We investigate how non-linear scalar field theories respond to point sources. Taking the symmetron as a specific example of such a theory, we solve the non-linear equation of motion in one spatial dimension for (i) an isolated point source and (ii) two identical point sources with arbitrary separation. We find that the mass of a single point source can be screened by the symmetron field, provided that its mass is above a critical value. We find that two point sources behave as independent, isolated sources when the separation between them is large, but, when their separation is smaller than the symmetrons Compton wavelength, they behave much like a single point source with the same total mass. Finally, we explore closely related behavior in a toy Higgs-Yukawa model, and find indications that the maximum fermion mass that can be generated consistently via a Yukawa coupling to the Higgs in 1+1 dimensions is roughly the mass of the Higgs itself, with potentially intriguing implications for the hierarchy problem.
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