Konishi operator at intermediate coupling

TL;DR

This work advances the numerical analysis of the AdS5×S5 spectral problem by solving the excited-state TBA equations for the Konishi descendant in the sl(2) sector up to λ ≈ 2046. It reveals that the Konishi energy is governed by a 1/4-power strong-coupling expansion, with leading coefficients c_{-1} ≈ 2 and c_1 ≈ 2, while c_4 remains nonzero, challenging a pure 1/√λ expansion. The study shows that first critical coupling values lie beyond current computational reach (λ_cr^{(1)} > 5300), and Y_Q contributions remain substantial in the intermediate regime. A key methodological contribution is a new integral representation for the BES dressing phase, significantly speeding up calculations. Collectively, these results refine our understanding of finite-size effects and the analytic structure of Y-functions in the AdS/CFT spectral problem, and they set the stage for more extensive numerical and analytic investigations at strong coupling.

Abstract

TBA equations for two-particle states from the sl(2) sector proposed by Arutyunov, Suzuki and the author are solved numerically for the Konishi operator descendent up to 't Hooft's coupling lambda ~ 2046. The data obtained is used to analyze the properties of Y-functions and address the issue of the existence of the critical values of the coupling. In addition we find a new integral representation for the BES dressing phase which substantially reduces the computational time.

Figures (25)

• Figure 1: Black dots represent the numerical solution of the TBA equations for the Konishi state energy $E_K(\lambda)$. The brown (upper) curve represents the solution of the Bethe-Yang equation, and the red (lower) curve is the graph of $2 \sqrt[4]{{\lambda}}$ which is the large $\lambda$ asymptote of ${E}_K(\lambda)$. The range of the coupling constant is from $g=0.1, \lambda=0.39$ to $g=7.2, \lambda=2046.56$.
• Figure 2: Here black dots, the brown and red curves are the same as in Figure 1, and the blue (upper) curve is the graph of $E_{\rm dis}(w)$ where $w=w(g)$ is the solution of the exact Bethe equations.
• Figure 3: This is the graph of the contribution of $E_{\rm Y}$ to the energy. It obviously shows a linear growth starting already with $g\sim 2$.
• Figure 4: The black dots represent our numerical solution of the exact Bethe equation, and the brown curve is the graph of the corresponding solution of the Bethe-Yang equation. The exact Bethe root $w(g)$ reaches its minimum at $g\approx 1$, and $g\approx 1.6$ is the inflection point.
• Figure 5: The graphs of the derivative of $E_{\rm dis}$ and $E_{\rm Y}$ with respect to $g$.