# Self-Labeling Refinement for Robust Representation Learning with Bootstrap
Your Own Latent

Siddhant Garg
<EMAIL_ADDRESS>
&Dhruval Jain
<EMAIL_ADDRESS>
(September 2021)

###### Abstract

In this work, we have worked towards two major goals. Firstly, we have
investigated the importance of Batch Normalisation (BN) layers in a non-
contrastive representation learning framework called Bootstrap Your Own Latent
(BYOL)[18]. We conducted several experiments to conclude that BN layers are
not necessary for representation learning in BYOL [39]. Moreover, BYOL only
learns from the positive pairs of images but ignore other semantically similar
images in same input batch. For the second goal, we have introduced two new
loss functions to determine the semantically similar pairs in the same input
batch of images and reduce the distance between their representations. These
loss functions are Cross-Cosine Similarity Loss (CCSL) and Cross-Sigmoid
Similarity Loss (CSSL). Using the proposed loss functions, we are able to
surpass the performance of Vanilla BYOL (71.04%) by training BYOL framework
using CCSL loss (76.87%) on STL10 dataset. BYOL trained using CSSL loss
performs comparably with Vanilla BYOL.

##  Introduction

Self-Supervised learning methods aim to leverage information from unlabeled
dataset and improve the representation of the embeddings or hidden layers by
learning semantic information about the inputs. Self-Supervised Learning
methods have become quite popular in the domain of Computer Vision and many
state-of-the-art methods, like SEER [17], SwAV [4], Contrastive Learning [33]
methods like SimCLR [5] and Momentum Contrast (MoCo) [21], and Non-
Constrastive Learning methods like BYOL[18] and RAFT[40] have surfaced over
past years that are outperforming the traditional supervised learning methods
on downstream tasks like Image Classification, Object Detection and Semantic
Segmentation.

Training deep neural networks with a Self-Supervised framework for Computer
Vision tasks require us to design various pretext tasks like Context
Prediction [9], or ordering of image grids [32] to learn various concepts. The
representation or features of similar concepts should be closer while the
features of semantically different concepts should be further apart. This is
achieved using $InfoNCE$ Loss function [34] and large number of negative
samples for a given input image. However, a particular Contrastive Learning
Method called Momentum Contrast (MoCo) [21], keeps memory bank of negative
samples from the past mini-batches that allows it to reduce the batch size
during training, and thus making it computationally feasible. Recenlty, a non-
contrastive method called Bootstrap Your Own Latent(BYOL)[18], was proposed
that does not require negative samples at all for learning features. The
method is also robust to the image augmentations and batch size while
training.

The negative pairs in contrastive learning were necessary to avoid a
representation collapse in which case the network outputs a trivial
representation for all the input images. However, the absence of the negative
pairs in BYOL stirred a lot of commotion in the community and several works
[39, 44, 43, 40] have tried to understand the phenomenon. It has been shown
that BYOL avoids representation collapse by using a predictor network at the
end of the online network and by updating its target network using slow-moving
averages and not by backpropagating the gradients through the target network.
We have also conducted several experiments to rule out the importance or
necessity of Batch Normalization [24] layers for the representation learning
in BYOL, as suggested by [44], by replacing them with Layer Normalization [1]
and Group Normalization layers [45].

Furthermore, the MoCo framework assumes that if the key and query pair (images
after applying data augmentations) are from the same image, then they are
treated as positive pairs and rest of the keys, in the mini-batch, are treated
as negative pair with the query. But, more than one images in a mini-batch can
be semantically similar and by treating them as negative pairs, the quality of
the representations are impaired. Zhou et al. [47] proposed a Self-Labeling
Refinement Method to improve the label quality and learning more generalized
representations. We will extend this method for BYOL by proposing two new loss
functions and compare the results of the vanilla training method with the
proposed Self-Label Refinement method 111Work done for neural networks course
project at UMass Amherst .

##  Related Work

Learning unsupervised representations can be seen as learning a energy
function or a contrast function that have low energy values on the data
distribution and high energies everywhere else. Probabilistic models can be
used to model such functions using Gibbs Distribution [14] which is
proportional to the exponential of the negative energy values. The
normalization constant, however, is intractable for high dimensional data and
they are approximated using MCMC methods [13, 6, 15]. Sparse Autoencoders
[31], and Latent Variable Models [3] are some of the examples that learn the
latent representations of the unlabeled data by minimizing some form of energy
function using encoder-decoder frameworks.

For Self-Supervised learning, several contrastive methods [21, 5, 35, 42, 22,
23, 2, 30, 29] have been proposed that achieved state-of-the-art results.
Contrastive learning methods bring the representations of the augmented views
(positive pair) of the same image closer and augmented views of different
images (negative pairs) further apart. SimCLR [5] and MoCo [21] are the most
popular contrastive learning methods that use a loss function for contrastive
prediction task called InfoNCE [34] loss which is defined using equation 9.

Self-Supervised learning can also be done by introducing some pretext tasks
and perform predictions using those auxiliary tasks. Auxiliary tasks can
include image jigsaw puzzle [32], in which the image is divided into grids and
the task is to predict the correct ordering of the grids, colorization of
grayscale images [46, 27], image inpainting [36], relative patch prediction
[11, 10], geometric transformations [12, 16] like image rotations, image super
resolution [28] or by using suitable architectures [26] like Prototypical
networks[29]. Though we can design variety of pretext tasks, the contrastive
methods are the state-of-the-art for Self-Supervised learning.

Though contrastive learning methods have been proven to achieve state-of-the-
art results for the Self-Supervised learning but they require high
computational resources. For a single positive pair, a large amount of
negative samples are required with high batch size. For example, in SimCLR
[5], for a single positive pair the batch size went up to 8192 and the model
was trained upto 128 TPU cores. Momentum Contrast (MoCo) [21] tries to
alleviate this issue by using a dictionary of negative samples. However,
Bootstrap Your Own Latent (BYOL) [18], addresses the issue of high
computational costs by removing the negative pairs for learning image
representations. Instead BYOL uses a slow-moving average called target network
to learn the image representations. Since, BYOL does not use negative pairs,
an extra predictor head is used in the online network encoder to avoid
representation collapse. BYOL outperformed several contrastive learning
methods including SimCLR and MoCo on ImageNet [8] dataset and it has been
shown to be robust to the batch size. MoCo also uses a slow moving target to
update the key encoder which is necessary for learning representations. Mean
Teacher(MT) [41] networks, that are used in semi-supervised learning, also use
slow-moving averages as the target network called teacher to train the online
network called student.

There have been several attempts at analysing BYOL to understand its dynamics
and to answer questions like why it does not collapse without the contrastive
negative pairs. Tian et al [43] performs several experiments and report that
the predictor network at the end of the online network and stop-gradients for
the target network are necessary to avoid the representation collapse. Some
works [44] also suggest that the Batch Normalization layer is responsible for
implicit contrastive learning but experiments from [39] shows that Batch
Normalization [24] layers are not necessary to avoid the representation
collapse. We conduct several experiments in this work by replacing the Batch
Normalization layers [24] with Layer Normalization [1] and Group Normalization
[45] along with weight standardization [38].

Contrastive Learning methods learn the representations from the unlabeled data
by considering the views of the same image as positive while taking all the
other views as negative in the input batch of images. This can introduce some
errors because some negative pairs in the same batch might be semantically
similar to the positive pairs. Zhou et al [47] proposed two modules called
Self-label refinery to generate accurate labels and Momentum Mixup to enhance
the similarity between an image and its positive. These two modules were
designed for MoCo. In this work, we proposed a novel Self-Labeling Refinement
for BYOL by introducing two loss functions that identifies the semantically
similar images in an input batch and minimize the representation loss between
them.

##  Technical Approach and Methods

In this section, we will describe the details of BYOL network and details of
the proposed methods.

###  Bootstrap Your Own Latent(BYOL)

Figure 1: The figure (from [18]) represents the framework of BYOL. Two views
of the same image are passed through online and target networks and their is
predictor at the end of online network, which prevents collapsed
representations. $sg$ stands for $stop-gradient$ because the gradients will
get backpropagated through the online network only, whereas the target network
will be updated using the slow-moving average of the online network. The loss
function minimizes the $L2$ norm between the outputs of the two networks.

####  Representation Learning

BYOL [18] uses two neural networks called online network, represented by
parameters $\theta$ and target network, represented by $\xi$. The online
network consists of an encoder $f_{\theta}$, projector $g_{\theta}$ and a
predictor $q_{\theta}$ and target network consists of and encoder $f_{\xi}$
and a projector $g(\xi)$. The representations from the target network are used
to obtain more enhanced representations by predicting the target
representations using the online networks. The weights of the online network
are back-propagated and the target network is updated using the slow moving
average of the online network with the decay rate of $\tau\in[0,1]$ using
equation 4.

Given a set of input images $\mathcal{D}$ and two image augmentations
$\mathcal{T}$ and $\mathcal{T^{\prime}}$, two different views of the same
image, called $v=t(x)$ and $v^{\prime}=t^{\prime}(x)$ can be generated using
$t\sim T$ and $t^{\prime}\sim T^{\prime}$. Using the two augmented views, the
online network produces an output $q_{\theta}(z_{\theta})$ and the target
network produces the output $z^{\prime}_{\xi}$. Both the outputs are
normalized (getting $\bar{q_{\theta}}(z_{\theta})$ and
$\bar{z^{\prime}}_{\xi}$) and the loss $\mathcal{L_{\theta,\xi}}$ is
calculated by minimizing the $L2$ loss between the normalized outputs.

$\displaystyle\mathcal{L_{\theta,\xi}}$
$\displaystyle=\left\|{\bar{q_{\theta}}(z_{\theta})-\bar{z^{\prime}}_{\xi}}\right\|_{2}=2-2.\frac{\langle
q_{\theta}(z_{\theta}),z^{\prime}_{\xi}\rangle}{\left\|{q_{\theta}(z_{\theta})}\right\|_{2}.\left\|{z^{\prime}_{\xi}}\right\|_{2}}$
(1)

Note that we only have a predictor $q_{\theta}$ in the online network
therefore, the architecture is asymmetric between the online network and the
target network. To symmetrize the loss, the inputs, $v$ and $v^{\prime}$ are
swapped and $v$ is fed to the target network and $v^{\prime}$ is fed to the
online network to get the loss $\tilde{\mathcal{L}}_{\theta,\xi}$. Combining
both the losses we get the final loss function, $\mathcal{L}^{byol}$.

