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We show that using the multi-splitting algorithm as a preconditioner for the domain wall Dirac linear operator, arising in lattice QCD, effectively reduces the inter-node communication cost, at the expense of performing more on-node floating point and memory operations. Correctly including the boundary \textit{snake} terms, the preconditioner is implemented in the QUDA framework, where it is found that utilizing kernel fusion and the tensor cores on NVIDIA GPUs is necessary to achieve a sufficiently performant preconditioner. A reduced-dimension (reduced-$L_s$) strategy is also proposed and tested for the preconditioner. We find the method achieves lower time to solution than regular CG at high node count despite the additional local computational requirements from the preconditioner. This method could be useful for supercomputers with more on-node flops and memory bandwidth than inter-node communication bandwidth., Add DOI

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→πtransition at zero momentum transfer, as well as the decay constant ratio$$f_K/f_\pi $$fK/fπand its consequences for the CKM matrix elements$$V_{us}$$Vusand$$V_{ud}$$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)Rand$$SU(3)_L\times SU(3)_R$$SU(3)L×SU(3)RChiral Perturbation Theory. We review the determination of the$$B_K$$BKparameter of neutral kaon mixing as well as the additional fourBparameters that arise in theories of physics beyond the Standard Model. For the heavy-quark sector, we provide results for$$m_c$$mcand$$m_b$$mbas well as those forD- andB-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.

Lattice QCD calculations including the effects of one or more non-degenerate sea quark flavors are conventionally performed using the Rational Hybrid Monte Carlo (RHMC) algorithm, which computes the square root of the determinant of $\mathscr{D}^{\dagger} \mathscr{D}$, where $\mathscr{D}$ is the Dirac operator. The special case of two degenerate quark flavors with the same mass is described directly by the determinant of $\mathscr{D}^{\dagger} \mathscr{D}$ --- in particular, no square root is necessary --- enabling a variety of algorithmic developments, which have driven down the cost of simulating the light (up and down) quarks in the isospin-symmetric limit of equal masses. As a result, the relative cost of single quark flavors --- such as the strange or charm --- computed with RHMC has become more expensive. This problem is even more severe in the context of our measurements of the $\Delta I = 1/2$ $K \rightarrow \pi \pi$ matrix elements on lattice ensembles with $G$-parity boundary conditions, since $G$-parity is associated with a doubling of the number of quark flavors described by $\mathscr{D}$, and thus RHMC is needed for the isospin-symmetric light quarks as well. In this paper we report on our implementation of the exact one flavor algorithm (EOFA) introduced by the TWQCD collaboration for simulations including single flavors of domain wall quarks. We have developed a new preconditioner for the EOFA Dirac equation, which both reduces the cost of solving the Dirac equation and allows us to re-use the bulk of our existing high-performance code. Coupling these improvements with careful tuning of our integrator, the time per accepted trajectory in the production of our 2+1 flavor $G$-parity ensembles with physical pion and kaon masses has been decreased by a factor of 4.2., Comment: 61 pages, 18 figures, 14 tables. v2: minor updates for journal submission

The RBC and UKQCD Collaborations have shown that light hadron masses and meson decay constants measured on 2+1 flavor Mobius DWF ensembles generated with the Iwasaki gauge action and a dislocation suppressing determinant ratio (DSDR) term show few percent O(a2) scaling violations for ensembles with a-1 = 1 GeV. We call this combination the ID+MDWF action and this scaling implies that, to a good approximation, these ensembles lie on a renormalization group trajectory, where the form of the action is unchanged and only the bare parameters need to be tuned to stay on the trajectory. Here we investigate whether a single-step APE-like blocking kernel can reproduce this trajectory and test its accuracy via measurements of the light hadron spectrum and non-perturbative renormalization. As we report, we find close matching to the renormalization group trajectory from this simple blocking kernel.

We review lattice results related to pion, kaon, D- and B-meson physics with the aim of making them easily accessible to the particle-physics community. More specifically, we report on the determination of the light-quark masses, the form factor f+(0) , arising in the semileptonic K→ π transition at zero momentum transfer, as well as the decay constant ratio fK/ fπ and its consequences for the CKM matrix elements Vu s and Vu d. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of SU(2) L× SU(2) R and SU(3) L× SU(3) R Chiral Perturbation Theory. We review the determination of the BK parameter of neutral kaon mixing as well as the additional four B parameters that arise in theories of physics beyond the Standard Model. The latter quantities are an addition compared to the previous review. For the heavy-quark sector, we provide results for mc and mb (also new compared to the previous review), 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. Finally, we review the status of lattice determinations of the strong coupling constant αs.

We have performed global fits of fπ and mπ , from a variety of RBC-UKQCD domain wall fermion ensembles, to SU(2) partially quenched chiral perturbation theory at NNLO. We report values for 9 NLO and 8 linearly independent combinations of NNLO partially quenched low energy constants, which we compare to other lattice and phenomenological determinations. We discuss the convergence of the expansion and use our large set of low energy constants to make predictions for the pion mass splitting due to QCD isospin breaking effects and the s-wave ππ scattering lengths. We conclude that, for the range of pseudoscalar masses explored in this work, 115 MeV . mPS . 430 MeV, the NNLO SU(2) expansion is quite robust and can fit lattice data with percent-scale accuracy.

