# Infrared spectroscopic confirmation of $z\sim 2$ photometrically-selected
obscured quasars

Yuzo Ishikawa,1 Ben Wang,2,3 Nadia L. Zakamska,1,4 Gordon T. Richards,5 Joseph
F. Hennawi,3,6 Angelica B. Rivera,7
1The William C. Miller III Department of Physics and Astronomy, Johns Hopkins
University, Baltimore, MD 21218, USA
2Department of Astronomy, Tsinghua University, Beijing 100084, China
3Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The
Netherlands
4Institute for Advanced Study, Princeton University, Princeton, NJ 08544, USA
5Department of Physics, Drexel University, 32 S. 32nd Street, Philadelphia, PA
19104, USA
6Department of Physics, Broida Hall, UC Santa Barbara, Santa Barbara, CA
93106-9530, USA
7Department of Physics, US Naval Academy, 572C Holloway Rd., Annapolis, MD
21402, USA
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The census of obscured quasar populations is incomplete, and remains a major
unsolved problem, especially at higher redshifts, where we expect a greater
density of galaxy formation and quasar activity. We present Gemini GNIRS near-
infrared spectroscopy of 24 luminous obscured quasar candidates from the Sloan
Digital Sky Survey’s Stripe 82 region. The targets were photometrically
selected using a WISE/W4 selection technique that is optimized to identify IR-
bright and heavily-reddened/optically-obscured targets at $z>1$. We detect
emission lines of ${\rm H\alpha}$, ${\rm H\beta}$, and/or [O iii] in 23
sources allowing us to measure spectroscopic redshifts in the range $1<z<3$
with bolometric luminosities spanning $L=10^{46.3}-10^{47.3}$ erg s-1. We
observe broad $10^{3}-10^{4}$ km s-1 Balmer emissions with large ${\rm
H\alpha}/{\rm H\beta}$ ratios, and we directly observe a heavily reddened
rest-frame optical continuum in several sources, suggesting high extinction
($A_{V}\sim 7-20$ mag). Our observations demonstrate that such
optical/infrared photometric selection successfully recovers high-redshift
obscured quasars. The successful identification of previously undetected red,
obscured high-redshift quasar candidates suggests that there are more obscured
quasars yet to be discovered.

###### keywords:

galaxies: high-redshift - quasars: general - quasars: supermassive black holes
- quasars: emission lines - infrared:galaxies

††pubyear: 2022††pagerange: Infrared spectroscopic confirmation of $z\sim 2$
photometrically-selected obscured quasars–Infrared spectroscopic confirmation
of $z\sim 2$ photometrically-selected obscured quasars

##  1 Introduction

According to the standard unification model, many of the observed properties
of active galactic nuclei (AGN) may be explained by differences in the viewing
angle with respect to the dusty torus (e.g., Antonucci, 1993). In this scheme,
there is a compact source of optical, ultraviolet (UV) and X-ray emitting
continuum associated with the accretion disk and the corona near the
supermassive black hole. On larger scales, other components include the broad
line region, the dusty obscuring torus, and the narrow line region of gas
photo-ionized by the quasar radiation (e.g., Krolik & Begelman, 1988; Lawrence
& Elvis, 2010). In this picture, Type-1 quasars are observed with a direct
line of sight to the central source revealing the broad emission lines. Type-2
quasars are observed sideways through the obscuring torus, so that only narrow
(and forbidden) emission lines and infrared (IR) emission produced by the
reprocessing in the torus are observed.

In addition to the purely orientation-based hypothesis for the difference
between obscured and unobscured sources, there are other possible
interpretations. Popular galaxy evolution models postulate that unobscured and
obscured quasars represent different phases of evolution. In this picture, the
obscured phase precipitates a phase of strong mechanical feedback, during
which the obscuring material is expelled to reveal the unobscured quasar (e.g
Sanders et al., 1988; Canalizo & Stockton, 2001; Hopkins et al., 2006; Glikman
et al., 2012). There is evidence that some obscured sources, although a
minority, may be caused by transient absorbers in the line of sight (e.g.,
LaMassa et al., 2015). Some of the most luminous obscured quasars at high
redshifts like extremely red quasars (ERQs) may be super-Eddington accretors
(e.g., Assef et al., 2015; Perrotta et al., 2019; Zakamska et al., 2019).
These quasars may be obscured by inflowing and outflowing materials.
Quantifying and characterizing obscured quasars remains an ongoing problem in
modern astrophysics.

A complete understanding of the growth of supermassive black holes requires a
complete census of both the unobscured and obscured quasar populations. The
unification models predict that the number density of unobscured and obscured
quasars should be comparable through cosmic time, peaking at $z\sim 2.5$
(Richards et al., 2006; Ross et al., 2013). A large population of obscured
quasars would affect our existing calculations of the accretion efficiency of
supermassive black holes (Soltan, 1982; Yu & Tremaine, 2002). Furthermore,
obscured quasars are a key component to the cosmic X-ray background (e.g.,
Treister & Urry, 2005; Lanzuisi et al., 2009). Moreover, $z\sim 2$ marks the
epoch of rapid growth of the supermassive black holes through mass accretion
in the most luminous quasars, so it is critical to characterize the quasars at
high redshift.

Decades of work have established that the number density of unobscured and
obscured quasars at low redshifts ($z<1$) are comparable (Lawrence & Elvis,
2010; Reyes et al., 2008). However, the census at high redshift remains
incomplete and debated (e.g., Polletta et al., 2008; Eisenhardt et al., 2012;
Lusso et al., 2013; Hennawi & Prochaska, 2013; Prochaska et al., 2013; Assef
et al., 2015). Some X-ray studies suggest that the obscured fraction may be a
function of luminosity with an anti-correlation between obscured fraction and
quasar luminosity, yet remain debated (e.g., Ueda et al., 2003; Glikman et
al., 2018). The apparent paucity may imply that obscured quasars are not very
common, possibly due to physical changes in their obscuring medium (Hasinger,
2008; Lawrence, 1991; Konigl & Kartje, 1994; Elitzur & Shlosman, 2006).
Alternatively, they may have eluded discovery due to incomplete selection.

It is important to highlight the difficulty of observing obscured quasars.
Quasars typically outshine their host galaxies by several orders of magnitude
in the rest-frame UV and optical wavelengths. However, a key characteristic of
obscured quasars is that dust obscuration in the line of sight can heavily
redden the quasar continuum. As a result, the largest catalogs of obscured
quasars are largely limited to low redshift targets that are optically
selected based on their extremely strong narrow emission lines which originate
outside of the obscuring material (e.g., Zakamska et al., 2003; Reyes et al.,
2008; Yuan et al., 2016). Alternatively, powerful X-ray detections can be
indicative of quasar activity. However, the X-ray selection is typically
limited to sources with modest column densities that allow hard X-rays to
travel unimpeded. At extremely high, Compton-thick column densities, the
absorption makes X-ray detections prohibitively difficult, despite high
intrinsic luminosities of the quasar hidden behind the obscuration (e.g.,
Goulding et al., 2018; Zappacosta et al., 2018; LaMassa et al., 2019; Ishikawa
et al., 2021). Therefore, a more robust strategy is to take advantage of the
dust obscuration by probing the mid-infrared (MIR) emission, which is thought
to be both more universal and isotropic, compared to UV/optical selections.
Even Compton-thick quasars should be MIR-bright due to the reprocessed
emission from the warm/hot dust in/around the obscuring torus. However,
caution is needed as this method has many contaminants. For example, MIR
continuum emission can also result from intense star-forming galaxies; the
clumpy nature of the torus, which can allow MIR emission to escape unimpeded;
and dust in the polar regions rather than the obscuring torus (Hickox &
Alexander, 2018).

MIR-based color selection using WISE (Wright et al., 2010) or Spitzer (Werner
et al., 2004) has had successes in identifying obscured quasars. Recent work
(Eisenhardt et al., 2012; Wu et al., 2012; Yan et al., 2013; Ross et al.,
2015; Glikman et al., 2018) demonstrated that a combined selection of bright
mid-IR sources that are optically faint efficiently identifies obscured
quasars both at $z<1$ and at $z\sim 2$. However, canonical mid-IR Spitzer/IRAC
color selection wedges, which rely on $8\micron$ (rest-frame $2.7\micron$)
fluxes, do not sufficiently probe the peak of the quasar-heated dust emission
at $\sim 10\micron$ (Lacy et al., 2004; Stern et al., 2005; Polletta et al.,
2008; Donley et al., 2012; Lacy et al., 2013) and may miss populations of high
redshift obscured quasars. Therefore, it would be ideal to probe even longer
wavelengths, such as with the all-sky WISE/W4 ($\sim 22\micron$) coverage,
complemented with Spitzer observations. In this paper, we take advantage of
color-selection strategies based on both WISE/W4 and Spitzer fluxes to select
luminous $z\sim 2$ obscured candidates over an area larger than those of the
previous searches.

This paper is structured as follows. In Section 2 we discuss the sample
selection and data reduction. In Section 3 we present the analysis of spectra
and photometry of our candidates. And Section 4 we discuss the implications of
our results, and we conclude in Section 5. We use the AB magnitude system
unless otherwise specified. We adopt the $h=0.7$, $\Omega_{M}=0.3$, and
$\Omega_{\Lambda}=0.7$ cosmology.

##  2 Target selection

###  2.1 Sample selection

Figure 1: The color-color distribution of the WISE/W4 selected objects. The
black points show the distribution of $f_{24\micron}>1\textrm{ mJy}$ point
sources detected in Spitzer/IRAC. The orange and purple circles indicate
spectroscopically confirmed $z<1$ and $z>1$ sources, respectively. The gray
contours indicate the SDSS non-detections. The red diamonds indicate our
sources. We stress that our selections in this study primarily rely on WISE/W4
and SDSS fluxes. Any suggestive color-color relations may serve as a guide for
future searches of obscured quasars. We also compare with the color cuts of
HotDOGs (Eisenhardt et al. 2012; black dashed line) and the observed WISE
colors of ERQs (Perrotta et al. 2019; blue squares): both of which rely on
color-based selections.

Our selection uses WISE, Spitzer, and SDSS data to identify sources that are
MIR luminous, yet optically faint. First, WISE/W4 bright sources are selected.
We require $5\sigma$ detection with $6<W4<8\textrm{ Vega mag}$ ($f_{W4}\geq
5\textrm{ mJy}$). The bright limit is applied to reduce stellar contamination.
Additional criteria are applied to ensure robust MIR detections and to limit
contamination from stars and other spurious sources: $5\sigma$ detection of
WISE/W3 ($\sim 12\micron$) and $W1\geq 12\textrm{ Vega mag}$. We also consider
the WISE data-quality flags to avoid artifacts ($\texttt{w4flg}=0$ and
$\texttt{w3flg}=0$) and to limit to point-sources ($\texttt{ext\\_flg}=0$ and
$\texttt{nb}\leq 2$). These WISE-selected sources are then cross-matched with
Spitzer surveys to obtain precise astrometry of $\sim 0.1"$ for narrow-slit
spectroscopic follow-up.

The MIR selections are further narrowed down to identify optically-faint
sources. The full search area includes fields from SHELA (Papovich et al.,
2016), SWIRE (Lonsdale et al., 2003), SERVS (Mauduit et al., 2012), COSMOS
(Sanders et al., 2007), SDWFS, and AllWISE surveys. For this Gemini/GNIRS
study, we limit the selection to optically faint sources from the Spitzer IRAC
Equatorial Survey (SpIES; PI Richards), which is a SWIRE-depth survey in the
SDSS Stripe 82 area (Timlin et al., 2016). Stripe 82 is the largest field
considered. The effective search area is $\gtrsim 164\textrm{ deg}^{2}$. We
define optical faintness as either having $r>23\textrm{ AB mag}$ or undetected
in the Sloan Digital Sky Survey (SDSS; York et al. 2000). If we combine the
SDSS and WISE+Spitzer selections, we obtain a $r-W4>13.5$ color cut. The
effective color cut implied by these criteria is somewhat more stringent than
that for the selection of Dust Obscured Galaxies (DOGS; Dey et al. 2008).
Previous studies (Brand et al., 2007; Ross et al., 2015) show that this
selection uncovers luminous obscured quasars at $z\sim 2$.

We identified 373 sources from the W4 selection over all fields considered.
This selection recovers 15 known $z<1$ sources and 22 known $z>1$ sources, of
which 18 are known $z>1$ Type-2 or reddened Type-1 quasars, including a
$z=3.106$ HotDOG (Lacy et al., 2013; Glikman et al., 2013; Banerji et al.,
2015; Tsai et al., 2015; Glikman et al., 2018). From the candidates, we
selected sources that lack spectroscopic confirmation. We find 26 optically
faint sources in the Stripe 82 region that are appropriate for Gemini/GNIRS.
Interestingly, only 3 sources are detected in SDSS photometry. We obtained
Gemini/GNIRS spectra for 24 of these sources to identify the ${\rm H\alpha}$
emission line to obtain spectroscopic redshifts.

