Resurgence of the Cusp Anomalous Dimension

TL;DR

This paper demonstrates that the cusp anomalous dimension ${\Gamma}_{\text{cusp}}(g)$ in planar ${\cal N}=4$ SYM admits a resurgent transseries at strong coupling, with perturbative and non-perturbative sectors interlinked through large-order relations. By solving the Beisert-Eden-Staudacher (BES) equation, the authors extract high-order coefficients and show that Borel resummation ambiguities are exactly canceled by non-perturbative contributions, yielding unambiguous, real results that agree with direct finite-coupling evaluations. A key outcome is the identification of a multi-parameter transseries structure, where non-perturbative scales are tied to the mass gap of the $O(6)$ sigma model and progressively enrich the transseries beyond a single instanton sector. The work also uncovers surprising relations between the cusp function and the mass gap, suggesting deeper structural ties within the AdS/CFT integrability framework and motivating extensions to related observables and theories such as ABJM and Konishi. Overall, the paper provides a concrete, highly nontrivial instance of resurgence in a quantum field theory context and demonstrates how perturbative data encode non-perturbative physics via transseries.

Abstract

We revisit the strong coupling limit of the cusp anomalous dimension in planar N=4 super Yang-Mills theory. It is known that the strong coupling expansion is asymptotic and non-Borel summable. As a consequence, the cusp anomalous dimension receives non-perturbative corrections, and the complete strong coupling expansion should be a resurgent transseries. We reveal that the perturbative and non-perturbative parts in the transseries are closely interrelated. Solving the Beisert-Eden-Staudacher equation systematically, we analyze in detail the large order behavior in the strong coupling perturbative expansion and show that the non-perturbative information is indeed encoded there. An ambiguity of (lateral) Borel resummations of the perturbative expansion is precisely canceled by the contributions from the non-perturbative sectors,

Figures (3)

• Figure 1: The difference between left and right resummation along the singular direction $\theta$ as a sum over Hankel contours.
• Figure 2: We show the singularities of the Padé approximant for the Borel transform $\widetilde{\mathcal{B}}[ \Gamma_\text{cusp}^{(n)} ](\zeta)$ ($n=0,1,2$) in the complex Borel plane. It is obvious to see that the Borel transforms for $n=0,1$ have the singularities at $\zeta=1, -4$, while the Borel transform for $n=2$ has the singularities at $\zeta=\pm 1$.
• Figure 3: (Left) We show the convergence behavior of the original sequence $a_\ell^{(1)}$, its first and fifth Richardson transforms by the blue, purple and red curves, respectively. The black dashed line is the expected convergence value $\mathcal{A}_1$. The Richardson extrapolation accelerates the convergence. (Right) The difference $|1 - \mathcal{R}_n [a_{180-n}^{(1)}]/\mathcal{A}_1|$ is shown as a function of $n$. In this case, the Richardson extrapolation is especially good for the range $20 \lesssim n \lesssim 60$.