Critical line of $2+1$ flavor QCD

We determine the curvature of the (pseudo)critical line of QCD with $n_f=2+1$ staggered fermions at nonzero temperature and quark density by analytic continuation from imaginary chemical potentials. Monte Carlo simulations are performed by adopting the highly improved staggered quarks /tree action discretization, as implemented in the code by the MILC Collaboration, suitably modified to include a nonzero imaginary baryon chemical potential. We work on a line of constant physics, as determined in Ref. [1], adjusting the couplings so as to keep the strange quark mass $m_s$ fixed at its physical value, with a light to strange mass ratio of $m_l/m_s=1/20$. In the present investigation, we set the chemical potential at the same value for the three quark species, ${\mu }_l={\mu }_s\equiv \mu$. We explore lattices of different spatial extensions, ${16}^3\times 6$ and ${24}^3\times 6$, to check for finite size effects, and present results on a ${32}^3\times 8$ lattice, to check for finite cutoff effects. We discuss our results for the curvature $\kappa$ of the (pseudo)critical line at $\mu =0$, which indicate $\kappa =0.018(4)$, and compare them with previous lattice determinations by alternative methods and with experimental determinations of the freeze-out curve.

Figure 1

The real part of the disconnected susceptibility of the light-quark chiral condensate for ${16}^3\times 6$ and ${24}^3\times 6$ (full circles and full squares, respectively) and for ${32}^3\times 8$ (full triangles) at $\mu /(\pi T)=0.2i$. Full lines are the fits to the peaks using a Lorentzian.

Figure 2

$T_c(\mu )/T_c(0)$ versus $((i\mu )/(\pi T){)}^2$ obtained on a ${16}^3\times 6$ lattice (full circles), on a ${24}^3\times 6$ lattice (full square), and on a ${32}^3\times 8$ lattice (full triangle). For the sake of readability, the abscissae at $((i\mu )/(\pi T){)}^2=-0.04$ for ${24}^3\times 6$ and ${32}^3\times 8$ data have been slightly shifted. The full line is a linear fit to the data on the ${16}^3\times 6$ lattice.

Figure 3

Comparison of different determinations of the curvature of the chemical freeze-out curve and of the (pseudo)critical line for QCD with $n_f=2+1$. From left to right: (i) analytic continuation, disconnected chiral susceptibility, this study; (ii) Taylor expansion, chiral susceptibility, Ref. [26]; (iii) Taylor expansion, chiral condensate, Ref. [25]; (iv) Taylor expansion, strange quark number susceptibility, Ref. [25]; (v) analytic continuation, Polyakov loop, Ref. [27]; (vi) freeze-out curvature, standard analysis, Ref. [2]; (vii) freeze-out curvature, revised analysis of Ref. [3].

Figure 4

$T_c({\mu }_B)$ versus ${\mu }_B$ (units in GeV). Experimental values of $T_c({\mu }_B)$ are taken from Fig. 1 of Ref. [2] (circles) and from Table I of Ref. [3] (squares), respectively, for the standard and the modified statistical hadronization model. The solid line is a parametrization corresponding to $T_c({\mu }_B)=T_c(0)-b{\mu }_B^2$, with $T_c(0)=0.154(9)\mathrm{GeV}$ and $b=0.117(27){\mathrm{GeV}}^{-1}$. The dashed lines represent the corresponding error band.