$\displaystyle\mathcal{L}^{byol}_{\theta,\xi}$
$\displaystyle=\mathcal{L}_{\theta,\xi}+\tilde{\mathcal{L}}_{\theta,\xi}$ (2)

The gradients of the loss function in equation 2 are calculated with respect
to $\theta$ only and not $\xi$. The complete optimization process can be
summarized in equations 3 and 4.

$\displaystyle\theta$
$\displaystyle\leftarrow\text{optimizer}(\theta,\nabla_{\theta}\mathcal{L}^{byol}_{\theta,\xi},\eta)$
(3) $\displaystyle\xi$ $\displaystyle\leftarrow\tau\xi+(1-\tau)\theta$ (4)

###  Batch Statistics

Contrastive learning focuses on minimizing the distance between intra-class
images (positive pairs) while maximizing that distance metric for inter-class
images (negative pairs). This paradigm has much gathered a lot of attention in
representation learning. However, BYOL framework does not use any contrastive
term explicitly in its loss function for negative pairs. This is well
illustrated from Eq (2). It is astonishing that still it is able to avoid
learning a collapse solution, where the networks predicts the same constant
representation for all samples. It has been hypothesised that Batch
Normalisation [25] (BN) provides an implicit contrastive(negative) term.
Inspired from [39], we perform an ablation study by checking the performance
of BYOL after removing all BN layers. Additionally, we have also applied
Weight standardization (WS) [37] after convolution layers that were followed
by BN layers in both the online and target networks. All the experiments are
described in detail in section 4.

####  Weight Standardization (WS)

The weight matrix $W$ from any convolution layer is normalised along the rows
to produce $\widehat{W}$ which is then used for training. However, during the
back-propogation, the gradient of the loss is computed only with respect to
the non-normalized weights $W$.

$\displaystyle\widehat{W}_{i,j}$
$\displaystyle=\frac{W-\mu_{i,j}}{\sigma_{i,j}},\hskip
5.69054pt\text{where}\hskip
2.84526pt\mu_{i}=\frac{1}{R}\sum_{j=1}^{R}W_{i,j}\hskip
5.69054pt\text{and}\hskip
5.69054pt\sigma_{i}=\sqrt{\epsilon+\frac{1}{R}\sum_{j=1}^{R}(W_{i,j}-\mu_{i})^{2}}$
(5)

where $R$ is the input dimension which is equal to the product of input
channel dimension and the filter size of the convolution layer. $\epsilon$ is
typically of the order of $10^{-4}$. WS was originally proposed for applying
BN with micro-batch training settings, but it also smoothens the loss
landscape. In their paper, [37] have proved that WS reduces the Lipschitz
constants of the loss and the gradients which results in faster convergence
and reduction in training time. There are no learning parameters in WS unlike
BN.

###  Self-Labeling Refinement

BYOL, does not rely on negative samples to learn robust image representations
but only minimizes the representational loss between the two different views
of the same image. But consider the case in which there are semantically
similar images (images containing dogs, or images containing only cats) in the
same input mini-batch. Can we somehow identify those and make their
representations closer? And whether doing this would improve the quality of
the representations or would it lead to representation collapse? In this
section, we propose two methods to identify the semantically similar images
from the learned representations of the BYOL and further train the network
using different loss functions. To identify the similar images we calculated
the dot product of the normalized representations of every image with every
other image using the representations of the online network and the target
network. Since, the target network is just the moving exponential average of
the online network we would expect that the similar images have higher dot
product and dissimilar images have small values of the dot products. We
defined thresholds $\theta_{p}$ and $\theta_{n}$ to keep most similar and
dissimilar images. If the dot product is higher than $\theta_{p}$, then the
images are similar, therefore, their representations should be closer and if
the dot product is lower than $\theta_{n}$, then the images are dissimilar and
there representations should be further apart.

Now we will describe the two proposed loss functions below. First loss
function just extends the $L2$ loss function to include the $L2$ loss between
the similar and dissimilar images. Our second loss function extends the
infoNCE [19] loss to calculate the similarity and dissimilarity between the
input images.

####  Cross-Cosine Similarity Loss

The Cross-Cosine Similarity Loss(CCSL) calculates the cosine loss between the
output representations of the online and target network of BYOL which was
trained earlier. For an input image $x_{i}$, the expressions of the loss
function is described in in Eq 6

$\begin{split}\mathcal{L}_{CCSL}(x_{i})&=\left\|{\bar{q}_{z_{\theta}}(x_{i})-\bar{z}^{\prime}_{\xi}(x_{i})}\right\|_{2}+\lambda\sum_{j\neq
i}\mathbbm{1}[\langle\bar{q}_{z_{\theta}}(x_{i}),\bar{z}^{\prime}_{\xi}(x_{j})\rangle\geq\theta_{p}]\left\|{\bar{q}_{z_{\theta}}(x_{i})-\bar{z}^{\prime}_{\xi}(x_{j})}\right\|_{2}\\\
&\hskip 170.71652pt-\lambda\sum_{j\neq
i}\mathbbm{1}[\langle\bar{q}_{z_{\theta}}(x_{i}),\bar{z}^{\prime}_{\xi}(x_{j})\rangle\leq\theta_{n}]\left\|{\bar{q}_{z_{\theta}}(x_{i})-\bar{z}^{\prime}_{\xi}(x_{j})}\right\|_{2}\end{split}$
(6)

Note that the Eq 6 consists of three terms. The first term is the standard
$L2$ loss between the online and target representations of the same image. The
second term is the cosine loss/distance between the highly similar images
identified by the indicator function that checks the dot product between the
$l2$-normalized outputs and if the distance is above a certain threshold
$\theta_{p}$, we try to bring the representations even closer by adding the
$L2$ loss between them. Similarly, when the cosine similarity between the
representations is lower than a certain threshold $\theta_{n}$, we try to
separate them out even more by adding the negative $L2$ loss between them.
However, adding the negative $L2$ loss between the representations led to the
representational collapse therefore, we removed this term and kept only the
first two terms. Therefore, the final loss function is calculated using
equation 7

$\begin{split}\mathcal{L}_{CCSL}(x_{i})&=\left\|{\bar{q}_{z_{\theta}}(x_{i})-\bar{z}^{\prime}_{\xi}(x_{i})}\right\|_{2}+\lambda\sum_{j\neq
i}\mathbbm{1}[\langle\bar{q}_{z_{\theta}}(x_{i}),\bar{z}^{\prime}_{\xi}(x_{j})\rangle\geq\theta_{p}]\left\|{\bar{q}_{z_{\theta}}(x_{i})-\bar{z}^{\prime}_{\xi}(x_{j})}\right\|_{2}\end{split}$
(7)

Note that we are weighted the second term of the loss with $\lambda$ which is
set to a small value to avoid the representation collapse otherwise without it
all the representations became equal which could also lead to a minimum loss.
We are calling this loss function as Cross-Cosine Similarity Loss because we
are calculating the loss between $\mathcal{O}(N^{2})$ terms, $N$ being the
mini-batch size. Finally, to make the loss function symmetric we will also
interchange the input values of the online network and target network and
compute the CCSL loss just like we did in the vanilla BYOL setup. Let this
value be called as $\tilde{\mathcal{L}}_{CCSL}$, then the final CCSL loss can
written as in Eq 8

$\mathcal{L}_{CCSL}^{BYOL}(x_{i})=\mathcal{L}_{CCSL}(x_{i})+\tilde{\mathcal{L}}_{CCSL}(x_{i})$
(8)

The loss over the mini-batch can be computed as the average of the loss in Eq
8

####  Cross-Sigmoid Similarity Loss

In popular contrastive representation learning methods like SimCLR [5], the
representation loss is calculated using the $NT-Xent$ (Normalized temperature
scaled cross-entropy loss Eq 9

$\ell(x_{i},\tilde{x}_{j})=-\log\frac{exp(sim(z_{i},\tilde{z}_{j})/\tau)}{\sum_{k=1}^{2N}\mathbbm{1}[k\neq
i]exp(sim(z_{i},\tilde{z}_{k})/\tau)}$ (9)

The $sim(.)$ in eq 9 is the cosine similarity, $\tau$ is the temperature
parameter and the loss is calculated such that the two views of the same image
should become similar and the views of the different images become different
in the terms of the cosine similarity function. Note that the Eq 9 is similar
to the popular cross entropy loss of multi-class image classification problem
where we calculate the class probabilities by using the softmax function and
then calculating the negative log loss over them. This works well for the
single label classification problems but it is not used for multi-label
classification problems where we have multiple labels of the same image and we
try to maximize the scores of each of the labels. Instead another loss
function called Binary Cross Entropy function (Eq 10)

$\ell(y,p)=-\sum_{i=1}^{C}y_{i}\log
p_{i}-\sum_{i=1}^{C}(1-y_{i})\log(1-p_{i})$ (10)

In the equation 10, $y$ is the true label, $p$ are the scores between $0$ and
$1$, and $C$ is the total number of classes. In our problem, since we want to
minimize the loss between many similar images, we extend the binary cross
entropy term to calculate the loss between all the image representations. The
scores $p(x_{i},\tilde{x}_{j})$ can be replaced the cosine similarity between
the $i^{th}$ and $j^{th}$ representations and the labels
$y(x_{i},\tilde{x}_{j})$ can be computed using the threshold values of the
similarity score as we did in Section 3.3.1. More specifically, let
$\bar{q}_{z_{\theta}}(x_{i})$ and $\bar{z}^{\prime}_{\xi}(x_{j})$ be the image
representations of images $x_{i}$ and $x_{j}$ from the online and target
networks respectively. Then the Cross-Sigmoid Similarity Loss(CSSL) can be
calculated using the equation 11

