# Asteroseismology sheds light on the origin of carbon-deficient red giants:
likely merger products and linked to the Li-rich giants

Sunayana Maben  CAS Key Laboratory of Optical Astronomy, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100101, Peoples Republic
of China  School of Astronomy and Space Science, University of Chinese Academy
of Sciences, Beijing 100049, Peoples Republic of China  Indian Institute of
Astrophysics, 100ft road Koramangala, Bangalore, 560034, India  Simon W.
Campbell  School of Physics and Astronomy, Monash University, Clayton,
Victoria, Australia  ARC Centre of Excellence for Astrophysics in Three
Dimensions (ASTRO-3D), Australia  Yerra Bharat Kumar  Indian Institute of
Astrophysics, 100ft road Koramangala, Bangalore, 560034, India  Bacham E.
Reddy  Indian Institute of Astrophysics, 100ft road Koramangala, Bangalore,
560034, India  Gang Zhao  CAS Key Laboratory of Optical Astronomy, National
Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101,
Peoples Republic of China  School of Astronomy and Space Science, University
of Chinese Academy of Sciences, Beijing 100049, Peoples Republic of China
Yerra Bharat Kumar and Gang Zhao<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Carbon-deficient red giants (CDGs) are a peculiar class of stars that have
eluded explanation for decades. We aim to better characterise CDGs by using
asteroseismology (Kepler, TESS) combined with spectroscopy (APOGEE, LAMOST),
and astrometry (Gaia). We discovered 15 new CDGs in the Kepler field, and
confirm that CDGs are rare, being only $0.15\%$ of our background sample.
Remarkably, we find that our CDGs are almost exclusively in the red clump (RC)
phase. Asteroseismic masses reveal that our CDGs are primarily low-mass stars
($M\lesssim$ 2 M⊙), in contrast to previous studies which suggested they are
intermediate mass ($M=2.5-5.0~{}\rm M_{\odot}$) based on HR diagrams. A very
high fraction of our CDGs ($50\%$) are also Li-rich giants. We observe a
bimodal distribution of luminosity in our CDGs, with one group having normal
RC luminosity and the other being a factor of two more luminous than expected
for their masses. We find demarcations in chemical patterns and luminosities
which lead us to split them into three groups: (i) normal-luminosity CDGs,
(ii) over-luminous CDGs, and (iii) over-luminous highly-polluted CDGs. We
conclude that a merger of a helium white dwarf with an RGB star is the most
likely scenario for the two groups of over-luminous stars. Binary mass-
transfer from intermediate-mass AGB stars is a possibility for the highly-
polluted over-luminous group. For the normal-luminosity CDGs, we cannot
distinguish between core He-flash pollution or lower-mass merger scenarios.
Due to the overlap with the CDGs, Li-rich giants may have similar formation
channels.

##  1 Introduction

When a star first ascends the red giant branch (RGB) the first dredge-up (FDU)
occurs. In the FDU, the convective envelope moves inwards (in mass) and mixes
material from the interior to the surface. This material has been exposed to
H-burning on the main-sequence and therefore the surface composition of the
star changes. There is good agreement between theory (e.g., Iben 1964, 1967;
Dearborn et al. 1976, 1978) and observation (e.g., Lambert & Ries 1977;
Kjaergaard et al. 1982; Shetrone et al. 1993, 2019) on this event. The main
surface abundance changes are an increase in the 4He, 14N and 13C abundances,
and a decrease in the 12C abundance by about 30% (Iben & Renzini, 1984).

Contrary to this picture, 44 giants have been found to have extremely low
carbon abundances (Bidelman 1951; Bidelman & MacConnell 1973; Bond 2019), well
below what is expected from FDU. These stars are known as the weak G-band
(wGb) stars, which are G and K giants whose spectra show very weak or absent
G-band absorption of the CH molecule at 4300 Å. Of the 44 wGb stars, only 29
stars have carbon abundances from high-resolution spectra (R $\approx$ 48,000
- 60,000; Adamczak & Lambert 2013; Palacios et al. 2016). When the high-
resolution spectra of these peculiar stars were analyzed in detail, in
addition to these stars being extremely carbon-deficient, their carbon
isotopic ratios $\rm{}^{12}C/^{13}C$ were found to be close to the equilibrium
value of 3 - 4. Also, N was found to be enhanced, and in some cases, they were
found to be overabundant in Li and Na as well. These studies suggested that
the wGb stars were probably intermediate-mass stars ranging in mass from about
2.5 $\rm M_{\odot}$ to 5.0 $\rm M_{\odot}$. Based on their position on the
Hertzsprung–Russell diagram (HRD), many of them were shown to be in the sub-
giant branch (SGB)/RGB phase and a few in the core He-burning phase (red
clump; RC). The first dedicated spectroscopic survey for C-deficient stars was
undertaken 50 years ago by Bidelman & MacConnell (1973). Recently, Bond (2019)
added 5 carbon-deficient giants (CDGs) based on spectroscopy to the initial
list of CDGs. Since then no new CDGs based on spectroscopy have been
identified.

To this day, there is no consensus on why these giants have extremely low C
and high N abundances. Some studies favor in-situ origin, that is internal
nucleosynthesis with extra mixing in stars (Adamczak & Lambert, 2013), and
others favor an external origin such as pollution of their stellar atmospheres
during the main sequence (MS) or pre-main sequence (PMS) by CN-processed
material (Palacios et al., 2016).

Two of the key factors that are required to decipher the possible origin of
the carbon anomaly and its connection with other elements are the mass and
evolutionary status of the wGb stars. Although there have been many attempts
at determining these two stellar characteristics, the results so far have been
inconclusive, since previous determinations have been based only on locations
in the HRD (Palacios et al. 2012; Adamczak & Lambert 2013; Palacios et al.
2016; Bond 2019).

In the current study, we aim to determine the mass and evolutionary status of
these peculiar stars. For the masses we use asteroseismology combined with
astrometry, photometry, and spectroscopy. For the evolutionary phase, we will
use the period spacing of the g-dominated mixed modes ($\Delta$P,
$\Delta\Pi_{1}$) to determine if the wGb stars are in the RGB phase or He-core
burning RC phase (Bedding et al. 2011; Mosser et al. 2014). These results,
combined with chemical composition information, will hopefully allow us to
better understand the origin of the carbon (and other) anomalies in these
stars.

To this end, in Section 2 we build a catalog by conducting a large systematic
search of wGb stars that have both asteroseismic data from the Kepler mission
(Borucki et al., 2010), and spectroscopic data from the APOGEE survey
(Majewski et al., 2017). The results are presented in Section 3. We provide
substantial discussion on the implications of our results in Section 4, and
our conclusions are summarized in Section 5.

##  2 Sample selection

###  2.1 APOGEE-KEPLER cross-match

We take the spectroscopic data from data release 17 (DR17; Abdurro’uf et al.
2022) of the Apache Point Observatory Galactic Evolution Experiment (APOGEE),
which is a large-scale near-IR, high-resolution (R $\approx$ 22,500) survey.
Stellar parameters and individual elemental abundances for up to 20 species
were derived by the APOGEE Stellar Parameters and Chemical Abundances Pipeline
(ASPCAP, García Pérez et al. 2016) for 733,901 stars across the Milky Way.

In the current study we focus on the Kepler field since we are interested in
having asteroseismic constraints. We explore the entire APOGEE sample in a
separate study (Maben et al. 2023, in preparation). We use the Kepler stellar
properties catalog by Mathur et al. (2017). This catalog is based on Kepler
data release 25, which contains data from Quarters 1 to 17. It is the largest
catalog of stars observed by the Kepler mission with 197,096 individual
targets.

To begin with, we cross-matched these two catalogs, which yielded a sample of
23,129 stars (APOGEE-KEPLER sample). Secondly, we checked how many of these
stars have global asteroseismic parameters derived from precision Kepler light
curves in the literature (Stello et al. 2013; Mosser et al. 2014; Vrard et al.
2016; Serenelli et al. 2017; Yu et al. 2018; Pinsonneault et al. 2018; Li et
al. 2020). We found 11,099 stars (i.e., 48% of the APOGEE-KEPLER sample) that
matched this criterion.

Further, we constrain our sample to have good-quality APOGEE spectroscopic
data. The ASPCAPFLAG is used to flag potential issues with an observation
and/or with specific stellar parameters. Hence, we remove stars with flags
STAR_BAD or STAR_WARN. Since we also require reliable metallicity and carbon
abundances, we limit our sample to stars that have FE_H_FLAG=0 and C_FE_FLAG=0
(Jönsson et al., 2020).

We restrict our sample to disk metallicities ([Fe/H] $\geq-0.8$ dex; Tomkin et
al. 1995). This is consistent with the known wGb stars. Carbon is more complex
at low metallicities since there is a significant dispersion in abundances
(e.g., Romano et al. 2019). In addition, there is a known carbon-depletion
mechanism – deep-mixing in red giants – that increases the carbon depletion
rate at low metallicities (e.g., Martell et al. 2008; Gerber et al. 2019).
These constraints give us 10,674 stars (i.e., 46% of the APOGEE-KEPLER
sample).

One of the primary purposes of this study is to determine the evolutionary
phase of CDGs. We therefore, select stars that have their evolutionary status
positively determined in the different asteroseismic studies mentioned above.
Applying this criterion we obtain a final common sample of 10,180 stars (i.e.,
44% of the APOGEE-KEPLER sample).

Figure 1: Hertzsprung–Russell diagram (left) and log(g)seis as a function of
[C/Fe] (right). Our APOGEE-KEPLER common sample forms the background in a
color scale that represents the number density of stars; darker colors
indicating higher density: Dwarfs and sub-giants (blue dots), inert He-core
giants ascending the RGB for the first time (grey dots) and He-core burning
giants (red dots). Kernel density histograms are included to show the log(g)
and [C/Fe] distributions for various parts of the sample, with the color
scheme following the background sample, and the dashed histogram showing our
final C-deficient sample. We superimpose model tracks in the log(g)-[C/Fe]
plane for comparison (see text for model details). Starting with an initial
[C/Fe] $=0.0$, the models show a reduction of [C/Fe] by $\approx 0.05$ to
$0.2$ dex, due to FDU dilution. As some dwarf stars have lower [C/Fe], which
may reflect a lower initial abundance, we offset the 2 M⊙ model (which has the
larger C-depletion) by $-0.15$ dex to match the lower envelope of those stars.
This indicates the expected lower limit of [C/Fe] for later phases of
evolution (dashed model line). The vertical dashed line at [C/Fe] $=-0.4$ dex
denotes the cut we use to define carbon-deficient stars (see text for
details). The larger symbols (filled circles) show our carbon-deficient
sample. Large red circles are RC stars and the two large grey circles are RGB
stars. Formal error bars are shown on the bottom right end of each subplot.
The top and right panel of the right subplot shows kernel density histograms
for the normal giants (solid lines) and RC CDGs (dashed red line). We use the
solar abundance of C as derived by Grevesse et al. (2007).

###  2.2 Defining and identifying carbon-deficient stars

Figure 1 shows the APOGEE-KEPLER common sample that made it through the
quality criteria discussed in Section 2.1. In the left panel of the figure, we
see that we have a sample that covers all the evolutionary phases, from dwarfs
and sub-giants to H-shell burning stars and core He-burning (CHeB) stars. The
distribution of the carbon abundance for the sample is shown in the central
panel. It can immediately be seen that there are very few outliers, with low
[C/Fe].

Here we want to make a quantitative definition of carbon-deficient stars. To
guide us, we use stellar models, which illustrate the theoretical expectations
for the carbon surface abundances. We calculated a series of models using the
MESA stellar code (Paxton et al. 2011, 2019; version 12778). RGB mass loss was
modeled using the Reimers (1975) formula ($\eta=0.3$). We used the standard
MESA nuclear network (‘basic.net’), standard equation of state (see Paxton et
al. 2019 for details), and $\alpha_{MLT}=2.0$. Convective boundary locations
were based on the Schwarzschild criterion, extended with exponential overshoot
(Herwig et al. 1997) during core helium burning ($f_{OS}=0.001$, following
Constantino et al. 2017). Models were run from the pre-main sequence to the
start of the thermally pulsing-asymptotic giant branch (TP-AGB) phase, so as
to cover the CHeB phase for which we have many stars in our sample. In Figure
1 we show two of our models, having masses of 1 M⊙ and 2 M⊙. These model
masses are representative of the bulk of the background observational sample.
There is a metallicity distribution in our sample with [Fe/H] ranging from
$-0.8$ to $+0.5$ dex and peaking at [Fe/H] $=0.0$ dex. Our models showed that
metallicity has a much smaller effect than mass on [C/Fe], so we only show
[Fe/H] $=0.0$ models for clarity. As can be seen in Figure 1, the models start
at [C/Fe] $=0.0$ then deplete C around log(g) $=3.2-3.5$, as the convective
envelope deepens during FDU early on the RGB. By log(g) $\approx 2.8-3.2$, the
surface C abundances have reached their minima, with a total [C/Fe] reduction
of $0.05$ dex and $0.18$ dex for the 1 M⊙ and 2 M⊙ models, respectively. From
then on the C remains constant.

