Finite size effects in strongly interacting matter at zero chemical potential from Polyakov loop Nambu-Jona-Lasinio model in the light of lattice data
EEur. Phys. J. C manuscript No. (will be inserted by the editor)
Finite size effects in strongly interacting matterat zero chemical potential from Polyakov loopNambu-Jona-Lasinio model in the light of lattice data
A. G. Grunfeld a,1,2 , G. Lugones b,3 CONICET, Godoy Cruz 2290, (C1425FQB) Buenos Aires, Argentina. Departamento de F´ısica, Comisi´on Nacional de Energ´ıa At´omica, Av.Libertador 8250, (1429) Buenos Aires, Argentina. Universidade Federal do ABC, Centro de Ciˆencias Naturais e Humanas, Avenida dos Estados 5001- Bang´u, CEP 09210-580,Santo Andr´e, SP, Brazil.Received: date / Accepted: date
Abstract
We study finite volume effects within thePolyakov loop Nambu-Jona-Lasinio model for two lightand one heavy quarks at vanishing baryon chemical po-tential and finite temperatures. We include three differ-ent Polyakov loop potentials and ensure that the pre-dictions of our effective model in bulk are compatiblewith lattice QCD results. Finite size effects are takeninto account by means of the Multiple Reflection Ex-pansion formalism. We analyze several thermodynamicquantities including the interaction measure, the speedof sound, the surface tension, and the curvature energyand find that they are sensitive to finite volume effects,specially for systems with radii below ∼
10 fm andtemperatures around the crossover one. For all sizes,the system undergoes a smooth crossover. The chiralcritical temperature decreases by around 5% and thedeconfinement temperature by less than a 2% whenthe radius goes from infinity to 3 fm. Thus, as thedrop’s size decreases, both temperatures become closer.The surface tension is dominated by the contributionof strange quarks and the curvature energy by u and d quarks. At large temperatures both quantities growproportionally to T / but saturate to a constant valueat low T . Keywords
Quark deconfinement · Chirality: particlephysics · Phase transitions in finite-size systems · Lattice QCD calculations
PACS · · · a e-mail: [email protected] b e-mail: [email protected] Understanding the hadron-quark phase transition is stilla challenge from both the theoretical and experimentalpoints of view. The framework for describing it is providedby Quantum Chromodynamics (QCD), which is thefundamental theory of strong interactions. However, thenonperturbative character of QCD at low energies makesextremely difficult to solve it in the regime of interme-diate temperatures and chemical potentials, althoughlattice methods had a huge progress in the last years [1–5]. In this context, effective models such as the Nambu-Jona-Lasinio (NJL) model [6–9] are very useful becausethey can address many aspects of the QCD phase dia-gram without computational shortcomings at finite chem-ical potentials. The NJL model has many similaritieswith the full QCD theory but does not take into ac-count the property of confinement, since quarks interacteach other via pointlike interactions without exchangedgluons. Thus, in order to obtain a more realistic descrip-tion, taking into account the quark confinement at lowenergies, the Polyakov loop was introduced in the NJLmodel [10], leading to the so called Polyakov loop NJL(PNJL) model (see also [11]). From this widely stud-ied effective QCD model, many properties of stronglyinteracting matter can be obtained, such as its phasediagram [12–14].On the other hand, a comprehension of finite sizeproperties is very important for situations where thedeconfinement transition occurs over a finite volume asin relativistic heavy ion collisions and neutron stars.The strongly interacting matter formed in a heavy-ioncollision is finite in volume, and its size depends onthe size of the colliding nuclei, the collision center ofmass energy, and the centrality of the collision. In neut- a r X i v : . [ h e p - ph ] A ug ron stars, a deconfinement transition to quark matter ispossible and a hybrid star or a strange quark star can beformed. The conversion of the star is expected to startwith the nucleation of small quark matter drops [15–18]which subsequently grow at the expenses of the gravita-tional energy extracted from the contraction of the ob-ject and/or through a strongly exothermic combustionprocess. Quark matter droplets with a variety of geo-metrical forms can also arise within the mixed hadron-quark phase that is expected to form inside hybridstars if global charge neutrality is allowed [19]. Also,the most external layers of a strange star may fragmentinto a charge-separated mixture, involving positively-charged strange droplets (strangelets) immersed in anegatively charged sea of electrons, forming a crystal-line solid crust [20].In the past years, many theoretical studies of finite-volume effects have been performed based on the NJLmodel [21–23]. However, studies within the PNJL modelare more recent [24–26]. To incorporate finite-size ef-fects different procedures have been employed, such asMonte Carlo simulations [24], a renormalization groupapproach [27], and the implementation of a low mo-mentum cutoff Λ on the integration of the thermody-namic potential density of the PNJL model [28].In the present work we use a different approach forthe inclusion of finite size effects, known as MultipleReflection Expansion (MRE) formalism [29]. First, dif-ferent thermodynamic quantities calculated within oureffective model in the bulk (including three differentPolyakov loop potentials) are compared to the corres-ponding lattice QCD results. This is necessary as astarting point to check the validity of our