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

TL;DR

The paper introduces the Higgs oblique parameter $\hat{H}$ as the Wilson coefficient of $\mathcal{O}_{\Box}=|\Box H|^2$ in a Universal EFT and develops a framework combining the Källén-Lehmann representation with EFT diagnostics to assess its theoretical consistency. It shows that $gg\to h^*\to VV$ is not sensitive to $\hat{H}$ due to cancellations, making off-shell Higgs observables essential; four-top production $pp\to t\bar{t}t\bar{t}$ is identified as a particularly promising high-energy probe of off-shell Higgs behavior, capable of constraining $\hat{H}$ for moderate to strong UV couplings. The authors construct UV completions illustrating how $\mathcal{O}_{\Box}$ can arise and discuss EFT validity via positivity, convergence, and unitarity constraints that bound the EFT cutoff and the energy range of validity. They provide quantitative collider projections: current and HL-LHC on-shell fits limit $\hat{H}$ at the level of ~0.1–0.2, while HL-LHC and future colliders like FCC-hh can substantially improve sensitivity through off-shell four-top channels, thus offering a complementary route to map universal UV theories in the Higgs sector.

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 Källén-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.

Figures (5)

• Figure 1: In the plane spanned by $c_1$ and $c_2$ (the two leading coefficients of the propagator derivative expansion) we show how the constraints from (i) positivity, (ii) convergence, (iii) perturbative unitarity single out a theoretically-allowed bounded region. An experimental measurement of $a_1$ and $a_2$ (the first two terms in a momentum expansion) selects the curve $c_2 = c_1^2\, a_2/a_1^2$. Examples of these curves (for different values of $a_2/a_1^2$) are shown by solid red lines, which are generated by varying the cutoff mass $M$. The value of $M$ increases along the direction of the arrows. The stronger bound on $M$ comes from convergence when $a_1^2/a_2 \lesssim 4\pi$ and from perturbative unitarity when $a_1^2/a_2 \gtrsim 4\pi$.
• Figure 2: Higgs self-energy correction from the two-point function of the operator $\mathcal{O}$.
• Figure 3: The magnitude of EFT corrections to the propagator in the flat extra dimension Universal theory. Here we plot the full EFT correction to the propagator $\Delta_{\text{EFT}}(x)$ as compared to the using the self-energy approach $\Delta^\Sigma_6(x)$, for which the derivative series diverges, and the approach advertised here, in which the propagator is expanded consistently at dimension-6 to find the correction $\Delta_6(x)$ wherein the derivative expansion must converge. We also show an envelope around the full correction within which the dimension-6 approximation is expected to fall.
• Figure 4: A sample of Feynman diagrams with an off-shell Higgs contribution to four-top production at the LHC ($p p \to t \bar{t} t \bar{t}$).
• Figure 5: The 3 ab$^{-1}$ HL-LHC and 30 ab$^{-1}$ FCC-hh sensitivity projections for the $\hat{H}$ parameter in four-top production ($p p \to t \bar{t} t \bar{t}$). The solid and dashed black curves show the expected sensitivity at 95% CL as a function of the kinematic variable $M_{\textrm{cut}}$ for a different systematic uncertainty $\delta_{sys}$. Superimposed to this plot are three dashed brown lines showing the corresponding values of $c_\Box$ assuming $M = M_{\textrm{cut}}$. The regions above the lines $c_\Box =4\pi$ and $c_\Box =(4\pi)^2$ are incompatible with the criteria of perturbative unitarity and naïve perturbativity, respectively.