HAWC Study of Very-High-Energy γ-ray Spectrum of HAWC J1844-034

C.D. Rho
Y. Son

0000-0003-0197-5646]A. Albert
Physics Division, Los Alamos National Laboratory, Los Alamos, NM, USA

0000-0001-8310-4486]C. Alvarez
Universidad Autónoma de Chiapas, Tuxtla Gutiérrez, Chiapas, México

0000-0002-4020-4142]D. Avila Rojas
Instituto de Física, Universidad Nacional Autónoma de México, Ciudad de Mexico, Mexico

0000-0002-2084-5049]H.A. Ayala Solares
Department of Physics, Pennsylvania State University, University Park, PA, USA

0000-0002-5529-6780]R. Babu
Department of Physics, Michigan Technological University, Houghton, MI, USA

0000-0003-3207-105X]E. Belmont-Moreno
Instituto de Física, Universidad Nacional Autónoma de México, Ciudad de Mexico, Mexico

0000-0003-0268-5122]M. Breuhaus
Max-Planck Institute for Nuclear Physics, 69117 Heidelberg, Germany

0000-0003-2158-2292]T. Capistrán
Instituto de Astronomía, Universidad Nacional Autónoma de México, Ciudad de Mexico, Mexico

0000-0002-8553-3302]A. Carramiñana
Instituto Nacional de Astrofísica, Óptica y Electrónica, Puebla, Mexico

0000-0002-6144-9122]S. Casanova
Institute of Nuclear Physics Polish Academy of Sciences, PL-31342 IFJ-PAN, Krakow, Poland

0000-0002-1132-871X]J. Cotzomi
Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, Puebla, Mexico

0000-0002-7747-754X]S. Coutiño de León
Department of Physics, University of Wisconsin-Madison, Madison, WI, USA

0000-0001-9643-4134]E. De la Fuente
Departamento de Física, Centro Universitario de Ciencias Exactase Ingenierias, Universidad de Guadalajara, Guadalajara, Mexico

0000-0002-2672-4141]D. Depaoli
Max-Planck Institute for Nuclear Physics, 69117 Heidelberg, Germany

0000-0001-8487-0836]R. Diaz Hernandez
Instituto Nacional de Astrofísica, Óptica y Electrónica, Puebla, Mexico

0000-0001-8451-7450]B.L. Dingus
Physics Division, Los Alamos National Laboratory, Los Alamos, NM, USA

0000-0002-2987-9691]M.A. DuVernois
Department of Physics, University of Wisconsin-Madison, Madison, WI, USA

0000-0003-2169-0306]M. Durocher
Physics Division, Los Alamos National Laboratory, Los Alamos, NM, USA

0000-0001-5737-1820]K. Engel
Department of Physics, University of Maryland, College Park, MD, USA

0000-0001-7074-1726]C. Espinoza
Instituto de Física, Universidad Nacional Autónoma de México, Ciudad de Mexico, Mexico

0000-0002-8246-4751]K.L. Fan
Department of Physics, University of Maryland, College Park, MD, USA

0000-0002-5387-8138]K. Fang
Department of Physics, University of Wisconsin-Madison, Madison, WI, USA

0000-0002-0173-6453]N. Fraija
Instituto de Astronomía, Universidad Nacional Autónoma de México, Ciudad de Mexico, Mexico

0000-0002-4188-5584]J.A. García-González
Tecnologico de Monterrey, Escuela de Ingeniería y Ciencias, Ave. Eugenio Garza Sada 2501, Monterrey, N.L., Mexico, 64849

0000-0002-5209-5641]M.M. González
Instituto de Astronomía, Universidad Nacional Autónoma de México, Ciudad de Mexico, Mexico

0000-0002-9790-1299]J.A. Goodman
Department of Physics, University of Maryland, College Park, MD, USA

S. Groetsch
Department of Physics, Michigan Technological University, Houghton, MI, USA

0000-0001-9844-2648]J.P. Harding
Physics Division, Los Alamos National Laboratory, Los Alamos, NM, USA

I. Herzog
Department of Physics and Astronomy, Michigan State University, East Lansing, MI, USA

0000-0002-1031-7760]J. Hinton
Max-Planck Institute for Nuclear Physics, 69117 Heidelberg, Germany

0000-0002-5447-1786]D. Huang
Department of Physics, Michigan Technological University, Houghton, MI, USA

0000-0002-5527-7141]F. Hueyotl-Zahuantitla
Universidad Autónoma de Chiapas, Tuxtla Gutiérrez, Chiapas, México

0000-0002-1432-7771]T.B. Humensky
NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA

0000-0002-3302-7897]P. Hüntemeyer
Department of Physics, Michigan Technological University, Houghton, MI, USA

0000-0003-4467-3621]V. Joshi
Erlangen Centre for Astroparticle Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, Germany

S. Kaufmann
Universidad Politecnica de Pachuca, Pachuca, Hgo, Mexico

0000-0002-2153-1519]J. Lee
University of Seoul, Seoul, Rep. of Korea

0000-0001-5516-4975]H. León Vargas
Instituto de Física, Universidad Nacional Autónoma de México, Ciudad de Mexico, Mexico

0000-0001-8825-3624]A.L. Longinotti
Instituto de Astronomía, Universidad Nacional Autónoma de México, Ciudad de Mexico, Mexico

0000-0003-2810-4867]G. Luis-Raya
Universidad Politecnica de Pachuca, Pachuca, Hgo, Mexico

0000-0001-8088-400X]K. Malone
Space Science and Applications Group, Los Alamos National Laboratory, Los Alamos, NM, USA

0000-0001-9052-856X]O. Martinez
Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, Puebla, Mexico

0000-0002-2824-3544]J. Martínez-Castro
Centro de Investigación en Computación, Instituto Politécnico Nacional, Ciudad de México, México

0000-0002-2610-863X]J.A. Matthews
Dept of Physics and Astronomy, University of New Mexico, Albuquerque, NM, USA

0000-0002-8390-9011]P. Miranda-Romagnoli
Universidad Autónoma del Estado de Hidalgo, Pachuca, Mexico

0000-0001-9361-0147]J.A. Morales-Soto
Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Mexico

0000-0002-1114-2640]E. Moreno
Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, Puebla, Mexico

0000-0002-7675-4656]M. Mostafá
Department of Physics, Pennsylvania State University, University Park, PA, USA

0000-0003-1059-8731]L. Nellen
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de Mexico, Ciudad de Mexico, Mexico

0000-0001-7099-108X]R. Noriega-Papaqui
Universidad Autónoma del Estado de Hidalgo, Pachuca, Mexico

0000-0002-9105-0518]L. Olivera-Nieto
Max-Planck Institute for Nuclear Physics, 69117 Heidelberg, Germany

0000-0002-5448-7577]N. Omodei
Department of Physics, Stanford University: Stanford, CA 94305–4060, USA

0000-0001-5998-4938]E.G. Pérez-Pérez
Universidad Politecnica de Pachuca, Pachuca, Hgo, Mexico

0000-0002-6524-9769]C.D. Rho
Department of Physics, Sungkyunkwan University, Suwon 16419, South Korea

0000-0003-1327-0838]D. Rosa-González
Instituto Nacional de Astrofísica, Óptica y Electrónica, Puebla, Mexico

0000-0001-6939-7825]E. Ruiz-Velasco
Max-Planck Institute for Nuclear Physics, 69117 Heidelberg, Germany

0000-0003-4556-7302]H. Salazar
Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, Puebla, Mexico

0000-0002-9312-9684]D. Salazar-Gallegos
Department of Physics and Astronomy, Michigan State University, East Lansing, MI, USA

0000-0001-6079-2722]A. Sandoval
Instituto de Física, Universidad Nacional Autónoma de México, Ciudad de Mexico, Mexico

0000-0001-8644-4734]M. Schneider
Department of Physics, University of Maryland, College Park, MD, USA

J. Serna-Franco
Instituto de Física, Universidad Nacional Autónoma de México, Ciudad de Mexico, Mexico

0000-0002-1012-0431]A.J. Smith
Department of Physics, University of Maryland, College Park, MD, USA

0000-0002-7214-8480]Y. Son
University of Seoul, Seoul, Rep. of Korea

0000-0002-1492-0380]R.W. Springer
Department of Physics and Astronomy, University of Utah, Salt Lake City, UT, USA

0000-0002-9074-0584]O. Tibolla
Universidad Politecnica de Pachuca, Pachuca, Hgo, Mexico

0000-0001-9725-1479]K. Tollefson
Department of Physics and Astronomy, Michigan State University, East Lansing, MI, USA

0000-0002-1689-3945]I. Torres
Instituto Nacional de Astrofísica, Óptica y Electrónica, Puebla, Mexico

0000-0002-7102-3352]R. Torres-Escobedo
Tsung-Dao Lee Institute & School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, China

0000-0003-1068-6707]R. Turner
Department of Physics, Michigan Technological University, Houghton, MI, USA

0000-0002-2748-2527]F. Ureña-Mena
Instituto Nacional de Astrofísica, Óptica y Electrónica, Puebla, Mexico

0000-0003-0715-7513]E. Varela
Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, Puebla, Mexico

0000-0001-6876-2800]L. Villaseñor
Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, Puebla, Mexico

0000-0001-6798-353X]X. Wang
Department of Physics, Michigan Technological University, Houghton, MI, USA

