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We calculate the fermion propagator and the quark-antiquark Green's functionsfor a complete set of ultralocal fermion bilinears, ${{\cal O}_\Gamma}$[$\Gamma$: scalar (S), pseudoscalar (P), vector (V), axial (A) and tensor (T)],using perturbation theory up to one-loop and to lowest order in the latticespacing. We employ the staggered action for fermions and the Symanzik Improvedaction for gluons. From our calculations we determine the renormalizationfunctions for the quark field and for all ultralocal taste-singlet bilinearoperators. The novel aspect of our calculations is that the gluon links whichappear both in the fermion action and in the definition of the bilinears havebeen improved by applying a stout smearing procedure up to two times,iteratively. Compared to most other improved formulations of staggeredfermions, the above action, as well as the HISQ action, lead to smaller tasteviolating effects. The renormalization functions are presented in the RI$'$scheme; the dependence on all stout parameters, as well as on the couplingconstant, the number of colors, the lattice spacing, the gauge fixing parameterand the renormalization scale, is shown explicitly.  We apply our results to a nonperturbative study of the magneticsusceptibility of QCD at zero and finite temperature. In particular, weevaluate the "tensor coefficient", $\tau$, which is relevant to the anomalousmagnetic moment of the muon.

Perturbative renormalization of staggered fermion operators with stout improvement: Application to the magnetic susceptibility of QCD