The initial C abundance of a star will affect its future surface abundance.
Our sample includes dwarf stars that have abundances as low as [C/Fe]
$\simeq-0.15$ dex. To roughly account for stars with lower initial C we offset
the 2 M⊙ model track (which has the largest C depletion) by $-0.15$ dex
(similar to Mishenina et al. 2006 and Tautvaišienė et al. 2010). Here, the
carbon depletion reaches as low as [C/Fe] $\simeq-0.33$ dex. Based on this,
and the observed distribution of the low-carbon tail, we make a conservative
cut at [C/Fe] $=-0.4$ dex, below which we consider all stars as carbon-
deficient (vertical dashed line in Figure 1). With this definition, we find a
total of 15 carbon-deficient stars, $0.15\%$ of our final APOGEE-KEPLER
sample. This highlights how rare carbon-deficient stars are. We will be
referring to these stars as the ‘CDGs’. All stars in our CDG sample have known
evolutionary phases. In Figure 1, we indicate their respective evolutionary
phases using different symbols. Thirteen of our CDGs are RC stars, and two are
RGB stars (although we later re-classify one of those as RC; see Section 3.2).

##  3 Analysis and Results

###  3.1 Evolutionary phase: Seismic diagram

It has been shown that RC and RGB stars can be separated by considering
measurements of their mixed-mode period spacing $\Delta$P (Bedding et al.,
2011) or equivalently, the asymptotic g-mode period spacing $\Delta\Pi_{1}$
(Mosser et al., 2014). Red giants with $\Delta\Pi_{1}\geq 80$ seconds are RC
stars and the giants in the very narrow strip (Figure 2) with
$\Delta\Pi_{1}\leq 80$ seconds are RGB stars (Vrard et al., 2016).

Out of our 15 CDGs, 6 stars have both the asteroseismic parameters
$\Delta\Pi_{1}$ and $\Delta\nu$ from Mosser et al. (2014), and 5 stars have
these parameters from Vrard et al. (2016)111The single RGB star (see Section
3.2) does not have a reported $\Delta\Pi_{1}$, but it is clearly of too low
luminosity to be a RC star.. In Figure 2 we plot these 11 RC stars (large
symbols) over the background sample of Mosser et al. (2014). We have limited
the background sample to match our RC sample mass range of 1.0 M⊙ $\leq M\leq$
2.5 M⊙ (Section 3.2.1) for better comparison.

As seen in Figure 2, all of our CDGs with $\Delta\Pi_{1}$ have large values of
this parameter. They also have small values of $\Delta\nu$ ($<$ 5 $\mu$Hz).
They clearly occupy the He-core burning phase region of the
$\Delta\Pi_{1}$-$\Delta\nu$ diagram. Two of the CDGs are classified as Helium
subflashing stars by Mosser et al. (2014) (see discussion in Section 4.4.1).

Figure 2: Seismic diagram of our RC CDGs which have known $\Delta\Pi_{1}$ and
$\Delta\nu$ (11 stars; large filled red circles). Giants classified based on
asteroseismic analysis form the background from Mosser et al. (2014) (small
filled circles; see key), in a color scale that represents the number density
of stars; darker colors indicate higher densities.

###  3.2 Luminosities

Luminosities of the CDGs were determined using the standard
formula222log($L/L_{\odot}$) = $-$0.4 [$V_{0}$ $-$ $(m-M)_{0}$ $+$ $BC$ $-$
$M_{bol,\odot}$]. Distances were taken from the catalog of Bailer-Jones et al.
(2021). The visual magnitudes and their errors were estimated from the color-
color transformations given in Riello et al. (2021) that relate the Gaia DR3
photometric system to the Johnson-Cousins system (Stetson, 2000). We used the
Green et al. (2019) three-dimensional dust map to estimate the reddening for
each of the stars. We applied bolometric corrections following Alonso et al.
(1999), adopting errors in $T_{\rm eff}$ and [Fe/H] of $50$ K and $0.05$ dex,
respectively. Our derived luminosities are listed in Table 1.

All of our stars, except one (KIC 8352953), have luminosities consistent with
being RC stars (Table 1, also see Kiel diagram in Figure 1). KIC 8352953 has a
luminosity that coincides with the lower-RGB phase and is classified as an RGB
star in Yu et al. (2018). All except one of the stars (KIC 8222189) that have
RC luminosities are classified as RC stars by asteroseismic studies (Stello et
al. 2013; Mosser et al. 2014; Vrard et al. 2016; Yu et al. 2018; Pinsonneault
et al. 2018). This one star has a stellar evolutionary phase of RGB (or AGB)
in Pinsonneault et al. (2018), is unclassified in Yu et al. (2018) and
Elsworth et al. (2017), but is classified as a RC star by Ting et al. (2018).
Given that KIC 8222189 has a luminosity and surface gravity that is consistent
with the core He-burning phase, we classify it as a RC star from now on, but
note that a $\Delta P$ measurement would be required to be certain. In
summary, our CDG sample has one RGB star and 14 RC stars.

Figure 3: Carbon abundance as a function of $\log(L/L_{\odot})$ for our RC
CDGs. As can be seen in the Gaussian kernel density histogram above, the
luminosity distribution is bimodal. This bimodality is discussed in detail in
Section 3.6. We use the solar abundance of C as derived by Grevesse et al.
(2007).

Looking at the distribution of the luminosities, we find there is a bimodality
(see Figure 3), with one group having higher luminosity
($\log(L/L_{\odot})=2.1\pm 0.1$ dex) than the other ($\log(L/L_{\odot})=1.8\pm
0.1$ dex). Interestingly the brighter RC stars tend to have the lowest C
abundances (see Figures 3 and 10). The bi-modality is also present in log(g)
but is not as clear (Figure 1). We discuss this bimodality in detail in
Section 3.6.

Table 1: Atmospheric, asteroseismic, kinematic parameters along with the CNO
abundance ratios and mass estimates from equations (1) - (5) of the CDGs. We
have adopted the errors in $T_{\rm eff}$ and [Fe/H] as 50 K and 0.05 dex
respectively for all the stars. $M_{avg}$ is the average of the asteroseismic
mass equations (1), (3), and (4).

KIC | $T_{\rm eff}$ | [Fe/H] | log(g)seisc | $\log(L/L_{\odot})$ |  $\nu_{\rm max}$c | $\Delta\nu$ | $\Delta\Pi_{1}$ | [C/Fe] | [N/Fe] | [O/Fe] | M1 | M2 | M3 | M4 | Mavg | M5 | Rseis | b | Z
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| (K) |  |  |  | ($\mu$Hz) | ($\mu$Hz) | (s) |  |  |  | (M⊙) | (M⊙) | (M⊙) | (M⊙) | (M⊙) | (M⊙) | (R⊙) | (deg) | (kpc)
5881715 | 4840 | -0.12 | 2.41$\pm$0.02 | 1.89$\pm$0.05 | 30.90 | 3.42a | 202a | -0.52 | 0.53 | -0.02 | 1.91$\pm$0.22 | 1.26$\pm$1.86 | 1.44$\pm$0.18 | 1.76$\pm$0.17 | 1.70 | 1.35$\pm$0.23 | 14.46$\pm$0.63 | 10.15 | 0.33
8879518 | 4832 | 0.08 | 2.58$\pm$0.01 | 1.73$\pm$0.04 | 46.18 | 4.65b | 268b | -0.47 | 0.61 | -0.01 | 1.87$\pm$0.14 | 1.35$\pm$2.12 | 1.50$\pm$0.15 | 1.74$\pm$0.11 | 1.70 | 2.11$\pm$0.32 | 11.66$\pm$0.35 | 14.46 | 0.32
4071012 | 4992 | 0.07 | 2.38$\pm$0.01 | 2.07$\pm$0.04 | 29.20 | 3.03b | 276b | -0.87 | 0.68 | -0.11 | 2.74$\pm$0.25 | 1.53$\pm$0.84 | 1.85$\pm$0.19 | 2.43$\pm$0.18 | 2.34 | 2.81$\pm$0.43 | 17.65$\pm$0.70 | 8.18 | 0.27
3355015 | 4846 | -0.15 | 2.34$\pm$0.02 | 1.81$\pm$0.05 | 26.98 | 3.39a | 292a | -0.49 | 0.57 | 0.03 | 1.31$\pm$0.14 | 0.94$\pm$1.31 | 1.04$\pm$0.13 | 1.22$\pm$0.11 | 1.19 | 1.06$\pm$0.18 | 12.76$\pm$0.51 | 8.15 | 0.29
3736289 | 4978 | -0.10 | 2.61$\pm$0.01 | 1.80$\pm$0.04 | 49.94 | 4.99a | 307a | -0.43 | 0.52 | -0.04 | 1.87$\pm$0.17 | 1.65$\pm$0.98 | 1.72$\pm$0.18 | 1.82$\pm$0.14 | 1.80 | 1.60$\pm$0.24 | 11.16$\pm$0.37 | 12.78 | 0.28
5446927 | 5107 | -0.74 | 2.26$\pm$0.01 | 2.05$\pm$0.05 | 21.88 | 2.89a | 313a | -0.49 | 0.47 | 0.15 | 1.46$\pm$0.12 | 1.12$\pm$0.39 | 1.22$\pm$0.15 | 1.38$\pm$0.10 | 1.36 | 1.34$\pm$0.22 | 14.80$\pm$0.51 | 11.36 | 0.50
4667911 | 4740 | 0.08 | 2.47$\pm$0.01 | 1.67$\pm$0.04 | 36.37 | 4.18b | 318b | -0.41 | 0.53 | -0.03 | 1.34$\pm$0.13 | 1.00$\pm$3.43 | 1.10$\pm$0.11 | 1.26$\pm$0.10 | 1.23 | 0.98$\pm$0.15 | 11.19$\pm$0.42 | 8.34 | 0.15
5000307 | 5018 | -0.29 | 2.54$\pm$0.01 | 1.75$\pm$0.04 | 42.16 | 4.74a | 324a | -0.62 | 0.63 | -0.01 | 1.39$\pm$0.09 | 1.20$\pm$0.57 | 1.26$\pm$0.13 | 1.35$\pm$0.07 | 1.33 | 1.49$\pm$0.23 | 10.43$\pm$0.26 | 13.20 | 0.33
11971123 | 4848 | -0.13 | 2.43$\pm$0.01 | 1.79$\pm$0.04 | 32.47 | 3.88b | 324b | -0.45 | 0.52 | 0.05 | 1.33$\pm$0.13 | 1.14$\pm$1.58 | 1.20$\pm$0.12 | 1.29$\pm$0.10 | 1.27 | 1.37$\pm$0.21 | 11.74$\pm$0.45 | 13.04 | 0.31
8110538 | 4975 | -0.59 | 2.38$\pm$0.02 | 1.98$\pm$0.05 | 28.77 | 3.62b | 334b | -0.70 | 0.80 | 0.49 | 1.29$\pm$0.14 | 1.62$\pm$1.00 | 1.50$\pm$0.19 | 1.35$\pm$0.12 | 1.38 | 1.12$\pm$0.19 | 12.19$\pm$0.52 | 9.74 | 0.62
2423824 | 5007 | -0.39 | 2.26$\pm$0.01 | 2.15$\pm$0.05 | 22.02 | 2.70a | 360a | -0.70 | 0.63 | 0.08 | 1.88$\pm$0.18 | 1.56$\pm$0.81 | 1.66$\pm$0.20 | 1.81$\pm$0.14 | 1.79 | 1.71$\pm$0.29 | 16.85$\pm$0.67 | 13.59 | 0.49
8352953 | 5101 | -0.18 | 3.29$\pm$0.01 | 1.10$\pm$0.04 | 230.27 | 16.74c |  | -0.42 | 0.63 | 0.24 | 1.46$\pm$0.07 | 1.45$\pm$0.48 | 1.45$\pm$0.15 | 1.46$\pm$0.07 | 1.45 | 1.66$\pm$0.25 | 4.56$\pm$0.08 | 16.27 | 0.26
7848354 | 5004 | -0.08 | 2.62$\pm$0.02 | 1.75$\pm$0.06 | 51.74 | 4.67c |  | -0.57 | 0.59 | 0.03 | 2.66$\pm$0.26 | 1.19$\pm$0.64 | 1.56$\pm$0.23 | 2.26$\pm$0.20 | 2.16 | 1.67$\pm$0.31 | 13.05$\pm$0.44 | 7.59 | 0.42
4830861 | 4959 | -0.12 | 2.33$\pm$0.01 | 2.14$\pm$0.05 | 25.94 | 2.96c |  | -0.51 | 0.56 | 0.01 | 2.06$\pm$0.18 | 1.94$\pm$1.29 | 1.98$\pm$0.24 | 2.03$\pm$0.15 | 2.02 | 2.67$\pm$0.45 | 16.26$\pm$0.59 | 11.81 | 0.43
8222189 | 4914 | -0.09 | 2.28$\pm$0.02 | 2.17$\pm$0.04 | 23.02 | 2.69c |  | -0.78 | 0.65 | -0.06 | 2.07$\pm$0.38 | 1.88$\pm$1.62 | 1.94$\pm$0.22 | 2.03$\pm$0.30 | 2.02 | 2.43$\pm$0.37 | 17.37$\pm$1.16 | 15.57 | 0.35

