"Where does \(G\) come from" ── that question was shaped so that it could never have an answer
A door closed in Episode 13 and another in Episode 14. What remained was \(G\) itself.
Going to dig, I hit a wall at once. And that wall was not built this time ── it was the one I had proved myself back in Episode 1. A quantity with dimensions changes value when you change units. So "why does \(G\) have this value" has no answer.
Fix the question and things move. And the fixed form has a surprisingly straightforward mechanism ── nineteen orders of smallness come out of the unremarkable number \(1/57\).
This episode is the fifth return to Episode 1. But this time we did not hit the wall; we found out what the wall is.
What Episode 1 did was brute-force dimensional analysis. It enumerated every dimensionless ratio buildable from 25 physical quantities (162,931 of them) and looked for ones landing on meaningful-looking values ── the result was \(p=0.67\): essentially all coincidence.
Of what that episode confirmed, this is the part that matters now.
The "value" of a quantity with dimensions carries no physical information. Because changing units changes it. Physics can only speak about dimensionless quantities.
Apply that to my own question.
| question | well posed? |
|---|---|
| why does \(G=6.674\times10^{-11}\) have this value | no. change units and it can be made 1 |
| why is the Planck length \(1.6\times10^{-35}\) m | no. same reason |
| why is \(Gm_p^2/\hbar c\) equal to \(5.9\times10^{-39}\) | yes |
At the end of Episode 14 I wrote "all that remains is where to get \(G\) itself," but that phrasing itself was not up to the job. What has to be got is not \(G\).
Electromagnetism has \(\alpha=e^2/4\pi\varepsilon_0\hbar c\simeq1/137\). Do the same for gravity ──
\(G=6.67430\times10^{-11}\), \(m_p=1.67262192369\times10^{-27}\,\mathrm{kg}\), \(\hbar=1.054571817\times10^{-34}\), \(c=2.99792458\times10^8\)
$$\alpha_G=5.9061\times10^{-39}$$So "why is gravity so weak" is the same question as "why is the proton nineteen orders lighter than the Planck mass". Not a story about \(G\) but a story about the hierarchy of masses.
And this form has a mechanism.
The proton's mass does not come from the quark masses. Almost all of it is the energy of the strong interaction. And that scale \(\Lambda_{\rm QCD}\) is a number that was never in the theory to begin with.
The QCD Lagrangian has no mass scale. All it has is the dimensionless coupling \(\alpha_s\). But \(\alpha_s\) runs with energy and diverges at some low energy. Where it diverges is \(\Lambda_{\rm QCD}\) ── a quantity with dimensions born from a dimensionless number. This is called dimensional transmutation.
Solve this for \(\Lambda\) ──
$$\boxed{\;\Lambda=\mu\,\exp\!\left(-\frac{2\pi}{b_0\,\alpha_s(\mu)}\right)}$$There is a \(1/\alpha_s\) inside the exponent. So even with \(\alpha_s\) an unremarkable number around \(1/50\), \(\Lambda/\mu\) can become fantastically small.
There is only one input ── the measured \(\alpha_s(M_Z)=0.1179\). From it we get \(\Lambda\) and run up to the Planck scale.
# input: alpha_s(M_Z) = 0.1179, M_Z = 91.1876 GeV, n_f = 5, one loop b0 = 11 - 2*5/3 = 7.6667 # downward: get Lambda Lambda = M_Z exp(-2pi/(b0 * 0.1179)) = 0.0873 GeV = 87.3 MeV # upward: extend the same straight line to the Planck scale ln(M_Pl/Lambda) = ln(1.2209e19 / 0.0873) = 46.39 alpha_s(M_Pl) = 2pi/(b0 * 46.39) = 0.01767 = 1/56.6 # read off the hierarchy Lambda / M_Pl = exp(-46.39) = 7.15e-21 observed m_p/M_Pl = 7.685e-20 offset = 10.7 ← m_p/Lambda = 938 MeV / 87.3 MeV = 10.7 => twenty orders of hierarchy came out of the number 1/56.6. the remaining offset is only "how many times Lambda the proton mass is". O(10) hadron physics.
