The gradient flow in simple field theories

The gradient flow is a valuable tool for the lattice community, with applications from scale-setting to implementing chiral fermions. Here I focus on the gradient flow as a means to suppress power-divergent mixing. Power-divergent mixing stems from the hypercubic symmetry of the lattice regulator and is a particular difficulty for calculations of, for example, high moments of parton distribution functions. The gradient flow removes power-divergent mixing on the lattice, provided the flow time is kept fixed in physical units, at the expense of introducing a new physical scale in the continuum. One approach to dealing with this new scale is the smeared operator product expansion, a formalism that systematically connects nonperturbative calculations of flowed operators to continuum physics. I study the role of the gradient flow in suppressing power-divergent mixing and present the first nonperturbative study in scalar field theory.