Perturbative determination of $c_{SW}$ with Symanzik improved gauge action and stout smearing

We determine the improvement factor $c_{SW}$ in one-loop lattice perturbation theory for the plaquette and Symanzik improved gauge actions. The fermionic action is $\mathcal{O}(a)$ clover improved with one-time stout smearing. $c_{SW}$ is derived from the one-loop correction to the quark-quark-gluon vertex in the off-shell regime. We give a first numerical value for the one-loop contribution to the non gauge-invariant improvement coefficient $c_{NGI}$ for the quark field using the plaquette action. A discussion of mean field improvement is included.