Interplay Between Chiral Dynamics and Repulsive Interactions

Interplay between chiral dynamics and repulsive interactions

Michal Marczenko,∗ Krzysztof Redlich, and Chihiro Sasaki Institute of Theoretical Physics, University of Wroc law, plac Maksa Borna 9, PL-50204 Wroc law, Poland (Dated: January 1, 2021) We investigate fluctuations of the net-baryon number density in hot hadronic matter. We discuss the interplay between chiral dynamics and repulsive interactions and their influence on the properties of these fluctuations near the chiral crossover. The chiral dynamics is modeled by the parity doublet Lagrangian that incorporates the attractive and repulsive interactions mediated via the exchange of scalar and vector mesons, respectively. The mean-field approximation is employed to account for chiral criticality. We focus on the properties and systematics of the cumulants of the net- baryon number density up to the sixth order. It is shown that the higher-order cumulants exhibit a substantial suppression in the vicinity of the chiral phase transition due to the presence of repulsive interactions. We find, however, that the characteristic properties of cumulants near the chiral crossover observed in lattice QCD results are entirely linked to the critical chiral dynamics and, in general, cannot be reproduced in phenomenological models, which account only for effective repulsive interactions via excluded-volume corrections or van-der-Waals type interactions. Consequently, a description of the higher-order cumulants of the net-baryon density in the chiral crossover requires a self-consistent treatment of the chiral in-medium effects and repulsive interactions.

I. INTRODUCTION of QCD is well-described by the hadron resonance gas (HRG) model [40, 41]. Since the fundamental quarks Establishing the thermodynamic properties of strongly and gluons are confined, it is to be expected that at interacting matter, described by quantum chromody- low temperatures the QCD partition function is domi- namics (QCD), is one of the key directions in modern nated by the contribution of hadrons. The HRG model high-energy physics. At vanishing density, the first- describes well the LQCD data below the crossover tran- principles calculations of lattice QCD (LQCD) provide sition to a quark-gluon plasma, as well as the hadron a reliable description of the equation of state (EoS) and yields in heavy-ion collisions [41]. Different extensions fluctuations of conserved charges [1–4]. There, the EoS of the HRG model have been proposed to quantify the exhibits a smooth crossover from confined hadronic mat- LQCD EoS and various fluctuation observable up to near ter to a deconfined quark-gluon plasma, which is linked chiral-crossover. They account for consistent implemen- to the color deconfinement and the restoration of chiral tation of hadronic interactions within the S-matrix ap- symmetry [5, 6]. However, the nature of the EoS at finite proach [42–46], a more complete implementation or a density is still not resolved by LQCD, owing to the sign continuously growing exponential mass spectrum and/or problem, and remains an open question. possible repulsive interactions among constituents [47– The LQCD results [7–9] exhibit a clear manifestation 53]. Recently, an interesting suggestion was made that of the parity doubling structure for the low-lying baryons deviations of the LQCD data on higher-order fluctuations around the chiral crossover. The masses of the positive- of net-baryon number density from the HRG baseline in parity ground states are found to be rather temperature- the near vicinity of the chiral transition can be attributed independent, while the masses of negative-parity states to repulsive interactions among constituent hadrons [54]. drop substantially when approaching the chiral crossover Fluctuations of conserved charges are known to be aus- temperature Tc. The parity doublet states become al- picious observable for the search of the chiral-critical be- most degenerate with a finite mass in the vicinity of the havior at the QCD phase boundary [55–57], and chem- arXiv:2012.15535v1 [hep-ph] 31 Dec 2020 chiral crossover. Even though these LQCD results are ical freeze-out of produced hadrons in heavy-ion colli- still not obtained in the physical limit, the observed be- sions [58–63]. In particular, fluctuations have been pro- havior of parity partners is likely an imprint of the chi- posed to probe the QCD critical point in the beam energy ral symmetry restoration in the baryonic sector of QCD. scan (BES) programs at the Relativistic Heavy Ion Col- Such properties of the chiral partners can be described in lider (RHIC) at BNL and the Super Proton Synchrotron the framework of the parity doublet model [10–12]. The (SPS) at CERN, as well as the remnants of the O(4) crit- model has been applied to hot and dense hadronic mat- icality at vanishing and finite baryon densities [63–66]. ter, neutron stars, as well as the vacuum phenomenology of QCD [13–39]. In this work, we analyze the properties and systematics It is already confirmed, that at small net-baryon num- of the fluctuations of conserved charges in the context of ber density, the thermodynamics of the confined phase the parity doublet model, which incorporates the chiral symmetry restoration and repulsive interactions via the exchange of the scalar and vector mesons, respectively. To account for critical behavior, the mean-field approxi- ∗ [email protected] mation is employed, which contains basic features of the 2