Note: a Mosser et al. (2014); b Vrard et al. (2016); c Yu et al. (2018)

####  3.2.1 Masses

The masses of the CDGs were determined using several combinations of the
seismic ($\nu_{max}$, $\Delta\nu$) and non-seismic parameters ($T_{\rm eff}$,
$L/L_{\odot}$, $g$) that were available. The global seismic parameters are
correlated with fundamental stellar properties; $\nu_{max}\propto gT_{\rm
eff}^{-1/2}$ and $\Delta\nu\propto\rho^{-1/2}$, where $g$ is the surface
gravity and $\rho$ is the mean density of the star (Ulrich 1986; Brown et al.
1991; Kjeldsen & Bedding 1995). Combining these relations with the
Stefan–Boltzmann luminosity law, L $\propto$ R2 $T_{\rm eff}^{4}$, allows the
derivation of the four seismic mass equations (Equations (1)-(4); e.g., Miglio
et al. 2016; Howell et al. 2022). Equation (5) is a standard, but non-seismic
mass formula, which uses the photometric and spectroscopic parameters of a
star. We determined the masses of the CDGs using all of these 5 mass
equations:

$\displaystyle\left(\frac{M}{M_{\odot}}\right)$
$\displaystyle\simeq\left(\frac{\nu_{max}}{\nu_{max,\odot}}\right)^{3}\left(\frac{\Delta\nu}{\Delta\nu_{\odot}}\right)^{-4}\left(\frac{T_{\rm
eff}}{T_{\rm eff,\odot}}\right)^{3/2}$ (1)
$\displaystyle\left(\frac{M}{M_{\odot}}\right)$
$\displaystyle\simeq\left(\frac{\Delta\nu}{\Delta\nu_{\odot}}\right)^{2}\left(\frac{L}{L_{\odot}}\right)^{3/2}\left(\frac{T_{\rm
eff}}{T_{\rm eff,\odot}}\right)^{-6}$ (2)
$\displaystyle\left(\frac{M}{M_{\odot}}\right)$
$\displaystyle\simeq\left(\frac{\nu_{max}}{\nu_{max,\odot}}\right)\left(\frac{L}{L_{\odot}}\right)\left(\frac{T_{\rm
eff}}{T_{\rm eff,\odot}}\right)^{-7/2}$ (3)
$\displaystyle\left(\frac{M}{M_{\odot}}\right)$
$\displaystyle\simeq\left(\frac{\nu_{\rm
max}}{\nu_{max,\odot}}\right)^{12/5}\left(\frac{\Delta\nu}{\Delta\nu_{\odot}}\right)^{-14/5}\left(\frac{L}{L_{\odot}}\right)^{3/10}$
(4) $\displaystyle\left(\frac{M}{M_{\odot}}\right)$
$\displaystyle\simeq\left(\frac{L}{L_{\odot}}\right)\left(\frac{g}{g_{\odot}}\right)\left(\frac{T_{\rm
eff}}{T_{\rm eff,\odot}}\right)^{-4}$ (5)

Here, we use $\nu_{max,\odot}=3090\pm 30~{}\mu$Hz, $\Delta\nu_{\odot}=135.1\pm
0.1~{}\mu$Hz (both from Huber et al. 2011), $T_{\rm eff,\odot}=5777$ K, and
$\log(g)_{\odot}=4.44$ (both from Morel et al. 2014) as our adopted solar
values.

For Equations (1), (2) and (4), it is known that one needs to apply a
correction to the observed $\Delta\nu$ for use in the scaling relations (e.g.,
White et al. 2011; Miglio 2012). We calculate $\Delta\nu$ corrections using
ASFGRID (Sharma et al., 2016) and apply these to the $\Delta\nu$ values of the
CDGs obtained from Mosser et al. (2014) and Vrard et al. (2016) only, as Yu et
al. (2018) provides corrected $\Delta\nu$.

Figure 4: Comparison of various mass estimates of the RC CDGs using Equations
(1)-(5). Figure 5: Top panel: Comparison of the average seismic masses of the
RC CDGs using Equations (1), (3), and (4) ($\rm{M}_{avg}$; see text for
details), versus the non-seismic masses from Equation (5). The horizontal
error bars show the $1\sigma$ standard deviation around the average seismic
masses for each star. Bottom panel: Comparison of the seismic radii and radii
calculated from the Stefan–Boltzmann law.

We list our mass determinations for all the 15 CDGs using each of the five
mass equations in Table 1. We also show them graphically in Figure 4, where we
compare the individual masses of the RC CDG stars calculated with each
equation.

As a summary, we show in Table 2 the average uncertainties on the masses for
each equation, along with the average masses for each equation. It can be seen
that the masses calculated with asteroseismic Equation (2) have the largest
uncertainties ($\overline{err}$ = 1.26 M⊙; also see error bars in Figure 4),
when compared to the uncertainties from asteroseismic equations (1), (3), and
(4), which all have similar uncertainties
($0.14~{}\rm{M}_{\odot}-0.17~{}\rm{M}_{\odot})$. The non-seismic Equation 5
has larger uncertainties ($0.27~{}\rm{M}_{\odot}$) than the asteroseismic
equations (apart from Eqn. 2), but is still of the same order of magnitude as
the more accurate seismic equations. From now on we do not use the Equation
(2) masses, due to the large uncertainties.

For our final seismic masses, we take an average of the three remaining
seismic mass equations for each individual star ($M_{avg}$ in Table 1). In
Figure 5 we show these masses versus the non-seismic mass determinations. The
seismic mass range of our RC CDGs is 1.2 M${}_{\odot}\leq M\leq
2.3~{}\rm{M}_{\odot}$ (with average standard deviation between mass equations
of $0.2~{}\rm{M}_{\odot}$). The non-seismic mass equation gives a slightly
expanded mass range of 1.0 M${}_{\odot}\leq M\leq 2.8~{}\rm{M}_{\odot}$, with
slightly higher average uncertainties ($0.3~{}\rm{M}_{\odot}$).

For the single RGB star, the seismic mass is
$1.5~{}\pm~{}0.1~{}\rm{M}_{\odot}$ and the non-seismic mass is
$1.7\pm\rm{0.3}~{}M_{\odot}$.

Table 2: Average uncertainties on the masses for each mass equation
($\overline{err}$; Eqn. (1)–(5)). Also shown are the average of the masses
given by each mass equation ($\overline{M}$). It can be seen that Eqn. (2) has
very large uncertainties. Units are M⊙.

Mass Eqn. | $\overline{err}\pm 1\sigma$ | $\overline{M}\pm 1\sigma$
---|---|---
1 | $0.17\pm 0.08$ | $1.77\pm 0.46$
2 | $1.26\pm 0.77$ | $1.39\pm 0.29$
3 | $0.17\pm 0.04$ | $1.50\pm 0.29$
4 | $0.14\pm 0.06$ | $1.68\pm 0.37$
5 (non-seis.) | $0.27\pm 0.09$ | $1.69\pm 0.55$

###  3.3 Radius determination

We calculated the seismic stellar radius for our CDGs using the equation
(Ulrich 1986; Kjeldsen & Bedding 1995):

$\displaystyle\left(\frac{R}{R_{\odot}}\right)$
$\displaystyle\simeq\left(\frac{\nu_{max}}{\nu_{max,\odot}}\right)\left(\frac{\Delta\nu}{\Delta\nu_{\odot}}\right)^{-2}\left(\frac{T_{\rm
eff}}{T_{\rm eff,\odot}}\right)^{1/2}$ (6)

We have used corrected $\Delta\nu$ values of the CDGs, along with the
$\nu_{max}$ and $T_{\rm eff}$ values as stated in Table 1. The seismic radii
estimates for all 15 CDGs are also provided in Table 1.

We compare the seismic radius to independent radius estimates calculated using
the Stefan–Boltzmann law in Figure 5. While having greater average
uncertainties ($\simeq 0.8~{}\rm{R}_{\odot}$) than the asteroseismic equation
($\simeq 0.5~{}\rm{R}_{\odot}$), the non-seismic radius equation is
nonetheless of the same order of magnitude as the more precise seismic
equation. Between these two methods, all of the RC CDG radii, except two,
agree within the $2\sigma$ uncertainties.

The seismic radius range of all the RC CDGs is about $10-18~{}\rm{R}_{\odot}$.
Ignoring the two outliers, there is a slight offset between the seismic and
non-seismic radii estimates of $\simeq 0.5~{}\rm{R}_{\odot}$, where the
seismic radii are larger.

For the RGB star, the seismic radius ($4.6~{}\pm~{}0.1~{}\rm{R}_{\odot}$) and
the non-seismic radius ($4.6\pm\rm{0.3}~{}R_{\odot}$) agree with each other
perfectly.

###  3.4 Kinematics

Here we compile the kinematic properties of our sample of CDGs and determine
what component(s) of the Galaxy they belong to.

We calculated the Galactic space velocity components (U, V, W) for each star
using astrometry with distances taken from the catalog of Bailer-Jones et al.
(2021) and proper motions from Gaia DR3 (Gaia Collaboration et al., 2016,
2022). The radial velocity data was taken from APOGEE DR17. We have adopted
the Sun’s distance from the Galactic center as $8.2$ kpc (Bland-Hawthorn &
Gerhard, 2016), and its distance above the Galactic plane as $25$ pc (Jurić et
al., 2008). The space velocity components were computed using the Astropy
Galactocentric package (Astropy Collaboration et al., 2013, 2018). Space
velocities are converted to the local standard of rest (LSR) frame using the
solar motion ($U_{\odot}$, $V_{\odot}$, $W_{\odot}$)= (11.1, 12.24, 7.25)
km$s^{-1}$ (Bland-Hawthorn & Gerhard, 2016). We adopt a Local Standard of Rest
velocity $V_{LSR}$ = 232.8 km s-1.

Figure 6: Our CDG sample is shown in the Toomre diagram. Dashed lines show
constant values of the total space velocity,
$v_{tot}=(U^{2}_{LSR}$+$V^{2}_{LSR}$+$W^{2}_{LSR}$)1/2 at 70 and 180 km s-1
demarcating thin disk and thick disk components, respectively (Nissen, 2004;
Venn et al., 2004).

In Figure 6, our CDG sample is plotted in the Toomre diagram. We apply limits
from Nissen (2004) (also Venn et al. 2004) that define the different Galactic
components. We find our CDGs are distributed in all components of the Galaxy –
the thin disk, thick disc, and one star in the halo.

###  3.5 Abundances

####  3.5.1 Carbon, Nitrogen and Oxygen

In Figure 7 we plot the C, N, O abundances, the [C+N/Fe] and [C+N+O/Fe] sums,
and the [C/N] ratio against [Fe/H] for a background sample of RC stars (Vrard
et al., 2016), along with our CDG stars. All abundances are from the APOGEE
catalog (DR17; Abdurro’uf et al. 2022; García Pérez et al. 2016). All stars
have good quality abundances (X_FE_FLAG=0; Jönsson et al. 2020). The
background sample has been limited in mass 1.0 M⊙ $\leq M\leq$ 2.5 M⊙, to
match the mass range of our RC CDGs (Section 3.2.1).

The nitrogen abundances of the CDGs are enhanced by an average of $+0.35$ dex
over the average [N/Fe] of the background RC sample. This is a smaller offset
than for [C/Fe] ($-0.52$ dex). Due to the C and N offsets being in different
directions, the average [C/N] offset is large, at $+0.86$ dex. In contrast,
the [O/Fe] values of the CDGs generally track the abundances of the background
sample. Most are close to scaled solar, except for three stars which have
higher [O/Fe]. Two of these stars are our most metal-poor objects. Looking at
Mg (Figure 7), it shows the same pattern as oxygen, so we suggest that these
three stars were likely $\alpha$-enhanced from birth.

When we sum C and N we see that most of the stars are scaled solar. The
exceptions are the three $\alpha$-enhanced stars. This again suggests that
these 3 stars had different (non-scaled-solar) initial abundances. This is
even more evident in the [C+N+O/Fe] plot.

The strong N over-abundance anti-correlated with the C under-abundance, along
with the [C+N+O/Fe] sum being mostly scaled solar, indicates that the material
in the atmospheres of CDGs has been processed through CNO cycle hydrogen
burning. We see no evidence for O depletion, and therefore infer that the ON
cycle was not operating significantly, i.e., only CN cycling occurred.

