Net Baryon Number Fluctuations Research Articles

The imprint of conservation laws on correlated particle production

The study of event-by-event fluctuations of net-baryon number in a subspace of full phase space is a promising direction for deciphering the structure of strongly interacting matter created in collisions of relativistic heavy nuclei. Such fluctuations are generally suppressed by exact baryon number conservation. Moreover, the suppression is stronger if baryon number is conserved locally. In this report we present a conceptually new approach to quantify correlations in rapidity space between baryon-antibaryon, baryon-baryon, and antibaryon-antibaryon pairs and demonstrate their impact on net-baryon number fluctuations. For the special case of Gaussian rapidity distributions, we use the Cholesky factorization of the covariance matrix, while the general case is introduced by exploiting the well-known Metropolis and Simulated Annealing methods. The approach is based on the use of the canonical ensemble of statistical mechanics for baryon number and can be applied to study correlations between baryons as well as strange and/or charm hadrons. It can also be applied to describe relativistic nuclear collisions leading to the production of multi-particle final states. One application of our method is the search for formation of proton clusters at low collision energies emerging as a harbinger of the anticipated first-order chiral phase transition. In a first step, the results obtained are compared to the recent measurements from the CERN ALICE collaboration. Such investigations are key to explore the phase diagram of strongly interacting matter and baryon production mechanisms at energy scales from several GeV to several TeV.

Anatomy of critical fluctuations in hadronic matter

Critical phenomena in phase transitions of strongly interacting matter, governed by quantum chromodynamics, are inherently encoded in the fluctuations of conserved charges. In this work, we study the net-baryon number density fluctuations, including the lowest-lying nucleon and the baryonic resonance Δ(1232), based on the parity doublet model in the mean-field approximation. We focus on the qualitative features of the second-order susceptibility of the net-baryon number density in dense hadronic matter and how the inclusion of Δ(1232) affects it. We demonstrate that the fluctuations of the individual baryons do not necessarily reflect the total net-baryon number fluctuations at finite density, due to the nontrivial correlations between different particle species. Our results highlight the role of baryonic correlations in the interpretation of data from heavy ion collision experiments. Published by the American Physical Society 2024

Fluctuations and correlations of baryonic chiral partners

The exploration of critical phenomena in phase transitions of strongly interacting matter governed by quantum chromodynamics (QCD) is one of the goals of present ultrarelativistic heavy-ion collision experiments at BNL and CERN. The key research direction is to locate the putative critical point on the phase diagram of QCD linked to the chiral symmetry restoration at finite temperature and/or density. One of the main theoretical tools used for this purpose is the fluctuations of conserved charges, such as the net-baryon number. However, due to experimental limitations, analyses of heavy-ion collision data suffer from a very doubtful basing of the net-proton number being a proxy for the total net-baryon number fluctuations. In this work, we use the parity doublet model to investigate the fluctuations of the net-baryon number density in hot and dense hadronic matter. The model accounts for chiral criticality within the mean-field approximation. We focus on the qualitative properties and systematics of the first- and second-order susceptibility of the net-baryon number density, and their ratios for nucleons of positive and negative parity, as well as their correlator. We show that the fluctuations of the positive-parity nucleon do not necessarily reflect the fluctuations of the total net-baryon number density at the phase boundary of the chiral phase transition. We also investigate the nontrivial structure of the correlator. Furthermore, we discuss and quantify the differences between the fluctuations of the net-baryon number density in the vicinity of the chiral and liquid-gas phase transition in nuclear matter. We indicate a possible relevance of our results with the interpretation of the experimental data on net-proton number fluctuations in heavy-ion collisions. Published by the American Physical Society 2024

Fifth- and sixth-order net baryon number fluctuations in nuclear matter at low temperature

