Theoretical and experimental constraints for the equation of state of dense and hot matter

Abstract

This review aims at providing an extensive discussion of modern constraints relevant for dense and hot strongly interacting matter. It includes theoretical first-principle results from lattice and perturbative QCD, as well as chiral effective field theory results. From the experimental side, it includes heavy-ion collision and low-energy nuclear physics results, as well as observations from neutron stars and their mergers. The validity of different constraints, concerning specific conditions and ranges of applicability, is also provided.

1 Introduction

Depending on conditions (thermodynamic variables), such as temperature and density, matter can appear in many forms (phases). Typical phases include solid, liquid, and gas; but many others can exist, such as plasmas, condensates, and superconducting phases (just to name a few). How matter transitions from one phase to another can also take many forms. A first-order phase transition is how water typically changes from solid to liquid or liquid to gas wherein the phase transition happens at a fixed temperature, free energy, and pressure, which leads to dramatic changes in certain thermodynamic properties (e.g., a jump in the density). At extremely large temperatures and pressures, for water, a crossover phase transition is reached between the liquid and gas phases: depending on what thermodynamic observable one looks at, the substance could look more like a liquid or a gas. In other words, the phase transition no longer takes place at a fixed temperature, free energy, and pressure, but rather across a range of them. Finally, bordering these two regimes, there exists a critical point that separates a crossover phase transition from a first-order one. To describe these different phases of matter, one requires an equation of state (EoS) that depends on the thermodynamic variables of the system. One should note, however, that the EoS is an equilibrium property, and, of course, out-of-equilibrium effects can also be quite relevant. For instance, imagine a body of water that is flowing and being cooled at the same time. In such a dynamical system, one also requires information about the transport coefficients in order to properly describe its behavior as it freezes.

In this work, we will concern ourselves with phases of matter that appear at high energy, relevant when studying the strong force. This is the force that binds together the nucleus, and leads to the generation of 99% of the visible matter in the universe. The theory that governs the strong force is quantum chromodynamics (QCD, Gross and Wilczek 1973; Politzer 1973). QCD describes the interactions of the smallest building blocks of matter (quarks and gluons). Quarks and gluons are normally not free (or “deconfined”) in Nature, but rather confined within hadrons. The latter comprise mesons (quark anti-quark pairs $q{\bar{q}}$), baryons (three-quark states qqq), or anti-baryons (three anti-quark states ${\bar{q}}{\bar{q}}{\bar{q}}$).¹ The quark content and their corresponding quantum numbers (see Table 1) yield the quantum numbers of the hadron itself. One can calculate the thermodynamic properties of strongly interacting matter using either lattice QCD in the non-perturbative regime, or perturbative QCD (pQCD) where the coupling is small (high temperatures and/or extremely high densities).

Table 1 Summary table of quarks including flavor, mass, and quantum numbers (Workman et al. 2022)

Protons (uud quark state), neutrons (udd quark state), and, in rare cases, hyperons (baryons with strange quark content) form nuclei, the properties of which depend on the number of nucleons A, as well as the number of protons Z and the number of neutrons $A-Z$ within the nucleus.² In principle, QCD also drives the properties of nuclei. However, in the vast majority of cases, it would not be convenient to calculate the properties of nuclei or nuclear matter (beyond densities and temperatures at which nuclei dissolve into a soup of hadrons) directly from the Lagrangian of QCD, both because of the numerical challenges but also because it would not be the most effective way (it would be akin to calculating the properties of a lake from the microscopic interactions of H$_2$O molecules). The objective of this review article is to put together the constraints derived from fundamental theories (that are gauge invariant and renormalizable) and observations. For this reason, we do not discuss relativistic mean-filed models. We also opted not to incorporate the different approaches used to describe nuclear matter, and refer instead to an excellent review on this subject (Oertel et al. 2017). Two different approaches are generally used to obtain the equation of state of nuclear matter from these constraints: ab-initio many-body methods using realistic interactions (these include Green Function methods, variational and Monte Carlo methods, (Dirac)–Brueckner–Hartree–Fock calculations and an example is the well known Akmal, Pandharipande, and Ravenhall EoS (Akmal et al. 1998)) or phenomenological approaches based in density functional theories applying effective interactions, including relativistic mean-field models with meson exchange forces and non-relativistic Skyrme and Gogny forces, see Oertel et al. (2017) for a review. Using these methods, one can calculate thermodynamic quantities at low temperatures (on the MeV scale) and around nuclear saturation density, $n_{\text{sat}}$, which represents the point on the saturation curve where the binding energy per nucleon in a nucleus is at its lowest, indicating a balance between the attractive and repulsive nuclear forces, therefore, maximal stability within the nuclear system (Haensel et al. 1981).

How can we solve QCD and study nuclear matter theoretically? How can we probe QCD and nuclear matter experimentally? What systems in Nature and in the laboratory are sensitive to quarks and gluons, hadrons, or nuclei? At large temperatures and vanishing net baryon densities $n_B=0$ (i.e., the same amount of baryons/quarks and anti-baryons/anti-quarks), the conditions are the same as those of the early universe and can be reproduced in the laboratory, at the Large Hadron Collider (LHC, Citron et al. 2019) and at the Relativistic Heavy Ion Collider (RHIC, STAR Collaboration 2014; Cebra et al. 2014) for top center-of-mass beam energies $\sqrt{s_{NN}}=200$ GeV. In equilibrium, lattice QCD can be used to calculate the EoS, which can be extended to finite $n_B$ using expansion schemes up to baryon chemical potentials (over temperature) of about $\mu _B/T\sim 3.5$. Medium to low energy RHIC collisions explored in the Beam Energy Scan (BES) phase I and II ($\sqrt{s_{NN}}=7.7-200$ GeV in collider mode), as well as existing and future fixed target experiments at RHIC (STAR Collaboration 2014; Cebra et al. 2014), SPS (Pianese et al. 2018), HADES (Galatyuk 2014, 2020), and FAIR (Friese 2006; Tahir et al. 2005; Lutz et al. 2009; Durante et al. 2019) can reach temperatures in the range $T\sim 50-350$ MeV and baryon chemical potentials $\mu _B\sim 20 -800$ MeV, using a range of center of mass beam energies $\sqrt{s_{NN}}\sim 2-11$ GeV. Therefore, these low-energy experiments provide a significant amount of information that can also be used to infer the EoS (Dexheimer et al. 2021c; Lovato et al. 2022b; Sorensen et al. 2024). However, these systems are probed dynamically and may be far from equilibrium, so one must not consider the EoS extracted from heavy-ions as data in the typical sense, but rather as a posterior model that is sensitive to priors and systematic uncertainties that may exist in that model. In the high temperature and/or chemical potential limit, systematic methods such as perturbative resummations can be used to calculate the EoS analytically directly form the QCD Lagrangian.

Low-energy nuclear experiments provide methods to extract key properties of nuclei. Most stable nuclei are composed of “isospin-symmetric nuclear matter”, i.e $Z=0.5\ A$, such that the number of protons and neutrons are equal in the nucleus. For simplicity, one defines the charge fraction $Y_Q=Z/A$, which can also be related to the charge density $n_Q$ (assuming a system of only hadrons, no leptons) over the baryon density $n_B$ such that $Y_Q=n_Q/n_B$ as well. Then, for symmetric nuclear matter $Y_Q=0.5$ and this is where most nuclear experiments provide information. However, heavy nuclei do become more neutron rich, such that $Y_Q\sim 0.4$. Note that, for the highest energies, heavy-ion experiments only probe $Y_Q=0.5$ as the nuclei basically pass through each other, and the fireball left behind cannot create net isospin ($Y_Q\ne 0.5$) or strangeness ($Y_S=S/A=n_S/n_B\ne 0$) during the very brief time of the collision (on the order of $\sim 10$ fm/c or $10^{-23}$ s).

