11institutetext:  Max-Planck-Institut für Radioastronomie, Auf-dem-Hügel 69,
53121 Bonn, Germany<EMAIL_ADDRESS>22institutetext: Astron. Dept.,
Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589,
Saudi Arabia 33institutetext: Xinjiang Astronomical Observatory, Chinese
Academy of Sciences, 830011 Urumqi, China 44institutetext: European Southern
Observatory, Alonso de Córdova, 3107, Vitacura, Santiago 763-0355, Chile
55institutetext: Joint ALMA Observatory, Alonso de Córdova, 3107, Vitacura,
Santiago 763-0355, Chile 66institutetext: National Radio Astronomy
Observatory, 520 Edgemont Road, Charlottesville, VA 22903-2475, USA
77institutetext: National Astronomical Observatory of Japan, 2-21-1 Osawa,
Mitaka, Tokyo 181-8588, Japan 88institutetext: Institute of Astronomy and
Astrophysics, Academia Sinica, 11F of AS/NTU Astronomy-Mathematics Building,
No.1, Sec. 4, Roosevelt Rd, Taipei 10617, Taiwan 99institutetext: Department
of Astronomy, School of Science, The Graduate University for Advanced Studies
(SOKENDAI), 2-21-1 Osawa, Mitaka, Tokyo, 181-1855 Japan 1010institutetext:
Department of Space, Earth and Environment, Chalmers University of Technology,
Onsala Space Observatory, SE-43992 Onsala, Sweden 1111institutetext:
Department of Physics, Faculty of Science and Technology, Keio University,
3-14-1 Hiyoshi, Yokohama, Kanagawa 223–8522 Japan 1212institutetext: Institute
of Astronomy, Graduate School of Science, The University of Tokyo, 2-21-1
Osawa, Mitaka, Tokyo 181-0015, Japan 1313institutetext: Argelander-Institut
für Astronomie, Universität Bonn, Auf dem Hügel 71, D-53121 Bonn, Germany
1414institutetext: Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer
de Magrans, E-08193 Barcelona, Spain 1515institutetext: IRAP, Université de
Toulouse, CNRS, UPS, CNES, Toulouse, France 1616institutetext: Leiden
Observatory, Leiden University, PO Box 9513, NL - 2300 RA Leiden, The
Netherlands 1717institutetext: Department of Physics and Astronomy, University
College London, Gower Street, London WC1E6BT, UK 1818institutetext: Centro de
Astrobiología (CSIC-INTA), Ctra. de Torrejón a Ajalvir km 4, 28850, Torrejón
de Ardoz, Madrid, Spain 1919institutetext: INAF Osservatorio Astrofisico di
Arcetri, Largo Enrico Fermi 5, I-50125 Firenze, Italy 2020institutetext: New
Mexico Institute of Mining and Technology, 801 Leroy Place, Socorro, NM 87801,
USA 2121institutetext: National Radio Astronomy Observatory, PO Box O, 1003
Lopezville Road, Socorro, NM 87801, USA

# Methanol masers in NGC 253 with ALCHEMI

P. K. Humire Methanol masers in NGC 253 with ALCHEMIMethanol masers in NGC 253
with ALCHEMI C. Henkel Methanol masers in NGC 253 with ALCHEMIMethanol masers
in NGC 253 with ALCHEMIMethanol masers in NGC 253 with ALCHEMIMethanol masers
in NGC 253 with ALCHEMIMethanol masers in NGC 253 with ALCHEMIMethanol masers
in NGC 253 with ALCHEMI A. Hernández-Gómez Methanol masers in NGC 253 with
ALCHEMIMethanol masers in NGC 253 with ALCHEMI S. Martín Methanol masers in
NGC 253 with ALCHEMIMethanol masers in NGC 253 with ALCHEMIMethanol masers in
NGC 253 with ALCHEMIMethanol masers in NGC 253 with ALCHEMI J. Mangum Methanol
masers in NGC 253 with ALCHEMIMethanol masers in NGC 253 with ALCHEMI N.
Harada Methanol masers in NGC 253 with ALCHEMIMethanol masers in NGC 253 with
ALCHEMIMethanol masers in NGC 253 with ALCHEMIMethanol masers in NGC 253 with
ALCHEMIMethanol masers in NGC 253 with ALCHEMIMethanol masers in NGC 253 with
ALCHEMI S. Muller Methanol masers in NGC 253 with ALCHEMIMethanol masers in
NGC 253 with ALCHEMI K. Sakamoto Methanol masers in NGC 253 with
ALCHEMIMethanol masers in NGC 253 with ALCHEMI K. Tanaka Methanol masers in
NGC 253 with ALCHEMIMethanol masers in NGC 253 with ALCHEMI Y. Yoshimura
Methanol masers in NGC 253 with ALCHEMIMethanol masers in NGC 253 with ALCHEMI
K. Nakanishi Methanol masers in NGC 253 with ALCHEMIMethanol masers in NGC 253
with ALCHEMIMethanol masers in NGC 253 with ALCHEMIMethanol masers in NGC 253
with ALCHEMI S. Mühle Methanol masers in NGC 253 with ALCHEMIMethanol masers
in NGC 253 with ALCHEMI R. Herrero-Illana Methanol masers in NGC 253 with
ALCHEMIMethanol masers in NGC 253 with ALCHEMIMethanol masers in NGC 253 with
ALCHEMIMethanol masers in NGC 253 with ALCHEMI D. S. Meier Methanol masers in
NGC 253 with ALCHEMIMethanol masers in NGC 253 with ALCHEMIMethanol masers in
NGC 253 with ALCHEMIMethanol masers in NGC 253 with ALCHEMI E. Caux Methanol
masers in NGC 253 with ALCHEMIMethanol masers in NGC 253 with ALCHEMI R.
Aladro Methanol masers in NGC 253 with ALCHEMIMethanol masers in NGC 253 with
ALCHEMI R. Mauersberger Methanol masers in NGC 253 with ALCHEMIMethanol masers
in NGC 253 with ALCHEMI S. Viti Methanol masers in NGC 253 with
ALCHEMIMethanol masers in NGC 253 with ALCHEMIMethanol masers in NGC 253 with
ALCHEMIMethanol masers in NGC 253 with ALCHEMI L. Colzi Methanol masers in NGC
253 with ALCHEMIMethanol masers in NGC 253 with ALCHEMIMethanol masers in NGC
253 with ALCHEMIMethanol masers in NGC 253 with ALCHEMI V. M. Rivilla Methanol
masers in NGC 253 with ALCHEMIMethanol masers in NGC 253 with ALCHEMIMethanol
masers in NGC 253 with ALCHEMIMethanol masers in NGC 253 with ALCHEMI M.
Gorski Methanol masers in NGC 253 with ALCHEMIMethanol masers in NGC 253 with
ALCHEMI K. M. Menten Methanol masers in NGC 253 with ALCHEMIMethanol masers in
NGC 253 with ALCHEMI K.-Y. Huang Methanol masers in NGC 253 with
ALCHEMIMethanol masers in NGC 253 with ALCHEMI S. Aalto Methanol masers in NGC
253 with ALCHEMIMethanol masers in NGC 253 with ALCHEMI P. P. van der Werf
Methanol masers in NGC 253 with ALCHEMIMethanol masers in NGC 253 with ALCHEMI
K. L. Emig Jansky Fellow of the National Radio Astronomy ObservatoryMethanol
masers in NGC 253 with ALCHEMIMethanol masers in NGC 253 with ALCHEMI

(Received xxx, yyy; accepted xxx, yyy)

###### Abstract

Context. Methanol masers of Class I (collisionally-pumped) and Class II
(radiatively-pumped) have been studied in great detail in our Galaxy in a
variety of astrophysical environments such as shocks and star-forming regions
and are helpful to analyze the properties of the dense interstellar medium.
However, the study of methanol masers in external galaxies is still in its
infancy.

Aims. Our main goal is to search for methanol masers in the central molecular
zone (CMZ; inner 500 pc) of the nearby starburst galaxy NGC 253.

Methods. Covering a frequency range between 84 and 373 GHz ($\lambda$ = 3.6 to
0.8 mm) at high angular (1$\aas@@fstack{\prime\prime}$6$\sim$27 pc) and
spectral ($\sim$8–9 km s-1) resolution with the ALMA large program ALCHEMI, we
have probed different regions across the CMZ of NGC 253. In order to look for
methanol maser candidates, we employed the rotation diagram method and a set
of radiative transfer models.

Results. We detect for the first time masers above 84 GHz in NGC 253, covering
an ample portion of the $J_{-1}\rightarrow(J-$ 1)${}_{0}-E$ line series (at
84, 132, 229, and 278 GHz) and the $J_{0}\rightarrow(J-$ 1)${}_{1}-A$ series
(at 95, 146, and 198 GHz). This confirms the presence of the Class I maser
line at 84 GHz, already reported but now being detected in more than one
location. For the $J_{-1}\rightarrow(J-$ 1)${}_{0}-E$ line series, we observe
a lack of Class I maser candidates in the central star-forming disk.

Conclusions. The physical conditions for maser excitation in the
$J_{-1}\rightarrow(J-$ 1)${}_{0}-E$ line series can be weak shocks and cloud-
cloud collisions as suggested by shock tracers (SiO and HNCO) in bi-symmetric
shock/active regions located in the outskirts of the CMZ. On the other hand,
the presence of photodissociation regions due to a high star-formation rate
would be needed to explain the lack of Class I masers in the very central
regions.

###### Key Words.:

galaxies: individual (NGC 253) – galaxies: starburst – masers – radio lines:
galaxies

##  1 Introduction

Methanol (CH3OH) is a molecule prone to population inversion under specific
excitation conditions in the interstellar medium (ISM) (e.g. Cragg et al.
1992), causing maser emission. In particular, methanol masers are unique tools
for studying physical properties of dense gas associated with Young Stellar
Objects (YSOs). Given their brightness and compactness (Menten 1991), their
positions can be determined with high precision astrometry (i.e., at milli-
arcsecond accuracy with very long baseline interferometry) and over vast
distances.

Thousands of such methanol masers have been detected in the Milky Way (Cotton
& Yusef-Zadeh 2016; Green et al. 2017; Yang et al. 2019). However, in nearby
galaxies we can only account for a handful of successful detections (e.g.
McCarthy et al. 2020). In particular, the brightest Galactic CH3OH maser
transition at 6.7 GHz (Breen et al. 2015), remains elusive in extragalactic
objects outside the Local Group (Ellingsen et al. 1994; Darling et al. 2003),
with the only exception of NGC 3079 (Impellizzeri et al. 2008), and possibly
Arp220 (Salter et al. 2008), where this line is detected in absorption.

The early discovery that CH3OH masers can be divided into two classes, a
collisionally-pumped Class I and a radiatively-pumped Class II (Batrla et al.
1987; Menten 1991), allows us to trace either stellar-induced outflows (Class
I) or ultra-compact H ii regions (Class II). Class I methanol masers have been
observed toward high and low-mass stars (Kalenskii et al. 2006, 2010;
Rodríguez-Garza et al. 2017), while Class II masers have been observed only
toward high-mass YSOs (Breen et al. 2013). Unlike H2O and OH masers, Class II
methanol masers seem to be exclusively correlated with star-forming regions
(Walsh et al. 2001; Breen et al. 2013).

Because Class II masers are usually brighter than Class I masers in our
Galaxy, the former have been studied in great detail, leading to surveys
targeting exclusively their relation with the surrounding conditions (Yang et
al. 2017; Billington et al. 2020). However, outside our Galaxy Class II masers
were only detected in the Magellanic Clouds (Sinclair et al. 1992; Green et
al. 2008; Ellingsen et al. 2010) and the Andromeda galaxy (Sjouwerman et al.
2010), with luminosities not surpassing those in our Galaxy. On the other
hand, extragalactic Class I masers can be more luminous than those of Class
II, and have been successfully observed beyond the Local Group, particularly
in nearby barred spiral galaxies like NGC 253, IC 342, or NGC 4945 (e.g.
Ellingsen et al. 2014; McCarthy et al. 2017; Gorski et al. 2018).

There are two types of methanol. For E-type methanol, one of the protons in
the hydrogen atoms of the methyl (CH3) group has an anti-parallel nuclear spin
with respect to the others, analogous to the case of para-NH3. In the A-type
methanol, the nuclear spins of the three protons in the methyl group are
parallel, as in the case of ortho-NH3. As the two methanol types have
different transition frequencies and may arise in different physical
environments, we decided to analyse them separately.

Hereafter we use the conventional notation for A+ and A- introduced by Lees &
Baker (1968), related to a combination between the A–CH3OH overall-parity and
Mulliken symbols111https://mathworld.wolfram.com/CharacterTable.html A1 and
A2. This is done to discriminate between splitted $+K$ and $-K$ levels
(doublets), with $K$ being the projection of the angular momentum along the
molecular symmetry axis. $+K$ and $-K$ levels are torsionally degenerate for
the case of A-type methanol, contrary to the case of E-type methanol (see also
Cragg et al. 1993).

As shown by Lees (1973), among E-type methanol transitions (E–CH3OH) Class I
population inversion is favoured in the $K=-$1 relative to the $K=0$ or $K=-$2
ladders. This leads to the prominence of the $J_{-1}\rightarrow(J-$
1)${}_{0}-E$ series (see e.g. Leurini et al. 2016, their Fig. 3). Indeed, the
$4_{-1}\rightarrow 3_{0}-E$ (36.2 GHz) and $5_{-1}\rightarrow 4_{0}-E$ (84.5
GHz) lines have been recently discovered to be masing in one extragalactic
object, NGC 253 (Ellingsen et al. 2014; McCarthy et al. 2018). For A-type
methanol, population inversion in the $K=0$ relative to the $K=1$ ladder is
favoured, playing out in the $J_{0}\rightarrow(J-$1)${}_{1}-A^{+}$ series, of
which the 44.1 GHz line, at $J=$7, has been also detected in NGC 253
(Ellingsen et al. 2017). The emission of Galactic Class II methanol masers are
more compact than those of their Galactic Class I cousins (Moscadelli et al.
2003; Matsumoto et al. 2014), and that could make Class II methanol masers
more difficult to detect at extragalactic distances. In addition, Class I
masers require lower densities and temperatures than Class II masers (Menten
2012), making them more numerous. Specifically for the $J_{-1}\rightarrow(J-$
1)${}_{0}-E$ series with $J=$ 4 and 5, their intensities were predicted to be
of the order of 50 mJy in the case of NGC 253 (Sobolev 1993).

Class I methanol masers may be associated with a variety of phenomena, such as
supernova remnants (Plambeck & Menten 1990; Pihlström et al. 2014), massive
protostellar induced outflows (Cyganowski et al. 2018), and interactions of
expanding H ii regions with surrounding molecular gas (Voronkov et al. 2010),
namely regions where shocks compress and heat the gas. In the central
molecular zone (CMZ) of our Galaxy (i.e., the inner $\sim$200 pc in radius;
Morris & Serabyn 1996), cosmic ray interactions with molecular clouds have
been claimed to be an additional source of methanol production. While a high
methanol abundance alone is certainly not sufficient to trigger maser
emission, there seems to be indeed a clear enhancement of Class I masers in
this region (Yusef-Zadeh et al. 2013; Cotton & Yusef-Zadeh 2016; Ladeyschikov
et al. 2019). In particular, extended strong emission showing characteristics
of maser action had been found by Haschick & Baan (1993) and Salii et al.
(2002) in the G1.6$-$0.025 region at the periphery of the CMZ and by
Szczepanski et al. (1989) and Liechti & Wilson (1996) in a region in which the
supernova remnant Sgr A East interacts with a Giant Molecular Cloud (GMC).
Noteworthy, CMZ conditions in the Milky Way should provide some guidance to
the central regions of starburst galaxies (Belloche et al. 2013).

Class I methanol masers at 36, 44, and 84 GHz have already been reported in
NGC 253 by Ellingsen et al. (2014), Ellingsen et al. (2017), and McCarthy et
al. (2018), respectively. The former, at 36 GHz, has been also observed in NGC
4945, IC 342, NGC 6946, and Maffei 2 (McCarthy et al. 2017; Gorski et al.
2018; Humire et al. 2020). A 84 GHz mega-maser ($\geq$106 times more luminous
than typical Galactic masers; Lo 2005) has been reported in NGC 1068 (Wang et
al. 2014), but a confirmation would be needed to put this onto a firm basis.
Although a tentative detection of methanol mega-maser emission was claimed in
Arp 220 (Chen et al. 2015), later studies ruled it out (Humire et al. 2020;
McCarthy et al. 2020). Making use of the unprecedented spectral coverage
available by the Atacama Large Millimeter/sub-millimeter Array (ALMA) Large
Program ALCHEMI (Martín et al. 2021), we have been able to make a
comprehensive study toward one of the best candidates to search for
extragalactic maser emission, the Sculptor galaxy NGC 253.

NGC 253 is a nearby (D $\sim$3.5 Mpc, Rekola et al. 2005) highly inclined
($i\sim 70^{\circ}$–79∘, Pence 1980; Iodice et al. 2014) SAB(s)c galaxy (de
Vaucouleurs et al. 1991) with a systemic heliocentric velocity ($v_{\rm sys}$)
of $\sim$258.8 km s-1 (Meyer et al. 2004). Its large-scale bar feeds the
nuclear region producing stars at an approximate rate of $\sim$1.7 M⊙ yr-1
(Bendo et al. 2015) within the central 20″$\times$10″ , where 1″ corresponds
to $\sim$17 pc. The position angle (PA) of the large-scale bar is 51∘ (Pence
1980). The PA continues till the central $\sim$170 parsecs. Further into the
core, the isovelocity contours of the gas change their orientation by about
90∘ due to the existence of a nuclear bar (Cohen et al. 2020). Like our
Galaxy, NGC 253 is characterized by a particularly strong and diverse
molecular emission in its central 500 pc (Sakamoto et al. 2006), which we
therefore identify as its CMZ. Previous studies do not suggest that an Active
Galactic Nucleus (AGN) is important for the properties of the molecular gas in
the CMZ of NGC 253 (Müller-Sánchez et al. 2010).

