Non-perturbative renormalization of tensor bilinears in Schrödinger Functional schemes

We present preliminary result for the study of the renormalization group evolution of tensor bilinears in Schrödinger Functional (SF) schemes for $N_f=0$ and $N_f=2$ QCD with non-perturbatively $\mathcal{O}(a)$-improved Wilson fermions. First $N_f=2+1$ results (proceeding in parallel with the ongoing computation of the running quark masses [1] are also discussed. A one-loop perturbative calculation of the discretisation effects for the relevant step scaling functions has been carried out for both Wilson and $\mathcal{O}(a)$-improved actions and for a large number of lattice resolutions. We also calculate the two-loop anomalous dimension in SF schemes for tensor currents through a scheme matching procedure with RI and $\overline{\rm MS}$. Thanks to the SF iterative procedure the non-perturbative running over two orders of magnitude in energy scales, as well as the corresponding Renormalization Group Invariant operators, have been determined.

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