 Equation-of-motion reduction of scalar operators

At first order in the suppression of an effective operator, a factor of the leading Euler-Lagrange field equation defines a redundant operator removable by a local field redefinition. For ${\mathcal{L}}_0=(\partial \varphi {)}^2/2-m^2{\varphi }^2/2-\lambda {\varphi }^4/24$, one obtains ${\varphi }^3\square \varphi \doteq -m^2{\varphi }^4-\lambda {\varphi }^6/6$ and $(\square \varphi {)}^2\doteq m^4{\varphi }^2+m^2\lambda {\varphi }^4/3+{\lambda }^2{\varphi }^6/36$. Lower-dimensional parameter shifts must accompany this reduction when $m\ne 0$; it is not an identity for untransformed off shell correlation functions.