# Course Concept Expansion in MOOCs with External Knowledge and Interactive
Game

Jifan Yu1,2, Chenyu Wang3, Gan Luo1,2 ,Lei Hou1,2, Juanzi Li1,2, Jie Tang1,2,
Zhiyuan Liu1,2
1Dept. of Computer SCi.& Tech., Tsinghua University, China 100084
2KIRC, Institute for Artificial Intelligence, Tsinghua University, China
100084
3Shenyuan Honors College, Beihang University, China 100083
{yujf18<EMAIL_ADDRESS>
{houlei@,lijuanzi@,jietang<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
corresponding author

###### Abstract

As Massive Open Online Courses (MOOCs) become increasingly popular, it is
promising to automatically provide extracurricular knowledge for MOOC users.
Suffering from semantic drifts and lack of knowledge guidance, existing
methods can not effectively expand course concepts in complex MOOC
environments. In this paper, we first build a novel boundary during searching
for new concepts via external knowledge base and then utilize heterogeneous
features to verify the high-quality results. In addition, to involve human
efforts in our model, we design an interactive optimization mechanism based on
a game. Our experiments on the four datasets from Coursera and XuetangX show
that the proposed method achieves significant improvements(+0.19 by MAP) over
existing methods. The source code and datasets111Source Codes & Datasets:
https://github.com/thukg/concept-expansion-kg have been published.

##  1 Introduction

Self-determination theory was first formally proposed by Deci and Ryan in Deci
et al. (1991), suggesting that educators should support students in
autonomously discovering and learning course-related knowledge. In fact, in
addition to the concepts taught in course, many related concepts are also
worthy of learning. Figure 1 shows a real example from Coursera in _Data
Structure_ course. When the concept _Binary Search Tree_ is taught, some other
concepts, including its similar structures (_Heap_), applications (_Sorting_
and _Priority Queue_) and advanced researches (_Tango Tree_ 222It is an online
binary search tree that achieves an $O(\log\log n)$ competitive ratio. Demaine
et al. (2007)) also benefit students for further course understanding.
However, these concepts are not available without specifical mention,
especially in the era of Massive Open Online Courses (MOOCs).

Figure 1: An example of “out-of-teaching” concepts in the course “ _Data
Structure and Algorithm_ ”.

In MOOCs, teachers need to keep a moderate length of the course to face with
thousands of students with various backgrounds Jordan (2015), making it
infeasible to manually pick out these helpful concepts. Therefore, there is a
clear need to automatically identify course-related concepts, so that the
students can easily acquire additional knowledge and achieve better
educational outcomes.

Although extracting course concepts from teaching materials Kay and Holden
(2002) or course subtitles Pan et al. (2017b) has attracted several attempts,
the research in finding the concepts absent in course materials, which we call
Course Concept Expansion, has not been explored. Despite abundant work on
related topics, including concept expansion or set expansion Wang and Cohen
(2007); Wang et al. (2015); Adrian and Manna (2018), it is far from sufficient
to directly apply these methods in the MOOC environments due to the following
challenges.

First, unlike the set expansion for a clear general category (e.g., country),
course concepts are often the combinations of multiple categories, which is
easy to cause semantic drift Curran et al. (2007) during exploring in
different domains (such as Structures: _Heap_ , _Binary Tree_ and _Algorithms_
: _Divide and Conquer_ , _Greedy Algorithm_). Second, the features for
manifesting course-related concepts are heterogeneous. As shown in Figure 1,
we regard _Heap_ as a course concept due to its similar structure while
_Binary Search Tree_ is a prerequisite concept of _Tango Tree_. Thus mere
context information is not enough for effective expansion. Third, as an
application-oriented task, it is beneficial to involve human interactions. How
to properly leverage the feedback from MOOC users to obtain a better
performance for concept expansion remains a challenging issue.

To address the above problems, we propose a three-stage course concept
expansion model. Inspired by the idea of concept space Hori (1997), we first
build an accurate boundary for a given course to alleviate the semantic drift
during candidate concept generation from an external knowledge base. Then we
transform the expansion into a binary classification problem as previous
positive unlabeled learning methods for set expansion Li et al. (2010); Wang
et al. (2017). Three types of features are proposed to incorporate
heterogeneous information into classifier to identify high-quality concepts
among candidates. Finally, we design a lightweight but attractive top-student
game to subtly collect MOOC users’ feedback and iteratively optimize the
expansion results.

Figure 2: Top-Student Game in course ”Introduction to Psychology” of XuetangX.

For evaluation, we compare the proposed method with 4 representative set
expansion methods on real courses from Coursera333https://www.coursera.org/
and XuetangX444http://www.xuetangx.com/, and further conduct online evaluation
in the game mechanism.

Contributions. Our contributions include: a) systematically investigate the
problem of course concept expansion in MOOCs; b) proposal of an effective
three-stage model for course concept expansion using an external knowledge
base and interactive game; c) four benchmark datasets using real courses from
Coursera and XuetangX.

##  2 Problem Formulation

In this section, we first give some necessary definitions and then formulate
the problem of course concept expansion.

A Course corpus is composed by $n$ courses in the same subject area, denoted
as $\mathcal{D}=\left\\{\mathcal{C}_{j}\right\\}_{j=1}^{n}$, where
$\mathcal{C}_{j}$ is one course. We assume that course
$\mathcal{C}_{j}=\left\\{v_{ij}\right\\}_{i=1}^{m_{j}}$ consists of $m_{j}$
course videos, where $v_{ij}$ stands for the $i$-th video. Following Pan et
al. (2017b), we define Course Concepts are the subjects taught in the course
denoted as
$\mathcal{M}=\left\\{c_{i}\right\\}_{i=1}^{\left|\mathcal{M}\right|}$.

Existing work can extract course concepts $\mathcal{M}$ from course corpus
$\mathcal{D}$, but $\mathcal{D}$ could inevitably miss some important course
concepts (as illustrated in Figure 1). Therefore, there is a clear need to
expand the course concepts beyond $\mathcal{M}$ using external resources. In
this paper, we focus on the expansion using external knowledge bases.

Knowledge Base is formally defined as $\mathcal{KB}=\left(E,R\right)$, where
$E=\left\\{e_{i}\right\\}_{i=1}^{\left|E\right|}$ represents all concepts,
$R=\left\\{r_{i}\right\\}_{i=1}^{\left|R\right|}$ represents the relationships
between concepts, and $(e_{i},r_{j},e_{k})$ is a triple in $\mathcal{KB}$
meaning $e_{i}$ has relationship $r_{j}$ with $e_{k}$.

Course Concept Expansion Using Knowledge Base in MOOCs is formally defined as
follows. Given the course concepts $\mathcal{M}$ and knowledge base
$\mathcal{KB}$, Course Concept Expansion returns a ranked list of expanded
concepts $E_{c}\subset E$, and outputs $s_{i}$ for $e_{i}\in E_{c}$ to
indicate its likelihood to be an expanded concept of $\mathcal{D}$.

##  3 Method

To appropriately expand course concepts in MOOCs, we need to address three
crucial problems. 1. How to alleviate semantic drift? 2. How to employ
heterogeneous information to identify high-quality expanded concepts? 3. How
to properly involve human efforts to optimize the expansion result? In this
section, we introduce our novel course concept expansion model in three
stages.

Figure 3: Framework of course concept expansion. Figure 4: The concept space
boundary is fitted in process of searching candidates.

(1)Candidate Generation: To reduce semantic drift, a dynamic boundary is set
during our searching for new concepts in $\mathcal{KB}$. We only admit the
concepts within the boundary as candidates.

(2)Concept Classification: To leverage heterogeneous information in expansion,
we propose three types of novel features to build a classifier, identifying
the high-quality expansion concepts among candidates and rerank the result
list.

(3)Game-based Optimization: To involve human efforts, we creatively design an
interactive online game named top-student game which has been applied in a
real MOOC platform to collect user feedback and cyclically optimize the
expansion process at multiple levels.

###  3.1 Candidate Generation

In this section, we present an unsupervised embedding based algorithm that
iteratively generates ranked concept candidates from an external knowledge
base $\mathcal{KB}$. In particular, we build a boundary to avoid semantic
drift based on the following concept space assumption.

###### Assumption 3.1

A course is a concept space which contains one or more concept clusters.

The idea of concept space was proposed and applied in digital teaching and
ontology engineering Hori (1997); Cassidy et al. (2006). We extend the
assumption by considering concepts’ polycentric clustering pattern (e.g., Data
Structure course’s concepts mainly gather in several clusters, such as graph
algorithms, binary trees, etc.). As shown in Figure 4, we initialize the
course concepts in one loose cluster and separate the gathering ones into new
clusters during candidate generation. An explicit boundary of course is
dynamically formed by its clusters to avoid semantic drift.

Given the course corpus $\mathcal{D}$ and knowledge base $\mathcal{KB}=(E,R)$,
we first utilize the method in Pan et al. (2017b, a) to extract course
concepts $\mathcal{M}$. Note that we remove the extracted results which do not
exist in $E$, i.e., $\mathcal{M}\subset E$, to facilitate following candidate
generation. We use bold-face letters to denote the embeddings of the
corresponding terms (i.e., $\mathbf{c_{i}}$ is the embedding of $c_{i}$).

Before introducing the algorithm details, we first define the concept cluster
as follows.

###### Definition 3.1

A concept cluster $H$ is formed by several semantically related course
concepts $\\{c_{i}\\}_{i=1,\ldots,|H|}$, and is formally represented as a
hypersphere $(\mathbf{o},\gamma)$ with $\mathbf{o}$ and $\gamma$ denoting its
center and radius respectively. Mathematically,

$\displaystyle\mathbf{o}$ $\displaystyle=\sum\nolimits_{c_{i}\in
H}\mathbf{c_{i}}/|H|$ $\displaystyle\gamma$
$\displaystyle=\max\nolimits_{c_{i}\in H}edis(\mathbf{o},\mathbf{c_{i}})$

where $edis(\cdot,\cdot)$ returns the Euclidean distance between the input
vectors.