Being inside an exponent means the sensitivity is exponential too. Try moving the figure.
| \(1/\alpha_s(M_{\rm Pl})\) | \(\ln(M_{\rm Pl}/\Lambda)\) | \(\Lambda\) | proton mass would be |
|---|---|---|---|
| 45 | 36.88 | 1180 GeV | a proton at the \(10^4\) GeV scale |
| 50 | 40.98 | 19.5 GeV | the 200 GeV scale |
| 56.6 | 46.39 | 87.3 MeV | 938 MeV ✓ |
| 60 | 49.17 | 5.39 MeV | the 58 MeV scale |
| 65 | 53.27 | 0.0895 MeV | the 1 MeV scale |
From \(1/50\) to \(1/60\) ── a 20% change moves \(\Lambda\) by four orders.
The mechanism's strength and its weakness come from the same place. It can create smallness because it is exponential. The value is sensitive because it is exponential. You cannot take only one.
Let us draw the line honestly.
| quantity | is there a mechanism? |
|---|---|
| proton mass / Planck mass (\(10^{-20}\)) | yes. dimensional transmutation. but \(1/\alpha_s\simeq57\) is an input |
| \(\alpha_s(M_{\rm Pl})\simeq1/57\) itself | no. that becomes a question about unification |
| Higgs mass / Planck mass | no ── this is the real hierarchy problem |
| the cosmological constant (\(10^{-122}\)) | no. worse still |
The third row matters. The proton is protected; the Higgs is not.
The proton mass is small because QCD's coupling runs only logarithmically, so exponential smallness appears automatically. Nothing was tuned. The Higgs mass squared, by contrast, receives corrections going as the cutoff squared, so keeping it small requires cancellations. There is no automatic answer to "why is it light" ── that is the hierarchy problem.
It does not. But the way it fails to is suggestive.
Asymptotic safety says the dimensionless coupling \(g=Gk^2\) runs to a fixed point at high energy. Its value in the Einstein–Hilbert truncation is ──
Since \(g_*=Gk^2\), \(G(k)=g_*/k^2\). As \(k\to\infty\), \(G\to0\).
What is fixed is not \(G\) but \(Gk^2\). A dimensionless quantity. The low-energy value of \(G\) is set by "where you exit the fixed point," and that is an IR input.
What asymptotic safety reduces is the number of free parameters. Infinitely many counterterms drop to finitely many (the number of relevant directions). But the one that sets the overall scale always remains.
This is not a defect; it is Episode 1's conclusion itself ── scales cannot be predicted. Only ratios can.
Let us count. How many times has this series returned to Episode 1 §08's "dimensions get you as far as \(S\propto A\), and only the \(1/4\) refuses"? ──
| episode | how it returned | the reading at the time |
|---|---|---|
| Episode 9 | while forming the conjecture | filling the coefficient needs something to count |
| Episode 11 | hunting for the central charge | maybe we are looking in the wrong place |
| Episode 12 | after the area law came out by itself | not even dimensional analysis, and the wall is the same height |
| Episode 14 | trying to count degrees of freedom | the coefficient can never come out of matter degrees of freedom |
| Episode 15 | trying to dig out \(G\) | this wall was the wall Episode 1 proved |
The \(1/4\) did not come out not because the tools were insufficient. The \(1/4\) is a number that only means anything paired with \(G\), and \(G\) carries dimensions.
The same reason the brute force of Episode 1 failed. It has taken fourteen episodes to apply what I proved back then to my own question.
The wall does not move, but we know what it is ── this is not a wall to climb but the edge of the map. If there is something past the edge, it is not "a way to compute the \(1/4\)" but "a theory that makes the list of dimensionless quantities shorter."
"Where does \(G\) come from" is not well posed. The value of a quantity with dimensions changes with units, so it carries no physical information ── something I had confirmed myself in Episode 1. Episode 12's "only \(\log q/a^2\) is determined" and Episode 14's "only the ratio of \(\sigma\) to \(G\) is determined" made three encounters with the same shape.
The fixed form is \(\alpha_G=Gm_p^2/\hbar c=5.9061\times10^{-39}\). This equals \((m_p/m_{\rm Pl})^2\) (verified numerically). That is, "why is gravity weak" = "why is the proton nineteen orders lighter than the Planck mass." Not a story about \(G\) but about the hierarchy of masses.
