Complete set of positivity constraints on the HEFT at NLO

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 ± , Z L , and the Higgs boson, h . We find two sets of constraints: (i) specific linear combinations of C P -even Wilson coefficients (WCs) must be positive, and (ii) the magnitudes of some WCs—including all C P -odd ones—must be smaller than products of other C P -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 s t -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 , h h → h h and V L V L , h h → V L h process, there are no reported experimental limits and our theoretical constraints provide the first bounds.