Fluctuation of strongly interacting matter in Polyakov Nambu Jona-Lasinio model in finite volume
aa r X i v : . [ h e p - ph ] M a r Fluctuation of strongly interacting matter in Polyakov Nambu Jona-Lasinio model infinite volume
Abhijit Bhattacharyya ∗ Department of Physics, University of Calcutta, 92, A. P. C. Road, Kolkata - 700009, INDIA
Rajarshi Ray † Department of Physics and Centre for Astroparticle Physics & Space Science,Bose Institute, EN-80, Sector V, Bidhan Nagar, Kolkata - 700091, INDIA
Subrata Sur ‡ Department of Physics, Panihati Mahavidyalaya,Barasat Road, Sodepur, Kolkata - 700110, INDIA
We estimate the susceptibilities of conserved charges for two flavor strongly interacting matterwith varying system sizes, using the Polyakov loop enhanced Nambu–Jona-Lasinio model. Thesusceptibilities for vanishing baryon densities are found to show a scaling with the system volume inthe hadronic as well as partonic phase. This scaling breaks down for a temperature range of about30-50 MeV around the crossover region. Simultaneous measurements of the various susceptibilitiesmay, thus, indicate how close to the crossover region the freeze-out occurs for the fireball created inheavy-ion collision experiments.
PACS numbers: 12.38.Aw, 12.38.Mh, 12.39.-x
Strongly interacting matter under extreme conditions of temperature and density is expected to show a rich phasestructure. In the early Universe, a few microsecond after the Big Bang, when the temperature was extremely high,exotic states namely quarks and gluons may have been prevalent [1]. On the other hand inside the core of a compactstar, where the baryon matter density is extremely high various exotic phases like color superconductor, color superfluidetc. may be present [2]. Experiments with heavy ions colliding with each other or with a target, at the facilities atCERN(France/Switzerland), BNL(USA) and GSI(Germany), are continuing the search for such exotic states of matterin the laboratory.The matter formed due to the energy deposition of the colliding particles obviously has a finite volume. It istherefore imperative to have a clear understanding of the finite size effects to fully contemplate the thermodynamicphases that may be created in the experiments. These effects depend on the size of the colliding nuclei, the center ofmass energy ( √ s ) and the centrality of collisions. There have been a large number of efforts to estimate the systemsize at freeze-out for different √ s and different centralities. A study using the measurement of HBT radii [3] indicatesthat the freeze out volume increases as the √ s increases. In the same work the freeze out volume has been estimatedto be 2000 f m to 3000 f m . On the other hand in Ref. [4] the volume of homogeneity has been calculated usingUrQMD model [5] and compared with the experimentally available results. The √ s considered was in the range of62.4 GeV to 2760 GeV for lead-lead collisions at different centralities. The system volume has been found to vary from50 f m to 250 f m . The effect of colliding particles and √ s has been further analyzed by the ALICE collaborationin [6]. Given that these are the volumes at the time of freeze-out, one may expect an even smaller system size at theinitial equilibration time [7, 8].The importance of finite size effects in the thermodynamics of strong interaction may be brought forward with thehelp of finite size scaling analysis [9]. In the context of heavy ion collisions such a possible analysis has been discussedin the literature (see e.g. [10–12]).Other theoretical studies of finite volume effects have been performed in various contexts. In Ref. [13] the effectivedegrees of freedom have been found to be reduced due to finite volume using a non-interacting bag model. The effectof finite volume has been studied also with a