# STRPM: A Spatiotemporal Residual Predictive Model for High-Resolution Video
Prediction

Zheng Chang1,2, Xinfeng Zhang1, Shanshe Wang3, Siwei Ma3, and Wen Gao1,3
1School of Computer Science and Technology,
University of Chinese Academy of Sciences, Beijing, China
2Institute of Computing Technology, Chinese Academy of Sciences, Beijing,
China
3National Engineering Research Center of Visual Technology,
School of Computer Science, Peking University, Beijing, China
<EMAIL_ADDRESS><EMAIL_ADDRESS>{sswang, swma,
<EMAIL_ADDRESS>Corresponding author: Shanshe Wang<EMAIL_ADDRESS>This
work was supported in part by the National Natural Science Foundation of China
(62025101, 62072008, 62071449, U20A20184), National Key Research and
Development Project of China (2019YFF0302703, 2021YFF0900503) and High-
performance Computing Platform of Peking University, which are gratefully
acknowledged.

###### Abstract

Although many video prediction methods have obtained good performance in low-
resolution (64$\sim$128) videos, predictive models for high-resolution
(512$\sim$4K) videos have not been fully explored yet, which are more
meaningful due to the increasing demand for high-quality videos. Compared with
low-resolution videos, high-resolution videos contain richer appearance
(spatial) information and more complex motion (temporal) information. In this
paper, we propose a Spatiotemporal Residual Predictive Model (STRPM) for high-
resolution video prediction. On the one hand, we propose a Spatiotemporal
Encoding-Decoding Scheme to preserve more spatiotemporal information for high-
resolution videos. In this way, the appearance details for each frame can be
greatly preserved. On the other hand, we design a Residual Predictive Memory
(RPM) which focuses on modeling the spatiotemporal residual features (STRF)
between previous and future frames instead of the whole frame, which can
greatly help capture the complex motion information in high-resolution videos.
In addition, the proposed RPM can supervise the spatial encoder and temporal
encoder to extract different features in the spatial domain and the temporal
domain, respectively. Moreover, the proposed model is trained using generative
adversarial networks (GANs) with a learned perceptual loss (LP-loss) to
improve the perceptual quality of the predictions. Experimental results show
that STRPM can generate more satisfactory results compared with various
existing methods.

##  1 Introduction

Video prediction is a key component of representation learning due to its
great ability in modeling meaningful representations for natural videos and
has been applied to various video processing applications, such as video
coding [14], precipitation nowcasting [20], robotic control [6], autonomous
driving [2] and so on. Different from video interpolation [17, 16], video
prediction (extrapolation) is more challenging by merely utilizing limited
information from previous frames to predict the unknown future frames.
Motivated by the advantages of deep learning technologies in extracting deep
features, in recent years, various learning-based methods have been proposed
for video prediction which can be summarized into three types.

Figure 1: Qualitative results between the proposed STRPM and the state-of-the-
art method CrevNet [29] on the SJTU4K dataset (4K: 2160$\times$3840
resolution, 4 frames $\rightarrow$ 1 frame). STRPM has generated much better
visual details compared with CrevNet.

The first type of methods [20, 22, 26, 25, 29, 27] utilize Recurrent neural
networks (RNN) to progressively predict video frames due to their unique
advantages in sequence learning and have obtained some satisfactory results.
However, the predictions from RNN-based methods are typically blurry due to
the standard mean square error based loss function. To solve this problem, the
second type of methods [1, 5, 23, 7, 28] utilize deep stochastic models to
predict different futures instead of an averaged future for different samples
and the third type of methods [8, 15, 13, 12, 4] employ generative adversarial
networks (GANs) [8] and additional perceptual loss functions to augment the
visual qualities of the predictions.

Although the above methods have obtained some satisfactory results, the
resolutions of video datasets utilized in the above methods are usually low
(64$\sim$128), and the performance in high-resolution (512$\sim$4K) videos is
still hardly satisfactory (shown in Figure 1), preventing their adaptability
and practicability into real scenarios. There are mainly two challenges
restricting the resolution of predictions. The first challenge is that high-
resolution videos usually contain more complex visual details. However,
limited by the computation resources, videos are usually encoded to low-
dimensional features and then decoded back to the video frames, during which,
lots of visual details can be abandoned. The second challenge is that the
motion information in high-resolution videos typically involves multiple
objects, which is much more complex and hard for traditional predictive memory
to predict. To deal with the above two problems in high-resolution video
prediction, the appearance information in the spatial domain and the motion
information in the temporal domain need to be carefully reconsidered.

In this paper, we propose a Spatiotemporal Residual Predictive Model (STRPM)
to deal with the above two challenges. Firstly, to predict more satisfactory
appearance details for each frame, we novelly propose the spatiotemporal
encoding-decoding scheme, which utilizes independent encoders to extract deep
features in both spatial and temporal domains. In this way, both the spatial
and temporal information can no longer affect each other and more visual
details can be preserved. Secondly, to accurately model the complex motion
information in high-resolution videos, we designed a Residual Predictive
Memory (RPM) to focus on modeling the inter-frame spatiotemporal residual
features (STRF) with a relatively low computation load and fewer parameters.
Moreover, since the encoded spatial and temporal features will be fed into the
spatial and temporal modules in RPM for transitions in vertical (spatial
domain) and horizontal (temporal domain) directions, the RPM can indirectly
supervise the spatial encoder and the temporal encoder to extract
corresponding features in the spatial domain and the temporal domain.

By jointly using the encoded spatiotemporal features and STRF, more reliable
spatiotemporal features for future frames can be predicted, which will be
further decoded back to the high-dimensional data space with the help of
spatiotemporal decoders. Furthermore, in the training stage, the standard MSE
loss, the adversarial loss as well as the learned perceptual loss are jointly
utilized to improve the visual quality of the predictions. Experimental
results show that the proposed model can achieve state-of-the-art performance
compared with other methods.

##  2 Related Work

In recent years, many learning-based predictive models have been applied in
video prediction. [18] first utilized language modeling for video prediction,
which was further improved by [22] using Long Short-Term Memories (LSTMs) [9],
denoted as FC-LSTM. To improve the model perception to visual data, [20]
integrated convolutional operations to FC-LSTMs (ConvLSTM) and achieved
significant improvements on the Moving MNIST dataset.

However, the above works only focus on the inter-frame temporal information
(motion information) and ignored intra-frame spatial information (appearance
information). To preserve the appearance information for videos, [26] designed
an appearance-preserving block for ConvLSTM (PredRNN). [24] further improved
PredRNN by solving the gradient propagation difficulties in deep predictive
models (PredRNN++) and integrating 3D convolution operations and RECALL gate
to enhance the ability to capture both long-term and short-term dependencies
of the predictive model (E3D-LSTM). To further improve the visual quality of
the predictions, [29] proposed a conditionally reversible network (CrevNet) to
preserve the spatiotemporal information for the inputs and [11] leveraged the
high-frequency information to preserve visual details for videos.

However, the above works can only generate an averaged future for all samples
due to the standard MSE loss function and the predictions are usually blurry.
To solve this problem, a variety of methods have been proposed. On the one
hand, some methods aim to predict different futures for different samples. [1]
proposed a stochastic variational video prediction (SV2P) method to predict a
different possible future for each sample based on the latent variables. [5,
28] proposed video generation models with a learned prior over stochastic
latent variables for video prediction. [7] proposed a stochastic temporal
model for video prediction whose dynamics are governed in a latent space by a
residual update rule.

Figure 2: The structure of the proposed spatiotemporal residual predictive
model (STRPM). The green arrows denote the input information flows and the red
arrows denote the predicted information flows.

On the other hand, some works aim to improve the standard MSE-based loss
function. [15] proposed three different and complementary feature learning
strategies to predict naturalistic videos. Besides, motivated by the great
power in generating naturalistic images, generative adversarial networks
(GANs) were employed by [13] to generate realistic results and [12] utilized
CycleGAN [31] to further improve the perceptual quality of the predictions.
Although some improvements have been achieved in above works and the
resolutions of the predicted videos have been improved in [29, 3]
(256$\sim$512), the unacceptable computation load and quality distortions
prevent them from predicting videos with higher resolution (512$\sim$4K). To
solve the above problems, we propose a spatiotemporal residual predictive
model (STRPM) for high-resolution video prediction with an acceptable
computation load. Moreover, by using the proposed learned perceptual loss,
more naturalistic videos can be generated from the proposed method.

##  3 The Spatiotemporal Residual Predictive Model

In this section, we introduce the proposed Spatiotemporal Residual Predictive
Model (STRPM) in detail. The overall structure of the proposed model is shown
in Figure 2. Different from low-resolution videos, high-resolution videos
contain more complex texture details and more variable motion information,
motivated by which, two problems for high-resolution video prediction urgently
need to be solved:

  * •

How to preserve more visual details for each frame?

  * •

How to predict more accurate motion information between frames?

We propose the Spatiotemporal Residual Predictive Model (STRPM) to solve the
above problems.

Figure 3: The structure of the proposed residual predictive memory: RPM. The
temporal module and the spatial module can indirectly supervise the temporal
encoder and the spatial encoder to extract different features in the temporal
domain and the spatial domain.

