Abstract
Neutrino oscillations in matter can be fully described by six effective parameters, namely, three neutrino mixing angles \( \left\{{\tilde{\theta}}_{12},{\tilde{\theta}}_{13},{\tilde{\theta}}_{23}\right\} \), one Dirac-type CP-violating phase \( \tilde{\delta} \), and two neutrino mass-squared differences \( {\tilde{\Delta}}_{21}\equiv {\tilde{m}}_2^2-{\tilde{m}}_1^2 \) and \( {\tilde{\Delta}}_{21}\equiv {\tilde{m}}_3^2-{\tilde{m}}_1^2 \). Recently, a complete set of differential equations for these effective parameters have been derived to characterize their evolution with respect to the ordinary matter term \( a\equiv 2\sqrt{2}{G}_{\mathrm{F}}{N}_eE \), in analogy with the renormalization-group equations (RGEs) for running parameters. Via series expansion in terms of the small ratio αc ≡ Δ21/Δc with Δc ≡ Δ31 cos2 θ12+Δ32 sin2θ12, we obtain approximate analytical solutions to the RGEs of the effective neutrino parameters and make several interesting observations. First, at the leading order, \( {\tilde{\theta}}_{12} \) and \( {\tilde{\theta}}_{13} \) are given by the simple formulas in the two-flavor mixing limit, while \( {\tilde{\theta}}_{23}\approx {\theta}_{23} \) and \( \tilde{\delta}\approx \delta \) are not changed by matter effects. Second, the ratio of the matter-corrected Jarlskog invariant \( \tilde{\mathcal{J}} \) to its counterpart in vacuum \( \mathcal{J} \) approximates to \( \tilde{\mathcal{J}}/\mathcal{J}\approx 1/\left({\widehat{C}}_{12}{\widehat{C}}_{13}\right) \), where \( {\widehat{C}}_{12}\equiv \sqrt{1-2{A}_{\ast } \cos 2{\theta}_{12}+{A}_{{}^{\ast}}^2} \) with A∗ ≡ a/Δ21 and \( {\widehat{C}}_{13}\equiv \sqrt{1-2{A}_c \cos 2{\theta}_{13}+{A}_c^2} \) with Ac ≡ a/Δc have been defined. Finally, after taking higher-order corrections into account, we find compact and simple expressions of all the effective parameters, which turn out to be in perfect agreement with the exact numerical results.
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ArXiv ePrint: 1901.10882
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Wang, X., Zhou, S. Analytical solutions to renormalization-group equations of effective neutrino masses and mixing parameters in matter. J. High Energ. Phys. 2019, 35 (2019). https://doi.org/10.1007/JHEP05(2019)035
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DOI: https://doi.org/10.1007/JHEP05(2019)035