An Elusive Population of Massive Disk Galaxies Hosting Double-lobed Radio-loud AGNs

Zihao Wu

0000-0002-8876-5248]Zihao Wu
Department of Astronomy, School of Physics, Peking University, Beijing 100871, People's Republic of China

0000-0001-6947-5846]Luis C. Ho
Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, People's Republic of China
Department of Astronomy, School of Physics, Peking University, Beijing 100871, People's Republic of China

0000-0001-5105-2837]Ming-Yang Zhuang
Department of Astronomy, School of Physics, Peking University, Beijing 100871, People's Republic of China
Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, People's Republic of China

It is commonly accepted that radio-loud active galactic nuclei are hosted exclusively by giant elliptical galaxies. We analyze high-resolution optical Hubble Space Telescope images of a sample of radio galaxies with extended double-lobed structures associated with disk-like optical counterparts. After systematically evaluating the probability of chance alignment between the radio lobes and the optical counterparts, we obtain a sample of 18 objects likely to have genuine associations. The host galaxies have unambiguous late-type morphologies, including spiral arms, large-scale dust lanes among the edge-on systems, and exceptionally weak bulges, as judged by the low global concentrations, small global  indices, and low bulge-to-total light ratios (median $B/T = 0.13$). With a median  index of 1.4 and low effective surface brightnesses, the bulges are consistent with being pseudo bulges. The majority of the hosts have unusually large stellar masses (median $M_* = 1.3\times 10^{11}\, M_\odot$) and red optical colors (median $g-r = 0.69$ mag), consistent with massive, quiescent galaxies on the red sequence. We suggest that black hole mass (stellar mass) plays a fundamental role in launching large-scale radio jets, and that the rarity of extended radio lobes in late-type galaxies is the consequence of the steep stellar mass function at the high-mass end. The disk radio galaxies have mostly Fanaroff-Riley type II morphologies yet lower radio power than sources of a similar type traditionally hosted by ellipticals. The radio jets show no preferential alignment with the minor axis of the galactic bulge or disk, apart from a possible mild tendency for alignment among the most disk-dominated systems.

§ INTRODUCTION

Massive black holes (BHs) lurk at the center of all massive galaxies [Kormendy & Ho, 2013] and even in a significant fraction of less massive systems [Greene et al., 2020]. Mass accretion onto the BH generates nuclear activity that gives rise to a plethora of energetic phenomena associated with active galactic nuclei (AGNs). One of the most enduring unsolved problems is the origin of radio jets, a distinctive feature of some but not all AGNs [Urry & Padovani, 1995]. AGNs are conventionally designated as either radio-loud or radio-quiet, even if there is strong motivation to abandon these labels in favor of more physical alternatives (for a recent perspective, see Padovani, 2017). For historical reasons, two operational definitions of radio-loudness have been commonly invoked. On the one hand, radio-loudness can be demarcated on the basis of an absolute radio power, such as $P_{\rm 6\,cm} > 10^{25}\,\rm W\,Hz^{-1}\,sr^{-1}$ [Miller et al., 1990]. This has the advantage of simplicity and independence from observations at other wavelengths. On the other hand, it has become popular to define radio-loudness not by an absolute but instead by the relative strength of the radio emission. For instance, [Kellermann et al., 1989] proposed the widely embraced convention of $R \equiv L_{\nu}({\rm 6\,cm})/L_{\nu}(B)$, with the boundary between radio-loud versus radio-quiet set at $R = 10$. By this criterion, $\sim 10\%-20\%$ of optically selected quasars are radio-loud (e.g., Visnovsky et al., 1992, Hooper et al., 1995). Both of these traditional criteria encounter difficulties once we confront AGNs of lower luminosity drawn from BHs of lower mass or lower accretion rate. For the vast majority of the galaxy population and for most of their lifecycle, the radio-loudness of an AGN cannot be measured accurately without taking into account host galaxy contamination (e.g., Ho & Peng, 2001). Notwithstanding these complications, a distinctive empirical trend regarding the nature of radio jets has remained largely intact: the most powerful radio-loud AGNs, especially those that sport large-scale, double-lobed hotspots that extend beyond the confines of the host galaxy, reside nearly universally in elliptical galaxies, while the host galaxies of the radio-quiet counterparts span a wide range of optical morphologies <cit.>. The dichotomy in the host morphology in terms of radio-loudness has long been recognized in radio galaxies [Matthews et al., 1964, Zirbel, 1996, McLure et al., 2004] and confirmed by high-resolution observations of the hosts of radio-loud quasars [McLure et al., 1999, Hamilton et al., 2002, Dunlop et al., 2003] and BL Lac objects [Urry et al., 2000]. Throughout this paper, we restrict our attention to the class of radio-loud AGNs typified by extended, double-lobed radio jets.

What is the physical basis for this observational trend? The leading model suggests that powerful, relativistic jets in radio-loud AGNs form by tapping into the rotational energy of a rapidly spinning BH [Blandford & Znajek, 1977]. In an attempt to link radio-loudness with galaxy morphology, [Wilson & Colbert, 1995] proposed that radio-loud AGNs are uniquely associated with elliptical galaxies because their central BHs have been spun up by galaxy-galaxy major mergers. Spiral galaxies, on the other hand, cannot spin up BHs because they evolve largely through internal secular processes. Wilson & Colbert's merger-driven mechanism of generating BH spin explains why spiral galaxies cannot host radio-loud AGNs. [Baum et al., 1995] further suggested that the spin paradigm may further be able to account for the difference in radio power between <cit.> type I and type II radio galaxies: FR IIs are more powerful than FR Is because they have more rapidly spinning BHs.

Recent advances in BH spin measurements have severely challenged this spin paradigm. Although the number of supermassive BHs with reliable spin measurements is still quite limited, the extant evidence suggests that many supermassive BHs, at least in the nearby Universe, are rapidly spinning objects [Reynolds, 2019, Reynolds, 2021], with a significant fraction hosted by spiral galaxies <cit.>. Notably, recent observations from the Event Horizon Telescope [Akiyama et al., 2022] reveal that the central BH of our own Galaxy, Sgr A*, itself has a high spin. In light of these developments, it is no longer tenable to invoke the spin paradigm to account for the morphology dichotomy between radio-loud and radio-quiet AGNs.

Perhaps BH mass, not spin, determines jet power. After all, the fraction of galaxies that hosts radio-loud AGNs depends strongly on stellar mass [Scarpa & Urry, 2001, Best et al., 2005, Mauch & Sadler, 2007]. Now, to the extent that BH mass is closely linked with stellar mass [Magorrian et al., 1998, Kormendy & Ho, 2013], it is tempting to conclude that more massive BHs are more prone to producing radio-loud AGNs. Moreover, the very power of the radio jet increases with BH mass <cit.>. As late-type galaxies generally host less massive BHs [Greene et al., 2020], likely an indirect manifestation of their typically lower stellar masses [Moffett et al., 2016], it would be natural that disk-dominated galaxies can only sustain less powerful, and hence less extended [Ledlow et al., 2002], radio jets. These expectations are merely indicative, at best, for they are based on statistical correlations with substantial scatter. Given the significant dispersion in the relation between BH mass and radio luminosity <cit.> and the similarly loose correlation between BH mass and galaxy stellar mass for late-type galaxies [Greene et al., 2020], we should remain open to the possibility that some late-type galaxies might host radio-loud AGNs.

What concrete evidence exists, then, that disk galaxies host radio-loud AGNs? Much attention has been cast on the so-called radio-loud narrow-line Seyfert 1 galaxies (NLS1s). NLS1s designate AGNs with relatively low BH masses and high accretion rates (e.g., Boroson, 2002), which normally tend to be radio-quiet [Ulvestad et al. , 1995, Greene et al., 2006]. Consistent with their low BH masses, the hosts of NLS1s are generally disk galaxies (e.g., Orban de Xivry et al., 2011, Kim et al., 2017). A minority of NLS1s, however, exhibit blazar-like characteristics at radio and even $\gamma$-ray energies (e.g., Abdo et al., 2009, Foschini et al., 2015), strongly suggesting that at least in this sub-population late-type galaxies can launch relativistic jets. If true, this would represent a fundamental paradigm shift. However, despite numerous efforts to characterize the structure and morphology of the host galaxies of radio-loud NLS1s, disentangling the host from the glare of the bright nucleus remains challenging using ground-based images, even under conditions of exceptionally good seeing and pushing the observing bandpass to the near-infrared. While some examples of disk hosts have emerged (e.g., Kotilainen et al. , 2016, Vietri et al., 2022), others are still open to interpretation (e.g., Léon-Tavares et al., 2014, Järvelä et al., 2018, Olguín-Iglesias et al. , 2020), raising doubts as to whether they are bona fide late-type disk galaxies [Tadhunter, 2016]. The situation is considerably simpler in the case of radio galaxies, whose orientation in the plane of the sky shields the bright nucleus from the viewer. To date, only a handful of cases of extended radio lobes have been reported to be hosted by disk galaxies [Ledlow et al., 1998, Hota et al., 2011, Bagchi et al., 2014, Mao et al., 2015, Singh et al., 2015, Mulcahy et al., 2016, Vietri et al., 2022]. However, most discoveries lack detailed chance alignment analysis to confirm the radio associations, and the number of objects is too marginal for meaningful statistical analysis.

In this study, we present a sample of radio-loud AGNs with classical double-lobed radio structures that we can associate with high confidence with host galaxies having clearly late-type, disk-dominated optical morphology. These objects originate from a parent sample of objects initially selected by the Gems of Galaxy Zoo <cit.> project, which obtained high-resolution Hubble Space Telescope (HST) optical images of a collection of sources previously identified through the Radio Galaxy Zoo program [Banfield et al., 2015] to be double radio lobes apparently associated with late-type galaxies. We systematically and quantitatively analyze the HST images and estimate the probability of chance alignment for each object. From the original sample of 32 sources presented by [Keel et al., 2022], we arrive at a high-confidence sample of 18 sources that we deem to be truly associated with the optical counterpart imaged with HST. We demonstrate that extremely late-type disk galaxies can indeed produce extended radio jets, but these galaxies turn out to be very massive and mostly red. We argue that BH mass, not only BH spin, is a key factor governing jet production.

The paper is structured as follows. Section <ref> summarizes the observations used in this paper. We evaluate the probability of chance alignment and establish a high-confidence sample in Section <ref>. We describe our measurements of the radio and optical images in Section <ref> and present the statistical properties of the host galaxies and radio sources in Section <ref>. Section <ref> discusses the connection between launching radio lobes and the physical properties of the host galaxies. This paper adopts a [Chabrier, 2003] stellar initial mass function and a $\Lambda$CDM cosmology with $H_0 = 69\,\mathrm{ km\, s^{-1} \,Mpc^{-1}}$, $\Omega_\mathrm{m} = 0.29$, and $\Omega_\Lambda = 0.71$ based on the final nine-year Wilkinson Microwave Anisotropy Probe observations [Hinshaw et al., 2013].

Basic Properties of the Sample

Name    Full Name     R. A. (J2000)     Decl. (J2000)     $z_\mathrm{phot}$     $z_\mathrm{spec}$     $A_V$     $m_g$    $m_r$   $m_i$

($^{\mathrm{h}}\;{}^{\mathrm{m}}\;{}^{\mathrm{s}}$)     ($^\circ\;{}^\prime\;{}^{\prime\prime}$)                  (mag)     (mag)     (mag)     (mag)
J0209+075     SDSS J020904.75+075004.5     02 09 04.75     07 50 04.5     $0.251\pm0.024$         0.15     18.80     17.47     16.97

J0219+015     UGC 1797     02 19 58.73     01 55 48.7     $0.033\pm0.016$     0.041     0.13     14.48     13.59     13.12

J0802+115     SDSS J080259.73+115709.7     08 02 59.73     11 57 09.7     $0.173\pm0.023$         0.07     18.79     17.92     17.42

J0806+062     SDSS J080658.46+062453.4     08 06 58.46     06 24 53.4     $0.112\pm0.055$         0.06     18.60     18.20     17.89

J0813+552     SDSS J081303.10+552050.7     08 13 03.10     55 20 50.7     $0.279\pm0.035$         0.12     20.39     19.51     19.13

J0823+033     SDSS J082312.91+033301.3     08 23 12.91     03 33 01.3     $0.122\pm0.011$         0.08     16.63     15.59     15.11

J0832+184     SDSS J083224.82+184855.4     08 32 24.82     18 48 55.4     $0.122\pm0.005$     0.114     0.09     17.11     16.05     15.60

J0833+045     SDSS J083351.28+045745.4     08 33 51.28     04 57 45.4     $0.246\pm0.03$         0.09     19.77     18.99     18.64

J0847+124     SDSS J084759.90+124159.3     08 47 59.90     12 41 59.3     $0.157\pm0.025$     0.175     0.07     18.38     17.15     16.61

J0855+420     B3 0852+422     08 55 49.15     42 04 20.1     $0.191\pm0.025$         0.08     19.45     18.14     17.55

J0901+164     SDSS J090147.17+164851.3     09 01 47.17     16 48 51.3     $0.232\pm0.026$         0.08     19.28     18.26     17.79

J0903+432     SDSS J090305.84+432820.4     09 03 05.84     43 28 20.4     $0.369\pm0.041$     0.373     0.05     20.70     19.17     18.46

J0914+413     B3 0911+418     09 14 45.54     41 37 14.3     $0.149\pm0.010$     0.140     0.05     16.35     15.22     14.72

J0919+135     SDSS J091949.07+135910.7     09 19 49.07     13 59 10.7     $0.417\pm0.049$         0.10     21.47     20.50     20.23

