Constraining the Higgs Trilinear Coupling from an $SU(2)$ Quadruplet with Bounded-from-Below Conditions

Integrating out a heavy scalar can cause the Higgs trilinear coupling to deviate from its Standard Model value: a good example is provided by an $SU(2)$ quadruplet. Constraints on the full theory, however, can limit the size of the deviation. We show that the bounded-from-below conditions for the Standard Model extended by an $SU(2)$ quadruplet strongly constrain the ${\mathbb{Z}}_2$-breaking Higgs portal and can bound the Higgs trilinear coupling close to its Standard Model value. For TeV-scale quadruplet masses in models with custodial symmetry violation, these constraints can be a few times stronger than constraints from electroweak precision measurements. For the custodial quadruplet, these are the strongest theoretical constraints available.