Neutrino Propagation in Matter

Neutrino Propagation in Matter

Mattias Blennow Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany
and
KTH Royal Institute of Technology, AlbaNova University Center, Roslagstullsbacken 21, 106 91 Stockholm, Sweden
   Alexei Yu. Smirnov The Abdus Salam International Centre for Theoretical Physics, I-34100 Trieste, Italy
Abstract

We describe the effects of neutrino propagation in the matter of the Earth relevant for 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 multi-layer medium, (iv) oscillations in thin layers of matter. We present the results of semi-analytic 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 multi-megaton scale detectors having low energy thresholds is explored. The potential of future accelerator experiments to establish the hierarchy is outlined.

I Introduction

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

Neutrino propagation in matter is vast area of research which covers 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 05-Wolfenstein:1977ue (). He marked that matter suppresses oscillations of the solar neutrinos propagating in the Sun and supernova neutrinos inside a star. He considered a 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 05-Barger:1980tf () have observed that matter can also enhance oscillations at certain energies. The work of Wolfenstein was expanded upon in papers by Mikheev and Smirnov 05-Mikheev:1986gs (); 05-Mikheev:1986wj (); 05-Mikheev:1986if (), 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 were 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 05-Notzold:1987ik (), 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 review 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, i.e., in usual electrically neutral and non-relativistic matter. We update existing results on effects of neutrino propagation in view of the recent determination of the 1-3 mixing.

The review is organized as follows: In Sec. II 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. Sec. III is devoted to various effects relevant for neutrino propagating in the Earth. We consider the properties of the oscillation/conversion probabilities in different channels. In Sec. IV 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 Sec. V.

Ii Neutrino properties in matter

We will consider the system of 3 flavor neutrinos, , mixed in vacuum:

(1)

Here is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix 05-Pontecorvo:1957cp (); 05-Maki:1962mu (); 05-Pontecorvo:1967fh () and is the vector of mass eigenstates with masses (i = 1, 2, 3). We will use the standard parameterization of the PMNS matrix,

(2)

which is the most suitable for describing usual matter effects. In Eq. (2) are the matrices of rotations in the planes with angles and .

In vacuum the flavor evolution of these neutrinos is described by the the Schrödinger-like equation

(3)

where is the neutrino mass matrix in the flavor basis and is the neutrino energy. Eq. (3) is essentially a generalization of the equation for a single ultra relativistic particle. According to Eq. (3), the Hamiltonian in vacuum can be written as

(4)

where and we take the masses to be real 111The term is omitted in (4) since it does not produce phase difference..

ii.1 Refraction and matter potentials

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

At low energies, the Hamiltonian is the effective four fermion Hamiltonian due to exchange of the and bosons:

(5)

where and are the vector and axial vector coupling constants.

In the Standard Model the matrix of the potentials in the flavor basis, is diagonal: .

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 non-relativistic and unpolarized medium (as well as for an isotropic distribution of ultra relativistic electrons) only the component of the vector current gives a non-zero result, which is proportional to the number density of the corresponding particles. Furthermore, due to conservation of the vector current (CVC), the couplings and 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, the NC contributions from electrons and protons cancel in electrically neutral medium. As a result, the potential for neutrino flavor is

(6)

where and are the densities of electrons and neutrons, respectively.

Only the difference of potentials has a physical meaning. Contribution of the neutral current scattering to is the same for all active neutrinos. Since ( or a combination thereof) is due to the neutral current scattering, in a normal medium composed of protons neutrons (nuclei) and electrons, . Furthermore, the difference of the potentials for and is due to the charged current scattering of on electrons () 05-Wolfenstein:1977ue ():

(7)

The difference of potentials leads to the appearance of an additional phase difference in the neutrino system: . This determines the refraction length, the distance over which an additional “matter” phase equals ,:

(8)

Numerically,

(9)

where is the nucleon mass. The corresponding column density is given by the Fermi coupling constant only.

For antineutrinos the potential has an opposite sign. Being zero in the lowest order the difference of potentials in the system appears at a level of due to the radiative corrections 05-Botella:1986wy (). Thus in the flavor basis in the lowest order in EW interactions the effect of medium on neutrinos is described by with given in Eq. (7).

The potential has been computed for neutrinos in different type of media, such as polarized or heavily degenerate electrons, in 05-Esposito:1995db (); 05-Nunokawa:1997dp (); 05-Lobanov:2001ar ().

ii.2 Evolution equation, effective Hamiltonian, and mixing in matter

ii.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 05-Wolfenstein:1977ue (); 05-Wolfenstein:1978ui (); 05-Wolfenstein:1979ni (); 05-Mikheev:1986gs (); 05-Mikheev:1986wj (); 05-Mikheev:1986if ():

(10)

In Eq. (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)

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 05-Akhmedov:2012mk () and references therein).

