# Collective Knowledge Graph Completion with Mutual Knowledge Distillation

Weihang Zhang
Imperial College London
<EMAIL_ADDRESS>
Ovidiu Serban
Imperial College London
<EMAIL_ADDRESS>
Jiahao Sun
Imperial College London
Royal Bank of Canada
<EMAIL_ADDRESS>
Yi-ke Guo
Imperial College London
<EMAIL_ADDRESS>

###### Abstract

Knowledge graph completion (KGC), the task of predicting missing information
based on the existing relational data inside a knowledge graph (KG), has drawn
significant attention in recent years. However, the predictive power of KGC
methods is often limited by the completeness of the existing knowledge graphs
from different sources and languages. In monolingual and multilingual
settings, KGs are potentially complementary to each other. In this paper, we
study the problem of multi-KG completion, where we focus on maximizing the
collective knowledge from different KGs to alleviate the incompleteness of
individual KGs. Specifically, we propose a novel method called CKGC-CKD that
uses relation-aware graph convolutional network encoder models on both
individual KGs and a large fused KG in which seed alignments between KGs are
regarded as edges for message propagation. An additional mutual knowledge
distillation mechanism is also employed to maximize the knowledge transfer
between the models of “global” fused KG and the “local” individual KGs.
Experimental results on multilingual datasets have shown that our method
outperforms all state-of-the-art models in the KGC task.

##  1 Introduction

Knowledge Graphs (KGs) are often viewed as large-scale semantic networks that
store facts as triples in the form of (subject entity, relation, object
entity). KGs have been widely adopted in many industrial applications because
they capture the multi-relational nature between real-world entities well.
However, real-world KGs are usually incomplete (Dong et al., 2014). To tackle
the incompleteness problem, there has been a surge of research interest in the
KGC task in recent years. KGC and many other KG-based tasks are usually based
on knowledge representation learning (KRL), in which entities and relations in
a KG are encoded into low-dimensional vectors. With the recent advances in
Graph Neural Network(GNN) (Scarselli et al., 2009), many recently published
methods like R-GCN (Schlichtkrull et al., 2018) and CompGCN (Vashishth et al.,
2020) all employed an encoder-decoder mechanism to tackle the KGC problem:
variations of Graph Convolutional Networks (GCN) (Kipf and Welling, 2017) are
used as encoders to generate embeddings for entities and relations in a KG,
and traditional shallow KG embedding methods like TransE (Bordes et al., 2013)
and DistMult (Yang et al., 2015) are used as decoders for the KGC task. With
the additional message propagation and aggregation mechanism of graph
convolution operation in the encoding stage, these methods have shown more
promising results on the KGC task than traditional knowledge graph embedding
methods. However, not all the missing information can be inferred based on the
information inside a KG. Even with a better encoding mechanism of GCNs, the
expressiveness and quality of trained models can still be limited by the
sparseness and incompleteness of the individual KG the model is trained on. At
the same time, real-world entities are usually captured in more than one KG
from either different sources or languages. The common entities in the
disjoint real-world KGs can potentially serve as bridges to better connect
them and transfer additional knowledge to one another to alleviate the
sparseness problem faced by almost all real-world KGs. The common entities
across different KGs are often known as seed alignments, which usually
originate from the manual annotation of human annotators. Because of the scale
and size of KGs nowadays, seed alignments are usually relatively scarce.

In this paper, we focus on the multi-KG completion problem and aim to
collectively utilize multiple KGs and seed alignments between them to maximize
the KGC task performance on each individual KG. We propose a novel method that
concurrently trains CompGCN-based (Vashishth et al., 2020) encoders on each
individual KGs and a fused KG in which seed alignments are regarded as edges
for connecting KGs together and augmented message propagation for “knowledge
transfer”. During the concurrent training, we also employ the mutual knowledge
distillation mechanism. CompGCN-based encoders on individual KGs and the fused
KG are trained to learn potentially complementary features from each other.
The intuition behind the mutual knowledge distillation process is that the
small encoders trained on individual KGs capture local semantic relationships
while the large encoder trained on the large fused KG captures the global
semantic relationships better because of the intra-KG message propagation. In
the mutual knowledge distillation process, the small and large encoders take
turns to become “teachers” in the knowledge distillation to encourage mutual
knowledge transfer between them. Lastly, we use an ensemble to combine the
predictions from the individual KG models and the fused KG model to produce
the KGC predictions on the test set for each individual KG. Figure 1 provides
an illustrative figure of the overall structure of CKGC-CKD.

The main contribution of this paper can be summarized as follows: 1) we
propose a novel augmented CompGCN encoder to facilitate intra-KG knowledge
transfer and tackle the multi-KG completion task; 2) we propose a novel mutual
knowledge distillation mechanism to encourage collaborative knowledge transfer
between the models trained on individual KGs and globally fused KG.
Experimental results on popular multilingual datasets show that our proposed
method outperforms all state-of-the-art models. Extensive ablation studies are
conducted on monolingual and multilingual datasets to demonstrate the
contribution of each component in the proposed method.

Figure 1: An illustrative figure of the proposed CKGC-CKD with 2 KGs.

##  2 Related work

###  2.1 Knowledge graph embeddings

The research on knowledge graph embeddings has gained significant attention in
recent years. This task aims to encode entities and relations of a KG into
low-dimensional vectors. Traditional translation-based methods like TransE
(Bordes et al., 2013), TransH (Wang et al., 2014), TransR (Lin et al., 2015),
as well as the semantic matching models like RESCAL (Nickel et al., 2011) and
DistMult (Yang et al., 2015), all achieved promising results on the KGC task.
These methods all served as decoders on shallow embeddings to exploit the
local structural information within triples and capture the semantics of
relation triples. Other than the triple-based embedding models, another stream
of recent works (Schlichtkrull et al., 2018; Vashishth et al., 2020; Yu et
al., 2021) all employed the graph structure to propagate information between
adjacent entities and encode them into embeddings. Specifically, variants of
the GCN model are used as encoders to embed entities and relations in the
heterogeneous graphs into vectors. Traditional knowledge graph embedding
methods like TransE are then used as decoders for various downstream tasks
like KGC and node classification.

###  2.2 KGC across multiple Knowledge graphs

Compared to KGC on a single KG, KGC across multiple KGs is a relatively under-
explored area. Wang et al. (2021) proposed ATransN, an adversarial embedding
transfer network which aims to facilitate the knowledge transfer from a pre-
trained embedding of a teacher KG to a student KG with a set of seed
alignments. Chen et al. (2020) was the first to propose multilingual KGC
problem setting and tackled the problem by generating ensemble results on
shared triples between KGs in different languages. On the same multilingual
problem setting, Singh et al. (2021) proposed AlignKGC, which employs a multi-
task strategy to jointly trains KGC, entity alignment and relation alignment
tasks. Most recently, Huang et al. (2022) proposed SS-AGA, which models seed
alignments as edges to fuse multiple knowledge graphs while using a generator
model to dynamically capture more potential alignments between entities and
iteratively add more edges to the graph. Additionally, Sourty et al. (2020)
proposed KD-MKB, which assumes the existence of both shared relations and
shared entities across individual KGs, and therefore tackles multi-KG
completion tasks from a knowledge distillation perspective.

##  3 Methods

###  3.1 Preliminaries

The framework of a multi-KG completion task involves two or more KGs. Without
loss of generality, we assume there are a total of $m$ KGs in the problem
setting. We formalize the $i$-th heterogeneous KG in the task as
$KG_{i}=\\{E_{i},R_{i},T_{i}\\}$, where ${E_{i},R_{i},T_{i}}$ respectively
represent the entity set, relation set, and fact triple set of $KG_{i}$. A set
of seed alignments between KGs, known before training, is denoted by
$S_{KG_{i},KG_{j}}=\\{(e_{i},\sim,e_{j}):(e_{i},e_{j})\in E_{i}\times
E_{j}\\}$, where $\sim$ denotes the equivalence relation. The complete set of
seed alignments can then be denoted by
$S_{align}=\cup_{i=1}^{m}\cup_{j=i+1}^{m}S_{KG_{i},KG_{j}}$. We can then
formalize the large fused KG connected by seed alignments as
$KG_{f}=\\{E_{f},R_{f},T_{f}|E_{f}=\cup_{i=1}^{m}E_{i},R_{f}=\cup_{i=1}^{m}R_{i},T_{f}=(\cup_{i=1}^{m}T_{i})\cup
S_{align}\\}$. Let $M_{i}$ and $M_{f}$ denote the encoder models for the
$i$-th individual KG and the fused KG, respectively.