0000-0003-2141-3413]I.J. Watson
University of Seoul, Seoul, Rep. of Korea

0000-0002-6623-0277]E. Willox
Department of Physics, University of Maryland, College Park, MD, USA

0000-0003-0513-3841]H. Zhou
Tsung-Dao Lee Institute & School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, China

76(HAWC collaboration)

Recently, the region surrounding eHWC J1842-035 has been studied extensively by gamma-ray observatories due to its extended emission reaching up to a few hundred TeV and potential as a hadronic accelerator.
In this work, we use 1,910 days of cumulative data from the High Altitude Water Cherenkov (HAWC) observatory to carry out a dedicated systematic source search of the eHWC J1842-035 region. During the search we have found three sources in the region, namely, HAWC J1844-034, HAWC J1843-032, and HAWC J1846-025. We have identified HAWC J1844-034 as the extended source that emits photons with energies up to 175 TeV.
We compute the spectrum for HAWC J1844-034 and by comparing with the observational results from other experiments, we have identified HESS J1843-033, LHAASO J1843-0338, and TASG J1844-038 as very-high-energy gamma-ray sources with a matching origin.
Also, we present and use the multi-wavelength data to fit the hadronic and leptonic particle spectra. We have identified four pulsar candidates in the nearby region from which PSR J1844-0346 is found to be the most likely candidate due to its proximity to HAWC J1844-034 and the computed energy budget. We have also found SNR G28.6-0.1 as a potential counterpart source of HAWC J1844-034 for which both leptonic and hadronic scenarios are feasible.

§ INTRODUCTION

The origin of cosmic rays has been one of the primary questions the astrophysical community has been asking for over a century. Although there has been a lot of advancement in observational techniques and technology, our understanding of cosmic rays is far from complete.
Cosmic ray sources cannot be traced directly due to deflections of cosmic ray particles, but $\gamma$-ray photons of the hadronic origin can be used to indirectly search for cosmic ray sources. However, $\gamma$-ray photons can also be produced from leptonic processes, hence a rigorous analysis is required to discern between hadronic-induced $\gamma$-ray sources and leptonic-induced $\gamma$-ray sources.

One particular feature of the cosmic ray spectrum is a break at around a few peta-electronvolts (PeVs) that is known as the “knee” [Amenomori et al., 2008].
Cosmic rays with energies less than the knee are believed to be accelerated by Galactic objects.
These sources that can accelerate particles up to a PeV energy scale without a break are known as PeVatrons [Tibet AS Collaboration et al., 2021].

If a Galactic gamma-ray source has been identified as a cosmic ray accelerator, we can study its spectrum to look for potential traces of a PeVatron.
The proton spectrum of a PeVatron should cut off after a few PeV, leading to a hard $\gamma$-ray spectrum below the cutoff region corresponding to $\sim$100 TeV [Celli et al., 2020]. Therefore, the spectrum of $\gamma$-rays near the 100 TeV energy range can be useful for identifying PeVatron candidates [Sudoh & Beacom, 2022].

eHWC J1842-035 is one of the nine sources that was observed at above 56 TeV by the High Altitude Water Cherenkov (HAWC) Observatory in its “High-Energy Catalog” [Abeysekara et al., 2020]. This result was later reinforced by the Large High Altitude Air Shower Observatory (LHAASO) in 2021 [Cao et al., 2021]. LHAASO observed a bright extended source at a similar location of $0\fdg14$ from eHWC J1842-035. LHAASO reported the $\gamma$-ray flux at 100 TeV. Also, in 2022 the Tibet AS-Gamma Experiment published observational results on an extended source, TASG J1844-038, with a spectrum reaching up to 100 TeV [Amenomori et al., 2022]. The observations by the three different multi-TeV $\gamma$-ray observatories are highly likely to be originating from the same source.

In this work, we report the updated HAWC results on this bright extended source (to be referred to as HAWC J1844-034) obtained from a rigorous multi-source modeling of the region (Section <ref>). In Section <ref>, we discuss the spectral studies on the results as well as the origin of the observed emission and systematic studies, providing evidence of counterpart sources in the region from other experiments.

§ INSTRUMENT AND DATA ANALYSIS

§.§ HAWC

HAWC is a ground-based particle sampling array located on the mountain Sierra Negra, Mexico at an altitude of 4,100 m. HAWC is designed to observe very-high-energy (VHE) $\gamma$-ray photons by detecting Cherenkov radiation produced from electromagnetic air showers as they propagate through the water Cherenkov detectors (WCDs). The main array of HAWC, composed of 300 WCDs, cover a geometrical area of 22,000 m${}^{2}$ [Abeysekara et al., 2023]. Each WCD consists of a bladder containing 180,000 l of purified water that is protected by a metal casing with a roof on top. A WCD has a height of 5.4 m and a diameter of 7.3 m. Four photomultiplier tubes (PMTs) are located at the bottom of each WCD to collect the Cherenkov light emitted by air shower particles. For more detailed information on the design and operation of HAWC, refer to Abeysekara et al., 2017, Abeysekara et al., 2023.

The gamma-ray events collected by HAWC are categorized into analysis bins based on the fraction of the PMTs used in the event reconstruction ($f_{\mathrm{hit}}$) and the estimated energies of the particular event [Abeysekara et al., 2017, Abeysekara et al., 2019]. We call these the “ground parameter” (GP) and “neural network” (NN) energy estimators.

The HAWC High-Energy Catalog released in 2020 used the GP dataset that contains 1,039 days of cumulative data [Abeysekara et al., 2020].
In this work, we have used $1,910$ days of the GP energy estimator data above 17.8 TeV and $f_{\mathrm{hit}}$ above 24.5 % that are within our region of interest (ROI) to compute the most sensitive results on the high energy spectrum of HAWC J1844-034. The energy resolution for the GP energy estimator is 50% at 20 TeV and 34% at 100 TeV [Abeysekara et al., 2019]. The angular resolution range for the GP energy estimator bins used in the analysis is $0\fdg1 - 0\fdg6$ [Abeysekara et al., 2019]. Our ROI is a rectangular region defined as $28\degr < l < 30\degr$ and $-5\degr < b < 5\degr$, where $l$ and $b$ are the Galactic longitude and latitude, respectively. The ROI is presented as the brighter region in Figure <ref>, which has been chosen to include the extended emission.

The multi-mission maximum likelihood (3ML) [3ML; <https://github.com/threeML/threeML>] framework and HAWC accelerated likelihood (HAL) [HAL; <https://github.com/threeML/hawc_hal>] plugin [Vianello et al., 2015, Abeysekara et al., 2021] are used to simultaneously fit multiple source models and their relevant free parameters, based on maximum likelihood estimations. Spectral and morphological models mentioned in this work can be found in astromodels [astromodels; <https://github.com/threeML/astromodels>].

In our analysis, a test statistic (TS) is used to evaluate a pre-trial statistical significance of a fitted model in a region with a predetermined number of free parameters.
TS provides a statistical measure of how well an alternative hypothesis performs over a null hypothesis. TS values can be used to compare between different nested models.
TS is defined as below:

\begin{equation}
\mathrm{TS} = 2\mathrm{ln}\left(\frac{L_{1}}{L_{0}}\right),
\end{equation}

where $L_{1}$ is the likelihood for an alternative hypothesis and $L_{0}$ is the likelihood for a null hypothesis. To compare two different alternative hypotheses, $\Delta \mathrm{TS} = \mathrm{TS}_{2} - \mathrm{TS}_{1} = 2\mathrm{ln}\left(L_{2}/L_{1}\right)$ can be used, where $L_{2}$ is the likelihood for a different alternative hypothesis [Abeysekara et al., 2017]. Using Wilks' theorem [Wilks, 1938], a pre-trial significance can be computed with $\sigma \simeq \sqrt{\mathrm{TS}}$, which is used to generate HAWC significance maps [Abeysekara et al., 2017].

In order to find a statistically optimal list of sources in the ROI, a systematic approach developed by the Fermi Large Area Telescope (Fermi-LAT) Collaboration is used [Ackermann et al., 2017].
For the source search, we recursively add a point source to the model in our ROI until the improvement of the overall TS becomes less than 16. Then, we test an extended source model for each of the found point sources. Starting from the brightest source, we accept an extended source model if $\Delta\mathrm{TS} > 16$.

During this process, we assumed a power-law spectral model for all of the sources involved:

\begin{equation}\label{eq:powerlaw}
\Phi_{\gamma}\left(E_{\gamma}\right) = \Phi_{0}\left(\frac{E_{\gamma}}{E_{\mathrm{piv}}}\right)^{-\Gamma},
\end{equation}

where $\Phi_{\gamma}\left(E_{\gamma}\right)$ is the $\gamma$-ray flux at the $\gamma$-ray energy $E_{\gamma}$, $\Phi_{0}$ is the flux normalization at a pivot energy $E_{\mathrm{piv}}$, and $\Gamma$ is the spectral index. $\Phi_{0}$ and $\Gamma$ are set as free parameters to be determined by the likelihood fit, and $E_{\mathrm{piv}}$ is fixed at 20 TeV. For the pivot energy, we have scanned through the entire energy range from which we have obtained minimum correlation between the free parameters at $20~\mathrm{TeV}$.

