11institutetext: Information Science, University of Regensburg, Regensburg,
Germany
11email<EMAIL_ADDRESS>
# Exploring Fake News Detection with Heterogeneous Social Media Context
Graphs††thanks: version as accepted at ECIR 2023 with author information,
Github link and Acknowledgements added

Gregor Donabauer  Udo Kruschwitz

###### Abstract

Fake news detection has become a research area that goes way beyond a purely
academic interest as it has direct implications on our society as a whole.
Recent advances have primarily focused on text-based approaches. However, it
has become clear that to be effective one needs to incorporate additional,
contextual information such as spreading behaviour of news articles and user
interaction patterns on social media. We propose to construct heterogeneous
social context graphs around news articles and reformulate the problem as a
graph classification task. Exploring the incorporation of different types of
information (to get an idea as to what level of social context is most
effective) and using different graph neural network architectures indicates
that this approach is highly effective with robust results on a common
benchmark dataset.

###### Keywords:

Fake News Detection Social Media Networks Graph Machine Learning

##  1 Introduction

Detecting online information disorders is an important step to guarantee free
expression and discourse, e.g. [7]. A traditional approach to address this
problem is manual fact-checking by experts, which is highly labour-intensive,
but more recently automated methods that make use of external knowledge bases
and natural language processing (NLP) have been proposed [29, 10], sometimes
as a tool to support the fact-checker [20]. Recent studies have demonstrated
the growing importance of social media as a source of information for many
people [21, 25]. In this context it has been pointed out that a range of
information types extracted from social media and the relations between such
entities are useful for detecting fake news, e.g., [26, 19]. Relations between
news items and related information signals have been modelled to extract
additional features by applying graph neural networks (GNNs) on graph
structures (as discussed below), but so far this has typically either been
limited to individual social media features or in combination with features
that are not taken from social media (which makes it hard to assess the
relevance of _social context only_).

To link different types of information they can be modelled as heterogeneous
graphs [21], and such networks have the capability of introducing multiple
types of relations around a news article within a _single data structure_ ,
which allows to see each graph as an independent data point. However it is not
clear if such data structures are superior compared to homogeneous graphs that
model only one relation type, as spreading and interaction behaviour between
true and fake news generally heavily differs on social media [28].

There is a substantial body of work that demonstrates the use of GNNs, but it
remains an open question as to how suitable they are for the use-case of fake
news detection with _social media context features only_. It is also unclear
which social media signals are most effective here and if there are
significant differences between GNN architectures. To investigate these
questions, our contributions can be summarized as follows: (1) We introduce
HetSMCG, a methodology to construct extensive Heterogeneous Social Media
Context Graphs around news articles fusing multiple types of information in a
single graph and reformulate fake news detection as a _graph classification
problem_ ; (2) We evaluate a range of experimental settings exploring
information sources taken from _social media only_ and different GNN
architectures to understand how the task can effectively be solved; (3) We
compare how performance between heterogeneous multi-relation graphs and
homogeneous single-relation graphs differs; (4) We make all our code and in-
depth experimental results available to the community to foster
reproducibility and sharing of resources.111https://github.com/doGregor/Graph-
FakeNewsNet

##  2 Related Work

One way of classifying the different research directions at a high level is to
distinguish _content-based_ methods that focus on the stories themselves,
_context-based_ approaches tapping into, e.g., social media signals and
_intervention-based_ approaches [23]. We limit our focus on the context-based
paradigm with graph-based representations in our brief contextualization
within existing related work.

The majority of context-based approaches for fake news detection use graph
structures to model relations, e.g. [4] created user-article graphs and
utilized the neighborhood structure to embed news nodes that can then be
classified. However, the number of information types is limited to users and
news items. In similar problem formulations, user-post interaction graphs are
considered [18, 22]. Mehta et al. [16] treat fake news detection as a node
classification task based on a large graph of users, news and publishers
(though they do not model differences in edge types and do not limit
information to social media context). In follow up work they also introduce
link prediction in this graph structure to find previously unavailable
connections [17]. Lu and Li [14] also consider user graphs to extract social
interactions using GNNs. In combination with recurrent networks, that leverage
retweet patterns, they classify related news articles. Similar approaches that
use temporal retweet patterns have been proposed [8, 27]. Formulating fake
news detection as graph classification has been applied by Dou et al. [8].
They calculated user representations by leveraging timeline tweet embeddings
and use the outcome as node features to model news propagation paths
associated with articles. Another propagation-based approach was introduced by
Song et al. [27]. In their work graph structured temporally evolving retweet
patterns have been used to classify whether an article is fake or not. Two
other approaches left are most similar to our work [21, 13]. Both are
constructing large heterogeneous graphs of articles, users, social media
features like tweets, and publishers to classify news nodes as either true or
fake (again not limited to social media context). Incoming news nodes need to
be integrated into this grown data structure to produce accurate results. With
our approach we aim at proposing a method where each article can be seen as an
independent data point.

We conclude that graph structures are a powerful way to include different
dimensions of information and thus to improve fake news detection. However,
linking many different entity types (and not just one) in independent data
structures has barely been investigated yet (and even less if we only consider
information types taken from social media in a heterogeneous way) and we aim
at contributing to filling this gap with our research.

