Selfinteracting Particle-Antiparticle System of Bosons
SSelfinteracting Particle-Antiparticle System of Bosons
D. Anchishkin,
1, 2
V. Gnatovskyy, D. Zhuravel, and V. Karpenko Bogolyubov Institute for Theoretical Physics, 03143 Kyiv, Ukraine Taras Shevchenko National University of Kyiv, 03022 Kyiv, Ukraine
Abstract
Thermodynamic properties of a system of interacting boson particles and antiparticles at finite tem-peratures are studied within the framework of the thermodynamically consistent Skyrme-like mean-field model. The mean field contains both attractive and repulsive terms. Self-consistency relationsbetween the mean field and thermodynamic functions are derived. We assume a conservation of theisospin density for all temperatures. It is shown that, independently of the strength of the attrac-tive mean field, at the critical temperature T c the system undergoes the phase transition of secondorder to the Bose-Einstein condensate, which exists in the temperature interval 0 ≤ T ≤ T c . It isobtained that the condensation represents a discontinuity of the derivative of the specific heat at T = T c and condensate occurs only for the component that has a higher particle-number density inthe particle-antiparticle system. PACS numbers: 12.40.Ee, 12.40.-yKeywords: relativistic bosonic system, Bose-Einstein condensation, second order phase transition a r X i v : . [ nu c l - t h ] F e b . INTRODUCTION The knowledge of the phase structure of the meson systems, in the regime of finite temper-atures and isospin densities is crucial for understanding of a wide range of phenomena fromnucleus-nucleus collisions to neutron stars and cosmology. This field is an essential part of inves-tigations of hot and dense hadronic matter which is a subject of active research [1]. Meanwhile,investigations of the meson systems has its own specifics due to a possibility of the Bose-Einsteincondensation of interacting bosonic particles. The problem of the Bose-Einstein condensationof pi-mesons has been studied previously, starting from the pioneer works of A.B. Migdal andcoworkers (see [2] for references). Later this problem was investigated by many authors usingdifferent models and methods. Formation of classical pion fields in heavy-ion collisions wasdiscussed in refs. [3–6] and the systems of pions and K-mesons with a finite isospin chemicalpotential have been considered in more recent studies [7–11]. First-principles lattice calcu-lations provide a solid basis for our knowledge of the finite temperature regime. Interestingnew results concerning dense pion systems have been obtained recently using lattice methods[12–14].In the present paper we consider interacting particle-antiparticle boson system at the con-served isospin density n I and finite temperatures. We name the bosonic particles as “pions” justconventionally. The preference is made because the charged π -mesons are the lightest hadronsthat couple to the isospin chemical potential. On the other hand, the pions are the lightestnuclear boson particles and thus, an account for “temperature creation” of particle-antiparticlepairs is a relevant problem on the basis of quantum-statistical approach.To account for the interaction between the bosons we introduce a phenomenological Skyrme-like mean field U ( n ), which depends only on the total meson density n . This mean field ratherreflects the presence of other strongly interacting particles in the system, for instance ρ -mesonsand nucleon-antinucleon pairs at low temperatures or gluons and quark-antiquark pairs at hightemperatures, T > T qgp ≈