We have performed fits of the pseudoscalar masses and decay constants, from a variety of RBC-UKQCD domain wall fermion ensembles, to SU(2) partially quenched chiral perturbation theory at next-to leading order (NLO) and next-to-next-to leading order (NNLO). We report values for 9 NLO and 8 linearly independent combinations of NNLO partially quenched low energy constants, which we compare to other lattice and phenomenological determinations. We discuss the size of successive terms in the chiral expansion and use our large set of low energy constants to make predictions for mass splittings due to QCD isospin breaking effects and the S-wave ?? scattering lengths. We conclude that, for the range of pseudoscalar masses explored in this work, 115 MeV?mPS?430 MeV, the NNLO SU(2) expansion is quite robust and can fit lattice data with percent-scale accuracy.

We present the first calculation of the kaon semileptonic form factor with sea and valence quark masses tuned to their physical values in the continuum limit of 2+1 flavour domain wall lattice QCD. We analyse a comprehensive set of simulations at the phenomenologically convenient point of zero momentum transfer in large physical volumes and for two different values of the lattice spacing. Our prediction for the form factor is f + π (0) = 0.9685(34)(14) where the first error is statistical and the second error systematic. This result can be combined with experimental measurements of K → π decays for a determination of the CKM-matrix element for which we predict |V us | = 0.2233(5)(9) where the first error is from experiment and the second error from the lattice computation.

We determine the scaling behavior of the autocorrelation times of observables constructed from the topological charge density on lattices with periodic and open boundary conditions using a series of high-statistics numerical simulations. The autocorrelation functions of such observables turn out to obey a simple differential equation which allows the motion of topological charge in hybrid Monte Carlo simulations to be understood in terms of only two processes: diffusion and tunneling. There is a characteristic lattice spacing at which open boundary conditions become worthwhile for reducing autocorrelations and we show how this lattice spacing is related to the diffusion constant, the tunneling rate, and the lattice Euclidean time extent. The subject of this talk was treated in more detail in a paper recently published by the authors [1].

We present new results for the amplitude $A_2$ for a kaon to decay into two pions with isospin $I=2$: Re$A_2 = 1.50(4)_\mathrm{stat}(14)_\mathrm{syst}\times 10^{-8}$ GeV; Im$A_2 = -6.99(20)_\mathrm{stat}(84)_\mathrm{syst}\times 10^{-13}$ GeV. These results were obtained from two ensembles generated at physical quark masses (in the isospin limit) with inverse lattice spacings $a^{-1}=1.728(4)$ GeV and $2.358(7)$ GeV. We are therefore able to perform a continuum extrapolation and hence largely to remove the dominant systematic uncertainty from our earlier results, that due to lattice artefacts. The only previous lattice computation of $K\to\pi\pi$ decays at physical kinematics was performed using an ensemble at a single, rather coarse, value of the lattice spacing ($a^{-1}\simeq 1.37(1)$ GeV). We confirm the observation that there is a significant cancellation between the two dominant contributions to Re$A_2$ which we suggest is an important ingredient in understanding the $\Delta I=1/2$ rule, Re$A_0$/Re$A_2\simeq 22.5$, where the subscript denotes the total isospin of the two-pion final state. Our result for $A_2$ implies that the electroweak penguin contribution to $\epsilon^\prime/\epsilon$ is Re($\epsilon^\prime/\epsilon)_\textrm{EWP}=-(6.6\pm 1.0)\times 10^{-4}$.

Quantum chromodynamics (QCD) is the widely accepted theory of the strong interactions of quarks and gluons. Only through large scale numerical simulation has it been possible to work out the predictions of this theory for a vast range of phenomena relevant to the US Department of Energy experimental program. Such simulations are essential to support the discovery of new phenomena and more fundamental interactions. With support from SciDAC the USQCD collaboration has developed software and prototyped custom computer hardware to carry out the required numerical simulations. We have developed a robust, portable data-parallel code suite. It provides a user-friendly basis for writing physics application codes for carrying out the calculations needed to predict the phenomenology of QCD. We are using this efficient and optimized code base to develop new physics application code, to improve the performance of legacy code, and to construct higher level tools, such as QCD-specific sparse matrix solvers. We give a brief overview of the design of the data parallel API and its various components. We describe performance gains achieved in the past year. Finally, we present plans for further improvements under SciDAC-2.

After a brief overview of quantum chromodynamics (QCD), the fundamental theory of the strong interactions, we describe the QCDOC computer, its architecture, construction, software and performance. Three 12K-node, 4 Teraflops (sustained) QCDOC computers have been constructed, two at the Brookhaven National Lab and one at the University of Edinburgh. The present status of these machines and their first physics results and objectives are discussed and the catalytic role of the SciDAC program in enabling the effective use of this new architecture by the US lattice QCD community outlined.

