Do 1/r potentials require massless particles?

Long-range 1/r potentials play a fundamental role in physics. Their ultimate origin is usually traced back to the existence of genuine massless particles as photons or gravitons related to fundamental properties of continuum quantum field theories such as gauge invariance. In this Letter, it is argued that, in principle, an asymptotic, infinitesimally weak 1/r potential might also occur in the cutoff version of a simple, one-component spontaneously broken Φ4 theory, after taking into account the peculiar nature of the zero-momentum limit of the connected scalar propagator. Physical interpretation, phenomenological implications and proposals for a new generation of lattice simulations are also discussed.