model. Then,we study the relevance of finite size effects on manyproperties of strongly interacting matter and analyzehow they deviate from the bulk case.A comparison with lattice QCD is always import-ant to calibrate effective models, that can be later ex-trapolated to a higher density regime. For example, theeffective model can be used to explore finite-volume ef-fects in a regime where they are known to be essential,such as in relativistic heavy ion collisions. Addition-ally, some results could be of interest for the analysisof the cosmological quark-hadron transition, which oc-curred in the early Universe about 10 µ s after the BigBang, when a hot unconfined quark-gluon plasma wasconverted, as the Universe expanded and cooled, into aconfined hadronic phase.The paper is organized as follows. In Sec. 2 we re-view the PNJL model in bulk for different Polyakovloop potentials and in Sec. 3 we introduce finite sizeeffects through the MRE formalism. Our results arepresented in Sec. 4 where we analyze the behavior of several thermodynamic quantities such as the chiralcritical temperature, the deconfinement temperature,the constituent masses, the interaction measure, thepressure, the energy density, the entropy density, thespeed of sound, the surface tension, and the curvatureenergy for different system sizes. Finally, we present ourconclusions in Sec. 5. The Lagrangian of the Polyakov loop extended SU (3) f NJL model including the six-quark ’t Hooft interactionreads L = ¯ q ( iD/ − ˆ m ) q + g S N f − (cid:88) a =0 (cid:2) (¯ qλ a q ) + (¯ qiγ λ a q ) (cid:3) + g D [det (¯ q (1 − γ ) q ) + det (¯ q (1 + γ ) q )] − U ( l, ¯ l ; T ) , (1)where q = ( u, d, s ) represents the three flavor quarkfield with three colors and ˆ m = diag( m u , m d , m s ) standsfor the current quark mass matrix. We assume the SU (2) V isospin symmetry limit in which m u = m d . The cov-ariant derivative in the fermion kinetic term couples atemporal background gauge field, the Polyakov loop, tothe quark fields through D µ = ∂ µ − iA µ with A µ = δ µ A in Polyakov gauge, and A = − iA . Here, we used thenotation A µ = gA aµ λ a / g the SU (3) c gauge coup-ling. The λ a s stand for the Gell-Mann matrices with λ = (cid:112) / in flavor space. The four-quark interac-tion coupling in the (pseudo)scalar channel is denotedby g S and the six-quark ’t Hooft interaction coupling,induced by instantons, is labeled by g D . The latter onebreaks the axial U A (1) symmetry. Finally, the aboveLagrangian includes an effective potential U ( l, ¯ l ; T ) thataccounts for gauge field self-interactions and is a func-tion of the temperature T and the normalized color-traced Polyakov loop expectation value and its Her-mitean conjugate, defined by l = (cid:104) tr c L (cid:105) /N c , ¯ l = (cid:104) tr c L † (cid:105) /N c , (2)where the Polyakov loop L is an N c × N c matrix incolor space, as a function of A . The explicit form ofthe Polyakov loop potential U ( l, ¯ l ; T ) will be discussedin Sec. 2.2.2.1 The thermodynamic potentialDifferent thermodynamic properties of our model canbe obtained from the thermodynamic potential in themean-field approximation (MFA). The thermodynamicgrand potential Ω ( T, µ ) of the PNJL model in the MFA has been largely considered in the literature, see e.g. [30,31]. Based on [30] we write the thermodynamic grandpotential per unit volume as follows Ω P NJL = Ω cond + Ω zero + Ω quark − Ω vac + U ( l, ¯ l ; T ) . (3)The first term is the condensation energy, that con-tains the contribution of the scalar four-quark inter-action proportional to g S plus the six-quark ’t Hooftinteraction, proportional to g D . In the MFA this termdepends on the three condensates (cid:104) ¯ uu (cid:105) , (cid:104) ¯ dd (cid:105) and (cid:104) ¯ ss (cid:105) as follows Ω cond = g S (cid:2) (cid:104) ¯ uu (cid:105) + (cid:104) ¯ dd (cid:105) + (cid:104) ¯ ss (cid:105) (cid:3) + 4 g D (cid:104) ¯ uu (cid:105)(cid:104) ¯ dd (cid:105)(cid:104) ¯ ss (cid:105) . (4)The zero point energy Ω zero = − N c (cid:90) Λ (cid:88) i d p (2 π ) (cid:15) i ( p ) (5)is clearly divergent. Since the PNJL model is non-renor-malizable, the zero-point energy contribution requiresan ultraviolet cutoff Λ . The quark quasiparticle energiesare denoted by (cid:15) i ( p ) = (cid:112) p + M i where the constitu-ent quark masses M i for flavors i = u, d, s are: M u = m u − g S (cid:104) ¯ uu (cid:105) − g D (cid:104) ¯ dd (cid:105)(cid:104) ¯ ss (cid:105) , (6) M d = m d − g S (cid:104) ¯ dd (cid:105) − g D (cid:104) ¯ ss (cid:105)(cid:104) ¯ uu (cid:105) , (7) M s = m s − g S (cid:104) ¯ ss (cid:105) − g D (cid:104) ¯ uu (cid:105)(cid:104) ¯ dd (cid:105) , (8)being m i the current quark masses.The Ω quark term is ultraviolet finite and hence nomomentum cutoff is imposed on it. It contains the coup-ling between the chiral condensates and the Polyakovloop L , and reads [30]: Ω quark = − T (cid:88) i (cid:90) d p (2 π ) ln det[1 + Le − (cid:15)i − µiT ] − T (cid:88) i (cid:90) d p (2 π ) ln det[1 + L † e − (cid:15)i + µiT ] . (9)As shown in [30], taking an average of the 3 × Ω quark = − T (cid:88) i (cid:90) d p (2 π ) [ln (cid:104) det f − i (cid:105) + ln (cid:104) det f + i (cid:105) ] , (10)where (cid:104) det f − i (cid:105) = 1 + e − (cid:15) i − µ i ) /T + 3 le − ( (cid:15) i − µ i ) /T +3¯ le − (cid:15) i − µ i ) /T , (11) (cid:104) det f + i (cid:105) = 1 + e − (cid:15) i + µ i ) /T + 3¯ le − ( (cid:15) i + µ i ) /T +3 le − (cid:15) i ( p )+ µ i ) /T . (12) The fourth contribution in Eq. (3) is a constant Ω vac ≡ − P vac , which is usually introduced in orderto obtain a vanishing pressure at vanishing temperat-ure and chemical potential. We