$\begin{split}\ell(x_{i},x_{j})&=\ell(\bar{q}_{z_{\theta}}(x_{i}),\bar{z}^{\prime}_{\xi}(x_{j}))\\\
\ell(\bar{q}_{z_{\theta}}(x_{i}),\bar{z}^{\prime}_{\xi}(x_{j}))&=-\mathbbm{1}\Big{[}\langle\bar{q}_{z_{\theta}}(x_{i}),\bar{z}^{\prime}_{\xi}(x_{j})\rangle\geq\theta_{p}\Big{]}\log
p\Big{(}\bar{q}_{z_{\theta}}(x_{i}),\bar{z}^{\prime}_{\xi}(x_{j})\Big{)}\\\
&\hskip
56.9055pt-\mathbbm{1}\Big{[}\langle\bar{q}_{z_{\theta}}(x_{i}),\bar{z}^{\prime}_{\xi}(x_{j})\rangle\leq\theta_{n}\Big{]}\log\Big{(}1-p(\bar{q}_{z_{\theta}}(x_{i}),\bar{z}^{\prime}_{\xi}(x_{j}))\Big{)}\end{split}$
(11)

In the equation 11, if the similarity of the representations are high then
they will get even more closer because of the first term, and if the
similarity is very less, then they will get further apart due to the second
term. Using the described loss term, the loss function for the input image
$x_{i}$ can be written as

$\mathcal{L}_{CSSL}(x_{i})=-\log\Big{(}p(\bar{q}_{z_{\theta}}(x_{i}),\bar{z}^{\prime}_{\xi}(x_{i}))\Big{)}-\lambda\sum_{i\neq
j}\ell(x_{i},x_{j})$ (12)

In the equation 12, the first term is the loss between the representations of
the online and target networks for the same image, and the second term is the
loss term of the online representation of $x_{i}$ with the target
representations of $x_{j}$, $\forall j$ such that $j\neq i$, in the input
mini-batch. The weighting parameter $\lambda$ is used to avoid the
representational collapse. Note that in this CSSL loss we have also taken the
loss from the dissimilar images which was not the case in the CCSL loss
function. Finally to keep the symmetry between the online and target networks,
we interchange their inputs and calculate the loss
$\tilde{\mathcal{L}}_{CSSL}(x_{i})$ and the final loss is given by the
equation 13

$\mathcal{L}_{CSSL}^{BYOL}(x_{i})=\mathcal{L}_{CSSL}(x_{i})+\tilde{L}_{CSSL}(x_{i})$
(13)

The loss across the mini-batch is the average loss over all the images.

Figure 2: Feature Loss across training epochs of the Vanilla L2 Loss Function
of BYOL

Figure 3: Feature Loss across training epochs of the CCSL Loss Function of
BYOL

Figure 4: Feature Loss across training epochs of the CSSL Loss Function of
BYOL

Figure 5: Linear Evaluation Accuracy of Vanilla BYOL.

Figure 6: Linear Evaluation Accuracy of CCSL BYOL.

Figure 7: Linear Evaluation Accuracy of CSSL BYOL.

##  Experiments and Results

Configuration | Accuracy
---|---
Without BN | 52.46%
Without BN + WS | 53.12%
LN | 60.03%
LN + WS | 69.65%
GN | 62.96%
GN + WS |  70.11%
Vanilla BYOL | 71.04%
CCSL BYOL |  76.87%
CSSL BYOL |  72.92%
Table 1: Accuracies obtained on STL10 dataset with different model
configurations. Bolded figures confirm expected high results.

Figure 8: Smoothened Train Loss in
LN+WS model

###  Dataset

We will use STL10[7] dataset for all our experiments. The dataset consists of
100,000 unlabeled RGB images with $96\times 96$ pixels. There is also small
labeled dataset with 500 training images divided into 10 classes and 800 test
images per class. The unlabeled partition of the dataset is used to train the
BYOL network and the labeled partition is used to evaluate the learned
representations using the Linear Evaluation protocol described Section 4.3

###  Network Architecture

We have used ResNet-18 [20] as online network $f_{\theta}$ and target network
$f_{\xi}$. The output representations of the ResNet-18 before the fully
connected layer is of size 512. The projector networks for both the online and
target networks are 2-layer neural networks with Batch Normalization and ReLU
activation function. The input size, hidden size, and the output sizes are all
equal to 512. The predictor head of the online network is 2-layer network with
same configurations as the projector networks. We have used SGD with momentum
of 0.9, learning rate of 0.008, and batch size of 512 to train the network. To
reduce the over fitting and regularizing the network we have also inserted
Dropout layers between the Bottleneck modules of the ResNet-18 layers with the
probability of dropping the weights as $0.1$.

###  Linear Evaluation Protocol

To evaluate the quality of the representations learned by the online network
of BYOL, we take the online network and freeze its weights. Then a randomly
initialized fully connected layer is trained on top of the online network
using the labeled dataset. After some training runs of BYOL, we train the
linear image classifier using random fully connected layer to convergence.
From Figure 7, we can see that as the training of BYOL progresses, the the
learned representations are able to achieve consistently higher accuracies.

Figure 9: The similarity scores between different images using the features
from the Vanilla BYOL representations.

Figure 10: The similarity scores between different images using the features
from the CCSL BYOL representations.

Figure 11: The similarity scores between different images using the features
from the CSSL BYOL representations.

###  Results

####  Batch Statistics Experiments

We begin by investigating the importance of BN in BYOL framework. For that
purpose, we removed all the BN layers in both encoder and target network. We
found that the performance is nearly random. However, keeping BN in the only
the encoder results in boost in performance as compared to the without-BN
version. We were keen to find out the whether is it the batch statistics which
induces contrastive learning implicitly within the Vanilla BYOL framework. So,
we replaced BN with Layer Normalisation (LN) [1] and Group Normalisation (GN)
[45] which do not require batch statistics. From Table 1, we see that
performance of BYOL with LN or GN has boosted up after adding WS in
convolutional layers followed by GN or LN. WS has also helped us reducing the
training time drastically. Comparing Fig. 2 and Fig. 8, we observe that the
spikes in Figure 2 in case of Vanilla BYOL have vanished and loss landscape
has become smooth. We also infer from Table 1 that GN+WS and LN+WS versions
are performing comparable to the vanilla BYOL. This shows that BYOL is able to
learn non-collapsed representation without BN.

####  Self-Labeling Refinement Results

In this section we describe the results of the experiments conducted in this
work and their analysis. Figures 4, 4, and 4 shows the representation loss and
figures 7, 7, and 7 shows the results for the linear evaluation protocol for
the vanilla BYOL, CCSL BYOL and CSSL BYOL respectively. From the Table 1, we
can see that the representations from the CCSL BYOL gives the best accuracy on
the STL10 validation dataset and CSSL BYOL gives similar results to the
Vanilla BYOL setting. For qualitative analysis of the learned image features,
we take the output representations of the online network and the target
network for various images and computed the similarity between them using the
cosine similarity or the dot product of the normalized representations. From
Figure 11, we can see that the similarity between the image representations is
highest along the diagonal, which can be expected because the online network
and target network is giving the representations for two different views of
the same image. However, in the same figure 11, we can see the high similarity
scores in the first row, between cat-1, and dog-1 which is 0.8551 but low
score between cat-1 and cat-2 which is 0.2580. Intuitively, the similarity
scores between the images of cat-1 and cat-2 should be higher than the
similarity score between cat-1 and dog-1. But in the case of vanilla BYOL, we
can see that there is a discrepancy. More examples of this discrepancy can be
seen in the same figure between the dog-1 and cat-1 in the $3^{rd}$ row and
between dog-2 and cat-1 in the $4^{th}$ row.

The proposed CCSL and CSSL loss functions try to mitigate this discrepancy and
improve the similarity scores between the semantically similar images. After
training the BYOL with CCSL loss function, we produced the similarity scores.
The similarity scores between the images are presented in figure 11. The
similarity scores along the diagonal is highest but ’regularized’ because the
scores are less than corresponding scores we got with vanilla BYOL
representations. From the $1^{st}$ row of figure 11, we can see that score
between cat-1 and dog-1 is low as compared to the similarity scores that we
got with the vanilla BYOL. Similarly, in figure 11, the scores in the $1^{st}$
row between the cat-1 and dog-1 decreased slightly as compared to the scores
with vanilla BYOL. Also note that while some scores are showing the expected
behavior, some of the scores are erratic. For instance, in figure 11, the
score between cat-1 and cat-2 in the $1^{st}$ row is negative when it should
be positive. We attribute this erratic behavior to the size of the unlabeled
dataset. If we increase the amount of the unlabeled images, we might see
expected scores between the semantically similar and dissimilar images.

###  Conclusion

In our project, we have analysed the representation learning in BYOL. We
conducted experiments to confirm that normalisation schemes other than BN like
GN and LN perform comparably when incorporated with weight standardization. We
were also show that the performance of BYOL with explicit constrastive terms
in the loss expression. The CSSL loss function gives better performance than
vanilla BYOL and CCSL performs comparably. To this end, we conclude that BYOL
benefits with the proposed loss functions. For the future work, we would want
to push the performance of BYOL with the proposed loss functions on the
ImageNet [8] dataset which is a standard benchmark. We would work to rectify
the failure cases in Fig. 10 and Fig. 11 by fine-tuning the neural network
using the different values of $\lambda$.

