The scaling dimension of a local operator is its eigenvalue under dilations. Under , a scalar operator of dimension transforms by .
An exactly marginal operator has scaling dimension equal to the spacetime dimension and a deformation coupling whose beta function vanishes to all orders. Varying that coupling moves along a family of conformal field theories without introducing a scale.
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The term "scaling dimension" can refer to different concepts depending on the context in which it is used, particularly in physics and mathematics. Here are a couple of relevant interpretations: 1. **In Physics (Statistical Mechanics and Quantum Field Theory)**: The scaling dimension is a property of operators in conformal field theories (CFTs). It describes how the correlation functions of those operators change under rescaling of the coordinates.