will discuss the proced-ure for fixing P vac and its effect on the thermodynamicquantities in the next section.Finally, the term U ( l, ¯ l ; T ) in Eq. (3), represents thepure gluonic effective potential in terms of the Polyakovloop variables, which will be presented below in detail.Notice that the U ( l, ¯ l ; T ) potential and Ω quark are in-variant under the simultaneous exchange of l ↔ ¯ l to-gether with − µ i ↔ + µ i . Let us remark that for threequark flavors the thermodynamic grand potential Ω ( T, µ i )generally depends on three independent quark chem-ical potentials µ i . As a consequence of the isospin sym-metry, the light quark chemical potentials are also de-generated. In the present work, we consider quark mat-ter to be symmetric and define a common chemical po-tential µ ≡ µ u = µ d = µ s . Moreover, since we want tocompare our results in the bulk with lattice QCD res-ults we will work at finite temperature and vanishingchemical potential.In order to obtain the dependence of the order para-meters on the temperature and the chemical potential,one has to solve the following set of coupled equations: ∂Ω P NJL ∂ (cid:104) ¯ uu (cid:105) = ∂Ω P NJL ∂ (cid:104) ¯ dd (cid:105) = ∂Ω P NJL ∂ (cid:104) ¯ ss (cid:105) = 0 , (13) ∂Ω P NJL ∂l = ∂Ω P NJL ∂ ¯ l = 0 . (14)These conditions are consequences from the fact thatthe thermodynamically consistent solutions correspondto the stationary points of Ω P NJL with respect to (cid:104) ¯ uu (cid:105) , (cid:104) ¯ dd (cid:105) , (cid:104) ¯ ss (cid:105) , l and ¯ l .2.2 Polyakov loop potentialsThe choice of the effective Polyakov loop potential U isnot unique. In general, it can be constructed from thecenter symmetry of the pure-gauge sector. The requiredparameters can be extracted from pure gauge latticedata at µ = 0 [32]. Among several possible choices, seee.g. [33], we will use the following effective Polyakovloop potentials:(i) Logarithmic potential : the logarithmic ansatz presen-ted in [33] is: U L T = − a ( T )2 l ¯ l + b ( T ) ln[1 − l ¯ l − l ¯ l ) +4( l + ¯ l )] , (15)where a ( T ) and b ( T ) are defined by [34]: a ( T ) = a + a ( T /T ) + a ( T /T ) , (16) b ( T ) = b ( T /T ) , (17) with a = 3 . a = − . a = 15 . b = − . Polynomial potential : Another choice is [31]: U P T = − b ( T )2 l ¯ l − b l + ¯ l ) + b l ¯ l ) , (18)where b ( T ) = a + a ( T /T ) + a ( T /T ) + a ( T /T ) , (19)with a = 6 . a = − . a = 2 . a = − . b = 0 .
75 and b = 7 .
5. In the absenceof dynamical quarks, in a pure gauge sector, oneexpects a deconfinement temperature T = 270MeV. Nevertheless, in [35] it has been shown thatin the presence of two light dynamical quarks anda massive strange one, this temperature is rescaledto about 187 MeV, with an uncertainty of about30 MeV. In fact, for N f = 2 + 1, T = 187 MeVand T = 190 MeV have been used in [33] and in[25] respectively. Here we use T = 185 MeV.(iii) Fukushima potential : Finally, we will use the strong-coupling inspired version of the effective Polyakovpotential with only two parameters a and b pro-posed by Fukushima [30] U F T = − bT [ 54 e − a/T l ¯ l + ln { − l ¯ l − l ¯ l ) + 4( l + ¯ l ) } ] . (20)The first term (proportional to l ¯ l ) reminds thenearest neighbor interaction in the effective actionat strong coupling and its temperature-dependentcoefficient controls the deconfinement phase trans-ition temperature. The logarithmic term comesfrom the Haar measure of the group integrationwith respect to the SU(3) Polyakov loop matrix.The parameters a and b are independent of thetemperature, the chemical potential and the num-ber of quark flavors N f . The parameter a con-trols only the deconfinement transition temperat-ure and can be determined by the condition thatthe first-order phase transition in pure gluody-namics takes place at T = 270 MeV, which resultsin a = 664 MeV. On the other hand, the para-meter b can be used to control the relative value ofthe deconfinement and chiral restoration crossovertemperatures. Since there is no established pre-scription for fixing b , we shall adopt here two dif-ferent values. First, we consider b = (196 . as suggested in [30, 33] leading to an almost sim-ultaneous crossover for deconfinement and chiralrestoration at a temperature of T (cid:39)
200 MeV(we call this case U F ). The second choice is b =(115 MeV) (we call this case U F ) which giveslower critical temperatures as we will see below. 2.3 ParametrizationIn order to fully specify the non-local model under con-sideration we fix the model parameters following Ref.[36]. For comparison with some recent results [37], wehave considered the parameters in [9], m u = m d = 5 . m s = 135 . Λ = 631 . g S · Λ = 3 . g D · Λ = − . Now we are ready to introduce the effects of finite size inthe thermodynamic potential. For doing so we considerthe MRE formalism (see Refs. [29, 38–40] and refer-ences therein) which takes into account the modifica-tion in the density of states resulting when the systemis restricted to a finite domain. For the case of a finitespherical droplet the density of states reads: ρ i, MRE ( p, m i , R ) = 1 + 6 π pR f i,S + 12 π ( pR ) f i,C (21)where the surface contribution to the density of statesis f i,S = − π (cid:18) − π arctan pm i (cid:19) , (22)and the curvature contribution is given by Madsen’sansatz [38] f i,C = 112 π (cid:20) − p m i (cid:18) π − arctan pm i (cid:19)(cid:21) , (23)which takes into account the finite quark mass contri-bution.The MRE density of states for massive quarks isreduced compared with the bulk one, and for a rangeof small momenta becomes negative. This non-physicalnegative values are removed by introducing an infrared(IR) cutoff in