All thermodynamic properties change as $Y_Q$ varies. This can be measured experimentally in low-energy nuclear experiments around $n_{\text{sat}}$ through the determination of the symmetry energy $E_{\text{sym}}$, which can be approximated as the difference between the energy per nucleon of $Y_Q=0$ (pure neutron matter $E_{{\text{PNM}}}$) and $Y_Q=0.5$ matter (symmetric nuclear matter $E_{{\text{SNM}}}$)³

$$\begin{aligned} E_{\text{sym}}\equiv \frac{E_{{\text{PNM}}}-E_{{\text{SNM}}}}{N_B}. \end{aligned}$$

(1)

The baryon number $N_B$ is more comprehensive than A, as it also includes quarks, with $N_B=1/3$. At saturation density, many other quantities can be determined such as the binding energy per nucleon, or the (in)compressibility of matter, in addition to $n_{\text{sat}}$ itself. At small $Y_Q$, matter in neutron stars provides information about both nuclear and QCD matter at low temperatures and medium-to-high densities. Matter in this case is necessarily charge-neutral, as $Y_Q=Y_{lep}$, the charge fraction of leptons (electrons and muons). On the other hand, weak(-force) equilibrium ensures $\mu _Q=-\mu _e=-\mu _\mu$, meaning that the charge chemical potential, the difference between the chemical potential of protons and neutrons (in the absence of hyperons), or up and down quarks, equals the ones of electrons and muons.

At saturation densities, a neutron star’s internal composition is primarily made up of nucleons and leptons. However, as the density increases, other baryonic species may appear due to the rapid rise in baryon chemical potential associated with a higher density and reduce the ground state energy of the dense nuclear matter phase by opening new Fermi channels. Due to the long time-scales involved (when compared to weak interactions), matter in neutron stars can also include particles with net strangeness, hyperons. Here on Earth, hyperons can be produced but are unstable and quickly decay in $\sim 10^{-8}$ seconds via weak interactions into protons and neutrons. In the high density regime in the core of neutron stars, hyperons cannot decay back to nucleons due to Pauli blocking, meaning that producing additional nucleons would increase the energy of the system (Joglekar et al. 2019; Alford et al. 2021; Gavassino et al. 2021; Celora et al. 2022; Most et al. 2022) as well as deconfinement to quark matter (Bauswein et al. 2019; Most et al. 2019; Weih et al. 2020; Blacker et al. 2020; Tootle et al. 2022; Constantinou et al. 2021).

2.8 Organization of the paper

This review paper aims at compiling up-to-date constraints from high-energy physics, nuclear physics, and astrophysics that relate to the EoS and are, therefore, fundamental to the understanding of current and future data from heavy-ion collisions to gravitational waves, making them relevant to a very broad community. Additionally, precise knowledge of the dense and hot matter EoS can help physicists to look beyond the standard model either for dark matter, which may accumulate in or around neutron stars, or for modified theories of gravity.

The paper is organized as follows: we first discuss the theoretical constraints of lattice (Sect. 3) and perturbative QCD (Sect. 4), followed by $\chi$EFT (Sect. 5). Then, we discuss experimental constraints from heavy-ion collisions (Sect. 6), (isospin symmetric and asymmetric) low-energy nuclear physics (Sect. 7), and astrophysical observations (Sect. 8). We provide a future outlook in Sect. 9, since a significant amount of new data is anticipated over the next decade.

3 Theoretical constraints: lattice QCD

Lattice QCD is the most suitable method to study strong interactions around and above the deconfinement phase transition region in the QCD phase diagram, due to its non-perturbative nature (Drischler et al. 2021b). As discussed in the introduction, due to the sign problem, first-principles lattice QCD results for the EoS at finite $\mu _B$ are currently restricted. Since direct lattice simulations at $\mu _B=0$ and imaginary $\mu _B$ are feasible, observables can be extrapolated using techniques involving zero or imaginary chemical potential simulations, i.e., analytical continuation, Taylor series and other alternative expansions. In this section, we present various constraints on the EoS, BSQ (baryon number, strangeness, and electric charge) susceptibilities, and partial pressures evaluated using lattice QCD.

3.1 Equation of state

In Borsanyi et al. (2014a) and Bazavov et al. (2014), the EoS was obtained in lattice QCD simulations at $\mu _B=0$. It was found that the rigorous continuum extrapolation results for 2+1 quark flavors are perfectly compatible with previous continuum estimates based on coarser lattices (Aoki et al. 2006; Borsanyi et al. 2010). The obtained pressure, entropy density, and interaction measure are displayed in the left panel of Fig. 3 alongside the predictions of the hadron resonance gas (HRG) model (Venugopalan and Prakash 1992) at low temperatures and the Stefan–Boltzmann (or conformal) limit of a non-interacting massless quarks gas at high T. They show full agreement with HRG results in the hadronic phase, and reach about 75% of the Stefan–Boltzmann limit at $T\simeq 400$ MeV.

Fig. 3

Images reproduced with permission from [left] Ratti (2018), copyright by IOP, and [right] from Borsányi et al. (2021), copyright by the author(s)

Left: comparison between the lattice QCD EoS at $\mu _B=0$ from the WB collaboration Borsanyi et al. (2014a) (colored points) and the HotQCD one (Bazavov et al. 2014) (gray points). Right: Baryonic density as a function of the temperature, for different values of $\mu _B/T$. At low temperature, full lines show results from the HRG model.

Furthermore, a Taylor series can be utilized to expand many observables to finite $\mu _B/T$

$$\begin{aligned} \frac{p(T,\mu _B,\mu _Q, \mu _S)}{T^4}=\sum _{i,j,k}\frac{1}{i!j!k!}\chi _{ijk}^{BQS} \left( \frac{\mu _B}{T}\right) ^i\left( \frac{\mu _Q}{T}\right) ^j\left( \frac{\mu _S}{T}\right) ^k\, \end{aligned}$$

(2)

where the susceptibilities $\chi _{ijk}^{BQS}$ are defined as follows

$$\begin{aligned} \chi _{lmn}^{BSQ}=\frac{\partial ^{l+m+n}(p/T^4)}{\partial (\mu _B/T)^l\partial (\mu _S/T)^m\partial (\mu _Q/T)^n}. \end{aligned}$$

(3)

They were obtained from lattice QCD calculations up to ${\mathcal {O}}(\mu _B/T)^4$ for the full series of coefficients and up to ${\mathcal {O}}(\mu _B/T)^8$ for some of the coefficients. The range of applicability of the Taylor expansion has recently been extended from $\mu _B/T\le 2$ (Guenther et al. 2017; Bazavov et al. 2017) to $\mu _B/T\le 2.5$ (Bollweg et al. 2022). Isentropic trajectories in the $T-\mu _B$ plane have been extracted in Guenther et al. (2017), for which the starting points are the freeze-out parameters at different collision energies at RHIC (Alba et al. 2014). Strangeness neutrality and electric charge conservation were enforced by tuning the strange and electric charge chemical potentials, $\mu _S(\mu _B, T)$ and $\mu _Q(\mu _B,T)$, to reproduce the conditions $Y_S=0$ and $Y_Q=0.4$ (Guenther et al. 2017).

