# A nebular origin for the persistent radio emission of fast radio bursts

Gabriele Bruni1 Luigi Piro1 Yuan-Pei Yang2,3 Salvatore Quai4,5 Bing Zhang6,7
Eliana Palazzi5 Luciano Nicastro5 Chiara Feruglio8,9 Roberta Tripodi10,8,9
Brendan O’Connor11 Angela Gardini12 Sandra Savaglio13 Andrea Rossi5 A. M.
Nicuesa Guelbenzu14 Rosita Paladino15

###### Abstract

Fast Radio Bursts (FRBs) are millisecond-duration, bright ($\sim$Jy)
extragalactic bursts, whose production mechanism is still unclear [1].
Recently, a persistent radio source (PRS) of non-thermal origin was discovered
to be physically associated to two of the repeating FRB sources [2, 3]. These
two sources have unusually large Rotation Measure (RM) values [2, 3], likely
tracing a dense magneto-ionic medium, consistent with a synchrotron radiation
originating from a nebula surrounding the FRB source [4, 5, 6, 7]. Recent
theoretical arguments predict that, if the observed RM mostly arises from the
PRS region, there should be a simple relation between the luminosity of the
PRS and the RM [8, 6]. We report here the detection of a third, less luminous
PRS associated with the nearby FRB 20201124A at a distance of 413 Mpc,
significantly expanding the predicted relation into the low luminosity – low
RM regime ($<$1000 rad/m2). At lower values of the RM, the radio luminosity
falls below the limit of detection threshold for nowadays radio telescopes.
These findings support the idea that the PRS is generated by a nebula in the
FRB environment, and that most FRBs do not show a PRS because of a weaker
magneto-ionic medium. This is generally consistent with models foreseeing a
young magnetar as the central engine of the FRB, where the surrounding ionized
nebula powers the PRS.

IAPS – Institute for Space Astrophysics and Planetology, INAF, via del Fosso
del Cavaliere, 100, Rome, 00133, Italy.

South-Western Institute for Astronomy Research, Yunnan University, University
City, Chenggong District, Kunming, 650504, Yunnan, China.

Purple Mountain Observatory, Chinese Academy of Sciences, 10 Yuanhua Road,
Qixia District, Nanjing, 210023, Jiangsu, China.

Dipartimento di Fisica e Astronomia ‘Augusto Righi’, Università degli Studi di
Bologna, UNIBO, via Piero Gobetti 93/3, Bologna, 40129, Italy.

Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, INAF, via Piero
Gobetti 93/2, Bologna, 40129, Italy.

Nevada Center for Astrophysics, University of Nevada, NV 89154-4002, Las
Vegas, Nevada USA.

Department of Physics and Astronomy, University of Nevada, NV 89154-4002, Las
Vegas, Nevada USA.

Osservatorio Astronomico di Trieste, INAF, via G.B. Tiepolo 11, Trieste,
34143, Italy.

IFPU–Institute for Fundamental Physics of the Universe, via Beirut 2, Trieste,
34151, Italy.

Dipartimento di Fisica, Università di Trieste, via G.B. Tiepolo 11, Trieste,
34143, Italy.

McWilliams Center for Cosmology, Department of Physics, Carnegie Mellon
University, 4000 Forbes Avenue, Pittsburgh, PA 15213, USA.

Instituto de Astrofísica de Andalucía, CSIC, Glorieta de la Astronomía s/n,
Granada, 18008, Spain.

Dipartimeto di Fisica, Universit‘a della Calabria, Arcavacata di Rende, 87036,
Italy.

Thüringer Landessternwarte Tautenburg, Sternwarte 5, Tautenburg, 35 07778,
Germany.

IRA – Istituto di Radioastronomia, INAF, via Piero Gobetti, 101, Bologna,
40129, Italy.

FRB 20201124A is a repeating fast radio burst which underwent a reactivation
during March 2021 [9, 10], allowing to locate it with an uncertainty of a few
milli-arcseconds through Very Long Baseline Interferometry (VLBI) observations
[11]. Its precise localization and close distance (redshift=0.098) make it an
ideal target to study the physical conditions in its surroundings. Following
the discovery of extended radio emission associated with star formation in the
FRB 20201124A environment [12], we targeted the FRB region with deep Very
Large Array (VLA) observations at sub-arcsec angular resolution, at 15 and 22
GHz. Disentangling a possible compact source from the previously detected,
diffuse (3-4 kpc) emission is indeed crucial to assess the presence of a PRS
associated with the FRB.

Observations at 15 GHz highlight a compact, unresolved emitting region, with a
position consistent with the FRB location (see Methods). Its flux density at
15 GHz outshines the diffuse component from the host galaxy at the same
frequency (see Fig. 1, top panel). This bright, compact PRS falls in the
South-West region of the extended emission previously discovered at higher
frequencies [12]. At 22 GHz, a compact component is still detected, although
at a lower significance level of 4-$\sigma$, at a position consistent with the
FRB location and the peak of the 15 GHz component (see Fig. 1 in Extended
data). The estimated projected physical size at 15 GHz is $\lesssim 700$ pc,
but a much smaller size cannot be excluded. As also noted in [13], the flux
densities measured up to 22 GHz imply that this compact source could not be
detected in previous VLBI observations because too faint for the image noise
level reached in those works – $\sim$10 $\mu$Jy/beam [14, 12], at the limit of
capabilities for nowadays VLBI arrays.

In order to derive the spectral shape of the source, we have reanalyzed the 6
GHz VLA archival data from [13], that have a resolution similar to our
observations. We could disentangle the compact PRS from the diffuse star-
formation emission, and estimate its flux density, that is consistent with the
one predicted by those authors on the basis of general image-quality
considerations. Considering the flux densities collected from multi-frequency
(6, 15, 22 GHz) VLA data, the PRS shows an inverted radio spectrum
$F_{\nu}\propto\nu^{1}$.

Figure 1: Top Panel: VLA image at 15 GHz of the host galaxy of FRB 20201124A
from this work. Black contours indicate 3, 4, 5, 7, 9 $\times\sigma$ levels.
The white contours are from the 6 GHz VLA image by [13] at similar resolution.
The red contours are from the previous 22 GHz VLA image by [12] at lower
resolution. The location of the FRB as measured by the EVN [11] is reported as
a yellow cross, while the center of the host galaxy as estimated from the
HST/UVIS image [13] ($F475X$ filter) is reported as a red cross. Bottom Panel:
HST/UVIS image of the host galaxy by [13], with contours reporting the 15 GHz
VLA image from this work. As for the previous image, the FRB position is
reported as a yellow cross, and the host galaxy’s center as a red cross.

It is difficult to associate such radio emission to a star-forming region.
First, the inverted spectrum is unlike the spectral shape observed in radio
from star formation, usually steep or at most flat (see Methods). Furthermore,
any star formation capable to account for the radio luminosity would leave
imprints at other wavelengths, that are not detected. Our integral field unit
observations with GTC/MEGARA (see Methods) confirms that the FRB falls at the
Northern edge of the host galaxy’s bar, as seen in previous HST images [13].
The star formation rate (SFR) map, as estimated from the H$\alpha$ emission
line (see Fig. 2, left panel), reveals that highest SFR regions are towards
the galaxy’s center, and match the UV emission as seen by HST (see Fig. 1,
right panel). Conversely, close to the FRB position a SFR of $\sim
0.4\,M_{\odot}/yr$ is measured. Additionally, we performed mm-band
observations with NOEMA centered on the FRB coordinates to independently
estimate the SFR through dust grey-body emission. These resulted in two non-
detections at 236.5 and 250 GHz, corresponding to $3\sigma$ upper limits of
130 $\mu$Jy/beam and 160 $\mu$Jy/beam, respectively. The angular resolution of
NOEMA images allowed to probe a physical scale of about 1.3$\times$0.7 kpc,
comparable with the one probed by the VLA. We modelled the cold dust spectral
energy distribution (SED) with a Modified Black Body (MBB) function in order
to derive upper limits for the dust mass and SFR (see details in Methods),
finding a dust mass of $M_{\rm dust}<7.9\times 10^{6}\ {\rm M_{\odot}}$ and a
star formation rate of SFR$<2.2\ \rm M_{\odot}$/yr at the location of the 15
GHz PRS. Similarly, from our radio images we could estimate that, in order to
account for the observed radio flux density, a ${\rm SFR}\gtrsim 3\
M_{\odot}$/yr would be needed (see Methods). This number is not consistent
with either the upper limit from obscured star formation derived from NOEMA
observations (SFR$<2.2\ \rm M_{\odot}$/yr), or the H$\alpha$-derived
measurement of SFR at the location of the compact source as estimated from
GTC/MEGARA spectroscopy ($\sim 0.4\,M_{\odot}$/yr), suggesting a different
origin.