Once the optimal number of point and extended sources is determined from the systematic source search, more realistic spectral models are applied and tested. Power-law with an exponential cutoff and log parabola models have been used:

\begin{eqnarray}
\begin{array}{cc} \label{eq:cutoffpowerlaw}
\Phi_{\gamma}\left(E_{\gamma}\right) = \Phi_{0}\left(\frac{E_{\gamma}}{E_{\mathrm{piv}}}\right)^{-\Gamma}\mathrm{exp}\left(-\frac{E_{\gamma}}{E_{c}}\right)
\end{array},
\\
\begin{array}{cc} \label{eq:logparabola}
\Phi_{\gamma}\left(E_{\gamma}\right) = \Phi_{0}\left(\frac{E_{\gamma}}{E_{\mathrm{piv}}}\right)^{-\Gamma-\beta\ln{\left(E_{\gamma}/E_{\mathrm{piv}}\right)}},\end{array}
\end{eqnarray}

where $E_{c}$ of Equation <ref> is the cutoff energy and $\beta$ of Equation <ref> describes the curvature of the power-law spectrum.

The radially symmetric Gaussian morphology [Abeysekara et al., 2019] is formulated as

\begin{equation} \label{eq:gaussian}
\frac{dN}{d\Omega}=\left(\frac{180}{\pi}\right)^2\frac{1}{2\pi\sigma^2}\exp{\left(-\frac{\theta^2}{2\sigma^2}\right)}
\end{equation}

where $\theta$ and $\sigma$ are the angular distance and Gaussian width as a measure of the size of the extended source model, respectively. $\sigma$ is set as a free parameter during likelihood fits.

§.§ Naima Framework

The Naima software [Naima; <https://github.com/zblz/naima>] [Zabalza, 2015] is a non-thermal $\gamma$-ray modeling framework based on Markov chain Monte Carlo methods. It is used to model and fit the spectra of accelerated particles in hadronic and leptonic scenarios, based on the observed electromagnetic data.
The Naima software provides numerical models for a hadronic scenario based on neutral pion decays from inelastic proton-proton collisions [Kafexhiu et al., 2014] and a leptonic scenario based on inverse Compton scattering of relativistic electrons with low-energy photons and synchrotron emission from relativistic electrons [Khangulyan et al., 2014, Aharonian et al., 2010]. We have used the Naima framework to generate particle spectra for the hadronic and leptonic scenarios of HAWC J1844-034. The results are presented in Sections <ref> and <ref>.

§ RESULTS

1,910 day HAWC significance map in Galactic coordinates highlighting the eHWC J1842-035 region. The brighter part of the map is the ROI used for the analysis. The labels indicate the known VHE $\gamma$-ray sources within the ROI. Note that 3HWC J1843-034 indicates the best-fit location of the source found from the HAWC all-energy catalog [Albert et al., 2020].

Figure <ref> shows the 1,910 day HAWC significance map of the eHWC J1842-035 region in Galactic coordinates, where our ROI has been indicated by the brighter rectangular region ($28\degr < l < 30\degr$ and $-5\degr < b < 5\degr$).
Figure <ref> also includes source labels from various $\gamma$-ray observatories such as the High Energy Stereoscopic System (H.E.S.S.), HAWC, Tibet AS-Gamma, and LHAASO [H.E.S.S. Collaboration et al., 2018, Abeysekara et al., 2020, Albert et al., 2020, Cao et al., 2021, Amenomori et al., 2022].

Best-fit parameters found from a multiple source likelihood fit of the nested model in the ROI

Source      1|c$\alpha$
$\delta$     $\Phi_{0}$     $\Gamma$     $\beta$     Size     TS      Signif.

[deg]     [(TeV cm$^{2}$ s)${}^{-1}$]               [deg]

HAWC J1844-034     $281.02^{+0.05}_{-0.05}$     $-3.64^{+0.05}_{-0.04}$     $1.32^{+0.10}_{-0.09}{} \times 10^{-14}$     $2.72^{+0.21}_{-0.22}{}$     $0.51^{+0.21}_{-0.21}$     $0.48^{+0.02}_{-0.02}$     $718.49$     $26.26$

HAWC J1843-032     $280.83^{+0.04}_{-0.03}$     $-3.31^{+0.04}_{-0.04}$     $9.31^{+2.12}_{-1.70} \times 10^{-16}$     $3.34^{+0.37}_{-0.37}$             $33.39$     $4.89$

HAWC J1846-025     $281.61^{+0.06}_{-0.06}$     $-2.81^{+0.08}_{-0.08}$     $4.33^{+1.87}_{-1.29} \times 10^{-16}$     $2.94^{+0.50}_{-0.48}$             $18.67$     $3.32$

The listed values are the best-fit parameters and their statistical uncertainties. $\alpha$ and $\delta$ are right ascension and declination, respectively. Signif. is the post-trial significance. $\Phi_{0}$ is the flux normalization at a pivot energy of 20 TeV. Size is the 1$\sigma$ radius of the extended Gaussian morphological model used to fit HAWC J1844-034.

model_G4paperMagma_deg.pdf(a) residual_G4paperMagma_deg.pdf(b)
(a): A significance map of the best-fit model described in Table <ref>. (b): A residual map produced by subtracting the best-fit model.

From a systematic source search in the ROI, we have found three sources as the statistical optimum, namely, HAWC J1843-032 and HAWC J1846-025 as point sources, and HAWC J1844-034 as an extended source fitted with a radially symmetric Gaussian morphology (Equation <ref>). We have used a power-law spectral model for each of the point sources and a log parabolic spectral model for HAWC J1844-034. The best-fit parameters at the pivot energy of 20 TeV along with their TS and post-trial significance values can be found in Table <ref>. Among the three sources we have found during the systematic source search, HAWC J1844-034 is by far the brightest and most dominant source with the $1\sigma$ upper energy bound of the observed photons reaching 175 TeV, which is comparable to 260 TeV photons observed by LHAASO [Cao et al., 2021]. Therefore, we have conducted further analyses on HAWC J1844-034, which will be discussed in Section <ref>. Note that the upper energy bound has been obtained by profiling the likelihood of the best-fit spectrum multiplied by a step function while having the bound energy as the only free parameter. The $1\sigma$ level corresponds to the estimate at $\Delta\mathrm{TS}=1$ [Albert et al., 2021]. On the other hand, both HAWC J1843-032 ($4.89\sigma$) and HAWC J1846-025 ($3.32\sigma$) have their significance $<5\sigma$. Hence, HAWC J1843-032 and HAWC J1846-025 are gamma-ray candidates that we have found during the modeling of HAWC J1844-034 region.

In Figure <ref> (b), a residual map is presented after fitting and subtracting the best-fit model (Figure <ref> (a)) from Figure <ref>.
Also, Figure <ref> is a significance histogram that shows the computed distribution of pixels in the ROI of Figure <ref> (b).
The histogram is used to look for obvious signatures of overfitting or underfitting of the $\gamma$-ray excess in the ROI, neither of which are present in the distribution. Also for the histogram, we have computed chi square per number of degrees of freedom of $92.85/77\approx1.20$ for the best-fit Gaussian curve in green.

A one dimensional significance distribution histogram collected per pixel of Figure <ref> (b). The expected background-only distribution is described by the red Gaussian curve. The green line is the best-fit Gaussian curve of the data present in the range between -2 and 2.

§ DISCUSSIONS

§.§ Spectral Properties of HAWC J1844-034

The best-fit spectrum of HAWC J1844-034.

Figure <ref> shows the spectrum of HAWC J1844-034. The gray and red bands represent the statistical and systematic uncertainties, respectively. The various systematic uncertainty components have been summed in quadrature (later discussed in Section <ref>). We have calculated the flux points of HAWC J1844-034 (black) by refitting $\Phi_{0}$ at each energy bin while maintaining the best-fit parameters for the spectral shape, $\Gamma$ and $\beta$ [Abeysekara et al., 2019].

On the plot, the TeV $\gamma$-ray flux data points from Tibet AS-Gamma (green) and LHAASO (purple) at 100 TeV have also been added for comparison [Cao et al., 2021, Amenomori et al., 2022].
The fitted HAWC spectrum is in good agreement with the results from both LHAASO and Tibet AS-Gamma. According to the HAWC results, the fall off in the spectrum is gradual, which may be consistent with 260 TeV photons observed by LHAASO (purple dashed line) [Cao et al., 2021]. The best-fit spectrum of TASG J1844-038 has been observed with a power-law function in $25~\mathrm{TeV} < E_{\gamma} < 130~\mathrm{TeV}$ [Amenomori et al., 2022]. In this energy range, TASG J1844-038 has four flux points exceeding $2\sigma$, which correspond well with the best-fit spectrum of HAWC J1844-034. The best-fit power-law spectrum for TASG J1844-038 has a spectral index of $\Gamma=3.26\pm0.30$, which falls more sharply than the index of HAWC J1844-034 $\Gamma=2.72^{+0.21}_{-0.22}$ at 20 TeV [Amenomori et al., 2022]. This may be due to the different conditions we are performing the fits in such as the energy range, pivot energy, and spectral model. However, there is sufficient overlap despite having different conditions as shown in Figure <ref>. Also, at the pivot energy of TASG J1844-038 ($E_{\mathrm{piv}}=40~\mathrm{TeV}$), the spectral index of HAWC J1844-034 is found to be $\Gamma=3.07^{+0.36}_{-0.37}$, which is consistent with the results from Tibet AS-Gamma. Furthermore, the sources found by the three gamma-ray observatories are spatially consistent given their angular resolutions: $\sim0\fdg1 - 0\fdg6$ for $\geq10~\mathrm{TeV}$ (HAWC), $0\fdg24 - 0\fdg3$ for $100~\mathrm{TeV}$ (LHAASO), and $\sim0\fdg28$ for $100~\mathrm{TeV}$ (Tibet AS-Gamma) [Abeysekara et al., 2019, Wang et al., 2021, Amenomori et al., 2022].

