# Meta-Point Learning and Refining for Category-Agnostic Pose Estimation

Junjie Chen1 Jiebin Yan1 Yuming Fang1 Li Niu2
1 Jiangxi University of Finance and Economics,
2 Shanghai Jiao Tong University
<EMAIL_ADDRESS><EMAIL_ADDRESS>Corresponding author

###### Abstract

Category-agnostic pose estimation (CAPE) aims to predict keypoints for
arbitrary classes given a few support images annotated with keypoints.
Existing methods only rely on the features extracted at support keypoints to
predict or refine the keypoints on query image, but a few support feature
vectors are local and inadequate for CAPE. Considering that human can quickly
perceive potential keypoints of arbitrary objects, we propose a novel
framework for CAPE based on such potential keypoints (named as meta-points).
Specifically, we maintain learnable embeddings to capture inherent information
of various keypoints, which interact with image feature maps to produce meta-
points without any support. The produced meta-points could serve as meaningful
potential keypoints for CAPE. Due to the inevitable gap between inherency and
annotation, we finally utilize the identities and details offered by support
keypoints to assign and refine meta-points to desired keypoints in query
image. In addition, we propose a progressive deformable point decoder and a
slacked regression loss for better prediction and supervision. Our novel
framework not only reveals the inherency of keypoints but also outperforms
existing methods of CAPE. Comprehensive experiments and in-depth studies on
large-scale MP-100 dataset demonstrate the effectiveness of our framework.
Code is avaiable at https://github.com/chenbys/MetaPoint

##  1 Introduction

Pose estimation is a fundamental and significant computer vision task, which
aims to produce the locations of pre-defined semantic part of object instance
in 2D image. Recently, it has received increasing attention in the computer
vision community due to its wide applications in virtual reality, augmented
reality, human-computer interaction, robot and automation. However, most pose
estimation methods are trained with category-specific data and thus cannot be
applied for novel classes, especially when they have different keypoint
classes. Therefore, category-agnostic pose estimation (CAPE) [48, 41] is
recently proposed to localize desired keypoints for arbitrary classes given
one or few support images annotated with keypoints.

Figure 1: Overview of existing methods and our method. (a): Existing methods
only rely on the local features at support keypoints to predict or refine the
keypoints. (b): Our method employs learnable embeddings to capture inherent
information and produces meta-points without support. Then our method assigns
and refines meta-points according to support keypoints.

Unlike the standard pose estimation task where the keypoint classes are pre-
defined, the desired keypoints in CAPE are defined by support keypoints.
Therefore, existing methods [48, 41] intuitively extract features at support
keypoints and then rely on them to predict or refine the keypoints on query
image, as shown in Fig. 1 (a). Although such straightforward pipeline has
achieved considerable success, the information provided by a few feature
vectors are local [45, 32] and inadequate, especially when support keypoints
are partially occluded or blurred. Furthermore, above methods focus on
learning pixel-level correspondences and neglect to mine inherent and
complementary information of keypoints, which is a fundamental issue of CAPE.

We conjecture that there are some inherent or universal points in each object,
based on the insight that human can quickly grasp the essentials of keypoints
of arbitrary objects without any reference or support image. We refer to such
class-agnostic potential points as meta-points, which may seem similar to
object proposal [1, 57, 22] or mask proposal [37, 23, 25] but actually have
significant characteristics. For single object, meta-points could provide
fine-grained and structural information with a sparse representation. For two
objects, meta-points of the same semantic part should have consistent contact.
For example, a lion and a bird may have consistent meta-points (_e.g._ , eye)
and individual meta-points (_e.g._ , wing). Therefore, meta-point learning is
a fundamental yet challenging issue.

In this paper, we propose a novel pipeline for CAPE, which first predicts
meta-points without support and then assign and refine them to desired
keypoints according to support, as illustrated in Fig. 1 (b). Specifically, we
maintain learnable meta-embeddings to store the inherent information of
various keypoints for producing meta-points. Given a query image, these meta-
embeddings will interact with its feature maps via a transformer decoder to
mine the inherent information and thus could produce meta-points without
support. To bridge the gap between inherent meta-points and desired keypoints,
we first employ bipartite matching-based assignment to assign optimal meta-
points for desired keypoints according to support. Afterwards, we employ the
support feature vectors together with the mined inherent information to refine
assigned meta-points via another transformer decoder. Thanks to the meta-
points and mined inherent information, our framework could produce more
precise keypoints for CAPE. In addition, we propose a progressive deformable
point decoder, which progressively mines fine-grained features to decode
points with last points as reference based on deformable attention [56]. We
also propose a slacked regression loss to reduce the immature gradients of
auxiliary decoder layers.

We evaluate our framework on the largest dataset for CAPE, _i.e._ , Multi-
category Pose (MP-100) dataset [48]. The in-depth studies demonstrate that our
method could learn meaningful meta-points for arbitrary classes without
support. And the comprehensive experiments indicate that our meta-points could
dramatically facilitate CAPE and lead to better performance against existing
methods. Our contributions could be summarized as: 1) To the best of our
knowledge, we are the first to learn class-agnostic potential keypoints for
CAPE. 2) We propose a novel framework for CAPE based on meta-point learning
and refining. 3) Two minor contributions are progressive deformable point
decoder and slacked regression loss for better prediction and supervision. 4)
Our method not only produces meaningful meta-points but also predicts more
precise keypoints against existing methods in CAPE.

##  2 Related Works

###  2.1 Category-Specific Pose Estimation

Pose estimation is a fundamental task in computer vision, which aims to
localize the pre-defined keypoints of objects in the input image. Most pose
estimation methods are class-specific, _e.g._ , estimating pose for human [2,
27], animals [4, 26], and vehicles [38, 43]. Technically, existing pose
estimation methods could be roughly categorized into regression-based methods
[36, 28, 19], heatmap-based methods [54, 11, 13], and query-based methods [40,
34, 50]. To name a few, Li _et al._ [28] proposed a novel regression paradigm
with Residual Log-likelihood Estimation to learn the change of the
distribution to facilitate the training process. Geng _et al._ [19] proposed
to employ multi-branch structure for disentangled keypoint regression, which
could be able to attend to the keypoint regions and thus improve the
performances. Luo _et al._ [33] proposed to estimate scale-adaptive heatmaps,
which adaptively adjusts the standard deviation for each keypoint and thus is
tolerant of various scales and ambiguities. Shi _et al._ [40] proposed to
learn multiple pose queries for reasoning a set of full-body poses and then
use a joint decoder to refine by exploring kinematic relations. Although above
works have achieved great success for estimating pose of specific classes,
they are inapplicable for novel classes, especially when the novel classes
have different number and kinds of keypoints. In this work, we focus on
estimating pose for arbitrary classes with a few support images annotated with
desired keypoints.

###  2.2 Category-Agnostic Pose Estimation

Few-shot learning [46] focuses on learning novel classes using only a few
samples and has been applied in extensive visual tasks, _e.g._ ,
classification [12, 6], object detection [16, 55], segmentation [53, 29, 9].
For keypoint estimation, previous few-shot methods mostly focus on specific
domains, _e.g._ , facial images [3, 47], clothing images [18], or X-ray images
[52, 10]. For wider classes, Lu _et al._ [30] explored a flexible few-shot
scenario including novel/base classes and novel/base keypoints. Later, Lu _et
al._ [31] proposed a saliency-guided transformer for above scenario and
transductive scenario. He _et al._ [21] explored to estimate keypoints using a
few examples combined with unlabeled images. Recently, POMNet [48] elaborated
a large-scale dataset including 100 classes and set the task of category-
agnostic pose estimation. Technically, POMNet [48] proposed a keypoint
matching framework, which extracts the features at support keypoints and
employs their matching similarities to every pixels to retrieve the desired
keypoints. Afterwards, CapeFormer [41] enhanced the similarity modeling in
above matching pipeline, and proposed to further refine each keypoint via a
elaborate transformer decoder. Although above methods [48, 41] have great
promoted the development of CAPE, they only rely on the local features at
support keypoints and neglect to mine inherent and universal information of
keypoints. In contrast, we propose to mine the inherent or universal points in
objects without any support, and propose a progressive deformable point
decoder to progressively mines fine-grained features to decode points more
precisely.

