Collective three-flavor oscillations of supernova neutrinos

# Collective three-flavor oscillations of supernova neutrinos

Basudeb Dasgupta and Amol Dighe Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
###### Abstract

Neutrinos and antineutrinos emitted from a core collapse supernova interact among themselves, giving rise to collective flavor conversion effects that are significant near the neutrinosphere. We develop a formalism to analyze these collective effects in the complete three-flavor framework. It naturally generalizes the spin-precession analogy to three flavors and is capable of analytically describing phenomena like vacuum/MSW oscillations, synchronized oscillations, bipolar oscillations and spectral split. Using the formalism, we demonstrate that the flavor conversions may be “factorized” into two-flavor oscillations with hierarchical frequencies. We explicitly show how the three-flavor solution may be constructed by combining two-flavor solutions. For a typical supernova density profile, we identify an approximate separation of regions where distinctly different flavor conversion mechanisms operate, and demonstrate the interplay between collective and MSW effects. We pictorialize our results in terms of the “ triangle” diagram, which is a tool that can be used to visualize three-neutrino flavor conversions in general, and offers insights into the analysis of the collective effects in particular.

###### pacs:
14.60.Pq, 97.60.Bw
preprint: TIFR/TH/07-36

## I Introduction

Neutrinos emitted from a core collapse supernova carry information about the primary fluxes, neutrino masses and mixing, and SN dynamics raffelt-0701677 (); amol-nufact (). Neutrinos, produced in the region of the neutrinosphere, freestream outwards and pass through the core, mantle and envelope of the star. The drastically different environments in these regions, consisting of varying densities of ordinary matter, neutrinos and antineutrinos, affect flavor conversions among neutrinos. Multiple shock fronts and turbulence generated during the explosion may also affect the neutrino flavor conversions.

The traditional picture of flavor conversions in a SN is based on the assumption that the effect of neutrino-neutrino interactions is small. In this picture, neutrinos that are produced approximately as mass eigenstates at very high ambient matter density in the core propagate outwards from the neutrinosphere. Flavor conversion proceeds most efficiently at the electron density corresponding to the MSW resonance wolfenstein (); mikheyev-smirnov (). The outcoming incoherent mixture of vacuum mass eigenstates is observed at a detector to be a combination of primary fluxes of the three neutrino flavors. This scenario of resonant neutrino conversions in a SN fuller-mayle-wilson-schramm-ApJ322 () has been studied extensively to probe neutrino mixings and SN dynamics. The work has focussed on the determination of mass hierarchy and signatures of a non-zero dighe-smirnov-9907423 (); lunardini-smirnov-0302033 (), earth matter effects on the neutrino fluxes when they pass through matter dighe-keil-raffelt-jcap0306005 (); dighe-keil-raffelt-jcap0306006 (); dighe-kachelriess-raffelt-tomas-jcap0401 (), shock wave effects on observable neutrino spectra and their model independent signatures schirato-fuller-0205390 (); takahashi-sato-dalhed-wilson-0212195 (); fogli-lisi-mirizzi-montanino-0304056 (); tomas-kachelreiss-raffelt-dighe-janka-scheck-0407132 (); fogli-lisi-mirizzi-montanino-0412046 (); huber (). Recently, possible interference effects for multiple resonances dasgupta-dighe-0510219 (), the role of turbulence in washing out shock wave effects fogli-lisi-mirizzi-montanino-0603033 (); choubey-harries-ross-0605255 (); friedland-gruzinov-0607244 (), and time variation of the signal kneller-mclaughlin-brockman-0705.3853 () have also been explored.

