11institutetext: Institute of High Performance Computing, A*STAR, Singapore
22institutetext: Centre for Frontier AI Research, A*STAR, Singapore
22email<EMAIL_ADDRESS>
22email<EMAIL_ADDRESS>
# Consistency Regularization for
Domain Adaptation

Kian Boon Koh 1122 Basura Fernando 1122

###### Abstract

Collection of real world annotations for training semantic segmentation models
is an expensive process. Unsupervised domain adaptation (UDA) tries to solve
this problem by studying how more accessible data such as synthetic data can
be used to train and adapt models to real world images without requiring their
annotations. Recent UDA methods applies self-learning by training on pixel-
wise classification loss using a student and teacher network. In this paper,
we propose the addition of a consistency regularization term to semi-
supervised UDA by modelling the inter-pixel relationship between elements in
networks’ output. We demonstrate the effectiveness of the proposed consistency
regularization term by applying it to the state-of-the-art DAFormer framework
and improving mIoU19 performance on the GTA5 to Cityscapes benchmark by 0.8
and mIou16 performance on the SYNTHIA to Cityscapes benchmark by 1.2.

##  1 Introduction

Semantic segmentation is a task which requires a lot of pixel level
annotations and obtaining these annotations is expensive and time consuming.
To overcome this issue, one solution is to obtain annotations from synthetic
data such as Games (e.g. GTA5) and train models on these synthetic data for
semantic segmentation. However, the problem is that even if modern synthetic
data is near photo realistic, still there is a distribution mismatch between
the synthetic data and real images. One solution is to develop models that can
overcome this distribution mismatch between models that are trained on
synthetic data and real data which is the topic of unsupervised domain
adaptation (UDA) [7, 8, 13, 10, 22, 2, 29, 9].

UDA for semantic segmentation has made significant progress in recent years.
One of the most recent method called DAFormer [16] obtained massive
improvement over prior methods by using a Transformer architecture and self-
training. However one of the challenges in self-training is that generated
pseudo labels can be wrong and that may result in poor transfer of information
from source domain to the target domain. Therefore, it is needed to further
regularize the self-training learning process.

In this work, we present a new consistency regularization method based on
correlation between pixel-wise class predictions. We enforce two models
(teacher and student) to have similar inter-pixel similarity structure and by
doing so we regularize the self-training process. This helps to improve the
generalization of the student network as well as the teacher network allowing
better transfer of information from the source domain to the target domain. We
demonstrate its effectiveness by applying it to DAFormer and improving mIoU19
performance on the GTA5 to Cityscapes benchmark by 0.8 and mIou16 performance
on the SYNTHIA to Cityscapes benchmark by 1.2. Implementation of our proposed
method is available at our GitHub repository111https://github.com/kw01sg/CRDA.

##  2 Related Work

###  2.1 Unsupervised Domain Adaptation

Domain adaptation is a field of techniques that aims to solve the domain shift
problem, when data distributions experience change between datasets. UDA is a
subset of the domain adaptation field that aims to utilize a labeled source
domain to learn a model that performs well on an unlabeled target domain.
Recent UDA methods can be grouped into either adversarial training or self-
supervised learning (SSL) approaches. Adversarial training methods aim to
reduce source and target distribution mismatch by aligning distributions at
either the pixel [14, 11, 3] or intermediate feature level [30, 15] using a
generative adversarial network (GAN).

SSL methods allow models to be trained directly on the target domain by
generating pseudo labels from the target domain. Recent advances focuses on
improving the quality of pseudo labels using various approaches, such as using
representative prototypes [37] or using more complex, Transformer-based
network architecture [16]. It is also possible for methods to adopt a hybrid
approach and use both adversarial training and SSL. Li et al. does so in their
bidirectional learning framework [18]. Adversarial training is first used to
obtain an image-to-image translation model and a segmentation model. Target
domain pseudo labels are then generated from high confidence predictions,
which are then subsequently used to fine tune the segmentation model. The
improved segmentation model can then be used in the first adversarial stage to
form a close loop.

###  2.2 Semantic Segmentation

Early methods on semantic segmentation problems were largely based on Fully
Convolutional Network (FCN) [26], which typically follows an encoder-decoder
architecture [1, 24]. Further improvements were made by using dilated
convolutions to overcome the loss of spatial resolution [34], and pyramid
pooling [38, 4] to enhance capturing of contextual information. Recent success
of attention-based Transformers [31] in natural language processing has seen
adaptations of Transformers for image segmentation [19, 33] that were able to
obtain state-of-the-art results.

###  2.3 Consistency Regularization

Consistency regularization is a regularization technique used to encourage
networks to make consistent predictions that are invariant to perturbations.
Tarvainen and Valpola improved model performance on the image classification
problem by using a student and teacher network pair in their Mean Teacher
model [28], where the weights of the teacher network are an exponential moving
average (EMA) of the student network. Consistent predictions between the two
networks are then promoted by optimizing a consistency loss between their
predictions. Interpolation consistency training by Verma et al. [32] combines
mixup [36] and the Mean Teacher model [28] to implement consistency
regularization. During training, unlabelled samples are interpolated to create
an augmented sample. Predictions by the student network on the augmented
sample are then optimized to be consistent with interpolated predictions by
the teacher network on the original non-interpolated samples. Kim et al. [17]
uses cosine similarity in their consistency regularization method for semantic
segmentation. They propose a structured consistency loss that optimizes
predictions to be consistent in not only pixel-wise classification, but also
inter-pixel relationship.

