Constrained Pad\'e Ensembles for Thermal $\mathcal{N}{=}4$ SYM with the Exact $\mathcal O(\lambda^{5/2})$ Coefficient

TL;DR

This paper refines the Padé ensemble approach for thermal $ ext{N}=4$ SYM by incorporating the exact $ ext{O}( ext{lambda}^{5/2})$ coefficient, leading to a more precise and unique interpolation of the theory's thermodynamics.

Contribution

It upgrades the weak-coupling truncation in the Padé ensemble method to the exact $ ext{O}( ext{lambda}^{5/2})$ coefficient, reducing uncertainties and collapsing the admissible set to a single curve.

Findings

Admissible set collapses from 9 to 1 curve.

Pointwise band width drops to zero within numerical resolution.

Hermite-Padé central curve differs from the LSTP survivor after upgrade.

Abstract

We revisit the constrained log-subtracted two-point Pad\'e (LSTP) ensemble for thermal $\mathcal{N}=4$ supersymmetric Yang--Mills (SYM) thermodynamics in four spacetime dimensions after upgrading the weak-coupling truncation from $\mathcal{O}(\lambda^2)$ to the exact $\mathcal{O}(\lambda^{5/2})$ coefficient. We keep the interpolation ansatz unchanged and shift the weak-side matching points to the regime where the new term is numerically significant. The admissible set collapses from $9$ nominal survivors ($3$ distinct curves) to a single distinct curve, the crossover range shrinks to a unique value, and the pointwise band width drops to zero within numerical resolution. The Hermite-Pad\'e (HP) central curve does not coincide with the unique LSTP survivor, so the exact weak-coupling coefficient removes the LSTP scan uncertainty but not the difference between the two routes. The next step is to compute the unknown $\mathcal{O}(\lambda^{-3})$ strong-coupling coefficient.