An Effective Two-Flavor Approximation for Neutrino Survival Probabilities in Matter

Abstract

It is known in vacuum that the three-flavor neutrino survival probability can be approximated by the effective two-flavor form to first orders in $\epsilon \equiv \Delta m_{21}^2/\Delta m_{31}^2$, with introduction of the effective $\Delta m_{\alpha \alpha }^2$ ($\alpha =e,\mu ,\tau$), in regions of neutrino energy $E$ and baseline $L$ such that $\Delta m_{31}^{2}L/2E\sim \pi$. Here, we investigate the question of whether the similar effective two-flavor approximation can be formulated for the survival probability in matter. Using a perturbative framework with the expansion parameters $\epsilon$ and $s_{13}\propto \sqrt{\epsilon }$, we give an affirmative answer to this question and the resultant two-flavor form of the probability is valid to order $\epsilon$. However, we observe a contrived feature of the effective $\Delta m_{\alpha \alpha }^2(a)$ in matter. It ceases to be a combination of the fundamental parameters and has energy dependence, which may be legitimate because it comes from the matter potential. But, it turned out that $\Delta m_{\mu \mu }^2(a)$ becomes $L$-dependent, though $\Delta m_{ee}^2(a)$ is not, which casts doubt on adequacy of the concept of effective $\Delta m^2$ in matter. We also find that the appearance probability in vacuum admits, to order $\epsilon$, the similar effective two-flavor form with a slightly different effective $\Delta m_{\beta \alpha }^2$ from the disappearance channel. A general result is derived to describe suppression of the matter effect in the oscillation probability.

Discussion on appearance effective $\Delta m^2$ added. Version to appear in JHEP. 26 pages, 3 figures

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Cited by in corpus (4)

• Compact Perturbative Expressions for Neutrino Oscillations in Matter: II
• The Effective $\Delta m_{ee}^2$ in Matter
• From ray to spray: augmenting amplitudes and taming fast oscillations in fully numerical neutrino codes
• The Smallness of Matter Effects in Long-Baseline Muon Neutrino Disappearance