##  3 Our Method

Given source domain images $x_{S}\in X_{S}$ with their annotations (labels)
$y_{S}\in Y_{S}$ and target domain images $x_{T}\in X_{T}$ without annotations
(labels), we want to learn a network $h$ that can correctly predict the
annotations for target images $X_{T}$ denoted by $\hat{Y}_{T}$. Typically,
there is a mismatch in the joint probability distributions of source domain
data $P(X_{S},Y_{S})$ and the target domain data $P(X_{T},Y_{T})$. Due to this
mismatch or the gap between source and target domains, an image segmentation
model $h$ that is trained on the source data usually results in a low
performance on target images. One common solution to address this issue is to
use self-training as also done in the prior works such as DAFormer [16].
However, semi-supervised self-training methods could easily over-fit to source
distribution and could generate inconsistent or wrong pseudo labels for the
target domain images. To overcome this limitation, we propose the addition of
consistency regularization to the DAFormer [16] framework during model
training to further improve model performance. Next, we explain the overall
training framework.

###  3.1 Overall Training

Overall training of the network is composed of three components: supervised
training using source images, self-training using target images, and
consistency regularization. Total loss $\mathcal{L}_{total}$ is given as

$\mathcal{L}_{total}=\mathcal{L}_{S}+\mathcal{L}_{T}+\lambda_{c}\mathcal{L}_{C}$
(1)

where $\mathcal{L}_{S}$ is supervised cross entropy loss using source images,
$\mathcal{L}_{T}$ is self-trained cross entropy loss using pseudo labels,
$\mathcal{L}_{C}$ is our consistency regularization term, and $\lambda_{c}$ is
a parameter we use to weigh $\mathcal{L}_{C}$. The following sections will
present each of the losses in detail.

####  3.1.1 Supervised Training

Supervised training on the source domain is conducted using cross entropy loss
for semantic segmentation. For a source image $x_{S}$ and its annotation
$y_{S}$, $\mathcal{L}_{S}$ can be defined as

$\mathcal{L}_{S}(x_{S},y_{S})=-\frac{1}{HW}\sum_{j=1}^{H\times
W}\sum_{c=1}^{C}y_{S}^{(j,c)}\log h(x_{S})^{(j,c)}$ (2)

where $C$ is the number of classes and $H$ and $W$ are the height and width of
the segmentation output. The notation $y_{S}^{(j,c)}$ denotes the presence of
class $c$ at pixel location $j$ (1 if present and 0 if not). Similarly,
$h(x_{S})^{(j,c)}$ denotes the predicted score for class $c$ at pixel location
$j$ using model $h$ for image $x_{S}$.

####  3.1.2 Self-Training

Self-training uses a teacher network $f(;\phi)$ to produce pseudo labels on
which the student network $h(;\theta)$ will be trained on. For a target image
$x_{T}$, its pseudo label $p_{T}$ is formally defined as

$p^{(j,c)}_{T}=\llbracket
c=\operatorname*{arg\,max}_{c^{\prime}}f(x_{T};\phi)^{(j,c^{\prime})}\rrbracket$
(3)

where $\llbracket\cdot\rrbracket$ denotes the Iverson bracket.

We follow the Mean Teacher model [28] where the weights of the teacher network
$f(;\phi)$ are the EMA of the weights of the student network $h(;\theta)$
after each training step $t$. The EMA weights used by the teacher model at
training step $t$ is formally defined as

$\phi_{t+1}=\alpha\phi_{t}+(1-\alpha)\theta_{t}$ (4)

where $\phi_{t+1}$ is the EMA of successive weights and $\alpha$ is a
smoothing coefficient hyperparameter. It should also be noted that no
gradients will be backpropagated into the teacher network from the student
network.

A confidence estimate for the pseudo labels, defined as the ratio of pixels
with maximum softmax probability exceeding a pre-defined threshold $\tau$, is
also used in the self-training loss. For a target image $x_{T}$, its
confidence estimate $q_{T}$ is formally defined as

$q_{T}=\frac{\sum^{H\times
W}_{j=1}[\max_{c^{\prime}}f(x_{T};\phi)^{j,c^{\prime}}>\tau]}{HW}$ (5)

Self-training loss of the student network $\mathcal{L}_{T}$ for a target image
$x_{T}$ can thus be defined as

$\mathcal{L}_{T}(x_{T})=-\frac{1}{HW}\sum_{j=1}^{H\times
W}\sum_{c=1}^{C}q_{T}\times p^{(j,c)}_{T}\times\log h(x_{T};\theta)^{(j,c)}$
(6)

We follow DAFormer’s [16] method of using non-augmented target images for the
teacher network $f$ to generate pseudo labels and augmented targeted images to
train the student network $h$ using Equation 6. We also follow their usage of
color jitter, Gaussian blur, and ClassMix [21] as data augmentations in our
training process.

