We perform the step-scaling investigation of the running coupling constant, using
the gradient-flow scheme, in SU(3) gauge theory with twelve massless fermions in the
fundamental representation. The Wilson plaquette gauge action and massless
unimproved staggered fermions are used in the simulations. Our lattice data are
prepared at high accuracy, such that the statistical error for the renormalised coupling,
Article funded by SCOAP3
0$~\cite{Luscher-ml-2011bx}. In particular ref.~\cite{Luscher-ml-2010iy} suggests using the action density to define a renormalised coupling at a scale $\mu = 1/\sqrt{8t}$, \begin{equation} g^{2}(\mu) = \frac{16\pi^2}{3}t^2\langle E(t)\rangle\,, \end{equation} with \begin{equation} E(t) = -\frac{1}{2}{\rm tr}(G_{\mu\nu}G_{\mu\nu})\,. \label{eq:energy_density} \end{equation} In the context of finite size scaling, the renormalisation scale is identified with the linear size of the box $\mu = \frac{1}{\sqrt{8t}}= 1/c_\tau L$ where $c_\tau = \sqrt{8t}/L$ is a constant that defines our renormalisation scheme. Several finite volume renormalisation schemes have been defined by using different boundary conditions: periodic~\cite{Fodor-ml-2012td}, Schr\"odinger functional~\cite{Fritzsch-ml-2013je}, open-SF~\cite{Luscher-ml-2014kea}, and more directly related with this work, twisted boundary conditions~\cite{Ramos-ml-2014kla}. There is quite some freedom when translating the flow equation eq.~\eqref{eq:flow} to the lattice. Since the r.h.s.\ of eq.~\eqref{eq:flow} is just the gradient of the Yang-Mills action, a straightforward option consists in evolving the lattice gauge links $V_{\mu}(t,x)$ according to the gradient of a lattice action\footnote{For the precise definition of the Lie-algebra valued derivative $\partial_{x,\mu}$ see~\cite{Luscher-ml-2010iy}.} \begin{equation} \frac{\partial V_{\mu} (t,x)}{\partial t} = - g_{0}^{2} \left \{ \partial_{x,\mu} S_{{\mathrm{latt}}} \left [ V_{\mu} \right ]\right \} V_{\mu} (t,x) , \mbox{ }\mbox{ } V_{\mu} (0,x) = U_{\mu} (x) . \label{eq:WF_equation} \end{equation} Moreover one has also the freedom to choose amongst different discretisations to define the observable $E(t)$, the most popular one being the clover and the plaquette. In general when evaluating the coupling these two choices of discretisations (i.e.\ the lattice flow equation and the observable) are, together with the action chosen to produce the configurations, the three sources of cutoff effects~\cite{Fodor-ml-2014cpa,Ramos-ml-2014kka}. Although recently a discretisation of the flow equation and observables free of $\mathcal O(a^2)$ effects has been proposed~\cite{Ramos-ml-2015baa}, in this work we have used the Wilson flow (i.e.\ eq.~\eqref{eq:WF_equation} with the Wilson plaquette action for $S_{\rm latt}$), and two different discretisations of the observable (clover and plaquette). \sloppy In full glory our coupling reads, \begin{equation} \label{eq:GF_scheme_def} \bar{g}^{2}_{\rm latt} (\beta,\hat{L}) = \hat{{\mathcal{N}}}^{-1} (c_{\tau}, a/L) t^{2} \la E^{({\rm latt})}(a,t)\ra \big{|}_{t = c_{\tau}^{2} L^{2} /8 } , \end{equation} where \begin{equation} \beta = \frac{6}{g_{0}^{2}}, \mbox{ } \hat{L} = L/a , \label{eq:beta_and_Lhat} \end{equation} $E^{({\rm latt})}(a,t)$ is the discretised version of $E(t)$ defined in eq.~(\ref{eq:energy_density}), and the constant $\hat{\mathcal N}(c_\tau, a/L)$ has been computed to tree-level with our choice of discretisations (see~\cite{Ramos-ml-2014kla} for the concrete expressions). \fussy ]]>
