Exploring the Critical Points in QCD with Multi-Point Padé and Machine Learning Techniques in (2+1)-flavor QCD

Aug 4, 2026

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Using lattice QCD simulations at multiple imaginary chemical potentials for (2+1)-flavor QCD, the work constructs multi-point Padé approximants and analyzes their singularities. The extracted pole structure is checked against universal scaling expectations for Lee-Yang edge singularities, including behavior consistent with the vicinity of the QCD critical endpoint. A Masked Autoregressive Density Estimator (MADE) is then used to model the temperature-dependent probability density of the singularities and interpolate between temperatures. Finally, a scaling ansatz enables extrapolation to estimate the QCD critical point, with preliminary values compatible with existing model and lattice results.

LanguageEnglish

CategoryResearch & Report

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Pages4

Reading Time10 min

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Introduction

LYEs and related scaling of the QCD critical point

Machine learning model for the LYEs and extrapolation to the critical point

arXiv :2401 .05651v1 [hep-lat] 11 Jan 2024 Exploring the Critical Points in QCD with Multi-Point Padé and Machine Learning Techniques in (2+1)-flavor QCD Jishnu Goswami 1 , ∗ , D. A. Clarke2 , P. Dimopoulos2 , F. Di Renzo3 , C. Schmidt4 , S. Singh4 , and K. Zambello5 1RIKEN Center for Computational Science, Kobe 650-0047, Japan 2Department of Physics and Astronomy, University of Utah, Salt Lake City, Utah 84112, United States 3Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma and INFN, Gruppo Collegato di Parma I-43100 Parma, Italy 4Fakultät für Physik, Universität Bielefeld, D-33615 Bielefeld, Germany 5Dipartimento di Fisica dell’Università di Pisa and INFN–Sezione di Pisa, Largo Pontecorvo 3, I-56127 Pisa, Italy. Abstract. Using simulations at multiple imaginary chemical potentials for (2 + 1)-flavor QCD, we construct multi-point Padé approximants. We determine the singularties of the Padé approximants and demonstrate that they are consistent with the expected universal scaling behaviour of the Lee-Yang edge singularities. We also use a machine learning model, Masked Autoregressive Density Estimator (MADE), to estimate the density of the Lee-Yang edge singularities at each temperature. This ML model allows us to interpolate between the temperatures. Finally, we extrapolate to the QCD critical point using an appropriate scaling ansatz. 1 Introduction Our ability to predict thermodynamic observables and determine the QCD critical point at real values of chemical potentials is severely limited by the infamous sign problem. To address this issue, there are two common approaches: expanding the QCD partition function ina Taylor series with respect to the charge chemical potentials (µB , µQ , µS ) [1, 2] or analytically continuing from imaginary chemical potentials [3, 4] . However, both methods have limitations, particularly for larger values of the baryon chemical potential. Recently, we proposed a multi-point Padé approach [5, 6] . In this proceeding, we will focus on the QCD critical point, which will emerge at a real value of the baryon chemical potential. We use the multi-point Padé approach to locate the Lee-Yang edge (LYE) singularities in the complex chemical potential plane, which are obtained from lattice QCD simulated data. The universal scaling of singularities in the vicinity of the QCD critical endpoint is also investigated. LYEs associated with the QCD critical point at real chemical potential for various temperatures are calculated. As the temperature decreases, the imaginary part of the singularities becomes smaller, hinting at the possible existence of a critical point at low temperature. A machine learning technique is used to model the probability density of the singularities and interpolate the real and imaginary parts of the ∗ e-mail: [jishnu.goswami@riken.jp](jishnu.goswami@riken.jp) Figure 1. Distribution of the poles at different temperature for 363 × 6 lattice. Figure 2. A schematic diagram of the MADE block. singularities between different temperatures. By employing a suitable scaling ansatz, the singularities are extrapolated towards the real axis, and the possible location of the QCD critical point is estimated. Preliminary results of (TCEPc, µCEPB) are consistent with model predictions and other lattice QCD calculations [7–9] . Recently, there are also other estimates for the QCD critical point from LYEs [10, 11] . In the following sections we present the LYEs obtained for (2+1)-flavor QCD at each temperature. We use two methods, one based on a universal scaling ansatz and one based on a machine learning ansatz, to interpolate the data between the temperatures. Finally, we extrapolate the LYEs to the QCD critical point. We performed simulations at four temperature values for the lattice size 363 × 6 with multiple imaginary chemical potential values. The temperature scales are chosen according to the parametrization given in [7] in the fK s

FAQs

Why are imaginary chemical potentials used in this study?

The approach circumvents the sign problem by relying on simulations at imaginary chemical potentials, enabling the reconstruction of relevant singularity information before extrapolating to real chemical potentials.

How are Lee-Yang edge singularities located and analyzed?

Multi-point Padé approximants are built from lattice data, and their poles/singularities are determined. The resulting behavior is compared with universal scaling expectations for Lee-Yang edge singularities near the critical endpoint.

What role does the MADE model play in the workflow?

MADE estimates the probability density of Lee-Yang edge singularities at each temperature, allowing interpolation of their real and imaginary parts across temperatures.

How is the QCD critical point estimated from the obtained singularities?

After interpolating temperature dependence using the machine-learning model, an appropriate scaling ansatz extrapolates the singularities toward the real axis to infer the critical point location.

Aug 4, 2026

Upload by Eliana