Phase diagram of interacting pion matter and isospin charge fluctuations
O. S. Stashko, O. V. Savchuk, R. V. Poberezhnyuk, V. Vovchenko, M. I. Gorenstein
PPhase diagram of interacting pion matter and isospin charge fluctuations
O. S. Stashko, O. V. Savchuk, R. V. Poberezhnyuk, V. Vovchenko, and M. I. Gorenstein Taras Shevchenko National University of Kyiv, 03022 Kyiv, Ukraine Frankfurt Institute for Advanced Studies, Giersch Science Center, D-60438 Frankfurt am Main, Germany Bogolyubov Institute for Theoretical Physics, 03680 Kyiv, Ukraine Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA (Dated: February 5, 2021)Equation of state and electric (isospin) charge fluctuations are studied for matter composed ofinteracting pions. The pion matter is described by self interacting scalar fields via a φ − φ typeLagrangian. The mean-field approximation is used, and interaction parameters are fixed by fittinglattice QCD results on the isospin density as a function of the isospin chemical potential at zerotemperature. Two scenarios for fixing the model parameters – with and without the first orderphase transition – are considered, both yielding a satisfactory description of the lattice data. Ther-modynamic functions and isospin charge fluctuations are studied and systematically compared forthese two scenarios, yielding qualitative differences in the behavior of isospin charge susceptibilities.These differences can be probed by lattice simulations at temperatures T (cid:46)
100 MeV.
Keywords: Bose-Einstein condensation, pion matter
I. INTRODUCTION
The Bose-Einstein condensation (BEC) [1, 2] is a fas-cinating phenomenon that occurs in a system of bosonswhen a macroscopic amount of particles occupies thezero-momentum state. This century-old phenomenon,observed experimentally in cold atomic gases [3–6], is pre-dicted to occur in very different physical systems, rang-ing from condensed matter physics to high-energy nuclearphysics, astrophysics, and cosmology (see, e.g., Refs. [7–15]). A theoretical description of the BEC appears to berather sensitive to delicate details of particle interactions[16–25].In the present work we study the BEC phenomenonin strongly interacting QCD matter. The effective low-energy degrees of freedom in QCD are pions – the threepseudo-Goldstone bosons in the confined phase. The pi-ons obey the Bose-Einstein statistics, thus an emergenceof the BEC of pions is possible and has been predictedto occur at large isospin chemical potentials, both in ef-fective QCD theories [26, 27] and in first-principle latticeQCD simulations [28, 29]. In nature, the pion BEC mayoccur during the cooling of the early Universe [30], in thegravitationally bound pion stars [29, 31, 32], or as a non-equilibrium phenomenon in heavy-ion collisions [8, 9, 33].The hypothetical boson stars [34–36] may exist and canbe a candidate for the dark matter in the Universe [37–42].Different effective QCD descriptions of the phasediagram of interacting pion matter with a BEC in-clude chiral perturbation theory [43, 44], Nambu-Jona-Lasinio model [45], Polyakov-loop extended quark mesonmodel [46, 47] etc. Recently, a possibility of the BEC inthe pion system at zero chemical potential was considered within a Skyrme-like model including both attractive andrepulsive interaction terms [14, 48, 49]. Effects of repul-sive interactions on the BEC of pions were studied inRef. [50] at non-zero chemical potential. The system ofpions at zero chemical potential was described [14] by aneffective Lagrangian with the attractive ( φ ) and repul-sive ( φ ) terms of a scalar field φ . In the present paperwe extend this model to the finite isospin chemical po-tential µ I .The phase diagram on the whole plane of isospin chem-ical potential µ I and temperature T is investigated. Mostmacroscopic systems with both repulsive and attractiveinteractions between constituents display the first orderliquid-gas phase transition (FOPT) which is ended by thecritical point (CP). Therefore, these phenomena can alsobe expected for the interacting pions in addition to theBEC.Lattice QCD results support an existence of the pionBEC at finite isospin chemical potential [28]. We use therecent lattice data at zero temperature to fix the repul-sive and attractive interaction parameters of the model.Then, thermodynamic functions and electric (isospin)charge fluctuations up to the fourth order are calcu-lated in the ( µ I , T )-plane. Two different scenarios areemployed and systematically compared. The first one in-cludes only the repulsive interactions via the φ term, butnot the attractive φ term. In this case no FOPT tran-sition is observed, only the BEC transition. The secondpossibility takes into account both the repulsive and at-tractive pion-pion interactions. In this case the FOPT is We use the common simplified terminology and call the thirdcomponent of isospin (electric charge) the isospin charge. a r X i v : . [ h e p - ph ] F e b observed at small T and generates a non-trivial interplaybetween the FOPT and BEC transitions on the phase di-agram. The measures of the isospin charge fluctuations– scaled variance, skewness, and kurtosis – appear to bevery sensitive to a presence of the CP and BEC phenom-ena. They are used to differentiate these two scenarios.The paper is organized as follows: the theoretical de-scription of interacting pion system is presented in Sec. II.The two choices of the interaction potential, the fixing ofthe model parameters, and the resulting phase diagramsare discussed. Section III is dedicated to fluctuationsof the isospin charge, in particular the scaled variance,skewness, and kurtosis are discussed in some detail. Sum-mary in Sec. IV closes the paper. II. INTERACTING PION SYSTEMA. Model formulation