Since AGB stars increase [C+N+O/Fe] through third dredge-up (e.g., Karakas &
Lattanzio 2014), they are unlikely to be polluters for our sample. More
generally, any scenario in which He-burning products are mixed into the
polluting material are not supported by our observations. This is discussed
further in Section 4.2.

Figure 7: Various CNO abundance ratios and sums along with sodium, magnesium,
and aluminium abundances against [Fe/H] for our CDGs (large symbols). A large
sample of RC stars from Vrard et al. (2016) form the background sample (small
circles). All abundances are from APOGEE DR17. Average error bars are shown on
the lower right of each subplot. Solar abundances are from Grevesse et al.
(2007).

####  3.5.2 Sodium, Magnesium and Aluminium

The Na, Mg, and Al abundances vs metallicity of our CDG stars is shown in
Figure 7, against a background sample of RC stars from Vrard et al. (2016).

Interestingly, Figure 7 shows that the sample splits into two groups in
[Na/Fe], one with enhanced [Na/Fe] and one with scaled-solar Na. Taking a cut
in [Na/Fe] at $\sim+0.10$ dex, for the Na-enhanced group we find an average
sodium abundance of [Na/Fe$]=0.3\pm 0.1$ dex, and the Na-normal group has
[Na/Fe$]=0.0\pm 0.1$ dex. We discuss the implications of this Na bimodality
(and its correlation with the luminosity bimodality; Section 3.6) in Section
4.

The [Mg/Fe] and [Al/Fe] abundances of the CDGs are consistent with those of
the background sample. As mentioned, the three stars that appeared to be
$\alpha$-enhanced in the [O/Fe]-[Fe/H] plane of Figure 7, are the ones
enhanced with the $\alpha$-element Mg as well. Thus it is likely that these 3
stars had different (non-scaled-solar) initial abundances, as suggested in
Section 3.5.1.

####  3.5.3 Lithium

We cross-matched our CDG sample with the LAMOST survey (Zhao et al. 2006,
2012), which contains the $6707.8~{}\AA$ Li line. We found 11 out of the 15
CDGs to have low-resolution spectra (R $\approx$ 1800; see top panel of Figure
8). Only one star had a medium-resolution spectrum (R $\approx$ 7500; see
bottom panel of Figure 8). This star (KIC 8879518) has been previously
identified to be a super-Li-rich star with A(Li)333A(Li) = log
($n$(Li)/$n$(H)) + 12, where $n$ is the number density of atoms. $=3.51\pm
0.12$ dex using this spectrum (Singh et al., 2021).

Figure 8: Top panel: LAMOST low-resolution spectra of the CDGs. Note that KIC
8352953 is the only RGB CDG in our sample. Bottom panel: Comparison of the
observed LAMOST medium-resolution spectra and synthetic spectra of KIC 8879518
for different Li abundances. Synthetic spectra (green solid line) of A(Li) =
3.2 dex best fits the observed spectra. The first five CDGs in the top panel
and the CDG in the bottom panel show strong Li absorption lines at 6707.8 Å.

We determine the Li abundances for the 12 stars by matching synthetic spectra
with the observed 6707.8 $\AA$ line. First, the radial-velocity corrected
spectra were continuum-fitted and normalized using the iSpec code (Blanco-
Cuaresma et al. 2014; Blanco-Cuaresma 2019). Stellar parameters ($T_{\rm
eff}$, $logg,[\rm{Fe/H}]$) were taken from the LAMOST catalog for stars with
low-resolution spectra and from the APOGEE catalog for the star with medium-
resolution spectrum. The microturbulent velocities ($\xi$) were derived using
an empirical relation for giants from Holtzman et al. (2018). We compiled a
line list with associated atomic and molecular data around the Li line using
the $linemake$ code444https://github.com/vmplacco/linemake (Placco et al.,
2021).

Local thermodynamic equilibrium (LTE) model atmospheres were generated using
the ATLAS9 code (Castelli & Kurucz, 2003) for the adopted atmospheric
parameters. A series of synthetic spectra were then generated for each star by
varying the Li abundance, using the Python wrapper of the 2013 version of
local thermal equilibrium (LTE) radiative transfer code MOOG (Sneden, 1973),
pyMOOGi555https://github.com/madamow/pymoogi. Finally, the synthetic spectra
were matched with the observed spectra. The Li abundance of the best-matched
(least $\chi$-square) synthetic spectrum was taken as the LTE abundance for
each star. The NLTE Li abundance was then computed following Lind et al.
(2009) 666Data obtained from the INSPECT database, version 1.0 (www.inspect-
stars.net).

We show the derived A(Li)LAMOST of the CDGs using the LAMOST spectra in Table
3. In the Table, we also present the comparison between A(Li)LAMOST and
A(Li)Literature of the CDGs. We see a good consistency between our
measurements and the results derived in the literature.

Table 3: Derived NLTE lithium abundances for our CDGs from LAMOST spectra.
Typical uncertainties are $\pm 0.2$ dex.

KIC | A(Li)Literature | ${\rm A(Li)_{LAMOST}}^{1}$
---|---|---
| (dex) | (dex)
5000307 | 2.73 | 2.6
5881715 | 3.4$\pm$0.14 | 3.4
7848354 | 3.74 | 3.3
11971123 | 2.94 | 2.9
3355015 | 1.94 | 2.1
5446927 |  |  $<$1.8
4071012 |  |  $<$1.8
3736289 |  |  $<$1.8
8110538 |  |  $<$1.8
8352953 |  |  $<$1.8
4830861 |  |  $<$1.8
88795182 | 3.5$\pm$0.15 | 3.2

Note: 1This work; 2Only has LAMOST medium-resolution spectrum; 3Silva Aguirre
et al. (2014); 4Yan et al. (2021); 5Singh et al. (2021).

In total we found 6 out of the 12 CDGs with LAMOST spectra to be Li-rich.
These stars are also identified as Li-rich in the Yan et al. (2021) study
(Table 3). Thus 50% of our sample for which we have spectra is Li-rich. This
can be compared to the expected fraction for field stars. First we note that
our detection limit, A(Li) $>1.8$ dex, is similar to the traditional Li-rich
definition, A(Li) $>1.5$ dex. Using a GALAH survey sample, Kumar et al. 2020
showed that the fraction of RC stars with A(Li) $>1.5$ dex is $\approx 3\%$.
Thus our finding of $50\%$ represents an extremely high fraction – $17$ times
the expected fraction – despite the selection criterion being on carbon, not
lithium. Further, for RC stars it has been shown that stars with A(Li) $>-0.9$
dex can be considered Li-enhanced, due to Li destruction in the preceding RGB
phase (at least for stars in the mass and metallicity range considered in
Kumar et al. 2020). Thus many more CDGs in our sample may be Li-enhanced if
studied with high-resolution spectra – our 50% is a lower limit in this
regard. Regardless of the exact proportion, it is clear that Li-richness is
highly correlated with C-deficiency. We discuss this further in Section 4.6.

###  3.6 Bimodality of the RC CDGs

####  3.6.1 Over-luminous and normal-luminosity stars

In Section 3.2 we reported a bimodality in the luminosity distribution of our
RC CDGs (see Figure 3). The fainter CDGs are in a region bounded by
$\log(L/L_{\odot})$ from $1.67$ dex to $1.89$ dex, which matches well with
theoretical predictions of the core He-burning phase for low-mass stars (i.e.,
$\log(L/L_{\odot})\simeq 1.55-1.85$ dex; Girardi 2016). This is also clear
from Figure 11a where we compare our RC CDG sample with a background RC sample
– the fainter RC CDGs fall on the normal RC. We will be referring to these
stars as ‘Normal-luminosity-RC CDGs’.

The brighter CDGs are more luminous than the typical RC stars, they have
$\log(L/L_{\odot})$ between $1.98$ dex and $2.17$ dex. RC models of masses
covering our seismic mass range do not have luminosities this high.
Observationally, we see that the bright CDGs are a factor of $\approx 2$
brighter than the normal RC for stars of the same mass ($\approx
60~{}\rm{L}_{\odot}$ vs $120~{}\rm{L}_{\odot}$; Figure 11a). We refer to this
group as ‘Over-luminous-RC CDGs’.

It is possible that the bimodality in the luminosity distribution is an
artefact of systematics in our luminosity determination. In Figures 10c and
10d we show the asteroseismic parameters for our sample. These show
correlations with luminosity, with the over-luminous stars having low
$\Delta\nu$ and $\nu_{max}$. Since luminosity and the seismic parameters are
totally independent measurements, this strongly suggests that the bimodality
is real. Moreover, in Figure 10h we see a very strong correlation with sodium,
with the over-luminous stars generally having higher Na than the normal-
luminosity-RC CDG stars, by an average of $+0.25$ dex. This further supports
that these are two distinct groups of RC stars.

Radius shows an expected correlation with luminosity. On the other hand, mass
does not show a significant correlation – both groups roughly cover the same
mass range ($\simeq 1.2$ to 2.3 M⊙), although the normal-luminosity group
appears biased to slightly lower masses on average, with $M_{avg}=1.5~{}\rm
M_{\odot}$ compared to $1.8~{}\rm M_{\odot}$ for the over-luminous stars.

In the bottom panel of Figure 10 we also show the C, N, and O abundances of
our sample versus luminosity. The over-luminous stars have slightly lower
carbon than the normal-luminosity stars. We find an average carbon abundance
of [C/Fe$]=-0.7\pm 0.1$ dex for the former and [C/Fe$]=-0.5\pm 0.1$ dex for
the latter. The nitrogen abundances of the over-luminous stars are enhanced by
an average of $+0.1$ dex more than the average [N/Fe] of the normal-luminosity
CDG stars. This is a smaller offset than the average for carbon ($-0.2$ dex).
In contrast, the [O/Fe] values of both groups of stars are close to scaled
solar, although 3 over-luminous stars appear to be $\alpha$-enhanced (Section
3.5).

####  3.6.2 Known-CDGs in context of luminosity bimodality

In the bottom panel of Figure 10 we also show 29 known-wGb stars from the
literature that have carbon abundances from high-resolution optical spectra
(Adamczak & Lambert, 2013; Palacios et al., 2016). Of these, only 6 stars have
$\nu_{max}$ reported in the literature, based on TESS data (Hon et al., 2021).
These 6 stars are plotted in the top panel of Figure 10. It can be seen that
these known-wGb stars have high luminosities and low $\nu_{max}$, similar to
our sample of over-luminous stars. They also have similar $T_{\rm eff}$ to our
luminous sample, and therefore lie in the same region of the HRD in Figure
11a. This suggests that they are also RC stars, and appear to be the same type
of stars as our over-luminous-RC CDGs, just with more extreme chemical
signatures (Figure 10).

However, despite being over-luminous, this literature sample have
spectroscopic log(g) values close to the normal-luminosity stars (average
log(g) $\simeq 2.7\pm 0.1$ dex; also see Figure 1), and thus appear to be
systematically high in log(g) given their luminosities. We note that the
log(g) values are from the literature, so may have a systematic offset from
our sample. No seismic log(g) is available for this known-wGb sample as a
cross-check. More data is required to determine the source of this
discrepancy. In contrast, the luminosities of these stars are on the same
scale as our sample.

Combining $\nu_{max}$ with the non-seismic parameters ($T_{\rm eff}$,
$L/L_{\odot}$), we estimated the seismic masses of these 6 stars using mass
Equation 3. The seismic mass range of these known-wGb stars is 1.6
M${}_{\odot}\leq M\leq 2.9~{}\rm{M}_{\odot}$ (with average $1\sigma$
uncertainties on the masses of $0.3~{}\rm{M}_{\odot}$). With the exception of
one star at $\simeq 3~{}\rm{M}_{\odot}$, the rest of the sample is consistent
with being low-mass (M $\leq 2~{}\rm{M}_{\odot}$; Figure 10b), similar to our
Kepler-APOGEE sample. This is in contrast to previous studies which have
reported that these 6 known-wGb stars are intermediate-mass stars ranging from
2.9 $\rm M_{\odot}$ to 3.3 $\rm M_{\odot}$, based on their positions on the
HRD (Adamczak & Lambert 2013; Palacios et al. 2016).

As a check, we also calculated the non-seismic masses (mass Equation 5) for
this sample. We found that these masses were substantially higher. In Figure 9
we compare these two mass estimates. We are unsure as to why there are such
large differences, but it may be related to the log(g)-luminosity tension
noted above. Further, the uncertainties on the non-seismic masses are
generally very large. In fact, 4 out of the 6 stars have non-seismic masses
within 1.5$\sigma$ of the seismically-determined masses (Figure 9). We note
that this discrepancy is not found for our Kepler-APOGEE sample, where the
seismic masses are consistent with the non-seismic masses (Figure 5).

Figure 9: Comparison of the seismic masses of the previously-known-wGb stars
with TESS data determined using Equation (3), versus their masses determined
using the non-seismic mass Equation (5).

