The Resurgence of the Cusp Anomalous Dimension

TL;DR

This work analyzes the strong-coupling expansion of the cusp anomalous dimension in $\mathcal{N}=4$ SYM, revealing factorial divergence and non-perturbative corrections tied to the $O(6)$ sigma-model mass gap via the BES equation. It develops a resurgent transseries with two non-perturbative sectors, uses Borel–Padé techniques to map the Borel-plane singularities, and verifies large-order relations that fix the Stokes constant, enabling a median-resummed, unambiguous result. The authors demonstrate that the resummed transseries accurately interpolates between strong and weak coupling and encodes the analytic structure of the observable, including Stokes phenomena. They also outline future work to extend the transseries to related observables and to higher non-perturbative orders, addressing resonance and broader analytic features of the BES system.

Abstract

This work addresses the resurgent properties of the cusp anomalous dimension's strong coupling expansion, obtained from the integral Beisert-Eden-Staudacher (BES) equation. This expansion is factorially divergent, and its first nonperturbative corrections are related to the mass gap of the $O(6)$ $σ$-model. The factorial divergence can also be analysed from a resurgence perspective. Building on the work of Basso and Korchemsky, a transseries ansatz for the cusp anomalous dimension is proposed and the corresponding expected large-order behaviour studied. One finds non-perturbative phenomena in both the positive and negative real coupling directions, which need to be included to address the analyticity conditions coming from the BES equation. After checking the resurgence structure of the proposed transseries, it is shown that it naturally leads to an unambiguous resummation procedure, furthermore allowing for a strong/weak coupling interpolation.

Figures (5)

• Figure 1: Poles of the diagonal Borel--Padé approximant of order 100 for the perturbative series of $\Gamma(2g)/(2g)-1$. There accumulation of poles in both positive and negative real directions, starting at $s=A\equiv1/2$ and at $s=-4A=-2$. Note the existence of spurious poles away from the real line: non-stable numerical effects of the Padé method, which move away by choosing different non-diagonal approximants.
• Figure 2: Convergence of the large order ratio of perturbative coefficients to the predicted result related to the first non-perturbative sector. In red is shown the original ratio, and in blue the corresponding Richardson transforms of order 2 (light blue) and 6 (dark blue). In light green the predicted value for the coefficient $c_{10}$ is shown.
• Figure 3: Convergence of the perturbative coefficients to the Stokes constant. In red the original series is shown, and in blue the second Richardson transform with increased convergence. In dark green the calculated value for the Stokes constant is shown.
• Figure 4: Order of magnitude $10^{-\alpha}$ (where $\alpha$ is shown on the y-axis) of the imaginary part of the transseries for the cusp anomalous dimension, when one adds only the perturbative series (dark brown), if one adds also the first non-perturbative correction (light brown) and if one takes all three calculated contributions (light purple).
• Figure 5: Resummation of the transseries result for different values of the coupling, shown in red. For large coupling the behaviour is dictated by the perturbative series alone, as shown by the dashed line -- the truncated summation of the perturbative expansion $\Gamma^{(0)}$, which diverges for small g. In this regime another line is shown in blue, corresponding to the small coupling expansion (up to 7 loops) of the cusp anomalous dimension. The red dots interpolate between these two regimes, starting to diverge around value $g=0.2$.