##  3 Methodology and Experiments

###  3.1 Dataset and Graph Construction Method

The only common benchmark dataset that provides rich social context
information in network-like structures is FakeNewsNet [24] which is why we
adopt it. FakeNewsNet consists of two different datasets, Politifact and
GossipCop. Both datasets are from different domains (PolitiFact for political
news and GossipCop for entertainment stories) which allows to distinguish
between domain-specific performances. Twitter’s terms and condition state that
tweets and other information cannot be publicly released
directly.222https://developer.twitter.com/en/developer-terms/agreement-and-
policy We therefore recrawl all data (thereby acquiring a dataset that will
slightly differ from what others have used so far).

We construct a heterogeneous social media context graph
$\mathcal{G}=(\mathcal{V},\mathcal{E})$ around each news article. Thus, each
snapshot consists of a set of disjoint vertex sets
$\mathcal{V}=\mathcal{V}_{N}\cup\mathcal{V}_{T}\cup\mathcal{V}_{U}$ where
$\mathcal{V}_{i}\cap\mathcal{V}_{j}=\emptyset,\forall i\neq j$.
$\mathcal{V}_{N}$ represents news articles and $|\mathcal{V}_{N}|=1$ for each
graph. Node features for this type of vertex are obtained using BERT-base
document embeddings, i.e. $\forall
v_{n}\in\mathcal{V}_{N}:v_{n}\in\mathbb{R}^{768}$. All embeddings are
generated using flairNLP333https://github.com/flairNLP/flair. Set
$\mathcal{V}_{T}$ includes all tweets that refer to the original news article,
all tweets retweeting those posts and the latest timeline tweets of each user.
Since in our experiments we are incrementally increasing the amount of
information types in the heterogeneous graphs, the number of nodes in
$\mathcal{V}_{T}$ varies depending on the experimental setup. Node features
are a concatenation of textual BERT-base document embeddings, as well as
retweet count and favorite count, i.e. $\forall
v_{t}\in\mathcal{V}_{T}:v_{t}\in\mathbb{R}^{770}$. Finally, $\mathcal{V}_{U}$
is the set of user nodes where features are a concatenation of BERT-base
profile description embeddings, follower count, friends count, favorites count
and statuses count, i.e. $\forall
v_{u}\in\mathcal{V}_{U}:v_{u}\in\mathbb{R}^{772}$. We also evaluate graphs
with text embeddings only and leave out other features (e.g. number of
favorite count) in that case. Edges connecting the nodes are satisfying
constraints according to the node types they link together. More specifically,
we use at most three types of edges: tweets citing news articles
($(v_{t},\tau_{TN},v_{n})\in\mathcal{E}\to
v_{t}\in\mathcal{V}_{T},v_{n}\in\mathcal{V}_{N}$), users posting tweets (which
also applies to users posting retweets and users posting timeline tweets)
($(v_{u},\tau_{UT},v_{t})\in\mathcal{E}\to
v_{u}\in\mathcal{V}_{U},v_{t}\in\mathcal{V}_{T}$) and tweets retweeting tweets
($(v_{t},\tau_{TT},u_{t})\in\mathcal{E}\to
v_{t}\in\mathcal{V}_{T},u_{t}\in\mathcal{V}_{T}$). The graphs fuse all types
of information (the news piece as well as a range of different social media
context features) without any prior aggregation step.

###  3.2 Experimental Setup

We construct graphs by incrementally increasing the amount of included data as
follows: (1) We start with only considering tweets related to the news
article; (2) We add user profiles of the people who posted the tweets to the
graphs; (3) We include the five latest timeline posts of those users; (4) We
add retweets related to the initial Twitter posts; (5) We combine all types of
information mentioned so far. For simplicity we set the number of timeline
tweets to five to keep the relation between number of all nodes and number of
timeline tweet nodes in the graph manageable. The in-depth exploration of
different settings is left as future work. To keep the structure of graphs
consistent we exclude samples where news data are no longer available. Graphs
with fewer than five vertices per node type are skipped to make sure that at
least some social media features are available (other studies sometimes even
set higher limits of e.g. a minimum of 15 tweets [15]). Furthermore, we make
all edges undirected to improve message passing during graph convolution. For
all experiments we use 5-fold cross-validation and report the average scores
as results. We keep the overall number of graphs and the cross-validation
split consistent for all setups and datasets to obtain comparable results. The
number of graphs per dataset amounts to 483 for PolitiFact (real: 235; fake:
248), 12,214 for GossipCop (real: 10,067; fake: 2,147) and thus 12,697 for the
full FakeNewsNet dataset (real: 10,302; fake: 2,395) which consists of a
mixture of the two subsets.

We are also flattening all graphs into non-heterogeneous structures to
understand how such a setup performs compared to the one described so far.
Since homogeneous graphs do not explicitly model disjoint sets of vertices and
use a single adjacency matrix, we have to unify all feature dimensions and we
do that by either pruning all nodes to a dimension of 768 (only text) or zero-
padding all nodes to a dimension of 772. We classify each graph using two-
layer vanilla graph neural networks and evaluate three different types of
graph convolution: (1) GraphSAGE; (2) Graph Attention Convolution (GAT); (3)
Heterogeneous Graph Transformer (HGT). For details on the models we refer to
our GitHub.

All models are trained with the same hyperparameter setting to obtain
comparable results. We use 20 train epochs, a batch size of 16 and a learning-
rate of $8e^{-5}$. As the number of real and fake graphs is unbalanced in
terms of GossipCop and the full FakeNewsNet dataset, we use class weights
during experiments with those data. Our implementations are based on PyTorch
Geometric444https://github.com/pyg-team/pytorch_geometric [9].Training and
testing are executed using a single NVIDIA GTX 1080 GPU with an overall
graphical memory size of 11GB. Even though we use high-dimensional node
features and a large number of graphs during our experiments, training and
inference in the described setup only take about 15 minutes on a single GPU
(with respect to the highest number of graphs).

