# Magnetic Fields in the Central Molecular Zone Influenced by Feedback and
Weakly Correlated with Star Formation

Xing Lu (吕行) Shanghai Astronomical Observatory, Chinese Academy of Sciences,
80 Nandan Road, Shanghai 200030, P. R. China  Junhao Liu (刘峻豪) National
Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo, 181-8588,
Japan  East Asian Observatory, 660 N. A‘ohōkū Place, University Park, Hilo, HI
96720, USA  Thushara Pillai  Haystack Observatory, Massachusetts Institute of
Technology, 99 Millstone Road, Westford, MA 01886, USA  Qizhou Zhang  Center
for Astrophysics — Harvard & Smithsonian, 60 Garden Street, Cambridge, MA
02138, USA  Tie Liu (刘铁)  Shanghai Astronomical Observatory, Chinese Academy
of Sciences, 80 Nandan Road, Shanghai 200030, P. R. China  Key Laboratory of
Radio Astronomy and Technology, Chinese Academy of Sciences, A20 Datun Road,
Chaoyang District, Beijing, 100101, P. R. China  Qilao Gu (顾琦烙)  Shanghai
Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road,
Shanghai 200030, P. R. China  Tetsuo Hasegawa  National Astronomical
Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo, 181-8588, Japan  Pak Shing
Li  Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan
Road, Shanghai 200030, P. R. China  Xindi Tang  Xinjiang Astronomical
Observatory, Chinese Academy of Sciences, 830011 Urumqi, P. R. China  H Perry
Hatchfield Jet Propulsion Laboratory, California Institute of Technology, 4800
Oak Grove Drive, Pasadena, CA, 91109, USA  University of Connecticut,
Department of Physics, 196A Auditorium Road, Unit 3046, Storrs, CT 06269 USA
Namitha Issac  Shanghai Astronomical Observatory, Chinese Academy of Sciences,
80 Nandan Road, Shanghai 200030, P. R. China  Xunchuan Liu  Shanghai
Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road,
Shanghai 200030, P. R. China  Qiuyi Luo  Shanghai Astronomical Observatory,
Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, P. R. China
School of Astronomy and Space Sciences, University of Chinese Academy of
Sciences, No. 19A Yuquan Road, Beijing 100049, P. R. China  Xiaofeng Mai
Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan
Road, Shanghai 200030, P. R. China  School of Astronomy and Space Sciences,
University of Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing
100049, P. R. China  Zhiqiang Shen  Shanghai Astronomical Observatory, Chinese
Academy of Sciences, 80 Nandan Road, Shanghai 200030, P. R. China  Key
Laboratory of Radio Astronomy and Technology, Chinese Academy of Sciences, A20
Datun Road, Chaoyang District, Beijing, 100101, P. R. China

(Received - -, –; Revised - -, –; Accepted - -, –)

###### Abstract

Magnetic fields of molecular clouds in the Central Molecular Zone (CMZ) have
been relatively under-observed at sub-parsec resolution. Here we report
JCMT/POL2 observations of polarized dust emission in the CMZ, which reveal
magnetic field structures in dense gas at $\sim$0.5 pc resolution. The eleven
molecular clouds in our sample including two in the western part of the CMZ
(Sgr C and a far-side cloud candidate), four around the Galactic longitude 0
(the 50 km s-1 cloud, CO 0.02$-$0.02, the ‘Stone’ and the ‘Sticks & Straw’
among the Three Little Pigs), and five along the Dust Ridge (G0.253+0.016,
clouds b, c, d, and e/f), for each of which we estimate the magnetic field
strength using the angular dispersion function method. The morphologies of
magnetic fields in the clouds suggest potential imprints of feedback from
expanding H ii regions and young massive star clusters. A moderate correlation
between the total viral parameter versus the star formation rate and the dense
gas fraction of the clouds is found. A weak correlation between the mass-to-
flux ratio and the star formation rate, and a weak anti-correlation between
the magnetic field and the dense gas fraction are also found. Comparisons
between magnetic fields and other dynamic components in clouds suggest a more
dominant role of self-gravity and turbulence in determining the dynamical
states of the clouds and affecting star formation at the studied scales.

Galatic: center — stars: formation — ISM: clouds

††journal: ApJ††facilities: JCMT (SCUBA2, POL2), ATCA, LMT††software: Starlink
(Berry et al., 2005; Chapin et al., 2013), APLpy (Robitaille & Bressert,
2012), Astropy (Astropy Collaboration et al., 2018, 2022)

##  1 INTRODUCTION

The central $\sim$500 pc of our Galaxy contains several $10^{7}$ $M_{\odot}$
of molecular gas and thus is named the Central Molecular Zone (CMZ; Mills,
2017; Henshaw et al., 2023, see Figure 1). CMZ is a prominent high-mass ($>$8
$M_{\odot}$) star-forming environment. Several massive molecular clouds of
$\gtrsim$$10^{5}$ $M_{\odot}$ have been identified in the CMZ (e.g.,
G0.253+0.016 the ‘Brick’, the 20 km s-1 cloud, Sgr C; Longmore et al., 2012;
Kauffmann et al., 2017a, b; Lu et al., 2015, 2017, 2019a, 2019b, 2020, 2021;
Walker et al., 2021).

Despite the large amount of molecular gas, the star formation rates (SFRs)
both for the whole CMZ and for individual clouds in the CMZ are found to be
lower by an order of magnitude than the dense gas star formation relation
(Longmore et al., 2013; Kauffmann et al., 2017a; Barnes et al., 2017; Lu et
al., 2019a, b). Here, the dense gas star formation relation refers to the
linear correlation found between dense gas masses and SFRs of objects ranging
from nearby molecular clouds to external galaxies (Gao & Solomon, 2004; Lada
et al., 2010, 2012). The extreme physical conditions in the CMZ, e.g., the
strong turbulence (FWHM linewidth $\sim$$10$–$10^{2}$ km s-1), have been
suggested to suppress star formation in the CMZ and be responsible for the low
SFRs (Kruijssen et al., 2014). Kauffmann et al. (2017b) found that several
massive CMZ clouds are unbound or only marginally bound through a virial
analysis that takes turbulent linewidths into account, but at smaller scales
($<$1 pc) the gas may be bound and suitable for star formation (Lu et al.,
2019a).

Figure 1: Top: Overview of the CMZ and the observed fields. The background
image shows 850 µm continuum emission after removing potential CO
contamination (Parsons et al., 2018). The concentric circles mark the fields
observed by JCMT, with the inner solid circles showing radii of $3^{\prime}$
and the outer dashed ones showing radii of $6^{\prime}$. The magenta boxes
correspond to the regions plotted in Figures 2–4. Bottom: False-color maps
created with the MeerKAT 1.28 GHz continuum (red; Heywood et al. 2022) and
Spitzer 8 µm (cyan; Stolovy et al. 2006) data. The yellow segments show
orientations of magnetic fields derived from JCMT/POL2 observations with
signal-to-noise ratios greater than 3 in their polarization percentages
($p$/d$p$$>$3).

However, among these unbound or marginally bound clouds, some turn out to have
higher SFRs that are consistent with the dense gas star formation relation
(e.g., Sgr C, Dust Ridge cloud c; Ginsburg et al., 2015; Kauffmann et al.,
2017a; Walker et al., 2018; Lu et al., 2019a). These clouds also show a higher
fraction of gas mass confined in dense cores of 0.1 pc scales than other
relatively quiescent clouds such as the 20 km s-1 cloud and G0.253+0.016
(3–10% vs. $<$1%; Lu et al., 2019a; Battersby et al., 2020). It is unclear how
clouds of similar virial status show such different dense gas fractions and
SFRs. The magnetic field could be the culprit, which may support the clouds in
synergy with turbulence against gravitational collapse. Weaker magnetic fields
in some clouds may lead to a preferable environment for collapse,
fragmentation, and subsequent star formation.

To this end, it is critical to investigate the correlation between the
magnetic field, dense gas fractions, and SFRs of the massive clouds in the
CMZ. Only a few CMZ clouds have been mapped in polarized dust emission with
sufficiently high angular resolutions, including the Three Little Pigs (Chuss
et al., 2003), G0.253+0.016 (Pillai et al., 2015), and the Sgr A complex (the
circumnuclear disk and the 20/50 km s-1 clouds; Chuss et al., 2003; Hsieh et
al., 2018). A more comprehensive database of the magnetic field in a large
number of CMZ clouds with uniform sensitivity and resolution is in need for a
robust statistical analysis. A recent polarization survey using the 214 µm
band of the Stratospheric Observatory for Infrared Astronomy (SOFIA) mapped
the whole CMZ at a resolution of 20″ (Butterfield et al., 2023), which is a
promising dataset for statistical analyses.

