Analytical solutions to renormalization-group equations of effective neutrino masses and mixing parameters in matter
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 ˜ $$ \tilde{\delta} $$ ,...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2019-05-01
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| Series: | Journal of High Energy Physics |
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| Online Access: | http://link.springer.com/article/10.1007/JHEP05(2019)035 |
| _version_ | 1818109284985077760 |
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| author | Xin Wang Shun Zhou |
| author_facet | Xin Wang Shun Zhou |
| author_sort | Xin Wang |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-11T02:28:49Z |
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| spelling | doaj.art-28d07e70de224993bcd63c439ee78e162022-12-22T01:23:52ZengSpringerOpenJournal of High Energy Physics1029-84792019-05-012019512010.1007/JHEP05(2019)035Analytical solutions to renormalization-group equations of effective neutrino masses and mixing parameters in matterXin Wang0Shun Zhou1Theoretical Physics Division, Institute of High Energy PhysicsTheoretical Physics Division, Institute of High Energy PhysicsAbstract 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.http://link.springer.com/article/10.1007/JHEP05(2019)035Neutrino PhysicsRenormalization Group |
| spellingShingle | Xin Wang Shun Zhou Analytical solutions to renormalization-group equations of effective neutrino masses and mixing parameters in matter Journal of High Energy Physics Neutrino Physics Renormalization Group |
| title | Analytical solutions to renormalization-group equations of effective neutrino masses and mixing parameters in matter |
| title_full | Analytical solutions to renormalization-group equations of effective neutrino masses and mixing parameters in matter |
| title_fullStr | Analytical solutions to renormalization-group equations of effective neutrino masses and mixing parameters in matter |
| title_full_unstemmed | Analytical solutions to renormalization-group equations of effective neutrino masses and mixing parameters in matter |
| title_short | Analytical solutions to renormalization-group equations of effective neutrino masses and mixing parameters in matter |
| title_sort | analytical solutions to renormalization group equations of effective neutrino masses and mixing parameters in matter |
| topic | Neutrino Physics Renormalization Group |
| url | http://link.springer.com/article/10.1007/JHEP05(2019)035 |
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