# Modular Neural Network Approaches for Surgical Image Recognition

Nosseiba Ben Salem1, Younès Bennani1, Joseph Karkazan2, Abir Barbara1, Charles
Dacheux2, Thomas Gregory2
1Université Sorbonne Paris Nord
LIPN, CNRS UMR 7030
La Maison des Sciences Numeriques
F-93430,
Villetaneuse - France
<EMAIL_ADDRESS>
2APHP-Hôpital Avicenne, Bobigny - France
<EMAIL_ADDRESS>
###### Abstract

Deep learning-based applications have seen a lot of success in recent years.
Text, audio, image, and video have all been explored with great success using
deep learning approaches. The use of convolutional neural networks (CNN) in
computer vision, in particular, has yielded reliable results. In order to
achieve these results, a large amount of data is required. However, the
dataset cannot always be accessible. Moreover, annotating data can be
difficult and time-consuming. Self-training is a semi-supervised approach that
managed to alleviate this problem and achieve state-of-the-art performances.
Theoretical analysis even proved that it may result in a better generalization
than a normal classifier. Another problem neural networks can face is the
increasing complexity of modern problems, requiring a high computational and
storage cost. One way to mitigate this issue, a strategy that has been
inspired by human cognition known as modular learning, can be employed. The
principle of the approach is to decompose a complex problem into simpler sub-
tasks. This approach has several advantages, including faster learning, better
generalization, and enables interpretability.

In the first part of this paper, we introduce and evaluate different
architectures of modular learning for Dorsal Capsulo-Scapholunate Septum
(DCSS) instability classification. Our experiments have shown that modular
learning improves performances compared to non-modular systems. Moreover, we
found that weighted modular, that is to weight the output using the
probabilities from the gating module, achieved an almost perfect
classification. In the second part, we present our approach for data labeling
and segmentation with self-training applied on shoulder arthroscopy images.

_K_ eywords Data Science $\cdot$ Machine Learning $\cdot$ Modular Neural
Networks $\cdot$ Self-training $\cdot$ Dorsal Capsulo-Scapholunate Septum
$\cdot$ Shoulder arthroscopy.

##  1 Introduction

Cognitive neuropsychology has been a great inspiration and the key behind the
success of data science. Our brains are the most powerful machines, that’s why
they have been trying to replicate how it works and how it solves a problem
into machines. This was the main reason to the idea of neural networks and
their ability to handle complex problems. Continuing to explore the human
mind, some researchers focused their attention on how our brain is subdivided
into independent components. Each subset of neurons specializes in a function,
and they interact with each other [12]. Thereby, the beginning of using
modular architecture in learning. The main idea is to decompose a complex task
into simple independent sub-tasks. It was first introduced in [20] when they
demonstrated that dividing processing produces better performance under
certain conditions. Later, other researchers revealed that adding a priori
knowledge helps the model improve the results compared to non-modular. And in
recent research [15] stated that modularity leads to better generalization,
faster learning, possesses scaling properties, and allows interpretability.
Therefore, the rising interest among researchers. The modular architecture is
a concept, therefore, it exists several strategies to decompose a problem. For
instance, models can compete with each other to complete the same task [1], or
they can cooperate [18], etc. However, it may not always be evident to
determine how and where to subdivide.

Another concept that has shown remarkable results, is the semi-supervised
algorithm self-training. With the urging need for a massive number of data to
train models and the difficulty or high cost of labeling data, semi-supervised
learning became an active research discipline. Self-training alleviates this
problem by relying on a small subset of labeled samples and a large subset of
unlabeled samples, and still achieves state-of-the-art results [5]. It assumes
that the decision boundary should lie in a low-density region to obtain better
generalization [7]. Pseudo-labeling, that is annotating the unlabeled samples
iteratively, we call them pseudo-labels, and adding them to the training
subset, is a common use of self-training.

In this work, we built systems using modular learning and self-training and
applied them in two contexts. The first application aims to classify
instability in a wrist structure named Dorsal Capsulo-Scapholunate Septum
(DCSS). We tested different strategies of modular learning and compared them
to the non-modular architecture. In the second application, we tested pseudo-
labeling on shoulder arthroscopy images to detect structures using image
segmentation.

To this end, we describe a brief overview of the background and investigate
the related works on both modular learning and self-training in the first
section. Then in the second section, we present our proposed approach for
classifying DCSS using modular learning and display the obtained results.
Finally, we present the self-training approach for data labeling and
segmentation.

##  2 Fundamental background of the proposed approach and related work

In this section, we will introduce some basic background on modular learning
and self-training methods and give an overview on the previous approaches in
the literature.

###  2.1 Modular learning

According to neuroscience, our brain is an amalgamation of several modules
operating independently and they cooperate and communicate with each other
[25]. This decomposition, inspired researchers to build a system composed of
different simple modules and then integrate their outputs. Each module is a
neural network with learnable parameters and output a probability of
prediction. Modules are trained independently and are controlled by a gating
module. The gating module is responsible to activate modules according to a
certain rule, either set by the user or found automatically. There exist
several ways to set the interaction between the different modules. For
instance, one can fragment a complex problem into smaller sub-tasks. Each
module is responsible for a fragment, and they cooperate between each other to
create a complementary network. Or modules can compete between each other by
learning to solve the same problem.

Using modular learning, offers major advantages comparing to non-modular
system [15]. For instance, using multiple simple networks might significantly
reduce the computational time, usually needed to train a complex model. It has
been shown that modular systems improves substantially the robustness of
neural networks[24]. Combining knowledge from different modules makes the
model more robust. It also reduces the model’s complexity, thus optimized
computation time in the training process. Moreover, the modularity limits the
propagation of biases, contrary to a non-modular model. In 1989, Rueckl, [20]
began to study the representation and location of objects of "what" and
"where" to separate. They used retinal images for classification and revealed
that modular learning is more efficient in computation and learned more easily
compared to non-modular network under certain circumstances. There exists
several strategies to build modular architecture depending on the type of
interaction between the structures. For instance, in [2], they use decoupled
modules where data is separated and each group is assigned to a module. The
final prediction is the maximum probability from all modules. In [18], they
used hierarchical network. Others [1], and [3], use ensemble networks where
all the modules are identical, and they learn to perform the same task. The
final prediction is a majority vote of prediction probability from all the
networks.

The most common structure is the Multiple-experts network [11]. It consists of
gating module and experts. The gating module controls the activation of the
networks and distributes the tasks. The networks do not need to be identical
it can be a hybrid system with a variety of techniques [4]. We used multiple
experts in our case.

###  2.2 Self-Training

Supervised learning using neural networks has seen success in several
applications. To ensure a better generalization, neural networks require to be
trained on a huge amount of labeled data. However, labeling data can be an
expensive and time-consuming task and in some cases not accessible. Thus, the
rise of a domain called semi-supervised. This domain aims to use both labeled
and unlabeled data from the same or similar distribution for a specific
problem. There exist several semi-supervised methods. Self-training is a
common semi-supervised method. Its increasing interest is due to its better
performance compared to supervised learning. This result is somehow
surprising, given that we don’t add any external information. Theoretical
works have proven that self-training engenders a better generalization bound
given certain conditions.

