# Neutrino Propagation in Matter

Authors: Mattias Blennow; Alexei Yu. Smirnov
Journal: Advances in High Energy Physics
 (2013)

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## 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.


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## Body

## 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 Section2 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,νfT≡(νe,νμ,ντ), mixed in vacuum:
(1)νf=UPMNSνm.
Here UPMNS is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix [7–9] and νmT≡(ν1,ν2,ν3) is the vector of mass eigenstates with masses mi (i=1,2,3). We will use the standard parameterization of the PMNS matrix,
(2)UPMNS=U23(θ23)IδU13(θ13)Iδ*U12(θ12),
which is the most suitable for describing usual matter effects. In (2) Uij(θij) are the matrices of rotations in the ij-planes with angles θij and Iδ≡diag(1,1,eδ).In vacuum the flavor evolution of these neutrinos is described by the Schrödinger-like equation(3)idνfdt=MM†2Eνf,
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
(4)H0=12EUPMNSMdiag2UPMNS†,
where Mdiag2≡M†M=diag(m12,m22,m32) and we take the masses mi 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 mediumVf can be computed as a forward scattering matrix element Vf=〈Ψ|Hint|Ψ〉. Here Ψ is the wave function of the system of neutrino and medium, and Hint is the Hamiltonian of interactions.At low energies, the HamiltonianHint is the effective four-fermion Hamiltonian due to exchange of the W and Z  bosons:
(5)Hint=GF2ν-γμ(1-γ5)ν×{e-γμ(gV+gAγ5)e+p-γμ(gVp+gApγ5)p+n-γμ(gVn+gAnγ5)n},
where gV and gA are the vector and axial vector coupling constants.In the Standard Model the matrix of the potentials in the flavor basis is diagonal:Vf=diag(Ve,Vμ,Vτ,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 theγ0 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 gVp and gVn 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 νa is
(6)Va=2GF(δaene-12nn),
where ne and nn are the densities of electrons and neutrons, respectively.Only the difference of potentials has a physical meaning. Contribution of the neutral current scattering toV is the same for all active neutrinos. Since Va (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, Vμ-Vτ=0. Furthermore, the difference of the potentials for νe and νa is due to the charged current scattering of νe on electrons (νee→νee) [1]:
(7)V=Ve-Va=2GFne.The difference of potentials leads to the appearance of an additional phase difference in the neutrino system:ϕmatter≡(Ve-Va)t≈Vx. This determines the refraction length, the distance over which an additional “matter” phase equals 2π:
(8)l0≡2πVe-Va=2πGFne.
Numerically,
(9)l0=1.6·109cm1g/cm3nemN,
where mN is the nucleon mass. The corresponding column density d≡l0ne=2π/GF 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 theνμ-ντ 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(Ve,0,0) with Ve 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]:
(10)Hf=12EUPMNSMdiag2UPMNS†+V^.
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
(11)U⟶U*,V⟶-V.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 HamiltonianHf describes evolution in time, with the connection x=vt≈x=ct, (12) can be rewritten as idνf/dx=(H0+V^)νf with V=V(x), so it can be used as an evolution equation in space.Due to the strong hierarchy ofΔm2 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, νfT=(νe,νa), in matter is
(12)idνfdt=[Δm24E(-cos2θsin2θsin2θcos2θ)+(12Ve00-12Ve)]νf,
where the Hamiltonian is written in symmetric form.


### 2.3. Mixing and Eigenstates in Matter

The mixing in matter is defined with respect toνim—the eigenstates of the Hamiltonian in matter Hf.As usual, the eigenstates are obtained from the equation(13)Hfνim=Himνim,
where Him are the eigenvalues of Hf. If the density and therefore Hf are constant, νim correspond to the eigenstates of propagation. Since Hf≠H0, the states νim differ from the mass states, νi. For low density n→0, the vacuum eigenstates are recovered: νim→νi. If the density, and thus Hf change during neutrino propagation, νim and Him should be considered as the eigenstates and eigenvalues of the instantaneous Hamiltonian: Hf=Hf(x), νim=νim(x), and Him=Him(x). For n→0 we have Him→mi2/2E.The mixing in matter is a generalization of the mixing in vacuum (1). Recall that the mixing matrix in vacuum connects the flavor neutrinos, νf, and the massive neutrinos, νmass. The latter are the eigenstates of Hamiltonian in vacuum: νH=νmass. Therefore, the mixing matrix in matter is defined as the matrix which relates the flavor states with the eigenstates of the Hamiltonian in matter νHT=(ν1m,ν2m,ν3m):
(14)νf=UmνH.From (13) we find that
(15)νjm†Hfνim=Himδji.
Furthermore, the Hamiltonian can be represented in the flavor basis as
(16)Hf=∑αβ‍Hαβνανβ†.
Inserting this expression as well as the relation νjm=Uαjm*να, which follows from (14), into (15) one obtains
(17)∑αβ‍Uαjm*HαβUβim=Himδji
or in matrix form Um†HfUm=Hdiag=diag(H1m,H2m,H3m). Thus, the mixing matrix Um can be found diagonalizing the full Hamiltonian. The columns of the mixing matrix, Ui≡(Ueim,Uμim,Uτim), are the eigenstates of the Hamiltonian Hf which correspond to the eigenvalues Him. Indeed, it follows from (17) that HfUm=UmHdiag.Equation (14) can be inverted to νH=Um†νf, or in components νim=Uαim*να,α=e,μ,τ. According to this, the elements of mixing matrix determine the flavor content of the mass eigenstates so that |Uαim|2 gives the probability to find να in a given eigenstate νim. 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 the2ν case, there is single mixing angle in matter θm and the relations between the eigenstates in matter and the flavor states read
(18)νe=cosθmν1m+sinθmν2m,νa=cosθmν2m-sinθmν1m.
The angle θm is obtained by diagonalization of the Hamiltonian (12) (see previous section):
(19)sin22θm=1Rsin22θ,R≡(cos2θ-2VEΔm2)2+sin22θ,
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 Him equals
(20)ωm≡H2m-H1m=Δm22ER.
This difference is also called the level splitting or oscillation frequency, which determines the oscillation length: lm=2π/ωm (see Section 3.2).The matter potential andΔm2 always enter the mixing angle and other dimensionless quantities in the combination
(21)2EVΔm2=lνl0,
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:
(22)Hf=ωm2(-cos2θmsin2θmsin2θmcos2θm).


#### 2.4.1. Resonance and Level Crossing

According to (19) the effective mixing parameter in matter, sin22θm, depends on the electron density and neutrino energy through the ratio (21) of the oscillation and refraction lengths, x=lν/l0∝EV. The dependence sin22θm(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 sin22θm on E has a resonant character [3]. At
(23)lν=l0cos2θ
the mixing becomes maximal: sin22θm=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/lν=Δm2/2E, coincides with the eigenfrequency of the medium, 2π/l0=1/V. The resonance condition (23) determines the resonance density
(24)neR=Δm22Ecos2θ2GF.
The width of resonance on the half of height (in the density scale) is given by 2ΔneR=2neRtan2θ. Similarly, for fixed ne one can introduce the resonance energy and the width of resonance in the energy scale. The width can be rewritten as ΔneR=n0sin2θ, where n0≡Δm2/22EGF. When the mixing approaches its maximalvalue: θ→π/4, the resonance shifts to zero density: neR→0, and the width of the resonance increases converging to the fixed value: ΔneR→n0.Resonance in neutrino mixing. The dependence ofsin22θmij on the product VE for vacuum mixing: sin22θ12=0.851, Δm212=7.59·10-5 eV2 (red) and sin2θ13=0.0241, Δm312=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.
(a)
(b)
(c)
(d)In a medium with varying density, the layer in which the density changes in the intervalneR±ΔneR is called the resonance layer. In this layer the angle θm varies in the interval from π/8 to 3π/8.ForV≪VR, the mixing angle is close to the vacuum angle: θm≈θ, while for V≫VR, the angle becomes θm≈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 Hf which relies on the strong hierarchy of the mass squared differences:
(25)rΔ≡Δm212Δm312≈0.03.
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 UPMNS=U23IδU13U12. Here we will describe the case of normal mass hierarchy: Δm312>0,Δm322>0. Subtracting from the Hamiltonian the matrix proportional to the unit matrix m12/2EI, we obtain
(26)Mdiag2=Δm312diag(0,rΔ,1).


#### 2.5.1. Propagation Basis

The propagation basis,ν~=(νe,ν~2,ν~3)T, which is most suitable for consideration of the neutrino oscillations in matter, is defined through the relation
(27)νf=U23Iδν~.
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
(28)H~=12EU13U12Mdiag2U12†U13†+V^.
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 HamiltonianH~ can be written as
(29)H~=Δm3122E×(s132+s122c132rΔ+2VeEΔm312s12c12c13rΔs13c13(1-s122rΔ)…c122rΔ-s12c12s13rΔ……c132+s122s132rΔ).Here all the off-diagonal elements contain small parametersrΔ and/or s13. Notice that, for the measured oscillation parameters, s132~rΔ.


#### 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 rΔ. After a 1–3 rotation
(30)ν~=U13(θ13m)ν′
over the angle θ13m determined by
(31)tan2θ13m=sin2θ13cos2θ13-2EV′/Δm312,whereV′=V1-s122rΔ,
the 1–3 element of (29) vanishes. The expression (31) differs from that for 2ν mixing in matter by a factor (1-s122rΔ), which increases the potential and deviates from 1 by
(32)ξ≡s122rΔ≈10-2.
After this rotation the Hamiltonian in the ν′ basis (30) becomes
(33)H′=Δm3122E×(h11s12c12rΔcos(θ13m-θ13)0…c122rΔs12c12rΔsin(θ13m-θ13)……h33),where(34)h11,33=12[{[cos2θ13(1-ξ)-x]2}(1+ξ+x)∓[cos2θ13(1-ξ)-x]2+sin22θ13(1-ξ)2],
and x≡2EV/Δm312. For ξ=0, these elements are reduced to the standard 2ν expressions. In the limit of zero density, x→0, h11=ξ=s122rΔ, and consequently the 11-element of the Hamiltonian equals H11′=s122Δm122/2E.In the lowestrΔ approximation one can neglect the nonzero 2-3 element in (33). The state ν3′ then decouples and the problem is reduced to a two-neutrino problem for (ν1′,ν2′). The eigenvalue of this decoupled state equals
(35)H3m≈Δm3122Eh33,h33≥1.
The diagonalization of the remaining 1-2 submatrix is given by rotation
(36)ν′=U12(θ12m)νm,
where θ12m is determined by
(37)tan2θ12m=sin2θ12rΔcos(θ13m-θ13)c122rΔ-h11.
Here h11 and θ13m are defined in (34) and (31), respectively. The eigenvalues equal
(38)H1m,2m=Δm3124E[{(c122rΔ-h11)2}c122rΔ+h11∓(c122rΔ-h11)2+sin22θ12rΔ2cos2(θ13m-θ13)].According to this diagonalization procedure in the lowest order inrΔ the mixing matrix in matter is given by
(39)Um=U23(θ23)IδU13(θ13m)U12(θ12m),
where mixing angles θ12m and θ13m 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 H1m and H2m are given in (38) and H3m is determined by (35).The 2-3 element of matrix (33) vanishes after additional 2-3 rotation by an angle θ23′~rΔ:
(40)tan2θ23′=sin2θ12rΔsin(θ13m-θ13)h33-c122rΔ,
which produces corrections of the next order in rΔ. With an additional 2-3 rotation the mixing matrix becomes
(41)Um=U23(θ23)IδU13(θ13m)U12(θ12m)U23(θ23′)≈U23(θ23m)IδmU13(θ13m)U12(θ12m),
where
(42)U23(θ23m)Iδm=U23(θ23)IδU23(θ-23),
and the last 2-3 rotation is on the angle θ-23 determined through sinθ-23=sinθ23′/cosθ13m. 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(θ23′). 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
(43)sinδmsin2θ23m=sinδsin2θ23,
that is, the combination sinδsin2θ23 is invariant under inclusion of matter effects. Furthermore, θ23m≈θ23 and δm≈δ up to corrections of the order O(rΔ). The results described here allowing understand the behavior of the mixing parameters sin22θmij in the EV region of the 1–3 resonance and above it (see Figure 1).In Figure2 we present dependence of the flavor content of the neutrino eigenstates on the potential. The energy level scheme, the dependence of the eigenvalues Him 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.The flavor contents of the eigenstates of the Hamiltonian in matter as functions ofEV. 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.The energy level scheme. We here show the dependence of the eigenvalues of the Hamiltonian in matter onEV. Note that we are plotting 2EHim, which goes to Δmi12 for low VE. The left (right) panel corresponds to normal (inverted) mass ordering.
(a)
(b)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 theνe-ντ′ channel, is associated to the 1–3 mixing and large mass splitting. According to (31) θ13m=π/4 at
(44)V13R=cos2θ13(1-s122rΔ)Δm3122E.(2)
The L resonance at low densities is associated to the small mass splitting and 1-2 mixing. It appears in theνe′-νμ′ channel, where νe′ and νe differ by small (at low densities) rotation given by an angle ~θ13 (see (31)). According to (37) the position of the L-resonance, θ12m=π/4, is given by c122rΔ=h11, where h11 is defined in (34). This leads to
(45)V12R=cos2θ12Δm2122E1c132.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:(46)H1m⟶-V,H2m⟶Δm212c1222Eν,H3m⟶Δm312c1322Eν.


## 2.1. Refraction and Matter Potentials

The effective potential for a neutrino in mediumVf can be computed as a forward scattering matrix element Vf=〈Ψ|Hint|Ψ〉. Here Ψ is the wave function of the system of neutrino and medium, and Hint is the Hamiltonian of interactions.At low energies, the HamiltonianHint is the effective four-fermion Hamiltonian due to exchange of the W and Z  bosons:
(5)Hint=GF2ν-γμ(1-γ5)ν×{e-γμ(gV+gAγ5)e+p-γμ(gVp+gApγ5)p+n-γμ(gVn+gAnγ5)n},
where gV and gA are the vector and axial vector coupling constants.In the Standard Model the matrix of the potentials in the flavor basis is diagonal:Vf=diag(Ve,Vμ,Vτ,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 theγ0 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 gVp and gVn 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 νa is
(6)Va=2GF(δaene-12nn),
where ne and nn are the densities of electrons and neutrons, respectively.Only the difference of potentials has a physical meaning. Contribution of the neutral current scattering toV is the same for all active neutrinos. Since Va (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, Vμ-Vτ=0. Furthermore, the difference of the potentials for νe and νa is due to the charged current scattering of νe on electrons (νee→νee) [1]:
(7)V=Ve-Va=2GFne.The difference of potentials leads to the appearance of an additional phase difference in the neutrino system:ϕmatter≡(Ve-Va)t≈Vx. This determines the refraction length, the distance over which an additional “matter” phase equals 2π:
(8)l0≡2πVe-Va=2πGFne.
Numerically,
(9)l0=1.6·109cm1g/cm3nemN,
where mN is the nucleon mass. The corresponding column density d≡l0ne=2π/GF 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 theνμ-ντ 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(Ve,0,0) with Ve 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]:
(10)Hf=12EUPMNSMdiag2UPMNS†+V^.
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
(11)U⟶U*,V⟶-V.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 HamiltonianHf describes evolution in time, with the connection x=vt≈x=ct, (12) can be rewritten as idνf/dx=(H0+V^)νf with V=V(x), so it can be used as an evolution equation in space.Due to the strong hierarchy ofΔm2 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, νfT=(νe,νa), in matter is
(12)idνfdt=[Δm24E(-cos2θsin2θsin2θcos2θ)+(12Ve00-12Ve)]νf,
where the Hamiltonian is written in symmetric form.


## 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]:
(10)Hf=12EUPMNSMdiag2UPMNS†+V^.
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
(11)U⟶U*,V⟶-V.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 HamiltonianHf describes evolution in time, with the connection x=vt≈x=ct, (12) can be rewritten as idνf/dx=(H0+V^)νf with V=V(x), so it can be used as an evolution equation in space.Due to the strong hierarchy ofΔm2 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, νfT=(νe,νa), in matter is
(12)idνfdt=[Δm24E(-cos2θsin2θsin2θcos2θ)+(12Ve00-12Ve)]νf,
where the Hamiltonian is written in symmetric form.


## 2.3. Mixing and Eigenstates in Matter

The mixing in matter is defined with respect toνim—the eigenstates of the Hamiltonian in matter Hf.As usual, the eigenstates are obtained from the equation(13)Hfνim=Himνim,
where Him are the eigenvalues of Hf. If the density and therefore Hf are constant, νim correspond to the eigenstates of propagation. Since Hf≠H0, the states νim differ from the mass states, νi. For low density n→0, the vacuum eigenstates are recovered: νim→νi. If the density, and thus Hf change during neutrino propagation, νim and Him should be considered as the eigenstates and eigenvalues of the instantaneous Hamiltonian: Hf=Hf(x), νim=νim(x), and Him=Him(x). For n→0 we have Him→mi2/2E.The mixing in matter is a generalization of the mixing in vacuum (1). Recall that the mixing matrix in vacuum connects the flavor neutrinos, νf, and the massive neutrinos, νmass. The latter are the eigenstates of Hamiltonian in vacuum: νH=νmass. Therefore, the mixing matrix in matter is defined as the matrix which relates the flavor states with the eigenstates of the Hamiltonian in matter νHT=(ν1m,ν2m,ν3m):
(14)νf=UmνH.From (13) we find that
(15)νjm†Hfνim=Himδji.
Furthermore, the Hamiltonian can be represented in the flavor basis as
(16)Hf=∑αβ‍Hαβνανβ†.
Inserting this expression as well as the relation νjm=Uαjm*να, which follows from (14), into (15) one obtains
(17)∑αβ‍Uαjm*HαβUβim=Himδji
or in matrix form Um†HfUm=Hdiag=diag(H1m,H2m,H3m). Thus, the mixing matrix Um can be found diagonalizing the full Hamiltonian. The columns of the mixing matrix, Ui≡(Ueim,Uμim,Uτim), are the eigenstates of the Hamiltonian Hf which correspond to the eigenvalues Him. Indeed, it follows from (17) that HfUm=UmHdiag.Equation (14) can be inverted to νH=Um†νf, or in components νim=Uαim*να,α=e,μ,τ. According to this, the elements of mixing matrix determine the flavor content of the mass eigenstates so that |Uαim|2 gives the probability to find να in a given eigenstate νim. 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 the2ν case, there is single mixing angle in matter θm and the relations between the eigenstates in matter and the flavor states read
(18)νe=cosθmν1m+sinθmν2m,νa=cosθmν2m-sinθmν1m.
The angle θm is obtained by diagonalization of the Hamiltonian (12) (see previous section):
(19)sin22θm=1Rsin22θ,R≡(cos2θ-2VEΔm2)2+sin22θ,
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 Him equals
(20)ωm≡H2m-H1m=Δm22ER.
This difference is also called the level splitting or oscillation frequency, which determines the oscillation length: lm=2π/ωm (see Section 3.2).The matter potential andΔm2 always enter the mixing angle and other dimensionless quantities in the combination
(21)2EVΔm2=lνl0,
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:
(22)Hf=ωm2(-cos2θmsin2θmsin2θmcos2θm).