Figure 2: Star formation rate (SFR) map of the FRB 20201124A host galaxy, as
derived from GTC/MEGARA integral field spectroscopy. We estimated the SFR from
the intrinsic H$\alpha$ luminosity, taking into account Galactic and intrinsic
dust extinction using the Balmer decrement (H$\alpha$/H$\beta$). Pixels with
an H$\alpha$ significance below a $3\sigma$ threshold are omitted. In cases
where the H$\beta$ significance falls below a $2\sigma$ threshold – so that no
accurate correction for dust extinction is possible – we give a lower limit of
the SFR based on the upper limits of H$\beta$, and represent these pixels in
greyscale. The white contours represent the VLA image at 15 GHz, whereas the
red and black crosses represent the FRB and Galactic centre, respectively. The
magenta rectangle define the galactic region encompassing six adjacent MEGARA
pixels surrounding the FRB and PRS zone. Figure 3: The relation between the
specific luminosity of persistent radio source and RM of FRBs [8, 6]. The red
dotted, dashed, and solid lines denote the predicted relation for
$\zeta_{e}\gamma_{\rm th}^{2}(R/10^{-2}~{}{\rm pc})^{2}=10,1,0.1$,
respectively. The orange star point denotes the data of FRB 20201124A. The
black down-triangle points denote FRB 20110523A with an upper limit of the
persistent emission and a measured value of RM, which gives one of the most
conservative constraints for non-localized sources in literature. The vertical
bar denotes the uncertainty in the upper limit. The circle points (FRB
20121102A and FRB 20190520B) denote the FRBs with measured values of the
persistent emission flux and RM. The blue points correspond to the FRBs with
precise localizations.

Large $|{\rm RM}|$ values found for some FRBs imply a dense and magnetized
environment in the vicinity of the sources [8, 6]. If relativistic electrons
constitute a significant fraction of the medium, the magnetized environment
near the FRB source could produce synchrotron radiation, powering a bright PRS
[15, 16, 8]. There should be a simple relation between the luminosity of the
PRS and the rotation measure, tracing the magneto-ionic medium in the FRB
environment [8, 6]. We consider that the electron distribution in the PRS
emission region has a thermal component and a non-thermal component, and the
RM is mainly contributed by the thermal component. As discussed in the Methods
section, the foreseen relation is:

$L_{\nu}\propto(\zeta_{e}\gamma_{\rm th}^{2}R^{2})\times|\rm RM|.$ (1)

where $\zeta_{e}$ is the number fraction between the non-thermal electrons
radiating synchrotron emission in the observed band and thermal electrons,
$\gamma_{\rm th}$ the typical Lorentz factor of thermal electrons, and $R$ is
the radius of the plasma screen that contributes to the PRS and the RM.

In Figure 3 we plot the relation between the specific luminosity (or upper
limit) of the PRS against the RM of known FRBs. The red dotted, dashed, and
solid lines correspond to the predicted relation for $\zeta_{e}\gamma_{\rm
th}^{2}(R/10^{-2}~{}{\rm pc})^{2}=10,1,0.1$, respectively (see Methods). The
observed specific PRS luminosities of FRB 20121102A, FRB 20190520B, and FRB
20201124A are close to the predicted value for $\zeta_{e}\gamma_{\rm
th}^{2}(R/10^{-2}~{}{\rm pc})^{2}\sim(0.1-10)$. In particular, the detection
of the PRS associated to FRB 20201124A allows for the first time to test the
predicted relation at lower radio luminosity, at the limits of nowadays radio
telescopes detection threshold, thus significantly broadening the probed
parameters space. The upper limit of FRB 20180916B is also close to the
predicted relation. As a caveat, we should notice that for the FRB with ${\rm
RM}\lesssim\text{a few }\times 100~{}{\rm rad~{}m^{-2}}$, the RM from the
Galactic component or the host component would dominate. This places FRB
20201124A, with a $|\rm RM|\sim 900~{}{\rm rad~{}m^{-2}}$, at the limits of
the explorable range.

These findings pose in favour of a nebular origin for the three detected PRS
associated with FRB so far. The possible nature of the FRB engine itself
remains unclear, however, the discussed nebula model is generally consistent
with the proposed scenarios involving a magnetar or a hyperaccreting X-ray
binary as the central engine, which already foresee an ionized nebula around
it [17, 7]. Thanks to the increasing census of FRB, and in particular repeater
ones, deep radio observations of a larger sample of these objects could put
further constraints on the nebula model, and on the physical conditions in the
surroundings of FRBs in general.

## Methods

###  0.1 VLA detection of the compact PRS

We detected a compact component both at 15 and 22 GHz. Figure 1 shows the
overplot of the image at 6-GHz that we have derived reanalyzing archival data
from [13] with contours from our 15 and 22 GHz images at similar resolution.
The emission comes from regions consistent with the EVN position at all
frequencies. Indeed, a Gaussian fit results in a compact component at 15 and
22 GHz, with an integrated flux density of $20.0\pm 3.5$ $\mu$Jy and $30.0\pm
9.7\;\mu$Jy, respectively. The peak position of the fitted Gaussian component
is RA 05:08:03.50270 $\pm 0.00097$ s, Dec +26:03:38.444 $\pm 0.014$ arcsec at
15 GHz, while 05:08:03.4944 $\pm 0.0052$ s, Dec +26:03:38.423 $\pm 0.080$
arcsec at 22 GHz. Considering a standard astrometric accuracy of $\sim 10$% of
the FWHM, and summing that in quadrature with the Gaussian fit error, we get a
final uncertainty of $0.04$ arcsec $\times 0.04$ arcsec at 15 GHz, and
0$\overset{\prime\prime}{.}$08 $\times$ 0$\overset{\prime\prime}{.}$09 at 22
GHz. The mentioned positional uncertainties imply that the detected compact
component is consistent within 3-$\sigma$ with the FRB location by [11]. The
measured flux density at 22 GHz accounts for $\sim 40$% of the value reported
at the same frequency, but at lower resolution, in [12], indicating that the
previously detected diffuse emission falls below the detection threshold per
angular resolution element of the new image.

The image at 6 GHz, and the 15 and 22 GHz ones presented here (see fig. 1)
have a consistent angular resolution and RMS level, and can then be used to
estimate the spectral index of the compact PRS. At 6 GHz, [13] discuss how a
possible PRS would be embedded in a diffuse emission region due to star
formation. Those authors thus provide a conservative upper limit of 10
$\mu$Jy. We re-imaged the 6 GHz data from [13] (program 22A-213, PI: Fong),
and performed an additional analysis to refine the flux density and
compactness estimate of the possible PRS. To account for the diffuse emission,
we estimated the mean flux density in the regions contiguous with the FRB
location, and outside a $\sim$1 beam area centered on the component peak at 6
GHz. In this way, we obtained an estimate of 8 $\mu$Jy/beam for the emission
plateau due to star formation, consistent with the one discussed by [13]. We
then used this value as an input for the offset parameter of the IMFIT task,
that allows to account for a zero-level offset when performing a Gaussian fit
on the image. This procedure revealed a compact component at 6 GHz with an
integrated flux density of 8.2$\pm$3.8 $\mu$Jy, and centered at RA
05:08:03.5007$\pm 0.0023$ s, Dec +26:03:38.5440 $\pm 0.0438$ arcsec. As
previously discussed, considering an astrometric accuracy of $\sim 10$% of the
FWHM we estimate a final uncertainty of $0.05$ arcsec $\times 0.06$ arcsec,
making again the peak position consistent with the FRB location within
3-$\sigma$. We note that, despite the uncertainty on the component flux
density is $\sim$45% of the value, it is still at a 4-$\sigma$ detection level
with respect to the image RMS (2 $\mu$Jy/beam), and also consistent with the
upper limit of 10 $\mu$Jy discussed by [13].

Considering the estimated flux densities at 6, 15, and 22 GHz, we obtain the
following spectral indices ($S\propto\nu^{\alpha})$ for the PRS:
$\alpha_{6}^{15}=0.97\pm 0.54$, $\alpha_{15}^{22}=1.06\pm 0.96$, and an
overall value $\alpha_{6}^{22}=1.00\pm 0.43$, indicating an inverted spectrum.

