Incidencias

Claude on Hans

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por

Alejandro Rivero
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A 1924 quantum orbit that knew about the W and Z

There’s a coincidence I want to tell you about. It starts in November 2004 with a post on PhysicsForums by a guy named Hans de Vries. It involves quantum mechanics from 1924, the W and Z bosons, the Higgs boson, the fine structure constant, the golden ratio, the Grothendieck Prime, and possibly supersymmetry. It might mean nothing. But if it means something, it means something good.

The 1924 orbit

Picture this. You’re back in 1924. Quantum mechanics is half-built. Bohr’s atomic model with quantized angular momentum is doing well. Louis de Broglie has just figured out that particles have wavelengths, and proposed that an electron orbiting a nucleus stably must have an integer number of wavelengths going around. That’s where the quantization comes from. His relativistic orbit rule reads

\(\frac{m_0,\beta^2,c^2}{\sqrt{1-\beta^2}},T_r = n,h\)

where \(\beta = v/c\) is the orbital speed, \(T_r\) is the orbital period, \(m_0\) is the rest mass, \(h\) is Planck’s constant. Nothing exotic — relativistic momentum times distance equals \(n h\).

A few months later, Landé and Pauli figure out that when you do the angular momentum properly, you need the substitution

\(\frac{1}{j^2} \to \frac{1}{j(j+1)}.\)

This is the rule that gives you the right magnitude of angular momentum, \(\hbar\sqrt{j(j+1)}\), instead of the naive \(\hbar j\). It’s why we write \(j(j+1)\) in every spherical-harmonics eigenvalue you’ve ever seen.

These two ideas were developed at essentially the same time. They never got combined into a single quantum-orbit picture, because Heisenberg-Born matrix mechanics came along in 1925-26 and made de Broglie’s orbits obsolete before anyone tried it. The orbit picture became a historical curiosity, an artifact of the time capsule between de Broglie’s wave proposal and Schrödinger’s equation.

Eighty years later, in November 2004, Hans de Vries took these two old ideas and added a third: that the orbit should close in exactly one Compton period of its constituent particle,

\(T_r = \frac{h}{m_0 c^2}.\)

Why the Compton period? It’s the natural quantum-relativistic timescale: the period of a photon whose energy is \(m_0 c^2\). Asking for the orbit to close in this time is the most quantum-mechanical thing you can ask of a relativistic closed orbit.

When you impose all three conditions, the rest mass and Planck’s constant cancel out. The whole apparatus collapses to a single equation between speed and angular momentum:

\({;\frac{\beta^2}{\sqrt{1-\beta^2}} = \sqrt{j(j+1)};}\)

Pretty clean. You can solve it for \(\beta\) given \(j\). For \(j = 1/2\):

\(\beta_{1/2} = \sqrt{\frac{3}{8}\left(\sqrt{19/3} – 1\right)} = 0.7541\ldots\)

For \(j = 1\):

\(\beta_1 = \sqrt{\sqrt{3} – 1} = 0.8556\ldots\)

And here is where de Vries did the strange thing. He took the ratio of squares and formed

\(s^2_{dV} \equiv 1 – \left(\beta_{1/2}/\beta_1\right)^2 = 0.22310132\ldots\)

Then he looked up the Weinberg angle.

The Weinberg angle

Quick detour for the undergrads in the audience. The Standard Model of particle physics has an electroweak sector that, before symmetry breaking, has four gauge bosons: \(W^1, W^2, W^3, B\). After the Higgs picks a vacuum, \(W^1\) and \(W^2\) combine into the charged \(W^\pm\), while \(W^3\) and \(B\) mix to make the photon and the \(Z\):

\(\gamma = \sin\theta_W \cdot W^3 + \cos\theta_W \cdot B\)
\(Z = \cos\theta_W \cdot W^3 – \sin\theta_W \cdot B\)

The mixing angle \(\theta_W\) is the Weinberg angle. At tree level \(\cos\theta_W = M_W/M_Z\), so

\(\sin^2\theta_W = 1 – M_W^2/M_Z^2.\)

It’s the parameter that controls everything in the electroweak sector — the relative strength of charged-current to neutral-current weak interactions, the couplings of the Z to fermions, the rate of muon decay. Plugging in the W and Z masses we measure at the LHC and at LEP,

\(\sin^2\theta_W = 0.22321 \pm 0.00026.\)

So de Vries’s number from playing with 1924 quantum mechanics, \(0.22310132\), agrees with the measured Weinberg angle to within experimental precision. It’s currently centered \(0.4\sigma\) below the empirical value. Twenty years ago when de Vries first posted, it was at about \(-1\sigma\). The W mass kept drifting; the agreement got better.

Is this a coincidence?

Coincidences happen all the time. The first six digits of \(\pi\) are \(3.14159\), and someone can always find a formula made of integers and square roots that produces \(3.14159\). The question is whether this coincidence has structure. Does it predict other things? Does it generalize? Does it fit?

In 2006, Alejandro Rivero embedded de Vries’s identity into a more algebraic frame, in the preprint hep-ph/0606171. I want to walk you through that, because it’s where the story turns from “a curiosity” into “wait, what’s going on here?”