##  4 Results

In line with common practice in NLP, we report precision, recall and macro F1
scores for all setups [12]. Detailed results are reported on our GitHub. As
previously mentioned, the values are the average results obtained by 5-fold
cross validation. In general, excluding social features like retweet count and
number of followers yields better results than concatenating them with text
embedding to initially represent the graph nodes. We observe the highest
scores for Politifact with setup (5) (0.979 macro F1 and accuracy), for
GossipCop with setup (3) (0.972 macro F1 and 0.983 accuracy) and the full
FakeNewsNet dataset with setup (5) (0.966 macro F1 and 0.979 accuracy), only
considering textual features. Interestingly, we get good results for
Politifact from the point of adding retweets. Timeline tweets instead are not
important. For GossipCop we observe very similar performance for all setups
after once including tweets and users. Here, the other features (retweets and
timeline tweets) do not significantly change the model performance. As the
full FakeNewsNet dataset mainly consists of GossipCop, we can observe very
similar results here. The best performing GNN convolution type is throughout
all setups HGT.

The results for the setups with homogeneous graphs are all significantly lower
at p < 0.05 compared to their equivalent with heterogeneous graphs. This
observation holds true for truncated and padded homogeneous graphs for all
three datasets. We demonstrate how modelling disjoint sets of nodes and
multiple types of edges improve the representation of social context and lead
to overall significantly better fake news detection performance.

For comparison, we use two recently published approaches as strong baselines.
As we excluded some of the articles from our experiments (due to missing
social context information), we rerun both baseline systems with the same data
used in our experiments. There are many other studies reporting results on
FakeNewsNet (e.g. [1, 8, 6, 27]) but we picked the two most competitive,
namely: (1) CMTR [11]: BERT classified texts that were preprocessed using
summarization techniques as well as additional social media features like
comments. (2) HetTransformer [13]: node classification in a heterogeneous
graph featuring user, post and news nodes using an encoder-decoder transformer
model. We use our results obtained by the same setup (HGT and all types of
information) for comparison against the baseline scores even if this setup
does not always reach the highest performance in each setup (to avoid cherry-
picking). Table 1 reports competitive and robust results on all datasets and a
new benchmark performance on the Politifact subset (though this not
statistically significant).

Approach | Politifact | Gossipcop | FakeNewsNet
---|---|---|---
| F1 | ACC | F1 | ACC | F1 | ACC
CMTR [11] | 0.965 | 0.965 | 0.854** | 0.922** | 0.859** | 0.920**
HetTransformer [13] | 0.900** | 0.900** |  0.994** |  0.997** |  0.985** |  0.991**
HetSMCG (our approach) | 0.979 | 0.979 | 0.969 | 0.982 | 0.966 | 0.979
Table 1: F1 and accuracy scores for Politifact, GossipCop and the full
FakeNewsNet dataset. Significant differences to our approach (at $p<0.05$ and
Bonferroni correction) are marked with with **.

##  5 Discussion

An interesting observation is that the results for the Politfact dataset
improve when we include retweet data in our network structures. It has already
been pointed out that fake news tend to have more retweets than real news
[24], and the results provide some support for the utility of this feature.
Moreover, it has also previously been reported that misinformation in general
spreads more effectively due to its emotionalizing content [2]. Due to the
small size of the dataset (real: 239; fake: 261), this could have a large
influence on classification performance. The much larger GossipCop and full
FakeNewsNet datasets (real: 12,610; fake: 3,185) seems to be less influenced
by such information type-specific characteristics. Since we assign retweets to
the same node type as article-related tweets and user timeline tweets we
address the problem of information type-specific performance gaps. An
observation that applies to every dataset is that including all types of
information leads to the best performing setup. For our experiments we also
find that there are only marginal differences between the different GNN
architectures while HGT gives the highest performance.

We should also point out some limitations of our work. Even though we compare
our results against strong, recently published baselines one needs to be
careful when reporting and interpreting improvements over existing systems,
e.g. [5]. Generalisability of insights is a related issue [3]. Providing
access to all code and data in our Github repository helps addressing this
concern.

##  6 Conclusion

We have demonstrated how including a variety of social media context
information can improve fake news detection. By modelling the problem as a
graph classification task using heterogeneous graph data structures we achieve
competitive results on a real-world dataset. Incorporating all the contextual
information at our disposal leads to the overall best performances.

## Acknowledgments

This work was supported by the project COURAGE: A Social Media Companion
Safeguarding and Educating Students funded by the Volkswagen Foundation, grant
number 95564.