§.§ Extended Emission from HAWC J1844-034

The radial profile from the best-fit location of HAWC J1844-034 as center. The dotted lines are radial distances of the best-fit positions of the sources from the center of HAWC J1844-034.

Figure <ref> displays the radial distribution of $\gamma$-ray excess profile from the best-fit location of HAWC J1844-034 (magenta points). The brown solid line represents the overall best-fit model that includes all three HAWC sources of the region. The black dashed, blue dot-dashed, and red dotted lines represent the best-fit models for each source, and the dotted vertical lines represent the distances of each labeled source from HAWC J1844-034. Note that TASG J1844-038 (green) and LHAASO J1843-0338 (purple) are not included in the inclusive model.
The inclusive model shows a good match with the excess. Also, TASG J1844-038 presents a good spatial agreement with HAWC J1844-034. Conversely, LHAASO J1843-0338 is positioned between the peak of HAWC J1844-034 and HAWC J1843-032, which corresponds to a plateau of the overall excess peak.

§.§ Systematic Uncertainties

Systematic uncertainties

4|cHAWC J1844-034     2|cHAWC J1843-032     2|cHAWC J1846-025

Uncertainty origin     1|c$\Phi_{0}$     $\Gamma$
$\beta$     Size     1|c$\Phi_{0}$     $\Gamma$     1|c$\Phi_{0}$     $\Gamma$

1|c[(TeV cm$^{2}$ s)${}^{-1}$]
[deg]     1|c[(TeV cm$^{2}$ s)${}^{-1}$]          1|c[(TeV cm$^{2}$ s)${}^{-1}$]

Late light     ${}^{+1.08}_{-2.21} \times 10^{-15}$     ${}^{+0.22}_{-0.08}$     ${}^{+0.10}_{-0.27}$     ${}^{+0.00}_{-0.00}$     ${}^{+3.87}_{-6.73} \times 10^{-17}$     ${}^{+0.01}_{-0.02}$     ${}^{+3.75}_{-7.01} \times 10^{-17}$     ${}^{+0.08}_{-0.03}$

Charge uncertainty     $-7.48 \times 10^{-16}$     $+0.08$     $-0.04$     $+0.00$     $-4.75 \times 10^{-17}$     $+0.03$     $-2.26 \times 10^{-17}$     $+0.03$

PMT efficiency     $+4.67 \times 10^{-16}$     ${}^{+0.02}_{-0.04}$     $-0.04$     ${}^{+0.00}_{-0.00}$     $+6.13 \times 10^{-17}$     ${}^{+0.01}_{-0.02}$     $+2.67 \times 10^{-17}$     ${}^{+0.02}_{-0.03}$

PMT threshold     ${}^{+1.36}_{-1.11} \times 10^{-16}$     ${}^{+0.09}_{-0.05}$     ${}^{+0.06}_{-0.07}$     ${}^{+0.00}_{-0.00}$     ${}^{+1.86}_{-1.00} \times 10^{-18}$     ${}^{+0.04}_{-0.02}$     $+9.06 \times 10^{-18}$     $+0.03$

DBE     ${-2.70} \times 10^{-15}$     ${-0.11}$     $+0.37$     ${-0.08}$     ${-1.52} \times 10^{-17}$     ${-0.02}$     ${-8.06 \times 10^{-19}}$     ${+0.06}$

ROI     ${+7.17 \times 10^{-16}}$     ${+0.00}$     ${-0.02}$     ${+0.03}$     ${-2.78 \times 10^{-18}}$     ${-0.02}$     ${+1.61 \times 10^{-16}}$     ${-0.34}$

Quadratic summed     ${}^{{+1.38}}_{{-3.57}} \times 10^{-15}$     ${}^{+0.25}_{{-0.15}}$     ${}^{+0.39}_{{-0.29}}$     ${}^{+0.03}_{{-0.08}}$     ${}^{+7.25}_{{-8.38}} \times 10^{-17}$     ${}^{+0.05}_{{-0.04}}$     ${}^{{+1.68}}_{{-0.74}} \times 10^{-16}$     ${}^{{+0.11}}_{-0.34}$

Factors that may contribute towards the systematics are listed in Table <ref>. The first four components in the table arise from various detector and simulation effects <cit.>.
The DBE is a systematic effect arising from any unresolved sources and the true Galactic diffuse emission. We have calculated this effect by eadding a Gaussian background model, along the Galactic equator with the fixed width of $1\degr$ [Albert et al., 2021, Rho et al., 2018], in the nested fit. We have compared the best-fit parameters of the three sources and added the differences as a systematic component. A power-law spectral model (Equation <ref>) with an index of $2.75$ is assumed for the DBE based on Abdo et al., 2008, Albert et al., 2022.

Finally, our choice of ROI has been considered as a systematic uncertainty. Our ROI has been chosen as the best compromise between considering a large enough region to include the extended gamma-ray excess of HAWC J1844-034 and minimizing the number of sources we would have to simultaneously fit. During the analysis, the ROI is predetermined before the systematic source search, hence we have used a larger ROI of $28\degr < l < 30.3\degr$ to see how much effect this would have on the source of our main interest, HAWC J1844-034. Although extending the ROI has shown a notable effect on HAWC J1846-025, there is a minimal effect on HAWC J1844-034.

§.§ Leptonic versus hadronic scenarios

The nature of the gamma-ray emission from HAWC J1844-034 has been modeled and studied to test if this source is a potential hadronic accelerator. In the hadronic scenario, the emission of $\gamma$-rays is dominated by the process $\pi^{0} \rightarrow 2\gamma$. If the neutral pions are produced from the collisions of ambient protons that are accelerated by HAWC J1844-034 [Kafexhiu et al., 2014], then the observed gamma-ray spectrum can be used to constrain the proton spectrum of HAWC J1844-034 [Celli et al., 2020]. The hadronic scenario is powered by the proton-proton scatterings of accelerated protons and protons of the ambient gas. During the fit, we have assumed the proton density of 1 cm$^{-3}$.

In the case of the leptonic scenario, the TeV $\gamma$-ray emission is produced mostly from the inverse Compton scattering between low-energy ambient photons and high-energy electrons accelerated by HAWC J1844-034 [Khangulyan et al., 2014]. Due to the Klein-Nishina effects, the $\gamma$-ray spectrum originating from the inverse Compton scattering has a spectral cutoff at tens of TeV [Aliu et al., 2014]. For the inverse Compton scattering of the leptonic scenario, we used cosmic microwave background (CMB) photons as seed photons that are up-scattered by the accelerated electrons.

A significance map of the HAWC J1844-034 region with TeV gamma-ray sources labeled at the top [H.E.S.S. Collaboration et al., 2018, Amenomori et al., 2022, Abeysekara et al., 2020, Albert et al., 2020, Cao et al., 2021], Fermi-LAT sources in red [Abdollahi et al., 2020], and known pulsars and SNR in green and blue, respectively [Manchester et al., 2005, Bamba et al., 2001, Ueno et al., 2003].

Figure <ref> presents the locations of the sources inside our ROI. The labels at the top of the figure are the VHE $\gamma$-ray sources found by H.E.S.S., Tibet AS-Gamma, LHAASO, and previously detected sources from HAWC catalogs [H.E.S.S. Collaboration et al., 2018, Amenomori et al., 2022, Cao et al., 2021]. The purple labels indicate the sources specifically found by this work. The lower red labels indicate GeV $\gamma$-ray sources observed by Fermi-LAT <cit.>, the green labels are for pulsars from the ATNF pulsar catalog [Manchester et al., 2005], and the blue label marks a known supernova remnant SNR G28.6-0.1 [Condon et al., 1998, Bamba et al., 2001, Ebisawa et al., 2005].

A list of known $\gamma$-ray sources near HAWC J1844-034

Source     1|c$\alpha$ [deg]     $\delta$ [deg]     $\Delta$ [deg]     Size [deg]

HAWC J1844-034     $281.02{^{+0.05}_{-0.05}}$     $-3.64{^{+0.05}_{-0.04}}$         $0.48^{+0.02}_{-0.02}$

LHAASO J1843-0338     $280.75$     ${-3.65}$     $0.27$     $0.3$

TASG J1844-038     $281.09{^{+0.10}_{-0.10}}$     $-3.76{^{+0.09}_{-0.09}}$     $0.14$     $0.34\pm0.12$

HESS J1843-033     $280.95$     $-3.55$     $0.11$     $0.24\pm0.06$

HESS J1844-030     $281.17$     $-3.10$     $0.56$     $0.02\pm0.013$

4FGL J1843.7-0326     $280.93$     $-3.44$     $0.22$

4FGL J1844.4-0345     $281.12$     $-3.75$     $0.15$

$\Delta$ is the angular distance from the center of HAWC J1844-034. The size of LHAASO J1843-0338 is referred from the extension template of $0\fdg3$ [Cao et al., 2021]. The size of TASG J1844-038 does not include the PSF contribution <cit.>. HESS J1843-033 consists of two Gaussian peaks [H.E.S.S. Collaboration et al., 2018]. The positional error of LHAASO J1843-0338 was not reported.

