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The shaded area shows the $a > 0.04\,\fm$ region of large volume results which dominate the present PDG and FLAG estimates~\cite{Olive:2016xmw,Aoki:2016frl} of $\alpha_\msbar(\mu)$ (curve evaluated to two loops for $\nf=3$ with $\Lambda^{(3)}_\msbar =341\,\MeV$). The data points on the left are finite-size scaling computations~\cite{Aoki:2009tf,Brida:2016flw,DallaBrida:2016kgh}.

Running couplings of $\nf=3$ QCD from integrating the nonperturbative $\beta$-functions in the SF and GF schemes~\cite{Brida:2016flw,DallaBrida:2016kgh}. They are matched nonperturbatively by defining $\gbar_{\rm SF}^2(\mu_0) = 2.012$ and computing $\gbar_{\rm GF}^2(\mu_0/2) = 2.6723(64)$.

Dependence of $\fpik$ at $\phi_4=1.11$ on the pion mass through $\phi_2=m_\pi^2/\muref^2$~\cite{Bruno:2016plf}. We normalized to $\fpik^{\rm sym}$ at the symmetric point $\mup=\mdown=\mstrange$. The ratio follows the parameter-free prediction of NLO chiral PT.

Continuum extrapolations described in the text. Extrapolated values are shown in proximity of $a=0$.

Interpolations $\gbar_{{\rm GF}}^{-2} = k_0 + k_1 (6/g_0^2) +k_2 (6/g_0^2)^2$, performed separately for each $L/a$, with $k_2=0$ for $L/a=20$.

Normalized autocorrelation function, $\rho(t_\mathrm{MC})$ of $a\muref^\star$, as defined in~\cite{Wolff:2003sm}, and error estimate including a tail contribution (dashed curve), starting from $t_\mathrm{W}$.

Fit of $\log(a\mu_{\rm ref}^\star)$ to the points marked CLS. This fit allows us to read the values of $a\mu_{\rm ref}^\star$ at the values of $g_0$ where $a\mu_{\rm had,1}$ is known, set {\tt B} (points labelled Interpolation).