The three pions species, ( π + , π − , π ), are representedas a triplet of interacting pseudo-scalar fields φ =( φ , φ , φ ) that are described by an effective relativis-tic Lagrangian density: L = 12 (cid:0) ∂ µ φ ∂ µ φ − m π φ (cid:1) + L int (cid:0) φ (cid:1) , (1)where m π is the vacuum pion mass and L int is the inter-action part of the Lagrangian. We omit here the electro-magnetic interactions. Consider now this system in sta-tistical equilibrium within the grand canonical ensemble(GCE). The independent variables are the temperature T and the isospin chemical potential µ I . The isospin chem-ical potential couples to the conserved isospin charge, thepion species π + , π − , and π carry the isospin charges of+1, -1, and 0, respectively.To proceed, we apply a relativistic mean-field approx-imation, i.e. series L int in terms of δσ = φ − σ , where σ = (cid:104) φ (cid:105) is the expectation value of the scalar field and (cid:104) ... (cid:105) denotes the GCE averaging. The effective mean-fieldLagrangian can then be represented as [14]: L ≈ (cid:2) ∂ µ φ ∂ µ φ − M ( σ ) φ (cid:3) + p ex ( σ ) , (2)where M ( σ ) is the effective pion mass and p ex ( σ ) is theso-called excess pressure, M ( σ ) = m π − d L int dσ , p ex ( σ ) = L int − σ d L int dσ . (3) We use the natural units, (cid:126) = c = k = 1, and assume equalmasses of all three pion species, m π = 140 MeV. The effective Lagrangian form of Eq. (2) implies that themain effect of interactions in our description leads to anappearance of a medium-dependent effective mass M ( σ ).The excess pressure p ex ( σ ) – the second term in the righthand side of Eq. (2) – ensures the proper counting of theinteraction energy.The details of the model formulation can be found inRefs. [14, 51]. This model was previously used to describethe pion system at zero chemical potential [14] and thesystem of interacting alpha particles [51]. In the presentstudy we apply this model to the new physical situationand consider the pion system at non-negative values ofthe isospin chemical potential µ I ≥
0. The results at µ I ≤ π + and π − . Values of µ I > n I ≡ n + − n − >
0, where n + and n − correspond to π + and π − particle number den-sities, respectively. We consider the possible BEC of thepositively charged pions in this regime. The expectationvalue of the scalar field σ is presented as (see Refs. [14, 51]for the derivation details) σ ( T, µ I , M ) = (cid:88) i σ th i ( T, µ i , M ) + σ bc+ , (4)where σ th i correspond to the contributions of the thermalpions, i = (+ , − , σ th i ( T, µ i , M ) = (cid:90) d k (2 π ) n k ( T, µ i , M ) √ k + M , (5)while σ bc+ corresponds to a possible contribution of theBose condensate (BC) of π + . Here µ + = µ I , µ − = − µ I , µ = 0, and n k ( T, µ i , M ) = (cid:34) exp (cid:32) √ k + M − µ i T (cid:33) − (cid:35) − . (6)We will use a Skyrme-like parameterization of the in-teraction term: L int ( σ ) = a σ − b σ , a ≥ , b > . (7)In Eq. (7), a ≥ b > M ( σ ) = (cid:112) m π − aσ + bσ , (8) p ex ( σ ) = − a σ + b σ . (9)Inverting Eq. (8) with respect to σ we obtain σ = a + (cid:112) a + 4 b ( M − m π )2 b . (10)At given T and µ I we use the system of self-consistentEqs. (4) and (8) to determine σ and M .The pressure p , the number densities of thermal pions n th i , and the isospin charge density n I can be calculatedas p = (cid:88) i (cid:90) d k (2 π ) k √ k + M n k ( T, µ i , M ) + p ex ( σ ) , (11) n th i = (cid:90) d k (2 π ) n k ( T, µ i , M ) , (12) n I = (cid:18) ∂p∂µ I (cid:19) T = n + − n − . (13)Here n i are the total number densities of pions. The n + density may include a contribution from a Bose conden-sate (BC). The condensation does not occur if µ I < M .In the case µ I < M only the thermal pions contribute tothe total number densities, n + = n th+ , n − = n th − , n = n th0 , µ I < M . (14)The BEC of π + occurs when their chemical potential µ + ≡ µ I reaches the value of the effective mass, i.e. µ I = M . In this case the number density n + may receivesa contribution n bc+ from the BC: n + = n th+ + n bc+ , µ I = M. (15)The number of densities of the two other pions speciesare unchanged: n − = n th − , n = n th0 . The number densityof π + in a condensate reads n bc+ = µ I σ bc+ = µ I (cid:32) σ − (cid:88) i σ th i (cid:33) . (16)In Eq. (16) the quantities σ and σ th i are calculated ac-cording to Eq. (10) and Eq. (5), respectively.An onset of the BEC takes place when µ I reaches theeffective mass M . This condition defines a line in thephase diagram – the BEC line. This line is calculatedby substituting M → µ I and σ bc+ → The second root of Eq. (8) corresponds to mechanically unstablestates. Note that chemical potential values µ I > M exceeding the effec-tive mass are forbidden as they would lead to negative occupancynumbers n k (6) for some k -states. equations (4), (8), and (12) and solving it with respectto µ I at given value of T . B. Fixing the parameters using lattice data at zerotemperature
