Spectral reconstruction based on dimensional reduction in high-temperature gauge theories

TL;DR

The work tackles the ill-posed problem of extracting real-time spectral information from Euclidean lattice QCD at high temperature by embedding a semi-classical real-time dynamics picture into standard lattice simulations through high-temperature dimensional reduction and static projection. It introduces a static-projection formalism that renders a computable $S^s(w)$ from a background of time-averaged gauge fields and then refines the full spectral function by reconstructing the difference with $G_E(\tau)$ using maximum-entropy-like methods, thereby achieving superior frequency resolution. Validation is performed in a solvable $(1+1)$-D $U(1)$ gauge theory with fermions, where $S^s(w)$ closely reproduces the full quantum $S(w)$ in the high-$T$, weak-coupling regime, and in high-temperature lattice QCD with Wilson-Dirac dynamics, where the approach yields physically sensible features such as a transport peak and rising high-$w$ behavior in vector channels, aided by quantum-typical stochastic evaluation and polynomial approximations. The method preserves non-perturbative lattice benefits while avoiding full real-time gauge evolution, offering a practical, scalable route to transport coefficients and spectral structures in hot QCD with potential extensions to higher Matsubara modes and bosonic sectors.

Abstract

We propose a numerical spectral reconstruction workflow for high-temperature gauge theories that incorporates elements of semi-classical real-time evolution directly into standard lattice QCD simulations via high-temperature dimensional reduction, thus counteracting the deterioration of Euclidean-time correlators at high temperatures. With a moderate numerical cost, our method allows to estimate spectral functions with parametrically better frequency resolution as compared with spectral reconstruction methods based on Euclidean-time correlators alone. The method is tested on a simple (1+1)-dimensional Abelian gauge theory with fermions, where our method precisely reproduces the full quantum spectral functions calculated using exact numerical diagonalization in the high-temperature, weak-coupling regime. We also demonstrate the feasibility of our approach by applying it to light-quark meson correlators in lattice QCD deep in the deconfinement regime.

Figures (8)

• Figure 1: Ratios of Euclidean-time correlators $G_E \left( \tau \right)$ of the operator $\hat{\Delta} = \sum_f q_f \, \left( \hat{\psi}^{\dag}_{f,1} \hat{\psi}_{f,1} - \hat{\psi}^{\dag}_{f,0} \hat{\psi}_{f,0} \right)$ and their static approximations $G^s_E \left( \tau \right)$ at different couplings $g$ and temperatures $T$ on the lattice with $L_s = 5$.
• Figure 2: Comparison between the full spectral functions $S \left( w \right)$ calculated from numerical exact diagonalization of the Hamiltonian (\ref{['eq:u1_Hamiltonian']}) (red line and dashed blue line for different values of electric flux cut-off $N_m$, see SM for algorithm details) and their static approximations $S^s \left( w \right)$ calculated in QMC simulations (green line) using (\ref{['eq:spectral_function_static']}). We also show numerical estimates of $S \left( w \right)$ obtained from MaxEnt method with a constant model function (grey band/line), and with $S^s \left( w \right)$ as a model function (magenta band/line). The inset shows retarded correlators $G_R \left( t \right)$ (red solid lines) and static approximations thereof (green dashed lines).
• Figure 3: Ratios of Euclidean vector-vector correlators calculated on full and static gauge field configurations at different temperatures. For comparison, dashed lines show ratios between free quark correlators and the full lattice QCD data.
• Figure 4: Static approximation $S^s \left( w \right)$ to the QCD spectral function in the vector meson channel. We also show the output of the MaxEnt method on the full lattice QCD data with $S^s \left( w \right)$ and a constant function as a model function (prior).
• Figure 5: Raw data for the full quantum Euclidean-time meson correlators with different quantum numbers and for different temperatures. The points with $\tau=0$ are not shown because of contact term divergences.