Integrating out heavy scalars with modified equations of motion: matching computation of dimension-eight SMEFT coefficients

The shift in focus towards searches for physics beyond the Standard Model employing model-independent effective field theory methods necessitates a rigorous approach to matching to guarantee the validity of the obtained results and constraints. The limits on the leading dimension-six effective field theory effects can be rather inaccurate for LHC searches that suffer from large uncertainties while exploring an extensive energy range. Similarly, precise measurements can, in principle, test the subleading effects of the operator expansion. In this work, we present an algorithmic approach to automatize matching computations for dimension-eight operators for generic scalar extensions with proper implementation of equations of motion. We devise a step-by-step procedure to obtain the dimension-eight Wilson coefficients in a nonredundant basis to arrive at complete matching results. We apply this formalism to a range of scalar extensions of the Standard Model and provide tree-level and loop-suppressed results. Finally, we discuss the relevance of the dimension-eight operators for a range of phenomenological analyses, particularly focusing on Higgs and electroweak physics