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# Photoproduction of the $B_{c}$ meson at the future $e^{+}e^{-}$ colliders

Xi-Jie Zhan<EMAIL_ADDRESS>Xing-Gang Wu<EMAIL_ADDRESS>Xu-Chang Zheng
<EMAIL_ADDRESS>Department of Physics, Chongqing Key Laboratory for
Strongly Coupled Physics, Chongqing University, Chongqing 401331, P. R. China

###### Abstract

In the present paper, we study the photoproduction of various $S$-wave $B_{c}$
states, i.e. $B_{c}(1{}^{1}S_{0})$, $B_{c}^{*}(1{}^{1}S_{0})$,
$B_{c}(2{}^{1}S_{0})$ and $B^{*}_{c}(2{}^{3}S_{1})$, at future $e^{+}e^{-}$
colliders within the framework of non-relativistic QCD. Two sources of the
initial photons can be realized in the production, i.e., the LBS photon and
the WWA photon. In addition to the direct photoproduction via the sub-process,
$\gamma+\gamma\rightarrow B_{c}+b+\bar{c}$, we also calculate the resolved
photoproduction via the sub-processes, $\gamma+g\rightarrow B_{c}+b+\bar{c}$
and $q+\bar{q}\rightarrow B_{c}+b+\bar{c}$ with $q=u,d,s,g$, respectively.
Numerical results indicate that the contributions from the single resolved
photoproduction are significant and even dominant at certain collision
configuration. At the future high-energy and high-luminosity $e^{+}e^{-}$
colliders, the $B_{c}$ meson generated via the photoproduction mechanism is
promisingly observable and can be well studied.

##  I Introduction

The $B_{c}(1{}^{1}S_{0})$ meson, including its excited states, are strong
interaction systems composed of two heavy quarks with different flavors. This
unique flavored bound-state system could be a good probe for studying the
strong interaction among quarks and gluons, and the quantum chromodynamics
(QCD). So far the $B_{c}$ mesons have been only observed at the hadronic
colliders. Its ground state was discovered in 1998 by the CDF collaboration
Abe  _et al._ (1998a, b), and its excited $2S$ states were observed by ATLAS
collaboration Aad  _et al._ (2014) in 2014 and by CMS and LHCb collaborations
Sirunyan  _et al._ (2019); Aaij  _et al._ (2019) in 2019. Many theoretical
studies have been done for $B_{c}$ production at hadron colliders Chang and
Chen (1993); Chang  _et al._ (1995); Kolodziej  _et al._ (1995); Berezhnoy
_et al._ (1995); Chang  _et al._ (1996); Berezhnoy  _et al._ (1996); Baranov
(1997a, b); Cheung (2000); Chang and Wu (2004); Chang  _et al._ (2004a, 2005a,
2005b, b, 2006); Wang and Wu (2012); Bi  _et al._ (2017); Chen  _et al._
(2018); Berezhnoy  _et al._ (2019); Zheng  _et al._ (2020a). A computer
program, BCVEGPY, for the direct hadronic production of $B_{c}$ meson has been
given in Refs.Chang  _et al._ (2004b, 2006); Wang and Wu (2012). The next
generation $e^{+}e^{-}$ colliders were proposed by some groups, such as the
FCC-ee Abada  _et al._ (2019), the CEPC Group (2018); Dong  _et al._ (2018),
the ILC Aarons  _et al._ (2007); Erler  _et al._ (2000) and so on. These
future $e^{+}e^{-}$ colliders are planed to have the ability to run at several
high collision energies with unprecedented luminosities. They are expected to
be great platforms for many subjects including heavy quarkonium physics Yang
_et al._ (2012); Chen  _et al._ (2014a, 2013); Sun  _et al._ (2013); Chen  _et
al._ (2014b); Sun  _et al._ (2014, 2015); He and Kniehl (2020); Zhan and Wang
(2020, 2021); Zhan  _et al._ (2022).

As for the $B_{c}$ meson, there are two production modes at the $e^{+}e^{-}$
collider, i.e., the production via the $e^{+}e^{-}$ annihilation Yang  _et
al._ (2012, 2013); Zheng  _et al._ (2016); Berezhnoy  _et al._ (2017); Zheng
_et al._ (2018, 2020b); Zhang  _et al._ (2022); Yang  _et al._ (2022a) and the
photoproduction mechanism Chen  _et al._ (2014b, 2020); Yang  _et al._
(2022b). A computer program, BEEC, for the first production mode has been
presented in Refs.Yang  _et al._ (2013, 2022a). As for the photoproduction
mechanism, the $B_{c}$ meson can be produced via the photon-photon fusion such
as $\gamma+\gamma\rightarrow B_{c}+b+\bar{c}$. Here the initial photons can
come from the bremsstrahlung effect, whose energy spectrum are well described
in Weizäcker-Williams approximation (WWA) Frixione  _et al._ (1993); moreover,
the laser back-scattering (LBS) can also provide high energy photon. In
addition to the above direct photoproduction, there are also channels of
resolved photoproduction, where the photons participate in the hard process
via their quark and gluon content Klasen  _et al._ (2002). As a result, the
process $e^{+}e^{-}\rightarrow e^{+}e^{-}B_{c}+X$ receives contributions from
three channels, i.e., the direct, single-resolved and double-resolved
photoproduction. All three channels should be considered in the calculation
because they are of the same order in the perturbative expansion. The resolved
photoproduction of heavy quarkonium have been investigated in some literature
Klasen  _et al._ (2002); Li and Chao (2009); Zhan and Wang (2020, 2021); Zhan
_et al._ (2022) and it is indicated that the single-resolved photoproduction
can give significant or dominant contributions both in the color-singlet and
color-octet channels. While for the $B_{c}$ meson, only the direct
photoproduction channel has been studied Chen  _et al._ (2014b, 2020); Yang
_et al._ (2022b) and it is worthy and reasonable to investigate the effects of
the resolved channels according to the previous studies.

