# Dimensional Analysis and Power Counting in $R$ and $R^2$ Gravity Perturbation Expansions

In the context of $$R$$ gravity, the perturbation expansion appears as:

$$S=\int \left( \partial \tilde{h} \partial \tilde{h} + X \tilde{h} \partial \tilde{h} \partial \tilde{h} + ... \right) d^4 x$$

To argue the non-renormalizability of this theory using the power counting argument, I've considered: $$[\partial]=1$$, $$[h]=1$$, and $$[X]=[-1]$$. Given these dimensions, all the terms in the perturbation expansion have a net dimension of 4, which with the factor $$[d^4x]=-4$$, renders the action dimensionless. Furthermore, because $$[X]<0$$, higher energy terms are enhanced rather than suppressed, leading to the conclusion that the theory is not power-counting renormalizable. Is my understanding correct?

For $$a R^2$$ gravity (where $$[a]=-2$$ and $$[R^2]=4$$), the perturbation expansion is:

$$S=\int \left( ( \partial \tilde{h} \partial \tilde{h} )^2 + 2\sqrt{X} \tilde{h}(\partial \tilde{h} \partial {h})^2 + ... \right) d^4 x$$

A particular paper (RG) I'm referencing (see equation 51 on page 10) posits that $$[X] = 0$$, which would imply the renormalizability of $$aR^2$$ gravity.

I'm struggling with applying the power counting argument here. If we maintain $$[h]=1$$ and $$[\partial]=1$$ as in the $$R$$ gravity scenario, then $$[( \partial \tilde{h} \partial \tilde{h} )^2]$$ ends up being 8, which does not make the action dimensionless. My subsequent deduction would be that $$[\sqrt{X}]$$ must cancel out the dimension of $$[h]$$, resulting in $$[\sqrt{X}]=-1$$. The only resolution I see is if the metric perturbation $$h$$ in $$aR^2$$ gravity does not retain the same dimensions as in $$R$$ gravity. Could $$h$$ in $$aR^2$$ be dimensionless, implying $$[X]=0$$? Is there a justification for $$h$$ having different dimensions across the two theories, even though both $$h$$'s represent metric perturbations?

In the case of GR, the conformal fixed point is the free theory, $$\sim (\partial h)^2$$. In this case, we don't really need to assume that the free theory is a fixed point; we know a massless free theory is conformal. Because the theory is conformal, we know the action can't change under a rescaling of the coordinates. Then $$h$$ inherits its scaling properties from the scaling of $$d^4 x$$, $$\partial$$, and the requirement that the action not change. Once we know the scaling of $$h$$, we can then determine the mass dimension of operators that deform the fixed point.
In the case of $$R^2$$ gravity, the authors are assuming that $$R^2$$ gravity is a consistent conformal fixed point. That is not something they can prove (and I would even say probably isn't true for reasons I'll get to in a moment). They then want to check that this is a consistent assumption by looking at various interactions. Again, assuming that we have a conformal fixed point, the scaling of $$h$$ can be deduced from the "kinetic term" $$\sim (\partial h)^4$$. Given the scaling rules for $$d^4 x$$, $$\partial$$, and that the action has to be invariant, we see that $$h$$ has to be dimensionless. Then one can consider additional operators, and one finds that the new terms in $$R^2$$ can be consistently added with no dimensionful operators. To directly answer your question: it is not a problem that we find $$h$$ scales differently in $$R^2$$ gravity vs GR (with $$R$$), because our starting conformal fixed point was different.
One major issue with this analysis is that one is assuming that there is a consistent fixed point using $$R^2$$. A reason to be skeptical of this claim is that the equations of motion are more than second order in $$h$$. This will lead to an Ostragradsky instability classically, or a ghost quantum mechanically. Relatedly, there are bounds like the Froissart bound coming from unitarity that essentially say that the propagator (two point function) cannot fall off faster than $$\sim k^{-2}$$ in momentum space at high energies. $$R^2$$ gravity will violate this bound. The moral of the story is that there is more to life than power counting renormalizabilty; the combination of constraints unitarity, renormalizability, and Lorentz invariance are very strong and hard to simultaneously satisfy.