A collection of spectra for $\gamma$-ray sources within the vicinity of HAWC J1844-034. The results have been collected from Fermi-LAT [Abdollahi et al., 2020], H.E.S.S. [H.E.S.S. Collaboration et al., 2018], Tibet AS-Gamma [Amenomori et al., 2022], LHAASO [Cao et al., 2021], and HAWC.

Figure <ref> shows the spectra of the GeV to TeV $\gamma$-ray sources within the extension of HAWC J1844-034 as shown in Figure <ref>. HESS J1843-033 has been considered as the counterpart of TASG J1844-038 [Amenomori et al., 2022]. The observed spectrum of HESS J1843-033 (brown) has a similar amplitude as that of HAWC J1844-034 (black) and TASG J1844-038 (green), so it has also been assumed as the sub-TeV counterpart in our analysis. The flux points of HESS J1844-030 (orange) and 4FGL J1843.7-0326 (blue) are smoothly connected, so it is likely that these two observations come from the same source. However, it is difficult to find an association of these two sources with HAWC J1844-034 (as well as TASG J1844-038 and LHAASO J1843-0338) because HESS J1844-030 has not been observed with significant emission at energies above 10 TeV [H.E.S.S. Collaboration et al., 2018]. The spectrum of 4FGL J1844.4-0345 (red) has been found to follow a log parabolic shape in the GeV range and cuts off sharply, not detected above 100 GeV from Fermi-LAT [Abdollahi et al., 2020].
Therefore, it is difficult to identify a source in the GeV energies that shows clear association with HAWC J1844-034 at this time.

The two scenarios have been fitted with the Naima framework using only the TeV gamma-ray flux data from HESS J1843-033 [H.E.S.S. Collaboration et al., 2018], TASG J1844-038 [Amenomori et al., 2022], LHAASO J1843-0338 [Cao et al., 2021], and HAWC J1844-034 (this work).
The particle spectra of each scenario have been assumed to be a power law with an exponential cutoff with a pivot energy of 7 TeV.

\begin{eqnarray}\label{eq:EnergyDist}
\rho_{\mathrm{P}}(E_{\mathrm{P}}) = \rho_{0,\mathrm{P}\left(1\mathrm{kpc}\right)}\left(\frac{E_{\mathrm{P}}}{7~\mathrm{TeV}}\right)^{-\Gamma_{\mathrm{P}}} \nonumber \\
\times \mathrm{exp}\left(-\frac{E_{\mathrm{P}}}{E_{c,\mathrm{P}}}\right)
\end{eqnarray}

where $\rho_{\mathrm{P}}(E_{\mathrm{P}})$ is the spectrum of the accelerated particles, either electrons or protons, $E_{\mathrm{P}}$ is energy of accelerated particles, $\rho_{0,\mathrm{P}\left(1\mathrm{kpc}\right)}$ is the value of $\rho_{\mathrm{P}}(E_{\mathrm{P}})$ at the distance of 1 kpc and the pivot energy of 7 TeV.

Best-fit parameters for the hadronic and leptonic scenarios

1|cHadronic Scenario     1|cLeptonic Scenario

$\rho_{0,\mathrm{P}\left(1\mathrm{kpc}\right)}$ [1/eV]     $1.85^{+0.34}_{-0.37}\times10^{34}$     $7.04^{+1.17}_{-1.13}\times10^{31}$

$\Gamma_{\mathrm{P}}$     $1.56^{+0.11}_{-0.17}$     $2.17^{+0.14}_{-0.20}$

$E_{c,\mathrm{P}}$ [TeV]     $169^{+32}_{-35}$     $72^{+13}_{-13}$

BIC     $85.9$     $84.4$

$\rho_{0,\mathrm{P}\left(1\mathrm{kpc}\right)}$ is normalization at a source distance of 1 kpc, $\Gamma_{\mathrm{P}}$ is spectral index, and $E_{c,\mathrm{P}}$ is cutoff energy.

Table <ref> shows the best-fit values for the proton spectrum of the hadronic scenario and the electron spectrum of the leptonic scenario. The three free parameters of Equation <ref> are listed for each scenario in the table. The obtained spectral index of 1.56 for the hadronic scenario may seem hard considering other hadronic spectra in general. However, hadronic scenarios with such extreme spectral indices are discussed in some cases [Abdalla, H. et al., 2021]. The best-fit cutoff energy for the hadronic scenario at 169 TeV is too low for HAWC J1844-034 to be defined as a PeVatron.
Note that the particle spectra have been fitted while assuming HAWC J1844-034 is located at a unit distance of 1 kpc since the exact counterpart source has not been identified. The source distance is later accounted for with a known source distance in Section <ref> when we try to identify a counterpart source.

A spectral distributions plot of the best-fit leptonic (blue) and hadronic (red) scenarios that have been fitted using the TeV gamma-ray flux points speculated of having the same original source [H.E.S.S. Collaboration et al., 2018, Amenomori et al., 2022, Cao et al., 2021]. The error bands indicate statistical errors.

In Figure <ref>, we have the data points from LHAASO J1843-0338 (purple), HESS J1843-033 (brown), TASG J1844-038 (green) and HAWC J1844-034 (black). The dashed purple vertical line is the maximum photon energy as observed by LHAASO on this source. The $\gamma$-ray spectra obtained from fitting the data points with the Naima framework are presented as blue and red lines for the leptonic and hadronic scenarios, respectively, with their statistical uncertainties shown as lighter bands. The overall trend of the spectra are very similar at energies below 100 TeV. At higher energies, the two spectra start to diverge as the leptonic spectrum cuts off at lower energies. From the Naima fit, the leptonic model is slightly more favorable than the hadronic model with a difference in BIC of 1.5.

Various properties of four pulsars found near HAWC J1844-034 used for calculations [Manchester et al., 2005, Devin, J. et al., 2021]

2|c     Spin-Down          1|cAngular     $\gamma$-ray

Pulsar     1|cRight Ascension     Declination     Power     Distance     1|cDistance     Luminosity

1|c$\alpha$ [deg]     $\delta$ [deg]     $\dot{E}$ [erg/s]     $d_{\mathrm{src}}$ [kpc]     1|c$\Delta$ [deg]     $L_{\gamma}^{\mathrm{HAWC}}$ [erg/s]

PSR J1841-0345     280.4112     -3.8118     $2.7 \times ~ 10^{35}$     $3.776$     $0.63$     $2.14^{+0.44}_{-0.27} \times ~ 10^{34}$

PSR J1844-0346     281.13704     -3.7752     $4.2 \times ~ 10^{36}$     $4.3$     $0.18$     $2.78^{+0.53}_{-0.35} \times ~ 10^{34}$

PSR J1843-0355     280.77776     -3.93239      $1.77 \times ~ 10^{34}$     $5.798$     $0.38$     $5.05^{+0.97}_{-0.64} \times ~ 10^{34}$

PSR J1844-0310     281.1895     -3.1770     $2.79 \times ~ 10^{33}$     $5.973$     $0.49$     $5.36^{+1.03}_{-0.67} \times ~ 10^{34}$

Angular distance is the distance from the center of HAWC J1844-034.

§ ORIGIN OF THE G-RAY EMISSION OF HAWC J1844-034

In this section, we analyze nearby pulsars and a supernova remnant to gauge their potentials as a counterpart to HAWC J1844-034.

§.§ Pulsars

There are three pulsars that are within the best-fit size of HAWC J1844-034: PSR J1843-0355, PSR J1844-0310, and PSR J1844-0346 [Manchester et al., 2005]. Also, PSR J1841-0345 is suggested as a possible counterpart source to LHAASO J1843-0338 in de Oñ a Wilhelmi et al., 2022.
Therefore, we have examined these four pulsars as a candidate counterpart source to HAWC J1844-034.
The properties of the pulsars used in the analysis have been referenced from the ATNF catalog [Manchester et al., 2005] and listed in Table <ref>.

In Table <ref>, we have computed a number of parameters on the four pulsars for comparison. $L_{\gamma}^{\mathrm{HAWC}}$ is the $\gamma$-ray luminosity estimated from integrating the spectrum of HAWC J1844-034 in Figure <ref> as shown below:

\begin{equation}
L_{\gamma}^{\mathrm{HAWC}} = 4\pi d_{\mathrm{src}}^2 \int_{~11~\mathrm{TeV}}^{202~\mathrm{TeV}} dE_{\gamma} ~ E_{\gamma} \Phi_{\gamma}.
\label{eq:luminosity}
\end{equation}

The angular distance is the distance to each pulsar from the best-fit location of HAWC J1844-034.

Assuming the spin-down of a pulsar powers the acceleration of the particles that then produce TeV $\gamma$-rays, the $\gamma$-ray luminosity is required to be less than the spin-down luminosity of a pulsar to be a possible counterpart source [Cataldo et al., 2020, Albert et al., 2023, de Oñ a Wilhelmi et al., 2022]. PSR J1843-0355 and PSR J1844-0310 both have their spin-down powers that are lower than their $\gamma$-ray luminosities if they were associated with HAWC J1844-034. Therefore, they are disfavored to be the counterpart source of HAWC J1844-034.

