11institutetext: Instituto de Estudios Astrofísicos, Facultad de Ingeniería y
Ciencias, Universidad Diego Portales, Santiago 8370191, Chile 22institutetext:
Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing
100871, China 33institutetext: Joint ALMA Observatory, Avenida Alonso de
Cordova 3107, Vitacura, Santiago 7630355, Chile 44institutetext: INAF –
Astronomical Observatory of Rome, Via Frascati 33, 00040 Monte Porzio Catone,
Italy 55institutetext: Department of Space, Earth and Environment, Chalmers
University of Technology, SE-412 96 Gothenburg, Sweden 66institutetext: RIKEN
Cluster for Pioneering Research, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
77institutetext: Department of Space, Earth and Environment, Chalmers
University of Technology, Onsala Space Observatory, 43992 Onsala, Sweden
88institutetext: X-Ray Astrophysics Laboratory, NASA Goddard Space Flight
Center, Code 662, Greenbelt, MD 20771, USA 99institutetext: Yale Center for
Astronomy & Astrophysics, 52 Hillhouse Avenue, New Haven, CT 06511, USA
1010institutetext: Instituto de Astrofísica and Centro de Astroingeniería,
Facultad de Física, Pontificia Universidad Católica de Chile, Campus San
Joaquín, Av. Vicuña Mackenna 4860, Macul Santiago 7820436, Chile
1111institutetext: Millennium Institute of Astrophysics, Nuncio Monseñor
Sótero Sanz 100, Of 104, Providencia, Santiago, Chile 1212institutetext: Space
Science Institute, 4750 Walnut Street, Suite 205, Boulder, Colorado 80301, USA
1313institutetext: NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
1414institutetext: Department of Astronomy, School of Physics, Peking
University, Beijing 100871, China 1515institutetext: Space Telescope Science
Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA 1616institutetext:
MIT Kavli Institute for Astrophysics and Space Research, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA 1717institutetext: Eureka
Scientific, 2452 Delmer Street Suite 100, Oakland, CA 94602-3017, USA
1818institutetext: Department of Physics and Astronomy, West Virginia
University, P.O. Box 6315, Morgantown, WV 26506, USA 1919institutetext: Center
for Research and Exploration in Space Science and Technology, NASA/GSFC,
Greenbelt, MD 20771, USA 2020institutetext: Department of Astronomy,
University of Maryland, College Park, MD 20742, USA 2121institutetext: Joint
Space-Science Institute, University of Maryland, College Park, MD 20742, USA
2222institutetext: Department of Astronomy, University of Geneva, ch. d’Écogia
16, CH-1290, Versoix, Switzerland 2323institutetext: National Radio Astronomy
Observatory, Charlottesville, VA 22903, USA 2424institutetext: Department of
Astronomy, University of Florida, Gainesville, FL, 32611, USA
2525institutetext: Department of Astronomy, University of Virginia,
Charlottesville, VA, 22904, USA 2626institutetext: Department of Physics and
Astronomy, Texas A&M University, College Station, TX 77845, USA
2727institutetext: Department of Physics, Southern Methodist University, 3215
Daniel Ave., Dallas, TX 75205, USA 2828institutetext: Physics Department, Tor
Vergata University of Rome, Via della Ricerca Scientifica 1, 00133 Rome, Italy
2929institutetext: INFN – Rome Tor Vergata, Via della Ricerca Scientifica 1,
00133 Rome, Italy 3030institutetext: School of Physics and Astronomy, Tel Aviv
University, Tel Aviv 69978, Israel

# Joint ALMA/X-ray monitoring of the radio-quiet type 1 AGN IC 4329A

E. Shablovinskaya<EMAIL_ADDRESS>11 C. Ricci 1122 C-S.
Chang 33 A. Tortosa 44 S. del Palacio 55 T. Kawamuro 66 S. Aalto 77 Z.
Arzoumanian 88 M. Balokovic 99 F. E. Bauer 101011111212 K. C. Gendreau 1313 L.
C. Ho 221414 D. Kakkad 1515 E. Kara 1616 M. J. Koss 17171212 T. Liu 1818 M.
Loewenstein 19192020 R. Mushotzky 20202121 S. Paltani 2222 G. C. Privon
232324242525 K. Smith 26262727 F. Tombesi 2828442929 B. Trakhtenbrot 3030

(Received …; accepted …)

The origin of a compact millimeter (mm, 100–250 GHz) emission in radio-quiet
active galactic nuclei (RQ AGN) remains debated. Recent studies propose a
connection with self-absorbed synchrotron emission from the accretion disk
X-ray corona. We present the first joint ALMA ($\sim$100 GHz) and X-ray
(NICER/XMM-Newton/Swift; 2–10 keV) observations of the unobscured RQ AGN, IC
4329A ($z=0.016$). The time-averaged mm-to-X-ray flux ratio aligns with
recently established trends for larger samples (Kawamuro et al., 2022; Ricci
et al., 2023), but with a tighter scatter ($\sim$0.1 dex) compared to previous
studies. However, there is no significant correlation on timescales of less
than 20 days. The compact mm emission exhibits a spectral index of $-0.23\pm
0.18$, remains unresolved with a 13 pc upper limit, and shows no jet
signatures. Notably, the mm flux density varies significantly (factor of 3)
within 4 days, exceeding the contemporaneous X-ray variability (37% vs. 18%)
and showing the largest mm variations ever detected in RQ AGN over daily
timescales. The high amplitude variability rules out scenarios of heated dust
and thermal free–free emission, pointing toward a synchrotron origin for the
mm radiation in a source of $\sim$1 light day size. While the exact source is
not yet certain, an X-ray corona scenario emerges as the most plausible
compared to a scaled-down jet or outflow-driven shocks.

###### Key Words.:

Galaxies: individual: IC 4329A – X-rays: galaxies – submm/mm: galaxies

##  1 Introduction

Most active galactic nuclei (AGN) do not show bright radio structures
associated with powerful relativistic jets launched from regions close to the
accreting supermassive black hole (SMBH). However, these radio-quiet (RQ)
sources exhibit faint radio emissions, often appearing in compact, parsec-
scale regions that remain unresolved even with VLBA observations (e.g.
Alhosani et al., 2022; Chen et al., 2023; Wang et al., 2023). While in radio-
loud AGN the emission is known to be dominated by non-thermal synchrotron
radiation from jets, the origin of compact radio emissions in RQ AGN is still
a subject of debate. Various hypotheses have been proposed, including small-
scale jets, nuclear star formation regions, thermal free–free radiation, and
the magnetised corona of the accretion disk (see Panessa et al., 2019, for a
review). Among these explanations, the accretion disk corona is one of the
most favoured. This arises from the RQ AGN following the radio/X-ray relation
$L_{\mathrm{5GHz}}/L_{\mathrm{0.2-20keV}}\sim 10^{-5}$ (Laor & Behar, 2008)
previously discovered for the coronally active stars (Guedel & Benz, 1993),
where the corona is magnetically heated similarly to what is expected for AGN
(e.g. Merloni & Fabian, 2001b, a). In this scenario, the population of high-
energy electrons upscattering UV/optical seed photons into X-ray energies also
produces optically thick synchrotron radio emission while moving in the
coronal magnetic field. Under certain assumptions, the synchrotron emission of
the structure with an approximately X-ray corona size can be self-absorbed at
frequencies lower than $\sim$100–300 GHz (i.e., millimeter-wave band, mm),
where the contribution from cold dust emission is minimal. Recent theoretical
calculations (e.g., Raginski & Laor, 2016; Inoue & Doi, 2018) also predict
that coronal emission can peak at 100–200 GHz, producing flat synchrotron
emission up to 300 GHz. Flux densities at 100 GHz have been measured for a few
dozen AGN and consistently exceed extrapolations from low-frequency power-law
slopes (e.g., Behar et al., 2015, 2018; Inoue & Doi, 2018), confirming the
presence of a compact, optically-thick core.

Figure 1: X-ray and mm-band light curves. From top to bottom: the X-ray (2–10
keV) flux with fitted power-law model, 2–10 keV flux deviations from the long-
term trend, and the 3 mm (97.5 GHz) flux density. The interval of the
contemporaneous X-ray and mm observations is marked with a grey region. All
data are listed in Appendix A.

Recent research has further supported the connection between mm and X-ray
luminosity: while Behar et al. (2018) found only a tentative correlation
between 100 GHz and 2–10 keV luminosity for 34 AGN with a large scatter111Here
and after we mean 1$\sigma$ scatter. ($\sim$0.5 dex) likely caused by the
sample heterogeneity and low angular resolution ($\gtrsim 1^{\prime\prime}$),
a recent high-spatial resolution study of a sample of 98 AGN with ALMA
(Kawamuro et al., 2022, 2023) measured a highly significant linear correlation
between the time-averaged hard X-ray (14–150 keV) and the instantaneous mm
(230 GHz) luminosity with a $\sim$0.36 dex scatter. Notably, higher spatial
resolution (1–23 pc) 100 GHz ALMA observations of nearby hard X-ray-selected
AGN have shown that the correlation between X-ray and mm emission (Ricci et
al., 2023) has a low scatter of 0.22 dex. However, to unequivocally establish
the common origin of mm and X-ray flux by the electrons in the X-ray corona,
it would be crucial to detect their correlated variability. Detecting
correlated variability would also significantly contribute to our
understanding of the X-ray origin, providing evidence that the magnetic field
plays an important role in relativistic particle acceleration and corona
heating (Merloni & Fabian, 2001b, a). Currently, only two observational
campaigns (Behar et al., 2020; Petrucci et al., 2023) have been conducted
using IRAM (with $\sim$17–28′′ resolution) and NOEMA ($\sim$1′′) at 100 GHz,
respectively, and found no definitive evidence of correlated variability (see
Section 4.2). In this context, sub-arcsecond observations with ALMA can play a
crucial role due to its sensitivity and angular resolution to localise the
compact source of the mm emission.