###  2.3 Category-Agnostic Proposal Learning

Category-agnostic proposal learning is a long-standing research topic in
computer vision and has extensive downstream tasks. Early methods rely on low-
level features (_e.g._ , edges, super-pixels, or saliency) to produce bounding
box proposals [1, 57] or mask proposals [23, 25, 8, 7]. Recent methods employ
various network to learn box proposals [24, 39] or mask proposals [49, 15]. To
name a few, one of the most impressive works is Faster R-CNN [39], which first
learns class-agnostic box proposals and then refines them to object bounding
boxes. Pinheiro _et al._ [37] proposed to learn the class-agnostic
segmentation masks and their object likelihood scores. Therefore, proposal
learning is a significant and foundational task and could greatly facilitate
downstream tasks in the second stage. Nevertheless, there are few works on
learning class-agnostic keypoints, and we propose to learn potential keypoints
and name our learned points as meta-points. Basically, our meta-point could
provide fine-grained and structural information with a sparse representation.
Furthermore, our meta-points have consistent contact for the same semantic
part of different objects. Therefore, such meta-points not only reveals the
inherency of keypoints but also serves as meaningful proposals for downstream
tasks. Although the coarse keypoints in [41] are named as “proposals”, their
“proposals” are predicted according to support keypoints and thus do not have
the class-agnostic properties.

##  3 Method

Class-agnostic pose estimation (CAPE) aims to estimate desired keypoints on
query image given a few support images annotated with keypoints for novel
classes. For $N$-shot setting, CAPE could be formulated as a function:

$\bm{P}_{q}=\mathcal{F}_{cape}(\bm{x}_{q},\\{\bm{x}_{s_{n}}\\}_{n=1}^{N},\\{\bm{P}^{*}_{s_{n}}\\}_{n=1}^{N})$
(1)

where $N$ support images $\\{\bm{x}_{s_{n}}\\}_{n=1}^{N}$ and their ground-
truth (GT) keypoints $\\{\bm{P}^{*}_{s_{n}}\\}_{n=1}^{N}$ are given, and the
task is to estimate the desired keypoints $\bm{P}_{q}$ on query image
$\bm{x}_{q}$. To evaluate the generalization capacity for arbitrary classes,
all classes in benchmark are split into non-overlapping base classes and novel
classes, and query and support images are sampled from only base (_resp._ ,
novel) classes for training (_resp._ , evaluating). For simplicity, we first
introduce our method in 1-shot setting, and then clarify the intuitive
extension with more support images. Specifically, the function is simplified
as $\bm{P}_{q}=\mathcal{F}_{cape}(\bm{x}_{q},\bm{x}_{s},\bm{P}^{*}_{s})$,
where $\bm{x}_{q},\bm{x}_{s}\in\mathbb{R}^{3\times H\times W}$ and
$\bm{P}_{q},\bm{P}^{*}_{s}\in\mathbb{R}^{K\times 2}$. For brevity of
description, we employ subscript to indicate the variable source and use
square bracket to show the index in variable, _e.g._ , $\bm{P}_{q}[k]$ for the
$k$-th keypoint on query image. To tackle such problem, we propose a novel
framework based on meta-point learning and refining, which not only reveals
the keypoint inherency but also produces precise keypoints. The detailed
framework is illustrated in Fig. 2, we first introduce the pipelines of two-
stages and then clarify the architecture of our point decoder used in
pipelines.

Figure 2: Our framework employs two stages to predict meta-points and desired
keypoints. In the first stage, the learnable meta-embeddings interact with
query feature maps via our progressive deformable point decoder to mine
inherent information to predict meta-points and their visibilities. In the
second stage, the meta-points are assigned with identities according to the
given support keypoints. After that, the assigned meta-points are refined to
desired keypoints based the support features and mined inherent information
via another point decoder.

###  3.1 Predicting Meta-Point without Support

Considering that human is able to quickly abstract keypoints for arbitrary
classes without support, we propose to model such fundamental ability to
facilitate CAPE. As shown in the upper part of Fig. 2, we maintains $M$
trainable meta-embeddings $\tilde{\bm{E}}\in\mathbb{R}^{M\times D}$ in our
model (_e.g._ , $M=100$), which are shared by all samples and store the
inherent and universal information of keypoints. For the input sample
$\bm{x}_{q}$, our model first employs ResNet-50 [20] backbone to extract
pyramid feature maps $\\{\bm{F}^{r}_{q}\\}_{r=1}^{R}$, where
$\bm{F}^{r}_{q}\in\mathbb{R}^{D^{r}\times H^{r}\times W^{r}}$ corresponds to
the $r$-th residual layer in deep-to-shallow order ($R=3$ by default).
Afterwards, the meta-embeddings interact with image feature maps to mine
inherent information, which could be summarized as:

$\left\\{\left[\tilde{\bm{P}}^{l}_{q};\tilde{\bm{E}}^{l}_{q}\right]\right\\}_{l=1}^{L}=\mathcal{D}^{L}_{meta}(\tilde{\bm{E}},\tilde{\bm{P}}_{init},\\{\bm{F}^{r}_{q}\\}_{r=1}^{R}).$
(2)

Specifically, $\mathcal{D}_{meta}$ is implemented with our Progressive
Deformable Point Decoder, which employs DeformAttn[56] to progressively decode
$L$ layers of points with last points as reference points and will be clearly
introduced in Sec. 3.3. $\tilde{\bm{P}}_{init}\in\mathbb{R}^{M\times 2}$ are
the initial points, and we adopt uniform grid points to launch the decoding
for meta-points. From the $l$-th decoder layer in $\mathcal{D}_{meta}$, we
obtain the query meta-points $\tilde{\bm{P}}^{l}_{q}\in\mathbb{R}^{M\times 2}$
and query meta-embeddings $\tilde{\bm{E}}^{l}_{q}\in\mathbb{R}^{M\times D}$.
We use the predictions from the last layer as the proposals for following
processes. Because not all meta-points are visible for arbitrary objects, we
employ a lightweight binary classifier to predict the visibilities
$\tilde{\bm{v}}_{q}\in\mathbb{R}^{M}$ according to the last query meta-
embeddings.

In this way, we are able to produce meta-points $\tilde{\bm{P}}^{L}_{q}$ and
their visibilities $\tilde{\bm{v}}_{q}$ without any support, which reveal the
inherency of keypoints and serve as meaningful proposals to facilitate
downstream tasks.

###  3.2 Refining Meta-Point with Support

Due to the inevitable gap between keypoint inherency and annotation, we have
to refine above meta-points to desired keypoints $\bm{P}_{q}$ with GT support
keypoints $\bm{P}^{*}_{s}$. The support information includes two aspects,
_i.e._ , desired identity and desired detail, which will be described as
follows.

####  3.2.1 Identity Assignment

Each annotated keypoint has its identity/index, and thus we need to determine
the index mapping $\delta(\cdot)$ between meta-points and desired keypoints,
_i.e._ , the $\delta(k)$ meta-point corresponds to the $k$-th keypoint.
Specifically, we employ the bipartite matching-based assignment as in [14, 5]
to solve the mapping according to a cost matrix
$\mathcal{C}\in\mathbb{R}^{M\times K}$, where each entry $\mathcal{C}[m,k]$
indicates the cost for assigning the $m$-th meta-point to the $k$-th annotated
keypoint. To solve the assignment according to support, we obtain the meta-
points $\\{\tilde{\bm{P}}^{l}_{s}\\}_{l=1}^{L}$ and visibilities
$\tilde{\bm{v}}_{s}$ on support image as the same pipeline in Sec. 3.1. Then,
we compute the cost by:

$\mathcal{C}_{s}[m,k]=\sum_{l=1}^{L}{\mathcal{L}_{sl_{1}}(\tilde{\bm{P}}^{l}_{s}[m],\bm{P}^{*}_{s}[k],l)-\alpha{\rm
log}(\tilde{\bm{v}}_{s}[m])},$ (3)

where $\mathcal{L}_{sl_{1}}$ measures the $l_{1}$ deviation with slacks on
auxiliary decoder layers and will be introduced in Sec. 3.4. The second term
considers the classification of meta-point visibility, with a trade-off hyper-
parameter $\alpha=0.5$. Finally, we use $\mathcal{C}_{s}$ to obtain the
optimal assignment $\delta(\cdot)$. In this way, each desired keypoint is
assigned with a meta-point, _i.e._ , the $k$-th keypoint is assigned with
$\tilde{\bm{P}}[\delta(k)]$. Because we set $K\leq M$, and the unassigned
meta-points are treated as ignore. For brevity, we employ
$\delta(\tilde{\bm{P}})\in\mathbb{R}^{K\times 2}$ and
$\delta(\tilde{\bm{E}})\in\mathbb{R}^{K\times D}$ to denote the assigned meta-
points and meta-embeddings re-ordered in desired sequence.

