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The apparent breakdown of unitarity in low order perturbation theory is oftenis used to place bounds on the parameters of a theory. In this work we give analgorithm for approximately computing the next-to-leading order (NLO)perturbativity bounds on the quartic couplings of a renormalizable theory whosescalar sector is $\phi^4$-like. By this we mean theories where either there areno cubic scalar interactions, or the cubic couplings are related to the quarticcouplings through spontaneous symmetry breaking. The quantity that tests whereperturbation 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. Wealso present a simple example to illustrate the effect of considering thesebounds at different orders in perturbation theory. For example, there is anoticeable difference in the viable parameter when the square of the NLO pieceis included versus when it is not.

NLO Perturbativity Bounds on Quartic Couplings in Renormalizable Theories with $\phi^4$-like Scalar Sectors