The QCD transition temperature: results with physical masses in the continuum limit

TL;DR

The paper analyzes the finite-temperature QCD transition using Symanzik-improved gauge action and stout-link improved staggered fermions with physical light and strange masses, performing continuum extrapolations from $N_t=4,6,8,10$. It demonstrates that the transition is a non-singular cross-over with no unique $T_c$, showing that different observables yield distinct continuum values: $T_c$ from the renormalized chiral susceptibility is $151(3)(3)$ MeV, while the strange quark number susceptibility and the Polyakov loop give higher values by $24(4)$ MeV and $25(4)$ MeV, respectively, each with non-vanishing widths ($28(5)(1)$, $42(4)(1)$, and $38(5)(1)$ MeV). The analysis uses a Line of Constant Physics fixed by $m_K/f_K$ and $m_K/m_\pi$, and cross-validates scale setting with multiple observables to ensure consistent continuum results. These findings highlight the importance of scale setting, the cross-over nature of the QCD transition, and the need for corroboration with alternative lattice formulations.

Abstract

The transition temperature ($T_c$) of QCD is determined by Symanzik improved gauge and stout-link improved staggered fermionic lattice simulations. We use physical masses both for the light quarks ($m_{ud}$) and for the strange quark ($m_s$). Four sets of lattice spacings ($N_t$=4,6,8 and 10) were used to carry out a continuum extrapolation. It turned out that only $N_t$=6,8 and 10 can be used for a controlled extrapolation, $N_t$=4 is out of the scaling region. Since the QCD transition is a non-singular cross-over there is no unique $T_c$. Thus, different observables lead to different numerical $T_c$ values even in the continuum and thermodynamic limit. The peak of the renormalized chiral susceptibility predicts $T_c$=151(3)(3) MeV, wheres $T_c$-s based on the strange quark number susceptibility and Polyakov loops result in 24(4) MeV and 25(4) MeV larger values, respectively. Another consequence of the cross-over is the non-vanishing width of the peaks even in the thermodynamic limit, which we also determine. These numbers are attempted to be the full result for the $T$$\neq$0 transition, though other lattice fermion formulations (e.g. Wilson) are needed to cross-check them.

Figures (5)

• Figure 1: The phase diagram of water around its critical point (CP). For pressures below the critical value ($p_c$) the transition is first order, for $p>p_c$ values there is a rapid crossover. In the crossover region the critical temperatures defined from different quantities are not necessarily equal. This can be seen for the temperature derivative of the density ($d\rho/dT$) and the specific heat ($c_p$). The bands show the experimental uncertainties (see Spang).
• Figure 2: Scaling of the mass of the $K^*(892)$ meson, the pion decay constant and $r_0$ towards the continuum limit. As a continuum value (filled boxes) we took the average of the continuum extrapolations obtained using our 2 and our 3 finest lattice spacings. The difference was taken as a systematic uncertainty, which is included in the shown errors. The quantities are plotted in units of the kaon decay constant. In case of the upper two panels the bands indicate the physical values of the ratios and their experimental uncertainties. For $r_0$ (lowest panel) in the absence of direct experimental results we compare our value with the $r_0f_K$ obtained by the MILC, HPQCD and UKQCD collaborations Aubin:2004fsGray:2005ur.
• Figure 3: Temperature dependence of the renormalized chiral susceptibility ($m^2\Delta \chi_{\bar{\psi}\psi}/T^4$), the strange quark number susceptibility ($\chi_s/T^2$) and the renormalized Polyakov-loop ($P_R$) in the transition region. The different symbols show the results for $N_t=4,6,8$ and $10$ lattice spacings (filled and empty boxes for $N_t=4$ and $6$, filled and open circles for $N_t=8$ and $10$). The vertical bands indicate the corresponding critical temperatures and its uncertainties coming from the T$\neq$0 analyses. This error is given by the number in the first parenthesis, whereas the error of the overall scale determination is indicated by the number in the second parenthesis. The orange bands show our continuum limit estimates for the three renormalized quantities as a function of the temperature with their uncertainties.
• Figure 4: Continuum limit of the transition temperatures obtained from the renormalized chiral susceptibility ($m^2\Delta \chi_{\bar{\psi}\psi}/T^4$), strange quark number susceptibility ($\chi_s/T^2$) and renormalized Polyakov-loop ($P_R$).
• Figure 5: Resolving the discrepancy between the critical temperature of Ref. Cheng:2006qk and that of the present work (see text). The major part of the difference can be traced back to the unreliable continuum extrapolation of Cheng:2006qk. Left panel: In Ref. Cheng:2006qk$r_0$ was used for scale setting (filled boxes), however using the kaon decay constant (empty boxes) leads to different critical temperatures even after performing the continuum extrapolation. Right panel: in our work the extrapolations based on the finer lattices are safe, using the two different scale setting methods one obtains consistent results.