On Quantum Corrections to Spinning Strings and Bethe Equations

TL;DR

We study one-loop energy shifts of spinning strings on $AdS_5\times S^5$ and their relation to quantum string Bethe equations. By carefully treating divergent sums and combining small-$n$ and large-$n$ analyses, we obtain the $1/\mathcal{J}$-expansion that reproduces the known even terms and uncovers new odd-power contributions starting at $\mathcal{O}(1/\mathcal{J}^5)$, which are shown to be captured by a quantum-corrected Bethe ansatz via a dressing phase. Introducing an interpolating function for the dressing phase, notably $c_2(\lambda)=1-\frac{16}{3\sqrt{\lambda}}+\cdots$, the authors demonstrate that these odd terms are consistent with a smooth interpolation between strong and weak coupling, potentially resolving the traditional three-loop string/gauge discrepancy as an order-of-limits effect. The results suggest a universal mechanism whereby quantum corrections to the Bethe equations account for these effects across sectors and near the plane-wave limit, and outline implications for finite-size and wrapping corrections as well as finite-$\lambda$-behavior of the dressing phase.

Abstract

Recently, it was demonstrated that one-loop energy shifts of spinning superstrings on AdS5xS5 agree with certain Bethe equations for quantum strings at small effective coupling. However, the string result required artificial regularization by zeta-function. Here we show that this matching is indeed correct up to fourth order in effective coupling; beyond, we find new contributions at odd powers. We show that these are reproduced by quantum corrections within the Bethe ansatz. They might also identify the "three-loop discrepancy" between string and gauge theory as an order-of-limits effect.