A new expansion scheme for extending the EoS of QCD to unprecedentedly large baryonic chemical potential up to $\mu _B/T<3.5$ has been proposed recently (Borsányi et al. 2021). The drawbacks of the conventional Taylor expansion approach, such as the challenges involved in carrying out such an expansion with a constrained number of coefficients and the low signal-to-noise ratio for the coefficients themselves, are significantly reduced in this new scheme (Borsányi et al. 2021). In the hadronic phase, a good agreement is found for the thermodynamic variables with HRG model results. This scheme is based on the following identity

$$\begin{aligned} \frac{\chi _1^B(T,\mu _B)}{\mu _B/T}=\chi _2^B(T',0)\, \end{aligned}$$

(4)

with

$$\begin{aligned} T'(T,\mu _B)=T(1+\kappa _2^{BB}(T)\left( \frac{\mu _B}{T}\right) ^2+\kappa _4^{BB}(T)\left( \frac{\mu _B}{T}\right) ^4+{\mathcal {O}}((\mu _B/T)^6. \end{aligned}$$

(5)

The baryonic density as a function of the temperature for different values of $\mu _B/T$ from Borsányi et al. (2021) is shown in the right panel of Fig. 3. This extrapolation method was then generalized to include the strangeness neutrality condition (Borsanyi et al. 2022), which requires $\mu _S\ne 0$. The extrapolation approach is devoid of the unphysical oscillations that afflict fixed order Taylor expansions at higher $\mu _B$, even in the strangeness neutral situation. Effects beyond strangeness neutrality are estimated by computing the baryon-strangeness correlator to strangeness susceptibility ratio $\frac{\chi ^{BS}_{11}}{\chi ^S_2}$ (discussed in the following subsection) at finite real $\mu _B$ on the strangeness neutral line. This permits a leading order extrapolation in the ratio $R={\chi ^S_1}/{\chi ^B_1}$ (Borsanyi et al. 2022).

3.2 BSQ susceptibilities

Fluctuations of different conserved charges have been postulated as a signal of the deconfinement transition because they are sensitive probes of quantum numbers and related degrees of freedom. In heavy-ion collisions, one needs to relate fluctuations of net baryon number, strangeness, and electric charge with the event-by-event fluctuations of particle species. Non-diagonal correlators of conserved charges, like fluctuations, are useful for studying the chemical freeze-out in heavy-ion collisions. In thermal equilibrium, these correlators may be estimated using lattice simulations, and linked to moments of event-by-event distributions of multiplicity (i.e., number of particles of a given species in some kinematic region, typically these are all charged particles) distributions. They are defined as derivatives of the pressure with respect to the chemical potentials according to Eq. (3). The quark number chemical potentials appear as parameters in the Grand Canonical partition function. The derivative of this function with respect to these chemical potentials yields the susceptibilities and the non-diagonal correlators of the quark flavors. Quark flavor chemical potentials can be related to the conserved charge ones through the following relationships: $\mu _u=\frac{1}{3}\mu _B+\frac{2}{3}\mu _Q$, $\ \mu _d=\frac{1}{3}\mu _B-\frac{1}{3}\mu _Q$, and $\ \mu _s=\frac{1}{3}\mu _B-\frac{1}{3}\mu _Q-\mu _S$.

For $T=125-400$ MeV and at $\mu _B=\mu _S=\mu _Q=0$, the Wuppertal-Budapest lattice QCD collaboration computed the non-diagonal (us) and diagonal (B,s,Q,I,u) susceptibilities for a system of 2+1 staggered quark flavors (Borsanyi et al. 2012), where I stands for isospin. Selected susceptibilities are shown in the left panel of Fig. 4. A Symanzik-improved gauge and a stout-link improved staggered fermion action were used in this analysis. The ratios of fluctuations were found, whose behavior may be recreated using hadronic observables, i.e. proxies, to compare either to lattice QCD findings or experimental observations (Bellwied et al. 2020).

Fig. 4

Images reproduced with permission from [left] Borsanyi et al. (2012), copyright by SISSA, and [right] from Alba et al. (2017), copyright by APS

Left: Baryon number, strange quark, electric charge, isospin number, and up quark susceptibilities as functions of the temperature at $\mu _B=0$. Right: compilation of partial pressures for different hadron families as functions of the temperature.

Continuum extrapolated lattice QCD findings for $\chi _{2,2}^{u,s},~\chi _{2,2}^{u,d},~\chi _{1,1}^{u,d},~\chi _4^u,~\chi _4^B$ were presented in Bellwied et al. (2015). Second and fourth-order cumulants of conserved charges were constructed in a temperature range spanning from the QCD transition area to the region of resummed perturbation theory. It was found that, in the hadronic phase ($T \sim 130$ MeV), the HRG model predictions accurately reflect the lattice data, whereas in the deconfined region ($T \gtrsim 250$ MeV), a good agreement was found with three loop hard-thermal-loop (HTL) outcomes (Bellwied et al. 2015). Different diagonal and non-diagonal fluctuations of conserved charges are estimated up to sixth-order on a lattice size of 48$^3 \times$ 12 (Borsanyi et al. 2018). Higher-order fluctuations at zero baryon/charge/strangeness chemical potential are calculated. The ratios of baryon-number cumulants as functions of T and $\mu _B$ are derived from these correlations and fluctuations, which fulfill the experimental criteria of proton/baryon ratio and strangeness neutrality and in turn, describe the observed cumulants as functions of collision energy from the STAR collaboration (Borsanyi et al. 2018). Ratios of fourth-to-second order susceptibilities for light and strange quarks were presented in Bellwied et al. (2013).

3.3 Partial pressures

Under the assumption that the hadronic phase can be treated as an ideal gas of resonances, and using lattice simulations, the partial pressures of hadrons were determined with various strangeness and baryon number contents. To explain the difference between the results of the HRG model and lattice QCD for some of them, the existence of missing strange resonances was proposed (Bazavov et al. 2013b; Alba et al. 2017). Note that partial pressures are only possible within the hadron resonance gas phase because (i) they require hadronic degrees-of-freedom and (ii) they are applicable under the assumption that the pressure can be written as separable components by the quantum number of the hadrons, i.e.,

$$\begin{aligned} P({\hat{\mu }}_B,{\hat{\mu }}_S)= & {} P^{BS}_{00}+P^{BS}_{10}\cosh ({\hat{\mu }}_B)+P^{BS}_{01} \cosh ({\hat{\mu }}_S) + P^{BS}_{11} \cosh ({\hat{\mu }}_B-{\hat{\mu }}_S) \nonumber \\{} & {} + P^{BS}_{12} \cosh ({\hat{\mu }}_B-2{\hat{\mu }}_S)+ P^{BS}_{13} \cosh ({\hat{\mu }}_B-3{\hat{\mu }}_S) \, \end{aligned}$$

(6)

where the coefficients $P^{BS}_{ij}$ indicate the baryon number i and strangeness j of the family of hadrons for which the partial pressure is being isolated, and the dimensionless chemical potentials are written as ${\hat{\mu }}=\mu /T$. The calculations were made feasible by taking imaginary values of strange chemical potential in the simulations. For strange mesons, more interaction channels should be incorporated into the HRG model, in order to explain the lattice data (Alba et al. 2017). The right panel of Fig. 4 shows a compilation of these partial pressures.