####  3.1.3 Consistency Regularization

As mentioned in Mean Teacher [28], cross entropy loss in Equation 6 between
predictions of the student model and pseudo labels (which are predictions from
the teacher model) can be considered as a form of consistency regularization.
However, different from classification problems, semantic segmentation
problems have a property where pixel-wise class predictions are correlated
with each other. Thus, we propose to further enhance consistency
regularization by focusing on this inter-pixel relationship. Inspired by the
method of Kim et al. [17], we use the inter-pixel cosine similarity of
networks’ predictions on target images to regularize the model. Formally, we
define the similarity between pixel $i$ and $j$ class predictions on a target
image $x_{T}$ as

$a_{i,j}=\frac{\textbf{p}^{T}_{i}\textbf{p}_{j}}{\norm{\textbf{p}_{i}}\cdot\norm{\textbf{p}_{j}}}$
(7)

where $a_{i,j}$ represents the cosine similarity between the prediction vector
of the ith pixel and the prediction vector of the jth pixel. Note that the
similarity between the probability vector $\textbf{p}_{i}$ and
$\textbf{p}_{j}$ can also be computed using Kullback-Leibler (KL) divergence
and cross entropy. We investigate these options in Section 4.3.3. The
consistency regularization term, $\mathcal{L}_{C}$ can then be defined as the
mean squared error (MSE) between the student network’s similarity matrix and
the teacher network’s similarity matrix

$\mathcal{L}_{C}=\frac{1}{(HW)^{2}}\sum^{H\times W}_{i=1}\sum^{H\times
W}_{j=1}\norm{a^{s}_{i,j}-a^{t}_{i,j}}^{2}$ (8)

where $a^{s}_{i,j}$ is the similarity obtained from the student network and
$a^{t}_{i,j}$ is the similarity obtained from the teacher network. We also
follow the method of Kim et al. [17] to restrict the number of pixels used in
the calculation of similarity matrices by performing a random sample of
$N_{pair}$ pixels for comparison. Thus, the consistency regularization in
Equation 8 is updated to the following equation

$\mathcal{L}_{C}=\frac{1}{(N_{pair})^{2}}\sum^{N_{pair}}_{i=1}\sum^{N_{pair}}_{j=1}\norm{a^{s}_{i,j}-a^{t}_{i,j}}^{2}$
(9)

This term $\mathcal{L}_{C}$ is particularly useful for domain adaptation as it
helps to minimize the divergence between the source representation and the
target representation by enforcing a structural consistency in the image
segmentation task.

##  4 Experiments

###  4.1 Implementation Details

####  4.1.1 Datasets

We use the Cityscapes street scenes dataset [6] as our target domain.
Cityscapes contains 2975 training and 500 validation images with resolution of
2048$\times$1024, and is labelled with 19 classes. For our source domain, we
use the synthetic datasets GTA5 [23] and SYNTHIA [25]. GTA5 contains 24,966
images with resolution of 1914$\times$1052, and is labelled with the same 19
classes as Cityscapes. For compatibility, we use a variant of SYNTHIA that is
labelled with 16 of the 19 Cityscapes classes. It contains 9,400 images with
resolution of 1280$\times$760\. Following DAFormer [16], we resize images from
Cityscapes to 1024$\times$512 pixels and images from GTA5 to 1280$\times$720
pixels before training.

####  4.1.2 Network Architecture

Our implementation is based on DAFormer [16]. Previous UDA methods mostly used
DeepLabV2 [5] or FCN8s [26] network architecture with ResNet [12] or VGG [27]
backbone as their segmentation model. DAFormer proposes an updated UDA network
architecture based on Transformers that was able to achieve state-of-the-art
performance. They hypothesized that self-attention is more effective than
convolutions in fostering the learning of domain-invariant features.

####  4.1.3 Training

We follow DAFormer [16] and train the network with AdamW [20], a learning rate
of $\eta_{base}=6\times 10^{-5}$ for the encoder and $6\times 10^{-4}$ for the
decoder, a weight decay of 0.01, linear learning rate warmup with
$t_{warm}=1500$, and linear decay. Images are randomly cropped to $512\times
512$ and trained for 40,000 iterations on a batch size of 2 on a NVIDIA
GeForce RTX 3090. We also adopt DAFormer’s training strategy of rare class
sampling and thing-class ImageNet feature distance to further improve results.
For hyperparameters used in self-training, we follow DAFormer and set
$\alpha=0.99$ and $\tau=0.968$. For hyperparameters used in consistency
regularization, we set $N_{pair}=512$, $\lambda_{c}=1.0$ when calculating
similarity using cosine similarity and $\lambda_{c}=0.8\times 10^{-3}$ when
calculating similarity using KL divergence.

