Tsallis statistics and thermofractals: applications to high energy and hadron physics

Tsallis statistics and thermofractals: applications to high energy and hadron physics Eugenio Meg´ıas1, Evandro Andrade II2, Airton Deppman3, Arnaldo Gammal3, D´ebora P. Menezes4, Tiago Nunes da Silva4 and Varese S. Tim´oteo5 1Departamento de F´ısica At´omica, Molecular y Nuclear and Instituto Carlos I de F´ısica Te´orica y Computacional, Universidad de Granada, Avenida de Fuente Nueva s/n, 18071 Granada, Spain 2Departamento de Ciˆencias Exatas e Tecnol´ogicas, Universidade Estadual de Santa Cruz, Ilh´eus, CEP 45662-900 Bahia, Brazil 3Instituto de F´ısica, Universidade de S˜ao Paulo, Rua do Mat˜ao 1371-Butant˜a, S˜ao Paulo-SP, CEP 05580-090, Brazil 4Departamento de F´ısica, CFM-Universidade Federal de Santa Catarina, Florian´opolis, SC-CP. 476-CEP 88.040-900, Brazil 5Grupo de ´ Optica e Modelagem Num´erica, Faculdade de Tecnologia GOMNI/FT - Universidade Estadual de Campinas - UNICAMP 13484-332, Limeira, SP, Brazil E-mail: [email protected], [email protected], [email protected], [email protected], [email protected], [email protected], [email protected] Abstract. We study the applications of non-extensive Tsallis statistics to high energy and hadron physics. These applications include studies of pp collisions, equation of state of QCD, as well as Bose-Einstein condensation. We also analyze the connections of Tsallis statistics with thermofractals, and address some of the conceptual aspects of the fractal approach, which are expressed in terms of the renormalization group equation and the self-energy corrections to the parton mass. We associate these wellknown concepts with the origins of the fractal structure in the quantum field theory. Keywords: Tsallis statistics, pp collisions, hadron physics, quark-gluon plasma, thermofractals, Bose-Einstein condensation 1. Introduction Important advances in the study of the phenomenology of Quantum Chromodynamics (QCD) in the hot and dense regimes, in particular in the quark-gluon plasma, have been developed in recent years. These studies have motivated the introduction of several approaches, including lattices studies [1], chiral quark models [2, 3], hadron resonance gas (HRG) models [4, 5, 6, 7], and holographic models [8, 9], among others. Motivated arXiv:2201.08771v1 [hep-ph] 21 Jan 2022 Tsallis statistics and thermofractals: applications to high energy and hadron physics 2 by the large amount of information that emerged from high energy physics (HEP) and heavy-ion physics experiments, the consequences of those advances are far-reaching. Let us summarize the three fundamental theories that will be used for the developments that will be discussed below: the Yang-Mills field (YMF) theory, the fractal geometry, and the non-extensive statistics proposed by Constantino Tsallis. YMF theory is a prototype theory that allows describing most of the physical phenomena [10]. It was incorporated in the electro-weak theory in the 1960s, and in QCD in the 1970s. One of the fundamental properties of physics laws is the renormalization group (RG) invariance, an aspect that plays an important role in the renormalization properties of YMF theories after the ultraviolet divergences are subtracted [11, 12]. Fractals are complex systems presenting a fine structure with an undetermined number of components that are also fractals similar to the original system but at a different scale [13]. This property is known as self-similarity. Fractal geometry has been used to describe many natural shapes that can be observed in everyday life. A direct consequence of the self-similarity is the power-law behavior of distributions observed for fractals. Tsallis statistics was introduced as a generalization of Boltzmann-Gibbs (BG) statistics by considering a non-additive form of the entropy [14]. Contrary to the exponential distribution of BG statistics, Tsallis distribution has a power-law behavior which has led in the last few years to a wide range of applications apart from HEP, see e.g. [15, 16]. However, the full understanding of this statistics has not been accomplished yet, one of the open questions being the physical origin of the power-law behavior in physical systems. The goal of this manuscript is to provide an overview of the main applications of Tsallis statistics to HEP and hadron physics, as well as to study the link between RG invariance of YMF theory, fractals and Tsallis statistics. The manuscript is organized as follows. In Sec. 2 we will introduce Tsallis statistics and explore the main properties of fractals. We will also investigate the connection between thermofractals and Tsallis statistics, and address the thermofractal description of YMF theory. We will study in Sec. 3 some of the recent applications of Tsallis statistics to HEP, including pp collisions, QCD thermodynamics, and Bose-Einstein condensation (BEC). Finally, in Sec. 4 we present our conclusions. 