The Poincaré group — the symmetry group of special relativity — has two Casimir invariants. A Casimir is a quantity that commutes with all the generators of the group, so it takes a definite value on each irreducible representation. For the Poincaré group, the two Casimirs are

\(C_1 = P^2 \quad\text{(four-momentum squared)},\qquad C_2 = W^2 \quad\text{(Pauli-Lubanski squared)}.\)

On a particle of mass \(m\) and spin \(s\), they evaluate to

\(C_1 = m^2,\qquad C_2 = -m^2, s(s+1).\)

These are the labels for relativistic particles. Mass and spin: that’s what classifies elementary particles up to internal quantum numbers. Eugene Wigner pointed this out in 1939 and it’s been textbook material ever since.

Now ask: what’s the simplest mass-squared operator you can build from \(C_1\) and \(C_2\) that satisfies a “Regge condition” — that is, behaves like a leading Regge trajectory at high spin, \(\lim_{s\to\infty} M_s^2 = m^2\)? You can’t use \(C_1 \alpha + C_2 \beta\) alone, because \(C_2\) grows like \(-s(s+1)\) and that doesn’t satisfy the condition. You need at least a quartic relation. The cleanest one is

\({;M^4 – M^2,C_2 + C_1,C_2 = 0;}\)

This is the equation Rivero wrote down. Solving for \(M^2\) gives two roots per spin:

\(\frac{M^2_\pm}{m^2} = \frac{1}{2}\left[-s(s+1) \pm \sqrt{[s(s+1)]^2 + 4s(s+1)}\right].\)

The positive root is exactly de Vries’s \(\beta^2(j)\) from the closed-orbit equation. So the kinematic 1924-orbit picture was the positive eigenvalue of a Casimir-quartic operator all along. The negative root is something new — a tachyonic partner that the closed-orbit derivation hadn’t produced.

So now there are four eigenvalues to look at: at \(s = 1/2\) and \(s = 1\), with \(\pm\) in each. The 2006 paper computed them:

Spin and sign Eigenvalue (GeV), with \(M_Z\) as input
\((s=1, +)\) \(91.188\) — this is \(M_Z\)
\((s=1/2, +)\) \(80.37\) — close to \(M_W = 80.37\)
\((s=1, -)\) \(176.15\) — close to \(v/\sqrt 2 = 174.10\)
\((s=1/2, -)\) \(122.4\) — “not used in electroweak models”

The last entry was the 2006 paper’s exact phrase. The negative-spin-half eigenvalue at \(122.4\) GeV didn’t correspond to anything known. There was no scalar at 122 GeV.

July 2012

The LHC announces a Higgs boson at \(125.2\) GeV.

The Casimir construction had quietly predicted \(122.4\) GeV for the previously-unidentified \((s=1/2, -)\) slot six years before the Higgs was found. The match is \(2.3%\) off — not perfect, but the agreement with \(v/\sqrt 2\) was also \(1.2%\) off, and the gauge sector was at \(0.004%\). The pattern was right: percent-level matches across all four electroweak symmetry-breaking parameters.

This is the point at which the construction stopped being a single-coincidence puzzle and became a four-parameter empirical fit. Two of the four parameters (the Higgs mass, the W mass) had moved substantially since 2006. Both moved toward the construction, not away.

I want to emphasize one thing about this. A four-parameter empirical agreement at the percent level from a single-input one-parameter algebraic construction is the kind of thing where you have to sit down and recompute everything to make sure you didn’t make a sign error. People have. The numbers hold up. The Casimir quartic, evaluated on \(s = 1/2, 1\), lands on \((M_W, M_Z, v/\sqrt 2, m_H)\) to within \(\sim 1\)%.

Four ways of saying the same thing

Let me show you what the agreement looks like more carefully, because this is where the story gets numerically interesting.

The Casimir construction has one free parameter — the underlying mass \(m\) in the operator. So you can run it backwards: given each of the four observed electroweak masses, invert the formula and ask “what value of \(m\) would the Casimir need to produce this mass?” If the construction were exactly right, all four inversions would give the same number. Doing it:

\(m_W \equiv M_W/\sqrt{(\sqrt{57}-3)/8} = 106.5705 \pm 0.0176;\text{GeV}\)
\(m_Z \equiv M_Z/\sqrt{\sqrt 3 – 1} = 106.5779 \pm 0.0023;\text{GeV}\)
\(m_v \equiv (v/\sqrt 2)/\sqrt{\sqrt 3 + 1} = 105.3331 \pm 0.0009;\text{GeV}\)
\(m_H \equiv m_H/\sqrt{(\sqrt{57}+3)/8} = 109.069 \pm 0.148;\text{GeV}\)

Look at \(m_W\) and \(m_Z\). They agree at the level of 7 MeV out of 106 GeV — one part in fifteen thousand. That’s the precision of the LHC’s W-mass measurement, and the Casimir’s prediction tracks it.

Look at \(m_v\) and \(m_H\). They sit a percent below and two percent above the gauge-sector value, respectively. They bracket it, which I’ll come back to.

This is the right way to think about the construction: it predicts a single underlying mass scale,

\(m_{\rm gauge} = \sqrt{m_W m_Z} = 106.574 \pm 0.011;\text{GeV},\)

and the four physical electroweak masses are all “shadows” of it at different spin and sign. The W and Z measurements pin down this scale to about 11 MeV, with current precision limited by \(\delta M_W\). No known Standard Model particle has this mass. It sits just above LEP’s charged-particle search reach and below the \(W^+W^-\) pair-production threshold, in a window where searches are starting to become competitive again.