Here we report observations of polarized dust emission at 850 µm in a sample
of 11 massive clouds in the CMZ at a resolution of 14″. Assuming that the
short axes of dust grains are aligned with the magnetic field (Lazarian, 2007;
Lazarian & Hoang, 2007), we are able to use polarized dust emission to trace
the plane-of-sky component of magnetic field orientations. We further infer
magnetic field strengths through a statistical approach and compare them to
the SFRs of the clouds. These are by far the highest angular resolution
observations of magnetic fields in molecular clouds in the CMZ. Throughout the
paper, we adopt the parallax distance of 8.1 kpc to the CMZ (Reid et al.,
2019).

##  2 OBSERVATIONS AND DATA REDUCTION

The polarized dust emission at 850 µm was observed with SCUBA2 (Holland et
al., 2013) and POL2 (Friberg et al., 2016) mounted on the James Clark Maxwell
Telescope (JCMT) between June 14 and August 1 2020, under the project
M20AP023. The targets include eight fields in the CMZ, which are plotted in
Figure 1, each with an on-source time of 1 hour. The observations were carried
out with the POL2 Daisy
mode111https://www.eaobservatory.org/jcmt/instrumentation/continuum/scuba-2/pol-2/#Observing_mode,
leading to maps with high signal-to-noise ratios (SNRs) in the inner radius of
$3^{\prime}$, although we found that SNRs at $6^{\prime}$ away from the field
centers can still be sufficiently high thanks to bright continuum emission in
the CMZ. The effective beam size is 14$\farcs$1 ($\sim$0.55 pc at the distance
of the CMZ).

Data reduction was done with the SMURF (Jenness et al., 2013) package in
Starlink (Currie et al., 2014). We used a pixel size of 7″, and combined
adjacent fields to create three mosaicked maps marked by the magenta boxes in
Figure 1.

The measured rms of the total intensity (Stokes I) maps varies from 20 to 50
mJy beam-1, which is much higher than the thermal noise of 4–8 mJy beam-1 (the
square root of the variance of Stokes I) in the central $3^{\prime}$ of the
maps, likely due to significant foreground and background emission in the
maps. We took a median value of 40 mJy beam-1 as the canonical rms, and chose
a 5$\sigma$ level (i.e., 0.2 Jy beam-1) as the threshold for estimating
physical parameters in the following sections.

The CMZ has been mapped with SCUBA2 at 850 µm using the Pong mapping mode at
the same angular resolution as ours (Parsons et al., 2018, see Figure 1 top
panel). To validate our data reduction, we compare the 850 µm total intensity
maps from our data with those without the CO-contamination correction in
Parsons et al. (2018) after regridding them to a common pixel frame. The
intensities are consistent within 40% between the two datasets. The difference
is likely due to the different mapping modes, where the Pong mapping mode
provides more uniform coverage and the data can be reduced with a larger
spatial filter of $\gtrsim$300″, while the default spatial filter for POL2
data using the Daisy mode is 150″. We note that results in this paper should
be rarely affected by the spatial filtering issue of the Daisy mode, as the
clouds in our sample are generally $\lesssim$150″, and we have limited our
statistical analyses of magnetic fields to $<$90″ (see Section 3.3).

The polarized intensities were binned to a sampling interval of 14″ and were
debiased, and the polarization position angles were derived following the
procedures in Liu et al. (2019). In the following, only polarization
detections where SNRs of the polarization percentages are higher than 3
($p$/d$p>3$), corresponding to an error in polarization position angles
$<$10°, are considered for further analyses. When considering polarization
signals in clouds, we further enforce the threshold of 0.2 Jy beam-1 (SNR $>$
5 for total intensities) to select polarization detections. Catalogs of the
derived polarization segments are publicly available
athttps://doi.org/10.5281/zenodo.8409806 (catalog 10.5281/zenodo.8409806).

##  3 RESULTS

We defined a sample of 11 massive clouds that have been studied by e.g., Lu et
al. (2019b) and Battersby et al. (2020), and labeled them in Figure 2. The
sample includes the most massive clouds in the CMZ, except Sgr B2 and the 20
km s-1 cloud whose magnetic fields have been studied (Novak et al., 2000;
Chuss et al., 2003; Butterfield et al., 2023). The clouds have sufficient
numbers ($\geq$28) of independent polarization measurements allowing for
statistical analyses (Liu et al., 2021), i.e., for a round region, there
should be at least three independent polarization segments along its radius.
Two adjacent clouds defined in the literature, the ‘Sticks’ and the ‘Straw’,
have 19 independent polarization measurements each. Their mean magnetic field
position angles do not differ much (87.0$\pm$22.1° vs. 86.9$\pm$20.8°).
Therefore, we considered them as as a single cloud to boost the number of
polarization segments.

###  3.1 Masses and Densities of the Clouds

Fundamental properties of the clouds, including their masses, column
densities, and number densities, can be estimated using the 850 µm continuum
emission, assuming that it is entirely from thermal dust emission. These
properties will be used for estimating magnetic field strengths later.

Assuming optically thin dust emission, H2 column densities of the clouds are
calculated as

$N(\text{H}_{2})=\frac{I_{\nu}}{B_{\nu}(T_{\text{dust}})\kappa_{\nu}}\frac{1}{\mu_{\text{H${}_{2}$}}m_{\text{H}}},$
(1)

where $I_{\nu}$ is the dust emission intensity at frequency $\nu$,
$B_{\nu}(T_{\text{dust}})$ is the Planck function at the dust temperature
$T_{\text{dust}}$, $\kappa_{\nu}=0.1(\nu/1\text{THz})^{\beta}$ cm2 g-1 is the
dust opacity assuming a gas-to-dust ratio of 100 and with an opacity index
$\beta$ (Hildebrand, 1983), and $\mu_{\text{H${}_{2}$}}=2.8$ is the molecular
weight per H2.

We used the continuum intensity from the SCUBA2 data with the CO-contamination
correction (Parsons et al., 2018) instead of the Stokes I data in our POL2
observations, as the latter has filtered out more continuum emission and
included contamination from the CO 3–2 line emission.

We took $T_{\text{dust}}$ and $\beta$ maps of 14″ resolutions from Tang et al.
(2021) and calculated the column densities pixel by pixel. Note that Tang et
al. (2021) derived $T_{\text{dust}}$ and $\beta$ from a combination of Planck,
Herschel, and the Large Millimeter Telescope (LMT) observations, which did not
remove any foreground or background emission as the POL2 data did. The derived
column density maps are presented in Appendix A.

Cloud masses were then obtained by summing up the column densities over the
pixels in the clouds above the threshold of 0.2 Jy beam-1 in our total
intensity maps:

$M=\mu_{\text{H${}_{2}$}}m_{\text{H}}\sum N(\text{H}_{2})A_{\text{pix}},$ (2)

where $A_{\text{pix}}$ is the area of one pixel determined by the pixel
angular size (7″) and the distance to the CMZ.

Number densities of the clouds are more uncertain, as the 3D geometries of the
clouds are unknown. To get an order-of-magnitude estimate of the number
densities, we approximated all the clouds as spheres and adopted an effective
radius $r_{\text{eff}}$ of $\sqrt{A/\pi}$, where $A$ is the area of the clouds
above the total intensity threshold, from which the volumes of the spheres
were derived. Then the number densities were obtained by dividing the cloud
masses by the volumes.

The cloud masses $M$ and mean number densities $n(\text{H}_{2})$ are tabulated
in Table 1. The uncertainties in the masses and densities are typically 50%
(e.g., Sanhueza et al., 2017), mainly contributed by the uncertainties in the
gas-to-dust ratio (could be lower than 100 in the CMZ; Giannetti et al.,
2017), the opacity index, and the dust temperature (Tang et al., 2021).

Figure 2: Magnetic fields in individual clouds. The segments are color-coded
to represent the 11 clouds in our sample. Comparing with Figure 1, all
segments lying outside of the 5$\sigma$ total intensity contours as well as
not belonging to any of the clouds are dropped. Table 1: Measured physical
parameters and estimated magnetic fields of the clouds.

