# Search for exotic physics at BESIII

Zhi-Jun Li  Zheng-Yun You on behalf of BESIII  School of Physics, Sun Yat-sen
University, Guangzhou 510275, China

###### Abstract

Exotic physics typically encompasses two categories of exotic particles:
“dark” particles beyond the standard model and exotic hadrons within the
extended standard model. We present a summary of the recent results on exotic
particles at BESIII, which could play a crucial role in the study of exotic
physics.

##  1 Introduction

The standard model (SM) has achieved significant success, identifying a series
of “fundamental” particles such as quarks, leptons, photon, Higgs boson, $Z$
boson, $W$ boson, and gluons, which have been extensively studied. However,
there are still unresolved puzzles, such as dark matter (DM), the strong CP
problem, the muon $g-2$ anomaly, the fermion mass hierarchy, and more. These
puzzles suggest the existence of a dark sector beyond the standard model,
potentially containing dark particles and dark mediators that serve as a
portal connecting the dark sector with the visible normal sector. The search
for these exotic dark particles provides an intriguing avenue for probing new
physics (NP) beyond the SM. Another category of exotic particles, known as
exotic hadrons in the extended SM, includes glueballs, multi-quark particles,
and hybrids, whose components differ from those of normal hadrons (mesons and
baryons). If the mass of these exotic particles falls within the MeV to GeV
range, they can be accessed through high-intensity $e^{+}e^{-}$ collider
experiments.

Beijing Spectrometer III (BESIII) [1, 2] is a general-purpose spectrometer for
$\tau$-charm physics study in the center-of-mass energy range from 2.0 to 4.7
GeV. BESIII records symmetric $e^{+}e^{-}$ collisions provided by the Beijing
Electron Positron Collider II (BEPCII) storage ring [3] and has collected
large data samples in this energy region [4], such as 10 billion $J/\psi$
events, 2.7 billion $\psi(2S)$ events, $20\leavevmode\nobreak\ \rm{fb}^{-1}$
data at 3.773 GeV and more than $20\leavevmode\nobreak\ \rm{fb}^{-1}$ data
above 4.0 GeV in total. With the world’s largest charmonium data samples,
significant baryon and meson data samples can also be generated from
charmonium decay. Leveraging these extensive data samples at BESIII and mature
analytical techniques [5], it becomes feasible to investigate exotic particles
and new physics beyond the standard model [6].

##  2 Exotic particles search in BESIII

###  2.1 Search for the invisible massive dark photon

An extra Abelian gauge group may cause the associated gauge boson, the dark
photon $\gamma^{\prime}$ [7]. If the symmetry of this extra group is broken
spontaneously, the dark photon will be massive. The dark photon always has a
kinetic mixing with SM photon, $\frac{1}{2}\epsilon
F^{\prime}_{\mu\nu}F_{\mu\nu}$, where $F_{\mu\nu}$ is the SM photon field
strength, $F^{\prime}_{\mu\nu}$ is the dark photon field strength and
$\epsilon$ is a small mixing parameter. Since the SM photon can couple with
the SM fermion, this kinetic mixing could be an effective coupling between the
SM fermion and the dark photon with coupling term
$\mathcal{L}=\frac{e\epsilon}{\sqrt{1-\epsilon^{2}}}J_{\mu}A^{\prime\mu}\sim
e\epsilon J_{\mu}A^{\prime\mu}$, where $J_{\mu}$ is the current of ordinary SM
matter and $A^{\prime}_{\mu}$ represents the dark photon [8]. The dark photon
with mass can be produced in any process by replacing SM photon, such as
$e^{+}e^{-}\to\gamma\gamma^{\prime}$ and
$J/\psi\to\gamma^{\prime}\eta^{(^{\prime})}$, where the dark photon could
decay to a lepton pair or be invisible. If $m_{\chi}<m_{\gamma^{\prime}}/2$
($\chi$ is the DM particle), the dark photon would predominately decay into a
pair of DM particles, which will be invisible at BESIII. Based on
$14.9\leavevmode\nobreak\ \rm{fb}^{-1}$ $e^{+}e^{-}$ annihilation data at
$\sqrt{s}=4.13\sim 4.60$ GeV, the invisible massive dark photon is searched
for via $e^{+}e^{-}\to\gamma\gamma^{\prime}$, where $\gamma^{\prime}$ is
reconstructed from the visible SM photon $\gamma$ with the relation
$E_{\gamma}=\frac{s-m^{2}_{\gamma^{\prime}}}{2\sqrt{s}}$. The signals are
extracted from the SM photon energy spectrum, and the maximum global
significance is $2.2\leavevmode\nobreak\ \sigma$ with no significant signals.
The constraint of the mixing parameter can be translated by [9]

$\displaystyle\sigma(e^{+}e^{-}\to\gamma\gamma^{\prime})=\frac{2\pi\alpha^{2}}{s}\epsilon^{2}(1-\frac{m^{2}_{\gamma^{\prime}}}{s})\times\left(\left(1+\frac{2m^{2}_{\gamma^{\prime}}/s}{(1-m^{2}_{\gamma^{\prime}}/s)^{2}}\right)\rm{log}\frac{(1+cos\theta_{c})^{2}}{(1-cos\theta_{c})^{2}}-2cos\theta_{c}\right),$
(1)

where $cos\theta_{c}=0.6$ is the cut for the signal photon polar angle. The
90% confidence level (CL) upper limit (UL) of the mixing parameter $\epsilon$
is set to be $(1.6\sim 5.7)\times 10^{-3}$ in the GeV mass region [10], which
is consistent with what already excluded by BaBar, as shown in Fig 1.

Figure 1:  The 90% CL UL of the mixing parameter with the dark photon mass
less than 10 GeV/$c^{2}$.

###  2.2 Search for the massless dark photon

If the symmetry of the extra Abelian gauge group is unbroken, the dark photon
will be massless and be significantly different from the massive dark photon.
The massless dark photon has no direct interaction with the SM particle in the
dimension-four operator but can be coupled with the SM particle in a higher-
dimension operator [11]

$\displaystyle\mathcal{L}_{\rm{NP}}=$
$\displaystyle\frac{1}{\Lambda^{2}_{\rm{NP}}}(C^{u}_{jk}\bar{q}_{j}\sigma^{\mu\nu}u_{k}\tilde{H}+C^{d}_{jk}\bar{q}_{j}\sigma^{\mu\nu}d_{k}H+C^{l}_{jk}\bar{l}_{j}\sigma^{\mu\nu}e_{k}H+h.c.)F^{\prime}_{\mu\nu},$
(2)

which includes the NP energy scale $\Lambda_{\rm{NP}}$ and dimensionless
coefficients $C_{jk}$. In this operator, it has a naturally flavor-violating
coupling, such as $cu\gamma^{\prime}$ coupling. Based on
$4.5\leavevmode\nobreak\ \rm{fb}^{-1}$ $e^{+}e^{-}$ annihilation data at
$\sqrt{s}=4.6\sim 4.7$ GeV, the first search for the $cu\gamma^{\prime}$
coupling is performed through the process $\Lambda_{c}\to p\gamma^{\prime}$,
where the invisible massless dark photon $\gamma^{\prime}$ is detected from
the recoiling. No significant signals are observed, and the UL of the
branching fraction (BF) is set to be $8.0\times 10^{-5}$ at the 90% CL [12],
corresponding to the NP energy scale related parameter
$|\mathcal{C}|^{2}+|\mathcal{C}_{5}|^{2}<9.6\times 10^{-16}\rm{GeV}^{-2}$,
where $\mathcal{C}=\Lambda_{\rm{NP}}^{-2}(C^{u}_{12}+C^{u*}{12})\nu/\sqrt{8}$,
$\mathcal{C}_{5}=\Lambda_{\rm{NP}}^{-2}(C^{u}_{12}-C^{u*}{12})\nu/\sqrt{8}$,
and $\nu$ is Higgs vacuum expected value [13].

###  2.3 Search for the muonphilic particle

Similar to the previous dark photon, an extra $U(1)$ group is added as a
minimal extension to the SM. There is a
$U(1)_{\mathcal{L}_{\mu}-\mathcal{L}_{\tau}}$ model predicts that a new
massive scalar boson $X_{0}$ or vector boson $X_{1}$ only couples to the
second and third generations of leptons ($\mu,\leavevmode\nobreak\
\tau,\leavevmode\nobreak\ \nu_{\mu},\leavevmode\nobreak\ \nu_{\tau}$) with the
operator $\mathcal{L}_{\mu}^{\rm{scalar}}=-g^{\prime}_{0}X_{0}\bar{\mu}\mu$
and
$\mathcal{L}_{\mu}^{\rm{vector}}=-g^{\prime}_{1}X_{1}\bar{\mu}\gamma^{\alpha}\mu$,
where $g^{\prime}_{0,1}$ is the coupling strength. The light muonphilic scalar
or vector particles can contribute to the muon anomalous magnetic moment and
explain the $(g-2)_{\mu}$ anomaly [14]. Based on $(8.998\pm 0.039)\times
10^{9}$ $J/\psi$ events, the search for the muonphilic particles $X_{0,1}$ is
performed via the process $J/\psi\to\mu^{+}\mu^{-}X_{0,1}$, where $X_{0,1}$ is
invisible in the analysis. Three cases of muonphilic particles are considered.
The first one is named the “vanilla” model with $m_{\chi}>m_{X_{1}}/2$, and
the muonphilic vector particle $X_{1}$ can only decay to the SM particle pair.
$X_{1}\to\nu\nu$ can be accessible in the invisible final state, where
$\mathcal{B}(X_{1}\to\nu\nu)=33\%\sim 100\%$ with different $m_{X_{1}}$. The
second one is named the “invisible” model with $m_{\chi}<m_{X_{1}}/2$, and the
muonphilic vector particle $X_{1}$ mainly decay to the DM particle pair with
the assumption of $g^{\prime}_{D}>>g^{\prime}_{1}$. The third one is named the
“scalar” model with the assumption that the muonphilic scalar particle $X_{0}$
is long-lived or only decay to invisible final states. The invisible
muonphilic particle is reconstructed from the recoiling of $\mu^{+}\mu^{-}$,
and the maximum local significance is 2.5 $\sigma$ at
$m(X_{0,1})=720\leavevmode\nobreak\ \rm{MeV}/c^{2}$. No evidence for signals
from $X_{0,1}$ invisible decays are observed, and the coupling constraint at
90% CL is given in Fig 2. For the “vanilla” model, Barbar, CMS, and Belle
measure with process of $X_{1}\to\mu^{+}\mu^{-}$, while Belle II and BESIII
measure with $X_{1}\to\nu\bar{\nu}$. For the “invisible” model, BESIII
provides better sensitivity in the range of 200 $\sim$ 860 $\rm{MeV}/c^{2}$.
For the “scalar” model, BESIII gives the first constraint of the “scalar”
invisible $X_{0}$ case. [15]