J0926+465     SDSS J092605.17+465233.9     09 26 05.17     46 52 33.9     $0.202\pm0.040$         0.04     19.46     18.21     17.74

J0941+312     B2 0938+31A     09 41 03.63     31 26 18.7     $0.366\pm0.037$         0.05     20.93     19.56     19.00

J0956+162     SDSS J095605.87+162829.9     09 56 05.87     16 28 29.9     $0.341\pm0.066$         0.09     20.43     19.18     18.77

J0958+561     SDSS J095833.44+561937.8     09 58 33.44     56 19 37.8     $0.247\pm0.021$         0.03     19.05     17.78     17.35

J1128+241     SDSS J112811.63+241746.9     11 28 11.63     24 17 46.9     $0.121\pm0.022$         0.05     17.88     17.34     17.04

J1136+125     SDSS J113648.57+125239.7     11 36 48.57     12 52 39.7     $0.059\pm0.021$     0.034     0.07     17.37     17.02     16.80

J1303+511     SDSS J130300.80+511954.7     13 03 00.80     51 19 54.7     $0.122\pm0.037$         0.03     19.09     18.35     17.95

J1322+270     IC 4234     13 22 59.87     27 06 59.1     $0.027\pm0.005$     0.034     0.06     14.26     13.63     13.31

J1328+571     SDSS J132809.31+571023.3     13 28 09.31     57 10 23.3     $0.032\pm0.029$         0.03     16.91     16.78     16.69

J1349+454     SDSS J134900.13+454256.5     13 49 00.13     45 42 56.5     $0.273\pm0.042$         0.04     20.83     19.45     18.85

J1354+465     B3 1352+471     13 54 36.02     46 57 01.5     $0.180\pm0.034$         0.05     20.65     19.97     19.62

J1509+515     SDSS J150903.21+515247.9     15 09 03.21     51 52 47.9     $0.575\pm0.031$     0.579     0.06     20.81     19.54     18.72

J1633+084     SDSS J163300.85+084736.4     16 33 00.85     08 47 36.4     $0.266\pm0.032$         0.17     18.96     17.69     17.20

J1636+243     SDSS J163624.97+243230.8     16 36 24.97     24 32 30.8     $0.118\pm0.040$         0.11     20.10     19.60     19.33

J1646+383     B2 1644+38     16 46 28.42     38 31 16.0     $0.098\pm0.021$     0.108     0.04     17.51     16.63     16.17

J1656+640     SDSS J165620.60+640752.9     16 56 20.60     64 07 52.9     $0.201\pm0.018$     0.212     0.07     17.98     16.90     16.40

J1721+262     SDSS J172107.89+262432.1     17 21 07.89     26 24 32.1     $0.147\pm0.011$     0.170     0.11     18.78     17.68     17.18

J2141+082     SDSS J214110.61+082132.6     21 41 10.61     08 21 32.6     $0.307\pm0.043$         0.15     20.22     19.11     18.43

Col. (1): Abbreviated object name. Col. (2): Full object name from [Keel et al., 2022]. Cols. (3)–(4): Coordinates. Col. (5): Photometric redshift from the DESI survey. Col. (6): Spectroscopic redshift; all except object J0219+015 [Huchra et al., 1999] derive from SDSS. Col. (7): Galactic extinction from [Schlafly & Finkbeiner, 2011]. Cols. (8)–(10): SDSS $gri$-band model magnitude after correction for Galactic extinction using the extinction curve of [Cardelli et al., 1989].

HST F475W images of the 18 high-confidence sources in our sample, overlaid with FIRST 1.4 GHz contours in the left panel and zoomed-in to highlight the optical morphology of the galaxy in the right panel. All images are centered on the galaxy with north up and east to the left. The restoring beam of the radio map is depicted as a hatched ellipse on the lower-right corner. Radio contours are ($-3, 3, 6, 12, 24, 48, 96, ...$) $\times$ rms of each image, where the values of the rms are listed in Table 3. Object J0847+124 shows an additional 1.5 rms contour in light-grey color. The scale bar in the lower-left corner of each panel indicates the proper distance at the redshift of each object. Figure A1 displays the low-confidence objects.

§ OBSERVATIONS

Our sample is based on the Radio Galaxy Zoo and the Zoo Gems projects. Radio Galaxy Zoo is an online citizen-science program that invites volunteers to identify infrared and optical counterparts of radio sources in the Faint Images of the Radio Sky at Twenty Centimeters <cit.> survey. The FIRST survey gives snapshots of the sky in the 1.4 GHz band with an exposure time of 3 minutes, which typically generates images with a full-width at half maximum (FWHM) beam size of 54, an astrometric accuracy of 50$\,$mas, and root mean square (rms) noise of 0.15$\,$mJy beam$^{-1}$. Radio Galaxy Zoo includes all spatially resolved radio sources in the FIRST survey with signal-to-noise ratios higher than 10. The radio images are fed to volunteers to locate the host galaxies by overlaying them with optical images from Data Release 10 [Ahn et al., 2014] and Data Release 12 [Alam et al., 2015] of the Sloan Digital Sky Survey (SDSS) and with near-infrared images from the Wide-Field Infrared Survey Explorer <cit.>.

Zoo Gems is a follow-up project of Radio Galaxy Zoo that makes use of short windows in the HST schedule to observe galaxies with peculiar morphologies of interest. Before the start of Zoo Gems, roughly half of the candidate FIRST sources had host identifications and were sent to the data pool of Zoo Gems for further analysis. Zoo Gems selected 32 double-lobed radio sources apparently associated with a host galaxy having a prominent disk component, according to analysis based on the SDSS pipeline and two-dimensional fits using  [Peng et al., 2002, Peng et al., 2010]. HST observations of 674 s duration were obtained in the F475W filter (close to the SDSS $g$ band) using the Wide-Field Camera mode of the Advanced Camera for Surveys [Ford et al., 1998]. A two-point dither pattern was adopted to better sample the point-spread function. The images have a pixel scale of $0\farcs05$, a resolution of ${\rm FWHM} \approx 0\farcs10-0\farcs14$, and an astrometry accuracy of $\lesssim 10\,\mathrm{mas}$. Using these 32 objects as our initial sample, we retrieved the radio images from the FIRST archive[http://sundog.stsci.edu/cgi-bin/searchfirst] and the optical HST images from the Mikulski Archive for Space Telescopes (MAST)[https://mast.stsci.edu/search/hst/ui/]. Figure <ref> displays the HST images overlaid with the FIRST radio contours, for the subset of 18 sources we regard as having genuine optical-radio association (see details in Section 3); the images for the objects with less confident association are presented in Appendix A.

All galaxies have SDSS multiband photometry, and for our purposes we collect their model magnitudes in the $g$, $r$, and $i$ band (Table <ref>). Only 11 objects have spectroscopic redshifts ($z_{\rm spec}$). For the remaining 21 objects, we adopt photometric redshifts ($z_{\rm phot}$) from [Duncan, 2022], which is based on the Dark Energy Spectroscopic Instrument (DESI) Legacy Imaging Surveys [Dey et al., 2019]. DESI covers most of the footprint of SDSS in $grz$, with $5\,\sigma$ point source depths reaching $m_g=24.7$, $m_r=23.9$, $m_z=23.0$ mag at $1\farcs5$ seeing, $\sim 1.5$ mag deeper than SDSS. [Duncan, 2022] derived photometric redshifts using a machine-learning technique, which is based on galaxy properties in the optical and mid-infrared bands, including color, magnitude, and size. Their photometric redshifts have a robust scatter compared with the spectroscopic redshift, with a normalized median absolute deviation $1.48 \times \mathrm{median}(|z_{\mathrm{phot}}- z_{\mathrm{spec}} |/\left(1+z_{\mathrm{spec}})\right) = 0.05$. Comparing $z_{\rm phot}$ with $z_{\rm spec}$ for the 11 objects with spectra yields good consistency with a scatter of only 0.01 for (1+$z$).

For completeness, we note that there is a mismatch between the target names and sky coordinates in Table 2 of [Keel et al., 2022]. The names always lead to correct targets, as judged by the ancillary information available in NASA/IPAC Extragalactic Database (NED). Some sources are named by radio programs (e.g., B3 0911+418), and their NED coordinates pertain to the center position of the radio emission instead of the optical center of the host galaxy. For consistency, all the updated coordinates in Table <ref> refer to the position of the optical center of the galaxy.

§ SAMPLE DEFINITION

The original sample of double-lobed radio sources from the Zoo Gems project was defined by their apparent association with an optically visible disk galaxy. To be fully convinced that the radio-optical association is real, we assess the probability that the galaxies we observe are aligned fortuitously with the radio lobes. The identification of the hosts of radio sources is usually based on the relative position between the radio emission and optically visible galaxies. However, because radio lobes lack emission lines to determine their redshift, the line-of-sight distance between the radio lobes and galaxies is unknown. Other sources along the line-of-sight might happen to be projected toward the radio center and be mistaken for the host galaxy. To statistically rule out such coincidences, we calculate the probability that the observed optical counterparts are chance-aligned, while the true host galaxies (assumed to be elliptical galaxies) are too faint to be detected. We use this chance alignment probability to define the final sample of galaxies included in this paper.

§.§ Probability of Disk Galaxies in Chance Alignment

We calculate the probability that an unrelated disk galaxy happens to be projected toward the center of the radio lobes. To begin with, the chance alignment probability depends on the number density of such galaxies. We use the [Schechter, 1976] luminosity function to calculate the number density of disk galaxies, which, in terms of magnitude, is [Loveday et al., 2012]

\begin{equation}
\phi(M) =0.4 \ln 10\;\phi^{*}\left\{10^{0.4\left(M^{*}-M\right)}\right\}^{\alpha+1} \exp \left\{-10^{0.4\left(M^{*}-M\right)}\right\},
\end{equation}

where $M^*$ is the characteristic absolute magnitude, $\alpha$ is the power index, and $\phi ^{*}$ is the normalization of the number density in units of $\mathrm{\;Mpc^{-3}}$. We adopt the $g$-band luminosity function of blue galaxies from [Loveday et al., 2012], for which $M^* - 5\log\, h = -19.6$, $\phi^* = 7.1 \times 10^{-3} \,h^{-3}\mathrm{\,Mpc^{-3}}$, and $\alpha = -1.42$, with $h$ the dimensionless Hubble constant. For galaxies brighter than a given apparent magnitude $m_0$, we calculate the surface number density $n(m \le m_0)$ by integrating over all redshift range in a unit solid angle, taking into account K-correction with a color $g-r=0.66\,$mag, typical of Sab galaxies [Fukugita et al., 1995].

Some galaxies have a radio core in their nucleus, but their number density is even smaller if their radio-loudness is high. Studying the radio-to-optical flux ratio

\begin{equation}
R_B = f_\mathrm{1.4}/(k\cdot 10^{-0.4m_B}),
\label{eq:loundness}
\end{equation}

where $f_\mathrm{1.4}$ is the 1.4 GHz flux density in units of mJy, $m_B$ is the $B$-band magnitude of the galaxy, and $k = 4.44 \times 10^6$ is a normalization factor, [Gavazzi & Boselli, 1999] found that only $\sim 0.4\%$ of late-type galaxies have $R_B\ge 10$ and $\sim 4\%$ have $R_B \approx 1-10$. We derive the galaxy $B$-band magnitude from SDSS $g$ and $r$ magnitudes following [Jester et al., 2005], after considering K-correction and Galactic extinction. We measure the nuclear radio emission with the Common Astronomy Software Application (; McMullin et al., 2007) and apply K-correction assuming a spectral index of $-0.8$ [Blundell et al., 1999] to derive the radio luminosity. Note that here we assume that the associations with the radio lobes are unknown and only take the core radio emission into consideration. Depending on the radio-loudness, we multiply the rarity factors (0.004 for $R_B\ge 10$ and 0.04 for $R_B = 1-10$) to their number density.

Finally, following [Bloom et al., 2002] and [Berger, 2010], we calculate the probability that a galaxy coincides with the radio center within a separation $r_\mathrm{offset}$:

\begin{equation}
p= 1- e^{-\pi r_\mathrm{offset}^2 n(m\le m_\mathrm{gal})}\approx \pi r_\mathrm{offset}^2 n(m\le m_\mathrm{gal}),
\end{equation}

where $n(m\le m_\mathrm{gal})$ in units of arcsec$^{-2}$ is the surface number density of disk galaxies of the same or brighter magnitude, and $r_\mathrm{offset}$ is in units of arcsec. As in [Laing et al., 1983], the radio center is defined as the midpoint of the two radio hotspots.

§.§ Probability of Observing No Elliptical Hosts

If the disk galaxies we observe are not the hosts of the radio lobes, the real hosts, which are usually giant elliptical galaxies, should appear in the deep HST images, unless they are too faint. Therefore, we estimate the probability distribution of the magnitude of possible elliptical hosts.

§.§.§ Probability Distribution of Host Magnitude

The probability distribution of luminosity for a galaxy is proportional to its luminosity function. [Scarpa & Urry, 2001] found that the luminosity function of radio elliptical galaxies correlates with that of normal elliptical galaxies as $A \phi(L)L^2$, with $L$ the galaxy luminosity, $\phi(L)$ the luminosity function for normal elliptical galaxies, and a normalization factor $A$. We adopt $\phi(L)$ of red galaxies from [Loveday et al., 2012] and derive the probability density distribution of absolute magnitude for radio elliptical galaxies (Figure <ref>a). Our order-of-magnitude estimates, which suffice for our purposes, do not consider evolutionary effects of the luminosity function. Section <ref> will show that this assumption provides a conservative estimate of chance-alignment probability.

