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The gradient flow is a valuable tool for the lattice community, withapplications from scale-setting to implementing chiral fermions. Here I focuson the gradient flow as a means to suppress power-divergent mixing.Power-divergent mixing stems from the hypercubic symmetry of the latticeregulator and is a particular difficulty for calculations of, for example, highmoments of parton distribution functions. The gradient flow removespower-divergent mixing on the lattice, provided the flow time is kept fixed inphysical units, at the expense of introducing a new physical scale in thecontinuum. One approach to dealing with this new scale is the smeared operatorproduct expansion, a formalism that systematically connects nonperturbativecalculations of flowed operators to continuum physics. I study the role of thegradient flow in suppressing power-divergent mixing and present the firstnonperturbative study in scalar field theory.

The gradient flow in simple field theories