Note that the center $\mathbf{o}$ may be a “virtual” concept, i.e., it does
not correspond to any known concept in $\mathcal{M}$ or $E$. To facilitate the
generation process, we introduce a special subset $S_{H}\subset H$ that
includes a fixed size $\tau$ ($\tau$ is experimentally set to 8) of
representative concepts as seeds. We always select the “actual” concepts
nearest to the center $\mathbf{o}$, which means that $S_{H}$ might change
dynamically during the generation process. The candidate generation algorithm
contains two phases: initialization and searching.

Initialization: We initialize a concept cluster $H_{0}$ with all the concepts
in $\mathcal{M}$ (as shown in Figure 4), calculate its center
$\mathbf{o}_{H_{0}}$ and radius $\gamma_{H_{0}}$, and select the
representative subset $S_{H_{0}}$. Then following a predefined order555The
course concepts are extracted with the method proposed in Pan et al. (2017b),
which also assigns a confidence score for each concept. Here we sort the
extracted concepts by the confidence score in descending order., we adapt the
single-pass online clustering Guha et al. (2003) to group the concepts into
potential clusters. The clustering algorithm sequentially processes the
concepts, one at a time, and grows clusters incrementally. A concept $c_{i}$
is absorbed by a previously-generated cluster $H_{i}$ if its Euclidean
distance to a concept in $H_{i}$ is below a predefined threshold666In
experiment, it is set to the minimal distance between representative concepts
and the center of $H_{0}$, i.e., $\min\nolimits_{c\in
S_{H_{0}}}edis(\mathbf{o}_{H_{0}},\mathbf{c})$; otherwise, the concept is
treated a new potential cluster. Finally, we successfully partition the course
concepts $\mathcal{M}$ into $L$ potential clusters $H_{1},H_{2},\ldots,H_{L}$.

Searching: For each concept $c_{ij}$ in a potential cluster $H_{i}$, we search
its directly-connected concepts in knowledge base $\mathcal{KB}$, e.g.,
$(c_{ij},r,e)$ with $e\in E$ and $r\in R$, and use the distance between $e$
and $c_{ij}$ to determine whether to merge it into $H_{i}$. Similar to the
single-pass clustering, and merge it into $H_{i}$ if
$edis(e,c_{ij})<\min\nolimits_{c\in
S_{H_{i}}}edis(\mathbf{o}_{H_{i}},\mathbf{c})$. During the process, we
separate a cluster from $H_{0}$ whenever its size reaches $\tau$ (i.e., it is
big enough to select representative concepts), update $H_{0}$ (including
$\mathbf{o}_{H_{0}}$, $\gamma_{H_{0}}$ and $S_{H_{0}}$) and calculate its own
parameters. For those potential clusters whose size are less than $\tau$, we
use $\mathbf{o}_{H_{0}}$ and and $S_{H_{0}}$ to make the above decision. When
$e$ is merged into $H_{i}$, we define the following confidence score $s_{e}$
to measure its likelihood to be a course concept,

$\displaystyle s_{e}=cos(\mathbf{e},\mathbf{c}_{ij})+\sum\limits_{c_{ik}\in
H_{i}}cos(\mathbf{c}_{ik},\mathbf{c}_{ij})\times
cos(\mathbf{e},\mathbf{c}_{ik})$ (1)

where $cos(\cdot,\cdot)$ returns the cosine similarity of the input vectors.

After the expansion for all concepts in $\mathcal{M}$, we obtain the expanded
concept set $E_{c}^{1}\in E$. Then we sort $E_{c}^{1}$ by $s_{e}$ in
descending order and iteratively repeat the search phase for $E_{c}^{1}$ to
obtain $E_{c}^{2}$. The algorithm stops when no concepts in $E$ could be
merged. Finally, we achieve $E_{c}=\bigcup\nolimits_{i}E_{c}^{i}$ and sort it
by $s_{e}$ in descending order as candidate set.

It’s worth noting that each final candidate $e\in E_{c}$ is directly or
indirectly related to a course concept $c\in\mathcal{M}$. $e$ and $c$ are
connected by a search path $c\rightarrow r_{1}\rightarrow
e_{1}\rightarrow\ldots\rightarrow e$, where $e_{i}\in E$ and $r_{i}\in R$ are
concept and relation in $\mathcal{KB}$, and we record such path (denoted as
$path(e)$) to hold more semantics. The whole process is summarized in
Algorithm 1. Specifically, due to the huge number of operations for finding
nearest concepts, we use K-D Tree777K-D tree is a useful data structure for
nearest neighbor searches (Wikipedia). to store concept vectors, which greatly
improves the time efficiency.

###  3.2 Concept Classification

To integrate heterogeneous information in expansion, we propose three features
from various sources and rerank the candidates after classification. As a
binary classification problem, all existing classifiers can be applied, and we
experimentally select XGboost Chen and Guestrin (2016). In this section, we
introduce the three types of features and how we partially rerank the
candidates.

Confidence Score. In accordance to our assumption, the confidence score
$s_{e}$ represents the degree of remoteness between the candidate concept $e$
and the concept cluster $H$ to which it belongs. Thus we select it as the
first feature to capture a candidate’s basic relevance to the course.

Search Path Encoding. During candidate generation, the search paths insinuate
the semantic relations between course concepts. Taking “ _Floyd Algorithm_ ”
as an example, its search path, “BFS $\rightarrow$ _InstanceOf_ $\rightarrow$
Graph Algorithms $\rightarrow$ _Instance_ $\rightarrow$ Floyd Algorithm”,
indicates that “Floyd Algorithm is a sibling of course concept “BFS”. To make
effective use of this semantic information, we employ an RNN encoder-decoder
neural network Cho et al. (2014) to encode $path(e)$ for candidate $e$.
Specifically, we train the neural network to take $path(e)$ as input and
output the same sequence. Thus, we can obtain a fixed-length vector
representation of $path(e)$ from the final hidden state of the RNN encoder.

Input: $\mathcal{M}$, $\mathcal{KB}=(E,R)$, $\tau$

Output: $E_{c}$

1

2Sort $\mathcal{M}$, initialize $H_{0}$ and further partition into
$H_{1},H_{2},\ldots,H_{L}$

3 $t=0;E_{c}^{0}=\mathcal{M}$

4 do

5 $E_{c}^{t+1}=\emptyset$

6 for  _$c_{ij}\in H_{i}\subset E_{c}^{t}$ has related concept $e\in E$_ do

7 if  _$edis(e,c_{ij}) <\min\nolimits_{c\in
S_{H_{i}}}edis(\mathbf{o}_{H_{i}},\mathbf{c})$_ then

8 $E_{c}^{t+1}=E_{c}^{t+1}\cup\\{e\\}$

9 Merge $e$ into $H_{i}$, calculate $s_{e}$, record $path(e)$ and update
$H_{i}$

10 if  _$|H_{i}|\geq\tau$_ then

11 Separate $H_{i}$ from $H_{0}$ and update $H_{0}$

12 end if

13

14 end if

15 $E_{c}=E_{c}\cup E_{c}^{t+1}$

16 end for

17 Sort $E_{c}^{t+1}$ by $s_{e}$ and $t+=1$

18while  _$E_{c}^{t}\neq\emptyset$_ ;

Sort $E_{c}$ by $s_{e}$

Algorithm 1 Candidate Generation

Prerequisite Features. The course concepts also have an unique relationship
called _Prerequisite_ Margolis and Laurence (1999). Prerequisite concept pair
$\left(A,B\right)$ means if someone wants to study A, he/she is better to
understand B in advance (e.g., _Binary Tree_ is a prerequisite concept to
_Black-Red Tree_), which indicates how concepts in the course are connected.
There are a few previous efforts to extract prerequisite relations from
Wikipedia Talukdar and Cohen (2012); Liang et al. (2015), textbooks Yosef et
al. (2011); Wang et al. (2016) and MOOCs Pan et al. (2017a). In this paper, we
select five features from Pan et al. (2017a) that only rely on the course
text, and $Pv(a,b)$ is the combination of these five features reflecting the
prerequisite likelihood of $a$ to $b$. Since these features can only measure
the relationship between the two concepts that exist in the course, we
calculate the prerequisite feature of $e$ using its search root phrase $c_{i}$
as follows:

$\displaystyle
Pf(e)=\frac{cos\left\langle\mathbf{e},\mathbf{c_{i}}\right\rangle\ast\sum_{c_{j}\in\mathcal{M}}Pv(c_{i},c_{j})}{\left|\mathcal{M}\right|}$
(2)

Partial Reranking. After feature extraction and classification, each candidate
is labeled with a tag P (positive) or N (negative). Then we partially adjust
the rankings in low-confidence interval to improve the recall. We define a
threshold $\alpha\in\left[0,1\right]$ to control the reranking range and
adjust rankings after $\alpha\ast\left|E_{c}\right|$. Specifically, we sort
positive and negative results separately according to their confidence score
$s$ and then place the positive results before negative ones. Eventually a
reranked expansion list is achieved.

###  3.3 Game-based Optimization

As an application-oriented task, it is beneficial to properly introduce human
efforts to monitor and optimize our expansion model. However, the design of
this human-model interaction faces several challenges. For human, we need to
ensure the quality and sufficiency of their feedback. For the model, we need
to effectively and fully utilize the provided data in model optimization.

Thanks to the multimedia and web platform of MOOCs Volery and Lord (2000), we
are able to attractively collect feedback from students with diverse
backgrounds using an online game, which far exceeds traditional modes in
amount. In this section, we introduce the game design and how the collected
feedback optimize our model.

Game Design. We design our feedback collector, a game named “Top-Student” by
considering its attractiveness and the quality of collected data.

To make game attractive, we place the game under each video $v_{ij}$ of course
$\mathcal{D}$, whose basic idea is that the player gains the score and
competes with other students of the course by deleting low-quality expanded
concepts. It allows users to quickly start with a simple click-to-delete
operation. Further, we only show expansion results that are most relevant to
the concepts in the video $v_{ij}$, which increases the affinity of the users
who just finished the video. Those design facilitates a wider collection of
feedback.

