Enforcing and Discovering Structure in Machine Learning
Francesco Locatello
MSc ETH in Computer Science, ETH Zurich

Prof. Dr. Gunnar Rätsch (ETH Zurich), examinerProf. Dr. Andreas Krause (ETH Zurich), co-examiner Prof. Dr. Bernhard Schölkopf (MPI Tübingen), co-examiner Prof. Dr. Volkan Cevher (EPFL), co-examiner

Constrained Optimization

Disentangled Representations

Concluding Remarks

[-1]Lists of Tableslot

[-1]Lists of Figureslof

[-1]Lists of Algorithmsloa

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titleDisentangled sequential autoencoder

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titleConditional Gradient Methods via Stochastic Path-Integrated
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titleA Conditional Gradient Framework for Composite Convex
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journaltitle2nd workshop on Fairness, Accountability, and
Transparency in Machine Learning


#  Search for synchrotron emission from secondary electrons of proton-proton
interaction in Galactic PeVatron candidate HESS J1641$-$463

Naomi Tsuji Faculty of Science, Kanagawa University, 3-27-1 Rokukakubashi,
Kanagawa-ku, Yokohama-shi, Kanagawa 221-8686, Japan  Interdisciplinary
Theoretical & Mathematical Science Program (iTHEMS), RIKEN, 2-1 Hirosawa,
Wako, Saitama 351-0198, Japan  Takaaki Tanaka  Konan University, Department of
Physics, 8-9-1 Okamoto, Higashinada, Kobe, Hyogo, Japan, 658- 8501  Samar
Safi-Harb  Department of Physics and Astronomy, University of Manitoba,
Winnipeg, MB R3T 2N2, Canada  Felix Aharonian  Dublin Institute for Advanced
Studies, School of Cosmic Physics, 31 Fitzwilliam Place, Dublin 2, Ireland
Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg,
Germany  Sabrina Casanova  Instytut Fizyki Ja̧drowej PAN, ul. Radzikowskiego
152, 31-342 Kraków, Poland  Roland Kothes Dominion Radio Astrophysical
Observatory, Herzberg Astronomy and Astrophysics Research Centre, National
Research Council Canada, P.O. Box 248, Penticton, British Columbia V2A 6J9,
Canada  Emmanuel Moulin  IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-
Yvette, France  Hiroyuki Uchida  Department of Physics, Kyoto University,
Kitashirakawa Oiwake-cho, Sakyo, Kyoto 606-8502, Japan  Yasunobu Uchiyama
Department of Physics, Rikkyo University, 3-34-1 Nishi Ikebukuro, Toshima-ku,
Tokyo 171-8501, Japan

###### Abstract

HESS J1641$-$463 is an unidentified gamma-ray source with a hard TeV gamma-ray
spectrum, and thus it has been proposed to be a possible candidate for cr
accelerators up to PeV energies (a PeVatron candidate). The source spatially
coincides with the radio snr G338.5$+$0.1, but has not yet been fully explored
in the X-ray band. We analyzed newly taken NuSTAR data, pointing at HESS
J1641$-$463, with 82 ks effective exposure time. There is no apparent X-ray
counterpart of HESS J1641$-$463, while nearby stellar cluster, Mercer 81, and
stray-light X-rays are detected. Combined with the archival Chandra data,
partially covering the source, we derived an upper limit of $\sim 6\times
10^{-13}$ ${\rm erg}~{}{\rm cm}^{-2}~{}{\rm s}^{-1}$ in 2–10 keV ($\sim
3\times 10^{-13}$ ${\rm erg}~{}{\rm cm}^{-2}~{}{\rm s}^{-1}$ in 10–20 keV). If
the gamma-ray emission is originated from decay of $\pi^{0}$ mesons produced
in interactions between cr protons and ambient materials, secondary electrons
in the proton-proton interactions can potentially emit synchrotron photons in
the X-ray band, which can be tested by our X-ray observations. Although the
obtained X-ray upper limits cannot place a constraint on the primary proton
spectrum, it will be possible with a future hard X-ray mission.

Galactic cosmic rays (567) — Diffuse radiation (383) — Gamma rays (637) —
Gamma-ray sources (633)

††journal: ApJ††facilities:  Chandra, NuSTAR††software:  HEAsoft (v6.29), CIAO
(v4.14, Fruscione et al. (2006)), NuSTARDAS (v2.1.1), nuskybgd
(https://github.com/NuSTAR/nuskybgd and https://github.com/achronal/nuskybgd-
py; Wik et al. (2014)), XSPEC (v12.12.0, Arnaud (1996)), naima
(https://naima.readthedocs.io/en/latest/; Zabalza (2015)), GAMERA
(http://libgamera.github.io/GAMERA/docs/documentation.html),

gde short = GDE , long = Galactic Diffuse Emission, cgb short = CGB , long =
cosmic gamma-ray background, grb short = GRB , long = gamma-ray burst, agn
short = AGN , long = active galactic nucleus , long-plural-form = active
galactic nuclei fsrq short = FSRQ , long = flat-spectrum radio quasar, lmxb
short = LMXB , long = low mass X-ray binary , long-plural-form = low mass
X-ray binaries , hmxb short = HMXB , long = high mass X-ray binary , long-
plural-form = high mass X-ray binaries , ic short = IC , long = inverse
Compton , nustar short = NuSTAR , long = Nuclear Spectroscopic Telescope Array
, bi short = BI , long = backside illumination , fi short = FI , long =
frontside illumination , fov short = FoV , long = field of view , sim short =
SIM , long = Science Instrument Module , hetg short = HETG , long = High
Energy Transmission Grating , letg short = LETG , long = Low Energy
Transmission Grating , hrc short = HRC , long = High Resolution Camera , acis
short = ACIS , long = Advanced CCD Imaging Spectrometer , hrma short = HRMA ,
long = High Resolution Mirror Assembly, compton short = Compton , long =
Compton Gamma Ray Observatory, hst short = HST , long = Hubble Space Telescope
, iact short = IACT , long = Imaging Atmospheric Cherenkov Telescope , wcd
short = WCD , long = Water Cherenkov Detector , hawc short = HAWC , long =
High-Altitude Water Cherenkov , cta short = CTA , long = Cherenkov Telescope
Array , em short = EM , long = electromagnetic , ism short = ISM , long =
interstellar medium , csm short = CSM , long = circumstellar medium , sne
short = SNe , long = supernovae , iss short = ISS , long = International Space
Station , uhecr short = UHECRs , long = ultra high energy cosmic rays , ta
short = TA , long = Telescope Array , auger short = Auger , long = Pierre
Auger Observatory , ams short = AMS , long = Alpha Magnetic Spectrometer ,
pamela short = PAMELA , long = Payload for Antimatter Matter Exploration and
Light-nuclei Astrophysics , cmb short = CMB , long = Cosmic Microwave
Background , sed short = SED , long = spectral energy distribution , mhd short
= MHD , long = magnetohydrodynamical , dof short = dof , long = degree of
freedom , cco short = CCO , long = central compact object , first-style =
default lmc short = LMC , long = Large Magellanic Cloud , smc short = SMC ,
long = Small Magellanic Cloud , hess short = H.E.S.S. , long = High Energy
Stereoscopic System , first-style = default snr short = SNR , long = supernova
remnant , pwn short = PWN , short-plural = e , long = pulsar wind nebula ,
long-plural = e , sn short = SN , short-plural = e , long = supernova , long-
plural = e , first-style = default nw short = NW , long = northwest , first-
style = default hxc short = HXC , long = hard X-ray component , first-style =
default cr short = CR , long = cosmic ray , psf short = PSF , long = point
spread function , hpd short = HPD , long = half power diameter , fwhm short =
FWHM , long = full width of half maximum , pic short = PIC , long = particle-
in-cell , tag = numerical , cxb short = CXB , long = Cosmic X-ray Background ,
grxe short = GRXE , long = Galactic Ridge X-ray Emission , pa short = PA ,
long = Positional Angle , dsa short = DSA , long = diffusive shock
acceleration , patch / longtable = false

