Neutrino Propagation in Matter

Abstract

We describe the effects of neutrino propagation in the matter of the Earth relevant to experiments with atmospheric and accelerator neutrinos and aimed at the determination of the neutrino mass hierarchy and CP violation. These include (i) the resonance enhancement of neutrino oscillations in matter with constant or nearly constant density, (ii) adiabatic conversion in matter with slowly changing density, (iii) parametric enhancement of oscillations in a multilayer medium, and (iv) oscillations in thin layers of matter. We present the results of semianalytic descriptions of flavor transitions for the cases of small density perturbations, in the limit of large densities and for small density widths. Neutrino oscillograms of the Earth and their structure after determination of the 1–3 mixing are described. A possibility to identify the neutrino mass hierarchy with the atmospheric neutrinos and multimegaton scale detectors having low energy thresholds is explored. The potential of future accelerator experiments to establish the hierarchy is outlined.

1. Introduction

Neutrinos are eternal travelers: once produced (especially at low energies) they have little chance to interact and be absorbed. Properties of neutrino fluxes are flavor compositions, lepton charge asymmetries, and energy spectra of encode information. Detection of the neutrinos brings unique knowledge about their sources, properties of medium, the space-time they propagated as well as about neutrinos themselves.

Neutrino propagation in matter is vast area of research which covers a variety of different aspects: from conceptual ones to applications. This includes propagation in matter (media) with (i) different properties (unpolarized, polarized, moving, turbulent, fluctuating, with neutrino components, etc.), (ii) different density profiles, and (iii) in different energy regions. The applications cover neutrino propagation in matter of the Earth and the Sun, supernova and relativistic jets as well as neutrinos in the early universe.

The impact of matter on neutrino oscillations was first studied by Wolfenstein in 1978 [1]. He marked that matter suppresses oscillations of the solar neutrinos propagating in the Sun and supernova neutrinos inside a star. He considered hypothetical experiments with neutrinos propagating through 1000 km of rock, something that today is no longer only a thought but actual experimental reality. Later Barger et al. [2] have observed that matter can also enhance oscillations at certain energies. The work of Wolfenstein was expanded upon in papers by Mikheev and Smirnov [3–5], in particular, in the context of the solar neutrino problem. Essentially two new effects have been proposed: the resonant enhancement of neutrino oscillations in matter with constant and nearly constant density and the adiabatic flavor conversion in matter with slowly changing density. It was marked that the first effect can be realized for neutrinos crossing the matter of the Earth. The second one can take place in propagation of solar neutrinos from the dense solar core via the resonance region inside the Sun to the surface with negligible density. This adiabatic flavor transformation, called later the MSW effect, was proposed as a solution of the solar neutrino problem.

Since the appearance of these seminal papers, neutrino flavor evolution in background matter was studied extensively including the treatment of propagation in media which are not consisting simply of matter at rest, but also backgrounds that take on a more general form. For instance, in a thermal field theory approach [6], effects of finite temperature and density can be taken readily into account. If neutrinos are dense enough, new type of effects can arise due to the neutrino background itself, causing a collective behavior in the flavor evolution. This type of effect could have a significant impact on neutrinos in the early universe and in central parts of collapsing stars.

There has been a great progress in treatments of neutrino conversion in matter, both from an analytical and a pure computational points of view. From the analytical side, the description of three-flavor neutrino oscillations in matter is given by a plethora of formulas containing information that may be hard to get a proper grasp of without introducing approximations. Luckily, given the parameter values inferred from experiments, various perturbation theories and series expansions in small parameters can be developed. In this paper we will explain the basic physical effects important for the current and next generation neutrino oscillation experiments and provide the relevant formalism. We present an updated picture of oscillations and conversion given the current knowledge on the neutrino oscillation parameters.

In this paper we focus mainly on aspects related to future experiments with atmospheric and accelerator neutrinos. The main goals of these experiments are to (i) establish the neutrino mass hierarchy, (ii) discover CP violation in the lepton sector and determination of the CP-violating phase, (iii) precisely measure the neutrino parameters, in particular, the deviation of 2-3 mixing from maximal, and (iv) search for sterile neutrinos and new neutrino interactions.

Accelerator and atmospheric neutrinos propagate in the matter of the Earth. Therefore we mainly concentrate on effects of neutrino propagation in the Earth, that is, in usual electrically neutral and nonrelativistic matter. We update existing results on effects of neutrino propagation in view of the recent determination of the 1–3 mixing.

The paper is organized as follows. In Section 2 we consider properties of neutrinos in matter, in particular, mixing in matter and effective masses (eigenvalues of the Hamiltonian); we derive equations which describe the propagation. Section 3 is devoted to various effects relevant to neutrino propagating in the Earth. We consider the properties of the oscillation/conversion probabilities in different channels. In Section 4 we explore the effects of the neutrino mass hierarchy and CP-violating phase on the atmospheric neutrino fluxes and neutrino beams from accelerators. Conclusions and outlook are presented in Section 5.

2. Neutrino Properties in Matter

We will consider the system of 3-flavor neutrinos, 𝜈^𝑇_𝑓≡(𝜈_𝑒,𝜈_𝜇,𝜈_𝜏), mixed in vacuum: 𝜈_𝑓=𝑈_PMNS𝜈_𝑚. (1) Here 𝑈_PMNS is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix [7–9] and 𝜈^𝑇_𝑚≡(𝜈₁,𝜈₂,𝜈₃) is the vector of mass eigenstates with masses 𝑚_𝑖 (𝑖=1,2,3). We will use the standard parameterization of the PMNS matrix, 𝑈_PMNS=𝑈₂₃𝜃₂₃𝐼_𝛿𝑈₁₃𝜃₁₃𝐼^∗_𝛿𝑈₁₂𝜃₁₂, (2) which is the most suitable for describing usual matter effects. In (2) 𝑈_𝑖𝑗(𝜃_𝑖𝑗) are the matrices of rotations in the 𝑖𝑗-planes with angles 𝜃_𝑖𝑗 and 𝐼_𝛿≡diag(1,1,𝑒^𝛿).

In vacuum the flavor evolution of these neutrinos is described by the Schrödinger-like equation 𝑖𝑑𝜈_𝑓=𝑑𝑡𝑀𝑀^†𝜈2𝐸^𝑓, (3) where 𝑀 is the neutrino mass matrix in the flavor basis and 𝐸 is the neutrino energy. Equation (3) is essentially a generalization of the equation 𝐸≈𝑝+𝑚²/2𝐸 for a single ultrarelativistic particle. According to (3), the Hamiltonian in vacuum can be written as 𝐻₀=1𝑈2𝐸^PMNS𝑀²_diag𝑈^†_PMNS, (4) where 𝑀²_diag≡𝑀^†𝑀=diag(𝑚²₁,𝑚²₂,𝑚²₃) and we take the masses 𝑚_𝑖 to be real (the term 𝑝𝐼 is omitted in (4) since it does not produce a phase difference).

2.1. Refraction and Matter Potentials

The effective potential for a neutrino in medium 𝑉_𝑓 can be computed as a forward scattering matrix element 𝑉_𝑓=⟨Ψ|𝐻_int|Ψ⟩. Here Ψ is the wave function of the system of neutrino and medium, and 𝐻_int is the Hamiltonian of interactions.

At low energies, the Hamiltonian 𝐻_int is the effective four-fermion Hamiltonian due to exchange of the 𝑊 and 𝑍 bosons: 𝐻_int=𝐺_𝐹√2𝜈𝛾^𝜇1−𝛾₅𝜈×𝑒𝛾_𝜇𝑔_𝑉+𝑔_𝐴𝛾₅𝑒+𝑝𝛾_𝜇𝑔^𝑝_𝑉+𝑔^𝑝_𝐴𝛾₅𝑝+𝑛𝛾_𝜇𝑔^𝑛_𝑉+𝑔^𝑛_𝐴𝛾₅𝑛, (5) where 𝑔_𝑉 and 𝑔_𝐴 are the vector and axial vector coupling constants.

In the Standard Model the matrix of the potentials in the flavor basis is diagonal: 𝑉_𝑓=diag(𝑉_𝑒,𝑉_𝜇,𝑉_𝜏,0,…).

For medium the matrix elements of vectorial components of vector current are proportional to velocity of particles of medium. The matrix elements of the axial vector current are proportional to spin vector. Therefore for nonrelativistic and unpolarized medium (as well as for an isotropic distribution of ultrarelativistic electrons) only the 𝛾⁰ component of the vector current gives a nonzero result, which is proportional to the number density of the corresponding particles. Furthermore, due to conservation of the vector current (CVC), the couplings 𝑔^𝑝_𝑉 and 𝑔^𝑛_𝑉 can be computed using the neutral current couplings of quarks. Thus, taking into account that, in the Standard Model, the neutral current couplings of electrons and protons are equal and of opposite sign, their NC contributionscancel in electrically neutral medium. As a result, the potential for neutrino flavor 𝜈_𝑎 is 𝑉_𝑎=√2𝐺_𝐹𝛿_𝑎𝑒𝑛_𝑒−12𝑛_𝑛, (6) where 𝑛_𝑒 and 𝑛_𝑛 are the densities of electrons and neutrons, respectively.

Only the difference of potentials has a physical meaning. Contribution of the neutral current scattering to 𝑉 is the same for all active neutrinos. Since 𝑉_𝑎 (𝑎=𝜇,𝜏, or a combination thereof) is due to the neutral current scattering, in a normal medium composed of protons neutrons (nuclei) and electrons, 𝑉_𝜇−𝑉_𝜏=0. Furthermore, the difference of the potentials for 𝜈_𝑒 and 𝜈_𝑎 is due to the charged current scattering of 𝜈_𝑒 on electrons (𝜈_𝑒𝑒→𝜈_𝑒𝑒) [1]: 𝑉=𝑉_𝑒−𝑉_𝑎=√2𝐺_𝐹𝑛_𝑒. (7)

The difference of potentials leads to the appearance of an additional phase difference in the neutrino system: 𝜙_matter≡(𝑉_𝑒−𝑉_𝑎)𝑡≈𝑉𝑥. This determines the refraction length, the distance over which an additional “matter” phase equals 2𝜋: 𝑙₀≡2𝜋𝑉_𝑒−𝑉_𝑎=√2𝜋𝐺_𝐹𝑛_𝑒. (8) Numerically, 𝑙₀=1.6⋅10⁹cm1g/cm³𝑛_𝑒𝑚_𝑁, (9) where 𝑚_𝑁 is the nucleon mass. The corresponding column density 𝑑≡𝑙₀𝑛_𝑒=√2𝜋/𝐺_𝐹 is given by the Fermi coupling constant only.

For antineutrinos the potential has an opposite sign. Being zero in the lowest order the difference of potentials in the 𝜈_𝜇-𝜈_𝜏 system appears at a level of 10⁻⁵𝑉 due to the radiative corrections [10]. Thus in the flavor basis in the lowest order in EW interactions the effect of medium on neutrinos is described by 𝑉=diag(𝑉_𝑒,0,0) with 𝑉_𝑒 given in (7).

The potential has been computed for neutrinos in different types of media, such as polarized or heavily degenerate electrons, in [11–13].

2.2. Evolution Equation, Effective Hamiltonian, and Mixing in Matter

2.2.1. Wolfenstein Equation

In the flavor basis, the Hamiltonian in matter can be obtained by adding the interaction term to the vacuum Hamiltonian in vacuum [1, 3–5, 14, 15]: 𝐻_𝑓=1𝑈2𝐸^PMNS𝑀²_diag𝑈^†_PMNS+𝑉. (10) In (10) we have omitted irrelevant parts of the Hamiltonian proportional to the unit matrix. The Hamiltonian for antineutrinos can be obtained by the substitution 𝑈⟶𝑈^∗,𝑉⟶−𝑉. (11)

There are different derivations of the neutrino evolution equation in matter, in particular, strict derivations starting from the Dirac equation or derivation in the context of quantum field theory (see [16] and references therein).

Although the Hamiltonian 𝐻_𝑓 describes evolution in time, with the connection 𝑥=𝑣𝑡≈𝑥=𝑐𝑡, (12) can be rewritten as 𝑖𝑑𝜈_𝑓/𝑑𝑥=(𝐻₀+𝑉)𝜈_𝑓 with 𝑉=𝑉(𝑥), so it can be used as an evolution equation in space.

Due to the strong hierarchy of Δ𝑚² and the smallness of 1–3 mixing, the results can be qualitatively understood and in many cases quantitatively described by reducing 3𝜈 evolution to 2𝜈 evolution. The reason is that the third neutrino effectively decouples and its effect can be considered as a perturbation. Of course, there are genuine 3𝜈 phenomena such as CP violation, but even in this case the dynamics of evolution can be reduced effectively to the dynamics of evolution of 2𝜈 systems. The evolution equation for two-flavor states, 𝜈^𝑇_𝑓=(𝜈_𝑒,𝜈_𝑎), in matter is 𝑖𝑑𝜈_𝑓=𝑑𝑡Δ𝑚²+14𝐸−cos2𝜃sin2𝜃sin2𝜃cos2𝜃2𝑉_𝑒010−2𝑉_𝑒𝜈_𝑓, (12) where the Hamiltonian is written in symmetric form.

2.3. Mixing and Eigenstates in Matter

The mixing in matter is defined with respect to 𝜈_𝑖𝑚—the eigenstates of the Hamiltonian in matter 𝐻_𝑓.

As usual, the eigenstates are obtained from the equation 𝐻_𝑓𝜈_𝑖𝑚=𝐻_𝑖𝑚𝜈_𝑖𝑚, (13) where 𝐻_𝑖𝑚 are the eigenvalues of 𝐻_𝑓. If the density and therefore 𝐻_𝑓 are constant, 𝜈_𝑖𝑚 correspond to the eigenstates of propagation. Since 𝐻_𝑓≠𝐻₀, the states 𝜈_𝑖𝑚 differ from the mass states, 𝜈_𝑖. For low density 𝑛→0, the vacuum eigenstates are recovered: 𝜈_𝑖𝑚→𝜈_𝑖. If the density, and thus 𝐻_𝑓 change during neutrino propagation, 𝜈_𝑖𝑚 and 𝐻_𝑖𝑚 should be considered as the eigenstates and eigenvalues of the instantaneous Hamiltonian: 𝐻_𝑓=𝐻_𝑓(𝑥), 𝜈_𝑖𝑚=𝜈_𝑖𝑚(𝑥), and 𝐻_𝑖𝑚=𝐻_𝑖𝑚(𝑥). For 𝑛→0 we have 𝐻_𝑖𝑚→𝑚²_𝑖/2𝐸.

The mixing in matter is a generalization of the mixing in vacuum (1). Recall that the mixing matrix in vacuum connects the flavor neutrinos, 𝜈_𝑓, and the massive neutrinos, 𝜈_mass. The latter are the eigenstates of Hamiltonian in vacuum: 𝜈_𝐻=𝜈_mass. Therefore, the mixing matrix in matter is defined as the matrix which relates the flavor states with the eigenstates of the Hamiltonian in matter 𝜈^𝑇_𝐻=(𝜈_1𝑚,𝜈_2𝑚,𝜈_3𝑚): 𝜈_𝑓=𝑈^𝑚𝜈_𝐻. (14)

From (13) we find that 𝜈^†_𝑗𝑚𝐻_𝑓𝜈_𝑖𝑚=𝐻_𝑖𝑚𝛿_𝑗𝑖. (15) Furthermore, the Hamiltonian can be represented in the flavor basis as 𝐻_𝑓=_𝛼𝛽𝐻_𝛼𝛽𝜈_𝛼𝜈^†_𝛽. (16) Inserting this expression as well as the relation 𝜈_𝑗𝑚=𝑈^𝑚∗_𝛼𝑗𝜈_𝛼, which follows from (14), into (15) one obtains _𝛼𝛽𝑈^𝑚∗_𝛼𝑗𝐻_𝛼𝛽𝑈^𝑚_𝛽𝑖=𝐻_𝑖𝑚𝛿_𝑗𝑖 (17) or in matrix form 𝑈^𝑚†𝐻_𝑓𝑈^𝑚=𝐻^diag=diag(𝐻_1𝑚,𝐻_2𝑚,𝐻_3𝑚). Thus, the mixing matrix 𝑈^𝑚 can be found diagonalizing the full Hamiltonian. The columns of the mixing matrix, 𝑈_𝑖≡(𝑈^𝑚_𝑒𝑖,𝑈^𝑚_𝜇𝑖,𝑈^𝑚_𝜏𝑖), are the eigenstates of the Hamiltonian 𝐻_𝑓 which correspond to the eigenvalues 𝐻_𝑖𝑚. Indeed, it follows from (17) that 𝐻_𝑓𝑈^𝑚=𝑈^𝑚𝐻^diag.

Equation (14) can be inverted to 𝜈_𝐻=𝑈^𝑚†𝜈_𝑓, or in components 𝜈_𝑖𝑚=𝑈^𝑚∗_𝛼𝑖𝜈_𝛼,𝛼=𝑒,𝜇,𝜏. According to this, the elements of mixing matrix determine the flavor content of the mass eigenstates so that |𝑈^𝑚_𝛼𝑖|² gives the probability to find 𝜈_𝛼 in a given eigenstate 𝜈_𝑖𝑚. Correspondingly, the elements of the PMNS matrix determine the flavor composition of the mass eigenstates in vacuum.

2.4. Mixing in the Two-Neutrino Case

In the 2𝜈 case, there is single mixing angle in matter 𝜃_𝑚 and the relations between the eigenstates in matter and the flavor states read 𝜈_𝑒=cos𝜃_𝑚𝜈_1𝑚+sin𝜃_𝑚𝜈_2𝑚,𝜈_𝑎=cos𝜃_𝑚𝜈_2𝑚−sin𝜃_𝑚𝜈_1𝑚. (18) The angle 𝜃_𝑚 is obtained by diagonalization of the Hamiltonian (12) (see previous section): sin²2𝜃_𝑚=1𝑅sin²2𝜃,𝑅≡cos2𝜃−2𝑉𝐸Δ𝑚²²+sin²2𝜃, (19) where 𝑅 is the resonance factor. In the limit 𝑉→0, the factor 𝑅→1 and the vacuum mixing are recovered. The difference of eigenvalues 𝐻_𝑖𝑚 equals 𝜔_𝑚≡𝐻_2𝑚−𝐻_1𝑚=Δ𝑚²√2𝐸𝑅. (20) This difference is also called the level splitting or oscillation frequency, which determines the oscillation length: 𝑙_𝑚=2𝜋/𝜔_𝑚 (see Section 3.2).

The matter potential and Δ𝑚² always enter the mixing angle and other dimensionless quantities in the combination 2𝐸𝑉Δ𝑚²=𝑙_𝜈𝑙₀, (21) where 𝑙₀ is the refraction length. This is the origin of the “scaling” behavior of various characteristics of the flavor conversion probabilities. In terms of the mixing angle in matter the Hamiltonian can be rewritten in the following symmetric form: 𝐻_𝑓=𝜔^𝑚2−cos2𝜃_𝑚sin2𝜃_𝑚sin2𝜃_𝑚cos2𝜃_𝑚. (22)

2.4.1. Resonance and Level Crossing

According to (19) the effective mixing parameter in matter, sin²2𝜃_𝑚, depends on the electron density and neutrino energy through the ratio (21) of the oscillation and refraction lengths, 𝑥=𝑙_𝜈/𝑙₀∝𝐸𝑉. The dependence sin²2𝜃_𝑚(𝑉𝐸) for two different values of the vacuum mixing angle, corresponding to angles from the full three-flavor framework, is shown in Figure 1. The dependence of sin²2𝜃_𝑚 on 𝐸 has a resonant character [3]. At 𝑙_𝜈=𝑙₀cos2𝜃 (23) the mixing becomes maximal: sin²2𝜃_𝑚=1 (𝑅=sin²2𝜃). The equality in (23) is called the resonance condition and it can be rewritten as 2𝐸𝑉=Δ𝑚²cos2𝜃. For small vacuum mixing the condition reads the following: oscillation length ≈ refraction length. The physical meaning of the resonance is that the eigenfrequency, which characterizes a system of mixed neutrinos, 𝜔=2𝜋/𝑙_𝜈=Δ𝑚²/2𝐸, coincides with the eigenfrequency of the medium, 2𝜋/𝑙₀=1/𝑉. The resonance condition (23) determines the resonance density 𝑛^𝑅_𝑒=Δ𝑚²2𝐸cos2𝜃√2𝐺_𝐹. (24) The width of resonance on the half of height (in the density scale) is given by 2Δ𝑛^𝑅_𝑒=2𝑛^𝑅_𝑒tan2𝜃. Similarly, for fixed 𝑛_𝑒 one can introduce the resonance energy and the width of resonance in the energy scale. The width can be rewritten as Δ𝑛^𝑅_𝑒=𝑛₀sin2𝜃, where 𝑛₀≡Δ𝑚²/2√2𝐸𝐺_𝐹. When the mixing approaches its maximalvalue: 𝜃→𝜋/4, the resonance shifts to zero density: 𝑛^𝑅_𝑒→0, and the width of the resonance increases converging to the fixed value: Δ𝑛^𝑅_𝑒→𝑛₀.

