Reparameterisation Invariance and RG equations: Extension of the Local Potential Approximation

Abstract

Equations related to the Polchinski version of the exact renormalisation group equations for scalar fields which extend the local potential approximation to first order in a derivative expansion, and which maintain reparameterisation invariance, are postulated. Reparameterisation invariance ensures that the equations determine the anomalous dimension eta unambiguously and the equations are such that the result is exact to O(epsilon^2) in an epsilon-expansion for any multi-critical fixed point. It is also straightforward to determine eta numerically. When the dimension d=3 numerical results for a wide range of critical exponents are obtained in theories with O(N) symmetry, for various N and for a ranges of eta, are obtained within the local potential approximation. The associated eta, which follow from the derivative approximation described here, are found for various N. The large N limit of the equations is also analysed. A corresponding discussion is also given in a perturbative RG framework and scaling dimensions for derivative operators are calculated to first order in epsilon.

30 pages, 4 figures, version 2 some arguments expanded, additional references

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• Essential renormalisation group
• Global Wilson-Fisher fixed points
• Functional perturbative RG and CFT data in the $\epsilon$-expansion
• Asymptotic safety of scalar field theories
• Functional Renormalization and $\overline{\text{MS}}$
• Spectrum of anomalous dimensions in hypercubic theories
• Functional flows for complex effective actions
• Revisiting the local potential approximation of the exact renormalization group equation
• A Wilsonian Energy-Momentum Tensor
• Universal scaling dimensions for highly irrelevant operators in the Local Potential Approximation