In Figure 1 we show a color-color plot of the sources to compare with other
known color-selected obscured quasars. We stress that our selections do not
rely on $W1-W2$ or $W1-W4$ colors. We discuss the implications of the
selection criteria in Section 4.1. We list our Gemini/GNIRS-observed targets
and their selection properties in Table 1. We summarize the selection criteria
here:

$\begin{split}&W4\geq 5\textrm{ mJy}\\\ &r>23\textrm{, or undetected in
SDSS}.\end{split}$ (1)

Table 1: Sample list of observed quasars. The target coordinates are determined from Spitzer+WISE and SDSS photometry. The targets are ordered by coordinates. Sources observed on different dates have been co-added to produce a single spectrum. We list the selection properties. We classify the sources as A/B/BB/C/D based on their continuum and emission line strengths. All continuum detections have a red slope; ‘BB’ marks sources with a blue excess. † denotes sources with no SDSS detections; SDSS r-band upper limit is $r>23\textrm{ AB mag}$ (York et al., 2000).  Target name | RA | DEC | Date(s) observed | W4 [22µm] | $r^{\prime}_{\textrm{SDSS}}$ | Category
---|---|---|---|---|---|---
| (J2000) | (J2000) | (yyyy-mm-dd) | (AB mag) | (AB mag) |
J002407.02$-$001237.2 | 00:24:07.02 |  $-$00:12:37.2 | 2017-10-06 | $7.39\pm 0.12$ | -† | C
J004157.77$-$002932.1 | 00:41:57.77 |  $-$00:29:32.1 | 2017-10-07 | $7.94\pm 0.21$ | -† | BB
J004729.25$+$000358.8 | 00:47:29.25 |  $+$00:03:58.8 | 2017-10-21, 2017-10-30 | $7.67\pm 0.16$ | -† | D
J005424.45$+$004750.2 | 00:54:24.45 |  $+$00:47:50.2 | 2017-10-21 | $7.21\pm 0.13$ | -† | C
J010552.86$-$002351.2 | 01:05:52.86 |  $-$00:23:51.2 | 2017-10-21, 2017-10-30 | $7.78\pm 0.16$ | -† | C
J011222.64$-$001633.0 | 01:12:22.64 |  $-$00:16:33.0 | 2017-11-05 | $7.73\pm 0.20$ | -† | C
J011314.49$+$002917.1 | 01:13:14.49 |  $+$00:29:17.1 | 2017-11-05 | $7.85\pm 0.19$ | -† | C
J013033.47$+$000950.4 | 01:30:33.47 |  $+$00:09:50.4 | 2017-11-05 | $7.98\pm 0.20$ | -† | C
J014939.96$+$005256.7 | 01:49:39.96 |  $+$00:52:56.7 | 2017-11-07 | $7.95\pm 0.17$ | -† | B
J015055.28$+$005600.2 | 01:50:55.28 |  $+$00:56:00.2 | 2017-11-06 | $7.98\pm 0.21$ | -† | D
J015235.29$-$002459.4 | 01:52:35.29 |  $-$00:24:59.4 | 2017-11-06 | $7.26\pm 0.09$ | $22.57\pm 0.18$ | A
J021345.44$+$002436.1 | 02:13:45.44 |  $+$00:24:36.1 | 2017-11-07 | $7.86\pm 0.16$ | -† | A
J021426.98$-$000021.3 | 02:14:26.98 |  $-$00:00:21.3 | 2017-11-07, 2017-11-09 | $7.56\pm 0.12$ | -† | C
J021514.76$+$004223.8 | 02:15:14.76 |  $+$00:42:23.8 | 2017-09-27 | $7.39\pm 0.12$ | -† | B
J022127.60$+$005024.6 | 02:21:27.60 |  $+$00:50:24.6 | 2017-09-29 | $7.70\pm 0.14$ | -† | C
J222920.83$+$002253.5 | 22:29:20.83 |  $+$00:22:53.5 | 2017-11-07 | $7.04\pm 0.11$ | -† | C
J223358.38$-$000414.9 | 22:33:58.38 |  $-$00:04:14.9 | 2017-11-05 | $7.81\pm 0.19$ | -† | D
J223904.01$-$003054.9 | 22:39:04.01 |  $-$00:30:54.9 | 2018-01-02 | $7.46\pm 0.16$ | -† | D
J223911.98$-$005422.3 | 22:39:11.98 |  $-$00:54:22.3 | 2017-09-29, 2017-09-30 | $7.75\pm 0.18$ | $23.31\pm 0.32$ | D
J224338.04$+$001749.9 | 22:43:38.04 |  $+$00:17:49.9 | 2017-09-30 | $7.51\pm 0.17$ | -† | A
J225851.90$-$002207.0 | 22:58:51.90 |  $-$00:22:07.0 | 2017-10-01 | $7.89\pm 0.20$ | -† | C
J225956.84$-$000918.4 | 22:59:56.84 |  $-$00:09:18.4 | 2018-01-03 | $7.76\pm 0.19$ | -† | B
J232925.01$+$002057.7 | 23:29:25.01 |  $+$00:20:57.7 | 2017-11-07 | $7.88\pm 0.19$ | -† | C
J233441.49$+$003114.0 | 23:34:41.49 |  $+$00:31:14.0 | 2017-11-06 | $7.62\pm 0.16$ | $23.10\pm 0.22$ | BB

###  2.2 Observation and reduction

The targets were observed with the Gemini/GNIRS spectrograph (Elias et al.,
2006a; Elias et al., 2006b) under GN-2017B-Q-51 (PI: Richards) over several
nights spanning 2017-09-27 to 2018-01-03. We used GNIRS with the
$0.15"/\textrm{pixel}$ camera and the $32\textrm{ l/mm}$ grating in the cross
dispersed mode for a coverage over 0.9-2.4 µm. We used the $0.45"$ slit for an
effective spectral resolution of $R\sim 1100$. Observations were performed
with the ABBA observing sequence. Each object has a total integration time of
2400s.

We reduce the data with the PypeIt pipeline (Prochaska et al., 2020b, c, a).
This reduction produces fully calibrated science spectra. For objects with
clear continuum, we run the automatic spectra extraction routine. For objects
with faint or no continuum detected, we manually look for emission lines to
select the extraction aperture in the 2D spectra for the spectral extraction.
Flux calibration is applied with standard stars observed on the same night.
Telluric corrections are also applied.

We examine the fully reduced and calibrated spectra to categorize the sources
as A/B/C/D based on the observed continuum and emission line strengths. The
sources are classified into the following categories: (A) strong line(s) with
strong continuum, (B) weak line(s) with weak continuum, (BB) ‘B’-type with a
blue-sloped excess, (C) weak line with no continuum, and (D) no line and no
continuum. We indicate these classifications in Table 1. We are able to
confidently categorize most sources, although some required additional
Keck/LRIS observations for confirmation (Wang et al., in prep). Roughly half
of the observed sources have significant emission lines and/or continuum. The
other half show emission with low equivalent widths, which made it difficult
to determine the spectroscopic redshift. Only one source (J0047+0003) showed
no features. We show a subset of the observed sources with significant
emission and/or continuum in Figure 2.

##  3 Results: spectral analysis

Table 2: Emission line properties of the targets listed by coordinates. ∗ indicates sources with significant telluric absorption of ${\rm H\alpha}$. † indicates sources with featureless spectra, but are spectroscopically confirmed by Keck/LRIS observations; $z_{\textrm{best}}$ is the best redshift estimate, and $z_{\textrm{GNIRS}}$ is the redshift estimate using only Gemini/GNIRS data. ${}^{\textrm{w}}$ indicate redshifts identified determined by Gemini/GNIRS and Keck/LRIS (Wang et al., in prep).  Target | $z_{\textrm{best}}$ | $z_{\textrm{G}}$ | Ref.line | $\log_{10}L_{H\alpha}$ | FWHMHα | $\log_{10}L_{H\beta}$ | FWHMHβ | $\log_{10}L_{[\textrm{O}~{}\textsc{iii}]}$ | FWHM${}_{[\textrm{O}~{}\textsc{iii}]}$ | $\log_{10}L_{\textrm{bol}}$
---|---|---|---|---|---|---|---|---|---|---
|  |  |  | ($\textrm{erg s}^{-1}$) | ($\textrm{km s}^{-1}$) | ($\textrm{erg s}^{-1}$) | ($\textrm{km s}^{-1}$) | ($\textrm{erg s}^{-1}$) | ($\textrm{km s}^{-1}$) | ($\textrm{erg s}^{-1}$)
J0024$-$0012 | 1.528${}^{\textrm{w}}$ | 0.925 | [O iii] | $42.01\pm 0.06$ | $2355\pm 193$ | $41.5\pm 2.0$ | $650\pm 300$ | $42.33\pm 0.15$ | $3300\pm 900$ | $46.97\pm 0.02$
J0041$-$0029 | 2.09${}^{\textrm{w}}$ | 1.495 | ${\rm H\alpha}$ | $42.61\pm 0.01$ | $2780\pm 1700$ | - | - | - | - | $46.71\pm 0.05$
J0047$+$0003 | - | - | - | - | - | - | - | - | - | -
J0054$+$0047 | 2.17 | 2.17 | ${\rm H\alpha}$ | - | - | - | - | - | - | $47.31\pm 0.03$
J0105$-$0023† | 1.865 | 1.865 | ${\rm H\alpha}$ | $44.33\pm 0.05$ | $200\pm 70$ | - | - | - | - | $46.81\pm 0.05$
J0112$-$0016 | 2.99${}^{\textrm{w}}$ | - | [O iii] | - | - | - | - | $41.88\pm 0.07$ | $2000\pm 1000$ | $47.72\pm 0.03$
J0113$+$0029 | 2.33 | 2.33 | ${\rm H\alpha}$ | $42.38\pm 0.21$ | $470\pm 175$ | - | - | - |  | $46.83\pm 0.10$
J0130$+$0009 | 2.5 | 2.5 | ${\rm H\alpha}$ | $42.70\pm 0.13$ | $1180\pm 255$ | - | - | - | - | $47.23\pm 0.04$
J0149$+$0052 | 1.85 | 1.85 | ${\rm H\alpha}$ | $42.06\pm 0.09$ | $2370\pm 450$ | - | - | $42.43\pm 0.05$ | $1840\pm 100$ | $46.95\pm 0.04$
J0150$+$0056 | 1.7 | 1.7 | ${\rm H\alpha}$ | $42.76\pm 0.39$ | $3300\pm 1000$ | - | - | - | - | $46.91\pm 0.04$
J0152$-$0024 | 2.775 | 2.775 | ${\rm H\alpha}$ | $43.37\pm 0.01$ | $3840\pm 300$ | - | - | $42.90\pm 0.18$ | $940\pm 100$ | $47.35\pm 0.04$
J0213$+$0024∗ | 1.805 | 1.805 | [O iii] | $43.05\pm 0.01$ | $6896\pm 300$ | $42.25\pm 0.05$ | $5000\pm 1000$ | $42.38\pm 0.20$ | $2386\pm 260$ | $47.02\pm 0.03$
J0214$-$0000 | 1.627 | 1.627 | ${\rm H\alpha}$ | - | - | - | - | - | - | $47.03\pm 0.02$
J0215$+$0042 | 0.88 | 0.88 | ${\rm H\alpha}$ | $42.38\pm 0.03$ | $2150\pm 100$ | $41.47\pm 0.39$ | $1690\pm 1400$ | $42.13\pm 0.12$ | $3000\pm 200$ | $46.27\pm 0.01$
J0221$+$0050 | 2.48 | 2.48 | ${\rm H\alpha}$ | $42.94\pm 0.02$ | $4400\pm 370$ | - | - | - | - | $46.25\pm 0.04$
J2229$+$0022 | 1.93 | 1.93 | ${\rm H\alpha}$ | $42.58\pm 0.12$ | $2840\pm 1190$ | - | - | $42.19\pm 0.13$ | $950\pm 197$ | $47.19\pm 0.03$
J2233$-$0004† | 1.602${}^{\textrm{w}}$ | - | ${\rm H\alpha}$ | - | - | - | - | - | - | $46.23\pm 0.16$
J2239$-$0030† | 1.905${}^{\textrm{w}}$ | - | ${\rm H\alpha}$ | - | - | - | - | - | - | $47.20\pm 0.03$
J2239$-$0054† | 2.085${}^{\textrm{w}}$ | - | ${\rm H\alpha}$ | - | - | - | - | - | - | $47.12\pm 0.04$
J2243$+$0017∗ | 1.905 | 1.905 | [O iii] | - | - | - | - | $44.7\pm 0.05$ | $1865\pm 100$ | $47.10\pm 0.04$
J2258$-$0022 | 2.42 | 2.42 | ${\rm H\alpha}$ | $42.29\pm 0.05$ | $2210\pm 400$ | - | - | - | - | $46.95\pm 0.08$
J2259$-$0009 | 1.885 | 1.885 | ${\rm H\alpha}$ | $42.09\pm 0.09$ | $2350\pm 115$ | - | - | $42.24\pm 0.12$ | $1750\pm 100$ | $47.03\pm 0.04$
J2329$+$0020 | 2.665 | 2.665 | ${\rm H\alpha}$ | $43.18\pm 0.03$ | $2340\pm 115$ | $42.24\pm 0.15$ | $677\pm 150$ | $42.62\pm 0.11$ | $1330\pm 280$ | $47.16\pm 0.06$
J2334$+$0031 | 2.095${}^{\textrm{w}}$ | 1.355 | ${\rm H\alpha}$ | $42.19\pm 0.09$ | $4350\pm 200$ | $42.17\pm 0.28$ | $1600\pm 000$ | $42.85\pm 0.02$ | $3500\pm 500$ | $46.95\pm 0.06$

###  3.1 Redshift estimates

|
---
Figure 2: Example spectra of A/B/BB/C sources with continuum and/or emission
lines detected. We do not show the featureless D-sources here. The shaded blue
regions mark telluric absorption, and the orange curve is the noise vector
plot. Raw spectra are in light gray and the smoothed spectra in bold black.
The vertical red lines indicate the emission lines identified.
---
Figure 3: Redshift distribution of the observed sources.

