Curvature of the chiral pseudocritical line in QCD

We determine the curvature of the pseudocritical line of strong interactions by means of numerical simulations at imaginary chemical potentials. We consider $N_f=2+1$ stout improved staggered fermions with physical quark masses and the tree level Symanzik gauge action, and explore two different sets of lattice spacings, corresponding to temporal extensions $N_t=6$ and $N_t=8$. Both the renormalized chiral condensate and the renormalized chiral susceptibility are used to locate the transition. The determinations obtained from the two quantities are in good agreement, a preliminary continuum extrapolation yields $\kappa =0.013(2)(1)$. We also investigate the impact of a nonzero strange quark chemical potential and compare our results to previous determinations in the literature, discussing the possible sources of systematic errors affecting the various procedures.

Figure 1

Sketched phase diagram in the $T-{\theta }_l$ plane for ${\mu }_s={\mu }_l$. Solid lines indicate the RW lines, while the dashed lines corresponds to the analytic continuation of the pseudocritical line.

Figure 2

The same as for Fig. 1, but for ${\mu }_s=0$. In this case the exact location of the RW-like lines, apart from the one at ${\theta }_l=\pi$, is temperature dependent and known analytically only in the high $T$ limit.

Figure 3

Imaginary part of the Polyakov loop as a function of ${\theta }_l$ at fixed $T\approx 208\mathrm{MeV}$ for ${\mu }_s={\mu }_l$ and for ${\mu }_s=0$.

Figure 4

Renormalized light chiral condensate for various values of $T$ and ${\mu }_l$ respectively on $24^3\times 6$ lattice with ${\mu }_s=0$ (top panel); $32^3\times 8$ lattice with ${\mu }_s=0$ (middle panel); $32^3\times 8$ lattice with ${\mu }_s={\mu }_l$ (bottom panel). In the top figure the data from the $32^3\times 6$ lattice at ${\mu }_l=0$ are also shown for comparison. Lines correspond to the best fit described in the text and the filled triangles denote the values at the pseudocritical temperature.

Figure 5

Renormalized light chiral susceptibility, divided by $m_{\pi }^4$, for various values of $T$ and ${\mu }_l$ respectively on $24^3\times 6$ lattice with ${\mu }_s=0$ (top panel); $32^3\times 8$ lattice with ${\mu }_s=0$ (middle panel); $32^3\times 8$ lattice with ${\mu }_s={\mu }_l$ (bottom panel). In the top figure the data from the $32^3\times 6$ lattice at ${\mu }_l=0$ are also shown for comparison.

Figure 6

Determinations of $T_c$ obtained from the renormalized chiral condensate $⟨\overline{\psi }\psi {⟩}_l^r$ for various values of the chemical potential and lattice sizes. The lines correspond to quadratic and quartic fits in ${\mu }_{l,I}$, as discussed in the text. Fit results are reported in Table 2.

Figure 7

Determinations of $T_c$ obtained from the renormalized chiral susceptibility ${\chi }_{\overline{\psi }\psi }^r$ for various values of the chemical potential and lattice sizes. The lines correspond to quadratic and quartic fits in ${\mu }_{l,I}$, as discussed in the text. Fit results are reported in Table 3.

Figure 8

Fitted values of the curvature $\kappa$ from the $N_t=6$ lattices as a function of the inverse spatial volume. Squares correspond to the determinations obtained by using the chiral condensate, triangles to the chiral susceptibility estimates.

Figure 9

Determinations of the critical line curvature $\kappa$ as a function of $N_t^{-2}$. Squares correspond to the determinations obtained by using the chiral condensate, triangles to the chiral susceptibility estimates. The points at zero abscissa correspond to a rudimentary continuum limit extrapolation, assuming corrections linear in $N_t^{-2}$.

Figure 10

Determinations of the critical line curvature $\kappa$ in different works. From bottom to top: (i) Analytic continuation, renormalized chiral condensate, this work; (ii) Analytic continuation, renormalized chiral susceptibility, this work; (iii) Analytic continuation, disconnected chiral susceptibility with ${\mu }_s={\mu }_l$, Ref. [20]; (iv) Taylor expansion, chiral susceptibility, Ref. [7]; (v) Taylor expansion, chiral condensate (different renormalization), Ref. [8].

Figure 11

Comparison between the zero and finite temperature chiral susceptibility at zero chemical potential. These terms are required to compute the renormalized chiral susceptibility, see Eq. (13). Data on the vertical axis are in lattice units. A linear fit of the $T=0$ data is also shown.

Figure 12

Effective Polyakov loop potential computed from Eq. (c1) for ${\mu }_u={\mu }_d$ and ${\mu }_s=0$.

Figure 13

Same as in Fig. 12, but for ${\mu }_s={\mu }_u={\mu }_d$: this is actually the same as Fig. 2 of Ref. [42]. The true vacuum potential from Fig. 12 is also reported to allow for a direct comparison of the two cases.