###  0.2 SFR estimates from NOEMA observations

In order to derive upper limits for the dust mass and SFR in this objects
based on the NOEMA fluxes, we modelled the SED of the dust emission with a
modified black-body (MBB) function given by

$S_{\nu_{\rm obs}}^{\rm obs}=S_{\nu/(1+z)}^{\rm
obs}=\dfrac{\Omega}{(1+z)^{3}}[B_{\nu}(T_{\rm dust}(z))-B_{\nu}(T_{\rm
CMB}(z))](1-e^{-\tau_{\nu}}),$ (2)

where $\Omega=(1+z)^{4}A_{\rm gal}D_{\rm L}^{-2}$ is the solid angle with
$A_{\rm gal}$ , and $D_{\rm L}$ is the surface area and luminosity distance of
the galaxy, respectively. The dust optical depth is

$\tau_{\nu}=\dfrac{M_{\rm dust}}{A_{\rm galaxy}}k_{0}\biggl{(}\dfrac{\nu}{250\
\rm GHz}\biggr{)}^{\beta},$ (3)

with $\beta$ the emissivity index and $k_{0}=0.45\ \rm cm^{2}\ g^{-1}$ the
mass absorption coefficient [18]. The solid angle is estimated using the size
of the NOEMA beam, that is similar for both observations, $0.7\times 0.4$
arcsec2. The effect of the CMB on the dust temperature is given by

$T_{\rm dust}(z)=((T_{\rm
dust})^{4+\beta}+T_{0}^{4+\beta}[(1+z)^{4+\beta}-1])^{\frac{1}{4+\beta}},$ (4)

with $T_{0}=2.73$ K. We also considered the contribution of the CMB emission
given by $B_{\nu}(T_{\rm CMB}(z)=T_{0}(1+z))$ [19], even if at this redshift
is almost negligible.

Therefore, this model has three free parameters: $T_{\rm dust}$, $M_{\rm
dust}$, and $\beta$. Since we were able to determine only upper limits in the
two NOEMA bands, we evaluated possible combination of $T_{\rm dust}$ and
$\beta$ that can reproduce the limits at 236 GHz and 250 GHz, leaving $M_{\rm
dust}$ as a free parameter. Fig. 3 presents 4 models built from a reasonable
combinations of dust temperature and emissivity: $T_{\rm dust}=20,30$ K and
$\beta=1.5,2.0$, that are typical values found in local star-forming galaxies.
We explored the one dimensional parameter space using a Markov chain Monte
Carlo (MCMC) algorithm implemented in the EMCEE package [20], and we assumed a
uniform prior for the dust mass, $10^{4}\ {\rm M_{\odot}}<M_{\rm dust}<10^{9}\
{\rm M_{\odot}}$. We obtained $M_{\rm dust}/{10^{6}\rm
M_{\odot}}=7.9,7.8,4.7,4.5$ with $[T_{\rm dust}(\rm
K),\beta]=[20,1.5],[20,2.0],[30,1.5],[30,2.0]$, from a MCMC with 25 chains,
1500 trials and a burn-in phase of $\sim 50$ for each model.

The SFR is estimated from the total infrared luminosity (TIR) as ${\rm
SFR}[{\rm M_{\odot}}]=10^{-10}\ L_{\rm TIR}\rm[L_{\odot}]$, assuming Chabrier
IMF [21]. To compute the TIR luminosity we integrated the MBB from $8$ to
$1000\ \mu$m rest-frame, and therefore we obtained SFR/$\rm
M_{\odot}~{}yr^{-1}$ = 0.13, 0.34, 0.69, and 2.24 for $[T_{\rm dust}(\rm
K),\beta]=[20,1.5],[20,2.0],[30,1.5],[30,2.0]$. Overall we obtained that
$M_{\rm dust}<7.9\times 10^{6}\ {\rm M_{\odot}}$ and SFR$<2.2\;\rm
M_{\odot}~{}yr^{-1}$.

In order to constrain the emission from the host galaxy (see also [13]), we
re-computed the $3\sigma$ upper limits at 236 GHz and 250 GHz considering the
region with significance $>3\sigma$ in the 6 GHz VLA map, which corresponds to
a physical region of $\sim 2.85$ kpc2. The $3\sigma$ upper limits are then
$0.49$ mJy and $0.55$ mJy at 236 GHz and 250 GHz, respectively. Fig. 4 shows
the best-fitting curves to the new upper limits for $[T_{\rm dust}(\rm
K),\beta]=[20,1.5],[20,2.0],[30,1.5],[30,2.0]$. We obtained $1.6<M_{\rm
dust}/{10^{7}\rm M_{\odot}}<2.9$, implying ${\rm SFR}<8.8\ \rm
M_{\odot}~{}yr^{-1}$.

###  0.3 SFR estimates from GTC/MEGARA observations

Since the emission lines observed in the FRB region and the entire galaxy
spectra are produced by UV radiation from stellar photospheres (see
Supplementary information), we can thus estimate SFRs directly from the
H$\alpha$ intrinsic luminosity. First, we corrected H$\alpha$ for intrinsic
dust attenuation based on the Balmer decrement (i.e., the H$\alpha$/H$\beta$
ratio). We show in Figure 5 the map of colour excess E(B - V) of the galaxy
obtained from the Balmer decrement. To perform a proper correction for dust
extinction, in pixels with S/N(H$\beta$) $\geq 2$, the colour excess E(B - V)
is derived adopting the [22] attenuation law and assuming the Case B
recombination and a Balmer decrement H$\alpha$/H$\beta=2.86$ (typical of H II
regions with electron temperatures Te = $10^{4}$ K and electron density ne
between $10^{2}$ \- $10^{4}$ cm-3 [23]. We assigned E(B - V)$=0$ to pixels
with negative E(B - V) values between about $-0.05$ and $\sim 0$ (i.e. pixels
showing $\sim 2.7\leq$H$\alpha$/H$\beta<2.86$). Moreover, we also assigned an
E(B-V) lower limit to pixels with S/N(H$\beta$)$<2$, using the upper limit in
H$\beta$ we defined earlier in this section. The dust-corrected fluxes are
converted to luminosity. Then, we derived the SFR using the dust-corrected
H$\alpha$ luminosity and adopted the [24] conversion factor converted to the
[21] IMF. We have already shown and discussed the resulting SFR map of the
galaxy in Figure 2. To summarize, we report that in the stacked spectrum of
the FRB region, we measured an SFR of $0.41^{+0.02}_{-0.01}$ M$\odot$/yr,
while the entire galaxy has an SFR of $4.11^{+0.59}_{-0.53}$ M$\odot$/yr.

###  0.4 Expected radio emission from a star forming region

Radio emission from SFR involves typically two components: a steep component
($\approx\nu^{-0.7}$) produced by synchrotron emission from electrons
accelerated in SN explosion, with a lifetime of the order of 100 Myr [25],
that dominates the emission at low frequencies (below 6 GHz); a thermal, flat
component by bremmstrahlung, that is a direct measure of the current
production of ionized photons by young and hot stars with lifetime $<10$ Myr
[26, 27, 28]. On the other hand, the inverted spectrum observed in the compact
radio source is not consistent with either SF-driven radio components.
Furthermore, the level of SFR required to account for the observed radio
emission would violate the limits derived from mm and Hα measurements. The SFR
needed to produce the radio thermal component is [29]:

$\frac{{\rm SFR}^{T}}{M_{\odot}yr^{-1}}=4.6\
10^{-28}\frac{T_{e}}{10^{4}K}\nu_{GHz}^{0.1}\frac{L(\nu_{GHz})}{erg\;s^{-1}\,Hz^{-1}}$
(5)

The observed $L_{15\rm{GHz}}=4.9\times 10^{27}\,$erg s${}^{-1}\,$Hz-1 would
then imply ${\rm SFR}^{T}\gtrsim 3M_{\odot}$ yr-1 to account for the radio
emission. This number is not consistent with either the upper limit from
obscured star formation derived from NOEMA and from the H$\alpha$ measurement
of SFR at the location of the compact source, suggesting another origin.

###  0.5 Nebular emission model

We consider that the electron distribution has a thermal component in the low-
energy regime, i.e., $n_{e,{\rm th}}(\gamma)\simeq n_{e,{\rm
th0}}(\gamma/\gamma_{\rm th})^{2}$ for $\gamma<\gamma_{\rm th}$ and $n_{e,{\rm
th}}(\gamma)\simeq 0$ for $\gamma>\gamma_{\rm th}$, where $n_{e,{\rm th0}}$ is
the total number density of thermal electrons and $\gamma_{\rm th}$ is the
thermal Lorentz factor, and a non-thermal component in the high-energy regime,
i.e., $n_{e}(\gamma)\simeq n_{e,0}(\gamma/\gamma_{m})^{-p}$ for
$\gamma>\gamma_{m}$ and $n_{e}(\gamma)\simeq 0$ for $\gamma<\gamma_{m}$, where
$n_{e,0}$ is the total number density of non-thermal electrons and
$\gamma_{m}$ is the minimum Lorentz factor of the non-thermal component. We
assume that the RM is mainly contributed by the thermal component,

$\displaystyle{\rm RM}\simeq\frac{e^{3}}{2\pi m_{e}^{2}c^{4}}\frac{n_{e,{\rm
th0}}B_{\parallel}}{\gamma_{\rm th}^{2}}\Delta R.$ (6)

where $B_{\parallel}$ is the mean parallel component of the magnetic field
along the line of sight, $\Delta R$ is the thickness of the plasma screen. For
the non-thermal component, the Lorentz factor of the electrons emitting
synchrotron radiation in the observation frequency $\nu$ is required to be

