Classical nonrelativistic effective field theories for a real scalar field

A classical nonrelativistic effective field theory for a real Lorentz-scalar field φ is most conveniently formulated in terms of a complex scalar field ψ. There have been two derivations of effective Lagrangians for the complex field ψ in which terms in the effective potential were determined to order (ψ*ψ)⁴. We point out an error in each of the effective Lagrangians. After correcting the errors, we demonstrate the equivalence of the two effective Lagrangians by verifying that they both reproduce T-matrix elements of the relativistic real scalar field theory and by also constructing a redefinition of the complex field ψ that transforms terms in one effective Lagrangian into the corresponding terms of the other.