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., Lattice2001(algorithms), 7 pages, 4 figures

We present results for several light hadronic quantities ($f_\pi$, $f_K$, $B_K$, $m_{ud}$, $m_s$, $t_0^{1/2}$, $w_0$) obtained from simulations of 2+1 flavor domain wall lattice QCD with large physical volumes and nearly-physical pion masses at two lattice spacings. We perform a short, O(3)%, extrapolation in pion mass to the physical values by combining our new data in a simultaneous chiral/continuum `global fit' with a number of other ensembles with heavier pion masses. We use the physical values of $m_\pi$, $m_K$ and $m_\Omega$ to determine the two quark masses and the scale - all other quantities are outputs from our simulations. We obtain results with sub-percent statistical errors and negligible chiral and finite-volume systematics for these light hadronic quantities, including: $f_\pi$ = 130.2(9) MeV; $f_K$ = 155.5(8) MeV; the average up/down quark mass and strange quark mass in the $\bar {\rm MS}$ scheme at 3 GeV, 2.997(49) and 81.64(1.17) MeV respectively; and the neutral kaon mixing parameter, $B_K$, in the RGI scheme, 0.750(15) and the $\bar{\rm MS}$ scheme at 3 GeV, 0.530(11)., Comment: 131 pages, 30 figures. Updated to match published version

We study the autocorrelations of observables constructed from the topological charge density, such as the topological charge on a time slice or in a subvolume, using a series of hybrid Monte Carlo simulations of pure SU(3) gauge theory with both periodic and open boundary conditions. We show that the autocorrelation functions of these observables obey a simple diffusion equation and we measure the diffusion coefficient, finding that it scales like the square of the lattice spacing. We use this result and measurements of the rate of tunneling between topological charge sectors to calculate the scaling behavior of the autocorrelation times of these observables on periodic and open lattices. There is a characteristic lattice spacing at which open boundary conditions become worthwhile for reducing autocorrelations and we show how this lattice spacing is related to the diffusion coefficient, the tunneling rate, and the lattice Euclidean time extent., Comment: 36 pages, 14 figures

We review lattice results related to pion, kaon, D- and B-meson physics with the aim of making them easily accessible to the particle physics community. More specifically, we report on the determination of the light-quark masses, the form factor f+(0), arising in semileptonic K -> pi transition at zero momentum transfer, as well as the decay constant ratio fK/fpi of decay constants and its consequences for the CKM matrix elements Vus and Vud. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of SU(2)LxSU(2)R and SU(3)LxSU(3)R Chiral Perturbation Theory and review the determination of the BK parameter of neutral kaon mixing. The inclusion of heavy-quark quantities significantly expands the FLAG scope with respect to the previous review. Therefore, for this review, we focus on D- and B-meson decay constants, form factors, and mixing parameters, since these are most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. In addition we review the status of lattice determinations of the strong coupling constant alpha_s., Several errors of transcription in the entries of Table 1 have been corrected. The entries of Table 1 now agree with the final results quoted in the main text. 324 pages, 26 figures, 118 tables, 697 references

We report on the first lattice calculation of the QCD phase transition using chiral fermions with physical 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 is calculated at a single lattice spacing corresponding to a temporal Euclidean extent of N(t) = 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 near 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 crossover for m(π) = 135 MeV. The peak location determines a pseudocritical temperature T(c) = 155(1)(8) MeV, in agreement with earlier staggered fermion results. However, the peak height is 50% greater than that suggested by previous staggered results. Chiral SU(2)(L) × SU(2)(R) symmetry is fully restored above 164 MeV, but anomalous U(1)(A) symmetry breaking is nonzero above T(c) and vanishes as T is increased to 196 MeV.

We present preliminary results for the calculation of the Kaon Bag parameter $B_K$ in $N_f=2+1$ lattice QCD, using M\"obius Domain Wall Fermion ensembles generated by the RBC-UKQCD collaboration. This computation is done directly at physical meson masses, so that we do not have to rely on chiral perturbation theory or any other mass extrapolation. In parallel, the four-quark operator is renormalised through the Rome-Southampton technique. Finally, we compare our value with previous results and draw some conclusions about the remaining dominant contributions in our error budget., Comment: presented at the 31st International Symposium on Lattice Field Theory (Lattice 2013), 29 July - 3 August 2013, Mainz, Germany

We present results from a 2+1 flavor DWF calculation at 1/a = 3 GeV and discuss strategies for similar calculations at finer lattice spacings which will target charm physics. At weak coupling the autocorrelation time of the global topological charge becomes very long because the HMC algorithm has trouble moving between topological sectors. We report the results of simulations that test two ideas for reducing the autocorrelation time of topological charge. In weak coupling quenched simulations we find that the open boundary conditions suggested by L\"uscher and Schaefer do not prevent the appearance of extremely long autocorrelation times for topological observables. We discuss the idea of a "dislocation-enhancing determinant" and show that it can produce an increase in topological tunneling., Comment: 7 pages, 4 figures. Presented at the 31st International Symposium on Lattice Field Theory (Lattice 2013), 29 July - 3 August 2013, Mainz, Germany

We present an update on the RBC/UKQCD collaboration’s results for the K ! π semileptonic vector form factor at zero momentum transfer from simulations of chiral fermions. Our results cover the whole range of light quark masses between the SU(3)-symmetric and the physical point for three lattice spacings and large physical volumes. Using partially twisted boundary conditions we calculate the form factor directly at zero momentum transfer. The comprehensive set of data points allows for turning the extrapolation in the quar k mass into an interpolation around the physical point thereby removing the dominant systematic uncertainty in previous results, the chiral extrapolation. We briefly discuss our prediction in view of Standard Model phenomenology.

