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.
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