two model equation of state and it has been found that the criticaltemperature looses its sharpness [14]. A few first principle study of pure gluon theory on space-time lattices were per-formed, showing the possibility of significant finite size effects [15, 16]. The meson properties show a significant volumedependence as found in Ref. [17, 18]. In the context of chiral perturbation theory the implications of finite system sizehave been discussed [19, 20]. There are also studies with four-fermi type interactions, like the Nambu − Jona-Lasinio ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] (NJL) [21] models [10, 22, 23], linear sigma models [11, 24, 25] and Gross-Neveu models [26]. While in Ref. [24]the scaling behavior of chiral phase transition for finite and infinite volumes has been studied, the character of phasediagram has been studied in Ref. [10, 11, 25, 26]. In refs. [22] and [23] the authors have studied the chiral properties asa function of the radius of a finite droplet of quark matter. The stability of such a droplet in the context of strangeletformation within the NJL model has been addressed in Ref. [27]. Size dependent effects of di-fermion states within2-dimensional NJL model has been studied in Ref. [28] and that of magnetic field is discussed in Ref. [29]. Recently ina 1+1 dimensional NJL model the induction of charged pion condensation phenomenon in dense baryonic matter dueto finite volume effects have been studied in [30]. Recently some of us have studied the thermodynamic properties ofstrongly interacting matter in a finite volume using Polyakov-Nambu-Jona–Lasinio (PNJL) model [31]. It has beenshown there that the critical temperature for the cross-over transition at zero baryon density decreases as the volumedecreases. Furthermore at low volume the critical end point is pushed towards the higher µ and lower T domain. At R = 2 f m , it was found that the critical end point (CEP) vanishes and the whole phase diagram becomes a cross-over.The possible chiral symmetry restoration in a color confined state has also been discussed.Though various thermodynamic properties have been studied to some extent in finite size systems, not much hasbeen done to estimate the fluctuations occurring in finite volumes. On the lattice, the Polyakov loop susceptibilityhas been calculated for a finite volume [32]. A similar work has been done in the PNJL model using Monte Carlosimulation [33] and also in the quark-meson model using a renormalisation group approach [34]. Fluctuations ofconserved quantum numbers are related to the respective susceptibilities via the fluctuation-dissipation theorem. Fora 2-flavor strongly interacting system one has the quark number susceptibility (QNS) and isospin number susceptibility(INS) etc. These fluctuations are sensitive indicators of the transition from hadronic matter to partonic state. Alsothe existence of the CEP may be signalled by the diverging behavior of fluctuations. Here we report our calculationsof quark and isospin number susceptibilities of strongly interacting matter using PNJL model up to sixth order. Thereport is organized as follows. First we give a very brief description of the PNJL model and the necessary methodology.Thereafter we present the results for the various susceptibilities. Finally we summarize and conclude.The PNJL model used here is based on a series of works [21, 31, 35–47]. For some recent progress in this modelsee e.g. [44, 48–57]. For a detailed overview see e.g. [58] and references therein. The PNJL model for 2 flavors isdescribed by the Lagrangian, L = X f = u,d ¯ ψ f γ µ iD µ ψ f − X f m f ¯ ψ f ψ f + X f µ f γ ¯ ψ f ψ f + g S X a =1 , , [( ¯ ψτ a ψ ) + ( ¯ ψiγ τ a ψ ) ] − U ′ (Φ[ A ] , ¯Φ[ A ] , T ) (1)where Polyakov loop potential U ′ (Φ[ A ] , ¯Φ[ A ] , T ) can be expressed as, U ′ (Φ[ A ] , ¯Φ[ A ] , T ) T = U (Φ[ A ] , ¯Φ[ A ] , T ) T − κ ln( J [Φ , ¯Φ]) (2)Here U (Φ , ¯Φ , T ) is a Landau-Ginsburg type potential as given in Ref. [38], U (Φ , ¯Φ , T ) T = − b ( T )2 ¯ΦΦ − b + ¯Φ ) + b , (3)where b ( T ) = a + a ( T T ) + a ( T T ) + a ( T T ) , (4) b and b being constants. The second term in Eqn.(2) is the Vandermonde term. J [Φ , ¯Φ] = (27 / π ) h −
