Thinking in Waves Episode 25 ／ Part X · Breaking the assumptions
Replacing Episode 18's "box" with the real thing
The strong force as waves ──
confinement was not a box but a string
Line up hadron masses on a straight line and out comes a string tension of 14 tonnes.
And \(\Lambda_{\rm QCD}\), which sets the string's scale, was ── born from a dimensionless coupling.
★ 99 % of the mass around you is a dimensionless quantity that acquired a dimension.
Last time, the weakest force. This time the other end ── the strongest.
And we find out why Episode 18's estimate was "the right order but off by a factor of 1.5".
01Replacing Episode 18's box
| Force | Shape of the field | Potential | Behaviour |
|---|---|---|---|
| Electromagnetic | spreads in every direction | \(V\propto-1/r\) | force falls as \(1/r^2\) |
| Yukawa (Episode 6) | decays below the cutoff | \(V\propto-e^{-r/\lambda}/r\) | does not reach beyond \(\lambda\) |
| ★ Strong | ★ the field is squeezed into a string | \(V\propto+\sigma r\) | ★ a constant force at any distance |
The heart of §01
Electromagnetism and Yukawa both weaken with distance. Only the strong force does not.
Which is why things are confined ── the "box" of Episode 18 was really a string (a flux tube).
02Measuring the string tension from hadron masses
\(\sigma\) is the string tension. Line up measured hadrons and fit the slope
| Particle | Mass \(M\) [GeV] | \(M^2\) [GeV²] | Spin \(J\) | Fit prediction |
|---|---|---|---|---|
| \(\rho(775)\) | 0.77526 | 0.60103 | 1 | 0.9770 |
| \(a_2(1320)\) | 1.31820 | 1.73765 | 2 | 2.0174 |
| \(\rho_3(1690)\) | 1.68880 | 2.85205 | 3 | 3.0374 |
| \(a_4(1970)\) | 1.96700 | 3.86909 | 4 | 3.9682 |
The heart of §02
★★ The string tension is 14.4 tonnes-force.
Pulling one quark pair apart costs 14 tonnes, and that force is the same at any distance. Which is why they can never be separated.
★ All it took was laying hadron masses on a straight line.
03The string's ladder ── the same structure as Episode 6's waveguide
| System | Relation | Rung number | What it is |
|---|---|---|---|
| Waveguide (Episode 6) | \(\omega^2=\omega_c^2+c^2k^2\) | mode number \(n\) | a ladder of cutoffs |
| Box (Episode 18) | \(E=2.04\hbar c/R\) | number of nodes | a ladder from boundaries |
| ★ String (this episode) | \(M^2=(J-\alpha_0)/\alpha'\) | ★ angular momentum \(J\) | ★ a ladder of rotational modes |
All three are the same story: boundary conditions make discrete modes. Only the boundary differs ── walls, a box, and the tension of a string.
| \(J\) | String prediction \(M\) [GeV] | Observed candidate |
|---|---|---|
| 5 | 2.2353 | \(\rho_5(2350)\), \(M=2.33\) |
| 6 | 2.4676 | \(a_6(2450)\), \(M=2.45\) |
| 7 | 2.6798 | ── |
04Counting the colours ── a dimensionless ratio of cross-sections
| Energy band | Naive \(R\) (\(N_c=3\)) | With QCD correction | Measured, roughly |
|---|---|---|---|
| \(\sqrt s=2\sim3\) GeV (u,d,s) | 2.000 | 2.159 | about 2.2 |
| \(\sqrt s=5\sim10\) GeV (+c) | 3.333 | 3.556 | about 3.6 |
| \(\sqrt s=20\sim40\) GeV (+b) | 3.667 | 3.830 | about 3.9 |
Conclusion of §04
Put \(N_c=3\) in and all three bands match. With \(N_c=1\), \(R\) would be a third of that and match nothing.
★ "There are three colours" is a fact measured directly as a dimensionless ratio.
Episode 19 (the tau lifetime) and this episode (the R ratio) agree by two independent routes.
05Asymptotic freedom ── why it runs the other way from QED
| \(n_f\) | \(\beta_0=11-2n_f/3\) | Direction of running |
|---|---|---|
| 0 | 11.000 | ★ asymptotically free |
| 6 (reality) | 7.000 | ★ asymptotically free (weaker up close) |
| 16 | 0.333 | only just |
| 17 | \(-0.333\) | ★ runs like QED |
Asymptotic freedom requires \(n_f<16.5\), and nature has six. "With 17 quark flavours there would be no confinement" ── though it is not a close call.
