Thermal freeze-out parameters and pseudo-entropy from charged hadron spectra in high energy collisions

We collected the transverse momentum (mass) spectra of charged hadrons ($\pi^{-}$, $\pi^{+}$, $K^{-}$, $K^{+}$, $\overline{p}$, and $p$) produced in collisions over a center-of-mass energy range from 2.70 to 200 GeV (per nucleon pair). The modified Tsallis--Pareto-type function (the TP-like function) with average transverse flow velocity is used to describe the contribution of participant or constituent quarks to transverse momentum of considered hadron. The experimental spectra of $\pi^{\mp}$ and $K^{\mp}$ (or $\overline{p}$ and $p$) are fitted by the convolution of two (or three) TP-like functions due to the fact that two (or three) constituent quarks are regarded as two (or three) energy resources in the formation of considered hadron. From the reasonable fits to the spectra, the thermal freeze-out parameters are extracted, and the pseudo-entropy is newly defined and extracted. Some parameters quickly change in the energy range of less than 7.7 GeV, and slowly change in the energy range of greater than 7.7 GeV, indicating the variation of collision mechanism at around 7.7 GeV.

As the basic gauge field theory [1][2][3] which is used for describing strong interactions, Quantum Chromodynamics (QCD) predicts that under the condition of high temperature and high density [4], partons can be released from the confined hadron phase to form a new form of substance that is called Quark-Gluon Plasma (QGP) [5][6][7][8]. The high-temperature and high-density fireball formed in relativistic heavy-ion collisions can take the shape of this kind of new form of strongly interacting substance. In the meantime, the particular system com-posed by this kind of substance is usually described by the QCD phase diagram [9,10]. Nevertheless, once the external conditions are changed, the strongly interacting substances described by QCD will undergo a phase transition [9,10]. On the one hand, if the thermodynamic properties of the system are expressed by the temperature and the chemical potential of baryons [11][12][13], the first-order phase transition will occur in the region where the chemical potential of baryons is higher and the temperature is lower. On the other hand, in the region where both the baryon chemical potential [11][12][13] and temperature are high, the system maintains dynamic balance which passes a smooth transition. In other words, there is a critical point [14][15][16][17][18][19] from hadronic to QGP phases in the deconfinement process, that is the end point of the first-order phase transition of the QCD matter.
One of the most important targets of relativistic heavy-ion collisions [20][21][22][23] is to explore the QCD phase diagram, understand the structure of QGP, and determine the phase boundary [24,25] between different phases. The Beam Energy Scan (BES) project [26][27][28][29] started by the Relativistic Heavy Ion Collider (RHIC) in 2010 is to study the phase diagram [9,10] of strongly interacting nuclear matter. The RHIC-BES program [30][31][32] is performed for mainly three objectives: The first one is to find the onset energy of deconfinement phase transition [9,10]. The second one is to identify the critical point of the QCD phase diagram [9,10,[14][15][16][17]. And the third one is to determine the characteristics of the first-order phase transition. Since the baryon chemical potential of matter is related to the collision energy [11][12][13], researchers expect to find the corresponding critical point of phase transition by the way of changing the collision energy. The RHIC-BES program including its fixed target experiments is able to vary the collision energy over a wide range in low energy region, thereby achieving an extensive range of baryon chemical potential, further expanding the search objective.
Hadrons are particles involved in strong interactions, including two kinds of particles: mesons (bosons) and baryons (fermions). The most common mesons measured in experiments include, but are not limited to, negatively and positively charged π mesons (π − and π + ), negatively and positively charged K mesons (K − and K + ), etc. The most common baryons measured in experiments include, but are not limited to, anti-protons and protons (p and p), etc. In the RHIC-BES program and at the previous and lower energy such as the Alternating Gradient Synchrotron (AGS) energy, π − , π + , K − , K + , p, and p are particularly important. In fact, the abundant transverse momentum (mass) spectra of the mentioned charged hadrons can be used to extract the common thermal parameters such as the freeze-out temperature and transverse flow velocity. Meanwhile, the novel pseudoentropy, which will be defined later in this paper, can be also extracted from the spectra.