In terms of chemical abundances, the 6 known-wGb stars have more extreme
chemical patterns than our sample of stars. Their C is more depleted and N
more enhanced. Sodium, which is scaled solar in the normal-luminosity group
and enhanced by an average of [Na/Fe$]=+0.2$ dex in our over-luminous group,
is further enhanced in the known-wGb sample ([Na/Fe$]=+0.4$ dex). Oxygen, in
contrast with our sample which is approximately scaled solar, is on average
slightly enhanced, with [O/Fe$]\sim+0.1$ dex. We are unsure whether this is
due to their initial composition being $\alpha$-enhanced due to their lower
average [Fe/H], or if it may indicate a further nucleosynthetic product of the
pollution episode. This is explored further in the Discussion.

####  3.6.3 Seismic data and the luminosity bimodality: Masses and Radii

In panels (b) and (c) of Figure 11 we plot seismically-derived parameters,
following the analysis of Li et al. (2022) into underluminous and stripped
stars. We note that in all panels of Figure 11, our over-luminous-RC CDGs are
concentrated in regions far from the normal (background) RC stars. This is
true for panel (a), which is based on photometric data, and also panel (c)
which is entirely based on seismic data. As mentioned above, the fact that the
over-luminous-RC CDGs stand out in diagrams with totally independent data (and
in chemistry; see Figure 10) is a very strong indication that the bimodality
is real.

The reason the over-luminous-RC CDGs stand out in panel (c) is because they
have low $\nu_{max}$ and low $\Delta\nu$ for their masses, given their phase
of evolution. This suggests that they have different structures compared to
normal RC stars in the same mass range. The middle panel of Fig 11 shows that
the majority of these stars also have abnormally large radii, with the 4
largest stars (out of 6) having an average radius of
$R_{avg}=17.0\pm\rm{0.8}~{}\rm{R}_{\odot}$ compared to the background sample
with $R_{avg}=11.5~{}R_{\odot}$ in the mass range of interest. Thus there is a
radius difference of $\simeq 5.5~{}\rm{R}_{\odot}$. Of the other two stars,
one has a high radius for its mass (although not as big a difference as the
others), and the other has a radius consistent with normal RC stars.

The normal-luminosity-RC CDGs (8 stars) mostly have normal RC radii
($R_{avg}=11.7\pm\rm{0.4}~{}R_{\odot}$)777This average ignores the outlier.,
although there is one outlier that has a large radius. Also, one star has
slightly lower radius than expected for the RC. This star (KIC 5000307) is
classed as an underluminous star (partially stripped) by Li et al. (2022).
Although it does stand out in the seismic diagram we find its luminosity to
match that of normal RC stars (Figure 11a).

We note that the over-luminous-RC CDGs in all panels of Figure 11 coincide
with the position of stars in the helium subflashing phase as defined by
Mosser et al. (2014). We discuss this in Section 4.4.1.

Finally, as mentioned above, the normal- and over-luminous groups cover
roughly the same mass range. However we see in Figure 11b that both groups
have higher average masses than the background RC sample. In particular, there
are no RC CDGs below about $1.2~{}\rm M_{\odot}$. Further, the over-luminous
stars appear biased to slightly higher masses ($M_{avg}=1.8~{}\rm M_{\odot}$),
while the normal-luminosity stars have an average mass of $1.5~{}\rm
M_{\odot}$. Being relatively massive is suggestive of a merger-product
population, which we discuss in detail in Sections 4.4.2 and 4.5.2.

Figure 10: Top panel: $\log(L/L_{\odot})$ as a function of asteroseismic
radius, mass, $\Delta\nu$, and $\nu_{max}$. Bottom panel: $\log(L/L_{\odot})$
as a function of the carbon, nitrogen, oxygen, and sodium abundances. Kernel
density histograms are included to show the distributions for the RC CDGs,
with the color scheme following the sample. The horizontal dashed line at
$\log(L/L_{\odot})~{}=1.925$ dex, denotes the cut we use to define two
different luminosity groups of our RC CDG sample (see text for details). The
large filled red triangles and open blue circles show our over-luminous-RC
CDGs and normal-luminosity-RC CDGs, respectively. Large black squares are the
known-wGb stars that have asteroseismic parameters from Hon et al. (2021) and
the small black circles are the rest of the known-wGb stars from the
literature. The abundances of the known-wGb stars are from high-resolution
optical spectra (Adamczak & Lambert, 2013; Palacios et al., 2016). In order to
compare the abundances of the known-wGb with that of our CDG sample, they need
to be on the same scale. We apply the average offset of $+0.02$, $-0.15$,
$-0.12$, $-0.11$, and $-0.03$ dex to the [C/Fe], [N/Fe], [O/Fe], [Na/Fe], and
[Fe/H] values of known-wGb respectively, to account for the systematic
difference between the abundances from IR and optical spectra (Jönsson et al.,
2020). The top and right panels of all the subplots show kernel density
histograms for the RC CDGs (dashed blue lines and large red dots) and known-
wGb stars with asteroseismic parameters (small black dots). We use the solar
abundance of C, N, O, and Na as derived by Grevesse et al. (2007). Figure 11:
(a) The Hertzsprung–Russell diagram. (b) The mass-radius diagram. (c) The
seismic quantity ${\rm\nu_{max}}^{0.75}/\Delta\nu$ versus $\rm\nu_{max}$. The
RC stars classified based on asteroseismic analysis form the background from
Mosser et al. (2014) and Yu et al. (2018) (small filled grey and silver
circles respectively). The helium subflash stars from Mosser et al. (2014) are
represented by small filled green circles. The underluminous stars from Li et
al. (2022) are shown by orange squares, the known-wGb stars by black squares,
the normal-luminosity-RC CDGs, and the over-luminous-RC CDGs from our study by
blue circles and red triangles respectively.

##  4 Discussion

The first dedicated spectroscopic survey for C-deficient stars was undertaken
50 years ago by Bidelman & MacConnell (1973), who found 34 wGb/CDG stars. This
sample has comprised 77% of the wGb stars studied through the decades (e.g.,
Sneden et al. 1978; Cottrell & Norris 1978; Parthasarathy & Rao 1980; Palacios
et al. 2012; Adamczak & Lambert 2013; Palacios et al. 2016). Bond (2019) added
5 more wGb stars identified via spectroscopy. The primary aim of the current
study was to better characterize the CDGs since they are still poorly
understood.

###  4.1 Overview of our results

To better characterize the CDGs we took advantage of the high-quality
asteroseismic data of the Kepler field, which allows mass and evolutionary-
phase determinations, combined with data from large spectroscopic surveys that
overlap with the Kepler field (APOGEE and LAMOST), which provide abundances
and stellar parameters. Here we enumerate our key results, and we discuss the
implications of these findings further below:

  1. 1.

To date, there has been no strict definition for CDGs in literature. We
developed a definition for CDGs based on stellar model predictions and a large
spectroscopic sample. Standard stellar models do not deplete carbon below
$\approx-0.3$ dex (taking into account lower than average initial abundances),
and we consider stars with [C/Fe] $<-0.4$ dex as carbon deficient.

  2. 2.

Our sample adds 15 new CDGs to the literature. All of our CDGs are in the
Kepler field. They have quality abundances, asteroseismic masses and
evolutionary phases.

  3. 3.

Considering our new seismic masses, most of our CDGs are low-mass stars, that
is, stars that are expected to have gone through the core helium flash
($M\lesssim 2~{}\rm{M}_{\odot}$). Taking into account mass uncertainties of
the 2 most massive stars in the sample, 100% of the sample is consistent with
being low-mass ($1~{}\rm M_{\odot}-2~{}\rm M_{\odot}$). This is at odds with
the previous understanding in the literature, where wGb stars were considered
to be of intermediate mass ($M=2.5~{}\rm M_{\odot}-5.0~{}\rm M_{\odot}$),
which do not experience the core helium flash.

  4. 4.

Remarkably, we conclude that 93% of our CDGs are in the RC phase – 14 out of
15 stars – with only one being an RGB star. This is strongly supported by our
subsample of 11 stars for which we have asymptotic g-mode values
$\Delta\Pi_{1}$, $100\%$ of which were found to be RC stars (Figure 2).

  5. 5.

Two out of the 11 stars (18%) for which we have $\Delta\Pi_{1}$ measurements
are in the helium subflashing phase according to Mosser et al. (2014). This is
4.5 times the expected percentage (3.8%) found in our background sample of RC
stars. Both stars are over-luminous-RC CDGs. Moreover, in the seismic diagram
of Figure 11 almost all of the over-luminous-RC CDGs are found in the same
region as the subflashing stars.

  6. 6.

We found that the CDGs are universally N-enhanced. Analysis of CNO sums showed
that the material in the envelopes of our CDG stars has been processed through
the CN cycle, and likely not the ON cycle. The fact that [C+N+O/Fe] is scaled-
solar indicates that no He-burning products have been involved in the
pollution event(s) (although this is not true for the literature sample in
Section 4.2.2).

  7. 7.

We find a strong correlation between stars being C-deficient and Li-rich, with
6 out of 12 CDGs (50%) for which we have low-resolution spectra being Li-rich.
This is 17 times higher than the expected fraction of 3%.

  8. 8.

We find a bimodality (roughly 60:40) in luminosity within our sample, with the
normal-luminosity-RC CDGs (60% of the sample) having an average luminosity of
$\log(L/L_{\odot})\simeq 1.8$ dex ($60~{}\rm{L}_{\odot}$) , and the over-
luminous-RC-CDGs having an average luminosity of $\log(L/L_{\odot})\simeq 2.1$
dex ($120~{}\rm{L}_{\odot}$). Given their low masses, the over-luminous-RC-
CDGs are significantly more luminous than expected, by a factor of about 2.

  9. 9.

On further investigation we found the luminosity groups to also be distinct in
Na abundance, with the normal-luminosity-RC CDGs having scaled solar Na but
the over-luminous-RC CDGs having on average [Na/Fe] $\sim 0.3$ dex. This is a
factor of about 1.5 greater than expected from FDU. The luminosity groups also
strongly correlate with radius and the asteroseismic parameters, with the more
luminous stars having lower $\Delta\nu$ and lower $\nu_{max}$. In addition,
the over-luminous-RC-CDGs have lower carbon and higher N than the normal-
luminosity-RC CDGs, indicating more extreme pollution.

  10. 10.

Comparing our sample with the previously-known wGb stars for which we have
some seismic data (TESS), we found that the latter are also primarily low-mass
stars and are over-luminous for their masses. Thus they appear to be members
of our over-luminous-RC group. They however show signs of having undergone
even more extreme pollution.

###  4.2 Characterisation of carbon-deficient stars

####  4.2.1 Three groups of CDGs

As reported in Section 3.6 (also see points 9 and 10 above), our CDGs appear
to fall into two distinct groups: normal-luminosity-RC and over-luminous-RC
stars. Before we discuss the possible formation theories for CDGs, here we
explore the differences between these groups further and compare to other,
previously-known CDGs in the literature.

In Figure 12 we show [Na/Fe] versus the sum [C+N+O/Fe] for various stellar
samples. We see here that our two luminosity groups are clearly separated in
Na. However, they are not separated in [C+N+O/Fe], with both showing scaled-
solar composition. By contrast, the known-wGb stars for which we have seismic
data (open squares, 6 stars) are enhanced in Na but also in [C+N+O/Fe] (by an
average of $+0.2$ dex). A larger literature sample, which has no seismic data
(filled circles, 23 stars), is also shown in Figure 12. These stars overlap
with our sample – they are also simultaneously Na-rich and CNO-rich. Thus it
appears that the CDGs fall into three groups.

To further investigate these possible groupings, we collate all key
information we have from the current study: abundances, masses, and
luminosities. The groupings do appear to be distinct – we display the
characteristics for each in Table 4, and include the non-seismic literature
sample as Group 3b. In this table, we see a clear progression from lower
luminosities to higher luminosities, along with lower masses to higher masses,
although there is a significant overlap between the groups. There is also a
progression of C-depletion (N-enhancement) through the groups. Oxygen is only
enhanced in Group 3. Lithium is anti-correlated with Na in the first two
groups, with the normal-luminosity-RC CDGs being Li-rich but the over-
luminous-RC CDGs not. At the more massive/luminous end (Group 3), Li-rich
stars appear again.