Let $N$ be the number of labeled data and $M$ the number of unlabeled data. An
initial model $f_{\bm{w}^{(0)}}$ is trained from
$\left\\{\bm{x}_{n},y_{n}\right\\}_{n=1}^{N}$. Given the unlabeled data
$\left\\{\widetilde{\bm{x}}_{m}\right\\}_{m=1}^{M}$ The obtained model plays
the role of pseudo-labeler by computing
$\hat{y}_{m}=\hat{f}_{\bm{w}^{(0)}}(\bm{x}_{m})$. Let $k$ be the number of
selected predicted labels and let $\ell$ be the number of iterations. Then
$f_{\bm{w}^{(\ell)}}$ is trained from
$\left\\{\bm{x}_{n},y_{n}\right\\}_{n=1}^{N}\cup\left\\{\bm{x}_{m},\hat{y}_{m}\right\\}_{m=1}^{k}$
and is used as pseudo-labeler for the next iteration. Reiterate until reaching
stopping criteria. The empirical risk to be minimized is as follows

$\displaystyle\hat{R}(\bm{W})$ $\displaystyle{}={}$
$\displaystyle\frac{\lambda}{2N}\sum_{n=1}^{N}\left(y_{n}-f\left(\bm{W};\bm{x}_{n}\right)\right)^{2}$
(1)
$\displaystyle+\frac{\tilde{\lambda}}{2M}\sum_{m=1}^{M}\left(\hat{y}_{m}-f\left(\bm{W};\hat{\bm{x}}_{m}\right)\right)^{2}$

where $\lambda+\tilde{\lambda}=1$.

Similar to modular training, self-training was used decades ago from 1976 to
train a model without external knowledge [9]. And in these recent years,
researchers have shown an interest in self-training given its ability to train
the model with few labeled data and a lot of unlabeled ones. The strategy of
choosing pseudo-labels differs from work to another. For instance, [14],
selected the data having the maximum prediction probability in the current
model tested on denoising auto-encoder,[22] selected the nearest observations
in feature space . Others [27], [23], selected the proportion of the most
confident data instead of setting a threshold.

The preceding research has supposed all the classes are equals and uses a
constant threshold to sort out the most confident unlabeled data to add to the
training set. That is what motivated the use of dynamic threshold to prevent
ignoring certain unlabeled data. [26], suggested adapting curriculum learning
in self-training and called it curriculum pseudo labeling (CPL). Curriculum
learning is an approach based on training the model starting with the data
easier to learn to the more complex ones. The strategy of CPL is to adapt the
threshold for each class and select unlabeled data depending on the model’s
learning status. They compute performances for different classes and adjust
the threshold accordingly. Meaning, if the results in a certain class are not
high, the threshold ought to be lower. The authors demonstrated that their
method converged faster than fixed threshold methods and achieved good results
on state-of-the-art benchmarks of semi-supervised learning. Another use of
curriculum labeling in [6] achieved significant results. They observed that
the distribution of the maximum probability prediction follows the Pareto
distribution, that is, a skewed distribution. The idea is to choose unlabeled
data by taking a percentile of the distribution of the maximum probability for
pseudo-labeled samples.

##  3 Proposed approaches: Modular learning for classification

###  3.1 Motivation

The first purpose of this work is to classify images of the Dorsal Capsulo-
Scapholunate Septum (DCSS) to identify the stage of scapholunate instability
using a modular architecture. DCSS is a structure in the wrist that ensures a
scapholunate articulation [16]. A common disease in DCSS is scapholunate
instability. It causes severe wrist dysfunction, making it difficult for the
patient to perform everyday activities [13]. Identifying the scapholunate
instability requires an accurate diagnosis in order to prevent it in the early
phases. We used images from wrist arthroscopy and the goal is to determine
whether the structure is healthy or if it exists instability in stage 1, stage
2, or stage 3. We exploited the nature of the problem to build a modular
system. We can divide the problem into, first, predicting if the structure is
healthy or pathological. Then, if the DCSS is pathological, we can then use a
second component to discriminate between the three stage.

###  3.2 System architecture

The system is composed of two modules; the gating module and the
discriminative module. The gating module is trained to detect if the DCSS is
healthy or not from the arthroscopy image. If the DCSS is healthy, we output
its prediction probability $P(y=\text{'healthy'}|X)$. If the DCSS is
pathological, we feed the image to the discriminative module. The latter can
be designed in different ways, and the final output can be computed
differently. We choose to test three architectures (modular, modular 1 Vs 1,
and weighted modular). In the following, we will detail the different cases.
Let $H$ be the event ’healthy’, $\bar{H}$ ’pathological’ and $S_{i}$,
$i\in\\{1,2,3\\}$ the stages of instability.

##### Modular (All)

In this case, the discriminative module is trained to distinguish between the
three stages of instability. Given the image predicted by the gating module to
be pathological, the discriminative module computes the prediction probability
$P(S_{i}|X,\bar{H}),\quad i=1,\ldots,3$. The discriminative module’s
prediction is as follows:

$\operatorname{arg\,max}_{i}P(S_{i}|X,\bar{H})$ (2)

##### Modular (1 Vs 1)

In this case, we built three experts competing with each other. The first
module is intended to discriminate between stage 1 and 2, The second for the
stage 1 and 3, and, finally, one for the stage 2 and 3. The output is the
maximum from each expert. The three modules computes the probabilities in
equation 3 with i= {1,2}, {1,3} and {2,3} respectively.

$P_{m}(S_{i}|X,\bar{H})\times P_{m}(\bar{H}|X)+P_{m}(S_{i}|X,H)\times
P_{m}(H|X).$ (3)

The discriminative module’s prediction is as follows:

$\operatorname{arg\,max}_{(i,k)}P_{m_{k}}(S_{i}|X),\quad i=1,\ldots,3\quad
k=1,\ldots,3.$ (4)

##### Weighted modular

This system is similar to the first modular case, except for computing the
output. $P(S_{i}|X,\bar{H})\times P(\bar{H}|X)+P(S_{i}|X,H)\times P(H|X),\quad
i=1,\ldots,3$. In this case, the gating module assigns a weight to each output
and picks the output that yields the highest prediction probability. The
advantage of this method is to take into account the error in the gating
module. In the first system, we supposed that the gating module is perfect
opposing to this architecture. For instance, if the difference between the
prediction probability of the gating expert is not large, it means that the
module is not confident. By assigning weights to the outputs with the gating
module probabilities, we add this information to the prediction. The figure 1
represents the weighted modular architecture. The discriminative module’s
prediction is as follows:

$\operatorname{arg\,max}_{i}P(S_{i}|X),\quad i=1,\ldots,3.$ (5)

Figure 1: Architecture of weighted modular learning.

####  3.2.1 Learning phase

In each module, the network is composed of a pretrained model ResNet18 and a
fully connected layer. The architecture of ResNet18 is illustrated in figure
2. It is composed of 18 deep layers. Every four layers are similar, and each
layer is composed of two blocks of deep residual network. The latter
characterizes by adding an identity connection, meaning if the output of the
layer is $F(x)$, the output of the block is $F(x)+x$ followed by the
activation function ReLU. The residual network is an attempt to solve the
problem of vanishing gradient, that is, when the gradient reaches zero quickly
[10]. And the final layer consists of average pooling followed by a fully
connected layer and a softmax.

Figure 2: ResNet18 architecture[17].

###  3.3 Experiments

####  3.3.1 Dataset

The images used in this work were sampled from arthroscopic videos on Dorsal
Capsulo-Scapholunate Septum (DCSS) performed at Avicenne Hospital . The
collected data includes 840 photos in total, including 105 healthy DCSS images
(12.5%), 173 (20,6%) in stage 1, 187 (22.2%) in stage 2, and 375 in stage 3
(44.64%). Entry from each patient was equally distributed. In order to
maintain a balance between the classes, we have expanded the training set
using image manipulations such as random rotations, adjusting the brightness,
zooming in on the image, etc. After transformation, we were able to obtain
3076 healthy DCSS images, 1111 in stage 1, 1126 in stage 2, and 1279 in stage
3. The image’s size was then resized to $224\times 224$. The figure 3 presents
samples from the dataset.

Figure 3: Visualization examples of DCSS images.

