# Discovering Relationships between Object Categories
via Universal Canonical Maps

Natalia Neverova  Artsiom Sanakoyeu  Patrick Labatut  David Novotny  Andrea
Vedaldi
Facebook AI Research
<EMAIL_ADDRESS>
11footnotetext: Both authors contributed equally to this work.

Figure 1: Inter-category correspondences emerge from dense pose prediction.
Our method discovers high-quality correspondences between different object
classes automatically, as a byproduct of learning category-specific dense pose
predictors. It does so by enforcing cycle consistency between reference 3D
templates as well as by a new type of consistency between images and
templates. This allows the model to transfer information between animal
classes (the location of the eyes).

Abstract

_We tackle the problem of learning the geometry of multiple categories of
deformable objects jointly. Recent work has shown that it is possible to learn
a unified dense pose predictor for several categories of related objects.
However, training such models requires to initialize inter-category
correspondences by hand. This is suboptimal and the resulting models fail to
maintain correct correspondences as individual categories are learned. In this
paper, we show that improved correspondences can be learned automatically as a
natural byproduct of learning category-specific dense pose predictors. To do
this, we express correspondences between different categories and between
images and categories using a unified embedding. Then, we use the latter to
enforce two constraints: symmetric inter-category cycle consistency and a new
asymmetric image-to-category cycle consistency. Without any manual annotations
for the inter-category correspondences, we obtain state-of-the-art alignment
results, outperforming dedicated methods for matching 3D shapes. Moreover, the
new model is also better at the task of dense pose prediction than prior
work._

##  1 Introduction

Algorithms can nowadays understand well the geometry of _specific object
categories_ such as humans: we have reliable methods for detecting and
segmenting them, extracting their 2D landmarks and dense surface coordinates,
as well as reconstructing them in 3D. In principle, these methods can be
applied to many other types of objects, such as any kind of animal, from pets
to wildlife. In practice, however, doing so is often prohibitively expensive.
The main bottleneck is data acquisition, especially for supervised training in
3D, and extensive manual annotation. High-quality 3D human models are
bootstrapped using specialized motion capture systems such as domes that are
difficult to apply to objects such as wild animals. Annotating 2D geometric
primitives such as segments and 2D keypoints can be done manually from raw
images, but it is costly and somewhat difficult to do for unfamiliar animal
anatomies. Thus, a naïve application of existing high-quality model
acquisition techniques cannot trivially scale to learning the massive variety
of object types that exist in the world, which include 6.5K mammal species,
7.7M animal species, and around 8.7M natural species overall [9, 34].

The key to scaling is to realize that, while there are indeed millions of
different types of objects, these are not independent. For instance, different
cat breeds are relatively similar, so a single ‘cat’ model is likely to work
well for all cats, just like a single ‘human’ model has been shown to work
well for many different human body shapes [32]. In fact, useful information
can likely be shared among fairly different types of objects, such as all
mammals or all animals. The limit is given by the ability of the model to
represent diverse information while capturing and eliminating redundancies
wherever possible. The hope is that such a model could learn the geometry of
different object types with a cost which is sub-linear in their number.

A similar idea was recently pursued in [35] for the task of _dense pose
prediction_ [17]. Just like 2D pose prediction estimates the location of a
small number of distinctive object landmarks, dense pose estimation does so
for a continuous set of landmarks, identified as the point of a 3D template of
the object (fig. 1). The goal is to learn a _canonical map_ , i.e. a function
that maps all relevant pixels in an image to the corresponding points in the
template, thus identifying them. For supervised learning, correspondences
between images and templates are collected manually, using a category-specific
template for each example object. As a result, annotations for different
object categories are unrelated, which makes it hard to learn a universal,
category-agnostic object representation.

In order to address this problem, the authors of [35] establish initial point-
to-point correspondences between different category-specific templates using a
mix of manual annotations and automated interpolation. However, as we show in
the experiments, their approach has two shortcomings. Firstly, their manual
correspondence initialization is somewhat arbitrary and thus likely
suboptimal. The second problem, which partially arises from the first, is that
their initial inter-category correspondences are not _maintained_ while the
model is trained, and are eventually ‘forgotten’.

In this paper we argue that, if the goal of the alignment is to facilitate
learning a multi-category object representation, an optimal alignment should
emerge spontaneously as part of the learning process, thus solving the two
issues above. Our key contribution is thus a new learning formulation for
universal canonical maps that induces _automatically_ high-quality intra-
category correspondences. The most important outcome is that the learned maps
solve the dense pose prediction problem accurately for several object
categories while at the same time putting those in correspondence, allowing to
transfer information between them.

We base our model on learning a single, universal embedding space to express
all required correspondences. Points in the different 3D templates as well as
image pixels are mapped to this common space, which allows to compute dense
template-to-template and image-to-template correspondences. Differently from
[35], the template embeddings in this work are _not_ initialized from manually
annotated inter-category correspondences. Instead, all embeddings are obtained
automatically while learning the canonical maps for individual categories
while satisfying certain consistency constraints.

For the constraints, we use simple but effective rules. Apart from the most
basic one, which encourages similarity of the embeddings of nearby template
points (smoothness), we contribute by introducing two types of cycle-
consistency for learning canonical surface mappings: The first one enforces
cycle consistency between different 3D templates, which encourages bijective
correspondences between them. Additionally, we note that canonical maps, by
establishing correspondences from images to templates, are not bijective but
injective, and we show that this can be exploited by an asymmetric form of
cycle consistency between images and templates. By using the common embedding
space, all such constraints are expressed as differentiable loss terms.

Empirically, we demonstrate several advantages of our new approach compared to
[35]. We show that our approach finds automatically high-quality
correspondences between different object categories _without any manual
supervision for this task_. This is compelling because it shows that, as we
hypothesized, there is a natural advantage in learning jointly the geometry of
different but related object types. In fact, the 3D correspondences we
discover in this manner outperform the ones discovered by state-of-the-art 3D
shape alignment methods. Finally, our method not only aligns canonical maps,
but also improves their quality, resulting in more accurate dense pose
prediction than the state of the art.

##  2 Related work

##### Human pose estimation.

Human pose prediction often starts by detecting 2D landmarks, usually
coinciding with the main joints of the body [31, 1, 23, 22]. For this task,
early shallow methods [15, 5, 23, 40] have been surpassed by deep
convolutional architectures [37, 48, 10]. Sparse landmarks can be replaced by
dense ones, identifiable with a reference 3D template of the object. The
resulting dense pose prediction problem was pioneered by DensePose [17] using
the SMPL [32] mesh as a canonical template. Parsing R-CNN [51] improved the
Dense Pose network by extending the popular R-CNN architecture [16]. More
recently, Slim DensePose [36] showed that a smaller number of keypoint
annotations is sufficient to learn competitive DensePose models, potentially
significantly reducing the effort required for learning new non-human
categories.

##### Animal pose estimation.

Several works also attempted to estimate the pose of various animal species.
Methods such as [52, 44, 25, 26, 38, 49] learned to detect [52, 44], match
[25] or reconstruct [26, 38] various birds from the CUB dataset [49]. 3D
reconstruction of the shape of a broader set of animal species has been
attempted by Zuffi et al. in [59, 60, 58]. Similar to monocular 3D human mesh
recovery models [24, 28, 27] that predict parameters of SMPL, [59, 60, 58]
utilize a parametric model of a mesh of an animal body (SMAL) in order to
constrain the set of possible animal reconstructions, with further
improvements in the work of Biggs et al. [4, 3]. Sanakoyeu et al. [43]
transfer DensePose from humans to proximal animal classes without extra labels
by a self-training approach.

The work most relevant to ours is Neverova et al. [35], which introduced the
idea of continuous surface embeddings (CSE) to tackle the dense pose
prediction problem for several animal categories together. They further
contributed a dataset of dense pose annotations for various animal species. We
improve on [35] by learning more accurate canonical maps that are more
consistent across different categories and by not requiring any manual
initialization for the correspondences between different object categories. We
also contribute with an extended dataset of dense animal poses for
experimentation.

##### Intrinsic 3D shape analysis.

Our work is also related to the analysis of the intrinsic properties of 3D
shapes, where the fundamental problem is to establish correspondences between
different shapes. Non-deep learning methods include embeddings of geodesic
distance matrices [13, 6] and various kinds of diffusion geometry [11]
descriptors — Heat Kernel Signature [45] and its scale-invariant follow-up
[8], Gromov-Hausdorff descriptors [7] and the Wave Kernel Signature [2]. One
of the main building blocks of the aforementioned descriptors are the
eigenfunctions of the Laplace-Beltrami operator (LBO) [42] that define a
smooth basis of a coordinate frame of a mesh surface. Ovsjanikov et al. [39]
proposed the functional map (FM) framework that establishes soft
correspondences between pairs of shapes by relating the mesh LBO
eigenfunctions with a simple linear mapping. The CSE method from [35] uses FMs
and ZoomOut [33] to interpolate an initial set of manually-established inter-
class correspondences. Differently from them, we only use LBO to express
smoothness, but we otherwise consider topological constraints such as
bijectivity and injectivity that are more appropriate for establishing non-
isometric correspondences, such as between different animal categories.

##### Cycle consistency.

Cycle consistency is a powerful paradigm that has been explored in many
different fields of computer vision: pixel-wise image matching [56, 55], image
translation [57], or category-specific 3D reconstruction [53]. Given a single
input image of an instance of an object category, Kulkarni et al. [30, 29]
enforce consistency between a rendered UV map of a 3D template shape of the
object category and the learned canonical map, while our method does not
require to fit/render the 3D model. In the context of 3D shape analysis, Huang
et al. [20] introduced a semi-definite programming formulation that factorized
a matrix of all point-to-point matches between pairs of meshes in order to
make the matches cycle-consistent. Similarly, Yang et al. [50] use the
Sinkhorn regularization (SH) to find the nearest cycle-consistent solution to
an initial matrix of noisy point-wise matches. Ren et al. [41] exploit the
spectral properties of correspondences and cycle consistence between shape
pairs. Our method is inspired by [50] in the sense that we utilize cycle
consistency in order to improve our dense pose labels by relating surfaces of
different category template shapes.

