We report the latest results on the search for the QCD critical point in the QCD phase diagram through high-energy heavy-ion collisions. The measurements discussed are based on the higher moments of the net-proton multiplicity distributions in heavy-ion collisions. A non-monotonic variation in the product of kurtosis times the variance of the net-proton distribution is observed as a function of the collision energy with 3\(\sigma \) significance. We also discuss the results of the thermal model in explaining the measured particle yield ratios in heavy-ion collisions and comparison of the different variants of hardon resonance gas model calculation to the data on higher moments of net-proton distributions. We end with a note that the upcoming programs in high baryon density regime at various experimental facilities will complete the search for the QCD critical point through heavy-ion collisions.

The appearance of large, non-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 chemical potential but perform also calculations of Taylor expansion coefficients of the pressure at purely imaginary chemical potentials.

We present results from lattice QCD calculations for \(2^{\mathrm {nd}}\) and \(4^{\mathrm {th}}\) order cumulants of conserved charge fluctuations and correlations, and compare these with various High Resonance Gas (HRG) model calculations. We show that differences between HRG and QCD calculations already show up in the second order cumulants close to the pseudo-critical temperature for the chiral transition in (\(2+1\))-flavor QCD and quickly grow large at higher temperatures. We also show that QCD results for strangeness fluctuations are enhanced over HRG model calculations which are based only on particles listed in the Particle Data Group tables as 3-star resonances. This suggests the importance of contributions from additional strange hadron resonances. We furthermore argue that additional (repulsive) interactions, introduced either through excluded volume (mean field) HRG models or the S-matrix approach, do not improve the quantitative agreement with \(2^{\mathrm {nd}}\) and \(4^{\mathrm {th}}\) order cumulants calculated in lattice QCD. HRG-based approaches fail to describe the thermodynamics of strongly interacting matter at or shortly above the pseudo-critical temperature of QCD.

Shear viscosity is a dynamical property of fluid systems close to equilibrium, describing resistance to shear flow. After reviewing the physics of viscosity and the reason why it is usually difficult to compute, I discuss its importance within the theory of QCD and the obstacles to carrying out such a computation. A diagrammatic analysis requires extensive resummations and even then convergence is poor at physically relevant couplings. Lattice approaches require a poorly controlled analytical continuation of data from the Euclidean to the Minkowski domain. At present, our best results for QCD shear viscosity come from the hydrodynamical interpretations of experiments, with first-principles calculations trailing behind.

We analyze the thermal profiles of the bulk to shear viscosity ratio in a quasiparticle framework which describes the interaction in a deconfined medium through dynamical masses of its constituents. The temperature dependence of the effective masses is specified by a running coupling deduced from the lattice QCD thermodynamics. To study the impact of dynamical quarks on the transport properties of the hot medium, we confront the results in \(N_\mathrm {f}=2+1\) QCD with the observations in pure Yang–Mills theory. We show that dynamical quarks modify the behavior of the bulk to shear viscosity ratio and delay the restoration of conformal invariance. Around the (pseudo)critical temperature in both theories, the bulk to shear viscosity ratio behaves linearly in the squared speed of sound, as found in the AdS/CFT approach. At high temperature, the behavior of the ratio becomes quadratic, which corresponds to the perturbative QCD scaling. Thus, we find that the quasiparticle model is capable of describing the transport properties of the QCD in the weak and strong coupling regimes.

We review recent results on the QCD phase transition line and second-order fluctuations and correlators of conserved charges calculated in lattice QCD. In the case of fluctuations, we compare them to the Hadron Resonance Gas model and construct proxies that allow a direct comparison between first principle simulations and measurements.

We present a lattice QCD based determination of the chiral phase transition temperature in QCD with two massless (up and down) and one strange quark having its physical mass. We propose and calculate two novel estimators for the chiral transition temperature for several values of the light-quark masses, corresponding to the Goldstone pion masses in the range of \(58~{\mathrm {MeV}}\lesssim m_\pi \lesssim 163\) MeV. The chiral phase transition temperature is determined by extrapolating to vanishing pion mass using universal scaling relations. After thermodynamic, continuum and chiral extrapolations, we find the chiral phase transition temperature \(T_{\mathrm {c}}^0=132^{+3}_{-6}\) MeV. We also show some preliminary calculations that use the conventional estimator for the pseudo-critical temperature and compare them with the new estimators for \(T_{\mathrm {c}}^0\). Furthermore, we show results for the ratio of the chiral order parameter and its susceptibility, and argue that this ratio can be used to differentiate between O\((N)\) and Z\(_2\) universality classes in a non-parametric manner.

