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

## TL;DR

This paper develops an algorithm to compute next-to-leading order perturbativity bounds on quartic couplings in scalar theories, improving the accuracy of unitarity-based parameter constraints.

## Contribution

It introduces a method for estimating NLO perturbativity bounds in $\

## Key findings

- NLO corrections significantly affect the bounds on quartic couplings.
- Including NLO terms improves the convergence assessment of the perturbative series.
- The method provides a more accurate determination of the viable parameter space.

## Abstract

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.