In this paper, we perform the first application of the hybrid method (exact low modes plus stochastically estimated high modes) for all-to-all propagators to the HAL QCD method. We calculate the HAL QCD potentials in the I = 2 ππ scattering in order to see how statistical fluctuations of the potential behave under the hybrid method. All of the calculations are performed with the 2+1 flavor gauge configurations on a 16 3 × 32 lattice at the lattice spacing a ≈ 0.12 fm and m π ≈ 870 MeV. It is revealed that statistical errors for the potential are enhanced by stochastic noises introduced by the hybrid method, which, however, are shown to be reduced by increasing the level of dilutions, in particular, that of space dilutions. From systematic studies, we obtain a guiding principle for a choice of dilution types/levels and a number of eigenvectors to reduce noise contamination to the potential while keeping numerical costs reasonable. We also confirm that we can obtain the scattering phase shifts for the I = 2 ππ system by the hybrid method within a reasonable numerical cost; these phase shifts are consistent with the result obtained with the conventional method. The knowledge that we obtain in this study will become useful for the investgation of hadron resonances that require quark annihilation diagrams such as the ρ meson by the HAL QCD potential with the hybrid method. arXiv:1904.09549v2 [hep-lat]
Bridge++ is a general-purpose code set for a numerical simulation of lattice QCD aiming at a readable, extensible, and portable code while keeping practically high performance. The previous version of Bridge++ is implemented in double precision with a fixed data layout. To exploit the high arithmetic capability of new processor architecture, we extend the Bridge++ code so that optimized code is available as a new branch, i.e., an alternative to the original code. This paper explains our strategy of implementation and displays application examples to the following architectures and systems: Intel AVX-512 on Xeon Phi Knights Landing, Arm A64FX-SVE on Fujitsu A64FX (Fugaku), NEC SX-Aurora TSUBASA, and GPU cluster with NVIDIA V100.
The approximated partial wave decomposition method to the discrete data on a cubic lattice, developed by C. W. Misner, is applied to the calculation of S-wave hadron-hadron scatterings by the HAL QCD method in lattice QCD. We consider the Nambu-Bethe-Salpeter (NBS) wave function for the spin-singlet Λ c N system calculated in the (2 + 1)-flavor QCD on a (32a fm) 3 lattice at the lattice spacing a 0.0907 fm and m π 700 MeV. We find that the l = 0 component can be successfully extracted by Misner's method from the NBS wave function projected to A + 1 representation of the cubic group, which contains small l ≥ 4 components. Furthermore, while the higher partial wave components are enhanced so as to produce significant comb-like structures in the conventional HAL QCD potential if the Laplacian applied to the NBS wave function is approximated by the usual second order difference, such structures are found to be absent in the potential extracted by Misner's method, where the Laplacian can be evaluated analytically. Despite the difference in the potentials, two methods give almost identical results on the central values and on the magnitude of statistical errors for the fits of the potentials, and consequently on the scattering phase shifts. This indicates that not only that Misner's method works well in lattice QCD but also the contaminations from higher partial waves in the study of S-wave scatterings are well under control even in the conventional HAL QCD method. It will be of interest to study interactions in higher partial wave channels in Misner's method, where the utility of this method may become clearer.
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