### 2.4.1. Resonance and Level Crossing

According to (19) the effective mixing parameter in matter, sin22θm, depends on the electron density and neutrino energy through the ratio (21) of the oscillation and refraction lengths, x=lν/l0∝EV. The dependence sin22θm(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 sin22θm on E has a resonant character [3]. At
(23)lν=l0cos2θ
the mixing becomes maximal: sin22θm=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/lν=Δm2/2E, coincides with the eigenfrequency of the medium, 2π/l0=1/V. The resonance condition (23) determines the resonance density
(24)neR=Δm22Ecos2θ2GF.
The width of resonance on the half of height (in the density scale) is given by 2ΔneR=2neRtan2θ. Similarly, for fixed ne one can introduce the resonance energy and the width of resonance in the energy scale. The width can be rewritten as ΔneR=n0sin2θ, where n0≡Δm2/22EGF. When the mixing approaches its maximalvalue: θ→π/4, the resonance shifts to zero density: neR→0, and the width of the resonance increases converging to the fixed value: ΔneR→n0.Resonance in neutrino mixing. The dependence ofsin22θmij on the product VE for vacuum mixing: sin22θ12=0.851, Δm212=7.59·10-5 eV2 (red) and sin2θ13=0.0241, Δm312=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.
(a)
(b)
(c)
(d)In a medium with varying density, the layer in which the density changes in the intervalneR±ΔneR is called the resonance layer. In this layer the angle θm varies in the interval from π/8 to 3π/8.ForV≪VR, the mixing angle is close to the vacuum angle: θm≈θ, while for V≫VR, the angle becomes θm≈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.


## 2.4.1. Resonance and Level Crossing

According to (19) the effective mixing parameter in matter, sin22θm, depends on the electron density and neutrino energy through the ratio (21) of the oscillation and refraction lengths, x=lν/l0∝EV. The dependence sin22θm(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 sin22θm on E has a resonant character [3]. At
(23)lν=l0cos2θ
the mixing becomes maximal: sin22θm=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/lν=Δm2/2E, coincides with the eigenfrequency of the medium, 2π/l0=1/V. The resonance condition (23) determines the resonance density
(24)neR=Δm22Ecos2θ2GF.
The width of resonance on the half of height (in the density scale) is given by 2ΔneR=2neRtan2θ. Similarly, for fixed ne one can introduce the resonance energy and the width of resonance in the energy scale. The width can be rewritten as ΔneR=n0sin2θ, where n0≡Δm2/22EGF. When the mixing approaches its maximalvalue: θ→π/4, the resonance shifts to zero density: neR→0, and the width of the resonance increases converging to the fixed value: ΔneR→n0.Resonance in neutrino mixing. The dependence ofsin22θmij on the product VE for vacuum mixing: sin22θ12=0.851, Δm212=7.59·10-5 eV2 (red) and sin2θ13=0.0241, Δm312=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.
(a)
(b)
(c)
(d)In a medium with varying density, the layer in which the density changes in the intervalneR±ΔneR is called the resonance layer. In this layer the angle θm varies in the interval from π/8 to 3π/8.ForV≪VR, the mixing angle is close to the vacuum angle: θm≈θ, while for V≫VR, the angle becomes θm≈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 Hf which relies on the strong hierarchy of the mass squared differences:
(25)rΔ≡Δm212Δm312≈0.03.
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 UPMNS=U23IδU13U12. Here we will describe the case of normal mass hierarchy: Δm312>0,Δm322>0. Subtracting from the Hamiltonian the matrix proportional to the unit matrix m12/2EI, we obtain
(26)Mdiag2=Δm312diag(0,rΔ,1).


### 2.5.1. Propagation Basis

The propagation basis,ν~=(νe,ν~2,ν~3)T, which is most suitable for consideration of the neutrino oscillations in matter, is defined through the relation
(27)νf=U23Iδν~.
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
(28)H~=12EU13U12Mdiag2U12†U13†+V^.
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 HamiltonianH~ can be written as
(29)H~=Δm3122E×(s132+s122c132rΔ+2VeEΔm312s12c12c13rΔs13c13(1-s122rΔ)…c122rΔ-s12c12s13rΔ……c132+s122s132rΔ).Here all the off-diagonal elements contain small parametersrΔ and/or s13. Notice that, for the measured oscillation parameters, s132~rΔ.


### 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 rΔ. After a 1–3 rotation
(30)ν~=U13(θ13m)ν′
over the angle θ13m determined by
(31)tan2θ13m=sin2θ13cos2θ13-2EV′/Δm312,whereV′=V1-s122rΔ,
the 1–3 element of (29) vanishes. The expression (31) differs from that for 2ν mixing in matter by a factor (1-s122rΔ), which increases the potential and deviates from 1 by
(32)ξ≡s122rΔ≈10-2.
After this rotation the Hamiltonian in the ν′ basis (30) becomes
(33)H′=Δm3122E×(h11s12c12rΔcos(θ13m-θ13)0…c122rΔs12c12rΔsin(θ13m-θ13)……h33),where(34)h11,33=12[{[cos2θ13(1-ξ)-x]2}(1+ξ+x)∓[cos2θ13(1-ξ)-x]2+sin22θ13(1-ξ)2],
and x≡2EV/Δm312. For ξ=0, these elements are reduced to the standard 2ν expressions. In the limit of zero density, x→0, h11=ξ=s122rΔ, and consequently the 11-element of the Hamiltonian equals H11′=s122Δm122/2E.In the lowestrΔ approximation one can neglect the nonzero 2-3 element in (33). The state ν3′ then decouples and the problem is reduced to a two-neutrino problem for (ν1′,ν2′). The eigenvalue of this decoupled state equals
(35)H3m≈Δm3122Eh33,h33≥1.
The diagonalization of the remaining 1-2 submatrix is given by rotation
(36)ν′=U12(θ12m)νm,
where θ12m is determined by
(37)tan2θ12m=sin2θ12rΔcos(θ13m-θ13)c122rΔ-h11.
Here h11 and θ13m are defined in (34) and (31), respectively. The eigenvalues equal
(38)H1m,2m=Δm3124E[{(c122rΔ-h11)2}c122rΔ+h11∓(c122rΔ-h11)2+sin22θ12rΔ2cos2(θ13m-θ13)].According to this diagonalization procedure in the lowest order inrΔ the mixing matrix in matter is given by
(39)Um=U23(θ23)IδU13(θ13m)U12(θ12m),
where mixing angles θ12m and θ13m 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 H1m and H2m are given in (38) and H3m is determined by (35).The 2-3 element of matrix (33) vanishes after additional 2-3 rotation by an angle θ23′~rΔ:
(40)tan2θ23′=sin2θ12rΔsin(θ13m-θ13)h33-c122rΔ,
which produces corrections of the next order in rΔ. With an additional 2-3 rotation the mixing matrix becomes
(41)Um=U23(θ23)IδU13(θ13m)U12(θ12m)U23(θ23′)≈U23(θ23m)IδmU13(θ13m)U12(θ12m),
where
(42)U23(θ23m)Iδm=U23(θ23)IδU23(θ-23),
and the last 2-3 rotation is on the angle θ-23 determined through sinθ-23=sinθ23′/cosθ13m. 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(θ23′). 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
(43)sinδmsin2θ23m=sinδsin2θ23,
that is, the combination sinδsin2θ23 is invariant under inclusion of matter effects. Furthermore, θ23m≈θ23 and δm≈δ up to corrections of the order O(rΔ). The results described here allowing understand the behavior of the mixing parameters sin22θmij in the EV region of the 1–3 resonance and above it (see Figure 1).In Figure2 we present dependence of the flavor content of the neutrino eigenstates on the potential. The energy level scheme, the dependence of the eigenvalues Him 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.The flavor contents of the eigenstates of the Hamiltonian in matter as functions ofEV. 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.The energy level scheme. We here show the dependence of the eigenvalues of the Hamiltonian in matter onEV. Note that we are plotting 2EHim, which goes to Δmi12 for low VE. The left (right) panel corresponds to normal (inverted) mass ordering.
(a)
(b)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 theνe-ντ′ channel, is associated to the 1–3 mixing and large mass splitting. According to (31) θ13m=π/4 at
(44)V13R=cos2θ13(1-s122rΔ)Δm3122E.(2)
The L resonance at low densities is associated to the small mass splitting and 1-2 mixing. It appears in theνe′-νμ′ channel, where νe′ and νe differ by small (at low densities) rotation given by an angle ~θ13 (see (31)). According to (37) the position of the L-resonance, θ12m=π/4, is given by c122rΔ=h11, where h11 is defined in (34). This leads to
(45)V12R=cos2θ12Δm2122E1c132.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:(46)H1m⟶-V,H2m⟶Δm212c1222Eν,H3m⟶Δm312c1322Eν.


## 2.5.1. Propagation Basis

The propagation basis,ν~=(νe,ν~2,ν~3)T, which is most suitable for consideration of the neutrino oscillations in matter, is defined through the relation
(27)νf=U23Iδν~.
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
(28)H~=12EU13U12Mdiag2U12†U13†+V^.
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 HamiltonianH~ can be written as
(29)H~=Δm3122E×(s132+s122c132rΔ+2VeEΔm312s12c12c13rΔs13c13(1-s122rΔ)…c122rΔ-s12c12s13rΔ……c132+s122s132rΔ).Here all the off-diagonal elements contain small parametersrΔ and/or s13. Notice that, for the measured oscillation parameters, s132~rΔ.


## 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 rΔ. After a 1–3 rotation
(30)ν~=U13(θ13m)ν′
over the angle θ13m determined by
(31)tan2θ13m=sin2θ13cos2θ13-2EV′/Δm312,whereV′=V1-s122rΔ,
the 1–3 element of (29) vanishes. The expression (31) differs from that for 2ν mixing in matter by a factor (1-s122rΔ), which increases the potential and deviates from 1 by
(32)ξ≡s122rΔ≈10-2.
After this rotation the Hamiltonian in the ν′ basis (30) becomes
(33)H′=Δm3122E×(h11s12c12rΔcos(θ13m-θ13)0…c122rΔs12c12rΔsin(θ13m-θ13)……h33),where(34)h11,33=12[{[cos2θ13(1-ξ)-x]2}(1+ξ+x)∓[cos2θ13(1-ξ)-x]2+sin22θ13(1-ξ)2],
and x≡2EV/Δm312. For ξ=0, these elements are reduced to the standard 2ν expressions. In the limit of zero density, x→0, h11=ξ=s122rΔ, and consequently the 11-element of the Hamiltonian equals H11′=s122Δm122/2E.In the lowestrΔ approximation one can neglect the nonzero 2-3 element in (33). The state ν3′ then decouples and the problem is reduced to a two-neutrino problem for (ν1′,ν2′). The eigenvalue of this decoupled state equals
(35)H3m≈Δm3122Eh33,h33≥1.
The diagonalization of the remaining 1-2 submatrix is given by rotation
(36)ν′=U12(θ12m)νm,
where θ12m is determined by
(37)tan2θ12m=sin2θ12rΔcos(θ13m-θ13)c122rΔ-h11.
Here h11 and θ13m are defined in (34) and (31), respectively. The eigenvalues equal
(38)H1m,2m=Δm3124E[{(c122rΔ-h11)2}c122rΔ+h11∓(c122rΔ-h11)2+sin22θ12rΔ2cos2(θ13m-θ13)].According to this diagonalization procedure in the lowest order inrΔ the mixing matrix in matter is given by
(39)Um=U23(θ23)IδU13(θ13m)U12(θ12m),
where mixing angles θ12m and θ13m 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 H1m and H2m are given in (38) and H3m is determined by (35).The 2-3 element of matrix (33) vanishes after additional 2-3 rotation by an angle θ23′~rΔ:
(40)tan2θ23′=sin2θ12rΔsin(θ13m-θ13)h33-c122rΔ,
which produces corrections of the next order in rΔ. With an additional 2-3 rotation the mixing matrix becomes
(41)Um=U23(θ23)IδU13(θ13m)U12(θ12m)U23(θ23′)≈U23(θ23m)IδmU13(θ13m)U12(θ12m),
where
(42)U23(θ23m)Iδm=U23(θ23)IδU23(θ-23),
and the last 2-3 rotation is on the angle θ-23 determined through sinθ-23=sinθ23′/cosθ13m. 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(θ23′). 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
(43)sinδmsin2θ23m=sinδsin2θ23,
that is, the combination sinδsin2θ23 is invariant under inclusion of matter effects. Furthermore, θ23m≈θ23 and δm≈δ up to corrections of the order O(rΔ). The results described here allowing understand the behavior of the mixing parameters sin22θmij in the EV region of the 1–3 resonance and above it (see Figure 1).In Figure2 we present dependence of the flavor content of the neutrino eigenstates on the potential. The energy level scheme, the dependence of the eigenvalues Him 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.The flavor contents of the eigenstates of the Hamiltonian in matter as functions ofEV. 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.The energy level scheme. We here show the dependence of the eigenvalues of the Hamiltonian in matter onEV. Note that we are plotting 2EHim, which goes to Δmi12 for low VE. The left (right) panel corresponds to normal (inverted) mass ordering.
(a)
(b)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 theνe-ντ′ channel, is associated to the 1–3 mixing and large mass splitting. According to (31) θ13m=π/4 at
(44)V13R=cos2θ13(1-s122rΔ)Δm3122E.(2)
The L resonance at low densities is associated to the small mass splitting and 1-2 mixing. It appears in theνe′-νμ′ channel, where νe′ and νe differ by small (at low densities) rotation given by an angle ~θ13 (see (31)). According to (37) the position of the L-resonance, θ12m=π/4, is given by c122rΔ=h11, where h11 is defined in (34). This leads to
(45)V12R=cos2θ12Δm2122E1c132.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:(46)H1m⟶-V,H2m⟶Δm212c1222Eν,H3m⟶Δm312c1322Eν.


## 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:
(47)ν(t)=S(t,t0)ν(t0).
Inserting this expression in the evolution equation (12), we find that S(t,t0) satisfies the same evolution equation as ν(t):
(48)idSdt=HS.
The elements S(t,t0)αβ of this matrix are the amplitudes of νβ→να transitions: S(t,t0)αβ≡A(νβ→να). The transition probability equals Pαβ=|S(t,t0)αβ|2. The unitarity of the evolution matrix, S†S=I, leads to the following relations between the amplitudes (matrix elements):
(49)|Sαα|2+|Sβα|2=1,|Sββ|2+|Sαβ|2=1,Sαα*Sαβ+Sβα*Sββ=0,Sαβ*Sαα+Sββ*Sβα=0.
The first and the second equations express the fact that the total probability of transition of να to everything is one, and the same holds for νβ. The third and fourth equations are satisfied if
(50)Sαα=Sββ*,Sβα=-Sαβ*.
With these relations the evolution matrix can be parametrized as
(51)S=(αβ-β*α*),|α|2+|β|2=1.The Hamiltonian for a2ν 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 Sβα→Sαβ, and the diagonal elements Sαα do not change. Therefore according to (50) the T invariance implies that Sβα=-Sβα*, or ReSβα=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:θm(E,n)=const. Consequently, the flavor composition of the eigenstates does not change and the eigenvalues Him 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
(52)idνmdx=Hdiagνm,
where Hdiag≡diag(H1m,H2m). This system of equations splits and the integration is trivial, νim(t)=e-iHimtνim(0). The corresponding S matrix is diagonal:
(53)S~(x,0)=(eiϕm(x)00e-iϕm(x)),
where ϕm≡(1/2)ωmx is the half-oscillation phase in matter and a matrix proportional to the unit matrix has been subtracted from the Hamiltonian.TheS matrix in the flavor basis (νe,νa) is therefore
(54)S(x,0)=UmS~(x,0)Um†=(cosϕm+icos2θmsinϕm-isin2θmsinϕm-isin2θmsinϕmcosϕ-icos2θmsinϕm).
Then, for the transition probability, we can immediately deduce
(55)Pea=|Sea|2=sin22θmsin2ϕm,
where ϕm=πx/lm, with
(56)lm=2πH2m-H1m=lνR
being the oscillation length in matter. The dependence of lm on the neutrino energy is shown in Figure 4. For small energies, VE≪Δm2, the length lm≃lν. It then increases with energy and for small θ reaches the maximum lmmax=l0/sin2θ at Emax=ER/cos22θ, that is, above the resonance energy. For E→∞ the oscillation length converges to the refraction length lm→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 theS matrix for a layer with constant density follows from (54):
(57)S(x,0)=cosϕmI-isinϕm(σ·n),
where σ is a vector containing the Pauli matrices and n≡(sin2θm,0,-cos2θm).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→sin22θm, lν→lm.


### 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, ψT=(ψe,ψa). The corresponding Hamiltonian can be written as
(58)H=(H·σ),
where σ=(σ1,σ2,σ3), H is the Hamiltonian vector H≡(2π/lm)·(sin2θm,0,cos2θm), and lm=2π/ΔHm is the oscillation length. The evolution equation then becomes
(59)iψ˙=(H·σ)ψ.
Let us define the polarization vector P(60)P≡ψ†σ2ψ.
In terms of the wave functions, the components of P equal
(61)(Px,Py,Pz)=(Reψe*ψa,Imψe*ψa,12(|ψe|2-|ψa|2)).
The z-component can be rewritten as Pz=|ψe|2-1/2; therefore Pe≡|ψe|2=Pz+1/2 and from unitarity Pa≡|ψa|2=1/2-Pz. Hence, Pz 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 ψ˙ and ψ˙† from evolution equation (59). As a result, one finds that
(62)ddtP=H×P.
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 phaseϕ (see Figure 5) the vector P moves on the surface of the cone having axis H. The cone angle θa, 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 νe, the angle equals θa=2θm in the initial moment.Figure 5
Graphic representation of neutrino oscillations. Neutrino polarization vectorP 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 vectorP are nothing but the elements of the density matrix ρ=σ·P. The evolution equation for ρ can be obtained from (62)
(63)idρdt=[H,ρ].
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 stateνμ with continuous energy spectrum. This flux then traverses a layer of length L with constant density ne. At the end of this layer a detector measures the νe component of the flux, so that oscillation effect is given by the transition probability Pμe. 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 sin22θm. The period of the oscillatory curve decreases with the length L. At the resonance energy,
(64)ER=Δm2cos2θ2V=Δm2cos2θ22GFne,
oscillations proceed with maximal depths. Oscillations are enhanced up to P>1/2 in the resonance range (ER±ΔER) where ΔER=tan2θER (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 probabilityνe→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
(65)S~(x,0)=(e-2iϕ1m(x)000e-2iϕ2m(x)000e-2iϕ3m(x)),
and for the elements of the S matrix in the flavor basis we obtain Sαβ=∑i‍Uαim*Uβime-2iϕim(x). Removing e-2iϕ2m and using the unitarity of the mixing matrix in matter we have
(66)Sαβ=δαβ+2ieϕ21m(x)Uα2m*Uβ2msinϕ21m(x)-2ie-iϕ32m(x)Uα3m*Uβ3msinϕ32m(x).
In particular, for the amplitudes in matter involving only νe and νμ, we obtain
(67)Seμcst=2ieiϕ21m[Ue1mUμ1m*sinϕ21m-e-iϕ31mUe3mUμ3m*sinϕ32m],Sμμcst=1+2ieiϕ21m|Uμ1m|2sinϕ21m-2ie-iϕ32m|Uμ3m|2sinϕ32m,Seecst=1+2ieiϕ21mcos2θ13mcos2θ12msinϕ21m-2ie-iϕ32msin2θ13msinϕ32m.