####  3.2.2 Details Enhancement

Besides the desired identities, we refine meta-points to desired coordinates
according to the support information. By default, we employ the prediction of
the last decoder layer in $\mathcal{D}_{meta}$ to refine and omit the decoder
layer index for brevity, _i.e._ , we use assigned meta-points
$\delta(\tilde{\bm{P}}_{q})$ and their embeddings $\delta(\tilde{\bm{E}}_{q})$
to launch the refinement. Basically, we derive support keypoint features
$\bm{F}_{s}$ via “soft” ROI pooling as in previous works [41, 48], which
contain information of local details in support image. We also collect the
meta-embeddings assigned to support keypoints $\delta(\tilde{\bm{E}}_{s})$,
which have absorbed the global inherent information in support image.
Afterwards, we squeeze the concatenation of above embeddings to obtain
comprehensive embeddings $\bm{E}_{q}\in\mathbb{R}^{K\times D}$ for keypoints,
_i.e._ ,
$\bm{E}_{q}=f_{sqz}([\delta(\tilde{\bm{E}}_{q});\delta(\tilde{\bm{E}}_{s});\bm{F}_{s}])$.
Then, the assigned meta-points $\delta(\tilde{\bm{P}}_{q})$ are progressively
refined according to $\bm{E}_{q}$ and image feature maps, as:

$\left\\{\left[\bm{P}^{l}_{q};\bm{E}^{l}_{q}\right]\right\\}_{l=1}^{L}=\mathcal{D}^{L}_{ref}(\bm{E}_{q},\delta(\tilde{\bm{P}}_{q}),\\{\bm{F}^{r}_{q}\\}_{r=1}^{R}).$
(4)

Specifically, we also employ our Progressive Deformable Point Decoder to
implement $\mathcal{D}_{ref}$ (same architecture but different parameters with
$\mathcal{D}_{meta}$) and leave the details to Sec. 3.3. In this way, our
decoder $\mathcal{D}_{ref}$ progressively refines assigned meta-points to
desired keypoints.

###  3.3 Progressive Deformable Point Decoder

Figure 3:  Illustration of decoder layer in our Progressive Deformable Point
Decoder. The input embeddings first interact with each other via a self-
attention module. After that, the offsets and weights are predicted to mine
fine-grained features on input feature maps. Finally, a deformable attention
module uses input points as reference to refine the embeddings and points.

Predicting precise keypoints requires fine-grained and high-resolution
features, and thus we propose a progressive deformable point decoder, which
makes full use of deformable attention module [56] to precisely refine
keypoints. Our decoder consists of $L$ decoder layers ($L=3$ by default), and
the architecture of decoder layer is shown in Fig. 3. Given points
$\bm{P}^{l-1}\in\mathbb{R}^{K\times 2}$ and their embeddings
$\bm{E}^{l-1}\in\mathbb{R}^{K\times D}$, each decoder layer performs self
attention [44] and deformable attention [56] to update points and embeddings
according to feature maps $\\{\bm{F}^{r}\\}_{r=1}^{R}$, as:

$[\bm{P}^{l};\bm{E}^{l}]=\mathcal{D}(\bm{E}^{l-1},\bm{P}^{l-1},\\{\bm{F}^{r}\\}_{r=1}^{R}).$
(5)

By stacking multiple such layers, our decoder could update the points in a
progressive and precise manner. Specifically, each decoder layer first employs
a self attention module to render each point aware of other points’ positions
and contents as in [41] (also with identity embeddings and position
embeddings). Afterwards, each embedding predicts the offsets
$\\{\Delta\bm{P}^{r}\\}_{r=1}^{R}$ and weights $\\{\bm{A}^{r}\\}_{r=1}^{R}$
for paying attention to each feature map. By using the points $\bm{P}^{l-1}$
as reference points, the deformable attention module aggregates sampled
features according to weights and obtain updated embeddings $\bm{E}^{l}$.
Then, the coordinates of points are refined by:

$\bm{P}^{l}=\sigma\left(\sigma^{-1}(\bm{P}^{l-1})+\mathcal{F}_{mlp}(\bm{E}^{l})\right),$
(6)

where $\mathcal{F}_{mlp}$ is a lightweight point head and $\sigma$ indicates
Sigmoid function. In this way, input points and their embeddings could be
progressively refined according to the nearby fine-grained features.

###  3.4 Training and Inference

Our full training objective consists of keypoint regression loss and
visibility classification loss. For the regression loss, strict supervision on
auxiliary decoder layers will trigger immature gradients for fitting keypoints
with inadequate capacities. And thus we adopt a slacked $L_{1}$ loss:

$\mathcal{L}_{sl_{1}}(\bm{P},\bm{P}^{*},l)=\sum_{k}\varepsilon\left(\left|{\bm{P}[k]-\bm{P}^{*}[k]}\right|_{1}-\lambda\cdot(L-l)\right),$
(7)

where $\varepsilon$ indicates ReLU function and $\lambda=0.1$ is a slack
hyper-parameter. And our total regression loss is

$\mathcal{L}_{reg}=\sum_{l=1}^{L}{\mathcal{L}_{sl_{1}}(\delta(\tilde{\bm{P}}^{l}_{q}),\bm{P}^{*}_{q},l)+\mathcal{L}_{sl_{1}}(\bm{P}^{l}_{q},\bm{P}^{*}_{q},l)},$
(8)

where the first part supervises the assigned meta-points and the second part
supervises the refined meta-points towards to GT keypoints $\bm{P}^{*}_{q}$.
For the visibility classification loss, we employ binary cross-entropy loss:

$\mathcal{L}_{vis}=BCE(\tilde{\bm{v}_{q}},\bm{1}_{\delta}),$ (9)

where $\bm{1}_{\delta}\in\mathbb{R}^{M}$ is a vector indicating assigned meta-
points with $1$, otherwise $0$. Therefore, our full training objective is
calculated as
$\mathcal{L}_{full}=\mathcal{L}_{reg}+\alpha\cdot\mathcal{L}_{vis}$, where we
reuse the balance hyper-parameter $\alpha$ in cost computation Eqn. 3.

In the inference stage, we employ the result of last decoder layer
$\bm{P}^{L}_{q}$ as the estimated keypoints. As for $N$-shot setting, we
average the support keypoint features from different shots as previous works
[41, 48]. Besides, we average their cost matrices to obtain better assignment
$\delta(\cdot)$, _i.e._ ,
$\mathcal{C}_{\bar{s}}=\frac{1}{N}\sum_{n}^{N}{\mathcal{C}_{s_{n}}}$. We also
average their assigned meta-embeddings, _i.e._ ,
$\delta(\tilde{\bm{E}}_{\bar{s}})=\frac{1}{N}\sum_{n}^{N}{\delta(\tilde{\bm{E}}_{s_{n}}})$.
In this way, we could obtain enhanced information to facilitate prediction.

##  4 Experiments

###  4.1 Dataset, Metric, and Implementation Details

Overall, we follow the experimental setting of previous work [41].
Specifically, we conduct experiments on MP-100 dataset [48], which covers
$100$ classes and $8$ super-classes and is the largest benchmark dataset for
CAPE. MP-100 contains over $18$K images and $20$K annotations, and the
keypoint numbers range from $8$ to $68$ across different classes. The
collected $100$ classes are split into non-overlapping train/val/test sets
with the ratio of $70:10:20$. There are five random splits to reduce the
impact of randomness. We use the Probability of Correct Keypoint (PCK) [51] as
the quantitative metric. Apart from the PCK of threshold $0.2$ as in [48, 41],
we also report the mPCK (of $[0.05,0.1,0.15,0.2]$) for a more comprehensive
evaluation.