160 MeV. Calculations for noninteracting hadron resonance gas showthat the particle densities may reach values (0 . − .
2) fm − at temperatures 100 −
160 MeV,which are below the deconfinement phase transition, see e.g. refs. [15, 16].The presented study is a development of the approach proposed in ref. [2], where the bo-son system was considered within the framework of the Grand Canonical Ensemble with zerochemical potential. Meanwhile, here we investigate the thermodynamic properties of the mesonsystem in the Canonical Ensemble, where the canonical variables are the temperature T andthe isospin density n I . We regard a selfinteracting many-particle system, which is being studied2s a toy-model that can help us understand Bose-Einstein condensation and phase transitionsover a wide range of temperatures and densities.So, in this work, in the formulation of the Canonical Ensemble, we calculate the ther-modynamic characteristics of a non-ideal hot “pion” gas with a fixed isospin density n I = n ( − ) π − n (+) π >
0, where n ( ∓ ) π are the particle-number densities of the π ∓ mesons, respectively. InSect. II we develop the formalism of the thermodynamic mean-field model [17] to describe theboson system of particles and antiparticles, which will be used in the presented calculations. InSect. III we introduce the Skyrme-like parametrization of the mean field and the correspondingthermodynamic functions are calculated. In Sect. IV we demonstrate the possibility of Bosecondensation when the attractive interaction is “weak”. Our conclusions are summarized inSect. V. II. THE MEAN-FIELD MODEL FOR THE SYSTEMOF BOSON PARTICLES AND ANTIPARTICLES
Our consideration in this section is based on the thermodynamic mean-field model whichwas introduced in refs. [19, 20], and then developed in ref. [17]. We limit our consideration tothe case where at a fixed temperature the interacting boson particles and boson antiparticlesare in the dynamical equilibrium with respect to annihilation and pair-creation processes. Thechemical potentials of boson particles µ p and boson antiparticles µ ¯ p are then, have oppositesigns: µ p = − µ ¯ p ≡ µ . (1)We are going to consider the system of bosonic particles and bosonic antiparticles with theconserved density of the isospin number n I = n ( − ) − n (+) , where n ( − ) is the particle-numberdensity of bosonic particles and n (+) is the particle-number density of bosonic antiparticles.Therefore, the Euler relation includes isospin number density only: ε + p = T s + µ n I . (2)The total particle-number density is n = n ( − ) + n (+) . Roughly speaking, in such a problem the chemical potential controls the difference of particleand antiparticle numbers µ → ( N ( − ) − N (+) ) whereas the total number of particles is controlled The dynamical conservation of the total number of pions in a pion-enriched system created on an intermediatestage of a heavy-ion collision was considered in refs. [21–23]
3y the temperature T → ( N = N ( − ) + N (+) ). Indeed, if some amount of particle-antiparticlepairs M has been created additionally to the existing particles N ( − ) and N (+) in a closed system,then approximately the same value µ is in correspondence µ → [( N ( − ) + M ) − ( N (+) + M )]but T (cid:48) → ( N ( − ) + M + N (+) + M ), where T (cid:48) > T . This qualitative consideration indicatesthe existence of one to one correspondence of independent pairs of variables ( T, µ ) ⇔ ( N, N I ).Actually, it is an easy task to show that the latter statement is valid in ideal quantum gas ofparticles and antiparticles. Meanwhile, it is not so simple the rigorous proof of the independenceof thermodynamic variables n and n I in a more general case where the mean fields, which dependon these variables, are present in the system (see [18]).In general the mean field U depends on both independent variables n, n I , i.e. U ( n, n I ).On the other hand, as proved in [18], the mean field can be separated into n -dependent and n I -dependent pieces where then, it reads respectively for particles and antiparticles as U ( − ) (cid:0) n, n I (cid:1) = U ( n ) − U I (cid:0) n I (cid:1) , (3) U (+) (cid:0) n, n I (cid:1) = U ( n ) + U I (cid:0) n I (cid:1) . (4)These signs in eqs. (3) and (4) are due to odd dependence on the isospin number n I .The total pressure in the two-component system reads p = − gT (cid:90) d k (2 π ) ln (cid:34) − exp (cid:32) − (cid:112) m + k + U ( n ) − U I ( n I ) − µT (cid:33)(cid:35) −− gT (cid:90) d k (2 π ) ln (cid:34) − exp (cid:32) − √ m + k + U ( n ) + U I ( n I ) + µT (cid:33)(cid:35) + P ( n, n I ) , (5)where P ( n, n I ) is the excess pressure. At the first step of investigation we neglect that part of the mean field which depends onisospin density, i.e. we assume U I ( n I ) = 0. Therefore, in this approximation, the excess pressurealso depends only on the total particle-number density, P ( n ).The thermodynamic consistency of the mean-field model can be obtained by putting incorrespondence of two expressions which must coincide in the result. These expressions, whichdetermine the isospin density, read n I = (cid:18) ∂p∂µ (cid:19) T , (6)where pressure is given by Eq.