### 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:ne=ne(t). Correspondingly, the Hamiltonian of system depends on time, H=H(t), and therefore the mixing angle changes during neutrino propagation: θm=θm(ne(t)). Furthermore, the eigenstates of the instantaneous Hamiltonian, ν1m and ν2m, are no longer the “eigenstates” of propagation. Indeed, inserting νf=U(θm)νm in the equation for the flavor states (c.f., (3)) we obtain the evolution equation for eigenstates νim:
(68)idνmdt=(H1m-iθ˙miθ˙mH2m)νm,
where θ˙m≡dθm/dt. The Hamiltonian for νim (68) is nondiagonal and, consequently, the transitions ν1m↔ν2m occur. The rate of these transitions is given by the speed with which the mixing angle changes with time. According to (68) [3, 25], |θ˙m| determines the energy of transition ν1m↔ν2m and |H2m-H1m| gives the energy gap between the levels.The off-diagonal elements of the evolution equation (68) can be neglected if θ˙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 θ˙m is
(69)θ˙m≪H2m-H1m.
This inequality implies a slow enough change of density and is called the adiabaticity condition. Defining the adiabaticity parameter [22, 25] as
(70)γ≡|θ˙mH2m-H1m|,
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]: ΔrR>lmR, where lmR≡lν/sin2θ is the oscillation length in resonance, and ΔrR≡(ne/(dne/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, ne(av), larger than the resonance density: ne(av)→nB>nR. Here nB=Δm2/22GFE 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 andθ˙m can be neglected, the Hamiltonian for the eigenstates becomes diagonal. Consequently, the equations for the instantaneous eigenstates νim 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 θ˙m in evolution equation but do not neglect the dependence of θm on density. The solution can be obtained immediately as
(71)S~(x,0)=(eiϕm00e-iϕm),ϕm=12∫0x‍(H2m-H1m)dx′,
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:(72)Sf(x,0)=Um(t)S~(x,0)Um†(0)=(cmcm0eiϕm+smsm0e-iϕm-cmsm0eiϕm+smcm0e-iϕm-smcm0eiϕm+cmsm0e-iϕmsmsm0eiϕm+cmcm0e-iϕm).
From this procedure we find, for example, the probability of νe-νe transition
(73)Pee=|Sf(x,0)ee|2=12[1+cos2θm(x)cos2θm(0)]+12sin2θm(x)sin2θm(0)cos2ϕm(x).
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:
(74)Pαβ=|∑i‍Uαim(0)Uβim*(0)e-iϕim(t,0)|
with the mixing angle taken at the borders (initial or final state). In particular, the survival probability equals Pαα=1-sin22θm(0)sin2ϕm(x).Averaging over the phase, which means that the contributions fromν1 and ν2add incoherently, gives
(75)P=(cosθmcosθm0)2+(sinθmsinθm0)2=sin2θm+cos2θmcos2θm0.
The mixing in the neutrino production point θm0 is determined by density in this point, ne0, 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=|sin2θmsin2θm0|. 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/nR 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 ont via the phase ϕm(t) and also via the mixing angle θm(t). Two degrees of freedom are operative and P dependence on time is an interplay of two effects: oscillations, associated with the phase ϕm(t), and the adiabatic conversion related to change of θm. Depending on initial condition ne0, the relative importance of the two effects is different. If neutrinos are produced far above the resonance, ne0≫neR, the initial mixing is strongly suppressed, θm0≈π/2. Consequently, the neutrino state, for example, νe, consists mainly of one eigenstate, ν2m, and furthermore, one-flavor νe dominates in ν2m. 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 ν2m) follows the density change. At zero density ν2m=ν2, and therefore the probability to find the electron neutrino (survival probability) equals [3]
(76)P=|〈νe∣ν(t)〉|2≈|〈νe∣ν2m(t)〉|2=|〈νe∣ν2〉|2≈sin2θ.
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, atne0>neR, the initial mixing is not suppressed. Although ν2m is the main component of the neutrino state, the second eigenstate, ν1m, 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; ifθm0=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,θ˙m is not negligible in (68) and the adiabaticity condition (70) is not satisfied. The transitions ν1m↔ν2m 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
(77)Sf(x,0)=Um(t)S~(x,0)Um†(0)=Um(t)(S11-S21*S21S11*)Um†(0),
where S~ is the evolution matrix in the basis of instantaneous eigenstates. Then the νe-νe transition probability Pee≡|Sf(x,0)ee|2 equals
(78)Pee=12[1+cos2θm(t)cos2θm(0)]-P21cos2θm(t)cos2θm(0)+Pint,
where P21≡|S21|2 is the probability of ν2m→ν1m transitions and Pint is an interference term
(79)Pint=14sin2θm(t)sin2θm(0)[S112+S11*2+S212+S21*2]+12sin[2θm(0)-2θm(x)][S11S21*+S11*S21],
which depends on the oscillation phase. The averaged probability (Pint=0) equals [30]
(80)Pee=12+(12-P21)cos2θm(t)cos2θm(0).
If the initial density is much larger than the resonance density, then θm(0)≈π/2 and cos2θm(0)=-1. In this case the averaged probability can be rewritten as
(81)Pee=sin2θm(t)+P21cos2θm(t).
Violation of adiabaticity weakens transitions if cos2θm(t)>0, thus leading to an increase of the survival probability. In the adiabatic case S11=eiϕm, S21=0, and therefore S112+S11*2=2cos2ϕm(x), so that (78) is reduced to (73).In the graphic representation (Figure5), 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 θa changes as a result of violation of the adiabaticity).There are different approaches to compute the flop probabilityP21. 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 H2m-H1m by a system with the kinetic energy dθm/dt. This leads to the Landau-Zener probability
(82)PLZ=exp(-π2κR)=exp(-πhΔm24Esin22θcos2θ),
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:(83)d=∫0L‍ne(x)dx.
We can define d1/2 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]
(84)d1/2≥dmin=π22GFtan2θ
for all density profiles. Furthermore, the minimum, dmin, 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, 2GFd, 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
(85)S=𝒯[exp(-i∫0L‍H(x)dx)],
where 𝒯 denotes time ordering of the exponential. For small values of L, it can be expanded as
(86)S=1-i∫0L‍H(x)dx+𝒪(L2).
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≪lm 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(87)P=sin22θmsin2ϕm=1Rsin22θsin2ϕR≈ϕ2sin22θ.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 isd=nx≪GF-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 lengthx. There are three possibilities.(i)
E≪ER (ER 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~lν.(ii)
E~ER—modification of oscillation parameters is strong; however lνR~lν/sin2θ~2π/(Vsin2θ). Consequently, x/lνR=xVsin2θ/2π≪1. Oscillations are undeveloped due to smallness of phase.(iii)
E≫ER—matter suppresses oscillation depth by a factor ER/E≪1. Since the oscillation length equals lm≈2π/V, one obtains x/lm=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 changeTn is an integer times the effective oscillation length lm [38]:
(88)∫lT‍dxlm=k,(k=1,2,3,…),
or lT/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 thecastle wall profile, for which the period lT is divided into the two parts l1 and l2 (l1+l2=lT) 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]:
(89)Φ1=Φ2=π.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≡2θ1m-2θ2m). 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
(90)ST=S2S1,
where Sk(k=1,2) is the evolution in layer k given by (57). For brevity we will write it as Sk=ckI-isk(σ·nk),k=1,2, where ck≡cosϕk, sk≡sinϕk, and ϕk is the half phase acquired in layer k:
(91)ϕk=12ΔHklk=Δm24ER(Vk)1/2lk,nk≡(sin2θmk,0,-cos2θmk).
Here θmk is the mixing angle in layer k.Insertion ofSk from (57) into (90) gives [37]
(92)ST=YI-i(σ·X),
where
(93)Y≡c1c2-s1s2(n1·n2),X=s1c2n1+s2c1n2-s1s2[n1×n2].
Explicitly, (n1·n2)=cos(2θm1-2θm2) and [n1×n2]=sin(2θm1-2θm2)ey. Using unitarity of ST, 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 ST can be written in the form ST=cosΦ-isinΦ(σ·X^)=e-i(σ·X^)Φ, where X^≡X/X. Consequently, the evolution matrix after n periods equals
(94)Sn=(ST)n=e-i(σ·X^)nΦ=cosnΦ-i(σ·X^)sinnΦ.
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′)Sn.The transition probability computed with (94) is
(95)Peμn=|Seμn|2=X12+X22X2sin2nΦ.
It has the form of the usual oscillation probability with phase nΦ and depth (X12+X22)/X2. The oscillations described by (95) are called the parametric oscillations. Under condition
(96)-X3=s1c2cos2θm1+s2c1cos2θm2=0,
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 theparametric 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,cos2θm=0, can be written according to (22) and (51) as α=α*, that is, S11(1)=S22(1), or equivalently, ImS11(1)=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≡ReS11(1)=0. The absolute maximum of the transition probability occurs when these conditions are satisfied simultaneously, that is, when S11(1)=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 S11(2)=S22(2). Let us find the conditions for extrema for density profiles consisting of two layers. We have S(2)=S2S1, where S11(2)=α2α1-β2β1*,S12(2)=α2β1+β2α1*, and αi,βi for each layer have been defined in (51). The sum of the two complex numbers in the transition amplitude S12(2) has the largest possible result if they have the same phase: arg(α2β1)=arg(β2α1*), which can also be rewritten as
(97)arg(α1α2β1)=arg(β2).
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 |βi| 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 matrixS(3) equal
(98)S12(3)=α3S12(2)+β3S11(2)*=α3α2β1+α3β2α1*+β3α2*α1*-β3β2*β1.
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=-β1*, and the expression for S12(3) can be written as
(99)S12(3)=α1α2β1-α1*α2*β1*+|α1|2β2+|β1|2β2*.
Note that β2 is pure imaginary because the core density profile is symmetric. Therefore the amplitude S12(3) 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 S12(3), 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 whenV>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+HI with
(100)H~0=ωm2(100-1),HI=sin2θΔ(-ϵ11ϵ),
where ωm is the oscillation frequency (20) and ϵ≡(2Δcos2θ-V+ωm)/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 θm: 2Δsin2θ/ωm=sin2θm. Therefore for sin2θm≪1 the term HI can be considered as a perturbation. Furthermore, ϵ~sin2θm, so the diagonal terms in HI can be neglected in the lowest approximation.The solution forS matrix can be found in the form S=S0·SI, where S0 is the solution of the evolution equation with H replaced by H0 (see (71)). The matrix SI then satisfies the equation
(101)idSIdx=S0-1HIS0SI=H~ISI,
where H~I≡S0-1HIS0 is the perturbation Hamiltonian in the “interaction” representation. Equation (101) can be solved by iterations: SI=I+SI(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,
(102)S(L)=S0(L)[I-iΔsin2θ∫0Ldx(0ei2ϕ(x)e-i2ϕ(x)0)].
The νe↔νa transition probability P2=[S(L)]ae is given by
(103)P2=Δ2sin22θ|∫0L‍dxe-i2ϕ(x)|2.For density profiles that are symmetric with respect to the center of the neutrino trajectory,V(x)=V(L-x), (103) gives
(104)P2=4(Δm24E)2sin22θ[∫0L/2‍dzcos2ϕ(z)]2,
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 forE≳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 ofH0 in (100) has to be flipped, and correspondingly one has to replace ωm→-ωm 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:
(105)H(x)=H-+ΔH(x),
where
(106)H-≡ω-(-cos2θ-sin2θ-sin2θ-cos2θ-),ΔH≡ΔV(x)2(100-1).
Here, θ-=θm(V-) is the mixing angle in matter and ω-=ωm(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 θm=θ- and ϕm(x)=ϕ(x)≡ω-x,  ω-=ωm(V-).The solution of the evolution equation with Hamiltonian (105) [46] is of the form
(107)S(x)=S-(x)+ΔS(x),ΔS(x)=-iS-(x)K1(x),
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):
(108)dK1(x)dx=S-†ΔH(x)S-=ΔV2[-cos2θ-(-cos2θ-sin2θ-sin2θ-cos2θ-)+sin2θ-cos2ϕG(θ-)+sin2θ-sin2ϕσ2{(-cos2θ-sin2θ-sin2θ-cos2θ-)}],
where G(θ-)≡cos2θ-σ1+sin2θ-σ3. The first term in (108) does not contribute to S≡S(L) since 〈ΔV〉≡∫‍ΔV(x)dx=0, and (108) can be immediately integrated:
(109)K1(L)=12sin2θ-[G(θ-)∫0L‍ΔV(x)cos2ϕ(x)dx+σ2∫0L‍ΔV(x)sin2ϕ(x)dx].
Introducing the distance from the midpoint of the neutrino trajectory z≡x-L/2, one obtains from (109)
(110)ΔS≡ΔS(L)=-isin2θ-[G(θ-)ΔI+σ2ΔJ],
where ΔI≡(1/2)∫-L/2L/2‍ΔV(z)cos(2ω-z)dz,ΔJ≡(1/2)∫-L/2L/2‍ΔV(z)sin(2ω-z)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 forS which is exactly unitary regardless of the size of the perturbation. For this we rewrite (110) as follows:
(111)ΔS=εS′,S′=-i[G(θ-)cosξ+σ2sinξ],
where sinξ=ΔJ/(ΔJ)2+(ΔI)2 and ϵ=sin2θ-·(ΔJ)2+(ΔI)2. Thus, S=S-+εS′ and we replace it by
(112)S=cosεS-+sinεS′.
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 termΔJ is absent. From (54), (110), and (112) we immediately get the transition probability
(113)P=[cosεsin2θ-sinϕ+sinεcos2θ-]2≈sin22θ-[sinϕ+ΔIcos2θ-]2,
where ε≡sin2θ-ΔI and ϕ≡ϕ(L)=ω-L. Here the first term in the square brackets describes oscillations in constant density matter with average potential V-1.


