In the context of the ongoing search for the QCD critical point at the Relativistic Heavy-Ion Collider, we study the equation of state near the critical point in the temperature and baryon chemical potential plane. We use the parametric representation introduced in earlier literature, which maps the universal 3D Ising equation of state onto the QCD phase diagram using several non-universal parameters. We focus on the quartic cumulant of the baryon number, or baryon number susceptibility~$χ_4^B$, which can be accessed experimentally via net-proton fluctuation kurtosis measurements. It was originally predicted, through universality arguments based on the {\em leading} singular contribution, that $χ_4^B$ and net-proton kurtosis should show a specific non-monotonic behavior due to the critical point. In particular, when following the freeze-out curve on the phase diagram by decreasing beam energy, the kurtosis is expected to dip, and then peak, when the beam energy scan passes close to the critical point. We study the effects of the non-universal and thus far unknown parameters of the Ising-to-QCD mapping on the behavior of~$χ_4^B$. We find that, while the peak remains a solid feature, the presence of the critical point does not necessarily cause a dip in $χ_4^B$ on the freezeout line {\em below} the transition temperature. The critical point contribution to the dip appears only for a narrow set of mapping parameters, when subleading singular terms are sufficiently suppressed.
We calculate the baryon factorial cumulants assuming arbitrary short-range correlations and the global baryon number conservation. The general factorial cumulant generating function is derived. Various relations between factorial cumulants subjected to baryon number conservation and the factorial cumulants without this constraint are presented. We observe that for $n$-th factorial cumulant, the short-range correlations of more than $n$ particles are suppressed with the increasing number of particles. The recently published [1] relations between the cumulants in a finite acceptance with global …
We calculate the baryon number cumulants within acceptance with short-range correlations and global baryon number conservation in terms of cumulants in the whole system without baryon conservation. We extract leading and next-to-leading order terms of the large baryon number limit approximation. Our results extend the findings of Refs. [1,2]. These approximations are checked to be very close to the exact results.
We evaluate the ratio of baryon and electric-charge cumulants at second order from the recent experimental results at $\sqrt{s_{_{NN}}}=200$ GeV by the STAR Collaboration. The baryon number cumulant is reconstructed from the proton number distribution, and effects of the finite acceptance on the transverse momentum are corrected assuming the independent particle emission. We show that the obtained ratio has a dependence on the rapidity window. Comparison of the result with the hadron resonance gas model and lattice QCD numerical simulations suggests that if the fluctuations are generated from …
The proton, antiproton and mixed proton-antiproton factorial cumulants originating from the global conservation of baryon number are calculated analytically up to the sixth order. Our results can be directly tested in experiments.
We present a dynamical description of (anti)proton number fluctuations cumulants and correlation functions in central Au-Au collisions at $\sqrt{s_{\rm NN}} = 7.7-200$ GeV by utilizing viscous hydrodynamics simulations. The cumulants of proton and baryon number are calculated in a given momentum acceptance analytically, via an appropriately extended Cooper-Frye procedure describing particlization of an interacting hadron resonance gas. The effects of global baryon number conservation are taken into account using a generalized subensemble acceptance method. The experimental data of the STAR col…
Higher-order cumulants of the distributions of conserved charges, such as net-baryon number, are sensitive to the quantum chromodynamics(QCD) phase transition and the QCD critical point. We calculate the cumulants and correlation functions of proton, antiproton, and net-proton multiplicity distributions in Au+Au collisions at $\sqrt{s_{NN}} = 7.7$ GeV using a multiphase transport model(AMPT). The AMPT model can basically describe the trends of cumulants, cumulant ratios, (normalized) correlation functions of the proton and net-proton measured by the STAR experiment. The multiproton (baryon) co…
We present new results on up to $6^{th}$ order cumulants of net baryon-number fluctuations at small values of the baryon chemical potential, $μ_B$, 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, $M_B/σ_B^2=χ_1^B(T,μ_B)/χ_2^B(T,μ_B)$, 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 skewn…
Within the ultra-relativistic quantum molecular dynamics (UrQMD) model, in which the Lorentz-covariant treatment of nuclear mean-field potential is considered, the fluctuations of net-baryon, net-charge and deuterons multiplicity distributions in Au+Au head-on collisions at $\sqrt{s_{NN}}=3.0-5.0$ GeV are calculated. The results show that the nuclear mean-field potential can significantly enhance the magnitude of baryon number fluctuations in narrow rapidity windows, and this enhancement rapidly weakens with increasing beam energy. However, for proton and net-charge number fluctuations, the me…