Figure 2:  The constraint of the coupling between muonphilic particle and the
SM muon, where (a) is the “vanilla” model, (b) is the “invisible” model, and
(c) is the “scalar” model.

###  2.4 Search for the QCD axion

The QCD axion ($a$) is originally predicted by the Peccei-Quinn (PQ) solution
to the strong CP problem, which is also an excellent cold dark matter
candidate. The mass of the QCD axion is related to the decay constant $f_{a}$
with the relation of $m_{a}=5.691(51)\leavevmode\nobreak\ \rm{\mu
eV}(\frac{10^{12}\leavevmode\nobreak\ \rm{GeV}}{f_{a}})$, and the decay
constant satisfies $f_{a}>>10^{6}$ GeV, making $m_{a}<\rm{eV}$, which is
“massless” compared to the resolution of BESIII. Such a small mass makes its
lifetime larger than the age of the universe and the axion invisible in the
detector. The QCD axion could be coupled with the SM fermions with the
operator
$\mathcal{L}_{a-f}=\partial_{\mu}a\bar{f}_{i}\gamma^{\mu}(\frac{1}{F^{V}_{ij}}+\frac{\gamma^{5}}{F^{A}_{ij}})$,
where $F^{V}_{ij}$ and $F^{A}_{ij}$ are the effective decay constants for the
vector coupling term and axial coupling term [16]. If lepton U(1) charges are
flavour non-universal, it will have naturally flavour-violating couplings.
Based on $(10084\pm 44)\times 10^{6}\leavevmode\nobreak\ J/\psi$ events decay
to $\Sigma^{+}\bar{\Sigma}^{-}$ pair, the QCD axion is searched for through
the process of $\Sigma^{+}\to pa$. A kinematic fit is performed to constrain
the invisible axion mass to zero, and the signals are extracted in the energy
spectrum of the extra shower in the electromagnetism counter. No significant
signals are found, and the UL of $\mathcal{B}(\Sigma^{+}\to pa)$ is set to be
$3.2\times 10^{-5}$ at the 90% CL. The constraint on the effective decay
constants can be given by [16]

$\displaystyle\Gamma(\Sigma^{+}\to
pa)=\frac{M^{3}_{\Sigma^{+}}}{16\pi}\left(1-\frac{M^{2}_{p}}{M^{2}_{\Sigma^{+}}}\right)\left(\frac{(-1)^{2}}{|F^{V}_{sd}|^{2}}+\frac{(0.34)^{2}}{|F^{A}_{sd}|^{2}}\right).$
(3)

As shown in Fig 3 (a), BESIII provides a competitive constraint on the axial-
vectorial effective decay constant $F^{A}_{sd}>2.8\times 10^{7}$ GeV. [17]

Figure 3:  (a) The constraint on the effective decay constant of QCD axion
associated with $sda$ coupling, where “this work” is the measurement from
$\Sigma^{+}\to pa$. (b) The constraint on the ALPs with the mass of ALP less
than 10 GeV/$c^{2}$.

###  2.5 Search for the axion like particle

The axion like particles (ALPs) have the same quantum numbers as the QCD axion
but have no strict relation between their couplings and mass, which means that
the ALPs can have arbitrary masses and couplings. ALPs can have interaction
with fermions, gluon or photons, and the operator of coupling with photons is
$\mathcal{L}=-\frac{1}{4}g_{a\gamma\gamma}aF^{\mu\nu}\tilde{F}_{\mu\nu}$ with
the coupling strength $g_{a\gamma\gamma}$, leading to the corresponding decay
width
$\Gamma_{a\to\gamma\gamma}=\frac{g^{2}_{a\gamma\gamma}m^{3}_{a}}{64\pi}$.
Taking $g_{a\gamma\gamma}\sim 10^{-4}\leavevmode\nobreak\ \rm{GeV}^{-1}$,
$m_{a}\sim\rm{GeV}/c^{2}$, the lifetime of ALP will be short in the detector
and can be visible via $a\to\gamma\gamma$. Based on $(2.71\pm 0.01)\times
10^{9}\leavevmode\nobreak\ \psi(2S)$ events
($\psi(2S)\to\pi^{+}\pi^{-}J/\psi$) and another $(10084\pm 44)\times
10^{6}\leavevmode\nobreak\ J/\psi$ events ($e^{+}e^{-}\to J/\psi$), the ALPs
are searched for via the process $J/\psi\to\gamma a,\leavevmode\nobreak\
a\to\gamma\gamma$. The signals are extracted from the $M_{\gamma\gamma}$
spectrum, and the maximum local significance is 2.6 $\sigma$ at
$M_{a}=2208\leavevmode\nobreak\ \rm{MeV}/c^{2}$ for $\psi(2S)$ data and the
maximum global significance is 1.6 $\sigma$ at $M_{a}=2786\leavevmode\nobreak\
\rm{MeV}/c^{2}$ for $J/\psi$ data, which means that no evidence for signals
from ALPs visible decays are observed. The ALPs-photon coupling constraints
are given by [18]

$\displaystyle\frac{\mathcal{B}(J/\psi\to\gamma a)}{\mathcal{B}(J/\psi\to
e^{+}e^{-})}=\frac{m^{2}_{J/\psi}}{32\pi\alpha}g^{2}_{a\gamma\gamma}\left(1-\frac{m^{2}_{a}}{m^{2}_{J/\psi}}\right)^{3}.$
(4)

As shown in Fig 3 (b), BESIII provides an improvement by a factor of 5 over
the previous Belle II measurement [19, 20].

###  2.6 Other exotic “dark” particles search

There are also some other published exotic “dark” particle searches at BESIII,
such as the full invisible decay
$\Lambda/\omega/\phi/\eta/\eta^{\prime}\to\chi\chi$, semi-invisible decay
$D^{0}\to\pi^{0}\chi\chi$, visible dark photon search $\gamma^{\prime}\to
e^{+}e^{-}$ via $J/\psi\to\gamma^{\prime}\eta^{(^{\prime})}$ and light Higgs
search $J/\psi\to\gamma A^{0},A^{0}\to\mu^{+}\mu^{-}$, which are summarized in
Fig 4.

Figure 4:  The summary of the exotic “dark” particles from hadron decays at
BESIII.

###  2.7 Recent glueball studies

$J/\psi$ radiative decays provide gluon rich environment and an ideal place
for glueball study. A partial wave analysis (PWA) of $J/\psi\to\gamma
K_{S}K_{S}\pi^{0}$ is performed at BESIII, and two pseudo-scalar states are
observed, $\eta(1405)$ and $\eta(1475)$ [21], as shown in Fig 5 (a). In the
Quark model prediction, there is only one pseudo-scalar meson near 1.4 GeV,
and $\eta(1475)$ is regarded as the first radial excitation of $\eta^{\prime}$
while $\eta(1405)$ may be a pseudo-scalar glueball candidate. This measurement
could be an important input for $0^{-+}$ glueball. In the LQCD prediction, the
mass of $0^{-+}$ glueball is in the range of 2.3 $\sim$ 2.4 $\rm{GeV}/c^{2}$.
Some theorists attempt to explain $\eta(1405)$/$\eta(1475)$ using a single
state [22] , while others propose more than one pole [23].

Figure 5:  (a) The PWA of $J/\psi\to\gamma K_{S}K_{S}\pi^{0}$, where MI is
Mass independent PWA, MD is Mass dependent PWA and the results from these two
methods are consistent. (b) The PWA of $J/\psi\to\gamma
K_{S}K_{S}\eta^{\prime}$.

Another pseudo-scalar glueball candidate is $X(2370)$. $X(2370)$ was firstly
observed in $J/\psi\to\gamma\pi^{+}\pi^{-}\eta^{\prime}$ [24], and a new PWA
of $J/\psi\to\gamma K_{S}K_{S}\eta^{\prime}$ is performed at BESIII, as shown
in Fig 5 (b), resulting that $M_{X(2370)}=(2395\pm 11\leavevmode\nobreak\
^{+26}_{-0.84})\leavevmode\nobreak\ \rm{MeV}/c^{2}$,
$\Gamma_{X(2370)}=(188\leavevmode\nobreak\ ^{+18}_{-17}\leavevmode\nobreak\
{}^{+124}_{-33})\leavevmode\nobreak\ \rm{MeV}/c^{2}$, and
$J^{PC}_{X(2370)}=0^{-+}$, where the spin-parity of $X(2370)$ is determined to
be $0^{-+}$ for the first time [25]. The measured mass and spin-parity of the
$X(2370)$ are all consistent with the predictions of a pseudo-scalar glueball.