### 3.13. Oscillation Probabilities and Their Properties

It is convenient to consider the neutrino flavor evolution in the propagation basisν~=(νe,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
(114)ν~=U13I-δU12ν.
Since the transformations, which connect ν~ and νf, 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(115)S~=(AeeAe2~Ae3~A2~eA2~2~A2~3~A3~eA3~2~A3~3~).
Then according to (27) the S matrix in the flavor basis equals
(116)S=U~S~U~†,U~≡U23Iδ.
In this part, we use the notation Aij for the amplitudes in the propagation basis and Sij 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
(117)S=(Aeec23Ae2~+s23e-iδAe3~-s23Ae2~+c23e-iδAe3~c23A2~e+s23eiδA3~ec232A2~2~+s232A3~3~+Kμμ-s23c23(A2~2~-A3~3~)+Kμτ-s23A2~e+c23eiδA3~e-s23c23(A2~2~-A3~3~)+Kτμs232A2~2~+c232A3~3~+Kττ),where(118)Kμμ≡s23c23(e-iδA2~3~+eiδA3~2~),Kμτ≡c232e-iδA2~3~-s232eiδA3~2~,Kτμ=Kμτ(δ⟶-δ,2~⟷3~),Kττ=-Kμμ.
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):
(119)Ae3~,A3~e~s13,Ae2~,A2~e~rΔ~s132,A3~2~,A2~3~~s13rΔ~s133,
that is, A2~3~ and A3~2~ 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
(120)A2~e=Ae2~,A3~e=Ae3~,A3~2~=A2~3~.
Consequently, for Kαβ we obtain
(121)Kμτ=A2~3~(cos2θ23cosδ-isinδ),Kτμ=Kμτ(δ⟶-δ),Kμμ=-Kττ=A2~3~sin2θ23cosδ.
These terms proportional to small amplitudes A2~3~ and A3~2~ are of the order O(s132).Figure 7
Scheme of transitions between the flavor states. Evolution is considered in the propagation basisν~. 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 Pαβ≡|Sβα|2:
(122)Pee=|Aee|2=1-|Ae2~|2-|Ae3~|2,(123)Pμe=c232|Ae2~|2+s232|Ae3~|2+2s23c23Re(e-iδAe2~*Ae3~),(124)Pτe=s232|Ae2~|2+c232|Ae3~|2-2s23c23Re(e-iδAe2~*Ae3~),(125)Pμμ=|c232A2~2~+s232A3~3~+2s23c23cosδA2~3~|2,(126)Pμτ=|s23c23(A3~3~-A2~2~)+(cos2θ23cosδ+isinδ)A2~3~|2.
For antineutrinos the amplitudes can be obtained from the results presented above substituting
(127)δ⟶-δ,Aij⟶A-ij,whereA-ij≡Aij(V⟶-V).
Notice that the amplitudes of transitions (123) and (124), that involve νe, 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 HamiltonianH~ in (29), in the limits Δm212→0 or/and s12→0 the state ν~2 decouples from the rest of the system, and consequently, the amplitude Ae2~ vanishes. In this limit, Ae3~ (as well as A3~3~ and See) is reduced to a 2ν amplitude which depends on the parameters Δm312 and θ13: AA(Δm312,θ13)≡Ae3~(Δm212=0). The corresponding probability equals PA≡|AA|2.In the limits13→0 the state ν~3 decouples while the amplitude Ae3~ vanishes and the amplitude Ae2~ reduces to a 2ν amplitude depending on the parameters of the 1-2 sector, Δm212 and θ12. Denoting this amplitude by AS we have AS(Δm212,θ12)≡Ae2~(θ13=0). We will use the notation PS≡|AS|2.This consideration implies that to the leading nontrivial order in the small parameterss13 and rΔ the amplitudes Ae2~ and A2~e are reduced to two neutrino probabilities and depend only on the “solar” parameters, whereas the amplitudes Ae3~ and A3~e only on the “atmospheric” parameters:
(128)Ae2~≃A2~e≃AS(Δm212,θ12),Ae3~≃A3~e≃AA(Δm312,θ13).
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 Ae2~ are of order s132, whereas the corrections to the “atmospheric” amplitude Ae3~ are of order rΔ [49], in agreement with our consideration for constant density. The amplitude Ae3~ does not in general have a 2-flavor form, once the corrections to the factorization approximation are taken into account.Using the expressions forUeim and Uμim in terms of the mixing angles in the standard parametrization, we can rewrite (67) as
(129)Seμcst=cosθ23mAe2~cst+sinθ23me-iδmAe3~cst,
where
(130)Ae2~cst≡-ieiϕ21mcosθ13msin2θ12msinϕ21m,Ae3~cst≡-ieiϕ21msin2θ13m[sinϕ32me-iϕ31m+cos2θ12msinϕ21m].
Here ϕ31m=ϕ32m+ϕ21m. Since to a good approximation θ23m≈θ23 and δm≈δ (see Section 2.5) [20, 50], the amplitudes Ae2~cst and Ae3~cst can be identified with Ae2~ and Ae3~ in (123) and (124).In terms of mixing angles,Uμ1m=-s12mc23m-c12ms13ms23meiδm, Uμ3m=c13ms23m, the amplitude Sμμcst can be rewritten as
(131)Sμμcst=cos2θ23mA2~2~cst+sin2θ23mA3~3~cst+sin2θ23mcosδmA2~3~cst,
where
(132)A2~2~cst≡1+2ieiϕ21msin2θ12msinϕ21m,A3~3~cst≡1-2ie-iϕ32mcos2θ13msinϕ32m+2ieiϕ21msin2θ13mcos2θ12msinϕ21m,A2~3~cst≡ieiϕ21msinθ13msin2θ12msinϕ21m.
Notice that A2~2~cst has exactly the form of the corresponding 2ν amplitude driven by the solar parameters. The amplitude A3~3~cst also coincides to a very good approximation with the corresponding 2ν amplitude driven by the atmospheric parameters. In the approximation θ23m≈θ23 and δm≈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 νe disappearance equals
(133)1-Pee=Peμ+Peτ=Pe2~+Pe3~.
The probabilityPee 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 Pee originates mainly from Pe3~≡|Ae3~|2. The survival probability then equals Pee=1-Peμ-Peτ=1-PA-PS. At high energies, where the effects of the 1-2 mixing and mass splitting in P are suppressed, the probability is Pee≈1-Peτ≈1-PA.
(2)
ν
e
-
ν
μ and νe-ντ channels: the transition probability Pμe≡P(νμ→νe) (see (123)) can be rewritten as
(134)Pμe=c232|Ae2~|2+s232|Ae3~|2+sin2θ23|Ae2~*Ae3~|cos(ϕ-δ),
whereϕ≡arg(Ae2~*Ae3~). Unlike 1-Pee, this probability contains the interference term between Ae2~ and Ae3~ which depends on the CP violation phase. Since the amplitude Ae2~ is suppressed at high energies due to the smallness of the 1-2 mixing in matter, in the lowest approximation we have
(135)Pμe≈sin2θ23|Ae3~|2≈sin2θ23|AA|2.
The maximal value of the probability equalsPμe≃s232.
According to (123) and (124) the oscillation probabilities Pτe and Peτ can be obtained from the corresponding probabilities Pμe and Peμ through the substitution s23→c23, c23→-s23 [51]. The interference term has the opposite signs for channels including ντ as compared with those with νμ, which can be obtained from the unitarity condition Pee+Pμe+Pτe=1 and the fact that Pee does not depend on δ.
(3)
Theνμ survival probability, Pμμ, for symmetric density profiles, (125), can be rewritten as
(136)Pμμ=|c232A2~2~+s232A3~3~|2+2sin2θ23cosδRe[A2~3~*(c232A2~2~+s232A3~3~)]+sin22θ23cos2δ|A2~3~|2.
SinceA2~3~=𝒪(rΔs13) is a small quantity, to a good approximation one can neglect the term ~cos2δ in (125), which is proportional to |A2~3~|2. For high energies in the limit Δm212→0 we have A2~2~=1, A2~3~=0. Then, parameterizing the 33-amplitude as A3~3~=1-PAe-iϕ3~3~m we obtain from (136)
(137)Pμμ(Δm212=0)=1-sin22θ23sin2ϕx-s234PA-0.5sin22θ23cos2ϕX(1-1-PA),
whereϕX=0.5arg[A3~3~*A2~2~]=ϕ2~2~m-ϕ3~3~m. The probability can be rewritten as
(138)Pμμ=1-0.5sin22θ23-s234PA+0.5sin22θ23(1-PA)cos2ϕX.
(4)
ν
μ
-
ν
τ channel: for symmetric matter density profiles the probability of νμ→ντ oscillations is given in (126). It can be rewritten as
(139)Pμτ=14sin22θ23|A2~2~-A3~3~|2+sin2θ23cos2θ23cosδRe[(A3~3~*-A2~2~*)A2~3~]-sin2θ23sinδIm[Ae2~*Ae3~]+(1-sin22θ23cos2δ)|A2~3~|2.The amplitude depends on δ through the terms proportional to cosδ and sinδ, and therefore Pμτ 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 Pμτδ=-Pμeδ-Pμμδ. In the limit Δm212→0 we obtain
(140)Pμτ=0.5sin22θ23-s232c232PA-0.5sin22θ23(1-PA)cos2ϕX.


## 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:
(47)ν(t)=S(t,t0)ν(t0).
Inserting this expression in the evolution equation (12), we find that S(t,t0) satisfies the same evolution equation as ν(t):
(48)idSdt=HS.
The elements S(t,t0)αβ of this matrix are the amplitudes of νβ→να transitions: S(t,t0)αβ≡A(νβ→να). The transition probability equals Pαβ=|S(t,t0)αβ|2. The unitarity of the evolution matrix, S†S=I, leads to the following relations between the amplitudes (matrix elements):
(49)|Sαα|2+|Sβα|2=1,|Sββ|2+|Sαβ|2=1,Sαα*Sαβ+Sβα*Sββ=0,Sαβ*Sαα+Sββ*Sβα=0.
The first and the second equations express the fact that the total probability of transition of να to everything is one, and the same holds for νβ. The third and fourth equations are satisfied if
(50)Sαα=Sββ*,Sβα=-Sαβ*.
With these relations the evolution matrix can be parametrized as
(51)S=(αβ-β*α*),|α|2+|β|2=1.The Hamiltonian for a2ν 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 Sβα→Sαβ, and the diagonal elements Sαα do not change. Therefore according to (50) the T invariance implies that Sβα=-Sβα*, or ReSβα=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:θm(E,n)=const. Consequently, the flavor composition of the eigenstates does not change and the eigenvalues Him 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
(52)idνmdx=Hdiagνm,
where Hdiag≡diag(H1m,H2m). This system of equations splits and the integration is trivial, νim(t)=e-iHimtνim(0). The corresponding S matrix is diagonal:
(53)S~(x,0)=(eiϕm(x)00e-iϕm(x)),
where ϕm≡(1/2)ωmx is the half-oscillation phase in matter and a matrix proportional to the unit matrix has been subtracted from the Hamiltonian.TheS matrix in the flavor basis (νe,νa) is therefore
(54)S(x,0)=UmS~(x,0)Um†=(cosϕm+icos2θmsinϕm-isin2θmsinϕm-isin2θmsinϕmcosϕ-icos2θmsinϕm).
Then, for the transition probability, we can immediately deduce
(55)Pea=|Sea|2=sin22θmsin2ϕm,
where ϕm=πx/lm, with
(56)lm=2πH2m-H1m=lνR
being the oscillation length in matter. The dependence of lm on the neutrino energy is shown in Figure 4. For small energies, VE≪Δm2, the length lm≃lν. It then increases with energy and for small θ reaches the maximum lmmax=l0/sin2θ at Emax=ER/cos22θ, that is, above the resonance energy. For E→∞ the oscillation length converges to the refraction length lm→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 theS matrix for a layer with constant density follows from (54):
(57)S(x,0)=cosϕmI-isinϕm(σ·n),
where σ is a vector containing the Pauli matrices and n≡(sin2θm,0,-cos2θm).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→sin22θm, lν→lm.


## 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, ψT=(ψe,ψa). The corresponding Hamiltonian can be written as
(58)H=(H·σ),
where σ=(σ1,σ2,σ3), H is the Hamiltonian vector H≡(2π/lm)·(sin2θm,0,cos2θm), and lm=2π/ΔHm is the oscillation length. The evolution equation then becomes
(59)iψ˙=(H·σ)ψ.
Let us define the polarization vector P(60)P≡ψ†σ2ψ.
In terms of the wave functions, the components of P equal
(61)(Px,Py,Pz)=(Reψe*ψa,Imψe*ψa,12(|ψe|2-|ψa|2)).
The z-component can be rewritten as Pz=|ψe|2-1/2; therefore Pe≡|ψe|2=Pz+1/2 and from unitarity Pa≡|ψa|2=1/2-Pz. Hence, Pz 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 ψ˙ and ψ˙† from evolution equation (59). As a result, one finds that
(62)ddtP=H×P.
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 phaseϕ (see Figure 5) the vector P moves on the surface of the cone having axis H. The cone angle θa, 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 νe, the angle equals θa=2θm in the initial moment.Figure 5
Graphic representation of neutrino oscillations. Neutrino polarization vectorP 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 vectorP are nothing but the elements of the density matrix ρ=σ·P. The evolution equation for ρ can be obtained from (62)
(63)idρdt=[H,ρ].
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 stateνμ with continuous energy spectrum. This flux then traverses a layer of length L with constant density ne. At the end of this layer a detector measures the νe component of the flux, so that oscillation effect is given by the transition probability Pμe. 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 sin22θm. The period of the oscillatory curve decreases with the length L. At the resonance energy,
(64)ER=Δm2cos2θ2V=Δm2cos2θ22GFne,
oscillations proceed with maximal depths. Oscillations are enhanced up to P>1/2 in the resonance range (ER±ΔER) where ΔER=tan2θER (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 probabilityνe→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
(65)S~(x,0)=(e-2iϕ1m(x)000e-2iϕ2m(x)000e-2iϕ3m(x)),
and for the elements of the S matrix in the flavor basis we obtain Sαβ=∑i‍Uαim*Uβime-2iϕim(x). Removing e-2iϕ2m and using the unitarity of the mixing matrix in matter we have
(66)Sαβ=δαβ+2ieϕ21m(x)Uα2m*Uβ2msinϕ21m(x)-2ie-iϕ32m(x)Uα3m*Uβ3msinϕ32m(x).
In particular, for the amplitudes in matter involving only νe and νμ, we obtain
(67)Seμcst=2ieiϕ21m[Ue1mUμ1m*sinϕ21m-e-iϕ31mUe3mUμ3m*sinϕ32m],Sμμcst=1+2ieiϕ21m|Uμ1m|2sinϕ21m-2ie-iϕ32m|Uμ3m|2sinϕ32m,Seecst=1+2ieiϕ21mcos2θ13mcos2θ12msinϕ21m-2ie-iϕ32msin2θ13msinϕ32m.


## 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:ne=ne(t). Correspondingly, the Hamiltonian of system depends on time, H=H(t), and therefore the mixing angle changes during neutrino propagation: θm=θm(ne(t)). Furthermore, the eigenstates of the instantaneous Hamiltonian, ν1m and ν2m, are no longer the “eigenstates” of propagation. Indeed, inserting νf=U(θm)νm in the equation for the flavor states (c.f., (3)) we obtain the evolution equation for eigenstates νim:
(68)idνmdt=(H1m-iθ˙miθ˙mH2m)νm,
where θ˙m≡dθm/dt. The Hamiltonian for νim (68) is nondiagonal and, consequently, the transitions ν1m↔ν2m occur. The rate of these transitions is given by the speed with which the mixing angle changes with time. According to (68) [3, 25], |θ˙m| determines the energy of transition ν1m↔ν2m and |H2m-H1m| gives the energy gap between the levels.The off-diagonal elements of the evolution equation (68) can be neglected if θ˙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 θ˙m is
(69)θ˙m≪H2m-H1m.
This inequality implies a slow enough change of density and is called the adiabaticity condition. Defining the adiabaticity parameter [22, 25] as
(70)γ≡|θ˙mH2m-H1m|,
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]: ΔrR>lmR, where lmR≡lν/sin2θ is the oscillation length in resonance, and ΔrR≡(ne/(dne/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, ne(av), larger than the resonance density: ne(av)→nB>nR. Here nB=Δm2/22GFE 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].


## 3.6.1. Equation for the Instantaneous Eigenvalues and the Adiabaticity Condition

In nonuniform media, the density changes along neutrino trajectory:ne=ne(t). Correspondingly, the Hamiltonian of system depends on time, H=H(t), and therefore the mixing angle changes during neutrino propagation: θm=θm(ne(t)). Furthermore, the eigenstates of the instantaneous Hamiltonian, ν1m and ν2m, are no longer the “eigenstates” of propagation. Indeed, inserting νf=U(θm)νm in the equation for the flavor states (c.f., (3)) we obtain the evolution equation for eigenstates νim:
(68)idνmdt=(H1m-iθ˙miθ˙mH2m)νm,
where θ˙m≡dθm/dt. The Hamiltonian for νim (68) is nondiagonal and, consequently, the transitions ν1m↔ν2m occur. The rate of these transitions is given by the speed with which the mixing angle changes with time. According to (68) [3, 25], |θ˙m| determines the energy of transition ν1m↔ν2m and |H2m-H1m| gives the energy gap between the levels.The off-diagonal elements of the evolution equation (68) can be neglected if θ˙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 θ˙m is
(69)θ˙m≪H2m-H1m.
This inequality implies a slow enough change of density and is called the adiabaticity condition. Defining the adiabaticity parameter [22, 25] as
(70)γ≡|θ˙mH2m-H1m|,
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]: ΔrR>lmR, where lmR≡lν/sin2θ is the oscillation length in resonance, and ΔrR≡(ne/(dne/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, ne(av), larger than the resonance density: ne(av)→nB>nR. Here nB=Δm2/22GFE 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 andθ˙m can be neglected, the Hamiltonian for the eigenstates becomes diagonal. Consequently, the equations for the instantaneous eigenstates νim 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 θ˙m in evolution equation but do not neglect the dependence of θm on density. The solution can be obtained immediately as
(71)S~(x,0)=(eiϕm00e-iϕm),ϕm=12∫0x‍(H2m-H1m)dx′,
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:(72)Sf(x,0)=Um(t)S~(x,0)Um†(0)=(cmcm0eiϕm+smsm0e-iϕm-cmsm0eiϕm+smcm0e-iϕm-smcm0eiϕm+cmsm0e-iϕmsmsm0eiϕm+cmcm0e-iϕm).
From this procedure we find, for example, the probability of νe-νe transition
(73)Pee=|Sf(x,0)ee|2=12[1+cos2θm(x)cos2θm(0)]+12sin2θm(x)sin2θm(0)cos2ϕm(x).
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:
(74)Pαβ=|∑i‍Uαim(0)Uβim*(0)e-iϕim(t,0)|
with the mixing angle taken at the borders (initial or final state). In particular, the survival probability equals Pαα=1-sin22θm(0)sin2ϕm(x).Averaging over the phase, which means that the contributions fromν1 and ν2add incoherently, gives
(75)P=(cosθmcosθm0)2+(sinθmsinθm0)2=sin2θm+cos2θmcos2θm0.
The mixing in the neutrino production point θm0 is determined by density in this point, ne0, 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=|sin2θmsin2θm0|. 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/nR 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 ont via the phase ϕm(t) and also via the mixing angle θm(t). Two degrees of freedom are operative and P dependence on time is an interplay of two effects: oscillations, associated with the phase ϕm(t), and the adiabatic conversion related to change of θm. Depending on initial condition ne0, the relative importance of the two effects is different. If neutrinos are produced far above the resonance, ne0≫neR, the initial mixing is strongly suppressed, θm0≈π/2. Consequently, the neutrino state, for example, νe, consists mainly of one eigenstate, ν2m, and furthermore, one-flavor νe dominates in ν2m. 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 ν2m) follows the density change. At zero density ν2m=ν2, and therefore the probability to find the electron neutrino (survival probability) equals [3]
(76)P=|〈νe∣ν(t)〉|2≈|〈νe∣ν2m(t)〉|2=|〈νe∣ν2〉|2≈sin2θ.
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, atne0>neR, the initial mixing is not suppressed. Although ν2m is the main component of the neutrino state, the second eigenstate, ν1m, 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; ifθm0=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,θ˙m is not negligible in (68) and the adiabaticity condition (70) is not satisfied. The transitions ν1m↔ν2m 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
(77)Sf(x,0)=Um(t)S~(x,0)Um†(0)=Um(t)(S11-S21*S21S11*)Um†(0),
where S~ is the evolution matrix in the basis of instantaneous eigenstates. Then the νe-νe transition probability Pee≡|Sf(x,0)ee|2 equals
(78)Pee=12[1+cos2θm(t)cos2θm(0)]-P21cos2θm(t)cos2θm(0)+Pint,
where P21≡|S21|2 is the probability of ν2m→ν1m transitions and Pint is an interference term
(79)Pint=14sin2θm(t)sin2θm(0)[S112+S11*2+S212+S21*2]+12sin[2θm(0)-2θm(x)][S11S21*+S11*S21],
which depends on the oscillation phase. The averaged probability (Pint=0) equals [30]
(80)Pee=12+(12-P21)cos2θm(t)cos2θm(0).
If the initial density is much larger than the resonance density, then θm(0)≈π/2 and cos2θm(0)=-1. In this case the averaged probability can be rewritten as
(81)Pee=sin2θm(t)+P21cos2θm(t).
Violation of adiabaticity weakens transitions if cos2θm(t)>0, thus leading to an increase of the survival probability. In the adiabatic case S11=eiϕm, S21=0, and therefore S112+S11*2=2cos2ϕm(x), so that (78) is reduced to (73).In the graphic representation (Figure5), 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 θa changes as a result of violation of the adiabaticity).There are different approaches to compute the flop probabilityP21. 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 H2m-H1m by a system with the kinetic energy dθm/dt. This leads to the Landau-Zener probability
(82)PLZ=exp(-π2κR)=exp(-πhΔm24Esin22θcos2θ),
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:(83)d=∫0L‍ne(x)dx.
We can define d1/2 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]
(84)d1/2≥dmin=π22GFtan2θ
for all density profiles. Furthermore, the minimum, dmin, 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, 2GFd, 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
(85)S=𝒯[exp(-i∫0L‍H(x)dx)],
where 𝒯 denotes time ordering of the exponential. For small values of L, it can be expanded as
(86)S=1-i∫0L‍H(x)dx+𝒪(L2).
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≪lm 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(87)P=sin22θmsin2ϕm=1Rsin22θsin2ϕR≈ϕ2sin22θ.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 isd=nx≪GF-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 lengthx. There are three possibilities.(i)
E≪ER (ER 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~lν.(ii)
E~ER—modification of oscillation parameters is strong; however lνR~lν/sin2θ~2π/(Vsin2θ). Consequently, x/lνR=xVsin2θ/2π≪1. Oscillations are undeveloped due to smallness of phase.(iii)
E≫ER—matter suppresses oscillation depth by a factor ER/E≪1. Since the oscillation length equals lm≈2π/V, one obtains x/lm=xV/2π≪1. Hence in this case the distance is very small and oscillation effect in the layer has double suppression.