Fluctuations of the baryon number in relativistic heavy-ion collisions are a promising observable to explore the structure of the QCD phase diagram. The cumulant ratios in heavy ion collisions at intermediate energies ($\sqrt{s_{\textrm{NN}}} < 7$ GeV) have not been studied to date. We investigate the effects of mean field potential and clustering on the cumulant ratios of baryon and proton number distributions in Au+Au collisions at beam energy of 1.23 GeV$/$nucleon as measured by the HADES Collaboration at GSI. Ultrarelativistic Quantum Molecular Dynamics (UrQMD) and the JAM model are used t…
The maximum entropy method (MEM) and the Gaussian process (GP) regression, which are both well-suited for the treatment of inverse problems, are used to reconstruct net-baryon number distributions based on a finite number of cumulants of the distribution. Baryon number distributions across the chiral phase transition are reconstructed. It is found that with the increase of the order of cumulants, distribution in the long tails, i.e., far away from the central number, would become more and more important. We also reconstruct the distribution function based on the experimentally measured cumulan…
The cumulants of baryon multiplicity distribution in relativistic heavy-ion collisions (HICs) have attracted considerable attention recently. It has been conjectured that they may serve as a promising observable to detect the critical end point in the QCD phase diagram, while the cumulants in HICs at intermediate energies have not been widely studied to date. How to interpret the cumulants data at intermediate energies and compare with the data at relativistic energies is now being actively discussed. Both meam-field potential and clustering are highly important to HICs at intermediate energie…
We study fluctuations in the canonical ensemble, where the net baryon number is exactly conserved. The focus is on cumulants and factorial cumulants linked to the baryon and antibaryon multiplicities and their sum or difference in full phase-space as well as in subsystems. In particular, we connect the fluctuations of the net baryon number in a subsystem, relevant for fluctuation studies in nucleus-nucleus collisions, with fluctuations of the baryon and antibaryon numbers of the total system. We derive analytic expressions for factorial cumulants of arbitrary order. Compact results are obtaine…
Canonical statistical hadronization with local baryon conservation for higher-order cumulants
Dipartimento di Fisica dell’Università and Sezione INFN（都灵大学物理系和INFN分部）；CERN（欧洲核子研究组织）；University of Houston（休斯顿大学）；Bogolyubov Institute for Theoretical Physics（博戈留博夫理论物理研究所）
We show that the values of the first three cumulants of the baryon number distribution can be used to calculate the isothermal speed of sound and its logarithmic derivative with respect to the baryon number density. We discuss applications of this result to heavy-ion collision experiments and address possible challenges, including effects due to baryon number conservation, differences between proton and baryon cumulants, and the influence of finite number statistics on fluctuation observables in both experiment and hadronic transport simulations. In particular, we investigate the relation betw…
We present a method that may allow an estimate of the value of the speed of sound as well as its logarithmic derivative with respect to the baryon number density in matter created in heavy-ion collisions. To this end, we utilize well-known observables: cumulants of the baryon number distribution. In analyses aimed at uncovering the phase diagram of strongly interacting matter, cumulants gather considerable attention as their qualitative behaviour along the explored range of collision energies is expected to aid in detecting the QCD critical point. We show that the cumulants may also reveal the…
We study the higher moments of the baryon number in the immediate neighbourhood of the QCD critical endpoint within the framework of Ising-QCD thermodynamics (N.~G. Antoniou {\it et al}, arXiv:1705.09124 [hep-ph]). We show that the kurtosis, as a function of the freeze-out baryon chemical potential, attains a sharp minimum very close to the critical point. We argue that the sharpness of this minimum is due to the narrowness of the critical region in the chemical potential direction. Our analysis reveals that the broad minimum of the kurtosis observed in Au+Au central collisions at STAR (in RHI…
We analyze the rapidity and transverse momentum dependence for the cumulants of the net-proton and net-baryon distributions in Au+Au collisions at $\sqrt{s_{\text{NN}}} = 5\,\text{GeV}$ with a microscopic hadronic transport (JAM) model. To study the effects of mean field potential and softening of equation of state (EoS) on the fluctuations of net-proton (baryon) in heavy-ion collisions, the calculations are performed with two different modes. The softening of EoS is realized in the model by implementing the attractive orbit in the two-body scattering to introduce a reduction pressure of the s…
Cumulants of baryon number are given considerable attention in analyses of heavy-ion collision experiments as possible signatures of the QCD critical point. In this work, we show that the values of the lowest three cumulants can also be utilized to recover information about the isothermal speed of sound and its logarithmic derivative with respect to the baryon number density. This result provides a new method for obtaining information about fundamental properties of nuclear matter studied in heavy-ion collisions, with consequences for both the search for the QCD critical point and neutron star…
