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Motivated by tremendous progress in neutrino oscillation experiments, we derive a new set of simple and compact formulas for three-flavor neutrino oscillation probabilities in matter of a constant density. A useful definition of the $eta$-gauge neutrino mass-squared difference $Delta^{}_* equiv eta Delta^{}_{31} + (1-eta) Delta^{}_{32}$ is introduced, where $Delta^{}_{ji} equiv m^2_j - m^2_i$ for $ji = 21, 31, 32$ are the ordinary neutrino mass-squared differences and $0 leq eta leq 1$ is a real and positive parameter. Expanding neutrino oscillation probabilities in terms of $alpha equiv Delta^{}_{21}/Delta^{}_*$, we demonstrate that the analytical formulas can be remarkably simplified for $eta = cos^2 theta^{}_{12}$, with $theta_{12}^{}$ being the solar mixing angle. As a by-product, the mapping from neutrino oscillation parameters in vacuum to their counterparts in matter is obtained at the order of ${cal O}(alpha^2)$. Finally, we show that our approximate formulas are not only valid for an arbitrary neutrino energy and any baseline length, but also still maintaining a high level of accuracy.
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We construct a new perturbative framework to describe neutrino oscillation in matter with the unique expansion parameter epsilon, which is defined as Delta m^2_{21} / Delta m^2_{ren} with the renormalized atmospheric Delta m^2_{ren} equiv Delta m^2_{31} - s^2_{12} Delta m^2_{21}. It allows us to derive the maximally compact expressions of the oscillation probabilities in matter to order epsilon in the form akin to those in vacuum. This feature allows immediate physical interpretation of the formulas, and facilitates understanding of physics of neutrino oscillations in matter. Moreover, quite recently, we have shown that our three-flavor oscillation probabilities P( u_alpha rightarrow u_beta) in all channels can be expressed in the form of universal functions of L/E. The u_e disappearance oscillation probability P( u_e rightarrow u_e) has a special property that it can be written as the two-flavor form which depends on the single frequency. This talk is based on the collaborating work with Stephen Parke [1].
In this paper we review different expansions for neutrino oscillation probabilities in matter in the context of long-baseline neutrino experiments. We examine the accuracy and computational efficiency of different exact and approximate expressions. We find that many of the expressions used in the literature are not precise enough for the next generation of long-baseline experiments, but several of them are while maintaining comparable simplicity. The results of this paper can be used as guidance to both phenomenologists and experimentalists when implementing the various oscillation expressions into their analysis tools.
In neutrino oscillations, a neutrino created with one flavor can be later detected with a different flavor, with some probability. In general, the probability is computed exactly by diagonalizing the Hamiltonian operator that describes the physical system and that drives the oscillations. Here we use an alternative method developed by Ohlsson & Snellman to compute exact oscillation probabilities, that bypasses diagonalization, and that produces expressions for the probabilities that are straightforward to implement. The method employs expansions of quantum operators in terms of SU(2) and SU(3) matrices. We implement the method in the code NuOscProbExact, which we make publicly available. It can be applied to any closed system of two or three neutrino flavors described by an arbitrary time-independent Hamiltonian. This includes, but is not limited to, oscillations in vacuum, in matter of constant density, with non-standard matter interactions, and in a Lorentz-violating background.
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 $epsilon equiv Delta m^2_{21} / Delta m^2_{31}$, with introduction of the effective $Delta m^2_{alpha alpha}$ ($alpha = e, mu, tau$), in regions of neutrino energy $E$ and baseline $L$ such that $Delta m^2_{31} L / 2E sim pi$. 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 $Delta m^2_{alpha alpha} (a)$ 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 $Delta m^2_{mu mu} (a)$ becomes $L$-dependent, though $Delta m^2_{ee} (a)$ 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 $Delta m^2_{beta alpha}$ from the disappearance channel. A general result is derived to describe suppression of the matter effect in the oscillation probability.
The existence of light sterile neutrinos is a long standing question for particle physics. Several experimental ``anomalies could be explained by introducing ~eV mass scaled light sterile neutrinos. Many experiments are actively hunting for such light sterile neutrinos through neutrino oscillation. For long baseline experiments, matter effect needs to be treated carefully for precise neutrino oscillation probability calculation. However, it is usually time-consuming or analytical complexity. In this manuscript we adopt the Jacobi-like method to diagonalize the Hermitian Hamiltonian matrix and derive analytically simplified neutrino oscillation probabilities for 3 (active) + 1 (sterile)-neutrino mixing for a constant matter density. These approximations can reach quite high numerical accuracy while keeping its analytical simplicity and fast computing speed. It would be useful for the current and future long baseline neutrino oscillation experiments.
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