Fermi Geometry of the Higgs Sector

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.

💡 Research Summary

The paper presents a comprehensive geometric formulation of scalar‑fermion effective field theories (EFTs) by treating the combined field space as a vector‑bundle supermanifold. The scalar fields define a real base manifold M, while the fermionic degrees of freedom live in a complex vector‑space fibre V, giving rise to a total bundle E → M. Field redefinitions are interpreted as coordinate transformations on this bundle: analytic scalar redefinitions correspond to diffeomorphisms of M, and fermion redefinitions correspond to ϕ‑dependent complex linear transformations τ(ϕ) acting on the fibre.

The kinetic terms of the Lagrangian naturally define a Riemannian metric g_{IJ}(ϕ) on M and a Hermitian metric k_{pr}(ϕ) on the fermion bundle. From these metrics one constructs the Levi‑Civita connection Γ^K_{IJ} and the bundle connection Γ^p_{Ir}, Γ^{\bar p}{I\bar r}. The latter are built from the fermion kinetic matrix k and the mixed fermion‑scalar term ω, and they generate curvature tensors R^p{rIJ} and R^{\bar p}_{\bar rIJ} that encode the non‑trivial geometry of the fermionic fibre. A unified graded metric G on the total space is also introduced, allowing a compact expression for all kinetic structures and facilitating the computation of Christoffel symbols and curvature on the full supermanifold.

A central technical advance is the construction of “Fermi normal coordinates” (FNC). Starting from the vacuum point (ϕ=0, χ=0) as the origin, a primary geodesic G is chosen and aligned with a primary coordinate ϕ⁰. Orthogonal secondary geodesics emanating from points on G define the remaining coordinates ϕ^A. In this system the connection components along ϕ⁰ vanish (Γ=0) and higher‑order covariant derivatives reduce to ordinary partial derivatives, as expressed in equations (2.13)–(2.19). Consequently, any tensor evaluated along G can be expanded in a Taylor series whose coefficients are directly related to physical scattering amplitudes. This makes FNC an ideal tool for probing geometric features that lie away from the vacuum, such as singularities or curvature hotspots, without being obscured by intermediate curvature effects.

Applying the formalism to the Standard Model Higgs sector, the authors treat the Higgs field h and the three Goldstone bosons as coordinates on M. In the custodial‑symmetric limit the base manifold possesses an SO(4) isometry, but explicit custodial‑symmetry breaking introduces anisotropic components into g_{IJ} and generates new curvature pieces R_{hIJK}. The physical Higgs direction becomes a distinguished scalar axis, and its curvature couplings encode deviations from the Standard Model.

In the fermionic sector, the top and bottom quarks are taken as representative flavours. The bundle connection Γ^p_{Ir} acquires custodial‑violating pieces, which feed into the field‑dependent fermion mass matrix M_{pr}(ϕ) and the four‑fermion operators c_{prst}. These geometric deformations manifest in scattering processes such as hh → tt or hh → bb, where curvature‑dependent form factors modify the energy dependence and angular distributions. The framework therefore translates abstract geometric data (metrics, connections, curvatures) into concrete, observable signatures in Higgs‑pair production and fermion‑Higgs interactions.

The appendices collect useful phase‑space integration formulas and a catalogue of geometric identities employed throughout the text.

In conclusion, the work establishes a realistic, systematic geometric language for scalar‑fermion EFTs up to second order in the derivative expansion. By introducing Fermi normal coordinates on the vector‑bundle supermanifold, the authors provide a clear bridge between field‑space geometry and measurable scattering amplitudes, especially highlighting how custodial‑symmetry violation reshapes the Higgs sector’s curvature and fermionic connections. The approach is readily extensible to include gauge‑boson longitudinal modes, multiple fermion generations, loop corrections, and color dynamics, offering a powerful toolkit for future phenomenological analyses of Higgs‑sector data.