##  1 Introduction

The origin of high-energy cr remains one of the most important topics in high-
energy and astroparticle physics. CRs up to the knee ($\sim$3 PeV) have been
widely believed to be produced in snr in the Galaxy through the diffusive
shock acceleration mechanism (e.g., Vink, 2011). One of the issues in this SNR
paradigm is that most of the snr detected in the gamma-ray range have spectra
with exponential cutoff of at most a few tens of TeV, which failed to give a
decisive answer to whether SNR shocks are capable of accelerating particles up
to the knee —PeVatrons (e.g., Funk, 2015). However, a remarkable discovery was
made by the High Energy Stereoscopic System (H.E.S.S.) collaboration:
Relatively hard spectra without any sign of cutoff were obtained from the
Galactic center (HESS Collaboration et al., 2016) and the unidentified source
HESS J1641$-$463 (Abramowski et al., 2014a), making them as the first
candidates for PeVatron sources. Recently, (sub-) PeV gamma-ray astronomy has
begun by extensive air shower arrays (EASA), such as HAWC, Tibet AS$\gamma$,
and LHAASO. These observations are accelerating the study of PeVatrons.

In this paper, we would like to revive the interest in the primary PeVatron
candidate, HESS J1641$-$463111Note that HESS J1641$-$463 is located at a
position where LHAASO and other EASAs on the north hemisphere cannot observe.
(Abramowski et al., 2014a). Since it is close to another gamma-ray bright
source HESS J1640$-$465 (Abramowski et al., 2014b), the confusion had been
preventing the detection of HESS J1641$-$463 until recently. Thanks to the
improved H.E.S.S. point spread function, HESS J1641$-$463 was discovered in
the tail of HESS J1640$-$465 with a slight spatial extension up to $\sim$3′
(Abramowski et al., 2014a; H. E. S. S. Collaboration et al., 2018). HESS
J1641$-$463 spatially coincides with the snr G338.5$+$0.1, which has roughly a
circular morphology with a diameter of 5′–9′ in the radio band (Whiteoak &
Green, 1996), a distance of $\approx$11 kpc (Kothes & Dougherty, 2007), and an
estimated age of 1.1–17 kyr (Abramowski et al., 2014a). HESS J1641$-$463 has
one of the hardest gamma-ray spectra with $\Gamma=2.07\pm 0.11_{\rm stat}\pm
0.20_{\rm sys}$ in 0.64–100 TeV without any indication of a cutoff, implying
that the cutoff energy of accelerated particles is higher than 100 TeV at 99%
C.L. (Abramowski et al., 2014a). Thus, HESS J1641$-$463 was proposed to be a
primary candidate for a PeVatron.

The origin of gamma-ray emission in the HESS J1641$-$463 and HESS J1640$-$465
system remains unsettled. In the GeV gamma-ray band, there exist counterparts
of HESS J1641$-$463 that has a point-like morphology and log-parabola spectrum
and of HESS J1640$-$465 that is characterized by
$\sim$3$.\mkern-4.0mu^{\prime}$2 spatial extension and power-law spectrum with
$\Gamma=1.7$ (Mares et al., 2021). As discussed by Mares et al. (2021), there
are a couple of scenarios for this complicated system, such as (1) hadronic
gamma-ray radiation (snr or illuminated molecular cloud), (2) pulsar, or (3)
PWN for the emission of HESS J1641$-$463\. None of them, however, are
conclusive for HESS J1641$-$463, while HESS J1640$-$465 is likely originated
from a PWN (e.g., Abdelmaguid et al., 2023).

The spectrum without a cutoff generally prefers a hadronic gamma-ray
production over leptonic emission (i.e., ic scattering of the ambient photons
by accelereated electrons) because the Klein-Nishina effect significantly
reduces the gamma-ray flux and inevitably makes a spectral cutoff at
$\gtrsim$10 TeV. The hadronic gamma-ray emission is attributed to decay of
neutral pions produced in interactions of cr protons with ambient protons. The
same proton-proton interactions produce charged pions, which decay into muons
and electrons/positrons in turn. These electrons and positrons are often
referred to as secondary electrons. If the energy of the parent proton is
sufficiently high, the secondary electrons are also energetic enough to emit
X-ray photons via synchrotron radiation, which can be potentially measured
with X-ray observations. Although there are some dedicated studies to predict
the secondary synchrotron component (e.g., Aharonian, 2004, 2013; Huang et
al., 2020), the emission has not been detected to date. In this paper, we aim
at searching for the secondary synchrotron emission in the X-ray data of HESS
J1641$-$463 and investigating how the synchrotron radiation from the secondary
electrons depends on physical parameters. If detected, we show that the X-ray
observation would be a new diagnostic tool to determine the maximum energy of
accelerated protons, while only lower limit ($>$100 TeV) is obtained by the
current gamma-ray observation.

We present X-ray observations of HESS J1641$-$463 in Section 2. Section 3
describes modeling of the broadband sed, including calculation of the
synchrotron radiation from secondary electrons. The discussion and conclusion
are presented in Sections 4 and 5, respectively.