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

The origin of mass is one of the deepest mysteries in science. Neutrons and protons, which account for almost all visible mass in the Universe, emerged from a primordial plasma through a cataclysmic phase transition microseconds after the Big Bang. However, most mass in the Universe is invisible. The existence of dark matter, which interacts with our world so weakly that it is essentially undetectable, has been established from its galactic-scale gravitational effects. Here we describe results from the first truly physical calculations of the cosmic phase transition and a groundbreaking first-principles investigation into composite dark matter, studies impossible with previous state-of-the-art methods and resources. By inventing a powerful new algorithm, "DSDR," and implementing it effectively for contemporary supercomputers, we attain excellent strong scaling, perfect weak scaling to the LLNL BlueGene/Q two million cores, sustained speed of 7.2 petaflops, and time-to-solution speedup of more than 200 over the previous state-of-the-art.

We present physical results for a variety of light hadronic quantities obtained via a combined analysis of three 2+1 flavour domain wall fermion ensemble sets. For two of our ensemble sets we used the Iwasaki gauge action with beta=2.13 (a^-1=1.75(4) GeV) and beta=2.25 (a^-1=2.31(4) GeV) and lattice sizes of 24^3 x 64 and 32^3 x 64 respectively, with unitary pion masses in the range 293(5)-417(10) MeV. The extent L_s for the 5^th dimension of the domain wall fermion formulation is L_s=16 in these ensembles. In this analysis we include a third ensemble set that makes use of the novel Iwasaki+DSDR (Dislocation Suppressing Determinant Ratio) gauge action at beta = 1.75 (a^-1=1.37(1) GeV) with a lattice size of 32^3 x 64 and L_s=32 to reach down to partially-quenched pion masses as low as 143(1) MeV and a unitary pion mass of 171(1) MeV, while retaining good chiral symmetry and topological tunneling. We demonstrate a significant improvement in our control over the chiral extrapolation, resulting in much improved continuum predictions for the above quantities. The main results of this analysis include the pion and kaon decay constants, f_\pi=127(3)_{stat}(3)_{sys} MeV and f_K = 152(3)_{stat}(2)_{sys} MeV respectively (f_K/f_\pi = 1.199(12)_{stat}(14)_{sys}); the average up/down quark mass and the strange-quark mass in the MSbar-scheme at 3 GeV, m_{ud}(MSbar, 3 GeV) = 3.05(8)_{stat}(6)_{sys} MeV and m_s(MSbar, 3 GeV) = 83.5(1.7)_{stat}(1.1)_{sys}; the neutral kaon mixing parameter in the MSbar-scheme at 3 GeV, B_K(MSbar,3 GeV) = 0.535(8)_{stat}(13)_{sys}, and in the RGI scheme, \hat B_K = 0.758(11)_{stat}(19)_{sys}; and the Sommer scales r_1 = 0.323(8)_{stat}(4)_{sys} fm and r_0 = 0.480(10)_{stat}(4)_{sys} (r_1/r_0 = 0.673(11)_{stat}(3)_{sys}). We also obtain values for the SU(2) ChPT effective couplings, \bar{l_3} = 2.91(23)_{stat}(7)_{sys}$ and \bar{l_4} = 3.99(16)_{stat}(9)_{sys}., Comment: 95 pages, 23 figures. Added missing author to metadata

With a substantial amount of simulations, we have explored the system across a wide range of lattice scales. We have located a lattice artifact, first order bulk transition, have studied its properties, and found that the flavor-singlet scalar meson mass vanishes at the critical endpoint. We will discuss the lattice phase diagrams and the continuum limits for both a spontaneous chiral symmetry breaking phase and an infrared conformal phase, and compare results with other groups., 7 pages, 7 figures; Contribution to SCGT12 "KMI-GCOE Workshop on Strong Coupling Gauge Theories in the LHC Perspective", 4-7 Dec. 2012, Nagoya University

We present results on both the restoration of the spontaneously broken chiral symmetry and the effective restoration of the anomalously broken $U(1{)}_{A}$ symmetry in finite temperature QCD at zero chemical potential using lattice QCD. We employ domain wall fermions on lattices with fixed temporal extent ${N}_{\ensuremath{\tau}}=8$ and spatial extent ${N}_{\ensuremath{\sigma}}=16$ in a temperature range of $T=139--195\text{ }\text{ }\mathrm{MeV}$, corresponding to lattice spacings of $a\ensuremath{\approx}0.12--0.18\text{ }\text{ }\mathrm{fm}$. In these calculations, we include two degenerate light quarks and a strange quark at fixed pion mass ${m}_{\ensuremath{\pi}}=200\text{ }\text{ }\mathrm{MeV}$. The strange quark mass is set near its physical value. We also present results from a second set of finite temperature gauge configurations at the same volume and temporal extent with slightly heavier pion mass. To study chiral symmetry restoration, we calculate the chiral condensate, the disconnected chiral susceptibility, and susceptibilities in several meson channels of different quantum numbers. To study $U(1{)}_{A}$ restoration, we calculate spatial correlators in the scalar and pseudoscalar channels, as well as the corresponding susceptibilities. Furthermore, we also show results for the eigenvalue spectrum of the Dirac operator as a function of temperature, which can be connected to both $U(1{)}_{A}$ and chiral symmetry restoration via Banks-Casher relations.

