#  Meson mass and width: Deep learning approach

M. Malekhosseini1, S. Rostami1, A. R. Olamaei2,3, R. Ostovar1, K. Azizi1,4
1 Department of Physics, University of Tehran, North Karegar Avenue, Tehran
14395-547, Iran
2 Department of Physics, Jahrom University, Jahrom, P. O. Box 74137-66171,
Iran
3 School of Physics, Institute for Research in Fundamental Sciences (IPM),
P. O. Box 19395-5531, Tehran, Iran
4 Department of Physics, Doğuş University, Dudullu-Ümraniye, 34775 Istanbul,
Türkiye
Corresponding author

It is fascinating to predict the mass and width of the ordinary and exotic
mesons solely based on their quark content and quantum numbers. Such
prediction goes beyond conventional methodologies traditionally employed in
hadron physics for calculating or estimating these quantities. The relation
between the quantum numbers and the properties of the mesons, such as the mass
and width, is complicated in the world of particle physics. However, the deep
neural network (DNN) as a subfield of machine learning techniques provides a
solution to this problem. By analyzing large datasets, deep learning
algorithms can automatically identify complex patterns among the particles’
quantum numbers, and their mass and width, that would otherwise require
complex calculations. In this study, we present two approaches using the DNNs
to estimate the mass of some ordinary and exotic mesons. Also for the first
time, the DNNs are trained to predict the width of ordinary and exotic mesons,
whose widths have not been experimentally known. Our predictions obtained
through the DNNs, will be useful for future experimental searches.

##  1 Introduction

Nature suggests that hadrons are composite subatomic particles containing two
or more quarks binding together via strong interaction. The realm of hadrons
is divided into baryons and mesons. Baryons are composed of three valence-
quarks and mesons are created as valence quark-antiquark bound states [1, 2,
3]. As we know, the true structure of a hadron may be more complex than a
simple composition of two or three quarks [4, 5]. Obtaining the quantum
numbers and physical structures of these hadronic subatomic particles provides
us with important pieces of information. Also, the mass and width are two
determinative quantities for theorists and experimentalists’ researches. The
origin of the hadron mass has been a challenging topic in particle physics for
a long time. The mass of a hadron is influenced by not only the quark content
but also the dynamics within the particle. Indeed, the creation and
annihilation of virtual quark-antiquark pairs and virtual gluons and their
interactions with the QCD vacuum (condensations) contribute to the hadronic
compositions [6, 7]. According to the quark model, the hadrons going beyond
$q\bar{q}$ and $qqq$ compositions are called exotic hadrons. For instance, the
four-quark mesons or tetraquarks are composed of $q\bar{q}q\bar{q}$. These
exotic states have been explored by theorists [8, 9, 10, 11, 12, 13, 14, 15,
16, 17] and observed by the LHCb and other Collaborations [18, 19, 20, 21, 22,
23, 24, 25, 26, 27, 28]. In recent years, various kinds of exotic particles
have been studied through theoretical methods such as QCD sum rules (QSR) [29,
30, 31, 32], potential models and lattice QCD [33, 34]. ATLAS experiment has
presented results on exotic resonances from the proton-proton collision [35].

To date, the Particle Data Group (PDG) [36] has documented hundreds mesonic
and baryonic states. After all, it is still a hot topic to probe structures of
the hadrons and exotic states, since the internal hadronic configurations of
quark and gluons can be obtained in high-energy reactions [37], thanks to the
progresses made at different hadron colliders. Moreover, it is important to
highlight that the new exotic particles reveal our limited knowledge of the
hadronic systems due to the gap between the theory and experiment [38].

Artificial Intelligence and the machine learning (AI/ML) techniques have the
potential to bridge this gap and aid the theory and experiment for improved
performance and accuracy [39, 40]. The traditional AI and ML methods have been
used in high-energy physics (HEP) since the 1990s [39, 40, 41, 42].
Afterwards, the ML approaches were mostly applied in particle and event
identification as well as reconstruction in the 2010s, and consequently played
a role in discovery of the Higgs boson at the Large Hadron Collider (LHC) in
2012 [39, 40, 41, 42, 43, 44, 45, 46].

The recent breakthroughs in the modern ML have been transfiguring particle
physics studies in both phenomenological and experimental areas. In other
words, the modern ML facilitates collaboration between LHC experimentalists,
phenomenologists, and the data science community [39, 40, 41, 42, 43, 44, 45,
46, 47, 48, 49, 50, 51, 52, 53, 54, 55]. Some established ML algorithms, such
as boosted decision trees (BDTs) and neural networks (NNs), have a long and
successful history in many experimental analyses. These powerful methods are
now considered standard tools and are extensively implemented by the ATLAS,
CMS, and LHCb Collaborations [54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65,
66, 67]. In the first observation of the Higgs boson, the ATLAS Collaboration
successfully leveraged the NN techniques [46, 61]. Similarly, the LHCb
Collaboration used the ML tools for the first observations of the exotic
structures referring to charmonium-pentaquark states [19, 61].

The NNs are a branch of the ML models that use learning algorithms inspired by
the human brain to recognize complex relations between various parameters
based on large numbers of examples and make intelligent decisions. The NNs
receive raw input data in the first layer (input layer), process it through at
least one hidden layer, and then present the result in the last layer (output
layer). The simplest form of a NN consists of an input layer, a hidden layer,
and an output layer. Deep neural networks (DNNs) are more advanced, with
several hidden layers. These networks are capable of solving more complex
problems [68, 69]. Application of the NNs to the HEP experiments and
phenomenology has been very successful [41, 70, 71, 72, 73, 74, 75, 76, 77,
78, 79, 80]. The DNNs are becoming indispensable tools in searching and
unraveling dark matter nature [81, 82, 83]. Graph neural network (GNN) models
are a further extension of NNs, able to process data in graph structures. The
GNNs show great promise in particle reconstruction problems [75]. One of the
most powerful types of deep learning models is Generative Adversarial Network
(GAN). A GAN is a type of artificial intelligence framework that consists of
two NNs, a generator and a discriminator which compete with each other via
deep learning algorithms. They are trained simultaneously through adversarial
training. The generator network generates new data samples, while the
discriminator network tries to distinguish between real data samples and fake
ones generated by the generator. Event generation, anomaly detection, data
augmentation, and background subtraction are just a few examples of how the
GANs can be applied in the field of particle physics to empower data analysis,
simulation, and detection capabilities [55, 84, 85, 86, 87, 88].

The DNNs are standard tools for solving classification and regression problems
[70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80]. In the classification tasks, the
output of the algorithm is restricted to specific values, or classes, such as
“signal” or “background” [51, 55]. An innovative DNN method has been
represented for jet identification in the CMS experiment at the LHC [70]. Ref.
[71] trained and developed the DNNs to judge whether the exotic states of the
$X(3872)$, $X(4260)$ and $Z_{c}(3900)$ can be considered as hadronic molecules
of special channels. In addition, the nature of hidden charm pentaquarks with
a NN approach was studied [73]. As previously explained, the NNs have the
potential to perform the regression analysis and predict continuous numerical
values. The regression algorithms can be implemented for reconstruction
techniques in the HEP, e.g. precise calculations of continuous quantities
like, hit positions and track momenta or the jet energies [51, 55, 56, 59, 89,
90]. The ATLAS collaboration has employed a regression DNN on a sample of
simulated $t\bar{t}$ events to reconstruct the top pair system invariant mass
[91]. Also, the NNs can be utilized to compute the mass of the exotic hadrons,
doubly charmed and bottom baryons. For this purpose, the original data was
extended by using artificial data augmentation methods [76]. Interestingly,
Ref. [77] uses basic information of the mesons spectrum to predict the masses
of the baryons, pentaquarks and other exotic hadrons. Inspired by these
applications and based on the search strategies performed in [77], it is a
fortunate time to extend this scheme and design the DNNs to precisely compute
the mass and width of some ordinary and exotic mesons. The quark content of
some famous mesons are still unclear and their identification has remained a
complicated experimental task. We estimate the mass of these challenging
mesons through the DNNs, considering the ordinary and exotic structures for
their quark contents separately. Furthermore, we examine the power of our
designed DNNs in prediction of the mass of some light and exotic mesons.
Moreover, the DNNs are implemented to predict the decay width of some ordinary
and exotic mesons whose decay widths have not been permanently confirmed by
the PDG [36].

In our study, the DNNs are trained based on two effective approaches in
organizing the dataset, according to the quark contents and global quantum
numbers of the meson such as the angular momentum, parity, isospin and charge
conjugation. The obtained results regarding the mass spectra of the normal and
exotic mesons are found to be in good agreement with the experimental results.
For the first time, we are able to predict the decay width of the mesonic
states with a good approximation based on their quark contents and the other
quantum numbers like the angular momentum, parity, etc. and also mass a new
input, through the deep learning algorithms. In this analysis, the encoded
data of the mesons are given to the DNNs. The DNN recognizes the mesons in
terms of their quark contents and the other global quantum numbers. The data
structure is examined in each hidden layer. Each layers’ output is a new
description of the input data for the next layer. Improving this process,
layer by layer, until the final predictions of the mass and width are
presented. At last, the DNN converts the encoded data into the practical
output.

