Renormalized quark masses using gradient flow

Abstract

We propose a new and simple method for determining the renormalized quark masses from lattice simulations. Renormalized quark masses are an important input to many phenomenological applications, including searching and modeling physics beyond the Standard Model. The non-perturbative renormalization is performed using gradient flow combined with the short-flow-time expansion that is improved by renormalization-group (RG) running to match to the $\overline{\text{MS}}$-scheme. Implementing the RG running perturbatively, we demonstrate this method works reliably at least up to the charm-quark mass and exhibits an easily-attainable ``windowing condition''. Using RBC/UKQCD's (2+1)-flavor Shamir domain-wall fermion ensembles with Iwasaki gauge action, we find $m_s^\overline{\text{MS}}(μ=2 \text{ GeV}) = 90(3)$ MeV and $m_c^\overline{\text{MS}}(μ=3 \text{ GeV}) = 972(16)$ MeV. These results predict the scale-independent ratio $m_c/m_s= 12.1(4)$. Generalization to other observables is possible, providing an efficient approach to determine non-perturbatively renormalized fermionic observables like form factors or bag parameters from lattice simulations.

Figures (11)

• Figure 1: The matching factor $\zeta_{AP}^{-1}$ as the function of the flow time $\tau$ for different values of the matching scale $\mu$. The left panel shows the (dotted lines) and (solid lines) predictions without, the right panel with improvement. The vertical dashed lines indicate $\tau_{\text{min}}$ used in our analysis. For consistency we use the three-loop $\gamma^_m$ function when evaluating , but only the two-loop function for . Using three-loop $\gamma^\text{GF}_m$ for both and , the differences would mostly disappear. Recalling that $\zeta_{AP}^{-1}|_=1$, these plots demonstrate the perturbative validity of the approach for our choice of parameters.
• Figure 2: On the left we present the ratios $R^P(t/a;\tau/a^2)$ defined in \ref{['eq:R']} and on the right $R^{AP}(t/a;\tau/a^2)$ introduced in Eq. \ref{['eq.RAP']} for selected flow times $\tau/a^2$ for the $D_s$ meson on the F1S ensemble. Statistical errors are within the symbol sizes. Due to the finite extent $T/a$ of our gauge-field configurations and the use of anti-periodic boundary conditions for fermions in the time direction, we need to explicitly account for the "around-the-world effect" in $R^P(t/a;\tau/a^2)$ which enters with a different sign for $\langle AP\rangle$ than for the $\langle PP\rangle$ correlators. We therefore obtain $\bar{R}$ in \ref{['eq:Rbar']} from the combination $R(t/a;\tau/a^2) \text{cosh}(M_\text{PS}(T/2-t))/\text{sinh}(M_\text{PS}(T/2-t))$. Since the pseudo-scalar mass $M_\text{PS}$ can be measured very precisely (see \ref{['Tab.M_PS']}), this combination shows a long, flat plateau. The alternative combination $R^{AP}(t/a;\tau/a^2)$ avoids the problem of different "around-the-world" contributions and directly shows long, flat plateaus, as illustrated on the right panel. For both quantities we perform correlated fits to the plateaus from time slice 36 to 46, extracting the values of $\bar{R}(\tau/a^2)$ shown in the legend. Magnified plateau regions are shown in Fig. \ref{['Fig.zoom']}.
• Figure 3: Extracting $\bar{R}$-ratios (see \ref{['eq:Rbar']}) for $D_s$ meson correlators. Top: dependence on the flow time $\tau/a$ in lattice units for our six ensembles. Bottom: converting to physical units and adding the $a\to 0$ continuum limit predictions.
• Figure 4: Details of the continuum-limit extrapolation at selected flow times $\tau=0.090,\, 0.181,\, 0.297\, \text{GeV}^2$ for the $D_s$ meson. Since the data points for M1, M2, M3 as well as C1 and C2 sit on top of each other, sea quark mass effects are not resolved. Hence we obtain the $a\to 0$ limit simply by fitting an ansatz linear in $a^2$.
• Figure 5: The large panel illustrates extracting $(m_c+m_s)/2$ at the renormalization scale $\mu_=4 \text{ GeV}$, converted to $\mu=3 \text{ GeV}$ using 4-loop $N_f=4$$\overline{\text{\scalefont{.9}MS}}$ running Chetyrkin:2000ytHerren:2017osy. Pink squares and light blue triangles denote to the and the values, respectively. Purple diamonds and dark blue circles show the corresponding improved and values. The $\tau\to 0$ limit is taken in a $(\tau_\text{min},\tau_\text{max})$ range as explained in the text. The shaded error bands show the maximal spread of quadratic or linear-log fits using the central values plus/minus their uncertainty. Only filled symbols enter the $\tau\to 0$ extrapolation. The small panel compares the final predictions using solid filled symbols for linear-log and shaded fillings for quadratic fits.