Although the Hamiltonian describes evolution in time, with the connection , Eq. (12) can be rewritten as with , so it can be used as an evolution equation in space.

Due to the strong hierarchy of and the smallness of 1-3 mixing, the results can be qualitatively understood and in many cases quantitatively described by reducing -evolution to -evolution. The reason is that the third neutrino effectively decouples and its effect can be considered as a perturbation. Of course, there are genuine phenomena such as CP-violation, but even in this case the dynamics of evolution can be reduced effectively to the dynamics of evolution of systems. The evolution equation for two flavor states, , in matter is

(12)

where the Hamiltonian is written in symmetric form.

ii.3 Mixing and eigenstates in matter

The mixing in matter is defined with respect to - the eigenstates of the Hamiltonian in matter .

As usually, the eigenstates are obtained from the equation

(13)

where are the eigenvalues of . If the density, and therefore , are constant, correspond to the eigenstates of propagation. Since , the states differ from the mass states, . For low density, , the vacuum eigenstates are recovered: . If the density, and thus , changes during neutrino propagation, and should be considered as the eigenstates and eigenvalues of the instantaneous Hamiltonian: , and . For we have .

The mixing in matter is a generalization of the mixing in vacuum (1). Recall that the mixing matrix in vacuum connects the flavor neutrinos, , and the massive neutrinos, . The latter are the eigenstates of Hamiltonian in vacuum: . Therefore, the mixing matrix in matter is defined as the matrix which relates the flavor states with the eigenstates of the Hamiltonian in matter :

(14)

From Eq. (13) we find that

(15)

Furthermore, the Hamiltonian can be represented in the flavor basis as

(16)

Inserting this expression as well as the relation , which follows from Eq. (14), into Eq. (15) one obtains

(17)

or in matrix form . Thus, the mixing matrix can be found diagonalizing the full Hamiltonian. The columns of the mixing matrix, , are the eigenstates of the Hamiltonian which correspond to the eigenvalues . Indeed, it follows from Eq. (17) that .

Equation (14) can be inverted to , or in components . According to this, the elements of mixing matrix determine the flavor content of the mass eigenstates so that gives the probability to find in a given eigenstate . Correspondingly, the elements of the PMNS matrix determine the flavor composition of the mass eigenstates in vacuum.

ii.4 Mixing in the two neutrino case

In the -case, there is single mixing angle in matter and the relations between the eigenstates in matter and the flavor states reads

(18)

The angle is obtained by diagonalization of the Hamiltonian (12) (see previous section):

(19)

where is the resonance factor. In the limit , the factor and the vacuum mixing is recovered. The difference of eigenvalues equals

(20)

This difference is also called the level splitting. or oscillation frequency, which determines the oscillation length: (see Sect. III.2).

The matter potential and always enter the mixing angle and other dimensionless quantities in the combination

(21)

where 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)

ii.4.1 Resonance and level crossing

According to Eq. (19) the effective mixing parameter in matter, , depends on the electron density and neutrino energy through the ratio (21) of the oscillation and refraction lengths, . The dependence for two different values of the vacuum mixing angle, corresponding to angles from the full three flavor framework, is shown in Fig. 1.

Figure 1: Resonance in neutrino mixing. The dependence of on the product for vacuum mixing: ,  eV (red) and ,  eV (green). The left semi-plane corresponds to antineutrinos. The behavior of with vacuum value 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 – the inverted hierarchy.

The dependence of on has a resonant character 05-Mikheev:1986gs (). At

(23)

the mixing becomes maximal: (). The equality in (23) is called the resonance condition and it can be rewritten as . For small vacuum mixing the condition reads: . The physical meaning of the resonance is that the eigenfrequency, which characterizes a system of mixed neutrinos, , coincides with the eigenfrequency of the medium, . The resonance condition (23) determines the resonance density

(24)

The width of resonance on the half of height (in the density scale) is given by . Similarly, for fixed one can introduce the resonance energy and the width of resonance in the energy scale. The width can be rewritten as , where . When the vacuum mixing approaches maximal value, the resonance shifts to zero density: , the width of resonance increases converging to fixed value: .

In a medium with varying density, the layer in which the density changes in the interval is called the resonance layer. In this layer the angle varies in the interval from to .

