The Second Flavor of hydrogenic atoms/ions and their applications in atomic physics, nuclear physics, and cosmology

While all papers – with one exception – referred to in this mini-review were published in the last five years, the exception is paper [1] of 2001, and it is a paper, from which the story started. Paper [1] was motivated by a really huge discrepancy between the experimental and theoretical distributions of the linear momentum in the ground state of hydrogen atoms. The ratio of the experimental and theoretical results was up to several thousands. Namely, the experimental high-energy tail of the linear momentum distribution fell off much-much slower than predicted by the theory. In search for the explanation, the author revisited the Dirac equation for hydrogen atoms, but with the allowance for the finite nuclear size, i.e., for the finite size of the proton charge distribution.

The Dirac “equation” is actually two coupled linear differential equations of the 1st order for the two components of the Dirac bispinor f(r) and g(r). By expressing f(r) via g(r) and its derivative from one equation, and then substituting into the other equation, one obtains a linear differential equation of the 2nd order for g(r).

Any linear differential equation of the 2nd order generally has two independent solutions. In the case of the ground state of hydrogen atoms, for relatively small r, the first solution g ₁(r) scales ∼ 1/r ^0.000027 and so does f ₁(r), thus exhibiting only a very weak singularity (we call it the “regular” solution for brevity). The second solution g ₂(r) scales ∼ 1/r ² and so does f ₂(r), thus exhibiting a strong singularity. For the model of the point-like nucleus, this second solution is usually rejected because the normalization integral diverges at r = 0.

However, the nucleus (the proton) is not point-like. The corresponding interaction potential, being the Coulomb potential V _ext(r) in the external region r ≥ R (where R is the proton size), has a different form V _int(r) in the internal region r ≤ R, V _int(r) being controlled by the Charge Density Distribution (CDD) ρ(r) inside the proton. In paper [1], there was derived a general condition imposed on V _int(r) and V _ext(r), under which the Wave Functions (WFs) and their derivatives in the interior and exterior regions can be matched at the boundary r = R. It was also shown that the experimental CDD inside protons generates V _int(r) satisfying that condition.

Thus, a regular solution of the Dirac equation inside the proton can be matched with the corresponding singular solution at the boundary. So, for the ground state of hydrogen atoms, the singular exterior solution is actually legitimate.

The corresponding WF for the outer region has been derived in paper [1]. Based on this, the above huge discrepancy between the experimental and the previous theoretical (based on the regular exterior solution) distributions of the linear momentum in the ground state of hydrogen atoms got completely eliminated. Physically this was because the much stronger rise of the singular exterior WF toward the proton (compared to the regular exterior WF) translates into the much slower fall-off of the corresponding WF in the linear momentum representation at large momentum p – according to the properties of the Fourier transform (by which the two representations of the WF are connected). Namely, it falls off ∼ 1/p⁴, just as the experimental distribution, while the previous theoretical one fell off ∼ 1/p⁶.

This constituted the first experimental evidence of the existence of hydrogen atoms of the second kind – the atoms described by the singular exterior solution. This is because no alternative explanation for the above huge discrepancy was ever provided.

In paper [2] of 2020 it was shown that with the allowance for the finite proton size, the singular exterior solution of the Dirac equation is legitimate not only for the ground state, but for all S-states of hydrogen atoms. Due to the selection rules of quantum mechanics, the atoms having only the S-states do not emit or absorb the electromagnetic radiation: they remain dark.

This second kind of hydrogen atoms was later called the Second Flavor of Hydrogen Atoms (SFHA) in paper [3]. Here is the reason.

Both the regular and singular solutions of the Dirac equation represent eigenfunctions corresponding to the same energy, meaning the additional degeneracy. Consequently, there should be an additional conserved quantity – according to the fundamental theorem of quantum mechanics. Thus, hydrogen atoms have two flavors differing by the eigenvalue of the additional conserved quantity: hydrogen atoms exhibit the flavor symmetry – called so by analogy with the flavor symmetry of quarks. Namely, for quarks there was introduced a conserved quantity called isospin I _q , whose eigenvalues are +1/2 for the up-quark and −1/2 for the down-quark. For hydrogen atoms, the additional conserved quantity could be called isohyspin I_H, whose eigenvalues are +1/2 for the 1st (usual) flavor of hydrogen atoms and −1/2 for the 2nd flavor of hydrogen atoms.

By now there are two more evidences from two different kinds of atomic experiments that the SFHA does exist: from the experiments on the electron impact excitation of hydrogen atoms and from the experiments on the electron impact excitation of hydrogen molecules. In both situations, the SFHA-based explanation removed large discrepancies – up to a factor of five – between the experimental results and the corresponding previous theories, while no alternative explanation for these large discrepancies was ever provided.