We calculate the density fluctuations of net baryon number up to sixth order induced by the interactions of nuclear matter, and explore their relationship with the nuclear liquid-gas phase transition (LGPT), including the stable and metastable phases as well as the region far from the phase transition. The results show that dramatic density fluctuations exist in the vicinity of LGPT, and the higher order density fluctuations are more sensitive than the lower order ones to the interactions and structural properties of nuclear matter. The study also indicates that, even far away from the critical region of LGPT, the hadronic interactions can still lead to larger fifth- and sixth-order density fluctuations. In combination with the chemical freeze-out line fitted from the experimental data, the derived results can be used to investigate the chiral phase transition and nuclear LGPT, as well as to analyze related experimental signals. Published by the American Physical Society 2024

Impact of hadronic interactions and conservation laws on cumulants of conserved charges in a dynamical model

Understanding the phase diagram of QCD by measuring fluctuations of conserved charges in heavy-ion collision is one of the main goals of the beam energy scan program at the BNL Relativistic Heavy Ion Collider (RHIC). Within this work, we calculate the role of hadronic interactions and momentum cuts on cumulants of conserved charges up to fourth order in a system in equilibrium within a hadronic transport approach (smash). In our model the net baryon, net charge, and net strangeness are perfectly conserved on an event-by-event basis and the cumulants are calculated as a function of subvolume sizes and compared to analytic expectations. We find a modification of the kurtosis due to charge annihilation processes in systems with simplified degrees of freedom. Furthermore the result of the full smash hadron gas for the net baryon and net proton number fluctuations is presented for systems with zero and finite values of baryon chemical potential. Additionally we find that due to dynamical correlations the cumulants of the net baryon number cannot easily be recovered from the net protons. Finally the influence of deuteron cluster formation on the net proton and net baryon fluctuations in a simplified system is shown. This analysis is important to better understand the relation between measurements of fluctuations in heavy-ion collisions and theoretical calculations which are often performed in a grand canonical ensemble.

Recent results on conserved charge fluctuations in ALICE

Recent results on the analysis of event-by-event net-baryon number fluctuations in Pb–Pb collisions at sNN=2.76 and 5.02 TeV are presented. The cumulants of the net-proton distributions, proxies for the net-baryon distributions, up to third order are discussed. The experimental results are compared with HIJING and EPOS model calculations and the dependence of fluctuation measurements on the phase-space coverage is addressed in the context of calculations from Lattice QCD (LQCD) and the Hadron Resonance Gas (HRG) model.

Exploration of the phase diagram and the thermodynamic properties of QCD at finite temperature and chemical potential with the PNJL effective model

The QCD transition from a hadronic to a quark-gluon plasma phase is a cross-over at vanishing/small baryo-chemical potential, while at higher chemical potentials it is argued that it becomes of first order, ending with a critical end-point. The present goal is the determination of the critical line and, possibly, the recognition of the critical endpoint. For this purpose, the effective model (Nambu-Jona-Lasinio model with Polyakov Loop) with 2+1 flavours is used. In Mean Field Approximation it is possible to obtain all thermodynamic quantities (pressure, entropy, energy and quark density) and their fluctuations, in particular, the generalized quark-number susceptibilities. We use the net baryon-number fluctuations and net-strangeness fluctuations to identify the critical line and the order of the phase transition in the chemical potential-temperature plane.

Skewness, kurtosis, and the fifth and sixth order cumulants of net baryon-number distributions from lattice QCD confront high-statistics STAR data

We present new results on up to $6^{th}$ order cumulants of net baryon-number fluctuations at small values of the baryon chemical potential, $\mu_B$, obtained in lattice QCD calculations with physical values of light and strange quark masses. Representation of the Taylor expansions of higher order cumulants in terms of the ratio of the two lowest order cumulants, $M_B/\sigma_B^2=\chi_1^B(T,\mu_B)/\chi_2^B(T,\mu_B)$, allows for a parameter free comparison with data on net proton-number cumulants obtained by the STAR Collaboration in the Beam Energy Scan at RHIC. We show that recent high statistics data on skewness and kurtosis ratios of net proton-number distributions, obtained at beam energy $\sqrt{s_{_{NN}}}=54.4$ GeV, agree well with lattice QCD results on cumulants of net baryon-number fluctuations close to the pseudo-critical temperature, $T_{pc}(\mu_B)$, for the chiral transition in QCD. We also present first results from a next-to-leading order expansion of $5^{th}$ and $6^{th}$ order cumulants on the line of pseudo-critical temperatures.