##  3 Method

We start by summarizing the continuous surface embedding (CSE) representation
of [35] (section 3.1) and then we explain how to extend it to learn high-
quality inter-category correspondences automatically (section 3.2).

###  3.1 Continuos surface embeddings

The _continuous surface embedding_ (CSE) of [35] allows us to express
correspondences between different 3D templates and between templates and
images in a homogeneous and differentiable manner. A CSE is a function
$e:S\rightarrow\mathbb{R}^{D}$ sending each point $X\in S$ of a 3D surface $S$
to a $D$-dimensional embedding vector. We assume that the surface $S$ is a
mesh with $K$ vertices $S=(X_{k})_{1\leq k\leq K}$ and we collect the
corresponding embedding vectors as the rows of a matrix
$E\in\mathbb{R}^{K\times D}$. The matrix $E$, which we learn from data, can be
fairly large, but smoothness222the fact that nearby vertices should have
similar embeddings. can help to reduce its dimensionality. This can be done by
considering a smooth functional basis $U\in\mathbb{R}^{K\times Q}$ on the
mesh, where $Q\ll D$, and define $E=U\hat{E}$. With this, we can work with the
compressed embedding parameterization $\hat{E}\in\mathbb{R}^{Q\times D}$. As
in [35], we take $U$ to be the lowest eigenvectors of the Laplace-Beltrami
operator (LBO) of the mesh $S$. While the LBO is often used in the literature
as a cue to match near-isometric shapes, our shapes are not at all isometric.
For this reason, we use the LBO only to encode a generic notion of smoothness,
but _not_ as a cue for matching.

##### Encoding correspondences via CSEs.

Embedding vectors can be used to define correspondences between any two sets
of objects $A=(a_{1},\dots,a_{K})$ and $B=(b_{1},\dots,b_{L})$. Namely, given
embedding functions $e:A\rightarrow\mathbb{R}^{D}$ and
$e:B\rightarrow\mathbb{R}^{D}$, we can compare embedding vectors to send
elements of set $B$ to elements of set $A$ probabilistically:

$p(a_{k}|b_{l},e)=\frac{\exp\left(-\langle
e_{a_{k}},e_{b_{l}}\rangle\right)}{\sum_{t=1}^{K}\exp\left(-\langle
e_{a_{t}},e_{b_{l}}\rangle\right)}.$ (1)

In our case, given two CSEs $E=U\hat{E}$ and
$E^{\prime}=U^{\prime}\hat{E}^{\prime}$ for two different meshes $S$ and
$S^{\prime}$, eq. 1 gives us distributions
$p(X_{k}|X^{\prime}_{l},\hat{E},\hat{E}^{\prime})$ and
$p(X^{\prime}_{l}|X_{k},\hat{E},\hat{E}^{\prime})$, encoding mappings
$S^{\prime}\rightarrow S$ and $S\rightarrow S^{\prime}$, respectively. We can
also express image-to-mesh and mesh-to-image maps. For this, let $I$ be an
image and consider a finite set $\Omega\subset\mathbb{R}^{2}$ of pixel
locations. We use a deep convolutional neural network $e_{x}=[\Phi(I)]_{x}$ to
compute the embedding vectors for all the pixels $x\in\Omega$. Then, given a
mesh $S$ together with its embedding matrix $E=U\hat{E}$, we can use eq. 1 to
obtain a distribution $p(X_{k}|x,\hat{E},\Phi(I))$ encoding a map
$\Omega\rightarrow S$ from the pixels to the mesh. Note that the latter is, by
definition, a canonical map, and as such it provides a solution to the dense
pose prediction task. We can also swap the roles of image and mesh in this
expression, obtaining a probability $p(x|X_{k},\hat{E},\Phi(I))$ encoding a
reverse map $S\rightarrow\Omega$. This map will be useful later.

Finally, we can, in an entirely analogous manner, define image-to-image
correspondences $\Omega\rightarrow\Omega^{\prime}$ by comparing embeddings
$\Phi_{x}(I)$ and $\Phi(I^{\prime})_{x^{\prime}}$. This is useful to transfer
information directly across images, as we demonstrate in the experiments for
keypoint transfer.

##### Working with multiple object categories.

The approach above can easily accommodate any number of categories and
corresponding templates. Each category $m=1,\dots,M$ is captured by a mesh and
its embedding $(S^{m},\hat{E}^{m})$. Each mesh can have a different number of
vertices $|S^{m}|=K^{m}$. Likewise, the LBO basis
$U^{m}\in\mathbb{R}^{Q^{m}\times D}$ is mesh-specific, including having
potentially a different number of basis elements $Q^{m}$. Crucially, however,
the dimensionality of the embedding space $D$ is the same for all templates,
as the embeddings must be comparable.

###  3.2 Dense pose and emerging correspondences

The use of a common embedding space for templates and images means that all
such objects can be put in correspondence by using the method of section 3.1.
However, this does not necessarily mean that the correspondences learned by
the model are meaningful. In more detail, manual annotations for the dense
pose task are of the type $(I,m,x,X)$ where $I$ is an image, $m$ a category,
$x$ a pixel, and $X\in S^{m}$ its corresponding vertex in the category-
specific template $S^{m}$ [17, 35]. By fitting such annotations, the model is
encouraged to learn good dense pose predictors for each category, but not
necessarily good inter-category correspondences. The latter may emerge because
the neural network $\Phi$ is shared in full or in part among different
categories, which means that similarly-looking images will naturally tend to
be embedded in similar ways. However, this is a weak effect. Below, we add
several constraints to improve the quality of the emerging correspondences.

##### Dense pose supervision.

Solving the dense pose prediction tasks means learning maps $\Omega\rightarrow
S^{m}$ sending the image region $\Omega$ that contains an occurrence of the
object to the template $S^{m}$ of the object itself. As noted above,
supervision for this task comes in the form of a dataset $\mathcal{D}$ of
tuples $(I,m,x,X)$ and is captured by the loss as follows:

$\mathcal{L}^{\text{sup}}=\frac{1}{|\mathcal{D}|}\sum_{(I,m,x,X)\in\mathcal{D}}\sum_{k=1}^{K^{m}}d_{S^{m}}(X_{k}^{m},X)\\\
\cdot p(X_{k}^{m}|x,\hat{E}^{m},\Phi^{m}(I)).$ (2)

In this expression, $d_{S^{m}}$ is the geodesic distance on the mesh $S^{m}$.
This loss is optimized w.r.t. the mesh embeddings and neural networks
$(\hat{E}^{m},\Phi^{m})_{1\leq m\leq M}$ (where the different networks share
most or all of their parameters).

##### Inter-category correspondences.

We assume that there exists sensible one-to-one correspondences
$S^{m}\leftrightarrow S^{n}$ between any pair of templates. In this case, the
cycle $S^{m}\rightarrow S^{n}\rightarrow S^{m}$ should approximate the
identity function. We can rewrite the cycle in terms of the probabilistic
correspondences described in section 3.1 by marginalizing the intermediate
step as follows:

$p(X^{m}_{k}|X^{m}_{t})=\sum_{l=1}^{K^{n}}p(X^{m}_{k}|X^{n}_{l})\,p(X^{n}_{l}|X^{m}_{t}).$
(3)

While we do not show it for compactness, note that all such probabilities
depend on the learned embeddings $\hat{E}^{m}$ and $\hat{E}^{n}$. If the cycle
is closed correctly, this probability should peak at $X^{m}_{t}=X^{m}_{k}$,
which is captured by the _mess-to-mesh_ loss (m2m):

$\mathcal{L}^{mn}=\frac{1}{K^{m}}\sum_{k=1}^{K^{m}}\sum_{t=1}^{K^{m}}d_{S^{m}}(X^{m}_{k},X^{m}_{t})\,p(X^{m}_{k}|X^{m}_{t}).$
(4)

To the loss $\mathcal{L}^{mn}$ we also add the symmetric loss
$\mathcal{L}^{nm}$. Cycle consistency has been exploited before in many
different contexts [30, 47, 50, 57, 54, 19, 21]. Here we use it to guide the
discovery of correspondences between different meshes.

##### Canonical map injectivity.

The signal (2) is only given at a sparse set of manually-labelled image
pixels. A denser constraint can be obtained by noting that the canonical maps
$\Omega\rightarrow S^{m}$ must be _injective_ , in the sense that all pixels
in the object region $\Omega$ should map to different vertices in the mesh
$S^{m}$. Injectivity means that the canonical map has a left inverse: if a
mesh vertex corresponds to at least one image pixel, then this correspondence
must be unique. We can thus close the cycle $\Omega\rightarrow
S^{m}\rightarrow\Omega$, resulting in the _image-to-mesh_ loss (i2m):

$\mathcal{L}^{Im}=\frac{1}{|\Omega|}\sum_{x\in\Omega}\,\sum_{y\in\Omega}d_{I}(y,x)p(y|x).$
(5)

where $d_{I}$ is a distance in image space (e.g. Euclidean) and, similar to
eq. 3,

$p(y|x)=\frac{1}{K^{m}}\sum_{k=1}^{K^{m}}p(y|X^{m}_{k})\\\ \,p(X^{m}_{k}|x).$
(6)

While not shown for compactness, all these probabilities depend on the mesh
embedding $E^{m}$ and the neural network $\Phi^{m}$ that we wish to learn.

