# Point-JEPA: A Joint Embedding Predictive Architecture for Self-Supervised
Learning on Point Cloud

Ayumu Saito
Graphics & Spatial Computing Lab
Saint Mary’s University, Canada
<EMAIL_ADDRESS>Jiju Poovvancheri
Graphics & Spatial Computing Lab
Saint Mary’s University, Canada
<EMAIL_ADDRESS>
###### Abstract

Recent advancements in self-supervised learning in the point cloud domain have
demonstrated significant potential. However, these methods often suffer from
drawbacks, including lengthy pre-training time, the necessity of
reconstruction in the input space, or the necessity of additional modalities.
In order to address these issues, we introduce Point-JEPA, a joint embedding
predictive architecture designed specifically for point cloud data. To this
end, we introduce a sequencer that orders point cloud tokens to efficiently
compute and utilize tokens’ proximity based on their indices during target and
context selection. The sequencer also allows shared computations of the
tokens’ proximity between context and target selection, further improving the
efficiency. Experimentally, our method achieves competitive results with
state-of-the-art methods while avoiding the reconstruction in the input space
or additional modality.

##  1 Introduction

Sef-supervised learning is a paradigm that allows the model to learn a
meaningful representation from unlabeled data. This allows the model to
utilize a vast amount of unlabeled data and learn a strong representation,
leading to advancements in natural language processing [9, 4, 20, 21] and 2D
computer vision [2, 8, 12, 5, 11].

Only recently, we have seen the successful applications of self-supervised
learning in the point cloud domain [17, 29, 27, 28, 7], achieving state-of-
the-art results on downstream tasks. However, our initial investigation found
that they require a significant amount of pre-training time as shown in Figure
1. This slow pre-training process can pose constraints in scaling to a larger
dataset or deeper and more complex models, hindering the key advantage of sel-
supervised learning; its capacity to learn a strong representation from a vast
amount of data.

The recently proposed Joint-Embedding Predictive Architecture [14] and
successful implementations of the architecture for pre-training a model [2, 3]
show JEPA’s ability to learn strong off-the-shelf semantic representations.
The idea behind JEPA is to learn a representation by predicting the embedding
of the input signal, called target, from another compatible input signal,
called context, with the help of a predictor network. This allows learning in
the representation space instead of the input space, leading to efficient
learning.

Figure 1: ModelNet40 Linear Evaluation. Pre-training time on NVIDIA RTX A5500
and overall accuracy with SVM linear classifier on ModelNet40 [26]. We compare
PointJEPA with previous methods utilizing standard Transformer architecture.

Inspired by I-JEPA [2], we aim to apply Joint-Embedding Predictive
Architecture in the point cloud domain. While this application introduces a
promising direction for self-supervised learning in the point cloud domain, it
also presents a unique challenge. Unlike an image, a point cloud is a set of
points with no specific order. Regardless of the permutations applied to the
set, the set presents represents the same object. This implies that the model
consuming $N$ points needs to be invariant to $N!$ permutations of the order
of the input data [18]. This unordered nature of the point cloud data makes
the context and target selection of the data difficult especially if we aim to
select spatially contiguous patches similar to I-JEPA [2]. In order to
overcome this challenge while utilizing the full potential of Joint-Embedding
Predictive Architecture for efficiency, we introduce Point-JEPA. To this end,
we introduce a sequencer that orders the input sequence such that elements
keep the spatial proximity when they are adjacent in the data. This allows the
shared computation for proximity between the target and context, allowing
efficiency while achieving a state of the performance with linear evaluation
on ModelNet40 [26] as shown in Figure 1. Our method also achieves promising
results in other downstream tasks showing the strong and transferable learned
representation.

##  2 Point-JEPA Architecture

Our aim is to bring Joint-Embedding Predictive Architecture to the point cloud
domain while measuring the efficient implementation. The overall framework, as
shown in Figure 2, first converts the point cloud to a set of tokens, then the
sequencer orders the tokens based on the spatial proximity, and Joint-
Embedding Predictive Architecture is applied to the ordered tokens. We utilize
a small PointNet [18] architecture for encoding the grouped points and
standard Transformer [25] architecture for the context and target encoder as
well as the predictor. It is important to note that our JEPA architecture
operates on the token instead of patches in order to share the point encoder
network between context and target encoder for efficiency similar to Point2Vec
[28].

#### Point Cloud Tokenizer.

Similar to previous studies utilizing standard Transformer architecture on
point cloud objects [17, 27, 7, 28], we adopt a point cloud tokenizer that
embeds groups of points. Given a point cloud object, $c$ center points are
first sampled using the farthest point sampling [19]. Then using the
$k$-nearest neighbors algorithm, we sample $k$ closest points to each of $c$
center points. These point patches are then normalized by subtracting the
center point coordinate from the coordinates of the points in the patches.
This allows the separation between local structural information and the
positional information of the patches. In order to embed the local point
patches, we utilize a small PointNet [18] architecture. This ensures that the
patch embedding, or token, remains invariant to any permutations of data
feeding order of points within the patch. Specifically, this PointNet contains
two sets of a shared MLP and a max-pooling layer. First, a shared MLP maps
each point into a feature vector. Then, we apply max-pooling to these vectors
and concatenate the result back to the original feature vector. Another shared
MLP then processes these concatenated vectors and a max-pooling layer is
applied to produce tokens.

#### Sequencer.

Because of the previously observed advantages of spatial proximity in patches
for targets and context, referred to as a block in I-JEPA [2], we aim to
sample tokens that are spatially close to each other. As previously mentioned,
point cloud data is permutation invariant to data feeding order, which implies
that even if the indices of tokens are sequential, they might not be spatially
adjacent. Moreover, our method involves selecting spatially contiguous M
blocks of encoded embedding vectors as the target while ensuring that the
context does not include the tokens corresponding to these embedding vectors
(details in the next paragraph). In order to overcome these challenges, we
introduce a sequencer that is applied after the tokenization of the points.
This sequencer orders tokens based on their associated center points. The
process begins with a chosen center point and its associated token. In each
subsequent step, the center point closest to the one previously chosen and its
associated token are selected. This is iterated until the sequencer visits all
of the center points. The resulting arrangement of tokens is in a sequence
where contiguous elements are also spatially contiguous in most cases. This
allows the shared computation of spatial proximity between context and target
selection. At the same time, this also allows simpler implementation for
context and target selection. It is worth noting, however, that selecting two
adjacent token indices does not always guarantee spatial proximity; there
might be a gap between them. While this is true, the experiment results show
that this iterative ordering is effective enough in our JEPA architecture.

Figure 2: Point-JEPA. The tokenization of point cloud patches is shown on the
left and the joint embedding architecture is shown on the right.

#### Context and Target.

Targets in Point-JEPA can be considered patch-level representations of the
point cloud object, which the predictor aims to predict. As illustrated in
Figure 2, the target encoder initially encodes the tokens conventionally, and
we randomly select M possibly overlapping target blocks, which are sets of
adjacent embedding vectors. We denote these sets as target blocks. Context, on
the other hand, is the representation of the point cloud object which is
passed to the predictor to facilitate the reconstruction of target blocks. In
order to avoid trivial learning, we remove the corresponding tokens of the
target blocks during the context selection step. Out of this subset of tokens,
we select a set of tokens that are adjacent in their indices and denote it as
a context block. Because of this, context may be formed from multiple sets of
spatially contiguous patches while targets often consist of one “block” of
contiguous patches as shown in Figure 3.

#### Predictor and Loss

The task of predictor given targets and context is analogous to the task of
supervised prediction. Given a context as input along with a certain
condition, it aims to predict the target representations. Here, the condition
involves the mask tokens, which are created from shared learned parameters, as
well as positional encoding created from center points associated with the
targets. Because the predictor’s task is to predict the representation
produced by the target encoder, the loss can be defined to minimize the
disagreement between the predictions and targets. To this end, we utilize
Smooth L1 loss to measure the dissimilarity between target embedding and
predicted embedding. The parameters of the target encoder, as well as the
point tokenizer, are updated via gradient-based optimization, while the
parameters of the target encoder are updated via the exponential moving
average of the context encoder.

Figure 3: Context and Targets. We visualize the corresponding grouped points
of context and target blocks. Here, we use (0.15, 0.2) for the target
selection ratio and (0.4, 0.75) for the context selection ratio.

##  3 Experiments

In this section, we first describe the details of self-supervised pre-
training. Following that, we compare the performance of the learned
representation to some of the top-performing self-supervised learning methods
in the point cloud domain, especially those that utilize the ShapeNet [6]
dataset in pre-training. We specifically evaluate the learned representation
using linear probing, end-to-end fine-tuning, and a few-shot learning setting.