In this work, based on the framework of non-relativistic QCD (NRQCD) Bodwin
_et al._ (1995), we study the photoproduction of
$B_{c},B_{c}^{*}(1{}^{3}S_{1}),B_{c}(2{}^{1}S_{0})$ and
$B^{*}_{c}(2{}^{3}S_{1})$ at future $e^{+}e^{-}$ colliders, considering two
sources of the initial photons, i.e., the LBS photon and the WWA photon. In
addition to the direct photoproduction $\gamma+\gamma\rightarrow
B_{c}+b+\bar{c}$, we specially calculate the sub-processes of the resolved
photoproduction, $\gamma+g\rightarrow B_{c}+b+\bar{c}$ and
$q+\bar{q}\rightarrow B_{c}+b+\bar{c}$ with $q=u,d,s,g$, respectively. In
Section II, we present the formulation of our calculation. Numerical results
and discussions are given in Section III and a brief summary is in Section IV.

##  II Formulation

Figure 1:  The energy spectra of the LBS photon and the WWA photon. Figure 2:  Some typical Feynman diagrams for calculating the partonic cross section $\hat{\sigma}$ of $B_{c}$ photoproduction at $e^{+}e^{-}$ collider. The diagrams are drawn by JaxoDraw Binosi and Theussl (2004). Table 1: The integrated cross sections (in unit of fb) for the photoproduction of $B_{c}$ via the LBS photon and the WWA photon (in brackets), respectively. Three typical collision energies are taken and the cut $p_{t}>1$ is imposed. Three channels of Eq. (6,7,8) have been summed up. $\sqrt{S}(\mathrm{GeV})$ | 250 | 500 | 1000
---|---|---|---
$\sigma_{B_{c}}$ | 33.50(0.73) | 38.74(1.59) | 55.99(3.04)
$\sigma_{B^{*}_{c}}$ | 175.06(7.14) | 177.49(14.09) | 240.69(24.82)
$\sigma_{B_{c}(2{}^{1}S_{0})}$ | 20.05(0.44) | 23.19(0.95) | 33.52(1.82)
$\sigma_{B^{*}_{c}(2{}^{3}S_{1})}$ | 104.80(4.27) | 106.26(8.43) | 144.09(14.86)
Table 2:  The integrated cross sections (in unit of fb) of different channels of the photoproduction of $B_{c}$ via the LBS photon. Three typical collision energies, $250(500,1000)\mathrm{~{}GeV}$, are taken and the cut $p_{t}>1$ is imposed. $channels$ | $\gamma+\gamma$ | $\gamma+g$ | $q+\bar{q}$
---|---|---|---
$\sigma_{B_{c}}$ | $20.28(10.74,4.92)$ | $12.89(27.10,48.70)$ | $0.33(0.90,2.37)$
$\sigma_{B^{*}_{c}}$ | $114.90(54.93,23.42)$ | $59.03(120.05,211.15)$ | $1.14(2.51,6.12)$
$\sigma_{B_{c}(2{}^{1}S_{0})}$ | $12.14(6.43,2.95)$ | $7.72(16.22,29.15)$ | $0.20(0.54,1.42)$
$\sigma_{B^{*}_{c}(2{}^{3}S_{1})}$ | $68.78(32.88,14.02)$ | $35.34(71.87,126.41)$ | $0.68(1.50,3.66)$
Table 3:  The integrated cross sections (in unit of fb) of different channels of the photoproduction of $B_{c}$ via the WWA photon. Three typical collision energies, $250(500,1000)\mathrm{~{}GeV}$, are taken and the cut $p_{t}>1$ is imposed. $channels$ | $\gamma+\gamma$ | $\gamma+g$ | $q+\bar{q}$
---|---|---|---
$\sigma_{B_{c}}$ | $0.68(1.33,2.37)$ | $0.05(0.19,0.58)$ | $0.001(0.005,0.017)$
$\sigma_{B^{*}_{c}}$ | $6.89(13.18,22.11)$ | $0.24(0.89,2.64)$ | $0.007(0.02,0.06)$
$\sigma_{B_{c}(2{}^{1}S_{0})}$ | $0.41(0.83,1.46)$ | $0.03(0.11,0.35)$ | $0.0008(0.003,0.01)$
$\sigma_{B^{*}_{c}(2{}^{3}S_{1})}$ | $4.13(7.89,13.24)$ | $0.14(0.53,1.58)$ | $0.004(0.013,0.036)$

Figure 3:  The $p_{t}$ distributions for $B_{c}$ photoproduction. (a,d):
$p_{t}$ distributions of four $B_{c}$ states and “prompt $B_{c}$” means
production of the ground $B_{c}$ after including the feed-down contributions
from excited states with $100\%$ decay probability to it. (b,c,e,f): $p_{t}$
distributions for the channels in Eq. (6,7,8) of the ground $B_{c}$
production.

Figure 4:  The rapidity ($y$) distributions for $B_{c}$ photoproduction.
(a,c,e,g): $y$ distributions of four $B_{c}$ states and “prompt $B_{c}$” means
production of the ground $B_{c}$ after including the feed-down contributions
from excited states with $100\%$ decay probability to it. (b,d,f,h): $y$
distributions for three channels in Eq. (6,7,8) of the ground $B_{c}$
production.

