# DRAM Errors and Cosmic Rays:
Space Invaders or Science Fiction?

1st Isaac Boixaderas  Barcelona Supercomputing Center
Barcelona, Spain
<EMAIL_ADDRESS>2nd Jorge Amaya  European Space Agency
Hesse, Germany
<EMAIL_ADDRESS>3rd Sergi Moré  4th Javier Bartolome  Barcelona
Supercomputing Center
Barcelona, Spain
<EMAIL_ADDRESS>Barcelona Supercomputing Center
Barcelona, Spain
<EMAIL_ADDRESS>5th David Vicente  Barcelona Supercomputing Center
Barcelona, Spain
<EMAIL_ADDRESS>6th Osman Unsal  7th Dimitris Gizopoulos  Barcelona
Supercomputing Center
Barcelona, Spain
<EMAIL_ADDRESS>University of Athens
Athens, Greece
<EMAIL_ADDRESS>8th Paul M. Carpenter  Barcelona Supercomputing Center
Barcelona, Spain
<EMAIL_ADDRESS>9th Petar Radojković  10th Eduard Ayguadé  Barcelona
Supercomputing Center
Barcelona, Spain
<EMAIL_ADDRESS>Barcelona Supercomputing Center
Barcelona, Spain
eduard.ayguade<EMAIL_ADDRESS>
###### Abstract

It is widely accepted that cosmic rays are a plausible cause of DRAM errors in
high-performance computing (HPC) systems, and various studies suggest that
they could explain some aspects of the observed DRAM error behavior. However,
this phenomenon is insufficiently studied in production environments.

We analyze the correlations between cosmic rays and DRAM errors on two HPC
clusters: a production supercomputer with server-class DDR3-1600 and a
prototype with LPDDR3-1600 and no hardware error correction. Our error logs
cover 2000 billion MB-hours for the MareNostrum 3 supercomputer and 135
million MB-hours for the Mont-Blanc prototype. Our analysis combines
quantitative analysis, formal statistical methods and machine learning.

We detect no indications that cosmic rays have any influence on the DRAM
errors. To understand whether the findings are specific to systems under
study, located at 100 meters above the sea level, the analysis should be
repeated on other HPC clusters, especially the ones located on higher
altitudes. Also, analysis can (and should) be applied to revisit and extend
numerous previous studies which use cosmic rays as a hypothetical explanation
for some aspects of the observed DRAM error behaviors.

###### Index Terms:

Memory system, Reliability, Cosmic rays

##  I Introduction

In large-scale compute clusters, main memory is one of the principal causes of
hardware failures [1, 2, 3, 4, 5]. These failures are especially costly in
high-performance computing (HPC) systems, where a single tightly-coupled job
may execute for days on thousands of nodes. If one of these nodes fails, the
whole job is terminated. It is therefore important to understand memory system
reliability, as it is an important limit on the ability to scale to larger
systems.

Many studies analyze DRAM errors in the field. One of the potential causes of
DRAM errors are cosmic rays. Various industrial studies performed between the
1970s and early 1990s [6, 7] demonstrate that some of these particles have
sufficient energy to penetrate Earth’s atmosphere down to the sea level,
through the ceiling of the multistory buildings and induce DRAM bit-flips [8].
Since then, the community has accepted that cosmic rays are a plausible cause
of DRAM errors in HPC systems [9, 3, 10, 2, 11, 12, 13, 14, 15].

Some studies go a step further and suggest that cosmic rays could explain some
aspects of the observed DRAM error behavior, such as the higher error rates in
the top rack chassis [16] or in the clusters geographically located at higher
altitudes [17, 16, 18]. The studies also suggest that some (but not all) DRAM
devices are susceptible to transient faults from cosmic ray strikes, and that
this susceptibility highly varies between the device manufacturers [18]. The
cosmic rays are also considered a potential cause of failures with no prior
symptoms and no relevant information in the system logs [12].

To the best of our knowledge, only one previous study has analyzed the
relationship between DRAM error rates and the cosmic rays intensity. El-Sayed
and Schroeder [19] analyze error logs from HPC clusters at Los Alamos National
Lab and conclude that rates of uncorrected DRAM errors are not associated with
cosmic rays intensity. The authors find this result unexpected and provide one
possible explanation for it: while increasing cosmic rays might lead to a
higher rates of DRAM corruptions, the types of corruption caused by those
events might usually be corrected with the built-in error correction codes.
This reasoning is aligned with the idea that transient errors typically lead
to a single-bit data corruption [10].

Our study describes the underlying sources of high-energy cosmic rays,
explains which particles should be observed, and points to the publicly-
available logs from reliable measurement facilities (Section II). We study the
DRAM errors of MareNostrum 3 large-scale production system and mid-scale Mont-
Blanc prototype, which does not support memory error correction in hardware.
The systems are located in the same geographical location, in Barcelona
(Spain), and the errors are logged in overlapping time-periods. The MareNostum
3 logs cover 2000 billion MB-hours of server-class DDR3-1600 DIMMs during
which we detected 4.5 million corrected and 71 uncorrected DRAM errors
(Section III). In the Mont-Blanc prototype, we collected over 25 million
memory errors during the 135 million MB-hours production of mobile LPDDR3-1600
chips (Section IV).

We explore a correlation between the cosmic rays intensity and number of
corrected, transient corrected and uncorrected DRAM errors (Section V and VI).
We also use statistical tests to study whether the error rates have different
distributions during periods of elevated cosmic ray activity, e.g. above
$90^{th}$ or $99^{th}$ neutron count percentile (Section VII). Our exploration
covers a vast number of tests with numerous error categories, time windows and
system scopes. We also examine whether machine learning methods for DRAM error
prediction would be enhanced if they considered the neutron count measurements
(Section VIII). Finally, we apply the presented methodology to verify and
question some findings from previous studies (Section IX).

We hope that the insights gained in this study and the presented methodology
will become a standard for any future analysis of the relationship between
cosmic rays and DRAM errors in the field. The analysis can (and should) be
also applied to revisit and extend numerous previous studies which use cosmic
rays as a hypothetical explanation for some aspects of the observed DRAM error
behaviors. Our study motives, guides and enables a follow-up research on this
frequently mentioned, but insufficiently studied phenomenon.

##  II Cosmic rays

###  II-A Background

Cosmic rays are high-energy particles that originate from outer space. They
normally come from supernovae or other energetic events in the galaxy, and
occasionally come from the Sun during periods of high solar activity. When
cosmic rays enter the Earth’s atmosphere, they produce a cascade of secondary
particles which can reach the Earth’s surface and even penetrate underground.
The particles hitting the DRAM chips can alternate the cell charge and corrupt
the stored data, causing DRAM errors.

The density of the atmosphere decreases exponentially with altitude, providing
less shielding at higher altitudes. This results in a greater flux of
secondary particles and an increased probability of errors caused by cosmic
rays at higher elevations [20, 8, 7]. A common belief is that cosmic rays are
also a plausible cause of DRAM errors in datacenters, especially if they are
located on high altitudes. This belief, however, is not supported by any
large-scale field study.

###  II-B Data collection

From all the cascading elements produced by cosmic rays, neutrons are the most
commonly observed at ground level. Their flux can be measured with specialized
neutron monitors, ground-based detectors that measure the neutrons that reach
the Earth’s surface. Rare solar events that produce very high-energy particles
can also generate secondary cascades in the atmosphere, which can be detected
by neutron monitors. The Neutron Monitor Database [21] provides public access
to neutron count logs from more than 50 neutron monitors located all over the
world, most of them in the northern hemisphere.

To quantify the impact of cosmic rays on HPC cluster failure rates, users can
correlate the system error logs with the neutron counts from a monitor. The
monitor should be selected based on the distance and the altitude difference
with respect to the HPC facility. If various nearby neutron monitors are
available, the users should compare their measurements to ensure that they are
representative of the cosmic ray activity in the region. The exact number of
neutron counts reported by these monitors can vary, but their correlation
should be high.

We considered the cosmic ray logs from two neutron monitors. The first one is
located in Guadalajara (Spain), 470 km southwest of Barcelona, situated at an
altitude of 708 meters [22]. The second one is in Rome (Italy), 870 km east of
Barcelona and located close to sea level [23], as the MareNostrum
supercomputer and the Mont-Blanc prototype. During the period of our DRAM
error logs, we observed a high correlation coefficient between neutron counts
from Guadalajara and Rome. This correlation, together with the geographic
positioning of Guadalajara and Rome neutron monitors, both relatively close to
Barcelona but on opposite sides, indicate that the considered neutron monitor
measurements are likely representative of the general cosmic rays in our
region.

Because of proximity, we used the Guadalajara cosmic ray logs. To ensure that
the measurements accurately reflect cosmic ray activity, we use data corrected
for pressure and instrumental factors.111In the Neutron Monitor logs, this is
referred to as efficiency correction. Corrections for instrumental changes,
such as modifications of the building and the measurement equipment, are
necessary to maintain data consistency. Also, the atmospheric pressure
correction is important because higher atmospheric pressure increases the air
mass above the detector, reducing the detected neutron flux.

Solar cycles are periodic fluctuations in solar activity, lasting
approximately 11 years. During the period of our logs, we were in a descending
phase of a Solar cycle which began in December 2008 and ended in December
2019. During periods of high solar activity, the solar wind acts as a shield
against galactic cosmic rays from sources beyond the Sun and prevent them to
reach the Earth [24]. A clear anti-correlation has been observed between the
activity of the solar cycle and measurements, on the ground and in space, of
galactic cosmic rays. This is visible in the neutron counts, which moderately
increase during the observation period (see Section V-A).