Meanwhile, we estimate the probability distribution of redshift based on the radio luminosity function and the observed radio flux density (Section <ref>). We describe the radio luminosity function as a double power law,

\begin{equation}
\Phi(L_\mathrm{1.4})=\frac{\Phi^*}{\left(L_\mathrm{1.4}/ L^*\right)^{\alpha}+\left(L_\mathrm{1.4}/ L^*\right)^{\beta}},
\end{equation}

where $L_\mathrm{1.4}$ is the 1.4 GHz luminosity in units of $\mathrm{W} \mathrm{\,Hz}^{-1}$, $L^*$ is the characteristic radio luminosity, and $\Phi^*$ is the characteristic number density. Following [Mauch & Sadler, 2007], we adopt $\Phi^*=10^{-5.5} \,\mathrm{mag}^{-1}\, \mathrm{Mpc}^{-3}$, $L^*=10^{24.6} \,\mathrm{W \, Hz}^{-1}$, $\alpha=1.27 $, and $\beta=0.49$. The probability that a radio source with a flux density $S$ is at redshift $z$ is proportional to the number of sources at that redshift:

\begin{equation}
p(z,S)\,dz\,dS=\frac{1}{\kappa}\,\Phi (L_\mathrm{1.4})\, d\log\, L_\mathrm{1.4}\, dV,
\end{equation}

where $\kappa$ is a normalization factor, $L_\mathrm{1.4}$ and $V$ are the corresponding absolute luminosity and comoving volume at redshift $z$, and $L_\mathrm{1.4}= S\cdot 4\pi d_L/(1+z)^{1+\alpha}$, with $d_L$ the luminosity distance and $\alpha=-0.8$ the spectral index [Blundell et al., 1999]. Figure <ref>b shows the probability density as a function of redshift, for a typical radio flux density in our sample, $S_\mathrm{tot}=50\,$mJy (Table <ref>). The probability peaks at $z\approx0.2$ and decreases rapidly toward both lower and higher redshift.

Finally, we calculate the probability distribution of the apparent magnitude $m$ by combining those of the absolute magnitude and the redshift, considering

\begin{equation}
\end{equation}

where $\mu(z)$ is the distance modulus as a function of redshift, defined as $\mu = 5 \log\,(d_L/10\,\rm{pc})$, and $K(z)$ is the K-correction for elliptical galaxies, for which we use the spectral energy distribution (SED) templates generated using a delayed star formation history with an e-folding time of 2.5$\,$Gyr. This set of templates is based on optical and near-infrared SEDs of elliptical and S0 galaxies in the Carnegie-Irvine Galaxy Survey <cit.>, which match well the typical star formation history of early-type galaxies from the GAMA survey [Bellstedt et al., 2020]. We compare our probability distribution (Figure <ref>c) with a large sample of galaxies from the complete radio survey[We transform their magnitudes in the $b_J$ band to the $g$ band assuming $b_J - g = 0.5\,\rm{mag}$ [Jester] et al., 2005, Chang et al., 2006]. of [Sadler et al., 2002]. We find excellent agreement between the observations and our model if we constrain the radio flux density to the same range ($S_\mathrm{tot} = 30-70\,$mJy). Moreover, we note that the probability distribution is insensitive to the value of the observed radio flux density in the range of this sample ($6-374\,$mJy). For instance, if $m_g < 25\,$mag, the typical flux density (50 mJy) gives a probability $\sim 0.1$, while the faintest case (6 mJy) gives a probability of $\sim 0.15$. Since we only need an order-of-magnitude estimate of the probability, we adopt the typical results for all objects.

(a) Probability density as a function of $g$-band absolute magnitude for red galaxies from [Loveday et al., 2012]. (b) Predicted probability density as a function of redshift for galaxies with radio lobes and $S_\mathrm{tot}=50\,\mathrm{mJy}$. (c) Cumulative probability distribution of a potential elliptical host of radio lobes as a function of $g$-band apparent magnitude. Histograms denote objects with detected radio lobes and $S_\mathrm{tot} = 30-70\,\mathrm{mJy}$ [Sadler et al., 2002].

§.§.§ Source Detection and Constraint on Host Magnitude

Might the hosts of the radio sources actually be elliptical galaxies that have simply escaped detection? We use the Python source detection package  [Bradley et al., 2020] to search a region centered at the midpoint of the radio hotspots. Defining $d$ as the angular distance between the two hotspots, [Laing et al., 1983] find that 96% of optical counterparts are distributed within $0.2\,d$ from the radio center. We thus restrict our search to a radius of $0.2\,d$. We detect sources with a threshold set to 3 times the rms noise per pixel, requiring that the source contains 40 connected (0.1$\,$arcsec$^2$) pixels, which is one-tenth of the minimal area of a galaxy of radius 5$\,$kpc across the entire redshift range (the angular-diameter scale peaks at $\mathrm{8.6\,kpc\,arcsec^{-1}}$ in our cosmology). We further crossmatch the detected sources with the DESI Legacy Imaging Surveys Data Release 8 [Duncan, 2022] to obtain their redshift and exclude local small galaxies from candidate giant ellipticals, according to their absolute magnitude and size. Appendix <ref> describes four positive detections of possible candidates. To be conservative, we view them as candidate hosts. For the other cases, we constrain the hypothetical elliptical galaxy host to be fainter than our detection limit, and we estimate the probability of it being so faint.

To estimate the detection limits, we randomly add simulated elliptical galaxies to the HST images and perform source detection as described above to determine the detection rate, which is defined as the percentage of detected cases. We use GALFIT to simulate single-component [Sérsic, 1968] models with index $n=4$, as typically seen in elliptical galaxies [de Vaucouleurs , 1948]. We set the effective radii to $r_e = 3\farcs0$, $1\farcs6$, and $1\farcs2$, which correspond to typical radio galaxies with $r_e \approx 10\,$kpc [McLure et al., 1999] at $z= 0.2$, 0.5, and 1.2. To examine our capability to detect sources with $m_g = 27\,$mag, we simulate 100 mock elliptical galaxies for each $r_e$. All the HST images have detection rates higher than 97% for each value of $r_e$, which suggests that the detection limit is $\gtrsim 27\,$mag. Nevertheless, we do not adopt $m_g = 27\,$mag as the detection limit in our analysis, for too many random sources ($> 1$) would fall into our detection regions with this magnitude limit. Instead, we only regard sources brighter than 25$\,$mag as candidate hosts. This much more conservative limit suffices for our chance-alignment probability analysis and minimizes the number of confusing objects ($\lesssim 0.3$) in our detection regions. We additionally consider the fact that our detection would be less sensitive if the sources overlap with a foreground disk galaxy, the likelihood of which depends on the size of the disk galaxy and the spatial distribution of the latent hosts. While 96% of optical counterparts are distributed within $0.2\,d$ from the radio center [Laing et al., 1983], the chance of overlap is $\sim 50\%$ when a disk galaxy with $r_e = 0.1\,d$ is located in the center. Therefore, the chance of overlap is significant for disk galaxies with $r_e>0.1 \,d$.

In the case of blended sources, we estimate the detection limit by adding simulated elliptical galaxies of increasing brightness to the region within the source's effective radius, until reaching the critical magnitude beyond which they are hardly discernible through visual inspection of the residual images from  decomposition (Section 4.2). Our tests indicate a limiting magnitude of $m_g \approx 22$. The probability of an elliptical host being fainter than the limit is 0.4, according to Figure <ref>c. Incidentally, the specific value of the detection limit does not impact substantially our probability analysis for limiting magnitudes in the range $m_g \approx 20-24$ (Figure <ref>c). Although our estimate of the detection limit is approximate, it is sufficient for our analysis. In the end, we conclude that the non-detection probability is $\sim 0.4$ when the chances of overlapping are high, and $\sim 0.1$ when the chances are low.

Sample Definition Based on Probability of Chance Alignment

Name    $r_\mathrm{offset}$     $R_B$    magnitude     Surface Density     Overlap     $P$    $N_c$

()          (mag)     (deg$^{-2}$)

(1)     (2)     (3)        (4)     (5)     (6)     (7)

3lHigh-confidence Sample

J0847+124     0.7     23.8     18.4     24     H     $4.6 \times 10^{-9}$     0.0002

J0219+015     1.1         14.5     0.17     H     $2.0 \times 10^{-8}$     0.0010

J0914+413     5.4     3.9     16.4     2.06     L     $5.8 \times 10^{-8}$     0.0029

J1633+084     3.7     21.5     19.0     49     L     $6.6 \times 10^{-8}$     0.0033

J1646+383     0.9         17.5     8.1     L     $1.6 \times 10^{-7}$     0.0080

J0956+162     2.7     56.1     20.4     248     L     $1.7 \times 10^{-7}$     0.0087

J0958+561     7.5     43.9     19.1     56     L     $3.0 \times 10^{-7}$     0.015

J1721+262     9.3     18.2     18.8     39     L     $3.3 \times 10^{-7}$     0.016

J0832+184     1.4         17.1     5.0     H     $9.4 \times 10^{-7}$     0.047

J0926+465     1.7     5.1     19.5     89     H     $1.0 \times 10^{-6}$     0.050

J1656+640     9.5     $\lesssim$2     18.0     15     L     $1.3 \times 10^{-6}$     0.065

J0806+062     0.7         18.6     31     H     $1.5 \times 10^{-6}$     0.073

J0855+420     0.5         19.4     79     H     $1.9 \times 10^{-6}$     0.096

J1128+241     2.6         17.9     13     L     $2.2 \times 10^{-6}$     0.11

J2141+082     1.1         20.2     198     L     $5.8 \times 10^{-6}$     0.29

J1328+571     4.0         16.9     3.9     H     $6.0 \times 10^{-6}$     0.30

J0209+075     2.7         18.8     39     L     $6.9 \times 10^{-6}$     0.34

J0802+115     1.5         18.8     39     H     $8.5 \times 10^{-6}$     0.42

3lLow-confidence Sample

J0833+045     2.4         19.8     126     L     $1.8 \times 10^{-5}$     0.88

J1354+465     2.3         20.7     345     L     $4.4 \times 10^{-5}$     2.2

J1636+243     3.8         20.1     177     L     $6.2 \times 10^{-5}$     3.1

J0901+164     2.0         19.3     70     L     $6.8 \times 10^{-5}$     3.4

J1136+125     7.2         17.4     7.2     L     $9.0 \times 10^{-5}$     4.5

J1509+515     3.8         20.8     384     L     $1.3 \times 10^{-4}$     6.7

J0813+552     1.6         20.4     248     L     $1.5 \times 10^{-4}$     7.7

J0903+432     4.4         20.7     345     L     $1.6 \times 10^{-4}$     8.1

J0919+135     18.1     $\lesssim$32     21.5     804     L     $2.6 \times 10^{-4}$     13

J0941+312     8.2         20.9     428     L     $7.0 \times 10^{-4}$     35

Col. (1): Object name. Col. (2): Offset between the galaxy center and the midpoint of the radio lobes. Col. (3): Radio-loudness of the radio emission in the nuclear region, defined by Equation <ref>. Here we assume that the association with the radio lobes is unknown. Col. (4): Surface number density of galaxies of the same or brighter magnitude. Col. (5): Overlapping chance that a latent elliptical host galaxy hides behind the foreground galaxy: “L” = low chance of overlap, corresponding to a 0.1 probability of non-detection; “H” = high chance of overlap, corresponding to a 0.4 probability of non-detection. Col. (6): Chance alignment probability for a single case provided the radio-optical relative location and luminosity. Col. (7): Expected number of chance-aligned cases if selecting with criteria as in this case in the entire FIRST survey, which has $N \approx 5 \times 10^4$ pairs of double-lobed sources [McMahon et al., 2002]; $N_c = P \times N$.

§.§ Total Probability of Chance Alignment

We calculate the total probability of chance alignment $P$ by multiplying the probability that a disk galaxy coincides with the radio center (Section <ref>) with the probability that an elliptical host galaxy is potentially present but too faint to be detected (Section <ref>). [McMahon et al., 2002] detected a total of $N\approx 5\times 10^4$ double-lobed sources in the FIRST survey. The expected number of chance-aligned events with properties specified for each object is therefore $N_c = P \times N$ (Table <ref>). We compile objects with $N_c < 0.5$ into the high-confidence sample and restrict our present analysis only to these. Assuming a Poisson distribution, the high-confidence sample has 90% probability of including no more than one dubious case. We note that there may be a few cases having real radio associations even in the low-confidence sample, as we have not taken into consideration other evidence such as AGN activity and jet orientation.

We conclude this section with a discussion of the evolutionary effects of the luminosity function on our probability  estimation. Our results, which are based on a non-evolving luminosity function, are roughly consistent with those in an evolutionary framework because the radio sources are most likely located at low redshifts (Figure <ref>b). For a more quantitative estimation, we consider the evolving radio and optical luminosity functions of [Ceraj et al., 2018] and [Loveday et al., 2012], respectively. We find that the chance-alignment probabilities decrease by $\sim$ 70%, which suggests that our previous estimation is very conservative. However, since the evolution parameters have large uncertainties, whose precise determination remains controversial [Loveday et al., 2012, Loveday et al., 2015, Prescott et al., 2016, Ocran et al., 2021], we conservatively choose to adopt the non-evolutionary results in our analysis.