## Acknowledgments

This work is supported in part by National Key Research and Development
Program of China under Contracts Nos. 2023YFA1606000, 2020YFA0406400,
2020YFA0406300; Joint Large-Scale Scientific Facility Funds of the National
Natural Science Foundation of China (NSFC) and Chinese Academy of Sciences
(CAS) under Contracts Nos. U1932101.

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# Self-Supervised Hypergraph Convolutional Networks for
Session-based Recommendation

Xin Xia1, Hongzhi Yin1, Junliang Yu1, Qinyong Wang1, Lizhen Cui2, Xiangliang
Zhang3
Corresponding author

###### Abstract

Session-based recommendation (SBR) focuses on next-item prediction at a
certain time point. As user profiles are generally not available in this
scenario, capturing the user intent lying in the item transitions plays a
pivotal role. Recent graph neural networks (GNNs) based SBR methods regard the
item transitions as pairwise relations, which neglect the complex high-order
information among items. Hypergraph provides a natural way to capture beyond-
pairwise relations, while its potential for SBR has remained unexplored. In
this paper, we fill this gap by modeling session-based data as a hypergraph
and then propose a hypergraph convolutional network to improve SBR. Moreover,
to enhance hypergraph modeling, we devise another graph convolutional network
which is based on the line graph of the hypergraph and then integrate self-
supervised learning into the training of the networks by maximizing mutual
information between the session representations learned via the two networks,
serving as an auxiliary task to improve the recommendation task. Since the two
types of networks both are based on hypergraph, which can be seen as two
channels for hypergraph modeling, we name our model DHCN (Dual Channel
Hypergraph Convolutional Networks). Extensive experiments on three benchmark
datasets demonstrate the superiority of our model over the SOTA methods, and
the results validate the effectiveness of hypergraph modeling and self-
supervised task. The implementation of our model is available via
https://github.com/xiaxin1998/DHCN.

## Introduction

Session-based recommendation (SBR) is an emerging recommendation paradigm,
where long-term user profiles are usually not available (Wang, Cao, and Wang
2019; Guo et al. 2019). Generally, a session is a transaction with multiple
purchased items in one shopping event, and SBR focuses on next-item prediction
by using the real-time user behaviors. Most of the research efforts in this
area regard the sessions as ordered sequences, among which recurrent neural
networks (RNNs) based (Hidasi et al. 2015; Jannach and Ludewig 2017; Hidasi
and Karatzoglou 2018) and graph neural networks (GNNs) (Wu et al. 2020) based
approaches have shown great performance.

In RNNs-based approaches, modeling session-based data as unidirectional
sequences is deemed as the key to success, since the data is usually generated
in a short period of time and is likely to be temporally dependent. However,
this assumption may also trap these RNNs-based models because it ignores the
coherence of items. Actually, unlike linguistic sequences which are generated
in a strictly-ordered way, among user behaviors, there may be no such strict
chronological order. For example, on Spotify111https://www.spotify.com/, a
user can choose to shuffle an album or play it in order, which generates two
different listening records. However, both of these two play modes serialize
the same set of songs. In other words, reversing the order of two items in
this case would not lead to a distortion of user preference. Instead, strictly
and solely modeling the relative orders of items and ignoring the coherence of
items would probably make the recommendation models prone to overfitting.

Recently, the effectiveness of graph neural networks (GNNs) (Wu et al. 2020;
Yu et al. 2020; Yin et al. 2019) has been reported in many areas including
SBR. Unlike the RNNs-based recommendation method, the GNNs-based approaches
(Wu et al. 2019b; Xu et al. 2019; Qiu et al. 2020b) model session-based data
as directed subgraphs and item transitions as pairwise relations, which
slightly relaxes the assumption of temporal dependence between consecutive
items. However, existing models only show trivial improvements compared with
RNNs-based methods. The potential reason is that they neglect the complex item
correlations in session-based data. In real scenarios, an item transition is
often triggered by the joint effect of previous item clicks, and many-to-many
and high-order relations exist among items. Obviously, simple graphs are
incapable of depicting such set-like relations.

To overcome these issues, we propose a novel SBR approach upon hypergraph to
model the high-order relations among items within sessions. Conceptually, a
hypergraph (Bretto 2013) is composed of a vertex set and a hyperedge set,
where a hyperedge can connect any numbers of vertices, which can be used to
encode high-order data correlations. We also assume that items in a session
are temporally correlated but not strictly sequentially dependent. The
characteristics of hyperedge perfectly fit our assumption as hyperedge is set-
like, which emphasizes coherence of the involved elements rather than relative
orders. Therefore, it provides us with a flexibility and capability to capture
complex interactions in sessions. Technically, we first model each session as
a hyperedge in which all the items are connected with each other, and
different hyperedges, which are connected via shared items, constitute the
hypergraph that contains the item-level high-order correlations. Figure 1
illustrates the hypergraph construction and the pipeline of the proposed
method.

By stacking multiple layers in the hypergraph channel, we can borrow the
strengths of hypergraph convolution to generate high-quality recommendation
results. However, since each hyperedge only contains a limited number of
items, the inherent data sparsity issue might limit the benefits brought by
hypergraph modeling. To address this problem, we introduce line graph channel
and innovatively integrate self-supervised learning (Hjelm et al. 2018) into
our model to enhance hypergraph modeling. A line graph is built based on the
hypergraph by modeling each hyperedge as a node and focuses on the
connectivity of hyperedges, which depicts the session-level relations. After
that, a Dual channel Hypergraph Convolutional Network (DHCN) is developed in
this paper with its two channels over the two graphs. Intuitively, the two
channels in our network can be seen as two different views that describe the
intra- and inter- information of sessions, while each of them knows little
information of the other. By maximizing the mutual information between the
session representations learned via the two channels through self-supervised
learning, the two channels can acquire new information from each other to
improve their own performance in item/session feature extraction. We then
unify the recommendation task and the self-supervised task under a
primary&auxiliary learning framework. By jointly optimizing the two tasks, the
performance of the recommendation task achieves decent gains.

Overall, the main contributions of this work are summarized as follows:

  * •

We propose a novel dual channel hypergraph convolutional network for SBR,
which can capture the beyond-pairwise relations among items through hypergraph
modeling.

  * •

We innovatively integrate a self-supervised task into the training of our
network to enhance hypergraph modeling and improve the recommendation task.

  * •

Extensive experiments show that our proposed model has overwhelming
superiority over the state-of-the-art baselines and achieves statistically
significant improvements on benchmark datasets.

Figure 1: The construction of hypergraph and the pipeline of the proposed DHCN
model.

## Related Work

### Session-based Recommendation

The initial exploration of SBR mainly focuses on sequence modeling, where
Markov decision process is the preferred technique at this phase. (Shani,
Heckerman, and Brafman 2005; Rendle, Freudenthaler, and Schmidt-Thieme 2010;
Zimdars, Chickering, and Meek 2013) are the representative works of this line
of research. The boom of deep learning provides alternatives to exploit
sequential data. Deep learning models such as recurrent neural networks
(Hochreiter and Schmidhuber 1997; Cho et al. 2014) and convolutional neural
networks (Tuan and Phuong 2017) have subsequently been applied to SBR and
achieved great success. (Hidasi et al. 2015; Tan, Xu, and Liu 2016; Li et al.
2017; Liu et al. 2018) are the classical RNNs-based models which borrow the
strengths of RNNs to model session-based data.

Graph Neural Networks (GNNs) (Wu et al. 2020; Zhou et al. 2018) recently have
drawn increasing attention and their applications in SBR also have shown
promising results (Wang et al. 2020b, c; Yuan et al. 2019; Chen and Wong
2020). Unlike RNNs-based approaches working on sequential data, GNNs-based
methods learn item transitions over session-induced graphs. SR-GNN (Wu et al.
2019b) is the pioneering work which uses a gated graph neural network to model
sessions as graph-structured data. GC-SAN (Xu et al. 2019) employs self-
attention mechanism to capture item dependencies via graph information
aggregation. FGNN (Qiu et al. 2019) constructs a session graph to learn item
transition pattern and rethinks the sequence order of items in SBR. GCE-GNN
(Wang et al. 2020c) conduct graph convolution on both the single session graph
and the global session graph to learn session-level and global-level
embeddings. Although these studies demonstrate that GNN-based models
outperform other approaches including RNNs-based ones, they all fail to
capture the complex and higher-order item correlations.

### Hypergraph Learning

Hypergraph provides a natural way to capture complex high-order relations.
With the boom of deep learning, hypergraph neural network also have received
much attention. HGNN (Feng et al. 2019) and HyperGCN (Yadati et al. 2019) are
the first to apply graph convolution to hypergraph. (Jiang et al. 2019)
proposed a dynamic hypergraph neural network and (Bandyopadhyay, Das, and
Murty 2020) developed the line hypergraph convolutional networks.