O(4) criticality, albeit with diﬀerent critical exponents. the vector ﬁeld, and the potentials read We study the behavior of the second- and higher-order cumulants of the net-baryon number density up to the λ2 λ4 2 λ6 3 Vσ = − Σ + Σ − Σ − σ, (4a) sixth order, as well as the bulk equation of state. It 2 4 6 is systematically examined to what extent the thermal m2 V = − ω ω ωµ. (4b) behaviors are dominated by the chiral criticality and re- ω 2 µ pulsive interactions. 2 2 2 4 2 This paper is organized as follows. In Sec. II, we in- where Σ = σ + π , λ2 = λ4fπ − λ6fπ − mπ, and 2 troduce the parity doublet model. In Sec. III, we discuss = mπfπ. mπ and mω are the π and ω meson masses, the structure of the higher-order cumulants of the net- respectively, and fπ is the pion decay constant. Note baryon number density. In Sec. IV, we present results on that the chiral symmetry is explicitly broken by the lin- the equation of state and the higher-order cumulants of ear term in σ in Eq. (4a). the net-baryon number density. Finally, Sec. V is devoted The full Lagrangian of the parity doublet model is then to summary and conclusions. L = LN + LM . (5)

In the diagonal basis, the masses of the chiral partners, II. PARITY DOUBLET MODEL N±, are given by q 1 2 2 2 In the conventional Gell-Mann–Levy model of mesons m± = (g1 + g2) σ + 4m0 ∓ (g1 − g2) σ . (6) and nucleons [67], the nucleon mass is entirely generated 2 by the non-vanishing expectation value of the sigma field. From Eq. (6), it is clear that, in contrast to the naive Thus, the nucleon inevitably becomes massless when the assignment under chiral symmetry, the chiral symmetry chiral symmetry gets restored. This is led by the partic- breaking generates only the splitting between the two ular chirality-assignment to the nucleon parity doublers, masses. When the symmetry is restored, the masses be- where the nucleons are assumed to be transformed in the come degenerate, m±(σ = 0) = m0. same way as the quarks are under chiral rotations. To investigate the properties of strongly-interacting More general allocation of the left- and right-handed matter, we adopt a mean-field approximation [68]. Ro- chiralities to the nucleons, the mirror assignment, was tational invariance requires that the spatial component proposed in [10]. This allows an explicit mass-term for #1 of the ωµ field vanishes, namely hωi = 0 . Parity the nucleons, and consequently, the nucleons stay mas- conservation on the other hand dictates hπi = 0. The sive at the chiral restoration point. For more details, see mean-field thermodynamic potential of the parity dou- Refs. [10–12]. blet model reads In the mirror assignment, under SU(2) × SU(2) ro- L R X tation, two chiral fields ψ1 and ψ2 are transformed as Ω = Ωx + Vσ + Vω, (7) follows: x=±

ψ1L → Lψ1L, ψ1R → Rψ1R, with (1) ψ2L → Rψ2L, ψ2R → Lψ2R, Z d3p Ω = γ T ln (1 − f ) + ln 1 − f¯ , (8) x x (2π)3 x x where ψi = ψiL + ψiR, L ∈ SU(2)L and R ∈ SU(2)R. The nucleon part of the Lagrangian in the mirror model where γ± = 2 × 2 denotes the spin-isospin degeneracy ¯ reads factor for both parity partners, and fx (fx) is the particle (antiparticle) Fermi-Dirac distribution function, ¯ ¯ ¯ ¯ LN = iψ1∂ψ/ 1 + iψ2∂ψ/ 2 + m0 ψ1γ5ψ2 − ψ2γ5ψ1 ¯ ¯ 1 + g1ψ1 (σ + iγ5τ · π) ψ1 + g2ψ2 (σ − iγ5τ · π) ψ2 fx = ∗ , β(Ex−µ ) ¯ ¯ 1 + e − gωψ1ωψ/ 1 − gωψ2ωψ/ 2, (9) ¯ 1 (2) fx = ∗ , 1 + eβ(Ex+µ ) where g1, g2, and gω are the baryon-to-meson coupling with β being the inverse temperature, the dispersion rela- p 2 2 constants and m0 is a mass parameter. tion Ex = p + mx and the eﬀective chemical potential ∗ The mesonic part of the Lagrangian reads µ = µB − gωω.