Figure 2 is the game layout in the course of “Introduction to Psychology”. The
blue concept “IQ” is a course concept $c$ in video, while the orange concepts
are its relevant expanded concepts $E_{c}$. Users delete low-quality concepts
among orange ones and gain different scores while we always record the total
deletion times of each expanded concept $e_{i}$’s (denoted as $del(e_{i})$).

To ensure data quality, we avoid users’ irresponsible deletion by employing a
group-vote scoring mechanism. Specifically, when a user deletes the expanded
concept $e_{i}$, he/she gets a score $Q=del(e_{i})/\max_{e_{j}\in
E_{c}}del(e_{j})-\frac{1}{2}$, i.e., every user operation corresponds to a
score $S\in\left[-\frac{1}{2},\frac{1}{2}\right]$ (In real application, the
value is enlarged to $\left[-5,5\right])$ ) based on all existing deletion
data, which means that irresponsible operations subject to a penalty.

We set up the game by calculating and storing the expanded concept $e\in
E_{c}$ with highest cosine similarity for each $c_{i}$ as inputs. Finally we
get total deletion $del(e)$ of each $e$ as outputs. The Top-Student Game has
been applied in several courses at one of largest Chinese MOOC websites,
XuetangX and collected over 10,000 records as of this writing.

Multi-level Optimization. The user feedback affects both candidate generation
in Section 3.1 and concept classification in Section 3.2 to perform a multi-
level optimization.

We first define and calculate the deletion ratio of a candidate $e_{i}$ as
$Dr(e_{i})=del(e_{i})/\max_{e_{j}\in E_{c}}(del(e_{j}))$. The value reflects
the acceptance of $e_{i}$ comparing with the other expansion results. Then we
present the optimization at two levels.

Candidate Generation: The confidence score $s_{e}$ in Equation 1 is updated,
reducing the likelihood from its directly related concept according to its
deletion ratio $Dr(e)$.

$\displaystyle s_{e}=$ $\displaystyle
cos(\mathbf{e},\mathbf{c}_{ij})\times(1-Dr(e))$ (3)
$\displaystyle+\sum\limits_{c_{ik}\in
H_{i}}cos(\mathbf{c}_{ik},\mathbf{c}_{ij})\times
cos(\mathbf{e},\mathbf{c}_{ik})$

Concept Classification: We regard deletion ratio $dr(e)$ as a new feature to
incorporate user insights into our classifier.

In this way, user feedback is applied to each process of the model, and new
results generated after optimization are also periodically entered into the
game to collect feedback again. Finally it iterates over and gets high-quality
expansion results.

##  4 Experimental Evaluation

###  4.1 Datasets

Since there is no publicly available dataset for course concept expansion in
MOOCs, we use two different domains of Chinese and English courses: “Data
Structure and Algorithm” and “Introduction to Psychology” to construct four
datasets through a three-stage process.

First, for each domain, we select its most relevant English courses from
Coursera and Chinese courses from XuetangX, e.g., for EN-DSA, we select 3
courses (The three courses: Algorithms (Princeton), Algorithms (Stanford),
Data Structure and Algorithm (UC San Diego)) of 3 universities and obtain a
total of 449 videos. Then, we use the method of Pan Pan et al. (2017b) to
extract the course concepts and manually select the high-quality ones as the
course concepts $\mathcal{M}$. Finally, we take XLORE Jin et al. (2018) as
$\mathcal{KB}$ to search for related course concepts and manually labeled the
reasonable expansion results. For each domain, we collect 100,000 related
concepts and record their search path to train the encoder in Section 3.2. But
the large amount requires arduous human labeling work, thus we only pick 800
expanded concepts with the highest average cosine similarity to the course
concepts to label. For each concept, two human annotators majoring in the
corresponding domain are asked to label them as “0: Not related” or “1:
Related ” based on their own knowledge. Thus, each dataset is doubly
annotated, and pearson _correlation_ coefficient is applied to assess inter-
annotator agreement. A candidate is labeled as a related concept only if the
two annotators are in agreement. For each dataset, we split it into training
(400), validation (200) and test set (200).

Table 1 presents the detailed statistics, where _#courses_ , _#videos_ ,
_$|\mathcal{M}|$_ , _1-Label_ and _0-Label_ are the number of courses, videos,
course concepts, positive and negative labels. We can only obtain _#deletions_
from game for Chinese datasets.

| DSA | PSY
---|---|---
ZH | EN | ZH | EN
_#courses_ | 1 | 3 | 1 | 1
_#videos_ | 490 | 465 | 57 | 478
_$|\mathcal{M}|$_ | 305 | 201 | 575 | 470
_1-Label_ | 398 | 232 | 237 | 246
_0-Label_ | 402 | 568 | 563 | 554
_correlation_ | 0.696 | 0.734 | 0.712 | 0.681
_#deletions_ | 6939 | - | 4920 | -
Table 1: Datasets Statistics

###  4.2 Experiment Settings

Basic Setting. We choose GloVe Pennington et al. (2014) as our English word
embedding, Li et al. (2018) as our Chinese word embedding. We follow the same
process of Cho et al. (2014) to train the path encoder and Pan et al. (2017a)
to get prerequisite features for classifier in Section 3.2.

Baseline Methods. We compare our models (simple candidate generation results
denoted as MOOC) with four typical methods which employ different similarity
metrics. MOOC-C means our model with Classification and MOOC-CG means the
whole model added game optimization.

$\bullet$ PR Graph based method: We build the candidates and course concepts
into a graph. When the similarity between two concepts exceeds a threshold
$\tau_{PR}$, there is a link between them. The PageRank score of each
candidate is finally used for sorting. A most famous method employing pagerank
is SEAL Wang and Cohen (2007).

$\bullet$ SEISA SEISAHe and Xin (2011) is an entity set expansion system
developed by Microsoft after SEAL and outperforms traditional graph-based
methods by an original unsupervised similarity metric. We implement its
Dynamic Thresholding algorithm to sort expanded concepts.

$\bullet$ EBM Embedding based method mainly utilizes context information to
examine the similarity between expanded concepts and seeds like Mamou et al.
(2018). For each expanded concept $e$, we calculate the pairwise cosine
similarity with course concepts $\mathcal{M}$ in word2vec and use the average
as golden standard to rank the expanded concept list.

$\bullet$ PUL PU learning is a semi-supervised learning model regarding set
expansion as a binary classification task. We employ the same setting as Wang
et al. (2017) to classify and sort concepts.

Evaluation Metrics. Our objective is to generate a ranked list of expanded
concepts. Thus, to evaluate the ranking result, we use the Mean Average
Precision(MAP) as our evaluation metric, which is the preferred metric in
information retrieval for evaluating ranked lists.

###  4.3 Overall Evaluation

| ZH-PSY | ZH-DSA | ZH-Avg | EN-PSY | EN-DSA | EN-Avg
---|---|---|---|---|---|---
PR | 0.849 | 0.519 | 0.684 | 0.833 | 0.822 | 0.827
SEISA | 0.877 | 0.448 | 0.663 | 0.814 | 0.890 | 0.852
EBM | 0.875 | 0.421 | 0.648 | 0.777 | 0.858 | 0.817
PUL | 0.855 | 0.680 | 0.768 | 0.734 | 0.795 | 0.765
MOOC | 0.894 | 0.785 | 0.839 | 0.781 | 0.933 | 0.857
MOOC-C | 0.939 | 0.804 | 0.872 | 0.922 | 0.976 | 0.949
MOOC-CG | 0.954 | 0.819 | 0.886 | - | - | -
Table 2: MAP of different methods on datasets. ($S_{H}=8,\alpha=0.4$).

Table 2 summarizes the comparing results of different methods on all datasets,
and -avg means the average MAP of datasets in same language. We find that our
method outperforms existing methods across all 4 datasets. The improvements
are all statistically significant tested with bootstrap re-sampling with 95%
confidence. For example, our whole model surpasses PageRank based method and
SEISA by about 0.10 on the average of English courses. Further, we observe the
performance in following aspects:

For different datasets, our methods stably perform at a competitive level
while existing methods fluctuate fiercely. All methods maintain a better
result in English than Chinese. Especially in ZH-DSA, existing methods meet a
sharp decline at over 0.17. To find out the reason, we calculate the average
pairwise similarity between the extracted concepts $\mathcal{M}$ in each
dataset. Results show that ZH-DSA contains the most scattered course concepts
at a pairwise distance of 0.60 (ZH-PSY, EN-PSY, EN-DSA at 0.49, 0.50, 0.36).
But our expansion achieves a fine result in ZH-DSA at 0.785, indicating it
effectively relieves the semantic drift after candidate generation.

For different components of our methods. The pure candidate generation (MOOC)
mainly improves the performance by obvious promotion in ZH-DSA. The governing
improvement exists after classification (at an average over 0.90), verifying
the effectiveness of heterogeneous features we proposed. Moreover, the game-
based optimization further improves the performance (+0.14), which proves the
power of human efforts and our feedback optimization.

###  4.4 Result Analysis

The size of seed set $\tau$. The seed set size $\tau$ controls how many
concepts of $E_{c}$ and $\mathcal{M}$ should be employed to calculate
confidence score. We adjust $\tau$ from 1 to 10 and explore the influence of
$\tau$ on Candidate Generation. Figure 5(a) shows the MAP transmutation.
Despite different setting of $\tau$, our model maintain a preeminent
competitive performance at an average MAP of 0.85 for English courses and 0.81
for Chinese courses.

(a) The MAP Curve of $\tau$ (when $\alpha=0.4$)

(b) The MAP Curve of $\alpha$ (when $\tau=8$)

Figure 5: Parameter analysis.

Feature Contribution Analysis. To evaluate the features proposed in Section
3.2 and 3.3,with highest average F1-score at 0.94 as our classifier and run
our approach 4 times on the 2 Chinese datasets, with one different feature
deleted in each test. Table 3 records the changes of $P$, $R$ and $F_{1}$ for
each setting. According to the decrement of F1-scores, we find that all the
proposed features are indispensable in classification. Especially, we observe
that search path encode $Pe$ plays the most important role, decreasing most
F1-score by 7.96%. Besides, user deletion $Dr$ from game also outstandingly
increases the precision of classifier by 8.21%.

