If you chase "What is the dimension of the weak force and the strong force? Can this formula give it?" without flinching
Up through last time we saw that \(F=1/(Cn)^D\) is not a law of force but a ruler for dimension. So what does that instrument read when we point it at nature's four forces? Electromagnetism gives \(D=2\) (\(C\!\approx\!137\)) — the inverse-square law. Then what is the \(D\) of the weak and strong forces? The conclusion, stated up front — you can't get it as a single number. But that isn't a failure; it becomes the cleanest concrete example of last time's "\(D\) runs." Put in a computer person's words, a force is "how far the message called a mediator particle reaches down the wiring called distance without fading." The way it fades differs across the four — that's all.
The definition of the ruler was this: as you increase size (here distance \(r\)), by what fraction does the quantity \(F\) shrink? The slope of that on a log–log plot is the dimension \(D\).
The important thing is that this formula returns \(D\) as a single number only when \(F\) is a clean power law. If \(F\) isn't a power, \(D\) takes a different value at each distance — that is, it runs. The differences among the four forces show up right here. \(C\) is "the reference where the needle points to 1" = the strength of the coupling (for electromagnetism, \(1/\alpha\!\approx\!137\)).
The mediator of electromagnetism (the photon) and of gravity (the graviton) are massless. A massless message reaches arbitrarily far and thins out honestly in 3D space — it spreads over the sphere \(4\pi r^2\), so \(F\propto 1/r^2\). This is a genuine power law, so the instrument answers without hesitation:
\(F=1/r^2\) → \(D=-d\ln F/d\ln r = \mathbf{2}\). No matter how you change the distance, the slope stays 2 — it does not run. This is the "honest inverse square," and your \(C\!=\!137,\ D\!=\!2\) (electromagnetism) is exactly this. Gravity is the same \(D=2\); only \(C\) differs (\(\sim\!10^{39}\), in the table below).
This is the baseline. "Massless mediator = constant \(D=2\)." The remaining two forces are determined by how they deviate and run from this baseline.
The mediators of the weak force (the W and Z particles) are heavy (about 90 times the proton). A heavy message can't fly far and dies off exponentially at a range \(\lambda=\hbar/(Mc)\approx10^{-18}\,\text{m}\) (the Yukawa potential). The force takes the form \(F\sim e^{-r/\lambda}/r^2\). This is not a power law. Even so, apply the instrument \(D=-d\ln F/d\ln r\):
\(\ln F = -\,r/\lambda - 2\ln r\). Differentiating with respect to \(\ln r\) gives \(d\ln F/d\ln r = -\,r/\lambda - 2\). Flip the sign and \(D=2+r/\lambda\).
The strong force is even more extreme. The force between quarks doesn't diminish even as you pull them apart (confinement). The potential is \(V(r)=-a/r+\sigma r\) (the Cornell form) — Coulomb-like \(-a/r\) up close, rising linearly as \(\sigma r\) far away. As a force this is \(F=a/r^2+\sigma\). Apply the instrument:
(\(r_0=\sqrt{a/\sigma}\) is the crossover scale, \(\sim\!0.2\,\text{fm}\).)
In the ruler of last time, \(D=0\) is the reading of zero dimension: "changing the size doesn't change the contents." The tube of force (the flux tube) stays twisted and taut; the more you stretch it, the more energy it stores, until it finally snaps and creates a new quark pair — which is why you can't extract a lone quark. The strong force's \(C\approx1\) (\(\alpha_s\!\sim\!1\)) is, literally, the origin of "strong."
| Force | D (short range → long range) | C ≈ 1/coupling | Reason for the falloff (computer-person vocabulary) |
|---|---|---|---|
| Gravity | 2 (constant) | ~10³⁹ | massless mediator → honest power law |
| Electromagnetism | 2 (constant) | 137 | massless mediator → honest power law |
| Weak force | 2 → ∞ | ~30 | heavy mediator (W/Z) → exponential cutoff |
| Strong force | 2 → 0 | ~1 | confinement → doesn't fall off with distance |
All four forces start from \(D=2\) at ultra-short range. Because in 3+1 dimensions the field a point source sprays out must begin as inverse-square (\(1/r^2\)). The difference is only what happens at long range — the mediator's mass (weak) or confinement from self-interaction (strong). So the correct answer to "what is the dimension of the weak and strong forces?" is "not a single number, but a function that runs from 2 to ∞ / from 2 to 0." You can get it from the formula — but only as a running \(D(r)\).
Here is the one line you must not blur. For the weak and strong forces, \(D\) does not settle into a single integer. That is not a defect of the instrument; it merely reflects, honestly, the fact that the target isn't a power law — indeed it is fully consistent with last time's "\(D\) runs" philosophy. But there is one more seam that must be stated honestly:
Last time \(C\) was "the reference scale where the needle points to 1 (the unit of resolution)." In this episode, though, \(C\!=\!137\) is used as "\(1/\alpha\) = the strength of the coupling." Scale and coupling are originally different things. In \(F=1/(Cr)^D\), the \(C\) is the prefactor (the place that absorbs the coupling) and \(D\) is the exponent (the geometry = how the force falls off with distance). Make this division of labor — "\(C\) = coupling / \(D\) = geometry" — explicit, and there's no confusion. Blur them and claim "one formula gave it all," and you are planting a flag.
And there is not a single piece of new physics in this episode. Inverse-square (massless field), the Yukawa potential (exponential cutoff from a massive mediator), asymptotic freedom and confinement (the Cornell form \(-a/r+\sigma r\), the running of \(\alpha_s\)) — all established physics. \(D=-d\ln F/d\ln r\) is the defining formula of dimension, and we simply read it by applying it to known force shapes. The holes that remain — "why does the mediator have that mass?", "where does \(\sigma\) come from (the mass-gap problem)?" — are open problems for physics as a whole, and we don't fill them here. Never say "solved" — that is the rule of the whole series.
Apply the instrument \(D=-d\ln F/d\ln r\) to the four forces and every one starts at \(D=2\) at ultra-short range. Gravity and electromagnetism have massless mediators and stay at constant \(D=2\), inverse-square. The weak force, with heavy mediators, runs \(D=2+r/\lambda\to\infty\) (which is why "short range" = "weak"). The strong force, with confinement, runs \(D=2/(1+(r/r_0)^2)\to0\) (which is why it "doesn't fall off with distance" = "strong").
So the honest answer to "what is the dimension of the weak and strong forces?" is not a single integer but a running function. You can get it from the formula — but as a concrete case of last time's "\(D\) runs." Make explicit that \(C\) (coupling) and \(D\) (geometry) are separate axes, and leave "why that mass, why that \(\sigma\)?" open as a hole in physics as a whole — keep those two disciplines, and you move forward with a signpost, not a flag.
Print / save as PDF: Ctrl+P (⌘+P on Mac). On screen, move the distance cursor with the slider to read the running dimension D of all four forces at once. Click "Show the answer" to open each solution.