1 1 1 L = (∂ σ)2 + (∂ π)2 − (ω )2 − V − V , (3) M 2 µ 2 µ 4 µν σ ω #1 Since ω0 is the only non-zero component in the mean-ﬁeld approximation, we simply denote it by ω ≡ ω. where ωµν = ∂µων − ∂ν ωµ is the ﬁeld-strength tensor of 0 3

2 m0 [MeV] m+ [MeV] m− [MeV] mπ [MeV] fπ [MeV] mω [MeV] mρ [MeV] λ4 λ6fπ gω g1 g2 850 939 1500 140 93 783 775 13.15 4.67 4.64 12.42 6.39

TABLE I. Physical vacuum inputs and the parity doublet model parameters used in this work. See Sec. II for details.

In-medium profiles of the mean fields are obtained We note that the six-point scalar interaction term in by extremizing the thermodynamic potential in Eq. (7), Eq. (4a) is essential in order to reproduce the empiri- leading to the following gap equations: cal value of the compressibility in Eq. (15c) [38]. The parameters used in this paper are tabulated in Table I. ∂Ω ∂Vσ ∂m+ ∂m− 0 = = + s+ + s− , For this set of parameters, we obtain the critical tem- ∂σ ∂σ ∂σ ∂σ (10) perature of the chiral crossover transition at vanishing ∂Ω ∂Vω 0 = = + gω (n+ + n−), chemical potential, Tc = 208.7 MeV. ∂ω ∂ω In the following, we will also compare our results where the scalar and vector densities are with the hadron resonance gas (HRG) [48, 49, 69–72] Z 3 d p m± ¯ model formulation of the thermodynamics of the con- s± = γ± 3 f± + f± (11) (2π) E± fined phase of QCD. The thermodynamic potential of the HRG model is given as a sum of uncorrelated ideal-gas and particles: Z 3 d p ¯ n± = γ± f± − f± , (12) HRG X (2π)3 Ω = Ωx, (16) x=± respectively. In the grand canonical ensemble, the thermodynamic with Ωx given by Eq. (8), where the masses of N± pressure reads are taken to be the vacuum masses (see Table I) and ∗ P = −Ω + Ω , (13) µ = µB. We will also consider a modification of the 0 HRG model, σHRG, where the thermodynamic poten- where Ω0 is the value of the thermodynamic potential in tial is given as in Eq. (16), but the vacuum masses of the vacuum, and the net-baryon number density can be N ± are substituted by the in-medium masses obtained calculated as follows: by solving the parity doublet model. In both models, ∂P (T, µB) the pressure and net-baryon density are obtained through nB = . (14) Eqs. (13) and (14), respectively. ∂µB In the next section, we discuss the general structure of The positive-parity state, N , corresponds to the nu- + the higher-order cumulants of the net-baryon number. cleon N(938). Its negative parity partner is identified with N(1535). Their vacuum masses are shown in Ta- ble I. The value of the parameter m0 has to be cho- III. HIGHER-ORDER CUMULANTS OF THE sen so that a chiral crossover transition is featured at NET-BARYON NUMBER finite temperature and vanishing chemical potential. The model predicts the chiral transition to be a crossover for The fluctuations of conserved charges reveal more in- m 700 MeV. Following the previous studies of the 0 & formation about the matter composition than the equa- parity-doublet-based models [13–39], as well as recent tion of state and can be used as probes of a phase bound- lattice QCD results [7–9], we choose a rather large value, ary. The critical properties of chiral models, within the m = 850 MeV. We note that, the results presented in 0 functional renormalization group (FRG) approach [73– Sec. IV qualitatively do not depend on the choice of m , 0 76], are governed by the same universality classes as as long as chiral crossover transition is featured. The QCD, i.e., the chiral transition belongs to O(4) univer- parameters g and g are determined by the aforemen- 1 2 sality class, which, at large values of the baryon chemical tioned vacuum nucleon masses and the chirally invariant potential, may develop a Z(2) critical point, followed by mass m via Eq. (6). The parameters g , λ and λ are 0 ω 4 6 the first-order phase transition [77–79]. This criticality is fixed by the properties of the nuclear ground state at zero naturally encoded in the hadronic parity doublet model, temperature, i.e., the saturation density, binding energy as well as in quark-based models [64, 80–83], although the and compressibility parameter at µ = 923 MeV. The B mean-field treatment yields different critical exponents. constrains are as follows: The main objective of the present studies is to ana- −3 nB = 0.16 fm , (15a) lyze and delineate the contribution to thermodynamics