We report on new data from additional zero temperature simulations of QCD with 12 flavors. This is a continuation of previous studies using the DBW2 gauge action and naive staggered fermions. With the use of the force gradient integrator and a multiple-quark-mass preconditioned HMC, we have done simulations with input quark masses from $m_q=0.003$ to $m_q=0.008$. We have observed a metastable, first order, bulk transition that occurs at small input quark masses. As the quark mass increases, this first order bulk transition ends at a second order critical point, and, for still heavier quark masses, becomes the cross-over we have previously reported. We present measurements of hadron masses, decay constants and other low energy observables in the small quark mass region on the weak coupling side of the bulk transition. Our results show that the behavior of the system is still consistent with spontaneously broken chiral symmetry. We also discuss a preliminary investigation into the behavior of the bulk transition itself. We have found that, as the system approaches the second order critical end point, the scalar singlet meson becomes lighter. Thus it appears that the critical endpoint corresponds to a continuum limit theory only involving scalars and, following known triviality arguments, this is likely a free field theory. The presence of this critical endpoint could influence scaling of lattice observables in the conventional continuum limit., Comment: 7 pages, 13 figures, 2 tables. Contribution to the XXIX International Symposium on Lattice Field Theory, July 10-16, 2011, Squaw Valley, Lake Tahoe, California

We determine the neutral kaon mixing matrix element BK in the continuum limit with 2+1 flavors of domain wall fermions, using the Iwasaki gauge action at two different lattice spacings. These lattice fermions have near exact chiral symmetry and therefore avoid artificial lattice operator mixing. We introduce a significant improvement to the conventional nonperturbative renormalization (NPR) method in which the bare matrix elements are renormalized nonperturbatively in the regularization invariant momentum scheme (RI-MOM) and are then converted into the MS? scheme using continuum perturbation theory. In addition to RI-MOM, we introduce and implement four nonexceptional intermediate momentum schemes that suppress infrared nonperturbative uncertainties in the renormalization procedure. We compute the conversion factors relating the matrix elements in this family of regularization invariant symmetric momentum schemes (RI-SMOM) and MS? at one-loop order. Comparison of the results obtained using these different intermediate schemes allows for a more reliable estimate of the unknown higher-order contributions and hence for a correspondingly more robust estimate of the systematic error. We also apply a recently proposed approach in which twisted boundary conditions are used to control the Symanzik expansion for off-shell vertex functions leading to a better control of the renormalization in the continuum limit. We control chiral extrapolation errors by considering both the next-to-leading order SU(2) chiral effective theory, and an analytic mass expansion. We obtain BKMS? (3??GeV)=0.529(5)stat(15)?(2)FV(11)NPR. This corresponds to B?KRGI? =0.749(7)stat(21)?(3)FV(15)NPR. Adding all sources of error in quadrature, we obtain B?KRGI? =0.749(27)combined, with an overall combined error of 3.6%.

We present physical results obtained from simulations using 2+1 flavors of domain wall quarks and the Iwasaki gauge action at two values of the lattice spacing $a$, ($a^{-1}$=\,1.73\,(3)\,GeV and $a^{-1}$=\,2.28\,(3)\,GeV). On the coarser lattice, with $24^3\times 64\times 16$ points, the analysis of ref.[1] is extended to approximately twice the number of configurations. The ensembles on the finer $32^3\times 64\times 16$ lattice are new. We explain how we use lattice data obtained at several values of the lattice spacing and for a range of quark masses in combined continuum-chiral fits in order to obtain results in the continuum limit and at physical quark masses. We implement this procedure at two lattice spacings, with unitary pion masses in the approximate range 290--420\,MeV (225--420\,MeV for partially quenched pions). We use the masses of the $\pi$ and $K$ mesons and the $\Omega$ baryon to determine the physical quark masses and the values of the lattice spacing. While our data are consistent with the predictions of NLO SU(2) chiral perturbation theory, they are also consistent with a simple analytic ansatz leading to an inherent uncertainty in how best to perform the chiral extrapolation that we are reluctant to reduce with model-dependent assumptions about higher order corrections. Our main results include $f_\pi=124(2)_{\rm stat}(5)_{\rm syst}$\,MeV, $f_K/f_\pi=1.204(7)(25)$ where $f_K$ is the kaon decay constant, $m_s^{\bar{\textrm{MS}}}(2\,\textrm{GeV})=(96.2\pm 2.7)$\,MeV and $m_{ud}^{\bar{\textrm{MS}}}(2\,\textrm{GeV})=(3.59\pm 0.21)$\,MeV\, ($m_s/m_{ud}=26.8\pm 1.4$) where $m_s$ and $m_{ud}$ are the mass of the strange-quark and the average of the up and down quark masses respectively, $[\Sigma^{\msbar}(2 {\rm GeV})]^{1/3} = 256(6)\; {\rm MeV}$, where $\Sigma$ is the chiral condensate, the Sommer scale $r_0=0.487(9)$\,fm and $r_1=0.333(9)$\,fm., Comment: 129 pages, 59 figures, Published version containing an extended discussion of reweighting and including a new appendix (Appendix C)