$\displaystyle\gamma_{\rm obs}\sim\left(\frac{2\pi
m_{e}c\nu}{eB}\right)^{1/2}\simeq 600\left(\frac{\nu}{1~{}{\rm
GHz}}\right)^{1/2}\left(\frac{B}{1~{}{\rm mG}}\right)^{-1/2}.$ (7)

Define $\zeta_{e}$ as the fraction between the electrons radiating synchrotron
emission in the observed band and thermal electrons, it is approximately given
by

$\displaystyle\zeta_{e}\sim\frac{\gamma_{\rm obs}n_{e}(\gamma_{\rm
obs})}{n_{e,{\rm th0}}}\sim\frac{n_{e,0}}{n_{e,{\rm
th0}}}\left(\frac{\gamma_{\rm obs}}{\gamma_{\rm th}}\right)^{1-p}$ (8)

for $\gamma_{\rm obs}>\gamma_{\rm th}$. Assuming that the magnetic field is
large-scale (leading to the parallel mean magnetic field of the order of the
total magnetic field) the specific luminosity of synchrotron radiation could
be written as [8, 6]

$\displaystyle L_{\nu}=\frac{64\pi^{3}}{27}\zeta_{e}\gamma_{\rm
th}^{2}m_{e}c^{2}R^{2}\left|{\rm RM}\right|\simeq 5.7\times 10^{28}~{}{\rm
erg~{}s^{-1}~{}Hz^{-1}}$ $\displaystyle\times\zeta_{e}\gamma_{\rm
th}^{2}\left(\frac{\left|{\rm RM}\right|}{10^{4}~{}{\rm
rad~{}m^{-2}}}\right)\left(\frac{R}{10^{-2}~{}{\rm pc}}\right)^{2}.$ (9)

where $R$ is the radius of the plasma screen that contributes to the PRS and
the RM.

In Table 2, we list 14 FRBs with the measured RM and the measured values (or
upper limits) of the flux densities of the PRSs, and only FRB 20121102A, FRB
20190520B, and FRB 20201124A have the measured values of both. Some sources
have precise localizations, so the redshifts in the top row of Table 2 are the
directly measured values. For other FRBs, due to the lack of a precise
localization, we estimate their redshifts and luminosity distances via the
extragalactic DM as performed by [8]. Notice that for redshift inferred from
the exceeded DM, the corresponding redshift error is attributed to the IGM
density fluctuation [30, 8].

Based on the data in Table 2, we plot the relation between the specific
luminosity of the PRS and the RM of FRBs in Figure 3. The red dotted, dashed,
and solid lines correspond to the predicted relation for $\zeta_{e}\gamma_{\rm
th}^{2}(R/10^{-2}~{}{\rm pc})^{2}=10,1,0.1$, respectively. It is interesting
that the observed specific luminosities of FRB 20121102A, FRB 20190520B, and
FRB 20201124A are close to the predicted value for $\zeta_{e}\gamma_{\rm
th}^{2}(R/10^{-2}~{}{\rm pc})^{2}\sim(0.1-10)$. The upper limit of FRB
20180916B is close to the predicted relation. However, we should notice that
for the FRB with ${\rm RM}\lesssim\text{a few }\times 100~{}{\rm
rad~{}m^{-2}}$, the RM from the Galactic component or the host component would
dominate.

###  0.6 Considerations on the observed radio spectrum

As pointed out above, the PRS spectrum of FRB 20201124A satisfies
$F_{\nu}\propto\nu^{1.0}$, which is much harder than those of FRB 20121102A
and FRB 20190520B. If the observed spectrum of the PRS is produced by the
synchrotron radiation, a spectral index larger than $1/3$ would imply that the
synchrotron absorption might be dominant at the observed bands $(6-22)\;{\rm
GHz}$. Such a high absorption frequency would lead to FRBs absorbed by the
PRS, which is inconsistent with the observation of FRB 20201124A. On the other
hand, we notice that the spectral index of the PRS is also consistent with
$1/3$ within $1.5\sigma$. In this case, to explain the spectral shape of the
PRS, the typical frequency $\nu_{m}$ that corresponds to the minimum Lorentz
factor of the accelerated non-thermal electrons is required to be

$\displaystyle\nu_{m}\simeq\frac{\gamma_{m}^{2}eB}{2\pi
m_{e}c}\gtrsim(22-240)~{}{\rm GHz},$ (10)

where $B$ is the magnetic field strength at the emission region. Therefore, we
obtain the following constraints:

$\displaystyle\left(\frac{\gamma_{m}}{10^{3}}\right)^{2}\left(\frac{B}{10~{}{\rm
mG}}\right)\gtrsim(0.8-8.6),$ (11)

which is consistent with a magnetized shocked nebula.

###  0.7 Considerations on a possible background source

In the GTC/MEGARA spectrum, we found a detection of a possible [OII] emission
line in correspondence of the FRB location, but at a redshift $z=0.549$. We
thus tested the scenario of a background source potentially responsible for
the compact radio component, that we instead interpreted as a PRS linked to
the FRB.

Deep radio surveys show that the radio sky at a flux level of $\approx
10\;\mu$Jy exhibit about $1.4\times 10^{-3}$ arcsec-2 sources [31]. The vast
majority have a steep spectrum, unlike the one displayed by the source we
observe here. Taking into account that only a fraction of $\lesssim 0.07$ has
a flat spectrum we derive that the probability of one of them falling within
the beam size at the position of the FRB is $\approx 10^{-5}$. This already
results in a very low chance that the radio emission is due to a serendipitous
background source. However, in the following we discuss the different classes
of sources that could produce the compact radio component studied in this
work, and why we can confidently rule them out.

Possible extragalactic sources producing radio emission are blazars, radio
galaxies, radio quiet AGN, or a star forming galaxy. At the mentioned
redshift, the radio luminosity as estimated from our VLA measurement is
Log($L_{R}$)$\sim$39 erg/s, thus indicating a radio-quiet regime. This already
rules out a possible background blazar or radio galaxy, although it is still
consistent with the one estimated for the faintest (radio quiet) Seyfert
galaxies observed to date in the radio band ($35<\rm{Log}(L_{R})<39$ erg/s
[32]). A radio quiet AGN could show a radio component due to a low-power jet,
a nuclear wind, or coronal emission [33]. In the first case, while the
physical scale subtended at $z=0.549$ by the compact radio component ($\sim 2$
kpc) would be enough to accommodate the presence of a jet, the expected radio
spectrum of its extended regions would be steep ($\alpha<-0.5$). The inverted
radio spectrum we found would thus require a collimated jet with an
orientation within a few degrees from the line of sight. This would in turn
result in a dominant, Doppler-boosted jet emission visible up to the optical
band. This feature has been commonly observed for powerful jets at different
scales, from radio-loud AGN to X-ray binaries, but not yet in radio-quiet AGN
where the flat/inverted radio spectrum is attributed to a coronal emission
mechanism ([34, 35], see also discussion below). Moreover, the NOEMA non-
detection in the mm-band put a strong constraint on that: a simple power-law
extrapolation of the radio spectrum up to 250 GHz results in an expected flux
density $>$300 $\mu$Jy, well above the NOEMA upper limit ($<$160 $\mu$Jy). In
the second case (nuclear wind), a steep radio spectrum is expected, thus not
consistent with the inverted one seen in our VLA measurements. In the third
case (AGN coronal emission), a ratio between radio and X-ray luminosity
$L_{R}/L_{X}\sim 10^{-5}$ is expected [36], while previous Chandra X-ray
measurements in a circular region with a 1” radius centered on the FRB
location [12] constrain this ratio to $\sim 10^{-3}$. The X-ray luminosity is
thus much lower than the one expected from a corona able to produce the radio
emission measured with the VLA. Finally, a possible radio emission from a
background star-forming galaxy is excluded by the expected relation between
[OII] and radio emission in SFR [24], foreseeing a radio luminosity $\sim$10
times lower than the one measured here.

###  0.8 Acknowledgments

B.O. gratefully acknowledges support from the McWilliams Postdoctoral
Fellowship at Carnegie Mellon University. Y.P.Y. is supported by the National
Natural Science Foundation of China grant No. 12003028 and the National SKA
Program of China (2022SKA0130100). The National Radio Astronomy Observatory is
a facility of the National Science Foundation operated under cooperative
agreement by Associated Universities, Inc. This work is partly based on
observations carried out under project number W22BS with the IRAM NOEMA
Interferometer. IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN
(Spain). Partly based on observations made with the Gran Telescopio Canarias
(GTC), installed at the Spanish Observatorio del Roque de los Muchachos of the
Instituto de Astrofísica de Canarias, on the island of La Palma. This work is
partly based on data obtained with MEGARA/MIRADAS instrument, funded by
European Regional Development Funds (ERDF), through Programa Operativo
Canarias FEDER 2014–2020. We wish to thank Armando Gil de Paz (Facultad
Ciencias Físicas, Universidad Complutense de Madrid) for his valuable support
to the MEGARA data analysis. This research made use of APLpy, an open-source
plotting package for Python hosted at http://aplpy.github.com. This research
made use of Astropy, a community-developed core Python package for Astronomy
[37].

