Overlap between Lattice QCD and HRG with inmedium effects
and parity doubling^{†}^{†}thanks: Presented at Strangeness in Quark Matter, Utrecht, the Netherlands, 1015 July 2017
Abstract
We investigate the fluctuations and correlations involving baryon number in hot hadronic matter with modified masses of negativeparity baryons, in the context of the hadron resonance gas. Temperaturedependent masses are adopted from the recent lattice QCD results and from a chiral effective model which implements the parity doubling structure with respect to the chiral symmetry. Confronting the baryon number susceptibility, baryoncharge correlation, and baryonstrangeness correlation and their ratios with the lattice QCD data, we show that the strong downward mass shift in hyperons can accidentally reproduce some correlation ratios, however it also tends to overshoot the individual fluctuations and correlations. This indicates, that in order to correctly account for the influence of the chiral symmetry restoration on the fluctuation observables, a consistent framework of inmedium effects beyond hadron mass shifts is required.
Overlap between Lattice QCD and HRG with inmedium effects
and parity doubling^{}^{}Presented at Strangeness in Quark Matter, Utrecht, the Netherlands, 1015 July 2017
Kenji Morita, Chihiro Sasaki, Pok Man Lo, and Krzysztof Redlich^{}^{}Speaker, email: redlich@ift.uni.wroc.pl
1Institute of Theoretical Physics, University of Wrocław, PL50204, Wrocław, Poland
\@textsuperscript2iTHES Research Group, RIKEN, Saitama 3510198, Japan
\@textsuperscript3ExtreMe Matter Institute EMMI, GSI, D64291 Darmstadt, Germany

Abstract.
1 Introduction
Fluctuations and correlations of conserved charges provide diagnostic tools for the nature of strongly interacting matter described by Quantum Chromodynamics (QCD). The firstprinciples calculations by lattice QCD (LQCD) have provided not only equation of state at the physical quark masses but also the fluctuations and correlations of the netbaryon, netelectric charge, and netstrangeness [1, 2, 3, 4, 5, 6]. The fluctuations and correlations can also be measured in heavyion collisions to identify the state of created matter. In particular, nonGaussian (higherorder) fluctuations have been expected to probe critical properties of the system, such as the QCD critical point in the beamenergy scan program at RHIC and a remnant of the criticality at small baryon density [7, 8].
For a physical interpretation of the thermodynamic properties, the hadron resonance gas (HRG) model [9, 10] has been used as a reference. The equation of state and fluctuations have been well described by the model below the chiral crossover temperature MeV [11, 12], and the experimental data of fluctuations have been analyzed in terms of a gas of hadronic states [13, 14, 15, 16]. At vanishing baryon density, the critical behavior due to the chiral phase transition appears at the sixth order of netbaryon fluctuations [8]. This is because the singular contribution to the free energy is suppressed in the lowerorder fluctuations even in the vicinity of the expected second order transition. Therefore, one expects that the lower order fluctuations and correlations may be well described by the hadronic degrees of freedom, and they would provide a reliable baseline for exploring the critical behavior [7].
According to the recent LQCD calculations, however, some correlations between conserved charges cannot be explained by a conventional HRG model. Particularly interesting quantities are those involving netbaryon number, baryoncharge (BQ) and baryonstrangeness (BS) correlations. Since mesons do not contribute to these quantities, one may gain access to a role of baryonic degrees of freedom and interactions which are not visible in the equation of state and other mesondominated quantities due to their heavy masses [17].
In the HRG model the interaction of hadrons is replaced by resonances, and in the first approximation their widths can be neglected. Then, the partition function of the interacting hadronic system can be written as a mixture of free gases of all stable and resonant hadrons [18]. The validity of the vanishingwidth approximation for the fluctuations has been recently examined based on the Smatrix formalism for – [19] and –– [20] systems. It has been found that an explicit treatment of the width can have a substantial effect on the fluctuations.