###  3.1 Self-Supervised Pre-training

We pre-train our model on training set of ShapeNet [6] following the previous
studies utilizing the standard Transformer [25] architecture such as Point-MAE
[17], Point-M2AE [29], PointGPT [7], and Point2Vec [28]. The dataset consists
of 41952 3D point cloud instances created from synthetic 3D meshes from 55
categories. As previously mentioned, standard Transformer [25] architecture is
used for the context and target encoder as well as the predictor. During pre-
training, we set the number of center points to 64 and the group size to 32.
The point tokenization is applied to the input point cloud containing 1024
points per object. We set the depth of the Transformer in the context and
target encoder to 12 with the embedding width of 384 and 6 heads. For the
predictor, we use the narrower dimension of 192 following I-JEPA [2]. The
depth of the predictor is set to 6, and the number of heads is set to 6.

###  3.2 Downstream Tasks

In this section, we report the performance of the learned representation on
several downstream tasks. Following the previous studies [27, 29, 7, 28], we
report the overall accuracy as a percentage. Unless specified otherwise, we
report the mean accuracy and standard deviation from 10 independent runs, each
with a different seed, in order to account for variability across independent
runs.

#### Linear Probing.

After pre-training on ShapeNet [6], we evaluate the learned representation via
linear probing on ModelNet40 [26]. Specifically, we freeze the learned context
encoder and place the SVM classifier on top. To enforce invariance to
geometric transformation, we utilize max and mean pooling on the output of the
Transformer encoder [17, 28]. Here, we utilize 1024 points for both training
and test sets. As shown in Table 1, our method achieves state-of-the-art
accuracy, providing $+0.8\%$ performance gain, showing the robustness of the
learned representation.

Table 1:  Linear Evaluation on ModelNet40 [26]. We compare Point-JEPA to self-
supervised learning methods pre-trained with point cloud data created from
ShapeNet [6].
* signifies that the result for linear evaluation is not available in the original paper. We cite the results from [29, 30].  methods | Overall Accuracy
---|---
Latent-GAN [1] | 85.7
3D-PointCapsNet [31] | 88.9
STRL [13] | 90.3
Sauder _et al_. [22] | 90.6
Fu _et al_. [10] | 91.4
Transformer-OcCo* [27] | 89.6
Point-Bert* [27] | 87.4
Point-MAE* [17] | 90.0
Point-M2AE [29] | 92.9
Point-JEPA (Ours) | $\bm{93.7}\scriptstyle\pm 0.2$

#### End-to-end Fine-Tuning

We also investigate the performance of the learned representation via end-to-
end fine-tuning. After pre-training, we utilize the context encoder to extract
the max and average pooled outputs. These outputs are then processed by a
three-layer MLP for classification tasks. This class-specific head as well as
the context encoder is fine-tuned end-to-end on ModelNet40 [26] and
ScanObjectNN [23]. ModelNet40 consists of 12311 synthetic 3D objects from 40
distinct categories, while ScanObjectNN contains objects from 15 classes, each
containing 2902 unique instances collected by scanning real-world objects. For
ModelNet40, we sub-sample 1024 points per object and sample 64 center points
with 32 points in each point patch. On the other hand, we utilize all 2048
points for the ScanObjNN dataset and sample 128 center points with 32 nearest
neighbors for the grouped points. As shown in Table 2, our method achieves
competitive results when compared to other state-of-the-art methods.
Especially, in the OBJ-BG variant of the ScanObjNN [23] dataset, which
presents a realistic representation of a point cloud that includes both the
object and its background, our method achieves an improvement of $+1\%$ over
the best-performing method. This shows the learned representation obtained
from pre-training with Point-JEPA can easily be transferred to a
classification task.

Table 2: End-to-End Classification. Overall accuracy on ModelNet40 [26] and ScanObjNN [23] with end-to-end fine-tuning. We specifically compare our methods to the method utilizing standard Transformer architecture pre-trained on ShapeNet )[6] with only point cloud (no additional modality.  | Overall Accuracy
---|---
| ModelNet40 | ScanObjNN
Method | +Voting | -Voting | OBJ-BG
Point-BERT [27] | 93.2 | 92.7 | 87.4
Point-MAE [17] | 93.8 | 93.2 | 90.0
Point-M2AE [29] | 94.0 | 93.4 | 91.2
Point2Vec [28] | 94.8 | 94.7 | 91.2
PointGPT-S [7] | 94.0 | – | 91.6
Point-JEPA (Ours) | $94.1\scriptstyle\pm 0.1$ | $93.8\scriptstyle\pm 0.2$ |  92.9 $\scriptstyle\pm 0.4$
Table 3:  Result of Few-Shot classification on ModelNet40 [26]. 10 independent trials are completed under one setting. We report mean and standard deviation over $10$ trials. | Overall Accuracy
---|---
| 5-way | 10-way
Method | 10-shot | 20-shot | 10-shot | 20-shot
Point-BERT [27] | $94.6\scriptstyle\scriptstyle\pm 3.1$ | $96.3\scriptstyle\pm 2.7$ | $91.0\scriptstyle\pm 5.4$ | $92.7\scriptstyle\pm 5.1$
Point-MAE [17] | $96.3\scriptstyle\pm 2.5$ | $97.8\scriptstyle\pm 1.8$ | $92.6\scriptstyle\pm 4.1$ | $95.0\scriptstyle\pm 3.0$
Point-M2AE [29] | $96.8\scriptstyle\pm 1.8$ | $98.3\scriptstyle\pm 1.4$ | $92.3\scriptstyle\pm 4.5$ | $95.0\scriptstyle\pm 3.0$
Point2Vec [28] | $97.0\scriptstyle\pm 2.8$ | $98.7\scriptstyle\pm 1.2$ | $93.9\scriptstyle\pm 4.1$ | $95.8\scriptstyle\pm 3.1$
PointGPT-S [7] | $96.8\scriptstyle\pm 2.0$ | $98.6\scriptstyle\pm 1.1$ | $92.6\scriptstyle\pm 4.6$ | $95.2\scriptstyle\pm 3.4$
Point-JEPA (Ours) | $\bm{97.4}\scriptstyle\pm 2.2$ | $\bm{99.2}\scriptstyle\pm 0.8$ | $\bm{95.0}\scriptstyle\pm 3.6$ | $\bm{96.4}\scriptstyle\pm 2.7$

#### Few-Shot Learning

We conduct few-shot learning experiments on Modelnet40 [26]. Experiments are
done in $m$-way, $n$-shot setting as shown in Table 3. Specifically, we
randomly sample $n$ instances of $m$ classes for training. We select 20
instances of $m$ support classes for evaluation. Under one setting, we run 10
independent runs under a fixed random seed on 10 different folds of the
dataset, and we report the mean and standard deviation of overall accuracy. As
shown in Table 3, our method exceeds the performance of the current state-of-
the-art in all settings. Our method yields a $+1.1\%$ improvement in the most
difficult 10-way 10-shot setting, showing the robustness of the learned
representation of Point-JEPA, especially in the low-data regime.

##  4 Conclusion

This work introduced Point-JEPA, a joint embedding predictive architecture
applied to point cloud objects. In order to efficiently select targets and
context blocks even under the invariance property of point cloud data, we
introduced a sequencer, which orders the center points and their corresponding
tokens by iteratively selecting the next closest center point. This eliminates
the necessity of computing spatial proximity between every pair of tokens or
embedding vectors when sampling the targets and context. Our method
experimentally achieves state-of-the-art performance in several downstream
tasks, showing a strong learned representation. Specifically, Point-JEPA
excels in few-shot learning as well as linear evaluation, making the method
highly useful when there is a large amount of unlabeled data and a limited
amount of labeled data. It is also worth noting that Point-JEPA converges much
faster during pre-training, offering a more efficient pre-training alternative
in the point cloud domain.

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Supplementary Material

##  A Further Pre-training Details

#### Optimization.

We utilize AdamW [16] optimizer with cosine learning decay [15]. Starting from
learning rate of $10^{-5}$, we increase it to $10^{-3}$ in the first 30 epochs
and decay it to $10^{-6}$. The batch size for pre-training is set to 512, and
the beta value for Smooth L1 loss is set to $2$, similar to Point2Vec [28].
The target encoder and context encoder initially have identical parameters.
The context encoder’s parameters are updated via backpropagation, while the
target encoder’s parameters are updated using the exponential moving average
of the context encoder parameters, that is
$\overline{\theta}_{t+1}\xleftarrow{}\tau\overline{\theta}_{t}+(1-\tau)\theta_{t}$
where $\tau\in[0,1]$ denotes the decay rate, $\overline{\theta}$ denotes the
parameter of the context encoder, and $\overline{\theta}$ denotes the
parameter of the target encoder. We gradually increase the decay rate of the
exponential moving average from 0.995 to 1.0 during pre-training.

#### Masking and Ordering.

To determine the sequence of tokens, we utilize the iterative ordering of
associated center points, as previously mentioned. We chose the starting point
in this sequence with the lowest sum of its coordinates. This method allows
the sequence to start from a point on the outer edge of the object rather than
from a point within the object’s interior. This consistency in selecting the
initial point is experimentally shown to deliver a slightly better-learned
representation than taking the first available index (See Sec. C for the
details).

