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.
Highlights
It has been realized recently that the four-point correlation function of the so-called stress-tensor multiplets in N = 4 super-Yang-Mills theory (SYM) has a new symmetry [1]
As we have shown in Refs. [2], this property alone combined with the correct asymptotic behaviour of the correlation function in the short-distance and the light-cone limits, allows us to completely determine F (l)(xi) up to six loops in the planar sector
In this paper we have developed a new efficient method for the computation of the Konishi anomalous dimension at higher loops
Summary
In the OPE context, the distinguishing feature of the Konishi operator is that it controls the leading asymptotic behaviour of the four-point correlation function at loop level in the shortdistance limit In this manner, we obtain an analytic result for the Konishi anomalous dimension at five loops in planar N = 4 SYM theory and observe perfect agreement with the prediction based on integrability in AdS/CFT [5, 6, 7, 8]. The Konishi operator provides the leading contribution to the asymptotic behaviour of the four-point correlation function at short distances G(1, 2, 3, 4) ∼ (x212)γK(a)/2 as x1 → x2 (with x12 ≡ x1 − x2) At weak coupling, this asymptotic behaviour implies that perturbative corrections to the correlation function at l loops are given by a sum of logarithmic singularities (ln x212)k with powers k ≤ l. In Appendix C, we perform the OPE analysis of the four-point correlation function and extract the values of three-loop anomalous dimensions of twist-two operators with Lorentz spin zero, two and four
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