###  3.1 Spatiotemporal Encoding-Decoding Scheme

To reduce the computation resources, video frames are typically encoded to
low-dimensional features using a single encoder in video prediction. However,
the temporal information and the spatial information will affect each other
and the predictive memories have to further extract the temporal and spatial
information to predict future frames, during which, lots of spatiotemporal
information may be lost, making it very difficult to reconstruct satisfactory
visual details for each frame. To solve this problem (the first problem), we
novelly utilize multiple spatiotemporal encoders to independently extract deep
features in both temporal and spatial domains. In this way, the spatial
information and the temporal information will no longer affect each other,
making it easier for the predictive memories to utilize the spatiotemporal
information for video prediction. The encoding process can be expressed as
follows,

$(T_{E},S_{E},O_{E})=(Enc_{T}(v_{t}),Enc_{S}(v_{t}),Enc_{O}(v_{t})),$ (1)

where $v_{t}$ denotes the $t^{th}$ frame in source video $V$.
$Enc_{T}(\cdot),Enc_{S}(\cdot),Enc_{O}(\cdot)$ denote the temporal, spatial
and spatiotemporal encoders respectively. $T_{E},S_{E},O_{E}$ denote the
encoded low-dimensional temporal, spatial and spatiotemporal features at time
step $t$, respectively.

In particular, the above encoded features $T_{E},S_{E},O_{E}$ will be fed into
the corresponding modules in the proposed residual predictive memory: RPM,
which will be detailedly introduced in section 3.2. In this way, the RPM can
indirectly supervise different encoders to extract different features in
different domains. And the predicted spatiotemporal features can be
represented as follows,

$(T_{P},S_{P},O_{P})=RPM(T_{E},S_{E},O_{E},\mathcal{T},\mathcal{S}),$ (2)

where $T_{P},S_{P},O_{P}$ denote the predicted temporal, spatial and
spatiotemporal features at time step $t$ from RPM , respectively.
$\mathcal{T},\mathcal{S}$ are the preserved temporal and spatial information.

Similar as the encoding process, to decode more spatiotemporal details, we
also utilize multiple spatiotemporal decoders to decode the predicted features
from low-dimensional feature space back to high-dimensional temporal and
spatial data space respectively, which can be expressed as follows,

$(T_{D},S_{D},O_{D})=(Dec_{T}(T_{P}),Dec_{S}(S_{P}),Dec_{O}(O_{P})),$ (3)

where $Dec_{T}(\cdot),Dec_{S}(\cdot),Dec_{O}(\cdot)$ denote the temporal,
spatial and spatiotemporal decoders, respectively. $T_{D},S_{D},O_{D}$ denote
the decoded high-dimensional temporal, spatial and spatiotemporal features,
respectively.

By jointly utilizing the decoded high-dimensional features, the predicted
frame at time step $t$ can be represented as follows,

$\hat{v}_{t+1}=O_{D}\odot\tanh(W_{1\times 1}\ast[T_{D},S_{D}]),$ (4)

where $\hat{v}_{t+1}$ denotes the predicted frame at time step $t$,
$\odot,\ast$ denote the hadamard product and convolutional operators.

###  3.2 The Residual Predictive Memory: RPM

Current predictive memories aim to predict future frames by learning a single
representation for the whole frame, containing both the spatial representation
and the temporal representation, which is efficient to predict simple motions.
However, high-resolution videos usually contained much more complex motion
information compared with low-dimensional videos. To deal with this special
characteristic in high-resolution videos (the second problem), we design a
Residual Predictive Memory (RPM) to focus on modeling inter-frame motion
information by predicting the spatiotemporal residual features (STRF) between
previous and future frames in the feature space, which is shown in Figure
3(a). In addition, compared with traditional ST-LSTM structure [26], the
proposed RPM also benefits from more efficient state-to-state transitions
(fewer gates) and wider spatiotemporal receptive field (simultaneously
utilizing multiple spatiotemporal states). To further extract more efficient
deep spatiotemporal features, multiple RPMs are typically stacked into a
single model, as shown in Figure 3(b). For RPM at time step $t$ in layer $k$,
the encoded features $T_{E},S_{E},O_{E}$ are fed into the corresponding
modules of RPM. In this way, the proposed RPM can indirectly supervise the
spatial encoder and the temporal encoder to extract different deep features in
the spatial domain and the temporal domain. In particular, for $k>1$, the
encoded features are represented with the hidden state from the previous
layer, i.e., $T_{E},S_{E},O_{E}=H_{t}^{k-1}$.

For each RPM, there are seven inputs: $T_{E}$, the encoded features for the
temporal module; $S_{E}$, the encoded features for the spatial module;
$O_{E}$, the encoded spatiotemporal features for the output module;
$H_{t-1}^{k}$, the hidden state from previous time step; $S_{t}^{k-1}$, the
previous spatial state from previous layer $k-1$;
$\mathcal{T}:T_{t-\tau:t-1}^{k}$, the previous $\tau$ temporal states;
$\mathcal{S}:S_{t}^{k-\theta:k-1}$, the previous $\theta$ spatial states. To
further improve local perception to videos, the input states are typically
preprocessed using convolutional layers:

$\displaystyle(TF_{t}^{k},SF_{t}^{k},OF_{t}^{k})$ $\displaystyle=$
$\displaystyle(W_{t}\ast T_{E},W_{s}\ast S_{E},W_{o}\ast O_{E}),$
$\displaystyle(HF_{t}^{k},MF_{t}^{k})$ $\displaystyle=$
$\displaystyle(W_{h}\ast H_{t-1}^{k},W_{m}\ast S_{t}^{k-1}),$ (5)

where $W$ denotes the parameters of the integrated convolutional layers.
$TF_{t}^{k}$, $SF_{t}^{k}$, $OF_{t}^{k}$, $HF_{t}^{k}$, $MF_{t}^{k}$ denote
the extracted deep features from $T_{E}$, $S_{E}$, $O_{E}$, $H_{t-1}^{k}$,
$S_{t}^{k-1}$, respectively. Then the extracted features will be fed into RPM
at time step $t$ in layer $k$.

For both the temporal and spatial module, two residual gates are designed to
model the inter-frame residual information, which is shown as follows,

$\displaystyle R_{T}$ $\displaystyle=\sigma(TF_{t}^{k}+HF_{t}^{k}),$
$\displaystyle R_{S}$ $\displaystyle=\sigma(SF_{t}^{k}+MF_{t}^{k}),$ (6)

where $R_{T},R_{S}$ denote the temporal and spatial residual gates,
respectively.

As shown in Figure 3(a), the temporal module (blue block) of RPM is utilized
to capture reliable motion information between frames. To preserve more useful
temporal information from the past, RPM jointly utilizes multiple temporal
states, and the transitions can be expressed as follows,

$\displaystyle T_{t}^{k}$ $\displaystyle=$ $\displaystyle
R_{T}\odot(\tanh(TF_{t}^{k}+HF_{t}^{k})+ATT_{T}(\mathcal{T})).$ (7)

The predicted temporal residual state $T_{t}^{k}$ consists of two terms, where
the first term $R_{T}\odot\tanh(TF_{t}^{k}+HF_{t}^{k})$ represents the encoded
features from current input and the second term $R_{T}\odot
ATT_{t}(\mathcal{T})$ represents the preserved temporal information from
previous $\tau$ time steps. In this way, more useful temporal information can
be kept from a longer past. In particular, $ATT_{T}(\cdot)$ denotes the
temporal attention network which is constructed with convolutional layers and
can help merge the multiple temporal states to a single one.

In the spatial module (orange block), by utilizing the multiple spatial states
$\mathcal{S}:S_{t}^{k-\theta:k-1}$, both low-level texture information and
high-level semantic information can be jointly utilized, and similar to the
temporal module, the state-to-state transitions can be represented as follows,

$\displaystyle S_{t}^{k}$ $\displaystyle=$ $\displaystyle
R_{S}\odot(\tanh(SF_{t}^{k}+MF_{t}^{k})+ATT_{S}(\mathcal{S})),$ (8)

where $S_{t}^{k}$ denotes the predicted spatial residual state and $ATT_{S}$
denotes the spatial attention network.

The predicted temporal residual state $T_{t}^{k}$ and the predicted spatial
residual state $S_{t}^{k}$ will be further aggregated to the final hidden
state in the output module (gray block):

$\displaystyle R_{O}$ $\displaystyle=$
$\displaystyle\sigma(OF_{t}^{k}+HF_{t}^{k}+W_{os}\ast S_{t}^{k}+W_{ot}\ast
T_{t}^{k}),$ $\displaystyle STRF_{t}$ $\displaystyle=$ $\displaystyle
R_{O}\odot\tanh(W_{1\times 1}\ast[T_{t}^{k},S_{t}^{k}]),$ $\displaystyle
STIF_{t}$ $\displaystyle=$ $\displaystyle W_{1\times 1}\ast[T_{E},S_{E}],$
$\displaystyle H_{t}^{k}$ $\displaystyle=$ $\displaystyle
STIF_{t}^{k}+STRF_{t}^{k},$ (9)

where $R_{O}$ denotes the output residual gate, which is utilized to aggregate
the predicted temporal and spatial residual information. $H_{t}^{k}$ denotes
the final hidden state. In particular, the hidden state $H_{t}^{k}$ consists
of two terms, where the first term $STIF_{t}^{k}$ denotes the spatiotemporal
input features and the second term $STRF_{t}^{k}$ denotes the predicted
spatiotemporal residual features between previous and future frames.

