1. Introduction
Broadly speaking, physics can be classified into three areas: the theoretical part, which, through mathematical formalism, explains and understands the physical phenomena; the experimental part, which groups disciplines related to the acquisition of data, the methods of data acquisition, and the design and performance of experiments; and finally, the aspect that links these two, known as phenomenology, which deals with (1) validating theoretical models, (2) investigating how to measure their parameters, (3) investigating how to distinguish one model from another, and (4) studying the experimental consequences of these models. This paper addresses a phenomenological approach by providing a solution to the validation of a theoretical model of physics, more specifically, from particle physics, to unveil the mechanisms of fermion mass generation and to reproduce the elements of the
(Cabibbo–Kobayashi–Maskawa unitary matrix), which describes the mixing of different types of quark during weak interactions by encapsulating the probabilities of transitions between these quark types (see, for example, pertinent introductory reviews [
1,
2,
3]), known as the mixing matrix.
The beginning of these models goes back to the first years of the 1970s, shortly after the establishment of the standard model (SM) of particle physics. Since then, different approaches have been developed in the context of theoretical and phenomenological models, which can be broadly classified as follows: Radiative mechanisms, where fermion masses come from contributions of processes generated by new particles added to SM [
4,
5]; Textures, where the mass matrix has zeros in some entries [
6,
7,
8]; Symmetries between families, where a mathematical group symmetry is added to the theory [
9,
10]; and Seesaw mechanisms, an elegant way of generating small masses for neutrinos [
11,
12,
13]. These approaches are phenomenological interrelated; for example, when we add an extra group symmetry to the theory it is common to arrive at texture structure to mass matrices [
14,
15], and in linear and inverse Seesaw models a texture structure is used to obtain the neutrino masses [
16].
The texture formalism was born by considering that certain entries of the mass matrix are zero, such that we can compute analytically the matrix that diagonalizes it and hence the matrix
. In 1977, Harald Fritzsch created this formalism by using 6-zero hermetic textures as a viable model [
17]; in 2005, with experimental data from that year, he found that 4-zero hermetic textures were viable to generate the quark masses and the mixing matrix [
18]. The analytical approach is very complex, so certain considerations are made to simplify the model. However, the results obtained are not accurate, which is why numerical approaches have recently been used to obtain more accurate solutions. Given the precision of the current experimental data, it is worthwhile assessing the feasibility of these texture models with early studies on the 4-zero mass matrix [
19], further exploring their potential to align with experimental data.
There are numerical works on 4-zero texture models; however, the detailed numerical descriptions of the techniques and algorithms used are not described by their authors [
20]. The numerical analysis of texture models requires a
criterion that establishes when the theoretical part of the model agrees with the experimental part [
21]. That is, to validate the model, a function (which we will call
) is built, and permissible values of the free parameters
X of the model must be found, such that this function takes the minimum value possible greater than 0 but less than 1. In other words, it is necessary to optimize said function under a certain threshold. For the particular case of the function
obtained from the texture formalism, the difficulty comes from the cumbersome algebraic manipulation of the expressions involved, which complicate and entangle the application of classic optimization techniques; thus, alternative optimization techniques are required.
To the best of the authors’ knowledge, the works where bio-inspired optimization algorithms have been used within particle physics are as follows: in experimental contexts, where particle swarm optimization algorithms (PSOs), as well as genetic algorithms (GAs), have been implemented within the hyperparameter optimization of machine-learning (ML) models used in the analysis of data obtained in high-energy physics experiments [
22]; in the optimization of the design of particle accelerators, where the differential evolution (DE) algorithm is quite effective [
23,
24]; and in phenomenology, where genetic algorithms have been used to discriminate models from supersymmetric theories [
25].
As can be seen, the incursion of bio-inspired optimization algorithms in particle physics has been very limited, even when the results are favorable. For this reason, further application of these techniques and algorithms in this type of particle physics area, especially in texture formalism, is of great interest.