There are also a few studies combining hypergraph learning with recommender
systems (Bu et al. 2010; Li and Li 2013). The most relevant work to ours is
HyperRec (Wang et al. 2020a), which uses hypergraph to model the short-term
user preference for next-item recommendation. However, it does not exploit
inter-hyperedge information and is not designed for session-based scenarios.
Besides, the high complexity of this model makes it impossible to be deployed
in real scenarios. Currently, there is no research bridging hypergraph neural
networks and SBR, and we are the first to fill this gap.

### Self-supervised Learning

Self-supervised learning (Hjelm et al. 2018) is an emerging machine learning
paradigm which aims to learn the data representation from the raw data. It was
firstly used in visual representation learning (Bachman, Hjelm, and Buchwalter
2019). The latest advances in this area extend self-supervised learning to
graph representation learning (Velickovic et al. 2019). The dominant paradigm
based on contrastive learning (Hassani and Khasahmadi 2020; Qiu et al. 2020a)
suggests that contrasting congruent and incongruent views of graphs with
mutual information maximization can help encode rich graph/node
representations.

As self-supervised learning is still in its infancy, there are only several
studies combining it with recommender systems (Zhou et al. 2020; Ma et al.
2020; Xin et al. 2020). The most relevant work to ours is S3-Rec (Zhou et al.
2020) for sequential recommendation, which uses feature mask to create self-
supervision signals. But it is not applicable to SBR since the session data is
very sparse and masking features cannot generate strong self-supervision
signals. Currently, the potentials of self-supervised learning for hypergraph
representation learning and SBR have not been investigated. We are the first
to integrate self-supervised learning into the scenarios of SBR and hypergraph
modeling.

## The Proposed Method

In this section, we first introduce the notions and definitions used
throughout this paper, and then we show how session-based data is modeled as a
hypergraph. After that, we present our hypergraph convolutional network for
SBR. Finally, we devise the line graph channel and integrate self-supervised
learning into the dual channel network to enhance hypergraph modeling.

### Notations and Definitions

Let $I=\\{i_{1},i_{2},i_{3},...,i_{N}\\}$ denote the set of items, where $N$
is the number of items. Each session is represented as a set
$s=[i_{s,1},i_{s,2},i_{s,3},...,i_{s,m}]$ and $i_{s,k}\in I(1\leq k\leq m)$
represents an interacted item of an anonymous user within the session $s$. We
embed each item $i\in I$ into the same space and let
$\mathbf{x}_{i}^{(l)}\in\mathbb{R}^{d^{(l)}}$ denote the vector representation
of item $i$ of dimension $d^{l}$ in the $l$-th layer of a deep neural network.
The representation of the whole item set is denoted as
$\mathbf{X}^{(l)}\in\mathbb{R}^{N\times d^{(l)}}$. Each session $s$ is
represented by a vector $\mathbf{s}$. The task of SBR is to predict the next
item, namely $i_{s,m+1}$, for any given session $s$.

Definition 1. Hypergraph. Let $G=(V,E)$ denote a hypergraph, where $V$ is a
set containing $N$ unique vertices and $E$ is a set containing $M$ hyperedges.
Each hyperedge $\epsilon\in E$ contains two or more vertices and is assigned a
positive weight $W_{\epsilon\epsilon}$, and all the weights formulate a
diagonal matrix $\mathbf{W}\in\mathbb{R}^{M\times M}$. The hypergraph can be
represented by an incidence matrix $\mathbf{H}\in\mathbb{R}^{N\times M}$ where
$H_{i\epsilon}$ = 1 if the hyperedge $\epsilon\in E$ contains a vertex
$v_{i}\in V$, otherwise 0. For each vertex and hyperedge, their degree
$D_{ii}$ and $B_{\epsilon\epsilon}$ are respectively defined as
$D_{ii}=\sum_{\epsilon=1}^{M}W_{\epsilon\epsilon}H_{i\epsilon};B_{\epsilon\epsilon}=\sum_{i=1}^{N}H_{i\epsilon}$.
$\mathbf{D}$ and $\mathbf{B}$ are diagonal matrices.

Definition 2. Line graph of hypergraph. Given the hypergraph $G=(V,E)$, the
line graph of the hypergraph $L(G)$ is a graph where each node of $L(G)$ is a
hyperedge in $G$ and two nodes of $L(G)$ are connected if their corresponding
hyperedges in $G$ share at least one common node (Whitney 1992). Formally,
$L(G)=(V_{L},E_{L})$ where $V_{L}=\\{v_{e}:v_{e}\in E\\}$, and
$E_{L}=\\{(v_{e_{p}}$, $v_{e_{q}}):e_{p}$, $e_{q}\in E,|e_{p}\cap e_{q}|\geq
1\\}$. We assign each edge $(v_{e_{p}}$, $v_{e_{q}})$ a weight $W_{p,q}$,
where $W_{p,q}=|e_{p}\cap e_{q}|/|e_{p}\cup e_{q}|$.

### Hypergraph Construction

To capture the beyond pairwise relations in session-based recommendation, we
adopt a hypergraph $G=(V,E)$ to represent each session as a hyperedge.
Formally, we denote each hyperedge as
$[i_{s,1},i_{s,2},i_{s,3},...,i_{s,m}]\in E$ and each item $i_{s,m}\in V$. The
changes of data structure before and after hypergraph construction are shown
in the left part of Figure 1. As illustrated, the original session data is
organized as linear sequences where two items $i_{s,m-1},i_{s,m}$ are
connected only if a user interacted with item $i_{s,m-1}$ before item
$i_{s,m}$. After transforming the session data into a hypergraph, any two
items clicked in a session are connected. It should be noted that we transform
the session sequences into an undirected graph, which is in line with our
intuition that items in a session are temporally related instead of
sequentially dependent. By doing so, we manage to concretize the many-to-many
high-order relations. Besides, we further induce the line graph of the
hypergraph according to Definition 2. Each session is modeled as a node and
different sessions are connected via shared items. Compared with the
hypergraph which depicts the item-level high-order relations, the line graph
describes the session-level relations that are also termed cross-session
information.

### Hypergraph Convolutional Network

After the hypergraph construction, we develop a hypergraph convolutional
network to capture both the item-level high-order relations.

#### Hypergraph Channel and Convolution

The primary challenge of defining a convolution operation over the hypergraph
is how the embeddings of items are propagated. Referring to the spectral
hypergraph convolution proposed in (Feng et al. 2019), we define our
hypergraph convolution as:

$\mathbf{x}_{i}^{(l+1)}=\sum_{j=1}^{N}\sum_{\epsilon=1}^{M}H_{i\epsilon}H_{j\epsilon}W_{\epsilon\epsilon}\mathbf{x}_{j}^{(l)}.$
(1)

Following the suggestions in (Wu et al. 2019a), we do not use nonlinear
activation function and the convolution filter parameter matrix. For
$W_{\epsilon\epsilon}$, we assign each hyperedge the same weight 1. The matrix
form of Eq. (1) with row normalization is:

$\mathbf{X}_{h}^{(l+1)}=\mathbf{D}^{-1}\mathbf{H}\mathbf{WB^{-1}}\mathbf{H}^{\mathrm{T}}\mathbf{X}_{h}^{(l)}.$
(2)

The hypergraph convolution can be viewed as a two-stage refinement performing
‘node-hyperedge-node’ feature transformation upon hypergraph structure. The
multiplication operation $\mathbf{H}^{\top}\mathbf{X}^{(l)}_{h}$ defines the
information aggregation from nodes to hyperedges and then premultiplying
$\mathbf{H}$ is viewed to aggregate information from hyperedges to nodes.

After passing $\mathbf{X}^{(0)}$ through $L$ hypergraph convolutional layer,
we average the items embeddings obtained at each layer to get the final item
embeddings $\mathbf{X}_{h}=\frac{1}{L+1}\sum_{l=0}^{L}\mathbf{X}^{(l)}_{h}$.
Although this work mainly emphasizes the importance of the coherence of a
session, the temporal information is also inevitable for better recommendation
results. Position Embeddings is an effective technique which was introduced in
Transformer (Vaswani et al. 2017) and has been applied in many situations for
the memory of position information of items. In our method, we integrate the
reversed position embeddings with the learned item representations by a
learnable position matrix
$\mathbf{P}_{r}=\left[\mathbf{p_{1}},\mathbf{p_{2}},\mathbf{p_{3}},...,\mathbf{p_{m}}\right]$,
where $m$ is the length of the current session. The embedding of $t$-th item
in session $s=[i_{s,1},i_{s,2},i_{s,3},...,i_{s,m}]$ is:

$\mathbf{x}^{*}_{t}=\tanh\left(\mathbf{W}_{1}\left[\mathbf{x}_{t}\|\mathbf{p}_{m-i+1}\right]+\mathbf{b}\right),$
(3)

where $\mathbf{W}_{1}\in\mathbb{R}^{d\times 2d}$, and $b\in\mathbb{R}^{d}$ are
learnable parameters.

