Five-loop Konishi in N=4 SYM

TL;DR

This paper develops a novel, diagram-free approach to compute the Konishi anomalous dimension in $\mathcal{N}=4$ SYM at weak coupling by exploiting the OPE of the four-point function of stress-tensor multiplets. By analyzing the double short-distance limit and regularizing in dimensional regularization, the authors relate the one- and higher-loop logarithmic singularities to finite lower-loop two-point integrals, enabling a loop-reduction to $(\ell-1)$-loops. They successfully obtain analytic results for $\gamma_{\mathcal K}(a)$ up to five loops in the planar limit and confirm exact agreement with AdS/CFT integrability predictions, while also predicting the non-planar four-loop correction in the form $r\zeta_5/N_c^2$ with $r=-135/2$. The method uses a universal representation of the four-point function, master integrals, and IBP reductions, and promises a feasible path to six-loop results, thereby strengthening the bridge between perturbative gauge theory and integrability.

Abstract

We present a new method for computing the Konishi anomalous dimension in N=4 SYM at weak coupling. It does not rely on the conventional Feynman diagram technique and is not restricted to the planar limit. It is based on the OPE analysis of the four-point correlation function of stress-tensor multiplets, which has been recently constructed up to six loops. The Konishi operator gives the leading contribution to the singlet SU(4) channel of this OPE. Its anomalous dimension is the coefficient of the leading single logarithmic singularity of the logarithm of the correlation function in the double short-distance limit, in which the operator positions coincide pairwise. We regularize the logarithm of the correlation function in this singular limit by a version of dimensional regularization. At any loop level, the resulting singularity is a simple pole whose residue is determined by a finite two-point integral with one loop less. This drastically simplifies the five-loop calculation of the Konishi anomalous dimension by reducing it to a set of known four-loop two-point integrals and two unknown integrals which we evaluate analytically. We obtain an analytic result at five loops in the planar limit and observe perfect agreement with the prediction based on integrability in AdS/CFT.

Figures (7)

• Figure 1: Diagrammatic representation of the chain relation. Solid line with index $\alpha$ stands for $1/(x^2)^\alpha$ and black dot denotes the integration point.
• Figure 2: Diagrammatic representation of the integrals in (\ref{['C2-fin']}). Solid lines without labels depict propagators $1/x_{ij}^2$, the label 2 refers to the square of the propagator. Dashed lines represent numerator factors $x_{ij}^2$, black and white dots represent integration and external points, respectively.
• Figure 3: Diagrammatic representation of the integral (\ref{['C3-np1']}) in the $x-$representation (left) and in the dual momentum representation (right). In what follows, all momentum integrals are shown in blue.
• Figure 4: The dual momentum master integrals defining the four-loop Konishi anomalous dimension.
• Figure 5: Diagrammatic representation of the planar basis integrals in the dual momentum representation. Blue line denote momentum propagators $1/k^2$ with the momentum $k=x_i-x_j$.