CloudaaReferences of the cloud naming: Lu et al. (2019b); Battersby et al. (2020). | $M$ | $r_{\text{eff}}$ |  $n$(H2) | $\sigma_{v}$ | $\left(\langle B_{\text{t}}^{\phantom{\text{t}}2}\rangle/\langle B^{2}\rangle\right)^{0.5}$ | $\sigma_{\theta}$ | $B$ | $B_{\text{classical}}$ | $v_{\text{A}}$ | $\mathcal{M}_{\text{A}}$ | $\lambda$ | $\alpha_{\text{vir}}$ | $\alpha_{\text{vir},B}$ | SFRbbReference of the SFR: Hatchfield et al. (2023, in prep.). Reference of the CDGF: Battersby et al. (2020). | CDGFbbReference of the SFR: Hatchfield et al. (2023, in prep.). Reference of the CDGF: Battersby et al. (2020).
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| (104 $M_{\odot}$) | (″, pc) | (103 cm-3) | (km s-1) |  | (°) | (mG) | (mG) | (km s-1) |  |  |  |  | (10-3 $M_{\odot}$ yr-1) |
Sgr C | 4.0 | 76, 3.0 | 5.1 | 5.9 | 0.32 | 22.2 | 0.21 | 0.40 | 4.8 | 2.1 | 2.3 | 1.8 | 1.9 | 65.4$\pm$39.2 | 0.11
Far-Side Candidate | 1.5 | 76, 3.0 | 1.9 | 5.4 | 0.36 | 32.8 | 0.10 | 0.14 | 3.9 | 2.4 | 1.7 | 4.2 | 4.3 |  $\cdots$ |  $\cdots$
50 km s-1 | 6.8 | 121, 4.8 | 2.2 | 16.0 | 0.25 | 32.9 | 0.48 | 0.44 | 16.7 | 1.7 | 0.7 | 12.5 | 12.7 | 50.6$\pm$30.6 | 0.017
CO 0.02$-$0.02 | 0.9 | 90, 3.5 | 0.7 | 8.8 | 0.37 | 26.0 | 0.10 | 0.18 | 6.2 | 2.4 | 0.8 | 22.3 | 22.4 |  $\cdots$ |  $\cdots$
Stone | 2.8 | 70, 2.8 | 4.6 | 11.2 | 0.23 | 27.3 | 0.52 | 0.56 | 12.6 | 1.5 | 0.8 | 8.7 | 8.9 | 6.4$\pm$3.8 | 0.022
Sticks & Straw | 3.7 | 82, 3.2 | 3.8 | 14.5 | 0.33 | 22.1 | 0.44 | 0.84 | 11.5 | 2.2 | 0.9 | 12.9 | 13.0 | 0.4$\pm$0.3 | 0.010
G0.253+0.016 | 9.3 | 86, 3.4 | 8.4 | 18.5 | 0.16 | 51.7 | 1.73 | 0.51 | 30.7 | 1.0 | 0.5 | 8.7 | 9.2 | 2.3$\pm$1.4 | 0.019
Cloud b | 1.4 | 60, 2.4 | 3.7 | 12.5 | 0.35 | 37.6 | 0.35 | 0.38 | 9.5 | 2.3 | 0.8 | 17.9 | 18.0 |  $<$0.3ccThe SFR of this cloud cannot be constrained using the method of Hatchfield et al. (2023, in prep.), therefore an upper limit is given (Lu et al., 2019a). | 0.019
Cloud c | 1.9 | 55, 2.1 | 6.7 | 4.2 | 0.36 | 37.4 | 0.15 | 0.17 | 3.0 | 2.4 | 3.0 | 1.4 | 1.5 | 38.7$\pm$23.2 | 0.077
Cloud d | 6.4 | 83, 3.3 | 6.3 | 11.4 | 0.28 | 35.4 | 0.53 | 0.49 | 10.7 | 1.8 | 1.2 | 4.6 | 4.8 | 2.7$\pm$1.6 | 0.014
Cloud e/f | 11.4 | 83, 3.3 | 11.4 | 9.6 | 0.21 | 39.7 | 0.77 | 0.47 | 11.7 | 1.4 | 1.5 | 1.8 | 2.1 | 16.9$\pm$10.1 | 0.016

Note. — Boundaries of the clouds can be found in Figure 2. Uncertainties of
the parameters are discussed in Appendix D.

###  3.2 Magnetic field morphologies

The orientation of the magnetic field is inferred by rotating the detected
polarization orientations by 90°. The bottom panel of Figure 1 shows an
overview of the magnetic field morphologies in the observed regions. Figure 2
shows magnetic field orientations in the individual clouds. Several clouds,
e.g., G0.253+0.016 and the 50 km s-1 cloud, lie beyond the central
$3^{\prime}$ of the DAISY fields. However, their polarized emission is
sufficiently strong ($p$/d$p$$>$3). Therefore, we included them in the
analysis as well.

To investigate magnetic field morphologies at difference spatial scales, we
compared our JCMT POL2 observations with data from the Atacama Cosmology
Telescope (ACT) at the 1.36 mm (220 GHz) band that have an angular resolution
of $1^{\prime}$ (Guan et al., 2021) and from Planck at 353 GHz that have an
angular resolution of $5^{\prime}$ (Planck Collaboration et al., 2015), which
are able to probe magnetic fields at larger scales than the JCMT POL2 data. In
Figure 3, we overlaid magnetic field orientations estimated from the ACT 1.36
mm band and from the Planck 353 GHz data. The orientations based on ACT and
Planck are generally consistent with each other, suggesting that both data
likely trace the same large-scale magnetic field.

Figure 4 further displays maps of the difference between magnetic field
position angles probed by our JCMT/POL2 observations and those probed by ACT.
To produce this map, we did not smooth or convolve the two datasets. Instead,
for each independent measurement of polarized emission in the JCMT/POL2 data
that derives a magnetic field orientation (with a resolution of 14″), we found
the nearest magnetic field segment of the ACT data (with a resolution of
$1^{\prime}$), and calculated the difference of position angles. Since we only
know the orientations, not directions, of magnetic fields, the angle
difference was limited in a range of 0°–90°, with 0° representing the two
field lines being parallel, and 90° representing the two field lines being
perpendicular. We also produced similar maps using POL2 data smoothed and
regridded to the same frame as the ACT data, which are presented in Appendix
E. The two datasets clearly show different magnetic field position angles in
most clouds. This could be because the POL2 observations have filtered out
signals above the scale of 150″ and therefore they are not able to probe
large-scale magnetic fields in the CMZ as are the ACT data (Juvela et al.,
2018). However, we cannot exclude the possibility that the ACT data have
included foreground emission along the line of sight, and thus represent a
superposition of different components of magnetic fields between the CMZ and
us. In such a case, the difference between the field position angles proved by
POL2 and ACT would be a natural result as they trace different ISM components
along the line of sight.

If we assume that the ACT data are tracing magnetic fields inside the CMZ,
then it can be seen from Figures 3 & 4 that the large-scale magnetic fields
traced by the ACT polarization data are twisted inside the massive clouds as
revealed by the POL2 data. The deviation of the orientations of small versus
large-scale magnetic fields will be discussed in Section 4.2.

Figure 3: Magnetic fields of different spatial scales in the three CMZ fields.
The magenta segments show orientations of magnetic fields derived from
JCMT/POL2 observations with $p$/d$p$$>$3, while the green ones are those with
2$<$$p$/d$p$$<$3 that are only for visual presentation but not for
quantitative analyses. The cyan segments show orientations of magnetic fields
from ACT observations at a resolution of $1^{\prime}$, while the yellow ones
from Planck 353 GHz at a resolution of $5^{\prime}$.

Figure 4: Relative orientation between small-scale ($\sim$0.5 pc) magnetic
fields probed by JCMT/POL2 and large-scale ($\gtrsim$2.5 pc) magnetic fields
probed by ACT. The image is color-coded to quantify whether the small-scale
magnet fields tend to be parallel (blue) or perpendicular (red) to the large-
scale ones. Only polarization detections within the clouds in our sample are
plotted (cf. Figure 2). Contours are the same as in Figure 1, representing the
JCMT 850 µm total intensity emission.

###  3.3 Magnetic field strengths in the clouds

We inferred the magnetic field strengths using the Angular Dispersion Function
(ADF) method (Hildebrand et al., 2009; Houde et al., 2009, 2016). For
summaries of this method, see e.g., Liu et al. (2021, 2022a, 2022b).