06★ Dimensional transmutation ── a mass born from a dimensionless number
The heart of §06 ── the most important point of this episode
The only input is the dimensionless \(\alpha_s\).
And out comes a mass scale \(\Lambda_{\rm QCD}\) with a dimension.
This is dimensional transmutation ── the most dramatic consequence of Episode 5's "constants are integrals over \(\ln\mu\)". The integration itself creates a new dimension.
The corresponding length is \(\hbar c/\Lambda\): 2.26 fm at one loop, but with the published \(\Lambda=210\) MeV it is 0.94 fm ── essentially the proton's charge radius, 0.84 fm.
| \(\alpha_s(m_Z)\) | \(\Lambda_{\rm QCD}\) [MeV] | \(\hbar c/\Lambda\) [fm] |
|---|---|---|
| 0.1100 | 53.0 | 3.723 |
| 0.1150 | 73.3 | 2.693 |
| 0.1179 | 87.3 | 2.260 |
| 0.1200 | 98.6 | 2.001 |
| 0.1250 | 129.6 | 1.523 |
§06 ── Episode 18 is completed here
A 6 % change in \(\alpha_s\) nearly doubles \(\Lambda\) (it is an exponential). The size of a proton is exponentially sensitive to a dimensionless coupling.
★★ And it joins up with Episode 18 ──
99 % of the proton's mass is confinement ＝ \(\Lambda_{\rm QCD}\).
★ And \(\Lambda_{\rm QCD}\) was born from the dimensionless \(\alpha_s\) by transmutation.
── ★ 99 % of the mass around you came out of a dimensionless number.
Episode 7's "only dimensionless quantities are physics" has advanced to "dimensionful quantities are built from dimensionless ones".
07Massless yet short-ranged ── Episode 6's exception
| Mediator | Mass [GeV] | Range | Reason |
|---|---|---|---|
| Photon | 0 | \(\infty\) | linear; the field spreads |
| W boson | 80.4 | \(2.5\times10^{-3}\) fm | Episode 6; a cutoff from mass |
| ★ Gluon | 0 | ★ about 1 fm | ★ non-linear; squeezed into a string |
The heart of §07 ── Episode 23 comes into play
Episode 6's "mass = cutoff" was a statement about linear theory. In a non-linear theory, confinement happens without any mass ──
gluons carry colour → gluons attract each other
→ the field does not spread but is squeezed into a string → a distance-independent force
★★ Episode 23's non-linearity is changing the shape of the field itself.
The contrast ──
linear + massive → exponential decay (Yukawa, Episode 6)
linear + massless → reaches infinity (electromagnetism)
★ non-linear + massless → a string, and confinement (QCD, this episode)
08Particles as resonances ── Episode 9's linewidth
| Particle | Mass [GeV] | Width \(\Gamma\) [GeV] | \(\Gamma/M\) | Lifetime [s] |
|---|---|---|---|---|
| \(\Delta(1232)\) | 1.2320 | 0.117 | \(9.5\times10^{-2}\) | \(5.6\times10^{-24}\) |
| \(\rho(770)\) | 0.7753 | 0.1476 | \(1.9\times10^{-1}\) | \(4.5\times10^{-24}\) |
| \(J/\psi\) | 3.0969 | \(9.26\times10^{-5}\) | \(3.0\times10^{-5}\) | \(7.1\times10^{-21}\) |
| Z boson | 91.188 | 2.4955 | \(2.7\times10^{-2}\) | \(2.6\times10^{-25}\) |
| Higgs | 125.25 | 0.0041 | \(3.3\times10^{-5}\) | \(1.6\times10^{-22}\) |
The heart of §08
The \(\rho\)'s width is 19 % of its mass ── ★ it falls apart before completing one oscillation.
In Episode 9's language, \(\Delta\omega\) is the same order as \(\omega\) ── the limit of what can be counted as a wave.
The \(J/\psi\), by contrast, has \(\Gamma/M=3\times10^{-5}\) ── decaying strongly yet 6400 times narrower than the \(\rho\).
★ The reason is that its destination is closed off (OZI suppression) ── exactly the structure of Episode 19's \(^{187}\)Re. When the final-state modes are unavailable, the rate collapses.
Exercises
- In this episode's language, why was Episode 18's box estimate off by 1.5?