Lattice QCD simulations at finite isospin density pro-vide constraints on the equation of state from first princi-ples [28]. In particular, the isospin density n I ( T = 0 , µ I )at zero temperature has been presented in [29]. Here wewill use these lattice data to constrain the parameters ofour model.In the limit of zero temperature, T = 0, the thermalpion excitations are absent, i.e. all thermal densities (12)vanish. In this case the system consists solely of the BCof π + -mesons, thus, the isospin density coincides with thenumber density of the condensed pions, n I ( T = 0 , µ I ) = n bc+ , and the total pressure equals to the excess pressure, p ( T = 0 , µ I ) = p ex . The explicit expression for n I ( T =0 , µ I ) in the the considered model follows from Eqs. (5),(10), and (16): n I ( T = 0 , µ I ) = µ I (cid:32) a + (cid:112) a + 4 b ( µ I − m π )2 b (cid:33) . (17)
1. Scenario I: Repulsive interactions only
In the first scenario we consider purely repulsive inter-actions between pions. To achieve this we set a = 0 and b > Equation (17) in this case is reduced to n I ( T = 0 , µ I ) = b − / µ I (cid:113) µ I − m π θ ( µ I − m π ) . (18)An onset of the BEC occurs at µ I = m π . The isospindensity is a continuous function of µ I since n I ( T =0 , µ I = m π ) = 0. On the other hand, the µ I -derivativeof n I exhibits a discontinuity at µ I = m π . Therefore,the transition between vaccuum and a pion-condensedphase at µ I = m π is a second-order phase transition at T = 0. Qualitatively, this is consistent with predictionsof many different theories, including for instance chiralperturbation theory [26, 43] or Polyakov-loop extendedquark meson model [46, 47].To fix the value of the parameter b we fit the latticeQCD data on n I ( T = 0 , µ I ) of Ref. [29] in the range ofchemical potentials µ I /m π <
2. We obtain b (cid:39) . /m π Another option would be to set b = 0 and take a <
0. The resultsin such a case are qualitatively similar to a = 0 and b > (cid:2) (cid:1) (cid:7) (cid:3) (cid:1) (cid:2) (cid:3) (cid:1) (cid:4) (cid:3) (cid:1) (cid:5) (cid:3) (cid:1) (cid:6) (cid:3) (cid:1) (cid:7) (cid:4) (cid:1) (cid:2)(cid:2) (cid:1) (cid:2)(cid:2) (cid:1) (cid:4)(cid:2) (cid:1) (cid:5)(cid:2) (cid:1) (cid:6)(cid:2) (cid:1) (cid:7)(cid:3) (cid:1) (cid:2) (cid:1) (cid:3) (cid:2) (cid:1) (cid:5) (cid:18) (cid:15) (cid:9) (cid:1) (cid:15) (cid:9) (cid:14) (cid:18) (cid:11)(cid:16) (cid:10)(cid:13) (cid:12)(cid:1) (cid:3) (cid:17) (cid:17) (cid:15) (cid:6) (cid:7) (cid:17) (cid:10)(cid:13) (cid:12) (cid:2) (cid:15) (cid:9) (cid:14) (cid:18) (cid:11)(cid:16) (cid:10)(cid:13) (cid:12)(cid:1) (cid:4) (cid:6) (cid:17) (cid:17) (cid:10)(cid:7) (cid:9) (cid:1) (cid:8) (cid:6) (cid:17) (cid:6) (cid:3) I (cid:1) (cid:1) (cid:2) (cid:1) p m I / (cid:5) p (cid:4) (cid:1) (cid:3) (cid:1) (cid:2) (cid:2) (cid:1) (cid:7) (cid:6) (cid:3) (cid:1) (cid:2) (cid:2) (cid:3) (cid:1) (cid:2) (cid:4)(cid:2)(cid:2) (cid:1) (cid:2) (cid:5)(cid:2) (cid:1) (cid:3) (cid:2) (cid:1) (cid:2) (cid:2) (cid:1) (cid:4) (cid:2) (cid:1) (cid:5) (cid:2) (cid:1) (cid:6) (cid:2) (cid:1) (cid:7) (cid:3) (cid:1) (cid:2) (cid:3) (cid:1) (cid:4) (cid:3) (cid:1) (cid:5)(cid:2) (cid:1) (cid:8) (cid:8)(cid:3)(cid:3) (cid:1) (cid:2) (cid:3) (cid:1) (cid:3) (cid:2) (cid:2) (cid:1) (cid:2) (cid:2) (cid:1) (cid:5) (cid:2) (cid:6) p m (cid:1) (cid:4) (cid:1) (cid:3) Figure 1. (a): A comparison of the pion condensate as a function of the isospin chemical potential, µ I , at zero temperature, T = 0, with lattice data of Ref. [29] is shown by solid line and dashed line for the Scenario I ( a = 0) and Scenario II ( a > n I ≈ .
022 fm − at µ I ≈ . m π for a > n I = 0 at µ I = m π for a = 0 are marked by, respectively, the red circle and the hollow green star. The inset shows the zoomed in picture closeto µ I = m π . (b): The ratio of the system’s pressure to the ideal gas pressure at zero isospin chemical potential, µ I = 0, as afunction of temperature. with χ /dof (cid:39) .
62. A comparison with the lattice datais shown in Fig. 1 (a) by blue dashed line.
2. Scenario II: Repulsion + attraction
Let us turn now to the more general case when both theattractive and repulsive interactions are present, a > b >
0. In this case the system undergoes a first-orderphase transition between vacuum (the gaseous phase) anda pion-condensed phase (the liquid phase), with the co-existence point being characterized by a vanishing pres-sure. To see this consider Eq. (9): this equation have twosolutions for p ex = p ( T = 0 , µ I ) = 0, defining the expec-tation values of the scalar field at the FOPT boundaries, σ g = 0 and σ l = 3 a/ b . This corresponds, via Eq. (8), tothe following values of the effective mass in the gaseousand liquid components: M g = m π , M l = (cid:114) m π − a b . (19)The FOPT takes place at µ = M l . The gaseous phaseat T = 0 corresponds to the vacuum, thus, n g = 0. Theisospin density jumps at µ I = µ from n g = 0 to n l ≡ n I ( T = 0 , µ I → µ + 0) = 3 a b (cid:114) m π − a b . (20) To fix the numerical values of a and b we again fit thelattice data on n I at T = 0. We obtain a (cid:39) .
93 and b (cid:39) . /m π with a fit quality of χ /dof (cid:39) .
35 – aslightly better fit compared to Scenario I.As discussed above, Scenario II predicts the FOPT.Using the numerical values of the a and b parametersfitted to the lattice data, one obtains the FOPT at µ ≈ . m π , where the isospin density jumps from n g = 0 to n l >
0. The values of the π + density n I and the bindingenergy per particle W at T = 0 and µ I = µ , n I ≈ . m π ≈ .
022 fm − , (21) W ≡ εn I − m π = M l − m π ≈ − , (22)obtained in Scenario II correspond to the ground stateof the pion matter. This density is about 7 timessmaller than the normal nuclear matter density of n =0 .
