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

TL;DR

This paper extends the local potential approximation in the exact renormalisation group framework to include reparameterisation invariance, enabling precise calculation of critical exponents and anomalous dimensions for scalar field theories.

Contribution

It introduces reparameterisation invariant equations in the local potential approximation, allowing unambiguous and exact determination of the anomalous dimension eta in scalar field theories.

Findings

Numerical critical exponents for O(N) models in d=3 are obtained.

The equations are analyzed in the large N limit.

Perturbative RG results for scaling dimensions are provided.

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.