Strangeness-neutral equation of state for QCD with a critical point

Abstract

We present a strangeness-neutral equation of state for QCD that exhibits critical behavior and matches lattice QCD results for the Taylor-expanded thermodynamic variables up to fourth order in $\mu _B/T$. It is compatible with the SMASH hadronic transport approach and has a range of temperatures and baryonic chemical potentials relevant for phase II of the Beam Energy Scan at RHIC. We provide an updated version of the software BES-EoS, which produces an equation of state for QCD that includes a critical point in the 3D Ising model universality class. This new version also includes isentropic trajectories and the critical contribution to the correlation length. Since heavy-ion collisions have zero global net-strangeness density and a fixed ratio of electric charge to baryon number, the BES-EoS is more suitable to describe this system. Comparison with the previous version of the EoS is thoroughly discussed.

Appendix A: smooth merging of correlation length

In order to obtain a result for the correlation length that is consistent with the scaling behavior of the 3D Ising model, we perform a smooth merging of the $\epsilon$-expansion and asymptotic forms of the correlation length, as shown in Eqs. (14), (15). We employ a hyperbolic tangent in order to achieve a well-behaved result, useful for hydrodynamic simulations:

$$\begin{aligned} \begin{aligned} \xi _{\text {final}}(T,\mu _B)&= \xi _{\text {asymp}}(T,\mu _B) \frac{1}{2} \Big [1 \pm \tanh {\Big (\frac{T-T_{\text {low/high}}'(\mu _B)}{\Delta T(\mu _B)}}\Big )\Big ] \\&\quad + \xi _{\epsilon }(T,\mu _B) \frac{1}{2} \Big [1 \mp \tanh {\Big (\frac{T-T_{\text {low/high}}'(\mu _B)}{\Delta T(\mu _B)}}\Big )\Big ], \end{aligned} \end{aligned}$$

(A.1)

Fig. 11

The curve $x=5$ in the QCD phase diagram over which the merging of the correlation length formulations is performed

where $T'(\mu _B)$ acts as the switching temperature and $\Delta T(\mu _B)$ is the overlap region between the two representations. The merging is performed along a curve in the $(T,\mu _B)$ plane that corresponds to a value of the scaling parameter $x=5$, as stated in Sect. 4. The shape which this curve follows in the QCD phase diagram is shown in Fig. 11. Since this is a multi-valued function of $\mu _B$, in order to fit this curve for the switching temperature, we separate it into two functions $T_{\text {low}}'(\mu _B)$ and $T_{\text {high}}'(\mu _B)$ for different temperature regimes. These curves were best fit with exponentials, the functional forms for which are

$$\begin{aligned} T_{\text {low}}'(\mu _B)= & {} T_{\text {M,low}} - (T_{\text {M,low}} - T_{0,\text {low}}) e^{-k_{\text {low}} \mu _B} \end{aligned}$$

(A.2)

$$\begin{aligned} T_{\text {high}}'(\mu _B)= & {} T_{0,\text {high}} e^{-k_{\text {high}} \mu _B}, \end{aligned}$$

(A.3)

where $T_{\text {M,low}}=195$ MeV, $T_{0,\text {low}}=58$ MeV, $k_{\text {low}}=0.00318462$, $T_{0,\text {high}}=282.594$ MeV, $k_{\text {high}}=0.00207197$. We additionally write the overlap region, $\Delta T$, as a function of $\mu _B$ in order to achieve the smoothest possible result for the correlation length. For this function, we also employ an exponential:

$$\begin{aligned} \begin{aligned} \Delta T(\mu _B) = T_{\text {0,merg}} + (T_{\text {M,merg}} - T_{0,\text {merg}}) e^{-k_{\text {merg}} \mu _B} \end{aligned} \end{aligned}$$

(A.4)

where $T_{\text {M,merg}}=17$ MeV, $T_{0,\text {merg}}=0.1$ MeV, $k_{\text {merg}}=0.0075$. This procedure for the smooth merging of the two representations of the correlation length through a hyperbolic tangent yields the critical contribution to the correlation length shown in Fig. 6.