We present lattice QCD results for ratios of net-baryon number cumulants along the pseudo-critical line and compare them with STAR measurements from the RHIC BES-II program. The ratio of first and second order cumulants, R_12^B, agrees well with corresponding net-proton number cumulants down to √(s_NN)=11.5 GeV or baryon chemical potentials μ_B/T ≤ 2. Likewise higher-order cumulant ratios, R_31^B and R_42^B, show no sign for the existence of a critical point in the parameter range explored with these cumulant ratios. A QCD critical point is unlikely to occur within the BES-II range in collider mode. Moreover, the results demonstrate that a non-interacting HRG description breaks down for μ_B/T > 1. We further analyze baryon-strangeness correlations normalized by strangeness fluctuations, finding consistency with STAR data at large beam energies but deviations at lower energies. Comparisons of electric-charge and strangeness correlations with STAR and ALICE data also show agreement at high energies, while the deviations at lower energies emphasize the role of unobserved strange resonances and the need for controlled feed-down corrections in baryon-strangeness correlations.

For quantifying the universal properties of the chiral phase transition in QCD through numerical calculations on a discrete space-time lattice, one needs to perform controlled extrapolations to the continuum and infinite-volume limits followed by an extrapolation to the limit of massless light quarks. We discuss here, the results on the latter two limits at still finite lattice spacings. We use here for chiral symmetry breaking, an improved order parameter free of additive and multiplicative divergences and we analyse its volume and quark mass dependence. Comparing to the expected universal behavior in the chiral limit, we quantify deviations from the universal finite-size scaling behavior as function of the light to strange quark mass ratio.

We demonstrate that conditional Masked Autoregressive Flows constitute a flexible interpolation tool for lattice QCD observables, conditioned on bare lattice parameters. As a benchmark, we use the chiral phase structure of QCD with five degenerate light quark flavours, which on coarse lattices exhibits a region of first-order chiral transitions terminating in a critical quark mass. The method successfully reproduces standard reweighting in the gauge coupling, and naturally extends to interpolation in quark mass and spatial volume, for which reweighting is computationally prohibitive or inapplicable, respectively. Once trained, the model generates samples across the full parameter space in minutes, which can be used to obtain consistent first estimates of the critical quark mass without simulating all intermediate parameter values. This offers a concrete reduction in the number of lattice ensembles required. Precision on the critical mass from learned distributions is so far prohibited by the mode-covering effect inherent to maximum-likelihood-based training, which introduces a systematic bias near first-order transitions. At the current stage, the method is well-suited for a range of practical applications: localising phase boundaries, identifying the universal scaling axes at a critical point, and accelerating informed determinations of parameter values ahead of high-precision Monte Carlo campaigns.

In this proceeding, we discuss the finite-size scaling analysis of the order parameter related to the chiral phase transition in QCD with two massless quarks. We use data obtained in lattice QCD calculations performed with highly improved staggered quarks (HISQ) for a range of light quark masses, 1/240 ≤ m_ℓ/m_s ≤ 1/27 for different spatial volumes (N_σ) on Euclidean lattices with temporal extent N_τ=8, satisfying 3 N_τ≤ N_σ≤ 10 N_τ. We observe that infinite volume extrapolated data for the order parameter agree reasonably well with the expected O(2) scaling behavior even for physical ratios of the light-to-strange quark mass ratio. We quantify deviations from asymptotic scaling and perform a detailed analysis of the influence of finite-size effects in terms of temperature and quark masses at a fixed lattice cutoff. This is crucial for improving the reliability of the infinite-volume extrapolated estimate of the chiral order parameter and for a more precise determination of chiral phase transition temperature from direct Lattice QCD simulations.

To quantify the universal properties of chiral phase transition in (2+1)-flavor QCD, we use an improved, renormalized order parameter for the chiral symmetry breaking. We construct ratios of this divergence-free order parameter from its values for different pairs of light quark masses. From this, we determine in a parameter-independent manner, the chiral phase transition temperature T_c and the associated critical exponent δ of the universality class. We present first numerical results of these calculations on N_τ=8 lattices, with staggered fermions.