####  4.2.2 Chemical signatures in the groups

Having [C+N+O/Fe] $>0.0$ is a signature of He-burning products being mixed up
to the envelope, whilst [C+N+O/Fe] $=0.0$ is a signature of CN(O) cycling
only, since in this case the CNO elements are just transmuted within the cycle
(primarily to N). Given a scaled-solar starting composition during a CNO burn,
there is a limit to which N can be enhanced via the CN cycle. As an estimate,
taking the extreme limit where all C is burned to N, we would expect [N/Fe] to
reach its upper limit at $\simeq+0.6$ dex. If the ON cycle were activated,
higher N abundances would be possible. Given that our sample of CDGs (Groups 1
and 2) generally has scaled-solar oxygen (Figure 7; Table 4) and [C+N+O/Fe]
$=0.0$, it appears the ON cycle was not activated in these stars. However, due
to the very high initial abundance of O, it would take very little ON burning
to increase the N. Taking a typical uncertainty on the O abundances as $\simeq
0.1$ dex, a rough calculation shows that a hidden (within uncertainties) 0.1
dex depletion in O would increase [N/Fe] by another 0.2 dex, to [N/Fe] $=0.8$
dex. We indicate this limit in Figure 13 (vertical shaded band, since it is
approximate), which shows the C and N distributions of all the CDG groups. The
limit matches well with the observed upper [N/Fe] limit of our CDGs. Any stars
with [N/Fe] above this limit must have been enhanced in CNO elements at some
stage.

The previously-known CDGs with TESS seismic data (open squares in Figs. 12 &
13; Group 3a in Table 4) have much higher N abundances than our Group 1 and 2
stars, starting at [N/Fe] $\simeq+1.0$ dex and increasing to $\simeq 1.3$ dex.
Thus there appears to be a nitrogen gap between Groups 1+2 and Group 3. Given
Group 3 is above the [N/Fe] limit of +0.8 dex for complete burning of C to N,
these stars must have had CNO elements added to their envelopes, likely
through dredge-up of He-burning products. This aligns with their
overabundances of [C+N+O/Fe] (Figure 12 and Table 4). However, the fact that
the N is so high (and C so low) in the CNO-enhanced group indicates that there
has been very strong hydrogen burning on top of this. This is supported by the
low $\rm{}^{12}C/^{13}C$ and very low [C/N] (see Table 4), both indicating
equilibrium CN(O) cycling. To reach CNO equilibrium typically takes about
$10^{4}$ years (Caughlan, 1965). This puts a time constraint on the minimum
burn time for the event that gave rise to the chemical pattern in Group 3.
Sodium is also enhanced, which is also expected for hot hydrogen burning. This
suggests the burn temperature was high enough to activate the NeNa chain at
around 50-60 MK (Arnould et al., 1999). We note that this chemical pattern is
very similar to that of Hot Bottom Burning (HBB) on top of third dredge-up
(TDU) in AGB stars (e.g., Karakas & Lattanzio 2014; see discussion in Section
4.4.3). Any pollution scenario would need to reproduce HBB conditions, at
least for Group 3 stars. This may be achieved, for example, by stellar
mergers, as discussed below (Section 4.4.2).

Figure 12: Trends of sodium abundance vs. the [C+N+O/Fe] abundance ratio. Giants classified based on asteroseismic analysis form the background from Mosser et al. (2014) and Yu et al. (2018) (small filled circles; see key). The underluminous stars from Li et al. (2022) are shown by orange squares. The abundance ratios of the normal giants and the underluminous stars are from APOGEE DR17. The normal-luminosity-RC CDGs, and the over-luminous-RC CDGs from our study are represented by blue circles and red triangles respectively. Large black squares are the previously-known-wGb stars that have asteroseismic data from Hon et al. (2021) and the small black circles are the rest of the known-wGb stars from literature with carbon abundances from high-resolution spectra. The Li-rich giants (black crosses) are from the Yan et al. (2021) study. These Li-rich giants are the low-spectral-resolution RC sample that have good-quality APOGEE DR17 data. In total, we have a sample of 59 Li-rich RC stars for comparison with the CDGs. Six of our normal-luminosity-RC CDGs are included in the Li-rich study of Yan et al. (2021). We use the solar abundance of C, N, O, and Na as derived by Grevesse et al. (2007). Figure 13: Seismic mass versus [C/Fe] (left panel); seismic mass versus [N/Fe] (right panel) for our CDGs, the previously-known-wGb stars, underluminous stars from Li et al. (2022), Li-rich giants from Yan et al. (2021) and a large sample of RC stars from Mosser et al. (2014) and Yu et al. (2018) with symbols having the same meaning as in Figure 12. We determine luminosities for the Li-rich giants in the same manner as our CDGs, detailed in Section 3.2. Using this, we estimated the seismic masses of these stars using mass Equation 3. Six of our normal-luminosity-RC CDGs are included in the Li-rich study of Yan et al. (2021). They overlap with the Li-rich giants with a slight offset in seismic mass. The ‘normal’ RC stars show a correlation between mass and nitrogen (and carbon) abundance. This reflects first dredge-up surface pollution from early on the RGB (e.g., Iben & Renzini 1984). The trend shows fairly sharp edges, allowing us to make a cut in N (or C) that varies with mass (dashed lines). This helps in distinguishing the chemically peculiar stars. To the right of the N line the stars can be considered N-rich, for their mass. The vertical shaded region at [N/Fe$]=+0.8\pm 0.05$ dex, denotes the upper limit of our CDG sample’s N enhancement as well as the highest N abundance possible if the ON cycle was activated after all C is burned to N (see text for details). Solar abundances are from Grevesse et al. (2007). Table 4: Comparison of our classifications of the CDGs. Approximate averages and ranges are given. See text for definition of each sample/group. | Group 1 | Group 2 | Group 3a | Group 3b
---|---|---|---|---
| (normal- | (over- |  |
| luminosity- | luminous- | (TESS) | (Lit.)
| RC) | RC) |  |
Lum. (L⊙) | $50-80$ | $95-150$ | $120-155$ | $110-760$
Mass (M⊙) | $1.2-2.2$ | $1.4-2.4$ | $1.6-3.0$ | ?
[Na/Fe] | $0.0$ | $+0.3$ | $+0.4$ | $+0.4$
[CNO/Fe] | $0.0$ | $+0.1$ | $+0.2$ | $+0.4$
Li-rich frac. | $90\%$ | $0\%$ | $85\%$ | $40\%$
[C/Fe] | $-0.5$ | $-0.7$ | $-1.3$ | $-1.1$
[N/Fe] | $+0.5$ | $+0.6$ | $+1.1$ | $+1.3$
[O/Fe] | $0.0$ | $+0.1$ | $+0.1$ | $+0.2$
[C/N] | $-1.1$ | $-1.3$ | $-2.4$ | $-2.4$
$\rm{}^{12}C/^{13}C$ | ? | ? | $6$ | $5$
[Fe/H] | $-0.1$ | $-0.3$ | $-0.2$ | $-0.2$

Turning to the Group 1 and 2 stars, they are distinguished by having
[C+N+O/Fe] approximately unchanged from their initial scaled-solar values.
This indicates that, unlike Group 3, the polluting material was not enriched
in He-burning products through core dredge-up. The degree of
N-enhancement/C-deficiency depends on the amount of burning that the material
underwent (along with any dilution that may have occurred if the pollution
from from an external source). Given their high [N/Fe] and low [C/Fe], these
stars show signs of substantially progressed CN(O) burning. We do not have
$\rm{}^{12}C/^{13}C$ for them, but the [C/N] is low (Table 4). However, as
shown by Clayton (1983) we expect [C/N] $\approx-2.5$ dex in equilibrium CN(O)
burning at around $30-50$ MK, which is much lower than the observed $-1.1$ to
$-1.3$ dex in these stars. Thus it appears that equilibrium was not quite
reached during the burning that gave rise to the patterns in these stars.
Since it takes about $10^{4}$ years to reach equilibrium (Caughlan, 1965),
this puts a time constraint on the maximum burn time for the Group 1+2
chemical patterns. The [C/N] ratio is higher in Group 1, which could be due to
a lower-temperature burn. Further, Na is not enhanced in Group 1 while it is
in Group 2. This suggests that the burn temperatures were around $20-40$ MK
(Adamczak & Lambert 2013) in Group 1 but around $50-60$ MK in Group 2. This
may have allowed Li to survive in the Group 1 stars, as opposed to Group 2
(Table 4; also see Li discussion in Section 4.6).

###  4.3 Revisiting previous theories on the origin of C-deficient stars

Before the current study, the general picture of the nature of wGb/CDGs was
that:

  * •

Most have uncertain evolutionary status (SGB, RGB, RC, EAGB).

  * •

They are intermediate-mass stars ($M\approx 2.5~{}\rm M_{\odot}-5.0~{}\rm
M_{\odot}$).

  * •

They may mainly be thick-disk objects given their large distances from the
Galactic plane.

  * •

All have N over-abundances.

  * •

Some are enhanced in Li.

  * •

Most are enhanced in Na.

  * •

All those measured have low $\rm{}^{12}C/^{13}C$ ratios.

For seven decades the precise evolutionary state of the CDGs/wGb stars has
been uncertain. This is due to the fact that they are found in a very crowded
area of the HRD where it is difficult to distinguish between various
evolutionary phases (SGB, RGB, RC, EAGB).

By taking advantage of the Kepler asteroseismic data our study shows
conclusively that almost all (14/15) of our sample of stars are in the RC
(core He-burning) phase. For our previously-known CDG sample with data, for
which we have only $\nu_{max}$, we find they are also consistent with being RC
stars (Figure 10). Considering the wider literature sample of CDGs, we find
that many of them are consistent with RC luminosities but there is also a
subset that has much higher luminosities (black dots in Figure 10).

The fact that there are very few stars below the RC luminosity – we have one
single example out of 44 stars (15 Kepler CDGs + 29 known-CDGs; Figures 1 and
10) – means that the CDG chemical signature could not have arisen before the
RGB bump (since the bump has roughly the same luminosity at the RC). Further,
if almost all CDGs are RC stars (we require more seismic data to determine
this), the pollution event could not have occurred before the RGB tip (except
in a merger scenario, see Section 4.4.2). This important new finding helps
constrain theories of the origin of the CDGs. Theories that suggest that the
chemical pattern was imprinted during the PMS or MS are not supported by this
new evidence. This applies to self-enrichment theories (e.g. self-enrichment
through rapid rotation; Adamczak & Lambert 2013) and also external pollution
theories, for example where the chemical pattern is thought to reflect the
birth composition of the stars, whereby they could have been born polluted or
have been polluted in their infancy by more massive stars that would have
ejected CNO cycle processed material (Palacios et al., 2016). Previous studies
did note that these theories did have their weaknesses, for example, there
were no known MS stars with the same chemical pattern as the CDGs that could
have been progenitors. We note that we also did not find any MS or SGB
C-deficient stars (Figure 1), further suggesting that the pollution event
doesn’t happen early in the evolution.

Another possible theory in the literature is that CDGs are products of mass
transfer or merger events (Bond, 2019). This is based on three pieces of
evidence, (i) the systematically high distances from the Galactic plane as
compared to normal red giants lying in the same location in the CMD, (ii)
their apparent higher masses ($2.0-4.5~{}\rm{M}_{\odot}$), which might imply
they are binaries or binary products, and (iii) a subset having high rotation
rates. In contrast, for our sample (for which we have high-quality seismology)
we find that (i) there is no preference for the Galactic location of these
CDGs (Figure 6; Table 1), and (ii) our CDGs are predominantly low-mass stars
($1.2-2.3~{}\rm{M}_{\odot}$; Figure 5). Due to these findings, it can be seen
that the Bond (2019) theory of CDGs belonging to a special stellar population
may not apply to our sample. Although the masses of our sample are relatively
low, we note that they still appear biased towards higher masses, just not as
strongly. We consider the merger scenario for our sample in light of our new
constraints in Sections 4.4.2 and 4.5.2.

As a caveat we note that in Section 3.6.1 we reported seismic masses for 6
stars from the literature. Five out of 6 of these stars were in the Bond
(2019) study. As mentioned in Section 3.6.1, there is a tension between our
seismic masses, which are generally low, and our non-seismic masses which are
generally higher (Figure 9). This adds some uncertainty around the formation
scenario(s), for this subset of stars.

###  4.4 Possible explanations for the origin of Group 2 and Group 3 CDG stars
based on our new findings

####  4.4.1 Over-luminous-RC CDGs (Group 2): Helium subflashing stars?

At the tip of the RGB, low-mass stars go through the core helium flash. After
this they undergo a series of weaker He-flashes as they descend down to the RC
luminosity (Bildsten et al. 2012; Singh et al. 2021), where they then spend
$\simeq 80-100$ Myr. As mentioned above, two out of the 11 stars (18%) for
which we have $\Delta\Pi_{1}$ measurements are in the helium subflashing phase
according to Mosser et al. (2014). This is 4.5 times the expected percentage
(3.8%) found in our background sample of RC stars. Also, both stars are over-
luminous-RC CDGs (Table 4 and Figure 10), and in the seismic diagram of Figure
11 almost all of the over-luminous-RC CDGs are found in the same region as the
subflashing stars. Thus it is tempting to associate our over-luminous-RC CDGs
with stars currently undergoing helium-burning subflashes.