## References

  * [1] Jimmy Lei Ba, Jamie Ryan Kiros and Geoffrey E Hinton  “Layer normalization”  In _arXiv preprint arXiv:1607.06450_ , 2016
  * [2] Philip Bachman, R Devon Hjelm and William Buchwalter  “Learning representations by maximizing mutual information across views”  In _arXiv preprint arXiv:1906.00910_ , 2019
  * [3] Christopher M Bishop  “Latent variable models”  In _Learning in graphical models_ Springer, 1998, pp. 371–403
  * [4] Mathilde Caron et al.  “Unsupervised learning of visual features by contrasting cluster assignments”  In _arXiv preprint arXiv:2006.09882_ , 2020
  * [5] Ting Chen, Simon Kornblith, Mohammad Norouzi and Geoffrey Hinton  “A simple framework for contrastive learning of visual representations”  In _International conference on machine learning_ , 2020, pp. 1597–1607  PMLR
  * [6] Siddhartha Chib and Edward Greenberg  “Understanding the metropolis-hastings algorithm”  In _The american statistician_ 49.4 Taylor & Francis, 1995, pp. 327–335
  * [7] Adam Coates, Andrew Ng and Honglak Lee  “An analysis of single-layer networks in unsupervised feature learning”  In _Proceedings of the fourteenth international conference on artificial intelligence and statistics_ , 2011, pp. 215–223  JMLR WorkshopConference Proceedings
  * [8] Jia Deng et al.  “Imagenet: A large-scale hierarchical image database”  In _2009 IEEE conference on computer vision and pattern recognition_ , 2009, pp. 248–255  Ieee
  * [9] Carl Doersch, Abhinav Gupta and Alexei A Efros  “Unsupervised visual representation learning by context prediction”  In _Proceedings of the IEEE international conference on computer vision_ , 2015, pp. 1422–1430
  * [10] Carl Doersch, Abhinav Gupta and Alexei A Efros  “Unsupervised visual representation learning by context prediction”  In _Proceedings of the IEEE international conference on computer vision_ , 2015, pp. 1422–1430
  * [11] Carl Doersch and Andrew Zisserman  “Multi-task self-supervised visual learning”  In _Proceedings of the IEEE International Conference on Computer Vision_ , 2017, pp. 2051–2060
  * [12] Alexey Dosovitskiy, Jost Tobias Springenberg, Martin Riedmiller and Thomas Brox  “Discriminative unsupervised feature learning with convolutional neural networks”  In _Advances in neural information processing systems_ 27, 2014, pp. 766–774
  * [13] Alan E Gelfand  “Gibbs sampling”  In _Journal of the American statistical Association_ 95.452 Taylor & Francis, 2000, pp. 1300–1304
  * [14] Stuart Geman and Donald Geman  “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images”  In _IEEE Transactions on pattern analysis and machine intelligence_ IEEE, 1984, pp. 721–741
  * [15] Charles J Geyer  “Practical markov chain monte carlo”  In _Statistical science_ JSTOR, 1992, pp. 473–483
  * [16] Spyros Gidaris, Praveer Singh and Nikos Komodakis  “Unsupervised representation learning by predicting image rotations”  In _arXiv preprint arXiv:1803.07728_ , 2018
  * [17] Priya Goyal et al.  “Self-supervised pretraining of visual features in the wild”  In _arXiv preprint arXiv:2103.01988_ , 2021
  * [18] Jean-Bastien Grill et al.  “Bootstrap your own latent: A new approach to self-supervised learning”  In _arXiv preprint arXiv:2006.07733_ , 2020
  * [19] Raia Hadsell, Sumit Chopra and Yann LeCun  “Dimensionality reduction by learning an invariant mapping”  In _2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’06)_ 2, 2006, pp. 1735–1742  IEEE
  * [20] Kaiming He, Xiangyu Zhang, Shaoqing Ren and Jian Sun  “Deep residual learning for image recognition”  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2016, pp. 770–778
  * [21] Kaiming He et al.  “Momentum contrast for unsupervised visual representation learning”  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2020, pp. 9729–9738
  * [22] Olivier Henaff  “Data-efficient image recognition with contrastive predictive coding”  In _International Conference on Machine Learning_ , 2020, pp. 4182–4192  PMLR
  * [23] R Devon Hjelm et al.  “Learning deep representations by mutual information estimation and maximization”  In _arXiv preprint arXiv:1808.06670_ , 2018
  * [24] Sergey Ioffe and Christian Szegedy  “Batch normalization: Accelerating deep network training by reducing internal covariate shift”  In _International conference on machine learning_ , 2015, pp. 448–456  PMLR
  * [25] Sergey Ioffe and Christian Szegedy  “Batch normalization: Accelerating deep network training by reducing internal covariate shift”  In _International conference on machine learning_ , 2015, pp. 448–456  PMLR
  * [26] Alexander Kolesnikov, Xiaohua Zhai and Lucas Beyer  “Revisiting self-supervised visual representation learning”  In _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_ , 2019, pp. 1920–1929
  * [27] Gustav Larsson, Michael Maire and Gregory Shakhnarovich  “Learning representations for automatic colorization”  In _European conference on computer vision_ , 2016, pp. 577–593  Springer
  * [28] Christian Ledig et al.  “Photo-realistic single image super-resolution using a generative adversarial network”  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2017, pp. 4681–4690
  * [29] Junnan Li, Pan Zhou, Caiming Xiong and Steven CH Hoi  “Prototypical contrastive learning of unsupervised representations”  In _arXiv preprint arXiv:2005.04966_ , 2020
  * [30] Ishan Misra and Laurens van der Maaten  “Self-supervised learning of pretext-invariant representations”  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2020, pp. 6707–6717
  * [31] Andrew Ng  “Sparse autoencoder”  In _CS294A Lecture notes_ 72.2011, 2011, pp. 1–19
  * [32] Mehdi Noroozi and Paolo Favaro  “Unsupervised learning of visual representations by solving jigsaw puzzles”  In _European conference on computer vision_ , 2016, pp. 69–84  Springer
  * [33] Aaron van den Oord, Yazhe Li and Oriol Vinyals  “Representation learning with contrastive predictive coding”  In _arXiv preprint arXiv:1807.03748_ , 2018
  * [34] Aaron van den Oord, Yazhe Li and Oriol Vinyals  “Representation learning with contrastive predictive coding”  In _arXiv preprint arXiv:1807.03748_ , 2018
  * [35] Aaron van den Oord, Yazhe Li and Oriol Vinyals  “Representation learning with contrastive predictive coding”  In _arXiv preprint arXiv:1807.03748_ , 2018
  * [36] Deepak Pathak et al.  “Context encoders: Feature learning by inpainting”  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2016, pp. 2536–2544
  * [37] Siyuan Qiao et al.  “Micro-batch training with batch-channel normalization and weight standardization”  In _arXiv preprint arXiv:1903.10520_ , 2019
  * [38] Siyuan Qiao et al.  “Weight standardization”, 2019
  * [39] Pierre H Richemond et al.  “BYOL works even without batch statistics”  In _arXiv preprint arXiv:2010.10241_ , 2020
  * [40] Haizhou Shi et al.  “Run away from your teacher: Understanding byol by a novel self-supervised approach”  In _arXiv preprint arXiv:2011.10944_ , 2020
  * [41] Antti Tarvainen and Harri Valpola  “Mean teachers are better role models: Weight-averaged consistency targets improve semi-supervised deep learning results”  In _arXiv preprint arXiv:1703.01780_ , 2017
  * [42] Yonglong Tian, Dilip Krishnan and Phillip Isola  “Contrastive multiview coding”  In _Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part XI 16_ , 2020, pp. 776–794  Springer
  * [43] Yuandong Tian, Xinlei Chen and Surya Ganguli  “Understanding self-supervised learning dynamics without contrastive pairs”  In _arXiv preprint arXiv:2102.06810_ , 2021
  * [44] Yuandong Tian, Lantao Yu, Xinlei Chen and Surya Ganguli  “Understanding self-supervised learning with dual deep networks”  In _arXiv preprint arXiv:2010.00578_ , 2020
  * [45] Yuxin Wu and Kaiming He  “Group normalization”  In _Proceedings of the European conference on computer vision (ECCV)_ , 2018, pp. 3–19
  * [46] Richard Zhang, Phillip Isola and Alexei A Efros  “Colorful image colorization”  In _European conference on computer vision_ , 2016, pp. 649–666  Springer
  * [47] Pan Zhou, Caiming Xiong, Xiao-Tong Yuan and Steven Hoi  “A Theory-Driven Self-Labeling Refinement Method for Contrastive Representation Learning”  In _arXiv preprint arXiv:2106.14749_ , 2021


# Femtoscopic signatures of the lightest $S$–wave scalar open-charm mesons

M. Albaladejo  0000-0001-7340-9235<EMAIL_ADDRESS>J. Nieves
0000-0002-2518-4606<EMAIL_ADDRESS>Instituto de Física Corpuscular
(centro mixto CSIC-UV), Institutos de Investigación de Paterna, C/Catedrático
José Beltrán 2, E-46980 Paterna, Valencia, Spain  E. Ruiz Arriola
0000-0002-9570-2552<EMAIL_ADDRESS>Departamento de Física Atómica, Molecular
y Nuclear and Instituto Carlos I de Física Teórica y Computacional,
Universidad de Granada, E-18071, Granada, Spain

###### Abstract

We predict femtoscopy correlation functions for $S$–wave $D_{(s)}\phi$ pairs
of lightest pseudoscalar open charm mesons and Goldstone bosons from next-to-
leading order unitarized heavy-meson chiral perturbation theory amplitudes.
The effect of the two-state structure around $2300\,\text{MeV}$ can be clearly
seen in the $(S,I)=(0,1/2)$ $D\pi$, $D\eta$, $D_{s}\overline{K}$ correlation
functions, while in the scalar-strange $(1,0)$ sector, the
$D^{\ast}_{s0}(2317)^{\pm}$ state lying below the $DK$ threshold produces a
depletion of the correlation function near threshold. These exotic states owe
their existence to the nonperturbative dynamics of Goldstone-boson scattering
off $D_{(s)}$. The predicted correlation functions could be experimentally
measured and will shed light into the hadron spectrum confirming that it
should be viewed as more than a collection of quark model states.

## Introduction.—

For hadrons containing charm quarks, there is no possibility of performing
traditional scattering experiments due to technical limitations imposed by the
extremely short life-time of these heavy particles, and experimental
information on the interactions involving charmed hadrons is commonly
extracted from reactions where they are produced in the final state. In
addition, lattice QCD (LQCD) is seen as a benchmark scheme to constrain and
validate the effective field theories (EFTs) employed to describe the relevant
final state interactions (FSI) in the analyzed reactions.

