Asymptotic Safety: The Quantum Gravity Fixed Point

RG Flow Parameters

Quantum correction C 1.50

Λ–G coupling a 0.80

Trajectories 30

Flow speed 0.60

Live Readouts

Fixed point g̃*

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Fixed point λ̃*

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Scale k / k_Pl

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Spread from FP

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How it works

In asymptotic safety, gravity is quantized by studying how its couplings change with the observation energy scale k — the same renormalization-group (RG) idea used for the other forces, applied to G and the cosmological constant Λ. Made dimensionless, g̃ = G(k)k² and λ̃ = Λ(k)/k², they obey flow equations of the schematic Wetterich-equation form used in truncated exact-RG calculations:

dg̃/dt  = 2g̃ − C·g̃²
dλ̃/dt = −2λ̃ + a·g̃·(1 − b·λ̃)
   t = ln(k / k_Pl),  b = 0.5 fixed

The canonical term 2g̃ (from G's mass dimension) pushes the coupling away from the trivial Gaussian point g̃=0 as t grows; the one-loop-type correction −C·g̃² bends the flow back, producing a second, non-Gaussian fixed point at g̃* = 2/C that is attractive in the ultraviolet (t → +∞) from any positive starting value — exactly the mechanism (dimension term + quantum correction ⇒ nontrivial UV fixed point) behind Reuter's asymptotic-safety conjecture. λ̃ relaxes onto its own fixed value λ̃* = a·g̃*/(2 + a·b·g̃*), sourced by g̃.

Trajectories — each line is one RG flow starting from a different low-energy (infrared) condition; the moving beads mark today's playback scale t.
C — strength of the quantum correction; moves the fixed point g̃*=2/C.
a — how strongly the cosmological constant is sourced by the gravitational coupling; moves λ̃*.
All trajectories funnel onto the same vertical line (g̃*, λ̃*) as k grows — a UV-complete theory needs only finitely many measured numbers, which is the whole point of asymptotic safety as a "theory of everything" candidate: no infinities, no extra dimensions, just an ordinary interacting fixed point of gravity itself.

This is a simplified, pedagogical logistic-flow model that reproduces the real qualitative structure (Gaussian IR fixed point at g̃=0, non-Gaussian UV-attractive fixed point at finite g̃*) rather than the full numerically-fitted Reuter–Saueressig coefficients.

RG trajectory (IR → UV) Non-Gaussian fixed point

x: g̃ y: λ̃ z: t = ln(k/k_Pl)

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