Rather than summing eq. 5 on the entire set $\Omega$, we consider a
_downsampled_ version $\bar{\Omega}\subset\Omega$ with
$|\bar{\Omega}|\ll|\Omega|$. This is done for computational efficiency (as
there can be a very large number of pixels in certain image regions). Compared
to using the full domain, the effect is to slightly relax eq. 5.

Note that, differently from the mesh-to-mesh cycle, this cycle is _not_
symmetric: while we can close the chain $\Omega\rightarrow
S^{m}\rightarrow\Omega$, we _cannot_ close the chain
$S^{m}\rightarrow\Omega\rightarrow S^{m}$. The first chain is valid because
all pixels in $\Omega$ correspond to _a unique point_ of the mesh $S^{m}$. On
the other hand, many of the points in the mesh $S^{m}$ will _not_ have a
corresponding image point in $\Omega$ for the simple fact that at least part
of the object cannot be visible in a given image.

###  3.3 Overall loss

To summarize, our model is trained by minimizing a combination of the losses
of eqs. 2, 4 and 5:

$\mathcal{L}^{\text{sup}}+\frac{1}{M(M-1)}\mathop{\sum_{m,n=1}^{M}}_{m\not=n}\mathcal{L}^{mn}+\frac{1}{|\mathcal{D}|}\sum_{(I,m)\in\mathcal{D}}\mathcal{L}^{Im}.$

##  4 Experiments

dataset | init. | train | $\textbf{AP}\downarrow$ | $\textbf{GErr}\downarrow$ | $\textbf{GPS}\uparrow$
---|---|---|---|---|---
DP-LVIS [35] | ZoomOut | – | 25.4 | 23.7 | 66.7
✓ | 35.1 | 28.0 | 68.7
| Random | ✓ | 34.4 | 34.1 | 63.7
DP-LVIS v1.0 | Random | ✓ | 37.4 | 20.7 | 77.1

Table 1:  Baselines (humans & animals). We train a universal canonical map
using DensePose-COCO and animal data and report DensePose performance on
animal categories (AP), as well as mesh alignment quality for animals and
people (GErr and GPS). The architecture is of [35], combined with multi-class
detection. ZoomOut initialization does not result in performance gains on a
larger dataset.  Figure 2: Qualitative results by the full model m2m+i2m-all.
The dog 3D model serves as a common reference for all classes. This setup is
significantly more challenging than in [35], where each class was visualised
with its own 3D reference.

method | $\textbf{GErr}\downarrow$ | $\textbf{GPS}\uparrow$ | AP | $\textbf{AP}_{50}$ | $\textbf{AP}_{75}$ | $\textbf{AP}_{M}$ | $\textbf{AP}_{L}$ | AR | $\textbf{AR}_{50}$ | $\textbf{AR}_{75}$ | $\textbf{AR}_{M}$ | $\textbf{AR}_{L}$
---|---|---|---|---|---|---|---|---|---|---|---|---
our baseline | 14.05 | 84.33 | 37.5 | 67.8 | 36.4 | 35.1 | 41.9 | 51.5 | 78.8 | 53.2 | 41.7 | 55.1
w/ m2m | 11.96 | 87.35 | 38.2 | 68.5 | 36.4 | 36.6 | 42.6 | 52.0 | 79.7 | 52.6 | 42.9 | 55.6
w/ i2m | 12.67 | 85.13 | 38.1 | 68.7 | 36.2 | 35.5 | 42.4 | 52.0 | 79.6 | 53.2 | 42.7 | 55.6
w/ i2m-all | 11.74 | 87.48 | 38.3 | 68.9 | 36.3 | 35.7 | 42.5 | 52.3 | 79.9 | 53.5 | 42.8 | 55.7
w/ m2m+i2m | 11.37 | 88.14 | 38.3 | 68.7 | 36.6 | 36.7 | 42.5 | 52.3 | 79.7 | 53.7 | 43.0 | 55.7
w/ m2m+i2m-all | 10.90 | 88.85 | 38.5 | 68.7 | 37.1 | 37.5 | 42.6 | 52.5 | 79.7 | 54.3 | 43.8 | 55.9

Table 2: Ablation of cycle-consistency loss terms (animals only): i2m
corresponds to comparing the image to the target mesh given the ground truth
class label, while i2m-all matches all object instances to all meshes in a
cross-category regime. Figure 3: Effect of the m2m+i2m-all term: improved
local consistency and smoothness in dense pose predictions (see outlined
regions), as well as more accurate instance masks.

method | $\textbf{GErr}\downarrow$ | $\textbf{GPS}\\!\uparrow$
---|---|---
ZoomOut [33] | 26.24 | 63.33
Deep Shells [12] | 17.91 | 78.68
our baseline | 14.05 | 84.33
\+ m2m+i2m-all (best $\mathbf{AP}$) | 10.90 | 88.85
\+ m2m+i2m-all* (best $\mathbf{GErr}$) | 9.42 | 91.14

Figure 4: 3D mesh alignment: 9 animals. *number obtained with a 10x increased
weight of the m2m+i2m-all term. Figure 5:  3D model alignment: dog-human. The
m2m+i2m-all term is critical for aligning dissimilar categories in 3D.

After discussing the experimental data and implementation details, we focus on
our key contributions: simultaneously discovering high-quality inter-class
correspondences while learning category-specific dense pose predictors
(sections 4.1 and 4.2). We also show (section 4.3) that learned embeddings in
the pixel space allow for effective retrieval of analogous points (body
landmarks) on the surfaces of objects belonging to the same or different
categories (a task that we call keypoint transfer).

##### Training datasets.

Following [35], we use the original people-centric DensePose-COCO [17] for
pre-training in all experiments. We also exploit this data for conducting
studies on a joint set of animal and human categories.

For the animal classes, we propose an extended version of the DensePose-LVIS
data of [35], which originally contained 9 animal categories from the LVIS
v0.5 dataset for instance segmentation [18]. Following a recent release of
LVIS v1.0, which extended the benchmark to 160k images and 2M instance
annotations, we also expand the DensePose annotation pool for the same animal
classes and call this benchmark DensePose-LVIS v1.0 (see sup. mat.). The
original DensePose-LVIS contains fairly sparse annotations (according to [35]
only $18\%$ of the vertices of the animal meshes have at least one ground
truth annotation, and each image contains no more than 3 annotated points).
While compared to DensePose-LVIS, we have 3.6$\times$ annotations, this is
still far less than the original DensePose-COCO (which annotates 5 million
points and obtained $96\%$ coverage of the SMPL mesh). At the same time,
quality of dense labels in DensePose-LVIS v1.0 has been further improved by
introducing an additional step of croud-sourced manual verification for all
annotations.

##### Implementation details.

Our architecture is similar to the R50 variant of [35], with the only
difference being the multi-class setting for object detection (in [35]
detection was implemented in a class-agnostic manner, and ground truth class
labels were required for inference).

We pre-train on the DensePose-COCO dataset for 130k iterations (following the
standard s1x schedule). All animal models are then trained on DensePose-LVIS
v1.0 for 16k iterations, with a 10x drop of learning rate after 12k and 14k
iterations, with the rest of hyperparameters being identical to [35]. For
experiments on the joint set of human and animal categories, we train our
models for 80k iterations (with a learning rate decrease after 60k and 70k
iterations).

##### Evaluation metrics.

The quality of learned _dense pose predictions_ is evaluated by a standard set
of AP/AR metrics [17] (higher is better). We also estimate the quality of
_inter-class mesh alignment_ by computing the Geodesic Error ($\mathbf{GErr}$,
lower is better) between the predicted and the ground truth vertices along the
surface of the target mesh, given a set of manually annotated semantic
keypoints. For this purpose, all vertex coordinates in each mesh are
normalised to have the maximum of geodesic distance $\mathbf{d}_{max}=2.27$
(analogously to [17, 35]). Finally, we report Geodesic Point Similarity
($\mathbf{GPS}$, as in [17], higher is better) as an alternative indicator of
the quality of cross-category mesh alignment.

###  4.1 Inter-class alignment and dense pose prediction

Compared to prior work such as [35], our most important contribution is to
discover automatically effective inter-category correspondences, without
manual input for this task, while simultaneously learning high-quality
canonical maps for each of the individual animal object categories.

In order to conduct a fair comparison, we start by rerunning the baseline of
[35] using the embeddings and the DensePose-LVIS v1.0 data (table 1). We also
compare using two different initializations for the embedding of the different
3D canonical shapes: random and ZoomOut. The latter follows [35], obtaining an
initial set of inter-class 3D mesh correspondences from sparse manual
annotations interpolated using the ZoomOut technique [33].

We observe a 2.3pp AP gain in DensePose performance on the new dataset (AP
$35.1\rightarrow 37.4$). While ZoomOut initialization of animal mesh
embeddings provided a clear advantage for DensePose in a lower data regime (AP
$25.4\rightarrow 35.1$), the quality of mesh alignment worsens as the network
diverges from its initialization point (GErr $23.7\rightarrow 28.0$). The
dynamic on animal-only categories is similar. On DensePose-LVIS v1.0 the
automatic alignment learned from random initialization is already of better
quality than ZoomOut (20.7 GErr), and the difference in DensePose performance
is no longer observed. Note that the latter is already a confirmation of our
key hypothesis that good inter-category correspondences should spontaneously
emerge by jointly modelling them.

In table 2 we report results on the animals-only benchmark, including
assessing the contributions of the m2m and i2m regularisers. Both m2m and
i2m-all terms significantly improve mesh alignment (GErr $14.05\rightarrow
12.67$, $11.74$, respectively) and also contribute to the dense pose
performance (AP $37.5\rightarrow 38.1$, $38.3$, respectively). Their
combination yeilds best results (GErr $14.05\rightarrow 10.90$, AP
$37.5\rightarrow 38.5$) and fixes certain typical errors, as shown in fig. 3.
Fig. 2 shows qualitative results.

