An approach of statistical corrections to interactions in hadron resonance gas
aa r X i v : . [ h e p - ph ] N ov An approach of statistical corrections to interactions in hadron resonance gas
Mahmoud Hanafy ∗ Physics Department, Faculty of Science, Benha University, 13518, Benha, Egypt
Muhammad Maher † Helwan University, Faculty of Science, Physics Department, 11795 Ain Helwan, Egypt (Dated: November 11, 2020)We propose a new model for hadrons with quantum mechanical attractive and repulsive in-teractions sensitive to some spatial correlation length parameter inspired by Beth-Uhlenbeckquantum mechanical non-ideal gas model [1]. We confront the thermodynamics calculatedusing our model with a corresponding recent lattice data at four different values of the baryonchemical potential, µ b = 0 , , ,
425 MeV over temperatures ranging from 130 MeV to200 MeV and for five values for the correlation length ranging from 0 to 0 . ≃
160 MeV,a decent fitting between the model and the lattice data is observed for different values of r ,especially at ( µ b , r ) = (170 , . , (340 , . , . µ b is in MeV and r is infm. For vanishing chemical potential, the uncorrelated model ( r = 0), which corresponds toideal hadron resonance gas model seems to offer the best fit. The quantum hadron correla-tions seem to be more probable at non-vanishing chemical potentials, especially within therange µ b ∈ [170 ,
340 MeV].
PACS numbers: 05.50.+q, 21.30.Fe, 05.70.CeKeywords: Lattice theory and statistics, Forces in hadronic systems and effective interactions,Thermodynamic functions and equations of state
I. INTRODUCTION
One of the key goals of the ultrarelativistic nuclear collisions is to gain information on the hadron-parton phase diagram, which is characterized by different phases and different types of the phasetransitions [2]. Quantum Chromodynamics (QCD), the gauge field theory that describes the stronginteractions of colored quarks and gluons and their colorless bound states, has two important inten- ∗ Electronic address: [email protected] † Electronic address: [email protected] sive state parameters at equilibrium, namely temperature T and baryon chemical potential µ b . Aremarkable world-wide theoretical and experimental effort has been dedicated to the study of stronglyinteracting matter under extreme condition of temperature and baryon chemical potential. The latticeQCD simulations provide an a priori non-perturbative regularization of QCD that makes it compliantwith analytic and computational methods with no model assumptions other than QCD itself beingneeded to formulate the theory. The temperature and density (chemical potential) dependence of thebulk thermodynamic quantities, commonly summarized as the equation of state (EoS), provides themost basic characterization of equilibrium properties of the strongly interacting matter. Its analysiswithin the framework of lattice QCD has been refined ever since the early calculations performedin pure SU ( N ) gauge theories [3]. The EoS at vanishing chemical potentials does already provideimportant input into the modeling of the hydrodynamic evolution of hot and dense matter created inheavy-ion collisions [4, 5]. While this is appropriate for the thermal conditions met in these collisionsat the LHC and the highest RHIC beam energies, knowledge of the EoS at non-vanishing baryon,strangeness and electric charge chemical potentials is indispensable for the hydrodynamic models ofthe conditions met in the beam energy scan (BES) at RHIC [6] and in future experiments at facilitieslike FAIR at GSI and NICA at JINR [7, 8].Bulk thermodynamic observables such as pressure, energy density, and entropy density as well asthe second order quantities such as the specific heat and velocity of sound have now been obtainedat vanishing chemical potentials for the three lightest quark flavors [9]. By the analysis of the chiraltransition temperature, T c ≃ ± ≤ µ b /T . . ≤ √ s NN ≤
200 GeV[17]. A promising approach in this quest is the investigation of hadron production. The hadronresonance gas (HRG) is customarily used in the lattice QCD calculations as a reference for the hadronicsector [18, 19]. At low temperatures, they are found to be in quite good agreement with the HRG modelcalculations [20], although some systematic deviations have been observed, which may be attributedto the existence of additional resonances which are not taken into account in HRG model calculationsbased on well established resonances listed by the particle data group [21] and perhaps to need toextend the model to incorporate interactions.In the HRG model, the thermodynamics of a strongly interacting system is conjectured to beapproximated as an ideal gas