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In 1961, Ernest Sternglass published a paper where, using what seems to be to be a combination of relativistic kinematics and Bohr’s old quantisation procedure, he looked at the energy levels of a set of metastable electron-positron states, and found the lowest of these to be a mass surprisingly close to the measured mass of the neutral pion. He also calculated its lifetime, through what looks to me to be a form of dimensional analysis, to be close to that of the neutral pion also.

We now know, of course, that this is not the correct model of the neutral pion, but how did his analysis manage to produce these curiously close results? Is it understandable in terms of our modern model of neutral pions, a mistake in the argument, a coincidence, or some combination of these?

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It is a coincidence. The claim in the article can be summarized as $$ \frac{\alpha}{2}\frac{m_\pi}{m_e}=0.96 $$ which is close to $1$. This relation doesn't have a deep origin, it is just a coincidence of numerical factors. Particle physics has dozens of numerical parameters, and thousands of possible ways to combine them into dimensionless expressions. Eventually, you will find many combinations that are close to $1$.

Note that $\alpha$ and $m_e$ are parameters of electromagnetism, and $m_\pi$ is a parameter of the strong force. As far as we know, these two forces are completely independent (at least at the energies relevant to the calculation; they may unify at larger energies but that would not matter as far as the calculation of $m_\pi$ is concerned). There is no fundamental principle of nature that relates $m_\pi$ to $m_e$ or $\alpha$. The pion mass depends on the quark masses and $SU(3)$ interactions alone; it can be predicted using lattice QCD without even introducing the electromagnetic $U(1)$ sector (of course, the mass splitting of $\pi^0$ and $\pi^\pm$ does care about EM, but this is a tiny subleading effect).

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  • $\begingroup$ I see... I think there must also be a mistake in the reasoning though, otherwise we would still observe positronium states with a mass coincidentally close to the neutral pion mass, which as far as I know, we don't. I haven't worked out where the problem lies, though. $\endgroup$ Aug 13, 2022 at 13:59
  • $\begingroup$ @turbodiesel4598 The problem is basically the whole paper -- it relies on extrapolating results from nonrelativistic quantum mechanics and classical electromagnetism far, far beyond their domain of validity. The electron and positron are assumed to be orbiting ultrarelativistically within closer than a Compton wavelength, yet the quantization of the electron field, the quantization of the electromagnetic field, and even basic radiative effects of the classical electromagnetic field are all neglected. So there is no reason to expect that the positronium state in the paper actually exists. $\endgroup$
    – knzhou
    Aug 13, 2022 at 18:20
  • $\begingroup$ Electromagnetism actually is an aspect of the neutral pion, since the quarks are charged... It's been pointed out physics.stackexchange.com/questions/426153/mass-of-the-electron that the electromagnetic self-energy of the electron contributes about 1/5 of the electron mass, could some of the pion mass come from the electromagnetic self-energy of its quarks?? $\endgroup$ Nov 5, 2022 at 19:08
  • $\begingroup$ @MitchellPorter Some of the pion mass does come from EM interactions, but the contribution is extremely tiny (this contribution explains why $\pi^0$ has slightly different mass from $\pi^\pm$, for example). $\endgroup$ Nov 10, 2022 at 16:12
  • $\begingroup$ Hmm... I know that in the chiral limit, pions are supposed to be massless... I assume that this is true even for charged quarks... I'll have to think where the electromagnetic self-energy goes in that case. Or I might just ask:) $\endgroup$ Nov 10, 2022 at 19:00
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I don't know about lifetime, but as for mass (and the following is just speculation), one of the (very rare) decay mode of the neutral pion is to gamma and positronium (P.A. Zyla et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020), p.38). As the decay is very rare, the relevant phase space volume must be very low, and it is possible that it corresponds to low gamma energy, so it is possible that the mass of the resulting positronium is close to that of the initial neutral pion, so this may be the reason for Sternglass' calculations for "relativistic positronium" giving a good estimate of neutral pion's mass.

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    $\begingroup$ “ As the decay is very rare, the relevant phase space volume must be very low” That’s not true though, there are many other reasons decays can be rare… in this, I think it’s because the electron and positron have to be emitted with almost zero relative velocity so that they’ll be bound to each other. In any case, we do know the mass of positronium perfectly well and it’s nowhere near the pion mass. $\endgroup$
    – knzhou
    Aug 14, 2022 at 2:29
  • $\begingroup$ @knzhou : My understanding is excited states of positronium ("relativistic positronium" from Sternglass' paper) can have much higher mass. $\endgroup$
    – akhmeteli
    Aug 14, 2022 at 2:38
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    $\begingroup$ But we've also measured the energy levels of positronium in great detail. And we measure the masses of pion decay products -- when we say a pion decays into positronium plus a photon, we know positronium was produced precisely because it has the mass we expect. If there was a positronium-like state with a mass near the pion, it would look like a new particle and be treated as such. $\endgroup$
    – knzhou
    Aug 14, 2022 at 5:31
  • $\begingroup$ @knzhou : So you don't think relativistic states of positronium exist? There is, e.g., this Nature (1966) article: doi.org/10.1038/211810a0 . Maybe it is also wrong... $\endgroup$
    – akhmeteli
    Aug 14, 2022 at 6:48
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    $\begingroup$ There are hundreds or even thousands of papers on positronium, and entire reference books — if this purported state isn’t mentioned except in some almost uncited 50 year old papers, then it almost certainly turned out to not exist. $\endgroup$
    – knzhou
    Aug 14, 2022 at 14:58
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Circular arguments, arguments based on existing assumptions, all demonstrating a reluctance to abandon what you have learned in school. There’s no coincidence here. The strong force is just compressed EM force. Think of this “coincidence” as one data point, Sternglass’ greatness lies in discovering all coincidences across the whole field of particle physics and cosmology. That’s why it’s no coincidence.

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