For masking, we define a range of ratios with both upper and lower limits
similar to I-JEPA [2]. To start with, we clarify that the term “block” refers
to a sequence of tokens and their corresponding encoded embedding vectors that
are contiguous in their indices. Because of the sequencing process applied
before the target and context selection, most contiguous tokens and encoded
embedding vectors are also spatially contiguous. For the target, we randomly
select 4 blocks of embedding vectors from within the 0.15 to 0.2 range. We
then remove the corresponding tokens of encoded embedding vectors that have
already been chosen as targets for further selection. Following this, we
sample tokens within the ratio range of 0.4 to 0.75 out of the available
tokens. Because some of the tokens are not available for context selection, we
note that context block usually consists of multiple sets of tokens that are
spatially contiguous.

##  B Further Results

Following previous studies [27, 29, 28, 17, 7], we report the performance of
Point-JEPA in part segmentation task. Here, we utilize the ShapeNetPart [6]
dataset, consisting of 16881 objects from 16 categories. We utilize the
identical architecture employed in Point2Vec [28] for this task. Specifically,
we take the embeddings from $4^{\text{th}}$, $8^{\text{th}}$, and
$12^{\text{th}}$ Transformer block and take the average of them. Then, we
apply mean and average pooling to this averaged output. The max and mean
pooled embedding along with a one-hot encoded class label of an object is used
as a global feature vector for the object. The original averaged output is
also up-sampled using the PontNet++ [19] feature propagation layer to create a
feature vector for each point. Then, each feature vector is concatenated with
the global feature vector. A shared MLP is utilized on this concatenated
vector to predict the segmentation label for the given point. Although the
Point-JEPA shows competitive results as shown in Table 4, its performance is
slightly worse than the state-of-the-art methods. This, coupled with the fact
that its learned presentation has shown to be superior in classification tasks
suggests that Point-JEPA’s learned representation emphasizes global features
over local features.

Table 4:  Part Segmentation on ShapeNetPart [6]. mIoUC is the mean IoU for all part categories, and mIoUI is the mean IoU for all instances.  Method |  mIoUC |  mIoUI
---|---|---
Transformer-OcCo [27] | 83.4 | 85.1
Point-Bert [27] | 84.1 | 85.6
Point-MAE [17] | 84.1 | 86.1
Point-M2AE [29] | 84.9 | 86.5
Point2Vec [28] | 84.6 | 86.3
PointGPT-S [7] | 84.1 | 86.2
Point-JEPA (Ours) | $83.9\pm 0.1$ | $85.8\pm 0.1$

##  C Ablation

In this section, we conducted thorough ablation studies to understand the
effect of moving parts of Point-JEPA. We pre-train Point-JEPA on the ShapeNet
[6] dataset under various settings and evaluate the learned representation
with linear probing on the ModelNet40 [26] dataset.

#### Masking Strategy.

We investigate the impact of the masking type on the performance. We consider
single-block masking and multi-block masking. For single-block masking
strategies, we consider random masking and contiguous masking. For random
masking, we randomly select the 60% of indices out of all encoded embedding
vectors. Similarly, for contiguous masking, embedding vectors that are
spatially contiguous are selected. In this setting, all tokens not
corresponding to the selected target blocks are used as context (denoted as
rest). On the other hand, in the muli-block masking setting, we sample
multiple possibly overlapping spatially contiguous embedding vectors as
targets, and we remove the corresponding tokens already utilized for targets
during context selection. In this setting, we set the ratio range of 0.15 to
0.2 for targets, while we set the ratio range of 0.4 to 0.75 for context. As
shown in Table 10, the single-block masking achieves sub-optimal performance
regardless of the spatial contiguity of the target embedding. It shows that
our method learns stronger representation by utilizing a smaller amount of
targets with a larger frequency.

#### Ratio of Targets.

We change the ratio of the selected embedding vectors for the target selection
while keeping the number of target blocks and the ratio of context tokens
fixed. As shown in Table 5, the performance increases when you increase the
ratio to a certain point. However, beyond this point, further increasing the
ratio results in decreased performance. This implies that Point-JEPA does not
require a large size for the target blocks and benefits from a sufficient
amount of available tokens for context selection.

Targets | Context |
---|---|---
Ratio | Freq. | Ratio | Modelnet40 Linear
(0.1, 0.2) | 4 | (0.85, 1.0) | 93.0
(0.15, 0.2) | 4 | (0.85, 1.0) | 93.3
(0.2, 0.25) | 4 | (0.85, 1.0) | 93.2
(0.25, 0.3) | 4 | (0.85, 1.0) | 92.4
(0.3, 0.35) | 4 | (0.85, 1.0) | 90.5
(0.35, 0.4) | 4 | (0.85, 1.0) | 84.6
Table 5:  Ratio Range for Target. The ratio of encoded embedding vectors
selected for each target.

#### Ratio of Context.

In this study, we change the ratio of tokens selected for context encoding
while keeping the number of targets and the ratio range for targets fixed. As
shown in Table 6, having a relatively large difference between the lower and
upper bound of the ratio can improve performance. In other words, Point-JEPA
learns a better representation when the number of selected context tokens
varies more between training iterations. Additionally, when the upper bound of
the ratio is somewhat constrained, we see increased performance.

Targets | Context |
---|---|---
Ratio | Freq. | Ratio | Modelnet40 Linear
(0.15, 0.2) | 4 | (0.85, 1.0) | 93.1
(0.15, 0.2) | 4 | (0.75, 1.0) | 92.8
(0.15, 0.2) | 4 | (0.65, 1.0) | 93.4
(0.15, 0.2) | 4 | (0.45, 1.0) | 93.6
(0.15, 0.2) | 4 | (0.6, 0.75) | 93.4
(0.15, 0.2) | 4 | (0.5, 0.75) | 93.1
(0.15, 0.2) | 4 | (0.4, 0.75) | 93.7
Table 6:  Ratio Range for Context. The ratio of tokens selected for context
encoding.

#### Number of Target Blocks.

We will also consider the effect of the number of blocks chosen for targets on
the performance of the learned representation while we keep the ratio for
targets and context fixed. As shown in Table 7, the performance increases as
we increase the number of targets. However, the performance decreases as you
increase the number of target blocks after a specific frequency. Similarly to
what we observed in Section C, we note that our method benefits from having a
sufficient amount of tokens available for context encoding.

Targets | Context |
---|---|---
Ratio | Freq. | Ratio | Modelnet40 Linear
(0.15, 0.2) | 1 | (0.85, 1.0) | 93.0
(0.15, 0.2) | 2 | (0.85, 1.0) | 93.5
(0.15, 0.2) | 3 | (0.4, 0.75) | 93.4
(0.15, 0.2) | 4 | (0.4, 0.75) | 93.7
(0.15, 0.2) | 5 | (0.4, 0.75) | 93.4
(0.15, 0.2) | 6 | (0.4, 0.75) | 93.2
Table 7:  Number of Target blocks. We change the number (frequency) of target blocks while keeping the other components fixed.  Predictor Depth | Modelnet40 Linear
---|---
2 | 92.5
3 | 92.8
4 | 93.2
5 | 93.4
6 | 93.7
Table 8:  Predictor Depth. Predictor depth and its effect on learned
representation.

#### Predictor Depth

We also study the effect of the predictor’s depth on the learned
representation. To this end, we vary the predictor depth and observe its
effect on the linear evaluation accuracy. As shown in Table 8, Point-JEPA
benefits from a deeper predictor.

#### Initial Point for Sequencer

In this section, we study the importance of the initial center point in the
sequencer. We compare a sequencer that takes the initial index of the center
point against another sequencer that takes the minimum sum of the coordinates
as the initial point.

Sequencer Initial Point | Modelnet40 Linear
---|---
Minimum index | 92.7
Minimum Coordinate Sum | 93.7
Table 9:  Initial Point For Sequencer. The initial selected point in the
sequencer and its effect on the learned representation.

As previously mentioned, selecting the first center point presented in the
sequencer does not guarantee the selection from the edge of the object. On the
other hand, selecting the point with minimum coordinate sum at least
guarantees that the starting center point in the sequencer lies near the edge
of the object. As shown in Table 9, taking the point near the edge of the
object helps with learning stronger representation.

#### Learned Representation

In order to qualitatively analyze the learned representation, we reduce the
dimension of the learned representation by utilizing t-SNE [24]. Specifically,
we introduce max and mean pooling on the output of the context encoder,
similar to the classification setup, and apply t-SNE on the pooled embedding.
We visualize the learned representation on ModelNet40 [26] with no fine-tuning
on the dataset. In other words, the context encoder has not been trained with
this data. Despite this, our context encoder produces discriminative features
as shown in Figure 4, showing the robustness of the learned representation.

