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We study the chiral properties of quenched domain wall fermions with several gauge actions. We demonstrate that the residual chiral symmetry breaking, which is present for a finite number of lattice sites in the fifth dimension ($L_s$), can be substantially suppressed using improved gauge actions. In particular the Symanzik action, the Iwasaki action, and a renormalization group improved gauge action, called doubly blocked Wilson (DBW2), are studied and compared to the Wilson action. All improved gauge actions studied show a reduction in the additive residual quark mass, $\mres$. Remarkably, in the DBW2 case $\mres$ is roughly two orders of magnitude smaller than the Wilson gauge action at $a^{-1}=2$ GeV and $L_s=16$. Significant reduction in $\mres$ is also realized at stronger gauge coupling corresponding to $a^{-1}=1.3$ GeV. As our numerical investigation indicates, this reduction is achieved by reducing the number of topological lattice dislocations present in the gauge field configurations. We also present detailed results for the quenched light hadron spectrum and the pion decay constant using the DBW2 gauge action., Comment: 50 pages, 25 figures, RevTeX formated, minor corrections

QCDOC is a supercomputer designed for high scalability at a low cost per node. We discuss the status of the project and provide performance estimates for large machines obtained from cycle accurate simulation of the QCDOC ASIC., Comment: 3 pages 1 figure. Lattice2002(machines)

A status report is given of the QCDOC project, a massively parallel computer optimized for lattice QCD using system-on-a-chip technology. We describe several of the hardware and software features unique to the QCDOC architecture and present performance figures obtained from simulating the current VHDL design of the QCDOC chip with single-cycle accuracy., Comment: Lattice2001(algorithms), 7 pages, 4 figures

We report the results of a calculation of the K --> pi pi matrix elements relevant for the $\DIhalf$ rule and $\epe$ in quenched lattice QCD using domain wall fermions at a fixed lattice spacing $a^{-1} \sim 2$ GeV. Working in the three-quark effective theory, where only the u, d and s quarks enter and which is known perturbatively to next-to-leading order, we calculate the lattice K --> pi and K --> |0> matrix elements of dimension six, four-fermion operators. Through lowest order chiral perturbation theory these yield K --> pi pi matrix elements, which we then normalize to continuum values through a non-perturbative renormalization technique. For the ratio of isospin amplitudes |A_0|/|A_2| we find a value of $25.3 \pm 1.8$ (statistical error only) compared to the experimental value of 22.2, with individual isospin amplitudes 10-20% below the experimental values. For $\epe$, using known central values for standard model parameters, we calculate $(-4.0 \pm 2.3) \times 10^{-4}$ (statistical error only) compared to the current experimental average of $(17.2 \pm 1.8) \times 10^{-4}$. Because we find a large cancellation between the I = 0 and I = 2 contributions to $\epe$, the result may be very sensitive to the approximations employed. Among these are the use of: quenched QCD, lowest order chiral perturbation theory and continuum perturbation theory below 1.3 GeV. We have also calculated the kaon B parameter, B_K and find $B_{K,\bar{MS}}(2 {\rm GeV}) = 0.532(11)$. Although currently unable to give a reliable systematic error, we have control over statistical errors and more simulations will yield information about the effects of the approximations on this first-principles determination of these important quantities., Comment: 158 pages, 51 tables, 41 figures. No numerical data or results changed in this revision, except for an improved analysis of B_K. Many improvements and modifications to the text and explanations. Table XLVII changed to include improved values for B_K. Submitted to Physical Review D

In the dilute instanton gas model of the QCD vacuum, one expects a strong spatial correlation between chirality and the maxima of the Dirac eigenvectors with small eigenvalues. Following Horvath, {\it et al.} we examine this question using lattice gauge theory within the quenched approximation. We extend the work of those authors by using weaker coupling, $\beta=6.0$, larger lattices, $16^4$, and an improved fermion formulation, domain wall fermions. In contrast with this earlier work, we find a striking correlation between the magnitude of the chirality density, $|\psi^\dagger(x)\gamma^5\psi(x)|$, and the normal density, $\psi^\dagger(x)\psi(x)$, for the low-lying Dirac eigenvectors., Comment: latex, 25 pages including 12 eps figures