§ DATA ANALYSIS

§.§ Properties of the Radio Core and Lobes

We use  to measure the basic radio properties of the sources in the FIRST images (Table <ref>). We determine the rms noise from the standard deviation of pixel values in the background regions far from the main source. We measure the integrated flux density of the radio lobes and radio core, if present, with polygons that enclose all the radio emission. Provided the redshift of the host galaxy, we infer the absolute radio luminosity $L_{1.4}$ after K-correction assuming a spectral index of $-0.8$ [Blundell et al., 1999]. We fit the hotspots and radio core with a two-dimensional Gaussian profile, respectively, to measure their positions and flux density. We determine the separation between the two hotspots, their mean position angle (PA) relative to the galactic center, and the projected proper distance of the hotspots. We visually classify the radio sources into two types following the precepts of [Fanaroff & Riley, 1974], according to whether the distance between the two hotspots is smaller than half of the total extent of the double lobes (FR I) or not (FR II).

Measurement of Radio Properties

Name     Beam     rms     FR Type      $d$     $D$     PA     $S_\mathrm{core}$     $S_\mathrm{tot}$     $\log \, L_\mathrm{1.4}$

( $\times$ )     (mJy beam$^{-1}$)           ()     (kpc)     ($^\circ$)     (mJy)     (mJy)     ($\mathrm{W\,Hz^{-1}}$)

(1)     (2)     (3)     (4)     (5)     (6)     (7)     (8)     (9)     (10)

J0209+075     $6.4\times5.4$     0.12     II     63 $\pm$ 4     $250\pm23$     121 $\pm$ 4         $223 \pm 22$     $25.62\pm0.10$

J0219+015     $6.4\times5.4$     0.13     II     39 $\pm$ 4     $31\pm3$     5 $\pm$ 6         $315 \pm 31$     $24.10\pm0.04$

J0802+115     $5.4\times5.4$     0.19     II     14 $\pm$ 5     $41\pm15$     3 $\pm$ 23         $21 \pm 4$     $24.24\pm0.14$

J0806+062     $5.4\times5.4$     0.17     II     12 $\pm$ 4     $24\pm12$     73 $\pm$ 17         $5.7 \pm 2.9$     $23.27\pm0.41$

J0813+552     $5.4\times5.4$     0.16     II     34 $\pm$ 4     $145\pm21$     54 $\pm$ 7         $30 \pm 5$     $24.85\pm0.13$

J0823+033     $6.4\times5.4$     0.16     II                 $16.3 \pm 0.5$     $147 \pm 15$     $24.76\pm0.09$

J0832+184     $5.4\times5.4$     0.11     II     14 $\pm$ 4     $29\pm8$     151 $\pm$ 15         $26 \pm 4$     $23.94\pm0.06$

J0833+045     $5.4\times5.4$     0.15     II     26 $\pm$ 10     $101\pm40$     21 $\pm$ 23         $25 \pm 5$     $24.65\pm0.14$

J0847+124     $5.4\times5.4$     0.16     II     28 $\pm$ 3     $84\pm9$     4 $\pm$ 6     $5.1 \pm 0.4$     $7.8 \pm 3.3$     $23.82\pm0.15$

J0855+420     $5.4\times5.4$     0.14     I     8.4 $\pm$ 2     $27\pm6$     38 $\pm$ 13         $100 \pm 12$     $25.01\pm0.13$

J0901+164     $5.4\times5.4$     0.14     II     43 $\pm$ 1     $160\pm13$     96 $\pm$ 2         $118 \pm 12$     $25.27\pm0.11$

J0903+432     $5.4\times5.4$     0.14     II     44 $\pm$ 4     $229\pm20$     116 $\pm$ 5         $7 \pm 4$     $24.51\pm0.20$

J0914+413     $5.4\times5.4$     0.18     II     68 $\pm$ 10     $169\pm24$     172 $\pm$ 8     $4.7 \pm 0.6$     $374 \pm 38$     $25.29\pm0.04$

J0919+135     $5.4\times5.4$     0.20     II     142 $\pm$ 7     $793\pm65$     160 $\pm$ 3     $1.8 \pm 0.6$     $233 \pm 25$     $26.14\pm0.12$

J0926+465     $5.4\times5.4$     0.14     II     36 $\pm$ 2     $121\pm19$     155 $\pm$ 3     $0.5 \pm 0.2$     $11 \pm 3$     $24.11\pm0.20$

J0941+312     $5.4\times5.4$     0.15     II     151 $\pm$ 7     $777\pm60$     53 $\pm$ 3         $143 \pm 16$     $25.80\pm0.11$

J0956+162     $5.4\times5.4$     0.15     II     37 $\pm$ 2     $182\pm23$     124 $\pm$ 3     $5.5 \pm 1.0$     $74 \pm 8$     $25.44\pm0.18$

J0958+561     $5.4\times5.4$     0.14     II     92 $\pm$ 13     $360\pm55$     140 $\pm$ 6     $6.1 \pm 0.5$     $186 \pm 19$     $25.53\pm0.09$

J1128+241     $5.4\times5.4$     0.18     II     53 $\pm$ 3     $116\pm19$     86 $\pm$ 3         $63 \pm 8$     $24.38\pm0.16$

J1136+125     $5.4\times5.4$     0.15     II     187 $\pm$ 7     $127\pm4$     94 $\pm$ 2     $2.6 \pm 0.4$     $40 \pm 7$     $23.03\pm0.07$

J1303+511     $5.4\times5.4$     0.22     II                     $174 \pm 18$     $24.83\pm0.25$

J1322+270     $5.4\times5.4$     0.16     II                 $2.9 \pm 0.3$     $2.9 \pm 0.3$     $21.89\pm0.04$

J1328+571     $5.4\times5.4$     0.16     II     13 $\pm$ 3     $8\pm7$     52 $\pm$ 12         $17 \pm 4$     $22.61\pm0.58$

J1349+454     $5.4\times5.4$     0.15     II                     $6.7 \pm 0.7$     $24.18\pm0.15$

J1354+465     $5.4\times5.4$     0.16     II     36 $\pm$ 3     $110\pm18$     94 $\pm$ 5         $96 \pm 12$     $24.94\pm0.17$

J1509+515     $5.4\times5.4$     0.17     II     23 $\pm$ 7     $153\pm46$     140 $\pm$ 17         $162 \pm 16$     $26.32\pm0.04$

J1633+084     $5.4\times5.4$     0.16     II     55 $\pm$ 2     $227\pm20$     148 $\pm$ 2     $2.8 \pm 0.4$     $36 \pm 6$     $24.89\pm0.13$

J1636+243     $5.4\times5.4$     0.15     II     76 $\pm$ 3     $163\pm46$     39 $\pm$ 2         $61 \pm 8$     $24.34\pm0.28$

J1646+383     $5.4\times5.4$     0.16     II     57 $\pm$ 11     $113\pm21$     140 $\pm$ 15         $195 \pm 21$     $24.77\pm0.04$

J1656+640     $5.4\times5.4$     0.13     II     150 $\pm$ 17     $524\pm59$     169 $\pm$ 7     $0.7 \pm 0.2$     $49 \pm 9$     $24.80\pm0.07$

J1721+262     $5.4\times5.4$     0.16     II     224 $\pm$ 3     $656\pm8$     68 $\pm$ 1     $2.5 \pm 0.6$     $68 \pm 10$     $24.73\pm0.06$

J2141+082     $6.4\times5.4$     0.15     II     9.6 $\pm$ 1.0     $44\pm6$     13 $\pm$ 7         $86 \pm 9$     $25.41\pm0.14$

Col. (1): Object name. Col. (2): Beam size (major $\times$ minor axis). Col. (3): Background noise. Col. (4): FR type. Col. (5): Angular distance between the two radio hotspots. Col. (6): Projected proper distance between the two radio hotspots. Col. (7): Position angle of the hotspots relative to the galaxy center, averaged over the hotspot on each side. Col. (8): Flux density of the radio core, if any, measured from two-dimensional Gaussian fits. Col. (9): Integrated flux density of the entire radio source. Col. (10): Absolute luminosity of the entire radio source at 1.4 GHz. Objects J0823+033, J1303+511, J1322+270, and J1349+454 have no well-defined double radio lobes and thus no lobe size and PA measurements.

Examples of two-dimensional bulge-disk decomposition using . The left column shows the surface brightness profile of the data (open circles with error bars),  bulge component (red), exponential disk component (blue), and total (bulge + disk) model (purple). The $\chi^2_\nu$ from  is shown in the lower-left corner. The lower panel gives the residuals between the data and the model. The right three columns show the original data, best-fit total model, and residuals, respectively. All images are displayed on a log stretch.

Measurement of Host Galaxy Properties of the High-confidence Sample

Name     $m_\mathrm{tot}$     $r_e$     $n_\mathrm{global}$     $\epsilon$     $C$     $m_{\mathrm{bulge}}$     $r_{\mathrm{bulge}}$     $\langle\mu_e\rangle_{\rm bulge}$     $n_{\mathrm{bulge}}$    PA$_{\mathrm{bulge}}$   $m_{\mathrm{disk}}$   PA$_\mathrm{disk}$   $B/T$   $\log\, M_*$     $g-r$

(mag)     ()                    (mag)     ()               ($^\circ$)    (mag)    ($^\circ$)        (${M_\odot}$)     (mag)

J0209+075     19.07      2.76         0.08     3.11       21.26        0.39       19.09        2.23         147       18.96         130        0.11     $11.42\pm0.18$     0.86

J0219+015     14.85     12.41        2.01     0.54     3.46       16.03        5.67       20.68        1.31         127       15.16         137        0.31     $11.36\pm0.15$     0.84

J0802+115     19.05      1.55        2.18     0.77     4.11                                     $10.82\pm0.20$     0.68

J0806+062     18.53      2.94        0.93     0.45     2.66       22.44        0.46       21.41        0.37         165       18.56         155        0.03      $9.81\pm0.52$     0.40

J0832+184     17.17      3.12        3.50     0.42     4.53       18.06        1.01       18.70        1.81          18       17.62         164        0.40     $11.27\pm0.15$     0.85

J0847+124     18.24      5.85        1.28     0.66     3.00       21.29        1.14       21.92        0.53          56       18.22         115        0.06     $11.33\pm0.15$     0.85

J0855+420     19.70      2.04        2.06     0.50     3.17       20.75        1.05       21.12        1.45           2       20.26         173        0.39     $11.07\pm0.20$     0.76

J0914+413     16.77     12.00        3.89     0.26     3.24       18.06        2.06       20.16        2.45           5       17.02          27        0.28     $11.85\pm0.15$     0.86

J0926+465     19.66      7.30        2.66     0.74     3.32       22.55        0.61       21.69        0.48          68       19.73          68        0.07     $10.92\pm0.25$     0.72

J0956+162     20.53      2.21        0.87     0.64     3.08       22.66        0.46       20.43        0.62         167       20.50          23        0.12     $10.78\pm0.25$     0.60

J0958+561     19.13      4.76        5.53     0.47     4.35       19.90        0.83       19.45        2.85          91       19.59          95        0.43     $11.17\pm0.17$     0.76

J1128+241     18.45      4.16        2.23     0.19     2.93       21.07        0.54       20.35        1.72           3       18.40          50        0.08     $10.30\pm0.23$     0.46

J1328+571     16.74      9.42        1.63     0.41     3.27                           17.09          70        0.00      $8.89\pm1.34$     0.20

J1633+084     19.68      3.53        1.66     0.49     2.99       21.86        0.57       20.42        0.37          45       19.64          47        0.11     $11.35\pm0.19$     0.85

J1646+383     17.75      2.28        2.76     0.35     3.76       19.63        0.50       18.80        0.16          26       18.06          32        0.19     $10.91\pm0.15$     0.72

J1656+640     18.33      4.68        2.13     0.56     3.43       19.91        1.43       20.83        1.08          54       18.50          56        0.21     $11.46\pm0.15$     0.70

J1721+262     18.63      3.44         0.15     4.47       19.45        1.52       20.71        6.24         176       18.29          16        0.34     $10.99\pm0.15$     0.78

J2141+082     20.45      2.22        3.19     0.67     3.56       22.61        0.18       18.45        1.41          25       20.60          16        0.14     $11.13\pm0.21$     0.53

Col. (1): Object name. Col. (2): Petrosian magnitude after correction for Galactic extinction. Col. (3): Effective radius from fitting a single- model. Objects J0209+075 and J1721+262 are not well fit by a single- model, and for them we adopt the half-light radii from non-parametric measurements. Col. (4): Global  index from the single- model fit. Col. (5): Ellipticity from non-parametric measurements, defined by $\epsilon = 1-b/a$. Col. (6): Concentration parameter, defined by $C=5\log\,({r_{80}/r_{20}})$. Cols. (7)–(13): Magnitude, effective radius, effective surface brightness ($R$ mag arcsec$^{-2}$),  index, and position angle (east of north) of the bulge and disk components derived from a two-component  model. Col. (14): Bulge-to-total luminosity ratio. Col. (15): Stellar mass. Col. (16): Rest-frame $g-r$ color of the entire galaxy. Object J0802+115 is an irregular galaxy without meaningful bulge-disk decomposition, while object J1328+571 is a dwarf (Magellanic spiral) galaxy without a bulge ($B/T=0$).

§.§ Properties of the Host Galaxy

We first estimate and subtract the sky background of the HST images using a sigma-clipping method that iteratively clips the images beyond $3\,\sigma$, where $\sigma$ is the standard deviation of the background pixel distribution. We use  to create a segmentation map, setting the segmentation threshold to $2\,\sigma$ above the background noise. The segmentation map allows us to improve the background measurement by calculating the pixel distribution in the sky regions. We create a mask image to remove unrelated sources according to the segmentation map.