The energy spectrum of the photon from the bremsstrahlung effect can be
described in WWA, which takes the form Frixione  _et al._ (1993),

$\displaystyle f_{\gamma/e}(x)$ $\displaystyle=$
$\displaystyle\frac{\alpha}{2\pi}\Bigg{[}\frac{1+(1-x)^{2}}{x}{\rm
log}\frac{Q^{2}_{\rm max}}{Q^{2}_{\rm min}}$ (1)
$\displaystyle+2m_{e}^{2}x\left(\frac{1}{Q^{2}_{\rm max}}-\frac{1}{Q^{2}_{\rm
min}}\right)\Bigg{]},$

where $x=E_{\gamma}/E_{e}$ is the fraction of the longitudinal momentum
carried by the photon, $\alpha$ is the electromagnetic fine structure
constant, $Q^{2}_{\rm min}=m_{e}^{2}x^{2}/(1-x)$ and $Q^{2}_{\rm
max}=(E\theta_{c})^{2}(1-x)+Q^{2}_{\rm min}$. $\theta_{c}=32\mathrm{~{}mrad}$
is the maximum scattered angular cut in order to ensure the photon to be real.
$E=E_{e}=\sqrt{s}/2$ with the collision energy $\sqrt{s}$.

And for the case of LBS photon, its spectrum function is Ginzburg  _et al._
(1983),

$\displaystyle
f_{\gamma/e}(x)=\frac{1}{N}\left[1-x+\frac{1}{1-x}-4r(1-r)\right],$ (2)

where $r=x/\left[x_{m}(1-x)\right]$, and the normalization factor,

$\displaystyle N$ $\displaystyle=$
$\displaystyle\left(1-\frac{4}{x_{m}}-\frac{8}{x_{m}^{2}}\right)\log(1+x_{m})$
(3) $\displaystyle+\frac{1}{2}+\frac{8}{x_{m}}-\frac{1}{2(1+x_{m})^{2}}.$

Here $x_{m}=4E_{e}E_{l}\cos^{2}\frac{\theta}{2}$, $E_{e}$ and $E_{l}$ are the
energies of incident electron and laser beams, respectively. $\theta$ is the
angle between them. The energy of the LBS photon is restricted by

$\displaystyle 0\leq x\leq\frac{x_{m}}{1+x_{m}},$ (4)

with optimal value of $x_{m}$ being $4.83$ Telnov (1990). These two spectra
have quite different behaviors as shown in Fig. 1.

Within the framework of NRQCD factorization, the cross section of $B_{c}$
photoproduction at the $e^{+}e^{-}$ collider can be factorized as 111For
simplicity, here only formulas for the ground-state $B_{c}$ meson are
presented and those for the excited states are similar.,

$\displaystyle\mathrm{d}\sigma\left(e^{+}e^{-}\rightarrow
e^{+}e^{-}B_{c}+b+\bar{c}\right)$
$\displaystyle=\int\mathrm{d}x_{1}f_{\gamma/e}\left(x_{1}\right)\int\mathrm{d}x_{2}f_{\gamma/e}\left(x_{2}\right)$
$\displaystyle\times\sum_{i,j,k}\int\mathrm{d}x_{i}f_{i/\gamma}\left(x_{i}\right)\int\mathrm{d}x_{j}f_{j/\gamma}\left(x_{j}\right)$
$\displaystyle\times\sum_{n}\mathrm{~{}d}\hat{\sigma}(ij\rightarrow
c\bar{b}[n]+b+\bar{c})\left\langle\mathcal{O}^{B_{c}}[n]\right\rangle.$ (5)

Here $f_{i/\gamma}$($i=\gamma,g,u,d,s$) represents the Glück-Reya-Schienbein
(GRS) distribution function of parton $i$ in photon Gluck  _et al._ (1999).
$f_{\gamma/\gamma}(x)=\delta(1-x)$ is for the direct photoproduction process.
$d\hat{\sigma}(ij\to c\overline{b}[n]+b+\bar{c})$ is the differential partonic
cross section, which can be calculated perturbatively. $c\overline{b}[n]$ is
the intermediate state with quantum number $n$. $\langle{\cal
O}^{B_{c}}[n]\rangle$ is the long distance matrix element(LDME) representing
the probability for a $c\bar{b}[n]$ pair hadronizing into $B_{c}$ meson. In
the lowest order approximation, only the color-singlet intermediate state
$c\bar{b}[{}^{1}S^{[\textbf{1}]}_{0}]$ is considered and the corresponding
LDME can be calculated by potential models. For definiteness, we consider the
following sub-processes for three production channels,

$\displaystyle\gamma+\gamma\rightarrow
B_{c}(B^{*}_{c},B_{c}(2{}^{1}S_{0}),B^{*}_{c}(2{}^{3}S_{1}))+b+\bar{c},$ (6)
$\displaystyle\gamma+g\rightarrow
B_{c}(B^{*}_{c},B_{c}(2{}^{1}S_{0}),B^{*}_{c}(2{}^{3}S_{1}))+b+\bar{c},$ (7)
$\displaystyle q+\bar{q}\rightarrow
B_{c}(B^{*}_{c},B_{c}(2{}^{1}S_{0}),B^{*}_{c}(2{}^{3}S_{1}))+b+\bar{c},$ (8)

with $q=g,u,d,s$. Some typical Feynman diagrams for calculating the partonic
cross sections are shown in Fig. 2. In the analytical and numerical
calculations, we use the well-established package Feynman Diagram Calculation
(FDC) Wang (2004), which employs the standard projection method Bodwin and
Petrelli (2002) to deal with the amplitudes.

