Geomagnetism and Aeronomy, 2021, Vol. 61, No. 7, pp. 1069–1074. DOI:
10.1134/S0016793221070057 Detection of Optical Flares on the Selected G-M
Dwarfs from Long-term Photometric Series N. I. Bondar’1,*, M. M. Katsova2,**,
and A. A. Shlyapnikov1,*** stars: low-mass stars – stars: activity – stars:
flares – techniquies: long-term photometry We have carried out a search for
flares from the analysis of light curves for 12 active G, K, and M dwarfs. As
sources of data we used ground-based observations in 2000–2020 from the
photometric databases ASAS, SuperWASP, KWS. Events of low-amplitude
brightening ($\Delta V<0.25\;$mag), which possibly could be flares, were
revealed for 11 stars. A large number of such increases in brightness were
found on K dwarfs. Events of increasing in $V$-magnitudes to 0.5 mag or more
were detected on light curves of one G star, BE Cet, and two M dwarfs. For
three flares we could follow their development with time. We have estimated
the duration of these flares; they lasted more than an hour, but less than 3
hours. In most cases we could not determine a lifetime of the suggested
flares, but we believe that most of the probable flares on the investigated
cool dwarfs are short-lived, on the order of several minutes. DOI
10.1134/S0016793221070057

##  1 INTRODUCTION

Starspots and flares in the atmospheres of low-mass G-M stars are proxies of
their magnetic activity manifestation. Since the 1960s, these structures have
been studied in different spectral ranges using ground-based and space
observations. Starspots are a cause of low-amplitude variations in brightness
with the rotation period of a star and the long-term variability, often
cyclical changes, by a few or even tens of percent relative to the quiescent
state of a star.

Although flares are associated with starspots, they are irregular phenomena
that produce significant rapid increases of radiation in the optical
continuum, emission lines, and in X-ray fluxes. On the UV Cet type stars
(flaring M dwarfs), most of the detected flares are classical, i.e. with a
fast rise in brightness up to several magnitudes and a subsequent gradual
decay. The energies of such flares reach $10^{30}-10^{34}\;$erg. A review of
studies concerning flare activity on the main-sequence G-M stars is given in
the recently published monograph by Gershberg et al. (2021). The efforts of
astronomers are currently aimed at studying flares in the visible range of
wavelengths and at searching for powerful stellar flares.

Table 1:  Sample of investigated G-M dwarfs and number of positive $\Delta V$
(events of possible flares)

Star | Spectral type |  $V$, mag | Number of $\Delta V<0.25^{m}$ | $\Delta V>0.3^{m}$
---|---|---|---|---
HD 1835, BE Cet | G3 V | 6.39 | 2 (K) | 0.3(K), 0.5 (K)
HD 4747, GJ 36 | G9 V | 7.16 | 4 (A) | –
HD 97334, MN UMa | G1 V | 6.41 | – | –
HD 168443, NSV 24398 | G6 V | 6.92 | 2(A), 2(K) | –
HD 17382, BC Ari | K1 V | 7.62 | 4 (K) | –
HD 61606, V869 Mon | K2 V | 7.17 | 8(A), 10(K) | –
HD 283750, V833 Tau | K2 V | 7.90 | 5(A), 4(S), 9(K) | –
HD 45088, OU Gem | K3 V | 6.75 | 1(A), 5(K) | –
HD 197481, AU Mic | M1 V | 8.63 | 7(K) | 0.32(K)
EV Lac, GJ 873 | M4 V | 10.26 | 1(K) | 1.08 (K)
GJ 1243, KIC 9726699 | M4 V | 12.7 $(m_{K\\!p})$ | 9(S) | –
YZ CMi, GJ 285 | M4.5 V | 11.22 | 3(A), 7(K) | 0.5(A), 0.67(K)

Letters A, K, S in brackets indicate catalogs ASAS, KWS, SW;
the magnitude $m_{K\\!p}$ of the star GJ 1243 is given according to the
$Kepler$ mission data.

The white-light flares were discovered on the Sun observing the intensity of
radiation from the Sun as a star. The contribution of optical flares can be
traced over all spectral regions; their energy distribution corresponds to
blackbody radiation with a temperature of 9000 K (Kretzschmar, 2011). Due to
observations from the $Kepler$ mission, the manifestations of activity have
been studied for a large number of solar-type and cool stars (for instance,
Davenport, 2016; Savanov and Dmitrienko, 2017; Namekata et al., 2019). As
shown by the results of the recent statistical analysis of flares on G dwarfs
based on data of the $Kepler$ mission and Gaia-DR2 catalog, non-steady events
on solar-like stars can reach energies within $10^{36}\;$erg (Okamoto et al.
(2021). On the modern Sun there occurred flares with total energies not
exceeding $3\times 10^{32}\;$erg, and only the Carrington flare on September
1, 1859 is estimated of several times powerful. By studying optical radiation
from stars in a certain range of physical parameters, it is possible to
determine both typical and maximum flare energies, to trace their development
on G-M dwarfs with different ages, and to investigate the distribution of
flare energies. This knowledge is necessary for studying the formation of
planetary systems and their environment. The study of solar-like stars is
important to understanding whether superflares were produced over the lifetime
of the Sun. Here we present the results of a search for flares on 12 G-M
dwarfs derived from the long-term ground-based photometry. Our sample includes
both poorly studied stars and well-known ones with high-level flare activity.
The brightness of the most selected objects was studied during the 20-year
time interval, but in most cases the data in catalogs of ground-based
observational programs have been obtained with a low-time resolution and a
short-time continuous monitoring during a night. Therefore, our work aims to
reveal events of increase in brightness, which may indicate the development of
flares, to determine the maximum $V$-values for possible flares and also their
number for dwarfs of different spectral types. Section 2 contains a list of
objects in our research and a description of the used databases, Sections 3
and 4 contain processing methods and results, a brief discussion of results is
given in Conclusions.

Figure 1:  Examples of the selected observational series from different
catalogs and possible flares for V869 Mon (KWS), V833 Tau (SW), open circles
mark data for a comparison star, and YZ CMi (ASAS).

##  2 OBJECTS FOR THE RESEARCH AND DATA SOURCES

Flares on stars located on the lower part of the main sequence are an
irregular, stochastic phenomenon, and their detection requires a systematic
many hour monitoring of stellar brightness during a night. The long-term
observational data for a large number of stars are contained in a convenient
form in photometric databases All-Sky Automated Survey (ASAS) (Pojmanski,
1997, 2002), the SuperWASP photometric survey (SW) (Pollacco et al., 2006;
Norton et al., 2007; Butters et al., 2010), the Kamogata Wide-field Survey
(KWS) (Maehara, 2014).

These databases are effective sources of data for studying starspots on the
surface of G-M dwarfs, for determining rotation periods of stars and
parameters of rotational brightness modulation, and also for searching for
cycles of photospheric activity. The light curves constructed from these data
often show short-term increases in brightness, which are usually considered as
outliers and are excluded from data series which are analyzed for systematic
variability. Some of these brightness increases are likely caused by real
flares.

Based on the available nearly 20-year data, we can explore whether a
significant increase in brightness of the star may be considered as flares
observed over this span. The ASAS archive can be used to study the brightness
behavior from 2000 to 2009\. The best photometric accuracy was derived for
stars with magnitudes in the range of $8<V<12\;$mag. The SuperWASP photometric
catalog contains the results of observations obtained in 2004–2008 for the
stars brighter than 15 mag. The accuracy of data for stars with a magnitude of
$8<V<11.5$ reaches 1%, the errors increase for brighter and fainter stars
(Pollacco et al., 2006; Norton et al., 2007).

The survey observations of Kamogata Wide-field Survey (KWS) started in 2010
and are currently being continued in bands $B$, $V$, $I_{c}$. Data series in
the $V$-filter are more numerous. The errors of observations are given for
each $V$-value, photometric precision is not less than 5% for stars with
magnitudes of $5<V<11.5$ (Maehara, 2014).

We selected 12 G-M dwarfs, out of 4 stars in each spectral group (Table 1),
taking into account indices of activity of stars, their stellar magnitudes,
the time span of data series in the $V$-filter in the indicated catalogs. As
we noted above, in the used catalogs errors of observations of $V$-values
increase for both faint and bright stars, therefore we have included in our
sample stars with magnitudes in the range of $6.4<V<13\;$mag, G dwarfs are the
brightest among them.

Two G stars, BE Cet and MN UMa, are young solar analogous (Radick et al.,
2018). The sample of K dwarfs includes active BY Dra-type stars – V833 Tau and
OU Gem. The group of M dwarfs includes the stars AU Mic, EV Lac, YZ CMi, which
are known by their high-level flare activity, also the star GJ 1243, for which
a large number of flares have been found from the extra-atmospheric
observations (Hawley et al., 2014; Silverberg et al., 2016).

##  3 SEARCH FOR POSSIBLE FLARES

For each star under study, the preliminary processing of data series was
performed. As a result, we obtained a series of $V$-magnitudes sorted by
Julian dates, from which the random significantly fainter $V$-values were
excluded. Records in the ASAS catalog contain for each of the five observation
cameras the observation times (Julian dates), $V$-magnitudes, and their
errors. We have compared records from different cameras, and for further
calculations there were chosen 2–3 cameras with smaller observational errors
and the best coincidence of light curves.

Then we determined average magnitudes at the moments of observations and
corresponding errors. The full data series obtained by this way contain up to
700 $V$-values, but in the observational date 1–2 measurements were taken
usually, sometimes up to 4. Data series in the KWS catalog include several
hundreds of measurements covering the seasonal intervals of observations well,
for a date of observation there are given 2–4 estimates of brightness, for
some dates two exposures were made at the beginning and at the end of the
40-min cadence.

The SW catalog contains a large number of data, up to several thousand,
obtained by several cameras with a time resolution of 6–8 min. The time of
monitoring was several hours over a night. We examined the data series from
the SW archive and selected the sets with the largest number of records from
one of the cameras for a more detailed analysis. Using the SW catalog we could
study only two stars, V833 Tau and GJ 1243, a research of other chosen stars
was performed based on data of the ASAS and KWS catalogs.

We constructed light curves from data of each catalog and found for each
continuous set of observations the mean magnitudes $V_{av}$, standard
deviation $\sigma$, and for the further flare search process we selected sets
with positive $\Delta V_{i}=V_{i}-V_{av}>2.5\sigma$. A similar simple
statistical analysis was made in studies of data series from the $Kepler$
mission and the ASAS-SN catalog (Hawley et al., 2014; Martinez et al., 2020).
In most cases, increases in brightness confirmed by two or more consecutive
records of observations were considered as possible flares. We have found
clear evidence for reality of the suspected flares for some stars, comparing
their light curves with ones for comparison stars observed simultaneously. In
addition, for the stars with the known rotation periods, phase curves were
also constructed. This allows us to take into account variations in the light
curve produced by rotational modulation due starspots, and also to determine
phases of an appearance of possible flares.

