# Three-body molecules $\bar{D}\bar{D}^{\ast}\Sigma_{c}$\- understanding the
nature of $T_{cc}$, $P_{c}(4312)$, $P_{c}(4440)$ and $P_{c}(4457)$

Ya-Wen Pan  School of Physics, Beihang University, Beijing 102206, China
Tian-Wei Wu  School of Fundamental Physics and Mathematical Sciences, Hangzhou
Institute for Advanced Study, UCAS, Hangzhou, 310024, China  Ming-Zhu Liu
<EMAIL_ADDRESS>School of Space and Environment, Beihang University,
Beijing 102206, China  School of Physics, Beihang University, Beijing 102206,
China  Li-Sheng Geng<EMAIL_ADDRESS>School of Physics, Beihang
University, Beijing 102206, China  Beijing Key Laboratory of Advanced Nuclear
Materials and Physics, Beihang University, Beijing 102206, China  School of
Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001,
China

###### Abstract

The nature of the three pentaquark states, $P_{c}(4312)$, $P_{c}(4440)$ and
$P_{c}(4457)$, discovered by the LHCb Collaboration in 2019, is still under
debate, although the $\bar{D}^{(\ast)}\Sigma_{c}$ molecular interpretation
seems to be the most popular. In this work, by adding a $\bar{D}$ meson into
the $\bar{D}^{\ast}\Sigma_{c}$ pair, we investigate the mass and decay width
of the three-body molecules $\bar{D}\bar{D}^{\ast}\Sigma_{c}$ and explore the
correlation between the existence of the $\bar{D}\bar{D}^{\ast}\Sigma_{c}$
molecules with the existence of $\bar{D}^{(\ast)}\Sigma_{c}$ and
$\bar{D}^{\ast}\bar{D}$ two-body molecules. The latter can be identified with
the doubly charmed tetraquark state $T_{cc}$ recently discovered by the LHCb
Collaboration. Based on the molecular nature of $P_{c}(4312)$, $P_{c}(4440)$,
$P_{c}(4457)$, and $T_{cc}$, our results indicate that there exist two three-
body bound states of $\bar{D}\bar{D}^{\ast}\Sigma_{c}$ with
$I(J^{P})=1(1/2^{+})$ and $I(J^{P})=1(3/2^{+})$, and binding energies $37.24$
MeV and $29.63$ MeV below the $\bar{D}\bar{D}^{\ast}\Sigma_{c}$ mass
threshold. In addition, we find that the mass splitting of these two three-
body molecules are correlated to the mass splitting of $P_{c}(4440)$ and
$P_{c}(4457)$, which offers a non-trivial way to reveal the molecular nature
of these states. The partial widths of two $\bar{D}\bar{D}^{\ast}\Sigma_{c}$
molecules decaying into $J/\psi p\bar{D}$ and $J/\psi p\bar{D}^{\ast}$ are
found to be several MeV. We recommend the experimental searches for the
$\bar{D}\bar{D}^{\ast}\Sigma_{c}$ molecules in the $J/\psi p\bar{D}$ and
$J/\psi p\bar{D}^{\ast}$ invariant mass distributions.

##  I Introduction

In terms of the constituent quark model proposed by Gell-mann Gell-Mann (1964)
and Zweig Zweig (1964a, b), hadrons can be classified either as mesons made of
a pair of quark and anti-quark or baryons made of three quarks, the property
of which can be well described in the conventional quark model Godfrey and
Isgur (1985); Capstick and Isgur (1985). However, more and more the so-called
exotic states beyond the traditional quark model have been discovered
experimentally, starting from $X(3872)$ in 2003 Choi et al. (2003). To clarify
the nature of these exotic states, a lot of theoretical interpretations were
proposed, such as hadronic molecules, compact multiplet quark sates, kinetic
effects, and so on (for recent reviews, see Refs. Chen et al. (2016); Hosaka
et al. (2017); Lebed et al. (2017); Oset et al. (2016); Guo et al. (2018);
Olsen et al. (2018); Ali et al. (2017); Brambilla et al. (2020); Liu et al.
(2019a); Guo et al. (2020) ). Among them, the hadronic molecular picture is
rather popular because many (if not all) of these states are located near the
mass threshold of a pair of conventional hadrons. Nevertheless, how to confirm
the molecular nature of these exotic states remains a big challenge for both
experiments and theory.

To confirm an exotic state as a hadronic molecule, one needs to be able to
describe its production rate, decay width, mass, spin-parity, and other
relevant properties consistently in the molecular picture. However, most
approaches can only describe part of the relevant properties. This motivates
us to find alternative methods to help achieve this goal. In nuclear physics,
the existence of light nuclei, such as triton or ${}^{3}H$ serves as a non-
trivial check on the two-body bound-state nature of the deuteron. Along this
line, assuming $D_{s0}^{*}(2317)$ as a $DK$ bound state, we have studied the
few-body systems of $DDK$ and $DDDK$, the existence of which indeed support
the molecular nature of $D_{s0}^{*}(2317)$ Wu et al. (2019). In this work,
assuming $T_{cc}$ and the three pentaquark states [$P_{c}(4312)$,
$P_{c}(4440)$ and $P_{c}(4457)$] as $DD^{\ast}$ and $\bar{D}^{\ast}\Sigma_{c}$
bound states, respectively, we investigate the related three-body system
$\bar{D}\bar{D}^{\ast}\Sigma_{c}$ and explore the correlation of the three-
body bound states $\bar{D}\bar{D}^{\ast}\Sigma_{c}$ with the related two-body
bound states.

The hidden charm pentaquark states, $P_{c}(4380)$ and $P_{c}(4450)$, were
firstly discovered by the LHCb Collaboration in 2015 Aaij et al. (2015). With
a statistics 10 times larger, the $P_{c}(4450)$ state splits into
$P_{c}(4440)$ and $P_{c}(4457)$, in addition a new state $P_{c}(4312)$
appears, all of which lie close to the mass thresholds of
$\bar{D}^{(\ast)}\Sigma_{c}$ Aaij et al. (2019). In our previous work Liu et
al. (2019b), we have employed a contact-range effective field theory (EFT) to
assign $P_{c}(4312)$, $P_{c}(4440)$ and $P_{c}(4457)$ as
$\bar{D}^{(\ast)}\Sigma_{c}$ hadronic molecules dictated by heavy quark spin
symmetry(HQSS), which was confirmed by many other groups Xiao et al. (2019a,
b); Sakai et al. (2019); Yamaguchi et al. (2020); Liu et al. (2021); Pavon
Valderrama (2019); Meng et al. (2019); Du et al. (2020); Burns and Swanson
(2019); Wu and Chen (2019); Azizi et al. (2021); Phumphan et al. (2021). Even
so, there still exist other explanations, such as, hadro-charmonium Eides et
al. (2020), compact pentaquark states Ali and Parkhomenko (2019); Mutuk
(2019); Wang (2020); Cheng and Liu (2019); Weng et al. (2019); Zhu et al.
(2019); Pimikov et al. (2020); Ruangyoo et al. (2021), virtual states
Fernández-Ramírez et al. (2019) and double triangle singularities Nakamura
(2021). The existence of three-body bound states
$\bar{D}\bar{D}^{\ast}\Sigma_{c}$ will further verify the molecular nature of
the pentaquark states, where $\bar{D}\bar{D}^{\ast}\Sigma_{c}$ system can be
viewed as a cluster of a $\bar{D}$ meson and the $\bar{D}^{\ast}\Sigma_{c}$
pair or a $\bar{D}^{\ast}$ meson and the $\bar{D}\Sigma_{c}$ pair.

In addition, the $\bar{D}\bar{D}^{\ast}\Sigma_{c}$ system can be regarded as a
cluster of a $\Sigma_{c}$ baryon and the $\bar{D}^{\ast}\bar{D}$ pair, which
is related to the doubly charmed tetraquark state ${T}^{+}_{cc}$ discovered by
the LHCb Collaboration Aaij et al. (2021a). The mass of $T^{+}_{cc}$ is below
the mass threshold of $D^{0}D^{\ast+}$ by only several hundred keV, and its
decay width from the unitary analysis is rather small, only a few tens of keV
Aaij et al. (2021b). Assuming that the $T_{cc}$ state is a $DD^{\ast}$ bound
state, its mass and decay width can be described in the hadronic molecular
model Meng et al. (2021); Dong et al. (2021); Chen et al. (2021); Ling et al.
(2022); Ren et al. (2022); Feijoo et al. (2021); Yan and Valderrama (2022);
Albaladejo (2021); Du et al. (2022); Ke et al. (2022). Although the molecular
interpretation seems to be the most popular, the interpretation of a compact
tetraquark state cannot be ruled out. The study of the three-body system
$\bar{D}\bar{D}^{\ast}\Sigma_{c}$ could also be helpful to verify its
molecular nature.

The three-body system $\bar{D}\bar{D}^{(\ast)}\Sigma_{c}$ is particularly
interesting for a number of reasons. First of all, there is no annihilation of
a pair of light quark and its anti-quark. This indicates that such a state, if
exists, has a minimum quark content $\bar{c}\bar{c}cqqqq$, which is explicitly
exotic. Second, the interactions of the sub-systems $\bar{D}\Sigma_{c}$,
$\bar{D}^{\ast}\Sigma_{c}$, and $\bar{D}^{\ast}\bar{D}$ can be precisely
determined by reproducing the masses of their corresponding molecular
candidates using the one-boson-exchange (OBE) potential, which largely reduces
the uncertainty of the so-obtained binding energy of the
$\bar{D}\bar{D}^{\ast}\Sigma_{c}$ state. Such exotic states, if discovered by
experiments, will help verify the molecular nature of $T_{cc}$ as well as
$P_{c}(4312)$, $P_{c}(4440)$ and $P_{c}(4457)$. In this work, we employ the
Gaussian Expansion Method(GEM) to study the three-body system,
$\bar{D}\bar{D}^{\ast}\Sigma_{c}$, and then use an effective Lagrangian
approach to evaluate its main strong decay modes.