E/A − m+ = −16 MeV, (15b) from chiral dynamics and repulsive baryon-baryon interactions. To this end, we analyze the ﬂuctuations of the ∂2 (E/A) 2 net-baryon number at ﬁnite temperature and vanishing K = 9nB 2 = 240 MeV. (15c) ∂nB chemical potential. 4

HRG HRG σHRG σHRG 0.4 0.4 Parity Doublet Parity Doublet

0.3 0.3 4 2 χ

P/T 0.2 0.2

0.1 0.1

0 0 0.7 0.8 0.9 1 1.1 1.2 1.3 0.7 0.8 0.9 1 1.1 1.2 1.3

T [Tc] T [Tc]

FIG. 1. Thermodynamic pressure (left), and the second-order cumulant of the net-baryon number density (right), calculated under diﬀerent schemes, see text. The temperature is expressed in the units of the chiral-critical temperature, Tc = Tc(µB = 0).

∗ id In the grand canonical ensemble, the cumulants of chemical potential is µ = 0. Thus, χ2 contains only the the net-baryon number, χn, are commonly defined as contribution from the σ mean field. Therefore, it encodes temperature-normalized derivatives w.r.t. the baryon the information about attractive interactions, while the chemical potential, information about repulsive interactions is contained in the suppression factor βrep. To some extent, such sep- n−1 n−4 ∂ nB (T, µB) aration is qualitatively similar to that of the excluded χn (T, µB) = T n−1 , (17) ∂µB volume approach. We note that the expression for the second-order cumulant in Eq. (20) is exact at vanishing where nB is defined in Eq. (14). chemical potential. In the mean-field approximation, the net-baryon In similar spirit, one derives the higher-order cumu- number density, as well as any other thermodynamic lants as quantity, contains explicit dependence on the mean id n−1 fields. Here, we consider only σ and ω mean fields χn = χn βrep + ... (21) (cf. Eq. (7)), thus nB = nB (T, µB, σ(T, µB), ω(T, µB)). id Consequently, from Eq. (17) one derives the following where χn is the ideal gas expression for the n’th order general form of the second-order cumulant, cumulant. For n > 2, Eq. (21) contains extra terms, as explained in Appendix A. Nevertheless, keeping the first id ∂nB ∂σ term provides a relatively good approximation to the full χ2 = χ2 βrep + , (18) #2 ∂σ ∂µB expression . We note that the general structure of χn derived in Eq. (21) is the same in any kind of σ−ω model id id where χ2 = χ2 (T, µB, σ(T, µB), ω(T, µB)) is the ideal under the mean-field approximation and is independent gas expression for the net-baryon number susceptibility, of the details of the model. and From Eq. (21), keeping only the first term, we may also estimate the ratio of the cumulants: ∂ω βrep = 1 − gω (19) id ∂µB χn χn n−m = id βrep + ... (22) χm χ is the suppression factor due to repulsive interactions. m From Eq. (18) it is clear that the non-interacting ideal Clearly, the separation of the attractive and repulsive gas result is retrieved when the mean-field contribution contributions persists in the approximation of the higher- is neglected. order cumulants, as well as in their ratios. This allows At vanishing chemical potential, Eq. (18) reduces to to precisely delineate the contribution of chiral symmetry restoration and repulsive interaction to the critical id χ2 = χ2 βrep. (20)

id We note that, depending on the details of the model, χ2 in Eqs. (18) and (20) contains also dependence on the σ #2 In Appendix A, we present the evaluation of the higher-order and ω mean ﬁelds. However, at vanishing µB, the ex- cumulants and discuss the comparison of the full expressions and pectation value of ω vanishes as well, i.e., the eﬀective approximations used in this study. 5

behavior of the cumulants in the vicinity of the chiral βrep 1 2 phase transition. βrep 4 In the following, we quantify and discuss the prop- βrep erties of the obtained equation of state, as well as the 0.8 higher-order cumulants of the net-baryon number density at vanishing chemical potential in order to identify the importance of the repulsive interactions near the chi- 0.6 ral crossover transition. 0.4