(5), and n I = g (cid:90) d k (2 π ) (cid:2) f (cid:0) E ( k, n ) , µ (cid:1) − f (cid:0) E ( k, n ) , − µ (cid:1)(cid:3) . (7) Here and below we adopt the system of units (cid:126) = c = 1, k B = 1 E ( k, n ) = ω k + U ( n ) with ω k = (cid:112) m + k and the Bose-Einstein distribution functionreads f (cid:0) E, µ (cid:1) = (cid:20) exp (cid:18) E − µT (cid:19) − (cid:21) − . (8)In order the expressions (6) and (7) to coincide in the result the following relation between themean field and the excess pressure arises n ∂U ( n ) ∂n = ∂P ( n ) ∂n . (9)It provides the thermodynamic consistency of the model. When both components of π − - π + system are in the thermal (kinetic) phase the pressure and energy density read p = g (cid:90) d k (2 π ) k ω k (cid:2) f (cid:0) E ( k, n ) , µ (cid:1) + f (cid:0) E ( k, n ) , − µ (cid:1)(cid:3) + P ( n ) , (10) ε = g (cid:90) d k (2 π ) ω k (cid:2) f (cid:0) E ( k, n ) , µ (cid:1) + f (cid:0) E ( k, n ) , − µ (cid:1)(cid:3) + nU ( n ) − P ( n ) . (11) III. SKYRME-LIKE PARAMETRIZATION OF THE MEAN FIELD
The thermodynamic mean-field model has been applied for several physically interestingsystems including the hadron-resonance gas [17] and the pionic gas [25]. This approach wasextended to the case of a bosonic system at µ = 0 which can undergo Bose condensation [2, 24].In the present study a generalized formalism given in section II is used to describe the particle-antiparticle system of bosons when the isospin density is kept constant. As was mentioned inthe previous section the mean field in general case splits into two pieces with dependence on thetotal particle density n and on the isospin density n I , respectively, see eqs. (3) and (4). At thefirst stage of our investigation we assume that the interaction between particles is described bythe Skyrme-like mean field which depends only on the total particle-number density n . Looselyspeaking we take into account just a strong interaction. So, we assume that the mean fieldreads U ( n ) = − A n + B n , (12)where A and B are the model parameters, which should be specified. Some additional contribu-tion to the attractive mean field at high temperatures, ( T ∝ −
160 MeV), may be providedby other hadrons present in the system, like ρ -mesons [26] or baryon-antibaryon pairs [27]. Aswas mentioned in the introduction, an investigation of the properties of a dense and hot piongas is well inspired by formation of the medium with low baryon numbers at midrapidity whatwas proved in the experiments at RHIC and LHC [28, 29].5y this reason, in our calculations we consider a general case of A >
0, to study a bosonicsystem with both attractive and repulsive contributions to the mean field (12). For the repulsivecoefficient B we use a fixed value, obtained from an estimate based on the virial expansion [30], B = 10 mv with v equal to four times the proper volume of a particle, i.e. v = 16 πr / v = 0 .
45 fm that corresponds to a “particle radius” r ≈ . m = 139 MeV,which we call “pions”. In this case the repulsive coefficient is B/m = 2 .
025 fm − and it iskept constant through all present calculations. (For instance, in Ref. [31] authors use the value B/m = 21 . − .) At the same time the coefficient A , which determines the intensity ofattraction of the mean field (12), will be varied. It is advisable to parameterize the coefficient A . We are going to do this with making use of solutions of equation U ( n ) + m = 0, similar toparametrization adopted in refs. [2, 24]. For the given mean field (12) there are two roots ofthis equation ( n , = ( A ∓ √ A − mB ) / B ) n = (cid:114) mB (cid:16) κ − √ κ − (cid:17) , n = (cid:114) mB (cid:16) κ + √ κ − (cid:17) , (13)where κ ≡ A √ m B . (14)Then, one can parameterize the attraction coefficient as A = κA c with A c = 2 √ mB . As we willshow below, the dimensionless parameter κ is the scale parameter of the model, that is, whenthe isospin density is fixed, the parameter κ determines the phase structure of the system. Asit is seen from eq. (13) for the values of parameter κ < A c is obtained when both roots coincide, i.e. when κ = 1, then A = A c = 2 √ mB .We consider two intervals of the parameter κ . 1) First interval corresponds to κ ≤