L_{{\rm ref}} > a , \end{equation} and work with fixed lattice spacing. To proceed, in the following discussion we will present our argument using the GF-scheme renormalised coupling, $\bar{g}_{{\rm latt}}^{2}(g_{0}^{2},\hat{L})$, as defined in eq.~(\ref{eq:GF_scheme_def}). Expressing all the length scales in lattice units, and identifying $l_{1}$ and $l_{2}$ in eq.~(\ref{eq:FSS_l1_l2}) with $L_{{\rm ref}}$ and $L$, one obtains \begin{equation} \bar{g}^{2}_{{\rm latt}} ( g_{0}^{2}, \hat{L} ) = g^{2}_{\ast} + \left [ \bar{g}^{2}_{{\rm latt}} ( g_{0}^{2}, \hat{L}_{{\rm ref}} ) - g^{2}_{\ast}\right ] \left ( \frac{\hat{L}_{{\rm ref}}}{\hat{L}}\right )^{\gamma_{\ast}} , \label{eq:FSS_Lref_L_near_IRFP} \end{equation} in the vicinity of the IRFP. This equation can be regarded as a finite-size scaling formula. Confronting it with lattice data enables us to confirm/exclude IR conformality, and it leads to the determination of $g_{\ast}$ and $\gamma_{\ast}$. In addition to fixing the bare coupling, we can further choose to work at a particular value of $\hat{L}_{\rm ref}$ in the analysis. It has to be stressed again that the renormalised couplings, $\bar{g}^{2}_{{\rm latt}} ( g_{0}^{2}, \hat{L} )$ and $\bar{g}^{2}_{{\rm latt}} ( g_{0}^{2}, \hat{L}_{{\rm ref}} )$, still contain lattice artefacts, and therefore one has to work in a regime where lattice artefacts are small compared with the statistical uncertainties. This can be checked in practice by using different discretisations and/or different lattice sizes to extract $\bar{g}^{2}_{{\rm latt}}$. We further notice, from figure~\ref{fig:coupling}, that in this work the change of $\bar{g}^{2}_{{\rm latt}} (g_{0}^{2},\hat{L})$ is small when varying $\hat{L}$ between 8 and 24 at fixed lattice spacing. When the coupling is very small, this is due to the effect of the Gaussian UVFP. At intermediate and strong couplings, such behaviour arises from the smallness of the $\beta{-}$function. Therefore, away from the asymptotic-freedom regime, we can fit our data, at a particular choice of $\hat{L}_{\rm ref}$ and $a$, with the formula, \begin{equation} \bar{g}^{2}_{{\rm latt}} ( g_{0}^{2}, \hat{L} ) = g^{2}_{\rm l} (g_{\rm ref}) + \left [ g^{2}_{\rm ref} - g^{2}_{l} (g_{\rm ref})\right ] \left ( \frac{\hat{L}_{{\rm ref}}}{\hat{L}}\right )^{\gamma (g_{\rm ref})} , \label{eq:FSS_Lref_L} \end{equation} where $g_{l}$ and $\gamma$ are the free parameters, with the definition, \begin{equation} g_{{\rm ref}} \equiv \bar{g}_{{\rm latt}} (g_{0}^{2}, \hat{L}_{{\rm ref}}) . \label{eq:g_ref_def} \end{equation} Equation~(\ref{eq:FSS_Lref_L}) can be regarded as the consequence of a ``locally linearised'' $\beta{-}$function, which is a good approximation only when one works with small variations of the coupling around $g_{{\rm ref}}$. This is the reason why $g_{l}$ and $\gamma$ depend on $g_{{\rm ref}}$. Nevertheless, when the theory is tuned to be close to an IRFP, this equation must converge to eq.