3.4 Pseudo-phase transition line

In a crossover, there is no sudden jump in the first derivatives of the pressure. Nevertheless, a (chiral) pseudo-phase transition line can be calculated based on where the order parameters change more rapidly. The exact location of the QCD transition line is a hot topic of research in the field of strong interactions. The most recent results are contained in Borsanyi et al. (2020). The transition temperature, obtained from the chiral condensate and its susceptibility, as a function of the chemical potential can be parametrized as

$$\begin{aligned} \frac{T_c(\mu _B)}{ T_c(\mu _B=0)}=1 - \kappa _2 \left( \frac{\mu _B}{T_c(\mu _B)}\right) ^2 - \kappa _4 \left( \frac{\mu _B}{T_c(\mu _B)}\right) ^4+\dots . \end{aligned}$$

(7)

The crossover or pseudo-critical temperature $T_c$ has been determined with extreme accuracy and extrapolated from imaginary up to real $\mu _B \approx 300$ MeV. Additionally, the width of the chiral transition and the peak value of the chiral susceptibility were calculated along the crossover line. Both of them are constant functions of $\mu _B$. This means that, up to $\mu _B=300$ MeV, no sign of criticality has been observed in lattice results. In fact, at the critical point the height of the peak of the chiral susceptibility would diverge and its width would shrink. The small error reflects the most precise determination of the $T-\mu _B$ phase transition line using lattice techniques. Besides $T_c=158 \pm 0.6$ MeV, the study provides updated results for the coefficients $\kappa _2=0.0153\pm 0.0018$ and $\kappa _4=0.00032\pm 0.00067$ (Borsanyi et al. 2020). Similar coefficients for the extrapolation of the transition temperature to finite strangeness, electric charge, and isospin chemical potentials were obtained in Bazavov et al. (2019), and are displayed in Table 2.

Table 2 Continuum-extrapolated values of $\kappa _2^X$ and $\kappa _4^X$ (with $\mu _Q=\mu _S=0$ for $X=B$, $\mu _B=\mu _Q=0$ for $X=S$ and $\mu _B=\mu _S=0$ for $X=Q,I$) from Bazavov et al. (2019)

3.5 Limits on the critical point location

As mentioned in the previous subsection, in Borsanyi et al. (2020), by extrapolating the proxy for the transition width as well as the height of the chiral susceptibility peak from imaginary to real $\mu _B$, the strength of the phase transition was evaluated and no indication of criticality was found up to $\mu _B \approx$ 300 MeV. On the other hand, a phase transition temperature at $\mu _B=0$ of $T_c=132^{+3}_{-6}$ MeV was found in the chiral limit by the HotQCD collaboration using lattice QCD calculations with “rooted” staggered fermions (Ding et al. 2019). This transition temperature is computed with two massless light quarks and a physical strange quark based on two unique estimators. Since the curvature of the phase diagram is negative, a critical point in the chiral limit would sit at a temperature smaller than this one. The expectation is that the temperature of the critical point at physical quark masses has to be smaller than the one of the critical point in the chiral limit, and therefore definitely smaller than $T_c=132^{+3}_{-6}$ MeV.

4 Theoretical constraints: perturbative QCD

It is possible to compute analytically the QCD EoS directly from the QCD Lagrangian using finite temperature/density perturbation theory. However, in thermal and chemical equilibrium, when $T \gg \mu _i$ (with quark chemical potentials $\mu _i$), one finds that the naive loop expansion of physical quantities is ill-defined and diverges beyond a given loop order, which depends on the quantity under consideration. In the calculation of QCD thermodynamics, this stems from uncanceled infrared (IR) divergences that enter the expansion of the partition function at three-loop order. These IR divergences are due to long-distance interactions mediated by static gluon fields and result in contributions that are non-analytic in the strong coupling constant $\alpha _s = g^2/4\pi$, e.g., $\alpha _s^{3/2}$ and $\log (\alpha _s)$, unlike vacuum perturbation expansions which involve only powers of $\alpha _s$.

It is possible to understand at which perturbative order terms that are non-analytic in $\alpha _s$ appear by considering the contribution of non-interacting static gluons to a given quantity. For simplicity, we now discuss the case of $\mu _B=0$ for this argument, but the same holds true at finite chemical potential. For the pressure of a gas of gluons one has $P_\text {gluons}\sim \int d^3p \,p\, f_B(E_p)$, where $f_B$ denotes a Bose–Einstein distribution function and $E_p$ is the energy of the in-medium gluons. The contributions from the momentum scales $\pi T$, gT and $g^2T$ can be expressed as

$$\begin{aligned} P_\text {gluons}^{p\sim \pi T}\sim & {} T^4f_B(\pi T)\;\sim \; T^4+{{\mathcal {O}}}(g^2) \,, \end{aligned}$$

(8)

$$\begin{aligned} P_\text {gluons}^{p\sim g T}\sim & {} (gT)^4 f_B(g T)\;\sim \; g^3T^4+{{\mathcal {O}}}(g^4) \,, \end{aligned}$$

(9)

$$\begin{aligned} P_\text {gluons}^{p\sim g^2 T}\sim & {} (g^2T)^4 f_B(g^2 T)\;\sim \; g^6T^4 \, \end{aligned}$$

(10)

where we have used the fact that $f_B(E)\sim T/E$ if $E\ll T$. This fact is of fundamental importance, since it implies that when the energy/momentum are soft, corresponding to electrostatic contributions $p_\text {soft} \sim g T$, one receives an enhancement of 1/g compared to contributions from hard momenta, $p_\text {hard} \sim T$, due to the bosonic nature of the gluon. For ultrasoft (magnetostatic) momenta, $p_\text {ultrasoft} \sim g^2 T$, the contributions are enhanced by $1/g^2$ compared to the naive perturbative order. As Eqs. (8)–(10) demonstrate, it is possible to generate contributions of the order $g^3 \sim \alpha _s^{3/2}$ from soft momenta and, in the case of the pressure, although perturbatively enhanced, ultrasoft momenta only start to play a role at order $g^6 \sim \alpha _s^3$.

Due to the infrared enhancement of electrostatic contributions, there is a class of diagrams called hard-thermal-loop (HTL) graphs that have soft external momenta and hard internal momenta that need to be resummed to all orders in the strong coupling (Braaten and Pisarski 1990a, b, c). There are now several schemes for carrying out such soft resummations (Arnold and Zhai 1994, 1995; Zhai and Kastening 1995; Braaten and Nieto 1995, 1996a; Kajantie et al. 1997; Andersen et al. 1999, 2000a, b; Blaizot et al. 1999a, b, 2001a, b; Andersen et al. 2002, 2004, 2010b, 2011c; Haque et al. 2014a, b). We note however, that even with such resummations, if one casts the result as a strict power series in the strong coupling constant the convergence of the perturbative series for the QCD free energy is quite poor. To address this issue, one must treat the soft sector non-perturbatively and re-sum contributions to all orders in the strong coupling constant. This can be done using effective field theory methods (Ghiglieri et al. 2020), approximately self-consistent two-particle irreducible methods (Blaizot et al. 1999a, b, 2001a, b), or the hard-thermal-loop perturbation theory reorganization of thermal field theory (Andersen et al. 1999, 2000b, 2002, 2004, 2010b, 2011c; Haque et al. 2014a, b).

Thus, the calculation of the QCD EoS requires all-orders resummation, which can be accomplished in a variety of manners. Despite the fact that different methods exist, they all rely fundamentally on the use of so-called hard-thermal- or hard-dense-loops, which self-consistently include the main physical effect of the generation of in-medium gluon and quark masses at the one-loop level. By reorganizing the perturbative calculation of the QCD EoS around the high-temperature hard-loop limit of quantum field theory, the convergence of the perturbative series can be extended to phenomenologically relevant temperatures and densities. Below we summarize the results that have been obtained and compared to lattice QCD calculations where available.