###  4.2 Results

Table 1: Comparison with other UDA methods on GTA5 to Cityscapes. Results for
DAFormer and our method using cosine similarity are averaged over 6 random
runs, while results for our method using KL Divergence are averaged over 3
random runs

Method | Road | Sidewalk | Build. | Wall | Fence | Pole | Tr.Light | Sign | Veget. | Terrain | Sky | Person | Rider | Car | Truck | Bus | Train | M.bike | Bike | mIoU19 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
BDL | 91.0 | 44.7 | 84.2 | 34.6 | 27.6 | 30.2 | 36.0 | 36.0 | 85.0 | 43.6 | 83.0 | 58.6 | 31.6 | 83.3 | 35.3 | 49.7 | 3.3 | 28.8 | 35.6 | 48.5 |
ProDA | 87.8 | 56.0 | 79.7 | 46.3 | 44.8 | 45.6 | 53.5 | 53.5 | 88.6 | 45.2 | 82.1 | 70.7 | 39.2 | 88.8 | 45.5 | 59.4 | 1.0 | 48.9 | 56.4 | 57.5 |
DAFormer | 95.5 | 68.9 | 89.3 | 53.2 | 49.3 | 47.8 | 55.5 | 61.2 | 89.5 | 47.7 | 91.6 | 71.1 | 43.3 | 91.3 | 67.5 | 77.6 | 65.5 | 53.6 | 61.2 | 67.4 |
Ours (Cosine) | 95.5 | 69.2 | 89.5 | 52.1 | 49.6 | 48.9 | 55.2 | 62.1 | 89.8 | 49.0 | 91.1 | 71.7 | 45.1 | 91.7 | 70.0 | 77.6 | 65.2 | 56.6 | 62.8 | 68.0 |
Ours (KL) | 96.1 | 71.6 | 89.5 | 53.2 | 48.6 | 49.5 | 54.7 | 61.1 | 90.0 | 49.4 | 91.7 | 70.7 | 44.0 | 91.6 | 70.0 | 78.1 | 68.9 | 55.1 | 62.9 | 68.2 |

Table 2: Comparison with other UDA methods on SYNTHIA to Cityscapes. Results
for DAFormer and our method using cosine similarity are averaged over 6 random
runs, while results for our method using KL Divergence are averaged over 3
random runs

Method | Road | Sidewalk | Build. | Wall | Fence | Pole | Tr.Light | Sign | Veget. | Sky | Person | Rider | Car | Bus | M.bike | Bike | mIoU16 | mIoU13
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
BDL | 86.0 | 46.7 | 80.3 | - | - | - | 14.1 | 11.6 | 79.2 | 81.3 | 54.1 | 27.9 | 73.7 | 42.2 | 25.7 | 45.3 | - | 51.4
ProDA | 87.8 | 45.7 | 84.6 | 37.1 | 0.6 | 44.0 | 54.6 | 37.0 | 88.1 | 84.4 | 74.2 | 24.3 | 88.2 | 51.1 | 40.5 | 40.5 | 55.5 | 62.0
DAFormer | 80.5 | 37.6 | 87.9 | 40.3 | 9.1 | 49.9 | 55.0 | 51.8 | 85.9 | 88.4 | 73.7 | 47.3 | 87.1 | 58.1 | 53.0 | 61.0 | 60.4 | 66.7
Ours (Cosine) | 86.3 | 44.2 | 88.3 | 39.2 | 7.5 | 49.2 | 54.7 | 54.7 | 87.2 | 90.7 | 73.8 | 47.3 | 87.4 | 55.9 | 53.7 | 60.7 | 61.3 | 68.1
Ours (KL) | 89.0 | 49.6 | 88.1 | 40.3 | 7.3 | 49.2 | 53.5 | 52.1 | 87.0 | 88.0 | 73.8 | 46.4 | 87.1 | 58.7 | 53.9 | 61.7 | 61.6 | 68.4

Image | DAFormer | Ours | Ground Truth
---|---|---|---
|  |  |
|  |  |
|  |  |
|  |  |
|  |  |
|  |  |
|  |  |
Figure 1: Qualitative results comparing predictions on validation data of
Cityscapes. From left: input image, predictions by DAFormer, predictions by
our method, and the ground truth. The last row provides an example where
DAFormer performed better compared to our method as it was able to correctly
predict the sidewalks

We compare the results of our method against other state-of-the-art UDA
segmentation methods such as BDL [18], ProDA [37] and DAFormer [16]. In table
1, we present our experimental results on the GTA5 to Cityscapes problem. It
can be observed that our method improved UDA performance from an mIoU19 of
67.4 to 68.0 when cosine similarity is used in Equation 7 and 68.2 when KL
divergence is used. Table 2 shows our experimental results on the SYNTHIA to
Cityscapes problem. Similarly, our method improved performance from an mIoU16
of 60.4 to 61.3 with cosine similarity and 61.6 with KL divergence.