In this paper, we study the X-ray and mm variability of the nearby RQ AGN IC
4329A ($z=0.016$) using, for the first time, ALMA observations with angular
resolution $<$0$\aas@@fstack{\prime\prime}$1\. Previous high-resolution
(0$\aas@@fstack{\prime\prime}$16–0$\aas@@fstack{\prime\prime}$4) ALMA
observations have shown that IC 4329A exhibits relatively bright ($\sim$8 mJy)
compact emission at 100 GHz, showing an excess compared to lower frequencies
(Inoue & Doi, 2018), which can be attributed to self-absorbed synchrotron
radiation. Additionally, IC 4329A stands out as the brightest unobscured AGN
in the southern sky ($F_{2-10\ \rm keV}\sim 10^{-10}$ $\rm erg\,\rm
cm^{-2}\,\rm s^{-1}$, Ricci et al., 2017), showing significant variability in
the X-ray band (from $\sim$8 $\times$ 10-11 $\rm erg\,\rm cm^{-2}\,\rm s^{-1}$
to $\sim$2 $\times$ 10-9 $\rm erg\,\rm cm^{-2}\,\rm s^{-1}$ in 2–10 keV band
on timescales as short as several hours, see Tortosa et al., 2023, for
details). All these characteristics, in both the X-ray and mm band, make IC
4329A the best target for the search for correlated X-ray/mm variability. In
2021, IC 4329A was the focus of an extensive observational campaign, with 45
X-ray observations carried out by XMM-Newton, NuSTAR, NICER and Swift and 10
ALMA observations over 10 consecutive days. The analysis of these high-quality
XMM-Newton and NuSTAR observations were described in Tortosa et al. (2023). In
this article, we report the analysis of the X-ray and mm variability on
timescales of days, with the goal of investigating the correlation between
these two bands. Throughout the paper we used the standard cosmological
parameters ($H_{0}=70$ km s-1 Mpc-1, $\Omega_{\rm m}=0.3$,
$\Omega_{\Lambda}=0.7$).

##  2 Observations and data reduction

###  2.1 X-ray

####  2.1.1 XMM-Newton

IC 4329A was observed once per day for ten consecutive days (from 2021-08-10
to 2021-08-19, P.I. C. Ricci) by the X-ray Multi-Mirror Mission (XMM-Newton,
Jansen et al. 2001) during XMM-Newton AO 19. The European Photon Imaging
Camera (EPIC hereafter) cameras were operated in the small window and thin
filter mode. The observation 0862090401 is not included in the analysis since,
due to a problem in the ground segment, the EPIC exposure was lost.

The EPIC camera event lists are extracted with the epproc (pn) and and emproc
(MOS) tools of the standard System Analysis Software (SAS v.18.0.0; Gabriel et
al. 2004). The latest calibration files, available at January 2023, were used.
The choice of optimal time cuts for the flaring particle background has been
performed by visually inspecting light curves created in the energy range
10–12 keV (EPIC-pn) with PATTERN=0 (single events). For the choice of source
and background extraction radii, we identified point-like sources in each
target field of view running the meta-task edetect-chain on the 0.5–2 keV
energy band EPIC images and performed an iterative process that maximises the
signal-to-noise ratio (SNR) as in Piconcelli et al. (2004). The resulting
optimal source extraction radius was $30\arcsec$, and the background spectra
were extracted from source-free circular regions with radii of $\sim$
$50\arcsec$ for each observation. Response matrices and auxiliary response
files were generated using the SAS tools rmfgen and arfgen, respectively.
EPIC-pn spectra were binned to oversample the instrumental energy resolution
by a factor larger than three and to have no less than 20 counts in each
background-subtracted spectral channel. No significant pile-up affected the
EPIC data, as indicated by the SAS task epatplot.

####  2.1.2 NICER

Data reduction of the 21 NICER (Gendreau et al., 2012; Arzoumanian et al.,
2014; Gendreau et al., 2016) observations (PI: C. Ricci) of our campaign was
performed following the same procedure reported in Ricci et al. (2021).

####  2.1.3 Swift

A total of 12 observations from the X-ray telescope (XRT, Burrows et al.,
2005) on board the Neil Gehrels Swift Observatory (Gehrels et al., 2004) (PI:
C. Ricci) were used here. We performed the Swift/XRT data analysis using the
xrtpipeline following the standard guidelines (Evans et al., 2009).

###  2.2 mm

The ALMA band-3 observations were taken for 10 consecutive days from August 28
to September 6, 2021 (Project code 2019.1.01181.S; PI C. Ricci) in four
channels of 1.985 GHz bandwidth with the central frequencies $\nu=$ 90.52,
92.42, 102.52, and 104.48 GHz. Most observations are taken within 3 hours from
the NICER observations. Data are processed using CASA version 6.1.1.15 and the
ALMA Pipeline version 2020.1.0.40 (Hunter et al., 2023). The spectral setup is
in Time-Division Mode (TDM) around 100 GHz, as we aim to detect the continuum
emission of IC 4329A. Our monitoring campaign was scheduled during the ALMA
long baseline configuration (C43-9/10) with the longest baseline of 13.8 km.
For each observing session, the on-source time is about 8 minutes. As a
result, the typical beam size is between 0′′.04 and 0′′.09 (i.e. $\sim$13 and
$\sim$29 pc). According to the ALMA proposer’s guide, the statistical flux
error for band-3 observations is 5%. By analyzing the flux measurements of the
phase calibrator J1351-2912, we derived consistent results. Therefore, we
applied the 5% error to our analysis.

To have an insight into the spectral variability of IC 4329A, we derived the
spectral index using the fluxes measured by the four spectral windows from the
10 epochs of observations. This analysis assumes that the spectral slope of
the flux calibrator J1337-1257 and the phase calibrator were stable during our
campaign. The ALMA calibration shows that the variation of the spectral index
for the flux calibrator is within 2% during our campaign.

##  3 Results

The X-ray (2–10 keV) and mm (100 GHz) light curves of IC 4329A obtained are
illustrated in Fig. 1. Note that the error bars are shown in the figure but
are typically smaller than the symbols. In both wavebands, the observations
were carried out with a daily cadence. The X-ray light curve, obtained with
XMM-Newton, NICER and Swift, spanning 40 days, is shown in the top panel of
Fig. 1. Contemporaneous ALMA 100 GHz observations were carried out over 10
days (bottom panel). In the following, we examine the behavior of both light
curves.

Figure 2: Wavelet analysis of the X-ray variability with the trend subtracted
and the data interpolated on the regular grid. Left: the magnitude of the
wavelet transform. The 95% confidence level is given in white contours. The
cone-of-influence is given in the solid black line; the region out of the cone
in the darker colour shows less reliable periods that may be affected by edge
effects (see text for details). Right: the wavelet transform profiles in grey
and their median value in black indicate the variability periods of 14.9 and
4.5 days. The dashed lines correspond to the maxima of the Lomb-Scargle
periodogram.

###  3.1 X-ray variability

During our campaign, the X-ray (2–10 keV) flux of IC 4329A showed significant
variability on timescales of days. To quantify the intensity of the
variability taking into account the measurement uncertainties we calculated
the fractional variability of the light curve $F_{\rm var}$ (Vaughan et al.,
2003):

$F_{\rm
var}=\sqrt{\frac{S^{2}-\overline{\sigma^{2}}}{\overline{x_{i}}^{2}}}\times
100\%,$ (1)

where $S^{2}$ is the sample variance, $\overline{\sigma^{2}}$ is the mean
square error, and $\overline{x_{i}}^{2}$ is the square mean of the data. For
the X-ray data, $F_{\rm var}^{\mathrm{X-ray}}=(17.5\pm 0.5)$%, while the
intraday variations are less than 5% (see Tortosa et al., 2023). Due to the
focus of the work on the variability of the longer (daily) timescale, we
ignored the intraday variations in the analysis.

During our monitoring campaign, the X-ray flux steadily increased (top panel
of Fig. 1), following a long-term trend of $\sim$10-12 $\rm erg\,\rm
cm^{-2}\,\rm s^{-1}$ per day. Once this trend has been subtracted (middle
panel of Fig. 1), the X-ray light curve shows variations on a timescale of
several days. Although the nature of variability in AGN is stochastic, the
observed fluctuations in the X-ray flux relative to the average may suggest
the presence of regular oscillations. To investigate the periodicity of these
variations, we calculated the Lomb-Scargle periodogram (Lomb, 1976; Scargle,
1982) used for the analysis of unevenly sampled data employing a least-squares
fitting procedure. The periodogram revealed two possible variability periods
of $14.4\pm 0.9$ and $4.5\pm 0.1$ days. However, when the light curve is
irregular and evolves over time, wavelet analysis (Morlet, 1983) is more
efficient than the periodogram. Unlike the Fourier transform, whose kernel is
not time-localised, wavelet analysis dissects the data series into distinct
frequency components and examines each segment with corresponding temporal
resolution. Therefore, the wavelet analysis provides both variability
frequency (or period) and its evolution during the observations. For IC 4329A,
the result of the wavelet transformation of the X-ray flux variability with
the subtracted trend is illustrated in Fig. 2. Here, the original epochs are
interpolated to the regular grid using the spline extrapolation. The wavelet
analysis revealed two variation periods corresponding to the periodogram
result: $4.5\pm 0.8$ and $14.9\pm 2.5$ days. The $14.9\pm 2.5$ days period is
stable during the observations but is greater than 1/3 of the monitoring
period and is located out of the cone-of-influence (see details in Torrence &
Compo, 1998), i.e., in the area where the analysis may be affected by data
limitations (such as the finite duration of observations) and edge effects,
thus it may be an artefact. The $4.5\pm 0.8$ days period is less stable but
can potentially reveal quasi-periodic oscillation in the AGN. Nevertheless,
the estimated periods characterize only the observed segment of the light
curve, the overall behaviour of which can be entirely stochastic.

Figure 3: Upper panel: mm light curve in four different frequencies: 90.52,
92.42, 102.52, and 104.48 GHz, shown in different colours and symbols. Bottom
panel: variations of the mm spectral slope. Grey dashed line corresponds to
the median spectral slope $-0.23$.

###  3.2 Millimeter variability

In the mm band, IC 4329A showed very strong variability during the 10-day
observational campaign: the flux changed significantly, up to a factor of 2.6
between maximum and minimum, with the day-to-day variations reaching a factor
of $\sim$1.5. The fractional variability for the mm data calculated using Eq.
(1) is two times higher than in the X-rays: $F_{\rm var}^{\rm mm}=(37.1\pm
0.5)$% including ALMA calibration uncertainties of $\sim$5%. Even scanning the
X-ray light curve with a 10-day window for a conservative comparison, $F_{\rm
var}^{\rm X-ray}$ was found to be significantly lower than in the mm band,
from 6.5% to 18.5%, and the maximum $F_{\rm var}^{\rm X-ray}$ was observed
during the 10 days of the simultaneous X-ray/ALMA observations. Within 10
days, the mm light curve showed two clear maxima separated by $>$8 days.

As ALMA observations were made at frequencies $\nu=$ 90.52, 92.42, 102.52, and
104.48 GHz, this could provide us with insight into the spectral variability
of IC 4329A. Assuming a power-law spectrum ($F_{\rm mm}\propto\nu^{\alpha}$),
for each epoch, we calculated the spectral slope $\alpha$. We found an average
spectral slope $-0.23$ with $\sigma=0.18$, with time variations below
$\sigma$.

