Why is Perturbative expansion of gravity in terms of $GE^2$?

Question

From General Relativity by Weinberg p.797 edited by Hawking & Israel:

This is to be used to generate a perturbation series in powers of $GE^2$ or $G/r^2$ (where $E$ and $r$ are an energy and a length that are characteristic of the process under study)

From Quantum Field Theory in a Nutshell by Zee p.172:

We come to the same morose conclusion that the theory of gravity, just like Fermi’s theory of weak interaction, is nonrenormalizable. To repeat the argument, if we calculate graviton-graviton scattering at energy $E$, we encounter the series $\sim[1+ G_NE^2 + (G_NE^2)^2 + ...]$.

I'm just looking for a brief explanation for this type of expansion or some resource where I can understand why it is in this specific form.

Answer 1

It is a dimensional analysis argument. In units with $\hbar = c = 1$ (natural in quantum field theory), $G$ has dimension $[G] = E^{-2}$ (where $E$ denotes dimension of energy). Zee and Hawking are making an argument with dimensionless quantities, so the natural way of performing this perturbative expansion is in powers of $GE^2$ (where $E$ now denotes a typical energy scale at hand). Of course, this is not exactly how the perturbative series looks like in its "pure form". For example, the first loop corrections to the action of general relativity look like $$S = \int \left(\frac{1}{16\pi G}R + \alpha R^2 + \beta R_{\mu\nu}R^{\mu\nu}\right) \sqrt{-g} \mathrm{d}^4x,$$ where $\alpha$ and $\beta$ are dimensionless constants to be determined and the $R$ term corresponds to the standard Einstein–Hilbert term.

Now make a dimensional analysis. $[\mathrm{d}^4x] = E^{-4}$, $[R] = E^2$. Hence, the Einstein–Hilbert term is proportional to $(G E^2)^{-1}$, while the one-loop corrections are proportional to $(G E^2)^{0} = 1$, and so on.

Answer 2

1. More generally in $d$ spacetime dimensions, the perturbative gravity Lagrangian density is schematically of the form$^1$ $$ {\cal L}~=~\frac{\sqrt{|g|}}{16\pi G}\frac{f(\ell_{\!P}^2 R)}{\ell_{\!P}^2}~=~(\partial h)^2+\frac{\hbar}{16\pi\ell_{\!P}^d}f_{\rm int}(\ell_P\partial,\kappa h),\tag{1}\label{eq:1}$$ where the metric field $$g_{\mu\nu}~=~\eta_{\mu\nu}+\kappa h_{\mu\nu},\qquad \kappa~:=~2\sqrt{8\pi G}, \tag{2}\label{eq:2}$$ is expanded around a background $\eta_{\mu\nu}$, typically the Minkowski spacetime. Here $R$ is a shorthand for either the Riemann, Ricci or scalar curvature tensor. The functions$^2$ in eq. \eqref{eq:1} take dimensionless arguments into dimensionless values. The Planck length $\ell_{\!P}$ satisfies $$ \hbar G~=~\ell_{\!P}^{d-2},\tag{3}\label{eq:3}$$ cf. e.g. Ref. 3.

2. The graviton-graviton vacuum-polarization/self-energy $\widetilde{\Pi}$ [which consists of amputated Feynman diagrams stripped of a delta function $\delta^d(k_1\!+\!k_2)$ with total wave number conservation] has dimension $$\begin{align} [\widetilde{\Pi}]~=~&\frac{\text{Length}^{-2}}{\text{Angular momentum}}\cr ~=~&\frac{\text{Energy}^{2}}{(\text{Angular momentum})^3},\end{align}\tag{4}\label{eq:4}$$ cf. e.g. this Phys.SE post. For dimensional reasons, it is of the form $$\widetilde{\Pi}(E)~=~\frac{E^2}{\hbar^3}f\left(\frac{E}{m_P}\right),\tag{5}\label{eq:5}$$ where $E$ is the CM energy, and $$m_P~=~\frac{\hbar}{\ell_{\!P}}~\stackrel{(3)}{=}~\hbar(\hbar G)^{-1/(d-2)}\tag{6}\label{eq:6}$$ is the Planck mass.

3. For $d=4$ the variable $$\frac{E}{m_P}~=~E\sqrt{\frac{G}{\hbar}}\tag{7}\label{eq:7}$$ in eq. \eqref{eq:5} leads to the perturbative series$^3$ mentioned in Refs. 1 & 2.

References:

1. S. Weinberg, UV divergences of quantum theories of gravitation. Published in GR: An Einstein centenary survey (Eds. S.W. Hawking & W. Israel), 1979; p.797.

2. A. Zee, QFT in a nutshell, 2010; section III.2 p.169-172 & section VIII.1 p.434-435.

3. B. Zwiebach, A first course in String Theory, 2nd edition, 2009; section 3.8, eq. (3.108).

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$^1$ We work in units where the speed of light $c=1$ is one.

$^2$ We call all such functions $f$, even though they are different.

$^3$ To only have even powers in the power series, we are implicitly assuming that each Feynman diagram with an even/odd number of external legs only comes with an even/odd power of the coupling constant $\kappa$, respectively. This is true in GR.