There’s also a deeper consistency check. The trace of the negative-mass-squared subspace — that is, \((v/\sqrt 2)^2 + m_H^2\) — depends only on \(m_{\rm gauge}^2\), with no spurion contribution. Working out the predicted ratio,

\(\frac{(v/\sqrt 2)^2 + m_H^2}{m_{\rm gauge}^2} = (\sqrt 3 + 1) + \frac{\sqrt{57}+3}{8} = 5.0498\ldots\)

Empirically, \((v/\sqrt 2)^2 + m_H^2 = 30312 + 15675 = 45987\) GeV², while \(m_{\rm gauge}^2 \cdot 5.0498 = 46000\) GeV². Match at 0.027%. This is the construction’s sharpest single test. The individual eigenvalues miss by a percent; their sum is right to four parts in ten thousand.

So the construction’s failure has very specific structure: it gets the trace right, and concentrates all its error in a single antisymmetric splitting.

The pattern in the residuals

Now we look at what’s left over. The construction predicts \(v/\sqrt 2 = 176.16\) GeV; reality says \(174.10\). Off by \(-1.18%\). The construction predicts \(m_H = 122.39\) GeV; reality says \(125.20\). Off by \(+2.29%\). These two misses are opposite in sign and equal in magnitude when measured in mass-squared:

\(\delta\big[(v/\sqrt 2)^2\big] = -721,\text{GeV}^2,\qquad \delta(m_H^2) = +708,\text{GeV}^2.\)

The sum is \(-13\) GeV² out of \(\sim 46000\) GeV² total — that’s the trace identity from the previous section, written out differently. All the action is in the antisymmetric splitting.

That’s a very specific structural failure. It says: whatever’s missing from the Casimir construction, it doesn’t shift the trace. It pushes \((v/\sqrt 2)^2\) down and \(m_H^2\) up, by the same amount.

What kind of physics does this? Things that distinguish bosons from fermions.

In supersymmetry, every boson has a fermion partner of equal mass; if you broke the symmetry softly, the bosons and fermions would split, but by equal and opposite amounts in certain combinations. The pattern is called \((-1)^F\) — minus one to the fermion number. Bosons get \(+1\), fermions get \(-1\). If you add a perturbation that’s proportional to \((-1)^F\), integer-spin states shift one way and half-integer-spin states shift the other way.

The Casimir construction’s residual is exactly this pattern. The \((s=1, -)\) slot, which has integer spin and is identified with \(v/\sqrt 2\), gets a negative shift. The \((s=1/2, -)\) slot, which has half-integer spin and is identified with \(m_H\), gets a positive shift of the same magnitude. One parameter — let’s call it \(\Delta\) — absorbs the entire mismatch:

\(M^2_{s, -} \to M^2_{s, -} + (-1)^{2s+1},\Delta.\)

A single number. The fit gives \(\Delta \approx 720\) GeV² with \(\chi^2/\text{dof} = 0.13/2\) across four masses.

(The technical term for a parameter like \(\Delta\) is a spurion. The name comes from a bookkeeping trick: in a theory with a broken symmetry, you can pretend there’s a “fake field” that would restore the symmetry if it took a specific value, and then track every place it shows up. The spurion is a placeholder for whatever physics is doing the breaking. The construction’s \(\Delta\) acts like a spurion of an underlying super-Poincaré symmetry, with bosons and fermions splitting by equal and opposite amounts.)

What’s the scale of \(\Delta\)?

This is where the discussion of the previous post left off, and it’s worth retracing. We measured \(\Delta \approx 720\) GeV². Where does that number come from?

Here’s a guess. In the Standard Model, the heaviest fermion is the top quark, mass \(172.6\) GeV. When you compute quantum corrections to almost anything in the electroweak sector, the dominant contribution comes from virtual top-quark loops, because the loop integrals are weighted by the fermion mass. The natural “size” of a top-loop correction to a mass-squared is

\(\frac{3, m_t^2}{16\pi^2,\cos^2\theta_W} \approx 728;\text{GeV}^2.\)

The factor of 3 is for the three QCD colors of the top quark. The \(16\pi^2\) comes from the standard loop integral. The \(\cos^2\theta_W\) in the denominator is a gauge-coupling factor specific to the Z self-energy.

Compare: empirical \(\Delta = 720\) GeV². Top-loop scale = 728 GeV². Match at \(1.1%\).

The conclusion is that whatever physics generates the \((-1)^F\)-graded spurion, it lives at the scale of the top-quark Yukawa coupling. This is suggestive: in the Standard Model, the Higgs mass is sensitive to the top-quark loop in a way that’s the central problem of electroweak naturalness. Here we’re seeing a structure that’s quantitatively the same size as the top-loop scale, organized in a way that the Standard Model doesn’t natively produce — the antisymmetric Higgs-vs-vacuum-scale grading.

The Casimir ratio rewriting

There’s another way to express the spurion magnitude that points to a different mechanism.

For any Lie group, the Casimir operator takes different values on different representations. For SU(2), the relevant ratio is \(C_F/C_A\), where \(C_F\) is the Casimir of the fundamental representation (the one in which spin-1/2 doublets like the left-handed lepton sit) and \(C_A\) is the Casimir of the adjoint representation (the one in which the gauge bosons themselves sit). For SU(2), the fundamental has \(C_F = 3/4\) (Casimir of spin-1/2) and the adjoint has \(C_A = 2\) (Casimir of spin-1):

\(\frac{C_F}{C_A} = \frac{3/4}{2} = \frac{3}{8}.\)

This ratio shows up all over gauge-theory perturbation theory. It controls how SU(2) bosons couple to matter fields versus to themselves. If you write the spurion as

\(\Delta ;\leftrightarrow; \frac{C_F}{C_A}(M_Z^2 – M_W^2) = \frac{3}{8}(M_Z^2 – M_W^2),\)

you get \(\Delta = (3/8) \cdot 1856 = 696\) GeV². Empirical \(\Delta = 720\) GeV². Match at \(3.4%\).