Ignored Feature | $P$ | $R$ | $F_{1}$
---|---|---|---
$s_{e}$ | -2.65% | -1.73% | -1.83%
$Pe$ | -3.97% | -10.6% | -7.96%
$Pf$ | -7.39% | -1.55% | -3.78%
$Dr$ | -8.21% | -3.10% | -4.93%
Table 3: Contribution analysis of different features. $s_{e}$, Pe, Pf, Dr are
respectively confidence score, search path encoding, prerequisite features and
deletion ratio.

The ranking threshold $\alpha$. The ranking threshold $\alpha$ is the
parameter controls how much ratio of results in Section 3.1 should be adjusted
by classification. As we increase $\alpha$, less candidates will be adjusted,
which weakens the role of the classifier. In Figure 5(b), we set $\alpha$ from
0 to 1 and find that the performance reaches a peak at an average 0.3 of
$\alpha$. The results demonstrate extra information provided by classifier
effectively lifts the recall in low-confidence area (latter 70% in average).

###  4.5 Online Evaluation

In particular, our model uses a gamified form to build a human interface. This
interactive design not only optimizes the model, but can also be used to
evaluate the effectiveness of the model in practical teaching applications.

By collecting deletion data of the expanded concepts, we can peep whether our
expansion results are really helpful to the MOOC users. In order to
quantitatively observe the user’s feedback on the model, we propose Average
Correction Rate($C_{r}$) as a game-to-model evaluation metric. This metric is
the ratio of the number of times the user deletes the top $n$ concepts to the
total deletions, and is formally denoted as
$C_{r}=\sum_{i=1}^{n}del(e_{i})/\sum_{j=1}^{\left|E_{c}\right|}del(e_{j})$. We
set $n$ to 10, 50, 100, and the $C_{r}$ of each methods are listed in Table 4.
Higher $C_{r}$ indicates less users think the expansion results are helpful.
From this perspective, our method is the most helpful to them. For $C_{r}$@10,
PUL performs a slight advantage at 0.002. However, once the range of
observation is broadened to 50 or 100, our method shows an obvious ascendancy.
Besides, after adding classifier and game, user satisfaction further
increases, reducing the $C_{r}$ by 0.004 and 0.049 at $C_{r}$@50 and
$C_{r}$@100, reiterating the improvement of these components.

|  $C_{r}$@10 |  $C_{r}$@50 |  $C_{r}$@100
---|---|---|---
PR | 0.034 | 0.202 | 0.452
SEISA | 0.028 | 0.210 | 0.486
EBM | 0.043 | 0.219 | 0.446
PUL | 0.026 | 0.181 | 0.424
MOOC | 0.028 | 0.166 | 0.437
MOOC-C | 0.028 | 0.162 | 0.412
MOOC-CG | 0.028 | 0.160 | 0.386
Table 4: Online evaluation results.

##  5 Related Works

Our work is based on phrase extraction in MOOCs Pan et al. (2017b) and is
relevant to the set expansion problem, which takes a set of seed entities as
input to expand a single category.

Google Sets was an early set expansion system. It spawned quite a few set
expansion techniques, such as Bayesian Sets Ghahramani and Heller (2006), SEAL
Wang and Cohen (2007), SEISA He and Xin (2011) and others Sarmento et al.
(2007); Wang and Cohen (2008); Wang et al. (2015). They mainly leverage the
similarity between entities measured by their co-occurrences in web texts,
wrappers and lists. For example, SEISA employs iterative similarity
aggregation and SEAL employs PageRank. Recently, SetExpan Shen et al. (2017)
extend previous works by selecting context features and Mamou et al. (2018)
skillfully employ five different type of context information and gain a very
competitive result.

Distinctively, PU-Learning methods Li et al. (2010); Wang et al. (2017)
transform set expansion into a two-class classification problem. A seed set is
regarded as a set $P$ of positive examples and candidate set is a set $U$
containing hidden positive and negative cases. The task of filtering the
candidate set turns to building a classifier to test if each candidate member
is positive or not and this inspires our classification work.

Our approach also benefits from theories of pedagogy. Concept space was first
proposed to benefit knowledge comprehension in educationHori (1997), and was
gradually employed in domain ontology representation; Online games were
already used in Kiili (2005); Threatt (2014) education and were also applied
for crowdsourcing information collectionYang et al. (2018). Both of them
significantly affected our design of model.

##  6 Conclusions and Future Work

We conduct a new investigation on automatically course concept expansion in
MOOCs. We precisely define the problem and propose an active model to search
external knowledge base for candidate concepts and detect high-quality ones
with a classifier. Moreover, we design a game-based mechanism to subtly
involve human efforts in model optimization. Experimental results on online
courses with different domains validate the effectiveness of the proposed
method. Promising future directions would be to investigate how to utilize
user interaction in MOOCs more adequately, as well as how attributes of course
concepts can help expanding.

### Acknowledgments

The work is supported by NSFC key project (U1736204, 61533018, 61661146007),
Ministry of Education and China Mobile Joint Fund (MCM20170301), and THUNUS
NExT Co-Lab.

Zhiyuan Liu is supported by the National Key Research and Development Program
of China (No. 2018YFB1004503).

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##  Appendix A Probe into the different performance across datasets

In our overall evaluation part, methods’ performance in different datasets is
undulating: 1) English datasets are tend to provide better performance. 2) ZH-
DSA is a roadblock of methods, i.e., each method meet a decline in ZH-DSA. To
explore the cause of these phenomenons, we take a further observation on the
situation of datasets in two aspects.

Looseness of Course concepts.

We calculate the average pairwise similarity of each dataset, which reflect
the alienation of course concepts. The results are shown in Figure 6.
Obviously the datasets in Chinses contains a more loose $\mathcal{M}$ than
English datasets. Thus, when expanding concepts in Chinese courses, the new
found concepts are easier to be admitted, for the radius of cluster may be too
large to intercept low-quality concepts. In another word, semantic drift is
more prone to happen in the two Chinese datasets. Therefore, it is necessary
to avoid semantic drifts in real concept expansion of MOOCs.

Figure 6: Average pairwise similarity of different datasets.

Specificity of Samples in test set.

Despite above observation, we still cannot explain the performance decline in
ZH-DSA, for ZH-PSY also provide a competitive result. A deeper investigation
is formed by finding out the sample distribution of two Chinese classification
results. We reduce the feature of all test concepts to a 2-Dimension vector
and differentiate their colors according to actual classification results.
When comparing the samples in the two test sets, we can obtain two main
observations. For one thing ZH-PSY contains more positive samples. For another
thing, the positive and negative samples are more blended in ZH-DSA. The
fundamental cause of these characteristics may be the nature of the courses.
The course of _Data Structure and Algorithm_ in Chinese is interdisciplinary
of Computer Science and Mathematics while ‘Intrduction to Psychology’ in
Chinese is much more ‘pure’, only containing one domain. Thus, heterogeneous
information is indispensable to some extent, for it can effectively utilize
different features to query proper expanded concepts.

(a) Sample distribution of ZH-DSA.

(b) Sample distribution of ZH-PSY.

Figure 7: The sample distribution of two Chinese datasets. TP, FP, TN, FN are
respectively _True Positive_ , _False Positive_ , _True Negative_ , _False
Negative_ samples.


-0.722483-0.0331869i&-0.636202-0.0203087i&-0.24586-0.106325i\\\
\end{array}\right),$ (212)

such the leptonic mixing matrix $V_{l}=V_{e_{L}}^{\dagger}V_{\nu_{L}}$ is
unitary and agrees with the constraints imposed by the allowed ranges of the
absolute values of the PMNS mixing matrix entries, given in Gonzalez-
Garcia:2021dve ; Workman:2022ynf .

On the one hand, the masses and eigenstates of exotic charged leptons
$E_{1,2}$ are supposed to be the same exotic quarks, i.e they do not mix with
SM charged leptons and are nearly degenerate in mass ($m_{E_{1}}\sim
m_{E_{2}}\sim\mathcal{O}(1)$ TeV). On the other hand, the mixing matrices of
right-handed neutrinos $\nu_{R}$ and neutral leptons $N_{R}$ are assumed to be
diagonal, i.e $V_{\nu_{R}}=I,V_{N_{R}}=I$, for simplicity. The active neutrino
masses are chosen in the normal hierarchy as follows

$\displaystyle m_{\nu_{1L}}=0.5\ \text{eV},\hskip
8.5359ptm^{2}_{\nu_{2L}}=m_{\nu_{1L}}^{2}+\Delta m_{21}^{2},\hskip
8.5359ptm^{2}_{\nu_{3L}}=m_{\nu_{1L}}^{2}+\Delta m_{31}^{2},$ (213)

where $\Delta m_{21}^{2},\Delta m_{31}^{2}$ are given in Gonzalez-
Garcia:2021dve . Besides, the masses of heavy neutral lepton $N_{R}$ and
right-handed neutrinos $\nu_{R}$ are obtained from the inverse seesaw
mechanism as $m_{N_{aR}}\simeq m_{\nu_{aR}}=\frac{v_{\sigma}}{\sqrt{2}}$
Hernandez:2021xet . With the remaining SM parameters used in our numerical
study, we list them in Table 6 6.