###  3.3 Training Details

In training stage, to predict more naturalistic results, the proposed model is
trained with the help of GANs and the whole model consists of two submodules:
predictor $P$ which is utilized to generate future frames and discriminator
$D$ which is utilized to judge whether the input frames are real or generated.
The adversarial loss for both modules can be expressed as follows,

$\displaystyle\mathcal{L}_{GAN}(D)$ $\displaystyle=$
$\displaystyle-\sum_{t=2}^{T}[\log(D(v_{t}))+\log(1-D(\hat{v}_{t}))],$
$\displaystyle\mathcal{L}_{GAN}(P)$ $\displaystyle=$
$\displaystyle-\sum_{t=2}^{T}[\log(D(\hat{v}_{t})),$ (10)

where $T$ denotes total number of the time steps. $v$ and $\hat{v}$ denote the
input and predicted frames respectively.

Since the discriminators in GANs can model the distribution of the input data
(fake or real), we utilize the feature map from the layer $k$ of the
discriminator as the learned perceptual representations for current input. And
a learned perceptual loss, which can indicate the perceptual distribution of
the inputs, is represented as follows (Figure 2),

$\mathcal{L}_{LP}=\sum_{t=2}^{T}\mathcal{L}_{2}[D_{k}(v_{t}),D_{k}(\hat{v}_{t})],$
(11)

where $D_{k}$ denotes the $k^{th}$ layer of the discriminator $D$ (the bottom
layer in our method). $\mathcal{L}_{2}(\cdot)$ denotes the standard MSE loss
function. By using the additional loss functions, more naturalistic results
can be predicted and the final loss function for the predictor can be
expressed as follows,

$\mathcal{L}_{P}=\mathcal{L}_{MSE}+\lambda_{1}\mathcal{L}_{LP}+\lambda_{2}\mathcal{L}_{GAN}(P),$
(12)

where $\lambda_{1},\lambda_{2}$ control the relative importance.

##  4 Experiments

In this section, we evaluate all models on three high-resolution datasets, UCF
Sports dataset ($480\times 720$) [19], Human3.6M dataset ($1000\times 1000$)
[10] and SJTU4K dataset ($2160\times 3840$) [21]. We stack 16 RPMs to the
proposed STRPM and the integrated convolutional operations are set with a
kernel size $5\times 5$. The stride is set to 1 for each dimension. We set the
number of previous spatiotemporal states $\tau$, $\theta$ to 5. The hidden
states for STRPM and the discriminator are set with 128 channels. All models
are implemented using Pytorch and trained with Adam optimizer. In the training
stage, models are trained to predict the next frame with 4 successive frames
as the input on all datasets. In the testing stage, models are evaluated to
predict multiple frames. The balance weights $\lambda_{1}$, $\lambda_{2}$ are
set to 0.01, 0.001 for UCF Sports and Human3.6M datasets, and 0.005, 0.0005
for SJTU4K dataset.

Figure 4: The generated examples on the UCF Sports test set (4 frames
$\rightarrow$ 6 frames).

###  4.1 UCF Sports Dataset

Table 1: Quantitative results of different methods on the UCF Sports (4 frames $\rightarrow$ 6 frames) and Human3.6M (4 frames $\rightarrow$ 4 frames) datasets. Lower LPIPS($10^{-2}$) and higher PSNR(dB) scores indicate better results. Method | UCF Sports | Human3.6M
---|---|---
$t=5$ | $t=10$ | $t=5$ | $t=8$
PSNR$\uparrow$ / LPIPS$\downarrow$ | PSNR$\uparrow$ / LPIPS$\downarrow$ | PSNR$\uparrow$ / LPIPS$\downarrow$ | PSNR$\uparrow$ / LPIPS$\downarrow$
BeyondMSE (ICLR2016) [15] | 26.42 / 29.01 | 18.46 / 55.28 | - | -
PredRNN (NeurIPS2017) [26] | 27.17 / 28.15 | 19.65 / 55.34 | 31.91 / 12.62 | 25.65 / 14.01
PredRNN++ (ICML2018) [24] | 27.26 / 26.80 | 19.67 / 56.79 | 32.05 / 13.85 | 27.51 / 14.94
SAVP (arXiv 2018) [13] | 27.35 / 25.45 | 19.90 / 49.91 | - | -
SV2P (ICLR2018) [1] | 27.44 / 25.89 | 19.97 / 51.33 | 31.93 / 13.91 | 27.33 / 15.02
HFVP (NeurIPS2019) [23] | - | - | 32.11 / 13.41 | 27.31 / 14.55
E3D-LSTM (ICLR2019) [25] | 27.98 / 25.13 | 20.33 / 47.76 | 32.35 / 13.12 | 27.66 / 13.95
CycleGAN (CVPR2019) [12] | 27.99 / 22.95 | 19.99 / 44.93 | 32.83 / 10.18 | 28.26 / 11.03
CrevNet (ICLR2020) [29] | 28.23 / 23.87 | 20.33 / 48.15 | 33.18 / 11.54 | 28.31 / 12.37
MotionRNN (CVPR2021) [27] | 27.67 / 24.23 | 20.01 / 49.20 | 32.20 / 12.11 | 28.03 / 13.29
STRPM |  28.54 / 20.69 |  20.59 / 41.11 |  33.32 / 9.74 |  29.01 / 10.44

The UCF Sports dataset contains a series of human actions collected from
various sports events and are typically captured on broadcast television
channels such as the BBC and ESPN. A total of 150 videos with resolution of
$480\times 720$ are contained in the UCF Sports dataset. We resize each frame
to $512\times 512$. 6,288 sequences are for training and 752 for testing.
Figure 4 shows the qualitative results generated from different methods, where
the proposed method obviously outperforms others with more naturalistic
predictions. In Table 1, we utilize the Peak Signal to Noise Ratio (PSNR) to
represent the objective quality and the Learned Perceptual Image Patch
Similarity (LPIPS) [30] to represent the perceptual quality. The quantitative
results show that the proposed method achieves the best PSNR score and LPIPS
scores.

Figure 5: The visualization results of the spatiotemporal encoding scheme and
the residual predictive memory. Temporal Features and Spatial Features denote
the encoded features from the temporal encoder and the spatial encoder. Input
Information and Residual Information denote $STIF$ and $STRF$ in Equation 3.2.
Figure 6: The generated examples on the Human3.6M dataset (4 frames
$\rightarrow$ 1 frame). Figure 7: The generated examples on the SJTU4K dataset
(4 frames $\rightarrow$ 1 frame).

To further evaluate the efficiency of the proposed spatiotemporal encoding
scheme and the residual predictive memory, we visualize the temporal features
$T_{E}$, the spatial features $S_{E}$, the input information and the
spatiotemporal residual information (STRF). The visualized results are shown
in Figure 5, where the temporal features contain a wider motion area while the
spatial features focus on the appearance area of the person with greater
weight values. The differences between the temporal features and the spatial
features indicate that the spatiotemporal encoding scheme can help extract
different features from the temporal domain and the spatial domain,
respectively. Moreover, compared with the temporal features, the weights of
the learned spatiotemporal residual features are much greater, indicating the
proposed residual structure can help the predictive memory pay more attention
to modeling the complex motion information instead of the appearance
information (the weights of the appearance information is less than the
spatial features).

###  4.2 Human3.6M and SJTU4K Datasets

The Human3.6M dataset consists of 3.6 million 3D human poses and corresponding
images acted by 11 professional actors in 17 scenarios, including discussion,
smoking, taking photos and so on. All videos are recorded with 4 calibrated
cameras with a resolution $1000\times 1000$, which are further resized to
$1024\times 1024$ in this paper. 73,404 sequences are for training and 8,582
for testing. Figure 6 shows the generated examples from the proposed method
and other state-of-the-art methods, where the proposed STRPM significantly
outperforms others and the predicted results are more naturalistic. In Table
1, the proposed method achieves the best PSNR and LPIPS scores compared with
other state-of-the-art methods.

The SJTU4K dataset consists of 15 ultra-high resolution 4K videos with a wide
variety of contents. The resolution for each video is $2160\times 3840$. To
evaluate the performance on ultra-high resolution videos, the inputs and
outputs are all 4K videos without being down-sampled. 3,873 sequences are for
training and 445 for testing. To the best of our knowledge, the proposed STRPM
is the first one predicting 4K videos. Figure 7 shows the predicted 4K video
frames from different methods. The quantitative results are summarized in
Table 2. As shown in Figure 7 and Table 2, the proposed method has achieved
the best qualitative and quantitative results on ultra-high-resolution videos
with a satisfactory inference speed.