###  3.4 Setting

We used 80% (6592) of the dataset for training and 20% (252) for testing. We
used stochastic gradient descent to optimize the model and a learning rate of
$0.001$. We trained for 10 epochs. We used the same hyperparameters for both
models. The first model was trained on all the training datasets labeled
healthy, not healthy and the second model was trained on only non-healthy
DCSS.

####  3.4.1 Results and discussion

We evaluated the three types of modular learning using accuracy with an
estimation of the confidence interval. The confidence interval was computed as
explained in [4]. They assume that the probability distribution of the number
of errors is binomial, so we calculate the interval according to the formula
below

$I_{\alpha}=\frac{P+\frac{Z_{\alpha}^{2}}{2N}\pm
Z_{\alpha}\sqrt{\frac{P(1-P)}{N}+\frac{Z_{\alpha}^{2}}{4N^{2}}}}{1+\frac{Z_{\alpha}^{2}}{N}},$
(6)

where $N$ is the number of observations in the test set, $P$ is the
probability that an observation is an error, and, $\alpha\%$ is the confidence
level, usually set to $95\%$, $Z_{\alpha}$ is a value given by the normalized
centered Gaussian distribution ($Z_{95\%}=1.96$ and $Z_{90\%}=2.48$).

The table 1 summarizes the results obtained. As we can see in the table,
modular learning increased the accuracy from $97.6\%$ to $98.4\%$ and a
tighter confidence interval of $[97.79\%;100\%]$ with an almost perfect gating
module. These results demonstrate that adding a priori knowledge from the
modular nature of the problem allowed a better generalization and a finer
decision boundary than the non-modular architecture. The decomposition of the
task makes the problem less complex, thus a bigger probability to converge
towards the ground truth function.

The experiments in 1, 2 also showed that weighted modular had better
performance than the first case modular architecture with $98.81\%$ accuracy.
This indicates that adding probabilities from the gating module calibrates the
results.

We investigated the distribution of the difference between the prediction
probability of the different classes in the discriminative module. We can see
in figure 4 that the median difference is almost one, which proves that the
discriminative module is confident in its decision. We can also notice a high
difference between the prediction probability of stage 2 and 3 meaning that
the model is able to distinguish between the two stages confidently. However
surprisingly, the probabilities between stages 1 and 3 are relatively small.
By analyzing the images further, we notice the presence of redness in both
stages, which could be the reason for such results.

Since the problem is medical, it is important to explain the choice made by
the model. However, neural network is a black box. Therefore, we attempted to
evaluate two methods to interpret the results using a heatmap on images. The
pixels with the higher contributions to the neural network’s prediction are
located in hotter areas. The first method is called GradCam (gradient-weighted
class activation mapping), which creates visual explanations [21]. Grad-CAM
creates a coarse localization map that highlights key areas in the image by
using the gradients of any target concept and flowing into the final
convolutional layer. This makes it possible to visualize the results of
various CNN model layers.

The second approach used for explainability is Sobol attribution method [8].
It relies on sensitivity analysis techniques commonly used in physics and
economics, in particular the Sobol indices. The method, not only does it
identify the contribution of each image regions to the neural network’s
decision but also, the sobol indices are able to estimate High-order
interactions between the regions in the image and their contributions. To
estimate the indices, the authors used Jansen estimator with Quasi-Monte Carlo
for sampling. First, they sample masks from the sequence, apply them to the
image with a perturbation operator, then obtain prediction scores by
forwarding the obtained image to the system. The explanations are then
computed using the masks and the prediction scores. In figure 5 and 6, we
evaluate displays the visual output of the discriminative model of the two
methods. We notice that the GradCam provides a continuous region, and we
notice a certain pattern for each class, while the Sobol attribution method
gives a finer heatmap. In both cases, both methods were able to locate the rip
in the DCSS in Stage 3.

Table 1: Average accuracy on different modular architecture Architecture | Accuracy (%) | Confidence interval(%)
---|---|---
Non-modular | 97.6 | [96.52 ; 100]
Modular (All) | 98.4 | [97.79 ; 100]
Modular (1 Vs 1) | 94.4 | [91.1 ; 100]
Weighted modular | 98.81 | [98.4 ; 100]
Table 2: Evaluation matrix of weighted modular on test set | Predicted result | Recall (%)
---|---|---
| Healthy | Stage 1 | Stage 2 | Stage 3 |
Ground truth | Healthy | $30$ | $0$ | $0$ | $0$ | $100$
| Stage 1 | $2$ | $60$ | $1$ | $0$ | $95.23$
| Stage 2 | $0$ | $0$ | $62$ | $0$ | $100$
| Stage 3 | $0$ | $0$ | $0$ | $97$ | $100$
Precision (%) |  | $93.75$ | $100$ | $98.41$ | $100$ |
Figure 4: Distribution of difference in prediction. Figure 5: Interpretation
of model using GradCam. Figure 6: Interpretation of model using Sobol
attribution method.

##  4 Proposed approaches: Self-training for data labeling and segmentation

###  4.1 Motivation

The main objective of this section is to, first, label arthroscopy images
using self-training and then detect and identify structures from arthroscopy
videos with a modular architecture. We used videos of shoulder arthroscopy. In
one second of video, we can obtain from 25 to 30 images per second. Therefore,
several minutes in a video can produce several images, thus labeling these
data can be difficult and time-consuming. The idea is to select few pictures
from each video and apply the self-training process for pseudo-labeling, that
is, producing predicted labels from a small set of labeled data and adding
them to the training dataset, of the rest of the video. This way, we exploit
all the available data we have.

###  4.2 System architecture

The model for the self-training (i.e. pseudo-labeler) is built using a
segmentation network. It takes as input an arthroscopic image and its ground
truth mask with 5 classes (background, long biceps tendon, subscapularis
tendon, supraspinatus tendon, cartilage (glenoid and humeral)), and it outputs
a prediction mask. We have found while training the segmentation task, that
the model was able to perform a binary segmentation, meaning identifying the
structure versus the background. However, it fails to discriminate between the
different classes. That’s what motivates the use of modular architecture in
learning. The first network learns to distinguish between the background and
the anatomical structure. Then, we remove the predicted background from the
initial image and feed it to the second network. The latter is trained to
learn to discriminate between the four classes.

The self-training phase, consists in, first, training the two models with the
available annotated images. Then, we predict the mask for all the images in
the unlabeled set. Afterward, we select only the images that produced a mean
prediction probability superior to a threshold. We set another threshold to
filter the selected outputs, pixel by pixel, according to their prediction
probability. The experiments have shown that adding the second filter to the
predicted images increases the accuracy. Let $\hat{y}_{m}$ be a predicted
image from the model. $\hat{y}_{m_{i,j}}$ represents the pixel at position
$(i,j)$. Let $T\in\mathopen{[}0,1\mathclose{]}$ be the threshold and the
Softmax function is defined as $S:\mathbb{R}^{K}\rightarrow(0,1)^{K}$ where
$K\geq 1$; $S(\mathbf{x}_{i})=\frac{e^{x_{i}}}{\sum_{j=1}^{K}e^{x_{j}}}\quad$
for $i=1,\ldots,K$ and
$\mathbf{x}=\left(x_{1},\ldots,x_{K}\right)\in\mathbb{R}^{K}$. The image is
filtered as follows;

$\hat{y}_{m_{i,j}}=\begin{cases}\hat{y}_{m_{i,j}},&\text{if
}\operatorname{arg\,max}_{(i,j)}S(\hat{y}_{m_{i,j}})\geq T\\\
0,&\text{otherwise.}\end{cases}$ (7)

We add the final outputs to the training set and reiterate until reaching the
stopping criteria; either all the images are annotated or the algorithm
reached the maximum number of iteration.