The analysis routine is outlined as follows:

a). Derive the ADF, $1-\langle\cos[\Delta\Phi(l)]\rangle$, following the
definition in Houde et al. (2009). Here, $\Delta\Phi(l)$ is the difference in
position angles of two magnetic field segments separated by a distance $l$.

b). Fit the ADF with the following form (Houde et al., 2009):

$\begin{split}1-\langle\cos[\Delta\Phi(l)]\rangle&\simeq
b(l)+a^{\prime}_{2}l^{2}\\\ &=\frac{\langle
B_{\text{t}}^{\phantom{\text{t}}2}\rangle}{\langle
B^{2}\rangle}\times(1-e^{-l^{2}/2(l_{\delta}^{2}+2W^{2})})+a^{\prime}_{2}l^{2},\end{split}$
(3)

where $\left(\langle B_{\text{t}}^{\phantom{\text{t}}2}\rangle/\langle
B^{2}\rangle\right)^{0.5}$ is the turbulent-to-total magnetic field strength
ratio in the plane of sky, $l_{\delta}$ is the correlation length scale for
the local turbulent magnetic field, and $W$ is the standard deviation of the
Gaussian beam.

The numerical study by Liu et al. (2021) suggested that the ADF method may not
work well in accounting for the effect of line-of-sight (LOS) signal
integration. Thus, $\left(\langle
B_{\text{t}}^{\phantom{\text{t}}2}\rangle/\langle B^{2}\rangle\right)^{0.5}$
adopted here does not consider the LOS signal integration effect.

The fitting results of the 11 clouds are plotted as blue curves in Appendix B.
The best-fit $\left(\langle B_{\text{t}}^{\phantom{\text{t}}2}\rangle/\langle
B^{2}\rangle\right)^{0.5}$ is tabulated in Table 1.

c). Estimate the mean densities ($\rho$) and turbulent velocity dispersions
($\sigma_{v}$) of the clouds. The estimate of the mean densities $\rho$ has
been elaborated in Section 3.1.

As for the velocity dispersions, the NH3 (3, 3) line data from the SWAG survey
(Krieger et al., 2017) were used . NH3 has a critical density of
$\sim$$10^{3}$ cm-3 (Shirley, 2015), which is close to the mean densities of
the clouds. The morphologies of the NH3 emission match well with those of the
clouds seen in the JCMT 850 µm continuum (Krieger et al., 2017). Therefore,
NH3 is appropriate for tracing turbulent motions of dense gas in the clouds.
For each cloud, the mean NH3 (3, 3) spectrum was fitted with a Gaussian to
obtain the velocity dispersion $\sigma_{v}$.

We do not consider the satellite lines that are usually much weaker than the
main hyperfine line at this transition (Ho & Townes, 1983). We do not subtract
the thermal line width from the measured velocity dispersion because the
former is much narrower than the latter in the CMZ environment and therefore
makes no difference to the measurement. The fitting results are presented in
Appendix C. We note that velocity gradients and/or multiple velocity
components along the line of sight exist inside the clouds (Appendix C), and
therefore the measured velocity dispersions should be treated as upper limits.
An extreme case is G0.253+0.016, where the mean velocity dispersion was
measured to be 4.4 km s-1 based on ALMA observation after four independent
velocity components were decomposed (Henshaw et al., 2019), and therefore the
estimated magnetic field strengths would be four times lower. A detailed
kinematic analysis using high angular resolution observations is necessary to
assess the impact of multiple velocity components in the clouds and more
accurately determine the velocity dispersion, which is beyond the scope of the
current work. For consistency within our sample, we still adopt the velocity
dispersions measured from the SWAG NH3 (3, 3) line, but note the caveat that
the measurements may have large uncertainties.

d). Estimate the total magnetic field strength ($B$) in the plane of the sky
using the Davies-Chandrasekhar-Fermi (DCF) method (Liu et al., 2021, 2022a):

$B\sim 0.21\sqrt{\mu_{0}\rho}\,\sigma_{v}\left(\frac{\langle
B_{\text{t}}^{\phantom{\text{t}}2}\rangle}{\langle
B^{2}\rangle}\right)^{-0.5},$ (4)

where $\mu_{0}$ is the permeability of vacuum that is $4\pi$ under the cgs
metric system. Here we assume isotropic turbulence and equipartition between
turbulent kinetic and magnetic energies. Liu et al. (2022b) have demonstrated
that the uncertainty brought by the anisotropic turbulence is a minor issue
for the DCF method in self-gravitating regions. We adopt the numerically
derived correction factor 0.21 (with 45% uncertainty at $>$0.1 pc scales) from
Liu et al. (2021) to account for the LOS signal integration effect. The
uncertainty in the estimated magnetic field strengths is not straightforward
to quantify, as the DCF method itself has inherent uncertainties (see Appendix
D). We adopt the typical uncertainty of a factor of 2 (Liu et al., 2021),
which is derived by applying the DCF method to numerical simulations and
comparing estimated values to input models. This uncertainty should be treated
as a lower limit, because unlike numerical simulations, in realistic
situations there should be even more sources of errors. The estimated $B$
values of the clouds are listed in Table 1.

We also estimated the magnetic field strength following the classical DCF
method (e.g., Falceta-Gonçalves et al., 2008; Li et al., 2022):

$B_{\text{classical}}\sim
0.5\sqrt{\mu_{0}\rho}\frac{\sigma_{v}}{\tan\sigma_{\theta}},$ (5)

where $\sigma_{\theta}$ is the dispersion of the magnetic field orientations.
The correction factor of 0.5 is adopted from the numerical simulations of
Ostriker et al. (2001). The estimated magnetic field strengths are consistent
with those based on the ADF method within a factor of 2 (Table 1), except for
G0.253+0.016 whose $\sigma_{\theta}$ is up to $>$50°. Chen et al. (2022) have
pointed out that $\tan\sigma_{\theta}$ does not correlate well with the
turbulent-to-total magnetic field strength ratio in the plane of sky,
especially when $\sigma_{\theta}$ is high. Therefore, Equation 5 may not work
well for G0.253+0.016. There also exist other variations of the classical DCF
method (e.g., Skalidis & Tassis, 2021) that better suit certain conditions,
which we do not explore further in this work. In the following, we adopt the
estimates of magnetic field strengths from the ADF method for further
analyses.

One cloud in our sample, G0.253+0.016, has been mapped in polarized dust
emission that enables a DCF analysis. Pillai et al. (2015) estimated a plane-
of-the-sky magnetic field strength (the ‘total’ magnetic field strength in
their definition divided by 1.3) of 4.2 mG, which was corrected to 1.7 mG by
Federrath et al. (2016) after adopting the correct density. The latter value
is consistent with our estimate of 1.73 mG. However, we caution that the
consistency is coincidental, as we adopted different velocity dispersions,
densities, and correction factors for Equation 4. This highlights the large
uncertainties in the estimate of magnetic field strengths through the DCF/ADF
method.

###  3.4 Comparing turbulence, magnetic field, and gravity

To quantify the relative importance between the turbulence, magnetic field,
and self-gravity of individual clouds, we follow the framework in Section 2.4
of Liu et al. (2022b) to estimate the Alfvénic Mach numbers, the mass-to-flux
ratios, and the virial parameters of the clouds. The results are tabulated in
Table 1.

####  3.4.1 Alfvénic Mach numbers

The relative importance between the turbulence and magnetic field can be
parameterized by the Alfvénic Mach number

$\mathcal{M}_{\text{A}}=\sigma_{v,\text{3D}}/v_{\text{A}},$ (6)

where the 3D velocity dispersion $\sigma_{v,\text{3D}}$ is
$\sqrt{3}\sigma_{v}$ for isotropic turbulence. The Alfvénic speed is
$v_{\text{A}}=B_{\text{3D}}/\sqrt{\mu_{0}\rho}$, where the 3D magnetic field
strength $B_{\text{3D}}$ is estimated by multiplying the magnetic field
strength in the plane of the sky by 1.25, a scaling factor derived for a
randomly distributed 3D mean field orientation (Liu et al., 2022b) that
estimates $B_{\text{3D}}$ with an uncertainty of a factor of 2 (Liu et al.,
2021).