## 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:(83)d=∫0L‍ne(x)dx.
We can define d1/2 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]
(84)d1/2≥dmin=π22GFtan2θ
for all density profiles. Furthermore, the minimum, dmin, 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, 2GFd, 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
(85)S=𝒯[exp(-i∫0L‍H(x)dx)],
where 𝒯 denotes time ordering of the exponential. For small values of L, it can be expanded as
(86)S=1-i∫0L‍H(x)dx+𝒪(L2).
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≪lm 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(87)P=sin22θmsin2ϕm=1Rsin22θsin2ϕR≈ϕ2sin22θ.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 isd=nx≪GF-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 lengthx. There are three possibilities.(i)
E≪ER (ER 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~lν.(ii)
E~ER—modification of oscillation parameters is strong; however lνR~lν/sin2θ~2π/(Vsin2θ). Consequently, x/lνR=xVsin2θ/2π≪1. Oscillations are undeveloped due to smallness of phase.(iii)
E≫ER—matter suppresses oscillation depth by a factor ER/E≪1. Since the oscillation length equals lm≈2π/V, one obtains x/lm=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 changeTn is an integer times the effective oscillation length lm [38]:
(88)∫lT‍dxlm=k,(k=1,2,3,…),
or lT/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 thecastle wall profile, for which the period lT is divided into the two parts l1 and l2 (l1+l2=lT) 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]:
(89)Φ1=Φ2=π.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≡2θ1m-2θ2m). 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
(90)ST=S2S1,
where Sk(k=1,2) is the evolution in layer k given by (57). For brevity we will write it as Sk=ckI-isk(σ·nk),k=1,2, where ck≡cosϕk, sk≡sinϕk, and ϕk is the half phase acquired in layer k:
(91)ϕk=12ΔHklk=Δm24ER(Vk)1/2lk,nk≡(sin2θmk,0,-cos2θmk).
Here θmk is the mixing angle in layer k.Insertion ofSk from (57) into (90) gives [37]
(92)ST=YI-i(σ·X),
where
(93)Y≡c1c2-s1s2(n1·n2),X=s1c2n1+s2c1n2-s1s2[n1×n2].
Explicitly, (n1·n2)=cos(2θm1-2θm2) and [n1×n2]=sin(2θm1-2θm2)ey. Using unitarity of ST, 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 ST can be written in the form ST=cosΦ-isinΦ(σ·X^)=e-i(σ·X^)Φ, where X^≡X/X. Consequently, the evolution matrix after n periods equals
(94)Sn=(ST)n=e-i(σ·X^)nΦ=cosnΦ-i(σ·X^)sinnΦ.
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′)Sn.The transition probability computed with (94) is
(95)Peμn=|Seμn|2=X12+X22X2sin2nΦ.
It has the form of the usual oscillation probability with phase nΦ and depth (X12+X22)/X2. The oscillations described by (95) are called the parametric oscillations. Under condition
(96)-X3=s1c2cos2θm1+s2c1cos2θm2=0,
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 theparametric 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,cos2θm=0, can be written according to (22) and (51) as α=α*, that is, S11(1)=S22(1), or equivalently, ImS11(1)=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≡ReS11(1)=0. The absolute maximum of the transition probability occurs when these conditions are satisfied simultaneously, that is, when S11(1)=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 S11(2)=S22(2). Let us find the conditions for extrema for density profiles consisting of two layers. We have S(2)=S2S1, where S11(2)=α2α1-β2β1*,S12(2)=α2β1+β2α1*, and αi,βi for each layer have been defined in (51). The sum of the two complex numbers in the transition amplitude S12(2) has the largest possible result if they have the same phase: arg(α2β1)=arg(β2α1*), which can also be rewritten as
(97)arg(α1α2β1)=arg(β2).
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 |βi| 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 matrixS(3) equal
(98)S12(3)=α3S12(2)+β3S11(2)*=α3α2β1+α3β2α1*+β3α2*α1*-β3β2*β1.
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=-β1*, and the expression for S12(3) can be written as
(99)S12(3)=α1α2β1-α1*α2*β1*+|α1|2β2+|β1|2β2*.
Note that β2 is pure imaginary because the core density profile is symmetric. Therefore the amplitude S12(3) 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 S12(3), 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).


## 3.10.1.  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 changeTn is an integer times the effective oscillation length lm [38]:
(88)∫lT‍dxlm=k,(k=1,2,3,…),
or lT/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 thecastle wall profile, for which the period lT is divided into the two parts l1 and l2 (l1+l2=lT) 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]:
(89)Φ1=Φ2=π.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≡2θ1m-2θ2m). 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
(90)ST=S2S1,
where Sk(k=1,2) is the evolution in layer k given by (57). For brevity we will write it as Sk=ckI-isk(σ·nk),k=1,2, where ck≡cosϕk, sk≡sinϕk, and ϕk is the half phase acquired in layer k:
(91)ϕk=12ΔHklk=Δm24ER(Vk)1/2lk,nk≡(sin2θmk,0,-cos2θmk).
Here θmk is the mixing angle in layer k.Insertion ofSk from (57) into (90) gives [37]
(92)ST=YI-i(σ·X),
where
(93)Y≡c1c2-s1s2(n1·n2),X=s1c2n1+s2c1n2-s1s2[n1×n2].
Explicitly, (n1·n2)=cos(2θm1-2θm2) and [n1×n2]=sin(2θm1-2θm2)ey. Using unitarity of ST, 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 ST can be written in the form ST=cosΦ-isinΦ(σ·X^)=e-i(σ·X^)Φ, where X^≡X/X. Consequently, the evolution matrix after n periods equals
(94)Sn=(ST)n=e-i(σ·X^)nΦ=cosnΦ-i(σ·X^)sinnΦ.
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′)Sn.The transition probability computed with (94) is
(95)Peμn=|Seμn|2=X12+X22X2sin2nΦ.
It has the form of the usual oscillation probability with phase nΦ and depth (X12+X22)/X2. The oscillations described by (95) are called the parametric oscillations. Under condition
(96)-X3=s1c2cos2θm1+s2c1cos2θm2=0,
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 theparametric 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,cos2θm=0, can be written according to (22) and (51) as α=α*, that is, S11(1)=S22(1), or equivalently, ImS11(1)=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≡ReS11(1)=0. The absolute maximum of the transition probability occurs when these conditions are satisfied simultaneously, that is, when S11(1)=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 S11(2)=S22(2). Let us find the conditions for extrema for density profiles consisting of two layers. We have S(2)=S2S1, where S11(2)=α2α1-β2β1*,S12(2)=α2β1+β2α1*, and αi,βi for each layer have been defined in (51). The sum of the two complex numbers in the transition amplitude S12(2) has the largest possible result if they have the same phase: arg(α2β1)=arg(β2α1*), which can also be rewritten as
(97)arg(α1α2β1)=arg(β2).
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 |βi| 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 matrixS(3) equal
(98)S12(3)=α3S12(2)+β3S11(2)*=α3α2β1+α3β2α1*+β3α2*α1*-β3β2*β1.
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=-β1*, and the expression for S12(3) can be written as
(99)S12(3)=α1α2β1-α1*α2*β1*+|α1|2β2+|β1|2β2*.
Note that β2 is pure imaginary because the core density profile is symmetric. Therefore the amplitude S12(3) 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 S12(3), 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 whenV>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+HI with
(100)H~0=ωm2(100-1),HI=sin2θΔ(-ϵ11ϵ),
where ωm is the oscillation frequency (20) and ϵ≡(2Δcos2θ-V+ωm)/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 θm: 2Δsin2θ/ωm=sin2θm. Therefore for sin2θm≪1 the term HI can be considered as a perturbation. Furthermore, ϵ~sin2θm, so the diagonal terms in HI can be neglected in the lowest approximation.The solution forS matrix can be found in the form S=S0·SI, where S0 is the solution of the evolution equation with H replaced by H0 (see (71)). The matrix SI then satisfies the equation
(101)idSIdx=S0-1HIS0SI=H~ISI,
where H~I≡S0-1HIS0 is the perturbation Hamiltonian in the “interaction” representation. Equation (101) can be solved by iterations: SI=I+SI(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,
(102)S(L)=S0(L)[I-iΔsin2θ∫0Ldx(0ei2ϕ(x)e-i2ϕ(x)0)].
The νe↔νa transition probability P2=[S(L)]ae is given by
(103)P2=Δ2sin22θ|∫0L‍dxe-i2ϕ(x)|2.For density profiles that are symmetric with respect to the center of the neutrino trajectory,V(x)=V(L-x), (103) gives
(104)P2=4(Δm24E)2sin22θ[∫0L/2‍dzcos2ϕ(z)]2,
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 forE≳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 ofH0 in (100) has to be flipped, and correspondingly one has to replace ωm→-ωm 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:
(105)H(x)=H-+ΔH(x),
where
(106)H-≡ω-(-cos2θ-sin2θ-sin2θ-cos2θ-),ΔH≡ΔV(x)2(100-1).
Here, θ-=θm(V-) is the mixing angle in matter and ω-=ωm(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 θm=θ- and ϕm(x)=ϕ(x)≡ω-x,  ω-=ωm(V-).The solution of the evolution equation with Hamiltonian (105) [46] is of the form
(107)S(x)=S-(x)+ΔS(x),ΔS(x)=-iS-(x)K1(x),
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):
(108)dK1(x)dx=S-†ΔH(x)S-=ΔV2[-cos2θ-(-cos2θ-sin2θ-sin2θ-cos2θ-)+sin2θ-cos2ϕG(θ-)+sin2θ-sin2ϕσ2{(-cos2θ-sin2θ-sin2θ-cos2θ-)}],
where G(θ-)≡cos2θ-σ1+sin2θ-σ3. The first term in (108) does not contribute to S≡S(L) since 〈ΔV〉≡∫‍ΔV(x)dx=0, and (108) can be immediately integrated:
(109)K1(L)=12sin2θ-[G(θ-)∫0L‍ΔV(x)cos2ϕ(x)dx+σ2∫0L‍ΔV(x)sin2ϕ(x)dx].
Introducing the distance from the midpoint of the neutrino trajectory z≡x-L/2, one obtains from (109)
(110)ΔS≡ΔS(L)=-isin2θ-[G(θ-)ΔI+σ2ΔJ],
where ΔI≡(1/2)∫-L/2L/2‍ΔV(z)cos(2ω-z)dz,ΔJ≡(1/2)∫-L/2L/2‍ΔV(z)sin(2ω-z)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 forS which is exactly unitary regardless of the size of the perturbation. For this we rewrite (110) as follows:
(111)ΔS=εS′,S′=-i[G(θ-)cosξ+σ2sinξ],
where sinξ=ΔJ/(ΔJ)2+(ΔI)2 and ϵ=sin2θ-·(ΔJ)2+(ΔI)2. Thus, S=S-+εS′ and we replace it by
(112)S=cosεS-+sinεS′.
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 termΔJ is absent. From (54), (110), and (112) we immediately get the transition probability
(113)P=[cosεsin2θ-sinϕ+sinεcos2θ-]2≈sin22θ-[sinϕ+ΔIcos2θ-]2,
where ε≡sin2θ-ΔI and ϕ≡ϕ(L)=ω-L. Here the first term in the square brackets describes oscillations in constant density matter with average potential V-1.


## 3.13. Oscillation Probabilities and Their Properties

It is convenient to consider the neutrino flavor evolution in the propagation basisν~=(νe,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
(114)ν~=U13I-δU12ν.
Since the transformations, which connect ν~ and νf, 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(115)S~=(AeeAe2~Ae3~A2~eA2~2~A2~3~A3~eA3~2~A3~3~).
Then according to (27) the S matrix in the flavor basis equals
(116)S=U~S~U~†,U~≡U23Iδ.
In this part, we use the notation Aij for the amplitudes in the propagation basis and Sij 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
(117)S=(Aeec23Ae2~+s23e-iδAe3~-s23Ae2~+c23e-iδAe3~c23A2~e+s23eiδA3~ec232A2~2~+s232A3~3~+Kμμ-s23c23(A2~2~-A3~3~)+Kμτ-s23A2~e+c23eiδA3~e-s23c23(A2~2~-A3~3~)+Kτμs232A2~2~+c232A3~3~+Kττ),where(118)Kμμ≡s23c23(e-iδA2~3~+eiδA3~2~),Kμτ≡c232e-iδA2~3~-s232eiδA3~2~,Kτμ=Kμτ(δ⟶-δ,2~⟷3~),Kττ=-Kμμ.
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):
(119)Ae3~,A3~e~s13,Ae2~,A2~e~rΔ~s132,A3~2~,A2~3~~s13rΔ~s133,
that is, A2~3~ and A3~2~ 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
(120)A2~e=Ae2~,A3~e=Ae3~,A3~2~=A2~3~.
Consequently, for Kαβ we obtain
(121)Kμτ=A2~3~(cos2θ23cosδ-isinδ),Kτμ=Kμτ(δ⟶-δ),Kμμ=-Kττ=A2~3~sin2θ23cosδ.
These terms proportional to small amplitudes A2~3~ and A3~2~ are of the order O(s132).Figure 7
Scheme of transitions between the flavor states. Evolution is considered in the propagation basisν~. 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 Pαβ≡|Sβα|2:
(122)Pee=|Aee|2=1-|Ae2~|2-|Ae3~|2,(123)Pμe=c232|Ae2~|2+s232|Ae3~|2+2s23c23Re(e-iδAe2~*Ae3~),(124)Pτe=s232|Ae2~|2+c232|Ae3~|2-2s23c23Re(e-iδAe2~*Ae3~),(125)Pμμ=|c232A2~2~+s232A3~3~+2s23c23cosδA2~3~|2,(126)Pμτ=|s23c23(A3~3~-A2~2~)+(cos2θ23cosδ+isinδ)A2~3~|2.
For antineutrinos the amplitudes can be obtained from the results presented above substituting
(127)δ⟶-δ,Aij⟶A-ij,whereA-ij≡Aij(V⟶-V).
Notice that the amplitudes of transitions (123) and (124), that involve νe, 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 HamiltonianH~ in (29), in the limits Δm212→0 or/and s12→0 the state ν~2 decouples from the rest of the system, and consequently, the amplitude Ae2~ vanishes. In this limit, Ae3~ (as well as A3~3~ and See) is reduced to a 2ν amplitude which depends on the parameters Δm312 and θ13: AA(Δm312,θ13)≡Ae3~(Δm212=0). The corresponding probability equals PA≡|AA|2.In the limits13→0 the state ν~3 decouples while the amplitude Ae3~ vanishes and the amplitude Ae2~ reduces to a 2ν amplitude depending on the parameters of the 1-2 sector, Δm212 and θ12. Denoting this amplitude by AS we have AS(Δm212,θ12)≡Ae2~(θ13=0). We will use the notation PS≡|AS|2.This consideration implies that to the leading nontrivial order in the small parameterss13 and rΔ the amplitudes Ae2~ and A2~e are reduced to two neutrino probabilities and depend only on the “solar” parameters, whereas the amplitudes Ae3~ and A3~e only on the “atmospheric” parameters:
(128)Ae2~≃A2~e≃AS(Δm212,θ12),Ae3~≃A3~e≃AA(Δm312,θ13).
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 Ae2~ are of order s132, whereas the corrections to the “atmospheric” amplitude Ae3~ are of order rΔ [49], in agreement with our consideration for constant density. The amplitude Ae3~ does not in general have a 2-flavor form, once the corrections to the factorization approximation are taken into account.Using the expressions forUeim and Uμim in terms of the mixing angles in the standard parametrization, we can rewrite (67) as
(129)Seμcst=cosθ23mAe2~cst+sinθ23me-iδmAe3~cst,
where
(130)Ae2~cst≡-ieiϕ21mcosθ13msin2θ12msinϕ21m,Ae3~cst≡-ieiϕ21msin2θ13m[sinϕ32me-iϕ31m+cos2θ12msinϕ21m].
Here ϕ31m=ϕ32m+ϕ21m. Since to a good approximation θ23m≈θ23 and δm≈δ (see Section 2.5) [20, 50], the amplitudes Ae2~cst and Ae3~cst can be identified with Ae2~ and Ae3~ in (123) and (124).In terms of mixing angles,Uμ1m=-s12mc23m-c12ms13ms23meiδm, Uμ3m=c13ms23m, the amplitude Sμμcst can be rewritten as
(131)Sμμcst=cos2θ23mA2~2~cst+sin2θ23mA3~3~cst+sin2θ23mcosδmA2~3~cst,
where
(132)A2~2~cst≡1+2ieiϕ21msin2θ12msinϕ21m,A3~3~cst≡1-2ie-iϕ32mcos2θ13msinϕ32m+2ieiϕ21msin2θ13mcos2θ12msinϕ21m,A2~3~cst≡ieiϕ21msinθ13msin2θ12msinϕ21m.
Notice that A2~2~cst has exactly the form of the corresponding 2ν amplitude driven by the solar parameters. The amplitude A3~3~cst also coincides to a very good approximation with the corresponding 2ν amplitude driven by the atmospheric parameters. In the approximation θ23m≈θ23 and δm≈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 νe disappearance equals
(133)1-Pee=Peμ+Peτ=Pe2~+Pe3~.
The probabilityPee 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 Pee originates mainly from Pe3~≡|Ae3~|2. The survival probability then equals Pee=1-Peμ-Peτ=1-PA-PS. At high energies, where the effects of the 1-2 mixing and mass splitting in P are suppressed, the probability is Pee≈1-Peτ≈1-PA.
(2)
ν
e
-
ν
μ and νe-ντ channels: the transition probability Pμe≡P(νμ→νe) (see (123)) can be rewritten as
(134)Pμe=c232|Ae2~|2+s232|Ae3~|2+sin2θ23|Ae2~*Ae3~|cos(ϕ-δ),
whereϕ≡arg(Ae2~*Ae3~). Unlike 1-Pee, this probability contains the interference term between Ae2~ and Ae3~ which depends on the CP violation phase. Since the amplitude Ae2~ is suppressed at high energies due to the smallness of the 1-2 mixing in matter, in the lowest approximation we have
(135)Pμe≈sin2θ23|Ae3~|2≈sin2θ23|AA|2.
The maximal value of the probability equalsPμe≃s232.
According to (123) and (124) the oscillation probabilities Pτe and Peτ can be obtained from the corresponding probabilities Pμe and Peμ through the substitution s23→c23, c23→-s23 [51]. The interference term has the opposite signs for channels including ντ as compared with those with νμ, which can be obtained from the unitarity condition Pee+Pμe+Pτe=1 and the fact that Pee does not depend on δ.
(3)
Theνμ survival probability, Pμμ, for symmetric density profiles, (125), can be rewritten as
(136)Pμμ=|c232A2~2~+s232A3~3~|2+2sin2θ23cosδRe[A2~3~*(c232A2~2~+s232A3~3~)]+sin22θ23cos2δ|A2~3~|2.
SinceA2~3~=𝒪(rΔs13) is a small quantity, to a good approximation one can neglect the term ~cos2δ in (125), which is proportional to |A2~3~|2. For high energies in the limit Δm212→0 we have A2~2~=1, A2~3~=0. Then, parameterizing the 33-amplitude as A3~3~=1-PAe-iϕ3~3~m we obtain from (136)
(137)Pμμ(Δm212=0)=1-sin22θ23sin2ϕx-s234PA-0.5sin22θ23cos2ϕX(1-1-PA),
whereϕX=0.5arg[A3~3~*A2~2~]=ϕ2~2~m-ϕ3~3~m. The probability can be rewritten as
(138)Pμμ=1-0.5sin22θ23-s234PA+0.5sin22θ23(1-PA)cos2ϕX.
(4)
ν
μ
-
ν
τ channel: for symmetric matter density profiles the probability of νμ→ντ oscillations is given in (126). It can be rewritten as
(139)Pμτ=14sin22θ23|A2~2~-A3~3~|2+sin2θ23cos2θ23cosδRe[(A3~3~*-A2~2~*)A2~3~]-sin2θ23sinδIm[Ae2~*Ae3~]+(1-sin22θ23cos2δ)|A2~3~|2.The amplitude depends on δ through the terms proportional to cosδ and sinδ, and therefore Pμτ 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 Pμτδ=-Pμeδ-Pμμδ. In the limit Δm212→0 we obtain
(140)Pμτ=0.5sin22θ23-s232c232PA-0.5sin22θ23(1-PA)cos2ϕX.


