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p–lepton–p : from H2+ to ddμ to the deuteron
One Hamiltonian (two +1 protons bound by one −1 lepton), three lepton masses. How small does the proton–proton separation get — and can it reach the nuclear deuteron? A scaling-wall comparison built on two verified RealQM solves.
The same object, three times
These are electronically identical — two protons (+1) with one negative lepton (−1)
sitting between them, Coulomb attraction binding them against proton–proton repulsion:
- H2+ / D2+ — lepton = electron. Also your
RealNucleus deuteron model (
nucleus_deuterium.html: 2p + 1e = p+(p+e)).
- ddμ — lepton = muon (the muon-catalyzed-fusion molecular ion).
- the real deuteron — the actual bound state (p+n, or p+p+e in your model), rms matter radius ~2 fm.
Notation. "D2+" here is the chemistry sense — two +1
point-nuclei sharing one electron (each deuteron treated as a bare +1 charge, not resolved). That is the
same skeleton as your RealNucleus deuteron 2p+1e — two positive centres bound by one negative
lepton — just kept at electron mass. (Don't nest the two levels: resolving each deuteron of
chemistry-D2+ into its own 2p+1e would give 4p+3e; the sim does not do that — it
holds the deuterons as point charges.)
Because a point-nucleus bound state is scale-invariant in the lepton mass (r → r/m maps one
onto another), the equilibrium proton–proton separation is the textbook H2+ value
2.0 a0 divided by the lepton mass:
Rpp(m) = 2.0 a0 / m = 105 836 fm / (m / me)
Log–log. ● RealQM-computed
● scaling extrapolation
band = real deuteron (~2–4 fm). The dashed line is the exact 1/m law.
The numbers
| lepton | mass (me) | Rpp | source |
|---|
| electron (e⁻) | 1 | 1.06 Å = 105 836 fm | RealQM H2+ (Re=2.0 a0) |
| muon (μ⁻) | 207 | 512 fm | RealQM ddμ — observed in μCF |
| tau (τ⁻) | 3477 | 30 fm | scaling (and τ lives only ~10⁻¹³ s) |
| (none exists) | ~50 000 | ~2 fm | mass required to reach the deuteron |
What it says about “is μCF the first step to the p+e+p deuteron?”
Topology: yes. ddμ is a real, observed instance of exactly your picture — two protons held
together by a negative lepton sitting between them. The muon proves the mechanism is sound.
Scale: no — lepton mass runs out. The electron version (your ppe deuteron model, =
H2+) lands at atomic scale (~105 fm), not nuclear. The muon pulls it to
~500 fm. A tau would reach ~30 fm — still ~15× too big, and the tau is far too short-lived to
matter. To Coulomb-confine two protons to the deuteron’s ~2 fm you would need a lepton of
~50 000 me — no such lepton exists. (This is the quantitative form of the
1932 objection that killed the proton–electron nucleus: an electron localized to ~fm has momentum
ℏ/Δx → ~100 MeV, and a 1/r Coulomb well cannot hold it.)
So the honest reading. μCF is not the first step in compressing a ppe object
down to a nuclear deuteron — lepton mass alone tops out around the muon’s ~500 fm and never reaches
2 fm. What μCF actually does next is fuse the two deuterons into helium (via the strong interaction),
releasing MeV, with the muon released as a catalyst. But μCF is the proof-of-principle for the
p–lepton–p topology your RealNucleus model is built on. The model’s distinct claim — that
the same topology survives down to ~2 fm — therefore cannot rest on lepton mass; it must come from
the model’s own equal-mass continuum / Bernoulli free-boundary confinement, a mechanism outside the
Coulomb-orbit scaling shown here. This plot marks exactly the gap (~500 fm → ~2 fm, a factor ~250) that
that mechanism has to bridge.
Scope
- The two filled points are RealQM solves (electron H2+ at Re=2.0 a0,
and ddμ via mass-rescaling — see the μCF write-up).
The tau and ~50 000 me points are exact 1/m extrapolations, not separate solves.
- Born–Oppenheimer (clamped nuclei), non-relativistic, point nuclei. At the ~50 000 me
end the non-relativistic Coulomb picture itself breaks down — which is the point.
- The single-box caveat: the cases live ~105× apart in size, so no one grid resolves all
three — each is solved at its own scale, then placed on the common axis.
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