This paper is organized as follows. In Sec. II, we briefly explain how to
solve the three-body Schrödinger equation by GEM. Next in Sec. III, we present
the binding energies of the three-body bound states
$\bar{D}\bar{D}^{\ast}\Sigma_{c}$ and calculate the partial widths of the
$\bar{D}\bar{D}^{\ast}\Sigma_{c}$ molecules decaying into $J/\psi
p\bar{D}^{(\ast)}$ and $\bar{T}_{cc}\Lambda_{c}\pi$. Finally, this paper is
ended with a short summary in Sec. IV.

##  II FORMALISM

To obtain the binding energy of the $\bar{D}\bar{D}^{(\ast)}\Sigma_{c}$
system, we need to solve the three-body Schrödinger equation by GEM, which has
been widely applied to investigate few-body systems in nuclear physics Hiyama
et al. (2003) and hadron physics Hiyama et al. (2006); Yoshida et al. (2015).
The three-body Schrödinger equation reads

$[T+V^{1}(r_{1})+V^{2}(r_{2})+V^{3}(r_{3})-E]\Psi_{JM}^{Total}=0,$ (1)

where $T$ is the kinetic-energy operator, $V^{i}(r_{i})$ is the potential
between the $j_{\rm{th}}$ and $k_{\rm{th}}$ particle pair $(i,j,k=1-3)$, and
the $1_{\rm{st}}$, $2_{\rm{nd}}$ and $3_{\rm{rd}}$ particle refer to the
$\bar{D}$ meson, $\bar{D}^{*}$ meson and $\Sigma_{c}$ baryon, respectively.
The total wave function $\Psi_{JM}^{Total}$ is expressed as a sum of three
component functions:

$\Psi_{JM}^{Total}=\sum_{i=1}^{3}C_{i,\alpha}\Phi_{JM,\alpha}^{c=i}(\textbf{r}_{i},\textbf{R}_{i}),$
(2)

where $C_{i,\alpha}$ are the expansion coefficients of relevant basis,
$i=1,2,3$ denotes the three channels of Fig. 1, and
$\alpha\equiv\\{nl,NL,\lambda,\Sigma,s,T,t\\}$. Here $l$ and $L$ are the
orbital angular momentum of the coordinates $r$ and $R$, $t$ and $s$ are the
isospin and spin of the two-body subsystem in each channel, and $\lambda$,
$\Sigma$ and $T$ are the total orbital angular momentum, spin and isospin,
respectively. The wave function of each channel is expressed as

$\Phi_{JM,\alpha}^{c}(\textbf{r}_{i},\textbf{R}_{i})=\left[\Phi_{lL,\lambda}^{c}\Omega_{\Sigma,s}^{c}\right]_{JM}H_{Tt}^{c},$
(3)

where $\Phi_{lL,\lambda}^{c}$ is the spatial wave function, and
$\Omega_{\Sigma,s}^{c}$ is the spin wave function. The total isospin wave
function $H_{Tt}^{c}$ in each channel are written as

$\displaystyle
H_{Tt_{1}}^{c=1}=[[\eta_{\frac{1}{2}}(\bar{D}^{\ast})\eta_{1}(\Sigma_{c})]_{t_{1}}\eta_{\frac{1}{2}}(\bar{D})]_{T},$
(4) $\displaystyle
H_{Tt_{2}}^{c=2}=[[\eta_{\frac{1}{2}}(\bar{D})\eta_{1}(\Sigma_{c})]_{t_{2}}\eta_{\frac{1}{2}}(\bar{D}^{\ast})]_{T},$
$\displaystyle
H_{Tt_{3}}^{c=3}=[[\eta_{\frac{1}{2}}(\bar{D}^{\ast})\eta_{\frac{1}{2}}(\bar{D})]_{t_{3}}\eta_{1}(\Sigma_{c})]_{T},$

where $\eta$ is the isospin wave function of each particle. The spatial wave
function $\Phi_{lL,\lambda}^{c}$ can be expanded as

$\displaystyle\Phi_{lL,\lambda}^{c}=[\phi_{n_{c}l_{c}}^{G}(\textbf{r}_{c})\psi_{N_{c}L_{c}}^{G}(\textbf{R}_{c})]_{\lambda},$
(5)
$\displaystyle\phi_{nlm}^{G}(\textbf{r})=N_{nl}r^{l}e^{-\nu_{n}r^{2}}Y_{lm}(\hat{\textbf{r}}),$
$\displaystyle\psi_{NLM}^{G}(\textbf{R})=N_{NL}R^{L}e^{-\lambda_{N}R^{2}}Y_{LM}(\hat{\textbf{R}}),$

where $N_{nl}(N_{NL})$ is the normalization constant, and the relevant
parameters $\nu_{n}$ and $\lambda_{N}$ are given by

$\displaystyle\nu_{n}=1/r^{2}_{n},\quad
r_{n}=r_{1}a^{n-1},\quad(n=1-n_{max}),$ (6)
$\displaystyle\lambda_{N}=1/R^{2}_{N},\quad
R_{N}=R_{1}A^{N-1},\quad(N=1-N_{max}),$

where $\\{n_{max},r_{min},a~{}\mbox{or}~{}$$r_{max}\\}$ and
$\\{N_{max},R_{min},A~{}\mbox{or}~{}$ $R_{max}\\}$ are Gaussian basis
parameters given in Table 1.

\begin{overpic}[scale={.153}]{DbarDbarstarSigmaC.png} \end{overpic} Figure 1:  Three Jacobi coordinates of the $\bar{D}\bar{D}^{*}\Sigma_{c}$ system  Table 1: Three-body angular-momentum space and the Gaussian range parameters for the $I(J^{P})=1(\frac{1}{2}^{+})$ and $1(\frac{3}{2}^{+})$ configurations of the $\bar{D}\bar{D}^{*}\Sigma_{c}$ system. Lengths are in units of fm. $I(J^{P})$ | $c$ | $l$ | $L$ | $\lambda$ | $s$ | $\Sigma$ | $t$ | $n_{max}$ | $r_{min}$ | $r_{max}$
---|---|---|---|---|---|---|---|---|---|---
$(N_{max})$ | $(R_{min})$ | $(R_{max})$
$1(\frac{1}{2}^{+})$ | 1 | 0 | 0 | 0 | 1/2 | 1/2 | 1/2(3/2) | 10 | 0.1 | 20.0
2 | 0 | 0 | 0 | 1/2 | 1/2 | 1/2(3/2) | 10 | 0.1 | 20.0
3 | 0 | 0 | 0 | 1 | 1/2 | 0(1) | 10 | 0.1 | 20.0
$1(\frac{3}{2}^{+})$ | 1 | 0 | 0 | 0 | 3/2 | 3/2 | 1/2(3/2) | 10 | 0.1 | 20.0
2 | 0 | 0 | 0 | 1/2 | 3/2 | 1/2(3/2) | 10 | 0.1 | 20.0
3 | 0 | 0 | 0 | 1 | 3/2 | 0(1) | 10 | 0.1 | 20.0

##  III Results and Discussions

First we discuss the quantum numbers of the $\bar{D}\bar{D}^{\ast}\Sigma_{c}$
system. Considering only $S$-wave interactions, the total angular momentum of
the $\bar{D}\bar{D}^{\ast}\Sigma_{c}$ system is either $J=1/2$ or $J=3/2$. The
isospin of $\bar{D}\bar{D}^{\ast}$ is either 0 or 1. In the OBE model, the
interaction in isospin 0 is much stronger than that in isospin 1, to such an
extent that $T_{cc}$ can be understood as an isospin 0 $\bar{D}\bar{D}^{\ast}$
bound state. As a result, the total isospin of the
$\bar{D}\bar{D}^{\ast}\Sigma_{c}$ system is taken to be 1. Therefore, in this
work we investigate the two $\bar{D}\bar{D}^{\ast}\Sigma_{c}$ configurations
with $I(J^{P})=1(\frac{1}{2}^{+})$ and $I(J^{P})=1(\frac{3}{2}^{+})$. The
relevant quantum numbers of the $\bar{D}\bar{D}^{\ast}\Sigma_{c}$ system are
given Table 1.

In this work, we employ the OBE model to construct the potentials of
$\bar{D}\Sigma_{c}$, $\bar{D}^{\ast}\Sigma_{c}$, and $\bar{D}^{\ast}\bar{D}$
through the effective Lagrangians describing the interactions between charmed
hadrons and light mesons $\pi$, $\rho$, $\sigma$ and $\omega$. For details, we
refer to Refs. Liu et al. (2019c, 2021). Since the $S$-wave interaction plays
a dominant role in forming hadronic molecules, we only consider the $S$-wave
interaction in this work. To estimate the impact of the finite size of hadrons
on the OBE potentials, we adopt a monopole form factor
$\frac{\Lambda^{2}-m^{2}}{\Lambda^{2}-q^{2}}$ for the relevant meson-baryon
vertices, which introduces an unknown parameter $\Lambda$. To decrease the
uncertainty of the OBE potential induced by the cutoff, we determine it by
reproducing the masses of some well known molecular candidates. Assuming
$P_{c}(4312)$, $P_{c}(4440)$, and $P_{c}(4457)$ as
$\bar{D}^{(\ast)}\Sigma_{c}$ bound states, the corresponding cutoff (denoted
by $\Lambda_{P}$) is fixed to be 1.16 GeV, while the cutoff of the
$\bar{D}\bar{D}^{\ast}$ system (denoted by $\Lambda_{T}$) is fixed to be
$0.998$ GeV if $T_{cc}$ is regarded as a $\bar{D}\bar{D}^{\ast}$ bound state.
Therefore, we take three sets of cutoff values to search for three-body bound
states in the $\bar{D}\bar{D}^{\ast}\Sigma_{c}$ system: Case I:
$\Lambda_{T}$=$\Lambda_{P}$=0.998 GeV; Case II: $\Lambda_{T}$=$0.998$ GeV,
$\Lambda_{P}$=1.16 GeV; and Case III: $\Lambda_{T}$=$\Lambda_{P}$=1.16 GeV. As
the OBE interaction increases with the cutoff, we anticipate that Case III
will yield the largest binding energies while Case I the smallest ones.

