Physics overview
The present code provides a four dimensional Taylor-expanded QCD equation of state (EoS) as a function of temperature \(T\) and baryon, strangeness and electric charge chemical potential \(\mu_B\), \(\mu_S\) and \(\mu_Q\), respectively. At \(\mu_i = 0\), where \(i = B/Q/S\), we use the equation of state produced by lattice QCD simulations, and perform a Taylor expansion in terms of \(\frac{\mu_i}{T}\) in order to go to finite value of these chemical potentials. Susceptibilities of \(B\), \(Q\) and \(S\) conserved charges at zero chemical potential \(\chi_{ijk}^{BQS}(T,\mu_i=0)\), which play the role of Taylor expansion coefficients, are described using a Padé functional form fitted to lattice susceptibilities, up to \(4^{th}\) order [1].
All lattice data used in this construction have been computed by the Wuppertal-Budapest collaboration, from a \(N_\tau=12\) lattice using 4stout action [2].
Taylor Expansion
The Taylor series of the pressure in terms of the three conserved charge chemical potentials is written as:
The first term, \(i = j = k = 0\), corresponds to the pressure itself as calculated on the lattice (at zero chemical potentials), and the subsequent Taylor expansion coefficients are the conserved charge susceptibilities with appropriate factorial coefficients corresponding to their order. The susceptibilities are defined as derivatives of the QCD pressure with respect to the various conserved charge chemical potentials:
Lattice QCD simulations allow to compute these coefficients at several finite lattice spacing, to then extrapolate them to infinite volume. Here, we employ susceptibilities up to \(4^{th}\) order, i.e. susceptibilities for which \(i+j+k \leqslant 4\), with \(i+j+k\) even due to symmetry between baryonic and antibaryonic matter. Using the Taylor expansion and the parametrized coefficients, we obtain pressure and derive all other quantities from thermodynamic relations.
Parameterizations
The susceptibilities, or Taylor expansion coefficients, are parameterized to obtain a smooth description of each one of them over the entire temperature range \(30 \leqslant T \leqslant 800 MeV\) [1]. This allow to get pressure as a smooth function of \(T\), \(\mu_B\), \(\mu_Q\) and \(\mu_S\). In order to achieve it, each susceptibility is parametrized by means of a ratio of polynomials in \(1/T\), up to \(9^{th}\) order:
with \(a_{0-9}\), \(b_{0-9}\) and \(c_0\) being coefficients of the parameterization, unique for each \(\chi_{ijk}^{BQS} (T)\). Only one exception applies to \(\chi_{2}^{B} (T)\), for which the optimal parametrization is given as:
In these equations, \(x = T/154 MeV\) and \(x' = T/200 MeV\). These correspond to normalized temperatures on which the fit functions depend. The normalization factor for the temperature in the polynomial parametrization was chosen to be close to the crossover transition temperature. For \(\chi_2^{B}\), the best fit is obtained by normalizing around the point at which the function begins to plateau.
These parametric forms are fitted to a combination of HRG calculations [3], for \(T<135\) MeV, and lattice data of the Wuppertal-Budapest collaboration obtained from simulations on \(48^3 \times 12\) lattices [2], for \(135<T<220\) MeV. In order to obtain the correct extrapolation up to high \(T\), the value of each different susceptibilities at \(T=800\) MeV has been assumed to be \(10\%\) away from their respective Stefan-Boltzmann limit (i.e. their limit at \(T \to \infty\)).