Figure 2:  Two flares detected from the SW catalog data for the star GJ 1243.
The left panel shows changes in brightness over time given in Julian dates
(the squares represent the comparison star), and the right panel shows the
development of flares with time in minutes from the moment of peak of
brightness (black circles). The open circles show the result of five-point
moving average smoothing. Levels of $2\sigma$ and $3\sigma$ are shown by
dashed lines.

##  4 RESULTS ON IDENTIFICATION OF FLARE EVENTS

The events of low-amplitude ($\Delta V<0.3\;$mag) and short-term increases in
brightness relative to the average value in the selected interval were found
for all the stars of our sample, except for MN UMa (G1 V). The number of such
events is given in Table 1. An increase in brightness up to $0.3-0.5\;$mag was
detected for the stars BE Cet (G3 V) and AU Mic (M1 V), possible flares of
0.5–0.67 mag were found in light curves of the star YZ CMi (M4.5 V). Analysis
of the KWS data for the star EV Lac (M4 V) allowed us to define one event of
an increase in its brightness by 0.18 mag ($4\sigma$) and one significant
brightening up to 1.08 mag detected from 4 measurements at the beginning of
observations on July 14, 2011. If this flare is real, its duration is not less
than 30 min. Figure 1 shows examples of the light curves from different
catalogs for some stars. For stars with the known rotation periods, we have
constructed phase light curves to resolve an issue whether the moments of
appearance of possible flares and development of starspots are consistent.
Flares on the stars BE Cet and V833 Tau have been observed at phases that are
distant from the minimum produced by the maximum contribution of cool
starspots.

Using the SW catalog, a search for flares was performed for two stars: for GJ
1243 (M4 V) we studied light curves from observations on June 4, 2007 and July
10, 2007 (Fig. 2), for V833 Tau (K2 V) we analyzed data on December 14, 2006
(Fig. 3).

Figure 3:  The same as in Fig. 2 for the star V833 Tau.

The presence of increases in brightness is confirmed by the light curves of
comparison stars. Figures 2 and 3 (right panels) show changes in the
$V$-values over time in minutes from the moment of maximum of increases
relative to the average value for the considered interval. The relative
intensity $f_{r}=f_{i}/f_{c}-1$, where $f_{i}$ was calculated by the formula
$-0.4m=logf_{i}$, $f_{c}$ is the average value, $m=V_{i}$. The increase in
brightness for both stars is no more than 15%. In order to determine the onset
and the end of a possible flare, the $f_{r}$-values were smoothed by moving
averages over 5 points. According to our estimates, flares on GJ 1243 have
duration of 1.4 h and 2.4 h, and duration of the flare on V833 Tau was 2.2
hours.

The considered photometric catalogs contain results of survey observations,
i.e. short-term brightness estimates, which allow one to register only those
rare flares that occur at the moment of observation. Nevertheless, such random
registrations characterize to a certain extent the flare activity of the star.
For each catalog and each star, we determined the frequency of brightenings
(possible flares) on the value of $\Delta V<0.25\;$mag. Its average value for
each spectral type can be considered as a characteristic of the flare activity
of stars of this spectral type.

The frequency of flares is $n=N_{f}/T_{obs}$, where $N_{f}$ is the number of
the registered flares in the observation interval, $T_{obs}$ is the
observation time of the star, $T_{obs}=t_{exp}N$, here $t_{exp}$ is the
exposure time, $N$ is the number of $V$-data. Table 2 lists the values of $n$
for stars studied from the KWS catalog data series. For G stars, the average
value $n$ is 0.26, for K and M dwarfs – 0.36 and 1.06, respectively.

##  5 CONCLUSIONS

We have undertaken a search for flares on 12 active G-M dwarfs, studying about
150 light curves constructed from long-term photometric series of the ASAS,
SuperWASP, and KWS catalogs. Short-term increases in brightness, which do not
exceed 0.3 mag, were detected for 11 stars, on three stars there were revealed
several possible flares with an increase in brightness of $0.3-0.6\;$mag, one
flare up to 1 mag was suspected for EV Lac.

The detected events of possible flares are few for the considered 20-year
spans and give poor statistics, nonetheless the obtained results confirm the
well known conclusion that the number of flares increases for cool stars, and
their amplitude increases also. On K dwarfs we observed mainly low-amplitude
flares. Among several dozen events on these dwarfs, there was no any potential
flare with high amplitude.

The result on the flare activity of BE Cet, a young solar analog on which, in
addition to low-amplitude ones, flares up to 0.5 mag are possible, deserves
further consideration. For each star, we have studied some series of
observations, and we can note that stars have epochs of weak and strong flare
activity. Long-term data from photometric surveys can be useful to study
changes in the level of flare activity.

Single estimates of brightness on the date of observations stored in the ASAS
and KWS catalogs do not allow us to determine the duration of the suspected
flares, but most of them are likely short-lived and lasting for a few minutes.
Using the SW database, we traced the development of two flares on the star GJ
1243 and one flare on V833 Tau. According to our estimates, their duration was
within 1.4–2.4 hours.

Table 2:  Estimates of the number of flares with $\Delta V<0.25$ from the KWS data series  Star | Spectral | Time interval | Number |  $T_{obs}$ of star, h | Number | Number
---|---|---|---|---|---|---
| type | JD2450000+ | of data |  | of flares | of flare/h
HD 1835 | G3 V | 5548–8841 | 703 | 6.75 | 2 | 0.30
HD 168443 | G6 V | 5661–9067 | 615 | 8.86 | 2 | 0.22
HD 17382 | K1 V | 5527–8866 | 979 | 14.10 | 4 | 0.28
HD 61606 | K2 V | 5891–8937 | 1285 | 18.50 | 10 | 0.54
HD 45088 | K3 V | 5516–8550 | 1214 | 14.57 | 5 | 0.34
HD 197481 | M1 V | 6457–8420 | 464 | 3.30 | 7 | 2.09
EV Lac | M4 V | 5757–8451 | 638 | 9.19 | 1 | 0.11
YZ CMi | M4.5 V | 6235–8573 | 608 | 5.73 | 7 | 0.87

##  6 ACKNOWLEDGMENTS

We have used information from the SIMBAD database, photometric data from the
Hipparcos, the International Variable Star Index (VSX) database, supported by
AAVSO, Cambridge, Massachusetts, USA, the Kamogata-Kiso-Kyoto Wide-field
Survey. The authors are thankful to all the staff providing the replenishment
of these databases and access to them. We would like to especially thank
referees for useful comments and suggestions.

##  7 FUNDING

These results were obtained with partial support by the Russian Foundation for
Basic Researches, project no. 19-02-00191, and by the grant 075-15-2020-780 of
the Ministry of Science and Higher Education of the Russian Federation.

##  8 CONFLICT OF INTEREST

The authors declare that they have no conflicts of interest.

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XXXX-XXXX

1]Research Center for Neutrino Science, Tohoku University, 6-3 Aoba, Aramaki,
Aoba-ku, Sendai, Miyagi 980-8578, Japan 2]Kamioka Observatory, Institute for
Cosmic Ray Research, The University of Tokyo, 456 Higashi-Mozumi, Kamioka,
Hida, Gifu 506-1205, Japan 3]Kavli Institute for the Physics and Mathematics
of the Universe (WPI), The University of Tokyo Institutes for Advanced Study,
The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan
4]Department of Physics and Astronomy, Faculty of Science and Technology,
Tokyo University of Science, 2641 Yamazaki, Noda, Chiba 278-8510, Japan
5]Department of Physics, Yokohama National University, 79-5 Tokiwadai,
Hodogaya-ku, Yokohama, Kanagawa, 240-8501, Japan 6]Department of Human
Science, Obihiro University of Agriculture and Veterinary Medicine, Inada-cho
Nishi 2-11, Obihiro, Hokkaido 080-8555, Japan *]E-mail:
<EMAIL_ADDRESS>
# Development of a low-background HPGe detector at Kamioka Observatory

K. Ichimura  H. Ikeda  Y. Kishimoto  M. Kurasawa  A.A. Suzuki  Y. Gando  M.
Ikeda  K. Hosokawa  H. Sekiya  H. Ito  A. Minamino  S. Suzuki  [  [  [  [  [
[  [

###### Abstract

A new ultra-low background high-purity germanium (HPGe) detector has been
installed at the Kamioka underground experimental site. The background count
rate in the energy range from 40 keV to 2700 keV is about 25% lower than that
of the first HPGe detector installed in 2016, which has the same detector
specification and similar shielding geometry. This paper describes the
shielding configuration, including the cleaning of the material surface, the
comparison of calibration data and simulation, the time variation of the
background spectra, the sample measurement procedure, and some results of the
radioactivity in the selected samples.

H20

##  1 Introduction

Material screening with high sensitivity is one of the important tasks for the
current and next-generation rare event search experiments. For example, the
SK-Gd project Beacom:2003nk  requires that the concentration of radioactive
impurities in Gd2(SO4)${}_{3}{\cdot}$8H2O should be less than 0.5 mBq kg-1 of
the latter segment of U-chain (226Ra) and 0.05 mBq kg-1 of the latter segment
of the Th-chain (228Ra) for the precise measurements of the solar neutrino
spectrum 10.1093/ptep/ptac170 . The KamLAND-Zen experiment
PhysRevLett.130.051801 , aiming to explore the neutrinoless double beta decay
of 136Xe, plans to use a scintillation film and a wavelength shifter to
enhance the sensitivity 10.1093/ptep/ptz064 . The radioactivities of the
scintillation film and the wavelength shifter are required to be equal to or
less than $\mathcal{O}$(1 ppt) for 238U and 232Th. In order to measure
radioactivity at the aforementioned required level, we developed the first
ultra-low background (BG) high-purity germanium (HPGe) detector in 2016,
hereafter referred to as Ge01. This detector has been used to screen materials
with high sensitivity 10.1093/ptep/ptac170 ; 10.1093/ptep/pty096 ;
10.1093/ptep/ptaa105 ; Abe_2020 ; 10.1093/ptep/ptaa119 . Then in 2020, we
developed a new ultra-low BG HPGe detector with more careful treatment of the
shielding material, to improve the sensitivity for increasing the capacity of
material screenings for the second Gd-loading to Super-Kamiokande for 0.03% Gd
concentration [K. Abe et al., manuscript in preparation] and for future
projects conducted at Kamioka.