(a)

(b)

(c)

(d)

Figure 1 

Resonance in neutrino mixing. The dependence of sin22𝜃𝑚𝑖𝑗 on the product 𝑉𝐸 for vacuum mixing: sin22𝜃12=0.851, Δ𝑚221=7.59⋅10−5 eV2 (red) and sin2𝜃13=0.0241, Δ𝑚231=2.47⋅10−3 eV2 (green). The left semiplane corresponds to antineutrinos. The behavior of 𝜃23 with vacuum value sin22𝜃23=0.953 is included for completeness. The dashed lines are the predictions from a strict two-flavor approximation while the solid thin lines are the results of numerical diagonalization of the full three-flavor system. The upper panels show the case of the normal mass hierarchy and the lower panels show the inverted hierarchy.

In a medium with varying density, the layer in which the density changes in the interval 𝑛^𝑅_𝑒±Δ𝑛^𝑅_𝑒 is called the resonance layer. In this layer the angle 𝜃_𝑚 varies in the interval from 𝜋/8 to 3𝜋/8.

For 𝑉≪𝑉_𝑅, the mixing angle is close to the vacuum angle: 𝜃_𝑚≈𝜃, while for 𝑉≫𝑉_𝑅, the angle becomes 𝜃_𝑚≈𝜋/2 and the mixing is strongly suppressed. In the resonance region, the level splitting is minimal [17, 18], therefore the oscillation length, as the function of density, is maximal.

2.5. Mixing of 3 Neutrinos in Matter

To a large extent, knowledge of the eigenstates (mixing parameters) and eigenvalues of the instantaneous Hamiltonian in matter allows the determination of flavor evolution in most of the realistic situations (oscillations in matter of constant density, adiabatic conversion, and strong breaking of adiabaticity). The exact expressions for the eigenstates and eigenvalues [19, 20] are rather complicated and difficult to analyze. Therefore approximate expressions for the mixing angles and eigenvalues are usually used. They can be obtained performing an approximate diagonalization of 𝐻_𝑓 which relies on the strong hierarchy of the mass squared differences: 𝑟_Δ≡Δ𝑚²₂₁Δ𝑚²₃₁≈0.03. (25) Without changing physics, the factor 𝐼_−𝛿 in the mixing matrix can be eliminated by permuting it with 𝑈₁₂ and redefining the state 𝜈₃. Therefore, in what follows, we use 𝑈_PMNS=𝑈₂₃𝐼_𝛿𝑈₁₃𝑈₁₂. Here we will describe the case of normal mass hierarchy: Δ𝑚²₃₁>0,Δ𝑚²₃₂>0. Subtracting from the Hamiltonian the matrix proportional to the unit matrix 𝑚²₁/2𝐸𝐈, we obtain 𝑀²_diag=Δ𝑚²₃₁diag0,𝑟_Δ,1. (26)

2.5.1. Propagation Basis

The propagation basis, 𝜈^∼=(𝜈_𝑒,𝜈^∼₂,𝜈^∼₃)^𝑇, which is most suitable for consideration of the neutrino oscillations in matter, is defined through the relation 𝜈_𝑓=𝑈₂₃𝐼𝜈_𝛿^∼. (27) Since the potential matrix is invariant under 2-3 rotations, the matrix of the potentials is unchanged and the the Hamiltonian in the propagation basis becomes 𝐻^∼=1𝑈2𝐸¹³𝑈₁₂𝑀²_diag𝑈^†₁₂𝑈^†₁₃+𝑉. (28) It does not depend on the 2-3 mixing or CP violation phase, and so the dynamics of the flavor evolution do not depend on 𝛿 and 𝜃₂₃. These parameters appear in the final amplitudes when projecting the flavor states onto propagation-basis states and back onto (27) the neutrino production and detection.

Explicitly, the Hamiltonian 𝐻^∼ can be written as 𝐻^∼=Δ𝑚²₃₁×𝑠2𝐸₂₁₃+𝑠²₁₂𝑐²₁₃𝑟_Δ+2𝑉_𝑒𝐸Δ𝑚²₃₁𝑠₁₂𝑐₁₂𝑐₁₃𝑟_Δ𝑠₁₃𝑐₁₃1−𝑠²₁₂𝑟_Δ…𝑐²₁₂𝑟_Δ−𝑠₁₂𝑐₁₂𝑠₁₃𝑟_Δ……𝑐²₁₃+𝑠²₁₂𝑠²₁₃𝑟_Δ. (29)

Here all the off-diagonal elements contain small parameters 𝑟_Δ and/or 𝑠₁₃. Notice that, for the measured oscillation parameters, 𝑠²₁₃∼𝑟_Δ.

2.5.2. Mixing Angles in Matter

The Hamiltonian in (29) can be diagonalized performing several consecutive rotations which correspond to developing the perturbation theory in 𝑟_Δ. After a 1–3 rotation 𝜈^∼=𝑈₁₃𝜃^𝑚₁₃𝜈^′ (30) over the angle 𝜃^𝑚₁₃ determined by tan2𝜃^𝑚₁₃=sin2𝜃₁₃cos2𝜃₁₃−2𝐸𝑉^′/Δ𝑚²₃₁,where𝑉^′=𝑉1−𝑠²₁₂𝑟_Δ, (31) the 1–3 element of (29) vanishes. The expression (31) differs from that for 2𝜈 mixing in matter by a factor (1−𝑠²₁₂𝑟_Δ), which increases the potential and deviates from 1 by 𝜉≡𝑠²₁₂𝑟_Δ≈10⁻². (32) After this rotation the Hamiltonian in the 𝜈^′ basis (30) becomes 𝐻^′=Δ𝑚²₃₁×ℎ2𝐸₁₁𝑠₁₂𝑐₁₂𝑟_Δcos𝜃^𝑚₁₃−𝜃₁₃0…𝑐²₁₂𝑟_Δ𝑠₁₂𝑐₁₂𝑟_Δsin𝜃^𝑚₁₃−𝜃₁₃……ℎ₃₃, (33)

where ℎ_11,33=12)∓(1+𝜉+𝑥cos2𝜃₁₃(1−𝜉)−𝑥²+sin²2𝜃₁₃(1−𝜉)², (34) and 𝑥≡2𝐸𝑉/Δ𝑚²₃₁. For 𝜉=0, these elements are reduced to the standard 2𝜈 expressions. In the limit of zero density, 𝑥→0, ℎ₁₁=𝜉=𝑠²₁₂𝑟_Δ, and consequently the 11-element of the Hamiltonian equals 𝐻^′₁₁=𝑠²₁₂Δ𝑚²₁₂/2𝐸.

In the lowest 𝑟_Δ approximation one can neglect the nonzero 2-3 element in (33). The state 𝜈^′₃ then decouples and the problem is reduced to a two-neutrino problem for (𝜈^′₁,𝜈^′₂). The eigenvalue of this decoupled state equals 𝐻_3𝑚≈Δ𝑚²₃₁ℎ2𝐸³³,ℎ₃₃≥1. (35) The diagonalization of the remaining 1-2 submatrix is given by rotation 𝜈^′=𝑈₁₂𝜃^𝑚₁₂𝜈_𝑚, (36) where 𝜃^𝑚₁₂ is determined by tan2𝜃^𝑚₁₂=sin2𝜃₁₂𝑟_Δcos𝜃^𝑚₁₃−𝜃₁₃𝑐²₁₂𝑟_Δ−ℎ₁₁. (37) Here ℎ₁₁ and 𝜃^𝑚₁₃ are defined in (34) and (31), respectively. The eigenvalues equal 𝐻_1𝑚,2𝑚=Δ𝑚²₃₁𝑐4𝐸²¹²𝑟_Δ+ℎ₁₁∓𝑐²₁₂𝑟_Δ−ℎ₁₁²+sin²2𝜃₁₂𝑟²_Δcos²𝜃^𝑚₁₃−𝜃₁₃. (38)

According to this diagonalization procedure in the lowest order in 𝑟_Δ the mixing matrix in matter is given by 𝑈^𝑚=𝑈₂₃𝜃₂₃𝐼_𝛿𝑈₁₃𝜃^𝑚₁₃𝑈₁₂𝜃^𝑚₁₂, (39) where mixing angles 𝜃^𝑚₁₂ and 𝜃^𝑚₁₃ are determined in (37) and (31), respectively. The 2-3 angle and the CP violation phase are not modified by matter in this approximation. The eigenvalues 𝐻_1𝑚 and 𝐻_2𝑚 are given in (38) and 𝐻_3𝑚 is determined by (35).

The 2-3 element of matrix (33) vanishes after additional 2-3 rotation by an angle 𝜃^′₂₃∼𝑟_Δ: tan2𝜃^′₂₃=sin2𝜃₁₂𝑟_Δsin𝜃^𝑚₁₃−𝜃₁₃ℎ₃₃−𝑐²₁₂𝑟_Δ, (40) which produces corrections of the next order in 𝑟_Δ. With an additional 2-3 rotation the mixing matrix becomes 𝑈^𝑚=𝑈₂₃𝜃₂₃𝐼_𝛿𝑈₁₃𝜃^𝑚₁₃𝑈₁₂𝜃^𝑚₁₂𝑈₂₃𝜃^′₂₃≈𝑈₂₃𝜃^𝑚₂₃𝐼_𝛿^𝑚𝑈₁₃𝜃^𝑚₁₃𝑈₁₂𝜃^𝑚₁₂, (41) where 𝑈₂₃𝜃^𝑚₂₃𝐼^𝑚_𝛿=𝑈₂₃𝜃₂₃𝐼_𝛿𝑈₂₃𝜃₂₃, (42) and the last 2-3 rotation is on the angle 𝜃₂₃ determined through sin𝜃₂₃=sin𝜃^′₂₃/cos𝜃^𝑚₁₃. The expression on the RH of (41) is obtained by reducing the expression on the LH side to the standard form by permuting the correction matrix 𝑈₂₃(𝜃^′₂₃). According to (42), it is this matrix that leads to the modification of 2-3 mixing and CP phase in matter. From (42) one finds sin𝛿^𝑚sin2𝜃^𝑚₂₃=sin𝛿sin2𝜃₂₃, (43) that is, the combination sin𝛿sin2𝜃₂₃ is invariant under inclusion of matter effects. Furthermore, 𝜃^𝑚₂₃≈𝜃₂₃ and 𝛿^𝑚≈𝛿 up to corrections of the order 𝑂(𝑟_Δ). The results described here allowing understand the behavior of the mixing parameters sin²2𝜃_𝑚𝑖𝑗 in the 𝐸𝑉 region of the 1–3 resonance and above it (see Figure 1).

In Figure 2 we present dependence of the flavor content of the neutrino eigenstates on the potential. The energy level scheme, the dependence of the eigenvalues 𝐻_𝑖𝑚 on matter density, is shown in Figure 3. The energy levels in matter do not depend on 𝛿 or 𝜃₂₃, but they do depend on the 1–3 and 1-2 mixing.

Figure 2 

The flavor contents of the eigenstates of the Hamiltonian in matter as functions of 𝐸𝑉. The vertical width of the band is taken to be 1, then the vertical sizes of the colored parts give |𝑈𝑒𝑖|2 (red), and |𝑈𝜇𝑖|2 (green), |𝑈𝜏𝑖|2 (blue). The right and left panels correspond to neutrinos and antineutrinos, respectively. We take the best fit values of [21] with 𝛿=0. Variations of 𝛿 change the relative 𝜈𝜇 and 𝜈𝜏contents. The dashed red line shows a shift of border between 𝜈𝜇 and 𝜈𝜏 flavors for 𝛿=𝜋. The upper (lower) panel corresponds to normal (inverted) mass ordering.

(a)

(b)

Figure 3 

The energy level scheme. We here show the dependence of the eigenvalues of the Hamiltonian in matter on 𝐸𝑉. Note that we are plotting 2𝐸𝐻𝑖𝑚, which goes to Δ𝑚2𝑖1 for low 𝑉𝐸. The left (right) panel corresponds to normal (inverted) mass ordering.

In the case of normal mass hierarchy, there are two resonances (level crossings) whose location is defined as the density (energy) at which the mixing in a given channel becomes maximal.(1)The H resonance, in the 𝜈_𝑒-𝜈^′_𝜏 channel, is associated to the 1–3 mixing and large mass splitting. According to (31) 𝜃^𝑚₁₃=𝜋/4 at 𝑉^𝑅₁₃=cos2𝜃₁₃1−𝑠²₁₂𝑟_ΔΔ𝑚²₃₁.2𝐸 (44)(2)The L resonance at low densities is associated to the small mass splitting and 1-2 mixing. It appears in the 𝜈^′_𝑒-𝜈^′_𝜇 channel, where 𝜈^′_𝑒 and 𝜈_𝑒 differ by small (at low densities) rotation given by an angle ∼𝜃₁₃ (see (31)). According to (37) the position of the L-resonance, 𝜃^𝑚₁₂=𝜋/4, is given by 𝑐²₁₂𝑟_Δ=ℎ₁₁, where ℎ₁₁ is defined in (34). This leads to 𝑉^𝑅₁₂=cos2𝜃₁₂Δ𝑚²₂₁12𝐸𝑐²₁₃. (45)

For antineutrinos (𝑉𝐸<0 in Figure 3), the oscillation parameters in matter can be obtained from the neutrino parameters taking 𝑉→−𝑉 and 𝛿→−𝛿. The mixing pattern and level scheme for neutrinos and antineutrinos are different both due to the possible fundamental violation of CP invariance and the sign of matter effect. Matter violates CP invariance and the origin of this violation stems from the fact that usual matter is CP asymmetric; in particular, there are electrons in the medium but no positrons.

In the case of normal mass hierarchy there are no antineutrino resonances (level crossings), and with the increase of density (energy) the eigenvalues have the following asymptotic limits: 𝐻_1𝑚⟶−𝑉,𝐻_2𝑚⟶Δ𝑚²₂₁𝑐²₁₂2𝐸_𝜈,𝐻_3𝑚⟶Δ𝑚²₃₁𝑐²₁₃2𝐸_𝜈. (46)

3. Effects of Neutrino Propagation in Different Media

3.1. The Evolution Matrix

The evolution matrix, 𝑆(𝑡,𝑡₀), is defined as the matrix which gives the wave function of the neutrino system 𝜈(𝑡) at an arbitrary moment 𝑡 once it is known in the initial moment 𝑡₀: 𝜈(𝑡)=𝑆𝑡,𝑡₀𝜈𝑡₀. (47) Inserting this expression in the evolution equation (12), we find that 𝑆(𝑡,𝑡₀) satisfies the same evolution equation as 𝜈(𝑡): 𝑖𝑑𝑆𝑑𝑡=𝐻𝑆. (48) The elements 𝑆(𝑡,𝑡₀)_𝛼𝛽 of this matrix are the amplitudes of 𝜈_𝛽→𝜈_𝛼 transitions: 𝑆(𝑡,𝑡₀)_𝛼𝛽≡𝐴(𝜈_𝛽→𝜈_𝛼). The transition probability equals 𝑃_𝛼𝛽=|𝑆(𝑡,𝑡₀)_𝛼𝛽|². The unitarity of the evolution matrix, 𝑆^†𝑆=𝐼, leads to the following relations between the amplitudes (matrix elements): 𝑆_𝛼𝛼²+𝑆_𝛽𝛼²𝑆=1,_𝛽𝛽2+𝑆_𝛼𝛽²𝑆=1,_∗𝛼𝛼𝑆_𝛼𝛽+𝑆^∗_𝛽𝛼𝑆_𝛽𝛽𝑆=0,_∗𝛼𝛽𝑆_𝛼𝛼+𝑆^∗_𝛽𝛽𝑆_𝛽𝛼=0. (49) The first and the second equations express the fact that the total probability of transition of 𝜈_𝛼 to everything is one, and the same holds for 𝜈_𝛽. The third and fourth equations are satisfied if 𝑆_𝛼𝛼=𝑆^∗_𝛽𝛽,𝑆_𝛽𝛼=−𝑆^∗_𝛼𝛽. (50) With these relations the evolution matrix can be parametrized as 𝑆=𝛼𝛽−𝛽^∗𝛼^∗,|𝛼|²+𝛽²=1. (51)

The Hamiltonian for a 2𝜈 system is T symmetric in vacuum as well as in medium with constant density. In medium with varying density the T symmetry is realized if the potential is symmetric. Under T transformations 𝑆_𝛽𝛼→𝑆_𝛼𝛽, and the diagonal elements 𝑆_𝛼𝛼 do not change. Therefore according to (50) the T invariance implies that 𝑆_𝛽𝛼=−𝑆^∗_𝛽𝛼, or Re𝑆_𝛽𝛼=0; that is, the off-diagonal elements of the 𝑆 matrix are pure imaginary.

3.2. Neutrino Oscillations in Matter with Constant Density

In a medium with constant density and therefore constant potential the mixing is constant: 𝜃_𝑚(𝐸,𝑛)=const. Consequently, the flavor composition of the eigenstates does not change and the eigenvalues 𝐻_𝑖𝑚 of the full Hamiltonian are constant. The two-neutrino evolution equation in matter of constant density can be written in the matter eigenstate basis as 𝑖𝑑𝜈_𝑚𝑑𝑥=𝐻^diag𝜈_𝑚, (52) where 𝐻^diag≡diag(𝐻_1𝑚,𝐻_2𝑚). This system of equations splits and the integration is trivial, 𝜈_𝑖𝑚(𝑡)=𝑒^{−𝑖𝐻𝑖𝑚𝑡}𝜈_𝑖𝑚(0). The corresponding 𝑆 matrix is diagonal: 𝑆^∼()=𝑒𝑥,0^{𝑖𝜙𝑚(𝑥)}00𝑒^{−𝑖𝜙𝑚(𝑥)}, (53) where 𝜙_𝑚≡(1/2)𝜔^𝑚𝑥 is the half-oscillation phase in matter and a matrix proportional to the unit matrix has been subtracted from the Hamiltonian.

The 𝑆 matrix in the flavor basis (𝜈_𝑒,𝜈_𝑎) is therefore 𝑆(𝑥,0)=𝑈𝑆^𝑚∼(𝑥,0)𝑈^𝑚†=cos𝜙_𝑚+𝑖cos2𝜃_𝑚sin𝜙_𝑚−𝑖sin2𝜃_𝑚sin𝜙_𝑚−𝑖sin2𝜃_𝑚sin𝜙_𝑚cos𝜙−𝑖cos2𝜃_𝑚sin𝜙_𝑚. (54) Then, for the transition probability, we can immediately deduce 𝑃_𝑒𝑎=𝑆_𝑒𝑎²=sin²2𝜃_𝑚sin²𝜙_𝑚, (55) where 𝜙_𝑚=𝜋𝑥/𝑙_𝑚, with 𝑙_𝑚=2𝜋𝐻_2𝑚−𝐻_1𝑚=𝑙_𝜈√𝑅 (56) being the oscillation length in matter. The dependence of 𝑙_𝑚 on the neutrino energy is shown in Figure 4. For small energies, 𝑉𝐸≪Δ𝑚², the length 𝑙_𝑚≃𝑙_𝜈. It then increases with energy and for small 𝜃 reaches the maximum 𝑙_𝑚^max=𝑙₀/sin2𝜃 at 𝐸^max=𝐸_𝑅/cos²2𝜃, that is, above the resonance energy. For 𝐸→∞ the oscillation length converges to the refraction length 𝑙_𝑚→𝑙₀.