The DE algorithm is one of the evolutionary algorithms that has stood out the most in recent times due to its simplicity, power, and efficiency [
26,
27]; however, like other evolutionary algorithms, it is likely to suffer premature convergence and stagnation in local minima [
26,
28]. The strategies implemented to solve these problems can be classified as follows [
29]:
Change the mutation strategy. The mutation phase of the DE algorithm is important since it allows the integration of new individuals into the existing population, thus influencing its performance. Algorithms such as CoDE [
30] have introduced new mutation strategies and implemented different mutation strategies, respectively, in order to improve the efficiency of the DE algorithm.
Change the parameter control. The DE algorithm is sensitive to the configuration of its main parameters: the scale factor
F and the crossover probability
[
26,
31]. Self-adaptive control of these parameters has been shown to improve the performance of the DE algorithm significantly. In this sense, the SHADE [
32] and SaDE [
33] algorithms represent two fairly well-known variants.
Incorporation of population mechanisms. The way in which the population is handled in the DE algorithm can improve its performance. Techniques such as genotypic topology [
34], opposition-based initialization [
35], and external population archives, as seen in JADE [
36], have shown positive effects.
Hybridization with other optimization algorithms. One way to improve the performance of the DE algorithm is to take advantage of the operators’ strengths from other algorithms and incorporate them into the structure of the DE algorithm through a hybridization process [
27,
37]. Hybridization with other computational intelligence algorithms, such as artificial neural networks (ANNs), ant colony optimization (ACO), and particle swarm optimization (PSO), has marked a trend within the last decade [
27,
38].
The self-adaptive differential evolution algorithm based on particle swarm optimization (DEPSO) [
39] is a recent variant of the DE algorithm that integrates the PSO mutation strategy within its structure. DEPSO employs a probabilistic selection technique to choose between two mutation strategies: a new mutation strategy with an elite file called DE/e-rand/1, a modification of the DE/rand/1 strategy, and the mutation strategy of the PSO algorithm. This probabilistic selection technique enables DEPSO to improve the balance between exploitation and exploration, resulting in significantly better performance compared to both DE and PSO on various single-objective optimization problems.
This paper proposes a new variant of the DEPSO algorithm called historical elite differential evolution based on particle swarm optimization (HE-DEPSO) to improve optimization performance in solving complex single-objective problems, with a specific focus on addressing the problem of optimizing the criteria for the 4-zero texture model of high energy physics. The proposed variant aims to explore the areas of opportunity encountered to enhance DEPSO, specifically introducing a new mutation strategy named DE/current-to-EHE/1, which utilizes information from the elite individuals of the population and incorporates historical data from the evolutionary process to improve the balance between exploration and exploitation by leveraging information from elite individuals and historical data, particularly during the early stages of the evolutionary process. Additionally, HE-DEPSO employs the self-adaptive parameter control mechanism from the SHADE algorithm to reduce the sensitivity of the algorithm’s parameters. To test the HE-DEPSO algorithm’s performance, it was compared against other optimization algorithms, including DE, PSO, CoDE, SHADE, and DEPSO, using the CEC 2017 single-objective benchmark function set. The HE-DEPSO algorithm outperformed the other algorithms in terms of solution quality. Finally, the validation of the 4-zero texture model was conducted, optimizing the function and, at the same time, comparing the performance of our proposal against DE, PSO, CoDE, SHADE, and DEPSO algorithms for this particular application. The results are encouraging and expand the use of bio-inspired methods in high-energy physics by integrating a metaheuristic approach.
The remainder of the paper is structured as follows:
Section 2 contains the definition of the problem to be solved, including the definition of the
criterion;
Section 3 reviews the versions of the PSO and DEPSO algorithms used here.
Section 4 explains our proposal, the HE-DEPSO algorithm;
Section 5 subjects our algorithm to benchmark tests, while
Section 6 presents the validation problem of the 4-zero texture model. Finally, in
Section 7, the conclusions are presented.
