Lattice results on the phase structure and equation of state in QCD at finite temperature

TL;DR

This work surveys lattice QCD results for the phase structure and equation of state at finite temperature, focusing on the transition temperature $T_c$ and the universality class of the chiral transition in 2+1 flavor QCD. It compares multiple fermion actions (improved staggered, Wilson-type, and chiral quarks) and scales, diagnosing the historical Tc discrepancy with the root cause being taste violations in staggered formulations and scale choices; improvements like HISQ, stout, and p4 mitigate these artifacts and bring Tc into a common range around $T_c\in[145,165]$ MeV, with chiral and strange-quark susceptibilities providing complementary determinations. The phase-structure analyses reveal $O(4)$ scaling for Wilson-type quarks in the chiral limit and, for staggered quarks, emergent $O(2)$ scaling near the physical point due to lattice artifacts, suggesting a second-order chiral transition in two-flavor QCD under certain conditions. EOS studies using the integration method and fixed-scale approaches show action-dependent trace anomalies and peak structures around $T\sim$200 MeV, with initial results for both staggered and Wilson quarks and ongoing work toward continuum and chiral extrapolations. Overall, the paper highlights significant progress in first-principles QCD thermodynamics, cross-checks across fermion formulations, and the path forward to precise, continuum EOS predictions relevant to heavy-ion phenomenology and the QGP regime.

Abstract

I review recent developments in the studies of the phase structure and equation of state in finite temperature QCD on the lattice.

Figures (6)

• Figure 1: Chiral observables in 2+1 flavor QCD with improved staggered quarks. (Left) Subtracted chiral condensate (difference of scaled light quark and $s$ quark condensates to remove a divergent renormalization factor) from the HISQ, p4, asqtad and stout actions BazavovSoeldnerLat10. (Center) Comparison of HISQ and asqtad results for the disconnected part of the chiral susceptibility BazavovSoeldnerLat10. In these panels, the HISQ, p4 and asqtad data are obtained at the bare quark mass ratio $m_{ud}/m_s = 0.05$ with $m_s$ around the physical value, using a scale fixed by $r_0$ . The stout data, which is obtained at the physical point using a $f_K$ scale WB_Tc3, is shifted in $T$ to correct the difference in the scale setting. Note that this procedure causes a slight deviation from the physical point at finite lattice spacings. (Right) Subtracted chiral condensate with stout and asqtad quarks at $N_t=8$WB_Tc3. The open rectangles are results of stout quark action at $m_\pi^{\rm pNG}=414$ MeV WB_Tc3, and the red curve is the result of asqtad quark action at $m_\pi^{\rm pNG}=220$ MeV BNLB_EOSt8. The stout quark mass for open rectangle is adjusted to reproduce $m_\pi^{\rm RMS}=587$ MeV of the asqtad data at $T\approx 135$ MeV on an $N_t=8$ lattice.
• Figure 2: Order of the finite temperature transition in 2+1 flavor QCD as a function of the degenerate $u$ and $d$ quark mass $m_{ud}$ and the $s$ quark mass $m_s$. (Left) The standard scenario with the second order chiral transition for two-flavor QCD. (Right) An alternative scenario when the two-flavor chiral transition is first order.
• Figure 3: (Left) O(4) scaling test for the subtracted chiral condensate in two-flavor QCD using the clover-improved Wilson quark action and the RG-improved Iwasaki gauge action on an $N_t=4$ lattice by the CP-PACS CollaborationCppacsPRD63. The dashed curve is the scaling function obtained in an O(4) spin model. Similar result supporting the O(4) scaling was obtained with the unimproved Wilson quark action and the Iwasaki gauge action too IwasakiPRL78. (Right) Rescaled chiral condensate $M_0 = m_s \langle \bar{\psi}\psi \rangle_{ud}/ T^4$ in 2+1 flavor QCD using the p4-improved staggered quark action and tree-level Symanzik gauge action on an $N_t=4$ lattice by the BNL-Bielefeld Collaboration BNLB_O4. The QCD data are fitted to the O(2) scaling function with correction terms modeling a deviation from the scaling.
• Figure 4: Trace anomaly in 2+1 flavor QCD. The scale is set by $r_0$, except for the stout data for which the scale is set by $f_K$. (Left) Comparison of HSQ, asqtad and p4 quarks at $m_{ud}/m_s = 0.05$ on $N_t=8$ lattices BazavovSoeldnerLat10. (Right) Comparison of stout, asqtad and p4 quarks on lattices $N_t \ge 8$WB_EOS10. The stout data is obtained at the physical point ($m_{ud}/m_s \approx 0.035$) while asqtad and p4 data are for $m_{ud}/m_s = 0.05$. The stout values are corrected by a tree-level improvement factor.
• Figure 5: EOS in 2+1 flavor QCD. (Left) p4 results at $m_\pi^{\rm pNG} \approx 154$ MeV ($m_{ud}/m_s = 0.05$) obtained on an $N_t=8$ lattice using the $r_0$ scale BNLB_EOSp. Results at $m_\pi^{\rm pNG} = 220$ MeV ($m_{ud}/m_s = 0.1$) are compared. (Right) Pressure with stout quarks using the $f_K$ scale WB_EOS10.