Variance reduction and cluster decomposition

It is a common problem in lattice QCD calculation of the mass of the hadron with an annihilation channel that the signal falls off in time while the noise remains constant. In addition, the disconnected insertion calculation of the three-point function and the calculation of the neutron electric dipole moment with the $\theta$ term suffer from a noise problem due to the $\sqrt{V}$ fluctuation. We identify these problems to have the same origin and the $\sqrt{V}$ problem can be overcome by utilizing the cluster decomposition principle. We demonstrate this by considering the calculations of the glueball mass, the strangeness content in the nucleon, and the $CP$ violation angle in the nucleon due to the $\theta$ term. It is found that for lattices with physical sizes of 4.5–5.5 fm, the statistical errors of these quantities can be reduced by a factor of 3 to 4. The systematic errors can be estimated from the Akaike information criterion. For the strangeness content, we find that the systematic error is of the same size as that of the statistical one when the cluster decomposition principle is utilized. This results in a 2 to 3 times reduction in the overall error.

Figure 1

Nucleon two-point functions at $t=9$ for three different valence quark masses as a function of the cutoff $R$.

Figure 2

The value and error of $C_3/C_2(R,\tau =5,t=10)$ are plotted in the upper panel as a function of $R$. The lower panel displays the three-point function in Eq. (7) as a function of $r$ without summing over it. The green band shows the signal and the blue band the error.

Figure 3

DI calculation for the strange scalar matrix element in the nucleon as a function of $\tau -t/2$. For each of the source-sink separations at 1.00 fm (upper panel) and 1.57 fm (lower panel), two results with a cutoff of $R_s=27$ and $R_s=12$ are plotted.

Figure 4

Scalar (operator $E^2$ and $B^2$) and pseudoscalar ($E·B$) glueball correlators at $t=4$ as a function of cutoff $R$.

Figure 5

The $CP$-violation phase ${\alpha }^1$ calculated on the 48I lattice as a function of cutoff $R$. For each $R$, the value is averaged from $t=6$ to 13.