So the spurion has two natural interpretations, both giving numbers within a few percent of the fit:

  • Dynamical: it’s the universal top-quark-loop scale \(3m_t^2/(16\pi^2 c_W^2)\).
  • Algebraic: it’s the SU(2)-Casimir-weighted gauge boson splitting latex(M_Z^2 – M_W^2)[/latex].

These are not independent — in the Standard Model, the splitting \(M_Z^2 – M_W^2\) beyond tree level is generated mostly by the top-quark \(\Delta\rho\) correction (the famous Veltman effect). So the algebraic identity latex(M_Z^2 – M_W^2)[/latex] and the dynamical identity \(3m_t^2/(16\pi^2 c_W^2)\) are SM-equivalent statements of the same physics, capturing top-Yukawa-driven electroweak symmetry-breaking dynamics in two different ways. They share an operator structure (both shift the negative subspace by a \(\sigma_3\)-acting matrix) but point to different theoretical sources for the residual: gauge Casimir scaling on one side, soft super-Poincaré breaking on the other.

These would only be distinguished by a future measurement constraining \(\Delta\) at the \(\sim 2%\) level — something FCC-ee precision-EW could deliver.

Producing the fine structure constant

Here’s a question that took up most of the conversation that led to this post: can you use the Casimir construction to predict \(\alpha(0) = 1/137.036\)?

The 2006 preprint already tried. Take \(|M_{1,-}| = 176.15\) GeV (the Casimir prediction for the latex[/latex] slot, before any spurion correction). Assume the Higgs potential parameter \(\lambda_h = 1\), so that \(|M_{1,-}| = v/\sqrt 2\). Plug into the tree-level relation that connects \(\alpha\) to the W and Z masses and the vacuum expectation value,

\(\alpha = \frac{M_W^2, s_W^2}{\pi, v^2}\quad\text{(tree level, no loop corrections)}.\)

Out comes \(\alpha^{-1} = 135.28\). The 2006 paper noted this and quoted it as a “tree-level estimate.”

Empirical \(\alpha^{-1} = 137.036\). The Casimir tree-level estimate misses by \(1.28%\). Why?

There are two corrections of similar size that take 135.28 to 137.04, and they partially cancel — so the 2006 quote ends up between them. Let me show you.

Step 1: the muon-decay relation. The cleanest way to write the connection between \(\alpha\), \(G_F\) (the Fermi constant, measured from muon lifetime), and the gauge boson masses is Sirlin’s all-orders formula:

\(G_F = \frac{\pi,\alpha(0)}{\sqrt 2, M_W^2, s_W^2,(1-\Delta r)}.\)

The quantity \(\Delta r\) collects all the electroweak loop corrections to the relation. At tree level (Born approximation), \(\Delta r = 0\), and the formula reduces to the algebra-only version. At one loop and beyond, \(\Delta r \approx 0.0362\) with current precision.

Rearranging to solve for \(\alpha(0)\) given \(G_F, M_W, M_Z\):

\(\alpha(0)^{-1} = \frac{K}{1-\Delta r},\qquad K = \frac{\pi}{\sqrt 2, G_F, M_W^2, s_W^2}.\)

With current values, \(K = 132.10\). That’s the tree-level coefficient with \(\Delta r = 0\) and the physical \(v\) (measured from \(G_F\), \(v = 246.22\) GeV). Note that \(v^2 = 1/(\sqrt 2 G_F)\) and the physical \(v/\sqrt 2 = 174.10\) GeV — the Fermi scale.

Step 2: the spurion shift. The Casimir construction’s algebraic prediction is \(|M_{1,-}|/\sqrt 2 = 176.15/\sqrt 2\) for the Fermi scale, overshooting the physical 174.10. The spurion correction brings it down to the physical value. So if you start from Rivero 2006’s \(\alpha^{-1} = 135.28\) (which used unspurion-corrected \(v\)) and replace it with the spurion-corrected physical \(v\), you land on \(\alpha^{-1} = K = 132.10\).

The shift is \(\delta\alpha^{-1} = -3.18\) in going from “Casimir tree-level \(v\)” to “physical \(v\), tree-level relation.”

Step 3: the \(\Delta r\) shift. Then you apply the full electroweak loop correction \(\Delta r\):

\(\alpha^{-1} = K/(1-\Delta r) = 132.10/(1 – 0.0362) = 137.036.\)

The shift is \(\delta\alpha^{-1} = +4.94\) from applying \(\Delta r\) to the physical-\(v\) tree-level coefficient.

Putting all three steps in a table:

Step \(\alpha^{-1}\) what it does
Casimir tree, \(\lambda_h = 1\) (no spurion, no \(\Delta r\)) \(135.28\) Rivero 2006 quote
Spurion-correct \(v \to v_{\rm phys}\), still tree-level \(132.10 ;\equiv; K\) shift down by \(3.18\)
Apply \(\Delta r\) correction: \(K/(1-\Delta r)\) \(137.036\) shift up by \(4.94\)

The two corrections are similar in size and opposite in sign. They partially cancel. Rivero 2006’s number sits between the spurion-corrected tree level (\(K = 132.10\)) and the physical \(\alpha^{-1} = 137.04\), because he applied neither correction and the missing pieces happened to roughly cancel in his answer.