We present results for screening masses of mesons built from light and strange quarks in the temperature range of approximately between 140 MeV to 800 MeV. The lattice computations were performed with 2 + 1 dynamical light and strange flavors of improved (p4) staggered fermions along a line of constant physics defined by a pion mass of about 220 MeV and a kaon mass of 500 MeV. The lattices had temporal extents N-tau = 4, 6 and 8 and aspect ratios of N-s/N-tau = 4. At least up to a temperature of 140 MeV the pseudo-scalar screening mass remains almost equal to the corresponding zero temperature pseudo-scalar (pole) mass. At temperatures around 3T(c) (T-c being the transition temperature) the continuum extrapolated pseudo-scalar screening mass approaches very close to the free continuum result of 2 pi T from below. On the other hand, at high temperatures the vector screening mass turns out to be larger than the free continuum value of 2pT. The pseudo-scalar and the vector screening masses do not become degenerate even for a temperature as high as 4T(c). Using these mesonic spatial correlation functions we have also investigated the restoration of chiral symmetry and the effective restoration of the axial symmetry. We have found that the vector and the axial-vector screening correlators become degenerate, indicating chiral symmetry restoration, at a temperature which is consistent with the QCD transition temperature obtained in previous studies. On the other hand, the pseudo-scalar and the scalar screening correlators become degenerate only at temperatures larger than 1.3T(c), indicating that the effective restoration of the axial symmetry takes place at a temperature larger than the QCD transition temperature.

We report on recent simulations of QCD with two flavors of staggered fermions done on the 6.4 Gflop Columbia parallel processor using lattices of size 16 3 × 8. Calculations have been performed with m u a = m d a = 0.004 for values of β between 5.33 and 5.63, using both ordered and disordered starts for each β. We give our results for various observables and consider in some detail the evolution at β = 5.48, where we find results consistent with a two-state signal.

We study the region of the QCD phase transition using $2+1$ flavors of domain wall fermions and a ${16}^{3}\ifmmode\times\else\texttimes\fi{}8$ lattice volume with a fifth dimension of ${L}_{s}=32$. The disconnected light quark chiral susceptibility, quark number susceptibility, and the Polyakov loop suggest a chiral and deconfining crossover transition lying between 155 and 185 MeV for our choice of quark mass and lattice spacing. In this region the lattice scale deduced from the Sommer parameter ${r}_{0}$ is ${a}^{\ensuremath{-}1}\ensuremath{\approx}1.3\text{ }\text{ }\mathrm{GeV}$, the pion mass is $\ensuremath{\approx}300\text{ }\text{ }\mathrm{MeV}$, and the kaon mass is approximately physical. The peak in the chiral susceptibility implies a pseudocritical temperature ${T}_{c}=171(10)(17)\text{ }\text{ }\mathrm{MeV}$ where the first error is associated with determining the peak location and the second with our unphysical light quark mass and nonzero lattice spacing. The effects of residual chiral symmetry breaking on the chiral condensate and disconnected chiral susceptibility are studied using several values of the valence ${L}_{s}$.

As part of our ongoing investigations of QCD with many flavors of quarks, here we report on studies of the finite temperature phase transition for eight-flavor QCD with the DBW2 gauge action and na\"ive staggered fermions. We find a clear first order phase transition between the chirally asymmetric phase at zero temperature and the chirally symmetric phase at finite temperature, signaled by a two-state signal for $\langle\bar{\psi}\psi\rangle$ at a non-zero temperature. We see this signal at a gauge coupling of $\beta=0.54$, where, to set the scale, the zero temperature value for $f_\pi$, in the chiral limit, is 0.06661(92). This strong, first-order signal is seen for two different values of the quark mass, $m_q=0.007$ and 0.0195, at $N_\tau=8$ and 6 respectively. Using $f_\pi(m_q)$ as the scale, the critical temperature is measured to be $T_c/f_\pi=1.638(93)$ at $m_\pi/f_\pi=3.329(30)$ for $m_q=0.007$, and $T_c/f_\pi=1.779(27)$ at $m_\pi/f_\pi=4.093(15)$ for $m_q=0.0195$. At a weaker coupling $\beta=0.56$, where at zero temperature and in the chiral limit we find $f_\pi=0.0312(10)$, the first order signal becomes numerically invisible to us for the $N_\tau \leq 14$ lattices we have investigated so far., Comment: 7 pages, 8 figures, 1 table, Talk presented at The XXVIII International Symposium on Lattice Field Theory, LATTICE 2010 - Villasimius, Sardinia Italy / June 14-19, 2010

QCD with eight flavors is studied on ${16}^{3}\ifmmode\times\else\texttimes\fi{}{N}_{t}$ lattices with ${N}_{t}=4,6, \mathrm{and} 8$, a dynamical quark mass $ma=0.015$, and lattice coupling $\ensuremath{\beta}=\frac{6}{{g}^{2}}$ between 4.5 and 5.0. For ${N}_{t}=16$, hadron masses and screening lengths are computed for a variety of valence quark masses. The previously observed, strong, first-order transition for ${N}_{t}=4,6, \mathrm{and} 8$ is seen, for ${N}_{t}=16$, to become a $\ensuremath{\beta}$-independent, zero-temperature transition characterized by a factor of \ensuremath{\cong} 3 change in lattice scale. This strong, first-order transition restores chiral symmetry, at least for ${N}_{t}=4,6, \mathrm{and} 8$, 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.