We use up to fourth-order charm fluctuations and their correlations with net baryon number, electric charge, and strangeness fluctuations, calculated within the framework of lattice QCD, to study the continuum partial pressure contributions of charmed baryons and mesons. We show that, at and below the chiral crossover temperature, these partial pressures receive enhanced contributions from experimentally unobserved charmed hadrons predicted by the Quark Model. Additionally, we demonstrate that at the chiral crossover, the Hadron Resonance Gas description breaks down, signaling the modification of open charm hadrons and thereby implying the onset of charm deconfinement. We present evidence for the survival of low-lying non-radial 1S and 1P hadron-like excitations above the chiral crossover, which hints at the sequential melting of charmed hadrons. Finally, we investigate the continuum partial pressure contribution of charm quark-like excitation that emerges at the chiral crossover and calculate its temperature-dependent in-medium mass.

We study the generalized charm susceptibilities in 2+1 flavor QCD on the lattice at several lattice spacings. We show that, below the chiral crossover, these susceptibilities are well described by the hadron resonance gas (HRG) model if charmed hadrons not listed in tables of the Particle Data Group are included. However, the HRG description abruptly breaks down just above the chiral crossover. To understand this, we use a model for the charm pressure in which it is expressed as the sum of partial pressures from charmed baryons, charmed mesons, and charm quarks. We present continuum estimates of these partial pressures and find that, while the partial pressures of charmed mesons and baryons drop below their respective HRG predictions, the charm quark pressure becomes non-zero above the chiral crossover.

The nonsinglet part of the chiral symmetry in quantum chromodynamics (QCD) with two light flavors is known to be restored through a crossover transition at a pseudocritical temperature. However, the temperature dependence of the singlet part of the chiral symmetry and whether it is effectively restored at the same temperature is not well understood. Using (2+1)-flavor QCD configurations generated using the Möbius domain-wall discretization on an Nτ=8 lattice, we construct suitable observables where the singlet and nonsinglet chiral symmetries are disentangled in order to study their temperature dependence across the crossover transition. From the peak of the disconnected part of the chiral susceptibility, we obtain a pseudocritical temperature Tpc=158.7−2.3+2.6 MeV where the nonsinglet part of the chiral symmetry is effectively restored. From a calculation of the topological susceptibility and its temperature dependence we find that the singlet UA(1) part of the chiral symmetry is not effectively restored at T≲186 MeV. Published by the American Physical Society 2025

Using an eighth-order Taylor expansion in baryon chemical potential, we recently obtained the (2+1)-flavor QCD equation of state (EoS) at non-zero conserved charge chemical potentials from the lattice. We focused on strangeness-neutral, isospin-symmetric QCD matter, which closely resembles the situation encountered in heavy-ion collision experiments. Using this EoS, we present here results on various QCD material parameters; in particular we compute the specific heat, speed of sound, and compressibility along appropriate lines of constant physics. We show that in the entire range relevant for the beam energy scan at RHIC, the specific heat, speed of sound, and compressibility show no indication for an approach to critical behavior that one would expect close to a possibly existing critical endpoint.

Phase diagrams of some important theories like quantum chromodynamics (QCD), are only accessible through numerical simulations with only finite degrees of freedom. Information about critical phenomena can then only inferred through finite size scaling studies of various moments of the order parameter. Alternatively, one can study the complex zeros of the grand canonical partition function, known as Lee-Yang zeros, which are accessible as poles of the thermodynamic variables. Moreover the temperature and volume scaling of these poles can be used as a probe to understand the nature of phase transitions in many systems. We will discuss here the recent progress that has been made in this direction with respect to the Roberge-Weiss transition, the 2D Ising model and the conjectured QCD critical point. We will further discuss results on the universal location of Lee-Yang edge singularity from continuum extrapolated scaling function for some $O(N)$ models in $3-d$.

We study the nature of charm degrees of freedom in hot strong interaction matter by performing lattice QCD calculations of the second and fourth-order cumulants of charm fluctuations, and their correlations with net baryon number, electric charge and strangeness fluctuations. We show that below the chiral crossover temperature thermodynamics of charm can be very well understood in terms of charmed hadrons. Above the chiral transition charm quarks show up as new degrees of freedom contributing to the partial charm pressure. However, up to temperatures as high as 175 MeV charmed hadron-like excitations provide a significant contribution to the partial charm pressure.