Let $P\in\mathopen{[}0,1\mathclose{]}$ be a threshold, then in each iteration
the set of selected images is as following:

$\hat{I}\coloneqq\\{\hat{f}_{\bm{w}^{(\ell)}}(\bm{x}_{m})\mid\operatorname{arg\,max}_{\bm{x}_{m}}S(\hat{f}_{\bm{w}^{(\ell)}}(\bm{x}_{m}))\geq
P\\}$ (8)

and the output prediction is defined as follows:

$\hat{y}_{m}=\delta_{\operatorname{arg\,max}_{(i,j)}S(I_{m_{i,j}})\geq
T}\hat{I}_{m};\quad 0\leq m\leq n(\hat{I}),$ (9)

where $n(\hat{I})$ is the cardinality of the set $\hat{I}$.

####  4.2.1 Learning phase

The architecture of each module is inspired by UNet [19]. The choice of this
particular pretrained model is motivated, first, by its common use in medical
applications. The second key is its ability to perform well with few labeled
images [19]. The architecture of UNet is U-shaped. It consists of an encoder
(contracting path) and a symmetric decoder (expanding path). The contracting
path consists of 2 blocks of $3\times 3$ convolution followed by the
activation function rectified linear unit (ReLU) then a $2\times 2$ max
pooling. These operations are repeated five times. The number of channels
doubles in each time (64, 128, 256, 512, 1024). The expanding path consists of
the same blocks and ReLU and uses $2\times 2$ for the up convolution. The
final layer is a $1\times 1$ convolution. The figure 7 displays the UNet
architecture.

Figure 7: UNet architecture[19].

###  4.3 Experiments

####  4.3.1 Dataset

The data of our second study were collected by video recording of shoulder
arthroscopies performed at Avicenne Hospital between November 2021 and June
2022. The videos of arthroscopies could be retrieved directly from the
arthroscopy video console in mp4 format. From these arthroscopies, a total of
459 images were extracted (121 of long biceps tendon, 41 of supraspinatus
tendon, 181 of subscapularis tendon, and 116 of glenoid or humeral cartilage),
and saved in png format. The images were annotated by a surgeon using the
software "LabelStudio". Labeling was made using polygon annotations. The
latter ensures a more precise annotation for irregular shapes. Figure 8
presents examples of shoulder arthroscopy images with their segmentation.

Figure 8: Visualization examples of segmentation on shoulder arthroscopy
image. Each color corresponds to the classes accordingly; red: Biceps, blue:
Cartilage ,green: Subscapularis , yellow: Supraspinatus.

###  4.4 Setting

Data was split into $70\%$ train set (321 images), and $30\%$ test set (138
images). We used $10\%$ of the training set (32 images) as labeled data to
train the first model in self-training and we supposed the $90\%$ of the
training set (289 images) unlabeled data. The final evaluation was made on the
test set. The images were resized to $224\times 224$. We used an Adam
optimizer and 15 number of epoch. After experiments, we set the threshold to
$0.7$.

####  4.4.1 Results and discussion

We use a threshold on the prediction probability to choose from the unlabeled
subset, and the samples to add to the training set (improved accuracy of self-
training from 76% to 80%). We varied the threshold to analyze its effect on
performance. We reported the results on the test set. As shown in table 3,
increasing the threshold improves the accuracy. However, for the threshold
$0.9$, the performance degrades. One reason can be that the strict selection
of samples engenders the model to train on the same samples, therefore
overfits. The optimal threshold value is $0.7$ with an accuracy of $80\%$. We
also show that the class Supraspinatus, presented in yellow in figure 8 has
the highest accuracy compared to other classes. We found that adding the
pixel-by-pixel threshold, explained in 4.2, improved the accuracy (from $76\%$
to $80\%$). We observed the effect of the ratio of the labeled data in the
first iteration on the performance in figure 9. We notice that the greater
percentage, the higher accuracy. We choose the ratio $10\%$. Samples of the
segmentation predictions on the test set are shown in figure 10.

Table 3: Comparison of threshold value in self-training. Threshold | Classes
---|---
|  | Mean value | Biceps | Cartilage | Subscapularis | Supraspinatus
0.6 | Accuracy | 71.2% | 64.48% | 37.25% | 25.95% | 84.35%
| IOU | 0.31 | 0.47 | 0.29 | 0.15 | 0.34
0.7 | Accuracy | 80% | 70.76% | 65.25% | 64.21% | 85.73%
| IOU | 0.3 | 0.3 | 0.4 | 0.4 | 0.11
0.8 | Accuracy | 79.55% | 44.91% | 62.35% | 50.83% | 92.32%
| IOU | 0.39 | 0.35 | 0.29 | 0.29 | 0.61
0.9 | Accuracy | 76.73% | 43.66% | 73.17% | 43.03% | 88.64%
| IOU | 0.3 | 0.38 | 0.39 | 0.16 | 0.39
Figure 9: Comparison of ratio of labelled data with respect to accuracy.
Figure 10: Examples of segmentation results predicted with self-training. Each
color corresponds to the classes accordingly; red: Biceps, blue: Cartilage
,green: Subscapularis , yellow: Supraspinatus.

##  5 Conclusion

In this paper, we first introduce a modular framework for Dorsal Capsulo-
Scapholunate Septum classification. We have shown that decomposing the problem
into a gating module and discriminative module improved the performance.
Moreover, including the contribution of the gating module as weights, achieved
the best results compared to the different strategies tested. Additionally, we
presented a modular self-training approach for data labeling and segmentation.
Our results validated the improvement of results by using self-training.
However, while modular learning outperformed non-modular systems in the first
problem, it was not the case for the second problem. The reason may be related
to the strategy of dividing the problem. In the classification problem, we
used a priori knowledge on the classification subject, contrary to the
segmentation problem.

In conclusion, modular learning is an efficient approach to reduce the
complexity of a model and improve model generalization. However, it is crucial
to manage when and where to divide a problem. In future work, a different
strategy should be developed and further analyzed for modular self-training
segmentation. Moreover, propose a new measure of feature selection for
interpretability on convolutional neural networks.

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# Refined determination of the weak mixing angle at low energy

M. Atzori Corona<EMAIL_ADDRESS>Dipartimento di Fisica,
Università degli Studi di Cagliari, Complesso Universitario di Monserrato -
S.P. per Sestu Km 0.700, 09042 Monserrato (Cagliari), Italy  Istituto
Nazionale di Fisica Nucleare (INFN), Sezione di Cagliari, Complesso
Universitario di Monserrato - S.P. per Sestu Km 0.700, 09042 Monserrato
(Cagliari), Italy  M. Cadeddu<EMAIL_ADDRESS>Istituto Nazionale di
Fisica Nucleare (INFN), Sezione di Cagliari, Complesso Universitario di
Monserrato - S.P. per Sestu Km 0.700, 09042 Monserrato (Cagliari), Italy  N.
Cargioli<EMAIL_ADDRESS>Dipartimento di Fisica, Università degli
Studi di Cagliari, Complesso Universitario di Monserrato - S.P. per Sestu Km
0.700, 09042 Monserrato (Cagliari), Italy  Istituto Nazionale di Fisica
Nucleare (INFN), Sezione di Cagliari, Complesso Universitario di Monserrato -
S.P. per Sestu Km 0.700, 09042 Monserrato (Cagliari), Italy  F. Dordei
<EMAIL_ADDRESS>Istituto Nazionale di Fisica Nucleare (INFN), Sezione
di Cagliari, Complesso Universitario di Monserrato - S.P. per Sestu Km 0.700,
09042 Monserrato (Cagliari), Italy  C. Giunti<EMAIL_ADDRESS>Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Torino, Via P. Giuria
1, I–10125 Torino, Italy

###### Abstract

The weak mixing angle is a fundamental parameter of the electroweak theory of
the standard model whose measurement in the low-energy regime is still not
precisely determined. Different probes are sensitive to its value, among which
atomic parity violation, coherent elastic neutrino-nucleus scattering and
parity-violating electron scattering on different nuclei. In this work, we
attempt for the first time to combine all these various determinations by
performing a global fit that also keeps into account the unavoidable
dependence on the experimentally poorly known neutron distribution radius of
the nuclei employed, for which a new measurement using proton-cesium elastic
scattering became available. By using all present direct determinations of the
neutron distribution radius of cesium we find
$\sin^{2}\\!\vartheta_{W}=0.2396^{+0.0020}_{-0.0019}$, which should supersede
the previous value determined from atomic parity violation on cesium. When
including electroweak only, but also indirect, determinations of the neutron
distribution radius of cesium the uncertainty reduces to 0.0017 maintaining
the same central value, showing an excellent agreement independently of the
method used.