Table 2: Binding energies (in units of MeV), expectation values of the Hamiltonian (potential and kinetic energies)(in units of MeV) and root-mean-square radii (in units of fm) of the three-body system $\bar{D}\bar{D}^{*}\Sigma_{c}$ obtained in the three cases detailed in the main text.  $I(J^{P})$ | $B$ | $T$ | $V_{\bar{D}^{*}\Sigma_{c}}$ | $V_{\bar{D}\Sigma_{c}}$ | $V_{\bar{D}\bar{D}^{*}}$ | $r_{\bar{D}^{*}\Sigma_{c}}$ | $r_{\bar{D}\Sigma_{c}}$ | $r_{\bar{D}\bar{D}^{*}}$
---|---|---|---|---|---|---|---|---
Case I $\Lambda_{P}=\Lambda_{T}=0.998$ GeV
$1(\frac{1}{2}^{+})$ | 10.86 | 65.41 | -19.64 | -21.69 | -34.94 | 1.42 | 1.41 | 1.36
$1(\frac{3}{2}^{+})$ | 7.06 | 52.18 | -19.66 | -10.46 | -29.12 | 1.62 | 1.81 | 1.64
Case II $\Lambda_{T}=0.998$ GeV $\Lambda_{P}=1.16$ GeV
$1(\frac{1}{2}^{+})$ | 37.24 | 116.16 | -41.53 | -72.44 | -39.43 | 1.00 | 0.88 | 1.03
$1(\frac{3}{2}^{+})$ | 29.63 | 92.50 | -81.32 | -21.67 | -19.15 | 0.91 | 1.36 | 1.40
Case III $\Lambda_{P}=\Lambda_{T}=1.16$ GeV
$1(\frac{1}{2}^{+})$ | 63.07 | 169.01 | -52.14 | -66.03 | -113.91 | 0.83 | 0.82 | 0.75
$1(\frac{3}{2}^{+})$ | 46.94 | 141.01 | -61.84 | -25.27 | -100.84 | 0.91 | 1.02 | 0.86

In case I, the cutoff of the $\bar{D}^{(\ast)}\Sigma_{c}$ potential is taken
the same as that of the $\bar{D}\bar{D}^{\ast}$ potential. For such
potentials, there exist two three-body bound states
$\bar{D}\bar{D}^{\ast}\Sigma_{c}$ with $I(J^{P})=1(\frac{1}{2}^{+})$ and
$I(J^{P})=1(\frac{3}{2}^{+})$, and binding energies, 10.9 MeV and 7.1 MeV,
respectively. For a cutoff of $\Lambda=0.998$ GeV, the OBE
$\bar{D}^{\ast}\Sigma_{c}$ potential does not support the existence of
$\bar{D}^{\ast}\Sigma_{c}$ bound states corresponding to $P_{c}(4440)$ and
$P_{c}(4457)$. As a result, case I indicates that there exist two three-body
$\bar{D}\bar{D}^{\ast}\Sigma_{c}$ bound states even the
$\bar{D}^{\ast}\Sigma_{c}$ system does not bind as long as $T_{cc}$ is a
$\bar{D}\bar{D}^{\ast}$ bound state.

In case II, we change the cutoff of the $\bar{D}^{(\ast)}\Sigma_{c}$ potential
from 0.998 GeV to 1.16 GeV, while keep the cutoff of the
$\bar{D}^{\ast}D^{\ast}$ potential unchanged. In this case, the strength of
the $\bar{D}^{(\ast)}\Sigma_{c}$ potential becomes stronger, resulting in two
three-body bound states with larger binding energies $37.2$ MeV and $29.6$
MeV. One can see that assuming $T_{cc}$ as a $\bar{D}\bar{D}^{\ast}$ bound
state and $P_{c}(4312)$, $P_{c}(4440)$ and $P_{c}(4457)$ as
$\bar{D}^{(\ast)}\Sigma_{c}$ bound states, we obtain two three-body bound
states below the $\bar{D}\bar{D}^{\ast}\Sigma_{c}$ mass threshold.

In case III, we change the cutoff of the $\bar{D}^{\ast}\bar{D}$ potential
from 0.998 GeV to 1.16 GeV, which naturally results in two bound states with
even larger binding energies as shown in Table 2. In Fig. 2, we present the
binding energies of the $\bar{D}\bar{D}^{\ast}\Sigma_{c}$ system as a function
of $\Lambda$. One can see that the three-body system
$\bar{D}\bar{D}^{\ast}\Sigma_{c}$ remains bound even when both $DD^{\ast}$ and
$\bar{D}^{(\ast)}\Sigma_{c}$ are unbound, with binding energies of the order
of several MeV. If the $\bar{D}\bar{D}^{\ast}\Sigma_{c}$ bound states are
observed experimentally in the future, it will help verify the molecular
nature of $P_{c}(4312)$, $P_{c}(4440)$, $P_{c}(4457)$ and $T_{cc}$ in terms of
the results shown in Fig. 2.

Figure 2: Binding energies of the $\bar{D}\bar{D}^{\ast}\Sigma_{c}$ molecules
as a function of $\Lambda$. The red and blue solid lines represent the
$J^{P}=1/2$ and $J^{P}=3/2$ states, respectively. The green dot lines
represent cutoffs that could reproduce the binding energy of
$T_{cc}$($\Lambda_{T}$) and the pentaquark states($\Lambda_{P}$).  Figure 3:
Mass splitting of the three-body $\bar{D}\bar{D}^{\ast}\Sigma_{c}$ doublet as
a function of the cutoff of the $\bar{D}\bar{D}^{\ast}$ potential for fixed
$\Lambda_{P}=1.16$ GeV.

It is interesting to note that the mass splitting of the three-body
$\bar{D}\bar{D}^{\ast}\Sigma_{c}$ doublet in case III is larger than that of
case II, which implies that the strength of the $\bar{D}\bar{D}^{\ast}$
potential affects the splitting. In Fig. 3, we present the mass splitting as a
function of the cutoff of the $\bar{D}\bar{D}^{\ast}$ potential. It is obvious
that the mass splitting increases with the strength of the
$\bar{D}\bar{D}^{\ast}$ potential. Interestingly, the mass splitting is
positive, which means that the mass of the spin 3/2
$\bar{D}\bar{D}^{\ast}\Sigma_{c}$ bound state is larger than that of its spin
1/2 counterpart. We note that in this case the $J^{P}=\frac{1}{2}^{-}$
$\bar{D}^{\ast}\Sigma_{c}$ system is more bound than the
$J^{P}=\frac{3}{2}^{-}$ $\bar{D}^{\ast}\Sigma_{c}$ system. It indicates that
the mass splitting of the three-body $\bar{D}\bar{D}^{\ast}\Sigma_{c}$ doublet
is oppositely correlated to the mass splitting of the two-body
$\bar{D}^{\ast}\Sigma_{c}$ bound states, which offers a non-trivial way to
check the molecular nature of the involved states. We note in passing that in
Ref. Pan et al. (2020), we found that the mass splitting of $P_{c}(4440)$ and
$P_{c}(4457)$ is correlated to the mass splitting of the
$\Xi_{cc}^{(\ast)}\Sigma_{c}^{(\ast)}$ doublet via heavy antiquark diquark
symmetry.

(a)

(b)

(c)

Figure 4:  Tree level diagrams for the three-body
$\bar{D}\bar{D}^{\ast}\Sigma_{c}$ bound states decaying into $J/\psi
p\bar{D}$(a), $J/\psi p\bar{D}^{\ast}$(b) and $\pi\Lambda_{c}\bar{T}_{cc}$(c).

From our above study we conclude that there exist two three-body molecules
$\bar{D}\bar{D}^{\ast}\Sigma_{c}$ with $I(J^{P})=1(\frac{1}{2}^{+})$ and
$I(J^{P})=1(\frac{3}{2}^{+})$. In the following, we denote $P_{c}(4312)$,
$P_{c}(4440)$, and $P_{c}(4457)$ as $P_{c1}$, $P_{c2}$, and $P_{c3}$, and the
$I(J^{P})=1(\frac{1}{2}^{+})$ and $I(J^{P})=1(\frac{3}{2}^{+})$
$\bar{D}\bar{D}^{\ast}\Sigma_{c}$ bound states as $Hq_{1}$ and $Hq_{2}$,
respectively. We will discuss their possible decay modes and calculate the
decay widths via the effective Lagrangian approach. Such
$\bar{D}\bar{D}^{\ast}\Sigma_{c}$ bound states can be regarded as three kinds
of quasi two-body bound states, $P_{c2}(P_{c3})\bar{D}$,
$P_{c1}\bar{D}^{\ast}$ and $\bar{T}_{cc}\Sigma_{c}$. One should note that the
particles, $P_{c2/c3}$, $P_{c1}$, and $\Sigma_{c}$, should be viewed as
unstable particles in contrast to the particles, $\bar{D}$, $\bar{D}^{\ast}$
and $\bar{T}_{cc}$ Zyla et al. (2020); Aaij et al. (2021b), and these unstable
particles can further decay into two other particles. As a result, the three-
body bound states $\bar{D}\bar{D}^{\ast}\Sigma_{c}$ can decay into $J/\psi
p\bar{D}$, $J/\psi p\bar{D}^{\ast}$, and $\Lambda_{c}\pi\bar{T}_{cc}$, as
shown in Fig. 4. Since the minimum number of valence quarks of the
$\bar{D}\bar{D}^{\ast}\Sigma_{c}$ states is 7, it will not couple to a pair of
traditional hadrons, which indicates that they can only decay into at least
three traditional hadrons. In other words, the decay mechanism shown in Fig. 4
should be the dominant ones.

