The $\widehat{H}$-Parameter: An Oblique Higgs View

We study, from theoretical and phenomenological angles, the Higgs boson oblique parameter $\widehat{H}$, as the hallmark of off-shell Higgs physics. $\widehat{H}$ is defined as the Wilson coefficient of the sole dimension-6 operator that modifies the Higgs boson propagator, within a Universal EFT. Theoretically, we describe self-consistency conditions on Wilson coefficients, derived from the K"{a}ll'{e}n-Lehmann representation. Phenomenologically, we demonstrate that the process $gg\to h^{\ast }\to VV$ is insensitive to propagator corrections from $\widehat{H}$, and instead advertise four-top production as an effective high-energy probe of off-shell Higgs behaviour, crucial to break flat directions in the EFT.