IV. RESULTS 0.2

In the left panel of Fig. 1, we show the numerical 0 results on the temperature-normalized thermodynamic 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 pressure at vanishing baryon chemical potential. The T [Tc] pressure obtained in the HRG model increases monotonically and does not resemble any critical behavior. This FIG. 2. The suppression factor βrep from Eq. (21) for differ- HRG is expected because Ω is just a sum of uncorrelated ent temperatures, expressed in the units of the chiral-critical particles (cf. Eq. (16)) with vacuum hadron masses. temperature, Tc = Tc(µB = 0). There are clear deviations of the parity doublet model result on thermodynamic pressure from the correspond- ing ideal HRG. The increase of the pressure is a bulk In the left panel of Fig. 3, we show the net-baryon consequence of an interplay between critical chiral dy- kurtosis χ4/χ2, and the ratio χ6/χ2. For the ideal HRG namics with in-medium hadron masses and repulsive in- model, these ratios are equal to unity due to the expected teractions. The influence of in-medium hadron masses Skellam probability distribution of the net-baryon den- is identified when considering the pressure P σHRG of the sity [40]. The chiral dynamics and repulsive interactions σHRG model which increases around Tc. This is evi- implemented in the parity doublet model imply strong dently linked to the in-medium shift of baryon masses due deviations of these fluctuation ratios from the Skellam to chiral symmetry restoration. We note, that P σHRG is baseline. The kurtosis exhibits a peak around the transi- systematically higher than the parity doublet pressure. tion temperature, after which it drastically drops below The reason is that the partition function of uncorrelated unity. This is in contrast to the σHRG result, where the particles does not contain the mean-field potentials which peak structure appears as well, however, the result con- provide a negative contribution to the pressure, as seen verges back to the Skellam distribution limit at higher in the parity doublet model from Eq. (7). All pressures temperatures. Thus, the appearance of the peak in the shown in Fig. 1 converge to the Stefan-Boltzmann limit kurtosis is attributed to remnants of the chiral symmetry at high temperatures. restoration, whereas strong suppression around T ' Tc In the right panel of Fig. 1, we show the second-order is due to repulsive interactions between baryons. We cumulant, χ2. We note that, at vanishing chemical po- note that at vanishing chemical potential and in the chi- id n tential the expectation value of ω is zero, thus χn are ral limit χ cumulants are non-critical up to n < 6 or- equivalent to the σHRG formulation. Similarly to the der [64]. Thus, the peak of the kurtosis stays finite at case of pressure, the result for σHRG lies systemati- the chiral phase transition and its strength is strongly cally above the ideal gas result. χ2 in HRG and σHRG model-dependent. It is known that when including quan- models converge to the Stefan-Boltzmann limit at high- tum and thermal fluctuations in effective chiral models temperatures. In contrast, the parity doublet result sat- within the FRG approach such a peak essentially disap- urates above Tc and monotonically decreases to zero at pears leaving the kurtosis nearly unaffected by the chiral high temperature. From Eq. (20), it is clear that the dif- symmetry restoration [64, 80–82]. The influence of repul- ference between σHRG and parity doublet results is due sive interactions, however, can imply suppression of kur- to the suppression originating from βrep. tosis near the chiral crossover below the Skellam baseline. In Fig. 2, we show the suppression factor, βrep. It In this model calculations, such suppression at T ' Tc is changes gradually from unity at low temperatures to zero of the order of 35%. The kurtosis of net baryon-number at high temperatures. This indicates that the repulsive density was introduced as an excellent probe of quark forces become more important with increasing temper- deconfinement, and outside the critical region was shown ature. At the critical temperature, βrep ' 0.8. Conse- to be quantified by the square of the baryon number car- id quently, χ2 is reduced by 20% due to repulsive inter- ried by medium constituents [84, 85]. Such an interpreta- actions. From Eq. (21) one may also estimate the sup- tion of kurtosis is not accessible in the present hadronic pression of the higher-order cumulants. At T ' Tc, the chiral model since it does not contain quarks degrees of suppression of χ4 and χ6 due to repulsive interactions freedom. Thus, the parity doublet model can be only amounts to 41% and 67%, respectively. applicable up to the near vicinity of the chiral crossover. 6