composed of hadron resonances with masses . T c . Therefore, the hadronic phasein the confined phase of QCD could be modeled as a non-interacting gas of the hadron resonances. Itis reported in recent literature that the standard performance of the HRG model seems to be unableto describe all the available data that is predicted by recent lattice QCD simulations [22, 23]. Theconjecture to incorporate various types of interactions has been worked out in various studies [5, 24–26].When comparing the thermodynamics calculated within the HRG framework with the correspondingdata obtained using lattice QCD methods, one has to decide how to incorporate interactions amongthe hadrons.Arguments based on the S-matrix approach [27–29] suggest that the HRG model includes attractiveinteractions between hadrons which lead to the formation of resonances. More realistic hadronic modelstake into account the contribution of both attractive and repulsive interactions between the componenthadrons. Repulsive interactions in the HRG model had previously been considered in the frameworkof the relativistic cluster and virial expansions [28], via repulsive mean fields [30, 31], and via excludedvolume (EV) corrections [32–37]. In particular, the effects of EV interactions between hadrons onHRG thermodynamics [38–45] and on observables in heavy-ion collisions [46–53] has extensively beenstudied in the literature. Recently, repulsive interactions have received renewed interest in the contextof lattice QCD data on fluctuations of conserved charges. It was shown that large deviations of severalfluctuation observables from the ideal HRG baseline could well be interpreted in terms of repulsivebaryon-baryon interactions [53–55].The present script is organized as follows: In Section , we review the detailed formalism of theconventional ideal (uncorrelated) HRG model, then we develop a non-ideal (correlated) statisticalcorrection to the ideal HRG model inspired by Beth-Uhlenbeck (BU) quantum theory of non-idealgases. The calculations of the HRG thermodynamics based on the proposed correction are discussedin Section . Section is devoted to the conclusions and outlook. II. MODEL DESCRIPTION
In this study, we use the particle interaction probability term originally implemented in the expres-sion for the second virial coefficient worked out in ref. [1] in order to suggest a statistical correctionto the uncorrelated HRG model. Beth and Uhlenbeck suggested a connection between the virial co-efficients and the probabilities of finding pairs, triples and so on, of particles near each other [56]. Inthe classical limit, which is usually designated by sufficiently high temperatures and/or low particledensities, it was shown that these probabilities (explicit expressions are to follow next section) can beexpressed by Boltzmann factors so long as the de Broglie wavelength, which is a common measure ofthe significance of the quantum non-localiy, is small enough compared with the particle spacial extentmeasured by the particle ”diameter” [56]. Such a particle diameter can be considered as a measure ofthe spatial extent within which a particle can undergo hardcore (classical) interactions. Based on acomparison of model with experimental results, it was concluded that at sufficiently low temperaturesfor which the thermal de Broglie wave length is comparable with the particle diameter, deviationsfrom the classical excluded volume model due quantum effects will be significant [56].An extension has been made to the quantum mechanical model of the particle interactions asproposed in ref. [56] by considering the influence of Bose or Fermi statistics in addition to the effectof the inclusion of discrete quantum states for a general interaction potential that is not necessarilycentral [1]. The expression for the second virial expansion developed in ref. [56] and then extended inref. [1] was later generalized using the cluster integral to describe particle interactions provided thatthose particles don’t form bound states [27, 28, 57].The quantum mechanical Beth-Uhlenbeck (BU) approach were quite recently used to model therepulsive interactions between baryons in a hadron gas [58]. The second virial coefficient or the“excluded volume” parameter was calculated within the BU approach and found to be temperaturedependent, and found also to differ dramatically from the classical excluded volume (EV) model result.At temperatures T = 100 −
200 MeV, the widely used classical EV model [59–61] underestimates theEV parameter for nucleons at a given value of the nucleon hard-core radius (assumed ≃ . Z ( T, µ, V )from which the thermodynamics of such a system can be deduced by taking the proper derivatives.
