BSM patterns in scalar-sector coupling modifiers

We consider what multiple Higgs interactions may yet reveal about the scalar sector. We estimate the sensitivity of a Feynman topology-templated analysis of weak boson Higgs pair production at present and future colliders - where the signal is a function of the Higgs coupling modifiers ${\kappa }_V$, ${\kappa }_{2V}$, and ${\kappa }_{\lambda }$. While measurements are statistically limited at the LHC, they are under general perturbative control at present and future colliders, departures from the SM expectation give rise to a significant future potential for BSM discrimination in ${\kappa }_{2V}$. We explore the landscape of BSM models in the space of deviations in ${\kappa }_V$, ${\kappa }_{2V}$, and ${\kappa }_{\lambda }$, highlighting models that have measurable order-of-magnitude enhancements in either ${\kappa }_{2V}$ or ${\kappa }_{\lambda }$, relative to their deviation in the single Higgs coupling ${\kappa }_V$.