The time spent in subflashes (with convective He-burning shells) is very short
($\simeq 10^{4}$ years; Bildsten et al. 2012; Singh et al. 2021), so it is
very unlikely to find stars in this phase. If these stars are in the
subflashing phase then they will soon evolve to the RC. Since our over-
luminous-RC CDG stars are C-deficient, this would mean that we would expect to
see large numbers of C-deficient RC stars, which we do not. Given this, it
appears that our over-luminous CDGs cannot be subflashing stars. Their
location in the seismic diagram (Figure 2) is likely degenerate with other
phases of evolution – or different stellar structures, as discussed in the
next subsection.

Another argument against these stars being subflashing stars is that,
theoretically, it is expected that any mixing at the core helium flash would
result in an increase in carbon at the surface (e.g., Deupree & Wallace 1987;
Izzard et al. 2007; Mocák et al. 2009). However it has been suggested that Li-
rich stars may provide evidence for flash-induced mixing without CNO
enrichment (Section 4.6).

####  4.4.2 Over-luminous-RC CDGs (Group 2): Merger products?

Our results show that our over-luminous-RC CDGs (Group 2 in Table 4) stand out
in a number of ways:

  * •

Critically, they are more luminous than expected for low-mass RC stars, by a
factor of about 2 ($\approx 60~{}\rm{L}_{\odot}$ vs $120~{}\rm{L}_{\odot}$;
Figure 11).

  * •

They have higher average masses than our background RC sample. In particular,
there are no stars below $1.35~{}\rm M_{\odot}$ (Figure 11).

  * •

The majority of these stars also have abnormally large seismically-measured
radii (middle panel of Figure 11) with the 4 largest stars (out of 6) having
an average radius of $17.0\pm\rm{0.8}~{}R_{\odot}$ which is $\simeq
5.5~{}\rm{R}_{\odot}$ larger than the background sample of RC stars
($R_{avg}=\rm 11.5~{}R_{\odot}$ in the mass range of interest).

  * •

These stars have more extreme chemical patterns than the normal-luminosity-RC
CDGs (Group 1 in Table 4). In particular, they show Na enrichment
([Na/Fe]${}_{\rm avg}=+0.3$ dex) whereas the normal CDGs have scaled-solar Na.
For C and N our overluminous-RC-CDGs are 0.2 dex more C-deficient and 0.1 dex
more N-rich (Table 4 and Figure 10).

  * •

They also stand out in the seismic diagram of Figure 11, well away from the
normal RC distribution.

The combination of their relatively high masses (compared to normal RC stars),
higher luminosities and radii (for their masses) suggests that these stars may
be merger products. This was proposed by Bond (2019) for their sample of CDGs.
Our seismic information strengthens the fact that these stars are quite
different to normal RC stars. In Figure 11 we have 5 of the Bond (2019) stars
(with seismic parameters). They appear to be equivalent to our over-luminous-
RC CDGs, although their C depletion is even stronger (and N, Na more
enhanced). The Bond (2019) sample being more chemically extreme may be due to
selection bias through how they were discovered. They were classified based on
the weak CH-band in the spectra recorded in objective prism plates (Bidelman
1951; Bidelman & MacConnell 1973).

As discussed in Section 4.2.2, the chemical pattern seen in these stars
reflects previous hydrogen burning through the CN (or CNO) cycle, and the
presence of Na indicates that the NeNa chain was also running. This suggests
the burn temperature was around 50-60 MK (Arnould et al., 1999). Adamczak &
Lambert (2013) also noted that some CDGs have unusually high Li abundances
indicating Li production presumably through the Cameron–Fowler mechanism. On
the other hand, our sample of overluminous-RC CDGs (Group 2 in Table 4) does
not contain any very Li-rich stars (all stars for which we have spectra have
A(Li) < 1.8 dex; Table 3). This is in stark contrast to our normal-luminosity-
RC CDG sample (Group 1) of which $\simeq 86\%$ are very Li-rich (6/7 stars; we
have spectra for 7 out of 8 of this sample; Table 3).

If we accept the merger scenario for these overluminous-RC CDGs and combine it
with our result that virtually all of the CDGs are at the RC luminosity or
brighter, the burning must have happened during or after the merger event.
That is, since the chemical pattern is not seen in earlier phases of
evolution, the most simple explanation is that it arises as a result of the
merger event.

Interestingly, in the low-mass merger models of Zhang & Jeffery (2013), there
is a case in which CN(O) and NeNa nucleosynthesis does indeed occur (their
model B2). Although this model did not match the chemical patterns of the
stars they were studying (early-type R and J stars; carbon-rich), they
suggested that this type of merger could help explain the globular cluster
abundance anomalies, which show a similar chemical pattern to the CDGs. Their
model produces stars with surface abundances [C/Fe]$=-1.15$ dex, [N/Fe]$=0.88$
dex, [O/Fe]$=-0.10$ dex, and [Na/Fe]$=1.42$ dex. Qualitatively this appears to
match the abundances of the over-luminous CDGs. They mention that Li is at
first produced, but quickly destroyed. Their model B2 is a merger between a
helium white dwarf ($0.2~{}\rm M_{\odot}$) and an RGB star with a He core mass
of $0.3~{}\rm M_{\odot}$. This is a common formation channel as shown by
binary population synthesis (e.g., Izzard et al. 2007). In the B2 model case
of Zhang & Jeffery (2013) accretion is from the HeWD onto the RGB helium core.

Figure 14: Schematic representation of the possible steps in a HeWD plus RGB
star merger leading to the formation of a red clump star with a carbon-poor
and nitrogen-rich surface. This illustration is based on Figure 5 in Postnov &
Yungelson (2014) and Figure 2 in Zhang & Jeffery (2013).

The full evolutionary picture is shown in Figure 14. Once a HeWD + RGB binary
has formed, the RGB star eventually expands and overflows its Roche lobe,
leading to a common envelope event. It is assumed that the HeWD merges with
the RG core, which means that spiral-in must occur faster than envelope
ejection. The resulting merged star has a He-rich core of $0.5~{}\rm
M_{\odot}$ with a H-rich envelope. The Zhang & Jeffery (2013) B2 model
accretes enough envelope mass to make a total merged-star mass of $2~{}\rm
M_{\odot}$. Evolving the model forward, it underwent some strong burning at
the base of the convective envelope, which reaches right down to the H-burning
shell (their Figure 8). The study did not report the temperature of the
envelope burning however it can be expected to be in the range 50-70 MK
judging from the nucleosynthesis that occurs. The burning lasts about 3
million years, at which point it undergoes a series of He-shell flashes before
settling into convective-core helium burning (RC), where it remains for
$\simeq 50$ million years. It is this RC phase that we associate with our
observed sample of over-luminous-RC CDG stars – they have higher than normal
luminosities, a similar chemical pattern, and are core-helium burners (RC, as
identified through Kepler asteroseismology, see Section 3.1). Although there
is not an exact quantitative match between our abundances and this particular
merger model, it can be expected that a change in model parameters (e.g., HeWD
mass, RG core mass), will lead to a variation in burning products due to
different thermal structures.

As for the previously-known wGb stars for which we have some seismology (Group
3a in Table 4; open squares in Figures 10 and 11), they appear to be members
of our over-luminous-RC group, with slightly more extreme chemical patterns
(Section 3.6.1), so they also fit this merger scenario.

In Figure 10 we also show an extended literature sample of CDGs (Group 3b in
Table 4), for which we do not have asteroseismic data. These stars show
similar extreme abundance patterns as the known sample for which we have data.
However they show a wide distribution in luminosity, reaching far above the RC
luminosity (and above the overluminous-RC). One interpretation of these much
brighter stars is that they are evolved versions of the RC stars (early-AGB).
However, if this were the case then we would expect to see bright CDGs with
C-depletion similar to the normal-luminosity and overluminous-RC CDGs (i.e.
lying directly above these stars in the left-most plot of the lower panel of
Figure 10), but we do not observe this. Lacking data to confidently assign
evolutionary status, we speculate that the bright extension of CDGs are also
merger-product RC stars, and the increase in luminosity is due to increasing
mass. More data is required to test this hypothesis. We note that these stars
may be over-represented in the literature sample due to their brightness.

We conclude that a merger between a helium white dwarf and a red giant can
explain the overluminous-RC CDGs (Group 2), and possibly the bright extension
seen in the literature sample of wGb stars (Group 3).

####  4.4.3 Previously-known-wGb stars (Group 3): AGB mass transfer pollution?

As mentioned in Section 4.2, AGB stars that are undergoing HBB on top of TDU
can qualitatively reproduce the chemical pattern in the Group 3 stars. In
particular, stars of $\simeq 6~{}\rm{M}_{\odot}$ produce Na and Li (the Li-
rich phase is transient; Karakas & Lattanzio 2014), as well as increasing
[C+N+O/Fe], as observed in these CDG stars (Table 4). However, the luminosity
range of our sample ($\sim 10^{2}-10^{3}$ L⊙) is not compatible with HBB AGB
stars ($\sim 10^{4}$ L⊙). Also, the CDG masses are not as high as
$6~{}\rm{M}_{\odot}$. Therefore, if the material did originate in an AGB star,
it must have been added to the surface of these stars through binary mass
transfer.

A mass-transfer scenario would imply the current Group 3 CDGs should be in
binary systems. We checked for binarity in the literature, consulting multiple
catalogues (Mason et al. 2001; Dommanget & Nys 2002; Kovaleva et al. 2015; El-
Badry et al. 2021) and found 12 giants either in multiple systems or in a
binary system. This is 41$\%$ of the 29 known-CDGs, which is broadly
consistent with the expected binary fraction in this mass range ($\sim 48\%$;
Parker & Meyer 2014). Thus there appears to be no bias towards a high binary
fraction that would expect from this scenario. We note however that this
result may be affected by incomplete data. A dedicated binary survey of these
stars would be needed to be certain of the true binary fraction. If the AGB
binary mass-transfer scenario were correct, we would also expect the primary
(donor) stars to have evolved to the WD by now, so we would expect to see
Group 3 stars with WD companions. We only find three of the 29 Group 3 stars
with evolutionary phase identifications in the literature. All of these are
identified as MS-MS binaries (El-Badry et al., 2021), at odds with the
scenario. More data on binarity is required to properly test the AGB binary
mass-transfer theory.

###  4.5 Possible explanations for the origin of Group 1 CDG stars based on
our new findings

####  4.5.1 Normal-luminosity-RC CDG stars (Group 1): Core-flash mixing?

Our sample of normal-luminosity-RC CDG stars (Group 1 in Table 4) have
luminosities consistent with the background sample of RC stars (hence the
name; Figure 11). They also mostly have normal RC radii although there is one
outlier that has a large radius. They have less extreme chemical patterns,
being less C-deficient and less N-enhanced. A striking difference with the
overluminous sample is that they do not have any Na overabundance – they all
have scaled-solar Na. This suggests the burning that led to the chemical
pattern of these stars was not at such high temperature, with just the CN
cycle operating ($\simeq 20-40$ MK; Adamczak & Lambert 2013). The masses of
the normal-luminosity-RC CDGs are similar to the overluminous sample but
biased to slightly lower masses (Figure 10; Table 4). Despite this, the sample
is also missing the lowest mass stars as compared to the background RC sample,
with no stars of mass less than $1.2~{}\rm{M}_{\odot}$. This could be evidence
of a merger scenario, discussed in Section 4.5.2. With respect to seismology,
they primarily follow the normal RC distribution (see seismic diagram in
Figure 11). Interestingly, most of this group is very Li-rich. Six out of the
7 stars ($\simeq 86\%$) for which we have spectra are Li-rich (Table 3). We
discuss the remarkable number of Li-rich CDGs in Section 4.6.

The fact that CDGs are generally not found at luminosities below the RC is
suggestive of a pollution event near the RGB tip. Given the mass range of our
Group 1 stars (Table 4), they should all have gone through degenerate ignition
of He in the core – the core He flash (CHeF). Although envelope pollution has
been suggested at the CHeF (e.g., Deupree & Wallace 1987; Izzard et al. 2007;
Mocák et al. 2009), it is expected that any mixing at the core helium flash
would result in an increase in carbon at the surface (e.g., Deupree & Wallace
1987; Izzard et al. 2007; Mocák et al. 2009), incompatible with the
C-deficient stars. However a ‘weaker’ pollution event has been suggested at
the CHeF to explain Li-rich giants (Kumar et al. 2011), which are primarily RC
stars too, and also to explain ‘Li-enhanced’ RC stars (Kumar et al. 2020;
Schwab 2020). One might imagine a flash-induced mixing scenario between these
two extremes, whereby CN(O) burning operates, producing N-enhancement and
C-deficiency. As mentioned, the fact that there is no Na enhancement in these
stars means that the hypothesized extra-mixing would not have to reach very
high burning temperatures. As far as we are aware CHeF extra-mixing in this
regime has not yet been investigated. Finally, since there are so few CDGs
($0.15\%$ of our sample; Section 2), this proposed extra-mixing event would
only occur in a very small fraction of stars.