###  4.2 Further inter-class alignment analysis

We compare the quality of inter-class mesh alignment produced by our networks
with state-of-the-art methods exploiting 3D geometry: namely, ZoomOut [33]
(initialized with the same manually keypoints, as we use for evaluation) and
Deep Sheels [12] (unsupervised). Qualitative and quantitative results on
animal classes are shown in fig. 4. Our method demonstrates consistently
stronger performance across all categories, and rather successfully handles
transfer between highly dissimilar categories, such as dog-giraffe and dog-
elephant, where state-of-the-art geometry-based methods tend to fail (GErr
ZoomOut: $26.24$, Deep Shells: $17.91$, and our best result: $9.42$).

An extreme case of human-dog alignment is shown in fig. 5: our method produces
meaningful correspondences in the 3D space (on the left) and consistent cross-
category predictions in the pixel space (on the bottom right, shown using the
dog 3D mesh as a reference for visualization).

method |  target class | supervision | animal category | mean
---|---|---|---|---
mask | points | 3D mesh |  horse | cow | sheep | cat | dog
Rigid-CSM [30] | single | ✓ | ✗ | ✓ | 31.2 | 26.3 | 24.7 | – | – | –
Dense-Equi [46] | single | ✓ | ✗ | ✗ | 23.3 | 20.9 | 19.6 | – | – | –
A-CSM [29] | single | ✓ | ✗ | ✓ | 32.9 | 26.3 | 28.6 | – | – | –
Rigid-CSM + keyp. [30] | single | ✓ | ✓* | ✓ | 42.1 | 28.5 | 31.5 | – | – | –
A-CSM + keyp. [29] | single | ✓ | ✓* | ✓ | 44.6 | 29.2 | 39.0 | – | – | –
our baseline | multi | ✓ | ✓ | ✗ | 58.1 | 49.9 | 43.9 | 41.6 | 41.9 | 47.1
w/ m2m | multi | ✓ | ✓ | ✗ | 57.1 | 49.5 | 45.1 | 40.0 | 42.5 | 46.8
w/ i2m | multi | ✓ | ✓ | ✗ | 59.0 | 51.1 | 46.2 | 45.9 | 45.7 | 49.7
w/ i2m-all | multi | ✓ | ✓ | ✗ | 59.2 | 51.5 | 46.3 | 46.5 | 45.9 | 49.9

Table 3: Keypoint transfer on PASCAL VOC, within each of training animal
categories. PCK-Transfer metric, higher is better. * – supervision on the same
set of keypoints that are used for evaluation, as opposed to random sampling
in DensePose.

| method | horse | cow | sheep | cat | dog | mean
---|---|---|---|---|---|---|---
(I) | our baseline | 47.7 | 45.7 | 43.5 | 41.8 | 40.0 | 43.8
w/ m2m | 47.6 | 45.0 | 45.0 | 41.4 | 40.5 | 43.9
w/ i2m | 49.5 | 47.4 | 47.0 | 44.4 | 44.1 | 46.5
(II) | our baseline | 52.0 | 49.1 | 43.0 | 34.6 | 42.1 | 44.2
w/ i2m | 54.6 | 49.5 | 44.7 | 37.7 | 43.7 | 46.0

Table 4: Keypoint transfer on PASCAL VOC: (I) across training categories, (II)
within new animal categories not observed during training for dense
correspondences (only boxes and masks are provided during training to ensure
robust detection). PCK-Transfer metric, higher is better. Figure 6: Keypoint
transfer on PASCAL VOC. One experiment – one column. Green marks indicate
categories included in training, red marks – a new, test only category.

###  4.3 Keypoint transfer

To evaluate learned transferability of surface embeddings within and across
training categories, as well as their ability to generalize to new animal
classes, we look at the problem of keypoint transfer. As per section 3.1, we
can in fact use our learned embeddings to establish correspondences between a
source image $I$ and a target image $I^{\prime}$ directly and use it to
transfer keypoints. To do this, in the target image, for each type of landmark
annotated as “visible”, we take the nearest neighbor of the embedding of the
pixel in the source image, corresponding to the same landmark.

##### Evaluation metric.

Following [29], we evaluate performance on this task by estimating the
Percentage of Correct Keypoint transfers (PCK-Transfer). The transfer is said
to be correct if the target landmark is localized within distance
$0.1\cdot\max(h,w)$ from the annotated location, where $h,w$ are the height
and the width of the predicted bounding box. Prior to that, we match predicted
object instances to ground truth objects by estimating the bounding box IoU.
Objects that are not retrieved are excluded from evaluation. We report
performance on animal categories from PASCAL VOC [14], overlapping with animal
categories in DensePose-LVIS v1.0 (horse, cow, sheep, cat, dog).

##### Experimental protocol.

We evaluate the ability of our model to perform keypoint transfer in three
distinct settings: (a) within each category observed at training time (Tab.
3); (b) across training categories (Tab. 4, I); (c) within new animal
categories (zero shot) not observed at training time. For (c), dense
correspondences for one class are removed from the training set, and only
bounding boxes and object instance masks are provided as supervision (Tab. 4,
II).

##### Discussion.

As shown in tables 3, 4 and 6, the learned embeddings work very well to
transfer keypoints between known as well as unknown animal classes,
demonstrating once more the power of generalization that comes from joint
modelling. For this experiment, the i2m regularization term significantly
improves the results (m2m vs i2m PCK: $46.8\rightarrow 49.9$ in table 4). This
might be expected since m2m works on the alignment between 3D templates,
whereas i2m works at the image level, which is more relevant for this
experiment. Note that methods [30, 29] re-project a 3D mesh template onto the
images to establish correspondences, which we do not do.

##  5 Conclusions

We have introduced a method to learn high-quality dense pose predictors for
multiple object categories while discovering automatically semantic
correspondences between them. The method represents correspondences via a
unified embedding and network predictor while enforcing reasonable topological
consistency constraints. Our encouraging results indicate that joint modelling
is not only just viable, but has significant positive effects on performance
and scalability of such neural dense predictors.

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Appendix

Project page: https://gdude.de/discovering-3d-obj-rel.

##  Appendix A Experiments

###  A.1 DensePose-LVIS v1.0 dataset details

We introduce DensePose-LVIS v1.0 dataset, an extended version of the
DensePose-LVIS data of [35]. We improve the quality of the existing labels and
expand the DensePose annotation pool for the same animal classes as in the
previous version of this dataset [35]. In Tab. 5 we report the number of train
and test instances annotated with DensePose in every category.

category | DensePose-LVIS | DensePose-LVIS v1.0
---|---|---
train, inst. | test, inst. | train, inst. | test, inst.
dog | 483 | 200 | 1607 | 316
cat | 586 | 200 | 1912 | 379
bear | 98 | 200 | 735 | 132
sheep | 257 | 200 | 1655 | 350
cow | 426 | 200 | 2105 | 340
horse | 605 | 200 | 2292 | 458
zebra | 665 | 200 | 2864 | 556
giraffe | 651 | 200 | 2709 | 534
elephant | 670 | 200 | 2839 | 539
all | 4441 | 1800 | 18718 | 3604

Table 5: DensePose-LVIS v1.0 dataset: 3.6x increase in a number of annotated
instances, better quality of labels.

###  A.2 3D mesh alignment

If not stated otherwise, we used cross-validation to find the m2m loss weight
that maximizes AP metric after training. Additionally we also cross-validated
the m2m loss weight to minimize the GErr, we denote this experiment as m2m*.
As mentioned in the caption of Fig. 4, the optimal weight of m2m term is
tenfold larger for GErr than for AP. Visual mappings in Fig. 4 correspond to
the m2m model.

###  A.3 Keypoint transfer

For keypoint transfer experiments we train our models on DensePose-LVIS v1.0
dataset and do not use any PASCAL VOC [14] images during training. We select
animal categories from PASCAL VOC [14], overlapping with animal categories in
DensePose-LVIS v1.0: horse, cow, sheep, cat, dog. Following Kulkarni et al.
[29], we randomly sample $100$ images for each category from PASCAL VOC
mentioned above and use them for evaluation. We average PCK-Transfer score
across all possible (source, target) image pairs.

method | animal category | mean
---|---|---
horse | cow | sheep | cat | dog
our baseline | 58.1 | 49.9 | 43.9 | 41.6 | 41.9 | 47.1
w/ m2m | 57.1 | 49.5 | 45.1 | 40.0 | 42.5 | 46.8
w/ i2m | 59.0 | 51.1 | 46.2 | 45.9 | 45.7 | 49.7
w/ i2m-all | 59.2 | 51.5 | 46.3 | 46.5 | 45.9 | 49.9
w/ m2m+i2m | 57.7 | 49.9 | 44.8 | 40.6 | 42.4 | 47.1
w/ m2m+i2m-all | 57.8 | 50.2 | 44.9 | 40.6 | 42.6 | 47.2

Table 6: Keypoint transfer on PASCAL VOC, within each of training animal
categories. PCK-Transfer metric, higher is better. m2m term is not helpful for
this task.