1, thereare no real roots of equation U ( n ) + m = 0. We associate these values of κ with a “weak”attractive interaction and in the present study we consider variations in the attraction coefficient A for values of κ only from this interval. 2) Second interval corresponds to κ >
1, there aretwo real roots of equation U ( n ) + m = 0. We associate this interval with a “strong” attractiveinteraction. This case will be considered elsewhere.If one assumes a possibility of the Bose-Einstein condensation in the two-component system,then it is instructive to classify a phase structure of the system in accordance with two basiccombinations which determine for the “weak” attraction the different thermodynamic states:(i) Both components, or the boson particles and boson antiparticles, i.e. π − and π + , are in the6hermal (kinetic) phase; (ii) Particles ( π − ) are in the condensate phase and antiparticles ( π + )are in the thermal (kinetic) phase - this combination can be named as the “cross” state.It is necessary to note, that expression “particles are in the condensate phase” is, of course,a conventional one, because in the essence it is a mixture phase, where at a fixed temperaturesome fraction of particles, i.e. a fraction of π − -mesons, is in thermal states with momentum | k | > π − -component belongs to the Bose-Einstein condensate, whereall π − -mesons have zero momentum, k = 0.We are going now to consider these basic thermodynamic states of the system using themean field (12). IV. THERMODYNAMIC PROPERTIES OF THE BOSONIC PARTICLE-ANTIPARTICLE SYSTEM AT “WEAK ATTRACTION”
In the mean-field approach the behavior of the particle-antiparticle bosonic system in thermal(kinetic) phase is determined by the set of two transcendental equations (we keep n I = const) n = (cid:90) d k (2 π ) (cid:2) f (cid:0) E ( k, n ) , µ (cid:1) + f (cid:0) E ( k, n ) , − µ (cid:1)(cid:3) , (15) n I = (cid:90) d k (2 π ) (cid:2) f (cid:0) E ( k, n ) , µ (cid:1) − f (cid:0) E ( k, n ) , − µ (cid:1)(cid:3) , (16)where the Bose-Einstein distribution function f (cid:0) E, µ (cid:1) is defined in (8) and E ( k, n ) = ω k + U ( n ). Equations (15)-(16) should be solved selfconsistently with respect to n and µ for a giventemperature T with account for n I = const. In the present we consider bosonical system in theCanonical Ensemble, where the independent canonical variables are T and n I , particles spinequal to zero. In this approach the chemical potential µ is a thermodynamic variable whichdepends on the canonical variables, i.e. µ ( T, n I ).In case of the cross state, when the particles, i.e. π − -mesons, are in the condensate phaseand antiparticles are still in the thermal (kinetic) phase, eqs. (15), (16) should be generalized toinclude condensate component n ( − )cond . Besides this we should take into account that the particles( π − or high-density component) can be in condensed state just under the necessary condition U ( n ) − µ = − m . (17)During decreasing of temperature from high values, where both π − and π + are in the thermalphase, the density of π − -component n ( − ) ( T, µ ) (high-density component) achieves first thecritical curve at temperature T ( − )c , where condition (17) is valid. This means that the curve7 (id)lim ( T ), which is defined as n (id)lim ( T ) = (cid:90) d k (2 π ) f (cid:0) ω k , µ (cid:1)(cid:12)(cid:12)(cid:12) µ = m , (18)is the critical curve for π − -mesons or for high-density component. Here f ( ω k , µ ) is the Bose-Einstein distribution function defined in (8). As we see function (18) represents the maximaldensity of thermal (kinetic) boson particles of the ideal gas at temperature T when µ = m .Hence, we obtain that the critical curve in the mean-field approach under consideration for theboson particles coincides with the critical curve for the ideal gas.With account for eqs. (17) and (18) we write the generalization of the set of eqs. (15), (16) n = n ( − )cond ( T ) + n (id)lim ( T ) + (cid:90) d k (2 π ) f (cid:0) E ( k, n ) , − µ (cid:1) , (19) n I = n ( − )cond ( T ) + n (id)lim ( T ) − (cid:90) d k (2 π ) f (cid:0) E ( k, n ) , − µ (cid:1) ; (20)Meanwhile, because of relation (17) between the mean field and chemical potential, i.e. E ( k, n ) − µ = ω k − m , this set of equations can