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In this work, we systematically assess the performance of a new method from [H. Shah et al., ] for locating the QCD critical point using constant-entropy contours by testing it against various effective QCD approaches. We demonstrate that, while the method yields spurious critical points in purely hadronic models (HRG) due to nonparabolic contour behavior at low temperatures ( $T\lesssim 120\mathrm{MeV}$ ), it accurately reproduces the critical point location in frameworks that feature a genuine phase transition and benchmarked against lattice QCD, such as holographic Einstein-Maxwell-dilaton, and functional QCD approaches. Building on our previous determination of constant entropy contours using lattice data, we extend that analysis to construct a complete lattice-based equation of state (EoS) at finite density, which features a critical point at $(T,{\mu }_B)\approx (114,602)\mathrm{MeV}$ . By integrating the extrapolated entropy density with respect to temperature, we reconstruct the pressure, baryon density, susceptibility, and speed of sound in the critical region, and analyze the focusing behavior of isentropic trajectories in the vicinity of the critical point. 

We propose a novel method to locate the QCD critical point by constructing an expansion along contours of constant entropy density. Applying two independent analyses of lattice QCD data at zero baryon chemical potential, we find a critical point atT_c=114 ± 7 MeV andμ_Bc= 602±62 MeV for an expansion truncated at orderμ_B². This approach is consistent with recent lattice QCD results up toμ_B/T≤ 3. A more precise determination of the required expansion coefficients from lattice simulations will be essential for reliably establishing the location of the QCD critical endpoint. 

We propose a new method to investigate the existence and location of the conjectured high-temperature critical point of strongly interacting matter via contours of constant entropy density. By approximating these lines as a power series in the baryon chemical potential ${\mu }_B$ , one can extrapolate them from first-principle results at zero net-baryon density, and use them to locate the quantum chromodynamics (QCD) critical point, including the associated first-order and spinodal lines. As a proof of principle, we employ currently available continuum-extrapolated lattice data from the Wuppertal-Budapest collaboration to find a critical point at a temperature and a baryon chemical potential of $T_c=114.3\pm 6.9$ MeV and ${\mu }_{B,c}=602.1\pm 62.1$ MeV, respectively, at expansion order $\mathcal{O}\left({\mu }_B^2\right)$ . We advocate for a more precise determination of the required expansion coefficients via lattice QCD simulations as a means of pinpointing the location of the critical endpoint in the phase diagram of strongly interacting matter. 

We present a Bayesian analysis, based on holography and constrained by lattice QCD simulations, which leads to a prediction for the existence and location of the QCD critical point. We employ two different parametrizations of the functions that characterize the breaking of conformal invariance and the baryonic charge in the Einstein-Maxwell-dilaton holographic model. They lead to predictions for the critical point that overlap at one sigma. While some samples of the prior distribution do not predict a critical point, or produce critical points that cover large regions of the phase diagram, all posterior samples present a critical point at chemical potentials µBc~550-630 MeV. 

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.