Operator mixing in N=4 SYM: The Konishi anomaly revisited

TL;DR

The paper analyzes operator mixing in N=4 SYM to revisit the Konishi anomaly. By treating the Konishi descendant K_{10} and the protected descendant O_{10} as mixtures of B (classical, trilinear) and F (quantum, bilinear) operators, it shows that superconformal symmetry restricts the right-hand side of the Konishi anomaly to the classical term, absorbing the quantum piece into the mixing with divergent renormalization factors. The authors perform an explicit two-loop calculation in the SSDR scheme, diagonalize the mixing in SU(3) channels, and demonstrate that the protected descendant remains unrenormalized while the Konishi descendant acquires loop-dependent corrections. They extend the discussion to BMN operators, derive operator identities that hold in SSDR, and provide concrete mixing coefficients, illustrating the utility of superconformal constraints for organizing operator renormalization in N=4 SYM. Overall, the work clarifies the nature of the Konishi anomaly in a fully superconformal context and shows how higher-loop mixing can be systematically analyzed.

Abstract

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.