Nonexistence of Multiple-Instanton Solution of the Massless 4D ϕ4 Theory

The massless ϕ⁴ theory in four Euclidean dimensions is shown not to have a multiple-instanton solution — even though it does have a single-instanton solution. Our criterion for a multiple-instanton solution of this nontopological theory is that the energy-momentum tensor of the solution be identically zero. Arguments are given that a similar result holds in any number of dimensions for a one-component scale-invariant scalar theory.