15 HRG HRG 2.5 σHRG σHRG exHRG exHRG Parity Doublet 10 Parity Doublet 2 5

2 1.5 2

/χ /χ 0 4 6 χ χ 1 −5

0.5 −10

0 −15 0.7 0.8 0.9 1 1.1 1.2 1.3 0.7 0.8 0.9 1 1.1 1.2 1.3

T [Tc] T [Tc]

FIG. 3. Ratio of different higher-order cumulants of the net-baryon number density fluctuations, χ4/χ2 (left) and χ6/χ2 (right), calculated under different schemes, see text. The temperature is expressed in the units of the chiral-critical temperature, Tc = Tc(µB = 0).

The ratio χ6/χ2 exhibits a strong sensitivity to dy- was extensively studied [42, 48, 49, 72, 87–89]. We con- namical effects related to chiral symmetry restoration, sider the common formulation of the excluded volume as shown in the right panel of Fig. 3. The characteristic effect, in which it is considered for the bulk pressure of S-type structure of this ratio obtained in the parity dou- the system. In Ref. [89], it was pointed out that this may blet model with a well-pronounced peak followed by a dip not be a robust gauge of the repulsion in the individual at negative values in the near vicinity of Tc is expected interaction channels. as an imprint of the chiral criticality [64]. The leading The thermodynamic pressure in the excluded volume role of the chiral symmetry restoration on the properties approach is given by a thermodynamically self-consistent of χ6/χ2 is also seen by comparing the full parity doublet equation: and σHRG model results in Fig. 3. In both cases, the S- ex ? X id type structure of this ratio is preserved, albeit with some P (T, µ ) = P (T, µk) , (23) quantitative differences which are linked to repulsive in- k teractions. Indeed, as already discussed in the context of ex ? id with µk = ±µB − v0P (T, µ ), and P is the ideal-gas the kurtosis, the presence of repulsive interactions sup- pressure. The (±) sign applies to particles and antiparti- presses χ6/χ2 when compared to the σHRG results. Nev- cles, respectively. The summation over k goes over par- ertheless, the qualitative structure of this ratio remains ticles and anti-particles. The constant eigenvolume, the same. The magnitude of the peak-dip structure observed in 16 3 v0 = πr0, (24) χ6/χ2 is a direct consequence of the mean-field approx- 3 imation employed in our calculations. We note that where r0 = 0.3 fm. Once the excluded volume, v0, the inclusion of mesonic fluctuations weakens the critical is specified, the pressure can be obtained by solving behavior. This was presented in other models exhibit- Eq. (23) self-consistently. From this, higher-order cu- ing O(4) chiral criticality, e.g., quark-meson (QM) [86] mulants are obtained numerically. and Polyakov loop-extended quark-meson (PQM) [64, 80, In Fig. 3, we show the ratios of cumulants, χ4/χ2 and 81, 83] models within FRG approach. In these mod- χ6/χ2, obtained in the excluded volume approach labeled els, the χ4/χ2 ratio decreases monotonously with tem- as exHRG. For consistency, the temperature is normal- perature and practically no peak structure is exhibited ized to the critical temperature, Tc, obtained in the par- neat Tc. In contrast, the general peak-dip structure of ity doublet model. The excluded volume model shows a χ6/χ2 obtained in the mean-field approximation prevails swift decrease from the ideal HRG behavior at low tem- when mesonic fluctuations are included. This highly non- perature and turns negative at around 1.35 Tc. Recently, monotonic behavior is also seen in lattice QCD simula- it was suggested that this behavior may call into question tions. In particular, the peak in χ6/χ2 is seen, despite the connection to deconfinement transition in QCD [54]. huge systematic error at low temperatures [2, 3]. For χ6/χ2 the excluded volume approach deviates from Lastly, we compare the properties of χ4/χ2 and χ6/χ2 the ideal HRG result, turns negative, and predicts a dip fluctuation ratios with the excluded volume formulation above Tc. Very similar behavior is also observed in mod- of the repulsive interactions. The effect of excluded vol- els incorporating Van-der-Waals type formulation of re- ume on thermodynamic properties of a hadronic medium pulsive and attractive interactions between hadrons [54]. 7