We compute the redshifts in two steps. First, we visually identify emission
lines of interest (e.g. ${\rm H\alpha}$ and ${\rm H\beta}$$+$[O iii]) to
determine the rough spectroscopic redshift. We set these as priors to fit
Gaussians to the observed emission lines. We then refine the spectroscopic
redshift with the best-fit line centroid shifts. To this end, we first
subtract out the underlying continuum. All sources with strongly detected
continuum flux have a red slope. We define the continuum by masking out the
emission lines and telluric contamination at $\sim 1.4\micron$ and $\sim
1.9\micron$ and interpolating over the masked spectrum. Finally, we fit single
Gaussians to the continuum-subtracted emission line spectrum.

For targets with sufficient GNIRS detections, we take the ${\rm H\alpha}$
and/or ${\rm H\beta}$$+$[O iii] lines to estimate the spectroscopic redshifts.
The observed equivalent widths of the emission lines vary. For sources with
heavy telluric contamination, we use other emission lines for spectroscopic
redshift determination. For example, J0213+0024 and J2243+0017 show heavy
$\sim 1.9\micron$ telluric contamination, making it difficult to accurately
find the centroid of the ${\rm H\alpha}$ line; instead we use the [O iii]
doublet and [O i]$\lambda 8446$Å. If we only detect one emission line (e.g.
J2334+0031), then we assume it is ${\rm H\alpha}$. We fit the [O iii] doublet
by kinematically tying the lines to [O iii]5009Å and setting the peak flux [O
iii] line ratio $\lambda 5009$Å/$\lambda 4959$Å=2.993. Some sources were
originally identified as featureless ‘D’-sources, but were confirmed with
Keck/LRIS observations (Wang et al., in prep). For example, the ${\rm
H\alpha}$ line for J0112-0016 is redshifted out of the GNIRS coverage, and the
one line visible in the GNIRS spectrum turns out to be [O iii].

In Table 2, we show the redshift estimates from Gemini/GNIRS-only data,
$z_{\textrm{GNIRS}}$, and the “best” estimates from the combined
Gemini/GNIRS+Keck/LRIS data, $z_{\textrm{best}}$. The differences in the
redshift estimates highlight how challenging it can be to identify obscured
quasars. In Figure 3 we show the redshift distribution of the
spectroscopically confirmed objects. We can see that the initial color
selection preferentially selects $1<z<3$ objects with the average redshift of
$\langle z_{\textrm{best}}\rangle=2.008\pm 0.45$. We do not see any redshift
dependence on the spectral types outlined in Section 2.2.

###  3.2 Emission line properties

---
Figure 4: Line kinematics and the line luminosity of ${\rm H\alpha}$ (left),
${\rm H\beta}$ (center), and [O iii] (right). The red symbols indicate sources
with sufficient signal-to-noise ratio (SNR) for line fits. The observed ${\rm
H\alpha}$ display broad widths similar to typical quasars. However, the width
distribution of [O iii] and ${\rm H\alpha}$ are inconsistent at $\sim 1\sigma$
using the KS-test. We also compare the [O iii] properties with values from
literature Harrison et al. (2012); Zakamska & Greene (2014); Shen (2016);
Bischetti et al. (2017); Perrotta et al. (2019); Temple et al. (2019). No
obvious correlation between the line luminosity and widths is seen.
---
Figure 5: Line luminosities (top row) and kinematics (bottom row) shown
against $L_{5\micron}$ for ${\rm H\alpha}$ (left), ${\rm H\beta}$ (center),
and [O iii] (right). The red symbols indicate sources with sufficient SNR for
line fits. Light-red diamonds indicate the $1\sigma$ upper limits based on the
average continuum flux. We also compare the ${\rm H\alpha}$, ${\rm H\beta}$
and [O iii] kinematics with values from Harrison et al. (2012); Zakamska &
Greene (2014); Shen (2016); Bischetti et al. (2017); Wu et al. (2018);
Perrotta et al. (2019). For the comparison with our sample, we converted the
published $w_{80}$ velocities to FWHM. Our sources show a wide range of
velocities with 3 reaching ERQ-like velocities (see bottom right).

After we obtain the best-fit line parameters, we calculate the line
luminosities. The line luminosity is calculated as the observed fluxes
multiplied by $4\pi D_{L}^{2}$ assuming that the emission is isotropic, where
$D_{L}$ is the luminosity distance at $z_{\textrm{best}}$. In Table 2 we show
the calculated line luminosity and the full-width-half-maximum (FWHM;
$2.355\sigma$). The ${\rm H\alpha}$ line is the brightest with luminosity
reaching $10^{42}\textrm{-}10^{43.4}\textrm{ erg s}^{-1}$, whereas the ${\rm
H\beta}$$+$[O iii] lines are nearly $\times 10$ fainter. The observed ${\rm
H\alpha}$ fluxes are commensurate with those of DOGS (Brand et al., 2007).
This is also at the limit of GNIRS sensitivities, based on GNIRS ITC
predictions of emission from a $z=2.3$ source with $1\times 10^{-16}\textrm{
erg s}^{-1}\textrm{ cm}^{-2}$ and $\textrm{FWHM}\times 1000\textrm{ km
s}^{-1}$ superposed on a continuum of $1\times 10^{-18}\textrm{ erg
s}^{-1}\textrm{ cm}^{-2}$Å-1.

In Figure 4, we show the relationship between emission line widths (FWHM) and
luminosities. We find that no obvious correlations between the line width and
the line luminosity. We further see that the ${\rm H\alpha}$ emission lines
consistently show broad profiles with FWHM of several $1,000\textrm{ km
s}^{-1}$. In contrast, the [O iii] width distribution is concentrated around
FWHM=$1,000-3,000\textrm{ km s}^{-1}$. For the 9 sources with both ${\rm
H\alpha}$ and [O iii] detections, the Kolmogorov-Smirnov (KS) test does not
indicate a strong correlation between the line widths. This lack of
correlation may suggest that the observed ${\rm H\alpha}$ and [O iii]
kinematics may originate from different mechanisms (e.g. BLR and NLR emission
vs. outflows). To better understand the observed line kinematics, we compare
our [O iii] measurements with those of other quasar samples: $z<1$ Type 2
(Zakamska & Greene, 2014), $z\sim 2$ Type 2 (Harrison et al., 2012),
$1.5<z<3.5$ Type 1 (Shen, 2016), $2.3\lesssim z\lesssim 3.5$ quasars
(Bischetti et al., 2017), and $z\sim 2$ ERQs (Perrotta et al., 2019). We can
see that the [O iii] emission lines in our sample are much fainter, yet are
broad. The observed line properties are mostly consistent with other known
Type-2 quasars. Zakamska & Greene (2014) show median [O iii] FWHM of $\sim
890\textrm{ km s}^{-1}$, ranging between $340-4390\textrm{ km s}^{-1}$, after
converting from $w_{80}$.

In addition to ${\rm H\alpha}$, we see various low ionization lines like [O
i]$\lambda 6300$Å and [O ii]$\lambda 3727$Å in certain sources (e.g.
J0213+0024, J2243+0017, J0149+0052, J0215+0042). [O i] may be suggestive of
shocks or outflows in the neutral medium. The non-detection of Fe ii further
supports that these targets are indeed obscured or red quasars. The focus of
this paper will be on the ${\rm H\alpha}$ and ${\rm H\beta}$$+$[O iii] lines.

###  3.3 Rest-frame optical vs. IR

Due to the nature of the color selection, all our targets have strong IR
fluxes, despite variations in the optical emission. We compare the
monochromatic IR luminosity at rest-frame $\lambda=5\micron$, which we denote
as $L_{5\micron}\equiv\lambda L_{\lambda}(5\micron)$. Since the observed
broadband WISE filters correspond to different rest-frame wavelengths, we must
interpolate over the observed WISE fluxes, $F_{\lambda,\textrm{obs}}$. We
assume the IR spectra take a power-law form
$L_{\lambda}\propto\lambda^{\alpha}$ and interpolate to compute
$L_{5\micron}$. We find an average slope of $\langle\alpha\rangle=1.05\pm 0.6$
around $5\micron$. All sources are very IR luminous with $L_{5\micron}\sim
10^{46}-10^{47}\textrm{erg s}^{-1}$, comparable to the brightest known
obscured quasars (e.g., Ross et al., 2015).

We also calculate the bolometric luminosity by using the IR-bolometric
correction $L_{\textrm{bol}}=8\times L_{3.45\micron}$ (Hamann et al., 2017;
Perrotta et al., 2019). We denote $L_{3.45\micron}$ as the monochromatic
luminosity at $\lambda=3.45\micron$, which is extrapolated from the computed
$L_{5\micron}$ value using the same piece-wise power-law interpolation
described above. $L_{\textrm{bol}}$ reaches and exceeds $10^{47}\textrm{erg
s}^{-1}$. We list the calculated $L_{5\micron}$ and $L_{\textrm{bol}}$ values
in Table 2. We calculate the source luminosities (emission line and
bolometric) assuming isotropic emission. This assumption means that if
obscuration makes the observed infrared emission anisotropic, as is the case
with low redshift Type-2 quasars (Liu et al., 2013), then the actual
$L_{\textrm{bol}}$ may even be higher as noted by Zakamska & Greene (2014).

In Figure 5 we show the ${\rm H\alpha}$ and [O iii] line luminosities and
kinematics, along with with the computed $L_{5\micron}$, which we use as a
proxy for $L_{\textrm{bol}}$. We do not see any obvious correlations between
the ${\rm H\alpha}$ and [O iii] line kinematics and $L_{5\micron}$.

###  3.4 Spectral energy distribution (SED) fitting

|
---|---
|
Figure 6: We show the co-added GNIRS+WISE SEDs overplotted in solid black line
and the best-fit SEDs in solid red line. (Top left) ‘A’ strong continuum and
emission lines; (top right) ‘B’ moderate continuum and emission lines; (bottom
left) ‘BB’ blue-sloped continuum with moderate emission lines; (bottom right)
‘C’+‘D’ weak continuum with single-line ${\rm H\alpha}$ detection. We indicate
the [O iii] and ${\rm H\alpha}$ emission lines (light blue dashed lines), the
other low ionization lines (light grey lines), and $L_{5\micron}$ (light red).
We also plot the SDSS upper limits (light diamonds). Figure 7: The model SEDs
from SWIRE (Polletta et al., 2007). The plots are normalized by $\lambda
L_{\lambda}\sim 10^{45}\textrm{ erg s}^{-1}$. The galaxy models are scaled at
$L_{B}\sim 10^{44.5}\textrm{ erg s}^{-1}$, and the quasar models are scaled at
$L_{5\mu m}\sim 10^{46.3}\textrm{ erg s}^{-1}$.

Taking advantage of the optical and IR multi-wavelength data, we construct
SEDs and perform SED fits to understand the nature of our objects. In Figure 6
we show the SEDs separated by Gemini/GNIRS spectral quality. Since many of the
individual spectra suffer from low SNR, we co-add the GNIRS spectra according
to the A/B/BB/C/D categories defined in Section 2.2. We show the co-added
spectra in Figure 6. Interestingly, we find that the sources that did not show
an obvious continuum individually now show a slight red slope after co-adding.
We use the co-added spectra for the SED fits to simplify the analysis.

