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We present a new equation of state for QCD in which the temperature $T$and the three chemical potentials for baryon number ${\mu }_B$, electric charge ${\mu }_Q$, and strangeness ${\mu }_S$can be varied independently. This result is based on a generalization of the $T^{\prime }$expansion scheme, thanks to which the diagonal ${\mu }_B$extrapolation was pushed up to a baryo-chemical potential ${\mu }_B/T\sim 3.5$for the first time. This considerably extended the coverage of the Taylor expansion, limited to ${\mu }_B/T<2.5–3$. As a consequence, we are able to offer a substantially larger coverage of the four-dimensional QCD phase diagram as well, compared to previously available Taylor expansion results. Our findings are based on new continuum estimated lattice data on the full set of second- and fourth-order fluctuations. 

Although calculations of QCD thermodynamics from first-principle lattice simulations are limited to zero net-density due to the fermion sign problem, several methods have been developed to extend the equation of state (EoS) to finite values of theB,Q,Schemical potentials. Taylor expansion aroundµ_i=0 (i = B,Q,S) enables to cover with confidence the region up toµ_i/T< 2.5. Recently, a new method has been developed to compute a 2D EoS in the (T,µ_B) plane. It was constructed through aT-expansion scheme (TExS), based on a resummation of the Taylor expansion, and is trusted up to densities aroundµ_B/T= 3.5. We present here the new 4D-TExS EoS, a generalization of the TExS to all 3 chemical potentials, expected to offer a larger coverage than the 4D Taylor expansion EoS. After explaining the basics of theT-Expansion Scheme and how it is generalized to multiple dimensions, we will present results for thermodynamic observables as functions of temperature and both finite baryon and strangeness chemical potentials. 

The BEST Collaboration equation of state combining lattice data with the 3D Ising critical point encounters limitations due to the truncated Taylor expansion up toμ_B/T~ 2.5. This truncation consequently restricts its applicability at high densities. Through a resummation scheme, the lattice results have been extended toμ_B/T= 3.5. In this article, we amalgamate these ideas with the 3D-Ising model, yielding a family of equations of state valid up toμ_B= 700MeV with the correct critical behavior. Our equations of state feature tunable parameters, providing a stable and causal framework-a crucial tool for hydrodynamics simulations. 

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.