NLO Unitarity Bounds on Quartic Couplings in a General Renormalizable Theory

The apparent breakdown of unitarity in low order perturbation theory is often is used to place bounds on the parameters of a theory. Typically, however, only a tree level calculation is done to see where this breakdown of perturbation theory occurs. A better approach would be to compute the bounds at multiple orders in perturbation theory and see how well they agree. In this work we give an algorithm for approximately computing the next-to-leading order (NLO) unitarity bounds on the quartic couplings in a general renormalizable theory. We also present a simple example to illustrate the effect of considering unitarity 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.