N. Ishizuka, a,b Y. Iwasaki, T. Izubuchi, e§ K. Kanaya, a T. Kaneko, e Y. Kuramashi, e V. Lesk, K.-I. Nagai, b¶ J. Noaki, a† M. Okawa, e Y. Taniguchi, a A. Ukawa and T. Yoshié Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan Center for Computational Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan Institute for Cosmic Ray Research, University of Tokyo, Kashiwa 277-8582, Japan High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan Institute of Theoretical Physics, Kanazawa University, Ishikawa 920-1192, Japan

We present results for the equation of state for pure SU(3) gauge theory obtained on anisotropic lattices with the anisotropy ξ ≡ as/at = 2. The pressure and energy density are calculated on Nt/ξ = 4, 5 and 6 lattices with the integral method. They are found to satisfy the leading 1/N t scaling from our coarsest lattice Nt/ξ = 4. This enables us to carry out well controlled continuum extrapolations. We find that the pressure and energy density agree with those obtained using the isotropic plaquette action, but have smaller and more reliable errors.

Renormalization constants of vector (ZV ) and axial-vector (ZA) currents are determined non-perturbatively in quenched QCD for a renormalization group improved gauge action and a tadpole improved clover quark action using the Schrödinger functional method. Non-perturbative values of ZV and ZA turn out to be smaller than the one-loop perturbative values by O(10%) at a ≈ 1 GeV. A sizable scaling violation of meson decay constants fπ and fρ observed with the one-loop renormalization factors remains even with non-perturbative renormalization.

We present our final results of the charmonium spectrum in quenched QCD on anisotropic lattices. Simulations are made with the plaquette gauge action and a tadpole improved clover quark action employing ξ = as/at = 3. We calculate the spectrum of Sand P-states and their excitation, and study the scaling behavior of mass splittings. Comparison is made with the experiment and previous lattice results. The issue of hyperfine splitting for different choices of the clover coefficients obtained by Klassen is discussed.

The non-perturbative csw determined by the Schrödinger functional (SF) method with the RG-improved gauge action in dynamical N f = 3 QCD shows a finite volume effect when the numerical simulations are carried out at a constant lattice size L/a.We remove the unwanted finite volume effect by keeping physical lattice extent L at a constant.The details of the method and the result obtained for non-perturbative csw with a constant L are reported.

Renormalization constants of vector (Z_V) and axial-vector (Z_A) currents are determined non-perturbatively in quenched QCD for an RG-improved gauge action and a tadpole-improved clover quark action using the Schrödinger functional method. Meson decay constants f_ρ and f_π show much better scaling when Z_V and Z_A estimated for infinite physical volume are used instead of Z-factors from tadpole-improved one-loop perturbation theory.

Renormalization constants (Z-factors ) of vector and axial-vector currents are determined nonperturbatively in quenched QCD for a renormalization group improved gauge action and a tadpole-improved clover quark action using the Schrodinger functional method. Nonperturbative values of Z-factors turn out to be smaller than 1-loop perturbative values by O(15%) at a lattice spacing of a(-1)approximate to 1 GeV. The pseudoscalar and vector meson decay constants calculated with the nonperturbative Z-factors show a much better scaling behavior compared to previous results obtained with tadpole-improved one-loop Z-factors. In particular, the nonperturbative Z-factors normalized at infinite physical volume show that the scaling violations of the decay constants are within about 10% up to the lattice spacing a(-1)similar to1 GeV. The continuum estimates obtained from data in the range a(-1)similar to 1-2 GeV agree with those determined from finer lattices (a(-1)similar to2-4 GeV) with the standard action.

We explore application of the domain wall fermion formalism of lattice QCD to calculate the $K\to\pi\pi$ decay amplitudes in terms of the $K\to\pi$ and $K\to 0$ hadronic matrix elements through relations derived in chiral perturbation theory. Numerical simulations are carried out in quenched QCD using domain-wall fermion action for quarks and an RG-improved gauge action for gluons on a $16^3\times 32\times 16$ and $24^3\times 32\times 16$ lattice at $\beta=2.6$ corresponding to the lattice spacing $1/a\approx 2$GeV. Quark loop contractions which appear in Penguin diagrams are calculated by the random noise method, and the $\Delta I=1/2$ matrix elements which require subtractions with the quark loop contractions are obtained with a statistical accuracy of about 10%. We confirm the chiral properties required of the $K\to\pi$ matrix elements. Matching the lattice matrix elements to those in the continuum at $\mu=1/a$ using the perturbative renormalization factor to one loop order, and running to the scale $\mu=m_c=1.3$ GeV with the renormalization group for $N_f=3$ flavors, we calculate all the matrix elements needed for the decay amplitudes. With these matrix elements, the $\Delta I=3/2$ decay amplitude shows a good agreement with experiment in the chiral limit. The $\Delta I=1/2$ amplitude, on the other hand, is about 50--60% of the experimental one even after chiral extrapolation. In view ofthe insufficient enhancement of the $\Delta I=1/2$ contribution, we employ the experimental values for the real parts of the decay amplitudes in our calculation of $\epsilon'/\epsilon$. We find that the $\Delta I=3/2$ contribution is larger than the $\Delta I=1/2$ contribution so that $\epsilon'/\epsilon$ is negative and has a magnitude of order $10^{-4}$. Possible reasons for these unsatisfactory results are discussed.

N. Tsutsui, c A. Ukawa, a,d T. Umeda, N. Yamada and T. Yoshié Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan Institute for Cosmic Ray Research, University of Tokyo, Kashiwa, Chiba 277-8582, Japan High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan Center for Computational Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan Department of Physics, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8526, Japan

We report on our study of two-flavor full QCD on anisotropic lattices using $O(a)$-improved Wilson quarks coupled with an RG-improved glue. The bare gauge and quark anisotropies corresponding to the renormalized anisotropy $\xi=a_s/a_t = 2$ are determined as functions of $\beta$ and $\kappa$, using the Wilson loop and the meson dispersion relation at several lattice cutoffs and quark masses.

We investigate a chiral property of the domain-wall fermion (DWF) system using the four-dimensional hermitian Wilson-Dirac operator HW. A formula expressing the Ward-Takahashi identity quark mass m5q with eigenvalues of this operator is derived, which well explains the N5 dependence of m5q observed in previous numerical simulations. We further discuss the chiral property of DWF in the large volume in terms of the spectra of HW.

Renormalization constants of vector (Z(V)) and axial-vector (Z(A)) currents are determined non-perturbatively in quenched QCD for a renormalization group improved gauge action and a tadpole improved clover quark action using the Schrodinger functional method. Non-perturbative values of Z(V) and Z(A) turn out to be smaller than the one-loop perturbative values by O(10%) at a(-1) approximate to 1 GeV. A sizable scaling violation of meson decay constants f(pi) and f(rho) observed with the one-loop renormalization factors remains even with non-perturbative renormalization.

We present our final results of the charmonium spectrum in quenched QCD on anisotropic lattices. Simulations are made with the plaquette gauge action and a tadpole improved clover quark action employing ξ = a_s/a_t = 3. We calculate the spectrum of S- and P-states and their excitation, and study the scaling behavior of mass splittings. Comparison is made with the experiment and previous lattice results. The issue of hyperfine splitting for different choices of the clover coefficients obtained by Klassen is discussed.