With the sky-subtracted image and corresponding segmentation map and mask image, we use the Python package  [Rodriguez-Gomez et al., 2019] to compute several non-parametric quantities of interest (Table <ref>), including the total magnitude ($m_{\rm tot}$) within the [Petrosian, 1976] radius, ellipticity $\epsilon \equiv 1-b/a$, with $a$ the semi-major and $b$ the semi-minor axes calculated from the second-order moments[Equations 24–25 from the SExtractor user manual at <https://readthedocs.org/projects/sextractor/downloads/>], and the concentration parameter $C\equiv5 \log\,{r_{80}/r_{20}}$ [Conselice, 2003], where $r_{80}$ and $r_{20}$ are the circular aperture radii enclosing 80% and 20% of the total flux, respectively.

We use  to model the two-dimensional light distribution of the galaxy. As in Zhuang & Ho, 2022, we construct a point-spread function using [https://reproject.readthedocs.io/en/stable/index.html] to stack unsaturated bright stars within the same image of each object. To reproject and stack the stars, we adopt the flux-conserving scheme , which treats pixels as four-sided spherical polygons and computes the exact overlap of pixels on the sky. We first fit the galaxy using a single- component to obtain a global model, yielding the effective radius $r_e$ and  index $n_{\rm global}$. We then fit two components to decompose the bulge and disk[Although Zoo Gems also used  to analyze the images, no details were given by [Keel] et al., 2022]., using a  component with a free index $n_{\rm bulge}$ to model the bulge and an exponential profile to model the disk. Obvious dust lanes were masked manually to mitigate their effects on the fits. The model residual images show that in general the two-component bulge-disk decompositions are quite successful. We find no evidence for any additional component that might arise from a prominent nucleus. Figure <ref> shows a couple of examples of the decompositions, and the fits for the full set of galaxies can be found in Appendix <ref>. Table <ref> summarizes the fitting results, which include quantitative parameters for the bulge and disk. Our fits do not take into account detailed features such as spiral arms. According to [Gao & Ho, 2017], the systematic uncertainties introduced by this over-simplification are 0.14 mag for $m_{\rm bulge}$, 10% for $r_{\rm bulge}$, and 14% for $n_{\rm bulge}$.

We estimate the stellar mass of the host galaxies using SDSS multiband photometry and the method of [Taylor et al., 2011], which is based on the rest-frame $i$-band absolute magnitude and $g-i$ color of $z < 0.65$ (median $z = 0.2$) galaxies. The $1 \,\sigma $ uncertainty in $\log\, M_*$ is $\sim 0.1$ dex. We use SDSS magnitudes in the $g$ and $i$ bands and apply K-correction [Chilingarian et al., 2010, Chilingarian & Zolotukhin, 2012] and Galactic extinction correction to convert to rest-frame magnitudes. The final error budget of the stellar mass has contributions from the redshift, K-correction, and the stellar mass estimation method. Seven objects overlap with the second release of the GALEX-SDSS-WISE Legacy Catalog <cit.>, whose stellar masses are derived from ultraviolet-to-infrared SED fitting. Comparing the stellar masses estimated in this paper and those from GSWLC-2, we find good agreement between the two methods, with maximum difference $\Delta \log\, M_*<0.3$ dex.

We compare the radio galaxies with normal galaxies in CGS [Ho et al., 2011], a statistically complete sample of 605 bright ($B_T<12.9\,$mag), southern ($\delta < 0\degr$) galaxies imaged in the optical with the facilities at Las Campanas Observatory. The images have a median seeing of $\sim 1\arcsec$, field-of-view of $8\farcm9\times 8\farcm9$, and median limiting surface brightness $\sim 27.5$, 26.9, 26.4, and 25.3 mag arcsec$^{-2}$ in the $B$, $V$, $R$, and $I$ bands, respectively. The flux fraction radii (thus concentration parameter) are provided in [Li et al., 2011], and [Gao et al., 2019] give bulge-to-total light ratios ($B/T$) and bulge  indices. We measure the global  index in the $B$ band with , as in Section <ref>. The stellar masses of the CGS galaxies are obtained from modeling with a delayed star formation history using CIGALE [Boquien et al., 2019] their complete optical and near-IR photometry (M.-Y. Zhuang et al., in preparation), when available, and otherwise from $B-V$ color and $K$ or $I$ (if $K$ is not available) magnitude using conversions from [Bell et al., 2003]. Figure <ref> shows parameter distributions of the radio disk galaxies compared to the control sample of CGS galaxies. Although our galaxies were not observed in the same bands and have slightly higher redshift ($z\approx0.2$) than the CGS nearby galaxy sample, the difference in structural parameters is expected to be mild [Häußler et al., 2013, Häußler et al., 2022, Conselice, 2014].

Distribution of (a) concentration parameter $C$, (b) bulge-to-total light ratio $B/T$, (c) global  index $n_\mathrm{global}$, and (d) bulge  index $n_\mathrm{bulge}$ for our sample of disk radio galaxies, in comparison with the control sample of normal galaxies from CGS <cit.>. The irregular galaxy J0802+115 and the dwarf Magellanic spiral galaxy J1328+571, which do not have a bulge, are not shown in panels (b) and (d). All quantities pertain to the $R$ band.

Six galaxies in the high-confidence sample have SDSS spectra, whose spectroscopic fiber covers the central 3 or $\sim 2-10\,$kpc of the galaxies. We obtain emission-line fluxes from the SDSS  catalog [Brinchmann et al., 2004]. For objects with signal-to-noise ratios larger than 2 for H$\beta$,  $\lambda$5007, H$\alpha$, and  $\lambda 6584$ or  $\lambda\lambda6716, 6731$, we show their [Baldwin et al., 1981] diagnostic diagrams in Figure <ref>. Five objects host AGNs or composite (AGN plus star-forming) nuclei according to the /H$\beta$–/H$\alpha$ diagram, while in the /H$\beta$–/H$\alpha$ diagram one object is a Seyfert and two are low-ionization nuclear emission-line regions (LINERs; Heckman, 1980). The two LINERs (J0914+413 and J1721+262) both have a compact radio core in their nucleus, a common feature in AGNs with low accretion rates [Ho, 2002, Ho, 2008].

The /H$\beta$ versus (a) /H$\alpha$ and (b) /H$\alpha$ diagnostic diagrams for objects with signal-to-noise ratio larger than 2 in their optical emission-line fluxes. The extreme starburst boundary (red solid line; Kewley et al., 2001), the pure star formation line (blue dashed line in panel a; Kauffmann et al., 2003), and the separation between Seyferts and LINERs (green dotted line in panel b; Kewley et al., 2006) are used to classify galaxies into star-forming (H$\,$II) galaxies, Seyferts, LINERs, and composite (star-forming and AGN) systems.

Effective radius ($r_e$) versus stellar mass ($M_*$) for the host galaxies of the double-lobed radio sources in our sample, with colors indicating their bulge-to-total light ratios ($B/T$). Blue and red solid lines represent the mass-size relations of star-forming and quiescent galaxies at $z \approx 0.2-0.4$ [Kawinwanichakij et al., 2021], with shaded areas indicating the $1\,\sigma$ intrinsic scatter of $\log\, r_e$.

Double-lobed Radio Sources Hosted by Disk Galaxies from the Literature

Name     R. A. (J2000)     Decl. (J2000)     $z$     $D$     $L_{\mathrm{1.4}}$     FR Type     $M_R$     $\log\,M_\mathrm{*}$     $\log\,M_\mathrm{BH}$     Ref.
($^{\mathrm{h}}\;{}^{\mathrm{m}}\;{}^{\mathrm{s}}$)     ($^\circ\;{}^\prime\;{}^{\prime\prime}$)          (kpc)     ($\mathrm{W\, Hz^{-1}}$)         (mag)     ($M_\odot$)     ($M_\odot$)

J0315$-$1906 (0313$-$192)     03 15 52.1     $-$19 06 44     0.067      200     1.0 $\times10^{24}$       I     $-21.28$     10.41     7.31      1

J0354$-$1340                  03 54 32.8     +13 40 07       0.076      240     $2.06\times 10^{23}$     II     $-21.91$     10.66     7.72     2, 3

J0836$+$0532                  08 36 55.9     +05 32 42       0.099      420     $1.53 \times10^{24}$     II     $-23.11$     11.36     8.85      4

J1159$+$5820                  11 59 05.8     +58 20 36       0.054      392     $2.26 \times10^{24}$     II     $-23.32$     11.36     8.85      4

J1352$+$3126                  13 52 17.8     +31 26 46       0.045      335     $2.26 \times10^{25}$     II     $-22.80$     11.32     8.79      4

J1409$-$0302 (Speca)          14 09 48.8     $-$03 02 32     0.138     1000     7.0 $\times10^{24}$      II     $-23.16$     11.57     9.19      5

J1649$+$2635                  16 49 23.9     $+$26 35 03     0.055       86     $1.07 \times10^{24}$     II     $-22.85$     11.42     8.94      4, 6

J2345$-$0449                  23 45 32.7     $-$04 49 25     0.076     1600     $2.5 \times10^{24}$      II     $-22.86$     11.04     8.33      7

MCG+07$-$47$-$10              23 18 32.7     +43 14 49       0.012      207     $1.12 \times 10^{22}$    II     $-20.67$     10.16     6.92      8, 9

Col. (1): Object name. Cols. (2)–(3): Coordinates. Col. (4): Redshift. Col. (5): Projected proper distance between the two radio hotspots; for J0354$-$1340, what is shown is the de-projected linear size. Col. (6): Radio luminosity at 1.4 GHz. Col. (7): FR type. They are predominantly classified as FR II but have much lower radio power than the lower limit of FR II sources hosted by elliptical galaxies <cit.>. Col. (8): Host galaxy $R$-band absolute magnitude. For objects J0836+0532, J1159+5820, J1352+3126, J1409$-$0302, and J1649+2635, we derive the $R$-band magnitude from the SDSS $gri$ magnitudes according to [Jester et al., 2005]. For the others with more limited photometry, we derive their $R$-band magnitude assuming the color of Sab galaxies, following the conversions in [Fukugita et al., 1995] and [Chilingarian et al., 2017]. Col. (9): Stellar mass of the host galaxy, computed as described in Section 4.2. Col. (10): Black hole mass, computed from $M_*$ as described in Section 5.2. Col. (11): References: (1) Ledlow et al. 1998; (2) Chen et al. 2020; (3) Vietri et al. 2022; (4) Singh et al. 2015; (5) Hota et al. 2011; (6) Mao et al. 2015; (7) Bagchi et al. 2014; (8) Mulcahy et al. 2016; (9) Bilicki et al. 2014.

Distribution of bulges of host galaxies in the Kormendy relation. The black solid line shows the Kormendy relation measured in the $R$ band from [Gao et al., 2020], with the shaded region showing the $3\,\sigma$ range. Classical bulges and ellipticals follow the Kormendy relation, while pseudo bulges are low-surface brightness outliers below the relation. The HST F475W magnitudes have been K-corrected to the $R$ band.

§ STATISTICAL PROPERTIES

§.§ Massive, Quiescent, Disk-dominated Hosts

Radio galaxies mostly consist of giant ellipticals with stellar mass exceeding $10^{11}\,M_\odot$ [Best et al., 2005]. The hosts of our sample of double-lobed radio galaxies are also predominantly massive systems (Figure <ref>), having a median stellar mass $M_* = 1.3 \times 10^{11}\,M_\odot$. As with other massive galaxies in the nearby Universe, they have optical colors ($\langle g-r \rangle = 0.69 \pm 0.18\,$mag, corrected for redshift and Galactic extinction) that place them on the red sequence [Blanton & Moustakas, 2009]. There are, however, two noteworthy outliers that have blue colors and unexpectedly low stellar masses: J0806+062 with $M_* = 6.5\times 10^9\,M_\odot$ and $g-r = 0.4\,$mag, and J1328+571 with $M_* = 7.8\times10^8\,M_\odot$ and $g-r = 0.13\,$mag. The extreme late-type, barred spiral morphology of J1328+571 bears a striking resemblance to the Large Magellanic Cloud, which is only $\sim 3$ times more massive ($M_* = 2.7\times 10^9\,M_\odot$; van der Marel et al., 2002). Nothing particularly unusual stands out in terms of the effective radii of the sample: most of the members obey the stellar mass-size relation of massive, quiescent galaxies at $z \approx 0.2-0.4$ (Figure <ref>; Kawinwanichakij et al., 2021).

However, completely counterintuitive to expectation, the galaxies in our sample have stellar morphologies and internal substructures of unmistakably disk-dominated, in many instances extremely late-type galaxies. Even a cursory inspection of the high-quality, optical HST images in Figure <ref> reveals that the hosts clearly have disks, as evidenced by spiral structure when viewed at low inclinations and large-scale dust lanes when seen edge-on. Six galaxies (J0209+075, J0806+062, J1128+241, J1328+571, J1646+383, J1656+640) have spiral features, and seven (J0219+015, J0847+124, J0855+420, J0914+413, J0926+465, J0956+162, J1633+084) have prominent dust lanes with dimensions comparable to that of the entire galaxy, which are common in spiral galaxies and are regarded as signatures of edge-on spiral disks [Holwerda et al., 2019]. In sum, we conclude that $\sim 70\%$ of the sample has spiral arms. The late-type nature of the hosts is further supported by more quantitative metrics. For instance, the sample galaxies have concentrations (median $C = 3.3 \pm 0.5$) consistent with late-type galaxies (Conselice, 2014; $C = 3.1 \pm 0.4$), which is further reinforced by their high global ellipticities (median $\epsilon = 0.48 \pm 0.19$) and low  indices (median $n_{\rm global} = 2.2 \pm 1.2$). Such low values of optical concentration and global  index deviate strongly from the majority of $M_* \approx 10^{11}\,M_\odot$ galaxies (Figures <ref>a and <ref>c), and elliptical galaxies rarely are flatter than $\epsilon = 0.7$ [Sandage et al., 1970].