However, neutrino and antineutrino densities near the neutrinosphere are extremely high ( per cm), which make the neutrino-neutrino interactions significant. Such a dense gas of neutrinos and antineutrinos is coupled to itself, making its evolution nonlinear. The flavor off-diagonal terms can be sizeable, and significant flavor conversion is possible pantaleone-PRD46 (); pantaleone-PLB287 (). A formalism to study flavor evolution of such dense relativistic neutrino gases was developed in thompson-mckellar-PLB259 (); raffelt-sigl-NPB406 (); thompson-mckellar-PRD49 (), where a set of quantum kinetic equations for their evolution were written down. These equations have been studied in detail, though mostly in the two-flavor approximation, and the nature of flavor evolution has been identified samuel-PRD48 (); kostecky-samuel-9506262 (); pantaleone-PRD58 (); samuel-9604341 (). It was eventually understood that a dense gas of neutrinos displays collective flavor conversion, i.e. neutrinos of all energies oscillate together, through synchronized oscillations pastor-raffelt-semikoz-0109035 () and/or bipolar oscillations hannestad-raffelt-sigl-wong-0608095 (); duan-fuller-carlson-qian-0703776 (). Another remarkable effect of these interactions is a partial or complete swapping of the energy spectra of two neutrino flavors, called step-wise spectral swapping or simply spectral splits, as the neutrinos transit from a region where collective effects dominate to a region where neutrino density is low raffelt-smirnov-0705.1830 (); raffelt-smirnov-0709.4641 ().

The nonlinear effects in the context of SNe were considered in pantaleone-9405008 (); qian-fuller-9406073 (); sigl-9410094 (); pastor-raffelt-0207281 (); balantekin-yuksel-0411159 (). Recent two-flavor simulations showed that the collective effects affect neutrino flavor conversions substantially duan-fuller-carlson-qian-0606616 (); duan-fuller-carlson-qian-0608050 (). Different collective flavor transformations seem to play a part in different regions of the star. Many features of the results of these simulations can be understood from the “single-angle” approximation, i.e. ignoring the dependence of the initial launching angle of neutrinos on the evolutions of neutrino trajectories fuller-qian-0505240 (); duan-fuller-qian-0511275 (); hannestad-raffelt-sigl-wong-0608095 (); duan-fuller-carlson-qian-0703776 (); raffelt-smirnov-0705.1830 (); duan-fuller-qian-0706.4293 (); duan-fuller-carlson-zhong-0707.0290 (); estebanpretel-pastor-tomas-raffelt-sigl-0706.2498 (); fogli-lisi-marrone-mirizzi-0707.1998 (); raffelt-smirnov-0709.4641 (). Angular dependence of flavor evolution can give rise to additional angle dependent features observed in two-flavor simulations duan-fuller-carlson-qian-0606616 (); duan-fuller-carlson-qian-0608050 (), or to decoherence effects pantaleone-PRD58 (); raffelt-sigl-0701182 (). For a realistic asymmetry between neutrino and antineutrino fluxes, such angle dependent effects are likely to be small estebanpretel-pastor-tomas-raffelt-sigl-0706.2498 (); fogli-lisi-marrone-mirizzi-0707.1998 (). Recently collective effects have also been numerically investigated in the context of the neutronization-burst phase of O-Ne-Mg supernovae duan-fuller-carlson-qian-0710.1271 (). The impact of these nonlinear interactions has also been studied in the context of cosmological neutrino flavor equilibration in the early Universe where the synchronized oscillations play a significant part kostelecky-pantaleone-samuel-PLB315 (); kostelecky-pantaleone-samuel-PLB318 (); kostelecky-pantaleone-samuel-PRD49 (); kostelecky-samuel-9507427 (); kostelecky-samuel-9610399 (); dolgov-hansen-pastor-petkov-raffelt-semikoz-0201287 (); wong-0203180 (); wong-talk (); abazajian-beacom-bell-0203442 ().

Most of the analytical results in this area are in the two-flavor approximation, where the equations describing flavor transformations are similar to the equations of motion of a gyroscope. As a result, two-flavor results are now fairly well understood analytically, with the exception of possibly two issues, viz. decoherence (or lack of it) for asymmetric systems estebanpretel-pastor-tomas-raffelt-sigl-0706.2498 (), and the existence of the antineutrino spectral split fogli-lisi-marrone-mirizzi-0707.1998 (). In this work we focus on the effects of the mixing of the third flavor. We present an analytical framework to study three-flavor effects and demonstrate an approximate factorization of the full three-flavor problem into simpler two-flavor problems, when and . We also develop the “ triangle” diagram, a tool that can be used to visualize the three neutrino flavor conversions in general, and allows one to gauge the extent of additional effects of the third flavor. We numerically study collective flavor transformations for a typical SN density profile, identify regions where different flavor conversion mechanisms operate, and explain the features of the spectra using our formalism.

