Machine Learning-Based Estimation of Cumulants of Chiral Condensate via Multi-Ensemble Reweighting with Deborah.jl

TL;DR

The results demonstrate that bias-corrected ML estimates can significantly reduce measurement overhead while preserving the stability of higher-order observables relevant for locating the QCD critical endpoint.

Abstract

We investigate a bias-corrected machine learning (ML) strategy for estimating traces of the inverse Dirac operator, $\text{Tr}\, M^{-n}$ ($n=1,2,3,4$), motivated by the need for higher-order cumulants of the chiral condensate near the finite-temperature QCD critical endpoint. Our supervised regression framework is trained on Wilson-clover ensembles with the Iwasaki gauge action, and we explore two input feature scenarios: one using $\text{Tr}\, M^{-1}$ and another relying solely on gauge observables (plaquette and rectangle), enabling a fully feature-based prediction pipeline. Using $\text{Tr}\, M^{-1}$ both as a physical input to cumulant construction and as a feature for predicting higher powers, we find that even with $\sim1\%$ labeled data, the resulting susceptibility, skewness, and kurtosis remain statistically consistent with fully measured baselines, reducing computational cost to about $26\%$. In the feature-only approach, where correlations rather than explicit stochastic traces drive the predictions, bias correction plays a more pronounced role. We quantify this impact through multi ensemble reweighting across nearby quark masses. Our results demonstrate that bias-corrected ML estimates can significantly reduce measurement overhead while preserving the stability of higher-order observables relevant for locating the QCD critical endpoint. Code for this work is available at https://github.com/saintbenjamin/Deborah.jl .

Figures (3)

• Figure 1: Correlation between physical observables.
• Figure 2: Bhattacharyya coefficient $C_\text{B}$ maps for the kurtosis at the transition point $K(\kappa_t)$ obtained from multi-ensemble reweighting. Panel \ref{['fig:kurt-kurt-heatmap-ML2-LBP-1-25']} shows the results for $\mathcal{P}1$ under the \ref{['it:plaquette-feature']} setup. Red cross marks indicate cells where the Newton--Raphson solver failed to converge within the maximum iteration count, while white diagonal marks indicate cases where convergence was achieved but required more than $10$ Newton iterations. Panel \ref{['fig:kurt-kurt-heatmap-ML1-LBP-1-25']} shows the corresponding $\mathcal{P}1$ results for the \ref{['it:trace-feature']} setup.
• Figure 3: Heatmap showing the $\mathcal{P}1$ results of $x$ and $r$ evaluations, where $x$ and $r$ are defined in Eq. \ref{['eq:x-r-1']}, for the estimation of the kurtosis at the phase transition point, $K(\kappa_t)$, based on the \ref{['it:plaquette-feature']} approach. Within the panel, the left half represents the $x$ results, while the right half corresponds to $r$. In the $x$ maps, lighter shades indicate smaller normalized mean separations $x$, corresponding to better agreement between the original and predicted means, whereas darker shades represent larger $x$ values, signaling poorer overlap. The kurtosis is computed using $\mathrm{Tr}\,M^{-n}$ predicted from plaquette and rectangle as the input feature.