An equally remarkable testimony to the exceptionally late-type nature of the host galaxies comes from the bulge-to-disk decomposition analysis described in Section 4.2 (Figure <ref>; Appendix C), which yields a sample median $B/T = 0.13$. Seven sources have $B/T \lesssim 0.1$ and can be deemed essentially bulgeless. Formally speaking, J1328+571 is truly bulgeless. As a consequence of their different evolutionary histories [Kormendy & Kennicutt, 2004, Kormendy & Ho, 2013], galactic bulges fall into two types—classical and pseudo—that broadly can be distinguished by their internal kinematics, structure, and radial light profile. Many authors (e.g., Fisher & Drory, 2008, Fisher & Drory, 2016) classify bulges according to the value of their  index: classical and pseudo bulges are defined by $n > 2$ and $n \leq 2$, respectively. By this criterion, the vast majority (12/16 or 75%) of our sample galaxies with successful bulge-to-disk decomposition qualify as having a pseudo bulge, not an unanticipated result in view of the low $B/T$ values that dominate the sample and the tendency for pseudo bulges to have $B/T \lesssim 0.35$ [Kormendy & Ho, 2013, Kormendy , 2016]. Others argue, however, that the  index can yield misleading bulge classifications (e.g., Gadotti, 2009, Gao et al., 2018), advocating, instead, that bulge classification should place greater reliance on the [Kormendy, 1977] relation between surface brightness and effective radius, to which classical bulges and ellipticals adhere but pseudo bulges do not [Neumann et al., 2017, Gao et al., 2020, Gao et al., 2022, Sachdeva et al., 2020]. Using the CGS $R$-band Kormendy relation of ellipticals and classical bulges as a reference, Figure <ref> illustrates that the bulges of our sample of disky radio galaxies systematically deviate to lower surface brightness at fixed effective radii, consistent with the expected behavior of pseudo bulges [Gao et al., 2020, Gao et al., 2022]. To enable this comparison, we K-corrected the HST F475W magnitudes to the $R$ band, assuming a bulge SED from [Kinney et al., 1996] and using Astrolib PySynphot [STScI Development Team, 2013].

It is quite unusual for spiral galaxies to have stellar masses as large as those in our sample, for the characteristic stellar mass of spiral galaxies is $\sim 3\times 10^{10}\,M_\odot$, above which the galaxy number density significantly drops [Kelvin et al., 2014, Ogle et al., 2016]. Spiral galaxies with $M_* > 10^{11}\,M_\odot$ are extremely uncommon and have been systematically studied only recently [Ogle et al., 2016, Ogle et al., 2019]. Rarer still are spiral galaxies that host powerful radio AGNs. The large fraction of massive objects in our sample suggests that they are not a random subset of normal galaxies. The nearest neighbors two-sample test [Rizzo, 2019], as implemented in the  package  [Crookston et al., 2008], yields $p$-values $<$ 0.02 when we compare their distributions with CGS spiral galaxies in the four diagrams of Figure <ref>. Prior to this study, nine secure cases of double-lobed radio sources hosted by disk had been reported (Table 5), among which only one (J0315$-$1906; Keel et al., 2006) had the benefit of high-resolution imaging from HST. Intriguingly, six of the nine cases also have stellar masses $M_* \gtrsim 10^{11} \,M_\odot$. Together with the statistics from our new sample, these trends suggest that the formation of large-scale radio lobes may be linked to the unusually high stellar mass of the hosts. We will revisit this topic in Section 6.

Radio–optical luminosity relation showing the FR I–FR II dichotomy. The solid line is the Owen-Ledlow relation that divides FR I (red) and FR II (blue) sources. Large filled circles are objects from our sample, stars are objects from the literature (Table 5), and small open circles are elliptical galaxies from [Owen & Ledlow, 1994].

Radio luminosity–BH mass relation showing the FR I–FR II dichotomy. The top $x$-axis gives the total stellar mass. The right $y$-axis converts radio luminosity to jet power following Equation 10. The solid line is the BH mass version of the Owen-Ledlow relation, where we converted the $R$-band absolute magnitude to BH mass as described in Section 5.2. Dashed lines show constant critical ratios of jet power over Eddington luminosity, $\alpha \equiv P_\mathrm{jet} / L_\mathrm{Edd} = 1$, 0.1, and 0.005. Large filled circles are objects from our sample, stars are objects from the literature (Table 5), and small open circles are elliptical galaxies from [Owen & Ledlow, 1994].

§.§ FR I–FR II Dichotomy

FR II radio sources are on average more powerful than FR I radio sources [Fanaroff & Riley, 1974], with the division occuring at a power of $\sim 10^{26}\, \mathrm{W \, Hz}^{-1}$ at 178 MHz, or, equivalently at 1.4 GHz, $L_{1.4}\approx 3\times 10^{25}\,\mathrm{W\,Hz^{-1}}$ [Tadhunter, 2016]. <cit.> suggested that the division line correlates with the absolute magnitude of the host galaxy,

\begin{equation}
\log\, \left(\frac{L_\mathrm{1.4}}{{\rm W\,Hz^{-1}}} \right) = -0.66 M_{R}+10.37,
\label{eq:owen}
\end{equation}

where $M_R$ is the Cousins $R$-band absolute magnitude. Figure <ref> shows where our objects lie with respect to the FR I–FR II dichotomy, as defined by the elliptical radio galaxy sample of [Owen & Ledlow, 1994]. We obtain the $R$-band magnitude of our objects from their SDSS magnitudes, adopting conversions from [Jester et al., 2005] with K-correction applied. To maximize the sample, we also add the previously known disk galaxies hosting double-lobed radio sources (Table <ref>).

Even though all but two of the combined sample of 27 objects have FR II morphologies, nearly all lie below the Owen-Ledlow relation because they tend to have much lower radio power than traditional FR II sources hosted by elliptical galaxies of the same optical luminosity. Violating the well-established FR I–FR II dichotomy, disk radio sources have radio powers characteristic of FR I systems but nevertheless display FR II morphologies. Moreover, the fraction of FR II sources among spiral hosts significantly exceeds that in elliptical radio galaxies, for which the proportion of FR I and FR II types is roughly equal [Capetti et al., 2017, Capetti et al., 2017]. [Singh et al., 2015] suggest that the radio lobes hosted by spiral galaxies may be in a late phase of evolution, caught during a period when the radio emission has subsided. However, if radio lobes expand slowly and fade rapidly, as theory predicts [Luo & Sadler, 2010], the fading stage lasts but a fleeting moment compared to the lifetime of the jet, rendering the fading scenario implausible to explain the low powers of all disk radio sources discovered to date.

What physically underpins the Owen-Ledlow relation? In view of the fundamental connection between BH mass and bulge stellar mass [Kormendy & Ho, 2013], [Ghisellini & Celotti, 2001] proposed that the Owen-Ledlow relation ultimately reflects a connection between radio power and the mass of the central BH. To revisit this idea, we estimate the BH masses of our combined sample of spiral hosts (Tables 4 and 5) with the aid of the empirical scaling relation between BH mass and galaxy total stellar mass [Greene et al., 2020]. Note that the slope, zero point, and intrinsic scatter of the $M_\mathrm{BH}-M_*$ relation vary considerably depending on galaxy morphological type. While the disk-dominated structure of the hosts tempts us to consider the relation for late-type galaxies, we are conflicted simultaneously by the large stellar masses of the hosts that more closely mimic early-type galaxies. For concreteness, we choose the relation calibrated using all galaxy types combined:

\begin{equation}
\log\, \left(\frac{M_\mathrm{BH}}{M_\odot} \right) = (7.43 \pm0.09) + (1.61 \pm 0.12)\log\, \left(\frac{M_*}{M_0} \right),
\label{equation8}
\end{equation}

with $M_0 = 3 \times 10^{10} \, M_\odot$ and an intrinsic scatter of 0.81 dex. At a fiducial stellar mass of $M_* = 10^{11}\,M_\odot$, the scaling relation of early-type galaxies overpredicts $M_\mathrm{BH}$ by 0.36 dex, while the late-type galaxy calibration underpredicts $M_\mathrm{BH}$ by 0.69 dex. We are even warier of relying on scaling relations based on bulge properties, in view of the poor link between BHs and pseudo bulges
[Kormendy & Ho, 2013]. Figure <ref> shows the Owen-Ledlow relation transformed into the reference frame of BH mass,

\begin{equation}
\log\, \left(\frac{L_\mathrm{1.4}}{{\rm W\,Hz^{-1}}} \right) = 1.35\log\, \left(\frac{M_\mathrm{BH}}{M_\odot} \right) +12.9,
\end{equation}

where we have converted $R$-band absolute magnitude to BH mass using the tight (scatter 0.30 dex) correlation between BH mass and $K$-band luminosity and $V-K=3.0$ mag [Kormendy & Ho, 2013], together with $V-R=0.61$ mag appropriate for ellipticals [Fukugita et al., 1995].

In the new diagram comparing radio luminosity versus BH mass, the FR I–FR II dichotomy is largely restored, although the introduction of the spiral hosts blurs the sharp boundary between the two types of radio sources as traditionally defined by elliptical hosts. It is possible that the FR I–FR II boundary depends on the morphological type, such that for a given BH mass spiral hosts generate systematically less radio power than their elliptical host counterparts. This can explain the paucity of FR I sources currently found among the new population of spiral hosts. The average radio power of FR Is is $\sim 2$ orders of magnitude weaker than that of FR IIs. Given the typical flux density of 50 mJy of the FR IIs currently detected in disk galaxies, all else being equal the FR I counterparts would have flux densities of merely 0.5 mJy, which would place them below the detection limit of FIRST, which is 1 mJy for individual sources and even worse for extended radio lobes [Becker et al., 1995]. Deeper, higher-resolution observations may yet reveal a more extensive population of FR I sources hosted in spiral galaxies. Indeed, [McCaffrey et al., 2022] detected complex, sub-galactic radio structures in radio-quiet quasars that are possibly FR I radio lobes, although little is known about the properties of the host galaxies.

[Ghisellini & Celotti, 2001] suggested that the division between the two types of radio sources may be associated with a transition in the accretion mode, from a radiatively inefficient, advection-dominated accretion flow [Narayan & Yi, 1994] in FR Is to the standard geometrically thin, optically thick accretion disk [Shakura & Sunyaev, 1973] in FR IIs. The transition occurs at a critical ratio $\alpha\equiv P_\mathrm{jet} /L_\mathrm{Edd} \approx 10^{-3}-10^{-2}$, where the Eddington luminosity of the BH $L_\mathrm{Edd}\equiv 1.3 \times 10^{38} \,(M_\mathrm{BH}/M_\odot)\, \mathrm{erg \, s^{-1}}$ and jet power [Cavagnolo et al., 2010]

\begin{equation}
%\log\, (P_{\mathrm{jet}}/P_\mathrm{1})=0.75(\pm 0.14) \log\, (P_\mathrm{radio}/P_2)+1.91(\pm 0.18),
\log\, \left(\frac{P_\mathrm{jet}}{P_1} \right) = (0.75\pm 0.14) \log\, \left(\frac{P_\mathrm{radio}}{P_2} \right) + (1.91\pm 0.18),
\end{equation}

with $P_1 = 10^{42}\, \mathrm{erg \, s^{-1}}$, $P_2 = 10^{40} \, \mathrm{erg \, s^{-1}}$, $P_\mathrm{radio}\equiv \nu L_\mathrm{\nu}$, and $\nu=1.4\,\mathrm{GHz}$. Figure <ref> shows that the traditional FR I–FR II dichotomy of elliptical hosts corresponds to $\alpha \approx 0.005$, consistent with the notion that the FR dichotomy arises from the transition of accretion modes. If a similar dichotomy exists in disk galaxies, we speculate that it would occur at an even lower value of $\alpha$. We note that the FR dichotomy in disk radio galaxies appears less distinct than that for the elliptical galaxies from [Owen & Ledlow, 1994]. We suspect that this is partly because our sample, owing to detection limits, lacks FR I objects with low radio luminosities. It is also possible that the dichotomy is intrinsically not sharp and breaks down at low radio power [Best & Heckman, 2012, Whittam et al., 2022] since the original [Owen & Ledlow, 1994] sample is affected by Malmquist bias.

The relative position angle ($\Delta$PA) between the radio lobes and the minor axes of the host galaxy (a) disk and (b) bulge. The upper panels show the correlation between $\Delta$PA and bulge-to-total ratio ($B/T$), where the color code indicates bulge  index ($n$). The lower panels give the distribution of $\Delta$PA in bins of $30\degr$.

§.§ Misalignment Between Radio and Optical Axes

It has long been realized that the orientation of radio lobes is not preferentially aligned with the minor axis of elliptical galaxies [Birkinshaw & Davies, 1985]. [Browne & Battye, 2010] found that the alignment is related to the ratio of radio to optical flux, with a higher probability of finding alignment in objects with lower ratios. Adopting their classification criterion, all our sources fall in the high-ratio class. The relative position angles ($\Delta$PA) between radio lobes and the minor axis of galactic disks and bulges span a broad distribution, with lobes ranging from aligned (0) to nearly perpendicular (90) relative to the galaxy (Figure <ref>). Notwithstanding a slight, apparent trend toward alignment directions, after accounting for measurement uncertainties and Poisson noise the distribution of $\Delta$PA is still consistent with being uniform, with $\chi^2=1.2$ for the disk (Figure <ref>a) and $\chi^2=1.4$ for the bulge (Figure <ref>b). Since the orientation of the jet is coupled with the spin of the central BH [Blandford & Znajek, 1977, Mirabel & Rodríguez, 1999], our results suggest that the spin of the BH is not necessarily aligned with the large-scale angular momentum of the host galaxy.

