  Physics beyond the Standard Model (BSM) can be described in a consistent and
general way through the Higgs Effective Field Theory (HEFT). Measurements of
model-independent HEFT coefficients allow one to constrain the parameter space
of BSM models via a matching procedure. In this work, we show that this
procedure is not unique and depends on the scalings of the parameters of the
Lagrangian. As examples, we consider three BSM models: the real singlet
extension of the SM with a $Z_2$ symmetry, the complex singlet extension (CSE)
of the SM and the 2 Higgs Doublet Model. We discuss several physical
observables, and show that different scalings of the model parameters with the
UV scale in the matching to the HEFT can yield quite different results. This
complicates the interpretation of HEFT measurements in terms of parameters of
BSM models. Additionally, as a by-product, we report the first matching of the
CSE to the HEFT.