###  0.9 Authors contribution

G.B. led the VLA and NOEMA observational campaigns, conducted the VLA data
calibration, analysis and interpretation, and led the paper writing. L.P.,
Y.P.Y., B.Z., S.S. worked on the interpretation of the results. E.P., L.N.,
S.Q, A.M.N.G., A.R. conducted the GTC/MEGARA observations, data analysis, and
interpretation. C.F., R.T. worked on the NOEMA data calibration and analysis.
B.O. realized the host galaxy broad-band SED fitting. A.G. led the GTC/MEGARA
proposal. R.P. contributed to the NOEMA proposal preparation. All authors
contributed to the discussion of the results presented and commented on the
manuscript.

###  0.10 References

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## Extended Data

Figure 1: Overplot of the 6 GHz VLA image from [13] (in colors) with contours
from the 15 (black) and 22 (red) GHz VLA images from this work. The purple
cross indicated the host galaxy center, while the blue cross is the FRB
position. Figure 2: Radio spectra of the 3 PRSs known so far. A power-law fit
is reported for each one. The green dashed lines indicate the maximum and
minimum slope consistent with measurements within errors. The yellow triangles
represent the NOEMA upper limits. The blue shaded area represents the region
not detectable by the VLA, in the range 1-50 GHz (we assume a representative
RMS=1 $\mu$Jy/beam, reachable in $\sim$10h at 6 GHz). Figure 3: SED of the
cold dust emission of FRB 20201124A based on the NOEMA upper limits at 236 GHz
and 250 GHz (cyan squares) computed in one beam towards the FRB position as
seen in the VLA 15 GHz map. The best fitting curves with fixed $[T_{\rm
dust}(\rm K),\beta]=[20,1.5],[20,2.0],[30,1.5],[30,2.0]$ are plotted as solid
violet, dashed violet, solid blue and dashed blue lines. Figure 4: SED of the
cold dust emission of FRB 20201124A based on the NOEMA upper limits at 236 GHz
and 250 GHz (cyan squares) computed for the region with $>3\sigma$ in the 6
GHz map. The best fitting curves with fixed $[T_{\rm dust}(\rm
K),\beta]=[20,1.5],[20,2.0],[30,1.5],[30,2.0]$ are plotted as solid violet,
dashed violet, solid blue and dashed blue lines. Figure 5: GTC /MEGARA: Map of
the intrinsic E(B-V) of the galaxy. We estimated the intrinsic E(B-V) taking
into account Galactic dust extinction and using the Balmer decrement
(H$\alpha$/H$\beta$). Pixels with an H$\alpha$ significance below a $3\sigma$
threshold are omitted. In cases where the H$\beta$ significance falls below a
$2\sigma$ threshold so that no accurate correction for dust extinction is
possible, we give a lower limit of intrinsic E(B-V), and represent these
pixels in greyscale. The red and black crosses represent the FRB and galactic
centre, respectively, as in Figure 1. The magenta rectangle define the
galactic region encompassing six adjacent MEGARA pixels surrounding the FRB
and PRS zone. Table 1: Journal of radio observations presented in this work,
including the estimated flux densities for the PRS.

Telescope | Date | Frequency | Bandwidth | FWHM | P.A. | RMS | Flux density
---|---|---|---|---|---|---|---
| dd/mm/yyyy | (GHz) | (GHz) | (arcsec) | (deg) | ($\mu$Jy/beam) | ($\mu$Jy)
VLA† | 06/03/2022 | 6 | 4 | 0.38$\times$0.36 |  $-$51 | 2.0 | 8.2$\pm$3.8
VLA | 07/10/2021 | 15 | 6 | 0.38$\times$0.36 | +3.6 | 2.3 | 20.0$\pm$3.5
VLA | 03-07/01/2022 | 22 | 8 | 0.37$\times$0.32 |  $-$53 | 3.0 | 30.0$\pm$9.7
NOEMA | 12/03/2023 | 236.5 | 7.6 | 0.7$\times$0.42 | +15 | 43 | $<130^{\ddagger}$
NOEMA | 12/03/2023 | 250 | 7.6 | 0.66$\times$0.39 | +15 | 53 | $<160^{\ddagger}$

Table 2:  FRB sample with measurements (or upper limits) of RM and flux
densities of the persistent radio source. aThe observed dispersion measures
(DMs) of FRBs. bThe Galactic DM contributions from the references or the
NE2001 from FRB catalog [38]. cThe measured/inferred redshifts of FRBs. The
redshifts in the top row are directly measured by the host galaxies of FRB.
The redshifts in the bottom row are inferred by the extragalactic DMs, see
[8]. The $\Lambda$CDM cosmological parameters are taken as $\Omega_{\rm
m}=0.315$, $\Omega_{b}h^{2}=0.02237$, and $H_{0}=67.36\,{\rm
km\,s^{-1}\,Mpc^{-1}}$ [39]. d The luminosity distance calculated by redshift.
e The observed RMs of FRBs. For the repeating sources, we take the largest
absolute value for the observed RMs. f The observed PRS flux densities. g The
measured frequencies of the persistent radio source. h The specific
luminosities of the PRS inferred by the observed flux density and distance.
References: (1) [40]; (2) [41]; (3) [42]; (4) [43]; (5) [44]; (6) [45]; (7)
[46]; (8) [47]; (9) [3]; (10) [48]; (11) [49]; (12) [50]; (13) [51]; (14)
[52]; (15) [53]; (16) [54]; (17) [55]; (18) [56]; (19) [57]; (20) [58].

FRB Name | ${\rm DM_{obs}}^{\rm a}$ | $\rm{DM_{MW}}^{\rm b}$ | $z^{\rm c}$ | $d_{\rm{L}}^{\rm d}$ | ${\rm RM}^{\rm e}$ | $F_{\nu}^{\rm f}$ | $\nu^{\rm g}$ | $L_{\nu}^{\rm h}$ | References
---|---|---|---|---|---|---|---|---|---
($\mathrm{pc\,cm^{-3}}$) | ($\mathrm{pc\,cm^{-3}}$) |  | ($\mathrm{Gpc}$) | ($\mathrm{rad\,m^{-2}}$) | ($\mu\mathrm{Jy}$) | ($\mathrm{GHz}$) | ($10^{29}\mathrm{erg\,s^{-1}Hz^{-1}}$) |  |
FRB 20121102A | $557$ | $188$ | $0.19273$ | $0.98$ | $1.4\times 10^{5}$ | 180 | $1.7$ | $2.1$ | 1,2,3,4
FRB 20180916B | $348.76$ | $200$ | $0.0337$ | $0.15$ | $-114.6$ | $<18$ | $1.6$ | $<0.0048$ | 5,6
FRB 20180924B | $361.42$ | $40.5$ | $0.3214$ | $1.74$ | $14$ | $<20$ | $6.5$ | $<0.72$ | 7
FRB 20181112A | $589.27$ | $102$ | $0.47550$ | $2.76$ | $10.9$ | $<21$ | $6.5$ | $<1.91$ | 8
FRB 20190520B | $1204.7$ | $113$ | $0.241$ | $1.25$ | $-3.6\times 10^{4}$ | $202$ | $3$ | $3.8$ | 9,10
FRB 20201124A | $413$ | $140$ | $0.0978$ | $0.47$ | $-889.5$ | $20$ | $15$ | $0.053$ | 11, this work
FRB 20210117A | $729.1$ | $34$ | $0.214$ | $1.10$ | $43$ | $<10$ | $6$ | $<0.15$ | 12
FRB 20110523A | $623.3$ | $43.52$ | $0.58^{+0.21}_{-0.21}$ | $3.5^{+1.6}_{-1.4}$ | $-186.1$ | $<40$ | $0.8$ | $<5.8^{+6.6}_{-3.7}$ | 13
FRB 20150215A | $1105.6$ | $427.2$ | $0.69^{+0.22}_{-0.22}$ | $4.3^{+1.7}_{-1.6}$ | $1.5$ | $<6.48$ | $10.1$ | $<1.4^{+1.4}_{-0.9}$ | 14
FRB 20150418A | $776.2$ | $188.5$ | $0.59^{+0.21}_{-0.21}$ | $3.6^{+1.6}_{-1.4}$ | $36$ | $<70$ | $1.4$ | $<11^{+12}_{-7}$ | 15
FRB 20150807A | $266.5$ | $36.9$ | $0.17^{+0.10}_{-0.11}$ | $0.85^{+0.57}_{-0.57}$ | $12$ | $<240$ | $5.5$ | $<2.1^{+3.7}_{-1.8}$ | 16
FRB 20160102A | $2596.1$ | $13$ | $3.04^{+0.51}_{-0.48}$ | $26.4^{+5.4}_{-4.9}$ | $-220.6$ | $<30$ | $5.9$ | $<249^{+112}_{-84}$ | 17,18
FRB 20180309A | $263.42$ | $44.69$ | $0.16^{+0.10}_{-0.10}$ | $0.79^{+0.57}_{-0.51}$ | $<150$ | $<105$ | $2.1$ | $<0.8^{+1.5}_{-0.7}$ | 19
FRB 20191108A | $588.1$ | $52$ | $0.53^{+0.20}_{-0.20}$ | $3.1^{+1.5}_{-1.3}$ | $474$ | $<213$ | $1.4$ | $<24^{+29}_{-16}$ | 20