###  3.2 Augmented CompGCN Link Prediction

We use CompGCN (Vashishth et al., 2020) as our encoders for the knowledge
graph embeddings. In the method, CompGCN encoders with non-parametric
composition operations are trained on each individual KGs and the fused KG
concurrently. In standard CompGCN, the update equation of CompGCN node
embeddings can be written as:

$\displaystyle h^{t}_{v}=f(\Sigma_{(u,r)\in N(v)}Me(u,r))$ (1) $\displaystyle
Me(u,r)=W_{\lambda(r)}\phi(h^{t-1}_{u},h^{t-1}_{r})$ (2)

, where $h^{t}_{v}$ denotes the updated embedding for node $v$ at $t$-th
layer, $N(v)$ denotes the set of neighboring entities and relations of node
$v$, $h^{t-1}_{u}$ and $h^{t-1}_{r}$ denotes the embeddings for node $u$ and
relation $r$ at ($t$-$1$)-th layer respectively, $\phi$ denotes the non-
parametric composition operation and $W_{\lambda(r)}$ denotes the direction
specific transformation matrix where $\lambda$ denotes the direction of
relation. In our method, the vanilla CompGCN encoder is used without
modification on individual KGs, while we decide to use an augmented CompGCN
encoder for better knowledge transfer on the fused $KG_{f}$. Specifically,
although seed alignments are viewed as regular relations in the fused KG, we
remove the composition operator for message propagation between the seed
alignments and instead use the standard non-relation-specific message passing
between KGs. The augmented message function in the fused KG can then be
written as:

$\displaystyle
Me(u,r)=\begin{cases}W_{align}h^{t-1}_{u}\qquad\quad\text{if}(u,\sim,v)\in
S_{align}\\\
W_{\lambda(r)}\phi(h^{t-1}_{u},h^{t-1}_{r})\quad\text{otherwise}\end{cases}$
(3)

, where $W_{align}$ denotes the transformation matrix specifically for intra-
KG message propagation. The composition operation is removed because we view
the cross-KG equivalence as a bi-directional relationship different from the
triples inside KGs. Additionally, many existing methods (Wang et al., 2021;
Singh et al., 2021) use a loss regularization to ensure the equivalent
entities in each KG have similar embeddings with or without transformation.
However, instead of imposing the regularization directly on the training loss
term, we impose a softer regularization in the message passing augmentation,
where the contextualized node embeddings of entities in each knowledge graph
are passed to their counterparts in other KGs during encoding. As a result,
the contextualized embedding of entities in each KG can be shared across the
KGs by the augmented message propagation in the encoding phase and optimized
during the training of the KGC task on fact triples.

The encoded entity and relation embeddings are then passed to the decoder,
which performs the link prediction task on triples in KG, and computes the
knowledge representation loss. The margin-based knowledge representation loss
can be written as:

$\displaystyle L_{T}=\Sigma_{t_{i}\in T_{i},t_{i}^{\prime}\in
T_{i}^{\prime}}f(t_{i})-f(t_{i}^{\prime})+\gamma$ (4)

, where $T_{i}^{\prime}$ denotes the negative samples created from corrupting
head or tail entity in triple $t_{i}$; $f(t_{i})$ denotes the scoring function
for TransE or similar knowledge embedding model; and $\gamma$ denotes the
margin, which is a hyperparameter describing the ideal distance between
positive triples and negative triples.

###  3.3 Mutual Knowledge Distillation

We employ the mutual knowledge distillation mechanism between each model on
individual KGs $M_{i}$ and the model on the fused KG $M_{f}$. At each training
step, each $M_{i}$ pairs with $M_{f}$ to conduct mutual knowledge
distillation, where $M_{i}$ and $M_{f}$ learn simultaneously from each other
via a mimicry loss that measures the difference between the posterior
predictions of each other on the KGC task on triples $T_{i}$ in the
corresponding $KG_{i}$. Three different KGC tasks are used for mutual
knowledge distillation: for a triple $t=(s,r,o)$, the task is to predict the
missing component given the other two in the triple, i.e., head prediction,
tail prediction and relation prediction. The distillation loss can be written
as:

$\displaystyle L_{D}^{i}=\sum_{t_{i}\in T_{i}}\sum_{\beta\in
Task}d_{KL}(t_{i},\beta)$ (5) $\displaystyle
d_{KL}(t_{i},\beta)=D_{KL}(P_{i}^{\beta}(t_{i}),P_{f}^{\beta}(t_{i}))$ (6)

, where $Task$ denotes three KGC tasks, $D_{KL}$ denotes the Kullback Leibler
(KL) Divergence, and $P$ denotes the categorical distribution predicted by the
knowledge graph embedding scoring function on task $\beta$. As an example, for
tail prediction of triple $t_{i}=(s_{i},r_{i},o_{i})$, the categorical
distribution can be written as softmax of tail prediction across all
candidates:

$\displaystyle
P_{i}(t_{i})=\frac{exp(f(M_{i}(s_{i}),M_{i}(r_{i}),M_{i}(o_{i})))}{\sum_{o_{j}\in
E_{i}}exp(f(M_{i}(s_{i}),M_{i}(r_{i}),M_{i}(o_{j})))}$ (7)

, where $M_{i}()$ denotes the embedding look-up operation for entities and
relations from encoder model $M_{i}$. In practice, predicting across all
candidates $E_{i}$ and comparing the categorical distribution across all
entities can be inefficient due to the size of KG. Therefore, we employ the
top-k sampling technique used in the work of Sourty et al. (2020) by using the
“teacher” model to select the top-k most confident candidates for the
categorical distribution comparison in the mutual distillation.

Additionally, we enabled the mutual distillation process to be performance-
aware as the performance of individual model $M_{i}$ and the fused model
$M_{f}$ could differ by some margin. A recent work (Xie and Du, 2022) on
knowledge distillation pointed out that a worse-performing model could
potentially generate negative knowledge transfer and lead to collective
failure. Therefore, we adopted a softer restriction than Xie and Du (2022) and
designed a hyperparameter $\theta$ such that the worse-performing model is
allowed to become a teacher and generate soft training target only if its mean
reciprocal rank metric on the validation set is less $\theta$ away from the
better-performing model.

###  3.4 Training

The overall loss term combines the knowledge representation and mutual
knowledge distillation loss:

$\displaystyle L=L_{T}+\alpha L_{D}$ (8)

, where $\alpha$ is a hyperparameter controlling the contribution of mutual
distillation loss to the overall loss term. The models $M_{i}$ and $M_{f}$ are
trained concurrently on KGC tasks while learning from the best-performing
model of each other via the mutual distillation process. In practice, the
training process is separated into two stages for better convergence and
faster training. In the first stage, individual and fused models are trained
independently with only knowledge representation loss, while in the second
stage, knowledge distillation losses are introduced so that models can
mutually learn from each other.

###  3.5 Ensemble Prediction

In the end, the output for KGC tasks is generated by combining predictions
from models $M_{i}$ and $M_{f}$ using an ensemble. Concretely, the for triple
$t_{i}\in T_{i}$, the final scoring function becomes:
$F(t_{i})=f(M_{i}(t_{i}))+f(M_{f}(t_{i}))$. The ensemble scores are then used
for further ranking and evaluation.

Table 1: Results on DBP-5L dataset.

| EL | JA | ES | FR | EN
---|---|---|---|---|---
| H@1/H@10/MRR | H@1/H@10/MRR | H@1/H@10/MRR | H@1/H@10/MRR | H@1/H@10/MRR
KenS | 28.1 / 56.9 / - | 32.1 / 65.3 / - | 23.6 / 60.1 / - | 25.5 / 62.9 / - | 15.1 / 39.8 / -
CG-MuA | 21.5 / 44.8 / 32.8 | 27.3 / 61.1 / 40.1 | 22.3 / 55.4 / 34.3 | 24.2 / 57.1 / 36.1 | 13.1 / 33.5 / 22.2
AlignKGC | 27.6 / 56.3 / 33.8 | 31.6 / 64.3 / 41.6 | 24.2 / 60.9 / 35.1 | 24.1 / 62.3 / 37.4 | 15.5 / 39.2 / 22.3
SS-AGA | 30.8 / 58.6 / 35.3 | 34.6 / 66.9 / 42.9 | 25.5 / 61.9 / 36.6 | 27.1 / 65.5 / 38.4 | 16.3 / 41.3 / 23.1
KGC-I | 28.9 / 66.8 / 41.6 | 30.3 / 61.7 / 41.4 | 24.8 / 61.2 / 37.5 | 25.8 / 64.1 / 39.1 | 20.5 / 58.6 / 33.5
KGC-A | 43.4 / 89.2 / 60.1 | 42.7 / 83.6 / 57.2 | 32.8 / 77.2 / 48.4 | 35.1 / 81.4 / 51.3 | 25.1 / 67.2 / 39.6
CKGC-CKD |  49.2 / 88.5 / 63.9 |  48.4 / 84.6 / 60.9 |  37.5 / 77.6 / 51.8 |  40.9 / 83.0 / 55.8 |  28.8 / 67.1 / 42.1

Table 2: Results on E-PKG dataset.

| EN | DE | FR | JA | ES | IT
---|---|---|---|---|---|---
| H@1/H@10/MRR | H@1/H@10/MRR | H@1/H@10/MRR | H@1/H@10/MRR | H@1/H@10/MRR | H@1/H@10/MRR
KenS | 26.2 / 69.5 / - | 24.3 / 65.8 / - | 25.4 / 68.2 / - | 33.5 / 73.6 / - | 21.3 / 59.5 / - | 25.1 / 64.6 / -
CG-MuA | 24.8 / 67.9 / 40.2 | 22.9 / 64.9 / 38.7 | 23.0 / 67.5 / 39.1 | 30.4 / 72.9 / 45.9 | 19.2 / 58.8 / 33.8 | 23.9 / 63.8 / 37.6
AlignKGC | 25.6 / 68.3 / 40.5 | 22.1 / 65.1 / 38.5 | 22.8 / 67.2 / 38.8 | 31.2 / 72.3 / 46.2 | 19.4 / 59.1 / 34.2 | 24.2 / 63.4 / 37.3
SS-AGA | 26.7 / 69.8 / 41.5 | 24.6 / 66.3 / 39.4 | 25.9 / 68.7 / 40.2 | 33.9 / 74.1 / 48.3 | 21.0 / 60.1 / 36.3 | 24.9 / 63.8 / 38.4
KGC-I | 44.5 / 81.7 / 58.6 | 36.3 / 75.4 / 50.4 | 36.5 / 77.4 / 51.2 | 44.5 / 83.5 / 59.3 | 30.2 / 67.3 / 43.6 | 37.4 / 75.1 / 51.6
KGC-A | 44.3 / 82.8 / 58.8 | 38.5 / 78.7 / 53.1 | 37.1 / 79.9 / 52.5 | 46.2 / 84.7 / 60.8 | 36.1 / 73.8 / 49.6 | 40.1 / 78.4 / 54.3
CKGC-CKD |  46.6 / 84.1 / 60.7 |  40.0 / 79.8 / 54.4 |  39.0 / 81.0 / 54.2 |  47.5 / 85.5 / 62.1 |  36.9 / 75.3 / 50.5 |  41.6 / 80.0 / 55.8

##  4 Experiments

###  4.1 Basic settings

All of our experiments are conducted on a Linux server running Ubuntu 20.04
with 64GB memory and a Nvidia A100 GPU.

