Transition temperature in QCD

Tuning the strange quark mass in lattice simulations

We investigate the criticality of chiral phase transition manifested in the first and second order derivatives of Dirac eigenvalue spectrum with respect to light quark mass in (2+1)-flavor lattice QCD. Simulations are performed at temperatures from about 137 MeV to 176 MeV on $N_{\tau}=8$ lattices using the highly improved staggered quarks and the tree-level improved Symanzik gauge action. The strange quark mass is fixed to its physical value $m_s^{\text{phy}}$ and the light quark mass is set to $m_s^{\text{phy}}/40$ which corresponds to a Goldstone pion mass $m_{\pi}=110$ MeV. We find that in contrast to the case at $T\simeq 205$ MeV $m_l^{-1} \partial \rho(\lambda, m_l)/\partial m_l$ is no longer equal to $\partial ^2\rho(\lambda, m_l)/\partial m_l^2$ and $\partial ^2\rho(\lambda, m_l)/\partial m_l^2$ even becomes negative at certain low temperatures. This means that as temperature getting closer to $T_c$ $\rho(\lambda, m_l)$ is no longer proportional to $m_l^2$ and thus dilute instanton gas approximation is not valid for these temperatures. We demonstrate the temperature dependence can be factored out in $\partial \rho(\lambda, m_l)/ \partial m_l$ and $\partial^2 \rho(\lambda, m_l)/ \partial m_l^2$ at $T \in [137, 153]$ MeV, and then we propose a feasible method to estimate the power $c$ given $\rho \propto m_l^{c}$.

When the quark masses are lighter than those in QCD, the standard lore is that a chiral transition of first order must emerge for three light flavors. Recently, however, numerical simulations on the lattice suggest that the chiral transition is of second order in the chiral limit. Using an extended linear sigma model in the mean field approximation, we study the relation between terms which break the anomalous, U(1)A symmetry and the order of the chiral phase transition, especially how a chiral transition of second order can arise for three massless flavors. We note that in an (unphysical) region of the “Columbia” phase diagram, when the strange quark mass is light and negative, corresponding to topological angle θ=π, the CP symmetry is spontaneously broken.

We have simulated QCD using 2+1 flavors of domain wall quarks on a $(2.74 {\rm fm})^3$ volume with an inverse lattice scale of $a^{-1} = 1.729(28)$ GeV. The up and down (light) quarks are degenerate in our calculations and we have used four values for the ratio of light quark masses to the strange (heavy) quark mass in our simulations: 0.217, 0.350, 0.617 and 0.884. We have measured pseudoscalar meson masses and decay constants, the kaon bag parameter $B_K$ and vector meson couplings. We have used SU(2) chiral perturbation theory, which assumes only the up and down quark masses are small, and SU(3) chiral perturbation theory to extrapolate to the physical values for the light quark masses. While next-to-leading order formulae from both approaches fit our data for light quarks, we find the higher order corrections for SU(3) very large, making such fits unreliable. We also find that SU(3) does not fit our data when the quark masses are near the physical strange quark mass. Thus, we rely on SU(2) chiral perturbation theory for accurate results. We use the masses of the $\Omega$ baryon, and the $\pi$ and $K$ mesons to set the lattice scale and determine the quark masses. We then find $f_\pi = 124.1(3.6)_{\rm stat}(6.9)_{\rm syst} {\rm MeV}$, $f_K = 149.6(3.6)_{\rm stat}(6.3)_{\rm syst} {\rm MeV}$ and $f_K/f_\pi = 1.205(0.018)_{\rm stat}(0.062)_{\rm syst}$. Using non-perturbative renormalization to relate lattice regularized quark masses to RI-MOM masses, and perturbation theory to relate these to $\bar{\rm MS}$ we find $ m_{ud}^{\bar{\rm MS}}(2 {\rm GeV}) = 3.72(0.16)_{\rm stat}(0.33)_{\rm ren}(0.18)_{\rm syst} {\rm MeV}$ and $m_{s}^{\bar{\rm MS}}(2 {\rm GeV}) = 107.3(4.4)_{\rm stat}(9.7)_{\rm ren}(4.9)_{\rm syst} {\rm MeV}$.

We investigate U A (1) breaking above T c in terms of the Dirac spectrum on configurations with (2+1)-flavors, using the HISQ action.The strange quark mass is at its physical value.We use several light quark masses corresponding to the Goldstone pion masses in the range of about 115 -230 MeV on lattices of size 32 3 ×8 and 48 3 ×8.We calculate the 100 lowest-lying Dirac eigenvalues at temperatures below and above T c .We investigate the volume dependence of the Dirac eigenvalue density to determine whether there is a gap around zero, which can appear if U A (1) symmetry is restored in the chiral symmetric phase.We also investigate the quark mass dependence of the Dirac eigenvalue density at zero and check whether there is a linear behavior that would signal the U A (1) breaking above T c .

Lattice QCD at the physical point: Light quark masses

A fully non-perturbative lattice determination of the up/down and strange quark masses is given for quenched QCD using both, $O(a)$ improved Wilson fermions and ordinary Wilson fermions. For the strange quark mass with $O(a)$ improved fermions we obtain $m^{\msbar}_s(\mu=2 {GeV}) = 105(4) {MeV}$, using the interquark force scale $r_0$. Due to quenching problems fits are only possible for quark masses larger than the strange quark mass. If we extrapolate our fits to the up/down quark mass we find for the average mass $m^{\msbar}_l(\mu=2 {GeV}) = 4.4(2) {MeV}$.

