The precise study of the Higgs boson properties -- in particular its mass and couplings -- is of the utmost importance for the investigation of BSM models with extended Higgs sectors. As New Physics currently seems to be driven to higher scales by experimental searches, the framework of Effective Field Theory is becoming necessary to address the large mass hierarchies that appear in calculations.
In this context, there is a growing interest for the precise matching of Higgs couplings, and its automation. I will present recent work  on the matching of renormalisable couplings -- in particular scalar quartic and Yukawa couplings -- between generic theories at one-loop order. I will first show how to avoid a potential loss of accuracy in the automated calculation of threshold corrections due to large logarithmic terms in both the low- and high-energy parts of the corresponding matching conditions. Then, I will discuss different possible choices of renormalisation schemes and procedures that can potentially simplify the matching, focusing in particular on the treatment of mixing between scalars. Finally, I show some examples of results, with a comparison of different schemes in a toy model, and expressions for the one-loop matching of the Higgs quartic coupling in Supersymmetric models with Dirac gauginos.
: JB, M.Goodsell, P.Slavich, arXiv:1810.09388