In this paper, the transverse momentum spectra of π − , π + , K − , K + , p, and p produced at mid-rapidity (mid-y) in gold-gold (Au-Au) collisions with different centralities at the RHIC and its BES energy [33][34][35] are collected. Meanwhile, the transverse mass spectra of the mentioned charged hadrons produced at mid-y in central Au-Au collisions at the AGS energy [36][37][38] are also col-lected, though the spectra of p are not available due to low energy. These spectra are used to extract the thermal freeze-out parameters and the pseudo-entropy.
The remainder of this paper is structured as follows. The formalism and method are briefly introduced in Section 2. The results and discussion are given in Section 3. In Section 4, we summarize our main observations and conclusions.

II. FORMALISM AND METHOD
Non-extensive thermodynamics is a new method for studying heavy-ion collisions at relativistic energy. In the collisions such as in Au-Au collisions, the Tsallis-Paretotype function [39][40][41][42][43][44] can fit the transverse momentum (p T ) spectra in low and intermediate regions, particularly in the final-state or hadronization process, demonstrating a strong relation among particles. However, in very lowp T region, the fit result is not ideal, and the meaning of Tsallis parameters remains an open question.
In order to better fit the spectra in very low-p T region, we express the information contained in the parameter more intuitively using the modified Tsallis-Pareto-type function (the TP-like function) as follows [45,46]: Here N is the number of particles, C is the normalization constant, m 0 is the rest mass of the considered particle, n is the power index that describes the degree of nonequilibrium, T is the effective temperature of the collision system, a 0 is the correction index, and m T = p 2 T + m 2 0 is the transverse mass of the particle. As an extention, the TP-like function is naturally converged to the Tsallis-Pareto-type function if we set a 0 = 1 in Eq. (1). In order to obtain the thermal or kinetic (or kinematic) freeze-out temperature T 0 and the average transverse flow velocity β t [47,48], one may fit firstly the p T spectra of particles to obtain the effective temperature T . The average p T ( p T ) and average energy (m) of particles in the source rest frame can be obtained by using the Monte Carlo algorithm. Then, one may extract the intercept as T 0 in the linear relation of T versus m 0 , and further obtain the slope as β t in the linear relation of p T versus m.
For purpose of simplifying this complicated solution process, based on the idea of other work [49], we modify m T and p T in Eq. (1). For clarity, we use m ′ T and p ′ T instead of m T and p T in Eq. (1) respectively. The Lorentzlike transformation is m ′ T = γ t (m T − p T β t ) and |p ′ T | = γ t |p T − m T β t |, where γ t = 1/ 1 − β t 2 is the Lorentz-like factor. The absolute value |p T − m T β t | is used due to the fact that p ′ T is positive and p T −m T β t is possibly negative in low-p T region. After the conversion, the new TP-like function f (p T ) certainly obeys the relation One can see that T in Eq. (1) is naturally converted to T 0 in Eq. (2).
Our exploratory research shows that although Eq. (2) is applicable in the fit of p T spectra, it is not flexible in some cases and explicable in depth at quark level due to m 0 being the rest mass of the considered particle. Empirically, Eq. (2) can be regarded as the probability density function obeyed by the transverse momentum p ti of the i-th constituent quark that contributes to p T of particle. Concretely, we have new TP-like function to be where m 0i is the constituent mass (0.31 MeV/c 2 for u and d quarks and 0.5 GeV/c 2 for s quark, as given in general textbook [50]) and m ti = p 2 ti + m 2 0i is the transverse mass of the i-th quark.
For a meson, p T = p t1 + p t2 due to two constituent quarks which contribute independently. The probability density function of meson's p T is the convolution of two TP-like functions. For a baryon, p T = p t1 + p t2 + p t3 due to three constituent quarks which also contribute independently. The probability density function of baryon's p T is the convolution of three TP-like functions. This analysis is at the quark level due to the fact that it is based on the constituent mass of quark. The present method is similar to the analysis at the particle level in terms of the similar convolution.
