Title:

Nonpolynomial scalar field potentials in the local potential approximation

We present the renormalisation group analysis of O(N) invariant scalar field theory in the local potential approximation. Linearising around the Gaussian fixed point, we find the same eigenoperators solutions exist for both the Wilsonian and the Legendre effective actions, given by solutions to Kummer’s equations. We find the usual polynomial eigenoperators and the Hilbert space they define are a natural subset of these solutions given by a specific set of quantised eigenvalues. Allowing for continuous eigenvalues, we find nonpolynomial eigenoperator solutions, the so called HalpernHuang directions, that exist outside of the polynomial Hilbert space due to the exponential field dependence. Carefully analysing the large field behaviour shows that the exponential dependence implies the Legendre effective action does not have a well defined continuum limit. In comparison, flowing towards the infrared we find that the nonpolynomial eigenoperators flow into the polynomial Hilbert space. These conclusions are based off RG flow initiated at an arbitrary scale, implying nonpolynomial eigenoperators are dependent upon a scale other than k. Therefore, the asymptotic field behaviour forbids selfsimilar scaling. These results hold when generalised from the HalpernHuang directions around the Gaussian fixed point to a general fixed point with a general nonpolynomial eigenoperator. Legendre transforming to results of the Polchinski equation, we find the flow of the Wilsonian effective action is much better regulated and always fall into the polynomial Hilbert space. For large Wilsonian effective actions, we find that the nonlinear terms of the Polchinski equation forbid any nonpolynomial field scaling, regardless of the fixed point. These observations lead to the conclusion that only polynomial eigenoperators show the correct, selfsimilar, scaling behaviour to construct a nonperturbatively renormalisable scalar QFT.