I discuss the deconfinement transition in (\(2+1\))-flavor QCD in terms of Polyakov loops as well as the hadron resonance gas for hadrons containing static quarks and charm quarks.

In the heavy, static quark mass regime of QCD, the Polyakov loop is well known to be an order parameter of the deconfinement phase transition; however, the sensitivity of the Polyakov loop to the deconfinement of light, dynamical quarks is less clear. On the other hand, from the perspective of an effective Lagrangian written in the vicinity of the chiral transition, the Polyakov loop is an energy-like operator and should hence scale as any energy-like operator would. We show here that the Polyakov loop and heavy-quark free energy are sensitive to the chiral transition, i.e. their scaling is consistent with energy-like observables in 3d, O\((N)\) universality classes.

A possible effective restoration of the anomalous U\(_{\mathrm {A}}(1)\) symmetry would have a non-trivial effect on the global phase diagram of QCD. In this work, we investigate the effective restoration of the U\(_{\mathrm {A}}(1)\) through the calculation of scalar and pseudo-scalar screening masses and corresponding susceptibilities, for physical and lower than physical pion masses. Calculations have been performed in (\(2+1\))-flavor HISQ discretization scheme with a physical value of the strange-quark mass. Preliminary calculations of the continuum extrapolated scalar and pseudo-scalar masses are presented, based on lattices with three different temporal extent. Non-trivial structure of the difference between scalar and pseudo-scalar susceptibilites are discussed for \(N_\tau =8\) lattices.

After combined character and hopping expansions and integration over the spatial gauge links, lattice QCD reduces to a three-dimensional SU\((3)\) Polyakov loop model with complicated interactions. A simple truncation of the effective theory is valid for heavy quarks on reasonably fine lattices and can be solved by linked cluster expansion in its effective couplings. This was used ealier to demonstrate the onset transition to baryon matter in the cold and dense regime. Repeating these studies for general \(N_{\mathrm {c}}\), one finds that for large \(N_{\mathrm {c}}\) the onset transition becomes first-order, and the pressure scales as \(p\sim N_{\mathrm {c}}\) through three consecutive orders in the hoppoing expansion. These features are consistent with the formal definition of quarkyonic matter given in the literature. We discuss the implications for \(N_{\mathrm {c}}=3\) and physical QCD.

The description of hadron production in relativistic heavy-ion collisions in the statistical hadronization model is very good, over a broad range of collision energy. We outline this both for the light (\(u, d, s\)) and heavy (charm) quarks and discuss the connection it brings to the phase diagram of QCD.

Non-critical contributions are discussed in the context of fluctuations of conserved charges, which are of particular importance for disentangling critical signals originating from second order phase transitions. The approach is based on a model-independent construction of canonical partition function, i.e. , by using measured mean multiplicities of baryons and anti-baryons. The experimental measurements of the STAR and ALICE collaborations are confronted with the presented model predictions. For this purpose, in line with the experimental observations, different acceptances are introduced for baryons and anti-baryons. It is demonstrated that nearly all measured experimental signals of net-proton cumulants, up to order four, can be described by accounting for global baryon number conservation. A dedicated Python package is developed in order to obtain analytical expressions for cumulants of any order of net-baryon (net-proton) distributions. Moreover, it is demonstrated that contributions due to local baryon number conservation at the LHC energies are negligible, which implies sensitivity of measured second order cumulants to early stages of collisions.

We report on recent progress concerning effects of global conservation laws on cumulants of conserved quantities. Specifically, we will relate — for an arbitrary equation of state — cumulants of a conserved charge measured in a subvolume of a thermal system with the corresponding grand-canonical susceptibilities, taking into account exact global conservation of that charge. Applications to actual measurement at the RHIC and LHC as well as extensions to multiple conserved charges will be discussed.