Learning by example and transfering data at each layer are known as the most
powerful aspects of neural network. Despite the broad deployment of the NNs in
strongly correlated systems and pattern recognition tasks, it has not been
clarified how the individual neurons work together to reach the final output.
The DNNs are typically considered as one of the most famous black box
algorithms in extracting knowledge from data. We have designed the DNNs which
are able to predict the mesons’ mass spectra and widths, whereas, most mesons
listed in the PDG are unstable and determination of their mass and total decay
width requires various complex methods in the theory and experiment. This is
the point that makes our research more attractive.

The structure of this paper is in the following form. In section 2, the DNN is
briefly introduced. The details of the data preprocessing is described in
section 3. In section 4, we present and discuss the results of our analyses.
Lastly, the section 5 is reserved for our concluding notes.

##  2 Neural Networks

The ML models aim to automatically learn to recognize complicated patterns and
make intelligent decisions based on data. The NNs, also known as artificial
NNs (ANNs) are famous framework to perform ML goals. In fact, the NN uses a
series of learning algorithms to analyse and function in a manner inspired by
the human brain to find patterns among vast amounts of data.

A typical NN is composed of highly interconnected processing elements called
neurons or nodes, organized in several layers. Indeed, the most fundamental
component of the NN is the neuron, which takes incoming information from the
neurons of the previous layer or directly from the input features. After
processing the information through some mathematical functions, the neuron
transmits the output value to the neurons of the next layer, i.e. the outputs
of the neurons in one layer will be the inputs for the next layer. There exist
three types of layers in the NN. An input layer, at least one hidden layer and
one output layer. The input layer is not responsible for any computation or
transformation. It just passes the data features to the next layers. The
process of passing input data through the NN in a forward direction, from the
input layer through any hidden layers and finally to the output layer is
called feedforward. A feedforward NN is one of most common and successful
learning algorithms where information moves only in one direction. Consider a
feedforward NN with $L$ layers, so we have one input layer, one output layer,
and $L-2$ hidden layer. A neuron transforms the input values $\mathbf{x}$ into
the output $\mathbf{y}$,

$\textbf{x}\rightarrow\textbf{x}^{1}\rightarrow\textbf{x}^{2}...\rightarrow\textbf{x}^{L}\equiv\textbf{y},$
(1)

where $\textbf{x}^{l}$ refers to the vector of x in layer $l$. The weights
between neurons in a NN are represented as a weight matrix W, where each
element $w_{ij}$ represents the weight between neuron $i$ in the previous
layer and neuron $j$ in the current layer. A bias vector can be defined as an
essential component that enables the model to fit data more effectively. The
bias vector is a set of constant values (one for each neuron) appended to the
weighted input. The bias term for each neuron is denoted as a vector b, where
each element $b_{i}$ indicates the bias for neuron $i$ in the current layer.
It should be noted that the number of neurons may be chosen different for each
layer.

The output of a neuron in a layer can be calculated with the input vector
dimension $n$, as follows:

$x^{(l)}_{j}=\sum_{i=1}^{n}(w^{(l)}_{ij}\,x_{i}^{(l-1)}+b^{(l)}_{j}).$ (2)

Then, the output of the neuron is passed through an activation function
$\sigma$ to produce the neuron of the next layer:

$x^{(l-1)}\rightarrow x^{(l)}:=\sigma(W^{(l)}\,x^{(l-1)}+b^{(l)}).$ (3)

An activation function is a mathematical function that calculates the weighted
sum of inputs and biases. Selection of the activation function depend on the
type of prediction problem and nature of the data. It can be linear or
nonlinear. Common activation functions include sigmoid:
$\sigma(x)=1/(e^{x}+1)$, ReLU: $\sigma(x)=max\\{0,x\\}$, softmax:
$\sigma(x_{i})=e^{x}_{i}/(\sum_{j}e^{x}_{j}+1)$, and hyperbolic tangent:
$\sigma(x)=\tanh(x)$.

Each layer in the network performs mathematical operations on the input data,
transforming it into a more useful representation for the next layer. This
process iterates over hidden layers, and continues until the output layer
produces a final result. During the training, weights and biases are updated.
In order to get the optimal set of weight and bias, we need to calculate the
error between the predicted output and the desired outcome and minimize it.
Thus, an optimization procedure is implemented to modify the weights and
biases iteratively in order to reduce the error. This process is called back-
propagation which is a crucial step in training the NN. A schematic of a
neural network architecture including input, hidden, and output layers is
illustrated in Fig.(1).

Figure 1: Schematic illustration of a feedforward NN architecture.

Training a NN with a great performance to new (unseen) data can be a
challenging problem. A model with too small and limited training data cannot
learn the problem. In contrast, a model with too much capacity of training
dataset can learn it too well and lead to overfiting. Specialized techniques
are needed to avoid them. The goal of a NN is to design a final model to fit
the training data and make reliable predictions on the new data (test data).

Validation is an effective technique in the ML to estimate performance of the
model during learning. It is done by splitting the input dataset into the
training and validating sets and then evaluating the predictive power of the
model (DNN in this case) on the validation sets. In this study, the input data
is randomly separated into two subsets as 90.0% and 10.0% for training and
validation data respectively.

It is obvious that there is no universal rule for the NN architecture to get
the best performance. Obtaining the appropriate number of neurons and the
hidden layers as well as choosing the sufficient activation functions depend
on the problem to be solved.

In this work, to prevent the overfitting, and to train the DNN in a reasonable
time, the number of hidden layers and nodes should not be too large. We found
good performance considering the following points. In fact, number of the
neurons on the hidden layers approximately equal to the average number of the
input and output layers. Also, the number of hidden neurons keep on decreasing
in subsequent layers.

###  2.1 Loss function and training the DNN:

When we train a DNN, we feed the data to the network, make predictions,
compare them with the actual output (the targets) and then calculate what is
known as a loss. A loss function plays a key role to evaluate the performance
of the NN model. It indicates how well the algorithm is learning the patterns
in the training data. The higher the loss the worse our model is performing.
Minimizing the loss function directly causes the model to make more accurate
predictions. The average, or expected loss is then given by

$\displaystyle\mathbb{E}[{\cal L}]=\int\int{\cal L}(\hat{y},y({\bf x}))p({\bf
x},\hat{y})d{\bf x}d\hat{y},$ (4)

where a specific estimate $y({\bf x})$ of the value of $\hat{y}$ for each
input ${\bf x}$, and $p({\bf x},\hat{y})$ is a joint distribution over ${\bf
x}$ and $\hat{y}$. We are not required to know the exact form of the
distribution. If we approximate it with the sum over the samples in the
training dataset:

$\displaystyle\mathbb{E}[{\cal L}]\approx\dfrac{1}{N}\sum_{i=1}^{N}{\cal
L}(\hat{y},y({\bf x})),$ (5)

where $N$ is the sample size.

As mentioned above, one way to determine the network parameters is minimizing
the loss function. Weights at the iteration step $t$ are changed and updated
along the direction of the derivative of the loss function with respect to the
parameters:

$\displaystyle\theta^{(t+1)}_{j}=\theta^{(t)}_{j}-\eta\dfrac{\partial{\cal
L}^{(t)}}{\partial\theta_{j}}\;\;\;\;\;\;\;\;\;\;with\;\;\;\;\;\;\;\;\theta_{j}\in\\{b,W\\},$
(6)

where $\eta$ is the learning rate, and the minus sign implies that our
optimization algorithm is navigating down the gradient. Optimization algorithm
is in charge of reducing the losses. This naive form has some advantages and
disadvantages. For example, this form is easy for computing, implementing, and
understanding but may trap at local minima. Weights and biases are changed and
updated after calculating the gradient over the entire training dataset. Thus,
if the dataset is too large, then it may take a lot of time to converge to the
minima. There are many different optimization algorithms that can be used to
solve this problem. Adam is one of the best and powerfull optimization
algorithms for the DNNs. It’s known for its effectiveness in training the
DNNs, particularly in high dimensional parameter spaces. We have examined
different optimization algorithms to train the DNNs, and found the Adam as the
best optimizer for our specific task.

An epoch can be explained as one complete passing of the training data through
the algorithm. The number of epochs is an important hyperparameter for the
training process. We realized that a range between 500 to 1500 epochs is
appropriate to balance between underfitting and overfitting in various
analyses of this project.

Dropout method is generally used to reduce the overfitting of the NNs.
Besides, batch normalization technique accelerates the training process and in
some cases may improve the model performance. However, these methods were not
efficient for our codes structure.

Generally, choosing the loss function is related to the activation function
used in the output layer of the DNN. In other words, these two main functions
are connected. For our setup, Tanh and Relu activation functions are
considered in hidden layers and the linear activation function $y=x$ for the
output layer. Mean Squared Logarithmic Error (MSLE) and Mean Squared Error
(MSE) are employed as the loss function and the metric respectively. We have
used a Python library, Keras [92] with TensorFlow [93].

