Two-loop dimension Six Effective Action: Integrating Out Heavy Scalar

TL;DR

This work develops a model-independent framework to compute the two-loop effective action up to dimension six after integrating out heavy scalar fields, using the Heat-Kernel method within SMEFT. The approach focuses on vacuum diagrams, derives the Lagrangian corrections in terms of a general potential $U$ and its derivatives, and organizes the results into poles and finite parts that map onto the SILH basis. The authors explicitly apply the formalism to two UV completions—the electroweak complex triplet and a second Higgs doublet—providing complete two-loop dimension-six bosonic and, where applicable, fermionic operator coefficients. The methodology provides a systematic, automatable route for EFT matching at two loops, with detailed tables of Wilson coefficients that connect high-energy scalars to low-energy observables and potential RG analyses.

Abstract

For the first time, we present the model-independent two-loop effective action up to dimension six after integrating out heavy scalar(s) employing the Heat-Kernel method. We compute the effective operators that emerge at two-loop for two example models: heavy electroweak complex Triplet and Doublet scalars. We present our results on the SILH basis. We also capture the effect in the fermion sector. For these two scenarios, we compute all the fermionic effective operators up to dimension six.

Figures (1)

• Figure 1: Left: Sunset Diagram, Middle: Infinity Diagram, Right: Counter-term Diagram