## Supplementary Methods

###  0.11 VLA observations

Observations with the VLA were performed in B-configuration at 15 GHz (Ku-
band) and 22 GHz (K-band), under DDT project 21B-330 [PI: Bruni]. The 22 GHz
observations were divided into two $\sim 2$-hours blocks, and then
concatenated in order to achieve a similar RMS to the 15 GHz ones. Data
calibration was performed with the CASA111https://casa.nrao.edu/, [59] VLA
pipeline, and images deconvolution through the TCLEAN task. The signal to
noise ratio at 15 GHz allowed us to apply the robust weighting scheme to
improve the angular resolution, obtaining a final value for the FWHM similar
to the 22 GHz one. The RMS was 2–3 $\mu$Jy/beam at both frequencies.

###  0.12 NOEMA observations

Observations with the NOrthern Extended Millimeter Array (NOEMA) were carried-
out through project W22BS [PI:Bruni] during March 2023 using the B array
configuration with 12 antennas. The phase center was set at RA, DEC =
05:08:03.470, 26:03:38.50. The Band 3 (1 mm) receiver was tuned at 244 GHz, so
that the Lower Side Band (LSB) was centered at 236.5 GHz and the Upper Side
Band (USB) at 250 GHz. Visibilities were calibrated using the CLID pipeline
within the GILDAS software 222www.iram.fr/IRAMFR/GILDAS. The absolute flux
scale accuracy is of the order 10%. Imaging and deconvolution were performed
using MAPPING applying natural weighting. The final continuum maps reach a
1$\sigma$ RMS of 4.3$\times 10^{-5}$ Jy/beam in LSB and 5.3$\times 10^{-5}$
Jy/beam in USB. The clean beams are $0.70\times 0.42$ arcsec2 and $0.66\times
0.39$ arcsec2 (PA = 15 deg) in LSB and USB, respectively.

No clear detection was found in either LSB or USB data towards the FRB
position, as derived from 15 GHz VLA data. We derived $3\sigma$ upper limits
of 0.13 mJy and 0.15 mJy/beam in LSB and USB, respectively.

###  0.13 GTC/MEGARA observations

The GTC/MEGARA instrument ([60]; [61]) collected a series of 6 observations on
October 26, 2022, using the Large Compact Bundle (LCB) Integral Field Unit
(IFU) mode. It provides a field of view (FoV) of $12.5\times 11.3$ arcsec2
using 567 fibres in a hexagonal tessellation with a spaxel size of
$0\overset{\prime\prime}{.}62$. In order to cover the main optical emission
lines, useful to derive the SFR spatial distribution within the galaxy, the
observations were carried out with two different low resolution (LR, resolving
power $\mathrm{R}\sim 6000$) volume-phase holographic (VPH) grating: VPH570-LR
(LR-V), and VPH675-LR (LR-R), whose specifications can be found in Table 1.
This produces a combined spectrum covering the spectral range from 4350 to
7288 Å (see Figure 3). This range, at the redshift of our target, includes
spectral emission lines such as H$\alpha$, H$\beta$, $[\mathrm{OIII}]$,
$[\mathrm{NII}]$, needed for deriving the spatial distribution of the galaxy
parameters like SFR, intrinsic extinction and velocity map. The measured
seeing during the observations were in the range
$0\overset{\prime\prime}{.}7$–$0\overset{\prime\prime}{.}9$.

Table 1: GTC/MEGARA observations journal. Date | MEGARA VPH | Band | Exposure | R.L.D.* | R
---|---|---|---|---|---
28 October 2022 | LR-V | 5165.6 – 6176.2 | 3 $\times$ 1800 | 0.27 | 6078
28 October 2022 | LR-R | 6158.3 – 7287.7 | 3 $\times$ 1200 | 0.31 | 6100
*Reciprocal linear dispersion. |  |  |  |
Figure 1: GTC/MEGARA: Map of the net counts of the galaxy. The white contours
represent the VLA image at 15 GHz, whereas the red and black crosses represent
the FRB and galactic centre, respectively. The magenta rectangle define the
galactic region encompassing six adjacent MEGARA pixels surrounding the FRB
and PRS zone.  Figure 2: Compilation of R+V grisms GTC/MEGARA spectra of the
six pixels surrounding the FRB region (see the magenta box in Figure 1. The
colored vertical dotted lines indicate the spectral positions of the lines at
the reference redshift (z=0.0978), revealing that the FRB region is
blueshifted by $60-80$ km/s relative to the reference redshift (see 7). We
observe an absorption line at approximately $5493\AA$ across all spaxels in
the region; however, the nature of this spectral feature remains unrecognized
by our analysis.

We used the MEGARA data reduction pipeline (DRP) v0.12.0 [62] to process the
data. We followed the same procedure as described in detail in [63]. The final
spectra where flux calibrated using the standard stars HR8634, available in
the ESO spectro-photometric standards database ([64]). In Figure 1, we present
the map of the net counts of the galaxy.

###  0.14 Host galaxy characterization with GTC/MEGARA

The continuum emission of the projected galaxy extends approximately 9 kpc
along the major axis ($a$) and around 5 kpc along the minor axis ($b$),
without accounting for PSF effects. By assuming a straightforward circular
galactic geometry, we calculated an inclination angle $i$ of approximately
56.3 degrees based on the ratio of $a/b$.

Figure 2 shows a compilation of optical spectra of the galaxy. In particular,
we show the spectra of H$\beta$, [O III] 5007 and H$\alpha$ corrected for
Galactic extinction for the six pixels surrounding the FRB and PRS (referred
to as the FRB region), from left to right. The FRB region is delineated by a
magenta rectangle in Figure 1. These spectra show blue-shifted emission lines
with respect to the reference redshift (z=0.0978), indicating rotational
velocities between $-60$ and $-80$ km/s, as shown in Figure 7. The H$\alpha$
line in the right panels of Figure 2 shows a clear and highly significant
detection. Despite its intrinsic weakness compared to H$\alpha$ by a factor of
about 3, H$\beta$ performs quite well, even in a spectral region with a RMS
roughly 2.7 times worse than that in the H$\alpha$ spectral region. Indeed,
the standard deviation of the continuum is about $1.7\times 10^{-17}$
erg/s/cm2/$\AA$ for H$\beta$ compared to $6.2\times 10^{-18}$ erg/s/cm2/$\AA$
for H$\alpha$. Conversely, the [O III] 5007 line shown in the middle panels of
Figure 2 is either absent or very weak in all spectra. We will analyse this
spectral feature in more detail later in this section. We have identified an
unexpected absorption feature, consistently appearing at an observed
wavelength of approximately $\lambda\sim 5493$ Åin all six pixels within the
FRB region. We anticipate that this feature is also present in the stacked
spectrum of the entire galaxy which we will introduce later in this session.

We obtained measurements of the emission lines H$\alpha$, H$\beta$, [O III]
5007, and [NII] 6584 at the spaxel level through the following procedure: as a
first step we corrected each spectrum for Galactic extinction (E(B-V) = 0.652)
using the [65] extinction law. Subsequently, we removed the underlying stellar
continuum by subtracting a pseudo-continuum, defined as a straight line
connecting adjacent bluer and redder regions to the emission component, devoid
of other relevant spectral features. We used a Python-Astropy routine based on
the Levenberg-Marquardt algorithm and least squares statistic [66] to fit up
to two Gaussian components to each line, selecting as the best-fit solution
the one that minimized the reduced chi-square statistic toward 1. We assessed
the validity of our Gaussian fits by directly integrating the total line flux
within a wavelength interval centered in the line centroid and spanning twice
the standard deviation of the best-fitted Gaussian for each line. Finally, we
used a bootstrapping technique to quantify uncertainties in the Gaussian
parameters for each emission line (i.e., amplitude, line’s centroid, and
standard deviation). We iteratively sampled and fitted the spectra $1000$
times, introducing Gaussian noise with an amplitude equal to the RMS of the
spectrum within the spectral regions corresponding to the pseudo-continuum of
the respective line. This bootstrapping process yields a Gaussian distribution
of parameter values, with the standard deviation providing a robust estimate
of the uncertainty associated with the fitted parameter.