Using high statistics datasets generated in (2+1)-flavor QCD calculations at finite temperature we present results for low order cumulants of net baryon-number fluctuations at non-zero values of the baryon chemical potential. We calculate Taylor expansions for the pressure (zeroth order cumulant), net baryon-number density (first order cumulant) and the variance of the distribution on net-baryon number fluctuations (second order cumulant). We obtain series expansions from an eighth order expansion of the pressure and compare these to diagonal Padé approximants. This allows us to estimate the r…
Using recent results on higher order cumulants of conserved charge fluctuations from lattice QCD, we construct mean, variance, skewness, kurtosis, hyper-skewness and hyper-kurtosis of net-baryon number distributions for small baryon chemical potentials $μ_B$. For the strangeness neutral case ($μ_S=0$) at fixed ratio of electric charge to baryon number density ($\frac{n_Q}{n_B}=0.4$), which is appropriate for a comparison with heavy ion collisions, we present results for $κ_B σ_B^2$, $S_B σ_B^3/M_B$, $κ^{H}_{B}σ_{B}^4$ and $S^{H}_{B}σ^5_{B}/M_{B}$ on the crossover line for the chiral transition…
We analyze the behavior of (net-)proton number cumulants in central collisions of heavy ions across a broad collision energy range by utilizing hydrodynamic simulations. The calculations incorporate essential non-critical contributions to proton fluctuations such as repulsive baryonic core and exact baryon number conservation. The experimental data are consistent with non-critical physics at collision energies $\sqrt{s_{\rm NN}} \geq 20$ GeV. The data from STAR and HADES Collaborations at lower collision energies indicate an excess of (multi-)proton correlations over the non-critical reference…
Fluctuations and correlations of conserved charges are sensitive observables for studying the QCD phase transition and critical point in high-energy heavy-ion collisions. We have studied the centrality and energy dependence of mixed-cumulants (up to fourth order) between net-baryon and net-strangeness in Au+Au collisions at $\sqrt{s_{NN}}$= 7.7, 11.5, 19.6, 27, 39, 62.4, 200 GeV from UrQMD model. To compare with other theoretical calculations, we normalize these mixed-cumulants by various order cumulants of net-strangeness distributions. We found that the results obtained from UrQMD calculatio…
Fluctuations of conserved quantities, such as baryon, electric charge and strangeness number, are sensitive observables in heavy-ion collisions to search for the QCD phase transition and critical point. In this paper, we performed a systematical analysis on the various cumulants and cumulant ratios of event-by-event net-strangeness distributions in Au+Au collisions at $\sqrt{s_{NN}}$=7.7, 11.5, 19.6, 27, 39, 62.4 and 200 GeV from UrQMD model. We performed a systematical study on the contributions from various strange baryons and mesons to the net-strangeness fluctuations. The results demonstra…
The effects of finite particle number sampling on the net baryon number cumulants, extracted from fluid dynamical simulations, are studied. The commonly used finite particle number sampling procedure introduces an additional Poissonian (or multinomial if global baryon number conservation is enforced) contribution which increases the extracted moments of the baryon number distribution. If this procedure is applied to a fluctuating fluid dynamics framework one severely overestimates the actual cumulants. We show that the sampling of so called test-particles suppresses the additional contribution…
We determine the net-baryon multiplicity distribution which reproduces all cumulants measured so far by lattice QCD. We present the dependence on the volume and temperature of this distribution. We find that for temperatures and volumes encountered in heavy ion reactions, the multiplicity distribution is very close to the Skellam distribution, making the experimental determination of it rather challenging. We further provide estimates for the statistics required to measure cumulants of the net-baryon and net-proton distributions.
We study the critical properties of net-baryon-number fluctuations at the chiral restoration transition in a medium at finite temperature and net baryon density. The chiral dynamics of quantum chromodynamics (QCD) is modeled by the Polykov-loop extended Quark-Meson Lagrangian, that includes the coupling of quarks to vector meson and temporal gauge fields. The Functional Renormalization Group is employed to properly account for the $O(4)$ criticality at the phase boundary. We focus on the properties and systematics of ratios of the net-baryon-number cumulants $χ_B^n$, for ${1\leq n\leq 6}$, nea…
We study the influence of global baryon number conservation on the non-critical baseline of net baryon cumulants in heavy-ion collisions in a given acceptance, accounting for the asymmetry between the mean-numbers of baryons and antibaryons. We derive the probability distribution of net baryon number in a restricted phase space from the canonical partition function that incorporates exact conservation of baryon number in the full system. Furthermore, we provide tools to compute cumulants of any order from the generating function of uncorrelated baryons constrained by exact baryon number conser…
Unveiling QCD Criticality with Cross-Rapidity Net-Baryon Cumulants
Universität Heidelberg（海德堡大学）；GSI Helmholtzzentrum für Schwerionenforschung（GSI重离子研究中心）；Tsinghua University（清华大学）；University of California, Berkeley（加州大学伯克利分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