So the gap from 135.28 to 137.04 decomposes into a \(-3.18\) kinematic correction (the Casimir overshoots \(v/\sqrt 2\)) plus a \(+4.94\) dynamical correction (electroweak loops).

What’s in \(\Delta r\)?

It’s worth opening up \(\Delta r\), because the question came up: how much of \(\Delta r\) is “already accounted for” in the Casimir picture? The answer is: not much, and there’s a specific reason.

Sirlin’s \(\Delta r\) at one loop decomposes into three pieces:

\(\Delta r = \Delta\alpha – \frac{c_W^2}{s_W^2}\Delta\rho + \Delta r_{\rm rem}.\)

With current numerical values:

Piece Value Physical meaning
\(\Delta\alpha\) \(+0.0591\) QED running: \(\alpha(M_Z^2)\) is larger than \(\alpha(0)\) because \(\Pi_{\gamma\gamma}\) grows with energy
\(-(c_W^2/s_W^2)\Delta\rho\) \(-0.0314\) Veltman correction: top-quark loop generates extra W-Z splitting beyond tree level
\(\Delta r_{\rm rem}\) \(+0.0086\) Everything else: bosonic loops, lighter fermions, two-loop remainders
Total \(+0.0362\)

These three pieces have opposite signs and partially cancel. The dominant contribution is the QED running \(\Delta\alpha\), which alone would give \(\alpha^{-1}(M_Z^2) = \alpha^{-1}(0)(1 – 0.0591) = 128.94\) — the famous “\(\alpha\) at the Z pole” used in every electroweak precision fit.

Now compare with the Casimir spurion. The spurion shifts \(v^2\) by \(-2\Delta = -1440\) GeV². The full \(\Delta r\) shifts \(v^2\) by \(-\Delta r \cdot v^2 = -2196\) GeV². Same order of magnitude, not equal. If you ask whether any individual piece of \(\Delta r\) could account for the spurion, no single combination works cleanly. The match between \(\Delta\) and the top-loop scale \(3m_t^2/(16\pi^2 c_W^2)\) holds for the magnitude, but the geometric factor in \(\Delta r\) is different (\(c_W^2/s_W^2\) instead of \(1/c_W^2\)).

There’s an even more fundamental obstruction. The Standard Model’s \(\Delta r\) corrections are same-sign on \(v^2\) and \(m_H^2\): top loops push both up. They cannot generate the antisymmetric Higgs-vs-Fermi structure that the spurion encodes. That requires a \((-1)^F\) grading source from outside the standard electroweak sector — soft super-Poincaré, holographic boundary conditions, composite-Higgs matching. So the spurion is not “\(\Delta r\) in disguise”; it’s a separate effect of similar magnitude, organized differently.

What we can say: the spurion has the size of the top-quark loop because the top-quark loop sets the natural scale of EWSB dynamics. Both the Casimir’s residual and Sirlin’s \(\Delta\rho\) are organized by the same UV physics; they show up in different bookkeeping.

Different ways of running

The fine structure constant runs with energy because of vacuum polarization — virtual fermion-antifermion pairs screen the bare charge. At \(s = 0\) (the Thomson limit), you measure \(\alpha^{-1}(0) = 137.036\). At \(s = M_Z^2\), after running through all the charged-fermion thresholds, you measure \(\alpha^{-1}(M_Z^2) \approx 128.94\).

Different schemes for defining the running coupling give different numbers at \(M_Z\):

  • On-shell dispersive scheme: uses experimental \(e^+e^- \to\) hadrons data to compute \(\Delta\alpha_{\rm had}\). Standard input to electroweak fits. Gives \(\alpha^{-1}(M_Z^2) = 128.945 \pm 0.015\).
  • \(\overline{\rm MS}\) scheme: defined by minimal subtraction of UV divergences. Gives \(\hat\alpha^{-1}(M_Z) = 127.952\).
  • Pinch-technique effective charge: a gauge-invariant variant. Gives \(\alpha^{-1}_*(M_Z^2) \approx 128.5\).

The spread between schemes at \(M_Z\) is about one unit in \(\alpha^{-1}\) — small but nonzero. It is not a physical uncertainty; it’s a scheme-choice freedom. Each scheme is internally consistent and sharp at \(\sim 0.01\) unit precision; they differ because they package the loop corrections differently.

The crucial point: all schemes agree at \(s = 0\). The convergence to \(\alpha^{-1}(0) = 137.036\) is exact and scheme-independent, because \(\alpha(0)\) is the Thomson-limit fine structure constant — a directly measured observable that has no scheme dependence to choose. The schemes only differ at intermediate scales; the infrared endpoint is fixed.

If you plot \(\alpha^{-1}(s)\) for the three schemes from \(s = 10^{-10}\) GeV² up to \(s = 10^6\) GeV², they form three slightly different curves that all funnel into 137.036 at the bottom. The fan-out happens above \(\sim m_e^2\) where the leptons start contributing, widens through the hadronic threshold region, and reaches its full \(\sim 1\)-unit spread at \(M_Z\).