Targets | Context |  |
---|---|---|---
Strategy | Ratio | Freq. | Strategy | Ratio | Modelnet40 Linear
random | (0.6, 0.6) | 1 | rest | – | 92.5
contiguous | (0.6, 0.6) | 1 | rest | – | 92.3
contiguous | (0.15, 0.2) | 4 | contiguous | (0.4, 0.75) | 93.7
Table 10:  Masking Strategies. Here, we apply multi-block and single-block
masking strategies and see their effect on the learned representation.  Figure
4:  Embedding Visualization on ModelNet40[26]. We visualize the context
encoder’s learned representation with t-SNE [24].


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# On order $\Lambda_{\rm QCD}^{2}/m$ renormalons in quarkonium system

Takuya Agemura and Yukinari Sumino  Department of Physics, Tohoku University,
Sendai, 980–8578 Japan

###### Abstract

For the heavy quarkonium system we examine ${\cal O}(\Lambda_{\rm QCD}^{2}/m)$
renormalons, which are expected to be included in the perturbative series of
the pole mass and $1/(mr^{2})$ interquark potential. We find indications of
existence and cancellation of these renormalons, from examinations of
stability and convergence properties of the perturbative series and their
resummations, as well as by comparison with the known properties of the ${\cal
O}(\Lambda_{\rm QCD})$ renormalons.

††journal: Physics Letters B

TU–1218

##  1 Introduction

The proper treatment of renormalons in theoretical calculations of QCD plays
an important role in recent and near-future high-precision particle physics.
For instance, cancellation of renormalons is incorporated in determinations of
the fundamental physical constants $m_{b}$, $m_{c}$, $\alpha_{s}(M_{Z})$,
$|V_{cb}|$, etc. Renormalons have been actively investigated particularly in
the context of heavy quarkonium systems. In fact, the discovery of the
cancellation of the ${\cal O}(\Lambda_{\rm QCD})$ renormalons between the
twice of the quark pole mass $2m_{\rm pole}$ and the static QCD potential
$V_{\rm{QCD}}(r)$ [1, 2, 3] led to a drastic improvement in the predictability
of the observables of the heavy quarkonium systems. This cancellation is
achieved by expressing the pole mass by a short-distance mass such as the
$\overline{\rm MS}$ mass.

In this Letter we focus on the next-to-leading order renormalons in the heavy
quarkonium system, i.e., ${\cal O}(\Lambda_{\rm QCD}^{2}/m)$ renormalons.
Possible existence of ${\cal O}(\Lambda_{\rm QCD}^{2}/m)$ renormalon in the
quark pole mass has been anticipated and discussed. Known properties are as
follows [4]. (a) It is induced by the non-relativistic kinetic energy operator
$\bm{D}^{2}/(2m)$; (b) It is not forbidden by any symmetry, and parametrically
it induces an ${\cal O}(\Lambda_{\rm QCD}^{2}/m)$ uncertainty; (c) It does not
appear in the large-$\beta_{0}$ approximation. So far, no strong positive
evidence has been reported on the size or effects of this renormalon. The
next-to-leading order renormalon contained in $V_{\rm{QCD}}(r)$ is of order
$\Lambda_{\rm QCD}^{3}r^{2}$ [5, 6] and no renormalon of order $\Lambda_{\rm
QCD}^{2}/m$ is included (since $V_{\rm{QCD}}(r)$ is independent of $m$). The
counterpart which cancels the ${\cal O}(\Lambda_{\rm QCD}^{2}/m)$ renormalon
of $2m_{\rm pole}$ is expected to be included in the potential
$V_{1/r^{2}}(r)\sim-1/(mr^{2})$, which is a part of the non-relativistic
Hamiltonian of the quarkonium system. In fact, by naive power counting, we
expect the size of the renormalon to be

$\displaystyle\delta V_{1/r^{2}}~{}\sim~{}\int_{q\lesssim\Lambda_{\rm
QCD}}\\!\\!\\!\\!d^{3}\bm{q}\,\,\frac{e^{i\bm{q}\cdot\bm{r}}}{m\,q}~{}\sim~{}\frac{{\Lambda_{\rm
QCD}}^{2}}{m}\,,$ (1)

in the case $\Lambda_{\rm QCD}\ll 1/r$, where we used the Fourier integral
representation of $V_{1/r^{2}}(r)$. Moreover, it has been pointed out that
$V_{1/r^{2}}(r)$ induces anomalously large corrections to the quarkonium
spectrum [7], which may be a sign of the ${\cal O}(\Lambda_{\rm QCD}^{2}/m)$
renormalon.

A frequently-used method to investigate renormalons in perturbative QCD is the
so-called “large-$\beta_{0}$ approximation” [8]. Unfortunately it is difficult
to apply this method to the analysis of the ${\cal O}(\Lambda_{\rm
QCD}^{2}/m)$ renormalons of $2m_{\rm pole}$ and $V_{\rm NA}(r)$, since it
would involve difficult calculations of bubble-chain insertions to the
diagrams shown in Fig. 1. Instead, we use the recently developed method for
estimating renormalons, the FTRS [9, 10] and DSRS [11] methods. These methods
utilize the property that the renormalons are suppressed in the Fourier space
or dual space: By renormalization-group (RG) improvement of the perturbative
series in these spaces we can separate the renormalons and the rest, and we
can estimate them individually and systematically as more terms of the
perturbative series are included; At all orders they coincide with the
conventional definition based on the regularized Borel resummation of the
perturbative series. Furthermore, for the reason explained below, we restrict
the analysis to the “maximally non-abelian (MNA) part” of the perturbative
series, corresponding to the part proportional to
$C_{F}{C_{A}}^{\\!\\!n}\alpha_{s}^{n+1}$.

Figure 1: Feynman diagrams relevant to the calculations of the ${\cal
O}(\Lambda_{\rm QCD}^{2}/m)$ renormalons of $2m_{\rm pole}$ and $V_{\rm
NA}(r)$, respectively, in the large-$\beta_{0}$ approximation. Bubble chain
insertions to the gluon propagators of these diagrams need to be computed.

In Sec. 2 we examine $V_{1/r^{2}}$ in the leading-logarithmic (LL)
approximation using the FTRS and DSRS methods. Secs. 3 and 4 deal with beyond-
the-LL approximation. In Sec. 3 we examine the total energy $2m_{\rm
pole}+V_{\rm QCD}+V_{1/r^{2}}$ of the heavy quarkonium system in the fixed
order of the perturbative expansion. In Sec. 4 we examine the sizes of the
${\cal O}(\Lambda_{\rm QCD}^{2}/m)$ renormalons as well as the total energy in
the FTRS and DSRS methods. We give summary and conclusion in Sec. 5.

##  2 ${\cal O}(\Lambda_{\rm QCD}^{2}/m)$ renormalon of $V_{\text{NA}}$ in LL
approximation

The $1/(mr^{2})$ potential in the non-relativistic Hamiltonian of the
quarkonium system is known up to the order $\alpha_{s}^{3}$ [12] within the
framework of potential-NRQCD effective field theory [13]. This potential
appears first at order $\alpha_{s}^{2}$ and its explicit form (in one operator
basis) is given by

$\displaystyle V_{1/r^{2}}=\frac{\alpha_{s}^{2}}{m_{\rm
pole}r^{2}}\Biggl{[}\frac{C_{F}(C_{F}-2C_{A})}{4}+\frac{C_{F}\alpha_{s}}{\pi}\Biggl{\\{}\frac{130C_{A}C_{F}-101C_{A}^{2}+(49C_{A}-8C_{F})T_{F}n_{l}}{72}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\frac{\beta_{0}}{4}(C_{F}-2C_{A})(\gamma_{E}+\log(\mu
r))-\frac{2}{3}C_{A}(C_{A}+2C_{F})(\log 2+2\gamma_{E}+2L_{\rm
US})\Biggr{\\}}\Biggr{]}\,,$ (2)
$\displaystyle\beta_{0}=\frac{11}{3}C_{A}-\frac{4}{3}T_{F}n_{l}\,,~{}~{}~{}~{}~{}~{}~{}L_{\rm
US}=\log(\mu_{\rm US}r)\,,$ (3)

where the color factors are given by $C_{F}=4/3$, $C_{A}=3$ and $T_{F}=1/2$;
$n_{l}$ denotes the number of light quark flavors;
$\alpha_{s}=\alpha_{s}(\mu)$ denotes the strong coupling constant of the
theory with $n_{l}$ quark flavors only in the $\overline{\rm MS}$ scheme.