Table 1: Results of 1-shot and 5-shot setting on MP-100 dataset. We summarize
<EMAIL_ADDRESS>results and report mPCK results in Tab. 2.

Method | 1-shot setting | 5-shot setting
---|---|---
Split1 | Split2 | Split3 | Split4 | Split5 | Average | Split1 | Split2 | Split3 | Split4 | Split5 | Average
ProtoNet [42] | 46.05 | 40.84 | 49.13 | 43.34 | 44.54 | 44.78 | 60.31 | 53.51 | 61.92 | 58.44 | 58.61 | 58.56
MAML [17] | 68.14 | 54.72 | 64.19 | 63.24 | 57.20 | 61.50 | 70.03 | 55.98 | 63.21 | 64.79 | 58.47 | 62.50
Fine-tune [35] | 70.60 | 57.04 | 66.06 | 65.00 | 59.20 | 63.58 | 71.67 | 57.84 | 66.76 | 66.53 | 60.24 | 64.61
POMNet [48] | 84.23 | 78.25 | 78.17 | 78.68 | 79.17 | 79.70 | 84.72 | 79.61 | 78.00 | 80.38 | 80.85 | 80.71
CapeFormer [41] | 89.45 | 84.88 | 83.59 | 83.53 | 85.09 | 85.31 | 91.94 | 88.92 | 89.40 | 88.01 | 88.25 | 89.30
1-13[1.5pt/3pt] MetaPoint | 83.45 | 79.88 | 77.59 | 76.53 | 79.09 | 79.31 | 84.13 | 80.30 | 78.02 | 77.12 | 79.58 | 79.83
MetaPoint+ | 90.43 | 85.59 | 84.52 | 84.34 | 85.96 | 86.17 | 92.58 | 89.63 | 89.98 | 88.70 | 89.20 | 90.02

Table 2: The evaluation results using mPCK metric.

(a) in 1-shot and 5-shot settings.

|  | Split1 | Split2 | Split3 | Split4 | Split5
---|---|---|---|---|---|---
1-shot | CapeFormer | 75.13 | 69.30 | 68.59 | 68.50 | 71.38
MetaPoint+ | 77.11 | 71.07 | 70.32 | 69.93 | 72.73
5-shot | CapeFormer | 78.05 | 74.43 | 74.87 | 73.80 | 76.11
MetaPoint+ | 79.22 | 75.51 | 76.20 | 75.92 | 77.65

(b) in cross super-category pose estimation.

| Human Body | Human Face | Vehicle | Furniture
---|---|---|---|---
CapeFormer | 60.21 | 58.04 | 27.34 | 35.60
MetaPoint+ | 62.42 | 60.72 | 30.78 | 38.52

###  4.2 Comparison with Prior Works

Following previous works [41], our comparison baselines include ProtoNet [42],
MAML [17], Fine-tune [35], POMNet [48], and CapeFormer [41]. Specifically,
ProtoNet, MAML, and Fine-tune are popular few-shot learning methods adapted in
[48]. POMNet [48] adopts a keypoint matching framework to match support
keypoints to the pixels of query image. CapeFormer [41] enhances above
similarity modeling and further refines keypoints via transformer modules. We
use two versions of our model for comparison, _i.e._ , MetaPoint and
MetaPoint+. MetaPoint directly employs the predicted meta-points as results,
where the support keypoints are only used to determine the identities for
evaluation. MetaPoint+ is the full-fledged version of our model and employs
the refined meta-points as results.

We summarize the results of<EMAIL_ADDRESS>in 1-shot setting and 5-shot setting in
Tab. 1. Firstly, we could see that our MetaPoint achieves satisfactory
performances without any support and refinement, which indicates the mined
meta-points could reveal the keypoints inherency and serve as high-quality
proposals. Our MetaPoint+ achieves the optimal performances against all
baselines, demonstrating the effectiveness of our framework on learning and
refining meta-points. We could also see that our methods have relatively
robust performances on various dataset splits and different number of support
images.

In addition, the performances of various methods approach to saturation,
probably because<EMAIL_ADDRESS>employs a coarse threshold. We also report the
results of mPCK (with the thresholds in [0.05, 0.1, 0.15, 0.2]) against the
most competitive baseline (_i.e._ , CapeFormer) for a more comprehensive
evaluation in Tab. 2. From the perspective of mPCK, we can see that our
MetaPoint+ outperforms dramatically against the most competitive baseline,
which more clearly demonstrates the effectiveness of our novel framework based
on meta-point learning and refining.

###  4.3 Cross Super-Category Pose Estimation

Table 3: Results of cross super-category pose estimation. We summarize the
<EMAIL_ADDRESS>results and leave mPCK results in Tab. 2.

Method | Human Body | Human Face | Vehicle | Furniture
---|---|---|---|---
ProtoNet | 37.61 | 57.80 | 28.35 | 42.64
MAML | 51.93 | 25.72 | 17.68 | 20.09
Fine-tune | 52.11 | 25.53 | 17.46 | 20.76
POMNet | 73.82 | 79.63 | 34.92 | 47.27
CapeFormer | 83.44 | 80.96 | 45.40 | 52.49
MetaPoint+ | 84.32 | 82.21 | 46.51 | 53.67

Figure 4:  In each row, we show the GT keypoints and predictions of a pair of
support image and query image. The left two columns show the GT keypoints and
estimated meta-points on support image. The middle two columns show the GT
keypoints and estimated meta-points on query image. The right two columns show
the final keypoints predicted by ours and CapeFormer. We employ the small
black arrows to indicate the deviations to GT. The radii of drew meta-points
indicate their visibilities, and the assigned meta-points are encircled with
red.  Figure 5:  In each row, we track one meta-embedding and visualize its
meta-point on various samples if the visibilities are greater than $0.5$.

In the standard setting of CAPE, although the training classes, validation
classes, and test classes have no overlap, some classes may have common
characteristics and thus easy to transfer, _e.g._ , the body of various kinds
of animals. We follow the experiments of cross super-category evaluation in
previous works [41, 48] to investigate the generalization ability on
significantly different classes of our method. Specifically, we respectively
employ one of four super-categories (_i.e._ , human face, human body, vehicle,
and furniture) in MP-100 dataset as the test categories, while the other
categories are used as the training categories. As shown in Tab. 3, our method
outperforms than all baselines, demonstrating that the our method could mine
more class-agnostic information via learning meta-points, and thus are more
robust against the distribution shift of categories. We also include the
evaluation results using mPCK metric in Tab. 2. The performances vary greatly
across super-classes, probably because different super-classes have diverse
kinds of keypoints as discussed in [41].

###  4.4 Qualitative Analyses

For a more intuitive understanding of our method, we visualize the predicted
meta-points and keypoints in 1-shot setting for novel classes in Fig. 4. As
shown in the second column and the fourth column, our model could predict
meaningful meta-points without any support, which have satisfactory
performances compared with assigned GT. Furthermore, the meta-points
correspond to consistent semantic parts in support images and query images.
Therefore, the predicted meta-points not only reveal the inherency of
keypoints but also could serve as favourable proposals for subsequent
refinement. As shown in the first row, the support keypoints of bird feet are
hard to provide clear support information. In such case, previous works may be
confused by the ambiguous support, while our method could predict precise
keypoints based on the inherent and complementary information mined via meta-
points. Similar cases could be found in other visualizations.

To further explore the properties of learned meta-points, we track one meta-
embedding and visualize its meta-point on various samples if the visibilities
are greater than $0.5$ in Fig. 5. As shown in the first row, we could find
that the meta-point could correspond to consistent semantic parts of various
classes, even if the parts have just high-level or potential relation. Note
that there is no annotation binding the points across classes, which
demonstrates that our model could mine and extract the inherent and universal
information of keypoints. In the second row, the meta-point may correspond to
the egoistic left-upper point of objects. Similar phenomena could be found in
other cases.