16 fm − , and the binding energy is about 16 timessmaller than that in the nuclear ground state.The behavior of n I in Scenario II at zero temperatureis shown in Fig. 1 (a) by the solid red line. A comparisonwith the lattice data and the predictions of Scenario Iare also shown. Overall, the behavior of n I in the bothscenarios is similar. Even though a nature of the phasetransition differs between the two scenarios, due to thesmall latent heat of the FOPT in Scenario II it is difficultto distinguish it from the second-order phase transitionin Scenario I using the presently available lattice data. InSec. III we discuss fluctuations as a possibility to makesuch a distinction. C. Phase diagram at finite temperatures
Model calculations at finite temperatures are straight-forward. Important constraints on the equation of stateof pion matter can be obtained at zero chemical po-tentials and large temperatures, 120 ≤ T ≤
160 MeV,where the QCD equation of state is known from latticeQCD [52, 53]. In this range, the pressure and energy den-sity are reasonably well described by the ideal hadron-resonance gas (see e.g., Ref. [54]). This indicates thateffects of pion interactions in this regime are small. Toverify this we plot in Fig. (b) a ratio of the pressure ofinteracting pions to the ideal pion gas baseline (i.e., at a = 0 and b = 0) for the two scenarios. For purely repul-sive interactions (scenario I) the pressure demonstratessmall suppression relative to the ideal gas. If attractiveeffects are included (scenario II) the pressure at smalltemperatures is higher than that of the ideal gas of pi-ons. However, at large T the repulsive effects becomedominant and the pion pressure is again suppressed. Inboth scenarios the corrections to the ideal gas pressureat µ I = 0 are small, not exceeding 1%. This is not thecase at non-zero values of µ I : the thermodynamics ofthe interacting pion gas differs drastically from that inthe ideal pion gas in the ( µ I , T )-region of the phase dia-gram where a BEC pions is formed, as discussed in thefollowing.In Scenario I, where the attractive pion interactionsare absent ( a = 0), there is no FOPT in the pion system.An onset of the π + BEC takes place when µ I reaches thevalue of the effective mass M . The BEC line can thusbe obtained by substituting M → µ I in the system ofequations (4) and (8) and solving it with respect to T .The resulting BEC line T bc ( µ I ) is shown in Fig. 2 (a) bythe dashed line. In the ideal gas limit one would obtaina vertical BEC line, µ I = m π . The deviation from theideal gas behavior thus becomes evident as T is increased.This is due to large particle number densities, and thusstronger effects of interactions, as the temperature is in-creased. Note that in the ideal Bose gas, a region of the( µ I , T )-plane with µ I > m π is forbidden, whereas in theinteracting system considered here this region is legiti-mate. It follows from Eq. (10) that the effective mass isalways larger than the vacuum mass, M ( T, µ I ) > m π ,the pure repulsion scenario ( a = 0). The ( n I , T ) phasediagram in the a = 0 scenario is shown in Fig. 2 (c). Thethermodynamic states below the dashed lines in Figs. 2(a) and (c) correspond to a non-zero density of the BC,i.e. to a macroscopic number of π + -mesons occupying the zero momentum level k = 0.In Scenario II, with both the repulsive ( b >
0) andattractive ( a >
0) pion interactions present, the FOPTphase transition takes place in addition to the BEC for-mation. The ( µ I , T ) and ( n I , T ) planes are presentedfor this scenario in Figs. 2 (b) and (d), respectively.The line of the FOPT is shown by a thick solid linein Fig. 2 (b). This line ends in a critical point (CP)at T = T c ≈ . m π , µ I = µ c ≈ . m π , and n I = n c ≈ . m π , which is shown by the greenstar. The µ c value can be expressed explicitly in termsof model parameters µ c = (cid:114) m π − a b . (23)An approximate analytical dependence of T c on a, b , and m π can also be obtained T c ≈ π √ b (cid:18) aζ (3 / (cid:19) / (cid:0) bm π − a (cid:1) − / . (24)Here ζ ( x ) is the Riemann zeta function. The relation (24)has been obtained assuming n − (cid:28) n (cid:28) n + as well asthe non-relativistic approximation in vicinity of the CP.Using the previously obtained parameters a and b fromfitting the lattice data one obtains T c ≈ . m π . Thisis within 6% of the numerical result obtained withoutapproximations. Note that the limit a → T c → µ c → m π .At the FOPT line in the ( µ I , T )-plane the pressuresof the gaseous and liquid phases are equal to each other.On the other hand, the isospin charge density n I has adiscontinuity. The mixed phase shown in Fig. 2 (d) isbounded by the gas-like (left) and liquid-like (right) bin-odals presented by solid lines that intersect each otherat the CP. The pion states inside the mixed phase cor-respond to linear combinations of the diluted (gaseous)and dense (liquid) states lying on the left and right bin-odals, respectively. The liquid component of the mixedphase always lies below the BEC line, thus it always con-tains a non-zero fraction of condensed π + -mesons. Thegaseous component, on the other hand, does not containthe BEC.A remarkable feature of the considered model is thatthe BEC line enters the mixed phase at the CP. Thisproperty of the model is robust with regard to variationsin the values of the a and b parameters. Another peculiarproperty is the non-smooth intersection of the left andright binodals at the CP.Scenarios I and II provide a similar picture of the phasediagram at T (cid:29) T c . At T (cid:46) T c , on the