##  III MareNostrum 3 DRAM errors

###  III-A MareNostrum 3 supercomputer

The MareNostrum 3 error logs cover a period of more than two years, from
October 2014 to November 2016, during which we detected 4.5 million corrected
and 71 uncorrected DRAM errors.222The log also contains 262 over-temperature
DRAM errors, but they are not analyzed because they are unrelated to the
cosmic ray activity. The supercomputer was located in Barcelona (Spain), 100
meters above the sea level, in a stone-walls chapel with a ceramic tiles roof.
At the time, MareNostrum 3 was one of the six Tier-0 (largest) HPC systems in
the Partnership for Advanced Computing in Europe (PRACE) [25]. It comprised
3056 compute nodes, each with two eight-core Intel Sandy Bridge-EP E5-2670
sockets with a 2.6 GHz nominal clock frequency. We use the logs from the
compute nodes only, excluding the login and test nodes which are not part of
the same monitoring infrastructure and whose failures do not impact large-
scale compute jobs.

The MareNostrum 3 compute nodes included more than 25,000 DDR3-1600 DIMMs from
all three major manufacturers: 6694, 5207 and 13,419 DIMMs from _Manufacturer
A_ ,  _B_ and  _C_ , respectively. The manufacturers have been anonymized to
protect the interested parties. MareNostrum 3 DIMMs were built in three
different DRAM technologies: $\overline{3x}$ nm, $\overline{2y}$ nm and
$\overline{2z}$ nm. We are allowed to show only the first of two digits of the
nanometer technology and their relative order: $\overline{3x}$ nm $>$
$\overline{2y}$ nm $>$ $\overline{2z}$ nm. During the observation period we
collected measurements on more than 2000 billion MB-hours. The main workloads
executed on MareNostrum 3 were large-scale scientific HPC applications and the
system utilization typically exceeded 95%.

MareNostrum 3 employed a Chipkill ECC scheme, which could correct all errors
coming from a single x4 device. For x8 devices, the ECC can correct up to
4-bit errors coming from the same DRAM chip. The ECC check is performed on
each application memory read and by a patrol scrubber which periodically
traverses the whole physical memory and performs an ECC check on each
location.

During production, any DIMM that showed early signs of failure was flagged by
a pre-failure alert and retired by the system administrators. This action was
recorded in the system log together with the date and time. Over the two-year
period analysed by this paper, 51 DIMMs were retired for this reason.

###  III-B Data collection

Corrected errors (CEs) were logged by a daemon, based on the mcelog Linux
kernel module [26], that periodically extracts information about corrected
errors from the CPU machine-check architecture (MCA) registers [26]. Each
corrected error was recorded in a log file, which specifies the error
timestamp, node and DIMM id, and the physical location of the error in the
DIMM including rank, bank, row and column. The log entry also indicates
whether correction was done during an application memory read or by patrol
scrubbing.

The daemon accessed the MCA registers with a period of 100 ms, which was the
shortest time interval that caused negligible overhead to the production
applications. If more than one error occurred in a 100 ms time interval, the
MCA registers record the number of errors, but only provide detailed
information for one error in the interval. Our logs therefore specify the
exact total number of corrected errors and provide detailed error information
for a subset of the errors. Increasing the sampling frequency would increase
the number of errors with detailed information, but also increase the
performance overhead of the error logging daemon. Previous studies perform
similar readings of the memory error registers with a period of a few seconds
[27, 16, 18] or once per hour [10].

Our analysis covers both, transient and non-transient CEs. Transient CEs are
frequently considered to be caused by temporary environmental factors, such as
cosmic ray strikes [10, 4, 27, 18]. We consider an error to be transient if,
during the entire observation period, the error occurred only once in a given
cell and no other errors occurred in the same row or column [4].

Uncorrected errors (UEs) were logged by IBM firmware [28], which is part of
the MareNostrum 3 monitoring software. For each uncorrected error, the log
specifies the DIMM that failed and the cause of the error, i.e. whether it
occurred during an application memory read or patrol scrubbing. Finally, the
log records UE warnings generated when memory modules are throttled to prevent
an over-temperature condition and when the correctable ECC logging limit has
been reached.

##  IV Mont-Blanc DRAM errors

###  IV-A Mont-Blanc prototype

The prototype was located in Barcelona (Spain), in the same area and altitude
as MareNostrum, 100 meters above the sea level. The system was deployed in a
brick building with a concrete roof. The Mont-Blanc prototype comprised 1080
nodes based on the Samsung Exynos 5250 mobile system-on-chip (SoC) [29]. Each
SoC integrated two 1.7 GHz Arm Cortex-A15 cores, one on-chip Mali-T604 GPU and
4 GB of LPDDR3-1600. All memory devices are from the same manufacturer. The
objective of the Mont-Blanc prototype was to explore the suitability of high-
performance computing on hardware initially targeting mobile devices.
Therefore, the deployed LPDDR3 main memory included no error detection or
correction mechanisms.

###  IV-B Data collection

Mont-Blanc error logging lasted for over a year, February 2015 to February
2016, which is within the time-frame of the MareNostrum logs. It monitored
over 4.2 million node-hours, 135 million MB-hours of LPDDR3-1600, and logging
over 25 million memory errors [30]. DRAM errors were logged with a software
daemon running an infinite loop that traverses the whole available system
memory. In each node, the daemon allocated approximately 3 GB of main memory,
while the rest was dedicated for the OS and other system software. The daemon
was first writing a specific value into each memory location, and then, in the
next loop iteration, was checking whether the memory still contained the
correct data. If this was not the case, the tool recorded the memory error
location and the number of corrupted bits. Since the daemon memory utilization
was significant, it was executed only when the nodes were not running any user
job. For this reason, the total amount of memory scanned depended on the
prototype utilization, which varied in time. To account for this, we normalize
the number of DRAM errors per amount of scanned memory in that specific time.
The Mont-Blanc error-logging framework was set by Bautista et al. [30]. Our
analysis is based on the data reported in the original study.

##  V Timeline analysis

We first examine the timelines of the DRAM error logs and neutron counts to
understand the general trends and overall behavior. This analysis helps to put
into context the correlation analysis of Section VI.

###  V-A Neutron counts

Figure 1 shows neutron counts per second recorded by the Guadalajara neutron
monitor from October 2014 to November 2016, the period corresponding to the
DRAM error logs of MareNostrum 3. The overall neutron count range in this
period is representative of the complete operational neutron monitor data
spanning from 2012 to 2024. We plot the neutron counts per second averaged by
hour and by month. The hourly data is the finest granularity that provides a
readable plot. We also see some variation within the cycle, and some peak
behavior, e.g. negative peaks in January and July 2015, and a positive peak in
November 2016. Finally, we also see a general trend of increasing monitor
counts over time, especially after August 2015. This increasing trend is
confirmed with the trend-line showing monthly neutron averages, and it may be
attributed to decreasing solar activity in this period, as discussed in
Section II-B.

###  V-B DRAM errors

Figure 1: Neutron counts per second. Hour and month averages.

(a) MareNostrum 3: Total corrected errors per hour and per month.

(b) MareNostrum 3: Total uncorrected errors per hour and per month.

(c) Mont-Blanc: Total errors per day and per month.

Figure 2: DRAM errors over time. MareNostrum 3 production supercomputer and
Mont-Blanc prototype.

Figure 2(a) shows the MareNostrum corrected errors (CEs) hourly and monthly
occurrences between October 2014 and November 2016. On most hours there are up
to 100 corrected errors, but on a few hours the number of errors is large, up
to about 10,000. The results suggest that we should further explore a
potential alignment between the neutron count and error peaks. Comparison of
the neutrons and CEs monthly trend-lines, Figures 1 and 2(a), detects no
obvious correlation between these two variables. We repeat this analysis
focused only on the transient CEs, and we reach the same conclusions.

Figure 2(b) shows the number of uncorrected MareNostrum DRAM errors (UEs) per
hour and per month, across all DIMMs. In most hours we detect no errors, on a
few hours we detect one error, and very occasionally we detect two or three
errors. In total during the observation period of 25 months we detect only 71
errors. Regarding the monthly values, we detect somewhat higher UE rates in
the period June 2015 to June 2016, which does correspond to above-average
neutron counts. However, we also detect some of the lowest UE numbers in the
last months of the study, which is the period of the highest cosmic rays
intensity. Overall, the visual comparison of the neutron and UE trend-lines
detects no obvious correlation.

Figure 2(c) depicts the number of errors per day and per month for the Mont-
Blanc system, normalized to the amount of scanned main memory. The timescale
($x$-axis) is aligned with the neutron counts measurements in Figure 1 and the
hatched section of the chart indicates the time period for which we do not
have Mont-Blanc error logs. From May to November 2015, we see a significant
monotonic increment of the error rates, well correlated with the neutron
counts. However, the correlation is low at the beginning and the end of the
Mont-Blanc logs. This is especially pronounced in the last three months,
November 2015 to February 2016, in which the neutron counts continue to
increase, while the error rates drop by an order of magnitude.

##  VI Correlations

Next, we further explore the correlation between the neutron counts and DRAM
error rates.

###  VI-A Methodology

First, we use a visual correlation between the neutron counts and the total
number of DRAM errors detected in the same time period. Second, we apply the
Kendall test [31] to explore correlation for a large number of error
categories.333Kendall’s test assesses the strength and direction of the
association between two variables by measuring the ordinal association between
them. We also considered Pearson correlation and Spearman’s rank correlation.
Pearson correlation is disregarded because it requires normal data
distribution and a linear relationship between the two variables. As Kendall’s
coefficient, Spearman’s correlation is a rank-based non-parametric test, so it
can be applied to our analysis. We selected the Kendall correlation measure
because it is more robust than Spearman’s rank correlation [32]. We perform
statistical tests on all combinations of the error categories, time windows,
and system scopes listed in Table I (MareNostrum 3) and Table II (Mont-Blanc).
For MareNostrum supercomputer, this leads to a vast number of correlation
tests: 1,764,096 for uncorrected and 10,584,576 for corrected errors.444For
example, for uncorrected errors, there are four time-windows, four
manufacturer categories, three error types, and four technologies for a total
of 192 combinations. The main increase in tests comes from 9188 different
system scopes. The tests are applied to the whole system, but also
individually for each of the 37 racks, 3050 nodes, and 6100 sockets
($1+37+3050+6100=9188$). In Mont-Blanc error logs contain much less
information, leading to only 21 error categories.