The kurtosis of net baryon number fluctuations from a realistic Polyakov\u2013Nambu\u2013Jona-Lasinio model along the experimental freeze-out line

Firstly we qualitatively analyze the formation of the dip and peak structures of the kurtosis kappa sigma ^2 of net baryon number fluctuation along imagined freeze-out lines and discuss the signature of the existence of the QCD critical end point (CEP) in the Nambu–Jona-Lasinio (NJL) model, Polyakov-NJL (PNJL) model as well as mu -dependent PNJL(mu PNJL) model with different parameter sets, and then we apply a realistic PNJL model with parameters fixed by lattice data at zero chemical potential, and quantitatively investigate its kappa sigma ^2 along the real freeze-out line extracted from experiments. The important contribution from gluodynamics to the baryon number fluctuations is discussed. The peak structure of kappa sigma ^2 along the freeze-out line is solely determined by the existence of the CEP mountain and can be used as a clean signature for the existence of CEP. The formation of the dip structure is sensitive to the relation between the freeze-out line and the phase boundary, and the freeze-out line starts from the back-ridge of the phase boundary is required. To our surprise, the kurtosis kappa sigma ^2 produced from the realistic PNJL model along the experimental freeze-out line agrees with BES-I data well, which indicates that equilibrium result can explain the experimental data. It is worth to point out that the extracted freeze-out temperatures from beam energy scan measurement are indeed higher than the critical temperatures at small chemical potentials, which supports our qualitative analysis.

Exploring chiral symmetry restoration in heavy-ion collisions with fluctuation observables

In the vicinity of the QCD phase boundary, the thermodynamic pressure receives contributions from a regular component Pr and a non-analytic component Ps. Pr describes the bulk properties of the QCD thermodynamics, whereas Ps carries information on chiral criticality. We discuss the computation of Pr in the hadronic phase within the S-matrix formulation of statistical mechanics, and the determination of Ps using high-order cumulants of net baryon number fluctuations. We show that the S-matrix approach provides a consistent description to both the lattice QCD results on the baryon electric charge correlation and on the proton production yield in heavy ion collisions at the LHC. In addition, it is suggested that chiral criticality is reflected in the energy dependence of certain ratios of net baryon number fluctuations.

The QCD crossover at zero and non-zero baryon densities from Lattice QCD

We map out the QCD crossover line Tc(μB)Tc(0)=1−κ2(μBTc(0))2−κ4(μBTc(0))+O(μB6) for the first time up to O(μB4) for a strangeness neutral system by performing a Taylor expansion of chiral observables in temperature T and chemical potentials μ. At vanishing chemical potential, we report a crossover temperature Tc(0)=(156.5±1.5)MeV defined by the average of several second-order chiral susceptibilities. For a system with thermal conditions appropriate for a heavy-ion collision, we determined a curvature from the subtracted condensate as κ2=0.0120(20) and from the disconnected susceptibility as κ2=0.0123(30). The next order κ4 is significantly smaller. We also report the crossover temperature as a function of the chemical potentials for: baryon-number, electric charge, strangeness and isospin. Additionally, we find that Tc(μB) is in agreement with lines of constant energy density and constant entropy density. Along this crossover line, we study net baryon-number fluctuations and show that their increase is substantially smaller compared to that obtained in HRG model calculations. Similarly, we analyze chiral susceptibility fluctuations along the crossover line and show that these are constant. We conclude that no signs for a narrowing of the crossover region can be found for baryon chemical potential μB<250MeV.

