Abstract
We study the renormalization group (RG) running of the quark mass, for ${N}_{f}=3$ 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 ($\ensuremath{\chi}\mathrm{SF}$) boundary conditions for valence quarks. This necessitates the tuning of the boundary factor ${z}_{\mathrm{f}}({g}_{0}^{2})$ of the $\ensuremath{\chi}\mathrm{SF}$ 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 ${Z}_{S}/{Z}_{P}$ and ${\mathrm{\ensuremath{\Sigma}}}_{S}/{\mathrm{\ensuremath{\Sigma}}}_{P}$ of the renormalization parameters and step scaling functions of the scalar and pseudoscalar densities. Where comparison is possible, our ${Z}_{S}/{Z}_{P}$ results are found to agree with previous determinations, based on a mass ratio method [G. M. de Divitiis et al. (ALPHA Collaboration), Eur. Phys. J. C 79, 797 (2019)] and Ward identities [ J. Heitger et al. (ALPHA Collaboration), Eur. Phys. J. C 80, 765 (2020); J. Heitger et al., Eur. Phys. J. C 81, 606 (2021)], with Schr\"odinger functional boundary conditions. The behavior of ${\mathrm{\ensuremath{\Sigma}}}_{S}/{\mathrm{\ensuremath{\Sigma}}}_{P}$ confirms the theoretical expectations of $\ensuremath{\chi}\mathrm{SF}$ 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 Schr\"odinger functional boundary conditions [I. Campos et al., Eur. Phys. J. C 78, 387 (2018)], and provides a strong universality test for the two lattice setups.
Highlights
We find that the behavior of ZS=ZP at high energies agrees with expectations from perturbation theory; at low energies it agrees with earlier nonperturbative determinations, based on other methods
We have analyzed the ensembles generated for the computation of the renormalization group (RG) running of the quark mass in Nf 1⁄4 3 massless QCD [10], imposing chirally rotated Schrödinger functional boundary conditions on the valence quarks
The data spans a few orders of magnitude, allowing a completely nonperturbative determination of the mass running function between the hadronic and very high-energy scales, where contact with fixed-order perturbation theory can be made
Summary
The chirally rotated Schrödinger functional (χSF) [1] is a variant of the Schrödinger functional (SF) renormalization scheme, which enables us to obtain renormalization parameters and lattice step-scaling functions, which are “automatically” OðaÞ improved. Comparison of results obtained with (i) unimproved SF Wilson fermions, (ii) SF clover fermions, and (iii) χSF fermions confirmed automatic improvement of the latter and the universality of the continuum limit. L denotes the lattice physical extension; note that it ranges from very small values of about L ≈ 10−3 to L ⪅ 1 fm On these ensembles we compute correlation functions with dimension-3 scalar and pseudoscalar bilinear operators in the bulk and χSF boundary conditions. This is a mixed action approach, as sea and valence quarks have different regularizations (SF and χSF respectively). We confirm that the ratio ΣS=P of the corresponding step-scaling functions in the χSF theory goes to unity in the continuum limit, in accordance with chiral symmetry restoration. Our results have been presented in preliminary form in Refs. [15,16]
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