## Brief Descriptions and Links to Recent Physics Papers

• Status of the QCDOC project

Authors: P.A. Boyle, D. Chen, N.H. Christ, C. Cristian, Z. Dong, A. Gara, B. Joó, C. Kim, L. Levkova, X. Liao, G. Liu, R.D. Mawhinney, S. Ohta, T. Wettig, A. Yamaguchi

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.

• Kaon Matrix Elements and CP-violation from Quenched Lattice QCD: (I) the 3-flavor
case

Authors: T. Blum, P. Chen, N. Christ, C. Cristian, C. Dawson, G. Fleming, R. Mawhinney, S. Ohta, G. Siegert, A. Soni, P. Vranas, M. Wingate, L. Wu, Y. Zhestkov, RBC Collaboration

We report the results of a calculation of the $K \to \pi \pi$ matrix elements relevant for the $\DIhalf$ rule and $\epe$ in quenched lattice QCD using domain wall fermions. 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 $K \to \pi$ and $K \to |0>$ matrix elements of dimension six, four-fermion operators. Through lowest order chiral perturbation theory these yield $K \to \pi \pi$ matrix elements, which we then normalize to continuum values through a non-perturbative renormalization technique. For the $\DIhalf$ rule 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 $B_K(2 {\rm GeV}) = 0.513(11)$. 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.

• Chirality Correlation within Dirac Eigenvectors from Domain Wall Fermions

Authors: T. Blum, N. Christ, C. Cristian, C. Dawson, X. Liao, G. Liu, R. Mawhinney, L. Wu, Y. Zhestkov

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.

• Non-perturbative Renormalisation of Domain Wall Fermions: Quark Bilinears

Authors: T. Blum, N. Christ, C. Cristian, C. Dawson, G. Fleming, G. Liu, R. Mawhinney, A. Soni, P. Vranas, M. Wingate, L. Wu, Y. Zhestkov

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.

• QCDOC: A 10-teraflops scale computer for lattice QCD

Authors: D. Chen, N. H. Christ, C. Cristian, Z. Dong, A. Gara, K. Garg, B. Joo, C. Kim, L. Levkova, X. Liao, R. D. Mawhinney, S. Ohta, T. Wettig

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''.

Authors: T. Blum, P. Chen, N. Christ, C. Cristian, C. Dawson, G. Fleming, A. Kaehler, X. Liao, G. Liu, C. Malureanu, R. Mawhinney, S. Ohta, G. Siegert, A. Soni, C. Sui, P. Vranas, M. Wingate, L. Wu, Y. Zhestkov

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.

Authors: P. Chen, N. Christ, G. Fleming, A. Kaehler, C. Malureanu, R. Mawhinney, G. Siegert, C. Sui, P. Vranas, L. Wu, Y. Zhestkov

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.

Authors: Robert D. Mawhinney

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.

Authors: Robert D. Mawhinney

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.

Authors: Ping Chen, Norman Christ, Geoerge Fleming, Adrian Kaehler, Catalin Malureanu, Robert Mawhinney, Gabriele Siegert, Chengzhong Sui, Pavlos Vranas, Yuri Zhestkov

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.

Authors: Ping Chen (1), Norman H. Christ (1), George R. Fleming (1), Adrian L. Kaehler (1), Catalin I. Malureanu (1), Robert D. Mawhinney (1), Gabriele U. Siegert (1), ChengZhong Sui (1), Pavlos M. Vranas (1), Yuri Zhestkov (1) ((1) Columbia University)

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.

Authors: Shailesh Chandrasekharan, Dong Chen, Norman Christ, Weonjong Lee, Robert Mawhinney, Pavlos Vranas

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.

Authors: Frank R. Brown, Hong Chen, Norman H. Christ, Zhihua Dong, Robert D. Mawhinney, Wendy Schaffer, Alessandro Vaccarino

QCD with eight flavors is studied on $16^3\times N_t$ lattices with $N_t=4$, 6, 8, 16 and 32, a dynamical quark mass $ma=0.015$ and lattice coupling $\beta=6/g^2$ between 4.5 and 5.0. For $N_t=16$ and 32, hadron masses and screening lengths are computed for a variety of valence quark masses. The previously observed, strong, first-order transition for $N_t=4$, 6 and 8 is seen, for $N_t=16$, to become a $\beta$-independent, zero-temperature transition characterized by a factor of $\approx 3$ change in lattice scale. This strong, first-order transition restores chiral symmetry, at least for $N_t=4$, 6 and 8, producing a chirally symmetric, weak-coupling phase. However, as $N_t$ increases to 16, the chiral symmetry properties of the weak-coupling side of the zero-temperature transition are unclear and offer a hint of a normal, finite-temperature, chiral symmetry breaking transition in the weak-coupling phase.