Table 6:  The numerical values of input parameters. Input parameters | Values | Input parameters | Values
---|---|---|---
$f_{B_{s}}$ |  $230.3(1.3)\ \text{MeV}$ FlavourLatticeAveragingGroupFLAG:2021npn  | $m_{B_{s}}$ |  $5366.88(11)\ \text{MeV}$ Workman:2022ynf
$m_{u}$ |  $1.24(22)\ \text{MeV}$ Xing:2020ijf ; Workman:2022ynf  | $m_{d}$ |  $2.69(19)\ \text{MeV}$ Xing:2020ijf ; Workman:2022ynf
$m_{c}$ |  $0.63(2)\ \text{GeV}$ Xing:2020ijf ; Workman:2022ynf  | $m_{s}$ |  $53.5(4.6)\ \text{MeV}$ Xing:2020ijf ; Workman:2022ynf
$m_{t}$ |  $172.9(4)\ \text{GeV}$ Xing:2020ijf  | $m_{b}$ |  $2.86(3)\ \text{GeV}$ Bona:2018
$N(E_{\gamma})$ |  $3.3\times 10^{-3}$ Misiak:2020vlo  | $C_{7}^{\text{SM}}(\mu_{b}=2.0\ \text{GeV})$ |  $-0.3636$Misiak:2006ab ; Czakon:2006ss ; Misiak:2020vlo
$C_{9}^{\text{SM}}(\mu_{b}=5.0\ \text{GeV})$ |  $4.344$ Beneke:2017vpq  | $C_{10}^{\text{SM}}(\mu_{b}=5.0\ \text{GeV})$ |  $-4.198$ Beneke:2017vpq
$y_{s}$ |  $0.0645(3)$ HFLAV:2022pwe  | $\kappa_{7}$ |  $0.408$ Buras:2011zb
$\kappa_{7}$ |  $0.408$ Buras:2011zb  | $\kappa_{8}$ | $0.129$
$m_{W}$ |  $80.385\ \text{GeV}$ Czakon:2015exa  | $m_{Z}$ |  $91.1876\ \text{GeV}$ Czakon:2015exa
$G_{F}$ |  $1.166379\times 10^{-5}\ \text{GeV}^{-2}$ Workman:2022ynf  | $s_{W}^{2}$ |  $0.2312$ Workman:2022ynf
$\lambda$ |  $0.22519(83)$ UTfit:2022hsi  | $A$ |  $0.828(11)$ UTfit:2022hsi
$\bar{\rho}$ |  $0.1609(95)$ UTfit:2022hsi  | $\bar{\eta}$ |  $0.347(10)$ UTfit:2022hsi

With all specified setup of input parameters, the free parameters used in the
analysis of the observables are only the couplings $f,B$ appearing in the
scalar potential, the masses of charged exotic fermions
$m_{E_{1}},m_{D_{1}},m_{U}$, the charged Higgs boson masses
$m_{H^{\pm}},m_{H_{1,2}^{\pm}}$, the mixing angle $\theta$ between
$m_{H_{1,2}^{\pm}}$. For our numerical analysis, we will randomly vary these
unknown parameters in the following ranges

$\displaystyle-f\in[10^{-3},10^{-1}],\hskip 8.5359pt\theta\in[0,\pi/2],\hskip
8.5359ptB\in[10^{-3},10^{-2}]\ \text{TeV},\hskip 8.5359ptm_{E_{1}}\in[1,4]\
\text{TeV},\hskip 8.5359ptm_{H^{\pm},H_{1,2}^{\pm}}\in[0.5,1.5]\ \text{TeV}$
(214)

Below, we provide a justification for the choice of these ranges. The ranges
of $f$ and $B$ are chosen based on the assumptions used in the diagonalization
of the scalar mass matrices, whereas the mass ranges of exotic fermions and
charged Higgs bosons are obtained from the experimental exclusion limits
arising from collider searches Sanyal:2019xcp ; CMS:2020osd . Additionally,
the mass range of charged Higgs bosons also satisfies the experimental
constraints resulting from the experimentally allowed ranges of the anomalous
magnetic moments of the electron and muon $\Delta a_{e,\mu}$, as shown in
Hernandez:2021xet .

###  VI.2 SM-like Higgs boson decays $h\to\bar{f}f$ and
$h\to\bar{f}f^{\prime}$

Firstly, we are interesting in the deviations factors $a_{h\bar{f}f}$ from the
couplings of SM like Higgs boson $h$ with fermions $f$ as follows

$\displaystyle
a_{h\tau\tau}=\frac{(g_{h\tau\tau})_{\text{theory}}}{(g_{h\tau\tau})_{\text{SM}}},\hskip
8.5359pta_{h\mu\mu}=\frac{(g_{h\mu\mu})_{\text{theory}}}{(g_{h\mu\mu})_{\text{SM}}},\hskip
8.5359pta_{h\bar{b}b}=\frac{(g_{h\bar{b}b})_{\text{theory}}}{(g_{h\bar{b}b})_{\text{SM}}},\hskip
8.5359pta_{h\bar{t}t}=\frac{(g_{h\bar{t}t})_{\text{theory}}}{(g_{h\bar{t}t})_{\text{SM}}},$
(215)

where $(g_{h\bar{f}f})_{\text{theory}}$ are the couplings predicted by the
model and are obtained from the last lines of Eq. (217) and Eq. (218. The
$(g_{h\bar{f}f})_{\text{SM}}=m_{f}/v$ are the SM predictions. With the help of
input parameters given in Sec. (VI.1), we find the predicted values of the
model for these factors and are shown them in the Table (7). We see that
factors $a_{h\tau^{+}\tau^{-}},a_{h\bar{b}b}$ and $a_{h\bar{t}t}$ are in
agreement with the $1\sigma$ experimentally allowed range reported by ALTAS
and CMS ATLAS:2022per ; CMS:2022dwd , whereas the model estimates
$a_{h\mu^{+}\mu^{-}}$ is in the $3\sigma$ experimentally allowed range of the
ATLAS result. This can be understood because the second and first SM fermion
families receive tree and one loop level masses from the inverse seesaw
mechanism, whereas the masses for the third generation of SM fermions are
generated at tree level by Yukawa interactions involving the Higgs doublets
$\phi_{1,2}$ ($\phi_{1}$ for the top and $\phi_{2}$ for the bottom and tau
lepton). Therefore, the couplings of the SM-like Higgs boson $h$ with the
first and second SM fermion generations are smaller than the coupling of the
$126$ GeV SM-like Higgs boson $h$ with the third family of the SM fermions.

Table 7:  The comparison between predicted values and experimental limits of deviation factors $a_{h\bar{f}f}$ Observables $a_{h\bar{f}f}$ | $a_{h\mu\mu}$ | $a_{h\tau\tau}$ | $a_{hbb}$ | $a_{htt}$
---|---|---|---|---
Predicted values | $\simeq 0.143$ | $\simeq 0.86$ | $\simeq 1.01$ | $\simeq 0.997$

Besides, the model also predicts the LFV decays of SM like Higgs boson $h$,
namely BR$(h\to\bar{l}_{b_{1}}l_{b_{2}})$, which are demonstrated in the below
Table (8)

Table 8:  The comparison between predicted values and experimental limits of BR$(h\to\bar{l}_{b_{1}}l_{b_{2}})$,  Branching ratios | Predicted values | Experimental upper limits
---|---|---
$h\to e\mu$ | $\simeq 2.7\times 10^{-10}$ |  $4.4\times 10^{-5}$ CMS:2023pte
$h\to e\tau$ | $\simeq 3.0\times 10^{-8}$ |  $2.0\times 10^{-3}$ ATLAS:2023mvd
$h\to\mu\tau$ | $\simeq 1.2\times 10^{-3}$ |  $1.8\times 10^{-3}$ATLAS:2023mvd

We see that all predicted branching ratios of LFV decays of SM like Higgs
boson $h\to e\mu,e\tau,\mu\tau$ satisfy the current upper experimental bounds
CMS:2023pte ; ATLAS:2023mvd . Specifically, the decays $h\to e\mu,e\tau$ are
significantly lower by several orders of magnitudes compared with
corresponding experiment results, whereas $h\to\mu\tau$ channel is quite close
to its measurement limit.

###  VI.3 Leptonic flavor observables

In what follows we perform a numerical study of the constraints imposed by the
upper limits of the branching ratios of cLFV decays BR$(l_{b_{1}}\to
l_{b_{2}}\gamma)$ as well as by the allowed experimental ranges of the
anomalous magnetic moments $\Delta a_{e(\mu)}$ via the Figs (3, 4).

Figure 3: The left and right panels respectively show the correlations of
couplings $B-f$ and $m_{E_{1}}-f$ satisfying the experimental limits of cLFV
and anomalous magnetic moments $\Delta a_{e(\mu)}$.

Figure 4: The left and right panels respectively show the correlations of
couplings $-f$ vs $m_{H_{1}^{\pm}}$ and $m_{E_{1}}$ vs$m_{H_{1}^{\pm}}$
satisfying the experimental limits of cLFV and anomalous magnetic moments
$\Delta a_{e(\mu)}$.

The left panel in Fig. (3) illustrates the correlation between couplings of
the scalar potential, denoted as $-f$ and $B$. There is almost no dependence
between these parameters; in other words, the change of $B$ does not affect
$f$, especially for $-f\geq 0.04$. This indicates that the values of the $B$
coupling are insignificant to the leptonic observables. From the left panel,
the values of $-f$ mainly range from $-f\geq 0.04$. On the other hand, the
right panel demonstrates stronger correlations between the $-f$ parameter and
the mass of exotic charged lepton $E_{1}$. We notice that the behavior of
$m_{E_{1}}$ and $-f$ is reversed; in other words, an increase of the charged
exotic lepton mass $m_{E_{1}}$ leads to a decrease of the $-f$ parameter and
vice versa. From the right panel, we find the limit $m_{E_{1}}\geq 2.2$ TeV.

Turning to Fig. (4), we illustrate the correlations between $-f$ or
$m_{E_{1}}$ and mass of charged Higgs $m_{H_{1}^{\pm}}$ in the left and right
panel, respectively. We observe that both panels indicate the irrelevance of
$m_{H}^{\pm}$ to leptonic observables. Moreover, this behavior is similar if
we consider other charged Higgs bosons $H_{1,2}^{\pm}$. Additionally, the
Figs. (3,4) demonstrate the remarkable impacts of the loop diagrams involving
the exotic charged lepton $E_{1}$ and neutral Higgs bosons $H(\mathcal{A})$ on
cLFV and anomalous magnetic moments $\Delta a_{e(\mu)}$.