Figure 4 

Dependence of the oscillation length in matter in units of the refraction length on neutrino energy for two different mixing angles in vacuum.

A useful representation of the 𝑆 matrix for a layer with constant density follows from (54): 𝑆(𝑥,0)=cos𝜙_𝑚𝐼−𝑖sin𝜙_𝑚(𝜎⋅𝐧), (57) where 𝜎 is a vector containing the Pauli matrices and 𝐧≡(sin2𝜃_𝑚,0,−cos2𝜃_𝑚).

The dynamics of neutrino flavor evolution in uniform matter are the same as in vacuum, that is, it has a character of oscillations. However, the oscillation parameters (length and depth) differ from those in vacuum. They are now determined by the mixing and effective energy splitting in matter: sin²2𝜃→sin²2𝜃_𝑚, 𝑙_𝜈→𝑙_𝑚.

3.3. Neutrino Polarization Vectors and Graphic Representation

It is illuminating to consider the dynamics of transitions in different media using graphic representation [22–24]. Consider the two-flavor neutrino state, 𝜓^𝑇=(𝜓_𝑒,𝜓_𝑎). The corresponding Hamiltonian can be written as 𝐻=(𝐇⋅𝜎), (58) where 𝜎=(𝜎₁,𝜎₂,𝜎₃), 𝐇 is the Hamiltonian vector 𝐇≡(2𝜋/𝑙_𝑚)⋅(sin2𝜃_𝑚,0,cos2𝜃_𝑚), and 𝑙_𝑚=2𝜋/Δ𝐻_𝑚 is the oscillation length. The evolution equation then becomes 𝑖𝜓̇=(𝐇⋅𝜎)𝜓. (59) Let us define the polarization vector 𝐏𝐏≡𝜓^†𝜎2𝜓. (60) In terms of the wave functions, the components of 𝐏 equal 𝑃_𝑥,𝑃_𝑦,𝑃_𝑧=Re𝜓^∗_𝑒𝜓_𝑎,Im𝜓^∗_𝑒𝜓_𝑎,12𝜓_𝑒²−𝜓_𝑎². (61) The 𝑧-component can be rewritten as 𝑃_𝑧=|𝜓_𝑒|²−1/2; therefore 𝑃_𝑒≡|𝜓_𝑒|²=𝑃_𝑧+1/2 and from unitarity 𝑃_𝑎≡|𝜓_𝑎|²=1/2−𝑃_𝑧. Hence, 𝑃_𝑧 determines the probabilities to find the neutrino in a given flavor state. The flavor evolution of the neutrino state corresponds to a motion of the polarization vector in the flavor space. The evolution equation for 𝐏 can be obtained by differentiating (60) with respect to time and inserting 𝜓̇ and 𝜓̇^† from evolution equation (59). As a result, one finds that 𝑑𝑑𝑡𝐏=𝐇×𝐏. (62) If 𝐇 is identified with the strength of a magnetic field, the equation of motion (62) coincides with the equation of motion for the spin of electron in the magnetic field. According to this equation 𝐏 precesses around 𝐇.

With an increase of the oscillation phase 𝜙 (see Figure 5) the vector 𝐏 moves on the surface of the cone having axis 𝐇. The cone angle 𝜃_𝑎, the angle between 𝐏 and 𝐇, depends both on the mixing angle and on the initial state, and, in general, on changes in process of evolution, for example, if the neutrino evolves through several layers of different density. If the initial state is 𝜈_𝑒, the angle equals 𝜃_𝑎=2𝜃_𝑚 in the initial moment.

Figure 5 

Graphic representation of neutrino oscillations. Neutrino polarization vector 𝐏 precesses around the Hamiltonian vector 𝐇 (or the vector of eigenstates of the Hamiltonian). The angle between 𝐏 and 𝐇 is given by the cone angle 𝜃𝑎, and the direction of axis of the cone is determined by the mixing angle in matter 2𝜃𝑚.

The components of the polarization vector 𝐏 are nothing but the elements of the density matrix 𝜌=𝜎⋅𝐏. The evolution equation for 𝜌 can be obtained from (62) 𝑖𝑑𝜌=.𝑑𝑡𝐻,𝜌 (63) The diagonal elements of the density matrix give the probabilities to find the neutrino in the corresponding flavor state.

3.4. Resonance Enhancement of Oscillations

Suppose a source produces flux of neutrinos in the flavor state 𝜈_𝜇 with continuous energy spectrum. This flux then traverses a layer of length 𝐿 with constant density 𝑛_𝑒. At the end of this layer a detector measures the 𝜈_𝑒 component of the flux, so that oscillation effect is given by the transition probability 𝑃_𝜇𝑒. In Figure 6 we show dependence of this probability on energy for thin and thick layers. The oscillatory curves are inscribed in the resonance envelope sin²2𝜃_𝑚. The period of the oscillatory curve decreases with the length 𝐿. At the resonance energy, 𝐸_𝑅=Δ𝑚²cos2𝜃=2𝑉Δ𝑚²cos2𝜃2√2𝐺_𝐹𝑛_𝑒, (64) oscillations proceed with maximal depths. Oscillations are enhanced up to 𝑃>1/2 in the resonance range (𝐸_𝑅±Δ𝐸_𝑅) where Δ𝐸_𝑅=tan2𝜃𝐸_𝑅 (see Section 2.4). This effect was called the resonance enhancement of oscillations.

Figure 6 

Resonance enhancement of neutrino oscillations in matter with constant density. Shown is the dependence of the transition probability 𝜈𝑒→𝜈𝜇 on energy for sin2𝜃13=0.0241 for three different sizes of layers: 𝐿=3𝑙𝑚(𝐸𝑅)/2, 𝑙𝑚(𝐸𝑅)/2, and 𝑙𝑚(𝐸𝑅)/6. The shaded area shows the resonance envelope: sin22𝜃𝑚(𝐸).

3.5. Three-Neutrino Oscillations in Matter with Constant Density

The oscillation probabilities in matter with constant density have the same form as oscillation probabilities in vacuum and the generalization of (53) is straightforward. In the basis of the eigenstates of the Hamiltonian the evolution matrix equals 𝑆^∼()=𝑒𝑥,0^{−2𝑖𝜙1𝑚(𝑥)}000𝑒^{−2𝑖𝜙2𝑚(𝑥)}000𝑒^{−2𝑖𝜙3𝑚(𝑥)}, (65) and for the elements of the 𝑆 matrix in the flavor basis we obtain 𝑆_𝛼𝛽=∑_𝑖𝑈^𝑚∗_𝛼𝑖𝑈^𝑚_𝛽𝑖𝑒^{−2𝑖𝜙𝑚𝑖(𝑥)}. Removing 𝑒^{−2𝑖𝜙2𝑚} and using the unitarity of the mixing matrix in matter we have 𝑆_𝛼𝛽=𝛿_𝛼𝛽+2𝑖𝑒^{𝜙𝑚21(𝑥)}𝑈^𝑚∗_𝛼2𝑈^𝑚_𝛽2sin𝜙^𝑚₂₁(𝑥)−2𝑖𝑒^{−𝑖𝜙𝑚32(𝑥)}𝑈^𝑚∗_𝛼3𝑈^𝑚_𝛽3sin𝜙^𝑚₃₂(𝑥). (66) In particular, for the amplitudes in matter involving only 𝜈_𝑒 and 𝜈_𝜇, we obtain 𝑆^cst_𝑒𝜇=2𝑖𝑒^{𝑖𝜙𝑚21}𝑈^𝑚_𝑒1𝑈^𝑚∗_𝜇1sin𝜙^𝑚₂₁−𝑒^{−𝑖𝜙𝑚31}𝑈^𝑚_𝑒3𝑈^𝑚∗_𝜇3sin𝜙^𝑚₃₂,𝑆^cst_𝜇𝜇=1+2𝑖𝑒^{𝑖𝜙𝑚21}𝑈^𝑚_𝜇1²sin𝜙^𝑚₂₁−2𝑖𝑒^{−𝑖𝜙𝑚32}𝑈^𝑚_𝜇3²sin𝜙^𝑚₃₂,𝑆^cst_𝑒𝑒=1+2𝑖𝑒^{𝑖𝜙𝑚21}cos²𝜃^𝑚₁₃cos²𝜃^𝑚₁₂sin𝜙^𝑚₂₁−2𝑖𝑒^{−𝑖𝜙𝑚32}sin²𝜃^𝑚₁₃sin𝜙^𝑚₃₂. (67)

3.6. Propagation in a Medium with Varying Density and the MSW Effect

3.6.1. Equation for the Instantaneous Eigenvalues and the Adiabaticity Condition

In nonuniform media, the density changes along neutrino trajectory: 𝑛_𝑒=𝑛_𝑒(𝑡). Correspondingly, the Hamiltonian of system depends on time, 𝐻=𝐻(𝑡), and therefore the mixing angle changes during neutrino propagation: 𝜃_𝑚=𝜃_𝑚(𝑛_𝑒(𝑡)). Furthermore, the eigenstates of the instantaneous Hamiltonian, 𝜈_1𝑚 and 𝜈_2𝑚, are no longer the “eigenstates” of propagation. Indeed, inserting 𝜈_𝑓=𝑈(𝜃_𝑚)𝜈_𝑚 in the equation for the flavor states (c.f., (3)) we obtain the evolution equation for eigenstates 𝜈_𝑖𝑚: 𝑖𝑑𝜈_𝑚=𝐻𝑑𝑡^1𝑚−𝑖𝜃̇_𝑚𝑖𝜃̇_𝑚𝐻_2𝑚𝜈_𝑚, (68) where 𝜃̇_𝑚≡𝑑𝜃_𝑚/𝑑𝑡. The Hamiltonian for 𝜈_𝑖𝑚 (68) is nondiagonal and, consequently, the transitions 𝜈_1𝑚↔𝜈_2𝑚 occur. The rate of these transitions is given by the speed with which the mixing angle changes with time. According to (68) [3, 25], |𝜃̇_𝑚| determines the energy of transition 𝜈_1𝑚↔𝜈_2𝑚 and |𝐻_2𝑚−𝐻_1𝑚| gives the energy gap between the levels.

The off-diagonal elements of the evolution equation (68) can be neglected if 𝜃̇_𝑚 is much smaller than other energy scales in the system. The difference of the diagonal elements of the Hamiltonian is, in fact, the only other energy quantity and therefore the criterion for smallness of 𝜃̇_𝑚 is 𝜃̇_𝑚≪𝐻_2𝑚−𝐻_1𝑚. (69) This inequality implies a slow enough change of density and is called the adiabaticity condition. Defining the adiabaticity parameter [22, 25] as 𝜃̇𝛾≡^𝑚𝐻_2𝑚−𝐻_1𝑚, (70) the adiabaticity condition can be written as 𝛾≪1.

For small mixing angle, the adiabaticity condition is most crucial in the resonance layer where the level splitting is small and the mixing angle changes rapidly. In the resonance point, it takes the physically transparent form [3]: Δ𝑟_𝑅>𝑙^𝑅_𝑚, where 𝑙^𝑅_𝑚≡𝑙_𝜈/sin2𝜃 is the oscillation length in resonance, and Δ𝑟_𝑅≡(𝑛_𝑒/(𝑑𝑛_𝑒/𝑑𝑟))_𝑅tan2𝜃 is the spatial width of the resonance layer. According to this condition at least one oscillation length should be obtained within the resonance layer.

In the case of large vacuum mixing, the point of maximal adiabaticity violation [26, 27] is shifted to density, 𝑛_𝑒(𝑎𝑣), larger than the resonance density: 𝑛_𝑒(𝑎𝑣)→𝑛_𝐵>𝑛_𝑅. Here 𝑛_𝐵=Δ𝑚²/2√2𝐺_𝐹𝐸 is the density at the border of resonance layer for maximal mixing. Outside the resonance and in the nonresonant channel, the adiabaticity condition has been considered in [28, 29].

3.7. Adiabatic Conversion and the MSW Effect

If the adiabaticity condition is fulfilled and 𝜃̇_𝑚 can be neglected, the Hamiltonian for the eigenstates becomes diagonal. Consequently, the equations for the instantaneous eigenstates 𝜈_𝑖𝑚 split as in the constant density case. The instantaneous eigenvalues evolve independently, but the flavor content of the eigenstates changes according to the change of mixing in matter. This is the essence of the adiabatic approximation; we neglect 𝜃̇_𝑚 in evolution equation but do not neglect the dependence of 𝜃_𝑚 on density. The solution can be obtained immediately as 𝑆^∼()=𝑒𝑥,0^𝑖𝜙𝑚00𝑒^{−𝑖𝜙𝑚},𝜙_𝑚=12^𝑥₀𝐻_2𝑚−𝐻_1𝑚𝑑𝑥^′, (71) in symmetric form. The only difference from the constant density case is that the eigenvalues now depend on time and therefore integration appears in the phase factors.

The evolution matrix in the flavor basis can be obtained by projecting back from the eigenstate basis to the flavor basis with the mixing matrices corresponding to initial and final densities: 𝑆_𝑓(𝑥,0)=𝑈^𝑚(𝑡)𝑆^∼(𝑥,0)𝑈^𝑚†(0)=𝑐_𝑚𝑐⁰_𝑚𝑒^𝑖𝜙𝑚+𝑠_𝑚𝑠⁰_𝑚𝑒^{−𝑖𝜙𝑚}−𝑐_𝑚𝑠⁰_𝑚𝑒^𝑖𝜙𝑚+𝑠_𝑚𝑐⁰_𝑚𝑒^{−𝑖𝜙𝑚}−𝑠_𝑚𝑐⁰_𝑚𝑒^𝑖𝜙𝑚+𝑐_𝑚𝑠⁰_𝑚𝑒^{−𝑖𝜙𝑚}𝑠_𝑚𝑠⁰_𝑚𝑒^𝑖𝜙𝑚+𝑐_𝑚𝑐⁰_𝑚𝑒^{−𝑖𝜙𝑚}. (72) From this procedure we find, for example, the probability of 𝜈_𝑒-𝜈_𝑒 transition 𝑃_𝑒𝑒=𝑆_𝑓(𝑥,0)_𝑒𝑒²=121+cos2𝜃_𝑚(𝑥)cos2𝜃_𝑚(0+1)2sin2𝜃_𝑚(𝑥)sin2𝜃_𝑚(0)cos2𝜙_𝑚(𝑥). (73) If the initial and final densities coincide, as in the case of neutrinos crossing the Earth, we obtain the same formulas as in constant density case: 𝑃_𝛼𝛽=_𝑖𝑈^𝑚_𝛼𝑖(0)𝑈^𝑚∗_𝛽𝑖(0)𝑒^{−𝑖𝜙𝑖𝑚(𝑡,0)} (74) with the mixing angle taken at the borders (initial or final state). In particular, the survival probability equals 𝑃_𝛼𝛼=1−sin²2𝜃_𝑚(0)sin²𝜙_𝑚(𝑥).

Averaging over the phase, which means that the contributions from 𝜈₁ and 𝜈₂add incoherently, gives 𝑃=cos𝜃_𝑚cos𝜃⁰_𝑚²+sin𝜃_𝑚sin𝜃⁰_𝑚²=sin²𝜃_𝑚+cos2𝜃_𝑚cos²𝜃⁰_𝑚. (75) The mixing in the neutrino production point 𝜃⁰_𝑚 is determined by density in this point, 𝑛⁰_𝑒, and the resonance density. Consequently, the picture of the conversion depends on how far from the resonance layer (in the density scale) a neutrino is produced. Strong transitions occur if the initial and final mixings differ substantially, which is realized when the initial density is much above the resonance density and the final one is below the resonance density and therefore neutrinos cross the resonance layer.

According to (73) the oscillation depth equals 𝐷=|sin2𝜃_𝑚sin2𝜃⁰_𝑚|. Both the averaged probability (75) and the depth (73) are determined by the initial and final densities and do not depend on the density distribution along the neutrino trajectory. Essentially they are determined by the ratios 𝑦≡𝑛/𝑛_𝑅 in the initial and final moments. This is a manifestation of the universality of the adiabatic approximation result.

In contrast, the phase does depend on the density distribution and the period of oscillations (the latter is given by the oscillation length in matter). So, it is the phase that encodes information about the density distribution.

The probability depends on 𝑡 via the phase 𝜙_𝑚(𝑡) and also via the mixing angle 𝜃_𝑚(𝑡). Two degrees of freedom are operative and 𝑃 dependence on time is an interplay of two effects: oscillations, associated with the phase 𝜙_𝑚(𝑡), and the adiabatic conversion related to change of 𝜃_𝑚. Depending on initial condition 𝑛⁰_𝑒, the relative importance of the two effects is different. If neutrinos are produced far above the resonance, 𝑛⁰_𝑒≫𝑛^𝑅_𝑒, the initial mixing is strongly suppressed, 𝜃⁰_𝑚≈𝜋/2. Consequently, the neutrino state, for example, 𝜈_𝑒, consists mainly of one eigenstate, 𝜈_2𝑚, and furthermore, one-flavor 𝜈_𝑒 dominates in 𝜈_2𝑚. Since the admixture of the second eigenstate is very small, oscillations (interference effects) are strongly suppressed. Thus, here the nonoscillatory flavor transition occurs when the flavor of whole state (which nearly coincides with 𝜈_2𝑚) follows the density change. At zero density 𝜈_2𝑚=𝜈₂, and therefore the probability to find the electron neutrino (survival probability) equals [3] 𝑃=𝜈_𝑒∣𝜈(𝑡)²≈𝜈_𝑒∣𝜈_2𝑚(𝑡)²=𝜈_𝑒∣𝜈₂²≈sin²𝜃. (76) The final probability, 𝑃=sin²𝜃, is the feature of the nonoscillatory transition (as pure adiabatic conversion). Deviation from this value indicates the presence of oscillations; see (73).

If neutrinos are produced not too far from resonance, for example, at 𝑛⁰_𝑒>𝑛^𝑅_𝑒, the initial mixing is not suppressed. Although 𝜈_2𝑚 is the main component of the neutrino state, the second eigenstate, 𝜈_1𝑚, has appreciable admixture; the flavor mixing in the neutrino eigenstates is significant, and the interference effect is not suppressed. Here we deal with the interplay of the adiabatic conversion and oscillations.

Production in the resonance is a special case; if 𝜃⁰_𝑚=45^∘, the averaged probability equals 𝑃=1/2 independently of the final mixing. This feature is important for determining the oscillation parameters. Strong transitions (𝑃>1/2) occur when neutrinos cross resonance layer. These features are realized for solar neutrinos when propagating from their production region inside the Sun to the surface of the Sun. The adiabatic propagation occurs also in a single layer of the Earth (e.g., in the mantle).

3.8. Adiabaticity Violation

For most of applications the adiabaticity is either well satisfied (neutrinos in the Sun or supernovae), or maximally broken due to sharp (instantaneous) density change (neutrinos in the Earth, neutrinos crossing the shock wave fronts in supernova). In the former case the evolution is described by the adiabatic formulas. In the latter case description is also simple; one just needs to match the flavor conditions at the borders between layers, find the flavor state before the density jump, and then use it as an initial state for the evolution after the jump. The intermediate case of the adiabaticity breaking can be realized for neutrinos in the mantle of the Earth, for high energy neutrinos propagating in the Sun (neutrinos from annihilation of hypothetical WIMPs) or for sterile neutrinos with very small mixing.