The effect of nuclear interactions on measurable net-proton number fluctuations in heavy ion collisions at the SIS18/GSI accelerator is investigated. The state of the art UrQMD model including interaction potentials is employed. It is found that the nuclear forces enhance the baryon number cumulants, as predicted from grand canonical thermodynamical models. The effect however is smeared out for proton number fluctuations due to iso-spin randomization and global baryon number conservation, which decreases the cumulant ratios. For a rapidity acceptance window larger than $Δy> 0.4$ the effects of…
We present lattice QCD calculations of higher order cumulants of electric charge distributions for small baryon chemical potentials $μ_B$ by using up to NNNLO Taylor expansions. Ratios of these cumulants are evaluated on the pseudo-critical line, $T_{pc}(μ_B)$, of the chiral transition and compared to corresponding measurements in heavy ion collision experiments by the STAR and PHENIX Collaborations. We demonstrate that these comparisons give strong constraints on freeze-out parameters. Furthermore, we use strangeness fluctuation observables to compute the ratio $μ_S/μ_B$ on the crossover line…
The appearance of large, none-Gaussian cumulants of the baryon number distribution is commonly discussed as a signal for the QCD critical point. We review the status of the Taylor expansion of cumulant ratios of baryon number fluctuations along the freeze-out line and also compare QCD results with the corresponding proton number fluctuations as measured by the STAR Collaboration at RHIC. To further constrain the location of a possible QCD critical point we discuss poles of the baryon number fluctuations in the complex plane. Here we use not only the Taylor coefficients obtained at zero chemica…
We present here continuum extrapolated results for all 2nd order cumulants using the most recent results obtained by the HotQCD collaboration in (2+1)-flavor QCD. We constrain the applicability of various HRG models by presenting a detailed comparison of our results based on different sets of hadron spectra as well as with virial expansion based model calculations. A comparison with our lattice QCD results for 2nd order cumulants with models that parametrize repulsive interactions among baryons and anti-baryons in a hadron resonance gas through a single excluded volume parameter (EVHRG) is als…
We develop a framework to relate proton number cumulants measured in heavy-ion collisions within a momentum space acceptance to the susceptibilities of baryon number, assuming that particles are emitted from a fireball with uniform distribution of temperature and baryochemical potential, superimposed on a hydrodynamic flow velocity profile. The rapidity acceptance dependence of proton cumulants measured by the HADES Collaboration in $\sqrt{s_{\rm NN}} = 2.4$ GeV Au-Au appears to be consistent with thermal emission of nucleons from a grand-canonical heat bath, with the extracted baryon number s…
Finite-size behavior of higher-order cumulant ratios near criticality in two-dimensional Potts models
Indian Institute of Science Education and Research Bhopal（印度科学教育研究所博帕尔分校）；School of Physical Sciences, National Institute of Science Education and Research（国家科学教育研究所物理科学学院）；Homi Bhabha National Institute（霍米·巴巴国立研究所）
By linearly parameterizing the QCD Landau free energy near the critical point in the baryon chemical potential and temperature plane, we study the fluctuations of the QCD chiral order parameter field (the $σ$ field) in the equilibrium case and dynamical phase transition, respectively. By setting the system size to the typical size of the QGP fireball ($\approx 10^3$ fm$^3$), we show that in the equilibrium case, the discontinuity of the order parameter in the first order phase transition region is replaced by smooth crossover, and the corresponding fluctuations are broadened. Meanwhile, the qu…
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 val…
The search for a possible critical point in the QCD phase diagram is ongoing in heavy ion collision experiments at RHIC which scan the phase diagram by scanning the beam energy; a coming upgrade will increase the luminosity and extend the rapidity acceptance of the STAR detector. In fireballs produced in RHIC collisions, the baryon density depends on rapidity. By employing Ising universality together with a phenomenologically motivated freezeout prescription, we show that the resulting rapidity dependence of cumulant observables sensitive to critical fluctuations is distinctive. The dependence…
We update lattice QCD results for second order cumulants of conserved charge fluctuations and correlations at non-zero temperature and vanishing values of the conserved charge chemical potentials. We compare these results to hadron resonance gas calculations with and without excluded volume terms as well as S-matrix results in the hadronic phase of QCD, and comment on their current limitations. We, furthermore, use these results to characterize thermal conditions in the vicinity of the pseudo-critical line of the chiral transition in QCD. We argue that the ratio of strange to baryon chemical p…
We analyze the behavior of cumulants of conserved charges in a subvolume of a thermal system with exact global conservation laws by extending a recently developed subensemble acceptance method (SAM) [V. Vovchenko et al., arXiv:2003.13905] to multiple conserved charges. Explicit expressions for all diagonal and off-diagonal cumulants up to sixth order that relate them to the grand canonical susceptibilities are obtained. The derivation is presented for an arbitrary equation of state with an arbitrary number of different conserved charges. The global conservation effects cancel out in any ratio …