The outline of this paper is as follows. In Sec. II, we review the equations of motion of a dense gas of neutrinos in steady state. We specialize these equations to spherical geometry in the single-angle half-isotropic approximation, and write the three-flavor analogue of the gyroscope equations, by introducing the eight-dimensional Bloch vectors. We recover the two-flavor limit of those equations, and recognize an approximate factorization of the three-flavor problem (reminiscent of the - factorization in the standard picture) into smaller two-flavor problems. We show that survival probabilities can be written down in a simple form, purely in terms of the solutions to the two-flavor problems, as long as the frequencies governing the oscillations are hierarchically separated. In Sec. III, we illustrate the above factorization for vacuum/MSW oscillations as well as collective synchronized oscillations. We also explain the three-flavor features of bipolar oscillations and spectral splits qualitatively and pictorially. In Sec. IV, we calculate the flavor conversion probabilities numerically for a typical SN matter profile, and identify the additional features due to the third neutrino. We conclude in Sec. V with a summary of our results and comments about directions of future investigation.

## Ii Three-Flavor Formalism

In this section we derive the steady state equations of motion for an ensemble of dense gas of three-flavor neutrinos, as a straightforward generalization of the corresponding equations in the two-flavor case.

### ii.1 Hamiltonian and Equations of Motion

We denote a neutrino of momentum at time at position by , and work in the modified flavor basis defined such that , where is the rotation matrix and the neutrino mixing angle in the - plane. 111This basis has also been denoted in the literature as dighe-smirnov-9907423 (). In this basis, the density matrix for neutrinos with momenta between and at any position between and may be written as

 ρνανβ(p,r,t) ≡ 1nν(p,r,t)∑|ν(p,r,t)⟩⟨ν(p,r,t)|αβ, (1)

where and the summation is over all neutrinos. Note that the density matrix is normalized to have unit trace, but the neutrino density itself is , which typically falls off as from the source. The number density of neutrinos with flavor is obtained through

 nνα(p,r,t)=nν(p,r,t)ρνανα(p,r,t) . (2)

We define the analogous quantities for antineutrinos similarly, but with a reversed order of flavor indices to keep the form of equations of motion identical raffelt-sigl-NPB406 (), and denote them with a “bar” over the corresponding variables for neutrinos.

The effective Hamiltonian in the modified flavor basis for neutrinos of energy in vacuum is

 Hvac(p)=UM2U†/2p, (3)

where the masses and the mixing matrix are parametrized as

 M ≡ Diag(m1,m2,m3) , (4) U ≡ R†23(θ23)R23(θ23)R13(θ13)R12(θ12) , (5)

with being the rotation matrices in the - plane. In this work, we take the value of the -violating phase in neutrino sector to be zero. Now may be explicitly written as

 Hvac(p) = Δm2132p⎛⎜ ⎜⎝s2130c13s13000c13s130c213⎞⎟ ⎟⎠ (6) +

where and other symbols have their usual meanings. In matter, neutrinos feel the Mikheyev-Smirnov-Wolfenstein (MSW) potential wolfenstein (); mikheyev-smirnov () due to charged leptons 222We assume that the density of and is negligible, and that and feel approximately identical potentials, which have been taken to be zero by convention. An analysis of collective effects including a potential has recently been carried out EstebanPretel:2007yq ().

 V(r,t)=√2GFne−(r,t)Diag(1,0,0) (7)

that adds to the Hamiltonian, where is the net electron number density at . The effective Hamiltonian also includes the effects of neutrino-neutrino interactions, which to the leading order in depend only on forward scattering and contribute thompson-mckellar-PLB259 (); raffelt-sigl-NPB406 (); thompson-mckellar-PRD49 ()

 Hνν(p,r,t)=√2GF∫d3q(2π)3κpq(nν(q,r,t)ρ(q,r,t)−¯nν(q,r,t)¯ρ(q,r,t)) . (8)

The interaction strength is dependent on the angular separation of the momenta of the interacting particles, and is given by , where is the angle between and .