In the following, we will show how to calculate the partial decay widths of
the $\bar{D}\bar{D}^{\ast}\Sigma_{c}$ bound states in the effective Lagrangian
approach. It should be noted that in the following study, we focus on Case II,
among the three cases studied, because it can reproduce the pentaquark states
and $T_{cc}$. The effective Lagrangians describing the interactions between
the three-body bound states and their constituents have the following form

$\displaystyle\mathcal{L}_{Hq_{1}P_{c3}\bar{D}}=$ $\displaystyle
g_{Hq_{1}P_{c3}\bar{D}}Hq_{1}(x)\int
dy\bar{D}(x+\omega_{P_{c3}}y)P_{c3}(x+\omega_{\bar{D}}y)\Phi(y^{2}),$ (7)
$\displaystyle\mathcal{L}_{Hq_{2}P_{c2}\bar{D}}=$ $\displaystyle
g_{Hq_{2}P_{c2}\bar{D}}Hq_{2}^{\mu}(x)\int
dy\bar{D}(x+\omega_{P_{c2}}y)P_{c2\mu}(x+\omega_{\bar{D}}y)\Phi(y^{2}),$
$\displaystyle\mathcal{L}_{Hq_{1}P_{c1}\bar{D}^{\ast}}=$ $\displaystyle
g_{Hq_{1}P_{c1}\bar{D}^{\ast}}Hq_{1}(x)\int
dy\bar{D}^{\ast\mu}(x+\omega_{P_{c1}}y)\gamma_{\mu}\gamma_{5}P_{c1}(x+\omega_{\bar{D}^{\ast}}y)\Phi(y^{2}),$
$\displaystyle\mathcal{L}_{Hq_{2}P_{c1}\bar{D}^{\ast}}=$ $\displaystyle
g_{Hq_{2}P_{c1}\bar{D}^{\ast}}Hq_{2\mu}(x)\int
dy\bar{D}^{\ast\mu}(x+\omega_{P_{c1}}y)P_{c1}(x+\omega_{\bar{D}^{\ast}}y)\Phi(y^{2}),$
$\displaystyle\mathcal{L}_{Hq_{1}T_{cc}\Sigma_{c}}=$ $\displaystyle
g_{Hq_{1}T_{cc}\Sigma_{c}}Hq_{1}(x)\int
dyT_{cc}^{\mu}(x+\omega_{\Sigma_{c}}y)\gamma_{\mu}\gamma_{5}\Sigma_{c}(x+\omega_{\bar{T}_{cc}}y)\Phi(y^{2}),$
$\displaystyle\mathcal{L}_{Hq_{2}T_{cc}\Sigma_{c}}=$ $\displaystyle
g_{Hq_{2}T_{cc}\Sigma_{c}}Hq_{2\mu}(x)\int
dy\bar{T}_{cc}^{\mu}(x+\omega_{\Sigma_{c}}y)\Sigma_{c}(x+\omega_{\bar{T}_{cc}}y)\Phi(y^{2}),$

where $\Phi(y^{2})$ denotes the Gaussian form factor, $g$ with different
subscripts represent the relevant coupling constants, and
$\omega_{i}=\frac{m_{i}}{m_{i}+m_{j}}$ is the kinematical parameter with
$m_{i}$ and $m_{j}$ being the masses of the involved hadrons. We rely on the
compositeness condition to estimate the above couplings, which is an effective
approach to estimate the couplings between bound states and their constituents
Weinberg (1963). The condition implies that the coupling constants can be
determined from the fact that the renormalization constant of the wave
function of a composite particle should be zero. Following our previous works
Ling et al. (2021, 2022), with the cutoff $\Lambda=1$ GeV and the masses of
the three-body bound states $\bar{D}\bar{D}^{\ast}\Sigma_{c}$ the couplings
are determined as shown in Table 3. The details about deriving the couplings
can be found in Ref. Ling et al. (2021).

The Lagraingian describing the secondary decay process are expressed as

$\displaystyle\mathcal{L}_{P_{c1}J/\psi p}$ $\displaystyle=$ $\displaystyle
g_{P_{c1}J/\psi p}P_{c1}\gamma_{\mu}\gamma_{5}J/\psi^{\mu}p,$ (8)
$\displaystyle\mathcal{L}_{P_{c2}J/\psi p}$ $\displaystyle=$ $\displaystyle
g_{P_{c2}J/\psi p}P_{c2\mu}J/\psi^{\mu}p,$
$\displaystyle\mathcal{L}_{P_{c3}J/\psi p}$ $\displaystyle=$ $\displaystyle
g_{P_{c3}J/\psi p}P_{c3}\gamma_{\mu}\gamma_{5}J/\psi^{\mu}p,$
$\displaystyle\mathcal{L}_{\pi\Lambda_{c}{\Sigma}_{c}}$ $\displaystyle=$
$\displaystyle\frac{g_{\pi\Lambda_{c}{\Sigma}_{c}}}{f_{\pi}}~{}\bar{\Lambda}_{c}\gamma^{\mu}\gamma_{5}\partial_{\mu}\vec{\phi}_{\pi}\cdot\vec{\tau}{\Sigma}_{c},$

where the pion decay constant $f_{\pi}=132$ MeV and the coupling
$g_{\Sigma_{c}\Lambda_{c}\pi}$ are determined as 0.55 by reproducing the decay
width of $\Sigma_{c}\to\Lambda_{c}\pi$ of 1.89 MeV Zyla et al. (2020). Since
the partial decay widths of the pentaquark states into $J/\psi p$ are unknown,
we can not determine them in terms of experimental data. In this work, we
determine the couplings of the pentaquark states to $J/\psi p$ in the contact-
range EFT approach, where the $\eta_{c}p$, $J/\psi p$, and
$\bar{D}^{(\ast)}\Sigma_{c}$ channels dictated by HQSS are taken into account.
By reproducing the masses and widths of the pentaquark states, the relevant
couplings are determined to be $g_{p_{c1}J/\psi p}=0.22$, $g_{p_{c2}J/\psi
p}=0.44$, and $g_{p_{c3}J/\psi p}=0.33$ Xie et al. (2022), consistent with the
chiral unitary approach Xiao et al. (2020).

Table 3: Couplings of the $\bar{D}\bar{D}^{\ast}\Sigma_{c}$ molecules to their components obtained with a cutoff of $\Lambda=$1 GeV.  Couplings | $g_{H_{q1}P_{c2}\bar{D}}$ | $g_{H_{q1}P_{c1}\bar{D}^{\ast}}$ | $g_{H_{q1}T_{cc}\Sigma_{c}}$
---|---|---|---
Value | 3.04 | 1.70 | 2.58
Couplings | $g_{H_{q2}P_{c3}\bar{D}}$ | $g_{H_{q2}P_{c1}\bar{D}^{\ast}}$ | $g_{H_{q2}T_{cc}\Sigma_{c}}$
Value | 1.94 | 2.64 | 4.15

With the above relevant Lagrangians the corresponding amplitudes of the strong
decays of Fig. 4 are

$\displaystyle\mathcal{M}_{a(J=1/2)}=$ $\displaystyle
ig_{Hq_{1}\bar{D}P_{c3}}g_{P_{c3}J/\psi
p}\bar{u}_{Hq_{1}}\frac{1}{{/\\!\\!\\!k}-m_{P_{c3}}}\gamma_{\mu}\gamma_{5}\varepsilon^{\mu}(p_{2})u_{p},$
(9) $\displaystyle\mathcal{M}_{a(J=3/2)}=$ $\displaystyle
ig_{Hq_{2}\bar{D}P_{c2}}g_{P_{c2}J/\psi
p}\bar{u}_{Hq_{2}}^{\mu}\frac{S_{\mu\nu}(k)}{{/\\!\\!\\!k}-m_{P_{c2}}}\varepsilon^{\mu}(p_{2})u_{p},$
$\displaystyle\mathcal{M}_{b(J=1/2)}=$ $\displaystyle
ig_{Hq_{1}\bar{D}^{\ast}P_{c1}}g_{P_{c1}J/\psi
p}\bar{u}_{Hq_{1}}\varepsilon_{\nu}(p_{3})\gamma^{\nu}\gamma^{5}\frac{1}{{/\\!\\!\\!k}-m_{P_{c1}}}\gamma_{\mu}\gamma_{5}\varepsilon^{\mu}(p_{2})u_{p},$
$\displaystyle\mathcal{M}_{b(J=3/2)}=$ $\displaystyle
ig_{Hq_{2}\bar{D}^{\ast}P_{c1}}g_{P_{c1}J/\psi
p}\bar{u}_{Hq_{1}}^{\nu}\varepsilon_{\nu}(p_{3})\frac{1}{{/\\!\\!\\!k}-m_{P_{c1}}}\gamma_{\mu}\gamma_{5}\varepsilon^{\mu}(p_{2})u_{p},$
$\displaystyle\mathcal{M}_{c(J=1/2)}=$ $\displaystyle
ig_{Hq_{1}T_{cc}\Sigma_{c}}\frac{g_{\pi\Lambda_{c}{\Sigma}_{c}}}{f_{\pi}}\bar{u}_{Hq_{1}}\varepsilon_{\nu}(p_{3})\gamma^{\nu}\gamma^{5}\frac{1}{{/\\!\\!\\!k}-m_{\Sigma_{c}}}\gamma_{\mu}\gamma_{5}p_{2}^{\mu}u_{\Lambda_{c}},$
$\displaystyle\mathcal{M}_{c(J=3/2)}=$ $\displaystyle
ig_{Hq_{2}T_{cc}\Sigma_{c}}\frac{g_{\pi\Lambda_{c}{\Sigma}_{c}}}{f_{\pi}}\bar{u}_{Hq_{2}}^{\nu}\varepsilon_{\nu}(p_{3})\frac{1}{{/\\!\\!\\!k}-m_{\Sigma_{c}}}\gamma_{\mu}\gamma_{5}p_{2}^{\mu}u_{\Lambda_{c}},$

where $u$ and $\bar{u}$ represent the corresponding spinor function denoted by
the subscripts, $\varepsilon$ is the polarization vector, and
$S_{\mu\nu}=g^{\mu\nu}-\frac{1}{3}\gamma^{\mu}\gamma^{\nu}-\frac{\gamma^{\mu}p^{\nu}-\gamma^{\nu}p^{\mu}}{3m}-\frac{2p^{\mu}p^{\nu}}{3m^{2}}$
with $m$ being the corresponding mass of an spin-3/2 particle.