For mesons, we have the convolution of two TP-like functions to be For baryons, we have the convolution of the first two TPlike functions to be The convolution of three TP-like functions is In some cases, the spectra in experiments are for m T . We need to convert the probability density func- For invariant p T and m T spectra which are used usually in experiments, we have Other forms of experimental spectra can be transformed from Eq. (8) conveniently if dy and the cross section are considered reasonably. According to statistical physics and information theory, the essence of entropy is an expression of the degree of chaos existed inherently in a system [51][52][53][54]. Entropy has a certain relationship with the probability density function in thermodynamics. In order to further study the role of color confinement in relativistic heavy-ion collisions, the quantum entanglement entropy (the entropy of gluons or partons) is proposed to explore the underlying mechanism. As we know, the relationship between the measured particle multiplicity distribution in the deep inelastic scattering process and the entropy of final hadrons is expressed as [51][52][53][54] S hadron = − P (N ) ln P (N ), where P (N ) is the multiplicity distribution of the charged particles measured in a given measurement area. The entropy of hadrons is assumed to be equal to the partonic entropy [51][52][53][54]. The TP-like function and the corresponding convolutions, which are used to describe the p ti and p T distributions respectively, are also the probability density functions. Analogous to the multiplicity distribution of the above formula, for the first time, we define the pseudoentropy as follows: in which the p T distribution f (p T ) replaced the multiplicity distribution P (N ) in Eq. (9). Different from P (N ) in which N is discrete, p T in f (p T ) is continuous. To give a suitable and uniform description, we set the bin width of p T to be 0.1 GeV/c in the considered energy range in this paper. Although the pseudo-entropy is the bin width dependent, we may compare the relative sizes in different cases if the bin width is beforehand fixed.

III. RESULTS AND DISCUSSION
A. Comparison with data Figure 1 shows the p T spectra, (1/2πp T )d 2 N/dp T dy, of π − (a), π + (b), K − (c), K + (d), p (e), and p (f) produced at mid-y (|y| < 0.1) in Au-Au collisions at centerof-mass energy √ s N N = 7.7 GeV (per nucleon pair) with different centrality intervals. The symbols represent the experimental data measured by the STAR Collaboration in the RHIC-BES project [33,34] and re-scaled by different amounts marked in the panels for clarity. The curves in Figures 1(a)-1(d) represent the results fitted by the convolution of two TP-like functions, for the spectra of π ∓ and K ∓ . The curves for the spectra of p (e) and p (f) are the results fitted by the convolution of three TPlike functions. We have used the method of least squares to determine the values of parameters, and the method of statistical simulation to determine the errors of parameters. The values of n, T 0 , a 0 , β t , χ 2 , and degrees of freedom (ndof) are listed in Table 1. It can be seen that the convolution of two (three) TP-like functions fits well the experimental spectra of π ∓ and K ∓ (p and p) produced in 7.7 GeV Au-Au collisions with different centrality intervals in the available p T range.
Similarly, the p T spectra of π − (a), π + (b), K − (c), K + (d), p (e), and p (f) produced at mid-y in Au-Au collisions with different centrality intervals at √ s N N = 11. 5, 14.5, 19.6, 27, and 39 GeV are demonstrated in Figures  2-6, respectively, where the data are cited from the STAR Collaboration [33,34]. The values of parameters, χ 2 , and ndof are listed in Tables 2-6 for different energies. One can see that the convolution of two (three) TP-like functions fits well the experimental spectra of π ∓ and K ∓ (p and p) produced in Au-Au collisions at 11.5, 14.5, 19.6, 27, and 39 GeV with different centrality intervals in the available p T range.
In order to better explore the change trends of parameters, besides the RHIC-BES energies, we have also conducted data collection and research on the p T spectra at other RHIC energies. Figures 7-9 show the p T spectra of π − (a), π + (b), K − (c), K + (d), p (e), and p (f) produced in Au-Au collisions at 62.4, 130, and 200 GeV, respectively. The data with |y| < 0.1 are also cited from the STAR Collaboration [35]. The centrality intervals at 62.4 and 200 GeV are the same as those at RHIC-BES energies. However, the centrality intervals at 130 GeV are somehow different from others. The related parameters are listed in Tables 7-9 for different energies. One can see again that the convolution of two (three) TP-like functions fits well the experimental spectra of π ∓ and K ∓ (p and p) produced in Au-Au collisions at 62.4, 130, and 200 GeV with different centrality intervals in the available p T range.
Except the above comparisons, we have studied the m T spectra of π − (a), π + (b), K − (c), K + (d), and p (e) produced at mid-y in 0-5% Au-Au collisions at √ s N N = 2.70, 3.32, 3.84, 4.30, and 4.88 GeV in Figure  10, where the five energies are only available in panel (d).