## 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(115)S~=(AeeAe2~Ae3~A2~eA2~2~A2~3~A3~eA3~2~A3~3~).
Then according to (27) the S matrix in the flavor basis equals
(116)S=U~S~U~†,U~≡U23Iδ.
In this part, we use the notation Aij for the amplitudes in the propagation basis and Sij 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
(117)S=(Aeec23Ae2~+s23e-iδAe3~-s23Ae2~+c23e-iδAe3~c23A2~e+s23eiδA3~ec232A2~2~+s232A3~3~+Kμμ-s23c23(A2~2~-A3~3~)+Kμτ-s23A2~e+c23eiδA3~e-s23c23(A2~2~-A3~3~)+Kτμs232A2~2~+c232A3~3~+Kττ),where(118)Kμμ≡s23c23(e-iδA2~3~+eiδA3~2~),Kμτ≡c232e-iδA2~3~-s232eiδA3~2~,Kτμ=Kμτ(δ⟶-δ,2~⟷3~),Kττ=-Kμμ.
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):
(119)Ae3~,A3~e~s13,Ae2~,A2~e~rΔ~s132,A3~2~,A2~3~~s13rΔ~s133,
that is, A2~3~ and A3~2~ 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
(120)A2~e=Ae2~,A3~e=Ae3~,A3~2~=A2~3~.
Consequently, for Kαβ we obtain
(121)Kμτ=A2~3~(cos2θ23cosδ-isinδ),Kτμ=Kμτ(δ⟶-δ),Kμμ=-Kττ=A2~3~sin2θ23cosδ.
These terms proportional to small amplitudes A2~3~ and A3~2~ are of the order O(s132).Figure 7
Scheme of transitions between the flavor states. Evolution is considered in the propagation basisν~. 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 Pαβ≡|Sβα|2:
(122)Pee=|Aee|2=1-|Ae2~|2-|Ae3~|2,(123)Pμe=c232|Ae2~|2+s232|Ae3~|2+2s23c23Re(e-iδAe2~*Ae3~),(124)Pτe=s232|Ae2~|2+c232|Ae3~|2-2s23c23Re(e-iδAe2~*Ae3~),(125)Pμμ=|c232A2~2~+s232A3~3~+2s23c23cosδA2~3~|2,(126)Pμτ=|s23c23(A3~3~-A2~2~)+(cos2θ23cosδ+isinδ)A2~3~|2.
For antineutrinos the amplitudes can be obtained from the results presented above substituting
(127)δ⟶-δ,Aij⟶A-ij,whereA-ij≡Aij(V⟶-V).
Notice that the amplitudes of transitions (123) and (124), that involve νe, 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 HamiltonianH~ in (29), in the limits Δm212→0 or/and s12→0 the state ν~2 decouples from the rest of the system, and consequently, the amplitude Ae2~ vanishes. In this limit, Ae3~ (as well as A3~3~ and See) is reduced to a 2ν amplitude which depends on the parameters Δm312 and θ13: AA(Δm312,θ13)≡Ae3~(Δm212=0). The corresponding probability equals PA≡|AA|2.In the limits13→0 the state ν~3 decouples while the amplitude Ae3~ vanishes and the amplitude Ae2~ reduces to a 2ν amplitude depending on the parameters of the 1-2 sector, Δm212 and θ12. Denoting this amplitude by AS we have AS(Δm212,θ12)≡Ae2~(θ13=0). We will use the notation PS≡|AS|2.This consideration implies that to the leading nontrivial order in the small parameterss13 and rΔ the amplitudes Ae2~ and A2~e are reduced to two neutrino probabilities and depend only on the “solar” parameters, whereas the amplitudes Ae3~ and A3~e only on the “atmospheric” parameters:
(128)Ae2~≃A2~e≃AS(Δm212,θ12),Ae3~≃A3~e≃AA(Δm312,θ13).
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 Ae2~ are of order s132, whereas the corrections to the “atmospheric” amplitude Ae3~ are of order rΔ [49], in agreement with our consideration for constant density. The amplitude Ae3~ does not in general have a 2-flavor form, once the corrections to the factorization approximation are taken into account.Using the expressions forUeim and Uμim in terms of the mixing angles in the standard parametrization, we can rewrite (67) as
(129)Seμcst=cosθ23mAe2~cst+sinθ23me-iδmAe3~cst,
where
(130)Ae2~cst≡-ieiϕ21mcosθ13msin2θ12msinϕ21m,Ae3~cst≡-ieiϕ21msin2θ13m[sinϕ32me-iϕ31m+cos2θ12msinϕ21m].
Here ϕ31m=ϕ32m+ϕ21m. Since to a good approximation θ23m≈θ23 and δm≈δ (see Section 2.5) [20, 50], the amplitudes Ae2~cst and Ae3~cst can be identified with Ae2~ and Ae3~ in (123) and (124).In terms of mixing angles,Uμ1m=-s12mc23m-c12ms13ms23meiδm, Uμ3m=c13ms23m, the amplitude Sμμcst can be rewritten as
(131)Sμμcst=cos2θ23mA2~2~cst+sin2θ23mA3~3~cst+sin2θ23mcosδmA2~3~cst,
where
(132)A2~2~cst≡1+2ieiϕ21msin2θ12msinϕ21m,A3~3~cst≡1-2ie-iϕ32mcos2θ13msinϕ32m+2ieiϕ21msin2θ13mcos2θ12msinϕ21m,A2~3~cst≡ieiϕ21msinθ13msin2θ12msinϕ21m.
Notice that A2~2~cst has exactly the form of the corresponding 2ν amplitude driven by the solar parameters. The amplitude A3~3~cst also coincides to a very good approximation with the corresponding 2ν amplitude driven by the atmospheric parameters. In the approximation θ23m≈θ23 and δm≈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 νe disappearance equals
(133)1-Pee=Peμ+Peτ=Pe2~+Pe3~.
The probabilityPee 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 Pee originates mainly from Pe3~≡|Ae3~|2. The survival probability then equals Pee=1-Peμ-Peτ=1-PA-PS. At high energies, where the effects of the 1-2 mixing and mass splitting in P are suppressed, the probability is Pee≈1-Peτ≈1-PA.
(2)
ν
e
-
ν
μ and νe-ντ channels: the transition probability Pμe≡P(νμ→νe) (see (123)) can be rewritten as
(134)Pμe=c232|Ae2~|2+s232|Ae3~|2+sin2θ23|Ae2~*Ae3~|cos(ϕ-δ),
whereϕ≡arg(Ae2~*Ae3~). Unlike 1-Pee, this probability contains the interference term between Ae2~ and Ae3~ which depends on the CP violation phase. Since the amplitude Ae2~ is suppressed at high energies due to the smallness of the 1-2 mixing in matter, in the lowest approximation we have
(135)Pμe≈sin2θ23|Ae3~|2≈sin2θ23|AA|2.
The maximal value of the probability equalsPμe≃s232.
According to (123) and (124) the oscillation probabilities Pτe and Peτ can be obtained from the corresponding probabilities Pμe and Peμ through the substitution s23→c23, c23→-s23 [51]. The interference term has the opposite signs for channels including ντ as compared with those with νμ, which can be obtained from the unitarity condition Pee+Pμe+Pτe=1 and the fact that Pee does not depend on δ.
(3)
Theνμ survival probability, Pμμ, for symmetric density profiles, (125), can be rewritten as
(136)Pμμ=|c232A2~2~+s232A3~3~|2+2sin2θ23cosδRe[A2~3~*(c232A2~2~+s232A3~3~)]+sin22θ23cos2δ|A2~3~|2.
SinceA2~3~=𝒪(rΔs13) is a small quantity, to a good approximation one can neglect the term ~cos2δ in (125), which is proportional to |A2~3~|2. For high energies in the limit Δm212→0 we have A2~2~=1, A2~3~=0. Then, parameterizing the 33-amplitude as A3~3~=1-PAe-iϕ3~3~m we obtain from (136)
(137)Pμμ(Δm212=0)=1-sin22θ23sin2ϕx-s234PA-0.5sin22θ23cos2ϕX(1-1-PA),
whereϕX=0.5arg[A3~3~*A2~2~]=ϕ2~2~m-ϕ3~3~m. The probability can be rewritten as
(138)Pμμ=1-0.5sin22θ23-s234PA+0.5sin22θ23(1-PA)cos2ϕX.
(4)
ν
μ
-
ν
τ channel: for symmetric matter density profiles the probability of νμ→ντ oscillations is given in (126). It can be rewritten as
(139)Pμτ=14sin22θ23|A2~2~-A3~3~|2+sin2θ23cos2θ23cosδRe[(A3~3~*-A2~2~*)A2~3~]-sin2θ23sinδIm[Ae2~*Ae3~]+(1-sin22θ23cos2δ)|A2~3~|2.The amplitude depends on δ through the terms proportional to cosδ and sinδ, and therefore Pμτ 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 Pμτδ=-Pμeδ-Pμμδ. In the limit Δm212→0 we obtain
(140)Pμτ=0.5sin22θ23-s232c232PA-0.5sin22θ23(1-PA)cos2ϕX.


## 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 energiesE>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:
(141)V(x)=V(L-x).
Here L=2R⊕|cosθz| is the length of the trajectory inside the Earth, R⊕=6371 km is the Earth radius, and θz is the zenith angle related to the nadir angle as Θν=π-θz. For 0≤Θν≤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 dEarth=∫‍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 theEν-cosθz plane. In Figure 8, we show the oscillograms for the oscillation probabilities Peμ and Pμμ, as well as the corresponding probabilities for antineutrinos [43, 46, 52–55].Neutrino oscillograms of the Earth. Shown are the lines of equal flavor conversion probability in theEν-cosΘν plane. Upper panels: νe→νμ (left) and ν-e→ν-μ (right); bottom panels: νμ→νμ (left) and ν-μ→ν-μ (right). Normal hierarchy is assumed.
(a)
(b)
(c)
(d)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)
sin22θm=1, the amplitude condition, which is nothing but the MSW resonance condition, and(ii)
ϕ=π/2+πk, the phase condition.AtE>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 atEν~(5–7)GeV. The position of the maximum is given by the MSW resonance condition:
(142)Eν=ER(Θν)=Δm312cos2θ132V-1(Θν),whereV-1(Θν) is the average value of the potential along the trajectory characterized by Θν. The phase condition becomes 2ϕ(Eν,Θν)=2ω(V-,Eν)L(Θν)=π and the intersection of the resonance and the phase condition lines gives the absolute maximum of PA. Combining these conditions gives the coordinates of the peak: cosΘν=0.77 and ER=6 GeV.(2)
Three parametric ridges in the domain of core-crossing trajectories|cosθz|>0.87 and Eν>3 GeV. The parametric ridges differ by the oscillation phase acquired in the core, ϕ2.
(i)
Ridge A lies between the core resonance (at Θν~0∘) and the mantle resonance regions, Eν≈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 Eν≥5 GeV. For the lowest energies in the ridge and Θν~0, the half phase in the core equals ϕ2~(1.2-1.3)π.(iii)
Ridge C is located at Eν>11 GeV in the matter dominated region, where the mixing and, consequently, oscillation depth are suppressed.(3)
The MSW resonance peak in core located atEν~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 Figure9, we show graphic representations of oscillations which correspond to salient features of the oscillograms.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 vectorH for the mantle (red) and core (green), respectively.
(a)
(b)
(c)For energiesEν<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-Pee=|Ae2~|2≡PS. The probabilities of the modes including νe are expressed in terms of a unique probability PS.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,lmR=lν/sin2θ12~lν. The resonance energy is shifted to smaller values both due to Δm212≪Δm312 and because of the factor cos2θ12≈0.4: E12R=(Δm212/2V-)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 lm 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 Θν=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 (Θν>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 (Θν≈82∘) corresponds to ϕ≈π/2, the middle (Θν=60∘) to ϕ≈3π/2, and the inner (Θν≈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
(143)E12R≈0.12GeV.
The ratio of the 1-2 and 1–3 resonance energies equals E12R/E13R≈1/50. The estimate (143) is valid for the two outer peaks. For the peak at Θν=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 Eν direction as compared to the oscillations governed by the 1–3 mixing and splitting.The resonance energy in the core isE12R≈0.04 GeV. Therefore for Eν>(0.10–0.15)GeV the 1-2 mixing in the core is substantially suppressed by matter. The peak at Eν≃0.2 GeV and Θν≃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 ϕmantle≈π/2 and ϕcore≈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 ofΔm212=0, the neutrino oscillograms for the inverted hierarchy coincide with the antineutrino oscillograms for the normal hierarchy and vice versa, provided that Δm312 is taken to be the same in both cases [56]: PAIH=P-ANH,ϕXIH=-ϕ-XNH. Therefore PαβIH=P-αβNH,P-αβIH=PαβNH. 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 probabilityPee 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 νe is not affected by this transformation. For oscillations in vacuum, or in matter with symmetric density profiles, the other two survival probabilities, Pμμ and Pττ, 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 phaseϕ≡arg(Ae2~*Ae3~) and omitting small terms proportional to |A2~3~|2=𝒪(s136) we obtain
(144)Pμeδ=sin2θ23cos(ϕ-δ)|Ae2~Ae3~|,(145)Pμμδ=-sin2θ23cosδcosϕ|Ae2~Ae3~|-D23,(146)Pμτδ=-sin2θ23sinδsinϕ|Ae2~Ae3~|+D23,
where D23≡(1/2)sin4θ23cosδRe[A2~3~*(A3~3~-A2~2~)] is proportional to the small deviation of the 2-3 mixing from maximal one. Notice that D23 enters Pμμδ and Pμτδ with opposite signs while Pμeδ does not depend on D23 at all. D23 is CP even. The sum of these interference terms is zero.For the other channels,Pαβδ=Pβα-δ. 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, |Ae2~Ae3~| and D23.


#### 4.2.2. Magic Lines and CP Domains

To better assess the effect ofδ, one can consider the difference of the oscillation probabilities for two different values of the CP phase ΔPαβCP(δ)≡Pαβ(δ)-Pαβ(δ0). In practice, this quantifies how well the phase δ fits with some assumed true value δ0. The structure of the oscillograms for ΔPαβCP(δ) can be understood in terms of the grids of magic lines and interference phase lines along which ΔPαβCP(δ)≈0.For theνμ→νe oscillation probability, the equality
(147)ΔPμeCP(δ)≡Pμe(δ)-Pμe(δ0)=Pμeδ(δ)-Pμeδ(δ0)
is exact and the condition ΔPμeCP=0 is equivalent to
(148)|Ae2~Ae3~|cos(ϕ-δ)=|Ae2~Ae3~|cos(ϕ-δ0).
This equality is satisfied if at least one of the following three conditions is fulfilled:
(149)Ae2~(Eν,Θν)=0,Ae3~(Eν,Θν)=0,ϕ(Eν,Θν)-δ0=-[ϕ(Eν,Θν)-δ]+2πl.
The last condition implies
(150)ϕ(Eν,Θν)=(δ+δ0)2+πl.
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 Ae2~ and Ae3~ are complex quantities, these conditions can be satisfied in isolated points of the (Θν,Eν) plane only. In contrast to this, in the factorization approximation Ae2~=AS and Ae3~=AA both the conditions are fulfilled along certain lines in the oscillograms. This occurs because the amplitudes AS and AA take a 2-flavor form. On the basis of neutrino states where the corresponding 2×2 Hamiltonians are traceless, both AA and AS are pure imaginary because of the symmetry of the Earth’s density profile [47].Let us consider the equalitiesAS=0 and AA=0 using the constant density approximation.(1)
The conditionAS(Eν,Θν)=0 is satisfied when sinϕS(Eν,Θν)=0, which leads to
(151)L(Θν)≈2πnω21m,n=1,2,….
At energiesEν≳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 ω21m≈V and the condition (151) becomes
(152)L(Θν)≃2πnV.
This expression is energy independent and determines the baselines for which the “solar” contribution to the probability vanishes [61]. In the plane (Θν,Eν) it represents nearly vertical lines at fixed Θν. There are three solar magic lines which correspond to n=1 (in the mantle domain) Θν≈54∘ and n=2,3 (in the core domain) [61] Θν≈30∘ and 12∘. The existence of a baseline (L≈7600 km) for which the probability of νe↔νμ oscillations in the Earth is approximately independent of the “solar” parameters (Δm212, 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 conditionAA(Eν,Θν)=0 [61]. Along these lines, the “atmospheric” contribution to the amplitudes of νμ↔νe and ντ↔νe transitions vanishes and the probabilities of oscillations involving νe or ν-e do not depend on CP phase. In the constant density approximation, the condition AA=0 is satisfied when sinϕA=0 (ϕA=πk, k=1,2,…) or explicitly
(153)L(Θν)≈2πkω31m,k=1,2,….
For energies which are not too close to the 1–3 MSW resonance, it reduces to(154)Eν≃Δm312L(Θν)|4πk±2VL(Θν)|,
which corresponds to the bent curves in the(Θν,Eν) plane. For very large energies, where Δm312/2E≪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 (Θν,Eν) plane. In the constant density approximation ϕ≈-ϕ31m. Consequently in the energy range between the two resonances we have
(155)ϕ31m≈Δm312L4Eν=ϕA0,
that is, in the first approximationϕ does not depend on the matter density. From (150) we then obtain
(156)Eν=Δm312L(Θν)4πl-2(δ+δ0).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 Pμe,Peμ,Pτe, and Peτ 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 probabilitiesΔPμeCP(δ)=Pμe(δ)-Pμe(δ0) with δ0=0∘. Shown are the solar (black), atmospheric (white), and interference phase condition (cyan) curves from [60].From Figure10, we can see that the magic lines described above do not coincide exactly with the lines of ΔPμeCP=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 Figure11 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 Pαβ 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 Pμe 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 forνe→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
(
θ
13
,
δ
)
degeneracy: for any combination of (θ13,δ), there exists a different combination (θ^13,δ^) that also predicts the same oscillation probabilities.
(3)
Octant
(
θ
23
)
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.Two different illustrations of parameter space degeneracies. Left panel: biprobability plot forL=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 P-μe (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.
(a)
(b)For known mass hierarchy (e.g., normal one) a given value ofθ13 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 θ13prime, which crosses this point and therefore θ13prime 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 probabilityPμe 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 Pμe 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
(157)sin22θ~=sin22θ1+3cos22θ≃14sin22θ,
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
(158)sin22θ~≃sin22θ[1+4VEΔm2cos2θ].
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 probabilityPμe 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 P-μe 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 ofVE 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