##  3 Data preprocessing strategy

Data preparation for the ML techniques including NN, starts with data
collection. It is a crucial step in every ML project, since it directly
affects the model’s performance and accuracy. In this paper, the dataset has
been collected from the PDG [36]. Indeed, information of 376 mesons as well as
exotic mesons has been precisely extracted. It must be emphasized that our
dataset is more completed than those prepared in previous study [77]. At first
approach, according to the Ref. [77], the dataset includes the quark content
of mesons, their isospin ($I$), angular momentum ($J$), and parity ($P$)
quantum numbers. The variable $I$ can be $0,1/2$ and $1$. $J$ varies from 0
through 6 in positive integers. $P$ only takes $-1$ or $1$ values. It should
be noted that, the mesons with the same quark structure and the identical
quantum numbers but different masses may cause ambiguity in the NN
performance. So, as to modify the input data and resolve ambiguities, another
feature is defined as higher state ($h$). Notice that $h$ is not a real
quantum number. This is a new parameter that just distinguishes the particles
with the same properties and various masses. For instance,$f_{0}(980)$ and
$f_{0}(1370)$ have the identical features in the input data and different
masses. Consequently, they are given the $h$ values of $0$ and $1$
respectively to be distinguishable as two separate entities for the DNN
algorithm. The range of $h$ can be changed from $0$ to $10$ depending to
numbers of similar mesons. The value of $h$ for mesons with unique properties
is $0$. Furthermore, $G$-parity, $C$-parity as well as the mass and the decay
width of the particles in units of MeV has been extracted. Finally, we have
the following input vector,

$\displaystyle\vec{v}=(d,\bar{d},u,\bar{u},s,\bar{s},c,\bar{c},b,\bar{b},I,J,P,h,G,C).$
(7)

To determine the valence quarks of the mesons with linear combinations of
pairs of quarks and antiquarks ($q\bar{q}$), we have implemented conventions
recommended in Ref.[77]. For instance, the feature values of valence quarks of
isotriplet mesons like $\pi_{0}$ and $\rho_{0}$ are
$(1,1,\vec{0}_{8},I,J,P,h,G,C)$. The values of $G$ or $C$ will be zero if a
meson is not a $G$-parity or $C$-parity eigenstate. It is noteworthy that, the
mass and the width values of the input mesons are scaled through common
scaling techniques in data science to constrain the range of possible outputs.
Since the mass and width parameters are dimensionful, it is divided by 1 MeV
before scaling. Finally, the preparation of the dataset is completely done.

First and foremost, we are going to compute the mass of some ordinary and
exotic mesons, whose mass values have been confirmed by the PDG [36]. The
results can demonstrate the strength of our designed DNNs in prediction of the
mass. The input dataset containing information of 341 mesons is randomly
splitted into two subsets as 90.0% and 10.0% for the training and validation
data respectively. As mentioned in section 2, the performances during learning
processes are monitored by the validation test. At the first step of
calculation, the input data including thirteen feature columns, (the number of
valence quarks,$I,J,P$) is fed into the DNN. The DNN processes the data from
the input node to the output node and returns the outputs (particles’ mass)
through deep learning algorithms. In the next step, to be more accurate, the
variable $h$ as the fourteenth feature, is added to the data and the deep
learning algorithms are implemented again. Last step is allocated to examine
the influence of adding the $G$ and $C$ parities to the data features. The
numerical results are demonstrated in tables (1) to (16). Our predictions
based on the first approach, for the mass along with a comparison to the
experiment [36] and the other NN estimation [77] are illustrated in tables
(1), (4), and (7). Interestingly, the tables (1) and (2) draw attention to the
performance of the DNN on predicting the mass of conventional mesons vs.
corresponding tetraquarks. Section 4 interprets the results in details.
Accordingly, we aim to predict the width of the normal and exotic mesons whose
widths have not been confirmed by the PDG. For this purpose, the mass of
particles is appended to the feature columns in the dataset and the same
procedure mentioned above, is performed. Our results for the width
predictions, based on the first approach, are given in tables (10) and (12),
and discussed in section 4.

The architecture of the dataset indicates how the information is accessed and
organized. Namely, designing an efficient and appropriate data structure,
significantly improves the DNN algorithm’s performance. In the first approach,
the data structure was categorized according to the way proposed in Ref. [77].
In fact, first ten columns of the dataset belong to the number of valence
quarks of each meson. Although, this procedure of organizing the data, looks
to be straightforward, works well just for mesons with $q\bar{q}$ structure.
There exist troubles in specifying the number of valence quarks of the mesons
with linear combinations of $q\bar{q}$ pairs in the dataset. For instance, the
flavor wave functions of the mesons of $\pi_{0}$ and $\eta$ are,

$\displaystyle\pi_{0}$ $\displaystyle=$
$\displaystyle\dfrac{(u\bar{u}-d\bar{d})}{\sqrt{2}},$ $\displaystyle\eta$
$\displaystyle=$
$\displaystyle\dfrac{(u\bar{u}+d\bar{d}-2s\bar{s})}{\sqrt{6}}.$ (8)

As mentioned before, in the first approach the procedure recommended in the
Ref. [77] were applied. For instance, the isotriplet mesons such as $\pi_{0}$
are encoded as $(1,1,\vec{0}_{8},I,J,P)$, and $\eta$ which is included in the
category of the lighter isosinglet mesons, encoded as
$(\vec{0}_{2},1,1,\vec{0}_{6},I,J,P)$. Although this approach is not exactly
describing the quark content of all kinds of mesons, the DNNs have proven to
be strong enough to predict the mass and width of particles with a fair
approximation. To protect the credibility and reliability of the data as well
as enhancing the accuracy, we propose a novel idea for preprocessing the
dataset. In this procedure, the quark content of mesons in association with
the values of corresponding Clebsch-Gordan coefficients will be considered. So
the first twenty five entries of the dataset are devoted to possible
$q\bar{q}$ states constructing the mesons’ structure, like $d\bar{d}$,
$d\bar{u}$, $c\bar{d}$, and etc. Consequently, every meson with linear
combinations of pairs of quarks and antiquarks, gets the corresponding
Clebsch-Gordan coefficient of $q\bar{q}$ state as the feature value, unlike
the previous approach that the number of valence quarks of a given type were
counted. The rest columns of the dataset is not changed and will be the same
as the first approach. Consequently, the dataset is optimized and becomes a
more effective framework. For example, the input vector of $\eta$ has the
following form,

$\displaystyle\eta=(\dfrac{1}{\sqrt{6}},\vec{0}_{4},\dfrac{1}{\sqrt{6}},\vec{0}_{6},\dfrac{-2}{\sqrt{6}},\vec{0}_{12},0,0,0,-1,1).$
(9)

Now, the modified dataset is ready for analyses. Same as the previous
approach, the DNN evaluates the input data to make predictions on the test
data in three levels. Firstly, the input data contains the Clebsch-Gordan
coefficient of $q\bar{q}$ states of the mesons in association with the $I,J,P$
as the features. In the second level, the higher state of $h$ is appended to
the data features. Thirdly, the $C$ and $G$ variables are considered too. Our
predictions based on the second approach, for the mass compared to the
experiment [36] and the other NN estimation [77] are illustrated in tables
(2), (5), and (8). It should be emphasized that the main network architecture,
including the number of layers, scaling methods, setting the optimizer, loss
function and metrics remains unchanged. Certainly, increasing the nodes of
each layer is inevitable due to the enhancing the features. Besides,
considering the mass as another feature in the dataset, we repeat this
procedure to compute the mesons’ width values. The numerical results are
summarized in tables (11) and (13). All the numerical results are disscused in
section 4 completely.

##  4  Interpretations of the results

In this section, we present and discuss our numerical results determined by
the DNNs. As pointed out above, we adopt two distinct approaches to construct
the dataset. In the both approaches, the data features are being completed in
three steps. So the data is fed into the DNN with three sets of input vectors.
For prediction of the mass, at the first step, the data are classified based
on the mesons’ quark content and their quantum numbers $I$, $J$, and $P$ (base
data). Next, the variable $h$ is appended to the base data (base data
$+~{}h$). At last, the quantum numbers $G$ and $C$ are added to the features
(base data $+~{}h\,G\,C$). Important to underline that, in the first approach
(A1), the number of valence quarks of each meson are entered in the first ten
columns of the dataset. Therefore, the input vector (7) denotes the most
complete version of the data features. In contrast, the second approach (A2)
considers the Clebsch-Gordan coefficients of all possible $q\bar{q}$
structures in the quark content of a typical meson, leading to a
25-dimensional vector in the dataset. Correspondingly, for prediction of the
width, the mass of particles is appended to the features in the dataset and
the same procedure mentioned above, are implemented.

We aim to predict the mass of some well-known, light and exotic mesons, as
well as the width of mesons that have not been experimentally determined. The
numerical results for prediction of the mass and decay width as well as the
corresponding mean errors are demonstrated in tables (1) to (16). The tables
(1) to (9) show the predicted mass and the tables (10) to (13) illustrate the
estimated width results. We also estimate the mass and width of four new
tetraquarks, through the DNNs, summarized in tables (14) to (16).

###  4.1 $q\bar{q}$ versus $q\bar{q}q\bar{q}$

The structure of some famous mesons have been actively discussing for years.
The question is whether the $a_{0}(980)$, $f_{0}(980)$,
$D^{*}_{s0}(2317)^{\pm}$ and $D_{s1}(2460)$ are grouped in the ordinary mesons
or they belong to the exotic states. We intend to perdict the mass of these
challenging mesons using the DNN, based on two assumptions. We first suppose
the $q\bar{q}$ state and then examine the $q\bar{q}q\bar{q}$ state for their
quark contents. Tables (1) and (2) show the numerical results considering both
quark contents, obtained according to A1 and A2 respectively. The DNNs are
supposed to predict the mass considering three sets of dataset (base data,
base data $+~{}h$ and base data $+~{}h\,G\,C$). As previously stated, the data
features become more complete step by step. Entirely, the predicted results
for the mass, assuming two different structures, based on two approaches, are
found to be compareble with the experimental data. Let’s compare the obtained
results according to the quark content, the approach and the dataset features.
We calculated the mean error of the predicted masses with the observed ones
too, shown in table (3). One can observe that the mean error with respect to
the observed results for the $q\bar{q}$ structure is much lower than the
tetraquark $q\bar{q}q\bar{q}$ one, suggesting the $q\bar{q}$ structure for the
corresponding mesons.

