Exploring the mirror TBA

TL;DR

This work develops a state-specific excited-state Thermodynamic Bethe Ansatz framework for the AdS$_5\times$S$^5$ mirror model in the $\mathfrak{sl}(2)$ sector, using a contour deformation trick to derive integral equations for two-particle states. It reveals an infinite tower of critical coupling values $g_{cr}^{(m)}$ (with the first Konishi-related value near $g_{cr}^{(1)}\approx4.429$ and $\lambda\approx774$) at which the TBA equations must be modified, and demonstrates how zeros of $Y_{M|vw}$-functions drive these changes; the analysis distinguishes Konishi-like states and generalizes to arbitrary two-particle states. The paper provides both canonical and simplified/ hybrid TBA formulations, relates asymptotic Y-functions to transfer matrices, and discusses the intricate Y-system with jump discontinuities on an infinite-genus Riemann surface, offering new insights into the finite-size spectrum and signaling caution for prior universal-excited-state prescriptions. The results help explain discrepancies between different approaches to Konishi scaling and outline a path to robustly determining finite-size energies across coupling, with implications for higher-particle sectors and strong-coupling behavior.

Abstract

We apply the contour deformation trick to the Thermodynamic Bethe Ansatz equations for the AdS_5 \times S^5 mirror model, and obtain the integral equations determining the energy of two-particle excited states dual to N=4 SYM operators from the sl(2) sector. We show that each state/operator is described by its own set of TBA equations. Moreover, we provide evidence that for each state there are infinitely-many critical values of 't Hooft coupling constant λ, and the excited states integral equations have to be modified each time one crosses one of those. In particular, estimation based on the large L asymptotic solution gives λ\approx 774 for the first critical value corresponding to the Konishi operator. Our results indicate that the related calculations and conclusions of Gromov, Kazakov and Vieira should be interpreted with caution. The phenomenon we discuss might potentially explain the mismatch between their recent computation of the scaling dimension of the Konishi operator and the one done by Roiban and Tseytlin by using the string theory sigma model.

Figures (11)

• Figure 1: These are the mirror and string regions on the $z$-torus. They are in one-to-one correspondence with the $u$-planes. The boundaries of the regions are mapped to the cuts.
• Figure 2: These are the plots of $u$ which solves the BY equation for the Konishi state.
• Figure 3: On the left and right pictures $Y_{1|vw}$, $Y_{2|vw}$ and $Y_{3|vw}$ are plotted for the Konishi state at $\bar{g}_{cr}^{(1)}\approx 4.5$ and $\bar{g}_{cr}^{(2)}\approx 11.5$, respectively. $Y_{2|vw}$ touches the $u$-axis at $g=\bar{g}_{cr}^{(1)}$, and has two real zeros for $\bar{g}_{cr}^{(1)}<g<\bar{g}_{cr}^{(2)}$.
• Figure 4: On the left and right pictures $Y_{1|vw}$, $Y_{2|vw}$, $Y_{3|vw}$ and $Y_{4|vw}$ are plotted for the Konishi state at $\bar{g}_{cr}^{(2)}\approx 11.5$ and $\bar{g}_{cr}^{(3)}\approx 21.6$, respectively. $Y_{1|vw}$ and $Y_{3|vw}$ touch the $u$-axis at $g=\bar{g}_{cr}^{(2)}$, and $Y_{2|vw}$ and $Y_{4|vw}$ touch it at $g=\bar{g}_{cr}^{(3)}$. $Y_{1|vw}$ has four real zeros for $g>\bar{g}_{cr}^{(2)}$.
• Figure 5: On the left picture and right pictures $Y_{2|vw}$, $Y_{3|vw}$, $Y_{4|vw}$ and $Y_{5|vw}$ are plotted for the Konishi state at $\bar{g}_{cr}^{(3)}\approx 21.6$ and $\bar{g}_{cr}^{(4)}\approx 34.9$, respectively. $Y_{2|vw}$ and $Y_{4|vw}$ touch the $u$-axis at $g=\bar{g}_{cr}^{(3)}$, and $Y_{3|vw}$ and $Y_{5|vw}$ touch it at $g=\bar{g}_{cr}^{(4)}$. $Y_{2|vw}$ has four real zeros for $g>\bar{g}_{cr}^{(3)}$.