Based on the bright WISE IR emission that reaches
$10^{45.5}\textrm{-}10^{47}\textrm{erg s}^{-1}$, we can assume that the SED is
most likely dominated by the quasar torus emission. It is unlikely for the MIR
emission to originate from the host galaxy. For the brightest sources, if the
observed light is dominated by the stellar light from the host galaxy, it
would require B band luminosities of least $\lambda L_{\lambda}\sim
10^{45}\textrm{erg s}^{-1}$ to produce $W4\sim 10^{46}\textrm{erg s}^{-1}$,
which is an order-of-magnitude larger than typical galaxies. For example, we
know that ERQ host galaxies can be up to a few $10^{11}L_{\astrosun}\approx
10^{44.5}\textrm{erg s}^{-1}$. Our sources are not more luminous than ERQs, so
they are unlikely to live in even more massive galaxies, as discussed in
Section 4.1.

In Figure 7 we show the different models considered. The MIR emission probed
with WISE is consistent with a quasar+torus dominated spectrum with a heavy
reddening. We account for these observed properties by constructing an SED
model consisting of: quasar (Type 1 or 2) + torus + (host galaxy, if needed)
with varying dust absorption. We use the SWIRE galaxy and quasar spectral
templates (Polletta et al., 2007). We do not include the SDSS upper limits
since they are too shallow to be constraining. Here, we determine the best SED
(quasar/host/torus emission strength) and the estimated $A_{V}$ continuum
reddening.

We assume a “cold screen of dust absorption” model with an extinction. For
simplicity, we use the Milky Way extinction model Cardelli et al. (1989);
Calzetti et al. (2000); Gordon et al. (2003). We fit the coadded SEDs
following $F_{\textrm{obs}}=F_{\textrm{int}}e^{-\tau_{\lambda}}$ to determine
the best $A_{V}$ extinction. We solve the following:

$\log F_{\textrm{obs}}=\log
k_{1}F(\textrm{QSO+torus+gal})-k_{2}\tau_{\lambda}$ (2)

where $k_{1}$ is the normalization at $5$$\mu$m for the template
quasar$+$torus SED and $k_{2}$ is the normalization for the extinction curve
$\tau_{\lambda}$. Initially, we attempted a simple single-phase absorption
model; however, it was quickly apparent that a multi-component absorption
model is necessary (e.g., Ricci et al., 2017), to describe the red-sloped
“bump” seen with our GNIRS data. Here, we apply separate cold-dust absorption
on each component (quasar, galaxy, and torus) and combine to produce a
“master” SED. Both the quasar and torus components required heavy absorption,
reaching $A_{V}\sim 20$, whereas the galaxy component requires moderate
absorption. We caution against over-interpreting the SEDs without detailed
radiative transfer modeling.

In Figure 6, we show the best SED fits. As expected, we find that the WISE-
detected MIR emission is best fit with a torus model. The GNIRS-detected rest-
frame optical spectra are best fit with a quasar spectrum reddened by at least
$A_{V}\sim 7-20$. The exception to this model is the blue excess ‘BB’ targets,
in which an additional non-reddened quasar spectrum is needed. We explore the
blue excess in Sections 4.1 and 4.3. The brightest ‘A’ sources require a
bright galaxy component to explain the optical continuum. The exact properties
of the host galaxy is unclear. What is clear is that the quasar spectrum
dominates. We explore the interpretation of the reddening and the blue excess
in the Discussion.

---
Figure 8: (Left) Luminosities of ${\rm H\alpha}$ and ${\rm H\beta}$, compared
with the theoretical expectation ${\rm H\alpha}/{\rm H\beta}\approx 2.9$ shown
with the blue dashed line. (Right) ${\rm H\alpha}/{\rm H\beta}$ Balmer
decrement shown against $L_{5\micron}$. We also show the $A_{V}$ calculated
with ${\rm H\alpha}/{\rm H\beta}$. We see no obvious trends with
$L_{5\micron}$. Since very few sources had sufficient ${\rm H\beta}$ SNR, we
place upper limits on $L_{H\beta}$, which translates to a lower bound on ${\rm
H\alpha}/{\rm H\beta}$.

##  4 Discussion

###  4.1 Efficiency of photometric selection

The redshift distribution in Figure 3 shows that our selection preferentially
selects $1<z<3$ objects. Unsurprisingly, the sources with stronger continuum
and stronger emission lines have more reliable redshift determinations, as was
noted by previous studies (e.g., Lacy et al., 2013).

Past studies of W4- and optically-bright sources failed to uncover a high
density of obscured quasars. In fact, the motivating color-selection of
optically faint, yet red objects are similar to those of previous studies like
ULIRGS, HotDOGs, and ERQs (i.e. $r-W4>14$). Instead, in this study we take a
step back by considering IR-bright sources with $W4\geq 5\textrm{ mJy}$ that
have more accurate positions from Spitzer and that lack SDSS detections for a
more robust selection.

Figure 1 serves as a diagnostic to compare the different color selection
criteria for red and obscured quasars. The key differences in color-selection
are as follows. HotDOGs primarily rely on W1 and W2 dropouts (Eisenhardt et
al., 2012): $W1>17.4\textrm{ Vega mag}$ and either $(W4<7.7\land
W2-W4>8.2\textrm{ Vega mag})$ or $(W3<10.6\land W2-W3>5.3\textrm{ Vega mag})$,
whereas we only require bright W4 detections. ERQs are defined by $i-W4>4.6$
colors and [C iv] equivalent width $>100$Å (Hamann et al., 2017; Perrotta et
al., 2019), whereas we strictly require weak $r$-band SDSS detections or non-
detections in the SDSS optical photometry. Temple et al. (2019) selection of
reddened quasars applied a $W1-W2>0.85\textrm{ Vega mag}$ along with
J/H/K-band photometric cuts, which covers a redder subset of the obscured
quasars discussed here. This means that our selections identifies a reddened
$1<z<3$ population that is both HotDOG-like (but bluer) and ERQ-like (but
redder), as demonstrated in Figure 1.

Despite the similarities in the color-selection, our sources may constitute a
distinct, yet similar population to HotDOGs and ERQs. For example, only 3
objects record any $i$-band SDSS fluxes, which means the ERQ $i-W4$ selection
would fail to identify most of our sources. All of our sources have brighter
WISE/W4 fluxes than ERQs. Since our sources probe a bluer selection than
HotDOGs, we also see a greater range in the observed rest-frame optical
luminosity, with 2 sources showing a distinct blue excess in their spectra.
The blue continuum may be indicative of scattered light. The current
optically-faint requirement selects against objects with a strong UV
continuum, so naturally, we find very few blue-sloped targets. However, the
detection of the ‘BB’ objects may display an analogous component to those of
ERQs and BHDs. We explore more in Section 4.3.

Failure modes of the color-selected targets would result in either low
redshift contaminants or featureless quasar spectra. However, low redshift
contaminants should be minimized by limiting to optically-faint SDSS dropouts.
Astrometry checks with WISE+Spitzer ensures that the Gemini/GNIRS pointing is
robust. Barring heavy telluric contamination that prevent us from seeing the
faint emission, the detection of featureless spectrum may simply reflect heavy
obscuration of the inherent quasar spectrum, whether they are at circumnuclear
or galactic scales. We explore more in Section 4.2.

###  4.2 Levels of obscuration

As evident from the optical vs. IR luminosity in Figure 6, the quasars in our
sample have substantial reddening. Here we characterize the origin of this
reddening. In Figure 8, we compare the ${\rm H\alpha}/{\rm H\beta}$ ratio to
characterize the Balmer decrement. Photoionization by the quasar is likely the
dominant mechanism that gives rise to Balmer emission in our targets. The
balance of photoionization and recombination sets the ratios of Balmer lines
at particular values – from Case B recombination of hydrogen, the lower
theoretical bound on ${\rm H\alpha}/{\rm H\beta}$ is $\approx 2.98$ in dust-
free gas at $T_{e}=10^{4}\textrm{ K}$ (Dopita & Sutherland, 2003; Osterbrock &
Ferland, 2006). Indeed, in unobscured Type-1 quasars, this value peaks at 3.3
(Kim et al., 2006). Strong enhancement in this ratio is likely due to dust
extinction. We estimate $A_{V}$ from the ${\rm H\alpha}/{\rm H\beta}$ ratio
following Riffel et al. (2021):

$A_{V}=7.22\log\bigg{(}\frac{F_{H\alpha}/F_{H\beta}}{2.86}\bigg{)}$ (3)

assuming a $R_{V}=3.1$ extinction (Cardelli et al., 1989). Balmer decrement is
typically a reliable technique to estimate dust extinction in galaxies (e.g.,
Kennicutt, 1992). However, this may not be the case if there is no reliable
measure of extinction due to the low SNR of ${\rm H\beta}$ (Assef et al.,
2015; Wu et al., 2018). We see an excess of ${\rm H\alpha}$ of up to $\sim
40$, which translates to $A_{V}\sim 20\textrm{ mag}$. The observed Balmer
decrement is consistent with the continuum reddening in Section 3.4. However,
this is a little misleading since ${\rm H\beta}$ is not detected for all
sources, as indicated in Table 2. Sources with significant ${\rm H\beta}$
emission have a median Balmer decrement of ${\rm H\alpha}/{\rm H\beta}\sim
3.5$, which correspond to $\langle A_{V}\rangle\sim 4.8\pm 2.8\textrm{ mag}$.
For sources with no ${\rm H\beta}$ detection, we indicate ${\rm H\alpha}/{\rm
H\beta}$ as lower limits. We do not see any correlation with the IR
luminosity.

Although we find extreme reddening in some of our sources with large ${\rm
H\alpha}/{\rm H\beta}$ values, we find that the average ${\rm H\alpha}/{\rm
H\beta}$ reddening is consistent with other Type-2 quasars. A handful show
much larger ${\rm H\alpha}/{\rm H\beta}$ than those of other known reddened
quasars (e.g., Glikman et al., 2018; Jun et al., 2020). HotDOGs also show
large $H\alpha/H\beta\gtrsim 4$ values.

We combine the cold dust extinction SED fits from Section 3.4 and the Balmer
decrement estimates. From the SED fits, these sources are likely quasar-
dominated, with a bright MIR component from the reprocessed emission by the
torus and a significantly reddened optical component. Considering the shape of
the MIR SED, which is due to warm dust, and the large continuum reddening of
the intrinsic quasar spectrum, circumnuclear obscuration is the most probable
scenario. Although we cannot completely rule out the possibility of galactic
scale obscuration, as Temple et al. (2019) found using forbidden [O iii] lines
produced on large scales, our sources would nonetheless still require a
circumnuclear torus to produce the observed MIR emission. While not impossible
to have both circumnuclear and galactic scale obscuration, it is contrived.
Our data do not have strong evidence in favor of galactic obscuration.

###  4.3 Properties of sources

As discussed earlier, the GNIRS observations revealed 3 types of sources with
varying levels of continuum strengths: strong, moderate, and weak with weak
lines. In Figure 6 we show the representative SED plots. Despite differences
in their rest-frame optical fluxes by up to 1.5 dexs, the IR fluxes show very
similar values. The shapes of their IR SEDs are also similar. The IR
luminosities reach up to $10^{46}-10^{47}\textrm{ erg s}^{-1}$. The best-fit
SEDs suggest that these are likely due to the quasar torus emission,
suggesting the dusty nature of these quasars, complementing the non-detections
in the rest-frame UV and the broad ${\rm H\alpha}$ and ${\rm H\beta}$ emission
lines observed. These observations suggest that photometric color selection is
an excellent metric for identifying obscured quasars at $z\sim 2$.

The typical line widths of the [O iii] emission in our sources are $\sim
1800\textrm{ km s}^{-1}$ at FWHM. [O iii] is a forbidden emission line that
must originate in low-density regions, typically well outside of the sphere of
influence of the black hole. These velocity widths are too high to be
contained by a realistic galaxy potential and are therefore indicative of
likely galactic wind activity. However, colors alone are insufficient for
identifying obscured quasars with extreme kinematics. ERQs show [O iii]
velocity widths in the range $1500-7400\textrm{ km s}^{-1}$ at FWHM, which are
significantly higher than in the population presented here (Figure 5, bottom
right). ERQs were identified both by colors and by line diagnostics like [C
iv] equivalent widths (Hamann et al., 2017; Perrotta et al., 2019; Zakamska et
al., 2019). Using additional line diagnostics may be necessary to identify
sources with potential quasar-driven galactic winds. We explore more in
Section 4.4.

Although black hole mass scaling relations from reverberation mapping (e.g.
using continuum emission and ${\rm H\beta}$ widths; Greene & Ho 2005;
Vestergaard & Peterson 2006) are commonly used to estimate $M_{\textrm{BH}}$,
the nature of the obscuration complicates the comparisons to obscured quasars,
as noted by Zakamska et al. (2019). Supermassive black holes that are heavily
obscured, have powerful winds, or are in the super-Eddington accretion limit
do not necessarily obey the scaling relations because the inner broad-line
region and the accretion disk are either not visible or are not in the same
dynamical equilibrium as the sources used in the scaling relations. Since we
have extremely low ${\rm H\beta}$ SNR and very little optical continuum, we
cannot obtain reliable black hole mass measurements. With these caveats in
mind, we assume that the quasars are hosted by $M_{\textrm{BH}}\sim
10^{9}M_{\astrosun}$ black holes. This mass corresponds to an Eddington-
limited ratio of $\lambda_{\textrm{Edd}}=L_{\textrm{bol}}/L_{\textrm{Edd}}\sim
0.7$, where $L_{\textrm{Edd}}\approx 1.3\times 10^{47}\textrm{ erg s}^{-1}$.
However, if $L_{\textrm{bol}}$ is underestimated due to the obscuration, it
may place these quasars closer to the Eddington limited regime. And if the
black holes are instead significantly more massive ($\sim
10^{10}M_{\astrosun}$), they may be in the accretion regime close to that of
normal quasars.

