The force of attraction between nucleons due to vacuum fluctuation

Abstract

We investigate quantum vacuum interactions arising from the zero-point fluctuations of a spatially confined massive scalar field. Deriving analytical expressions for the planar interaction energy and vacuum pressure, we identify a fundamental transition from the canonical power-law scaling of massless fields to a distinct quantum saturation regime. We prove that in the macroscopic limit, where the boundary separation far exceeds the field's Compton wavelength ($a \gg λ_c$), the interaction energy does not vanish; instead, it asymptotes to a persistent constant, $-\hbar c / (24πλ_c^3)$. This reveals a cohesive zero-point energy reservoir inherent to massive vacua. Applying this formalism to the femtometer scale of the deuteron ($^2$H) nuclei, we demonstrate that confining massive pion fluctuations generates an attractive force between nucleons.