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A non-perturbative exploration of the high energy regime in $N_\text{f}=3$ QCD - Dalla Brida, Mattia et al - arXiv:1803.10230CERN-TH-2018-060DESY 18-044WUB/18-01DESY-18-044WUB-18-01
 
The step scaling function $\sigma(u)$, a discrete version of the $\beta$-function, defined in \eq{eq:sigdef}. The combination shown here yields directly the lowest order cofficient, $b_0$ of the $\beta$-function as $(\sigma(u)-u)/u^2 = 2 b_0\ln\,2 +\rmO(u)$. The dashed lines show the perturbative 2-loop behavior. The purple 1-sigma band shows our result (fit~C in table~\ref{tab:Sigfits}). Data points for $\nf=0,2,3,4$ are taken from the literature \cite{Capitani:1998mq,DellaMorte:2004bc,Aoki:2009tf,Tekin:2010mm}.
The step scaling function $\sigma(u)$, a discrete version of the $\beta$-function, defined in \eq{eq:sigdef}. The combination shown here yields directly the lowest order cofficient, $b_0$ of the $\beta$-function as $(\sigma(u)-u)/u^2 = 2 b_0\ln\,2 +\rmO(u)$. The dashed lines show the perturbative 2-loop behavior. The purple 1-sigma band shows our result (fit~C in table~\ref{tab:Sigfits}). Data points for $\nf=0,2,3,4$ are taken from the literature \cite{Capitani:1998mq,DellaMorte:2004bc,Aoki:2009tf,Tekin:2010mm}.
Coefficients $c^\nu_1(s)$ and $c^\nu_2(s)$ for two different SF$_\nu$ schemes. Left $\nu=0$, right $\nu=-0.5$. The values $s^\star$ defined by the condition $c_1(s^\star)=0$ are approximately $s^\star\approx 3$ (left) and $s^\star\approx 5$ (right).
The step-scaling function for the $\nu=0$ SF-coupling. The band shows our result (fit C, cf.~table~\ref{tab:Sigfits}). The data points are the approximations at finite $L/a=6,8,12$ taken from table~\ref{tab:rawdata2} with errors from Eq.~(\ref{eq:ssf-error}).
Determination of $L_0\Lambda_{\overline{\rm MS}}$ at different physical scales (parametrized by the value of $\alpha$ in the $x$-axes), and using different renormalization scales (value of $s$) to match with the $\MSbar$ scheme. The left (right) panel uses the SF$_\nu$-scheme with $\nu=0$ ($\nu=-0.5$), cf.~text.
The bands show the continuum fit functions for fits of type A and B to one-loop improved data for $L/a\ge 8$ and the data points are one-loop improved data with the cutoff effects subtracted using the models $\rho^{(1)}(u,a/L)$ and $\tilde\rho^{(1)}(u,a/L)$ from the type A fit.
The left panel shows the dependence on $\ct$ of the coupling $u=\gbar^2(L)$. The improvement coefficient is varied around its 2-loop value and the value of the coupling is $u\approx 2$. The right panel shows the slopes vs.~$a/L$.
Determination of the $\Lambda$-parameter in units of $L_0$ at different values of $\alpha$. We compare the extraction in different schemes ($\nu=-0.5,0,0.3$), and show a comparison with our final result Eq.~\eqref{eq:final_result}. As the reader can see when the extraction is performed at high enough energies ($\alpha\sim 0.1$), all schemes nicely agree. See the text for a full discussion.
The step-scaling function for the $\nu=0$ SF-coupling. The band shows our result (fit C, cf.~table~\ref{tab:Sigfits}). The data points are the approximations at finite $L/a=6,8,12$ taken from table~\ref{tab:rawdata2} with errors from Eq.~(\ref{eq:ssf-error}).
Coefficients $c^\nu_1(s)$ and $c^\nu_2(s)$ for two different SF$_\nu$ schemes. Left $\nu=0$, right $\nu=-0.5$. The values $s^\star$ defined by the condition $c_1(s^\star)=0$ are approximately $s^\star\approx 3$ (left) and $s^\star\approx 5$ (right).
