The Potential of HEFT and the scale of New Physics

TL;DR

This work develops a field-space geometric framework to study the leading high-energy behaviour of tree-level amplitudes with $N$ Nambu–Goldstone bosons and a single Higgs-like scalar with a general potential $V$. By working in Riemann normal coordinates and exploiting covariant derivatives, the authors derive closed-form expressions for the leading $n$-point amplitudes and, from them, inclusive cross sections, decay rates, and perturbative unitarity bounds, all expressed in terms of geometric derivatives of $V$ and field-space curvature. They apply the formalism to the SM scalar sector and contrast HEFT with SMEFT, identifying decoupling and non-decoupling scenarios through three benchmark models, including a dilaton EFT that can exhibit a smooth HEFT-to-SM path without passing through SMEFT. A key result is that, for non-analytic potentials with finite radius of convergence, the sums over intermediate states yield hypergeometric-resummed unitarity constraints that illuminate possible UV completions and “backdoors” to the SM. Overall, the paper provides a powerful, reparameterisation-invariant toolkit for connecting ultraviolet structure to infrared scattering observables in scalar EFTs and offers concrete insights into the viability of HEFT versus SMEFT in describing electroweak symmetry breaking.

Abstract

We employ a geometric framework to compute the leading high-energy behaviour of tree-level scattering amplitudes in theories containing $N$ Nambu-Goldstone bosons and a single Higgs-like scalar with an arbitrary potential $V$. Using these methods, we obtain closed-form expressions for the leading contribution to the full infinite set of tree amplitudes involving any number of Goldstone and Higgs-like external states. These results are then used to derive total cross sections, decay rates, and perturbative unitarity bounds. We then apply our general formalism to the Standard Model scalar sector using the equivalence theorem, working in the regime of small field-space curvature, and use it to characterise the relation between Higgs Effective Field Theory (HEFT) and the Standard Model Effective Field Theory (SMEFT). We analyse three representative classes of models. Two reproduce previously known behaviours, while the third, based on a dilaton effective theory, exhibits a noteworthy feature within the HEFT framework: a smooth decoupling limit that connects HEFT directly to the Standard Model without passing through a SMEFT regime, providing a possible backdoor to the Standard Model.

Figures (13)

• Figure 1: Diagrammatic depiction of the quadratic term in amplitude for unitarity constraints; we have $\bar{a}$ Higgs-like and $\bar{b}$ NGB's coming in and going out while the sum is over $a$ Higgs-like and $b$ NGB particles.
• Figure 2: Unitarity constraint in the Argand plane; the real and imaginary parts of the $2\bar{a}$-Higgs and $2\bar{b}$ NGB derivatives of the potential should lie within the blue shaded circle of radius $r\sqrt{1-\kappa}$.
• Figure 3: Complex $J$ plane showing the singular point of the potential as $\otimes$ and the would-be symmetry restoring point as $\odot$, each at a distance of $\ell^2$ and $\ell_\odot^2$ respectively of the vacuum.
• Figure 4: Ratio of cut-off estimates $E_*^\pi/E_*$ vs order of the polynomial $n$ in $I$ (i.e. operator of dimension $2n$) assessing how much the sum over NGB only overestimates the cut-off compared to the sum over NGBs and Higgses which seems to tend to $1$.
• Figure 5: Left panel: unitarity-bound-derived cut-off $E_*$ as a function of $\ell_S$ for $f_\lambda=10^{-5}$ for the decoupling model of Section \ref{['sec:Dec']}. Right panel: $s_\star\, R_{\pi,h}$ as a function of $\ell_S$ showing that for our parameter choice curvature effects are subdominant.