On the other hand, at finite temperature and density one expects changes in the spectral property of hadrons. In particular, a search for those modifications because of the partial restoration of the QCD chiral symmetry has been one of the central subjects in heavyion collisions [21]. Since the vacuum masses are used in the HRG model, the observed agreement of the equation of state in the HRG and LQCD suggests no need for a strong mass reduction in the dominant degrees of freedom such as light mesons. Although chiral symmetry predicts substantial medium modifications of lowlying mesons [22, 23], their reliable estimates in LQCD have been limited to the screening masses [24].
Recently, masses of the nonstrange and strange baryons with positive and negative parity have been extracted from the temporal correlation functions by the FASTSUM collaboration [25, 26]. The negativeparity states clearly show downward mass shifts, whereas the positiveparity baryons stay insensitive to temperature. The obtained spectra follow an expectation from the parity doublet picture of the chiral symmetry, indicating that the masses of a negative and positiveparity partners tend to degenerate when approaching the chiral crossover. The above medium modification was used to possibly explain a missing contribution in the correlations between the conserved charges [26].
In the following we investigate the fluctuations and correlations of the netbaryon with netcharge and netstrangeness on the basis of the HRG model implementing inmedium mass modifications. We employ temperature dependent masses of the negativeparity octet and decouplet baryons from the lattice QCD and from a chiral effective model with parity doubling [27]. We show that the strong downward mass shift in hyperons can accidentally reproduce some correlation ratios, however it also tends to overshoot the individual fluctuation and correlation observables. This indicates that to quantify fluctuation and correlation of conserved charges in the presence of chiral symmetry restoration it is not sufficient to implement inmedium hadron masses in the statistical sum of the hadron resonance gas.
2 Baryon masses in parity doublet picture
2.1 Lattice QCD
P State  

Mass LGT [25, 26]  1779  1899  1823  1917  2138  2131  2164  2193 
[MeV]  1254  1172  1329  1295  1405  1398  1426  1383 
0.338  0.369  0.257  0.275  0.312  0.257  0.246  0.213  
Assignment A  2250  
Assignment B  1700  
Mass [MeV]  1535  1790  1880  2090  1710  1930  2150  2380 
Assignment C  1535 
The FASTSUM collaboration presented the temperature dependent masses of , , and states, extracted from imaginary time correlators of the corresponding interpolating operators in lattice simulations [25]. Their calculations were performed for heavier lightquark mass than the physical one; MeV, while the strange quark is set to the physical one. Thus, the mass of baryons except are heavier than the physical ones. This yields the nucleon with positive parity has MeV (939 MeV in PDG), while the positive parity has MeV (1672 MeV in PDG). The results of other octet and decuplet states have been shown in [26]. Here we use the masses at , , , and . Note that owing to the heavier pion mass, the crossover temperature MeV, which was determined from the renormalized Polyakov loop, is also higher than the physical one, MeV.
In Table 1, we list the masses of the octet and decuplet negativeparity states at . They are well parameterized, for the given the state , by [25, 26]
(1)  
(2) 
where is fixed to be the value obtained at and and are the fitting parameters fixed for each channel. The data are given as a function of ; the value of does not affect the fit but later it is set to the physical value 154 MeV. The parameter corresponds to the width of the chiral crossover. The results of the fits are shown in Table 1.
To correct the unphysical effect from the heavy up and down quarks, is set to its PDG mass in the following. The value of needs to be corrected as well, and we shall rescale it by multiplying the factor with .
It is not clear how to assign the negativeparity states to the observed ones because of unknown quantum numbers of some of the candidates. We follow the suggestion in [26] and further adopt two different assignments, A and B. The set A takes the lighter state for , , and while the set B does the heavier ones.
The obtained temperaturedependent masses are displayed in Figs. 1 and 2. The masses of negativeparity states start to drop around , then approach those of their positiveparity partners. In comparing the nonstrange baryons with the hyperons, one finds the correction of significantly affects the baryons with more light quarks, as expected.
2.2 Chiral model
The paritydoubled baryons can be modeled in an effective chiral approach [28], where a chiralinvariant mass is naturally introduced. In [27], the mass relations to the light quark and strange quark condensates are given for the octet and decuplet states as ^{1}^{1}1 In [27] the same was assumed common to the octet and decuplet baryons. Here we lift this constraint and introduce two independent masses. Their splitting can be deduced from the spinspin interaction.