other hand, thedifferences are significant. We argue that these differ-ences can be most clearly seen by studying the behav- (cid:5) (cid:5) (cid:4) (cid:5) (cid:7) (cid:5) (cid:4) (cid:5) (cid:9) (cid:5) (cid:4) (cid:5) (cid:11)(cid:5)(cid:5) (cid:4) (cid:6)(cid:5) (cid:4) (cid:7)(cid:5) (cid:4) (cid:8)(cid:5) (cid:4) (cid:9)(cid:5) (cid:4) (cid:10)(cid:5) (cid:4) (cid:11) (cid:5) (cid:5) (cid:4) (cid:5) (cid:7) (cid:5) (cid:4) (cid:5) (cid:9) (cid:5) (cid:4) (cid:5) (cid:11) (cid:5) (cid:4) (cid:5) (cid:12)(cid:5) (cid:4) (cid:13) (cid:12) (cid:5) (cid:4) (cid:13) (cid:13) (cid:6) (cid:6) (cid:4) (cid:5) (cid:6)(cid:5)(cid:5) (cid:4) (cid:6)(cid:5) (cid:4) (cid:7)(cid:5) (cid:4) (cid:8)(cid:5) (cid:4) (cid:9)(cid:5) (cid:4) (cid:10)(cid:5) (cid:4) (cid:11) (cid:2) (cid:1) (cid:3)(cid:2) (cid:1) (cid:2) (cid:6)(cid:2) (cid:1) (cid:2) (cid:5)(cid:2) (cid:1) (cid:2) (cid:4) (cid:2) (cid:1) (cid:3) p m I (cid:1) (cid:2) p (cid:2) (cid:17) (cid:3) (cid:5) (cid:4) (cid:13) (cid:12) (cid:5) (cid:4) (cid:13) (cid:13) (cid:6) (cid:6) (cid:4) (cid:5) (cid:6) (cid:6) (cid:4) (cid:5) (cid:7) (cid:2) (cid:1) (cid:3) (cid:4)(cid:2) (cid:1) (cid:3)(cid:2) (cid:1) (cid:2) (cid:6)(cid:2) (cid:1) (cid:2) (cid:5)(cid:2) (cid:1) (cid:2) (cid:4) m I (cid:1) (cid:2) p (cid:3)(cid:3) (cid:2) (cid:3) (cid:5)(cid:3) (cid:2) (cid:3) (cid:6)(cid:3) (cid:2) (cid:3) (cid:7)(cid:3) (cid:2) (cid:3) (cid:8)(cid:3) (cid:2) (cid:4) (cid:3)(cid:3) (cid:2) (cid:4) (cid:5)(cid:3) (cid:2) (cid:4) (cid:6) (cid:3) I (cid:1) (cid:2) (cid:3) p (cid:2) (cid:18) (cid:3) (cid:1) (cid:4) (cid:2) (cid:2) (cid:1) (cid:3) p (cid:3) I (cid:1) (cid:2) (cid:1) p (cid:14) (cid:16) (cid:15) (cid:1) (cid:5) (cid:2) (cid:1) (cid:14) (cid:18)(cid:20) (cid:21) (cid:15) (cid:13) (cid:19)(cid:1) (cid:9) (cid:11) (cid:10) (cid:1) (cid:19)(cid:18)(cid:20) (cid:16)(cid:1) (cid:10) (cid:12)(cid:1) (cid:17) (cid:22) (cid:21) (cid:25) (cid:20) (cid:15) (cid:1) (cid:23) (cid:24) (cid:13) (cid:24) (cid:16) (cid:3) I (cid:1) (cid:2) (cid:1) p (cid:25) (cid:23) (cid:20) (cid:19) (cid:1) (cid:26) (cid:22) (cid:17) (cid:28) (cid:20) (cid:24) (cid:23) (cid:27) (cid:30) (cid:23) (cid:19) (cid:1)(cid:31) (cid:23) (cid:29) (cid:22) (cid:1) (cid:14) (cid:16) (cid:15)(cid:21) (cid:17) (cid:28) Figure 2. Phase diagrams of the pion matter in the ( µ I /m π , T /m π ) and ( n I /m π , T /m π ) planes, at a = 0 [(a) and (c)], and a > n I /m π . ior of isospin charge fluctuations. This is discussed inSec. III. III. FLUCTUATIONS
The two presented descriptions of the isospin chargedensity from the lattice results at T = 0 both containthe BEC. Within the second description, the FOPT at T < T c leads to the n I discontinuity. We argue that the (cid:4) (cid:3) (cid:12) (cid:11) (cid:4) (cid:3) (cid:12) (cid:12) (cid:5) (cid:5) (cid:3) (cid:4) (cid:5)(cid:4)(cid:4) (cid:3) (cid:5)(cid:4) (cid:3) (cid:6)(cid:4) (cid:3) (cid:7)(cid:4) (cid:3) (cid:8)(cid:4) (cid:3) (cid:9)(cid:4) (cid:3) (cid:10) (cid:1) (cid:13) (cid:2) (cid:3) (cid:1) (cid:5)(cid:3) (cid:2)(cid:4)(cid:2) (cid:1)(cid:5)(cid:3) (cid:4) (cid:2) (cid:1) (cid:3) p m I (cid:1) (cid:2) p (cid:4) (cid:3) (cid:12) (cid:11) (cid:4) (cid:3) (cid:12) (cid:12) (cid:5) (cid:5) (cid:3) (cid:4) (cid:5) (cid:5) (cid:3) (cid:4) (cid:6)(cid:1) (cid:14) (cid:2) (cid:3) (cid:1)(cid:5)(cid:4)(cid:3) (cid:2) (cid:1) (cid:5) (cid:4) (cid:3)(cid:2) m I (cid:1) (cid:2) p (cid:1)(cid:2)(cid:2) (cid:1) w Figure 3. The scaled variance of the isospin charge fluctuations of the pion matter in the ( µ I /m π , T /m π ) plane is shown for(a) the pure repulsion case (Scenario I) and (b) the full potential case (Scenario II). Colors show values of the scaled variance.Dashed lines correspond to the onset of the BEC. Black solid lines correspond to the first order phase transition. The CP ismarked by the green star. difference between the two scenarios can be probed byconsidering isospin charge fluctuations.In the GCE, the j -th order susceptibility of the isospincharge is determined by a j -th order partial derivative ofthe pressure p with respect to the chemical potential µ I : χ j = ∂ j ( p/T ) ∂ ( µ I /T ) j . (25)Ratios of susceptibilities given by (25) can be particularlyuseful as such quantities are intensive in the thermody-namic limit. Some of the most well known such quantitiesinclude the scaled variance ω , skewness Sσ , and kurtosis κσ (see, e.g., Ref. [55]): ω = χ χ , Sσ = χ χ , κσ = χ χ . (26)Using Eq. (25) together with Eq. (11) the scaled vari-ance ω can be written as ω = Tn I (cid:18) ∂n I ∂µ I (cid:19) T . (27)In the ideal pion gas the scaled variance diverges at the BEC line [8], i.e., ω id → ∞ at µ I → m π − ω id = T m / π √ πn I ( m π − µ I ) − / → ∞ . (28)Due to the repulsive interactions in the considered modelthe scaled variance remains finite. On the BEC line, M = µ I , one finds: ω = M Tn I (cid:18) ∂M∂σ (cid:19) − = µ I Tn I (cid:112) b ( µ I − µ c ) , (29)where µ c is given by Eq. (23). The value of ω remainsalso finite in a presence of the BC, n bc+ >
0. In Sce-nario II ( a > ω exhibits singular behavior at the CP, T = T c , µ I = µ c , where it diverges. A systematic ex-pansion of the thermodynamic functions in a vicinity ofthe CP allows to obtain the critical exponents. We ex-pect that the critical exponents of the considered systemare different from those in the mean-field class universal-ity. This is due to a presence of the two order parameters, n lI − n gI > n bc+ >
0, which disappear simultaneouslyat the CP (see, e.g., Ref. [56]). A detailed discussion ofthis subject is however outside of the scope of the presentstudy.
Scaled variance.