A. Non-correlated ideal HRG
In a grand canonical ensemble, the partition function reads [4, 5, 20, 65–67] Z ( T, V, µ ) = Tr (cid:20) exp (cid:18) µN − HT (cid:19)(cid:21) , (1)where H is Hamiltonian combining all relevant degrees of freedom and N is the number of constituentsof the statistical ensemble. Eq. (1) can be expressed as a sum over all hadron resonances taken fromrecent particle data group (PDG) [21] with masses up to 2 . Z ( T, V, µ ) = X i ln Z i ( T, V, µ ) = V X i g i π Z ∞ ± p dp ln (cid:20) ± λ i exp (cid:18) − ε i ( p ) T (cid:19)(cid:21) , (2)where the pressure can be derived as T ∂ ln Z ( T, V, µ ) /∂V , ± stands for fermions and bosons, respec-tively. ε i = (cid:0) p + m i (cid:1) / is the dispersion relation and λ i is the fugacity factor of the i -th particle[20], λ i ( T, µ ) = exp (cid:18) B i µ b + S i µ S + Q i µ Q T (cid:19) , (3)where B i ( µ b ), S i ( µ S ), and Q i ( µ Q ) are baryon, strangeness, and electric charge quantum numbers (theircorresponding chemical potentials) of the i -th hadron, respectively. From phenomenological point ofview, the baryon chemical potential µ b - along the chemical freezeout boundary, where the productionof particles is conjectured to cease - can be related to the nucleon-nucleon center-of-mass energy √ s NN [68] µ b = a b √ s NN , (4)where a = 1 . ± .
049 GeV and b = 0 . ± .
028 GeV − . In addition to pressure, the numberand energy density, respectively, and likewise the entropy density and other thermodynamics canstraightforwardly be derived from the partition function by taking the proper derivatives n i ( T, µ ) = X i g i π Z ∞ p dp h µ i − ε i ( p ) T i ± , (5) ρ i ( T, µ ) = X i g i π Z ∞ p dp − ε i ( p ) ± µ i exp h µ i − ε i ( p ) T i ± . (6)It should be noticed that both T and µ = B i µ b + S i µ S + · · · are related to each other and to √ s NN [20]. As an overall thermal equilibrium is assumed, µ S is taken as a dependent variable to be estimateddue to the strangeness conservation, i.e. at given T and µ b , the value assigned to µ S is the one assuring h n S i− h n ¯ S i = 0. Only then, µ S is combined with T and µ b in determining the thermodynamics, such asthe particle number, energy, entropy, etc. The chemical potentials related to other quantum charges,such as the electric charge and the third-component isospin, etc. can also be determined as functionsof T , µ b , and µ S and each of them must fulfill the corresponding conservation laws. B. Quantum-statistically correlated HRG
As introduced in the previous section, for a quantum gas of fermions and bosons with mass m i and correlation (interaction) distance r , at temperature T and vanishing µ b , a two-particle interactionprobability of the form 1 ± exp (cid:0) − π m i T r (cid:1) (7)was first introduced by Beth and Uhlenbeck [1] in an attempt to model the interactions of a quantumgas of particles assuming a general potential and neglecting the possibility for bound states formation.The Boltzmann-like term exp (cid:0) − π m i T r (cid:1) remains in effect even for an ideal gas, which is a typicalapproximation at sufficiently high temperatures. The ± sign expresses the apparent attraction (re-pulsion) between bosons (fermions) due to change of statistics [1]. Inspired by such a correction, weintroduce a correction for the probability term in the expression for the the ideal hadron gas partitionfunction given in Eq. (1). We propose a new probability term of the form1 ± λ i exp (cid:18) − ε i ( p ) T (cid:19) (cid:2) ± exp (cid:0) − π m i T r (cid:1)(cid:3) . (8)This corrected probability function obviously incorporates interactions in the the hadron reso-nance gas in the sense of Beth and Uhlenbeck quantum correlations [1] with r being