##  2 X-ray analysis

This section describes X-ray observations and analysis of the gamma-ray source
HESS J1641$-$463\. In the soft X-ray band ($\lesssim 10$ keV), on-axis
observations pointing at this source have not been performed. There are
archival observations with Chandra (Obs IDs 11008 and 12508) and XMM-Newton
(Obs ID 0302560201), which partially cover the gamma-ray extent of HESS
J1641$-$463\. We made use of these Chandra data in this paper.222We do not use
the XMM-Newton observation because of its large ($\sim 14$′) off-axis
pointing. In addition to the soft X-ray data, we also used newly retrieved
hard X-ray data by NuSTAR (Obs ID 40401004002). Table 1 summarizes information
on these X-ray data analyzed in this paper.

The Chandra data were processed using CALDB version 4.9.6 in CIAO version
4.14. The NuSTAR data were calibrated and screened by using nupipeline of
NuSTAR Data Analysis Software (NuSTARDAS version 2.1.1 with CALDB version
20220118) included in HEAsoft version 6.29. We used the strictest mode (i.e.,
SAAMODE$=$STRICT and TENTACLE$=$YES cut) for screening the NuSTAR data. We
present the analysis and results of X-ray images and spectra in Sections 2.1
and 2.2, respectively.

Table 1:  Dataset of X-ray observations  Satellite | Target name | Obs ID | Effective exposure | Date | RA | Dec | Roll angle
---|---|---|---|---|---|---|---
|  |  | (ks) |  | (deg) | (deg) | (deg)
Chandra | GLIMPSE 81 | 11008 | 39.6 | 2010-06-19 | 250.1 |  $-$46.4 | 321.7
Chandra | Norma Region | 12508 | 18.5 | 2011-06-06 | 250.2 |  $-$46.5 | 342.2
NuSTAR | HESS J1641$-$463 | 40401004002 | 81.6 | 2018-10-11 | 250.2 |  $-$46.3 | 357.8

###  2.1 Image

We used fluximage to produce Chandra flux images. For NuSTAR images, we
generated count maps and exposure maps by Xselect and nuexpomap, respectively,
and then produced flux images by dividing the count maps by the exposure maps.
Background was not taken into account for the X-ray flux images.

Figure 1 shows flux images with Chandra in the 0.5–7 keV band, and Figure 2
shows flux images with NuSTAR in the 3–20 and 20–79 keV bands. There is no
apparent counterpart of HESS J1641$-$463 (see Figures 1 and 2). Another
source, stellar cluster candidate Mercer 81, clearly appeared in both the
Chandra and NuSTAR images. The Chandra images contain the other point-like
sources, reported by Abramowski et al. (2014a).

NuSTAR data are sometimes contaminated by nearby bright sources located
outside the fov, known as stray-light contamination. As shown in Figure 2,
unfortunately our observation was heavily contaminated by stray-light sources,
since the target region is crowded in the Galactic plane with a low latitude
of $b=0\hbox{$.\\!\\!^{\circ}$}1$. The FPMA image included stray light from AX
J1631.9$-$4752 and Big Dipper, while the FPMB image was contaminated by stray
light from 2MASS J16480656$-$4512068 (see the panels (a) and (d) in Figure 2).
Because the TeV gamma-ray emitting region was entirely overwrapped with the
stray-light component in the FPMB image, it would be better to use local
background in the same fov for estimation of the background rather than using
Nuskybgd (Wik et al., 2014). It should be noted that we checked consistency
between the two background methods. For the spectral analysis in Section 2.2,
the stray-light region of AX J1631.9$-$4752 in the FPMA data was excluded, and
we extracted the background in the same region contaminated by stray-light
2MASS J16480656$-$4512068 in the FPMB data.

Figure 1:  Flux images of HESS J1641$-$463 in 0.5–7 keV with Chandra ObsIDs
11008 (b), 12508 (c), and merged (a). (a): The contour indicates the
significance map of H.E.S.S. (Abramowski et al., 2014a). The yellow cross and
circle show positions of HESS J1641$-$463 and HESS J1640$-$465 obtained by the
Fermi-LAT data (Mares et al., 2021), respectively. (b) and (c): The large
green circle shows the TeV gamma-ray extent of HESS J1641$-$463 with its
radius of 3′. Source and background regions for spectral analysis are
indicated by green solid and green dashed lines, respectively.  Figure 2:
Left (a and d): 3–20 keV count maps of NuSTAR, shown with stray-light sources,
the contour of H.E.S.S. significance map (Abramowski et al., 2014a), and a
yellow cross of the best-fit position of the GeV gamma-ray source (Mares et
al., 2021). Middle (b and e): Flux images with NuSTAR in 3–20 keV. Right (c
and f): Flux images with NuSTAR in 20–79 keV. The FPMA and FPMB images are
respectively shown in the top and bottom panels. Source and background regions
for spectral analysis are indicated by green solid and green dashed lines,
respectively.

###  2.2 Spectrum

This section presents the spectral analysis of the Chandra and NuSTAR data. We
used specextract and nuproducts with ‘extended = yes’, respectively, to
produce Chandra and NuSTAR spectra. Xspec (Arnaud, 1996, version 12.12.0) was
used in spectral fitting. Errors are shown in 1$\sigma$ confidence level.
Source and background regions are shown in Figure 1 and Figure 2.

Besides HESS J1641$-$463, we first conducted spectral analysis of Mercer 81,
which was clearly detected by both Chandra and NuSTAR, and thus it is helpful
to check consistency between the Chandra and NuSTAR analyses. For the Chandra
data, the source spectrum was extracted from a circle with a center at (RA,
Dec) $=(16\hbox{${}^{\rm h}$}40\hbox{${}^{\rm m}$}29\hbox{$.\\!\\!^{\rm
s}$}4,-46\hbox{${}^{\circ}$}23\hbox{${}^{\prime}$}32\hbox{$.\\!\\!^{\prime\prime}$}1)$
and a radius of 0$.\mkern-4.0mu^{\prime}$9, while the background region was
taken from an annulus with 1′–2′. We used only the data with ObsID 11008,
which fully covered the entire emission of Mercer 81. For the NuSTAR data, the
source region was the same as the Chandra data, while the background was
extracted from an annulus with
1$.\mkern-4.0mu^{\prime}$5–2$.\mkern-4.0mu^{\prime}$5\. The obtained spectra
of Mercer 81 are shown in Figure 3. We fit the spectra with two models;
nonthermal (i.e., power law plus Gaussian that accounts for $\sim$6 keV line
emission) and thermal (apec) models (Table 2). For both models, absorption was
taken into account by using the TBabs model (Wilms et al., 2000). When fitting
with the nonthermal model, we obtained the best-fit parameters of
$\hbox{$N_{\rm H}$}=(17\pm 1)\times 10^{22}~{}{\rm cm}^{-2}$, $\Gamma=3.9\pm
0.2$, and Gaussian line with center energy of $6.66\pm 0.03$ keV,
$\sigma=(9.3\pm 3.7)\times 10^{-2}$ keV, and normalization of $(3.8\pm
0.6)\times 10^{-6}$ photons cm-2 s-1. The fitting result with the thermal
model is as follows: $\hbox{$N_{\rm H}$}=(14\pm 1)\times 10^{22}~{}{\rm
cm}^{-2}$, $kT=2.1\pm 0.2$ keV, and the abundance of $1.0\pm 0.2$. The thermal
model gave a smaller value of $\chi^{2}$: $\chi^{2}$ was 267 (d.o.f. 229) and
258 (d.o.f. 231) for the nonthermal and thermal models, respectively. We found
that the normalization of the Chandra and NuSTAR spectra was in good
agreement, as shown in Figure 3, thus our estimation of background (local
background method) for the NuSTAR data is reliable to some extent.