TABLE I: MareNostrum 3 error categories, time-windows and system scopes.
MareNostrum uncorrected and corrected errors
---
Manufacturers |  $A$, $B$, $C$, All
DIMM technology |  $\overline{3x}$ nm, $\overline{2y}$ nm, $\overline{2z}$ nm, All
Time window |  Hour, day, week, month
System scope |  Whole system, 37 racks, 3050 nodes, 6100 sockets
MareNostrum corrected errors
Transient _vs._ Non-transient |  Transient, Non-transient, All
Error type |  Memory read, Patrol scrub, All
Single/multi-cell |  Single-cell, Multi-cell, All
CEs _vs._ DIMMs |  Total number of CEs, Number of DIMMs with at least one CE
MareNostrum uncorrected errors
Error types |  Uncorrected ECC, Scrub failed, All
TABLE II: Mont-Blanc error categories and time-windows. n-bit error |  1, 2, 3, 4, 5, 6+ bits, All
---|---
Time window |  Day, week, month

For each experiment, the Kendall’s test returns a correlation coefficient
between -1 (strong inverse correlation) and +1 (strong direct correlation).
The test also detects cases in which the correlation cannot be tested. If all
values of one variable are identical, e.g. zero detected DRAM errors, it is
impossible to rank them meaningfully, and there is no basis to determine
concordance or discordance with the cosmic rays intensity. Finally, the
Kendall’s test returns a p-value that quantifies the correlation significance.
A large number of applied tests increases the probability that some tests
incorrectly suggest correlation significance (i.e. low p-values). To minimize
this likelihood we apply the Benjamini–Yekutieli false discovery rate method
[33].

###  VI-B Visual correlation

(a) MareNostrum: Corrected errors

(b) MareNostrum: Uncorrected errors

(c) Mont-Blanc errors (normalized)

Figure 3: Neutron counts ($x$-axis) and the number of DRAM errors in the
corresponding time period ($y$-axis). Bare-eye analysis detects no
correlations.

Figure 3 shows averaged neutron counts per second ($x$-axis) and the number of
memory errors recorded in the corresponding time period ($y$-axis). As in the
rest of the paper, the Mont-Blanc error counts are normalized with the amount
of scanned memory. The figures plot the data for the whole system, 3056
MareNostrum 3 nodes and 1080 Mont-Blanc nodes. We use a gray heat-map to show
the number of observations with a given error and neutron count. To cover a
large range of MareNostrum CEs per hour, the $y$-axis and the heatmap in
Figure 3(a) use a log scale.

Overall, we find no correlation that can be observed with the naked eye. We
repeat the experiments for MareNostrum transient corrected errors, and we
reach the same conclusion. All three charts show a higher density between 70
and 72 neutron counts per second because this is the most frequent neutron
count range during the time of the study.

###  VI-C Statistical tests

(a) MareNostrum: Corrected errors

(b) MareNostrum: Uncorrected errors

(c) Mont-Blanc errors

Figure 4: Most of the correlation tests report low coefficients.

In MareNostrum analysis, numerous error categories, time windows and system
scopes lead to a vast overall exploration space. However, the nature of the
DRAM errors and some practical limitations, hugely reduce the number of
practical correlation tests: from 1,764,096 to 2352 for uncorrected, and from
31,753,728 to 671,404 for corrected errors. There are three main reasons for
this reduction. First, most of the nodes experienced no DRAM errors: 62%
experienced no CEs, and 99% no UEs. For these nodes, there is no basis to
determine DRAM error concordance or discordance with the neutron counts. Also,
not all possible categorical combinations are present in practice. For
example, each MareNostrum node comprises DIMMs from the same manufacturer and
technology. If a given node experiences DRAM errors, it can be tested only for
this specific manufacturer–technology pair, not for all possible combinations
of these two parameters. Finally, many of the error log inputs are incomplete,
with some fields that are omitted.

Figure 4 shows the Kendall’s correlation coefficient for all DRAM error
categories, time windows and system scopes. The coefficients are sorted in
ascending order. To increase the visibility in MareNostum experiments, we plot
distinct charts for negative and positive ($\geq 0$) correlation coefficients,
while the gap between them indicates the tests that could not be deployed in
practice. The corrected error coefficients range are low, with only a handful
of experiments that exceed a value of $0.3$, see Figure 4(a). For the
uncorrected errors, depicted in Figure 4(b), the coefficients are even lower.
Also, in both charts we observe a roughly equal proportion of the positive
correlations, and the unexpected negative ones.

For the Mont-Blanc prototype, we test 21 correlations, for three time-windows
and seven error types. Their correlation coefficients range from $-0.24$ to
$+0.49$.

To consider the significance of the Kendall’s correlation coefficients, we
analyze the corresponding p-values. Figure 5 depicts the histogram of the
p-values provided directly by the Kendall’s test and after the
Benjamini–Yekutieli false discovery rate correction [33].

(a) MareNostrum: Corrected errors

(b) MareNostrum: Uncorrected errors

(c) Mont-Blanc errors

Figure 5: After the false discovery rate correction, $p\text{-value}=1$ in a
vast majority of correlation tests.

In MareNostrum tests, after the correction, $p\text{-value}=1$ in a vast
majority of the CE tests and all UE tests. As the final step of this analysis,
we explore the tests that have statistically significant (p-value below 0.05)
and moderate or high correlation coefficient. We detect no such tests for
transient corrected errors. This is surprising because cosmic ray strikes are
frequently considered a plausible cause of these errors [10, 4, 27, 18]. When
we analyze the whole CE sample, we find no statistically significant tests
with inverse correlations below $-0.5$, and only 40 tests with positive
correlations above $+0.5$. All of them correspond to the 1 month time window
in the 25-months logging period, i.e. the sample of only 25 observations. Out
of the 40 tests, 36 are from three DIMMs located in two nodes. All three DIMMs
experience higher error rates in the months of the logging period, coinciding
with higher neutron counts. However, the recorded errors are repeated, up to
thousands of times, on a small number of distinct physical locations (memory
cells). Therefore, it is highly unlikely that their underlying cause is an
external factor, such as cosmic rays.

After the false discovery rate correction, only four Mont-Blanc tests have
p-values below 0.05. All of them correspond to correlation coefficients of
0.13 and 0.14, indicating a low correlation with cosmic rays.

The correlation tests consider the overall trend across all data points and
may not reveal a significant association if DRAM errors are more likely to
occur only above a certain neutron count threshold. To address this
limitation, in the next section we explore the error rates during periods of
elevated cosmic ray activity.

##  VII Periods with the highest neutron counts

###  VII-A Methodology

To explore whether DRAM errors are more affected by cosmic rays above a
certain neutron count threshold, we partition the DRAM errors into two
samples: those observed during high neutron-count periods (above a certain
threshold) and the remaining errors. Then we use the Kolmogorov–Smirnov (KS)
test [34] to determine whether two DRAM error samples come from the same
distribution. Since there is no predefined neutron count threshold which is
known to impact DRAM errors, we consider the $90^{th}$, $95^{th}$, $99^{th}$,
and $99.9^{th}$ neutron count percentiles. As in the previous section, we
apply the test to all combinations of the error categories, time windows and
system scopes listed in Table I and II, and then use the Benjamini–Yekutieli
p-value correction.

###  VII-B Results

(a) MareNostrum: Corrected errors

(b) MareNostrum: Uncorrected errors

(c) Mont-Blanc errors

Figure 6: After the false discovery rate correction, $p\text{-value}=1$ in a
vast majority of correlation tests.

Figure 6 shows the histogram of the p-values provided directly by the
Kolmogorov–Smirnov test and after the false discovery rate correction. After
the correction, $p\text{-value}=1$ in all Mont-Blanc and MareNostrum
uncorrected error tests. Out of 127,014,912 MareNostrum corrected error tests,
we detect 40 statistically significant instances (p-values below 0.05) that
exhibit a surprising behavior: lower error rates during the higher cosmic ray
activity. The expected behavior, higher error rates during periods with
neutron counts above $90^{th}$ and $95^{th}$ percentile, is detected in 184
tests. Most of these 184 tests are triggered by only five DIMMs located in
three nodes. These DIMMs and the nodes are different from the ones that had a
statistically significant correlation with the neutron counts (see Section
VI-C). However, they show a similar behavior: they experience higher error
rates in the months with the highest cosmic rays activity, but the errors are
periodically repeated on a small number of distinct physical locations.
Therefore, it is highly unlikely that their underlying cause are cosmic rays.

##  VIII DRAM error prediction

We also explore whether machine learning methods for DRAM error prediction
would be enhanced by considering the neutron count measurements.

###  VIII-A Methodology

We start from random forest model used by Boixaderas et al. [35]. The model
predicts DRAM uncorrected errors (UEs) based on system metrics and error log
information from each DIMM: number of CEs, UEs and UE warnings, number of
affected ranks, banks, rows and columns, etc. The model checks the system logs
every minute and then predicts whether a UE will occur in a DIMM within the
next day. The evaluation is based on a cost–benefit analysis, which compares
the system resources needed for training, failure prediction and failure
mitigation against the saved compute time due to successful failure prediction
and mitigation.

We also use the same random forest model, set of features and methodology to
predict the forthcoming corrected errors (CEs). To account for the higher
frequency of CEs (compared to UEs) we reduce the prediction window to one
hour. The objective of our CE predictions is not to improve the HPC system
utilization,555The CEs do not lead to HPC job failures and lost compute time
[35]. but to investigate whether the neutron counts data enhance the model’s
predictive accuracy. If this were the case, it would indicate that the
deployed random forest model is capable of detecting subtle relationships
between cosmic ray intensity and CE rates; a relationship not captured by the
statistical methods analyzed in the previous sections. The CE evaluation is
based on the Area Under the Curve (AUC), a standard performance metric used to
evaluate the quality of a binary classification model [36]. The metric
considers the model’s true positive and false positive rates across various
decision threshold settings.