We perform experiments and compare the performance of the proposed CKGC-CKD
method with the state-of-the-art models on the existing multilingual dataset
DBP-5L (Chen et al., 2020) and E-PKG (Huang et al., 2022). DBP-5L is a dataset
sampled from DBpedia (Auer et al., 2007); it contains five KGs from different
languages: English (EN), French (FR), Spanish (ES), Japanese (JA) and Greek
(EL). E-PKG is an industrial multilingual E-commerce product KG dataset, which
describes phone-related products across six different languages: English (EN),
German (DE), French(FR), Japanese (JA), Spanish (ES) and Italian (IT). In this
work, we follow the evaluation scheme of previous works (Chen et al., 2020;
Singh et al., 2021; Huang et al., 2022): for a test triple $(h,r,t)$, use the
embedding model to rank all possible answers to tail prediction query
$(h,r,?)$; and apply the MRR(mean reciprocal ranks), Hit@1 and Hit@10 metrics
under filtered settings (Wang et al., 2014; Yang et al., 2015) to evaluate the
performance of the models. The reported CKGC-CKD model uses a 1-layer encoder,
with TransE as a knowledge embedding decoder and an embedding dimension of
100. However, CKGC-CKD can be easily extended to use other knowledge-embedding
decoders. Deeper encoder can also be explored if more hardware resources
become available.

Table 3: Ablation study results on DBP-5L and D-W-15K-LP.

|  | DBP-5L | D-W-15K-LP |
---|---|---|---|---
| Metric | EL | JA | ES | FR | EN | DBpedia | Wikidata |
KGC-I |  | H@1
---
H@10
MRR
| 22.0
---
49.1
31.3
| 23.1
---
44.7
30.7
| 19.0
---
44.1
27.9
| 21.1
---
45.3
29.5
| 17.2
---
45.5
26.9
| 29.9
---
54.2
38.4
| 25.5
---
49.2
34.2
KGC-C |  | H@1
---
H@10
MRR
| 36.4
---
69.5
47.9
| 32.7
---
63.6
43.6
| 27.0
---
57.1
37.6
| 28.7
---
59.7
39.5
| 20.2
---
50.8
30.7
| 29.9
---
55.3
38.8
| 26.8
---
50.4
35.4
KGC-A |  | H@1
---
H@10
MRR
| 37.9
---
71.5
49.9
| 35.5
---
67.0
46.5
| 28.0
---
59.5
39.1
| 29.4
---
62.2
40.9
| 21.0
---
53.8
32.3
| 30.7
---
55.6
39.3
| 27.5
---
50.8
35.8
KGC-I-D |  | H@1
---
H@10
MRR
| 37.8
---
66.0
48.1
| 35.4
---
62.4
44.9
| 28.6
---
55.4
38.0
| 30.5
---
58.8
40.6
| 22.3
---
50.3
31.8
| 31.2
---
54.9
39.4
| 29.2
---
49.8
36.5
KGC-A-D |  | H@1
---
H@10
MRR
| 40.2
---
71.8
51.3
| 37.8
---
66.8
47.9
| 30.2
---
59.5
40.6
| 31.9
---
63.1
42.8
| 23.0
---
52.7
33.3
| 31.7
---
55.7
40.0
| 29.2
---
50.7
36.9
CKGC-CKD |  | H@1
---
H@10
MRR
| 41.4
---
70.5
52.1
| 38.7
---
67.2
48.6
| 31.3
---
59.5
41.2
| 32.7
---
63.4
43.6
| 23.2
---
53.4
33.6
| 31.7
---
55.8
40.1
| 29.3
---
50.8
37.0

###  4.2 Results

In table 1 and 2, we present the experiment results on the DBP-5L dataset
(Chen et al., 2020) and E-PKG dataset (Huang et al., 2022) respectively 111We
directly report the benchmarking results from the work of Huang et al. (2022)
for the first four rows in the table. However, we discovered that the filtered
setting used in the benchmark was slightly different from the traditional
filtered setting: Huang et al. (2022) assumes the candidate space during
testing excludes all positive triples from the training set. At the same time,
the traditional filtered setting also excludes validation and test set. For
fairness of comparison, all results we report in the table adopted the
filtered setting by Huang et al. (2022).. In the tables, performances of two
extra baseline models are reported: KGC-I refers to the standard CompGCN
encoder model trained on individual KG, and KGC-A refers to the augmented
message propagation encoder trained on the fused KG. It can be observed that
the proposed CKGC-CKD method outperforms all baseline and state-of-the-art
models on both datasets. Compared to the previous cross-lingual models, the
individually trained KGC-I model on each language can already achieve similar
performance in most languages, which verifies the effectiveness of the CompGCN
encoder. The KGC-A model trained on the fused KG provided a large margin over
the KGC-I and the previous models. This implies that the inclusion of multiple
KGs truly helps the KGC task of each other and also verifies the benefit of
the augmented cross-KG message propagation even without explicit data leakage
(direct swapping of relations between aligned entities). In the end, with
mutual knowledge distillation between KGC-I and KGC-A enabled, the CKGC-CKD
model uses the ensemble predictions from both distilled models. This achieves
the best performances in the table across almost all of the metrics. Overall,
compared to the KGC-I model trained on monolingual KG, CKGC-CKD achieves a
significant gain on the lower-resource languages like Greek(EL) in DBL-5L and
Japanese(JA) in E-PKG, which verifies the common belief that lower-resource
KGs would typically benefit from the multi-KG learning setting. Moreover, we
also observed performance improvement on relatively rich languages like
French(FR) and English(EN) also in the experiments.

##  5 Ablation study

###  5.1 Contribution of each component

In table 3, we report the results of our ablation studies to analyze how each
of the components in the proposed method affects the results. We report the
ablation study results on the multilingual DBP-5L dataset and a monolingual
self-generated D-W-15K-LP dataset. D-W-15K-LP is a dataset generated from the
entity alignment benchmarking datasets D-W-15K (Sun et al., 2020). To mimic a
more real-life setting, we employed the sampling strategy proposed in the work
of Sun et al. (2021) to create dangling entities (entities without alignment
across KGs) in the KGs. In the sampling process, triples containing removed
entities are excluded by removing part of the alignments from KGs. This
results in a more sparse dataset with dangling entities in each individual KG.
In addition to the KGC-I and the KGC-A models reported in the section 4, we
additionally report the performance of several ablation models: KGC-C refers
to the ablation model trained on fused KG without augmented message
propagation, KGC-I-D and KGC-A-D respectively represent the ablation models
with mutual distillation enabled for KGC-I and KGC-A. Therefore, the reported
CKGC-CKD is the ensemble results of KGC-A-D and KGC-I-D. For a more complete
and universal comparison, in the ablation study, we use the traditional “link
prediction” task that includes both head prediction and tail prediction with
the traditional filtered setting used in the works of Wang et al. (2014) and
Yang et al. (2015).

On both datasets, we can observe a clear margin that the KGC-A model created
over the KGC-C model, which verifies the effectiveness of augmented message
propagation. Additionally, on both datasets, the distillation-enabled KGC-I-D
and KGC-A-D models have shown superior performance in almost all metrics over
the KGC-I and KGC-A models, respectively. This has shown that the mutual
knowledge distillation process benefits individual and fused models. Lastly,
CKGC-CKD achieves the best performances in most metrics, which verifies the
gains provided by the ensemble technique. An interesting observation is that
even after the mutual knowledge distillation, the KGC-I-D models still perform
slightly worse than the fused model KGC-A-D, and the difference in performance
also varies across different KG. One possible reason behind this observation
is that we used a constant $\alpha$ for all KGs in one dataset to control the
trade-off between knowledge distillation loss and knowledge representation
loss. Limited by the hardware resources, we did not explore possibilities of
assigning different $\alpha$ for each KG and decided to leave that for future
work that possibly explores a fine-tuning stage of the model to better
reconcile the difference and imbalance of resources in each KG.

Table 4: Results of CKGC-CKD models w/ and w/o extra triples introduced by
alignment-informed meta-paths, metrics reported on link prediction task under
standard filtered settings.