This paper describes the performance of the newly installed HPGe detector,
including its analysis procedure for measuring radioactivity. Sect. 2 provides
details of the detector configuration, data acquisition system, and
calibration results. Sect. 3 describes the data analysis, the time variation
of the BG spectra, the comparison of the BG count rate with Ge01, and the
sample measurement procedure. A summary is presented in Sect. 4.

##  2 Germanium Detector

###  2.1 Detector configuration

Figure 1: Schematic view of the new HPGe detector. The HPGe detector with its
cold finger and dewar (a), 6N grade copper (b), OFHC copper (c), lead layers
with 210Pb activities of 5$\pm$3 Bq kg-1 (d), 35$\pm$4 Bq kg-1 (e),
$\mathcal{O}(100)$ Bq kg-1 (f), and $\mathcal{O}(500)$ Bq kg-1 (g, only for
the top part) can be seen. The top part circled with the red line can be
floated and moved.

The new HPGe detector, hereafter referred to as Ge02, has been manufactured by
Mirion Technologies France MirionTechnologies  with an ultra-low BG
specification. Ge02 is a p-type coaxial detector with a crystal mass of 1.7 kg
and a relative efficiency of 80%111Efficiency is defined relative to a 3-inch
diameter, 3-inch long NaI(Tl) crystal, for 1333 keV photopeak from a 60Co
source positioned 25 cm away from the detector., the same specification as
Ge01. The detectors are installed at the Kamioka Observatory in the Kamioka
Mine under Mt. Ikenoyama. The average rock overburden is 2700 m water
equivalent, which reduces the cosmic ray muon rate by 5 orders of magnitude.
Fig. 1 illustrates the schematic view of Ge02. The sample chamber has
dimensions of 23$\times$23$\times$48 cm3, with the HPGe detector positioned
centrally within it. The Ge crystal is housed in a cylindrical endcap made of
ultra-low BG aluminum. The spacious sample chamber allows the measurement of
large sample quantities, such as 13$\times$3-inch photomultiplier tubes
Abe_2020  and 10 kg of Gd2(SO4)${}_{3}\cdot$8H2O 10.1093/ptep/ptac170 . To
mitigate the BG caused by radon (Rn) gas emanating from the shielding
materials mentioned below, the sample chamber is continuously purged with Rn-
free air at a rate of 5 standard liters per minute in which the concentration
of 222Rn is $<1$ mBq/m3 NAKANO2017108 . The chamber is enveloped by 1 cm
thick, 6N grade copper ($>$ 99.9999% purity) supplied by Mitsubishi Materials
Corporation Mitsubishi . Surrounding the 6N grade copper is a 5 cm layer of
oxygen-free-high-conductivity (OFHC) copper that has been stored at the
Kamioka mine for over a decade. These layers of copper act as a shield against
bremsstrahlung radiation generated by the beta-rays of 210Bi decay in the lead
layers mentioned below. The OFHC copper is surrounded by a 2.5 cm lead layer
with a 210Pb activity of 5$\pm$3 Bq kg-1. Then there is a 5 cm lead layer for
the bottom (10 cm for the side) with a 210Pb activity of 35$\pm$4 Bq kg-1,
purchased from SEIKO EG$\&$G Co Ltd. SeikoEGandG . Outside of these low 210Pb
lead layers are covered by lead with a 210Pb activity of $\mathcal{O}(100)$ Bq
kg-1.

The upper part of the 5 cm OFHC copper layer and the lead layers can be
floated and moved laterally using a frame guide. The innermost 2.5 cm of the
lead layer in the top section has a 210Pb activity of 5$\pm$3 Bq kg-1 and the
remaining 17.5 cm lead layer has that of $\mathcal{O}(500)$ Bq kg-1. In total,
the lead layers have a combined thickness of 22.5 cm (20 cm for the top part)
to shield against environmental gamma-rays. Ge01 and Ge02 have the same
thickness of the Cu layer and the combined lead layers. The difference in the
shield configuration between Ge01 and Ge02 is that Ge01 has a 210Pb activity
of O(100) Bq kg-1 lead for the layer marked as (e, that of 35$\pm$4 Bq kg-1 )
and [g, that of O(500 Bq kg-1)] in Fig. 1. The newly purchased 6N grade copper
and lead bricks with a 210Pb activity of 5$\pm$3 Bq kg-1 underwent a cleaning
process involving immersion in 4% dilute nitric acid for 20 minutes, followed
by rinsing with ultra-pure water for an additional 20 minutes to remove
surface contamination. These materials were then wiped with a clean cloth and
dried in a chamber supplied with nitrogen gas flowing at a rate of 5 standard
liters per minute for a day. Other copper and lead bricks were cleaned with
ethanol prior to installation. As for the cleaning process for 6N copper and
lead bricks with a 210Pb activity of 5$\pm$3 Bq kg-1 used in Ge01 shield
layers, they were only cleaned with ethanol using an ultrasonic cleaning
machine. Typically, normal samples are placed on a 5 mm thick acrylic stage.
The distance between the top of the acrylic stage and the end cap of the Ge
detector is 9 mm. However, for measuring a large amount of sample, such as the
10 kg Gd2(SO4)${}_{3}{\cdot}$8H2O mentioned in Sect. 3, the acrylic stage can
be removed, allowing the sample piece to be placed next to the Ge detector.

###  2.2 Data acquisition system

The signal from the Ge02 detector preamplifier was fed into a Canberra Model
2026 main amplifier. The signal from both the preamplifier and the main
amplifier was recorded by a CAEN DT5724B waveform digitizer with a sampling
rate of 100 MHz. The coarse and fine gain settings of the main amplifier were
adjusted to avoid saturation of the Analog-to-Digital Converter (ADC) of the
waveform digitizer by the 208Tl 2614 keV photopeak. Fig. 2 shows the typical
waveform of the 137Cs 662 keV photopeak event. The threshold of the main
amplifier waveform is 30 ADC counts above the baseline, corresponding to an
energy threshold of about 10 keV. For energy estimation, the maximum amplitude
of the pulse from the main amplifier and a baseline calculated from the first
2 $\mu$s of the waveform were used. Pulse information such as the baseline
fluctuation and the channel at the maximum amplitude facilitates the
identification and rejection of unphysical electric noise events and pile-up
events.

Figure 2: Typical preamplifier and main amplifier waveforms for a 137Cs 662
keV photopeak event. The hosizontal dashed line represents the baseline level
calculated from the first 2 $\mu$s of the waveform. The vertical dashed line
corresponds to the amplitude of the maximum peak observed in the output signal
of the main amplifier. This value is obtained by subtracting the baseline ADC
from the peak height and is used to estimate the energy associated with the
detected gamma-ray event.

###  2.3 Calibration measurement and efficiency determination

Detector calibrations have been performed regularly using radioactive sources
of 60Co, 133Ba, and 137Cs. Fig. 3 shows the linearity of the energy response
and the energy resolution as a function of the energy. Energy is calibrated
within 0.06% uncertainty. The full width at half maximum (FWHM) of the 1333
keV photopeak of 60Co was measured to be 1.82 keV. In the comparison between
the energy spectra of the experimental data and the Monte Carlo (MC)
simulations, the energy resolution function shown in Fig. 3 (right) is used.

Figure 3: (Left) Linearity of the energy response and (right) energy
resolution, defined as the $\sigma$ divided by the energy, obtained from
calibration sources and sample measurements. The dashed curve represents a
fitted result of $\sigma^{2}(E)=E^{2}[(4.95\pm 1.07)\times 10^{-8}]+E[(2.31\pm
0.24)\times 10^{-4}]+[(2.09\pm 0.10)\times 10^{-1}]$ using calibration
sources.

The MC simulations based on Geant4 AGOSTINELLI2003250  are performed to
compute the detection efficiencies for different gamma-lines used to assess
radioactivities, as described in Sect. 3.2. These simulations take into
account the detailed geometry of the detector, including the shielding layers.
Fig. 4 shows a comparison between the spectra obtained from the 60Co, 133Ba,
and 137Cs calibrations and those obtained from the simulations which trace the
full decay chain of the isotopes using G4RadioactiveDecay. Except for an
unknown continuous component excess of data below 60 keV, the simulation
reproduces the calibration data well. The peak count rates observed in the
calibration data, with the source placed at different positions, agree within
10% with those obtained from the MC simulation. Following the approach
outlined in Refs. ABE2019171 ; Abe_2020 , since the systematic uncertainties
associated with the detection efficiencies derived from the simulation are
common to all measurements, the results presented in Sect. 3.2 are given only
with their respective statistical uncertainties.

Figure 4: Comparison of the energy spectrum from 60Co (top), 137Cs (middle),
and 133Ba (bottom) calibration data with that of a Monte Carlo simulation. In
the simulation, the energy spectrum is normalized by the measurement time and
the activity of the source, and the energy deposition in the HPGe detector is
smeared with the energy resolution function described in Fig. 3. The
differences of the count rate (MC $-$ Data)/Data in the $\pm$3$\sigma$ region,
subtracting the continuous count rate derived from the sideband region, are
$-$16.5$\pm$6.9% (133Ba, 53 keV), 7.6$\pm$1.1% (133Ba, 81 keV), 1.8$\pm$2.3%
(133Ba, 277 keV), 5.1$\pm$1.4% (133Ba, 303 keV), 4.5$\pm$0.8% (133Ba, 356
keV), 8.3$\pm$2.1% (133Ba, 384 keV), 11.4$\pm$0.9% (137Cs, 662 keV),
9.5$\pm$1.3% (60Co, 1173 keV), 9.7$\pm$1.3% (60Co, 1333 keV), respectively.
Here error values correspond to the statistical uncertainties.

##  3 Data analysis methods

###  3.1 Background spectra

Data acquisition with the shielding described in Sect. 2.1 started in December
2021. During the sample measurements described in Sect. 3.2, three long-term
BG data acquisitions were performed. Fig. 5 shows the time variation of the BG
spectra. The continuous component and the distinct peaks due to cosmogenic
activation in germanium and copper, as mentioned in Ref. CEBRIAN2010316 , such
as 57Co (122 keV, T1/2 = 271.74 d) and 58Co (811 keV, T1/2 = 70.86 d), show a
decreasing trend with time. The peaks attributed to 210Pb and 137Cs, which
were observed in the energy spectra of Ge01 (Fig. 6 in Ref.
10.1093/ptep/ptac170 ), also show a significant decrease from Ge01.