With the so-obtained amplitudes , one can easily calculate the partial decay
width

$d\Gamma=\frac{1}{(2\pi)^{3}}\frac{1}{2J+1}\frac{\overline{|\mathcal{M}|^{2}}}{32m_{Hq_{1(2)}}^{3}}dm_{12}^{2}dm_{23}^{2}.$
(10)

In Table 4, we present the partial decay widths of the two three-body bound
states, $Hq_{1}$ and $Hq_{2}$. We find that the bound state with $J=1/2$
dominantly decays into $J/\psi p\bar{D}$, much larger than that of the $J=3/2$
state. Therefore, the $J/\psi p\bar{D}$ mode is a golden channel to
discriminate the spin of $Hq_{1}$ and $Hq_{2}$ molecules. $Hq_{1}$ and
$Hq_{2}$ decay almost equally into $J/\psi p\bar{D}^{\ast}$, while the decay
into $\bar{T}_{cc}\Lambda_{c}\pi$ is rather small in contrast to the other two
decay modes. One should note that the partial decay widths of the three
pentaquark states decaying into $J/\psi p$ are not very precisely known Xiao
et al. (2020), leading to some uncertainties about the partial decay widths
given in Table 4. Nonetheless, we suggest to search for them in the $J/\psi
p\bar{D}^{\ast}$ or $J/\psi p\bar{D}$ mass distributions.

Table 4: Partial decay widths of the $\bar{D}\bar{D}^{\ast}\Sigma_{c}$ molecules.  Modes of Fig. 4(a) | $Hq_{1}\to J/\psi p\bar{D}$ | $Hq_{2}\to J/\psi p\bar{D}$
---|---|---
Value (MeV) | 12.3 | 0.9
Modes of Fig. 4(b) | $Hq_{1}\to J/\psi p\bar{D}^{\ast}$ | $Hq_{2}\to J/\psi p\bar{D}^{\ast}$
Value (MeV) | 3.7 | 3.9
Modes of Fig. 4(c) | $Hq_{1}\to\bar{T}_{cc}\Lambda_{c}\pi$ | $Hq_{2}\to\bar{T}_{cc}\Lambda_{c}\pi$
Value (keV) | 0.2 | 3.3

##  IV Summary and conclusion

The exotic states, $T_{cc}$, $P_{c}(4312)$, $P_{c}(4440)$, and $P_{c}(4457)$,
discovered by the LHCb Collaboration recently, have been suggested to be
$DD^{\ast}$ and $\bar{D}^{(\ast)}\Sigma_{c}$ hadronic molecules. However,
their molecular nature is difficult to be confirmed either experimentally or
theoretically. In this work, we have investigated these exotic states in the
three-body $\bar{D}\bar{D}^{\ast}\Sigma_{c}$ system, which is equivalent to
adding a $\bar{D}$ meson into the $\bar{D}^{\ast}\Sigma_{c}$ system. The OBE
interactions of the sub-systems, $\bar{D}^{(\ast)}\Sigma_{c}$ and
$\bar{D}\bar{D}^{\ast}$, are determined by reproducing the masses of the
molecular candidates, i.e., the three pentaquatk states [$P_{c}(4312)$,
$P_{c}(4440)$ and $P_{c}(4457)$] and $\bar{T}_{cc}$. After solving the three-
body schrödinger equation, we obtained two three-body bound states,
$I(J^{P})=1(\frac{1}{2}^{+})$ $\bar{D}\bar{D}^{\ast}\Sigma_{c}$ and
$I(J^{P})=1(\frac{3}{2}^{+})$ $\bar{D}\bar{D}^{\ast}\Sigma_{c}$, with binding
energies 37.2 MeV and 29.6 MeV, respectively. In particular, we explored the
correlation between the existence of $\bar{D}\bar{D}^{\ast}\Sigma_{c}$
molecules with the existence of $\bar{D}^{(\ast)}\Sigma_{c}$ and
$\bar{D}^{\ast}\bar{D}$ molecules. If the $\bar{D}\bar{D}^{\ast}\Sigma_{c}$
bound states can be observed experimentally in the future, the correlation can
help to test the molecular nature of $P_{c}(4312)$, $P_{c}(4440)$,
$P_{c}(4457)$ and $\bar{T}_{cc}$. The mass splitting of the three-body doublet
is found to be correlated to that of the $\bar{D}^{\ast}\Sigma_{c}$ doublet.
Assuming $P_{c}(4457)$ and $P_{c}(4440)$ as $J=1/2$ and $J=3/2$
$\bar{D}^{\ast}\Sigma_{c}$ bound states, respectively, we find that the mass
splitting between $I(J^{P})=1(\frac{1}{2}^{+})$ and
$I(J^{P})=1(\frac{3}{2}^{+})$ $\bar{D}\bar{D}^{\ast}\Sigma_{c}$ bound states
is positive.

At last, we employed the effective Lagrangian approach to calculate the
partial decay widths of $\bar{D}\bar{D}^{\ast}\Sigma_{c}$ bound states. We
find that $J=1/2$ and $J=3/2$ $\bar{D}\bar{D}^{\ast}\Sigma_{c}$ bound states
mainly decay into $J/\psi p\bar{D}$ and $J/\psi p\bar{D}^{\ast}$,
respectively, while the decay into $\bar{T}_{cc}\Lambda_{c}\pi$ is small. We
strongly recommend experimental searches for such three-body bound states in
the $J/\psi p\bar{D}$ and $J/\psi p\bar{D}^{\ast}$ mass distributions, which
can help verify the molecular nature of $T_{cc}$, $P_{c}(4312)$,
$P_{c}(4440)$, and $P_{c}(4457)$.

##  V Acknowledgments

This work is supported in part by the National Natural Science Foundation of
China under Grants No.11975041, No.11735003, and No.11961141004. Ming-Zhu Liu
acknowledges support from the National Natural Science Foundation of China
under Grant No.1210050997. Tian-Wei Wu acknowledges support from the National
Natural Science Foundation of China under Grant No.12147152.

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# Rapid expansion of red giant stars during core helium flash by waves
propagation to the envelope and implications to exoplanets

Ealeal Bear  Department of Physics, Technion – Israel Institute of Technology,
Haifa 3200003, Israel<EMAIL_ADDRESS><EMAIL_ADDRESS>Ariel
Merlov  Department of Physics, Technion – Israel Institute of Technology,
Haifa 3200003, Israel<EMAIL_ADDRESS><EMAIL_ADDRESS>Yarden Arad  Department of Physics, Technion – Israel Institute of Technology,
Haifa 3200003, Israel<EMAIL_ADDRESS><EMAIL_ADDRESS>Noam
Soker Department of Physics, Technion – Israel Institute of Technology, Haifa
3200003, Israel<EMAIL_ADDRESS><EMAIL_ADDRESS>Guangdong
Technion Israel Institute of Technology, Guangdong Province, Shantou 515069,
China

###### Abstract

We assume that the strong convection during core helium flash of low mass red
giant branch (RBG) stars excite waves that propagate to the envelope, and find
that the energy that these waves deposit in the envelope causes envelope
expansion and brightening. We base our assumption and the estimate of the
waves’ energy on studies that explored such a process due to the vigorous core
convection of massive stars just before they experience a core collapse
supernova explosion. Using the stellar evolutionary code mesa we find that the
waves’ energy causes an expansion within few years by tens to hundreds solar
radii. Despite the large brightening, we expect the increase in radius and
luminosity to substantially enhance mass loss rate and dust formation. The
dust shifts the star to become much redder (to the infrared), and the star
might actually become fainter in the visible. The overall appearance is of a
faint red transient event that lasts for months to few years. We suggest that
in some cases envelope expansion might lead stars that are about to leave the
RGB to engulf exoplanets. The extended envelope has a smaller binding energy
to a degree that allows planets of several Jupiter masses or more and brown
dwarfs to survive the common envelope evolution. We suggest this scenario to
account for the planet orbiting the white dwarf (WD) WD 1856+534 (TIC
267574918) and for the WD - brown dwarf binary system ZTFJ003855.0+203025.5.

planet-star interactions – binaries: close – white dwarfs – binaries: close –
planets and satellites: individual: WD 1856+534 b

##  1 Introduction

The presence of exoplanets in close orbits around evolved stars, i.e.,
horizontal branch stars or low mass white dwarfs (WDs) that are descendant of
red giant branch (RGB) stars and WDs that are descendant of asymptotic giant
branch (AGB) stars, is not easy to explain. By close orbits we refer to orbits
with semi-major axes of $a\ll R_{\rm G}$ where $R_{\rm G}$ is the maximum
radius that the giant progenitor has attained. Such close orbits imply that
either the planet survived a common envelope evolution (CEE) or that it
acquired its close orbit by dynamical interaction with a third body in the
system.