For , the mixing angle is close to the vacuum angle: , while for the angle becomes and the mixing is strongly suppressed. In the resonance region, the level splitting is minimal 05-cabbibo (); 05-Bethe:1986ej (), therefore the oscillation length, as the function of density, is maximal.

ii.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, strong breaking of adiabaticity). The exact expressions for the eigenstates and eigenvalues 05-Bueno:2000fg (); 05-Freund:2001pn () 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 which relies on the strong hierarchy of the mass squared differences:

(25)

Without changing physics, the factor in the mixing matrix can be eliminated by permuting it with and redefining the state . Therefore, in what follows, we use . Here we will here describe the case of normal mass hierarchy: . Subtracting from the Hamiltonian the matrix proportional to the unit matrix , we obtain

(26)

ii.5.1 Propagation basis

The propagation basis, , which is most suitable for consideration of the neutrino oscillations in matter is defined through the relation

(27)

Since the potential matrix is invariant under 2-3 rotations the matrix of the potentials is unchanged and the Hamiltonian the propagation basis becomes

(28)

It does not depend on the 2-3 mixing or CP-violation phase, and so the dynamics of the flavor evolution does not depend on and . These parameters appear in the final amplitudes when projecting the flavor states onto propagation basis states and back (27) at the neutrino production and detection.

Explicitly, the Hamiltonian can be written

(29)

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

ii.5.2 Mixing angles in matter

The Hamiltonian in Eq. (29) can be diagonalized performing several consecutive rotations which correspond to developing the perturbation theory in . After a 1-3 rotation

(30)

over the angle determined by

(31)

the 1-3 element of (29) vanishes. The expression (31) differs from that for mixing in matter by a factor , which increases the potential and deviates from 1 by

After this rotation the Hamiltonian in the basis (30) becomes

(32)

where

(33)

and . For , these elements are reduced to the standard expressions. In the limit of zero density, , and consequently the 11 element of the Hamiltonian equals .

In the lowest approximation one can neglect the non-zero 2-3 element in Eq. (32). The state then decouples and the problem is reduced to a two neutrino problem for . The eigenvalue of this decoupled state equals

(34)

The diagonalization of the remaining 1-2 sub-matrix is given by rotation

(35)

where is determined by

(36)

Here and are defined in Eqs. (33) and (31), respectively. The eigenvalues equal

(37)

According to this diagonalization procedure in the lowest order in the mixing matrix in matter is given by

(38)

where mixing angles and are determined in Eqs. (36) and (31), respectively. The 2-3 angle and the CP-violation phase are not modified by matter in this approximation. The eigenvalues and are given in Eq. (37) and is determined by Eq. (34).

The 2-3 element of matrix (32) vanishes after additional 2-3 rotation by an angle :

(39)

which produces corrections of the next order in . With an additional 2-3 rotation the mixing matrix becomes

(40)

where

(41)

and the last 2-3 rotation is on the angle determined through . The expression on the RH of Eq. (40) is obtained by reducing the expression on the LH side to the standard form by permuting the correction matrix . According to Eq. (41), it is this matrix that leads to the modification of 2-3 mixing and CP phase in matter. From Eq. (41) one finds

i.e., the combination is invariant under inclusion of matter effects. Furthermore, and up to corrections of the order . The results described here allow to understand behavior of the mixing parameters in the region of the 1-3 resonance and above it (see Fig. 1).

Figure 2: The flavor contents of the eigenstates of the Hamiltonian in matter as functions of . The vertical width of the band is taken to be 1, then the vertical sizes of the colored parts give (red), (green), (blue). The right and left panels correspond to neutrinos and anti-neutrinos, respectively. We take the best fit values of 05-Fogli:2012ua () with . Variations of change the relative and content. The dashed red line shows a shift of border between and flavors for . The upper (lower) panel corresponds to normal (inverted) mass ordering.
Figure 3: The energy level scheme. We here show the dependence of the eigenvalues of the Hamiltonian in matter on . Note that we are plotting , which goes to for low . The left (right) panel corresponds to normal (inverted) mass ordering.

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

In the case of normal mass hierarchy, there are two resonances (level crossings). whose location is defined as the density (energy) at which the mixing in a given channel becomes maximal.

1. The H-resonance, in the channel, is associated to the 1-3 mixing and large mass splitting. According to Eq. (31) at

(42)

2. The L-resonance at low densities is associated to the small mass splitting and 1-2 mixing It appears in the channel, where and differ by small (at low densities) rotation given by an angle (see eq. (31)). According to Eq. (36) the position of the L-resonance, is given by , where is defined in Eq. (33). This leads to

(43)

For antineutrinos ( in Fig. 3), the oscillation parameters in matter can be obtained from the neutrino parameters taking and . 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 is no antineutrino resonances (level crossings), and with the increase of density (energy) the eigenvalues have the following asymptotic limits:

(44)

Iii Effects of neutrino propagation in different media

iii.1 The evolution matrix

The evolution matrix, , is defined as the matrix which gives the wave function of the neutrino system at an arbitrary moment once it is known in the initial moment :

(45)

Inserting this expression in the evolution equation (12), we find that satisfies the same evolution equation as :

(46)

The elements of this matrix are the amplitudes of transitions: . The transition probability equals . The unitarity of the evolution matrix, , leads to the following relations between the amplitudes (matrix elements)

