Vector operators in the BMN correspondence

TL;DR

The paper probes BMN operators with a mixed scalar-vector impurity in N=4 SYM, focusing on planar and genus-one (torus) corrections in the BMN limit. It develops a q-variation formalism to define vector BMN operators and demonstrates that their anomalous dimensions match those of scalar BMN operators at both planar and torus levels, with a SUSY-based argument providing a protective mapping. The torus result reveals a nonzero genus-one mass renormalization arising from non-contractible diagrams, provoking discussion about string-field-theory prescriptions and possible contact-term contributions. Overall, the work strengthens the field-theory to string-theory map in the pp-wave/BMN context and points to subtle higher-genus effects and SUSY structure as key features.

Abstract

We consider a BMN operator with one scalar, phi, and one vector, D_{m}Z, impurity field and compute the anomalous dimension both at planar and torus levels. This "mixed" operator corresponds to a string state with two creation operators which belong to different SO(4) sectors of the background. The anomalous dimension at both levels is found to be the same as the scalar impurity BMN operator. At planar level this constitutes a consistency check of BMN conjecture. Agreement at the torus level can be explained by an argument using supersymmetry and supression in the BMN limit. The same argument implies that a class of fermionic BMN operators also have the same planar and torus level anomalous dimensions. Implications of the results for the map from N=4 SYM theory to string theory in the pp-wave background are discussed.

Figures (24)

• Figure 1: A typical torus digram. Dashed line represents $\phi$ and arrow on a solid line is $\partial_{\mu}Z$. The derivative $\partial_{\nu}$ can be placed on any line.
• Figure 2: Combination of $g_{YM}^2$ corrections under a total vertex.
• Figure 3: Planar interactions of chiral primaries can be obtained by placing the total vertex between all adjacent pairs. To find the vector correlator one simply dresses this figure by $\partial_{\mu}^q$ and $\partial_{\nu}^{\bar{r}}$.
• Figure 4: Two-loop diagrams of vacuum polarization in scalar QCD. Treatment of other four diagrams obtained by replacing the scalar lines with gluons can be separately and do not affect our argument.
• Figure 5: First class of ${\cal O}(g_{YM}^2)$ diagrams involving external gluons. These do not yield anomalous dimension as there are no internal vertices.