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The gradient flow in non-abelian gauge theories on R^4 is defined by a localdiffusion equation that evolves the gauge field as a function of the flow timein a gauge-covariant manner. Similarly to the case of the Langevin equation,the correlation functions of the time-dependent field can be expanded inperturbation theory, the Feynman rules being those of a renormalizable fieldtheory on R^4 x [0,oo). For any matter multiplet and to all loop orders, weshow that the correlation functions are finite, i.e. do not require additionalrenormalization, once the theory in four dimensions is renormalized in theusual way. The flow thus maps the gauge field to a one-parameter family ofsmooth renormalized fields.

Perturbative analysis of the gradient flow in non-abelian gauge theories