To explore the influence of cosmic rays, we extend the feature set of the
original random forest model with the neutron counts data. We include the
average, standard deviation, and percentage variation of neutron counts over 1
hour, 5 hours, 10 hours, 1 day, 1 week, and 1 month. We split the feature data
into 60% for training, 20% for validation and 20% for testing. After training
and selecting the best model through hyperparameter tuning, we retrain it
using both the training and validation sets before evaluating its final
performance on the test set. To determine the significance of neutron count
features, we train and evaluate the primary and reference models. The primary
models use the actual neutron counts information. The reference models use the
same data, but with permuted neutron count features. This is a standard
procedure for assessing feature importance because it maintains the
distribution of feature values and ensures the consistency between primary and
reference models [37]. We analyze the predictive capabilities of the models
for corrected and uncorrected errors, and use the Gini importance [38] to
analyze the significance of the neutron count features.

###  VIII-B Results

The models that include the actual neutron counts show practically the same
prediction performance as the reference models with permuted neutron count
features. The primary and reference CE prediction models achieve an AUC of
0.92 and 0.93, respectively. The cost–benefit evaluation of their UE
predictions suggest savings of 7451 and 7449 node–hours.666 We use the same
HPC job logs and cost–benefit calculation as the original study of Boixaderas
et al. [35]. The only difference is that we train the model with the majority
of the logging data, and test with UEs from the final 20% of the logs. The
original study uses time series cross-validation across the entire dataset,
starting with smaller datasets at the beginning of the logs and incrementally
increasing their size. For this reason, we report a different number of saved
node–hours.

(a) MareNostrum: Corrected errors

(b) MareNostrum: Uncorrected errors

Figure 7: Gini importance indicates which features are most influential in the
model’s predictions.

Figure 7 shows the importance of different feature categories, as assigned by
the prediction models. Our analysis focuses on the neutron counts group (in
bold). In the CE prediction model with actual neutron counts, this feature
category is the third most important one (Figure 7(a)). The reference model
appropriately assigns zero Gini importance to these permuted features. The UE
models assign even higher importance to the neutron count features (Figure
7(b)). In the primary model, this is the second most important category,
exceeded only by the CE features. Even the reference model puts significant
importance to these features. This unexpected model behavior could be related
to the instability of the UE prediction model. This is because a small number
of uncorrected errors leads to a significant class imbalance that requires
considerable undersampling [35], reducing the available information during
training and leading to increased model instability.

Overall, CE and UE prediction models using actual neutron counts assign some
importance to these features, but they do not perform better than the
reference models in which the neutron counts are permuted. The assigned
importance could be attributed to the inherent randomness of the models, where
the neutron counts may occasionally appear significant in random subsets of
the data. This can be observed in Figure 7(b), where even randomly permuted
neutron counts are given certain importance. Another possibility could be that
neutron counts are correlated with another feature, causing the importance to
shift to that feature when neutron counts are permuted. However, we discard
this possibility by performing correlation tests between each neutron count
feature and every other feature, confirming that no such correlation exists.
Finally, it is also possible that neutron counts are only important in a very
limited set of circumstances, making their impact imperceptible when
evaluating the overall model performance.

##  IX The signal and the noise

(a) Mont-Blanc errors. Reproduced from the original study [30].

(b) Cosmic rays

(c) MareNostrum: Uncorrected errors

(d) MareNostrum: Corrected errors

(e) MareNostrum: Transient corrected errors

Figure 8: The neutron counts do not vary between different hours. DRAM error
variations cannot be explained with cosmic rays intensity.

In this section we demonstrate why the analysis and methodology we advocate
are so important, and what can happen if they are not followed.

Previous study on the Mont-Blanc DRAM errors _“presents evidence which
suggests that multi-bit errors are more likely to occur during day time with a
high peak when the Sun is at the highest point in the sky”_ [30]. The authors
therefore _“suggest that multi-bit memory errors are mostly caused by cosmic
rays”_. These conclusions are based on the results presented in Figure 8(a),
which is reproduced from the original paper. The figure plots the total number
of multi-bit memory corruptions at a given hour of the day ($x$-axis), during
the observation period from February 2015 to February 2016.

Figure 8(b) plots the neutron counts recorded by the Guadalajara neutron
monitor in the same period. Unsurprisingly, we detect no variation in cosmic
ray intensity depending on the time of the day. We get the same results for
the complete operational neutron monitor data spanning from 2012 to 2024. This
is simply because the position of the Sun on the sky does not impact the rates
of cosmic rays that could cause DRAM errors.

For comparison, we also plot the hour-of-the-day for MareNostrum errors
between February 2015 and February 2016. In this period, we detected 29
uncorrected errors (Figure 8(c)). We also plot the hour-of-the-day chart for
all corrected errors (Figure 8(d)) and the transient ones (Figure 8(e)). For
all CEs, we detect a clear peak at 9 h, while the largest number of transient
CEs is detected at 18 h. To illustrate how a small number of DIMMs can change
the general trend of the data, we remove 1% of the DIMMs with the highest
number of errors. These results are shown with darker narrow bars in Figures
8(d) and 8(e). We still detect some variation between the time of the day, but
some of the highest error rates are now recorded in different hours. We
repeated the same analysis with our full logs covering the period October 2014
to November 2016. These results are not displayed, but the conclusions are the
same.

Our results show that DRAM error rates can be volatile even after long periods
of logging on large-scale systems [9]. Therefore, although in many research
areas bare-eye analysis of the charts might be sufficient, in this particular
domain, it will almost certainly lead to unreliable results and incorrect
conclusions. The best way to overcome this problem is to use formal
statistical methods. The methods that we advocate and deploy are simple to
understand and implement, reliable and widely accepted in the statistical
community. In addition to the extensive methodology, we describe the
underlying sources of cosmic rays, explain which particles should be observed,
and provide pointers to publicly-available logs from reliable measurement
facilities. We hope that the shared resources and presented methodology will
become a standard for any future analysis of the relationship between cosmic
rays and DRAM errors in the field. Also, numerous past studies on DRAM errors
can (and should) be revisited and extended with this analysis.

##  X Related Work

Many recent studies have presented extensive quantitative analyses of DRAM
errors in the field. The studies analyze error rates and correlate them with
chip capacity, temperature, utilization, aging and DIMM generation [3]. The
papers also distinguish between the transient (soft) and non-transient (hard)
errors [10, 4], and consider the position of the DRAM device in the data-
center [16] and the error position in the chip [27, 16, 18, 39, 9, 5].

It is widely accepted that cosmic rays are a plausible cause of DRAM errors in
datacenters [9, 35, 40, 3, 10, 2, 11, 12, 13, 14, 15]. Also, cosmic rays are
frequently used as an explanation for some aspects of the observed DRAM error
behaviors [17, 30, 12, 16, 18]. Schroeder and Gibson [17] analyze error logs
of 22 HPC systems that were in production use at Los Alamos National
Laboratory between 1996 and 2005. The authors detect high DRAM error rates,
and suggest that part of the problem could be high altitude of the cluster,
which makes it more exposed to cosmic rays. Das et al. [12] analyze the root
causes of failures in peta-scale HPC clusters between 2014 and 2016. For some
failures, the system logs have insufficient information and show no prior
symptoms. The authors, therefore, suspect that a potential cause of these
failures could be solar flares or rare cosmic radiations. Sridharan et al.
[16] analyze DRAM and SRAM errors from two supercomputers located at Los
Alamos National Lab (7320 feet altitude) and Oak Ridge National Lab (850 feet
altitude). The high-altitude Los Alamos facility reports significantly higher
SRAM error rates. The authors consider that this finding is sufficient to
conclude that the majority of observed SRAM faults are caused by cosmic ray
strikes. The authors also suggest that cosmic rays are another possible cause
of the elevated fault rates in the higher rack chassis. The follow-up study
[18] from the same HPC facility concludes that some (but not all) DRAM devices
are susceptible to transient faults from cosmic rays, and that this
susceptibility highly varies by device manufacturer. All these studies
consider cosmic rays to be an intuitive hypothetical cause of DRAM errors in
the field. However, none of them tests this hypothesis.

A quantitative correlation between DRAM error rates and cosmic rays intensity
is performed only by El-Sayed and Schroeder [19]. The study analyzes error
logs from HPC clusters at Los Alamos National Lab with the objective to
discover the factors that influence the HPC system reliability. One out of
many factors under study is the cosmic radiation. The authors visually analyze
monthly probability of a uncorrected DRAM error as a function of the monthly
average neutron counts reported by a neutron monitor located 400 km north from
the national lab. The results suggest that cosmic rays are not associated with
higher uncorrected DRAM error rates. Actually, the results suggest the
opposite trend — the higher the neutron counts, the lower the number of
errors. The authors find this unexpected and provide one possible explanation
for it: while increased rates of cosmic rays might lead to a higher number of
corrupted bits, the types of corruption caused by those events might usually
be corrected with the error correction codes. This reasoning is aligned with
the idea that transient errors typically lead to a single-bit data corruption
[10]. We agree that contemporary error correcting codes, capable to correct up
to 4 corrupted bits in a 72-bit word, would probably correct a vast majority
of the transient errors caused by an external radiation. For this reason, we
extend the study of El-Sayed and Schroeder with analysis of the corrected and
transient corrected errors. Also, apart from the visual correlation, we apply
formal statistical tests and analyze the reported correlation coefficients and
their statistical significance. We also use statistical tests to explore
whether the error rates have different distributions during periods of
elevated cosmic ray activity, e.g. above $90^{th}$ or $99^{th}$ neutron count
percentile. We perform a vast number of tests that cover all combinations of
the error categories, time windows and system scopes. Finally, we analyze in
detail all tests that showed statistical significance.

Machine learning methods predict DRAM errors based on the error history,
device characteristics (e.g. manufacturer and technology), sensors that
monitor the system (e.g. temperature), and workload properties (e.g. memory
bandwidth utilization) [41, 42, 43, 44, 45, 46, 47, 2, 48, 49, 35, 40, 15,
50]. To the best of our knowledge, our study is the first one to explore
whether considering the neutron count measurements would enhance these
prediction methods.