In the case of the $a_{0}(980)$ meson, pursuant to A1, the DNN predictions for
the mass, are getting close to the experimental value ($980\pm 20$ MeV) with
completing the data features. More precisely, when the $q\bar{q}$
configuration is considered, the DNN, forecasts a mass of $998\pm{94}$ MeV by
importing the base data $+~{}h\,G\,C$. While the $a_{0}(980)$ with
$q\bar{q}q\bar{q}$ structure is given to the DNN, the mass is relatively well
estimated at $1069\pm{224}$ MeV, in the most complete version of the data.
Similarly, if we notice the results based on A2, the DNN predictions for the
$a_{0}(980)$ mass, are improving step by step with enhacing the data features.
Supposing the $q\bar{q}$ picture, the predicted mass is $989\pm 93$ MeV, and
considering $a_{0}(980)$ as the tetraquark, the mass is obtaind $1152\pm 97$
MeV, in the most complete version of the data features. These observations
highlight critical importance of the model selection in theoretical
predictions and underscore the need for empirical validation to refine these
models further. According to the PDG [36], the mass of the $f_{0}(980)$ is
$990\pm 20$ MeV. Whenever the $f_{0}(980)$ is supposed to be an ordinary
meson, the predicted mass via our DNN is more compatible with the experimental
data, per the A1 and A2. Best prediction for the mass value is obtained
$966\pm 51$ MeV, when the base data $+~{}h$, according to the A1, is fed into
the DNN. This assertion holds true for the mesons $D_{s_{0}}^{*}(2317)^{\pm}$
and $D_{s_{1}}(2460)^{\pm}$. Our DNNs mass predictions exhibit diminished
discrepancies when juxtaposed with experimental measurements, surpassing the
precision of the estimates presented in Ref. [77]. This confluence of accuracy
across multiple mesonic cases not only bolsters the credibility of our DNNs
performance, but also provides compelling evidence for its superiority in
capturing the nuances of mesonic mass spectra.

Furthermore, incorporating the parameter $h$ into the dataset (base data
$+~{}h$) markedly reduces the mean error more substantially than the base data
alone, while the inclusion of $C$ and $G$ quantum numbers does not enhance the
model’s accuracy. The minimal mean error achieved is $1.74\%$, realized
through the A1 within the base data $+h$ dataset. The A2 also serves as a
competent predictor, yields a mean error of $3.49\%$. These results underscore
the efficacy of integrating higher states into the dataset for precision
modelling in particle physics.

Lastly, we find that, our DNN predictions for the mass of these controversial
mesons are more compatible to the experimental ones, when they are classed as
the ordinary $q\bar{q}$ mesons for both approaches. Nevertheless, the actual
quark content of these mesons still has remained ambiguous.

Meson | $I^{G}\,(J^{PC})$ | Exp. Mass (MeV) [36] | quark content | Ref.[77] | base data | base data $+~{}h$ | base data $+~{}h\,G\,C$
---|---|---|---|---|---|---|---
$a_{0}(980)$ | $1^{-}\,(0^{++})$ | $980\pm 20$ | $u\bar{u}$ | $1277\pm 246$ | $1312\pm{54}$ | $1020\pm{69}$ | $998\pm{94}$
$u\bar{s}\bar{u}s$ $(K\overline{K})$ | $2172\pm 466$ | $1340\pm{104}$ | $1100\pm{104}$ | $1069\pm{224}$
$f_{0}(980)$ | $0^{+}\,(0^{++})$ | $990\pm 20$ | $d\bar{d}$ | $921\pm 117$ | $1457\pm{63}$ | $966\pm{51}$ | $883\pm{45}$
$d\bar{s}\bar{d}s$ $(K\overline{K})$ | $1592\pm 401$ | $1723\pm{125}$ | $1155\pm{104}$ | $1086\pm{68}$
$D_{s0}^{*}(2317)^{\pm}$ | $0\,(0^{+})$ | $2317.8\pm 0.5$ | $c\bar{s}$ | $2640\pm 433$ | $2289\pm 183$ | $2322\pm{142}$ | $2343\pm{169}$
$c\bar{u}u\bar{s}$ $(DK)$ | $4326\pm 925$ | $3091\pm{361}$ | $2471\pm{284}$ | $2511\pm{334}$
$D_{s1}(2460)^{\pm}$ | $0\,(1^{+})$ | $2459.5\pm 0.6$ | $c\bar{s}$ | $2547\pm 39$ | $2356\pm{121}$ | $2453\pm{128}$ | $2442\pm{218}$
$c\bar{u}u\bar{s}$ $(D^{*}K)$ | $3431\pm 544$ | $2845\pm{251}$ | $2527\pm{289}$ | $2748\pm{504}$

Table 1: Our DNN predictions for the mass of four well-known conventional
mesons vs. corresponding tetraquark structures (in units of MeV), based on the
A1, in comparison with the experimental [36] and the other NN results in
Ref.[77].

Meson | $I^{G}\,(J^{PC})$ | Exp. Mass (MeV)[36] | quark content | Ref.[77] | base data | base data $+~{}h$ | base data $+~{}h\,G\,C$
---|---|---|---|---|---|---|---
$a_{0}(980)$ | $1^{-}\,(0^{++})$ | $980\pm 20$ | $u\bar{u}$ | $1277\pm 246$ | $1079\pm 91$ | $987\pm 12$ | $989\pm 93$
$u\bar{s}\bar{u}s$ $(K\overline{K})$ | $2172\pm 466$ | $1490\pm 108$ | $1041\pm 200$ | $1152\pm 97$
$f_{0}(980)$ | $0^{+}\,(0^{++})$ | $990\pm 20$ | $d\bar{d}$ | $921\pm 117$ | $1793\pm 276$ | $1051\pm 175$ | $1111\pm 40$
$d\bar{s}\bar{d}s$ $(K\overline{K})$ | $1592\pm 401$ | $1792\pm 250$ | $1251\pm 226$ | $1216\pm 216$
$D_{s0}^{*}(2317)^{\pm}$ | $0\,(0^{+})$ | $2317.8\pm 0.5$ | $c\bar{s}$ | $2640\pm 433$ | $2975\pm 79$ | $2456\pm 107$ | $2308\pm 112$
$c\bar{u}u\bar{s}$ $(DK)$ | $4326\pm 925$ | $3997\pm 314$ | $3912\pm 184$ | $3819\pm 210$
$D_{s1}(2460)^{\pm}$ | $0\,(1^{+})$ | $2459.5\pm 0.6$ | $c\bar{s}$ | $2547\pm 39$ | $2537\pm 84$ | $2432\pm 87$ | $2490\pm 81$
$c\bar{u}u\bar{s}$ $(D^{*}K)$ | $3431\pm 544$ | $3779\pm 250$ | $3466\pm 274$ | $3386\pm 253$

Table 2: Our DNN predictions for the mass of four well-known conventional
mesons vs. corresponding tetraquarks structures (in units of MeV), based on
the A2, in comparison with the experimental [36] and the other NN results in
Ref.[77].

$q\bar{q}$ vs $q\bar{q}q\bar{q}$ |  $q\bar{q}$ A1% |  $q\bar{q}$ A2% |  $q\bar{q}$ Ref.[77] % |  $q\bar{q}q\bar{q}$ A1% |  $q\bar{q}q\bar{q}$ A2% |  $q\bar{q}q\bar{q}$ Ref.[77] %
---|---|---|---|---|---|---
base | 21.63 | 30.68 | 13.68 | 39.95 | 64.78 | 77.15
base$+\,h$ | 1.74 | 3.44 | 28.70 | 35.57
base$+\,h\,G\,C$ | 3.61 | 3.70 | 9.71 | 35.70

Table 3:  The mean errors of the predicted mass of four well-known mesons
based on the A1 and A2, in comparison with Ref.[77].

###  4.2 Light mesons

We present predictions for the mass of some light mesons with $q\bar{q}$
structure per A1 and A2, in tables (4) and (5) respectively, which have not
been predicted in Ref. [77]. The mass of the $f_{0}(500)$ meson has not been
exactly determined and the PDG [36] only gives a range of possible values.
Firstly, we feed the dataset prepared accordance with the A1 into the DNN.
Using only the base data, the DNNs prediction falls outside the experimental
range. However, when we include the effects of the higher state $h$ and the
quantum numbers $C$ and $G$, our prediction agrees with the PDG range. This
shows the importance of taking into account these symmetries in the mass
spectrum of light mesons [94, 95]. Subsequently, when the data features are
modified based on the A2, considering each set of dataset (base data, base
data $+~{}h$ and base data $+~{}h\,G\,C$), the DNNs exhibit remarkable
performance in estimating the mass of the $f_{0}(500)$.

For the $K_{4}(2500)$ meson, predictions based solely on the base data per two
approaches, are close and slightly smaller than the expected value,
considering the range of uncertainties. Once the data features are optimized
based on the A2, considering the base data $+~{}h$ or the base data
$+~{}h\,G\,C$, the predictions coincide fairly well with the observed mass
values within the uncertainty range. In fact, involving the Clebsch-Gordan
coefficients in the data features, yielding more consistent results across the
experimental values.

