SMEFT as a slice of HEFT's parameter space

The Standard Model Effective Field Theory (SMEFT) is the parametrization chosen to interpret many modern measurements. We have recently discussed, building on the work of other groups, that its overall framework can be experimentally tested, beyond simply constraining its parameters. This is because the Higgs Effective Field Theory (HEFT) is somewhat more general, as it does not assume that the Higgs boson $h$ needs to be embedded in a complex doublet $H$ on which the Standard Model (SM) and SMEFT are built. As a result, the HEFT parameter spaces for the various relevant channels contains hypersurfaces over which one may use SMEFT to describe data. If experimental measurements of HEFT's parameters in any of those various channels yield a point outside of any of the hypersurfaces, SMEFT is falsified; meanwhile, its framework remains appropriate (in particular, as long as the SM remains compatible with data). A common necessity of the various possible tests is that processes involving different number of Higgs bosons (maintaining the number and nature of other particles unchanged) need to be contrasted.