Although $\Delta$PA is randomly distributed for the whole sample, it appears to be mildly correlated with bulge prominence or bulge type. Host galaxies with the least conspicuous ($B/T<0.2$), low- index ($n\leq 1$) bulges have radio jets that tend to show alignment with the disk and bulge minor axis. In other words, the spin of the central BH of galaxies that evolved secularly through internal gas inflows instead of via mergers may bear the imprint of the angular momentum of the large-scale disk.

§ DISCUSSION

Although historically double-lobed radio sources have been found exclusively in giant elliptical galaxies [Best et al., 2005], to date no clear consensus exists as to how such an association arises. [Wilson & Colbert, 1995] posited that the galaxy-galaxy mergers that classically are invoked to form ellipticals (e.g., Toomre & Toomre, 1972) may also be conducive to spinning up the central BH, a prerequisite for producing strong radio jets if they are powered by BH spin [Blandford & Znajek, 1977]. This popular explanation has lost some persuasion, however, because rapidly spinning BHs evidently can inhabit galaxies of diverse morphological types, not only ellipticals [Reynolds, 2021].

This study adds a new twist to the narrative. Analyzing HST images acquired as part of the Zoo Gems project [Keel et al., 2022], we demonstrate unambiguously that at least some double-lobed radio sources can originate from late-type galaxies. Unlike previous reports of a similar nature that were largely based on heterogeneous ground-based images (as summarized in Section 1), the evidence introduced here is incontrovertible because of the benefits afforded by high-resolution HST images of uniform quality. Besides having been systematically selected [Banfield et al., 2015], our final high-confidence sample is twice as large as all previously studied cases of a similar nature combined (Table 5). Our quantitative analysis firmly establishes the late-type morphology of the host galaxies through multiple lines of evidence, including the direct detection of spiral arms, large-scale dust lanes, which are likely edge-on projections of spiral arms, high ellipticity, low global  index, low concentration, and, of course, low bulge-to-total ratio. The bulges are not merely modest, but, judging from their low  indices and low effective surface brightnesses relative to the Kormendy relation, they can be categorized as pseudo bulges, not classical bulges. These characteristics are extraordinary, but not unprecedented, for the hosts of extended radio sources. The closest analog is J2345$-$0449, a galaxy with a spectacular 1.6 Mpc-scale “double-double” radio jet that hosts a pseudo bulge ($n_{\rm bulge} \approx 1$) with $B/T \approx 0.14-0.18$ [Bagchi et al., 2014]. [Singh et al., 2015] suppose that radio galaxies with late-type optical morphology may originate from ellipticals that recently acquired a disk component after merging with a disk galaxy. We do not believe that this explanation is viable. An elliptical galaxy that swallows a gas-rich companion—the nearby example of Centaurus A comes to mind [Baade & Minkowski, 1954, Graham , 1979]—can gain a disk, but it cannot lose a gigantic bulge. The sources discussed in this study, which, unlike the majority of previous examples in the literature, have highly robust morphologies and structural parameters derived from high-quality HST images, decidedly do not have a substantial classical bulge component.

The unusual nature of the above-mentioned characteristics becomes even more acute when we consider the fact that the majority of the spiral hosts have very large stellar masses. More than 60% (11/18) of our main HST sample has $M_* \gtrsim 10^{11}\,M_\odot$, and within the literature sample, the percentage is similar (56% or 5/9; Table 5). Curiously, despite the clear presence of spiral structure in their disks[This is true at least for the main Zoo Gems sample, which enjoys the benefits of HST imaging that the literature sample, apart from $0313-192$ [Keel et al., 2006], does not.], the optical colors are consistent with those of passive galaxies on the red sequence [Blanton & Moustakas, 2009]. The subset of galaxies with $M_* \gtrsim 10^{11}\,M_\odot$ has a median optical color, corrected for redshift and Galactic extinction, of $g-r = 0.84\,$mag for the HST sample and $g-r = 0.73\,$mag for the literature sample. This population of supermassive spiral galaxies is qualitatively reminiscent of the superluminous spirals and lenticulars highlighted by [Ogle et al., 2016, Ogle et al., 2019]. Extremely massive disk galaxies in the nearby Universe may have arisen from gas-rich minor mergers [Jackson et al., 2022], or perhaps from having experienced an anomalously quiet merger history [Jackson et al., 2020, Zeng et al., 2021].

If galaxy morphology no longer can be regarded as a unique signpost of a galaxy's ability to generate large-scale radio jets, then what physical parameter is responsible? The key factor seems to be the galaxy's stellar mass, or, equivalently, BH mass, to the extent that the two are closely related (Equation 9). Notwithstanding a few outliers (see below), the vast majority of the radio galaxies have stellar masses $\gtrsim 10^{11}\,M_\odot$ or BH masses $\gtrsim 10^{8}\,M_\odot$. This is a longstanding, familiar result in the context of the standard paradigm that radio galaxies are exclusively ellipticals [Dunlop et al., 2003, McLure et al., 2004, Best et al., 2005]. The results of this study show that this traditional view must be modified. The hosts of double-lobed radio sources encompass not only elliptical galaxies but also a rare population of spiral/disk galaxies, whose common characteristic is that they, too, have unusually large stellar masses (Figure <ref>). Of course, not all massive galaxies produce extended radio lobes, but the probability that they do increases with galaxy stellar luminosity or mass [Scarpa & Urry, 2001, Best et al., 2005]. As a class, radio-loud AGNs are predominantly hosted by massive galaxies [Mauch & Sadler, 2007]. Moreover, among the massive galaxies that launch large-scale radio lobes, their radio luminosity or jet power scales roughly with galaxy stellar mass (Figure <ref>), which was already implicit in the Owen-Ledlow relation (Figure <ref>). Combining Equations 9 and 10, we note that the jet power is almost exactly linearly proportional to BH mass: $P_{\rm jet} \propto M_{\rm BH}^1$.  Again, we now know that these trends hold irrespective of galaxy morphology. So, too, can be said of the FR I–FR II dichotomy, except that in disk galaxies the transition between the two FR types may occur at a lower value of the critical ratio ($\alpha$) between jet power and Eddington luminosity. This remains to be verified with future, deeper observations capable of detecting the expected weaker emission of FR I sources.

If stellar mass—and, by extension, BH mass—is the primary factor that determines the probability that a galaxy launches radio jets, while, at the same time, it also controls the power of the jet that ultimately gets launched, then perhaps we can understand why extended jet structures are so rarely found in spiral galaxies. The stellar mass function of disk-dominated galaxies is weighted toward substantially lower masses than that of bulge-dominated galaxies, and even more so still when compared to that of ellipticals (e.g., Moffett et al., 2016). Thus, extended radio jets are a priori expected to be both rare and weak in spiral galaxies. Even when present, the jets will be more compact, as jet size correlates with jet power (e.g., Ledlow et al., 2002). If the jet does not extend significantly beyond the boundaries of the stellar disk, it is likely to be confused with the native synchrotron emission from star formation. For example, the AGN components of the radio emission in nearby Seyfert galaxies have 1.4 GHz powers $\lesssim 10^{23}\,\mathrm{W\,Hz^{-1}}$ and linear source sizes $\lesssim 10\,$kpc [Ho & Ulvestad , 2001, Ulvestad & Ho, 2001].

We close with some cautionary notes and suggestions for future work. Our study hinges on the key assumption that we have identified the correct optical counterpart of the host galaxy of the radio lobes. While we have devoted concerted efforts to cull a sample for which spurious chance alignment is low (Section 3), of course, such statistical arguments cannot completely guarantee that all the associations are real. Two sources in our sample stand out as glaring outliers. As discussed in Section 5.1, J0806+062 has a stellar mass of only $M_* = 6.5\times 10^9\,M_\odot$, and J1328+571, with $M_* = 7.8\times10^8\,M_\odot$, is more extreme still. Low-mass galaxies can host nuclear central BHs [Filippenko & Ho, 2003, Greene & Ho, 2004, Greene et al., 2020], and a minority even have radio cores and compact jets (e.g., Greene et al., 2006, Wrobel & Ho, 2006), but not classical double-lobed radio structures. Although J0806+062 and J1328+571 have the lowest radio powers in the sample ($L_{1.4} = 2\times 10^{23}$ and $4 \times 10^{22}$ W Hz$^{-1}$), their radio sources, while also small compared to the rest ($D = 24$ and 8 kpc), are clearly not confined to the nucleus (Figure 1). Follow-up observations are urgently needed to verify the reality of these two baffling sources.

The preceding discussion often presupposes that BH mass truly traces galaxy stellar mass. While this has been established for nearby inactive galaxies for which BH mass can be measured by dynamical methods and for active galaxies for which BH mass can be estimated through their broad emission lines (see reviews in Kormendy & Ho, 2013, Greene et al., 2020), we acknowledge the inherent uncertainty and ambiguity of applying the $M_{\rm BH}-M_*$ relation to our sample. Which $M_{\rm BH}$–host galaxy scaling relation, if any, is appropriate for these unusual galaxies? An important next step should secure more complete and higher quality optical spectra of the candidate optical counterparts, to fully delineate the ionization source of their nuclei. Are they all AGNs? If so, what type? X-ray observations <cit.> would be particularly helpful to disentangle possible confusion arising from dust obscuration or contamination by stellar energy sources. The present sample was originally selected from the FIRST survey, whose sensitivity ($\sim 1\,$mJy) and angular resolution ($\sim 5\arcsec$) can be vastly improved with follow-up observations with the Jansky Very Large Array.

§ SUMMARY

We report the existence of disk galaxies as hosts of double-lobed radio sources, using high-resolution optical (F475W) HST images acquired as part of the Zoo Gems program, which targeted 32 sources originally selected from the Radio Galaxy Zoo project that identified double-lobed radio sources associated with disk galaxies from cross-matching the FIRST survey and SDSS. To examine the fidelity and reliability of the physical association between the apparent optical counterparts and radio lobes, we systematically calculate the probability of chance alignment. For a subset of 18 high-confidence objects for which chance alignment is unlikely, we derive the optical morphologies and global and bulge structural parameters, which are combined with the physical properties of the host and radio sources to understand the nature of jet production.

Our main results are as follows:

* The host galaxies have unambiguous disk-dominated morphologies, as judged by the presence of spiral arms, large-scale dust lanes among the edge-on systems, and low global  indices and optical concentrations. Two-dimensional image decomposition yields bulge-to-total light ratios of $B/T = 0 - 0.43$, with a median value of 0.13. The bulges have low  indices (median $n_{\rm bulge} = 1.4$) and low effective surface brightnesses that are consistent with pseudo bulges.

* Despite the obvious morphological and structural properties of late-type galaxies, the majority of the hosts have very large stellar masses (median $M_* = 1.3\times 10^{11}\,M_\odot$) and red optical colors (median $g-r = 0.69\,$mag), consistent with massive, quiescent galaxies on the red sequence. A literature sample of nine radio sources previously found to be hosted by disk galaxies shares strikingly similar characteristics.

* As with the dominant population of elliptical radio galaxies, among disk radio galaxies the jet power scales with stellar mass. In terms of black hole mass, $P_{\rm jet} \propto M_{\rm BH}^1$.

* Elliptical radio galaxies display a dichotomy in the $P_{\rm jet}-M_{\rm BH}$ plane, such that sources with an FR II radio morphology produce systematically more powerful jets than FR I sources of the same black hole mass. The separation line corresponds to $P_\mathrm{jet}/L_\mathrm{Edd} \approx 0.005$, which may be related to a transition in accretion mode from a standard disk to a radiatively inefficient accretion flow. Nearly all of the currently known disk radio galaxies are FR II sources. We suggest that among the new population of disky hosts the critical threshold for the accretion mode transition occurs at a lower value of $P_\mathrm{jet}/L_\mathrm{Edd}$, which has yet to be reached by the sensitivity of the current radio observations.

* The axis of the radio jets is uncorrelated with the minor axis of the host galaxy or its bulge, although galaxies with the smallest bulges ($B/T < 0.2$, $n_{\rm bulge} \le 1$) may show a mild preference to be aligned with the jet, suggesting that the angular momentum on nuclear scales bears the imprint of the angular momentum of the large-scale galactic disk, plausibly as a consequence of gas inflows through secular evolution.

We are grateful to the anonymous referee for helpful comments. We thank Subo Dong and Kejia Lee for helpful discussions, and Hua Gao for advice on CGS data. We received financial support from the National Science Foundation of China (11721303, 11991052, 12011540375, 12192220, 12192222) and the China Manned Space Project (CMS-CSST-2021-A04, CMS-CSST-2021-A06). This research is based on observations (program 15445) made with the NASA/ESA Hubble Space Telescope obtained from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5–26555. We made use of the NASA/IPAC Infrared Science Archive, which is funded by the National Aeronautics and Space Administration and operated by the California Institute of Technology.

