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