Wrapping at four loops in N=4 SYM

TL;DR

The paper computes the planar four-loop anomalous dimension of the length-four Konishi-descendant operator in the SU(2) sector of ${\cal N}=4$ SYM, where wrapping effects first appear. The authors employ a two-step strategy: subtract the range-five part of the known four-loop dilatation operator $D_4$ to isolate non-wrapping contributions for a length-4 state, and then evaluate wrapping corrections using ${\cal N}=1$ superspace techniques together with GPXT and MINCER for the loop integrals. A key finding is the appearance of a $\zeta(5)$ term in the wrapping sector, increasing the transcendentality of the result and contradicting earlier conjectures based on the Hubbard model or BFKL constraints. The final explicit planar four-loop anomalous dimension is $\gamma = 4 + 12 g^2 - 48 g^4 + 336 g^6 + g^8(-2496 + 576 \zeta(3) - 1440 \zeta(5))$, illustrating the nontrivial role of wrapping at this order and providing a concrete benchmark for integrability-based approaches.

Abstract

We present the planar four-loop anomalous dimension of the composite operator tr(phi[Z,phi]Z) in the flavour SU(2) sector of the N=4 SYM theory. At this loop order wrapping interactions are present: they give rise to contributions proportional to zeta(5) increasing the level of transcendentality of the anomalous dimension. In a sequel of this paper all the details of our calculation will be reported.

Figures (1)

• Figure 1: Classes of four-loop wrapping graphs and their quantitative contributions. For the classes $W_i$, $i=1,\dots,6$ only the underlying chiral structure is shown. It has to be completed to the wrapping graphs by appropriately adding the indicated number of gauge boson lines. For the chiral classes $C_j$, $j=1,\dots3$ all interactions are present.