If the density changes rapidly, 𝜃̇_𝑚 is not negligible in (68) and the adiabaticity condition (70) is not satisfied. The transitions 𝜈_1𝑚↔𝜈_2𝑚 become noticeable and therefore the admixtures of the eigenstates in a given propagating state change. The 𝑆 matrix in the flavor basis is given by 𝑆_𝑓(𝑥,0)=𝑈^𝑚(𝑡)𝑆^∼(𝑥,0)𝑈^𝑚†(0)=𝑈^𝑚(𝑡)𝑆₁₁−𝑆^∗₂₁𝑆₂₁𝑆^∗₁₁𝑈^𝑚†(0), (77) where 𝑆^∼ is the evolution matrix in the basis of instantaneous eigenstates. Then the 𝜈_𝑒-𝜈_𝑒 transition probability 𝑃_𝑒𝑒≡|𝑆_𝑓(𝑥,0)_𝑒𝑒|² equals 𝑃_𝑒𝑒=121+cos2𝜃_𝑚(𝑡)cos2𝜃_𝑚(0)−𝑃₂₁cos2𝜃_𝑚(𝑡)cos2𝜃_𝑚(0)+𝑃_int, (78) where 𝑃₂₁≡|𝑆₂₁|² is the probability of 𝜈_2𝑚→𝜈_1𝑚 transitions and 𝑃_int is an interference term 𝑃_int=14sin2𝜃_𝑚(𝑡)sin2𝜃_𝑚(0)𝑆²₁₁+𝑆^∗2₁₁+𝑆²₂₁+𝑆^∗2₂₁+12sin2𝜃_𝑚(0)−2𝜃_𝑚(𝑥)𝑆₁₁𝑆^∗₂₁+𝑆^∗₁₁𝑆₂₁, (79) which depends on the oscillation phase. The averaged probability (𝑃_int=0) equals [30] 𝑃_𝑒𝑒=12+12−𝑃₂₁cos2𝜃_𝑚(𝑡)cos2𝜃_𝑚(0). (80) If the initial density is much larger than the resonance density, then 𝜃_𝑚(0)≈𝜋/2 and cos2𝜃_𝑚(0)=−1. In this case the averaged probability can be rewritten as 𝑃_𝑒𝑒=sin²𝜃_𝑚(𝑡)+𝑃₂₁cos2𝜃_𝑚(𝑡). (81) Violation of adiabaticity weakens transitions if cos2𝜃_𝑚(𝑡)>0, thus leading to an increase of the survival probability. In the adiabatic case 𝑆₁₁=𝑒^𝑖𝜙𝑚, 𝑆₂₁=0, and therefore 𝑆²₁₁+𝑆^∗2₁₁=2cos2𝜙_𝑚(𝑥), so that (78) is reduced to (73).

In the graphic representation (Figure 5), the neutrino vector moves on the surface of the cone (phase change) and the axis of the cone rotates according to the density change. The cone angle 𝜃_𝑎 changes as a result of violation of the adiabaticity).

There are different approaches to compute the flop probability 𝑃₂₁. In the adiabatic regime the probability of transition between the eigenstates is exponentially suppressed 𝑃₁₂∼exp(−𝜋/2𝛾) with 𝛾 given in (70) [30, 31]. One can consider such a transition as penetration through a barrier of height 𝐻_2𝑚−𝐻_1𝑚 by a system with the kinetic energy 𝑑𝜃_𝑚/𝑑𝑡. This leads to the Landau-Zener probability 𝑃_𝐿𝑍=exp−𝜋²𝜅_𝑅=exp−𝜋ℎΔ𝑚²4𝐸sin²2𝜃,cos2𝜃 (82) where ℎ≡𝑛(𝑑𝑛/𝑑𝑟)⁻¹ [32]. In the case of weak adiabaticity violation, one can develop an adiabatic perturbation theory which gives the results as a series expansion in the adiabaticity parameter [33].

3.9. Theory of Small Matter Effects

3.9.1. Minimal Width Condition

If the vacuum mixing angle is small, there exists a lower limit on the amount of matter needed to induce significant flavor change due to matter effect. The amount of matter is characterized by the column density of electrons along the neutrino trajectory: 𝑑=^𝐿₀𝑛_𝑒(𝑥)𝑑𝑥. (83) We can define 𝑑_1/2 as the column density for which the oscillation transition probability surpasses 1/2 for the first time in the course of propagation. Then it is possible to show that [34] 𝑑_1/2≥𝑑_min=𝜋2√2𝐺_𝐹tan2𝜃 (84) for all density profiles. Furthermore, the minimum, 𝑑_min, is realized for oscillations in a medium of constant density equal to the resonance density. The relation (84) is known as the minimal width condition. This condition originates from an interplay between matter effects and vacuum mixing. The acquired matter phase, √2𝐺_𝐹𝑑, must be large. At the same time, since matter effects by themselves are flavor conserving, also vacuum mixing is required in order to induce flavor conversion. The smaller the vacuum mixing is, the larger the width that is required.

3.9.2. Vacuum Mimicking

Vacuum mimicking [35], which states that regardless of the matter density, the initial flavor evolution of neutrino state is similar to that of vacuum oscillations. Consequently for small baselines, 𝐿, it is not possible to see matter effect and any such effect appearing in higher order of 𝐿. Indeed, consider the evolution matrix 𝑆=𝒯exp−𝑖^𝐿₀𝐻(𝑥)𝑑𝑥, (85) where 𝒯 denotes time ordering of the exponential. For small values of 𝐿, it can be expanded as 𝑆=1−𝑖^𝐿₀𝐻(𝑥)𝑑𝑥+𝒪𝐿². (86) If initial neutrino state has definite flavor, the amplitude of flavor transition is given by the off-diagonal element of 𝐻(𝑥) which does not depend on matter potential. The matter contribution to 𝐻(𝑥) is diagonal. Therefore the flavor transitions depend on the matter density only at higher order in 𝐿. This result holds true as long as 𝐿≪𝑙_𝑚 or when the phase of oscillation is small [36].

This can be seen explicitly in the case of medium with constant density where, expanding the oscillatory factor for small oscillation phase, we have the transition probability 𝑃=sin²2𝜃_𝑚sin²𝜙^𝑚=1𝑅sin²2𝜃sin²𝜙√𝑅≈𝜙²sin²2𝜃. (87)

Note that vacuum mimicking only occurs if the initial neutrino state is a flavor eigenstate [36]. If the initial neutrino is in a flavor-mixed state, for example, in a mass eigenstate, then matter will affect this state already at lowest order in 𝐿. This situation is realized in several settings involving astrophysical neutrinos propagating through the Earth, for example, solar and supernova neutrinos, where the neutrinos arrive at the Earth as mass eigenstates. The mimicking is not valid if there are nonstandard flavor changing interactions, so that matter effect appears in the off-diagonal elements of the Hamiltonian.

3.9.3. Effects of Small Layers of Matter

If the minimal width condition is not satisfied, that is 𝑑=𝑛𝑥≪𝐺_𝐹⁻¹, the matter effect on result of evolution is small. This inequality can be written as 𝑉𝑥≪1 which means that the oscillation phase is small. In this case the matter effect can be considered as small perturbation of the vacuum oscillation result even if the MSW resonance condition is satisfied.

The reasons for the smallness of the matter effect are different depending on the energy interval. Consider a layer of constant density with the length 𝑥. There are three possibilities.(i)𝐸≪𝐸_𝑅 (𝐸_𝑅 is the resonance density)—nearly vacuum oscillations in low density medium take place. Matter effect gives small corrections to the oscillation depth and length which are characterized by 2𝑉𝐸/Δ𝑚²=𝑉𝑥/2𝜋≪1, here 𝑥∼𝑙_𝜈.(ii)𝐸∼𝐸_𝑅—modification of oscillation parameters is strong; however 𝑙^𝑅_𝜈∼𝑙_𝜈/sin2𝜃∼2𝜋/(𝑉sin2𝜃). Consequently, 𝑥/𝑙^𝑅_𝜈=𝑥𝑉sin2𝜃/2𝜋≪1. Oscillations are undeveloped due to smallness of phase.(iii)𝐸≫𝐸_𝑅—matter suppresses oscillation depth by a factor 𝐸_𝑅/𝐸≪1. Since the oscillation length equals 𝑙_𝑚≈2𝜋/𝑉, one obtains 𝑥/𝑙_𝑚=𝑥𝑉/2𝜋≪1. Hence in this case the distance is very small and oscillation effect in the layer has double suppression.

3.10. Propagation in Multilayer Medium

3.10.1. Parametric Effects in the Neutrino Oscillations

The strong transitions discussed in the previous sections require the existence of large effective mixing, either in the entire medium (constant density) or at least in a layer (adiabatic conversion). There is a way to get strong transition without large vacuum or matter mixings. This can be realized with periodically or quasiperiodically changing density [24, 37] when the conditions of parametric resonance are satisfied. Although the flavor conversion in a layer which corresponds to one period is small, strong transitions can build up over several periods. For large mixing even a small number of periods are enough to obtain strong flavor transitions.

The usual condition of parametric resonance is that the period of density change 𝑇_𝑛 is an integer times the effective oscillation length 𝑙_𝑚 [38]: _𝑙𝑇𝑑𝑥𝑙_𝑚=𝑘,(𝑘=1,2,3,…), (88) or 𝑙_𝑇/𝑙_𝑚=𝑘. Such an enhancement has been considered first for modulation of the profile by sine function [39]. This may have some applications for intense neutrino fluxes when neutrino-neutrino interactions become important.

The solvable case, which has simple physical interpretation, is provided by the castle wall profile, for which the period 𝑙_𝑇 is divided into the two parts 𝑙₁ and 𝑙₂ (𝑙₁+𝑙₂=𝑙_𝑇) with the densities 𝑛₁ and 𝑛₂, respectively (𝑛₁≠𝑛₂ and, in general, 𝑙₁≠𝑙₂). Thus, the medium consists of alternating layers with two different densities [37, 40–45].

For the “castle wall” profile, the simplest realization of the parametric resonance condition is reduced to equality of the oscillation phases acquired by neutrinos over the two parts of the periods [41]: Φ₁=Φ₂=𝜋. (89)

The enhancement of transition depends on the number of periods and on the amplitude of perturbation, which determines the swing angle (the difference of the mixing angles in the two layers, Δ𝜃≡2𝜃_1𝑚−2𝜃_2𝑚). For small Δ𝜃 a large transition probability can be achieved after many periods. For large “swing” angle, even a small number of periods are sufficient.

3.10.2. Parametric Enhancement: General Consideration

In general the condition (89) is not necessary for the enhancement or even for maximal enhancement. First, consider the oscillation effect over one period. The corresponding evolution matrix is given by the product 𝑆_𝑇=𝑆₂𝑆₁, (90) where 𝑆_𝑘(𝑘=1,2) is the evolution in layer 𝑘 given by (57). For brevity we will write it as 𝑆_𝑘=𝑐_𝑘𝐼−𝑖𝑠_𝑘(𝜎⋅𝐧_𝑘),𝑘=1,2, where 𝑐_𝑘≡cos𝜙_𝑘, 𝑠_𝑘≡sin𝜙_𝑘, and 𝜙_𝑘 is the half phase acquired in layer 𝑘: 𝜙_𝑘=12Δ𝐻_𝑘𝑙_𝑘=Δ𝑚²𝑅𝑉4𝐸^𝑘1/2𝑙_𝑘,𝐧_𝑘≡sin2𝜃_𝑚𝑘,0,−cos2𝜃_𝑚𝑘. (91) Here 𝜃_𝑚𝑘 is the mixing angle in layer 𝑘.

Insertion of 𝑆_𝑘 from (57) into (90) gives [37] 𝑆_𝑇=𝑌𝐈−𝑖(𝜎⋅𝐗), (92) where 𝑌≡𝑐₁𝑐₂−𝑠₁𝑠₂𝐧₁⋅𝐧₂,𝐗=𝑠₁𝑐₂𝐧₁+𝑠₂𝑐₁𝐧₂−𝑠₁𝑠₂𝐧₁×𝐧₂. (93) Explicitly, (𝐧₁⋅𝐧₂)=cos(2𝜃_𝑚1−2𝜃_𝑚2) and [𝐧₁×𝐧₂]=sin(2𝜃_𝑚1−2𝜃_𝑚2)𝐞_𝑦. Using unitarity of 𝑆_𝑇, which gives 𝑋²+𝑌²=1, one can parametrize 𝑋 and 𝑌 with a new phase Φ as 𝑌≡cosΦ and 𝑋≡sinΦ. Then the evolution matrix 𝑆_𝑇 can be written in the form 𝑆_𝑇=cosΦ−𝑖sinΦ(𝜎⋅𝐗)=𝑒^{−𝑖(𝜎⋅}𝐗_)Φ, where 𝐗≡𝐗/𝑋. Consequently, the evolution matrix after 𝑛 periods equals 𝑆^𝑛=𝑆_𝑇^𝑛=𝑒^{−𝑖(𝜎⋅}𝐗_)𝑛Φ=cos𝑛Φ−𝑖𝜎⋅𝐗sin𝑛Φ. (94) It is simply accounted for by an increase of the phase: Φ→𝑛Φ. This is the consequence of the fact that the evolution matrices over all periods are equal and therefore commute. If the evolution ends at some instant 𝑡 which does not coincide with the end of a full period, that is, 𝑡=𝑛𝑇+𝑡^′, then 𝑆(𝑡)=𝑆(𝑡^′)𝑆_𝑛.

The transition probability computed with (94) is 𝑃^𝑛_𝑒𝜇=𝑆^𝑛_𝑒𝜇²=𝑋²₁+𝑋²₂𝑋²sin²𝑛Φ. (95) It has the form of the usual oscillation probability with phase 𝑛Φ and depth (𝑋²₁+𝑋²₂)/𝑋². The oscillations described by (95) are called the parametric oscillations. Under condition −𝑋₃=𝑠₁𝑐₂cos2𝜃_𝑚1+𝑠₂𝑐₁cos2𝜃_𝑚2=0, (96) which is called the parametric resonance condition, the depth of oscillations (95) becomes 1 and the transition probability is maximal when 𝑛Φ=𝜋/2+𝜋𝑘, where 𝑘 is an integer. There are different realizations of the condition (96) which imply certain correlations among the mixing angles and phases. The simplest one, 𝑐₁=𝑐₂=0, coincides with (89).

3.10.3. Parametric Enhancement in Three Layers

For small number of layers an enhancement of flavor transition can occur due to certain relations between the phases and mixing angles in different layers. This in turn imposes certain conditions on the parameters of the layers: their densities and widths. The conditions are similar to the parametric resonance condition and this enhancement is called the parametric enhancement of flavor transitions. These conditions can be satisfied for certain energies and baselines for neutrinos propagating in the Earth.

Consider conditions for maximal enhancement of oscillations for a different number of layers. It is possible to show [46] that they are generalizations of the conditions in one layer which require that (i) the depth of oscillations is 1 (we call it the amplitude condition) and (ii) the oscillation phase is 𝜙=𝜋/2+𝜋𝑘—the phase condition.

Consider first the case of one layer with (in general) varying density (it can correspond to the mantle crossing trajectories in the Earth). The resonance condition for constant density case, cos2𝜃_𝑚=0, can be written according to (22) and (51) as 𝛼=𝛼^∗, that is, 𝑆⁽¹⁾₁₁=𝑆⁽¹⁾₂₂, or equivalently, Im𝑆⁽¹⁾₁₁=0, where the superscript indicates the number of layers. This generalization goes beyond the original MSW resonance condition (even for constant density). The phase condition can be rewritten in terms of the elements of the evolution matrix (c.f., (54)) as Re𝛼≡Re𝑆⁽¹⁾₁₁=0. The absolute maximum of the transition probability occurs when these conditions are satisfied simultaneously, that is, when 𝑆⁽¹⁾₁₁=0.

The parametric resonance condition (96) can be generalized to the case of nonconstant densities in the layers although the generalization is not unique. Indeed, according to (92) the condition 𝑋₃=0 can be written in terms of the elements of the evolution matrix for the two layers as the equality of the diagonal elements 𝑆⁽²⁾₁₁=𝑆⁽²⁾₂₂. Let us find the conditions for extrema for density profiles consisting of two layers. We have 𝑆⁽²⁾=𝑆₂𝑆₁, where 𝑆⁽²⁾₁₁=𝛼₂𝛼₁−𝛽₂𝛽^∗₁,𝑆⁽²⁾₁₂=𝛼₂𝛽₁+𝛽₂𝛼^∗₁, and 𝛼_𝑖,𝛽_𝑖 for each layer have been defined in (51). The sum of the two complex numbers in the transition amplitude 𝑆⁽²⁾₁₂ has the largest possible result if they have the same phase: arg(𝛼₂𝛽₁)=arg(𝛽₂𝛼^∗₁), which can also be rewritten as arg𝛼₁𝛼₂𝛽₁=arg𝛽₂. (97) This condition is called the collinearity condition [46]. It is an extremum condition for the two-layer transition probability under the constraint of fixed transition probabilities in the individual layers. In other words, if the absolute values |𝛽_𝑖| of the transition amplitudes are fixed while their arguments are allowed to vary, then the transition probability reaches an extremum when these arguments satisfy (97).

The conditions for maximal transition probability for three layers can be found in the following way. The 1-2 elements of the evolution matrix 𝑆⁽³⁾ equal 𝑆⁽³⁾₁₂=𝛼₃𝑆⁽²⁾₁₂+𝛽₃𝑆^(2)∗₁₁=𝛼₃𝛼₂𝛽₁+𝛼₃𝛽₂𝛼^∗₁+𝛽₃𝛼^∗₂𝛼^∗₁−𝛽₃𝛽^∗₂𝛽₁. (98) In the case of neutrino oscillations in the Earth, the third layer is just the second mantle layer, and its density profile is the reverse of that of the first layer. The evolution matrix for the third layer is therefore the transpose of that for the first one [47]; that is, 𝛼₃=𝛼₁, 𝛽₃=−𝛽^∗₁, and the expression for 𝑆⁽³⁾₁₂ can be written as 𝑆⁽³⁾₁₂=𝛼₁𝛼₂𝛽₁−𝛼^∗₁𝛼^∗₂𝛽^∗₁+𝛼₁²𝛽₂+𝛽₁²𝛽^∗₂. (99) Note that 𝛽₂ is pure imaginary because the core density profile is symmetric. Therefore the amplitude 𝑆⁽³⁾₁₂ in (99) is also pure imaginary, as it must be because the overall density profile of the Earth is symmetric as well. If the collinearity condition for two layers (97) is satisfied, then not only the full amplitude 𝑆⁽³⁾₁₂, but also each of the four terms on the right-hand side of (99) is pure imaginary. If the collinearity condition is satisfied for two layers, then it is automatically satisfied for three layers. This is a consequence of the facts that the density profile of the third layer is the reverse of that of the first layer and that the second layer has a symmetric profile. The conditions described here allow reproducing very precisely all the main structures of the oscillograms of the Earth (see Section 4.1).

3.11. Oscillations of High Energy Neutrinos

At high energies or in high density medium when 𝑉>Δ𝑚²/2𝐸, we can use Δ/𝑉≡Δ𝑚²/4𝐸𝑉 as a small parameter and develop a perturbation theory using its smallness. However, in most situations of interest, the neutrino path length in matter 𝐿 is so large that Δ⋅𝐿≳1. Therefore the vacuum part of the Hamiltonian cannot be considered as a small perturbation in itself and the effect of Δ on the neutrino energy level splitting should be taken into account. For this reason we split the Hamiltonian as 𝐻=𝐻^∼₀+𝐻_𝐼 with 𝐻^∼₀=𝜔^𝑚2,𝐻100−1_𝐼=sin2𝜃Δ,−𝜖11𝜖 (100) where 𝜔^𝑚 is the oscillation frequency (20) and 𝜖≡(2Δcos2𝜃−𝑉+𝜔^𝑚)/2Δsin2𝜃≈(Δ/𝑉)sin2𝜃≪1. The ratio of the second and the first terms in the Hamiltonian (100) is given by the mixing angle in matter 𝜃_𝑚: 2Δsin2𝜃/𝜔^𝑚=sin2𝜃_𝑚. Therefore for sin2𝜃_𝑚≪1 the term 𝐻_𝐼 can be considered as a perturbation. Furthermore, 𝜖∼sin2𝜃_𝑚, so the diagonal terms in 𝐻_𝐼 can be neglected in the lowest approximation.