Astrolib PySynphot [STScI Development Team, 2013],  [Astropy Collaboration et al., 2018, Astropy Collaboration et al., 2013],  [McMullin et al., 2007],  [Peng et al., 2002, Peng et al., 2010],  [Harris et al., 2020],  [Bradley et al., 2020],  (<https://reproject.readthedocs.io/en/stable/index.html>),  [Crookston et al., 2008]

§ IMAGES OF THE LOW-CONFIDENCE SAMPLE

We present the images of the low-confidence sample in Figure A1. The radio lobes and optical galaxies have a considerable probability of being chance-aligned, given the large offsets between galaxies and radio centers. However, some may also have genuine associations because our probability estimation of chance alignment is conservative. Moreover, we have not exploited every piece of information to constrain the probabilities. Future observations, including optical spectroscopy, X-ray observations, and radio observations of higher angular resolution and better sensitivity are needed to confirm their associations.

HST F475W images of the 14 low-confidence sources in our sample, overlaid with FIRST 1.4 GHz contours in the left panel and zoomed-in to highlight the optical morphology of the galaxy in the right panel. All images are centered on the galaxy with north up and east to the left. The restoring beam of the radio map is depicted as a hatched ellipse on the lower-right corner. Radio contours are ($-3, 3, 6, 12, 24, 48, 96, ...$) $\times$ rms of each image, where the values of the rms are listed in Table 3. The scale bar in the lower-left corner of each panel indicates the proper distance at the redshift of each object.

§ NOTES ON INDIVIDUAL SOURCES

J0209$+$075: This is a face-on galaxy with a luminous bulge and loose spiral arms, which harbor several star-forming clumps. The southwestern arm looks disjoint from the galaxy, indicating a low stellar density in the inter-arm region. The bright point-like source northwest of the galaxy is a foreground star identified by SDSS.

J0219$+$015: This local ($z_\mathrm{spec}=0.04$) galaxy has a fairly prominent bulge and prominent dust structures. The radio source is classified as winged or X-shaped [Yang et al., 2019].

J0802$+$115: We do not attempt to decompose this interacting system, which has an asymmetric light distribution and long tidal tails.

J0806$+$062: This galaxy shows several spiral arms and a luminous bar-like region. A galaxy to the northeast with $z_\mathrm{phot}=0.11\pm 0.08$ might be a genuine companion. A star to the east of the galaxy with $m_g=20.7 \,\mathrm{mag}$ has been identified by SDSS and Gaia[https://irsa.ipac.caltech.edu/cgi-bin/Gator/nph-dd] [Gaia Collaboration et al., 2021].

J0813$+$552: We assigned the galaxy to the low-confidence sample because it is too faint ($m_g=20.4$ mag), and the number density at that magnitude is large. In addition, we detect a few sources to the north of the galaxy that might be candidates for the host.

J0823$+$033: This is a complicated case because the double radio lobes have a wide opening angle relative to the galaxy. Our chance-alignment analysis is invalid for this complicated object, and thus we cannot confidently determine the association between the galaxy and the radio lobes.

J0832$+$184: This galaxy has one of the most prominent bulges in our sample ($B/T=0.4$), and it exhibits a large degree of lopsidedness, suggesting possible signs of interactions. The northern radio lobe is much brighter than the southern one.

J0833$+$045: This object has a high probability of chance alignment because the galaxy is faint and off-center from the radio lobes.

J0847$+$124: This nearly edge-on galaxy has a long dust lane with complicated substructures. The double radio lobes are asymmetric; while the northern lobe is luminous, the southern one is only barely detected. A radio core coincides with the galaxy.

J0855$+$420: This edge-on late-type galaxy, bisected by a prominent, $\sim 10$ kpc dust lane, has a large stellar mass ($M_*=1.2\times 10^{11}\,M_\odot$). The radio structure has an FR I morphology because the hotspots are much closer to the galaxy than the lobes. The double lobes have a total length of $\sim 400$ kpc and a width of $\sim 100$ kpc (projected). The lobes have sinuous shapes near the hotspots.

J0901$+$164: A massive spiral with $M_* = 8\times 10^{10}\,M_\odot$, this galaxy has prominent arms and star-forming clumps. The eastern radio lobe has a tail to the southwest instead of toward the direction of the other radio lobe. We crossmatched the peak coordinate of the tail with the AllWISE catalog [Wright et al., 2010] but found no source. A spiral galaxy with $z_\mathrm{phot}=0.22\pm 0.05$ and a prominent bar is located southeast of the main object. Another faint object with $z_\mathrm{phot}=1.35\pm 0.25$ is located to the northwest of the main object and close to the center of the radio contours. To obtain a conservative estimate of the chance-alignment probability, we regard this faint object as a possible radio source.

J0903$+$432: This is a faint ($m_g=20.7$ mag), very massive ($M_*=2\times 10^{11}\,M_\odot$), edge-on galaxy with a prominent dust lane. The radio lobes are perpendicular to the galaxy plane. It has a high probability of being chance-aligned with the radio lobes because the number density of galaxies of this magnitude is large.

J0914$+$413: The galaxy is bisected by a prominent, $\sim 10$ kpc dust lane. The complicated dust lane introduces significant uncertainties to the bulge-to-disk decomposition. The northern lobe has a second peak with an intensity half of that of the main one, but no counterpart is found in the HST image and AllWISE catalog. In addition to the two lobes, a radio core coincides with the galaxy.

J0919$+$135: This is likely a chance-aligned case because the galaxy is too faint ($m_g=21.5$ mag) and significantly offset from the center of the radio lobes ($r_\mathrm{offset}=18\farcs1$). Moreover, many other objects are detected near the center of the radio lobes that may be candidate hosts.

J0926$+$465: This is an edge-on galaxy with a dust lane $\sim 20$ kpc long. In addition to the two lobes, faint radio emission also coincides with the galaxy.

J0941$+$312: We classified this faint edge-on galaxy as a low-confidence object. However, it is located in the center of radio emission, and no other sources are detected nearby. The radio lobes have an angular size of $\sim 2\farcm5$. If they are associated with the galaxy, which has $z_\mathrm{phot}=0.366\pm 0.037$, the physical dimension would be $\sim 1.4$ Mpc, similar to J2345$-$0449 [Bagchi et al., 2014].

J0956$+$162: This edge-on galaxy with a prominent ($\sim 15$ kpc) dust lane has intense radio emission from the nucleus, with peak intensity reaching $\sim 1.1$ mJy beam$^{-1}$. The two radio lobes are roughly equidistant from the center of the galaxy. We find an object with $z_\mathrm{phot}=0.46\pm 0.08$ in the direction of the eastern lobe, $\sim 2\arcsec$ offset from the radio peak; it is unlikely to be associated with the radio lobe because of its small size ($0\farcs91 \pm 0\farcs04$) and faintness ($m_{\rm F475W}=22.5 \pm 0.1 \, \mathrm{mag}$).

J0958$+$561: The galaxy has a prominent bulge and an extended disk. Its nuclear region is complicated in the HST image. The GALFIT residuals reveal a compact source near the center, which might be a candidate secondary nucleus (see Figure <ref>).  In addition to the two lobes, a radio core coincides with the galaxy.

J1128$+$241: This is a nearly face-on spiral galaxy with obvious star-forming clumps along its arms, rendering its optical color ($g-r = 0.46$ mag) bluer than those of others in the sample.

J1136$+$125: This is a low-mass galaxy ($M_*=2\times 10^9\,M_\odot$) with a peculiar morphology and distortions in its outer regions. It has a star formation rate of $0.4\,M_\odot\,\mathrm{year}^{-1}$ according to GSWLC-2. We do not consider it a likely host for the radio source. We note that there is a bright point source with $m_\mathrm{F475W}$ = 22.1 mag located only 1 west of the galactic center, which should be investigated further as a possible quasar candidate responsible for the radio lobes.

J1303$+$511: The radio morphology is too complicated to interpret, and thus we do not attempt to link it to the optical galaxy.

J1322$+$270: This is a nearby galaxy without detected radio lobes in the FIRST images. We find no reports of radio lobes in the literature either.

J1328$+$571: This low-mass ($M* = 8\times 10^8\,M_\odot$), low-redshift ($z_\mathrm{phot}=0.032\pm 0.029$) galaxy has many star-forming regions, at least two prominent arms, and a strong bar. With an estimated BH mass of $\lesssim 10^5\,M_\odot$, it qualifies as one of the very few intermediate-mass BHs known to have strong radio emission, and the first of its kind to have a double-lobed, extended jet structure. [Greene et al., 2020] suggest that intermediate-mass BHs may not have sufficient time to sink into the galactic center by dynamical friction. This may explain why the center of the radio lobes is offset from the galactic nucleus by $\sim 4\farcs0$ ($\sim 2.5$ kpc).

J1349$+$454: We assign this galaxy to the low-confidence sample because only one lobe is detected. It does not satisfy our sample definition of double-lobed radio sources. Further observation may help to verify the existence of the other lobe.

J1354$+$465: This is a late-type galaxy with a stellar mass of $5\times 10^9 \,M_\odot$. Although classified into the low-confidence sample, it is still a strong candidate for radio association due to its proximity to the center of two radio lobes. Further high-resolution radio observations and optical spectroscopy to secure a better redshift would help to determine the optical-radio association.

J1509$+$515: This galaxy has two grand-design spiral arms, a long bar that reaches $\sim$10 kpc, and an enormous stellar mass of $5\times 10^{11}\,M_\odot$. We consider it a low-confidence object because it is offset from the center of the radio lobes. However, we regard this galaxy as still a strong candidate, given its unusually large stellar mass.

J1633$+$084:  This massive ($M_* = 2\times 10^{11}\, M_{\sun}$), edge-on galaxy has a prominent, large-scale dust lane. It is quite unexpected that a galaxy this massive would have such a small bulge ($B/T=0.11$), which is likely a pseudo bulge given its very small  index ($n_\mathrm{bulge}=0.37$). In addition to the two lobes, a prominent radio core is coincident with the optical galaxy. Since both radio lobes are barely resolved, one might suspect that they are two unrelated point sources instead of radio lobes. However, the chance is extremely small for three unrelated sources to be so well aligned. We estimate the probability that three random radio sources with a projected distance smaller than 50 to be aligned by chance into a straight line with an accuracy of 5. Since the FIRST survey has $\sim 220,000$ resolved sources spanning $ 10,000 \, \mathrm{deg}^2$ of the sky with S/N $> 10$ [Banfield et al., 2015], the total number of such alignments is given by [Edmunds & George, 1981]:

\begin{equation}
N=\frac{2 \pi}{3} \Omega\, d^{3} \,n^{3} \,P,
\end{equation}

where $\Omega$ is the sky coverage of the survey, $d$ is the maximum distance between the three sources, $n$ is the source surface density, and $P$ is the alignment accuracy. For this object, we find $N=0.8$. Moreover, the actual probability is likely much lower since we find no optical counterparts for the two radio lobes. Therefore, it is unlikely that the three sources have no physical association, and we consider the two compact, symmetrically aligned radio sources to be radio lobes.

J1636$+$243: This faint galaxy ($m_g=20.1$ mag) has a stellar mass of $ 3\times 10^{9} \,M_\odot$. We classify it as a low-confidence source mainly because the number density of galaxies of that magnitude is large.

J1646$+$383: The galaxy has a prominent dust lane with a peculiar, arc-like shape. Curiously, we find a star projected against a radio contour, located northeast of the galaxy at ($\alpha, \delta) = (16^{\mathrm{h}}46^{\mathrm{m}}29.7^{\mathrm{s}}$, $+38^\circ31{}^\prime36\farcs10$). The parallax measurement from Gaia of $0.44 \pm 0.05$ mas rules out the possibility that it is a quasar, and its SDSS colors ($u-g=1.7$, $g-r=0.6$) are consistent with those of a  K-type star [Covey et al., 2007]. Thus, it is unlikely to be associated with the large-scale, intense radio emissions. Moreover, its Gaia proper motion of $v_{\rm N,E}=(-3.3 \pm 0.1,2.1 \pm 0.1)$ mas year$^{-1}$ suggests that it is moving from northeast to southwest toward, instead of away from, the center of the radio lobes.

J1656$+$640: This extremely massive ($M_* = 3\times 10^{11}\,M_\odot$) spiral galaxy, classified as star-forming from its narrow emission lines [Baldwin et al., 1981], has a star formation rate of $10 \, M_{\odot}\, \mathrm{year^{-1}}$ according to the GSWLA-2 catalog. A radio core coincides with the galactic nucleus, with an intensity slightly above $3\,\sigma$. We note three other galaxies near the radio center, which are relatively low-redshift objects with $z_\mathrm{phot}<0.3$, not likely distant giant elliptical galaxies.

J1721$+$262: The galaxy has a luminous bulge and an extended disk with spiral arms. A star, identified by SDSS and Gaia, is projected to the south. The two radio hotspots have similar projected distances to the galaxy, with an extended tail emanating from the eastern one. A radio core coincides with the galaxy.

J2141$+$082: The galaxy has an interesting “eye-like” shape with a large stellar mass in excess of $ 10^{11}\,M_\odot$.

§ TWO-DIMENSIONAL BULGE-DISK DECOMPOSITION IMAGES

We present the GALFIT bulge-to-disk decomposition results for 16 of the 18 sources in the high-confidence sample in Figure C1. Two galaxies are omitted: J0802+115 is a highly disturbed interacting system, for which no meaningful decomposition can be done, and J1328+571, which is a dwarf Magellanic spiral that has no detectable bulge.

Two-dimensional bulge-disk decomposition using  of 16 of the 18 high-confidence sources in our sample. The left column shows the surface brightness profile of the data (open circles with error bars),  bulge component (red), exponential disk component (blue), and total (bulge + disk) model (purple). The $\chi^2_\nu$ from   is shown in the lower-left corner. The lower panel gives the residuals between the data and the model. The right three columns show the original data, best-fit total model, and residuals, respectively. All images are displayed on a log stretch.

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