The light curves of IC 4329A in different frequencies are shown in Fig. 3
(upper panel). The variability pattern is the same for all wavebands. For
comparison, we illustrated the variations of the spectral slope $\alpha$ in
the bottom panel, Fig. 3 that did not show a correlation with the flux.
Additionally, we examined the difference $\Delta$ between the lower and higher
frequencies: $\Delta=(F_{91}-F_{103})/F_{103}\times 100$%, where $F_{91}$ is
the mean flux between 90.52 and 92.42 GHz, and $F_{103}$ is the mean flux
between 102.52 and 104.48 GHz. Variations in this difference in time are
statistically significant at the 95% significance level, changing between
$\sim$6% (during the first flare) and $\sim$0% (before the second flare).
However, this value is also not correlated with the flux.

The exceptional resolution and sensitivity of ALMA allowed us to resolve a
faint diffuse structure to the east of the central bright source in the
continuum maps of IC 4329A (Fig. 4). Notably, this structure is observed
throughout the entire campaign, though due to beam size and shape, it is
clearly distinguishable in only five of the 10 epochs. We measured the flux of
this diffuse spot whenever possible and found that it is consistent, within
the uncertainties in all epochs, with a flux density of $\sim$0.6 mJy. Given
that this value is 10 times less than the flux emitted by the bright variable
core even at its minimum, we ignored the contribution of this constant fainter
elongated structure to the mm flux.

Figure 4: ALMA image of IC 4329A (median band 3) 9/3/2021. Beam size is
0$\aas@@fstack{\prime\prime}$05 $\times$ 0$\aas@@fstack{\prime\prime}$04 (i.e.
$\sim$16 $\times$ 13 pc). The coordinates are given for the J2000 epoch.

###  3.3 X-ray – mm correlation

To quantitatively investigate a possible correlation and time delay between
the X-ray and mm light curves, we applied several approaches commonly used in
reverberation mapping campaigns (as, e.g., SDSS-RM project, see Shen et al.,
2023). Here and in the rest of the paper, positive time lags correspond to the
X-ray flux variability preceding the mm one. For the analysis, we used the
full X-ray and mm light curves without trend subtraction. One of the most
commonly used approaches to look for correlated variability is the
interpolated cross-correlation function (CCF), which involves the linear
interpolation of unevenly sampled light curves. Its modernised version PyMCCF
(Gaskell & Sparke, 1986; Oknyanskii, 1993; Peterson et al., 1998; Oknyansky &
Oknyansky, 2022) optimizes interpolation to reduce noise occurring from
interpolation errors minimising the number of interpolated points used. The
result of the PyMCCF analysis is shown as a grey curve in Fig. 5. The
calculated CCF shows two peaks corresponding to time delays of
$15.1^{+1.4}_{-0.7}$ and of $0.5^{+0.1}_{-0.1}$ days (vertical dashed lines).
The errors are estimated by calculating 84.13th (+1$\sigma$) and 15.87th
($-$1$\sigma$) percentiles of a dataset.

It has been shown that CCF-based methods can be less stable and efficient when
the observational cadence is irregular (Li et al., 2019). This is the reason
why tools such as, e.g., JAVELIN (Zu et al., 2016; Yu et al., 2020) have been
widely used. JAVELIN constructs a model of the flux variability utilizing the
damped random walk (DRW), extracting posterior distributions for critical DRW
parameters through MCMC sampling. Another iteration of MCMC allows JAVELIN to
determine the posterior time delay distribution between the light curves
comparing the constructed models. Applying the JAVELIN analysis to the IC
4329A data, we found two peaks in the posterior distribution of the time lag
(see the histogram of $10^{5}$ sampling iterations in Fig. 5), which are not
identical to those identified by PyMCCF. We found two possible cases: a
negative time lag $\tau_{1}=-11.5\pm$0.2 days and a positive time lag
$\tau_{2}=15.5\pm$0.9 days. The posterior distribution shows a larger and
narrower peak at $-11.5$ days; however, the mm light curve shifted to $-11.5$
days does not fully overlap the X-ray one, so, $\tau_{1}$ comes partially from
the modelled variability behaviour. Also, JAVELIN analysis generally provides
one with a constraint on the width of a top-hat function. This quantity could
be used to have an insight into the size of the structure responding to
variable emission. However, for our dataset, we found no peak in the
distribution of this parameter.

We also used the PyROA tool (Donnan et al., 2021) based on a similar approach
as JAVELIN, with the main difference being that in this case the variability
is described using a running optimal average. However, it appeared that for
the given data series the PyROA code provides unstable results, dependent on
the initial parameters of the model, and no clear estimation of a time lag can
be obtained.

Generalizing the results obtained by various methods, we obtain three possible
time delays: -11.5, 0.5, and 15.5. Note that 15 and 11 days are longer than
the duration of the ALMA campaign, and 0.5 days is less than the cadence of
the observations. Thus, the data is distorted by windowing and aliasing
effects. While we will continue to analyze the light curves considering all
three time lags, their reliability is questionable. Therefore, we will
prioritize focusing on $\tau=0$.

Figure 5: Results of the cross-correlation analysis of X-ray and mm data. The
posterior distribution of the time lag provided by JAVELIN is illustrated by
the red histogram. Two peaks at $\tau_{1}=-11.5$ days (for the mm flux
preceding the X-ray one) and $\tau_{2}=15.5$ days are marked with vertical
dashed lines. The result of PyMCCF is shown in grey with the 0.5-day time lag
marked with a vertical dashed line.  Figure 6: Variations of the mm/X-ray
ratio during our observational campaign. The X-ray light curves shifted to the
three time-lags: $\tau=0$, $-11.5$ and +15.5 days (see columns from left to
right) are shown in panel (1), and the mm light curve is in panel (2). The
ratio $\log(F_{\rm mm}/F_{\rm X-ray})$, where the X-ray light curve was
interpolated to the ALMA epochs, is given in panel (3) for each of the time-
lags; the black dash-dotted line is the median value, and the red solid line
corresponds to the relation between 100 GHz and 2–10 keV fluxes from (Ricci et
al., 2023) with 1$\sigma$ scatter given by the dashed red lines. The IC 4329A
epoch-to-epoch scatter ($\sigma$) is also presented in the figure: the scatter
relative to the median is shown in black, while the one relative to the
predicted value is in red. Panel (4) contains the same values as panel (3) but
as a function of the Eddington ratio $\lambda_{\rm Edd}$. The colour of the
circles illustrates the date of the observation.

##  4 Discussion

###  4.1 Evolution of mm/X-ray ratio & intrinsic scatter

Recently, for a volume-limited sample of radio-quiet hard X-ray selected AGN,
Ricci et al. (2023) found a tight correlation between the observed X-ray (2–10
keV) and mm (100 GHz) emission, described by

$\log F_{\rm mm}=(-0.8\pm 0.4)+(1.37\pm 0.04)\log F_{\rm X-ray},$ (2)

where the fluxes are given in units of erg s-1 cm-2, with a scatter of 0.22
dex. From this correlation, the typical ratio between the 100 GHz continuum
and the 2–10 keV emission is $\log(F_{\rm mm}/F_{\rm X-ray})=-4.63\pm 0.06$.
Also, a significant correlation between 230 GHz and 14–150 keV luminosities
for 98 AGN was found by Kawamuro et al. (2022) with a scatter of $\sim$0.36
dex. In both cases, non-simultaneous X-ray and mm observations were used, thus
the scatter observed in these correlations can be (at least partly) due to the
intrinsic variability of the AGN. To investigate intrinsic scatter in the
correlation, we studied the variations of the mm to X-ray ratio in our IC
4329A data given in the left column on panels 1 and 2 of Fig. 6. Note that
while the original X-ray and mm data were obtained with a minor time
difference of a few hours, sometimes the data points cannot be compared with
each other unambiguously. Therefore, we interpolated the X-ray data to match
the ALMA epochs. The evolution of $\log(F_{\rm mm}/F_{\rm X-ray})$ over time
during the simultaneous mm and X-ray monitoring is shown in Fig. 6 (panel 3,
left column). The scatter was estimated relative to two average values: the
first is the one expected from the Eq. (2) calculated for the mean X-ray flux
$\log(F_{\rm mm}/F_{\rm X-ray})=-4.5$, and the second is the time-averaged
value of the flux ratio was found to be $\log(F_{\rm mm}/F_{\rm X-ray})=-4.4$.
The average ratio found is in good agreement with the X-ray/100 GHz
correlation. We estimated the scatter to be 0.13 dex and 0.11 dex,
respectively. In both cases, the scatter is about two times smaller than one
from Ricci et al. (2023). This suggests that non-simultaneous observations can
introduce significant scatter in the mm/X-ray correlation due to the intrinsic
variability of AGN, as discussed in Kawamuro et al. (2022).

Since we have found a possible time delay between the mm and X-ray light
curves, in Fig. 6 we show the mm/X-ray ratio both for the zero time lag and
two more cases: $\tau=-11.5$ and 15.5 days (middle and right column,
respectively). In panel 1, the X-ray light curve is shifted according to the
different $\tau$, and the unshifted mm light curve is in panel 2. We found
that the scatter of the mm/X-ray ratio (panel 3) changes when we correct the
light curves for the possible time shift. For $\tau=+15.5$, the scatter
relative to the median ratio $\log(F_{\rm mm}/F_{\rm X-ray})=-4.3$ is the same
as for $\tau=0$, but twice larger, 0.24 dex, relative to the value calculated
with Eq. (2). For the $\tau=-11.5$ days, the scatter becomes very small
($\sigma=0.05$), and the median ratio coincides with that predicted by the
correlation. The values of scatter calculated as the standard deviation
relative to the mm/X-ray average and the value predicted from mm/X-ray
correlation from Ricci et al. (2023) are illustrated in Fig. 6 for each time
lag.