Session embeddings can be represented by aggregating representation of items
in that session. We follow the strategy used in SR-GNN (Wu et al. 2019b) to
refine the embedding of session $s=[i_{s,1},i_{s,2},i_{s,3},...,i_{s,m}]$:

$\displaystyle\alpha_{t}=\mathbf{f}^{\top}\sigma\left(\mathbf{W}_{2}\mathbf{x}^{*}_{s}+\mathbf{W}_{3}\mathbf{x}^{*}_{t}+\mathbf{c}\right),\mathbf{\theta}_{h}=\sum_{t=1}^{m}\alpha_{t}\mathbf{x}_{t}$
(4)

where $\mathbf{x}^{*}_{s}$ is the embedding of session $s$ and here it is
represented by averaging the embeddings of items it contains, which is
$\mathbf{x}^{*}_{s}=\frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{m}$, and
$\mathbf{x}^{*}_{t}$ is the embedding of the $t$-th item in session $s$.
User’s general interest embedding $\mathbf{\theta}_{h}$ across this session is
represented by aggregating item embeddings through a soft-attention mechanism
where items have different levels of priorities.
$\mathbf{f}\in\mathbb{R}^{d}$, $\mathbf{W}_{2}\in\mathbb{R}^{d\times d}$ and
$\mathbf{W}_{3}\in\mathbb{R}^{d\times d}$ are attention parameters used to
learn the item weight $\alpha_{t}$. Note that, following our motivation in
Section I, we abandon the sequence modeling techniques like GRU units and
self-attention used in other SBR models. The position embedding is the only
temporal factor we use, and hence our model is very efficient and lightweight.

#### Model Optimization and Recommendation Generation

Given a session $s$, we compute scores $\hat{\mathbf{z}}$ for all the
candidate items $i\in I$ by doing inner product between the item embedding
$\mathbf{X}_{h}$ learned from hypergraph channel and $\mathbf{\theta}_{h}$:

$\hat{\mathbf{z}}_{i}=\mathbf{\theta}_{h}^{T}\mathbf{x}_{i}.$ (5)

After that, a softmax function is applied to compute the probabilities of each
item being the next one in the session:

$\hat{\mathbf{y}}=\operatorname{softmax}(\hat{\mathbf{z}}).$ (6)

We formulate the learning objective as a cross entropy loss function, which
has been extensively used in recommender systems and defined as:

$\mathcal{L}_{r}=-\sum_{i=1}^{N}\mathbf{y}_{i}\log\left(\hat{\mathbf{y}}_{i}\right)+\left(1-\mathbf{y}_{i}\right)\log\left(1-\hat{\mathbf{y}}_{i}\right),$
(7)

where $\mathbf{y}$ is the one-hot encoding vector of the ground truth. For
simplicity, we leave out the $L_{2}$ regularization terms. By minimizing
$\mathcal{L}_{r}$ with Adam, we can get high-quality session-based
recommendations.

### Enhancing SBR with Self-Supervised Learning

The hypergraph modeling empowers our model to achieve significant performance.
However, we consider that the sparsity of session data might impede hypergraph
modeling, which would result in a suboptimal recommendation performance.
Inspired by the successful practices of self-supervised learning on simple
graphs, we innovatively integrate self-supervised learning into the network to
enhance hypergraph modeling. We first design another graph convolutional
network based on the line graph of the session-induced hypergraph to generate
self-supervision signals. Then by maximizing the mutual information between
the session representations learned via the two channels through contrastive
learning, the recommendation model can acquire more information and the
recommendation performance can be improved. Since the two types of networks
both are based on hypergraph, which can be seen as two channels for hypergraph
modeling, we name our model as DHCN (Dual Channel Hypergraph Convolutional
Networks).

#### Line Graph Channel and Convolution

The line graph channel encodes the line graph of the hypergraph. Fig. 1 shows
how we transform the hypergraph into a line graph of it. The line graph can be
seen as a simple graph which contains the cross-session information and
depicts the connectivity of hyperedges. As there are no item involved in the
line graph channel, we first initialize the channel-specific session
embeddings $\mathbf{\Theta}_{l}^{(0)}$ by looking up the items belonged to
each session and then averaging the corresponding items embeddings in
$\mathbf{X}^{(0)}$. An incidence matrix for $L(G)$ is defined as
$\mathbf{A}\in\mathbb{R}^{M\times M}$ where $M$ is the number of nodes in the
line graph and $A_{p,q}=W_{p,q}$ according to Definition 2. Let
$\hat{\mathbf{A}}=\mathbf{A}+\mathbf{I}$ where $\mathbf{I}$ is an identity
matrix. $\hat{\mathbf{D}}\in\mathbb{R}^{M\times M}$ is a diagonal degree
matrix where $\hat{\mathbf{D}}_{p,p}=\sum_{q=1}^{m}\hat{\mathbf{A}}_{p,q}$.
The line graph convolution is then defined as:

$\mathbf{\Theta}^{(l+1)}_{l}=\mathbf{\hat{D}}^{-1}\mathbf{\hat{A}}\mathbf{\Theta}^{(l)}.$
(8)

In each convolution, the sessions gather information from their neighbors. By
doing so, the learned $\mathbf{\Theta}$ can capture the cross-session
information. Likewise, we pass $\mathbf{\Theta}_{l}^{(0)}$ through $L$ graph
convolutional layer, and then average the session embeddings obtained at each
layer to get the final session embeddings
$\mathbf{\Theta}_{l}=\frac{1}{L+1}\sum_{l=0}^{L}\mathbf{\Theta}^{(l)}_{l}$.

#### Creating self-supervision signals.

So far, we learn two groups of channel-specific session embeddings via the two
channels. Since each channel encodes a (hyper)graph that only depicts either
of the item-level (intra-session) or the session-level (inter-session)
structural information of the session-induced hypergraph, the two groups of
embeddings know little about each other but can mutually complement. For each
mini-batch including $n$ sessions in the training, there is a bijective
mapping between the two groups of session embeddings. Straightforwardly, the
two groups can be the ground-truth of each other for self-supervised learning,
and this one-to-one mapping is seen as the label augmentation. If two session
embeddings both denote the same session in two views, we label this pair as
the ground-truth, otherwise we label it as the negative.

#### Contrastive learning.

Following (Velickovic et al. 2019; Bachman, Hjelm, and Buchwalter 2019), we
regard the two channels in DHCN as two views characterizing different aspects
of sessions. We then contrast the two groups of session embeddings learned via
the two views. We use a noise-contrastive type objective with a standard
binary cross-entropy (BCE) loss between the samples from the ground-truth
(positive) and the corrupted samples (negative) as our learning objective and
defined it as:

$\mathcal{L}_{s}=-\log\sigma(f_{\mathrm{D}}(\theta^{h}_{i},\theta^{l}_{i}))-\log\sigma(1-f_{\mathrm{D}}(\tilde{\theta}^{h}_{i},\theta^{l}_{i})),$
(9)

where $\tilde{\theta}^{h}_{i}$ (or $\tilde{\theta}^{l}_{i}$) is the negative
sample obtained by corrupting $\Theta_{h}$ ($\Theta_{l}$) with row-wise and
column-wise shuffling, and
$f_{\mathrm{D}}(\cdot):\mathbb{R}^{d}\times\mathbb{R}^{d}\longmapsto\mathbb{R}$
is the discriminator function that takes two vectors as the input and then
scores the agreement between them. We simply implement the discriminator as
the dot product between two vectors. This learning objective is explained as
maximizing the mutual information between the session embeddings learned in
different views (Velickovic et al. 2019). By doing so, they can acquire
information from each other to improve their own performance in item/session
feature extraction through the convolution operations. Particularly, those
sessions that only include a few items can leverage the cross-session
information to refine their embeddings.

Finally, we unify the recommendation task and this self-supervised task into a
primary&auxiliary learning framework, where the former is the primary task and
the latter is the auxiliary task. Formally, the joint learning objective is
defined as:

$\mathcal{L}=\mathcal{L}_{r}+\beta\mathcal{L}_{s},$ (10)

where $\beta$ controls the magnitude of the self-supervised task.

## Experiments

### Experimental Settings

#### Datasets.

We evaluate our model on two real-world benchmark datasets:
Tmall222https://tianchi.aliyun.com/dataset/dataDetail?dataId=42,
Nowplaying333http://dbis-nowplaying.uibk.ac.at/#nowplaying and
Diginetica444http://cikm2016.cs.iupui.edu/cikm-cup/. Tmall dataset comes from
IJCAI-15 competition, which contains anonymized user’s shopping logs on Tmall
online shopping platform. Nowplaying dataset describes the music listening
behavior of users. For both datasets, we follow (Wu et al. 2019b; Li et al.
2017) to remove all sessions containing only one item and also remove items
appearing less than five times. To evaluate our model, we split both datasets
into training/test sets, following the settings in (Wu et al. 2019b; Li et al.
2017; Wang et al. 2020c). Then, we augment and label the dataset by using a
sequence splitting method, which generates multiple labeled sequences with the
corresponding labels
$([i_{s,1}],i_{s,2}),([i_{s,1},i_{s,2}],i_{s,3}),...,([i_{s,1},i_{s,2},...,i_{s,m-1}],i_{s,m})$
for every session $s=[i_{s,1},i_{s,2},i_{s,3},...,i_{s,m}]$. Note that the
label of each sequence is the last click item in it. The statistics of the
datasets are presented in Table 1.