We also observed that our method was able to make notable improvements on the
”Road” and ”Sidewalk” categories. This is especially so on the SYNTHIA to
Cityscapes problem, where we improved UDA performance on ”Road” from 80.5 to
89.0 and ”Sidewalk” from 37.6 to 49.6. We further verify this improvement in
our qualitative analysis presented in Figure 1, where we observed that our
method had better recognition on the ”Sidewalk” and ”Road” categories. We
attribute this improvement to our method’s effectiveness in generating more
accurate pseudo labels. We present pseudo labels generated during the training
process in Figure 2, where we observed more accurate pseudo labels for the
”Road” and ”Sidewalk” categories.

Target Image | DAFormer | Ours
---|---|---
|  |
|  |
Figure 2: Qualitative results on pseudo labels generated from the training
data of Cityscapes. From left: target image, pseudo labels generated by
DAFormer, and pseudo labels generated by our method using cosine similarity

It should be noted that experimental results obtained using the DAFormer
method in Tables 1 and 2 were obtained by averaging 6 random runs using the
official DAFormer implementation222https://github.com/lhoyer/DAFormer. Even
though we were unable to reproduce the exact numbers published in the DAFormer
paper, we believe our experimental results for DAFormer are comparable.

###  4.3 Ablation Study

####  4.3.1 Number of Pixels Sampled

Table 3: Influence of $N_{pair}$ on UDA performance. Results for all experiments were averaged over 3 random runs except for $N_{pair}=512$, which was an average over 6 runs $N_{pair}$ | 4 | 16 | 64 | 256 | 512 | 1024
---|---|---|---|---|---|---
mIoU16 | 61.4 | 60.8 | 61.4 | 61.1 | 61.3 | 60.6

We conducted additional experiments on SYNTHIA to Cityscapes to observe the
effect of $N_{pair}$ (from Equation 9) on model performance. Theoretically,
sampling more pixels for similarity calculation (i.e. a larger $N_{pair}$)
allows us to have a more complete model of the inter-pixel relationship
between predictions. However, empirical results in table 3 suggests that
$N_{pair}$ does not have significant influence on UDA performance. We observe
that very small samples, such as $N_{pair}=4$, were able to obtain comparable
results with larger sample sizes.

Additional experiments using $N_{pair}=4$ were conducted to observe the
locations of sampled pixels. Visualization of our ablation study is presented
in Figure 3. We found that after 40,000 training iterations, sampled pixels
covered approximately 45.73% of the 512$\times$512 images the network was
trained on despite the small sampling size. This suggests that if a reasonable
image coverage can be obtained during the training process, a small $N_{pair}$
is sufficient to model the inter-pixel relationship between predictions,
allowing us to minimize computational cost of our consistency regularization
method. The influence of sampling coverage and sampling distribution on the
effectiveness of consistency regularization is an interesting study that can
be explored in the future.

|  |  |
---|---|---|---
(a) | (b) | (c) | (d)
Figure 3: Visualization of pixels sampled in experiments using $N_{pair}=4$.
(a) Target image cropped to $512\times 512$; (b), (c) and (d) visualizes
sampled pixels in three separate runs

####  4.3.2 Proximity of Sampled Pixels

Kim et al. adopted cutmix augmentation [35] in their consistency
regularization method [17] to limit sampled pixel pairs to within a local
region. They theorized that pixel pairs that are in close proximity to each
other have high correlation, and hence have more effect on UDA performance. We
tested this theory on SYNTHIA to Cityscapes by performing $N_{box}$ crops and
sampling $N_{pair}$ pixels from each crop. This localizes sampled pixels and
restricts them to have closer proximity. Sampled pixels are then used to
compute inter-pixel similarity to obtain a $N_{box}\times N_{pair}\times
N_{pair}$ similarity matrix which is used for loss calculation in Equation 8.

Table 4: Influence of $N_{box}$ and $N_{pair}$ on UDA performance. Total number of sampled pixels i.e. $N_{box}\times N_{pair}$ is kept at 512 for a fair comparison Crop Size | $N_{box}$ | $N_{pair}$ | mIoU16
---|---|---|---
256 | 32 | 16 | 61.2
128 | 32 | 16 | 61.1
64 | 32 | 16 | 60.2

We present the experimental results in Table 4 where three different crop
sizes were varied to restrict the proximity of sampled pixels. We did not
observe an improvement in UDA performance compared to results presented in
Table 2, suggesting that proximity of sampled pixels perhaps may not be that
influential for consistency regularization.

####  4.3.3 Measuring Inter-Pixel Similarity

In Section 3.1.3, we adopted the method of Kim et al. to use cosine similarity
in the measure of inter-pixel similarity [17]. In this section, we conducted
additional experiments on SYNTHIA to Cityscapes to observe the influence
different methods of measuring inter-pixel similarity have on UDA performance.

Table 5: Comparison of UDA performance using different methods to calculate inter-pixel similarity. We also provide the optimal $\lambda_{c}$ obtained using hyperparameter tuning Method | $\lambda_{c}$ | mIoU16
---|---|---
Cosine Similarity | 1.0 | 61.3
Cross Entropy | $1.0\times 10^{-3}$ | 61.2
KL Divergence | $0.8\times 10^{-3}$ | 61.6

We tested the usage of cross entropy and KL divergence to measure inter-pixel
similarity instead of cosine similarity in Equation 7. Results from our
empirical experiments are presented in Table 5. We observed that all three
methods provided comparable results with each other, with KL divergence
providing slightly better results.

