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

Question

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?

Answer 1

The starting assumption of this kind of scaling analysis is that you are working around a conformal fixed point.

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.