Five-loop anomalous dimension at critical wrapping order in N=4 SYM

TL;DR

The paper computes the planar five-loop anomalous dimension of a length-five operator in the SU(2) sector of ${\cal N}=4$ SYM at the critical wrapping order, using ${\cal N}=1$ superspace methods and a detailed subtraction of range-six diagrams. It constructs the complete five-loop asymptotic dilatation operator, includes the dressing phase, and evaluates wrapping diagrams with Gegenbauer polynomial x-space techniques, yielding $\gamma_5 = 6664+1152\zeta(3)+3840\zeta(5)-2240\zeta(7)$. The authors verify this result against the Y-system prediction extended to five loops, which gives an identical total anomalous dimension, thereby providing a strong field-theoretic confirmation of the twist-three five-loop formula derived from maximal transcendentality. These results substantiate the consistency between perturbative calculations and integrability-based approaches in short operators and highlight wrapping effects as essential at five loops in the planar ${\cal N}=4$ theory.

Abstract

We compute the anomalous dimension of a length-five operator at five-loop order in the SU(2) sector of N=4 SYM theory in the planar limit. This is critical wrapping order at five loops. The result is obtained perturbatively by means of N=1 superspace techniques. Our result from perturbation theory confirms explicitly the formula conjectured in arXiv:0901.4864 for the five-loop anomalous dimension of twist-three operators. We also explicitly obtain the same result by employing the recently proposed Y-system.

Figures (21)

• Figure A.1: Completely chiral range-six diagrams
• Figure A.2: Completely chiral range-five diagrams
• Figure A.3: Range-six diagrams with structure $\chi(1,2,4,5)$
• Figure A.4: Range-six diagrams with structure $\chi(1,5,4,3)$
• Figure A.5: Range-six diagrams with structure $\chi(1,3,4,5)$