Determination of the $\Lambda$-parameter in units of $L_0$ at different values of $\alpha$. We compare the extraction in different schemes ($\nu=-0.5,0,0.3$), and show a comparison with our final result Eq.~\eqref{eq:final_result}. As the reader can see when the extraction is performed at high enough energies ($\alpha\sim 0.1$), all schemes nicely agree. See the text for a full discussion.
Coefficients $c^\nu_1(s)$ and $c^\nu_2(s)$ for two different SF$_\nu$ schemes. Left $\nu=0$, right $\nu=-0.5$. The values $s^\star$ defined by the condition $c_1(s^\star)=0$ are approximately $s^\star\approx 3$ (left) and $s^\star\approx 5$ (right).
The left panel shows the dependence on $\ct$ of the coupling $u=\gbar^2(L)$. The improvement coefficient is varied around its 2-loop value and the value of the coupling is $u\approx 2$. The right panel shows the slopes vs.~$a/L$.
Determination of $L_0\Lambda_{\overline{\rm MS}}$ at different physical scales (parametrized by the value of $\alpha$ in the $x$-axes), and using different renormalization scales (value of $s$) to match with the $\MSbar$ scheme. The left (right) panel uses the SF$_\nu$-scheme with $\nu=0$ ($\nu=-0.5$), cf.~text.
Determination of $L_0\Lambda_{\overline{\rm MS}}$ at different physical scales (parametrized by the value of $\alpha$ in the $x$-axes), and using different renormalization scales (value of $s$) to match with the $\MSbar$ scheme. The left (right) panel uses the SF$_\nu$-scheme with $\nu=0$ ($\nu=-0.5$), cf.~text.
Determination of $L_0\Lambda_{\overline{\rm MS}}$ at different physical scales (parametrized by the value of $\alpha$ in the $x$-axes), and using different renormalization scales (value of $s$) to match with the $\MSbar$ scheme. The left (right) panel uses the SF$_\nu$-scheme with $\nu=0$ ($\nu=-0.5$), cf.~text.
Coefficients $c^\nu_1(s)$ and $c^\nu_2(s)$ for two different SF$_\nu$ schemes. Left $\nu=0$, right $\nu=-0.5$. The values $s^\star$ defined by the condition $c_1(s^\star)=0$ are approximately $s^\star\approx 3$ (left) and $s^\star\approx 5$ (right).
Statistical (interior error band) and total (exterior error band) uncertainties in the determination of $L_0\Lambda_{\overline{\rm MS}}$. The total uncertainty is computed by adding in quadratures the statistical and systematic uncertainties. The latter are computed varying the renormalization scale by a factor 2 above and below the value $s^\star$. See text for more details.
Statistical (interior error band) and total (exterior error band) uncertainties in the determination of $L_0\Lambda_{\overline{\rm MS}}$. The total uncertainty is computed by adding in quadratures the statistical and systematic uncertainties. The latter are computed varying the renormalization scale by a factor 2 above and below the value $s^\star$. See text for more details.
The left panel shows the dependence on $\ct$ of the coupling $u=\gbar^2(L)$. The improvement coefficient is varied around its 2-loop value and the value of the coupling is $u\approx 2$. The right panel shows the slopes vs.~$a/L$.
The bands show the continuum fit functions for fits of type A and B to one-loop improved data for $L/a\ge 8$ and the data points are one-loop improved data with the cutoff effects subtracted using the models $\rho^{(1)}(u,a/L)$ and $\tilde\rho^{(1)}(u,a/L)$ from the type A fit.
The left panel shows the dependence on $\ct$ of the coupling $u=\gbar^2(L)$. The improvement coefficient is varied around its 2-loop value and the value of the coupling is $u\approx 2$. The right panel shows the slopes vs.~$a/L$.