Clearly, the results of the excluded volume and the tention is to be given to account for possible repulsive parity doublet model are qualitatively different. How- interactions between baryons. ever, our consistent chiral model calculations confirmed A frequently used approach to account for repulsive that indeed repulsive interactions between hadrons imply hadronic interactions is the hard-core repulsion or van- suppression of χ4/χ2 and χ6/χ2 fluctuation ratios near der-Waals type interaction model. We have compared the chiral crossover as pointed out in Ref. [54]. Thus, the our results for the nth-order cumulants of net-baryon phenomenological hadronic models that account for re- number fluctuations with an excluded volume formula- pulsive and attractive interactions between constituents tion of the repulsive interactions. Such phenomenological can be successful in describing some deviations of net- model provides suppression of cumulants with increasing charge fluctuations from the HRG baseline observed in temperature due to hadronic repulsion. In particular, the LQCD. However, this is not the case if such fluctuation kurtosis χ4/χ2 in this model is reduced from the Skellam observables are affected by the chiral criticality. This is limit towards the chiral crossover, as observed in LQCD very transparent when considering χ6/χ2 ratio shown in results. However, when considering the χ6/χ2 fluctua- Fig. 3. The lack of in-medium effects due to the chiral tion ratio, which exhibits a dominant contribution from symmetry restoration in the excluded volume approach the chiral criticality, such phenomenological model fails directly implies that such a model is not capable of re- to capture the characteristic properties of this ratio. Con- producing a characteristic structure of this ratio near the sequently, a description of the higher-order cumulants of chiral crossover. This indicates that in order to fully de- the net-baryon density in the chiral crossover requires a scribe the properties of cumulants of net-baryon number consistent framework that accounts for a self-consistent fluctuations near the chiral crossover, it is not sufficient treatment of the chiral in-medium effects and repulsive to account only for repulsive interactions, but it is essen- interactions. tial to formulate a consistent framework that implements At low temperature, the parity doublet model predicts the chiral in-medium effects and repulsive interactions si- sequential liquid-gas and chiral phase transitions in the multaneously. baryon-rich matter, with critical endpoints of both transitions at moderate temperatures of tens of MeV [18, 31]. It is challenging to identify the role of in-medium effects V. SUMMARY and hadronic interactions on the properties of higher- order cumulants near these distinct phase transitions. Work in this direction is in progress. We have discussed the role of attractive and repulsive nucleon-nucleon interactions on the thermodynamic and chiral-critical properties of a strongly interacting medium ACKNOWLEDGMENTS at finite temperature. To this end, we have used the parity doublet model in the mean-field approximation. This work was partly supported by the Polish Na- We have analyzed the thermodynamic pressure and the tional Science Center (NCN), under OPUS Grant No. cumulants of the net-baryon number density, χn, up to 2018/31/B/ST2/01663 (K.R. and C.S.) and Preludium the sixth order, as well as their ratios. Grant No. UMO-2017/27/N/ST2/01973 (M.M.). K.R. We have shown that, at vanishing chemical potential, also acknowledges the support of the Polish Ministry of the second-order cumulant factorizes as a product of a Science and Higher Education. M.M. acknowledges help- term that is directly linked to attractive scalar interac- ful comments from Pok Man Lo. tions and a suppression factor due to repulsive interactions. Furthermore, we have found that to a good approximation, a similar separation also holds for higher-order Appendix A: Evaluation of the higher-order cumulants. This allowed to consistently delineate differ- cumulants ent in-medium effects and to identify the role of repulsive interactions near the chiral crossover transition. From the definition of the nth order cumulant of the We have pointed out that even a moderate influence of net-baryon number density, the repulsive interactions between hadrons on the equation of state becomes more readily exposed in the quantitative structure of the cumulants with increasing their ∂n−1n χ = T n−4 B , (A1) order. Consequently, in the phenomenological descrip- n n−1 ∂µB tion of cumulants of net baryon density fluctuations calculated in LQCD or measured in heavy-ion collisions at- one can derive the following relation: 8

Full result Full result 0.25 Approximation 0.5 Approximation

0.2 0

−0.5 0.15 4 6 χ χ −1 0.1 −1.5

0.05 −2

0 −2.5 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

T [Tc] T [Tc]

FIG. 4. Fourth- and sixth-order cumulants of the net-baryon number obtained in the parity doublet model and their approximations at µB = 0, see text. The temperature is expressed in the units of the chiral-critical temperature, Tc = Tc(µB = 0).

