We present results on the chiral and deconfinement properties of the QCD transition at finite temperature. Calculations are performed with 2 + 1 flavors of quarks using the p4, asqtad and HISQ/tree actions. Lattices with temporal extent Nτ = 6, 8 and 12 are used to understand and control discretization errors and to reliably extrapolate estimates obtained at finite lattice spacings to the continuum limit. The chiral transition temperature is defined in terms of the phase transition in a theory with two massless flavors and analyzed using O(N ) scaling fits to the chiral condensate and susceptibility. We find consistent estimates from the HISQ/tree and asqtad actions and our main result is Tc = 154 ± 9 MeV.

We present results for the equation of state in (2+1)-flavor QCD using the highly improved staggered quark action and lattices with temporal extent Nτ = 6, 8, 10, and 12. We show that these data can be reliably extrapolated to the continuum limit and obtain a number of thermodynamic quantities and the speed of sound in the temperature range (130-400) MeV. We compare our results with previous calculations, and provide an analytic parameterization of the pressure, from which other thermodynamic quantities can be calculated, for use in phenomenology. We show that the energy density in the crossover region, 145 MeV ≤ T ≤ 163 MeV, defined by the chiral transition, is c = (0.18 − 0.5) GeV/fm 3 , i.e., (1.2 − 3.1) nuclear . At high temperatures, we compare our results with resummed and dimensionally reduced perturbation theory calculations. As a byproduct of our analyses, we obtain the values of the scale parameters r0 from the static quark potential and w0 from the gradient flow.

We calculate the equation of state in 2+1 flavor QCD at finite temperature with physical strange quark mass and almost physical light quark masses using lattices with temporal extent Nτ = 8. Calculations have been performed with two different improved staggered fermion actions, the asqtad and p4 actions. Overall, we find good agreement between results obtained with these two O(a 2 ) improved staggered fermion discretization schemes. A comparison with earlier calculations on coarser lattices is performed to quantify systematic errors in current studies of the equation of state. We also present results for observables that are sensitive to deconfining and chiral aspects of the QCD transition on Nτ = 6 and 8 lattices. We find that deconfinement and chiral symmetry restoration happen in the same narrow temperature interval. In an Appendix we present a simple parametrization of the equation of state that can easily be used in hydrodynamic model calculations. In this parametrization we also incorporated an estimate of current uncertainties in the lattice calculations which arise from cutoff and quark mass effects. We estimate these systematic effects to be about 10 MeV.

Scalar and tensor interactions were once competitors to the now well-established V − A structure of the Standard Model weak interactions. We revisit these interactions and survey constraints from low-energy probes (neutron, nuclear, and pion decays) as well as collider searches. Currently, the most stringent limit on scalar and tensor interactions arise from 0 + → 0 + nuclear decays and the radiative pion decay π → eνγ, respectively. For the future, we find that upcoming neutron beta decay and LHC measurements will compete in setting the most stringent bounds. For neutron beta decay, we demonstrate the importance of lattice computations of the neutronto-proton matrix elements to setting limits on these interactions, and provide the first lattice estimate of the scalar charge and a new average of existing results for the tensor charge. Data taken at the LHC is currently probing these interactions at the 10 −2 level (relative to the standard weak interactions), with the potential to reach the < ∼ 10 −3 level. We show that, with some theoretical assumptions, the discovery of a charged spin-0 resonance decaying to an electron and missing energy implies a lower limit on the strength of scalar interactions probed at low energy.

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We review lattice results related to pion, kaon, D-meson, B-meson, and nucleon physics with the aim of making them easily accessible to the nuclear and particle physics communities. More specifically, we report on the determination of the light-quark masses, the form factor $$f_+(0)$$f+(0) arising in the semileptonic $$K \rightarrow \pi $$K→π transition at zero momentum transfer, as well as the decay constant ratio $$f_K/f_\pi $$fK/fπ and its consequences for the CKM matrix elements $$V_{us}$$Vus and $$V_{ud}$$Vud. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of $$SU(2)_L\times SU(2)_R$$SU(2)L×SU(2)R and $$SU(3)_L\times SU(3)_R$$SU(3)L×SU(3)R Chiral Perturbation Theory. We review the determination of the $$B_K$$BK parameter of neutral kaon mixing as well as the additional four B parameters that arise in theories of physics beyond the Standard Model. For the heavy-quark sector, we provide results for $$m_c$$mc and $$m_b$$mb as well as those for D- and B-meson decay constants, form factors, and mixing parameters. These are the heavy-quark quantities most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. We review the status of lattice determinations of the strong coupling constant $$\alpha _s$$αs. Finally, in this review we have added a new section reviewing results for nucleon matrix elements of the axial, scalar and tensor bilinears, both isovector and flavor diagonal.

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 present high statistics results for the isovector charges g u−dA , g u−dS and g u−dT of the nucleon. Calculations were carried out on eleven ensembles of gauge configurations generated by the MILC collaboration using highly improved staggered quarks action with 2 þ 1 þ 1 dynamical flavors. These ensembles span four lattice spacings a ≈ 0.06, 0.09, 0.12 and 0.15 fm and light-quark masses corresponding to M π ≈ 135, 225 and 315 MeV. Excited-state contamination in the nucleon three-point correlation functions is controlled by including up to three-states in the spectral decomposition. Remaining systematic uncertainties associated with lattice discretization, lattice volume and light-quark masses are controlled using a simultaneous fit in these three variables. Our final estimates of the isovector charges in the MS scheme at 2 GeV are T with precision low-energy nuclear experiments, and find them comparable to those from the ATLAS and the CMS experiments at the LHC.