# On the range of validity of General Relativity and Quantum Field Theory in terms of energy and impact parameter (from Rovelli & Vidotto's book)

In Fig. 1.1 on page 5 in Rovelli & Vidotto's 2015 book Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory (PDF), there is this graph giving a general notion of why someone would pursue research in quantum gravity.

Regimes for the gravitational scattering of neutral particles in Plank [sic] units $$c = \hbar = G = 1$$. $$E$$ is the energy in the center of mass reference system and $$b$$ the impact parameter (how close to one another come the two particles). At low energy, effective QFT is sufficient to predict the scattering amplitude. At high energy, classical general relativity is generally sufficient. In (at least parts of the) intermediate region (colored wedge) we do not have any predictive theory.

In short, I'm confused about how to understand this graph.

What I understand: It makes sense for me that for small masses (let's say $$E \ll M_P$$, where $$M_P$$ is Planck's mass, as is the case for the gravitational attraction between two neutrons at rest, for example), we can use an EFT treatment of quantum gravity to get a description of what is going on, while for large masses ($$E \gg M_P$$, such as the case for a black hole collision) we can use classical GR. Somewhere in the middle we don't have any reliable prediction, so we can't really compute gravitational scattering of two particles at energies close to the Planck scale, for example. So far so good, I believe. If I got anything wrong, please let me know.

What I do not understand: I'm confused about the dependence on impact parameter. The graph suggests that if we fix some energy (regardless of this energy being close or far from the Planck scale), GR or QFT will apply depending on the impact parameter. From the graph, it seems I could describe the gravitational attraction between neutrons by using GR if the impact parameter is low enough (which is highly counterintuitive, since a low impact parameter suggests quantum effects should kick in), while GR should fail at large impact parameter and somehow quantum effects should kick in, so I can't describe a black hole collision within GR if they are too far apart, which is once again counterintuitive.

Question: How should one interpret this graph?

• "GR or QFT will apply depending on the impact parameter." I think the italics is what is confucisng. It is not "either or", but the the solutions of GR and QFT are valid in the white region but undeterminable in the colored (not considering whether the solutions are trivial or non measurable, just that there are solutions,) Commented Apr 29, 2022 at 5:58
• In the text when listing situations where both GR and QFT are needed he says "... or simply the scattering amplitude of two neutral particles at small impact parameter and high energy. See Figure 1.1." However, in the figure this would be the top left. I am guessing that the x-axis was mislabeled and should have read 1/b. Commented Apr 29, 2022 at 15:15

Yes, OP is right:

• From the HUP the region below the hyperbola $$E b \lesssim \hbar c$$ in the $$(b,E)$$-diagram is in the quantum regime.

• For constant COM energy $$E$$ and growing impact parameter $$b$$, the tree-level scattering/Born-approximation should become better and better (except for an IR divergence of soft-gravitons for spacetime dimension $$D\leq 4$$).

• For constant impact parameter $$b$$ and growing COM energy $$E$$, classical black holes should form.

For a more detailed discussion, see e.g. Ref. 1.

$$\uparrow$$ Fig. 3 from Ref. 1. Note that the axis are logarithmic and exchanged as compare to Fig. 1.1. $$M_D$$ is the Planck mass in $$D$$ spacetime dimensions. "NR" denotes an uncontrolled, non-renormalizable Planck regime.

References:

1. S.B. Giddings, The gravitational S-matrix: Erice lectures, arXiv:1105.2036; Fig. 3 on p. 8.
• Massless scalar propagator $\frac{\hbar}{k^2}$. Massive scalar propagator $\frac{\hbar}{\bar{p}\cdot k}$. Scalar 3-vertex: QED $\frac{kQ}{\hbar}$ & GR $\frac{k^2\sqrt{8\pi G}}{\hbar}$. $t$-channel $I$-tree-diagram: scalar GR eq. (G3.11) + eq. (DV2.9): $\frac{8\pi G\bar{s}^2}{\hbar k^2}$ $\quad\Rightarrow\quad$ ${\cal M}\sim \ldots \times\left(\frac{\hbar}{\sqrt{\bar{s}}}\right)^2$ $=\frac{8\pi G\hbar\bar{s}}{k^2}$ $=\frac{8\pi Gs}{\hbar k^2}$. Scalar/spinor QED: $\frac{Q_1Q_2}{\hbar k^2}$ $\quad\Rightarrow\quad$ ${\cal M}\sim(2\omega_1\hbar)\frac{Q_1Q_2}{\hbar k^2}(2\omega_2\hbar)$. Commented May 5, 2022 at 10:54
• The impact parameter $b$ only makes sense for $D\geq 2$. [The transverse space ($\perp$ 2 ext. initial $D$-mom=temporal & longitudinal direction) has a 1D overlap with the scattering plane.] We need $D\geq 3$ to have gravitons. Ladder diagram: Each ladder step adds $\hbar^{I-V}\int d^Dk \frac{\bar{p}\cdot k~e_2^2}{\bar{p}\cdot k~\bar{p}\cdot k~k^2}=\int d^Dk \frac{e^2}{p\cdot k~k^2}$, where $e_2=\frac{e}{\hbar c}$. Classical electron radius: $\quad r_e=\frac{k_ce^2}{mc^2}$. arxiv.org/abs/2112.12243 Commented Jun 4 at 8:13
• Eqs. (G3.11+16+20). arxiv.org/abs/2306.16488 eq. (DV2.24). WKB/clas/eikonal$\quad Gsb^{4-D}\gg\hbar$ ($\left(\frac{R_S}{b}\right)^{D-3}\gg\frac{\ell_c}{b}$) $Gs\bar{q}_{\perp}^{D-4}\gg\hbar$ $\quad\wedge$ $\quad G\sqrt{s}b^{3-D}\ll 1$ ($R_S\ll b$) $G\sqrt{s}\bar{q}_{\perp}^{D-3}\ll 1$ point particle limit/weak gravity/no black holes. Commented Jun 30 at 14:06
• $\quad\Rightarrow$ $\quad b\sqrt{s}\gg\hbar$ ($\ell_c\ll b$) $\sqrt{s}\gg\hbar\bar{q}_{\perp}$ (near) forward limit/quantum limit. arxiv.org/abs/0908.0004 Commented Jul 2 at 14:51
• $\quad L-\pi_0=I_q+I_m-V_3$; $\quad 3V_3=2(I_q+I_m)+E$; Eikonal Approx: Levy & Sucher doi.org/10.1103/PhysRev.186.1656 Abarbanel & Itzykson doi.org/10.1103/PhysRevLett.23.53 Eikonal scattering=(possibly non-planar) gen. ladder +self-energy diagrams. $\quad\Delta I_q=\Delta L-\Delta \pi_0$; $\quad I_q=L-\pi_0+\frac{E}{2}$; $\quad V_3=2I_q=2(L-\pi_0)+E$; $\quad\Delta I_m=\Delta V_3$; $\quad I_m=V_3-\frac{E}{2}=2(L-\pi_0)+\frac{E}{2}$; Commented Jul 4 at 11:47