Hadronic physics from a Wilson fermion mixed-action approach: Setup and scale setting

TL;DR

This work develops and validates a lattice QCD mixed-action approach with $N_f=2+1$ Wilson sea quarks and Wilson twisted-mass valence quarks at maximal twist, establishing a precise matching strategy along a renormalised chiral trajectory and performing a universality test against a unitary Wilson setup. The authors implement a comprehensive scale-setting program using the gradient-flow scale $t_0$ in conjunction with the flavour-averaged decay constant $f_{\pi K}$ to determine $t_0^{\mathrm{ph}}$ and the lattice spacing, incorporating multiple parameterisations of lattice artefacts and a model-averaging procedure to robustly estimate systematic uncertainties. They demonstrate consistency between mixed-action and unitary continuum limits and show that combining both data sets enhances control over extrapolations, yielding a precise determination of $t_0^{\mathrm{ph}}$ and a reliable lattice scale. The methodology is framed to support high-precision hadronic observables and future applications, including charm-quark physics, by leveraging automatic $O(a)$-improvement at maximal twist and rigorous matching in a line of constant physics. Overall, the paper provides a transparent, quantitatively controlled pathway to accurate scale setting and universality checks in mixed-action lattice QCD, with implications for precision SM phenomenology.

Abstract

We introduce a lattice QCD mixed action approach that employs Wilson-type quarks in the sea and valence sectors. The sea sector is based on gauge ensembles with $N_{\rm f}=2+1$ flavours of non-perturbatively O($a$)-improved Wilson fermions generated by the Coordinated Lattice Simulations (CLS) initiative. The parameter space of the considered ensembles encompasses five values of the lattice spacing, a range of pion masses extending down to the physical point, and large physical volumes. In the valence sector, we employ Wilson twisted-mass fermions at maximal twist, using the same massless Wilson-Dirac operator in both the sea and valence sectors. We describe the strategy applied for the required matching of the sea and valence quark masses along the target renormalised chiral trajectory. A precise universality test is then conducted by comparing the continuum-limit results of the mixed-action approach and of the unitary setup, in which the same Wilson fermion regularisation is employed in the sea and in the valence. As a key application, we conduct a scale setting procedure based on lattice determinations of the masses and decay constants of the pion and kaon, as well as the gradient flow scale $t_0$. The scale setting can consequently be performed in three distinct ways, utilising the unitary setup, the mixed action approach, and their combination. We observe that the latter combination results in enhanced control of the systematic uncertainties, thereby yielding a precise determination of the physical value of $t_0$.

Figures (33)

