Anomalous scaling dimension

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In theoretical physics, by anomaly one usually means that the symmetry remains broken when the symmetry-breaking factor goes to zero. When the symmetry which is broken is scale invariance, then true power laws usually cannot be found from dimensional reasoning like in turbulence or quantum field theory. In the latter, the anomalous scaling dimension of an operator is the contribution of quantum mechanics to the classical scaling dimension of that operator.

The classical scaling dimension of an operator O is determined by dimensional analysis from the Lagrangian (in 4 spacetime dimensions this means dimension 1 for elementary bosonic fields including the vector potentials, 3/2 for elementary fermionic fields etc.). However if one computes the correlator of two operators of this type, one often finds logarithmic divergences arising from one-loop Feynman diagrams. The expansion in the coupling constant has the schematic form

O(x)O(y) \sim \frac{1}{|x-y|^{2\Delta_0}} - 2 g^2 A \log(\Lambda|x-y|) + \cdots

where g is a coupling constant, Δ0 is the classical dimension, and Λ is an ultraviolet cutoff (the maximal allowed energy in the loop integrals). A is a constant that appears in the loop diagrams. The expression above may be viewed as a Taylor expansion of the full quantum dimension.

O(x)O(y)\sim\frac{1}{|x-y|^{2\Delta}};\qquad\Delta=\Delta_0 + g^2 A + \cdots

The term g2A is the anomalous scaling dimension while Δ is the full dimension. Conformal field theories are typically strongly coupled and the full dimension cannot be easily calculated by Taylor expansions. The full dimensions in this case are often called critical exponents. These operators describe conformal bound states with a continuous mass spectrum.

In particular, 2Δ = d − 2 + η for the critical exponent η for a scalar operator. We have an anomalous scaling dimension when η ≠ 0.

An anomalous scaling dimension indicates a scale dependent wavefunction renormalization.

Anomalous scaling appears also in classical physics.

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