BMN Correlators and Operator Mixing in N=4 Super Yang-Mills Theory

TL;DR

This work analyzes correlation functions of BMN operators in N=4 SYM, focusing on four-point, two-point with operator mixing, and three-point functions in the BMN limit. It shows non-extremal four-point functions are ill-defined in this limit and that planar one-loop corrections diverge, supporting the view that only two-point functions map to pp-wave strings. By performing genus-one operator mixing, the authors construct correctly redefined BMN operators and compute their torus anomalous dimensions, discovering degeneracy of Δ_n across symmetric, antisymmetric, and singlet two-impurity sectors. They further demonstrate that operator mixing modifies three-point functions and can restore conformal invariance at one loop, challenging simple identifications with string vertices and highlighting the non-uniqueness of the operator dictionary for protected modes.

Abstract

Correlation functions in perturbative N=4 supersymmetric Yang-Mills theory are examined in the Berenstein-Maldacena-Nastase (BMN) limit. We demonstrate that non-extremal four-point functions of chiral primary fields are ill-defined in that limit. This lends support to the assertion that only gauge theoretic two-point functions should be compared to pp-wave strings. We further refine the analysis of the recently discovered non-planar corrections to the planar BMN limit. In particular, a full resolution to the genus one operator mixing problem is presented, leading to modifications in the map between BMN operators and string states. We give a perturbative construction of the correct operators and we identify their anomalous dimensions. We also distinguish symmetric, antisymmetric and singlet operators and find, interestingly, the same torus anomalous dimension for all three. Finally, it is discussed how operator mixing effects modify three point functions at the classical level and, at one loop, allow us to recover conformal invariance.

Figures (10)

• Figure 1: Spherical diagrams for the four-point correlator. The charges of the depicted operators are $J_1=4$, $J_2=9$, $K_1=5$, $K_2=8$. In the left diagram the number of lines, $a=2$, between $J_1$ and $K_1$ is not fixed. In the middle diagram the distribution of lines between $J_2$ and $K_2$ is not fixed. The right diagram is unique. Furthermore, permutations of the four operators must be taken into account.
• Figure 2: Planar diagrams of four-point correlators with leading radiative corrections. The wiggly line represents the sum of all radiative contributions within a face of the diagram, as explained in detail in appendix \ref{['app:4pt']}. Diagrams like the ones on the top line contribute while diagrams on the bottom line do not.
• Figure 3: Cutting the torus of a correlator between two single trace operators yields double trace operators. The two impurities can either be on the same branch or on different branches corresponding to ${\cal T}^{J,r}_{ij,n}$ and ${\cal T}^{J,r}_{ij}$ of \ref{['double']}, respectively.
• Figure 4: Algebraic structure of gluon vertex and scalar self-energy.
• Figure 5: Gluon exchange between two scalar lines.