Together with the evolution of the mm/X-ray ratio in time, we also studied its
variation with the Eddington ratio ($\lambda_{\rm Edd}$). To calculate
$\lambda_{\rm Edd}$, we used the SMBH mass recently estimated for IC 4329A by
reverberation mapping $M_{\rm BH}=6.8^{+1.2}_{-1.1}\times 10^{7}M_{\odot}$
(Bentz et al., 2023), while the bolometric luminosity was calculated from the
X-ray flux in the 2–10 keV range. We considered a comoving distance $D=69.3$
Mpc (Koss et al., 2022). Recently, it has been shown that the 2–10 keV
bolometric correction ($\kappa_{2-10}$) is primarily a function of the
Eddington ratio (Gupta et al. 2024, in prep.), and it follows:

$\log(\kappa_{2-10})=C\times\log(\lambda_{\rm
Edd})^{2}+B\times\log(\lambda_{\rm Edd})+A,$

where $C=0.054\pm 0.034$, $B=0.309\pm 0.095$ and $A=1.538\pm 0.063$. Using
this approach we found that during joint ALMA/X-ray monitoring, the Eddington
ratio changes from $\log(\lambda_{\rm Edd})=-1.01$ to $-0.73$ with a median
value $\log(\lambda_{\rm Edd})=-0.89$ for $\tau=0$. The median bolometric
correction is $\kappa_{2-10}=20.2$. The $\kappa_{2-10}$ could be tentative due
to the large intrinsic dispersion, the obtained value of the bolometric
correction is very close to the commonly used value $\kappa=20$ from Vasudevan
& Fabian (2009). In panel 4 of Fig. 6 we illustrate the relation between the
mm/X-ray ratio and the Eddington ratio. The figure does not show a clear
correlation between the mm/X-ray ratio and $\lambda_{\rm Edd}$ for the three
different time lags.

It is important to note that while our calculated time delays show lower
scatters compared to previous studies, we have not observed a clear
correlation between X-ray and mm emission over the campaign duration, which is
not surprising given the short duration of ALMA observations and coarse
sampling. This suggests that unstable processes might dominate on the scale of
days, leading to varied flux behaviours. The discovered tentative
characteristic timescales selected by the periodogram and wavelet analysis
also do not shed light on this correlation: the observed mm flares are spaced
8 days apart, while the most probable X-ray oscillations are of the 5 days.
Despite the roughly constant flux ratio between mm and X-rays, it seems
different processes influence their variability on short timescales. With
longer ALMA observations, we could better understand this flux correlation.

###  4.2 Comparison with other mm/X-ray monitoring campaigns

Simultaneous mm and X-ray studies of the variability of RQ AGN are still
scarce, and only two such monitoring campaigns have been carried out before.
The first of these was dedicated to the RQ Seyfert 1 NGC 7469 located at the
same distance as IC 4329A ($z=0.016$) and having lower mass $M_{\rm
SMBH}\approx 9.1\times 10^{6}M_{\odot}$ (Peterson et al., 2014). This object
was known for a significant variability at 95 GHz, by a factor of two within
4–5 days as found in CARMA (Combined Array for Research in Millimeter-wave
Astronomy) observations by Baldi et al. (2015). Later, Behar et al. (2020)
provided the simultaneous mm/X-ray monitoring of NGC 7469. The observations
were taken in the 0.3–10 keV range with Swift/XRT and in the mm band at 95 and
143 GHz frequencies with the Institut de Radioastronomie Millimetrique (IRAM)
30-m single-dish radio telescope. This monitoring lasted longer than our
campaign ($\sim$70 days in X-rays, of which $\sim$50 days of contemporaneous
mm and X-ray observations) with a cadence of 1–2 days, but both the typical mm
uncertainties ($\sim$0.8 mJy) and beam sizes ($\sim$17–28′′) are considerably
larger than those of our ALMA observations. Yet Behar et al. (2020) stated a
marginal correspondence of the measured mm flux with earlier observations with
a smaller beam size $\sim$2.2′′ at CARMA interferometer (Baldi et al., 2015),
the mm/X-ray ratio observed for NGC 7469 is unexpectedly large: $\log(F_{\rm
95GHz}/F_{\rm 2-10keV})\approx-3.5$, which could be, at least in part,
explained by the low resolution of both IRAM and CARMA data. The scatter
associated with variability ($\sigma=0.12$) appears consistent with that found
for the current dataset from IC 4329A.

While Behar et al. (2020) provides a comprehensive analysis of the X-ray and
mm variability in NGC 7469, we are reanalyzing this data using the same
approaches applied to our IC 4329A data. This ensures a homogeneous analysis
of both datasets for a more accurate comparison. Using the wavelet analysis,
we do not find any evidence of a periodicity in the X-ray light curve of NGC
7469. Concerning the time delay between X-ray and mm emission in the
simultaneous NGC 7469 observations, Behar et al. (2020) cautiously stated that
the mm variability appears to precede the X-ray flux by $\sim$14 days. We have
repeated the correlation analysis of NGC 7469 data using the same tools as
described in Sect. 3.3. The PyROA and PyMCCF methods did not reveal any time
lag, while the analysis done with the JAVELIN code confirmed the $-14$ days
time lag. Moreover, we also discovered two more peaks in the posterior
distribution, corresponding to 4.1 and 13.8 days. The correlation peaks at
$-14$ and 13.8 days showed an amplitude approximately 7 times lower than the
peak at 4.1 days. Thus, as in the case of IC 4329A, we cannot clearly
determine the presence and magnitude of the time delay between mm and X-rays.
While in NGC 7469, this uncertainty could result from insufficient
observational accuracy, a similar outcome for IC 4329A indicates that in both
cases, the relation between the bands is intricate, and shorter timescales of
the variability need to be probed.

Behar et al. (2020) also examined the X-ray hardness ratios as a proxy of
possible changes in the level of photo-electric absorption, which could give
rise to the observed X-ray variability. However, neither the ”harder when
dimmer” trend, predicted by Behar et al. (2020) following Mehdipour et al.
(2017), nor a significant correlation between mm flux and hardness ratio was
observed in NGC 7469. Following Behar et al. (2020), we calculated the
hardness ratio in terms of count rates for the NICER data for IC 4329A: HR =
$(H-S)/(H+S)$, where $H$ is the count rate (count s-1) in the hard (2.0–10.0
keV) band, and $S$ is the count rate in the soft (0.3–2.0 keV) band. We found
that the HR does not vary significantly, and it does not show any correlation
with the mm flux or mm spectral index changes.

A second simultaneous mm/X-ray campaign of an RQ AGN was recently reported by
Petrucci et al. (2023) for MCG+08$-$11$-$11 ($z=0.02$), a Seyfert 1 galaxy
with a black hole mass comparable to that in IC 4329A, $M_{\rm SMBH}\approx
2.8\times 10^{7}M_{\odot}$. The variability in the 3–10 keV X-ray band
obtained by XMM-Newton and the 100 GHz flux inferred by NOEMA were studied on
time scales of 14 hours. Both fluxes showed slight increases, corresponding to
$\sim$1.06 times and $\sim$1.19 times for the mm and X-ray flux, respectively,
but no correlated variability was found. Similarly to NGC 7469 data, the
mm/X-ray ratio is significantly larger than the mean value found for nearby
AGN by Ricci et al. (2023): $\log(F_{\rm 100GHz}/F_{\rm 2-10keV})\approx-3.4$
with $\sigma=0.12$. Moreover, the mm/X-ray ratio previously measured in Behar
et al. (2018) with comparable angular resolution in mm also appeared an order
of magnitude larger than Ricci et al. (2023) predicted, yet the mm flux was
2.4 times lower ($\sim$7.5 mJy instead of $\sim$18.3 mJy). It is likely that
such a large mm/X-ray ratio is due to the relatively low resolution of the mm
observations ($\sim$1′′ according to the data archive), for which other
components emitting in mm may contribute to the measured flux. Therefore, the
observed mm variability might be more contaminated by non-variable extended
emission. This highlights the critical importance of utilizing ALMA, not only
for achieving higher accuracy of the flux measurement (twice better than NOEMA
data for MCG+08$-$11$-$11 and 50 times better than IRAM data for NGC 7469) but
also for isolating the flux coming from the compact structure in the core
thanks to the $<0$$\aas@@fstack{\prime\prime}$1 resolution.

Interestingly, both simultaneous monitoring campaigns presented in (Behar et
al., 2020) and (Petrucci et al., 2023), as well as our results for IC 4329A,
while having different measured mm/X-ray ratios, showed a similar scatter in
the variations of this ratio ($\sim$0.12 dex). In all cases, this scatter is a
factor $\sim$2 lower than that found by Ricci et al. (2023) for the non-
simultaneous mm and X-ray data. Thus, we can argue that half of the scatter in
Ricci et al. (2023) is due to the non-simultaneity of the mm and X-ray
observations, while the rest of the scatter could be intrinsic, and associated
with the different physical drivers of variability on the short (daily)
timescale, or due to the possible time lag between X-rays and mm. However,
unlike monitoring IC 4329A, a smaller scatter in the NGC 7469 and MCG
data+08$-$11$-$11 observations may also be related to the contribution from
extended non-variable mm radiation, which is why a direct comparison of the
scatters is not entirely correct. This once again shows the importance of
using high spatial resolution mm observations.

While no clear evidence of correlated mm/X-ray variability was found for NGC
7469 and MCG+08$-$11$-$11, both AGN showed significant variations in both
bands during the monitoring campaigns: $F_{\rm var}^{\rm X-ray}\approx 25$%
and $F_{\rm var}^{\rm mm}\approx 13$% for NGC 7469 (with the maximum amplitude
of a factor of 2 between days), and $F_{\rm var}^{\rm X-ray}\approx 7$% and
$F_{\rm var}^{\rm mm}\approx 2$%222Note here that fractional uncertainty of
the NOEMA observation is comparable, $\sim$2%. for MCG+08$-$11$-$11\. In both
cases, the X-ray fractional variability was larger than the one in the mm
band. However, our data for IC 4329A showed the opposite: $F_{\rm var}^{\rm
mm}$ is approximately twice as large as $F_{\rm var}^{\rm X-ray}$. Also, the
mm flux of IC 4329A showed a significant flare, during which the flux
increased by a factor $\sim$3 within 4–5 days, which was not observed before
in other RQ AGN. Due to the limited amount of data, it is still challenging to
define whether the observed variability of IC 4329A is a specific
characteristic of the source or is typical but not observed in the existing mm
observations of NGC 7469 and MCG+08$-$11$-$11 for some reasons. In any case,
it is evident that further high spatial resolution variability studies in the
mm band are necessary to identify, at the very least, the characteristics of
mm variability in RQ AGN.

###  4.3 Extended mm structure

As discussed in Sect. 3.2 and shown in Fig. 4, high spatial resolution ALMA
observations of IC 4329A revealed a distant structure besides the bright,
variable unresolved core in the observed mm region. Due to the faintness of
the extended mm structure (signal-to-noise ratio is $\sim$ 20), we cannot
examine its properties in detail, but our observations show that its flux is
constant at $\sim$0.6 mJy within the errors during the 10-day campaign. Since
the bright variable core and the faint extended structure are clearly resolved
in the ALMA maps, the linear distance between them projected onto the
celestial plane is of the order of 10 pc.