##  I Introduction

The standard model (SM) of the electroweak interactions is described by the
gauge group SU(2)$\times$U(1), with the $i=1,2,3$ gauge bosons $W^{i}_{\mu}$
and $B_{\mu}$ for the SU(2) and U(1) groups, respectively, and the
corresponding gauge coupling constants $g$ and $g^{\prime}$. After spontaneous
symmetry breaking, the physical $Z$ boson and photon mediators are obtained
from a rotation of the basis of the two gauge bosons, $B_{\mu}$ and
$W_{\mu}^{3}$. The angle of this rotation is known as the weak mixing angle,
$\vartheta_{\text{W}}\equiv\mathrm{tan}^{-1}(g^{\prime}/g)$, also referred to
as the Weinberg angle [1]. In practice, the quantity
$\sin^{2}\\!\vartheta_{\text{W}}\,$is usually quoted instead of the weak
mixing angle itself.

The experimental determination of $\sin^{2}\\!\vartheta_{\text{W}}\,$and its
dependence on the energy scale of the process, so-called running, provides a
direct probe of physics phenomena beyond the SM (BSM). Its value is extracted
from neutral-current processes and $Z$-pole observables. More in detail, at
the LEP Collider [2], it was possible to achieve the most precise measurements
of $\sin^{2}\\!\vartheta_{\text{W}}\,$in the high-energy electroweak (EW)
sector, in perfect agreement with other collider determinations [1] (Tevatron,
LHC and SLC). In the mid-energy range, the most precise result has been
derived from the measurement of the weak charge of the proton, $Q^{p}_{W}$,
performed by the $Q_{\text{weak}}$ Collaboration and found to be
$Q^{p}_{W}=0.0719\pm 0.0045$ [3], showing an excellent agreement with the
predicted SM running. Moving to the low-energy sector [4], the most precise
weak mixing angle measurement so far belongs to the so-called atomic parity
violation (APV) experiments, also known as parity nonconservation (PNC), using
caesium atoms [5, 6], namely $\sin^{2}\\!\vartheta_{W}=0.2367\pm 0.0018$. This
value is slightly smaller than the SM prediction at near zero momentum
transfer, $\mathrm{Q}=0$, calculated in the so-called modified minimal
subtraction ($\overline{MS}$) renormalization scheme,
$\sin^{2}\\!\vartheta_{W}^{\rm SM}(\mathrm{Q}=0)=0.23863\pm 0.00005$ [1, 7,
8]. Atomic parity violation is caused by the weak interaction, and it is
manifested in P-violating atomic observables [9]. Other targets have also been
used, even if with less precise outcomes, and of interest for this work is the
measurement of APV in lead [10, 11]. Such experiments play a unique role
complementary to those at high-energy [12]. In particular, APV is highly
sensitive to extra light $Z^{\prime}$ bosons predicted by BSM theories,
underscoring the need for improved experimental determinations of
$\sin^{2}\\!\vartheta_{\text{W}}\,$in the low-energy regime [12, 13].

Figure 1:  Variation of $\sin^{2}\\!\vartheta_{\text{W}}$ with scale Q. The SM
prediction is shown as the solid curve, together with experimental
determinations in black at the $Z$-pole [1] (Tevatron, LEP1, SLC, LHC), from
APV on cesium [5, 6] (APV(Cs)), Møller scattering [14] (E158), deep inelastic
scattering of polarized electrons on deuterons [15] (PVDIS) and the result
from the proton’s weak charge [3] ($Q_{\text{weak}}$). For illustration
purposes, the Tevatron and LHC points have been shifted horizontally to the
left and right, respectively. In pink the result derived in this paper when
combining APV(Cs) with CE$\nu$NS COHERENT CsI data [16, 17] and the neutron
skin determination at the CSRe facility [18]. A similar result, slightly more
precise, is obtained considering electroweak probes only and is shown in red.

A summary of the most precise weak mixing angle measurements as a function of
the scale, Q, is shown in Fig. 1, along with the SM predicted running of
$\sin^{2}\\!\vartheta_{\text{W}}$, calculated in the $\overline{MS}$
renormalization scheme [1, 7, 8].

In the low-energy sector, there are two other electroweak probes mildly
sensitive to the weak mixing angle. They are the coherent elastic neutrino-
nucleus scattering (CE$\nu$NS) [19] and measurements of parity-violation in
electron scattering (PVES) on nuclei. The first process has been observed so
far in three targets, namely in cesium iodide (CsI) [16, 17], and in argon
(Ar) [20] by the COHERENT Collaboration using a spallation neutron source and
in germanium (Ge) [21] using neutrinos from the Dresden-II reactor. The
CE$\nu$NS cross section depends on the value of $\sin^{2}\\!\vartheta_{W}$
through the neutrino-proton coupling $g_{V}^{p}$ [22, 19], but being the
proton contribution subdominant with respect to the neutron one, only broad
constraints on $\sin^{2}\\!\vartheta_{\text{W}}\,$can be obtained [23, 24, 25,
26, 27, 28, 29]. The most recent updated result is
$\sin^{2}\\!\vartheta_{W}=0.231^{+0.027}_{-0.024}$ [29], obtained from the
latest COHERENT CsI data [17].
PVES consists of polarized electron-nucleus scattering, e.g. in lead, that
happens through both the weak and the electromagnetic currents. Isolating the
first contribution, it provides an interesting way to assess the nuclear
structure, but it can also be used to put constraints on
$\sin^{2}\\!\vartheta_{W}$. This has been recently suggested in Ref. [30],
using the latest PVES measurements on lead released by the PREX-II
Collaboration [31].

Historically, the APV measurement in cesium has moved significantly over the
years (see the inset of Fig. 9 of Ref. [29]), being mostly lower than the SM
prediction, motivating a further investigation of all the inputs entering this
measurement. Moreover, the extraction of the weak mixing angle value using
electroweak probes (APV, CE$\nu$NS and PVES) is always affected by the limited
knowledge of the so-called neutron skin of the nuclei used as a target [32].
The latter is defined as $\Delta R_{\rm{np}}\equiv R_{\rm{n}}-R_{\rm{p}}$, and
quantifies the difference between the neutron and the proton nuclear
distribution radii, $R_{\rm{n}}$ and $R_{\rm{p}}$, respectively, where the
latter is experimentally well known from electromagnetic measurements [33].
The usage of an extrapolated or imprecise value of the neutron radius of
cesium or lead would bias the extraction of
$\sin^{2}\\!\vartheta_{\text{W}}\,$and vice-versa, misinterpreting potential
signs of BSM physics. It is thus of pivotal importance to exploit all
available inputs on $\Delta R_{\rm{np}}$ and
$\sin^{2}\\!\vartheta_{\text{W}}\,$in a combined measurement, in order to take
advantage of possible correlations and minimize external assumptions.

