in the $D_{s}^{+}\pi^{-}$ invariant mass
distribution is more significant.

##  IV  Conclusions

Recently, the LHCb Collaboration has reported their amplitude analysis of the
processes $B^{0}\to\bar{D}^{0}D_{s}^{+}\pi^{-}$ and $B^{+}\to
D^{-}D_{s}^{+}\pi^{+}$, where two states $T_{c\bar{s}0}(2900)^{0}$ and
$T_{c\bar{s}0}(2900)^{++}$ were observed in the $D_{s}\pi$ invariant mass
distributions. The resonance parameters of these two resonances indicate that
they are two of the isospin triplet. Motivated by those observations of the
LHCb, we propose to search for the state $T_{c\bar{s}0}(2900)^{0}$ in the
process $B^{-}\to D_{s}^{+}K^{-}\pi^{-}$.

In the picture of $T_{c\bar{s}0}(2900)$ as a $D^{*}K^{*}$ molecular state, we
have investigated the process $B^{-}\to D_{s}^{+}K^{-}\pi^{-}$ by taking into
account the $S$-wave $D_{s}^{*}\rho$ and $D^{*}K^{*}$ interaction, and the
$S$-wave pseudoscalar meson-pseudoscalar meson interactions, which dynamically
generate the resonance $D_{0}^{*}(2300)$. We have found that there is a near-
threshold enhancement in the $D_{s}^{+}K^{-}$ invariant mass distribution,
which is in good agreement with the Belle measurements. Indeed, this
enhancement structure is mainly due to the high pole of the $D_{0}^{*}(2300)$.
In addition, a clear peak structure appears around 2900 MeV in the
$D_{s}^{+}\pi^{-}$ invariant mass distribution, which should be associated to
the $T_{c\bar{s}0}(2900)$.

Considering that our predictions for the $D_{s}^{+}K^{-}$ invariant mass
distribution are lower than the Belle measurements in the high energy region,
we take the decay width $\Gamma_{T_{c\bar{s}0}^{0}\to D_{s}^{+}\pi^{-}}$ to be
free parameter, and obtain $\Gamma_{T_{c\bar{s}0}^{0}\to
D_{s}^{+}\pi^{-}}=12.84$ MeV by fitting to Belle measurements. Our new results
show a more significant peak of $T_{c\bar{s}0}(2900)$ in the
$D_{s}^{+}\pi^{-}$ invariant mass distribution. In summary, our predictions
could be tested by the LHCb and Belle II experiments in future, and the more
precise measurements of the process $B^{-}\to D_{s}^{+}K^{-}\pi^{-}$ would
shed light on the nature of the $T_{c\bar{s}0}(2900)^{0}$ resonance.

## Acknowledgements

This work is supported by the National Natural Science Foundation of China
under Grant Nos. 11775050, 12175037, and 12192263, the Natural Science
Foundation of Henan under Grand Nos. 222300420554, 232300421140, the Project
of Youth Backbone Teachers of Colleges and Universities of Henan Province
(2020GGJS017), the Youth Talent Support Project of Henan (2021HYTP002), and
the Open Project of Guangxi Key Laboratory of Nuclear Physics and Nuclear
Technology, No. NLK2021-08.

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