Table 2:  Fitting results of Mercer 81  | Nonthermal | Thremal
---|---|---
Parameter | (PL+Gauss) | (Apec)
$N_{\rm H}$ (1022 cm-2) | 17 $\pm$ 1 | 14 $\pm$ 1
$\Gamma$ / $kT$ (keV) | 3.9 $\pm 0.2$ | 2.1$\pm 0.2$
$F_{2-10}$ (10-13 ${\rm erg}~{}{\rm cm}^{-2}~{}{\rm s}^{-1}$) | 2.9 ${}^{+0.2}_{-0.5}$ | 2.9 ${}^{+0.2}_{-0.4}$
Gaussian line energy (keV) | 6.66 $\pm$ 0.03 | —
Gaussian line width (keV) | $(9.3\pm 3.7)\times 10^{-2}$ | —
Gaussian normalization (photons cm-2 s-1) | $(3.8\pm 0.6)\times 10^{-6}$ | —
Abundance | — | 1.0 $\pm$ 0.2
$\chi^{2}$ (dof) | 267 (229) | 258 (231)

The source region of HESS J1641$-$463 was set to a circle with a center at
(RA, Dec) $=(16\hbox{${}^{\rm h}$}41\hbox{${}^{\rm m}$}2\hbox{$.\\!\\!^{\rm
s}$}1,\
-46\hbox{${}^{\circ}$}18\hbox{${}^{\prime}$}13\hbox{$.\\!\\!^{\prime\prime}$}0)$
and a radius of 3′, which is comparable to the TeV gamma-ray extent
(Abramowski et al., 2014a). For the Chandra data, the background was extracted
from an annulus region of
3$.\mkern-4.0mu^{\prime}$3–4$.\mkern-4.0mu^{\prime}$5, centered at the
aforementioned position of HESS J1641$-$463\. For the NuSTAR data, the
background was taken from an annulus region of
3$.\mkern-4.0mu^{\prime}$5–4$.\mkern-4.0mu^{\prime}$1 for both FPMA and FPMB.
We show the spectra of HESS J1641$-$463 in Figure 4. In the case of a partial
coverage, we scaled the obtained normalization to the TeV gamma-ray size of
$r=3$′ on the assumption of a uniform distribution. This normalization scaling
was applied to the following data: the Chandra data with ObsID 11008, where
the region outside the fov (46% of the full coverage) was masked, and the
NuSTAR FPMA data, where the region contaminated by stray-light (36%) was
excluded.

Since there was no apparent X-ray signal from the HESS J1641$-$463 region, we
assumed a fitting model of an absorbed power law and fixed a column density to
the Galactic H I value of $\hbox{$N_{\rm H}$}=2\times 10^{22}~{}{\rm cm}^{-2}$
(Dickey & Lockman, 1990; Kalberla et al., 2005; Mares et al., 2021) and a
photon index to $\Gamma=2$. On this assumption, 2$\sigma$ upper limits were
derived (Table 3). We checked how the choice of the $N_{\rm H}$ value depends
the flux: the normalization of the NuSTAR spectrum, which is not largely
affected by the absorption, remains within 50% in a range of $N_{\rm H}$ from
$0.7\times 10^{22}~{}{\rm cm}^{-2}$ to 20 $\times 10^{22}~{}{\rm cm}^{-2}$,
while that of Chandra is consistent within a factor of 3 in the same range.
The flux with Chandra ObsID 12508 in the 2–10 keV energy band was $6.3\times
10^{-13}$ ${\rm erg}~{}{\rm cm}^{-2}~{}{\rm s}^{-1}$, which is reconciled with
that with NuSTAR of (6.2–7.1)$\times 10^{-13}$ ${\rm erg}~{}{\rm
cm}^{-2}~{}{\rm s}^{-1}$. The scaled flux obtained with Chandra ObsID 11008 is
$2.8\times 10^{-13}$ ${\rm erg}~{}{\rm cm}^{-2}~{}{\rm s}^{-1}$, which is in
agreement with $2.31\times 10^{-13}$ ${\rm erg}~{}{\rm cm}^{-2}~{}{\rm
s}^{-1}$ in the literature (Mares et al., 2021). It should be noted that these
values are all consistent within the systematic uncertainty caused by the
choice of $N_{\rm H}$.

We evaluated X-ray detection significance of HESS J1641$-$463 as follows.
Background-subtracted count rates are $(1.26\pm 0.48)\times 10^{-2}$ cts s-1,
$(1.87\pm 1.37)\times 10^{-3}$ cts s-1, and $(6.19\pm 2.16)\times 10^{-3}$ cts
s-1 for Chandra, NuSTAR FPMA, and FPMB, respectively. This implies
1.4–2.9$\sigma$ significance above statistical fluctuation. We also found that
$\chi^{2}$ reduced from 501 (d.o.f. 435) to 457 (d.o.f. 434) when fitting the
Chandra \+ NuSTAR spectra with the normalization being fixed to 0 and free,
respectively. This gave us F-test statistic value of 15.7 and probability of
$1.38\times 10^{-4}$, corresponding to a 3.5$\sigma$ detection. Because the
detection significance was as low as $\sim 3\sigma$ and there was no apparent
counterpart seen in the images, we conservatively adopt the upper limits in
this paper.

Table 3:  Fitting results of HESS J1641$-$463

Region | Size | Detector | $F_{2-10}$ | $F_{10-20}$ | Scaled $F_{2-10}$ | Scaled $F_{10-20}$
---|---|---|---|---|---|---
Stray light excluded | 18.1 | NuSTAR FPMA | 4.0 | 1.7 | 6.2 | 2.7
Full | 28.3 | NuSTAR FPMB | 7.1 | 3.0 | 7.1 | 3.0
Full (ObsID 12508) | 28.3 |  Chandra ACIS-S | 5.1 | — | 6.3 | —
South half (ObsID 11008) | 15.35 |  Chandra ACIS-I | 1.5 | — | 2.8 | —

Note. —

Size is in units of arcmin2.