The solution for 𝑆 matrix can be found in the form 𝑆=𝑆₀⋅𝑆_𝐼, where 𝑆₀ is the solution of the evolution equation with 𝐻 replaced by 𝐻₀ (see (71)). The matrix 𝑆_𝐼 then satisfies the equation 𝑖𝑑𝑆_𝐼𝑑𝑥=𝑆₀⁻¹𝐻_𝐼𝑆₀𝑆_𝐼=𝐻^∼_𝐼𝑆_𝐼, (101) where 𝐻^∼_𝐼≡𝑆₀⁻¹𝐻_𝐼𝑆₀ is the perturbation Hamiltonian in the “interaction” representation. Equation (101) can be solved by iterations: 𝑆_𝐼=𝐼+𝑆_𝐼⁽¹⁾+…, which leads to the standard perturbation series for the 𝑆 matrix. For neutrino propagation between 𝑥=0 and 𝑥=𝐿 we have, to the lowest non-trivial order, 𝑆(𝐿)=𝑆₀(𝐿)𝐼−𝑖Δsin2𝜃^𝐿₀𝑑𝑥0𝑒^{𝑖2𝜙(𝑥)}𝑒^{−𝑖2𝜙(𝑥)}0. (102) The 𝜈_𝑒↔𝜈_𝑎 transition probability 𝑃₂=[𝑆(𝐿)]_𝑎𝑒 is given by 𝑃₂=Δ²sin²2𝜃^𝐿₀𝑑𝑥𝑒^{−𝑖2𝜙(𝑥)2}. (103)

For density profiles that are symmetric with respect to the center of the neutrino trajectory, 𝑉(𝑥)=𝑉(𝐿−𝑥), (103) gives 𝑃₂=4Δ𝑚²4𝐸²sin²2𝜃₀^𝐿/2𝑑𝑧cos2𝜙(𝑧)², (104) where 𝑧=𝑥−𝐿/2 is the distance from the midpoint of the trajectory and 𝜙(𝑧) is the phase acquired between this midpoint and the point 𝑧. The transition probability 𝑃₂ decreases with the increase of neutrino energy essentially as 𝐸⁻². The accuracy of (103) also improves with energy as 𝐸⁻².

Inside the Earth, the accuracy of the analytic formula is extremely good already for 𝐸≳8 GeV. When neutrinos do not cross the Earth’s core (cosΘ>−0.837) and so experience a slowly changing potential 𝑉(𝑥), the accuracy of the approximation (103) is very good even in the MSW resonance region 𝐸∼(5–8) GeV.

The above formalism applies in the low energy case as well, with only minor modifications: the sign of 𝐻₀ in (100) has to be flipped, and correspondingly one has to replace 𝜔^𝑚→−𝜔^𝑚 in the definition of 𝜖. The expressions for the transition probability in (103) and (104) remain unchanged.

3.12. Effects of Small Density Perturbations

Let us consider perturbation around smooth profile for which exact solution is known. The simplest possibility that has implications for the Earth matter profile is the constant density with additional perturbation: 𝑉(𝑥)=𝑉+Δ𝑉(𝑥). Correspondingly, the Hamiltonian of the system can be written as the sum of two terms: 𝐻(𝑥)=𝐻+Δ𝐻(𝑥), (105) where 𝐻≡𝜔−cos2𝜃sin2𝜃sin2𝜃cos2𝜃,(𝑥)Δ𝐻≡Δ𝑉2.100−1 (106) Here, 𝜃=𝜃_𝑚(𝑉) is the mixing angle in matter and 𝜔=𝜔^𝑚(𝑉) is half of the energy splitting (half-frequency) in matter, both with the average potential 𝑉. We will denote by 𝑆(𝑥) the evolution matrix of the system for the constant density case 𝐻(𝑥)=𝐻. The expression for 𝑆(𝑥) is given in (54) with 𝜃_𝑚=𝜃 and 𝜙_𝑚(𝑥)=𝜙(𝑥)≡𝜔𝑥,  𝜔=𝜔^𝑚(𝑉).

The solution of the evolution equation with Hamiltonian (105) [46] is of the form 𝑆(𝑥)=𝑆(𝑥)(𝑥),(𝑥)+Δ𝑆Δ𝑆=−𝑖𝑆(𝑥)𝐾₁(𝑥), (107) where 𝐾₁(𝑥) satisfies |𝐾₁(𝑥)_𝑎𝑏|≪1. Inserting (107) into the evolution equation, one finds the following equation for 𝐾₁(𝑥) to the first order in Δ𝐻(𝑥) and 𝐾₁(𝑥): 𝑑𝐾₁(𝑥)=𝑑𝑥𝑆^†Δ𝐻(𝑥)𝑆=Δ𝑉2−cos2𝜃−cos2𝜃sin2𝜃sin2𝜃cos2𝜃+sin2𝜃cos2𝜙𝐺𝜃+sin2𝜃sin2𝜙𝜎₂, (108) where 𝐺(𝜃)≡cos2𝜃𝜎₁+sin2𝜃𝜎₃. The first term in (108) does not contribute to 𝑆≡𝑆(𝐿) since ⟨Δ𝑉⟩≡∫Δ𝑉(𝑥)𝑑𝑥=0, and (108) can be immediately integrated: 𝐾₁(𝐿)=12sin2𝜃𝐺𝜃^𝐿₀Δ𝑉(𝑥)cos2𝜙(𝑥)𝑑𝑥+𝜎₂^𝐿₀Δ𝑉(𝑥)sin2𝜙(𝑥)𝑑𝑥. (109) Introducing the distance from the midpoint of the neutrino trajectory 𝑧≡𝑥−𝐿/2, one obtains from (109) Δ𝑆≡Δ𝑆(𝐿)=−𝑖sin2𝜃𝐺𝜃Δ𝐼+𝜎₂Δ𝐽, (110) where Δ𝐼≡(1/2)∫^𝐿/2_−𝐿/2Δ𝑉(𝑧)cos(2𝜔𝑧)𝑑𝑧,Δ𝐽≡(1/2)∫^𝐿/2_−𝐿/2Δ𝑉(𝑧)sin(2𝜔𝑧)𝑑𝑧. In these integrals, Δ𝑉(𝑧)≡Δ𝑉(𝑥(𝑧)) and 𝑥(𝑧)=𝑧−𝐿/2. The integral Δ𝐽 vanishes if the perturbation Δ𝑉(𝑧) is symmetric with respect to the midpoint of the trajectory. Analogously, Δ𝐼 vanishes if Δ𝑉(𝑧) is antisymmetric. The expression for 𝑆 defined in (107) is equivalent to (13)–(16) obtained in [48] in the context of solar neutrino oscillations.

For practical purposes it is useful to have an expression for 𝑆 which is exactly unitary regardless of the size of the perturbation. For this we rewrite (110) as follows: Δ𝑆=𝜀𝑆^′,𝑆^′=−𝑖𝐺𝜃cos𝜉+𝜎₂sin𝜉, (111) where sin𝜉=Δ𝐽/(Δ𝐽)²+(Δ𝐼)² and 𝜖=sin2𝜃⋅(Δ𝐽)²+(Δ𝐼)². Thus, 𝑆=𝑆+𝜀𝑆^′ and we replace it by 𝑆=cos𝜀𝑆+sin𝜀𝑆^′. (112) Here both 𝑆^′ and 𝑆 are unitary matrices, and due to their specific form the combination on the right-hand side of (112) is exactly unitary.

For a symmetric density profile with respect to the midpoint of the trajectory, the term Δ𝐽 is absent. From (54), (110), and (112) we immediately get the transition probability 𝑃=cos𝜀sin2𝜃sin𝜙+sin𝜀cos2𝜃²≈sin²2𝜃sin𝜙+Δ𝐼cos2𝜃², (113) where 𝜀≡sin2𝜃Δ𝐼 and 𝜙≡𝜙(𝐿)=𝜔𝐿. Here the first term in the square brackets describes oscillations in constant density matter with average potential 𝑉₁.

3.13. Oscillation Probabilities and Their Properties

It is convenient to consider the neutrino flavor evolution in the propagation basis 𝜈^∼=(𝜈_𝑒,𝜈^∼₂,𝜈^∼₃)^𝑇, defined in (27). In this basis propagation is not affected by the 2-3 mixing and CP violation. The dependence on these parameters appears when one projects the initial flavor state on the propagation basis and the final state back onto the original flavor basis. The propagation-basis states are related to the mass states as 𝜈^∼=𝑈₁₃𝐼_−𝛿𝑈₁₂𝜈. (114) Since the transformations, which connect 𝜈^∼ and 𝜈_𝑓, do not depend on matter potential and therefore distance, the states 𝜈^∼ satisfy the evolution equation 𝑖(𝑑𝜈^∼/𝑑𝑡)=𝐻^∼𝜈^∼, with the Hamiltonian 𝐻^∼ defined in (28).

3.13.1. 𝑆 Matrix and Oscillation Amplitudes

A number of properties of the oscillation probabilities can be obtained from general consideration of matrix of the oscillation amplitudes. We introduce the evolution matrix (the matrix of amplitudes) in the propagation basis as 𝑆^∼=𝐴_𝑒𝑒𝐴_𝑒2𝐴_𝑒3𝐴_2𝑒𝐴₂₂𝐴₂₃𝐴_3𝑒𝐴₃₂𝐴₃₃. (115) Then according to (27) the 𝑆 matrix in the flavor basis equals 𝑆=𝑈^∼𝑆^∼𝑈^∼†,_∼𝑈≡𝑈₂₃𝐼_𝛿. (116) In this part, we use the notation 𝐴_𝑖𝑗 for the amplitudes in the propagation basis and 𝑆_𝑖𝑗 for the amplitudes in the flavor basis. In terms of the propagation-basis amplitudes (115) the 𝑆 matrix in the flavor basis can be written as 𝐴𝑆=^𝑒𝑒𝑐₂₃𝐴_𝑒2+𝑠₂₃𝑒^−𝑖𝛿𝐴_𝑒3−𝑠₂₃𝐴_𝑒2+𝑐₂₃𝑒^−𝑖𝛿𝐴_𝑒3𝑐₂₃𝐴_2𝑒+𝑠₂₃𝑒^𝑖𝛿𝐴_3𝑒𝑐²₂₃𝐴₂₂+𝑠²₂₃𝐴₃₃+𝐾_𝜇𝜇−𝑠₂₃𝑐₂₃𝐴₂₂−𝐴₃₃+𝐾_𝜇𝜏−𝑠₂₃𝐴_2𝑒+𝑐₂₃𝑒^𝑖𝛿𝐴_3𝑒−𝑠₂₃𝑐₂₃𝐴₂₂−𝐴₃₃+𝐾_𝜏𝜇𝑠²₂₃𝐴₂₂+𝑐²₂₃𝐴₃₃+𝐾_𝜏𝜏, (117)

where 𝐾_𝜇𝜇≡𝑠₂₃𝑐₂₃𝑒^−𝑖𝛿𝐴₂₃+𝑒^𝑖𝛿𝐴₃₂,𝐾_𝜇𝜏≡𝑐²₂₃𝑒^−𝑖𝛿𝐴₂₃−𝑠²₂₃𝑒^𝑖𝛿𝐴₃₂,𝐾_𝜏𝜇=𝐾_𝜇𝜏3,𝐾𝛿⟶−𝛿,2⟷_𝜏𝜏=−𝐾_𝜇𝜇. (118) The scheme of transitions is shown in Figure 7. There is certain hierarchy of the amplitudes which can be obtained immediately from the form of the Hamiltonian in the propagation basis (29): 𝐴_𝑒3,𝐴_3𝑒∼𝑠₁₃,𝐴_𝑒2,𝐴_2𝑒∼𝑟_Δ∼𝑠²₁₃,𝐴₃₂,𝐴₂₃∼𝑠₁₃𝑟_Δ∼𝑠³₁₃, (119) that is, 𝐴₂₃ and 𝐴₃₂ are the smallest amplitudes. In the propagation basis there is no fundamental CP or T violation. Therefore for a symmetric density profile with respect to the middle point of trajectory (as in the case of the Earth) the neutrino evolution is T invariant which yields 𝐴_2𝑒=𝐴_𝑒2,𝐴_3𝑒=𝐴_𝑒3,𝐴₃₂=𝐴₂₃. (120) Consequently, for 𝐾_𝛼𝛽 we obtain 𝐾_𝜇𝜏=𝐴₂₃cos2𝜃₂₃,𝐾cos𝛿−𝑖sin𝛿_𝜏𝜇=𝐾_𝜇𝜏(),𝐾𝛿⟶−𝛿_𝜇𝜇=−𝐾_𝜏𝜏=𝐴₂₃sin2𝜃₂₃cos𝛿. (121) These terms proportional to small amplitudes 𝐴₂₃ and 𝐴₃₂ are of the order 𝑂(𝑠²₁₃).

Figure 7 

Scheme of transitions between the flavor states. Evolution is considered in the propagation basis 𝜈∼. The lines which connect the flavor states and the propagation-basis states indicated projection of one basis onto another. The lines connecting the states of propagation basis 𝜈∼ show transitions between them.

For a symmetric density profile, from (117), (120), and (121) one finds for the probabilities 𝑃_𝛼𝛽≡|𝑆_𝛽𝛼|²: 𝑃_𝑒𝑒=𝐴_𝑒𝑒²=1−𝐴_𝑒2²−𝐴_𝑒3²,𝑃_𝜇𝑒=𝑐²₂₃𝐴_𝑒2²+𝑠²₂₃𝐴_𝑒3²+2𝑠₂₃𝑐₂₃Re𝑒^−𝑖𝛿𝐴^∗_𝑒2𝐴_𝑒3,𝑃_𝜏𝑒=𝑠²₂₃𝐴_𝑒2²+𝑐²₂₃𝐴_𝑒3²−2𝑠₂₃𝑐₂₃Re𝑒^−𝑖𝛿𝐴^∗_𝑒2𝐴_𝑒3,𝑃_𝜇𝜇=𝑐²₂₃𝐴₂₂+𝑠²₂₃𝐴₃₃+2𝑠₂₃𝑐₂₃cos𝛿𝐴₂₃²,𝑃_𝜇𝜏=𝑠₂₃𝑐₂₃𝐴₃₃−𝐴₂₂+cos2𝜃₂₃cos𝛿+𝑖sin𝛿𝐴₂₃². (122)(123)(124)(125)(126) For antineutrinos the amplitudes can be obtained from the results presented above substituting 𝛿⟶−𝛿,𝐴_𝑖𝑗⟶𝐴_𝑖𝑗,where𝐴_𝑖𝑗≡𝐴_𝑖𝑗(𝑉⟶−𝑉). (127) Notice that the amplitudes of transitions (123) and (124), that involve 𝜈_𝑒, are given by linear combinations of two propagation-basis amplitudes. The other flavor amplitudes depend on three propagation-basis amplitudes.

3.13.2. Factorization Approximation and Amplitudes for Constant Density

As follows immediately from the form of the Hamiltonian 𝐻^∼ in (29), in the limits Δ𝑚²₂₁→0 or/and 𝑠₁₂→0 the state 𝜈^∼₂ decouples from the rest of the system, and consequently, the amplitude 𝐴_𝑒2 vanishes. In this limit, 𝐴_𝑒3 (as well as 𝐴₃₃ and 𝑆_𝑒𝑒) is reduced to a 2𝜈 amplitude which depends on the parameters Δ𝑚²₃₁ and 𝜃₁₃: 𝐴_𝐴(Δ𝑚²₃₁,𝜃₁₃)≡𝐴_𝑒3(Δ𝑚²₂₁=0). The corresponding probability equals 𝑃_𝐴≡|𝐴_𝐴|².

In the limit 𝑠₁₃→0 the state 𝜈^∼₃ decouples while the amplitude 𝐴_𝑒3 vanishes and the amplitude 𝐴_𝑒2 reduces to a 2𝜈 amplitude depending on the parameters of the 1-2 sector, Δ𝑚²₂₁ and 𝜃₁₂. Denoting this amplitude by 𝐴_𝑆 we have 𝐴_𝑆(Δ𝑚²₂₁,𝜃₁₂)≡𝐴_𝑒2(𝜃₁₃=0). We will use the notation 𝑃_𝑆≡|𝐴_𝑆|².

This consideration implies that to the leading nontrivial order in the small parameters 𝑠₁₃ and 𝑟_Δ the amplitudes 𝐴_𝑒2 and 𝐴_2𝑒 are reduced to two neutrino probabilities and depend only on the “solar” parameters, whereas the amplitudes 𝐴_𝑒3 and 𝐴_3𝑒 only on the “atmospheric” parameters: 𝐴_𝑒2≃𝐴_2𝑒≃𝐴_𝑆Δ𝑚²₂₁,𝜃₁₂,𝐴_𝑒3≃𝐴_3𝑒≃𝐴_𝐴Δ𝑚²₃₁,𝜃₁₃. (128) The approximate equalities in (128) are called the factorization approximation.

Due to the level crossing phenomenon the factorization approximation (128) is not valid in the energy range of the 1–3 resonance where the 1–3 mixing in matter is enhanced. In the case of a matter with an arbitrary density profile, one can show, using simple power counting arguments, that the corrections to the factorization approximation for the amplitude 𝐴_𝑒2 are of order 𝑠²₁₃, whereas the corrections to the “atmospheric” amplitude 𝐴_𝑒3 are of order 𝑟_Δ [49], in agreement with our consideration for constant density. The amplitude 𝐴_𝑒3 does not in general have a 2-flavor form, once the corrections to the factorization approximation are taken into account.

Using the expressions for 𝑈^𝑚_𝑒𝑖 and 𝑈^𝑚_𝜇𝑖 in terms of the mixing angles in the standard parametrization, we can rewrite (67) as 𝑆^cst_𝑒𝜇=cos𝜃^𝑚₂₃𝐴^cst_𝑒2+sin𝜃^𝑚₂₃𝑒^{−𝑖𝛿𝑚}𝐴^cst_𝑒3, (129) where 𝐴^cst_𝑒2≡−𝑖𝑒^{𝑖𝜙𝑚21}cos𝜃^𝑚₁₃sin2𝜃^𝑚₁₂sin𝜙^𝑚₂₁,𝐴^cst_𝑒3≡−𝑖𝑒^{𝑖𝜙𝑚21}sin2𝜃^𝑚₁₃sin𝜙^𝑚₃₂𝑒^{−𝑖𝜙𝑚31}+cos²𝜃^𝑚₁₂sin𝜙^𝑚₂₁. (130) Here 𝜙^𝑚₃₁=𝜙^𝑚₃₂+𝜙^𝑚₂₁. Since to a good approximation 𝜃^𝑚₂₃≈𝜃₂₃ and 𝛿^𝑚≈𝛿 (see Section 2.5) [20, 50], the amplitudes 𝐴^cst_𝑒2 and 𝐴^cst_𝑒3 can be identified with 𝐴_𝑒2 and 𝐴_𝑒3 in (123) and (124).

