  The apparent breakdown of unitarity in low order perturbation theory is often
is used to place bounds on the parameters of a theory. In this work we give an
algorithm for approximately computing the next-to-leading order (NLO)
perturbativity bounds on the quartic couplings of a renormalizable theory whose
scalar sector is $\phi^4$-like. By this we mean theories where either there are
no cubic scalar interactions, or the cubic couplings are related to the quartic
couplings through spontaneous symmetry breaking. The quantity that tests where
perturbation theory breaks down itself can be written as a perturbative series,
and having the NLO terms allows one to test how well the series converges. We
also present a simple example to illustrate the effect of considering these
bounds at different orders in perturbation theory. For example, there is a
noticeable difference in the viable parameter when the square of the NLO piece
is included versus when it is not.