PSR J1841-0345 is located $0\fdg63$ away from HAWC J1844-034, which is outside of the extension of HAWC J1844-034. Therefore, we rule out PSR J1841-0345 from the counterpart source candidates.

Finally, PSR J1844-0346 has its spin-down power that is approximately two orders of magnitude higher than the gamma-ray luminosity. Also, it is located closest to the best-fit positions of HAWC J1844-034, HESS J1843-033, and TASG J1844-038 with the angular distances of $0\fdg18$, $0\fdg28$, and $0\fdg05$, respectively. The angular resolutions for HAWC, H.E.S.S., and Tibet AS-Gamma are $0\fdg1 - 0\fdg6$ for $\geq10~\mathrm{TeV}$, $\sim0\fdg08$ for $\geq0.2~\mathrm{TeV}$, and $\sim0\fdg28$ for $100~\mathrm{TeV}$, respectively [Abeysekara et al., 2019, H.E.S.S. Collaboration et al., 2018, Amenomori et al., 2022]. Here, the angular resolution of H.E.S.S. is smaller than the angular distance between HESS J1843-033 and PSR J1844-0346. However, HESS J1843-033 is an extended source with the best-fit size of $0\fdg24$ (Table <ref>). Furthermore, although the angular distance of $0\fdg18$ is larger than the positional uncertainties given in Table <ref>, the $0\fdg18$ separation is contained within the $0\fdg48$ extension of HAWC J1844-034. Therefore, of the four pulsars found near the region of HAWC J1844-034, PSR J1844-0346 is considered to be the most favorable candidate according to this work.

§.§ SNR G28.6-0.1

SNR G28.6-0.1 is a shell-type supernova remnant [Bamba et al., 2001] that is another candidate counterpart source to HAWC J1844-034. It is located at an angular distance of $0\fdg28$ away from HAWC J1844-034, which is within the best-fit size of HAWC J1844-034. SNR G28.6-0.1 was suggested to be a potential candidate counterpart source for HESS J1843-033 with VHE gamma-rays originating from leptonic processes due to the observed synchrotron emission [Devin, J. et al., 2021, Bamba et al., 2001]. Assuming SNR G28.6-0.1 is responsible for the observed VHE $\gamma$-rays, by using the distance to SNR G28.6-0.1 of $9.6\pm0.3$ kpc [Ranasinghe & Leahy, 2018], the $\gamma$-ray luminosity and the proton density can be estimated as presented in Table <ref>.

We have calculated the energy budgets corresponding to each scenario at the known distance of SNR G28.6-0.1 to compare between leptonic and haronic scenarios for the SNR. The proton density in the ambient gas of HAWC J1844-034, $n_{h,d_{\mathrm{src}}}$, is required for the calculation. We use $N\left(\mathrm{HI}\right) = 1.8 \times 10^{21} ~\mathrm{cm}^{-2}$ as the HI column density and $N\left(\mathrm{H}_{2}\right) = 6.7 \times 10^{21} ~\mathrm{cm}^{-2}$ as the ${\mathrm{H}}_{2}$ column density, collected from the HI4PI survey and FUGIN data, respectively, at the center of HAWC J1844-034 [Bekhti et al., 2016, Umemoto et al., 2017]. The integration range of the radial velocity that corresponds to the location of SNR G28.6-0.1 that has been used to calculate the column densities is found to be $77-88~\mathrm{km/s}$ based on Reid et al., 2019. Assuming the density is spherically uniform within the best-fit size of HAWC J1844-034, the proton density of the ambient gas can be estimated as

\begin{eqnarray}
n_{h,d_{\mathrm{src}}} = \frac{\Theta}{V} = \frac{3(N\left(\mathrm{HI}\right)+2N\left(\mathrm{H}_{2}\right))}{4} \nonumber \\
\times \frac{180\degr}{0\fdg48~\pi d_{\mathrm{src}}},
\label{eq:hadron_density}
\end{eqnarray}

where $\Theta$ is the total number of protons, $V$ is the spherical volume of the emission region, and $0\fdg48\pi/180\degr$ is the best-fit size of HAWC J1844-034 in radians. The estimated proton density for SNR G28.6-0.1 is $\sim46~\mathrm{cm^{-3}}$.

The energy budget for each scenario is computed by integrating the particle spectrum $\rho_{\mathrm{P}}(E_{\mathrm{P}})$ provided in Table <ref>,

\begin{equation}
W_{\mathrm{P}} = \int_{~E_{\mathrm{min},\mathrm{P}}}^{E_{\mathrm{max},\mathrm{P}}} dE_{\mathrm{P}}~ E_{\mathrm{P}}\rho_{\mathrm{P}}(E_{\mathrm{P}}),
\label{eq:energybudget}
\end{equation}

where $E_{\mathrm{min},\mathrm{P}}$ and $E_{\mathrm{max},\mathrm{P}}$ are the minimum and maximum energies of the particle spectrum, respectively.

As for the hadronic scenario, we have used 1.22 GeV for the minimum proton energy $E_{\mathrm{min},p}$ since the inclusive $\pi^{0}$ cross section of a proton-proton collision starts to become meaningful at energies above 1.22 GeV <cit.>. For the maximum proton energy $E_{\mathrm{max},p}$, we have used 2.6 PeV, which is ten times the maximum photon energy observed by LHAASO [Cao et al., 2021].

Computed values for SNR G28.6-0.1

2|cHadronic     5|cLeptonic

$L_{\gamma}^{\mathrm{HAWC}}$ [erg/s]     1|c$n_{h,9.6{\mathrm{kpc}}}$ [cm${}^{-3}$]     $W_{p}$ [erg]     1|c$\rho_{0,e\left(9.6\mathrm{kpc}\right)}$ [1/eV]     $\Gamma_{e}$     $E_{c,e}$ [TeV]     $B$ [$\mu $G]     $W_{e}$ [erg]

$1.38^{+0.27}_{-0.17} \times ~ 10^{35}$     ${46}$     $2.37^{+1.59}_{-0.73}\times10^{49}$     $8.03^{+0.54}_{-0.52}\times10^{33}$     $2.37^{+0.01}_{-0.01}$     $92.20^{+6.23}_{-6.70}$     $2.85^{+0.17}_{-0.16}$     $2.60^{+0.54}_{-0.45} \times ~ 10^{50}$

The best-fit parameters for the proton spectrum are same as Table <ref>. The best-fit parameters for the electron spectrum ($\rho_{0e\left(9.6\mathrm{kpc}\right)}$, $\Gamma_{e}$, $E_{c,e}$) have been refitted at 9.6 kpc to include the synchrotron emission spectrum.

For the modeling of SNR G28.6-0.1, radio and infrared data are available as well as the $\gamma$-ray data. Hence, the Naima framework has computed both the synchrotron and inverse Compton scattering spectrum for the leptonic scenario [Castelletti et al., 2021, Lee et al., 2019].
Despite the near-infrared observations at Lee et al., 2019, the near-infrared photons are not considered as seed photons for inverse Compton scattering because photons with energies higher than the CMB are suppressed via the Klein-Nishina effect [Moderski et al., 2005].
With the additional data points, we are able to model and fit the synchrotron regime as well as the inverse Compton scattering regime for the leptonic scenario. Table <ref> shows the best-fit values of the newly defined leptonic scenario. Note that since we use radio data to fit the leptonic model for SNR G28.6-0.1, we have extended $E_{\mathrm{min},e}$ to 10 MeV. Here, we assume $E_{\mathrm{max},e}=740$ TeV, which is calculated in de Oñ a Wilhelmi et al., 2022, Khangulyan et al., 2014 based on the maximum photon energy of LHAASO J1843-0338.

The obtained magnetic field strength of $2.85^{+0.17}_{-0.16}~\mu$G from the Naima fit is comparable to that of the average Galactic magnetic field strength of $2-3~\mu$G [Longair, 2011]. The hadronic energy budget has been derived from the proton spectrum in Table <ref> since the hadronic model has been fitted with only the $\gamma$-ray data.

A spectral distributions plot with the best-fit leptonic (blue) and hadronic (red) scenarios for SNR G28.6-0.1.

The photon spectra obtained from the best-fit scenarios of SNR G28.6-0.1 are presented in Figure <ref>.
The gamma-ray data points are same as those of Figure <ref>. The pink and orange data points are from radio and infrared observations, respectively [Castelletti et al., 2021, Lee et al., 2019], used for the modeling of synchrotron emission.
The final leptonic and hadronic spectra for SNR G28.6-0.1 as a counterpart to HAWC J1844-034 is displayed as blue and red lines, respectively.
The leptonic spectrum (solid) consists of two sub-spectra, synchrotron emission (dash-dotted) and inverse Compton scattering (dotted). The synchrotron and the inverse Compton scattering spectra have been fitted simultaneously while assuming a single population of electrons is responsible for the production of both emissions.

We have not used any X-ray data in the fit because the reported X-ray flux values for SNR G28.6-0.1 from Bamba et al., 2001, Ueno et al., 2003 do not contain any flux uncertainties. Since the X-ray band is located in the keV cutoff region of the synchrotron emission, having the X-ray data in the fit model can alter the spectral break parameter and directly influence the spectral break of the inverse Compton scattering as well. Therefore, incorporating reliable X-ray data should help us obtain a more complete understanding of SNR G28.6-0.1 in the future.