The most comprehensive theoretical description of the SFHA was given in paper [4] of 2025. In particular, it showed that the WF inside the proton is non-analytic and its explicit form was derived.

In addition to the above applications of the SFHA to atomic physics, there are also applications to nuclear physics and to cosmology. Let us start with nuclear physics.

There was a long-standing puzzle of the neutron lifetime. Typically, free neutron undergoes the 3-body decay into free proton, free electron, and antineutrino. In the “beam” experiments there was measured the number of produced protons and claimed that it was equal to the number of decayed protons (thus asserting that there was only the 3-body decay). In the “trap” experiments, there was measured the number of surviving neutrons and thus, by subtraction, the number of decayed neutrons. It turned out that the neutron lifetime deduced from the beam experiments is noticeably longer than the one deduced from the trap experiments, the difference being well beyond the experimental error margins.

In fact, there was a theoretical prediction that sometimes free neutron can undergo the 2-body decay into hydrogen atom plus antineutrino. The beam experiments did not take this into account. However, for the quantitative explanation of the difference in the measured lifetimes, the Branching Ratio (BR) for the 2-body decay (compared to the 3-body decay) should have been ∼1 %, while the previous theoretical estimate was the BR ∼ 4 × 10⁻⁶. Then in papers [5], 6] it was shown that the 2-body decay of neutrons should produce overwhelmingly the SFHA rather than the usual hydrogen atoms. This is because the probability of the 2-body decay is controlled by the probability of finding the atomic electron at the proton surface. So, since for the SFHA, the WF rises toward the proton much faster than for the usual hydrogen atoms, the outcome should be overwhelmingly the SFHA. This increased the BR to be about 1 %: thus, the puzzle got resolved. In paper [6] there was also proposed the conceptual design of experiments confirming the above results. Such experiments, serving as an additional test of the existence of the SFHA, are now underway at some neutron research centers.

The cosmological application of the SFHA has to do with explaining the baffling observation by Bowman et al. [7] of the anomalous absorption in the redshifted 21 cm line from the early Universe: the absorption signal was 2 to 3 times stronger than predicted by the standard cosmology. This meant that the hydrogen gas was significantly cooler than predicted. Barkana [8] and McGaugh [9] suggested that some unspecified Dark Matter (DM) particles cooled the hydrogen gas by collisions. The quantitative explanation required the mass of those particles to be of the order of the baryons masses. In paper [2] we showed that if those unspecified DM particles were the SFHA, then this would explain the anomalous absorption signal both qualitatively and quantitatively.

This made the SFHA a leading candidate for baryonic DM, which according to astrophysical observations constitutes about 1/6 of the total dark matter. Moreover, in paper [10] we showed that from the comparison of astrophysical observations with atomic experiments, it follows that the SFHA constitutes the majority of baryonic DM.

For completeness: there were alternative hypotheses for explaining the anomalous absorption in 21 cm line, as follows. The SFHA-based explanation is more advantageous than suggesting a possible cooling of baryons by some exotic DM particles having the charge of the million times lesser than the electron charge, as in papers [11], 12]. Besides, in paper [12] it was estimated that if there would be charged DM particles, they could only represent ∼10⁻⁸ of the total DM energy density. The most important is the following: exotic DM particles of the charge of the million times lesser than the electron charge have been never discovered experimentally, while the existence of the SFHA is evidenced by three different kinds of atomic/molecular experiments, plus it provided the resolution of the long-standing puzzle of the lifetime of free neutrons.

Also, the SFHA-based explanation does not require an additional hypothetical radio background proposed in papers [13], [14], [15], [16]. Besides, in paper [17] it was demonstrated (already in 2018) and in paper [18] it was reconfirmed (in 2024) that an additional radio background is not capable to explicate the observed anomalous absorption signal.

The theory of the SFHA is based on the standard quantum mechanics: the Dirac equation. It does not go beyond the Standard Model of particle physics and does not resort to changing the physical laws – in distinction to the overwhelming majority of dark matter hypotheses. Therefore, it is favored by the Occam razor principles, which states that when theories compete, the one making the least number of assumptions is most probably to correspond to reality.

Last but not least: in paper [5] it was demonstrated that neutron stars slowly but continuously generate a new baryonic dark matter in the form of the SFHA. It was also pointed out that there is astrophysical evidence of this process.

For completeness: in paper [19] it was shown that hydrogenlike ions, whose nuclei are doubly-magic and thus spherically symmetric, can also have the second flavor. Examples are the following seven nuclei: ⁴⁰Ca₂₀, ⁴⁸Ca₂₀, ⁴⁸Ni₂₈, ⁵⁶Ni₂₈, ⁷⁸Ni₂₈, ¹⁰⁰Sn₅₀, ¹³²Sn₅₀.

In the near future, the author will analyze in detail whether the ionized helium (He⁺) can also have the second flavor.