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تسجيل مستخدم جديدOperator mixing in N = 4 SYM: The Konishi anomaly re-re-visited
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B. Eden
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The supersymmetry transformation relating the Konishi operator to its lowest descendant in the 10 of SU(4) is not manifest in the N=1 formulation of the theory but rather uses an equation of motion. On the classical level one finds one operator, the unintegrated chiral superpotential. In the quantum theory this term receives an admixture by a second operator, the Yang-Mills part of the Lagrangian. It has long been debated whether this anomalous contribution is affected by higher loop corrections. We present a first principles calculation at the second non-trivial order in perturbation theory using supersymmetric dimensional reduction as a regulator and renormalisation by Z-factors. Singular higher loop corrections to the renormalisation factor of the Yang-Mills term are required if the conformal properties of two-point functions are to be met. These singularities take the form determined in preceding work on rather general grounds. Moreover, we also find non-vanishing finite terms. The core part of the problem is the evaluation of a four-loop two-point correlator which is accomplished by the Laporta algorithm. Apart from several examples of the T1 topology with two lines of non-integer dimension we need the first few orders in the epsilon expansion of three master integrals. The approach is self-contained in that all the necessary information can be derived from the power counting finiteness of some integrals.
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In the context of the superconformal N=4 SYM theory the Konishi anomaly can be viewed as the descendant $K_{10}$ of the Konishi multiplet in the 10 of SU(4), carrying the anomalous dimension of the multiplet. Another descendant $O_{10}$ with the same
quantum numbers, but this time without anomalous dimension, is obtained from the protected half-BPS operator $O_{20}$ (the stress-tensor multiplet). Both $K_{10}$ and $O_{10}$ are renormalized mixtures of the same two bare operators, one trilinear (coming from the superpotential), the other bilinear (the so-called quantum Konishi anomaly). Only the operator $K_{10}$ is allowed to appear in the right-hand side of the Konishi anomaly equation, the protected one $O_{10}$ does not match the conformal properties of the left-hand side. Thus, in a superconformal renormalization scheme the separation into classical and quantum anomaly terms is not possible, and the question whether the Konishi anomaly is one-loop exact is out of context. The same treatment applies to the operators of the BMN family, for which no analogy with the traditional axial anomaly exists. We illustrate our abstract analysis of this mixing problem by an explicit calculation of the mixing matrix at level g^4 (two loops) in the supersymmetric dimensional reduction scheme.
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-p
oint 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.
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