##  5 Conclusion

In this work we presented a new consistency regularization method for UDA
based on relationships between pixel-wise class predictions from semantic
segmentation models. Using this technique we were able to improve the
performance of the state-of-the-art DAFormer method. We also observed that
even with smaller number of sampled pixel pairs $N_{pair}$, this
regularization method was still able to be effective. Therefore, with minimal
computational cost, we are able to improve the results of self-training
methods for unsupervised domain adaptation.

Acknowledgment This research is supported by the Centre for Frontier AI
Research (CFAR) and Robotics-HTPO seed fund C211518008.

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# Gluon mass from charmonium radiative decay and its impact on the hybrid
meson properties

Azzeddine Benhamida<EMAIL_ADDRESS>Lahouari Semlala  Faculty of Exact
Science, LPTO, University of Oran 1-Ahmed Ben Bella, Algeria  Ecole Supérieure
en Génie Electrique et Energétique d’Oran, Algeria

###### Abstract

We study the photon spectrum in the radiative decays of the charmonium
$\frac{1}{\Gamma}\frac{d\Gamma(J/\psi\rightarrow\gamma gg)}{dz}$ as an
excellent scene for hunting the exotic mesons (hybrids $q\bar{q}g$ and
glueballs $gg$) and an ideal frame for the theoretical and phenomenological
investigations of the gluon nature in IR, by subtle estimation of its non-zero
effective mass which seems required in Quark-Gluon Constituent Model (QGCM)
for identifying the nature of these exotic species.

###### keywords:

QCD , Quarkonium , Exotic mesons , QGCM , Gluon mass , Hybrid meson.

contents=  Submitted to the Proceedings of the African Conference on
Fundamental and Applied Physics Second Edition, ACP2021, March 7–11, 2022 —
Virtual Event

, scale=1,placement=top,opacity=1,color=black,position=3.25in,1.2in

##  1 Introduction

The radiative decay of heavy quarkonium $J/\psi\rightarrow\gamma gg$ is the
most promising process to search for glueballs $gg$ and to a lesser extent
hybrids $q\bar{q}g$. Its advantage is that the $c\bar{c}$ system decay into
light quarks is suppressed by the OZI rule [1]. Therefore, in most decays, the
$J/\psi$ undergoes a transition into three gluons which then decay to form
hadrons. But $J/\psi$ can also decay into two gluons and a photon. The photon
can be experimentally detected, while the two gluons “theoretically” can bind
together to form a glueball (see Figure 1), or form a hybrid via quark pair
creation (QPC) by the annihilation of one of the gluons, and decay
subsequently to light exotic mesons.

Figure 1: (a, b) Glueball and hybrid meson production in the decay
$J/\psi\rightarrow\gamma(X_{G},X_{H})$, (c) a production of hybrid meson
through $J/\psi\rightarrow(\omega,\rho)X_{H}$.

The main salient feature of these exotic mesons is the excitation gluon
picture, as it is presumed by many potential models as a massive quasi-
constituent particle ( see [2] and references therein) as it plays a double
role inside the hadron; since it propagates the interaction between color
sources (valence quarks) and, being itself colored it undergoes the
interaction. But the exploration remains neither complete nor definitive since
there is a slight problem with the concept “constituent”. Furthermore, it is
not clear yet whether the number of gluons is conserved, and do they have a
non-vanishing rest mass in the infrared (IR) [3].

##  2 Phenomenological gluon mass in $J/\psi$ radiative decay

First we simulate the photon spectrum in $J/\psi\rightarrow\gamma gg$ to
extract $\alpha_{s}$ from the ratio

$R_{\gamma}=\frac{\Gamma_{\gamma gg}}{\Gamma_{ggg}}=\frac{N_{\gamma
gg}}{N_{ggg}}\quad\sim\frac{\alpha}{\alpha_{s}}$

(here $N_{\gamma gg}$ and $N_{ggg}$ represent the number of event) and then
estimate the Phenomenological gluon mass by fitting the photon spectrum
$\frac{1}{\Gamma}\frac{d\Gamma}{dz}$ (where $z=2E_{\gamma}/M_{J/\psi}$) taking
into account all possible theoretical corrections:

### relativistic corrections

Since heavy quarkonium contains at least heavy (c or b) quark, the simulations
using non-relativistic QCD or heavy quark effective theory, which expanses the
QCD Lagrangian in powers of the heavy quark velocity, or the heavy quark mass.

The NRQCD factorization involves separating the short-distance physics of the
annihilation process at scales $\sim 1/m$ that are calculated perturbatively
in $\alpha_{s}(m_{q})$ and encoded in matching coefficients, from the non-
relativistic physics of the quarkonium system over scales $\sim 1/(mv)$ ($v$
being the heavy-quark velocity in the quarkonium rest frame).

