Towards the HEFT-hedron: the complete set of positivity constraints at NLO

TL;DR

This paper derives the complete set of positivity constraints on the Higgs Effective Field Theory at NLO, constraining 15 operators and providing the first theoretical bounds for certain vector boson scattering processes.

Contribution

It identifies all positivity bounds on HEFT operators at NLO, linking them to known SMEFT bounds and deriving new theoretical constraints for vector boson scattering.

Findings

Only about 5% of HEFT parameter space satisfies positivity bounds.

Derived stronger bounds than current experimental limits for vector boson scattering.

Provided first theoretical bounds for processes with no experimental limits.

Abstract

We present the complete set of positivity bounds on the Higgs Effective Field Theory (HEFT) at next-to-leading order (NLO). We identify the 15 operators that can be constrained by positivity, as they contribute to $s^2$-growth in the amplitude for longitudinal gauge-Higgs scattering, that is to all possible 2-to-2 scattering processes involving longitudinal gauge bosons, $V_L = W_L^\pm, Z_L$, and the Higgs boson, $h$. We find two sets of constraints: (i) specific linear combinations of CP-even Wilson coefficients (WCs) must be positive, and (ii) the magnitudes of some WCs -- including all CP-odd ones -- must be smaller than products of other CP-even WCs. We present our final constraints on the 15 dimensional HEFT space and show how known positivity bounds on the 3 dimensional space of dimension 8 SMEFT can be recovered from them. We find that only about $5\%$ of the parameter space for WCs of HEFT operators at NLO complies with these positivity constraints. Additionally, we obtain double-sided bounds on these WCs by fully exploiting the implications of unitarity and $st$-crossing symmetry. For WCs contributing to the vector boson scattering process our final constraints are in most cases significantly stronger than the experimental ones. For the $V_L V_L, hh \to hh$ and $V_LV_L, hh \to V_Lh$ process, there are no reported experimental limits and our theoretical constraints provide the first bounds.