The Bajnok-Janik formula and wrapping corrections

TL;DR

This work demonstrates that, for minimal energy twist-two operators in the $sl(2)$ sector of $AdS_5 \times S^5$, the wrapping corrections from the linearized TBA equations precisely reproduce the leading generalized Lüscher (Bajnok-Janik) corrections at order $O(g^8)$. By recasting the TBA into a linear problem around the asymptotic solution and focusing on the $M|vw$-string sector, the authors derive the BJ wrapping term in the form $\delta\mathcal{R}_k = \frac{1}{2\pi} \sum_{m=1}^\infty \int du\, \partial_k Y_m^{o}(u)$, with the key identification $Y_m^{o}(u) = j_m(u)$. The results establish a robust cross-check between the TBA/nlIE framework and generalized Lüscher corrections for finite-size effects, and they generalize to other symmetric states and, via Appendix C, to relativistic integrable models. Overall, the paper reinforces the BJ formula as the leading finite-size correction across a broad class of integrable theories and strengthens confidence in the AdS/CFT finite-volume spectrum program.

Abstract

We write down the simplified TBA equations of the $AdS_5 \times S^5$ string sigma-model for minimal energy twist-two operators in the sl(2) sector of the model. By using the linearized version of these TBA equations it is shown that the wrapping corrected Bethe equations for these states are identical, up to O(g^8), to the Bethe equations calculated in the generalized Lüscher approach (Bajnok-Janik formula). Applications of the Bajnok-Janik formula to relativistic integrable models, the nonlinear O(n) sigma models for n=2,3,4 and the SU(n) principal sigma models, are also discussed.