# Symmetry of the Non-Analytic Solution of the Dirac Equation Inside the Proton of Hydrogen Atoms
## 1 Introduction  
There was a long-standing huge disagreement between the experimental [1] and theoretical [2] results concerning one of the features of hydrogen atoms: the Distribution of the Linear Momentum (DLM) in the lowest (ground) state. Namely, for relatively large momenta p, the theoretical DLM decreased much more rapidly than the experimental DLM, the ratio of the two reaching _tens of thousands_. In paper [3], this discrepancy was resolved by engaging the second solution of the Dirac equation, for which the corresponding radial wave function R\({}_{0,-1}\) (r) is proportional (at relatively small r) to 1/r q with q = 1 + (1 - \(\alpha^{2}\))\({}^{1/2}\), \(\alpha\) being the fine structure constant(The second subscript of the wave function R\({}_{0,-1}\) is the eigenvalue of the operator K = b(2 Ls +1), which is the conserved quantity. Here, L is the orbital angular momentum, s is the spin, and b is the Dirac matrix 4 \(\times\) 4). In paper [3], it was shown that after allowing for the finite size of the proton and for the fact that the charge density distribution inside the proton has the maximum at the origin and then decreases toward the proton periphery, the second solution becomes legitimate--because at the proton boundary, both the interior solution and the second exterior solution, as well as their derivatives, are continuous, as required by quantum mechanics. The central point was the following. The much more robust increase of the second exterior wave function toward the proton boundary compared to the first exterior wave function corresponded--in the momentum representation--to the significantly more moderate decrease of the second exterior wave function for large momenta compared to the 1 st exterior wave function, according to the Fourier transform properties. As a result, the long-standing huge disagreement between the experimental and theoretical results concerning the DLM was totally removed. Later, in paper [4], it was noted that the legitimacy of the second exterior solution in paper [3] was based on the fact that the quantum number k (shown as the second subscript of R\({}_{0,-1}\) (r)) was equal to \(-1\). For this reason, the second exterior solution of the Dirac equation is legitimate for any state (of hydrogen atoms) characterized by k = \(-1\). The states of k = \(-1\) are the S-states, i.e., the states of zero orbital momentum. This second kind of hydrogen atom, which does not have states of the non-zero orbital momentum, was called the Second Flavor of Hydrogen Atoms (SFHA) in a subsequent paper [5]. This is because both the first and second exterior solutions correspond to the _same energy_: consequently, the _additional degeneracy_--consequently, an _additional conserved quantity_--consequently, the _flavor symmetry_ (just as in the case of quarks). The above solution of the DLM puzzle, together with the absence of any alternative explanation, represented _the 1 st experimental evidence that the SFHA exists_. By now there are several pieces of additional experimental/observational evidence regarding the existence of the SFHA, as follows. The experimental [6; 7; 8] and theoretical [9] cross sections of the excitation of the triplet states of the H\({}_{2}\) molecule by the electron impact showed a disagreement of 3 to 5 times. The possible presence of the SFHA in the experimental gas removed this disagreement [10], while no alternative explication was given. The experimental [11; 12] and theoretical [13] cross sections of the excitation of the 2S state of hydrogen atoms by the electron impact demonstrated a disagreement of 20% (while the error margins were just 9%). The possible presence of the SFHA in the experimental gas removed this disagreement [14], while no alternative explication was given. One of the primary latest applications of the SFHA is in resolving the neutron lifetime puzzle, as briefly presented in paper [15]. The puzzle was the following.