The symbols for π ∓ in |y| < 0.05, K ∓ in |y| < 0.25, and p in |y| < 0.05 represent the experimental data measured by the E895 [36], E866 [37], and E895 [38] Collaborations, respectively, where some sets of data are re-scaled by different amounts for clarity. The data for p is not available in the same or similar experiments due to low energy. The related parameters are listed in Table 10. One can see that the convolution of two (three) TP-like functions fits approximately the experimental spectra of π ∓ and K ∓ (p) produced in 0-5% Au-Au collisions at 2.70, 3.32, 3.84, 4.30, and 4.88 GeV.   Figure 1, but showing the results for 11.5 GeV Au-Au collisions. The symbols represent the STAR data [33]. Table 2. Values of n, T0, a0, βt , χ 2 , and ndof corresponding to the curves in Figure 2.             Table 7. Values of n, T0, a0, βt , χ 2 , and ndof corresponding to the curves in Figure 7.  Table 8. Values of n, T0, a0, βt , χ 2 , and ndof corresponding to the curves in Figure 8.  Table 9. Values of n, T0, a0, βt , χ 2 , and ndof corresponding to the curves in Figure 9. We now analyze the tendencies of parameters on collision energy √ s N N and centrality interval (percentage). In Figure 11, the dependence of parameter n on √ s N N and centrality is shown, where panels (a)-(f) are for the results from the spectra of π − , π + , K − , K + , p, and p, respectively. It is not difficult to find out that with the increase of √ s N N , the values of n decrease significantly. Meanwhile, with the decrease of centrality from central to peripheral collisions, the values of n also decrease. As we know, n is used to describe the degree of non-equilibrium of the system. The larger the value of n is, the closer to equilibrium the system is. This work shows that the system at lower energy and in central collisions corresponds to larger n and hence to higher degree of equilibrium, compared with the system at higher energy and in peripheral collisions.
As one of the core concepts in thermodynamics and statistical mechanics, temperature is also the attention of our research. Generally, we can directly extract the temperature information of the system from particle spectra. Panels (a)-(f) in Figure 12 show the dependence of thermal freeze-out temperature T 0 on √ s N N extracted from the spectra of six kinds of charged hadrons in different centrality intervals. It can be clearly seen that in the available central collisions, T 0 increases rapidly in √ s N N = 2.7-7.7 GeV range. When √ s N N is larger than 7.7 GeV, T 0 increases slowly. The higher the temperature is, the higher the excitation degree is. As √ s N N increases, the excitation degree of the system also increases. The change trend of T 0 with √ s N N can well reflect this change phenomenon of the excitation degree. At nine collision energies and for six kinds of particles, T 0 decreases with the decrease of the centrality from central to peripheral collisions.
Similar to Figures 11 and 12, the dependence of the dimensionless correction index a 0 on √ s N N is shown in Figures 13(a)-13(f) for different centrality intervals and charged hadrons. From Figure 13 one can see that with the increase of √ s N N , in the available central collisions, a 0 increases quickly and then decreases significantly, and the boundary energy is 7.7 GeV. The values of a 0 from π ∓ spectra are negative, which is significantly less than those from K ∓ and p(p) spectra. From central to peripheral collisions, a 0 increases significantly for different kinds of particles in the available energy range. Figure 14 demonstrates the relationship between the average transverse flow velocity β t and √ s N N for different particles in panels (a)-(f), where the results for different centrality intervals are shown. With increasing √ s N N , one can see a significantly increasing β t for different hadrons in most cases. At different energies, β t for each kind of particles from central to peripheral collisions decreases gradually. Observing the values of β t , one can clearly find that the order of flow velocity is p(p) > K ∓ > π ∓ .
In order to represent the chaos degree of the final state particle information, we show the dependence of pseudoentropy S ′ hadron on √ s N N and centrality in Figure 15, where panels (a)-(f) are orderly for the results of π − , π + , K − , K + , p, and p. One can see that S ′ hadron of the six kinds of hadrons in different centrality intervals increases with the increase of √ s N N . In particular, for π ∓ and K ∓ and in central collisions, S ′ hadron increases quickly and then slowly with the increase of √ s N N , and the boundary energy is 7.7 GeV. From central to peripheral collisions, S ′ hadron decreases gradually. The chaos degree of particle information at low energy is less than that at high energy. In addition, the chaos degree in central collisions is larger than that in peripheral collisions. Observing the values of S ′ hadron , one can clearly find that the order of chaos degree is p(p) > K ∓ > π ∓ .
