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 $SU(2) \times U(1)$ 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 $SO(4)$ residual symmetry of the Higgs sector is mildly broken.