Resurgence of the Tilted Cusp Anomalous Dimension

TL;DR

This work shows that resurgent extrapolation can reconstruct the analytic structure of the tilted cusp anomalous dimension $\Gamma_a$ from purely perturbative data, enabling accurate interpolation between weak and strong coupling. By analyzing the weak-coupling series and performing advanced Borel techniques on the strong-coupling expansion, the authors identify leading Borel singularities at $\zeta_{leading}=(1-2a)$ and a distant $\zeta=-2$, and reveal a second independent singularity at $\zeta_{new}=(1+2a)$ with an infinite tower of repetitions. Singularity elimination further improves precision, yielding highly accurate Stokes constants and confirming the predicted non-perturbative scales from BES-based insights. The results connect resurgent structures to physical non-perturbative data, substantiating the claim that perturbative expansions encode detailed non-perturbative physics for the tilted cusp. This approach generalizes prior analyses of the usual cusp and demonstrates a robust path to extracting transseries data from perturbative inputs.

Abstract

We use resurgent extrapolation and continuation methods to extract detailed analytic information about the tilted cusp anomalous dimension solely from its weak coupling and strong coupling expansions. This enables accurate and smooth interpolation between the weak and strong coupling limits, and identifies the relevant singularities governing the finite radius of convergence of the weak coupling expansion and the asymptotic nature of the strong coupling expansion. The input data is purely perturbative, generated from the BES equations, and these resurgent methods extract accurate non-perturbative information which matches the underlying physical structure.

Figures (12)

• Figure 1: Padé poles for the weak coupling expansions for the cusp [left] and hex [right], in the complex $g^2$ plane. In each case the poles accumulate to the branch point at $g^2=-\frac{1}{16}$.
• Figure 2: Extrapolation of the weak coupling expansion of the cusp ($a=\frac{1}{4}$) based on 24 terms of the weak coupling expansion. The left-hand plot shows the extrapolation out to $g^2=1$, and the right-hand plot extends out to $g^2=10$. The blue curve is the 24-term weak coupling series expansion, whose breakdown at the radius of convergence, $g^2=\frac{1}{16}$, can be clearly seen. The red curves plot the first 3 terms of the (divergent) strong coupling expansion \ref{['eq:cusp-strong']}. The orange curves are Padé approximants of the 24-term weak coupling series. The black dots show the Padé-Conformal approximation, which extrapolates much further towards strong coupling.
• Figure 3: Extrapolation of the weak coupling expansion of the hex ($a=\frac{1}{3}$) based on 24 terms of the weak coupling expansion. The left-hand plot shows the extrapolation out to $g^2=1$, and the right-hand plot extends out to $g^2=10$. The blue curve is the 24-term weak coupling series expansion, whose breakdown at the radius of convergence, $g^2=\frac{1}{16}$, can be clearly seen. The red curves plot the first 3 terms of the (divergent) strong coupling expansion \ref{['eq:hex-strong']}. The orange curves are Padé approximants of the 24-term weak coupling series. The black dots show the Padé-Conformal approximation, which extrapolates much further towards strong coupling.
• Figure 4: The Darboux test ratios for the cusp [left] and hex [right], showing the ratio of the weak-coupling coefficients to the form in \ref{['eq:darboux']}. The blue dots show the raw ratio; the orange dots and green curve show the 4th and 5th order Richardson acceleration, respectively.
• Figure 5: The ratio on the left-hand-side of \ref{['eq:c-ratio']} for the cusp [left] with $a=\frac{1}{4}$, and the tilted cusp [right] with $a=\frac{1}{8}$, determining the constant $A$ to be as in \ref{['eq:a-value']}. The blue dots show the raw ratio \ref{['eq:c-ratio']}, and the black dots show the 5th order Richardson acceleration, converging rapidly to $1/(1-2a)$.