The Geometric $\nu$SMEFT: Operators and Connections

We write down a geometric realization of the Standard Model Effective Field Theory (SMEFT) extended by $n_f$ flavours of light sterile neutrinos, a so-called geo$\nu$SMEFT. As with the geoSMEFT introduced by Helset, Martin and Trott, we show that a refactorization of the $\nu$SMEFT's operator product expansion is possible, such that two- and three-point composite operator forms are dressed with field-space connections composed of towers of Higgs dressings and symmetry generators, valid at \emph{all-orders} in the $\overline{v}_T/\Lambda$ expansion parameter of the EFT ($\overline{v}_T \equiv \sqrt{2\langle H^\dagger H\rangle}$) . These connections are parameterized by real Higgs coordinates and contribute to the field-space geometry of the ($\nu$)SM, with structure linked to the strength of Beyond-the-($\nu$)Standard Model physics encoded in $\overline{v}_T/\Lambda$. In addition to enumerating the relevant composite operators and associated connections, we briefly outline the route to calculating all-$\overline{v}_T/\Lambda$-orders amplitudes, including the flavor-invariant theory required to understand the neutrino mass-eigenstate basis geometrically.