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

TL;DR

We address vacuum energy and topological mass generation in a magnetized, confined, two-scalar system by formulating a coupled real and complex scalar theory with Dirichlet and mixed boundary conditions between parallel plates. The authors develop a renormalized effective potential via zeta-function regularization that preserves magnetic contributions, performing renormalization in the Minkowski limit without forcing $B=0$, and compute the vacuum energy per unit area together with first-order coupling corrections up to two-loop order. They find a topological mass $m_T^2$ arising from boundary and magnetic effects, with its sign controlled by the competition between Dirichlet and magnetic terms; the strong-field regime yields exponential suppression while the weak-field regime shows polynomial and logarithmic behavior. The results elucidate how external fields and finite size modify vacuum fluctuations in coupled scalar theories, with implications for magnetized condensed-matter systems and high-energy contexts, and lay groundwork for extensions to finite temperature and more complex field content.

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.

Figures (6)

• Figure 1: Schematic view of two perfectly reflecting parallel plates in the $x$-$y$ plane separated by $L$ along the $z$-axis, with a uniform magnetic field $\mathbf{B}$ directed along $z$. The real scalar field $\psi$ satisfies Dirichlet boundary conditions, while the gauge-coupled complex scalar field $\phi$ obeys mixed boundary conditions.
• Figure 2: Ratio $\frac{E_R}{E_D}$ obtained from Eq. \ref{['VErenormalized']} plotted against $B_L$ for fixed $mL=1$, denoted as $m_L$. The curves illustrate the influence of the magnetic field contribution on the total vacuum energy.
• Figure 3: Vacuum energy contributions per unit area for Dirichlet part $\frac{E_D}{A}$ (left) and massless magnetic part $\frac{E_B}{A}$ (right). These plots illustrate the individual roles of boundary conditions and magnetic effects on the vacuum energy.
• Figure 4: Comparison between the expressions for $\frac{E_B}{A}$ given in Eqs. \ref{['VErenormalized_alternative']} and \ref{['VErenormalized']}, for $\mu_L=0.1$, 0.5 and 1.0.
• Figure 5: Topological mass given by Eq. \ref{['TM']} as a function of $B_L$, for $mL = 1$ and different values of $\mu_L$. The plots correspond to the cases $\lambda_{\psi} = 1.0$, $g = 0.2$ (left) and $\lambda_{\psi} = 0.02$, $g = 100$ (right).