Using the eigenmode of the Dirac operator 6 D ¼ D in quantum chromodynamics (QCD), we develop a manifestly gauge-covariant expansion and projection of the QCD operators such as the Wilson loop and the Polyakov loop. With this method, we perform a direct analysis of the correlation between confinement and chiral symmetry breaking in lattice QCD Monte Carlo calculation on 6 4 at ¼ 5:6. Even after removing the low-lying Dirac modes, which are responsible for chiral symmetry breaking, we find that the Wilson loop obeys the area law, and the slope parameter corresponding to the string tension, or confinement force, is almost unchanged. We find also that the Polyakov loop remains to be almost zero even without the low-lying Dirac modes, which indicates the Z 3 -unbroken confinement phase. These results indicate that one-to-one correspondence does not hold between confinement and chiral symmetry breaking in QCD.

The energy density and the pressure of SU(3) gauge theory at finite temperature are studied by direct lattice measurements of the renormalized energy-momentum tensor obtained by the gradient flow. Numerical analyses are carried out with β = 6.287-7.500 corresponding to the lattice spacing a = 0.013-0.061 fm. The spatial (temporal) sizes are chosen to be Ns = 64, 96, 128 (Nτ = 12,16,20,22,24) with the aspect ratio, 5.33 ≤ Ns/Nτ ≤ 8. Double extrapolation, a → 0 (the continuum limit) followed by t → 0 (the zero flow-time limit), is taken using the numerical data. Above the critical temperature, the thermodynamic quantities are obtained with a few percent precision including statistical and systematic errors. The results are in good agreement with previous high-precision data obtained by using the integral method.

The ΞΞ interaction in the 1 S 0 channel is studied to examine the convergence of the derivative expansion of the non-local HAL QCD potential at the next-to-next-to-leading order (N 2 LO). We find that (i) the leading order potential from the N 2 LO analysis gives the scattering phase shifts accurately at low energies, (ii) the full N 2 LO potential gives only small correction to the phase shifts even at higher energies below the inelastic threshold, and (iii) the potential determined from the wall quark source at the leading order analysis agrees with the one at the N 2 LO analysis except at short distances, and thus, it gives correct phase shifts at low energies. We also study the possible systematic uncertainties in the HAL QCD potential such as the inelastic state contaminations and the finite volume artifact for the potential and find that they are well under control for this particular system.

The S-wave ΛΛ and N Ξ interactions are studied on the basis of the (2+1)-flavor lattice QCD simulations close to the physical point (m π 146MeV and m K 525MeV). Lattice QCD potentials in four different spin-isospin channels are extracted by using the coupled-channel HAL QCD method and are parametrized by analytic functions to calculate the scattering phase shifts. The ΛΛ interaction at low energies shows only a weak attraction, which does not provide a bound or resonant dihyperon. The N Ξ interaction in the spin-singlet and isospin-singlet channel is most attractive and lead the N Ξ system near unitarity. Relevance to the strangeness=−2 hypernuclei as well as to two-baryon correlations in proton-proton, proton-nucleus and nucleus-nucleus collisions is also discussed.

The possible exotic meson Zc(3900), found in e + e − reactions, is studied by the method of coupledchannel scattering in lattice QCD. The interactions among πJ/ψ, ρηc andDD * channels are derived from (2+1)-flavor QCD simulations at mπ = 410-700 MeV. The interactions are dominated by the off-diagonal πJ/ψ-DD * and ρηc-DD * couplings, which indicates that the Zc(3900) is not a usual resonance but a threshold cusp. Semiphenomenological analyses with the coupled-channel interaction are also presented to confirm this conclusion.PACS numbers: 12.38. Gc, 14.40.Rt, 13.75.Lb One of the long-standing problems in hadron physics is to identify the existence of exotic hadrons different from the quark-antiquark states (mesons) and three-quark states (baryons). Candidates of such exotic hadrons include the pentaquark states P In particular, Z c (3900) appears as a peak in both the π ± J/ψ and DD * invariant mass spectra in the reaction,Its quantum numbers are then identified as I G (J P C ) = 1 + (1 +− ), so that at least four quarks, ccud (or its isospin partners), are involved. (See the level structure and the decay scheme in Fig.1.) So far, there have been various phenomenological attempts to characterize the Z c (3900) as a hadrocharmonium, a compact tetraquark, a hadronic molecule (e.g., Refs. [5,6]) as well as a kinematical threshold effect (e.g., Refs. [7,8]). However, due to the lack of information of the diagonal and off-diagonal interactions among different channels (such as πJ/ψ, ρη c , andDD * ), the predictions of those models are not well under theoretical control. On the other hand, the direct lattice QCD studies with the standard method of temporal correlations show no candidate for the Z c (3900) eigenstate [9,10], which indicates that the Z c (3900) may not be an ordinary resonance state. Under these circumstances, it is most desirable to carry out manifestly coupled-channel analyses with the first-principles QCD inputs.The purpose of this Letter is to report a first attempt to determine the nature of the Z c (3900) on the basis of the HAL QCD method [11][12][13][14]. We consider three two-body channels below Z c (3900) (πJ/ψ, ρη c andDD * ) which couple with each other. The interactions among these channels faithful to the QCD S matrix are derived from the equal-time Nambu-Bethe-Salpeter (NBS) wave functions on the lattice according to the coupled-channel formulation of the HAL QCD method [15][16][17]. The s-wave interactions and the S matrix thus obtained are used to search for the complex poles in the πJ/ψ andDD * scattering amplitudes to unravel the nature of the Z c (3900). We note here that the conventional resonances such as the ρ meson and the ∆ baryon have not yet been analyzed in the HAL QCD method, and the comparison with the Lüscher's method [18] in these channels is one of the important future subjects. (Such comparison in the nonresonant ππ channel has been done in Refs. [19,20]; See also Ref. [21]). Note also that the coupled-channel HAL QCD method has not been experimentally tested in othe...