Table 2: Quantitative results of different methods on the SJTU4K test set (4 frames $\rightarrow$ 4 frames). The inference time over 10 samples have also been summarized. Method | $t=5$ | $t=8$ | Inference
---|---|---|---
PSNR$\uparrow$/LPIPS$\downarrow$ | PSNR$\uparrow$/LPIPS$\downarrow$ | Time
ConvLSTM [20] | 22.74 / 67.81 | 17.91 / 86.84 | 39.38s
PredRNN [26] | 23.25 / 66.60 | 18.20 / 87.04 | 40.06s
PredRNN++ [24] | 23.43 / 64.07 | 18.55 / 86.34 | 53.11s
SAVP [13] | 23.41 / 61.44 | 18.63 / 80.45 | 100.23s
CrevNet [29] | 24.35 / 62.31 | 19.61 / 80.91 | 52.98s
MotionRNN | 23.47 / 65.21 | 19.72 / 81.39 | 61.87s
STRPM |  24.37 / 57.12 |  19.77 / 66.68 | 39.84s
Table 3: Ablation studies on the proposed residual predictive memory and the spatiotemporal encoding-decoding scheme (STED) on the Human3.6M dataset (4 frames $\rightarrow$ 4 frames). PSNR and LPIPS scores are averaged over all 4 predictions. For a fair comparison, the encoders and decoders for all models are with the same structure and the number of the hidden state channels for all the memories is set to 128. We stack 16 memories into each model. All models are trained with MSE loss functions. The floating point operations (FLOPs) are recorded over 1 sample. Method | PSNR$\uparrow$ | LPIPS$\downarrow$ | Parameters | FLOPs
---|---|---|---|---
Casual-LSTM [24] | 29.61 | 14.42 | 131.35M | 36.19G
E3D-LSTM [25] | 29.93 | 13.23 | 501.37M | 128.82G
Reversible-PM [29] | 30.13 | 12.62 | 114.35M | 35.07G
RPM w/o residual | 30.11 | 12.64 | 108.29M | 30.55G
RPM ($\theta=1,\tau=1$) | 30.14 | 12.58 | 45.95M | 14.96G
RPM ($\theta=5,\tau=1$) | 30.56 | 11.98 | 77.09M | 22.75G
RPM ($\theta=1,\tau=5$) | 30.32 | 12.31 | 77.09M | 22.75G
RPM | 31.10 | 11.89 | 108.29M | 30.55G
RPM + STED | 31.81 | 11.72 | 109.13M | 34.24G

###  4.3 Ablation Study

In this section, a series of ablation studies are conducted. Table 3 shows the
results of different models with different structures. For a fair comparison,
all models without STED (the spatiotemporal encoding-decoding scheme) are
built with the same structure except the predictive memory. Experimental
results show that the proposed residual predictive memory outperforms other
state-of-the-art memories with the lowest computation load and fewest
parameters. In addition, the residual structure, the broadened temporal
receptive field ($\tau>1$), and the broadened spatial receptive field
($\theta>1$) can help improve the performance of RPM. Moreover, RPM with STED
also help improves the model performance.

Table 4: Ablation studies on STRPM with different loss functions. Performance scores are averaged over all predictions. Method | UCF sports | Human3.6M
---|---|---
$4\rightarrow 6$ | $4\rightarrow 4$
PSNR$\uparrow$ | LPIPS$\downarrow$ | PSNR$\uparrow$ | LPIPS$\downarrow$
$\mathcal{L}_{MSE}$ | 24.89 | 38.81 | 31.81 | 11.72
$\mathcal{L}_{MSE}$+$\mathcal{L}_{GAN}$ | 23.98 | 35.01 | 30.53 | 10.98
$\mathcal{L}_{MSE}$+$\mathcal{L}_{GAN}$+$\mathcal{L}_{LP}$ | 24.30 | 31.50 | 31.00 | 10.11

Furthermore, results from methods trained with different loss functions are
summarized in Table 4, where the proposed perceptual loss can help obtain a
better trade-off between the objective quality (PSNR) and the perceptual
quality (LPIPS).

##  5 Conclusion and Discussions

We proposed a Spatiotemporal Residual Predictive Model (STRPM) for High-
Resolution Video Prediction. We designed the Spatiotemporal Encoding-Decoding
Scheme and the Residual Predictive Memory (RPM) to model the much more complex
appearance information and motion information in high-resolution videos. In
addition, we proposed a Learned Perceptual loss to generate more naturalistic
frames compared with the standard MSE loss. Experimental results showed that
the proposed model can predict high-resolution videos with the best objective
and subjective quality compared with various existing methods.

Although the model performance is better than current methods, the
practicality is still far from satisfactory, especially for videos with ultra-
high resolutions ($\geq$1080p). In addition, the model efficiency also needs
to be improved for multi-step predictions. Considering the above limitations,
current predictive models maybe not reliable enough to be applied into
decision-making systems that require data with high accuracy and real-time
interactions, such as autonomous driving, robot control, etc. Further works
are highly-encouraged to solve the above potential problems.

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# Chiral phase transition and spin alignment of vector meson in the Polarized-
Polyakov-loop Nambu–Jona-Lasinio model under rotation

Fei Suna,b,c<EMAIL_ADDRESS>Jingdong Shaob,d,e
<EMAIL_ADDRESS>Rui Wenb<EMAIL_ADDRESS>Kun Xub
<EMAIL_ADDRESS>Mei Huangb<EMAIL_ADDRESS>aDepartment of Physics,
China Three Gorges University, Yichang, 443002, China
bSchool of Nuclear Science and Technology, University of Chinese Academy of
Sciences, Beijing 100049, China
cCenter for Astronomy and Space Sciences, China Three Gorges University,
Yichang 443002, China
$d$ School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China
$e$ Institue of High Energy Physics, Chinese Academy of Sciences,Beijing,
100049, China

###### Abstract

By using the extrapolation method, a polarized Polykov-loop potential at
finite real angular velocity is constructed from the lattice results at finite
imaginary angular velocity. The chiral and deconfinement phase transitions
under rotation have been simultaneously investigated in the Polarized-
Polyakov-loop Nambu-Jona-Lasinio (PPNJL) model. It is observed that both
critical temperatures of deconfinement and chiral phase transition increase
with the angular velocity, which is in consistent with lattice results. The
spin alignment of vector meson has the negative deviation of $\rho_{00}-1/3$
under rotation, and the deviation in the PPNJL model is much more significant
than that in the NJL model and the quark coalescence model, which revealing
the important role of rotating gluons on the quark polarization.

##  I Introduction

For nearly a century, it has been recognized that the coupling between spin
and orbital angular momentum plays a crucial role, for example in the Barnett
effect[1, 2, 3]. This effect, which manifests as an increase in magnetization
due to rotation, is influenced by the intricate interplay between these two
types of angular momentum. In recent two decades, spin-orbit interactions in
non-central heavy-ion collisions has attracted lots of interests.

In non-central ultra relativistic heavy ion collisions, the angular momentum
with the magnitude of $\mathcal{O}(\text{10}^{4}-\text{10}^{5})\hbar$ [4] with
local angular velocity in the range of $0.01\sim 0.1$ GeV and the angular
velocity of order $\omega\sim 10^{21}\text{s}^{-1}$ [5] can be created. The
spin-orbit coupling of quantum chromodynamics (QCD) could give rise to
numerous intriguing phenomena. For example, Liang and Wang in 2005 predicted
the global spin polarization for the final-state hyperons [6] as well as
vector meson spin alignment [7]. The STAR Collaboration observed the global
polarization of $\Lambda$ and $\bar{\Lambda}$ hyperons in non-central heavy-
ion collisions in 2017 [8, 9, 10]. In 2020, the ALICE Collaboration reported
spin alignment of $\phi$ and $K^{*0}$, and $\rho_{00}<1/3$ for both $\phi$ and
$K^{*0}$ vector mesons with small transverse momenta [11]. The STAR
Collaboration also reported the measurement of spin alignment for $\phi$ and
$K^{*0}$ vector mesons, with the global spin alignment for $\phi$ yielding
unexpected results that $\rho_{00}>1/3$ with a large deviation from $1/3$,
while $\rho_{00}\approx 1/3$ for $K^{*0}$ [12] in 2023. The reasons for the
varied results in the spin alignments of the two vector mesons remain unclear,
prompting a series of extensive studies [13, 14, 15, 16, 17, 18, 19, 20, 21,
21, 22, 23, 24, 25, 26].

Meanwhile, QCD matter under rotation has also attracted much attention in
recent years. The chiral symmetry breaking and its restoration, and
confinement-deconfinement phase transition are most important topics in QCD.
In non-central heavy-ion collisions, the system carries large magnitude of
orbital angular momentum to create a state of rotating Quark-Gluon Plasma
(QGP). This raises the natural question whether rotation affects the QCD phase
transition and how the transition evolves with rotation. Currently, there has
been considerable theoretical research dedicated to this subject.
Specifically, in relation to the chiral phase transition, there have been
numerous comprehensive investigations in Refs. [27, 28, 29, 30, 31, 32, 33,
34, 35]. These studies have revealed that rotation suppresses the chiral
critical temperature. In contrast, the impact of rotation on the deconfinement
phase transition is less understood. Using the holographic QCD approach, it
has been found in Ref. [36] that the deconfinement critical temperature
decreases with increasing angular velocity, which is supported by other
holography studies [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49]. A
hadron resonance gas model obtained results are in consistent with those of
holographic QCD models [50]. However, recent lattice QCD simulations [51, 52,
53] have suggested that the critical temperature of the confinement phase
transition in gluon dynamics rises with increasing angular velocity, which is
opposite to above results obtained in effective models.

The inconsistency between the effective model calculations and lattice results
on rotation could be due to the absence of non-perturbative gluonic effects in
the model calculations. The quarks have the spin of $1/2$, and gluons have the
spin of $1$, in principle, gluons are more sensitive to rotation due to the
coupling between the spin and orbital angular momentum. It has been observed
in [54] that the spin-1 vector meson shows Zeeman-like behavior under
rotation, and the mass of the spin component $s_{z}=1$ vector meson decreases
linearly with $\omega$, while constituent quark mass almost does not change
under small angular velocity. This indicates that spin-1 vectors are more
sensitive to rotation than that of spin-$1/2$ quarks. It is reasonable to
expect that spin-1 gluons exhibit similar behavior to the vector mesons under
rotation.

In [27, 28, 29, 30, 31, 32, 33, 34], the investigation of chiral phase
transition under rotation didn't take into account the effect from rotating
gluodynamics. Therefore, in this work, we try to investigate how rotating
gluodynamics would affect the chiral phase transition. Recently, the rotating
gluon system has been considered in [55, 56]. In this work, we will construct
the polarized Polyakov-loop potential induced by rotation based on the lattice
results of Ref. [51], and add the 2-flavor NJL model on this polarized
Polyakov-loop potential, we call this model as PPNJL. Then investigate the
chiral phase transition of QCD and the spin alignment of the vector meson
$\rho$. The PPNJL model was chosen for its effectiveness in describing chiral
symmetry breaking and the confinement mechanism [57, 58, 59, 60, 61, 62, 63].