The difficulty in measuring $\Delta R_{\rm{np}}$ is that the nuclear neutron
distribution can be probed only by exploiting the strong or weak forces. The
effects of the weak neutral-current interactions, embodied by the weak charge
of the nucleus, are known with good approximation thus making these
measurements systematically clean. However, the statistical uncertainty is
still quite limited. On the contrary, the results of experiments with hadron
probes are more precise but their interpretation is difficult since the
effects of strong-force interactions cannot be calculated with sufficient
approximation and the interpretation can be done only by assuming a strong-
interaction model with all its limitations [32]. On top of that, the cesium
neutron radius determination with hadronic probes has been historically
experimentally challenging due to the low melting point and spontaneous
ignition in air, resulting up to now for the APV
$\sin^{2}\\!\vartheta_{\text{W}}\,$determination in the utilization of an
extrapolated $R_{\rm{n}}(\mathrm{Cs})$ value from antiprotonic atom x-ray data
[34]. However, recently, a new direct measurement of the cesium neutron skin,
$0.12\pm 0.21\,\mathrm{fm}$, appeared [18], obtained using proton-cesium
elastic scattering at low momentum transfer and an in-ring reaction technique
at the Cooler Storage Ring (CSRe) at the Heavy Ion Research Facility in
Lanzhou, which can be included in the derivation of
$\sin^{2}\\!\vartheta_{W}$. The authors employed this value to re-extract the
COHERENT $\sin^{2}\\!\vartheta_{\text{W}}\,$value by fitting the CE$\nu$NS CsI
dataset, finding $\sin^{2}\\!\vartheta_{W}=0.227\pm 0.028$.

Taking into account all these recent developments, in this work we combine all
the available measurements of $R_{\rm{n}}(\mathrm{Cs})$,
$R_{\rm{n}}(\mathrm{Pb})$ and $\sin^{2}\\!\vartheta_{\text{W}}\,$in a global
fit to extract the most up to date and precise determination of the weak
mixing angle at low energy. Moreover, to better check the consistency among
the different inputs and techniques, we also compare the electroweak-only
determination with the averages obtained using strong probes.

##  II Results

To start with, we combine all available measurements using cesium atoms,
namely atomic parity violation on cesium, APV(Cs), CE$\nu$NS on CsI, referred
to as COH, and the recent determination of $R_{\rm{n}}(\mathrm{Cs})$ at the
CSRe facility. The APV observable is the weak charge of the nucleus
$Q_{W}(\mathrm{Cs})$, which is extracted by means of the experimental
determination of the ratio of the parity-violating amplitude,
$E_{\mathrm{PNC}}$, the Stark vector transition polarizability, $\beta$, and
by calculating theoretically $E_{\mathrm{PNC}}$ in terms of $Q_{W}$. For the
latter, the Particle Data Group (PDG) uses the theoretical prediction of the
PNC amplitude $({\rm Im}\,E_{\rm PNC})_{\rm th.}^{\rm w.n.s.}=(0.8995\pm
0.0040)\times 10^{-11}|e|a_{B}\frac{Q_{W}}{N}$ of Ref. [6], referred hereafter
as APV PDG, where Im stands for the imaginary part, $a_{B}$ is the Bohr
radius, $N$ is the number of neutrons in the nucleus and $|e|$ is the absolute
value of the electric charge. The apex w.n.s. means that the neutron skin
correction has not been already implemented, given that we want to extract
this correction from the combined fit using external inputs. In this work, for
the PNC amplitude, we use the more precise value recently calculated in Ref.
[35], referred to as APV 21, which is in slight tension with that used by the
PDG and equal to $({\rm Im}\,E_{\rm PNC})_{\rm th.}^{\rm w.n.s.}=(0.8931\pm
0.0027)\times 10^{-11}|e|a_{B}\frac{Q_{W}}{N}$. Besides being more precise, it
is also in better agreement with those reported in Refs. [36, 37]. For
completeness, in Appendix B we repeat all the results reported in this work
using APV PDG.

Figure 2:  Constraints on the weak mixing angle $\sin^{2}\\!\vartheta_{\text{W}}\,$and the Cs neutron radius $R_{\rm{n}}(^{133}\rm{Cs})$ obtained from a combined APV(Cs) 21 + COH + CSRe fit at different CLs ($1-2-3\sigma$), together with their marginalizations in the side panels. The blue line indicates the theoretical low-energy value of the weak mixing angle, $\sin^{2}\\!\vartheta_{W}^{\rm SM}(\mathrm{Q}=0)$.  | $\sin^{2}\vartheta_{W}$ | $\mathrm{R}_{\rm{n}}(^{133}\rm{Cs})[fm]$
---|---|---
APV(Cs)+COH+CSRe | $0.2396^{+0.0020}_{-0.0019}$ | $5.04\pm 0.19$
EW combined | $0.2396\pm 0.0017$ | $5.04\pm 0.06$
Global fit | $0.2387\pm 0.0016$ | $4.952\pm 0.009$
Table 1: Summary of the constraints at 1$\sigma$ CL obtained in this work on
the weak mixing angle $\sin^{2}\\!\vartheta_{\text{W}}\,$and on the Cs neutron
radius $R_{\rm{n}}(^{133}\rm{Cs})$. The different labels refer to the COHERENT
CsI data (COH), APV (Cs) data using the PNC amplitude of Ref. [35], and the
CSRe determination of $R_{\rm{n}}(^{133}\rm{Cs})$. The electroweak result (EW
combined) combines APV(Cs)+COH with PREX-II and APV determinations on lead.
The global fit includes all of the above plus the non-EW determinations of
$R_{\rm{n}}$ on lead.

To combine APV(Cs) and COHERENT CsI we follow the technique initially
developed in Ref. [38] and the latest prescriptions detailed in Ref. [29],
adding a prior on $R_{\rm{n}}(\mathrm{Cs})=4.94\pm 0.21\,\mathrm{fm}$111The
latter radius has been obtained starting from the skin measured in Ref. [18]
and using the rms proton radius of cesium
$R_{\rm{p}}(\rm{Cs})=4.821(5)\,\mathrm{fm}$ [39, 33, 29] and the neutron
radius $\langle r_{\rm{n}}^{2}\rangle\simeq\langle
r_{\rm{p}}^{2}\rangle=0.708\,\mathrm{fm}^{2}$ [40]. coming from CSRe. The only
mild assumptions behind this combination are that
$\sin^{2}\\!\vartheta_{\text{W}}\,$is constant between the corresponding
experimental momentum transfers, $2.4\lesssim Q\lesssim 100\ \mathrm{MeV}$,
which is true in the absence of BSM effects, and that the
${}^{133}\mathrm{Cs}$ and ${}^{127}\mathrm{I}$ neutron skins are the same in
order to isolate the contribution of $R_{\rm{n}}(\mathrm{Cs})$ when analysing
the COHERENT data. Given the fact that the neutron skin difference for these
two nuclei is expected to be small compared to the current precision of
experimental data, this choice is a fair approximation. We also checked that
fitting for an average value of the rms neutron radii of ${}^{133}\mathrm{Cs}$
and ${}^{127}\mathrm{I}$ gives the same output. The result is shown in Fig. 2
at different confidence levels (CLs), while the numerical values can be found
in Tab. 1.