##  III Numerical results and discussions

The input parameters in the calculation are taken as follows. The fine
structure constant is fixed as $\alpha=1/137$. $m_{b}=4.8\mathrm{~{}GeV}$,
$m_{c}=1.5\mathrm{~{}GeV}$ and $M_{B_{c}}=m_{b}+m_{c}$. The one-loop running
strong coupling constant is employed. The renormalization scale is set to be
the transverse mass of the $B_{c}$ meson, $\mu=\sqrt{M^{2}_{B_{c}}+p^{2}_{t}}$
with $p_{t}$ being its transverse momentum. The LDMEs
$\left\langle\mathcal{O}^{B_{c}}[n]\right\rangle$ are related to the wave
function at the origin, e.g.,
$\left\langle\mathcal{O}^{B_{c}}[n]\right\rangle\approx
N_{c}|R_{S}(0)|^{2}/(2\pi)$, with $|R_{1S}(0)|^{2}=1.642\mathrm{~{}GeV}^{3}$
and $|R_{2S}(0)|^{2}=0.983\mathrm{~{}GeV}^{3}$ Eichten and Quigg (1994, 1995).

Table 1 lists the integrated cross section of the photoproduction of $B_{c}$
under three typical collision energies, both for the LBS photon and WWA
photon. It is shown from the table that all cross sections become larger with
the increment of collision energy. The cross sections via LBS photon are much
larger than those of WWA photon. This is due to the quite different spectra
functions of them as shown in Fig. 1. We can also see that the production rate
of the vector meson $B^{*}_{c}$ are much larger than those of the scalar
$B_{c}$. Since the excited states decay to the ground state with almost
$100\%$ probability, they will greatly increase the production of $B_{c}$.
Taking as example the integrated luminosity of future $e^{+}e^{-}$ collider to
be $\mathcal{O}(10^{4})\mathrm{~{}fb^{-1}}$ and considering the feed-down
contributions from excited $B_{c}$ states, we shall have about $3.3\times
10^{6}$ ($1.3\times 10^{5}$) $B_{c}$ mesons to be generated via LBS (WWA)
photons under collision energy $\sqrt{s}=250\mathrm{~{}GeV}$. Thus the
photoproduction of $B_{c}$ at future $e^{+}e^{-}$ colliders provides good
opportunity to study $B_{c}$ meson.

Table 2 shows the contributions from different channels for the LBS photon.
With the increase of $\sqrt{s}$, the cross section of direct photoproduction
channel (Eq. (6)) decreases, while those of the other two channels become
larger. At $\sqrt{s}=250\mathrm{~{}GeV}$, the $\gamma+\gamma$ channel provides
largest production and the single-resolved photoproduction $\gamma+g$ also
gives significant contributions. With the increment of the collision energy,
the channel $\gamma+g$ becomes dominant. Consequently for the LBS photon, the
resolved photoproduction channels of $B_{c}$ at future $e^{+}e^{-}$ colliders
should be taken into account.

The situations are quite different for the WWA photon, as shown in Table 3.
The cross sections of all the three channels become larger when increasing the
collision energy. The $\gamma+\gamma$ channels are always dominant while
contributions of other two channels are very small or even negligible.

Fig. 3 presents the transverse momentum distributions of $B_{c}$
photoproduction. All the distributions have a peak around several
$\mathrm{~{}GeV}$ of $p_{t}$ and decrease logarithmically in the large region.
From Fig. 3(b,c), we can see that the single resolved channel $\gamma+g$ give
important contributions in relative small $p_{t}$ region. In real experiments,
there maybe not enough $B_{c}$ events in large $p_{t}$ region to make well
measurements. Thus the single resolved photoproduction should be included in
the calculation of $B_{c}$ photoproduction. For the WWA photon, the direct
photoproduction channel $\gamma+\gamma$ is always primarily dominant in whole
$p_{t}$ region.

Fig. 4 presents the photoproduction in terms of the rapidity($y$)
distributions of final $B_{c}$ mesons. There are wide plateaux within
$|y|<2.5$ for the LBS photoproduction at $\sqrt{s}=250\mathrm{~{}GeV}$, while
the curves carved in at $\sqrt{s}=500\mathrm{~{}GeV}$. This is because with
the increase of collision energy, the contribution of $\gamma+g$ channel
becomes dominant. The rapidity distribution of WWA photoproduction look
ordinary compared with the LBS case.

Table 4: Variations of the integrated cross sections (in unit of fb) by $m_{c}$ for the photoproduction of $B_{c}$ via the LBS photon and WWA photon (in brackets) at $\sqrt{s}=250\mathrm{~{}GeV}$, respectively. The cut $p_{t}>1$ is imposed and three channels of Eq. (6,7,8) have been summed up. $m_{c}(\mathrm{GeV})$ | $1.4$ | $1.5$ | $1.6$
---|---|---|---
$\sigma_{B_{c}}$ | $41.09(0.92)$ | $33.50(0.73)$ | $27.64(0.59)$
$\sigma_{B^{*}_{c}}$ | $211.81(8.73)$ | $175.06(7.14)$ | $146.46(5.90)$
$\sigma_{B_{c}(2{}^{1}S_{0})}$ | $24.60(0.55)$ | $20.05(0.44)$ | $16.55(0.35)$
$\sigma_{B^{*}_{c}(2{}^{3}S_{1})}$ | $126.80(5.23)$ | $104.80(4.27)$ | $87.68(3.53)$
Table 5: Variations of the integrated cross sections (in unit of fb) by $m_{b}$ for the photoproduction of $B_{c}$ via the LBS photon and the WWA photon (in brackets) at $\sqrt{s}=250\mathrm{~{}GeV}$, respectively. The cut $p_{t}>1$ is imposed and three channels of Eq. (6,7,8) have been summed up. $m_{b}(\mathrm{GeV})$ | $4.6$ | $4.8$ | $5.0$
---|---|---|---
$\sigma_{B_{c}}$ | $38.72(0.87)$ | $33.50(0.73)$ | $29.16(0.61)$
$\sigma_{B^{*}_{c}}$ | $201.84(8.59)$ | $175.06(7.14)$ | $152.76(5.97)$
$\sigma_{B_{c}(2{}^{1}S_{0})}$ | $23.18(0.52)$ | $20.05(0.44)$ | $17.45(0.37)$
$\sigma_{B^{*}_{c}(2{}^{3}S_{1})}$ | $120.83(5.14)$ | $104.80(4.27)$ | $91.45(3.57)$