2. Tsallis statistics, thermofractals and Yang-Mills fields In this section, we will provide an introduction to Tsallis statistics and the formalism of thermofractals, and explore the link between these two descriptions. 2.1. Tsallis statistics Tsallis statistics is a generalization of BG statistics, with entropy given by [14] Sq≡ −kBX i p(xi)qln(−) qp(xi),(1) Tsallis statistics and thermofractals: applications to high energy and hadron physics 3 where p(x) is the probability of xto be observed, kBis the Boltzmann constant, and q is the entropic index that quantifies how Tsallis entropy departs from the extensive BG statistics. Tsallis statistics is defined in terms of the q-exponential and q-logarithmic functions, given by e(±) q(x) = [1 ±(q−1)x]±1 q−1,ln(±) q(x) = ±x±(q−1) −1 q−1,(2) respectively. A consequence of Eq. (1) is that the entropy of the system is non-additive, i.e. for two independent systems Aand B[14] SA+B=SA+SB+k−1 B(q−1)SASB.(3) Notice that e(±) q(x)−→ q→1exand ln(±) q(x)−→ q→1ln(x), so that as q→1 the BG statistics is recovered and the entropic form becomes additive. 2.2. Fractals and self-similarity We will introduce now the main concepts related to fractals. Fractals are defined by their self-similar properties at different scales. A scaling transformation changes the size of a system by a scaling factor, λ. On the other hand, a self-similar system is a system which is similar to a part of itself. A typical example of a fractal is the Sierpi´nski triangle. If length is reduced by the scaling factor as `(λ) = `0/λ, then a system of dimension Dcan be filled by N(λ) = N0λDsmaller self-similar systems. Then, one can define the fractal dimension as D≡lim λ→∞ ln N(λ) ln λ.(4) This definition is valid for both fractal and non-fractal systems. Another example of a fractal is the length of coastlines, as it depends on the resolution. If L0=N0`0is the measured length at some initial resolution, `0, then the measured length at a better resolution `(λ) = `0/λ turns out to be L(λ) = N(λ)`(λ) = L0·λD−1.(5) One can see that an increase of Lwith λis indicative of a fractal dimension D > 1. For the coastline of Great Britain, it is D≃1.25, but generically different shapes induce a fractal spectrum of dimensions. 2.3. Thermofractals The emergence of the non-extensive behavior in physical systems has been attributed in the literature to different causes: i) long-range interactions and correlations [17]; ii) temperature fluctuations; and iii) finite size of the system. We will explore below a natural derivation of non-extensive statistics in terms of thermofractals. These are systems in thermodynamical equilibrium presenting the following properties [18]: •The total energy of the system is given by U=F+E, where Fis the kinetic energy, and Eis the internal energy of Nconstituent subsystems, so that E=PN i=1 ε(1) i. Tsallis statistics and thermofractals: applications to high energy and hadron physics 4 •The constituent subsystems are thermofractals, which means that the energy distribution PTF(E) is self-similar or self-affine, i.e. at level nof the hierarchy of subsystems, PTF(n)(ε) is equal to the distribution in any other level PTF(n)(ε)∝ PTF(n+n0)(ε). •At any level nof the fractal structure, the phase space is so narrow that one can consider PTF(En)dEn=ρ dEn. This means that the internal energy fluctuations are small enough to be disregarded, and then the internal energy can be considered to be equal to the component mass m. Using these properties, it is possible to show that thermofractals result in energy distributions of the kind [18, 19, 20] P(±) TF(n)(ε) = A(n)·e(±) q−ε kBτ,(6) so that the energy distribution of thermofractals obeys Tsallis statistics. The positive(negative) version of the