2. Problem Definition
Within the scope of the SM, the masses of the quarks come from hermetic
matrices known as mass matrices (one for u-type quarks and another for d-type quarks) [
40,
41,
42], as the absolute values of their eigenvalues and the matrix
as the product of the matrix that diagonalizes the u-type quarks and the matrix that diagonalizes the d-type quarks; however, due to the mathematical formalism used, the mass matrices remain entirely unknown and, consequently, the masses of the quarks and the mixing matrix
cannot be theoretically predicted. The experiment provides us with the numerical value of these quantities.
The mixing matrix
can be written in a general way as:
and is a unitary matrix containing information about the probability of transition between u-type quarks (left-hand index) and d-type quarks (right-hand index), through the weak interaction [
43]. That is,
quantifies the transition probability between the quark
u and the quark
s through the interaction with the boson
. Over many years and different collaborations, the magnitudes of the elements of the mixing matrix
have been experimentally measured with an accuracy up to
, and it is well known that four quantities (three angles and one phase) are needed to have a parametrization that adequately describes the matrix
[
3,
44]. In this work, we choose the Chau–Keng parametrization [
3], and the three corresponding angles
,
, and
are obtained by choosing the three magnitudes of the elements
,
, and
, and the phase
is obtained from the Jarslog invariant (
J) through the following relations:
where
. Hence, we choose
,
,
, and
J as independent quantities.
Within the SM context, the mixing matrix,
, is defined by [
3]:
where
is the matrix that diagonalizes the mass matrix of the u-type quarks and
is the matrix that diagonalizes the d-type quarks. It is here where the texture formalism is born, which consists of proposing structures with zeros in some entries of the mass matrix in such a way that we can find the matrices that diagonalize it and be able to calculate analytically the mixing matrix
and validate the chosen texture structure.
Without loss of generality, the mass matrices
and
are considered hermitian so that the general mass structure has the form:
where the index
q runs over the labels
. The elements
,
, and
are real, while
,
, and
are complex and are usually written in their polar form
, where
is the magnitude and
is its angular phase (
).
A matrix of the type 4-zero texture [
18] is formed from the above matrix by taking the entries
,
, and
equal to zero. Thus, we arrive at the following matrix structure:
In references [
6,
18], it is shown that this matrix can be diagonalized by a unitary matrix as follows:
where
denotes a diagonal matrix and
denotes each of the three eigenvalues of
. The matrices
and
are given by:
and
Taking
and
, the relations between
and the physical masses of the quarks are:
where
is the u quark mass,
is the c quark mass,
is the t quark mass,
is the d quark mass,
is the s quark mass, and
is the b quark mass, and its experimental value is presented in
Appendix A. In this work, the same 4-zero texture structure is considered for the u-quark mass matrix and the d-type quark mass matrix (a parallel mass structure). The
q-index takes two values,
and
.
From Equation (
9), it is noticed that the elements of the matrix
depend on the free parameters
and
; the first parameter takes values of
and
and tells us that the eigenvalue of the mass matrix is negative, when
the first eigenvalue
is negative and the second eigenvalue
is positive, and when
the first eigenvalue
is positive and the second eigenvalue
is negative. The combinations of signs of the
and
parameters define the different cases of study to be considered (see
Table 1). The second pair of free parameters are
and
, whose values are restricted to the intervals
and
to ensure that the elements of the
and
matrices are real.
From the above, the mixing matrix
, predicted by the 4-zero texture model, is given by:
in an explicit form:
where the phases
and
are defined as:
These are considered to be measured in radians with principal argument
, and the indices
i and
j correspond, respectively, to the indices
and
.
The magnitude of the elements of the mixing matrix is given by:
At this point, it is essential to emphasize that analytical expressions are obtained for the elements of the mixing matrix predicted by the 4-zero texture formalism; with this information, it is possible to construct a more complete theory of the standard model where the origin of the mixing matrix is explained.