We find the renormalisation coefficients of the quark field and the flavour non-singlet fermion bilinear operators for the domain wall fermion action, in the regularisation independent (RI) renormalisation scheme. Our results are from a quenched simulation, on a 16^3x32 lattice, with beta=6.0 and an extent in the fifth dimension of 16. We also discuss the expected effects of the residual chiral symmetry breaking inherent in a domain wall fermion simulation with a finite fifth dimension, and study the evidence for both explicit and spontaneous chiral symmetry breaking effects in our numerical results. We find that the relations between different renormalisation factors predicted by chiral symmetry are, to a good approximation, satisfied by our results and that systematic effects due to the (low energy) spontaneous chiral symmetry breaking and zero-modes can be controlled. Our results are compared against the perturbative predictions for both their absolute value and renormalisation scale dependence., Comment: 53 pages, 21 figures, revtex

The architecture of a new class of computers, optimized for lattice QCD calculations, is described. An individual node is based on a single integrated circuit containing a PowerPC 32-bit integer processor with a 1 Gflops 64-bit IEEE floating point unit, 4 Mbyte of memory, 8 Gbit/sec nearest-neighbor communications and additional control and diagnostic circuitry. The machine's name, QCDOC, derives from ``QCD On a Chip''., Comment: Lattice 2000 (machines) 8 pages, 4 figures

We are using domain wall fermions to study $K \to \pi \pi$ matrix elements by measuring $K \to \pi$ and $K \to 0$ matrix elements on the lattice and employing chiral perturbation theory to relate these to the desired physical result. The residual chiral symmetry breaking of domain wall fermions with a finite extent in the fifth dimension impacts these measurements. Using the Ward-Takahashi identities, we investigate residual chiral symmetry breaking effects for divergent quantities and study pathologies of the quenched approximation for small quark mass. We then discuss the $\Delta S = 1$ operator $O_2$, where chiral symmetry is vital for the subtraction of unphysical effects., Comment: 4 pages, 3 figures, Lattice 2000 (Hadronic Matrix Elements), RBC Collaboration, corrected equations 2 and 7

Quenched QCD simulations on three volumes, $8^3 \times$, $12^3 \times$ and $16^3 \times 32$ and three couplings, $\beta=5.7$, 5.85 and 6.0 using domain wall fermions provide a consistent picture of quenched QCD. We demonstrate that the small induced effects of chiral symmetry breaking inherent in this formulation can be described by a residual mass ($\mres$) whose size decreases as the separation between the domain walls ($L_s$) is increased. However, at stronger couplings much larger values of $L_s$ are required to achieve a given physical value of $\mres$. For $\beta=6.0$ and $L_s=16$, we find $\mres/m_s=0.033(3)$, while for $\beta=5.7$, and $L_s=48$, $\mres/m_s=0.074(5)$, where $m_s$ is the strange quark mass. These values are significantly smaller than those obtained from a more naive determination in our earlier studies. Important effects of topological near zero modes which should afflict an accurate quenched calculation are easily visible in both the chiral condensate and the pion propagator. These effects can be controlled by working at an appropriately large volume. A non-linear behavior of $m_\pi^2$ in the limit of small quark mass suggests the presence of additional infrared subtlety in the quenched approximation. Good scaling is seen both in masses and in $f_\pi$ over our entire range, with inverse lattice spacing varying between 1 and 2 GeV., Comment: 91 pages, 34 figures