(47)

The first and the second equations express the fact that the total probability of transition of to everything is one, and the same holds for . The third and fourth equations are satisfied if

(48)

With these relations the evolution matrix can be parametrized as

(49)

The Hamiltonian for a 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 and the diagonal elements do not change. Therefore according to (48) the T-invariance implies that or i.e., the off-diagonal elements of the matrix are pure imaginary.

iii.2 Neutrino oscillations in matter with constant density

In a medium with constant density and therefore constant potential the mixing is constant: . Consequently, the flavor composition of the eigenstates do not change and the eigenvalues 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

(50)

where . This system of equations splits and the integration is trivial, . The corresponding -matrix is diagonal:

(51)

where is the half-oscillation phase in matter and a matrix proportional to the unit matrix has been subtracted from the Hamiltonian.

The matrix in the flavor basis is therefore

(52)

Then, for the transition probability, we can immediately deduce

(53)

where with

(54)

being the oscillation length in matter. The dependence of on the neutrino energy is shown in Fig. 4. For small energies, , the length . It then increases with energy and for small reaches the maximum at , i.e., above the resonance energy. For the oscillation length converges to the refraction length .

Figure 4: Dependence of the oscillation length in matter in units of the refraction length on neutrino energy for two different mixing angles in vacuum.

A useful representation of the matrix for a layer with constant density follows from Eq. (52):

(55)

where is a vector containing the Pauli matrices and .

The dynamics of neutrino flavor evolution in uniform matter are the same as in vacuum, i.e., 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: , .

iii.3 Neutrino polarization vectors and graphic representation

It is illuminating to consider dynamics of transitions in different media using graphic representation 05-Smirnov:1986ij (); 05-Bouchez:1986kb (); 05-Ermilova:1986ab (). Consider the two flavor neutrino state, . The corresponding Hamiltonian can be written as

(56)

where , is the Hamiltonian vector and is the oscillation length. The evolution equation then becomes

(57)

Let us define the polarization vector

(58)

In terms of the wave functions, the components of equal

(59)

The -component can be rewritten as , therefore and from unitarity . Hence, determines the probabilities to find the neutrino of 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 can be obtained by differentiating Eq. (58) with respect to time and inserting and from evolution equation (57). As a result, one finds that

(60)

If is identified with the strength of a magnetic field, the equation of motion (60) coincides with the equation of motion for the spin of electron in the magnetic field. According to this equation precesses around .

With an increase of the oscillation phase (see Fig. 5) the vector moves on the surface of the cone having axis . The cone angle , the angle between and depends both on the mixing angle and on the initial state, and in general, changes in process of evolution, e.g., if the neutrino evolves through several layers of different density. If the initial state is , the angle equals in the initial moment.

The components of the polarization vector are nothing but the elements of the density matrix . The evolution equation for can be obtained from (60)

(61)

The diagonal elements of the density matrix give the probabilities to find the neutrino in the corresponding flavor state.

Figure 5: Graphic representation of neutrino oscillations. Neutrino polarization vector precesses around the Hamiltonian vector (or the vector of eigenstates of the Hamiltonian). The angle between and is given by the cone angle , and the direction of axis of the cone is determined by the mixing angle in matter .

iii.4 Resonance enhancement of oscillations

Suppose a source produces flux of neutrinos in the flavor state with continuous energy spectrum. This flux then traverse a layer of length with constant density . At the end of this layer a detector measures the component of the flux, so that oscillation effect is given by the transition probability . In Fig. 6 we show dependence of this probability on energy for thin and thick layers. The oscillatory curves are inscribed in to the resonance envelope . The period of the oscillatory curve decreases with the length . At the resonance energy,

(62)

oscillations proceed with maximal depths. Oscillations are enhanced up to in the resonance range where (see Sec. II.4.1). This effect was called the resonance enhancement of oscillations.

Figure 6: Resonance enhancement of neutrino oscillations in matter with constant density. Shown is the dependence of the transition probability on energy for for three different sizes of layers: , and . The shaded area shows the resonance envelope: .

iii.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 Eq. (51) is straightforward. In the basis of the eigenstates of the Hamiltonian the evolution matrix equals

(63)

and for the elements of the matrix in the flavor basis we obtain . Removing and using the unitarity of the mixing matrix in matter we have

(64)

In particular, for the amplitudes in matter involving only and , we obtain

(65)
(66)
(67)

[[do we use this? add more?]]

iii.6 Propagation in a medium with varying density and the MSW effect

iii.6.1 Equation for the instantaneous eigenvalues and the adiabaticity condition

In non-uniform media, the density changes along neutrino trajectory: . Correspondingly, the Hamiltonian of system depends on time, , and therefore the mixing angle changes during neutrino propagation: . Furthermore, the eigenstates of the instantaneous Hamiltonian, and