The behavior of the scaled variance ω in the plane of temperature and isospin chemical po- (cid:4)(cid:4) (cid:3) (cid:5)(cid:4) (cid:3) (cid:6)(cid:4) (cid:3) (cid:7)(cid:4) (cid:3) (cid:8)(cid:4) (cid:3) (cid:9)(cid:4) (cid:3) (cid:10) (cid:1) (cid:13) (cid:2) (cid:1) (cid:4) (cid:2) (cid:1) (cid:3) (cid:2) (cid:1) (cid:5)(cid:6) (cid:2)(cid:4) (cid:2)(cid:3) (cid:2) (cid:2) (cid:1) (cid:3) p (cid:5) (cid:1) (cid:14) (cid:2) (cid:1) (cid:5)(cid:1) (cid:6) (cid:2) (cid:1) (cid:4) (cid:2)(cid:6) (cid:2)(cid:4) (cid:2)(cid:3) (cid:2) (cid:5) (cid:2)(cid:3)(cid:3) (cid:2)(cid:3) (cid:2) (cid:2) (cid:1) s (cid:1) (cid:3) (cid:2) (cid:1) (cid:3)(cid:1) (cid:3) (cid:2)(cid:1) (cid:3) (cid:2) (cid:2) (cid:4) (cid:3) (cid:12) (cid:11) (cid:4) (cid:3) (cid:12) (cid:12) (cid:5) (cid:5) (cid:3) (cid:4) (cid:5)(cid:4)(cid:4) (cid:3) (cid:5)(cid:4) (cid:3) (cid:6)(cid:4) (cid:3) (cid:7)(cid:4) (cid:3) (cid:8)(cid:4) (cid:3) (cid:9)(cid:4) (cid:3) (cid:10) (cid:3) (cid:2) (cid:2) (cid:3) (cid:2) (cid:1) (cid:3) (cid:2) (cid:1) (cid:3) (cid:2) (cid:2)(cid:3) (cid:2) (cid:3) (cid:2) (cid:2)(cid:3) (cid:2) (cid:2) (cid:1) (cid:3) p m I (cid:1) (cid:2) p (cid:1) (cid:15) (cid:2) (cid:4) (cid:3) (cid:12) (cid:11) (cid:4) (cid:3) (cid:12) (cid:12) (cid:5) (cid:5) (cid:3) (cid:4) (cid:5) (cid:5) (cid:3) (cid:4) (cid:6) (cid:3) (cid:2) (cid:2) (cid:3) (cid:2) (cid:2) (cid:3) (cid:2) (cid:1) (cid:3) (cid:2) (cid:1) (cid:1) (cid:16) (cid:2) (cid:3) (cid:2) (cid:2)(cid:3) (cid:2) (cid:3) (cid:2) (cid:2) m I (cid:1) (cid:2) p (cid:2)(cid:3)(cid:3) (cid:2)(cid:3) (cid:2) (cid:2)(cid:3) (cid:2) (cid:2) (cid:2)(cid:3) (cid:2) (cid:2) (cid:2) (cid:2) k s (cid:3) (cid:2) Figure 4. The skewness (a),(b) and kurtosis (c),(d) of the isospin charge fluctuations in ( µ I /m π , T /m π ) are shown for (a),(c)the pure repulsion case (Scenario I) and (b),(d) the full potential case (Scenario II). tential is shown in Fig. 3. In Scenario I (pure repul-sion), ω is a continuous function, in particular across theBEC boundary [see Fig. 3 (a)]. In Scenario II (full po-tential), on the other hand, ω exhibits a jump disconti-nuity over the FOPT line an becomes divergent at theCP [Fig. 3 (b)]. It is still a continuous function acrossthe BEC line, however.Note that in the limit a → T c → µ c → m π . Therefore, the point µ I = m π at zero temperature retains some of the propertires of the CP and exhibits large fluctuations in its vicin- ity. One can observe ω of any magnitude in the vicinityof this point, the exact magnitude depending on the pathof approach. In particular, approaching this point alongthe BEC line one finds ω → ∞ at T → Skewness.
The skewness, Sσ , for Scenario I ( a = 0)and II ( a >
0) is shown in Figs. 4 (a) and (b), respec-tively. At small µ I (cid:28) m π values, where the pion densitiesare small, both the pion interactions and Bose statisticseffects can be neglected, thus, Sσ ≈
1. The skewnessattains positive values in those regions of the phase dia-gram where there is no BC. Sσ is discontinuous along theBEC line, jumping from positive values outside the BECphase to negative values in the phase with a BC. Theabove observations are valid for both scenarios. In Sce-nario II ( a > Sσ shows singular behavior at the CP.The skewness can reach both −∞ and + ∞ at the CPdepending on the path of approach. When crossing theFOPT in Scenario II Sσ undergoes a jump discontinuity. Kurtosis. κσ presented in Fig. 4. In both the sce-narios it also always attains positive values everywhere onthe phase diagram. Kurtosis can strongly deviate fromthe baseline κσ = 1 of an ideal Boltzmann gas. Thisis due to the presence of interactions and Bose statistics.The largest values of the kurtosis are generally obtainedin the vicinity of the BEC line. The kurtosis exhibits anon-monotonic behavior as a function of µ I at both theBEC-line and the FOPT-line, where it jumps down as µ I is increased. κσ is an increasing function of µ I else-where on the phase diagram. The values of κσ remainlarge even far away from the CP and the Bose conden-sation boundary. This is due to its large sensitivity tointeractions in the system. κσ diverges at the CP. Themodel does not predict negative values of κσ anywhereon the phase diagram. This is in contrast to the univer-sal behavior of fluctuations in the Ising model [57, 58],as well as various model calculations [59–63], where neg-ative values of κσ are observed in the so-called analyticcrossover region above the critical temperature. In thepresent work the negative values of κσ are not observedbecause of the Bose-Einstein condensation. The BEC-line, which itself corresponds to a phase transition of ahigher order, crosses the CP, thus no region in the vicin-ity of the CP can be identified as an analytic crossover.We would like note that κσ exhibits a singular behav-ior also in Scenario I ( a = 0), at a point ( T = 0 , µ I = m π ) [see Fig. 4 (c)]. Approaching this point along theBEC line one finds κσ → ∞ at T → IV. SUMMARY