The domain wall formulation of lattice fermions is expected to support accurate chiral symmetry, even at finite lattice spacing. Here we attempt to use this new fermion formulation to simulate two-flavor, finite temperature QCD near the chiral phase transition. In this initial study, a variety of quark masses, domain wall heights and domain wall separations are explored using an 8^3 x 4 lattice. Both the expectation value of the Wilson line and the chiral condensate show the temperature dependence expected for the QCD phase transition. Further, the desired chiral properties are seen for the chiral condensate, suggesting that the domain wall fermion formulation may be an effective approach for the numerical study of QCD at finite temperature., Comment: 44 pages, 15 figures

The QCDSP computer (Quantum Chromodynamics on Digital Signal Processors) is an inexpensive, massively parallel computer intended primarily for simulations in lattice gauge theory. Currently, two large QCDSP machines are in full-time use: an 8,192 processor, 0.4 Teraflops machine at Columbia University and an 12,288 processor, 0.6 Teraflops machine at the RIKEN-BNL Research Center at Brookhaven National Laboratory. We describe the design process, architecture, software and current physics projects of these computers., Comment: 19 pages, 3 figures

One of the major frontiers of lattice field theory is the inclusion of light fermions in simulations, particularly in pursuit of accurate, first principles predictions from lattice QCD. With dedicated Teraflops-scale computers currently simulating QCD, another step towards precision full QCD simulations is underway. In addition to ongoing staggered and Wilson fermion simulations, first results from full QCD with domain wall fermions are available. After some discussion of work toward better algorithms, simulations completed to date will be discussed., Comment: LATTICE99 (plenary) - 10 pages, 11 figures

This paper has been withdrawn., Comment: This paper has been withdrawn

A serious difficulty in conventional lattice field theory calculations is the coupling between the chiral and continuum limits. With both staggered and Wilson fermions, the chiral limit cannot be realized without first taking the limit of vanishing lattice spacing. In this talk, we report on extensive studies of the domain wall formulation of lattice fermions, which avoids this difficulty at the expense of requiring that fermion propagators be computed in five dimensions. A variety of results will be described for quenched and dynamical simulations at both zero and finite temperature. Conclusions about the benefits of this new method and some new physical results will be presented. These results were obtained on the QCDSP machines recently put into operation at Columbia and the RIKEN Brookhaven Research Center., Comment: Talk presented at ICHEP'98, July 23-29, 1998, Vancouver, Canada. 7 pages, 9 figures Omission from electronic author list corrected

We report on simulations of quenched QCD using domain wall fermions, where we focus on basic questions about the formalism and its ability to produce expected low energy hadronic physics for light quarks. The work reported here is on quenched $8^3 \times 32$ lattices at $\beta = 5.7$ and 5.85, using values for the length of the fifth dimension between 10 and 48. We report results for parameter choices which lead to the desired number of flavors, a study of undamped modes in the extra dimension and hadron masses., Comment: Contribution to Lattice '98. Presented by R. Mawhinney. 3 pages, 3 figures

We examine the chiral limit of domain wall fermions in quenched QCD. One expects that in a quenched simulation, exact fermion zero modes will give a divergent, 1/m behavior in the chiral condensate for sufficiently small valence quark masses. Unlike other fermion formulations, domain wall fermions clearly demonstrate this behavior., Comment: LATTICE98(spectrum), G. R. Fleming presented talk, 5 pages, 3 figures, corrected typos in printed version

We describe the completed 8,192-node, 0.4Tflops machine at Columbia as well as the 12,288-node, 0.6Tflops machine assembled at the RIKEN Brookhaven Research Center. Present performance as well as our experience in commissioning these large machines is presented. We outline our on-going physics program and explain how the configuration of the machine is varied to support a wide range of lattice QCD problems, requiring a variety of machine sizes. Finally a brief discussion is given of future prospects for large-scale lattice QCD machines., Comment: LATTICE98(machines), 3 pages, 1 picture, 1 figure

We present results from numerical simulations of full, two flavor QCD thermodynamics at N_t=4 with domain wall fermions. For the first time a numerical simulation of the full QCD phase transition displays a low temperature phase with spontaneous chiral symmetry breaking but intact flavor symmetry and a high temperature phase with the full SU(2) x SU(2) chiral flavor symmetry., Comment: LATTICE98(hightemp)

