[Paper Review] Multi-fermion interaction models in curved spacetime

M Hayashi, Tomohiro Inagaki+1 more|ArXiv.org|Dec 4, 2008

Black Holes and Theoretical Physics1 references3 citations

TL;DR

This paper investigates multi-fermion interaction models, particularly an eight-fermion interaction in curved spacetime, using the auxiliary field method and Riemann normal coordinate expansion to compute the effective potential in the 1/N expansion. It finds that chiral symmetry is always dynamically broken in negative curvature spacetime, while curvature-induced phase transitions—first-order or crossover—occur for positive curvature, with the eight-fermion coupling enhancing symmetry breaking.

ABSTRACT

A model with a scalar type eight-fermion interaction is investigated in curved spacetime. The ground state of the model can be obtained by observing the effective potential. Applying the Riemann normal coordinate expansion, we calculate an effective potential of the model in a weakly curved spacetime. The result is extended to models with multi-fermion interactions. We numerically show the behavior of the effective potential and find the phase structure of the model.

Motivation & Objective

• To understand dynamical chiral symmetry breaking in multi-fermion models under curved spacetime conditions.
• To investigate how higher-dimensional interactions, such as eight-fermion couplings, affect phase structure and symmetry breaking.
• To extend the effective potential calculation beyond four-fermion models using Riemann normal coordinates and the 1/N expansion.
• To determine the role of spacetime curvature in triggering phase transitions, especially in the context of early universe critical phenomena.
• To examine the stability of the trivial vacuum and the emergence of non-trivial ground states in curved backgrounds.

Proposed method

• Formulates a four- and eight-fermion interaction model in curved spacetime using the action with Dirac fermions, curvature-coupled covariant derivatives, and scalar-type interactions.
• Applies the auxiliary field method to decouple the four- and eight-fermion interactions into quadratic terms involving auxiliary fields σ₁ and σ₂.
• Uses Riemann normal coordinate expansion to compute the effective potential up to linear order in the Ricci scalar R, valid in weakly curved spacetime.
• Performs the 1/N expansion to resum leading-order contributions and derive the effective potential in the large-N limit.
• Solves the gap equation numerically to determine the expectation values of auxiliary fields and phase boundaries.
• Analyzes the second derivative of the effective potential at σ₁ = σ₂ = 0 to assess vacuum stability and symmetry breaking.

Experimental results

Research questions

• RQ1 How does the inclusion of an eight-fermion interaction modify the phase structure of a four-fermion model in curved spacetime?
• RQ2 What is the behavior of the effective potential and vacuum stability in negative curvature spacetime?
• RQ3 Under what conditions does curvature induce a first-order or crossover phase transition in the chiral symmetry breaking pattern?
• RQ4 How does the eight-fermion coupling strength influence the critical curvature R_cr for symmetry restoration?
• RQ5 Can the mean field approximation be recovered as a limiting case of the derived effective potential?

Key findings

• The trivial vacuum with σ₁ = σ₂ = 0 is unstable for negative Ricci curvature R < 0, indicating that chiral symmetry is always dynamically broken in such spacetimes.
• For positive curvature, chiral symmetry is restored at a critical curvature R_cr ≥ 0, with the transition being first-order for strong four-fermion coupling and crossover for weak coupling.
• The eight-fermion coupling G₂ enhances chiral symmetry breaking, leading to a larger mass gap and shifting the phase boundary to lower R_cr values.
• Numerical evaluation reveals two local minima in the effective potential for large positive or negative curvature, indicating coexistence of multiple phases.
• The phase boundary separating symmetric and broken phases is independent of G₂ in the crossover regime, but shifts with G₂ in the first-order transition region.
• The effective potential calculation is valid up to linear order in R, but higher-order corrections may alter the phase structure in strongly curved spacetimes.