Determination of $α_S$ in the $SU(3)$ Yang-Mills theory

Abstract

The decoupling strategy allows one to obtain the value of the strong coupling in QCD from the running in pure gauge. Here we present our strategy to determine the running in the $SU(3)$ Yang-Mills theory. We use a finite-volume scheme with twisted boundary conditions and a step-scaling approach based on a gradient-flow coupling. We show preliminary results for the continuum extrapolation of the step-scaling function. Compared with other finite-volume approaches, we expect a reduced statistical error and absence of linear cutoff effects due to the translational invariance of the boundary conditions.

Figures (4)

• Figure 1: Tuning process to extract lines of constant physics for the continuum extrapolations of $\Sigma$ and $J_1$.
• Figure 2: Preliminary continuum extrapolations for $\Sigma$ and $J_1$, tuning at the central value of the coupling at $L/a = 48$, $\beta = 7.6$ at $c=0.3$ and $c=0.2$ respectively. A linear (quadratic) fit is shown in blue (green), in the range $L/a \geq 8$ ($L/a \geq 16$) for $\Sigma$ and $L/a \geq 12$ ($L/a \geq 20$) for $J_1$. Although both extrapolations are good, in the case of $J_1$ the arm of the extrapolation is shorter and more points are available in the linear scaling region (see Table \ref{['tab::quality_extrap']}).
• Figure 3: Tuning process and continuum extrapolation for the extraction of $\mathcal{J}_2$.
• Figure 4: Preliminary continuum extrapolations of $J_1$ and $J_2^{-1}$. To make errors more visible the combination $1/{u_t}-1/J_i$, with $J_i = J_1, J_2^{-1}$ is plotted, where $u_t$ fixes the line of constant physics.