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

We study, from theoretical and phenomenological angles, the Higgs boson oblique parameter Ĥ, as the hallmark of off-shell Higgs physics. Ĥ 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 Källén-Lehmann representation. Phenomenologically, we demonstrate that the process gg → h$^{∗}$ → V V is insensitive to propagator corrections from Ĥ, 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.