##  XI Conclusions

In this paper we describe the underlying sources of cosmic rays, explain which
particles should be observed, and point to publicly-available logs from
reliable measurement facilities. Then we analyze cosmic rays and DRAM errors
of two HPC clusters located in Barcelona, 100 meters above the sea level. We
explore correlations between cosmic ray intensity and the number of DRAM
errors, and apply statistical tests to study whether the error rates have
different distributions during periods of elevated cosmic ray activity, e.g.
above $90^{th}$ or $99^{th}$ percentile. The exploration covers a vast number
of tests that include numerous error categories, time windows and system
scopes. We also examine whether machine learning methods for DRAM error
prediction would be enhanced if they considered the neutron count
measurements.

For the two HPC systems under study, we detect no apparent correlation between
DRAM errors and cosmic rays, and no error rate increment even during periods
with the highest cosmic ray intensity. We also see that considering neutron
counts does not improve DRAM error predictions. In addition to this, we apply
the presented methodology to verify the findings from previous studies which
claim that some DRAM errors are mostly caused by cosmic rays. We show that
these results are unreliable and the conclusions incorrect. Overall, we detect
no indications that cosmic rays have any influence on DRAM errors. To
understand whether the findings are specific to the systems under study, the
analysis should be repeated on other HPC clusters, especially the ones located
on higher altitudes. Additionally, the analysis can (and should) be applied to
revisit and extend numerous previous studies that use cosmic rays as a
hypothetical explanation for certain aspects of the observed DRAM error
behaviors. Overall, our study motives, guides and enables a future research on
the frequently mentioned, but insufficiently studied relationship between
cosmic rays and DRAM errors in the field.

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# Searching for the open flavor tetraquark $T^{++}_{c\bar{s}0}(2900)$ in the
process $B^{+}\to K^{+}D^{+}D^{-}$

Man-Yu Duan1 En Wang2<EMAIL_ADDRESS>Dian-Yong Chen1,3111Corresponding
author<EMAIL_ADDRESS>1School of Physics, Southeast University, Nanjing
210094, China
2School of Physics and Microelectronics, Zhengzhou University, Zhengzhou,
Henan 450001, China
3Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou 730000,
P. R. China

###### Abstract

Inspired by recent observations of $T_{c\bar{s}0}(2900)^{0}$ in the
$D_{s}^{+}\pi^{-}$ invariant mass distribution of
$B^{0}\to\bar{D}^{0}D_{s}^{+}\pi^{-}$ decay and $T_{c\bar{s}0}(2900)^{++}$ in
the $D_{s}^{+}\pi^{+}$ invariant mass distribution of $B^{+}\to
D^{-}D_{s}^{+}\pi^{+}$ decay, we investigate the $T_{c\bar{s}0}(2900)^{++}$
contribution to the $B^{+}\to K^{+}D^{+}D^{-}$ decay in a molecular scenario,
where we consider $T_{c\bar{s}0}(2900)^{++}$ as a $D^{\ast+}K^{\ast+}$
molecular state. Our estimations indicate that the fit fraction of
$T_{c\bar{s}0}(2900)^{++}$ in the $B^{+}\to K^{+}D^{+}D^{-}$ is about
$12.5\%$, and its signal is visible in the $D^{+}K^{+}$ invariant mass
distribution. With the involvement of $T_{c\bar{s}0}(2900)^{++}$, the fit
fractions of $\chi_{c0}(3915)$ and $\chi_{c2}(3930)$ may be much different
with the ones obtained by the present amplitude analysis [Phys. Rev. D 102,
112003 (2020)], which may shed light on the long standing puzzle of
$\chi_{c0}(3915)$ as the conventional charmonium.

###### pacs:

14.40.Pq, 13.20.Gd, 12.39.Fe

##  I INTRODUCTION

The $B$ meson decay process is the most productive and important platform of
searching for the QCD exotic states. Two typical types of exotic candidates
could be observed in this process. One is the charmonium-like state observed
in the invariant mass distributions of a charmonium plus one or more light
meson, such as the first observed charmonium-like state, $X(3872)$, which was
first 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 Belle:2003nnu , and then confirmed by the
BaBar Aubert:2004fc ; Aubert:2004ns ; Aubert:2005eg ; Aubert:2005zh ;
Aubert:2005vi ; Aubert:2006aj ; Aubert:2007rva ; Aubert:2008gu ; Aubert:2008ae
; delAmoSanchez:2010jr , CDF Acosta:2003zx ; Abulencia:2005zc ;
Abulencia:2006ma ; Aaltonen:2009vj , D0 Abazov:2004kp , CMS CMS:2011yra ;
Vesentini:2012lea ; Chatrchyan:2013cld ; DallOsso:2013rtt ; DallOsso:2014cmg ;
Sirunyan:2020qir , and LHCb Aaij:2011sn ; LHCb:2011bia ; LHCb:2011cra ;
Aaij:2013zoa ; Aaij:2013rha ; Aaij:2014ala ; Aaij:2015eva ; Aaij:2016kxn ;
Aaij:2017tzn ; Aaij:2019zkm ; Durham:2020zuw ; Aaij:2020qga ; Aaij:2020xjx ;
Aaij:2020tzn  in the $B$ decay process, as well as the BESIII Ablikim:2013dyn
; Ablikim:2019soz ; Ablikim:2019zio ; Ablikim:2020xpq  Collaboration in the
electron-positron annihilation process. Besides the charmonium-like states,
another type of exotic candidates observed in the $B$ decay processes is the
open-charm states with strangeness observed in the invariant mass spectra of a
charmed meson and a (anti-)kaon meson or $D_{s}\pi$, such as
$D_{s0}^{\ast}(2317)$ and $D_{s1}(2460)$, which were first observed by BaBar
BaBar:2003oey  and CLEO CLEO:2003ggt  Collaborations, respectively.

In the year of 2020, the LHCb Collaboration performed the amplitude analysis
of the process $B^{+}\to D^{-}D^{+}K^{+}$ LHCb:2020bls ; LHCb:2020pxc , and
two new structures with spin-0 (named $X_{0}(2900)$) and spin-1 (named
$X_{1}(2900)$), were reported in the $D^{-}K^{+}$ invariant mass distribution.
The masses and widths of these two states are measured to be LHCb:2020pxc ;
LHCb:2020bls

$\displaystyle m_{X_{0}(2900)}$ $\displaystyle=$ $\displaystyle(2866\pm 7\pm
2)~{}\mathrm{MeV}\ ,$ $\displaystyle\Gamma_{X_{0}(2900)}$ $\displaystyle=$
$\displaystyle(57\pm 12\pm 4)~{}\mathrm{MeV}\ ,$ $\displaystyle
m_{X_{1}(2900)}$ $\displaystyle=$ $\displaystyle(2904\pm 5\pm
1)~{}\mathrm{MeV}\ ,$ $\displaystyle\Gamma_{X_{1}(2900)}$ $\displaystyle=$
$\displaystyle(110\pm 11\pm 4)~{}\mathrm{MeV}\ ,$ (1)

respectively.

It is interesting to notice that both $X_{0}(2900)$ and $X_{1}(2900)$ are
fully open-flavor states and their minimal quark components are
$\bar{c}\bar{s}ud$ , which indicates that $X_{0}(2900)$ and $X_{1}(2900)$
could be good candidates of tetraquark states Agaev:2022eeh ; Ozdem:2022ydv ;
Agaev:2021knl ; Xue:2020vtq ; Wang:2020xyc ; Zhang:2020oze ; He:2020jna ;
Lu:2020qmp . In addition, the observed masses of $X_{0}(2900)$ and
$X_{1}(2900)$ are close to the threshold of $D^{\ast}\bar{K}^{\ast}$, then the
$D^{\ast}\bar{K}^{\ast}$ molecular interpretations have been proposed
Ke:2022ocs ; Bayar:2022wbx ; Chen:2021tad ; Wang:2021lwy ; Kong:2021ohg ;
Xiao:2020ltm ; Agaev:2020nrc ; Hu:2020mxp ; He:2020btl ; Liu:2020nil .

Recently, the LHCb Collaboration reported two new tetraquark states
$T_{c\bar{s}0}(2900)^{0}$ and $T_{c\bar{s}0}(2900)^{++}$ in the
$D_{s}^{+}\pi^{-}$ and $D_{s}^{+}\pi^{+}$ mass distributions of the
$B^{0}\to\bar{D}^{0}D_{s}^{+}\pi^{-}$ and $B^{+}\to D^{-}D_{s}^{+}\pi^{+}$,
respectively LHCb:2022xob ; LHCb:2022bkt . The masses and widths of the
$T_{c\bar{s}0}(2900)^{0}$ and $T_{c\bar{s}0}(2900)^{++}$ are measured to be
LHCb:2022xob ; LHCb:2022bkt

$\displaystyle m_{T_{c\bar{s}0}(2900)^{0}}$ $\displaystyle=$
$\displaystyle(2892\pm 14\pm 15)~{}\mathrm{MeV}\ ,$
$\displaystyle\Gamma_{T_{c\bar{s}0}(2900)^{0}}$ $\displaystyle=$
$\displaystyle(119\pm 26\pm 12)~{}\mathrm{MeV}\ ,$ $\displaystyle
m_{T_{c\bar{s}0}(2900)^{++}}$ $\displaystyle=$ $\displaystyle(2921\pm 17\pm
19)~{}\mathrm{MeV}\ ,$ $\displaystyle\Gamma_{T_{c\bar{s}0}(2900)^{++}}$
$\displaystyle=$ $\displaystyle(137\pm 32\pm 14)~{}\mathrm{MeV}\ .$ (2)

The resonance parameters of these two states are consistent with each other,
which indicates that they are two of isospin triplet. When taking the isospin
relationship into consideration, the mass and width of $T_{c\bar{s}0}(2900)$
are fitted to be LHCb:2022xob ; LHCb:2022bkt ,

$\displaystyle m_{T_{c\bar{s}0}(2900)}$ $\displaystyle=$
$\displaystyle(2908\pm 11\pm 20)~{}\mathrm{MeV}\ ,$
$\displaystyle\Gamma_{T_{c\bar{s}0}(2900)}$ $\displaystyle=$
$\displaystyle(136\pm 23\pm 11)~{}\mathrm{MeV}\ .$ (3)

In addition, the amplitude analysis indicates the quantum numbers of
$T_{c\bar{s}0}$ are $J^{P}=0^{+}$.

