Dynamical Symmetry Breaking in Fractal Space

TL;DR

This paper develops field theories in fractal space, analyzing phase diagrams of dynamical symmetry breaking in models like GN and QED4, revealing critical lines and system-size effects related to the fractal dimension.

Contribution

It introduces a novel formulation of field theories in fractal spaces and derives phase diagrams for symmetry breaking, extending known models to fractal dimensions.

Findings

Phase diagram of GN model in fractal space resembles that of d-dimensional models.

Critical lines for symmetry breaking depend on fractal dimension and system size.

Consistent results with known cases at d=2 and d=4 for QED4.

Abstract

We formulate field theories in fractal space and show the phase diagrams of the coupling versus the fractal dimension for the dynamical symmetry breaking. We first consider the 4-dimensional Gross-Neveu (GN) model in the (4-d)-dimensional randomized Cantor space where the fermions are restricted to a fractal space by the high potential barrier of Cantor fractal shape. By the statistical treatment of this potential, we obtain an effective action depending on the fractal dimension. Solving the 1/N leading Schwinger-Dyson (SD) equation, we get the phase diagram of dynamical symmetry breaking with a critical line similar to that of the d-dimensional (2<d<4) GN model except for the system-size dependence. We also consider QED4 with only the fermions formally compactified to d dimensions. Solving the ladder SD equation, we obtain the phase diagram of dynamical chiral symmetry breaking with a linear critical line, which is consistent with the known results for d=4 (the Maskawa-Nakajima case) and d=2 (the case with the external magnetic field).