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Indirect probes of the trilinear Higgs coupling: $gg \to h$ and $h \to γγ$

In the framework of the Standard Model effective field theory, we examine the indirect constraints on the trilinear Higgs coupling $λ$ that arise from Higgs production in gluon-gluon-fusion and diphoton Higgs decays. We calculate 2-loop contributions to the $gg \to h$ and $h \to γγ$ amplitudes that are affected by modifications of the trilinear Higgs-boson vertex. This calculation involves both the computation of anomalous dimensions and finite matching corrections. Based on our new results, we analyse the sensitivity of present and future measurements of the $hgg$ and $h γγ$ couplings to shifts in $λ$. Under the assumption that $O_6 = - λ\left (H^\dagger H \right )^3$ is the only dimension-6 operator that alters the trilinear Higgs interactions, we find that at present the considered loop-level probes provide stronger constraints than $pp \to 2h$. At future high-energy colliders indirect ${\cal O} (5)$ determinations of the trilinear Higgs coupling may be possible, making precision measurements of $gg \to h$ and $h \to γγ$ a useful addition to direct extractions of $λ$ through double-Higgs production.