This determination of $\sin^{2}\\!\vartheta_{\text{W}}\,$depends on the CSRe
determination of $R_{\rm{n}}(\mathrm{Cs})$, which dominates the COHERENT and
APV(Cs) sensitivity on the neutron distribution radius. In order to check the
impact of relying so heavily on a measurement of the neutron distribution
radius obtained using strong probes, we perform a further determination of
$\sin^{2}\\!\vartheta_{\text{W}}\,$using electroweak probes only. To do so, we
exploit two additional EW probes, namely PREX-II and APV on lead. The former
determines the weak form factor value at the experimental mean momentum
transfer, $Q_{\mathrm{PREX-II}}\simeq 78\ \mathrm{MeV}$ [31], which depends on
both the neutron distribution radius of lead and
$\sin^{2}{\vartheta_{\text{W}}}$. A simultaneous fit of these two parameters
can be achieved following the method developed in Ref. [30] and produces an
almost fully degenerate oblique band in the
$R_{\rm{n}}(\mathrm{Pb})-\sin^{2}{\vartheta_{\text{W}}}$ plane. To break the
degeneracy, the PREX-II result can be combined with the APV experiment on
${}^{208}\mathrm{Pb}$, which is sensitive to the nuclear weak charge at a
momentum transfer of $Q_{\mathrm{APV(\rm{Pb})}}\sim 8\ \mathrm{MeV}$. In this
case, we assume $\sin^{2}\\!\vartheta_{\text{W}}\,$to be constant between the
corresponding experimental momentum transfers, $8\lesssim Q\lesssim 78\
\mathrm{MeV}$. Furthermore, a practically model-independent extrapolation can
be performed, following the method developed in Ref. [13] and briefly
summarised in Appendix A, to translate the $\mathrm{R}_{\rm{n}}(\mathrm{Pb})$
determination into a measurement of $\mathrm{R}_{\rm{n}}(\mathrm{Cs})$. In
this way, the green contour shown in Fig. 3 is obtained at the 1$\sigma$ CL.
In the same figure, it is possible to judge the good agreement between the
different EW probes available nowadays, namely APV(Cs), APV(Pb)+PREX-II, and
COHERENT CsI. All these probes can be combined together to get a fully EW
determination of $\sin^{2}\\!\vartheta_{\text{W}}\,$and
$\mathrm{R}_{\rm{n}}(\mathrm{Cs})$, as shown by the red contour at 1$\sigma$
in Fig. 3.

Figure 3: Individual and combined contours at $1\sigma$ CL of the available
electroweak probes. Namely, APV(Cs) (orange dash-dotted line), APV(Pb)+PREX-II
already converted into $\mathrm{R}_{\rm{n}}(\mathrm{Cs})$ (dotted green line),
and COH CsI (light-blue dashed line). The red solid contour is the combination
of all these EW probes, with the red cross indicating the best-fit values.
Figure 4: Combined EW fit at different CLs ($1-2-3\sigma$) and its
marginalized $\Delta\chi^{2}$’s curves in the side panels considering APV(Cs)
21. The pink and purple bands indicate non-EW measurements of the cesium
radius, coming from proton scattering on cesium (CSRe), and from the average
of non-EW measurements on lead converted into
$\mathrm{R}_{\rm{n}}(\mathrm{Cs})$ using the method explained in the Appendix
A. In green the result of the combined APV(Cs)21+COH+CSRe+PREX-II+APV(Pb)+non-
EW(on Pb) fit is shown at $1\sigma$ CL. The blue line shows the SM value of
the weak mixing angle, $\sin^{2}\\!\vartheta_{W}^{\rm SM}(\mathrm{Q}=0)$.

The EW combination is also shown at different CLs and together with the
marginalized $\Delta\chi^{2}$’s curves in Fig. 4. Here, we also compare the EW
fit to other non-EW measurements of $\mathrm{R}_{\rm{n}}(\mathrm{Cs})$, namely
the direct one derived at CSRe using proton-cesium scattering [18] and a more
precise determination that is obtained from a conversion of the non-EW
measurements of $\mathrm{R}_{\rm{n}}(\mathrm{Pb})$. The latter average is
retrived by considering all non-electroweak $\mathrm{R}_{\rm{n}}(\mathrm{Pb})$
determinations in Tab. 4 of Ref. [41], Tab. I of Ref. [42] and a recent
measurement performed at the LHC [43]. A summary of all the measurements
considered is shown in Fig. 5, where it is possible to see that a rather good
agreement among all the different techniques is obtained. The average is
$\Delta\mathrm{R}^{\mathrm{non-EW}}_{{\rm{np}}}(\mathrm{Pb})=0.16\pm
0.01~{}\mathrm{fm}$.

Figure 5: Summary of available measurements of the lead neutron skin
considering different processes [44, 45, 46, 30]. The black data points
indicate non-EW probes, while in green we report the combined APV(Pb)+PREX-II
EW measurement. The purple band indicates the $1\sigma$ CL value obtained by
averaging over all the available non-EW measurements.

For comparison, in the same figure, we also show the EW determination coming
from our combined fit of APV(Pb)+PREX-II, namely
$\Delta\mathrm{R}^{\mathrm{EW}}_{{\rm{np}}}(\mathrm{Pb})=0.262\pm
0.136~{}\mathrm{fm}$, which also underlines the rather good agreement that can
be obtained between EW and non-EW probes as long as in the former the
dependence on the $\sin^{2}\\!\vartheta_{\text{W}}\,$is taken into account. In
addition, all these experimental values are in rather good agreement with the
theoretical expected range
$0.13<\Delta\mathrm{R}_{{\rm{np}}}(\mathrm{Pb})<0.19~{}\mathrm{[fm]}$ [47, 48,
49] and the first ab-initio estimate
$0.14<\Delta\mathrm{R}_{{\rm{np}}}(\mathrm{Pb})<0.20~{}\mathrm{[fm]}$ [50].

Interestingly, there is an extraordinary agreement among the central values
obtained for $\sin^{2}\\!\vartheta_{\text{W}}\,$when using the strong probe
determination at CSRe of $\mathrm{R}_{{\rm{n}}}(\mathrm{Cs})$
(APV(Cs)+COH+CSRe) and that obtained using exclusively EW probes, as visible
in Tab. 1, with the latter being slightly more precise. This achievement gives
confidence that an overall excellent agreement is emerging among EW and strong
probe determinations of $R_{\rm{n}}$, such that the
$\sin^{2}\\!\vartheta_{\text{W}}\,$result obtained is not too sensitive to the
particular dataset or method used. These two determinations, which overlap
besides the uncertainty, have also been added to Fig. 1 and should be compared
with the previous PDG determination of APV(Cs) [1], the shift being largely
due to the different PNC amplitude used. We would like to underline that our
two determinations have uncertainties that are comparable with that of the PDG
one.

For completeness, we checked what happens if a global fit of all the
measurements shown in Fig. 4 is performed and it is indicated by the green
contour in the same figure. Numerical values are also listed in Tab. 1.
Clearly, the central value of $\sin^{2}\\!\vartheta_{\text{W}}\,$is dominated
by the non-EW determinations of $\mathrm{R}_{{\rm{n}}}(\mathrm{Pb})$, but it
is possible to see that there is no much gain on the uncertainty, being the
latter dominated by the uncertainty of APV(Cs). Thus, there is no clear
advantage of performing such an aggressive global fit.