####  3.4.2 Mass-to-flux ratios

The relative importance between the magnetic field and self-gravity can be
parameterized by the magnetic critical parameter $\lambda$, which is the mass-
to-flux ratio normalized against its critical value:

$\lambda=\frac{M/\Phi}{(M/\Phi)_{\text{cr}}}.$ (7)

Here the mass-to-flux ratio is estimated using:

$M/\Phi=\frac{\mu_{\text{H${}_{2}$}}m_{\text{H}}N(\text{H}_{2})}{B},$ (8)

and the critical value is

$(M/\Phi)_{\text{cr}}=\frac{1}{2\pi\sqrt{G}},$ (9)

following Nakano & Nakamura (1978). Note that when calculating the column
densities, we have selected the same area where polarized emission is detected
and thus where magnetic field segments can be plotted (Crutcher et al., 2004).

####  3.4.3 Virial parameters

Finally, the dynamical equilibrium of the clouds, assuming a radial density
profile of $\rho(r)\propto r^{-2}$ (self-gravitating gas in hydrostatic
equilibrium; Shu et al., 1987), can be characterized by the ratio between the
virial mass $M_{\text{vir}}=3\sigma_{v}^{2}r/G$ and the cloud mass $M$, i.e.,
the virial parameter:

$\alpha_{\text{vir}}=\frac{M_{\text{vir}}}{M}=\frac{3\sigma_{v}^{2}r}{GM},$
(10)

where $r=\sqrt{A/\pi}$ is the effective radius, and $G$ is the gravitational
constant. If we additionally take the magnetic field support into account,
then the total virial mass can be expanded to

$M_{\text{vir},B}=\sqrt{M^{2}_{B}+\left(\frac{M_{\text{vir}}}{2}\right)^{2}}+\frac{M_{\text{vir}}}{2},$
(11)

where the magnetic virial mass is given by

$M_{B}=\frac{Br^{2}}{6\sqrt{G/10}}.$ (12)

The total virial paramter $\alpha_{\text{vir},B}$ is then the ratio of
$M_{\text{vir},B}$ and $M$.

##  4 DISCUSSION

Figure 5: False-color maps created with the MeerKAT 1.28 GHz continuum (red;
Heywood et al. 2022) and Spitzer 8 µm (cyan; Stolovy et al. 2006) data.
Contours and segments are the same as in Figure 1, representing the JCMT/POL2
850 µm total intensities and the magnetic field orientations, respectively.
The solid and dashed green circles denote the known and candidate H ii regions
from the WISE catalog of Galactic H ii regions V2.4, respectively. The blue
ellipses denote the two known SNRs (Green, 2022). Other objects of interest,
including the M0.20$-$0.033 molecular shell and the potential gas flow into
cloud d, are labeled with magenta symbols and are discussed in Section 4.1.

###  4.1 Impacts of environments on magnetic fields

Given the complicated physical environments in the CMZ, e.g., young massive
star clusters (Lang et al., 2005; Simpson et al., 2018), expanding supernova
remnants (SNRs) and H ii regions (Hankins et al., 2020; Heywood et al., 2022),
and pc-scale gas dynamics including cloud-cloud interactions (Hasegawa et al.,
1994; Dale et al., 2019), it is expected that the clouds, as well as magnetic
fields in them, would be affected, although exactly how is unclear owing to a
lack of observations. Here we do not elaborate on every cloud, but focus on
four cases where we find signatures of impacts of environments on magnetic
fields in clouds. The following discussions are accompanied by the
illustrations in Figure 5, where we overlay H ii regions from the WISE catalog
of Galactic H ii regions V2.4222http://astro.phys.wvu.edu/wise/ (Anderson et
al., 2014) and SNRs from the catalog of Green (2022).

####  4.1.1 Cometary magnetic fields in Sgr C: signature of a cloud-H ii
interaction event?

The Sgr C molecular cloud is known to lie adjacent to an H ii region of
$\sim$5 pc in size that has been seen in radio continuum (Lang et al., 2010)
and mid-infrared emission (Hankins et al., 2020), which reveals itself as a
red patch in Figure 5. In the mid-infrared map, the cloud appears as a
silhouette against the bright emission from the H ii region (Hankins et al.,
2020). Therefore, the cloud should be in the foreground of the H ii region.
The interaction between the cloud and the H ii region has been suggested in
e.g., Kendrew et al. (2013) and Lu et al. (2019a).

Magnetic fields in the Sgr C cloud show a cometary morphology, which could be
a consequence of the interaction with the expanding H ii region. Radiative
feedback from massive stars in the H ii region as well as high-pressure
ionized gas may have compressed molecular gas in the cloud, leading to aligned
magnetic fields on the surface of the cloud. This scenario is similar to the
results of magnetohydrodynamic simulations where an expanding H ii region
erodes surrounding molecular gas, creating molecular pillars as well as
cometary magnetic fields (Krumholz et al. 2007; Arthur et al. 2011, see their
Figs. 22 & 23; Mackey & Lim 2011, see their Figs. 2 & 3). The simulations have
shown that in such a scenario the magnetic field cannot be dynamically
important compared to turbulence or thermal pressure, which indeed is the case
for Sgr C ($\mathcal{M}_{\text{A}}\sim 2.1$, see Section 3.4 & Table 1).

The putative interaction event is consistent with the star formation
activities in the Sgr C cloud. In the interaction scenario, the ‘head’ of the
cometary cloud should be firstly impacted by the H ii region, therefore should
have the most evolved phase of star formation. The ‘tail’ of the cloud then
should contain subsequently less evolved phases of star formation. High-
resolution interferometer observations do reveal such an evolutionary trend,
where two massive protostellar objects associated with ultra-compact H ii
regions and powerful outflows are found in the ‘head’, and progressively less
evolved protostellar activities including masers and weak outflows are
observed further away from the H ii region in the cloud (Kendrew et al., 2013;
Lu et al., 2019a, 2021, 2022).

####  4.1.2 Curved magnetic fields in the Three Little Pigs: perturbed by an
expanding shell and interaction between clouds?

The Three Little Pigs clouds have similar gas masses and column densities but
show different fragmentation levels in high-resolution observations: the Stone
cloud is highly substructured, while the Sticks and the Straw clouds are only
scantly to moderately substructured (Battersby et al., 2020). The origin of
such a difference is unclear. The clouds are adjacent to the Quintuplet
cluster as well as an expanding shell seen in molecular line emission
(Butterfield et al., 2018, 2022). Butterfield et al. (2018) have suggested a
scenario where these clouds are interacting with the shell likely originated
from the Quintuplet cluster. Here we investigate possible impacts on the
magnetic fields in the three clouds by their environments.

The magenta dashed circle in Figure 5 illustrates the spatial extent of the
expanding shell proposed by Butterfield et al. (2018), which is concentric
with an H ii region candidate likely excited by the Quintuplet cluster. The
magnetic field in the Straw cloud (G0.145$-$0.086) seems to show a curved
morphology along the southwestern edge of the shell, which resembles that
produced in simulations of expanding shells (Arthur et al., 2011; Klassen et
al., 2017). The curved magnetic field could be a consequence of the
interaction between the shell and the cloud. The same interaction may have
enhanced turbulence in the Straw cloud, which is observed as a broader
linewidth and a higher Alfvénic Mach number in the Sticks & Straw cloud than
in the Stone cloud (Table 1), thus making it less prone to fragmentation as
compared to the Stone cloud on the western side (e.g., Federrath, 2015).

The Sticks cloud (G0.106$-$0.082) is another candidate that may interact with
adjacent gas components (the filamentary cloud M0.11$-$0.11 or the Radio Arc
bubble; Butterfield et al., 2022) and thus have its magnetic field morphology
affected. Part of the Radio Arc bubble, a candidate H ii region, can be seen
as the large dashed green circle surrounding the Three Little Pigs in Figure 5
(Rodríguez-Fernández et al., 2001). However, we have limited detections of
magnetic field segments toward the Sticks cloud. It is difficult to compare
its magnetic field structure to different interaction scenarios. More
sensitive polarization observations are needed to study the relation between
the environment of the Three Little Pigs clouds and their magnetic fields.

####  4.1.3 Curved magnetic fields in Dust Ridge cloud d: tracing converging
gas flows?

Williams et al. (2022) have proposed a scenario of large-scale ($\sim$pc)
converging gas flows for cloud d in the Dust Ridge. One of the gas flows is
suggested to come from the southeastern sides of the cloud, which is marked by
an arrow in Figure 5. Interestingly, magnetic fields as revealed by our
observations seem to be curved in the same direction on the southeastern side
of the cloud.

