| Summary: | Abstract Neutrino oscillations in matter can be fully described by six effective parameters, namely, three neutrino mixing angles θ ˜ 12 θ ˜ 13 θ ˜ 23 $$ \left\{{\tilde{\theta}}_{12},{\tilde{\theta}}_{13},{\tilde{\theta}}_{23}\right\} $$ , one Dirac-type CP-violating phase δ ˜ $$ \tilde{\delta} $$ , and two neutrino mass-squared differences Δ ˜ 21 ≡ m ˜ 2 2 − m ˜ 1 2 $$ {\tilde{\Delta}}_{21}\equiv {\tilde{m}}_2^2-{\tilde{m}}_1^2 $$ and Δ ˜ 21 ≡ m ˜ 3 2 − m ˜ 1 2 $$ {\tilde{\Delta}}_{21}\equiv {\tilde{m}}_3^2-{\tilde{m}}_1^2 $$ . 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 ≡ 2 2 G F N e E $$ a\equiv 2\sqrt{2}{G}_{\mathrm{F}}{N}_eE $$ , in analogy with the renormalization-group equations (RGEs) for running parameters. Via series expansion in terms of the small ratio α c ≡ Δ21/Δc with Δc ≡ Δ31 cos2 θ12+Δ32 sin2 θ 12, we obtain approximate analytical solutions to the RGEs of the effective neutrino parameters and make several interesting observations. First, at the leading order, θ ˜ 12 $$ {\tilde{\theta}}_{12} $$ and θ ˜ 13 $$ {\tilde{\theta}}_{13} $$ are given by the simple formulas in the two-flavor mixing limit, while θ ˜ 23 ≈ θ 23 $$ {\tilde{\theta}}_{23}\approx {\theta}_{23} $$ and δ ˜ ≈ δ $$ \tilde{\delta}\approx \delta $$ are not changed by matter effects. Second, the ratio of the matter-corrected Jarlskog invariant J ˜ $$ \tilde{\mathcal{J}} $$ to its counterpart in vacuum J $$ \mathcal{J} $$ approximates to J ˜ / J ≈ 1 / C ^ 12 C ^ 13 $$ \tilde{\mathcal{J}}/\mathcal{J}\approx 1/\left({\widehat{C}}_{12}{\widehat{C}}_{13}\right) $$ , where C ^ 12 ≡ 1 − 2 A ∗ cos 2 θ 12 + A ∗ 2 $$ {\widehat{C}}_{12}\equiv \sqrt{1-2{A}_{\ast } \cos 2{\theta}_{12}+{A}_{{}^{\ast}}^2} $$ with A ∗ ≡ a/Δ21 and C ^ 13 ≡ 1 − 2 A c cos 2 θ 13 + A c 2 $$ {\widehat{C}}_{13}\equiv \sqrt{1-2{A}_c \cos 2{\theta}_{13}+{A}_c^2} $$ 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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