Because the Hamiltonian itself is not a direct physical observable, there
arise two problems in analyzing this potential. One problem is that the
$1/(mr^{2})$ potential mixes with other operators in the Hamiltonian under
unitary transformations which do not affect physical observables [14].111
Since the leading-order Hamiltonian includes the Coulomb potential
$-C_{F}\alpha_{s}/r$, the potential proportional to
$(C_{F}\alpha_{s})^{2}/(mr^{2})$ changes its coefficient, e.g., by the unitary
transformation with a generator proportional to
$iC_{F}\alpha_{s}\bm{p}\cdot\bm{x}/(mr)$.  This feature gives a certain
ambiguity to the potential to be analyzed for the renormalon cancellation. The
same problem does not occur for the MNA part of the $1/(mr^{2})$ potential
(the part proportional to $C_{F}{C_{A}}^{\\!\\!n}\alpha_{s}^{n+1}$).222 This
is the case when the unitary transformation keeps the leading-order
Hamiltonian unchanged and the generator of the transformation is not enhanced
by $C_{A}$. There is hardly any reason to consider such an enhanced
transformation, since the tree-level quark-antiquark scattering amplitude does
not have a part proportional to $C_{A}$ in the color-singlet channel.  Below,
we analyze the MNA part, since this part remains invariant under unitary
transformations and is expected to reflect most vividly the characteristics of
the IR renormalons. We call the MNA part of $V_{1/r^{2}}(r)$ as the “non-
abelian (NA) potential” for short and denote it by $V_{\rm NA}(r)$.

The other problem is the existence of an IR divergence. $V_{1/r^{2}}$ or
$V_{\rm NA}(r)$ is IR divergent at order $\alpha_{s}^{3}$ and beyond. In the
above explicit form, the IR divergence has been subtracted in the
$\overline{\rm MS}$ scheme in momentum space. Associated to the IR divergence,
it has an IR logarithmic term represented by $L_{\rm US}$. This term is
canceled by the UV logarithmic term associated with the UV divergence of the
ultra-soft (US) contribution. Thus, in a physical observable, $L_{\rm US}$
does not appear but is replaced by logarithms of typical US scales. In the
case of the perturbative series of the quarkonium spectrum, the physical
logarithmic terms include the “QCD Bethe logarithm” [15] which is analogous to
the Bethe logarithm in the Lamb shift. Up to now, there are no known examples
where those US contributions give large effects to physical observables;
rather their contributions are always with moderate size. Hence, in our
analysis below, we either replace $\mu_{\rm US}$ by a typical IR scale and
vary it in a reasonable range, or replace the US logarithm by 1 or 0 for the
sake of an order-of-magnitude estimate.

Now we estimate the size of the ${\cal O}(\Lambda_{\rm QCD}^{2}/m)$ renormalon
of the NA potential using the FTRS method and the LL approximation. Then we
demonstrate that the convergence of the perturbative series of the NA
potential improves by subtraction of the renormalon.

The FTRS method utilizes the fact that IR renormalons are suppressed by
Fourier transformation. As a result, in momentum space convergence is expected
to be good. For this reason, it would make sense to approximate the NA
potential in momentum space by its LL approximation. Thus, we set

$\displaystyle\tilde{V}_{\text{NA}}^{\text{LL}}(q)$
$\displaystyle=-\frac{\pi^{2}C_{A}C_{F}}{m_{\text{pole}}q}{\alpha_{s}(q^{2})}^{2}$
(4)
$\displaystyle\simeq-\frac{\pi^{2}C_{A}C_{F}}{m\,q}\sum_{n=0}^{\infty}(n+1)\left[\frac{\beta_{0}}{4\pi}\log\biggl{(}\frac{\mu^{2}}{q^{2}}\biggr{)}\right]^{n}\alpha_{s}(\mu^{2})^{n+2}\,,$

where (only in this section)
$\alpha_{s}(q^{2})=4\pi/[\beta_{0}\log(q^{2}/\Lambda_{\overline{\rm
MS}}^{2})]$ denotes the one-loop running coupling constant and we set
$n_{l}=0$ to take the MNA part. Within our approximation it does not make a
difference whether we use the pole mass $m_{\rm pole}$ or the $\overline{\rm
MS}$ mass $m\equiv m_{\overline{\rm MS}}(m_{\overline{\rm MS}})$, and we use
the latter in the second line. The NA potential in position space is obtained
by inverse Fourier transformation of each term of the series expansion in
$\alpha_{s}(\mu^{2})$:

$\displaystyle V_{\text{NA}}^{\text{LL}}$
$\displaystyle=\sum_{n=0}^{\infty}C_{n+1}\alpha_{s}(\mu^{2})^{n+2}\,,$ (5)

where

$\displaystyle C_{n+1}$
$\displaystyle=-\frac{\pi^{2}C_{A}C_{F}}{m}{\biggl{(}\frac{\beta_{0}}{4\pi}\biggr{)}}^{n}\int\frac{d^{3}\bm{q}}{(2\pi)^{3}}\frac{e^{i\bm{q}\cdot\bm{r}}}{q}(n+1)\left[\log\left(\frac{\mu^{2}}{q^{2}}\right)\right]^{n}.$
(6)

The Borel transform of $V_{\text{NA}}^{\text{LL}}$ is given by

$\displaystyle B_{V_{\rm NA}^{\rm
LL}}(u)=\sum_{n=0}^{\infty}\frac{C_{n+1}}{(n+1)!}\left(\frac{4\pi
u}{\beta_{0}}\right)^{n+1}=-\frac{4\pi^{3}C_{A}C_{F}}{m\beta_{0}}u\int\frac{d^{3}\bm{q}}{(2\pi)^{3}}\frac{e^{i\bm{q}\cdot\bm{r}}}{q}\left(\frac{\mu^{2}}{q^{2}}\right)^{u}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}=-\frac{2\pi^{3/2}C_{A}C_{F}}{m\beta_{0}r^{2}}\,\left(\frac{\mu^{2}r^{2}}{4}\right)^{u}\hskip
3.0pt\cfrac{u\,\Gamma(1-u)}{\Gamma(\frac{1}{2}+u)}.$ (7)

Thus, there are poles at $u=1,2,3,\cdots$ in the Borel plane. The standard
prescription for the Borel resummation is to regularize the inverse Borel
transform by deforming the integral contour of the Borel integral into the
upper or lower half $u$-plane. The size of the ${\cal O}(\Lambda_{\rm
QCD}^{2}/m)$ renormalon is defined as the imaginary part of the regularized
Borel integral around $u=1$. Hence, it can be calculated from the residue of
the pole at $u=1$ as

$\displaystyle[\delta V_{\text{NA}}^{\text{LL}}]_{u=1}^{\mathrm{Borel}}$
$\displaystyle=\frac{2\pi}{i\beta_{0}}\oint_{-C_{1}}du\left(-\frac{2\pi^{3/2}C_{A}C_{F}}{m\beta_{0}r^{2}}\right)\left(\frac{\mu^{2}r^{2}}{4}\right)^{u}\hskip
3.0pt\cfrac{\Gamma(2-u)}{\Gamma(\frac{1}{2}+u)}\frac{u}{1-u}e^{-\tfrac{4\pi}{\beta_{0}\alpha_{s}}u}$
(8)
$\displaystyle=-\frac{4\pi^{3}C_{A}C_{F}}{{\beta_{0}}^{2}}\frac{{\Lambda_{\overline{\mathrm{MS}}}}^{2}}{m}.$

Thus, the $u=1$ renormalon of the NA potential is independent of $r$.

In the FTRS method, there is a shortcut to calculate the same quantity
(regularized Borel resummation) bypassing the Borel transform. In the inverse
Fourier transformation, we deform the integral path to circumvent the pole of
$\alpha_{s}(q)$. Then the renormalon can be extracted as the imaginary part of
the regularized inverse Fourier integral. Thus,

$\displaystyle[\delta V_{\text{NA}}^{\text{LL}}]^{\text{FTRS}}=-\frac{\pi
C_{A}C_{F}}{2mr}\oint_{C}\frac{dq}{2\pi i}\sin(qr)\alpha_{s}(q)^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=-\frac{\pi
C_{A}C_{F}}{2mr}\oint_{C}\frac{dq}{2\pi
i}\left[qr-\frac{1}{6}(qr)^{3}+\cdots\right]\alpha_{s}(q)^{2}\,,$ (9)
$\displaystyle[\delta
V_{\text{NA}}^{\text{LL}}]^{\text{FTRS}}_{u=1}=-\frac{\pi
C_{A}C_{F}}{2m}\oint_{C}\frac{dq}{2\pi
i}\,q\,\alpha_{s}(q)^{2}=-\frac{4\pi^{3}C_{A}C_{F}}{{\beta_{0}}^{2}}\frac{{\Lambda_{\overline{\mathrm{MS}}}}^{2}}{m}\,,$
(10)

where $C$ is the loop surrounding the pole of $\alpha_{s}(q)$ at
$q=\Lambda_{\overline{\rm MS}}$. The linear term in the expansion of
$\sin(qr)$ is taken to obtain the $u=1$ renormalon. The result in eq. (10)
indeed matches the calculation by the Borel resummation method in eq. (8).