In order to quantify the universal properties of the chiral phase transition in (2+1)-flavor QCD, we make use of an improved, renormalized order parameter for chiral symmetry breaking which is obtained as a suitable difference of the 2-flavor light quark chiral condensate and its corresponding light quark susceptibility. Having no additive ultraviolet as well as multiplicative logarithmic divergences, we use ratios of this order parameter constructed from its values for two different light quark masses. We show that this facilitates determining in a parameter-independent manner, the chiral phase transition temperature T_c and the associated critical exponent δ which, for sufficiently small values of the light quark masses, controls the quark mass dependence of the order parameter at T_c. We present first results of these calculations from our numerical analysis performed with staggered fermions on N_τ=8 lattices.

We present some lattice QCD results on first (chi(i)(1)) and second (chi(i)(2)) cumulants of and correlations (chi(ij)(11)) among net baryon-number (B), strangeness (S) and electric charge (Q) along the pseudocritical line [T-pc(mu(B))] in the temperature (T)-baryon chemical potential (mu(B)) phase diagram of (2+1)-flavor QCD. We point out that violations of sum rules among second order cumulants, which hold in the isospin symmetric limit of vanishing electric charge chemical potential, are small along the T-pc(mu(B)) for the entire range of mu(B) covered in the RHIC beam energy scan. For the strangeness neutral matter produced in heavy-ion collisions this leads to a close relation between chi(BS)(11) and chi(QS)(11). We compare lattice QCD results for chi(BS)(11)/chi(S)(2) along the T-pc(mu(B)) line with preliminary experimental measurements of chi(BS)(11)/chi(S)(2) for collision energies 7.7 GeV <= root s(NN) <= 62.4 GeV. While we find good agreements for root s(NN) >= 39 GeV, differences are sizeable at smaller values of root s(NN). Moreover, we compare lattice QCD results for the ratio of the strangeness (mu(S)) to baryon (mu(B)) chemical potentials, which define a strangeness neutral system with fixed electric charge to baryon number density, with experimental results obtained by the STAR collaboration for mu(S)/mu(B) using strange baryon yields on the freeze-out line. Finally, we determine the baryon chemical potential at the freeze-out (mu(f)(B)) by comparing chi(B)(1)/chi(B)(2) along the T-pc(mu(B)) with the experimentally measured net-proton cumulants chi(p)(1)/chi(p)(2). We find that {mu(f)(B), T-pc(mu(f)(B))} are consistent with the freeze-out parameters of the statistical-model fits to experimentally measured hadron yields for root s(NN) >= 11.5 GeV.

We discuss the analytic continuation of scaling function in the 3-dimensional Z(2),O(2) andO(4) universality classes using the Schofield representation of the magnetic equation of state. We show that a determination of the location of Lee-Yang edge singularities and, in the case of Z(2), also the Langer edge singularity yields stable results. Results for the former are in good agreement with Functional Renormalization Group calculations. We also present results for the location of the Langer edge singularity in the 3-d,Z(2) universality class. We find that in terms of the complex scaling variable z the distance of the Langer edge singularity to the critical point agrees within errors with that of the Lee-Yang edge singularity. Furthermore the magnitude of the discontinuity along the Langer branch cut is an order of magnitude smaller than that along the Lee-Yang branch cut.

