Field sparsening for the construction of the correlation functions in lattice QCD

Two field-sparsening methods, namely the sparse-grid method and the random field selection method, are used in this paper for the construction of the 2-point and 3-point correlation functions in lattice QCD. We argue that, due to the high correlation among the lattice correlators at different field points associated with source, current, and sink locations, one can save a lot of computational time by performing the summation over a subset of the lattice sites. Furthermore, with this strategy, one only needs to store a small fraction of the full quark propagators. It is found that the number of field points can be reduced by a factor of $\sim 100$ for the point-source operator and a factor of $\sim 1000$ for the Gaussian-smeared operator, while the uncertainties of the correlators only increase by $\sim 15\%$. Therefore, with a modest cost of the computational resources, one can approach the precision of the all-to-all correlators using the field-sparsening methods.

Figure 1

Effective masses for the pion (left panel) and the proton (right panel) with full set ${\mathrm{\Lambda }}_{\text{full}}$ (blue), sparse-grid (dark green) and random-field selection (cyan). For the two field-sparsening methods, we use $N_{\mathrm{\Lambda }}=2^3$. The green and cyan data points are shifted horizontally for an easier comparison.

Figure 2

Effective masses for the pion (left panel) and proton (right panel) from the correlated fit as a function of $N_{\mathrm{th}}$. For each $N_{\mathrm{th}}$, the number of field points is given in Eq. (6). The data points from sparse-grid method and random field selection are printed in blue and green color, respectively.

Figure 3

Effective masses for the pion and proton from the smeared-source smeared-sink (s-s) and point-source point-sink (p-p) correlation functions. In the left panel, the effective masses as a function of $N_{\mathrm{th}}$ are shown. Here we use the random field selection. In the right panel, the ratio between the statistical uncertainty and the effective mass $m_{\alpha }^{\text{full}}$ for $\alpha =\pi$ and $p$ is shown.

Figure 4

Lattice results of $C(r,t)$ at $t=10$ together with a fit to Eq. (17).

Figure 5

Uncertainties for the three types of 2-point functions at $t=10$, 15 and 20. The blue bar indicates the uncertainty of the correlation function with random field selection for $N_{\mathrm{\Lambda }}=1$. The orange bar shows the uncertainty for the data using ${\mathrm{\Lambda }}_{\text{full}}$. The green bar shows the ${\delta }_{\text{rand}}$ only.

Figure 6

The left panel shows the ratio between 3-point and 2-point functions, $C^{3\mathrm{pt}}/C^{2\mathrm{pt}}$, as a function of $N_{{\mathrm{\Lambda }}_s}$ (the $x$-axis is labeled by $N_{\mathrm{th}}$). The 3-point functions are constructed using the sequential-source propagators. The time separation $t_i$ and $t_s$ are fixed as $t_i=5$ and $t_s=10$. In the right panel, the uncertainty of the ratio as a function of $N_{\mathrm{th}}$ is shown.

Figure 7

The left panel shows the results of $C^{3\mathrm{pt}}/C^{2\mathrm{pt}}$ as a function of $t_i-t_s/2$, where $t_s$ is chosen to be 10. The right panel shows the uncertainty of the ratio (with $t_s=10$ and $t_i-t_s/2=0$) as a function of $N_{{\mathrm{\Lambda }}_0}=N_{{\mathrm{\Lambda }}_s}$. In these two plots, we set $N_{\mathrm{\Lambda }}=L^3$.

Figure 8

The ratio of $C^{3\mathrm{pt}}/C^{2\mathrm{pt}}$ as a function of $t_i-t_s/2$ together with a two-state fit. The shadowed band indicates the result of $g_A$ obtained from the fit. We use $N_{{\mathrm{\Lambda }}_s}=N_{{\mathrm{\Lambda }}_0}=32$, $N_{\mathrm{\Lambda }}=L^3$, and $N_{{\mathrm{\Lambda }}_t}=24$.

Figure 9

The proton axial charge $g_A$ as a function of $N_{\mathrm{th}}$. Here $N_{\mathrm{th}}$ is related to the number of field points at the location of current insertion. We use $N_{{\mathrm{\Lambda }}_s}=N_{{\mathrm{\Lambda }}_0}=32$ and $N_{{\mathrm{\Lambda }}_t}=24$.