This paper is organized as follows: our observations are described in Sect. 2.
In Sect. 3 we show the selected regions to be studied in the CMZ of NGC 253.
In Sect. 4 we introduce the two methods used to identify methanol masers. In
Sect. 5 we discuss the results and provide a number of conditions to explain
maser emission. We finally draw the conclusions of our findings in Sect. 6.

##  2 Observations

We use ALMA observations of NGC 253 taken in Cycles 5 and 6 as part of the
large program ALCHEMI (ALMA Comprehensive High-resolution Extragalactic
Molecular Inventory, 2017.1.00161.L, followed up by program 2018.1.00162.S,
see Martín et al. 2021). The central region of the galaxy ($50\arcsec\times
20\arcsec$ = $850\times 340$ pc, with a position angle of $65^{\circ}$) was
covered in bands 3 to 7 (84.2 – 373.2 GHz), with both the 12 m and 7 m antenna
arrays, achieving a final homogeneous angular resolution of
1$\aas@@fstack{\prime\prime}$6\. The mentioned region was covered with a
single pointing in band 3, with a primary beam ranging between 57$\arcsec$ and
68$\arcsec$, and Nyquist-sampled mosaic patterns of 5 up to 19 pointings for
the remaining bands, ensuring a homogeneous sensitivity across the selected
region222ALMA sets up by default a hexagonal Nyquist sampled pattern to ensure
homogeneous sensitivity across the mapped region. Furthermore, mosaic images
are primary beam corrected, in our case this includes the studied $850\times
340$ pc region. For more information about mosaicing, we refer the reader to
Sect 7.7 in the ALMA technical Handbook
(https://almascience.eso.org/documents-and-tools/cycle9/alma-technical-
handbook). The phase center of the observations is
$\alpha=00^{h}47^{m}33^{s}.26$,
$\delta=-25^{\circ}17^{\prime}17\aas@@fstack{\prime\prime}7$
(ICRS333International Celestial Reference System, equivalent to J2000 within a
few milli–arcseconds (Ma et al. 1998)).

The absolute flux calibration accuracy is of the order of 15% for all ALMA
bands, since most percentages are lower across individual bands. The flux
density RMS noise ranges from 0.18 to 5.0 mJy beam-1, and the averaged
sensitivity is 14.8 mK. For more details about the data reduction procedures
and image processing, see Martín et al. (2021).

##  3 Selected positions

In Fig. 1, we show the integrated intensity map of the
6${}_{-1}\rightarrow$5${}_{0}-E$ methanol emission (132.9 GHz rest frequency)
using a 3$\sigma$ clip threshold and applying the formalism suggested by
Mangum & Shirley (2015, their Appendix C) to obtain the uncertainties. The
selection of this methanol line is motivated by its high signal-to-noise (S/N)
ratio, the absence of notable blending candidates, and maser emission in some
regions (see below). In NGC 253, the inner Lindblad resonance (ILR) was found
to be co-spatial with its circumnuclear ring (CNR) by Iodice et al. (2014).
The total ILR extension measured by them is 0.3$\pm$0.1–0.4$\pm$0.1 kpc
(deprojected size). We draw concentric ellipses at 0.2 and 0.5 kpc to denote
the ILR limits in Fig. 1 (dash-dotted grey ellipses) as well as its center at
0.35 kpc (solid black ellipse). Along the text, we use the ILR position as a
proxy to the x2 orbit in NGC 253 (see Sect. 5).

Figure 1: Integrated intensity map for the $6_{-1}\rightarrow 5_{0}-E$
methanol line at 132.9 GHz (within a velocity interval of $v_{\rm
sys}\lesssim\pm$200 km s-1). Selected positions for spectral extraction (Sect.
3) are encircled in beam-sized apertures and numbered from 1 to 10. The center
and edges of the inner Lindblad resonance from Iodice et al. (2014) are
denoted with a black ellipse and dash-dotted grey ellipses, respectively.
Superbubbles identified by Sakamoto et al. (2006) are indicated as orange
circles. The ALMA beam (1$\aas@@fstack{\prime\prime}$6) is shown in blue in
the bottom left corner. Table 1: Selected regions.

Region | $\alpha_{\rm ICRS}$ | $\delta_{\rm ICRS}$ | $v_{\rm LSR}$
---|---|---|---
| [00h:47m:–s] | [-25∘:17’:–”] | [km s-1]
R1 | 32.02 | 28.3 | 304
R2 | 32.28 | 20.2 | 330
R3 | 32.81 | 21.6 | 286
R4 | 32.97 | 20.0 | 252
R5 | 33.21 | 17.4 | 231
R6 | 33.33 | 15.8 | 180
R7 | 33.64 | 13.3 | 174
R8 | 34.02 | 11.4 | 205
R9 | 34.17 | 12.3 | 201
R10 | 34.24 | 7.8 | 144
444The Local Standard of Rest (LSR) velocities were obtained by fitting a
local thermodynamic equilibrium model (LTE) to the spectra and averaging
between the A– and E–CH3OH symmetry species. Our spectral resolution of
$\sim$8–9 km s-1 dominates the velocity uncertainty.

The presence of methanol masers depends on the physical conditions prevailing
across the CMZ of NGC 253. We decided to select ten distinct positions within
the CMZ (see Figure 1 and Table 1). These positions were established based on
intensity peaks of the CS J=2–1 and H13CN J=1–0 transitions, applying a GMC
identification approach based on Leroy et al. (2015), i.e., using the CPROPS
software (Rosolowsky & Leroy 2006). Our region coordinates are not exactly the
same as those used by Leroy et al. (2015), even though our numbering is close
to theirs. We preferred to use those GMC locations instead of the peak
intensities of our CH3OH lines at a given frequency, because the peak location
varies depending on the chosen transition (see also, Zinchenko et al. 2017;
McCarthy et al. 2018). We list the coordinates and velocities of all these
positions in Table 1. The velocities were obtained from preliminary LTE
radiative transfer models, similar to the ones described in Sect. 4.2, and
averaged over each methanol symmetric type.

For each region, we extracted the full spectrum (84–373 GHz, see Sect. 4.2)
from the data convolved to a common circular beam of
1$\aas@@fstack{\prime\prime}$6 diameter, equivalent to the maximum angular
resolution of the observations (see Sect. 2). At higher resolution, however,
GMCs further divide into molecular clumps with a size range of
0$\aas@@fstack{\prime\prime}$07–0$\aas@@fstack{\prime\prime}$25 (1.2–4.3 pc;
Leroy et al. 2018). Contrary to the spectrum of the other nine studied
regions, we have found a very complex spectrum in region 5 located next to the
dynamical center of the galaxy (Müller-Sánchez et al. 2010). Radio
recombination line observations revealed an S-shaped pattern with complex
kinematics including a counter-rotating core in the inner 2$\arcsec$
suggestive of a secondary bar (Anantharamaiah & Goss 1996) and evidencing a
black hole mass of $\sim$107 M$\odot$ (Cohen et al. 2020). This highly-
perturbed environment is likely causing the crowded spectrum observed in
Region 5. In this position, the broad line emission, with a full width at half
maximum (FWHM) above 100 km s-1, prevents an accurate Gaussian line fitting
for the rotation diagram analysis in Sect. 4.1, but can be used for the
radiative transfer modelling in Sect. 4.2. In addition, toward region 5 and
mostly in the low frequency range ($\sim$84–163 GHz, ALMA bands 3 and 4) we
observe absorption components.

Methanol lines in absorption or self-absorption (in the case of the line at
358.6 GHz) along the entire ALCHEMI spectral coverage are observed in at least
one region for the following four transitions: the 3${}_{1}\rightarrow
4_{0}-A^{+}$ line at 107.0 GHz, the 4${}_{-2}\rightarrow 4_{-0}-E$ line at
190.1 GHz, the 5${}_{-2}\rightarrow 5_{-0}-E$ line at 190.3 GHz (in absorption
in region 10 only), and the 4${}_{1}\rightarrow 3_{0}-E$ line at 358.6 GHz.
They are shown in Figure 6.

Figure 2: Methanol lines observed in absorption or self-absorption in one or
more of the selected regions (Table 1). Red and blue lines are synthetic
spectra for methanol A- and E- type, respectively (see Sect. 4.2). Line
frequencies are labeled in the top of the Figure, while the velocity range (in
km s-1, after applying the radio convention666https://web-
archives.iram.fr/IRAMFR/ARN/may95/node4.html and subtracting the $v_{\rm LSR}$
velocities from Table 1) is indicated in the bottom-left corner. Regions 1 to
10 are ordered from top to bottom as indicated in the leftmost panel (y-axis:
R from ”Region” plus the corresponding number). For the location of the
regions, see Fig. 1 and Table 1.

##  4 Methanol maser emission identification

In some cases, it can be difficult to assess whether a given line is excited
under thermal or maser conditions. Methanol maser lines in Galactic star-
forming regions are usually very bright (in the sense of brightness
temperature) and narrow. However, in the case of extragalactic objects, beam
dilution makes them apparently weaker and less distinguishable from thermal
emission due to line blending from surrounding gas. In the spectral dimension
this also complicates a proper line identification since the contribution of
other species might affect the true brightness of the putative maser lines
(see the last column of Table 5).

In this section we explore two methods to identify methanol emission lines
outside local thermodynamic equilibrium (LTE) as an indicator of potential
maser emission, namely rotation diagram analysis and comparisons with
synthetic spectra from radiative transfer modelling. For both methods, we
constrained the upper energy above the ground level ($E_{\rm up}$/k) of the
lines to $\leq$150 K, since the entirety of lines above that limit are blended
with other methanol transitions below that limit, and are likely not
contributing significantly to the overall emission (see the analysis in Sect.
4.2).

In this study, we consider as LTE conditions those reached when a single
excitation temperature, $T_{\rm ex}$, characterizing the energy level
population according to the Boltzmann distribution, is sufficient to explain
the line emission of methanol along all its observed transitions. Namely, an
equivalency between kinetic ($T_{\rm kin}$) and excitation temperatures is not
strictly required.

###  4.1 Rotation diagram

As a first approach, we use the rotation diagram method to compare the
relative intensities of methanol lines in NGC 253. The details of this method
can be found in Goldsmith & Langer (1999) and are also summarized in our
Appendix A.

Since this method assumes that the gas is under LTE conditions and that the
lines are optically thin, any transition deviating from a straight-line fit in
the rotation diagram indicates that, for values below the fitted line,
opacities are not low and that, for values above the fitted line, maser
emission may be a suitable explanation.

If a given line is affected by blending, we could overestimate its integrated
intensity, which would lead to a wrongly classified maser line. Therefore,
after identifying all the methanol transitions in our observations and before
performing the rotation diagrams, we have initially searched both in the CDMS
(Müller et al. 2005) and JPL (Pickett et al. 1998) databases for potentially
contaminating lines from other molecular species (last column in Table 5).
Special care was taken on previously reported molecules in NGC 253 (e.g.
Martín et al. 2006; Meier et al. 2015; Ando et al. 2017), as well as
preliminary line identification performed on the
1$\aas@@fstack{\prime\prime}$6 resolution ALCHEMI data (Martín et al. in
prep.). Blended non-methanol line candidates are required to fall inside the
FWHM of the given methanol line, which varies for each symmetric type and
region.

To produce the rotation diagrams we have used the
CASSIS777http://cassis.irap.omp.eu software. In particular, we have used the
spectroscopic VASTEL database, which comes from the JPL
catalog888http://cassis.irap.omp.eu/?page=catalogs-vastel, since it
distinguishes between A and E methanol forms. Covering transitions with
$E_{\rm up}$/k$<$150 K, we did not consider methanol lines separated from each
other by less than their FWHM. Because of that, we did not include in our
analysis the $J_{K}\rightarrow(J-1)_{K}$ line series at $\sim$96.7 GHz
($J=2$), $\sim$145.1 GHz ($J=$3), $\sim$193.5 GHz ($J=$4), $\sim$241.8 GHz
($J=$5), and $\sim$290.1 GHz ($J=$6).

A summary of the considered transitions for each region is shown in Table 5,
where line parameters were taken from the CDMS database (the same holds for
the other tables throughout this work). For each of the selected lines, we
have obtained the integrated line intensity $\int T_{\rm{mb}}$d$v$ by fitting
a single Gaussian profile. The resulting rotation diagrams are presented in
Figure 3, for A– and E–CH3OH symmetry species separately and for each studied
region (Table 1), where we conservatively assumed a flux calibration
uncertainty of 15%, as recommended by Martín et al. (2021). This uncertainty
plus Gaussian fitting uncertainties were added in quadrature for each line.

The rotation diagram method assumes the Rayleigh–Jeans (RJ) approximation (see
Appendix A). In our case, given the low $T_{\rm ex}$ measured by the rotation
diagrams in certain regions (see Table 6), the RJ approximation is not a good
assumption for high-frequency transitions, leading to an overestimation of
$\sim$40% for the upper level population of some transitions in the worst case
scenario. As explained in Appendix A, applying a correction factor to all our
transitions, as an attempt to go beyond the RJ approximation, we obtain slight
changes in $T_{\rm ex}$ but total column densities decrease by up to $\sim$40%
in extreme cases. The value that would be derived from the Planck function for
the corresponding transition assuming the same $T_{\rm ex}$ originally derived
with the RJ approximation for each region and methanol symmetric type, is
shown inside the error bars in Fig. 3. The application of the Planck formula
increases the lower-limit uncertainty as a function of the $T_{\rm ex}$ (see
Eq. 3) found for each region through RJ and the individual frequency of each
transition and therefore mostly affects the regions where $T_{\rm ex}$ is
lower than 12 K, i.e., regions 1, 2, and 8 to 10 in E–CH3OH. The parameters
derived after applying this correction are also included in the uncertainties
presented in Table 6.

On Figure 3 we also indicate the critical densities of the different
transitions shown in the rotation diagrams. To this end, we have used the most
recently available collisional rates (Cij), interpolating their values, given
in steps of 10 K by Rabli & Flower (2010), by the corresponding $T_{\rm ex}$
obtained through the rotation diagrams in each region and for each methanol
symmetric type. The critical densities thus obtained, in units of cm-3 are
then indicated by colors in the rotation diagrams (Fig. 3; transitions with
unavailable Cij are in black).

We started including all the considered transitions (Table 5) below a certain
$E_{\rm up}$/k (see below) to perform our rotation diagrams and obtain the
best linear fit. However, we soon realized that some transitions did not fall
on the linear fit line derived from the ensemble of CH3OH integrated
intensities. We excluded these lines that do not follow LTE conditions but are
not likely to correspond to masers, see Appendix B, to perform, in a second
iteration, a linear fit to the remaining data in order to estimate the column
density and $T_{\rm ex}$. Our fit results are shown at the bottom of each
rotation diagram in Fig. 3 with outliers indicated in the legends when they
correspond to maser candidates (see below). Column densities and $T_{\rm ex}$
with their uncertainties are listed in Table 6. In general, the distribution
of the data points can be well described by a single $T_{\rm ex}$.

Without considering the outliers, the standard deviations (1$\sigma$) of the
residuals of the data points from the fitted lines are of a difference
(ln($N_{\rm{up,\,data\,points}}$) - ln($N_{\rm{up,\,bestfit}})$) between 0.2
(E–CH3OH in region 10) and 0.5 (A–CH3OH in region 4). These values translate
into a factor ($N_{\rm{up,\,data\,points}}/N_{\rm{up,\,bestfit}}$) between
$\exp(0.2)=$1.2 and $\exp(0.5)=$1.7.

We conservatively rounded up the average standard deviation ($\sigma=$0.35 to
0.4) and obtained a factor of 3.3 scatter, in the exponential scale of our
rotation diagrams (exp(3$\sigma$)). We then classify as outliers all methanol
transitions surpassing by more than a factor of 3.3 the expected value from
LTE conditions.

These outliers are listed in Table 2, where their transitional quantum
numbers, frequencies, symmetry types, maser classes, $E_{\rm up}$/k, Aij, and
optical depths are indicated in columns 1 to 7, respectively. In this table we
also provide information about possible line blending with other methanol
transitions falling within the FWHM of the main transition; they possess
higher $E_{\rm up}$/k and lower Aij, implying a lower contribution to the
observed spectrum due to the need for more extreme conditions to be emitted.
In Table 2, the information given for these lines is the transition,
frequency, symmetric type, $E_{\rm up}$/k, and Aij along columns 8 to 12,
respectively.

Deviations from LTE are expected in the ISM, in particular for complex level
diagrams and if radiative and collisional excitation and de-excitation
compete. But in general, given the success in the fitting, it can be inferred
that potential blending lines in Table 5 are, in most cases, not significantly
contaminating.

We wish to emphasize that a sophisticated model, including hundreds or
thousands of transitions from many molecules, would be needed to determine
methanol line blending with transitions of other species in a thorough way,
providing a percentage of contamination to methanol made from other species.
Since this is clearly beyond the scope of this paper (but we will sometimes
quote initial results from region 5, obtained from Martin et al., in
preparation), we refer in the following to the case of blending of methanol
with other methanol lines, unless contamination with other species is
explicitly mentioned.