The typical energy of a type Ia supernova explosion is expected to reach $\sim10^{51}$ erg [Khokhlov et al., 1993]. According to our fitted model, the energy budget for the leptonic scenario is $\sim30~\%$ of the typical explosion energy. As for the hadronic scenario, the energy budget is estimated to be $\sim2$ % of the explosion energy, which is more than sufficient [Tatischeff, 2008]. Furthermore, from Amenomori et al., 2022, the diffusion timescale for protons can be estimated using,

\begin{eqnarray}\label{ProtonDiffusionTimescale}
\tau_{\mathrm{diff}}\sim1.2\times10^{4}~\chi^{-1}\left(\frac{R_{\mathrm{sys}}}{20~\mathrm{pc}}\right)^{2} ~~~~~~~~~~\nonumber \\
\times \left(\frac{E_{p}}{\mathrm{GeV}}\right)^{-0.5}\left(\frac{B}{10~\mu\mathrm{G}}\right)^{0.5}~\mathrm{yr},
\end{eqnarray}

where $\chi$ is the factor of diffusion suppression, $R_{\mathrm{sys}}$ is the radius of the emission region.
For the calculation, $\chi$ of 0.1 and $B$ of $10~\mu$G are assumed [Amenomori et al., 2022]. Using $R_{\mathrm{sys}} = (0\fdg48~\pi d_{\mathrm{src}})/180\degr \sim 80$ pc and $E_{p} = 169$ TeV (cutoff energy from the best-fit proton spectrum), we can obtain $\tau_{\mathrm{diff}}$ of $\sim5$ kyr, which is indeed smaller than the known age of SNR G28.6-0.1 of 19 kyr [Ranasinghe & Leahy, 2018].

§ CONCLUSION

In this work, the previous eHWC J1842-035 region has been thoroughly studied with improved dataset based on a systematic approach that precisely models the region. The best-fit model of the region includes three sources within the ROI. Of the three sources, HAWC J1844-034 has been found to have the highest TS with the best-fit location of ($\alpha,~\delta$)=($281\fdg02^{+0.05}_{-0.05},~-3\fdg64^{+0.05}_{-0.04}$) and a best-fit extension of $0\fdg48^{+0.02}_{-0.02}$. The highest $\gamma$-ray photon energy of HAWC J1844-034 is measured at 175 TeV with a differential flux $1.32^{+0.10}_{-0.09}\times10^{-14}$ (TeV cm${}^{2}$ s)${}^{-1}$ at 20 TeV. By increasing the statistics from 1,039 days to 1,910 days and lowering the energy threshold from 56 TeV to 17.8 TeV, HAWC J1844-034 could be studied in great depth.

We have confirmed that HAWC J1844-034 has spectral and spatial coherence with the previous multi-TeV observations, LHAASO J1843-0338 and TASG J1844-038. The most-likely sub-TeV counterpart has been determined to be HESS J1843-033, which has a similar flux magnitude to HAWC J1844-034. There is no plausible GeV counterpart for HAWC J1844-034 at this time.

We have fitted particle spectra based on two scenarios to find the origin of the $\gamma$-ray emission from HAWC J1844-034. The hadronic scenario shows the proton cutoff region at $\sim169$ TeV, so classifying HAWC J1844-034 as a PeVatron would be challenging.

Furthermore, to identify the nature of HAWC J1844-034, we have examined four nearby pulsars and a supernova remnant. We have found PSR J1844-0346 as the most favorable candidate among the nearby pulsars since it is the only pulsar that satisfies both the spin-down power budget and having the pulsar contained within the extension of HAWC J1844-034.

SNR G28.6-0.1 is another potential counterpart source that has spatial coherence with HAWC J1844-034 since it is also located within the extension of HAWC J1844-034. For the hadronic case of SNR G28.6-0.1, the diffusion timescale of the protons is reasonable when compared with the estimated age of the SNR, and it has a plausible energy budget to be a counterpart of HAWC J1844-034. The leptonic scenario is also viable for SNR G28.6-0.1. It would require $\sim30$ % of the total explosion energy used as the acceleration of particles.

§ ACKNOWLEDGEMENTS

We acknowledge the support from the US National Science Foundation (NSF); the US Department of Energy Office of High-Energy Physics; the Laboratory Directed Research and Development (LDRD) program of Los Alamos National Laboratory; Consejo Nacional de Ciencia y Tecnología (CONACyT), México, grants 271051, 232656, 260378, 179588, 254964, 258865, 243290, 132197, A1-S-46288, A1-S-22784, cátedras 873, 1563, 341, 323, Red HAWC, México; DGAPA-UNAM grants IG101320, IN111315, IN111716-3, IN111419, IA102019, IN112218; VIEP-BUAP; PIFI 2012, 2013, PROFOCIE 2014, 2015; the University of Wisconsin Alumni Research Foundation; the Institute of Geophysics, Planetary Physics, and Signatures at Los Alamos National Laboratory; Polish Science Centre grant, DEC-2017/27/B/ST9/02272; Coordinación de la Investigación Científica de la Universidad Michoacana; Royal Society—Newton Advanced Fellowship 180385; Generalitat Valenciana, grant CIDEGENT/2018/034; Chulalongkorn Universitys CUniverse (CUAASC) grant; Instituto de Física Corpuscular, Universitat de València grant E-46980; and National Research Foundation of Korea grant NRF-2021R1A2C1094974. Thanks to Scott Delay, Luciano Díaz, and Eduardo Murrieta for technical support.

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# London-like tensor modes of gravitational waves in cosmic string cosmology

Claudio Aravena-Plaza<EMAIL_ADDRESS>(corresponding author)
Departamento de Física, Facultad de Ciencias, Universidad de Chile, Santiago
7800003, Chile.  Víctor Muñoz  Departamento de Física, Facultad de Ciencias,
Universidad de Chile, Santiago 7800003, Chile.  Felipe A. Asenjo  Facultad de
Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Santiago 7491169, Chile.

###### Abstract

From a classical analysis, we show that gravitational waves in a cosmological
medium with equation of state $\omega=-1/3$ can follow a London-like equation,
implying that some gravitational wave solutions present a decay for certain
wavelengths. This scenario, corresponding to a cosmic string cosmology,
induces an attenuation temporal scale on the gravitational wave propagation.
We discuss on how these solutions impose a limit on the wavelength of the
waves that can propagate, which depends on the type of spatial curvature and
the energy density content of this type of cosmology.

##  I Introduction

From the linearization of the Einstein equations for a
Friedmann–Lemaître–Robertson–Walker (FLRW) cosmological background metric, the
scalar, vector and tensor modes for gravitational perturbations are
straightforwardly obtained weinberg . The tensor modes are of particular
importance because they describe gravitational waves on this cosmology
weinberg . The detection of these waves is important because of the idea of an
inflationary primitive Universe producing primordial gravitational waves that,
if measured, would be a substantial contribution to understanding the
evolution of our Universe osti_981884 .

In general, the study of tensor modes for a FLRW background metric can be
performed exactly. Their dynamical evolution is determined by the equation of
state of the cosmological content of the Universe, mainly described by a
perfect fluid with pressure $p$ and its energy density $\rho$, and thus,
equation of state $\omega=p/\rho$. The different values of $\omega$
characterize the evolution of the scale factor of the universe. The
distinction is usually made between oscillatory solutions with a cutoff value
of $\omega>-1/3$, where the solutions oscillate and damp, and with
$\omega<-1/3$, where the oscillation occurs and then freezes at some value.

On the other hand, the characterization of the Universe for $\omega=-1/3$ lies
within the so-called dark energy cosmologies, and consists of lineal-like
forms of energy, called the cosmic strings. A cosmic string could arise as a
topological defect that occurred during a phase transition early in the
Universe Kibble_1976 ; Bormotova ; vilenkin . Even if the strings formed
before the last epoch of inflation, they were thought to have been diluted by
inflation and would be too few to be cosmologically observable. However strong
evidence for a common spectrum process has recently been reported by the
NANOGrav Collaboration, which was interpreted as a stochastic gravitational
wave background (SGWB) in the framework of cosmic strings PhysRevD.107.042003
. The importance of this measurement opens the field of observation of
primordial background waves that would allow us to obtain information about
the Universe at this stage nanograv . Cosmic strings can produce primordial
gravitational waves, such as gravitational wave bursts and SGWBs Siemens_2007
; https://doi.org/10.48550/arxiv.2207.03510 ;
https://doi.org/10.48550/arxiv.2006.13872 ; CMB ; Emond_2022 ; Caprini_2018 .
In general, to show that a universe with a cosmic string can produce
primordial gravitational waves, models described by some action are used, for
example, using the Nambu-Goto action auclair2021window , or Lagrangian models
Emond_2022 .

Unlike those works, here we carry out a first order tensor mode analysis for
gravitational waves in a Universe filled with a perfect fluid with equation of
state $\omega=-1/3$. We show that dynamics follows a London equation. This
implies that when the medium has this equation of state, there exist
conditions for which gravitational waves cannot propagate through it, as they
fully damp on a specific temporal scale. London equations have been studied
for linear corrections of gravitational fields in flat spacetime Ummarino ,
but not in the context of cosmological gravitational waves.