It only remains to validate the theoretical model of textures with 4 zeros, that is, to find the range of the free parameters
,
,
, and
that agree with the experimental values of the matrix
, and for this, we define a function
and use a chi-square criterion [
45,
46] established by:
where
and
The super-indexes “
” are given by Equation (
15) and the quantities without super-index are the experimental data with uncertainty
(see Equations (
A1) and (
A2)).
Although, at first sight, the mathematically constructed function
turns out to have a simple structure and the amount of free parameters is small, the difficulty of finding the numerical range of the same ones that fulfill the condition given in Equation (
16) comes from the cumbersome composition of functions that constitute it. In order to have a notion of the topographic relief of this function, different projections of the function
in different planes are shown in
Figure 1.
Figure 1a shows the projection onto the planes
and
(held fixed values for
and
); that is, the dependence of the function
on the variables
and
is shown (right graph) and the corresponding contour lines are illustrated (left graph). Similarly,
Figure 1b corresponds to the projection of the function
onto
and
(setting
and
). Analogously in the graphical displays, we show the behavior of the function
in the planes
and
(
Figure 1c), in the planes
and
(
Figure 1d), in the planes
and
(
Figure 1e), and in the
and
planes (
Figure 1f).
The following color code has been established; regions towards the intense red color mean larger values of
, while regions towards the intense blue color correspond to smaller values of
. We point out that the function to optimize
is defined on (
17) with variables
,
,
,
bounded on the boundaries
,
,
,
, and as can be observed from the graphs (
Figure 1), the topography that it is intended to optimize is rugged. With this, one sees the opportunity to explore the use of alternative optimization techniques such as bio-inspired algorithms.
7. Conclusions
This paper explored the feasibility of the 4-zero texture model using the most recent experimental values. To gather data that can inform decisions on the model’s viability, we employed a chi-square fit to compare the theoretical expressions of the model with the experimental values of well-measured physical observables. We developed a single-objective optimization model by defining a function to identify the allowed regions for the free parameters of the 4-zero texture model that are consistent with the experimental data. To address the challenge of optimizing the function within the chi-square fit, we proposed a new DEPSO algorithm variant, HE-DEPSO.
The proposed algorithm has demonstrated its ability to efficiently optimize different functions, particularly those from the CEC 2017 single-objective benchmark functions set. The convergence properties of HE-DEPSO show a good balance between solution precision and convergence speed in most cases. Regarding optimizing the function, our findings indicate that HE-DEPSO and DEPSO are more than adequate for solving the optimization problem, with HE-DEPSO providing the best solution quality and consistency. At the same time, DEPSO maintains a faster convergence rate. This part of the investigation also highlights the difficulty that some algorithms can face when optimizing the function, such as algorithms like SHADE and CoDE, despite exhibiting good performance on the CEC 2017 test set. We can also conclude that optimization algorithms such as SHADE, CoDE, and DE would be worth considering for optimization problems in high-energy physics. Finally, with the use of the HE-DEPSO algorithm we show that the 4-zero texture model is compatible with current experimental data, and thus we can affirm that this model is physically feasible. Importantly, our proposed approach, with the help of HE-DEPSO, allows a more accurate exploration of the full parameter space of the elements of the 4-zero texture quark mass matrix, an aspect that is often overlooked in analytical approaches. While aligning with the latest experimental data, this approach could enhance the structural features of 4-zero quark mass matrices. It may serve as a valuable starting point for future model building.
Future work will focus on enhancing the convergence speed of the HE-DEPSO algorithm and evaluating its effectiveness across diverse problem domains, exploring its ability to tackle more complex optimization challenges. An extended comparative analysis against other bio-inspired algorithms or other advanced DE variants could also be conducted. Furthermore, this research could be expanded by analyzing additional texture models like 2- and 5-zero texture models and determining their validity based on current experimental data. Additionally, the use of this method could have potential applications in any phenomenological fermion mass model SM extension, for example, nearest-neighbor textures [
59], neutrino mass models [
60], and discrete flavor symmetries models [
14,
15].