Based on the line width, CASSIS is able to discriminate between unblended
methanol lines and those which are contaminated by adjacent lines of CH3OH. In
Fig. 3 we represent unblended lines with circles and blended lines with stars.
Since line widths change per region, the same transition may be denoted by
circles or stars depending on the region. Besides of that, we have found that
some methanol lines presented in Fig. 3 are blended with other highly
energetic methanol lines ($E_{\rm up}$/k$\geq$1000 K, see Table 2) and
therefore significant blending is highly unlikely. However, we decided to show
them also as blended lines (stars) in order to be impartial along all the
regions and transitions. In addition, we note that lines with $E_{\rm
up}$/k$\leq$150 K, being blended with CH3OH transitions with $E_{\rm
up}$/k$>$150 K, are plotted with the lower $E_{\rm up}$/k value in Fig. 3.
Finally, the LTE slope is adjusted exclusively considering unblended and non-
masing methanol lines (for a definition of the latter, see below).

Figure 3: Rotation diagrams based on our selected methanol transitions (Table
5) and separated by methanol symmetric type (E–CH3OH and A–CH3OH). Straight
black lines represent our best fits to the data considering only methanol
lines not blended with other methanol lines (see Sect. 4.1) and following LTE
conditions. Dotted grey lines are the slopes of the complementary methanol
symmetric type in the same region. Maser candidates (see Sect. 4.1) are
labelled with symbols. Points are color-coded depending on their critical
densities (in units of cm-3), as indicated by the color bar to their right
(transitions with unavailable critical densities are in black). In the bottom
left corner of each panel, the column density ($N_{\rm E-CH_{3}OH}$ and
$N_{\rm A-CH_{3}OH}$, in cm-2) and the excitation temperature ($T_{\rm ex}$,
in Kelvin) derived from the fit are indicated. Non-blended lines in LTE are
marked by circles, while methanol lines blended with other methanol lines (at
$E_{\rm up}$/k$>$150 K) are assigned by stars. Shaded areas represent
1$\sigma$ uncertainties (see Sect. 4.1). Table 2: Outliers in the rotation
diagrams.

Maser candidate |  |  |  |  |  $\tau$aa$a$Optical depth according to our non-LTE model of region 8 (see Appendix C |  blendedbb$b$Methanol lines next to the frequency given in column (2). Their frequency is given in column (9) |  |  |  |  |  masercc$c$Maser candidates based on the discussion in Sect. 4.1.1 |  locationdd$d$Regions where the transition is masing
---|---|---|---|---|---|---|---|---|---|---|---|---
Transition |  $\nu$ [GHz] | A/E | Class |  $E_{\rm up}$/k[K] |  Aij[s-1] |  [$\times$10-4] | Transition |  $\nu$ [GHz] | A/E |  $E_{\rm up}$/k[K] |  Aij[s-1] | [K] |
5${}_{-1}\rightarrow$40 | 84.521172 | E | I | 32.49 |  1.97$\times$10-6 |  $-$236.6 |  12${}_{10}\rightarrow$1211 | 84.531123 | E | 1085.97 |  7.72$\times$10-7 | yes | 1,2,7–10
2${}_{1}\rightarrow$11 | 95.914310 |  A+ | – | 21.44 |  2.49$\times$10-6 | 12.5 | not-blended | – | – | – | – | no |
2${}_{1}\rightarrow$11 | 97.582798 |  A- | – | 21.56 |  2.63$\times$10-6 | 3.8 | not-blended | – | – | – | – | no |
6${}_{-1}\rightarrow$50 | 132.890759 | E | I | 46.41 |  7.74$\times$10-6 |  $-$67.7 | not-blended | – | – | – | – | yes | 1,2,7–10
3${}_{1}\rightarrow$21 | 143.865795 |  A+ | – | 28.35 |  1.07$\times$10-5 | 7.8 | not-blended | – | – | – | – | no |
3${}_{1}\rightarrow$21 | 146.368328 |  A- | – | 28.59 |  1.13$\times$10-5 | 6.5 | not-blended | – | – | – | – | no |
9${}_{0}\rightarrow$81 | 146.618697 |  A+ | I | 104.41 |  8.03$\times$10-6 |  $-$0.1 |  14${}_{1}\rightarrow$132 | 146.617429 |  A+ | 256.02 |  6.6$\times$10-6 | yes | 1,7–9
2${}_{1}\rightarrow$30 | 156.602395 |  A+ | – | 21.44 |  1.78$\times$10-5 | 162.3 | not-blended | – | – | – | – | no |
10${}_{0}\rightarrow$91 | 198.403067 |  A+ | I | 127.60 |  4.10$\times$10-5 |  $-$0.01 | not-blended | – | – | – | – | yes | 3,6,7
8${}_{-1}\rightarrow$70 | 229.758756 | E | I | 81.20 |  4.19$\times$10-5 |  $-$2.1 | not-blended | – | – | – | – | yes | 2,7–9
2${}_{-1}\rightarrow$1-0 | 261.805675 | E | – | 28.01 |  5.57$\times$10-5 | 402.3 |  20${}_{5}\rightarrow$213 | 261.800288 | E | 611.06 |  9.85$\times$10-9 | no |
6${}_{2}\rightarrow$51 | 315.266861 | E | – | 63.10 |  1.18$\times$10-4 | 6.7 |  37${}_{-6}\rightarrow$36-7 | 315.237555 | E | 1808.32 |  5.73$\times$10-5 | no |
8${}_{1}\rightarrow$80 | 318.318919 |  A-+ |  IIee$e$According to the classification of the $J_{1}-J_{0}A^{-+}$ line series in Zinchenko et al. (2017). The dash (–) symbol indicates no information/detection. For more information about the columns see Sect. 4.1. | 98.82 |  1.79$\times$10-4 | 1.5 | not-blended | – | – | – | – | no |
7${}_{-0}\rightarrow$6-0 | 338.124488 | E | – | 78.08 |  1.70$\times$10-4 | 10.9 |  38${}_{12}\rightarrow$3710 | 338.117653 | E | 2425.29 |  7.38$\times$10-8 | no |
4${}_{-0}\rightarrow$31 | 350.687662 | E | – | 36.33 |  8.67$\times$10-5 | 1205.0 |  18${}_{-3}\rightarrow$18-2 | 350.723878 | E | 812.46 |  4.17$\times$10-5 | no |
4${}_{1}\rightarrow$30 | 358.605799 | E | – | 44.26 |  1.32$\times$10-4 | 224.1 | not-blended | – | – | – | – | no |
7${}_{2}\rightarrow$61 | 363.739868 | E | – | 79.36 |  1.70$\times$10-4 | 1.7 |  23${}_{6}\rightarrow$229 | 363.680652 | E | 1327.80 |  1.78$\times$10-7 | no |
999). . . .

####  4.1.1 Rotation diagram results

To distinguish maser candidates among outliers, that is, lines located beyond
a 3$\sigma$ scatter; the respective line must belong to known methanol maser
line series already discovered in NGC 253 at lower frequencies. This is
because maser action is more prominent at lower frequencies than the ones
covered in this work: as Aij is proportional to the cube of the frequency,
lower Aij leads to a longer time lapse to accumulate inverted populations. In
addition, we also require that the candidate line shows maser behaviour in
more than one region.

In the following we measure the departure from LTE of the maser lines as the
ratio between their nominal upper level column densities in the rotation
diagrams ($N_{\rm up,maser}$) over the expected upper level column density in
LTE ($N_{\rm up,LTE}$). In the computation of $N_{\rm up}$ one generally
assumes a proportional relation with the integrated intensity (see Eq. 2). For
maser lines, however, intensities and abundances are not related, as their
negative opacity amplifies radiation from the background. Therefore this
$N_{\rm up,maser}$/$N_{\rm up,LTE}$ factor must be taken as an intensity
difference instead of abundance difference.

####  4.1.2 E–type methanol masers

Among the E–CH3OH maser line candidates belonging to the
$J_{-1}\rightarrow(J-$ 1)${}_{0}-E$ series (at 84.5, 132.9, and 229.8 GHz),
none of them were detected out of LTE in regions 3 to 6 (see Table 2), which
are therefore excluded from the following analysis.

We present the spectra and velocity integrated intensity maps of the two
transitions with the highest S/N ratios ($5_{-1}\rightarrow 4_{0}-E$ at 84.5
GHz and $6_{-1}\rightarrow 5_{0}-E$ at 132.9 GHz) in Figures 4 and 5.

The $\mathbf{5_{-1}\rightarrow 4_{0}-E}$ line at 84.5 GHz departs from LTE by
factors ($N_{\rm up,maser}$/$N_{\rm up,LTE}$) ranging from 4.2$\pm$1.3 to
13.2$\pm$1.2 in regions 10 and 1, respectively (see Fig. 3). This is the first
detection of the 84.5 GHz maser line in more than one position, after its
first discovery by McCarthy et al. (2018). The higher $J$ masers in the
$J_{-1}\rightarrow(J-1)_{0}-E$ series are usually observed to originate in the
same regions, suggesting similar distributions and pumping mechanism.

The $\mathbf{6_{-1}\rightarrow 5_{0}-E}$ line at 132.9 GHz departs from its
expected integrated intensity in LTE by factors ranging from 4.6$\pm$1.2 to
10.6$\pm$1.2, in regions 2 and 1, respectively. Contrary to the previous
transition at 84.5 GHz, it is not blended with other methanol lines.

The $\mathbf{8_{-1}\rightarrow 7_{0}-E}$ line at 229.8 GHz is observed to mase
in the same regions as the ones at 84.5 and 132.9 GHz except in regions 2 and
10, which display the lowest LTE fitted $T_{ex}$ in the E-type methanol
species and also the lowest S/N ratios. In these couple of regions, the
emission is too weak to be classified as a maser. The 8${}_{-1}\rightarrow
7_{0}-E$ line is the highest frequency transition detected as a maser by the
rotation diagram method; its intensity departs from LTE by 9.8$\pm$1.2, in
region 7 and up to 32.0$\pm$1.2 in region 1.

Figure 4: Central panel: Velocity integrated intensity of the
$5_{-1}\rightarrow 4_{0}-E$ methanol line at 84.5 GHz obtained by integrating
the channels inside the colored areas shown in the side panels. Regions where
we detect Class I maser emission in the $J_{-1}\rightarrow(J-1)_{0}-E$ series
(see Sect. 4.1.2) are labeled in red and with slightly larger numbers. Side
panels: methanol spectra of the different regions as labeled in the top right
corner of each panel (a $v_{\rm sys}$ of 258.8 km s-1 was subtracted).
Vertical lines indicate the peak velocity of the 13CO $J=$1–0 line (110.2 GHz)
extracted in each region, as a reference for the systemic velocity at that
position. Coordinate system, beam size and other parameters are the same as in
Fig. 1. Figure 5: Same as for Figure 4 but for the $6_{-1}\rightarrow 5_{0}-E$
methanol line at 132.9 GHz.

####  4.1.3 A–type methanol masers

With respect to A–CH3OH, maser line candidates are part of the
$J_{0}\rightarrow(J-1)_{1}-A^{+}$ series (9${}_{0}\rightarrow 8_{1}-A^{+}$ and
10${}_{0}\rightarrow 9_{1}-A^{+}$ transitions). We note that a lower $J$ (and
lower frequency) transition in this series is the 7${}_{0}\rightarrow
6_{1}-A^{+}$ line at 44.1 GHz, the strongest Galactic Class I maser.

The $\mathbf{9_{0}\rightarrow 8_{1}-A^{+}}$ line at 146.618 GHz departs from
LTE by factors in the range of 8.4$\pm$1.2 (in region 8) to 29.8$\pm$1.2 (in
region 1), excluding region 4 because of line blending (see Appendix B). This
maser transition has been suggested to be part of the Class I family of
methanol masers (e.g. Yang et al. 2020, and references therein).

The $\mathbf{10_{0}\rightarrow 9_{1}-A^{+}}$ line at 198.4 GHz departs from
LTE in regions 3, 6, and 7, being the only masing line candidate for regions 3
and 6. In those regions, this line departs from LTE by factors of 4.3$\pm$1.3,
3.6$\pm$1.2, and 8.5$\pm$1.2, respectively.

We have found that all our maser candidates, except the $10_{0}\rightarrow
9_{1}-A^{+}$ line at 198.4 GHz, are masers at the outskirts of the CMZ of NGC
253 (see Table 2). The physical conditions in the regions giving rise to those
masers are characterized (see Fig. 6) by total column densities for A–CH3OH
lower than 6$\times$1014 cm-2, an excitation temperature lower than 15 K in
E–CH3OH, and differences between A– and E–CH3OH excitation temperatures larger
than 5.0 K. Conversely, we do not see a correlation between the presence of
masers and total column densities in E-type methanol, the temperature
described by A–CH3OH, or differences between column densities of the two
methanol types.

With the exception of the line at 198.4 GHz, it is worth noting that all the
regions where we detect Class I maser emission lie inside the Lindblad
resonances (see e.g. Fig. 4), with the only exception of region 7. We will
further explore this point in Sect. 5.2.

Based on LTE modelling, we have also found other transitions that are possibly
experiencing maser activity. They will be described below.

###  4.2 Radiative transfer modelling

In comparison with the rotation diagram method, synthetic spectra offer a
number of improvements, like the possibility to reproduce line profiles both
in LTE and out of LTE, which may also subtly indicate the number of gas
components present in the observed source (see e.g. Fig. 14). Also, as the
lines are observed on a linear scale, the effect of slight changes in the
assumed gas conditions are noticed in more detail compared with the
logarithmic scale used in the rotation diagram method. Apart from those
advantages, radiative transfer modelling allows us to fit not only single
methanol lines, but also blended ones, as their spectra are summed up when
trying to obtain the desired intensities. Alternatively, this provides an
estimate for the degree of contamination caused by other species. This allows
us to include many more lines, increasing the sample size. In fact, as we will
see below, there are 57 lines used for the fitting in our radiative transfer
models, in contrast to the 39 finally used in the rotation diagrams (see Sect.
4.1.1).

Figure 6: Physical conditions for the methanol maser candidates (see Table 2)
derived from our rotation diagrams. The x–axis indicates the number of the
region. The y–axis represents, from top to bottom, (1) the ratio between the
nominal upper level column densities of the lines $N_{\rm up,MASER}$ and the
expected values from our best fit to LTE conditions $N_{\rm up},LTE$, (2) the
total column density, (3) the difference between total E– and A–type column
densities, (4) the LTE excitation temperatures and (5) the difference between
the A– and E–type excitation temperatures. Shaded areas highlight regions were
maser emission in the $J_{-1}\rightarrow(J-$ 1)${}_{0}-E$ line series (at 84,
132, and 229 GHz), and the $J_{0}\rightarrow(J-1)_{1}-A^{+}$ line series (at
146 and 198.4 GHz), are observed, according to the first panel.

The possibility to perform non-LTE modelling helps us to discard maser line
candidates in case negative optical depths are not required to reproduce their
emission. Finally, the nature of blended methanol lines with other methanol
lines at known maser frequencies, such as the transitions at 95.169 and
278.305 GHz, could not be unveiled through the rotation diagram method. As
mentioned in Sect. 4.1, we use the term blending to refer to contamination of
methanol lines with other methanol transitions, unless contamination by other
species is explicitly mentioned.

We have computed synthetic spectra for each of the 10 selected regions in NGC
253, covering the entire ALCHEMI frequency range ($\sim$84–373 GHz, see Sect.
2), by using the CASSIS software, capable of producing LTE and non-LTE
spectral modelling. For the non-LTE case, CASSIS is used as a wrapper of RADEX
(van der Tak et al. 2007), a one-dimensional non-LTE radiative transfer code
based on the escape probability formulation. In CASSIS it is possible to
create a physical model defined by 6 parameters: column density of the species
($N(Sp)$), excitation ($T_{ex}$, LTE case) or kinetic ($T_{kin}$, non-LTE
case) temperature, full width at half maximum (FWHM) of the lines, velocity of
the source in the Local Standard of Rest ($V_{\rm LSR}$) system, size of the
source in arcseconds, and the H2 volumetric density ($n_{\rm H2}$) in the case
of non-LTE modelling. For simplicity, in this section we have considered that
only a single physical component for each methanol type is responsible for the
observed emission; for a more detailed analysis, see Appendix C. CASSIS makes
use of the Monte Carlo Markov Chain (MCMC) method (Hastings 1970) to explore a
user-predefined range of values for each of the parameters previously
mentioned. By means of the $\chi^{2}$ minimization method, CASSIS is able to
find the best ensemble of solutions. Computing the synthetic spectra we
assumed a beam filling factor of one, by selecting a source size of
1$\aas@@fstack{\prime\prime}$6 ($\sim$27 pc), and a slab geometry (appropriate
for shocks, e.g. Leurini et al. 2016).

Model solutions could be strongly influenced by the initial conditions.
Therefore we start by selecting unblended methanol lines to be fitted and then
add blended lines whose total line profile is successfully reproduced with the
starting models. This was achieved by exploring the success of CASSIS in
reproducing the lines throughout our selected regions.

After our initial attempts to fit methanol lines along the entire ALCHEMI
spectral coverage, assuming either LTE or non-LTE conditions, it became clear
from our 10 regions that it is impossible to properly fit all the lines with a
single physical set of parameters, even after separating between A– and
E–CH3OH flavours. This is especially important at frequencies below $\sim$156
GHz, and it is possibly due to a couple of factors: the presence of a series
of lines out of LTE, and the higher number of maser candidates.

Using our LTE model, we found that the $J_{K}\rightarrow(J-1)_{K}$ transitions
(for both methanol symmetric types), which were avoided in Sect 4.1 due to
line blending, are not following LTE conditions. Within the ALCHEMI frequency
coverage, these line series have the following frequencies: $\sim$96.7 GHz
($J=$2), $\sim$145.1 GHz ($J=$3), $\sim$193.5 GHz ($J=$4), $\sim$241.8 GHz
($J=$5), and $\sim$290.1 GHz ($J=$6). Performing a non-LTE model in region 8,
they are satisfactorily reproduced, as can be seen in Fig. 15 of Appendix C
without invoking the presence of masers.