The asymptotic behavior of the perturbative series induced by the $u=1$
renormalon can be extracted from the pole at $u=1$ of the Borel transform in
eq. (7) as

$\displaystyle B_{V_{\rm NA}^{\rm
LL}}(u)\sim-\frac{2\pi^{3/2}C_{A}C_{F}}{m\beta_{0}r^{2}}\,\frac{\mu^{2}r^{2}}{4}\,\cfrac{1}{\Gamma({3}/{2})}\,\frac{u}{1-u}\
=-\frac{\pi C_{A}C_{F}}{m\beta_{0}}\mu^{2}\sum_{n=0}^{\infty}u^{n+1}.$ (11)

Therefore, the order ${\alpha_{s}^{n+2}}$ term of $V^{\text{LL}}_{\text{NA}}$
behaves asymptotically as

$\displaystyle V_{\text{NA}}^{\text{LL},(n)}$
$\displaystyle\sim-\frac{C_{A}C_{F}{\alpha_{s}}^{2}}{4m}\mu^{2}\left(\frac{\beta_{0}\alpha_{s}}{4\pi}\right)^{n}(n+1)!.$
(12)

This $r$-independent (constant-potential-like) behavior represents the
contribution from the $u=1$ renormalon. Fig. 2(a) shows the truncated
perturbative expansion of the NA potential in the LL approximation [eq. (5)],
and Fig. 2(b) shows the potential after subtracting the renormalon
contribution eq. (12). Each line represents the NA potential up to order
${\alpha_{s}^{N+2}}$, with the input parameters $\mu=3$ GeV,
$\Lambda_{\overline{\rm MS}}=0.298$ GeV,333 This corresponds to
$\alpha_{s}(2\,{\rm GeV})=0.3$, which we use throughout this Letter as a
reference effective coupling close to that of the bottomonium states.
$n_{l}=0$, and $m=4.19$ GeV [$\alpha_{s}(\mu)=0.247$].

(a) (b)

Figure 2: (a) NA potential in the LL approximation by the FTRS method,
expanded in $\alpha_{s}(\mu)$ and truncated at order ${\alpha_{s}^{N+2}}$. (b)
The same potential after subtraction of the $u=1$ renormalon contributions.
Note that the scale of the vertical axis in (b) is magnified by a factor of
two compared to (a).

In Fig. 2(a), as the perturbative order increases, the potential shifts
downwards almost by a constant independent of $r$, indicating a lack of
convergence by the renormalon contributions [cf., eq. (12)]. On the other
hand, in Fig. 2(b), after subtracting the $u=1$ renormalon contributions, the
convergence is notably improved. This demonstrates that removing the $u=1$
renormalon enhances the convergence of the perturbative expansion. These
features are qualitatively similar to the behaviors of the QCD potential
before and after subtraction of the ${\cal O}(\Lambda_{\rm QCD})$ ($u=1/2$)
renormalon. (See Figs. 11 of Ref. [16].)

Next, we use the DSRS method to calculate the renormalon of
$V_{\text{NA}}^{\text{LL}}(r)$ defined by eqs. (5) and (6). While at all
orders the DSRS method gives the same result as the regularized Borel
resummation or the FTRS method, its systematic improvement is different from
the FTRS method as higher-order terms are included. We choose the parameters
of the DSRS method as $(a,u^{\prime})=(1,-1)$ which suppress the renormalons
at $u=1,2,3,\cdots$ in the dual space. Following the prescription of Ref.
[11], we obtain

$\displaystyle V_{\text{NA}}^{\text{LL}}(r)$
$\displaystyle=\int_{0}^{\infty}dp^{2}\,e^{-r^{2}p^{2}}\,\,\overline{V}_{\text{NA}}^{\text{LL}}(p^{2})\,,$
(13)

where the NA potential in the dual space is given by

$\displaystyle\overline{V}_{\text{NA}}^{\text{LL}}(p^{2})=-\frac{2\pi
C_{A}C_{F}}{\beta_{0}m}\sum_{n=0}^{\infty}(n+1)!\,\tilde{a}_{n+1}\,\left(\frac{\beta_{0}}{4\pi}\right)^{n+1}\alpha_{s}(p^{2})^{n+2}\,,$
(14)
$\displaystyle\sum_{n=0}^{\infty}\tilde{a}_{n+1}u^{n+1}=\frac{u\,\Gamma(1+u)}{\Gamma(1+2u)}\,.$
(15)

After truncating the series expansion in eq. (14) at an arbitrary order, we
deform the integral path of eq. (13) into the upper or lower-half complex
$p^{2}$ plane to circumvent the pole of $\alpha_{s}(p^{2})$. Similarly to the
FTRS method, the size of the $u=1$ renormalon is obtained as

$\displaystyle[\delta V^{\text{LL}}_{\text{NA}}]^{\text{DSRS}}_{u=1}$
$\displaystyle=-\frac{8\pi^{3}C_{A}C_{F}}{{\beta_{0}}^{2}m}\sum_{n=0}^{\infty}\tilde{a}_{n+1}(n+1)!\,\oint_{C}\frac{dp^{2}}{2\pi
i}\,\frac{1}{[\log(p^{2}/{\Lambda_{\overline{\mathrm{MS}}}}^{2})]^{n+2}}$ (16)
$\displaystyle=-\frac{8\pi^{3}C_{A}C_{F}}{{\beta_{0}}^{2}}\frac{{\Lambda_{\overline{\mathrm{MS}}}}^{2}}{m}\sum_{n=0}^{\infty}\tilde{a}_{n+1}.$

The sum over $n$ converges. By setting $u=1$ in eq. (15), we find
$\sum_{n=0}^{\infty}\tilde{a}_{n+1}=\frac{1}{2}$ and the previous result is
reproduced.

Let us examine how the size of the renormalon obtained by the DSRS method
converges towards the result of the Borel resummation or the FTRS method. Fig.
3(a) compares the values obtained by truncating the summation in eq. (16) at
$n=k$ with the $u=1$ renormalon calculated by the FTRS method (normalized to
unity). At low orders, the truncated sum oscillates rather rapidly. As the
order increases, the sum converges towards the value of FTRS slowly. From the
order $n=7$ the truncated sum starts to approximate the FTRS value with better
than 10 per cent accuracy. For comparison, Fig. 3(b) shows the similar
calculation of the $u=1/2$ renormalon in the QCD potential in the LL
approximation using the DSRS method. The size of the $u=1/2$ renormalon is
also normalized to unity. Note that the vertical-axis scales differ by a
factor of 4 between the two figures. It can be observed that the convergence
speed of the $u=1$ renormalon in $V^{\text{LL}}_{\text{NA}}$ is considerably
slower compared to the $u=1/2$ renormalon of the QCD potential.

(a) (b)

Figure 3: (a) Size of the $u=1$ renormalon of $V_{\rm NA}^{\rm LL}$ in the
DSRS method as we vary the truncation order. (b) Size of the $u=1/2$
renormalon of $V_{\rm QCD}^{\rm LL}$ in the DSRS method as we vary the
truncation order. Both series are normalized by the exact values. Note that
the scale of the vertical axis in (b) is magnified by a factor of 4 compared
to (a).

It can be shown that the expansion coefficients of the $u=u_{*}$ renormalon in
the DSRS method includes the factor $u_{*}^{n}$ as $n$ is varied. Hence, in
general the convergence tends to become worse as the renormalon is located
further away from the origin in the Borel plane.

We have also examined the real part of the resummed potential $V_{\rm NA}^{\rm
LL}$ in the FTRS and DSRS methods. The FTRS method gives the Borel resummed
potential in the principal-value (PV) prescription and the DSRS method gives a
series which converges to it, with our current construction of the potential.
The former potential approximates well the $N=7$ truncated potential in Fig.
2(b) (up to an additive constant) in the displayed range, which corresponds to
the most convergent line in this figure (truncation at the minimal term). This
feature is consistent with the similar property observed for the QCD potential
[17]. For the DSRS potential, the convergence is fairly slow in the displayed
range. The convergence is better at smaller $r$. More explicitly, evidently
converging behavior can be seen only at $r\lesssim 1~{}{\rm GeV}^{-1}$ and for
$k\geq 8$. We have to calculate up to higher $k$ to observe a sign of
convergence at larger $r$. On the other hand, we observe a better convergence
at smaller distances $r<0.5~{}{\rm GeV}^{-1}$.

It is difficult to perform a similar analysis for the $u=1$ renormalon of the
pole mass. This is because the $u=1$ renormalon has a subleading structure
$\alpha_{s}^{2}\times[\beta_{0}\alpha_{s}\log(\mu/m)]^{n}$, which resides in
between the more dominant leading logarithmic structure
$\alpha_{s}\times[\beta_{0}\alpha_{s}\log(\mu/m)]^{n}$, and it is difficult to
extract the $u=1$ renormalon as clearly as the NA potential. Instead we will
perform a more involved analysis in Sec. 4 to assess the $u=1$ renormalon of
the pole mass.

##  3 Beyond LL approximation: Fixed-order analysis of $2m_{\rm pole}$,
$V_{\rm QCD}$ and $V_{\rm NA}$

In this section we examine the perturbative series of the total energy of the
quarkonium system, $E_{\text{tot}}=2m_{\rm pole}+V_{\rm QCD}+V_{\rm NA}$, and
discuss cancellation of renormalons among $2m_{\rm pole}$, $V_{\rm QCD}$ and
$V_{\rm NA}$. We examine the MNA part of their exact perturbative expansions
in $\alpha_{s}(\mu^{2})$ up to order $\alpha_{s}^{3}$ (fixed-order analysis).

