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In lattice gauge theories, the gradient flow has been used extensively both,for scale setting and for defining finite volume renormalization schemes forthe gauge coupling. Unfortunately, rather large cutoff effects have beenobserved in some cases. We here investigate these effects to leading order inperturbation theory, considering various definitions of the lattice observable,the lattice flow equation and the Yang Mills lattice action. Theseconsiderations suggest an improved set- up for which we perform a scaling testin the pure SU(3) gauge theory, demonstrating strongly reduced cutoff effects.We then attempt to obtain a more complete understanding of the structure ofO($a^2$) effects by applying Symanzik's effective theory approach to the 4+1dimensional local field theory with flow time as the fifth dimension. Fromthese considerations we are led to a fully O($a^2$) improved set-up the studyof which is left to future work.

On O($a^2$) effects in gradient flow observables