At frequencies below $\sim$156 GHz we also cover four maser candidates, three
of them reported in Subsect. 4.1.2 plus an additional one that we will see
below (Subsect. 4.2.1). Therefore, at lower frequencies our models fail to
reproduce an important quantity of available, not blended, methanol lines.

We discard possible software issues by doing a sanity check with another
radiative-transfer code capable of producing LTE models, MADCUBA (Martín et
al. 2019). It shows similar results including convergence primarily toward
lines above 156 GHz.

Having said the above, we do not include the mentioned
$J_{K}\rightarrow(J-1)_{K}$ transition series nor the maser candidates in our
LTE modelling. Even with this restriction, we observe that these series are
well reproduced ($>$50%) in region 4 by merely fitting the other transitions,
and this can be improved with a two-component LTE model (see Appendix C).

When inspecting and comparing the results for all regions between the LTE and
non-LTE models, considering a single component for each methanol symmetric
type, we observe that they are in agreement within an uncertainty of about 15%
(although this agreement is not observed when we consider a two-component non-
LTE model, see Appendix C). This is not surprising since high H2 densities of
$>$107 cm-3 are needed in the non-LTE models to reproduce most line profiles.
At such densities collisions play a major role, tending to constrain the
spread in excitation temperatures between different lines. Therefore, we
conclude that the simpler LTE conditions are sufficient to represent the
observed spectra in NGC 253. From now on, we will therefore mainly refer to
our LTE modelling, except in a few exceptional cases where this is explicitly
mentioned.

As described in Sect. 4, we set an upper $E_{\rm up}$/k threshold of 150 K for
the synthetic spectra. This was determined through comparing observations to
model fits with and without the higher energy levels ($E_{\rm up}$/k$>$ 150
K). A total of 600 LTE models for each individual spectrum, one per selected
region (see Table 3), were computed, reaching a convergence after 300–400
iterations. We find that we only require models that include $E_{\rm up}$/k$<$
150 K to fit the observed spectra. Thermal line emission with $E_{\rm
up}$/k$>$ 150 K may be there, but is too faint to affect model fits or to be
separated from line blends. In the Galaxy, methanol masers with levels around
150 K above the ground state are scarce. Within the ALCHEMI frequency coverage
we can mention the case of the $11_{-1}\rightarrow 10_{-2}-E$ line at 104.3
GHz (Leurini & Menten 2018). However, this Class I maser is rarely seen
(Voronkov et al. 2012). We do not obtain any strong emission at this frequency
in NGC 253.

Best fit parameters determined from our LTE models for each region and
methanol species are listed in Table 3. Differences between A– and E–CH3OH
symmetry species are present in terms of temperature and density. In velocity
they differ by a few km s-1, although this discrepancy never exceeds our
spectral resolution of $\sim$8–9 km s-1. Thus, the averaged $V_{\rm LSR}$ of
these two species is adopted as the velocity of the region, in the same way as
it has been previously established in Table 1.

Table 3: Best fit parameters from our LTE modelling.

|  E–CH3OH |  A–CH3OH |  |
---|---|---|---|---
Region |  $N$(Sp) | $T_{ex}$ | FWHM |  VLSR |  $N$(Sp) | $T_{ex}$ | FWHM |  VLSR |  mean VLSR |
|  [$\times$1014 cm-3] | [K] |  [km s-1] |  [km s-1] |  [$\times$1014 cm-3] | [K] |  [km s-1] |  [km s-1] |  [km s-1] |
R1 |  3.7${}_{-0.1}^{+0.1}$ |  14.8${}_{-0.6}^{+0.4}$ |  59.7${}_{-2.5}^{+1.0}$ |  305.4${}_{-1.0}^{+0.8}$ |  0.76${}_{-0.04}^{+0.05}$ |  18.8${}_{-0.7}^{+0.8}$ |  61.3${}_{-0.4}^{+1.7}$ |  302.9${}_{-0.8}^{+1.4}$ |  304$\pm$9 |
R2 |  5.0${}_{-0.2}^{+0.3}$ |  9.3${}_{-0.4}^{+0.4}$ |  43.6${}_{-1.5}^{+2.5}$ |  328.1${}_{-0.7}^{+0.2}$ |  1.0${}_{-0.1}^{+0.1}$ |  14.8${}_{-0.9}^{+0.5}$ |  37.4${}_{-2.9}^{+2.6}$ |  327.2${}_{-1.1}^{+1.6}$ |  328$\pm$9 |
R3 |  19.5${}_{-0.5}^{+0.6}$ |  21.4${}_{-0.5}^{+0.4}$ |  53.8${}_{-0.9}^{+0.6}$ |  283.2${}_{-0.4}^{+0.4}$ |  9.3${}_{-0.2}^{+0.1}$ |  26.6${}_{-0.6}^{+0.3}$ |  55.0${}_{-0.3}^{+0.9}$ |  284.7${}_{-0.5}^{+0.3}$ |  283$\pm$9 |
R4 |  10.16${}_{-0.03}^{+0.04}$ |  31.0${}_{-1.2}^{+0.6}$ |  60.2${}_{-1.5}^{+1.7}$ |  252.0${}_{-0.6}^{+0.5}$ |  9.0${}_{-0.2}^{+0.2}$ |  34.2${}_{-0.9}^{+0.8}$ |  60.1${}_{-1.3}^{+0.7}$ |  251.5${}_{-0.6}^{+0.2}$ |  252$\pm$9 |
R5 |  6.2${}_{-0.4}^{+0.5}$ |  32.2${}_{-1.7}^{+1.8}$ |  136.0${}_{-7.9}^{+8.4}$ |  217.2${}_{-3.3}^{+4.5}$ |  3.2${}_{-0.2}^{+0.3}$ |  32.2${}_{-1.7}^{+1.8}$ |  136.0${}_{-7.9}^{+8.4}$ |  217.2${}_{-3.3}^{+4.5}$ |  217$\pm$10 |
R6 |  22.32${}_{-0.03}^{+0.07}$ |  24.8${}_{-0.4}^{+0.5}$ |  65.3${}_{-0.4}^{+0.9}$ |  179.9${}_{-0.5}^{+0.4}$ |  14.1${}_{-0.2}^{+0.2}$ |  31.0${}_{-0.6}^{+0.4}$ |  66.0${}_{-0.9}^{+0.7}$ |  182.1${}_{-0.3}^{+0.7}$ |  181$\pm$9 |
R7 |  17.4${}_{-0.4}^{+0.6}$ |  19.5${}_{-0.5}^{+0.4}$ |  82.8${}_{-1.4}^{+1.3}$ |  169.3${}_{-0.9}^{+0.6}$ |  5.5${}_{-0.1}^{+0.2}$ |  22.9${}_{-0.5}^{+0.4}$ |  83.2${}_{-2.9}^{+3.0}$ |  173.6${}_{-1.0}^{+1.2}$ |  171$\pm$9 |
R8 |  5.9${}_{-0.1}^{+0.2}$ |  14.8${}_{-0.4}^{+0.3}$ |  63.7${}_{-1.8}^{+0.9}$ |  204.2${}_{-1.9}^{+1.3}$ |  1.2${}_{-0.1}^{+0.1}$ |  19.5${}_{-0.7}^{+0.4}$ |  59.1${}_{-3.3}^{+3.7}$ |  205.2${}_{-1.6}^{+1.3}$ |  205$\pm$9 |
R9 |  5.6${}_{-0.1}^{+0.3}$ |  13.9${}_{-0.4}^{+0.5}$ |  57.4${}_{-1.3}^{+2.2}$ |  202.7${}_{-0.9}^{+1.1}$ |  1.5${}_{-0.1}^{+0.1}$ |  19.2${}_{-0.4}^{+0.6}$ |  56.3${}_{-1.9}^{+2.6}$ |  201.4${}_{-0.7}^{+1.1}$ |  202$\pm$9 |
R10 |  1.0${}_{-0.1}^{+0.1}$ |  12.3${}_{-1.1}^{+0.7}$ |  24.9${}_{-1.3}^{+1.8}$ |  143.9${}_{-0.8}^{+1.0}$ |  0.27${}_{-0.02}^{+0.03}$ |  27.36${}_{-3.1}^{+3.1}$ |  22.8${}_{-3.3}^{+3.9}$ |  146.4${}_{-1.5}^{+1.1}$ |  145$\pm$9 |
101010From left to right columns are: Selected region of the CMZ of NGC 253
(see Table 3), E-type methanol parameters: column density of the species
($N$(Sp)), excitation temperature ($T_{\rm ex}$), FWHM of the modeled methanol
lines, and the local standard of rest velocity of the region $V_{\rm LSR}$.
Columns 6–9 provide the same gas parameters for A–CH3OH. Uncertainties are
1$\sigma$. An extra uncertainty of 15% should be added to $N$(Sp) (due to the
flux calibration accuracy, see Sect. 2). In the last column we present the
average velocity of the two methanol symmetric types. In this case our
1$\sigma$ uncertainty is summed in quadrature with our velocity resolution of
$\sim$8–9 km s-1, which dominates.

A comparison between synthetic and observed spectra allows us to perform a
deep scan of the methanol lines, highlighting some lines that slightly deviate
from LTE conditions in the rotation diagrams or that are blended with other
methanol lines and that were previously not discussed. The result of this
inspection is henceforth described.

####  4.2.1 LTE modelling results

As can be seen in our Fig. 7, our synthetic spectra fit reasonably well the
observed spectra in most of the regions. Region 5 is the most difficult to
reproduce due to the large FWHM ($\sim$140 km s-1) of the lines, with its
spectrum almost reaching the confusion limit. Fortunately, based on the
remaining regions, we selected a large number of lines that are reproduced
(see Tables 7 and 8, and Figure 7). We attempted to fit the same lines in
region 5, preventing in this way false line identifications and subsequent
erroneous fitting attempts.

The best solution for Region 5 was reached by fitting both methanol symmetric
types simultaneously. When this is done in CASSIS, the whole set of parameters
between A- and E-type methanol is forced to be equal, allowing to change only
the ratio between their column densities. As can be seen in Table 3, the
column density of E–CH3OH is 2.1 times higher than that of A–CH3OH. A lower
column density E/A ratio leads us always to a worse fitting and is therefore
not attempted to reach. We have also discarded from the initial fit the
A–CH3OH transitions between 303.3 and 309.2 GHz, as their inclusion always
leads to an overestimation of a number of lines (e.g. transitions at 239.7,
241.9, 338.6, and 350.9 GHz, see Tables 7 and 8).

All the proposed maser candidates listed in Table 2 that were initially
unveiled through the rotation diagrams were confirmed by our models.

In addition to the outliers previously detected (see Table 2), we found
several transitions with intensities not reproduced by our synthetic spectra.
Among those outliers, listed in Table 4, there are a couple of maser
candidates. We describe them below.

####  4.2.2 Maser line candidates

Contrary to the remaining outliers found through the radiative transfer
modelling (shown in Fig. 8 and described at some level in Appendix D), our
maser candidates have negative optical depths, depart significantly from LTE
(see below), and belong to the same transition series as the maser candidates
detected previously through the rotation diagram method (Subsect. 4.1.3),
being Class I methanol masers.

The $\mathbf{8_{0}\rightarrow 7_{1}-A^{+}}$ transition line at 95.2 GHz shows
intensities $>$12.8 times stronger than predicted by the LTE modelling in all
the regions but region 4, where it is three times stronger. This large
departure in the inner regions of the CMZ remind us of the case of the
$10_{0}\rightarrow 9_{1}-A^{+}$ methanol line at 198.4 GHz, which shows
emission 3.3 times larger than expected in regions 3 and 6.

Table 4: Outliers in the LTE model.

Transition |  $\nu$ [GHz] | A/E |  $E_{\rm up}$/k[K] |  Aij[s-1] |  $\tau$dd$d$Optical depth according to our non-LTE model in region 8 (see Appendix C) [$\times$10-5] |  maseree$e$Maser candidates based on the discussion in Sect. 4.2.2. If it is a candidate, then we add the classification Class I or II, according to Menten (1991).
---|---|---|---|---|---|---
8${}_{0}\rightarrow$71 | 95.169391 |  A+ | 83.54 |  2.13$\times$10-6 |  $-$6.5 | I
3${}_{1}\rightarrow$40 |  107.013831aa$a$Blended with the methanol 34${}_{5}\rightarrow$35${}_{3}-A^{-}$ transition at 107.015820 GHz ($E_{\rm up}$/k$=$1515.904 K, A${}_{ij}=$1.00$\times$10-9 s-1) |  A+ | 28.35 |  3.06$\times$10-6 | 1021 | no
0${}_{0}\rightarrow$1-1 | 108.893945 | E | 13.21 |  1.47$\times$10-6 | 16590 | no
1${}_{1}\rightarrow$10 | 165.050175 | E | 15.47 |  2.35$\times$10-5 | 3423 | no
2${}_{1}\rightarrow$20 | 165.061130 | E | 20.11 |  2.35$\times$10-5 | 3801 | no
3${}_{1}\rightarrow$30 | 165.099240 | E | 27.08 |  2.35$\times$10-5 | 2579 | no
5${}_{3}\rightarrow$52 | 251.811956 |  A-+ | 37.96 |  1.65$\times$10-4 |  $-$0.1 | no
2${}_{0}\rightarrow$1-1 | 254.015377 | E | 12.19 |  1.90$\times$10-5 | 9522 | no
9${}_{-1}\rightarrow$80 |  278.304512bb$b$Blended with the methanol 2${}_{-2}\rightarrow$3${}_{-1}-E$ transition at 278.342261 GHz ($E_{\rm up}$/k$=$24.96 K, A${}_{ij}=$1.65$\times$10-5 s-1) | E | 102.07 |  7.67$\times$10-5 |  $-$2.5 | I
7${}_{-1}\rightarrow$6-1 | 338.344588 | E | 62.65 |  1.66$\times$10-4 | 1301 | no
$J_{K}\rightarrow(J-1)_{K}$cc$c$The $J_{K}\rightarrow(J-1)_{K}$ line series is described in Appendix C and D | 97–290 | A,E | – | – |  $>$0.0 | no
111111List of lines with intensities that exceed our LTE model. Their line
intensities cannot be reproduced without a non-LTE model and there are no
other lines subject to produce an important contribution if blended. . . . .

Figure 7: Zoom into successfully fitted lines for each region, which can be
considered to follow LTE conditions. Red and blue colors indicate methanol A-
and E- type, respectively, while the superposition of the two types, the final
fit, is indicated in green. Line frequencies (in GHz units) are labeled at the
top of the figure and are also indicated as a green dashed vertical line
inside panels, while the velocity range (in km s-1 with respect to the
systemic velocity of the individual regions, see Fig. 4) is indicated in the
bottom-left corner. Regions 1 to 10 are ordered from top to bottom as
indicated in the leftmost panel (R from ”Region” plus the corresponding
number).

Figure 8: Outliers to our LTE modelling (see Appendix D) with positive optical
depths. The only exceptions are the $J_{1}\rightarrow J_{0}A^{-+}$ line
series, recently proposed to present methanol maser Class II activity by
Zinchenko et al. (2017). Line frequencies are labeled at the top of the
Figure. Most of the lines are unclassified so far. Labels and colors are the
same as for Figure 7. Figure 9: Our proposed methanol masers belonging to
previously known maser transitions. Labels and colors are the same that for
Figure 7. Figure 10: Diagram of the energy levels for A-type (left) and E-type
(right) methanol transitions covered by ALCHEMI. The x-axis gives the quantum
number K, while the number for each level gives J. The transitions used in the
analysis are indicated with a color code corresponding to ALMA bands, as given
in the top right corner. The maser candidate lines are presented as dashed
lines, with their frequency in GHz. Note the energy difference of 7.9 K
between the ground state of A and E-type.

The $\mathbf{9_{-1}\rightarrow 8_{0}-E}$ line at 278.305 GHz is the last
methanol transition in the $J_{-1}\rightarrow(J-$ 1)${}_{0}-E$ line series
covered in this work. As previously mentioned in Sect. 4.1.2, it is strongly
contaminated by the 2${}_{-2}\rightarrow 3_{-1}-E$ line at 278.342 GHz and was
therefore excluded from the analysis with rotation diagrams. In regions 1, 7,
8, and 9, however, its line profile is clearly distinguished from the
contaminating line when being checked by the synthetic models. According to
our radiative transfer modelling, the companion line at 278.342 GHz is the
only one that should be observed (while the masing line intensity should be
negligible under LTE, see Fig. 9), with its peak velocity about 40 km s-1
lower than the maser line in region 9, a difference four times larger than our
spectral resolution.

In summary, all the outliers in the LTE models have intensities above the
expected one under LTE conditions, but most of them are likely able to be
reproduced under non-LTE conditions if the results obtained for region 8
(Appendix C) are maintained for the other regions. The only clear maser
candidates are the lines at 95.2 and 278.3 GHz.

Given the spatial resolution and software limitations (see van der Tak et al.
2007, their Sect. 3.6), we cannot account for a proper model to our maser
candidates. Instead, we reproduce their line profiles and check whether
negative optical depths are required or not. When no other results are
plausible, we can assure that the transitions are effectively experiencing a
population inversion. Unfortunately, as Class I maser emission arises from
spot sizes of the order of 10-5–10-3 pc (Voronkov et al. 2014; Matsumoto et
al. 2014), any attempt to model our observations will be devoid of a real
physical meaning. Negative optical depths are determined in the lines
belonging to the $J_{-1}\rightarrow(J-$ 1)${}_{0}-E$ and $J_{0}\rightarrow(J-$
1)${}_{1}-A^{+}$ series (see Fig. 9) and they constitute our maser candidates.
All of them are Class I methanol masers and, for the case of those belonging
to the $J_{-1}\rightarrow(J-$ 1)${}_{0}-E$ line series, depart from LTE at the
outskirts of the CMZ of NGC 253.