4.1 The resummed perturbative QCD EoS

The QCD EoS of deconfined quark matter at high chemical potential can be evaluated in terms of perturbative series in the running coupling constant $\alpha _s$. As a result, the neutron-star EoS can be studied using the weak coupling expansion (Kurkela et al. 2014; Annala et al. 2018; Shuryak 1978; Zhai and Kastening 1995; Braaten and Nieto 1996a, b; Arnold and Zhai 1995, 1994; Toimela 1985; Kapusta 1979; Annala et al. 2020; Kurkela et al. 2014, 2010; Freedman and McLerran 1977b, c; Ecker and Rezzolla 2022a; Altiparmak et al. 2022; Ecker and Rezzolla 2022b). The EoS and trace anomaly of deconfined quark matter have been calculated to three-loop order using HTL perturbation theory framework at small $\mu _B$ and arbitrary T. Renormalization of the vacuum energy, the HTL mass parameters, and $\alpha _s$ eliminate all UV divergences. The three-loop results for the thermodynamic functions are observed to be in agreement with lattice QCD data for $T \gtrsim 2-3 T_c$ after choosing a suitable mass parameter prescription (Andersen et al. 2011c). Furthermore, the QCD thermodynamic potential at nonzero temperature and chemical potential(s) has been calculated using N2LO at three-loop HTL perturbation theory which was used further to calculate the pressure, entropy density, trace anomaly, energy density, and speed of sound, $c_s$, of the QGP (Haque et al. 2014b). These findings were found to be in very good agreement with the data obtained from lattice QCD using the central values of the renormalization scales. This is illustrated in Figs. 5 and 6, which present comparisons of the resummed perturbative results with lattice data for the pressure and fourth-order baryonic and light-quark susceptibilities. In these figures, HTLpt corresponds to the N2LO hard thermal loop perturbation theory calculation of the EoS and EQCD corresponds to a resummed N2LO electric QCD effective field theory calculation of the same. The shaded bands indicate the size of the uncertainty due to the choice of renormalization scale.

Fig. 5

Left: the resummed QCD pressure for $\mu _B=0$ obtained using the three-loop EQCD and HTL perturbation theory results with lattice data from the Wuppertal-Budapest (WB) collaboration (Borsanyi et al. 2010). Right: the second-order light quark (and baryon) number susceptibilities. Lattice data are from the WB (Borsanyi 2013; Borsanyi et al. 2013) and BNLB collaborations (Bazavov et al. 2013a)

Fig. 6

Left: the 4th baryon number susceptibility. Right: the 4th light quark number susceptibility. Lattice data sources are the same as in Fig. 5

4.2 The curvature of the QCD phase transition line

In another study, for the second- and fourth-order curvatures of the QCD phase transition line, the N2LO HTL perturbation theory predictions were shown. In all three situations, (i) $\mu _{s}=$ $\mu _{l}=\mu _{B} / 3$, (ii) $\mu _{s}=0, \mu _{l}=\mu _{B} / 3$, and (iii) $S=0, Q / B=0.4, \mu _{l}=\mu _{B} / 3$, it was shown that N2LO HTL perturbation theory is compatible with the already available lattice computations of $\kappa _2$ and $\kappa _4$ as defined in Eq. (7) (Haque and Strickland 2021). This is illustrated in Fig. 7, which presents comparisons of the resummed perturbative results with lattice data for the coefficients $\kappa _2$ and $\kappa _4$.

Fig. 7

Left: filled circles are lattice calculations of the quadratic curvature coefficient $\kappa _2$ (Cea et al. 2016; Bonati et al. 2015, 2018; Borsanyi et al. 2020; Bazavov et al. 2019), from top to bottom, respectively. Red-filled circles are results obtained using the imaginary chemical potential method and blue-filled circles are results obtained using Taylor expansions around $\mu _B=0$. Black open circles are the N2LO HTL perturbation theory predictions. Right: filled circles are lattice calculations of quartic coefficient $\kappa _4$ from Borsanyi et al. (2020), Bazavov et al. (2019), from top to bottom, respectively. The error bars associated with the HTL perturbation theory predictions result from variations of the assumed renormalization scale

4.3 Application at high density

Gorda et al. (2021b), at $T=0$, calculated the N3LO contribution emerging from non-Abelian interactions among long-wavelength, dynamically screened gluonic fields using the weak-coupling expansion of the dense QCD EoS. In particular, they used the HTL effective theory to execute a comprehensive two-loop computation that is valid for long-wavelength, or soft, modes. In the plot of the EoS, unlike at high temperatures, the soft sector behaves well within cold quark matter, and the novel contribution reduces the renormalization-scale dependence of the EoS at high density (Gorda et al. 2021b). Working at exactly zero temperature is often a good approximation for fully evolved neutron stars but for the early stages of neutron-star evolution and neutron-star mergers, it is essential to incorporate temperature effects (Shen et al. 1998). However, the inclusion of finite temperature in high-$\mu _B$ quark matter gives rise to a technical difficulty for weak coupling expansions. It is no longer sufficient under this regime to handle simply the static sector of the theory nonperturbatively, but the $T=0$ limit’s accompanying technical simplifications are also unavailable. In the EoS plots (see Fig. 8), the breakdown of the weak coupling expansion is observed by a rapid increase in the uncertainty of the result with an increase in temperature for tiny values of $\mu _B$ (Kurkela and Vuorinen 2016).

Fig. 8

Image reproduced with permission from Kurkela and Vuorinen (2016), copyright by the author(s)

EoS of deconfined quark matter as a function of $\mu _B$ at four different temperatures. The new result is shown by the red bands, with the widths resulting from a change in the renormalization scale ${\tilde{\Lambda }}$ (Kurkela and Vuorinen 2016). The corresponding ${\mathcal {O}}(g^4)$ result at absolute zero is shown by the dashed blue lines (Freedman and McLerran 1977c; Baluni 1978; Vuorinen 2003).

The most up-to-date pQCD results at $T=0$ and finite densities can be found in Gorda et al. (2021a). The EoS derived in these calculations was applicable starting at $n_B\sim 40~n_{\text{sat}}$ and above. However, there is an overall renormalization scale parameter, X, that is unknown. One can extrapolate down to lower densities assuming that the speed of sound squared should be bounded by causality and stability, i.e., $0\le c_s^2 \le 1$. The results were shown in Komoltsev and Kurkela (2022) where they varied X in the range $1\le X \le 4$. The results of the constrained regime can be seen in Fig. 9. Various groups have then used these constraints in their neutron star EoS analyses (Marczenko et al. 2023; Somasundaram et al. 2023).

Fig. 9

Image reproduced with permission from Komoltsev and Kurkela (2022), copyright by APS

The extracted EoS constraints from pQCD and $\chi$EFT(CET in the figure) as a function of energy density are shown (see Sect. 5 for details). The excluded regions are in pink.

4.4 Transport coefficients at finite T and $\mu _B$

The quark-gluon plasma probed in heavy-ion collisions is not in equilibrium and viscous effects from shear and bulk viscosities are important for the evolution of the system. At the moment it is not yet possible to reliably compute the shear and bulk viscosities using first principle calculations (Meyer 2011). However, it is possible to perform calculations of these coefficients in the weak-coupling limit of QCD. The shear viscosity $\eta$ and relaxation time $\tau _\pi$ (the timescale within which the system relaxes towards its Navier–Stokes regime, Denicol and Rischke 2021) are usually related through

$$\begin{aligned} \tau _\pi =C \frac{\eta }{\varepsilon +p}\, \end{aligned}$$

(11)

where C is a constant determined by the theory. Calculations of $\eta /s$ in QCD have been completed up to NLO (next-to leading order, Ghiglieri et al. 2018a) and the constant C of the relaxation time at NLO (Ghiglieri et al. 2018b) for $\mu _B=0$, as shown in Fig. 10.

Fig. 10

Images reproduced with permission from [left] Ghiglieri et al. (2018a) and from [right] Ghiglieri et al. (2018b), copyright by the author(s)

Left: shear viscosity to entropy density ratio versus temperature derived from pQCD at LO (leading order) and NLO (next-to leading order) for two different choices of the running coupling for 3 flavors. Right: coefficient of the relaxation time for shear viscosity at leading order and next-to-leading order as a function of the Debye mass over temperature for QCD with 3 light flavors.