Dataset | Tmall | Nowplaying | Diginetica
---|---|---|---
training sessions | 351,268 | 825,304 | 719,470
test sessions | 25,898 | 89,824 | 60,858
# of items | 40,728 | 60,417 | 43,097
average lengths | 6.69 | 7.42 | 5.12
Table 1: Dataset Statistics

Method | Tmall | Nowplaying | Diginetica
---|---|---|---
P@10 | M@10 | P@20 | M@20 | P@10 | M@10 | P@20 | M@20 | P@10 | M@10 | P@20 | M@20
Item-KNN | 6.65 | 3.11 | 9.15 | 3.31 | 10.96 | 4.55 | 15.94 | 4.91 | 25.07 | 10.77 | 35.75 | 11.57
FPMC | 13.10 | 7.12 | 16.06 | 7.32 | 5.28 | 2.68 | 7.36 | 2.82 | 15.43 | 6.20 | 26.53 | 6.95
GRU4REC | 9.47 | 5.78 | 10.93 | 5.89 | 6.74 | 4.40 | 7.92 | 4.48 | 17.93 | 7.33 | 29.45 | 8.33
NARM | 19.17 | 10.42 | 23.30 | 10.70 | 13.6 | 6.62 | 18.59 | 6.93 | 35.44 | 15.13 | 49.70 | 16.17
STAMP | 22.63 | 13.12 | 26.47 | 13.36 | 13.22 | 6.57 | 17.66 | 6.88 | 33.98 | 14.26 | 45.64 | 14.32
SR-GNN | 23.41 | 13.45 | 27.57 | 13.72 | 14.17 | 7.15 | 18.87 | 7.47 | 36.86 | 15.52 | 50.73 | 17.59
FGNN | 20.67 | 10.07 | 25.24 | 10.39 | 13.89 | 6.8 | 18.78 | 7.15 | 37.72 | 15.95 | 50.58 | 16.84
DHCN | 25.14* | 13.91* | 30.43* | 14.26* | 17.22 | 7.78 | 23.03 | 8.18 | 39.87 | 17.53 | 53.18 | 18.44
$S^{2}$-DHCN | 26.22 | 14.60 | 31.42 | 15.05 | 17.35 | 7.87 | 23.50 | 8.18 | 40.21 | 17.59 | 53.66 | 18.51
Improv. (%) | 10.71 | 7.87 | 12.25 | 8.84 | 18.32 | 9.15 | 19.70 | 8.68 | 6.19 | 9.32 | 5.46 | 4.97

  * *

The reported results on Tmall are not the best here. Refer to the ablation
study in Section 4 for the best results.

Table 2: Performances of all comparison methods on three datasets.

#### Baseline Methods.

We compare DHCN with the following representative methods:

  * •

Item-KNN(Sarwar et al. 2001) recommends items similar to the previously
clicked item in the session, where the cosine similarity between the vector of
sessions is used.

  * •

FPMC (Rendle, Freudenthaler, and Schmidt-Thieme 2010) is a sequential method
based on Markov Chain.

  * •

GRU4REC (Hidasi et al. 2015) utilizes a session-parallel mini-batch training
process and adopts ranking-based loss functions to model user sequences.

  * •

NARM (Li et al. 2017): is a RNN-based model that models the sequential
behavior to generate the recommendations.

  * •

STAMP (Liu et al. 2018): employs the self-attention mechanism to enhance
session-based recommendation.

  * •

SR-GNN (Wu et al. 2019b): applies a gated graph convolutional layer to learn
item transitions.

  * •

FGNN (Qiu et al. 2019): formulates the next item recommendation within the
session as a graph classification problem.

#### Evaluation Metrics.

Following (Wu et al. 2019b; Liu et al. 2018), we use P@K (Precision) and MRR@K
(Mean Reciprocal Rank) to evaluate the recommendation results.

#### Hyper-parameters Settings.

For the general setting, the embedding size is 100, the batch size for mini-
batch is 100, and the $L_{2}$ regularization is $10^{-5}$. For DHCN, an
initial learning rate 0.001 is used. The number of layers is different in
different datasets. For Nowplaying and Diginetica, a three-layer setting is
the best, while for Tmall, one-layer setting achieves the best performance.
For the baseline models, we refer to their best parameter setups reported in
the original papers and directly report their results if available, since we
use the same datasets and evaluation settings.

### Experimental Results

#### Overall Performance.

The experimental results of overall performance are reported in Table 2, and
we highlight the best results of each column in boldface. Two variants of DHCN
are evaluated, and $S^{2}$-DHCN denotes the self-supervised version. The
improvements are calculated by using the difference between the performance of
$S^{2}$-DHCN and the best baseline to divide the performance of the latter.
Analyzing the results in Table 2, we can draw the following conclusions.

  * •

The recently proposed models that consider the sequential dependency in the
sessions (i.e., GRU4REC, NARM, STAMP, SR-GNN and DHCN) significantly
outperform the traditional models that do not (i.e., FPMC). This demonstrates
the importance of sequential effects for session-based recommendation.
Furthermore, the fact that GRU4REC, NARM, STAMP, SR-GNN and DHCN all employ
the deep learning technique confirms its key role in session-based
recommendation models.

  * •

For the baseline models based on deep recurrent neural structure (e.g., RNN,
LSTM and GRU), NARM obtains higher accuracy in all settings. This is because
that GRU4REC only takes the sequential behavior into account and may have
difficulty in dealing with the shift of user preference. By contrast, NARM and
STAMP uses recurrent units to encode user behaviors and exerts an attention
mechanism over the items in a session, improving the recommendation results by
a large margin. The superior performance of NARM and STAMP proves that
assigning various importance value on different items within the session help
formulate user intent more accurately. Besides, STAMP outperforms NARM by
incorporating the short-term priority over the last item in a session, further
demonstrating that directly using RNN to learn user representations may lead
to recommendation bias but this can be avoided by replacing it with the
attention mechanism.

  * •

The GNNs-based models: SR-GNN and FGNN outperform RNNs-based models. The
improvements can be owed to the great capacity of graph neural networks.
However, the improvements are also trivial compared with the improvements
brought by DHCN.

  * •

Our proposed DHCN shows overwhelming superiority over all the baselines on all
datasets. Compared with SR-GNN and FGNN, our model has two advantages: (1) It
uses hypergraph to capture the beyond pairwise relations. By modeling each
hyperedge as a clique whose items are fully connected, the connections between
distant items can be exploited. (2) Also, our DHCN is lightweight than the SR-
GNN and FGNN because we use very limited parameters in hypergraph convolution
of the two channels, showing the efficicency of DHCN.

  * •

Although not as considerable as those brought by hypergraph modeling, the
improvements brought by self-supervised learning are still decent. In
particular, on the two datasets which have shorter average length of sessions,
self-supervised learning plays a more important role, which is line with our
assumption that the sparsity of session data might hinder the benefits of
hypergraph modeling, and maximizing mutual information between the two views
in DHCN could address it.

Figure 2: Contribution of each component.

#### Ablation Study.

The overwhelming superiority of DHCN shown in the last section can be seen as
the result of the joint effect of hypergraph modeling, and temporal factor
exploitation. To investigate the contributions of each module in DHCN, we
develop two variants of DHCN: DHCN-P and DHCN-NA. DHCN-P represents the
version without the reversed position embeddings, and DHCN-NA represents the
version without the soft attention mechanism. We compare them with the full
DHCN on Tmall and Diginetica.

As can be observed in Figure 2, the contributions of each component are
different on the two datasets. For Tmall, to our surprise, when removing the
reversed position embeddings or soft attention, the simplified version
achieves a performance increase on both metrics and the performance is even
better than that of the the full version. Considering that the Tmall dataset
is collected in a real e-commerce situation, this finding, to some degree,
validates our assumption that coherence may be more important than strict
order modeling. By contrast, in Diginetica, the reversed position embeddings
and soft attention are beneficial. When removing reversed position embedding
or soft attention, there is a performance drop on both metrics. Soft attention
contributes more on Diginetica, demonstraing the importance of different
priorities of items when generating recommendation.

#### Impact of Model Depth.

To study the impacts of hypergraph convolutional network’s depth in session-
based recommendation, we range the numbers of layers of the network within {1,
2, 3, 4, 5}. According to the results presented in Figure 3, DHCN is not very
sensitive to the number of layers on Diginetica and a three-layer setting is
the best. However, on Tmall, a one-layer network achieves the best
performance. Besides, with the number of layer increases, the performance on
MRR@20 drops. The possible cause could be the increasingly over-smoothed
representations of items.

Figure 3: The impacts of the number of layer. Figure 4: The impact of the
magnitude of self-supervised learning.

#### Impact of Self-Supervised Learning.

We introduce a hyper-parameter $\beta$ to $S^{2}$-DHCN to control the
magnitude of self-supervised learning. To investigate the influence of the
self-supervised task based on two-view contrastive learning, we report the
performance of $S^{2}$-DHCN with a set of representative $\beta$ values
{0.001, 0.01, 0.02, 0.03, 0.05}. According to the results presented in Figure
4, recommendation task achieves decent gains when jointly optimized with the
self-supervised task. For both datasets, learning with smaller $\beta$ values
can boost both Prec@20 and MRR@20, and with the increase of $\beta$, the
performance declines. We think it is led due to the gradient conflicts between
the two tasks. Besides, with lager $beta$, performance declines obviously on
MRR@20, which means that in some cases, it is important to make a trade-off
between the hit ratio and item ranks when choosing the value of $\beta$.

## Conclusion

Existing GNNs-based SBR models regard the item transitions as pairwise
relations, which cannot capture the ubiquitous high-order correlations among
items. In this paper, we propose a dual channel hypergraph convolutional
network for SBR to address this problem, Moreover, to further enhance the
network, we innovatively integrate self-supervised into the training of the
network. Extensive empirical studies demonstrate the overwhelming superiority
of our model, and the ablation study validates the effectiveness and rationale
of the hypergraph convolution and self-supervised learning.

## Acknowledgments

This work was supported by ARC Discovery Project (GrantNo.DP190101985,
DP170103954).