We summarize the observed methanol transitions along the entire ALCHEMI
coverage in Fig. 10, where LTE lines are labeled in straight lines and maser
candidates are in dashed lines. We indicate with colors the ALMA band for each
transition. From Fig. 10 it becomes clear that the 7${}_{-1}\rightarrow
6_{0}-E$ line at 181.295 GHz should be pumped by the same conditions than the
other maser candidates in the $J_{-1}\rightarrow(J-$ 1)${}_{0}-E$ series. Our
non-LTE models performed in region 8 yield a negative optical depth (-2.4) for
this transition suggesting maser emission. Unfortunately, our spectral and
angular resolution is not sufficient to discriminate between the 181.295 GHz
line and the $J=$2–1 HNC line at 181.324 GHz.

##  5 Discussion

###  5.1 Conditions for maser emission

Among all the lines proposed to be masers, the ones detected initially by our
rotation diagrams are the most plausible ones. Here we will focus on the Class
I methanol masers in the $J_{-1}\rightarrow(J-$ 1)${}_{0}-E$ series (at 84,
132, 229, and 278 GHz), as they describe a very clear difference in maser
occurrence: a complete LTE behaviour in the central regions (3 to 6) and
strong maser activity at the outskirts of the CMZ of NGC 253, where important
$T_{ex}$ differences between A- and E- type methanol take place (see Fig. 6),
and also where Lindblad resonances are located (with the only exception of
region 7).

Unfortunately, based on the rotation diagram method we do not have enough
evidence to account for the presence of Class II masers. The only exception
might be the 3${}_{1}\rightarrow 4_{0}-A^{+}$ transition line at 107 GHz
($E_{\rm up}$/k$=$28 K), that follows LTE conditions in regions 2, 5, and 10
($\sim$1.5, $\sim$1.5, and 2$\sigma$, respectively). It is slightly above the
fit in regions 3, 4, 6, and 7, and is present in absorption in regions 1 and 9
(3 and 2$\sigma$, respectively), and maybe also in region 8 (2$\sigma$);
although in this region the line seems to exhibit emission in the middle of
the absorption feature.

Maser activity by the line at 107 GHz was first discovered by Val’tts et al.
(1995). They found this line either in absorption or showing quasi-thermal
emission. In the first case, this line is spatially correlated with the
5${}_{1}\rightarrow 6_{0}-A^{+}$ Class II maser line at 6.7 GHz, that belongs
to the same family of lines. Other lines in the
$J_{1}\rightarrow(J+1)_{0}-A^{+}$ series are the transitions at 57 GHz
($J=$4), 156.6 GHz ($J=$2), and 205.8 GHz ($J=$1). By means of the rotation
diagram method, we note that the transition at 156.6 GHz is above the LTE
trend in the same regions where the line at 107 GHz surpasses the LTE
modelling (regions 3, 4, 6, and 7). However, as indicated in Appendix B, the
line at 156.6 GHz may also follow the LTE conditions of A–CH3OH.

In NGC 253, absorption against the continuum in region 5 is noticeably seen in
the rotational ground state transitions of dense gas tracers such as the
formyl cation (H13CO+; Harada et al. 2021). Absorption features are also very
prominent in the $J$=2–1 (e.g. Meier et al. (2015)) and $J$=3–2 transitions of
silicon monoxide (SiO). All those features are uniquely observed in region 5,
contrary to the case of the 3${}_{1}\rightarrow 4_{0}-A^{+}$ transition line
at 107 GHz, that presents absorption mostly in regions 1 and 9.

Therefore, the edges of the CMZ, namely regions 1, 2, 7, 8, 9, and 10, appear
to provide suitable conditions for Class I maser emission in the
$J_{-1}\rightarrow(J-$ 1)${}_{0}-E$ line series. We are aware that these
conclusions only refer to the average conditions in our selected regions, as
our linear resolution is of the order of 27 pc, larger than typical clump
sizes by a factor of 6–23 (see e.g. Leroy et al. 2018, for estimations of
clump sizes in the CMZ of NGC 253) but usually small enough to resolve GMCs.

Although Lindblad resonances (e.g. ellipses in Fig. 11) have been claimed as a
good candidate for shocks (García-Burillo et al. 2000; Ellingsen 2018), for
galaxies with a strong bar there is theoretical support against the existence
of Lindblad resonances (Regan & Teuben 2003) or their relationship with the
circumnuclear ring (CNR) position, which is more accurately defined by x2
orbits (Kim et al. 2012; Li et al. 2015; Schmidt et al. 2019). The interplay
between the nuclear dust/gas lanes and the CNR looks as a more likely
explanation for the production of bi-symmetric shock/active regions: at the
outskirts of the CMZ, where we find higher levels of HNCO as compared to SiO
(regions 1,2, and 7 to 10; see Fig. 11), in a similar way as in M 83 (Harada
et al. 2019), leading to the appearance of methanol masers in the
$J_{-1}\rightarrow(J-$ 1)${}_{0}-E$ line series. We should see maser action
encircling the CMZ if they were caused by the ILR.

As an attempt to account for gas disturbances in the CMZ of NGC 253, we
obtained line intensity ratios from two well-known shock tracers, SiO and
HNCO. Their moment 0 maps, along with data reduction and a thorough analysis
will be described in a forthcoming paper by Huang et al. (in prep.). Here our
only interest is to relate the presence of shocks to methanol maser emission
in the $J_{-1}\rightarrow(J-$ 1)${}_{0}-E$ line series, which is mostly
observed toward intersecting areas between the circumnuclear ring and leading-
edge dust/gas lanes of the bar.

We selected transitions in the same ALMA bands (3 and 6) in order to reduce
instrumental uncertainties. Based on previous studies, here we consider SiO as
a tracer of strong shocks and HNCO as a tracer of weak shocks (e.g. Meier et
al. 2015; Yu et al. 2018), and therefore the SiO/HNCO ratio may be used as an
indicator of the shock strength (Kelly et al. 2017).

We note that HNCO (40,4–30,3), at 87.9 GHz, is enhanced in practically the
same regions where we detect Class I maser activity in the
$J_{-1}\rightarrow(J-$ 1)${}_{0}-E$ line series (at 84, 132, 229, and 278
GHz); this is also true for the previously detected methanol transition in
this series, at 36 GHz, if we dismiss the weak maser candidate at the very
center shown in Gorski et al. (2018). The resulting SiO $J=$2–1/HNCO
(40,4–30,3) ratios (see Fig. 11, left panel) are enhanced in the central
regions, especially in regions 4 and 6, where no maser activity, in the
$J_{-1}\rightarrow(J-$ 1)${}_{0}-E$ line series, is detected.

Considering the SiO and HNCO transitions at frequencies $\sim$220 GHz (ALMA
band 6), the largest SiO $J=$5–4/HNCO (100,10–90,9) line ratios come from
region 1 (see Fig. 11, right panel), the region where we see the largest
departure between maser Class I emission in the $J_{-1}\rightarrow(J-$
1)${}_{0}-E$ line series (and also in the A-type methanol line at 146.6 GHz)
and LTE conditions (see Fig. 6). On the other hand, the lowest SiO
$J=$5–4/HNCO (100,10–90,9) ratios are observed in region 3 and this coincides
with the lowest LTE departures of potential Class I masers in E–CH3OH lines
(see Fig. 6).

One, however, also needs to take into consideration the varying gas properties
likely causing the highly varying trend of these SiO/HNCO ratios. A more
complete investigation and further discussion upon the gas properties traced
by HNCO and SiO is covered in a forthcoming paper (Huang et al. in prep. ). In
addition, there is something missing in describing the intermediate regions.
For example, regions 4 and 6, where we do not see Class I maser emission in
E–CH3OH, present SiO $J=$5–4/HNCO (100,10–90,9) ratios similar to regions 1,
2, 7, 8, and 9, where we do observe Class I maser emission in E–CH3OH.

Figure 11: Strong over weak shock tracers as accounted by SiO $J$=2–1/HNCO
(40,4–30,3) (left) and SiO $J$=5–4/HNCO (100,10–90,9) (right) line ratios
obtained from ALCHEMI data (Martín et al. 2021). A sigma clip of 3.0 was
applied for both SiO and HNCO before obtaining the line ratios. Regions where
we detect Class I maser emission in the $J_{-1}\rightarrow(J-1)_{0}-E$ series
(see Sect. 4.1.2) are labeled in red and with slightly larger numbers. The
center and edges of the inner Lindblad resonance from Iodice et al. (2014) are
denoted with a black ellipse and dash-dotted grey ellipses, respectively. A
square root stretch has been applied for an easy visualization. Figure 12: CN
($N=$1–0)/C17O ($J=$1–0) (left) and CN ($N=$2–1)/C17O ($J=$2–1) (right) line
ratios obtained from ALCHEMI data (Martín et al. 2021), indicating the
presence of photodissociation regions. A 3$\sigma$ clipping was applied. Red
contours indicate the Hα emission in logarithmic scale from MUSE archival data
(ID:0102.B-0078(A), PI:Laura Zschaechner). Regions where we detect Class I
maser emission in the $J_{-1}\rightarrow(J-1)_{0}-E$ series (see Sect. 4.1.2)
are labeled in red and with slightly larger numbers. The center and edges of
the inner Lindblad resonance from Iodice et al. (2014) are denoted with a
black ellipse and dash-dotted grey ellipses, respectively. Higher ratios
indicate a higher photodissociation level. A square root stretch has been
applied for an easy visualization.

The missing element to be considered for the presence or absence of methanol
masers can be the occurrence of photodissociation regions (PDRs), since
photodissociation is the main mechanism for methanol destruction (see e.g.
Hartquist et al. 1995). We can use PDR tracers such as CN, whose abundance is
found to be high in PDR regions (e.g. Fuente et al. 1993; Kim et al. 2020), to
investigate whether PDRs may be dominant. As mentioned by Meier et al. (2015),
the high rate of star production in the central regions of NGC 253 can be
identified by means of e.g. CN/C17O line ratios, where C17O traces dense
molecular gas (e.g. Thomas & Fuller 2008).

We obtained the CN/C17O line ratios by taking the ratio of moment 0 maps of
the chosen molecules. A 3$\sigma$ clipping was applied in the creation of
those moment 0 and the resulting line ratio maps (see Fig. 12).

In Fig. 12 we also plot the H$\alpha$ emission from recent MUSE121212Multi
Unit Spectroscopic Explorer (MUSE; Bacon et al. 2017) observations obtained
from archival data (ID:0102.B-0078(A), PI:Laura Zschaechner) after continuum
subtraction by means of the STATCONT software (Sánchez-Monge et al. 2018). The
stronger H$\alpha$ emission is related to the starburst induced outflow
emerging from the nuclear bar, its morphology resembles the inner parts of
previously observed outflows in X-rays (Strickland et al. 2000) as well as the
OH plume to the north-west (Turner 1985). We created six red contours, on a
linear scale, covering emission in the range (1–7)$\times$10-14 erg s-1 cm-2.
The CN/C17O line ratios were obtained from ALMA band 3 and 6, namely CN
($N=$1–0)/C17O ($J=$1–0) and CN ($N=$2–1)/C17O ($J=$2–1) ratios, respectively.
These ratios are plotted in Fig. 12, where high values correspond to a high
rate of photodissociation.

It can be noted from Fig. 12 that high values of CN/C17O line intensity ratios
are closely following the root of the large-scale outflowing gas. Overall, we
note the highest level of photodissociation at the central regions of the CMZ
(regions 4, 5, and 6), coinciding with regions where we observe a lack of
methanol masers in the $J_{-1}\rightarrow(J-1)_{0}-E$ line series, and also
the A-type transition at 146.6 GHz, belonging to the
$J_{1}\rightarrow(J+1)_{0}-A^{+}$ line series. The only exception involves the
comparison between regions 3 and 7. Masers are observed in region 7, even
though the CN/C17O ratio is slightly higher than in region 3, where no maser
emission is encountered.

Although region 7 is located away from the nuclear ring, it is associated with
a hot core cluster indicating an active star formation activity, as deduced
from an increment of Complex Organic Molecules (COMs) such as CH3COOH (Ando et
al. 2017). As tracers of YSO outflows, Class I methanol masers are expected to
be detected in environments like our region 7. Additionally, region 7 is not
showing a high level of photodissociation as compared to regions 4 or 5, where
the star formation activity is further confirmed by the presence of the
H40$\alpha$ line (Bendo et al. 2015).

In conclusion, the PDRs at some level can destroy methanol molecules in the
central regions of NGC 253. This is one of the scenarios suggested by
Ellingsen et al. (2017). Either starburst induced outflows and/or shocks where
the leading-edge dust/gas lanes on the bar are connected to the circumnuclear
ring, are causing the prevalence of Class I methanol masers in the
$J_{-1}\rightarrow(J-$ 1)${}_{0}-E$ line series (at 84, 132, 229, and 278 GHz)
to regions farther away from the core (regions 1, 2, 7, 8, 9, and 10). Region
7, although away from the resonances, presents a strong star formation but a
rather moderate photodissociation level, giving way to the formation of
methanol masers both in the $J_{-1}\rightarrow(J-$ 1)${}_{0}-E$ and the
$J_{0}\rightarrow(J-1)_{1}-A^{+}$ line series.

###  5.2 Maser emission distribution

Based on our rotation diagrams, we selected a couple of E–CH3OH lines that
follow LTE conditions in all the studied regions and compare their averaged
integrated intensities with the ones observed in the maser line at 84.5 GHz
($E_{\rm up}$/k$=$32.5 K) in each spaxel. The selected lines in LTE are the
4${}_{2}\rightarrow 3_{1}-E$ and 3${}_{1}\rightarrow 2_{0}-E$ transitions at
218.4 ($E_{\rm up}$/k$=$37.6 K) and 310.1 GHz ($E_{\rm up}$/k$=$27.1 K),
respectively (see Table 5). They are not blended with other methanol
transitions and do not appear to be significantly contaminated by blending
lines from other species (Table 5), given their good fit in the rotation
diagrams. Additionally, these two lines have $E_{\rm up}$/k values around the
one of the maser line at 84.5 GHz. The result of dividing the integrated
intensity of the 84.5 GHz maser line by the averaged intensity of the two
lines proposed to be in LTE (at 218.4 and 310.2 GHz) is presented in the left
panel of Fig. 13. In this Figure the spaxels with higher intensity ratios are
in line with regions where we previously identified maser emission through the
rotation diagram method. The distribution of maser emission we found is also
in good agreement with latest detections of methanol maser emission at 36.2
GHz in NGC 253 (see McCarthy et al. 2020, their Fig. 2). We have further
checked an equivalency between the intensity ratios and upper level column
density ratios in our Appendix F. We find that a line intensity ratio of
¿0.1244 corresponds to regions with maser emission in the
$J_{1}\rightarrow(J-1)_{0}-E$ line series (upper level column density ratios
¿3.3 with respect to LTE (Fig. 6, first panel)). This intensity ratio may
appear low, but it is only because the spatially widespread thermal lines have
higher intensities than the likely much less widespread non-thermal methanol
transition at 84.5 GHz. Furthermore, it is important to emphasize that the
maser’s column densities are nominal and do not reflect column densities in a
linear way due to amplification effects.

Taking into account the previous subsection 5.1, we noticed that, in order to
explain the lack of methanol maser emission in the $J_{-1}\rightarrow(J-$
1)${}_{0}-E$ line series (Class I) in the central regions of NGC 253, it is
necessary to invoke a combination of strong photodissociation in region 5 as
well as strong rather than weak shocks in regions 3, 4, and 6. In other words,
suitable conditions of Class I maser emission in these line series are weak
shocks and low rates of photodissociation, the latter preventing methanol
destruction. In the right panel of Fig. 13, we empirically determine a
threshold for the two conditions previously mentioned as SiO/HNCO ratios lower
than 1.05 and CN/C17O ratios lower than 60 in ALMA band 3 data.

In the right panel of Fig. 13, we also show the average values of the 10
selected regions (Table 3) derived from band 3 and they are distributed as
expected: central regions fall where no maser emission is expected. We
performed a similar diagram for ALMA band 6 data (not shown); in this case
region 7 is difficult to separate from locations where LTE conditions are
predominant, and regions 1, 2, and 8 to 10, are located where SiO/HNCO ratios
are in the 0.4 to 1.2 range and CN/C17O ratios are lower than 4.75 (both
ratios obtained from ALMA band 6 data).

A similar distribution for Class I methanol maser emission was previously
found by Ellingsen et al. (2017) and Gorski et al. (2017, 2019) for the 36.2
GHz line, which also belongs to the $J_{-1}\rightarrow(J-$ 1)${}_{0}-E$ line
series.

Figure 13: Left: Maser emission distribution per spaxel as observed by
dividing integrated intensities of the maser methanol line at 84.5 GHz by the
mean intensity value of the surrounding (in terms of $E_{\rm up}$/k) methanol
lines in LTE at 218.4 and 310.1 GHz (see Sect. 5.2). A 3$\sigma$ clipping was
applied in all the transitions involved to produce the figure. The center and
edges of the inner Lindblad resonance from Iodice et al. (2014) are denoted
with a black ellipse and dash-dotted grey ellipses, respectively. Regions
where we observe methanol maser emission in the $J_{-1}\rightarrow(J-$
1)${}_{0}-E$ line series are labeled in red. Right: Same spaxels (with their
values as colors) as in the figure to the left, but this time distributed
according to their ratios in the SiO/HNCO (y-axis) and CN/C17O line ratio maps
in ALMA band 3. A logarithmic stretch has been applied in both panels for an
easy visualization. Colors are in common for both panels. Our threshold of 3.3
above the rotation diagram fit given in Fig. 6 (first panel) corresponds to an
intensity ratio of ¿0.1244 (see Appendix F).

