A new derivation of Luscher F-term and fluctuations around the giant magnon

TL;DR

The paper derives a generalized Luscher $F$-term from a summation over quadratic fluctuations around a soliton, valid for diagonal scattering and arbitrary dispersion $\varepsilon(p)$, and demonstrates its equivalence to a Poisson-resummed fluctuation sum. It then applies this formalism to the giant magnon to compute the leading quantum finite-size correction, yielding an explicit $F$-term expression with a characteristic $e^{-2\pi J/\sqrt{\lambda}}$ suppression and a $p$-dependent prefactor. This work unifies fluctuation-based energy shifts with Luscher-type finite-size formulas and clarifies the role of stability angles in moving versus stationary particles. The results provide a robust framework for 1-loop finite-size corrections in integrable AdS/CFT setups and offer insights for connecting to semiclassical and WKB analyses.

Abstract

In this paper we give a new derivation of the generalized Luscher F-term formula from a summation over quadratic fluctuations around a given soliton. The result is very general providing that S-matrix is diagonal and is valid for arbitrary dispersion relation. We then apply this formalism to compute the leading finite size corrections to the giant magnon dispersion relation coming from quantum fluctuations.