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#  Properties of Thorne-Żytkow object explosions

Takashi J. Moriya1,2 and Sergei I. Blinnikov3,4,5
1National Astronomical Observatory of Japan, National Institutes of Natural
Sciences, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
2School of Physics and Astronomy, Faculty of Science, Monash University,
Clayton, Victoria 3800, Australia
3 National Research Center "Kurchatov institute", Institute for Theoretical
and Experimental Physics (ITEP), Moscow 117218, Russia
4 Sternberg Astronomical Institute, Moscow State University, Moscow 119234,
Russia
5 Kavli Institute for the Physics and Mathematics of the Universe (WPI), The
University of Tokyo Institutes for Advanced Study, The University of Tokyo,
5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan E-mail:
<EMAIL_ADDRESS>(TJM)

(Accepted 2021 September 7. Received 2021 September 7; in original form 2021
July 26)

###### Abstract

Thorne-Żytkow objects are stars that have a neutron star core with an extended
hydrogen-rich envelope. Massive Thorne-Żytkow objects are proposed to explode
when the nuclear reactions sustaining their structure are terminated by the
exhaustion of the seed elements. In this paper, we investigate the
observational properties of the possible Thorne-Żytkow object explosions. We
find that Thorne-Żytkow object explosions are observed as long-duration
transients lasting for several years. If the accretion disk triggering the
explosions does not last for a long time, Thorne-Żytkow object explosions have
a luminosity plateau with about $10^{39}~{}\mathrm{erg~{}s^{-1}}$ lasting for
a few years, and then they suddenly become faint. They would be observed as
vanished stars after a bright phase lasting for a few years. If the accretion
disk is sustained for long time, the Thorne-Żytkow object explosions become as
bright as supernovae. They would be observed as supernovae with rise times of
several hundred days. We found that their photospheric velocities are
$2000~{}\mathrm{km~{}s^{-1}}$ at most, much smaller than those found in
supernovae. Supernovae with extremely long rise times such as HSC16aayt and SN
2008iy may be related to the explosions of Thorne-Żytkow objects.

###### keywords:

accretion, accretion discs – stars: neutron – stars: peculiar – supernovae:
general – supergiants

††pubyear: 2021††pagerange:  Properties of Thorne-Żytkow object explosions –
Properties of Thorne-Żytkow object explosions

##  1 Introduction

Thorne-Żytkow objects (TŻOs) are hypothetical stars that have a neutron star
core with an extended hydrogen-rich envelope (Thorne & Zytkow, 1975, 1977).
Such stars can be formed through the spiral-in of a neutron star into a
companion star following unstable mass transfer (Taam et al., 1978), a
supernova (SN) kick (Leonard et al., 1994) or an unsuccessful SN explosion
(Utrobin & Chugai, 2008).

Recent observations start to identify TŻO candidates. Levesque et al. (2014)
identified a chemical anomaly in a supergiant star HV2112 in Small Magellanic
Cloud (Worley et al., 2016; McMillan & Church, 2018) which is consistent with
a TŻO (Tout et al., 2014). It is also suggested that HV2112 may be an
asymptotic-giant-branch star rather than a TŻO (Tout et al., 2014; Beasor et
al., 2018; O’Grady et al., 2020). There are ongoing observational efforts to
discover TŻOs (DeMarchi et al., 2021).

Massive (more than about $16~{}\mathrm{M}_{\odot}$) TŻOs, which are supported
by the nuclear reactions near the neutron star surface (Cannon, 1993; Biehle,
1991), are expected to collapse when they burn all the seed elements required
for the nuclear reactions (Bisnovatyi-Kogan & Lamzin, 1984; Podsiadlowski et
al., 1995). Moriya (2018) suggested that massive TŻO collapses may lead to
their explosion. In this study, we investigate the outcomes of the potential
TŻO explosions and show their expected observational properties.

The rest of this paper is organized as follows. First, we show our assumptions
and methods to investigate the TŻO explosion properties in Section 2. Then we
present the expected TŻO explosion properties in Section 3. We discuss the
observational properties and possible TŻO explosion candidates in Section 4.
We summarize this paper in Section 5.

##  2 Methods

In this section, we briefly introduce our TŻO explosion model and our method
to estimate the properties of TŻO explosions.

###  2.1 Progenitor and explosion trigger

We take the 16 $\mathrm{M}_{\odot}$ TŻO model of Biehle (1991) as we did in
our previous study (Moriya, 2018). Assuming a typical TŻO angular velocity,
Moriya (2018) showed that the accretion disk around the central neutron star
is expected to appear at around 1 day after the onset of collapse when
$10^{-3}~{}\mathrm{M}_{\odot}$ is accreted. The subsequent accretion time-
scale becomes about 10 times longer than the free-fall time-scale, if we
assume a typical viscosity parameter of 0.1. The estimated accretion rate
$\dot{M}_{\mathrm{acc}}$ after 1 d since the onset of the collapse is
presented in Fig. 1.

Once the accretion disk is formed, a fraction ($\eta$) of accreted energy
could be released at the central region of the collapsing TŻO through an
accretion disk wind or jet. When the energy sufficient to unbind the TŻO has
been released an explosion of TŻO can be triggered. The energy input rate
$\dot{E}_{\mathrm{in}}$ at the centre is expressed as

$\dot{E}_{\mathrm{in}}=\eta\dot{M}_{\mathrm{acc}}c^{2},$ (1)

where $c$ is the speed of light. We adopt an efficiency $\eta=10^{-3}$ in this
study (e.g., Dexter & Kasen, 2013). Fig. 2 shows the total accreted mass
$M_{\mathrm{acc}}$ after the disk formation at 1 day after the collapse and
the total injected energy $\eta M_{\mathrm{acc}}c^{2}$. In this work, we
mainly consider the cases where the central energy injection occurs at wide
angles, not in collimated jets.

Once the energy injection from the accretion disk has released more energy
than the binding energy of the TŻO, the TŻO can explode. However, the
accretion energy injection could be terminated by the outflows that push back
the accreting materials. It is not clear when the accretion is terminated and
so we introduce another parameter, $t_{\mathrm{acc}}$, the time for which the
accretion continues. Because the binding energy of the 16 $\mathrm{M}_{\odot}$
TŻO is $5\times 10^{47}~{}\mathrm{erg}$, the accretion needs to last at least
for about 5 d to cause the TŻO to explode (Fig. 2). The TŻO explosion energy
depends on $t_{\mathrm{acc}}$. For $t_{\mathrm{acc}}=10,100$ and 1000 d the
explosion energies are $1.1\times 10^{48}~{}\mathrm{erg}$, $3.5\times
10^{49}~{}\mathrm{erg}$ and $3.3\times 10^{51}~{}\mathrm{erg}$, respectively.

Figure 1:  Accretion rate $\dot{M}_{\mathrm{acc}}$ towards the central compact
object after accretion disk formation that is estimated to occur at around 1
day after the onset of the TŻO collapse. The time $t$ is after the onset of
the TŻO collapse.  Figure 2: Mass accreted on to the central compact object
after accretion disk formation. The time $t$ is after the onset of the TŻO
collapse. The right axis shows the injected energy to the collapsing TŻO
envelope through the accretion with an efficiency $\eta=10^{-3}$. The dashed
line shows the binding energy ($5\times 10^{47}~{}\mathrm{erg}$) of the TŻO
modelled.

###  2.2 Radiation hydrodynamics calculations

In order to estimate the observational properties of TŻO explosions, we use
the one-dimensional multi-frequency radiation hydrodynamics code STELLA. We
refer to Blinnikov et al. (1998, 2000, 2006) for the full details of the code.
Briefly, STELLA calculates time-dependent equations of hydrodynamics and the
angular moments of intensity averaged over a frequency bin with the variable
Eddington method (Mihalas & Mihalas, 1984). Spectral energy distributions
(SEDs) are numerically evaluated at every time-step. When we define a
photosphere, we take the location where the Rosseland-mean optical depth
becomes $2/3$. In this work, we put thermal energy with the rate of Eq. 1 at 1
$\mathrm{M}_{\odot}$ from the centre. This is the surface of the central
neutron star supporting the TŻO. The energy injection begins 1 d after the
onset of the collapse and continues until $t_{\mathrm{acc}}$. Though we use a
one-dimensional code to investigate the TŻO explosion properties, the energy
injection from the accretion disk is not necessarily spherically symmetric.
The TŻO explosion properties that we show here are applicable when the energy
injection occurs on a large angular scale. Our models are not applicable if
the energy injection occurs on a small angular scale through such as in
collimated jets.

##  3 Results

Fig. 3 shows the bolometric light curves of the TŻO explosions with
$t_{\mathrm{acc}}=10,100$ and 1000 d. Their photospheric temperature and
velocity are presented in Fig. 4. The synthetic g, r and i band light curves
are presented in Fig. 5. Before the forward shock reaches the surface of the
progenitor, the bolometric luminosity stays at the TŻO luminosity of around
$5\times 10^{38}~{}\mathrm{erg~{}s^{-1}}$. The forward shock reaches the
surface at 105 d ($t_{\mathrm{acc}}=10~{}\mathrm{d}$) and 51 d
($t_{\mathrm{acc}}=100$ and 1000 d). The models with $t_{\mathrm{acc}}=100$
and 1000 d have the same shock-appearance date because the energy deposition
is the same in the two models at the beginning.