the correlation(interaction) length between any two hadrons at equilibrium temperature T . Based on our proposedcorrected probability function, we modify the non-correlated HRG partition function Z ( T, µ, V ) tohave the following formln Z ′ ( T, V, µ ) = X i V g i π Z ∞ ± p dp ln (cid:20) ± λ i exp (cid:18) − ε i ( p ) T (cid:19) (cid:2) ± exp (cid:0) π m i T r (cid:1)(cid:3)(cid:21) , (9)which apparently sums over all hadron resonances following the same recipe described in motivatingEq. (2) for the case of non-correlated HRG. The thermodynamics of the correlated HRG can thusbe calculated by taking the proper derivatives of ln Z as explicitly stated in the corresponding non-correlated HRG case discussed above. III. CALCULATION RESULTS
We confront the data of the thermodynamics calculated using our statistically corrected HRG modelbased on Eq. (9) with the corresponding lattice thermodynamics data from [69, 70] in the temperaturerange T ∈ [130 ,
200 MeV]. These temperatures are rather typical for the phenomenological applicationsin the context of heavy-ion collisions and lattice QCD equation-of-state. In refs. [69, 70], the authorscalculated the QCD equation of state using Taylor expansions that include contributions from up tosixth order in the baryon, strangeness and electric charge chemical potentials. Calculations have beenperformed with a highly improved staggered quark action in the temperature range T ∈ [130 ,
330 MeV]using up to four different sets of lattice cut-offs. The lattice data we are confronting to our model areshown on Figure 7. of ref. [69], (Left) the total pressure in (2+1)-flavor QCD and (Right) the totalenergy density in (2+1)-flavor QCD for several values of µ b /T .Figure 1 of this letter depicts the temperature dependence of the normalized pressure P/T , nor-malized energy density ρ/T , and trace anomaly ( ρ − P ) /T (dashed curves) calculated using ourstatistically corrected HRG model based on Eq. (9). Moreover, in Figure 1, our model data are con-fronted to the corresponding lattice data taken from ref. [69] (symbols with error bars) at vanishingbaryon chemical potential µ b = 0 M eV . Comparison is made for four different values of the correlationlength r . Table I lists χ /dof statistic for the normalized pressure P/T , trace anomaly ( ρ − P ) /T and normalized energy density ρ/T calculated in our statistically corrected hadron resonance gas(HRG) model and confronted to the corresponding lattice data from [69] for four values of baryonchemical potential µ b = 0 , , χ /dof ≃ T c ⋍
160 MeV). In the range T ∈ [130 ,
200 MeV], the discrepancybetween our model thermodynamics and the corresponding lattice data is amplified. Generally, it isobvious that increasing the correlation length emphasizes the mismatch between our model and latticedata.For the case µ b = 170 MeV and as it appears in Fig. 2, the best fit generally occurs for r = 0
60 80 100 120 140 160 180 200
T [MeV] =0 Mev b µ r=0 fm & LQCD_p/T 0.00065 ± /dof=0.020 χ , HRG_p/T /T ∈ LQCD_ 0.0034 ± /dof=0.080 χ , /T ∈ HRG_ -3p)/T ∈ LQCD_( 0.0025 ± /dof=0.061 χ , -3p)/T ∈ HRG_(
60 80 100 120 140 160 180 200
T [MeV] = 0 Mev b µ r = 0.05 fm & LQCD_p/T 0.00037 ± /dof=0.013 χ , HRG_p/T /T ∈ LQCD_ 0.0012 ± /dof=0.089 χ , /T ∈ HRG_ -3p)/T ∈ LQCD_( 0.0011 ± /dof=0.097 χ , -3p)/T ∈ HRG_(
60 80 100 120 140 160 180 200
T [MeV] = 0 Mev b µ r = 0.1 fm & LQCD_p/T 0.00051 ± /dof=0.009 χ , HRG_p/T /T ∈ LQCD_ 0.0033 ± /dof=0.174 χ , /T ∈ HRG_ -3p)/T ∈ LQCD_( 0.0021 ± /dof=0.091 χ , -3p)/T ∈ HRG_(
60 80 100 120 140 160 180 200
T [MeV] = 0 Mev b µ r = 0.15 fm & LQCD_p/T 0.0010 ± /dof=0.102 χ , HRG_p/T /T ∈ LQCD_ 0.0052 ± /dof=0.370 χ , /T ∈ HRG_ -3p)/T ∈ LQCD_( 0.0018 ± /dof=0.024 χ , -3p)/T ∈ HRG_(
Fig.