A non-AGN origin of the extended structure is not plausible. Given the edge-on
orientation of the host galaxy of IC 4329A, the observed emission may
originate in structures within the galaxy outside the AGN, like HII regions or
supernova remnants. Nonetheless, the mm observations from the literature
suggest that none of them can yield substantial flux. The anticipated flux
from HII regions at redshift $z=0.016$ is merely a few $\mu$Jy when scaling
data from (Sabbatini et al., 2005). Similarly, the expected flux from recent
supernovae is $<$ $\mu$Jy, as derived from the SN 1987A (Lakićević et al.,
2011) and Cas A (Wright et al., 1999; Loinard et al., 2003) data.

The extended structure is possibly associated with the radio jet of IC 4329A,
which is detected at lower frequencies (see Inoue & Doi, 2018). The VLA images
with a resolution of 1$\aas@@fstack{\prime\prime}$5 $\times$
0$\aas@@fstack{\prime\prime}$9 at 1.5 GHz and 1$\aas@@fstack{\prime\prime}$3
$\times$ 1$\aas@@fstack{\prime\prime}$2 at 4.9 GHz from Unger et al. (1987)
revealed a bright radio flux. However, the jet extends $\sim$ 6′′ to the west
– $180^{\circ}$ opposite to our current mm band observations. Moreover,
according to Inoue & Doi (2018), the expected mm emission from the jet is
considerably higher. Extrapolating from the radio data yields a flux density
of $\sim$3.3 mJy at 100 GHz, which is five times higher than the observed
emission.

It turned out that the observed emission in the extended mm structure closely
matches the expected emission from the photoionized gas in the ionisation cone
of the narrow-line region. In this region, mm emission is anticipated as a
result of free–free emission (see Panessa et al., 2019; Baskin & Laor, 2021).
Previous studies on IC 4329A have identified features of the ionisation cone,
which align with the extended low-frequency radio jet observed in VLA data
(Colbert et al., 1996; Thomas et al., 2017). Following Baskin & Laor (2021),
we estimate the mm flux using the relation $\log\nu L_{\rm 100GHz}=\log
L_{\rm[OIII]}-3.36$, where the [OIII] flux $F_{\rm[OIII]}=2.34\cdot 10^{-13}$
and the factor 9.5 intrinsic reddening factor are from Bentz et al. (2023). We
find the apparent flux $F_{\rm 100GHz}\approx 1$ mJy. To provide a conclusive
assessment, checking the spectral slope is essential. Unfortunately, the
structure appeared too faint to evaluate its spectral parameters, emphasizing
the need for deeper mm observations.

###  4.4 mm variability and its origin

One of the key findings of the monitoring is the remarkable variability
exhibited by the compact mm source in IC 4329A. Previous studies lacked mm
monitoring data for IC 4329A, with only limited ALMA observations. Imanishi et
al. (2016) reported a continuum flux of 13.0 mJy at 260 GHz with a $1\times
0$$\aas@@fstack{\prime\prime}$5 beam in April 2014. Subsequent observations in
October 2016 published by Inoue & Doi (2018) across the 90.5–231 GHz range
revealed a frequency-dependent flux, ranging from 8 to $\sim$5 mJy, with
spatial resolution between 0.45 and 0.14′′. Thus, during 10 days of our ALMA
monitoring, we captured the full historical range of flux variations, with a
flux change of 2.6 within just 4–5 days.

The observed mm changes are quite surprising. Traditionally, the largest
amplitude of variability in non-blazar AGN on timescales less than a month has
been observed in the X-ray range, as demonstrated by numerous multiwavelength
variability campaigns (e.g. Edelson et al., 2000, 2015). However, during our
campaign, IC 4329A exhibited more variability in the mm band than in X-rays,
as discussed in our comparison of $F_{\rm var}$ in Sect. 4.2. Based on the
factor $\sim$3 variability, we can rule out two of the most stable mechanisms
of the mm origin – thermal dust and free–free emission.

The dust contribution in IC 4329A is constrained to be at least two orders of
magnitude lower than the observed flux at 100 GHz (Fig. 6), less than 1.6
$\times\ 10^{-2}$ mJy (Inoue & Doi, 2018). What is more important, the
observed mm spectral slope $\sim-0.2$ is inconsistent with the dust scenario,
where a spectral index of $\sim$3.5 would be expected (e.g. Mullaney et al.,
2011). The estimated brightness temperature $T_{\rm b}$ from the mm flux
density following Eq. (1.34) from Wilson et al. (2013): $T_{\rm b}\approx
200\pm 100$ K is also higher than predicted from the SED fitting by Mehdipour
& Costantini (2018).

In some AGN such as NGC 1068, a significant fraction of mm emission is
described by the free–free emission coming from the winds or X-ray-heated disk
gas (Gallimore et al., 2004; Inoue et al., 2020). While the observed spectral
slope in IC 4329A is consistent with that expected for free–free processes,
the flux density and its rapid variations exclude this mechanism. Laor & Behar
(2008) demonstrated that the contribution of thermal free–free emission from
hot X-ray plasma in the corona is negligible. The free–free emission can
originate from the dusty torus in AGN. The torus size measured in IC 4329A is
of the order of a few hundred light days (ld; 0.15–0.19 pc Kishimoto et al.,
2011; Gravity Collaboration et al., 2023), and as Baskin & Laor (2021)
suggested, the expected variations of the free–free emission originating there
should occur on a yearly timescale rather than within days.

Therefore, the preferable mm emission mechanism is non-thermal synchrotron
produced in a compact region: since the flux changed by more than 3$\sigma$
from one day to another, we can assume the mm source size is smaller than 1 ld
($\sim$0.0008pc). The observed nearly flat spectrum with a slope of $-0.23$
suggests that the observed frequency is near the turnover between optically
thick ($\alpha=2.5$) and optically thin ($\alpha<-0.5$) regime. Further, let
us discuss the estimation of the synchrotron source size and the emission
origin.

###  4.5 Synchrotron emission

Figure 7: The dependence of the mm flux density at 100 GHz on the source size
and magnetic field strength for $z=0.016$. A dashed horizontal line
illustrates the region of source sizes $\geq$1 ld. A dashed vertical line
delineates the range where the magnetic field $\leq B_{\text{lim}}$. The
region between the refined values of $R$ and $B$ for the optically thick
synchrotron regime, and the observed flux of IC 4329A is highlighted in
yellow.

In Sect. 4.4, we estimated the mm emitter size as $\sim$1 ld. This estimation
aligns perfectly with the optically thick synchrotron source size derived
under the assumption of equipartition between magnetic energy density and
photon energy density, as introduced by Laor & Behar (2008):

$R=5.6\times 10^{2}L_{\rm 100GHz}^{0.4}L_{\rm bol}^{0.1}\nu^{-1}\ \ {\rm ld},$
(3)

where $L_{\rm 100GHz}$ is the mm luminosity in units of $10^{30}$ erg s-1
Hz-1, $L_{\rm bol}$ is bolometric luminosity in units of $10^{30}$ erg s-1,
and $\nu$ is the observed frequency in GHz. Using the bolometric luminosity
calculated in Sect. 4.1 we obtained the source size $1.0\pm 0.1$ ld. In the
more common case, the physical size of a self-absorbed synchrotron source $R$
is the function of the radio flux density and the magnetic field strength $B$
(Laor & Behar, 2008; Petrucci et al., 2023):

$R\simeq 3.05\times 10^{3}F_{\rm mm}^{0.5}\nu^{-1.25}B^{0.25}z\ \ {\rm ld},$
(4)

where $F_{\rm mm}$ is in mJy, and $z$ is the source redshift. In Fig. 7, we
present this relation as the $R$–$B$ diagram. The mm flux of IC 4329A varied
within the range of $\sim$3–10 mJy. Assuming the emission region size does not
exceed 1 ld, we estimate the constraint on the permissible maximum magnetic
field strength of 200 G for the minimum flux. The minimum magnetic field value
can be estimated following Laor & Behar (2008), assuming that synchrotron
cooling dominates other cooling processes. Thus,

$B>6.8\cdot 10^{4}t^{-2/3}_{\text{var}}\nu^{-1/3}\ \ {\rm G},$ (5)

where $t_{\text{var}}$ day is the variability timescale in seconds. We
obtained the limiting value $B_{\rm lim}=7.5$ G, leading to $R>0.4$ ld for the
minimum flux. These constraints significantly limit the permissible parameter
space of the mm emitter. In particular, if the source flux is 10 mJy, its size
must lie within 0.8–1 ld, and the magnetic field must range between 7.5–18 G.
We caution that there is additional uncertainty to these numbers as the source
is not deep in the optically thick regime at 100 GHz (as the observed spectral
slope is $-0.23$), but this should still provide a valid zero-order estimate
of the emitter size and magnetic field.

(a) MJD=59461.41 (b) MJD=59464.32

Figure 8: Multifrequency SED fitting of IC 4329A during the flux minimum and
maximum. The black points represent the high-resolution ALMA data (this work),
whereas the red arrows are the fluxes from low-resolution radio and infrared
observations (Inoue & Doi, 2018, and references therein), which in the context
of the fitting procedure represent strict upper limits to the total flux
coming from the nuclear region probed by ALMA. The different emission
components in the model and their 1$\sigma$ confidence intervals are marked,
together with the total emission. The coronal component dominates the 100 GHz
emission, and the diffuse synchrotron and dust emission components are poorly
constrained. The bottom subpanels show the fitting residuals.

Further, let us discuss several possible synchrotron mechanisms of mm emission
origin following Kawamuro et al. (2022). There we address a more detailed
modelling that does not require the assumption of an optically thick spectrum.

####  4.5.1 Synchrotron emission from outflow-driven shocks

One of the possible explanations for the tight mm/X-ray correlation observed
in RQ AGN is synchrotron emission from electrons accelerated in the shock
produced by an AGN outflow colliding with the surrounding interstellar medium.
Kawamuro et al. (2022) showed that the expected conversion of the kinetic
energy of the outflow into synchrotron emission of the relativistic particles
(Nims et al., 2015) is consistent with the results found, specifically in the
case of AGN with ultrafast X-ray outflows.

