Sparse modeling study of extracting charmonium spectral functions from lattice QCD at finite temperature

TL;DR

This work addresses the problem of extracting real-time charmonium spectral functions $\rho(\omega)$ from Euclidean lattice QCD correlators $G(\tau)$ by applying sparse modeling (SpM). SpM uses an intermediate representation (IR) basis with SVD-based dimensionality reduction, L1 regularization, and ADMM to enforce sparsity and positivity, while explicitly incorporating the covariance of $G$ to stabilize the inverse problem. Mock-data tests show that SpM can recover resonance structures but struggles to resolve transport peaks without extra modeling or priors, guiding the interpretation of its spectral outputs. Applying SpM to quenched lattice QCD data below and above $T_c$, the authors observe resonance-like peaks around 4 GeV at low temperature that broaden at higher temperature, with transport features remaining elusive; these findings are broadly consistent with MEM analyses, illustrating the method’s potential to capture key physical features while highlighting its limitations in resolving sharp transport structures.

Abstract

We present charmonium spectral functions extracted from Euclidean-time correlation functions using sparse modeling (SpM). SpM solves inverse problems by considering only the sparsity of the target solution. To assess the applicability of the method, we first test it with mock data designed to mimic charmonium correlation functions. We demonstrate that while resonance peaks in the spectral functions can be reconstructed using this method, transport peaks are difficult to resolve without introducing further assumptions beyond sparsity. We then apply the method to charmonium correlation functions obtained from lattice QCD at temperatures below and above the critical temperature. The results are found to be qualitatively consistent with those obtained using the maximum entropy method, although the transport peak is not clearly resolved. This indicates that, even when relying solely on the assumption of sparsity, the method can capture some relevant features of the underlying physics.

Figures (11)

• Figure 1: Spectral functions obtained by using $K^{(0)}_{W}(\hat{\tau}_{i},\hat{\omega}_{j})$ in the mock-data tests for $T<T_{\mathrm{c}}$. Figure (a) shows the results with a fixed noise level of $\varepsilon=5\times 10^{-3}$. The blue dashed, green dotted and red dash-dotted lines represent the output results with $N_{\tau}=48$, 64 and 96, respectively. Figure (b) shows the results with a fixed temporal extent of $N_{\tau}=96$. The blue dashed, green dotted and red dash-dotted lines represent the output results with $\varepsilon=10^{-2}$, $5\times 10^{-3}$ and $10^{-5}$, respectively. In both figures, the black solid line represents the input spectral function.
• Figure 2: Same as Fig. \ref{['fig:spf0_mock-data_tests_rho_bTc']} but obtained by using $K^{(1)}_{W}(\hat{\tau}_{i},\hat{\omega}_{j})$. Figure (a) and (b) show the results with $\varepsilon=5\times 10^{-3}$ and $N_{\tau}=96$, respectively.
• Figure 3: Same as Fig. \ref{['fig:spf0_mock-data_tests_rho_bTc']} but for $T>T_{\mathrm{c}}$. Figure (a) and (b) show the results with $\varepsilon=10^{-2}$ and $N_{\tau}=48$, respectively.
• Figure 4: $\hat{\rho}/(\hat{\omega}\hat{T})$ as a function of $\hat{\omega}$, which is an enlarged view of the plot in Fig. \ref{['fig:spf0_mock-data_tests_rho_aTc']}(b) in the very small value of $\hat{\omega}$. Note that $\hat{T}=N_{\tau}^{-1}$.
• Figure 5: Spectral functions obtained by using $K^{(0)}_{W}(\hat{\tau}_{i},\hat{\omega}_{j})$ in (a) pseudoscalar and (b) vector channels extracted from actual lattice QCD data for $T\simeq 0.73T_{\mathrm{c}}$. The parameters $\mu$ and $\mu^{\prime}$ in ADMM algorithm are set to $10^{2}$. The blue shaded areas represent the statistical errors of the spectral functions from Jackknife analyses, the blue solid lines represent the mean values, and the black horizontal error bars represent the uncertainty of the location of the first peak for each spectral function.