Dispersive determination of neutrino mass ordering

Abstract: We argue that the mixing phenomenon of a neutral meson formed by a fictitious massive quark will disappear, if the electroweak symmetry of the Standard Model (SM) is restored at a high energy scale. This disappearance is taken as the high-energy input for the dispersion relation, which must be obeyed by the width difference between two meson mass eigenstates. The solution to the dispersion relation at low energy, i.e., in the symmetry broken phase, then connects the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements to the quark masses involved in the box diagrams responsible for meson mixing. It is demonstrated via the analysis of the $D$ meson mixing that the typical $d$, $s$ and $b$ quark masses demand the CKM matrix elements in agreement with measured values. In particular, the known numerical relation $V_{us}\approx \sqrt{m_s/m_b}$ with the $s$ ($b$) quark mass $m_s$ ($m_b$) can be derived analytically from our solution. Next we apply the same formalism to the mixing of the $\mu^- e^+$ and $\mu^+ e^-$ states through similar box diagrams with intermediate neutrino channels. It is shown that the neutrino masses in the normal hierarchy (NH), instead of in the inverted hierarchy or quasi-degenerate spectrum, match the observed Pontecorvo-Maki-Nakagawa-Sakata matrix elements. The lepton mixing angles larger than the quark ones are explained by means of the inequality $m_2^2/m_3^2\gg m_s^2/m_b^2$, $m_{2,3}$ being the neutrino masses in the NH. At last, the solution for the $\tau^-e^+$-$\tau^+e^-$ mixing specifies the mixing angle $\theta_{23}\approx 45^\circ$, leading to the $\mu$-$\tau$ reflection symmetry. Our work suggests that the fermion masses and mixing parameters are constrained dynamically, and the neutrino mass orderings can be discriminated by the internal consistency of the SM.