The kinetic energy of the outflow is derived from the bolometric luminosity of
the AGN, and it is anticipated that more luminous AGN would expel more
energetic outflows (see Kawamuro et al., 2022, for references). Thus, a
correlation between the mm/X-ray ratio and the Eddington ratio is expected.
Kawamuro et al. (2022) reported the absence of such a correlation for the AGN
sample. In IC 4329A, where the ultra-fast outflow was found (Tombesi et al.,
2012; Tortosa et al., 2023), we also did not find any relation between the
mm/X-ray ratio and $\lambda_{\rm Edd}$ for either $\tau=0$ or $-11.5$ days.
Nevertheless, in the case of $\tau=+15.5$ days, a faint correlation can be
assumed with the Pearson coefficient of $\sim$0.65 (see Fig. 6). However,
$+15.5$ days time lag seems inconsistent, as Tombesi et al. (2012) and Tortosa
et al. (2023) discovered the outflow location in the range of 1.5–12 ld from
the core. Moreover, 1.5–12 ld scales of the outflow exceed the expected size
of the mm source and, as we show further, should produce optically thin
emission at 100 GHz (see details in Sect. 4.5.3).

####  4.5.2 Synchrotron emission from jet

Kawamuro et al. (2022) investigated the potential contribution of the jet in
the mm emission and found no discernible difference between type 1 and type 2
AGN, suggesting that the jet does not dominate the mm band. However, in IC
4329A, the jet still appears to contribute. As previously noted, extrapolating
data from Inoue & Doi (2018) yielded a mm flux from the jet at 100 GHz of
$\sim$3.3 mJy. Interestingly, the source flux has never been observed below
this value, implying a significant contribution from the jet in the flux
minimum.

However, the jet is unlikely to contribute to the mm variability
significantly. The main argument here is that the observed radio jet in IC
4329A exhibits a spectral slope of approximately $-0.59\pm 0.09$ (in the
1.4–43.3 GHz range Inoue & Doi, 2018). The spectral slope of the mm emission
is much flatter and never exhibits a spectral index as negative as the jet
component. Furthermore, there is no indication that the spectral slope becomes
more negative as the flux increases (see Fig. 3). Additionally, the daily-
scale variability seems inconsistent with the activity of the radio jet with
the typical scale exceeding several kpc. However, it is worth noting that
rapid variability may originate from a compact jet even in non-blazar AGN
(e.g. Smith & Sartori, 2023), thus this requires additional studies. These
findings suggest that the contribution of the jet to the mm emission is
limited and possibly exhibits slower changes compared to the observed mm
variability.

At present, the jet scenario is not favoured. However, determining the exact
contribution of the jet to the mm flux and whether this contribution is
variable remains an intriguing open question, necessitating simultaneous
observations at lower radio frequencies and in the mm band.

####  4.5.3 Synchrotron emission from X-ray corona

As shown by Behar et al. (2018), the mm excess observed in many RQ AGN cannot
be extrapolated from the radio data. In Sect. 4.5.2, we demonstrated that, in
the case of IC 4329A, the minimum of the flux aligns well with the radio
observations. However, this cannot fully account for the observed variability
and the mm excess in IC 4329A during its maximum.

Inoue & Doi (2014) proposed a scenario in which non-thermal relativistic
electrons produce mm emission in the magnetised X-ray corona. The model was
further investigated by Inoue & Doi (2018), predicting that synchrotron
emission peaks in the mm band, at $\sim$100–300 GHz, due to synchrotron self-
absorption (SSA). The synchrotron self-absorption scenario was also preferred
by Kawamuro et al. (2022, see their Fig. 25). For IC 4329A, we found a flatter
spectral slope than the average of the sample in (Kawamuro et al., 2022)
($\alpha_{\rm Kawamuro}^{\rm ave}=-0.5\pm 1.2$), which could be explained by
observations conducted at 100 GHz, closer to the SSA peak, and not at 230 GHz.

In this scenario, the synchrotron emissivity is a function of the electron
spectrum and the magnetic field strength and size of the corona. Thus, to
produce the observed rapid variability on timescales of days, fast changes in
the physical properties of the corona are required. To explore this more
carefully, we introduce a corona emission model based on Inoue & Doi (2018)
(see details in del Palacio et al. 2024, in prep.). This model has many
parameters to characterise the coronal properties and the non-thermal electron
distribution in it. Specifically, the parameters include the temperature,
electron density, and size of the corona, the fraction of energy in the
magnetic field and non-thermal electrons, and the spectral index of the
relativistic electron energy distribution. Most of these parameters affect the
SED in a similar way, increasing both the peak frequency and the peak flux
(due to increasing SSA opacity and synchrotron emissivity). The only parameter
that increases the flux while decreasing the peak frequency is the size of the
corona. As usually done, we parameterize this as
$R_{\mathrm{c}}=r_{\mathrm{c}}R_{\mathrm{Sch}}$, where
$R_{\mathrm{Sch}}\propto M_{\mathrm{BH}}$ is the Schwarzschild radius. Thus,
if the flux varies and the spectral index remains flat, the size must change.
Otherwise, the emission would get a more positive spectral index when it is
brighter, which is not observed (Fig. 3).

Not all parameters allow the reproduction of the observed flux density and
flat spectral index. For example, fixing $r_{\mathrm{c}}=40$ for the brightest
epoch and varying the remaining parameters, the model fails to reproduce the
spectral index and requires unphysical parameters (100 times more energy in
non-thermal electrons than in thermal ones). This is because the only factor
that can reduce the high SSA opacity without leading to a low flux density is
the corona size, with a larger size yielding a more diluted electron
population in the corona. For this reason, we allow the parameters
$r_{\mathrm{c}}$ and $\delta=U_{\mathrm{nt,e}}/U_{\mathrm{th,e}}$ (fraction of
the energy in non-thermal electrons w.r.t. thermal electrons) to vary, while
we fix the remaining parameters of the corona, $kT=200$ keV,
$\tau_{\mathrm{T}}=0.43$,
$\epsilon_{B}=U_{\mathrm{B}}/U_{\mathrm{th,e}}=10^{-2}$ (fraction of energy in
the magnetic field), and $p=2.1$ (relativistic electron distribution); we also
fix the dust parameters $\beta=1.6$ and $\nu_{\tau=1}=800$ GHz, and the
spectral index of the diffuse synchrotron emission to $\alpha=-0.6$. We note
that the diffuse components are very poorly constrained due to the lack of
observations with comparable resolution at other frequencies. Given that low-
resolution observations probe larger volumes in the galaxy and thus capture
more emission, the fluxes from archival low-resolution observations (Inoue &
Doi, 2018, and references therein) represent strict upper limits to the flux
coming from the inner region probed by the high-resolution ALMA observations.
In Fig. 8, we present the SED fitting of IC 4329A during two days that show
the highest variability. We obtain that the corona expanded from $r_{c}\approx
116\pm 54$ ($R_{c}\approx 0.91$ ld) during the minimum (MJD 59461.4) to
$r_{c}\approx 166\pm 25$ ($R_{c}\approx 1.3$ ld) during the maximum (MJD
59464.3) with the magnetic field strength varying from 6.6 to 5.6 G and the
fraction of non-thermal electrons increasing from $\log\delta=-3.18\pm 0.3$ to
$\log\delta=-2.87\pm 0.1$. The change in the size of the corona of 0.4 ld in 3
days would require a very fast expansion velocity of $\sim 0.1\,c$, which
could be consistent with the outflowing corona scenario (e.g. Kylafis et al.,
2023).

In summary, our preferred scenario attributes the mm variability and tight
X-ray/mm ratio to a compact X-ray corona. However, in this scenario, as well
as in the jet and outflow scenarios, no time lag is expected. The 11.5 and
15.5 days time lags pose challenges, as we struggle to envision how the source
could accumulate and later reradiate energy in mm or X-rays within such time
frames. Treating these time lags as light travel times is not a satisfactory
explanation, as it would require the sources to be separated by a distance
approximately equivalent to the broad-line region (BLR), which is about 16 ld
(Bentz et al., 2023). Within our current understanding of the mm source, a
zero time lag ($\tau=0$) seems to be more plausible.

Additionally, none of the discussed mechanisms accounts for the 5-day
timescale variations observed in X-rays and their absence in mm. The distinct
variability patterns over several days hint at the potential for common X-ray
and mm variability on longer timescales (weeks), while the daily timescale
variations suggest differences in short-term processes between X-rays and mm,
possibly involving distinct electron populations. Furthermore, investigating
X-ray variability showing local periodic changes independently holds interest,
given that X-rays are generally known for stochastic variability. However,
comparing X-ray periods with known timescales in IC 4329A fails to provide
insights into their physical origin.

##  5 Conclusions

We observed the RQ unobscured type 1 AGN IC 4329A to investigate the origin
and properties of the compact mm emission in RQ AGN. The source was observed
daily for 40 days in the 2–10 keV X-ray band using XMM-Newton, Swift, and
NICER. Within this campaign, we contemporaneously obtained 10 epochs of high-
resolution mm observations with daily cadence using ALMA at $\sim$100 GHz
frequency. Unlike previous contemporaneous mm/X-ray campaigns (Behar et al.,
2020; Petrucci et al., 2023), the outstanding capabilities of ALMA enable us
to isolate the compact mm emission from the potential contributions of other
components of the active galaxy and to observe the surprising behaviour of the
mm emission in the RQ AGN.

The recent results of the tight mm/X-ray correlation in RQ AGN by Ricci et al.
(2023) and Kawamuro et al. (2022) suggested the common origin of the mm and
X-ray emission most likely in the X-ray corona. Thus, we expected to discover
a correlated behaviour of both fluxes on a daily timescale. However, we did
not find statistically robust evidence of the correlated variability, and
distinct variability patterns were observed in each band.

  * •

The observed ratio between X-ray (2–10 keV) and mm (100 GHz) aligns with the
tight correlation reported in Kawamuro et al. (2022); Ricci et al. (2023).
Moreover, our simultaneous observations indicate a reduced scatter in this
ratio, measuring $\sim$0.1 dex instead of the 0.22 dex (Ricci et al., 2023).
This suggests that at least half of the previously observed scatter is due to
the non-simultaneity of the observations, while the rest may arise from
intrinsic processes in the AGN.

  * •

During the 10-day campaign, the compact mm emission exhibited significant
variability by a factor of three within 4–5 days, with an amplitude
surprisingly larger than in X-rays. This rules out the heated dust and thermal
free–free emission origin, indicating a synchrotron origin for the mm
radiation. The in-band spectral slope $-0.23\pm 0.18$ suggests that the
observed emission is close to the SSA turnover frequency. The SED fitting
modelling suggests a source size of $\sim$1 ld ($\sim$120 gravitational
radii), consistent with size estimations from the variability timescale, and a
magnetic field strength of $\sim$5–7 G. The observed variability and spectral
indices require the emitter size to change in time.