The equation of motion for the density matrix is

 ddtρ(p,r,t) = −i[H(p,r,t),ρ(p,r,t)]+∂∂tρ(p,r,t). (9)

In the steady state (no explicit time dependence in the Hamiltonian and initial conditions) and ignoring external forces (terms depending on ), we can drop the time dependence in the problem. Writing we have the equations of motion for and as Cardall:2007zw ()

 v⋅∂rρ(p,r) = −i[+Hvac(p)+V(r)+Hνν(p,r),ρ(p,r)] , (10) v⋅∂r¯ρ(p,r) = −i[−Hvac(p)+V(r)+Hνν(p,r),¯ρ(p,r)] . (11)

The effect of ordinary matter can be “rotated away” by working in the interaction picture duan-fuller-qian-0511275 (); duan-fuller-carlson-qian-0606616 (). We employ an operator under which a matrix transforms to

 Aint(r)=O(r)AO−1(r) , (12)

where

 O(r)=exp(i∫r0dr′V(r′)) . (13)

This choice simplifies the equations of motion by removing the matter term, giving us

 v⋅∂rρint(p,r) = −i[+Hintvac(p,r)+Hintνν(p,r),ρint(p,r)] , (14) v⋅∂r¯ρint(p,r) = −i[−Hintvac(p,r)+Hintνν(p,r),¯ρint(p,r)] . (15)

The transformation by leaves diagonal entries of and unchanged, but the off-diagonal entries become -dependent. For example, if varies adiabatically and only in the radial direction, the vacuum Hamiltonian changes according to Eq. (12) as

 Hintvac(p,r)=Hvac(p)+ir[V(r),Hvac(p)]+(ir)22![V(r),[V(r),Hvac(p)]]+... . (16)

We know that is a diagonal matrix, so only the off-diagonal elements of are affected by the transformation. The final observables we are going to be interested in, the number fluxes of neutrino flavors, involve only diagonal elements of the density matrix [see Eq. (2)], so the interaction basis is well suited for our purposes.

### ii.2 Spherical Symmetry and Single-angle Equations of Motion

The interaction term in Eq. (8) depends on , i.e. the angle between the momenta of interacting neutrinos. Thus while performing the angular integrals therein, the dependence of the neutrino flux on all angular variables must be taken into account. This makes the problem quite complicated, and an approximate treatment is needed in order to gain useful insights. Two levels of approximation have been considered in literature, viz. multi-angle and single-angle. In the multi-angle approximation, azimuthal symmetry about the axis defined by the source and observer is usually assumed, but not complete spherical symmetry. This essentially captures the effects of correlations between trajectories with different initial launching angles. The effects of such correlations can have interesting consequences which have been explored in detail duan-fuller-carlson-qian-0606616 (); duan-fuller-carlson-qian-0608050 (); pantaleone-PRD58 (); raffelt-sigl-0701182 (); estebanpretel-pastor-tomas-raffelt-sigl-0706.2498 (); fogli-lisi-marrone-mirizzi-0707.1998 (). In the single-angle approximation, it is assumed that the flavor evolution does not significantly depend on any of the angular coordinates (i.e. the evolution is spherically symmetric), and thus we can integrate over all angular degrees of freedom trivially. One must then choose a representative value for , which we take to be .

We assume half-isotropic emission from a source of radius , as defined in estebanpretel-pastor-tomas-raffelt-sigl-0706.2498 (), and write

 nν(p,r) = nν(p,r)=nν(p,r0) r20/r2 , (17) ρ(p,r) = ρ(p,r) . (18)

In the steady state, the fluxes of neutrinos and antineutrinos can be written as

 Φν = ∫dp2πp24πr20nν(p,r0) , (19) Φ¯ν = ∫dp2πp24πr20¯nν(p,r0) , (20)

the total flux being .

