  It is commonly believed that a massive real scalar field $\phi$ only mediates
short-range interactions on the scale of its Compton wavelength via the Yukawa
potential. However, in the nonperturbative regime of nonlinear self coupling,
$\phi$ can also mediate larger scale interactions. Moreover, the classical
potential, namely, the static configuration of $\phi$ in the presence of an
external source, is not always unique for given boundary conditions. In this
paper, a complete set of finite-energy potentials (FEPs) induced by a Gaussian
source is identified in one, two, and three spatial dimensions when the
nonlinearity is of the Mexican-hat type, which is often prescribed to induce
spontaneous symmetry breaking. In the size-strength parameter space of the
source, phase boundaries are mapped out, across which the number of FEPs
differ. Additionally, softer phase transitions are delineated according to
whether the potential exhibits a light horizon at which $\phi$ vanishes. The
light horizon is of physical significance when $\phi$ couples with other
particles. For example, when $\phi$ is the Higgs field, all elementary
particles become massless at the light horizon. It is remarkable that white
dwarfs and neutron stars are potentially in a phase where light horizons exist,
whose outer radii are a few times the star sizes. Moreover, suppose elementary
fermions of mass greater than $\sim10^3$ GeV exist, then they may also be
surrounded by light horizons with radii comparable to the Higgs Compton
wavelength. Finally, nonperturbative states may also be realized in condensed
matter systems, wherein phase transitions are controllable using localized
sources.