Quantum Chromodynamics, the theory of quarks and gluons, whose interactions can be described by a local SU(3) gauge symmetry with charges called “color quantum numbers”, is reviewed; the goal of this review is to provide advanced Ph.D. students a comprehensive handbook, helpful for their research. When QCD was “discovered” 50 years ago, the idea that quarks could exist, but not be observed, left most physicists unconvinced. Then, with the discovery of charmonium in 1974 and the explanation of its excited states using the Cornell potential, consisting of the sum of a Coulomb-like attraction and a long range linear confining potential, the theory was suddenly widely accepted. This paradigm shift is now referred to as the November revolution. It had been anticipated by the observation of scaling in deep inelastic scattering, and was followed by the discovery of gluons in three-jet events. The parameters of QCD include the running coupling constant, α _s(Q^2) , that varies with the energy scale Q^2 characterising the interaction, and six quark masses. QCD cannot be solved analytically, at least not yet, and the large value of α _s at low momentum transfers limits perturbative calculations to the high-energy region where Q^2≫ _QCD ^2≃ (250 MeV) ^2 . Lattice QCD (LQCD), numerical calculations on a discretized space-time lattice, is discussed in detail, the dynamics of the QCD vacuum is visualized, and the expected spectra of mesons and baryons are displayed. Progress in lattice calculations of the structure of nucleons and of quantities related to the phase diagram of dense and hot (or cold) hadronic matter are reviewed. Methods and examples of how to calculate hadronic corrections to weak matrix elements on a lattice are outlined. The wide variety of analytical approximations currently in use, and the accuracy of these approximations, are reviewed. These methods range from the Bethe–Salpeter, Dyson–Schwinger coupled relativistic equations, which are formulated in both Minkowski or Euclidean spaces, to expansions of multi-quark states in a set of basis functions using light-front coordinates, to the AdS/QCD method that imbeds 4-dimensional QCD in a 5-dimensional deSitter space, allowing confinement and spontaneous chiral symmetry breaking to be described in a novel way. Models that assume the number of colors is very large, i.e. make use of the large N_c -limit, give unique insights. Many other techniques that are tailored to specific problems, such as perturbative expansions for high energy scattering or approximate calculations using the operator product expansion are discussed. The very powerful effective field theory techniques that are successful for low energy nuclear systems (chiral effective theory), or for non-relativistic systems involving heavy quarks, or the treatment of gluon exchanges between energetic, collinear partons encountered in jets, are discussed. The spectroscopy of mesons and baryons has played an important historical role in the development of QCD. The famous X,Y,Z states – and the discovery of pentaquarks – have revolutionized hadron spectroscopy; their status and interpretation are reviewed as well as recent progress in the identification of glueballs and hybrids in light-meson spectroscopy. These exotic states add to the spectrum of expected qq̅ mesons and qqq baryons. The progress in understanding excitations of light and heavy baryons is discussed. The nucleon as the lightest baryon is discussed extensively, its form factors, its partonic structure and the status of the attempt to determine a three-dimensional picture of the parton distribution. An experimental program to study the phase diagram of QCD at high temperature and density started with fixed target experiments in various laboratories in the second half of the 1980s, and then, in this century, with colliders. QCD thermodynamics at high temperature became accessible to LQCD, and numerical results on chiral and deconfinement transitions and properties of the deconfined and chirally restored form of strongly interacting matter, called the Quark–Gluon Plasma (QGP), have become very precise by now. These results can now be confronted with experimental data that are sensitive to the nature of the phase transition. There is clear evidence that the QGP phase is created. This phase of QCD matter can already be characterized by some properties that indicate, within a temperature range of a few times the pseudocritical temperature, the medium behaves like a near ideal liquid. Experimental observables are presented that demonstrate deconfinement. High and ultrahigh density QCD matter at moderate and low temperatures shows interesting features and new phases that are of astrophysical relevance. They are reviewed here and some of the astrophysical implications are discussed. Perturbative QCD and methods to describe the different aspects of scattering processes are discussed. The primary parton–parton scattering in a collision is calculated in perturbative QCD with increasing complexity. The radiation of soft gluons can spoil the perturbative convergence, this can be cured by resummation techniques, which are also described here. Realistic descriptions of QCD scattering events need to model the cascade of quark and gluon splittings until hadron formation sets in, which is done by parton showers. The full event simulation can be performed with Monte Carlo event generators, which simulate the full chain from the hard interaction to the hadronic final states, including the modelling of non-perturbative components. The contribution of the LEP experiments (and of earlier collider experiments) to the study of jets is reviewed. Correlations between jets and the shape of jets had allowed the collaborations to determine the “color factors” – invariants of the SU(3) color group governing the strength of quark–gluon and gluon–gluon interactions. The calculated jet production rates (using perturbative QCD) are shown to agree precisely with data, for jet energies spanning more than five orders of magnitude. The production of jets recoiling against a vector boson, W^± or Z, is shown to be well understood. The discovery of the Higgs boson was certainly an important milestone in the development of high-energy physics. The couplings of the Higgs boson to massive vector bosons and fermions that have been measured so far support its interpretation as mass-generating boson as predicted by the Standard Model. The study of the Higgs boson recoiling against hadronic jets (without or with heavy flavors) or against vector bosons is also highlighted. Apart from the description of hard interactions taking place at high energies, the understanding of “soft QCD” is also very important. In this respect, Pomeron – and Odderon – exchange, soft and hard diffraction are discussed. Weak decays of quarks and leptons, the quark mixing matrix and the anomalous magnetic moment of the muon are processes which are governed by weak interactions. However, corrections by strong interactions are important, and these are reviewed. As the measured values are incompatible with (most of) the predictions, the question arises: are these discrepancies first hints for New Physics beyond the Standard Model? This volume concludes with a description of future facilities or important upgrades of existing facilities which improve their luminosity by orders of magnitude. The best is yet to come!