Formally at all orders of perturbative expansion, the dependence of an
observable on the renormalization scale $\mu$ disappears. Therefore, if the
perturbative series converges, $\mu$ dependence decreases as the order of the
perturbative expansion increases. This is not necessarily the case if the
perturbative expansion shows poor convergence. In this way, convergence and
stability with respect to the scale $\mu$ of the perturbative expansion are
closely related. Below, we examine the $\mu$ dependence of the total energy
$E_{\text{tot}}$ and consider the effects of renormalon cancellations.

It is believed that, in the quarkonium system, the $u=1/2$ renormalons cancel
between $2m_{\rm pole}$ and $V_{\rm QCD}$. We consider it plausible that also
the $u=1$ renormalons cancel out in the MNA part of the entire expression of
$E_{\text{tot}}=2m_{\text{pole}}+V_{\text{QCD}}+V_{\text{NA}}$. Since the
$u=1$ renormalon is subdominant, to observe the effects of the cancellation of
this renormalon, it is necessary to cancel the dominant $u=1/2$ renormalons.
To clarify the role of each renormalon contribution, we compare the following
three cases: (I) $E_{\text{pole}}=2m_{\rm pole}+V_{\rm QCD}+V_{\rm NA}$
without rewriting $m_{\rm pole}$, (II) $E_{{1/r}}=2m_{\rm pole}+V_{\rm QCD}$
after rewriting $m_{\rm pole}$ by the $\overline{\rm MS}$ mass $m$, (III)
$E_{\text{full}}=2m_{\rm pole}+V_{\rm QCD}+V_{\rm NA}$ after rewriting $m_{\rm
pole}$ by the $\overline{\rm MS}$ mass $m$. It is understood that the MNA part
is taken, and in addition we also retain the order $\alpha_{s}^{0}$ term
($2m_{\rm pole}$ in $E_{\text{pole}}$ or $2m$ in $E_{1/r}$, $E_{\text{full}}$)
for convenience. We expect that the following renormalons remain uncanceled in
each case: (I) ${\cal O}(\Lambda_{\rm QCD})$ and ${\cal O}(\Lambda_{\rm
QCD}^{2}/m)$, (II) ${\cal O}(\Lambda_{\rm QCD}^{2}/m)$, (III) None, up to
$u=1$.

In the cases (II) and (III) we use the MNA part of the pole mass expressed by
the $\overline{\rm MS}$ mass [18],

$\displaystyle m_{\rm pole}^{\rm
MNA}=m\,\Biggl{[}1+\frac{C_{F}\alpha_{s}(m^{2})}{\pi}\biggl{\\{}1+\frac{C_{A}\alpha_{s}(m^{2})}{\pi}\left(\frac{1111}{384}-\frac{\pi^{2}}{12}+\frac{\pi^{2}}{4}\log
2-\frac{3}{8}\zeta(3)\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\Bigl{(}\frac{C_{A}\alpha_{s}(m^{2})}{\pi}\Bigr{)}^{2}\biggl{(}\frac{1322545}{124416}+\frac{1955\pi^{2}}{3456}+\frac{179\pi^{4}}{3456}+\frac{115\pi^{2}}{72}\log
2-\frac{11\pi^{2}}{36}\log^{2}2$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\frac{11}{72}\log^{4}2-\frac{11}{3}\rm{Li}_{4}\Bigl{(}\frac{1}{2}\Bigr{)}-\frac{1343}{288}\zeta(3)-\frac{51\pi^{2}}{64}\zeta(3)+\frac{65}{32}\zeta(5)\biggr{)}\biggr{\\}}\Biggr{]}\,.$
(17)

We re-emphasize that $\alpha_{s}$ denotes the coupling constant of the theory
with $n_{l}$ quark flavors only. For $V_{\rm NA}$ we use the MNA part of eq.
(2). The MNA part of the QCD potential is given by [19]

$\displaystyle V_{\rm QCD}^{\rm
MNA}=-\frac{C_{F}\alpha_{s}(r^{-2})}{r}\Biggl{[}1+\frac{C_{A}\alpha_{s}(r^{-2})}{\pi}\left(\frac{31}{36}+\frac{11}{6}\gamma_{E}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\Bigl{(}\frac{C_{A}\alpha_{s}(r^{-2})}{\pi}\Bigr{)}^{2}\biggl{(}\frac{4343}{2592}+\frac{247}{54}\gamma_{E}+\frac{121}{36}\gamma_{E}^{2}+\frac{229\pi^{2}}{432}-\frac{\pi^{4}}{64}+\frac{11}{24}\zeta(3)\biggr{)}\Biggr{]}\,.$
(18)

We rewrite $\alpha_{s}(m^{2})$ and $\alpha_{s}(r^{-2})$ by
$\alpha_{s}(\mu^{2})$ using the two-loop relation and re-expand the
perturbative series by $\alpha_{s}(\mu^{2})$, in which we set $n_{l}=0$ to
maintain consistency with the MNA part.

Using the above expressions, we compare the $\mu$ dependences of
$E_{\text{pole}}$, $E_{1/r}$ and $E_{\text{full}}$ in Figs. 4(a) and (b). The
input parameters are taken as $\alpha_{s}(2\,{\rm GeV})=0.3$, $m_{\rm pole}=5$
GeV in $E_{\text{pole}}$, and $m=4.19$ GeV in $E_{\text{full}}$. We vary
$\mu_{\rm US}=0.5$ GeV and $1$ GeV in $E_{\text{full}}$. The interquark
distance $r$ is fixed at $0.6~{}\text{GeV}^{-1}$ and $1~{}\text{GeV}^{-1}$ in
Figs. 4(a) and (b), respectively.

(a) (b)

Figure 4:  The total energies $E_{\text{pole}}$, $E_{1/r}$ and
$E_{\text{full}}$ vs. $\mu$ at (a) $r=0.6~{}\text{GeV}^{-1}$, (b)
$r=1~{}\text{GeV}^{-1}$. The green (red) line represents $E_{\text{full}}$
with $\mu_{\rm US}=0.5~{}{\rm GeV}^{-1}$ ($1~{}{\rm GeV}^{-1}$). Horizontal
dashed lines are shown as a guide.

We observe that the $\mu$ dependence decreases in the order $E_{\text{pole}}$,
$E_{1/r}$ and $E_{\text{full}}$. This is reasonable with regard to the
expectation that the $u=1/2$ and $u=1$ renormalons get canceled in this order.
This order of the $\mu$ dependence of the three total energies is unchanged
for $0.5~{}{\rm GeV}^{-1}\lesssim r\lesssim 2.5~{}{\rm GeV}^{-1}$. At shorter
distances, however, the order of $\mu$ dependence between $E_{1/r}$ and
$E_{\text{full}}$ is reversed and the $\mu$ dependence of $E_{1/r}$ becomes
the smallest. These features agree with the following consideration. The
effects of the renormalons are expected to be more prominent at IR, so that
the ordering of the $\mu$ dependence according to the expected residual
renormalons would be most naturally realized at larger $r$. On the other hand,
at short distances $r\ll\Lambda_{\overline{\rm MS}}^{-1}$, $V_{\rm QCD}$ and
$V_{\rm NA}$ are mostly perturbative. Hence, $V_{\rm NA}\propto 1/r^{2}$ is
most enhanced and its $\mu$ dependence is also magnified at the same time.
This explains why at shorter distances the order of $\mu$ dependence between
$E_{1/r}$ and $E_{\text{full}}$ is reversed.

Let us fix $\mu$ to the scale where $E_{\text{full}}$ becomes least sensitive
to $\mu$ (the minimal sensitivity scale) and compare the perturbative series
of the three total energies. For $r=0.6~{}{\rm GeV}^{-1}$,
$dE_{\text{full}}/d\mu$ vanishes at $\mu=4.28$ GeV, where the perturbative
series are given by

$\displaystyle E_{\text{pole}}=10.00-0.447-0.358-0.423\hskip
10.0pt\text{GeV}\,,$ (19) $\displaystyle
E_{1/r}\,\,=8.380+0.268+0.150-0.085\hskip 10.0pt\text{GeV}\,,$ (20)
$\displaystyle E_{\rm full}\,\,=8.380+0.268+0.096-0.012\hskip
10.0pt\text{GeV}\,.$ (21)

We set $\mu_{\rm US}=1$ GeV in $E_{\text{pole}}$ and $E_{\text{full}}$. Thus,
the convergence becomes better in the order $E_{\text{pole}}$, $E_{1/r}$,
$E_{\text{full}}$, as expected. We also confirm that the $\mu$ dependence of
$E_{\text{full}}$ is reduced in a healthy manner as more terms of the
perturbative expansion are included. The qualitative features are the same
also at $r=1~{}{\rm GeV}^{-1}$ and/or with the choice $\mu_{\rm US}=0.5$ GeV.