###  4.5 Ablation Study

Table 4: Results (mPCK) of various component combinations.

| $+\tilde{\bm{v}}$ | $+\bm{F}_{s}$ | $+\delta(\tilde{\bm{E}_{s}})$ | $+\mathcal{L}_{sl_{1}}$ | Split1
---|---|---|---|---|---
#1 | - | - | - | - | 68.18
#2 | ✓ | - | - | - | 70.37
#3 | ✓ | ✓ | - | - | 74.70
#4 | ✓ | ✓ | ✓ | - | 76.43
#5 | ✓ | ✓ | ✓ | ✓ | 77.11
|  |  |  |  |

Table 5: Results (mPCK) of various configurations of our model.

| Version | Split1 | Split2 | Split3 | Split4
---|---|---|---|---|---
#1 | Standard | 77.11 | 71.07 | 70.32 | 69.93
#2 | Dual-Asm | 73.21 | 66.90 | 65.78 | 65.61
#3 | $R=2$ | 76.73 | 70.66 | 70.02 | 69.64
#4 | $R=4$ | 77.18 | 71.12 | 70.40 | 69.97
#5 | $L=2$ | 76.30 | 70.22 | 69.38 | 69.21
#6 | $L=4$ | 77.20 | 71.15 | 70.44 | 69.98
|  |  |  |  |

In this section, we investigate the performances of component combinations and
various configurations of our method. Firstly, we gradually enable various
components in our model and report the mPCKs in Tab. 4. The row #1 is a plain
version to directly predict and refine meta-points without extra information,
which shows the basic performance of learned meta-points. In row #2, we
further enable visibility $\tilde{\bm{v}}$, which avoids the wrong but
accidental assignment to non-existent meta-points (_e.g._ , occluded) and thus
prompts the performance. In row #3 and row #4, we enable support features
$\bm{F}_{s}$ and support meta-embeddings $\delta(\tilde{\bm{E}_{s}})$ in
refinement, which provide critical and complementary information for refining
meta-points to desired keypoints. Row #5 indicating the contribution of our
slacked regression loss $\mathcal{L}_{sl_{1}}$, which could slack the
supervision on immature predictions. In addition, we explore the impact of
various configurations in Tab. 5. Specifically, Row #1 indicates the standard
performance of our model. Row #2 solves the assignment $\delta(\cdot)$ based
on the sum of cost matrices of query and support images but obtain worse
results, probably because only the cost matrix of support image is available
in inference. Row #3 and row #4 explore the impact of feature map number $R$,
and our default configuration (_i.e._ , $R=3$) could mine effective features
from pyramid feature maps via our point decoder, but too shallow feature map
only improves negligibly (_i.e._ , $R=4$) probably because the features are
too low-level. The last two rows investigate the impact of decoder layer
number, and our point decoder could progressively refine keypoints via
multiple decoder layers. The performances are gradually saturated, and our
default choice (_i.e._ , $L=3$) is appropriate and enough.

##  5 Conclusion

In this paper, we have proposed a novel framework for CAPE based on meta-point
learning and refining. Such meta-points not only reveal the inherency of
keypoints but also serve as meaningful proposals for downstream tasks. We also
have proposed a progressive deformable point decoder and a slacked regression
loss for better prediction and supervision. The effectiveness of our framework
has been demonstrated by in-depth experiments on MP-100 dataset.

## Acknowledgements

This work was supported in part by the National Natural Science Foundation of
China under Grants 62132006, in part by the Natural Science Foundation of
Jiangxi Province of China under Grants 20223AEI91002, 20232BAB202001, and
20224BAB212012, and in part by the Project funded by China Posdoctoral Science
Foundation under Grant 2022M721417.

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# Heavy quarkonium with finite three momentum near $T_{c}$

HyungJoo Kim<EMAIL_ADDRESS>Department of Physics and Institute of
Physics and Applied Physics, Yonsei University, Seoul 03722, Korea  Seokwoo
Yeo<EMAIL_ADDRESS>Department of Physics and Institute of Physics and
Applied Physics, Yonsei University, Seoul 03722, Korea  Sungtae Cho
<EMAIL_ADDRESS>Division of Science Education, Kangwon National
University, Chuncheon 24341, Korea  Su Houng Lee<EMAIL_ADDRESS>Department of Physics and Institute of Physics and Applied Physics, Yonsei
University, Seoul 03722, Korea

###### Abstract

We investigate the non-trivial 3-momentum effects on the masses of heavy
quarkonium states that are moving in a hot medium using QCD sum rules. For all
charmonium states, we observe a negative mass shift near $T_{c}$ that is less
than 3$\%$ at a momentum of 1$\rm{GeV}$. Specifically, we first investigate
the difference between the longitudinal and transverse modes of both $J/\psi$
and $\chi_{c1}$. We find that the transverse mode of the $J/\psi$ experiences
larger modification than the longitudinal mode, while the $\chi_{c1}$ has the
opposite behavior. By comparing the $\eta_{c}$ and $\chi_{c0}$, and also the
unpolarized $J/\psi$ and $\chi_{c1}$, we recognize that the P-wave particles
have stronger momentum dependencies on their masses than the S-wave ones. We
also find $\Upsilon$(1S) has negligible 3-momentum dependence compared to the
charmonium states, e.g. less than 0.01$\%$ even at 1.4$T_{c}$ and at a
momentum of 4$\rm{GeV}$.

##  I Introduction

The study of quarkonium in a thermal medium is of great importance in our
understanding of the physics of strongly-interacting matter that existed in
the early universe. In particular, heavy quarkonium in a hot medium has long
been of interest ever since the seminal work by Matsui and SatzMatsui and Satz
(1986) that suppression of quarkonium yields in heavy-ion collisions could be
a signal for the formation of a deconfined quark-gluon plasma. As temperature
increases, quarkonium states undergo spectral modification, such as mass shift
and broadening, and eventually will melt and merge with the continuum. Over
the past decade, many theoretical approaches, such as lattice simulations,
effective field theories, and spectral reconstruction, are developed and
employed to understand the properties of quarkonium in a hot medium. Moreover,
the recent recognition that the in-medium interquark potential is complex-
valuedLaine  _et al._ (2007) stressed the necessity of dynamical description
for quarkonium melting and led to a new framework of the open quantum system.
See Refs.Brambilla  _et al._ (2011); Rothkopf (2020) for comprehensive
reviews.

Meanwhile, most theoretical studies assume quarkonium is at rest when
computing its in-medium properties. Because Lorentz symmetry is broken in the
presence of the medium, finite 3-momentum makes any computation more
complicated and thus often omitted. However, most particles produced in real
experiments propagate in a medium with non-zero momentum. For a more realistic
analysis, therefore, it is necessary to take into account non-trivial effects
coming from finite 3-momentum. In general, it is expected that two types of
effects appear for a particle moving in a medium. First, a particle will not
follow the standard energy-momentum dispersion relation, i.e.
$E^{2}-\vec{q}\,^{2}=m^{2}$, where $E$, $\vec{q}\,$ and $m$ denote the energy,
3-momentum and invariant mass of a particle, respectively. Instead, it will
follow a modified version, i.e. $E^{2}-\vec{q}\,^{2}=m(\vec{q}\,)^{2}$, which
indicates that the mass of a particle moving in a medium is not a invariant
quantity any more but possibly depends on the 3-momentum. Second, if a
particle has spin, its different polarization states that have the same
dispersion relation in a vacuum may behave differently. For example, the
transverse and longitudinal modes of massive spin-1 particles will follow
separate modified dispersion relations in a medium. In other words, by
tracking the 3-momentum dependence on the mass spectrum we might be able to
distinguish the polarization states of particlesPark  _et al._ (2022).
Consequently, the overall mass shift can vary depending on the polarization
states as well as the size of 3-momentum. Therefore, when we read off the mass
shift from the spectral change of quarkonium in a medium, the finite momentum
effect should be identified as distinct from the pure mass shift typically
defined at zero momentum.

There are some lattice simulations that have studied non-trivial momentum
effects on the dispersion relation of heavy quarkonium at finite
temperatureOktay and Skullerud (2010); Ding (2013); Aarts  _et al._ (2013);
Ikeda  _et al._ (2017); Ding  _et al._ (2018). All of them are based on the
maximum entropy method(MEM) to reconstruct the spectral function from lattice
simulation for the euclidean correlator. The $\eta_{c}$ and $J/\psi$ were
studied in Refs.Oktay and Skullerud (2010); Ding (2013); Ikeda  _et al._
(2017). The $\Upsilon$(1S) and $\eta_{b}$ are studied in Ref.Aarts  _et al._
(2013) using lattice NRQCD. In Ref.Ding  _et al._ (2018), S-wave states for
both charmonium and bottomonium are studied. In spite of these many efforts,
however, none of them made any definite conclusion on the non-trivial effects
because of the significant size of uncertainties.