The studied gravitational wave modes in this work correspond to spacetime
perturbations of the FLRW background metric, which is described by the
spacetime interval

$\displaystyle
ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=a^{2}\left(-d\eta^{2}+\gamma_{ij}dx^{i}dx^{j}\right)\,,$
(1)

where $a=a(t)$ is the scale factor of the Universe, and $x^{\mu}=(t,x^{i})$
(with $\mu,\nu=0,1,2,3$, and $i,j=1,2,3$). We have chosen the speed of light
$c=1$. In the following we describe the dynamics in terms of the cosmological
time $\eta=\int dt/a$. The three-dimensional spatial metric $\gamma_{ij}$ can
be represented in spherical coordinates as
$\gamma_{ij}dx^{i}dx^{j}=dD^{2}+D_{A}^{2}d\Omega$, where $D$ is the comoving
distance and $D_{A}=K^{-1/2}\sin(K^{1/2}D)$ is the angular diameter distance.
$K$ is a constant representing the spatial curvature of the Universe ($K=0$
for spatially flat, $K=1$ for closed Universe, and $K=-1$ for open Universe).

By linearizing the Einstein equations with this cosmological background
metric, the equation for the tensor mode, in absence of corrections to the
stress anisotropy, is obtained from Eq. (10) of Appendix A. By defining the
effective gravitational wave $y$ by the relation $H_{T}=y/a$, then we get

$\ddot{y}+\left(k^{2}+2K-\frac{\ddot{a}}{a}\right)y=0\,,$ (2)

where the overdot stands for the conformal time derivative, and we have
assumed the spatial behavior of the wave is sinusoidal $\nabla^{2}y=-k^{2}y$,
with $k$ the wavenumber of the gravitational wave. Also, the background FLRW
metric evolves following the equation ${\ddot{a}}/{a}=({4\pi
G}/{3})a^{2}\rho-K$, whereas the fluid content satisfies the continuity
equation $\dot{\rho}/\rho=-3\left(1+\omega\right)\left(\dot{a}/a\right)$,
whose solution is $\rho=\rho_{0}a^{-3(1+\omega)}$, where $\rho_{0}$ is initial
energy density. Substituting in Eq. (2), we finally obtain

$\ddot{y}+\left(k^{2}+3K-\frac{4\pi
G}{3}a^{-1-3\omega}\rho_{0}(1-3\omega)\right)y=0\,.$ (3)

By knowing the type of content of the Universe, and thus, how it evolves, by
means of $a$, diverse forms of gravitational propagation can be found from
this equation Caldwell_1993 ; grishchuk1974amplification ;
Peter_K_S_Dunsby_1997 .

##  II London-like behavior

Historically, the London equation is a phenomenological classical, way to
relate the current in a superconductor with the electric and magnetic fields
inside and outside of it. The idea is that electrons move freely inside a
superconductor, following equations detailed in Appendix B. Thereby, we obtain
that the magnetic field $\mathbf{B}$ can penetrate inside a superconductor,
and satisfies the London equation $\nabla^{2}\mathbf{B}=\mathbf{B}/{l^{2}}$,
where $l=\sqrt{m/4\pi ne^{2}}$ is called the penetration length of the
magnetic field. Here, $m$ is electron mass, and $n$ and $e$ are the electron
density and charge respectively. The solution of this equation describes
classically the Meissner effect, for an exponentially screened magnetic field
inside a superconductor, depending on the medium through the penetration
length london ; tinkham .

In this section we show that there exists a cosmological scenario where
gravitational wave behavior resembles this exponentially screened behavior,
i.e., Eq. (3) acquires a London-like form. In Eq. (3), let us consider the
cosmic string equation of state $\omega=-1/3$. In that case, there is no
explicit dependence on $a$, and it becomes simply

$\ddot{y}+\left(k^{2}+3K-\frac{8\pi G}{3}\rho_{0}\right)y=0\,.$ (4)

This equation presents a London-like behavior for gravitational waves as it
can be written as

$\ddot{y}=\frac{1}{\Omega^{2}}y\,,$ (5)

where

$\Omega=\left(\frac{8\pi G}{3}\rho_{0}-k^{2}-3K\right)^{-1/2}\,$ (6)

is an attenuation conformal cosmological temporal-scale, that depends on the
initial energy density content of this cosmic string universe $\rho_{0}$, the
spatial curvature, and the wavelength of the gravitational field wave. This
attenuation temporal-scale is valid only when the gravitational wave
wavenumber satisfies

$\frac{8\pi G}{3}\rho_{0}-3K>k^{2}\,.$ (7)

Instead of propagating, now the gravitational wave field is exponentially
screened in certain cosmological time $\Omega$. This is seen from the physical
solution of Eq. (5), which reads

$y(\eta)=y_{0}\exp\left(-\frac{\eta}{\Omega}\right)\,$ (8)

where $y_{0}$ is an integration constant, representing the initial amplitude
of the wave. This behavior for cosmological gravitational wave fields is the
temporal analogue to the London behavior of superconductivity. However, the
temporal attenuation of the gravitational perturbation does not only depend on
the medium (as in the electromagnetic London case) but also on its own wave
features. In effect, similar to the reflection of long wavelength waves in
magnetized plasmas, gravitational waves with small wavenumbers can satisfy Eq.
(7), and thus it is expected that no long wavelength gravitational waves
propagate in this era; they are all exponentially screened in a conformal
cosmological temporal scale of the order of $\Omega$.

##  III Discussion

The above conditions for tensor modes, obtained for a cosmology with a
Universe filled with matter with $\omega=-1/3$ equation of state, determine
the passage of gravitational waves through it or not. This is the typical
result derived from a London-like equation, such as Eq. (5). Under condition
(7), the modes gets damped in conformal time. This only occurs for
gravitational waves. For example, electromagnetic waves always propagate as
plane waves in isotropic cosmology, suffering no damping from the medium.

As a proper estimation, let us consider the spatially flat Universe case.
Then, any gravitational wave perturbation with wavelength $\lambda=2\pi/k$
larger than $\sqrt{3\pi/(2G\rho_{0})}$, becomes time screened from
propagation. The energy density of cosmic strings at formation can be put in
the form $\rho_{0}\sim\mu/L_{0}^{2}$, where $\mu$ is the linear mass density
of the string and $L_{0}$ is a characteristic length scale of their pseudo-
Brownian trajectories vilenkin . Thereby, any temporal attenuated
gravitational wave in this cosmology will have a wavelength that satisfies

$\frac{\lambda}{L_{0}}>\sqrt{\frac{3\pi}{2G\mu}}\,.$ (9)

If the linear mass density of cosmic strings lie between $\mu\sim 10^{-4}$
g/cm on the electroweak scale, to $\mu\sim 10^{22}$ g/cm for grand unification
strings vilenkin , then, we have that $\lambda/L_{0}>2.5\times
10^{16}$–$10^{3}$, from the electroweak scale to grand unification strings. On
the contrary, when $k^{2}$ does not satisfy condition (7), then cosmlogical
gravitational waves can travel through this medium.
Recently, it has been argued that gravitational waves originating from cosmic
cues of the SGWB type have been detected PhysRevLett.126.041304 ;
BUCHMULLER2020135914 . Although this interpretation is controversial
PhysRevD.103.103512 , this goes in the direction of advancing the measurement
of an analogue of the microwave background, but with primordial gravitational
waves auclair2021window ; PhysRevD.103.043023 . Our result imposes wavelength
conditions that establish which gravitational waves are damped, and therefore
detectable or undetectable. An estimate of the value of the density $\rho_{0}$
can even be obtained by knowing the specific wavelength constraints and the
respective attenuation time scale for this specific type of Universe.

###### Acknowledgements.

CAP thanks to ANID No. 7322/2020. This project has been financially supported
by FONDECyT under contracts No. 1201967 (VM), and No. 1230094 (FAA).

##  IV APPENDIX

###  IV.1 Tensorial gravitational modes in FLRW cosmology

The gravitational tensor modes $H_{T}$ for a spacetime perturbation of the
FLRW metric, without dissipative corrections to the inertia tensor, follow the
wave equation (weinberg, )

$\nabla^{2}H_{T}-{a^{2}}\left(\dfrac{d^{2}}{dt^{2}}H_{T}\right)-3{a}\left(\dfrac{d}{dt}a\right)\left(\dfrac{d}{dt}H_{T}\right)=0\,.$
(10)

The evolution of these modes as a function of the cosmological time $\eta$ can
be obtained by noticing that $\dot{a}\equiv{da}/{d\eta}=a\,{da}/{dt}$.
Incorporating curvature constant $K$ of the Universe, we obtain

$-\nabla^{2}H_{T}+\ddot{H}_{T}+2\frac{\dot{a}}{a}\dot{H}_{T}+2KH_{T}=0,$ (11)

where $\dot{H}_{T}\equiv{d}H_{T}/d\eta$.

###  IV.2 London equation

The London equation was formulated to have a classical phenomenological
explanation of superconductivity. It is derived from the equations

$\nabla\times\mathbf{j}_{s}=-\frac{n_{s}e^{2}}{m}\mathbf{B}\,,\qquad\nabla\times{\mathbf{B}}=4\pi{\mathbf{j}}_{s}\,,$
(12)

where $\mathbf{j}_{s}$ is the current in a superconductor, $\mathbf{B}$ is the
magnetic field, $e$ is the electron charge, $m$ is the electron mass, and
$n_{s}$ is the number density of particles moving within the superconductor.
By combining them, we obtain the London equation.

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