The study of the lightest scalar open-charm $D^{*}_{0}(2300)$ and
$D^{*}_{s0}(2317)^{\pm}$ provides an example of the performance of this
combined strategy [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. These states are exotic
in the sense that they cannot be accommodated in simple constituent quark
models (see for instance the discussion in Ref. [12]). However, both
resonances are dynamically generated from the S-wave scattering of Goldstone
bosons ($\phi$) off the lightest pseudoscalar open charm mesons [$D_{(s)}$].
The $T$–matrix is obtained after solving the Bethe-Salpeter equation with
irreducible amplitudes evaluated at next-to-leading (NLO) in Heavy Meson
Chiral Perturbation Theory (HMChPT) [13, 14] and adopting the on-shell
renormalization scheme [15, 16, 17, 18, 19]. All the unknown low-energy
constants (LECs) are determined from the LQCD $D_{(s)}\phi$ scattering lengths
computed for different pion masses in Ref. [1]. The scattering amplitudes
fulfill elastic unitarity in coupled-channels and successfully describe [2, 5]
the $D_{(s)}\phi$ LQCD energy levels obtained by the Hadron Spectrum [20] and
RQCD [21] Collaborations for various pion masses111The finite-volume scheme of
Ref. [2] provides predictions that also compare well with results from earlier
[22, 23] and more recent [24, 25, 26] LQCD simulations carried out for
different no-physical Goldstone boson masses.. Moreover the theoretical
framework is strongly supported [4, 8, 9] by recent high quality
$[D(\overline{D})\phi]$ FSI data on the $B_{(s)}\to
D(\overline{D})\phi\phi^{\prime}$ decays provided by the LHCb experiment [27,
28, 29, 30, 31]. In the case of the $D^{*}_{0}(2300)$, the theory predicts a
double-pole pattern which provides a natural explanation to various puzzles in
the charm-meson spectrum [4], in particular to why a single broad
$D^{*}_{0}(2300)$ resonance would have a mass almost equal or even higher than
its strange sibling $D^{*}_{s0}(2317)^{\pm}$. Actually, it was only in the
2018 edition of the Review of Particle Physics (RPP) [32] when the possible
existence of a lighter state associate to the $D^{*}_{0}(2300)$ was first
suggested based on the findings of Refs. [2, 4, 6].

On the other hand, femtoscopy has been traditionally utilized to measure the
size of the quark-gluon plasma fireball created in relativistic heavy-ion
collisions. However, since it is sensitive to the correlations between the
particles in the final state, the parameters of the strong interaction can be
probed as well [33, 34]. In high-multiplicity events of $pp$, $pA$ and $AA$
collisions, the hadron production yields are well described by the statistical
models, thereby leaving the correlations between outgoing particles relying
upon the FSI, from which the corresponding scattering parameters can be
extracted [35]. Femtoscopy techniques consist in measuring the correlation in
momentum space for any hadron-hadron pair, and this information is encoded in
the so-called two-particle correlation function (CF). The latter can be
computed as the quotient of the number of pairs of combined particles with the
same relative momentum produced in the same collision event over the reference
distribution of pairs from mixed events. On the theoretical side, CF comes in
terms of the product of the so-called source function, which can be seen as a
kind of form factor that gives the probability that the two particles forming
a pair are emitted at a relative distance of each other, times the squared of
the absolute value of the wave function of the considered channel, built from
the scattering amplitudes [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48,
49, 50, 51, 52].

Additional independent information on the charm-hadron spectrum would be most
welcome to further learn about its properties and nature. Femtoscopic CFs with
the observation of channels involving charmed hadrons in heavy ion collisions
should be then most valuable. There exist experimental studies in the
strangeness sector [53, 54, 55, 56, 57, 58, 33, 59, 60], but importantly the
ALICE collaboration measurement of the $pD^{-}$ and $\overline{p}D^{+}$ CFs in
high-multiplicity $pp$ collisions at 13 TeV [61] paves the way to access the
charm quark sector. This is precisely the goal of this letter, where we will
predict the visible signatures that the $D^{*}_{s0}(2317)^{\pm}$ and the two-
pole structure of the $D^{*}_{0}(2300)$ will produce in the $D^{0}K^{+}$,
$D^{+}K^{0}$ and $D_{s}^{+}\eta$, and the $D^{+}\pi^{0}$, $D^{0}\pi^{+}$
$D^{+}\eta$ and $D_{s}^{+}\overline{K}^{0}$ CFs, respectively. None of these
channels involve the Coulomb interaction and therefore the unitarized NLO
HMChPT scheme, constrained and tested by data and LQCD input, should perform
very well and provide very accurate results which could be confronted to
experiment or used to validate femtoscopic techniques and models.

## Correlation functions.—

The fundamental quantities of our study are the CFs, which can be written as:

$C_{i}(s)=1+\int_{0}^{\infty}\\!\\!\\!\\!\\!\mathrm{d}r\,S(r)\Big{(}\sum_{j}\left\lvert\psi_{i}^{\,j}(s,r)\right\rvert^{2}-j_{0}\left(p_{i}\,r\right)^{2}\Big{)}\,,$
(1a)
$\psi_{i}^{\,j}(s,r)=j_{0}\left(p_{i}\,r\right)\delta_{ij}+T_{ji}(s)\widetilde{G}_{j}(s,r)\,,$
(1b)

where the usual Mandelstam variable $s$ denotes the c.m. energy squared. The
index $i$ denotes the coupled hadron pair considered, and the sum in the index
$j$ runs over all possible ones. Here we consider the channels $D\pi$,
$D\eta$, and $D_{s}\overline{K}$ in the $(S,I)=(0,1/2)$ sector , and
$D_{s}^{+}\pi^{0}$, $D^{0}K^{+}$, $D^{+}K^{0}$, and $D_{s}^{+}\eta$ for
strangeness $S=1$ and isospin $I=0$ and $I=1$. The c.m. momentum of each
channel is denoted by $p_{i}(s)$, and $j_{0}(x)=\sin(x)/x$ is the zeroth order
spherical Bessel function. The functions $\widetilde{G}_{i}(s,r)$, given in
Ref. [51], can be written as:

$\widetilde{G}_{i}(s,r)=\frac{1}{\pi}\int_{s_{\text{th},i}}^{\infty}\\!\\!\\!\\!\mathrm{d}s^{\prime}\,\frac{p_{i}(s^{\prime})}{8\pi\sqrt{s^{\prime}}}\,\frac{j_{0}(p_{i}(s^{\prime})\,r)}{s-s^{\prime}+i\epsilon}\,\theta\left(\Lambda-
p_{i}(s^{\prime})\right).$ (2)

The sharp momentum cutoff $\Lambda$ is introduced to regularize the $r\to 0$
behaviour of the wave functions $\psi_{i}^{\,j}(s,r)$. In the work of Ref.
[51], it is inherited from the regularization of the amplitude $T(s)$, and
allows the on-shell factorization of the latter out of the integral. As it
will be shown, its effect is very small for natural $\Lambda$ values in the
range $\left[600,900\right]\,\text{MeV}$.

The source function is generally taken as a Gaussian, which in the radial
direction reads $S(r)=4\pi r^{2}/(4\pi R^{2})^{3/2}\exp(-r^{2}/4R^{2})$, with
a source size $R$. A factor $4\pi r^{2}$ coming from the differential volume
element $\mathrm{d}^{3}\vec{r}$ has been absorbed into $S(r)$.

Figure 1: Modulus squared of the diagonal elements of the $I=1/2$
$J^{P}=0^{+}$ $T$-matrix in Eq. (3).

The key ingredient in the CFs are the the $S$-wave unitarized partial wave
amplitudes $T_{ij}(s)$, which encode the meson–meson interactions. In this
work, they are taken from Ref. [1] and written as:

$T^{-1}(s)=V^{-1}(s)-G(s)~{},$ (3)

with elementary interactions $V(s)$ computed from NLO HMChPT, and $G(s)$ are
two-meson loop functions regularized with a subtraction constant. As mentioned
in the Introduction, the LECs as well as the subtraction constants have been
successfully determined in Ref. [1] through the computation of $D_{(s)}\phi$
scattering lengths for different pion masses. Therefore, these amplitudes are
completely determined (no free parameters) and, in this sense, the
calculations presented in this manuscript are genuine predictions for future
experiments. The amplitudes, in turn, have been used in Refs. [2] and [5] to
compute energy levels in a finite volume for the $D\pi$, $D\eta$, and
$D_{s}\overline{K}$ and $DK$ systems, that are successfully compared with the
LQCD simulations in Refs. [20] and [21], respectively. Importantly, in Ref.
[2] it is shown that these amplitudes contain two poles in two different
Riemann sheets, corresponding to two different $D^{\ast}_{0}$ states in the
$2300\,\text{MeV}$ region, and not just one (previously known as
$D^{\ast}_{0}(2400)$, see Refs. [32, 62]), as traditionally thought. Later on,
Ref. [4] showed that this two-state structure is also compatible with the
experimental LHCb data on $B^{-}\to D^{+}\pi^{-}\pi^{-}$ [31] and
$B^{0}_{s}\to\overline{D}^{\,0}K^{-}\pi^{+}$ [27]. As already mentioned, this
two-state picture has been further tested using FSI data from LHCb experiment,
and is now considered as a strong possibility in the current edition of the
RPP [62]. For later reference, the modulus squared of the diagonal matrix
elements are shown in Fig. 1, where a clear peak around $2135\,\text{MeV}$
produced by the lightest $D^{\ast}_{0}$ state can be seen, and the structure
of the second resonance can be appreciated also at $2450\,\text{MeV}$,
together with a strong interference with the $D\eta$ and $D_{s}\overline{K}$
thresholds.