Recently, the first calculations of shear viscosity at leading-log at finite $\mu _B$ were performed in QCD in Danhoni and Moore (2023), as shown in Fig. 11.

Fig. 11

Image reproduced with permission from Danhoni and Moore (2023), copyright by the author(s)

Shear viscosity times temperature divided by the enthalpy versus chemical potential over temperature in 3 flavor QCD.

Note, however, that at finite $\mu _B$ the most natural dimensionless quantity involves the enthalpy ($w=\varepsilon +p$), such that $\eta T/w$ is the relevant quantity (the factor of T is to ensure that it remains dimensionless) to be used (Liao and Koch 2010). In the limit of vanishing baryon chemical potential, then

$$\begin{aligned} \lim _{\mu _B\rightarrow 0} \frac{\eta T}{\varepsilon +p}= \frac{\eta }{s}\, \end{aligned}$$

(12)

such that these results should be smoothly connected regardless of $\mu _B$. The relaxation time has not yet been calculated in QCD at finite $\mu _B$. Finally we note that, in typical relativistic viscous hydrodynamics simulations performed in heavy-ion collisions, a number of other transport coefficients are also needed. For example, using perturbative QCD, the bulk viscosity has been computed in Arnold et al. (2006), conductivity and diffusion in Arnold et al. (2000), and some second-order transport coefficients can be found in York and Moore (2009). In practice, these perturbatively-determined expressions are not the ones used in simulations, which often rely on simple formulas involving the transport coefficients determined from, for instance, kinetic theory models (Denicol et al. 2012, 2014) or holography (Kovtun et al. 2005; Finazzo et al. 2015; Rougemont et al. 2017; Grefa et al. 2022), see Everett et al. (2021).

5 Theoretical constraints: chiral effective field theory

In the opposite regime of low density and temperature, Chiral Effective Field Theory ($\chi$EFT) is used to calculate the EoS relevant around $n_B\sim n_{\text{sat}}$ of neutron stars. For ab initio $\chi$EFT calculations, it is possible to study the EoS at arbitrary isospin asymmetry at both zero and nonzero temperature within a many-body perturbation theory (Drischler et al. 2014; Wellenhofer et al. 2016; Wen and Holt 2021; Somasundaram et al. 2021), or a many-body Brueckner–Hartree–Fock approach (Logoteta et al. 2016b, a). Recently, several benchmark calculations have been performed, considering the first and second generation of $\chi$EFT Norfolk NN and 3N interactions, to assess the possible error which is associated with the chosen method when solving the many-body Schrödinger equation (Piarulli et al. 2020; Lovato et al. 2022a). The authors have obtained a good agreement among the many-body techniques tested up to approximately the nuclear saturation density.

However, in practice it is convenient to first compute the EoS for symmetric nuclear matter ($Y_Q=0.5$) and pure neutron matter ($Y_Q=0$) and then interpolate between the two using the quadratic approximation for the isospin-asymmetry dependence of the EoS. From the density-dependent symmetry energy, one can extract the coefficients $E_{\text{sym}}$, L, $K_{\text{sym}}$ from $\chi$EFT calculations

$$\begin{aligned} E_{\text{NS}}=E_{Y_Q=0.5} + N_B\left[ E_{\text{sym},\text{sat}}+\frac{L_{\text{sat}}}{3}\left( \frac{n_B}{n_{\text{sat}}}-1\right) +\frac{K_{\text{sym},\text{sat}}}{18}\left( \frac{n_B}{n_{\text{sat}}}-1\right) ^{2}\right] \big (1-2Y_{Q}\big )^2, \end{aligned}$$

(13)

where $E_{\text{NS}}$ is the ground-state energy at a given density and isospin asymmetry, $E_{Y_Q=0.5}$ is the total energy for isospin-symmetric nuclear matter, $E_{\text{sym},\text{sat}}=\left( \frac{E_{Y_Q=0}-E_{Y_Q=0.5}}{N_B}\right) _{n_{\text{sat}}}$ is the symmetry energy at saturation, $L_{\text{sat}}=3n_{\text{sat}}\left( \frac{dE_{\text{sym}}}{dn_B}\right) _{n_{\text{sat}}}$ is the slope of the symmetry energy at saturation, and $K_{\text{sym},\text{sat}}=9n_{\text{sat}}^2\left( \frac{d^2E_{\text{sym}}}{dn_B^2}\right) _{n_{\text{sat}}}$ is the symmetry energy curvature at saturation. Using this expansion scheme, it is possible to obtain the neutron star outer core EoS with quantified uncertainties from $\chi$EFT. The properties of the low-density crust (Lim and Holt 2017; Grams et al. 2022) and the high-density inner core (Hebeler et al. 2010; Tews et al. 2018; Lim et al. 2021; Drischler et al. 2021a; Brandes et al. 2023) require additional modeling assumptions.

The energy per baryon of symmetric nuclear matter, $E_{Y_Q=0.5}/N_B$, and pure neutron matter, $E_{Y_Q=0}/N_B$, has been computed from $\chi$EFT at different orders in the chiral expansion and different approximations in many-body perturbation theory. As a representative example, in Holt and Kaiser (2017) the EoS was computed up to third order in many-body perturbation theory, including self-consistent second-order single-particle energies. Chiral nucleon–nucleon interactions were included up to N3LO, while three-body forces were included up to N2LO in the chiral expansion. Using the above approach, the authors gave error bands on the EoS (including $E_{\text{sym},\text{sat}}$ and slope parameter $L_{\text{sat}}$), taking into account uncertainties from the truncation of the chiral expansion and the choice of resolution scale in the nuclear interaction. The incorporation of third-order particle-hole ring diagrams (frequently overlooked in EoS computations) helped to reduce theoretical uncertainties in the neutron matter EoS at low densities, but beyond $n_B\gtrsim 2n_{\text{sat}}$ the EoS error bars become large due to the breakdown in the chiral expansion. Recent advances in automated diagram and code generation have enabled studies at even higher orders in the many-body perturbation theory expansion (Drischler et al. 2021c, 2019). In the following we focus on selected results obtained in many-body perturbation theory and refer the reader to Lynn et al. (2019), Carlson et al. (2015), Gandolfi et al. (2020), Tews (2020), Rios (2020), Hagen et al. (2014) for comprehensive review articles on many-body calculations in the frameworks of quantum Monte Carlo, self-consistent Green’s functions method, and coupled cluster theory.

The left panel of Fig. 12 shows the correlation between the symmetry energy $E_{\text{sym}}$ and slope parameter L at saturation for different choices of the high-momentum regulating scale (shown as different data points) and order in the chiral expansion (denoted by different colors) all calculated at 3rd order in many-body perturbation theory including self-consistent (SC) nucleon self-energies at second order (Holt and Kaiser 2017). The ellipses show the 95% confidence level at orders NLO, N2LO, and the N3LO$^*$ (where the star denotes that the three-body force is included only at N2LO) (Holt and Kaiser 2017). Interestingly, one finds that even EoS calculations performed at low order in the chiral expansion produce values of $E_{\text{sym}}$ and L at saturation that tend to lie on a well-defined correlation line. The N3LO* (red) ellipse illustrates the range of symmetry energy $28\, \text{MeV}< E_{\text{sym},\text{sat}}< 35\, \text{MeV}$ and slope parameter $20\, \text{MeV}<L_{\text{sat}}< 65\, \text{MeV}$, both of which are quite close to the findings of prior microscopic computations (Hebeler et al. 2010; Gandolfi et al. 2012) that also used neutron matter calculations plus the empirical saturation properties of symmetric nuclear matter to deduce $E_{\text{sym}}$ and L. All three sets of results are shown in the right panel of Fig. 12 and labeled ‘H’ (Hebeler et al. 2010), ‘G’ (Gandolfi et al. 2012), and ‘HK’ (Holt and Kaiser 2017) respectively. In contrast, a recent work (Drischler et al. 2020) that analyzed correlated $\chi$EFT truncation errors in the EoS for neutron matter and symmetric nuclear matter using Bayesian statistical methods found $E_{\text{sym}} = 31.7\pm 1.1$ MeV and $L = 59.8 \pm 4.1$ MeV at saturation, shown as ‘GP-B 500’ in the right panel of Fig. 12. The obtained value of $E_{\text{sym}}$ was similar to those from Gandolfi et al. (2012), Hebeler et al. (2010), Holt and Kaiser (2017), but L was systematically larger. The results, however, are in good agreement with standard empirical constraints (Drischler et al. 2020; Li et al. 2019) discussed in Sect. 7.2.1.

