Strong coupling for planar ${\cal N}=4$ SYM theory: an all-order result

TL;DR

This work determines the all-order strong-coupling structure of the first generalized scaling function $f_1(g)$ in the planar ${\cal N}=4$ SYM $sl(2)$ sector under a double scaling $s\to\infty$, $L\to\infty$ with $j=L/\ln s$ fixed. By recasting the Bethe Ansatz into a nonlinear integral framework and solving the resulting linear system, the authors derive exact strong-coupling expressions for the density coefficients and show that $f_1(g)\doteq -1$ with nonperturbative exponential corrections of order $g^{1/4} e^{-\pi g/\sqrt{2}}$, compatible with a mass-gap interpretation from the ${\cal O}(6)$ sigma model. Numerical solution up to $m=30$ confirms the asymptotic results and reveals the non-asymptotic, exponential contributions, which are fitted to yield $\kappa\approx 2.257$ and suggest $c=1$. A consistent picture emerges with the string side, supporting the interpretation that the nonperturbative corrections encode finite-size string effects and maturing AdS/CFT correspondence beyond the leading scaling function. These results pave the way to analogous treatments of higher generalized scaling functions $f_n(g)$ across all loops.

Abstract

We propose a scheme for determining a generalised scaling function, namely the Sudakov factor in a peculiar double scaling limit for high spin and large twist operators belonging to the $sl(2)$ sector of planar ${\cal N}=4$ SYM. In particular, we perform explicitly the all-order computation at strong 't Hooft coupling regarding the first (contribution to the) generalised scaling function. Moreover, we compare our asymptotic results with the numerical solutions finding a very good agreement and evaluate numerically the non-asymptotic contributions. Eventually, we illustrate the agreement and prediction on the string side.

Figures (1)

• Figure 1: Comparison between numerical solution (dots), fit as in formula (\ref{['f1exp']}) (dark grey line) and integral representation (\ref{['f1int']}) (black line). Detail in the small window: comparison between numerical solution (dots), weak coupling expansion of FRSBFR2 (light grey line) and integral representation (\ref{['f1int']}) (black line).