The domain wall approach to lattice fermions employs an additional dimension, in which gauge fields are merely replicated, to separate the chiral components of a Dirac fermion. It is known that in the limit of infinite separation in this new dimension, domain wall fermions have exact zero modes, even for gauge fields which are not smooth. We explore the effects of finite extent in the fifth dimension on the zero modes for both smooth and non-smooth topological configurations and find that a fifth dimension of around ten sites is sufficient to clearly show zero mode effects. This small value for the extent of the fifth dimension indicates the practical utility of this technique for numerical simulations of QCD., Comment: Updated fig. 3-7, small changes in sect. 3, added fig. 8, added more references

We study the anomalous breaking of U_A(1) symmetry just above the QCD phase transition for zero and two flavors of quarks, using a staggered fermion, lattice discretization. The properties of the QCD phase transition are expected to depend on the degree of U_A(1) symmetry breaking in the transition region. For the physical case of two flavors, we carry out extensive simulations on a 16^3 x 4 lattice, measuring a difference in susceptibilities which is sensitive to U_A(1) symmetry and which avoids many of the staggered fermion discretization difficulties. The results suggest that anomalous effects are at or below the 15% level., Comment: 10 pages including 2 figures and 1 table

The QCDSP machine at Columbia University has grown to 2,048 nodes achieving a peak speed of 100 Gigaflops. Software for quenched and Hybrid Monte Carlo (HMC) evolution schemes has been developed for staggered fermions, with support for Wilson and clover fermions under development. We provide an overview of the runtime environment, the current status of the QCDSP construction program and preliminary results not presented elsewhere in these proceedings., Comment: 3 pages, 1 figure, LaTeX with espcrc2.sty, talk presented at LATTICE97(Poster) by G. R. Fleming

We have done simulations of lattice QCD with different numbers of light dynamical quarks. Since we cannot reach the continuum limit with our current computers, we have done comparisons with 0 (quenched), 2 and 4 light quark flavors with the physical volume and lattice spacing constant, when these are determined from the mass of the rho. We find a 7% (2 standard deviation) difference in the nucleon to rho mass ratio for 2 and 4 quarks. More importantly, the effects of chiral symmetry breaking are dramatically decreased for the case of 4 light quarks., Comment: 15 pages, 6 figures. Presented at RHIC Summer Study '96

We present a summary of the progress on QCDSP in the last year. QCDSP, Quantum Chromodynamics on Digital Signal Processors, is an inexpensive computer being built at Columbia that can achieve 0.8 teraflops for three million dollars., Comment: 4 pages, 1 figure

We have measured the hadron spectrum in lattice QCD, using staggered fermions, for 0 (the quenched approximation), 2 and 4 light degenerate dynamical quarks. In addition to earlier results involving extrapolations in valence quark masses for fixed dynamical mass, we also report results where we extrapolate in the dynamical mass for 4 flavors. We see a marked difference in the hadron spectrum for 2 and 4 flavors; the hadron spectrum is nearly parity doubled for 4 flavors, indicating smaller effects of chiral symmetry breaking. This pronounced effect in the hadron spectrum cannot be removed by a simple change in scale as the number of light quark flavors is changed. Further simulations at larger volume are needed to rule out finite volume effects., Comment: 5 pages, 4 figures. Presented at Lattice QCD on Parallel Computers, Tsukuba, Japan, march 1997

We have studied the hadron spectrum while varying the number of light dynamical quarks when the physical lattice spacing and volume are held fixed relative to the rho mass. For two and zero flavors of staggered fermions, we find the nucleon to rho mass ratios (extrapolated to zero valence quark mass) are very similar. However, for four flavors the ratio is 7% (2 standard deviations) above the two flavor result., Comment: 3 pages, 3 figures