Massless Lüscher Terms and the Limitations of the AdS3 Asymptotic Bethe Ansatz

TL;DR

This work shows that in $AdS_{3}\times S^{3}\times T^{4}$ the massless sector induces Lüscher F-terms that are not exponentially suppressed, unlike in $AdS_{5}\times S^{5}$. By combining worldsheet semiclassical results with an extended Bethe Ansatz framework that includes multiparticle and multi-wrapping massless corrections, the authors explain the persistent one-loop mismatch observed for the $su(2)$ circular string. They derive a generalized massless Lüscher formula, implement it for the circular string, and show that summing all wrappings reproduces the mismatch up to a factor of 2 when using the standard AFS-like mixed-mass dressing phase. The findings indicate that massless wrapping effects must be incorporated into the massive sector's spectral description in $AdS_{3}$ backgrounds, with broader implications for macroscopic massless solutions and related integrable systems.

Abstract

In AdS5/CFT4 integrability the Bethe ansatz gives the spectrum of long strings, accurate up to exponentially small corrections. This is no longer true in AdS3, as we demonstrate here by studying Luscher F-terms with a massless particle running in the loop. We apply this to the classic test of Hernandez & Lopez, in which the su(2) sector Bethe equations (including one-loop dressing phase) should match the semiclassical string theory result for a circular spinning string. These calculations did not agree in AdS3xS3xT4, and we show that the sum of all massless Luscher F-terms can reproduce the difference.

Figures (1)

• Figure 1: On the left, the simplest F-term Feynman diagram. On the right, the generalisations we consider allow a multiparticle physical state ($K=4$ shown) and multiple wrappings by the virtual particle ($n=2$ shown).