More accurate $\sigma(\mathcal{G} \,\mathcal{G}\rightarrow h)$, $\Gamma(h \rightarrow \mathcal{G} \,\mathcal{G}, \mathcal{A} \mathcal{A}, \bar{\Psi} {\Psi})$ and Higgs width results via the geoSMEFT

We develop Standard Model Effective Field Theory (SMEFT) predictions of $\sigma(\mathcal{G} \,\mathcal{G}\rightarrow h)$, $\Gamma(h \rightarrow \mathcal{G} \,\mathcal{G})$, $\Gamma(h \rightarrow \mathcal{A} \mathcal{A})$ to incorporate full two loop Standard Model results at the amplitude level, in conjunction with dimension eight SMEFT corrections. We simultaneously report consistent $\Gamma(h \rightarrow \bar{\Psi} {\Psi})$ results including leading QCD corrections and dimension eight SMEFT corrections. This extends the predictions of the former processes $\Gamma, \sigma$ to a full set of corrections at $\mathcal{O}(\bar{v}_T^2/\Lambda^2 (16 \pi^2)^2)$ and $\mathcal{O}(\bar{v}_T^4/\Lambda^4)$, where $\bar{v}_T$ is the electroweak scale vacuum expectation value and $\Lambda$ is the cut off scale of the SMEFT. Throughout, cross consistency between the operator and loop expansions is maintained by the use of the geometric SMEFT formalism. For $\Gamma(h \rightarrow \bar{\Psi} {\Psi})$, we include results at $\mathcal{O}(\bar{v}_T^2/\Lambda^2 (16 \pi^2))$ in the limit where subleading $m_\Psi \rightarrow 0$ corrections are neglected. We clarify how gauge invariant SMEFT renormalization counterterms combine with the Standard Model counter terms in higher order SMEFT calculations when the Background Field Method is used. We also update the prediction of the total Higgs width in the SMEFT to consistently include some of these higher order perturbative effects.