Vacuum Energy and Topological Mass from a Constant Magnetic Field and Boundary Conditions in Coupled Scalar Field Theories

TL;DR

This paper studies how a uniform magnetic field and boundary conditions influence vacuum energy and topological mass in coupled scalar fields, using zeta-function regularization to derive effective potentials and analyze different magnetic regimes.

Contribution

It introduces a renormalization scheme that retains magnetic effects and computes vacuum energy and topological mass corrections up to two-loop order in a coupled scalar field system.

Findings

Magnetic field induces Landau levels affecting vacuum energy.

Topological mass emerges from boundary and magnetic contributions.

Asymptotic behaviors differ in weak and strong magnetic fields.

Abstract

We investigate the combined effects of a uniform magnetic field and boundary conditions on vacuum energy and topological mass generation in a coupled scalar field theory. The system consists of a real scalar field, subject to Dirichlet boundary conditions, interacting via self- and cross-couplings with a gauge-coupled complex scalar field obeying mixed boundary conditions between two perfectly reflecting parallel plates. The magnetic field induces Landau quantization, leading to novel contributions. Employing zeta-function regularization within the effective potential formalism, we derive the renormalized effective potential up to second order in the coupling constants without imposing a vanishing magnetic field in the renormalization scheme. Our renormalization approach preserves magnetic contributions while properly removing divergences, enabling a consistent treatment of finite-size corrections, magnetic effects, and interaction terms. We compute the vacuum energy per unit area of the plates, analyze the emergence of a topological mass from boundary and magnetic contributions, and evaluate the first-order coupling-constant corrections at two-loop order. Detailed asymptotic analysis are presented for both weak- and strong-field regimes, revealing exponential suppression at high magnetic fields and nontrivial polynomial and logarithmic behavior in the weak-field limit.