be reduced just to one equation with respectto n (+) and it reads n (+) = (cid:90) d k (2 π ) f (cid:0) E ( k, n ) , − µ (cid:1)(cid:12)(cid:12)(cid:12) µ = U ( n )+ m with E ( k, n ) = ω k + U (cid:0) n (+) + n I (cid:1) . (21)Solution of eq. (21) for temperatures T from the interval T < T ( − )c provides the density n (+) ( T )of π + -mesons.One can see from eqs. (19), (20) that the particle density n (+) is provided only by thermal(kinetic) antiparticles ( π + -mesons). Whereas, the density n ( − ) of π − -mesons is provided by twofractions: (1) the condensed particles ( π − -mesons at k = 0) with the particle-number density n ( − )cond ( T ), and (2) thermal π − -mesons at | k | > n (id)lim ( T ). Theparticle-density sum rule for these phase of π − -mesons in the interval T < T ( − )c reads n ( − ) = n ( − )cond ( T ) + n (id)lim ( T ) . (22) A. Numerical calculations
At high temperatures, i.e. T ≥ T ( − )c , both components of the bosonic particle-antiparticlesystem are in the thermal phase and thermodynamic properties of the system are determinedby the set of eqs. (15) and (16). Solving this set for given values T and n I we obtain thefunctions µ ( T, n I ) and n ( T, n I ) and then other thermodynamic quantities.8hen we decrease temperature, after crossing the value T = T ( − )c the particles which belongto the high-density component (or π − -mesons) start to “drop down” into the condensate state,which is characterized by the value of momentum k = 0. In the limit, when T = 0, allparticles of the high-density component n ( − ) , i.e. π − -mesons, are in condensed state. At thesame time, the particles of the low-density component or π + -mesons being in the thermalphase lose the density n (+) with decrease of temperature and it becomes rigorously zero at T = 0. For the temperature interval T < T ( − )c equations (15), (16) should be generalizedand now thermodynamic properties of the system are determined by eq. (21), where we takeinto account that µ = − U ( n ) + m for all temperatures of this interval unless the high-densitycomponent n ( − ) is in condensed state. Otherwise it is necessary to solve the set of eqs. (15)and (16) for the region where n ( − ) appears again in the thermal (kinetic) phase.For parameters n I = 0 . − , κ = 0 . κ = 1 . n (+) of π + -mesons and the density n ( − ) of π − -mesons are depicted in Fig. 1. In this figure wedepicted as well the behavior of the total meson density n = n (+) + n ( − ) as functions of thetemperature (in the figure field it is notated as n tot ).Analyzing the behavior of the condensate creation (see Fig. 1) it is necessary to note, that justhigh-density component of the particle-antiparticle gas undergoes the phase transition to theBose-Einstein condensate. If we apply our consideration to pion gas with n I = n ( − ) π − n (+) π > π − -component undergoes the phase transition to the Bose-Einstein condensateand the low-density component or π + mesons exist only in the thermal phase for whole rangeof temperatures. Hence, it makes sense to look for the Bose-Einstein condensate of π − mesonsonly in an experiment, for instance in heavy-ion collisions.Equation (19) can be used to determine the critical temperature T ( − )c . Indeed, let us takeinto account that at the crossing point with the critical curve the density of condensate is zeroso far, n ( − )cond (cid:0) T ( − )c (cid:1) = 0, and the density of thermal π − particles becomes equal to n ( − ) (cid:0) T ( − )c (cid:1) = n (id)lim (cid:0) T ( − )c (cid:1) . Then, at this temperature T = T ( − )c on the l.h.s. of eq. (19) we have n =2 n (id)lim (cid:0) T ( − )c (cid:1) − n I , and now at this temperature point on the critical curve eq. (19) with respectto T reads as: n (id)lim ( T ) − n I = (cid:90) d k (2 π ) f (cid:0) E ( k, n ) , − µ (cid:1)(cid:12)(cid:12)(cid:12) µ = U ( n )+ m with E ( k, n ) = ω k + U (cid:16) n (id)lim − n I (cid:17) . (23)Solving eq. (23) at n I = 0 . − , for κ = 0 . κ = 1 . T ( − )c = 128 . T ( − )c = 251 MeV, respectively. These results are depicted in Fig. 1 in the left and right panels,9 n ( + ) n ( - ) n [fm-3] T [ M e V ] n t o t n ( i d )l i m n I = 0 . 1 f m - 3 T ( - )c n ( + ) n ( - ) n ( - ) n ( + ) n [fm-3] T [ M e V ] n t o t n ( i d )l i m T n n I = 0 . 1 f m - 3 Figure 1:
Left panel:
The particle-number densities n (+) , n ( − ) and n tot = n (+) + n ( − ) versus temper-ature for the interacting π + - π − pion gas in the mean-field model. The total isospin density is keptconstant, n I = 0 . − , and the attraction parameter is κ = 0 .