From the observed processes, one can find the minimal quark components of
$T_{c\bar{s}0}(2900)^{0}$ and $T_{c\bar{s}0}(2900)^{++}$ are
$c\bar{s}\bar{u}d$ and $c\bar{s}\bar{d}u$, respectively, which indicates that
both $T_{c\bar{s}0}(2900)^{0}$ and $T_{c\bar{s}0}(2900)^{++}$ are also fully
open flavor tetraquark states, and in addition, $T_{c\bar{s}0}(2900)^{++}$ is
the first observed doubly charged tetraquark state. These particular
properties have stimulated theorists’ great interests. In the framework of the
QCD sum rules, the authors in Ref. Yang:2023evp ; Lian:2023cgs ; Jiang:2023rcn
; Liu:2022hbk ; Dmitrasinovic:2023eei  assigned $T_{c\bar{s}0}(2900)$ as the
scalar $c\bar{s}q\bar{q}$ tetraquark state. In addition, the observed mass of
$T_{c\bar{s}0}(2900)$ is close to the threshold of $D^{\ast}K^{\ast}$.
Together with $D_{s0}^{*}(2317)$ close to the $DK$ threshold and
$D_{s1}(2460)$ close to the $D^{\ast}K$ threshold, the observation of
$T_{c\bar{s}0}(2900)$ enrich the exotic candidate near the threshold of a
charmed meson and a strange meson. Similar to the case of $D_{s0}^{*}(2317)$
and $D_{s1}(2460)$, $T_{c\bar{s}0}(2900)$ has also been proposed to be
$D^{\ast}K^{\ast}$ molecular state with isospin $I=1$. By means of the QCD
two-point sum rule method, the mass and decay width could be reproduced in the
$D^{\ast}K^{\ast}$ molecular scenario Agaev:2022eyk . In the one-boson-
exchange model, the authors in Ref. Chen:2022svh  found that the masses of
$D_{s0}^{\ast}(2317)$, $D_{s1}(2460)$ and $T_{c\bar{s}0}(2900)$ could be
reproduced. In an effective Lagrangian approach, the decay properties of
$T_{c\bar{s}0}(2900)$ were also investigated in Ref. Yue:2022mnf . Besides the
resonance interpretations, the $T_{c\bar{s}0}(2900)$ was interpreted as the
threshold effect from the interaction of the $D^{\ast}K^{\ast}$ and
$D_{s}^{\ast}\rho$ channels Molina:2022jcd  or the triangle singularity
Ge:2022dsp .

On the experimental side, searching for more decay modes of
$T_{c\bar{s}0}(2900)$ can help us to reveal its internal structure. In the
$B^{+}\to D^{-}D^{+}K^{+}$ process where the tetraquark states $X_{0}(2900)$
and $X_{1}(2900)$ were observed, the LHCb Collaboration also present the
$D^{+}K^{+}$ invariant mass distribution LHCb:2020bls ; LHCb:2020pxc . From
the measured data, one find that the $D^{+}K^{+}$ invariant mass distribution
can not be well described in the vicinity of 2.9 GeV222More detail can be
found in Fig.10-(c) of Ref. LHCb:2020pxc , which indicates that there could be
some contributions from additional resonances. To further analyse the
resonance contributions to $B^{+}\to D^{-}D^{+}K^{+}$ process, we find,

  * •

Besides the resonance parameters of $T_{c\bar{s}0}(2900)$, the LHCb
Collaboration also reported the fit fraction of $T_{c\bar{s}0}(2900)^{++}$
component in the $B^{+}\to D^{-}D_{s}^{+}\pi^{+}$, which is $(1.96\pm 0.87\pm
0.88)\%$ LHCb:2022xob ; LHCb:2022bkt . In other words, the cascaded decay
process, $B^{+}\to D^{-}T_{c\bar{s}0}(2900)^{++}\to D^{-}D_{s}^{+}\pi^{+}$ are
sizable.

  * •

In the $D^{\ast}K^{\ast}$ molecular scenario, the decay properties of the
$T_{c\bar{s}0}(2900)^{0}$ were investigated in Ref. Yue:2022mnf . Our
estimations indicate that the $T_{c\bar{s}0}(2900)^{0}$ dominantly decays into
$D^{0}K^{0}$, and accordingly $T_{c\bar{s}0}(2900)^{++}$ should dominantly
decay into $D^{+}K^{+}$ on account of the isospin symmetry.

Based on the above experimental measurements and theoretical estimations, one
can anticipate that the tetraquark state $T_{c\bar{s}0}(2900)^{++}$ should
have non-negligible contribution to the process $B^{+}\to D^{-}(K^{+}D^{+})$.

In addition, the involvement of $T_{c\bar{s}0}(2900)^{++}$ in the process
$B^{+}\to K^{+}D^{+}D^{-}$ may also shed light on another long standing puzzle
for $\chi_{c0}(3930)$ as conventional charmonium Zhou:2015uva ; Chen:2012wy ;
Olsen:2014maa ; Guo:2012tv . The measurements from the BaBar Collaboration
indicated that the branching fraction of $B^{+}\to K^{+}\chi_{c0}(3930)\to
K^{+}J/\psi\omega$ is $(3.0^{+0.7+0.5}_{-0.6-0.3})\times 10^{-5}$
BaBar:2010wfc , while the branching fraction of $B^{+}\to
K^{+}\chi_{c0}(3930)\to K^{+}D^{+}D^{-}$ is reported to be $(8.1\pm 3.3)\times
10^{-6}$ LHCb:2020pxc . Thus, one can conclude that the branching fraction for
$\chi_{c0}(3930)\to J/\psi\omega$ is several times larger than the one of
$\chi_{c0}(3930)\to D^{+}D^{-}$, which is inconsistent with the expectations
of the conventional charmonium assignment of $\chi_{c0}(3930)$.

If carefully checking the $D^{+}K^{+}$ invariant mass distribution of
$B^{+}\to K^{+}D^{+}D^{-}$ in Ref. LHCb:2020pxc , one can find that the
charmonium $\chi_{c2}(3930)$ has significant contribution to the structure
near 2.9 GeV in the $D^{+}K^{+}$ invariant mass distribution. While both the
$\chi_{c0}(3930)$ and $\chi_{c2}(3930)$ are responsible for the peak in the
vicinity of 3.93 GeV in the $D^{+}D^{-}$ mass spectrum of $B^{+}\to
K^{+}D^{+}D^{-}$, then, the involvement of $T_{c\bar{s}0}(2900)^{++}$ in the
$B^{+}\to K^{+}D^{+}D^{-}$ may lead to a rather different fit fractions of
$\chi_{c0}(3930)$ and $\chi_{c2}(3930)$ with the present one. Thus, in the
present work, we investigate the possible contribution of
$T_{c\bar{s}0}(2900)^{++}$ in the process $B^{+}\to D^{-}D^{+}K^{+}$ in the
framework of the molecular scenario, where $T_{c\bar{s}0}(2900)^{++}$ is
considered as a $D^{\ast+}K^{\ast+}$ molecular state.

This paper is organized as follows. After the introduction, we will show the
formalism used in Sec. II. Our calculated results and related discussions will
be presented in Sec. III, and Sec. IV will devote to a short summary.

Figure 1: Diagrammatic decay at the quark level for the $B^{+}\to
D^{-}D^{\ast+}K^{\ast+}$ reaction.

Figure 2: A sketch diagram of the rescatting of $D^{\ast+}K^{\ast+}$ to give
the resonance $T_{c\bar{s}0}(2900)^{++}$ (diagram (a)), and further decay of
$T_{c\bar{s}0}(2900)^{++}$ to $D^{+}K^{+}$ (diagram (b)).

##  II FORMALISM

In the molecular scenario, the $T_{c\bar{s}0}(2900)^{++}$ is considered as a
molecular composed of $D^{\ast+}K^{\ast+}$, which is,

$\displaystyle\left|T_{c\bar{s}0}(2900)^{++}\rangle\right.=\left|D^{\ast+}K^{\ast+}\rangle\right..$
(4)

Thus, the primary reaction that could produce $T_{c\bar{s}0}(2900)^{++}$ is
$B^{+}\to D^{-}D^{\ast+}K^{\ast+}$. As shown in Fig. 1, this reaction proceeds
via the $W^{+}$ internal emission, where the $\bar{b}$ quark transits into
$\bar{c}$ quark by emitting a $W^{+}$ boson, while the $W^{+}$ boson couples
to the $c\bar{s}$ quarks pair. The $\bar{s}$ quark and the $u$ quark from the
initial $B^{+}$ meson form a $K^{\ast+}$ meson, while the rest $c\bar{c}$ and
$d\bar{d}$ created from vacuum hadronize into $D^{-}$ and $D^{\ast+}$ mesons.
In the hadron level, one can construct the $S$ wave component of the
transition amplitude by matching the angular momentum of $B^{+}$ meson
Dai:2022htx ; Dai:2022qwh , which is,

$\displaystyle-
it_{1}=-iC_{1}{\bm{\epsilon}(D^{*+})}\cdot{\bm{\epsilon}(K^{*+})},$ (5)

where the $\epsilon(D^{\ast+})$ and $\epsilon(K^{\ast+})$ are the polarization
vectors of the $D^{\ast+}$ and $K^{\ast+}$, respectively. $C_{1}$ is an
unknown coupling constant, which will be discussed later. Then the $D^{\ast+}$
and $K^{\ast+}$ couple to the molecular $T_{c\bar{s}0}(2900)^{++}$ with
$I(J^{P})=1(0^{+})$ as presented in Fig. 2-(a). As indicated in Ref.
Molina:2008jw , the spin of the $D^{\ast+}K^{\ast+}$ system could be projected
into different angular momentum, for example, the vertex for $R_{J}\to
D^{\ast+}K^{\ast+}$ with $J=0,1,2$ could be constructed as,

