On the gradient flow of the Lagrangian density in non-Abelian gauge theories

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. Moreover, $\beta_{wb}$ violates two-loop universality and may develop spurious zeroes both in the confined and the conformal phase, despite $g_{wb}^2(t)$ is asymptotic to a true RGI running coupling in a neighborhood of the asymptotically free fixed point. Our analysis allows us to reinterpret the contradictory lattice results based on this method, specifically those for the $N_f$=12 theory, explain the origin of their discrepancies and suggest a new strategy to discriminate between the confined phase and the conformal window. In this respect, we disagree with a recent claim that attributes the same contradictory results to staggered fermions being in the wrong universality class.