Imaginary-part distribution and signal improvement of lattice quantum chromodynamics data

Abstract

Understanding the statistical fluctuations of lattice observables over the gauge configurations is important both theoretically and practically. It provides a physical insight into tackling the famous signal-to-noise problem and the sign problem, and inspires new thoughts in developing methods to improve the signal of lattice calculations. Among many efforts, exploring the relationship between the real part and imaginary part of lattice numerical result is a new method to understand lattice signal and error, because both the real part and imaginary part come from the same sample of gauge field and their distributions on the gauge sample are related in principle. Specifically, by analyzing the distributions of the real part and imaginary part of quenched lattice two-point function with high statistics and non-zero momentum, this work proposes a possible quantitative formula connecting these two distributions as <inline-formula><tex-math id="M1">\begin{document}$R(x)=\displaystyle\int {\rm{d}}y S(y-x) \left[I(y) K(U_y)\right]$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230869_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230869_M1.png"/></alternatives></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$R(x)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230869_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230869_M2.png"/></alternatives></inline-formula> denotes the real-part distribution, <inline-formula><tex-math id="M3">\begin{document}$I(x)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230869_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230869_M3.png"/></alternatives></inline-formula> the imaginary-part distribution, <inline-formula><tex-math id="M4">\begin{document}$S(x)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230869_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230869_M4.png"/></alternatives></inline-formula> the underlying signal distribution and <inline-formula><tex-math id="M5">\begin{document}$K(U_x)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230869_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230869_M5.png"/></alternatives></inline-formula> a kernel function of the gauge field. This theoretical assumption has universal validity because 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 part and the kernel-function-modified imaginary part under the further assumption of the kernel function. It is found that the most naïve guess of <inline-formula><tex-math id="M6">\begin{document}$K(U_x)=1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230869_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230869_M6.png"/></alternatives></inline-formula> does not work, which leads to no statistically significant correlation. Meanwhile, the assumption that <inline-formula><tex-math id="M7">\begin{document}$K(U_x)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230869_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230869_M7.png"/></alternatives></inline-formula> is only a sign function works well, giving rise to <inline-formula><tex-math id="M8">\begin{document}$\sim70\%$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230869_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230869_M8.png"/></alternatives></inline-formula> correlation. Then, through the process of adding random distortions to the absolute values of the imaginary part, it is found that even a slight distortion, of around 1% could result in a significant reduction in the correlation between the real part and imaginary part down to less than 50% or lower. This essentially proves that the observed <inline-formula><tex-math id="M9">\begin{document}$\sim70\%$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230869_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230869_M9.png"/></alternatives></inline-formula> correlation is highly non-trivial and the hypothesis that <inline-formula><tex-math id="M10">\begin{document}$K(U_x)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230869_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230869_M10.png"/></alternatives></inline-formula> is 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 ratio in lattice calculation. Further research on the ability to use machine learning on various more accurate lattice data will hopefully give better instructions and constraint on the form of the kernel function.