Modified renormalization group method applied to the $O(1)$ ghost model with periodic condensate

TL;DR

This paper introduces a modified renormalization group method to analyze the emergence of a periodic condensate in the 3D ghost $O(1)$ model, providing new insights into its phase structure and fixed points.

Contribution

A novel Fourier-Wetterich RG approach is developed to incorporate periodic condensates into the effective average action framework.

Findings

Identified characteristics of the Wilson-Fisher fixed point.

Mapped the phase structure of the model with a periodic condensate.

Demonstrated the feasibility of the modified RG scheme with preliminary results.

Abstract

In order to discuss the occurrence of a periodic condensate in the Euclidean 3-dimensional ghost $O(1)$ model, a modified version of the effective average action (EAA) renormalization group (RG) method is developed, called by us Fourier-Wetterich RG approach. It is proposed to start with an ansatz for the EAA, that contains terms, in addition to the usual ones, induced by the various Fourier-modes of the periodic condensate and to expand the EAA in functional Taylor-series around the periodic background. The RG flow equations are derived in the next-to-next-to-leading order of the gradient expansion (GE). No field-dependence of the derivative couplings have been taken into account and $Z_2$ symmetry of the EAA is preserved. Preliminary numerical results have been obtained under various additional simplifying assumptions. The characteristics of the Wilson-Fisher fixed point and the phase structure of the model have been determined numerically in the local potential approximation and in the next-to-leading order of the GE, when the periodic condensate has been modelled by a single cosine mode in one spatial direction. From the preliminary results important information is gained on further possibilities to improve the proposed RG scheme.