Another question is the origin of the strong MIR emission. For high redshift
sources, we can be confident that these are quasar dominated. However, dusty,
star-forming galaxies may also produce large MIR luminosities. In Figure 7, we
compare the different SED templates, including Type-1 QSO, Type-2 QSO, Torus,
Starbursts, and ULIRGs. Fortunately, dusty star-forming galaxies like ULIRGs
typically have cooler SEDs compared to the quasar torus emission. Also,
considering the extreme optical faintness of our objects, it is unlikely to
simultaneously produce high IR emission with low optical emission. As explored
in Section 3.4, a more likely scenario is dominant torus emission with a faint
host galaxy with $L_{B}\sim 10^{44.5}\textrm{ erg s}^{-1}$ with extreme
reddening of the UV/optical quasar spectrum.

There are curious ‘BB’ sources like J2334+0031 and J0041-0029 that display a
noticeable blue continuum. These 2 sources also recorded SDSS fluxes at the
survey’s limiting magnitude, which is consistent with a UV quasar spectrum
(Figure 6). A blue excess in obscured quasars is observed when light from the
quasar continuum is scattered off the surrounding gas into our line of sight.
This phenomenon has been directly confirmed in ERQs with spectropolarimetric
observations (Alexandroff et al., 2018). Zakamska et al. (2006); Alexandroff
et al. (2018); Assef et al. (2022) have shown that polarization and blue
excess reveal scattering from dust within the ionization cone. Assef et al.
(2016, 2022) also identified a subset of the HotDOG population: so-called
“Blue HotDOGs” (BHDs) with a blue/UV excess observed with SDSS photometry.
Although we cannot ascertain the origin of the blue slope with the current
Gemini/GNIRS data, scattered light is a promising possibility and would
strengthen the case that these sources are indeed obscured quasars. The
discovery of these blue-slope sources and other obscured quasar candidates
reveals that the population of obscured quasars has diverse properties. Also,
these detections show that caution is needed when applying strict UV/optical
dropout requirements to identify obscured quasars, further supporting our
hypothesis that obscured quasars are simply missed. Using physically
motivated, strategic color-selections, such as ours, can uncover the diverse
populations of obscured quasars.

###  4.4 Comparing/linking with other obscured quasars

As discussed earlier, the colors of our targets overlap with other types of
obscured quasars. However, we see some key spectral differences, which may
probe different physics. Perhaps we are probing the transitional link between
the two populations. Assef et al. (2022) provided a physical model suggesting
that the transitional link between HotDOGs, BHDs, and ERQs may be the
interaction between the quasar-driven outflow and the obscuring medium. In
this model, (1) the central quasar of HotDOGs is completely enshrouded by the
obscuring material; (2) then massive, fast-moving outflows pierce through and
“blow out” the obscuration as observed with ERQs (and possibly BHDs); and (3)
finally, the column density of the obscuration is lowered, which is observed
as heavily reddened Type-1 quasars.

In the standard unification model of active nuclei, we expect the kinematics
of [O iii] to be similar in obscured and unobscured sources, since this line
is produced on extended scales and reflects the kinematics of the gas in the
host galaxy (or at least not in the sphere of influence of the black hole due
to its low critical density). Indeed, Temple et al. (2019) find that there are
no statistical differences in the [O iii] kinematics between obscured and
unobscured quasars. One exceptional population of obscured quasars is ERQs,
which show extremely high widths of [O iii], unmatched by any other quasar
population (Zakamska et al., 2016; Perrotta et al., 2019). Our sample shows a
broad range of [O iii] velocities, similar to those of typical powerful
quasars and somewhat below those in ERQs, as seen in Figures 4 and 5. We do
find 3 sources (J0024-0012, J0215+0042, and J2334+0031) reaching ERQ-like
velocities and IR luminosities, as shown in Figure 5.

Here, we suggest that the transitional link may be observed in the [O iii]
luminosity and velocity vs. $L_{5\micron}$ space, as shown in Figures 4 and 5.
An interesting observation is that the ERQ selection requires a high [C iv]
equivalent width (Hamann et al., 2017), which may preferentially select
reddened quasars with extreme [O iii] emission and winds. Our [O iii] emission
lines (luminosity and kinematics) are nearly $\times 10$ weaker than those
seen in ERQs, yet likely significantly larger than HotDOGs. Since our targets
show weak [O iii], it is difficult to determine if winds are present, although
they are unlikely to have extreme winds like ERQs. According to the Assef et
al. (2022) model, our selections may identify sources in the transition
between stage (1), appearing bluer than HotDOGs, and stage (2), appearing
redder than ERQs with weaker [O iii]. By adjusting the detection criteria of
optical and IR emission and nuclear line kinematics criteria, we may be able
to recover more obscured populations.

If our targets are indeed obscured quasars, then a literature review for
spectral data on our broader samples of 373 objects suggests that the number
density of obscured quasars is between $0.17-0.93\textrm{ deg}^{2}$. This
density is potentially comparable to the confirmed unobscured density to the
same depth in W4 but does not exceed the unobscured number density. A
conservative interpretation is that unobscured quasars may only outnumber
obscured quasars by a small factor. However, there is a caveat that this
selection is biased toward bright $W4<8\textrm{ Vega mag}$ sources, which may
fail to identify fainter quasars. Although more comprehensive studies are
necessary to resolve the number density estimates, our spectroscopic
confirmation of these targets as obscured quasars support our hypothesis that
obscured quasars have been missed by previous studies.

##  5 Conclusions

The census of obscured quasars at high redshift remains incomplete. We present
Gemini/GNIRS spectra of 24 luminous, red quasar candidates at $z\sim 2$
selected using WISE/W4 photometry from the SDSS Stripe 82 field. These sources
have little to no detectable optical flux, yet have high WISE/W4 fluxes
($\textrm{W4}\geq 5\textrm{ mJy}$). This selection aims to probe the peak of
the quasar-heated dust emission at rest-frame $\sim 10\micron$ while requiring
high optical obscuration with SDSS dropouts. The use of the Spitzer data
ensures precise astrometry for these optically faint sources. This study
probes a slightly different, yet complementary selection regime compared to
that of other known red, obscured quasars like HotDOGs and ERQs.

From spectroscopic analysis, we find a diverse range of spectral types. Two
sources show strong continuum and strong emission lines, six show moderate
continuum and emission lines, and others show no continuum with weak emission
lines if detected. All sources with continuum detection have a red slope. We
detect moderately strong ${\rm H\alpha}$ and [O iii] emission lines, which we
use to determine the redshift of these sources. We find that our WISE-selected
IR bright color-selection method selects $1<z<3$ objects with the average
redshift of $\langle z_{\textrm{best}}\rangle=2.008\pm 0.45$.

Due to the low SNR of the emission lines detected, we fit single Gaussians to
the continuum-subtracted spectra and we cannot study the details of line
kinematics. From these fits, we find broad ${\rm H\alpha}$ emission lines with
FWHM ranging several $10^{3}\textrm{ km s}^{-1}$ and luminosities reaching
$10^{42}-10^{43.4}\textrm{erg s}^{-1}$. Very little ${\rm H\beta}$ is
detected, whereas the [O iii] doublet show clear detections. [O iii]$\lambda
5009$Å show moderate line strengths with luminosities
$10^{42}-10^{43}\textrm{erg s}^{-1}$ and FWHM of several $10^{3}\textrm{ km
s}^{-1}$.

The red slope and the lack of ${\rm H\beta}$ detections with large ${\rm
H\alpha}/{\rm H\beta}$ ratios are indicative that these sources are
experiencing significant extinction with $\langle
A_{V}(H\alpha/H\beta)\rangle\sim 4.8\pm 2.8\textrm{ mag}$. We estimate the
$A_{V}$ extinction from the SED to probe the origin of the observed
obscuration. We find typical values of $A_{V,\textrm{SED}}\sim 7-20$. The SED
modeling indicate that these are indeed obscured quasars with a heavily-
reddened quasar+torus dominated spectrum.

As expected from the large WISE/W4 fluxes, we find that these sources have
large $L_{\textrm{bol}}$ reaching $\sim 10^{46}-10^{47}\textrm{erg s}^{-1}$,
which is comparable to some of the brightest known quasars. Compared to known
unobscured and obscured quasars at the similar redshift and
$L_{\textrm{bol}}$, we find that the [O iii] line widths of our sources are
similar to Type-1 quasars. Our sources do not display any signs of powerful
quasar-driven winds such as those seen in ERQs (Zakamska et al., 2016;
Perrotta et al., 2019). This suggests that the “red-ness” of the quasar does
not necessarily ensure the presence of strong winds.

The population of quasars we have discovered and characterized in this paper
appears to be an intermediate one between HotDOGs and ERQs, with HotDOGs
likely the most obscured. The differences in the properties are apparent from
the shapes of the optical-to-IR SED and from the relative strengths of the
emission lines. The red quasars of Temple et al. (2019) are significantly less
obscured. It would be interesting to determine whether the column densities of
X-ray absorbing gas in comparison to those of ERQs (Goulding et al., 2018) and
HotDOGs (Vito et al., 2018) support this picture.

## Acknowledgements

This work was enabled by observations made from the Gemini North telescope,
located within the Maunakea Science Reserve and adjacent to the summit of
Maunakea. We are grateful for the privilege of observing the Universe from a
place that is unique in both its astronomical quality and its cultural
significance.

Based on observations obtained at the international Gemini Observatory, a
program of NSF’s NOIRLab, which is managed by the Association of Universities
for Research in Astronomy (AURA) under a cooperative agreement with the
National Science Foundation. on behalf of the Gemini Observatory partnership:
the National Science Foundation (United States), National Research Council
(Canada), Agencia Nacional de Investigación y Desarrollo (Chile), Ministerio
de Ciencia, Tecnología e Innovación (Argentina), Ministério da Ciência,
Tecnologia, Inovações e Comunicações (Brazil), and Korea Astronomy and Space
Science Institute (Republic of Korea).

This research made use of PypeIt,111https://pypeit.readthedocs.io/en/latest/ a
Python package for semi-automated reduction of astronomical slit-based
spectroscopy (Prochaska et al., 2020c, a). The authors would especially like
to thank the PypeIt team members for their assistance with running the
pipeline. This work also made use of Astropy222http://www.astropy.org (Astropy
Collaboration et al., 2022).

Y.I. acknowledges support by Space@Hopkins Graduate Fellowship. N.L.Z. is
supported at the IAS by the J. Robert Oppenheimer Visiting Professorship and
the Bershadsky Fund. N.L.Z and G.T.R. acknowledge support of NASA ADAP
NNX12AI49G and NNX17AF04G. The authors also thank Jan-Torge Schindler and
Riccardo Nanni for their contributions.

## Data Availability

Data are publicly available from the Gemini Observatory Archive (GOA) located
at https://archive.gemini.edu under GN-2017B-Q-51 (PI: Richards).