DBP-5L | EL | JA | ES | FR | EN
---|---|---|---|---|---
H@1/H@10/MRR | H@1/H@10/MRR | H@1/H@10/MRR | H@1/H@10/MRR | H@1/H@10/MRR
w/ extra triples |  49.8 / 80.8 / 61.0 |  44.6 / 74.8 / 55.51 |  39.3 / 69.0 / 50.0 |  39.9 / 70.9 / 51.7 |  28.3 / 59.4 / 39.2
w/o extra triples | 41.4 / 70.5 / 52.1 | 38.7 / 67.2 / 48.6 | 31.3 / 59.5 / 41.2 | 32.7 /63.4 / 43.6 | 23.2 / 53.4 / 33.6

E-PKG | EN | DE | FR | JA | ES | IT
---|---|---|---|---|---|---
H@1/H@10/MRR | H@1/H@10/MRR | H@1/H@10/MRR | H@1/H@10/MRR | H@1/H@10/MRR | H@1/H@10/MRR
w/ extra triples | 42.4 / 61.8 / 49.3 | 21.8 / 42.3 / 29.4 | 24.9 / 43.3 / 32.0 | 22.9 / 42.9 / 30.5 | 22.9 / 44.5 / 31.1 | 28.4 / 46.4 / 35.4
w/o extra triples |  46.0 / 66.8 / 53.6 |  23.6 / 46.0 / 31.9 |  26.1 / 46.4 / 33.9 |  24.5 / 45.5 / 32.5 |  24.1 / 46.1 / 32.5 |  29.3 / 48.5 / 36.7

###  5.2 Discussion on the alignment-informed meta-path

With multilingual datasets sharing the same relation schema, there are obvious
mechanisms to introduce additional triples information via an alignment-
informed meta-path. For example, a meta-path of
($E_{a}\xrightarrow{SAMEAS}E_{b}\xrightarrow{R_{a}}E_{c}\xrightarrow{SAMEAS}E_{d}$)
implies a new triple $(E_{a}\xrightarrow{R_{a}}E_{d})$, this is usually
referred as parameter-swapping (Sun et al., 2020). When the dataset involves
more than two KGs, additional alignments can potentially also be inferred via
meta-path: $(E_{a}\xrightarrow{SAMEAS}E_{b}\xrightarrow{SAMEAS}E_{c})$ implies
$(E_{a}\xrightarrow{SAMEAS}E_{c})$. Additional triples information can be
generated or shared using the two alignment-informed meta-paths above.
However, this relies heavily on the quality annotation of seed alignments
between the KGs, which is not usually guaranteed, especially in scenarios
involving multiple KGs. In table 4, we report the link prediction performances
(incl. both head prediction and tail prediction) of our method with and
without alignment-informed meta-path included in training on the two
multilingual datasets, in which we have observed fairly contradictory
phenomenons: DBP5L dataset benefits massively from the additional triples from
alignment-informed meta-path, while on the E-PKG dataset we have observed a
decrease in performance when triples swapping are enabled. From further
analysis of the dataset, we hypothesized that the inconsistent observations on
both datasets mainly originate from the quality of seed alignment annotations.
Specifically, when we connect all entities with seed alignments annotations to
form an “alignment graph”, we discovered several connected components of size
over 1000 in such a graph on the E-PKG dataset, suggesting that over 1000
entities across the six KGs are annotated to be aligned to each other. This
could either be due to incorrect and noisy annotations or potentially
duplicate entities within monolingual KGs. A similar effect was not observed
in DBP-5L. The effect of lower quality KGs and seed alignments was enlarged
when triples generated from alignment-informed meta-path were introduced,
generating a prohibitively large number of inferred triples, resulting in the
inconsistent observations in table 4. Compared to DBP-5L, which was generated
from the sampling of the widely adopted and verified DBpedia (Auer et al.,
2007), E-PKG was a recently constructed industrial E-commerce product KG
dataset (Huang et al., 2022), which represents the quality of real-life
industrial dataset to some extent. Therefore, we have decided not to make any
assumptions about the dataset’s quality in our settings and report the main
results without including triples generated from alignment-informed meta-paths
in our training.

Figure 2: Relation correlation heatmap of Greek KG v.s. fused KG.

###  5.3 Knowledge transferred across KG

In this section, we discuss the knowledge being transferred across KGs.
Compared to the individual models trained on monolingual KGs, the model on
fused KG implicitly captures the transferable triples via alignment
information discussed previously while avoiding the error propagation of the
potential noisy seed alignments to some extent (as shown in table 4).
Additionally, relations embeddings are shared across individual KGs within the
connected model. As a result, relationships between the KG relations are also
better captured. In figure 2, we visualize the correlations of relation
embeddings learnt on Greek KG and the fused KG on the DBP-5L dataset. The
correlations of learnt KG relations in the Greek KG differ from those
discovered on the fused KG. Specifically, pairs of relatable connections like
(“birthplace”, “nationality”), (“birthplace”, “deathplace”), as well as
(“homearena”, “homestadium”), all exhibit a higher correlation in the fused KG
compared to the individual Greek KG. This implies that the connected model can
capture more accurate dynamics between relations with more triples information
and shared relation embeddings. At the same time, the low-resource Greek KG
converged to a suboptimal region mainly due to the lack of information. The
learnt knowledge is then transferred to the models on individual KGs via a
knowledge distillation mechanism and further optimized during training.

##  6 Conclusions

In this paper, we proposed a novel method CKGC-CKD that focuses on the KGC
task across multiple KGs. The proposed method uses an augmented CompGCN
encoder for message propagation across different KGs via seed alignments in a
fused KG. In addition, the proposed model employs additional mutual knowledge
distillations between individual KGs and the fused KG to maximize knowledge
transfer. CKGC-CKD outperforms the state-of-the-art models by a significant
margin on the KGC task on multilingual datasets DBP-5L and E-PKG. We also
demonstrate the performance gains provided by each component of the proposed
method. One limitation of this work lies in the assumption of sufficient seed
alignments between KGs. When seed alignments are more limited than current
experiment settings, it is fairly obvious that our method has potentials to
generate probalistic predictions of alignments aross KGs. As a result, we have
planned future work to specifically focus on including more
probabilistic/noisy alignments predictions iteratively while limiting the
propagation of error in the iterative process.

Algorithm 1 Pseudocode of the training process of CKGC-CKD.

$\triangleright$ Stage 1: Train each model $M_{i}$ and $M_{f}$ with the
Knowledge Representation loss.

for $i\in 1..m+1$ do

while $M_{i}$ not converged do

$L^{i}\leftarrow T_{i}$

$M_{i}\leftarrow$ Update w.r.t $L^{i}$

end while

end for

$\triangleright$ Stage 2: Train each model $M_{i}$ and $M_{f}$ with the
Knowledge Representation and the Knowledge Distillation loss.

while not converged do

$batch_{f}\leftarrow$ sample from triple set $T_{f}$

$L^{f}_{T}\leftarrow$ calculate loss of $batch_{f}$ base on equation 4

for $i\in 1..m$ do

$batch_{i}\leftarrow$ sample from triple set $T_{i}$

$L^{i}_{T}\leftarrow$ calculate loss of $batch_{i}$ based on equation 4

$L^{i}_{D},L^{f}_{D}\leftarrow$ calculate distillation losses between $M_{i}$
and $M_{f}$ on $batch_{i}$ based on equation 5 with top-k sampling to select
candidates space of distillation

$L^{i}\leftarrow L^{i}_{T}+\alpha L^{i}_{D}$

$L^{f}\leftarrow L^{f}_{T}+\alpha L^{f}_{D}$

$M_{i}\leftarrow$ Update w.r.t $L^{i}$

end for

$M_{f}\leftarrow$ Update w.r.t $L^{f}$

end while

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# Can the nature of $a_{0}(980)$ be tested in the
$D_{s}^{+}\to\pi^{+}\pi^{0}\eta$ decay?

Xi-Zhe Ling  School of Physics, Beihang University, Beijing 100191, China
Ming-Zhu Liu  School of Space and Environment, Beihang University, Beijing
100191, China  School of Physics, Beihang University, Beijing 100191, China
Jun-Xu Lu  School of Physics, Beihang University, Beijing 100191, China  Li-
Sheng Geng<EMAIL_ADDRESS>School of Physics, Beihang University,
Beijing 100191, China  Beijing Key Laboratory of Advanced Nuclear Materials
and Physics, Beihang University, Beijing 100191, China  School of Physics and
Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001, China  Ju-Jun
Xie<EMAIL_ADDRESS>Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  School of Nuclear Science and Technology,
University of Chinese Academy of Sciences, Beijing 101408, China  School of
Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001,
China

###### Abstract

From the amplitude analysis of the $D^{+}_{s}\to\pi^{+}\pi^{0}\eta$ decay, the
BESIII Collaboration firstly observed the $D^{+}_{s}\to a_{0}(980)^{+}\pi^{0}$
and $D^{+}_{s}\to a_{0}(980)^{0}\pi^{+}$ decay modes, which are expected to
occur through the pure $W$-annihilation processes. The measured branching
fraction $\mathcal{B}[D_{s}^{+}\to
a_{0}(980)^{+(0)}\pi^{0(+)},a_{0}(980)^{+(0)}\to\pi^{+(0)}\eta]$ is, however,
found to be larger than those of known $W$-annihilation decays by one order of
magnitude. This apparent contradiction can be reconciled if the two decays are
induced by internal $W$-conversion or external $W$-emission mechanisms instead
of $W$-annihilation mechanism. In this work, we propose that the $D^{+}_{s}$
decay proceeds via both the external and internal $W$-emission instead of
$W$-annihilation mechanisms. In such a scenario, we perform a study of the
$D^{+}_{s}\to\pi^{+}\pi^{0}\eta$ decay by taking into account the
contributions from the tree diagram
$D^{+}_{s}\to\rho^{+}\eta\to\pi^{+}\pi^{0}\eta$ and the intermediate
$\rho^{+}\eta$ and $K^{*}\bar{K}/K\bar{K}^{*}$ triangle diagrams. The
intermediate $a_{0}(980)$ state can be dynamically generated from the final
state interactions of coupled $K\bar{K}$ and $\pi\eta$ channels, and it is
shown that the experimental data can be described fairly well, which supports
the interpretation of $a_{0}(980)$ as a molecular state.

