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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 $\delta^d(k_1\!+\!k_2)$ of 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).

$^1$ 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.

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).

$^1$ 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.

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. For dimensional reasons, the graviton-graviton vacuum-polarization/self-energy $\widetilde{\Pi}$ [which consists of amputated Feynman diagrams stripped of a delta function $\delta^d(k_1\!+\!k_2)$ of total wave number conservation] has dimension $$ [\widetilde{\Pi}]~=~\text{(Angular momentum)}\times\text{Length}^{-2} ~=~\frac{\text{Energy}^{2}}{\text{Angular momentum}},\tag{4}\label{eq:4}$$ cf. e.g. this Phys.SE post. Therefore it is of the form $$\widetilde{\Pi}(E)~=~\frac{E^2}{\hbar}f\left(\frac{E}{m_P}\right),\tag{5}\label{eq:5}$$ 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).

$^1$ 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.

$^3$ Here 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 constants $\kappa$, respectively. This is true in GR.

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 $\delta^d(k_1\!+\!k_2)$ of 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).

$^1$ 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.

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

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

3. For $d=4$ the variable $$\frac{E}{m_P}~=~E\sqrt{\frac{G}{\hbar}}\tag{7}$$ in 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).

$^1$ 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.

$^3$ Here 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.

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. For dimensional reasons, the graviton-graviton vacuum-polarization/self-energy $\widetilde{\Pi}$ [which consists of amputated Feynman diagrams stripped of a delta function $\delta^d(k_1\!+\!k_2)$ of total wave number conservation] has dimension $$ [\widetilde{\Pi}]~=~\text{(Angular momentum)}\times\text{Length}^{-2} ~=~\frac{\text{Energy}^{2}}{\text{Angular momentum}},\tag{4}\label{eq:4}$$ cf. e.g. this Phys.SE post. Therefore it is of the form $$\widetilde{\Pi}(E)~=~\frac{E^2}{\hbar}f\left(\frac{E}{m_P}\right),\tag{5}\label{eq:5}$$ 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).

$^1$ 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.

$^3$ Here 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 constants $\kappa$, respectively. This is true in GR.