A further “unification” in the notation for neutrinos and antineutrinos is possible by noting that their equations of motion, i.e. Eqs. (10) and (11), differ only in the sign of . This suggests a change of variables from to

 ω=|Δm213|/(2p). (21)

Using the same convention as raffelt-smirnov-0705.1830 (), we define for neutrinos

 nν(ω,r)≡nν( p(ω), r),ρ(ω,r)≡ρ( p(ω), r), (22)

and for antineutrinos

 nν(−ω,r)≡¯nν( p(ω), r),ρ(−ω,r)≡¯ρ( p(ω), r). (23)

The negative values of thus correspond to antineutrinos. Then we need to solve only for , albeit at the cost of extending the domain of to both positive and negative values. This simplifies the term in Eq. (8) to 333 Note that depended on only through the direction of . This dependence no longer survives in the single-angle approximation.

 Hνν(r)=μ(r)∫∞−∞dωf(ω)ρ(ω,r)sgn(ω) . (24)

in terms of the distribution function

 f(ω)=1Φ|Δm213|3π2r20ω4nν(ω,r0) , (25)

normalized as , and the “collective potential”

 μ(r)=μ0 g(r). (26)

Here is the collective potential at the neutrinosphere:

 μ0≡μ(r0)=3√2GFΦ128π4r20, (27)

and the “geometric dilution factor” is given by

 g(r) ≡ 4r203r2∫1√1−(r0/r)2d(cosθq)(1−cosθqcosθp)∣∣∣cosθp=1/2 (28) = 4r203r2⎛⎝1−√1−r20r2−r204r2⎞⎠.

The geometric dilution factor equals unity for , whereas at large , it decreases as . The decrease of neutrino densities from a finite source accounts for a factor of , whereas the additional dilution factor of approximately arises from the integral in Eq. (28), due to the decreasing angle subtended by the source and reduced collinearity, which are encoded in the limits and the integrand respectively fuller-qian-0505240 (). Note that the exact numerical factors depend on the choice of .

The total flux remains conserved as long as there is no explicit temporal variation of the overall luminosity. We work in the steady state approximation and assume the luminosity to be constant in time. Slow variations in it may be taken into account by including an additional time dependent factor. Note that is independent of , which embodies the fact that the normalized neutrino spectrum does not change. Using Eq. (2), we can also write the flavor dependent -spectra as

 fνα(ω,r)=f(ω)ρνανα(ω,r) . (29)

Note that contains the spectra of both and , and depends on only through . It would be a constant on the trajectory if there were no flavor evolution of . For later use, we define the energy integrated neutrino fluxes for each flavor as

 Φνe(r) = Φ∫∞0dωfνe(ω,r) , (30) Φν = Φνe(r)+Φνx(r)+Φνy(r) , (31) Φ¯νe(r) = Φ∫0−∞dωfνe(ω,r) , (32) Φ¯ν = Φ¯νe(r)+Φ¯νx(r)+Φ¯νy(r) . (33)

With these approximations, the problem is reduced to an ordinary one dimensional problem along the radial direction. We denote the derivative with respect to using a “dot”, and using Eqs. (10) and (11), arrive at the single-angle equations of motion

 vr˙ρ(ω,r)=−i[+Hvac(ω,h)+V(r)+Hνν(r),ρ(ω,r)] , (34)

where encodes the dependence on mass hierarchy. Here

 vr=√1−r20r2sin2θp(r0) (35)

is the radial velocity of the neutrino. Note that for , we have . Since the flavor conversions due to collective effects start becoming significant only for estebanpretel-pastor-tomas-raffelt-sigl-0706.2498 (); fogli-lisi-marrone-mirizzi-0707.1998 (), for our analytic approximations we take , making Eq. (34)

 ˙ρ(ω,r)=−i[+Hvac(ω,h)+V(r)+Hνν(r),ρ(ω,r)] . (36)

We have thus used the spherical symmetry of the problem, and the simple energy dependence, to rephrase the equations of motion in a somewhat simpler form. This single-angle approximation is probably crude, but it has been shown in numerical simulations (for two flavors) that this approximation seems to work reasonably well fogli-lisi-marrone-mirizzi-0707.1998 (). It also seems that the multi-angle effects are suppressed when the neutrino and antineutrino spectra are not identical estebanpretel-pastor-tomas-raffelt-sigl-0706.2498 (). We assume the above results to hold true for three flavors as well, and ignore multi-angle effects in this work. Thus, for an analytical understanding of various flavor conversion phenomena associated with this system, we confine our discussion to the steady-state single-angle half-isotropic approximation that we have outlined above.