Figure 2: Correlation function for the $D\pi$ (a), $D\eta$ (b), and
$D_{s}\overline{K}$ (c) channels with $I=I_{z}=1/2$ as a function of their
c.m. momentum $k$ for different values of the source size $R=1\,\text{fm}$
(red), $2\,\text{fm}$ (blue), and $5\,\text{fm}$. The dark, inner bands
represent the variation of $\Lambda$ in Eq. (2) in a range
$\left[600,900\right]\,\text{MeV}$. The light, outer bands represent the
addition in quadrature of the 68% confident-level (CL) uncertainties in the
CFs inherited from those affecting the LECs of the NLO unitarized scattering
amplitudes. In panel (d) we present together the CFs, as a function of the
c.m. energy $\sqrt{s}$, for the three channels with a source of size
$R=1\,\text{fm}$, and with only the full uncertainty band.

## Non-strange sector and the two $D^{\ast}_{0}$ states.—

The CFs for the $I=1/2$ channels are shown in Fig. 2 for different, typical
sizes $R=1$, $2$, and $5\,\text{fm}$ of the source. One would expect the clear
peak at $\sqrt{s}\simeq 2135\,\text{MeV}$ in the $D\pi$ amplitude (see Fig.
1), produced by the lightest $D^{\ast}_{0}$ state, to leave a strong imprint
in the $C_{D\pi}(k)$ CF at $k\simeq 215\,\text{MeV}$. Instead of a bump, a
depletion and a minimum can be found there for the $R=1\,\text{fm}$ source,
which might come as a surprise. However, one can show (see Appendix) from the
Lednicky-Lyuboshits model [37] that this result can be expected for a
Breit–Wigner-like resonance with a width $\Gamma\simeq 200\,\text{MeV}$ and
for a source size
$R\mathrel{\ensurestackMath{\stackengine{-.4ex}{>}{\rotatebox{25.0}{$\sim$}}{U}{l}{F}{T}{S}}}1\,\text{fm}$.
For $R=2$ and $5\,\text{fm}$, the minimum is still present, but certainly
diluted. One would need a much narrower state
($\Gamma\mathrel{\ensurestackMath{\stackengine{-.4ex}{<}{\rotatebox{-25.0}{$\sim$}}{U}{r}{F}{T}{S}}}50\,\text{MeV}$)
or smaller source size
($R\mathrel{\ensurestackMath{\stackengine{-.4ex}{<}{\rotatebox{-25.0}{$\sim$}}{U}{r}{F}{T}{S}}}0.5\,\text{fm}$)
to observe a clear peak. Nonetheless, the position of the minimum closer to
$215\,\text{MeV}$ rather than to $400\,\text{MeV}$ (as expected for a nominal
mass around $2340\,\text{MeV}$ [62]) would be a clear indication of the
existence of a lower pole.

The strongest effect of the second (higher) pole can be seen in the
$D_{s}\overline{K}$ amplitude depicted in Fig. 1, although it is located below
the threshold. Indeed, one can observe in $C_{D_{s}\overline{K}}(k)$ [Fig. 2,
panel (c)] a clear dip for low momentum $k$, i.e. for energies $\sqrt{s}$
slightly above the $D_{s}\overline{K}$ threshold.

All these features can be better appreciated in Fig. 2, panel (d), where the
CFs of the three pairs are shown for $R=1\,\text{fm}$ as a function of
$\sqrt{s}$. There, the two different minima at $\sqrt{s}\simeq
2135\,\text{MeV}$ ($D\pi$ CF) and $2475\,\text{MeV}$ ($D_{s}\overline{K}$ CF),
produced by the two different $D^{\ast}_{0}$ states, can be observed. The
experimental observation of these features would constitute a strong
additional support of the two-state pattern predicted by the HMChPT unitary
amplitudes employed.

Finally, we note that we have considered the CF for the $I=1/2$ $D\pi$ system,
which, for $I_{z}=+1/2$, is a linear combination of the physical
$D^{+}\pi^{0}$ and $D^{0}\pi^{+}$.222Note that, by considering $I_{z}=+1/2$,
we avoid the interference with Coulomb interaction. The latter is present in
the case $I_{z}=-1/2$, which mixes $D^{+}\pi^{-}$ and $D^{0}\pi^{0}$, and
would be also present in $D_{s}^{+}K^{-}$ channel. Note also that the isospin
breaking effect between the $D^{+}\pi^{0}$ and $D^{0}\pi^{+}$ states induced
by the masses is expected to be small, since their thresholds are just
$1\,\text{MeV}$ apart. To obtain the truly measurable CFs for the physical
$D\pi$ states, one would need to consider also the $I=3/2$ components.
Introducing the appropriate isospin combinations in Eq. (1) one finds:

$C_{D^{+}\pi^{0}}=\frac{2}{3}C^{D\pi}_{3/2}+\frac{1}{3}C^{D\pi}_{1/2}~{},\quad
C_{D^{0}\pi^{+}}=\frac{1}{3}C^{D\pi}_{3/2}+\frac{2}{3}C^{D\pi}_{1/2}~{}.$ (4)

However, since the interaction in $I=3/2$ is very weak, one finds (as we have
checked) $C^{D\pi}_{3/2}\simeq 1$. Therefore, both physical CFs can be
obtained from our computed $C^{D\pi}_{1/2}(k)$ displayed in Fig. 2, and in
particular we also predict the relation
$2\,C_{D^{0}\pi^{+}}(k)-C_{D^{+}\pi^{0}}(k)\simeq 1$.

Figure 3: $D^{0}K^{+}$ correlation function for different source sizes,
$R=1\,\text{fm}$ (red), $2\,\text{fm}$ (blue), and $5\,\text{fm}$.

## Strange sector and the $D^{\ast}_{s0}(2317)^{\pm}$ state.—

In the $S=1$ sector we consider the $D_{s}^{+}\pi^{0}$, $D^{0}K^{+}$,
$D^{+}K^{0}$, and $D_{s}^{+}\eta$ channels, with $I=0$ and $I=1$ components.
The CF for the $D^{0}K^{+}$ channel (the lowest lying of the two $DK$ states
with $I_{z}=0$) is shown in Fig. 3 for different source sizes $R$. The
$D^{+}K^{0}$ threshold lies $\simeq 9\,\text{MeV}$ above the $D^{0}K^{+}$ one,
which implies a momentum $k_{D^{0}K^{+}}\simeq 83\,\text{MeV}$, where a cusp
effect can be seen in Fig. 3. Furthermore, and unlike the $D\pi$ CFs mentioned
before, the $DK$ CFs cannot be filtered, because:

$C_{D^{0}K^{+}}=C_{D^{+}K^{0}}=\frac{C^{DK}_{0}+C^{DK}_{1}}{2}\,$ (5)

with $C^{DK}_{I}$ the CFs for definite isospin $I$. This fact, together with
the larger mass difference between the $D^{0}K^{+}$ and $D^{+}K^{0}$
thresholds, prompts one to use physical channels from the beginning instead of
definite isospin ones333The interaction in the $DK$ isovector channel, though
smaller than in the scalar sector which generates the exotic
$D^{\ast}_{s0}(2317)^{\pm}$, is not negligible in contrast to the case of the
$I=3/2$ $D\pi$ channel..

A clear depletion in $C_{D^{0}K^{+}}$ at the origin can be seen in Fig. 3,
being the suppression larger for the smaller source sizes. This effect is due
to the presence in our amplitudes $T_{ij}(s)$ of the
$D^{\ast}_{s0}(2317)^{\pm}$ bound-state pole444In the isospin limit the $I=0$
and $I=1$ sectors decouple, and the $D^{\ast}_{s0}(2317)^{\pm}$ is a bound
state in the $I=0$ amplitudes below the $DK$ threshold. This state lies above
the $I=1$ $D_{s}^{+}\pi^{0}$ channel but has a negligible coupling to it when
the latter is coupled to the physical $DK$ ones. In this sense, we still call
the $D^{\ast}_{s0}(2317)^{\pm}$ a bound state. below the $DK$ threshold, with
a binding energy around $45\,\text{MeV}$, which translates into a large
binding momentum $p=i\gamma_{b}$, $\gamma_{b}\simeq 190\,\text{MeV}$. The
observation of this behavior at the $D^{0}K^{+}$ threshold in the measurement
of these CFs, and in particular their dependence on the source size $R$, could
provide important feedback on the exotic state $D^{\ast}_{s0}(2317)^{\pm}$.

The $D^{0}K^{+}$ CF has been recently computed in Ref. [52]. There, the chiral
amplitudes were computed only at LO and the $T_{ij}$ amplitudes were
regularized with Gaussian-like form factors, inducing a different off-shell
behavior of the $T$–matrix which also produces variations in the wave-
functions. The cutoff was fixed by adjusting the position of the generated
pole to the mass of the $D^{\ast}_{s0}(2317)^{\pm}$, but this does not
guaranty that the $DK$ $I=1/2$ scattering length obtained in Ref. [52] is
equal to that calculated in our scheme. These differences could explain the
changes that can be seen in the CFs, in particular at $k\simeq 0$ where, for
instance, the scheme of Ref. [52] predicts $C_{D^{0}K^{+}}(0)\simeq 0.4$ for
$R=1.2\,\text{fm}$. However, the trend is similar for both Ref. [52] and the
present work, which is reassuring. Moreover, the CF uncertainty bands in our
results might partially account for the variations.

## Conclusions.—

We have presented the first joint prediction of the CFs of the open-charm
$D_{(s)}\phi$ pairs in the $(S,I)=(0,1/2)$ and $S=1$ sectors based on well
determined NLO unitarized chiral amplitudes. For $I=1/2$ two distinct minima
are observed in the correlation functions $C_{D\pi}$ and
$C_{D_{s}\overline{K}}$, which are produced by the lower and higher
$D^{\ast}_{0}$ poles, respectively. The measurement of these features of the
CFs could shed light on this otherwise elusive resonance(s). We also predict
$2\,C_{D^{0}\pi^{+}}(k)-C_{D^{+}\pi^{0}}(k)\simeq 1$. For the $D^{0}K^{+}$
pair, a sizeable depletion is observed at threshold in the CF (as in Ref.
[52]), due to the presence of the $D^{\ast}_{s0}(2317)^{\pm}$ bound state
originated from the $I=0$ interaction.