Flux is 2$\sigma$ upper limit and in units of $10^{-13}$ ${\rm erg}~{}{\rm
cm}^{-2}~{}{\rm s}^{-1}$. $F_{2-10}$ and $F_{10-20}$ indicate the flux upper
limit in the 2–10 keV and 10–20 keV bands, respectively.

‘Scaled’ means the scaled value to the TeV gamma-ray extension with
$r=3\hbox{${}^{\prime}$}$.

Figure 3:  Chandra (shown in green) and NuSTAR (FPMA in black and FPMB in red)
spectra of Mercer 81, fitted with the thermal model (left) and the nonthermal
model (right). The fitting results are shown in Table 2.  Figure 4:  Chandra
(shown in green) and NuSTAR (FPMA in black and FPMB in red) spectra of HESS
J1641$-$463, fitted with the absorbed power-law model.

##  3 Broadband SED modeling

Figure 5 shows the broadband sed of HESS J1641$-$463, including the X-ray
upper limits in this paper as well as the radio and gamma-ray flux in the
literature.

Figure 5:  sed of HESS J1641$-$463\. All data in the literature (H. E. S. S.
Collaboration et al., 2018; Abramowski et al., 2014a; Mares et al., 2021;
Green, 2019) are also shown.

SNR G338.5$+$0.1, which is a counterpart candidate of HESS J1641$-$463, was
identified as a radio SNR by Shaver & Goss (1970) in an investigation of
extended radio sources with the Parkes telescope at 5000 MHz and the Molonglo
radio telescope at 408 MHz. However, because of the complex region in the
vicinity, the association is uncertain, and the SNR itself has an undefined
shape, preventing precise determination of its radio flux and size. It was
reported that G338.5$+$0.1 has a diameter of 5′–9′ and $\sim$12 Jy at 1 GHz
(Green, 2019; Whiteoak & Green, 1996). Although the radio flux of the entire
SNR is roughly 12 Jy at 1 GHz, Mares et al. (2021) calculated the radio
emission directly associated with the gamma-ray emitting region of HESS
J1641$-$463 (i.e., the size in FWHM of 3′) based on the Southern Galactic
Plane Survey (SGPS) data (Haverkorn et al., 2006), resulting in an upper limit
of $\sim$2 Jy at 1.4 GHz. The former value is used in this paper, otherwise
mentioned. Note that the usage of the radio flux affects only a leptonic
modeling, not a hadronic component that is our main point in this paper.

GeV gamma rays are detected with Fermi-LAT in the region of HESS J1641$-$463
(Mares et al., 2021). The GeV gamma-ray source is point-like and has a
spectrum with a curvature, implying that it would likely be a pulsar. Since
there is no counterpart of the source at the other wavelength, we do not
include the GeV gamma-ray emission in our models.

In the H.E.S.S. Galactic plane survey (HGPS), the obtained spectrum of HESS
J1641$-$463 was steeper than the one from the analysis dedicated for HESS
J1641$-$463 by Abramowski et al. (2014a): the photon index was 2.47$\pm$0.11
(H. E. S. S. Collaboration et al., 2018). See Figure 5 for comparison of these
two spectra. In this paper, we mainly use the spectrum of Abramowski et al.
(2014a).

The upper limits of the X-ray flux obtained in Section 2, combined with the
gamma-ray and radio flux in the literature, enable us to model the broadband
sed (Figure 5) in order to interpret the origin of the very-high-energy gamma-
ray emission. In the case of a leptonic origin, ic scattering of high-energy
electrons is accounted for the gamma-ray radiation, and synchrotron radiation
from the same electrons produce radio and X-ray photons. In a hadronic
scenario, decays of $\pi^{0}$ produced in proton-proton interactions are
applied to the gamma-ray data. Other channels in the proton-proton
interactions yield secondary electrons and positrons, which are potentially
energetic enough to produce synchrotron radiation in the X-ray energy band. We
summarize derivation of the secondary electrons in Section 3.1 and apply it to
the observation in Section 3.2.

###  3.1 Secondary electrons in proton-proton interaction

When cr protons collide with protons in ambient medium (proton-proton
interaction), pions, $\pi^{0}$ and $\pi^{\pm}$, are produced. They decay into
the following products:

$\displaystyle\pi^{0}$ $\displaystyle\rightarrow$ $\displaystyle 2\gamma$ (1)
$\displaystyle\pi^{\pm}$ $\displaystyle\rightarrow$
$\displaystyle\mu^{\pm}+\nu_{\mu}(\bar{\nu}_{\mu})$ (3)
$\displaystyle\mu^{\pm}\rightarrow
e^{\pm}+\nu_{e}(\bar{\nu}_{e})+\bar{\nu}_{\mu}(\nu_{\mu})$

$\pi^{0}$ decay yields gamma-ray photons (Equation 1). Electrons and positrons
($e^{\pm}$) are also produced from $\pi^{\pm}$ through Equations 3 and 3, and
these electrons and positrons are referred to as secondary electrons
hereafter. The secondary electrons have $\sim$10% energy of the primary
protons. The cross section of the $\pi^{0}$-decay photons,
$\frac{d\sigma(E_{p},E_{\gamma})}{dE_{\gamma}}$, is well formulated in the
literature (e.g., Kelner et al., 2006; Kafexhiu et al., 2014), and
Geant4-based cross section in Kafexhiu et al. (2014) is adopted for in this
paper. We make use of a library of
LibppGam333https://github.com/ervinkafex/LibppGam for calculating the gamma-
ray cross section. Kelner et al. (2006) derived the spectral ratio of the
products of proton-proton interactions, $F_{i}(E_{i}/E_{p},E_{p})$, where the
subscript $i$ indicates gamma rays, electrons, or neutrinos and $E_{p}$ is the
energy of the primary proton. The spectrum for the secondary electrons,
$F_{e}(E_{e}/E_{p},E_{p})$, is given by Equation (62) in Kelner et al. (2006).
Thus, the cross section of the secondary electrons is described by

$\displaystyle\frac{d\sigma(E_{p},E_{e})}{dE_{e}}=\frac{d\sigma(E_{p},E_{\gamma})}{dE_{\gamma}}\frac{F_{e}(\frac{E_{e}}{E_{p}},E_{p})}{F_{\gamma}(\frac{E_{\gamma}}{E_{p}},E_{p})}.$
(4)

Figure 6 compares the cross section of the secondary electrons by Equation 4
with that of the gamma rays in Kafexhiu et al. (2014).

Figure 6:  Differential cross section, $\frac{d\sigma}{dE}$, of gamma-rays
(solid lines) and secondary $e^{\pm}$ (dashed lines) for
$E_{e}=E_{\gamma}=100$ GeV, 1 TeV, and 10 TeV.

