This should be compared with Eq. (\[Pi\_usual\_QFT\]) and (\[eq:usual\_qft\_quad\]).

At this point it is worth reminding ourselves that the theory that we are dealing with is a non-renormalizable theory so that the countererm (CT) needed is not necessarily one of the terms we have already written down in the action Eq. (\[S\_SZ2010\]). From Eq. (\[gen\_exp\]), it is clear that the loop correction shall have a factor of $(H/\Lambda_U)^4$. Again, by dimensional analysis, it is clear that a single vertex of dimension $+8$ operator can give this factor. But since we want the CT to renormalize the two-point function, it better have two external lines. Thus, counterterm shall be a dimension $+8$ quadratic operator. On canonical normalization, $\pi_c$ has dimension $+1$ and shift symmetry forbids any polynomials in $\pi_c$ to be present in the action. Thus, we can only take derivatives, hence CTs can only be operators of the form $(\partial^3 \pi_c)^2$. The derivatives that we can take are either w.r.t. time or w.r.t. space, since we want to write a rotationally invariant action, the only options are the square of $\partial_t \partial_t \partial_t \pi $, the square of $\partial_t \partial_i \partial_i \pi$, and $\partial_t \partial_t \partial_i \pi \partial_t \partial_t \partial_i 