The explosion with $t_{\mathrm{acc}}=10~{}\mathrm{d}$ is observed as a Type
IIP SN-like transient with a very long plateau duration (Figs 3 and 5), as
analytically predicted by Moriya (2018). The plateau duration is about 500 d
and its bolometric luminosity is around $6\times
10^{39}~{}\mathrm{erg~{}s^{-1}}$, although the luminosity continues to
increase slowly until the end of the plateau. The absolute magnitudes in the
g, r and i bands are around $-7.8$, $-9.5$, and $-10.3$, respectively, but
they slightly depend on time. The photospheric velocity is relatively small
(50 to $70~{}\mathrm{km~{}s^{-1}}$, Fig. 4). The plateau phase is caused by
the hydrogen recombination as in the case of Type IIP SNe. Once the
recombination wave reaches the bottom of the envelope the plateau phase ends
and the luminosity drops.

The explosion model with $t_{\mathrm{acc}}=100~{}\mathrm{d}$ has a light curve
with a steady luminosity increase from around 150 d (Figs. 3 and 5). The
bolometric luminosity reaches a peak at 400 d with $3\times
10^{41}~{}\mathrm{erg~{}s^{-1}}$. The luminosity is sustained even after the
termination of the energy injection at 100 d because of the hydrogen
recombination as in the case of Type IIP SNe. In this model, the recession of
the recombination wave in the Lagrangian frame is slower than the ejecta
expansion in the Eulerian frame. Thus, the luminosity increases as the
recombination wave recedes towards the centre of the expanding envelope. The
time of the luminosity peak corresponds to the moment when the recombination
wave reaches at the bottom of the ejecta. Right after that the luminosity
suddenly declines. The photospheric velocity is around
$500~{}\mathrm{km~{}s^{-1}}$ (Fig. 4).

When the accretion energy input is sustained for 1000 d
($t_{\mathrm{acc}}=1000~{}\mathrm{d}$), the luminosity increases for 1000 d as
seen in Figs 3 and 5. The luminosity peak is reached at 1000 d when the energy
injection is terminated. The peak bolometric luminosity is $8\times
10^{43}~{}\mathrm{erg~{}s^{-1}}$. We find the peak optical magnitudes are
around $-20.7$. The long-lasting luminosity source is the steady accretion
towards the centre. The photospheric velocity is 1000 to
$2000~{}\mathrm{km~{}s^{-1}}$ (Fig. 4).

Figure 3: Synthetic bolometric luminosity $L$ of the TŻO explosions with
$t_{\mathrm{acc}}=10,100$ and 1000 d. The time $t$ is after the onset of the
TŻO collapse. The bolometric light curves of SN 1987A (Hamuy et al., 1988) and
SN 1999em (Bersten & Hamuy, 2009) are shown for comparison.

Figure 4: Photospheric temperature $T_{\mathrm{ph}}$ (top) and velocity
$v_{\mathrm{ph}}$ (bottom) evolution of the TŻO explosions. The time $t$ is
after the onset of the TŻO collapse.

Figure 5: Synthetic light curves of the TŻO explosions in the g, r, and i
bands. The time $t$ is after the onset of the TŻO collapse.

##  4 Discussion

In the previous section, we have found that TŻO outbursts should be observed
as long-lasting transients with durations of more than 100 d. The expected
luminosity range is diverse, ranging from those that are similar to massive
stars (about $10^{39}~{}\mathrm{erg~{}s^{-1}}$) to SNe (about
$10^{43}~{}\mathrm{erg~{}s^{-1}}$). The faintest TŻO explosions with
$t_{\mathrm{acc}}=10~{}\mathrm{d}$ are difficult to discover because of their
subtle luminosity change as well as their faintness. They could be discovered
as vanishing stars after a short luminosity increase lasting for a few years.
Such an event could be discovered during a survey for disappearing stars
(Kochanek et al., 2008; Adams et al., 2017b, a; Neustadt et al., 2021).

If the energy input from the accretion is sustained long enough, the TŻO
explosions become as bright as SNe and we expect to find them among SN
candidates. We predict that TŻO explosions have much longer time-scales (more
than $100~{}\mathrm{d}$) than typical SNe (less than $100~{}\mathrm{d}$).
Thus, TŻO explosions can be discovered among unusually long-lasting SNe. In
Fig. 6, we compare our TŻO light curve models with some long-lasting SNe.
HSC16aayt (Moriya et al., 2019) and SN 2008iy (Miller et al., 2010) are Type
IIn SNe showing narrow emission lines that are interpreted to originate from
the interaction between dense circumstellar media and SN ejecta. The
photospheric velocities of the TŻO explosions are predicted to be lower than
those of SNe and so relatively narrow spectral features may actually originate
from small photospheric velocities. It is also possible that the extended TŻO
progenitors experience mass loss that leads to the narrow spectral features.
The long-lasting SNe with the rise times of more than 100 d are promising
candidates for TŻO explosions.

Another well-known mysterious long-lasting SN with an extremely long duration
is iPTF14hls (Arcavi et al., 2017; Sollerman et al., 2019). Its origin is
still not clear (e.g., Woosley, 2018; Moriya et al., 2020). While the duration
is consistent with our TŻO explosion models, iPTF14hls has a photospheric
velocity of $4000~{}\mathrm{km~{}s^{-1}}$ which is much higher than what we
predict for our TŻO explosions. In addition, we do not expect to have the
bumpy light curves as found for iPTF14hls (Fig. 6). Thus, iPTF14hls is not
likely related to a TŻO explosions.

With an estimated Galactic TŻO birth rate of about
$10^{-4}~{}\mathrm{yr^{-1}}$ (Podsiadlowski et al., 1995; Ablimit et al.,
2021) and a Galactic SN rate of about $10^{-2}~{}\mathrm{yr^{-1}}$ (Li et al.,
2011), about 1% of SNe may come from TŻO explosions111We note that the event
rate estimate of Moriya (2018) has typos.. SNe lasting for more than 100 d are
rare and their event rates are not likely as high as 1% of SNe. This may mean
that most TŻO explosions are faint and difficult to discover. If the energy
injection from the accretion disk is not usually sustained for a long time,
most TŻO explosions would be observed only as faint transients. Thus,
searching for disappearing stars that accompany faint transients is more
likely a promising way to discover exploding TŻOs. Other possibilities are
that the TŻO birth rate is lower than $10^{-4}~{}\mathrm{yr^{-1}}$ or TŻOs do
not explode at all.

Distinguishing TŻO explosions from similar explosions that are triggered by
iron core collapse is quite challenging. Although core collapse of massive
stars around 16 $\mathrm{M}_{\odot}$ usually ends up with successful SN
explosions, it is possible that some of them fail to explode (Ugliano et al.,
2012; Sukhbold et al., 2016). If they form an accretion disk in a similar way
to that of the TŻO collapse, we can have a similar transient. The
identification of the progenitors is important to check if the explosions are
of TŻOs or not.

Figure 6:  Comparison of synthetic TŻO light curves having
$t_{\mathrm{acc}}=500$ and 700 d with long-lasting SNe, HSC16aayt (Moriya et
al., 2019), SN 2008iy (Miller et al., 2010), and iPTF14hls (Arcavi et al.,
2017; Sollerman et al., 2019). Time is arbitrary shifted to match the
observations and models.

We have so far approximated that the energy injection from the accretion disk
is spherically symmetric. It is possible, however, that the accretion disk
launches collimated jets. In such cases, the jets may lead to gamma-ray bursts
with a very long duration as proposed for Swift 1644+57 (Quataert & Kasen,
2012), although several different progenitors may lead to such gamma-ray
bursts with a very long duration (e.g., Soker, 2021).

##  5 Summary

We here investigated the observational properties of TŻO explosions. We found
that TŻO explosions lead to transients with the durations of 100 to
$1000~{}\mathrm{d}$. The possible luminosity range is quite diverse $10^{39}$
to $10^{44}~{}\mathrm{erg~{}s^{-1}}$. This depends on the duration of the
energy input from the accretion disk triggering the TŻO explosions. The
faintest TŻO explosions, which appear when the accretion energy injection is
not sustained for long, have similar luminosities to those of massive stars.
As in the case of Type IIP SNe, the light curves are expected to have a long
plateau phase. The plateau phase of TŻO explosions lasts for a few years.
Their photospheric velocity is 50 to $70~{}\mathrm{km~{}s^{-1}}$. They may be
observed as vanishing stars after brightening for several years. The bright
TŻO explosions, which appear when the accretion energy injection lasts a long
time, can be observed as SNe with rise times of several hundred days. We found
that the expected light curves are similar to those of some long-lasting SNe
such as HSC16aayt and SN 2008iy, although they are Type IIn SNe. The
photospheric velocity is expected to be of the order of
$100~{}\mathrm{km~{}s^{-1}}$.

While we only have a few SNe that have long-duration light curves similar to
our synthetic light curves from TŻO explosions, many long-duration SNe will be
found in the coming era of the Rubin Observatory’s Legacy Survey of Space and
Time (LSST). Since the faintest TŻO explosions are expected to have
luminosities that are similar to massive stars, they can be eventually found
as disappeared stars.

## Acknowledgements

We thank the referee, Christopher Tout, for providing a constructive report
that improved this manuscript. T.J.M. thanks Emily Levesque for organizing the
workshop “An Exploration of Thorne-Żytkow Objects” during the 238th meeting of
the American Astronomical Society which initiated this work. T.J.M. is
supported by the Grants-in-Aid for Scientific Research of the Japan Society
for the Promotion of Science (JP18K13585, JP20H00174, JP21K13966, JP21H04997).
S.I.B. is supported by grant RSF 19-12-00229 for the development of STELLA
code. This research has been supported in part by the RFBR (19-52-50014)-JSPS
bilateral program. Numerical computations were in part carried out on PC
cluster at Center for Computational Astrophysics (CfCA), National Astronomical
Observatory of Japan.

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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