Moreover, we consider the branching ratios of three body leptonic decays
BR$(l\to 3l^{\prime})$, Mu-$\overline{\text{Mu}}$ transition, and coherent
conversion $\mu\to e$ in a muonic atom, which only depends on the coupling
$f$, as shown in Fig . (5) and Fig (6). We observe that the model predicts
these observables to be much lower by several orders of magnitude than their
corresponding upper experimental bounds, as given in Table (4). Therefore, the
new physics contributions in the considering model for these observables are
negligible and can be ignored.

Figure 5: The figure shows the dependence of branching ratios of three body
leptonic decays as the function of coupling $-f$.

Figure 6: The left and right panels respectively show the dependence of
Mu-$\overline{\text{Mu}}$ transition in Eq. (131) and branching ratio
BR$(\mu\text{Au}\to e\text{Au})$ in Eq. (136) as the function of coupling
$-f$.

###  VI.4 Quark flavor observables

Let’s now focus on the phenomenology of flavor quark observables. First, we
will consider the FCNC observables in down type quark transitions $d_{a}\to
d_{b}$. It is important to note that only meson mass splittings, such as
$\Delta m_{K,B_{s},B_{d}}$, BR$(B_{s}\to\mu^{+}\mu^{-})$ and BR$(B^{+}\to
K^{+}\tau^{+}\tau^{-})$ do depend on the new neutral gauge boson mass
$m_{Z^{\prime}}$ and the coupling $g_{X}$ of $U(1)_{X}$ symmetry. Other quark
flavor observables, however, are independent of these parameters. Thus, our
initial focus is to analyze the constraints on $m_{Z^{\prime}}$ and $g_{X}$
that satisfy the experimental limits of $\Delta m_{K,B_{s},B_{d}}$,
BR$(B_{s}\to\mu^{+}\mu^{-})$ and BR$(B^{+}\to K^{+}\tau^{+}\tau^{-})$ by the
Fig. 7

Figure 7: The left and right panels respectively show the correlations of
$m_{Z^{\prime}}-g_{X}$ and $m_{Z^{\prime}}-f$ satisfying the constraints given
in Eq. (175) for BR$(B_{s}\to\mu^{+}\mu^{-})$ , Eqs. (181, 179) for $\Delta
m_{K,B_{s},B_{d}}$,and Workman:2022ynf  for BR$(B^{+}\to
K^{+}\tau^{+}\tau^{-})$

In the left panel, we observe that the range of $m_{Z^{\prime}}$ varies with
the values of $g_{X}$. For instance, with $|g_{X}|\simeq 0.2$, the mass of
$Z^{\prime}$ boson is about $\simeq[1,1.4]$ TeV. However, if $|g_{X}|>0.2$,
$m_{Z^{\prime}}$ increase and can reach up to $6.5$ TeV for $|g_{X}|\simeq 1$.
However, it should be noted that the LHC searches of the $Z^{\prime}$ gauge
boson set a lower mass limit of $m_{Z^{\prime}}\geq 4$ TeV Workman:2022ynf ,
implying the coupling $g_{X}$ should satisfy $|g_{X}|\geq 0.65$. These
constraints for $g_{X}$ and $m_{Z^{\prime}}$ are stronger than those obtained
from studying meson oscillation in Hernandez:2021xet . On the one hand, both
observables are also affected by the $f$ coupling and we depict the right
panel to demonstrate the correlations between $f$ and $m_{Z^{\prime}}$. We
observe that the correlation here is weaker compared to the left panel. This
suggests that the (pseudo)scalar WCs $C^{(^{\prime})}_{S(P)}$, which involve
CP-even or CP-odd Higgs bosons provide very small and subleading contributions
to these observables. This can be understood because when we impose the limit,
$-f\geq 0.04$ obtained from the analysis of leptonic flavor observables, such
WCs become $C^{(^{\prime})\mu\mu}_{S(P)}\sim\mathcal{O}(10^{-2}-10^{-4})$,
which are extremely small compared to WCs $C_{9,10}\sim\mathcal{O}(1)$. It is
worth noting that the lower bound $-f\geq 0.045$ obtained in the right panel
is relatively higher than that obtained in Fig. (3).

On the other hand, for the branching ratios of (semi)leptonic decays
BR$(B^{+}\to K^{+}\tau^{+}\mu^{-})$, BR$(B_{s}\to\tau^{+}\mu^{-})$ influenced
by (pseudo)scalar WCs $C_{S(P)}^{(^{\prime})\tau\mu}$ depending on the
parameter $f$, we plot the Fig .8 and realize that the model evaluates
BR$(B^{+}\to K^{+}\tau^{+}\mu^{-})_{\text{theory}}\sim 10^{-10}$,
BR$(B_{s}\to\tau^{+}\mu^{-})_{\text{theory}}\sim 10^{-9}-10^{-8}$. Compared
with relative experimental bounds given in Table 5, the predicted values are
much lower by several orders of magnitude.

Figure 8: The left and right panels respectively show the dependence of
BR$(B^{+}\to K^{+}\tau^{+}\mu^{-})$ and BR$(B_{s}\to\tau^{+}\mu^{-})$ as the
function of coupling $-f$.

For remaining observables related to the FCNC $d_{a}\to d_{b}$ transitions,
namely BR$(\bar{B}\to X_{s})\gamma$, which are contributed by WCs induced by
charged Higgs $H^{\pm},H_{1,2}^{\pm}$, and neutral Higgs bosons
$H(\mathcal{A})$, we plot the Fig. 9 to demonstrate the relationship between
parameters fulfilling the constraints given in Eq. (177).

Figure 9: The first three panels (from left to right) and last panel
respectively show the correlations between mass of exotic up type quark
$m_{U}$ with masses of charged Higgs bosons $H^{\pm}$, $H_{1,2}^{\pm}$ and
coupling $-f$ satisfying the constraints of BR$(\bar{B}\to X_{s}\gamma)$ in
Eq. (177).

In the first three panels (from left to right) of Fig. 9, we illustrate the
correlation between the mass of new up quark $m_{U}$ and the masses of three
charged Higgs bosons $m_{H^{\pm}},m_{H_{1,2}^{\pm}}$ satisfying the constraint
of BR$(\bar{B}\to X_{s}\gamma)$ in Eq. (175). We observe that the entire
ranges of $m_{H^{\pm}},m_{H_{1,2}^{\pm}}$ in all panels meet the constraint,
whereas the range of $m_{U}$ is more stringent, ranging up to approximately
$\simeq 2.5$ TeV but dominantly distribute with the ranges $m_{U}\in[1,2]$
TeV. It is worth noting that this observable also depends on $f$ through the
WCs induced by the neutral CP even (odd) Higgs bosons such as
$C_{7}^{(^{\prime})H(\mathcal{A})\bar{D}_{1}d}$. We plot the correlation
between $-f$ and mass of down type quark $m_{D_{1}}$ in the fourth panel of
Fig. (9). We observe that this correlation is not as strong compared to the
others. This can be understood because with $-f\geq 0.04$ and
$m_{D_{1}}\sim\mathcal{O}(1)$ TeV, we numerically estimate the magnitude of
WCs as
$|C_{7,8}^{H(\mathcal{A})\bar{D}_{1}d}|\sim\mathcal{O}(10^{-6}-10^{-4})\ll 1$,
which is much lower by 3 to 5 orders of magnitude than the corresponding SM
predictions $|C_{7}^{\text{SM}}|\sim\mathcal{O}(10^{-1})$. Consequently, this
implies that WCs induced by new quark $U$ and charged Higgs bosons
$H^{\pm},H_{1,2}^{\pm}$ dominantly contribute compared to those containing
$D_{1}$ and $H_{p}$, and therefore we can ignore these latter contributions in
such observables.

Turning to up quark transition observables $t\to u(c)$, we first study
branching ratios of tree-level top quark decays BR$(t\to u(c)h)$, as the
couplings of these observables are fixed and do not depend on free parameters.
The comparison between the predictions of the model and the upper experimental
bounds for such decays is shown in Table 9

Table 9:  The comparison between predicted values and experimental limits of BR$(t\to u(c)h)$ Branching ratios | Predicted values | Experimental limits ATLAS:2023ujo
---|---|---
$t\to uh$ | $\simeq 3.97\times 10^{-10}$ | $<3.8\times 10^{-4}$
$t\to ch$ | $\simeq 9.71\times 10^{-10}$ | $<4.3\times 10^{-4}$

We observe that the obtained branching ratios BR$(t\to u(c)h)$ in the model
are on the order of $10^{-9}-10^{-10}$, significantly larger by several orders
of magnitude than the corresponding SM values. Besides that, these results for
the decays satisfy the upper experimental bounds, notably as they are lower
than 5 to 6 orders of magnitude than the measurement ones. Thus, the NP
contributions of NP in these decays are small and safe under the experimental
constraints.

On the one hand, with branching ratios of radiative decays BR$(t\to
u(c)\gamma)$ are induced by loop diagrams containing charged Higgs bosons
$H^{\pm},H_{1,2}^{\pm}$ and exotic down-type quark $D_{1}$, as well as by
neutral CP even(odd) Higgs bosons $H(A)$ with up-type quark $U$ or SM quarks
$u,c,t$. The Figs. (10) are plotted using parameters obtained from $b\to s$
studies, enhancing the understanding of these transitions.

Figure 10: The first three panels (from left to right) and last panel
respectively show the correlations between mass of exotic down type quark
$m_{D_{1}}$ with masses of charged Higgs bosons $H^{\pm}$, $H_{1,2}^{\pm}$ and
coupling $-f$ satisfying the constraints of BR$(t\to u(c)\gamma)$
ATLAS:2022per .

Comparing the two panels, we observe that almost the entire range of the first
three panels satisfies the constraint, whereas the last panel demonstrates a
tighter correlation. This indicates that, for this kind of observable, the
contribution of WCs associated with charged Higgs bosons is not as strong as
those associated with neutral CP even (odd) Higgs bosons. This behavior is
opposite to the governing BR$(\bar{B}\to X_{s}\gamma)$.

