Dependence of the static quark free energy on ${\mu }_B$ and the crossover temperature of $N_f=2+1$ QCD

We study the dependence of the static quark free energy on the baryon chemical potential for $N_f=2+1$ QCD with physical quark masses, in a range of temperature spanning from 120 MeV up to 1 GeV and adopting a stout staggered discretization with two different values of the Euclidean temporal extension, $N_t=6$ and $N_t=8$. In order to deal with the sign problem, we exploit both Taylor expansion and analytic continuation, obtaining consistent results. We show that the dependence of the free energy on ${\mu }_B$ is sensitive to the location of the chiral crossover, in particular the ${\mu }_B$ susceptibility, i.e., the linear term in ${\mu }_B^2$ in the Taylor expansion of the free energy, has a peak around 150 MeV. We also discuss the behavior expected in the high temperature regime based on perturbation theory, and obtain a good quantitative agreement with numerical results.

Figure 1

Qualitative structure of the QCD phase diagram of QCD in the $T-{\mu }_{B,I}$ plane. The vertical lines are the RW transitions, while the dashed line is the analytic continuation of the pseudocritical line.

Figure 2

Square module of the Polyakov loop as a function of the imaginary chemical for several temperatures below (top) and above (bottom, normalized to the value at $\mu =0$) the pseudocritical temperature $T_c\simeq 155\mathrm{MeV}$ [50], measured on the $24^3\times 6$ lattice. Curves are the results of the fit using, respectively, the cosine expansion in Eq. (12) and the polynomial Ansatz in Eq. (13).

Figure 3

Susceptibility ${\chi }_{Q,{\mu }_B^2}$ as a function of the temperature $T$ extracted from two different lattices $24^3\times 6$ and $32^3\times 8$. The pattern of the dots indicates the method used for the computation, with empty and full data points corresponding, respectively, to the Taylor expansion method and to analytic continuation. For some values of the temperature, see e.g., the inset, both procedures have been used, so as to check the consistency of the results. Data points have been slightly shifted for the sake of readability.

Figure 4

${\chi }_{Q,{\mu }_B^2}$ as a function of $T$ in the region near the peak. Curves are the result of best fits to the Lorentzian form in Eq. (14), where bands are the 68% CIs plotted over the fit range. For temperatures where determinations obtained by both methods were available, we have used results from the Taylor expansion method, which typically leads to more conservative estimates. Reasonable values of ${\tilde{\chi }}^2$ have been obtained for both datasets: ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.=12.4/7$ and ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.=7.2/5$ respectively for the $24^3\times 6$ and the $32^3\times 8$ lattice.

Figure 5

Values of ${\chi }_{Q,{\mu }_B^2}$ in the high temperature regime. Curves represent best fits to Eq. (20), while bands are confidence intervals at 68% C.L. The value of the ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}$. test is 0.5 and 0.7 respectively for the $24^3\times 6$ and the $32^3\times 8$ lattice.