(3) 
The parameters are related via the GellMann–Okubo relation as
(4) 
The parameters in the above expressions are determined at zero temperature as in Table 2.
[GeV]  [GeV]  [GeV]  

The masses at are also summarized in Table 1. Thermal modifications of the baryon masses are driven by the quark condensates. Following [27], we shall use the inmedium condensates measured in LQCD by the HotQCD collaboration [12]. One readily finds that some of the hyperon states have larger masses than those expected from the lattice baryonspectrum in [25, 26]. Thus, we assign these hyperons to the PDG states as in the bottom line of Table 1 and refer to set C. For a comparison with LQCD, we also apply the set C to the latticemotivated parametrization (1).
The nonstrange baryons exhibit the thermal behavior as in Fig. 1. To some extent, their trends are similar to the results by the FASTSUM collaboration, whereas the parametrization (2) leads to a sizable difference in the mass dropping near . This is particularly evident when the rescaling of is made.
Figure 3 displays the mass of the hyperons in the assignment C. The most distinct difference from the nonstrange baryons lies in the mass of the hyperons near ; the model calculation shows a rather moderate downwardshift, while the lattice QCD results lead to a strong mass drop. Comparing (), , and states between the model and the latticeinspired scaling, one clearly finds that the difference becomes more significant as the baryon contains more strangeness. This may be attributed to the difference in the masses between the light and strange quarks ; The FASTSUM setup with a large pion mass MeV and the physical kaon mass is rather close to the flavor SU(3) limit, while the model calculation done with the physical pion mass is dominated by SU(2). Therefore, it will be intriguing to see whether the strong mass reduction of and would still persist in the lattice simulations with a lighter pion.
3 Fluctuations and correlations from inmedium HRG
We employ the hadron resonance gas model to explore the fluctuations and correlations in the hot hadronic matter. In this work we entirely rely on the zerowidth approximation; all the resonances are treated as pointlike particles and their spectral functions are given by the delta function. Consequently, the thermodynamic pressure at temperature and chemical potentials is given by the summation of ideal gas pressure of each particle species as
(5) 
where is the chemical potential of particle with baryon number , electric charge , and strangeness and the ideal gas pressure with degeneracy reads
(6) 
The signs are negative for mesons and positive for baryons. The fluctuations and the correlations at vanishing chemical potentials can be expressed as the generalized susceptibilities
(7) 
Although heavy particles () are thermally suppressed, the existence of missing states are expected likely in the strange sectors [29, 30]. In fact, the assignment C in Table 1 contains several 1star states whose spins are undetermined in the latest PDG [31]. For those state we shall assume the same spin with its partner. For consistency we also include other unconfirmed state; we have in total 28 nucleons, 22 baryons, 18 baryons, 21 baryons, 8 baryons, 3 baryons, and their isospin multiplets. We also include a deuteron, triton, He and He as they are considered to be thermal ingredients in QCD thermodynamics and heavy ion collisions [10].
In this work we focus on , , and at . Since , mesons do not contribute to these susceptibilities. Thus they are good measures of inmedium effects in the baryon sector on the QCD thermodynamics. In general, particle masses in a medium can depend not only on temperature but also on chemical potentials. We neglect such an intrinsic chemicalpotential dependence in the baryon masses in the present calculations. This is well justified at small chemical potential since the quark condensates are not much affected.
We display , , and in Fig. 4. In the baryon number susceptibility , the HRG model agrees with the LQCD results below . ^{2}^{2}2The present HRG leads to a slightly larger than the conventional one because of including the 1star states and the multibaryons. With the inmedium mass shift, a decrease in the negativeparity baryon masses lead to an enhancement of the susceptibility. One sees a moderate enhancement in all the three scenarios, Lattice (A) (B) and (C), and it becomes stronger when is corrected, since the correction further reduces the masses just below . On the other hand, the different assignments of the paritypartners lead to a minor change. The results with the masses from the chiral effective model follows the same trend, but the amount of the enhancement is much smaller than the parameterization, owing to the weaker downward mass shifts in the hyperon sectors (see Fig. 3.) In any case, all the results with mass reduction overshoot the lattice data of .