## References

  * Alexandroff et al. (2018) Alexandroff R. M., et al., 2018, MNRAS, 479, 4936
  * Antonucci (1993) Antonucci R., 1993, ARA&A, 31, 473
  * Assef et al. (2015) Assef R. J., et al., 2015, ApJ, 804, 27
  * Assef et al. (2016) Assef R. J., et al., 2016, ApJ, 819, 111
  * Assef et al. (2022) Assef R. J., et al., 2022, ApJ, 934, 101
  * Astropy Collaboration et al. (2022) Astropy Collaboration et al., 2022, ApJ, 935, 167
  * Banerji et al. (2015) Banerji M., Alaghband-Zadeh S., Hewett P. C., McMahon R. G., 2015, MNRAS, 447, 3368
  * Bischetti et al. (2017) Bischetti M., et al., 2017, A&A, 598, A122
  * Brand et al. (2007) Brand K., et al., 2007, ApJ, 663, 204
  * Calzetti et al. (2000) Calzetti D., Armus L., Bohlin R. C., Kinney A. L., Koornneef J., Storchi-Bergmann T., 2000, ApJ, 533, 682
  * Canalizo & Stockton (2001) Canalizo G., Stockton A., 2001, ApJ, 555, 719
  * Cardelli et al. (1989) Cardelli J. A., Clayton G. C., Mathis J. S., 1989, ApJ, 345, 245
  * Dey et al. (2008) Dey A., et al., 2008, ApJ, 677, 943
  * Donley et al. (2012) Donley J. L., et al., 2012, ApJ, 748, 142
  * Dopita & Sutherland (2003) Dopita M. A., Sutherland R. S., 2003, Astrophysics of the diffuse universe, doi:10.1007/978-3-662-05866-4.
  * Eisenhardt et al. (2012) Eisenhardt P. R. M., et al., 2012, ApJ, 755, 173
  * Elias et al. (2006a) Elias J. H., Rodgers B., Joyce R. R., Lazo M., Doppmann G., Winge C., Rodríguez-Ardila A., 2006a, in McLean I. S., Iye M., eds, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Vol. 6269, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series. p. 626914, doi:10.1117/12.671765
  * Elias et al. (2006b) Elias J. H., Joyce R. R., Liang M., Muller G. P., Hileman E. A., George J. R., 2006b, in McLean I. S., Iye M., eds, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Vol. 6269, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series. p. 62694C, doi:10.1117/12.671817
  * Elitzur & Shlosman (2006) Elitzur M., Shlosman I., 2006, ApJ, 648, L101
  * Glikman et al. (2012) Glikman E., et al., 2012, ApJ, 757, 51
  * Glikman et al. (2013) Glikman E., et al., 2013, ApJ, 778, 127
  * Glikman et al. (2018) Glikman E., et al., 2018, ApJ, 861, 37
  * Gordon et al. (2003) Gordon K. D., Clayton G. C., Misselt K. A., Landolt A. U., Wolff M. J., 2003, ApJ, 594, 279
  * Goulding et al. (2018) Goulding A. D., et al., 2018, ApJ, 856, 4
  * Greene & Ho (2005) Greene J. E., Ho L. C., 2005, ApJ, 630, 122
  * Hamann et al. (2017) Hamann F., et al., 2017, MNRAS, 464, 3431
  * Harrison et al. (2012) Harrison C. M., et al., 2012, MNRAS, 426, 1073
  * Hasinger (2008) Hasinger G., 2008, A&A, 490, 905
  * Hennawi & Prochaska (2013) Hennawi J. F., Prochaska J. X., 2013, ApJ, 766, 58
  * Hickox & Alexander (2018) Hickox R. C., Alexander D. M., 2018, ARA&A, 56, 625
  * Hopkins et al. (2006) Hopkins P. F., Hernquist L., Cox T. J., Di Matteo T., Robertson B., Springel V., 2006, ApJS, 163, 1
  * Ishikawa et al. (2021) Ishikawa Y., Goulding A. D., Zakamska N. L., Hamann F., Vayner A., Veilleux S., Wylezalek D., 2021, MNRAS, 502, 3769
  * Jun et al. (2020) Jun H. D., et al., 2020, ApJ, 888, 110
  * Kennicutt (1992) Kennicutt Robert C. J., 1992, ApJ, 388, 310
  * Kim et al. (2006) Kim M., Ho L. C., Im M., 2006, ApJ, 642, 702
  * Konigl & Kartje (1994) Konigl A., Kartje J. F., 1994, ApJ, 434, 446
  * Krolik & Begelman (1988) Krolik J. H., Begelman M. C., 1988, ApJ, 329, 702
  * LaMassa et al. (2015) LaMassa S. M., et al., 2015, ApJ, 800, 144
  * LaMassa et al. (2019) LaMassa S. M., Yaqoob T., Boorman P. G., Tzanavaris P., Levenson N. A., Gandhi P., Ptak A. F., Heckman T. M., 2019, ApJ, 887, 173
  * Lacy et al. (2004) Lacy M., et al., 2004, ApJS, 154, 166
  * Lacy et al. (2013) Lacy M., et al., 2013, ApJS, 208, 24
  * Lanzuisi et al. (2009) Lanzuisi G., Piconcelli E., Fiore F., Feruglio C., Vignali C., Salvato M., Gruppioni C., 2009, A&A, 498, 67
  * Lawrence (1991) Lawrence A., 1991, MNRAS, 252, 586
  * Lawrence & Elvis (2010) Lawrence A., Elvis M., 2010, ApJ, 714, 561
  * Liu et al. (2013) Liu G., Zakamska N. L., Greene J. E., Nesvadba N. P. H., Liu X., 2013, MNRAS, 436, 2576
  * Lonsdale et al. (2003) Lonsdale C. J., et al., 2003, PASP, 115, 897
  * Lusso et al. (2013) Lusso E., et al., 2013, ApJ, 777, 86
  * Mauduit et al. (2012) Mauduit J. C., et al., 2012, PASP, 124, 714
  * Osterbrock & Ferland (2006) Osterbrock D. E., Ferland G. J., 2006, Astrophysics of gaseous nebulae and active galactic nuclei
  * Papovich et al. (2016) Papovich C., et al., 2016, ApJS, 224, 28
  * Perrotta et al. (2019) Perrotta S., Hamann F., Zakamska N. L., Alexandroff R. M., Rupke D., Wylezalek D., 2019, MNRAS, 488, 4126
  * Polletta et al. (2007) Polletta M., et al., 2007, ApJ, 663, 81
  * Polletta et al. (2008) Polletta M., Weedman D., Hönig S., Lonsdale C. J., Smith H. E., Houck J., 2008, ApJ, 675, 960
  * Prochaska et al. (2013) Prochaska J. X., et al., 2013, ApJ, 776, 136
  * Prochaska et al. (2020a) Prochaska J. X., et al., 2020a, pypeit/PypeIt: Release 1.0.0, doi:10.5281/zenodo.3743493
  * Prochaska et al. (2020b) Prochaska J. X., Hennawi J. F., Westfall K. B., Cooke R. J., Wang F., Hsyu T., Davies F. B., Farina E. P., 2020b, arXiv e-prints, p. arXiv:2005.06505
  * Prochaska et al. (2020c) Prochaska J. X., et al., 2020c, Journal of Open Source Software, 5, 2308
  * Reyes et al. (2008) Reyes R., et al., 2008, AJ, 136, 2373
  * Ricci et al. (2017) Ricci C., et al., 2017, ApJ, 835, 105
  * Richards et al. (2006) Richards G. T., et al., 2006, AJ, 131, 2766
  * Riffel et al. (2021) Riffel R., et al., 2021, MNRAS, 501, 4064
  * Ross et al. (2013) Ross N. P., et al., 2013, ApJ, 773, 14
  * Ross et al. (2015) Ross N. P., et al., 2015, MNRAS, 453, 3932
  * Sanders et al. (1988) Sanders D. B., Soifer B. T., Elias J. H., Madore B. F., Matthews K., Neugebauer G., Scoville N. Z., 1988, ApJ, 325, 74
  * Sanders et al. (2007) Sanders D. B., et al., 2007, ApJS, 172, 86
  * Shen (2016) Shen Y., 2016, ApJ, 817, 55
  * Soltan (1982) Soltan A., 1982, MNRAS, 200, 115
  * Stern et al. (2005) Stern D., et al., 2005, ApJ, 631, 163
  * Temple et al. (2019) Temple M. J., Banerji M., Hewett P. C., Coatman L., Maddox N., Peroux C., 2019, MNRAS, 487, 2594
  * Timlin et al. (2016) Timlin J. D., et al., 2016, ApJS, 225, 1
  * Treister & Urry (2005) Treister E., Urry C. M., 2005, ApJ, 630, 115
  * Tsai et al. (2015) Tsai C.-W., et al., 2015, ApJ, 805, 90
  * Ueda et al. (2003) Ueda Y., Akiyama M., Ohta K., Miyaji T., 2003, ApJ, 598, 886
  * Vestergaard & Peterson (2006) Vestergaard M., Peterson B. M., 2006, ApJ, 641, 689
  * Vito et al. (2018) Vito F., et al., 2018, MNRAS, 474, 4528
  * Werner et al. (2004) Werner M. W., et al., 2004, ApJS, 154, 1
  * Wright et al. (2010) Wright E. L., et al., 2010, AJ, 140, 1868
  * Wu et al. (2012) Wu J., et al., 2012, ApJ, 756, 96
  * Wu et al. (2018) Wu J., et al., 2018, ApJ, 852, 96
  * Yan et al. (2013) Yan L., et al., 2013, AJ, 145, 55
  * York et al. (2000) York D. G., et al., 2000, AJ, 120, 1579
  * Yu & Tremaine (2002) Yu Q., Tremaine S., 2002, MNRAS, 335, 965
  * Yuan et al. (2016) Yuan S., Strauss M. A., Zakamska N. L., 2016, MNRAS, 462, 1603
  * Zakamska & Greene (2014) Zakamska N. L., Greene J. E., 2014, MNRAS, 442, 784
  * Zakamska et al. (2003) Zakamska N. L., et al., 2003, AJ, 126, 2125
  * Zakamska et al. (2006) Zakamska N. L., et al., 2006, AJ, 132, 1496
  * Zakamska et al. (2016) Zakamska N. L., et al., 2016, MNRAS, 459, 3144
  * Zakamska et al. (2019) Zakamska N. L., et al., 2019, MNRAS, 489, 497
  * Zappacosta et al. (2018) Zappacosta L., et al., 2018, A&A, 618, A28


$\displaystyle\hat{g}_{\hat{\Delta},s,r}(z,\bar{z},\omega)=\left(2\chi\right)^{-\hat{\Delta}}{}_{2}F_{1}\left(\frac{\hat{\Delta}+1}{2},\frac{\hat{\Delta}}{2};\hat{\Delta};\chi^{-2}\right)U_{s}\left(\cos(\phi)\right)U_{r}\left(-\psi\right)\,.$
(B.10)

Expressing the cross-ratios in terms of $z,\bar{z},\omega$ and using the
identity

$\displaystyle
U_{r}\left(\frac{\omega+\omega^{-1}}{2}\right)=\frac{\omega^{r+1}-\omega^{-(r+1)}}{\omega-\omega^{-1}}\,.$
(B.11)

we find the conformal blocks (2.42).

###  B.3 Table of superconformal blocks

Finally we tabulate the various superconformal blocks appearing in the OPE.
Here we list the conformal blocks content of each superconformal blocks.

The short multiplets $B[r]$ (for $r=1,2$) have superconformal blocks

$\displaystyle\hat{{\mathcal{G}}}_{B[r]}=\hat{g}_{2r,0,r}+\hat{g}_{2r+1,1,r-1}+\hat{g}_{2r+2,0,r-2}\,.$
(B.12)

Note that for $B[0]$ is the defect identity. The special case $B[1]$ is the
displacement operator supermultiplet and has a shortened superconformal block.
It contains only the blocks $g_{2,0,1}$ and $g_{3,1,0}$, and correspondingly
we can check that the last term in the ansatz above vanishes identically.

The semishort multiplets $A[r]_{s}$ (defined for $r=0,1$) respectively have
superblocks

$\displaystyle\hat{\mathcal{G}}_{A[0]_{s}}$
$\displaystyle=\hat{g}_{2+s,s,0}+\hat{g}_{3+s,s+1,1}+\hat{g}_{4+s,s+2,0}\,,$
(B.13)

and

$\displaystyle\hat{{\mathcal{G}}}_{A[1]_{s}}$
$\displaystyle=\hat{g}_{4+s,s,1}+\hat{g}_{5+s,s-1,0}+\hat{g}_{5+s,s+1,0}+\hat{g}_{5+s,s+1,2}+\hat{g}_{6+s,s,1}+\hat{g}_{6+s,s+2,1}+\hat{g}_{7+s,s+1,0}\,.$
(B.14)

Again for the special case $s=0$ some of these conformal blocks vanish
identically, which reflects the shortening of the multiplet.

Finally the long supermultiplet have superconformal blocks

$\displaystyle\hat{\mathcal{G}}_{L[s,0]_{\hat{\Delta}}}$
$\displaystyle=\hat{g}_{\hat{\Delta},s,0}+\hat{g}_{\hat{\Delta}+1,s+1,1}+\hat{g}_{\hat{\Delta}+1,s-1,1}+\hat{g}_{\hat{\Delta}+2,s-2,0}+\hat{g}_{\hat{\Delta}+2,s,0}$
(B.15) $\displaystyle\phantom{=}\
+\hat{g}_{\hat{\Delta}+2,s+2,0}+\hat{g}_{\hat{\Delta}+2,s,2}+\hat{g}_{\hat{\Delta}+3,s+1,1}+\hat{g}_{\hat{\Delta}+3,s-1,1}+\hat{g}_{\hat{\Delta}+4,s,0}\,.$

Notice that the blocks $\hat{g}$ appear, for our normalization, all with unit
coefficient.