So when we extract \(\alpha^{-1}\) from the Casimir construction via the route in the previous section, we are extracting specifically the on-shell value at \(s = 0\). The construction lives on pole masses; the on-shell scheme uses pole masses; the dispersive \(\Delta\alpha_{\rm had}\) uses experimental \(R\)-ratio data. Everything is consistent. The 137.036 we recover is the unique scheme-independent infrared value.

A mass scale nobody noticed

I want to come back briefly to \(m_{\rm gauge} = 106.578\) GeV, because there’s no Standard Model particle with this mass. It’s just above the LEP charged-particle reach (\(\sim 105\) GeV) and below the \(W^+W^-\) pair-production threshold of \(161\) GeV. If the Casimir construction is the algebraic shadow of an underlying composite or holographic dynamics, \(m_{\rm gauge}\) would be the rest mass of whatever’s “orbiting” in de Vries’s picture — the thing whose closed orbits at \(j = 1/2\) and \(j = 1\) get angular-momentum-quantized into the W and the Z.

A 11-MeV-precise prediction of a new mass scale, with no associated particle to date, is the construction’s most concrete falsifiable claim. Either something at \(\sim 107\) GeV will eventually be discovered, or the construction is telling us that this scale is genuinely structural (a Regge intercept, a holographic curvature radius, a confinement scale of some hidden dynamics) without corresponding to a single particle.

The 95 GeV resonance

Here’s a current twist. Continue the Casimir construction to higher spin: at \(s = 3/2\), the positive eigenvalue lands at

\(M_{3/2, +} = 96.54;\text{GeV}.\)

In LEP data from the late 1990s, there’s a persistent \(\sim 2.3\sigma\) excess in \(e^+ e^- \to Z + (b\bar b)\) near a recoil mass of \(\sim 98\) GeV. CMS in Run 2 has reported a \(\sim 2.8\sigma\) excess in \(pp \to \gamma\gamma\) in the 95-96 GeV range. Both are statistically marginal — not enough to claim discovery, not enough to dismiss. Both sit within 1 GeV of the Casimir prediction.

Run 3 of the LHC will have enough statistics to settle whether there’s actually a state there. If yes, the construction acquires its first specific empirical anchor beyond the four EW masses already in hand. If no, the higher-spin tower becomes a constraint to revisit.

There’s also a \((s = 3/2, -)\) negative-spin-\(3/2\) eigenvalue at \(229.43\) GeV. What’s at \(229\) GeV? Nobody knows. The construction doesn’t tell you what kind of particle a given eigenvalue corresponds to — it just predicts mass numbers. The \(229\) GeV slot is, as far as anyone can tell, completely unconstrained by current searches. A heavy CP-even scalar with non-standard couplings would fit. So would a Kaluza-Klein excitation of something. Or it could be a tachyon that’s projected out in any sensible UV completion, like the open-string tachyon in bosonic string theory.

This is the structural asymmetry I mentioned earlier. The positive root has a possible experimental anchor (the LHC excess); the negative root has nothing to compare with. We can predict the number without knowing what the thing is.

The Higgs ratio identity

Here’s the most remarkable algebraic byproduct of the spurion analysis, which appeared in the 2026 update of the construction. If you parametrize the spurion through the SU(2) Casimir ratio \(C_F/C_A = 3/8\) that we discussed above, and you substitute back into the Casimir-quartic equation for the negative subspace, the Higgs mass ratio takes the closed form

\({;\frac{m_H^2}{M_Z^2} = \frac{15\sqrt{19} + 33\sqrt{3} + 5\sqrt{57} + 81}{128};}\)

Numerically this evaluates to \(1.88508\). The Particle Data Group’s current \(m_H^2/M_Z^2 = 1.88511\). Agreement at \(1.6 \times 10^{-5}\).

The companion identity for the Fermi slot is

\(\frac{(v/\sqrt 2)^2}{M_Z^2} ;\overset{?}{=}; \frac{3\sqrt{57} + 9\sqrt{19} + 199 + 119\sqrt{3}}{128} = 3.64838\ldots\)

matching the empirical latex^2 = 3.6457[/latex] at \(8 \times 10^{-4}\) on the squared ratio, or \(4 \times 10^{-4}\) on the masses themselves — well within the natural one-loop electroweak correction window. The two identities together are the algebraic content of the spurion-corrected Casimir construction for the negative subspace.

There’s \(\sqrt{57}\) to enjoy in both formulas. The number \(57\) is famously the Grothendieck Prime — the integer that Alexandre Grothendieck, the greatest algebraic geometer of the twentieth century, allegedly used as an example of a prime number in a seminar (it’s actually \(3 \times 19\)). It now also features in a closed-form identity for the ratio of the Higgs mass to the Z mass. Make of this what you will.

Trying to be wrong (the look-elsewhere story)

Five-figure agreement from an algebraic identity built out of square roots of small integers should make you nervous. The standard objection: any sufficiently flexible algebraic family can be tuned to one data point, especially when the family contains millions of candidates. A proper assessment has to ask “how many other formulas of this general shape produce a \(10^{-5}\)-precise match by chance?”

