Finite-size effects in the superconformal beta-deformed N=4 SYM

TL;DR

This paper investigates finite-size (wrapping) effects in the $SU(2)$ sector of the $β$-deformed ${\cal N}=4$ SYM theory, focusing on a single magnon. Using an asymptotic dilatation operator derived from the undeformed theory via a deformed basis, the authors compute wrapping corrections up to four loops for short one-impurity states and analyze higher-order extensions. They show that at three loops the wrapping corrections cancel against subtractions, leaving the same result as the asymptotic calculation, while at four loops the wrapping contributions yield nontrivial transcendental terms involving $\zeta(3)$ and $\zeta(5)$. In the two-impurity and higher-order analyses, the deformed spectrum depends on $β$ through $\Delta(β)$ and exhibits a pattern where the transcendentality increases with loop order, reminiscent of the dressing phase. Overall, the work strengthens the connection between wrapping effects in weak coupling and finite-size corrections in the dual string description and points to the need for a wrapped Bethe Ansatz consistent with the $β$-deformation.

Abstract

We study finite size effects for composite operators in the SU(2) sector of the superconformal beta-deformed N=4 SYM theory. In particular we concentrate on the spectrum of one single magnon. Since in this theory one-impurity states are non BPS we compute their anomalous dimensions including wrapping contributions up to four loops and discuss higher order effects.

Figures (4)

• Figure 1: Completely chiral diagrams (wrapping and subtraction)
• Figure 2: Range-five diagram
• Figure A.1: Wrapping diagrams with chiral structure $\boldsymbol{\chi}(2,1)$
• Figure A.2: Wrapping diagrams with chiral structure $\boldsymbol{\chi}(1)$