On two fermion BMN operators

TL;DR

The paper develops a systematic superspace-based approach to compute two-loop anomalous dimensions of BMN operators with two impurities, including scalar-scalar and scalar-fermion mixes in the antisymmetric SO(4) sector. By differentiating in N=4 harmonic superspace, the author relates gamma2 to the square of one-loop two-fermion admixtures, effectively elevating the loop order from one to two. The work classifies operators into singlet, antisymmetric, and symmetric representations, derives protected vs. unprotected structures, and demonstrates consistency with the two-loop dilatation operator, including a detailed treatment of the Konishi anomaly. Explicit results for J=0,1,2 (SU(N)) enumerate the mixing matrices and anomalous dimensions, and the analysis provides a framework to extend to higher J and cross-check integrability-based predictions. Overall, the study clarifies the interdependence of operator mixing, superspace descendants, and anomalies in determining higher-loop spectra of BMN-like operators in N=4 SYM.

Abstract

We show how to determine the lowest order mixing of all scalar with two-fermion two impurity BMN operators in the antisymmetric representation of SO(4). Differentiation on harmonic superspace allows one to derive two-loop anomalous dimensions of gauge invariant operators from this knowledge: the value for the second anomalous correction to the dimension is essentially the square of the two-fermion admixture. The method effectively increases the loop order by one. For low J we find agreement to all orders in N with results obtained upon diagonalisation of the N=4 dilation operator. We give a formula for the generalised Konishi anomaly and display its role in the mixing. For J=2 we resolve the mixing up to order $g^2$ in the singlet representation. The sum of the anomaly and the naive variation of the leading two-fermion admixtures to the singlets is exactly equal to the two-fermion terms in the antisymmetric descendants.