The organization of this work is as follows. We construct a polarized Polykov-
loop potential based on lattice results in Section II. In Section III, we add
2-flavor NJL model on Polarized Polykov-loop potential, and construct the
2-flavor PPNJL model, and introduce how to calculate the quark polarization as
well as the spin alignment of the vector meson. Then in Section IV, we present
numerical results and discuss the effective quark mass, Polyakov-loop, the
chiral and deconfinement phase transitions, as well as the spin alignment of
the $\rho$ meson under rotation. Finally, we give summary and discussion at
the end of the paper.

##  II Polarized Polyakov-loop potential under rotation

The first lattice study of rotating QCD matter was conducted in Ref. [64].
Recently, a series of studies on the properties of SU(3) gluodynamics under
rotation using lattice simulations have been undertaken [52, 65, 66, 53, 67,
68, 69]. Due to the sign problem, the Monte Carlo simulations are conducted
with the imaginary angular velocity and the deconfinement phase transition is
found shifted to lower temperature as the imaginary angular velocity increases
[52, 65, 66, 53].

Figure 1: (Color online) The Polyakov-loop as a function of the temperature
scaled by $T_{0}$ for different values of angular velocity $\omega$. The
triangle dots correspond to lattice data with respect to different imaginary
angular velocity in Ref. [51], the hollow dots correspond the analytical
continuation to real angular velocity and the solid lines correspond the
fitting results with Eq. (2).

In order to effectively take into account the rotating gluon background, we
construct the polarized Polyakov-loop potential induced by rotation based on
the lattice results of Ref. [51]. To investigate the influence of rotation on
gluon dynamics properties, we take the rigid cylinder rotating around the
z-axis with constant angular velocity $\omega$. The rotational system must
satisfy the causality condition $\omega r<1$, and thus, in principle, the
boundary conditions should be taken into account. In Ref. [51], three types of
boundary conditions including open boundary conditions (OBC), periodic
boundary conditions (PBC) and Dirichlet boundary conditions (DBC)) were
considered. However, the results indicate that the qualitative behavior does
not depend on the boundary conditions. In our calculations, we choose the
lattice result in Ref. [51] in the case of PBC.

In Fig. 1, the triangle dots are the Polyakov-loop $\Phi$ in [51] as a
function of the scaled temperature $T/T_{0}$ with different $\Omega_{I}$. In
order to do the analytical continuation, we employ $\Omega_{I}\rightarrow
i\omega$ and expand the Polyakov-loop $\Phi$ by rescaling of temperatures

$\displaystyle\Phi(T,\omega)=\Phi(T^{\prime},\omega=0),\quad
T=T^{\prime}+k_{2}(T^{\prime})\omega^{2}+k_{4}(T^{\prime})\omega^{4}.$ (1)

Due to the small angular velocities, we only consider first 4-order of
$\omega$. With Eq. (1), we extrapolate the Polyakov-loop $\Phi$ from imaginary
angular velocity to real angular velocity, which are shown as hollow dots in
Fig. 1. We introduce an analytical function $f\left({T,\omega}\right)$ to fit
the Polyakov-loop empirically in a certain temperature interval

$\displaystyle\Phi\left(T,\omega\right)=\left(\frac{T}{T_{0}}\right)^{2}f\left({T,\omega}\right),$
(2)

with

$\displaystyle
f\left({T,\omega}\right)=a\left(\omega\right)\tanh\left({b\left(\omega\right)\left({\frac{T}{{{T_{0}}}}-c\left(\omega\right)}\right)}\right)+d\left(\omega\right),$
(3)

here $x(\omega)=x_{0}+x_{2}\omega^{2}+x_{4}\omega^{4}$ for $x\in\\{a,b,c,d\\}$
with $\omega$ values in unit of MeV. The corresponding coefficients are
determined by setting Polyakov-loop $\Phi$ consistent with the analytical
continuation results, which are listed in TABLE. 1. We plot them as solid
lines in Fig. 1. It is observed that indeed, though the critical temperature
for deconfinement phase transition is shifted to lower temperature as the
imaginary angular velocity increases, while the critical temperature is
shifted to higher temperature as the angular velocity increases.

Figure 2: (Color online) The 3D plot of the Polyakov-loop as functions of the
angular velocity and the scaled temperature $T/T_{0}$.

In Fig. 2, we present a 3D plot of the Polyakov-loop $\Phi$ as a function of
the real angular velocity $\omega$ and the scaled temperature $T/T_{0}$. It is
evident from this figure that the critical temperature of the confinement-
deconfinement phase transition increases with the growing angular velocity.

Here, it should be mentioned that in Ref. [51], the normalization of the
Polyakov-loop was not performed. In realistic calculations, a normalization
factor needs to be introduced, resulting in a rescaling of the Polyakov-loop
value, i.e, $\Phi_{nor}=C\cdot\Phi_{Latt}$. In our study, the introduction of
a normalization factor has been shown through numerical calculations not to
alter the qualitative behavior under investigation. For simplicity, we set the
normalization factor $C=4$, which provides a more reliable value for the
Polyakov-loop when the angular velocity is very small and the temperature is
high. In the remainder of this article, $\Phi$ denotes the normalized
Polyakov-loop.

Then, we construct a polarized Polyakov-loop potential ${\cal
U}(\Phi,\bar{\Phi},T,\omega)$ of the globally rotating pure gluon system,
which satisfies the $Z(3)$ center symmetry as in the pure gauge QCD
Lagrangian. The normalized Polyakov-loop should be the solution of the gap
equations

$\displaystyle\frac{\partial{\cal
U}(\Phi,\bar{\Phi},T,\omega)}{\partial\Phi}=\frac{\partial{\cal
U}(\Phi,\bar{\Phi},T,\omega)}{\partial\bar{\Phi}}=0.$ (4)

The expression of the potential is given as:

$\displaystyle\frac{{\cal
U}\left({\Phi,\bar{\Phi},T,\omega}\right)}{T^{4}}=-{Cf\left({T,\omega}\right)}{\left({\frac{T}{{{T_{0}}}}}\right)^{2}}\Phi\bar{\Phi}-\frac{1}{3}\left({{\Phi^{3}}+{{\bar{\Phi}}^{3}}}\right)+{C^{-1}f^{-1}\left({T,\omega}\right)}{{{\left({\frac{T}{{{T_{0}}}}}\right)}^{-2}}}{\Phi^{2}}{{\bar{\Phi}}^{2}},$
(5)

with the parameters listed in TABLE. 1.

$i$ |  0 |  2 |  4
---|---|---|---
$a_{i}$ |  0.0454431 |  -1.27942$\times 10^{-5}$ |  -5.43339$\times 10^{-9}$
$b_{i}$ |  46.8263 |  -0.0210165 |  -2.15394$\times 10^{-5}$
$c_{i}$ |  1.00298 |  1.55157$\times 10^{-4}$ |  -5.99032$\times 10^{-8}$
$d_{i}$ |  0.0600157 |  -5.74388$\times 10^{-6}$ |  -8.24192
Table 1: The parameters of the Polyakov-loop potential in Eq. (3) and Eq. ( 5)
by fitting lattice data from Ref. [51].

##  III The 2-flavor PPNJL model

To explore the influence of rotation on QCD matter, we consider a rigid
cylinder rotating around the $z$ axis with constant angular velocity $\omega$.
In this rotating frame of reference, the rotation can be characterized in
relation to an external gravitational field. The metric tensor can be utilized
to describe the structure of space-time reads

$\displaystyle
g_{\mu\nu}=\left(\begin{array}[]{cccc}1-{\vec{v}}^{\,2}&-v_{1}&-v_{2}&-v_{3}\\\
-v_{1}&-1&0&0\\\ -v_{2}&0&-1&0\\\ -v_{3}&0&0&-1\\\ \end{array}\right),$ (10)

where $v_{i}$ is the velocity and $v=\sqrt{v_{1}^{2}+v_{2}^{2}+v_{3}^{2}}$. In
the context of a system with an angular velocity $\omega$ along the fixed
$z$-axis, we have $\vec{v}=\vec{\omega}\times\vec{x}$, thus $v_{3}=0$ in the
metric tensor.

The Lagrangian in the two-flavor NJL model under rotation can be written as
follows:

${\cal L}_{\text{NJL}}=\\\
\sum_{f}\bar{\psi}_{f}\biggl{[}{i\bar{\gamma}^{\mu}(\partial_{\mu}+\Gamma_{\mu})-m+\gamma^{0}\mu}\biggr{]}\psi_{f}+G{\left({\bar{\psi}\psi}\right)^{2}},$
(11)

here, $\psi$ is the quark field, $\bar{\gamma}^{\mu}=e_{a}^{\mu}\gamma^{a}$
with $e_{a}^{\mu}$ being the tetrads for spinors and $\gamma^{a}$ represents
the gamma matrix, $\Gamma_{\mu}$ is defined as
$\Gamma_{\mu}=\frac{1}{4}\times\frac{1}{2}[\gamma^{a},\gamma^{b}]\
\Gamma_{ab\mu}$ which is the spinor connection, where
$\Gamma_{ab\mu}=\eta_{ac}(e^{c}_{\
\sigma}G^{\sigma}_{\mu\nu}e_{b}^{\nu}-e_{b}^{\nu}\partial_{\mu}e^{c}_{\nu})$,
and $G^{\sigma}_{\mu\nu}$ is the affine connection determined by $g^{\mu\nu}$,
$m$ is the bare quark mass matrix, $\mu$ denotes the chemical potential, and
$G$ represents the coupling constants in the scalar channel. Here we only
consider the scalar part, since we are mainly interested in the chiral phase
transition.

