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  1. More generallyin $d$ spacetime dimensions, the perturbative gravity Lagrangian density is schematically of the form$^1$ $$ {\cal L}~=~\frac{\sqrt{|g|}}{16\pi G}\frac{f(\ell_P^2 R)}{\ell_P^2}~=~(\partial h)^2+\frac{\hbar}{16\pi\ell_P^d}f_{\rm int}(\ell_P\partial,\kappa h),\tag{1}$$ where the metric field $$g_{\mu\nu}~=~\eta_{\mu\nu}+\kappa h_{\mu\nu},\qquad \kappa~:=~2\sqrt{8\pi G}, \tag{3}$$ 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. (1) takes dimensionless arguments to a dimensionless value. The Planck length $\ell_P$ satisfies $$ \hbar G~=~\ell_P^{d-2}.\tag{3}$$

  2. For dimensional reasons, the graviton-graviton vacuum-polarization/self-energy is of the form $$\Pi~=~E^2f(E/m_P),\tag{4}$$ where $E$ is the CM energy, and $$m_P~=~\frac{\hbar}{\ell_P}~\stackrel{(3)}{=}~\hbar(\hbar G)^{-1/(d-2)}\tag{5}$$ is the Planck mass.

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.


$^1$ 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.

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