5. The maximum density n (id)lim ofthe ideal gas of thermal pions at µ = m π is shown by the red dashed line. The shaded area showsthe possible states of condensed particles. The Bose-Einstein condensation of π − mesons occurs atthe temperature T = T ( − )c . Right panel:
The same as on the left panel, but with the parameter κ = 1 .
0. Here n tot = n = n ≡ n , (see eq. (13)), n ( − )1 , = ( n , + n I ) / n (+)1 , = ( n , − n I ) / T = T ≡ T , . respectively.It turns out that the temperature T ( − )c determines the phase transition to BEC for wholepion system because the antiparticles ( π + -mesons) from the low-density component n (+) ( T ) arecompletely in thermal state for all temperatures and thus, the condensate is created just bythe particles of high-density component n ( − ) ( T ). Then, the total density of condensate in thetwo-component pion system is created by π − -mesons only, i.e. n cond = n ( − )cond , and this particle-number density plays the role of the order parameter. The condensate density as functionof temperature obtained in the framework of our model for three values of the attractionparameter, κ = 0 . , . , . , .
0, at n I = 0 . − , is depicted in Fig. 2, left panel. We record avery small difference in the critical temperature T ( − )c when the attraction parameter κ changes,the difference does not exceed 4 MeV. Then it would be useful to define only one average valueof T c as T c = (cid:10) T ( − )c (cid:11) , (24)what gives for the particular choice of parameters T c ≈
129 MeV. This is a temperature of the10 (cid:7) (cid:1) (cid:6) (cid:1) (cid:3) T c ncond [1/fm3] T [ M e V ] (cid:7) (cid:1) (cid:6) (cid:1) (cid:3) (cid:2) (cid:5)(cid:7) (cid:1) (cid:6) (cid:1) (cid:4) (cid:7) (cid:1) (cid:6) (cid:1) (cid:3)(cid:7) (cid:1) (cid:6) (cid:1) (cid:4)(cid:7) (cid:1) (cid:6) (cid:1) (cid:3) (cid:2) (cid:5) m /m T [ M e V ] (cid:7) (cid:1) (cid:6) (cid:1) (cid:3) (cid:2) (cid:4) T c Figure 2:
Left panel:
The density of condensate versus temperature in the particle-antiparticleselfinteracting system when the isospin density is kept constant, n I = 0 . − . The curves aremarked by the attraction parameter κ . Right panel:
The chemical potential versus temperature atvalues of the attraction parameter κ = 0 . , . , . , .
0. The marked points on the curves correspondto the critical temperature T ( − )c . In both panels we set T c = (cid:68) T ( − )c (cid:69) . second order phase transition which “signals” of the creation of condensate when temperaturedecreases and crosses the value T = T c . Note that the critical temperature T c practically doesnot depend on the attraction parameter A of the mean field (12). In other words, the averageattraction between particles in the system has almost no effect on the critical temperature.The dependence of the chemical potential on temperature is depicted in Fig. 2 in the rightpanel for three values of the attraction parameter, κ = 0 . , . , . , .
0. First of all, we noticethat the chemical potential is almost independent of temperature when condensate exists inthe system, i.e. in the interval 0 < T ≤ T c . Value of µ changes from 1 . m π at the absenceof attraction, κ = 0 .
0, to µ = 0 . m π for the critical attraction parameter κ = 1 .