An example is the recently reported system WD 1856+534 (TIC 267574918) where a
planet orbits a WD with $a\simeq 0.02{~{}\rm AU}$ and an orbital period of
$P_{\rm orb}=1.4{~{}\rm days}$ (Vanderburg et al., 2020). Vanderburg et al.
(2020) claim that the large orbital separation makes the dynamical origin more
likely that the CEE origin for this system. There are other claims for
exoplanets orbiting WDs (e.g., Gänsicke et al. 2019; Manser et al. 2019).
Jones & Jenkins (2014) refuted the claim of Setiawan et al. (2010) for an
exoplanet orbiting a low metalicity horizontal branch star with an orbital
period of $P_{\rm orb}=16.2{~{}\rm days}$ (for some other similar refuted
claims see, e.g., Krzesinski et al. 2020). Nonetheless, the refuted claim of
Setiawan et al. (2010) for an exoplanet orbiting a metal poor horizontal
branch star that maintained a relatively massive envelope of $\simeq
0.3M_{\odot}$ and at a relatively large orbital separation of $a\simeq
25R_{\odot}$, promoted Bear et al. (2011) to speculate on a scenario where a
metal-poor RGB star suffered a large expansion following its core helium flash
and engulfed an exoplanet (section 2).

Systematic studies earlier than 2011 have already established the notion that
planets can influence the evolution on the RGB and beyond (e.g., Soker 1998;
Nelemans & Tauris 1998; Siess & Livio 1999a; Carney et al. 2003; Denissenkov &
Herwig 2004; Nordhaus & Blackman 2006; Massarotti 2008; Schröder & Smith 2008;
Carlberg et al. 2009; Villaver & Livio 2009; Nordhaus et al. 2010; for later
studies on RGB stars engulfing planets see, e.g., Kunitomo et al. 2011;
Mustill & Villaver 2012; Nordhaus & Spiegel 2013; Villaver et al. 2014;
Aguilera-Gómez et al. 2016; Carlberg et al. 2016; Geier et al. 2016; Guo et
al. 2016; Privitera et al. 2016; Rao et al. 2018; Schaffenroth et al. 2019;
Hegazi et al. 2020; Jimenez et al. 2020; Kramer et al. 2020). However, without
lowering the envelope binding energy just at the termination of the RGB, the
planet that the RGB star engulfs spirals to very small orbits and either the
RGB core tidally destroys the planet or the final orbit is very small and the
star maintains only a very light envelope.

Regarding the planet-WD system WD 1856+534, one group of models for its
formation considers the scattering of the planet to a close orbit around the
WD by other planet(s) (Maldonado et al., 2021), or by a another star i.e., the
Lidov-Kozai effect (e.g., Muñoz & Petrovich 2020; O’Connor et al. 2021;
Stephan et al. 2020; Vanderburg et al. 2020). Another group of models
considers the CEE. Lagos et al. (2021) study a CEE that takes place during the
AGB phase of the WD progenitor. Chamandy et al. (2021) take the extra energy
source that allows a planet to survive to come from the orbital energy that an
inner planet releases as it enters the giant envelope first and causes its
expansion (e.g., Siess & Livio 1999a, b; Staff et al. 2016). Inner planets can
indeed help outer planets to survive the evolution of their parent star (e.g.,
Bear et al. 2011; Lagos et al. 2021).

We describe the basic scenario and its assumptions in section 2. We then
present our numerical scheme (section 3) and the possible results of waves
that the core helium flash excite in the core and propagate to the envelope
(sections 4). We summarize our results and discuss implications to exoplanets
orbiting horizontal branch stars and WDs in section 5.

##  2 The proposed scenario

Bear et al. (2011) considered the energy source that causes envelope expansion
during the core helium flash to be the ignition of hydrogen at the base of the
hydrogen-rich envelope in metal poor stars. They based their speculative
scenario on the results of Mocák et al. (2010) who calculated hydrogen
ignition by the core helium flash. In the calculations of Mocák et al. (2010)
the hydrogen burning provides $\approx 1\times 10^{48}{~{}\rm erg}$ during the
first year, i.e., an average luminosity of $L_{\rm H}\approx 10^{7}L_{\odot}$
(their Figure 1). After a year this luminosity decreases to $L_{\rm H}\approx
10^{6}L_{\odot}$, still much larger than the RGB luminosity. The huge energy
production by the core convection and by the hydrogen burning decay in a time
scale of $\approx 10-100{~{}\rm yr}$ (e.g., Mocák et al. 2010). Bear et al.
(2011) justified the mixing of hot core material with the hydrogen-rich
envelope by the metal-poor star they studied and/or by rapid core rotation due
to an inner planet that spun-up the core.

Bear et al. (2011) manually added an energy of $E_{\rm in}=8.5\times
10^{46}{~{}\rm erg}$ just above the hydrogen-burning shell of their stellar
model. This amounts to $7\%$ of the energy that the hydrogen burning releases
in the model of Mocák et al. (2010). Bear et al. (2011) injected the energy in
a time period of 7 years at an average power of $L_{\rm in}=10^{5}L_{\odot}$.
Bear et al. (2011) found in their calculation that the outer radius of the
convective zone in the envelope increases by a factor of about 4. After about
$100{~{}\rm yr}$ the star shrinks back to its original radius.

We raise here the possibility for another energy source. Based on the
calculations of Quataert & Shiode (2012) and Shiode & Quataert (2014) for pre-
supernova massive stars, we consider the possibility that the vigorous
convection during the core helium flash excites waves that propagate into the
envelope. Quataert & Shiode (2012) and Shiode & Quataert (2014) proposed a
process by which a fraction of the energy in the gravity waves that the core
convection excites in pre-supernova massive stars converts into sound waves as
the gravity waves propagate into the envelope. The sound waves dissipate in
the envelope. This energy deposition leads to envelope expansion (e.g., Mcley
& Soker 2014; Fuller 2017). Mcley & Soker (2014) study one case and find the
pre-supernova model they use, of an initial mass of $15M_{\odot}$, to expand
by a factor of two, from about $1000R_{\odot}$ to about $2000R_{\odot}$. In
that respect we note that Mocák et al. (2010) find that the convection during
the core helium flash does excite gravity waves in the core. They also find
that convection carries most of the energy that the nuclear reactions
liberate.

There are earlier studies that consider other roles that the internal gravity
waves that the core helium flash excites play in stellar evolution. Schwab
(2020), for example, consider the extra mixing that these waves induce near
the core-envelope boundary during the helium flash. Miller Bertolami et al.
(2020), as another recent example, study how the waves can cause periodic
photometric variabilities in hot subdwarf stars. In this study we consider a
different role that these propagating waves might play.

We do not calculate the propagation of waves from the core to the envelope, as
this requires a separate study. We simply take the same wave luminosity
(power) as Shiode & Quataert (2014) take. This wave energy is (Lecoanet &
Quataert, 2013),

$\displaystyle\begin{aligned} L_{\rm wave,0}&\approx\mathcal{M}^{5/8}L_{\rm
conv}=2.7\times 10^{6}\\\
&\times\left(\frac{\mathcal{M}}{0.001}\right)^{5/8}\left(\frac{L_{\rm
conv}}{2\times 10^{8}L_{\odot}}\right)L_{\odot},\end{aligned}$ (1)

where

$L_{\rm conv}(r)=4\pi r^{2}v^{3}_{\rm conv}(r)\rho(r)$ (2)

is the luminosity that the convection carry at radius $r$ and
$\mathcal{M}=v_{\rm conv}/c_{\rm s}$ is the Mach number of the convective
motion of velocity $v_{\rm conv}$. In what follows we take the maximum value
of $L_{\rm wave,0}$ at each time. For scaling we use typical values from Mocák
et al. (2009) and Mocák et al. (2009), in addition to our simulations that we
describe later.

In what follows we will consider much lower wave energies than what equation
(1) gives, $L_{\rm w}\ll L_{\rm wave,0}$.

##  3 Numerical setting

We use mesa-single version 10398 (Paxton et al., 2011, 2013, 2015, 2018,
2019). We follow the example of $1M~{}pre~{}ms~{}to~{}wd$ and only change the
mass and the stopping condition. The mass was changed to either
$M=1.6M_{\odot}$ that we focus our study on, or to $M=1.2M_{\odot}$ or
$M=2M_{\odot}$. We set the termination condition of this phase to the time
when carbon nucleosynthesis starts ($star~{}C~{}mass~{}max~{}limit=0.02$)
which signifies that the He flash has already occurred.

After determining the maximum amount of energy and the duration of the wave
that we assume convection excited (using the profile files produced by mesa,
we apply equation 1). We manually insert a fraction of this maximum energy in
the src folder in the run-star-extra.f in the subroutine: subroutine energy-
routine file, when we set the pointer of $other~{}energy$ to true.

We insert the wave luminosity $L_{\rm W}$ during four years in the outer zone
of the envelope, either outer $0.2M_{\rm env}$, or $0.5M_{\rm env}$, or
$0.8M_{\rm env}$. For each of these three cases we find the appropriate mass
coordinate, m($0.2M_{\rm env})=0.8619M_{\odot}$, $m(0.5M_{\rm
env})=0.6546M_{\odot}$ and $m(0.8M_{\rm env})=0.4473M_{\odot}$ and insert the
energy above that mass coordinate at a constant power per unit mass. We
examine four cases of wave power as we describe in section 4.

