Understand
We observe that the would-be running coupling on the lattice defined by means of the gradient-flow method in order to identify the conformal window of QCD is not renormalization-group invariant (RGI).
- Indeed, we show that the would-be running coupling, $g_{wb}^2(t)\propto t^2\langle E(t)\rangle$, -- with $\langle E(t)\rangle$ the expectation value of the Lagrangian density smeared by means of the gradient flow -- has an anomalous dimension associated to the multiplicative renormalization factor of $t^2\langle E(t)\rangle$.
- As a consequence, at a nontrivial infrared (IR) fixed point with nonvanishing anomalous dimension, $\gamma_*$, in the conformal window, the would-be running coupling vanishes asymptotically as $g_{wb}^2(t)\propto t^2\langle E(t)\rangle\sim t^{-\gamma_*/2}$ and does not scale as $g_{wb}^2(t)\propto t^2\langle E(t)\rangle\sim g_{wb}^{*2}\neq 0$, with $g_{wb}^{*}$ the nonvanishing would-be coupling at the nontrivial fixed point, as postulated in the literature.
- The associated would-be beta function, $\beta_{wb}({g}_{wb}^2(t))$, is not proportional to a true RGI beta function, and it also vanishes asymptotically in the IR for nonvanishing $\gamma_*$ at the IR fixed point.