In this work, we investigated the non-trivial 3-momentum effects on the masses
of heavy quarkonium states that are moving in a hot medium using QCD sum
rules. This method has already been applied to study light vector mesons, such
as $\rho$, $\omega$, and $\phi$, that are moving in a dense mediumLee (1998);
Leupold and Mosel (1998); Kim and Gubler (2020), so we basically follow the
same strategy discussed there. Specifically, we included the Wilson
coefficients which are responsible for the non-trivial 3-momentum dependence
to the typical QCD sum rules for the quarkonium at rest. We first studied the
difference between the longitudinal and transverse modes of $J/\psi$ and
$\chi_{c1}$. To contrast the difference between S-wave and P-wave particles,
we then compared the $\eta_{c}$ and $\chi_{c0}$, and also the unpolarized
$J/\psi$ and $\chi_{c1}$. We also studied $\Upsilon$(1S) as a representative
of bottomonium states while distinguishing the longitudinal and transverse
modes. We hope that our work will provide a guideline on how much uncertainty
must be reduced in future lattice simulations to reveal the genuine non-
trivial effects on the dispersion relation of heavy quarkonium at finite
temperature.

This paper is organized as follows. In Section II, we give a brief description
of the formalism of QCD sum rules for a particle moving in a medium. Section
III is devoted to the detailed results obtained in this study. The paper is
summarized and concluded in Section IV. In the Appendix, we display the
explicit forms of Wilson coefficients which are responsible for the non-
trivial 3-momentum dependence.

##  II QCD sum rules with finite momentum

To study heavy quarkonium states in the pseudoscalar(P), scalar(S), vector(V),
and axialvecotr(A) channels, we consider a two-point correlator,

$\displaystyle\Pi^{J}(\omega,\vec{q}\,)=i\int d^{4}xe^{iq\cdot
x}\langle{T\\{j^{J}(x)j^{J}(0)\\}}\rangle,$ (1)

where the superscript $J$ indicates each channel that has the following
current structure: $j^{P}=i\bar{h}\gamma_{5}h$, $j^{S}=\bar{h}h$,
$j_{\mu}^{V}=\bar{h}\gamma_{\mu}h$, and
$j_{\mu}^{A}=(q_{\mu}q_{\nu}/q^{2}-g_{\mu\nu})\bar{h}\gamma_{5}\gamma^{\nu}h$.
Here, $h$ represents the heavy quark field that can be either a $c$ or $b$
quark. For convenience, we define the following dimensionless functions,

$\displaystyle\widetilde{\Pi}{{}^{P,S}(\omega^{2},\vec{q}\,^{2})}$
$\displaystyle=\frac{1}{q^{2}}\Pi^{P,S},$ (2)
$\displaystyle\widetilde{\Pi}^{V,A}_{L}(\omega^{2},\vec{q}\,^{2})$
$\displaystyle=\frac{1}{\vec{q}\,^{2}}\Pi^{V,A}_{00},$ (3)
$\displaystyle\widetilde{\Pi}^{V,A}_{T}(\omega^{2},\vec{q}\,^{2})$
$\displaystyle=-\frac{1}{2}(\frac{1}{\vec{q}\,^{2}}\Pi_{\mu}^{\mu,V,A}+\widetilde{\Pi}^{V,A}_{L}),$
(4)

where the V and A channels are decomposed into the longitudinal($L$) and
transverse($T$) modes.

In the deep Euclidean region where $\omega^{2}\ll\infty$ with finite
$|\vec{q\,}|$, one can compute these functions using the operator product
expansion(OPE). As stressed in Ref.Kim and Gubler (2020), it is more
convenient to express the OPE results by changing variables from
$(\omega^{2},\vec{q\,}^{2})$ to $(Q^{2},\vec{q\,}^{2})$ where
$Q^{2}\equiv-\omega^{2}+\vec{q\,}^{2}$. After this substitution, it becomes
transparent that $\vec{q\,}^{2}$ absorbed in $Q^{2}$ does not violate Lorentz
symmetry while the remaining $\vec{q\,}^{2}$, which only appears in the Wilson
coefficients of non-scalar operators, is responsible for the non-trivial
3-momentum effects on the dispersion relation. In this work, OPE is considered
up to dimension 4 gluon operators, i.e. scalar($G_{0}$) and twist-2($G_{2}$)
gluon condensates, which are defined as

$\displaystyle\langle{\frac{\alpha_{s}}{\pi}G_{\mu\alpha}^{a}G_{\nu}^{a,\alpha}}\rangle=\frac{1}{4}g_{\mu\nu}G_{0}+(u_{\mu}u_{\nu}-\frac{1}{4}g_{\mu\nu})G_{2},$
(5)

where the medium four-velocity, denoted as $u_{\mu}$, is taken to be at rest.
Because Wilson coefficients of these operators are already given in a
covariant form for all relevant channels Reinders  _et al._ (1981, 1985);
Klingl  _et al._ (1999); Song  _et al._ (2009), we can simply extract
3-momentum dependent terms after the change of variables.

In the conventional analysis, we typically apply the Borel transform to the
correlator with respect to $Q^{2}$,

$\displaystyle\mathcal{M}^{J}(M^{2},\vec{q\,}^{2})=\lim_{\mathclap{\begin{subarray}{c}n,Q^{2}\to\infty,\\\
{Q^{2}/n=M^{2}}\end{subarray}}}\frac{(Q^{2})^{n+1}}{n!}\left(-\partial_{Q^{2}}\right)^{n}\widetilde{\Pi}^{J}(Q^{2},\vec{q\,}^{2}),$
(6)

where $\mathcal{M}^{J}(M^{2},\vec{q\,}^{2})$ indicates the Borel transformed
correlator with $M$ being the Borel mass parameter. After this transform, the
final expression of OPE result can be written as

$\displaystyle\mathcal{M}^{J}_{\text{OPE}}(M^{2},\vec{q\,}^{2})=$
$\displaystyle
e^{-\nu}A^{J}(\nu)\Big{[}1+\alpha_{s}a^{J}(\nu)+b^{J}(\nu)\phi_{b}$
$\displaystyle+\Big{\\{}c^{J}(\nu)+\frac{\vec{q\,}^{2}}{m_{h}^{2}}d^{J}(\nu)\Big{\\}}\phi_{c}\Big{]},$
(7)

where $\nu=4m_{h}^{2}/M^{2}$,
$\phi_{b}=\frac{4\pi^{2}}{9(4m_{h}^{2})^{2}}G_{0}$, and
$\phi_{c}=\frac{4\pi^{2}}{3(4m_{h}^{2})^{2}}G_{2}$. For input parameters, we
use $m_{c}(p^{2}=-m_{c}^{2})=1.262\,\rm{GeV}$, $\alpha_{s}(8m_{c}^{2})=0.21$
for charmonium states and $m_{b}(p^{2}=-m_{b}^{2})=4.12\,\rm{GeV}$,
$\alpha_{s}(8m_{b}^{2})=0.158$ for bottomonium statesMorita and Lee (2010).
The values of $G_{0}$ and $G_{2}$ at finite temperature are taken from pure
SU(3) lattice gauge theoryBoyd  _et al._ (1996); Morita and Lee (2010). The
Wilson coefficients, $A^{J}(\nu)$, $a^{J}(\nu)$, $b^{J}(\nu)$, and
$c^{J}(\nu)$, have no distinction depending on the polarization states and
their explicit forms can be found in Refs.Bertlmann (1982); Morita and Lee
(2010). In this work, we derive $d^{J}(\nu)$ for the first time and display
their explicit forms in the appendix. The difference in the functional form
between $d^{V,A}_{L}(\nu)$ and $d^{V,A}_{T}(\nu)$ explicitly show why the
longitudinal and transverse modes of spin-1 quarkonium should have different
behaviors in a medium.