momentum space [40]. Thus, we have toperform the following replacement in order to obtainthe thermodynamic quantities (cid:90) Λ, ∞ · · · d p (2 π ) −→ (cid:90) Λ, ∞ Λ i, IR · · · ρ i, MRE d p (2 π ) . (24)The upper integration limit is either infinity or given bya cutoff Λ . The IR cut-off Λ i, IR is the largest solution ofthe equation ρ i, MRE ( p, m i , R ) = 0 with respect to themomentum p .After the above replacement, the full thermody-namic potential Ω MRE for a finite size spherical dropletreads: Ω MRE V = Ω cond + U ( l, ¯ l ; T ) − N c (cid:88) i (cid:90) ΛΛ i, IR (cid:15) i ( p ) ρ i, MRE d p (2 π ) − T (cid:88) i (cid:90) ∞ Λ i, IR (cid:2) ln (cid:104) det f − i (cid:105) + ln (cid:104) det f + i (cid:105) (cid:3) ×× ρ i, MRE d p (2 π ) + P vac . (25)Multiplying on both sides of the last equation by thevolume of the quark matter drop, replacing the area S = 4 πR and the curvature C = 8 πR for a sphericaldrop, and rearranging terms we arrive to the followingform for Ω MRE Ω MRE = − P V + αS + γC, (26)where the pressure P , the surface tension α and thecurvature energy density γ , are defined as in Ref. [41]: P ≡ − ∂ Ω MRE ∂V (cid:12)(cid:12)(cid:12)(cid:12) T,µ,S,C (27)= − Ω cond − U ( l, ¯ l ; T ) + 2 N c (cid:88) i (cid:90) ΛΛ i, IR (cid:15) i ( p ) dp (2 π ) +2 T (cid:88) i (cid:90) ∞ Λ i, IR [ln (cid:104) det f − i (cid:105) + ln (cid:104) det f + i (cid:105) ] dp (2 π ) − P vac ,α ≡ ∂ Ω MRE ∂S (cid:12)(cid:12)(cid:12)(cid:12) T,µ,V,C (28)= − N c (cid:88) i (cid:90) ΛΛ i, IR (cid:15) i ( p ) f i,S pdp − T (cid:88) i (cid:90) ∞ Λ i, IR [ln (cid:104) det f − i (cid:105) + ln (cid:104) det f + i (cid:105) ] f i,S pdp,γ ≡ ∂ Ω MRE ∂C (cid:12)(cid:12)(cid:12)(cid:12) T,µ,V,S (29)= − N c (cid:88) i (cid:90) ΛΛ i, IR (cid:15) i ( p ) f i,C dp − T (cid:88) i (cid:90) ∞ Λ i, IR [ln (cid:104) det f − i (cid:105) + ln (cid:104) det f + i (cid:105) ] f i,C dp. As we previously mentioned, the value of Λ IR is thelargest root when solving ρ i, MRE ( p, m i , R ) = 0 withrespect to the momentum p , i.e. Λ IR changes with m i and with the drop’s radius R .Finally, we will address some aspects of the presentmodel that deserve a more detailed discussion:(i) In the present treatment finite-size effects enterthe fermion loop integral only; i.e. these effects are not considered in the pure Yang-Mills sector. As aconsequence, the Polyakov loop potential remainsunchanged and feels volume effects only implicitlythrough the saddle point equations. A more de-tailed analysis is left for future work.(ii) The conventional procedure for fixing P vac is toimpose that the grand thermodynamic potential Ω must vanish at zero temperature and vanish-ing chemical potential for matter in bulk. For theabove quoted parametrization, this assumption leadsto the value P vac = 5080 MeV fm − . Nevertheless,it has been emphasized in previous works [42–44]that this prescription is no more than an arbitraryway to uniquely determine the EOS of the NJLmodel without any further assumptions. A changein the value of P vac has no influence on the fittingsof the vacuum values for the meson masses and de-cay constants and thus the standard prescriptionfor P vac is not related to experimental values. Infact, different prescriptions for determining P vac have been adopted [43], including the alternativeof taking it as a free parameter [44] as it is usu-ally done within the MIT bag model for the bagconstant. When studying finite size systems, thestandard choice for P vac has an additional issue.If Ω vanishes at T = µ = 0 for matter in bulk itwill not do so for a finite size, due to the contri-bution of surface and curvature effects (as can beseen from Eq. (26)).As in previous works [45, 46], we will fix P vac inthe standard way, i.e. setting Ω = 0 at T = µ = 0for matter in bulk, and will use this value for anysystem’s size. Nonetheless, it must be emphasizedthat most of the thermodynamic quantities of rel-evance here (such as the critical temperatures, theentropy density, the sound speed, the specific heat,the surface tension and the curvature energy) areindependent of the choice P vac since they are re-lated to derivatives of the grand thermodynamicpotential Ω . The influence of the P vac choice onother thermodynamic quantities will be discussedbelow. In this section we present our numerical results for somethermodynamic properties of bulk and finite size quarkmatter systems. We will show the dependence of ourresults on the size of the system as well as for differentchoices of the Polyakov loop potential. We work at zerochemical potential to compare our numerical results forthe bulk with those from lattice QCD for (2+1)-flavors
Table 1
Using the polynomial Polyakov loop potential andtaking different values for the drop’s radius R , we show thechiral critical temperature T χ of the u and d condensates, thecritical deconfinement temperature T d of the Polyakov loopexpectation value, and the temperature T ∗ below which thedrop’s pressure becomes negative. T χ and T d are independentof the choice of the vacuum pressure P vac . T ∗ is calculated forthe standard value P vac = 5080 MeV fm − . R [fm]3 5 10 ∞ T χ [MeV] 177 182 184 186 T d [MeV] 160 161 162 162 T ∗ [MeV] 155 141 124 0 Table 2
Same as in Table 1 but for the logarithmic Polyakovloop potential. R [fm]3 5 10 ∞ T χ [MeV] 181 187 190 192 T d [MeV] 150 151 152 152 T ∗ [MeV] 157 149 132 0 Table 3
Same as in Table 1 but for the Polyakov loop po-tential of Fukushima, version U F . R [fm]3 5 10 ∞ T χ [MeV] 197 201 203 204 T d [MeV] 190 192 193 194 T ∗ [MeV] 175 158 137 0 Table 4