In terms of mixing angles, 𝑈^𝑚_𝜇1=−𝑠^𝑚₁₂𝑐^𝑚₂₃−𝑐^𝑚₁₂𝑠^𝑚₁₃𝑠^𝑚₂₃𝑒^𝑖𝛿𝑚, 𝑈^𝑚_𝜇3=𝑐^𝑚₁₃𝑠^𝑚₂₃, the amplitude 𝑆^cst_𝜇𝜇 can be rewritten as 𝑆^cst_𝜇𝜇=cos²𝜃^𝑚₂₃𝐴^cst₂₂+sin²𝜃^𝑚₂₃𝐴^cst₃₃+sin2𝜃^𝑚₂₃cos𝛿^𝑚𝐴^cst₂₃, (131) where 𝐴^cst₂₂≡1+2𝑖𝑒^{𝑖𝜙𝑚21}sin²𝜃^𝑚₁₂sin𝜙^𝑚₂₁,𝐴^cst₃₃≡1−2𝑖𝑒^{−𝑖𝜙𝑚32}cos²𝜃^𝑚₁₃sin𝜙^𝑚₃₂+2𝑖𝑒^{𝑖𝜙𝑚21}sin²𝜃^𝑚₁₃cos²𝜃^𝑚₁₂sin𝜙^𝑚₂₁,𝐴^cst₂₃≡𝑖𝑒^{𝑖𝜙𝑚21}sin𝜃^𝑚₁₃sin2𝜃^𝑚₁₂sin𝜙^𝑚₂₁. (132) Notice that 𝐴^cst₂₂ has exactly the form of the corresponding 2𝜈 amplitude driven by the solar parameters. The amplitude 𝐴^cst₃₃ also coincides to a very good approximation with the corresponding 2𝜈 amplitude driven by the atmospheric parameters. In the approximation 𝜃^𝑚₂₃≈𝜃₂₃ and 𝛿^𝑚≈𝛿 the amplitudes (132) can be identified with the corresponding amplitudes in the propagation basis.

3.13.3. Properties of the Flavor Oscillation Probabilities

(1)𝜈_𝑒-𝜈_𝑒 channel: the total probability of the 𝜈_𝑒 disappearance equals 1−𝑃_𝑒𝑒=𝑃_𝑒𝜇+𝑃_𝑒𝜏=𝑃_𝑒2+𝑃_𝑒3. (133) The probability 𝑃_𝑒𝑒 does not depend on the CP-violating phase and the 2-3 mixing in the standard parametrization. The interference of the solar and atmospheric modes in 𝑃_𝑒𝑒 originates mainly from 𝑃_𝑒3≡|𝐴_𝑒3|². The survival probability then equals 𝑃_𝑒𝑒=1−𝑃_𝑒𝜇−𝑃_𝑒𝜏=1−𝑃_𝐴−𝑃_𝑆. At high energies, where the effects of the 1-2 mixing and mass splitting in 𝑃 are suppressed, the probability is 𝑃_𝑒𝑒≈1−𝑃_𝑒𝜏≈1−𝑃_𝐴. (2)𝜈_𝑒-𝜈_𝜇 and 𝜈_𝑒-𝜈_𝜏 channels: the transition probability 𝑃_𝜇𝑒≡𝑃(𝜈_𝜇→𝜈_𝑒) (see (123)) can be rewritten as 𝑃_𝜇𝑒=𝑐²₂₃𝐴_𝑒2²+𝑠²₂₃𝐴_𝑒3²+sin2𝜃₂₃𝐴^∗_𝑒2𝐴_𝑒3cos𝜙−𝛿, (134) where 𝜙≡arg(𝐴^∗_𝑒2𝐴_𝑒3). Unlike 1−𝑃_𝑒𝑒, this probability contains the interference term between 𝐴_𝑒2 and 𝐴_𝑒3 which depends on the CP violation phase. Since the amplitude 𝐴_𝑒2 is suppressed at high energies due to the smallness of the 1-2 mixing in matter, in the lowest approximation we have 𝑃_𝜇𝑒≈sin²𝜃₂₃𝐴_𝑒3²≈sin²𝜃₂₃𝐴_𝐴². (135) The maximal value of the probability equals 𝑃_𝜇𝑒≃𝑠²₂₃. According to (123) and (124) the oscillation probabilities 𝑃_𝜏𝑒 and 𝑃_𝑒𝜏 can be obtained from the corresponding probabilities 𝑃_𝜇𝑒 and 𝑃_𝑒𝜇 through the substitution 𝑠₂₃→𝑐₂₃, 𝑐₂₃→−𝑠₂₃ [51]. The interference term has the opposite signs for channels including 𝜈_𝜏 as compared with those with 𝜈_𝜇, which can be obtained from the unitarity condition 𝑃_𝑒𝑒+𝑃_𝜇𝑒+𝑃_𝜏𝑒=1 and the fact that 𝑃_𝑒𝑒 does not depend on 𝛿.(3) The 𝜈_𝜇 survival probability, 𝑃_𝜇𝜇, for symmetric density profiles, (125), can be rewritten as 𝑃_𝜇𝜇=𝑐²₂₃𝐴₂₂+𝑠²₂₃𝐴₃₃²+2sin2𝜃₂₃cos𝛿Re𝐴^∗₂₃𝑐²₂₃𝐴₂₂+𝑠²₂₃𝐴₃₃+sin²2𝜃₂₃cos²𝛿𝐴₂₃². (136) Since 𝐴₂₃=𝒪(𝑟_Δ𝑠₁₃) is a small quantity, to a good approximation one can neglect the term ∼cos²𝛿 in (125), which is proportional to |𝐴₂₃|². For high energies in the limit Δ𝑚²₂₁→0 we have 𝐴₂₂=1, 𝐴₂₃=0. Then, parameterizing the 33-amplitude as 𝐴₃₃=1−𝑃_𝐴𝑒^{−𝑖𝜙𝑚33} we obtain from (136) 𝑃_𝜇𝜇Δ𝑚²₂₁=0=1−sin²2𝜃₂₃sin²𝜙_𝑥−𝑠⁴₂₃𝑃_𝐴−0.5sin²2𝜃₂₃cos2𝜙_𝑋1−1−𝑃_𝐴, (137) where 𝜙_𝑋=0.5arg[𝐴^∗₃₃𝐴₂₂]=𝜙^𝑚₂₂−𝜙^𝑚₃₃. The probability can be rewritten as 𝑃_𝜇𝜇=1−0.5sin²2𝜃₂₃−𝑠⁴₂₃𝑃_𝐴+0.5sin²2𝜃₂₃1−𝑃_𝐴cos2𝜙_𝑋. (138)(4)𝜈_𝜇-𝜈_𝜏 channel: for symmetric matter density profiles the probability of 𝜈_𝜇→𝜈_𝜏 oscillations is given in (126). It can be rewritten as 𝑃_𝜇𝜏=14sin²2𝜃₂₃𝐴₂₂−𝐴₃₃²+sin2𝜃₂₃cos2𝜃₂₃cos𝛿Re𝐴^∗₃₃−𝐴^∗₂₂𝐴₂₃−sin2𝜃₂₃sin𝛿Im𝐴^∗_𝑒2𝐴_𝑒3+1−sin²2𝜃₂₃cos²𝛿𝐴₂₃². (139)The amplitude depends on 𝛿 through the terms proportional to cos𝛿 and sin𝛿, and therefore 𝑃_𝜇𝜏 contains both CP- and T-even and -odd terms. One can show that the 𝛿-dependent interference terms, which are proportional to sin𝛿 and cos𝛿, satisfy the relation 𝑃^𝛿_𝜇𝜏=−𝑃^𝛿_𝜇𝑒−𝑃^𝛿_𝜇𝜇. In the limit Δ𝑚²₂₁→0 we obtain 𝑃_𝜇𝜏=0.5sin²2𝜃₂₃−𝑠²₂₃𝑐²₂₃𝑃_𝐴−0.5sin²2𝜃₂₃1−𝑃_𝐴cos2𝜙_𝑋. (140)

4. Matter Effects and Determination of Neutrino Mass Hierarchy

4.1. Propagation of Neutrinos through the Earth

Flavor neutrino evolution in the Earth is essentially oscillations in a multi-layer medium with slowly changing density in the individual layers and sharp density change on the borders of layers. For energies 𝐸>0.1 GeV, possible short-scale inhomogeneities of the matter distribution can be neglected and the density profile experienced by neutrinos is symmetric with respect to the midpoint of the trajectory: 𝑉(𝑥)=𝑉(𝐿−𝑥). (141) Here 𝐿=2𝑅_⊕|cos𝜃_𝑧| is the length of the trajectory inside the Earth, 𝑅_⊕=6371 km is the Earth radius, and 𝜃_𝑧 is the zenith angle related to the nadir angle as Θ_𝜈=𝜋−𝜃_𝑧. For 0≤Θ_𝜈≤33.1^∘ neutrinos cross both the mantle and the core of the Earth, whereas for larger values of the nadir angle they only cross the Earth’s mantle. The column density of the Earth along the diameter equals 𝑑_Earth=∫𝑛(𝑥)𝑑𝑥, which is bigger than the minimal width; the size of the Earth is comparable with the neutrino refraction length.

For the 1-2 channel, the adiabaticity is well satisfied for all energies. We can therefore use the adiabatic approximation. The results of the evolution are determined by the mixing at the surface of the Earth and by the adiabatic phase. In the 1–3 channel the adiabaticity is broken at the resonance. Thus, the constant density approximation with the average density works well in this regime. For energies below the resonance the matter effect becomes small and the constant density approximation and the adiabatic approximation give very similar results.

For the core-crossing trajectories, the profile consists of three layers in the first approximation: (i) mantle (with increasing density), (ii) core (with a symmetric profile), and (iii) second mantle layer (with decreasing density). This second mantle layer is T-inverted with respect to the first. In this approximation the profile can be considered as three layers of constant effective densities. As such, it looks like a part (1.5 period) of the castle wall profile. Consequently, the parametric enhancement of oscillations, and in particular, the parametric resonance can be realized.

4.1.1. Neutrino Oscillograms of the Earth

A comprehensive description of effects of neutrino passage through the Earth can be obtained in terms of neutrino oscillograms. The oscillograms are defined as lines of equal probabilities (or certain combinations of probabilities) in the 𝐸_𝜈-cos𝜃_𝑧 plane. In Figure 8, we show the oscillograms for the oscillation probabilities 𝑃_𝑒𝜇 and 𝑃_𝜇𝜇, as well as the corresponding probabilities for antineutrinos [43, 46, 52–55].

(a)

(b)

(c)

(d)

Figure 8 

Neutrino oscillograms of the Earth. Shown are the lines of equal flavor conversion probability in the 𝐸𝜈-cosΘ𝜈 plane. Upper panels: 𝜈𝑒→𝜈𝜇 (left) and 𝜈𝑒→𝜈𝜇 (right); bottom panels: 𝜈𝜇→𝜈𝜇 (left) and 𝜈𝜇→𝜈𝜇 (right). Normal hierarchy is assumed.

The structure of the oscillograms is well defined and unique and reflects the structure of the Earth as well as the properties of the neutrinos themselves. In a sense, the oscillograms are the neutrino images of the Earth. In contrast to usual light, there are several different images in different flavors as well as in neutrinos and antineutrinos.

The positions of all main structures of the oscillograms are determined by different realizations of the amplitude condition and the phase condition. These are generalizations of the condition for maximal flavor transitions in the case of vacuum oscillations or oscillation in uniform matter. Recall that, in the latter case, 𝑃=1 requires (i)sin²2𝜃_𝑚=1, the amplitude condition, which is nothing but the MSW resonance condition, and (ii)𝜙=𝜋/2+𝜋𝑘, the phase condition.

At 𝐸>1 GeV the main structures of oscillograms are generated by the 1–3 mixing. They include the following. (1) The MSW resonance pattern (resonance enhancement of the oscillations) for trajectories crossing only the mantle, with the main peak at 𝐸_𝜈∼(5–7)GeV. The position of the maximum is given by the MSW resonance condition: 𝐸_𝜈=𝐸_𝑅Θ_𝜈=Δ𝑚²₃₁cos2𝜃₁₃2𝑉₁Θ_𝜈, (142) where 𝑉₁(Θ_𝜈) is the average value of the potential along the trajectory characterized by Θ_𝜈. The phase condition becomes 2𝜙(𝐸_𝜈,Θ_𝜈)=2𝜔(𝑉,𝐸_𝜈)𝐿(Θ_𝜈)=𝜋 and the intersection of the resonance and the phase condition lines gives the absolute maximum of 𝑃_𝐴. Combining these conditions gives the coordinates of the peak: cosΘ_𝜈=0.77 and 𝐸_𝑅=6 GeV.(2) Three parametric ridges in the domain of core-crossing trajectories |cos𝜃_𝑧|>0.87 and 𝐸_𝜈>3 GeV. The parametric ridges differ by the oscillation phase acquired in the core, 𝜙₂.(i)Ridge A lies between the core resonance (at Θ_𝜈∼0^∘) and the mantle resonance regions, 𝐸_𝜈≈3–6 GeV. The phase in the core is 𝜙₂≲𝜋. This ridge merges with the MSW resonance peak in the mantle.(ii)Ridge B is situated at 𝐸_𝜈≥5 GeV. For the lowest energies in the ridge and Θ_𝜈∼0, the half phase in the core equals 𝜙₂∼(1.2-1.3)𝜋.(iii)Ridge C is located at 𝐸_𝜈>11 GeV in the matter dominated region, where the mixing and, consequently, oscillation depth are suppressed. (3)The MSW resonance peak in core located at 𝐸_𝜈∼2.5–2.8 GeV.(4) The regular oscillatory pattern at low energies with “valleys” of zero probability and ridges in the mantle domain and a more complicated pattern with local maxima and saddle points in the core domain.

In Figure 9, we show graphic representations of oscillations which correspond to salient features of the oscillograms.

(a)

(b)

(c)

Figure 9 

Graphic representation of transition in different points of oscillogram: peak due to MSW resonance in the mantle (left), peak due to parametric enhancement of transition driven by 1–3 mixing (middle), and peak due to parametric enhancement of transition driven by 1-2 mixing (right). In the left panel, neutrinos traverse only the mantle layer. In the right panel, neutrinos traverse the mantle (red), the core (green), and again the mantle (blue). The dashed lines correspond to the Hamiltonian vector 𝐇 for the mantle (red) and core (green), respectively.

For energies 𝐸_𝜈<1 GeV the main structures are induced by the 1-2 mixing with small corrections due to 1–3 vacuum oscillations. Neglecting effect of 𝜃₁₃ we have 1−𝑃_𝑒𝑒=|𝐴_𝑒2|²≡𝑃_𝑆. The probabilities of the modes including 𝜈_𝑒 are expressed in terms of a unique probability 𝑃_𝑆.

The 1-2 pattern differs from the pattern for the 1–3 mixing due to the large value of the 1-2 mixing. The oscillation length at the resonance is smaller than that for the 1–3 mixing, 𝑙^𝑅_𝑚=𝑙_𝜈/sin2𝜃₁₂∼𝑙_𝜈. The resonance energy is shifted to smaller values both due to Δ𝑚²₂₁≪Δ𝑚²₃₁ and because of the factor cos2𝜃₁₂≈0.4: 𝐸^𝑅₁₂=(Δ𝑚²₂₁/2𝑉)cos2𝜃₁₂. Here 𝑉 is the average value of the potential. The adiabaticity is better satisfied than for the 1–3 mixing case and therefore the oscillation probability in the mantle is determined by the potential near the surface of the Earth 𝑉 averaged over a distance of the order of the first oscillation length. The oscillation length in matter 𝑙_𝑚 monotonically increases with energy, approaching the refraction length 𝑙₀≡2𝜋/𝑉 (c.f. Figure 4). The jump of the mixing angle at the mantle-core boundary is small. Thus, the sudden distortion of the oscillation patterns at Θ_𝜈=33^∘ is not as significant as it is for the 1–3 mixing, in particular below the 1-2 resonance energy. These features allow understanding the structure of the oscillograms. In the mantle domain (Θ_𝜈>33^∘) the oscillation pattern for neutrinos is determined by the resonance enhancement of oscillations. There are three MSW resonance peaks above 0.1 GeV, which differ from each other by value of the total oscillation phase. The outer peak (Θ_𝜈≈82^∘) corresponds to 𝜙≈𝜋/2, the middle (Θ_𝜈=60^∘) to 𝜙≈3𝜋/2, and the inner (Θ_𝜈≈40^∘) to 𝜙=5𝜋/2. Recall that such a large phase can be acquired due to the smaller resonance oscillation length in comparison to that of the 1–3 mixing case, for which only one peak with 𝜙=𝜋/2 can be realized. The resonance energy is given by (45), and for the surface potential we find 𝐸^𝑅₁₂≈0.12GeV. (143) The ratio of the 1-2 and 1–3 resonance energies equals 𝐸^𝑅₁₂/𝐸^𝑅₁₃≈1/50. The estimate (143) is valid for the two outer peaks. For the peak at Θ_𝜈=40^∘, 𝑉 is larger and, accordingly, the resonance energy is slightly smaller. The width of the 1-2 resonance is large and therefore the regions of sizable oscillation probability are more extended in the 𝐸_𝜈 direction as compared to the oscillations governed by the 1–3 mixing and splitting.

The resonance energy in the core is 𝐸^𝑅₁₂≈0.04 GeV. Therefore for 𝐸_𝜈>(0.10--0.15)GeV the 1-2 mixing in the core is substantially suppressed by matter. The peak at 𝐸_𝜈≃0.2 GeV and Θ_𝜈≃25^∘ is due to the parametric enhancement of the oscillations. It corresponds to the realization of the parametric resonance condition when the oscillation half phases equal approximately 𝜙_mantle≈𝜋/2 and 𝜙_core≈3𝜋/2 (note that the total phase is ≈5𝜋/2 and this parametric ridge is attached to the 5𝜋/2 MSW peak in the mantle domain).

4.1.2. Oscillograms for the Inverted Mass Hierarchy

The main change compared to the normal hierarchy is that the 1–3 resonance structure now appears in the antineutrino channel. The level crossing scheme is modified in comparison to NH. In the neutrino channel there is only the 1-2 resonance.

In the approximation of Δ𝑚²₂₁=0, the neutrino oscillograms for the inverted hierarchy coincide with the antineutrino oscillograms for the normal hierarchy and vice versa, provided that Δ𝑚²₃₁ is taken to be the same in both cases [56]: 𝑃_𝐴^𝐼𝐻=𝑃_𝐴^𝑁𝐻,𝜙_𝑋^𝐼𝐻=−𝜙_𝑋^𝑁𝐻. Therefore 𝑃^𝐼𝐻_𝛼𝛽=𝑃^𝑁𝐻_𝛼𝛽,𝑃^𝐼𝐻_𝛼𝛽=𝑃^𝑁𝐻_𝛼𝛽. The inclusion of the 1-2 mixing and mass splitting breaks this symmetry.

4.2. CP Violation Effects

4.2.1. Interference and CP violation

The survival probability 𝑃_𝑒𝑒 does not depend on the CP-violating phase 𝛿 neither for oscillations in vacuum nor in matter [57, 58]. This is the consequence of the facts that 𝛿 can be removed by transforming to the propagation basis and that 𝜈_𝑒 is not affected by this transformation. For oscillations in vacuum, or in matter with symmetric density profiles, the other two survival probabilities, 𝑃_𝜇𝜇 and 𝑃_𝜏𝜏, are T-even quantities dependent on 𝛿 only through terms proportional to cos𝛿 and cos2𝛿 [59]. In contrast to this, for oscillations in a matter with nonsymmetric density profiles, these probabilities also acquire terms proportional to sin𝛿 and sin2𝛿.

Introducing the phase 𝜙≡arg(𝐴^∗_𝑒2𝐴_𝑒3) and omitting small terms proportional to |𝐴₂₃|²=𝒪(𝑠⁶₁₃) we obtain 𝑃^𝛿_𝜇𝑒=sin2𝜃₂₃cos𝜙−𝛿𝐴_𝑒2𝐴_𝑒3,𝑃^𝛿_𝜇𝜇=−sin2𝜃₂₃cos𝛿cos𝜙𝐴_𝑒2𝐴_𝑒3−𝐷₂₃,𝑃^𝛿_𝜇𝜏=−sin2𝜃₂₃sin𝛿sin𝜙𝐴_𝑒2𝐴_𝑒3+𝐷₂₃, (144)(145)(146) where 𝐷₂₃≡(1/2)sin4𝜃₂₃cos𝛿Re[𝐴^∗₂₃(𝐴₃₃−𝐴₂₂)] is proportional to the small deviation of the 2-3 mixing from maximal one. Notice that 𝐷₂₃ enters 𝑃^𝛿_𝜇𝜇 and 𝑃^𝛿_𝜇𝜏 with opposite signs while 𝑃^𝛿_𝜇𝑒 does not depend on 𝐷₂₃ at all. 𝐷₂₃ is CP even. The sum of these interference terms is zero.