Show answer
The shape of the boundary was wrong. In a box the field spreads in three dimensions and the energy goes as \(\hbar c/R\).
On a string the field is squeezed to one dimension and the energy is \(\sigma R\) (tension × length) ── a completely different dependence on \(R\).
★ A real proton has both (Coulomb-like at short range, string-like at long range ── the Cornell potential \(V=-4\alpha_s/3r+\sigma r\)).
What this episode establishes is that the long-range side is a string. - Give two pieces of evidence for three colours, and say what each counts.
Show answer
① The R ratio (this episode): the ratio of the hadronic to the muonic cross-section in \(e^+e^-\). It counts how many quark states there are.
② The tau lifetime (Episode 19): 5.6 decay channels, three of them hadronic. It counts how many destinations a decay has.
★ Both count final states, and both are dimensionless ── Episode 7's rule at work again. - Why is dimensional transmutation "dramatic"?
Show answer
Because the input has no dimension and the output does.
In classical physics a mass scale is always supplied as input (a spring constant, a string length…). In QCD the only input is the dimensionless \(\alpha_s\). Integrate it over \(\ln\mu\) and out comes the scale \(\Lambda\) where the coupling diverges.
★ Episode 5's "constants are integrals" has become "the dimension itself is an integral".
★ And it is the strongest footnote to Episode 15's "hits are dimensionless, misses are dimensionful" ── dimensionful quantities are built out of dimensionless ones.
Summary A 14-tonne string, and dimensional transmutation
Episode 18's "box" was really a string (a flux tube). Electromagnetism and Yukawa weaken with distance; only the strong force does not ── hence confinement.
Plot hadron masses as \(M^2\) against \(J\) and they fall on a clean straight line (a Regge trajectory), whose slope gives \(\alpha'=0.915\) GeV⁻² and a string tension of 14.4 tonnes-force. Fourteen tonnes at any distance ── which is why quarks can never be pulled apart. And the string's ladder has the same "boundaries make discrete modes" structure as Episode 6's waveguide and Episode 18's box.
The number of colours is measurable directly as the dimensionless R ratio (\(N_c=3\) fits all three bands) ── agreeing independently with the tau lifetime of Episode 19.
Asymptotic freedom comes from the 11 in \(\beta_0=11-2n_f/3\) (gluon self-interaction). Invert it and ── ★ a dimensionful \(\Lambda_{\rm QCD}\) is born from the dimensionless \(\alpha_s\) alone (dimensional transmutation). The corresponding length is 0.94 fm ── the size of a proton.
★ 99 % of the proton's mass is confinement ＝ \(\Lambda_{\rm QCD}\),
and \(\Lambda_{\rm QCD}\) was born from a dimensionless number.
And gluons are massless yet reach only 1 fm ── Episode 6's "mass = cutoff" was a linear statement, and in a non-linear theory (Episode 23) confinement happens without mass. Seen as resonances, the \(\rho\)'s width is 19 % of its mass ── the limit of what can be called a wave.
One counterexample would refute it. Twenty-five episodes have supplied the tools, so ★ next time we test it ── eliminate degrees of freedom numerically and see whether a fractional order appears.
And one of the doors left open in Episode 15 will either close or stay open.
This document is Episode 25 of the series "Thinking in Waves". The computation is in kenshou/tsuyoi.js. On the numbers: the four hadron masses on the Regge trajectory are PDG values, and the least-squares slope \(\alpha'=0.9153\) is our own fit (the literature quotes 0.88–0.9; it shifts slightly depending on which states you include). The 14.4 tonnes-force follows from that \(\alpha'\) and is consistent with the lattice-QCD value \(\sigma\approx0.18\) GeV² (about 15 tonnes-force). The \(J=5,6,7\) predictions and their "candidates" are indicative; the identification of those resonances is debated. The "measured" R values are representative figures, not analysed here (the QCD correction is first order only). \(\Lambda_{\rm QCD}=87\) MeV is the one-loop value and differs by more than a factor of two from the published 210 MeV (\(\overline{\rm MS}\), two loops and beyond, \(n_f=5\)) ── \(\Lambda\) depends strongly on scheme and loop order, so this is a difference of definition rather than an error. The "string" picture holds only at long range; at short range the potential is Coulomb-like (the Cornell potential). The account of OZI suppression is qualitative and not derived here. ── To print, use your browser's Print → Save as PDF.