Following Equation (25) in Kafexhiu et al. (2014), the energy distribution of
the secondary electrons can be expressed as

$\displaystyle Q_{e}(E_{e},t)=nc\int
dE_{p}\beta\epsilon(E_{p})\frac{d\sigma(E_{p},E_{e})}{dE_{e}}\frac{dN_{p}}{dE_{p}}.$
(5)

Here, $n$ is the ambient density, $c$ the speed of light, and
$\frac{dN_{p}}{dE_{p}}$ the spectrum of protons. Technically, we made use of a
library of GAMERA444http://libgamera.github.io/GAMERA/docs/documentation.html
to solve Equation 6. We multiplied the cross section by the so-called nuclear
enhancement factor $\epsilon(E_{p})$ to account for proton-nucleus
interactions (Kafexhiu et al., 2014). In this paper, the nuclear enhancement
factor was taken from Sihver et al. (1993).

The secondary electrons, injected with Equation 5, suffer radiation cooling.
Therefore, we took the cooling effect into consideration in the following
equation of time evolution:

$\displaystyle\frac{\partial N_{e}(E_{e},t)}{\partial
t}=\frac{\partial}{\partial
E_{e}}\left[b(E_{e})N_{e}(E_{e},t)\right]+Q_{e}(E_{e},t),$ (6)

where $N_{e}$ is the spectrum of electrons, and $b(E)$ indicates the energy
loss rate. Although $b(E)$ consists of synchrotron radiation, ic scattering,
Bremsstrahlung, and Coulomb collision, energy loss due to synchrotron
radiation is dominant for $>$1 GeV electrons for a parameter range we consider
in the paper. In the case of synchrotron radiation, the break energy is
derived by equating the radiation timescale to the age of the source ($T$),

$E_{b}=125~{}\mathrm{TeV}~{}\left(\frac{T}{\rm
kyr}\right)^{-1}\left(\frac{B}{10~{}\mu\mathrm{G}}\right)^{-2},$ (7)

where $B$ indicates the strength of the magnetic field.

Figure 7 shows an example of the model using parameters summarized in Table 4.
We assumed a power-law spectrum with an exponential cutoff for primary
protons;

$\displaystyle\frac{dN_{p}}{dE_{p}}=K\left(\frac{E_{p}}{E_{\rm
ref}}\right)^{-s_{p}}\exp\left[-\left(\frac{E_{p}}{E_{c}}\right)\right],$ (8)

where $K$ is the normalization, $E_{\rm ref}$ (=1 TeV) is the reference
energy, $s_{p}$ is the spectral index, and $E_{c}$ is the cutoff energy.

In Figure 7 (right), we show an example of our radiation models, overlaid with
the observation data of the TeV gamma ray (Abramowski et al., 2014a), our
X-ray upper limits, and the radio flux (Green, 2019; Mares et al., 2021). We
calculated radiation spectra of $\pi^{0}$-decay gamma rays from the primary
protons and synchrotron X-rays from the secondary electrons by using
Naima555https://naima.readthedocs.io/en/latest/ (Zabalza, 2015). It should be
noted that the ic component from the secondary electrons is smaller than the
pion-decay emission by two orders of magnitude.

Figure 7:  Baseline model of particles (left) and photons (right) with
parameters listed in Table 4. Left: the injection spectrum of the protons is
shown with the blue line, the injection spectrum of the secondary electrons
(Equation 5 multiplied by the age $T$) is indicated by the pink line, and the
spectrum of the cooled secondary electrons is illustrated by the red line.
Right: See Figure 5 for the observational data.

###  3.2 Application to HESS J1641-463

We adopt the hadronic model to the sed of HESS J1641$-$463\. The hadronic
interpretation is supported by the presence of a dense molecular cloud with
$n\sim 100$ ${\rm cm}^{-3}$ observed by NANTEN (Abramowski et al., 2014a).
Computation of our model requires parameters of $n,~{}d$ (distance),
$T,~{}B,~{}s_{p},~{}E_{c,p}$, and $W_{p}$. Although these parameters are
poorly constrained, we defined a baseline model as
$(n,~{}d,~{}T,~{}B,~{}s_{p},~{}E_{c,p})=$ (100 ${\rm cm}^{-3}$, 11 kpc, 5 kyr,
300 $\mu\mathrm{G}$, 2.0, 1 PeV), listed in Table 4. From the literature, we
adopted the ambient density of 100 ${\rm cm}^{-3}$, the distance of 11 kpc,
and the age of 5 kyr (Dickey & Lockman, 1990; Kalberla et al., 2005;
Abramowski et al., 2014a). We set $B$ = 300 $\mu\mathrm{G}$, assuming that the
gamma rays are produced in a molecular cloud where the magnetic field could be
from tens to hundreds of $\mu\mathrm{G}$ (see Figure 8 for dependency on these
values). $s_{p}=2$ and $E_{c,p}$ = 1 PeV are obtained to reproduce the TeV
gamma-ray spectrum (Abramowski et al., 2014a). First, we fit the TeV gamma-ray
data with the $\pi^{0}$-decay model to determine the normalization of protons,
$K$ in Equation 8, which can be transferred to the total energy of protons
($W_{p}$). $W_{p}$ is 1.5$\times 10^{48}$ erg for the baseline model. Then, we
calculated the secondary synchrotron component with the parameters in Table 4.
The result of the baseline model is shown in Figure 7.

In order to explore how the secondary synchrotron emission depends on the
parameters, we performed the aforementioned computations with the parameters
different from the baseline model. We changed $T$ in a range of 1–11 kyr, $B$
in 100–2000 $\mu\mathrm{G}$, $s_{p}$ in 1.5–2.5, and $E_{c,p}$ in 100–3000
TeV. The results are shown in Figure 8. The age, $T$, has an effect only on
the break energy of the electrons (i.e., the energy where the age and the time
scale of sycnhrotron cooling are balanced in Equation 7) and does not largely
affect the model in the X-ray band (Figure 8a). The magnetic field, $B$,
slightly modifies the synchrotron model (Figure 8b). However, the choice of
$B$ does not largely change the normalization of the synchrotron radiation,
since large $B$ makes the energy loss large and the electron spectrum steep,
but, at the same time, makes the synchrotron power high, canceling out the
steep spectrum of electrons. The spectral shape of the primary protons,
$s_{p}$ and $E_{c,p}$, largely change the X-ray spectrum (Figure 8c and d). It
is worth noting that $n$ and $W_{p}$ (or $K$) are degenerated, and they do not
affect the X-ray spectrum.