Figure 3: Stacking GTC/MEGARA R+V grisms (from left to right) for the
H$\beta$, [O III] 5007, and H$\alpha$ regions is presented. (Top) Stacked
spectrum from the six spectra in the FRB region (as shown in Figure 2);
(Bottom) The stacked spectrum of the entire galaxy. Individual spectra were
shifted to align with the reference redshift (z=0.0978) before stacking to
compensate for any H$\alpha$-related velocity effect. Colored vertical dotted
lines mark the centroids of the three lines at the reference redshift. The
previously unidentified absorption line at approximately 5490 Å, observed in
the individual spectra of the FRB region (see Figure 2), is also evident in
the stacked spectra of the entire galaxy. Figure 4: GTC /MEGARA: Map of
H$\alpha$. Pixels with an H$\alpha$ significance below a $3\sigma$ threshold
are omitted. The red and black crosses represent the FRB and galactic centre,
respectively. The magenta rectangle define the galactic region encompassing
six adjacent MEGARA pixels surrounding the FRB and PRS zone.

In conjunction with our pixel-level analysis of the galaxy, this study
includes two additional spectra, shown in Figure 3. In the top panel we show
the composite spectrum derived by stacking the R and V grism spectra of the
six pixels defining the FRB region previously described in Figure 2. This
composite spectrum has been carefully corrected to a reference redshift of z =
0.0978 to compensate for any H$\alpha$-related velocity effects. In addition,
in the bottom panel of Figure 3 we present the global spectrum of the whole
galaxy obtained by stacking the R and V spectra of all pixels within the
rectangular region delimited by RA 5h 08m 3.4s - 3.6s and DEC 26° 03’ 35” -
40”. The fluxes and detection significance (expressed in signal-to-noise of
the emission line) for both stacked spectra are listed in Table 2.

Table 2: Emission lines in the stacked spectra. Fluxes are in units of [erg/s/cm2]. Line | Param. | FRB reg. | Global
---|---|---|---
H$\alpha$ (1st comp.) | flux | $3.03\times 10^{-15}$ | $1.53\times 10^{-14}$
| S/N | $102.1$ | $151.5$
H$\alpha$ (2nd comp.) | flux |  | $1.01\times 10^{-14}$
| S/N |  | $75.3$
H$\beta$ | flux | $1.19\times 10^{-15}$ | $7.96\times 10^{-15}$
| S/N | $15.6$ | $19.3$
[N II] 6584 | flux | $1.20\times 10^{-15}$ | $9.69\times 10^{-15}$
| S/N | $38.7$ | $57.9$
[O III] 5007 | flux | $5.63\times 10^{-16}$ | $3.9\times 10^{-15}$
| S/N | $1.1$ | $12.3$

The motivation behind the creation of these stacked spectra is to increase the
RMS values within the spectral regions under investigation, thereby improving
the statistical significance of weak spectral lines, including the H$\beta$
and [O III] 5007 lines. This enhancement allows us to investigate the
comprehensive properties of the galaxy, which are discussed in the following
sections of this appendix.

Figure 5: GTC/MEGARA: Maps, from top to bottom, of the emission lines [N II]
6584, H$\beta$, and [O III] 5007. Pixels with an H$\alpha$ significance below
a $3\sigma$ threshold are omitted. In cases where the line significance falls
below a $2\sigma$ we give a line upper limit (see 0.13), and represent these
pixels in grey-scale. The red and black crosses represent the FRB and galactic
centre, respectively. The magenta rectangle define the galactic region
encompassing six adjacent MEGARA pixels surrounding the FRB and PRS zone.

#### Emission line maps

Figure 4 shows the galaxy’s H$\alpha$ emission line map, which has been
corrected for Galactic extinction. In our analysis, we need reliable measures
of the H$\alpha$ line, hence we exclude pixels with H$\alpha$ signal-to-noise
(S/N) $<3$. In this map, only pixels with a signal-to-noise ratio (S/N) of
H$\alpha\geq 3$ are shown. This H$\alpha$ map serves as the starting point for
our analysis of the GTC/MEGARA optical spectra, enabling us to derive crucial
physical properties of the gas phase in the galaxy, including gas kinematics
and dynamics, ionization levels, star formation rate (SFR), and gas-phase
metallicity.

Figure 5 shows maps of three emission lines within the wavelength range
covered by our GTC/MEGARA data, specifically, [N II] 6584, H$\beta$, and [O
III] 5007 lines, from top to bottom. Each panel in Figure 5 uses two color
schemes: one represents pixels where the line is detected with a signal-to-
noise ratio (S/N) of $\geq 2$ (colored scale pixels), and the other represents
pixels with flux upper limits. We determined upper limits for the lines at a
$2\sigma$ significance level for all pixels where the signal-to-noise ratio of
the line fell below $2$. Here, $\sigma$ represents the standard deviation of
the Gaussian best-fit of the stacked spectrum of the whole galaxy, as shown in
the lower panel of Figure 3. The [N II] 6584 and H$\beta$ line maps exhibit
similar flux distribution patterns as observed in the H$\alpha$ map, albeit
with fainter emissions towards the galaxy’s outer regions (although H$\beta$
is intrinsically fainter, approximately three times less intense than
H$\alpha$). The [O III] 5007 line, instead, is undetected at a $2\sigma$ level
across nearly the entire galaxy. This faint emission in the [O III] lines may
be attributed to either high gas metallicity or a low ionization state
resulting from diminished UV radiation [67]. We will explore this ionization-
metallicity degeneracy in the next subsection.

#### Derived quantities from optical spectra

Figure 6: The BPT diagram [68] correlating log([O III] 5007/H$\beta$) and
log([N II] 6584/H$\alpha$). The red star marks the position of the entire
galaxy’s stacked spectrum, while the magenta circle represents the position of
the FRB region’s stacked spectrum. Error bars represent $1\sigma$ standard
errors for the emission line ratios. The arrow in the FRB region symbol
indicates an upper limit in the log([O III] 5007/H$\beta$) ratio, as the [O
III] line remains undetected at a $2\sigma$ level. The green curve, derived
from [69], serves as an empirical criterion to distinguish ionization sources,
with both data points lying below it. This positioning demonstrates that the
ionization source in the galaxy is predominantly radiation from massive and
young stars.

In the previous section, we established that the [O III] emission is lacking
in the FRB region and generally weak and undetectable throughout the entire
galaxy. This intriguingly suggests that the faintness of [O III] lines might
be the sign of recent star formation quenching. Following a sharp decline in
star formation, the high-energy UV radiation emitted by short-lived, massive
O-type stars decreases significantly within a few million years. This would
lead to a rapid decline in the luminosity of lines with high ionisation, such
as the [O III] emission lines. However, emission lines associated with lower
ionisation energies, such as the Balmer lines, would continue to be detectable
as long as late O and early B-type stars with longer lifetimes, remain in the
main sequence phase. However, [O III] 5007 emission can be depressed in a
similar way also by high gas metallicity, leading to the so called ionization-
metallicity degeneracy [67].

The ionization-metallicity degeneracy on the [O III] emission can be mitigated
by using pairs of emission line ratios, each related to either metallicity or
ionization. For example, the [N II] 6584/[O II] 3727 or [S II]/H$\alpha$
ratios can serve as proxies for metallicity, whereas [O III] 5007/H$\alpha$ or
[Ne III] 3869/[O II] 3727 ratios can serve as proxies for ionization levels
[67, 70, 71, 72]. Our GTC/MEGARA wavelength coverage is limited to the four
aforementioned lines, thus limiting us to disentangling the two underlying
physical properties responsible for the faintness of the [O III] 5007 line in
the galaxy. Nevertheless, the four aforementioned emission lines allow us to
explore the diagnostic diagram introduced by [68], hereafter referred to as
BPT, which relates log([O III] 5007/H$\beta$ and log([N II] 6584/H$\alpha$ to
the ionization level and metallicity contempt in the galaxy.

The location of a galaxy (or galactic region) in the BPT plane is very
sensitive to the source of UV radiation responsible for ionising its gas.
These UV photon sources can be radiation from the hot stellar photospheres of
recently formed massive stars, AGN activity, shock processes, or combinations
of these. Figure 6 shows the BPT diagram of the galaxy for the two stacked
spectra of the FRB region and the whole galaxy, as well as the distribution of
the SDSS-DR8 [73] Main galaxies as a reference for low redshift galaxies. Note
that the data point of the stacked spectrum representative of the FRB region
is an upper limit for [O III] 5007/H$\beta$ since the [O III] line is not
detected at least at the $2\sigma$ level in the stacked spectrum. We have
adopted the criterion of [69] to distinguish between ionisation due to star
formation (i.e. data points lying below the green curve in Figure 6), and
other radiation sources (e.g., ionization due to active galactic nuclei).
According to the [69], both the stacked FRB spectrum and the entire galaxy
have radiation dominated by star formation.