Figure 5: The energy spectrum of the background data taken on December 2021,
July 2022, and April 2023 for the energy ranges of [20,2700] keV (top left, 20
keV bin, with statistical error bar), [20, 500] keV (top right), [500, 1000]
keV (bottom left), and [1000,1500] keV (bottom right). The bin width for the
linear-scale figures is 2 keV. Clear peaks induced by cosmogenic activation in
germanium and copper are listed in the linear-scale figures.

Table 1 shows the BG count rate for the integral in the energy range from 40
to 2700 keV, as well as for several gamma-lines. After the decrease of
radioactivity due to cosmogenic activation, the integral count rate in the
energy range from 40 keV to 2700 keV of Ge02 becomes 25% lower than that of
Ge01. Table 2 summarizes the crystal mass, energy resolution and BG count rate
in comparison with other ultra-low BG HPGe detectors used for material
screening at the other experimental site. The BG level of Ge02 is comparable
to that of the HPGe detectors used for material screening at the most advanced
experiments.

Table 1: The time variation of the background count rates for the integral in the 40–2700 keV region and integral count rates in the $\pm$3$\sigma$ region for several gamma-lines. For the Ge01 detector, the count rates are calculated from the background data used in Ref. 10.1093/ptep/ptac170 . Detector | Ge01 | Ge02
---|---|---
Date | Dec. 2019 | Dec. 2021 | Jul. 2022 | Apr. 2023
Measurement time (d) | 23.0 | 19.0 | 47.2 | 86.2
Count rate (kg${}_{Ge}^{-1}$ day-1)
Integral 40 – 2700 keV | 112.6 | 140.2 | 100.0 | 84.3
208Tl, 2614 keV |  0.08$\pm$0.04 |  0.25$\pm$0.09 |  0.16$\pm$0.05 |  0.13$\pm$0.03
214Bi, 609 keV |  0.39$\pm$0.10 |  0.25$\pm$0.09 |  0.38$\pm$0.07 |  0.23$\pm$0.04
60Co, 1333 keV |  0.41$\pm$0.10 |  0.66$\pm$0.14 |  0.48$\pm$0.08 |  0.68$\pm$0.07
40K, 1461 keV |  0.44$\pm$0.11 |  0.31$\pm$0.10 |  0.44$\pm$0.07 |  0.42$\pm$0.05
137Cs, 662 keV |  1.29$\pm$0.18 |  0.53$\pm$0.13 |  0.38$\pm$0.07 |  0.32$\pm$0.05
210Pb, 46.5 keV |  3.24$\pm$0.29 |  0.69$\pm$0.14 |  0.64$\pm$0.09 |  0.59$\pm$0.06
Table 2: Comparison of crystal mass, relative efficiency, energy resolution and background rate with other HPGe detectors used for material screening at the other underground experimental site. The HPGe detectors described in Refs. 10.1093/ptep/ptac170 ; Araujo_2022  are listed.  Site | Detector | Crystal | Relative | FWHM at | BG rate
---|---|---|---|---|---
mass | efficiency | 1333 keV | 60 – 2700 keV
[kg] | [%] | [keV] |  [kg${}_{Ge}^{-1}$ day-1)]
Kamioka | Ge02 (This work) | 1.68 | 80 | 1.82 |  81.3$\pm$0.7
Ge01 10.1093/ptep/ptac170  | 1.68 | 80 | 2.39 | 104.5
LNGS |  Gator Araujo_2022  | 2.2 | 100.5 | 1.98 |  89.0$\pm$0.7
GeMPI Araujo_2022  | 2.2 | 98.7 | 2.20 |  24$\pm$1
BUGS |  Belmont 10.1093/ptep/ptac170  | 3.2 | 160 | 1.92 | 90.0
Merrybent 10.1093/ptep/ptac170  | 2.0 | 100 | 1.87 | 145.0
LSC |  GeOroel 10.1093/ptep/ptac170  | 2.31 | 109 | 2.22 | 128.7
Asterix 10.1093/ptep/ptac170  | 2.13 | 95.1 | 1.92 | 171.3
GeAnayet 10.1093/ptep/ptac170  | 2.26 | 109 | 1.99 | 461.2
BHUC |  Maeve LZ_Screening  | 2.0 | 85 | 3.19 | 956.1
LVdA |  GeMSE Araujo_2022 ; GeMSE  | 2.0 | 107.7 | 1.96 |  88$\pm$1

###  3.2 Sample measurements

Samples are flushed with Rn-free air, evacuated, and sealed in ethylene-vinyl
alcohol (EVOH) bags. EVOH bags retain Rn emanated from samples. Fig. 6 shows
photographs of a 440 g PEN pellet sample and a 10 kg of
Gd2(SO4)${}_{3}{\cdot}$8H2O sample.

Figure 6: Pictures of a 440 g PEN pellet sample on the acrylic sample
stage(left) and Ge02 detector with 10 kg of Gd2(SO4)3$\cdot$8H2O sample
(right).

Data collected within the first 12 hours from sample deployment were not used
in the analysis to mitigate the effects of Rn which was contaminated into the
chamber during the sample deployment. A noisy period associated with the
supply of liquid nitrogen was also excluded from the analysis. Figs. 7 and 8
show the energy spectra of the sample and the BG measurements.

Figure 7: The energy spectra for the PEN pellet sample measurement and the
background. Figure 8: The energy spectra for a typical 10 kg of
Gd2(SO4)${}_{3}{\cdot}$8H2O sample measurement and the background.

The determination of the radioactivity of the sample is based on the net
signal count $S_{net}$, which is defined as:

$S_{net}=S_{peak}-S_{comp}-(B_{peak}-B_{comp})\cdot T_{s}/T_{b},$ (1)

Here, $S_{peak}$ represents the counts in the photopeak region. The photopeak
region is defined as the $\pm$4 keV region from the peak, that is sufficiently
larger than the energy scale uncertainty and $\pm$3$\sigma$ of the energy
resolution. $S_{comp}$ corresponds to the Compton BG counts and is estimated
from the $\pm$4 keV regions adjacent to the photopeak region. $B_{peak}$ and
$B_{comp}$ denote the counts in the photopeak and Compton regions of the BG
data, respectively. $T_{s}$ and $T_{b}$ are the measurement times for the
sample and the BG, respectively.

The specific activity A [Bq kg-1] for a given gamma-line is determined using
the equation:

$A~{}{\rm[Bq~{}kg^{-1}]}=\frac{S_{net}}{I_{\gamma}\cdot\epsilon\cdot m\cdot
T_{s}}.$ (2)

In this equation, $I_{\gamma}$ represents the branching ratio of the gamma-
ray, $\epsilon$ is the photopeak detection efficiency and $m$ is the mass of
the sample in kilograms. The term $\epsilon$ is evaluated by the MC simulation
using regular-shaped geometry closest to that of the sample. The density and
composition of the sample are also taken into account by the MC to consider
self-absorption in the samples. The simulated monochromatic gamma-rays are
generated uniformly and isotropically within the sample volume. If several
prominent gamma-lines originate from a specific radioisotope (RI), the
radioactivity is determined using the weighted average of the evaluations for
each gamma-line. For example, gamma-lines from 214Pb (295 keV) and 214Bi (609,
1120, and 1764 keV) are used to assess the radioactivity of 226Ra. The effect
of the coincidence sum of multiple gamma-rays in the decay cascade was
simulated for these samples and found to be negligible compared to the
statistical error. Following Refs. ABE2019171 ; Abe_2020 , if the central
value does not exceed twice the statistical error or is negative, the result
is considered to be consistent with zero. In such cases, an upper limit at the
90% confidence level (C.L.) is calculated as max{0, the central value} + 1.28
$\times$ the statistical error. Table 3 shows the measured RIs for the PEN
pellet and Gd2(SO4)${}_{3}{\cdot}$8H2O samples. With the exception of 176Lu in
the Gd2(SO4)${}_{3}{\cdot}$8H2O sample, no significant RIs were observed.
Similar to Ge01, Ge02 demonstrates sensitivity to less than 0.5 mBq kg-1 of
226Ra in the Gd2(SO4)${}_{3}{\cdot}$8H2O sample at a 90% C.L. with about 10
days of measurement.

Table 3: Summary of the measurement results [mBq kg-1] for PEN pellet and Gd2(SO4)${}_{3}{\cdot}$8H2O sample. Upper limits correspond to the 90% confidence level.  Sample | PEN Pellet |  Gd2(SO4)${}_{3}{\cdot}$8H2O
---|---|---
Measurement time [d] | 13.4 | 10.2
Early part of 238U chain | $<$ 17.4 | $<$ 6.2
234Th 63, 93 keV
234mPa 1001 keV
middle part of 238U chain | $<$ 2.60 | $<$ 0.43
214Pb 295 keV
214Bi 609, 1120, 1764 keV
late part of 238U chain | $<$ 137 | $<$ 91
210Pb 46.5 keV
Early part of 232Th chain | $<$ 2.10 | $<$ 0.25
228Ac 338, 911, 969 keV
late part of 232Th chain | $<$ 1.96 | $<$ 0.16
212Bi 727 keV
208Tl 583, 861, 2614 keV
Early part of 235U chain | $<$ 5.56 | $<$ 1.79
235U 144, 163, 205 keV
late part of 235U chain | $<$ 3.47 | $<$ 0.66
223Ra 154, 269 keV
219Rn 271, 402 keV
211Pb 405, 427, 832 keV
40K 1461 keV |  $<$ 7.63 |  $<$ 0.86
60Co 1173, 1333 keV |  $<$ 0.66 |  $<$ 0.07
176Lu 202, 307 keV |  $<$ 0.52 |  0.46 $\pm$ 0.09

Almost all Gd2(SO4)${}_{3}{\cdot}$8H2O lots used in the second Gd-loading to
Super-Kamiokande were screened with the Ge01 and Ge02 detectors. Further
details on the screening of Gd2(SO4)${}_{3}{\cdot}$8H2O will be found in K.
Abe et al. [manuscript in preparation].

##  4 Conclusion

We have successfully deployed a new low-background HPGe detector at the
Kamioka Observatory. The integral background count rate in the energy range
from 40 keV to 2700 keV is about 25% lower than that of Ge01 and comparable to
that of the HPGe detectors used for material screening at the most advanced
experiments. The HPGe detector demonstrates a sensitivity of less than 0.5 mBq
kg-1 of 226Ra in a 10 kg Gd2(SO4)${}_{3}{\cdot}$8H2O sample at a 90% C.L. with
an approximate measurement time of 10 days. This detector is currently playing
essential roles for material screenings for the SK-Gd, KamLAND-Zen, and other
research and development experiments.

