EFT operator basis

Question

Let us suppose to have a simple theory with only 1 real scalar $\phi$, the most general Lagrangian is: $$\mathcal{L} = \frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi + \frac{1}{2}m^2\phi^2 + \frac{\lambda_3}{3!}\phi^3 + \frac{\lambda_4}{4!}\phi^4.$$
In writing all the EFT operators up to dimension 8, should we consider also operators with highly-order derivatives like $\mathcal{O}^{(5)} = \Box^2\phi$ or $\mathcal{O}^{(7)} = \Box^3\phi$?
Moreover, when we account for equations of motion redundancies by removing all operators of dimension 3 or higher that contain $\Box\phi$, where are the equations of motion calculated on? Do we need to compute the equations of motion using the 4-dim Lagrangian, namely $$\mathcal{L} = \frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi + \frac{1}{2}m^2\phi^2 + \frac{\lambda_3}{3!}\phi^3 + \frac{\lambda_4}{4!}\phi^4,$$ or should we use the Lagrangian which contains all the operators up to dimension 8? And why?

Answer 1

1. The Lagrangian you wrote for the scalar is not the most general Lagrangian, it is the most general (perturbatively) renormalizable Lagrangian.

2. When writing all EFT operators up to some dimension, you should include all operators consistent with the symmetries of your system. So, yes, you should include derivative operators.

3. There's a good discussion of redundant operators around Eq 21 of https://arxiv.org/abs/hep-th/0701053. The main idea is that any operator proportional to the lowest order equations of motion, can be removed with a field redefinition, that pushes the operator to higher order. Here, the "order" could refer to the loop expansion, or an expansion in some energy scale $E/M$. The equations of motion would be associated with the free Lagrangian (the most relevant operators), or $\square \phi - m^2 \phi = 0.$ You can check this explicitly using the general form of the field redefinition given in the reference I cited.