We have implemented a procedure, which we call extremization, that deterministically evolves a configuration of lattice field theory towards a solution of the lattice field equations. The solution obtained need not be a local minimum of the action. A Fourier accelerated version of the algorithm is used in three-dimensional SU (2) lattice gauge theory to generate saddle-point solutions of the lattice field equations from Monte Carlo generated lattices. We find that the string tension persists under moderate extremization, during which the average action decreases by a factor of about 100 and the lattices show localized peaks in the action density, as seen in cooling. Continued extremization removes the plateau in the Creutz ratios.

We compare extensive simulations of QCD with 8 and 12 flavors of degenerate quarks, using the DBW2 gauge action, naive staggered fermions, and the rational hybrid Monte Carlo algorithm. A variety of values of the coupling constant, quark mass, and lattice size have been used. Our data suggests that, as the bare coupling is decreased, a rapid cross-over, which dramatically changes the lattice scale, exists with both 8 and 12 flavors. The scale change across this cross-over is much larger with 12 flavors than it is with 8 flavors. All of the observables we have measured, in both the zero and finite temperature systems, are consistent with a chiral symmetry breaking phase for the zero temperature theory on the weak coupling side of the rapid cross-over., Comment: 7 pages, 8 figures, Talk presented at The XXVII International Symposium on Lattice Field Theory, LATTICE 2009 - Peking University, Beijing, China / July 26 - 31, 2009

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 Nt=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 Nt=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, 31 pages, 24 EPS-figures

We use cooling to probe the dynamics of confinement in three-dimensional gauge theory. We first observe the persistence of a plateau in the Creutz ratios under cooling for three-dimensional Georgi-Glashow models with both Wilson and Villain actions and comment on the structure of the cooled lattices. Since cooling moves a lattice configuration closer to a stationary point of the action, averages of an observable over many cooled configurations may be expected to give the leading semiclassical contribution. We explicitly demonstrate this for the three-dimensional Georgi-Glashow model (with Villain action) in the pure U(1) limit and report on the status of current work to extend these results away from this limit.

The quenched chiral condensate is expected to diverge because of the presence of topologically induced zero modes in the spectrum of the Dirac operator. We explain why this effect has not been observed in simulations using the staggered formalism, and show that the domain wall formalism is better suited to the problem. We present a preliminary finding of divergent behavior in quenched domain wall thermodynamics above Tc.

We analyze baryon number, strangeness and electric charge fluctuations as well as their correlations in QCD at high temperature. We present results obtained from lattice calculations performed with an improved staggered fermion action (p4-action) at two values of the lattice cut-off with almost physical up and down quark masses and a physical value for the strange quark mass. We compare these results, with an ideal quark gas at high temperature and a hadron resonance gas model at low temperature. We find that fluctuations and correlations are well described by the former already for temperatures about 1.5 times the transition temperature. At low temperature qualitative features of the lattice results are quite well described by a hadron resonance gas model. Higher order cumulants, which become increasingly sensitive to the light pions, however show deviations from a resonance gas in the vicinity of the transition temperature., 11 pages, 8 figures; revised and published version

We calculate the spatial string tension in (2+1) flavor QCD with physical strange quark mass and almost physical light quark masses using lattices with temporal extent N_tau=4,6 and 8. We compare our results on the spatial string tension with predictions of dimensionally reduced QCD. This suggests that also in the presence of light dynamical quarks dimensional reduction works well down to temperatures 1.5T_c., 8 pages ReVTeX, 4 figures

We have simulated QCD using 2+1 flavors of domain wall quarks on a $(2.74 {\rm fm})^3$ volume with an inverse lattice scale of $a^{-1} = 1.729(28)$ GeV. The up and down (light) quarks are degenerate in our calculations and we have used four values for the ratio of light quark masses to the strange (heavy) quark mass in our simulations: 0.217, 0.350, 0.617 and 0.884. We have measured pseudoscalar meson masses and decay constants, the kaon bag parameter $B_K$ and vector meson couplings. We have used SU(2) chiral perturbation theory, which assumes only the up and down quark masses are small, and SU(3) chiral perturbation theory to extrapolate to the physical values for the light quark masses. While next-to-leading order formulae from both approaches fit our data for light quarks, we find the higher order corrections for SU(3) very large, making such fits unreliable. We also find that SU(3) does not fit our data when the quark masses are near the physical strange quark mass. Thus, we rely on SU(2) chiral perturbation theory for accurate results. We use the masses of the $\Omega$ baryon, and the $\pi$ and $K$ mesons to set the lattice scale and determine the quark masses. We then find $f_\pi = 124.1(3.6)_{\rm stat}(6.9)_{\rm syst} {\rm MeV}$, $f_K = 149.6(3.6)_{\rm stat}(6.3)_{\rm syst} {\rm MeV}$ and $f_K/f_\pi = 1.205(0.018)_{\rm stat}(0.062)_{\rm syst}$. Using non-perturbative renormalization to relate lattice regularized quark masses to RI-MOM masses, and perturbation theory to relate these to $\bar{\rm MS}$ we find $ m_{ud}^{\bar{\rm MS}}(2 {\rm GeV}) = 3.72(0.16)_{\rm stat}(0.33)_{\rm ren}(0.18)_{\rm syst} {\rm MeV}$ and $m_{s}^{\bar{\rm MS}}(2 {\rm GeV}) = 107.3(4.4)_{\rm stat}(9.7)_{\rm ren}(4.9)_{\rm syst} {\rm MeV}$., Comment: 133 pages, 25 figures