6Φ ¯Φ + 4(Φ + ¯Φ ) − i The parameters a i , b i were fitted from Lattice results of pure gauge theory. The set of values chosen here are, a = 6 . a = − . a = 2 . a = − . b = 0 . b = 7 . T = 190 M eV , κ = 0 . p min = π/R = λ where R is the lateral size of a cubic volume V = R . In principle one should sum over discrete momentum values but forsimplification we integrate over continuous values of momentum. Also we neglect surface and curvature effects. Otherparameters of the model were not modified.We note here that in the NJL model discussions of using a lower momentum cut-off exists in the literature (see [59, 60]and references therein). The motivation of introducing this IR cut-off there has been to mimic confining effects ofstrong interaction which helps to remove spurious poles in the quark loop diagrams so that unphysical decay of hadronsto quarks do not take place. Since in the PNJL model such unphysical decays are restricted due to the vanishingof the Polyakov loop for low temperatures [38, 61] no IR cut-off is necessary in the PNJL model. However, for 2flavour PNJL model, the unphysical decay does not completely vanish as pointed out in Ref. [62] where a very smallbut non-zero sigma meson decay is observed at low temperatures. On the other hand, for 2+1 flavour the σ -mesonbecomes a true bound state for small temperatures [63]. P / T T/T c R=2fmR=2.5fmR=4fmR= ∞ FIG. 1: (Color online) Variation of pressure with temperature for different system sizes. c T/T c R=2fmR=2.5fmR=4fmR= ∞ c T/T c R=2fmR=2.5fmR=4fmR= ∞ -0.15-0.1-0.05 0 0.05 0.1 0.15 0.6 0.8 1 1.2 1.4 1.6 1.8 2 c T/T c R=2fmR=2.5fmR=4fmR= ∞ FIG. 2: (Color online) Variation of quark number susceptibilities with temperature for different system sizes.
Our starting point is the thermodynamic potential given by,Ω = U ′ [Φ , ¯Φ , T ] + 2 g S X f = u,d σ f − X f Z Λ λ d p (2 π ) E p f Θ(Λ − | ~p | ) − X f T Z ∞ λ d p (2 π ) ln (cid:20) − ( E p f − µ f ) T )) exp( − ( E p f − µ f ) T ) + exp( − E p f − µ f ) T ) (cid:21) − X f T Z ∞ λ d p (2 π ) ln (cid:20) − ( E p f + µ f ) T )) exp( − ( E p f + µ f ) T ) + exp( − E p f + µ f ) T ) (cid:21) (5)where E p f = q p + M f is the single quasi-particle energy. In the above expression σ f is given as σ f = h ¯ ψ f ψ f i = − M f π Z Λ λ p q p + M f dp, (6)We first obtain the mean fields σ , Φ and ¯Φ from the extremization conditions: ∂ Ω ∂σ = 0 , ∂ Ω ∂ Φ = 0 , ∂ Ω ∂ ¯Φ = 0. Thefield values so obtained are then put back into Ω to obtain the thermodynamic potential, which is then used to obtainvarious thermodynamic quantities. Some of which have been reported by us in Ref. [31]. For example, the pressurein the finite volume system is given by, P ( T, µ q , µ I ) = − ∂ (Ω( T, µ q , µ I ) V ) ∂V (7)where T is the temperature and µ q and µ I are the quark and isospin chemical potentials respectively. The variationof scaled pressure with T /T c is shown in figure 1. The critical temperature T c is dependent on the system size. Wehave considered different system sizes corresponding to R = 2 f m , R = 2 . f m , R = 4 f m and infinite volume. Thecorresponding values of T c are 167 MeV, 171 MeV, 183 MeV and 186 MeV respectively. c I T/T c R=2fmR=2.5fmR=4fmR= ∞ c I T/T c R=2fmR=2.5fmR=4fmR= ∞ -0.15-0.1-0.05 0 0.05 0.1 0.15 0.6 0.8 1 1.2 1.4 1.6 1.8 2 c I T/T c R=2fmR=2.5fmR=4fmR= ∞ FIG. 3: (Color online) Variation of isospin number susceptibilities with temperature for different system sizes.
