Conformal Dimensions of Two-Derivative BMN Operators

TL;DR

This work analyzes BMN operators with two covariant derivative impurities in the planar BMN limit, computing their one-loop anomalous dimensions to first order in λ'. The key result is δΔ_n = λ' n^2, identical to scalar and mixed impurities, validating the BMN correspondence for derivative insertions. A central technical challenge is the nonvanishing overlap of derivative impurities with the background field Z, necessitating a sequence of operator redefinitions to obtain primaries and orthogonality between zero and nonzero modes. The analysis highlights the special role of two-derivative vertices in the BMN limit and clarifies how derivative impurities differ conceptually from scalar ones, offering a path toward constructing derivative-impurity effective vertices in the planar regime.

Abstract

We compute the anomalous dimensions of BMN operators with two covariant derivative impurities at the planar level up to first order in the effective coupling lambda'. The result equals those for two scalar impurities as well as for mixed scalar and vector impurities given in the literature. Though the results are the same, the computation is very different from the scalar case. This is basically due to the existence of a non-vanishing overlap between the derivative impurity and the ``background'' field Z. We present details of these differences and their consequences.