We analyze scaling functions in the 3D $Z(2)$, $O(2)$, and $O(4)$ universality classes and their finite-size dependence using Monte Carlo simulations of improved ${\ensuremath{\phi}}^{4}$ models. Results for the scaling functions are fitted to the Widom-Griffiths form, using a parametrization also used in analytic calculations. We find good agreement on the level of scaling functions and the location of maxima in the universal part of susceptibilities. We also find that an earlier parametrization of the $O(4)$ scaling function, using 14 parameters, is well reproduced when using the Widom-Griffiths form with only three parameters. We furthermore show that finite-size corrections to the scaling functions are distinctively different in the $Z(2)$ and $O(N)$ universality classes and determine the volume dependence of the peak locations in order parameter and mixed susceptibilities.

We analyze scaling functions in the $3$-$d$, $Z(2)$, $O(2)$ and $O(4)$ universality classes and their finite size dependence using Monte Carlo simulations of improved $ϕ^4$ models. Results for the scaling functions are fitted to the Widom-Griffiths form, using a parametrization also used in analytic calculations. We find good agreement on the level of scaling functions and the location of maxima in the universal part of susceptibilities. We also find that an earlier parametrization of the $O(4)$ scaling function, using 14 parameters, is well reproduced when using the Widom-Griffiths form with only three parameters. We furthermore show that finite size corrections to the scaling functions are distinctively different in the $Z(2)$ and $O(N)$ universality classes and determine the volume dependence of the peak locations in order parameter and mixed susceptibilities.

Phase transitions in a non-perturbative regime can be studied by ab initio Lattice Field Theory methods. The status and future research directions for LFT investigations of Quantum Chromo-Dynamics under extreme conditions are reviewed, including properties of hadrons and of the hypothesized QCD axion as inferred from QCD topology in different phases. We discuss phase transitions in strong interactions in an extended parameter space, and the possibility of model building for Dark Matter and Electro-Weak Symmetry Breaking. Methodological challenges are addressed as well, including new developments in Artificial Intelligence geared towards the identification of different phases and transitions.

Phase transitions in a non-perturbative regime can be studied by ab initio Lattice Field Theory methods. The status and future research directions for LFT investigations of Quantum Chromo-Dynamics under extreme conditions are reviewed, including properties of hadrons and of the hypothesized QCD axion as inferred from QCD topology in different phases. We discuss phase transitions in strong interactions in an extended parameter space, and the possibility of model building for Dark Matter and Electro-Weak Symmetry Breaking. Methodological challenges are addressed as well, including new developments in Artificial Intelligence geared towards the identification of different phases and transitions.

We update results on the QCD equation of state in (2+1)-flavor QCD with non-zero conserved charge chemical potentials obtained from an eighth-order Taylor series. We present results for basic bulk thermodynamic observables of strangeness-neutral strong-interaction matter, i.e. pressure, number densities, energy and entropy density, and resum Taylor series results using Pad\'e approximants. Furthermore, we calculate the speed of sound as well as the adiabatic compression factor of strangeness-neutral matter on lines of constant entropy per net baryon number. We show that the equation of state ($P(n_B), \epsilon (n_B)$) is already well described by the $4^{\rm th}$-order Taylor series in almost the entire range of temperatures accessible with the beam energy scan in collider mode at the Relativistic Heavy Ion Collider.