We conducted keypoint transfer experiments using three distinct settings:

  * •

Within each category observed at training time, when source and target images
are from the same category (Tab. 3 in the main paper);

  * •

Across training categories (Tab. 4, I in the main paper). In this case source
and target images are from different training categories. For example,
keypoints from dog images are transferred to images from horse, cow, sheep,
and cat categories;

  * •

Zero shot scenario: Within new animal categories not observed at training time
(Tab. 4 II in the main paper). In this case, we remove ground truth dense
correspondences for one class from the training set and evaluate keypoint
transfer on the images within the removed class. Note that we do not remove
bounding boxes and object instance masks from training set and still use them
to train our detection and segmentation heads.

method | target class | horse | cow | sheep | cat | dog | mean
---|---|---|---|---|---|---|---
our baseline | single | 28.8 | 27.3 | 33.7 | 31.1 | 29.4 | 30.0
w/ i2m | single | 30.1 | 28.2 | 34.1 | 31.9 | 29.7 | 30.8

Table 7: Effect of i2m loss term when trained and evaluated on individual
categories of DensePose-LVIS v1.0. We report DensePose AP score.

method | target class | horse | cow | sheep | cat | dog | mean
---|---|---|---|---|---|---|---
Rigid-CSM + keyp. [30] | single | 42.1 | 28.5 | 31.5 | – | – | –
A-CSM + keyp. [29] | single | 44.6 | 29.2 | 39.0 | – | – | –
our baseline | single | 53.4 | 48.0 | 38.8 | 40.9 | 34.0 | 43.0
w/ i2m | single | 54.0 | 49.1 | 39.2 | 34.7 | 44.2 | 44.2

Table 8: Keypoint transfer on PASCAL VOC, when trained and evaluated on
individual categories. PCK-Transfer metric, higher is better.

In Tab. 6 we show results for all combinations of the loss terms on keypoint
transfer within each category. The i2m term enforces alignment on image level
which is more important for the task of within-category keypoint transfer,
while the m2m loss term is not helpful in this case as it enforces the
alignment between 3D templates.

###  A.4 Effect of i2m loss in a single-class training scenario

In contrast to our approach, Rigid-CSM [30] and A-CSM [29] cannot learn
multiple animal categories in a single model and have to train separate models
for every animal class. To make our setup closer to those in [30, 29], we
trained our models on individual categories from DensePose-LVIS v1.0 as well
(i.e., trained a new model for each class). Then we evaluated each model on
the corresponding individual categories: (a) on the test set of DensePose-LVIS
v1.0 by computing DensePose AP score (see Tab. 7); and (b) on PASCAL VOC by
computing PCK-Transfer metric for Keypoint Transfer task (see Tab. 8). From
the tables 7, 8 we can see that i2m improves the performance even when trained
and evaluated on individual categories and outperforms methods [30, 29] (we
used the results reported in the corresponding papers). However, a part of the
strength of our method comes from training on several classes jointly, which
results in even stronger performance of our models (see Tab. 3 in the main
paper).

####  A.4.1 Can the i2m loss be used in combination with Rigid-CSM [30] or
A-CSM [29] models?

Our i2m loss requires every pixel and every vertex of the mesh to be embedded
in a common embedding space. However, models [30, 29] directly predict $(u,v)$
coordinates for every pixel. Therefore our loss cannot be applied during
training of Rigid-CSM [30] and A-CSM [29].

###  A.5 Evaluation metrics

For completeness, we provide brief descriptions of AP/AR metrics used for
evaluation of learned dense pose predictions on DensePose-LVIS v1.0 dataset
and cross-category mesh alignment metrics $\mathbf{GErr}$, $\mathbf{GPS}$.

  * •

$\mathbf{GPS}$ (Geodesic Point Similarity) [17] is a correspondence matching
score indicator of the quality of aligning of two sets of vertices
$A=(a_{1},\dots,a_{N})$ and $B=(b_{1},\dots,a_{N})$ on a mesh.

$\mathbf{GPS}(A,B)=\frac{1}{N}\sum\limits_{i=1}^{N}\exp\left(\frac{-g(a_{i},b_{i})^{2}}{2\kappa^{2}}\right),$

where $N$ is the number of vertices in each set, $g(\cdot,\cdot)$ is the
geodesic distance between two surface points, and $\kappa$ is a normalization
constant. To make our $\mathbf{GPS}$ score comparable to the $\mathbf{GPS}$
score used for Human DensePose evaluation in Guler et al. [17], we normalize
all vertex coordinates in every animal mesh to have the maximum geodesic
distance $d_{max}=2.27$, which is equal to the maximal geodesic distance in
the SMPL [32] mesh of a human utilized in [17]. We set $\kappa=0.255$ so that
a single point has a $\mathbf{GPS}$ value of $0.5$ if its geodesic distance
from the ground truth equals the average half-size of a body segment. When we
evaluate cross-category mesh alignment quality, we compute $\mathbf{GPS}$
between a set of the ground truth semantic keypoints on a target mesh and the
estimated locations of these keypoints obtained by transferring keypoints from
a source mesh of other category. The mean $\mathbf{GPS}$ score is then
computed as an average across all possible (source, target) pairs of
categories.

  * •

Similarly to Geodesic Point Similarity, we define $\mathbf{GErr}$ (Geodesic
Error), the error between two sets of vertices along the surface of a mesh.
Before computing this error, all vertex coordinates are normalised to have the
maximum of geodesic distance $d_{max}=2.27$ (similar to [17, 35]).

$\mathbf{GErr}(A,B)=\frac{1}{|N|}\sum\limits_{i=1}^{N}g(a_{i},b_{i}).$

Analogously to $\mathbf{GPS}$, we use $\mathbf{GErr}$ to estimate the quality
of inter-category mesh alignment by comparing the ground truth semantic
keypoints on a target mesh and the estimated locations of these keypoints
obtained by transferring keypoints from a source mesh of other category.

  * •

To evaluate the quality of mapping from image pixels to 3D vertices on the
category-specific mesh, we use AP (Average Precision) and AR (Average Recall)
[17]. The location of the vertices on the mesh corresponding to image pixels
are estimated by finding for every pixel the most similar mesh vertex in the
learned embedding space. After that we compare estimated vertex locations with
the ground truth using $\mathbf{GPS}$ metric. Then we calculate AP and AR at
different $\mathbf{GPS}$ thresholds ranging from $0.5$ to $0.95$, following
the COCO challenge protocol [31]. We separately report Average Precision and
Average Recall at $\mathbf{GPS}$ thresholds equal to $0.5$ and $0.75$, denoted
as $\textbf{AP}_{50},\textbf{AP}_{75},\textbf{AR}_{50},\textbf{AR}_{75}$. In
addition to this we separately compute Average Precision and Average Recall
for instances with _medium_ and _large_ sizes
($\textbf{AP}_{M},\textbf{AR}_{M}$ for medium size and
$\textbf{AP}_{L},\textbf{AR}_{L}$ for large).

Note, that we report $\mathbf{GPS}\times 100$ and $\mathbf{GErr}\times 100$ in
all tables in the main paper.


# Production of the $B_{c}$ meson at the CEPC

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

###### Abstract

In this paper, we make a detailed study on the production of the $B_{c}$,
$B_{c}^{*}$, $B_{c}(2\,^{1}S_{0})$, and $B^{*}_{c}(2\,^{3}S_{1})$ mesons via
the three planned running modes ($Z$, $W$, and $H$) at the future Circular
Electron-Positron Collider (CEPC). The fragmentation-function and full
nonrelativistic QCD approaches are adopted to calculate the production cross
sections. Considering the excited states shall decay to the ground $B_{c}$
state (e.g. $1^{1}S_{0}$-state) with almost $100\%$ probability, our numerical
results show that up to next-to-leading order QCD corrections, there are about
$1.4\times 10^{8}$ $B_{c}$ events to be accumulated via the $Z$ mode
($\sqrt{s}=m_{{}_{Z}}$) of the CEPC, but only about $1.6\times 10^{4}$ and
$1.1\times 10^{4}$ $B_{c}$ events to be accumulated via the $W$
($\sqrt{s}=160\,{\rm GeV}$) and $H$ ($\sqrt{s}=240\,{\rm GeV}$) modes,
respectively. Since the $Z$ mode is the best mode among the three planned
modes of the CEPC for studying the production of the $B_{c}$, $B_{c}^{*}$,
$B_{c}(2\,^{1}S_{0})$, and $B^{*}_{c}(2\,^{3}S_{1})$ mesons and the
differential distributions of these mesons may be measured precisely at this
mode, we further present the differential cross sections $d\sigma/(dz\,d{\rm
cos}\theta)$ and $d\sigma/dz$ via the $Z$ mode of the CEPC.

##  I Introduction

The $B_{c}$ meson, which usually stands for the ground $1^{1}S_{0}$-state of
$(c\bar{b})$-quarkonium and carries two different heavy flavors, provides a
unique bound-state system for testing the Standard Model (SM) and has
attracted lots of interests. The study on the production of the $B_{c}$ meson
is important for the researches of the $B_{c}$ meson, since it tells us how
many $B_{c}$ events can be produced in a collider platform. On the other hand,
the theoretical models for the $B_{c}$ production can be tested after fixing
its production rate by measuring its cascade decays. Similar to the heavy
quarkonia such as $\eta_{c}$, $J/\psi$, $\eta_{b}$, $\Upsilon$, and etc., the
$B_{c}$ meson production cross section can also be factorized through the
nonrelativistic QCD (NRQCD) nrqcd  factorization theory and a lot of
information for the production of $B_{c}$ can be calculated perturbatively.
Compared with the quarkonium production where color-octet contribution is
usually important Brambilla:2010cs ; Andronic:2015wma ; Lansberg:2019adr ;
Chen:2021tmf , the production mechanism of the $B_{c}$ meson is simpler due to
the fact that the color-octet contribution is always suppressed compared to
the color-singlet contribution in the $B_{c}$ production processes.