Figure13 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 νμ-νe oscillation probability (the “golden channel”) can be expanded in the small quantity Δm212L/2E which gives [69]
(159)Peμ≈s232P2f+c13sin2θ13sin2θ12sin2θ23Δm2122EV×sin(VL2)sin(Δm312L4E)cos(δ-Δm312L4E),
where P2f is the two-flavor oscillation probability discussed earlier. In (159) we have neglected terms of the second (and higher) order in Δm212L/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 Δm312. 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 ofνe, νμ and the corresponding antineutrinos, while the original ντ flux is negligible. We introduce Φe0 and Φμ0, the electron and muon neutrino fluxes, as well as Φ-e0 and Φ-μ0, the electron and muon antineutrino fluxes, at the detector in the absence of oscillations. The flavor ratios
(160)r≡Φμ0Φe0,r-≡Φ-μ0Φ-e0
increase with energy.There is a mild neutrino-antineutrino asymmetry: the neutrino fluxΦ-μ0/Φμ0≈0.8-0.9. All the fluxes (at E>1 GeV) decrease rapidly with energy Φα0∝E-k, k=k(E)=3–5, and an azimuthal dependence shows up at low energies.The flux of neutrinos of flavorνα at a detector, with oscillations taken into account, is given by
(161)Φα=Φe0Peα+Φμ0Pμα=Φe0[Peα+r(E,Θν)Pμα],α=e,μ,τ.
Similar expressions hold for the antineutrino fluxes. Inserting the analytic expressions for the probabilities (122)–(126), one finds
(162)ΦeΦe0=1+(rs232-1)Pe3~+(rc232-1)Pe2~+rPμeδ,ΦμΦμ0≈1-2s232c232[1-Re(A2~2~*A3~3~)]-s232r(rs232-1)Pe3~-c232r(rc232-1)Pe2~+Pμμδ+1rPeμδ,ΦτΦμ0≈2s232c232[1-Re(A2~2~*A3~3~)]-c232r(rs232-1)Pe3~-s232r(rc232-1)Pe2~+Pμτδ+1rPeτδ,
where Pe3~≡|Ae3~|2 and Pe2~≡|Ae2~|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(163)|A2~2~|2=1-|A2~e|2-|A2~3~|2≈1-|A2~e|2=1-Pe2~,
where the terms proportional to |A3~2~|2 have been neglected we can approximate
(164)Re(A2~2~*A3~3~)≈(1-Pe3~)(1-Pe2~)cosψ.
Here ψ≡argA33A22* is the relative phase between the two amplitudes. For the νe flux, we then obtain
(165)ΦeΦe0≈1+(rs232-1)Pe3~+(rc232-1)Pe2~+rsin2θ23Pe3~Pe2~cos(ϕ-δ).
The oscillated fluxes satisfy the sum rule
(166)Φe+Φμ+Φτ=Φe0+Φμ0,
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]: Pe3~ with (rs232-1) and Pe2~ with (rc232-1). The contribution of the “atmospheric mode” vanishes along the line r(E,Θν)=1/s232, whereas the contribution of the “solar mode” vanishes along r(E,Θν)=1/c232. For maximal mixing both contributions vanish along the same line, r(E,Θν)=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: s232 or c232<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 theνμ flux, the contributions of the 1-2 and 1–3 modes are suppressed by additional factors s232/r and c232/r, respectively. There is no suppression of the interference terms, which depend on the CP violation phase. Furthermore, in the νe 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. Theνμ-like events correspond to interactions νμ+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 ντ which produce τ with subsequent decay into μ. The number of νμ-like events in the ij-bin in the Eν-cosθz plane equals
(167)Nij,μNH=2πNAρT∫Δicosθz‍dcosθz×∫ΔjEν‍dEνVeff(Eν)Dμ(Eν,θz),
where T is the exposure time, NA is the Avogadro number, ρ is the density of ice, Veff(Eν,θz) is the effective volume of the detector, and the number density of events per unit time per target nucleon is given by
(168)Dμ(Eν,θz)=[σCC(Φμ0Pμμ+Φe0Peμ)+σ-CC(Φ-μ0P-μμ+Φ-e0P-eμ)].
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 Eν=2 GeV to 20 Mt at Eν=20 GeV. The pattern of the event number distribution follows the oscillatory picture due to the νμ-νμ mode of oscillations with a certain distortion in the resonance region. The maxima and minima are approximately along the lines of equal oscillation phases Eν~ϕ32Δm322|cosθz|R⊕ (where R⊕ is the Earth radius), with distortion in the resonance region Eν=(4–10) GeV. In the high density bins, the number of events reaches 200 and the total number of events is about 105.Left: the binned distribution of the number ofμ 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].
(a)
(b)The expression for the density of events (168) can be written as
(169)DμNH=σCC(Eν)Φμ0[(Pμμ+1rPeμ)+κμ(P-μμ+1r-P-eμ)],
where
(170)κμ≡σ-CCΦ-μ0σCCΦμ0.
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 (Eν-cosθz) plane as
(171)Aμ,ijN-I≡Nμ,ijIH-Nμ,ijNHNμ,ijNH.
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: Sij=|Aij|.The strongest effect of hierarchy change is in the strips along the constant phase lines in the energy intervalEν=(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 ντ→τ→μ events can be considered as background events and treated within ~5% systematic errors.


#### 4.4.3. Measurements

According to Figure14, the hierarchy asymmetry of the νμ events has opposite signs in different parts of the oscillogram. Thus, the integration over Eν and cosθz 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 σE and angle σθ 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 Eνr and angle θzr can be obtained by smearing of the Eν-cosθz oscillograms with the reconstruction functions of the width σE and angle σθ.Small uncertaintiesσE and σθ require rather precise measurements of the energy Eμ and direction θμ of the muon, as well as energy of the accompanying hadron cascade Eh. Then the neutrino energy equals Eνr=Eμ+Eh. The reconstruction of the neutrino direction is more involved. In the first approximation, one can use θν≈θμ with a spread which decreases with energy: σθ~Amp/Eν(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 Figure14, 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 105νμ 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 (within1σ interval) variations of Δm322 and θ23. The effect of an unknown CP phase is small.High statistics would allow resolving the degeneracy problem by selecting specific regions in theEν-cosθz for the analysis, where effects of Δm322 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 Δm322 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.


## 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 energiesE>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:
(141)V(x)=V(L-x).
Here L=2R⊕|cosθz| is the length of the trajectory inside the Earth, R⊕=6371 km is the Earth radius, and θz is the zenith angle related to the nadir angle as Θν=π-θz. For 0≤Θν≤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 dEarth=∫‍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 theEν-cosθz plane. In Figure 8, we show the oscillograms for the oscillation probabilities Peμ and Pμμ, as well as the corresponding probabilities for antineutrinos [43, 46, 52–55].Neutrino oscillograms of the Earth. Shown are the lines of equal flavor conversion probability in theEν-cosΘν plane. Upper panels: νe→νμ (left) and ν-e→ν-μ (right); bottom panels: νμ→νμ (left) and ν-μ→ν-μ (right). Normal hierarchy is assumed.
(a)
(b)
(c)
(d)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)
sin22θm=1, the amplitude condition, which is nothing but the MSW resonance condition, and(ii)
ϕ=π/2+πk, the phase condition.AtE>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 atEν~(5–7)GeV. The position of the maximum is given by the MSW resonance condition:
(142)Eν=ER(Θν)=Δm312cos2θ132V-1(Θν),whereV-1(Θν) is the average value of the potential along the trajectory characterized by Θν. The phase condition becomes 2ϕ(Eν,Θν)=2ω(V-,Eν)L(Θν)=π and the intersection of the resonance and the phase condition lines gives the absolute maximum of PA. Combining these conditions gives the coordinates of the peak: cosΘν=0.77 and ER=6 GeV.(2)
Three parametric ridges in the domain of core-crossing trajectories|cosθz|>0.87 and Eν>3 GeV. The parametric ridges differ by the oscillation phase acquired in the core, ϕ2.
(i)
Ridge A lies between the core resonance (at Θν~0∘) and the mantle resonance regions, Eν≈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 Eν≥5 GeV. For the lowest energies in the ridge and Θν~0, the half phase in the core equals ϕ2~(1.2-1.3)π.(iii)
Ridge C is located at Eν>11 GeV in the matter dominated region, where the mixing and, consequently, oscillation depth are suppressed.(3)
The MSW resonance peak in core located atEν~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 Figure9, we show graphic representations of oscillations which correspond to salient features of the oscillograms.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 vectorH for the mantle (red) and core (green), respectively.
(a)
(b)
(c)For energiesEν<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-Pee=|Ae2~|2≡PS. The probabilities of the modes including νe are expressed in terms of a unique probability PS.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,lmR=lν/sin2θ12~lν. The resonance energy is shifted to smaller values both due to Δm212≪Δm312 and because of the factor cos2θ12≈0.4: E12R=(Δm212/2V-)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 lm 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 Θν=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 (Θν>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 (Θν≈82∘) corresponds to ϕ≈π/2, the middle (Θν=60∘) to ϕ≈3π/2, and the inner (Θν≈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
(143)E12R≈0.12GeV.
The ratio of the 1-2 and 1–3 resonance energies equals E12R/E13R≈1/50. The estimate (143) is valid for the two outer peaks. For the peak at Θν=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 Eν direction as compared to the oscillations governed by the 1–3 mixing and splitting.The resonance energy in the core isE12R≈0.04 GeV. Therefore for Eν>(0.10–0.15)GeV the 1-2 mixing in the core is substantially suppressed by matter. The peak at Eν≃0.2 GeV and Θν≃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 ϕmantle≈π/2 and ϕcore≈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 ofΔm212=0, the neutrino oscillograms for the inverted hierarchy coincide with the antineutrino oscillograms for the normal hierarchy and vice versa, provided that Δm312 is taken to be the same in both cases [56]: PAIH=P-ANH,ϕXIH=-ϕ-XNH. Therefore PαβIH=P-αβNH,P-αβIH=PαβNH. The inclusion of the 1-2 mixing and mass splitting breaks this symmetry.


## 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 theEν-cosθz plane. In Figure 8, we show the oscillograms for the oscillation probabilities Peμ and Pμμ, as well as the corresponding probabilities for antineutrinos [43, 46, 52–55].Neutrino oscillograms of the Earth. Shown are the lines of equal flavor conversion probability in theEν-cosΘν plane. Upper panels: νe→νμ (left) and ν-e→ν-μ (right); bottom panels: νμ→νμ (left) and ν-μ→ν-μ (right). Normal hierarchy is assumed.
(a)
(b)
(c)
(d)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)
sin22θm=1, the amplitude condition, which is nothing but the MSW resonance condition, and(ii)
ϕ=π/2+πk, the phase condition.AtE>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 atEν~(5–7)GeV. The position of the maximum is given by the MSW resonance condition:
(142)Eν=ER(Θν)=Δm312cos2θ132V-1(Θν),whereV-1(Θν) is the average value of the potential along the trajectory characterized by Θν. The phase condition becomes 2ϕ(Eν,Θν)=2ω(V-,Eν)L(Θν)=π and the intersection of the resonance and the phase condition lines gives the absolute maximum of PA. Combining these conditions gives the coordinates of the peak: cosΘν=0.77 and ER=6 GeV.(2)
Three parametric ridges in the domain of core-crossing trajectories|cosθz|>0.87 and Eν>3 GeV. The parametric ridges differ by the oscillation phase acquired in the core, ϕ2.
(i)
Ridge A lies between the core resonance (at Θν~0∘) and the mantle resonance regions, Eν≈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 Eν≥5 GeV. For the lowest energies in the ridge and Θν~0, the half phase in the core equals ϕ2~(1.2-1.3)π.(iii)
Ridge C is located at Eν>11 GeV in the matter dominated region, where the mixing and, consequently, oscillation depth are suppressed.(3)
The MSW resonance peak in core located atEν~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 Figure9, we show graphic representations of oscillations which correspond to salient features of the oscillograms.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 vectorH for the mantle (red) and core (green), respectively.
(a)
(b)
(c)For energiesEν<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-Pee=|Ae2~|2≡PS. The probabilities of the modes including νe are expressed in terms of a unique probability PS.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,lmR=lν/sin2θ12~lν. The resonance energy is shifted to smaller values both due to Δm212≪Δm312 and because of the factor cos2θ12≈0.4: E12R=(Δm212/2V-)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 lm 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 Θν=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 (Θν>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 (Θν≈82∘) corresponds to ϕ≈π/2, the middle (Θν=60∘) to ϕ≈3π/2, and the inner (Θν≈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
(143)E12R≈0.12GeV.
The ratio of the 1-2 and 1–3 resonance energies equals E12R/E13R≈1/50. The estimate (143) is valid for the two outer peaks. For the peak at Θν=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 Eν direction as compared to the oscillations governed by the 1–3 mixing and splitting.The resonance energy in the core isE12R≈0.04 GeV. Therefore for Eν>(0.10–0.15)GeV the 1-2 mixing in the core is substantially suppressed by matter. The peak at Eν≃0.2 GeV and Θν≃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 ϕmantle≈π/2 and ϕcore≈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 ofΔm212=0, the neutrino oscillograms for the inverted hierarchy coincide with the antineutrino oscillograms for the normal hierarchy and vice versa, provided that Δm312 is taken to be the same in both cases [56]: PAIH=P-ANH,ϕXIH=-ϕ-XNH. Therefore PαβIH=P-αβNH,P-αβIH=PαβNH. 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 probabilityPee 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 νe is not affected by this transformation. For oscillations in vacuum, or in matter with symmetric density profiles, the other two survival probabilities, Pμμ and Pττ, 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 phaseϕ≡arg(Ae2~*Ae3~) and omitting small terms proportional to |A2~3~|2=𝒪(s136) we obtain
(144)Pμeδ=sin2θ23cos(ϕ-δ)|Ae2~Ae3~|,(145)Pμμδ=-sin2θ23cosδcosϕ|Ae2~Ae3~|-D23,(146)Pμτδ=-sin2θ23sinδsinϕ|Ae2~Ae3~|+D23,
where D23≡(1/2)sin4θ23cosδRe[A2~3~*(A3~3~-A2~2~)] is proportional to the small deviation of the 2-3 mixing from maximal one. Notice that D23 enters Pμμδ and Pμτδ with opposite signs while Pμeδ does not depend on D23 at all. D23 is CP even. The sum of these interference terms is zero.For the other channels,Pαβδ=Pβα-δ. 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, |Ae2~Ae3~| and D23.


### 4.2.2. Magic Lines and CP Domains

To better assess the effect ofδ, one can consider the difference of the oscillation probabilities for two different values of the CP phase ΔPαβCP(δ)≡Pαβ(δ)-Pαβ(δ0). In practice, this quantifies how well the phase δ fits with some assumed true value δ0. The structure of the oscillograms for ΔPαβCP(δ) can be understood in terms of the grids of magic lines and interference phase lines along which ΔPαβCP(δ)≈0.For theνμ→νe oscillation probability, the equality
(147)ΔPμeCP(δ)≡Pμe(δ)-Pμe(δ0)=Pμeδ(δ)-Pμeδ(δ0)
is exact and the condition ΔPμeCP=0 is equivalent to
(148)|Ae2~Ae3~|cos(ϕ-δ)=|Ae2~Ae3~|cos(ϕ-δ0).
This equality is satisfied if at least one of the following three conditions is fulfilled:
(149)Ae2~(Eν,Θν)=0,Ae3~(Eν,Θν)=0,ϕ(Eν,Θν)-δ0=-[ϕ(Eν,Θν)-δ]+2πl.
The last condition implies
(150)ϕ(Eν,Θν)=(δ+δ0)2+πl.
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 Ae2~ and Ae3~ are complex quantities, these conditions can be satisfied in isolated points of the (Θν,Eν) plane only. In contrast to this, in the factorization approximation Ae2~=AS and Ae3~=AA both the conditions are fulfilled along certain lines in the oscillograms. This occurs because the amplitudes AS and AA take a 2-flavor form. On the basis of neutrino states where the corresponding 2×2 Hamiltonians are traceless, both AA and AS are pure imaginary because of the symmetry of the Earth’s density profile [47].Let us consider the equalitiesAS=0 and AA=0 using the constant density approximation.(1)
The conditionAS(Eν,Θν)=0 is satisfied when sinϕS(Eν,Θν)=0, which leads to
(151)L(Θν)≈2πnω21m,n=1,2,….
At energiesEν≳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 ω21m≈V and the condition (151) becomes
(152)L(Θν)≃2πnV.
This expression is energy independent and determines the baselines for which the “solar” contribution to the probability vanishes [61]. In the plane (Θν,Eν) it represents nearly vertical lines at fixed Θν. There are three solar magic lines which correspond to n=1 (in the mantle domain) Θν≈54∘ and n=2,3 (in the core domain) [61] Θν≈30∘ and 12∘. The existence of a baseline (L≈7600 km) for which the probability of νe↔νμ oscillations in the Earth is approximately independent of the “solar” parameters (Δm212, 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 conditionAA(Eν,Θν)=0 [61]. Along these lines, the “atmospheric” contribution to the amplitudes of νμ↔νe and ντ↔νe transitions vanishes and the probabilities of oscillations involving νe or ν-e do not depend on CP phase. In the constant density approximation, the condition AA=0 is satisfied when sinϕA=0 (ϕA=πk, k=1,2,…) or explicitly
(153)L(Θν)≈2πkω31m,k=1,2,….
For energies which are not too close to the 1–3 MSW resonance, it reduces to(154)Eν≃Δm312L(Θν)|4πk±2VL(Θν)|,
which corresponds to the bent curves in the(Θν,Eν) plane. For very large energies, where Δm312/2E≪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 (Θν,Eν) plane. In the constant density approximation ϕ≈-ϕ31m. Consequently in the energy range between the two resonances we have
(155)ϕ31m≈Δm312L4Eν=ϕA0,
that is, in the first approximationϕ does not depend on the matter density. From (150) we then obtain
(156)Eν=Δm312L(Θν)4πl-2(δ+δ0).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 Pμe,Peμ,Pτe, and Peτ 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 probabilitiesΔPμeCP(δ)=Pμe(δ)-Pμe(δ0) with δ0=0∘. Shown are the solar (black), atmospheric (white), and interference phase condition (cyan) curves from [60].From Figure10, we can see that the magic lines described above do not coincide exactly with the lines of ΔPμeCP=0 which bound the CP-domains. Furthermore, interconnections of the latter occur. This is due to the breakdown of the factorization approximation.