The long-distance effects are parameterized by a nonperturbative matrix
element $\langle\mathcal{O}^{V}_{n}\rangle$ which is subject to the power
counting rules of NRQCD and can be evaluated numerically using (NRQCD) lattice
simulations.

### High order QCD corrections

The perturbative HO QCD correction basically split into two types of
contributions, direct and fragmentation [4] ;

$\frac{d\Gamma}{dz}=\frac{d\Gamma^{dir}}{dz}+\frac{d\Gamma^{frag}}{dz}$

where

$\frac{d\Gamma^{dir}}{dz}=\sum_{i}C_{i}(M,z)\langle\Psi_{J/\psi}|\mathcal{O}_{i}|\Psi_{J/\psi}\rangle,\hskip
28.45274pt\frac{d\Gamma^{frag}}{dz}=\sum_{a=q,\bar{q},g}\int_{z}^{1}\frac{dx}{x}C_{a}(x)D_{a\gamma}\left(\frac{z}{x},M\right),$

where $C_{a}(x)$ represents the partonic kernels and $D_{a\gamma}(z/x,M)$
represents the fragmentation functions, and $M$ is the mass of the vector
quarkonium $J/\psi$. The partonic kernels can be expanded in powers of $v$.
And $C_{i}(z,M)$ are short distance Wilson coefficients, calculable in
perturbation theory, while $\langle\Psi_{V}|\mathcal{O}_{i}|\Psi_{V}\rangle$
are the NRQCD matrix elements with the $J/\psi$ quantum states
$|\Psi_{V}\rangle$ ( $V=J/\psi$).

Many theoretical considerations have been done in this context [4, 5], and
showed that, different approximations are needed in the different regions of
the photon spectrum , and the spectrum has been described quite well in the
case of bottomonium $\Upsilon$. While in the case of charmonium $J/\psi$ this
NRQCD formalism is not able to describe the experimental data [4, 6], which
infers that the gluon mass still needs to be incorporated especially in the
upper end-point region of the spectrum as suggested in [3, 7, 5] and advised
also in [4].

The complete formula of the spectrum up to NLO [8] can be written as

$\frac{d\Gamma}{dz}=\frac{d\Gamma_{LO}}{dz}+\frac{d\Gamma_{NLO}}{dz}+\frac{d\Gamma_{LO,\alpha_{s}}}{dz}$

the first term consists of the gluon mass correction factors through the
formula obtained by Liu and Wetzel [9]

$\begin{split}\frac{1}{\Gamma_{0}}\frac{d\Gamma}{dz}=&\frac{1}{\pi^{2}-9}[\frac{x_{+}-x_{-}}{z^{2}}+\frac{\ln(x_{+}/x_{-})}{z^{2}(-2+4\eta+z)^{3}}[8(2\eta-1)^{2}\\\
&(2-4\eta+7\eta^{2})+8(2\eta-1)(5-12\eta+10\eta^{2}+2\eta^{3})z\\\
&+2(2\eta-1)(-17+10\eta+6\eta^{2})^{2}+2(-5+2\eta+2\eta^{2})z^{3}]\\\
&+\frac{(1/x_{-}1/x_{+})}{z^{2}(-2+4\eta+z)^{2}}[4(2\eta-1)^{2}(1+3\eta^{2})+4(2\eta-1)\\\
&(3-4\eta+2\eta^{2}+2\eta^{3})z+2(7-18\eta+10\eta^{2}+10\eta^{3})z^{2}\\\
&+4(2+\eta)(2\eta-1)z^{3}+(2+\eta)z^{4}]]\end{split}$

where $z_{\pm}=1-2\eta-\frac{z}{2}\left[1\mp\sqrt{1-\frac{4\eta}{1-z}}\right]$
and $\eta=2m_{g}/M_{j/\psi}$, while the second and the third terms are
detailed in refs [8, 4].

Our initial results of the gluon mass are approximate $m_{g}\approx 0.67\pm
0.18$ GeV, which is simulated with a method similar to one in [5] by fitting
the improved photon spectrum with the theoretical corrections given in [4, 8]
with the experimental data obtained by CLEO collaboration [6]. However, the
result is not worth discussing here as it still needs more enhancements to
reproduce the photon spectrum since the errors bar seems rather large.

##  3 Search for hybrid mesons with exotic $J^{PC}$

The criteria to identify hybrid mesons such as X(1870) with quantum numbers
$0^{+}0^{-+}$, $0^{+}1{{}^{+}+}$, or $0^{+}2^{-+}$ as closest candidates are
through the systematical study of its mass spectrum and strong decay
properties, the results are not yet given here but we give an overview of
calculations. However, the first-principle computations, directly from the QCD
Lagrangian are extremely difficult, due to the failure of perturbation
expansions for QCD at low energies. As a result, our calculations are based on
the quark-gluon constituent model (QGCM) [2] as a QCD-based phenomenological
potential model. This model requires two main external ingredients (the gluon
mass $m_{g}$ and the strong coupling constant $\alpha_{s}$) that are extracted
phenomenological (this work) or through continuum methods [3].