In the above discussion, we have used a particular parameterization for the p T distribution and obtained the tendencies of parameters, but many other parameterizations are available and used in the field. Our conclusions on the behaviors of main parameters n [or q = (n+1)/n], T 0 , and β t with increasing the collision energy, event centrality, and particle mass are similar to others if one uses different parameterizations [48], though the absolute sizes may be different. The parameter a 0 is a new one introduced by us recently [45], which is hard to compare with others at present due to the lack of related work, though the relative sizes of a 0 on particle mass are understandable. The parameterization used in this paper is also suitable for smaller system studied in our previous work [45,46], where the dependences of parameters on the collision energy, event centrality (or multiplicity), and particle mass are similar, which is not very sensitive to the system size.    We now discuss this topic further. As we know, high energy collisions contain abundant information which are related to the productions of QGP and particles [55][56][57][58][59][60][61][62]. Of course, after a very short time, QGP finally decays into particles. The final-state products are naturally various kinds of particles. Indeed, understanding the characteristics of particle productions is particularly necessary for researchers to study the evolution of collision system and interactions among various particles [63][64][65][66][67][68][69][70]. By studying the particle spectra, this paper has extracted some quantities.
From the above comparison, one can see that this paper has used the transverse mass which contains the average transverse flow velocity β t through the Lorentz-like transformation, m ′ T → γ t (m T − p T β t ) [49]. Considering the influence of transverse flow, the effective temperature T in Eq. (1) has been changed to the thermal freeze-out temperature T 0 in Eq. (2). In this way, T 0 and β t can be directly extracted [49]. In this work, T 0 and β t are quantities at the quark level, but not the quantities at the particle level. This renders that the values obtained at the two levels are different.
There is a relation between the power index n and the entropy index q, n = 1/(q − 1). It shows indeed that a larger n corresponds to a smaller q that is also closer to 1. As we know, an exact and concrete q that separates an approximate equilibrium and non-equilibrium states is not available in literature. Empirically, we may use q = 1.25, i.e. n > 4, as the boundary between the approximate equilibrium and non-equilibrium states. The present work shows that n > 4 in most cases. This means that the system in terms of quarks or partons discussed here is at the approximate equilibrium or local equilibriums. The concept of temperature can be approximately used.
At lower energy and in central collisions, the system consisted of quarks or partons is at larger degree of equilibrium. This is understood in view of the longer reaction time in lower collision energy and more participant partons in central collisions. These two factors can affect the system to be at more equilibrium due to more cascade collisions happening. At the present energy at the LHC, the system is approximately at the equilibrium. It is possible that at very high energy the system will be at non-equilibrium due to very short reaction time. In most cases, the system in peripheral collisions is approximately at the equilibrium. It is not preclusive that in very peripheral collisions the system is possible at nonequilibrium due to very less cascade collisions in very limited collision volume, where the multiplicity is also low.
There are possible scenarios for single, double, or multiple kinetic freeze-out in literature. The single (two or multiple) kinetic freeze-out means that a set (two or multiple sets) of parameters can fit the spectra. There is not a sold conclusion for a definite scenario. The present work shows that some parameters from different particle spectra are different. This reflects the scenario for multiple kinetic freeze-out in high-energy collisions. According to the present work, we may say that different particles are produced in the system at different times. The decay of the system is not all at once, but successive.
The value of T 0 reflects the excitation degree of the system at the parton level, which is somehow larger than that at the particle level, due to the former happening earlier. The higher the value of T 0 is, the higher the excitation degree of the system is, and the larger the deposited energy is. The value of β t reflects impact and squeeze between the projectile and target at parton level. In high energy collisions, a lot of energy are deposited in the process of parton interactions, though some partons go through the system as spectators.
Some soft excitation and hard scattering processes among partons have happened. The contribution of soft component from resonance in π ∓ spectra is significantly larger than those in K ∓ and p(p) spectra. This renders that the spectra of π ∓ have a significant increase in lowp T region, comparing with the spectra of other particles. This change in low-p T region is mainly described by the value of a 0 in this work. A small a 0 describes a significant increase and a large a 0 describes a decrease. Naturally, this change also affects the parameters n, T 0 , and β t .
