Abstract
We go beyond a systematic review of the semiclassical approaches for determining the scaling dimensions of fixed-charge operators in $U(1)$ and $O(N)$ models by introducing a general strategy apt at determining the relation between a given charge configuration and the associated operators for more involved symmetry groups such as the $U(N)\ifmmode\times\else\texttimes\fi{}U(M)$. We show how, varying the charge configuration, it is possible to access anomalous dimensions of different operators transforming according to a variety of irreducible representations of the non-Abelian symmetry group without the aid of diagrammatical computations. We illustrate our computational strategy by determining the anomalous dimensions of several composite operators to the next-to-leading order in the semiclassical expansion for the $U(N)\ifmmode\times\else\texttimes\fi{}U(M)$ conformal field theory (CFT) in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions. Thanks to the powerful interplay between semiclassical methods and group theory we can, for the first time, extract scaling dimensions for a wide range of operators.
Highlights
There has been a flurry of interest in studying conformal field theories with continuous global symmetries in the limit of a large conserved charge Qin order to access nonperturbative corners of quantum field theories (QFT)s
Consequence 2: In a conformal field theory (CFT), operators that transform in the same irreducible representations of G but correspond to different weights have identical scaling dimensions, if they do not mix with operators that do not belong to their carrier space under renormalization
O1⁄2m corresponds to an irreducible representation of OðNÞ and has a definite scaling dhiamveenthsieosna.mTehevaalurgeuomf ePntii1⁄41⁄4aN1ls=o2 shows that operators that jmij and minimal classical scaling dimension (MCSD) all belong to the same irreducible OðNÞ representation and have the same scaling dimension, in agreement with the expectation that by an OðNÞ rotation we can associate all charges to a single Cartan generator
Summary
There has been a flurry of interest in studying conformal field theories with continuous global symmetries in the limit of a large conserved charge Qin order to access nonperturbative corners of quantum field theories (QFT)s. Another, yet unexplored possibility, is to introduce two independent small parameters, one that takes into account the deformation of the number of space-time dimensions and the other that controls the original fixed point [21] This last case has not yet been investigated in the literature and will be considered elsewhere. Compared to conventional perturbation theory according to which one chooses a specific composite operator and diagrammatically determines its scaling dimension, in the semiclassical fixed-charged framework one needs to reverse engineer the given charge configuration to determine the irreducible representation of the related composite operator.
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