  * •

The combined analysis of the 40-day X-ray (2–10 keV) and 10-day mm (100 GHz)
light curves suggests no reliable time lag. Additionally, the tentative
oscillations on the 5-day timescale observed in X-rays at the 95% confidence
level are not evident in the mm variability.

  * •

The high resolution of ALMA (up to $\sim$0$\aas@@fstack{\prime\prime}$05)
allowed us to resolve the variable bright core together with the faint and
non-variable mm structure 10 pc away from the nucleus. The measured flux and
the position of the non-variable component indicate that its emission likely
originates in the ionization cone of the narrow-line region.

The evidence (flux density, flat/negative spectral index, daily variability)
strongly suggests that the observed mm radiation has a non-thermal synchrotron
nature and originates from a compact source. This is best consistent with the
idea that, as suggested by Inoue & Doi (2014) and mm/X-ray correlations, the
mm emission is produced in a compact X-ray corona. Future simultaneous
radio/mm/X-ray observations, possibly covering more ALMA bands and longer
timescales, as well as mm polarization observations, will help to elucidate
the physical processes in the compact mm emitting region observed in RQ AGNs.

###### Acknowledgements.

We are grateful to the ALMA, NICER and XMM-Newton teams for their invaluable
help in coordinating the joint mm/ALMA observations. We sincerely thank Ari
Laor for the comments and fruitful discussion. ES acknowledges ANID BASAL
project FB210003 and Gemini ANID ASTRO21-0003. CR acknowledges support from
Fondecyt Regular grant 1230345 and ANID BASAL project FB210003. SdP and SA
gratefully acknowledge funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (grant
agreement No 789410, PI: S. Aalto). We gratefully acknowledge funding for this
work by ANID through the Millennium Science Initiative Program - ICN12_009
(FEB), CATA-BASAL - FB210003 (FEB), and FONDECYT Regular - 1200495 (FEB). LCH
was supported by the National Science Foundation of China (11991052,
12011540375, 12233001), the National Key R&D Program of China
(2022YFF0503401), and the China Manned Space Project (CMS-CSST-2021-A04, CMS-
CSST-2021-A06). The material is based upon work supported by NASA under award
number 80GSFC21M0002 (ML). FT acknowledges funding from the European Union -
Next Generation EU, PRIN/MUR 2022 (2022K9N5B4). The National Radio Astronomy
Observatory is a facility of the National Science Foundation operated under
cooperative agreement by Associated Universities, Inc.

## References

  * Alhosani et al. (2022) Alhosani, A., Gelfand, J. D., Zaw, I., et al. 2022, ApJ, 936, 73
  * Arzoumanian et al. (2014) Arzoumanian, Z., Gendreau, K. C., Baker, C. L., et al. 2014, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9144, Space Telescopes and Instrumentation 2014: Ultraviolet to Gamma Ray, ed. T. Takahashi, J.-W. A. den Herder, & M. Bautz, 914420
  * Baldi et al. (2015) Baldi, R. D., Behar, E., Laor, A., & Horesh, A. 2015, MNRAS, 454, 4277
  * Baskin & Laor (2021) Baskin, A. & Laor, A. 2021, MNRAS, 508, 680
  * Behar et al. (2015) Behar, E., Baldi, R. D., Laor, A., et al. 2015, MNRAS, 451, 517
  * Behar et al. (2020) Behar, E., Kaspi, S., Paubert, G., et al. 2020, MNRAS, 491, 3523
  * Behar et al. (2018) Behar, E., Vogel, S., Baldi, R. D., Smith, K. L., & Mushotzky, R. F. 2018, MNRAS, 478, 399
  * Bentz et al. (2023) Bentz, M. C., Onken, C. A., Street, R., & Valluri, M. 2023, ApJ, 944, 29
  * Burrows et al. (2005) Burrows, D. N., Hill, J. E., Nousek, J. A., et al. 2005, Space Sci. Rev., 120, 165
  * Chen et al. (2023) Chen, S., Kharb, P., Sasikumar, S., et al. 2023, arXiv e-prints, arXiv:2312.13351
  * Colbert et al. (1996) Colbert, E. J. M., Baum, S. A., Gallimore, J. F., et al. 1996, ApJS, 105, 75
  * Donnan et al. (2021) Donnan, F. R., Horne, K., & Hernández Santisteban, J. V. 2021, MNRAS, 508, 5449
  * Edelson et al. (2015) Edelson, R., Gelbord, J. M., Horne, K., et al. 2015, ApJ, 806, 129
  * Edelson et al. (2000) Edelson, R., Koratkar, A., Nandra, K., et al. 2000, ApJ, 534, 180
  * Evans et al. (2009) Evans, P. A., Beardmore, A. P., Page, K. L., et al. 2009, MNRAS, 397, 1177
  * Gabriel et al. (2004) Gabriel, C., Denby, M., Fyfe, D. J., et al. 2004, in Astronomical Society of the Pacific Conference Series, Vol. 314, Astronomical Data Analysis Software and Systems (ADASS) XIII, ed. F. Ochsenbein, M. G. Allen, & D. Egret, 759
  * Gallimore et al. (2004) Gallimore, J. F., Baum, S. A., & O’Dea, C. P. 2004, ApJ, 613, 794
  * Gaskell & Sparke (1986) Gaskell, C. M. & Sparke, L. S. 1986, ApJ, 305, 175
  * Gehrels et al. (2004) Gehrels, N., Chincarini, G., Giommi, P., et al. 2004, ApJ, 611, 1005
  * Gendreau et al. (2016) Gendreau, K. C., Arzoumanian, Z., Adkins, P. W., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9905, Space Telescopes and Instrumentation 2016: Ultraviolet to Gamma Ray, ed. J.-W. A. den Herder, T. Takahashi, & M. Bautz, 99051H
  * Gendreau et al. (2012) Gendreau, K. C., Arzoumanian, Z., & Okajima, T. 2012, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 8443, Space Telescopes and Instrumentation 2012: Ultraviolet to Gamma Ray, ed. T. Takahashi, S. S. Murray, & J.-W. A. den Herder, 844313
  * Gravity Collaboration et al. (2023) Gravity Collaboration, Amorim, A., Bourdarot, G., et al. 2023, A&A, 669, A14
  * Guedel & Benz (1993) Guedel, M. & Benz, A. O. 1993, ApJ, 405, L63
  * Hunter et al. (2023) Hunter, T. R., Indebetouw, R., Brogan, C. L., et al. 2023, PASP, 135, 074501
  * Imanishi et al. (2016) Imanishi, M., Nakanishi, K., & Izumi, T. 2016, AJ, 152, 218
  * Inoue & Doi (2014) Inoue, Y. & Doi, A. 2014, PASJ, 66, L8
  * Inoue & Doi (2018) Inoue, Y. & Doi, A. 2018, ApJ, 869, 114
  * Inoue et al. (2020) Inoue, Y., Khangulyan, D., & Doi, A. 2020, ApJ, 891, L33
  * Jansen et al. (2001) Jansen, F., Lumb, D., Altieri, B., et al. 2001, A&A, 365, L1
  * Kawamuro et al. (2022) Kawamuro, T., Ricci, C., Imanishi, M., et al. 2022, ApJ, 938, 87
  * Kawamuro et al. (2023) Kawamuro, T., Ricci, C., Mushotzky, R. F., et al. 2023, ApJS, 269, 24
  * Kishimoto et al. (2011) Kishimoto, M., Hönig, S. F., Antonucci, R., et al. 2011, A&A, 527, A121
  * Koss et al. (2022) Koss, M. J., Ricci, C., Trakhtenbrot, B., et al. 2022, ApJS, 261, 2
  * Kylafis et al. (2023) Kylafis, N. D., Reig, P., & Tsouros, A. 2023, A&A, 679, A81
  * Lakićević et al. (2011) Lakićević, M., van Loon, J. T., Patat, F., Staveley-Smith, L., & Zanardo, G. 2011, A&A, 532, L8
  * Laor & Behar (2008) Laor, A. & Behar, E. 2008, MNRAS, 390, 847
  * Li et al. (2019) Li, I-Hsiu, J., Shen, Y., Brandt, W. N., et al. 2019, ApJ, 884, 119
  * Loinard et al. (2003) Loinard, L., Lequeux, J., Tilanus, R. P. T., & Lagage, P. O. 2003, in Revista Mexicana de Astronomia y Astrofisica Conference Series, Vol. 15, Revista Mexicana de Astronomia y Astrofisica Conference Series, ed. J. Arthur & W. J. Henney, 267–269
  * Lomb (1976) Lomb, N. R. 1976, Ap&SS, 39, 447
  * Mehdipour & Costantini (2018) Mehdipour, M. & Costantini, E. 2018, A&A, 619, A20
  * Mehdipour et al. (2017) Mehdipour, M., Kaastra, J. S., Kriss, G. A., et al. 2017, A&A, 607, A28
  * Merloni & Fabian (2001a) Merloni, A. & Fabian, A. C. 2001a, MNRAS, 321, 549
  * Merloni & Fabian (2001b) Merloni, A. & Fabian, A. C. 2001b, MNRAS, 328, 958
  * Morlet (1983) Morlet, J. 1983, Sampling Theory and Wave Propagation (Springer Berlin Heidelberg), 233–261
  * Mullaney et al. (2011) Mullaney, J. R., Alexander, D. M., Goulding, A. D., & Hickox, R. C. 2011, MNRAS, 414, 1082
  * Nims et al. (2015) Nims, J., Quataert, E., & Faucher-Giguère, C.-A. 2015, MNRAS, 447, 3612
  * Oknyanskii (1993) Oknyanskii, V. L. 1993, Astronomy Letters, 19, 416
  * Oknyansky & Oknyansky (2022) Oknyansky, V. & Oknyansky, R. 2022, PyMCCF: Python Modernized Cross Correlation Function for reverberation mapping studies, Astrophysics Source Code Library, record ascl:2212.007
  * Panessa et al. (2019) Panessa, F., Baldi, R. D., Laor, A., et al. 2019, Nature Astronomy, 3, 387
  * Peterson et al. (2014) Peterson, B. M., Grier, C. J., Horne, K., et al. 2014, ApJ, 795, 149
  * Peterson et al. (1998) Peterson, B. M., Wanders, I., Horne, K., et al. 1998, PASP, 110, 660
  * Petrucci et al. (2023) Petrucci, P. O., Piétu, V., Behar, E., et al. 2023, A&A, 678, L4
  * Piconcelli et al. (2004) Piconcelli, E., Jimenez-Bailón, E., Guainazzi, M., et al. 2004, MNRAS, 351, 161
  * Raginski & Laor (2016) Raginski, I. & Laor, A. 2016, MNRAS, 459, 2082
  * Ricci et al. (2023) Ricci, C., Chang, C.-S., Kawamuro, T., et al. 2023, ApJ, 952, L28
  * Ricci et al. (2021) Ricci, C., Loewenstein, M., Kara, E., et al. 2021, ApJS, 255, 7
  * Ricci et al. (2017) Ricci, C., Trakhtenbrot, B., Koss, M. J., et al. 2017, ApJS, 233, 17
  * Sabbatini et al. (2005) Sabbatini, L., Cavaliere, F., dall’Oglio, G., et al. 2005, A&A, 439, 595
  * Scargle (1982) Scargle, J. D. 1982, ApJ, 263, 835
  * Shen et al. (2023) Shen, Y., Grier, C. J., Horne, K., et al. 2023, arXiv e-prints, arXiv:2305.01014
  * Smith & Sartori (2023) Smith, K. L. & Sartori, L. F. 2023, ApJ, 958, 188
  * Thomas et al. (2017) Thomas, A. D., Dopita, M. A., Shastri, P., et al. 2017, ApJS, 232, 11
  * Tombesi et al. (2012) Tombesi, F., Cappi, M., Reeves, J. N., & Braito, V. 2012, MNRAS, 422, L1
  * Torrence & Compo (1998) Torrence, C. & Compo, G. P. 1998, Bulletin of the American Meteorological Society, 79, 61
  * Tortosa et al. (2023) Tortosa, A., Ricci, C., Shablovinskaia, E., et al. 2023, arXiv e-prints, arXiv:2312.00783
  * Unger et al. (1987) Unger, S. W., Lawrence, A., Wilson, A. S., Elvis, M., & Wright, A. E. 1987, MNRAS, 228, 521
  * Vasudevan & Fabian (2009) Vasudevan, R. V. & Fabian, A. C. 2009, MNRAS, 392, 1124
  * Vaughan et al. (2003) Vaughan, S., Edelson, R., Warwick, R. S., & Uttley, P. 2003, MNRAS, 345, 1271
  * Wang et al. (2023) Wang, A., An, T., Zhang, Y., et al. 2023, MNRAS, 525, 6064
  * Wilson et al. (2013) Wilson, T. L., Rohlfs, K., & Hüttemeister, S. 2013, Tools of Radio Astronomy
  * Wright et al. (1999) Wright, M., Dickel, J., Koralesky, B., & Rudnick, L. 1999, ApJ, 518, 284
  * Yu et al. (2020) Yu, Z., Kochanek, C. S., Peterson, B. M., et al. 2020, MNRAS, 491, 6045
  * Zu et al. (2016) Zu, Y., Kochanek, C. S., Kozłowski, S., & Peterson, B. M. 2016, ApJ, 819, 122