By using analyticity of the correlator, one can connect Eq.(7) to the integral
of the spectral function,

$\displaystyle\mathcal{M}^{J}_{OPE}(M^{2},\vec{q}\,^{2})=\int^{\infty}_{-\vec{q}\,^{2}}dse^{-s/M^{2}}\rho^{J}(s,\vec{q}\,^{2}).$
(8)

The spectral function, $\rho^{J}(s,\vec{q}\,^{2})$, is often modeled to have a
single ground state pole and perturbative continuum,

$\displaystyle\rho^{J}(s,\vec{q}\,^{2})\approx$ $\displaystyle
f(\vec{q\,}^{2})\delta(s-m_{g}^{2}(\vec{q}\,^{2}))$
$\displaystyle+\frac{1}{\pi}\text{Im}\widetilde{\Pi}^{J,\text{pert}}(s)\theta(s-s_{0}(\vec{q}\,^{2})),$
(9)

where all non-trivial 3-momentum dependence is assumed to be involved in the
three spectral parameters, $f(\vec{q\,}^{2})$, $m_{g}(\vec{q\,}^{2})$, and
$s_{0}(\vec{q\,}^{2})$, which denote the residue, ground state mass, and
threshold, respectivelyKim and Gubler (2020). The explicit forms of
$\text{Im}\widetilde{\Pi}^{J,\text{pert}}(s)$ can be found in Refs.Reinders
_et al._ (1981); Morita and Lee (2010).

From Eq.(8) and the simple model of spectral function, the ground state mass
for a given temperature and 3-momentum can be expressed as,

$\displaystyle
m_{g}(M^{2},s_{0})=\sqrt{-\frac{\frac{\partial}{\partial(1/M^{2})}\bar{\mathcal{M}}^{J}(M^{2},s_{0})}{\bar{\mathcal{M}}^{J}(M^{2},s_{0})}},$
(10)

where $\bar{\mathcal{M}}^{J}$ $(M^{2},s_{0})$ $=\mathcal{M}^{J}_{OPE}$
$(M^{2},\vec{q}\,^{2})$ $-\int^{\infty}_{s_{0}}$ $ds$ $e^{-s/M^{2}}$
$\rho^{J}(s,\vec{q}\,^{2})$. This equation is reliable only inside a so-called
Borel window, $(M_{min},M_{max})$, which is determined by competition between
convergence of the OPE series and dominance of the pole contribution.
Specifically, we assume the following two conditions:

$\displaystyle M_{min}:\,$
$\displaystyle\Big{|}\alpha_{s}a^{J}(\nu)+b^{J}(\nu)\phi_{b}$
$\displaystyle+\Big{\\{}c^{J}(\nu)+\frac{\vec{q\,}^{2}}{m_{h}^{2}}d^{J}(\nu)\Big{\\}}\phi_{c}\Big{|}\leq
0.3$ (11) $\displaystyle M_{max}:\,$
$\displaystyle\frac{\int^{\infty}_{s_{0}}dse^{-s/M^{2}}\rho^{J}(s,\vec{q}\,^{2})}{\mathcal{M}^{J}_{OPE}(M^{2},\vec{q}\,^{2})}\leq
0.4$ (12)

Furthermore, we define the average value of the mass($\overline{m}_{g}$) and
its uncertainty($\chi^{2}$) within the given Borel window as,

$\displaystyle\overline{m}_{g}(s_{0})$
$\displaystyle=\int_{M_{min}}^{M_{max}}dM\frac{m_{g}(M^{2},s_{0})}{M_{max}-M_{min}},$
(13) $\displaystyle\chi^{2}(s_{0})$
$\displaystyle=\int_{M_{min}}^{M_{max}}dM\frac{(m_{g}(M^{2},s_{0})-\overline{m}_{g}(s_{0}))^{2}}{M_{max}-M_{min}}.$
(14)

Finally, we find a threshold parameter that minimizes the uncertainty by
scanning some range of threshold values. We then take the average mass at this
threshold value as the final result for the ground state mass. This process is
repeated while varying temperature, 3-momentum, channels, polarization states,
and quark flavors.

##  III Results

The ground state mass extracted from the above analysis depends on both the
temperature($T$) and 3-momentum($\vec{q}\,$). In the non-relativistic
limit($|\vec{q}\,|/m_{g}\ll 1$), the energy-momentum dispersion relation can
be expressed as,

$\displaystyle E^{2}-\vec{q}\,^{2}$ $\displaystyle=m_{g}^{2}(T,\vec{q}\,^{2})$
$\displaystyle\approx m_{g}^{2}(T,0)-\alpha(T)\vec{q}\,^{2}+\cdots,$ (15)

where $m_{g}(T,0)$ denotes the rest mass at finite temperature and $\alpha(T)$
indicates the temperature dependent deviation parameter corresponding to the
first-order coefficient of the non-trivial 3-momentum dependence on the ground
state mass. Therefore, we are mainly interested in the signs and magnitudes of
the deviation parameters for various quarkonium states that are moving at
finite temperature. To more intuitively illustrate how much the dispersion
relation is modified in a medium, we also presented the energy ratio which is
defined as

$\displaystyle\frac{E(v)}{E(0)}\approx
1+\frac{1}{2}(1-\alpha(T))v^{2}+\mathcal{O}(v^{4}),$ (16)

where $E(v)$ denotes the energy of a particle moving with finite velocity,
$v=|\vec{q}\,|/E(0)$, with $E(0)$ being equal to the rest mass $m_{g}(T,0)$.
In this work, we studied the $J/\psi$, $\eta_{c}$, $\chi_{c0}$, and
$\chi_{c1}$ of charmonium states, while only the $\Upsilon(1S)$ is studied as
a representative of bottomonium states. For charmonium states, the 3-momentum
is considered up to around 1$\rm{GeV}$, corresponding to $v^{2}\approx 0.1$.
The maximum temperature of $J/\psi$ is 1.05$T_{c}$, while the others are
considered up to 1.03$T_{c}$. In the case of $\Upsilon(1S)$, the temperature
is considered up to 1.4$T_{c}$ and the maximum 3-momentum is 4$\rm{GeV}$,
corresponding to $v^{2}\approx 0.18$.

###  III.1 $J/\psi$ and $\chi_{c1}$ : Longitudinal vs Transverse

Firstly, we examined the difference between the longitudinal and transverse
modes of $J/\psi$. Their energy ratios are shown as a function of $v^{2}$ in
Fig.1. As the black points$(0.9T_{c})$ indicate the vacuum result, it is
confirmed that both modes follow the trivial line, i.e. $1+v^{2}/2$, in the
non-relativistic limit. But, they start to deviate from this line as the
temperature increases. We found that the energies of both modes decrease with
increasing momentum while the transverse mode deviates more than the
longitudinal one. Similarly, it is also observed that the longitudinal mode of
$J/\psi$ experiences smaller medium modification in Ref.Oktay and Skullerud
(2010), but the authors could not see a definite signal for deviation due to
large uncertainties. Furthermore, we can also extract the deviation parameter
from fitting the energy ratio results with Eq.(16). In Fig.2 we plot deviation
parameters of both modes together as a function of temperature. It can be seen
that the difference between the two polarization modes increases with
increasing temperature. In addition, the deviation parameter of $\chi_{c1}$
was extracted in a similar manner as shown in Fig.3. Unlike the $J/\psi$,
however, we observed the opposite behavior for the $\chi_{c1}$, i.e. the
transverse mode has the smaller deviation parameter than the longitudinal one.

Figure 1: Energy ratio of the transverse and longitudinal modes of $J/\psi$ at
various temperatures and velocities. The dashed line shows $1+v^{2}/2$.
Figure 2: Deviation parameter $\alpha(T)$ of the $J/\psi$ Figure 3: Deviation
parameter $\alpha(T)$ of the $\chi_{c1}$

###  III.2 Charmonium : S-wave vs. P-wave

For the next analysis, we compared the magnitude of deviation parameters of
$\eta_{c}({}^{1}S_{0})$ and $\chi_{c0}({}^{3}P_{0})$, and also unpolarized
$J/\psi({}^{3}S_{1})$ and $\chi_{c1}({}^{3}P_{1})$. The comparison results are
shown together in Fig.4. Overall, under the same total spin, it is apparent
that the deviation parameters of P-wave particles$\,(\chi_{c0},\chi_{c1})$ are
greater than those of S-wave particles$\,(\eta_{c},J/\psi)$. This feature
might be originated from the property that the P-wave particles experience
stronger modification than the S-wave particles near $T_{c}$. The similar
property can also be observed in the typical QCDSR analysis for the ground
state masses at zero momentum as illustrated in Fig.5. When comparing rest
masses of the four charmonium states near $T_{c}$, we also observed that the
P-wave particles exhibit larger mass shifts than the S-wave ones. Therefore,
these results suggest that as the rest mass is more susceptible to temperature
changes, the dependence on 3-momentum will also increase.