### ii.3 Bloch Vector Notation

In the single-angle approximation, it is useful to re-express the density matrices and the Hamiltonian as Bloch vectors. The idea, analogous to the two-flavor case, is to express the matrices in a basis of hermitian matrices, and to study the motion of the vectors constructed out of the expansion coefficients (which are called the Bloch vectors). In our problem, we choose the basis consisting of the 33 identity matrix , and the Gell-Mann matrices given by

 Λ1 = ⎡⎢⎣010100000⎤⎥⎦, Λ2=⎡⎢⎣0−i0i00000⎤⎥⎦, Λ3=⎡⎢⎣1000−10000⎤⎥⎦, Λ4 = ⎡⎢⎣001000100⎤⎥⎦, Λ5=⎡⎢⎣00−i000i00⎤⎥⎦, Λ6=⎡⎢⎣000001010⎤⎥⎦, Λ7 = ⎡⎢⎣00000−i0i0⎤⎥⎦, Λ8=1√3⎡⎢⎣10001000−2⎤⎥⎦, (37)

which satisfy the Lie algebra

 [Λa,Λb]=ifabcΛc , (38)

where are integers from 1 to 8. Note that the normalization for the matrices is chosen such that

 Tr(ΛaΛb)=2δab . (39)

The structure constants are antisymmetric under exchange of any two indices and are specified by

 f123=2;f147,f165,f246,f257,f345,f376=1;f678,f458=√3. (40)

Note that basis of traceless matrices can be expressed as a semi-direct sum of

 \mathbbmK={Λ1,Λ2,Λ3,Λ8}and\mathbbmQ={Λ4,Λ5,Λ6,Λ7} , (41)

i.e. for and we have

 [Ka,Qb]∈\mathbbmK , [Qa,Qb]∈\mathbbmK and [Qa,Kb]∈\mathbbmQ . (42)

In fact this is not the only choice of and that has this property. In addition to the decomposition

 \mathbbmKex={Λ1,Λ2,Λ3,Λ8}and\mathbbmQex={Λ4,Λ5,Λ6,Λ7} , (43)

as above, we could also choose

 \mathbbmKey = {Λ3,Λ4,Λ5,Λ8}and\mathbbmQey={Λ1,Λ2,Λ6,Λ7}or (44) \mathbbmKxy = {Λ3,Λ6,Λ7,Λ8}and\mathbbmQxy={Λ1,Λ2,Λ4,Λ5} , (45)

which satisfy the conditions in Eq. (42). The meaning of the superscripts () on and will become clear later.

Using the basis matrices and , we now express any hermitian matrix as a vector in the generator space (with unit vectors ) as

 X=13X0I+12X⋅Λ . (46)

We call the Bloch vector corresponding to the matrix . The vector must lie completely within an eight-dimensional compact volume, called the Bloch ball, whose various two-dimensional sections are shown in Fig. 1. We say that the vector is contained in (, if the matrix can be expressed solely as a linear combination of (, .

We reparametrize our equations using Eq. (46), and define the Bloch vectors corresponding to the density matrices as

 ρ(ω,r)=13P0(ω,r)I+12P(ω,r)⋅Λ . (47)

Note that is an eight-vector of matrices. The scalar and the polarization vector encode the flavor content of neutrinos of energy at a distance for . The negative values of encode the same information for antineutrinos. Since has been normalised to have unit trace by definition, is equal to one. We will therefore not worry about the zeroth component of the polarization vector henceforth. For a pure state, lies on the boundary of the shaded region in Fig. 1, and has the magnitude . For a mixed state, the magnitude of is smaller and the vector lies within the shaded region.