## 4.2.1. Interference and CP violation

The survival probabilityPee 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 νe is not affected by this transformation. For oscillations in vacuum, or in matter with symmetric density profiles, the other two survival probabilities, Pμμ and Pττ, 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 phaseϕ≡arg(Ae2~*Ae3~) and omitting small terms proportional to |A2~3~|2=𝒪(s136) we obtain
(144)Pμeδ=sin2θ23cos(ϕ-δ)|Ae2~Ae3~|,(145)Pμμδ=-sin2θ23cosδcosϕ|Ae2~Ae3~|-D23,(146)Pμτδ=-sin2θ23sinδsinϕ|Ae2~Ae3~|+D23,
where D23≡(1/2)sin4θ23cosδRe[A2~3~*(A3~3~-A2~2~)] is proportional to the small deviation of the 2-3 mixing from maximal one. Notice that D23 enters Pμμδ and Pμτδ with opposite signs while Pμeδ does not depend on D23 at all. D23 is CP even. The sum of these interference terms is zero.For the other channels,Pαβδ=Pβα-δ. 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, |Ae2~Ae3~| and D23.


## 4.2.2. Magic Lines and CP Domains

To better assess the effect ofδ, one can consider the difference of the oscillation probabilities for two different values of the CP phase ΔPαβCP(δ)≡Pαβ(δ)-Pαβ(δ0). In practice, this quantifies how well the phase δ fits with some assumed true value δ0. The structure of the oscillograms for ΔPαβCP(δ) can be understood in terms of the grids of magic lines and interference phase lines along which ΔPαβCP(δ)≈0.For theνμ→νe oscillation probability, the equality
(147)ΔPμeCP(δ)≡Pμe(δ)-Pμe(δ0)=Pμeδ(δ)-Pμeδ(δ0)
is exact and the condition ΔPμeCP=0 is equivalent to
(148)|Ae2~Ae3~|cos(ϕ-δ)=|Ae2~Ae3~|cos(ϕ-δ0).
This equality is satisfied if at least one of the following three conditions is fulfilled:
(149)Ae2~(Eν,Θν)=0,Ae3~(Eν,Θν)=0,ϕ(Eν,Θν)-δ0=-[ϕ(Eν,Θν)-δ]+2πl.
The last condition implies
(150)ϕ(Eν,Θν)=(δ+δ0)2+πl.
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 Ae2~ and Ae3~ are complex quantities, these conditions can be satisfied in isolated points of the (Θν,Eν) plane only. In contrast to this, in the factorization approximation Ae2~=AS and Ae3~=AA both the conditions are fulfilled along certain lines in the oscillograms. This occurs because the amplitudes AS and AA take a 2-flavor form. On the basis of neutrino states where the corresponding 2×2 Hamiltonians are traceless, both AA and AS are pure imaginary because of the symmetry of the Earth’s density profile [47].Let us consider the equalitiesAS=0 and AA=0 using the constant density approximation.(1)
The conditionAS(Eν,Θν)=0 is satisfied when sinϕS(Eν,Θν)=0, which leads to
(151)L(Θν)≈2πnω21m,n=1,2,….
At energiesEν≳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 ω21m≈V and the condition (151) becomes
(152)L(Θν)≃2πnV.
This expression is energy independent and determines the baselines for which the “solar” contribution to the probability vanishes [61]. In the plane (Θν,Eν) it represents nearly vertical lines at fixed Θν. There are three solar magic lines which correspond to n=1 (in the mantle domain) Θν≈54∘ and n=2,3 (in the core domain) [61] Θν≈30∘ and 12∘. The existence of a baseline (L≈7600 km) for which the probability of νe↔νμ oscillations in the Earth is approximately independent of the “solar” parameters (Δm212, 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 conditionAA(Eν,Θν)=0 [61]. Along these lines, the “atmospheric” contribution to the amplitudes of νμ↔νe and ντ↔νe transitions vanishes and the probabilities of oscillations involving νe or ν-e do not depend on CP phase. In the constant density approximation, the condition AA=0 is satisfied when sinϕA=0 (ϕA=πk, k=1,2,…) or explicitly
(153)L(Θν)≈2πkω31m,k=1,2,….
For energies which are not too close to the 1–3 MSW resonance, it reduces to(154)Eν≃Δm312L(Θν)|4πk±2VL(Θν)|,
which corresponds to the bent curves in the(Θν,Eν) plane. For very large energies, where Δm312/2E≪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 (Θν,Eν) plane. In the constant density approximation ϕ≈-ϕ31m. Consequently in the energy range between the two resonances we have
(155)ϕ31m≈Δm312L4Eν=ϕA0,
that is, in the first approximationϕ does not depend on the matter density. From (150) we then obtain
(156)Eν=Δm312L(Θν)4πl-2(δ+δ0).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 Pμe,Peμ,Pτe, and Peτ 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 probabilitiesΔPμeCP(δ)=Pμe(δ)-Pμe(δ0) with δ0=0∘. Shown are the solar (black), atmospheric (white), and interference phase condition (cyan) curves from [60].From Figure10, we can see that the magic lines described above do not coincide exactly with the lines of ΔPμeCP=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 Figure11 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 Pαβ 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 Pμe 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 forνe→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
(
θ
13
,
δ
)
degeneracy: for any combination of (θ13,δ), there exists a different combination (θ^13,δ^) that also predicts the same oscillation probabilities.
(3)
Octant
(
θ
23
)
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.Two different illustrations of parameter space degeneracies. Left panel: biprobability plot forL=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 P-μe (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.
(a)
(b)For known mass hierarchy (e.g., normal one) a given value ofθ13 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 θ13prime, which crosses this point and therefore θ13prime 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 probabilityPμe 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 Pμe 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
(157)sin22θ~=sin22θ1+3cos22θ≃14sin22θ,
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
(158)sin22θ~≃sin22θ[1+4VEΔm2cos2θ].
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 probabilityPμe 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 P-μe 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 ofVE 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

Figure13 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 νμ-νe oscillation probability (the “golden channel”) can be expanded in the small quantity Δm212L/2E which gives [69]
(159)Peμ≈s232P2f+c13sin2θ13sin2θ12sin2θ23Δm2122EV×sin(VL2)sin(Δm312L4E)cos(δ-Δm312L4E),
where P2f is the two-flavor oscillation probability discussed earlier. In (159) we have neglected terms of the second (and higher) order in Δm212L/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 Δm312. 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.


## 4.3.1. CP Violation Effects and the Mass Hierarchy

Figure13 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 νμ-νe oscillation probability (the “golden channel”) can be expanded in the small quantity Δm212L/2E which gives [69]
(159)Peμ≈s232P2f+c13sin2θ13sin2θ12sin2θ23Δm2122EV×sin(VL2)sin(Δm312L4E)cos(δ-Δm312L4E),
where P2f is the two-flavor oscillation probability discussed earlier. In (159) we have neglected terms of the second (and higher) order in Δm212L/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 Δm312. 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 ofνe, νμ and the corresponding antineutrinos, while the original ντ flux is negligible. We introduce Φe0 and Φμ0, the electron and muon neutrino fluxes, as well as Φ-e0 and Φ-μ0, the electron and muon antineutrino fluxes, at the detector in the absence of oscillations. The flavor ratios
(160)r≡Φμ0Φe0,r-≡Φ-μ0Φ-e0
increase with energy.There is a mild neutrino-antineutrino asymmetry: the neutrino fluxΦ-μ0/Φμ0≈0.8-0.9. All the fluxes (at E>1 GeV) decrease rapidly with energy Φα0∝E-k, k=k(E)=3–5, and an azimuthal dependence shows up at low energies.The flux of neutrinos of flavorνα at a detector, with oscillations taken into account, is given by
(161)Φα=Φe0Peα+Φμ0Pμα=Φe0[Peα+r(E,Θν)Pμα],α=e,μ,τ.
Similar expressions hold for the antineutrino fluxes. Inserting the analytic expressions for the probabilities (122)–(126), one finds
(162)ΦeΦe0=1+(rs232-1)Pe3~+(rc232-1)Pe2~+rPμeδ,ΦμΦμ0≈1-2s232c232[1-Re(A2~2~*A3~3~)]-s232r(rs232-1)Pe3~-c232r(rc232-1)Pe2~+Pμμδ+1rPeμδ,ΦτΦμ0≈2s232c232[1-Re(A2~2~*A3~3~)]-c232r(rs232-1)Pe3~-s232r(rc232-1)Pe2~+Pμτδ+1rPeτδ,
where Pe3~≡|Ae3~|2 and Pe2~≡|Ae2~|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(163)|A2~2~|2=1-|A2~e|2-|A2~3~|2≈1-|A2~e|2=1-Pe2~,
where the terms proportional to |A3~2~|2 have been neglected we can approximate
(164)Re(A2~2~*A3~3~)≈(1-Pe3~)(1-Pe2~)cosψ.
Here ψ≡argA33A22* is the relative phase between the two amplitudes. For the νe flux, we then obtain
(165)ΦeΦe0≈1+(rs232-1)Pe3~+(rc232-1)Pe2~+rsin2θ23Pe3~Pe2~cos(ϕ-δ).
The oscillated fluxes satisfy the sum rule
(166)Φe+Φμ+Φτ=Φe0+Φμ0,
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]: Pe3~ with (rs232-1) and Pe2~ with (rc232-1). The contribution of the “atmospheric mode” vanishes along the line r(E,Θν)=1/s232, whereas the contribution of the “solar mode” vanishes along r(E,Θν)=1/c232. For maximal mixing both contributions vanish along the same line, r(E,Θν)=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: s232 or c232<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 theνμ flux, the contributions of the 1-2 and 1–3 modes are suppressed by additional factors s232/r and c232/r, respectively. There is no suppression of the interference terms, which depend on the CP violation phase. Furthermore, in the νe 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. Theνμ-like events correspond to interactions νμ+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 ντ which produce τ with subsequent decay into μ. The number of νμ-like events in the ij-bin in the Eν-cosθz plane equals
(167)Nij,μNH=2πNAρT∫Δicosθz‍dcosθz×∫ΔjEν‍dEνVeff(Eν)Dμ(Eν,θz),
where T is the exposure time, NA is the Avogadro number, ρ is the density of ice, Veff(Eν,θz) is the effective volume of the detector, and the number density of events per unit time per target nucleon is given by
(168)Dμ(Eν,θz)=[σCC(Φμ0Pμμ+Φe0Peμ)+σ-CC(Φ-μ0P-μμ+Φ-e0P-eμ)].
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 Eν=2 GeV to 20 Mt at Eν=20 GeV. The pattern of the event number distribution follows the oscillatory picture due to the νμ-νμ mode of oscillations with a certain distortion in the resonance region. The maxima and minima are approximately along the lines of equal oscillation phases Eν~ϕ32Δm322|cosθz|R⊕ (where R⊕ is the Earth radius), with distortion in the resonance region Eν=(4–10) GeV. In the high density bins, the number of events reaches 200 and the total number of events is about 105.Left: the binned distribution of the number ofμ 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].
(a)
(b)The expression for the density of events (168) can be written as
(169)DμNH=σCC(Eν)Φμ0[(Pμμ+1rPeμ)+κμ(P-μμ+1r-P-eμ)],
where
(170)κμ≡σ-CCΦ-μ0σCCΦμ0.
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 (Eν-cosθz) plane as
(171)Aμ,ijN-I≡Nμ,ijIH-Nμ,ijNHNμ,ijNH.
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: Sij=|Aij|.The strongest effect of hierarchy change is in the strips along the constant phase lines in the energy intervalEν=(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 ντ→τ→μ events can be considered as background events and treated within ~5% systematic errors.


### 4.4.3. Measurements

According to Figure14, the hierarchy asymmetry of the νμ events has opposite signs in different parts of the oscillogram. Thus, the integration over Eν and cosθz 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 σE and angle σθ 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 Eνr and angle θzr can be obtained by smearing of the Eν-cosθz oscillograms with the reconstruction functions of the width σE and angle σθ.Small uncertaintiesσE and σθ require rather precise measurements of the energy Eμ and direction θμ of the muon, as well as energy of the accompanying hadron cascade Eh. Then the neutrino energy equals Eνr=Eμ+Eh. The reconstruction of the neutrino direction is more involved. In the first approximation, one can use θν≈θμ with a spread which decreases with energy: σθ~Amp/Eν(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 Figure14, 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 105νμ 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 (within1σ interval) variations of Δm322 and θ23. The effect of an unknown CP phase is small.High statistics would allow resolving the degeneracy problem by selecting specific regions in theEν-cosθz for the analysis, where effects of Δm322 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 Δm322 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.


## 4.4.1. Neutrino Fluxes

The original flux of atmospheric neutrinos contains incoherent components ofνe, νμ and the corresponding antineutrinos, while the original ντ flux is negligible. We introduce Φe0 and Φμ0, the electron and muon neutrino fluxes, as well as Φ-e0 and Φ-μ0, the electron and muon antineutrino fluxes, at the detector in the absence of oscillations. The flavor ratios
(160)r≡Φμ0Φe0,r-≡Φ-μ0Φ-e0
increase with energy.There is a mild neutrino-antineutrino asymmetry: the neutrino fluxΦ-μ0/Φμ0≈0.8-0.9. All the fluxes (at E>1 GeV) decrease rapidly with energy Φα0∝E-k, k=k(E)=3–5, and an azimuthal dependence shows up at low energies.The flux of neutrinos of flavorνα at a detector, with oscillations taken into account, is given by
(161)Φα=Φe0Peα+Φμ0Pμα=Φe0[Peα+r(E,Θν)Pμα],α=e,μ,τ.
Similar expressions hold for the antineutrino fluxes. Inserting the analytic expressions for the probabilities (122)–(126), one finds
(162)ΦeΦe0=1+(rs232-1)Pe3~+(rc232-1)Pe2~+rPμeδ,ΦμΦμ0≈1-2s232c232[1-Re(A2~2~*A3~3~)]-s232r(rs232-1)Pe3~-c232r(rc232-1)Pe2~+Pμμδ+1rPeμδ,ΦτΦμ0≈2s232c232[1-Re(A2~2~*A3~3~)]-c232r(rs232-1)Pe3~-s232r(rc232-1)Pe2~+Pμτδ+1rPeτδ,
where Pe3~≡|Ae3~|2 and Pe2~≡|Ae2~|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(163)|A2~2~|2=1-|A2~e|2-|A2~3~|2≈1-|A2~e|2=1-Pe2~,
where the terms proportional to |A3~2~|2 have been neglected we can approximate
(164)Re(A2~2~*A3~3~)≈(1-Pe3~)(1-Pe2~)cosψ.
Here ψ≡argA33A22* is the relative phase between the two amplitudes. For the νe flux, we then obtain
(165)ΦeΦe0≈1+(rs232-1)Pe3~+(rc232-1)Pe2~+rsin2θ23Pe3~Pe2~cos(ϕ-δ).
The oscillated fluxes satisfy the sum rule
(166)Φe+Φμ+Φτ=Φe0+Φμ0,
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]: Pe3~ with (rs232-1) and Pe2~ with (rc232-1). The contribution of the “atmospheric mode” vanishes along the line r(E,Θν)=1/s232, whereas the contribution of the “solar mode” vanishes along r(E,Θν)=1/c232. For maximal mixing both contributions vanish along the same line, r(E,Θν)=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: s232 or c232<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 theνμ flux, the contributions of the 1-2 and 1–3 modes are suppressed by additional factors s232/r and c232/r, respectively. There is no suppression of the interference terms, which depend on the CP violation phase. Furthermore, in the νe 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. Theνμ-like events correspond to interactions νμ+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 ντ which produce τ with subsequent decay into μ. The number of νμ-like events in the ij-bin in the Eν-cosθz plane equals
(167)Nij,μNH=2πNAρT∫Δicosθz‍dcosθz×∫ΔjEν‍dEνVeff(Eν)Dμ(Eν,θz),
where T is the exposure time, NA is the Avogadro number, ρ is the density of ice, Veff(Eν,θz) is the effective volume of the detector, and the number density of events per unit time per target nucleon is given by
(168)Dμ(Eν,θz)=[σCC(Φμ0Pμμ+Φe0Peμ)+σ-CC(Φ-μ0P-μμ+Φ-e0P-eμ)].
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 Eν=2 GeV to 20 Mt at Eν=20 GeV. The pattern of the event number distribution follows the oscillatory picture due to the νμ-νμ mode of oscillations with a certain distortion in the resonance region. The maxima and minima are approximately along the lines of equal oscillation phases Eν~ϕ32Δm322|cosθz|R⊕ (where R⊕ is the Earth radius), with distortion in the resonance region Eν=(4–10) GeV. In the high density bins, the number of events reaches 200 and the total number of events is about 105.Left: the binned distribution of the number ofμ 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].
(a)
(b)The expression for the density of events (168) can be written as
(169)DμNH=σCC(Eν)Φμ0[(Pμμ+1rPeμ)+κμ(P-μμ+1r-P-eμ)],
where
(170)κμ≡σ-CCΦ-μ0σCCΦμ0.
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 (Eν-cosθz) plane as
(171)Aμ,ijN-I≡Nμ,ijIH-Nμ,ijNHNμ,ijNH.
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: Sij=|Aij|.The strongest effect of hierarchy change is in the strips along the constant phase lines in the energy intervalEν=(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 ντ→τ→μ events can be considered as background events and treated within ~5% systematic errors.


## 4.4.3. Measurements

According to Figure14, the hierarchy asymmetry of the νμ events has opposite signs in different parts of the oscillogram. Thus, the integration over Eν and cosθz 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 σE and angle σθ 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 Eνr and angle θzr can be obtained by smearing of the Eν-cosθz oscillograms with the reconstruction functions of the width σE and angle σθ.Small uncertaintiesσE and σθ require rather precise measurements of the energy Eμ and direction θμ of the muon, as well as energy of the accompanying hadron cascade Eh. Then the neutrino energy equals Eνr=Eμ+Eh. The reconstruction of the neutrino direction is more involved. In the first approximation, one can use θν≈θμ with a spread which decreases with energy: σθ~Amp/Eν(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 Figure14, 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 105νμ 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 (within1σ interval) variations of Δm322 and θ23. The effect of an unknown CP phase is small.High statistics would allow resolving the degeneracy problem by selecting specific regions in theEν-cosθz for the analysis, where effects of Δm322 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 Δm322 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 mixingωij=Δij2/2E 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 Δm212 and Δm312 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,rΔ (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 aroundER. 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<Δmij2/2E, and high density, V≫Δmij2/2E. 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 Δm322 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.



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*Source: 972485-2013-03-19.xml*