Vacuum energy density and pressure inside a soft wall

TL;DR

This paper develops an approximation scheme to compute vacuum energy density and pressure inside a soft wall modeled by a potential, extending previous methods to a specific case with , and verifies results through consistency checks.

Contribution

It introduces a novel approximation approach for the Green function in soft wall models, enabling calculation of vacuum energy and pressure for  potential, and validates the method against known results.

Findings

Computed energy density and pressure for  soft wall model.

Validated approximation scheme through consistency checks and trace anomaly.

Reproduced known results for quadratic wall with improved accuracy.

Abstract

In the study of quantum vacuum energy and the Casimir effect, it is desirable to model the conductor by a potential of the form $V(z)=z^\alpha$. This "soft wall" model was proposed so as to avoid the violation of the principle of virtual work under ultraviolet regularization that occurs for the standard Dirichlet wall. The model was formalized for a massless scalar field, and the expectation value of the stress tensor has been expressed in terms of the reduced Green function of the equation of motion. In the limit of interest, $\alpha \gg 1$, which approximates a Dirichlet wall, a closed-form expression for the reduced Green function cannot be found, so piecewise approximations incorporating the perturbative and WKB expansions of the Green function, along with interpolating splines in the region where neither expansion is valid, have been developed. After reviewing this program, in this article we apply the scheme to the wall with $\alpha=6$ and use it to compute the renormalized energy density and pressure inside the cavity for various values of the conformal parameter. The consistency of the results is verified by comparison to their numerical counterparts and verification of the trace anomaly and the conservation law. Finally, we use the approximation scheme to reproduce the energy density inside the quadratic wall, which was previously calculated exactly but with some uncertainty.