Same as in Table 1 but for the Polyakov loop po-tential of Fukushima, version U F . R [fm]3 5 10 ∞ T χ [MeV] 173 178 181 184 T d [MeV] 146 146 149 150 T ∗ [MeV] 149 135 118 0 using the highly improved staggered quark action ex-trapolated to the continuum limit [47] (see also [48]).Then we describe our predictions for finite size systems.4.1 Chiral and deconfinement transitionsHere we will focus on the order parameters for bothchiral and deconfinement transitions showing that, asthe temperature is increased at zero baryon chemicalpotential, the PNJL model presents a smooth crossovertransition at T ∼ −
200 MeV depending on thesize. Our results for the bulk are compatible with latticeQCD ones for N f = 2 + 1, as shown in Ref. [47] wherethe authors find a critical temperature of 154 ± χ uT ≡ ∂ (cid:104) ¯ uu (cid:105) /∂T and χ lT ≡ ∂l/∂T . The peak positions givethe inflexion points at the chiral critical temperature T χ and the critical deconfinement temperature T d of thecondensates and the Polyakov loop expectation valuerespectively. As discussed in [30], it is convenient totake the crossover temperature in the u − sector, be-cause the crossover temperature in the s − sector is lar-ger and would be far from the deconfinement transition.As also discussed in [30], the chiral and deconfinementtransitions do not take place at the same temperat-ure as long as we treat the chiral condensates and thePolyakov loop as independent variables. Anyway, theidea is exploring different parameters in the Polyakovloop potential to force as much as possible the prox-imity of both critical temperatures. Also, as shown in[30, 49, 50] the peak of χ lT occurs at a lower temperat-ure than the one of χ uT , in coincidence with our resultspresented in Tables 1 − T χ and T d for differentradii of the system. We also display the temperature T ∗ below which the drop’s pressure P would become negat-ive for the standard prescription of P vac . Below T ∗ theseresults would be unphysical if that P vac is adopted. Ifanother choice of P vac is used the curves in Fig. 1 wouldshift upwards or downwards and a better coincidence ofthe model curves with lattice results could be achievedfor the bulk case. From Tables 1, 2, 3 and 4 we see that(except for the cases U L and U F with R = 3 fm) T ∗ isalways below T χ . Since we are interested in the physicsaround the critical temperature (relevant for heavy ioncollisions and the early universe) we will not show ourresults for T < T ∗ . We emphasize that the values of T χ and T d are independent of the choice of P vac .In Table 1, we show our results for the polyno-mial Polyakov loop potential. The chiral critical tem-perature T χ has a significant dependence on the systemsize; it varies from 186 MeV to 177 MeV as the radiusshrinks from infinity to 3 fm. This effect is also appar-ent in the left panel of Fig. 2 where we see that thepeaks of ∂ (cid:104) ¯ uu (cid:105) /∂T move towards smaller temperaturesas the radius reduces. As seen in the right panel of Fig.2, ∂l/∂T is less sensitive to finite size effects. Thus,the critical deconfinement temperature T d varies overa narrower range than T χ , as can be verified in Table1. This behavior could have been anticipated because l feels volume changes only indirectly through the gapequations, and the Polyakov loop potential does notdepend explicitly on the size of the system. As a con-sequence, T χ and T d get closer to each other as thedrop’s size decreases. Figure 1
We show
P/T as a function of temperature fordifferent drop sizes and different Polyakov loop potentials.The gray band are the results for the equation of state in(2+1)-flavor QCD using the highly improved staggered quarkaction extrapolated to the continuum limit [47] (see also [48]). The critical temperatures for the model with a log-arithmic Polyakov loop potential can be seen in Table2. In this case T χ varies from 192 MeV to 181 MeVas R decreases. Here the deconfinement temperaturesare slightly smaller than in the previous case, and thechiral ones, larger. For R = 3 fm, we find that T d liesin the negative pressure region for the standard choiceof P vac .In the cases with U L and U P , the choice of T affectsboth, the deconfinement and the chiral critical temper-atures. Here we use T = 185 MeV in agreement withthe values used in [25, 33, 35] for N f = 2 + 1. For lar-ger T , T χ and T d approach each other but both valuesincrease, spoiling the coincidence with lattice results.On the other hand, for smaller T , T χ and T d are closerto lattice data but there is larger separation betweenthem.Finally, we show the critical temperatures for thePolyakov loop potential of Fukushima. Here we con-sidered two different examples, as discussed in the pre-vious section. In Table 3 we show the results for b =(196 . , and in Table 4 for b = (115 MeV) . Thefirst case gives higher T χ and T d but both temperat-ures are closer to each other. In the second case weobtain smaller critical temperatures (closer to latticeresults for 2+1 flavors) but there is a larger separationbetween them.Summing up, in all cases discussed above, we seeas a common behavior that T χ decreases with the sizeof the system by around 5% when the radius goes from R = ∞ to R = 3 fm. We also note that T d varies by lessthan 2% in the same size interval. As a consequence, as the drop’s size decreases, T χ becomes closer to T d . Thisbehavior is in agreement with the results presented in[24, 25].In Fig. 3 we show the temperature variation of theconstituent masses M u , M d and M s for different dropsizes and for all the Polyakov loop potentials presentedin Sect. 2.2. In the chirally broken phase, we find thatthe constituent quark masses are somewhat smaller fordrops with smaller radii. In this region, the volume de-pendence of the effective masses is stronger than in thechirally restored phase. Also, M u and M d show a steepslope around the crossover temperature while for M s the slope is smoother. As shown in in Tables 1 −
