# Symmetry of the Non-Analytic Solution of the Dirac Equation Inside the Proton of Hydrogen Atoms
## 1 Introduction
In the experiments using the beam technique, the measured neutron lifetime \(\tau_{\rm beam}\) turned out to be greater than the neutron lifetime \(\tau_{\rm trap}\) measured in the experiments using the trap technique, significantly beyond the margins of error (see, e.g., paper [16] and the references therein). The primary channel of limiting the neutron lifetime is the 3-body decay, producing a proton, an electron, and an antineutrino. In the beam experiments, there were counted only the protons from the 3-body decay. Theoretically, there should be also the 2-body decay producing a hydrogen atom and an antineutrino. To explain the neutron lifetime puzzle via 2-body decay, missed in the beam experiments, it would require the branching ratio (BR) for the 2-body decay (with respect to the 3-body decay) to be \(\sim\)1%. However, from paper [17] of 1961, it was known that the corresponding BR is just 4 \(\times\) 10\({}^{-6}\)In paper [15], there was outlined that the 2-body decay of neutrons should produce overwhelmingly the SFHA rather than the usual hydrogen atoms. It was roughly estimated that this can increase the corresponding BR to \(\sim\)1%, which would resolve the puzzle. We note that paper [15] was focused primarily on neutron stars, so that the 2-body decay of neutrons was just a \"by-product\" presented sketchily. The most important feature of the SFHA is that, because of having only S-states, and in view of the selection rules of quantum mechanics, the SFHA does not couple to the electromagnetic radiation, including the dipole and higher multipole radiation, as well as multi-photon transitions. The SFHA _cannot be excited or ionized by a static electric field or by a laser field_, though it can be excited or ionized, e.g., by an electron beam. In connection with the above most important feature of the SFHA, let us briefly reiterate astrophysical evidence of the existence of the SFHA. [PERSON] et al. [18], while observing the 21 cm line from the early Universe, found that the absorption signal in this line is 2 to 3 times stronger than expected from standard cosmology, meaning that the primordial hydrogen gas was significantly cooler than expected. The most plausible cooling agent was some unspecified baryonic dark matter (DM) particles that cooled the hydrogen gas by collisions--first according to [PERSON] [19] and then confirmed by [PERSON] [20]. Then, in paper [4], it was shown that if the role of the unspecified baryonic DM particles was played by the SFHA, then this would explain the perplexing observation made by [PERSON] et al. [18], not only qualitatively, but also quantitatively; this made the SFHA a compelling candidate for being the baryonic DM. Moreover, the comparison of the observed cosmological data on the ratio of the total DM to the usual matter and on the ratio of baryonic DM to non-baryonic DM with the data from atomic experiments that evidenced the existence of the SFHA, led to the following conclusion. The SFHA seems to comprise most of the baryonic DM in the current epoch [21]. It is significant that the SFHA is established as the solution of the Dirac equation, i.e., within the standard quantum mechanics. Thus, first, it is within the Standard Model of particle physics and second, no change of the physical laws is required. This is the clear difference of the SFHA compared to the vast majority of the models of DM (For completeness, we note the possibility of the existence of the second flavor of heavy hydrogenic ions, whose nuclei are doubly-magic and therefore spherical--as presented in our previous paper [22]). In the present paper, we provide the following _new results_. First, we derive the explicit form of the solution of the Dirac equation inside the proton for the S-states--the solution that at the proton boundary matches the corresponding singular solution, as required by quantum mechanics--both the wave function and its derivative are continuous at the boundary.