In 1979, the $K_{2}(1580)$ meson was detected via a partial-wave analysis of
the $K^{-}\pi^{+}\pi^{-}$ system. The mass spectrum of this meson was
empirically approximated to be in the vicinity of $1580$ MeV Ref. [96].
However, this estimation has yet to receive confirmation from the PDG [36].
Our DNNs provide a reasonable estimation for the mass of the $K_{2}(1580)$
meson as well. Employing the modified dataset based on the A2, which includes
the base data $+~{}h\,G\,C$, the DNNs forecast a mass of $1634\pm 52$ MeV for
the $K_{2}(1580)$ meson. This prediction align closely with the experimental
findings within the margin of uncertainty. Moreover, alternative mass
estimations for the $K_{2}(1580)$, particularly those based on the base data
within the A2, are in concordance with theoretical projections. Ref. [97]
notably cites a mass of approximately $1.75$ GeV for this meson.

We calculate the mean errors of the predicted mass of these light mesons with
$q\bar{q}$ structure relative to their experimental values. For the
$f_{0}(500)$, we use mean of the mass range as the input. The results are
shown in table (6). We observe that, the contribution of the Clebsch-Gordan
coefficients in the quark content, gives a better agreement with the
experimental data than the A1. This demonstrates the significance of the
Clebsch-Gordan coefficients in the mass formula of light mesons. The best fit
is obtained when we also consider the higher state number $h$ and the quantum
numbers $C$ and $G$. In this case, the mean error of the masses is reduced to
$4.14\%$.

Meson | $I^{G}\,(J^{PC})$ | Exp. Mass (MeV) [36] | quark content | base data | base data $+\,h$ | base data $+\,h\,G\,C$
---|---|---|---|---|---|---
$f_{0}(500)$ | $0^{+}\,(0^{++})$ | $400-800$ | $d\bar{d}$ | $1748\pm{16}$ | $668\pm{61}$ | $759\pm{134}$
$K_{4}(2500)_{0},\bar{K_{4}}(2500)_{0}$ | $1/2\,(4^{-})$ | $2490\pm{20}$ | $d\bar{s},s\bar{d}$ | $2232\pm{60}$ | $2278\pm{57}$ | $2308\pm{35}$
$K_{4}^{\pm}(2500)$ | $1/2\,(4^{-})$ | $2490\pm{20}$ | $u\bar{s},s\bar{u}$ | $2206\pm{28}$ | $2283\pm{51}$ | $2298\pm{25}$
$K_{2}(1580)_{0},\bar{K_{2}}(1580)_{0}$ | $1/2\,(2^{-})$ | $1580$ | $d\bar{s},s\bar{d}$ | $1787\pm{31}$ | $1645\pm{25}$ | $1646\pm{20}$
$K_{2}^{\pm}(1580)$ | $1/2\,(2^{-})$ | $1580$ | $u\bar{s}$ | $1661\pm{28}$ | $1687\pm{31}$ | $1653\pm{23}$

Table 4: Our DNN predictions for the mass of some light mesons (in units of
MeV), based on the A1, in comparison with the experimental [36] and the other
NN results in Ref.[77].

Meson | $I^{G}\,(J^{PC})$ | Exp. Mass (MeV) [36] | quark content | base data | base data $+\,h$ | base data $+\,h\,G\,C$
---|---|---|---|---|---|---
$f_{0}(500)$ | $0^{+}\,(0^{++})$ | $400-800$ | $d\bar{d}$ | $754\pm 278$ | $448\pm 285$ | $554\pm 128$
$K_{4}(2500)_{0},\bar{K_{4}}(2500)_{0}$ | $1/2\,(4^{-})$ | $2490\pm{20}$ | $d\bar{s},s\bar{d}$ | $2185\pm 125$ | $2445\pm 135$ | $2510\pm 112$
$K_{4}^{\pm}(2500)$ | $1/2\,(4^{-})$ | $2490\pm{20}$ | $u\bar{s},s\bar{u}$ | $2286\pm 114$ | $2459\pm 135$ | $2558\pm 128$
$K_{2}(1580)_{0},\bar{K_{2}}(1580)_{0}$ | $1/2\,(2^{-})$ | $1580$ | $d\bar{s},s\bar{d}$ | $1754\pm 78$ | $1696\pm 66$ | $1676\pm 57$
$K_{2}^{\pm}(1580)$ | $1/2\,(2^{-})$ | $1580$ | $u\bar{s}$ | $1789\pm 83$ | $1662\pm 42$ | $1634\pm 52$

Table 5: Our DNN predictions for the mass of some light mesons (in units of
MeV), based on the A2, in comparison with the experimental [36] and the other
NN results in Ref.[77].

light meson | A1% | A2%
---|---|---
base | 46.27 | 14.07
base$+\,h$ | 7.81 | 8.18
base$+\,h\,G\,C$ | 10.06 | 4.14

Table 6: The mean errors of the predicted masses of the light mesons, based on
the A1 and A2.

###  4.3 Exotic Mesons

The DNN are trained to predict the mass of some exotic mesons with
$q\bar{q}q\bar{q}$ structure, whose masses are confirmed by the PDG [36]. The
DNN predictions regarding the A1 and A2 are presented in tables (7) and (8)
respectively. We compare the DNN output values with the experimental ones and
calculate the mean errors, which are shown in table (9). We find that our deep
learning algorithms outperforms the previous model of Ref. [77] in terms of
accuracy. We also observe that the A2 gives better results than the A1. This
implies the significance of considering the Clebsch-Gordan coefficients in the
quark content of ordinary mesons, which enhance the performance of the model
even for the prediction of mass values of exotic states. The best fit will be
achieved, when we include the effects of the higher state number $h$ and the
quantum numbers $C$ and $G$. In this case, the mean error of the masses is
reduced to $16.01\%$.

Overall, our DNNs exhibit a higher precision in predicting the mass of the
ordinary mesons relative to the exotics, compared to the current experimental
data, as evidenced by the errors detailed in tables (3), (6) and (9). This
outcome is anticipated, given that our training dataset is exclusively
composed of ordinary mesons.

The enhancement of DNN performance through the integration of internal
structures and quantum numbers into the dataset is a testament to the
intricate relationship between detailed physical attributes and machine
learning efficacy. By embedding these fundamental aspects, our model can
achieve a more nuanced understanding and prediction, leading to more accurate
and insightful outcomes. For instance, in the case of the $\psi(4660)$ meson,
which has an experimentally observed mass of $(4630\pm 6)~{}\mathrm{MeV}$, our
initial approach (A1) using only the base data predicts a mass of $(3442\pm
137)~{}\mathrm{MeV}$, deviating from the observed value. However, by
incorporating the higher state $h$ into the dataset based on the A2, the
predicted mass improves to $(4186\pm 671)~{}\mathrm{MeV}$, aligning within the
experimental uncertainty range. Once more, this underscores the importance of
involving the Clebsch-Gordan coefficients in the data features as well as
contribution of higher state $h$ in raising the accuracy of mass predictions
for exotic mesons.

In the analysis of certain exotic mesons such as $R_{c0}(4240)$, importing the
base data $+~{}h\,G\,C$, based on the A1, yields predictions that concur with
the observed mass values within their uncertainty bounds. Conversely, the mass
predictions for $R_{c0}(4240)$, based on the A2, lied lower than the
experimental ones, despite the inclusion of the higher state $h$ and quantum
numbers $C$ and $G$. This discrepancy indeed emphasizes the intricate
challenges inherent in modeling exotic mesons. It raises the intriguing
possibility that alternative structures, such as sexaquarks, may be more
appropriate assignments for these mesons. Consequently, there is a pressing
need to refine the training set with data specific to exotic mesons. Such
targeted data enrichment would enable the models to more accurately
encapsulate and reflect the unique properties of these mesons, thereby
enhancing the fidelity of theoretical predictions within the quantum
chromodynamics framework.

