We study the Renormalisation Group (RG) running of the non-singlet tensor operator, for $N_\mathrm{ \scriptstyle f}=3$ QCD with Wilson fermions in a mixed action setup, with standard Schr\"odinger Functional (SF) boundary conditions for sea quarks and chirally rotated Schr\"odinger Functional ($\chi$SF) boundary conditions for valence quarks. Based on a recursive finite-size scaling technique we compute non-perturbatively the tensor step-scaling function for an energy range between a hadronic scale and an electroweak scale, above which perturbation theory may be safely applied. Our result is expressed as the RG-running factor $T^{ \mathrm{ RGI}}/[ T(\mu_{ \mathrm{ had}})]_{ \scriptstyle \rm R}$, where the numerator is the scale independent (Renormalisation Group Invariant - RGI) tensor operator and the denominator is its renormalised counterpart at a hadronic scale $\mu_{ \mathrm{ had}} = 233(8)$~MeV in a given scheme. We determine the step-scaling function in four distinct renormalisation schemes. We also compute the renormalisation parameters of these schemes at $\mu_{ \mathrm{ had}}$ which, combined with the RG-running factor, gives the scheme-independent quantity $Z^{ \mathrm{ RGI}}_{ \mathrm T}(g_0^2)$ in four schemes and for a range of bare gauge couplings in which large volume hadronic matrix element simulations are performed by the CLS consortium in $N_\mathrm{ \scriptstyle f}=2+1$ QCD. All four results are compatible and also agree with a recent determination based on a unitary setup for Wilson quarks with Schr\"odinger Functional boundary conditions~arXiv:2309.04314 . This provides a strong universality test.
Categories: hep-lat