Finally, we examine the LFUV ratios $R_{D^{(*)}}$, which are generated at the
tree level via the exchange of the SM charged gauge boson $W^{\pm}$ and the
charged Higgs bosons $H^{\pm},H_{1,2}^{\pm}$. In Fig. (11), we plot the
correlation between the mass of the charged Higgs bosons that satisfy the
measurement of $R_{D^{(*)}}$ LHCb:2023zxo  with input parameters enhanced by
the above studies. The figure shows that many points satisfy the constraint in
Eq . (178), but they are dominantly linearly distributed. This behavior can be
interpreted for the following reasons. Firstly, the ratios are depend on WCs
$\tilde{C}^{(^{\prime})}_{S}$ induced by charged Higgs $H_{1,2}^{\pm}$ which
are proportional $g_{L,R}^{H_{1,2}^{+}\bar{u}_{2}d_{3}}$. In addition, these
couplings respectively relate to elements of mixing charged scalar matrix
$(U^{\dagger}_{c})_{13(14)}$ and
$(U^{\dagger}_{c})_{23(24)}\sim\frac{1}{\epsilon_{1(2)}}\sim\frac{1}{m_{H_{1,2}^{\pm}}^{2}-m_{H^{\pm}}^{2}}$
given in Eq. (44). Therefore, the WCs induced by $H_{1,2}^{\pm}$ will be
enhanced significantly if there is slight degeneration in charged Higgs masses
$m_{H_{1,2}^{\pm}}\sim m_{H^{\pm}}$. This results in almost all points being
distributed linearly in Fig . (11). For $m_{H_{1,2}^{\pm}}\simeq
m_{H^{\pm}}+\mathcal{O}(10^{1})$ GeV, and with obtained parameter in above
studies, we can estimate the magnitude of these scalar WCs which can attain
the maximum value $|\tilde{C}^{(^{\prime})}_{S}|\sim\mathcal{O}(10^{-4})$,
which is much smaller than WC of SM $\tilde{C}_{V}$. This suggests that the
charged Higgs boson contributes insignificantly compared to the SM
contributions. Finally to close this section it is worth mentioning that this
scenario of nearly degenerate charged scalar masses is favored by the
constraints arising from electroweak precision observables, which is a generic
feature of multi-Higgs doublet models CarcamoHernandez:2015smi , like the one
analyzed in this work.

Figure 11: The figure shows the correlations between mass of charged Higgs
bosons $H^{\pm}$, $H_{1}^{\pm}$ satisfying the constraints of $R_{D^{(*)}}$ in
Eq. (178).

##  VII Conclusions

In this paper, we have investigated several flavor observables in both the
lepton and quark sectors in the context of an extended 2HDM with the inverse
seesaw mechanism. We have performed a detailed analysis of the scalar mass
spectrum using the perturbation method, based on certain assumptions for the
Higgs potential couplings. Additionally, we have obtained the benchmark points
fulfilling the SM lepton masses and mixing.

We found that the contributions to cLFV decays and anomalous magnetic moments
observables, resulting from loop diagrams involving the virtual exchange of CP
even (odd) Higgs bosons $H(\mathcal{A})$ and charged exotic leptons, are
significantly larger than those arising from the exchange of charged Higgs
boson. In addition, the constraints for coupling $f$ and $m_{E_{1}}$ are
derived as $-f\geq 0.04$ and $m_{E_{1}}\geq 2.2$ TeV. With the obtained bound
on $f$, the new physics contributions to the branching ratios of three-body
leptonic decays BR$(l\to 3l^{\prime})$, Mu-$\overline{Mu}$ transition, as well
as BR$(\mu\text{Au}\to e\text{Au})$ conversion, are shown to be remarkably
smaller than upper experimental limits. On the other hand, the study of quark
flavor observables is more complicated. Specifically, for the FCNC $d_{a}\to
d_{b}$ observables, the WCs induced by charged Higgs bosons
$H^{\pm},H_{1,2}^{\pm}$ and exotic quark $U$ exchange contribute the most to
BR$(\bar{B_{s}}\to X_{s}\gamma)$, whereas the remaining WCs of
$H(\mathcal{A})$ are insignificant. The range of $m_{U}\geq[1,2]$ TeV is
obtained. Furthermore, the meson oscillations are revisited with all
contributions and comprehensive quark couplings, combining with studies for
BR$(B_{s}\to\mu^{+}\mu^{-})$ and BR$(B^{+}\to K^{+}\tau^{+}\mu^{-})$, we
obtain tighter constraints on the $U(1)_{X}$ coupling $g_{X}$ and on the
$Z^{\prime}$ gauge boson which read $|g_{X}|\geq 0.65$ and $m_{Z^{\prime}}\geq
4$ TeV, respectively. The obtained lower bound of $m_{Z^{\prime}}$ is
consistent with LHC searches of the $Z^{\prime}$ gauge boson Workman:2022ynf .

The numerical values for the up-type quark flavor observables corresponding to
FCNC decays $t\to u(c)h$ are found to be much lower than their corresponding
upper experimental limits ATLAS:2023ujo , but several orders of magnitude
larger than SM predictions Eilam:1990zc ; Aguilar-Saavedra:2004mfd .
Concerning radiative decays $t\to u(c)\gamma$, their obtained branching ratios
are primarily influenced by WCs induced by CP even (odd) Higgs bosons
interacting with exotic up-type quark $U$ compared to WCs containing charged
Higgs bosons. This behavior is opposite to that governing BR$(\bar{B}\to
X_{s}\gamma)$. The FCCC $b\to s$ LFUV ratios $R_{D^{(*)}}$ are calculated to
agree with their experimental constraints, with the new physics contributions
arising from charged Higgs bosons at the tree-level shown to be negligible.

The properties of the SM-like Higgs boson $h$ are also discussed.
Particularly, the model predicts the deviation factors
$a_{h\tau\bar{\tau}},a_{hb\bar{b}}$ and $a_{ht\bar{t}}$ of the SM Higgs
couplings to $\tau\bar{\tau}$, $b\bar{b}$ and $t\bar{t}$ pairs, which are
found to agree with their $1\sigma$ experimentally allowed ranges
ATLAS:2022vkf ; CMS:2022dwd . Furthermore, the factor $a_{h\mu\bar{\mu}}$ of
the SM Higgs coupling to $\mu\bar{\mu}$ pair is estimated to agree with its
corresponding $3\sigma$ bound. This is because the first and second fermion
generations acquire masses via the tree and one-loop inverse seesaw
mechanisms, respectively. Additionally, the obtained values for the branching
ratios for the LFV decays of $h$ such as $h\to e\mu,h\to e\tau,h\to\mu\tau$,
are shown to be consistent with their corresponding experimental upper limits
ATLAS:2023mvd ; Workman:2022ynf .

## Acknowledgement

N.T. Duy was funded by the Postdoctoral Scholarship Programme of Vingroup
Innovation Foundation (VINIF), code VINIF.2023.STS.65; AECH was funded by
Chilean grants ANID-Chile FONDECYT 1210378, ANID PIA/APOYO AFB230003, ANID
Programa Milenio code ICN2019$\\_$044; D.T.Huong acknowledges the financial
support of the Vietnam Academy of Science and Technology under Grant
No.NVCC05.05/24-25.

##  Appendix A  Coefficients in Lagrangian for lepton and quark flavor
violating processes

The coefficients mentioned in Eq. (119) are given as follows

$\displaystyle
g_{R}^{H^{+}\bar{\nu}_{aL}l_{b}}=(U_{c}^{\dagger})_{22}\sum_{i=1}^{3}y_{i}^{(l)}(V^{*}_{\nu_{L}})_{ia}(V_{e_{R}})_{3b},\hskip
8.5359ptg_{R}^{H_{1,2}^{+}\bar{\nu}_{aL}l_{b}}=(U_{c}^{\dagger})_{23(24)}\sum_{i=1}^{3}y_{i}^{(l)}(V^{*}_{\nu_{L}})_{ia}(V_{e_{R}})_{3b},$
$\displaystyle
g_{R}^{H^{+}\bar{N}_{a}l_{b}}=(U_{c}^{\dagger})_{42}\sum_{i=1}^{3}\sum_{n=1}^{2}z_{in}^{(l)}(V^{*}_{N_{R}})_{ia}(V_{e_{R}})_{nb},\hskip
8.5359ptg_{R}^{H_{1,2}^{+}\bar{N}_{a}l_{b}}=(U_{c}^{\dagger})_{43(44)}\sum_{i=1}^{3}\sum_{n=1}^{2}z_{in}^{(l)}(V^{*}_{N_{R}})_{ia}(V_{e_{R}})_{nb},$
$\displaystyle
g_{L}^{H^{+}\bar{\nu}_{aR}l_{b}}=(U_{c}^{\dagger})_{22}\sum_{i=1}^{3}\sum_{j=1}^{3}y^{*(\nu)}_{ij}(V^{*}_{\nu_{R}})_{ja}(V_{e_{L}})_{ib},\hskip
8.5359ptg_{L}^{H_{1,2}^{+}\bar{\nu}_{aR}l_{b}}=(U_{c}^{\dagger})_{23(24)}\sum_{i=1}^{3}\sum_{j=1}^{3}y^{*(\nu)}_{ij}(V^{*}_{\nu_{R}})_{ja}(V_{e_{L}})_{ib},$
$\displaystyle
g_{L}^{H_{p}\bar{E}_{1}l_{b}}=\frac{1}{\sqrt{2}}\sum_{i=1}^{3}y_{i}^{*(E)}(V_{e_{L}})_{ib}(U_{S}^{\dagger})_{2p},\hskip
8.5359ptg_{R}^{H_{p}\bar{E}_{1}l_{b}}=\frac{1}{\sqrt{2}}\sum_{n=1}^{2}x_{n}^{(E)}(V_{e_{R}})_{nb}(U_{S}^{\dagger})_{3p},$
$\displaystyle
g_{L}^{\mathcal{A}_{p}\bar{E}_{1}l_{b}}=-\frac{1}{\sqrt{2}}\sum_{i=1}^{3}y_{i}^{*(E)}(V_{e_{L}})_{ib}(U_{A}^{\dagger})_{2p},\hskip
8.5359ptg_{R}^{\mathcal{A}_{p}\bar{E}_{1}l_{b}}=-\frac{1}{\sqrt{2}}\sum_{n=1}^{2}x_{n}^{(E)}(V_{e_{R}})_{nb}(U_{A}^{\dagger})_{3p},$
$\displaystyle
g_{R}^{H_{p}\bar{l}_{a}l_{b}}=\frac{1}{\sqrt{2}}\sum_{i=1}^{3}y_{i}^{(l)}(V^{*}_{e_{L}})_{ia}(V_{e_{R}})_{3b}(U_{S}^{\dagger})_{2p},\hskip
8.5359ptg_{R}^{\mathcal{A}_{p}\bar{l}_{a}l_{b}}=\frac{1}{\sqrt{2}}\sum_{i=1}^{3}y_{i}^{(l)}(V^{*}_{e_{L}})_{ia}(V_{e_{R}})_{3b}(U_{A}^{\dagger})_{2p}.$
(216)