####  4.5.2 Normal-luminosity-RC CDG stars (Group 1): Mergers?

As mentioned above, the normal-luminosity RC CDGs are low mass but show a bias
against the lowest masses relative to the background RC sample (Figure 11).
This is similar to our overluminous sample (Group 2 in Table 4), for which we
concluded the merger scenario to be the most likely. In particular, for the
normal-luminosity sample, there are no stars below $\simeq 1.2~{}\rm
M_{\odot}$, and the average mass of the stars is $1.5~{}\rm M_{\odot}$. This
could be the signature of a merger product population since mergers tend to
produce more massive stars.

Moreover, in the seismic diagram (Figure 11c), although on the whole, this
sample overlaps the background RC sample, it does show significant dispersion.
Half of the sample (4/8 stars) are on the extremes of the background
distribution. Further, two of them clearly fall above the general trend
(having relatively high ${\rm\nu_{max}}^{0.75}/\Delta\nu$ and $\nu_{max}$),
near the position of the overluminous-RC CDGs. This is further suggestive of
the merger scenario, and the dispersion may be related to variation in the
progenitors. We note that these stars do not appear to be stripped stars as
identified by Li et al. (2022) (orange symbols in Figure 11).

We suggest that the fact that these stars are not overluminous does not
necessarily rule out the merger scenario. This is because there may be a
bifurcation in core-mass post merger; lower-mass mergers will undergo the
CHeF, whereas higher-mass mergers may not (depending on the resultant merged
core mass). Alternatively, other merger scenarios or physics (e.g., Zhang &
Jeffery 2013) could produce the variation.

As mentioned, the chemical signature of these normal-luminosity RC CDGs is
relatively limited, compared to the overluminous sample. This could be
consistent with the lower masses of the normal-luminosity stars (on average
$0.3~{}\rm{M}_{\odot}$ lower), such that lower-mass mergers are likely to
result in lower-temperature burning which would give less extreme chemical
patterns.

###  4.6 Link to the Li-rich giants

Considering the rarity of Li-rich giants ($\simeq 3\%$ on the RC; Kumar et al.
2020), there are a remarkable number of Li-rich C-deficient stars. In our
sample for which we have spectra, $50\%$ (6/12 stars) have A(Li) $>1.8$ dex
(1.8 dex is our detection limit; Section 3.5.3). In our overluminous
literature sample with spectroscopic data (black open squares in Figure 11)
about $80\%$ (5/6 stars) are Li-rich (also see Table 4).

We compiled a comprehensive literature sample of previously-known CDGs for
which Li abundances are available (Adamczak & Lambert 2013; Palacios et al.
2016) and find that $\sim 50\%$ of those stars (14/29 stars) are Li-rich
(A(Li) $>1.5$ dex). Clearly there is an extremely strong bias towards
C-deficient giants also being Li-rich giants. This suggests that they may be
related in some way, so formation scenarios for the CDGs may also apply to the
Li-rich giant population (and vice-versa).

As discussed above, we have two main theories for our CDGs: CHeF mixing and
low-mass stellar mergers. For our overluminous-RC CGDs we believe the merger
scenario is the most likely, and for the normal-luminosity-RC CDGs we cannot
differentiate between the CHeF-mixing and merger scenarios.

Li-rich giants are generally (normal-luminosity) RC stars (Kumar et al. 2011;
Figure 9 of Zhang et al. 2020; Singh et al. 2021). That both Li-rich giants
and CDGs are primarily RC stars is another striking parallel between these two
chemically-peculiar populations, only just revealed by the current study.

As we suggested in Section 4.5.1 for our CDGs, it has also been suggested that
CHeF mixing may be responsible for the Li enrichment in Li-rich giants (Kumar
et al. 2011; Kumar et al. 2020; Schwab 2020).

The merger scenario has also been suggested for Li-rich giants by Zhang et al.
(2020) who provide population synthesis models. Yan et al. (2021) further
investigate this merger scenario. They are able to explain most features of
the Li-rich giants, however, they cannot explain the nitrogen enhancements in
many of the stars (see Figure 2 in their Extended Data section). Nitrogen
enhancement implies that the material has experienced CN(O) burning, so we
would expect C to have been depleted in some of the Li-rich giants,
particularly the most N-rich stars. To check this, in Figure 13 we compare the
C and N abundances (and masses) for a sample of Li-rich giants to various
samples of C-deficient giants. Indeed, taking Li-rich giants with strong
N-enhancement ([N/Fe] $>+0.4$ dex), we find $\sim 85\%$ of them to be
C-deficient giants as per our definition given in Section 2.2 ([C/Fe] $<-0.4$
dex). In fact, 6 of our CDGs are included in the Li-rich study of Yan et al.
(2021). This again shows the significant overlap in Li-rich and C-deficient
giants, which can be seen visually in Figure 13.

Li-rich giants have low $\rm{}^{12}C/^{13}C$ ratios, being mostly in the range
5-10 (although in some cases it is close to 30; Kumar et al. 2011). This
compares well with the CDGs for which $\rm{}^{12}C/^{13}C$ has been measured,
which also show low values (Table 4). As mentioned above, we do not have
$\rm{}^{12}C/^{13}C$ for our Group 1 and Group 2 stars but their very low
[C/N] indicates they have also undergone significant burning, albeit not to
equilibrium (Section 4.2.2). Thus it is expected that their
$\rm{}^{12}C/^{13}C$ must be fairly low888Also see the C-poor Li-rich giant in
Silva Aguirre et al. 2014 which has an upper limit for $\rm{}^{12}C/^{13}C$ of
20..

Given the aforementioned similarities/overlap between Li-rich giants and the
CDGs, we speculate that we are seeing a spectrum of merger products, varying
in progenitor mass. In Table 4 we see that Li-rich giants appear in Groups 1
and 3, which we have identified as having distinct chemical enrichments in CNO
and Na. This shows that Li is not the best tracer for identifying chemically
peculiar stars. This is likely due to its easily-produced and easily-destroyed
nature. We note that the majority of Li-rich giants do not show signs of CNO
burning in their envelopes (Figure 13). If we extend the merger hypothesis to
these stars, it may indicate lower-temperature burning and therefore lower-
mass progenitors. Alternatively, their Li may have been formed at the core
helium flash (Kumar et al., 2020; Schwab, 2020).

###  4.7 Globular cluster link

As noted by Zhang & Jeffery (2013), the chemical pattern of their B2 merger
model is similar to the abundance anomaly patterns found in second-generation
globular cluster populations: Low C, high N, and enhanced Na. We also
identified the B2 model as a possible match for many of our CDGs, since this
pattern also qualitatively matches them. However, as with the Zhang & Jeffery
(2013) model, the CDGs do not match the oxygen depletion seen in GCs (Gratton
et al., 2000), which gives rise to the Na-O anti-correlation. As speculated by
Zhang & Jeffery (2013), this may be a metallicity effect – metal-poor CDGs (or
merger models) may deplete O. This is an enticing scenario for GCs since it is
expected that stellar interactions are relatively common in such dense stellar
environments, with mergers being a key binary destruction pathway (see e.g.,
Figure 4 in Ivanova et al. 2005).

##  5 Conclusion

In this study, we addressed the long-standing questions related to the mass
and evolutionary phase of the carbon-deficient giants (CDGs) with the help of
asteroseismology (Kepler and TESS missions), spectroscopy (APOGEE and LAMOST
surveys), and astrometry (Gaia). We list our wide-ranging results in Section
4.1 (also see Table 4). We briefly summarise the most important points here.

We found 15 newly-identified CDGs in the Kepler field. As a fraction of our
Kepler-APOGEE sample, CDGs represent only $0.15\%$ of the stars and hence are
very rare.

For the first time, we unambiguously identify the evolutionary state of CDGs.
Remarkably, we found 93% of our sample to be in the red clump (RC) core He-
burning phase. This puts strong constraints on formation scenarios. The lack
of MS, SGB and RGB carbon-deficient stars rules out theories that posit the
chemical pollution occurred early in CDG evolution, be it through self-
pollution or external pollution scenarios.

In contrast to previous literature, where wGb stars were considered to be of
intermediate mass ($M\approx 2.5~{}\rm M_{\odot}-5.0~{}\rm M_{\odot}$) based
on their position on the HRD, the Kepler-based asteroseismic masses reveal
that our sample of CDGs are primarily low-mass stars ($M\lesssim$ 2 M⊙). We
also determined asteroseismic masses for a small sample of previously-known
wGb stars, finding that their seismic masses are also primarily low. This
finding puts further constraints on any formation scenario.

We find definite demarcations in the chemical patterns of CDGs, which enabled
us to split them into three groups, in increasing order of degree of
pollution. The first, Group 1, is characterized by non-equilibrium CN-cycle
burning pattern, whilst Group 2 is similar but slightly more extreme, but with
an additional enhancement of Na. The third Group has an even more extreme
chemical pattern and is distinct in that its pattern shows dredge-up of He-
burning products has occurred, with strong hydrogen burning on top of this.
The chemical patterns through the three groups also suggest an increasing
temperature of hydrogen burning.

We find a bimodality (roughly 60:40) in luminosity within our sample, with one
group (Group 1) having normal RC luminosity and the other group (Group 2)
being about a factor of two more luminous than expected for their masses
($\approx 60~{}\rm{L}_{\odot}$ vs $120~{}\rm{L}_{\odot}$). We concluded that
stars in the more luminous group are likely merger products, having formed
through the merger of a HeWD and an RGB star. One of the low-mass merger
models of Zhang & Jeffery (2013) is a good qualitative match for these stars.
Previously-known wGb stars for which we have TESS seismology are also over-
luminous, and we suggest they were formed through mergers as well. They do
have more extreme chemical patterns and, importantly, are enriched in CNO
elements, most likely from the dredge-up of He-burning products. In the merger
scenario, this suggests that the progenitor stars would have been more
massive, leading to more extreme mixing and burning (Zhang et al., 2020).

For the normal-luminosity-RC CDGs (Group 1), we cannot distinguish between the
two remaining formation scenarios. The first is a possible pollution event
near the end of the RGB, likely at the core He-flash. This has also been
suggested for the Li-rich giants (Kumar et al. 2011; Kumar et al. 2020; Schwab
2020; Singh et al. 2021). Interestingly, about 90% of our normal-luminosity-RC
CDGs are also Li-rich giants. The second suggested scenario is that these
stars are low-mass merger products like the over-luminous groups, just with
less extreme chemical pollution, which might indicate less massive progenitor
masses.

A significant fraction of our CDGs (50%) are also Li-rich giants. This is 17
times higher than the expected fraction of 3%. Thus there is a strong overlap
between CDGs and Li-rich giants. That said, there is variation between our
three proposed CDG groups – one group is $90\%$ Li-rich, the second has zero
Li-rich members, and the third group is $50\%$ Li-rich ($85\%$ in our small
TESS sample, which is biased to lower luminosities). As is well recognized,
lithium is a special element – it is easily produced but also easily
destroyed. Thus it is not surprising to see this variation. In the merger
models of Zhang & Jeffery (2013) and Zhang et al. (2020) they also find Li to
sometimes be produced and sometimes to be produced then destroyed.

To summarise, based on our new data on carbon-deficient giants, we suggest the
following scenarios for their formation, for the three groups we have defined:

  * •

Group 1 (normal-luminosity-RC CDGs): Core He-flash pollution event, or low-
mass mergers between HeWDs and RGB stars.

  * •

Group 2 (over-luminous-RC CDGs): Higher-mass mergers between HeWDs and RGB
stars.

  * •

Group 3 (previously-known wGb stars, many over-luminous): Mergers between
HeWDs and RGB stars, or binary mass-transfer from intermediate-mass AGB stars.

As can be seen, stellar mergers feature as a possibility across all groups. It
is tempting to see the different groups of CDGs as being produced by mergers
with different progenitor masses. More observations are required to confirm
(or disprove) these proposed scenarios.

## Acknowledgements

This study is supported by the National Natural Science Foundation of China
under grant Nos. 11988101, 11890694, and 11873052 and National Key R&D Program
of China No.2019YFA0405500. S.M. acknowledges the support by CAS-TWAS
Presidents fellowship for International Doctoral Students. S.M. thanks
Raghubar Singh for helpful conversations. We thank Evgenii Neumerzhitckii for
the use of his plotting routine. Funding for LAMOST (www.lamost.org) has been
provided by the Chinese NDRC. LAMOST is operated and managed by the National
Astronomical Observatories, CAS. This work has made use of data from the
European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia). We
made use of the SIMBAD database and the VizieR catalog access tool, CDS,
Strasbourg, France. S.W.C. acknowledges federal funding from the Australian
Research Council through a Future Fellowship (FT160100046) and Discovery
Projects (DP190102431 & DP210101299). Parts of this research was supported by
the Australian Research Council Centre of Excellence for All Sky Astrophysics
in 3 Dimensions (ASTRO 3D), through project number CE170100013. This research
was supported by use of the Nectar Research Cloud, a collaborative Australian
research platform supported by the National Collaborative Research
Infrastructure Strategy (NCRIS). S.W.C. thanks Carolyn Doherty for interesting
and helpful discussions.

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