We now discuss the various susceptibilities of quark number and iso-spin number. These are defined as, c n ( T ) = 1 n ! ∂ n (cid:0) Ω( T, µ q , µ I ) /T (cid:1) ∂ (cid:0) µ X T (cid:1) n (cid:12)(cid:12)(cid:12) µ X =0 . (8)where µ X = µ q or µ I . For an expansion around µ X = 0, the odd order terms vanish due to CP symmetry. Manyof these susceptibilities have been measured for infinite volume systems in first principle QCD calculations on thelattice [64–72] as well as hard thermal loop calculations [73–82]. At the same time various QCD inspired models havealso made suitable estimates of these fluctuations for infinite systems (see e.g. [40, 41, 58, 83–93]) Here we presentthe first computation of finite size effects on these fluctuations.For each system volume considered, we have calculated Ω at chemical potentials spaced by 0.1 MeV at a giventemperature. These have been fitted it to an eighth order polynomial in µ X using the GNU plot program. Wehave chosen the maximum range of µ X to be 200 MeV. From the fit we have extracted the coefficients c , c and c both for quark number and isospin number susceptibilities. This procedure has been repeated for different values oftemperature.The variation of quark number susceptibilities with T /T c are shown in Fig. (2). The general features for thesesusceptibilities in finite volumes are quite similar to that for infinite volume. However quantitatively we observesignificant volume dependence. With increase in system size there is an enhancement of all the susceptibilities. Forthe isospin number susceptibilities shown in figure 3, we find almost identical behavior. It may be noted that the mostsignificant finite size effects are seen in the higher order susceptibilities close to the cross-over region. Given that thedetectors in QGP search experiments are expected to observe the system frozen close to the cross-over region, onemay find an estimate of the system volume from the measurement of various higher order fluctuations.Alternatively, it is important to realize that for a comparison of fluctuations calculated theoretically with thatmeasured experimentally, one needs to be confident about the measured system volumes. Since measuring the systemsize is quite a difficult task in the experiments, and one usually resorts to consider ratios of fluctuations to eliminatethe volume factor [66]. However this assumption is valid when interactions are small and the volume factor scalesout. Therefore in the purely hadronic or partonic phases one may observe such a scaling of the fluctuations withsystem size. However, close to the cross-over region such an assumption may not hold as large scale fluctuations aredominant and the system deviates from a stable thermodynamic phase. Now that we have the actual calculations ofsystem size effects we can easily check the how the ratio of fluctuations behave. For this purpose we present the ratios c /c (kurtosis) and c /c for both the quark number and isospin number susceptibilities. In this case we obviouslyneed to plot the variation with temperature rather than with T /T c . The variations are shown in Fig. (4). We observe c / c T (MeV) R=2fmR=2.5fmR=4fmR= ∞ -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 100 150 200 250 300 350 400 c / c T (MeV) R=2fmR=2.5fmR=4fmR= ∞ c I / c I T (MeV) R=2fmR=2.5fmR=4fmR= ∞ -0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 100 150 200 250 300 350 400 c I / c I T (MeV) R=2fmR=2.5fmR=4fmR= ∞ FIG. 4: (Color online) Variation of ratios of fluctuations with temperature for different system sizes. that for low and high temperatures the ratios of fluctuations show the expected scaling with the system volume, whilein the cross-over region there is significant volume dependence. Thus if the system created in heavy-ion experimentsfreeze-out much below T c , the ratios of different susceptibilities would show the corresponding values for the hadronicphase. The amount of deviation of these ratios from the hadronic phase results would indicate the closeness of thesystem to the cross-over region.To summarize, we have studied the fluctuations of strongly interacting matter in a finite volume using the PNJLmodel. The susceptibilities in the quark number and isospin number are obtained up to sixth order for different systemsizes. We find a significant volume dependence in these quantities, which may be useful in analyzing the experimentaldata and obtain the size of the fireball formed in the heavy-ion collision experiments. The volume dependence showsan expected scaling behavior in the hadronic and partonic phases. In the cross-over region the system size scalingbreaks down and may be use to estimate the closeness of the created fireball to the cross-over region. 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