##  III Conclusions

Motivated by the lack of a precise determination of the weak mixing angle at
low energies we thoroughly investigate how to exploit correlations among the
different probes available in order to maximize the reliability and
significance of the $\sin^{2}\\!\vartheta_{\text{W}}\,$value that is
extracted. In particular, we combine atomic parity violation experiments on
cesium and lead nuclei, coherent elastic neutrino-nucleus scattering on cesium
iodide and parity-violating electron scattering on lead by performing a fit
that keeps also into account the unavoidable dependence on the experimentally
poorly known neutron distribution radius of the nuclei employed. For the
latter, we also exploit a recent measurement of the cesium neutron
distribution radius, obtained using proton-cesium elastic scattering at the
CSRe facility. To check the consistency of the results obtained, we compare
the weak-mixing angle values obtained using electroweak-only determinations of
the neutron distribution radius on cesium and lead nuclei (EW combined) with
that obtained using cesium-only determinations of $R_{\rm{n}}(\mathrm{Cs})$
including also strong probes (APV(Cs)+COH+CSRe). Respectively, we find

$\sin^{2}\\!\vartheta_{W}=\left\\{\begin{array}[]{l}0.2396\pm
0.0017\,(\mathrm{EW\,combined})\\\ \\\
0.2396^{+0.0020}_{-0.0019}\,(\mathrm{APV(Cs)+COH+CSRe})\end{array}\right.$

where an excellent agreement is visible, with the first method giving a
slightly more precise result. These findings underscore the fact that an
overall consistent picture is emerging between the values extracted using EW
and strong probes, as long as the correlation with the neutron skin is
properly taken into account. Finally, given that the latter combination of
APV(Cs), COHERENT CsI and $R_{\rm{n}}(\mathrm{Cs})$ from CSRe uses direct
determinations of $R_{\rm{n}}(\mathrm{Cs})$ it should supersede the
$\sin^{2}\\!\vartheta_{\text{W}}\,$value obtained exploiting APV(Cs)-only with
an indirect extrapolation of $R_{\rm{n}}(\mathrm{Cs})$ [1].

###### Acknowledgements.

The work of C. Giunti is supported by the PRIN 2022 research grant “Addressing
systematic uncertainties in searches for dark matter”, Number 2022F2843L,
funded by MIUR.

##  Appendix A Nuclear model predictions for
$\mathbf{\Delta\mathrm{R}_{{\rm{np}}}(\mathrm{Pb/Cs})}$

Figure 6: Correlation between the nuclear model prediction of the lead and
cesium neutron skin. The purple-shaded region corresponds to the mean of the
neutron skin measured through non-EW probes on lead, and its translation into
cesium.

In Fig. 6 we show the values of the point neutron skins222The physical proton
and neutron radii $R_{{\rm{p,n}}}$ can be retrieved from the corresponding
point-radii $R_{{\rm{p,n}}}^{\text{point}}$ adding in quadrature the
contribution of the rms nucleon N radius $\langle
r_{{\rm{N}}}^{2}\rangle^{1/2}\simeq 0.84\,\text{fm}$, that is considered to be
approximately equal for the proton and the neutron. Namely,
$R_{{\rm{p,n}}}^{2}=(R_{{\rm{p,n}}}^{\text{point}})^{2}+\langle
r_{{\rm{N}}}^{2}\rangle$. of ${}^{208}\mathrm{Pb}$ and ${}^{133}\mathrm{Cs}$
obtained with various nonrelativistic Skyrme-Hartree-Fock (SHF) [51, 52, 53,
54, 55, 56] and relativistic mean-field (RMF) [57, 58, 59, 60, 61, 62, 63, 64,
65, 66] nuclear models. A clear model-independent linear correlation [64, 67,
68, 47, 69, 24] is present between the two neutron skins within the
nonrelativistic and relativistic models with different interactions, with a
Pearson’s correlation coefficient $\rho\simeq 0.999$. Namely, we find $\Delta
R_{{\rm{np}}}^{\mathrm{point}}(\mathrm{Cs})=0.707\times\Delta
R_{{\rm{np}}}(\mathrm{Pb})+0.016$ fm that can be translated into a physical
$\Delta R_{{\rm{np}}}(\mathrm{Cs})$ determination. We exploit this powerful
linear correlation to translate the PREX-II and APV(Pb) combined measurement
of the lead neutron skin into a cesium one. Moreover, we also use it to
translate the mean of the neutron skin measured through non-EW probes on lead
into a determination of the cesium one. The latter is shown by the purple
extrapolation in Fig. 6.

##  Appendix B Results with ImEPNC PDG

Figure 7: Constraints on the weak mixing angle
$\sin^{2}\\!\vartheta_{\text{W}}\,$and the Cs neutron radius
$R_{\rm{n}}(^{133}\rm{Cs})$ obtained from a combined APV (Cs) PDG + COH + CSRe
fit at different CLs ($1-2-3\sigma$), together with their marginalizations in
the side panels. The blue line indicates the theoretical low-energy value of
the weak mixing angle, $\sin^{2}\\!\vartheta_{W}^{\rm SM}(\mathrm{Q}=0)$.

In this appendix, we report the results using for the atomic parity violation
determination in cesium the PNC amplitude $E_{\mathrm{PNC}}$ from Ref. [6],
referred to as APV(Cs) PDG. The combination of APV(Cs) PDG and COHERENT CsI
adding a prior on $R_{\rm{n}}(\mathrm{Cs})=4.94\pm 0.21\,\mathrm{fm}$ coming
from CSRe is shown in Fig. 7 at different CLs, while the numerical values can
be found in Tab. 2.

| $\sin^{2}\vartheta_{W}$ | $R_{\rm{n}}(^{133}\rm{Cs})[fm]$
---|---|---
APV(Cs)+COH+CSRe | $0.2372\pm 0.0022$ | $5.04\pm 0.19$
EW combined | $0.2372\pm 0.002$ | $5.03\pm 0.06$
Global fit | $0.2363^{+0.0018}_{-0.0019}$ | $4.951\pm 0.009$
Table 2: Summary of the constraints at 1$\sigma$ CL obtained in this work on
the weak mixing angle $\sin^{2}\\!\vartheta_{\text{W}}\,$and on the Cs neutron
radius $R_{\rm{n}}(^{133}\rm{Cs})$. The different labels refer to the COHERENT
CsI data (COH), APV (Cs) PDG data using the PNC amplitude of Ref. [6], and the
CSRe determination of $R_{\rm{n}}(^{133}\rm{Cs})$. The electroweak result (EW
combined) combines APV(Cs) +COH with PREX-II and APV determinations on lead.
The global fit includes all of the above plus the non-EW determinations of
$R_{\rm{n}}$ on lead.  Figure 8: Combined EW fit at different CLs
($1-2-3\sigma$) and its marginalized $\Delta\chi^{2}$’s curves in the side
panels considering APV(Cs) PDG. The pink and purple bands indicate non-EW
measurements of the cesium radius, coming from proton scattering on cesium
(CSRe), and from the average of non-EW measurements on lead converted into
$\mathrm{R}_{\rm{n}}(\mathrm{Cs})$ using the method explained in the Appendix
A. In green the result of the combined APV(Cs) PDG+COH+CSRe+PREX-
II+APV(Pb)+non-EW(on Pb) fit is shown at $1\sigma$ CL. The blue line shows the
SM value of the weak mixing angle, $\sin^{2}\\!\vartheta_{W}^{\rm
SM}(\mathrm{Q}=0)$.

All the EW probes available nowadays, namely APV(Cs) PDG, APV(Pb)+PREX-II, and
COHERENT CsI, are combined to get a fully EW determination of
$\sin^{2}\\!\vartheta_{\text{W}}\,$and $\mathrm{R}_{\rm{n}}(\mathrm{Cs})$, as
shown at different CLs and together with the marginalized $\Delta\chi^{2}$’s
curves in Fig. 8. Here, we also compare the EW fit to other non-EW
measurements of $\mathrm{R}_{\rm{n}}(\mathrm{Cs})$, namely the direct one
derived at CSRe using proton-cesium scattering [18] and a more precise
determination that is obtained from a conversion of the non-EW measurements of
$\mathrm{R}_{\rm{n}}(\mathrm{Pb})$, as explained in Appendix A.

For completeness, in Fig. 8 we also show the result of a global fit of all
these determinations. It is possible to notice, by comparing Fig. 8 and Fig.
4, that using the PNC amplitude $E_{\mathrm{PNC}}$ from Ref. [6] (APV(Cs) PDG)
results in a $\sin^{2}\\!\vartheta_{\text{W}}\,$value that is slightly smaller
than the SM prediction as well as less precise.

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