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.
The figure's caption reads
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?
Best Answer
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.
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