~(\ref{eq:FSS_Lref_L_near_IRFP}), and $g_{l}$ and $\gamma$ will approach constant values, $g_{\ast}$ and $\gamma_{\ast}$. In the numerical analysis, we always fix $\hat{L}_{\rm ref}$ to be 8, and use data at $\hat{L} \!=\! 10, 12, 16, 20, 24$ for fitting with eq.~(\ref{eq:FSS_Lref_L}).\footnote{We have also tried taking $\hat{L}_{\rm ref} = 10$, and using data at $\hat{L} = 12, 16, 20, 24$ for fitting. However, this leads to a significant increase in the error for the results.} For each fit, we specify a value for $g_{0}^{2}$ (hence $g_{{\rm ref}}$), and extract $g_{l}$ and $\gamma$. When conducting this procedure in a region without IR conformality, $g_{l}$ and $\gamma$ will show dependence on the input $g_{\rm ref}$. On the other hand, when the theory is engineered to be in the neighbourhood of an IRFP by tuning the bare coupling, these two quantities should show a clear trend to converge to $g_{\ast}$ and $\gamma_{\ast}$. In summary, we can utilise our data and perform the fit to eq.~(\ref{eq:FSS_Lref_L}) at fixed $\hat{L}_{\rm ref} = 8$, and scan though many values of $g_{0}$ (hence $g_{\rm ref}$) in the strong-coupling regime. At each choice of $g_{0}$, we carry out a fit. If our data indicate the existence of an IRFP, both $g_{l}$ and $\gamma$ should show plateau behaviour when plotted against $g_{\rm ref}$, and different discretisations for $\bar{g}_{\rm latt}$ will lead to consistent results. Figure~\ref{fig:FSS_test} is the outcome for $\gamma$ determined using the above analysis procedure at $c_{\tau} = 0.5$. In this plot, we notice that the largest $g_{\rm ref}$ for the clover discretisation is slightly smaller than the largest input $g_{\rm GF}(L)$ value which is $5.8$ for the same $c_{\tau}$ in the step-scaling analysis presented in section~\ref{sec:continuum_extrap}. This is because $\bar{g}_{\rm latt} (g_{0}^{2}, \hat{L})$ grows with $\hat{L}$ at fixed $g_{0}^{2}$ as a general trend in our data, and we follow the principle that no extrapolation in $g_{0}^{2}$ is implemented in this work. For the finite-size scaling test discussed in this section, the largest $g_{\rm ref}$ must be chosen to be the value of $\bar{g}_{\rm latt} (g_{0}^{2}, \hat{L}=8)$ at the largest $g_{0}^{2}$ where we have data for the $\hat{L}=24$ lattice. Therefore, according to table~\ref{tab:bare_g_summary}, the maximal bare coupling in the analysis leading to the result in figure~\ref{fig:FSS_test} is $g_{0}^{2} = 1.442$. On the other hand, the largest bare coupling for computing the input $g_{\rm GF}(L)$ in the step-scaling analysis is the maximal value of $g_{0}^{2}$ for the $\hat{L}=16$ lattice. It can be seen in table~\ref{tab:bare_g_summary} that this is at $g_{0}^{2} = 1.449$. This small difference in $g_{0}^{2}$ can produce minor but visible difference in the renormalised coupling, since in this regime $\bar{g}_{\rm latt}$ increases rapidly with $g_{0}^{2}$, as demonstrated by the plots in figure~\ref{fig:coupling}. \begin{figure} \centering \scalebox{0.65}{{\includegraphics[trim = 0 15 0 12, clip, ]{Gamma_refL8_cTau0point500.pdf}}} ]]>
6$. This is beyond the scope of this project. On the other hand, it will be interesting to examine the reliability of the procedure in ref.~\cite{Cheng-ml-2014jba} by carrying out the same analysis using the plaquette discretisation. ]]>