Non-polynomial scalar field potentials in the local potential approximation

Non-polynomial 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 non-polynomial eigenoperator solutions, the so called Halpern-Huang 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 non-polynomial eigenoperators flow into the polynomial Hilbert space. These conclusions are based off RG flow initiated at an arbitrary scale, implying non-polynomial eigenoperators are dependent upon a scale other than k. Therefore, the asymptotic field behaviour forbids self-similar scaling. These results hold when generalised from the Halpern-Huang directions around the Gaussian fixed point to a general fixed point with a general non-polynomial 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 non-linear terms of the Polchinski equation forbid any non-polynomial field scaling, regardless of the fixed point. These observations lead to the conclusion that only polynomial eigenoperators show the correct, self-similar, scaling behaviour to construct a non-perturbatively renormalisable scalar QFT.

University of Southampton

Bridle, Ismail, Hamzaan

5f017bbb-e2b8-4a67-9f66-42b25325153a

April 2017

Bridle, Ismail, Hamzaan

5f017bbb-e2b8-4a67-9f66-42b25325153a

Morris, Timothy

a9927d31-7a12-4188-bc35-1c9d3a03a6a6

Bridle, Ismail, Hamzaan
(2017)
Non-polynomial scalar field potentials in the local potential approximation.
*University of Southampton, Doctoral Thesis*, 137pp.

Record type:
Thesis
(Doctoral)

## Abstract

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 non-polynomial eigenoperator solutions, the so called Halpern-Huang 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 non-polynomial eigenoperators flow into the polynomial Hilbert space. These conclusions are based off RG flow initiated at an arbitrary scale, implying non-polynomial eigenoperators are dependent upon a scale other than k. Therefore, the asymptotic field behaviour forbids self-similar scaling. These results hold when generalised from the Halpern-Huang directions around the Gaussian fixed point to a general fixed point with a general non-polynomial 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 non-linear terms of the Polchinski equation forbid any non-polynomial field scaling, regardless of the fixed point. These observations lead to the conclusion that only polynomial eigenoperators show the correct, self-similar, scaling behaviour to construct a non-perturbatively renormalisable scalar QFT.

## More information

Published date: April 2017

Organisations:
University of Southampton, Physics & Astronomy

## Identifiers

Local EPrints ID: 410270

URI: http://eprints.soton.ac.uk/id/eprint/410270

PURE UUID: f2a7b75c-3ad6-43bb-a448-a7e52d198401

## Catalogue record

Date deposited: 06 Jun 2017 04:03

Last modified: 12 Dec 2021 02:36

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## Contributors

Author:
Ismail, Hamzaan Bridle

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