Meson | $I^{G}\,(J^{PC})$ | Exp. Mass (MeV) [36] | quark content | Ref.[77] | base data | base data $+\,h$ | base data $+\,h\,G\,C$
---|---|---|---|---|---|---|---
$\chi_{c1}(3872)$ | $0^{+}\,(1^{++})$ | $3871.65\pm 0.06$ |  $c\bar{u}\bar{c}u$ $(D^{0}\,\bar{D}^{*0})$ | $4815\pm 786$ | $3309\pm{174}$ | $2949\pm{159}$ | $2944\pm{177}$
$\psi(4230)$ | $0^{-}\,(1^{--})$ | $4222.5\pm 2.4$ |  $c\bar{s}\bar{c}s$ $(D_{s}\,\bar{D}_{s})$ | $(5.4\pm 1.1)\times 10^{3}$ | $3200\pm{176}$ | $3358\pm{246}$ | $3303\pm{175}$
$\psi(4360)$ | $0^{-}\,(1^{--})$ | $4374\pm 7$ |  $c\bar{u}\bar{c}u$ $(D_{1}\,\bar{D}^{*})$ | $4940\pm 903$ | $3442\pm{137}$ | $3422\pm{240}$ | $3190\pm{184}$
$\psi(4660)$ | $0^{-}\,(1^{--})$ | $4630\pm 6$ |  $c\bar{u}\bar{c}u$ $(f_{0}(980)\,\psi^{\prime})$ | $3480\pm{253}$ | $3300\pm{79}$
$Z_{c}(3900)^{\pm}$ | $1^{+}\,(1^{+-})$ | $3887.1\pm 2.6$ |  $\bar{c}uc\bar{d}$ $(D\,\bar{D}^{*})$ | $4991\pm 815$ | $3574\pm{113}$ | $3304\pm{164}$ | $3676\pm{183}$
$Z_{c}(4200)^{\pm}$ | $1^{+}\,(1^{+-})$ | $4196^{+35}_{-32}$ | $\bar{c}uc\bar{d}$ | $3494\pm{136}$ | $3981\pm{195}$
$Z_{c}(4430)^{\pm}$ | $1^{+}\,(1^{+-})$ | $4478^{+15}_{-18}$ |  $\bar{c}uc\bar{d}$ $(D_{1}D^{*}\,,\,D_{1}^{\prime}D^{*})$ | $3611\pm{126}$ | $4052\pm{197}$
$Z_{b}(10610)^{\pm}$ | $1^{+}\,(1^{+-})$ | $10607.2\pm 2.0$ |  $b\bar{d}\bar{b}u$ $(B\bar{B}^{*})$ | $(1.47\pm 0.17)\times 10^{4}$ | $8693\pm{293}$ | $8481\pm{380}$ | $8918\pm{447}$
$Z_{b}(10650)^{\pm}$ | $1^{+}\,(1^{+-})$ | $10652.2\pm 1.5$ |  $b\bar{d}\bar{b}u$ $(B^{*}\bar{B}^{*})$ | $8550\pm{271}$ | $9103\pm{159}$
$Z_{cs}(4220)^{+}$ | $1/2\,(1^{+})$ | $4216^{+50}_{-40}$ | $u\bar{c}\bar{s}c$ | $----$ | $3408\pm{234}$ | $3267\pm{136}$ | $3054\pm{182}$
$R_{c0}(4240)$ | $1^{+}\,(0^{--})$ | $4239^{+50}_{-21}$ | $c\bar{u}\bar{c}u$ | $----$ | $2790\pm{271}$ | $3379\pm{471}$ | $3760\pm{469}$

Table 7: Our DNN predictions for the mass of of some exotic mesons (in units
of MeV), based on the A1, compared to the experimental [36] and the other NN
results in Ref.[77].

Meson | $I^{G}\,(J^{PC})$ | Exp. Mass (MeV) [36] | quark content | Ref.[77] | base data | base data $+\,h$ | base data $+\,h\,G\,C$
---|---|---|---|---|---|---|---
$\chi_{c1}(3872)$ | $0^{+}\,(1^{++})$ | $3871.65\pm 0.06$ |  $c\bar{u}\bar{c}u$ $(D^{0}\,\bar{D}^{*0})$ | $4815\pm 786$ | $3004\pm 540$ | $3466\pm 673$ | $3345\pm 629$
$\psi(4230)$ | $0^{-}\,(1^{--})$ | $4222.5\pm 2.4$ |  $c\bar{s}\bar{c}s$ $(D_{s}\,\bar{D}_{s})$ | $(5.4\pm 1.1)\times 10^{3}$ | $3613\pm 294$ | $3313\pm 280$ | $3277\pm 379$
$\psi(4360)$ | $0^{-}\,(1^{--})$ | $4374\pm 7$ |  $c\bar{u}\bar{c}u$ $(D_{1}\,\bar{D}^{*})$ | $4940\pm 903$ | $3740\pm{609}$ | $3831\pm 681$ | $3659\pm{606}$
$\psi(4660)$ | $0^{-}\,(1^{--})$ | $4630\pm 6$ |  $c\bar{u}\bar{c}u$ $(f_{0}(980)\,\psi^{\prime})$ | $4186\pm 671$ | $3767\pm 701$
$Z_{c}(3900)^{\pm}$ | $1^{+}\,(1^{+-})$ | $3887.1\pm 2.6$ |  $\bar{c}uc\bar{d}$ $(D\,\bar{D}^{*})$ | $4991\pm 815$ | $2754\pm 316$ | $2969\pm 108$ | $3455\pm 340$
$Z_{c}(4200)^{\pm}$ | $1^{+}\,(1^{+-})$ | $4196^{+35}_{-32}$ | $\bar{c}uc\bar{d}$ | $3012\pm 228$ | $3587\pm 315$
$Z_{c}(4430)^{\pm}$ | $1^{+}\,(1^{+-})$ | $4478^{+15}_{-18}$ |  $\bar{c}uc\bar{d}$ $(D_{1}D^{*}\,,\,D_{1}^{\prime}D^{*})$ | $3280\pm 367$ | $3666\pm 732$
$Z_{b}(10610)^{\pm}$ | $1^{+}\,(1^{+-})$ | $10607.2\pm 2.0$ |  $b\bar{d}\bar{b}u$ $(B\bar{B}^{*})$ | $(1.47\pm 0.17)\times 10^{4}$ | $9548\pm 554$ | $9634\pm 526$ | $9541\pm 417$
$Z_{b}(10650)^{\pm}$ | $1^{+}\,(1^{+-})$ | $10652.2\pm 1.5$ |  $b\bar{d}\bar{b}u$ $(B^{*}\bar{B}^{*})$ | $9856\pm 500$ | $9494\pm 484$
$Z_{cs}(4220)^{+}$ | $1/2\,(1^{+})$ | $4216^{+50}_{-40}$ | $u\bar{c}\bar{s}c$ | $----$ | $3610\pm 227$ | $3975\pm 198$ | $3927\pm 201$
$R_{c0}(4240)$ | $1^{+}\,(0^{--})$ | $4239^{+50}_{-21}$ | $c\bar{u}\bar{c}u$ | $----$ | $2568\pm 436$ | $2994\pm 554$ | $2816\pm 553$

Table 8: Our DNN predictions for the mass of of some exotic mesons (in units
of MeV), based on the A2, compared to the experimental [36] and the other NN
results in Ref.[77].

Exotic | A1% | A2% | Ref.[77] %
---|---|---|---
base | 19.87 | 22.43 | 23.03
base$+\,h$ | 20.42 | 16.76
base$+\,h\,G\,C$ | 17.36 | 16.01

Table 9: The mean errors of the predicted mass of the exotic mesons based on
the A1 and A2 in comparison with Ref.[77].

###  4.4 Meson width

We have trained the well-built DNNs to estimate the mass of various mesons,
both the ordinary and exotic, based on their quark content and quantum
numbers. Our DNNs have achieved noticeable results in prediction of the mass.
Now, we are going to estimate the decay width of mesons and tetraquarks whose
widths have not been experimentally verified or measured. For this goal, the
mesons’ mass will be appended to the feature columns in the dataset and the
same procedure mentioned above, are performed. Tables (10) to (13) present the
predicted widths of some selected mesons according to the A1 and A2
respectively, along with the available experimental data for comparison.

As mentioned earlier, the quark content of the $a_{0}(980)$, $f_{0}(980)$,
$D^{*}_{s0}(2317)^{\pm}$ and $D_{s1}(2460)$ mesons have turned out to be
mysterious. The DNN is designed to perdict the decay width of these
challenging mesons too. We first suppose the $q\bar{q}$ structure and then
examine the $q\bar{q}q\bar{q}$ structure for their quark contents. The
predicted widths are shown in tables (10) and (11) for the A1 and A2
respectively. The enhanced accuracy in estimating the width of the
$a_{0}(980)$ meson, in both the A1 and A2, when modeled as an ordinary meson
rather than an exotic state, lends credence to the conventional $q\bar{q}$
structure over the more complex tetraquark configuration. The DNN offers that,
the $a_{0}(980)$ meson is more aptly described within the traditional mesonic
framework, highlighting the importance of structure in determining meson
characteristics.

The PDG estimates a range from $10$ to $100$ MeV for the $f_{0}(980)$ width.
The peak width in $\pi\pi$ is reported about $50$ MeV, but it might be much
larger. The predicted width of $f_{0}(980)$ by our DNN, coincides nicely with
the estimated range in the PDG, if we regard it the ordinary meson as well.
Likewise, The width estimations for the mesons $D^{*}_{s0}(2317)$ and
$D_{s1}(2460)$, which align with the conventional $q\bar{q}$ structure, fall
within the experimentally reported upper bounds. In contrast, the predictions
based on exotic structures significantly exceed these bounds. Just find it
interesting that, our DNNs strongly predict these mesons’ decay width values
close to expected ones when the $q\bar{q}$ configuration is considered. This
disparity suggests that the $q\bar{q}$ structure might be more plausible for
these mesons, reinforcing the traditional mesonic framework over exotic
configurations.

The enhanced precision of the second approach in predicting the widths of
mesons, such as $K^{*}_{0}(700)$ and $K^{*}_{0}(1430)$, whose measurements
have been experimentally confirmed, is noteworthy. The predictions align most
closely with the observed data when the model incorporates the higher state
$h$ and the quantum numbers $C$ and $G$. This suggests that a comprehensive
model that includes these factors is crucial for accurate theoretical
estimations of the meson properties.

The mass and the decay width spectra of some mesons and tetraquarks as well as
corresponding uncertainty regions estimated through the DNNs compared to the
current review of the PDG [36] are illustrated in Fig. (2). Left and right
panels are based on the A1 and A2, respectively. It seems obvious that our
results are found to be in fairly good agreement with the experimental data
reported by PDG [36]. Especially, when the Clebsch-Gordan coefficients
contribute in modifying the dataset (A2), the DNNs predict more accurate
results for the mass and width of mesons. As a consequence, the dataset is
optimized in an effective way in the A2.