Net baryon-number fluctuations in magnetized quark matter

The kurtosis and skewness of net baryon-number fluctuations are studied for the magnetized phase diagram of three-flavor quark matter within the Polyakov extended Nambu--Jona-Lasinio model. Two models with magnetic catalysis and inverse magnetic catalysis are considered. Special attention is given to their behavior in the neighborhood of the light and strange critical end points (CEPs). Several isentropic trajectories that come close the CEPs are studied in order to analyze possible signatures of a CEP in the presence of external magnetic fields. The effect of the magnetic field on the velocity of sound, ${v}_{s}^{2}$, when both the light and strange CEPs are approached from the crossover region is also investigated by calculating their temperature and baryon chemical potential dependencies at fixed distances from these CEPs. Regions with large fluctuations but no CEP in nonmagnetized matter develop a CEP under the action of a strong magnetic field. Besides, the Landau quantization of the quark trajectories may result in the appearance of extra CEPs, in particular, in the strange sector for strong magnetic fields, identifiable by the net baryon-number fluctuations. Stiffer (smoother) fluctuations in the region of the CEP are characteristic of models that do not predict (do predict) the inverse magnetic catalysis at zero chemical potential. Particularly interesting is the ratio ${\ensuremath{\chi}}_{B}^{4}/{\ensuremath{\chi}}_{B}^{2}$ that has a more pronounced peak structure, indicating that it is eventually a more convenient probe for the search of a CEP. The speed of sound shows a much richer structure in magnetized quark matter and allows one to identify both chiral and deconfinement transitions.

Net-baryon number fluctuations in the hybrid quark-meson-nucleon model at finite density

We study the mean-field thermodynamics and the characteristics of the net-baryon number fluctuations at the phase boundaries for the chiral and deconfinement transitions in the Hybrid Quark-Meson-Nucleon model. The chiral dynamics is described in the linear sigma model, whereas the quark confinement is manipulated by a medium-dependent modification of the particle distribution functions, where an additional scalar field is introduced. At low temperature and finite baryon density, the model predicts a first-, second-order chiral phase transition, or a crossover, depending on the expectation value of the scalar field, and a first-order deconfinement phase transition. We focus on the influence of the confinement over higher-order cumulants of the net-baryon number density. We find that whereas the cumulants show a substantial enhancement around the chiral phase transition, they are not as sensitive to the deconfinement transition.

Baryon number fluctuations and the phase structure in the PNJL model

We investigate the kurtosis and skewness of net-baryon number fluctuations in the Polyakov loop extended Nambu–Jona-Lasinio (PNJL) model, and discuss the relations between fluctuation distributions and the phase structure of quark-gluon matter. The calculation shows that the traces of chiral and deconfinement transitions can be effectively reflected by the kurtosis and skewness of net-baryon number fluctuations not only in the critical region but also in the crossover region. The contour plot of baryon number kurtosis derived in the PNJL model can qualitatively explain the behavior of net-proton number kurtosis in the STAR beam energy scan experiments. Moreover, the three-dimensional presentations of the kurtosis and skewness in this study are helpful to understand the relations between baryon number fluctuations and QCD phase structure.

Magnetized QCD Phase Diagram: Net-baryon Susceptibilities

Employing the Polyakov extended Nambu-Jona-Lasinio model, we determine the net-baryon number fluctuations of magnetized three-flavor quark matter. We show that the magnetic field changes the nature of the strange quark transition from crossover to first-order at low temperatures. In fact, the strange quark undergoes multiple first-order phase transitions and several critical end points emerge in the phase diagram.

Critical point in the phase diagram of primordial quark-gluon matter from black hole physics

Strongly interacting matter undergoes a crossover phase transition at high temperatures $T\sim 10^{12}$ K and zero net-baryon density. A fundamental question in the theory of strong interactions, Quantum Chromodynamics (QCD), is whether a hot and dense system of quarks and gluons displays critical phenomena when doped with more quarks than antiquarks, where net-baryon number fluctuations diverge. Recent lattice QCD work indicates that such a critical point can only occur in the baryon dense regime of the theory, which defies a description from first principles calculations. Here we use the holographic gauge/gravity correspondence to map the fluctuations of baryon charge in the dense quark-gluon liquid onto a numerically tractable gravitational problem involving the charge fluctuations of holographic black holes. This approach quantitatively reproduces ab initio results for the lowest order moments of the baryon fluctuations and makes predictions for the higher order baryon susceptibilities and also for the location of the critical point, which is found to be within the reach of heavy ion collision experiments.