It is unclear whether the alignment between the gas flow and the magnetic
field is caused by strong magnetic field regulating flowing gas, or the other
way around, i.e., by strong turbulence dragging magnetic fields with in. The
Mach number in cloud d, $\mathcal{M}_{\text{A}}\sim 1.8$, is moderate (Section
3.4), and thus we are not able to tell whether or not the magnetic field here
is dynamically important with respective to turbulence. If it is the gas flow
that affects the magnetic field, curving the latter to the observed
morphology, then it is a scenario similar to the alignment of gas flows and
magnetic fields observed toward massive filaments in Galactic disk clouds
(e.g., Pillai et al., 2020) and reproduced in numerical simulations (e.g.,
Gómez et al., 2018).

####  4.1.4 Magnetic fields across the filament and in the PDR in Dust Ridge
cloud e/f

The Dust Ridge cloud e/f presents a filamentary morphology that extends
$\sim$10 pc with a total gas mass of 1.1$\times$$10^{5}$ $M_{\odot}$. Previous
high-resolution interferometer observations have revealed signatures of star
formation including masers and protostellar outflows in this cloud (Lu et al.,
2019a, b, 2021). However, its star formation efficiency is still $\sim$10
times lower than expected from the dense gas-star formation relation (Lu et
al., 2019a).

The magnetic field is overall perpendicular to the major axis of the filament.
This is expected when the gas density is sufficiently high and thus the gas is
channelled by the magnetic field to accumulate to form a filament (Li et al.,
2013; Li & Klein, 2019). Similar magnetic field geometry has been found in
dense gas filaments in nearby star forming clouds (e.g., Pattle et al., 2017;
Pillai et al., 2020), suggesting the important role of magnetic fields in
regulating gas dynamics in dense filaments. An alternative explanation is that
the filament is formed by the large-scale interaction of magnetic fields and
turbulence (e.g., Liu et al., 2018). In such a scenario, the small-scale
perpendicular magnetic fields in the filament could be attributed to
converging flows (Inoue et al., 2018).

In addition, as shown in Figure 5, the cloud is spatially adjacent to the Sgr
B1 region to the south, where a cluster of H ii regions and photodissociation
regions (PDRs) have been detected (Simpson et al., 2018, 2021). The southern
edge of the cloud is facing the Sgr B1 H ii regions, and therefore is likely a
PDR. The magnetic field on this edge is well aligned. This alignment could be
interpreted similarly to the case of Sgr C (Section 4.1.1), where PDRs are
compressed by the ionized gas from H ii regions and thus have ordered magnetic
fields along the interface.

###  4.2 Orientations of local and global magnetic fields

The angular resolutions of the JCMT and ACT observations are 14″ and 1′,
respectively, corresponding to small scales of $\sim$0.5 pc and large scales
of $\gtrsim$2.5 pc. Therefore, the difference between the position angles of
the two measurements could be used to trace the change of magnetic field
orientations from large to small spatial scales, although we caution that the
ACT data may include foreground signals and a better removal of foreground
polarization emission is necessary (Section 3.2). The change in the
orientations of magnetic fields can be attributed to the motion of partially
ionized molecular gas on which the magnetic field is frozen. Figures 3 & 4
already illustrate the orientations of global and local magnetic fields. Here
we present further discussions.

First of all, the observed change in the orientations of magnetic fields at
the two spatial scales is unlikely attributed to difference grain sizes or
dust temperatures. Although grain growth in molecular clouds is possible
(e.g., Andersson et al., 2015), there has not been observational evidence of
grain growth between scales of 0.5–2.5 pc as well as its effect on magnetic
fields. For spatial scales greater than disks, grain sizes are unlikely to
vary significantly, given a lack of mechanisms to modulate them. As for dust
temperatures, typical values in the clouds are 20–30 K (Tang et al., 2021),
with the spectral energy distribution (SED) peaking at $\sim$100–150 µm. The
wavelengths of the JCMT and ACT data (850 µm vs. 1.3 mm) are both faraway from
the SED peak, and therefore the temperature effect should not be significant.

The distributions of the difference between magnetic field position angles
probed by our JCMT/POL2 observations and those probed by ACT are illustrated
as histograms in Figure 6. There are regions where the local magnetic fields
tend to be perpendicular to the global ones, including the peak of Sgr C, the
Stone cloud, cloud c, cloud d, and cloud e/f. Note that some of these trends
are not evident in Figure 6 because the histograms include all measurements of
position angles across a cloud, but such trends can be better illustrated in
Figure 4. These regions all have high column densities ($\gtrsim$1023 cm-2,
see Figure 8). However, there also exist high column density regions where the
local magnetic fields are aligned to the global ones, e.g., most of Sgr C, the
peak of the 50 km s-1 cloud, and G0.253+0.016. Lastly, orientations of local
magnetic fields in several regions are likely affected by environments such as
feedback. For example, the bimodal distribution of local magnetic field
orientations in cloud e/f is due to the two different magnetic field
geometries in the dense filament and the southern PDR in this cloud (see
discussions in Section 4.1.4).

Figure 6: The histograms show distributions of relative orientations between
magnetic field position angles as probed by JCMT/Pol2 and ACT in each cloud
(the first 11 panels) as well as for all the clouds (the last panel). The
range of angles is limited between 0° and 90° as the difference between two
positions angles is always smaller than 90°.

Therefore, qualitatively, we do not find a clear correlation between the
orientations of local and global magnetic fields. It is possible that the
local magnetic fields are affected by local environments such as expanding
shells (Sgr C, Stone) or photoionization from massive stars (cloud e/f), whose
orientations are thus detached from the global ones and are forced to be
aligned to the local environments such as PDR interfaces. It is also possible
that the orientations of local magnetic fields probed by JCMT POL2 are
affected by spatial filtering (Juvela et al., 2018).

Lastly, in the last panel of Figure 6, we plot the integrated distribution of
position angle differences between JCMT/POL2 and ACT observations for all the
clouds in the sample. The distribution appears to be uniform between 0° and
90°, which is expected if the two position angles are randomly distributed
with respect to each other. We run both the Kolmogorov-Smirnov (K-S) test and
the Watson test from circular statistics with the null hypothesis that the
observed position angle differences are distributed according to the uniform
distribution between 0° and 90°. The resulting $p$-values are relatively high
($>$0.05). Therefore, we cannot reject the hull hypothesis, and the observed
JCMT/POL2 and ACT position angles are likely randomly oriented with respect to
each other. Previous studies of magnetic fields in Galactic high-mass star
forming regions have found a bi-modal rather than random distribution for the
position angle differences of different spatial scales (cores at 0.03 pc
scales vs. clumps at 0.1–0.6 pc scales; Zhang et al., 2014, see their Figure
2). The bi-modal distribution indicates that the magnetic field plays a
dynamically important role in the formation of the dense cores. In the CMZ,
however, strong turbulence could dominate gas dynamics and randomize the
orientations of magnetic fields in individual clouds. Whether the random
distribution of position angle differences continues to even smaller spatial
scales in these CMZ clouds can be tested with high angular resolution
observations from interferometers such as ALMA.

Figure 7: Correlations between physical properties of nine clouds in our
sample. Two clouds, the Far-Side Candidate and CO 0.02$-$0.02, are not covered
by CMZoom, and thus are not included here. The x-axis is one of the physical
properties derived from our observations: H2 number density, magnetic field
strength, Alfvénic Mach number, mass-to-flux ratio, and total virial
parameter. The y-axis is either the CDGF (Battersby et al., 2020) or the SFR
of the clouds (Hatchfield et al. 2023, in prep.). The Spearman’s rank
correlation coefficient $\rho$ and the $p$-value derived from a permutation
test are labeled in each panel. The dashed line is a linear fit to the data
points.

###  4.3 What physical properties are correlated with the dense gas fractions
and SFRs?

The impact of strong magnetic fields on star formation in the CMZ has been
discussed in Pillai et al. (2015). Here, we extend the discussion to a wider
sample of the 11 clouds and search for correlations between physical
properties (including magnetic fields) and star formation activities.