Figure 2: The mass and width of some mesons and tetraquarks. Colorful values
with their gray uncertainties belong to our DNN predictions and gray ones
(having the same symbols with colorful ones for corresponding particles ) are
taken from the PDG [36]. Left panel (Right panel) corresponds to the results,
based on the A1 (A2). The uncertainties for our predictions as well as the
experimental data are all shown by the gray color to make the figure more
readable.

Besides, tables (12) and (13) display the DNN predictions for the width of
some mesons, whose widths have not been experimentally known, based on the A1
and A2 respectively. These results may help experimental groups in the course
of their search for the corresponding resonances and measure their width.

At the end, tables (14) and (15) demonstrate the predicted mass and width for
some new tetraquark states compared to the experiment. The congruence of both
approaches with the experimentally confirmed masses and widths for the mesons
$T^{a++}$ and $T^{a0}$ underscores the reliability of the predictive models
within the bounds of uncertainty. Similarly, the consistency of the mass and
width predictions for the $Z_{V}^{+}$ meson with the theoretical calculations
presented in Ref. [103], despite the lack of experimental observation, further
validates the robustness of these approaches in the field of particle physics.
This alignment between theory and experiment, or calculation in the case of
$Z_{V}^{+}$, is crucial for the ongoing refinement and validation of mesonic
models. The second approach again provides a better estimation for the mass
and the width. The mean error determined, $4.35\%$ and $36.36\%$, respectively
for the mass and width in the second approach compared to the first approach
with $8.60\%$ and $40.70\%$ for the errors of mass and width is depicted in
table (16).

Meson | $I\,(J^{PC})$ | Width (MeV) | base data | base data $+\,h$ | base data $+\,h\,G\,C$
---|---|---|---|---|---
$a_{0}(980)$ | $1^{-}\,(0^{++})$ |  $97\pm 1.9\pm 5.7$ [98] | $158\pm 48$ | 106$\pm 34$ | $113\pm 28$
$a_{0}(980)_{exotic}$ | $1^{-}\,(0^{++})$ |  $97\pm 1.9\pm 5.7$ [98] | $371\pm 90$ | $320\pm 40$ | $179\pm 84$
$f_{0}(980)$ | $0^{+}\,(0^{++})$ |  $10-100$ [36] | $114\pm 29$ | $122\pm 30$ | $105\pm 34$
$f_{0}(980)_{exotic}$ | $0^{+}\,(0^{++})$ |  $10-100$[36] | $253\pm 72$ | $288\pm 70$ | $120\pm 58$
$f_{0}(1370)$ | $0^{+}\,(0^{++})$ |  $200-500$ [36] | $144\pm 24$ | $132\pm 36$ | $107\pm 40$
$D^{*}(2007)^{0}$ | $1/2\,(1^{-})$ |  $<2.1$ ($CL=90\%$) [99] | $1.6\pm 0.8$ | $4.4\pm 1.8$ | $4.6\pm 1.2$
$D_{s_{0}}^{*}(2317)^{\pm}$ | $0\,(0^{+})$ |  $<3.8$ ($CL=95\%$) [100] | $4.2\pm 2$ | $3.4\pm 1.5$ | $3.1\pm 1.7$
$D_{s_{0}}^{*}(2317)^{\pm}_{exotic}$ | $0\,(0^{+})$ |  $<3.8$ ($CL=95\%$) [100] | $100\pm 46$ | $23\pm 11$ | $47\pm 23$
$D_{s1}(2460)^{\pm}$ | $0\,(1^{+})$ |  $<3.5$ ($CL=95\%$) [100] | $3.4\pm 2.1$ | $3.4\pm 2.1$ | $3.4\pm 1.6$
$D_{s1}(2460)^{\pm}_{exotic}$ | $0\,(1^{+})$ |  $<3.5$ ($CL=95\%$) [100] | $85\pm 39$ | $25\pm 14$ | $31\pm 21$
$\psi_{2}(3823)$ | $0\,(2^{--})$ |  $<2.9$ ($CL=90\%$) [101] | $5.9\pm 2$ | $3.7\pm 2$ | $6\pm 4$
$\eta_{b}(2s)$ | $0^{+}\,(0^{-+})$ |  $<24(CL=90\%)$ [102] | $40\pm 18$ | $38\pm 21$ | $54\pm 24$
$K_{0}^{*}(700)$ | $1/2\,(0^{+})$ |  $468\pm 30$ [36] | $177\pm 52$ | $363\pm 120$ | $328\pm 140$
$K_{0}^{*}(1430)$ | $1/2\,(0^{+})$ |  $270\pm 80$ [36] | $191\pm 40$ | $247\pm 48$ | $254\pm 84$

Table 10: Our DNN predictions for the width of some mesons (in units of MeV),
based on the A1, compared to the experimental results.

Meson | $I\,(J^{PC})$ | width (MeV) | base data | base data $+\,h$ | base data $+\,h\,G\,C$
---|---|---|---|---|---
$a_{0}(980)$ | $1^{-}\,(0^{++})$ |  $97\pm 1.9\pm 5.7$ [98] | $112\pm 68$ | $99\pm 44$ | $101\pm 26$
$a_{0}(980)_{exotic}$ | $1^{-}\,(0^{++})$ |  $97\pm 1.9\pm 5.7$ [98] | $389\pm 115$ | $552\pm 300$ | $234\pm 97$
$f_{0}(980)$ | $0^{+}\,(0^{++})$ |  $10-100$[36] | $115\pm 37$ | $124\pm 42$ | $72\pm 44$
$f_{0}(980)_{exotic}$ | $0^{+}\,(0^{++})$ |  $10-100$[36] | $303\pm 83$ | $545\pm 285$ | $315\pm 185$
$f_{0}(1370)$ | $0^{+}\,(0^{++})$ |  $200-500$ [36] | $116\pm 16$ | $119\pm 20$ | $95\pm 49$
$D^{*}(2007)^{0}$ | $1/2\,(1^{-})$ |  $<2.1$ ($CL=90\%$) [99] | $4.8\pm 1.4$ | $4.9\pm 1.3$ | $4.3\pm 1.7$
$D_{s_{0}}^{*}(2317)^{\pm}$ | $0\,(0^{+})$ |  $<3.8$ ($CL=95\%$) [100] | $0.6\pm 0.4$ | $1.75\pm 1$ | $1.9\pm 1.1$
$D_{s_{0}}^{*}(2317)^{\pm}_{exotic}$ | $0\,(0^{+})$ |  $<3.8$ ($CL=95\%$) [100] | $185\pm 80$ | $99\pm 52$ | $140\pm 73$
$D_{s1}(2460)^{\pm}$ | $0\,(1^{+})$ |  $<3.5$ ($CL=95\%$) [100] | $0.94\pm 0.3$ | $3.7\pm 1.9$ | $6.1\pm 4.6$
$D_{s1}(2460)^{\pm}_{exotic}$ | $0\,(1^{+})$ |  $<3.5$ ($CL=95\%$) [100] | $83\pm 34$ | $37\pm 19$ | $43\pm 20$
$\psi_{2}(3823)$ | $0\,(2^{--})$ |  $<2.9$ ($CL=90\%$) [101] | $2.4\pm 1.5$ | $0.8\pm 0.5$ | $0.6\pm 0.1$
$\eta_{b}(2s)$ | $0^{+}\,(0^{-+})$ |  $<24(CL=90\%)$ [102] | $10\pm 0.9$ | $7.8\pm 2.7$ | $11\pm 3$
$K_{0}^{*}(700)$ | $1/2\,(0^{+})$ |  $468\pm 30$ [36] | $318\pm 38$ | $404\pm 44$ | $446\pm 34$
$K_{0}^{*}(1430)$ | $1/2\,(0^{+})$ |  $270\pm 80$ [36] | $239\pm 50$ | $242\pm 31$ | $270\pm 40$

Table 11:  Our DNN predictions for the width of some mesons (in units of MeV),
based on the A2, compared to the experimental results.

Meson | $I\,(J^{PC})$ | base data | base data $+\,h$ | base data $+\,h\,G\,C$
---|---|---|---|---
$B^{*}$ | $1/2\,(1^{-})$ | $(2.3\pm 1.5)\times 10^{-2}$ | $5.8\pm 4.4$ | $0.84\pm 0.56$
$B^{*}_{s_{0}}$ | $0\,(1^{-})$ | $1.39\pm\times 10^{-2}$ | $2.8\pm 1$ | $0.63\pm 0.2$
$B_{c}(2s)^{\pm}$ | $0\,(0^{-})$ | $0.02\pm 0.014$ | $2.3\pm 0.8$ | $2.4\pm 1$
$\chi_{b_{0}}(1p)$ | $0^{+}\,(0^{++})$ | $49\pm 22$ | $40\pm 19$ | $45\pm 20$
$\chi_{b_{0}}(2p)$ | $0^{+}\,(0^{++})$ | $61\pm 34$ | $30\pm 23$ | $50\pm 28$
$\chi_{b_{1}}(1p)$ | $0^{+}\,(1^{++})$ | $39\pm 19$ | $19\pm 10$ | $16\pm 9$
$\chi_{b_{1}}(2p)$ | $0^{+}\,(1^{++})$ | $47\pm 25$ | $15\pm 6$ | $20\pm 10$
$\chi_{b_{1}}(3p)$ | $0^{+}\,(1^{++})$ | $51\pm 23$ | $16\pm 6$ | $29\pm 19$
$\chi_{b_{2}}(1p)$ | $0^{+}\,(2^{++})$ | $32\pm 19$ | $21\pm 11$ | $21\pm 7$
$\chi_{b_{2}}(2p)$ | $0^{+}\,(2^{++})$ | $48\pm 20$ | $16\pm 12$ | $23\pm 11$
$\chi_{b_{2}}(3p)$ | $0^{+}\,(2^{++})$ | $42\pm 22$ | $17\pm 6$ | $39\pm 19$
$h_{b}(1p)$ | $0^{-}\,(1^{+-})$ | $44\pm 12$ | $19\pm 10$ | $33\pm 16$
$h_{b}(2p)$ | $0^{-}\,(1^{+-})$ | $52\pm 15$ | $15\pm 6$ | $34\pm 12$
$K_{0},\bar{K}_{0}$ | $1/2\,(0^{-})$ | $(2.5\pm 1.2)\times 10^{-5}$ | $(2.24\pm 1.7)\times 10^{-6}$ | $(3.34\pm 1.8)\times 10^{-6}$
$\Upsilon_{2}(1D)$ | $0^{-}\,(2^{--})$ | $41\pm 8$ | $34\pm 14$ | $47\pm 21$

Table 12: Our DNN predictions for the width of some mesons (in units of MeV),
whose widths have not been experimentally known, based on the A1.

