Integrating out a heavy Higgs singlet: on the edge between SMEFT and HEFT

We use a functional approach based on the background-field formalism and the expansion by regions to integrate out the heavy Higgs field (associated with the mass eigenstate H) at the one-loop level in a singlet extension of the SM, which features an additional real scalar singlet and a spontaneously broken $Z_2$ symmetry. We obtain an effective Lagrangian to $O(1/M_H^2)$ in the limit of large Higgs mass ($M_H\gg M_h=125$GeV) providing a consistent treatment of effects from Higgs mixing and the renormalization of the underlying model. The scaling behaviour of the model parameters in the large-$M_H$ limit determines whether the effective Lagrangian can be accommodated in the SM Effective Field Theory (SMEFT) or involves non-SMEFT operators within the more general Higgs Effective Field Theory (HEFT) framework. We choose a limit that ensures decoupling of beyond-SM (BSM) effects at $O(M_H^0)$ by demanding that the Higgs mixing angle $\alpha$ is of $O(M_h/M_H)$ and putting minimal constraints on the other input parameters. The considered model is restricted to massless fermions. Bottom-up (diagrammatic) matching with only bosonic SMEFT operators at $O(1/M_H^2)$ necessarily fails, although the BSM sector does not directly couple to massless fermions. However, it is possible to match SMEFT with bosonic and fermionic operators to the SESM in the considered large-$M_H$ limit. For the emerging Effective Field Theory (EFT) we give two alternative Lagrangians: one that includes only bosonic BSM EFT operators, but necessarily involves operators of non-SMEFT type, and another one that is of SMEFT form, but involves also fermionic EFT operators. We validate our results for the effective Lagrangians at NLO in the coupling expansion by verifying that the difference between EFT and full-theory predictions vanishes faster than $1/M_H^2$ for several electroweak observables in the large-$M_H$ limit.