A ${\mathbb{CP}}^2$ SMEFT

An extension of the Standard Model is proposed, where the Higgs field is valued in the complex projective plane ${\mathbb{CP}}^2$, rather than ${\mathbb{C}}^2$. Its geometry is consistent with $U(2) \simeq (SU(2) \times U(1))/ \mathbb{Z}_2$ electroweak gauge symmetry. The leading terms in the Lagrangian, beyond those of the Standard Model, are much more tightly constrained than in general SMEFTs, and the custodial $SO(4)$ symmetry of the Higgs sector is mildly broken. The predicted tree-level deviations from Standard Model phenomenology depend on a single large mass parameter $M$. Current experimental data imply that $M$ is a few TeV or larger.