Fermi Geometry of the Higgs Sector

TL;DR

This work develops a geometric framework for scalar-fermion EFTs by modeling field space as a vector bundle over the scalar base manifold and by employing Fermi normal coordinates to connect curvature data with scattering amplitudes. It provides a covariant description of the SM Higgs sector, detailing the base and fiber metrics, their curvatures, and how custodial-symmetry violation manifests as geometric deformations in both scalar and fermion directions. By showing that high-energy amplitudes encode covariant curvature data, the approach offers a practical, parameterization-invariant route to diagnosing new physics through Higgs phenomenology. The framework generalizes to more fermion content and gauge interactions, outlining clear directions for future extensions and experimental probes of field-space geometry.

Abstract

We develop the field space geometry of scalar-fermion effective field theories as a vector bundle supermanifold. We further establish a Fermi normal coordinate system on the bundle that clarifies the geometric content in scattering amplitudes, particularly the imprints of field space non-analyticities. Specializing to the Standard Model Higgs sector, we examine the geometric consequences of custodial symmetry violation, including implications for the physical Higgs field as a distinguished scalar axis and deformations in the fermionic sector. Our results enable a systematic and realistic geometric interpretation of Higgs sector phenomenology.

Figures (1)

• Figure 1: An illustration of the vector bundle field space $\mathcal{E}$ of scalars and fermions in Fermi normal coordinates $(\phi^{\bm{I}}, \chi^{\bm{r}})$. The scalar field space constitutes the base space $\mathcal{M}$ over which a complex vector space $\mathcal{V}$ for fermion flavor is bundled. A connection $\Gamma$ specifies how parallel transport is performed between fibers over nearby points and makes $\mathcal{E}$ curved over $\mathcal{M}$. The Fermi normal coordinate system is determined by connecting the origin to another point of interest, such as a singularity (denoted here by $\mathcal{S}$), using a geodesic $\mathcal{G}$ which becomes the axis for the primary coordinate $\phi^{\bm{0}}$. In these coordinates, the geodesic is locally flat and an ending sequence of covariant derivatives in $\phi^{\bm{0}}$ is equivalent to partial ones.