  We present preliminary results for the renormalization functions (RFs) of a
number of quark and gluon operators studied in lattice QCD using a
gauge-invariant renormalization scheme (GIRS). GIRS is a variant of the
coordinate-space renormalization prescription, in which Green's functions of
gauge-invariant operators are calculated in position space. A novel aspect is
that summations over different time slices of the positions of the operators
are employed in order to reduce the statistical noise in lattice simulations.
We test the reliability of this scheme by calculating RFs for the vector
one-derivative quark bilinear operator, which enters the average momentum
fraction of the nucleon. We use $N_f=4$ degenerate twisted mass clover-improved
fermion ensembles of different volumes and lattice spacings. We also present
first results of applying GIRS when operator mixing occurs: the mixing
coefficients of the gluon and quark singlet energy-momentum tensor operators
are evaluated by imposing appropriate renormalization conditions on the
lattice.
