  We apply a sum rule for the forward light-by-light scattering process within
the context of the $\phi^4$ quantum field theory. As a consequence of the sum
rule a stringent causality criterion is presented and the resulting constraints
are studied within a particular resummation of graphs. Such resummation is
demonstrated to be consistent with the sum rule to all orders of perturbation
theory. We furthermore show the appearance of particular non-perturbative
solutions within such approximation to be a necessary requirement of the sum
rule. For a range of values of the coupling constant, these solutions manifest
themselves as a physical bound state and a $K$-matrix pole. For another domain
however, they appear as tachyon solutions, showing the inconsistency of the
approximation in this region.