##  4 Core helium flash

We follow the evolution of stellar models with initial masses of $M_{\rm
ZAMS}=1.2M_{\odot}$, $1.6M_{\odot}$, and $2M_{\odot}$, but focus on the
$M_{\rm ZAMS}=1.6M_{\odot}$ model. We follow each stellar model evolution in
more details in the time period around its core helium flash. We define the
time scale $t_{w}$ that we set to zero at the peak of the wave luminosity (see
below). At the beginning of the core helium flash (just before the flash) of
the $M_{\rm ZAMS}=1.6M_{\odot}$ model (now an RGB star) its luminosity,
radius, core mass, and envelope mass are, $L_{\rm b}=2027.74L_{\odot}$,
$R_{\rm b}=128.4R_{\odot}$, $M_{\rm core,b}=0.45M_{\odot}$, and $M_{\rm
env,b}=1.01M_{\odot}$, respectively. We use the subscript ‘b’ to indicate
values just before the core helium flash.

###  4.1 The energy in the waves

The relevant properties of the core helium flash in our proposed scenario are
the convective luminosity and the Mach number of the convective velocity
(equation 1). In Fig. 1 we present the convective luminosity $L_{\rm conv}(r)$
according to equation (2) (upper panel) the mach number of the convection
${\mathcal{M}}(r)=v_{\rm conv}/c_{\rm s}$ (lower panel) as function of radius
in the core and at several times as indicated for a stellar model with an
initial mass of $M_{\rm ZAMS}=1.6M_{\odot}$.

Figure 1: The convective luminosity (upper panel) and convective Mach number
(middle panel) in the core of a stellar model with an initial mass of $M_{\rm
ZAMS}=1.6M_{\odot}$ versus radius, and at several times around the peak of the
core helium flash. The times in the insets are $t_{\rm W}$ that is measured
relative to the peak of the wave energy. Solid lines are for pre-peak and
dotted line for post-peak values. The line get thinner away from the peak,
both before and after the peak.

In Fig. 1 we see the well known properties of the core helium flash that it
has a distinct peak in time that last for several months and that the nuclear
burning occurs off-center due to neutrino cooling in the center. In Fig. 2 we
plot the variation of $L_{\rm wave,0}(t)$ as function of time for the time
period when $L_{\rm wave,0}>10^{4}L_{\odot}$. For comparison we also plot
$L_{\rm wave,0}(t)$ for the two other stellar models that we simulate here.

Figure 2: The wave luminosity according to equation (1) as function of time
$t_{\rm W}$ measure from the peaks of eh wave energy for the three models of
$M_{\rm ZAMS}=1.2M_{\odot}$, $M_{\rm ZAMS}=1.6M_{\odot}$, and $M_{\rm
ZAMS}=2M_{\odot}$. We show only the time period when $L_{\rm
wave,0}>10^{4}L_{\odot}$. The times of core helium flash for the three models
are $t_{\rm He}=6.25\times 10^{9}{~{}\rm yr}$, $t_{\rm He}=2.39\times
10^{9}{~{}\rm yr}$, and $t_{\rm He}=1.08\times 10^{9}{~{}\rm yr}$ for $M_{\rm
ZAMS}=1.2M_{\odot}$, $M_{\rm ZAMS}=1.6M_{\odot}$, and $M_{\rm
ZAMS}=2M_{\odot}$, respectively. The masses of the cores at these times are
$M_{\rm core,b}=0.415M_{\odot}$, $M_{\rm core,b}=0.45M_{\odot}$, and $M_{\rm
core,b}=0.43M_{\odot}$, respectively.

Integrating over the wave power during the time period $\Delta t_{4}$ when
$L_{\rm wave,0}>10^{4}L_{\odot}$ we find the total wave energy for the $M_{\rm
ZAMS}=1.6M_{\odot}$ stellar model to be

$E_{\rm wave,0}=\int_{\Delta t_{4}}L_{\rm wave,0}dt=2.1\times 10^{47}{~{}\rm
erg}=1.7\times 10^{6}L_{\odot}{~{}\rm yr}.$ (3)

There is a large uncertainty in the variation of the power that the jets
deposit to the envelope because the propagation time of the waves through the
envelope is about the dynamical time, which is several months. For the present
model $\Delta t_{4}\simeq 4{~{}\rm yr}$. For that we consider the average
luminosity over four years

$L_{\rm W,0}\equiv\frac{E_{\rm wave,0}}{4{~{}\rm yr}}=4.3\times
10^{5}L_{\odot}$ (4)

For the cases of $M_{\rm ZAMS}=1.2M_{\odot}$ and $M_{\rm ZAMS}=2M_{\odot}$ the
total energy in the waves are about 20% large and 20% smaller than the value
we give in equation (3), respectively. Since the envelope mass of the $M_{\rm
ZAMS}=1.2M_{\odot}$ model is lower, it will suffer a much larger envelope
expansion when waves dissipate their energy in the envelope with respect to
the case we study here for $M_{\rm ZAMS}=1.6M_{\odot}$.

###  4.2 Wave energy dissipation in the envelope

We do not know where exactly in the envelope the wave will deposit their
energy. Quataert & Shiode (2012) and Shiode & Quataert (2014) proposed that
the location of wave-energy deposition is in the outer regions where the
maximum convective flux the envelope can carry, $L_{\rm max,conv}$ becomes
below that of the wave power

$L_{\rm max,conv}=4\pi r^{2}c^{3}_{\rm s}\rho<L_{\rm wave}.$ (5)

In Fig. 3 we plot the variation of $L_{\rm max,conv}$ in the envelope and mark
the values of $L_{\rm W,0}$ and of $L_{\rm W}=2\times 10^{4}L_{\odot}$ by
horizontal lines. We plot the later value as we assume that the actual energy
of the waves is much lower than $L_{\rm W,0}$. According to equation (5) and
as we see in teh Fig. 3 wave dissipation occurs in the outer parts of the
envelope. By mass coordinate wave dissipation occurs in the zone $M>M_{\rm
d}(L_{\rm W,0})=1.378M_{\odot}$ for a wave power of $L_{\rm W,0}=4.3\times
10^{5}L_{\odot}$, and in the zone $M>M_{\rm d}(L_{\rm W})=1.445M_{\odot}$ for
a wave power of $L_{\rm W,0}=2\times 10^{4}L_{\odot}$. The two zones
correspond to envelope mass fractions of $7.5\%$ and $0.9\%$, respectively. We
will deposit the wave energy into much larger envelope zone, the outer
$20\%-80\%$ of the envelope mass, to take into account the pressure that the
wave exert as they propagate through the envelope before the radius of
dissipation (Mcley & Soker, 2014).

Figure 3: The maximum wave luminosity $L_{\rm max,conv}$ that convection can
carry in the envelope according to equation (5) as function of radius for the
$M_{\rm ZAMS}=1.6M_{\odot}$ stellar model (red line). The blue line represents
a wave energy of $L_{\rm W,0}$ according to equation (4) and the cyan line
represents a wave energy of $L_{\rm W}=2\times 10^{4}L_{\odot}$. Strong wave
energy dissipation takes place in the outer zone where the wave energy is
larger than $L_{\rm max,conv}$. We indicate the mass coordinates where wave
energy equals $L_{\rm max,conv}$ and the total stellar mass.

Because of the above uncertainty in the distribution of energy deposition, in
section 4.3 we present results for three simple prescriptions, where in each
we spread the wave luminosity over the outer region of the envelope with a
constant power per unit mass. We spread the energy in either the outer
$\xi=80\%$, $\xi=50\%$, or $\xi=20\%$ of the envelope mass.

###  4.3 Outcomes of wave-energy deposition

In section 4.1 we describe the possible properties of the waves that the
convection excite during the core helium flash and their possible propagation
and energy deposition to the envelope. Following the discussion there, we here
present the results of energy deposition with a power of $L_{\rm W}\ll L_{\rm
W,0}$ during a time period of $\Delta t_{\rm dep}=4{~{}\rm yr}$ for the model
of $M_{\rm ZAMS}=1.6M_{\odot}$. We deposit the energy to the envelope outer
$\xi M_{\rm env,b}$ mass, with $\xi=80\%$, $\xi=50\%$, or $\xi=20\%$, and with
a constant power per unit mass. We examine the response of the envelope for
four different values of $L_{\rm W}$. Overall we have 12 energy deposition
prescriptions.

In Fig. 4 we present the evolution of the luminosity and the stellar radius
during and after the energy deposition for the 12 energy deposition
prescriptions. In Fig. 5 we present the evolution of the effective temperature
for the cases with $L_{\rm W}=2\times 10^{4}L_{\odot}$. We see that the star
relaxes towards its previous state on a time scale of several years. We will
discuss the implications of this behavior below and in section 5.

Figure 4: The luminosity (upper two rows) and radius (lower two rows) as
function of time during ($t_{\rm W}=0-4{~{}\rm yr}$) and after energy
deposition as function of time $t_{\rm W}$ (set to zero at maximum wave
luminosity). Each panel presents results for three different values of the
outer envelope mass to which we deposit the energy. We indicate above each
panel the power of the waves that we use. Note that the vertical scales and
minimum values change from one panel to another.  Figure 5: Similar to the
left panels in the second and fourth rows of Fig. 4 with $L_{\rm W}=2\times
10^{4}L_{\odot}$, but for the effective temperature.

In Fig. 6 we present the density profiles of the star just before the flash,
at the end of the energy deposition time $t_{\rm W}=4{~{}\rm yr}$, and at
$t_{\rm W}=4{~{}\rm yr}$ but without energy deposition. The upper panel
emphasizes the changes in the core which are mainly due to the core helium
flash and do not depend on wave energy deposition, while the lower panel
emphasize the changes in the envelope that are due to wave energy deposition.

Figure 6: The density (in units of ${~{}\rm g}{~{}\rm cm}^{-3}$) profiles of
the case with $M_{\rm ZAMS}=1.6M_{\odot}$, $L_{\rm W}=2\times 10^{4}L_{\odot}$
and energy deposition in the outer $\xi=50\%$ envelope mass. We present the
density profile before wave energy deposition (red line), at the end of wave
energy deposition at $t_{\rm W}=4{~{}\rm yr}$, and at $t_{\rm W}=4{~{}\rm yr}$
but with no wave energy deposition. The upper panel with logarithmic radius
scale emphasizes the changes in the core due to the core helium flash, and the
lower panel with a linear radius scale emphasizes the changes in the envelope
due to wave energy deposition.

