Abstract
Four-fermion interaction models are often used as simplified models of interacting fermion fields with the chiral symmetry. The chiral symmetry is dynamically broken for a larger four-fermion coupling. It is expected that the broken symmetry is restored under extreme conditions. In this paper, the finite size effect on the chiral symmetry breaking is investigated in the four-fermion interaction model. We consider the model on a flat spacetime with a compactified spatial coordinate, M D − 1 ⊗ S 1 and obtain explicit expressions of the effective potential for arbitrary spacetime dimensions in the leading order of the 1 / N expansion. Evaluating the effective potential, we show the critical lines which divide the symmetric and the broken phase and the sign-flip condition for the Casimir force.
Highlights
Fundamental theories of particle physics are constructed based on several types of symmetry.It is expected that a fundamental theory with a higher symmetry is realized at the early universe.The symmetry of the theory is partly broken on the ground state
Since the physical state depends on the environment, the broken symmetry can be restored under extreme conditions, small size, high temperature, high density, strong curvature, strong electromagnetic field, and so on
The finite size effect on the chiral symmetry has been studied in four-fermion interaction models with periodic or anti-periodic boundary conditions for the fermion fields
Summary
Fundamental theories of particle physics are constructed based on several types of symmetry. The finite size effect on the chiral symmetry has been studied in four-fermion interaction models with periodic or anti-periodic boundary conditions for the fermion fields. The finite size effect with a U(1)-valued boundary condition can be realized in the presence of a gauge field. It is well-known that a constant magnetic field enhances the chiral symmetry breaking. We study four-fermion interaction models on M D−1 ⊗ S1 and develop the procedure to calculate the stable environment and the Casimir effect. Evaluating the effective potential with a zero-point energy, we obtain the dynamically generated fermion mass and the critical length for a fixed boundary condition.
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