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.

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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.

We present results on both the restoration of the spontaneously broken chiral symmetry and the effective restoration of the anomalously broken U (1) A symmetry in finite temperature QCD at zero chemical potential using lattice QCD. We employ domain wall fermions on lattices with fixed temporal extent N τ = 8 and spatial extent N σ = 16 in a temperature range of T = 139 − 195 MeV, corresponding to lattice spacings of a ≈ 0.12 − 0.18 fm. In these calculations, we include two degenerate light quarks and a strange quark at fixed pion mass m π = 200 MeV. The strange quark mass is set near its physical value. We also present results from a second set of finite temperature gauge configurations at the same volume and temporal extent with slightly heavier pion mass. To study chiral symmetry restoration, we calculate the chiral condensate, the disconnected chiral susceptibility, and susceptibilities in several meson channels of different quantum numbers. To study U (1) A restoration, we calculate spatial correlators in the scalar and pseudo-scalar channels, as well as the corresponding susceptibilities. Furthermore, we also show results for the eigenvalue spectrum of the Dirac operator as a function of temperature, which can be connected to both U (1) A and chiral symmetry restoration via Banks-Casher relations.

We report a direct lattice calculation of the K to ππ decay matrix elements for both the ∆I = 1/2 and 3/2 amplitudes A 0 and A 2 on 2+1 flavor, domain wall fermion, 16 3 × 32 × 16 lattices. This is a complete calculation in which all contractions for the required ten, four-quark operators are evaluated, including the disconnected graphs in which no quark line connects the initial kaon and final two-pion states. These lattice operators are non-perturbatively renormalized using the RomeSouthampton method and the quadratic divergences are studied and removed. This is an important but notoriously difficult calculation, requiring high statistics on a large volume. In this paper we take a major step towards the computation of the physical K → ππ amplitudes by performing a complete calculation at unphysical kinematics with pions of mass 422 MeV at rest in the kaon rest frame. With this simplification we are able to resolve Re(A 0 ) from zero for the first time, with a 25% statistical error and can develop and evaluate methods for computing the complete, complex amplitude A 0 , a calculation central to understanding the ∆ = 1/2 rule and testing the standard model of CP violation in the kaon system.