Maximum Entropy Analysis of the Spectral Functions in Lattice QCD

TL;DR

The paper surveys a first-principles approach to QCD spectral functions extracted from lattice data using Maximum Entropy Method (MEM). Framed within Bayesian inference and utilizing Shannon-Jaynes entropy, MEM provides a nonparametric way to recover A(ω) from imaginary-time data D(τ) via the kernel K(τ,ω), with a provable path to unique solutions and quantitative error estimates. The authors validate MEM with mock data and apply it to quenched lattice simulations, successfully uncovering low-energy resonances (like the π and ρ) and the high-energy continuum, while discussing the impact of lattice kernels, discretization, and default-model choices on the results. The work demonstrates MEM as a powerful tool for studying hadronic properties in QCD beyond conventional lattice analyses and outlines future directions for finite-T studies, excited states, and more complex channels.

Abstract

First principle calculation of the QCD spectral functions (SPFs) based on the lattice QCD simulations is reviewed. Special emphasis is placed on the Bayesian inference theory and the Maximum Entropy Method (MEM), which is a useful tool to extract SPFs from the imaginary-time correlation functions numerically obtained by the Monte Carlo method. Three important aspects of MEM are (i) it does not require a priori assumptions or parametrizations of SPFs, (ii) for given data, a unique solution is obtained if it exists, and (iii) the statistical significance of the solution can be quantitatively analyzed. The ability of MEM is explicitly demonstrated by using mock data as well as lattice QCD data. When applied to lattice data, MEM correctly reproduces the low-energy resonances and shows the existence of high-energy continuum in hadronic correlation functions. This opens up various possibilities for studying hadronic properties in QCD beyond the conventional way of analyzing the lattice data. Future problems to be studied by MEM in lattice QCD are also summarized.

Figures (17)

• Figure 1: Input SPF with two Gaussian peaks (the dashed lines) and output SPF obtained by MEM (the solid lines) for different values of data-points $N$ and noise level $b$.
• Figure 2: Probability distribution $P[\alpha|DHm]$ for three different sets of $(N, b)$ in the case of Schematic SPF.
• Figure 3: Mock data $D_{mock}(\tau_i)$ in the case of the realistic SPF with Gaussian error attached.
• Figure 4: Input SPF with a resonance + continuum (the dashed lines) and output SPF obtained by MEM (the solid lines) for different values of $N$ and $b$.
• Figure 5: Probability distribution $P[\alpha|DHm]$ for three different sets of $(N, b)$ in the case of the realistic SPF.