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We study the Renormalisation Group (RG) running of the quark mass, for$N_f=3$ QCD with Wilson fermions in a mixed action setup, with standardSchr\"odinger Functional (SF) boundary conditions for sea quarks and chirallyrotated Schr\"odinger Functional ($\chi$SF) boundary conditions for valencequarks. This necessitates the tuning of the boundary factor $z_f(g_0^2)$ of the$\chi$SF valence action, in order to ensure that QCD symmetries are fullyrecovered in the continuum. The properties of this novel setup are monitoredthrough the ratios $Z_S/Z_P$ and $\Sigma_S/\Sigma_P$ of the renormalisationparameters and step scaling functions of the scalar and pseudoscalar densities.Where comparison is possible, our $Z_S/Z_P$ results are found to agree withprevious determinations, based on a mass ratio method arXiv:1906.03445 and Wardidentities arXiv:2005.01352, arXiv:2101.10969, with Schr\"odinger Functionalboundary conditions. The behaviour of $\Sigma_S/\Sigma_P$ confirms thetheoretical expectations of $\chi$SF QCD, related to the restoration of thetheory's symmetries in the continuum limit. From the step scaling function ofthe pseudoscalar density we obtain the quark mass RG-running function fromhadronic to perturbative energy scales. This is fully compatible with theearlier result obtained in a similar setup for Wilson quarks with Schr\"odingerFunctional boundary conditions arXiv:1802.05243 and provides a stronguniversality test for the two lattice setups.

Nonperturbative running of the quark mass for $N_f=3$ QCD from the chirally rotated Schr\"odinger Functional