## Acknowledgment

We thank Mirion Technologies staff for fruitful discussions and the
development of an ultra-low BG HPGe detector. We would like to thank Toyobo
Co., Ltd. and Nippon Yttrium Co., Ltd. for providing the samples. We thank the
technical staffs of the RCNS and ICRR for their assistance with the detector
installation work. This work was supported by the Japanese Ministry of
Education, Culture, Sports, Science and Technology, Grant-in-Aid for
Scientific Research, JSPS KAKENHI Grant No. 19H05807, 19H05808, 21H01105, the
joint research program of the Institute for Cosmic Ray Research (ICRR), the
University of Tokyo.

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# Magnetic moment of the $X_{1}(2900)$ state in the diquark-antidiquark
picture

U. Özdem<EMAIL_ADDRESS>Health Services Vocational School of Higher
Education, Istanbul Aydin University, Sefakoy-Kucukcekmece, 34295 Istanbul,
Türkiye  K. Azizi<EMAIL_ADDRESS>Department of Physics, University of
Tehran, North Karegar Avenue, Tehran 14395-547, Iran  Department of Physics,
Doǧuş University, Dudullu-Ümraniye, 34775 Istanbul, Türkiye

###### Abstract

Motivated by the discovery of fully open-flavor tetraquark states
$X_{0}(2900)$ and $X_{1}(2900)$ by the LHCb Collaboration, the magnetic dipole
moment of the $X_{1}(2900)$ state with the quantum numbers $J^{P}=1^{-}$ is
determined in the diquark-antidiquark picture using the light-cone sum rules.
The numerical result is obtained as
$\mu_{X_{1}}=0.79^{+0.36}_{-0.39}\,\mu_{N}$. The magnetic moments of hadrons
encompasses useful knowledge on the distributions of charge and magnetization
their inside, which can be used to better understand their geometrical shapes
and quark-gluon organizations. The observation of the $X_{0}(2900)$ and
$X_{1}(2900)$ as the first two fully open-flavor multiquark states has opened
a new window for investigation of the exotic states. The obtained results in
the present study may shed light on the future experimental and theoretical
searches on the properties of fully open-flavor multiquark states.

Magnetic dipole moment, $X_{1}(2900)$ state, Light-cone sum rules

##  I Introduction

In 2021, the LHCb Collaboration observed two clear peaks in the $D^{-}K^{+}$
invariant mass spectrum of the $B^{+}\to D^{+}D^{-}K^{+}$ decay Aaij  _et al._
(2020a, b). Their spectroscopic parameters are measured to be

$\displaystyle X_{0}(2900):M$ $\displaystyle=$ $\displaystyle 2866\pm 7\pm
2~{}\text{MeV},$ $\displaystyle\Gamma$ $\displaystyle=$ $\displaystyle 57\pm
12\pm 4~{}\text{MeV};$ $\displaystyle X_{1}(2900):M$ $\displaystyle=$
$\displaystyle 2904\pm 5\pm 1~{}\text{MeV},$ $\displaystyle\Gamma$
$\displaystyle=$ $\displaystyle 110\pm 11\pm 4~{}\text{MeV}.$

The $D^{-}K^{+}$ configuration suggests that the quark constituents of
$X_{0}(2900)$ and $X_{1}(2900)$ should be $[\bar{c}\bar{s}][ud]$, which means
that they are fully open-flavor exotic hadrons. The possible spin-parity
quantum numbers of $X_{0}(2900)$ and $X_{1}(2900)$ are estimated to be
$J^{P}=0^{+}$ and $1^{-}$, respectively.

This observation triggered interesting phenomenological studies on these new
resonances in the context of various approaches and models aiming to elucidate
their nature, quantum numbers and substructure. These studies are assigned
different schemes for these particles: Molecular forms of
$\bar{D}^{\ast}K^{\ast}$ and $\bar{D}_{1}K$ interactions in Refs. Chen  _et
al._ (2020); He and Chen (2021); Liu  _et al._ (2020a); Hu  _et al._ (2021);
Agaev  _et al._ (2021a); Chen  _et al._ (2021a), the diquark-antidiquark
picture in Refs. Chen  _et al._ (2020); Karliner and Rosner (2020); He  _et
al._ (2020); Wang (2020); Zhang (2021); Wang  _et al._ (2021); Agaev  _et al._
(2021b), and kinematic effects from the triangle singularities in Refs. Liu
_et al._ (2020b); Burns and Swanson (2021a). The production and decay
properties of these states were investigated in Refs. Huang  _et al._ (2020);
Chen  _et al._ (2021b); Burns and Swanson (2021b); Xiao  _et al._ (2021), as
well. One can also consult Refs. Albuquerque  _et al._ (2021); Lü  _et al._
(2020); Mutuk (2021); Tan and Ping (2021); Abreu (2021); Qi  _et al._ (2021);
Chen (2022); Hsiao and Yu (2021); Duan  _et al._ (2021); Kong  _et al._
(2021); Dong and Zou (2021); Bondar and Milstein (2020) for other pertinent
studies on parameters of these states.

Despite the above experimental and theoretical investigations, their
properties remain dubious and determination of their exact nature and
substructure is still problematic. Indeed, the properties of the $X_{0}(2900)$
and $X_{1}(2900)$ tetraquark states have been suggested differently in
different studies. To resolve these ambiguities, their parameters are needed
to be further investigated both in theory and experiment. These studies shall
include complementary investigations of their spectroscopy as well as their
various reactions with other known particles and/or their strong,
electromagnetic and weak decay modes.

Inspired by this, in the present study, we are going to consider the
interaction of $X_{1}(2900)$ ($X_{1}$ for short) tetraquark state with photon
and examine the magnetic dipole moment of this state within the framework of
the light-cone sum rules (LCSR). In the LCSR method, a two-point correlation
function is calculated in two different steps. In the first step, it is
obtained in terms of hadronic parameters such as form factors, electromagnetic
multipole moments, etc., which is called hadronic representation. In the
second step, it is calculated in terms of quark-gluon degrees of freedom,
which is called the QCD representation. Then, the correlation functions
obtained from these two different ways are related to each other using quark-
hadron duality assumption. Finally, the Borel transform and continuum
subtraction are performed to suppress the contributions of the possible higher
states and continuum. By this way, one obtains the sum rules for the desired
physical quantities. In the calculations we use the distribution amplitudes
(DAs) of the on-shell photon. When calculating the magnetic dipole moment of
$X_{1}$, we will consider that this state has a diquark-antidiquark
configuration. There are few studies in the literature where the magnetic
dipole moments of the open-flavor exotic states have been investigated, see
for instance the Refs. Azizi and Özdem (2018a, b, 2021).

The paper is organized as follows. In Section II, some details of the
calculations of the magnetic dipole moment of $X_{1}$ in LCSR method is given.
In Sec. III, we present our numerical results and discussions. In Section IV,
we discuss obtained results and conclude with brief notes.

##  II  formalism

As we have mentioned above, at the beginning of the analytic calculations of
the magnetic dipole moment, it is necessary to select a sufficient two-point
correlation function in the background electromagnetic field, which plays an
important role in the LCSR method. It is written as

$\Pi_{\mu\nu}(p,q)=i\int d^{4}xe^{ip\cdot x}\langle
0|\mathcal{T}\\{J_{\mu}(x)J_{\nu}^{\dagger}(0)\\}|0\rangle_{\gamma},$ (1)

where $J_{\mu}(x)$ and $\gamma$ represent the interpolating current of $X_{1}$
state and the external electromagnetic field, respectively. We need explicit
expression of $J_{\mu}(x)$ to proceed in the calculations. In the diquark-
antidiquark picture, $J_{\mu}(x)$ can be written in the following form

$\displaystyle J^{X_{1}}_{\mu}(x)$ $\displaystyle=$
$\displaystyle\varepsilon^{abc}\varepsilon^{amn}\big{[}u^{Tb}(x)C\gamma_{5}d^{c}(x)\big{]}$
(2)
$\displaystyle\times\big{[}\bar{c}^{m}(x)\gamma_{\mu}\gamma_{5}C\bar{s}^{Tn}(x)\big{]},$

where $C$ is the charge conjugation matrix; and $a,b,c,m,n$ are color indices.

In the hadronic representation, two complete sets of the initial and final
hadronic states are inserted into the correlation function. By isolating the
lowest $X_{1}$ state contribution, we obtain,

$\displaystyle\Pi_{\mu\nu}^{Had}(p,q)$ $\displaystyle={\frac{\langle 0\mid
J_{\mu}(x)\mid X_{1}(p,\varepsilon^{\theta})\rangle}{p^{2}-m_{X_{1}}^{2}}}$
$\displaystyle\times\langle X_{1}(p,\varepsilon^{\theta})\mid
X_{1}(p+q,\varepsilon^{\delta})\rangle_{\gamma}$
$\displaystyle\times\frac{\langle
X_{1}(p+q,\varepsilon^{\delta})\mid{J_{\nu}^{\dagger}}(0)\mid
0\rangle}{(p+q)^{2}-m_{X_{1}}^{2}}+\cdots,$ (3)

where dots denote the effects of the higher states and continuum. The matrix
elements in Eq. (II) are expressed as

$\displaystyle\langle
X_{1}(p+q,\varepsilon^{\delta})\mid{J_{\nu}^{\dagger}}(0)\mid
0\rangle=\lambda_{X_{1}}\varepsilon_{\nu}^{\delta}\,,$ (4)
$\displaystyle\langle 0\mid J_{\mu}(x)\mid
X_{1}(p,\varepsilon^{\theta})\rangle=\lambda_{X_{1}}\varepsilon_{\mu}^{\theta}\,,$
(5) $\displaystyle\langle X_{1}(p,\varepsilon^{\theta})\mid
X_{1}(p+q,\varepsilon^{\delta})\rangle_{\gamma}=-\varepsilon^{\tau}(\varepsilon^{\theta})^{\alpha}(\varepsilon^{\delta})^{\beta}\Big{\\{}G_{1}(Q^{2})$
$\displaystyle\times(2p+q)_{\tau}~{}g_{\alpha\beta}+G_{2}(Q^{2})g_{\tau\beta}~{}q_{\alpha}-g_{\tau\alpha}~{}q_{\beta})$
$\displaystyle-\frac{1}{2m_{X_{1}}^{2}}G_{3}(Q^{2})~{}(2p+q)_{\tau}q_{\alpha}q_{\beta}\Big{\\}},$
(6)

where $\lambda_{X_{1}}$ is the residue of $X_{1}$; and
$\varepsilon_{\mu}^{\theta}\ $, $\varepsilon_{\nu}^{\delta}\ $ and
$\varepsilon^{\tau}$ are the initial and final polarization vectors of the
$X_{1}$ and photon polarization vector, respectively. Here, $G_{1}(Q^{2})$,
$G_{2}(Q^{2})$ and $G_{3}(Q^{2})$ are electromagnetic form factors, with
$Q^{2}=-q^{2}$.

