Dynamical Symmetry Breaking in Spaces with Constant Negative Curvature

TL;DR

This paper investigates how constant negative curvature spaces influence dynamical symmetry breaking, revealing an effective dimensional reduction that allows symmetry breaking consistent with theoretical constraints.

Contribution

It demonstrates that negative curvature induces an effective dimensional reduction, enabling dynamical symmetry breaking without conflicts from scalar field corrections.

Findings

Critical coupling $g_c$ equals zero in negatively curved spaces.

Effective reduction of spacetime dimension occurs in the infrared region.

Symmetry breaking remains consistent with the Mermin-Wagner-Coleman theorem.

Abstract

By using the Nambu-Jona-Lasinio model, we study dynamical symmetry breaking in spaces with constant negative curvature. We show that the physical reason for zero value of critical coupling value $g_c = 0$ in these spaces is connected with the effective reduction of dimension of spacetime $1 + D \to 1 + 1$ in the infrared region, which takes place for any dimension $1 + D$. Since the Laplace-Beltrami operator has a gap in spaces with constant negative curvature, such an effective reduction for scalar fields is absent and there are not problems with radiative corrections due to scalar fields. Therefore, dynamical symmetry breaking with the effective reduction of the dimension of spacetime for fermions in the infrared region is consistent with the Mermin-Wagner-Coleman theorem, which forbids spontaneous symmetry breaking in (1 + 1)-dimensional spacetime.