##  I Introduction

The charmed meson weak decays into light mesons provide a very good channel to
study meson-meson interactions at low energies and the nature of the low-lying
scalar mesons Klempt:2008ux ; Bhattacharya:2010id ; Liang:2016hmr ;
Debastiani:2016ayp . We refer to Ref. Oset:2016lyh  for a review about the
study on the interactions of light hadrons from the weak decays of the $D$
mesons. Very recently, in Ref. Ablikim:2019pit , the BESIII Collaboration
firstly reported on the observation of the decay modes of $D_{s}^{+}\to
a_{0}(980)^{+}\pi^{0}$ and $D_{s}^{+}\to a_{0}(980)^{0}\pi^{+}$ in the
amplitude analysis of the $D_{s}^{+}\to\pi^{+}\pi^{0}\eta$ decay. The
$D_{s}^{+}\to a_{0}(980)^{+(0)}\pi^{0(+)}$ decays are claimed as
$W$-annihilation dominant processes. However, the measured absolute branching
fraction of $\mathcal{B}[D_{s}^{+}\to a_{0}(980)^{+(0)}\pi^{0(+)}$,
$a_{0}(980)^{+(0)}\to\pi^{+(0)}\eta]=(1.46\pm 0.15_{\mathrm{sta.}}\pm
0.23_{\mathrm{sys.}})\%$ is found to be larger than those of normal
$W$-annihilation processes by at least one order of magnitude. In Ref.
Molina:2019udw  it was proposed that the $D^{+}_{s}\to\pi^{+}\pi^{0}\eta$
decay actually occurs via an internal $W$-conversion process, where $\pi
K\bar{K}$ was produced at the first step, and then the $\pi\eta$ state is
generated from final state interaction of $K\bar{K}$ in $S$-wave and isospin
$I=1$, and the $a_{0}(980)$ resonance is dynamically generated via $\bar{K}K$
and $\pi\eta$ coupled channel interactions as described in the unitary chiral
theory Oller:1997ti ; Oller:2000fj . The main purpose of Ref. Molina:2019udw
is to get the $a_{0}(980)$ signal, and thus, only the experimental data with
the cut $M_{\pi^{+}\pi^{-}}>1$ GeV were studied, for which the tree diagram
$D^{+}_{s}\to\rho^{+}\eta\to\pi^{+}\pi^{0}\eta$ does not contribute. On the
other hand, in Ref. Hsiao:2019ait  it was shown that the experimental
measurements can also be described as an external $W$-emission process
$D^{+}_{s}\to\rho^{+}\eta$. Then the $\rho$ meson decays into a pair of
$\pi\pi$ and the $\pi\eta$ pair fuses into $a_{0}(980)$, which subsequently
decays into $\pi\eta$ again. As both the $W$-internal conversion and
$W$-external emission processes are believed to be larger than the
$W$-annihilation process, the puzzle seems to be resolved, though the two
theoretical studies seem to give conflicting results regarding the responsible
weak decay mechanism.

In the present work, we revisit this issue and argue that the two theoretical
works are not necessarily contradicting with each other. As a matter of fact,
the triangle mechanism of Ref. Hsiao:2019ait  may offer a way to estimate the
unknown weak decay coupling between $D_{s}^{+}$ and
$\bar{K}K^{*}(\bar{K}^{*}K)$, i.e., the first vertex of the triangle diagrams
for $D^{+}_{s}\to K^{+}\bar{K}^{*0}/K^{*+}\bar{K}^{0}\to\pi^{+}\pi^{0}\eta$.

At the quark level, the decay of $D_{s}^{+}\to\rho^{+}\eta$ proceeds through
external $W$-emission $c\to\bar{s}W^{+}$ as shown in Fig 1(a). According to
the review of the Particle Data Group Zyla:2020zbs , the absolute branching
fractions of the decay modes $D_{s}^{+}\to K^{\ast+}\bar{K}^{0}$ and
$D_{s}^{+}\to K^{+}\bar{K}^{*0}$ are $(5.4\pm 1.2)\%$ and $(5.16\pm 0.16)\%$
respectively, which are comparable to the absolute branching fraction of
$D_{s}^{+}\to\rho^{+}\eta$ that is $(8.9\pm 0.8)\%$. As a result, if
$D^{+}_{s}\to\rho^{+}\eta$ can contribute to the $D^{+}_{s}\to
a_{0}^{+(0)}\pi^{0(+)}$ [$a_{0}\equiv a_{0}(980)$] process via the triangle
diagrams as in Ref. Hsiao:2019ait , the $D^{+}_{s}\to
K^{+}\bar{K}^{*0}/\bar{K}^{0}K^{*+}$ processes shown in Fig 1(b) can also
contribute via the triangle diagrams, where the $K^{*}$ decays into $K\pi$ and
$K\bar{K}$ produce the $\pi\eta$ pair through final state interactions, from
which the $a_{0}(980)$ resonance can be produced. The latter has not been
considered either in Ref. Molina:2019udw  or in Ref. Hsiao:2019ait . 111As we
will show later, the anomalously large $a_{0}\pi\eta^{\prime}$ coupling
adopted in Ref. Hsiao:2019ait  helps to increase the branching fraction to
meet the experimental number. As a result, in this work we will consider both
$D_{s}^{+}\to\rho^{+}\eta$ and $D_{s}\to K^{+}\bar{K}^{*0}/K^{*+}\bar{K}^{0}$
induced triangle diagrams, which lead to $\pi^{+}a_{0}^{0}$ and
$\pi^{0}a_{0}^{+}$ final states.

Figure 1: (a) External $W$-emission mechanism for $D^{+}_{s}\to\rho^{+}\eta$
and (b) internal $W$-conversion mechanisms for $D^{+}_{s}\to
K^{+}\bar{K}^{*0}/K^{*+}\bar{K}^{0}$.

Compared to Ref. Hsiao:2019ait , we make a further improvement. It is well
known that the $a_{0}(980)$ state does not behave like a normal Breit-Wigner
resonance, because of the closeness of the $K\bar{K}$ threshold. It can be
dynamically generated as a molecular state from the $K\bar{K}$ and $\pi\eta$
coupled channel interactions in the chiral unitary approach Oller:1997ti ;
Oller:1997ng ; Kaiser:1998fi ; Locher:1997gr ; Nieves:1999bx . In this work,
we then investigate whether with the chiral unitary amplitudes one can
describe the BESIII data Ablikim:2019pit .

The article is organized as follows. In Sec. II, we lay out the theoretical
formalism. In Sec. III we show our theoretical results and discussions are
also given comparing with the experimental data from Ref. Ablikim:2019pit . We
summarize in Sec. IV.

##  II Theoretical FORMALISM

Figure 2: Tree-level diagram for
$D_{s}^{+}\to\eta\rho^{+}\to\eta\pi^{0}\pi^{+}$.

To calculate the decay width of $D_{s}^{+}\to\eta\pi^{0}\pi^{+}$, we consider
the contribution from both the tree-level diagram of Fig. 2 and the triangle
diagrams of Fig. 3 and Fig. 4. In the process described by the tree-level
diagram, $D_{s}^{+}$ first decays into $\rho^{+}$ and $\eta$, then $\rho^{+}$
decays to $\pi^{+}\pi^{0}$ as shown in Fig 2. As pointed out in Ref.
Hsiao:2019ait , the $\pi^{+}$/$\pi^{0}$ meson can interact with the $\eta$
meson to form $a_{0}(980)^{+}/a_{0}(980)^{0}$ via the triangle diagrams shown
in Fig. 3. On the other hand, if the processes depicted in Fig. 3 can occur,
those depicted in Fig. 4 can also occur, because 1) the branching fractions of
$D_{s}$ decaying into $K^{*}\bar{K}/K\bar{K}^{*}$ are comparable to that of
$D_{s}$ decaying into $\rho\eta$ and 2) $K\bar{K}$ is a dominant channel to
which the $a_{0}(980)$ state couples. As a result, their contributions cannot
be neglected. In this work, therefore, we also consider the contribution from
$D_{s}^{+}\to K^{\ast+}\bar{K}^{0}/K^{+}\bar{K}^{*0}$, which can proceed
through the triangle mechanisms shown in Fig 4.

Figure 3: Triangle rescattering diagrams for
$D_{s}^{+}\to(\rho^{+}\eta\to)\pi^{+}\pi^{0}\eta$.
Figure 4: Triangle rescattering diagrams for
$D_{s}^{+}\to(K^{*+}\bar{K}^{0}\to)\pi^{+}\pi^{0}\eta$ and
$D_{s}^{+}\to(K^{+}\bar{K}^{*0}\to)\pi^{+}\pi^{0}\eta$ .

