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

TL;DR

This paper develops a functional approach to integrate out a heavy Higgs singlet in an extended Standard Model, analyzing the resulting effective theories and their relation to SMEFT and HEFT frameworks.

Contribution

It provides a one-loop level derivation of the effective Lagrangian for a heavy Higgs singlet, clarifying conditions under which SMEFT or HEFT descriptions are appropriate.

Findings

Effective Lagrangian obtained at order 1/M_H^2 in the large Higgs mass limit.

Bottom-up matching with only bosonic SMEFT operators at this order fails.

Validation shows EFT and full-theory predictions converge faster than 1/M_H^2 for electroweak observables.

Abstract

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) in a singlet extension of the Standard Model (SM) at the one-loop level. In this way, we obtain an effective Lagrangian to $\mathcal{O}(1/M_H^2)$ in the limit of large Higgs mass ($M_H \gg M_h\approx125$GeV) providing a consistent treatment of effects from Higgs mixing and the renormalization of the underlying model. In particular, we address how the choice of 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. For our calculation, we choose a limit that ensures decoupling of beyond-SM effects at $\mathcal{O}(1/M_H^0)$ by demanding that the Higgs mixing angle $\alpha$ is of $\mathcal{O}(M_h/M_H)$ and putting minimal constraints on the other input parameters, which represents a phenomenologically viable scenario. The considered model is restricted to massless fermions, and the emerging Effective Field Theory (EFT) involves only bosonic EFT operators in addition to the SM Lagrangian. We explicitly show that any attempt of a bottom-up (diagrammatic) matching with only bosonic SMEFT operators at $\mathcal{O}(1/M_H^2)$ necessarily fails. We validate our results for the effective Lagrangian at next-to-leading order 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 precision observables in the large-$M_H$ limit. Moreover, both full-theory and EFT predictions asymptotically approach the corresponding SM results for very large values of $M_H$ as required by decoupling.