The wave energy dissipation in the envelope leads to a brightening of the
star. The luminosity increase depends on the wave energy $L_{\rm W}$ and the
mass of the envelope into which we deposit the wave energy. Here we deposit
the energy into the outer zone, mass fraction $\xi$, of an envelope of mass
$M_{\rm env,b}=1.01M_{\odot}$ as we indicate in the different panels of Fig.
4. The luminosity increases from about $L_{\rm b}=2\times 10^{3}L_{\odot}$ to
$L(T_{\rm W}=4{~{}\rm yr})=2.2\times 10^{3}L_{\odot}$ in the case of $(L_{\rm
W},\xi)=(5\times 10^{3}L_{\odot},80\%)$ and to $L(T_{\rm W}=4{~{}\rm
yr})=5\times 10^{4}L_{\odot}$ in the case of $(L_{\rm W},\xi)=(5\times
10^{4}L_{\odot},20\%)$. Other values are in between. In the later case the
envelope has reached a steady state and in the last three years of energy
deposition ($t_{\rm W}\simeq 1{~{}\rm yr}$ to $t_{\rm W}=4{~{}\rm yr}$) it
emits all the wave energy. We see this also from the evolution of the radius
of the star that reaches a constant value in the last three years of wave
energy deposition.

Consider the case of $(L_{\rm W},\xi)=(2\times 10^{4}L_{\odot},20\%)$ as an
example. Luminosity increases to $L=1.7\times 10^{4}L_{\odot}$ over four
years. However, we do not expect the observation of such an increase in the
visible band. After one year the luminosity increases from its initial value
of $L_{\rm b}=2\times 10^{3}L_{\odot}$ to $L(1)\simeq 4\times 10^{3}L_{\odot}$
and the radius increases from $R_{\rm b}=128R_{\odot}$ to about
$R(1)=180R_{\odot}$. This must lead to a substantial increase in mass loss
rate and a copious amount of dust formation. The large amounts of dust and
possible increase of the molecular opacity in the cooler upper atmosphere (as
Kravchenko et al. 2021 found for Betelgeuse) can even lead to a decrease of
the stellar brightness in the visible. The RGB star turns to a luminous red
transient. This event might be classified as a weak and red ‘gap event’, i.e.,
a transient event with a peak luminosity between those of classical novae and
typical supernovae. In this case it would be on the lower boundary of gap
transients. The onset of a common envelope evolution of a low mass companion
($M_{2}\simeq 0.3-0.5M_{\odot}$) that enters the envelope of and RGB star or
an AGB star might lead to a similar transient event that have an increase of
luminosity by $\approx 10^{4}L_{\odot}$ at peak and an event that lasts for
several years, and accompanied by enhanced mass loss rate and dust formation.
We expect present and future sky surveys to detect such weak-red gap
transients, a small fraction of which might be RGB stars experiencing a core
helium flash.

##  5 Discussion and Summary

The basic assumption of our study is that the gravity waves that the strong
convection excite during the core helium flash drive waves that propagate all
the way to the envelope. We based this assumption on a similar process that
Quataert & Shiode (2012) and Shiode & Quataert (2014) proposed and studied for
massive stars just before they experience a core collapse supernova explosion
(section 2). Using their wave power relation to the core convection (equation
1) we find the average wave power over $\Delta_{4}\simeq 4{~{}\rm yr}$ to be
$L_{\rm W,0}=4.3\times 10^{5}L_{\odot}$ (equation 4 for the $M_{\rm
ZAMS}=1.6M_{\odot}$ model, and Fig. 2 for the three models). Under their
assumption the waves deposit their energy in the outer envelope where the
envelope convection cannot transport the wave power (Fig. 3). We did not study
the wave propagation, but only their power and the effect of energy deposition
into the envelope of the $M_{\rm ZAMS}=1.6M_{\odot}$ stellar model over four
years during its core helium flash.

We implemented a more conservative approach and deposited much lower energies
than $E_{\rm wave,0}$ according to equation (3) and in a more extended
envelope zone than what Fig. 3 suggests. We present the response of the
$M_{\rm ZAMS}=1.6M_{\odot}$ envelope to energy deposition in Fig. 4
(luminosity and radius), in Fig. 5 (effective temperature) and in Fig. 6
(density profile). We found the degree of envelope expansion and luminosity
increase, and that the star relaxes back on a time scale of several years.

We discuss two consequences of wave energy deposition that lead to envelope
expansion and luminosity increase. These are the appearance of a gap transient
event (with peak luminosity between classical novae and typical supernovae)
and the possible engulfment of orbiting planets.

In section 4.3 we discussed our expectation that a small fraction of weak and
red gap transients be RGB stars experiencing a core helium flash. Because of
the large amount of dust that we expect the expanding star to form at the
beginning of the event, the object might not increase its luminosity in the
visible, and might actually become fainter in the visible. Such events are
better observed in the infrared.

We turn to discuss the possible engulfment of planets, that we further
elaborate on in an accompanying paper (Merlov et al., 2021). As we show in
Merlov et al. (2021), the expanding envelope as a result of wave energy
deposition at the core helium flash might engulf planets in the right orbital
separation that without the envelope expansion would survive the entire RGB
phase of their parent star. This formation of a common envelope evolution
brings the planet to spiral-in towards the core of the RGB star. Because of
the inflated envelope its binding energy is lower, and the planet might unbind
most of the envelope and therefore might survive the common envelope
evolution.

Consider the case of $(L_{\rm W},\xi)=(2\times 10^{4}L_{\odot},50\%)$ for the
$M_{\rm ZAMS}=1.6M_{\odot}$ stellar model. The binding energy of the envelope
residing above radius $r=1R_{\odot}$ at the end of energy deposition at
$t_{\rm W}=4{~{}\rm yr}$ is $E_{\rm e,bind}(1R_{\odot})=7\times 10^{45}{~{}\rm
erg}$. We can compare this energy to the orbital energy that a planet of mass
$M_{\rm p}$ releases as it spirals-in to an orbital separation of
$a=1R_{\odot}$, $E_{\rm orbit}=8\times 10^{45}(M_{\rm p}/10M_{\rm J}){~{}\rm
erg}$, where $M_{\rm J}$ is Jupiter mass. Because $E_{\rm orbit}\gtrsim E_{\rm
e,bind}$ it is possible for a planet to expel most of the envelope and survive
such an evolution. Only more accurate numerical simulations of common envelope
evolution with planets (e.g., Kramer et al. 2020) can determine whether the
planet survives or not.

In Merlov et al. (2021) we suggest that the general scenario that we discussed
in this paper might explain the system WD 1856+534 (TIC 267574918) where a
planet orbits a WD of mass $M_{\rm WD}\simeq 0.52M_{\odot}$ with $a\simeq
0.02{~{}\rm AU}$ and an orbital period of $P_{\rm orb}=1.4{~{}\rm days}$
(Vanderburg et al., 2020). We discussed some other scenarios for the formation
of this system in section 1, and present the evolution towards a common
envelope in Merlov et al. (2021). Here we note that in our models of $(L_{\rm
W},\xi)=(2\times 10^{4}L_{\odot},50\%)$ and $(L_{\rm W},\xi)=(2\times
10^{4}L_{\odot},20\%)$ for the $M_{\rm ZAMS}=1.6M_{\odot}$ stellar model the
binding energies of the envelope that resides above mass coordinate
$m=0.52M_{\odot}$ at the end of wave energy deposition is $E_{\rm
e,bind}(0.52M_{\odot})=3.9\times 10^{45}{~{}\rm erg}$ and $E_{\rm
e,bind}(0.52M_{\odot})=6.4\times 10^{45}{~{}\rm erg}$, respectively.
Depositing the wave energy in a lower envelope mass fraction (in our study 20%
of the envelope mass) leads to a larger envelope expansion that allows the
envelope to engulf more exoplanets. However, because of larger emission the
decrease in the envelope binding energy is less pronounced. The binding energy
without wave energy deposition is much larger $E_{\rm
e,bind,b}(0.52M_{\odot})=1.2\times 10^{46}{~{}\rm erg}$. The orbital energy
that the planet releases as it spirals-in to an orbital separation of
$a=4R_{\odot}$ around that WD is $E_{\rm orbit}=2.5\times 10^{45}(M_{\rm
p}/10M_{\rm J}){~{}\rm erg}$. For the above scaling, the planet might directly
unbind a large fraction of the envelope mass. A lower mass progenitor, e.g.,
$M_{\rm ZAMS}\simeq 1.2-1.4M_{\odot}$, makes the scenario more favorable.

If the RGB star engulfs inner planets to the planet that eventually survives,
the inner plant(s) can increase the likelihood of the planet to survive. Siess
& Livio (1999a, b) show that accretion process of a planet onto the core is
accompanied by a substantial expansion of the star that can lead to high mass
ejection. Namely, an inner planet or two can substantially reduce the mass of
the RGB envelope when the surviving planet enters the envelope during the core
helium flash.

Finally, we also suggest that the scenario we studied here where convection-
excited waves cause large envelope expansion during core helium flash might
explain the formation of the system ZTFJ003855.0+203025.5 that van Roestel et
al. (2021) discovered, where a brown dwarf of mass $\simeq 0.059M_{\odot}$ and
a WD of mass $\simeq 0.5M_{\odot}$ orbit each other with a semi-major axis of
$2.0R_{\odot}$.

This research was supported by a grant from the Israel Science Foundation
(769/20) Data availability The data underlying this article will be shared on
reasonable request to the corresponding author.

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