Towards determination of the strong coupling $α_s(m_Z)$ from four-flavor lattice QCD using the continuous $β$-function method

Abstract

The precise value of the strong coupling $α_s(m_{Z})$ at the $Z$-boson mass $m_{Z}$ is essential for high-energy phenomenology and precision tests of quantum chromodynamics (QCD). We present the status of a program targeting a $\sim 0.3\%$ determination of $α_s(m_{Z})$ using the renormalization group $β$-function in the infinite volume gradient flow scheme based on lattice QCD simulations of degenerate four-flavor highly improved staggered quark (HISQ) ensembles. In particular, we analyze both tree-level cutoff effects and finite-mass effects. We also outline the next steps of the analysis, including the infinite-volume and continuum extrapolations required for a precise determination of $α_s(m_Z)$.

Figures (5)

• Figure 1: The gradient flow $\beta$-function $\beta_{\rm GF}(\tau; m_f, L, \beta_{b})$ on $32^3\times 64$ lattices is shown as a function of the running coupling $g^2_{\rm GF}(\tau; m_f, L, \beta_{b})$ over a broad range of bare couplings ($7.00 \leq \beta_{b} \leq 20.0$), spanning the weak- to strong-coupling regime $g^2_{\rm GF}\in(0.75,18)$. Each color corresponds to a given bare gauge coupling $\beta_{b}$ and covers flow times $\tau/a^2 \in [2.0, 5.0]$. The width of each band represents the associated statistical error. Simulations with $\beta_b\geq 8.0$ were carried out directly in the massless limit $am_f=0$, whereas the three strongest couplings ($\beta_{b}=7.00, 7.25$, and $7.50$) employ $am_{f} = 0.001, 0.0025$, and $0.005$, distinguished by different shades. All data have been corrected with tree level normlization (TLN). For comparison, we overlay the universal one-loop (solid) and two-loop (dashed) as well as the three-loop (dash-dotted) perturbative GF-scheme $\beta$-functions from Ref. HarlanderNeumann2016, all shown as gray curves. The renormalized coupling at the reference scale $\tau_{0}$ ($g^2_{\rm GF}(\tau_0) = 0.3 \times 16 \pi^2/3 \approx 15.79$), is marked by the black vertical dashed line.
• Figure 2: The gradient-flow $\beta$-function, $\beta_{\rm GF}(\tau; L, \beta_{b})$, divided by $g^4_{\rm GF}(\tau; L, \beta_{b})$, on $L/a=32$ volume as a function of the running coupling $g^2_{\rm GF}(\tau; L, \beta_{b})$ over the bare-coupling range $8.00 \leq \beta_{b} \leq 20.0$. Each color corresponds to a fixed bare gauge coupling $\beta_{b}$. For a given color (i.e., fixed $\beta_{b}$), the points trace out a curve as the flow time is varied over $\tau/a^2 \in [2.0, 5.0]$. The width of each band represents the associated statistical error. All data sets are obtained at $am_{f}=0.0$ and include tree-level normalization (TLN) corrections. The resulting $\beta$-function is compared with the universal one-loop (solid) and two-loop (dashed) perturbative predictions, as well as the three-loop (dash-dotted) perturbative $\beta$-function HarlanderNeumann2016, all shown as gray curves.
• Figure 3: Gradient flow running coupling $g^2_{\mathrm{GF}}(\tau;m_{f},L,\beta_{b})$ against the flow time in lattice units $\tau/a^2$ at $\beta_{b}=20.0$, $am_f=0.00$ without tree-level corrections (no TLN, left panel) and with tree-level corrections (TLN, right panel). Each color represents a fixed flow-discretization combination: WW (blue), WC (orange), and WS (green).
• Figure 4: Same as Figure \ref{['tln_weak']} but at $\beta_b=7.0$, $am_f = 0.001$. The lattice data are extrapolated to the massless limit.
• Figure 5: Chiral ($am_{f} \rightarrow 0$) extrapolation of the renormalized coupling $g^2_{\rm GF}(\tau; m_f, L, \beta_b)$ in $am_{f}$ at fixed flow times for strong bare couplings. Left panel shows $\beta_{b}=7.00$, middle panel $\beta_{b}=7.25$, and right panel $\beta_{b}=7.50$. Each color represents a fixed flow time $2.0\leq \tau/a^2 \leq 4.5$ (yellow to purple). The result of extrapolation indicated by a band, the width of the band indicating the statistical error and the central value of the band indicated by a dashed line. The data entering the extrapolation for each band is indicated by an error bar with a circular marker.