We studied thermodynamic properties of interactingpion matter in the framework of a mean-field model witha φ - φ type Lagrangian. The phase structure has beenstudied at non-zero isospin chemical potential µ I thatcorresponds to the conserved 3rd component of isospin.Parameters of the repulsive and attractive interactionswere fixed using lattice QCD data on the isospin den-sity as the function of the chemical potential µ I at zerotemperature. The lattice data can be reasonably fittedwithin the two qualitatively different scenarios: Scenario I with only repulsive interactions, and Scenario II withboth repulsive and attractive interactions. In both sce-narios a phase with a Bose condensate of π + pions wasfound to occur at sufficiently large µ I , the transition be-tween ordinary pion matter and matter with a BC takingplace along the so-called BEC lines. The presence of theattractive interactions in Scenario II leads, in addition tothe BEC, also to a first order liquid-gas phase transitionof pions with a CP at T c ≈ . m π and µ c ≈ . m π .A notable qualitative feature of the model, present for abroad range of values of parameters a and b . is the factthat the BEC line merges with the FOPT line at the CP.The system is characterized by two order parameters: (i)the difference n l − n g > n bc+ of the Bose condensed π + pionsthat characterizes the BEC transition. This makes themodel qualitatively different from the usual systems witha CP and FOPT where only a single order parameter ispresent.The susceptibilities of isospin charge fluctuations upto the 4-th order studied in the paper can serve as arobust observable to distinguish between the two differ-ent scenarios. In the both scenarios, the scaled vari-ance ω = χ /χ , skewness Sσ = χ /χ , and kurtosis κσ = χ /χ remain finite on the BEC line. This hap-pens due to the repulsive interactions in the pion system,in contrast to the ideal pion gas where these measures be-come infinite on the BEC line. All three fluctuation mea-sures demonstrate anomalous properties approaching theCP: ω → ∞ , κσ → ∞ , and Sσ can reach both + ∞ and −∞ depending on a path to the CP. Note a significance ofthe higher order susceptibilities (e.g., skewness and kur-tosis fluctuation measures) which are highly sensitive toa presence of the CP. In the scenario I the CP is absent.In this case the anomalous fluctuations take place in thepoint T = 0 and µ I = m π . Approaching this point: ω and κσ can reach any value from 0 to ∞ , and Sσ canreach any value between −∞ and + ∞ depending on apath to the ( µ I = m π , T = 0)–point. The susceptibili-ties χ i can be computed in lattice QCD which are freeof sign problem at finite µ I . Analysis of their behaviorcan be used to establish a point (region) of anomalouslylarge fluctuations. Determining whether this point corre-sponds to zero or finite temperatures will allow to distin-guish Scenarios I ( T = 0) and II ( T = T c > ω near T = 0 in Scenario I and near T = T c inScenario II: ω → ∞ at T → T c in Scenario II, and ω canreach any value from 0 to ∞ depending on the path ofapproaching to T = 0 in Scenario I.The results obtained in this paper can be used in sys-tems where the pion densities are large and a BC of pionsmay occur. This can happen, for example, in heavy-ion0or proton-proton collisions where the pion condensationmay occur as a chemical non-equilibrium effect. Otherpossibilities include pion stars as well as the Early Uni-verse which may have passed through a pion-condensedphase if the lepton flavor asymmetries during its evolu-tion were large. ACKNOWLEDGMENTS
We are grateful to D.V. Anchishkin, I.N. Mishustin,L.M. Satarov, and H. Stoecker for fruitful discussions.This work is supported by the Target Program of Funda-mental Research of the Department of Physics and As-tronomy of the National Academy of Sciences of Ukraine(N 0120U100857). The work of O.S.St. was partially sup-ported by the National Research Foundation of Ukraineunder Project No. 2020.02/0073. [1] S. N. Bose, Z. Phys. , 178 (1924).[2] A. Einstein, Kgl. Preuss. Akad. Wiss , 137 (1925).[3] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E.Wieman, and E. A. Cornell, Science , 198 (1995).[4] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G.Hulet, Phys. Rev. Lett. , 1687 (1995).[5] K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. vanDruten, D. S. Durfee, D. M. Kurn, and W. Ketterle,Phys. Rev. Lett. , 3969 (1995).[6] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari,Rev. Mod. Phys. , 463 (1999).[7] L. Satarov, M. Gorenstein, A. Motornenko,V. Vovchenko, I. Mishustin, and H. Stoecker, J.Phys. G , 12 (2017), arXiv:1704.08039 [nucl-th].[8] V. Begun and M. I. Gorenstein, Phys. Lett. B , 190(2007), arXiv:hep-ph/0611043.[9] V. Begun and M. Gorenstein, Phys. Rev. C , 064903(2008), arXiv:0802.3349 [hep-ph].[10] G. C. Strinati, P. Pieri, G. R¨opke, P. Schuck, and M. Ur-ban, Physics Reports , 1 (2018), the BCS–BECcrossover: From ultra-cold Fermi gases to nuclear sys-tems.[11] P. Nozieres and S. Schmitt-Rink, J. Low. Temp. Phys. , 195 (1985).[12] Y. Funaki, T. Yamada, H. Horiuchi, G. R¨opke, P. Schuck,and A. Tohsaki, Phys. Rev. Lett. , 082502 (2008).[13] P.-H. Chavanis and T. Harko, Phys. Rev. D86 , 064011(2012), arXiv:1108.3986 [astro-ph.SR].[14] I. Mishustin, D. Anchishkin, L. Satarov, O. Stashko,and H. Stoecker, Phys. Rev. C , 022201 (2019),arXiv:1905.09567 [nucl-th].[15] L. E. Padilla, J. A. V´azquez, T. Matos, and G. Germ´an,Journal of Cosmology and Astroparticle Physics ,056–056 (2019).[16] J. I. Kapusta and C. Gale,
