Abstract
We elucidate aspects of the one-loop anomalous dimension of so(6)-singlet multi-trace operators in N=4SU(Nc) SYM at finite Nc. First, we study how 1/Nc corrections lift the large Nc degeneracy of the spectrum, which we call the operator submixing problem. We observe that all large Nc zero modes acquire anomalous dimensions starting at order 1/Nc2 with a non-positive coefficient and they mix only among the operators with the same number of traces at leading order. Second, we study the lowest one-loop dimension of operators of length equal to 2Nc. The dimension of such operators becomes more negative as Nc increases, which will eventually diverge in the double scaling limit. Third, we examine the structure of level-crossing at finite Nc in view of unitarity. Finally we find out a correspondence between the large Nc zero modes and completely symmetric polynomials of Mandelstam variables.
Highlights
Large Nc gauge theories have been intensively studied for decades since the work of ’t Hooft, who discovered that mesons are described by a string in the planar limit [1]
Maldacena argued that a d-dimensional gauge theory can be described by a (d + 1)-dimensional gravity theory, which is called the AdS/CFT correspondence [2]
We focus on the degenerate eigenstate having the zero anomalous dimension at large Nc, which
Summary
Large Nc gauge theories have been intensively studied for decades since the work of ’t Hooft, who discovered that mesons are described by a string in the planar limit [1]. It was observed that the one-loop mixing is highly constrained on these bases — two operators can mix if their Young diagrams are related by moving a single box of the Young diagram [18, 19, 20, 21] This property makes the problem tractable, and in certain situations it is possible to obtain the spectrum of the dilatation operator explicitly [22, 23, 24, 25, 26, 27]. The one-loop dimension of determinant-like operators generally receives non-planar corrections even in the large Nc limit, because their operator length is of order Nc. The proper way to take the large Nc limit is to study their dimension at a finite Nc ∼ L and extrapolate the results to Nc 1.
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