##  4 Beyond LL approximation: FTRS and DSRS analysis

|  | $\delta V_{\mathrm{NA}}$ | $2\,\delta m_{\rm pole}^{\rm MNA}$ | $\delta V_{\mathrm{NA}}+2\,\delta m_{\rm pole}^{\rm MNA}$ | $R$
---|---|---|---|---|---
FTRS | ${\cal O}(\alpha_{s}^{2})$ | $0.0091$ | $-0.0045$ | $~{}\,0.0046$ | $0.66$
| ${\cal O}(\alpha_{s}^{3})$ | $0.0062/0.0164$ | $-0.0243$ | $-0.0182/-0.0079$ | $0.41/0.81$
DSRS | ${\cal O}(\alpha_{s}^{2})$ | $0.1091$ | $-0.1005$ | $\,~{}0.0086$ | $0.96$
| ${\cal O}(\alpha_{s}^{3})$ | $0.172/0.294$ | $-0.0667$ | $~{}\,0.105/0.228$ | $0.56/0.37$
Table 1:  Estimates of the $u=1$ renormalons in GeV units at each order of the
expansions in the FTRS and DSRS methods. The input parameters are $m=4.19$ GeV
and $\alpha_{s}(2\,{\rm GeV})=0.3$. The parameters of the FTRS and DSRS
methods are chosen as $(a,u^{\prime})=(2,-1/2)$. For ${\cal
O}(\alpha_{s}^{3})$ of $\delta V_{\mathrm{NA}}$, $\log(\mu_{\rm
US}r)+\gamma_{E}$ is set to one or zero. $R$ represents the level of
cancellation of the $u=1$ renormalons [eq. (22)].

We estimate the sizes of the $u=1$ renormalons of $V_{\mathrm{NA}}$ and
$2\,m_{\rm pole}^{\rm MNA}$ using the FTRS and DSRS methods with their exact
perturbative series up to order $\alpha_{s}^{3}$. Namely, we use the order
$\alpha_{s}(q^{2})^{k}$ and $\alpha_{s}(p^{2})^{k}$ ($k=2,3$) expressions of
the perturbative series (with three-loop running coupling constant) in the
Fourier space and dual space, respectively. We choose the parameters of the
FTRS and DSRS methods as $(a,u^{\prime})=(2,-1/2)$ which suppress the
renormalons at $u=1/2,1,3/2,2,\cdots$ in the Fourier space and dual space,
since the pole mass includes the $u=1/2$ renormalon as well. The result is
summarized in Tab. 1. For estimating the US effects, $\log(\mu_{\rm
US}r)+\gamma_{E}$ is set to one or zero before taking the Fourier transform or
dual transform. In the last column,

$\displaystyle R=1-\frac{|\delta V_{\mathrm{NA}}+2\,\delta m_{\rm pole}^{\rm
MNA}|}{|\delta V_{\mathrm{NA}}|+|2\,\delta m_{\rm pole}^{\rm MNA}|}$ (22)

represents the level of cancellation of the $u=1$ renormalons between
$V_{\mathrm{NA}}$ and $2\,m_{\rm pole}^{\rm MNA}$. Because we can use only the
first two terms for $V_{\rm NA}$ and because the convergence is expected to be
slow (see Sec. 2), we expect to observe only a limited signal of renormalons
and their cancellation. In fact, in both methods we hardly see any sign of
convergence for the estimates of the sizes of the renormalons. Moreover, the
results are fairly dependent on our treatment of the US effects of $V_{\rm
NA}$. On the other hand, we see a tendency of partial cancellation between
$\delta V_{\mathrm{NA}}$ and $2\delta m_{\rm pole}^{\rm MNA}$ at each order.
This tendency is expected in the case that the convergence improves by the
cancellation.444 In Tab. 1 we choose common $(a,u^{\prime})$ for $\delta
V_{\mathrm{NA}}$ and $2\,\delta m_{\rm pole}^{\rm MNA}$, because only in that
case partial cancellation at each order is expected.  (The high cancellation
rate for the DSRS order $\alpha_{s}^{2}$ case is presumably accidental,
considering the size of each term and the convergence behavior in the LL
approximation.)

Next we calculate the real part (PV part) of $V_{\rm QCD}^{\rm MNA}+V_{\rm
NA}$ in the FTRS method and compare it with the fixed-order total energy
$E_{\rm full}$ calculated in the previous section. Since by looking into small
$r$ region we can detect UV region compared to the renormalons (typically at
$\Lambda_{\rm QCD}$ scale), we expect a better convergence behavior than
above. The comparison is shown in Fig. 5.

Figure 5:  Comparison of $V_{\rm QCD}^{\rm MNA}+V_{\rm NA}$ in the FTRS method
(black lines) and $E_{\text{full}}$ in the fixed-order perturbative series
(colored lines) as functions of $r$. An arbitrary $r$-independent constant is
added to the former. The latter lines are for $\mu=4,3,2,1.5$ GeV from left to
right. In the former potential $\log(\mu_{\rm US}r)+\gamma_{E}$ is set to one
or zero and the result changes only slightly. In the latter $\mu_{\rm US}$ is
set to 1 GeV.

We choose $(a,u^{\prime})=(1,-1/2)$ for $V_{\rm QCD}^{\rm MNA}$ and
$(a,u^{\prime})=(1,-1)$ for $V_{\rm NA}$ such that the renormalons of the
individual potentials are suppressed minimally. (These correspond to the
ordinary Fourier transformation of both potentials.) We add an $r$-independent
constant to the FTRS result to facilitate the comparison. For estimating the
US effects, $\log(\mu_{\rm US}r)+\gamma_{E}$ is set to one or zero before the
Fourier transform, by which $V_{\rm QCD}^{\rm MNA}+V_{\rm NA}$ changes the
shape only slightly in the displayed range. We see that, at short distances,
the FTRS result reproduces well the shape of $E_{\rm full}$ for a larger
$\mu$, while at larger distances it reproduces the shape of $E_{\rm full}$ for
a smaller $\mu$. In general, the convergence of the potentials are better at
smaller $r$ both for the FTRS and fixed-order calculations, and
$r\Lambda_{\overline{\rm MS}}$ would be reasonably smaller than unity in the
displayed range. Hence, we observe a plausible feature in this comparison.

We have also examined $V_{\rm QCD}^{\rm MNA}+V_{\rm NA}$ in the DSRS method.
At $r\lesssim 0.2~{}{\rm GeV}^{-1}$, the DSRS result agrees well with the FTRS
result, while at larger distances it deviates from the FTRS result (and also
from the fixed-order result). In view of the analysis in Sec. 2 this indicates
that the convergence of the DSRS result is quite slow compared to the FTRS
result.

##  5 Summary and Conclusion

In Sec. 2 we have seen that in the LL approximation of the FTRS method $V_{\rm
NA}(r)$ includes the ${\cal O}(\Lambda_{\rm QCD}^{2}/m)$ renormalon. In Sec. 3
we have seen that the fixed-order perturbative series of the total energy of
the quarkonium system improves stability against scale variation and
convergence as we incorporate possible cancellation of the ${\cal
O}(\Lambda_{\rm QCD})$ and ${\cal O}(\Lambda_{\rm QCD}^{2}/m)$ renormalons
successively. In Sec. 4 we have seen that the FTRS and DSRS estimates of the
${\cal O}(\Lambda_{\rm QCD}^{2}/m)$ renormalons of $V_{\mathrm{NA}}$ and
$2\,m_{\rm pole}^{\rm MNA}$ from the known first two terms do not show a sign
of convergence, while there is a tendency of partial cancellation between
them. On the other hand, the FTRS estimate of the PV part of $V_{\rm QCD}^{\rm
MNA}(r)+V_{\rm NA}(r)$ shows a reasonable agreement with the fixed-order total
energy at $r<2.5~{}{\rm GeV}^{-1}$.

These analysis results are mutually consistent and also consistent with the
hypothesis that both $2m_{\rm pole}$ and $V_{1/r^{2}}$ include the ${\cal
O}(\Lambda_{\rm QCD}^{2}/m)$ renormalons and that they cancel each other.
Since there is no argument which forbids existence of the ${\cal
O}(\Lambda_{\rm QCD}^{2}/m)$ renormalons, we interpret our analysis results to
be positive indications for their existence and cancellation.

The analysis in Sec. 3 indicates that already in the current status there is a
possibility to improve the accuracy of the theoretical prediction for the
quarkonium energy by incorporating cancellation of the ${\cal O}(\Lambda_{\rm
QCD}^{2}/m)$ renormalons. Note that the current calculations of the heavy
quarkonium spectrum do not incorporate this cancellation mechanism due to the
use of the $\epsilon$ expansion [20]. A proper account of this cancellation
may improve accuracy of the charm and bottom quark mass determinations from
the charmonium and bottomonium $1S$ energy levels [21, 22], for instance.

## Acknowledgement

The work of Y.S. was supported in part by JSPS KAKENHI Grant Number
JP23K03404.

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