Analytic structure of $\phi^4$ theory using light-by-light sum rules

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.