##  6 Summary and Conclusions

Searching for methanol (CH3OH) transitions, we have performed a spectral
survey toward the archetypical starburst galaxy NGC 253 with the ALMA
interferometer covering a frequency range between 84–373 GHz. We focus on ten
regions inside the CMZ of this galaxy; those regions are centered at the
position of giant molecular clouds.

After limiting our search to methanol lines with $E_{\rm up}$/k$<$150 K, since
above that limit lines are too weak to be detected, we identified all the
methanol transitions of each region separately and used the rotation diagram
method in order to determine which transitions deviate from LTE conditions.
Assuming LTE conditions, we found that E–CH3OH is more abundant at lower
$E_{\rm up}$/k and follows predominantly lower excitation temperatures than
A–CH3OH. We also find that including methanol lines with $E_{\rm up}$/k$<$150
K is sufficient to fit the observed spectra.

We also performed LTE and non-LTE model calculations with CASSIS-RADEX to find
maser candidates. A moderate difference of only about 15% in intensities
between the LTE and non-LTE predictions is found in our data when we consider
a single component for each methanol symmetric type (but this difference
becomes important when we consider two components, see Appendix C).

Although most of the observed methanol lines can be reproduced by our LTE
models, we found a number of outliers, of which 7 show maser properties. We
have confidently identified a total of 3 A-type and 4 E-type methanol maser
transitions, all of them classified as Class I.

We have also performed a more detailed non-LTE model in region 8 (Appendix C),
where the $J_{K}\rightarrow(J-1)_{K}$ line series is better reproduced. From
this model we obtained optical depths for the covered methanol transitions and
these support the presence of masers indicating negative opacities for our
best candidates.

Using rotation diagrams, we have confidently detected all but the last of the
covered methanol lines in the $J_{-1}\rightarrow(J-1)_{0}-$E series to be
masing, namely, methanol lines at 84.5, 132.9, and 229.8 GHz. The last
available line in this series, at 278.3 GHz, is detected to depart from LTE
when we compare the observed spectrum with the LTE synthetic spectrum, showing
maser characteristics at the edges of the CMZ in NGC 253. Maser action in the
84.5 GHz line, previously reported by McCarthy et al. (2018), is now detected
in more than one location for the first time.

An absorption line at 107 GHz was identified in the spectra of regions 1, 8
and 9, which are located at the outskirts of the CMZ of NGC 253.
Interestingly, the position of these regions coincides with the position of
the circumnuclear ring (co-spatial with the Lindblad resonances according to
Iodice et al. (2014), but see Sect. 5) and with Class I methanol maser
emission in the $J_{-1}\rightarrow(J-$ 1)${}_{0}-E$ line series.

The increment of weak shocks in the interplay between dust/gas lanes and the
circumnuclear ring is proposed to harbor favourable conditions to produce
Class I masers in the $J_{-1}\rightarrow(J-1)_{0}-$E line series. This happens
in regions 1, 2 and 8–10. These resonances are expected to create density
waves and locally increase the amount of shocks where they are located,
possibly creating favourable conditions to produce methanol masers. On the
other hand, considering the Class I masers belonging to the
$J_{0}\rightarrow(J-$ 1)${}_{1}-A$ line series (at 95, 146, and 198 GHz), we
found the first of them ($J=$8, at 95 GHz) departing at least by a factor of 3
(in region 4) from LTE in all the regions, and above a factor of 12 without
considering region 4. The levels giving rise to the transition at 146 GHz are
less populated, showing maser emission in regions 1, 7, 8 and 9. The last in
the $J_{0}\rightarrow(J-$ 1)${}_{1}-A$ series, at $J=$10, departs from LTE by
a factor higher than 3.3 in regions 3, 6, and 7, being the only one masing in
the nuclear parts of the CMZ detected through the rotation diagram method.

We computed the SiO $J=$5–4/HNCO (100,10–90,9) intensity ratios as a tracer of
strong over weak shocks for all regions and found that region 1 and 3 have the
highest and lowest values, respectively. When looking at the
$J_{-1}\rightarrow(J-1)_{0}-E$ series for those particular regions, we find
that in region 1 maser lines deviate the most from the expected LTE behaviour
in the rotation diagram. This indicates that the transitions in region 1 are
likely masing due to collisional excitation by shocks. For region 3, the
emission seems to be completely in LTE since the transitions align almost
perfectly in the rotation diagrams. When considering the SiO $J=$2–1/HNCO
(40,4–30,3) ratios, which cover much more spaxels and trace colder gas, we
find an opposite picture, where HNCO is stronger than SiO at the outskirts of
the CMZ, exactly where we observe methanol maser emission in the
$J_{-1}\rightarrow(J-1)_{0}-E$ line series, with the only exception of region
7, whose pumping mechanism might be dominated by star-forming processes, in
agreement with latest findings in NGC 253 targeting the methanol
4${}_{-1}\rightarrow$3${}_{0}-E$ transition at 36 GHz (Gorski et al. 2019)

For the regions in the center of the CMZ of NGC 253, we compared CN/C17O
ratios as a PDR tracer. We found out that regions 3, 4, and 5 show the highest
levels of photodissociation. The implied strong ultraviolet radiation may lead
to the destruction of methanol molecules in the central regions, preventing
the appearance of methanol masers.

Although several scenarios are proposed to explain the presence or absence of
methanol masers in all the regions in NGC 253, we cannot conclude which of
them is dominant over all the regions along the CMZ. Higher angular resolution
observations are needed to spatially resolve smaller regions and to
differentiate between the proposed mechanisms that give place to the formation
of methanol masers.

###### Acknowledgements.

PH is a member of and received financial support for this research from the
International Max Planck Research School (IMPRS) for Astronomy and
Astrophysics at the Universities of Bonn and Cologne. PH is grateful to Arnaud
Belloche, Dario Colombo, Yu Gao, and Sudeep Neupane, for their constructive
advice and fruitful discussions along a variety of aspects covered in this
work. VMR is funded by the Comunidad de Madrid through the Atracción de
Talento Investigador (Doctores con experiencia) Grant (COOL: Cosmic Origins Of
Life; 2019–T1/TIC–15379), and from the Agencia Estatal de Investigación (AEI)
through the Ramón y Cajal programme (grant RYC2020-029387-I). L.C. has
received partial support from the Spanish State Research Agency (AEI; project
number PID2019-105552RB-C41). KS thanks MOST grant 109-2112-M-001-020. N.H.
acknowledges support from JSPS KAKENHI Grant Number JP21K03634. This work
makes use of the following ALMA data: ADS/JAO.ALMA#2017.1.00161.L and
ADS/JAO.ALMA#2018.1.00162.S. ALMA is a partnership of ESO (representing its
member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST
and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the
Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and
NAOJ.

## References

  * Anantharamaiah & Goss (1996) Anantharamaiah, K. R. & Goss, W. M. 1996, ApJ, 466, L13
  * Ando et al. (2017) Ando, R., Nakanishi, K., Kohno, K., et al. 2017, ApJ, 849, 81
  * Araya et al. (2005) Araya, E., Hofner, P., Kurtz, S., Bronfman, L., & DeDeo, S. 2005, ApJS, 157, 279
  * Bacon et al. (2017) Bacon, R., Conseil, S., Mary, D., et al. 2017, A&A, 608, A1
  * Batrla et al. (1987) Batrla, W., Matthews, H. E., Menten, K. M., & Walmsley, C. M. 1987, Nature, 326, 49
  * Belloche et al. (2019) Belloche, A., Garrod, R. T., Müller, H. S. P., et al. 2019, A&A, 628, A10
  * Belloche et al. (2013) Belloche, A., Müller, H. S. P., Menten, K. M., Schilke, P., & Comito, C. 2013, A&A, 559, A47
  * Bendo et al. (2015) Bendo, G. J., Beswick, R. J., D’Cruze, M. J., et al. 2015, MNRAS, 450, L80
  * Billington et al. (2020) Billington, S. J., Urquhart, J. S., König, C., et al. 2020, MNRAS, 499, 2744
  * Breen et al. (2011) Breen, S. L., Ellingsen, S. P., Caswell, J. L., et al. 2011, ApJ, 733, 80
  * Breen et al. (2013) Breen, S. L., Ellingsen, S. P., Contreras, Y., et al. 2013, MNRAS, 435, 524
  * Breen et al. (2015) Breen, S. L., Fuller, G. A., Caswell, J. L., et al. 2015, MNRAS, 450, 4109
  * Chen et al. (2015) Chen, X., Ellingsen, S. P., Baan, W. A., et al. 2015, ApJ, 800, L2
  * Cohen et al. (2020) Cohen, D. P., Turner, J. L., & Consiglio, S. M. 2020, MNRAS, 493, 627
  * Cotton & Yusef-Zadeh (2016) Cotton, W. D. & Yusef-Zadeh, F. 2016, ApJS, 227, 10
  * Cragg et al. (1992) Cragg, D. M., Johns, K. P., Godfrey, P. D., & Brown, R. D. 1992, MNRAS, 259, 203
  * Cragg et al. (1993) Cragg, D. M., Mikhtiev, M. A., Bettens, R. P. A., Godfrey, P. D., & Brown, R. D. 1993, MNRAS, 264, 769
  * Cyganowski et al. (2018) Cyganowski, C. J., Hannaway, D., Brogan, C. L., Hunter, T. R., & Zhang, Q. 2018, in Astrophysical Masers: Unlocking the Mysteries of the Universe, ed. A. Tarchi, M. J. Reid, & P. Castangia, Vol. 336, 281–282
  * Darling et al. (2003) Darling, J., Goldsmith, P., Li, D., & Giovanelli, R. 2003, AJ, 125, 1177
  * de Vaucouleurs et al. (1991) de Vaucouleurs, G., de Vaucouleurs, A., Corwin, Herold G., J., et al. 1991, Third Reference Catalogue of Bright Galaxies
  * Ellingsen (2018) Ellingsen, S. P. 2018, in Astrophysical Masers: Unlocking the Mysteries of the Universe, ed. A. Tarchi, M. J. Reid, & P. Castangia, Vol. 336, 117–120
  * Ellingsen et al. (2010) Ellingsen, S. P., Breen, S. L., Caswell, J. L., Quinn, L. J., & Fuller, G. A. 2010, MNRAS, 404, 779
  * Ellingsen et al. (2017) Ellingsen, S. P., Chen, X., Breen, S. L., & Qiao, H. H. 2017, MNRAS, 472, 604
  * Ellingsen et al. (2014) Ellingsen, S. P., Chen, X., Qiao, H.-H., et al. 2014, ApJ, 790, L28
  * Ellingsen et al. (1994) Ellingsen, S. P., Norris, R. P., Whiteoak, J. B., et al. 1994, MNRAS, 267, 510
  * Fuente et al. (1993) Fuente, A., Martin-Pintado, J., Cernicharo, J., & Bachiller, R. 1993, A&A, 276, 473
  * García-Burillo et al. (2000) García-Burillo, S., Martín-Pintado, J., Fuente, A., & Neri, R. 2000, A&A, 355, 499
  * Goldsmith & Langer (1999) Goldsmith, P. F. & Langer, W. D. 1999, ApJ, 517, 209
  * Gorski et al. (2017) Gorski, M., Ott, J., Rand, R., et al. 2017, ApJ, 842, 124
  * Gorski et al. (2018) Gorski, M., Ott, J., Rand, R., et al. 2018, ApJ, 856, 134
  * Gorski et al. (2019) Gorski, M. D., Ott, J., Rand, R., et al. 2019, MNRAS, 483, 5434
  * Green et al. (2017) Green, J. A., Breen, S. L., Fuller, G. A., et al. 2017, MNRAS, 469, 1383
  * Green et al. (2008) Green, J. A., Caswell, J. L., Fuller, G. A., et al. 2008, MNRAS, 385, 948
  * Harada et al. (2021) Harada, N., Martín, S., Mangum, J. G., et al. 2021, ApJ, 923, 24
  * Harada et al. (2019) Harada, N., Sakamoto, K., Martín, S., et al. 2019, ApJ, 884, 100
  * Hartquist et al. (1995) Hartquist, T. W., Menten, K. M., Lepp, S., & Dalgarno, A. 1995, MNRAS, 272, 184
  * Haschick & Baan (1993) Haschick, A. D. & Baan, W. A. 1993, ApJ, 410, 663
  * Hastings (1970) Hastings, W. K. 1970, Biometrika, 57, 97
  * Holdship et al. (2021) Holdship, J., Viti, S., Martín, S., et al. 2021, A&A, 654, A55
  * Humire et al. (2020) Humire, P. K., Henkel, C., Gong, Y., et al. 2020, A&A, 633, A106
  * Impellizzeri et al. (2008) Impellizzeri, C. M. V., Henkel, C., Roy, A. L., & Menten, K. M. 2008, A&A, 484, L43
  * Iodice et al. (2014) Iodice, E., Arnaboldi, M., Rejkuba, M., et al. 2014, A&A, 567, A86
  * Kalenskii et al. (2010) Kalenskii, S. V., Johansson, L. E. B., Bergman, P., et al. 2010, MNRAS, 405, 613
  * Kalenskii et al. (2006) Kalenskii, S. V., Promyslov, V. G., Slysh, V. I., Bergman, P., & Winnberg, A. 2006, Astronomy Reports, 50, 289
  * Kalenskii et al. (2002) Kalenskii, S. V., Slysh, V. I., & Val’tts, I. E. 2002, in Cosmic Masers: From Proto-Stars to Black Holes, ed. V. Migenes & M. J. Reid, Vol. 206, 191
  * Kang et al. (2015) Kang, H., Kim, K.-T., Byun, D.-Y., Lee, S., & Park, Y.-S. 2015, ApJS, 221, 6
  * Kelly et al. (2017) Kelly, G., Viti, S., García-Burillo, S., et al. 2017, A&A, 597, A11
  * Kim et al. (2020) Kim, W. J., Wyrowski, F., Urquhart, J. S., et al. 2020, A&A, 644, A160
  * Kim et al. (2012) Kim, W.-T., Seo, W.-Y., & Kim, Y. 2012, ApJ, 758, 14
  * Ladeyschikov et al. (2019) Ladeyschikov, D. A., Bayandina, O. S., & Sobolev, A. M. 2019, AJ, 158, 233
  * Lees (1973) Lees, R. M. 1973, ApJ, 184, 763
  * Lees & Baker (1968) Lees, R. M. & Baker, J. G. 1968, J. Chem. Phys., 48, 5299
  * Leroy et al. (2015) Leroy, A. K., Bolatto, A. D., Ostriker, E. C., et al. 2015, ApJ, 801, 25
  * Leroy et al. (2018) Leroy, A. K., Bolatto, A. D., Ostriker, E. C., et al. 2018, ApJ, 869, 126
  * Leurini & Menten (2018) Leurini, S. & Menten, K. M. 2018, in Astrophysical Masers: Unlocking the Mysteries of the Universe, ed. A. Tarchi, M. J. Reid, & P. Castangia, Vol. 336, 17–22
  * Leurini et al. (2016) Leurini, S., Menten, K. M., & Walmsley, C. M. 2016, A&A, 592, 31
  * Li et al. (2015) Li, Z., Shen, J., & Kim, W.-T. 2015, ApJ, 806, 150
  * Liechti & Wilson (1996) Liechti, S. & Wilson, T. L. 1996, A&A, 314, 615
  * Lo (2005) Lo, K. Y. 2005, ARA&A, 43, 625
  * Ma et al. (1998) Ma, C., Arias, E. F., Eubanks, T. M., et al. 1998, AJ, 116, 516
  * Mangum & Shirley (2015) Mangum, J. G. & Shirley, Y. L. 2015, PASP, 127, 266
  * Martín et al. (2021) Martín, S., Mangum, J. G., Harada, N., et al. 2021, A&A, 656, A46
  * Martín et al. (2019) Martín, S., Martín-Pintado, J., Blanco-Sánchez, C., et al. 2019, A&A, 631, A159
  * Martín et al. (2006) Martín, S., Mauersberger, R., Martín-Pintado, J., Henkel, C., & García-Burillo, S. 2006, ApJS, 164, 450
  * Matsumoto et al. (2014) Matsumoto, N., Hirota, T., Sugiyama, K., et al. 2014, ApJ, 789, L1
  * McCarthy et al. (2018) McCarthy, T. P., Ellingsen, S. P., Breen, S. L., Voronkov, M. A., & Chen, X. 2018, ApJ, 867, L4
  * McCarthy et al. (2020) McCarthy, T. P., Ellingsen, S. P., Breen, S. L., et al. 2020, MNRAS, 491, 4642
  * McCarthy et al. (2017) McCarthy, T. P., Ellingsen, S. P., Chen, X., et al. 2017, ApJ, 846, 156
  * Meier et al. (2015) Meier, D. S., Walter, F., Bolatto, A. D., et al. 2015, ApJ, 801, 63
  * Menten (1991) Menten, K. M. 1991, ApJ, 380, L75
  * Menten (2012) Menten, K. M. 2012, in Cosmic Masers - from OH to H0, ed. R. S. Booth, W. H. T. Vlemmings, & E. M. L. Humphreys, Vol. 287, 506–515
  * Meyer et al. (2004) Meyer, M. J., Zwaan, M. A., Webster, R. L., et al. 2004, MNRAS, 350, 1195
  * Morris & Serabyn (1996) Morris, M. & Serabyn, E. 1996, ARA&A, 34, 645
  * Moscadelli et al. (2003) Moscadelli, L., Menten, K. M., Walmsley, C. M., & Reid, M. J. 2003, ApJ, 583, 776
  * Müller et al. (2005) Müller, H. S. P., Schlöder, F., Stutzki, J., & Winnewisser, G. 2005, Journal of Molecular Structure, 742, 215
  * Müller-Sánchez et al. (2010) Müller-Sánchez, F., González-Martín, O., Fernández-Ontiveros, J. A., Acosta-Pulido, J. A., & Prieto, M. A. 2010, ApJ, 716, 1166
  * Oike et al. (2004) Oike, T., Kawaguchi, K., Takano, S., & Nakai, N. 2004, PASJ, 56, 431
  * Pence (1980) Pence, W. D. 1980, ApJ, 239, 54
  * Pickett et al. (1998) Pickett, H. M., Poynter, R. L., Cohen, E. A., et al. 1998, J. Quant. Spec. Radiat. Transf., 60, 883
  * Pihlström et al. (2014) Pihlström, Y. M., Sjouwerman, L. O., Frail, D. A., et al. 2014, AJ, 147, 73
  * Plambeck & Menten (1990) Plambeck, R. L. & Menten, K. M. 1990, ApJ, 364, 555
  * Rabli & Flower (2010) Rabli, D. & Flower, D. R. 2010, MNRAS, 406, 95
  * Regan & Teuben (2003) Regan, M. W. & Teuben, P. 2003, ApJ, 582, 723
  * Rekola et al. (2005) Rekola, R., Richer, M. G., McCall, M. L., et al. 2005, MNRAS, 361, 330
  * Rodríguez-Garza et al. (2017) Rodríguez-Garza, C. B., Kurtz, S. E., Gómez-Ruiz, A. I., et al. 2017, ApJS, 233, 4
  * Rosolowsky & Leroy (2006) Rosolowsky, E. & Leroy, A. 2006, PASP, 118, 590
  * Sakamoto et al. (2006) Sakamoto, K., Ho, P. T. P., Iono, D., et al. 2006, ApJ, 636, 685
  * Salii et al. (2002) Salii, S. V., Sobolev, A. M., & Kalinina, N. D. 2002, Astronomy Reports, 46, 955
  * Salter et al. (2008) Salter, C. J., Ghosh, T., Catinella, B., et al. 2008, AJ, 136, 389
  * Sánchez-Monge et al. (2018) Sánchez-Monge, Á., Schilke, P., Ginsburg, A., Cesaroni, R., & Schmiedeke, A. 2018, A&A, 609, A101
  * Schmidt et al. (2019) Schmidt, E. O., Mast, D., Díaz, R. J., et al. 2019, AJ, 158, 60
  * Sinclair et al. (1992) Sinclair, M. W., Carrad, G. J., Caswell, J. L., Norris, R. P., & Whiteoak, J. B. 1992, MNRAS, 256, 33P
  * Sjouwerman et al. (2010) Sjouwerman, L. O., Murray, C. E., Pihlström, Y. M., Fish, V. L., & Araya, E. D. 2010, ApJ, 724, L158
  * Sobolev (1993) Sobolev, A. M. 1993, Astronomy Letters, 19, 293
  * Strickland et al. (2000) Strickland, D. K., Heckman, T. M., Weaver, K. A., & Dahlem, M. 2000, AJ, 120, 2965
  * Szczepanski et al. (1989) Szczepanski, J. C., Ho, P. T. P., Haschick, A. D., & Baan, W. A. 1989, in The Center of the Galaxy, ed. M. Morris, Vol. 136, 383
  * Thomas & Fuller (2008) Thomas, H. S. & Fuller, G. A. 2008, A&A, 479, 751
  * Turner (1985) Turner, B. E. 1985, ApJ, 299, 312
  * Val’tts et al. (1995) Val’tts, I. E., Dzura, A. M., Kalenskii, S. V., et al. 1995, A&A, 294, 825
  * van der Tak et al. (2007) van der Tak, F. F. S., Black, J. H., Schöier, F. L., Jansen, D. J., & van Dishoeck, E. F. 2007, A&A, 468, 627
  * Voronkov et al. (2012) Voronkov, M. A., Caswell, J. L., Ellingsen, S. P., et al. 2012, in Cosmic Masers - from OH to H0, ed. R. S. Booth, W. H. T. Vlemmings, & E. M. L. Humphreys, Vol. 287, 433–440
  * Voronkov et al. (2014) Voronkov, M. A., Caswell, J. L., Ellingsen, S. P., Green, J. A., & Breen, S. L. 2014, MNRAS, 439, 2584
  * Voronkov et al. (2010) Voronkov, M. A., Caswell, J. L., Ellingsen, S. P., & Sobolev, A. M. 2010, MNRAS, 405, 2471
  * Walmsley et al. (1988) Walmsley, C. M., Batrla, W., Matthews, H. E., & Menten, K. M. 1988, A&A, 197, 271
  * Walsh et al. (2001) Walsh, A. J., Bertoldi, F., Burton, M. G., & Nikola, T. 2001, MNRAS, 326, 36
  * Wang et al. (2014) Wang, J., Zhang, J., Gao, Y., et al. 2014, Nature Communications, 5, 5449
  * Wilson (2009) Wilson, T. L. 2009, arXiv e-prints, arXiv:0903.0562
  * Yang et al. (2019) Yang, K., Chen, X., Shen, Z.-Q., et al. 2019, ApJS, 241, 18
  * Yang et al. (2017) Yang, K., Chen, X., Shen, Z.-Q., et al. 2017, The Astrophysical Journal, 846, 160
  * Yang et al. (2020) Yang, W., Xu, Y., Choi, Y. K., et al. 2020, ApJS, 248, 18
  * Yu et al. (2018) Yu, N.-P., Xu, J.-L., & Wang, J.-J. 2018, Research in Astronomy and Astrophysics, 18, 015
  * Yusef-Zadeh et al. (2013) Yusef-Zadeh, F., Cotton, W., Viti, S., Wardle, M., & Royster, M. 2013, ApJ, 764, L19
  * Zinchenko et al. (2017) Zinchenko, I., Liu, S. Y., Su, Y. N., & Sobolev, A. M. 2017, A&A, 606, L6