Given the expected precision that can be achieved in the measurements of CFs
at ALICE, and other experiments that could join the femtoscopy venture, a
clear comparison with the CFs predicted in this work should be feasible in a
near future. In particular, the observation of the features produced by the
two-state pattern in the $D\pi$ and $D_{s}\overline{K}$ CFs would constitute a
further strong support of this picture. In addition to the experimental
consequences of the measurements of these correlation functions, the
comparison with the predictions presented in this work would certainly help to
reduce the theoretical uncertainties of the chiral amplitudes.

## Acknowledgements.—

The authors would like to thank Otón Vázquez Doce for valuable discussions.
This work was supported by the Spanish Ministerio de Ciencia e Innovación
(MICINN) under contracts Nos. PID2020-112777GB-I00 and PID2020-114767GB-I00,
by Generalitat Valenciana under contract PROMETEO/2020/023 and Junta de
Andalucía grant FQM-225. This project has received funding from the European
Union Horizon 2020 research and innovation programme under the program
H2020-INFRAIA-2018-1, grant agreement No. 824093 of the STRONG-2020 project.
M. A. is supported through Generalitat Valenciana (GVA) Grants No.
CIDEGENT/2020/002 and thanks the warm support of ACVJLI.

##  Appendix A Lednicky-Lyuboshits model and resonances

In the Lednicky-Lyuboshits approximation for a single-channel CF [37], one
substitutes the full wave function $\psi(s,r)$ in Eq. (1a) with its non-
relativistic, asymptotic ($r\to\infty$) form:

$\phi_{\text{asy}}(k,r)\simeq\frac{\sin{kr}}{kr}+f(k)\frac{e^{ikr}}{r}\,,$ (6)

where for clarity the momentum $k$ instead of the Mandelstam variable $s$ has
been taken as a variable. Introducing Eq. (6) into Eq. (1a), and assuming
still a gaussian source $S(r)$, the resulting CF is:

$C_{\text{LL}}(k)=1+\frac{\left\lvert
f(k)\right\rvert^{2}}{2R^{2}}+\frac{2\operatorname{Re}{f(k)}}{\sqrt{\pi}R}F_{1}(x)-\frac{\operatorname{Im}{f(k)}}{R}F_{2}(x)~{},$
(7)

where $x=2kR$, $F_{1}(x)=\int_{0}^{x}\mathrm{d}te^{(t^{2}-x^{2})}/x$, and
$F_{2}(x)=\left(1-e^{-x^{2}}\right)/x$. Above $f(k)$ is the standard quantum
mechanics amplitude, $\operatorname{Im}{f(k)}=-k$. It can be written as:

$f^{-1}(k)=k\cot\delta(k)-ik~{},$ (8)

where $\delta(k)$ is the phase shift, and one gets:

$C_{\text{LL}}(k)=1+\frac{2\sin^{2}\delta(k)}{x^{2}}\left(e^{-x^{2}}+\frac{2xF_{1}(x)}{\sqrt{\pi}}\cot\delta(k)\right)\,.$
(9)

At threshold,

$C_{\text{LL}}(k=0)=1+\frac{a\left(a+\frac{4R}{\sqrt{\pi}}\right)}{2R^{2}}\,,$
(10)

where $a$ is the scattering length, $a=\lim_{k\to 0}k\cot{\delta(k)}$. For
$a>0$ or $a<-\frac{4}{\sqrt{\pi}}R\simeq-2.26\,R$, one has
$C_{\text{LL}}(k=0)>1$. The lowest value $C_{\text{LL}}(k=0)$ can achieve is
for $a=-\frac{2}{\sqrt{\pi}}R\simeq-1.13\,R$, when one has
$C_{\text{LL}}(k=0)=1-\frac{2}{\pi}\simeq 0.36$. Incidentally, the $I=0$ $DK$
scattering length is $a=-0.84\,\text{fm}$, thus not far from this value for
$R=1\,\text{fm}$. The depletion at threshold in the $D^{0}K^{+}$ CF is thus
caused by its $I=0$ component.

On a different note, if there is a resonance for a momentum $k_{R}$, then
$\delta(k_{R})=\pi/2$, and one has
$C_{\text{LL}}(k_{R})=1+\frac{2e^{-x_{R}^{2}}}{x_{R}^{2}}$, with
$x_{R}=2k_{R}R$. Therefore, for large $x_{R}$, say $x_{R}\gtrsim 2$, the
correlation function is largely suppressed, $C_{\text{LL}}(k_{R})\simeq 1$,
slightly above unity. Furthermore, with a regular Breit-Wigner shape for the
phase shift $\delta(k)$, it can also be proven that $C^{\prime}(k_{R})<0$,
i.e. the CF is decreasing at $k=k_{R}$. For $k>k_{R}$ one has
$\cot\delta(k)<0$ ($\delta(k)>\pi/2$), which is a necessary condition for the
bracket in Eq. (9) to change sign. It is then possible for $C_{\text{LL}}(k)$
to go below one for $k>k_{R}$, specially taking into account that the first
term in the bracket is very suppressed with respect to the coefficient of
$\cot\delta(k)$ in the second term. Since asymptotically for large momenta one
has $C_{\text{LL}}(k)\to 1$, then a minimum will appear if the CF becomes
smaller than one for some momentum region above $k_{R}$.

Figure 4: Correlation function in the Lednicky-Lyuboshits approximation
($C_{\text{LL}}(k)$), in terms of the reduced variable $x=2kR$, for different
values of the reduced width $x_{\Gamma}^{2}=4\mu\Gamma R^{2}$ (see text for
further details).

Certainly, whether this minimum is pronounced or diluted, or a maximum can
appear for $k<k_{R}$, depends on the details of the resonance ($k_{R}$ and its
width, $\Gamma$) and the source size $R$. For illustrative purposes, let us
assume a non-relativistic BW-shape for $\cot\delta(k)$,

$\cot\delta(k)=-\frac{k_{R}}{k}\frac{k^{2}-k_{R}^{2}}{2\mu}\frac{2}{\Gamma}=-\frac{x_{R}}{x}\frac{x^{2}-x_{R}^{2}}{x_{\Gamma}^{2}}\,,$
(11)

with $x_{\Gamma}^{2}=4\,\mu\,\Gamma\,R^{2}$, and $\mu$ the reduced mass of the
hadron pair. We show in Fig. 4 $C_{\text{LL}}(k)$ (as a function of the
reduced variable $x$) obtained for different values of $x_{\Gamma}$, taking,
for definiteness, $x_{R}=2.2$ (for $R=1\,\text{fm}$, this would correspond to
$k_{R}\simeq 215\,\text{MeV}$, and $\sqrt{s}=2105\,\text{MeV}$ for a $D\pi$
system). As discussed earlier, since $x_{R}\gtrsim 2$, all the curves are very
close to one and decreasing around $x\simeq x_{R}$. As can be seen, for a
narrow resonance ($x_{\Gamma}=0.6$, red dashed line) there appears a maximum
before $x_{R}$, but there is still a minimum after. In the case of a broader
resonance with $x_{\Gamma}=1.8$ (blue dashed line) (which would correspond to
$\Gamma\simeq 200\,\text{MeV}$ for $R=1\,\text{fm}$ and $D\pi$ reduced mass
$\mu$) only a minimum can be appreciated. For larger $x_{\Gamma}$, the minimum
can no longer be observed.