The follwing effective Hamiltonian is needed to describe the
$D_{s}^{+}\to\rho^{+}\eta$ and $D_{s}^{+}\to
K^{\ast+}\bar{K}^{0}/K^{+}\bar{K}^{*0}$ processes,

$\mathcal{H}_{eff}=\frac{G_{F}}{\sqrt{2}}V_{cs}V_{ud}[c_{1}^{eff}(\bar{u}d)(\bar{s}c)+c_{2}^{eff}(\bar{s}d)(\bar{u}c)],$
(1)

where $G_{F}=1.166\times 10^{-5}{\rm GeV}^{-2}$ is the Fermi constant,
$V_{cs}$ and $V_{ud}$ are the CKM matrix elements, $c_{1,2}^{eff}$ are the
effective Wilson coefficients, and $\bar{q_{1}}q_{2}$ stand for vector and
axial vector currents, $\bar{q_{1}}\gamma_{\mu}(1-\gamma_{5})q_{2}$
Buras:1998raa ; Geng:2017esc ; Geng:2018upx ; Geng:2020ofy ; Han:2021azw . The
amplitude of $D_{s}^{+}\to\rho^{+}\eta$ and $D_{s}^{+}\to
K^{\ast+}\bar{K}^{0}/K^{+}\bar{K}^{*0}$ can be written as the products of two
hadronic matrix elements Ali:1998eb ; Li:2013xsa

$\displaystyle\mathcal{A}\left(D_{s}^{+}\to\eta\rho^{+}\right)$
$\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cs}V_{ud}a_{1}\left\langle\rho^{+}|(\bar{u}d)|0\right\rangle\left\langle\eta|(\bar{s}c)|D_{s}^{+}\right\rangle$
(2) $\displaystyle\mathcal{A}\left(D_{s}^{+}\to K^{*+}\bar{K}^{0}\right)$
$\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cs}V_{ud}a_{2}\left\langle\bar{K}^{0}|(\bar{s}d)|0\right\rangle\left\langle
K^{*+}|(\bar{u}c)|D_{s}^{+}\right\rangle,$ (3)
$\displaystyle\mathcal{A}\left(D_{s}^{+}\to\bar{K}^{*0}K^{+}\right)$
$\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cs}V_{ud}a_{2}^{\prime}\left\langle\bar{K}^{*0}|(\bar{s}d)|0\right\rangle\left\langle
K^{+}|(\bar{u}c)|D_{s}^{+}\right\rangle.$ (4)

where $a_{1}=c_{1}^{eff}+c_{2}^{eff}/N_{c}$ and
$a_{2}=c_{1}^{eff}/N_{c}+c_{2}^{eff}$ with $N_{c}$ the number of colors. It
should be noted that $a_{1}$ can be calculated in the naive factorization
approach, but $a_{2}$ and $a_{2}^{\prime}$ cannot be easily obtained within
the factorization approach Beneke:2003zv ; Ali:1998gb . In the present work we
will fix all of them by fitting directly to data.

The current matrix elements between a pseudoscalar meson or vector meson and
the vacuum have the following form:

$\displaystyle\left\langle\rho^{+}|(\bar{u}d)|0\right\rangle=m_{\rho}f_{\rho}\epsilon_{\mu}^{*},~{}~{}\left\langle\bar{K}^{0}|(\bar{s}d)|0\right\rangle=-f_{\bar{K}^{0}}p_{\bar{K}^{0}}^{\mu},~{}~{}\left\langle\bar{K}^{*0}|(\bar{s}d)|0\right\rangle=m_{\bar{K}^{*0}}f_{\bar{K}^{*0}}\epsilon_{\mu}^{*},$
(5)

where $f_{K}$, $f_{K^{\ast}}$, and $f_{\rho}$ are the decay constants for $K$,
$K^{\ast}$, and $\rho$ mesons, respectively, and $\epsilon_{\mu}^{*}$ is the
polarization vector of $\rho$ or $K^{*}$ meson. In this work, we take
$f_{\rho}=210$ MeV, $f_{K}=158$ MeV, $f_{K^{*}}=214$ MeV as in Refs.
Ali:1998eb ; Wang:2008rk ; Ali:2007ff .

The hadronic matrix elements can be written in terms of form factors as
follows Soni:2018adu

$\displaystyle\left\langle\eta|(\bar{s}c)|D_{s}^{+}\right\rangle$
$\displaystyle=$
$\displaystyle\left(p_{D_{s}}+p_{\eta}\right)^{\mu}F_{+}(q^{2})+q^{\mu}F_{-}(q^{2}),$
(6) $\displaystyle\left\langle K^{*+}|(\bar{u}c)|D_{s}^{+}\right\rangle$
$\displaystyle=$
$\displaystyle\frac{\epsilon_{\alpha}^{*}}{m_{D_{s}}+m_{K^{*+}}}\left[-g^{\mu\alpha}P\cdot
q^{\prime}A_{0}\left(q^{\prime 2}\right)+P^{\mu}P^{\alpha}A_{+}\left(q^{\prime
2}\right)\right.$ (8)
$\displaystyle\left.+q^{\prime\mu}P^{\alpha}A_{-}\left(q^{\prime
2}\right)+i\varepsilon^{\mu\alpha\beta\gamma}P_{\beta}q^{\prime}_{\gamma}V\left(q^{\prime
2}\right)\right],$ $\displaystyle\left\langle
K^{+}|(\bar{u}c)|D_{s}^{+}\right\rangle$ $\displaystyle=$
$\displaystyle\left(p_{D_{s}}+p_{K^{+}}\right)^{\mu}F_{+}(q^{\prime\prime
2})+q^{\prime\prime}_{\mu}F_{-}(q^{\prime\prime 2}),$ (9)

where $q^{\mu(\prime,\prime\prime)}$ represent the momentum of $\rho$, $K$,
and $K^{*}$ mesons, respectively, and $P^{\mu}=(p_{D_{s}}+p_{K^{*}})^{\mu}$.
The form factors of $F_{\pm}(t)$, $A_{0}(t)$, $A_{+}(t)$, $A_{-}(t)$, and
$V(t)$ with $t\equiv q^{(\prime,\prime\prime)2}$ can be parameterized as
Soni:2018adu

$X(t)=\frac{X(0)}{1-a\left(t/m_{D_{s}}^{2}\right)+b\left(t^{2}/m_{D_{s}}^{4}\right)}.$
(10)

In this work, we take these form factors $F_{\pm}$, $A_{0,\pm}$ and $V$ from
Ref. Soni:2018adu : $(F_{+}(0),a,b)^{D_{s}\to\eta}=(0.78,0.69,0.002)$,
$(F_{+}(0),a,b)^{D_{s}\to K}=(0.60,1.05,0.18)$, $(A_{+}(0),a,b)^{D_{s}\to
K^{*}}=(0.57,1.13,0.21)$, $(A_{0}(0),a,b)^{D_{s}\to K^{*}}=(1.53,0.61,-0.11)$,
and $(A_{-}(0),a,b)^{D_{s}\to K^{*}}=(-0.82,1.32,0.34)$. Note that the terms
containing $V(q^{\prime 2})$ and $F_{-}(q^{2}(q^{\prime\prime 2}))$ do not
contribute to the processes we study here.

For the strong decays $\rho^{+}\to\pi^{+}\pi^{0}$ and $K^{*+}\to K\pi$
($\bar{K}^{*0}\to\bar{K}\pi$), the amplitudes are

$\displaystyle\mathcal{A}\left(\rho^{+}\to\pi^{+}\pi^{0}\right)=g_{\rho\pi\pi}\epsilon\cdot\left(p_{\pi^{+}}-p_{\pi^{0}}\right),~{}~{}~{}\mathcal{A}\left(K^{*}\to
K\pi\right)=g_{K^{*}K\pi}\epsilon\cdot\left(p_{\pi}-p_{K}\right),$ (11)

where $g_{\rho\pi\pi}$ and $g_{K^{*}K\pi}$ denote the $\rho$ coupling to
$\pi\pi$ and the $K^{*}$ coupling to $K\pi$. With the masses of these
particles and the partial decay widths of $\rho\to\pi\pi$ and $K^{*}\to K\pi$
quoted in the PDG Zyla:2020zbs , we obtain $g_{\rho\pi\pi}=6.0$,
$g_{K^{*+}K\pi}=3.26$, and $g_{K^{*0}K\pi}=3.12$. On the other hand, the
couplings 222It should be stressed that the partial decay widths determine
only the absolute value of the corresponding coupling constants, but not their
phases. In this work, we assume that they are real and positive, which seems
to be a reasonable choice given the reasonable description of the experimental
data as shown below. $a_{1}=0.96$, $a_{2}=1.50$ and $a_{2}^{\prime}=1.51$ are
determined by fitting them to the experimental branching fractions
$\mathcal{B}(D_{s}^{+}\to\rho^{+}\eta)=(8.9\pm 0.8)\%$,
$\mathcal{B}(D_{s}^{+}\to K^{*+}\bar{K}^{0})=(5.4\pm 1.2)\%$ and
$\mathcal{B}(D_{s}^{+}\to K^{+}\bar{K}^{*0},\bar{K}^{*0}\to
K^{-}\pi^{+})=(2.58\pm 0.08)\%$ quoted in the PDG Zyla:2020zbs .

Putting all the pieces together, we obtain the decay amplitude of the
$D^{+}_{s}\to\pi^{+}\pi^{0}\eta$ from the tree-level diagram of Fig. 2 as

$\mathcal{A}^{\rm
tree}_{1}=i~{}\frac{\mathcal{A}(D_{s}^{+}\to\eta\rho^{+})\mathcal{A}\left(\rho^{+}\to\pi^{+}\pi^{0}\right)}{m_{13}^{2}-m_{\rho^{+}}^{2}+im_{\rho^{+}}\Gamma_{\rho^{+}}},$
(12)

with $m_{13}^{2}=(p_{\pi^{0}}+p_{\pi^{+}})^{2}$ the invariant mass squared of
the $\pi^{+}\pi^{0}$ system.