##  Appendix A Rotation diagrams

We make use of rotation diagrams as our first method to unveil the presence of
methanol maser lines. This procedure assumes the Rayleigh-Jeans (RJ)
approximation131313see e.g. the derivation presented in Araya et al. (2005),
between Equations A9 and A10., valid when the following condition is
fulfilled: $\nu$[GHz]$\ll$20.84 [GHz/K] $T_{\rm ex}$[K] (see e.g. Wilson
2009). The rotation diagram method also assumes a negligible background
continuum (see e.g. Belloche et al. 2019, their Chapter 4.4), and relates the
total column density $N_{\rm tot}$ of a given species with the excitation
temperature $T_{\rm ex}$141414Under LTE, the underlying assumption for the use
of the rotation diagram method, the excitation temperature of any transition
is equal to the rotation temperature that describes the populations of the
rotational levels and the integrated intensity of the line profiles as

$\frac{N_{\rm tot}}{N_{\rm up}}=\frac{Q_{\rm rot(\textit{T}_{\rm ex})}}{g_{\rm
up}}\exp{\Bigg{(}\frac{E_{\rm up}}{kT_{\rm ex}}\Bigg{)}},$ (1)

where $Q_{\rm rot}(T)$ is the partition function of the species, which is the
multiplication factor needed to go from a column density in a single state to
the entire column density of a molecular species summed over all states
(Mangum & Shirley 2015). $g_{\rm up}$ is the statistical weight of the upper
level, $E_{\rm up}$/k is the energy of the upper level above the ground state,
and $k$ is the Boltzmann constant. In the optically thin case, the beam-
averaged $N_{\rm up}$ is given by

$N_{\rm up}=\frac{8\pi k\nu^{2}}{hc^{3}A_{\rm ul}}\int T_{\rm mb}dv,$ (2)

where $\nu$ is the frequency, $h$ is the Planck constant, $c$ is the speed of
light, $A_{\rm ul}$ is the Einstein coefficient for spontaneous emission,
$T_{\rm mb}$ is the intensity in Kelvin units and $v$ denotes the radial
velocity over which the integral is calculated, covering the entire observed
range of a specific line emission. Thus, the term $\int T_{\rm mb}dv$
corresponds to the integrated intensity of the line, assuming that the
emitting region completely covers the beam. When plotting equation 1 with
$N_{\rm up}$ on a logarithmic scale, we obtain a straight line, with a slope
proportional to the negative inverse value of $T_{\rm ex}$.

Table 5: Selected methanol transitions.

Transition |  $\nu$ [GHz] | A/E | R1 | R2 | R3 | R4 | R5 | R6 | R7 | R8 | R9 | R10 | potential blending lines
---|---|---|---|---|---|---|---|---|---|---|---|---|---
5${}_{-1}\rightarrow$40 | 84.521172 | E |  |  |  |  | – |  |  |  |  |  | –
2${}_{1}\rightarrow$11 | 95.914310 |  A+ |  |  |  |  | – |  |  |  |  |  |  C2H5OH
2${}_{1}\rightarrow$11 | 97.582798 |  A- |  |  |  |  | – |  |  |  |  | – |  CH3COCH3
3${}_{1}\rightarrow$40 | 107.013831 |  A+ | – | – |  |  | – |  |  | – | – | – |  CH3COCH3
0${}_{0}\rightarrow$1-1 | 108.893945 | E |  |  |  |  | – |  |  |  |  |  | –
6${}_{-1}\rightarrow$50 | 132.890759 | E |  |  |  |  | – |  |  |  |  |  |  CH3OCHO
3${}_{1}\rightarrow$21 | 143.865795 |  A+ | * |  |  |  | – |  |  |  |  | – |  C2H5OH
3${}_{1}\rightarrow$21 | 146.368328 |  A- | * |  |  |  | – |  |  |  |  | bN |  cis–DCOOH, n–C3H7CN
9${}_{0}\rightarrow$81 | 146.618697 |  A+ |  | – | – |  | – | – | * |  |  | – |  SO2
8${}_{0}\rightarrow$8-1 | 156.488902 | E | – | – | – |  | – | – | – | – | – | – |  C2H3CN
2${}_{1}\rightarrow$30 | 156.602395 |  A+ | – | N |  |  | – |  |  | – |  | – |  C2H5C–13–N
7${}_{0}\rightarrow$7-1 | 156.828517 | E | – | – | – |  | – |  |  | – | – | – |  C2H5CN, NH2CHO
6${}_{0}\rightarrow$6-1 | 157.048617 | E | – | – | – |  | – |  |  | – | – | – | HCOOH
5${}_{0}\rightarrow$5-1 | 157.178987 | E | – | – | – |  | – |  |  | – | – | – |  CH3COCH3, 33SO
5${}_{1}\rightarrow$50 | 165.369341 | E |  |  |  |  | – |  |  |  |  | * |  C2H5CN
4${}_{1}\rightarrow$31 | 191.810503 |  A+ | – | * |  |  | – |  |  | bN |  | * |  HCOOC2H5, n–C3H7CN
4${}_{1}\rightarrow$31 | 195.146790 |  A- | * | * |  |  | – |  |  |  |  | N |  CH3OCHO, CH3OCH3
10${}_{0}\rightarrow$91 | 198.403067 |  A+ | * | * |  |  | – |  |  | N | bN | * | –
1${}_{1}\rightarrow$20 | 205.791270 |  A+ | * |  |  |  | – |  |  |  |  | bN |  C2H5OH, CH3OCHO
1${}_{1}\rightarrow$00 | 213.427061 | E |  |  |  |  | – |  |  |  |  |  |  c–C3H
5${}_{1}\rightarrow$42 | 216.945521 | E | – | – | – |  | – |  |  | – | – | – | –
4${}_{2}\rightarrow$31 | 218.440063 | E | * |  |  |  | – |  | * |  |  |  |  c–C3H2
8${}_{-1}\rightarrow$70 | 229.758756 | E |  | – | – |  | – |  |  |  |  | – |  CH3CHO
5${}_{1}\rightarrow$41 | 239.746219 |  A- |  | – |  |  | – |  |  |  |  | bN | HCOOD
5${}_{1}\rightarrow$41 | 243.915788 |  A+ | * | – |  |  | – |  |  | * |  | bN |  D2 34S, 34SO2
4${}_{2}\rightarrow$51 | 247.228587 | A | – | – | bN |  | – | – |  | – | – | – |  C2H5OH
2${}_{0}\rightarrow$1-1 | 254.015377 | E |  |  |  |  | – |  |  |  |  |  |  c–C3D2, C2H5OH
2${}_{1}\rightarrow$10 | 261.805675 | E |  |  |  |  | – |  |  |  |  |  |  C2H, c–C3H2
6${}_{1}\rightarrow$51 | 287.670767 |  A+ |  | – |  |  | – |  |  |  |  | – | CCS
6${}_{0}\rightarrow$50 | 289.939377 | E | – | – |  |  | – |  | * |  |  | – |  S17O, HNC18O
6${}_{1}\rightarrow$51 | 292.672889 |  A- |  | – |  |  | – |  |  | * |  | – |  CH3COCH3, CH3CHO
3${}_{0}\rightarrow$2-1 | 302.369753 | E |  |  |  |  | – |  |  |  |  |  |  c–HCOOH, CH3OCHO
1${}_{1}\rightarrow$10 | 303.366921 |  A-+ |  |  |  |  | – |  |  | * |  |  |  H2CN
2${}_{1}\rightarrow$20 | 304.208348 |  A-+ |  |  |  |  | – |  |  |  |  |  |  NH2CHO, C2H5OH
3${}_{1}\rightarrow$30 | 305.473491 |  A-+ |  |  |  |  | – |  |  |  |  | bN |  C2H5CN, CH3COCH3
4${}_{1}\rightarrow$40 | 307.165924 |  A-+ |  |  |  |  | – |  |  |  |  |  |  CH3OCHO
5${}_{1}\rightarrow$50 | 309.290360 |  A-+ | * |  |  |  | – |  |  |  |  | bN |  33SO
3${}_{1}\rightarrow$20 | 310.192994 | E |  |  |  |  | – |  |  |  |  |  |  CH3OCHO, CH3COCH3
6${}_{1}\rightarrow$60 | 311.852612 |  A-+ | * | bN |  |  | – |  |  |  |  |  |  c–C3H2, CH3COCH3
7${}_{1}\rightarrow$70 | 314.859528 |  A-+ | * | – |  |  | – |  |  | * |  | – |  C2H3CN
6${}_{2}\rightarrow$51 | 315.266861 | E |  | – |  |  | – |  |  | * | * | – |  C2H3CN
8${}_{1}\rightarrow$80 | 318.318919 |  A-+ | – | – |  |  | – |  |  | bN | bN | – |  c–C3H2, C2H5OH
7${}_{1}\rightarrow$61 | 335.582017 |  A+ |  | – |  |  | – |  |  | – | – | – |  n–C3H7CN, l–C3H2
7${}_{0}\rightarrow$60 | 338.124488 | E | – |  |  |  | – |  |  |  |  | – |  CH3OCHO
7${}_{1}\rightarrow$61 | 341.415615 |  A- | – | – |  |  | – |  |  | – | – | – |  NH2CHO, CH3OCHO
4${}_{0}\rightarrow$3-1 | 350.687662 | E |  |  |  |  | – |  |  |  |  |  | NO
1${}_{1}\rightarrow$00 | 350.905100 |  A+ |  |  |  |  | – |  |  |  |  |  |  CH3COCH3, 33SO2
4${}_{1}\rightarrow$30 | 358.605799 | E |  | * |  |  | – |  |  |  |  | – |  CH3OCHO, n–C3H7CN
7${}_{2}\rightarrow$61 | 363.739868 | E | – | – |  |  | – |  |  | – | – | – | –
TNoL |  |  | 25(35) | 23(29) | 41(42) | 49 | – | 46 | 45(48) | 31(38) | 35(39) | 16(26) |
Hα |  |  | – | 1147 | 58303 | 58579 | 17814 | 36998 | 36291 | 8150 | 7073 | 2603 |
151515Selected methanol lines to perform our rotation diagrams. Possible
contamination with other molecules is mentioned in the rightmost column. The
symbol * indicates bad-shaped lines. The letter N indicates lines embedded in
a noisy continuum but with intensities above the 3.5$\sigma$ level. bN denotes
lines tentatively detected (2–3$\sigma$).  indicates well-
distinguished/confidently-detected lines, and the dash (–) indicates a non-
detection. The third last row provides the total number of lines (TNoL)
detected toward each region, considering lines confidently observed (check-
mark). The TNoL considering the ones not clearly detected (*, N, bN), are
given in parenthesis. The last row presents the Hα mean intensity for each
region considering an angular extent of 1$\aas@@fstack{\prime\prime}$6 (the
beam size), obtained from MUSE archival data (ID:0102.B-0078(A), PI:Laura
Zschaechner) in units of 10-20 erg/s/cm2/$\mathrm{\SIUnitSymbolAngstrom}$. For
an alternative approach on line blending, see Tables 7 and 8.

