Constrained Padé Ensembles for Thermal N=4 SYM: Quantified Uncertainties and Next-Order Predictions

TL;DR

The paper develops an admissible ensemble of log-aware Padé approximants to interpolate the thermal ${\rm N}=4$ SYM equation of state between weak and strong coupling, replacing single-curve estimates with a central curve and a quantified uncertainty band for $f(\\lambda)$. Two independent routes, Hermite-Padé (HP) and log-subtracted Padé (LSTP), are employed to enforce the known weak- and strong-coupling expansions, including nonanalytic terms, while a set of admissibility filters yields a pole-free, bounded, monotone band. The central results include a crossover at ${\\lambda_c^{\mathrm{center}}}\approx 3.52$ with ${f(\\lambda_c)}\approx 0.854$ and a crossover window ${\\lambda_c}\in[2.95,6.73]$, plus model-independent predictions ${A_{5/2}=0.476\pm0.095}$ and ${S_{3}\in[-71.27,262.06]}$; the method is extended to transport observables, notably ${\\eta/s}$, ${\\hat{q}/T^3}$, and ${2\\pi TD_s}$, yielding consistent, testable predictions and curvature-based crossover markers. The framework provides a robust, falsifiable interpolation between weak and strong coupling with potential applications to QCD and other gauge theories, including joint constraints across multiple observables to sharpen intermediate-coupling inferences.

Abstract

We quantify the transition between weak and strong coupling in thermal $\mathcal N=4$ supersymmetric Yang-Mills (SYM) theory in four space-time dimensions by constructing an \emph{admissible ensemble} of log-aware Padé approximants that exactly reproduce the weak- and strong-coupling expansions through $\mathcal O(λ^2)$ and $\mathcal O(λ^{-3/2})$ (where $λ$ is the 't Hooft coupling), including the nonanalytic $λ^{3/2}$ and $λ^{2}\logλ$ terms. This replaces single-curve estimates with a reproducible uncertainty band and a well-defined central curve across the intermediate regime. Applying the same construction to transport, the $η/s$ band connects perturbative behavior to the Kovtun-Son-Starinets limit. The framework is \emph{predictive}, yielding $A_{5/2}=0.476\pm0.095$ on the weak side and a model-independent bound on the next strong-coupling term, thereby setting testable benchmarks for forthcoming perturbative and holographic calculations.

Figures (5)

• Figure 1: Admissible Padé band for $f(\lambda)=\mathcal{S}/\mathcal{S}_0$ in $\mathcal{N}=4~\mathrm{SYM}$. Shaded: band; solid: central curve. Also shown are the weak truncations $\mathcal{O}(\lambda)$, $\mathcal{O}(\lambda^{3/2})$, $\mathcal{O}(\lambda^{2})$ (including the exact $\lambda^{2}\log\lambda$ term) and the strong truncation $\mathcal{O}(\lambda^{-3/2})$.
• Figure 2: All admissible individual curves (HP and LSTP) overlaid, together with the weak $\mathcal{O}(\lambda^{2})$ and strong $\mathcal{O}(\lambda^{-3/2})$ truncations. The spread defines the admissible band.
• Figure 3: (a) Admissible Padé ensemble for $\eta/s$ in $\mathcal{N}{=}4$ SYM (log--log axes). (b) Central curve with perturbative and holographic asymptotes. (c) Curvature diagnostic: peak of $d^2\ln(\eta/s)/d(\ln\lambda)^2$ at $\lambda_c\simeq4.81$.
• Figure 4: (left) Admissible ensemble for $\hat{q}/T^3$ in $\mathcal{N}{=}4$ SYM (log–log axes). Shaded band: scan over $q_{\max}/T\in\{6,8,10,12,15\}$; solid: central ($q_{\max}/T{=}10$). (right) Curvature diagnostic for $\hat{q}/T^3$: extremum of $d^2\ln(\hat{q}/T^3)/d(\ln\lambda)^2$ gives $\lambda_c\simeq 4.36$; $\lambda_\pm$ are half–depth crossings.
• Figure 5: (left) Admissible ensemble for $2\pi T D_s$ (log–log axes). Shaded band: bump–modulated inverse–harmonic interpolation preserving both asymptotes; solid: central (baseline) curve; dashed: weak asymptote; dotted: strong asymptote $4/\sqrt{\lambda}$. (right) Curvature diagnostic for $2\pi T D_s$: peak at $\lambda_c\simeq 11.88$; $\lambda_\pm$ from half–depth.