We assume that all neutrinos are produced as flavor eigenstates, i.e. the primary flux consists of and with energy . The initial density matrix is therefore , and similarly for antineutrinos. The initial polarization vector may be written as

 P(ω,r0)=fνe(ω,r0)−fνx(ω,r0)f(ω)^e3+fνe(ω)+fνx(ω)−2fνy(ω,r0)√3f(ω)^e8. (48)

The polarization vector , when projected onto the plane, must lie within the triangle in Fig. 2, where we show a representative projected on the plane. The pure electron flavor is represented by

 ee=^e3+^e8√3. (49)

The or content with energy at position is given by

 ρνeνe(p,r)=nνe(p,r)nν(p)=fνe(ω,r)f(ω)=13+P⋅ee2=de√3 . (50)

The projection of on is thus related to as above. The same quantity can be easily visualized from the figure as , where is the distance of the tip of from the side of the triangle that is perpendicular to (as shown in the figure). The number of and are also similarly calculated. Negative values of encode the same information for the antineutrinos. This gives a simple pictorial way to represent the flavor content of the ensemble by plotting the tip of the projection of on the plane. 444Note that probability conservation in this representation corresponds to the theorem that the sum of the lengths of perpendiculars dropped from any point inside an equilateral triangle to the three sides is a constant.

For the mass term in the Hamiltonian, we have

 Hvac(ω,h) = hω(13B0I+12B⋅Λ) , (51)

where

 h B = ϵc13sin2θ12^e1+(s213−ϵ(c212−c213s212))^e3+(1−ϵs212)sin2θ13^e4 (52) −ϵs13sin2θ12^e6+((−2+ϵ)(3c213−1)+3ϵs213(2c212−1))/(2√3)^e8 .

Note that for neutrinos is always positive in this convention, and the negative sign of for inverted hierarchy is absorbed into . The terms involving arise from the mixing of the third flavor, and the three-flavor effects enter through them. The sign of is positive if the mass hierarchy is normal () and negative otherwise. This, along with the overall sign due to , guarantees that the contributions from always have the same sign. Note that vanish in the absence of -violation.

The MSW potential defined in Eq. (7) may be represented as

 V(r) = λ(r)(13L0I+12L⋅Λ) , (53)

where . The vector parameterizes the effect of background electrons, and is given by

 L=^e3+^e8/√3 . (54)

The term defined in Eq. (24) can also be simply written as

 Hνν(r) = μ(r)(13D0I+12D(r)⋅Λ) , (55)

where is defined as

 D(r)=∫dωf(ω)P(ω,r)sgn(ω) . (56)

In the next section, we shall express the evolution equation, i.e. Eq. (36) in terms of the Bloch vectors and .

### ii.4 Generalized Gyroscope Equations

We have expressed our problem in terms of the eight-dimensional Bloch vectors, and now we shall see that the equations of motion formally resemble the equations of a gyroscope. To make this apparent, we define as a generalized “cross product” kim-kim-sze () with as the structure constants, instead of the usual that appears in the two-flavor approximation, e.g.

 B×P≡8∑a,b=1fabcBaPb^ec . (57)

This makes it possible to write the equations of motion, i.e. Eq. (36), compactly as

 ˙P(ω,r)=(ωB+λ(r)L+μ(r)D(r))×P(ω,r) ≡H(ω,r)×P(ω,r) , (58)

where are defined in Eqs. (47), (52), (54) and (56) respectively. The couplings , and are defined in Eqs. (21), (28) and (53) respectively. Equation (58) resembles the equation of a spin in an external magnetic field, or equivalently, that of a gyroscope. We must remember that this similarity is purely formal, because unlike in the two-flavor case, we cannot write an arbitrary Bloch vector as a linear combination of two Bloch vectors and their cross product. We shall show in Sec. II.5 that under certain approximations these generalized gyrosope equations can be given a geometrical interpretation.