$\displaystyle\mathcal{V}^{(0)}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\epsilon_{l}(D^{*+})\epsilon_{l}(K^{*+})\delta_{ij},$
$\displaystyle\mathcal{V}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\epsilon_{i}(D^{*+})\epsilon_{j}(K^{*+})-\epsilon_{j}(D^{*+})\epsilon_{i}(K^{*+})\right],$
$\displaystyle\mathcal{V}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\epsilon_{i}(D^{*+})\epsilon_{j}(K^{*+})+\epsilon_{j}(D^{*+})\epsilon_{i}(K^{*+})\right]$
(6)
$\displaystyle-\frac{1}{3}\epsilon_{l}(D^{*+})\epsilon_{l}(K^{*+})\delta_{ij}.$

The experimental analysis indicated that the angular momentum of
$T_{c\bar{s}0}(2900)$ is $0$. Thus, one can obtain the transition amplitude of
$B^{+}\to D^{-}T_{c\bar{s}0}(2900)^{++}$ corresponding to Fig. 2-(a), which
is,

$\displaystyle-it_{2a}$ $\displaystyle=$ $\displaystyle-
iC_{1}\epsilon_{\alpha}(D^{*+})\epsilon_{\beta}(K^{*+})\delta^{\alpha\beta}G_{D^{*}K^{*}}(M_{\rm
inv}(D^{+}K^{+}))$ (7)
$\displaystyle\times\frac{1}{3}\epsilon_{l}^{*}(D^{*+})\epsilon_{l}^{*}(K^{*+})\delta_{ij}g_{T^{++}_{c\bar{s}0}D^{*}K^{*}}$
$\displaystyle=$ $\displaystyle-iC_{1}\delta_{ij}G_{D^{*}K^{*}}(M_{\rm
inv}(D^{+}K^{+}))g_{T^{++}_{c\bar{s}0}D^{*}K^{*}},$

where $\sum\limits_{pol}\epsilon_{i}(R)\epsilon_{j}^{*}(R)=\delta_{ij}$,
$R=D^{*+}$ or $K^{*+}$, and the sum over the same indices of the Kronecker
delta function is equal to 3, i.e., $\sum\limits_{ij}|\delta_{ij}|^{2}=3$.
$G_{D^{*}K^{*}}(M_{T_{c\bar{s}0}})$ is the loop function of the two-meson
$D^{\ast}$ and $K^{\ast}$, which will be discussed later.

Similarly, one can obtain the transition amplitude of $B^{+}\to
D^{-}T_{c\bar{s}0}(2900)^{++}\to D^{-}D^{+}K^{+}$ corresponding to Fig. 2-(b),
which is,

$\displaystyle-it_{2b}$ $\displaystyle=$ $\displaystyle-
iC_{1}\delta_{ij}G_{D^{*}K^{*}}\left(M_{\rm inv}(D^{+}K^{+})\right)$ (8)
$\displaystyle\times\frac{g_{T^{++}_{c\bar{s}0}D^{*}K^{*}}g_{T^{++}_{c\bar{s}0}DK}}{M^{2}_{\rm
inv}(D^{+}K^{+})-m^{2}_{T^{++}_{c\bar{s}0}}+im_{T^{++}_{c\bar{s}0}}\Gamma_{T^{++}_{c\bar{s}0}}},$

and then the square of the transition amplitude is,

$\displaystyle\sum|t_{2b}|^{2}$ $\displaystyle=$ $\displaystyle 3C_{1}^{2}\
\Big{|}G_{D^{*}K^{*}}\left(M_{\rm inv}(D^{+}K^{+})\right)\Big{|}^{2}\ $ (9)
$\displaystyle\times\frac{\Big{|}g_{T^{++}_{c\bar{s}0},D^{*}K^{*}}\Big{|}^{2}\Big{|}g_{T^{++}_{c\bar{s}0},DK}\Big{|}^{2}}{\left[M^{2}_{\rm
inv}(D^{+}K^{+})-m^{2}_{T^{++}_{c\bar{s}0}}\right]^{2}+m_{T^{++}_{c\bar{s}0}}^{2}\Gamma^{2}_{T^{++}_{c\bar{s}0}}},$

with $M^{2}_{\rm inv}(D^{+}K^{+})=(P_{D^{+}}+P_{K^{+}})^{2}$, and two-meson
loop function is given by,

$G=i\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{q^{2}-m_{1}^{2}+i\epsilon}\frac{1}{(q-P)^{2}-m_{2}^{2}+i\epsilon}\
,$ (10)

with $m_{1}$ and $m_{2}$ the masses of the two mesons involved in the loop.
$q$ is the four-momentum of the meson in the centre of mass frame, and $P$ is
the total four-momentum of the meson-meson system. In the present work, we use
the dimensional regularization method as indicated in Refs. Duan:2022upr ;
Duan:2021pll ; Duan:2020vye , and in this scheme, the two-meson loop function
$G$ can be expressed as,

$\displaystyle G$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\left[\alpha+\log\frac{m_{1}^{2}}{\mu^{2}}+\frac{m_{2}^{2}-m_{1}^{2}+s}{2s}\log\frac{m_{2}^{2}}{m_{1}^{2}}\right.$
(11)
$\displaystyle+\frac{|\vec{q}\,|}{\sqrt{s}}\left(\log\frac{s-m_{2}^{2}+m_{1}^{2}+2|\vec{q}\,|\sqrt{s}}{-s+m_{2}^{2}-m_{1}^{2}+2|\vec{q}\,|\sqrt{s}}\right.$
$\displaystyle+\left.\left.\log\frac{s+m_{2}^{2}-m_{1}^{2}+2|\vec{q}\,|\sqrt{s}}{-s-m_{2}^{2}+m_{1}^{2}+2|\vec{q}\,|\sqrt{s}}\right)\right],$

where $s=P^{2}=M^{2}_{\rm inv}(D^{+}K^{+})$, and $\vec{q}\,$ is the three-
momentum of the meson in the centre of mass frame, which reads,

$|\vec{q}\,|=\frac{\sqrt{\left[s-(m_{1}+m_{2})^{2}\right]\left[s-(m_{1}-m_{2})^{2}\right]}}{2\sqrt{s}},$
(12)

here we take $\mu=1500$ MeV and $\alpha=-1.474$, which are the same as those
in the study of the $D^{*}\bar{K}^{*}$ interaction Dai:2022qwh ; Dai:2022htx .

Besides the two-meson loop function, two coupling constants
$g_{T^{++}_{c\bar{s}0},D^{*}K^{*}}$ and $g_{T^{++}_{c\bar{s}0},DK}$ are
unknown. As for $g_{T^{++}_{c\bar{s}0},D^{*}K^{*}}$, it refers to the coupling
between $T_{c\bar{s}0}(2900)^{++}$ and its components $D^{\ast+}K^{\ast+}$,
which could be related to the binding energy by Weinberg:1965zz ; Baru:2003qq
; Wu:2023fyh ,

$g_{T^{++}_{c\bar{s}0},D^{*}K^{*}}^{2}=16\pi(m_{D^{*}}+m_{K^{*}})^{2}\tilde{\lambda}^{2}\sqrt{\frac{2\Delta
E}{\mu}},$ (13)

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.

As for $g_{T^{++}_{c\bar{s}0},DK}$, we tried to obtain its value by the
corresponding partial width of $T_{c\bar{s}0}(2900)^{++}\to D^{+}K^{+}$, with
an effective Lagrangian approach, the partial width of
$T_{c\bar{s}0}(2900)^{++}\to D^{+}K^{+}$ could be obtained as,

$\displaystyle\Gamma_{T^{++}_{c\bar{s}0}}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\frac{1}{m^{2}_{T^{++}_{c\bar{s}0}}}|g_{T^{++}_{c\bar{s}0},DK}|^{2}|\vec{q}_{K^{+}}|,$
(14)

with

$\displaystyle|\vec{q}_{K^{+}}|$ $\displaystyle=$
$\displaystyle\frac{\lambda^{1/2}(m^{2}_{T^{++}_{c\bar{s}0}},m^{2}_{D^{+}},m^{2}_{K^{+}})}{2m_{T^{++}_{c\bar{s}0}}},$
(15)

to be the momentum of $K^{+}$ in the $T_{c\bar{s}0}(2900)^{++}$ rest frame,
and $\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2yz-2xz$ is the
K$\ddot{\textrm{a}}$llen function. In Ref. Yue:2022mnf , our estimations
indicated that the $T_{c\bar{s}0}(2900)^{++}$ dominantly decay into $DK$, and
the partial width of $DK$ channel was estimated to be $(52.6\sim 101.7)$ MeV
in the considered parameter range. In the present work, we take the partial
width of $T_{c\bar{s}0}(2900)^{++}\to D^{+}K^{+}$ to be 80 MeV to estimate the
coupling constant $g_{T_{c\bar{s}0}^{++}DK}$.