Using Eqs. (II)-(6) and after performing some necessary calculations, the
final form of the correlation function is obtained as

$\displaystyle\Pi_{\mu\nu}^{Had}(p,q)=\frac{\varepsilon_{\rho}\,\lambda_{X_{1}}^{2}}{[m_{X_{1}}^{2}-(p+q)^{2}][m_{X_{1}}^{2}-p^{2}]}\Big{\\{}G_{2}(Q^{2})$
$\displaystyle\times\Big{(}q_{\mu}g_{\rho\nu}-q_{\nu}g_{\rho\mu}-\frac{p_{\nu}}{m_{X_{1}}^{2}}\big{(}q_{\mu}p_{\rho}-\frac{1}{2}Q^{2}g_{\mu\rho}\big{)}$
$\displaystyle+\frac{(p+q)_{\mu}}{m_{X_{1}}^{2}}\big{(}q_{\nu}(p+q)_{\rho}+\frac{1}{2}Q^{2}g_{\nu\rho}\big{)}-\frac{(p+q)_{\mu}p_{\nu}p_{\rho}}{m_{X_{1}}^{4}}\,Q^{2}\Big{)}$
$\displaystyle+\mbox{other independent structures}\Big{\\}}\,+\cdots.$ (7)

To calculate the magnetic dipole moment, we need to calculate only the form
factor $G_{2}(Q^{2})$, which is called the magnetic form factor,

$\displaystyle F_{M}(Q^{2})=G_{2}(Q^{2})\,.$ (8)

At static limit, $Q^{2}=0$, $F_{M}(Q^{2}=0)$ is proportional to the magnetic
dipole moment $\mu_{X_{1}}$ for real photon:

$\displaystyle\mu_{X_{1}}=\frac{e}{2\,m_{X_{1}}}\,F_{M}(Q^{2}=0).$ (9)

The correlation function, on the other hand, is determined in terms of the QCD
degrees of freedom and the photon distribution amplitudes in the second window
called the QCD side. In the QCD representation, we use the Wick’s theorem to
contract the corresponding quark fields to get the correlation function in
terms of the quark propagators and DAs of the photon. After replacing the
explicit expression of the interpolating current in the correlation function
and applying the Wick’s theorem, we get

$\displaystyle\Pi_{\mu\nu}^{\mathrm{QCD}}(p,q)$ $\displaystyle=$
$\displaystyle i\int
d^{4}xe^{ipx}\varepsilon^{abc}\varepsilon^{amn}\varepsilon^{a^{\prime}b^{\prime}c^{\prime}}\varepsilon^{a^{\prime}m^{\prime}n^{\prime}}$
$\displaystyle\times\mathrm{Tr}[\gamma_{5}\widetilde{S}_{u}^{bb^{\prime}}(x)\gamma_{5}S_{d}^{cc^{\prime}}(x)]\mathrm{Tr}[\gamma_{\mu}\gamma_{5}\widetilde{S}_{s}^{n^{\prime}n}(-x)$
$\displaystyle\times\gamma_{5}\gamma_{\nu}S_{c}^{m^{\prime}m}(-x)]\mid
0\rangle_{\gamma},$

where

$\widetilde{S}_{c(q)}^{ij}(x)=CS_{c(q)}^{ij\mathrm{T}}(x)C,$

with $S_{q(c)}(x)$ being the quark propagators. In the $x$-space for the
light-quark propagator we use

$\displaystyle S_{q}(x)$
$\displaystyle=S_{q}^{free}-\frac{\langle\bar{q}q\rangle}{12}\Big{(}1-i\frac{m_{q}x\\!\\!\\!/}{4}\Big{)}-\frac{\langle\bar{q}q\rangle}{192}m_{0}^{2}x^{2}\Big{(}1$
$\displaystyle-i\frac{m_{q}x\\!\\!\\!/}{6}\Big{)}-\frac{ig_{s}}{32\pi^{2}x^{2}}~{}G^{\mu\nu}(x)\Big{[}\hbox
to0.0pt{/\hss}{x}\sigma_{\mu\nu}+\sigma_{\mu\nu}\hbox
to0.0pt{/\hss}{x}\Big{]},$ (11)

where,

$\displaystyle
S_{q}^{free}=i\frac{{x\\!\\!\\!/}}{2\pi^{2}x^{4}}-\frac{m_{q}}{4\pi^{2}x^{2}}.$
(12)

The charm-quark propagator is given, in terms of the second kind Bessel
functions $K_{i}(x)$, as

$\displaystyle S_{c}(x)$
$\displaystyle=S_{c}^{free}-\frac{g_{s}m_{c}}{16\pi^{2}}\int_{0}^{1}dv\,G^{\mu\nu}(vx)\Bigg{[}\big{(}\sigma_{\mu\nu}{x\\!\\!\\!/}+{x\\!\\!\\!/}\sigma_{\mu\nu}\big{)}$
$\displaystyle\times\frac{K_{1}\Big{(}m_{c}\sqrt{-x^{2}}\Big{)}}{\sqrt{-x^{2}}}+2\sigma_{\mu\nu}K_{0}\Big{(}m_{c}\sqrt{-x^{2}}\Big{)}\Bigg{]}.$
(13)

where,

$\displaystyle
S_{c}^{free}=\frac{m_{c}^{2}}{4\pi^{2}}\Bigg{[}\frac{K_{1}\Big{(}m_{c}\sqrt{-x^{2}}\Big{)}}{\sqrt{-x^{2}}}+i\frac{{x\\!\\!\\!/}~{}K_{2}\Big{(}m_{c}\sqrt{-x^{2}}\Big{)}}{(\sqrt{-x^{2}})^{2}}\Bigg{]}.$
(14)

The correlation function in QCD representation includes two different
contributions: Perturbative and non-perturbative. Practically, the
perturbative contribution, in which the photon interacts with one of the
quarks perturbatively, can be computed by the replacing one of the light or
c-quark propagators by

$\displaystyle S^{free}\rightarrow\int d^{4}z\,S^{free}(x-z)\,\hbox
to0.0pt{/\hss}{\\!A}(z)\,S^{free}(z)\,,$ (15)

and the other three propagators with their perturbative or free parts. In the
last equation we also use the perturbative parts of the propagators in right
hand side as is seen.

For the non-perturbative contribution, in which the photon is radiated at long
distances, the correlation function can be computed by replacing one of the
light quark propagators by

$\displaystyle
S_{\alpha\beta}^{ab}\rightarrow-\frac{1}{4}(\bar{q}^{a}\Gamma_{i}q^{b})(\Gamma_{i})_{\alpha\beta},$
(16)

where
$\Gamma_{i}=I,\gamma_{5},\gamma_{\mu},i\gamma_{5}\gamma_{\mu},\sigma_{\mu\nu}/2$,
and the remaining light and heavy propagators with their full expressions. We
perform all the possible permutations in the perturbative and non-perturbative
parts of the correlation function. When Eq. (16) is employed in computation of
the non-perturbative effects, we observe that matrix elements of the forms
$\langle\gamma(q)|\bar{q}(x)\Gamma_{i}q(0)|0\rangle$ and
$\langle\gamma(q)|\bar{q}(x)\Gamma_{i}G_{\mu\nu}q(0)|0\rangle$ appear. These
matrix elements are written in terms of the photon distribution amplitudes
(see Ref. Ball  _et al._ (2003)). Using these matrix elements in terms of
photon’s DAs and the expressions of the propagators given above, the QCD
representation of the correlation function in coordinate space is obtained. We
perform Fourier transformation to carry the calculations to the momentum
space.

The LCSR for the magnetic dipole moment of $X_{1}$ state can be acquired by
matching the functions $\Pi_{\mu\nu}^{QCD}(p)$ and $\Pi_{\mu\nu}^{Had}(p)$
from both the QCD and hadronic sides. We choose the $\varepsilon_{\nu}q_{\mu}$
structure and equate the coefficients of this structure from both sides to
each other. We apply a double Borel transformation with respect to -$p^{2}$
and -$(p+q)^{2}$ and also continuum subtraction procedure based on the
standard prescriptions of the method in order to suppress the contributions of
the higher states and continuum (fore details see for instance Refs. Azizi
_et al._ (2018); Agaev  _et al._ (2016)). As a results, we get the desired
LCSR for the magnetic dipole moment as

$\displaystyle\mu_{X_{1}}=\frac{e^{\frac{m_{X_{1}}^{2}}{M^{2}}}}{\lambda_{X_{1}}^{2}}\,\,\Delta(M^{2},s_{0}),$
(17)

where the explicit expression of $\Delta(M^{2},s_{0})$ function, which
represents the final form of the QCD side of the calculations, is presented in
the Appendix. In obtaining the last result, we use
$\frac{1}{M^{2}}=\frac{1}{M_{1}^{2}}+\frac{1}{M_{2}^{2}}$ with $M_{1}^{2}$ and
$M_{2}^{2}$ being the Borel parameters in the initial and final channels,
respectively. We set $M_{1}^{2}=M_{2}^{2}=2M^{2}$ as the initial and final
particles are the same. We will fix $M^{2}$ and continuum threshold ($s_{0}$),
coming from the continuum subtraction procedure and appearing inside the
$\Delta(M^{2},s_{0})$ function, based on the standard criteria of the method
in next section.

##  III Numerical analysis

In this section, we present our numerical prediction for the magnetic dipole
moment of the $X_{1}$ state. To get our numerical results, the main input
parameters are the photon distribution amplitudes. The distribution amplitudes
of the photon, which depend on various non-perturbative parameters are
borrowed from Ref. Ball  _et al._ (2003). We use the values of some other
input parameters as follow: $m_{u}=m_{d}=0$, $m_{s}=96^{+8}_{-4}\,\mbox{MeV}$,
$m_{c}=(1.275\pm 0.025)\,$GeV, $m_{X_{1}}=2904\pm 5\pm 1~{}\text{MeV}$,
$f_{3\gamma}=-0.0039~{}GeV^{2}$ Ball  _et al._ (2003),
$\langle\bar{s}s\rangle$= $0.8\langle\bar{u}u\rangle$ with
$\langle\bar{u}u\rangle$=$(-0.24\pm 0.01)^{3}\,$GeV3 Ioffe (2006),
$m_{0}^{2}=0.8\pm 0.1$ GeV2 Ioffe (2006),
$\langle\frac{\alpha_{s}}{\pi}G^{2}\rangle=(0.012\pm 0.004)$
$~{}\mathrm{GeV}^{4}$ Belyaev and Ioffe (1983) and
$\lambda_{X_{1}}=m_{X_{1}}f_{X_{1}}$ with $f_{X_{1}}=(2.1\pm 0.4)\times
10^{-3}$ GeV4 Agaev  _et al._ (2021b).