##  Appendix C Calculation of $a_{2}$

We can calculate the coefficient $a_{2}$ appearing in (2.3) by relating it to
the anomaly coefficients $c,d$. The transformation rules for operators of the
stress tensor multiplet are given in [57]

$\displaystyle\delta T^{\mu\nu}=$ $\displaystyle\
\frac{1}{2}\varepsilon\gamma^{\rho(\mu}\partial_{\rho}J^{\nu)}\,,$
$\displaystyle\delta J^{\mu}=$ $\displaystyle\
2\varepsilon\gamma_{\nu}T^{\mu\nu}+\frac{2\alpha_{2}}{5\alpha_{3}}\left(6\eta^{\rho\mu}\left(\gamma^{\nu\sigma\lambda}+3\eta^{\sigma\nu}\gamma^{\lambda}\right)-\eta^{\mu\nu}\gamma^{\rho\sigma\lambda}\right)\check{\gamma}_{I}\partial_{\nu}H_{\rho\sigma\lambda}^{I}$
$\displaystyle+\frac{1}{10}\varepsilon\left(\gamma^{\mu\nu\rho}-4\eta^{\mu\rho}\gamma^{\nu}\right)\check{\gamma}^{IJ}\partial_{\nu}j_{\rho
IJ}\,,$ $\displaystyle\delta j^{\mu}_{IJ}=$ $\displaystyle\
{-\frac{1}{2}}\varepsilon\check{\gamma}_{IJ}J^{\mu}+\frac{1}{5\alpha_{3}}\varepsilon\gamma^{\mu\nu}\partial_{\nu}\check{\gamma}_{[I}\chi_{J]}\,,$
$\displaystyle\delta H^{I}_{\mu\nu\rho}=$ $\displaystyle\
\frac{\alpha_{3}}{8\alpha_{2}}\varepsilon\check{\gamma}^{I}\gamma_{[\mu\nu}J_{\rho]}+\frac{1}{120\alpha_{2}}\varepsilon\gamma_{\sigma}\bar{\gamma}_{\mu\nu\rho}\partial^{\sigma}\chi^{I}\,,$
$\displaystyle\delta\chi^{I}=$ $\displaystyle\
\alpha_{2}\varepsilon\gamma^{\mu\nu\rho}\left(\check{\gamma}^{IJ}+4\delta^{IJ}\right)H^{J}_{\mu\nu\rho}+\alpha_{3}\varepsilon\gamma_{\mu}\left(\check{\gamma}^{IJK}+3\delta^{IJ}\check{\gamma}^{K}\right)j^{\mu}_{JK}$
$\displaystyle+\frac{1}{\alpha_{1}}\varepsilon\gamma^{\mu}\check{\gamma}^{J}\partial_{\mu}O^{IJ}\,,$
$\displaystyle\delta O^{IJ}=$ $\displaystyle\
\alpha_{1}\varepsilon\check{\gamma}^{(I}\chi^{J)}\,.$ (C.1)

The constants $\alpha_{1},\alpha_{2},\alpha_{3}$ are arbitrary constants that
can absorbed in the normalisation for the operators ${\mathcal{O}},\chi,H$.

The coefficient of the 1-point function of $T^{\mu\nu}$ in the presence of $V$
is known to be related to the anomaly coefficient $d$ [103, 104, 57]. Acting
with supersymmetry, we find that the 1-point function of the superprimary
${\Phi}$ has coefficient [57]

$\displaystyle a_{2}=\frac{5\alpha_{1}\alpha_{3}d}{4\pi^{3}}\,.$ (C.2)

We can fix the coefficients $\alpha_{1}\alpha_{3}$ by requiring ${\Phi}$ to be
normalised as in (2.2). A simple way to do so is to compare the 2-point
function of ${\Phi}$ with the 2-point function of R-symmetry currents
$j^{\mu}$, which is fixed by conformal symmetry to take the form

$\displaystyle\left\langle j^{\mu ij}(x_{1})j^{\nu
kl}(x_{2})\right\rangle=C_{j}\left(\delta^{ik}\delta^{jl}-\delta^{il}\delta^{jk}\right)\frac{I_{\mu\nu}(x_{12})}{|x_{12}|^{10}}\,,\qquad
I_{\mu\nu}(x)=\delta_{\mu\nu}-2\frac{x_{\mu}x_{\nu}}{x^{2}}\,.$ (C.3)

The constant $C_{j}$ is related to the anomaly coefficient $c$ as [105]

$\displaystyle C_{j}=\frac{5c}{2\pi^{6}}\,.$ (C.4)

Acting twice with supersymmetry, a short calculation shows that

$\displaystyle\left\langle{\Phi}(x_{1},u_{1}){\Phi}(x_{2},u_{2})\right\rangle=\frac{25\alpha_{1}^{2}\alpha_{3}^{2}c}{32\pi^{6}}\frac{(2u_{1}\cdot
u_{2})^{2}}{|x_{12}|^{8}}\,.$ (C.5)

Matching with the normalisation of ${\Phi}$ (2.2) fixes
$(\alpha_{1}\alpha_{3})^{2}$, and we should take the negative branch to match
the supergravity calculation

$\displaystyle\alpha_{1}\alpha_{3}=-\frac{4\sqrt{2}\pi^{3}}{5\sqrt{c}}\,.$
(C.6)

Plugging back into (C.2) gives the result (1.7).

