Nonperturbative running of the quark mass for $N$$_{f}$ = 3 QCD from the chirally rotated Schrödinger functional

We study the renormalization group (RG) running of the quark mass, for <math display="inline"><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>3</mn></math> QCD with Wilson fermions in a mixed action setup, with standard Schrödinger functional (SF) boundary conditions for sea quarks and chirally rotated Schrödinger functional (<math display="inline"><mrow><mi>χ</mi><mi>SF</mi></mrow></math>) boundary conditions for valence quarks. This necessitates the tuning of the boundary factor <math display="inline"><msub><mi>z</mi><mi mathvariant="normal">f</mi></msub><mo stretchy="false">(</mo><msubsup><mi>g</mi><mn>0</mn><mn>2</mn></msubsup><mo stretchy="false">)</mo></math> of the <math display="inline"><mrow><mi>χ</mi><mi>SF</mi></mrow></math> valence action, in order to ensure that QCD symmetries are fully recovered in the continuum. The properties of this novel setup are monitored through the ratios <math display="inline"><msub><mi>Z</mi><mi>S</mi></msub><mo>/</mo><msub><mi>Z</mi><mi>P</mi></msub></math> and <math display="inline"><msub><mi mathvariant="normal">Σ</mi><mi>S</mi></msub><mo>/</mo><msub><mi mathvariant="normal">Σ</mi><mi>P</mi></msub></math> of the renormalization parameters and step scaling functions of the scalar and pseudoscalar densities. Where comparison is possible, our <math display="inline"><msub><mi>Z</mi><mi>S</mi></msub><mo>/</mo><msub><mi>Z</mi><mi>P</mi></msub></math> results are found to agree with previous determinations, based on a mass ratio method [G. M. de Divitiis (ALPHA Collaboration), Eur. Phys. J. C 79, 797 (2019)] and Ward identities [ J. Heitger (ALPHA Collaboration), Eur. Phys. J. C 80, 765 (2020); J. Heitger , Eur. Phys. J. C 81, 606 (2021)], with Schrödinger functional boundary conditions. The behavior of <math display="inline"><msub><mi mathvariant="normal">Σ</mi><mi>S</mi></msub><mo>/</mo><msub><mi mathvariant="normal">Σ</mi><mi>P</mi></msub></math> confirms the theoretical expectations of <math display="inline"><mrow><mi>χ</mi><mi>SF</mi></mrow></math> QCD, related to the restoration of the theory’s symmetries in the continuum limit. From the step-scaling function of the pseudoscalar density we obtain the quark mass RG-running function from hadronic to perturbative energy scales. This is fully compatible with the earlier result obtained in a similar setup for Wilson quarks with Schrödinger functional boundary conditions [I. Campos , Eur. Phys. J. C 78, 387 (2018)], and provides a strong universality test for the two lattice setups.