Curvature of the pseudocritical line in QCD: Taylor expansion matches analytic continuation

We determine the curvature of the pseudocritical line of $N_f=2+1$ QCD with physical quark masses via Taylor expansion in the quark chemical potentials. We adopt a discretization based on stout improved staggered fermions and the tree level Symanzik gauge action; the location of the pseudocritical temperature is based on chiral symmetry restoration. Simulations are performed on lattices with different temporal extent ($N_t=6$, 8, 10), leading to a continuum extrapolated curvature $\kappa =0.0145(25)$, which is in very good agreement with the continuum extrapolation obtained via analytic continuation and the same discretization, $\kappa =0.0135(20)$. This result eliminates the possible tension emerging when comparing analytic continuation with earlier results obtained via Taylor expansion.

Figure 1

Renormalized chiral condensate $⟨\overline{\psi }{\psi }_{r1}⟩$ for $N_t=6$ and $N_t=8$ lattices.

Figure 2

Renormalized chiral condensate $⟨\overline{\psi }{\psi }_{r2}⟩$ for $N_t=6$ and $N_t=8$ lattices.

Figure 3

First derivative with respect to ${\mu }_B^2$ of $⟨\overline{\psi }{\psi }_{r1}⟩$ for $N_t=6$ and $N_t=8$ lattices.

Figure 4

First derivative with respect to ${\mu }_B^2$ of $⟨\overline{\psi }{\psi }_{r2}⟩$ for $N_t=6$ and $N_t=8$ lattices.

Figure 5

First derivative with respect to ${\mu }_B^2$ of the renormalized condensate: comparison between the results obtained on the $16^3$ and the $24^3$ lattices for the two different definitions. Data points have been slightly shifted horizontally to improve readability.

Figure 6

Values obtained for ${\kappa }_1$ on different values of $N_t$ and continuum extrapolation.

Figure 7

Curvature coefficient $\kappa$: comparison with previous determinations. From the left to the right: estimate by this work, Refs. [29, 27, 28, 23, 22, 25]. Bars marked with circles and triangles indicate that estimates have been obtained, respectively, by Taylor expansion and analytic continuation. More precisely, the red, magenta, blue, cyan and green colors indicate that the estimate has been obtained, respectively, by Taylor expansion $+$ chiral condensate, Taylor expansion $+$ chiral susceptibility, analytic continuation $+$ chiral condensate, analytic continuation $+$ chiral susceptibility and analytic continuation $+$ combined analysis of various observables.

Figure 8

$T_c$ for different values of the imaginary chemical potential obtained on the $24^3\times 6$ lattice by looking at the temperature for which the renormalized chiral condensate achieves the same value as for $T=T_c(0)$. The black circles (${\mu }_{l,I}/(\pi T)=0.24$ and 0.275) are those already present in Ref. [23]; the red triangles (${\mu }_{l,I}/(\pi T)=0.08$, 0.12, 0.155, 0.185 and 0.21) have been obtained by new simulations. The continuous and dashed lines are the results of the best fits described in the text and in the legend.