For the other channels, 𝑃^𝛿_𝛼𝛽=𝑃^−𝛿_𝛽𝛼. For antineutrinos, according to (127), the probabilities have the same form as the corresponding probabilities derived above with a changed sign of 𝛿 and the amplitudes computed with the opposite sign of the potential. Thus, the 𝛿-dependent parts in all the channels are expressed in terms of two combinations of the propagation-basis amplitudes, |𝐴_𝑒2𝐴_𝑒3| and 𝐷₂₃.

4.2.2. Magic Lines and CP Domains

To better assess the effect of 𝛿, one can consider the difference of the oscillation probabilities for two different values of the CP phase Δ𝑃^CP_𝛼𝛽(𝛿)≡𝑃_𝛼𝛽(𝛿)−𝑃_𝛼𝛽(𝛿₀). In practice, this quantifies how well the phase 𝛿 fits with some assumed true value 𝛿₀. The structure of the oscillograms for Δ𝑃^CP_𝛼𝛽(𝛿) can be understood in terms of the grids of magic lines and interference phase lines along which Δ𝑃^CP_𝛼𝛽(𝛿)≈0.

For the 𝜈_𝜇→𝜈_𝑒 oscillation probability, the equality Δ𝑃^CP_𝜇𝑒(𝛿)≡𝑃_𝜇𝑒(𝛿)−𝑃_𝜇𝑒𝛿₀=𝑃^𝛿_𝜇𝑒(𝛿)−𝑃^𝛿_𝜇𝑒𝛿₀ (147) is exact and the condition Δ𝑃^CP_𝜇𝑒=0 is equivalent to 𝐴_𝑒2𝐴_𝑒3=𝐴cos𝜙−𝛿_𝑒2𝐴_𝑒3cos𝜙−𝛿₀. (148) This equality is satisfied if at least one of the following three conditions is fulfilled: 𝐴_𝑒2𝐸_𝜈,Θ_𝜈𝐴=0,_𝑒3𝐸_𝜈,Θ_𝜈𝜙𝐸=0,_𝜈,Θ_𝜈−𝛿₀=−𝜙𝐸_𝜈,Θ_𝜈−𝛿+2𝜋𝑙. (149) The last condition implies 𝜙𝐸_𝜈,Θ_𝜈=𝛿+𝛿₀2+𝜋𝑙. (150) Under the conditions (149), the equality (148) is satisfied identically for all values of 𝛿. In these cases the transition probability does not depend on the CP phase. Since the amplitudes 𝐴_𝑒2 and 𝐴_𝑒3 are complex quantities, these conditions can be satisfied in isolated points of the (Θ_𝜈,𝐸_𝜈) plane only. In contrast to this, in the factorization approximation 𝐴_𝑒2=𝐴_𝑆 and 𝐴_𝑒3=𝐴_𝐴 both the conditions are fulfilled along certain lines in the oscillograms. This occurs because the amplitudes 𝐴_𝑆 and 𝐴_𝐴 take a 2-flavor form. On the basis of neutrino states where the corresponding 2×2 Hamiltonians are traceless, both 𝐴_𝐴 and 𝐴_𝑆 are pure imaginary because of the symmetry of the Earth’s density profile [47].

Let us consider the equalities 𝐴_𝑆=0 and 𝐴_𝐴=0 using the constant density approximation.

(1) The condition 𝐴_𝑆(𝐸_𝜈,Θ_𝜈)=0 is satisfied when sin𝜙_𝑆(𝐸_𝜈,Θ_𝜈)=0, which leads to 𝐿Θ_𝜈≈2𝜋𝑛𝜔^𝑚₂₁,𝑛=1,2,…. (151) At energies 𝐸_𝜈≳0.5 GeV which are much higher than the 1-2 mixing MSW resonance in the mantle and in the core of the Earth one has 𝜔^𝑚₂₁≈𝑉 and the condition (151) becomes 𝐿Θ_𝜈≃2𝜋𝑛𝑉. (152) This expression is energy independent and determines the baselines for which the “solar” contribution to the probability vanishes [61]. In the plane (Θ_𝜈,𝐸_𝜈) it represents nearly vertical lines at fixed Θ_𝜈. There are three solar magic lines which correspond to 𝑛=1 (in the mantle domain) Θ_𝜈≈54^∘ and 𝑛=2,3 (in the core domain) [61] Θ_𝜈≈30^∘ and 12^∘. The existence of a baseline (𝐿≈7600 km) for which the probability of 𝜈_𝑒↔𝜈_𝜇 oscillations in the Earth is approximately independent of the “solar” parameters (Δ𝑚²₂₁, 𝜃₁₂) and of the CP phase 𝛿 was first pointed out in [62] and later discussed in, for example, [61, 63–68]. This baseline was dubbed “magic” in [63].(2) The atmospheric magic lines are determined by the condition 𝐴_𝐴(𝐸_𝜈,Θ_𝜈)=0 [61]. Along these lines, the “atmospheric” contribution to the amplitudes of 𝜈_𝜇↔𝜈_𝑒 and 𝜈_𝜏↔𝜈_𝑒 transitions vanishes and the probabilities of oscillations involving 𝜈_𝑒 or 𝜈_𝑒 do not depend on CP phase. In the constant density approximation, the condition 𝐴_𝐴=0 is satisfied when sin𝜙_𝐴=0 (𝜙_𝐴=𝜋𝑘, 𝑘=1,2,…) or explicitly 𝐿Θ_𝜈≈2𝜋𝑘𝜔^𝑚₃₁,𝑘=1,2,…. (153) For energies which are not too close to the 1–3 MSW resonance, it reduces to 𝐸_𝜈≃Δ𝑚²₃₁𝐿Θ_𝜈4𝜋𝑘±2𝑉𝐿Θ_𝜈, (154) which corresponds to the bent curves in the (Θ_𝜈,𝐸_𝜈) plane. For very large energies, where Δ𝑚²₃₁/2𝐸≪𝑉, the atmospheric lines approach the same vertical lines as the solar magic lines (152).(3) The condition (150) determines the interference phase lines in the (Θ_𝜈,𝐸_𝜈) plane. In the constant density approximation 𝜙≈−𝜙^𝑚₃₁. Consequently in the energy range between the two resonances we have 𝜙^𝑚₃₁≈Δ𝑚²₃₁𝐿4𝐸_𝜈=𝜙⁰_𝐴, (155) that is, in the first approximation 𝜙 does not depend on the matter density. From (150) we then obtain 𝐸_𝜈=Δ𝑚²₃₁𝐿Θ_𝜈4𝜋𝑙−2𝛿+𝛿₀. (156)

Thus, in the factorization approximation, the conditions (149) and (150) define three sets of lines (grid of magic lines) in the oscillograms (see Figure 10), which play crucial roles in understanding the CP violation effects. Along the lines, the probabilities 𝑃_𝜇𝑒,𝑃_𝑒𝜇,𝑃_𝜏𝑒, and 𝑃_𝑒𝜏 do not depend on the CP phase in the first order approximation. The other probabilities depend on the phase weakly.

Figure 10 

Oscillograms for the difference of probabilities Δ𝑃CP𝜇𝑒(𝛿)=𝑃𝜇𝑒(𝛿)−𝑃𝜇𝑒(𝛿0) with 𝛿0=0∘. Shown are the solar (black), atmospheric (white), and interference phase condition (cyan) curves from [60].

From Figure 10, we can see that the magic lines described above do not coincide exactly with the lines of Δ𝑃^CP_𝜇𝑒=0 which bound the CP-domains. Furthermore, interconnections of the latter occur. This is due to the breakdown of the factorization approximation.

4.3. Determination of Hierarchy with Accelerator Experiments

An accelerator neutrino experiment has a fixed baseline which corresponds to a vertical line with the length determined by the available energy spectrum. In the oscillogram of Figure 11 we have included such lines for a handful of accelerator experiments. Furthermore, this energy spectrum is usually peaked at certain energy (or narrow energy range) resulting in the experiment being most sensitive to the oscillation probability at that specific energy. An accelerator neutrino experiment would typically run for several years in neutrinos or antineutrinos before switching polarity and therefore getting information both on 𝑃_𝛼𝛽 and 𝑃_𝛼𝛽. The goal of such a search is to observe in which channel the oscillation probability is suppressed and in which it is enhanced. If a neutrino experiment could run at energy similar to the resonant one and at a baseline of several thousand kilometers, then this determination would be quite simple. However, as can be seen from the oscillogram, accelerator neutrino experiments are confined to relatively shallow trajectories with rather poor oscillatory pattern, and this severely limits their capabilities leading to various degeneracies. In particular, lack of knowledge of the mass hierarchy is part of the famous eightfold degeneracy, which arises as follows. Assume we have access to the values of oscillation probabilities 𝑃_𝜇𝑒 and 𝑃_𝜇𝑒 at a given baseline 𝐿 and energy 𝐸 only. Then there exist three types of ambiguities that give rise to the same values of the probabilities in different parts of the parameter space (mixing angles, CP phase, and signs of mass differences).

Figure 11 

Physics reach of the accelerator and atmospheric neutrino experiments. Shown are areas in the oscillogram for 𝜈𝑒→𝜈𝜇 channel which can be covered by different accelerator (vertical sections) and atmospheric neutrino experiments (sensitive to the area above the corresponding horizontal lines).

(1)Sign (hierarchy) degeneracy: this is the degeneracy due to the unknown neutrino mass hierarchy. Changing the mass hierarchy, it is often possible to find a point in parameter space that predicts the same oscillation probabilities.(2)Intrinsic (𝜃₁₃,𝛿) degeneracy: for any combination of (𝜃₁₃,𝛿), there exists a different combination (𝜃₁₃,𝛿) that also predicts the same oscillation probabilities.(3)Octant (𝜃₂₃) degeneracy: changing the octant of 𝜃₂₃ also leads to a degeneracy due to 𝜇-𝜏 symmetry. If 𝜃₂₃ is close to maximal, the effects of this degeneracy are less pronounced.

Since each of these degeneracies is twofold, an overall degeneracy is eightfold: 2³=8. The first two of these degeneracies can be illustrated in a biprobability plot of Figure 12. As follows from this figure, even if both the probabilities (for a given neutrino energy) are known with infinite accuracy, we cannot identify the hierarchy within the pink region.

(a)

(b)

Figure 12 

Two different illustrations of parameter space degeneracies. Left panel: biprobability plot for 𝐿=295km and 𝐸=0.65 GeV. The red band indicates the possible values of the probabilities for normal hierarchy, the blue for inverted, and the pink for the intersection of the two. The black and white ellipses represent the possible values of the probabilities for two different fixed values of 𝜃13. Right panel: probability isocontours of 𝑃𝜇𝑒 (black) and 𝑃𝜇𝑒 (red). The values of the probabilities correspond to those of the intersections between the black and white ellipses in the left panel (with the thick lines representing the upper left intersection). The intersections are where the parameter values reproduce the oscillation probabilities for both neutrinos and antineutrinos.

For known mass hierarchy (e.g., normal one) a given value of 𝜃₁₃ fixes ellipse in the plot along which the CP phase varies. Increasing 𝜃₁₃ moves the ellipse up and to the right in the plot. Therefore for every point on an ellipse, there will be another ellipse corresponding another value 𝜃^prime₁₃, which crosses this point and therefore 𝜃^prime₁₃ reproduces the same oscillation probabilities. For example, in the left intersection of the black and white ellipse (Figure 12) both combinations of 𝜃₁₃ and 𝛿 correspond to those precise oscillation probabilities and there are also values of 𝜃₁₃ and 𝛿 that will reproduce them in the inverted hierarchy. For the right intersection, the intrinsic degeneracy is still present, while the sign degeneracy is resolved. It should be remembered that this type of figure is just an illustration. In real experiment the neutrino energy spans over wide range, the oscillation probabilities would not be exactly known, and strictly this type of consideration becomes invalid.

In order to see how these degeneracies manifest themselves in an experimental setup, we show the oscillation probability 𝑃_𝜇𝑒 as a function of the baseline length in Figure 13. While the 295 km baseline is too short for matter effects to be very significant, as the baseline increases matter effects start being more and more important. In particular, when the oscillation phase maximum occurs at an energy similar to that of the matter resonance, as is the case of 7500 km baseline, we can see the enhancement of the transition probability in the neutrino channel for the normal hierarchy and the suppression in the inverted. In a simple two-flavor scenario, the amplitude of 𝑃_𝜇𝑒 at the resonance is one by definition in the normal mass hierarchy case. At the same time, the oscillation amplitude in the inverted hierarchy at the same energy is given by sin²2𝜃^∼=sin²2𝜃1+3cos²≃12𝜃4sin²2𝜃, (157) where the last equality holds for small 𝜃. On the other hand, if the neutrino energy is far below the resonance in order to accumulate a significant oscillation phase, such as in the left and middle panels, then the oscillation amplitude will be effectively given by sin²2𝜃^∼≃sin²2𝜃1+4𝑉𝐸Δ𝑚²cos2𝜃. (158) The reason that the 810 km baseline separates the hierarchies better than the 295 km one is based mainly on the fact that the oscillation maximum can be reached for higher energies due to the longer baseline, and thus, the relative difference between probabilities for the two hierarchies increases. Also note that the oscillation probabilities for the 7500 km baseline are not very dependent on the CP-violating phase 𝛿. This is due to the so-called magic baseline effect, which has been discussed before.

Figure 13 

The neutrino oscillation probability 𝑃𝜇𝑒 at baselines of 295 (left), 810 (middle), and 7500 km (right) as a function of the neutrino energy. The red (blue) band corresponds to the normal (inverted) mass hierarchy and the band width is obtained by varying the value of 𝛿. The probabilities for 𝑃𝜇𝑒 look similar with the hierarchies interchanged. Note the different scales of the axes.

In order to successfully determine the neutrino mass hierarchy in a single accelerator experiment, two conditions are of major importance. (1) The baseline must be long enough to allow for a significant value of 𝑉𝐸 in order to separate the neutrino and antineutrino oscillation probabilities. To separate the mass hierarchy determination from the effects of the CP phase, this separation must be large enough to avoid overlap of the probabilities within the experimental uncertainties. (2) The statistics must be high enough and the systematics low enough in order to make the split statistically significant. The literature contains several proposals for long baseline experiments with baselines of several thousands of kilometers in order to satisfy these conditions. However, as we will discuss later, the large value of 𝜃₁₃ also provides us with an opportunity to pin down the value of 𝛿. Such measurements require the presence of interference terms which will be small at the very long baselines, and instead medium long baselines around 1000 km, such as the 810 km baseline shown in Figure 13, may be preferable due to the significant 𝛿 dependence of probabilities.

4.3.1. CP Violation Effects and the Mass Hierarchy

Figure 13 shows a significant dependence of the probabilities on the CP-violating phase 𝛿, especially at small baselines. We are mainly interested in the oscillation probability at the first or second oscillation maximum, where an experiment would typically be placed. In these baselines 𝐿 the 𝜈_𝜇-𝜈_𝑒 oscillation probability (the “golden channel”) can be expanded in the small quantity Δ𝑚²₂₁𝐿/2𝐸 which gives [69] 𝑃_𝑒𝜇≈𝑠²₂₃𝑃^2𝑓+𝑐₁₃sin2𝜃₁₃sin2𝜃₁₂sin2𝜃₂₃Δ𝑚²₂₁2𝐸𝑉×sin𝑉𝐿2sinΔ𝑚²₃₁𝐿4𝐸cos𝛿−Δ𝑚²₃₁𝐿,4𝐸 (159) where 𝑃^2𝑓 is the two-flavor oscillation probability discussed earlier. In (159) we have neglected terms of the second (and higher) order in Δ𝑚²₂₁𝐿/2𝐸 (while the first neglected term is not suppressed by 𝜃₁₃, for the value of 𝜃₁₃ measured by reactor experiments the suppression by the solar mass square splitting is about 6 times stronger) as well as the matter effect on Δ𝑚²₃₁. It is the second term that is responsible for creating the band of different oscillation probabilities displayed in Figure 13, and hence, for creating the sign degeneracy in accelerator neutrino experiments. The appearance of the sin(𝑉𝐿/2) term is an inheritance from the magic baseline oscillations and will vanish the 𝛿-dependent term when 𝑉𝐿=2𝜋. Furthermore, we can observe that this term contains all of the mixing angles in the same way as the Jarlskog invariant, which is expected due to the CP dependence of the term.

4.4. Determination of Hierarchy with Atmospheric Neutrinos

4.4.1. Neutrino Fluxes

The original flux of atmospheric neutrinos contains incoherent components of 𝜈_𝑒, 𝜈_𝜇 and the corresponding antineutrinos, while the original 𝜈_𝜏 flux is negligible. We introduce Φ⁰_𝑒 and Φ⁰_𝜇, the electron and muon neutrino fluxes, as well as Φ⁰_𝑒 and Φ⁰_𝜇, the electron and muon antineutrino fluxes, at the detector in the absence of oscillations. The flavor ratios Φ𝑟≡^0𝜇Φ⁰_𝑒,𝑟≡Φ⁰_𝜇Φ⁰_𝑒 (160) increase with energy.

There is a mild neutrino-antineutrino asymmetry: the neutrino flux Φ⁰_𝜇/Φ⁰_𝜇≈0.8-0.9. All the fluxes (at 𝐸>1 GeV) decrease rapidly with energy Φ⁰_𝛼∝𝐸^−𝑘, 𝑘=𝑘(𝐸)=3–5, and an azimuthal dependence shows up at low energies.

The flux of neutrinos of flavor 𝜈_𝛼 at a detector, with oscillations taken into account, is given by Φ_𝛼=Φ⁰_𝑒𝑃_𝑒𝛼+Φ⁰_𝜇𝑃_𝜇𝛼=Φ⁰_𝑒𝑃_𝑒𝛼+𝑟𝐸,Θ_𝜈𝑃_𝜇𝛼,𝛼=𝑒,𝜇,𝜏. (161) Similar expressions hold for the antineutrino fluxes. Inserting the analytic expressions for the probabilities (122)–(126), one finds Φ_𝑒Φ⁰_𝑒=1+𝑟𝑠²₂₃−1𝑃_𝑒3+𝑟𝑐²₂₃−1𝑃_𝑒2+𝑟𝑃^𝛿_𝜇𝑒,Φ_𝜇Φ⁰_𝜇≈1−2𝑠²₂₃𝑐²₂₃1−Re𝐴^∗₂₂𝐴₃₃−𝑠²₂₃𝑟𝑟𝑠²₂₃−1𝑃_𝑒3−𝑐²₂₃𝑟𝑟𝑐²₂₃−1𝑃_𝑒2+𝑃^𝛿_𝜇𝜇+1𝑟𝑃^𝛿_𝑒𝜇,Φ_𝜏Φ⁰_𝜇≈2𝑠²₂₃𝑐²₂₃1−Re𝐴^∗₂₂𝐴₃₃−𝑐²₂₃𝑟𝑟𝑠²₂₃−1𝑃_𝑒3−𝑠²₂₃𝑟𝑟𝑐²₂₃−1𝑃_𝑒2+𝑃^𝛿_𝜇𝜏+1𝑟𝑃^𝛿_𝑒𝜏, (162) where 𝑃_𝑒3≡|𝐴_𝑒3|² and 𝑃_𝑒2≡|𝐴_𝑒2|² are defined in Section 3.13. In the factorization approximation they correspond to the atmospheric and solar oscillation modes. The 𝛿-dependent terms have been introduced in (144).