We present lattice QCD calculations of higher order cumulants of electric charge distributions for small baryon chemical potentials \(\mu _{B}\) by using up to NNNLO Taylor expansions. Ratios of these cumulants are evaluated on the pseudo-critical line, \(T_{\mathrm {pc}}(\mu _{B})\), 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 \(\mu _S/\mu _{B}\) on the crossover line and compare it to \(\mu _S/\mu _{B}\) at freeze-out stemming from fits to strange-baryon yields measured by the STAR Collaboration.

We study the signs of criticality in conserved charge fluctuations and related observables of the finite temperature QCD at vanishing chemical potential, as we approach the chiral limit of two light quarks. Our calculations have been performed on gauge ensembles generated using the Highly Improved Staggered Quark (HISQ) fermion action, with pion masses ranging from 140 MeV to 55 MeV.

Extending the successes of lattice quantum chromodynamics (QCD) at zero as well as nonzero temperatures to nonzero density is extremely desirable in view of the quest for the QCD phase diagram both theoretically and experimentally. It turns out though to give rise to some conundrums whose resolution may assist progress in this exciting but difficult area, and should therefore be sought actively.

In these proceedings, we discuss the natural connection between the reduction of neutral pion mass in the vacuum, and the magnetic catalysis as well as the reduction of transition temperature in the external magnetic field. We also present the first results on fluctuations of and correlations among conserved charges in strong magnetic fields from lattice QCD computations.

We show that external magnetic fields increase the strength of explicit center symmetry breaking induced by dynamical quarks. To study the consequences on deconfinement, we perform a schematic mean-field calculation of the Polyakov loop and its fluctuations, and find that the transition is shifted towards lower temperatures. We also show qualitatively how light quarks affect the magnetic field dependence of the deconfinement temperature in an effective model.

We present a recent development towards a unified description of quark–hadron matter in the QCD phase diagram that is based on a cluster decomposition of the generalized Beth–Uhlenbeck approach to quark matter, self-consistently coupled to Polyakov-loop and mesonic background fields. The Mott dissociation of hadrons under extreme conditions of temperature and density is triggered by chiral symmetry restoration and confining aspects are modeled by the coupling to the background mean fields. First results for the QCD phase diagram with the capability to describe critical endpoints as well as crossover-all-over are presented and an excellent agreement with lattice QCD on the temperature axis is obtained.

Recently, the ALICE Collaboration has observed an interesting systematic behavior of ratios of identified particles to pions yields at the LHC, showing that they depend solely on the charged-particle multiplicity \({\mathrm {d}}N_{\mathrm {ch}}/{\mathrm {d}}\eta \), in \(p\)–\(p\), \(p\)–Pb and Pb–Pb collisions. In particular, the yields of (multi)strange particles, relative to pions increase with \({\mathrm {d}}N_{\mathrm {ch}}/{\mathrm {d}}\eta \) and the enhancement becomes more pronounced with increasing strangeness content. We will argue that such a pattern of strangeness enhancement is arising naturally in the thermal model accounting for exact strangeness conservation. Furthermore, extending the thermal model by including hadron interactions within the S-matrix approach, the ALICE data can be well quantified by the thermal particle yields at the chiral-crossover temperature, as previously found in central Pb–Pb collisions.

We present the recent progress of the hybrid quark–meson–nucleon model for multi-messenger astronomy that unifies the thermodynamics of quark and hadronic degrees of freedom. The obtained equation of state is in accordance with nuclear matter properties at saturation density and with the flow constraint from heavy-ion collision experiments. The mass–radius relations and tidal deformabilities for compact stars are calculated and compared with the latest astrophysical observations. The phase diagram of the isospin-symmetirc matter is studied as well in terms of the higher-order cumulants of the net-baryon number.

We review the key steps of the relativistic fluid dynamics formalism with spin degrees of freedom initiated recently. We obtain equations of motion of the expansion of the system from the underlying definitions of quantum kinetic theory for the equilibrium phase-space distribution functions. We investigate the dynamics of spin polarization of the system in the Bjorken hydrodynamical background.