The transition temperature in QCD

TL;DR

This study determines the QCD transition temperature for ($2+1$)-flavor QCD using improved staggered fermions on lattices with $N_ au=4,6$, performing a chiral and continuum extrapolation to the physical point. By combining zero-temperature scales from the static quark potential with finite-temperature susceptibilities (Polyakov-loop and chiral condensates), the authors extract $T_c r_0=0.457(7)$ and, using $r_0=0.469(7)$ fm, obtain $T_c=192(7)(4)$ MeV; the chiral limit yields a slightly smaller $T_c$. A scaling Ansatz with $d\approx1.08$ governs the quark-mass dependence, and a $1/N_ au^2$ term handles lattice cut-off effects, supporting a consistent continuum picture. The work emphasizes the impact of scale setting on the absolute value of $T_c$ and discusses universal scaling behavior of thermodynamic observables near the chiral transition. Overall, the results reinforce a crossover transition in physical QCD with a transition temperature in the mid-200 MeV range, subject to systematic uncertainties in scale setting and discretization.

Abstract

We present a detailed calculation of the transition temperature in QCD with two light and one heavier (strange) quark mass on lattices with temporal extent N_t =4 and 6. Calculations with improved staggered fermions have been performed for various light to strange quark mass ratios in the range, 0.05 <= m_l/m_s <= 0.5, and with a strange quark mass fixed close to its physical value. From a combined extrapolation to the chiral (m_l -> 0) and continuum (aT = 1/N_t -> 0) limits we find for the transition temperature at the physical point T_c r_0 = 0.457(7) where the scale is set by the Sommer-scale parameter r_0 defined as the distance in the static quark potential at which the slope takes on the value, (dV_qq(r)/dr)_r=r_0 = 1.65/r_0^2. Using the currently best known value for r_0 this translates to a transition temperature T_c = 192(7)(4)MeV. The transition temperature in the chiral limit is about 3% smaller. We discuss current ambiguities in the determination of T_c in physical units and also comment on the universal scaling behavior of thermodynamic quantities in the chiral limit.

Figures (7)

• Figure 1: Time history of the light and strange quark chiral condensates for the smallest quark masses used in our simulations on lattices of temporal extent $N_\tau =4$ and $6$ and for values of the gauge coupling in the vicinity of the critical coupling of the transition on these lattices. The upper figure shows a run at $\beta= 3.305$ with $\hat{m}_l=0.05 \hat{m}_s$ on a $16^3\times 4$ lattice and the lower figure is for $\beta=3.46$ and $\hat{m}_l=0.1 \hat{m}_s$ on a $16^3\times 6$ lattice.
• Figure 2: The light quark chiral condensate in units of $a^{-3}$ (left) and the Polyakov loop expectation value (right) as function of the bare light quark mass in units of the temperature, $m_l/T\equiv \hat{m}_l N_\tau$ for fixed $\beta$ and $\hat{m}_s=0.065$ on lattices of size $8^3\times 4$ (circle) and $16^3\times 4$ (triangles). Shown are results for various values of $\beta$ ranging from $\beta = 3.28$ to $\beta = 3.4$ (top to bottom for $\langle \bar{\psi}\psi\rangle$ and bottom to top for $\langle L \rangle$). Full and open symbols show results obtained from direct simulations and Ferrenberg-Swendsen interpolations, respectively.
• Figure 3: The disconnected part of the light quark chiral susceptibility on lattices of size $8^3\times 4$ (squares) and $16^3\times 4$ (circles) for four different values of the light quark mass. The curves show Ferrenberg-Swendsen interpolations of the data points obtained from multi-parameter histograms with an error band coming from Ferrenberg-Swendsen reweightings performed on different jackknife samples.
• Figure 4: The disconnected part of the light quark chiral susceptibility (left) and the Polyakov loop susceptibility (right) on lattices of size $16^3\times 6$ for three different values of the light quark mass. Curves show Ferrenberg-Swendsen interpolations as discussed in the caption of Fig. 2
• Figure 5: The difference of gauge couplings at the location of peaks in the Polyakov loop and chiral susceptibilities, $\beta_{c,L}-\beta_{c,l}$. Shown are results from calculations on $8^3\times 4$ (left), $16^3\times 4$ (middle) and $16^3\times 6$ (right).