Matching the 2HDM to the HEFT and the SMEFT: Decoupling and Perturbativity

We consider the 2 Higgs Doublet Model (2HDM) and compare two effective field theory (EFT) approaches to it, according to whether the heavy degrees of freedom are integrated out before (SMEFT) or after (HEFT) spontaneous symmetry breaking. %We show that, in the HEFT, an inconsistent EFT is obtained if one considers an expansion simply in inverse powers of the heavy masses. %We consider a HEFT expansion consistent with perturbativity %We show that, if the HEFT approach is required to comply with perturbativity, it ends up obeying the same power counting as the SMEFT one. % By requiring decoupling and perturbativity in the 2HDM, we define a consistent EFT expansion in inverse powers of the heavy masses which is applied to both the SMEFT and the HEFT matchings to the 2HDM. We organize this expansion with a dimensionless parameter $\xi$, and investigate the tree-level scatterings $hh\rightarrow hh$ and $WW\rightarrow hh$ up to $\mathcal{O}(\xi^2)$. We find no differences between the HEFT and the SMEFT approaches at this order. We show scenarios where even including dimension-8 operators of the SMEFT is insufficient to obtain an accurate matching to the 2HDM.
Comment: The HEFT matching to the 2HDM is contained in the supplemental material