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

## TL;DR

This paper investigates the Higgs oblique parameter $ ilde{H}$, linking theoretical consistency conditions to phenomenological implications, and proposes four-top production as a key process to probe off-shell Higgs effects beyond traditional methods.

## Contribution

It introduces the $ ilde{H}$ parameter as a unique EFT observable, derives self-consistency conditions from fundamental principles, and identifies four-top production as a novel probe of off-shell Higgs physics.

## Key findings

- $gg o h^* o VV$ is insensitive to $ ilde{H}$ corrections.
- Four-top production effectively probes off-shell Higgs behavior.
- Self-consistency conditions constrain Wilson coefficients in the EFT.

## Abstract

We study, from theoretical and phenomenological angles, the Higgs boson oblique parameter $\hat{H}$, as the hallmark of off-shell Higgs physics. $\hat{H}$ 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 $\hat{H}$, 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.