#### The gas phase metallicity

We estimated the gas phase metallicity of the galaxy using the strong line
metallicity indicator O3N2, as initially defined by [74] as log([O III]
5007/H$\beta$/[N II] 6584/H$\alpha$). We adopted the metallicity calibration
by [75] with a solar abundance value of 12+log(O/H)${}_{\odot}=8.69$ [76]. We
measured slightly super solar values of 12+log(O/H)$=8.75$ and $8.77$ in the
stacked spectra of the entire galaxy and the FRB region, respectively. To
mitigate the ionization-metallicity degeneracy in the [O III] line, we used
predictions from the [67] photoionization models using the four emission lines
defining the BPT diagram. We found that the stacked spectrum of the entire
galaxy is compatible with emission from HII regions with ionization level
U$\sim-3.4$, typical of a star-forming galaxy, and solar metallicity. For the
FRB region, we obtained a metallicity compatible with solar metallicity, and
an upper limit in the ionization level U $\leq-3.5$. the upper limit in the
parameter U is due to the low significance level of the [O III] line of the
stacked spectrum in the FRB region. It is worth noting that the upper limit on
the parameter U arises from the lower significance level of the [O III] line
in the stacked spectrum of the FRB region. Consequently, given the current
depth and Signal-to-Noise Ratio (SNR) of our optical data, along with the
absence of information from other emission lines capable of independently
sensing changes in the ionization parameter and gas phase metallicity, we
cannot conclusively confirm or exclude the possibility that the FRB region has
recently experienced a decline in star formation.

Figure 7: GTC/MEGARA: Circular velocity map (left) and velocity dispersion map
(corrected for instrumental resolution) derived from H$\alpha$ emission.

#### Kinematics and dynamics of the gas-phase

The left panel of Figure 7 shows the circular velocity map (velcirc) for the
galaxy. We derived this map by de-projecting the H$\alpha$ radial velocity
($\text{vel}_{\text{los}}$), corrected for the relative velocity at the
reference redshift (z=0.0978), onto the galactic plane using the equation:

$\text{vel}_{\text{circ}}=\frac{\text{vel}_{\text{los}}}{\sin(i)},$ (12)

where $i=56.3^{\circ}$ represents the inclination angle of the galaxy, which
we estimated based on the net-counts map (Figure 1). The resulting velocity
map demonstrates that the galaxy exhibits well-ordered gas kinematics
predominantly characterized by circular orbits. Rotation velocities range from
0 to $-100$ km/s (at the galactic center and a radial distance of $\sim 3$
kpc, respectively) in the northern region, and from $0$ to approximately 150
km/s (at a radial distance of $\sim 3.5$ kpc) in the southern region.

In the right panel of Figure 7 we show the H$\alpha$ velocity dispersion map
($\sigma(\text{H}\alpha)$) for the galaxy. To account for instrumental
resolution effects, we subtracted in quadrature the sigma value (in units of
Å) derived from the Gaussian fit of the strong sky line at 7276.405 Å from
that of the measured H$\alpha$. With this approach we obtain a reliable
correction as the reference sky line is nearby to the observed H$\alpha$ line
and clearly detected. The galaxy shows dynamics typical of a local star-
forming galaxy [77] with velocity dispersion values reaching up to 80 km/s in
the center and lower values of only $10-20$ km/s in the outskirts.

We report here a summary of our findings regarding the GTC/MEGARA data. We
found a star-forming galaxy with an inclination angle of approximately 56 deg.
The H$\alpha$ velocity map reveals predominantly circular gas kinematics with
velocities ranging from $-100$ km/s (north) to 150 km/s (south). Emission line
measurements, including H$\alpha$, H$\beta$, [O III] 5007, and [NII] 6584,
corrected for extinction, provide insights into the gas metallicity and star
formation rates (SFR). Gas metallicity is slightly super-solar, and the BPT
diagram indicates radiation dominated by star formation. The SFR map shows a
peak near the galactic center as identified by the HST images, with
$\Sigma\;$SFR of $\sim 0.32$ to $0.64$ M$\odot$/yr/kpc2. The FRB region
exhibits lower SFR levels, and analysis of stacked spectra yields an estimated
SFR of $0.41^{+0.02}{-0.01}$ M$\odot$/yr, significantly lower than SFR
obtained from the radio emissions.

###  0.15 Host galaxy SED Analysis

We performed a re-analysis of the host galaxy SED to include the recently
acquired NOEMA data. The NOEMA limits set a stringent upper bound to the flux
in the mm range, and are over predicted by the model presented in Paper I. We
note that these limits seem to be marginally consistent with the best-fit
model presented by [78], while [79] does not present their SED out to these
wavelengths for comparison.

In order to determine how the NOEMA data at far-infrared wavelengths (FIR)
changes the results, we re-modeled the SED using prospector [80] with the
methods utilized in Paper I. We similarly include data spanning the $ugrizJHK$
bands, and in this case we include all four WISE bands ($W1$, $W2$, $W3$,
$W4$) in order to properly constrain the mid-infrared (MIR) SED. In order to
constrain the shape of the MIR SED, we likewise included the contribution of
dust emission from polycyclic aromatic hydrocarbons (PAHs), which uses the
model presented by [81]. We additionally test for the presence of an AGN
contributing to the MIR. These effects were not included in Paper I.

The dust emission model has three free parameters: the fraction of dust mass
in PAHs $Q_{\textrm{PAH}}$, lower limit to the strength of starlight in units
of the Milky Way $U_{\textrm{min}}$, and the fraction $\gamma_{e}$ of dust
irradiated by starlight stronger than $U_{\textrm{min}}$, where the power-law
distribution of starlight is a power-law with slope $\alpha=2$. We adopted
broad uniform priors for these parameters: $0.1<Q_{\textrm{PAH}}<10.0$,
$0.1<U_{\textrm{min}}<25$ (bounded by the wavelengths included in the fit),
and $0.0<\gamma_{e}<0.15$, as outlined in [82]. The AGN model utilizes the
templates from [83] and is parametrized by the fraction of bolometric stellar
luminosity $f_{\textrm{AGN}}$ emitted by the AGN, and the optical depth of the
torus $\tau_{\textrm{AGN}}$. We utilize log uniform priors of
$0<f_{\textrm{AGN}}<3$ and $5<\tau_{\textrm{AGN}}<150$, as in [82].

The best-fit SED is shown in Figure 8. We derive the following best-fit
parameters: a significant intrinsic extinction $A_{V}=0.81^{+0.17}_{-0.14}$, a
near solar metallicity $Z/Z_{\odot}=0.7\pm 0.2$, a stellar mass
$\log(M_{*}/M_{\odot})=9.98^{+0.07}_{-0.08}$, an old mass-weighted stellar age
$t_{m}=3.5\pm 1.2$ Gyr, an e-folding time $\tau=4.1^{+2.9}_{-2.4}$, a star
formation rate SFR=$2.4\pm 0.3$ $M_{\odot}$ yr-1. The dust parameters, in
particular $Q_{\textrm{PAH}}=7\pm 2\%$, are relatively unconstrained, besides
strongly favoring $\gamma<0.25$ and a larger value of $U_{\textrm{min}}>1.5$
($3\sigma$ confidence level). The preference towards a higher value of
$U_{\textrm{min}}=17^{+5}_{-9}$ is due to the fact that a larger value of
irradiance leads to a higher dust temperature, pushing the peak of the SED
away from the NOEMA limits. We further find that an AGN is likely only
contributing at the $\sim 1\%$ level of the total bolometric luminosity, and
at most $<5\%$, in agreement with the previous value ($\sim 4\%$) derived by
[78].

In comparison to Paper I, we find a smaller SFR and an older stellar
population, closer to the values reported by [78], but still with a slightly
smaller stellar mass. A similar analysis not including an AGN returns nearly
identical parameters for $A_{V}$, $Z$, $M_{*}$, $t_{m}$, $\tau$, and SFR
compared to Paper I (i.e., consistent within $1\sigma$ errors), while only
changing the dust parameters, similarly favoring a high
$U_{\textrm{min}}\approx 20-25$.

Following [84], we can convert the dust model parameters [81] to a
temperature, for better comparison to our analytical constraints on the NOEMA
data. We find a dust temperature of $T_{\textrm{dust}}=29.5^{+1.3}_{-3.4}$ K.
We note that the analytical constraints derived previously in §0.12 are based
on the local limits to the PRS, while here we use the global limits of the
entire galaxy.

Figure 8: SED of the host galaxy of FRB 20201124A from radio to X-rays. The
NOEMA limits are shown as dark brown downward triangles. We have only included
the VLA data obtained in the D configuration (reported in Paper I).