4, thechiral critical temperature T χ shifts to smaller valuesas the volume decreases. Such behavior is also appar-ent in Fig. 3 where we see that, for smaller systems,the constituent mass tends to the current value at lowertemperatures. A similar behavior has been reported in[25].4.2 Interaction measureA thermodynamic quantity of special interest is thethermal expectation value of the trace of the energymomentum tensor: Θ µµ ( T ) ≡ (cid:15) ( T ) − P ( T ) . (30)This quantity is known as trace anomaly or equivalentlyas interaction measure ∆ ( T ) ≡ (cid:15) ( T ) − P ( T ) since itis very sensitive to the non-perturbative effects in thequark-gluon plasma. Specifically, it measures the devi-ation from the equation of state of an ideal gas (cid:15) = 3 P due to interactions and/or finite quark masses.Here we focus on the quantity ∆ ( T ) T = (cid:15) ( T ) − P ( T ) T (31)which allows a straightforward assessment of deviationsfrom the Stefan-Boltzmann behavior.Within the present model, the energy density (cid:15) ( T )at zero chemical potential is given by (cid:15) ( T ) = Ω MRE ( T ) V + T s ( T ) (32)where the entropy density is given by: s ( T ) = − V ∂ Ω MRE ( T ) ∂T (33)The interaction measure is sensitive to the finitedrop’s volume, because the energy density has an expli-cit dependence on the surface tension and the curvatureenergy: (cid:15) ( T ) = − P ( T ) + α ( T ) SV + γ ( T ) CV + T s ( T ) , (34) Figure 2
We show χ uT ≡ ∂ (cid:104) ¯ uu (cid:105) /∂T and χ lT ≡ ∂l/∂T as a function of temperature for the polynomial Polyakov loop potential. Figure 3
Constituent masses M u , M d and M s as a function of temperature for different drop sizes and different Polyakov looppotentials. We do not show the branch of each curve corresponding to temperatures for which the drop’s pressure becomesnegative for the standard choice of P vac . In addition, as apparent from Eqs. (28), (29) and (30),there is an additional dependence on finite size effectsthrough the infrared cutoff Λ i, IR in the integrals for P , α and γ .In Fig. 4 we show our results for the bulk and for fi-nite size systems together with lattice QCD simulationdata in the continuum limit [47]. In general, we observethat the predictions of our effective model in bulk arein qualitative agreement with lattice QCD results. Thepeak heights are somewhat larger that in lattice QCD;nonetheless, the peak positions are in good coincidencewith lattice. As a global feature, common to all finitesizes models that include different Polyakov loop po-tentials U , the interaction measure presents a peak thatmoves towards decreasing temperatures as the radii de-crease. Note that, even though the interaction measureis explicitly dependent on P vac , the temperatures at which the peaks take place are not affected by the P vac choice.For the chirally broken phase, i.e. for temperaturesbelow the one in the peak, the curves for the bulk caseare in qualitative agreement with lattice data. Close to T ∗ , for finite sizes, the curves have a local minimumand start to increase at lower temperatures due to thecontribution of the surface tension and the curvatureenergy.Now let us concentrate on the peak of the curves.For R = ∞ , the peak position of the curves with U L , U P and U F are in better coincidence with lattice results.The one with U F is shifted to higher temperatures.For finite systems the position of the peaks is shiftedto lower temperatures in all cases. As a global feature,the peak heights with R = ∞ for all models are largerthan lattice results. We get a better agreement for U F Figure 4
We show ∆/T as a function of temperature for different drop sizes and different Polyakov loop potentials. We alsoinclude lattice QCD simulations data from [47] (gray band). which is a ∼
25% higher than lattice. For finite systemswe see that the height of the peaks increase as the radiidecrease, and they shift to smaller temperatures.For high enough temperatures, in the chirally re-stored phase, there is a reasonable agreement betweenthe bulk models and lattice results, specially for the U L and U P Polyakov potentials. Results for the U F and U F potentials, are somewhat below the lattice data.For finite sizes our results are superposed with the cor-responding bulk case.4.3 Energy density and entropy densityIn the bulk case, our results for the energy density andthe entropy density are in qualitative agreement withlattice QCD results (see Fig. 5) and with Ref. [33], ascan be seen from their Fig. 3. As previously mentionedfor the results of Fig. 1, different choices of P vac wouldlead only to a vertical shift of the curves for the energydensity but will not change the temperature of the in-flexion points. One could take advantage from this fea-ture and introduce a different procedure to fix P vac insuch a way that our predictions for the bulk case areas close as possible to lattice data. Since our focus hereis not centered on the equation of state we shall notexplore such strategy in the present work.For high enough temperatures, our curves for allthermodynamic quantities approach to the Stefan-Boltz-mann limit. The Stefan-Boltzmann limit for the pres-sure is given by p SB T = ( N c − π
45 + N c N f π , (35)where N c and N f are the number of colors and flavors.The first term represents the gluonic contribution and the second, the quark’s contribution. For N c = 3 and N f = 3 we have: p SB T = 8 π
45 + 7 π
20 = 1 .