0. Hence,for 0 . ≤ κ ≤ . ≤ µ ≤
138 MeV. It is intriguingto remind that already first attempts to fit the p T spectra of π − -mesons in O+Au collisions at200 AGeV/nucleon (at midrapidity) by the ideal-gas Bose-Einstein distribution results in thevalues µ ≈
126 MeV, T ≈
167 MeV and in S+S collisions at 200 AGeV/nucleon it results inthe values µ ≈
118 MeV, T ≈
164 MeV [32]. So, the fit of data required the pion chemicalpotential in the range µ ≈ −
130 MeV what we can just formally compare with the valuesof the chemical potential obtained in our model.The derivative of the chemical potential on temperature has a jump in points, which are11arked on the curves as small black circles, see Fig. 2 right panel. These points on the curves µ ( T ) correspond to T ( − )c , which values differ from one another not more than ∆ T = 4 MeV. Aswe concluded before, this is the temperature of phase transition, see eq. (24), which practicallydoes not depend on intensity of attraction. To prove that this is indeed a phase transition ofthe second order, we first calculate the heat capacity c v as c v = − T ∂ f∂T , (25)where f ( T, n I ) = − p ( T, n I )+ n I µ ( T, n I ) is the density of free energy, and s ( T, n I ) is the entropydensity. We are going to calculate f ( T, n I ) for two thermodynamic scenarios, when T ≥ T c andwhen T < T c .Having solved eqs. (15), (16) then, using eq. (10) one can calculate pressure for the casewhen particles and antiparticles are both in the thermal phase, i.e. T ≥ T c . In this case thedensity of free energy looks like f = n I µ ( T, n I ) − (cid:90) d k (2 π ) k ω k (cid:2) f (cid:0) E ( k, n ) , µ (cid:1) + f (cid:0) E ( k, n ) , − µ (cid:1)(cid:3) − P ( n ) , (26)where functions n ( T, n I ) and µ ( T, n I ) are known. The excess pressure P ( n ) is obtained byintegrating eq. (9) for the Skyrme-like parametrization of the mean field (12): P ( n ) = − A n + 2 B n , (27)where P ( n = 0) = 0 is taking into account.For temperatures less than T c , when the high-density component of the pion gas ( π − mesons)is in the condensate phase and low-density component ( π + mesons) is in the thermal phase,the density of free energy reads f = n I [ U ( n )+ m ] − (cid:90) d k (2 π ) k ω k f (cid:0) ω k , µ (cid:1)(cid:12)(cid:12)(cid:12) µ = m − (cid:90) d k (2 π ) k ω k f (cid:0) E ( k, n ) , − µ (cid:1)(cid:12)(cid:12)(cid:12) µ = U ( n )+ m − P ( n ) . (28)Here the total pion density is n = 2 n (+) + n I , µ = U ( n ) + m as in eq. (21), E ( k, n ) = ω k + U ( n )and n (+) ( T, n I ) is solution of eq. (21).Using the density of free energy (26) to the right of T c and (28) to the left of T c , respectively,we calculate the heat capacity normalized to T , as function of temperature at n I = 0 . − forfour values of the attraction parameter κ = 0 . , . , . , .
0. These dependencies are depicted As a matter of fact, here we calculate the volumetric heat capacity which is the heat capacity C V of a systemdivided by the volume V , i.e c v = C V /V . c (cid:72) (cid:45) (cid:76) T c (cid:72) (cid:45) (cid:76) Κ (cid:61)
Κ (cid:61)
Κ (cid:61)
Κ (cid:61) (cid:235)(cid:235) (cid:235)(cid:235)
60 80 100 120 140 160 1800123456 T (cid:64)
MeV (cid:68) c v (cid:144) T Κ (cid:61)
Κ (cid:61)
Κ (cid:61)
Κ (cid:61) (cid:230) T c
60 80 100 120 140 160 180050100150 T (cid:64)
MeV (cid:68) Ε (cid:64) M e V (cid:144) f m (cid:68) Figure 3:
Left panel:
Heat capacity normalized to T as a function of temperature in a self-interacting π − − π + meson system. The isospin density is kept constant, n I = 0 . − . The curves are markedwith the attraction parameter κ . Right panel:
Energy density versus temperature for the same systemand conditions as in the left panel. We set T c = (cid:68) T ( − )c (cid:69) . in Fig. 3 in the left panel. The temperature dependence of the heat capacity is a continuousfunction, but the derivative of this function has a finite discontinuity, which indicates a second-order phase transition, where the condensate density is an order parameter (strictly speaking,this is a third-order phase transition). To make sure that this is indeed a second-order phasetransition without the release of latent heat at the temperature T c , we calculate the energydensity ε for the same set of parameters κ , the functions ε ( T ) are shown in Fig. 3 in the rightpanel. Indeed, it is seen that these functions are not discontinuous at T = T c , what provesthat the system actually undergoes a second-order phase transition at this temperature. It isinteresting to note that the energy density in the temperature range 0 < T ≤ T c is practicallyindependent of the “weak” attraction (0 ≤ κ ≤
