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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 = − ! ! !µ. (4b) behaviors are dominated by the chiral criticality and re- ! 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 m! are the π and ! 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 !µ field vanishes, namely h!i = 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 1 and 2 are transformed as Ω = Ωx + Vσ + V!, (7) follows: x=± 1L ! L 1L; 1R ! R 1R, with (1) 2L ! R 2L; 2R ! L 2R, Z d3p Ω = γ T ln (1 − f ) + ln 1 − f¯ , (8) x x (2π)3 x x where i = iL + iR, L 2 SU(2)L and R 2 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 = i 1@/ 1 + i 2@/ 2 + m0 1γ5 2 − 2γ5 1 ¯ ¯ 1 + g1 1 (σ + iγ5τ · π) 1 + g2 2 (σ − iγ5τ · π) 2 fx = ∗ , β(Ex−µ ) ¯ ¯ 1 + e − g! 1!/ 1 − g! 2!/ 2, (9) ¯ 1 (2) fx = ∗ , 1 + eβ(Ex+µ ) where g1, g2, and g! 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 − g!!. 1 1 1 L = (@ σ)2 + (@ π)2 − (! )2 − V − V , (3) M 2 µ 2 µ 4 µν σ ! #1 Since !0 is the only non-zero component in the mean-field ap- proximation, we simply denote it by ! ≡ !. where !µν = @µ!ν − @ν !µ is the field-strength tensor of 0 3 2 m0 [MeV] m+ [MeV] m− [MeV] mπ [MeV] fπ [MeV] m! [MeV] mρ [MeV] λ4 λ6fπ g! g1 g2 850 939 1500 140 93 783 775 13.15 4.67 4.64 12.42 6.39 TABLE I.
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