In the LCSR method, in addition to the DAs and above parameters, there are two
extra auxiliary parameters as mentioned before: The Borel mass parameter
$M^{2}$ and the continuum threshold $s_{0}$. The continuum threshold is not
totally arbitrary but it shows the energy scale at which, the excited states
and continuum begin to contribute to the correlation function. The physical
quantities like the magnetic dipole moment are expected to be independent of
these helping parameters. In practice, however, there appear some residual
dependence on these parameters which are entered as the uncertainties to the
presented results. To find the working intervals of these auxiliary
parameters, we demand that both the continuum and higher states contributions
have to be sufficiently suppressed and series of the operator product
expansion (OPE) in QCD side converge. To this end, in technique language, we
define the pole contribution (PC) as

$\displaystyle\mbox{PC}=\frac{\Delta(M^{2},s_{0})}{\Delta(M^{2},\infty)},$
(18)

and require that it should exceed $20\%$ of the total contributions, which is
typical for the multiquark states. We also demand that the series of light-
cone expansion converges and contributions of the higher twist and higher
condensate terms are less than $10\%$ of the total contribution. These
considerations lead to the following working windows for $M^{2}$ and $s_{0}$:

$\displaystyle 3.0~{}\mbox{GeV}^{2}\leq M^{2}\leq 3.5~{}\mbox{GeV}^{2}$
$\displaystyle 11.0~{}\mbox{GeV}^{2}\leq s_{0}\leq 12.5~{}\mbox{GeV}^{2}.$

Our numerical analyses show that, by considering these working regions for the
auxiliary parameters, for the magnetic dipole moment of the $X_{1}$ state $PC$
varies within the interval $30\%\leq PC\leq 64\%$ corresponding to the upper
and lower limits of the Borel mass parameter. When we analyze the OPE
convergence, we see that the contribution of the higher twist and higher
dimensional terms in OPE is $2\%$ of the total and the series show a good
convergence. It is worth mentioning that the above interval for the continuum
threshold corresponds to $s_{0}\simeq(m_{X_{1}}+0.5^{+0.1}_{-0.1})^{2}$ GeV2,
which is typical in hadronic spectrum. Therefore, the chosen working windows
for $M^{2}$ and $s_{0}$ well fulfill the requirements of the light-cone sum
rules method.

In Fig.1, we depict the dependence of the magnetic dipole moment of the
$X_{1}$ state on $M^{2}$ at three fixed values of $s_{0}$. As one can see from
this figure, the magnetic dipole moment show a good stability with respect to
the variation of the Borel mass parameter. Although the dependence on $s_{0}$
is considerable, however, it remains within the limits allowed by the method
to calculate the magnetic dipole moment.

Figure 1: The magnetic dipole moment of the $X_{1}$ state versus $M^{2}$ at
three fixed values of $s_{0}$.

Considering all the input parameters, DAs of the photon and the working
intervals of auxiliary parameters, our prediction for the magnetic dipole
moment of the $X_{1}$ state, both in its natural unit
($\frac{e}{2\,m_{X_{1}}}$) and nuclear magneton
($\mu_{N}=\frac{e}{2\,m_{N}}$), is

$\displaystyle\mu_{X_{1}}=2.43^{+1.13}_{-1.21}\frac{e}{2\,m_{X_{1}}}=0.79^{+0.36}_{-0.39}\,\mu_{N}.$
(19)

The order of magnitude for the magnetic dipole moment shows that measurement
of $\mu_{X_{1}}$ is accessible in the future experiments.

##  IV Summary and Conclusion

Motivated by the discovery of fully open-flavor tetraquark states
$X_{0}(2900)$ and $X_{1}(2900)$ by the LHCb Collaboration, the magnetic dipole
moment of the $X_{1}(2900)$ tetraquark state have been determined using the
light-cone sum rules assigning the diquark-antidiquark structure with the
quantum numbers $J^{P}=1^{-}$ for this state. The magnetic moments of hadrons
encompasses useful knowledge about the distribution of charge and
magnetization their inside, which helps us to understand their nature, quark-
gluon organization and geometrical shape. The existing theoretical predictions
on the spectroscopic parameters of $X_{1}(2900)$ tetraquark and their
comparison with the experimental data have also given rise to different
assignments on the internal structure of this state. Calculations of
electromagnetic parameters of the exotic states like their magnetic dipole
moment can be useful in establishing the nature of these states. The
observation of the $X_{0}(2900)$ and $X_{1}(2900)$ by LHCb as the first two
fully open-flavor multiquark states has opened a new platform for
investigation of the exotic states. More experimental and theoretical research
is required to fully understand the properties of this class of hadrons. The
magnitude obtained for the magnetic dipole moment of $X_{1}(2900)$ shows a
possibility to measure it in future experiments. The obtained result in the
present study may be useful for analyses of the future experimental data on
parameters of fully open-flavor multiquark states.

## ACKNOWLEDGMENTS

K. Azizi is thankful to Iran Science Elites Federation (Saramadan) for the
partial financial support provided under the grant number ISEF/M/400150.

### Appendix: Explicit expression of $\Delta(M^{2},s_{0})$

In this appendix, we present the explicit expression of the function
$\Delta(M^{2},s_{0})$ entering into the sum rule for the magnetic moment of
the $X_{1}$ state. It is obtained as

$\displaystyle\Delta(M^{2},s_{0})$
$\displaystyle=-\frac{e_{c}}{36864m_{c}^{2}\pi^{6}}\Bigg{[}3m_{c}^{12}\big{(}3I[-6,4]-4I[-5,3]\big{)}-48m_{c}^{10}\big{(}I[-5,4]+I[-4,3]\big{)}-48m_{c}^{9}m_{s}I[-4,3]$
$\displaystyle+3m_{c}^{8}\Big{(}30I[-4,4]+P_{1}(I[-4,2]-2I[-3,1])+96m_{s}P_{3}\pi^{2}\big{(}I[-4,2]-2I[-3,1]\big{)}-24I[-3,3]\Big{)}+$
$\displaystyle
144m_{c}^{7}\big{(}4P_{3}\pi^{2}I[-3,2]-m_{s}I[-3,3]\big{)}-12m_{c}^{6}\Big{(}-32m_{0}^{2}m_{s}P_{3}\pi^{2}I[-3,1]+512P_{2}^{2}\pi^{4}I[-3,1]+P_{1}I[-3,2]$
$\displaystyle+96m_{s}P_{3}\pi^{2}I[-3,2]+6I[-3,4]+P_{1}I[-2,1]+96m_{s}P_{3}\pi^{2}I[-2,1]+4I[-2,3]\Big{)}$
$\displaystyle-12m_{c}^{5}\Big{(}48P_{3}\pi^{2}(m_{0}^{2}I[-2,1]+2I[-2,2])+m_{s}(P_{1}I[-2,1]+12I[-2,3])\Big{)}+3m_{c}^{4}\Big{(}64m_{0}^{2}m_{s}P_{3}\pi^{2}I[-2,1]$
$\displaystyle-1024P_{2}^{2}\pi^{4}I[-2,1]+3P_{1}I[-2,2]+288m_{s}P_{3}\pi^{2}I[-2,2]+7I[-2,4]-2P_{1}I[-1,1]-192m_{s}P_{3}\pi^{2}I[-1,1]$
$\displaystyle-4I[-1,3]\Big{)}+8\Big{(}-192P_{2}^{2}\pi^{4}(m_{0}^{2}I[0,0]+2I[0,1])+m_{s}P_{3}\pi^{2}\big{(}P_{1}I[0,0]+24m_{0}^{2}I[0,1]\big{)}\Big{)}$
$\displaystyle-192m_{c}^{2}I[0,3]-16m_{c}\Big{(}P_{1}P_{3}\pi^{2}I[0,0]+24m_{s}\big{(}-8P_{2}^{2}\pi^{4}I[0,0]+I[0,3]\big{)}\Big{)}+12m_{c}^{3}\big{(}48P_{3}\pi^{2}\big{(}I[-1,2]$
$\displaystyle-
m_{0}^{2}I[1,0]\big{)}-m_{s}\big{(}4I[-1,3]+P_{1}I[1,0]\big{)}\Big{)}\Bigg{]}$
$\displaystyle+\frac{e_{d}}{442368m_{c}^{2}\pi^{6}}\Bigg{[}f_{3\gamma}\pi^{2}\Bigg{(}-11P_{1}\big{(}m_{c}^{6}I[-3,1]+3m_{c}^{4}I[-2,1]+2I[0,1]\big{)}I_{1}[\mathcal{A}]+12\Big{(}5m_{c}^{10}I[-5,3]$
$\displaystyle-12m_{c}^{8}I[-4,3]-36m_{c}^{7}m_{s}I[-3,2]+m_{c}^{6}\big{(}48m_{s}P_{3}\pi^{2}I[-3,1]-9I[-3,3]\big{)}+24m_{c}^{5}\big{(}8P_{3}\pi^{2}I[-2,1]$
$\displaystyle+3m_{s}I[-2,2]\big{)}-2m_{c}^{4}\big{(}24m_{s}P_{3}\pi^{2}I[-2,1]+5I[-2,3]\big{)}-36m_{c}^{3}m_{s}I[-1,2]+48m_{0}^{2}m_{c}P_{3}\pi^{2}I[0,0]$
$\displaystyle+8m_{s}P_{3}\pi^{2}\big{(}m_{0}^{2}I[0,0]-12I[0,1]\big{)}-20I[0,3]\Big{)}I_{1}[\mathcal{V}]\Bigg{)}+4\Bigg{(}1152m_{c}^{4}P_{2}^{2}\pi^{4}I_{4}[\mathcal{S}]I[-2,1]$
$\displaystyle-6m_{c}^{5}P_{1}\big{(}m_{c}(2m_{c}(m_{c}+m_{s})I[-3,1]+I[-3,2])+(3m_{c}+2m_{s})I[-2,1]\big{)}+9m_{c}^{4}P_{1}I[-2,2]-6m_{c}^{4}P_{1}I[-1,1]$
$\displaystyle-6m_{c}^{3}m_{s}P_{1}I[-1,1]+12m_{c}^{3}m_{s}P_{1}I[0,0]+16m_{c}P_{1}P_{3}\pi^{2}I[0,0]+8m_{s}P_{1}P_{3}\pi
I[0,0]$
$\displaystyle-1152m_{c}m_{s}P_{2}^{2}\pi^{4}I_{4}[\mathcal{S}]I[0,0]-48m_{c}m_{s}P_{1}I[0,1]+1152P_{2}^{2}\pi^{4}I_{4}[\mathcal{S}]I[0,1]-144P_{2}^{2}\pi^{4}\big{(}6m_{c}^{6}I[-3,1]$
$\displaystyle+4m_{c}^{4}I[-2,1]+m_{0}^{2}I[0,0]+2I[0,1]\big{)}I_{1}[\mathcal{S}]+9P_{1}I[0,2]+18m_{c}^{3}m_{s}P_{1}I[1,0]+2f_{3\gamma}P_{1}\pi^{2}\big{(}3m_{c}^{6}I[-3,1]$
$\displaystyle+2m_{c}^{4}I[-2,1]+3I[0,1]+2m_{c}^{3}m_{s}I[1,0]\big{)}I_{6}[\psi_{a}]+4f_{3\gamma}P_{1}\pi^{2}\big{(}3m_{c}^{6}I[-3,1]+2m_{c}m_{s}I[0,0]-I[0,1]\big{)}\psi^{a}[u_{0}]\Bigg{)}\Bigg{]}$
$\displaystyle+\frac{e_{u}}{442368m_{c}^{2}\pi^{6}}\Bigg{[}f_{3\gamma}\pi^{2}\Bigg{(}12\Big{(}5m_{c}{10}I[-5,3]-12m_{c}^{8}I[-4,3]-36m_{c}^{7}m_{s}I[-3,2]+m_{c}^{6}\big{(}48m_{s}P_{3}\pi^{2}I[-3,1]$
$\displaystyle-9I[-3,3]\big{)}+24m_{c}^{5}\big{(}8P_{3}\pi^{2}I[-2,1]+3m_{s}I[-2,2]\big{)}-2m_{c}^{4}\big{(}24m_{s}P_{3}\pi^{2}I[-2,1]+5I[-2,3]\big{)}$
$\displaystyle-36m_{c}^{3}m_{s}I[-1,2]+48m_{0}^{2}m_{c}P_{3}\pi^{2}I[0,0]+8m_{s}P_{3}\pi^{2}\big{(}m_{0}^{2}I[0,0]-12I[0,1]\big{)}-44I[0,3]\Big{)}I_{2}[\mathcal{V}]$
$\displaystyle+P_{1}\Big{(}11\big{(}m_{c}^{6}I[-3,1]+3m_{c}^{4}I[-2,1]+2I[0,1]\big{)}I_{2}[\mathcal{A}]+8\big{(}3m_{c}^{6}I[-3,1]+2m_{c}^{4}I[-2,1]+3I[0,1]$
$\displaystyle+2m_{c}^{3}m_{s}I[1,0]\big{)}I_{6}[\psi^{a}]+16\big{(}3m_{c}^{6}I[-3,1]+2m_{c}m_{s}I[0,0]-I[0,1]\big{)}\psi^{a}[u_{0}])\Bigg{)}-24m_{c}^{5}(3m_{c}+2m_{s})P_{1}I[-2,1]$
$\displaystyle-4608P_{2}^{2}\pi^{4}\big{(}m_{c}^{4}I[-2,1]-m_{c}m_{s}I[0,0]+I[0,1]\big{)}I_{3}[\mathcal{S}]+576P_{2}^{2}\pi^{4}\big{(}6m_{c}^{6}I[-3,1]+4m_{c}^{4}I[-2,1]+m_{0}^{2}I[0,0]$
$\displaystyle+2I[0,1]\big{)}I_{2}[\mathcal{S}]+4P_{1}\Big{(}m_{c}^{4}(9I[-2,2]-6I[-1,1])+8m_{s}P_{3}\pi^{2}I[0,0]+16m_{c}\big{(}P_{3}\pi^{2}I[0,0]-3m_{s}I[0,1]\big{)}$
$\displaystyle+9I[0,2]-6m_{c}^{3}m_{s}(I[-1,1]-2I[0,0]-3I[1,0])\Big{)}\Bigg{]}$