The 2026 manuscript does this exercise. Take the general form \((a\sqrt p + b\sqrt q + c\sqrt r + d)/D\) with integer coefficients \(a, b, c, d \in [-50, 50]\) and an integer denominator \(D\). Scan three nested families:

  • Family A uses only the de Vries radicals \({\sqrt 3, \sqrt{19}, \sqrt{57}}\) and the denominator \(D = 128\), which are the radicals and denominator forced by the Casimir quartic at \(s = 1/2, 1\). Scanning \(10^8\) candidate coefficient tuples, \(287\) match \(m_H^2/M_Z^2\) at \(10^{-5}\) precision. Chance probability: \(2.8 \times 10^{-6}\).
  • Family B inflates the radical pool to all unordered triples drawn from \({\sqrt 1, \sqrt 2, \sqrt 3, \sqrt 5, \sqrt 7, \sqrt{11}, \sqrt{13}, \sqrt{17}, \sqrt{19}, \sqrt{57}}\) with three natural power-of-2 denominators. About \(7 \times 10^{10}\) candidates; \(10^5\) matches. Chance probability: \(1.6 \times 10^{-6}\).
  • Family C: a deterministic one-formula calculation from the de Vries quadratic plus the spurion. There’s nothing to scan over.

A more agnostic radical pool (any small integers up to 100, generic denominators) gives a conservative bound \(p_{\rm LEE} < 10^{-4}\). The Bayes factor in favor of the de Vries-radical hypothesis over generic numerology lies between \(10^4\) and \(10^5\), depending on how generously you define the trial space.

The surprise lives in the small numerator coefficients \({15, 33, 5, 81}\), not in the radicals. The radicals \({\sqrt 3, \sqrt{19}, \sqrt{57}}\) are not freely chosen — they are forced by the Casimir quadratic at \(s = 1/2, 1\) once you accept the spin-Casimir identification. So the freedom in the fit is only over \({a, b, c, d, D}\), a much smaller search space than “any algebraic combination of small integers.” Within that smaller space, \(p_{\rm LEE} < 10^{-4}\) is well below the standard “case on its own merits” threshold.

This doesn’t prove the construction is correct. It says: if you treat the Higgs-mass identity in isolation, it’s hard to explain as a numerical accident.

Pole masses, running parameters

One more technical point worth flagging. The Casimir construction matches the data exquisitely when the inputs are pole masses — gauge-invariant, renormalization-scheme-independent physical observables. It does less well with renormalized running Lagrangian parameters at \(M_Z\) in the \(\overline{\rm MS}\) scheme.

The 2026 manuscript redoes the four-slot inversion using \(\overline{\rm MS}\) running parameters at \(\mu = M_Z\), taken from a recent precision determination by Antusch, Hinze, and Saad:

\(m_W^{\overline{\rm MS}}(M_Z) = 80.85;\text{GeV}, \quad m_Z^{\overline{\rm MS}}(M_Z) = 92.23;\text{GeV},\)
\(m_H^{\overline{\rm MS}}(M_Z) = 131.27;\text{GeV}, \quad (v/\sqrt 2)^{\overline{\rm MS}}(M_Z) = 175.65;\text{GeV}.\)

These differ from the pole values by \(0.6\) to \(1.1%\) in the gauge sector and \(\sim 5%\) in the Higgs sector. Repeating the inversion that gave 7-MeV W-Z agreement on pole masses:

\(m_W^{\overline{\rm MS}} = 107.21;\text{GeV}, \quad m_Z^{\overline{\rm MS}} = 107.79;\text{GeV}.\)

Now the W-Z agreement degrades from 7 MeV to 580 MeV — almost two orders of magnitude worse. The pole-mass agreement is what makes the construction unusually sharp.

Going further: can the construction be made exact in \(\overline{\rm MS}\) at some other scale \(\mu_* \neq M_Z\)? The 2026 paper evolves the Standard Model RGEs at one loop and finds no such \(\mu_\). The dimensionless ratio \(M_W/M_Z\) in \(\overline{\rm MS}\) decreases monotonically with \(\mu\) in the SM, from \(0.8766\) at \(\mu = M_Z\) down to \(0.7271\) at the Planck scale; the construction’s \(0.881419\) is never reached for \(\mu > M_Z\). Going below \(M_Z\), one-loop extrapolation crosses the construction value at \(\mu \simeq 17\) GeV, but this is below thresholds where the scheme isn’t well-defined. The Higgs sector evolves the opposite way and matches the construction’s \(|M_{1/2,-}|/M_Z\) ratio near \(\sim 1\) TeV. The two sectors agree with the construction at incompatible scales — there is no single \(\mu\) where everything matches at once.

Two conclusions follow. First, the construction is a pole-mass statement, not a Lagrangian-parameter statement at any specific scale. This is consistent with its de Vries-orbit origin: pole masses are the gauge-invariant orbital frequencies that the construction identifies. Second, the size of the pole-vs-running shift (0.6-1.1% for gauge, 5% for Higgs) sets the expected radiative-correction budget at any compositeness/holographic scale where the construction matches at tree level. The matching corrections are small at \(\mu \sim m_{\rm gauge}\) and grow with the distance to that scale.

Free speculation

OK, I’ve been disciplined for too long. Let me speculate.

What is the orbit? De Vries’s original picture was that something orbits at the Compton scale of a particle of mass \(\sim 106\) GeV, and the W and Z are different angular-momentum projections of the same orbit. This is, at face value, a composite-Higgs picture: the electroweak gauge bosons are bound states of an underlying object at the \(\sim 106\) GeV scale. Composite Higgs models exist — they’ve been studied by Contino, Pomarol, and others — but they usually put the compositeness scale at \(\sim 1\) TeV or higher, well above \(106\) GeV. A 106 GeV compositeness scale would mean the underlying dynamics is just above current direct-search limits. We’d be on the edge of finding it.