The NJL model lacks gluon degrees of freedom. To effectively include the
contribution from rotating gluodynamics, we add the NJL model on the rotating
gluonic background given by Eq. (5). The Lagrangian of the PPNJL model takes
the following form:

$\displaystyle{\cal L}_{\text{PPNJL}}={{\cal
L}_{\text{NJL}}}+\bar{\psi}{\gamma^{\mu}}{A_{\mu}}\psi-{\cal
U}(\Phi,\bar{\Phi},T,\omega),$ (12)

here, we have included the coupling between the fermion fields and the gauge
fields, and the effective polarized Polyakov-loop potential ${\cal
U}(\Phi,\bar{\Phi},T,\omega)$, where the Polyakov-loop is a function of
temperature and angular velocity given in Eq. (5), which effectively takes
into account the roation effect on gluodynamics. In this sense, the PPNJL
model is different from the PNJL model,

By selecting
$e^{a}_{\mu}=\delta^{a}_{\mu}+\delta^{a}_{i}\delta^{0}_{\mu}v_{i}$ and
$e_{a}^{\mu}=\delta_{a}^{\mu}-\delta_{a}^{0}\delta_{i}^{\mu}v_{i}$ (further
details can be found in Refs. [70, 71]), the Lagrangian can be expanded to the
first order of angular velocity, yielding the following expression:

$\displaystyle{\cal
L}_{\text{PPNJL}}=\bar{\psi}\left[{i\gamma^{\mu}D_{\mu}-m+\gamma^{0}\mu+\left({\gamma^{0}}\right)^{-1}\left({\left({\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{\omega}}\times\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{x}}}\right)\cdot\left({-i\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{\partial}}}\right)+\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{\omega}}\cdot\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{S}}_{4\times
4}}\right)}\right]\psi+G{\left({\bar{\psi}\psi}\right)^{2}}-{\cal
U}(\Phi,\bar{\Phi},T,\omega),$ (13)

here the spin operator is given by $\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{S}}_{4\times
4}=\frac{1}{2}\left({\begin{array}[]{*{20}c}{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{\sigma}}}&0\\\
0&{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{\sigma}}}\\\
\end{array}}\right)$. From the equation above, it is evident that the free
Dirac Lagrangian is modified due to the presence of the gluon field and
rotation. The first term corresponds to the coupling between the quark field
and gluon field, while the fourth term accounts for the orbital-rotation
coupling effect and the spin-rotation coupling effect. The last term in the
Lagrangian represents the effective Polyakov-loop potential. Here the
covariant derivative is defined as ${D_{\mu}}={\partial_{\mu}}-i{A_{\mu}}$,
with $A^{\mu}=\delta_{0}^{\mu}$, which determines the coupling between the
Polyakov-loop and quarks. The Polyakov-loops $\Phi$ are obtained as follows,

$\displaystyle\Phi=\frac{1}{{{N_{c}}}}\left\langle{{\rm{tr}}({\cal P}{\rm
exp}[i{\int_{0}}^{\beta}d\tau A_{4}{({\bar{x}},\tau)}])}\right\rangle,$ (14)

where $\beta=\frac{1}{T}$, $\cal P$ denotes path ordering, $A_{4}=iA_{0}$ is
the temporal component of the Euclidean gauge field $(\bar{A},A_{4})$, and
$\bar{\Phi}=\Phi^{{\dagger}}$. In this model, the quarks couple to a
background (temporal) gauge field representing Polyakov-loop dynamics, and the
Polyakov-loop serves as an order parameter for confinement. When performing
the mean field approximation and employing the technique of path integral
formulation for Grassmann variables theory, the grand thermodynamic potential
reads,

$\displaystyle\Omega_{\text{PPNJL}}$ $\displaystyle=$ $\displaystyle
G\langle\bar{q}q\rangle^{2}-\frac{3}{2\pi^{2}}\sum_{n=-\infty}^{\infty}\int_{0}^{\Lambda}p_{t}dp_{t}\int_{-\sqrt{\Lambda^{2}-p_{t}^{2}}}^{\sqrt{\Lambda^{2}-p_{t}^{2}}}dp_{z}\biggl{(}J_{n+1}(p_{t}r)^{2}+J_{n}(p_{t}r)^{2}\biggr{)}\epsilon_{n}$
(15) $\displaystyle-$
$\displaystyle\frac{T}{2\pi^{2}}\sum_{n=-\infty}^{\infty}\int
p_{t}dp_{t}dp_{z}\Biggl{[}\biggl{(}J_{n+1}(p_{t}r)^{2}+J_{n}(p_{t}r)^{2}\biggr{)}\text{log}\biggl{(}1+3\Phi\text{exp}(-\frac{\epsilon_{n}-\mu}{T})+3\bar{\Phi}\text{exp}(-2\frac{\epsilon_{n}-\mu}{T})+\text{exp}(-3\frac{\epsilon_{n}-\mu}{T})\biggr{)}\Biggr{]}$
$\displaystyle-$ $\displaystyle\frac{T}{2\pi^{2}}\sum_{n=-\infty}^{\infty}\int
p_{t}dp_{t}dp_{z}\Biggl{[}\biggl{(}J_{n+1}(p_{t}r)^{2}+J_{n}(p_{t}r)^{2}\biggr{)}\text{log}\biggl{(}1+3\bar{\Phi}\text{exp}(-\frac{\epsilon_{n}+\mu}{T})+3\Phi\text{exp}(-2\frac{\epsilon_{n}+\mu}{T})+\text{exp}(-3\frac{\epsilon_{n}+\mu}{T})\biggr{)}\Biggr{]}$
$\displaystyle+$ $\displaystyle{\cal U}(\Phi,\bar{\Phi},T,\omega).$

Here, for simplicity, we have introduced the quark quasiparticle energy
${\varepsilon_{n}}=\sqrt{M^{2}+p_{t}^{2}+p_{z}^{2}}-\left({\frac{1}{2}+n}\right)\omega$
with the dynamic quark mass ${M}={m}-2G\left\langle{\bar{q}q}\right\rangle$.
From the grand thermodynamic potential, the equations of motion for the mean
fields $\left\langle\bar{q}q\right\rangle$, $\Phi$, and $\bar{\Phi}$ can be
derived as

$\displaystyle\frac{{\partial\Omega_{\text{PPNJL}}}}{{\partial\left\langle{\bar{q}q}\right\rangle}}=0,~{}~{}~{}\frac{{\partial\Omega_{\text{PPNJL}}}}{{\partial\Phi}}=0,~{}~{}~{}\frac{{\partial\Omega_{\text{PPNJL}}}}{{\partial\bar{\Phi}}}=0.$
(16)

These coupled equations are then solved as functions of temperature $T$, quark
chemical potential $\mu$, and angular velocity $\omega$. The detailed
expressions for these stationary condition equations are provided below:

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
2G\langle\bar{q}q\rangle+\frac{3}{2\pi^{2}}\sum_{n=-\infty}^{\infty}\int_{0}^{\Lambda}p_{t}dp_{t}\int_{-\sqrt{\Lambda^{2}-p_{t}^{2}}}^{\sqrt{\Lambda^{2}-p_{t}^{2}}}dp_{z}\biggl{(}J_{n+1}(p_{t}r)^{2}+J_{n}(p_{t}r)^{2}\biggr{)}\left({\frac{{2GM}}{{\sqrt{p_{t}^{2}+p_{z}^{2}+M^{2}}}}}\right)$
$\displaystyle-$ $\displaystyle\frac{1}{2\pi^{2}}\sum_{n=-\infty}^{\infty}\int
dp_{t}dp_{z}p_{t}\biggl{(}J_{n+1}(p_{t}r)^{2}+J_{n}(p_{t}r)^{2}\biggr{)}\left(\frac{6GM}{\sqrt{p_{t}^{2}+p_{z}^{2}+M^{2}}}\right)$
$\displaystyle\times$
$\displaystyle\left(\frac{\text{exp}(-3\frac{\epsilon_{n}-\mu}{T})+\Phi\text{exp}(-\frac{\epsilon_{n}-\mu}{T})+2\bar{\Phi}\text{exp}(-2\frac{\epsilon_{n}-\mu}{T})}{1+\text{exp}(-3\frac{\epsilon_{n}-\mu}{T})+3\Phi\text{exp}(-\frac{\epsilon_{n}-\mu}{T})+3\bar{\Phi}\text{exp}(-2\frac{\epsilon_{n}-\mu}{T})}+\frac{\text{exp}(-3\frac{\varepsilon_{n}+\mu}{T})+2\Phi{\text{exp}(-2\frac{\varepsilon_{n}+\mu}{T})}+\bar{\Phi}{\text{exp}(-\frac{{\varepsilon_{n}+\mu}}{T})}}{1+\text{exp}(-3\frac{\varepsilon_{n}+\mu}{T})+3\Phi{\text{exp}(-2\frac{\varepsilon_{n}+\mu}{T})}+3\bar{\Phi}{\text{exp}(-\frac{\varepsilon_{n}+\mu}{T})}}\right),$

