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New vacuum boundary effects of massive field theories

Manuel Asorey, Fernando Ezquerro, Miguel Pardina

TL;DR

This work investigates how boundary conditions shape the Casimir energy of a massive scalar field in 3+1 dimensions, focusing on low-temperature and large-distance limits. Using a zeta-function formalism with a $U(2)$ boundary-condition parametrization, it derives explicit expressions for the Casimir energy and shows two distinct exponential decays governed by $\mathrm{tr}(U\sigma_1)$. It provides exact results for canonical cases (Periodic, Dirichlet, Neumann, Anti-periodic, Zaremba) and establishes a general criterion for the asymptotic decay rate. The findings offer a theoretical bridge to the infrared behavior of non-Abelian gauge theories and guide future nonperturbative comparisons in 3+1 dimensions, potentially shedding light on the mass-gap problem.

Abstract

Analytical arguments suggest that the Casimir energy in 2+1 dimensions for gauge theories exponentially decays with the distance between the boundaries. The phenomenon has also been observed by non-perturbative numerical simulations. The dependence of this exponential decay on the different boundary conditions could help to better understand the infrared behavior of these theories and in particular their mass spectrum. A similar behavior is expected in 3+1 dimensions. Motivated by this feature we analyze the dependence of the exponential decay of Casimir energy for different boundary conditions of massive scalar fields in 3+1 dimensional spacetimes. We show that the boundary conditions classify in two different families according on the rate of this exponential decay of the Casimir energy. If the boundary conditions on each boundary are independent (e.g. both boundaries satisfy Dirichlet boundary conditions), the Casimir energy has a exponential decay that is two times faster than when the boundary conditions interconnect the two boundary plates (e.g. for periodic or antiperiodic boundary conditions). These results will be useful for a comparison with the Casimir energy in the non-perturbative regime of non-Abelian gauge theories.

New vacuum boundary effects of massive field theories

TL;DR

This work investigates how boundary conditions shape the Casimir energy of a massive scalar field in 3+1 dimensions, focusing on low-temperature and large-distance limits. Using a zeta-function formalism with a boundary-condition parametrization, it derives explicit expressions for the Casimir energy and shows two distinct exponential decays governed by . It provides exact results for canonical cases (Periodic, Dirichlet, Neumann, Anti-periodic, Zaremba) and establishes a general criterion for the asymptotic decay rate. The findings offer a theoretical bridge to the infrared behavior of non-Abelian gauge theories and guide future nonperturbative comparisons in 3+1 dimensions, potentially shedding light on the mass-gap problem.

Abstract

Analytical arguments suggest that the Casimir energy in 2+1 dimensions for gauge theories exponentially decays with the distance between the boundaries. The phenomenon has also been observed by non-perturbative numerical simulations. The dependence of this exponential decay on the different boundary conditions could help to better understand the infrared behavior of these theories and in particular their mass spectrum. A similar behavior is expected in 3+1 dimensions. Motivated by this feature we analyze the dependence of the exponential decay of Casimir energy for different boundary conditions of massive scalar fields in 3+1 dimensional spacetimes. We show that the boundary conditions classify in two different families according on the rate of this exponential decay of the Casimir energy. If the boundary conditions on each boundary are independent (e.g. both boundaries satisfy Dirichlet boundary conditions), the Casimir energy has a exponential decay that is two times faster than when the boundary conditions interconnect the two boundary plates (e.g. for periodic or antiperiodic boundary conditions). These results will be useful for a comparison with the Casimir energy in the non-perturbative regime of non-Abelian gauge theories.

Paper Structure

This paper contains 12 sections, 59 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Renormalized effective action
  3. Zeta function in low temperature regime
  4. Contribution of the modes with $l\not =0$
  5. Free energy
  6. Casimir energy in the asymptotic limit
  7. Particular cases of boundary conditions
  8. Asymptotic behavior of Casimir energy
  9. Conclusions
  10. Explicit calculations of the free energy
  11. Dirichlet boundary conditions
  12. Periodic boundary conditions

Figures (2)

  • Figure 1: Behavior of the adimensional Casimir energy (in logarithmic scale) for different boundary conditions as a function of the adimensional distance $mL$.
  • Figure 2: Behavior of the adimensional free energy in logarithmic scale for four different boundary conditions as a function of the effective distance $mL$ with a fixed temperature $m\beta=1$.