Non-perturbative corrections to the Casimir energy for a scalar field theory with non-linear boundary conditions

TL;DR

The paper addresses how nonlinear boundary conditions on plates can induce non-perturbative corrections to the Casimir energy in a scalar field theory, serving as a toy model for non-Abelian boundary dynamics. It introduces auxiliary boundary fields to enforce the nonlinear conditions, derives the quantum effective action, and computes the Casimir energy using a Gaussian (mean-field) approximation and Gel'fand–Yaglom techniques. A boundary mass generation of order $1/g^2$ emerges, with distinct outcomes for Neumann-like and Dirichlet-like conditions: the Neumann case features two vacua separated by infinite energy barriers and a Casimir force that can switch sign with $g$, while the Dirichlet case yields a non-perturbative minimum for massive scalars at large plate separation, adding a dominant boundary contribution to the energy. These findings shed light on possible non-perturbative boundary phenomena in Yang–Mills theories and have potential implications for lattice YM Casimir studies and the emergence of light boundary mass scales.

Abstract

We consider the case of a free real massive bulk scalar in D=4 dimensions, and embed two parallel plates as interfaces on which we impose non-linear boundary conditions, either Dirichlet- or Neumann-like, parameterized by a new coupling constant g. This mimics a non-Abelian gauge theory supplemented with boundary conditions on surfaces embedded in the bulk. We present the first evidence for a non-perturbative 1/g^2 boundary mass generation and its ensuing correction to the standard Casimir energy. This becomes possible by incorporating dynamical corrections to the effective boundary fields, which are used to build in the boundary conditions directly at the action level.

Figures (1)

• Figure 1: (Left) $E_\text{Cas, N}^{(1)}$ from \ref{['evacmassless0']} for $g=\frac{1}{2}$, $L=1$.; (Right) The corresponding non-linear Neumann Casimir force.