$\displaystyle 0$ $\displaystyle=$
$\displaystyle-\frac{{1}}{{2{\pi^{2}}}}T\sum_{n=-\infty}^{\infty}\int
dp_{t}dp_{z}p_{t}\biggl{(}J_{n+1}(p_{t}r)^{2}+J_{n}(p_{t}r)^{2}\biggr{)}$ (18)
$\displaystyle\times$
$\displaystyle\left({\frac{3{\text{exp}(-\frac{{\varepsilon_{n}}-{\mu}}{T})}}{1+\text{exp}(-3\frac{\epsilon_{n}-\mu}{T})+3\Phi\text{exp}(-\frac{\epsilon_{n}-\mu}{T})+3\bar{\Phi}\text{exp}(-2\frac{\epsilon_{n}-\mu}{T})}+\frac{{3\text{exp}(-2\frac{\varepsilon_{n}+\mu}{T})}}{{1+\text{exp}(-3\frac{\varepsilon_{n}+\mu}{T})+3\Phi{\text{exp}(-2\frac{\varepsilon_{n}+\mu}{T})}+3\bar{\Phi}{\text{exp}(-\frac{\varepsilon_{n}+\mu}{T})}}}}\right)$
$\displaystyle+$ $\displaystyle
T^{4}\left(-{Cf\left({T,\omega}\right)}{\left({\frac{T}{{{T_{0}}}}}\right)^{2}}{\bar{\Phi}}-\Phi^{2}+2{C^{-1}f^{-1}\left({T,\omega}\right)}{{{\left({\frac{T}{{{T_{0}}}}}\right)}^{-2}}}{\Phi}{{\bar{\Phi}}^{2}}\right),$

$\displaystyle 0$ $\displaystyle=$
$\displaystyle-\frac{{1}}{{2{\pi^{2}}}}T\sum_{n=-\infty}^{\infty}\int
dp_{t}dp_{z}p_{t}\biggl{(}J_{n+1}(p_{t}r)^{2}+J_{n}(p_{t}r)^{2}\biggr{)}$ (19)
$\displaystyle\times$
$\displaystyle\left(\frac{3\text{exp}(-2\frac{{\varepsilon_{n}}-{\mu}}{T})}{1+\text{exp}(-3\frac{\epsilon_{n}-\mu}{T})+3\Phi\text{exp}(-\frac{\epsilon_{n}-\mu}{T})+3\bar{\Phi}\text{exp}(-2\frac{\epsilon_{n}-\mu}{T})}+\frac{{3\text{exp}(-\frac{\varepsilon_{n}+\mu}{T})}}{{1+\text{exp}(-3\frac{\varepsilon_{n}+\mu}{T})}+3\Phi\text{exp}(-2\frac{\varepsilon_{n}+\mu}{T})+3\bar{\Phi}\text{exp}(-\frac{\varepsilon_{n}+\mu}{T})}\right)$
$\displaystyle+$ $\displaystyle
T^{4}\left(-{Cf(T,\omega)}\frac{T^{2}}{T_{0}^{2}}\Phi-\bar{\Phi}^{2}+2{C^{-1}f^{-1}\left({T,\omega}\right)}\Phi^{2}\bar{\Phi}\right).$

##  IV Spin alignment of vector meson $\rho$

In the following section, we study on the spin alignment of the vector meson
$\rho$ under rotation. We will briefly discuss the phenomenon of vector meson
spin alignment under rotation. The spin states of vector mesons are described
by the $3\times 3$ spin-density matrix $\rho_{ij}$. However, the polarization
of vector mesons cannot be directly measured, due to the parity-conserving of
their strong decay. It is challenging to ascertain whether the meson spin
aligns parallel or anti-parallel to the angular momentum, and with only
experimental access to $\rho_{00}$.

Experimentally, the spin alignment of vector mesons can be measured by the
angular distribution of decay products in the vector meson decaying to two
spinless particles [72]:

$\displaystyle\frac{{{\rm{d}}N}}{{{\rm{d}}\cos{\theta^{*}}}}=\frac{3}{4}[1-{\rho_{00}}+\left({3{\rho_{00}}-1}\right){\cos^{2}}{\theta^{*}}],$
(20)

where $\theta^{*}$ is the polar angle between the quantization axis and the
momentum direction of the decay particle. In a simple recombination picture,
the $\rho$ vector meson is produced by the simple coalescence of a quark and
antiquark with polarizations $P_{q}$ and $P_{\bar{q}}$ respectively. The meson
spin alignment is given by [7]:

$\displaystyle{\rho_{00}}=\frac{{1-{P_{q}}{P_{\bar{q}}}}}{{3+{P_{q}}{P_{\bar{q}}}}},$
(21)

and when ${P_{q}}{P_{\bar{q}}}$ is small, the spin alignment of $\rho$ can be
approximated as:

$\rho_{00}\approx\frac{1}{3}-\frac{4}{9}P_{q}P_{\bar{q}},$ (22)

where $P_{q}$ is the spin polarization of quark, and we simply take the
following definition:

$P_{q}=\frac{{N_{\uparrow}^{+}-N_{\downarrow}^{+}}}{{N_{\uparrow}^{+}+N_{\downarrow}^{+}}},$
(23)

here, $N^{+/-}_{\uparrow/\downarrow}$ denote the quark/anti-quark number
density with spin up/down, respectively, which can be extracted by taking the
partial derivative of $\Omega_{\text{PPNJL}}$ Eq. (15) with respect to $\mu$.
The detailed expressions for
$N_{\uparrow}^{+},N_{\downarrow}^{+},N_{\uparrow}^{-},N_{\downarrow}^{-}$ are
listed as follows:

$\displaystyle N_{\uparrow}^{+}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi^{2}}\sum_{n=-\infty}^{\infty}\int
dp_{t}dp_{z}p_{t}J_{n}(p_{t}r)^{2}\frac{3\Phi\text{exp}(-\frac{\epsilon_{n}-\mu}{T})+6\bar{\Phi}\text{exp}(-2\frac{\epsilon_{n}-\mu}{T})+3\text{exp}(-3\frac{\epsilon_{n}-\mu}{T})}{1+3\Phi\text{exp}(-\frac{\epsilon_{n}-\mu}{T})+3\bar{\Phi}\text{exp}(-2\frac{\epsilon_{n}-\mu}{T})+\text{exp}(-3\frac{\epsilon_{n}-\mu}{T})},$
(24) $\displaystyle N_{\downarrow}^{+}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi^{2}}\sum_{n=-\infty}^{\infty}\int
dp_{t}dp_{z}p_{t}J_{n+1}(p_{t}r)^{2}\frac{3\Phi\text{exp}(-\frac{\epsilon_{n}-\mu}{T})+6\bar{\Phi}\text{exp}(-2\frac{\epsilon_{n}-\mu}{T})+3\text{exp}(-3\frac{\epsilon_{n}-\mu}{T})}{1+3\Phi\text{exp}(-\frac{\epsilon_{n}-\mu}{T})+3\bar{\Phi}\text{exp}(-2\frac{\epsilon_{n}-\mu}{T})+\text{exp}(-3\frac{\epsilon_{n}-\mu}{T})},$
(25) $\displaystyle N_{\uparrow}^{-}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi^{2}}\sum_{n=-\infty}^{\infty}\int
dp_{t}dp_{z}p_{t}J_{n}(p_{t}r)^{2}\frac{-3\bar{\Phi}\text{exp}(-\frac{\epsilon_{n}+\mu}{T})-6\Phi\text{exp}(-2\frac{\epsilon_{n}+\mu}{T})-3\text{exp}(-3\frac{\epsilon_{n}+\mu}{T})}{1+3\bar{\Phi}\text{exp}(-\frac{\epsilon_{n}+\mu}{T})+3\Phi\text{exp}(-2\frac{\epsilon_{n}+\mu}{T})+\text{exp}(-3\frac{\epsilon_{n}+\mu}{T})},$
(26) $\displaystyle N_{\downarrow}^{-}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi^{2}}\sum_{n=-\infty}^{\infty}\int
dp_{t}dp_{z}p_{t}J_{n+1}(p_{t}r)^{2}\frac{-3\bar{\Phi}\text{exp}(-\frac{\epsilon_{n}+\mu}{T})-6\Phi\text{exp}(-2\frac{\epsilon_{n}+\mu}{T})-3\text{exp}(-3\frac{\epsilon_{n}+\mu}{T})}{1+3\bar{\Phi}\text{exp}(-\frac{\epsilon_{n}+\mu}{T})+3\Phi\text{exp}(-2\frac{\epsilon_{n}+\mu}{T})+\text{exp}(-3\frac{\epsilon_{n}+\mu}{T})}.$
(27)

Then, the spin alignment of $\rho$ can be approximately expressed as:

$\displaystyle\rho_{00}=\frac{1}{3}-\frac{4}{9}\frac{{N_{\uparrow}^{+}-N_{\downarrow}^{+}}}{{N_{\uparrow}^{+}+N_{\downarrow}^{+}}}\frac{{N_{\uparrow}^{-}-N_{\downarrow}^{-}}}{{N_{\uparrow}^{-}+N_{\downarrow}^{-}}}.$
(28)

##  V Numerical results and discussions

Figure 3: (Color online) (a)The light quark effective mass and (b)Polyakov-
loop as functions of temperature $T$ at zero quark chemical potential $\mu=0$
GeV for different angular velocities, $\omega=5,10,15$ MeV; (c)Polyakov-loop
as a function of temperature $T$ at nonzero quark chemical potential
$\mu=0.01$ GeV for different angular velocities, $\omega=5,15$ MeV.