Table 4:  Baseline model parameter  Parameter |  | Value
---|---|---
Ambient density | $n$ | 100 ${\rm cm}^{-3}$
Distance | $d$ | 11 kpc
Age | $T$ | 5 kyr
Magnetic field | $B$ | 300 $\mu\mathrm{G}$
Proton spectral index | $s_{p}$ | 2.0
Proton cutoff energy | $E_{c,p}$ | 1 PeV
Proton total energy |  $W_{p}~{}(E>1$ TeV) | 1.5$\times 10^{48}$ erg
Figure 8:  Models of Table 4, but changing the age $T$ (a), the magnetic field
$B$ (b), the spectral index $s_{p}$ (c), and the maximum energy $E_{c,p}$ (d).

##  4 Discussion

###  4.1 Gamma-ray origin of HESS J1641-463

####  4.1.1 Hadronic scenario

The hadronic gamma rays trace a molecular cloud (or dense gas) illuminated by
cosmic rays. There are two cases proposed: (1) SNR-cloud interaction —
particles are accelerated at the SNR shock and interacts with the molecular
clouds in the immediate vicinity of the SNR and (2) CR-cloud interaction —
accelerated particles have escaped the acceleration site, diffusively
propagated, and hit nearby molecular clouds. In the former case, the higher-
energy CRs can penetrate into clouds deeply, resulting in the relatively hard
CR spectrum (e.g., Inoue et al., 2012; Gabici & Aharonian, 2014; Celli et al.,
2019; Higurashi et al., 2020). The SNR shock at such dense environment would
produce thermal X-rays. In the CR-cloud scenario, the spectrum of the escaping
particles generally depends on assumptions, such as a time from the escape and
a distance from the source and the cloud (e.g., Aharonian & Atoyan, 1996;
Gabici & Aharonian, 2007; Ohira et al., 2010). Particularly, a self-consistent
calculation for the HESS J1641$-$463/HESS J1604$-$465 system was conducted in
detail by Tang et al. (2015), showing that the distance affects the low-energy
part of the spectrum of the escaping particles.

Since all of the models are lower than the X-ray upper limit obtained in this
paper, we cannot place tight constraints by the X-ray observations at this
time. As illustrated in Figure 8, we found that the synchrotron radiation from
the secondary electrons is variable in the X-ray energy band, depending
strongly on $s_{p}$ and $E_{c,p}$ and slightly on $B$. On the other words, a
precise measurement of the secondary component can constrain the proton
spectrum, which will be possible with the future missions (Section 4.2).

The hard TeV gamma-ray spectrum indicates particle acceleration up to the
multi-TeV range, $E_{c,p}>$100 TeV (Abramowski et al., 2014a, and Figure 8).
Given the SNR age of $\sim$5 kyr, the shock velocity should slow down to tens
of ${\rm km}~{}{\rm s}^{-1}$, making it unlikely to produce such energetic
particles at present. Thus, it implies the acceleration had been achieved in
the past in the SNR-cloud scenario. The CR-cloud scenario, on the other hand,
does not require such on-site and present acceleration. In both cases, these
high energy CRs can be hardly confined in the SNR shell and escape the shock
first.

Our modeling revealed that the total proton energy, $W_{p}$ ($E>1$ TeV), is
roughly $\sim 10^{48}$ erg on the assumption of $n=100$ ${\rm cm}^{-3}$.
Typically, an SN explosion releases an energy of $E_{\rm SN}\sim 10^{51}$ erg,
and a fraction of the energy is converted into acceleration of cr (e.g.,
$E_{\rm CR}\sim 10^{50}$ erg, assuming $\sim$10% for the conversion factor).
$W_{p}\sim 10^{48}$ erg in HESS J1641$-$463 is smaller than the typical value
for SNRs. In the SNR-cloud scenario, this may imply the conversion fraction
from $E_{\rm SN}$ to $E_{\rm CR}$ is lower for this source. If this SNR is
indeed interacting with a molecular cloud, parts of it may be radiative, in
which case a lot of energy could have been radiated away already.
Alternatively, the small proton energy might support the escape scenario.

It should be noted that the lack of bright thermal X-ray emission, which is
expected for such SNR with dense gas ($n\sim 100$ ${\rm cm}^{-3}$), may
disfavor the SNR scenario. However, the SNR with the age of $\sim$5 kyr might
be too old to heat up the gas to keV because of the decelerated shock speed.
On the other hand, in the escape (CR-cloud) scenario, the accelerated
particles have already left the source, which naturally explains the absence
of the thermal emission. The existence of the thermal emission can be verified
by deep observation in the soft X-ray band in the future.

Here, we note the result when we use the steep spectrum obtained in HGPS (H.
E. S. S. Collaboration et al., 2018). With $n$ being fixed to 100 ${\rm
cm}^{-3}$, we derived $s_{p}=2.44\pm 0.13$, $E_{p}>$200 TeV, and
$W_{p}=(1.4\pm 0.17)\times 10^{48}$ erg. Using these best-fit parameters of
the protons and the others of the baseline model, the synchrotron radiation
from the secondary electrons became about one order of magnitude lower than
the model optimized for the harder one in Abramowski et al. (2014a).

####  4.1.2 Other scenarios

##### Leptonic scenario

The leptonic-dominated model consists of ic scattering and synchrotron
radiation from primary cr electrons. Assuming a cutoff power-law model of
electrons (Equation 8) and typical values for the photon fields of CMB, NIR,
and FIR (i.e., temperatures of 2.72 K, 30 K, and 3000 K, and energy densities
of 0.261, 0.5, and 1 eV cm-3, respectively (Zabalza, 2015)), fitting the
multiwavelength sed results in the spectral index ($s_{e}$) of $2.9\pm 0.1$,
the magnetic field of $2.5^{+0.8}_{-0.5}$ $\mu\mathrm{G}$, and the poorly
determined cutoff energy of $E_{c,e}>100$ TeV. Adopting the upper limit of the
radio flux derived from the SGPS data (Haverkorn et al., 2006; Mares et al.,
2021), $s_{e}$ and $E_{c,e}$ cannot be well constrained while $B$ should be
less than 2 $\mu\mathrm{G}$ to surpress the synchrotron radiation.

Because the TeV gamma-ray flux is compatible with the X-ray upper limits, the
magnetic field should be less than $\sim$3 $\mu\mathrm{G}$ in the leptonic
case. This B-field value is comparable to the typical value of the
interstellar magnetic field, implying that no shock compression and/or
amplification are required and that the leptonic case seems very unlikely.
Such a small value of the magnetic field may imply the TeV gamma-ray origin is
a pulsar halo. The possible presence of the pulsar is also supported from the
curved spectrum from the GeV gamma-ray point-like source, although there are
no counterparts of the pulsar at the other wavelengths. Furthermore, the
electron total energy ($W_{e}$ for $E>$1 TeV) is $(3.6\pm 0.6)\times 10^{47}$
erg. If the electron to proton ratio, $K_{\rm ep}$, is 0.01, the proton energy
would be $\sim 10^{49}$ erg, which is larger then the hadronic case. For the
leptonic component to be dominant, the ambient density should be $<$10 ${\rm
cm}^{-3}$, which conflicts with the observed value of 100 ${\rm cm}^{-3}$. In
conclusion, the leptonic-dominated case is disfavored, although it is not
excluded.