Using unitarity relations 𝐴₂₂²=1−𝐴_2𝑒²−𝐴₂₃²≈1−𝐴_2𝑒²=1−𝑃_𝑒2, (163) where the terms proportional to |𝐴₃₂|² have been neglected we can approximate Re𝐴^∗₂₂𝐴₃₃≈1−𝑃_𝑒31−𝑃_𝑒2cos𝜓. (164) Here 𝜓≡arg𝐴₃₃𝐴^∗₂₂ is the relative phase between the two amplitudes. For the 𝜈_𝑒 flux, we then obtain Φ_𝑒Φ⁰_𝑒≈1+𝑟𝑠²₂₃−1𝑃_𝑒3+𝑟𝑐²₂₃−1𝑃_𝑒2+𝑟sin2𝜃₂₃𝑃_𝑒3𝑃_𝑒2cos𝜙−𝛿. (165) The oscillated fluxes satisfy the sum rule Φ_𝑒+Φ_𝜇+Φ_𝜏=Φ⁰_𝑒+Φ⁰_𝜇, (166) which simply reflects the unitarity of transitions and, consequently, conservation of the total flux in oscillations.

The formulas (161) also show the screening effect. Terms with oscillation probabilities driven by the 1-2 and 1–3 mixings appear with the “screening” factors [70, 71]: 𝑃_𝑒3 with (𝑟𝑠²₂₃−1) and 𝑃_𝑒2 with (𝑟𝑐²₂₃−1). The contribution of the “atmospheric mode” vanishes along the line 𝑟(𝐸,Θ_𝜈)=1/𝑠²₂₃, whereas the contribution of the “solar mode” vanishes along 𝑟(𝐸,Θ_𝜈)=1/𝑐²₂₃. For maximal mixing both contributions vanish along the same line, 𝑟(𝐸,Θ_𝜈)=2. For the neutrino energies above 0.1 GeV, 𝑟>1.8-1.9, and only one of these contributions can vanish for substantial deviation of the 2-3 mixing from maximal: 𝑠²₂₃ or 𝑐²₂₃<0.45. Thus, both the effects of 1-2 and 1–3 mixing turn out to be subleading and the oscillation effects are well described by the first order approximation of 2-3 vacuum oscillations.

In the 𝜈_𝜇 flux, the contributions of the 1-2 and 1–3 modes are suppressed by additional factors 𝑠²₂₃/𝑟 and 𝑐²₂₃/𝑟, respectively. There is no suppression of the interference terms, which depend on the CP violation phase. Furthermore, in the 𝜈_𝑒 flux the interference term is enhanced by the flux ratio 𝑟. There is no suppression of the interference terms of the 1-2 and 1–3 modes in the 𝜇-𝜏 mode.

4.4.2. Sensitivity to Mass Hierarchy

Let us discuss the sensitivity of large water or ice detectors of atmospheric neutrinos to the neutrino mass hierarchy. The 𝜈_𝜇-like events correspond to interactions 𝜈_𝜇+𝑁→𝜇+𝑋, 𝜈_𝜇+𝑁→𝜇⁺+𝑋 and can be observed as events with muon tracks and hadron cascades. There are also some contributions from 𝜈_𝜏 which produce 𝜏 with subsequent decay into 𝜇. The number of 𝜈_𝜇-like events in the 𝑖𝑗-bin in the 𝐸_𝜈-cos𝜃_𝑧 plane equals 𝑁^𝑁𝐻_{𝑖𝑗,𝜇}=2𝜋𝑁_𝐴𝜌𝑇_{Δ𝑖cos𝜃𝑧}𝑑cos𝜃_𝑧×_{Δ𝑗𝐸𝜈}𝑑𝐸_𝜈𝑉_eff𝐸_𝜈𝐷_𝜇𝐸_𝜈,𝜃_𝑧, (167) where 𝑇 is the exposure time, 𝑁_𝐴 is the Avogadro number, 𝜌 is the density of ice, 𝑉_eff(𝐸_𝜈,𝜃_𝑧) is the effective volume of the detector, and the number density of events per unit time per target nucleon is given by 𝐷_𝜇𝐸_𝜈,𝜃_𝑧=𝜎^𝐶𝐶Φ⁰_𝜇𝑃_𝜇𝜇+Φ⁰_𝑒𝑃_𝑒𝜇+𝜎^𝐶𝐶Φ⁰_𝜇𝑃_𝜇𝜇+Φ⁰_𝑒𝑃_𝑒𝜇. (168) It is assumed here that experiments do not distinguish the neutrino and antineutrino events and corresponding signals are summed up.

The fine-binned distribution of events (166) is shown in Figure 14. For illustration we use the effective volume of PINGU with 22 additional strings [72]. which increases from ~2 Mt at 𝐸_𝜈=2 GeV to 20 Mt at 𝐸_𝜈=20 GeV. The pattern of the event number distribution follows the oscillatory picture due to the 𝜈_𝜇-𝜈_𝜇 mode of oscillations with a certain distortion in the resonance region. The maxima and minima are approximately along the lines of equal oscillation phases 𝐸_𝜈∼𝜙₃₂Δ𝑚²₃₂|cos𝜃_𝑧|𝑅_⊕ (where 𝑅_⊕ is the Earth radius), with distortion in the resonance region 𝐸_𝜈=(4--10) GeV. In the high density bins, the number of events reaches 200 and the total number of events is about 10⁵.

(a)

(b)

Figure 14 

Left: the binned distribution of the number of 𝜇 events in PINGU after 1 year under the assumption that the neutrino hierarchy is normal. Right: the 𝑁-𝐼 hierarchy asymmetry of 𝜈𝜇 events in the 𝐸𝜈-cos𝜃𝑧 plane. The absolute value of the asymmetry in a given bin determines the statistical significance of the difference of the numbers of events for the inverted and normal mass hierarchies. Both figures from [56].

The expression for the density of events (168) can be written as 𝐷_𝜇^𝑁𝐻=𝜎^𝐶𝐶𝐸_𝜈Φ⁰_𝜇𝑃_𝜇𝜇+1𝑟𝑃_𝑒𝜇+𝜅_𝜇𝑃_𝜇𝜇+1𝑟𝑃_𝑒𝜇, (169) where 𝜅_𝜇≡𝜎^𝐶𝐶Φ⁰_𝜇𝜎^𝐶𝐶Φ⁰_𝜇. (170) Similarly one can determine the number of events for inverted mass hierarchy. Let us introduce the 𝑁-𝐼 hierarchy asymmetry for the 𝑖𝑗-bin in the (𝐸_𝜈-cos𝜃_𝑧) plane as 𝐴^𝑁-𝐼_{𝜇,𝑖𝑗}≡𝑁^𝐼𝐻_{𝜇,𝑖𝑗}−𝑁^𝑁𝐻_{𝜇,𝑖𝑗}𝑁^𝑁𝐻_{𝜇,𝑖𝑗}. (171) The moduli of the asymmetry (171) are the measures of statistical significance of the difference of the number of events for the normal and inverted mass hierarchies: 𝑆_𝑖𝑗=|𝐴_𝑖𝑗|.

The strongest effect of hierarchy change is in the strips along the constant phase lines in the energy interval 𝐸_𝜈=(4–12) GeV, where these lines are distorted by the matter effect. Here the asymmetry changes sign with the zenith angle, and the number of intervals with the same sign asymmetry increases with the decrease of energy. The 𝜈_𝜏→𝜏→𝜇 events can be considered as background events and treated within ~5% systematic errors.

4.4.3. Measurements

According to Figure 14, the hierarchy asymmetry of the 𝜈_𝜇 events has opposite signs in different parts of the oscillogram. Thus, the integration over 𝐸_𝜈 and cos𝜃_𝑧 substantially reduces the sensitivity to the hierarchy. Due to this, a relatively good reconstruction of the neutrino energy and direction is required to identify the hierarchy. The uncertainties of the reconstruction of energy 𝜎_𝐸 and angle 𝜎_𝜃 should be comparable to or smaller than the sizes of the domains with the same sign of the asymmetry. The oscillograms for the reconstructed neutrino energy 𝐸^𝑟_𝜈 and angle 𝜃^𝑟_𝑧 can be obtained by smearing of the 𝐸_𝜈-cos𝜃_𝑧 oscillograms with the reconstruction functions of the width 𝜎_𝐸 and angle 𝜎_𝜃.

Small uncertainties 𝜎_𝐸 and 𝜎_𝜃 require rather precise measurements of the energy 𝐸_𝜇 and direction 𝜃_𝜇 of the muon, as well as energy of the accompanying hadron cascade 𝐸_ℎ. Then the neutrino energy equals 𝐸^𝑟_𝜈=𝐸_𝜇+𝐸_ℎ. The reconstruction of the neutrino direction is more involved. In the first approximation, one can use 𝜃_𝜈≈𝜃_𝜇 with a spread which decreases with energy: 𝜎_𝜃∼𝐴𝑚_𝑝/𝐸_𝜈(𝐴=𝑂(1)). Knowledge of the hadron cascade energy allows reducing this uncertainty. Further improvements could be possible if some information about geometry of the cascade is available. A possibility to separate (at least partially) the neutrino and antineutrino samples would significantly improve sensitivity to the mass hierarchy, as well as to CP violation.

All this imposes conditions on the detector characteristics. According to Figure 14, the most sensitive region to the hierarchy is around the resonance and above: 𝐸=(5–15) GeV. The number of events in Super-Kamiokande is too small, but (upgraded) ice and underwater detectors of the multimegaton (~10 Mt) scale could collect around the order of 10⁵𝜈_𝜇 events a year in this range so that a high statistics study becomes possible.

A small enough spacing between the PMTs (~10–20 m between strings and 3–5 m in the vertical direction) will allow the reduction of the threshold down to a few GeV and perform reasonably good measurements of the muon and hadron cascade characteristics. Very high statistics will also allow resolving the problem of parameter degeneracy; effects qualitatively similar to the mass hierarchy effect can be obtained by small (within 1𝜎 interval) variations of Δ𝑚²₃₂ and 𝜃₂₃. The effect of an unknown CP phase is small.

High statistics would allow resolving the degeneracy problem by selecting specific regions in the 𝐸_𝜈-cos𝜃_𝑧 for the analysis, where effects of Δ𝑚²₃₂ are suppressed in comparison to the hierarchy effects or averaged out as a result of specific integration. High statistics also allow performing an analysis of the data using Δ𝑚²₃₂ and 𝜃₂₃ as fit parameters. This will open a possibility to determine the mass hierarchy and measure these parameters simultaneously.

Note that other experimental techniques using atmospheric neutrinos may also prove valuable for determination of the mass hierarchy. In particular, experiments that can separate neutrinos from antineutrinos on an event basis need a significantly lower number of events to obtain the same sensitivity. Thus, such detectors can be smaller in size as compared to the neutrino telescopes. In this context, a magnetized iron calorimeter, such as the India-based Neutrino Observatory [73], could also provide an important contribution to the determination of mass hierarchy. The capabilities of detectors using charge identification were studied in [74].

4.4.4. Interplay between Accelerator and Atmospheric Neutrinos

The atmospheric neutrino data can also be used to complement the data from accelerator neutrino experiments in order to extract the most information possible. As was demonstrated in [74], the atmospheric neutrino determination of the neutrino mass hierarchy can be significantly affected by the addition of external priors and, in particular, may lead to different sensitivity to the neutrino mass hierarchy in the cases of true normal or inverted hierarchy. However, once external input on the neutrino oscillation parameters is included by considering also other experiments, the room to mimic the true oscillation pattern in the wrong hierarchy becomes much more restricted and the sensitivity to the hierarchy increases. Adding the accelerator experiments’ own sensitivity to the mass hierarchy, a measurement may be possible even for the current generation of accelerator experiments by the addition of detector capable of lepton charge identification. This has been discussed in [75] and the prospects of using a magnetized iron calorimeter detector to augment the current generation of accelerator experiments are a 2–4𝜎 determination of the mass hierarchy within 10 years of data taking, depending on the true value of the oscillation parameters and the characteristics of the detector.

5. Discussion and Conclusions

In this paper, we have described the effects of neutrino propagation in matter relevant to experiments with atmospheric and accelerator neutrinos and aimed at the determination of the neutrino mass hierarchy and CP violation. Thus, to a large extent, we have focused on neutrino propagation in the Earth matter.(1) At relatively low energies, the dominant effect of neutrino interactions with matter is the elastic forward scattering, which is described by an effective potential. Neutrino evolution in matter is then described by a Schrödinger-like equation including this effective potential. The potential differences for neutrinos of different types influence the flavor evolution of the system of mixed neutrinos. In the majority of realistic situations, neutrinos propagate in normal (unpolarized nonrelativistic) matter with nearly constant or slowly changing density.(2) Matter modifies the neutrino flavor mixing and changes the eigenvalues of the Hamiltonian of propagation. This is equivalent to a modification of the dispersion relations of neutrinos. The influence of matter on mixing of neutrinos has a resonance character. At energies or densities for which the eigenfrequency of the neutrino system with mixing 𝜔_𝑖𝑗=Δ²_𝑖𝑗/2𝐸 equals approximately the eigenfrequency of the medium 2𝜋/𝑙₀, the mixing in matter becomes maximal. Large mixing shifts the position of the resonance to lower values of the potential. At usual densities, there are two resonances related to the two mass squared differences Δ𝑚²₂₁ and Δ𝑚²₃₁ between the neutrino mass eigenstates. The resonances are realized in oscillation channels involving electron neutrinos.(3) In many practical situations, knowledge of neutrino mixing in matter and the eigenstates of the Hamiltonian in matter allows finding the results of the neutrino flavor evolution immediately. This includes neutrino oscillations in matter with constant density and also adiabatic conversion of neutrinos, where the averaged oscillation results can be written down immediately. In the nonaveraged case, the problem is reduced to finding the oscillation phase (integrating the energy splittings over distance). In this sense the Nature has implemented the most (computationally) simple setups. The very convenient presentation of mixing in matter can be obtained as series expansion in the ratio of the two mass squared differences, 𝑟_Δ (perturbative diagonalization of the effective Hamiltonian), which allows to understanding a number of subtle results. The simplest and physically transparent description of dynamics of neutrino flavor evolution can be obtained in the propagation basis (in the case of the standard parameterization). In this basis, the CP-violating phase and 2-3 mixing do not influence the evolution and the amplitudes of transitions do not depend on 𝛿 or 𝜃₂₃. The dependence on these parameters appears as a result of projecting the states of the propagation basis back to the flavor states at production and detection. In many practical cases the 3𝜈 evolution can be reduced to evolution of two neutrino systems with certain corrections.(4) There are two practically important cases: (i) neutrino propagation in matter with constant or nearly constant density and (ii) neutrino propagation in matter with slowly (adiabatically) changing density.(5) In the case of constant density, flavor evolution has a character of oscillations with parameters determined by mixing and mass splitting in matter. The oscillations are an effect of a phase difference increase in the course of neutrino propagation. The resonance enhancement of oscillations is realized in an energy region around 𝐸_𝑅. If the density is approximately constant, then the results can be obtained by using perturbation theory in the deviation of the density distribution from a constant one. The accuracy improves if the density profile is symmetric with respect to the middle point of the neutrino trajectory, as is realized for neutrinos crossing the Earth. A simple and rather precise semianalytical description of neutrino oscillations in matter with varying density can be obtained in the limits of small density, 𝑉<Δ𝑚²_𝑖𝑗/2𝐸, and high density, 𝑉≫Δ𝑚²_𝑖𝑗/2𝐸. The latter gives a very accurate description of neutrino flavor evolution in the Earth at 𝐸>(8--10) GeV.(6) In a medium with slowly changing density, adiabatic conversion takes place. This effect is related to the change of mixing in matter due to density change. Adiabaticity implies that there are no transitions among the eigenstates of the instantaneous Hamiltonian during propagation. The strongest flavor transformation is realized when the initial density is much larger, and the final one is much lower than the resonance density. In this case, the initial state (and due to adiabaticity, the state at any other moment of evolution) practically coincides with one of the eigenstates. Therefore, oscillation effects are absent and nonoscillatory flavor conversion takes place. This is realized for supernova neutrinos and approximately for high energy solar neutrinos. In general, if the initial mixing is not strongly suppressed, an interplay of adiabatic conversion and oscillations occurs. Adiabatic transformations are also realized for neutrinos with energy ≤1 GeV propagating in the mantle of the Earth. In particular, this means that the oscillation depth at the detector is determined by mixing at the surface of the Earth and not by mixing at average density. Until now, the mater effects have been observed in solar neutrinos and, indirectly, in atmospheric neutrinos and there is good chance that they will be observed by new generation of the accelerator and atmospheric neutrino experiments.(7) Strong flavor transition can be realized without enhancement of mixing. This occurs in matter with periodic or quasiperiodic density change when the parametric resonance condition is fulfilled. For small mixing strong transition requires a large number of periods. A similar enhancement can take place in matter with several layers of different densities. Here the enhancement occurs when a certain correlation between the oscillation phases in each layer and amplitudes of oscillations determined by mixing is present. The case of a medium with 3 layers (1.5 periods) is of practical interest for neutrinos crossing both the mantle and the core of the Earth. For a multilayer medium two conditions must be satisfied to have strong transitions: the amplitude (collinearity) and the phase conditions.(8) For neutrinos crossing a small amount of matter, such as accelerator experiments with baselines up to (1-2)⋅10³ km, the column density of matter is small and, according to the minimal width condition, the matter effect on oscillations is small regardless of energy, vacuum mass splitting, and neutrino mixing. Furthermore, if the oscillation phase is small, then mimicking of vacuum oscillations occurs.(9) A comprehensive description of the neutrino flavor transitions in the Earth is given in terms of neutrino oscillograms of the Earth. After the recent determination of the 1–3 mixing, the structure of oscillograms is well fixed. The salient features of oscillograms at high energies (due to 1–3 mixing) are the MSW resonance peak in the mantle domain, three parametric ridges, and the MSW peak in the core domain. At low energies (due to 1-2 mixing), there are three peaks, due to the MSW resonance, and the parametric ridge. The positions of all these and other structures are determined by the generalized phase and amplitude conditions. In the case of normal mass hierarchy, the resonance peaks induced by the 1–3 mixing are in the neutrino channels. For inverted mass hierarchy they are in the antineutrino channels. This is the foundation for determining the neutrino mass hierarchy. The resonance structures due to the 1-2 mixing are always in the neutrino channels, since the sign of the small mass square difference has been fixed.(10) The CP properties of the oscillograms (their dependence on CP phase) are determined by the CP domains, areas in which the CP violation effect has the same sign. The borders of these domains are approximately determined by the grids of the magic lines (solar and atmospheric magic lines) and the lines where the oscillation phase condition is fulfilled.(11) Measurements of matter effects in neutrino oscillations provide a good opportunity to determine the neutrino mass hierarchy. The 1-2 ordering has been determined due to the matter effect of solar neutrinos. The 1–3 ordering can be identified by studying the matter effects in accelerator and atmospheric neutrino experiments. There is a good chance that future studies of the atmospheric neutrinos with multi-megaton underwater (ice) detectors will be able to establish the mass hierarchy. With a threshold of a few GeV, these detectors will be sensitive to the resonance region (∼6--10)GeV, where the difference of probabilities for the normal and inverted mass hierarchies is maximal. The challenges here are the accuracy of reconstruction of the neutrino energies and directions. Integration over the energy and angle, as well as summation of neutrino and antineutrino signals, diminishes the sensitivity to the hierarchy. Another problem is the degeneracy of the hierarchy effects with the effects of other neutrino parameters, in particular with Δ𝑚²₃₂ and 𝜃₃₂.(12) In accelerator experiments, many of the problems mentioned above are absent. However, existing and proposed accelerator experiments will cover only peripheral regions of oscillograms where enhancement of oscillations is very weak and oscillatory structures are rather poor. As a consequence the problem of degeneracy here is even more severe.