q-exponential function corresponds to type-I (typeII) thermofractals. The main difference between the two kinds of thermofractals is the character of the distribution: type-I requires a cut-off because of the negative sign in the argument, while type-II presents a distribution without a cut-off. 2.4. Fractal structures in Yang-Mills fields We have seen in Sec. 2.3 that thermofractals obey Tsallis statistics. On the other hand, as we will see in Sec. 3, the phenomenology of QCD can be successfully described by this statistics. Then, a natural question arises: Is it possible a thermofractal description of YMF theories? We will address below this question. Partons are considered fundamental particles without internal structure, therefore, in principle, they cannot be fractals. This statement holds until the QCD vacuum is not considered, though. We know that vacuum polarization is an essential part of the interaction not only in QCD but also in Quantum Electrodynamics and YMF theories in general [21]. The vacuum structure is an important component of partons interactions and the parton self-energy [22]. We will describe below how the fractal structure appears in parton dynamics. The YMF theory was shown to be renormalizable in Ref. [23], which means that the regularized vertex functions are related to the renormalized ones, to which the renormalized parameters, ¯mand ¯g, are associated by [11, 12] Γ(p, m, g) = λ−DΓ(p, ¯m, ¯g),(7) where λis the scale transformation parameter, i.e. pµ→p0µ=λpµ. This property is described by the RG equation, also known as Callan-Symanzik (CS) equation [24, 25] M∂ ∂M +β¯g ∂ ∂¯g+ ¯γΓ = 0 ,(8) where Mis the scale parameter, the beta function is defined as β¯g=M∂¯g ∂M , and ¯γis the anomalous dimension. RG invariance in YMF theory means that, after proper scaling, Tsallis statistics and thermofractals: applications to high energy and hadron physics 5 (a) (b) Figure 1. Pictorial representation of the effective parton and their interactions. (a) The vacuum polarization represented as the internal structure of the effective parton. (b) In the effective parton interaction the vacuum structure participates in a complex way. the loop in a higher-order graph in perturbative expansion is identical to a loop in lower orders. This is a direct consequence of the CS equation, and it is indicative of the selfsimilar properties of gauge fields. These properties have important consequences for the dynamics of partons, as we will see below. The partonic dynamics can be described by a Dyson-Schwinger expansion [11], leading to an effective parton which includes the self-energy interaction in the propagation of the elementary parton. Then, RG invariance is responsible for a complex structure of the effective parton which is depicted in Fig. 1 (a). In this figure, the vacuum polarization is represented by the + and −signs surrounding the elementary parton, represented by the central circle. In this sense, we say that the effective parton has an internal structure. The complexity of this structure can be evaluated by the number of Feynman graphs necessary to describe the self-interaction contributions even in low-orders of calculation. The proper-vertex interaction is still more complex, as can be observed in Fig. 1 (b). The interaction is mediated by another parton (boson) which has its own self-energy contributions. The detailed description of all possible configurations is a huge challenge to perturbative QCD. The present situation has led some authors to claim that the perturbative QCD approach will not be able to provide an accurate calculation of the running coupling constant at low energies, and that the renormalization procedure just exchanged the infinities of the vertex functions by an infinity number of parameters in the calculation of this constant. By using the thermofractal ideas introduced in Sec. 2.3, it has been derived in Refs. [18, 19] an effective description of YMF theory. We will summarize below the main results. Let us consider that the system with energy E, in which the parton with energy εjis one among Nconstituents, is itself a parton inside a larger system. Then, the power-law distribution of Eq. (6) describes how the energy received by the initial parton flows to its internal d.o.f., i.e. to partons at higher perturbative orders. This Tsallis statistics and