The shows a different tendency in Fig. 4(b); the results of Lattice (A)(C) without the correction of do not exhibit any enhancement from the HRG, even below the “Model” result. In the , states do not contribute because neutral particles do not contribute and the positivelycharged states cancel with the negativelycharged states. Charged states and doubly charged states contribute positively, while negativelycharged , and states contribute oppositely. Without a strong massshift of these multistrange states, the becomes larger due to the smallness of the hyperon contribution suppressed by the Boltzmann factor, as in the “Model” case. With the strong mass shift of the hyperons, they tend to cancel a part of the positive contribution, and suppress the . The strong enhancement with the corrected clearly signals that the effect of the correction is smaller for states and larger for and states, as seen in Figs. 13.
The behavior of in Fig. 4(c) can be understood in a similar way to the ; the “Model” result does not differ much from the HRG, because of a weaker mass reduction. Owing to the different trend in the hyperon sectors, the curves deviate more sensitively depending on the fate of the masses.
The observed differences can be more pronounced by taking the ratio between the susceptibilities. In Fig. 5(a), we display the together with the LQCD data. The results from Lattice (A) and (B) reproduce the one shown in [26]; they follows the trend seen in the HotQCD data. As explained above, however, the individual susceptibilities cannot be reproduced. The coincidence comes from the reduction of owing to the strong mass reductions in the charge asymmetric states. The “Model” case shows the opposite trend that the ratio is above the HRG, due to the much milder massreduction in the hyperon sectors. Therefore, the data cannot be explained solely by the mass reduction.
Also shown in Fig. 5(b) is the . Again, while the sets (A) and (B) follow the trend of the HotQCD data, the “Model” shows the opposite. This supports the observation that naive insertion of the inmedium masses of parity doublers into the HRG results in fluctuation observables which are inconsistent with LGT data. Nevertheless, our results do not rule out the manifestation of parity doublers. Several studies [19, 20] have revealed that the conventional treatment of the particles in the HRG is not sufficient and a proper inclusion of the width improves the thermodynamics. Although those analysis have been limited to the matrix approaches where interactions are incorporated via the twobody scattering phase shift in the vacuum, a consistent treatment of the resonance widths at finite temperature and densities should clarify the consequences of the partial restoration of the chiral symmetry in the baryonic sector of correlations and fluctuations.
4 Concluding remarks
We have studied the fluctuations and correlations of conserved charges by making use of the hadron resonance gas model with inmedium masses of the lowlying paritydoubled baryons. Motivated by the recent LQCD result on the baryon masses at finite temperature, we have examined the consequences of several scenarios with inmedium masses for the negativeparity octet and decouplet baryons, adopting a latticemotivated parametrization and that from a chiral effective theory. We have computed the baryon number fluctuation , baryoncharge correlation , and baryonstrangeness correlation . The results have been understood on the basis of the presence or absence of the strong mass reduction in the hyperon sectors. We have pointed out that the reproduction of the LQCD results of the ratios and is just accidental when the inmedium masses are naively introduced in the conventional HRG approach. Since the strong mass shifts of the hyperons observed in LQCD can be regarded as a consequence of the approximate SU(3) nature due to a heavy pion mass, it is a critical issue to be examined whether the hyperon massshift persists for the physical pion mass.
Although our results clarify the interplay of the different particle species in the behavior of the baryoncharge and baryonstrangeness correlations, we emphasize that the treatment of the resonances in the conventional HRG model is insufficient to quantify correlation and fluctuations of conserved charges. This indicates, that in order to correctly account for the influence of the chiral symmetry restoration on the fluctuation observables, a consistent framework of inmedium effects beyond hadron mass shifts is required.
Acknowledgments
We acknowledge stimulating discussions with Gábor Almási and Bengt Friman. This work has been partly supported by the Polish Science Center NCN under Maestro grant 2013/10/A/ST2/00106 and RIKEN iTHES project.
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