###  3.1 Mass spectrum

In this case we shall use the following lowest-lying state $q\bar{q}$-cluster
wave function for spin-space representation (for more details see [2]and
references therein);

$\begin{split}\Psi_{JM}^{\textrm{PC}}(\overrightarrow{\rho},\overrightarrow{\lambda})&=\left(\left(\left(\mathbf{e_{\mu_{g}}}\otimes\psi_{l_{g}}^{m_{g}}\right)_{j_{g}M_{g}}\otimes\psi_{l_{q\bar{q}}}^{m_{q\bar{q}}}\right)_{Lm}\otimes\chi_{\mu_{q\bar{q}}}^{S_{q\bar{q}}}\right)_{JM}^{\textrm{PC}}\\\
&=\sum_{\left(L\,l_{g}j_{g}l_{q\bar{q}}S_{q\bar{q}};\textrm{PC}\right)}\Psi_{JM;Ll_{g}j_{g}l_{q\bar{q}}S_{q\bar{q}}}(\overrightarrow{\rho},\overrightarrow{\lambda}),\end{split}$
(1)

Thus the hybrid meson mass can be calculated through.

$M_{J^{PC}}\simeq En(M_{i}^{0},\beta_{l_{q\bar{q}}l_{g}}^{0})_{J^{PC}}+\Delta
E(M_{i}^{0},\beta_{l_{q\bar{q}}l_{g}}^{0})_{J^{PC}}.$ (2)

while

$En(M_{i}^{0},\beta_{l_{q\bar{q}}l_{g}}^{0})_{J^{PC}}=\frac{\langle\Psi^{PC}_{JM}|H_{R}|\Psi^{PC}_{JM}\rangle}{\langle\Psi^{PC}_{JM}|\Psi^{PC}_{JM}\rangle},\hskip
28.45274pt\Delta
E(M_{i}^{0},\beta_{l_{q\bar{q}}l_{g}}^{0})_{J^{PC}}=\frac{\langle\Psi^{PC}_{JM}|\Delta
V_{SD}|\Psi^{PC}_{JM}\rangle}{\langle\Psi^{PC}_{JM}|\Psi^{PC}_{JM}\rangle}$

where $H_{R}$ and $\Delta V_{SD}$ are the Hamiltonian of the hybrid meson
state and spin-dependent corrections term respectively (we can not show all
the details here since it will blow up the notation considerably, for more
details the reader may take a look at ref [2]and references therein).

$H_{R}=\frac{\vec{p}^{2}_{\rho}}{2\mu_{\rho}}\frac{\vec{p}^{2}_{\lambda}}{2\mu_{\lambda}}+\frac{M_{q}}{2}+\frac{m_{q}^{2}}{2M_{q}}+\frac{M_{\bar{q}}}{2}+\frac{m_{\bar{q}}^{2}}{2M_{\bar{q}}}+\frac{M_{g}}{2}+\frac{m_{g}^{2}}{2M_{g}}+V_{eff}$

###  3.2 Strong decay

To lowest order the decay of an hybrid state A to two light meson stats B and
C is described by the matrix element of the Hamiltonian of quark pair creation

$\left\langle
BC|H|A\right\rangle=g\,f(A,B,C)\,\left(2\pi\right)^{3}\delta_{3}\left(\vec{p_{A}}-\vec{p_{B}}-\vec{p}_{C}\right).$
(3)

the Hamiltonian of the pair creation $g\rightarrow q\bar{q}$ is defined as

$H=\sum_{ss^{\prime}\lambda}\int\frac{d\vec{p}d\vec{p^{\prime}}d\vec{k}}{(2\pi)^{3}\sqrt{2\omega}}\delta^{(3)}(\vec{p}-\vec{p^{\prime}}-\vec{k})\bar{u}_{\vec{p}s}\gamma_{\mu}\frac{\lambda^{a}}{2}u_{-\vec{p^{\prime}}s^{\prime}}b^{{\dagger}}_{\vec{p}s}d^{{\dagger}}_{-\vec{p^{\prime}}s^{\prime}}a_{\vec{k}\lambda}\varphi_{a}\varepsilon_{\vec{k}\lambda}^{\mu}.$
(4)

The partial width is given by:

$\Gamma_{A\rightarrow
BC}=4\alpha_{s}\left|f(A,B,C)\right|^{2}\frac{P_{B}E_{B}E_{C}}{M_{A}},$

##  4 Summary

The advantage of the charmonium meson is that it is a multiscale system that
can probe all regimes of QCD since its mass scale falls in between those low
and high-energy scales. This makes it an excellent testing laboratory for the
interplay between perturbative and non-perturbative QCD and an excellent scene
for hunting the exotic mesons as well as the theoretical and phenomenological
investigations of the gluon nature in the infrared, which may shed some light
on the long-distance QCD dynamics since seemingly, gluons effective mass is a
true QCD property.

## Acknowledgments

This work was supported by the PRFU research program (under No.
B00L02EP310220190001).

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