The effects of the matter term in Eq. (58) can be rotated away by going to the interaction frame as described in Eq. (12), where a matrix becomes . In order to determine the Bloch vector corresponding to , we equate

 A03I+8∑a=1AaΛa2=OAO−1. (59)

Multiplying both sides by and taking trace, we get

 Ainta=Tr(ΛaOAO−1), (60)

where we have used . In particular, the Bloch vector may be written using Eqs. (13) and (60) as

 Bint(r) = B1cosζ(r) ^e1+B1sinζ(r) ^e2+B3 ^e3 (61) +B4cosζ(r) ^e4+B4sinζ(r) ^e5+B6^e6+B7^e7+B8^e8 ,

where . In dense matter, oscillates rapidly with the frequency , mimicking a suppression in the relevant mixing angles as in the two-flavor case hannestad-raffelt-sigl-wong-0608095 ().

We also define the “signed” and “unsigned” moments (with ) of as

 D(n)(r) = ∫dωωnf(ω)P(ω,r)sgn(ω) , (62) S(n)(r) = ∫dωωnf(ω)P(ω,r) . (63)

Note that is same as , and we will therefore refer to as . The evolution of these moments are governed by

 ˙D(n)(r) = B×D(n+1)(r)+(λ(r)L+μ(r)D(r))×D(n)(r) , (64) ˙S(n)(r) = B×S(n+1)(r)+(λ(r)L+μ(r)D(r))×S(n)(r) . (65)

We see that the higher moments turn up in equations of motion the lower moments. If we take the dot product of Eq. (64) with , and of Eq. (65) with , we get

 ∂r|D(n)(r)|2 = D(n)(r)⋅B×D(n+1)(r) , ∂r|S(n)(r)|2 = S(n)(r)⋅B×S(n+1)(r) . (66)

The above dependence of the moments on implies that there is likely to be a redistribution of flavor as a function of . It will be interesting to investigate if these moment equations can be used to predict the nature of the redistribution of flavor spectra.

### ii.5 Heavy-Light factorization of dynamics

The three-flavor dynamics in the traditional matter-driven scenario can be factorized into the so-called “heavy” and “light” MSW resonances that occur at densities corresponding to and respectively. Appropriate combination of the effective two-flavor dynamics in these two sectors approximates the three-flavor result reasonably well. We now proceed to illustrate a similar simplification for collective effects as well Let us first introduce the notion of “heavy” and “light” subspaces of the Bloch-sphere. In the - decomposition shown in Eq. (44), the vectors contained in are termed “heavy” () whereas those contained in are termed “light” (). A general vector may be decomposed as

 X=XH+XL . (67)

In particular, in Eq. (52) may be expressed as , with

 hBH = +((−2+ϵ)(3c213−1)+3ϵs213(2c212−1))/(2√3)^e8, hBL = ϵc13sin2θ12^e1−ϵs13sin2θ12^e6. (68)

The component appears primarily due to , and the other component vanishes if . Note that for two-flavors, or equivalently in the limit, is completely contained in . Now, note the following structure in the equations of motion of a polarization vector:

 ˙PH(ω,r) = HH(ω,r)×PH(ω,r)+HL(ω,r)×PL(ω,r) , (69) ˙PL(ω,r) = HL(ω,r)×PH(ω,r)+HH(ω,r)×PL(ω,r) . (70)

It is clear from the above set of equations that if and one begins with contained in , then always remains in , i.e. identically. To investigate this case more closely, we write Eq. (69) for each component of as 555In the following sections, the dependence of the Bloch vectors and the parameters on and is implicit.

 ˙P3 = H4P5−H5P4, (71) ˙P4 = H5P3−H3P5+√3(H5P8−H8P5), (72) ˙P5 = H3P4−H4P3+√3(H8P4−H4P8), (73) ˙P8 = √3(H4P5−H5P4). (74)

Note that . This suggests that we could rotate our coordinates in the plane by , so that in the rotated frame becomes a constant of motion. While going to the rotated frame, the components and of any Bloch vector transform as

 (˜X3˜X8)=(−1/2−√3