Otherwise, the coefficients used in Eq. (140) and Eq. (141) read

$\displaystyle
g_{L}^{H^{+}\bar{u}_{a}d_{b}}=-(U_{c}^{\dagger})_{12}\sum_{i=1}^{3}y_{i}^{*(u)}(V^{*}_{u_{R}})_{ia}(V_{d_{L}})_{3b},\hskip
8.5359ptg_{R}^{H^{+}\bar{u}_{a}d_{b}}=(U_{c}^{\dagger})_{22}\sum_{i=1}^{3}y_{i}^{(d)}(V^{*}_{u_{L}})_{3a}(V_{d_{R}})_{ib},$
$\displaystyle
g_{L}^{H_{1(2)}^{+}\bar{u}_{a}d_{b}}=-(U_{c}^{\dagger})_{13(14)}\sum_{i=1}^{3}y_{i}^{*(u)}(V^{*}_{u_{R}})_{ia}(V_{d_{L}})_{3b},\hskip
8.5359ptg_{R}^{H_{1(2)}^{+}\bar{u}_{a}d_{b}}=(U_{c}^{\dagger})_{23(24)}\sum_{i=1}^{3}y_{i}^{(d)}(V^{*}_{u_{L}})_{3a}(V_{d_{R}})_{ib},$
$\displaystyle
g_{L}^{H^{+}\bar{U}d_{b}}=-(U_{c}^{\dagger})_{22}\sum_{n=1}^{2}x_{n}^{*(U)}(V_{d_{L}})_{nb},\hskip
8.5359ptg_{R}^{H^{+}\bar{U}d_{b}}=(U_{c}^{\dagger})_{32}\sum_{i=1}^{3}[w^{(d)}_{k}i(V_{d_{R}})_{ib}],$
$\displaystyle
g_{L}^{H_{1(2)}^{+}\bar{U}d_{b}}=-(U_{c}^{\dagger})_{23(24)}\sum_{n=1}^{2}x_{n}^{*(U)}(V_{d_{L}})_{nb},\hskip
8.5359ptg_{R}^{H_{1(2)}^{+}\bar{U}d_{b}}=(U_{c}^{\dagger})_{33(34)}\sum_{i=1}^{3}[w^{(d)}_{k}i(V_{d_{R}})_{ib}],$
$\displaystyle
g_{L}^{H_{p}\bar{D}_{1}d_{b}}=\frac{1}{\sqrt{2}}\sum_{n=1}^{2}[x_{n}^{*(D)}(V_{d_{L}})_{nb}](U^{\dagger}_{S})_{1p},\hskip
8.5359ptg_{R}^{H_{p}\bar{D}_{1}d_{b}}=\frac{1}{\sqrt{2}}\sum_{i=1}^{3}[x_{i}^{(d)}(V_{d_{R}})_{ib}](U^{\dagger}_{S})_{3p},$
$\displaystyle
g_{L}^{\mathcal{A}_{p}\bar{D}_{1}d_{b}}=-\frac{1}{\sqrt{2}}\sum_{n=1}^{2}[x_{n}^{*(D)}(V_{d_{L}})_{nb}](U^{\dagger}_{A})_{1p},\hskip
8.5359ptg_{R}^{\mathcal{A}_{p}\bar{D}_{1}d_{b}}=\frac{1}{\sqrt{2}}\sum_{i=1}^{3}[x_{i}^{(d)}(V_{d_{R}})_{ib}](U^{\dagger}_{A})_{3p},$
$\displaystyle
g_{R}^{H_{p}\bar{d}_{a}d_{b}}=\frac{1}{\sqrt{2}}\sum_{i=1}^{3}[y_{i}^{(d)}(V^{*}_{d_{L}})_{3a}(V_{d_{R}})_{ib}](U_{S}^{\dagger})_{2p},\hskip
8.5359ptg_{R}^{\mathcal{A}_{p}\bar{d}_{a}d_{b}}=\frac{1}{\sqrt{2}}\sum_{i=1}^{3}[y_{i}^{(d)}(V^{*}_{d_{L}})_{3a}(V_{d_{R}})_{ib}](U_{A}^{\dagger})_{2p},$
(217) $\displaystyle
g_{L}^{H^{-}\bar{D}_{1}u_{b}}=(U_{c}^{\dagger})_{12}\sum_{n=1}^{2}x_{n}^{*(D)}(V_{u_{L}})_{nb},\hskip
8.5359ptg_{R}^{H^{-}\bar{D}_{1}u_{b}}=(U_{c}^{\dagger})_{32}\sum_{i=1}^{3}[w^{(U)}_{i}(V_{u_{R}})_{ib}],$
$\displaystyle
g_{L}^{H_{1(2)}^{-}\bar{D}_{1}u_{b}}=(U_{c}^{\dagger})_{13(14)}\sum_{n=1}^{2}x_{n}^{*(D)}(V_{u_{L}})_{nb},\hskip
8.5359ptg_{R}^{H_{1(2)}^{-}\bar{D}_{1}u_{b}}=(U_{c}^{\dagger})_{33(34)}\sum_{i=1}^{3}[w^{(U)}_{i}(V_{u_{R}})_{ib}],$
$\displaystyle
g_{L}^{H_{p}\bar{U}u_{b}}=\frac{1}{\sqrt{2}}\sum_{n=1}^{2}[x_{n}^{*(U)}(V_{u_{L}})_{nb}](U^{\dagger}_{S})_{1p},\hskip
8.5359ptg_{R}^{H_{p}\bar{U}u_{b}}=\frac{1}{\sqrt{2}}\sum_{i=1}^{3}[x_{i}^{*(u)}(V_{u_{R}})_{ib}](U^{\dagger}_{S})_{3p},$
$\displaystyle
g_{L}^{\mathcal{A}_{p}\bar{U}u_{b}}=-\frac{1}{\sqrt{2}}\sum_{n=1}^{2}[x_{n}^{*(U)}(V_{u_{L}})_{nb}](U^{\dagger}_{A})_{1p},\hskip
8.5359ptg_{R}^{\mathcal{A}_{p}\bar{U}u_{b}}=-\frac{1}{\sqrt{2}}\sum_{i=1}^{3}[x_{i}^{*(u)}(V_{u_{R}})_{ib}](U^{\dagger}_{A})_{3p},$
$\displaystyle
g_{R}^{H_{p}\bar{u}_{a}u_{b}}=\frac{1}{\sqrt{2}}\sum_{i=1}^{3}[y_{i}^{(u)}(V^{*}_{u_{L}})_{3a}(V_{u_{R}})_{ib}](U_{S}^{\dagger})_{1p},\hskip
8.5359ptg_{R}^{\mathcal{A}_{p}\bar{u}_{a}u_{b}}=\frac{1}{\sqrt{2}}\sum_{i=1}^{3}[y_{i}^{(u)}(V^{*}_{u_{L}})_{3a}(V_{u_{R}})_{ib}](U_{A}^{\dagger})_{1p}$
(218)

##  Appendix B Loop functions

The loop functions used in Eq .(148) are given as follow

$\displaystyle
f_{\gamma}^{W}(x)=\frac{-4x^{3}+45x^{2}-33x+10}{12(x-1)^{3}}-\frac{3x^{3}}{2(x-1)^{4}}\ln{x}$
$\displaystyle
f_{\gamma}(x)=\frac{(-1+5x+2x^{2})}{12(x-1)^{3}}-\frac{x^{2}}{2(x-1)^{4}}\ln{x},\hskip
8.5359ptf^{\prime}_{\gamma}(x)=\frac{-2-5x+x^{2}}{12(x-1)^{3}}+\frac{x}{2(x-1)^{4}}\ln{x},$
$\displaystyle
f^{\prime\prime}_{\gamma}(x)=\frac{-7+5x+8x^{2}}{36(x-1)^{3}}+\frac{(2-3x)x}{6(x-1)^{4}}\ln{x},\hskip
8.5359ptf^{\prime\prime\prime}_{\gamma}(x)=\frac{5-10x-7x^{2}}{36(x-1)^{3}}+\frac{x(3x-1)}{6(x-1)^{4}}\ln{x},$
$\displaystyle
h^{\prime}_{\gamma}(x)=\frac{x-3}{2(x-1)^{2}}+\frac{1}{(x-1)^{3}}\ln{x},\hskip
8.5359pth^{\prime\prime}_{\gamma}(x)=\frac{-3+5x}{6(x-1)^{2}}+\frac{(2-3x)}{3(x-1)^{3}}\ln{x},\hskip
8.5359pth^{\prime\prime\prime}_{\gamma}(x)=-\frac{2x}{3(x-1)^{2}}+\frac{3x-1}{3(x-1)^{3}}\ln{x},$
$\displaystyle
f^{\prime}_{g}(x)=f^{\prime\prime}_{g}(x)=\frac{-2-5x+x^{2}}{12(x-1)^{3}}+\frac{x}{2(x-1)^{4}}\ln{x},\hskip
8.5359pth^{\prime}_{g}(x)=h^{\prime\prime}_{g}(x)=\frac{x-3}{2(x-1)^{2}}+\frac{1}{(x-1)^{3}}.$
(219)

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