[Paper Review] Dynamical Symmetry Breaking in Curved Spacetime : Four-Fermion Interactions

Tomohiro Inagaki, T. Muta+1 more|arXiv (Cornell University)|Sep 12, 1997

Black Holes and Theoretical Physics6 references73 citations

TL;DR

This paper investigates dynamical chiral symmetry breaking in four-fermion models within curved spacetime using the 1/N expansion method. It analytically computes the effective potential in weakly curved, de Sitter, anti-de Sitter, and Einstein universes, demonstrating curvature-induced phase transitions with dynamically generated fermion masses, and extends results to gauged, higher-derivative, and supersymmetric NJL models, showing conditions for chiral symmetry breaking and the interplay of gravity, topology, and external fields.

ABSTRACT

This review deals with the theory of four-fermion interactions in curved spacetime. Starting with the D-dimensional Minkowski spacetime (2� D � 4) the effective potential in the leading order of 1/N-expansion is calculated and the phase structure of the theory is investigated. Using the same technique the effective potential for composite operatorin four-fermion models is calculated under the following circumstances: a) D-dimensional weakly curved spacetime (in linear curvature approximation), b) D-dimensional de Sitter and anti-de Sitter universe, c) D-dimensional Einstein universe. The phase structure of the theory is investigated analytically as well as numerically. Curvature induced phase transitions are discussed where fermion masses are dynamically generated. As an extension of four-fermion models we consider the gauged Nambu-Jona-Lasinio (NJL) model, higher derivative NJL model and supersymmetric NJL model in weakly curved spacetime where the effective potential is analytically evaluated. The phase structure of the models is again analyzed and the condition for the chiral symmetry breaking in the gauged NJL model is given in an analytical form. Finally the influence of two external effects (non-zero temperature and gravitational field, non- trivial topology and gravitational field as well as magnetic and gravitational field) to the phase structure of four-fermion models is analyzed. The possibility of curvature and temperature-induced or curvature- and topology-induced phase transitions is discussed. It is also argued that the chiral symmetry broken by a weak magnetic field may be restored due to the presence of gravitational field. Some applications of four-fermion models in quantum gravity are also briefly investigated. This paper was published in Prog. Theor. Phys. Suppl. 127 (1997) 93.

Motivation & Objective

• To analyze dynamical symmetry breaking in four-fermion interactions within curved spacetime using the 1/N expansion technique.
• To investigate how spacetime curvature—specifically in weakly curved, de Sitter, anti-de Sitter, and Einstein universes—affects the phase structure of four-fermion models.
• To extend the analysis to gauged, higher-derivative, and supersymmetric Nambu-Jona-Lasinio models in curved spacetime.
• To examine the combined effects of non-zero temperature, gravitational fields, non-trivial topology, and magnetic fields on phase transitions.
• To determine conditions under which chiral symmetry is broken or restored due to gravitational and external field influences.

Proposed method

• Calculates the effective potential in D-dimensional Minkowski spacetime (2D ≤ 4) using the leading-order 1/N expansion.
• Applies the same 1/N method to compute the effective potential for composite operators in weakly curved spacetime, linear curvature approximation.
• Evaluates the effective potential in de Sitter and anti-de Sitter spacetimes by incorporating curvature effects into the 1/N framework.
• Analyzes the Einstein universe case by considering compactified spatial dimensions and curvature effects in the 1/N expansion.
• Extends the formalism to gauged, higher-derivative, and supersymmetric NJL models in weakly curved spacetime, computing the effective potential analytically.
• Investigates combined effects of temperature, gravitational fields, non-trivial topology, and magnetic fields on phase transitions using analytical and numerical techniques.

Experimental results

Research questions

• RQ1 How does spacetime curvature induce phase transitions in four-fermion models, and can it lead to dynamical fermion mass generation?
• RQ2 What are the conditions for chiral symmetry breaking in the gauged Nambu-Jona-Lasinio model in curved spacetime?
• RQ3 How do non-zero temperature and gravitational fields jointly influence the phase structure of four-fermion models?
• RQ4 Can a weak magnetic field restore chiral symmetry that was previously broken, in the presence of a gravitational field?
• RQ5 What role does non-trivial spacetime topology play in modifying phase transitions in four-fermion systems?

Key findings

• Curvature-induced phase transitions lead to dynamical generation of fermion masses in four-fermion models across various curved spacetimes, including de Sitter and anti-de Sitter universes.
• The effective potential in weakly curved spacetime is analytically computed using the 1/N expansion, revealing curvature-dependent corrections to the phase structure.
• In the gauged NJL model, an analytical condition for chiral symmetry breaking is derived, showing explicit dependence on curvature and gauge coupling.
• The interplay between gravitational fields and magnetic fields can lead to the restoration of chiral symmetry that was previously broken by the magnetic field.
• Non-trivial spacetime topology combined with gravity induces distinct phase transitions, indicating topology as a key factor in symmetry breaking dynamics.
• Applications to quantum gravity suggest that four-fermion models in curved spacetime may provide insights into the low-energy behavior of quantum gravity.