K → π and K → 0 weak matrix elements of ∆S = 1 operators have been measured in 2+1 flavor domain wall fermion (DWF) QCD on (3 fm)3 lattices with a−1 = 1.73(3) GeV. As is well known, using these matrix elements and chiral perturbation theory allows a determination of the K → ππ matrix elements that enter in the quantitative value for the ∆I = 1/2 rule and e ′/e . Two light dynamical sea quark masses have been used, along with six valence quark masses, with the lightest valence quark mass ≈ 1/10 the physical strange quark mass. We report our results for lattice matrix elements in the SU(3)L × SU(3)R (27,1), (8,1), and (8,8) representations, paying particular attention to the statistical errors achieved after measurements on 75 configurations. We also report on our calculation of the non-perturbative renormalization coefficients for these ∆S = 1 weak operators, using the Rome-Southampton method.

We present results for the dependence of the residual mass of domain wall fermions (DWF) on the size of the fifth dimension and its relation to the density and localization properties of low-lying eigenvectors of the corresponding hermitian Wilson Dirac operator relevant to simulations of 2+1 flavor domain wall QCD. Using the DBW2 and Iwasaki gauge actions, we generate ensembles of configurations with a $$16^3\times 32$$ space-time volume and an extent of 8 in the fifth dimension for the sea quarks. We demonstrate the existence of a regime where the degree of locality, the size of chiral symmetry breaking and the rate of topology change can be acceptable for inverse lattice spacings $$a^{-1} \ge 1.6$$ GeV.

Configurations of SU(2) lattice gauge theory in three dimensions are generated by Monte Carlo and then cooled. These cooled configurations give area law behavior for Wilson loops (for single configurations and averages over all configurations), are composed of fields which are smooth (in Landau gauge) functions, and are seen to contain localized lumps of large action connected by tails of small action. Further cooling of only the tail regions tends to give well defined plateaus in the Creutz ratios and sometimes leads to plateaus even if the lumps undergo independent global color rotations.

We present a calculation of the renormalization coefficients of the quark bilinear operators and the K-Kbar mixing parameter B_K. The coefficients relating the bare lattice operators to those in the RI/MOM scheme are computed non-perturbatively and then matched perturbatively to the MSbar scheme. The coefficients are calculated on the RBC/UKQCD 2+1 flavor dynamical lattice configurations. Specifically we use a 16^3 x 32 lattice volume, the Iwasaki gauge action at beta=2.13 and domain wall fermions with L_s=16., 80 pages and 33 figures

We summarize the latest results from performing Lattice QCD calculations with two different discretization schemes, asqtad and p4fat3, on the LLNL BG/L supercomputer, and present a plan for continued running to complete the calculation of the QCD transition temperature and equation of state.

We present results on the equation of state in QCD with two light quark flavors and a heavier strange quark. Calculations with improved staggered fermions have been performed on lattices with temporal extent Nt =4 and 6 on a line of constant physics with almost physical quark mass values; the pion mass is about 220 MeV, and the strange quark mass is adjusted to its physical value. High statistics results on large lattices are obtained for bulk thermodynamic observables, i.e. pressure, energy and entropy density, at vanishing quark chemical potential for a wide range of temperatures, 140 MeV < T < 800 MeV. We present a detailed discussion of finite cut-off effects which become particularly significant for temperatures larger than about twice the transition temperature. At these high temperatures we also performed calculations of the trace anomaly on lattices with temporal extent Nt=8. Furthermore, we have performed an extensive analysis of zero temperature observables including the light and strange quark condensates and the static quark potential at zero temperature. These are used to set the temperature scale for thermodynamic observables and to calculate renormalized observables that are sensitive to deconfinement and chiral symmetry restoration and become order parameters in the infinite and zero quark mass limits, respectively., 22 pages, 17 EPS-figures; revised version, updated references, data added in Tab.1, several smaller changes

We present results for light meson masses and pseudoscalar decay constants from the first of a series of lattice calculations with 2+1 dynamical flavors of domain wall fermions and the Iwasaki gauge action. The work reported here was done at a fixed lattice spacing of about 0.12 fm on a 16^3\times32 lattice, which amounts to a spatial volume of (2 fm)^3 in physical units. The number of sites in the fifth dimension is 16, which gives m_{res} = 0.00308(4) in these simulations. Three values of input light sea quark masses, m_l^{sea} \approx 0.85 m_s, 0.59 m_s and 0.33 m_s were used to allow for extrapolations to the physical light quark limit, whilst the heavier sea quark mass was fixed to approximately the physical strange quark mass m_s. The exact rational hybrid Monte Carlo algorithm was used to evaluate the fractional powers of the fermion determinants in the ensemble generation. We have found that f_\pi = 127(4) MeV, f_K = 157(5) MeV and f_K/f_\pi = 1.24(2), where the errors are statistical only, which are in good agreement with the experimental values.