QCD Crossover at Finite Chemical Potential from Lattice Simulations

We provide the most accurate results for the QCD transition line so far. We optimize the definition of the crossover temperature $T_c$, allowing for its very precise determination, and extrapolate from imaginary chemical potential up to real ${\mu }_B\approx 300\mathrm{MeV}$. The definition of $T_c$ adopted in this work is based on the observation that the chiral susceptibility as a function of the condensate is an almost universal curve at zero and imaginary ${\mu }_B$. We obtain the parameters ${\kappa }_2=0.0153(18)$ and ${\kappa }_4=0.00032(67)$ as a continuum extrapolation based on $N_t=10$, 12, 16 lattices with physical quark masses. We also extrapolate the peak value of the chiral susceptibility and the width of the chiral transition along the crossover line. In fact, both of these are consistent with a constant function of ${\mu }_B$. We see no sign of criticality in the explored range.

Figure 1

Renormalized chiral condensate $⟨\overline{\psi }\psi ⟩$ (left) and chiral susceptibility $\chi$ (middle) as functions of the temperature for the intermediate lattice spacing in this study. The black curves correspond to vanishing baryon density, while results for various imaginary values of the chemical potential are shown in other colors. Finally, in the right panel we show the susceptibility as a function of the condensate. In this representation the chemical potential dependence is very weak.

Figure 2

Compilation of ${\kappa }_4$ (left) and ${\kappa }_2$ (right) coefficients from recent lattice studies. We only include those papers where physical quark masses were used, a controlled continuum extrapolation was performed, and either strangeness neutrality or ${\mu }_s=0$ was considered. [Note that while ${\mu }_s=0$ means ${\mu }_S={\mu }_B/3$ for all values of ${\mu }_B$, strangeness neutrality implies ${\mu }_S\approx {\mu }_B/4$ for small ${\mu }_B$.] The colors encode the numerical approach. Blue points indicate simulations at ${\mu }_B=0$ only, where the ${\mu }_B$ dependence of $T_c$ was extracted using a Taylor expansion. If a study used simulations with imaginary chemical potentials in addition, we plot the results with green points, instead. Top data points represent this work, the further references in order: [16, 24, 25, 30].

Figure 3

Top: Transition line extrapolated from lattice simulations at imaginary chemical potential using an analytical continuation with the ansatz used in step iv) of our analysis (green band) compared with an extrapolation using the formula in Eq. (1) up to the order of ${\kappa }_4$ (red band) or up to ${\kappa }_2$ (blue band). The proximity of the full and NLO result suggests that the higher order corrections are small in the range of ${\mu }_B$ considered here. Note that considering only the error bar of ${\kappa }_2$ underestimates the full error. The numerical values for the final analytical continuation, together with its error, are tabulated in the Supplemental Material [41]. Bottom: Crossover line from the lattice compared with a prediction from truncated Dyson–Schwinger equations [47] and some estimates of the chemical freeze-out parameters in heavy ion collisions [48, 49, 50, 51, 52]. Note that the width of the green band is not a representation of the width of the crossover region; it depicts the statistical and systematic errors achievable with the particular definition of the crossover temperature $T_c$ adopted in this work. Note also that the definition of the crossover temperature adopted in Ref. [47] is different from the one used in this work.

Figure 4

Top: Half-width $\sigma$ of the transition defined in Eq. (4) using the temperature difference of the contours $⟨\overline{\psi }\psi ⟩=0.355$ and $⟨\overline{\psi }\psi ⟩=0.215$. In the insert we show a plot of the $\chi (⟨\overline{\psi }\psi ⟩)$ peak, where the shaded region corresponds to $⟨\overline{\psi }\psi {⟩}_c\pm \mathrm{\Delta }⟨\overline{\psi }\psi ⟩/2$. Both are extrapolated to real ${\mu }_B$. Bottom: Result of a ${\mu }_B$ by ${\mu }_B$ analysis for the value of the chiral susceptibility at the crossover temperature after continuum extrapolation and including the systematic errors for $L{T}_c=4$. The green band shows a linear extrapolation in ${\widehat{\mu }}_B^2$.