And the nineteen orders have a mechanism ── dimensional transmutation. Because \(\Lambda=\mu\exp(-2\pi/b_0\alpha_s)\) puts \(1/\alpha_s\) in the exponent, an unremarkable coupling of order \(1/50\) yields fantastic smallness. From the single measured \(\alpha_s(M_Z)=0.1179\): \(\Lambda=87.3\) MeV, hence \(\alpha_s(M_{\rm Pl})=1/56.6\) and \(\Lambda/M_{\rm Pl}=e^{-46.39}=7.15\times10^{-21}\). The offset of 10.7 against the observed \(m_p/M_{\rm Pl}=7.685\times10^{-20}\) is exactly accounted for by \(m_p/\Lambda=10.7\).
But this is translation, not prediction. That the offset equals \(m_p/\Lambda\) is an identity, not an independent check. What was done is rewriting "why \(10^{-20}\)" as "why \(1/\alpha_s\simeq57\)." It still means something ── the thing to be explained dropped from exponential to linear.
The lever of the exponent. Move \(1/\alpha_s(M_{\rm Pl})\) 20%, from \(50\) to \(60\), and \(\Lambda\) moves four orders. The ability to create smallness and the sensitivity of the value come from the same exponent. You cannot take only one.
Protected quantities and unprotected ones. The proton mass is automatically light thanks to logarithmic running. The Higgs mass receives cutoff-squared corrections and therefore needs cancellations ── the real hierarchy problem. The same picture as anomaly cancellation in Episode 2, with the series' "binary target" appearing this time as "protected by renormalization, or not."
Asymptotic safety does not give \(G\) either. What is fixed is \(g_*=Gk^2\) (\(=0.911\), truncation dependent; other schemes report \(58.73\)). Only dimensionless quantities are fixed, and the one that sets the overall scale always remains. What is reduced is the number of parameters, not the scale.
Episode 1, a fifth time ── and this time it was the identity of the wall. The \(1/4\) did not come out not from a lack of tools but because the \(1/4\) only means anything paired with \(G\), and \(G\) carries dimensions. Not a wall to climb but the edge of the map. What lies past the edge is not "a way to compute the \(1/4\)" but a theory that makes the list of dimensionless quantities shorter.
This document is Episode 15 of the "Lattice We Build" series, a reading piece for high-school and university students who love physics. Where the sister series "That Clicks" explains known physics, this series shows the work itself ── including the realization that I had set the question up wrongly.
Established material: the general facts about dimensional analysis and dimensionless quantities; \(\alpha_G=Gm_p^2/\hbar c\) and the Planck-mass numbers (computed from CODATA 2018 \(G,m_p,\hbar,c\)); the one-loop QCD beta function and \(b_0=11-\frac23n_f\); the generation of \(\Lambda_{\rm QCD}\) by dimensional transmutation; \(\alpha_s(M_Z)=0.1179\) (PDG); the quadratic divergence of the Higgs mass and the hierarchy problem; and the Reuter fixed point of asymptotic safety with the values of \(g_*,\lambda_*\) in the Einstein–Hilbert truncation and their truncation dependence.
The calculations in this article (\(\alpha_G\), the one-loop \(\Lambda\) and \(\alpha_s(M_{\rm Pl})\), the comparison of \(e^{-46.39}\) with observation, the sensitivity table) were carried out and checked by the author from the above. Because a one-loop approximation is used, the value of \(\Lambda\) differs from the standard two-loop-and-beyond value (\(\Lambda^{(5)}_{\overline{\rm MS}}\simeq210\) MeV). This article prioritized internal consistency using only the measured \(\alpha_s(M_Z)\) and the one-loop formula.
The line that "gravity is weak thanks to dimensional transmutation" is itself a known argument (one Wilczek and others have made repeatedly) and is not this article's original claim. On the other hand, the reading that "the \(1/4\) did not come out as a corollary of Episode 1's conclusion" is this series' author's framing. That the horizon's Carrollian theory itself is unconstructed is as stated in Episodes 8, 11 and 12.
Main series: Episode 1｜Episode 2｜Episode 3｜Episode 4｜Episode 5｜Episode 6｜Episode 7｜Episode 8｜Episode 9｜Episode 10｜Episode 11｜Episode 12｜Episode 13｜Episode 14 ｜ bonus: ①／②／③ ── to print, use your browser's "Print" → "Save as PDF."
Print / PDF: ⌘+P (Ctrl+P on Windows). In the figure you can confirm that moving the coupling slightly moves the hierarchy by orders of magnitude.