With the above preparation, one can obtain the $D^{+}K^{+}$ invariant mass
distribution, which is,

$\displaystyle\frac{d\Gamma}{dM_{\rm
inv}(D^{+}K^{+})}=\frac{1}{(2\pi)^{3}}\frac{1}{4m^{2}_{B^{+}}}p_{D^{-}}\tilde{p}_{K^{+}}\sum|t_{2b}|^{2},$
(16)

with

$\displaystyle p_{{D}^{-}}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{1/2}\left(m^{2}_{B^{+}},m^{2}_{D^{-}},M^{2}_{\rm
inv}(D^{+}K^{+})\right)}{2m_{B^{+}}},$ $\displaystyle\tilde{p}_{K^{+}}$
$\displaystyle=$ $\displaystyle\frac{\lambda^{1/2}\left(M^{2}_{\rm
inv}(D^{+}K^{+}),m^{2}_{D^{+}},m^{2}_{K^{+}}\right)}{2M_{\rm
inv}(D^{+}K^{+})}.$ (17)

In addition, we would like to compare the above mass distribution with the one
of the background for the reaction $B^{+}\to D^{-}D^{+}K^{+}$. By analogy to
Eq. (5), we can obtain the transition matrix for $B^{+}\to D^{-}D^{+}K^{+}$,
which is

$\displaystyle-it_{3}=-iC_{3}.$ (18)

where $C_{3}$ is the coupling constant, which will be discussed in the
following section. With the above transition matrix, we can give the
background distribution for the $B^{+}\to D^{-}D^{+}K^{+}$ reaction, which is

$\displaystyle\frac{d\Gamma_{\rm bac}}{dM_{\rm
inv}(D^{+}K^{+})}=C_{3}^{2}\frac{1}{(2\pi)^{3}}\frac{1}{4m^{2}_{B^{+}}}p_{D^{-}}\tilde{p}_{K^{+}}.$
(19)

##  III Numerical RESULTS AND DISCUSSIONS

To calculate the $D^{\ast+}K^{\ast+}$ invariant mass distribution of $B^{+}\to
K^{+}D^{+}D^{-}$ as presented in Eq. (16), the coupling constant $C_{1}$ is
needed. However, the experimental measurement of $B^{+}\to
K^{\ast+}D^{\ast+}D^{-}$ is not available to date. Similar to $B^{+}\to
K^{+}D^{+}D^{-}$, the process $B^{+}\to K^{\ast+}D^{\ast+}D^{-}$ should also
occur via $W^{+}$ internal emission process. One can obtain the diagrammatic
decay at the quark level for the $B^{+}\to K^{+}D^{+}D^{-}$ by replacing
$K^{\ast+}$ and $D^{\ast+}$ in Fig. 1 with $K^{+}$ and $D^{+}$, which
indicates some similarities between the processes $B^{+}\to K^{+}D^{+}D^{-}$
and $B^{+}\to K^{\ast+}D^{\ast+}D^{-}$. However, there are also some
differences between these two processes. As indicated in the amplitude
analysis of $B^{+}\to K^{+}D^{+}D^{-}$ in Ref. LHCb:2020pxc , the typical
resonance contributions to this process are $B^{+}\to K^{+}(c\bar{c})\to
K^{+}D^{+}D^{-}$, where the charmonia include $\psi(3770)$, $\chi_{c0}(3930)$,
$\chi_{c2}(3930)$, $\psi(4040)$, $\psi(4160)$, and $\psi(4415)$. These
charmonia contributions should be suppressed due to phase space. In addition
to the charmonia contributions, the LHCb Collaboration also observed the
signals of $X_{0}(2900)$ and $X_{1}(2900)$ in the $D^{-}K^{+}$ invariant mass
spectra, these contributions also vanish in the $B^{+}\to
K^{\ast+}D^{\ast+}D^{-}$ process.

Besides the resonance contributions, the amplitude analysis also indicates
sizable nonresonant contribution, which should be the same for both $B^{+}\to
K^{+}D^{+}D^{-}$ and $B^{+}\to K^{\ast+}D^{\ast+}D^{-}$, thus, in the present
work, we first estimate the background distribution of $B^{+}\to
K^{+}D^{+}D^{-}$ with the branching fraction of the nonresonant contribution
from LHCb analyze, which is $(5.3\pm 1.8)\times 10^{-5}$ Workman:2022ynf .
From Eq. (19), the coupling constant $C_{3}$ could be determined. Considering
the similarity between $B^{+}\to K^{+}D^{+}D^{-}$ and $B^{+}\to
K^{\ast+}D^{\ast+}D^{-}$, we take $C_{1}=C_{3}$ to roughly estimate the
$D^{+}K^{+}$ invariant mass distribution resulted from
$T_{c\bar{s}0}(2900)^{++}$.

Figure 3: The $D^{+}K^{+}$ invariant mass distribution for the $B^{+}\to
D^{-}D^{+}K^{+}$ reaction.

With the above formalism, we have calculated the $D^{+}K^{+}$ invariant mass
distribution by assuming the values of $C_{1}$ and $C_{3}$ are the same, as
presented in Fig. 3. To further compare with the experimental measurements, we
normalized the background contribution estimated by Eq. (19) to the LHCb
experimental nonresonant contribution in Fig. 3, where the magenta-dash-dotted
and blue-dotted curves are the nonresonant contribution determined by the LHCb
amplitude analysis and our estimated background, respectively. The red-solid
curve is the resonant contribution form $T_{c\bar{s}0}(2900)^{++}$, which is
obtained with the resonance parameters of $m_{T_{c\bar{s}0}^{++}}=2885\
\mathrm{MeV}$ and $\Gamma_{T_{c\bar{s}0}^{++}}=136\ \mathrm{MeV}$. While the
blue band corresponds to the uncertainties of the $T_{c\bar{s}0}(2900)^{++}$
width. From Fig. 3, one can find that the $D^{+}K^{+}$ invariant mass
distribution around 2.9 GeV can not be well described by LHCb fit LHCb:2020pxc
, which indicates that there should be an additional resonance. Our results
show that the $T_{c\bar{s}0}(2900)^{++}$ plays an important role in this
region, thus we suggest that contribution from the $T_{c\bar{s}0}(2900)^{++}$
should be considered in the future amplitudes analysis.

Furthermore, we can integrate the invariant mass $M_{\rm inv}(D^{+}K^{+})$
over the whole invariant mass range for the signal and background, and their
ratio is given by,

$\displaystyle\frac{\int\frac{d\Gamma}{dM_{\rm
inv}(D^{+}K^{+})}}{\int\frac{d\Gamma_{bac}}{dM_{\rm inv}(D^{+}K^{+})}}\simeq
0.52.$ (20)

With the nonresonant fit fraction obtained by the amplitude analyze, we can
roughly estimate the fit fraction of $T_{c\bar{s}0}(2900)^{++}$ to be about
$12.5\%$, which is greater than the ones of $\chi_{c0}(3930)$ and
$\chi_{c2}(3930)$. Thus, the involvement of $T_{c\bar{s}0}(2900)^{++}$ will
certainly influence the fit fractions of $\chi_{c0}(3930)$ and
$\chi_{c2}(3930)$.

##  IV Summary

Recently, the LHCb Collaboration reported their amplitude analysis of the
decays $B^{0}\to\bar{D}^{0}D_{s}^{+}\pi^{-}$ and $B^{+}\to
D^{-}D_{s}^{+}\pi^{+}$, where two tetraquark states $T_{c\bar{s}0}(2900)^{0}$
and $T_{c\bar{s}0}(2900)^{++}$ were reported in the $D_{s}\pi$ invariant mass
distributions. The resonance parameters of these two resonances indicate that
they are two of the isospin triplet. Similar to $T_{c\bar{s}0}(2900)$, the
LHCb Collaboration reported another two tetraquark candidates $X_{0,1}(2900)$
in the $D^{-}K^{+}$ invariant mass distribution in the $B^{+}\to
D^{-}D^{+}K^{+}$ reaction in the year of 2020 LHCb:2020bls ; LHCb:2020pxc . In
the $D^{+}K^{+}$ invariant mass distribution of the $B^{+}\to D^{-}D^{+}K^{+}$
reaction, we find that the experimental data of the $D^{+}K^{+}$ invarinat
mass distribution around 2.9 GeV can not be well described, which indicates
that there should be an additional resonance. Inspired by the recent
observation of the $T_{c\bar{s}0}(2900)$ LHCb:2022xob ; LHCb:2022bkt  and the
decay properties of $T_{c\bar{s}0}(2900)$, we find that
$T_{c\bar{s}0}(2900)^{++}$ is likely to contribute to the $D^{+}K^{+}$
invariant mass distribution. Thus, in the present work we study the role of
$T_{c\bar{s}0}(2900)^{++}$ in the $D^{+}K^{+}$ invariant mass distribution of
the process $B^{+}\to D^{-}D^{+}K^{+}$.

In the present work, we estimate $T_{c\bar{s}0}(2900)^{++}$ contribution to
the process $B^{+}\to D^{-}D^{+}K^{+}$ in a molecular scenario, where we have
considered $T_{c\bar{s}0}(2900)^{++}$ as a $D^{\ast+}K^{\ast+}$ molecular
state. However, due to the lack of the experimental information of $B^{+}\to
D^{-}D^{\ast+}K^{\ast+}$, we have made an assumption that the coupling
constant for $B^{+}\to D^{-}D^{\ast+}K^{\ast+}$ is the same as the one for
nonresonant contribution in $B^{+}\to D^{-}D^{+}K^{+}$. Based on this
assumption, our estimation indicates that the contribution from
$T_{c\bar{s}0}(2900)^{++}$ is significant in the process $B^{+}\to
D^{-}D^{+}K^{+}$, and the $T_{c\bar{s}0}(2900)^{++}$ signal in the
$D^{+}K^{+}$ invariant mass distribution is visible. In addition, the fit
fraction of $B^{+}\to D^{-}T_{c\bar{s}0}(2900)^{++}\to K^{+}D^{+}D^{-}$ is
roughly estimated to be $12.5\%$, which could be tested by further
experimental analysis by the LHCb Collaboration.

Before the end of this work, it is worth to mention that the branching
fractions of $B^{0}\to D^{-}D^{0}K^{+}$ and $B^{0}\to D^{-}D^{+}K^{0}$ decays
are $(1.07\pm 0.07\pm 0.09)\times 10^{-3}$ and $(0.75\pm 0.12\pm 0.12)\times
10^{-3}$, respectively Workman:2022ynf . In the $D^{0}K^{+}$ invariant mass
distributions of these process, there should be the signal of
$T_{c\bar{s}0}(2900)^{+}$, which may be accessible for the LHCb Collaboration.

## Acknowledgement

This work is supported by the National Natural Science Foundation of China
under Grant Nos. 11775050, 12175037, and 12192263. This work is also supported
by the Natural Science Foundation of Henan under Grand Nos. 222300420554 and
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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