Finite-Temperature Field The-ory: Principles and Applications , 2nd ed., CambridgeMonographs on Mathematical Physics (Cambridge Uni-versity Press, 2006).[17] J. O. Andersen, Rev. Mod. Phys. , 599 (2004),arXiv:cond-mat/0305138 [cond-mat].[18] A. Griffin, D. W. Snoke, and S. Stringari, Bose-einsteincondensation (Cambridge University Press, 1996).[19] S. Watabe and Y. Ohashi, Phys. Rev. A , 053633(2013). [20] S. Watabe, Acta Phys. Polon. A , 1222–1230 (2019).[21] G. Baym, J.-P. Blaizot, M. Holzmann, F. Lalo¨e, andD. Vautherin, Phys. Rev. Lett. , 1703 (1999).[22] G. Baym, J.-P. Blaizot, and J. Zinn-Justin, EPL , 150(2000).[23] M. Holzmann and W. Krauth, Phys. Rev. Lett. , 2687(1999), arXiv:cond-mat/9905198 [cond-mat.stat-mech].[24] M. Holzmann, G. Baym, J.-P. Blaizot, and F. Laloe,Phys. Rev. Lett. , 120403 (2001), arXiv:cond-mat/0103595 [cond-mat.stat-mech].[25] K. Huang, Phys. Rev. Lett. , 3770 (1999).[26] D. Son and M. A. Stephanov, Phys. Rev. Lett. , 592(2001), arXiv:hep-ph/0005225.[27] H. Abuki, T. Brauner, and H. J. Warringa, Eur. Phys.J. C , 123 (2009), arXiv:0901.2477 [hep-ph].[28] B. Brandt, G. Endrodi, and S. Schmalzbauer, Phys. Rev.D , 054514 (2018), arXiv:1712.08190 [hep-lat].[29] B. B. Brandt, G. Endrodi, E. S. Fraga, M. Hippert,J. Schaffner-Bielich, and S. Schmalzbauer, Phys. Rev.D , 094510 (2018), arXiv:1802.06685 [hep-ph].[30] V. Vovchenko, B. B. Brandt, F. Cuteri, G. Endr˝odi,F. Hajkarim, and J. Schaffner-Bielich, Phys. Rev. Lett. , 012701 (2021), arXiv:2009.02309 [hep-ph].[31] M. Mannarelli, Particles , 411–443 (2019).[32] J. O. Andersen and P. Kneschke, “Bose-einstein conden-sation and pion stars,” (2018), arXiv:1807.08951 [hep-ph].[33] V. Begun and W. Florkowski, Phys. Rev. C , 054909(2015), arXiv:1503.04040 [nucl-th].[34] F. E. Schunck and E. W. Mielke, Classical and QuantumGravity , R301–R356 (2003).[35] S. L. Liebling and C. Palenzuela, Living Reviews in Rel-ativity , 5 (2017).[36] E. Braaten, A. Mohapatra, and H. Zhang, Phys-ical Review Letters (2016), 10.1103/phys-revlett.117.121801.[37] A. Su´arez, V. H. Robles, and T. Matos, AcceleratedCosmic Expansion , 107–142 (2013).[38] T. Bernal, V. H. Robles, and T. Matos, Monthly No-tices of the Royal Astronomical Society , 3135–3149(2017).[39] L. Visinelli, Journal of Cosmology and AstroparticlePhysics , 009–009 (2016).[40] S. HajiSadeghi, S. Smolenski, and J. Wudka, Physical Review D (2019), 10.1103/physrevd.99.023514.[41] J. Barranco and A. Bernal, Phys. Rev. D , 043525(2011), arXiv:1001.1769 [astro-ph.CO].[42] A. Gavrilik, M. Khelashvili, and A. Nazarenko, PhysicalReview D (2020), 10.1103/physrevd.102.083510.[43] P. Adhikari and J. O. Andersen, Phys. Lett. B ,135352 (2020), arXiv:1909.01131 [hep-ph].[44] P. Adhikari, J. O. Andersen, and M. A. Mojahed,(2020), arXiv:2010.13655 [hep-ph].[45] L.-y. He, M. Jin, and P.-f. Zhuang, Phys. Rev. D ,116001 (2005), arXiv:hep-ph/0503272.[46] P. Adhikari, J. O. Andersen, and P. Kneschke, Phys.Rev. D , 074016 (2018), arXiv:1805.08599 [hep-ph].[47] A. Folkestad and J. O. Andersen, Phys. Rev. D ,054006 (2019), arXiv:1810.10573 [hep-ph].[48] D. Anchishkin, I. Mishustin, and H. Stoecker, J. Phys.G , 035002 (2019), arXiv:1806.10857 [nucl-th].[49] O. S. Stashko, D. V. Anchishkin, O. V. Savchuk, andM. I. Gorenstein, “Thermodynamic properties of inter-acting bosons with zero chemical potential,” (2020),arXiv:2007.06321 [hep-ph].[50] O. Savchuk, Y. Bondar, O. Stashko, R. V. Poberezhnyuk,V. Vovchenko, M. I. Gorenstein, and H. Stoecker, Phys.Rev. C , 035202 (2020), arXiv:2004.09004 [hep-ph].[51] L. M. Satarov, R. V. Poberezhnyuk, I. N. Mishustin, andH. Stoecker, (2020), arXiv:2009.13487 [nucl-th].[52] S. Borsanyi, Z. Fodor, C. Hoelbling, S. D. Katz,S. Krieg, and K. K. Szabo, Phys. Lett. B , 99 (2014),arXiv:1309.5258 [hep-lat]. [53] A. Bazavov et al. (HotQCD), Phys. Rev. D , 094503(2014), arXiv:1407.6387 [hep-lat].[54] V. Vovchenko, D. V. Anchishkin, and M. I. Gorenstein,Phys. Rev. C , 024905 (2015), arXiv:1412.5478 [nucl-th].[55] F. Karsch and K. Redlich, Phys. Lett. B , 136 (2011),arXiv:1007.2581 [hep-ph].[56] I. P. Ivanov, Phys. Rev. E , 021116 (2009).[57] M. A. Stephanov, Phys. Rev. Lett. , 052301 (2011),arXiv:1104.1627 [hep-ph].[58] A. Bzdak, V. Koch, and N. Strodthoff, Phys. Rev. C ,054906 (2017), arXiv:1607.07375 [nucl-th].[59] V. Vovchenko, D. Anchishkin, M. Gorenstein, andR. Poberezhnyuk, Phys. Rev. C , 054901 (2015),arXiv:1506.05763 [nucl-th].[60] V. Vovchenko, R. V. Poberezhnyuk, D. V. Anchishkin,and M. I. Gorenstein, J. Phys. A , 015003 (2016),arXiv:1507.06537 [nucl-th].[61] J.-W. Chen, J. Deng, H. Kohyama, and L. Labun, Phys.Rev. D , 034037 (2016), arXiv:1509.04968 [hep-ph].[62] A. Mukherjee, J. Steinheimer, and S. Schramm, Phys.Rev. C , 025205 (2017), arXiv:1611.10144 [nucl-th].[63] A. Motornenko, J. Steinheimer, V. Vovchenko,S. Schramm, and H. Stoecker, Phys. Rev. C101