Fig. 12

Adapted from Holt and Kaiser (2017), Drischler et al. (2021c). Left: correlation between the symmetry energy $E_{\text{sym}}$ and its slope, L at saturation, from $\chi$EFT calculations at different orders in the chiral expansion (Holt and Kaiser 2017). Right: microscopic constraints on the $E_{\text{sym}} - L$ correlation from Hebeler et al. (2010), Gandolfi et al. (2012), Holt and Kaiser (2017), Drischler et al. (2020)

Chiral effective field theory has also been employed to study the liquid–gas phase transition and thermodynamic EoS at low temperatures ($T<25$ MeV) in isospin-symmetric nuclear matter (Wellenhofer et al. 2014). The EoS has been computed using $\chi$EFT nuclear potentials at resolution scales of 414, 450, and 500 MeV. The results from this study are tabulated in Table 3. In particular, the values of the liquid–gas critical endpoint in temperature, pressure, and density agree well with the empirical multifragmentation and compound nuclear decay experiments discussed in Sect. 7.3. In addition, at low densities and moderate temperatures, the pure neutron matter EoS is well described within the virial expansion in terms of neutron-neutron scattering phase shifts. The results from chiral effective field theory have been shown (Wellenhofer et al. 2015) to be in very good agreement with the model-independent virial EoS. Since finite-temperature effects are difficult to reliably extract empirically, chiral effective field theory calculations have been used to constrain the temperature dependence of the dense-matter EoS in recent tabulations Du et al. (2019, 2022) for astrophysical simulations.

Table 3 The N3LO contributions for binding energy per nucleon at saturation, saturation density, compressibility at saturation, and liquid–gas phase-transition critical values of temperature, density, and pressure at different resolution scales. Table adapted from Wellenhofer et al. (2014)

6 Experimental constraints: heavy-ion collisions

Given that hot and dense matter can be created experimentally in heavy-ion collisions, constraints on its EoS can be extracted via experimental measurements obtained from such collisions. As explained previously in Sect. 2.5, we will especially focus in this paper on the data themselves, and avoid citing quantities inferred from the data, as the latter come with an associated model dependence. We will mention particle production yields and their ratios, as well as fluctuation observables of particle multiplicities. Then, we will review experimental results on flow harmonics, to end with Hanbury–Brown–Twiss (HBT) interferometry measurements, also referred to in the field as femtoscopy.

We remind the reader that, as a general rule of thumb, high center of mass energy collisions, $\sqrt{s_{NN}}\gtrsim 200$ GeV, are in the regime of the phase diagram where $\mu _B \ll T$, such that the numbers of particles and anti-particles are approximately equal (i.e., $n_B\sim 0$). As one lowers $\sqrt{s_{NN}}$, baryons are stopped within the collision such that higher $n_B$ is reached. At sufficiently low beam energies, $\sqrt{s_{NN}}\lesssim 4-7$ GeV,⁵ matter is dominated by the hadron gas phase, such that lowering $\sqrt{s_{NN}}$ leads to lower temperatures and lower $n_B$.

6.1 Particle yields

Particle production spectra are part of the simplest experimental observables used in heavy-ion collisions, to access thermodynamic properties and characteristics of the hot and dense matter. Starting with the integrated production yields of identified hadrons, they can be measured to help determine properties from the evolution of the system, in particular at chemical and kinetic freeze-out. These steps designate the ending of inelastic collisions between formed hadrons (what fixes the chemistry of the system) and in turn the ceasing of all elastic collisions (after which particles stream freely to the detectors) in the evolution of a heavy-ion collision. Statistical hadronization models (Hagedorn 1965; Dashen et al. 1969; Becattini and Passaleva 2002; Wheaton et al. 2011; Petran et al. 2014; Andronic et al. 2018, 2019; Vovchenko and Stoecker 2019) are fitted to these yields and ratios by varying over T and $\mu _B$, in order to extract the respective chemical freeze-out values. Despite their ability to reproduce particle yields successfully, those models are limited in scope since they do not reproduce the dynamics of a collision and hinge on the assumption of thermal equilibrium, which is not necessarily achieved within the short time scales of heavy-ion collisions. Similar information can also be inferred for kinetic freeze-out, using a so-called blast-wave model (Schnedermann et al. 1993). The data from low transverse momentum particles (i.e., $p_T \lesssim 2-3$ GeV/c) measured at mid-rapidity (approximately transverse to the beam line direction) is used for statistical hadronization fits, because these particles spend the longest time within the medium (so they are more likely to be thermalized) and they have low enough momentum to avoid contributions from jet physics.

Measurement of yields for the most common light hadron species (namely $\pi ^\pm$, $K^\pm$, $p/{\overline{p}}$) and strange hadrons ($\Lambda /{\overline{\Lambda }}$, $\** ^-/{\overline{\** }}^+$ and $\Omega ^-/{\overline{\Omega }}^+$) was achieved by STAR at RHIC and ALICE at the LHC. As part of the BES program, STAR has measured these hadron yields in Au+Au collisions at center-of-mass energies of $\sqrt{s_{NN}}=$7.7, 11.5, 14.5, 19.6, 27, 39, 62.4 and 200 GeV (Abelev et al. 2009; Adamczyk et al. 2017; Adam et al. 2020b, d) and in U+U collisions at $\sqrt{s_{NN}} = 193$ GeV (Abdallah et al. 2023d). Motivated by results from the past SPS-experiment NA49 obtained from Pb+Pb collisions at $\sqrt{s_{NN}} < 20$ GeV (Alt et al. 2008), the NA61/SHINE experiment at CERN has conducted a scan in system-size and energy (in the same energy range as NA49). The diagram of all collided systems as a function of collision energy and nuclei is displayed in the left panel of Fig. 13. They published data for light hadrons in Ar+Sc (Acharya et al. 2021b) and Be+Be collisions (Acharya et al. 2021a) so far, while results from Xe+La and Pb+Pb collisions should become available in the next few years (Kowalski 2022). At LHC energies, the higher $\sqrt{s_{NN}}$ produces significantly more particles, allowing more precise measurements of the species. For this reason, the ALICE experiment has measured yields not only for light and (multi)strange hadrons, but also for light nuclei and hyper-nuclei, in Pb+Pb collisions at 2.76 TeV/A in particular (Abelev et al. 2013a, c, 2014a; Adam et al. 2016f; Acharya et al. 2018e; Adam et al. 2016a), and more recently in Pb+Pb collisions at 5.02 TeV/A too (Acharya et al. 2020c, 2023c), as well as Xe+Xe collisions at 5.44 TeV/A (Acharya et al. 2021e).

Fig. 13

Images reproduced with permission from [left] Kowalski (2022), copyright by the author(s), and [right] Adamczyk et al. (2017), copyright by APS

Left: diagram of different collided systems as a function of collision energy. Right: yields of hadrons measured by STAR in central Au+Au collisions at 7.7 and 39 GeV/A, compared with results from a grand canonical statistical hadronization model.

