${\bf I=2} \boldsymbol{\pi\pi}$ potential in the HAL QCD method with all-to-all propagators

Abstract

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]

Glueball scattering cross section in lattice SU(2) Yang-Mills theory

“…We expect the lattice calculation of the interglueball scattering to yield a quantitative relation between the cross section and the unknown scale parameter Λ of the dark YMT, which will be bounded by the observational data. In this work, we use the HAL QCD method [432][433][434][435][436][437][438][439][440][441][442][443][444][445][446][447][448], which is quite successful in the determination of the interhadron potential on lattice, to quantify the glueball scattering.…”

Study of the pion-pion scatterings with a combination of all-to-all propagators and the HAL QCD method

“…We first employ the setup of the case0 in Tab.1, same to the "case3" in Ref. [6], which gave the sufficiently precise I = 2 ππ potential. The result of the I = 1 ππ potential is shown in Figure 2(left), where we observe extremely large statistical fluctuations.…”

“…See Ref. [6] for more details.) Statistical errors from N conf = 60 are estimated by the jackknife method with bin-size 6.…”

“…We have also systematically studied how the statistical fluctuations of the potentials are affected by the choice of the parameters of the hybrid method. For more details, see [6].…”

“…Under such circumstances, we have been testing the hybrid method [5], which evaluates all-to-all quark propagators by the spectral decomposition for the low-mode part plus the stochastic estimation for the high-mode part. In this paper, we will report two results, the I = 2 S-wave ππ scattering study [6], which is intended to study how the statistical errors of the potential behaves with the hybrid method, and the I = 1 P-wave ππ scattering study, which is a preparatory research for future ρ resonance studies. From these results, we confirmed that the hybrid method works well in the HAL QCD method as far as we reduce additional statistical fluctuations due to the stochastic estimation in the hybrid method.…”