At present, most studies of the $B_{c}$ production focus on the hadronic
production due to the high production rate at a hadron collider, such as
Tevatron or LHC Chang:1992jb ; Chang:1994aw ; Kolodziej:1995nv ;
Berezhnoy:1994ba ; Chang:1996jt ; Berezhnoy:1996ks ; Baranov:1997wy ;
Baranov:1997sg ; Cheung:1999ir ; Chang:2003cr ; Chang:2004bh ; Chang:2005bf ;
Chang:2005wd ; Chang:2003cq ; Chang:2005hq ; Wang:2012ah ; Chen:2018obq ;
Berezhnoy:2019yei . Indeed, the $B_{c}$ meson has been only observed at hadron
colliders up to now CDF:1998ihx ; CDF:1998axz ; CDF:2012ksy ; D0:2008bqs ;
D0:2008thm ; LHCb:2012ihf ; LHCb:2013xlg ; LHCb:2013hwj ; LHCb:2014ilr ;
LHCb:2014mvo ; LHCb:2017lpu ; CMS:2021vuz . In addition to hadron colliders, a
high luminosity $e^{+}e^{-}$ collider can be a potentially good platform for
studying the $B_{c}$ meson. Compared with the hadron collider, there are less
backgrounds at the $e^{+}e^{-}$ collider, thus it is suitable for studying the
$B_{c}$ meson precisely. Moreover, the production mechanism of the $B_{c}$
meson at the $e^{+}e^{-}$ collider is much simpler than that at the hadron
collider. Recent studies on the $B_{c}$ production at the $e^{+}e^{-}$
colliders can be found in Refs.Yang:2011ps ; Yang:2013vba ; Chen:2014xka ;
Berezhnoy:2016etd ; Zheng:2015ixa ; Zheng:2017xgj ; Zheng:2018fqv ;
Chen:2020dtu .

The Chinese particle physics community has proposed to construct a high
luminosity $e^{+}e^{-}$ collider in China which is called the Circular
Electron-Positron Collider (CEPC) CEPCStudyGroup:2018ghi . The CEPC will
operate at three modes: $H(e^{+}e^{-}\to ZH)$ at around $\sqrt{s}=240\,{\rm
GeV}$, $Z(e^{+}e^{-}\to Z)$ at around $\sqrt{s}=m_{{}_{Z}}\simeq 91.2\,{\rm
GeV}$, and $W(e^{+}e^{-}\to W^{+}W^{-})$ at around $\sqrt{s}=160\,{\rm GeV}$.
The integrated luminosities of those modes are planned as $5.6\,ab^{-1}$,
$16\,ab^{-1}$ and $2.6\,ab^{-1}$, respectively An:2018dwb . As programmed, in
addition to studying the Higgs boson properties, the CEPC also provides a good
platform for studying the heavy flavor physics. Therefore, as one of the hot
topics in heavy flavor physics, the study of the $B_{c}$ production at the
CEPC is very interesting. In this paper, we will devote ourselves to studying
the properties of the inclusive $B_{c}$ production at the CEPC.

Since the $B_{c}$ meson carries two different heavy flavors, its excited
states below the BD threshold will directly or indirectly decay to the ground
state via the electro-magnetic or strong interactions with a probability of
almost $100\%$. These excited states are important sources of the $B_{c}$
meson. Moreover, the $2S$ excited states $B_{c}(2\,^{1}S_{0})$ and
$B^{*}_{c}(2\,^{3}S_{1})$ have been recently observed by the CMS and the LHCb
collaborations CMS:2019uhm ; LHCb:2019bem . Thus, the study on the production
of the two excited states is interesting by itself. In this paper, besides the
production of the ground state $B_{c}$ meson, we will also consider the
production of the excited states $B^{*}_{c}(1\,^{3}S_{1})$,
$B_{c}(2\,^{1}S_{0})$, and $B^{*}_{c}(2\,^{3}S_{1})$.

##  II Calculation technology

According to the NRQCD factorization formalism, the production cross section
of the $B_{c}$ meson in $e^{+}e^{-}$ collisions can be written as 111For
simplicity, we only present the formulas for the ground-state $B_{c}$ meson,
the formulas for the $B^{*}_{c}$, $B_{c}(2\,^{1}S_{0})$, and
$B^{*}_{c}(2\,^{3}S_{1})$ states are similar to those of the $B_{c}$ meson.

$\displaystyle d\sigma_{e^{+}e^{-}\to B_{c}+X}=$
$\displaystyle\sum_{n}d\tilde{\sigma}_{e^{+}e^{-}\to(c\bar{b})[n]+X}\langle{\cal
O}^{B_{c}}(n)\rangle.$ (1)

where $d\tilde{\sigma}$ denotes the production cross section for the
perturbative state $(c\bar{b})[n]$ with the quantum numbers $n$, which can be
calculated using perturbative QCD (pQCD). $\langle{\cal O}^{B_{c}}(n)\rangle$
denotes the long-distance matrix element (LDME) which is proportional to the
transition probability for a $(c\bar{b})[n]$ pair into the corresponding
$B_{c}$ meson state. In the lowest nonrelativistic approximation, only the
color-singlet LDME need to be considered, which can be related to the wave
function at the origin [i.e., $\langle{\cal
O}^{B_{c}}(^{1}S_{0}^{[1]})\rangle\approx
N_{c}|R_{S}(0)|^{2}/(2\pi)=\langle{\cal
O}^{c\bar{b}[^{1}S_{0}^{[1]}]}(^{1}S_{0}^{[1]})\rangle|R_{S}(0)|^{2}/(4\pi)$]
and can be calculated through phenomenological potential models. In addition
to the factorization formula (1), when the electron-position collision energy
$\sqrt{s}\gg m_{B_{c}}$, the production cross section can also be factorized
as

$\displaystyle\frac{d^{2}\sigma_{e^{+}e^{-}\to B_{c}+X}}{dz\,d{\rm
cos}\theta}=$
$\displaystyle\sum_{i}\int\frac{dy}{y}\frac{d^{2}\hat{\sigma}_{e^{+}e^{-}\to
i+X}}{dy\,d{\rm cos}\theta}(y,\mu_{F})$ (2) $\displaystyle\times D_{i\to
B_{c}}(z/y,\mu_{F}),$

where $d^{2}\hat{\sigma}_{e^{+}e^{-}\to i+X}/(dy\,d{\rm cos}\theta)$ denotes
the partonic cross section, $D_{i\to B_{c}}(z/y,\mu_{F})$ denotes the
fragmentation function for a parton $i$ into the $B_{c}$ meson,
$z=2p_{B_{c}}\cdot q/q^{2}$ (here, $q$ is the sum of initial-state $e^{+}$ and
$e^{-}$ momenta) is the momentum fraction carried by the $B_{c}$ meson,
$\theta$ is the angle between the momenta of the $B_{c}$ meson and the
initial-state $e^{-}$, and $\mu_{F}$ is the factorization scale. This
factorization formalism is known as the fragmentation-function approach.
Unlike the fragmentation functions for the light hadrons which are
nonperturbative in nature, the fragmentation function for the $B_{c}$ meson
contains much information which is perturbatively calculable, e.g., based on
the NRQCD factorization,

$\displaystyle D_{i\to
B_{c}}(z,\mu_{F})=\sum_{n}d_{i\to(c\bar{b})[n]}(z,\mu_{F})\langle{\cal
O}^{B_{c}}(n)\rangle,$ (3)

where $d_{i\to(c\bar{b})[n]}(z,\mu_{F})$ is the calculable short-distance
coefficient. The leading-order (LO) fragmentation functions for the $B_{c}$
meson and its excited states have been calculated in the 1990s Chang:1992bb ;
Braaten:1993jn ; Ma:1994zt ; Chen:1993ii ; Yuan:1994hn ; Cheung:1995ir . The
next-to-leading-order (NLO) fragmentation functions $D_{\bar{b}\to
B_{c}}(z,\mu_{F})$ and $D_{c\to B_{c}}(z,\mu_{F})$ can be found in
Ref.Zheng:2019gnb .

One advantage of the fragmentation-function approach is that the large
logarithms of $s/m_{Q}^{2}$ ($m_{Q}$ is the heavy quark mass) appearing in the
perturbative expansion of the cross section can be resummed through solving
the evolution equations of the fragmentation functions. For example, the
differential cross section up to NLO and next-to-leading logarithmic (NLL)
accuracy can be calculated through

$\displaystyle\frac{d^{2}\sigma^{\rm Frag,NLO+NLL}_{e^{+}e^{-}\to
B_{c}+X}}{dz\,d{\rm cos}\theta}=$
$\displaystyle\sum_{i=c,\bar{b}}\int\frac{dy}{y}\frac{d^{2}\hat{\sigma}^{\rm
NLO}_{e^{+}e^{-}\to i+X}}{dy\,d{\rm cos}\theta}(y,\mu_{F})$ (4)
$\displaystyle\times D^{\rm NLO+NLL}_{i\to B_{c}}(z/y,\mu_{F}),$

where the expression of the partonic cross section up to NLO accuracy can be
found in Ref.Nason:1993xx . To avoid the large logarithms appearing in the
partonic cross section, we shall implicitly set the factorization scale and
renormalization scale to be the same, i.e., $\mu_{F}=\mu_{R}=\sqrt{s}$. Then
the large logarithms of $s/m_{Q}^{2}$ appear in the fragmentation functions.
We derive the fragmentation functions $D^{\rm NLO+NLL}_{\bar{b}\to
B_{c}}(z,\mu_{F}=\sqrt{s})$ and $D^{\rm NLO+NLL}_{c\to
B_{c}}(z,\mu_{F}=\sqrt{s})$ by solving the DGLAP equation Dokshitzer:1977sg ;
Gribov:1972ri ; Altarelli:1977zs , and the NLO fragmentation functions $D^{\rm
NLO}_{\bar{b}\to B_{c}}(z,\mu_{F0}=m_{b}+2m_{c})$ and $D^{\rm NLO}_{c\to
B_{c}}(z,\mu_{F0}=2m_{b}+m_{c})$ with $\mu_{R}=\mu_{F0}$ obtained in
Ref.Zheng:2019gnb  are used as the boundary condition 222To avoid the large
logarithms appearing in the initial fragmentation functions, the factorization
and renormalization scales of the initial fragmentation functions are set to
be the minimal invariant mass of the fragmentation quark, i.e.,
$\mu_{F0}=\mu_{R}=m_{b}+2m_{c}$ for the $\bar{b}$-quark fragmentation channel
and $\mu_{F0}=\mu_{R}=2m_{b}+m_{c}$ for the $c$-quark fragmentation channel..
For solving the DGLAP equation, we adopt the Mellin transformation method, and
the NLO splitting function $P_{QQ}(y)$ Curci:1980uw ; Furmanski:1980cm ;
Floratos:1978ny ; Gonzalez-Arroyo:1979qht ; Floratos:1981hs  is used as the
evolution kernel. The related formulas for the Mellin transformation of the
NLO splitting function can be found in Ref.Mele:1990cw , and the inverse
Mellin transformation can be carried out numerically Graudenz:1995sk . Through
the DGLAP equation, the large logarithms of $s/m_{Q}^{2}$ in the fragmentation
functions are resummed to NLL accuracy.

