Luescher formula for GKP string

TL;DR

The paper develops a hole-based, Lüscher-type framework to compute finite-size corrections to the twist-two anomalous dimensions in planar N=4 SYM by placing the GKP string at the center of the analysis. By reformulating the problem in terms of the long-range Baxter equation and the BES equation, it derives a universal Lüscher formula for excitations propagating on the GKP background and separates the twist-two energy into bulk and finite-size (or alternating) pieces, enabling a quantitative match with known five-loop results. The work further clarifies the weak-to-strong coupling transition, showing that finite-size corrections scale as powers of 1/S at weak coupling and as 1/log S at strong coupling, and demonstrates consistency with the O(6) sigma model description in the strong-coupling regime. Importantly, it establishes an E_FS = E_alt + E_wrap relation, validating that the finite-size effects encode the correct wrapping-like contributions and providing a robust bridge between gauge-theory calculations and string-theory predictions for subleading large-spin corrections.

Abstract

We investigate finite-size corrections to anomalous dimensions of large-spin twist-two operators in the planar maximally supersymmetric Yang-Mills theory. We develop a framework for analysis of these corrections, that is complementary to the conventional spin-chain approach, by making use of the hole rather than the magnon picture. From the dual string theory perspective where the large-spin operator is identified with the Gubser-Klebanov-Polyakov (GKP) string, our approach is equivalent to constructing the first Luescher correction to the energy of the GKP string by incorporating the contribution of virtual excitations propagating on it. It allows us to propose a formula that controls a particular class of large-spin corrections to the twist-two anomalous dimension and holds at any value of the coupling constant. Compared to wrapping corrections computed with magnons propagating on the spin chain, the finite-size corrections that are encoded in our formalism start at a lower loop level. Our formalism thus calls for modification of the asymptotic contributions which are conventionally incorporated within the Asymptotic Bethe Ansatz. An educated guess allows us to remedy this pitfall and successfully confront our predictions with known results up to five loop accuracy at weak coupling. Finally, our formula sheds light on the weak-to-strong coupling transition for the subleading large-spin corrections under study and confirms stringy expectations at strong coupling where they are found to be identical to the first Luescher correction to the vacuum energy of the O(6) sigma model.

Figures (8)

• Figure 1: (a) Vacuum corresponds to twist-two operators. (b) Real one-particle excitation represents twist-three operators. (c) Corrections from finite size of the system. (d) Virtual bulk correction renormalizing the background.
• Figure 2: Singularity structure of solutions $Q_\pm (u)$ of the truncated Baxter equation. The $Q_\pm (u)$ are holomorphic functions in the upper/lower half of the complex plane at leading order of the large spin expansion and possess an infinite number of (blue) cuts for $\Im{\rm m} (u) \lessgtr 0$, respectively. However, they develop (red) cuts in physical region once subleading, i.e., $1/\eta^2$, effects are taken into account.
• Figure 3: Rapidity plane for energy and momentum of a scalar excitation. The double Wick rotation is implemented by the shift $u\rightarrow u+i$ and depicted by the arrow. Composing two such rotations yields the particle to antiparticle transformation $(E, p) \rightarrow (-E, -p)$ related to the crossing symmetry. The 'mirror' line in the lower-half plane is associated to the transformation $(E, p) \rightarrow (-ip, -iE)$. Up to the presence of the cuts, this is the same pattern as the one observed for energy and momentum of a relativistic particle in the $\theta$-plane, with $\theta \equiv \pi u/2$.
• Figure 4: Double Wick rotation for the gauge fields and bound states.
• Figure 5: Double Wick rotation for the fermion in the strong coupling regime.