Indeed, for π ∓ spectra in low-p T region, the production of resonances cannot be ignored. More contributions from the resonances in π ∓ productions render that the shape of the spectra of the considered different kinds of particles are different, in particular in low-p T region. This also renders the different sets of parameters for different particles. In addition, multiple kinetic freeze-out also results in multiple sets of parameters. The earlier the particles are emitted, the wider the spectrum is, and the higher the temperature is.
At given collision energy, the nuclear environments (the sizes of participant or spectator regions) in central and peripheral Au-Au collisions have some influences on the transverse momentum spectra. We attribute this phenomenon to the hadronic cascade collisions (or multiple scattering) in hot and dense participant zone. The larger number of participants in central collisions means more cascade collisions and, hence, the larger transverse momentum and temperature. In the considered transverse momentum region, the changing laws of transverse momentum spectra for given particles are independent of isospin. Compared with strong interactions, contribution of electromagnetic interactions is very small and negligible.
The purpose of the RHIC-BES program is to search for the critical point in the QCD phase diagram. In the RHIC-BES and the related AGS energy regions, from the energy dependences of related parameters in the available central collisions, we found that T 0 and S ′ hadron increase quickly and then slowly at around 7.7 GeV with the increase of collision energy. This special energy is the knee point of the equation of state (EoS) of the hot and dense matter, which is related to the critical energy, if not equal. Meanwhile, we found that a 0 increases and then decreases at around 7.7 GeV in the available central collisions. Generally, different parameters may reveal different knee points. The most probable knee point from the energy dependences of various parameters can be regarded as the critical point.
In the available central collisions, both the boundary energies of the rapid-slow increase in the energy dependence of T 0 and S ′ hadron and of the increase-decrease change in the energy dependence of a 0 are 7.7 GeV. We argue that this energy is possibly the critical energy of deconfinement phase transition from hadronic matter to QGP. Because of the complexity of the collision process, the conclusion about the critical energy in this work is not solid. A shift of 2-3 GeV is possiable. To make a solid conclusion, one needs a comprehensive analysis of various parameters. At least, we may state that the interaction mechanisms or strengths at the energy below and above about 7.7 GeV are different.

IV. SUMMARY AND CONCLUSIONS
In this paper, the TP-like function has been used to analyze the transverse momentum spectra of the charged particles [π ∓ , K ∓ , and p(p)] produced in Au-Au collisions at the RHIC-BES energies and the related AGS energies. The contribution of each constituent quark to the transverse momentum of charged particle is assumed to satisfy the TP-like function. Each constituent quark is also regarded as an energy resource. Therefore, the transverse momentum spectrum of mesons (baryons) is the convolution of two (three) TP-like functions. According to the fitting results, we have obtained four parameters: the power index n, the thermal freeze-out temperature T 0 , the correction index a 0 , and the average transverse flow velocity β t . The dependences of these parameters on collision energy and centrality have been also investigated.
It is obvious that with the increase of collision energy, n decreases, which indicates that the collision system is farther away from the equilibrium, though it is still approximately at the equilibrium, at higher energy. In other words, at lower energy, the longer the collision evolves, the closer to the equilibrium the system is. More importantly, with the increase of collision energy, T 0 extracted from meson spectra in central Au-Au collisions increases quickly in the range of less than 7.7 GeV and slowly in the range of greater than 7.7 GeV. Meanwhile, β t increases with the increase of collision energy. From central to peripheral collisions, all of n, T 0 , and β t decrease, and a 0 increases. Even in peripheral collisions, the system is approximately at the equilibrium.
We have introduced the concept and definition of a new quantity, the pseudo-entropy. The pseudo-entropy is carried out a more in-depth study on the transverse momentum spectra of charged particles. With the increase of collision energy, the pseudo-entropy extracted from the meson spectra in central Au-Au collisions increases quickly in the range of less than 7.7 GeV and slowly in the range of greater than 7.7 GeV. This behavior is similar to that for T 0 . We argue that 7.7 GeV is a special energy at which the phase transition of deconfinement from hadronic matter to QGP can possibly happen. At least, the interaction mechanism in the system or the production mechanism of charged particles at the energy below and above about 7.7 GeV are different.

Data Availability
The data used to support the findings of this study are included within the article and are cited at relevant places within the text as references.

Ethical Approval
The authors declare that they are in compliance with ethical standards regarding the content of this paper.

Disclosure
The funding agencies have no role in the design of the study; in the collection, analysis, or interpretation of the data; in the writing of the manuscript; or in the decision to publish the results.

Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.