As a comparison, we also calculate the differential cross section at the NLO
level alone under the fragmentation-function approach, i.e., without doing the
resummation of the large logarithms of $s/m_{Q}^{2}$,

$\displaystyle\frac{d^{2}\sigma^{\rm Frag,NLO}_{e^{+}e^{-}\to
B_{c}+X}}{dz\,d{\rm cos}\theta}=$
$\displaystyle\sum_{i=c,\bar{b}}\int\frac{dy}{y}\frac{d^{2}\hat{\sigma}^{\rm
NLO}_{e^{+}e^{-}\to i+X}}{dy\,d{\rm cos}\theta}(y,\mu_{F})$ (5)
$\displaystyle\times D^{\rm NLO}_{i\to B_{c}}(z/y,\mu_{F}),$

where the factorization and renormalization scales of the partonic cross
section and the fragmentation functions are set as the minimal invariant mass
of the fragmentation quark, i.e., $\mu_{F}=\mu_{R}=m_{b}+2m_{c}$ for the
$\bar{b}$-quark fragmentation channel and $\mu_{F}=\mu_{R}=2m_{b}+m_{c}$ for
the $c$-quark fragmentation channel.

##  III Numerical results and discussions

To do the numerical calculation, the necessary input parameters are taken as
follows:

$\displaystyle m_{c}=1.5\,{\rm GeV},\,m_{b}=4.9\,{\rm
GeV},\,m_{{}_{Z}}=91.1876\,{\rm GeV},$ $\displaystyle{\rm
sin}^{2}\theta_{w}=0.231,\;\alpha=1/128,\;\Gamma_{{}_{Z}}=2.4952\,{\rm GeV},$
$\displaystyle|R_{1S}(0)|^{2}=1.642\,{\rm
GeV}^{3},\,|R_{2S}(0)|^{2}=0.983\,{\rm GeV}^{3}.$ (6)

Here $\alpha=\alpha(m_{{}_{Z}})$ is the electromagnetic coupling constant at
the scale $m_{{}_{Z}}$, $|R_{1S}(0)|$ and $|R_{2S}(0)|$ are radial wave
functions at the origin for the color-singlet $1S$-level and $2S$-level
$(c\bar{b})$-quarkonium states, which are taken from the calculation based on
the Buchmuller-Tye potential model Eichten:1994gt . For the strong coupling
constant, we adopt the two-loop formula:

$\alpha_{s}(\mu)=\frac{4\pi}{\beta_{0}~{}{\rm ln}(\mu^{2}/\Lambda^{2}_{\rm
QCD})}\left[1-\frac{\beta_{1}~{}{\rm ln}~{}{\rm ln}(\mu^{2}/\Lambda^{2}_{\rm
QCD})}{\beta_{0}^{2}~{}{\rm ln}(\mu^{2}/\Lambda^{2}_{\rm QCD})}\right],$ (7)

where $\beta_{0}=11-\frac{2}{3}n_{f}$ and $\beta_{1}=102-\frac{38}{3}n_{f}$
($n_{f}$ being the active flavor numbers) are the one-loop and two-loop
$\\{\beta_{i}\\}$-functions. According to $\alpha_{s}(m_{{}_{Z}})=0.1185$
ParticleDataGroup:2016lqr , we obtain $\Lambda_{\rm QCD}^{n_{f}=5}=0.233\,{\rm
GeV}$.

###  III.1 Basic results for the three production modes at the CEPC under the
fragmentation-function approach up to NLO+NLL level

States |  $\sqrt{s}=m_{{}_{Z}}$ |  $\sqrt{s}=160\,{\rm GeV}$ |  $\sqrt{s}=240\,{\rm GeV}$
---|---|---|---
$B_{c}$ | 2.58 | $1.71\times 10^{-3}$ | $5.34\times 10^{-4}$
$B^{*}_{c}$ | 3.04 | $2.05\times 10^{-3}$ | $6.40\times 10^{-4}$
$B_{c}(2^{1}S_{0})$ | 1.54 | $1.03\times 10^{-3}$ | $3.20\times 10^{-4}$
$B^{*}_{c}(2^{3}S_{1})$ | 1.82 | $1.23\times 10^{-3}$ | $3.83\times 10^{-4}$
Table 1: The cross sections (in unit: $pb$) for the production of the $B_{c}$,
$B_{c}^{*}$, $B_{c}(2\,^{1}S_{0})$, and $B_{c}^{*}(2\,^{3}S_{1})$ states at
$\sqrt{s}=m_{{}_{Z}}$, $\sqrt{s}=160\,{\rm GeV}$, and $\sqrt{s}=240\,{\rm
GeV}$ under the fragmentation-function approach up to NLO+NLL level.

The cross sections for the production of $B_{c}$, $B_{c}^{*}$,
$B_{c}(2\,^{1}S_{0})$, and $B_{c}^{*}(2\,^{3}S_{1})$ states via the three
modes of the CEPC under the fragmentation-function approach up to NLO+NLL
level are given in TABLE 1. From the table, we can see that the cross sections
for the production of the $B_{c}$, $B_{c}^{*}$, $B_{c}(2\,^{1}S_{0})$, and
$B_{c}^{*}(2\,^{3}S_{1})$ states at $\sqrt{s}=m_{{}_{Z}}$ are larger than the
corresponding cross sections at $\sqrt{s}=160\,{\rm GeV}$ and
$\sqrt{s}=240\,{\rm GeV}$ by about three to four orders of magnitude. This is
because the cross sections are enhanced by the $Z$-boson resonance effect when
$\sqrt{s}=m_{{}_{Z}}$. More explicitly, the squared amplitude of the $Z$-boson
exchange diagrams is proportional to a factor of
$1/[(s-m_{{}_{Z}}^{2})^{2}+m_{{}_{Z}}^{2}\Gamma_{Z}^{2}]$. This factor has a
peak at the $Z$ pole, and it decreases rapidly when $\sqrt{s}$ deviates from
$m_{{}_{Z}}$. It has the value of $1.93\times 10^{-5}\,{\rm GeV}^{-4}$ (as a
comparison, the corresponding factor $1/s^{2}$ of the $\gamma$-exchange
diagrams is $1.45\times 10^{-8}\,{\rm GeV}^{-4}$) at the $Z$ pole, which
quickly decreases to $3.35\times 10^{-9}\,{\rm GeV}^{-4}$ and $4.12\times
10^{-10}\,{\rm GeV}^{-4}$ when $\sqrt{s}=160\,{\rm GeV}$ and
$\sqrt{s}=240\,{\rm GeV}$, respectively.

States |  $\sqrt{s}=m_{{}_{Z}}$ |  $\sqrt{s}=160\,{\rm GeV}$ |  $\sqrt{s}=240\,{\rm GeV}$
---|---|---|---
$B_{c}$ | $4.13\times 10^{7}$ | $4.45\times 10^{3}$ | $2.99\times 10^{3}$
$B^{*}_{c}$ | $4.86\times 10^{7}$ | $5.33\times 10^{3}$ | $3.58\times 10^{3}$
$B_{c}(2^{1}S_{0})$ | $2.46\times 10^{7}$ | $2.68\times 10^{3}$ | $1.79\times 10^{3}$
$B^{*}_{c}(2^{3}S_{1})$ | $2.91\times 10^{7}$ | $3.20\times 10^{3}$ | $2.14\times 10^{3}$
Table 2: The expected numbers of the $B_{c}$, $B_{c}^{*}$,
$B_{c}(2\,^{1}S_{0})$, and $B_{c}^{*}(2\,^{3}S_{1})$ events to be generated
via the three planned modes of the CEPC.

With the planned luminosities of the three running modes at the CEPC, the
numbers of the events of these states to be generated via the three modes of
the CEPC are estimated and presented in Table 2. We can see that there are
${\cal O}(10^{7})$ [${\cal O}(10^{3})$, ${\cal O}(10^{3})$] $B_{c}$,
$B_{c}^{*}$, $B_{c}(2\,^{1}S_{0})$, and $B_{c}^{*}(2\,^{3}S_{1})$ events to be
produced at the $Z$ ($W$, $H$) mode of the CEPC. Therefore, the CEPC
(especially the planned $Z$ mode) will provide a new opportunity for studying
the properties of the $B_{c}$ meson.

###  III.2 More studies on the $Z$ mode of the CEPC

Since the $Z$-boson resonance effect shall raise the production rates by
several orders of magnitude at the $Z$ pole, the total production cross
sections as well as the differential distributions of the $B_{c}$,
$B_{c}^{*}$, $B_{c}(2\,^{1}S_{0})$, and $B_{c}^{*}(2\,^{3}S_{1})$ states may
be measured precisely via the $Z$ mode of the CEPC. Therefore, it is important
to give a more detailed study on the production of those states at the $Z$
pole.

