  We study a notion of operator growth known as Krylov complexity in free and
interacting massive scalar quantum field theories in $d$-dimensions at finite
temperature. We consider the effects of mass, one-loop self-energy due to
perturbative interactions, and finite ultraviolet cutoffs in continuous
momentum space. These deformations change the behavior of Lanczos coefficients
and Krylov complexity and induce effects such as the "staggering" of the former
into two families, a decrease in the exponential growth rate of the latter, and
transitions in their asymptotic behavior. We also discuss the relation between
the existence of a mass gap and the property of staggering, and the relation
between our ultraviolet cutoffs in continuous theories and lattice theories.
