The Imaginary-Part Distribution of Lattice QCD Data and Signal Improvement

Understanding the statistical fluctuations of lattice observables over the gauge configurations is important both theoretically and practically. It provides physical insights to tackle the famous signal-to-noise problem and the sign problem, and inspires new thoughts in developing methodologies to improve the signal of lattice calculations. Among many efforts, exploring the connections between the real and imaginary parts of lattice numerical results is a novel way to learn about the lattice signal and error, since both the real and imaginary parts originate from the same sampling of gauge fields and their distributions over the gauge samples are in principle related. Specifically, by analyzing the distributions of the real and imaginary parts of quenched lattice two-point functions with high statistics and non-zero momentum, this work proposes a possible quantitative formula connecting these two distributions as R(x) = ∫dyS(y - x) [I(y)K(Uy)], where R(x) stands for the real-part distribution, I(x) the imaginary-part distribution, S(x) the underlying signal distribution and K(Ux) a kernel function of the gauge field. This theoretical assumption is of general validity since the kernel function contains the gauge field information that determines all the distributions. The formula is numerically verified by calculating the non-trivial statistical correlations of the real parts and the kernel-function-modified imaginary parts with further assumptions on the kernel function. It is found that the most naïve guess of K(Ux) = 1 does not work, which leads to no statistically significant correlation. Meanwhile, the assumption that K(Ux) is only a sign function works well, giving rise to ~ 70% correlation. Then, random distortions on the absolute values of the imaginary parts are added with different strength and the results show that even a small distortion, say 1%, would reduce the correlation between the real and imaginary parts down to less than 50%. This essentially proves that the observed ~ 70% correlation is highly non-trivial and the hypothesis of K(Ux) being a sign function captures at least some of the physical mechanisms behind the scenes. Employing this correlation, the variance of lattice results can be improved by around 40%. It is not a significant improvement in practice; however, this study offers an innovative strategy to understand the source of statistical uncertainties in lattice QCD and to improve the signal-to-noise ratios in lattice calculations. Further studies utilizing the power of machine learning on a variety of more precise lattice data will hopefully give better indication and constraint on the form of the kernel function.