Thinking in Waves Episode 33 ／ Part XIII · Frontier physics as waves
Is there any relation among the lepton masses?
Rereading the Koide formula as a wave ──
the AC and DC parts have equal RMS
Treat the three \(\sqrt m\) as a sinusoid sampled at 120° intervals and the Koide formula becomes one line.
★ \(K=2/3\ \Longleftrightarrow\ A=\sqrt2\ \Longleftrightarrow\) the AC RMS equals the DC part (measured ratio 0.99999077, 0 dB).
★ But run the masses and it degrades 200-fold ── this is a relation that holds only in the pole-mass scheme.
Episode 32 found that the Higgs distributes one dimensionful quantity \(v\) ── but the ratios it distributes in (the Yukawa couplings) went unexplained: \(y_{\rm top}/y_e=3.4\times10^5\).
★ So is there no relation at all among the masses? For the three charged leptons there is exactly one known candidate.
01Check it first ── and find the range \(K\) can take
It agrees to one part in \(10^5\). First, is \(K\) free to be anything? With \(x_i=\sqrt{m_i}\), \(K=|x|^2/(\sum x)^2\), and the range is fixed.
| Configuration | \(K\) |
|---|---|
| all three equal | 0.333333 (minimum) |
| one overwhelmingly largest | 1.000000 (maximum) |
| ★ Koide | 0.666667 ── exactly the midpoint |
The heart of §01
\(K\) can only lie between \(1/3\) and \(1\), and \(2/3\) is its midpoint. So "2/3 came out" is not in itself so surprising.
★★ What is surprising is the precision (\(10^{-5}\)). §07 measures exactly that.
02Reading it geometrically ── 45 degrees
| Quantity | Value |
|---|---|
| ★ Measured angle \(\theta\) | 44.999735° |
| \(K=2/3\) requires | exactly 45° |
| Range \(\theta\) can take | 0 – 54.7356° |
The heart of §02 ── ★ using the habit from Episodes 29 and 30
It matches 45° to five digits. So is this 45° the same as the "half order's 45° phase" of Episode 1?
★★ Rejected. They are different things.
Episode 1's 45° is a phase in the complex plane (a delay in time). This 45° is the direction of a vector in real three-space.
They live in different spaces. Equal numbers share no meaning.
03★★ Reading it as a wave ── a sinusoid sampled at 120° intervals
DC \(\mu\), AC amplitude \(\mu A\), phase \(\delta\) ── a 3-point DFT, literally
| Quantity | From measurement |
|---|---|
| DC \(\mu=(\sum\sqrt m)/3\) | 17.715562 \(\sqrt{\rm MeV}\) |
| ★ Amplitude ratio \(A\) | 1.41420051 |
| Phase \(\delta\) | 132.7328° |
Now use the three-phase identities ── \(\sum_k\cos(\delta+2\pi k/3)=0\) and \(\sum_k\cos^2=3/2\). Vary \(\delta\) and it stays \(3/2\) (confirmed to 12 digits). This is the same identity that makes three-phase electrical power constant.
The heart of §03
★★★ So the Koide formula \(K=2/3\) simply means \(A=\sqrt2\).
\(\dfrac{1+A^2/2}{3}=\dfrac23\ \Longleftrightarrow\ A^2=2\). Measured \(A=1.41420051\) against \(\sqrt2=1.41421356\) ── rel. diff. \(-9.2\times10^{-6}\).
04★★★ What \(A=\sqrt2\) means ── the AC and DC parts have equal RMS
A sinusoid of amplitude \(A\) has RMS \(A/\sqrt2\). So ── \(A=\sqrt2\ \Longleftrightarrow\) RMS ＝ 1 ＝ the DC part.
| Component | Value [\(\sqrt{\rm MeV}\)] |
|---|---|
| DC \(\mu\) | 17.715562 |
| ★ AC RMS \(\mu A/\sqrt2\) | 17.715398 |
| ★★ Ratio | 0.99999077 ── \(-8.0\times10^{-5}\) dB |
The heart of §04 ── ★ this episode's conclusion
★★★ The Koide formula says ──
"the AC and DC parts of the \(\sqrt{\text{mass}}\) wave have equal RMS."
That shows what is being claimed far better than the bare number 2/3. ★ The "power" carried by the oscillating part exactly equals the "power" carried by the constant part.
