Symmetry of the Non-Analytic Solution of the Dirac Equation Inside the Proton of Hydrogen Atoms

In one of our previous papers, we obtained a general class of potentials inside the nucleus, such that the singular solution of the Dirac equation for the S-states of hydrogen atoms outside the nucleus can be matched with the corresponding regular solution inside the nucleus (the proton) at the boundary. The experimental charge density distribution inside the proton generates a particular case of such potentials inside the proton. In this way, the existence of the second kind of hydrogen atom was predicted: atoms having only S-states. This theoretical prediction was then evidenced by several different types of atomic experiments and by astrophysical observations. In the present paper we provide the following new results. First, we show that the solution of the Dirac equation inside the proton can be (and is) found within the class of functions that are non-analytic at r = 0—in distinction to the traditional practice of limiting the search for the solution by the class of analytic functions. We demonstrate that this non-analytic solution inside the proton can be matched at the proton boundary R with the corresponding singular solution outside the proton regardless of the particular value of R. Second, we show that the interior and exterior solutions are scale-invariant with respect to the change of the boundary R between these two regions. Such invariance is the manifestation of a new symmetry—in addition to the previously discussed symmetries of the Dirac equation in the literature. Third, based on the new, more accurate results for the wave function inside and outside the proton, we revisit the resolution of the neutron lifetime puzzle initially outlined in our previous papers. On the basis of the more accurate calculations, we reconfirm that (A) the 2-body decay of neutrons produces overwhelmingly the SFHA (rather than the usual hydrogen atoms) and (B) the strengthened-in-this-way branching ratio for the 2-body decay of neutrons (compared to the 3-body decay) is in excellent agreement with the branching ratio required for reconciling the neutron lifetime values measured in the trap and beam experiments, so that the neutron lifetime puzzle seems to be indeed resolved in this way.