Finite-Size Effects for Dyonic Giant Magnons

TL;DR

This work computes finite-size corrections to dyonic giant magnons via two complementary approaches: (i) analyzing the $k\to1$ asymptotics of helical strings to extract finite-$J$ effects, and (ii) applying a generalized Lüscher $\mu$-term formula to magnon boundstates using the $su(2|2)^2$ S-matrix. By carefully identifying relevant poles and incorporating the dressing phase, the authors demonstrate exact agreement between the two methods across regimes $Q\sim O(g)$ and $Q\sim O(1)$, enabling the leading finite-size correction to be predicted to all orders in the 't Hooft coupling $\lambda$. The work also provides a finite-gap interpretation of the $k\to1$ limit, clarifying the role of the rapidity parameter $\omega_2$ and its geometric origin. Overall, the results substantiate the generalized Lüscher framework for boundstates and illuminate the connection between classical string finite-size effects and quantum S-matrix data, with implications for wrapping corrections and integrability-based techniques at finite volume.

Abstract

We compute finite-size corrections to dyonic giant magnons in two ways. One is by examining the asymptotic behavior of helical strings of hep-th/0609026 as elliptic modulus k goes to unity, and the other is by applying the generalized Luscher formula for mu-term of arXiv:0708.2208 to the situation in which incoming particles are boundstates. By careful choice of poles in the su(2|2)^2-invariant S-matrix, we find agreement of the two results, which makes possible to predict the (leading) finite-size correction for dyonic giant magnons to all orders in the 't Hooft coupling.

Figures (8)

• Figure 1: Diagrams for the leading finite-size corrections. The left is called $\mu$-term, and the right $F$-term. $a$ is an incoming physical particle, and $b,\,c$ are virtual (but on-shell) particles.
• Figure 2: Left: Type $(i)$ helical spinning string solution with two spins, where the $(x,y,z)$ axes show $\left( {\rm Re} \, \xi_1 \,, {\rm Im} \, \xi_1 \,, \left| \xi_2\right|\right)$. Right: The same string solution with $(x,y,z) = \left( {\rm Re} \, \xi_2 \,, {\rm Im} \, \xi_2 \,, \left| \xi_1\right|\right)$.
• Figure 3: The splitting process of box type.
• Figure 4: (i): Self-energy diagram of $I_{abc}$-type. (ii): Diagram of $ab \to ab$ scattering made from the diagram (i). (iii): The diagram (ii) can be viewed in two ways: $s$-type diagram as shown in the left, and $t$-type diagram as shown in the right.
• Figure 5: The scattering processes which correspond to (i) $y^- = X^+$, (ii) $y^+ = X^+$, (iii) $y^- = 1/X^-$, (iv) $y^+ = 1/X^-$. We follow the convention of the diagrams in DHM07.