Non-linear corrections to Lagrangians predicted by causal set theory: Flat space bosonic toy model

A while ago a proposal have been made regarding Klein Gordon and Maxwell Lagrangians for causal set theory. These Lagrangian densities are based on the statistical analysis of the behavior of field on a sample of points taken throughout some "small" region of spacetime. However, in order for that sample to be statistically reliable, a lower bound on the size of that region needs to be imposed. This results in "unwanted contributions" from higher order derivatives to the Lagrangian density, as well as non-trivial curvature effects on the latter. It turns out that both gravitational and non-gravitational effects end up being highly non-linear. In the previous papers we were focused on leading order terms, which allowed us to neglect these nonlinearities. We would now like to go to the next order and investigate them. In the current paper we will exclusively focus on the effects of higher order derivatives in the flat-space toy model. The gravitational effects will be studied in another paper which is currently in preparation. Both papers are restricted to bosonic fields, although the issue probably generalizes to fermions once Grassmann numbers are dealt with in appropriate manner.