75 + 3 .
45 = 5 .
20 (36)which results in (cid:15) SB /T = 5 .
26 + 10 .
36 = 15 .
62 and s SB /T = 7 + 13 . . U F and U F tend to the Stefan-Boltzmann limit for quarks only(no gluons) while models with the potentials U P and U L tend to the Stefan-Boltzmann limit including quarksand gluons.This behavior is already known from previous works[25, 30, 33]. In the case of the U P and U L potentials,both the unconfined transverse gluons as well as thePolyakov loop, which corresponds to longitudinal gluons,contribute to the thermodynamic quantities [30]. But,since the Polyakov-loop potentials are fitted to puregauge lattice data, they thus reproduce the total pres-sure, energy density, and entropy density of both thelongitudinal and the transverse gluons, overcountingthe degrees of freedom in the chirally symmetric phase[30, 33]. However, the potential ansatz by Fukushimaexcludes these transverse gluon contributions at hightemperatures leading to the differences found in Fig.5. Nonetheless, at temperatures around and below thetransition temperature such differences tend to disap-pear.It is worth to remark that in Fig. 5 there is a widerange of temperatures in which our results for U F arein a quantitatively good agreement with lattice results.For finite systems, we see that in all cases the curvesconverge to the bulk ones at high temperatures. Close tothe transition region, the curves for different radii start Figure 5
Energy density and entropy density as functions of temperature. Lattice data (gray band) are taken from [47]. to separate each other as the temperature decreases.In coincidence with Ref. [25], we find that the smallerthe radius the higher the temperature at the inflexionpoint. Nonetheless, in Ref. [25] the results for R = 5 fmand for the bulk case are coincident for all temperaturesbut in our case are not. In the chirally broken phase,the energy density and the entropy density change verylittle with the drop’s size.4.4 Specific heat and speed of soundThe specific heat at constant volume is given by c V = − T ∂ Ω MRE ∂T (cid:12)(cid:12)(cid:12)(cid:12) V , (37)and the corresponding results are summarized in Fig.6. At low temperatures c V grows with T , then shows apeak at the transition temperature, and approaches thecorresponding Stefan-Boltzmann limit for high enough T . For the bulk case our results are in agreement withRef. [33] and with lattice data [47]. In fact, in the leftpanel of Fig. 6 we note that lattice data show a softundulation around the critical temperature, whose po-sition is close to the peaks considering U F and U P . For U F and U L the peaks are shifted to higher temperat-ures. In general, the best agreement with lattice data(up to the critical temperature and somewhat above it)is obtained with U F .For finite size drops, we find that c V doesn’t changesignificantly with the change in volume, except in thecrossover region. In fact, we find that the height of thepeaks decreases as the volume shrinks, in agreementwith [25]. Also, the peak position shifts to smaller tem-peratures as the volume decreases, as in [25]. As for other thermodynamic quantities, we find that the spe-cific heat for the models with U F and U F tend tothe Stefan-Boltzmann limit for quarks while the mod-els with U P and U L tend to the Stefan-Boltzmann limitfor quarks and gluons, due to the differences in the con-tributing gluon degrees of freedom.In Fig. 6 we show our results for the speed of sound[33] c s = ∂p∂(cid:15) (cid:12)(cid:12)(cid:12)(cid:12) S = sc V . (38)The behavior of c s is associated directly with the role ofinteractions in the system. The strength of interactionscan be quantified through the interaction measure ∆ calculated in Sec. 4.2. A comparison between ∆ presen-ted in Fig. 4 and c s depicted in Fig. 6, shows that thesequantities are correlated. At large temperatures, as thevalue of ∆ goes to zero, the speed of sound tends tothe ultarelativistic limit of an ideal gas, c s = 1 / √
3. Atlower temperatures, interactions become relevant andtherefore ∆ grows and c s decreases significantly.Except for U F , all minima positions lie close to thelattice QCD one. In the chirally restored phase our res-ults for all Polyakov potentials are in good agreementwith lattice QCD data, except for the U F case thatapproaches lattice at higher temperatures.Contrary to previous findings [25], our results showthat the speed of sound doesn’t depend too much onthe system’s size. In fact, small variations are observedonly in the transition region.4.5 Surface tension and curvature energyIn Fig. 7 we show the total surface tension α TOT andthe total curvature energy γ TOT for drops with different Figure 6
Specific heat and speed of sound. Lattice data (gray band) are taken from [47].
Figure 7
Total surface tension and total curvature energy as a function of temperature; the scales of panels (a) and (b) aredifferent. The minimums of α TOT and of γ TOT occur at different temperatures. sizes. α TOT = (cid:80) i α i and γ TOT = (cid:80) i γ i include thecontribution of u , d and s quarks. We have checkedthat α s is more than 10 times larger than α u and α d ,in qualitative accordance with results for cold quarkmatter at very high densities [45, 46] that show that thetotal surface tension α TOT is largely dominated by thecontribution of strange quarks. On the contrary, γ u and γ d are typically ∼ − γ s and thus the behaviorof γ TOT is controlled mainly by u and d quarks.Both, α TOT and γ TOT show a significant variationwith R at all temperatures, specially for small dropswith radii below 10 fm. There is also a considerabledependence on the Polyakov loop potential.At large temperatures α TOT and γ TOT are mono-tonically increasing functions of T . Moreover, for T (cid:38)
