Massless Lüscher terms and the limitations of the ${\mathrm{AdS}}_3$ asymptotic Bethe ansatz

In ${\mathrm{AdS}}_5/{\mathrm{CFT}}_4$ integrability the Bethe ansatz gives the spectrum of long strings, accurate up to exponentially small corrections. This is no longer true in three-dimensional anti–de Sitter (${\mathrm{AdS}}_3$) space, as we demonstrate here by studying Lüscher F-terms with a massless particle running in the loop. We apply this to the classic test of Hernández and López, in which the $su(2)$ sector Bethe equations (including the one-loop dressing phase) should match the semiclassical string theory result for a circular spinning string. These calculations do not agree in ${\mathrm{AdS}}_3\times S^3\times T^4$, and we show that the sum of all massless Lüscher F-terms can reproduce the difference.

Figure 1

(Left panel) The simplest F-term Feynman diagram. (Right panel) The generalizations we consider allow a multiparticle physical state ($K=4$ shown) and multiple wrappings by the virtual particle ($n=2$ shown).