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This paper presents a set of exact solutions for the classical field equation of a quartic massless scalar field. These solutions have the property to satisfy a dispersion relation of a massive field even if one started from a massless field. They can be used as the expectation value on the vacuum state of a quantum field theory representing oscillations around a constant vacuum expectation value. Indeed, such solutions can be used to describe a scalar field theory that develops a mass gap with a tower of oscillations, those of a harmonic oscillator, being completely trivial in the infrared limit. To understand the formation of the mass gap the Green function is also computed and it is shown that a zero mode appears. So, the theory is seen to break conformal invariance.