thermofractals: applications to high energy and hadron physics 6 Figure 2. Vertex functions at scale λ0(left) and λ(right). suggests that at each vertex, this distribution plays the role of an effective coupling ¯g=G ˜ N Y i=1 h1+(q−1) εi kτ i−1 q−1,(9) where ˜ Nis the number of particles created or annihilated at each interaction, and G is the overall strength of the interaction. Within this picture, the entropic index qis related to the number of internal d.o.f. in the fractal structure. The renormalized vertex functions together with the CS equation were used to derive the beta function of QCD, leading to the 1-loop result [26] βQCD =−1 16π211 3c1−4 3c2¯g3,(10) where c1=Ncand c2=Nf/2. The beta function can be derived as well by using the effective thermofractal description introduced above. To do this, one should consider a vertex at two different scales λoand λ. As it is depicted in Fig. 2, the vertex function at scale λcontains one additional loop, from which one can identify the effective coupling ¯g. By using Eq. (9) with λ=λo/µ, where µis a scaling factor, the 1-loop beta function turns out to be β¯g=µ∂¯g ∂µ =−1 16π2 1 q−1g˜ N+1 ,(11) with ˜ N= 2 in YMF theory. Finally, from a comparison with the QCD result of Eq. (10), one can relate the entropic index qwith the gauge field parameters, leading to [19, 20] q= 1 + 3 11Nc−2Nf .(12) This leads to q≃1.14 when using Nc= 3 and Nf= 6, in excellent agreement with the experimental data analyses as we will see in the next section. 3. Tsallis statistics: applications to high energy and hadron physics We will discuss in this section some of the recent applications of Tsallis statistics to QCD phenomenology, including HEP, hadron physics and BEC. Tsallis statistics and thermofractals: applications to high energy and hadron physics 7 Figure 3. Left panel: Fitting of the experimental data for p⊥distribution of the abundance of different hadron species in pp collisions, by considering the NESCT distribution of Eq. (14). Right panel: Cumulative hadron spectrum, as a function of the hadron mass. The dots stand for the PDG result [30]. We display also the result by using the NESCT (blue) and the result predicted by Hagedorn (red), cf. Ref. [28]. 3.1. Transverse momentum distribution in pp collisions R. Hagedorn proposed a self-consistent thermodynamical approach to QCD formulated in terms of BG statistics, known as the HRG approach, allowing a description of the confined phase as a multi-component gas of non-interacting massive stable and pointlike particles [4, 5]. When Hagedorn’s theory was applied to pp collisions, it predicted the transverse momentum distribution of the particle production of hadrons given by d2N dp⊥dy =gV p⊥m⊥ (2π)2e−βm⊥,(13) where gis a constant, β≡1/(kBT), Vis the volume of the system, m⊥= (p2 ⊥+m2)1/2, and yis the rapidity. However, this exponential distribution turned out to be in disagreement with experimental data, as these behave instead as a power-law, cf. Fig. 3 (left). This was the motivation to consider the extension of Hagedorn’s theory to nonextensive statistics, within the so-called non-extensive self-consistent thermodynamics (NESCT) [27]. In this formalism, the p⊥distribution of particle species in pp collision turns out to be d2N dp⊥dy =gV p⊥m⊥ (2π)2e(−) q(−βm⊥).(14) This extended theory allows to reproduce the distribution of all the hadron species with high accuracy over 15 orders of magnitude, leading to q= 1.14(1) and T= 62(5) MeV [28, 29]. A second prediction of the NESCT is a power-law behavior for the hadron spectrum, with a density of hadron species given by [27] ρ(m) = ρo·e(+) q(βm).(15) Tsallis statistics and thermofractals: applications to high energy and hadron physics 8 We display in Fig. 3 (right) the cumulative number of hadrons, defined as the number of hadronic states below some mass m, i.e. N(m)≡Rm 0d˜m ρ( ˜m). It is found that the distribution of Eq. (15) leads to an excellent description of the hadron spectrum taken from the review by the Particle Data Group (PDG) [30], as compared to the exponential distribution ρ(m) = ρo·em/THproposed by Hagedorn, specially for the lightest hadrons, cf. Ref. [28]. 