##  Appendix A Data

Table 1: ALMA data. Start Time (UTC) | End Time (UTC) | beam size (arcsec) |  $F$ (mJy) |  $\sigma$ (mJy/beam)
---|---|---|---|---
8/28/2021 19:55 | 8/28/2021 20:27 | 0.0666 $\times$ 0.0576 | 3.61 | 0.0238
8/29/2021 16:08 | 8/29/2021 16:40 | 0.0916 $\times$ 0.0576 | 3.87 | 0.0229
8/30/2021 21:23 | 8/30/2021 21:54 | 0.0674 $\times$ 0.0627 | 5.10 | 0.0280
8/31/2021 16:09 | 8/31/2021 16:39 | 0.0811 $\times$ 0.0552 | 4.30 | 0.0266
9/01/2021 20:05 | 9/01/2021 20:36 | 0.0604 $\times$ 0.0504 | 3.51 | 0.0249
9/02/2021 17:04 | 9/02/2021 17:34 | 0.0539 $\times$ 0.0452 | 3.59 | 0.0244
9/03/2021 21:18 | 9/03/2021 21:48 | 0.0513 $\times$ 0.0441 | 4.00 | 0.0217
9/04/2021 16:22 | 9/04/2021 16:52 | 0.0546 $\times$ 0.0420 | 4.45 | 0.0229
9/05/2021 19:57 | 9/05/2021 20:28 | 0.0457 $\times$ 0.0438 | 6.38 | 0.0216
9/06/2021 19:14 | 9/06/2021 19:46 | 0.0649 $\times$ 0.0422 | 9.23 | 0.0277
Table 2: X-ray data in 2–10 keV range (except NICER epoch marked with the asterisk, where the data is given for 0.7–7 keV range). Obs. ID | Start Time | Telescope | Exposure | Flux | Flux range
---|---|---|---|---|---
| UTC |  | sec |  $10^{-11}$ $\rm erg\,\rm cm^{-2}\,\rm s^{-1}$ | 68% confidence
00014460001 | 2021-08-07 02:22:34 | Swift | 5571 | 10.8 | (1.04e$-$10 – 1.11e$-$10)
00014460002 | 2021-08-08 00:34:34 | Swift | 5631 | 8.12 | (7.82e$-$11 – 8.39e$-$11)
00014460003 | 2021-08-09 05:14:52 | Swift | 7729 | 8.70 | (8.45e$-$11 – 8.93e$-$11)
0862090101 | 2021-08-10 16:50:25 | XMM-Newton | 10290 | 11.0 | 0.24e$-$11
00089304001 | 2021-08-10 20:55:34 | Swift | 1573 | 11.6 | (1.08e$-$10 – 1.24e$-$10)
0862090201 | 2021-08-11 18:36:37 | XMM-Newton | 10540 | 8.84 | 0.38e$-$11
0862090301 | 2021-08-12 16:45:45 | XMM-Newton | 13070 | 7.79 | 0.53e$-$11
0862090501 | 2021-08-14 16:45:15 | XMM-Newton | 14440 | 9.09 | 0.29e$-$11
0862090601 | 2021-08-15 18:02:00 | XMM-Newton | 7455 | 9.18 | 0.42e$-$11
0862090701 | 2021-08-16 16:42:33 | XMM-Newton | 18750 | 8.64 | 0.36e$-$11
00089304002 | 2021-08-16 22:02:35 | Swift | 1808 | 8.94 | (8.44e$-$11 – 9.39e$-$11)
0862090801 | 2021-08-17 17:41:23 | XMM-Newton | 12360 | 8.76 | 0.29e$-$11
0862090901 | 2021-08-18 16:35:14 | XMM-Newton | 18030 | 8.49 | 0.27e$-$11
0862091001 | 2021-08-19 17:08:37 | XMM-Newton | 13620 | 8.90 | 0.46e$-$11
00014460004 | 2021-08-20 02:45:36 | Swift | 7399 | 9.57 | (9.28e$-$11 – 9.87e$-$11)
00014460005 | 2021-08-21 05:23:34 | Swift | 7062 | 10.6 | (1.03e$-$10 – 1.09e$-$10)
00014460006 | 2021-08-22 00:38:36 | Swift | 8666 | 9.71 | (9.43e$-$11 – 9.98e$-$11)
00014460007 | 2021-08-23 00:24:34 | Swift | 7684 | 11.3 | (1.09e$-$10 – 1.16e$-$10)
00014460008 | 2021-08-24 01:52:36 | Swift | 3389 | 12.0 | (1.15e$-$10 – 1.24e$-$10)
00014460009 | 2021-08-25 03:29:34 | Swift | 7657 | 10.7 | (1.04e$-$10 – 1.11e$-$10)
3599010101 | 2021-08-25 15:57:00 | NICER | 777 | 9.87 | (9.69e$-$11 – 1.00e$-$10)
00014460010 | 2021-08-26 00:03:34 | Swift | 10040 | 8.32 | (8.09e$-$11 – 8.53e$-$11)
3599010102 | 2021-08-26 04:21:00 | NICER | 2497 | 8.18 | (8.09e$-$11 – 8.26e$-$11)
3599010103* | 2021-08-27 19:05:23 | NICER | 1361 | 10.6 | (1.04e$-$10 – 1.08e$-$10)
3599010104 | 2021-08-28 10:35:23 | NICER | 2817 | 10.9 | (1.08e$-$10 – 1.10e$-$10)
3599010105 | 2021-08-29 20:41:25 | NICER | 1651 | 10.7 | (1.06e$-$10 – 1.08e$-$10)
3599010106 | 2021-08-30 19:56:04 | NICER | 1524 | 9.06 | (8.93e$-$11 – 9.18e$-$11)
3599010107 | 2021-08-31 19:11:05 | NICER | 1387 | 9.72 | (9.59e$-$11 – 9.84e$-$11)
3599010108 | 2021-09-01 19:59:29 | NICER | 1541 | 11.8 | (1.17e$-$10 – 1.20e$-$10)
3599010109 | 2021-09-02 19:14:29 | NICER | 1411 | 11.7 | (1.16e$-$10 – 1.18e$-$10)
3599010110 | 2021-09-03 20:02:30 | NICER | 1513 | 9.68 | (9.56e$-$11 – 9.81e$-$11)
3599010111 | 2021-09-04 19:18:10 | NICER | 1401 | 11.7 | (1.16e$-$10 – 1.18e$-$10)
3599010112 | 2021-09-05 20:02:00 | NICER | 1618 | 14.6e | (1.45e$-$10 – 1.48e$-$10)
3599010113 | 2021-09-06 10:01:42 | NICER | 854 | 14.9 | (1.47e$-$10 – 1.52e$-$10)
3599010114 | 2021-09-06 23:58:41 | NICER | 1983 | 15.2 | (1.50e$-$10 – 1.53e$-$10)
3599010115 | 2021-09-08 00:47:16 | NICER | 2174 | 13.0 | (1.29e$-$10 – 1.32e$-$10)
3599010116 | 2021-09-09 00:06:16 | NICER | 1980 | 12.2 | (1.21e$-$10 – 1.23e$-$10)
3599010117 | 2021-09-10 01:11:20 | NICER | 1219 | 11.4 | (1.12e$-$10 – 1.15e$-$10)
3599010118 | 2021-09-11 00:04:36 | NICER | 2094 | 13.4 | (1.32e$-$10 – 1.35e$-$10)
3599010119 | 2021-09-12 00:52:56 | NICER | 1839 | 12.1 | (1.20e$-$10 – 1.23e$-$10)
3599010120 | 2021-09-13 00:08:37 | NICER | 1229 | 12.0 | (1.18e$-$10 – 1.21e$-$10)
3599010121 | 2021-09-14 19:56:00 | NICER | 403 | 11.3 | (1.11e$-$10 – 1.16e$-$10)