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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 equation of state of Quantum Chromodynamics has been in recent years the focus of intense effort from first principle methods, mostly lattice simulations, with particular interest to the finite baryon density regime. Because of the sign problem, various extrapolation methods have been used to reconstruct bulk properties of the theory up to as far asμ_B=T≃ 3:5. However, said efforts rely on the equation of state at vanishing baryon density as an integration constant, which up toμ_B=T≃ 2 - 2:5 proves to be the dominant source of uncertainty at the level of precision currently available. In this contribution we present the update of our equation of state at zero net baryon density from 2014, performing a continuum limit from lattices with Nτ = 8; 10; 12; 16. We show how the improved precision is translated in a lower uncertainty on the extrapolated equation of state at finite chemical potential. 

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. 

In this contribution we present a resummation of the Quantum Chromodynamics (QCD) equation of state from lattice simulations at imaginary chemical potentials. We generalize the scheme introduced in a previous work [1], to the case of non-zero strangeness chemical potential. We present continuum extrapolated results for thermodynamic observables in the temperature range 130MeV ≤ T ≤ 280 MeV, for chemical potentials up to μ B / T = 3:5, along the strangeness neutral line. Furthermore, we relax the constraint of strangeness neutrality, by extrapolating to small values of the strangeness-to-baryon-number ratio R = n S / n B . 

The equation of state (EoS) of QCD is a crucial input for the modeling of heavy-ion-collision (HIC) and neutron-star-merger systems. Calculations of the fundamental theory of QCD, which could yield the true EoS, are hindered by the infamous Fermi sign problem which only allows direct simulations at zero or imaginary baryonic chemical potential. As a direct consequence, the current coverage of the QCD phase diagram by lattice simulations is limited. In these proceedings, two different equations of state based on first-principle lattice QCD (LQCD) calculations are discussed. The first is solely informed by the fundamental theory by utilizing all available diagonal and non-diagonal susceptibilities up to O(µ 4 B) in order to reconstruct a full EoS at finite baryon number, electric charge and strangeness chemical potentials. For the second, we go beyond information from the lattice in order to explore the conjectured phase structure, not yet determined by LQCD methods, to assist the experimental HIC community in their search for the critical point. We incorporate critical behavior into this EoS by relying on the principle of universality classes, of which QCD belongs to the 3D Ising Model. This allows one to study the effects of a singularity on the thermodynamical quantities that make up the equation of state used for hydrodynamical simulations of HICs. Additionally, we ensure that these EoSs are valid for applications to HICs by enforcing conditions of strangeness neutrality and fixed charge-to-baryonnumber ratio.