Given the relatively broad frequency range covered by ALCHEMI, it can be
expected that the RJ approximation will not be correct for all transitions and
temperatures measured in the CMZ of NGC 253. It is impossible to know a priori
whether the $T_{\rm ex}$ will be low enough to cause a significant distortion
between the Planck function and the RJ approximation. Given the results
summarized in our Table A.2, this approximation has an appreciable bias and we
have decided to calculate a correction factor ($CF$) meant to be multiplied by
the upper level column density of each methanol transition in the rotation
diagrams. This $CF$ is equal to the Planck function over the RJ approximation:

$CF=\frac{h\nu}{kT_{\rm ex}}\frac{1}{\exp(\frac{h\nu}{kT_{\rm ex}})-1}$ (3)

Considering methanol transitions with frequencies of 200 and 300 GHz inside a
region with a temperature of 10–20 K, we obtained correction factors in the
range of 0.6–0.8 and 0.45–0.7, respectively. After applying this correction to
our rotation diagrams (assuming RJ) the resulting $T_{\rm ex}$ obtained for
each region vary mostly by less than 1 K. The total column densities, on the
other hand, can decrease by $\sim$50% in the coldest regions, such as regions
2 and 10 for E–CH3OH (see Table 6), and by $\sim$20% in warmer environments.
The lowest values obtained in this way are included in the uncertainties shown
in Table 6.

We could do an iterative process from the temperature obtained from RJ to the
one obtained after applying the correction factor until reaching a certain
$T_{\rm ex}$ asymptotically. However, since $T_{\rm ex}$ hardly varies, we
decided to include the new $N_{\rm{up}}$ values inside the uncertainties after
a single correction is applied. Indeed, the change of the $T_{\rm ex}$ assumed
when applying $CF$ will not have an important effect on the slope derived in
the rotation diagrams. It might slightly increase the dispersion of the
individual $N_{\rm{up}}$ values with respect to the LTE slope (solid black
lines in Fig. 3).

This dispersion is inversely proportional to $T_{\rm ex}$, and $T_{\rm ex}$ is
higher for the A-CH3OH species with respect to the E-CH3OH species (see Table
6). One might suspect that an increased dispersion may discard the
10${}_{0}\rightarrow 9_{1}-A^{+}$ transition at 198.4 GHz as a maser candidate
because of its departure from the LTE slope of a factor of 3.6$\pm$1.2 in
region 6, as indicated in Sect. 4.1.3, given our threshold of 3.3 (3$\sigma$;
Sect. 4.1). However, this region has a $T_{\rm ex}$ of 24.3 K indicating that
RJ is a good approximation for the Planck formula. All the other methanol
maser candidates depart from LTE by a factor higher than four times the LTE
expected value.

The results from our rotation diagrams fitting to the LTE lines are listed in
Table 6. The effect of having a strong continuum might affect our synthetic
spectra, potentially leading to an overestimation of the emission (e.g.
Belloche et al. 2019). However, comparing our data with the models, this can
be ruled out. Indeed, we noticed that considering our deviation from the
Planck function in the total column density uncertainties, the rotation
diagram results are in better agreement with the modelling results (see Table
3). This is especially critical for the colder regions. It can also be
appreciated that sometimes the lower limits are placed much below the ones
derived by the radiative transfer models, as in the case of region 4, where
the temperatures derived from the rotation diagram method seem to be
underestimated by $\sim$5 K with respect to the LTE models, generating an
artificial scatter and lower total column densities. The latter support our
predicament of including the newly derived upper-level column density values
obtained after applying the $CF$ inside an error bar instead of replacing the
values obtained from RJ. Accounting for the excitation temperatures, they are
always higher in the modelling as compared to the rotation diagrams for
E-methanol the nuclear regions (3 to 6) in A-methanol, and the opposite is
observed at the outskirts of the NGC 253’s CMZ for the A-methanol case
(regions 1 and 8 to 10).

Table 6: Fit parameters from our rotation diagrams.

Region |  $N$(Sp) | $T_{\rm ex}$ |  $N$(Sp) |  Tex
---|---|---|---|---
|  [$\times$1013 cm-2] | [K] |  [$\times$1013 cm-2] | [K]
|  E–CH3OH |  |  A–CH3OH |
R1 |  67.3${}^{+10.1}_{-38.4}$ |  9.7${}^{+1.0}_{-2.1}$ |  9.9${}^{+1.5}_{-3.8}$ |  25.0${}^{+3.1}_{-2.9}$
R2 |  78.1${}^{+11.7}_{-30.6}$ |  8.4${}^{+0.8}_{-1.9}$ |  10.0${}^{+1.5}_{-5.0}$ |  14.5${}^{+1.9}_{-2.0}$
R3 |  263.0${}^{+39.4}_{-105.0}$ |  19.0${}^{+1.6}_{-2.7}$ |  84.2${}^{+12.6}_{-32.1}$ |  23.1${}^{+3.1}_{-3.3}$
R4 |  91.2${}^{+13.7}_{-28.8}$ |  27.5${}^{+3.9}_{-4.0}$ |  76.2${}^{+11.4}_{-26.5}$ |  27.7${}^{+5.3}_{-5.5}$
R6 |  253.7${}^{+38.1}_{-92.1}$ |  20.3${}^{+2.9}_{-1.9}$ |  115.4${}^{+17.3}_{-36.6}$ |  24.3${}^{+3.1}_{-4.8}$
R7 |  275.2${}^{+41.3}_{-134.3}$ |  12.3$\pm$1.3 |  58.4${}^{+8.8}_{-21.1}$ |  25.7${}^{+4.6}_{-5.2}$
R8 |  77.8${}^{+11.7}_{-44.4}$ |  9.9${}^{+0.9}_{-1.9}$ |  12.6${}^{+1.9}_{-4.1}$ |  30.8${}^{+7.3}_{-7.1}$
R9 |  103.3${}^{+11.7}_{-57.2}$ |  10.4${}^{+1.0}_{-1.9}$ |  17.9${}^{+2.7}_{-5.6}$ |  29.3${}^{+7.7}_{-7.6}$
R10 |  11.6${}^{+1.74}_{-6.8}$ |  9.6${}^{+0.8}_{-2.7}$ |  3.0${}^{+0.4}_{-1.1}$ |  23.8${}^{+14.5}_{-13.5}$
161616Total column densities ($N$(Sp)) and excitation temperatures ($T_{\rm
ex}$) for each methanol symmetric type derived from the best LTE fit to the
rotation diagrams (see Fig. 3). Uncertainties correspond to the standard
deviation (1$\sigma$).

##  Appendix B Rotation diagram: non-masing outliers

Making use of the rotation diagram method (Sec. 4.1), we have found a variety
of methanol lines that do not follow LTE conditions but that, at the same
time, do not correspond to masers. These lines are placed beyond a 3$\sigma$
scatter, surpassing by more than a factor of 3.3 the expected $N_{\rm up}$
value from LTE conditions. We list them in Table 2 and provide a description
below, in frequency order.

We realized that the five A–CH3OH lines, at 95.9 GHz, 97.6, 143.9, 146.4, and
156.6 GHz, that usually depart from the LTE conditions of A-type methanol,
actually follow the LTE trend of the E–CH3OH species. Since the conditions
between A- and E-methanol types differ towards the edge of the CMZ, these five
transitions are clearly apart from LTE conditions in regions 1, 2, and 7 to 9.
Therefore we decided to not include them in the LTE fitting in any region.

With the exception of the line at 156.6 GHz, the other four can be considered
as members of the same family, although varying between symmetric labels (A+
and A-), corresponding to the $(J+1)_{1}\rightarrow J_{1}-A$ series. In
general, the mentioned five transitions have Einstein coefficients for
spontaneous emission (Aij) below 1.78$\times$10-5 s-1, unlike the other
A–CH3OH lines in LTE.

As can be seen in Fig 3, these five A–CH3OH transitions with low Aij also have
60–600 times lower critical densities than transitions at similar $E_{\rm
up}$/k and this is not significantly changed if $T_{\rm ex}$ is moderately (by
less than a factor of five) lower than $T_{\rm kin}$. This means that those
transitions require a particularly small critical density to emerge, pointing
out that they are tracing conditions of cooler gas located in layers farther
away from the GMC cores as compared to the rest of the A-methanol lines. The
latter is also based on the overall higher temperatures in A-type methanol
compared to E-type methanol in the nine analyzed regions (see fourth panel in
Fig. 6).

The line at 261.8 GHz is located beyond LTE in region 6 only. This can be
caused by line blending with C2H and c–C3H2, molecules previously observed in
the central regions of NGC 253 (Oike et al. 2004; Martín et al. 2006; Holdship
et al. 2021).

In regions 1 and 8, the $6_{2}\rightarrow 5_{1}-E$ line at 315.3 GHz ($E_{\rm
up}$/k$=$63.1 K) seems to act as a maser. However, this line is not clearly
observed as an outlier in the other 7 regions. Additionally, if methanol is
tracing two different temperatures, the line at 315.2 GHz is related to the
temperature better described by A–CH3OH and is therefore likely not a maser.

The line at 318.3 GHz ($E_{\rm up}$/k$=98.8\,K$) strongly departs from LTE in
the inner regions (regions 3 to 6). Similar to the case of the line at 261.8
GHz, this departure is likely due to the contribution of c–C3H2.

The line at 338.1 GHz ($E_{\rm up}$/k$=78.8$ K) falls beyond LTE in region 8.
In regions 6 to 9 this transition is placed above the LTE trend described by
E-type methanol, being closer to the conditions described by A-type methanol
instead. In this case we can not argue for a higher critical density for this
behavior, contrary to other E–CH3OH transitions (see below).

The line at 350.7 GHz ($E_{\rm up}$/k$=36.3$ K) appears to be out of LTE in
region 6. Performing a simple LTE model for NO in region 6, we find that NO
contributes to 70% to the measured blended feature. We also find that NO is
stronger in region 6 than in any other studied region (by doing a moment 0 map
of this line, not shown), explaining why we do not see the 350.7 GHz methanol
line seemingly departing from LTE in any other region.

The E–CH3OH transitions at 358.605 ($E_{\rm up}$/k$=36.4$ K) and 363.739 GHz
($E_{\rm up}$/k$=79.4$ K) are emerging at the highest densities, tracing
conditions of A–CH3OH. They clearly depart from LTE in region 7, and serve as
a complementary case with respect to the five A–CH3OH lines mentioned above.

###  B.1 lines blended with maser candidates

The $\mathbf{5_{-1}\rightarrow 4_{0}-E}$ line at 84.52 GHz occupies a similar
frequency window as the $12_{10}\rightarrow 12_{11}-E$ methanol line at 84.53
GHz in all selected regions except region 10, where the line profiles are
narrow enough to be disentangled. Line blending, however, is unlikely since
the potential companion line connects levels 1085 K above the ground state and
has an Einstein coefficient $\sim$2.6 times lower than the masing line (see
Table 2). In addition to that, there is no observed emission from the line at
84.53 GHz in region 10 (only the maser line is observed). Furthermore, as
mentioned in Sections 4.1 and 4.2, in general there is no methanol emission
involving energy levels $>$150 K above the ground state.

The $\mathbf{7_{-1}\rightarrow 6_{0}-E}$ line at 181.295 GHz which, to our
knowledge, was not yet reported as a (Galactic) maser line (e.g. Leurini et
al. 2016), is dramatically contaminated by the $J=$2–1 HNC line at 181.324
GHz, impeding a proper line inspection. This methanol transition is also
affected by its proximity to telluric contamination (H2O at $\sim$183 GHz)
that decreases the quality of our data near this frequency.

The $\mathbf{9_{-1}\rightarrow 8_{0}-E}$ transition at 278.305 GHz is the
highest excited line ($E_{\rm up}$/k$\sim$102 K) covered by our observations
in the $J_{-1}\rightarrow(J-$ 1)${}_{0}-E$ series. It is strongly contaminated
by the 2${}_{-2}\rightarrow 3_{-1}-E$ line at 278.342 GHz. Due to this
blending, this methanol line is not considered in the rotation diagrams.
However, as presented in Sect. 4.2, a maser line profile emerges in regions 8,
9 and 10.

The $\mathbf{9_{0}\rightarrow 8_{1}-A^{+}}$ line at 146.618 GHz appears to be
masing in region 4, with a departure from LTE of 3.3$\pm$1.2 times. However,
this line is considerably contaminated by SO2 in region 5 ($\sim$80% of the
total integrated intensity of the blended feature, Martín et al. in prep.) and
therefore, due to the proximity to this region, we expect an important
contamination in region 4 and also some contribution in region 7 (after
performing a moment 0 map for SO2, not shown). The above influences its
departure from LTE in the mentioned regions, producing a larger displacement
than for the lines at 84.5 GHz and 132.9 GHz, which are not contaminated. The
$9_{0}\rightarrow 8_{1}-A^{+}$ line at 146.618 GHz is also blended with the
14${}_{1}\rightarrow 13_{2}-A^{+}$ methanol line at 146.617 GHz, which
requires twice the $E_{\rm up}$/k (256 K, see Table 2) to emerge. However, the
contribution of this companion line at 146.617 GHz to the maser line at
146.618 GHz is not higher than 2% (Martín et al. in prep.).

##  Appendix C Detailed model analysis

In Sect. 4.2, we limited our LTE and non-LTE modelling to a single component
for each methanol symmetric type and then we preferred to continue with the
LTE model only. This is enough to characterize our selected regions for the
scope of this study, resulting in synthetic spectra that adjust to most of the
observed methanol emission lines, as can be seen in our Tables 7 and 8. The
comparison between the observed and the resulting synthetic spectra clearly
unveil lines with intensities far above the LTE trend.

Table 7: E-type methanol lines used as input for our LTE model.

Transition |  $\nu$ [GHz] | R1 | R2 | R3 | R4 | R5 | R6 | R7 | R8 | R9 | R10 | Blending lines
---|---|---|---|---|---|---|---|---|---|---|---|---
6${}_{-2}\rightarrow$7-1 | 85.568131 | NL | NL | F | F | N | N | $\sim$ | NL | NL | NL | –
5${}_{-2}\rightarrow$6-1 | 133.605439 | $\sim$ | NL | F | F | $\sim$ | F | F | $\sim$ | F | N | –
8${}_{0}\rightarrow$8-1 | 156.488902 | $\sim$ | N | F | F | F | F | F | N | $\sim$ | NL |  C2H5OH
7${}_{0}\rightarrow$7-1 | 156.828517 | F | NL | F | F | F | F | F | N | $\sim$ | NL | –
6${}_{0}\rightarrow$6-1 | 157.048617 | $\sim$ | NL | F | F | F | F | F | F | F | N |  C3H
5${}_{0}\rightarrow$5-1 | 157.178987 | F | F | F | F | F | F | F | F | F | $\sim$ | –
4${}_{0}\rightarrow$4-1 | 157.246062 | F | F | F | $\sim$ | F | F | F | F | F | F | –
1${}_{0}\rightarrow$1-1 | 157.270832 | F | F | $\sim$ | F | F | $\sim$ | F | $\sim$ | F | F | –
3${}_{0}\rightarrow$3-1 | 157.272338 | F | F | $\sim$ | F | F | $\sim$ | F | F | F | F | –
2${}_{0}\rightarrow$2-1 | 157.276019 | F | F | $\sim$ | F | F | $\sim$ | F | $\sim$ | F | F | –
5${}_{1}\rightarrow$50 | 165.369341 | F | $\sim$ | $\sim$ | $\sim$ | F | $\sim$ | F | $\sim$ | F | N | –
5${}_{1}\rightarrow$42 | 216.945521 | N | $\sim$ | F | F | F | F | F | F | F | $\sim$ |  C2H3CN, CH3OCHO
8${}_{0}\rightarrow$71 | 220.078561 | NL | NL | F | F | N | F | F | NL | F | NL |  HC3N
3${}_{-2}\rightarrow$4-1 | 230.027047 | F | F | $\sim$ |  | F | N | F | F | F | F |  CH3NH2
5${}_{0}\rightarrow$40 | 241.700159 | $\sim$ | F | F | F | F | F | F | F | F | F | –
5${}_{2}\rightarrow$42 | 241.904643 | F | $\sim$ | F | F | F | F | F | F | F | $\sim$ | –
6${}_{1}\rightarrow$52 | 265.289562 | F | $\sim$ | F | F | F | F | F | F | F | F | –
5${}_{2}\rightarrow$41 | 266.838148 | F | $\sim$ | F | F | F | $\sim$ | F | $\sim$ | $\sim$ | N | –
6${}_{0}\rightarrow$50 | 289.939377 | F | F | F | F | F | F | F | F | F | F | –
6${}_{2}\rightarrow$52 | 290307.738 | F | N | F | F | F | F | F | F | F | N | –
3${}_{0}\rightarrow$2-1 | 302.369773 | F | F | F | $\sim$ | F | $\sim$ | F | F | F | F | –
3${}_{1}\rightarrow$20 | 310.192994 | $\sim$ | F | F | F | F | F | F | F | F | F | –
6${}_{2}\rightarrow$51 | 315.266861 | F | N | F | F | F | F | F | $\sim$ | $\sim$ | N |  C2H3CN
7${}_{0}\rightarrow$60 | 338.124488 | F | N | F | F | F | F | F | $\sim$ | $\sim$ | F | –
7${}_{1}\rightarrow$61 | 338.614936 | F | $\sim$ | F | $\sim$ | F | $\sim$ | F | $\sim$ | F | N |  CH3NH2, HC3N
7${}_{2}\rightarrow$62 | 338.721693 | F | $\sim$ | F | F | F | F | F | $\sim$ | F | N | –
7${}_{-2}\rightarrow$6-2 | 338.722898 | F | $\sim$ | F | F | F | F | F | $\sim$ | F | N | –
4${}_{0}\rightarrow$3-1 | 350.687662 | F | F | F | N | N | N | F | F | F | F | NO
7${}_{2}\rightarrow$61 | 363.739868 | $\sim$ | N | $\sim$ | F | $\sim$ | $\sim$ | F | $\sim$ | $\sim$ | N | –
171717The upper energy level was limited to 150 K and the catalog used inside
CASSIS was VASTEL (see Sect. 4.1). The frequency ranges approximately from 84
to 373 GHz. We indicate with letter N the lines that were not fitted
(reproduced by less than 40% or over-predicted by more than 40% of its