Next, we can write the total decay amplitude of
$D^{+}_{s}\to\pi^{+}\pi^{0}\eta$ for those triangle diagrams shown in Fig. 3,

$\displaystyle\mathcal{A}_{2}^{\rho\eta}$
$\displaystyle=\mathcal{A}_{a}^{\rho\eta}+\mathcal{A}_{b}^{\rho\eta},$ (13)
$\displaystyle\mathcal{A}_{a}^{\rho\eta}$
$\displaystyle=t_{\pi\eta\to\pi\eta}(m_{12}^{2})\int\frac{d^{4}q_{3}}{(2\pi)^{4}}\frac{{\rm
i}\mathcal{A}(D_{s}^{+}\to\eta\rho^{+})\mathcal{A}\left(\rho^{+}\to\pi^{+}\pi^{0}\right)}{\left(q_{1}^{2}-m_{\rho^{+}}^{2}+im_{\rho^{+}}\Gamma_{\rho^{+}}\right)\left(q_{2}^{2}-m_{\eta}^{2}+i\epsilon\right)\left(q_{3}^{2}-m_{\pi^{0}}^{2}\right)},$
(14) $\displaystyle\mathcal{A}_{b}^{\rho\eta}$
$\displaystyle=t_{\pi\eta\to\pi\eta}(m_{23}^{2})\int\frac{d^{4}q_{3}}{(2\pi)^{4}}\frac{{\rm
i}\mathcal{A}(D_{s}^{+}\to\eta\rho^{+})\mathcal{A}\left(\rho^{+}\to\pi^{0}\pi^{+}\right)}{\left(q_{1}^{2}-m_{\rho^{+}}^{2}+im_{\rho^{+}}\Gamma_{\rho^{+}}\right)\left(q_{2}^{2}-m_{\eta}^{2}+i\epsilon\right)\left(q_{3}^{2}-m_{\pi^{+}}^{2}\right)},$
(15)

where $m_{12}^{2}=\left(p_{\pi^{0}}+p_{\eta}\right)^{2}$,
$m_{23}^{2}=\left(p_{\pi^{+}}+p_{\eta}\right)^{2}$, and the momenta
$(q_{1},q_{2},q_{3})$ are those of $(\rho^{+},\eta,\pi^{0(+)})$, respectively.
The $t_{\pi\eta\to\pi\eta}$ stands for the two-body $\pi\eta\to\pi\eta$
scattering amplitude, which depends on the invariant mass of the $\pi\eta$
system.

The decay amplitudes of $D_{s}^{+}\to\pi^{+}\pi^{0}\eta$ via triangle diagrams
shown in Fig. 4 are written as

$\displaystyle\mathcal{A}_{3}^{K^{*}K}$ $\displaystyle=$
$\displaystyle\mathcal{A}_{a}^{K^{*}K}+\mathcal{A}_{b}^{K^{*}K}+\mathcal{A}_{c}^{K^{*}K}+\mathcal{A}_{d}^{K^{*}K},$
(16) $\displaystyle\mathcal{A}_{a}^{K^{*}K}$ $\displaystyle=$
$\displaystyle\frac{i}{\sqrt{2}}\int\frac{d^{4}q_{3}}{(2\pi)^{4}}\frac{\mathcal{A}\left(D_{s}^{+}\to
K^{*+}\bar{K}^{0}\right)\mathcal{A}\left(K^{*+}\to\pi^{+}K^{0}\right)t_{K\bar{K}\to\pi\eta}(m_{12}^{2})}{\left(q_{1}^{2}-m_{K^{*+}}^{2}+im_{K^{*+}}\Gamma_{K^{*+}}\right)\left(q_{2}^{2}-m_{K^{0}}^{2}+i\epsilon\right)\left(q_{3}^{2}-m_{K^{0}}^{2}\right)},$
(17) $\displaystyle\mathcal{A}_{b}^{K^{*}K}$ $\displaystyle=$
$\displaystyle-i\int\frac{d^{4}q_{3}}{(2\pi)^{4}}\frac{\mathcal{A}\left(D_{s}^{+}\to
K^{*+}\bar{K}^{0}\right)\mathcal{A}\left(K^{*+}\to\pi^{0}K^{+}\right)t_{K\bar{K}\to\pi\eta}(m_{23}^{2})}{\left(q_{1}^{2}-m_{K^{*+}}^{2}+im_{K^{*+}}\Gamma_{K^{*+}}\right)\left(q_{2}^{2}-m_{K^{0}}^{2}+i\epsilon\right)\left(q_{3}^{2}-m_{K^{+}}^{2}\right)},$
(18) $\displaystyle\mathcal{A}_{c}^{K^{*}K}$ $\displaystyle=$
$\displaystyle-\frac{i}{\sqrt{2}}\int\frac{d^{4}q_{3}}{(2\pi)^{4}}\frac{\mathcal{A}\left(D_{s}^{+}\to\bar{K}^{*0}K^{+}\right)\mathcal{A}\left(\bar{K}^{*0}\to\pi^{+}K^{-}\right)t_{K\bar{K}\to\pi\eta}(m_{12}^{2})}{\left(q_{1}^{2}-m_{K^{*0}}^{2}+im_{K^{*0}}\Gamma_{K^{*0}}\right)\left(q_{2}^{2}-m_{K^{+}}^{2}+i\epsilon\right)\left(q_{3}^{2}-m_{K^{-}}^{2}\right)},$
(19) $\displaystyle\mathcal{A}_{d}^{K^{*}K}$ $\displaystyle=$
$\displaystyle-i\int\frac{d^{4}q_{3}}{(2\pi)^{4}}\frac{\mathcal{A}\left(D_{s}^{+}\to\bar{K}^{*0}K^{+}\right)\mathcal{A}\left(\bar{K}^{*0}\to\pi^{0}\bar{K}^{0}\right)t_{K\bar{K}\to\pi\eta}(m_{23}^{2})}{\left(q_{1}^{2}-m_{K^{*0}}^{2}+im_{K^{*0}}\Gamma_{K^{*0}}\right)\left(q_{2}^{2}-m_{K^{+}}^{2}+i\epsilon\right)\left(q_{3}^{2}-m_{K^{0}}^{2}\right)},$
(20)

with momenta $(q_{1},q_{2},q_{3})$ for
$(K^{*}(\bar{K}^{*}),\bar{K}(K),K(\bar{K}))$, respectively. It is worth
mentioning that one needs to include the isospin factor $-\sqrt{\frac{1}{3}}$
and $\sqrt{\frac{1}{3}}$ for Figs. 4 (a) and (c) and Figs. 4 (b) and (d),
respectively. The $t_{K\bar{K}\to\pi\eta}$ stands for the two-body
$K\bar{K}\to\pi\eta$ scattering amplitude, which depends on the invariant mass
of the $\pi\eta$ system. It should be noted that in the present work, for the
$\rho$ and $K^{*}$ vector meson propagators, we take

$\displaystyle
G^{\mu\nu}_{V}(q^{2}_{V})=\frac{i(-g^{\mu\nu}+q^{\mu}_{V}q^{\nu}_{V}/q^{2}_{V})}{q_{V}^{2}-m_{V}^{2}+im_{V}\Gamma_{V}},$
(21)

where $m_{V}$ and $\Gamma_{V}$ are the mass and width of the vector mesons.

The triangle loop integrals in these above amplitudes are ultraviolet
divergent, in general one needs to include phenomenological form factors to
prevent ultraviolet divergence, as shown in Refs. Li:2013zcr ; Shen:2016tzq ;
Wu:2016ypc ; Zhang:2018eeo ; Liu:2019dqc ; Xie:2019iwz ; Wang:2020duv .
However, as discussed in Refs. Hsiao:2019ait ; Du:2019idk , the ultraviolet
divergences in the triangle loop diagrams integrals cancel out (for more
details see Ref. Achasov:2015uua ), thus we do not need to introduce these
form factors in this work.

In Ref. Hsiao:2019ait  the two-body scattering amplitude
$t_{\pi\eta\to\pi\eta}$ is parameterized with the Breit-Wigner form. In this
work, we describe the final state interaction between $\pi$ and $\eta$ as well
as the interaction between $K$ and $\bar{K}$ with the chiral unitary approach.
The scattering amplitudes $t_{\pi\eta\to\pi\eta}$ and $t_{K\bar{K}\to\pi\eta}$
can be obtained by solving the following Bethe-Salpeter equation

$t=[1-VG]^{-1}V,$ (22)

where $G$ is the loop function of two mesons and $V$ is the transition
potential. The loop function can be regularized by either the dimensional
regularization scheme or the cutoff regularization scheme. In this work we
employ the dimensional regularization scheme. The potential $V$ is a $2\times
2$ matrix of coupled channels $K\bar{K}$ and $\pi\eta$. At the leading chiral
order, the transition potential $V$ can be explicitly written as Oller:1997ti
; Xie:2014tma ; Liang:2014ama ; Liang:2015qva ; Xie:2018rqv ; Duan:2020vye

$\displaystyle V_{K\bar{K}\to
K\bar{K}}=-\frac{1}{4f^{2}}s,~{}V_{K\bar{K}\to\pi\eta}=\frac{\sqrt{6}}{12f^{2}}\left(3s-\frac{8}{3}m_{K}^{2}-\frac{1}{3}m_{\pi}^{2}-m_{\eta}^{2}\right),~{}V_{\pi\eta\to\pi\eta}=-\frac{1}{3f^{2}}m_{\pi}^{2},$

where we take the isospin multiplets as $K=(K^{+},K^{0})$,
$\bar{K}=(\bar{K^{0}},-K^{-})$, and $\pi=(-\pi^{+},\pi^{0},\pi)$. Then,
solving the Bethe-Salpeter equation with $\mu=1$ GeV,
$a(\mu)_{\pi\eta}=-1.522$, $a(\mu)_{\bar{K}K}=-1.499$ Oller:2000fj , we obtain
a resonance with mass $m=983.2$ MeV and width $\Gamma=105.6$ MeV. This can be
associated with the $a_{0}(980)$ state.

