Obtain analytic expressions for the free-scalar BCFT entropy and free-energy densities

Derive closed-form analytic expressions for the dimensionless entropy density and dimensionless free-energy density of a free massless scalar BCFT with Neumann or Dirichlet boundary conditions in four dimensions.

Background

The paper computes the dimensionless energy density for a free massless scalar with Neumann and Dirichlet boundary conditions and defines the entropy and free-energy densities through thermodynamic relations. Although the energy density is known analytically, the entropy and free-energy densities require integrations that the authors evaluate numerically.

Obtaining analytic formulas would clarify the pointwise behavior of these thermodynamic quantities and their behavior under the boundary renormalization-group flow from Neumann to Dirichlet boundary conditions.

References

We have not been able to find analytic expressions for f(z) and s(z).

— Boundaries and defects in conformal field theories at finite temperature  (2609.38130 - Artico et al., 29 Sep 2026) in Section 4.1.1, subsection “Thermal one-point function of T_{\mu\nu}”