Figure 4: Deviation parameters compared between the $\eta_{c}$ and $\chi_{c0}$
(above), and between the unpolarized $J/\psi$ and $\chi_{c1}$ (below) Figure
5: Temperature dependencies of rest masses $m_{g}(T,0)$ of the four charmonium
states

###  III.3 Bottomonium : $\Upsilon(1S)$

To estimate an order of magnitude of the deviation parameter for bottomonium
states, we investigated the $\Upsilon(1S)$ as a representative. The energy
ratio of the unpolarized case is shown in Fig.6, but the deviation is hardly
observed even at temperatures much higher than $T_{c}$. This behavior can be
understood from OPE structure. Because $d^{J}(\nu)$ terms in Eq.(7) are highly
suppressed by the heavy quark mass squared, we can expect that bottomonium
states have very small non-trivial 3-momentum dependence compared to the
charmonium states. In Ref.Aarts  _et al._ (2013) the energy ratios of
$\Upsilon(1S)$ and $\eta_{b}$ were studied in a range of $v^{2}\lesssim 0.04$
using lattice NRQCD, but the authors found no clear evidence for the non-
trivial 3-momentum dependence up to 2.09$T_{c}$ within uncertainties. In order
to extract the deviation parameter of $\Upsilon(1S)$, it requires much effort
than charmonium case throughout the numerical analysis because of tiny
variation in masses. The deviation parameters for transverse and longitudinal
modes are shown in Fig.7. As in the $J/\psi$ case, the transverse mode has
larger deviation parameter than the longitudinal one. But the magnitude of
extracted deviation parameter is very small. For example, even at 1.4$T_{c}$
and a momentum of 4$\rm{GeV}$($v^{2}\approx 0.2$), it is expected that the
mass shift caused by the 3-momentum is less than 0.01$\%$. Therefore, future
lattice simulation may need much higher precision in order to detect
meaningful 3-momentum effects on bottomonium states.

Figure 6: Energy ratio of the unpolarized $\Upsilon(1S)$ at various
temperatures and velocities. The dashed line shows $1+v^{2}/2$. Figure 7:
Deviation parameter $\alpha(T)$ of the $\Upsilon(1S)$

##  IV Summary and Conclusions

We investigated the non-trivial 3-momentum effects on the masses of
$\eta_{c}$, $J/\psi$, $\chi_{c0}$, $\chi_{c1}$, and $\Upsilon(1S)$ that are
moving in a hot medium using QCD sum rule approach. This study is achieved by
including the Wilson coefficients listed in the appendix to the typical QCD
sum rule analyses for the quarkonium states at rest. The newly considered
Wilson coefficients are responsible for the non-trivial 3-momentum dependence
on the ground state mass and explicitly show why the longitudinal and
transverse modes in the vector or axialvector channel behave differently in a
medium. The non-trivial 3-momentum effects are characterized by the deviation
parameter $\alpha(T)$ and our analyses revealed that all quarkonium states
experience negative mass shifts as the 3-momentum increases. Specifically, we
found that the transverse polarization mode in the vector channel has a
slightly larger deviation parameter than the longitudinal one, while the
axialvector channel has the opposite behavior. We also found that the P-wave
charmonium states($\chi_{c0}$ and $\chi_{c1}$) experience stronger non-trivial
effects than the S-wave ones($\eta_{c}$ and $J/\psi$). From the OPE structure,
we recognized the non-trivial 3-momentum dependence is suppressed by the heavy
quark mass squared. Indeed, even if we consider a much higher 3-momentum than
used in Ref.Aarts  _et al._ (2013), the $\Upsilon(1S)$ has a very small
deviation parameter compared to charmonium states, e.g. less than a 0.01$\%$
mass shift even at 1.4$T_{c}$ and at a momentum of 4$\rm{GeV}$.

Meanwhile, it should be noted that our analyses relied on the assumption in
which broadening effects are not considered. In fact, since the sum rule gives
a constraint only on the integral of the spectral function, the mass shift and
width broadening have a complementary relationshipLeupold  _et al._ (1998).
Therefore, the mass changes computed in this work could be the maximum values
that can be expected from the QCD sum rule. But the present work directly
observed the non-trivial 3-momentum effects that have not seen yet in the
previous lattice simulations, so it will provide a guideline to judge how much
uncertainty should be reduced in the future lattice simulations to reveal the
genuine 3-momentum effects on the dispersion relation of heavy quarkonium
states at finite temperature.

## Acknowledgements

This work was supported by Samsung Science and Technology Foundation under
Project Number SSTFBA1901-04, and by the Korea National Research Foundation
under the grant number No.2020R1F1A1075963 and No.2019R1A2C1087107.

## Appendix

In this appendix, we provide explicit forms of the Wilson coefficients,
$d^{J}(\nu)$, which are responsible for the non-trivial 3-momentum dependence
for twist-2 gluon operator. The results are represented by the Whittaker
function, $G(b,c,\nu)$, which is defined by

$\displaystyle
G(b,c,\nu)=\frac{1}{\Gamma(c)}\int^{\infty}_{0}dte^{-t}t^{c-1}(\nu+t)^{-b}.$
(17) $\displaystyle d^{S}$
$\displaystyle=\frac{\nu}{18G\left(\frac{3}{2},\frac{5}{2},\nu\right)}\left\\{6G\left(\frac{1}{2},\frac{7}{2},\nu\right)+8G\left(\frac{1}{2},\frac{5}{2},\nu\right)-9G\left(-\frac{1}{2},\frac{7}{2},\nu\right)\right\\},$
(18) $\displaystyle d^{P}$
$\displaystyle=\frac{\nu}{12G\left(\frac{1}{2},\frac{3}{2},\nu\right)}\left\\{2G\left(\frac{1}{2},\frac{7}{2},\nu\right)+8G\left(\frac{1}{2},\frac{5}{2},\nu\right)-G\left(-\frac{1}{2},\frac{7}{2},\nu\right)-2G\left(-\frac{3}{2},\frac{7}{2},\nu\right)\right\\},$
(19) $\displaystyle d_{L}^{V}$
$\displaystyle=-\frac{\nu}{6G\left(\frac{1}{2},\frac{5}{2},\nu\right)}\left\\{6G\left(\frac{1}{2},\frac{7}{2},\nu\right)-6G\left(-\frac{1}{2},\frac{7}{2},\nu\right)+G\left(-\frac{3}{2},\frac{7}{2},\nu\right)\right\\},$
(20) $\displaystyle d_{T}^{V}$
$\displaystyle=\frac{\nu}{12G\left(\frac{1}{2},\frac{5}{2},\nu\right)}\left\\{-6G\left(\frac{1}{2},\frac{7}{2},\nu\right)+8G\left(\frac{1}{2},\frac{5}{2},\nu\right)+3G\left(-\frac{1}{2},\frac{7}{2},\nu\right)-2G\left(-\frac{3}{2},\frac{7}{2},\nu\right)\right\\},$
(21) $\displaystyle d_{L}^{A}$
$\displaystyle=\frac{\nu}{3G\left(\frac{3}{2},\frac{5}{2},\nu\right)}\left\\{G\left(\frac{1}{2},\frac{7}{2},\nu\right)-2G\left(-\frac{1}{2},\frac{7}{2},\nu\right)\right\\},$
(22) $\displaystyle d_{T}^{A}$
$\displaystyle=\frac{\nu}{12G\left(\frac{3}{2},\frac{5}{2},\nu\right)}\left\\{-2G\left(\frac{1}{2},\frac{7}{2},\nu\right)+8G\left(\frac{1}{2},\frac{5}{2},\nu\right)-5G\left(-\frac{1}{2},\frac{7}{2},\nu\right)\right\\}.$
(23)

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