05Making a prediction ── the tau mass
Feed in only \(m_e\) and \(m_\mu\) and solve \(K=2/3\) for \(m_\tau\).
| Quantity | Value [MeV] |
|---|---|
| ★ Predicted \(m_\tau\) | 1776.969 |
| Measured \(m_\tau\) | 1776.86 ± 0.12 |
| ★ In sigmas | 0.91 σ |
Moving an input by 1 ppm shifts the prediction by 0.156 keV for \(m_e\) and 1.62 keV for \(m_\mu\). The muon mass dominates, but its current precision (2.3e−8) is ample. ★ So the formula is falsifiable ── a tenfold improvement in the tau measurement settles it.
06★ It breaks when you run it ── a pole-mass relation
Episodes 5, 18 and 28 kept repeating that masses run. The Koide ratio is dimensionless (it passes Episode 7's test), so running it is meaningful.
| Scheme | \(m_e\) | \(m_\mu\) | \(m_\tau\) | \(K\) | Off from 2/3 |
|---|---|---|---|---|---|
| Pole masses | 0.51100 | 105.658 | 1776.86 | 0.666661 | −0.0009 % |
| ★ \(\overline{\rm MS}\) @ \(M_Z\) | 0.48657 | 102.718 | 1746.24 | 0.667929 | ★ +0.1894 % |
In amplitude terms \(A\) goes from \(1.414201\) to \(1.416889\) (\(\sqrt2=1.414214\)). ★★ The discrepancy worsens by a factor of 200.
The heart of §06 ── ★ an important limitation
The Koide relation holds only in the pole-mass scheme.
A fundamental relation would normally be expected to hold at some physical scale. Pole masses live at a different scale for each particle, which is theoretically awkward.
★ Two readings are possible (this article cannot decide between them):
① It is a coincidence (§07 measures the probability)
② Pole masses mean something ── as Episode 28 put it, the pole mass is "the mass a particle has when it actually flies". On that reading the Koide formula would be a relation among leptons each measured at its own scale. How to test that is currently unclear.
07★ How surprising is it really? ── a control experiment
\(K\) lies in \([1/3,1]\) with \(2/3\) at the midpoint. So how often does a random triple land near \(2/3\)? We drew \(2\times10^7\) triples with \(\sqrt m\) log-uniform over 0.7–42 (the actual leptonic spread).
| Closer than | Hits | Fraction | One in |
|---|---|---|---|
| \(10^{-2}\) | 315474 | 1.58e−2 | 63 |
| \(10^{-3}\) | 31675 | 1.58e−3 | 631 |
| \(10^{-4}\) | 3192 | 1.60e−4 | 6270 |
| \(10^{-5}\) | 306 | 1.53e−5 | ★ 65,000 |
The heart of §07 ── an honest assessment
★ "2/3 came out" is not that unusual (it is the midpoint of the range).
★★ What is unusual is the precision ── the measurement sits at \(9.2\times10^{-6}\), which random triples reach about once in 65,000.
★ But as §06 showed, it breaks when run, so it may be an approximation to something.
Applying the same formula to quarks ──
| Triple | \(K\) | \(A\) | Off from 2/3 |
|---|---|---|---|
| Charged leptons | 0.666661 | 1.414201 | −0.001 % |
| Up-type quarks (u,c,t) | 0.848981 | 1.758944 | +27.3 % |
| Down-type quarks (d,s,b) | 0.731428 | 1.545499 | +9.7 % |
★ It fails for quarks. It is a charged-lepton relation only ── and that too is unexplained.
08A footnote ── the value of the phase \(\delta\)
The amplitude \(A\) is pinned to \(\sqrt2\), but the phase \(\delta\) is pinned to nothing. Here is its measured value.
The heart of §08 ── ★ held in reserve
★ Stated honestly: \(\delta\) is one free parameter fitted to the data.
A free parameter landing within four digits of a simple fraction is roughly a \(10^{-4}\) event. Intriguing, but not evidence. A mechanism is required.
★ And as §06 showed, \(\delta\) is scheme-dependent too (it moves when run).
Exercises
- Why \(\sqrt m\) rather than \(m\)? Can the wave language say?
Show the answer
It cannot ── that is exactly what is unsolved. There are hints though: ★ Episode 32 gave \(m_f=y_fv/\sqrt2\), so \(\sqrt m\propto\sqrt y\). The Koide formula is therefore a relation among square roots of Yukawa couplings.
★ Another reading uses amplitude versus intensity: for waves, intensity ∝ amplitude². If \(m\) is an "intensity" then \(\sqrt m\) is an "amplitude", and ★ the Koide formula lines three amplitudes up in three phases.
★ But this does not rise above wordplay, because nothing explains why mass should be an intensity. Episode 32's "the values are not explained" applies here too. - §07 gave "one in 65,000". Is that evidence that Koide is real?