The yields of particle production can also be used as an indicator of the onset of deconfinement, notably thanks to the strangeness enhancement: strange quark-antiquark pairs are expected to be produced at a much higher rate in a hot and dense medium than in a hadron gas. Hence, one should expect in particular an increase of multi-strange baryons compared to light-quark-compound hadrons in collision systems where the QGP has been formed, which has been observed experimentally in heavy-ion collisions at several energies (Antinori et al. 2006; Abelev et al. 2008b, 2014a). Moreover, the distinctive non-monotonic behavior of the $K^+/\pi ^+$ ratio as a function of the collision energy can also be considered as a sign of the onset of deconfinement, according to some authors (Gazdzicki and Gorenstein 1999; Poberezhnyuk et al. 2015). This so-called “horn” in the $K^+/\pi ^+$ ratio has been notably observed in Pb+Pb (Afanasiev et al. 2002; Alt et al. 2008) and Au+Au collisions (Akiba et al. 1996; Ahle et al. 2000; Abelev et al. 2009, 2010b; Adamczyk et al. 2017), but absent from p+p data (Aduszkiewicz et al. 2017; Abelev et al. 2010b; Aamodt et al. 2011c; Abelev et al. 2014c) and Be+Be collisions results (Acharya et al. 2021a). Recent results from Ar+Sc collisions (Kowalski 2022) have, however, stirred up doubts regarding the interpretation of this observable, as the value of this ratio from such collision system is closer to the one measured in big systems, while no horn structure is seen, similar to small systems. All these results for different systems can be seen in Fig. 14.

Fig. 14

Image reproduced with permission from Kowalski (2022), copyright by the author(s)

Preliminary results for the ratio of $K^+/\pi ^+$ yields at mid-rapidity as a function of the collision energy, compared for different collision systems.

6.2 Fluctuation observables

In heavy-ion collisions, observables measuring fluctuations are among the most relevant for the investigation of the QCD phase diagram. Within the assumption of thermodynamic equilibrium, cumulants of net-particle multiplicity distributions become directly related to thermodynamic susceptibilities, and can be compared to results from lattice QCD (see Sect. 3.2) to extract information on the chemical freeze-out line (Alba et al. 2014, 2015, 2020). Moreover, large, relatively long-range fluctuations are expected in the neighborhood of the conjectured QCD critical point, making fluctuation observables very promising signatures of criticality (Stephanov et al. 1999; Stephanov 2009; Athanasiou et al. 2010). It has also been proposed that the finite-size scaling of critical fluctuations could be employed to constrain the location of the critical point (Palhares et al. 2011; Fraga et al. 2010; Lacey 2015; Lacey et al. 2016). In Fraga et al. (2010), finite-size scaling arguments were applied to mean transverse-momentum fluctuations measured by STAR (Adams et al. 2005c, 2007) to exclude a critical point below $\mu _B \lesssim 450$ MeV.

Fluctuations of the conserved charges B, S, and Q are of particular importance. As mentioned already in Sect. 3.2, these fluctuations can be used to probe the deconfinement transition, as well as the location of the critical endpoint. Calculated via the susceptibilities expressed in Eq. (3), which can be evaluated via lattice QCD simulations or HRG model calculations, they can also be related to the corresponding cumulants of conserved charges $C^{BSQ}_{lmn}$,⁶ following the relation

$$\begin{aligned} C^{BSQ}_{lmn} = VT^3 \times \chi ^{BSQ}_{lmn} \, \end{aligned}$$

(14)

with the volume V and temperature T of the system, and $l,m,n \in {\mathbb {N}}$ (Luo and Xu 2017). The cumulants are also theoretically related to the correlation length of the system $\xi$, which is expected to diverge in the vicinity of the critical endpoint. In particular, the higher order cumulants are proportional to higher powers of $\xi$, making them more sensitive to critical fluctuations (Stephanov et al. 1999; Stephanov 2009; Athanasiou et al. 2010).

In heavy-ion collisions, however, it is impossible to measure the fluctuations of conserved charges directly, because one cannot detect all produced particles (e.g., neutral particles are not always possible to measure, so that baryon number fluctuations do not include neutrons). Nevertheless, it is common to measure the cumulants of identified particles’ net-multiplicity distributions, using some hadronic species as proxies for conserved charges (Koch et al. 2005). Net-proton distributions are used as a proxy for net-baryons (Aggarwal et al. 2010; Adam et al. 2019b; Abdallah et al. 2021b), net-kaons (Adamczyk et al. 2018c; Ohlson 2018; Adam et al. 2019c) or net-lambdas (Adam et al. 2020a) are used as a proxy for net-strangeness, and net-pions+protons+kaons (Adam et al. 2019c) has been recently used as a proxy for net-electric charge, instead of the actual net-charged unidentified hadron distributions. Mixed correlations have also been measured (Adam et al. 2019c), although alternative ones have been suggested, that would provide more direct comparisons to lattice QCD susceptibilities (Bellwied et al. 2020).

These net-particle cumulants can be used to construct ratios, as they are connected with usual statistic quantities characterizing the net-hadron distributions $N_\alpha = n_\alpha - n_{{\overline{\alpha }}}$ (with $n_{\alpha /{\overline{\alpha }}}$ being respectively the number of hadrons or anti-hadrons of hadronic specie $\alpha$). Hence, relations between such ratios and the mean $\mu _\alpha$, variance $\sigma _\alpha$, skewness $S_\alpha$ or kurtosis $\kappa _\alpha$ can be expressed as follows

$$\begin{aligned} \frac{\sigma _\alpha ^2}{\mu _\alpha }&= \frac{C_2^\alpha }{C_1^\alpha } = \frac{\langle (\delta N_\alpha )^2 \rangle }{\langle N_\alpha \rangle } \ , \end{aligned}$$

(15)

$$\begin{aligned} S_\alpha \sigma _\alpha&= \frac{C_3^\alpha }{C_2^\alpha } = \frac{\langle (\delta N_\alpha )^3 \rangle }{\langle (\delta N_\alpha )^2 \rangle } \ , \end{aligned}$$

(16)

$$\begin{aligned} \kappa _\alpha \sigma _\alpha ^2&= \frac{C_4^\alpha }{C_2^\alpha } = \frac{\langle (\delta N_\alpha )^4 \rangle }{\langle (\delta N_\alpha )^2 \rangle } - 3{\langle (\delta N_\alpha )^2 \rangle } \ , \end{aligned}$$

(17)

$$\begin{aligned} \frac{\kappa _\alpha \sigma _\alpha }{S_\alpha }&= \frac{C_4^\alpha }{C_3^\alpha } = \frac{\langle (\delta N_\alpha )^4 \rangle - 3{\langle (\delta N_\alpha )^2 \rangle }^2}{\langle (\delta N_\alpha )^3 \rangle } \ , \end{aligned}$$

(18)

with $\delta N_\alpha = N_\alpha - \langle N_\alpha \rangle$, and $\langle \, \rangle$ denoting an average over the number of events in a fixed centrality class at a specific beam energy (Luo and Xu 2020), copyright by Elsevier

Left: energy dependence of the $C_4/C_2 (=\kappa \sigma ^2)$ ratio from net-p distribution for $|y|<0.5$ and $0.4< p_T < 2.0$ GeV/c, measured in 0–5% Au-Au collisions by the STAR collaboration. Image adapted from Abdallah et al. (2021b). Right: expected behavior of the $C_4/C_1(=\omega _4)$ ratio for net-p, in the case of a freeze-out line passing through the critical region near the critical endpoint.