Meson | $I\,(J^{PC})$ | base data | base data $+\,h$ | base data $+\,h\,G\,C$
---|---|---|---|---
$B^{*}$ | $1/2\,(1^{-})$ | $1.7\pm 0.9$ | $2.4\pm 0.8$ | $2.0\pm 0.5$
$B^{*}_{s_{0}}$ | $0\,(1^{-})$ | $0.14\pm 0.06$ | $0.058\pm 0.025$ | $0.18\pm 0.05$
$B_{c}(2s)^{\pm}$ | $0\,(0^{-})$ | $1.79\pm 0.6$ | $1.8\pm 0.5$ | $2.5\pm 0.9$
$\chi_{b_{0}}(1p)$ | $0^{+}\,(0^{++})$ | $35\pm 12$ | $39\pm 10$ | $23\pm 9$
$\chi_{b_{0}}(2p)$ | $0^{+}\,(0^{++})$ | $37\pm 11$ | $37\pm 8$ | $21\pm 11$
$\chi_{b_{1}}(1p)$ | $0^{+}\,(1^{++})$ | $57\pm 20$ | $4724\pm$ | $57\pm 38$
$\chi_{b_{1}}(2p)$ | $0^{+}\,(1^{++})$ | $52\pm 17$ | $48\pm 27$ | $65\pm 34$
$\chi_{b_{1}}(3p)$ | $0^{+}\,(1^{++})$ | $49\pm 19$ | $42\pm 20$ | $48\pm 17$
$\chi_{b_{2}}(1p)$ | $0^{+}\,(2^{++})$ | $81\pm 37$ | $49\pm 18$ | $43\pm 17$
$\chi_{b_{2}}(2p)$ | $0^{+}\,(2^{++})$ | $79\pm 34$ | $43\pm 22$ | $41\pm 23$
$\chi_{b_{2}}(3p)$ | $0^{+}\,(2^{++})$ | $75\pm 31$ | $75\pm 24$ | $40\pm 30$
$h_{b}(1p)$ | $0^{-}\,(1^{+-})$ | $57\pm 19$ | $47\pm 12$ | $33\pm 15$
$h_{b}(2p)$ | $0^{-}\,(1^{+-})$ | $52\pm 17$ | $43\pm 14$ | $20\pm 10$
$K_{0},\bar{K}_{0}$ | $1/2\,(0^{-})$ | $1.4\pm 0.9$ | $0.7\pm 0.3$ | $0.6\pm 0.4$
$\Upsilon_{2}(1D)$ | $0^{-}\,(2^{--})$ | $92\pm 17$ | $83\pm 16$ | $63\pm 30$

Table 13: Our DNNs predictions for the width of some mesons (in units of MeV),
whose widths have not been experimentally known, based on the A2.

Meson | $I^{G}\,(J^{PC})$ | quark content | Exp. mass (MeV) | Exp. width (MeV) | mass | width
---|---|---|---|---|---|---
$Z_{V}^{++}$ | $1\,(1^{-})$ | $c\bar{d}u\bar{s}$ |  $3515\pm 125$ [103]* |  $156^{+56}_{-30}$ [103]* | $2925\pm 269$ | $190\pm 90$
$T^{a++}$ | $1\,(0^{+})$ | $c\bar{d}u\bar{s}\,(D^{*+}K^{*+})$ |  $2921\pm 17\pm 20$[22] |  $137\pm 32\pm 17$[22] | $2788\pm 390$ | $176\pm 81$
$T^{a0}$ | $1\,(0^{+})$ | $c\bar{u}d\bar{s}\,(K^{0}D^{0})$ |  $2892\pm 14\pm 15$ [22] |  $119\pm 26\pm 13$[22] | $2526\pm 419$ | $182\pm 82$
$T^{a+}$ | $1\,(0^{+})$ | $c\bar{d}d\bar{s}\,(K^{0}D^{+})$ | $----$ | $----$ | $2446\pm 2601$ | $186\pm 79$

Table 14: Our DNN predictions for the mass and width of four new tetraquarks
based on A1. The asterisk (*) signifies that the data is derived from
theoretical calculations in the absence of experimental results.

Meson | $I^{G}\,(J^{PC})$ | quark content | Exp. mass (MeV) | Exp. width (MeV) | mass | width
---|---|---|---|---|---|---
$Z_{V}^{++}$ | $1\,(1^{-})$ | $c\bar{d}u\bar{s}$ |  $3515\pm 125$ [103]* |  $156^{+56}_{-30}$ [103]* | $3195\pm 280$ | $187\pm 89$
$T^{a++}$ | $1\,(0^{+})$ | $c\bar{d}u\bar{s}\,(D^{*+}K^{*+})$ |  $2921\pm 17\pm 20$ [22] |  $137\pm 32\pm 17$[22] | $3158\pm 415$ | $171\pm 77$
$T^{a0}$ | $1\,(0^{+})$ | $c\bar{u}d\bar{s}\,(K^{0}D^{0})$ |  $2892\pm 14\pm 15$ [22] |  $119\pm 26\pm 13$[22] | $2875\pm 536$ | $176\pm 78$
$T^{a+}$ | $1\,(0^{+})$ | $c\bar{d}d\bar{s}\,(K^{0}D^{+})$ | $----$ | $----$ | $2704\pm 338$ | $182\pm 77$

Table 15: Our DNN predictions for the mass and width of four new tetraquarks
based on the A2. The asterisk (*) signifies that the data is derived from
theoretical calculations in the absence of experimental results.

New exotic | A1 % | A2 %
---|---|---
Mass | 8.60 | 4.35
Width | 40.35 | 36.36

Table 16: The mean errors of the predicted mass and width of the new
tetraquarks based on the A1 and A2.

##  5 Summary and conclusions

Deep learning algorithms have created a unique opportunity of cooperation
between the ML and the particle physics communities. The application of the
DNN in the hadronic physics is progressing well.

In this study, the DNNs have been trained to predict the mass and width of
some ordinary and exotic mesons based on their quark contents and
corresponding quantum numbers. The quark content of some famous mesons are
still controversial, whether if they are ordinary mesons or exotic states. We
tried to estimate their mass through the DNN, considering the $q\bar{q}$
structure and then the $q\bar{q}q\bar{q}$ one for their quark contents
separately. We found that when the ordinary structure of $q\bar{q}$ is
employed, our DNN predictions for these challenging mesons’ mass are more
consistent with the experimental values. Furthermore, we evaluated the power
of our designed DNNs in computing the mass of some light and exotic mesons.

For the first time, the DNNs are implemented to predict the width of some
ordinary and exotic mesons whose widths have not been permanently confirmed by
the PDG [36]. We precisely extracted the dataset containing the light and
heavy mesons as well as some tetraquarks. The mesons have been categorized
based on their quark content structures and quantum numbers like the angular
momentum, charge conjugation, isospin and parity. It is important to stress
that, we have presented and tested two approaches for organizing the dataset.
In general, the obtained results are in reasonable agreement with the
experimental values verified by the PDG. Especially, when the Clebsch-Gordan
coefficients contribute in modifying the dataset in the second approach, the
DNNs predict more accurate results for the mass and decay width spectra of the
mesons and tetraquarks. As a consequence, the dataset has been optimized in an
effective way in the second approach.

The DNN performance seems to be more stable compared to the common fitting
methods, particularly, for low-statistic data. This presented work has the
potential to be upgraded in order to accurately predict the mass spectra and
decay widths of other tetraquarks as well as hybrid mesons. A next important
step will be to explore prominent features of the baryons, pentaquarks and
possible molecular dibaryons through deep learning techniques. Recently, many
new exotic hadrons have been discovered by the experiments. Comperihensive
studies have been made to specify their properties and structures and it is
still a hot topic and an active area of research. It would be a worthwhile
idea to probe and predict the quantum numbers of exotic hadronic states via
constructing appropriate DNNs.

Ultimately, calculation and prediction of basic properties of particles
including the mass, width and the quantum numbers, using the statistical
learning algorithms, create new path forward toward robust and reliable
methods for discovering the fundamental structure of nature.

## ACKNOWLEDGEMENTS

M. M and S. R are grateful to the organizers of the MITP Summer School on
“Machine Learning in Particle Theory” for their support. K. A. thanks Iran
national science foundation (INSF) for the partial financial support provided
under the elites Grant No. 4025036.

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