Skewness and kurtosis of net baryon-number distributions at small values of the baryon chemical potential

We present results for the ratios of mean ($M_B$), variance ($\sigma_B^2$), skewness ($S_B)$ and kurtosis ($\kappa_B$) of net baryon-number fluctuations obtained in lattice QCD calculations with a physical light to strange quark mass ratio. Using next-to-leading order Taylor expansions in baryon chemical potential we find that qualitative features of these ratios closely resemble the corresponding experimentally measured cumulants ratios of net proton-number fluctuations for beam energies down to $\sqrt{s_{_{NN}}} \ge 19.6$ GeV. We show that the difference in cumulant ratios for the mean net baryon-number, $M_B/\sigma_B^2=\chi_1^B(T,\mu_B)/\chi_2^B(T,\mu_B)$ and the normalized skewness, $S_B\sigma_B=\chi_3^B(T,\mu_B)/\chi_2^B(T,\mu_B)$, naturally arises in QCD thermodynamics. Moreover, we establish a close relation between skewness and kurtosis ratios, $S_B\sigma_B^3/M_B=\chi_3^B(T,\mu_B)/\chi_1^B(T,\mu_B)$ and $\kappa_B\sigma_B^2=\chi_4^B(T,\mu_B)/\chi_2^B(T,\mu_B)$, valid at small values of the baryon chemical potential.

Conserved Charge Fluctuations from Lattice QCD and the Beam Energy Scan

We discuss the next-to-leading order Taylor expansion of ratios of cumulants of net-baryon number fluctuations. We focus on the relation between the skewness ratio, SBσB≡χ3B/χ1B, and the kurtosis ratio, κBσB2≡χ4B/χ2B. We show that differences in these two cumulant ratios are small for small values of the baryon chemical potential. The next-to-leading order correction to κBσB2 however is approximately three times larger than that for SBσB. The former thus drops much more rapidly with increasing beam energy, sNN. We argue that these generic patterns are consistent with current data on cumulants of net-proton number fluctuations measured by the STAR Collaboration at sNN≥19.6 GeV.

Net-baryon number fluctuations across the chiral phase transition at finite density in strong-coupling lattice QCD

We investigate the net-baryon number fluctuations across the chiral phase transition at finite density in the strong coupling and chiral limit. Mesonic field fluctuations are taken into account by using the auxiliary field Monte-Carlo method. We find that the higher-order cumulant ratios, $S\sigma$ and $\kappa\sigma^2$, show oscillatory behavior around the phase boundary at $\mu/T\gtrsim 0.2$, and there exists the region where the higher-order cumulant ratios are negative. The negative region of $\kappa\sigma^2$ is found to shrink with increasing lattice size. This behavior agrees with the expectations from the scaling analysis.

Momentum scale dependence of the net quark number fluctuations near chiral crossover

We investigate properties of the net baryon number fluctuations near chiral crossover in a hot and dense medium of strongly interacting quarks. The chirally invariant quark–antiquark interactions are modeled by an effective quark–meson Lagrangian. To preserve remnants of criticality in the |$O(4)$| universality class, we apply the functional renormalization group method to describe thermodynamics near chiral crossover. Our studies are focused on the influence of the momentum cuts on the critical behavior of different cumulants of the net quark number fluctuations. We use the momentum scale dependence of the flow equation to examine how the suppression of the momentum modes in the infrared and ultraviolet regimes modifies generic properties of fluctuations expected in the |$O(4)$| universality class. We show that the pion mass |$m_\pi$| is a natural soft momentum scale at which cumulants are saturated at their critical values, whereas for scales larger than |$2m_\pi$| the characteristic |$O(4)$| structure of the higher-order cumulants gets lost. These results indicate that when measuring fluctuations of the net baryon number in heavy ion collisions to search for a partial restoration of chiral symmetry or critical point, special care must be taken when introducing kinematical cuts on the fluctuation measurements.