Figure 1: The differential cross section $d^{2}\sigma/(dz\,d{\rm cos}\theta)$
for the production of the $B_{c}(1\,^{1}S_{0})$ meson at the $Z$ pole under
the fragmentation-function approach up to NLO+NLL level, where $\theta$ is the
angle between the momenta of the final $B_{c}$ meson and the initial electron.
Figure 2: The differential cross section $d^{2}\sigma/(dz\,d{\rm cos}\theta)$
for the production of the $B_{c}^{*}(1\,^{3}S_{1})$ meson at the $Z$ pole
under the fragmentation-function approach up to NLO+NLL level, where $\theta$
is the angle between the momenta of the final $B_{c}^{*}$ meson and the
initial electron. Figure 3: The differential cross section
$d^{2}\sigma/(dz\,d{\rm cos}\theta)$ for the production of the
$B_{c}(2\,^{1}S_{0})$ meson at the $Z$ pole under the fragmentation-function
approach up to NLO+NLL level, where $\theta$ is the angle between the momenta
of the final $B_{c}(2\,^{1}S_{0})$ meson and the initial electron. Figure 4:
The differential cross section $d^{2}\sigma/(dz\,d{\rm cos}\theta)$ for the
production of the $B_{c}^{*}(2\,^{3}S_{1})$ meson at the $Z$ pole under the
fragmentation-function approach up to NLO+NLL level, where $\theta$ is the
angle between the momenta of the final $B_{c}^{*}(2\,^{3}S_{1})$ meson and the
initial electron.

We present differential cross sections $d^{2}\sigma/(dz\,d{\rm cos}\theta)$ by
using the fragmentation-function approach up to NLO+NLL level for the
production of $B_{c}$, $B_{c}^{*}$, $B_{c}(2\,^{1}S_{0})$, and
$B_{c}^{*}(2\,^{3}S_{1})$ states at the $Z$ pole of CEPC in Figs.(1, 2, 3, 4).
Results for several typical momentum fractions (e.g. $z=0.2$, 0.4, 0.6 and
0.8) are presented. From those figures, we can see that the differential
distributions $d^{2}\sigma/(dz\,d{\rm cos}\theta)$ are asymmetric. Those
forward-backward asymmetries arise from the parity violation due to the
exchange of $Z$ boson.

Figure 5: The differential cross sections $d\sigma/dz$ for the production of
the $B_{c}$ meson at the $Z$ pole under different approaches. Figure 6: The
differential cross sections $d\sigma/dz$ for the production of the $B_{c}^{*}$
meson at the $Z$ pole under different approaches. Figure 7: The differential
cross sections $d\sigma/dz$ for the production of the $B_{c}(2\,^{1}S_{0})$
meson at the $Z$ pole under different approaches. Figure 8: The differential
cross sections $d\sigma/dz$ for the production of the
$B_{c}^{*}(2\,^{3}S_{1})$ meson at the $Z$ pole under different approaches.

By further integrating the differential cross sections $d^{2}\sigma/(dz\,d{\rm
cos}\theta)$ over ${\rm cos}\theta$, we obtain the differential cross sections
$d\sigma/dz$, which are presented in Figs.(5, 6, 7, 8). As a comparison of the
NLO+NLL fragmentation-function result (labelled as “Frag, NLO+NLL”), we also
present the results for the full NLO NRQCD approach (labelled as “Full NRQCD,
NLO”), the fragmentation-function approach up to NLO level but without doing
the resummation of the large logarithms of $m_{Q}^{2}/s$ (labelled as “Frag,
NLO”), and the combined results of the fragmentation-function approach and the
full NRQCD approach (labelled as “Combined, NLO+NLL”).

In the fragmentation-function approach, because of the approximation
$\sqrt{s}\gg m_{Q}$, some terms which are suppressed by various powers of
$m_{Q}^{2}/s$ have been neglected. Those neglected terms can be recovered with
the help of Eqs.(1, 5), and we then obtain the combined result of the
fragmentation-function approach and the full NRQCD approach up to NLO level,
i.e.

$\displaystyle\frac{d^{2}\sigma^{\rm Combined,NLO+NLL}_{e^{+}e^{-}\to
B_{c}+X}}{dz\,d{\rm cos}\theta}$ (8) $\displaystyle=$
$\displaystyle\frac{d^{2}\sigma^{\rm Frag,NLO+NLL}_{e^{+}e^{-}\to
B_{c}+X}}{dz\,d{\rm cos}\theta}$ $\displaystyle+\left(\frac{d^{2}\sigma^{\rm
Full\,NRQCD,NLO}_{e^{+}e^{-}\to B_{c}+X}}{dz\,d{\rm
cos}\theta}-\frac{d^{2}\sigma^{\rm Frag,NLO}_{e^{+}e^{-}\to
B_{c}+X}}{dz\,d{\rm cos}\theta}\right),$

in which the terms given in the parentheses count the contribution neglected
in the fragmentation-function formula (4). As a subtle point, to be self-
consistent with $d^{2}\sigma^{\rm Frag,NLO}_{e^{+}e^{-}\to B_{c}+X}/(dz\,d{\rm
cos}\theta)$, the renormalization scale of the differential cross section
$d^{2}\sigma^{\rm Full\,NRQCD,NLO}_{e^{+}e^{-}\to B_{c}+X}/(dz\,d{\rm
cos}\theta)$, which is calculated through the formula (1) up to NLO accuracy,
is taken as the same as that of the dominant $\bar{b}$-fragmentation channel,
i.e., $\mu_{R}=m_{b}+2m_{c}$. The combined result (8) is more accurate than
the results under the fragmentation-function approach or the full NRQCD
approach. From those figures, one can see that the curves from the “Frag, NLO”
are close to the corresponding curves from the “Full NRQCD, NLO”. Compared
with the “Full NRQCD, NLO” results, the “Frag, NLO” results neglect all the
non-leading power contributions. Thus, one can conclude that those
contributions are small for the production of the $B_{c}$, $B_{c}^{*}$,
$B_{c}(2\,^{1}S_{0})$, and $B_{c}^{*}(2\,^{3}S_{1})$ at the $Z$ pole, and the
fragmentation-function approach can give a good approximation to the full
NRQCD calculations. The effect of resumming the leading and next-to-leading
logarithms of $s/(m_{b}+2m_{c})^{2}$ or $s/(2m_{b}+m_{c})^{2}$ can be seen
through comparing the “Frag, NLO+NLL” curves with the “Frag, NLO” curves. One
may observe that after resumming the leading and next-to-leading logarithms of
$s/(m_{b}+2m_{c})^{2}$ or $s/(2m_{b}+m_{c})^{2}$, the $z$-distributions become
softer. However, the effects of the DGLAP evolution for the NLO calculations
are not as large as the LO case Zheng:2015ixa . This is because the “Frag,
NLO” calculations contain the first term of the leading logarithmic series,
and the effect of the DGLAP evoluntion for the NLO calculation is to resum the
leading and next-to-leading logarithms from the corrections beyond the NLO QCD
calculations. The “Combined, NLO+NLL” results not only include the power
suppressed terms which have been neglected in the fragmentation-function
approach but also resum the leading and next-to-leading logarithms to all
orders. Therefore, the “Combined, NLO+NLL” results are the most accurate
results among those predictions shown in Figs.(5, 6, 7, 8).

Total production cross sections at the $Z$ pole under the “Combined, NLO+NLL”
approach can be obtained through integrating the differential cross sections
over $z$. And we obtain $\sigma^{\rm Combined,NLO+NLL}_{e^{+}e^{-}\to
B_{c}+X}=2.51\,{\rm pb}$, $\sigma^{\rm Combined,NLO+NLL}_{e^{+}e^{-}\to
B_{c}^{*}+X}=3.02\,{\rm pb}$, $\sigma^{\rm Combined,NLO+NLL}_{e^{+}e^{-}\to
B_{c}(2^{1}S_{0})+X}=1.49\,{\rm pb}$, and $\sigma^{\rm
Combined,NLO+NLL}_{e^{+}e^{-}\to B_{c}^{*}(2^{3}S_{1})+X}=1.80\,{\rm pb}$ at
the $Z$ pole.

##  IV Summary

As a summary, we have studied the production of the $B_{c}$, $B_{c}^{*}$,
$B_{c}(2\,^{1}S_{0})$, and $B_{c}^{*}(2\,^{3}S_{1})$ mesons at the CEPC. The
cross sections for these states via the three planned modes ($Z$, $W$, and
$H$) of the CEPC are presented, and the numbers of the events are estimated.
If assuming the $B_{c}^{*}$, $B_{c}(2\,^{1}S_{0})$, and
$B_{c}^{*}(2\,^{3}S_{1})$ states decay to the ground-state $B_{c}$ with a
probability of 100% via electromagnetic or hadronic interactions, then there
are about $1.4\times 10^{8}$ ($1.6\times 10^{4}$, $1.1\times 10^{4}$) $B_{c}$
events to be produced via the $Z$ ($W$, $H$) mode of the CEPC. Thus we may
have a good opportunity to study the properties of the $B_{c}$ meson via the
$Z$ mode of the CEPC. The differential angle and energy distributions for
these states via the $Z$ mode are further analyzed. There are relatively fewer
$B_{c}$ events to be produced via the other two ($W$ and $H$) modes. If one
wants to study the properties of the $B_{c}$ meson via the two modes, it is
better to improve the luminosities of the two modes properly.

Acknowledgments: This work was supported in part by the Natural Science
Foundation of China under Grants No. 11625520, No. 12005028, No. 12175025, and
No. 12147102, by the China Postdoctoral Science Foundation under Grant No.
2021M693743, by the Fundamental Research Funds for the Central Universities
under Grant No. 2020CQJQY-Z003, and by the Chongqing Graduate Research and
Innovation Foundation under Grant No. ydstd1912.

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