Interplay of New Physics Effects in $(g-2)_\ell$ and $h\to\ell^+\ell^-$ -- Lessons from SMEFT

We explore the interplay of New Physics (NP) effects in $(g-2)_\ell$ and $h \to \ell^+ \ell^-$ within the Standard Model Effective Field Theory (SMEFT) framework, including one-loop Renormalization Group (RG) evolution of the Wilson coefficients as well as matching to the observables below the electroweak symmetry breaking scale. We include both the leading dimension six chirality flipping operators including a Higgs and $SU(2)_L$ gauge bosons as well as four-fermion scalar and tensor operators, forming a closed operator set under the SMEFT RG equations. We compare present and future experimental sensitivity to different representative benchmark scenarios. We also consider two simple UV completions, a Two Higgs Doublet Model and a single scalar LeptoQuark extension of the SM, and show how tree level matching to SMEFT followed by the one-loop RG evolution down to the electroweak scale can reproduce with high accuracy the $(g-2)_\ell$ and $h \to \ell^+ \ell^-$ contributions obtained by the complete one- and even two-loop calculations in the full models.