∂χn (T, µB, σ(T, µB), ω(T, µB)) ∂χn ∂χn ∂ω ∂χn ∂σ χn+1 = = + + = ∂µB ∂µB ∂ω ∂µB ∂σ ∂µB ∗ ∗ (A2) ∂χn ∂µ ∂χn ∂µ ∂ω ∂χn ∂σ id id ∂ω ∂χn ∂σ id ∂χn ∂σ ∗ + ∗ + = χn+1 − gωχn+1 + = χn+1βrep + , ∂µ ∂µB ∂µ ∂ω ∂µB ∂σ ∂µB ∂µB ∂σ ∂µB ∂σ ∂µB

id th where χn+1 is the ideal gas expression for the (n+1) or- cumulant as der cumulant, β = 1 − g ∂ω/∂µ , and µ∗ = µ − g ω rep ω B B ω ∂n ∂σ is the eﬀective chemical potential. Applying Eq. (A2) id B χ2 = χ2 βrep + . (A3) to the net-baryon density, one obtains the second-order ∂σ ∂µB

At µB = 0, nB vanishes, owing to particle-antiparticle symmetry. Hence, the last term in Eq. (A3) vanishes and one obtains Eq. (20). Higher-order cumulants are obtained by applying Eq. (A2) iteratively. In particular, the third- and fourth- order cumulants, read

id 2 2 2 2 id 2 ∂χ2 ∂σ id ∂ ω ∂ nB ∂σ ∂nB ∂ σ χ3 =χ3 βrep + 2 βrep − gωχ2 2 + 2 + 2 , (A4a) ∂σ ∂µB ∂µB ∂σ ∂µB ∂σ ∂µB id 2 2 id 2 id 3 ∂χ3 ∂σ 2 id ∂ ω ∂ χ2 ∂σ χ4 =χ4 βrep + 3 βrep − 3gωχ3 2 βrep + 3 2 βrep ∂σ ∂µB ∂µB ∂σ ∂µB id 2 2 3 3 3 2 2 3 ∂χ2 ∂ σ ∂σ ∂ ω id ∂ ω ∂ nB ∂σ ∂ nB ∂σ ∂ σ ∂nB ∂ σ + 3 2 βrep − gω 2 − gωχ2 3 + 3 + 3 2 2 + 3 . (A4b) ∂σ ∂µB ∂µB ∂µB ∂µB ∂σ ∂µB ∂σ ∂µB ∂µB ∂σ ∂µB

We note, that σ(T, µB) = σ(T, −µB) and ω(T, µB) = Consequently, at µB = 0, each term in χ3 vanishes −ω(T, −µB). Thus, the odd derivatives of σ and even separately, thus χ3(T, µB = 0) = 0. On the other hand, derivatives of ω w.r.t µB vanish, i.e.,

∂2k+1σ ∂2kω = 0, = 0. (A5) 2k+1 2k ∂µB ∂µB µB =0 µB =0 9 the even-order cumulants, χ2 and χ4, are reduced to used in Eqs. (20) and (21) are obtained by dropping all terms except the ﬁrst one. In Fig. 4, we present a di- id χ2 = χ2 βrep, (A6a) rect comparison of the exact results for the fourth- and id 2 3 sixth-order cumulants and their approximations given in id 3 ∂χ2 ∂ σ id ∂ ω χ4 = χ4 βrep + 3 2 βrep − gωχ2 3 . (A6b) Eq. (21). The agreement is quite satisfactory for tem- ∂σ ∂µB ∂µB peratures T ≤ 1.1 Tc, keeping the basic properties of the The higher-order cumulants are calculated following a cumulants. similar method as explained above. The approximations

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