## References

  * Liu  _et al._ [2013] L. Liu, K. Orginos, F.-K. Guo, C. Hanhart, and U.-G. Meissner, Phys. Rev. D 87, 014508 (2013), arXiv:1208.4535 [hep-lat] .
  * Albaladejo  _et al._ [2017] M. Albaladejo, P. Fernandez-Soler, F.-K. Guo, and J. Nieves, Phys. Lett. B 767, 465 (2017), arXiv:1610.06727 [hep-ph] .
  * Guo  _et al._ [2018a] F.-K. Guo, C. Hanhart, U.-G. Meißner, Q. Wang, Q. Zhao, and B.-S. Zou, Rev. Mod. Phys. 90, 015004 (2018a), arXiv:1705.00141 [hep-ph] .
  * Du  _et al._ [2018] M.-L. Du, M. Albaladejo, P. Fernández-Soler, F.-K. Guo, C. Hanhart, U.-G. Meißner, J. Nieves, and D.-L. Yao, Phys. Rev. D 98, 094018 (2018), arXiv:1712.07957 [hep-ph] .
  * Albaladejo  _et al._ [2018] M. Albaladejo, P. Fernandez-Soler, J. Nieves, and P. G. Ortega, Eur. Phys. J. C 78, 722 (2018), arXiv:1805.07104 [hep-ph] .
  * Guo  _et al._ [2018b] X.-Y. Guo, Y. Heo, and M. F. M. Lutz, Phys. Rev. D 98, 014510 (2018b), arXiv:1801.10122 [hep-lat] .
  * Sugiura and Hyodo [2019] T. Sugiura and T. Hyodo, Phys. Rev. C 99, 065201 (2019), arXiv:1904.00589 [hep-ph] .
  * Du  _et al._ [2019] M.-L. Du, F.-K. Guo, and U.-G. Meißner, Phys. Rev. D 99, 114002 (2019), arXiv:1903.08516 [hep-ph] .
  * Du  _et al._ [2021] M.-L. Du, F.-K. Guo, C. Hanhart, B. Kubis, and U.-G. Meißner, Phys. Rev. Lett. 126, 192001 (2021), arXiv:2012.04599 [hep-ph] .
  * Mai  _et al._ [2023] M. Mai, U.-G. Meißner, and C. Urbach, Phys. Rept. 1001, 1 (2023), arXiv:2206.01477 [hep-ph] .
  * Asokan  _et al._ [2022] A. Asokan, M.-N. Tang, F.-K. Guo, C. Hanhart, Y. Kamiya, and U.-G. Meißner, (2022), arXiv:2212.07856 [hep-ph] .
  * Ortega  _et al._ [2016] P. G. Ortega, J. Segovia, D. R. Entem, and F. Fernandez, Phys. Rev. D 94, 074037 (2016), arXiv:1603.07000 [hep-ph] .
  * Guo  _et al._ [2009] F.-K. Guo, C. Hanhart, and U.-G. Meissner, Eur. Phys. J. A 40, 171 (2009), arXiv:0901.1597 [hep-ph] .
  * Geng  _et al._ [2010] L. S. Geng, N. Kaiser, J. Martin-Camalich, and W. Weise, Phys. Rev. D 82, 054022 (2010), arXiv:1008.0383 [hep-ph] .
  * Oller and Oset [1997] J. A. Oller and E. Oset, Nucl. Phys. A 620, 438 (1997), [Erratum: Nucl.Phys.A 652, 407–409 (1999)], arXiv:hep-ph/9702314 .
  * Nieves and Ruiz Arriola [1999] J. Nieves and E. Ruiz Arriola, Phys. Lett. B 455, 30 (1999), arXiv:nucl-th/9807035 .
  * Oller and Oset [1999] J. A. Oller and E. Oset, Phys. Rev. D 60, 074023 (1999), arXiv:hep-ph/9809337 .
  * Nieves and Ruiz Arriola [2000] J. Nieves and E. Ruiz Arriola, Nucl. Phys. A 679, 57 (2000), arXiv:hep-ph/9907469 .
  * Oller and Meissner [2001] J. A. Oller and U. G. Meissner, Phys. Lett. B 500, 263 (2001), arXiv:hep-ph/0011146 .
  * Moir  _et al._ [2016] G. Moir, M. Peardon, S. M. Ryan, C. E. Thomas, and D. J. Wilson, JHEP 10, 011 (2016), arXiv:1607.07093 [hep-lat] .
  * Bali  _et al._ [2017] G. S. Bali, S. Collins, A. Cox, and A. Schäfer, Phys. Rev. D 96, 074501 (2017), arXiv:1706.01247 [hep-lat] .
  * Mohler  _et al._ [2013] D. Mohler, C. B. Lang, L. Leskovec, S. Prelovsek, and R. M. Woloshyn, Phys. Rev. Lett. 111, 222001 (2013), arXiv:1308.3175 [hep-lat] .
  * Lang  _et al._ [2014] C. B. Lang, L. Leskovec, D. Mohler, S. Prelovsek, and R. M. Woloshyn, Phys. Rev. D 90, 034510 (2014), arXiv:1403.8103 [hep-lat] .
  * Gayer  _et al._ [2021] L. Gayer, N. Lang, S. M. Ryan, D. Tims, C. E. Thomas, and D. J. Wilson (Hadron Spectrum), JHEP 07, 123 (2021), arXiv:2102.04973 [hep-lat] .
  * Cheung  _et al._ [2021] G. K. C. Cheung, C. E. Thomas, D. J. Wilson, G. Moir, M. Peardon, and S. M. Ryan (Hadron Spectrum), JHEP 02, 100 (2021), arXiv:2008.06432 [hep-lat] .
  * Gregory  _et al._ [2021] E. B. Gregory, F.-K. Guo, C. Hanhart, S. Krieg, and T. Luu, (2021), arXiv:2106.15391 [hep-ph] .
  * Aaij  _et al._ [2014] R. Aaij _et al._ (LHCb), Phys. Rev. D 90, 072003 (2014), arXiv:1407.7712 [hep-ex] .
  * Aaij  _et al._ [2015a] R. Aaij _et al._ (LHCb), Phys. Rev. D 91, 092002 (2015a), [Erratum: Phys.Rev.D 93, 119901 (2016)], arXiv:1503.02995 [hep-ex] .
  * Aaij  _et al._ [2015b] R. Aaij _et al._ (LHCb), Phys. Rev. D 92, 032002 (2015b), arXiv:1505.01710 [hep-ex] .
  * Aaij  _et al._ [2015c] R. Aaij _et al._ (LHCb), Phys. Rev. D 92, 012012 (2015c), arXiv:1505.01505 [hep-ex] .
  * Aaij  _et al._ [2016] R. Aaij _et al._ (LHCb), Phys. Rev. D 94, 072001 (2016), arXiv:1608.01289 [hep-ex] .
  * Tanabashi  _et al._ [2018] M. Tanabashi _et al._ (Particle Data Group), Phys. Rev. D 98, 030001 (2018).
  * ALICE Collab. [2020] ALICE Collab., Nature 588, 232 (2020), [Erratum: Nature 590, E13 (2021)], arXiv:2005.11495 [nucl-ex] .
  * ALICE Collab. [2022] ALICE Collab., (2022), arXiv:2211.02491 [physics.ins-det] .
  * Fabbietti  _et al._ [2021] L. Fabbietti, V. Mantovani Sarti, and O. Vazquez Doce, Ann. Rev. Nucl. Part. Sci. 71, 377 (2021), arXiv:2012.09806 [nucl-ex] .
  * Koonin [1977] S. E. Koonin, Phys. Lett. B 70, 43 (1977).
  * Lednicky and Lyuboshits [1981] R. Lednicky and V. L. Lyuboshits, Yad. Fiz. 35, 1316 (1981).
  * Pratt [1986] S. Pratt, Phys. Rev. D 33, 1314 (1986).
  * Pratt  _et al._ [1990] S. Pratt, T. Csorgo, and J. Zimanyi, Phys. Rev. C 42, 2646 (1990).
  * Bauer  _et al._ [1992] W. Bauer, C. K. Gelbke, and S. Pratt, Ann. Rev. Nucl. Part. Sci. 42, 77 (1992).
  * Morita  _et al._ [2015] K. Morita, T. Furumoto, and A. Ohnishi, Phys. Rev. C 91, 024916 (2015), arXiv:1408.6682 [nucl-th] .
  * Ohnishi  _et al._ [2016] A. Ohnishi, K. Morita, K. Miyahara, and T. Hyodo, Nucl. Phys. A 954, 294 (2016), arXiv:1603.05761 [nucl-th] .
  * Morita  _et al._ [2016] K. Morita, A. Ohnishi, F. Etminan, and T. Hatsuda, Phys. Rev. C 94, 031901 (2016), [Erratum: Phys.Rev.C 100, 069902 (2019)], arXiv:1605.06765 [hep-ph] .
  * Hatsuda  _et al._ [2017] T. Hatsuda, K. Morita, A. Ohnishi, and K. Sasaki, Nucl. Phys. A 967, 856 (2017), arXiv:1704.05225 [nucl-th] .
  * Mihaylov  _et al._ [2018] D. L. Mihaylov, V. Mantovani Sarti, O. W. Arnold, L. Fabbietti, B. Hohlweger, and A. M. Mathis, Eur. Phys. J. C 78, 394 (2018), arXiv:1802.08481 [hep-ph] .
  * Haidenbauer [2019] J. Haidenbauer, Nucl. Phys. A 981, 1 (2019), arXiv:1808.05049 [hep-ph] .
  * Morita  _et al._ [2020] K. Morita, S. Gongyo, T. Hatsuda, T. Hyodo, Y. Kamiya, and A. Ohnishi, Phys. Rev. C 101, 015201 (2020), arXiv:1908.05414 [nucl-th] .
  * Kamiya  _et al._ [2020] Y. Kamiya, T. Hyodo, K. Morita, A. Ohnishi, and W. Weise, Phys. Rev. Lett. 124, 132501 (2020), arXiv:1911.01041 [nucl-th] .
  * Kamiya  _et al._ [2022a] Y. Kamiya, K. Sasaki, T. Fukui, T. Hyodo, K. Morita, K. Ogata, A. Ohnishi, and T. Hatsuda, Phys. Rev. C 105, 014915 (2022a), arXiv:2108.09644 [hep-ph] .
  * Kamiya  _et al._ [2022b] Y. Kamiya, T. Hyodo, and A. Ohnishi, Eur. Phys. J. A 58, 131 (2022b), arXiv:2203.13814 [hep-ph] .
  * Vidana  _et al._ [2023] I. Vidana, A. Feijoo, M. Albaladejo, J. Nieves, and E. Oset, (2023), arXiv:2303.06079 [hep-ph] .
  * Liu  _et al._ [2023] Z.-W. Liu, J.-X. Lu, and L.-S. Geng, (2023), arXiv:2302.01046 [hep-ph] .
  * Acharya  _et al._ [2019a] S. Acharya _et al._ (ALICE), Phys. Rev. C 99, 024001 (2019a), arXiv:1805.12455 [nucl-ex] .
  * Acharya  _et al._ [2020a] S. Acharya _et al._ (ALICE), Phys. Rev. Lett. 124, 092301 (2020a), arXiv:1905.13470 [nucl-ex] .
  * Acharya  _et al._ [2022a] S. Acharya _et al._ (ALICE), Phys. Lett. B 833, 137272 (2022a), arXiv:2104.04427 [nucl-ex] .
  * Acharya  _et al._ [2020b] S. Acharya _et al._ (ALICE), Phys. Lett. B 805, 135419 (2020b), arXiv:1910.14407 [nucl-ex] .
  * Acharya  _et al._ [2019b] S. Acharya _et al._ (ALICE), Phys. Lett. B 797, 134822 (2019b), arXiv:1905.07209 [nucl-ex] .
  * Acharya  _et al._ [2019c] S. Acharya _et al._ (ALICE), Phys. Rev. Lett. 123, 112002 (2019c), arXiv:1904.12198 [nucl-ex] .
  * Acharya  _et al._ [2021] S. Acharya _et al._ (ALICE), Phys. Rev. Lett. 127, 172301 (2021), arXiv:2105.05578 [nucl-ex] .
  * Acharya  _et al._ [2022b] S. Acharya _et al._ (ALICE), Phys. Lett. B 829, 137060 (2022b), arXiv:2105.05190 [nucl-ex] .
  * Acharya  _et al._ [2022c] S. Acharya _et al._ (ALICE), Phys. Rev. D 106, 052010 (2022c), arXiv:2201.05352 [nucl-ex] .
  * Workman  _et al._ [2022] R. L. Workman _et al._ (Particle Data Group), PTEP 2022, 083C01 (2022).