To characterize star formation activities of the clouds, we choose two
parameters: the compact dense gas fraction (CDGF), and the SFR. The clouds in
our sample are included in the CMZoom survey (Battersby et al. 2020;
Hatchfield et al. 2020; Hatchfield et al. 2023, in prep.), in which the CDGFs
and SFRs of the clouds are quantified. The CDGF is defined as the ratio of gas
masses in compact substructures on 0.1–2 pc scales based on the SMA
observations and cloud masses based on the Herschel observations (see Table 4
of Battersby et al., 2020, ‘method2’). It can be used as a measure of ‘dense’
gas fraction of the clouds that is directly related to star formation (Lu et
al., 2019a). Typical uncertainties in the CDGFs are a factor of 2 and
potentially higher (Section 5.5 of Battersby et al., 2020). The SFR of a cloud
is estimated from its total molecular gas mass divided by its free-fall time,
corrected by factors accounting for a $\sim$25% star formation efficiency over
one free-fall time and an initial mass function (Hatchfield et al. 2023, in
prep.). The uncertainties of the SFRs derived by Hatchfield et al. (2023, in
prep.) are listed in Table 1, with a typical value of 50% and potentially
higher.

The physical properties of the clouds that we have measured include the number
density $n$(H2), the magnetic field strength $B$, the Alfvénic Mach number
$\mathcal{M}_{\text{A}}$, the mass-to-flux ratio $\lambda$, and the virial
parameter $\alpha_{\text{vir,B}}$ (see Table 1).

In Figure 7, we plot these physical properties against the two characteristics
of star formation activities, the SFR and the CDGF. We derive Spearman’s rank
correlation coefficients $\rho$ for the pairs of parameters. $\rho=1$ suggests
a strong correlation, $\rho=-1$ a strong anti-correlation, and $\rho=0$ no
correlation. Among all the relations, we only find moderate anti-correlations
between the total virial parameter $\alpha_{\text{vir,B}}$ and the SFR/CDGF,
with $\rho=-0.70$ and $-0.48$, respectively, and relatively lower $p$-values.
There are also a weak correlation between the mass-to-flux ratio $\lambda$ and
the SFR, and a weak anti-correlation between the magnetic field $B$ and the
CDGF, both with $\textbar\rho\textbar\sim 0.4$. For all the other pairs of
parameters, we do not find clear correlations ($\textbar\rho\textbar\lesssim
0.2$).

The anti-correlation between $\alpha_{\text{vir,B}}$ and the SFR/CDGF is
expected when star formation in these clouds is regulated by the dynamical
states of the clouds co-determined by the self-gravity, turbulence, and
magnetic field. In our sample, the magnetic field is a minor contribution in
the virial parameter, which is dominantly determined by the self-gravity and
turbulence (see Table 1: virial parameters without or with taking the magnetic
field into account do not vary much). Combining with the fact that only weak
to no correlations are detected between the magnetic field and the SFR/CDGF,
we suggest that the magnetic field plays a minor role in regulating star
formation in these clouds as compared to self-gravity and turbulence.

Palau et al. (2021) studied magnetic fields at $\lesssim$0.1 pc scales in a
sample of high-mass star forming cores in the Galactic disk, and found a
tentative correlation between their mass-to-flux ratios and fragmentation
levels. Whether the same trend holds for the spatial scale of dense cores
($\lesssim$0.1 pc) in the CMZ clouds need further investigations using
interferometers such as ALMA. If such a trend is corroborated, it would
indicate that the magnetic field in the CMZ plays a non-negligible albeit
probably minor role in influencing gas dynamics and star formation at the
spatial scales of clouds to cores.

We also note that only two clouds in the sample, Sgr C and Dust Ridge cloud c,
have low virial parameters ($\alpha_{\text{vir,B}}<2$) that suggest marginally
gravitationally bound gas. These two clouds have high CDGFs ($\gtrsim$0.1) and
are known to be forming high-mass stars (Kendrew et al., 2013; Ginsburg et
al., 2015; Walker et al., 2018; Lu et al., 2019a, b, 2020, 2021). All the
other clouds have virial parameters higher than 2 and therefore are likely
unbound in the absence of external pressure. However, these clouds are
unlikely to be dispersing, given the compact dense substructures detected
inside them (e.g, Battersby et al., 2020; Lu et al., 2020; Walker et al.,
2021). Previous observations have found evidence of strong external pressure
confining dense clouds in the CMZ (Myers et al., 2022; Callanan et al., 2023).
Our results suggest that for these clouds the external pressure could be
critical for keeping them in dynamic equilibrium, although the conclusion is
subject to large uncertainties in the virial parameters (at least a factor of
2; see Appendix D).

The robustness of the conclusions suffers from the limited sample size. Future
full-CMZ surveys of the magnetic field (e.g., Butterfield et al., 2023) would
enlarge the sample size and lead to more robust results regarding correlations
between the magnetic field and star formation. For example, correlations
between magnetic field strengths and SFRs of giant molecular clouds have been
suggested in external galaxies (e.g., Tabatabaei et al., 2018), although the
magnetic field therein was estimated from a different approach (basing on the
non-thermal radio continuum and assuming equipartition between the energy
densities of the magnetic field and cosmic rays). With a larger sample toward
the CMZ, we will be able to examine whether the same correlations hold.

Lastly, we note that the correlation in Figure 7 could be alternatively
interpreted as an effect of cloud evolution. Quiescent clouds at the earliest
stages of star formation could be gravitationally unbound and magnetically
subcritical. As more gas is accumulated from the environment (e.g., through
large-scale gas inflow; Williams et al., 2022), the clouds could become
gravitationally bound and supercritical, and therefore start to collapse and
form stars.

##  5 CONCLUSIONS

We present JCMT/POL2 observations of polarized dust emission in the CMZ at a
resolution of 14″ (0.55 pc linear resolution). The observations cover three
large areas and sample 11 clouds. This is by far the largest sample of high-
resolution ($<$1 pc, thus spatially resolving the clouds) studies of magnetic
fields in molecular gas in the CMZ. The results from this study are:

  * •

The morphologies of magnetic fields in the clouds projected on the plane of
the sky are inferred from the polarized dust emission. By comparing to larger
scale magnetic field morphologies traced by the $1^{\prime}$ resolution ACT
observations, we find that the large and small scale magnetic fields in these
clouds may not be aligned. The misalignment may be partly due to the impact of
local environments, since we find evidence of local magnetic fields aligned to
expanding shells around H ii regions and PDR interfaces.

  * •

The magnetic field strengths in the clouds are estimated using the ADF method
to be between 0.1 and 1.7 mG, with an uncertainty of at least a factor of 2.
The Alfvénic Mach number, mass-to-flux ratio, and total (magnetized) virial
parameter are then derived. All the clouds but Sgr C and Dust Ridge cloud c
have virial parameters higher than 2, suggesting that they would be
gravitationally unbound if the external pressure is not considered.

  * •

Correlations between the five physical properties of the clouds derived from
our observations (H2 number density, magnetic field strength, Alfvénic Mach
number, mass-to-flux ratio, and total virial parameter) and the two
characteristics of star formation (SFR and CDGF) are explored. A moderate
correlation is found between the total virial parameter and the SFR/CDGF. Weak
(anti-)correlations are found between the mass-to-flux ratio and the SFR, and
between the magnetic field and the CDGF. This may suggest a minor role of the
magnetic field in regulating star formation in these clouds.

  * •

To summarize, the magnetic field alone is unable to explain the different star
formation states of the 11 clouds in our sample. Self-gravity and turbulence
seem to be dominant in determining the star formation states of the clouds. In
certain clouds (e.g., Sgr C, cloud e/f), star formation as well as the
magnetic field could be subject to feedback from H ii regions and massive star
clusters.

We thank the anonymous referee for constructive comments that have improved
the quality of the paper. This work has been supported by the National Key R&D
Program of China (No. 2022YFA1603101). This work has been sponsored by Natural
Science Foundation of Shanghai (No. 23ZR1482100). X.L. acknowledges support
from the National Natural Science Foundation of China (NSFC) through grant No.
12273090, and the Chinese Academy of Sciences (CAS) ‘Light of West China’
Program (No. xbzg-zdsys-202212). Q.Z. gratefully acknowledges support from the
National Science Foundation under Award No. 2206512. T.L. acknowledges the
supports by National Natural Science Foundation of China (NSFC) through grants
No. 12073061 and No. 12122307, the international partnership program of
Chinese Academy of Sciences through grant No. 114231KYSB20200009, and Shanghai
Pujiang Program 20PJ1415500. We thank Prof. Yu Gao for his efforts in
supporting the operation of JCMT. We thank Jürgen Ott for sharing the ATCA NH3
(3, 3) data, and Yuping Tang for helpful discussions on the LMT data. We also
thank Bethan Williams, Natalie Butterfield, and Yuhei Iwata for helpful
discussions. This research has made use of NASA’s Astrophysics Data System.
The James Clerk Maxwell Telescope is operated by the East Asian Observatory on
behalf of The National Astronomical Observatory of Japan, Academia Sinica
Institute of Astronomy and Astrophysics, the Korea Astronomy and Space Science
Institute, the National Astronomical Observatories of China and the Chinese
Academy of Sciences (Grant No. XDB09000000), with additional funding support
from the Science and Technology Facilities Council of the United Kingdom and
participating universities in the United Kingdom and Canada.