In this section, we present the numerical results obtained from the PPNJL
model under rotation. For the Fermionic sector, we have selected the
parameters $m=0.005$ GeV, $\Lambda=0.65$ GeV, and $G=4.93~{}\text{GeV}^{-2}$,
as reported in Ref. [73], to match physical observations. Additionally, we
denote the $z$-angular-momentum quantum number as $n=0,\pm 1,\pm 2...$. In
principle, the sum over all values of $n$ is necessary; however, the rapid
convergence of these expressions allows us to limit the sum over $n$ from $-5$
to $5$. It is important to note that in the PPNJL model, the value of $T_{0}$
can be rescaled over a wide range, and here we set $T_{0}=0.22$ GeV. The
radius is set to $r=0.1$ GeV-1, ensuring that $\omega r<1$ in all
calculations.

Fig. 3 (a) and (b) depict the light quark effective mass and Polyakov-loop as
functions of temperature $T$ at quark chemical potential $\mu=0$ GeV for
different angular velocities, respectively. The quark condensate is related to
the spontaneous breaking of chiral symmetry, and in the 2-flavor PPNJL model,
there exists a simple relationship between the condensate and the quark
effective mass, given by ${M}={m}-2G\left\langle{\bar{q}q}\right\rangle$.
Meanwhile, the Polyakov-loop is associated with confinement-deconfinement.
Within the temperature range considered, we observe that at low temperatures,
both the quark effective mass and the Polyakov-loop exhibit weak dependence on
temperature for the different angular velocities selected. As the temperature
continues to increase and exceeds the transition region, the light quark
effective mass tends to approach its current mass, and the value of Polyakov-
loop tends to approach $1$. This indicates the restoration of chiral symmetry
and the gradual recovery of deconfinement at high temperatures. Surprisingly,
when considering the influence of the gluonic rotational contribution, we
observe that the effective quark mass is enhanced by the angular velocity in
the chiral phase transition region, while the Polyakov-loop is suppressed by
the angular velocity in the confinement-deconfinement phase transition region.
It is worth mentioning that in the general NJL model or in the PNJL model
under rotation, where the gluonic rotational contribution is not taken into
account, rotation tends to suppress the quark condensate. In contrast to the
majority of effective model predictions, when considering the polarization of
the gluodynamics induced by the rotation, our results in the PPNJL model
indicate an opposite behavior. In the transition region, there is an increase
in the chiral condensate and a decrease in the Polyakov-loop with rotation.
This suggests that rotation acts as a catalyst for chiral symmetry breaking
and confinement. Next, we turn to the situation of non-zero chemical
potential, Fig. 3 (c) shows the Polyakov-loop as a function of temperature $T$
at nonzero quark chemical potential $\mu=0.01$ GeV for different angular
velocities, $\omega=5,15$ MeV. When we introduce a non-zero chemical
potential, we observe that $\Phi$ and $\bar{\Phi}$ differ from each other and
for a fixed angular velocity $\bar{\Phi}>\Phi$.

Figure 4: (Color online) The scaled pseudocritical temperatures for chiral and
deconfinement phase transition , determined by the quark condensate and
Polyakov-loop, as functions of the angular velocity at zero chemical
potential, and both the Polyakov-loop potential obtained with and without
explicit rotational dependence are considered, which corresponding to the
PPNJL and PNJL models under rotation, respectively. Figure 5: (Color online)
The $\rho$ meson spin alignment as a function of temperature $T$ at $\mu=0.05$
GeV, and $\mu=0.15$ GeV for $\omega=5,15$ MeV, respectively.

The impact of the rotational contribution of gluons on the phase transition is
illustrated in Fig. 4, which shows the scaled pseudocritical temperatures of
the chiral phase transition and deconfinement phase transition as a function
of the angular velocity at zero chemical potential. The figure considers the
cases of Polyakov-loop potential with and without explicit rotational
dependence, denoted as ${\cal U}={\cal U}(\Phi,\bar{\Phi},T,\omega)$ and
${\cal U}={\cal U}(\Phi,\bar{\Phi},T)$, respectively (here the exact
expression for $\mathcal{U}(\Phi,\bar{\Phi},T)$ is taken from Ref. [62]). It
is evident that with the explicit rotational dependence, the pseudocritical
temperatures of both the chiral phase transition and confinement-deconfinement
phase transition increase with the angular velocity, which is consistent with
lattice QCD simulations results [52, 53]. When considering the Polyakov-loop
potential without explicit rotational dependence, the dashed line and dotted
line almost overlap, to further illustrate the rotation dependence of pseudo-
critical temperatures in this case, an enlarged sub-figure is included in Fig.
4. It can be clearly seen from the sub-figure that the pseudocritical
temperatures of both chiral and deconfinement phase transitions decrease with
increasing angular velocity. The chiral phase transition has been extensively
studied in the NJL model and holographic QCD models, and it has been observed
that the critical temperature of chiral phase transition decreases as the
angular velocity increases. However, when considering the contribution from
the polarized gluodynamics under rotation, it has the opposite effect on the
critical temperatures. The opposite effect indicates that the rotating gluons
has important effect on rotating quarks, and it is necessary to take into
account the contribution of rotating gluons to fully understand the
characteristics of a rotating QCD medium.

Fig. 5 depicts the influence of rotating gluons on the spin alignment of the
$\rho$ meson as a function of temperature $T$ at quark chemical potentials
$\mu=0.05$ GeV and $\mu=0.15$ GeV for different angular velocities. Any
deviation of the spin alignment from $1/3$ indicates the polarization of the
vector mesons along a particular direction. It is evident that at high
temperatures, the spin alignment approaches $1/3$, while at low temperatures,
rotation can significantly enhance the deviation from $1/3$. When comparing
the results corresponding to different quark chemical potentials at a fixed
angular velocity, a larger quark chemical potential can lead to a larger
deviation from $1/3$. However, compared to the contribution from the angular
velocity (note that here, the value of $\omega$ is in MeV, while $\mu$ is in
GeV), the deviation induced by the chemical potential is less obvious. Our
results indicate that in the 2-flavor PPNJL model under rotation, the rotating
gluon may play a significant role in the spin alignment of the $\rho$ meson.

From [74], it has been summarized that the spin alignment of vector meson (for
both $\rho$ and $\phi$) in the NJL model at $T=150~{}{\rm MeV}$ has the
relation of

$\rho_{00}(\omega)=\frac{1}{3}-5.10\omega^{2}+39.62\omega^{4},$ (29)

and in the quark coalescence model [75] the spin alignment of vector meson has
the form of

$\rho_{00}(\omega)=\frac{1}{3}-\frac{1}{9}(\beta\omega)^{2},$ (30)

with $\beta=1/T$. In Fig. 6 we show the spin alignment of $\rho$ meson as a
function of $\omega$ in the PPNJL model, NJL model and quark coalescence
model. It is noticed that the spin alignment of vector meson has the negative
deviation of $\rho_{00}-1/3$ under rotation, in the case of considering the
effect of rotating gluons, the spin alignment of vector meson is more
significantly deviated from $1/3$.

Figure 6: (Color online) The $\rho$ meson spin alignment as a function of
angular velocity $\omega$ at $T=0.15$ GeV and $\mu=0.0$ GeV in the PPNJL model
and comparing with the NJL model and quark coalescence model. The gray dashed
line is $\rho_{00}=\frac{1}{3}$.

##  VI CONCLUSIONS

In this work, in order to effectively take into account the rotating gluon
background, by using the extrapolation method, we construct the polarized
Polyakov-loop potential induced by rotation based on the lattice results at
finite imaginary angular velocity, and add the 2-flavor NJL model on the
rotating gluonic background. We explore the QCD phase transitions and spin
alignment of vector mesons using this 2-flavor Polarized-Polyakov-loop Nambu-
Jona-Lasinio (PPNJL) model under rotation. It is found that the critical
temperatures for both chiral phase transition and deconfinement phase
transition increase with the angular velocity, which is consistent with
lattice results, and opposite to results in the NJL/PNJL model and holographic
QCD model. Our findings reveal that the quark condensate and critical
temperature of the phase transition are sensitive to the effects of rotating
gluons.

Furthermore, we calculate the pin alignment of vector $\rho$ meson, and find
that the spin alignment of vector meson has a negative deviation of
$\rho_{00}-1/3$ under rotation, and the deviation in the PPNJL model is much
more significant than that in the NJL model and the quark coalescence model,
indicating the significant contribution from the rotating gluons.

Our study suggests that the contribution of rotating gluons is essential for
correctly understanding the properties of rotating QCD. The effect of rotating
gluons is also anticipated to make significant contributions to various
theoretical investigations. Recent studies in Refs. [76, 56] have demonstrated
that in the NJL model, the influence of rotating gluons can be effectively
taken into account by the running of the effective coupling, resulting in a
modification of the critical temperature behavior in the rotating system.
Alternative simulation method, e.g., Ref. [55] discussed the reliability of
the analytic continuation of the phase diagram from imaginary rotation to real
rotation. These investigations would help determine the Polyakov-loop
potential and unravel other related puzzles, thus greatly enhancing our
understanding of the properties of rotating QCD. We hope that our study will
stimulate further theoretical and lattice QCD simulation studies in the
future.

## Acknowledgements

We would like to thank Xu-Guang Huang, Ji-Chong Yang and Anping Huang for
useful discussions. The work has been supported by the National Natural
Science Foundation of China (NSFC) with Grant No.s: 12235016, 12221005 and the
Strategic Priority Research Program of Chinese Academy of Sciences under Grant
No XDB34030000.

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