1) between the particles.It has long been known, see ref. [33], that the Bose-Einstein condensation is indeed a third-order phase transition according to the first classification of general types of transitions betweenphases of matter, introduced by Paul Ehrenfest in 1933 [34, 35]. Therefore, the obtainedtemperature T c is really the temperature of the phase transition of the second order (accordingto modern terminology) and the density of condensate n cond = n ( − )cond provided by π − mesons isthe order parameter. 13 . CONCLUDING REMARKS In this paper we have presented a thermodynamically consistent method to describe at finitetemperatures a dense bosonic system which consists of interacting particles and antiparticles ata fixed isospin density n I . We considered the system of meson particles with m = m π and zerospin, which we named conventionally as “pions” because the charged π -mesons are the lightestnuclear particle and the lightest hadrons that couple to the isospin chemical potential.It turns out that the introduced dimensionless quantity κ = A/ √ mB , which is itself acombination of the mean-field parameters, U ( n ) = − An + Bn , and the particle mass, is thescale parameter of the model. The parameter κ determines the different possible phase scenarioswhich occur in the particle-antiparticle boson system. Attraction coefficient A = κA c , where A c ≡ √ mB , was parameterized by κ with κ = 1 as the critical value that separates the regimeof a “weak attraction” ( κ ≤
1) from the regime of a “strong attraction” ( κ > n I isconserved, there is a Bose-Einstein condensate in the system in the temperature interval 0 ≤ T ≤ T c , which is the result of a second-order phase transition that occurs at a temperature T c and condensate density is an order parameter. This statement is in contrast to the conclusiongiven in Refs. [2, 24, 31, 36], where the system with zero chemical potential, µ = 0, wasinvestigated. Indeed, in these works it was shown that in the case of a sufficiently strongattractive mean field ( κ > n I = n ( − ) π − n (+) π >
0, the π − mesons only undergo the phase transitionto the Bose-Einstein condensate. At the same time, the π + mesons exist only in the thermalphase for whole range of temperatures. Then, for the experimental efforts it makes sense tolook for the Bose condensate, which is created just by π − mesons.Description of thermodynamic properties of the system was performed employing the Canon-ical Ensemble formulation, where the chemical potential µ is a thermodynamic quantity whichdepends on the canonical variables ( T, n I ). We calculated dependence of the chemical poten- Note, the chiral perturbation theory predicts that transition between the vacuum and the BEC state is of thesecond order with universality class O (2) [7]. κ which show that µ ≈ const in the“condensate” interval of temperatures 0 ≤ T ≤ T c , where these constant values depend on theintensity of attraction. Meanwhile the temperature T ( − )c of the phase transition to the Bose-Einstein condensate of π − mesons (high-density component) exhibits very weak dependence on κ , as it is evidently seen in Fig. 2 in the left panel. For wide range of the values of κ , from 0to 1 .
0, these critical temperatures differ from one another not more than 4 MeV, this inspiresan introduction of the mean value T c = (cid:68) T ( − )c (cid:69) of the phase transition to the Bose-Einsteincondensate.The results obtained are in correspondence with known peculiar property of the ideal Bosegas: the Bose-Einstein condensation represents the phase transition of third order or a dis-continuity of the derivative of the specific heat [33]. In the framework of the presented modelwe obtained that in the same way the derivative of the specific heat undergoes a break atthe temperature T c , as it is evidently seen in the left panel in Fig. 3. A discontinuity of theenergy-density on temperature and an absence of the release of latent heat at T c one can see inthe right panel in Fig. 3 what proves that the system indeed undergoes a second-order phasetransition at this temperature.The role of neutral pions is left beyond the scope of the present paper. The present analysiscan be improved by addressing these issue in more detail and also by generalizing the calculationto nonzero contribution to the mean field which depends on n I . Authors plan to consider theseproblems elsewhere. Acknowledgements
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