Locating the QCD critical point through contours of constant entropy density

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 $μ_B$, one can extrapolate them from first-principle results at zero net-baryon density, and use them to locate the QCD critical point, including the associated first-order and spinodal lines. As a proof of principle, we employ currently available continuum-extrapolated first-principle results 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 $μ_{B,c} = 602.1 \pm 62.1$ MeV, respectively. 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.

💡 Research Summary

The authors introduce a novel strategy for locating the conjectured critical point (CP) of quantum chromodynamics (QCD) by exploiting contours of constant entropy density, s, in the temperature–baryon‑chemical‑potential (T, μ_B) plane. The key observation is that, in the presence of a first‑order phase transition, the entropy density becomes multivalued: for a given value of s there can be several temperatures at the same μ_B, corresponding to metastable and unstable branches. Consequently, the curves defined by s = const will intersect in the (T, μ_B) diagram precisely in the region where spinodal lines and the coexistence line meet, i.e., at the CP.

To make this idea quantitative, the authors define a function T_s(μ_B; T_0) such that s(T_s, μ_B) = s(T_0, 0) ≡ s_0. By fixing s_0 at μ_B = 0 they can use lattice QCD results, which are only available at zero chemical potential, as the anchor for the whole construction. They then expand T_s in even powers of μ_B:

 T_s(μ_B; T_0) = T_0 + ∑{n=1}^{N} α{2n}(T_0) μ_B^{2n}/(2n)!.

The coefficients α_{2n} are evaluated at μ_B = 0 and expressed in terms of thermodynamic derivatives that are directly accessible from lattice calculations: the entropy density s(T), the baryon‑number susceptibility χ_B^2(T) = ∂^2(p/T)/∂(μ_B/T)^2, and mixed temperature–chemical‑potential derivatives. At leading order (N = 1) the coefficient reads

 α_2(T_0) = −2 T_0 χ_B^2(T_0) + T_0^2 χ_B^{2 ′}(T_0) s′(T_0),

where primes denote temperature derivatives at fixed μ_B. This formulation differs fundamentally from the conventional Taylor expansion of the pressure because it expands the temperature required to keep s constant rather than the pressure itself, thereby allowing the constant‑entropy contours to cross already at O(μ_B^2).

The CP is identified by the simultaneous fulfillment of two conditions derived from the geometry of the contours:

 (∂T_s/∂T_0)_μB = 0 and (∂^2T_s/∂T_0^2)_μB = 0.

At O(μ_B^2) these translate into α_2′(T_0,c) = 0 (which fixes T_0,c) and a relation between μ_B,c and α_2′(T_0,c) that yields μ_B,c. The critical temperature follows from inserting these values into the expansion for T_s.

For the numerical implementation the authors use continuum‑extrapolated lattice data from the Wuppertal–Budapest collaboration: the temperature dependence of the entropy density s(T) (Ref.