For example, the effective field theory Lagrangian with cutoff $\Lambda$ for the renormalizable $\varphi^4$ theory is $$\mathcal L_{\mathrm{eff}}(\varphi;\Lambda)=\frac{1}{2}Z(\Lambda)\partial_\mu\varphi\partial_\mu\varphi+\frac{1}{2}m^2(\Lambda)\varphi^2+\frac{1}{24}\lambda(\Lambda)\varphi^4+\sum_{d\geq6}\sum_ic_{d,i}(\Lambda)\mathcal O_{d,i}$$ The operators $\mathcal O_{d,i}$ consist of all terms that have mass dimension $d\geq6$ and respect the original symmetry of the renormalizable $\varphi^4$ theory.(reference: Srednicki's QFT Chapter 29)

Then, terms like $\partial_\mu\varphi\partial_\mu\varphi\,\varphi^2$ are allowed in the operators $\mathcal O_{d,i}$. Can higher derivative terms like $(\partial_\mu\varphi\partial_\mu\varphi)^2\varphi^2$ be allowed? If these higher derivative terms are included, the effective field theory will be different from the original renormalizable $\varphi^4$ theory which only has second order derivative.

For example, the effective field theory Lagrangian with cutoff $\Lambda$ for the renormalizable $\varphi^4$ theory is $$\mathcal L_{\mathrm{eff}}(\varphi;\Lambda)=\frac{1}{2}Z(\Lambda)\partial_\mu\varphi\partial_\mu\varphi+\frac{1}{2}m^2(\Lambda)\varphi^2+\frac{1}{24}\lambda(\Lambda)\varphi^4+\sum_{d\geq6}\sum_ic_{d,i}(\Lambda)\mathcal O_{d,i}$$ The operators $\mathcal O_{d,i}$ consist of all terms that have mass dimension $d\geq6$ and respect the original symmetry of the renormalizable $\varphi^4$ theory.  (Reference: Srednicki's QFT Chapter 29)

Then, terms like $\partial_\mu\varphi\partial_\mu\varphi\,\varphi^2$ are allowed in the operators $\mathcal O_{d,i}$. Can higher derivative terms like $(\partial_\mu\varphi\partial_\mu\varphi)^2\varphi^2$ be allowed? If these higher derivative terms are included, the effective field theory will be different from the original renormalizable $\varphi^4$ theory which only has second order derivative.