250 MeV the surface tension grows approximately as α TOT = C α T / , (39) being C α ≈ . − .
034 MeV − / fm − , while the curvatureenergy grows as γ TOT = C γ T / , (40)being C γ ≈ . − .
035 MeV − / fm − .At lower temperatures both α TOT and γ TOT havelocal minimums. In the case of the total curvature en-ergy the minimum falls around the chiral critical tem-perature T χ of the u and d condensates, which evidencesthe fact that γ TOT is controlled mostly by up and downquarks and is sensitive to their chiral transition. Onthe other hand, the total surface tension is sensitive tothe chiral transition of strange quarks and therefore itsminimum falls at a larger temperature.At temperatures below that of the minimum there isa narrow interval where α TOT and γ TOT are decreasingfunctions of T . For even smaller temperatures, α TOT and γ TOT tend to a constant value which is of the same order of the values obtained within the NJL model forcold quark matter ( T = 0) at finite chemical potentials( µ = 0 −
450 MeV) [46]. In some cases such constantvalue is not shown in the figures because the pressurebecomes negative for the standard choice of P vac . In this work we studied the thermodynamic propertiesof finite systems composed by quark matter contain-ing two light and one heavy quark within the frame ofthe PNJL model. We have considered vanishing baryonchemical potential and finite temperatures. We com-pared our numerical results for the bulk case with thosefrom lattice QCD simulations, and then we studied thefinite size deviations from the bulk case. We includedfinite size effects through the Multiple Reflection Ex-pansion formalism and explored the effect of using dif-ferent Polyakov loop potentials. Finite size effects wereincorporated in the fermion integrals but not in thePolyakov loop potentials. However, if the pure Yang-Mills theory were formulated with a finite radius, thedeconfinement phase transition could be affected andpresumably the first-order transition would turn into asmooth crossover for small enough radii. This is beyondthe scope of the present work.As the temperature is increased at zero baryon chem-ical potential, the order parameters for both chiral anddeconfinement transitions indicate that the PNJL modelpresents a smooth crossover transition, in accordancewith lattice QCD results. For different radii of the sys-tem and different choices of the Polyakov loop poten-tial, we determined the chiral critical temperature T χ of the u and d condensates and the critical deconfine-ment temperature T d of the Polyakov loop expectationvalue (see Table 1). In general, T χ depends on the sys-tem’s size, decreasing by around 5% when the radiusgoes from infinity to 3 fm, while T d varies by less than2% in the same interval. Thus, as the drop’s size de-creases, T χ becomes closer to T d , in accordance with[24, 25].Then we focused on the interaction measure ∆ ( T ) ≡ (cid:15) ( T ) − P ( T ), which evaluates the deviation from anideal gas behavior ( (cid:15) = 3 P ) due to interactions and/orfinite quark masses. ∆/T goes to zero at low and largetemperatures and presents a peak around the trans-ition density. In the bulk case, our results for ∆/T are in qualitative agreement with lattice QCD results.Moreover, for U F we obtain a good quantitative agree-ment with lattice data up to temperatures around 250MeV. For different Polyakov loop potentials U , we findthat as the radii decrease the peak moves towards lowertemperatures and its height increases. At temperatures below that of the peak the results show a stronger de-pendence on the system’s size and on the choice of thePolyakov loop potential.In the bulk case, our results for the energy dens-ity (cid:15) , the entropy density s and the specific heat c V are in qualitative agreement with previous calculationspresented in Ref. [33] and with lattice QCD results [47].At high temperatures, the curves for all system’s radiiconverge to the bulk ones and approach to the Stefan-Boltzmann limit. However, models with the Polyakovloop potentials of Fukushima tend to the Stefan-Boltzmannlimit for quarks only (without gluons) while modelswith the polynomial and logarithmic Polyakov loop po-tentials tend to the Stefan-Boltzmann limit includingquarks and gluons, as already known from previousworks [25, 30, 33]. In general, (cid:15) , s and c V don’t changesignificantly with the change in volume, except for c V in the transition region and for (cid:15) at temperatures belowthe transition region.At high temperatures the speed of sound tends tothe ultarelativistic limit of an ideal gas, c s = 1 / √ c s decreases significantly. Again, for the bulkcase we find a qualitative agreement with lattice QCDresults. Notwithstanding, contrary to previous findings[25], our results show that the speed of sound doesn’tdepend too much on the system’s size. In fact, smallvariations are observed only in the transition region.Two very relevant quantities for finite systems arethe surface tension and the curvature energy which havebeen calculated for drops with different sizes. We findthat α TOT is largely dominated by the contribution ofstrange quarks (in coincidence with previous results forcold quark matter at very high densities [45, 46]), while γ TOT is controlled mainly by the behavior of u and d quarks. Both, α TOT and γ TOT change significantlywith R at all temperatures, specially for small dropswith radii below 10 fm. There is also a considerabledependence on the Polyakov loop potential. For T (cid:38)
250 MeV, α TOT and γ TOT grow proportionally to T / .At lower temperatures α TOT has a minimum relatedto the chiral transition of s quarks and γ TOT has aminimum associated with the u and d quarks chiraltransition. For smaller temperatures, α TOT and γ TOT tend to constant values of the same order of the onesobtained for very dense cold quark matter [46].In summary, our main conclusion is that severalthermodynamic quantities are sensitive to finite sizeeffects, particularly for temperatures around the cros-sover transition and for systems with radii below ∼
10 fm. These results can be potentially relevant forthe study of the QCD transition at the early Universe[51, 52] and should be extended to other regions of the QCD phase diagram, specially the region of hightemperatures and moderate baryon chemical potentialswhere heavy ion collisions take place. Acknowledgements
G.L. acknowledges the Brazilian agen-cies Conselho Nacional de Desenvolvimento Cient´ıfico e Tec-nol´ogico (CNPq) and Funda¸c˜ao de Amparo `a Pesquisa doEstado de S˜ao Paulo (FAPESP) for financial support.
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