Fixing dilaton boundary value and potential normalization without ad hoc entropy matching

Determine a principled procedure to fix the dilaton boundary value φ_b and the normalization U_0 of the dilaton potential U(φ) in the constructed dilaton–gravity model with action S = ∫_M √g (φ R + U(φ)) + 2 ∫_{∂M} √γ φ_b K and metric ds^2 = (1 − 16π^2 r^2/β^4) dr^2 + r^2 dθ^2, without resorting to ad hoc matching to the Heisenberg model’s thermal entropy; specify how φ_b and U_0 should be determined from first principles or boundary conditions within the dilaton–gravity framework.

Background

After reconstructing the effective bulk metric from geodesic lengths and solving for the corresponding dilaton potential, the authors compute thermodynamics of the resulting dilaton–gravity model. They note that reproducing the Heisenberg model’s thermal entropy requires choosing φ_b and U_0, but they lack a principle to fix these parameters and instead match to the entropy by hand.

Establishing a first-principles method to determine φ_b and U_0 would strengthen the correspondence between the Heisenberg model and the effective dilaton–gravity description and clarify the physical interpretation of these parameters.