Take $\sqrt{s}=250\mathrm{~{}GeV}$ as example, we estimate theoretical
uncertainties induced by the heavy quark masses and the renormalization scale.
Table 4 presents the uncertainties from the variation of charm quark mass
where we take $m_{c}=1.5\pm 0.1\mathrm{~{}GeV}$ with
$m_{b}=4.8\mathrm{~{}GeV}$ and $\mu=\sqrt{M^{2}_{B_{c}}+p^{2}_{t}}$. Table 5
shows the uncertainties from $m_{b}=4.8\pm 0.2\mathrm{~{}GeV}$ with
$m_{c}=1.5\mathrm{~{}GeV}$ and $\mu=\sqrt{M^{2}_{B_{c}}+p^{2}_{t}}$. Table 6
is for $\mu={\cal C}\sqrt{M^{2}_{B_{c}}+p^{2}_{t}}({\cal C}=0.5,1,2)$ with
$m_{c}=1.5\mathrm{~{}GeV}$ and $m_{b}=4.8\mathrm{~{}GeV}$. We see that slight
variation of the heavy quark mass can result in substantial change of the
cross sections. A strong dependence on the renormalization scale indicates it
is maybe large that the next-to-leading order and higher orders of corrections
in $\alpha_{s}$.

Table 6: Variations of the integrated cross sections (in unit of fb) by $\mu={\cal C}\sqrt{M^{2}_{B_{c}}+p^{2}_{t}}$ with ${\cal C}=0.5,1,2$, for the photoproduction of $B_{c}$ via the LBS photon and WWA photon (in brackets) at $\sqrt{s}=250\mathrm{~{}GeV}$, respectively. The cut $p_{t}>1$ is imposed and three channels of Eq. (6,7,8) have been summed up. ${\cal C}$ | $0.5$ | $1.0$ | $2.0$
---|---|---|---
$\sigma_{B_{c}}$ | $45.69(1.07)$ | $33.50(0.73)$ | $25.77(0.53)$
$\sigma_{B^{*}_{c}}$ | $243.06(10.68)$ | $175.06(7.14)$ | $133.30(5.15)$
$\sigma_{B_{c}(2{}^{1}S_{0})}$ | $27.35(0.64)$ | $20.05(0.44)$ | $15.43(0.32)$
$\sigma_{B^{*}_{c}(2{}^{3}S_{1})}$ | $145.51(6.39)$ | $104.80(4.27)$ | $79.80(3.08)$

##  IV Summary

In this work, we have investigated the photoproduction of
$B_{c},B_{c}^{*},B_{c}(2{}^{1}S_{0})$ and $B^{*}_{c}(2{}^{3}S_{1})$ at future
$e^{+}e^{-}$ colliders, where two sources of initial photons can be realized,
i.e., the LBS photon and the WWA photon. Besides the direct photon-photon
fusion, we specially consider the resolved photoproduction mechanisms that are
generalized via the channels of Eq. (7,8), which are lack in previous studies.
Numerical results show that the single resolved photoproduction channel (Eq.
(7)) can give sizable and even dominant contributions under certain collision
configurations, e.g., the LBS photoproduction at
$\sqrt{s}=500\mathrm{~{}GeV}$. Considering the excited states of $B_{c}$ could
decay to the ground state with almost $100\%$ probability, they shall be
important sources of $B_{c}$ production. If setting the integrated luminosity
of future $e^{+}e^{-}$ collider to be $\mathcal{O}(10^{4})\mathrm{~{}fb^{-1}}$
and considering the feed-down contributions from the excited $B_{c}$ states,
we shall have about $3.3\times 10^{6}$ ($1.3\times 10^{5}$) $B_{c}$ meson
events to be generated via the LBS (WWA) photons under the collision energy
$\sqrt{s}=250\mathrm{~{}GeV}$. However, the leading-order calculations in
$\alpha_{s}$ of the cross section have large dependence on the heavy quark
masses and the renormalization scale, which means corrections of higher orders
maybe substantial. We would like to leave them for future study. Nevertheless,
we could expect that future $e^{+}e^{-}$ colliders to be good platforms for
investigating the $B_{c}$ meson properties.

###### Acknowledgements.

This work was supported in part by the Natural Science Foundation of China
under Grants No. 12147116, No. 12175025, No. 12005028 and No. 12147102, by the
China Postdoctoral Science Foundation under Grant No. 2021M693743 and by the
graduate research and innovation foundation of Chongqing, China under Grant
No.ydstd1912.

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