$\displaystyle-\frac{e_{s}}{221184m_{c}^{2}\pi^{4}}\Bigg{[}m_{c}^{12}(6I[-6,4]+8I[-5,3])-24m_{c}{10}(I[-5,4]-I[-4,3])+m_{c}^{8}\big{(}P1I[-4,2]+30I[-4,4]$
$\displaystyle+2P_{1}I[-3,1]+24I[-3,3]\big{)}+2m_{c}^{6}\big{(}512P_{2}^{2}\pi^{4}I[-3,1]-P_{1}I[-3,2]-6I[-3,4]+P_{1}I[-2,1]+4I[-2,3]\big{)}$
$\displaystyle+1024P_{2}^{2}\pi^{4}I[0,1]-3P_{1}I[0,2]+64m_{c}^{2}I[0,3]+P_{3}\Bigg{(}864\big{(}m_{c}^{6}I[-3,2]-m_{c}^{4}I[-2,2]\big{)}\mathcal{A}[u_{0}]-432m_{c}^{2}\big{(}I_{4}[\mathcal{S}]+I_{4}[\mathcal{T}_{1}]$
$\displaystyle+I_{4}[\mathcal{T}_{2}]-I_{4}[\mathcal{T}_{3}]-I_{4}[\mathcal{T}_{4}]-I_{4}[\mathcal{\tilde{S}}]\big{)}\big{(}m_{c}^{4}I[-3,2]-2m_{c}^{2}I[-2,2]+I[-1,2]\big{)}+P_{1}\Big{(}23I_{4}[\mathcal{S}]+23I_{4}[\mathcal{T}_{1}]+23I_{4}[\mathcal{T}_{2}]$
$\displaystyle-12\big{(}I_{4}[\mathcal{T}_{3}]+I_{4}[\mathcal{T}_{4}]+I_{4}[\mathcal{\tilde{S}}]\big{)}\Big{)}I[0,0]+24\Big{(}36m_{c}^{4}(m_{c}^{4}I[-4,2]-2m_{c}^{2}I[-3,2]+I[-2,2])-P_{1}I[0,0]\Big{)}I_{6}[h_{\gamma}]\Bigg{)}$
$\displaystyle-
f_{3\gamma}\Bigg{(}144m_{c}^{8}I_{1}[\mathcal{A}]I[-4,3]+144m_{c}^{8}I_{1}[\mathcal{V}]I[-4,3]+288m_{c}^{6}I_{1}[\mathcal{A}]I[-3,3]+288m_{c}^{6}I_{1}[\mathcal{V}]I[-3,3]+23m_{c}^{4}P_{1}I_{1}[\mathcal{A}]I[-2,1]$
$\displaystyle+23m_{c}^{4}P_{1}I_{1}[\mathcal{V}]I[-2,1]+144m_{c}^{4}I_{1}[\mathcal{A}]I[-2,3]+144m_{c}^{4}I_{1}[\mathcal{V}]I[-2,3]+23P_{1}I_{1}[\mathcal{A}]I[0,1]+23P_{1}I_{1}[\mathcal{V}]I[0,1]$
$\displaystyle+576I_{1}[\mathcal{A}]I[0,3]+576I_{1}[\mathcal{V}]I[0,3]+24\Big{(}m_{c}^{6}(12m_{c}^{4}I[-5,3]+24m_{c}^{2}I[-4,3]+P_{1}I[-3,1]+12I[-3,3])-P_{1}I[0,1]$
$\displaystyle+48I[0,3]\Big{)}I_{6}[\psi^{a}]+96\Big{(}m_{c}^{6}(12m_{c}^{4}I[-5,3]+24m_{c}^{2}I[-4,3]+P_{1}I[-3,1]+12I[-3,3])+P_{1}I[0,1]+48I[0,3]\Big{)}I_{6}[\varphi_{\gamma}]$
$\displaystyle+576m_{c}^{10}I[-5,3]\psi^{a}[u_{0}]+48m_{c}^{6}P_{1}I[-3,1]\psi^{a}[u_{0}]-576m_{c}^{6}I[-3,3]\psi^{a}[u_{0}]+48P_{1}I[0,1]\psi^{a}[u_{0}]+48\Big{(}m_{c}^{6}\big{(}12m_{c}^{4}I[-5,3]$
$\displaystyle+24m_{c}^{2}I[-4,3]+P_{1}I[-3,1]+12I[-3,3]\big{)}+P_{1}I[0,1]+48I[0,3]\Big{)}\varphi_{\gamma}[u_{0}]\Bigg{)}\Bigg{]},$
(20)

where $P_{1}=\langle g_{s}^{2}G^{2}\rangle$ is gluon condensate,
$P_{2}=\langle\bar{q}q\rangle$ stands for u/d quark condensate, and
$P_{3}=\langle\bar{s}s\rangle$ represents the s-quark condensate. The
functions $I[n,m]$, $I_{1}[\mathcal{A}]$, $I_{2}[\mathcal{A}]$,
$I_{3}[\mathcal{A}]$, $I_{4}[\mathcal{A}]$, $I_{5}[\mathcal{A}]$ and
$I_{6}[\mathcal{A}]$ are defined as

$\displaystyle I[n,m]$
$\displaystyle=\int_{m_{c}^{2}}^{s_{0}}ds\int_{m_{c}^{2}}^{s}dl~{}e^{-s/M^{2}}~{}\frac{(s-l)^{m}}{l^{n}}$
$\displaystyle I_{1}[\mathcal{A}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{A}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta^{\prime}(\alpha_{q}+\bar{v}\alpha_{g}-u_{0}),$
$\displaystyle I_{2}[\mathcal{A}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{A}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta^{\prime}(\alpha_{\bar{q}}+v\alpha_{g}-u_{0}),$
$\displaystyle I_{3}[\mathcal{A}]$
$\displaystyle=\int_{0}^{1}du~{}A(u)\delta^{\prime}(u-u_{0}),$ $\displaystyle
I_{4}[\mathcal{A}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{A}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta(\alpha_{q}+\bar{v}\alpha_{g}-u_{0}),$
$\displaystyle I_{5}[\mathcal{A}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{A}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta(\alpha_{\bar{q}}+v\alpha_{g}-u_{0}),$
$\displaystyle I_{6}[\mathcal{A}]$ $\displaystyle=\int_{0}^{1}du~{}A(u),$

where $\mathcal{A}$ stands for the corresponding photon DAs and
$D_{\alpha_{i}}$ is the measure defined as

$\int{\cal
D}\alpha_{i}=\int_{0}^{1}d\alpha_{\bar{q}}\int_{0}^{1}d\alpha_{q}\int_{0}^{1}d\alpha_{g}\delta(1-\alpha_{\bar{q}}-\alpha_{q}-\alpha_{g}).\\\
$

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