The region of the Columbia plot with two light quark flavors is not yet conclusively understood. Non-perturbative effects, e.g. the magnitude of the anomalous U(1) axial symmetry breaking, decides on the nature of the phase transition in this region. We report on our study of this region of the Columbia plot using lattice techniques. We use gauge field ensembles generated within the Highly Improved Staggered Quark discretization scheme, with the strange quark mass fixed at its physical value and the light quark mass varied such that $m_l=m_s/27$ and $m_s/40$, where $m_l=m_s/27$ corresponds to the physical light quark mass. We study the eigenvalue spectrum of QCD using the overlap Dirac operator on these gauge field ensembles at finite temperature around the chiral transition temperature $T_c$, as the light quark masses approach the chiral limit, and infer about the fate of the anomalous $U_A(1)$ symmetry breaking.

Light quenched hadron spectrum and decay constants on different lattices

Phase structure and phase transitions in dense QCD are studied using the Schwinger-Dyson (SD) method in the improved ladder approximation. We construct the Cornwall-Jackiw-Tomboulis (CJT) effective potentials at finite temperature for two types of pairing ansatz, namely the color-flavor locking (CFL) state and the two-flavor superconducting (2SC) state. Strong coupling effects at low densities, such as the off-Fermi surface and the antiquark contribution to the pairing gap due to the large effective coupling, cause the gap, critical temperature and their ratio to deviate from the weak coupling values. Nevertheless, the ratio of the physical quantity for the CFL state and that for the 2SC state does not differ so much from the weak coupling value, as long as the pairing interaction is taken to be the same for both cases. As a consequence, the CFL state always dominates the 2SC state and the critical temperatures of the transitions to the quark-gluon plasma (QGP) phase from both states coincide in the chiral limit. The energy gain in the CFL state relative to the 2SC state becomes smaller as the critical line dividing (μ, T) plane into the QGP and CFL phases is approached, and thus small perturbations can lift the degeneracy of these critical lines. As one of the perturbations, the effect of the strange quark mass ms on the quark-pairing is examined. In particular, using a simple kinematical criterion, we discuss the behavior of the ‘ color-flavor unlocking line’, on which the CFL phase turns into the 2SC phase, through the variation of ms.

We present recent results on the QCD equation of state with 2+1+1 flavors of highly improved staggered quarks (HISQ). We focus on three sets of ensembles with temporal extent 6, 8 and 10, that reach up to temperatures of 967, 725 and 580 MeV, respectively. The strange and charm quark masses are tuned to the physical values and the light quarks mass is set to one fifth of the strange. This corresponds to a Goldstone pion of about 300 MeV.

Since gluons in QCD are interacting fundamental constituents just as quarks are, we expect that in addition to mesons made from a quark and an antiquark, there should also be glueballs and hybrids (bound states of quarks, antiquarks and gluons). In general, these states would mix strongly with the conventional q-bar-q mesons. However, they can also have exotic quantum numbers inaccessible to q-bar-q mesons. Confirmation of such states would give information on the role of "dynamical" color in low energy QCD. In the quenched approximation we present a lattice calculation of the masses of mesons with exotic quantum numbers. These hybrid mesons can mix with four quark (q-bar-q-bar-q-q) states. The quenched approximation partially suppresses this mixing. Nonetheless, our hybrid interpolating fields also couple to four quark states. Using a four quark source operator, we demonstrate this mixing for the 1-+ meson. Using the conventional Wilson quark action, we calculate both at reasonably light quark masses, intending to extrapolate to small quark mass, and near the charmed quark mass, where we calculate the masses of some c-bar-c-g hybrid mesons. The hybrid meson masses are large --- over 4 GeV for charmonium and more than twice the vector meson mass at our smallest quark mass, which is near the strange quark mass.

The QCD phase transition with 2, 3, and 4 flavors of dynamical quarks

Baryon number and charge fluctuations from lattice QCD

The infrared regularized and the large Nc standard U(3) chiral perturbation theory (ChPT) predicted scaling of the pion, kaon decay constants fπ, fK, and Mη2=mη2+mη′2, has been used in the renormalized 2+1 flavor quark-meson (RQM) model to find a consistent path to the chiral limit as the π and K meson masses approach zero. Using the inputs from the above two prescriptions of the ChPT, the Columbia plots are drawn and compared in the mπ−mK and the light strange quark mass mud−ms planes, free from any ambiguity or heuristic adjustment in the model parameter fixing away from the physical point. The on-shell renormalization of the parameters and consistent treatment of the divergent quark one-loop vacuum fluctuations in the RQM model make the axial UA(1) anomaly significantly stronger, and the light (strange) explicit chiral symmetry breaking strength becomes weaker by a small (relatively large) amount. The first-order chiral transition region in the mπ−mK and μ−mK planes of the Columbia plots generated using the inputs from both versions of the ChPT increases due to the above novel features of the RQM model. Comparing our results (in the presence of the axial UA(1) anomaly) regarding the softening effect of the quark one-loop vacuum fluctuations on the chiral transition, it looks overestimated in a recent functional renormalization group study. The UA(1) symmetry and the SUA(3) chiral symmetry restoration in the chiral limit takes place at the same transition temperature.