With the so-obtained decay amplitudes, one can calculate the invariant mass
distributions of $D^{+}_{s}\to\eta\pi^{+}\pi^{0}$ as a function of
$m_{12}^{2}$ and $m_{23}^{2}$ Zyla:2020zbs :

$d\Gamma=\frac{1}{(2\pi)^{3}}\frac{|\mathcal{A}|^{2}}{32m_{D_{s}}^{3}}dm_{12}^{2}dm_{23}^{2},$
(23)

where ${\cal A}$ is the total decay amplitude, which is ${\cal A}={\cal
A}^{\rm tree}_{1}+{\cal A}_{2}^{\rho\eta}+{\cal A}_{3}^{K^{*}K}$.

One can easily obtain the single differential invariant mass distribution
$d\Gamma/dm_{\pi\pi}$ and $d\Gamma/dm_{\pi\eta}$ by integrating over
$m_{\pi\eta}$ and $m_{\pi\pi}$ with the limits of the Dalitz Plot,
respectively.

##  III Numerical Results and Discussion

We first show the theoretical results for the $\pi^{+}\pi^{0}$ invariant mass
distribution in Fig. 5 in comparison with the BESIII data Ablikim:2019pit .
The solid curve stands for the total contributions from the tree diagram and
the triangle diagrams, while the dashed curve stands for the contribution from
only the tree diagram. The solid curve has been adjusted to the strength of
the experimental data of BESIII at its peak Ablikim:2019pit . Since we have
considered the tree diagram contribution from the $\rho^{+}$ meson, one can
see that the $\rho^{+}$ peak can be well reproduced. Furthermore, the high
energy points for the $\pi^{+}\pi^{0}$ invariant mass distributions can also
be well reproduced by including the contributions from the triangle diagrams.
It is interesting to mention that the interference between the tree diagram
and triangle diagrams is destructive below $m_{\pi^{+}\pi^{0}}=0.8$ GeV, while
above that energy point, the interference is constructive. Besides, with
$a_{1}$, $a_{2}$, and $a_{2}^{\prime}$ determined as specified above, we
obtain an absolute branching ratio of
$\mathcal{B}\left(D_{s}^{+}\to\pi^{+(0)}\left(a_{0}^{0(+)}\to\right)\pi^{0(+)}\eta\right)=1.45\%$.
This is in nice agreement with the BESIII measurement.

Figure 5: Invariant mass distribution of $\pi^{+}\pi^{0}$ for the
$D_{s}^{+}\to\eta(\rho^{+}\to)\pi^{0}\pi^{+}$ decay, in comparison with the
experimental data taken from Ref. Ablikim:2019pit .

In Fig. 6 and Fig. 7, we show the $\pi^{0}\eta$ and $\pi^{+}\eta$ invariant
mass distributions of the decay $D_{s}^{+}\to\pi^{+}\pi^{0}\eta$ without and
with the cut of $m_{\pi^{+}\pi^{0}}>1$ GeV, respectively. From Fig. 6, one can
see that the contribution from the tree diagram is predominant. The
theoretical results can describe the experimental data rather well,
particularly the shoulder around $m_{\pi\eta}\sim 0.8$ GeV. In addition, our
theoretical results do not show a pronounced asymmetric peak around
$m_{\pi\eta}\sim 0.9$ GeV as in Ref. Hsiao:2019ait  (see Fig. 4 of that
reference).

Figure 6: Invariant mass distributions of $\pi\eta$ for the
$D_{s}^{+}\to\pi(a_{0}(980)\to)\pi^{0}\pi^{+}\eta$ decay, in comparison with
the BESIII data Ablikim:2019pit .

In Fig. 7, the dashed curves represent the contributions from the $\rho\eta$
triangle diagrams shown in Fig. 3, while the blue-dashed curves represent the
contributions from the $\bar{K}K^{*}$ triangle diagrams shown in Fig. 4, and
the red-solid curves stand for the sum of the two contributions. From Fig. 7,
one can see that after the $m_{\pi^{+}\pi^{0}}>1$ GeV cut, the $a_{0}(980)$
signal is well reproduced, where it is dynamically generated from the
$\pi\eta$ and $K\bar{K}$ coupled channel interactions. However, our results at
the $a_{0}(980)$ peak position are somehow larger than the experimental data,
especially for the case of $a_{0}(980)^{+}$. It should be noted that in Ref.
Molina:2019udw , the $\pi K\bar{K}$ final states were produced at the first
step with the internal $W$-emission mechanism, and then the final state
interaction of $K\bar{K}$ produces $a_{0}(980)$, which then decays to
$\pi\eta$. Clearly, Ref. Molina:2019udw  and the present work share the same
mechanism for the final state interactions. As a result, both can describe the
$\pi\eta$ line shapes, but the present work also determines the global
strength of the $D_{s}$ decay. In principle, both weak mechanisms may play a
role. However, a quantitative consideration of the mechanism of Ref.
Molina:2019udw  inevitably introduces additional free parameters for the weak
interaction, which cannot yet be determined. Hence, we will leave such a study
to a future work when more precise experimental data become available.

Figure 7: Invariant mass distributions of $\pi\eta$ with the cut of
$m_{\pi^{+}\pi^{0}}>1$ GeV for the decay
$D_{s}^{+}\to\pi(a_{0}(980)\to)\pi^{0}\pi^{+}\eta$, in comparison with the
BESIII data Ablikim:2019pit .

It is worthwhile mentioning that in our framework that the $K\bar{K}^{*}$
contribution is larger than the $\rho\eta$ contribution, while the former was
not considered in Ref. Hsiao:2019ait , where the $\rho\eta^{\prime}$ channel
plays an important role and a large coupling for $a_{0}(980)$ to
$\pi\eta^{\prime}$ is used. However, both in the unitary chiral approach and
from the experimental information, it is known that the $\pi\eta^{\prime}$
coupling to the $a_{0}(980)$ resonance is small.

Figure 8: Invariant $K^{+}\bar{K}^{0}$ mass distribution for the
$D_{s}^{+}\to\pi^{0}K^{+}\bar{K}^{0}$ decay.

In addition, we study the $a_{0}(980)$ state in the $K\bar{K}$ channel from
the $D^{+}_{s}\to\pi K\bar{K}$ decay by including the contributions from the
triangle diagrams, which can be easily obtained with the replacement of the
$\pi\eta$ final state by $K\bar{K}$ in Figs. 3 and 4. The resulting
predictions for the $K^{+}\bar{K}^{0}$ invariant mass distributions are shown
in Fig. 8, which can serve as a highly non-trivial check of the mechanism
proposed in this work.

##  IV Summary

We studied the $D^{+}_{s}\to\pi^{+}\pi^{0}\eta$ decay recently analysed by the
BESIII Collaboration, where the $D^{+}_{s}\to\pi^{0}a_{0}(980)^{+}$ and
$D^{+}_{s}\to\pi^{+}a_{0}(980)^{0}$ decay modes are claimed as the
$W$-annihilation dominant processes observed for the first time, and their
branching fractions, however, are one order of magnitude larger than those of
known $W$-annihilation decays. Inspired by Ref. Hsiao:2019ait , we proposed
that the anomalously large branching ratios of these decay modes can be
understood via triangle diagrams. At first, the $D^{+}_{s}$ meson decays
weakly into either $\rho^{+}\eta$ or $K\bar{K}^{*}/K^{*}\bar{K}$. The vector
mesons then decay into a pair of pseudoscalar mesons, $\pi\pi$ or $K\pi$. One
of them interacts with the pseudoscalar meson from the weak decay of
$D^{+}_{s}$, and generates dynamically the $a_{0}(980)$ state. With the weak
decay couplings determined by fitting to the experimental branching fractions,
our method predicted both the absolute branching ratio of $D_{s}\to a_{0}\pi$
and the $\pi\eta$ invariant mass distributions, which are in nice agreement
with the BESIII data. For the $D^{+}_{s}\to\pi^{+}\pi^{0}\eta$ decay, the
contribution from the tree diagram as shown in Fig. 2 is the most dominant.
After the cut of $M_{\pi^{+}\pi^{0}}>1$ GeV, the contributions of the triangle
diagrams are crucial to produce the $a_{0}(980)$ resonance. In addition, we
predicted the $K\bar{K}$ invariant mass distributions of the $D_{s}\to\pi
K\bar{K}$ decay, which can be checked by future experimental measurements.

Furthermore, the present work provides a way to estimate the effective weak
coupling appearing in the $D_{s}\to K^{*}\bar{K}\to K\bar{K}\pi$ vertex. The
same mechanism might also be relevant to those of similar processes, such as
the $D^{+}\to\pi^{+}K\bar{K}$ and $D^{+}\to\pi^{+}\pi^{0}\eta$ reactions
Duan:2020vye ; Ikeno:2021kzf .

## Acknowledgments

This work is partly supported by the National Natural Science Foundation of
China under Grants No. 11735003, No. 11975041, No. 11775148, No. 12075288, and
No. 11961141004. It is also supported by the Youth Innovation Promotion
Association CAS (2016367).

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