We report on the first lattice calculation of the QCD phase transition using chiral fermions at physical values of the quark masses. This calculation uses 2+1 quark flavors, spatial volumes between (4 fm) 3 and (11 fm) 3 and temperatures between 139 and 196 MeV . Each temperature was calculated using a single lattice spacing corresponding to a temporal Euclidean extent of Nt = 8. The disconnected chiral susceptibility, χ disc shows a pronounced peak whose position and height depend sensitively on the quark mass. We find no metastability in the region of the peak and a peak height which does not change when a 5 fm spatial extent is increased to 10 fm. Each result is strong evidence that the QCD "phase transition" is not first order but a continuous cross-over for mπ = 135 MeV. The peak location determines a pseudo-critical temperature Tc = 155(1)(8) MeV. Chiral SU (2)L ×SU (2)R symmetry is fully restored above 164 MeV, but anomalous U (1)A symmetry breaking is non-zero above Tc and vanishes as T is increased to 196 MeV.PACS numbers: 11.15. Ha, 12.38.Gc As the temperature of the QCD vacuum is increased above the QCD energy scale Λ QCD = 300 MeV, asymptotic freedom implies that the vacuum breaking of chiral symmetry must disappear and the familiar chirally-asymmetric world of massive nucleons and light pseudoGoldstone bosons must be replaced by an SU (2) L × SU (2) R symmetric plasma of nearly massless up and down quarks and gluons. Predicting, observing and characterizing this transition has been an experimental and theoretical goal since the 1980's. General principles are consistent with this being either a first-order transition for sufficiently light pion mass or a second-order transition in the O(4) universality class at zero pion mass with cross-over behavior for non-zero m π . While second order behavior is commonly expected, first-order behavior may be more likely if anomalous U (1) A symmetry is partially restored at T c resulting in an effectiveThe importance of the SU (2) L × SU (2) R chiral symmetry of QCD for the phase transition has motivated the widespread use of staggered fermions in lattice studies of QCD thermodynamics because this formulation possesses one exact chiral symmetry at finite lattice spacing, broken only by the quark mass. However, the flavor symmetry of the staggered fermion formulation is complicated showing an SU L (4) × SU R (4) "taste" symmetry that is broken by lattice artifacts and made to resemble the physical SU (2) L × SU (2) R symmetry by taking the square root of the Dirac determinant, a procedure believed to have a correct but subtle continuum limit for non-zero quark masses.
We report on a study of the finite-temperature QCD transition region for temperatures between 139 and 196 MeV, with a pion mass of 200 MeV and two space-time volumes: 24 3 × 8 and 32 3 × 8, where the larger volume varies in linear size between 5.6 fm (at T=139 MeV) and 4.0 fm (at T=195 MeV). These results are compared with the results of an earlier calculation using the same action and quark masses but a smaller, 16 3 ×8 volume. The chiral domain wall fermion formulation with a combined Iwasaki and dislocation suppressing determinant ratio gauge action are used. This lattice action accurately reproduces the SU (2) L × SU (2) R and U (1) A symmetries of the continuum. Results are reported for the chiral condensates, connected and disconnected susceptibilities and the Dirac eigenvalue spectrum. We find a pseudo-critical temperature, T c , of approximately 165 MeV consistent with previous results and strong finite volume dependence below T c . Clear evidence is seen for U (1) A symmetry breaking above T c which is quantitatively explained by the measured density of near-zero modes in accordance with the dilute instanton gas approximation.
At stronger gauge-field couplings, the domain wall fermion (DWF) residual mass (m res ), a measure of chiral symmetry breaking (χ SB ), grows rapidly as it is largely due to near zero fermion eigenmodes of log(T ), where T is the 4D transfer matrix along the fifth dimension. The number of these eigenmodes increase rapidly at strong coupling. To suppress such near-zero eigenmodes, we have added to the DWF path integral a multiplicative weighting factor consisting of a ratio of determinants of Wilson-Dirac fermions having a chirally twisted mass with a large negative real component and a small imaginary chiral component. Numerical results show that this weighting factor with an appropriate choice of twisted masses significantly suppresses m res while allowing adequate topological tunneling.
We study the region of the QCD phase transition using 2+1 flavors of domain wall fermions (DWF) and a 16 3 × 8 lattice volume with a fifth dimension of L s = 32. The disconnected light quark chiral susceptibility, quark number susceptibility and the Polyakov loop suggest a chiral and deconfining crossover transition lying between 155 and 185 MeV for our choice of quark mass and lattice spacing. In this region the lattice scale deduced from the Sommer parameter r 0 is a −1 ≈ 1.3 GeV, the pion mass is ≈ 300 MeV and the kaon mass is approximately physical. The peak in the chiral susceptibility implies a pseudo critical temperature T c = 171 (10)(17) MeV where the first error is associated with determining the peak location and the second with our unphysical light quark mass and non-zero lattice spacing. The effects of residual chiral symmetry breaking on the chiral condensate and disconnected chiral susceptibility are studied using several values of the valence L s .
At stronger gauge-field couplings, the domain wall fermion (DWF) residual mass (m res ), a measure of chiral symmetry breaking (χ SB ), grows rapidly as it is largely due to near zero fermion eigenmodes of log(T ), where T is the 4D transfer matrix along the fifth dimension. The number of these eigenmodes increase rapidly at strong coupling. To suppress such near-zero eigenmodes, we have added to the DWF path integral a multiplicative weighting factor consisting of a ratio of determinants of Wilson-Dirac fermions having a chirally twisted mass with a large negative real component and a small imaginary chiral component. Numerical results show that this weighting factor with an appropriate choice of twisted masses significantly suppresses m res while allowing adequate topological tunneling.
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