Dynamical symmetry breaking in fractal space

Abstract

We formulate field theories in fractal space and show the phase diagrams of the coupling versus the fractal dimension for dynamical symmetry breaking. We first consider the four-dimensional Gross-Neveu (GN) model in $\left(4\mathrm{}-d\right)$-dimensional randomized Cantor space where the fermions are restricted to a fractal space by the high potential barrier of the 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 four-dimensional QED 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 $d=4\mathrm{}$ (the Maskawa-Nakajima case) and $d=2\mathrm{}$ (the case with the external magnetic field).