1: Normalized pressure
P/T , normalized energy density ρ/T and trace anomaly ( ρ − P ) /T (dashedcurves) calculated using our statistically corrected HRG model and confronted to the corresponding lattice datataken from ref. [69] (symbols with error bars), at µ b = 0 MeV. Comparison is made for four different values ofthe correlation length r . and for r = 0 .
05 fm. This good-fit temperature range extends from temperatures well below T c till T & T c ⋍
160 MeV. It is quite obvious here that the model fits the lattice data better comparedto the corresponding vanishing chemical potential case(s). However, in the temperature range T ∈ [170 ,
200 MeV], the mismatch of our model with the lattice data becomes more pronounced comparedto the corresponding range of the vanishing chemical potential case(s).For the case of µ b = 340 MeV, see Fig. 3, the only interesting observation is that for r = 0 .
15 fm,the model data well below and in the vicinity of T c and up to T ⋍
170 MeV significantly approachesthe corresponding lattice data. The data mismatch then diverges for higher temperatures.For the case of µ b = 425 MeV, Fig. 4, the data mismatch is generally too large to suggest anyplausible correlation at any of values of r and for all temperatures of interest.
60 80 100 120 140 160 180 200
T [MeV] =170 Mev b µ r=0 fm & LQCD_p/T 0.00020 ± /dof=0.002 χ , HRG_p/T /T ∈ LQCD_ 0.0010 ± /dof=0.036 χ , /T ∈ HRG_ -3p)/T ∈ LQCD_( 0.0050 ± /dof=0.3698 χ , -3p)/T ∈ HRG_(
60 80 100 120 140 160 180 200
T [MeV] = 170 Mev b µ r = 0.05 fm & LQCD_p/T 0.00041 ± /dof=0.008 χ , HRG_p/T /T ∈ LQCD_ 0.002 ± /dof=0.062 χ , /T ∈ HRG_ -3p)/T ∈ LQCD_( 0.0045 ± /dof=0.3190 χ , -3p)/T ∈ HRG_(
60 80 100 120 140 160 180 200
T [MeV] = 170 Mev b µ r = 0.1 fm & LQCD_p/T 0.00062 ± /dof=0.010 χ , HRG_p/T /T ∈ LQCD_ 0.0035 ± /dof=0.075 χ , /T ∈ HRG_ -3p)/T ∈ LQCD_( 0.0011 ± /dof=0.1452 χ , -3p)/T ∈ HRG_(
60 80 100 120 140 160 180 200
T [MeV] = 170 Mev b µ r = 0.15 fm & LQCD_p/T 0.0009 ± /dof=0.028 χ , HRG_p/T /T ∈ LQCD_ 0.0051 ± /dof=0.145 χ , /T ∈ HRG_ -3p)/T ∈ LQCD_( 0.0017 ± /dof=0.069 χ , -3p)/T ∈ HRG_(
Fig.
2: The same as in Fig. 1 but at µ b = 170 MeV. IV. CONCLUSIONS
We confronted a novel statistical correction of the HRG model with recent lattice data [69, 70]. Allour model calculations considered in this study doesn’t seem to satisfactorily mimic the correspondinglattice data in the full temperature range under investigation, T ∈ [130 ,
200 MeV]. However, thebest matching occurs locally in the vicinity of T c in the range T ∈ [140 ,
170 MeV] for the case of µ b = 170 MeV at zero and 0 . f m correlation radii, r , respectively. Another remarkable matchingbetween our model data with the corresponding lattice data occurs for the the case of µ b = 340 MeV,at 0 . .
