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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?

tomdodd4598
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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).

AccidentalFourierTransform
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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.

akhmeteli
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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.

Xiang He
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