3.2. QCD thermodynamics Tsallis statistics has been applied also to study the thermodynamics of QCD. The grandcanonical partition function for a non-extensive ideal quantum gas is given by [31, 32] ln Zq(V, T, µ) = −ξV Zd3p (2π)3X r=± Θ(rx) ln(−r) q e(r) q(x)−ξ e(r) q(x)!,(16) where x=β(εp−µ), the particle energy is εp=pp2+m2,µis the chemical potential, ξ=±1 for bosons(fermions), and Θ(z) is the step function. The partition function for bosons is defined only for the case µ≤m, therefore the term with r=−in the integrand is applied only for fermions, and it only contributes if µ>m. The thermodynamics of QCD in the confined phase has been widely studied within the HRG approach in which physical observables are described in terms of hadronic degrees of freedom [5]. These are usually taken as the conventional hadrons listed in the PDG [30]. In this approach, the partition function is given by ln Zq(V, T, {µQa}) = X i∈hadrons ln Zq(V, T, µQai ),(17) where µQai ≡µaQai refers to the chemical potential of charge Qa≡ {u, d, s}for the i-th hadron, while µais the chemical potential associated to charge Qa‡. From that, one can compute the thermodynamic quantities by using the standard thermodynamics relations. The thermal expectation value for the charge Qais given by hQai=1 β ∂ ∂µa ln Zqβ=QahNqi,(18) where hNqiis the average number of particles. By using that the baryon number for (anti)quarks is Bquarks = 1/3 and Bantiquarks =−1/3, the baryon density turns out to be ρB=hBi V=1 3V(hNquarksi−hNantiquarksi).(19) The thermodynamical relations for the pressure P, energy density ε, and entropy S, involve derivatives of ln Zqwith respect to V,µBand β[32, 33]. Using the arguments of Refs. [34, 35], the chemical freeze-out line T=T(µB) can been determined by the conditions hEi/hNi ≃ 1 GeV or s/T3≃5. The results, displayed in Fig. 4 (left), show ‡While we are considering the flavour basis {u, d, s}of the Nf= 3 flavor sector of QCD, where urefers to the number of constituent quarks minus antiquarks of type u(and similarly for dand s), we could work equivalently in the basis of conserved charges formed by the baryon number B, electric charge Q, and strangeness S. Tsallis statistics and thermofractals: applications to high energy and hadron physics 9 à à à à à à à à à à è è è è è è è è è è è è è è è è è è è è è è è ôô òò èè àà òò ôô 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.05 0.10 0.15 0.20 ΜB@GeVD T@GeVD Boltzmann-Gibbs 8 EN=1 GeV sT3=5 q=1.14 8 EN=1 GeV sT3=5 RHICSPSAGSSIS RHICSPSAGS HADES FOPI ---- ---- 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 ΜB@GeVD NbaryonsNtotal NantibaryonsNtotal NmesonsNtotal Figure 4. Left panel: Chemical freeze-out line T=T(µB). We plot the result by using BG statistics, and Tsallis statistics. Right panel: Number of (anti)baryons/mesons inside a volume Vproton, as a function of µB. We have considered in both panels q= 1.14. an inflection for µB≃mproton related to a sharp increase of the baryon density in this regime. The region below the freeze-out line refers to the confined regime. One important aspect to study is the limits of temperature and chemical potential within which the proton can exist as a confined system. To address this point, we can consider the MIT-bag model criterion, i.e. the proton exists only if the total energy inside a volume Vproton is smaller or equal to the proton mass, ε·Vproton ≤mproton [33]. Fig. 4 (right) shows the number of baryons, antibaryons and mesons normalized to the total number of hadrons, along the line ε·Vproton =mproton. According to this figure, the proton exists close to µB≃mproton, and at this value of chemical potential, the proton is completely baryonic in content. In order to evaluate the effects of non-extensivity in the thermodynamic quantities, we present in Fig. 5 (left) a plot of the pressure as a function of µBfor different values of the entropic index q. As it was discussed in Refs. [32, 33], the equation of state P=P(ε) becomes harder for q > 1 as compared to BG statistics. This has important implications for neutron stars, in particular, the non-extensive effects turn out to be enough to produce stars with higher maximum masses [36]. Finally, we display in Fig. 5 (right) the energy density in the (µB, T) plane. The curve for which Tand µBresults in total energy equal to the proton mass is indicated by red points. The system seems to behave close to the conformal limit in this regime, so that the trace anomaly is vanishing, i.e. ε−3P≃0. 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