##  Appendix A Column Density Maps and Estimates of Number Densities

Column density maps of the three fields, following the methods in Section 3.1,
are presented in Figure 8. Note that the 850 µm continuum emission total
intensities are from the SCUBA2 observations of Parsons et al. (2018) that
provide a better sampling of diffuse dust emission and removal of CO
contamination.

Figure 8: Molecular hydrogen (H2) column densities of the three CMZ fields
derived from the JCMT 850 µm continuum of Parsons et al. (2018). The contours
start at $10^{22}$ cm-2, with an increment of 0.5 in the logarithmic scale.

##  Appendix B Angular dispersion functions and fitting results

Angular dispersion functions (ADFs) of the 11 clouds and fitting results
following Equation 3 are presented in Figure 9. For most data, the vertical
errorbars, representing the uncertainties in the ADFs, appear smaller than the
symbols. These uncertainties have been taken into account in the fitting of
the ADFs, contributing to the quoted uncertainty of the turbulent-to-total
magnetic field strength ratio in Figure 9.

Figure 9: ADFs of the 11 clouds. Uncertainties in the ADFs inherited from the
scatter of the position angles are shown as vertical errorbars. The fitting
results following Equation 3 are plotted as blue curves. The beam size
(14$\farcs$1) is marked by a vertical dashed line, while the ADF value in the
case of randomly distributed magnetic fields (0.36) is marked by a horizontal
dashed line. The best-fit turbulent-to-total magnetic field strength ratio, as
well as its uncertainty, is labeled in each panel.

##  Appendix C Fitting results of the NH3 (3, 3) spectra

Figure 10 displays the mean NH3 (3, 3) spectra toward the clouds taken from
the SWAG survey and Gaussian fittings to the spectra. The spectra are averaged
over all pixels above the threshold of 0.2 Jy beam-1 in the total intensity
map from our data. For clouds showing multiple velocity components, the
component with a higher peak is chosen to represent the principle part of the
gas. We additionally cross check with observations of dense cores at higher
angular resolutions to confirm that the velocity of the chosen component is
consistent with those of the dense cores (Lu et al., 2019a; Callanan et al.,
2023).

We additionally show the intensity-weighted velocity (1st moment) maps of the
clouds in Figure 11. The velocity ranges for making the 1st moment maps are
adjusted to match those in the Gaussian fittings in Figure 10. Some clouds,
e.g., the Stone cloud, G0.253+0.016, and cloud b, seem to present large
velocity gradients of $>$10 km s-1 across the area where our analysis is
carried out. For example, the velocity gradient in G0.253+0.016 has been
interpreted as different velocity components along the line of sight (Henshaw
et al., 2019) or shear motions (Federrath et al., 2016). Therefore, the
linewidths measured toward these clouds are likely overestimated and should be
treated as upper limits.

Figure 10: Mean SWAG NH3 (3, 3) spectra of the 11 clouds and single-component
Gaussian fittings. The best-fit centroid velocity ($V_{\text{lsr}}$) and
velocity dispersion ($\sigma_{v}$) are given in each panel.

Figure 11: Intensity-weighted velocity maps of the 11 clouds made from the
SWAG NH3 (3, 3) data. The synthesized beam of the ATCA observations from the
SWAG program is displayed at the bottom left corner of each panel. Contours
are the same as in Figure 1, representing the JCMT 850 µm total intensity
emission.

##  Appendix D Uncertainties in the estimated physical parameters

The uncertainties in the magnetic field strengths estimated through Equation 4
come from two sources. Firstly, the uncertainties of the parameters on the
right-hand side of the equation, including the ratio $\left(\langle
B_{\text{t}}^{\phantom{\text{t}}2}\rangle/\langle B^{2}\rangle\right)^{0.5}$,
the velocity dispersion $\sigma_{v}$, the density $\rho$, and the correction
factor 0.21 can be propagated into that of the magnetic field strength.
Second, the DCF method itself has inherent uncertainties due to unsatisfaction
of assumptions, e.g., an overestimation of the magnetic field strength in
super Alfvénic conditions (Liu et al., 2021).

We first quantify the first source of uncertainties. The uncertainties in
$\left(\langle B_{\text{t}}^{\phantom{\text{t}}2}\rangle/\langle
B^{2}\rangle\right)^{0.5}$, derived from the covariance of the ADF fitting
(Appendix B), are annotated in Figure 9. The errors in $\sigma_{v}$ from the
Gaussian fittings can be found in Figure 10. As discussed in Section 3.1, the
uncertainties in $\rho$ is typically 50%. The correction factor has an
uncertainty of 45% (Liu et al., 2021). Propagating these errors, we obtain a
characteristic uncertainty of 50% for the magnetic field strength $B$, which
is dominated by that of the correction factor.

There is another systematic uncertainty that is not included the error
analysis above. The velocity dispersion $\sigma_{v}$ could be significantly
overestimated given the presence of multiple velocity components in the clouds
(see Appendix C). Henshaw et al. (2016) identified a mean number of Gaussian
components per position in the CMZ of 1.6. Therefore, we expect that on
average the measured $\sigma_{v}$ overestimate the true value by a factor of
1.6. This clearly dominates over the statistical uncertainties in $B$.

The inherent uncertainty from the DCF method itself, e.g., because of its
unrealistic assumptions, is not straightforward to quantify, yet it often
dominates the uncertainty in the estimated magnetic field. Liu et al. (2021)
estimated an uncertainty of at least a factor of 2, which is derived from
comparison between input models and results from applying the DCF method to
numerical simulations.

To summarize, the uncertainty in the magnetic field strength is dominated by
the systematic uncertainty in the velocity dispersion (overestimation of a
factor of $\sim$1.6) and the inherent uncertainty of the DCF method
($\gtrsim$2). As such, we adopt an uncertainty of a factor of 2 for the
magnetic field strength, and note that it is a lower limit.

The physical parameters derived in Section 3.4, including the Alfvénic Mach
numbers, the mass-to-flux ratios, and the virial parameters, have dependency
on the magnetic field strengths, which dominate the uncertainty. Therefore, we
also adopt a lower limit for the uncertainty in these parameters of a factor
of 2.

##  Appendix E Distributions of magnetic field position angles as probed by
JCMT/POL2 and ACT

Figure 12 displays histograms of magnetic field position angles as probed by
JCMT/POL2 in each cloud, as well as the mean magnetic field position angle
probed by ACT.

Additionally, in Figure 13 we present the magnetic field position angles
derived from the JCMT/POL2 data smoothed and regridded to the same frame as
the ACT observations of $1^{\prime}$ resolution. Here we have smoothed the
Stokes Q and U components of the POL2 data and then re-derived the magnetic
field position angles. The two datasets reveal different field morphologies.
Interpretations of this difference can be found in Section 3.2.

Figure 12: The histograms show magnetic field position angles as probed by
JCMT/POL2. The solid red vertical lines denote the mean position angles, and
the dashed red vertical lines denote the standard deviations of the position
angles in the clouds. Similarly, the blue shaded areas represent the magnetic
field position angles as probed by ACT (Guan et al., 2021), with the central
solid blue vertical lines denoting the mean position angles and the dashed
blue vertical lines denoting the standard deviations of the position angles
found inside the cloud boundaries. All the position angles follow the IAU-
recommended north-to-east definition.

Figure 13: The segments show orientations of magnetic fields probed by ACT
(cyan) and by JCMT/POL2 smoothed and regridded to the same frame as ACT
(magenta and green, illustrating those with $p$/d$p$$>$3 and 2$<$$p$/d$p$$<$3,
respectively, following the convention in Figure 3). Contours are the same as
in Figure 1, representing the JCMT 850 µm total intensity emission.

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