• Figure 1: The unshifted determinations of $\phi_4^{\mathrm{W}}$, as defined in Eq. (\ref{['eqn:phi4']}), for the ensembles considered in Table \ref{['tab:CLS_ens']} are shown as a function of $\phi_2^{\mathrm{W}}$. The grey horizontal band represents the physical value, as determined from the scale setting analysis below (cf. Eq. (\ref{['ch_ss:eq:phi4ph']})), to which all observables will be mass-shifted, while the dashed vertical line marks the physical value of $\phi_{2}$ (cf. Eq. (\ref{['ch_ss:eq:phi2ph']})). These mass-corrections will also result in a slight displacement of the data points on the horizontal axis, i.e. on the light-quark mass proxied by $\phi_2$.
• Figure 2: Consistency checks of the mass-shifting procedure, in which direct measurements carried out on two CLS ensembles, H400 and H402, differing only by their sea quark masses, are compared to the Taylor expansion procedure in Eq. (\ref{['eqn:mass_shift']}). More specifically, we consider two observables, $t_0/a^2$ and $\sqrt{8t_0}f_{\pi K}$, from the unitary setup as a function of the quark mass-shift $\Delta m_i^{\mathrm{ (sea)}}=m_i^{\prime\,{\mathrm{ (sea)}}}-m_i^{\mathrm{ (sea)}}$, starting from the H400 ensemble. The ensembles H400 and H402 share similar parameters, in particular $\beta=3.46,\;L/a=32,\;T/a=96$, and both lie on the SU(3) symmetric line where $m_\pi=m_K$. The H400 ensemble, indicated by the filled black circle, has a pion mass $m_\pi \approx 420\,\mathrm{MeV}$. In the case of the H402 ensemble (black square) with $m_\pi \approx 450\,\mathrm{MeV}$, the measurements reported in Ref. Bruno:2016plf are employed. The green band represents the mass-shift starting from the H400 point, summing over the three quarks flavours, $i=u,d,s$, in the last term on the right-hand side of Eq. (\ref{['eqn:mass_shift']}). The red cross symbol illustrates the position of the mass-shift from the H400 measurement to the physical value $\phi_4=\phi_4^{\mathrm{ph}}$, as determined in Eq. (\ref{['ch_ss:eq:phi4ph']}), required to fix the chiral trajectory. In this regard, it is noteworthy that the H400 ensemble requires the largest shift among all the considered ensembles. We observe that a first order Taylor expansion allows to carry out these small mass-corrections at the expense of increased statistical uncertainty.
• Figure 3: Top row: Derivative $d\left(\sqrt{8t_0}f_{\pi K}\right)/d\phi_4^{\mathrm{W}}$, defined in Eq. (\ref{['eqn:dOdphi4']}), for the unitary "W" setup (left panel), and the mixed-action "Wtm" setup (right panel) as a function of $\phi_2^{\mathrm{W}}$. The coloured bands represent the result of the fit in Eq. (\ref{['eqn:md_1']}) projected to each value of $\beta$. The grey bands correspond to the continuum-limit mass-dependence. The corresponding values of the fit parameters are collected in Table \ref{['tab:md']}. Both p-values are approximately $0.2$. In the unitary setup, the derivative involves all the terms in the right-hand side of Eq. (\ref{['eqn:mass_shift2']}). In contrast, in the mixed-action regularisation, the derivative is a function of the sea contribution alone, which arises from the terms depending on the action $S$. In this latter case, the derivative is employed to impose the physical value of $\phi_{4}$ in the sea sector, whereas the corresponding condition in the valence sector is implemented through the matching procedure described in Sec. \ref{['sec:match']}. Bottom row: As with the top row, these panels display the corresponding derivatives of the observable $\phi_2$. Both p-values in this case are approximately $0.9$.
• Figure 4: Comparison of the unshifted and shifted determinations of $\sqrt{8t_0}f_{\pi K}$ for the unitary setup, where $f_{\pi K}$ is defined in Eq. (\ref{['eq:fpik']}). As discussed in the text, the mass-shifts to $\phi_4^\mathrm{ph}$ based on Eq. (\ref{['eqn:mass_shift']}) and Eq. (\ref{['ch_ss:eq:phi4ph']}) are only applied to the strange quark mass. An empty symbol represents the unshifted determination, whereas the corresponding shifted result, indicated by a filled symbol, is connected by a black dashed line. The vertical dashed line represents the value of $\phi_2^{\mathrm{ph}}$ as determined from the scale-setting analysis (cf. Eq. (\ref{['ch_ss:eq:phi2ph']})).
• Figure 5: Illustration of the determination of $t_0/a^2$ for the ensemble J501 with OBC. Left panel: The Euclidean time dependence of $t^2E(x_0;t)$ is shown, with the time slice action density defined in Eq. (\ref{['eqn:Ex0']}), for a flow time $t$ in the neighbourhood of $t_0$. The green horizontal band corresponds to the model-average result for the extraction of the plateau in the bulk of the Euclidean time direction $x_0$. Right panel: The blue squared points correspond to the determination of $t^2 E(t)$ for the three values of the flow time $t$ closest to $t_0$. From these three points, a small interpolation (blue band) in the flow time $t$ is performed in order to identify the value $t=t_0$ which satisfies the condition $t^2 E(t) = 0.3$ in Eq. (\ref{['eqn:t0_def']}). The interpolated value of $t_{0}/a^{2}$ is indicated by the amber circular data point.