On Casimir effect in Yang-Mills theories in three and four dimensions

TL;DR

The paper develops a gauge-invariant framework to compute the Casimir energy for two parallel plates coupled to Yang–Mills fields in 2+1 and 3+1 dimensions by introducing a boundary action that enforces chromoelectric boundary conditions. The core result is that the Casimir energy is equivalent to the exchange of dim G massive gauge modes (one polarization in 2+1, two in 3+1), with the gluon mass m determined nonperturbatively and tied to magnetic screening in the high-temperature limit. In 2+1 the authors obtain a parameter-free expression that agrees exceptionally well with lattice data for SU(2) and SU(3), while in 3+1 they obtain a consistent but data-dependent description that depends on the dynamical mass and potential lattice-system prefactors. Overall, the work provides a gauge-invariant, first-principles account of Casimir energies in YM theories and reinforces the interpretation of the energy in terms of massive gluon exchange, with quantitative agreement to lattice results within the expected uncertainties.

Abstract

We calculate the Casimir energy for the configuration of two parallel plates coupled to nonabelian gauge fields with a Yang-Mills action. We consider both 2+1 and 3+1 dimensions in the manifestly gauge-invariant formalism we have pursued over the last several years which allows us to factor out the gauge degrees of freedom. A boundary action in the functional integral, equivalent to the insertion of operators representing the plates, is used to enforce the required boundary conditions for the gauge fields. The result is for a kinematic regime corresponding to the exchange of gluons with a dynamically generated mass. We find good agreement in 2+1 dimensions and reasonable agreement in 3+1 dimensions with lattice-based numerical evaluations.

Figures (4)

• Figure 1: Comparison of lattice-based results from chern1 (red,dashed), from ngwenya(green, dotted) with our formula (\ref{['13']}) for $SU(2)$ on the left and $SU(3)$ on the right.
• Figure 2: Comparison of lattice-based results for $SU(2)$ in 4 dimensions from ngwenya (green, dotted) with our formula (\ref{['4d-16']}) for $SU(2)$, with $m = 1.067 \sqrt{\sigma_{\rm F}}$ and $C = 1.126$ (red, dashed) and $C = 1$ (blue, solid).
• Figure 3: Comparison of lattice-based results for $SU(3)$ in 4 dimensions from chern2 (red, dashed) and from ngwenya (green, dotted) with our formula (\ref{['4d-16']}) for $SU(3)$, with $m = 1.0\sqrt{\sigma_{\rm F}}$ and $C = 1$ (blue, solid).
• Figure 4: Comparison of lattice-based results for $SU(3)$ in 4 dimensions from chern2 (red, dashed) and from ngwenya (green, dotted) with our formula (\ref{['4d-16']}) for $SU(3)$, with $m = 1.41\sqrt{\sigma_{\rm F}}$ and $C = 1$ (blue, solid).