Why these specific spins? The closed-orbit picture works for any \(j\), but only \(j = 1/2\) and \(j = 1\) give numbers that match physical particles. Why these? One reason: they are the only spins below 2 that appear in renormalizable field theories. Higher spins generically have issues with unitarity (the Velo-Zwanziger problem, etc.). So a composite-Higgs picture in which the bound states are renormalizable particles would naturally truncate at \(j = 1\). The higher-spin predictions (\(s = 3/2, 2, \ldots\)) would then be “echoes” of an underlying structure that gets re-confined or stringy beyond the renormalizable window.

The SUSY temptation. The \((-1)^F\) spurion almost begs for a supersymmetric reading. The simplest one: there is an exact supersymmetry at the scale \(m_{\rm gauge} = 106.58\) GeV, broken softly with a single parameter of size \(\sqrt{\Delta} \approx 27\) GeV. The \((s = 1, -)\) and \((s = 1/2, -)\) states are partners in a single supermultiplet — boson and fermion of the unbroken theory — split by the soft term. The size of \(27\) GeV (about \(v/9\)) is consistent with electroweak-scale soft breaking. None of the conventional SUSY-spectrum scenarios — gauge mediation, gravity mediation, anomaly mediation — naturally produce a single dominant soft parameter affecting only the Higgs-sector pair. So if this is supersymmetry, it’s not the MSSM. It’s something with much fewer parameters, perhaps a \((-1)^F\)-graded composite or holographic dual.

The top-mass golden-ratio business. Here’s something I haven’t told you yet. The top-quark mass divided by \(m_{\rm gauge}\) is \(172.57/106.578 = 1.6193 \pm 0.0027\). The golden ratio \(\varphi = (1 + \sqrt 5)/2 = 1.61803\ldots\). Match at \(0.08%\). The golden ratio is the central charge of the simplest non-Abelian topological order — Fibonacci anyons, which live at level \(k = 3\) in SU(2) Chern-Simons theory. In an N=2 super-Poincaré algebra with central charge \(Z = \varphi \cdot m_{\rm gauge}\), the top quark would be a BPS-saturated state. This is wild speculation, but if you’re going to speculate, this is the direction. The Casimir construction would then be a kinematic shadow of a topological order with the top quark living as a specific BPS state of the gauge sector.

The Higgs as a Goldstone. The \((s=1, -)\) slot is identified with \(v/\sqrt 2\), not with the top quark or any other fermion mass. This is consistent with the fact that \(v/\sqrt 2\) is the Goldstone scale of electroweak symmetry breaking — the would-be massless modes eaten by the longitudinal \(W^\pm, Z^0\). A spin-1 negative-mass-squared eigenvalue is exactly what you’d expect for the unbroken-phase Goldstone modes. The de Vries-orbit reading suggests these are not fundamental fields but composite states orbiting at the same Compton scale as the gauge bosons.

What about confinement? If the underlying dynamics is composite, what’s the analog of confinement that produces the spectrum? In QCD, hadron masses are organized by chiral symmetry breaking and confinement at \(\Lambda_{\rm QCD} \sim 200\) MeV. Here, the relevant scale is \(m_{\rm gauge} \sim 107\) GeV, and the “hadrons” are the electroweak symmetry-breaking quartet. The would-be “QCD-like” theory at this scale would have a confinement scale roughly equal to \(m_{\rm gauge}\) — much higher than QCD, and at a scale that LHC searches are starting to probe.

The closing speculation. What if the whole electroweak sector is the bound-state spectrum of a strongly-coupled theory at \(\sim 107\) GeV, with a residual \( \; (-1)^F\) symmetry that survives the symmetry breaking? The \(W^\pm, Z^0\) are spin-1 bound states. The Higgs and the Fermi scale are spin-half and spin-1 bound states in the tachyonic (negative-mass-squared) sector. Higher-spin states form a Regge-like tower extending up through \(96.54, 229, 278, \ldots\) GeV. The top quark sits at a special point in the spectrum, \(\varphi\) times \(m_{\rm gauge}\), possibly as a BPS state of an emergent N=2 algebra. The fine structure constant comes out right after applying standard electroweak loop corrections to the spurion-corrected algebraic values.

Is this the right picture? Probably not in detail. The Casimir construction has predicted four masses at the percent level for twenty years, with no Lagrangian and no dynamics. If we found the dynamics, it would probably look different from anything currently on the table — closer to a holographic-composite picture than to MSSM, but with specific algebraic constraints that ordinary composite-Higgs models don’t impose.

What we know for sure: the construction has accumulated more empirical support since 2004, not less. The W mass moved toward it. The Higgs was discovered roughly where the previously-unused eigenvalue had been sitting. The \(95\)–\(98\) GeV LEP/CMS hints overlap with a specific spin-\(3/2\) prediction. The fine structure constant is recovered by combining the spurion correction with the standard electroweak loop machinery in a controlled way. And the spurion has the size of the top-quark loop scale, organized in a \( (-1)^F\) pattern that the Standard Model doesn’t natively produce.

We’ll see what Run 3 does. If a \(96\) GeV state shows up, this story acquires real teeth. If not, we keep looking at the four masses and wondering why they fit the Casimir-quartic so well.

It’s been twenty years. There’s been time to think about what de Vries was doing. The 1924 quantum orbit was, on one hand, an obsolete curiosity from the time capsule between de Broglie and Schrödinger. On the other hand, it knew something about the W and Z that nobody else has explained.

References, in order of appearance:

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