15 fm correlation radii, respectively for temperatures T . T c and up to T ⋍
170 MeV. Inthe lower temperature range, T ∈ [130 ,
160 MeV], most of the cases investigated in this research showreasonable match with the corresponding lattice data for different correlation lengths except for thecase in which µ b = 425 MeV where no good fitting is observed for any correlation length. [1] G. Uhlenbeck and E. Beth, Physica , 915 (1937).[2] T. Banks and A. Ukawa, Nuclear Physics B , 145 (1983).
60 80 100 120 140 160 180 200
T [MeV] =340 Mev b µ r=0 fm & LQCD_p/T 0.0014 ± /dof=0.020 χ , HRG_p/T /T ∈ LQCD_ 0.0040 ± /dof=0.241 χ , /T ∈ HRG_ -3p)/T ∈ LQCD_( 0.0028 ± /dof=0.853 χ , -3p)/T ∈ HRG_(
60 80 100 120 140 160 180 200
T [MeV] = 340 Mev b µ r = 0.05 fm & LQCD_p/T 0.0016 ± /dof=0.010 χ , HRG_p/T /T ∈ LQCD_ 0.011 ± /dof=1.206 χ , /T ∈ HRG_ -3p)/T ∈ LQCD_( 0.0062 ± /dof=2.46 χ , -3p)/T ∈ HRG_(
60 80 100 120 140 160 180 200
T [MeV] = 340 Mev b µ r = 0.1 fm & LQCD_p/T 0.0010 ± /dof=0.005 χ , HRG_p/T /T ∈ LQCD_ 0.0093 ± /dof=0.399 χ , /T ∈ HRG_ -3p)/T ∈ LQCD_( 0.0075 ± /dof=1.31 χ , -3p)/T ∈ HRG_(
60 80 100 120 140 160 180 200
T [MeV] = 340 Mev b µ r = 0.15 fm & LQCD_p/T 0.0007 ± /dof=0.015 χ , HRG_p/T /T ∈ LQCD_ 0.005 ± /dof=0.101 χ , /T ∈ HRG_ -3p)/T ∈ LQCD_( 0.0029 ± /dof=0.335 χ , -3p)/T ∈ HRG_(
Fig.
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60 80 100 120 140 160 180 200
T [MeV] = 425 Mev b µ r=0 fm & LQCD_p/T 0.0012 ± /dof=0.065 χ , HRG_p/T /T ∈ LQCD_ 0.15 ± /dof=2.12 χ , /T ∈ HRG_ -3p)/T ∈ LQCD_( 0.0007 ± /dof=4.28 χ , -3p)/T ∈ HRG_(
60 80 100 120 140 160 180 200
T [MeV] = 425 Mev b µ r = 0.05 fm & LQCD_p/T 0.0030 ± /dof=0.351 χ , HRG_p/T /T ∈ LQCD_ 0.024 ± /dof=3.3 χ , /T ∈ HRG_ -3p)/T ∈ LQCD_( 0.0019 ± /dof=6.0 χ , -3p)/T ∈ HRG_(
60 80 100 120 140 160 180 200
T [MeV] = 425 Mev b µ r = 0.1 fm & LQCD_p/T 0.0091 ± /dof=0.035 χ , HRG_p/T /T ∈ LQCD_ 0.017 ± /dof=1.68 χ , /T ∈ HRG_ -3p)/T ∈ LQCD_( 0.0012 ± /dof=2.77 χ , -3p)/T ∈ HRG_(
60 80 100 120 140 160 180 200
T [MeV] = 425 Mev b µ r = 0.15 fm & LQCD_p/T 0.0015 ± /dof=0.058 χ , HRG_p/T /T ∈ LQCD_ 0.0082 ± /dof=0.376 χ , /T ∈ HRG_ -3p)/T ∈ LQCD_( 0.0038 ± /dof=1.44 χ , -3p)/T ∈ HRG_(
Fig.
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