## References

  * [1] J. M. Maldacena, “The large $N$ limit of superconformal field theories and supergravity,” Int. J. Theor. Phys. 38 (1999) 1113–1133, hep-th/9711200.
  * [2] E. Witten, “Baryons and branes in anti-de Sitter space,” JHEP 07 (1998) 006, arXiv:hep-th/9805112.
  * [3] O. Aharony, Y. Oz, and Z. Yin, “M-theory on $AdS_{p}\times S^{(11-p)}$ and superconformal field theories,” Phys. Lett. B 430 (1998) 87–93, arXiv:hep-th/9803051.
  * [4] S. Ferrara, A. Grillo, and R. Gatto, “Tensor representations of conformal algebra and conformally covariant operator product expansion,” Annals Phys. 76 (1973) 161–188.
  * [5] S. Ferrara, A. Grillo, G. Parisi, and R. Gatto, “Covariant expansion of the conformal four-point function,” Nucl. Phys. B 49 (1972) 77–98. [Erratum: Nucl.Phys.B 53, 643–643 (1973)].
  * [6] A. Polyakov, “Nonhamiltonian approach to conformal quantum field theory,” Zh. Eksp. Teor. Fiz. 66 (1974) 23–42.
  * [7] R. Rattazzi, V. S. Rychkov, E. Tonni, and A. Vichi, “Bounding scalar operator dimensions in 4D CFT,” JHEP 12 (2008) 031, arXiv:0807.0004.
  * [8] C. Beem, M. Lemos, P. Liendo, W. Peelaers, L. Rastelli, and B. C. van Rees, “Infinite chiral symmetry in four dimensions,” Commun. Math. Phys. 336 no. 3, (2015) 1359–1433, arXiv:1312.5344.
  * [9] C. Beem, L. Rastelli, and B. C. van Rees, “$\mathcal{W}$ symmetry in six dimensions,” JHEP 05 (2015) 017, arXiv:1404.1079.
  * [10] S. M. Chester and E. Perlmutter, “M-theory reconstruction from (2,0) CFT and the chiral algebra conjecture,” JHEP 08 (2018) 116, arXiv:1805.00892.
  * [11] C. Beem, M. Lemos, L. Rastelli, and B. C. van Rees, “The $(2,0)$ superconformal bootstrap,” Phys. Rev. D 93 no. 2, (2016) 025016, arXiv:1507.05637.
  * [12] L. Rastelli and X. Zhou, “Holographic four-point functions in the (2, 0) theory,” JHEP 06 (2018) 087, arXiv:1712.02788.
  * [13] X. Zhou, “On superconformal four-point Mellin amplitudes in dimension $d>2$,” JHEP 08 (2018) 187, arXiv:1712.02800.
  * [14] P. Heslop and A. E. Lipstein, “M-theory beyond the supergravity approximation,” JHEP 02 (2018) 004, arXiv:1712.08570.
  * [15] L. F. Alday, S. M. Chester, and H. Raj, “6d (2,0) and M-theory at 1-loop,” JHEP 01 (2021) 133, arXiv:2005.07175.
  * [16] G. Arutyunov and E. Sokatchev, “Implications of superconformal symmetry for interacting (2,0) tensor multiplets,” Nucl. Phys. B 635 (2002) 3–32, arXiv:hep-th/0201145.
  * [17] P. J. Heslop, “Aspects of superconformal field theories in six dimensions,” JHEP 07 (2004) 056, arXiv:hep-th/0405245.
  * [18] L. F. Alday and X. Zhou, “All tree-level correlators for M-theory on $AdS_{7}\times S^{4}$,” Phys. Rev. Lett. 125 no. 13, (2020) 131604, arXiv:2006.06653.
  * [19] M. Lemos, B. C. van Rees, and X. Zhao, “Regge trajectories for the (2, 0) theories,” JHEP 01 (2022) 022, arXiv:2105.13361.
  * [20] G. Kántor, V. Niarchos, C. Papageorgakis, and P. Richmond, “$6D$ (2,0) bootstrap with soft-Actor-Critic,” arXiv:2209.02801.
  * [21] J. Penedones, “Writing CFT correlation functions as $AdS$ scattering amplitudes,” JHEP 03 (2011) 025, arXiv:1011.1485.
  * [22] E. Witten, “Some comments on string dynamics,” in STRINGS 95: Future Perspectives in String Theory, pp. 501–523.  7, 1995.  arXiv:hep-th/9507121.
  * [23] A. Strominger, “Open $p$-branes,” Phys. Lett. B 383 (1996) 44–47, arXiv:hep-th/9512059.
  * [24] O. J. Ganor, “Six-dimensional tensionless strings in the large $N$ limit,” Nucl. Phys. B489 (1997) 95–121, hep-th/9605201.
  * [25] P. S. Howe, N. D. Lambert, and P. C. West, “The selfdual string soliton,” Nucl. Phys. B 515 (1998) 203–216, arXiv:hep-th/9709014.
  * [26] D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, “Generalized global symmetries,” JHEP 02 (2015) 172, arXiv:1412.5148.
  * [27] M. Del Zotto, J. J. Heckman, D. S. Park, and T. Rudelius, “On the defect group of a 6d SCFT,” Lett. Math. Phys. 106 no. 6, (2016) 765–786, arXiv:1503.04806.
  * [28] L. Bhardwaj and S. Schäfer-Nameki, “Higher-form symmetries of 6d and 5d theories,” JHEP 02 (2021) 159, arXiv:2008.09600.
  * [29] F. Apruzzi, L. Bhardwaj, D. S. W. Gould, and S. Schafer-Nameki, “2-Group symmetries and their classification in 6d,” SciPost Phys. 12 no. 3, (2022) 098, arXiv:2110.14647.
  * [30] J. M. Maldacena, “Wilson loops in large $N$ field theories,” Phys. Rev. Lett. 80 (1998) 4859–4862, hep-th/9803002.
  * [31] N. Drukker and M. Trepanier, “M2-doughnuts,” JHEP 02 (2022) 071, arXiv:2111.09385.
  * [32] N. Drukker and M. Trépanier, “Ironing out the crease,” JHEP 08 (2022) 193, arXiv:2204.12627.
  * [33] E. D’Hoker, J. Estes, M. Gutperle, and D. Krym, “Exact half-BPS flux solutions in M-theory II: Global solutions asymptotic to $AdS_{7}\times S^{4}$,” JHEP 12 (2008) 044, arXiv:0810.4647.
  * [34] C. Bachas, E. D’Hoker, J. Estes, and D. Krym, “M-theory solutions invariant under $D(2,1;\gamma)\oplus D(2,1;\gamma)$,” Fortsch. Phys. 62 (2014) 207–254, arXiv:1312.5477.
  * [35] P. Liendo, L. Rastelli, and B. C. van Rees, “The bootstrap program for boundary CFTd,” JHEP 07 (2013) 113, arXiv:1210.4258.
  * [36] M. Billò, V. Gonçalves, E. Lauria, and M. Meineri, “Defects in conformal field theory,” JHEP 04 (2016) 091, arXiv:1601.02883.
  * [37] P. Liendo and C. Meneghelli, “Bootstrap equations for $\mathcal{N}$ = 4 SYM with defects,” JHEP 01 (2017) 122, arXiv:1608.05126.
  * [38] M. Lemos, P. Liendo, M. Meineri, and S. Sarkar, “Universality at large transverse spin in defect CFT,” JHEP 09 (2018) 091, arXiv:1712.08185.
  * [39] S. Caron-Huot, “Analyticity in spin in conformal theories,” JHEP 09 (2017) 078, arXiv:1703.00278.
  * [40] J. Barrat, A. Gimenez-Grau, and P. Liendo, “Bootstrapping holographic defect correlators in $\mathcal{N}$ = 4 super Yang-Mills,” JHEP 04 (2022) 093, arXiv:2108.13432.
  * [41] A. Gimenez-Grau and P. Liendo, “Bootstrapping monodromy defects in the Wess-Zumino model,” JHEP 05 (2022) 185, arXiv:2108.05107.
  * [42] A. Gimenez-Grau, “Probing magnetic line defects with two-point functions,” arXiv:2212.02520.
  * [43] L. Bianchi, D. Bonomi, and E. de Sabbata, “Analytic bootstrap for the localized magnetic field,” arXiv:2212.02524.
  * [44] M. S. Costa, J. Penedones, D. Poland, and S. Rychkov, “Spinning conformal correlators,” JHEP 11 (2011) 071, arXiv:1107.3554.
  * [45] E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. 2 (1998) 253–291, arXiv:hep-th/9802150.
  * [46] K. Jensen, A. O’Bannon, B. Robinson, and R. Rodgers, “From the Weyl anomaly to entropy of two-dimensional boundaries and defects,” Phys. Rev. Lett. 122 no. 24, (2019) 241602, arXiv:1812.08745.
  * [47] M. Henningson and K. Skenderis, “The holographic Weyl anomaly,” JHEP 07 (1998) 023, arXiv:hep-th/9806087.
  * [48] K. A. Intriligator, “Anomaly matching and a Hopf-Wess-Zumino term in 6d $\mathcal{N}=(2,0)$ field theories,” Nucl. Phys. B 581 (2000) 257–273, arXiv:hep-th/0001205.
  * [49] A. A. Tseytlin, “$R^{4}$ terms in 11 dimensions and conformal anomaly of (2,0) theory,” Nucl. Phys. B 584 (2000) 233–250, arXiv:hep-th/0005072.
  * [50] M. Beccaria, G. Macorini, and A. A. Tseytlin, “Supergravity one-loop corrections on AdS7 and AdS3, higher spins and AdS/CFT,” Nucl. Phys. B 892 (2015) 211–238, arXiv:1412.0489.
  * [51] A. Chalabi, A. O’Bannon, B. Robinson, and J. Sisti, “Central charges of 2d superconformal defects,” JHEP 05 (2020) 095, arXiv:2003.02857.
  * [52] C. R. Graham and E. Witten, “Conformal anomaly of submanifold observables in $AdS$/CFT correspondence,” Nucl. Phys. B546 (1999) 52–64, hep-th/9901021.
  * [53] N. Drukker, M. Probst, and M. Trépanier, “Surface operators in the 6d $\mathcal{N}=(2,0)$ theory,” J. Phys. A 53 no. 36, (2020) 365401, arXiv:2003.12372.
  * [54] N. Drukker, S. Giombi, A. A. Tseytlin, and X. Zhou, “Defect CFT in the 6d $(2,0)$ theory from M2 brane dynamics in $AdS_{7}\times S^{4}$,” JHEP 07 (2020) 101, arXiv:2004.04562.
  * [55] J. Barrat, A. Gimenez-Grau, and P. Liendo, “A dispersion relation for defect CFT,” arXiv:2205.09765.
  * [56] L. Bianchi and D. Bonomi, “Conformal dispersion relations for defects and boundaries,” arXiv:2205.09775.
  * [57] N. Drukker, M. Probst, and M. Trépanier, “Defect CFT techniques in the 6d $\mathcal{N}=(2,0)$ theory,” JHEP 03 (2021) 261, arXiv:2009.10732.
  * [58] C. Cordova, D. Gaiotto, and S.-H. Shao, “Surface defects and chiral algebras,” JHEP 05 (2017) 140, arXiv:1704.01955.
  * [59] F. A. Dolan, L. Gallot, and E. Sokatchev, “On four-point functions of 1/2-BPS operators in general dimensions,” JHEP 09 (2004) 056, arXiv:hep-th/0405180.
  * [60] P. Liendo, C. Meneghelli, and V. Mitev, “On correlation functions of BPS operators in 3d ${\mathcal{N}}$ = 6 superconformal theories,” Commun. Math. Phys. 350 no. 1, (2017) 387–419, arXiv:1512.06072.
  * [61] M. Lemos, P. Liendo, C. Meneghelli, and V. Mitev, “Bootstrapping $\mathcal{N}=3$ superconformal theories,” JHEP 04 (2017) 032, arXiv:1612.01536.
  * [62] P. Liendo, C. Meneghelli, and V. Mitev, “Bootstrapping the half-BPS line defect,” JHEP 10 (2018) 077, arXiv:1806.01862 [hep-th].
  * [63] F. A. Dolan and H. Osborn, “Conformal four point functions and the operator product expansion,” Nucl. Phys. B 599 (2001) 459–496, arXiv:hep-th/0011040.
  * [64] M. Isachenkov, P. Liendo, Y. Linke, and V. Schomerus, “Calogero-Sutherland approach to defect blocks,” JHEP 10 (2018) 204, arXiv:1806.09703.
  * [65] P. S. Howe, G. Sierra, and P. K. Townsend, “Supersymmetry in six-dimensions,” Nucl. Phys. B 221 (1983) 331–348.
  * [66] B. Eden, S. Ferrara, and E. Sokatchev, “(2,0) superconformal OPEs in $d=6$, selection rules and nonrenormalization theorems,” JHEP 11 (2001) 020, arXiv:hep-th/0107084.
  * [67] S. Ferrara and E. Sokatchev, “Universal properties of superconformal OPEs for 1/2 BPS operators in $3\leq d\leq 6$,” New J. Phys. 4 (2002) 2, arXiv:hep-th/0110174.
  * [68] A. Gimenez-Grau and P. Liendo, “Bootstrapping Coulomb and Higgs branch operators,” JHEP 01 (2021) 175, arXiv:2006.01847.
  * [69] M. Gunaydin and R. J. Scalise, “Unitary lowest weight representations of the noncompact supergroup $OSp(2m^{*}|2n)$,” J. Math. Phys. 32 (1991) 599–606.
  * [70] N. B. Agmon and Y. Wang, “Classifying superconformal defects in diverse dimensions part I: superconformal lines,” arXiv:2009.06650.
  * [71] G. M. T. Watts, “$\mathcal{W}$-algebras and their representations,” in Conformal Field Theories and Integrable Models, Z. Horváth and L. Palla, eds., pp. 55–84.  Springer Berlin Heidelberg, Berlin, Heidelberg, 1997.
  * [72] P. Bouwknegt and K. Schoutens, “$\mathcal{W}$ symmetry in conformal field theory,” Phys. Rept. 223 (1993) 183–276, arXiv:hep-th/9210010.
  * [73] H.-C. Kim, S. Kim, S.-S. Kim, and K. Lee, “The general M5-brane superconformal index,” arXiv:1307.7660.
  * [74] M. Bullimore and H.-C. Kim, “The superconformal index of the (2,0) theory with defects,” JHEP 05 (2015) 048, arXiv:1412.3872.
  * [75] A. Campoleoni, S. Fredenhagen, and S. Pfenninger, “Asymptotic $\mathcal{W}$-symmetries in three-dimensional higher-spin gauge theories,” JHEP 09 (2011) 113, arXiv:1107.0290.
  * [76] F. Bastianelli and R. Zucchini, “Three point functions of chiral primary operators in $d=3,\mathcal{N}=8$ and $d=6,\mathcal{N}=(2,0)$ SCFT at large $N$,” Phys. Lett. B 467 (1999) 61–66, arXiv:hep-th/9907047.
  * [77] R. Corrado, B. Florea, and R. McNees, “Correlation functions of operators and Wilson surfaces in the $d=6$, (0,2) theory in the large $N$ limit,” Phys. Rev. D 60 (1999) 085011, arXiv:hep-th/9902153.
  * [78] F. Bonetti, C. Meneghelli, and L. Rastelli, “VOAs labelled by complex reflection groups and 4d SCFTs,” JHEP 05 (2019) 155, arXiv:1810.03612.
  * [79] L. Bianchi and M. Lemos, “Superconformal surfaces in four dimensions,” JHEP 06 (2020) 056, arXiv:1911.05082.
  * [80] P. C. Argyres, M. Lotito, and M. Weaver, “Vertex algebra of extended operators in 4d $\mathcal{N}=2$ superconformal field theories,” arXiv:2211.04410.
  * [81] N. Drukker, D. Gaiotto, and J. Gomis, “The virtue of defects in 4d gauge theories and 2d CFTs,” JHEP 06 (2011) 025, arXiv:1003.1112.
  * [82] V. A. Fateev and A. B. Zamolodchikov, “Conformal quantum field theory models in two-dimensions having $\mathbb{Z}_{3}$ symmetry,” Nucl. Phys. B 280 (1987) 644–660.
  * [83] N. Afkhami-Jeddi, K. Colville, T. Hartman, A. Maloney, and E. Perlmutter, “Constraints on higher spin CFT2,” JHEP 05 (2018) 092, arXiv:1707.07717.
  * [84] A. R. Linshaw, “Universal two-parameter $\mathcal{W}_{\infty}$-algebra and vertex algebras of type $\mathcal{W}(2,3,\dots,N)$,” Compos. Math. 157 no. 1, (2021) 12–82, arXiv:1710.02275.
  * [85] V. A. Fateev and S. L. Lukyanov, “The models of two-dimensional conformal quantum field theory with $\mathbb{Z}_{N}$ symmetry,” Int. J. Mod. Phys. A 3 (1988) 507.
  * [86] L. F. Alday, F. Benini, and Y. Tachikawa, “Liouville/Toda central charges from M5-branes,” Phys. Rev. Lett. 105 (2010) 141601, arXiv:0909.4776.
  * [87] C. Cordova and D. L. Jafferis, “Toda theory from six dimensions,” JHEP 12 (2017) 106, arXiv:1605.03997.
  * [88] L. F. Alday and S. Caron-Huot, “Gravitational $S$-matrix from CFT dispersion relations,” JHEP 12 (2018) 017, arXiv:1711.02031.
  * [89] E. Lauria, M. Meineri, and E. Trevisani, “Radial coordinates for defect CFTs,” JHEP 11 (2018) 148, arXiv:1712.07668.
  * [90] J. Maldacena, D. Simmons-Duffin, and A. Zhiboedov, “Looking for a bulk point,” JHEP 01 (2017) 013, arXiv:1509.03612.
  * [91] C. Beem, L. Rastelli, and B. C. van Rees, “The $\mathcal{N}=4$ superconformal bootstrap,” Phys. Rev. Lett. 111 (2013) 071601, arXiv:1304.1803.
  * [92] P. Liendo, Y. Linke, and V. Schomerus, “A lorentzian inversion formula for defect CFT,” JHEP 08 (2020) 163, arXiv:1903.05222.
  * [93] A. Erdélyi and H. Bateman, Higher transcendental functions.  Bateman Manuscript Project. Krieger, 1981.  https://books.google.ca/books?id=h7UTzQEACAAJ.
  * [94] O. Aharony, L. F. Alday, A. Bissi, and E. Perlmutter, “Loops in $AdS$ from conformal field theory,” JHEP 07 (2017) 036, arXiv:1612.03891.
  * [95] G. Mack, “$d$-independent representation of conformal field theories in $d$ dimensions via transformation to auxiliary dual resonance models. Scalar amplitudes,” arXiv:0907.2407.
  * [96] L. Rastelli and X. Zhou, “How to succeed at holographic correlators without really trying,” JHEP 04 (2018) 014, arXiv:1710.05923.
  * [97] V. Goncalves and G. Itsios, “A note on defect Mellin amplitudes,” arXiv:1803.06721.
  * [98] P. Ferrero and C. Meneghelli, “Bootstrapping the half-BPS line defect CFT in $\mathcal{N}=4$ supersymmetric Yang-Mills theory at strong coupling,” Phys. Rev. D 104 no. 8, (2021) L081703, arXiv:2103.10440.
  * [99] V. K. Dobrev, V. B. Petkova, S. G. Petrova, and I. T. Todorov, “Dynamical derivation of vacuum operator product expansion in euclidean conformal quantum field theory,” Phys. Rev. D 13 (1976) 887.
  * [100] V. K. Dobrev, G. Mack, V. B. Petkova, S. G. Petrova, and I. T. Todorov, Harmonic analysis on the $n$-dimensional Lorentz group and its application to conformal quantum field theory, vol. 63.  1977\.
  * [101] F. A. Dolan and H. Osborn, “Conformal partial waves and the operator product expansion,” Nucl. Phys. B 678 (2004) 491–507, arXiv:hep-th/0309180.
  * [102] D. M. McAvity and H. Osborn, “Conformal field theories near a boundary in general dimensions,” Nucl. Phys. B 455 (1995) 522–576, arXiv:cond-mat/9505127.
  * [103] A. Lewkowycz and E. Perlmutter, “Universality in the geometric dependence of Rényi entropy,” JHEP 01 (2015) 080, arXiv:1407.8171.
  * [104] L. Bianchi, M. Meineri, R. C. Myers, and M. Smolkin, “Rényi entropy and conformal defects,” JHEP 07 (2016) 76, arXiv:1511.06713.
  * [105] R. Manvelyan and A. C. Petkou, “A Note on $R$ currents and trace anomalies in the (2,0) tensor multiplet in $d=6$ $AdS$/CFT correspondence,” Phys. Lett. B 483 (2000) 264–270, arXiv:hep-th/0003017.