##### Mixed scenario (one-zone)

Here, we consider adding primary electrons that have the same spectral model
(Equation 8) with the protons, as one-zone model. Radiation cooling of
electrons is taken into account. In the ranges of $E_{p}=100$–1000 TeV,
$B=10$–1000 $\mu\mathrm{G}$, and $s=$2–2.5, no parameter set was found to be
in agreements with all the observations: (1) $K_{\rm ep}$ should be $<0.01$
for not exceeding X-ray ULs, but then the synchrotron flux becomes much lower
than the radio data, and (2) if we determine the normalization of the
electrons to reproduce the radio flux, the model exceeds the X-ray upper
limits or the Fermi-LAT data. $E_{c,p}$ (proton maximum energy) should be
$\gtrsim$100 TeV in order to reproduce the hard spectrum in the TeV gamma-ray
energy band, resulting in higher cutoff energy in the spectrum of the primary
synchrotron radiation, even though cooling is taken into consideration. In
conclusion, one-zone model is unlikely acceptable. Adopting the upper limit of
the radio flux derived from the SGPS data (Haverkorn et al., 2006; Mares et
al., 2021) does not largely change the aforementioned result.

##### Mixed scenario (two-zone)

In addition to the protons and the secondary electrons fixed to the baseline
model (Table 4), we add primary electrons with different spectral shape (i.e.,
the primary electron index of $s_{e}\neq s_{p}$, and the primary electron
cutoff energy of $E_{c,e}\neq E_{c,p}$). The magnetic field for the primary
electrons to emit synchrotron photons (labeled as $B_{e}$ hereafter) is
assumed to be independent of that of the secondary electrons (i.e., $B_{e}\neq
B$). After the synchrotron radiation from the primary electron is normalized
to the radio data, we searched for parameters ($E_{c,e}$ and $B_{e}$) with
which the total flux of all the radiation components does not exceed the
observations of the X-ray upper limits and the GeV emission from the point-
like source (Mares et al., 2021). Assuming $s_{e}=2.5$, $B_{e}$ should be
larger than 4 $\mu\mathrm{G}$ to suppress the IC component and $E_{c,e}$
should be less than $\sim$10 TeV not to exceed the X-ray upper limit. If we
adopt for $K_{\rm ep}$=0.01, $E_{c,e}$ and $B_{e}$ are constrained to be
$\lesssim$4 TeV and 20–40 $\mu\mathrm{G}$, respectively.

###  4.2 Future prospects

In the hadronic scenario, our models (Figure 8) suggest that the flux of the
secondary synchrotron component is $>1.8\times 10^{-14}$ ${\rm erg}~{}{\rm
cm}^{-2}~{}{\rm s}^{-1}$ in 2–10 keV and $>4.9\times 10^{-15}$ ${\rm
erg}~{}{\rm cm}^{-2}~{}{\rm s}^{-1}$ in 10–50 keV for $E_{c,p}>100$ TeV. This
flux level is sufficiently detectable with X-ray telescopes. In order to avoid
a contamination of the primary synchrotron radiation which generally has a
cutoff at the soft X-ray band, hard X-ray detectors would be needed. Proposed
missions, such as HEX-P666https://hexp.org/ (Madsen et al., 2019, 2023; Mori
et al., 2023; Reynolds et al., 2023) or FORCE (Mori et al., 2016) will be able
to measure the emission and determine the proton maximum energy by the X-ray
observation for the first time. Long-exposure observations with the existing
X-ray telescopes would also be helpful to search for the secondary
synchrotoron radiation, and we are planning an XMM-Newton observation.

Besides the synchrotron emission from the secondary electrons, neutrino is
also a smoking gun to evidence hadronic acceleration. Although the neutrino
amount from this single source would be much smaller than the detection
threshold of the existing neutrino observatories such as IceCube, neutrinos
accumulated by the hadronic PeVatron candidates may have a contribution to the
neutrino diffuse flux detected from the Galactic plane (ICECUBE COLLABORATION,
2023) as unresolved sources.

##  5 Conclusion

We analyzed newly taken NuSTAR data and the archival Chandra data of the
unidentified TeV gamma-ray source, HESS J1641$-$463, to search for the
synchrotron radiation from the secondary electrons in the hadronic
interpretation. No X-ray counterpart was detected in the gamma-ray extension,
and we obtained 2$\sigma$ upper limits of $\sim 6\times 10^{-13}$ ${\rm
erg}~{}{\rm cm}^{-2}~{}{\rm s}^{-1}$ in the 2–10 keV band and $\sim 3\times
10^{-13}$ ${\rm erg}~{}{\rm cm}^{-2}~{}{\rm s}^{-1}$ in the 10–20 keV band.
Combined with the published radio and gamma-ray spectra, we performed modeling
of the broadband sed. Although the X-ray upper limit could not constrain the
hadronic model, we found the flux of the secondary synchrotron component is
larger than $\sim 10^{-14}$ ${\rm erg}~{}{\rm cm}^{-2}~{}{\rm s}^{-1}$, which
would be detectable with a deep X-ray observation with existing telescopes or
with future mission.

We thank the anonymous referee for the helpful comments. We are greatful to
Kaya Mori for the fruitful discussion and Hiroya Yamaguchi, Hiroyasu Tajima,
and Benson Guest for the preparation of the X-ray proposals. This work has
made use of data from the NuSTAR mission, a project led by the California
Institute of Technology, managed by the Jet Propulsion Laboratory, and funded
by the National Aeronautics and Space Administration. This work is supported
by the NuSTAR Cycle 4 observation program. We appreciate the NuSTAR
Operations, Software, and Calibration teams for support with the execution and
analysis of these observations. This paper employs a list of Chandra datasets,
obtained by the Chandra X-ray Observatory, contained in
https://doi.org/10.25574/cdc.217 (catalog DOI:10.25574/cdc.217). This work was
supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant
Numbers JP22K14064 (N.T.), JP19H01936, and JP21H04493 (T.T.). S.S.H.
acknowledges support from the Natural Sciences and Engineering Research
Council of Canada (NSERC) through the Canada Research Chairs and the Discovery
Grants programs, from the Canadian Institute for Theoretical Astrophysics and
from the Canadian Space Agency.

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