Show the answer
Weak evidence at best, because of the look-elsewhere effect. Countless dimensionless combinations can be built from three masses, and the chance that some one of them lands within \(10^{-5}\) of a simple fraction is far higher than for a single pre-specified formula.
★ It does count in Koide's favour that the formula was published in 1981, before the precise tau measurements ── it is not a post-hoc fit.
★★ What will actually settle it is §05's falsifiability ── a tenfold improvement in \(m_\tau\) turns 0.9 σ into either 9 σ or still 0.9 σ. Not probability arguments, but the next measurement. - §06 showed it breaks when run. Is the formula still worth chasing?
Show the answer
Yes, but what needs explaining changes. Breaking under running means that if it is real, the pole-mass scheme carries physical meaning ── a strong claim, and a valuable one if it can be tested.
★ Conversely, a relation that holds only in one particular scheme is usually an approximation or an accident. This article cannot decide which.
★ ★ All this episode added is a restatement ── but "why do the AC and DC parts have equal RMS?" gives a much clearer shape to the mechanism one should be looking for than "why 2/3?".
Summary The Koide formula means AC = DC
Following Episode 32's unexplained Yukawa hierarchy, we reread the one known relation among charged-lepton masses as a wave.
Taking the three \(\sqrt m\) as a sinusoid sampled at 120° intervals, \(\sqrt{m_k}=\mu(1+A\cos(\delta+2\pi k/3))\), the three-phase identity \(\sum\cos^2=3/2\) (the same one that makes three-phase power constant) yields \(K=(1+A^2/2)/3\) by hand.
★★★ So \(K=2/3\) simply means \(A=\sqrt2\). And since a sinusoid of amplitude \(A\) has RMS \(A/\sqrt2\) ──
★ Koide ＝ "the AC and DC parts of the \(\sqrt{\text{mass}}\) wave have equal RMS"
measured ratio 0.99999077 (\(-8\times10^{-5}\) dB)
Geometrically, \((\sqrt m)\) makes 44.999735° with the diagonal (45° to five digits). But this 45° is not Episode 1's "half-order phase of 45°" ── a direction in real three-space and a phase in the complex plane live in different spaces (the habit of Episodes 29 and 30).
It also predicts ── from \(m_e,m_\mu\), \(m_\tau=1776.969\) MeV against the measured 1776.86 ± 0.12, 0.91 σ. A tenfold better tau measurement settles it.
★ The limitations are equally clear. Run to \(M_Z\) and \(K\) sits 0.19 % off \(2/3\) ── a 200-fold worse discrepancy ── so this is a pole-mass relation. It fails for quarks by 10–27 %. In the control experiment, random triples reach \(10^{-5}\) about once in 65,000: impressive but not decisive. The \(\delta\approx2/9\) coincidence is one free parameter matching to four digits, and stays in reserve.
All this episode added is a restatement, not an explanation. Why \(A=\sqrt2\) remains unknown. But "AC = DC" makes the thing to be explained far clearer than "2/3" does.
★ The one difference from Maxwell is this ── the wave is its own source. A wave that scatters off itself.
★★ Classically that already makes it chaotic (we measure the Lyapunov exponent), and the gluon propagator violates positivity ── ★ "a wave that cannot be a free particle" is what confinement looks like in the language of waves.
This document is Episode 33 of the "Thinking in Waves" series. The computations are in kenshou/koide.js. On the numbers here: the value of \(K\), the angle, \(\mu,A,\delta\) from the 3-point DFT, the three-phase identity check, the \(m_\tau\) prediction and its sensitivity, the comparison under running, and the random control experiment were all computed for this article. The lepton pole masses are PDG values; the \(\overline{\rm MS}\) masses at \(M_Z\) and the quark masses are representative literature values (which move by a few percent between sources, so read §06 and §07 with that latitude). The Koide relation itself, the parametrization \(\sqrt{m_k}=\mu(1+\sqrt2\cos(\delta+2\pi k/3))\), the 45°-to-\((1,1,1)\) geometric reading and the \(\delta\approx2/9\) coincidence are all already known and are not discoveries of this article ── what this article did was translate them into the wave language of "the AC and DC parts of a 3-point DFT", recast the content as \(A=\sqrt2\Leftrightarrow\) equal RMS, and actually measure both the degradation under running and the random control. The probability in §07 assumes \(\sqrt m\) is log-uniform; other assumptions give other numbers, and no look-elsewhere correction has been applied. ── To print, use your browser's Print → Save as PDF.